Advances in Steel Structures Vol.1

ADVANCES IN STEEL STRUCTURES

Proceedings of The Second International Conference on Advances in Steel Structures

15-17 December 1999, Hong Kong, China

Volume I

ADVANCES IN STEEL STRUCTURES



Volume I

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A D V A N C E S IN STEEL S T R U C T U R E S



Volume I

Edited by SLChan and JGTeng

The Hong Kong Polytechnic University

Organised by Department of Civil and Structural Engineering

The Hong Kong Polytechnic University

Sponsored by The Hong Kong Institution of Engineers

1 9 9 9

ELSEVIER A M S T E R D A M �9 L A U S A N N E �9 N E W Y O R K �9 O X F O R D �9 S H A N N O N . S I N G A P O R E . T O K Y O

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Preface

These two volumes of proceedings contain 9 invited keynote papers and 126 contributed papers presented at the Second International Conference on Advances in Steel Structures held on 15-17 December 1999 in Hong Kong. The conference was a sequel to the International Conference on Advances in Steel Structures held in Hong Kong in December 1996.

The conference provided a forum for discussion and dissemination by researchers and designers of recent advances in the analysis, behaviour, design and construction of steel structures. The papers presented at the conference cover a wide spectrum of topics and were contributed from over 15 countries around the world. They report the current state-of-the-art and point to future directions of structural steel research.

The organization of a conference of this magnitude would not have been possible without the support and contributions of many individuals and organizations. The strong support from the Hong Kong Polytechnic University, Professor M. Anson, Dean of the FaTculty of Construction and Land Use, and Professor J.M. Ko, Head of the Department of Civil and Structural Engineering, has been pivotal in the organization of this conference. We also wish to express our gratitude to the Hong Kong Institution of Engineers for sponsoring the conference and the Local Advisory Committee for mobilizing support froTm the construction industry and government departments.

Thanks are due to all the contributors for their careful preparation of the manuscripts and all the keynote speakers for their special support. Reviews of papers were carried out by members of the International Scientific Committee and the Local Organizing Committee. To all the reviewers, we are most grateful.

We would also like to thank all those involved in the day-to-day running of the organization work, including members of the Local Organizing Committee and secretarial staff in the Department of Civil and Structural Engineering.

Finally, we gratefully acknowledge our pleasant cooperation with Dr. J. Milne and Mrs R. Davies at Elsevier Science Ltd in the UK.

S.L. Chan J.G. Teng

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INTERNATIONAL SCIENTIFIC COMMITTEE

H. Adeli, H. Akiyama, M. Anson, P. Ansourian, J. Arbocz, R. Bjorhovde, M.A. Bradford, R.Q. Bridge, C.R. Calladine, S.F. Chen, W.F. Chen, Y.K. Cheung, C.K. Choi,

M. Chryssanthopoulos, A. Combescure, S.L. Dong, P.J. Dowling, M. Farshad, Y. Fukumoto, Y. Goto, P.L. Gould, R. Greiner, P. Grundy, G.J. Hancock, J.E. Harding, K.M. Hsiao, J.F. Jullien, S. Kato, S. Kitpornchai, V. Krupka, T.T. Lan, S.F. Li, R. Liew, Xila Liu, Xiliang Liu, L.W. Lu, Z.T. Lu, E. Lui, P. Marek, S. Morino, D.A. Nethercot, G.W. Owens,

The Ohio State University University of Tokyo The Hong Kong Polytechnic University The University of Sydney Delft University of Technology University of Pittsburgh University of New South Wales University of Western Sydney University of Cambridge Xi'an University of Architecture and Technology Purdue University The University of Hong Kong Korea Advanced Institute of Science and Technology Imperial College Laboratoire de Mechanique et Technologie Zhejiang University University of Surrey EMPA Fukuyama University Nagoya Institute of Technology Washington University, St Louis Technisce Universitat, Graz Monash University University of Sydney University of Surrey National Chao Tung University INSA Lyon Toyohashi University of Technology University of Queensland Institute of Applied Mechanics, Vitkovice Chinese Academy of Building Research Tsinghua University National University of Singapore Tsinghua University Yianjin University Lehigh University South-East University Syracuse University Academy of Science of the Czech Republic Mie University University of Nottingham Steel Construction Institute

USA Japan Hong Kong, China Australia Netherlands USA Australia Australia UK China USA Hong Kong, China Korea

UK France China UK Switzerland Japan Japan USA Austria Australia Australia UK Taiwan, China France Japan Australia Czech Republic China China Singapore China China USA China USA Czech Republic Japan UK UK

vii

J.A. Packer, S. Pellegrino, E.P. Popov, J. Rhodes, J.M. Rotter, H. Schmidt, G. Sedlacek, S.Z. Shen,

Z.Y. Shen, T.T. Soong, S.S. Sridharan, N.S. Trahair, T. Usami, A. Wada, F. Wald, E. Walicki, D. White, F. Williams, Y.B. Yang, R. Zandonini, S.T. Zhong,

University of Toromo University of Cambridge University of California, Berkeley University of Strathclyde University of Edinburgh Universitat Essen RWTH Aachen Harbin University of Civil Engineering and Architecture Tongji University Suny, Buffalo, NY Washington University, St Louis University of Sydney Nagoya University Tokyo Institute of Technology Czech Technical University Technical University of Zielona Gora Georgia Institute of Technology University of Wales National Taiwan University University of Trento Harbin University of Civil Engineering and Architecture

Canada UK USA UK UK Germany Germany China

China USA USA Australia Japan Japan Czech Republic Poland USA UK Taiwan, China Italy China

viii

LOCAL ADVISORY COMMITTEE

Chairman:

Members:

A.S. Beard F.S.Y. Bong A.K.C. Chan H.C. Chan Y.L. Choi W.K. Fung M. Harman I. Kimura M.H.C. Kwong C.K. Lau S.H. Ng W. Tang H. Wu

J.M. Ko The Hong Kong Polytechnic University

Mott MacDonald Hong Kong Ltd. Maunsell Consultants Asia Ltd. Ove Amp and Partners (HK) Ltd. The University of Hong Kong Buildings Department, HKSAR Government Architectural Services Department, HKSAR Government British Steel (Asia) Ltd. Nippon Steel Corporation Scott Wilson (Hong Kong) Ltd. Highways Department, HKSAR Government Hong Kong Housing Society The Hong Kong University of Science and Technology Kowloon Canton Railway Corporation

ix

LOCAL ORGANISING COMMITTEE

Chairman:

Vice-Chairman:

Members:

C.M. Chan W.T. Chan T.H.T. Chan K.F. Chung J.C.L. Ho W.M.G. Ho C.M. Koon M.K.Y. Kwok S.S. Lam J.C.W. Lau Y.W. Mak A.D.E. Pan A.K. Soh K.Y. Wong Y.L. Wong Y.L. Xu L.H. Yam

S. L. Chan The Hong Kong Polytechnic University

J.G. Teng The Hong Kong Polytechnic University

The Hong Kong University of Science and Technology Buildings Department, HKSAR Government The Hong Kong Polytechnic University The Hong Kong Polytechnic University Scott Wilson (Hong Kong) Ltd. Ove Amp and Partners (HK) Ltd. Buildings Department, HKSAR Government Ore Amp and Partners (HK) Ltd. The Hong Kong Polytechnic University James Lau & Associates Ltd. Housing Department, HKSAR Government The University of Hong Kong The University of Hong Kong Highways Department, HKSAR Government The Hong Kong Polytechnic University The Hong Kong Polytechnic University The Hong Kong Polytechnic University

C O N T E N T S

VOLUME I

Preface

International Scientific Committee vii

Local Advisory Committee

Local Organising Committee

Keynote Papers

Unbraced Composite Frames: Application of the Wind Moment Method D.A. Nethercot and J.S. Hensman

A Cumulative Damage Model for the Analysis of Steel Frames under Seismic Actions Z.- Y. Shen

13

Recent Research and Design Developments in Cold-Formed Open Section and Tubular Members

G.J. Hancock 25

Behaviour of Highly Redundant Multi-Storey Buildings under Compartment Fires J.M. Rotter

39

Design Formulas for Stability Analysis of Reticulated Shells S.Z. Shen

51

Ductility Issues in Thin-Walled Steel Structures T. Usami, Y. Zheng and H.B. Ge

63

High-Performance Steel Structures: Recent Research L.W. Lu, R. Sause and J.M. Ricles

75

A Unified Principle of Multiples for Lateral Deflection, Buckling and Vibration of Multi-Storey, Multi-Bay, Sway Frames

W.P. Howson and F.W. Williams 87

Beams and Columns

Three-Dimensional Hysteretic Modeling of Thin-Walled Circular Steel Columns L. Jiang and Y. Goto

101

xi

xii Contents

Local Buckling of Thin-Walled Polygonal Columns Subjected to Axial Compression or Bending

J.G. Teng, S.T. Smith and L. Y. Ngok

Ultimate Load Capacity of Columns Strengthened under Preload H. Unterweger

Chaotic Belt Phenomena in Nonlinear Elastic Beam Z. Nianmei, Y. Guitong and X. Bingye

Frames and Trusses

109

117

125

Investigation of Rotational Characteristics of Column Bases of Steel Portal Frames T.C.H. Liu and L.J. Morris

135

Ultimate Strength of Semi-Rigid Frames under Non-Proportional Loads B.H.M. Chan, L.X. Fang and S.L. Chan

Second-Order Plastic Analysis of Steel Frames P. P.-T. Chui and S.-L. Chan

Study on the Behaviour of a New Light-Weight Steel Roof Truss P. Ma'keldinen and O. Kaitila

145

151

159

A Proposal of Generalized Plastic Hinge Model for the Collapse Behavior of Steel Frames Governed by Local Buckling

S. Motoyui and T. Ohtsuka

Advanced Inelastic Analysis of Spatial Structures J.Y.R. Liew, H. Chen and L.K. Tang

Stability Analysis of Multistory Framework under Uniformly Distributed Load C. Haojun and W. Jiqing

Space Structures

Studies on the Methods of Stability Function and Finite Element for Second-Order Analysis of Framed Structures

S.L. Chan and J.X. Gu

167

175

183

193

Dynamic Stability of Single Layer Reticulated Dome under Step Load C. Wang and S. Shen

Experimental Study on Full-Sized Models of Arched Corrugated Metal Roof L. Xiliang, Z. Yong and Z. Fuhai

Quasi-Tensegric Systems and Its Applications L. Yuxin and L. Zhitao

201

209

217

Connections

The Design of Pins in Steel Structures R.Q. Bridge

229

Contents xiii

Finite Element Modelling of Eight-Bolt Rectangular Hollow Section Bolted Moment End Plate Connections

A.T. Wheeler, M.J. Clarke and G.J. Hancock

Finite Element Modelling of Double Bolted Connections Between Cold-Formed Steel Strips under Static Shear Loading

K.F. Chung and K.H. Ip

Analytical Model for Eight-Bolt Rectangular Hollow Section Bolted Moment End Plate Connections

A.T. Wheeler, M.J. Clarke and G.J. Hancock

Predictions of Rotation Capacity of RHS Beams Using Finite Element Analysis T. Wilkinson and G.J. Hancock

237

245

253

261

Failure Modes of Bolted Cold-Formed Steel Connections under Static Shear Loading K.H. Ip and K.F. Chung

Design Moment Resistance of End Plate Connections Y. Shi and J. Jing

Threaded Bar Compression Stiffening for Moment Connections T.F. Nip and J.O. Surtees

Experimental Study of Steel I-Beam to CFT Column Connections S.P. Chiew and C.W. Dai

269

277

283

291

Behaviour of T-End Plate Connections to RHS Part I: Experimental Investigation M. Saidani, M.R. Omair and J.N. Karadelis

The Behaviour of T-End Plate Connections to SHS. Part II: A Numerical Model J.N. Karadelis, M. Saidani and M. Omair

Cyclic Behaviour of Beam-To-Column Welded Connections E. Mele, L. Calado and A. De Luca

Advanced Methods for Modelling Hysteretic Behaviour of Semi-Rigid Joints Y.Q. Ni, J.Y. Wang and J.M. Ko

305

313

323

331

Cold-Formed Steel

Behaviour and Design of Cold-Formed Channel Columns B. Young and K.J.R. Rasmussen

Section Moment Capacity of Cold-Formed Unlipped Channels B. Young and G.J. Hancock

Web Crippling Tests of High Strength Cold-Formed Channels B. Young and G.J. Hancock

Local and Distortional Buckling of Perforated Steel Wall Studs J. Kesti and J.M. Davies

341

349

357

367

xiv Contents

An Experimental Investigation into Cold-Formed Channel Sections in Bending V. Enjily, M.H.R. Godley and R.G. Beale

375

Composite Construction

Flexural Strength for Negative Bending and Vertical Shear Strength of Composite Steel Slag-Concrete Beams

Q.-L. Wang, Q.-L. Kang and P.-Z. Cao 385

Concrete-Filled Steel Tubes as Coupling Beams for RC Shear Walls J.G. Teng, J.F. Chen and Y.C. Lee

391

Experimental Study of High Strength Concrete Filled Circular Steel Columns Y.C. Wang

401

Strength and Ductility of Hollow Circular Steel Columns Filled with Fibre Reinforced Concrete

G. Campione, N. Scibilia and G. Zingone 413

Axial Compressive Strength of Steel and Composite Columns Fabricated with High Strength Steel Plate

B. Uy 421

Concrete Filled Cold-Formed C450 RHS Columns Subjected to Cyclic Axial Loading X.L. Zhao, R.H. Grzebieta, P. Wong and C. Lee

429

Research on the Hysteretic Behavior of High Strength Concrete Filled Steel Tubular Members under Compression and Bending

Z. Wang and Y. Zhen 437

Design of Composite Columns of Arbitrary Cross-Section Subject to Biaxial Bending S.F. Chen, J.G. Teng and S.L. Chan

443

Effects of Loading Conditions on Behaviour of Semi-Rigid Beam-to-Column Composite Connections

Y.L. Wong, J.Y. Wang and S.L. Chan 451

Steel-Concrete Composite Construction with Precast Concrete Hollow Core Floor D. Lam, K.S. Elliott and D.A. Nethercot

459

Testing and Numerical Modelling of Bi-Steel Plate Subject to Push-Out Loading S.K. Clubley and R.Y. Xiao

467

Rectangular Two-Way RC Slabs Bonded with a Steel Plate J.W. Zhang, J.G. Teng and Y.L. Wong

477

Bridges

Structural Performance Measurements and Design Parameter Validation for Tsing Ma Suspension Bridge

C.K. Lau, W.P. Mak, K.Y. Wong, W.Y. Chan, K.L. Man and K.F. Wong 487

Contents

Wind Characteristics and Response of Tsing Ma Bridge During Typhoon Victor L.D. Zhu, Y.L. Xu, K.Y. Wong and K. W.Y. Chan

Structural Performance Measurement and Design Parameter Validation for Kap Shui Mun Cable-Stayed Bridge

C.K. Lau, W.P. Mak, K.Y. Wong, K.L. Man, W.Y. Chan and K.F. Wong

Free and Forced Vibration of Large-Diameter Sagged Cables Taking into Account Bending Stiffness

Y.Q. Ni, J.M. Ko and G. Zheng

Stability Analysis of Curved Cable-Stayed Bridges Y.-C. Wang, H.-S. Shu and J. Ermopoulos

Expert System of Flexible Parametric Study on Cable-Stayed Bridges with Machine Learning

B. Zhou and M. Hoshino

Parameter Studies of Moving Force Identification in Laboratory T.H.T. Chan, L. Yu, S.S. Law and T.H. Yung

Seismic Analysis of Isolated Steel Highway Bridge X.-S. Li and Y. Goto

Shear Analysis for Asphalt Concrete Deck Pavement of Steel Bridges X. Zha and Q. Xiao

VOLUME II

Preface

International Scientific Committee

Local Advisory Committee

Local Organising Committee

XV

497

505

513

521

529

537

545

553

vii

ix

Plates

Strength and Ductility of Plates in Shear T. Usami, H.B. Ge and M. Amano

Post-Buckling of Unilaterally Constrained Mild Steel Plates S.T. Smith, M.A. Bradford and D.J. Oehlers

Postbuckling Analysis of Plate with General Initial Imperfection by Finite Strip Method T.H. Lui and S.S.E. Lam

Post-Buckling Analysis of Web Plates of Girders by Three Dimensional Degenerated Shell Element Method

H. Qinghua, Y. Yue and L. Xiliang

563

571

579

587

xvi Contents

Shells

Buckling Interaction Strength of Cylindrical Steel Shells under Axial Compression and Torsion

H. Schmidt and T. A. Winterstetter 597

Shell Buckling Design of Austenitic Stainless Steel Cylinders under Elevated Temperatures up to 500~

H. Schmidt and K.T. Hautala 605

Cylindrical Shells Buckling under External Pressure--Influence of Localized Thickness Variation

J.F. Jullien, A. Limam and G. Gusic 613

Stability and Strength of Conical Shells Subject to Axial Load and External Pressure N. Panzeri and C. Poggi

621

The Nonlinear Stability of Semi-Thin Spherical Shell Joints under Uniformly Symmetric Circular Line Loads

Y.F. Luo, K.S. Huang and Q.Z. Li

The Influence of Circumferential Weld-Induced Imperfections on the Buckling of Silos and Tanks

M. Pircher and R. Bridge

631

639

Experimental Techniques for Steel Silo Transition Junctions J.G. Teng and Y. Zhao

647

Buckling Strength of T-Section Ringbeams in Steel Silos J.G. Teng and F. Chan

655

Abnormal Behaviour of a Steel Silo Caused by Paddy Rice Storage M.P. Luong

663

Bifurcation Buckling of Aboveground Oil Storage Tanks under Internal Pressure S. Yoshida

671

Buckling of Cylindrical Shells Subjected to Edge Vertical Deformation M. Jonaidi and P. Ansourian

679

On the Nonlinear Analysis of Shells with Eigenmode-Affine Imperfections J.G. Teng and C.Y. Song

687

Postbuckling Analysis of Shells of Revolution Considering Mode Switching and Interaction

T. Hong and J.G. Teng 697

Transition of Plastic Buckling Modes in Cylindrical Shells Y. Goto, C. Zhang and N. Kawanishi

705

Are the Static Postbuckling Predictions Conservative? A. Combescure

713

Contents xvii

Plastic Stability of Cylindrical Shells Taking Account of Loading History V.S. Hudramovych

721

Design and Construction

Prestressing and Loading Tests on Full-Scale Roof Truss of Shanghai Pudong International Airport Terminal

Z. Xiangzhong, C. Yiyi, S. Zuyan, C. Yangji, W. Dasui and Z. Jian

Air Mail Centre at Chek Lap Kok P.H. Lam

731

739

Composite Design and Construction of a Tall Building--Cheung Kong Center D. Scott, G.W.M. Ho and H. Nuttall

747

The Tallest Building in Mexico City: Torre Mayor, Mexico City, Mexico A. Rahimian and E.M. Romero

755

The Use of Triangular Added Damping and Stiffness (TADAS) Devices in the Design of the Core Pacific City Shopping Centre

K.L. Chang, S.J.W. Rees, C. Carroll and K. Clandening

Site Measurement of Vibration Characteristics of Shanghai Jin Mao Tower W. Shi, X. Lu and J. Shen

775

783

Design of Steel Scaffolding by Nonlinear Integrated Design and Analysis (NIDA) and the Stability Function

A. Y.T. Chu and S.L. Chan 791

Experimental Assessment for Aluminium Alloy Sections in Glass Curtain Walls of Shanghai Jinmao Building

L. Tong, Y. Luo, Z. Shen and Y. Wang

Dynamics and Seismic Design

Transverse Dynamic Characteristic and Seismic Responses of Large-Scale Tall-Pier Aqueduct

Y. Li

799

809

Dynamic Characteristic and Seismic Response of Semirigid Jointed Frames W.S. Zhang and Y.L. Xu

Nonlinear Seismic Analysis of Flexibly Connected Steel Buildings P. P.-T. Chui and S.-L. Chan

815

823

The Response Analysis of the Transversely Stiffening Single Curvature Cable- Suspended Roof to the Fluctuating Wind

X. Zhao, X. Liu and Y. Dou 831

Transient Analysis of Stiffened Panel Structure by a Finite Strip-Mode Superposition Method

J. Chen 839

xviii Contents

Dynamic Performance of Steel Lightweight Floors M.M. Alikhail, X.L. Zhao and L. Koss

Coupled Truss Walls with Damped Link Elements A. Rahimian

849

857

Galloping of Cables with Moving Rivulet L.Y. Wang and Y.L. Xu

Free Vibration Analysis of Thin-Walled Members with Shell Type Cross Sections M. Ohga, T. Shigematsu and T. Hara

A Simple Formulation for Free Vibration of Frame-Shear Wall Tall Building Q. Wang and L. Wang

Flexure-Torsion Coupled Vibrations for Tall Building Structures Considering the Effects of Vertical Loads

S.H. Bao and S.C. Yi

873

881

889

897

The Computational Time Efficient Finite Element Method for Large Amplitude Vibrations of Composite Plates

Y.-Y. Lee and C.-F. Ng

Determination of Model Order for Thin Steel Plate Systems Using Vibration Test Data Y.Y. Li and L.H. Yam

905

913

Prestressing

Study on Tendon Profile on the Analysis and Design of Prestressed Steel Beams G.N. Ronghe and L.M. Gupta

Long Term Analysis of Externally Prestressed Composite Beams A.D. Asta, L. Dezi and G. Leoni

Flexible Connection Influence on Ultimate Capacity of Externally Prestressed Composite Beams

A.D. Asta, L. Dezi and G. Leoni

A Fracture Criterion for Prestressing Steel Cracked Wires J. Toribio and M. Toledano

921

931

939

947

Failure Analysis of Prestressing Steel Wires J. Toribio and A. Valiente

955

Fatigue and Fracture

Experimental Study on Static and Fatigue Behavior of Steel-Concrete Preflex Prestressed Composite Beams

K. Zhang, S. Li and K. Liu

Object-Oriented Fatigue Reliability Analysis for the Offshore Steel Jacket C. Wang, Y. Shi and S. Li

965

975

Contents xix

Fatigue Strength of Thin-Walled Tube-To-Plate T-Joints under In-Plane Bending F.R. Mashiri, X.L. Zhao and P. Grundy

Failure Assessment of Beam-to-Column Steel Joints via Low-Cycle Fatigue Approaches C. Bernuzzi and R. Zandonini

983

991

A Method to Estimate P-S-N Curve for Misaligned Welded Joints G. Deqing

Reliability Analysis of Draw Bar of Large-Scale Lock Mitre Gate Z.G. Xu, C.Y. Bian and R.L. Wang

999

1005

Computation of Stress Intensity Factor for Surface Crack in Welded Joint G. Deqing and Y. Yong

Numerical Approach to the Ductile Fracture of Steel Members M. Obata, A. Mizutani and Y. Goto

1013

1021

Fire Performance

The First Code on Fire Safety of Steel Structures in China G.Q. Li, S.C. Jiang and J.L. He

Fire Resistance of Concrete Filled Steel Tubes in China L.-H. Han

1031

1039

Elevated Temperature Testing of Composite Columns N.L. Patterson, X.-L. Zhao, M.B. Wong, J. Ghojel and P. Grundy

1047

Full Scale Fire Test on the New UK Slim Floor System C.G. Bailey, T. Lennon and D.B. Moore

Mechanical Properties of an Austenitic Stainless Steel at Elevated Temperatures J. Outinen and P. Mdkeldinen

1055

1063

Optimization

Optical Design of Steel Frames with Non-Uniform Members A. Mu'ller, F. Werner and P. Osterrieder

1073

Optimal Sizing/Shape Design of Steel Portal Frames Using Genetic Algorithms P. Liu, C.-M. Chan and Z.-M. Wang

1081

Study on Optimization of Particular and Multi-Variable Structures by Wavelet Analysis L. Liu, Y. Zhai and H. Lin

1089

Optimal Analysis of Large Span Double-Layer Barrel Vaults L. Shan and H. Yan

1099

xx Contents

Analysis

Determination of Section Properties of Complicated Structural Members Z.X. Li, J.M. Ko, T.H.T. Chan and Y.Q. Ni

Adaptive Finite Element Buckling Analysis of Folded Plate Structures C.K. Choi and M.K. Song

Hoop Stress Reduction by Using Reinforced Rivets in Steel Structures K.T. Chau, S.L. Chan and X.X. Wei

1109

1117

1125

Safety Analysis and Design Consideration for Oil and Gas Pipelines A.N. Kumar

1133

Prediction of Residual Stresses: Comparison Between Experimental and Numerical Results

Y. Vincent, J.F. Jullien and V. Cano 1141

Soil Structure Interaction

Composite Foundation of Deep Mixing Piles for Large Steel Oil Tanks on Soft Ground X. Xie, X. Zhu and Q. Pan

An Analytical Study on Seismic Response of Steel Bridge Piers Considering Soil- Structure Interaction

A. Kasai and T. Usami

1151

1157

Late Papers

Modelling Hysteresis Loops of Composite Joints Using Neural Networks J.Y. Wang, Y.L. Wong and S.L. Chan

New Design Methods for Concrete Filled Steel Tubular Columns Y.C. Wang

Keynote Paper

The Implications of the Information Society on the Practice of and Training for Steelwork Construction

G. Owens

1167

1175

1187

Index of Contributors II

Keyword Index 15

Keynote Papers


UNBRACED COMPOSITE FRAMES: APPLICATION OF THE WIND MOMENT METHOD

D A Nethercot 1 and J S Hensman 2

ISchool of Civil Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, UK

2Caunton Engineering Limited, Moorgreen Industrial Park, Moorgreen, Nottingham NG16 3QU, UK

ABSTRACT

Proposals are given to extend the simplified design technique known as the Wind Moment Method to cover a limited range of composite frames. The range represents that of most interest in practice in the UK. Justification is by comparison with the findings from an extensive numerical study.

K E Y W O R D S : Composite Construction, Connections, Frames, Joints, Steel Structures, Structural Design

I N T R O D U C T I O N

The Wind Moment Method (WMM) has long been established as a simple, intuitively based, design approach for unbraced frames. More recently, it has been the subject of scientific study, designed to provide a more fundamental understanding of the link between actual frame behaviour and the inherent design simplifications. This work has, until now, been restricted to bare steel construction.

In a recent study Hensman, (1998), the authors have examined the case for an extension of the WMM to cover composite steel/concrete frames. Although the approach adopted resembles that used for bare steelwork, a number of particular features have had to be addressed. This paper summarizes the main outcomes from that study.

The basis for the extension was numerical modelling, utilising the available body of knowledge on the performance of composite connections, the previous application of the WMM to bare steelwork and the capabilities of the ABAQUS package. It was also found necessary to conduct a detailed examination of the role of column bases - a feature not previously addressed in WMM

D.A. Nethercot and J.S. Hensman

investigations. Several of the findings therefore have relevance to potential improvements in the WMM for bare steel frames. This paper covers: appraisal of the basic source data, outline of the numerical studies, presentation of the key findings and an indication of the resulting design approach. This last item will be presented in a fashion suitable for direct use by designers in a forthcoming Steel Construction Institute Design Guide.

KEY FEATURES OF THE WIND MOMENT METHOD

The approach was originally devised in the pre-computer era, when overall structural analysis of unbraced frames represented an extremely challenging and potentially tedious task. It therefore sought an acceptable simplification so that the labour involved in the structural analysis might be minimised. This was achieved by recognising that some simplification in the representation of the actual behaviour would be necessary. Although it is now quite widely accepted that the true behaviour of all practical forms of beam to column connection in steel and concrete construction function in a semi-rigid and partial strength fashion- with the ideals of pinned and rigid only occasionally being approached- early methods of structural analysis could only cater for one or other of these ideals. Thus the basic WMM uses the principle of superposition to combine the internal moments and forces obtained from a gravity load analysis that assumes all beams to be simply supported and a wind load analysis that assumes beam to column connections to be rigid with points of contraflexure at the mid-span of the beams and the mid-height of the columns as illustrated in Figure 1. This second assumption permits use of the so-called portal method of frame analysis.

Once it became possible to conduct full range analyses of steel frames allowing for material and geometrical non-linear effects and including realistic models of joint behaviour, studies were undertaken to assess the actual performance of frames designed according to the WMM principles. The findings permitted observations to be made of the two key behavioural measures:

�9 That the load factor at ultimate was satisfactory �9 That drift limits at serviceability were achieved.

This second point is of importance because, when estimating sway deflections at working load, the WMM normally involves taking the results of an analysis that assumes rigid connections and then applying a suitable scaling factor. Important contributions in the area of bare steel construction are those of Ackroyd and Gerstle, (1982), Ackroyd, (1987), and Anderson and his co-workers at Warwick, Reading, (1989), Kavianpour, (1990), Anderson, Reading and Kavianpour,(1991)

NUMERICAL APPROACH

All the numerical work was undertaken using the ABAQUS package. Whilst this contained sufficient functionality to cover many of the necessary behavioural features, three items required particular attention:

�9 Representation of the composite beams �9 Representation of composite beam to column connections �9 Inclusion of column base effects

4,4' 4'4, ,1,4, 4'4' 4'4' 4'4' 4'4' 4'4'4'

~ 4 ' 4'4' 4'4' 4'4' 4'4' 4'4' 4'4'4'

(a)

t 7" r 77

t t

Unbraced Compos i te Frames: Appl icat ion o f the Wind M o m e n t M e t h o d

Figure 1 Superposition of gravity and lateral load analyses

For the first of these the approach previously utilised by Ye, Nethercot and Li, (1996), that is based on moment curvature relationships developed by Li, Nethercot and Choo, (1993), was employed. Since composite endplates were assumed for the beam to column connections, the work of Ahmed and Nethercot,(1997), in predicting moment-rotation response under hogging moment was directly employed.

Data on the performance of composite beam to column connections under sagging i.e. opening, moments was, however, almost non-existent. Previous experience with the Wind Moment Method had, however, suggested that reversal in the sign of the rotation at any connection might be a rather unusual event. An approximate model for composite connection behaviour under sagging moments was therefore devised by examining test data for such connections when subject to cyclic loading.

All previous studies of the WMM have assumed rigid i.e. fully fixed column bases. Enquiries among practitioners had, however, already revealed that such an option was not attractive. In addition, there was a widely held belief that all practical forms of "pin" column bases were capable of supplying quite significant amounts of rotational restraint. Accordingly, all relevant information on column base effects - particularly previous experimental studies - was carefully reviewed in an attempt to identify suitable minimum restraint levels likely to be supplied by notionally pinned bases, Hensman and Nethercot, (2000a). The findings were then incorporated in the full parametric study. This point is regarded as particularly important as attempts to justify the WMM approach using truly pinned column bases, Hensman,(1998), had shown that it was almost impossible to satisfy realistic drift limitations due to the greatly enhanced overall frame flexibility resulting from the loss of column base restraint (as compared with the usual WMM assumption of fixed bases). It is believed that the exercise should be r epea t ed - since bare steel columns were assumed throughout, it would merely be a case of conducting appropriate analyses on bare steel frames - as a way of similarly relaxing an unattractive restriction in the application of the WMM to bare steel construction.

6 D.A. Nethercot and J.S. Hensman

Because of concem over the adequacy of the modelling of composite beam to column connections under sagging moments, particular attention was paid in an initial study, Hensman, J S (1998), to the occurrence (or not) of reversal in the sign of the connection rotations. Initial studies using the sub-frame of Figure 2, that was specially configured to represent a typical intermediate floor in a more extensive structure, showed that for realistic arrangements of frame layout, member sizes and levels of gravity and wind loading reversal of rotations, even at the potentially most vulnerable windward connections was extremely unlikely. It was therefore concluded that the full parametric study need not concern itself with further refinement of this feature.

PARAMETRIC STUDY

Figure 3 illustrates the basic frame layouts considered and Tables 1 and 2 list the range of variables considered within the numerical study. Although this was based on the equivalent set of restrictions given in Anderson, Reading and Kavianpour (1991) it has been adapted somewhat, both to recognise important differences between bare steel and composite construction e.g. the likely use of longer span beams, and to reflect certain preferences from the industry and recent changes in the UK design environment e.g. issue of a new Code for wind loading. A more detailed explanation of the arrangement of the study, including justification for decisions on joint types, load combinations etc., is available, Hensman and Nethercot (2000b). Full details of the 300 cases investigated covering 45 different frame arrangements, including summary results for each, are available in reference 1. In all cases the approach adopted was to first design the frame using the proposed WMM technique and then to conduct a full range computer analysis to check its condition at the SLS and ULS stages.

MAIN FINDINGS

Undoubtedly the most significant overall outcome of the parametric study was the finding that every frame design using the proposed WMM approach was essentially satisfactory in terms of providing an adequate margin of safety against ULS load combinations. This was despite the fact that the actual distributions of intemal forces and moments within the frames often differed significantly from those presumed by the WMM analyses. Only in an extremely small number of cases was any degree of column overstress observed (and then less than 4%) - a comforting feature given that actual end restraint moments obtained from the rigorous analyses were often significantly higher than the assumed 10% of the WMM. The actual values of up to 30% in certain cases might suggest that where gravity loads are high beam sections could be reduced by assuming a larger-say 20% - end restraint moment. Before so doing, however, it would be important to check the effect on overall lateral frame stiffness as it might well prove difficult to satisfy drift limitations with this inherently more flexible system.

For all cases of frames designed for maximum gravity load and minimum wind load the SLS conditions were met. However, if higher wind loads were introduced, particularly for frames with short bay widths, some difficulty in ensuring adequate serviceability performance might well be experienced.

Unbraced Composite Frames." Application of the Wind Moment Method 7

A general discussion on the findings from the numerical study in terms of possible future modifications to the WMM and links between flame features and observed behaviour is available in Hensman and Nethercot (2000b).

Figure 2: Typical subframe arrangement used for preliminary study (Beam spans vary between 6m and 12m)


Figure 3 �9 Schematic diagram of alternative flame layouts used in parametric study

Unbraced Composite Frames: Application of the Wind Moment Method

TABLE 1

RANGE OF VARIABLES CONSIDERED WITHIN THE PARAMETRIC STUDY

Minimum Maximum Number of storeys 2 4 Number of bays 2 4"1 Bay width (m) 6.0 12.0 Bottom storey height (m) 4.5 6.0 Storey height elsewhere (m) 3.5 5.0 Dead load on floors (kN/m 2) 3.50 5.00 Imposed load on floors (kN/m 2) 4.00 7.50 Dead load on roof (kN/m 2) 3.75 3.75 Imposed load on roof (kN/m 2) 1.50 1.50 Wind loads (kN) 10 .2 40 *2

*' frames can have more than 4 bays, but a core of 4 bays is the maximum that can be considered to resist the applied wind load.

,2 Wind loads = concentrated point load on plane frame at each floor level

TABLE 1

RELATIVE DIMENSIONS CONSIDERED WITHIN THE PARAMETRIC STUDY

Bay width: storey height (bottom storey) Bay width: storey height (above bottom storey) Greatest bay width: Smallest bay width

Minimum Maximum

1.33 2.67

1.33 3.43

1 1.5

R E C O M M E N D E D D E S I G N A P P R O A C H

The basic design approach is outlined in the chart of Figure 4. This presents all the relevant steps, including those intended to identify arrangements for which the W M M is not suitable. Some key details for certain of the steps in the actual design procedure are discussed below.

Once an initial frame arrangement has been decided upon, global analyses for the three load combinations:

�9 1.4DE + 1.6IL + Notional Horizontal Forces

�9 1 .2(DL+IL+WL)

�9 1.4 (DE+WE)

should be undertaken. Notional horizontal forces should be taken as 0.5% of the factored dead +

imposed load as specified by BS5950: Part 1. Pattern loading should be considered; it may well be


STEP 1 Define frame geometry

STEP 2 Define load types and magnitude

(1) Gravity (2) wind

NOTE: This flow chart is not a design procedure. It should be used only as a 'first check', to determine if the wind-moment method outlined in this document is a suitable design method for the frame in question.

STEP 3 Design composite beams as

simply supported with a capacity of 0.9Mp

STEP 4 Estimate required column

sections

I . . . . . . . . .

. . . . . . . . .

I , I

. . . . . . . . . I

The frame design is likely to be controlled by SLS sway. However, a suitable frame

design may still be achieved using the wind-moment method.

Consider increasing the member sizes.

l STEP 5

Predict the SLS sway using the method in Section 5 Design the frame as rigid, or include

vertical bracing.

~ ~ ~ Using the wind-moment method to ~ " Is the t~alframe ~ Is the to!al frame ~ design this frame for ULS is likely

sway <.n:.~uu: / ~ sway <h:200? J " to result in unsatisfactory SLS ~ sway b e h a v i o u r ~

~ Yes

~ ] Using the wind-moment method to ~ . ~ ~ _ _ ~ _. [design this frame for the ULS is likely]

to result in unsatisfactory inter-storey I sway behaviour. Design frame as I

rigid at the 1st storey level, or include I ~ e ~ vertical bracing.

s ~ es I I

/

Design the frame using the I The bottom storey SLS sway is likely wind-moment method,as ] to control the frame design. detailed in this document. Increasing the column section sizes

may resolve this problem; if not then it may be appropriate to use an

alternative method.

Figure 4 Steps in design approach

Unbraced Composite Frames." Application of the Wind Moment Method 11

critical for the design of internal columns when heavy imposed loads are present on long span beams.

For beam design under gravity loading an end restraint moment of 10% of the maximum sagging moment in the beam should be assumed. For horizontal loading, frame analysis should be by the "portal method".

Composite beams should be Class 1 designed for 90% of their plastic moment of resistance at mid- span. This provision has been introduced so as to ensure that adequate rotation capacity is present in the composite connections to develop the required span moment. Previous studies, Li, Choo and Nethercot, (1995), Nethercot, Li and Choo, (1995), have shown both that the available rotation capacity of composite connections is limited and that the non-linear relationship between beam span design moment and the amount of moment redistribution necessary to achieve this substantially reduces the rotational demands on the connections.

Columns, which are assumed to be of bare steel, should be designed by the usual interaction formula approach. Effective lengths for in-plane and out of plane checks should be taken as 1.5L and 1.0L respectively. Column end moments should allow for both the end restraint moment due to partial fixity when considering gravity loading and the moments calculated due to horizontal loading.

Connections must be designed for both maximum hogging and minimum sagging loads in recognition of the fact that wind loading can reverse.

The parametric study indicated deflections under serviceability loading to be of the order of 30% greater than those calculated assuming rigid joints due to the greater overall flexibility of the frames with semi-rigid connections. Rather than permit the use of any method for the determination of sway deflections, a development of that proposed by

Wood and Roberts, (1975), that employed a simple graphical technique is proposed. In this way the common drift limit of h/300 recommended by BS5950: Part 1 and EC3 may be achieved. In addition to considering the behaviour of the complete frame, it is important to check each individual story. The first story is likely to be the most critical, typically accounting for the percentage of total frame sway indicated in Table 3.

CONCLUSIONS

Based on the findings of a careful numerical study that employed a synthesis of the best currently available scientific evidence, proposals for the application of the Wind Moment Method to the design of a restricted range of unbraced composite frames have been made. In application, these closely follow the established procedure of the SCI Design Guide for bare steel frames. The background study has, however, recognised the need to properly consider the behaviour of both the composite beams and the composite connections; it has also recognised the desirability of using less than fully rigid column bases. In deriving the design procedure, account has been taken of industry wishes, important practical differences in the likely configuration of composite frames from steel frames and recent changes in the general structural design climate in the UK.

12

REFERENCES

D.A. Nethercot and J.S. Hensman

Ackroyd, M (1987), Design of flexibly connected unbraced steel building frames, Journal Constructional Steel Research, No 8, pp 261-286.

Ackroyd, M H and Gerstle, K H (1982), Behaviour of type II steel frames, Journal of the Structural Division ASCE, Vol. 108, No 7, pp 1541-1556.

Ahmed, B and Nethercot, D A (1997), Design of composite flush end-plate connections, The Structural Engineer, Volume 75, No. 14, pp 233-244.

Anderson, D, Reading S, J and Kavianpour, K (1991), Wind-moment design for unbraced frames, The Steel Construction Institute, Publication No 082.

Hensman, J S (1998), Investigation of the wind moment method for unbraced composite frames, MPhil thesis, Department of Civil Engineering, University of Nottingham.

Hensman, J S and Nethercot, D A, (2000a) Utilisation of test data for column bases when addressing overall frame response: a review, Advances in Structural Engineering, to be published

Hensman, J S and Nethercot, D A, (2000b) "Numerical studies of composite sway frames: Generation of Data to Validate Wind Moment Method of design", to be published.

Kavianpour, K (1990), Design and analysis of unbraced steel frames, PhD thesis, University of Warwick.

Li, T Q, Choo, B S and Nethercot, D A (1995), Determination of Rotation Capacity Requirements for Steel and Composite Beams, Journal of Constructional Steel Research, Vol. 32, pp. 303-332.

Li, T Q, Nethercot, D A and Choo, B S (1993), Moment curvature relations for steel and composite beams, Steel Structures, Journal of Singapore Structural Steel Society, Volume 4, No. 1, pp 35-51.

Nethercot, D A, Li, T Q and Choo, B S (1995), Required Rotations and Moment-Redistribution for Composite Frames and Continuous Beams, Journal of Constructional Steel Research, Vol. 35. No. 2, pp. 121-164.

Reading, S J (1989), Kavianpour, K (1990), Anderson, D, Reading S, J and Kavianpour, K (1991) Investigation of the wind connection method, MSc thesis, University of Warwick,

Wood, R H and Roberts, E H (1975), A graphical method of predicting sidesway in multistorey buildings, Proceedings Institution of Civil Engineers, Part 2, Vol. 59, pp 353-272.

Ye Mei-Xin, Nethercot, D A and Li, T Q (1996), Non-linear finite element analysis of composite frames, Proceedings Institution of Civil Engineers, Structures and Buildings, Volume 116, pp 244- 247.

A CUMULATIVE DAMAGE MODEL FOR THE ANALYSIS OF STEEL FRAMES

UNDER SEISMIC ACTIONS*

Zu-Yan Shen I

~State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, 200092, China

ABSTRACT

Based on a series of experiments and theoretical analysis, a cumulative damage mechanics model of steel under cyclic loading, hysteresis models for plane and space steel members with damage cumulative effects and a cumulative damage model for steel frames under seismic actions are presented. Using these cumulative damage models, the elasto-plastic response of steel framed structures sustained the major shock and successive aftershocks of an earthquake can be calculated and the performance of the structures can be predicted more precisely and more realistically. In order to verify the theoretical results and the acceptability of the cumulative damage models, shaking table tests were conducted by the author in the State Key Laboratory for Disaster Reduction in Civil Engineering of Tongji University.

KEYWORDS

Cumulative damage model, Steel framed structures, Seismic response, Shaking table test

INTRODUCTION

The collapse of structures is usually due to the cumulation of damage to certain extent. In order to take the cumulation of damage into account in analysis, damage mechanics has been developed, Kachnov(1986). But up to now few of the research results have been used in the seismic analysis of steel framed structures. Recently Shen and Dong (1997) suggested an experiment-based cumulative damage mechanics model for steel subjected to cyclic loading. Shen et al (1998) derived a hysteresis model for plane steel members with damage cumulative effects. Since these research results can take the damage cumulative effects into consideration, it becomes possible to establish an analysis approach for calculating the elasto-plastic response of steel framed structures sustained the major shock and successive aftershocks of an earthquake.

~ National Key Projects on Basic Research and Applied Research: Applied Research on Safety and Durability of Major Construction Projects

13

14 Z . - Y. Shen

In the paper, a cumulative damage model for steel framed structures under seismic actions is established. The performance of the structures subjected to a major shock and successive aftershocks of an earthquake can be analyzed and the extent of damage including the damage cumulation of the structures after each shock can be calculated. Shaking table tests were conducted by the author in the State Key Laboratory for Disaster Reduction in Civil Engineering of Tongji University for verifying the theoretical analysis.

CUMUALTIVE DAMAGE MECHANICS MODEL OF STEEL UNDER CYCLIC LOADING

A cumulative damage mechanics model of steel under cyclic loading was established by the author, Shen and Dong (1997). The model can be expressed as follows.

Damage index D is calculated by

" ~ C - 6g D = ( l - i l l e ' - e g + f l ~ (11

C - 6g . = 6=" - 6g

where N is the number of half cycles which cause plastic strain, fl is the weight value, 6 p is the

P is the plastic strain during the ith half cycle, 6 p is the ultimate plastic strain of the material and 6 .

largest plastic strain during all halfcycles.

The effects of damage on elastic modulus, yield strength and strain hardening coefficient are

E o = ( 1 - 4 D ) E

tro = (1 - ~2D)tr, (2)

where D is the damage index, D = 0 means no damage and D = 1 means complete failure of the

material, E and E n are the elastic modulus in respect of D = 0 and D, respectively, tr s is the

initial yield stress when D = O, try is the yield stress in respect of D, k 0 is the original strain

hardening Cbefficient and k (") is the strain hardening coefficient after the nth halfcycle. ~:~, ~:2 and ~3

are three material parameters.

And the cumulative damage mechanics model of the steel under cyclic loading is

for first half cycle

(3)

from the second half cycle,

Cumulative Damage Model for the Analysis of Steel

f ! = erA.+, + ED(')(C-- CA.+, ) - - a c 2 -~- b g + c

- - O ' C n + l -I- k(')ED(')(c - Cc.+,)

I~176 - ~ to? '~

?'or. ~ < [era,,+ , -o" I < (2 + r/)cr~ <')

IOta.,+,- or] > (2 + r/)cr~ ~

(4)

15

where ~s and es are the initial yield stress and strain, CA~+, and eAn+~ axe the stress and strain at

unloading point A of the (n+ 1)th halfcycle, ~cn+, and ecn+, are the stress and strain at stress hardening point C of the (n+l)th halfcycle, y and r/ are two material parameters and a, band c are the constants of the parabolic curve connecting the yielding point B and the stress hardening point C.

The cumulative damage mechanics model can be illustrated by Figure 1.

Figure 1: Hysteretic model of steel considering damage cumulation

For steel Q235, all the material parameters can be adopted according to Table 1.

TABLE 1 MATERIAL PARAMETERS FOR DAMAGE

MECHANIC MODEL OF STEEL Q235

0.0081 0.227 0.119 Eqn. 5 0.000073 1.44 0.041

- 0.014-016 t,m i+ 1 l:l'm I - I (5)

PRECISE HYSTERESIS MODEL OF STEEL MEMBERS WITH DAMAGE CUMULATION EFFECTS

Shen and Lu have developed a powerful integration method to calculate the behavior of steel members, Shen and Lu (1983). The method can analyze strength problems and stability problems as well, taking into account the effects of initial geometrical and physical imperfections including residual stresses of the steel member and can give the complete load-deflection relationship of the member including both the ascending and descending branches.

16 Z.- Y. Shen

Since the basic required input of the integration method is the stress-strain relationship of material, the method can be used to obtain the hysteresis model of steel members with damage cumulation effects, if we take the cumulative damage mechanics model of steel under cyclic loading as the input of the method.

Two experiments conducted by the author, Li et al. (1999), were calculated for verifying the proposed method. The material properties and the sectional dimension of the H section columns are shown in Table 2. or, and 6, denote yield stress and yield strain, respectively, crb and 6,, are ultimate stress

and ultimate strain, respectively, b and h are the width and height of the section and t, and t f are

the thickness of the web and the flange, respectively.

TABLE 2 MATERIAL PROPERTIES AND SECTIONAL DIMENSION

OF H-SECTION COLUMNS

E crs crb 6~ 6= h b tf t w

(MPa) (MPa) (MPa) (mm) (mm) (mm) (mm)

1.972x 105 290.08 440.67 0.00147 0.19978 176 160 8 8

The column specimens are cantilevers with length 1100mm. For specimen A, there are only two horizontal forces acting at the top, and for specimen B, there are two horizontal forces and one constant vertical force with magnitude 300kN acting at the top. The loading path of the specimens are shown in Figure 2 and the comparison between calculation and tests can be seen in Figure 3, where Px and Py are the horizontal forces applied on the top ends of the tested column specimens, and Dx

and Dy are the corresponding horizontal displacements. In Figure 3, the test results are shown by

solid lines, and the calculated results by dashed lines. The comparison shows that using the method proposed, a precise hysteresis curve can be obtained by theoretical analysis.

Figure 2: Loading path of the tested specimens

Cumulative Damage Model for the Analysis of Steel 17

Figure 3" Comparison of measured and calculated results

SIMPLIFIED HYSTERESIS MODEL OF STEEL MEMBERS WITH DAMAGE CUMULATION EFFECTS

In order to put the model into practice, the author has established a simplified hysteresis model for plane steel members with damage cumulation effects, Shen et al. (1998). The model can be extended to spatial steel members with damage cumulation effects.

The hysteresis parameter for a spatial member at nth loading can be expressed as the same as the plane member, Shen at al. (1998), if the ~o denotes the yield function of the spatial member.

[k(1-~,D)

(for (P < (Ps,n)

1 ~o - fps,, + ( k - l ) ] (for ~os., <(p < ~Op.,) (6)

J (for q) > ~Op, n )

where D is called equivalent damage index of the cross-section to be substituted for the actual damage,

.~ IDidA~ D - (7)

A is the area of the ith subsection, D, is the damage index of the ith subsection. ~o is the yield

function for the spatial member. ~os, . and ~Op,, denote the value of initial yield and the perfect yield

during the nth loading, respectively, and k is the strain hardening coefficient. The yield function for the spatial member has been developed by many authors, Chen and Austra (1976), Duan and Chen (1990), Kitipomchai et al. (1991 ).

When damage and plastic yield occur at both ends of the spatial member, the elasto-plastic tangent stiffness matrix can be expressed by the following equation

18 Z.- Y. Shen

[KpD] = [Ke]-[K,][G][E][L][E]r[G]r[K,] (8)

where [Ke] is the elastic stiffness matrix of the spatial member element, [G],[E] and [L] can be obtained from the reference, Li et al. (1999) The same experiments mentioned in the above section can be used for verifying the simplified hysteresis model for spatial members with damage cumulation effects. Figure 4 shows the comparison of the test results and the calculated results using the simplified hysteresis model, in which the test results are shown by solid lines and the calculated results by the dashed lines.

Figure 4: Comparison of tested results and calculated results

The comparison illustrates that the accuracy of the simplified hysteresis model is quite good and the model is acceptable for practical analysis.

After every half cycle of loading the strain of every subsection can be calculated, the damage index

D~ of the subsection is obtained by using Eqn. 1 and hence the equivalent damage index /9 of the

member is also obtained by Eqn. 7. Summing up the equivalent damage of all previous half cycles of loading, the cumulated damage index of the member can be obtained.

At the end of loading the cumulated damage index of specimen A and B are 0.121 and 0.305, respectively.

The simplified hysteresis model not only can be used to imitate the actual hysteresis curves of spatial members but also has several unique advantages.

First, it can be used to analyze the hysteresis behavior of a member with initial damage, if the initial damage index is known. Second, it can get the damage index and emulated damage index of a member after loading. And third, it can be used to analyze the behavior of members subjected to the seismic loading more than one times.

NONLINEAR DYNAMIC ANALYSIS OF STEEL FRAMES WITH CUMULATIVE DAMAGE EFFEXTS UNDER SEISMIC ACTION


From the ealsto-plastic tangent stiffness matrix, Eqn. 8, for spatial steel member with damage cttmualtion effects, the global tangent stiffness matrix [Kpo ] of the spatial steel frames can be easily

assembled. Then the deferential equation of dynamic equilibrium for the frames can be built as follows

[M] {A6} + [C] {AS} + [Kpo] {A6} = [MI[E]{A6g } (9)

where [M] is the mass matrix of the structure, [C] is the damping matrix of the structure, {6} is

the displacement vector of the structure and {6g } is the acceleration vector of ground movement.

Solving the above equation, the displacement response of the structure can be determined. Through the displacement, the damage index of each member can be calculated.

In order to verify the proposed method, a shaking table test was conducted by the author. Figure 5 shows the sketch of the spatial steel frame model used for shaking table test. All of the members are H-typed cross-sections. Node numbers are given in Figure 5.

A3 ,1 ? / ~ //~x,2

5

77

2400

Figure 5 Sketch of the frame model.

TABLE 3 MATERIAL PROPERTIES

E(MPa) o- s (MPa) o- b (MPa) 6s 6= ~' 6 2.03x 105 228.44 369.51 0.00113 0.204 64.76% 34.45%

TABLE 4 SECTIONAL DIMENSION OF H-SECTION BEAMS AND COLUMNS

b(mm) h(mm) tw (mm) t y (mm)

Beams 100 150 5 5 Columns 100 120 5 5

The material properties and the sectional dimension of the H-section members are shown in Table 3

20 Z.- Y. Shen

and Table 4, respectively. ~ and 6 denote the sectional reduction and elongation of the material, respectively.

The loading series of the shaking table test are listed in Table 5. The input ground movement in the x- direction and y-direction of the shaking table are El-Centro N-S and El-Centro E-W, respectively, taking the time ratio t / / ~ .

/~/.~

TABLE 5 LOADING SERIES OF THE TEST

Series No. 1 2 3 4 5 Amplitude

of the acceleration

X 0.30g 0.30g 0.50g 0.50g 0.70g Direction (0.309g) (0.311 g) (0.4976) (0.499g) (0.7046)

Y 0.15g 0.15g 0.25g 0.25g 0.25g direction (0.163g) (0.1556) (0.258g) (0.2546) (0.2526)

Series No. 6 7

Amplitude X 0.70g 0.60g of the Direction (0.703g) (0.6036)

acceleration Y 0.25g 0.30g Direction (0.2546) (0.3036)

( �9 ) is the actual amplitude of the shaking table.

8 9 10

0.60g 0.80g 0.80g (0.605g) (0.790g) (0.792g)

0.30g 0.35g 0.35g (0.305g) (0.358g) (0.359~;)

The natural frequencies of the free vibration of the model measured from the experiment are listed in Table 6. There is no different between the initial frequencies and the frequencies after 4th loading, that indicates the seismic action did not damage the steel frame model and the model was still in the elastic range. After the 4th loading, the frequencies decreased successively after subsequent loading due to the damage of the model. Using the simplified hysteresis model, the natural frequencies were calculated and also listed in Table 6 by parentheses. In Table 6, the errors 6 between tested and calculated frequencies are given as well. From Table 6 it can be seen that the frequencies of a damaged structure can be calculated by using the proposed simplified hysteresis model with sufficient accuracy.

TABLE 6 NATURAL FREQUENCIES TESTED AND CALCULATED OF THE FRAME MODEL

1st f (Hz) 2nd f2 (Hz) 3rd f3 (Hz) 4th f4 (Hz) Initial

After 4th Loading After 6th Loading After 8th Loading

After 10th Loading

3.174(3.181) (g =0.22%)

3.174(3.162) (6 =0.38%)

3.163(3.146) (6=0.54%) 3.135(3.111) (6 =0.77%)

3.092(3.089) (6=0.10%)

4.883(5.264) (6=7.23%)

4.883(5.242) (6 =6.85%)

4.863(5.221) (6=6.85%)

4.833(5.179) (6 =6.68%)

4.800(5.155) (6=6.87%)

9.115(9.243) (6=1.38%) 9.115(9.220) (6=1.14%)

9.081(9.202) (6=1.31%)

9.046(9.164) (6=1.28%) . 8.870(9.097) (6=2.49%)

15.462(17.730) (6=12.79%)

15.462(17.668) (8=12.49%)

15.339(17.618) (6=12.93%)

15.304(17.509) (6=12.59%)

15.200(17.390) (6=12.60%)


The dynamic displacements Aland A3 of the model during the 10th loading are shown in Figure 6. The calculated results considering and not considering the cumulative damage effects are illustrated by Figure 7 and Figure 8, respectively.

Figure 6 The tested displacement curves for points A1 and A3 during 10th loading

22 Z.- Y. Shen

Figure 7 The calculated displacement curves for points A1 and A3 during 10th loading considering the cumulative damage

Figure 8 The calculated displacement curves for points A1 and A3 during 10th loading assuming no damage(D=-0)


Table 7 are the maximum and minimum displacements of points A1 and A3 during different loadings. There are three results corresponding to the measured values, the calculated values taking the cumulative damage into account and the calculated values not considering the damage (/9=-0).

TABLE 7 THE MAXIMUM AND MINIMUM DISPLACEMENTS OF POINTS

A1 AND A3 DURING DIFFERENT LOADING

~ - - - . - u . e . . ~ n g ~ No. No. 5 No. 6 No. 7 No. 8 No. 9 No. 10 Displacement A1 Max Tested 2.42 2.45 3.30 3.39 3.61 3.82

(cm) Cal. with D 2.20 2.38 2.96 3.16 3.35 3.59 Cal. D=-0 2.19 2.24 2.79 2.87 3.08 3.11

.

Min Tested -2.44 -2.49 -3.01 -3.26 -3.57 -3.65 (cm) Cal. with D -2.37 -2.38 -2.63 -2.84 -3.09 -3.29

Cal. D=0 -2.36 -2.36 -2.50 -2.67 -2.97 -3.09 A3 Max Tested 1.29 1.31 1.08 1.10 1.39 1.43

(cm) Cal. with D 1.26 1.28 1.02 1.05 1.30 1.34 Cal. D=-0 1'.24 1.25 0.99 1.00 1.23 1.25

, ,

Min Tested -1.30 -1.31 -1.09 -1.09 -1.37 -1.42 (cm) Cal. withD -1.13 -1.14 -1.03 -1.05 -1.31 -1.34

Cal. D=-0 -1.10 -1.11 -1.02 -1.03 -1.28 -1.30

The cumulative damages of the columns of the steel frame model after each loading of the loading series are shown in Table 8. In the Table the two digits of the end member indicate the column (Figure 5) and the first digit means the end where damage occurs.

TABLE 8 CUMULATIVE DAMAGES OF THE COLUMNS OF THE STEEL FRAME MODEL

Loading End Number No. 1-5 5-1 2-6 6-2 3-7 7-3 4-8 8-4

4 0.083 0.020 0.080 0.024 0.085 0.020 0.081 0.025 5 0.089 0.023 0.153 0.029 0.091 0.024 0.155 0.029 6 0.112 0.037 0 . 1 7 1 0.032 0.165 0.034 0.172 0.033 7 0.155 0.051 0.196 0.065 0.201 0.063 0.198 0.066 8 0.219 0.073 0.348 0.068 0.265 0.088 0.364 0.068 9 0.242 0.081 0.371 0.080 0.288 0.096 0.387 0.080 10 0.287 0.095 0.395 0.097 0.334 0.111 0.432 0.109

Loading End Number No. 5-9 9-5 6-10 10-6 7-11 11-7 8-12 12-8

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

0 0.021 0.005 0.057 0.017 0.021 0.005 . . . . 0.042 0.013 0.026 0.128 0.042 0.013 0.084 0.007

From Figures 6 to 8 and Tables 7, 8 the following points can be drawn. First, a severe seismic action

24 Z.- Y. Shen

will cause structures damaged. Second, the damage in a structure will cumulate during successive seismic actions. Third, the cumulative damage will deduce the resistance capacity of structures to the seismic action. Fourth, the dynamic behavior of steel framed structures can be analyzed with acceptable accuracy by using the proposed cumulative damage hysteresis model.

CONCLUSION

Based on a series of experiments and theoretical analysis mentioned in the previous sections, the following main conclusions can be drawn:

(1) The cumulative damage mechanics model of steel under cyclic loading suggested by the author is easy to be used in structural analysis with satisfactory accuracy.

(2) Hysteresis curves of steel planar and spatial members can be precisely imitated by using Shen & Lu's integration method with cumulative damage mechanics model as the input of the steel hysteresis characters.

(3) The simplified hysteresis model of steel members with damage cumulation effects and the elasto- plastic tangent stiffness matrix of the spatial members derived by the author can put the analysis of steel framed structures subjected to more than one time's earthquakes into practice.

(4) Using the method proposed in the paper, the analysis of initially damaged structures becomes practical and the damage of structures due to loading can be calculated in a practical way.

REFERENCES

Chen W. F. and Ausuta T. (1976). Theory of Beam-Columns, vol. 2, MeGraw-Hill, New York.

Duan L. and Chen W. F. (1990). A Yield Surface Equation for Doubly Symmetrical Sections. Engineering Structures 12:4, 114-118.

Li G. Q. et al. (1999). Spatial Hysteretic model and Elasto-plastic Stiffness of Steel Columns. Journal of Constructional Steel Research 50:, 283-303.

Kachanov L. M.(1986). Introduction to Continuum Damage Mechanics, Martinus Nijhoff Publishers, Dordrecht.

Kitipomchai S. et al. (1991). Single-equation Yield Surfaces for Monosymmetric and Asymmetric Sections, Engineering Structures 13:10, 366-370.

Shen Z. Y. and Dong B.(1997). An Experiment-based Cumulative Damage Mechanics Model of Steel under Cyclic Loading. Advances in structural Engineering 1:1, 39-46.

Shen Z. Y., Dong B. and Cao W. w. (1998). A Hysteresis Model for Plane Steel Members with Damage Cumulation Effects. Journal of Constructional Steel Research 48:2/3, 79-87.

Shen Z. Y. and Lu L.W. (1983). Analysis of Initially Crooked, End Restrained Steel Columns, Journal of Constructional Steel Research, 3:1, 40-48.

RECENT RESEARCH AND DESIGN DEVELOPMENTS IN COLD- FORMED OPEN SECTION AND TUBULAR MEMBERS

Gregory J. Hancock

Department of Civil Engineering, University of Sydney NSW, 2006, Australia

ABSTRACT

A major research program has been performed for 20 years at the University of Sydney on cold- formed open section and tubular structural members. This research has included both members and connections and has been performed predominantly for high strength steel sections. The open section members include mainly angles, channels (with and without lips) and zeds, and the tubular members include mainly rectangular (RHS) and square (SHS) hollow sections. The research has been mainly incorporated in the Australian Steel Structures Standard AS 4100-1998 and the Australian/New Zealand cold-formed steel structures standard AS/NZS 4600. The paper summarises the recent developments in the research and points to on-going and future research needs.

KEYWORDS

Cold-formed, Steel Structures, Structural Design, Open Sections, Tubular Sections, Standards

INTRODUCTION

Cold-formed structural members are being used more widely in routine structural design as the world steel industry moves from the production of hot-rolled section and plate to coil and strip, often with galvanised and/or painted coatings. Steel in this form is more easily delivered from the steel mill to the manufacturing plant where it is usually cold-rolled into open and closed section members. In Australia, of the approximately one million tonnes of structural steel used each year, 125,000 tonnes is used for cold-formed open sections such as purlins and girts and 400,000 tonnes is used for tubular members. Tubular members are normally produced by cold-forming with an electric resistance weld (ERW) to form the tube. In most applications of open sections, the coil is coated by zinc or aluminium/zinc as part of the steel supply process. In some applications of tubular members, the sections are in-line galvanised with a subsequent enhancement of the tensile properties. The resulting product is called DuraGal (BHP (1996)). In Australia, the total quantity of cold-formed products now exceeds the total quantity of hot-rolled products.

The open section members are normally produced from steel manufactured to AS 1397 (Standards Australia, 1993). This steel is cold-reduced and galvanised and typically has yield stress values of

25

26 G.J. Hancock

450 MPa for steel greater than 1.2 mm (called G450), 500 MPa for steel in the range 1.0 to 1.2 mm (called G500) and 550 MPa for steel less than 1.0 mm (called G550). Hence the majority of the sections are constructed from high strength cold-reduced steel. Structural steel hollow sections are normally produced to the Australian Standard AS 1163 (1991). They are all cold-formed and usually have stress grades of 250 MPa (called C250), 350 MPa (called C350) and 450 MPa (called C450). The most common grade is C350 which has the yield strength enhanced from 300 MPa to 350 MPa during the forming process. The C450 grade is often achieved by in-line galvanising (BHP, 1996).

The Australian Standard for the design of steel structures AS 4100 was first published in limit states format in 1990 and permitted the use of cold-formed tubular members to AS 1163. Cold-formed tubular members had been permitted to be designed to the permissible stress steel structures design standard AS 1250 (Standards Australia 1981) since an amendment in 1982. However, the research on cold-formed tubular members was limited in many areas, particularly flexural members and connections, and so a significant research program was undertaken. Much of this research which was incorporated in the most recent edition of AS 4100 (Standards Australia, 1998) is described in Zhao, Hancock and Sully (1996).

The Australian/New Zealand Standard AS/NZS 4600 (Standards Australia 1996) for the limit states design of cold-formed open section members was published in 1996 and was based mainly on the American Iron and Steel Institute Specification (AISI, 1997). However, the Australian/New Zealand Standard permitted the use of high strength steel to AS 1397 and so research data was incorporated for this purpose.

This paper summarises the most recent research in the following areas:

�9 High strength angle sections in compression �9 Lipped and unlipped channel sections in compression �9 Unlipped channel sections in bearing �9 Lateral buckling of channel sections �9 Bolted and screwed connections in G550 steel �9 Tubular beam-columns �9 Bolted moment end-plate connections �9 Plastic design of cold-formed square and rectangular hollow sections

OPEN-SECTION MEMBERS

Axial Compression of Cold-Formed Angles

A major research program was performed on cold-formed angles formed by cold-rolling and in-line galvanising so that the final product had a yield stress of 450 MPa (BHP (1996)). Sections ranging from slender (EA 50*50*2.4 mm) to non-slender (EA 50"50"4.7mm) were tested in pin-ended concentric compression such that flexural buckling could occur about the minor principal axis. Detailed measurements of the stress-strain characteristics of the material forming the sections, the residual stresses and overall geometric imperfections were taken. The results are reported in Popovic, Hancock and Rasmussen (1999).

The results of the tests are compared with the design rules of AS 4100 (Standards Australia 1998) and AS/NZS 4600 (Standards Australia 1996) in Figs 1 and 2. Comparison of the angle tests is shown with AS 4100 in Fig. 1 and AS/NZS 4600 in Fig. 2 which only includes the slender sections. The slender sections failed in a combination of flexural and flexural-torsional buckling. For the sections

Recent Developments in Cold-Formed Open Section and Tubular Members 27

tested, it can be concluded that the design procedure in AS 4100 is not satisfactory if the design yield stress is taken from stub column strengths as shown in Fig. 1 but it is satisfactory if it is based on coupons taken from the flats. The design procedure does not include specific rules for flexural- torsional buckling. Higher design curves than recommended by AS 4100 can be used for the non- slender sections which did not include torsional deformations in the failure mode. As demonstrated in Fig. 2, the design procedure in AS/NZS 4600 is conservative for short length sections where torsional buckling is included twice by virtue of an effective section for local buckling and torsional buckling stresses in the column design. For longer length columns, the additional required moment equal to a load eccentricity of L/1000 need only be applied for slender sections as shown in Fig. 2. Non-slender (fully effective) sections do not need this additional eccentricity as demonstrated in Popovic, Hancock and Rasmussen (1999).

1.4

Long Column Tests - Pinned Ends AS 4100 Column Curves (Ns = Stub Column Strength)

1.2

1.0

(Xb = " 1.0 SSRC 1

(Xb = - 0.5 AISC-LRFD

(Xb= 0.0 SSRC2

(Xt)= 0.5

(Xb= 1.0 SSRC3

�9 L50x50x2.5 2~ 0.6 / �9 L50x50x4.0

�9 L50x50x5.0 !

0.4 , ~ = r-~.~ .~Kf-~

_:~ o.~ k:--~ 0.0 I I I I I I . . . .

0 20 40 60 80 100 120 140 160 180 200 ~n

Fig. 1 Comparison of angle section test strengths with hot-rolled design standards

1.4

1.2

1.0

0.8 Z

3 0.6 z

0.4

0.2

Long Column Tests L50x50x2.5 AS 4600 and AISI Column Curves (fy = 396 MPa)

l - Pin-Ended f-t buckling controls �9 Fixed-Ended

" / . . ff-t X A e NC

/ " �9 / . ,L f lexure controls

[ - - ~ 0 0 0 .

�9 I

0 20 40 60 80 1 O0 120 140 160 180 200 Le/r

0 . 0

Fig. 2 Comparison of angle section test strengths with cold-formed design standards

28 G.J. Hancock

Lipped and Unlipped Channels in Compression

A test program on unlipped (plain) and lipped channels in compression was performed where the channels were compressed between fixed ends and pinned ends (Young and Rasmussen, 1998a, 1998b). Whereas it is well-known that local buckling of pin-ended channel columns induces overall bending, this phenomenon does not occur in fixed-ended channel columns that remain straight after local buckling and only bend when overall buckling occurs. These fundamentally different effects of local buckling on the behaviour of pin-ended and fixed-ended channel sections lead to inconsistencies in traditional design approaches. The research program investigated these phenomena and compared the results with the design approach in AS/NZS 4600.

Results for plain channels compressed between fixed ends and pinned ends are shown in Figs. 3 and 4. The fixed ended tests (Fig. 3) clearly show that the formulae for column strength alone accurately predict the test results and the sections carry loads well in excess of the local buckling load. However, the pin-ended tests (Fig. 4) show that the loads carried are not significantly greater than the local buckling load for the pin-ended shorter length test specimens. By comparison, the design predictions accounting for the shift in effective centroid are very conservative. Similar results are achieved for lipped channels but the differences are not so marked as shown in Young and Rasmussen (1998b).

Fig. 3 Comparison of fixed-ended plain channel test strengths with design strengths

Recent Developments & Cold-Formed Open Section and Tubular Members 29

Fig. 4 Comparison of pin-ended plain channel test strengths with design strengths

Unlipped Channel Sections in Bearing

An experimental investigation of cold-formed unlipped channels subject to web crippling has been described in Young and Hancock (1998). The concentrated loading forces were applied by means of bearing plates which acted across the full flange widths of the channels. The web crippling results were compared with AS/NZS 4600. The design web crippling strength predictions given by the standard were found to be very unconservative for the unlipped channel sections tested which had web slenderness values ranging from 16.9 to 38.3. These slenderness values are fairly stocky compared with those for the test data base used for the American Iron and Steel Institute Specification (AISI, 1996) on which AS/NZS 4600 was based. A simple plastic mechanism model for the web crippling strength of unlipped channels was proposed. The plastic mechanism model is most appropriate for the stocky web sections which fail as a mechanism due to the load eccentricity resulting from the rounded comers.

Lateral Buckling of Channel Sections

A research program on the lateral buckling capacities of cold-formed lipped channel-section beams (CFCs) was undertaken and published in Put, Pi and Trahair (1999a). It has been argued that the design approximations based on hot-rolled beams may be inappropriate for CFCs, because of the very different cross-sectional shape and method of manufacture. The paper describes lateral buckling tests on simply supported unbraced CFCs of two different cross-sections which were undertaken to resolve the issue. However, the lateral buckling tests showed that the CFCs failed catastrophically by local and distortional buckling of the compressed element of the cross-section after quite large deformations. The failure moments were lower when the beam lateral deflection increased the compression in the compression lip, and higher when they increased the compression in the flange- web junction.

The results in Fig. 5, which are taken from Put, Pi and Trahair (1999a), show some interesting features when compared with the predictions of AS 4100 and AS/NZS 4600. The stockier section C10019 is fairly accurately predicted by AS/NZS 4600 although it is slightly conservative at longer lengths.

30 G.J. Hancock

Both the distortional buckling strength Md and section strength Ms reasonably accurately predict the test results at shorter lengths as well as the longer lengths specimens which fail in the lip buckling mode. AS 4100 provides an unconservative estimate of the section strength. However, for the more slender C10010 section, the distortional buckling strength Md and section strength Ms predictions of AS/NZS 4600 are unconservative. By comparison, AS 4100 is more accurate although this may be coincidental since the design method in AS 4100 was not developed for local and distortional buckling of such slender sections and the prediction is based on a very simple model of local buckling. Further, there seems to be a significant interaction between the lateral buckling mode and lip buckling at longer lengths with both AS 4100 and AS/NZS 4600 providing unconservative predictions of the strength. Further investigations of this phenomenon are required for slender sections.

A separate paper on the bending and torsion of cold-formed channel beams loaded concentrically and eccentrically at mid-span has been published (Put, Pi and Trahair, 1999b). The tests show that the beam strengths decrease as the load eccentricity increases and that the strength is higher when the load acts on the centroid side of the shear centre than when it acts on the side away from the shear centre. Good agreement is demonstrated between the test results and analytical predictions of the strengths. An extended series of analytical expressions was used to develop simple interaction equations that can be used in the design of eccentrically loaded cold-formed channel beams.

Fig. 5 Lateral buckling tests of cold-formed channels compared with design strengths

Bolted and Screwed Connections in G550 Sheet Steels

Cold-formed structural members are usually fabricated from sheet steels which must meet various material requirements prescribed in applicable national design standards. AS/NZS 4600 allows the use of thin (t< 0.9 mm), high strength (fy = 550 MPa) sheet steels in all structural sections. However, in the design the engineer must use a value of yield stress and ultimate strength reduced to 75% of the minimum specified values, due to lack of ductility exhibited by sheet steels which are cold reduced to thickness. Three papers investigating the ductility (Rogers and Hancock, 1997), bolted connection capacity (Rogers and Hancock, 1998) and screwed connection capacity (Rogers and Hancock, 1999) have recently been published summarising research investigating thin G300 and G550 sheet steels.


Fig. 6 Bearing strength of bolted connections in thin sheet steels compared with design strength

32 G.J. Hancock

In general, the problems with these steels were not a reduction in section strength due to the low ductility, but a problem in the bearing capacity of thin sections. This can be clearly seen in Fig. 6 where the bearing capacity of bolted connections in 0.42 mm G550 steel and 0.60 mm G550 and G300 steel are well below the predictions of AS/NZS 4600 and other design standards. The only standard to provide a reasonable prediction of this phenomenon was the Canadian standard for cold- formed steel structural members (CSA, 1994) which had a bearing coefficient which varied with the d/t ratio of the bolt and sheet. Proposals have been made for the Australian standard and American specification to adopt this approach.

Similar characteristics were discovered for screwed connections as reported in Rogers and Hancock (1999). The recommended beating coefficients also depend on the screw diameter to sheet thickness ratio and are shown in Fig. 7.

Fig. 7 Existing and Proposed Bearing Coefficients for Screw Connections

TUBULAR MEMBERS

Tubular Beam-Columns

A test program was conducted into the behaviour of cold-formed square hollow section (SHS) beam- columns of slender cross-section (Sully and Hancock, 1998). The experimental program follows an earlier test program on compact SHS beam-columns (Sully and Hancock, 1996). The tests were conducted in a purpose built testing rig capable of applying load and moment in a constant ratio. The tests specimens were pin-ended and were loaded at two different ratios of end moment. The results of the testing program have been compared in Sully and Hancock (1998) with the current design rules in AS 4100-1998, the American Institute of Steel Construction Specification and Eurocode 3.

From the interaction tests, it is clear that the slender sections collapse more suddenly as a result of inelastic local buckling than do compact sections. The long yielding plateau and associated high curvatures observed in the compact tests (Sully and Hancock, 1996) were not evident for the slender sections. Local imperfections are more easily formed in the slender sections particularly from the welding of the connection components. These local imperfections can have a detrimental effect on the section bending capacity of the member causing premature collapse through local instability. The possibility of this type of failure occurring is of particular concern in structures where maximum moments occur at the member connections. Further research is required in this area.

For the long length interaction tests where the maximum load was reached prior to local instability, the design rules in AS 4100 for compact doubly-symmetric sections are applicable. However, this


does not preclude the case of more slender sections than those tested which may locally buckle before reaching maximum load. Further investigation is required to determine if the AS 4100 compact section interaction rules are appropriate for non-uniform moment. Short length interaction tests indicated that local instability affects the beam-column strength more severely for short length specimens. Again, further investigation is required to determine if the AS 4100 interaction rule is appropriate for non-uniform moment. The simple linear interaction rule for non-compact sections in AS 4100 appears satisfactory for all the sections tested.

Bolted Moment End Plate Connections

Moment end plate connections joining 1-section members are used extensively and considerable documentation on their behaviour exists in the literature. In contrast, research on moment end plate connections joining rectangular and square hollow sections is limited and satisfactory design models are not widely available. The research on tubular end plate connections that has been conducted has concentrated on pure tensile loading or combined compression and bending. An analytical model to predict the serviceability limit moment and ultimate moment capacities of end plate connections joining rectangular hollow sections has been presented in Wheeler, Clarke, Hancock and Murray (1998). The connection geometry considered utilises two rows of bolts, one of which is located above the tension flange and the other of which is positioned symmetrically below the compression flange. Using a so-called modified stub-tee approach, the model considers the combined effects of prying action caused by flexible end plates and the formation of yield lines in the end plates as shown in Fig. 8. The model has been calibrated against experimental data from an extended test program forming part of the research project (Wheeler, Clarke and Hancock, 1995).

Of the three types of end plate behaviour considered in the stub-tee model (thick, thin and intermediate), the paper recommends that the end plate connections be designed to behave in an intermediate fashion, with the connection strength being govemed by tensile bolt failure. Thin plate behaviour results in connections that are of very ductile and exhibit extremely high rotations, while connections exhibiting thick plate behaviour are very brittle and may be uneconomical.

M M M

O O

O O

O O

(a) Mode 1 (b) Mode 2 (c) Mode 3

Fig. 8 Yield line mechanisms for bolted moment end plate connection

34 G.J. Hancock

Plastic Design of Cold-Formed Square and Rectangular Hollow Sections

Plastic design of cold-formed members has been limited by design standards such as AS 4100 since plastic design methods were verified by tests on hot-rolled steel members, which have notably different material properties compared to cold-formed hollow sections. To investigate the suitability of cold-formed hollow sections for plastic design, a series of bending tests examined the influence of web slenderness on the rotation capacity of cold-formed rectangular hollow sections (Wilkinson and Hancock 1998a). The results indicate that the plastic (Class 1) web slenderness limits in design standards, which are based on tests of I-sections, are not conservative for RHS. Some sections, which are classified as compact or Class 1 by current steel specifications, do not demonstrate rotation capacity suitable for plastic design. The common approach in which the flange and web slenderness limits are given independently is inappropriate for RHS. There is considerable interaction between the webs and the flange, which influences the rotation capacity, as shown by the approximate iso- rotation curves in Fig. 9. A proposal for a bilinear interaction formula between the web and flange slenderness limits for compact RHS is also shown in Fig. 9.

~" 50 r ~- 45

~40

N 35 II ~ 30

~ 25

s 20

"~ 10

~ 5

~ o

20

Possible new AS 410D I I Compact Limit Compa~:t

I Limit ~ < 70- 5~/6

~ ~ ~ ' ~ ~f< 30

s ,

Web Slenderness (AS 4100) ~ - (d-2t)/t-~(J'~/250)

30 40 50 60 70 80 90

Fig. 9 Iso-rotation curves and proposed compact limit for webs of rectangular hollow sections

Further research (Wilkinson and Hancock, 1998b,c,d) has recently been completed investigating the plastic behaviour and design of portal frames and connections within the frames. These papers described tests of different types of column-rafter knee connections, and tests of 3 large scale portal frames manufactured from cold-formed Grade C350 and Grade C450 cold-formed RHS. Some welded connections experienced fracture near the heat affected zone caused by welding, before adequate plastic rotation was achieved. A plastic mechanism was formed in each frame and plastic collapse occurred. The ultimate loads of the frames can be predicted by plastic analyses although second order effects and the shape of the stress-strain curve may be important.

Recent Developments in Cold-Formed Open Section and Tubular Members

CONCLUSIONS

35

A wide ranging research program on cold-formed members which has been performed at the University of Sydney over more than 15 years has been summarised. Emphasis has been placed on test data and comparison of the test results with design standards, particularly the Australian Standard AS 4100-1998 Steel Structures and the Australian/New Zealand Standard AS/NZS 4600:1996 Cold- Formed Steel Structures. The research has been performed mainly on high strength steels with the strength typically ranging from 350 MPa to 550 MPa. Both members and connections have been investigated.

There are several general conclusions that can be reached:

1. Open sections such as angles and channels in compression often suffer from structural instability in the elastic range due to the slender nature of the sections and the high yield strength of the sections. Torsional modes or torsional modes combined with flexure can become dominant. Care has to be taken with loading conditions such as fixed or pinned ends and assumptions regarding the line of action of axial load since it can have a large effect on axial load capacity.

2. Laterally unbraced flexural members may undergo lateral buckling with significant interaction with local and distortional modes. Clearly, more research is required in this area as the project described has found certain unconservative behaviour when compared with existing design standards for slender sections. Bearing failure may also be important in flexural members because the cold-formed sections have rounded comers and unstiffened webs.

3. Ductility was not found to be a problem in any of the members or connections tested even with high strength (G550) cold-reduced steel. Of greater importance is the thinness of the material and the types of bearing failures that can occur in bolted and screwed connections. New design rules have been proposed for these cases.

4. Slender tubular members are more likely to undergo inelastic local buckling in compression or combined compression and bending. The design rules for these types of members are included in AS 4100-1998. Care needs to be taken with welded connections to slender cold-formed tubes. Section distortion may occur and aggravate inelastic local buckling of the slender cold-formed sections.

5. Proposals for the design of bolted moment end plates in cold-formed tubular members have been made. This type of connection can be designed for satisfactory performance provided the welding of the tubes to the end plates is carried out to rigorous welding standards.

6. The plastic design of cold-formed tubular (RHS and SHS) members is possible provided the aspect ratio of the sections used for plastic design is chosen carefully. The existing Class 1 section web slenderness limits, which are based on I-section members, are unconservative for RHS members. Revised design rules have been proposed. Care also needs to be taken when designing moment resisting connections in cold-formed tubular members to ensure they have adequate rotation capacity for plastic design.

ACKNOWLEDGEMENTS

This paper has been prepared based on the research of many people. Permission to use their test data and resulting graphs is appreciated. They were all supplied in electronic form from the original authors which explains the slight change in format between the different figures. The following

36 G.J. Hancock

people are gratefully acknowledged: Emeritus Professor NS Trahair, Associate Professor Kim Rasmussen, Dr Murray Clarke, Dr Ben Young, Dr Andrew Wheeler, Dr Colin Rogers, Mr Bogdan Put and Mr Tim Wilkinson.

REFERENCES

American Iron and Steel Institute (1997). Specification for the Design of Cold-Formed Steel Structural Members, Washington, DC.

BHP Structural and Pipeline Products (1997). DuraGal Design Capacity Tables for Structural Steel Angles, Channels and Flats, BHP, Sydney.

Canadian Standards Association (1994). "Cold Formed Steel Structures Members", Toronto, Canadian Standards Association.

Popovic, D, Hancock, GJ and Rasmussen, KJR (1999). "Axial Compression Tests of Cold-Formed Angles", Journal of Structural Engineering, ASCE, 24:5, 515-523.

Pi, Y-L, Put, BM and Trahair, NS (1999a). "Lateral Buckling Tests of Cold-Formed Channel Beams", Journal of Structural Engineering, ASCE, 125: 5, 532-539.

Put, BM, Pi, Y-L and Trahair, NS (1999b). "Bending and Torsion of Cold-Formed Channel Beams", Journal of Structural Engineering, ASCE, 125-5, 540-546.

Rogers, CA and Hancock, GJ (1997). "Ductility of G550 Sheet Steel in Tension", Journal of Structural Engineering, ASCE, 123:12, 1586-594.

Rogers, CA and Hancock, GJ (1998). "Bolted Connection Tests of Thin G550 and G300 Sheet Steels", Journal of Structural Engineering, ASCE, 124:7, 798-808.

Rogers, CA and Hancock, GJ (1999). "Screwed Connection Tests of Thin G550 and G300 Sheet Steels", Journal of Structural Engineering, ASCE, 125:2, 128-136.

Standards Association of Australia (1991), Structural Steel Hollow Sections, AS 1163-1991.

Standards Australia (1993). Steel Sheet and Strip - Hot Dipped Zinc-Coated or Aluminium/Zinc- Coated, AS 1397-1993.

Standards Association of Australia. (1998). Steel Structures, AS 4100-1998.

Standards Association of Australia/Standards New Zealand (1998). Cold-Formed Steel Structures, AS/NZS 4600:1996.

Sully, R and Hancock, GJ (1996). "Behaviour of Cold-Formed SHS Beam Columns", Journal of Structural Engineering, ASCE 122:3, 326-336.

Sully, RM and Hancock, GJ (1998). "The Behaviour of Cold-Formed Slender Square Hollow Section Beam-Columns", Proceedings of the Eighth International Symposium on Tubular Structures, Singapore, 445-454.

Recent Developments in Cold-Formed Open SeCtion and Tubular Members 37

Wheeler AT, Clarke MJ & Hancock G J, (1995), "Tests of Bolted Moment End Plate Connections in Tubular Members", Proceedings, 14th Australasian Conference on Structures and Materials, University of Tasmania, Hobart, Tasmania, 331-336.

Wheeler, AT, Clarke, MJ, Hancock, GJ and Murray, TM (1998). "Design Model for Bolted Moment End Plate Connections Joining Rectangular Hollow Sections", Journal of Structural Engineering, 124:2, 164-173.

Wilkinson, T and Hancock, GJ. (1998a). "Tests to Examine Compact Web Slenderness of Cold- Formed RHS", Journal of Structural Engineering, ASCE, 124:10, 1166-174.

Wilkinson T and Hancock GJ (1998b). "Tests of Stiffened and Unstiffened Knee Connections in Cold-Formed RHS", Tubular Structures VIII, Proceedings, 8th International Symposium on Tubular Structures, Singapore, 177-186.

Wilkinson T and Hancock GJ (1998c)."Tests of Bolted and Intemal Sleeve Knee Connections in Cold-Formed RHS", Tubular Structures VIII, Proceedings, 8th International Symposium on Tubular Structures, Singapore, 187-195.

Wilkinson T and Hancock GJ (1998d). "Tests of Portal Frames in Cold-Formed RHS", Tubular Structures VIII, Proceedings of the 8th International Symposium on Tubular Structures, Singapore, 521-529.

Young, B and Rasmussen, KJR (1998a). "Tests of Fixed-Ended Plain Channel Columns", Journal of Structural Engineering, ASCE, 124-2, 131-139.

Young, B and Rasmussen, KJR (1998b). "Design of Lipped Channel Columns", Journal of Structural Engineering, ASCE, 124-2, 140-148.

Young, B and Hancock, GJ (1998). "Web Crippling Behaviour of Cold-Formed Unlipped Channels", 14 th International Specialty Conference on Cold-Formed Steel Structures, St Louis, October, 127-150.

Zhao, X-L, Hancock, GJ and R Sully (1996). "Design of Tubular Members and Connections using Amendment No 3 to AS 4100", Steel Construction, Australian Institute of Steel Construction, 30:4, 2- 15.

Wheeler, AT, Clarke, MJ and Hancock, GJ (1995)."Tests of Bolted Moment End Plate Connections in Tubular Members", Proceedings, 14 th Australasian Conference on Mechanics of Structures and Materials, University of Tasmania, 331-336.


BEHAVIOUR OF HIGHLY ~ D U N D A N T M U L T I - S T O ~ Y BUILDINGS UNDER COMPARTMENT F I ~ S

J.M. Rotter

School of Civil and Environmental Engineering, University of Edinburgh, Edinburgh EH9 3JN, UK

ABSTRACT

In current design practice, structural members under fire are treated as if each member is isolated and determinate, with the strength controlled by material property degradation at high temperature. This treatment might well seem appropriate for compartment fires where only the structural members in the compartment are affected. However, it is seriously misguided for large redundant composite multi- storey building structures, because the major influence of the adjacent cool structure on the behaviour of elements under extreme heating is ignored. The interactions between adjacent parts can completely transform the structural response and invalidate the design assumptions. Key features of the behaviour of a structural element under fire within a highly redundant structure are examined in this paper. The surrounding cool structural components restrain thermal expansion and provoke other displacements. Several examples are presented of the behaviour of quite simple structures which illustrate the roles of thermal expansion, loss of material strength, the relative stiffness of adjacent parts of the structure, the development of large deflections, post-buckling and temperature gradients. Although simple, the relevance of these examples to complete structures is clear. Several counter-intuitive phenomena are noted. From these discoveries, some significant implications are drawn for the philosophy of design to be used for large buildings under fire.

KEYWORDS

Compartment fires, composite, fire, floor systems, large deflections, membrane effects, multi-storey, non-linear response, plasticity, post-buckling, restraint, thermal buckling, thermal expansion.

INTRODUCTION

For fire control reasons, the spaces within large buildings have long been divided into compartments to ensure that the fire does not spread and that its effects can be contained locally. The consequence for the structure is that only a local part is severely heated, whilst its surroundings remain comparatively cool. The result is that a very hot weakening and expanding local region is contained within a large cool mass. The interaction between these two regions is the subject of this paper. The full scale fire tests on the composite building at Cardington (Kirby, 1997; Moore, 1997) showed that very high temperatures could be sustained in the steel joists. Since the temperatures were so high that the steel strength was effectively destroyed, and yet runaway failures did not occur, researchers are presented with a significant task to explain why; this paper sets out some fundamental parts of that explanation.

Current assessment methods for the fire resistance of a building structure (ENV 1994-1-2, 1995) are based on the fire testing of single elements, evaluated in terms of the time to failure. Naturally, these

39

40 J.M. Rotter

tests are supported by a good understanding of the phenomena and by calculation of the effects of fire in reducing the member's strength, which extend the scope and confidence of the assessment far beyond the conditions actually tested. However, the structural environment of a member in such a fire test is not well related to the situation in the complete structure in a real compartment fire. It has long been recognised that the thermal scenario is unrealistic, but the greater shortcomings of the structural idealisation have not been properly identified.

In a determinate structure, the pattem of internal forces and stresses can be determined using only equilibrium considerations, provided the displacements are small. Most fire tests on isolated members match this condition. By contrast, in a redundant structure, the pattern of internal forces and stresses depends on the relative stiffnesses of parts of the structure. In the training of structural engineers, the significance of lack of fit and imposed displacements in redundant structures is not strongly emphasised, and building structures are often portrayed as dominated by bending actions, accompanied by axial forces in the columns which are rather easily determined. There are good reasons for these choices, based on the theorems of plasticity. Whilst these ideas are effective in ambient temperature design, they do not carry over very well into the fire scenario.

Figure 1: Runaway failure in determinate structure under fire

At collapse, determinate and redundant structures are more sharply differentiated than the above simple definitions suggest. The determinate structure collapses when the most highly stressed region reaches the local strength, and this strength may be reduced by elevated temperatures. The concept of "runaway" failure in a structure under fire derives from this situation (Fig. 1) where the rapid deterioration of the properties of the material causes deflections to increase very rapidly when the temperature reaches the appropriate value (which naturally depends on the load level).

However, in the redundant structure with adequate ductility and without instability, different stress paths may support additional load when the local strength is reached at a single location. This effect is classically defined as "plastic redistribution", but it is open to wider interpretation if different load carrying mechanisms can come into operation. Where a structure is very redundant and there are many alternative load paths, large deformations can develop without a loss of its capacity to carry the imposed loads, and it may be difficult to decide how to define "failure". The question of how to define failure is faced in structural engineering fields apart from fire; researchers in pressure vessels and rectangular storage structures are faced with the need for new failure definitions which can incorporate survival under large displacements. It should be noted that the theorems of plasticity on which structural engineering design depends so much depend not only on ductility and lack of instability, but they are strictly only valid for small displacements.

Behaviour of Highly Redundant Multi-Storey Buildings 41

Structural engineering practice for the design of frames at ambient temperature is chiefly based on the concept that the forces in individual members can be found from a global elastic analysis, but the members are subsequently proportioned according to an ultimate strength assessment for each member alone. Thus, the inelastic and large deformation behaviour which may affect the member when alone is deemed to have little effect on the response of the complete structure. This design procedure cannot capture the phenomena which occur in highly restrained structural elements under fire.

The studies described in this paper arose from attempts to understand the complex behaviours seen in calculations (Sanad et al., 1999) to model the Cardington full scale fire tests on a composite building (Kirby, 1997; Moore, 1997). Many conclusions concerning behaviour could be drawn directly from the tests (Martin, 1995; Newman, 1997), but those presented here are more difficult to extract from the experimental record. The paper is particularly concerned with the development of large displacements, since these permit the new load-carrying mechanism of tensile membrane action to come into play.

EFFECTS OF IN-PLANE RESTRAINT IN COMPARTMENT FIRES

When a compartment fire occurs in a large building, the effects are felt on the floor system above the fire and the columns of the fire floor. The columns are critical to the building's survival, and need fire protection; they are not discussed further here. For the floor system, the compartment boundaries effectively isolate the surrounding structure from really high temperatures, and the floor' s continuity in its own plane means that differential thermal expansions play a dominant role (Fig. 2).

Figure 2: Plan view of floor with heated compartment under tire

Under fire conditions, temperatures of the order of 800 or 1000~ are achieved, and the thermal strains are extremely large. In such highly redundant structures, the consequent lack of fit means that the cold structure imposes huge forces on the heated region, but these are relieved by two mechanisms, which are the subject of this paper: plastic straining (with decreasing material strength) and post-buckling

42 J.M. Rotter

large displacements. The hot zone covers a limited area, determined by the compartment size, and the compressive stresses which develop within it are governed by the lack of fit, the in-plane stiffness of the floor system around it, and the stress-relieving mechanisms of plasticity and post-buckling. Most importantly, the deflections which develop within the hot region are not controlled by material degradation, as was the case for a determinate structure (Fig. 1) but by restrained thermal expansion. No "runaway" collapse conditions occur, provided the building has adequate in-plane restraint. The development of large deflections limits the damage to the structure, and these large deflections permit different load carrying mechanisms to develop (other than small deflection bending).

The differential thermal environment is not simply a contrast between the heated zone and the cold surroundings. Exposed steel members (low mass and high thermal conductivity) rapidly achieve high temperatures, but the concrete slab (high mass and low thermal conductivity) develops significant temperature gradients through its thickness, and with its high indeterminacy as a plate structure, acts as a major restraint against thermal expansion. As the slab is heated, its expansion must also be accommodated by the mechanisms described above, but its slenderness means that buckling, rather than plasticity, is the dominant phenomenon. Thermal gradients, both in the two dimensional horizontal plane, and vertically through the slab, strongly affect the deflections of the structure.

Yielding under thermal expansion

The floor system of a building is designed to carry load by bending and shear. The slab often spans between beams in something like one way action, and its behaviour is most easily understood by considering beam behaviours. As noted above, significant axial forces develop in a beam or slab if it is heated and fully or partially restrained against axial expansion (or contraction during cooling). Depending on the surroundings, these forces can be either beneficial or deleterious to the performance of the structure. When floor slabs expand, they can exert enormous forces on the surrounding structure. The first key aspect of the floor system behaviour under fire is therefore in the plane of the floor.

If the floor system provides stiff restraint, the thermal expansion forces can become very large. A fully restrained steel element under thermal expansion reaches compressive yield at a temperature of only:

% (1) ATe = Ec~

in which ATy is the temperature rise to cause yield, tx is the thermal expansion coefficient and E is the elastic modulus of steel. This relationship shows that a temperature change of 102~ for 250 grade steel and 142~ for 350 grade steel (ignoring any material degradation) is needed to achieve yield. Compared with the 800 or 1000~ which the fire may achieve, these temperatures are so low that there is plenty of scope for high stress development in real fires even when the restraint is only partial.

Key understanding of responses to thermal expansion

When heated, a structure displays a variety of responses. Structural engineers, trained at the outset to relate deflections to structural stiffnesses, stresses to deflections, and growing deflections to material degradation, are often surprised by the more complex responses arising from thermal expansion. Indeed, because the structural fire literature is mostly concerned with determinate structures in which these connections are valid, the importance of thermal expansion strains is often lost.

To understand redundant structural behaviours under fire, attention should be focused on the strain state, since this is where thermal expansion, stress-strain relationships, and strain-displacement relationships can all be brought together. The key relationships needed for understanding are:


~total -" ~thermal + ~mechanical ( 2 )

with: ~mechanical --> 6 stresses (3)

e,otat --> 8 deflections (4) The total strains govern the deformed shape of the structure ~5, through kinematics or compatibility. The stress state in the structure cr (elastic or plastic) depends only on the mechanical strains.

In a structure whose displacements are not restrained, thermal strains are free to develop in an unrestricted manner. If there are no external loads, axial expansion or thermal bowing results from:

~,o,al = ~,h~.n~ ( 5 )

[Stherma ! ~ 5 deflections (6)

By contrast, if thermal strains are fully restrained without external loads, thermal stresses and plastification result from:

0 = Etherma I + [;mechanical (7)

E:mechanical ~ Cr stresses (8)

In real structures under fire, most situations have a complex mix of mechanical strains due to applied loading and mechanical strains due to restrained thermal expansion. These lead to combined mechanical strains (Eqn 8) which often far exceed the yield values, resulting in extensive plastification. The deflections of the structure, by contrast, depend only on the total strains, so these may be quite small if there is high restraint, but they are associated with extensive plastic straining. Alternatively, where less restraint exists, larger deflections may develop, but with a lesser demand for plastic straining and so less destruction of the stiffness properties of the materials.

These relationships show that larger deflections may reduce material damage and may simultaneously correspond to higher structural stiffnesses. Alternatively, they show that high restraint may lead to smaller deflections with lower stiffnesses due to material damage. Thus they cause structural situations which appear to be quite counter-intuitive for most structural engineers.

small transverse load

P~I + + + + + + + + + + + + ~ + + + + + + + + + + + + ~P

prebuckling state: expansion develops axial compression ~ecr L

endsr n c, against axial ~ -- translation ~ postbuckled state: expansion produces deflections

Figure 3: Beam with rigid axial end restraint subjected to increasing temperature

Thermal buckling and post-buckling

When an elastic beam with rigid axial restraint at its ends is uniformly heated (Figure 3), compressive stresses develop following Eqns 2 & 3. If the modulus E and thermal expansion coefficient ot are deemed independent of temperature, the beam reaches a bifm'cation point when the thermal thrust attains the classical Euler buckling load:

44 J .M. Rot ter

: ~2 I~! 2 E1 EA ot AT = n 2 7 EA (9)

where g is the effective length of the beam and depends on the end flexural restraint conditions. The critical buckling temperature rise ATer for unchanging elastic modulus E is thus (Figure 3)

~2 (~/2 (AT)r = g (10)

For structural elements of the slenderness commonly found in slabs, this critical temperature can easily be as low as 100 or 200~ The phenomenon is thus also likely to occur in most fires. Any material degradation (deterioration with temperature rise or yielding or cracking) reduces the temperature.

If the elastic modulus and/or expansion coefficient are accepted as temperature dependent, the relationship is not so simply defined, since the thrust is a nonlinear integral of the thermal expansion and elastic modulus, whilst the stability is governed by a tangent modulus condition:

~cr = a(T) E,r(T,g) dT = n 2 E.r(Tcr,g,~r) (11) To

in which E.r(T,~) is the tangent modulus which varies with the temperature T and stress state cr and or(T) is the thermal expansion coefficient which changes with the phase of the material.

Whilst buckling may occur at quite a low temperature, the phenomenon is unlike that in a classical column; the force in the beam is controlled by constrained thermal expansion (the beam is too long), not by an imposed force. Thus, the large displacements which rapidly follow bifurcation phenomena under ambient temperature static loading do not occur here. The post-buckling axial shortening 8x and transverse deflection ~y of an axially loaded pinned beam may be approximated (Euler, 1744) by:

8x = L ~-~+ 2 - 1 (12)

~)y 2 " X f - 2 L ~ P = rc P-'~E- 1 (13)

Figure 4: Deflection of heated axially restrained elastic beam


Applying Eqn 10 and the thermal expansion Lo~AT in the post-buckled state (Figure 3), the post- buckling transverse deflection 8y becomes:

2@L ]o~AT - (nr I g)2 2@L ~ AT/ATcr- 1 8y = n " ~ 2 +- (-~rl-g)2 = ~ __ {2/(otATcr)} +1 (14)

The prediction of Eqn 14 for the post-buckling deflection of the beam is shown in Figure 4, together with the prediction from a large displacement finite element calculation using ABAQUS (1997) for the response in the presence of a small transverse load. The load smoothes the bifurcation phenomenon slightly, but the critical temperature can be clearly identified (matching Eqn 10), and the post-buckling response involving rapid growth of deflections into a large deformation state matches Eqn 14 The key feature of this behaviour is that the increasing deflections in the post-buckling state permit the thermal expansion to be accommodated in member curvature, thus reducing the stresses present but inducing large deflections. Here, post-buckling is not, in any sense, an unstable condition. The magnitudes of the deflections are very substantial compared with the length of the beam.

Figure 5: Axial force development in heated axially restrained elastic beam

The axial force developing in the beam under increasing temperature is shown in Figure 5. As assumed above, this force almost constant in the post-buckling region and additional thermal expansion is all absorbed in additional deflection, instead of causing increased stresses (Eqns 2-4). For local fires in real structures, this is a helpful effect as it limits the additional forces generated by the restrained thermal expansion and thus reduces damage to adjacent parts of the structure. Thus, buckling is good for this structure! This is perhaps a rather unexpected conclusion.

Figure 6: Response of heated axially restrained elastic-plastic beam

46 J.M. Rotter

In real structures, the elastic modulus and yield stress are affected by temperature rise. Thus, steel yielding and concrete cracking may be expected to damage the simple responses seen above. It is then no longer easy to perform algebraic analyses, but the corresponding ABAQUS calculation is shown in Figure 6. The axial force developed in the beam declines, controlled by development of plastic hinges at midspan and the ends. Increasing deflections mean that the axial force must fall even if the moment were to remain constant, but the yield surface permits an increasing moment with falling axial force. However, the magnitude of the deflection and its rapid growth are little changed. Initially, this is another surprising result for structural engineers, but it is easily understood in terms of thermal expansion; the expanded beam length is very precisely proscribed, and is accommodated by deflection.

This effect indicates that plasticity in the expanding structure may be a good phenomenon, since it reduces the forces to which other parts of the structure are subjected and thus reduces mechanical damage. This point is raised again later.

Finite axial restraint against thermal expansion

Rigid axial restraint is generally impossible to achieve, so the above represents only a limit; real structures offer only finite axial restraint. Assuming that the restraint to axial expansion can be represented by a linear translational spring of stiffness kt (Figure 7), the compressive axial stress developed by thermal expansion in an elastic beam with unchanging modulus becomes:

E ct AT ~=(I+EA-k~ ) (15)

The critical buckling temperature increment (AT)c r is modified from the Eqn 10 value to (Figure 8):

( A ~ . = ~ 1 + (16)

From this relationship it can be seen that buckling and post-buckling phenomena should be observable at moderate fire temperatures (say 300~ in structures with translational restraint stiffnesses kt which are quite comparable with the axial stiffness of the member (EA/L). This axial stiffness itself is reduced by heating through the reduction in ET, so these post-buckling phenomena should be observed in slabs and beams in typical fires.

p length L, effective length gefr properties E, A, I k p

t prebuckling state: expansion develops axial compression

~"

__ ~ with stiffness k postbuckled state: expansion produces deflections against axial

translation

Figure 7: Elastic axial restraint to beam expansion under increasing temperature

Not only are the buckling temperatures reduced by elastic-plastic material degradation, but as shown in Figure 6a, the forces imposed by the post-buckled beam decline rapidly. Because these forces become smaller, even relatively modest elastic restraint stiffnesses become effective and act in a manner similar to rigid axial restraints. For this reason, large post-buckling deflections can be expected in large buildings under compartment fires, even when the compartment is in a edge or comer position.


The moments developing in the beam have not been shown for space reasons, but it should be noted that the thermal expansion effects rapidly swamp the load-carrying primary bending effects.

Figure 8: Bifurcation temperature for partially axially restrained beams

THERMAL GRADIENTS THROUGH THE THICKNESS

The above discussion assumed a uniform temperature distribution through the slab or beam thickness. The concrete slab is heated from below and a high temperature gradient develops through its thickness. Temperature gradients also produce some surprising consequences. For clarity, it is helpful first to study the gradient alone, before recombining the effects of the gradient and a uniform rise into a realistic distribution. The temperature differential leads to thermally induced bending or to thermal bowing (Eqns 2-4 apply again). The differential induces either bending moments or additional deflections or both in the slab. Where the bending moment causes cracking, the stiffness again declines and encourages post-buckling large displacement effects.

8 ~ - ~ cold: c o n t r a ~ ~hermal] .. ~ a i n s ]

hot: expansion xxxxx

cold: contraction 8 = 0 ,~

hot: expansion

"q ai a'

Figure 9: Thermal gradient, and the effect of rotational boundary conditions

The natural starting point is a simply supported beam (axially and rotationally flee), subject to a linear dT dT

through-thickness thermal gradient ~ (Figure 9). A uniform curvature d~ = tx~--~y is caused by thermal

expansion (and contraction). No stresses develop and the hot lower surface leads to downward bowing. If instead, the beam is rigidly restrained against end rotations (but axially free to translate), no deflections develop at all in the beam! It remains perfectly straight. Instead, a constant bending

hogging moment is induced throughout the beam (Eqns 2-4 again), given by M = E1 c t i dy" The hot

48 J.M. Rotter

lower surface is thus in compression, and first cracking in concrete occurs on the top unheated surface (a counter-intuitive result for most structural engineers). More importantly, where the beam is composite, the steel joist at the bottom can become fully yielded throughout its length in compression under extreme fire conditions, causing engineers trained in conventional design to ask how the composite beam structure can possibly still carry its loads.

The thermal curvature qb due to a uniform gradient (with no net temperature rise), causes a deflection gy in an axially free beam of:

I~y = ~" 1 - cos 7 (17)

and, in a large displacement evaluation, this causes the distance between the supports to reduce by:

~Sx = L - 2 ~ (~-~) (sin txL dTyy~ ) - ~ - (18)

If the beam ends are now axially restrained, the loss of length in arc shortening 6x must be replaced by a stress-related extension, which requires a uniform axial tension closely modelled by (EA/L) 6x. Thus, for axially restrained but rotationally free beams (close to real conditions), a thermal gradient produces axial tension. By contrast, a uniform temperature rise produces axial compression.

Thus, the observed deformed shape of the structure is a poor indicator of whether part of the structure is in axial tension or compression, and a real temperature distribution with both thermal gradient and centroidal temperature rise can cause either axial tension or axial compression, with quite similar deformations. Some of these forces participate in load-carrying mechanisms (under large displacement regimes), whilst others are purely self-stressing in character. The effects of reduced flexural restraint at the ends of the beam is discussed by Rotter et al. (1999).

In a composite building, an expanding heated steel joist beneath a slab is restrained by the colder slab (a vertical thermal gradient) throughout the fire period and can become severely plastified in compression (Fig. 9) if large deflections do not occur. The slab is a major cause of thrust developing in the steel joist. The thermal expansion strains are absorbed as large compressive plastic strains, causing significant shortening of the joist. On cooling, this length reduction is not easily recovered, especially because the cooling steel gains stiffness and strength faster than the tensile stresses develop. Thus, very high tensile stresses develop during cooling, which can cause rupture damage to the connections unless these are designed to be ductile under joist tension, even though they occur in positions where the ambient temperature designer believes that hogging bending is occurring. In the design of highly redundant buildings, fire design should not ask "How is the load being carried?", but "Can large deflections develop well?", and "Must greater ductility be provided for the cooling phase?".

LARGE DEFLECTIONS AND MEMBRANE ACTION

Two separate structural stress pattems in slabs are termed "membrane action". Both involve axial forces in the plane of the slab (membrane forces). Both require the boundaries of the slab to be restrained in the plane of the slab (this was termed axial restraint above). At small displacements, compressive membrane action occurs (Figure 10). When cracking occurs in concrete, the neutral axis or zero strain axis is displaced in the direction of the compression face. The middle plane of the slab is thus effectively subjected to an expansion. Such an effect can occur at both midspan in sagging bending and at supports in hogging bending, giving additive expansive displacements. Where these expansions are resisted by a stiff boundary, additional compressive forces develop, and where the slab is thick, the eccentricity of the compressive force transmission produces an arching action which can


carry a greatly increased load. This mechanism is present in steel-concrete composite beams in highly redundant structures even under ambient conditions, due to the very large disparity between their hogging and sagging neutral axes. However, this action is more powerfully demonstrated in the thermally expanding slab of the composite structure, because the thermal expansions are very large and can cause major changes in the load-carrying mechanism. The load-carrying mechanism is promoted by any effect which assists the development of large deflections.

At large displacements, tensile membrane action begins (Figure 11). In tensile membrane action, the large deformations lead to a new load-carrying mechanism by change of geometry; effectively a small component of the tension carries transverse load directly. Under ambient conditions, such large displacements mean that large mechanical strains have developed, and there is a danger of rupture due to loss of ductility. Under fire conditions, thermal expansion provides much of the required deformation (Eqns 2-4), reducing the need for mechanical strains and ductility. The post-buckling deformations described above promote large displacements, and the 2D slab with a continuous displacement field, permits tensile membrane action to develop even adjacent to zones in a post- buckling compressive state. Because buckling restricts the compression forces and promotes increased deformations, tensile membrane action is more readily achieved. These membrane mechanisms make the floor slabs the strongest elements in the building since, under extreme conditions, they possess considerably greater strength than is required to carry the design loads in bending.

], sagging neutral axis high ~ i

end . . . . . . . ined ~-- ~ ~ }1 ~ against axial b ~ i translation ] hogging neutral axis low [

a x i a l l y restrained: compression due to changing NA location

Figure 10: Compressive membrane action

~ Shear, V

Axial tension, T

Figure 11: Tensile membrane action at large deflections

The worst scenario for a fire in a composite frame building structure is compartment breach. Structural fire design should define compartment breach as an "ultimate limit state" and ensure that it is prevented. The only structural member in a composite frame that acts as a compartment boundary is the composite floor slab. A compartment breach of the slab is unlikely because it is mostly in membrane compression throughout the fire. Appropriate reinforcement should be provided to ensure that through thickness cracks cannot develop in the slab.

CONCLUSIONS

Composite multi-storey building structures are highly redundant, and their floor systems exhibit high in-plane stiffness. When a compartment fire occurs beneath the floor, the behaviour of the floor system is dominated by restraint to thermal expansion, with middle surface heating and through thickness gradients causing quite different effects. The restraint to thermal expansion can easily lead to buckling and large post-buckling displacements, which are both stable and beneficial. Runaway failures are not seen in these redundant structures because the large displacements permit compressive and tensile membrane action to carry the loads in place of bending. Almost all the phenomena

50 J.M. Rotter

described derive from large displacements; small displacement ideas and small displacement analysis lead to serious misinterpretation of both test results and appropriate design measures.

Under fire loading, the dominant phenomenon in determinate structures is material degradation. In highly redundant structures, the single most important factor is the effect of thermal expansion. Where this leads to high stresses, damage occurs to the material (plasticity or concrete cracking). Where instead, large displacements develop in a post-buckling mode, the expansion is accommodated without so much damage, loads are carried by membrane action and the performance is considerably improved. Large displacements are commonly associated with bending failures, but here they may be beneficial, occurring with membrane thrusts, or with membrane tensions, depending on the thermal regime. A key conclusion is that the design criteria must not be based on limitation of deflections during the fire.

The effects of high temperatures on structures are best interpreted in the context of Eqns 2-4, which permit the roles of expansion and material degradation to be properly identified and which decouple the displacement and stress fields. Thermal expansion often couples with large displacements to produce effects which appear counter-intuitive to the conventionally trained structural engineer.

These findings are of fundamental importance to our understanding of composite frames in fire. They have major implications for the development of design philosophies and procedures.

ACKNOWLEDGEMENTS The support of DETR for funding this research through the PIT scheme is gratefully acknowledged. The author is most grateful for many discussions and calculations provided by Dr Asif Usmani and Dr Abdel Sanad of Edinburgh University and Dr Mark O'Connor and Dr Xiu Feng of British Steel.

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3. ENV 1994-1-2 (1995) "Design of Composite Steel and Concrete Structures: Structural Fire Design", Eurocode 4 Part 1.2, CEN, Brussels.

4. Kirby, B.R. (1997) "British steel technical European fire test programme - Design, construction and results", in Fire, Static and Dynamic Tests of Building Structures, eds G.S.T. Armer and T. O'Dell, Spon, London, ppl 11-126.

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6. Moore, D.B. (1997) "Full scale fire tests on complete buildings", in Fire, Static and Dynamic Tests of Building Structures, eds G.S.T. Armer and T. O'Dell, Spon, London, pp3-15.

7. Newman, G.M. (1997) "Design implications of the Cardington fire research programme", in Fire, Static and Dynamic Tests of Building Structures, eds G. Armer and T. O'Dell, Spon, pp 161-168.

8. Rotter, J.M., Sanad, A.M., Usmani, A.S. and Gillie, M. (1999) "Structural performance of redundant structures under local fires" Proc., Interflam '99,8th Int. Fire Science and Engg Conf., Edinburgh, 29 June- 1 July, Vol. 2, pp 1069-1080.

9. Sanad, A.M., Rotter, J.M., Usmani, A.S. and O'Connor, M.A. (1999) "Finite element modelling of fire tests on the Cardington composite building" Proc., Interflam '99,8th Int. Fire Science and Engg Conf., Edinburgh, 29 June - 1 July, Vol. 2, pp 1045-1056.

Design Formulas for Stability Analysis

of Reticulated Shells

S. Z. Shen

Harbin University of Civil Engineering and Architecture 202 Haihe Road, Harbin 150090, China

ABSTRACT

The aim of the paper is to propose some kind of design formulas for stability analysis of single-layer reticulated shells, reflecting the recent advances in theoretical study but simple in form for the convenience of practical application. For this purpose a comprehensive parametrical analysis of stability behaviors of single-layer reticulated shells of different types with various geometric and structural parameters has been carried out based upon complete load-deflection response analysis with consideration of the effects of initial imperfections and unsymmetrical distribution of loads.. More than 2800 examples of reticulated shells of prototype were analyzed, and the plentiful results obtained were thoroughly studied. As a result, practical formulas for predicting limit loads of reticulated domes, reticulated vaults with different supporting conditions, as well as reticulated shallow shells, obtained by regression analysis, were proposed.

KEYWORDS

Stability analysis, Complete load-deflection analysis, Limit load, Design formula, Reticulated shells, Reticulated domes, Reticulated vaults, Reticulated shallow shells, Reticulated saddle shells..

INTRODUCTION

The stability analysis is known as the key problem for the design of reticulated shells. The stability character of a complicated structure with numerous degrees of freedom such like reticulated shells can be revealed clearly and accurately by complete load-deflection response analysis, in which the

51

52 S.Z. Shen

structural response under loading is regarded as a continuous process rather than some individual structural properties such as critical load, buckling mode and etc.. The complete load-deflection curves give a more perfect picture about the behaviors of the structure. The deformation shapes varying with the loading process, and the possible buckling of different orders and of different characters (over-all or local buckling, bifurcation or limit point), with the corresponding critical loads and buckling modes, can be revealed in their proper order by the complete-process analysis. With the

development of non-linear finite element analysis and methods for tracing equilibrium path, it can be

said that the problem of stability evaluation of reticulated shells on the basis of complete load- deflection response analysis has been well solved from the viewpoint of theoretical side.

However, engineers working in design practice still feel puzzled when dealing with stability problems of reticulated shells. The theoretical method as discussed above seems to them too complicated for direct application. So it's desirable to propose some kind of design formulas, reflecting the recent advances of theoretical study but simple in form for the convenience of practical application. For this purpose a comprehensive parametric analysis of stability behaviors of different types of single-layer reticulated shells with varying geometric and structural parameters has been carried out based upon complete load-deflection analysis with consideration of the effects of initial

geometric imperfections and unsymmetrical distribution of loads. The "Consistent Mode Method" is

proposed for the imperfection analysis. This method assumes the geometric imperfection of a

reticulated shell to be distributed in consistence with the buckling mode of first order of the structure, which is supposed to be very likely the most unfavorable for the expected limit load of the reticulated shell. More than 2800 examples of reticulated shells of prototype were analyzed, and the plentiful results obtained were thoroughly studied. As a result, practical formulas for predicting limit loads, obtained by regression analysis respectively for different types of reticulated shells, rather simple for application but based upon accurate theoretical procedure as described, were proposed.

The complete-process analysis was carried out on the basis of geometrically non-linear finite element method, without consideration of material non-linearity, because it would be too time-consuming and hence very difficult at present to carry out such a large-scale parametric analysis with consideration of both geometric and material non-linearity. Besides, the reticulated shells under service condition are working in elastic range, and the material non-linearity would lead to some decrease of safety reserve in load-capacity of the structure; the latter effect could be assessed by some independent study [Wang]. A special computer program for complete load-deflection response analysis of complicated structures based upon non-linear finite element method, compiled by the author's team, was used for the parametric analysis and was proved to be effective.

THE PLAN OF PARAMETRIC ANALYSIS

The parametric analysis was carried out for single-layer reticulated domes, vaults, elliptical paraboloid shells ( EP shells, or shallow shells ) and hyperbolic paraboloid shells ( HP shells, or saddle shells ). For the purpose of practical application, all the reticulated shells analyzed are of prototype with member sections determined by calculation as in practical design. As the usual case in China, circular steel tube members and welded hollow spherical joints are used for these structures.

Design Formulas for Stability Analysis of Reticulated Shells 53

The reticulated domes analyzed have net systems of Kiewitt type ( K8 and K6 ), Schwedler type and geodesic type. For the Kiewitt dome K8, which was taken as the typical system to be studied, four different spans ( L = 40, 50, 60 and 70m ) and four different raise-span ratios ( f/L = 1/5, 1/6, 1/7 and 1/8 ) with four sets of member sections for each size, i.e. 64 different domes were analyzed. The effects of initial imperfections of consistent mode and with a maximum value equal to L/1000 were analyzed for each of the domes; besides, for part of the domes the effects of imperfections with

different values from L/1000 to L/100 were systematically studied. It's assumed that the dead load

( g ) is uniformly distributed over full span, while the live load ( p ) could also be distributed over

half-span (uniformly as well ); three different ratios of live load to dead load were considered: p/g = 0, 1/4 and 1/2. According to this plan, near 500 examples of K8 domes were analyzed, and, if

including the similar study for K6 domes, Schwedler domes and geodesic domes, the non-linear

complete-process analysis was carried out for 840 reticulated domes.

The reticulated vaults might have three kinds of supporting conditions: supported along the boundary,

supported along two longitudinal edges, or supported at two end cross-sections by means of rigid

diaphragms. The triangular net system, consisting of longitudinal and two sets of diagonal members,

as the most popular one is assumed for the reticulated vaults. The ratio of length to wave-span ( width ) of the vault ( L/b ) is a main factor effecting the structural behavior, and different ratios: L/b

= 1.0, 1.4, 1.8, 2.0, 2.2, 2.6 and 3.0 were considered in the parametric analysis, keeping the wave-

span of the vaults unchanged" b - 15m. Different raise-span ratios (f/b) and several sets of member

sections were assumed, and effects of different initial imperfections and unsymmetrical load distributions were studied. Besides, relatively long vaults supported at two ends may be provided

with intermediate diaphragms, and the effects of these diaphragms were analyzed. In sum, 1220

examples of complete load-deflection response analysis were carried out for single-layer reticulated vaults, including 350 examples for vaults with boundary supporting, 54 examples for vaults

supported along two longitudinal edges and 816 examples for vaults supported at two ends.

The elliptical paraboloid reticulated shells are usually used for rectangular or square plans, supported

along four sides by means of rigid diaphragms. The surface of a elliptical parapoloid is formed by a vertical parabola (as the generatrix), moving along another vertical parabola in the transverse direction. In engineering practice the parabolas are usually replaced with circle arcs, and the EP shells are often called as known as the shallow shells. Three kinds of plan dimensions (30"30m, 40"40m and 30"45m), three different raise-span ratios (f/L = 1/6, 1/7 and 1/8) and four sets of member sections for each size were considered. The raise-span ratio f/L is defined for each of the two directions, and equal ratios are assumed for both directions. Two kinds of net systems: triangular system and orthogonal system with diagonals were compared. As before, the effects of initial imperfections and unsymmetrical distributions of loads were studied. There were in all 783 examples of reticulated shallow shells to be analyzed.

The complete load-deflection behavior of hyperbolic paraboloid reticulated shells has its specific

characteristic. In this paper 14 HP shells of regular rhombic (square) plan with diagonal length equal

to 60m (taken as the span of the shell) were analyzed with consideration of the effects of different net

systems, different raise-span ratios and different rigidities of edge beams.

According to the plan of parametrical analysis as described above, more than 2800 examples of

54 S.Z. Shen

reticulated shells of different types were analyzed. For each of the examples the load-deflection curve drawn for the joint with maximum deflection at the end of iteration was taken to represent the

analyzed structure. From the viewpoint of practical application, the critical point of first order and

the related structural properties (critical load, buckling mode, and etc.), as well as the effects of

different factors to these properties, are of primary interest. So it's usually sufficient to take the

beginning part of the load-deflection curve (just ensuring a certain post-buckling path to be reserved )

for investigation. After this part, the load-deflection curve could be varied and colorful, theoretically

very interesting but less practical significance because of the too large deflections. Due to the limited

length of the paper just some examples of the curves obtained will be shown in the later sections.

STABILITY OF RETICULATED DOMES

The buckling of reticulated domes in most cases has a form of local concave on the surface as shown

in Fig.l, starting from snap-through of some joint and gradually expanding its area to become a

concave. The concave emerges at different place for different type of reticulated domes: it starts from

some joint of a main rib for Kiewitt domes, from some joint of the third ring (from bottom) for

Schwedler domes, and from some joint on the triangular surface for geodesic domes. The first

buckling of a dome is characterized as a limit point of the load-deflection curve, and the

corresponding critical load is taken as the limit load of the dome.

Figure 1: Buckling modes of reticulated domes

Because of the excellent 3-dimensional behavior of dome structure the unsymmetrical distribution of load shows very little effect to the limit load. For comparison, the load-deflection curves for three different distributions of loads ( p/g = 0, 1/4 and 1/2 ), taking the total load ( p+g ) as the ordinate,

have been put together for each of the domes. It's surprise to find that these three curves nearly

coincide one with another.

Meanwhile, the reticulated domes are very sensitive to the initial geometric imperfections. As an

example, the load-deflection curves for a Kiewitt dome with L=60m,f/L=l/8 and with nine different

values of initial imperfections ( the maximum value of imperfections r = 0, 3, 6, 10, 20, 30, 40, 50

and 60 cm, respectively ) are shown in Fig.2a. It can be indicated that the imperfections studied

attain a rather big value (up to L/100), and the presented study is primarily of theoretical interest. The

nine corresponding curves are put together for comparison. It's noticed that the curves vary with the

increase of imperfections in a good regularity. Then, if studying the load capacity of the domes, the


variation of limit load with the increase of imperfection values is shown in Fig.2b. It's seen that the

limit load drops rapidly at beginning, reaches a minimum value (approximately 50% of the limit load

of the corresponding perfect dome) as r=20 cm (i.e. L/300). Afterwards, the curve somewhat lifts

again, which seems inconsistent with the normal idea we might have. In fact, as the initial

imperfections go beyond some limit, the dome seriously deviates from its spherical shape and would become a " distorted" structure somewhat different from the original one. It can be seen from Fig.2a

the character of the load-deflection curves gradually varies with the increase of imperfection values:

the limit buckling for normal domes changes into bifurcation buckling for the domes with overlarge

imperfections. Besides, the" distorted" domes are less rigid, the deflections develop rapidly, and the

possible increase of critical load is meaningless in practice.

Figure 2 : a. Load-deflection curves of a dome with different imperfection values

b. Limit loads varying with increase of initial imperfection

The Schwedler domes with initial imperfections behave very similarly to Kiewitt domes, only the

limit load reaches the minimum value more rapidly ( as r = L/1000 - L/500 ). The response of

geodesic domes is somewhat different: the limit load, as well as the rigidity of the dome, drops

continuously with the increase of imperfection value within the studied range ( up to L/100 ), which

demonstrates the special significance of error control in erecting geodesic domes.

For practical purpose, it seems suitable to appoint a value of L/500 - L/300 as the acceptable maximum error of erection for reticulated domes, and to assume the limit load of the practical domes

with imperfections equal to 50% of that of the corresponding perfect structures. The geodesic domes

can also satisfy such an agreement.

How to make use of the large number of results obtained from the parametrical analysis for the

purpose of practical design? As one of the possible ways, it's considered preferable to propose some

appropriate formulas for predicting limit loads of reticulated shells by regression analysis of the data

obtained from the parametrical analysis. For reticulated domes such a formula is perhaps not so

difficult to work out, because there exists analytical formula of linear theory for predicting limit

loads of continual thin domes, the form of which could be taken as a reference. The formula for

predicting limit loads of reticulated domes is then suggested in the form as follows :

56 S.Z . Shen

~/BD q c r = g ~ ( 1 ) R 2

in which: R---radius of curvature of the dome ( m ); B---the equivalent membrane rigidity of the dome ( kN/m ); D---the equivalent bending rigidity of the dome ( kN.m ); and K--- coefficient,

determined by regression analysis.

The rigidity of reticulated shells is not uniform over the surface. So the proper position for calculating the value of B and D should be in consistence with the buckling mode of domes. For example, the buckling of Kiewitt dome occurs, as described above, at some joint of a main .rib, i.e., the limit load of the dome is primarily determined by the rigidity of the area round this joint. So B and D should be calculated according to the net size and member sections in this area. Similarly, for

Schwedler dome or geodesic dome the joint of the third ring or the joint on the triangular surface

should be taken as the calculated position, respectively. The formulas for calculating B and D are given in the Appendix to the paper. Besides, the reticulated shells are usually an-isotropic, and B and D in Eqn. 1 could be considered as the mean value of the rigidities in both main directions.

Due to the limited length of the paper the process of regression analysis is neglected, just indicating that the coefficients K calculated for different types of reticulated domes are very close one to another. This demonstrates that the formula in the form of Eqn.1 really reflects the characteristic features of the stability behavior of reticulated domes, and that it's correct to select the position for calculating B and D according to the buckling mode of different domes. It's finally suggested that the

limit load of practical reticulated domes of different types with initial imperfections to be controlled

within a limit less than L/500 can be determined by a unified formula as follows:

~ B D qcr = 1.05 R--- T- ( 2 )

STABILITY OF RETICULATED VAULTS

Vaults supported along the boundary

a. Vault supported along boundary b. Vault supported on longitudinal edges

Figure 3 : Buckling modes of reticulated vaults


The buckling mode of reticulated vaults supported along the boundary in most cases has the form of

a concave with three half-waves in the cross-section as shown in Fig.3a. For relatively long vaults

(L/b >~ 2.6) unsymmetrical mode with two half-waves (Fig.3b) is also possible, as in the case of

vaults supported on two longitudinal edges. It demonstrates the restricting effect of the end

diaphragms for vaults with L/b<2.6. For short vaults with L/b~< 1.4, such restricting effect becomes rather strong, and the buckling may have a mode of even higher order with four half-waves in the

cross-section.

The effect of length-span ratio L/b to the limit load of reticulated vaults supported along the boundary is very obvious, as shown in Fig.4. The limit load drops rapidly with the increase of L/b at

beginning, but gradually reaches a limit, in most cases as L/b = 2.6, but for high vaults with f/b = 1/2

the curve becomes even more slowly, usually as L/b>~ 3.0.

Figure 4 : Limit load of reticulated vaults supported along boundary with increase of L/b

The reticulated vaults supported along the boundary are not so sensitive to the initial imperfections.

Systematical analysis shows that the reduction in limit load at most consists of 20%, even as the

range of initial imperfections studied approaches a value as big as b/100.

The unsymmetrical distribution of loads nearly does not affect the stability behavior of reticulated

vaults of this type. As revealed by comparative analysis, the limit load defined as the total load p+g

does not decrease under unsymmetrical loading, only with an exception for short vaults of L/b~< 1.2.

For practical application, the effect of unsymmetrical loading to the limit loads of these short vaults

can be considered by a coefficient K2 calculated as:

K2 = 0.6 + 0.4 / ( 1 +2 p/g ) (applicable as p/g = 0-~2) (3)

It is somewhat difficult to derive the regression formula for the limit loads of reticulated vaults,

because there does not exist any theoretical form that could be referred to like the case with domes.

Anyway, some preliminary forms can be assumed based upon the ideas obtained from the

parametrical analysis. After repeated comparison by trial and error method, the following formula is

finally suggested for predicting the limit loads of reticulated vaults supported along the boundary:

58 S.Z. Shen

911 0-4 B22 029 qcr = 72.0 R3 L/b) 3 + 1.95 x 1 R(L/b) + 75.0 (R + 3f)b 2 ( 4 )

in which the indexes 11 and 22 indicate the longitudinal and transverse direction, respectively. The effect of initial imperfections has been considered in the formula. For short vaults with L/b 1.2 coefficient K2 as given by Eqn.3 should be multiplied to consider the effect of possible

unsymmetrical distribution of loads.

Vaults supported on longitudinal edges

To study the vault supported along the boundary, it can be imagined that, with the increase of length

of the vault, the effect of the end diaphragms to the behavior of center part of the vault would decrease, and the behavior of the vault in general is gradually close to that of a vault supported only on two longitudinal edges. It's seen now from Eqn.4 that the limit load decreases with L/b increasing, and only the third term of the formula will be retained as L/b approaches infinitive. It leads to a very

interesting question: if the third term of the formula can be used to evaluate the limit load of the vault supported on two longitudinal edges. This theoretical deduction was proved by the complete-process analysis of 54 examples of reticulated vaults of such kind. It's then concluded that the limit load of

reticulated vaults supported on two longitudinal edges can be predicted by the formula as follows:

qcr =75.0 D22 (R +3f)b2 ( 5 )

The effect of unsymmetrical distribution of loads need not be considered for vaults of this type.

Vaults supported at two ends

The vault supported at two ends has free longitudinal edges, but strengthened by edge beams with certain rigidity. Such a vault is behaving like a huge beam with curve cross-section supported at two end diaphragms. With the increase of length of the vault, the member forces in the vault, and hence the cross-section of the members, increase as well, that is not like the vault supported along the boundary. So, if keeping the other parameters unchanged, the member sections determined by calculation as in practical design are different for vaults with different length. Under this condition, the parametrical analysis shows that the limit loads of the vaults are rather stable for different values of length-width ratio L/b. That is, the limit load does not depend evidently upon the ratio L/b.

The buckling mode of the vaults has more likely a form of overall deformation of the surface

together with the bending and torsion of edge beams. The raise-width ratio f/b has obvious effect to

the limit load of the vaults: the vault with bigger f/b ratio shows higher stability load-capacity.

The vaults supported at two ends are not so sensitive to initial imperfections. As revealed by

systematical analysis, if taking b/300 as the acceptable maximum value of initial imperfection, the reduction in limit load does not exceed 18%.

The limit load is evidently affected by unsymmetrical distribution of loads. Such effect becomes sufficiently developed as early as p/g = 0.5, and the further reduction in limit load is not evident for


the bigger values of ratio p/g. For practical application the effect of unsymmetrical loading can be

considered by the coefficient K2 determined as follows:

K2 = 1 .0- 0.2 L/b ( L/b = 1.0-2.5 ) ( 6 )

K2 = 0.5 ( L/b = 2.5"`3.0 )

The intermediate diaphragms for relatively long vaults supported at two ends may be arranged at an

interval roughly equal to the width b in order to increase the overall rigidity of the surface and hence

to raise the limit load of the vaults. On the bases of systematical comparison it's suggested for

practical application that the effect of intermediate diaphragms can be considered by a coefficient K3

calculated by Eqn.7. For vaults with intermediate diaphragms unsymmetrical distribution of loads

does not affect the limit load any more.

K3 = 1.52- 0.12 L/b (applicable as L/b = 1.4--3.0) ( 7 )

After repeated comparative analysis the regression formula for predicting the limit load of reticulated

vaults supported at the ends is proposed as follows:

qcr - 0.063 + 0.138 + 0.083 ~ ( 8 )

in which the factor CL-- 0.96 + 0.16(1.8 - L/b )4 ; ih and Iv ---the horizontal and vertical linear rigidity

of the edge beam, respectively, which can be calculated as ( for latticed beams as usually used ): Ih,v = E(Alr~2+A2rRR)/L, in which A~ and A2 are the cross-section areas of two chords of the latticed beam, r~

and r2 are the corresponding radiuses of inertia.

The effect of initial imperfections has been included in the formula. The effect of unsymmetrical

loading should be considered by the coefficient K2 given by Eqn.6. For vault with intermediate

diaphragms the limit load determined by Eqn.8 should be multiplied by coefficient K 3 given by

Eqn.7, but without consideration of coefficient K2 �9

STABILITY OF RETICULATED SHALLOW SHELLS

The stability behaviors of shallow shells with triangular net system and with orthogonal net system are somewhat different each from other. In the comparative analysis the corresponding shells of these

two kinds were designed to have equal weight. Under this condition, the limit load of the shells with

triangular system is higher than that of the other. The buckling of the shells with orthogonal system

more likely has a form of local concave on the surface, but for shells with triangular system there appears more evident character of overall deformation, i.e., more obvious deformations arise in a

much wider range of the surface. The shells with triangular system show higher sensitivity to initial

imperfections. According to the comparative analysis, if the maximum value of initial imperfection

is controlled as L/500"`L/300, the reduction in limit load of shells with triangular system and with

orthogonal system can be taken in practical application as 35% and 25%, respectively.

The shallow shells are very sensitive to unsymmetrical distribution of loads. As an example, the limit

loads of a shell with a plan of 30"30m and with orthogonal system varying with the increase of ratio

60 S.Z. Shen

p/g are shown in Fig.5. It's seen that the curves sustainedly go down, do not approach a limit even as

p/g=2, meanwhile the limit load has dropped to a rather low value of about 30% of the case of

symmetrical loading. Based upon regression analysis the coefficient K2 of considering the effect of

unsymmetrical loading can be given as Eqn.9. This formula is applicable for both net systems.

Kz = 1 / [ 1 + 0.956 p/g + 0.076 (p/g)2 ] (applicable for p/g=0-2.0) ( 9 )

" . . . . . . . .

~ moveable hinge 15 N . . . . . . fixed hinge

z lO

- . , . . . , ~ . ~ . ~ . ~ ' - ~ . . .

5 - f/L = 1/6 ~ " - " - - ~ ~ . ~ ~ ~

f/L = 1/7

f/L= 1/8 ....... p / g 0 0 . . . . . . 015 i ' ' 1.5 2

Figure 5 : Limit loads of reticulated shallow shells with increase of p/g

In reference to the analytical formula of linear theory for domes and based upon regression analysis,

the design formula for predicting limit load of reticulated shallow shell can adopt rather simple form:

x/BD For triangular system qcr = 1.29K2 ( 10 )

R1R2

4"D For orthogonal system qcr = 1"07K2 ( 11 )

R1R 2

In which R1 and R 2 are the radiuses of curvature in two directions, respectively, and the coefficient K 2

is given in Eqn.9. The effect of initial imperfections has been considered in the formulas.

STABILITY OF RETICULATED SADDLE SHELLS

The complete load-deflection response of the reticulated HP shells are varied with variation of

geometrical and structural parameters such as the raise-span ratio, the net system, the rigidity of edge

beams and etc.. Some load-deflection curves may rise sustainedly with no critical point emerging.

Some curves may have bifurcation point appearing, but the load continues up with the rigidity matrix

of the structure keeping positive definite. And in some other curves buckling of limit-point type may

occur, but the load rises again after certain post-buckling path downwards. HoweVer, there exists a

common character for the load-deflection curves of HP shells: the load has a general tendency to

keep going up, and from practical viewpoint the load-capacity of the shells is maintained. As an

example, the load-deflection curves of HP shells of L=60m with different raises (H = 6,9,12,15 and

18m) are shown in Fig.6a. It demonstrates the specific characteristics of the shells of negative

Gaussion curvature. Further more, it can be supposed that the feature of monotonous rise would be

revealed more obviously for load-deflection curves of the practical shells with initial imperfections.


It seems rational to conclude that the stability problem is not significant for reticulated HP shells, and

as a necessary substitutive measure the rigidity of the shell should betaken as a main structural property to be checked in practical design. The maximum deflections of the shells with different raises under service load (2kN/m 2) are shown in Fig.6b. It's seen that the rigidity of HP shells with H=9m and 6m is obviously not enough.

Figure 6 : a. Load-deflection curves of reticulated HP shells with different raises

b. Maximum deflections under service load of these shells

CONCLUSIONS

1. Based upon the complete load-deflection analysis for more than 2800 examples of reticulated

shells of prototype the varied and colorful structural behaviors developing with the loading

process, the practical mechanism of structural instability and the complex effects of different

factors were revealed rather thoroughly for different types of reticulated shells. 2. Based upon the regression analysis of the plentiful data obtained from the parametrical analysis as

described above design formulas for predicting limit loads of reticulated domes, reticulated vaults

with different supporting conditions, as well as reticulated shallow shells, rather simple for application but obtained on the basis of accurate theoretical procedure, were proposed.

3. For reticulated saddle shells it's suggested just to carry out routine rigidity check instead of the complicated stability analysis.

REFERENCES

Chen X. and Shen S.Z (1993). Complete Load-Deflection Response and Initial Imperfection

Analysis of Single-Layer Lattice Dome. International Journal of Space Structures 8:4, 271-278

Wang N., Chen X. and Shen S.Z. (1993). Geometric and Material Non-linear Analysis of Latticed Shells of Negative Gaussion Curvature. Space Structures 4. London. 649-655

Shen S.Z. and Chen X. (1999). Stability of Reticulated Shells. The Science Publisher, Beijing, China

62 S.Z. Shen

APPENDIX: Formulas for Equivalent Rigidities of Reticulated Shells

The net systems used for reticulated shells can be classified into three basic types as shown in the

attached figure.

Attached Figure: Three typical net systems

The equivalent rigidities in two main directions can be calculated as follows:

1 .For net system ( a ) and system ( b ) with single diagonal

B11 EA1 EAc EI1 EIc = + sin4 a Dll = + sin4 a A 1 A c A 1 A c

B22 EA 2 EA~ E12 EZc 4 = + c o s 4 a 022 = + cos a A 2 A~ A 2 Ac

2.For net system ( b ) with double diagonals

EA~ EAc E11 El c Bll = + 2 sin 4 ct D~ = + 2 sin 4 a

A 1 Ac A1 Ac

B22 EA 2 EA c EI 2 EI = + 2 cos4 ct D22 = + 2 cos 4 a A 2 A~ A 2 Ac

3.For net system ( c )

EA 1 EAc Bll + 2 sin4 a Dll E11 Elc = = + 2 sin 4 ct A 1 Ac A1 A c

B22 2 EAc cos4 a 022 = 2 EI c = COS 4 a

Ac Ac

In the formulas A1,A 2 and A c are the cross-section areas of members in direction 1 and 2 and of diagonals, respectively, I l, I 2 and Ic are the corresponding moments of inertia, the intervals between

members A 1 , A 2 and A c , as well as the inclination angle a are as shown as in the figure.

DUCTILITY ISSUES IN THIN-WALLED STEEL STRUCTURES

T. Usami 1, Y. Zheng 1, and H.B. Ge 1

1Department of Civil Engineering, Nagoya University, Nagoya, 464-8603, JAPAN

ABASTRACT

The ductility of thin-walled steel box stub-columns under compression and bending is studied in this paper through extensive parametric analyses, and empirical ductility equations are developed. The equations for isolated plates and pipe stub-columns proposed in the previous studies are also presented. On this basis, a simplified ductility evaluation procedure is proposed for practical steel structures with thin-walled box or pipe sections. An inelastic pushover analysis is employed and a failure criterion is introduced. The implementation of the proposed procedure is demonstrated by application to some cantilever columns and a one-story frame. Moreover, the computed results are compared with the ductility estimations through cyclic analyses reported in the literature, which leads to the validation of the proposed method.

KEYWORDS

Thin-walled steel structure, Ductility, Pushover analysis, Stub-column, Residual stress, Initial deflection, Box section, Pipe section, Frame, Cyclic loading.

INTRODUCTION

Thin-walled steel columns and frames have been widely used as substructures in urban highway bridges, suspension and cable-stayed bridge towers in Japan as well as some other countries. But the need for evaluating the seismic performance, such as the ductility capacity, of such structures has come into focus following the damage and collapse observed d r ~ the 1995 Hyogoken-nanbu earthquake (Fukumoto 1997; Galambos 1998). Steel beam-column members employed in bridge structures are characterized by the use of relatively thin plates, which makes these structures vulnerable to damages caused by the local and overall interaction buckling. However, the task of accounting for such buckling can be formidable for a practical use where the balance between reliability and simplicity is required.

63

64 T. Usami et al.

A simplified ductility evalUation method for steel columns and frames composed of box sections was previously proposed by the authors (Usami et al. 1995). An inelastic pushover analysis is utilized in the method and the structural ultimate state is assumed to be attained when the compressive flange strain of the most critical part reaches its failure strain. However, the method employs an empirical equation based on isolated, simply supported plate under compression (Usami et al. 1995) to calculate the failure strain, and consequently leads to somewhat conservative predictions for structures composed of moderately thin plates. This is for the reason that the interactive effects between adjacent component plates at their junctions are neglected.

In this paper, aiming at proposing more refined empirical equations for failure strains, thin-walled steel box stub-columns are studied under combined action of compression and bending. Extensive parametric analyses are carried out to investigate the effects of some parameters on the behavior of stub-columns with and without longitudinal stiffeners. An elasto-plastic large deformation FEM analysis is employed. Based on the parametric analyses, empirical equations for the ductility of box stub-columns are developed. Besides, the ductility equations for isolated plates in compression and short cylinders in compression and bending proposed in the previous studies (Usami et al. 1995; Gao et al. 1998a) are also presented. By using the equations based on stub-columns, the previous ductility evaluation procedure for box-sectioned structures (Usami et al. 1995) is refined and meanwhile, is extended to both box and pipe-sectioned structures. A one-story frame with stiffened box sections and several cantilever columns with unstiffened box sections, stiffened box sections, and pipe sections are investigated as examples to demonstrate the application of the procedure. Moreover, the computed results are compared with previous results obtained through cyclic tests or numerical analyses (Usami 1996; Gao et al. 1998b; Nishikawa et al. 1999). The comparison illustrates the validity of the proposed method.

DUCTILITY OF BOX STUB-COLUMNS

Numerical Analytical Model

Both the box stub-columns with and without longitudinal stiffeners are studied. The analytical models of such stub-columns are shown in Fig. 1, which represent a part of a long column between the diaphragms. Due to the symmetry of geometry and loading, only a half or a quarter of the stub-column is analyzed. A simply supported boundary condition is assumed along the column end plate boundaries to simulate the local buckling mode of a long column, which would deforms into several waves along the length. To

Y Y

P P

W e b I

(a) Unstiffened (b) Stiffened [ s.s.: simply Supported Edge

Figure 1" Analytical model of box stub-columns

Ductility Issues in Th&-Walled Steel Structures 65

Figure 2: Residual stresses

impose a rotation of the edge, the end sections are constrained as rigid planes by using the multi-point constraint (MPC) boundary conditions, and the rotation displacement is applied at any node on the sections. The bending moment is obtained as the reaction force of the node. The general FEM program ABAQUS (1998) and a type of four-node doubly curved shell element (S4R) included in its package are employed in the elasto-plastic large deformation analysis.

An idealized rectangular form of residual stress distribution in each unstiffened panel, stiffened panel, and stiffener plate, is adopted due to the welding (see Fig. 2). The initial geometrical deflections are also considered. For unstiffened stub-columns, the shape is assumed to be sinusoidal in both flange and web plates (see Fig. 3(a)). The maximum values ofthe initial deflections in the flange and web are assumed to be B/500 and D/500 (where B and D are the breadth and depth of the box section), respectively. The directions of the initial deflections are assumed inward for flange plates while outward for web plates. The assumed initial deflection shape in the flange plate of stiffened stub-columns (Fig. 3(b)) are given by following equations:

where

8=~5a+8 L (1)

a ,000 sinI;

1 ( ) = s in ~Z y c o s • Z 150 ~ ~ (3) in which cY c denotes the global initial deflections; CYL represents the local initial deflections; a is the length of the stiffened stub-columns; n is the number of the subpanels divided by the stiffeners; m is the number of half-waves of the local initial deflections in the longitudinal direction, which is assumed as an integer giving the lowest failure strain and will be further discussed below. The initial deflections in the web plates are calculated by replacing B and z in Eqs. (2) and (3) by D and x, respectively, but assumed in opposite direction (outward).

A kind of steel stress-strain relation including a strain hardening part, proposed by Usami et al. (1995), is utilized in this study to define the material characteristics (see Fig. 4). Here, % and 6y denote the yield stress and strain, respectively; E is the elastic modulus (i.e., Young's modulus); 6,, is the strain at the onset of strain hardening; E~ is the initial strain hardening modulus; and E ' is the strain hardening modulus assumed as

E ' = E,~ e x p ( - ~ s - o%t ) ( 4 ) 6y

66 T. Usami et al.

Figure 3: Initial deflections

where 2j is a material coefficient. Mild steel SS400 (equivalent to ASTM A36) is utilized in the analysis of stub-columns, for which Cry = 235 MPa, E = 206 GPa, t,' = 0.3, e n = 10 e y, 2j = 0.06, and En=E/40.

In this study, the ductility of the stub-column is evaluated by using the failure strain, 6u/zy, which is defined as a point corresponding to 95% of the maximum strength after the peak in the bending moment versus average compressive strain curve (Usami and Ge 1998).

Parametric Study

Figure 4: Material model

The behavior of thin-walled box stub-columns subjected to compression and bending is considerably affected by the magnitude of axial load, P/Py (Py is the squash load), and the flange width-thickness ratio, RI, which is defined as

a ~ B I 1 2 ( 1 - v2) IO'y Ry : : t 4n2x 2 E (5)

in which O'cr is the elastic buckling stress; n is the number of subpanels (for unstiffened plate, n = 1). For stub-columns with stiffeners, the stiffener's slenderness ratio, 2 s , is another key parameter, given by:

- 1 a 1 ~-~y A" = x/-Q r, n 3r E (6)

1 Q -- 2-~f [13 - ~/13z - 4Rf ] (7)

13 - 1 .33Rf + 0 .868 (8)

in which rs is the radius of gyration of a T-shape cross section consisting of one longitudinal stiffener and the adjacent subpanel and Q is the local buckling strength of the subpanel plate (Structural Stability 1997). An alternative parameter reflecting the characteristics of the stiffener plate is the stiffener's relative flexural rigidity, y, which is interdependent on 2,. Thus, in the present study, only 2 s is considered in the ductility equations.

Ductility Issues in Thin-Walled Steel Structures

TABLE 1 TABLE 2 Parameters of Thin-walled Parameters of Stiffened

Unstiffened Box Stub-columns Box Stub-columns

D/B at = a/B Rf D/B a = a/B Rf ?" / y *

3/4, 1.0, 0.5, 0.7, 0.2, 0.4, 0.45, 0.5, 0.67, 1.00, 0.5, 0.7, 0.3, 0.35, 0.4, 0.45, 1.0, 4/3 1.0 0.55, 0.6, 0.8 1.33 1.0, 1.5 0.5, 0.55, 0.6, 0.7 3.0

67

~s 0.180 "~ 0.751

Nevertheless, to propose ductility equations for comprehensive applications, the influence of box cross-sectional shape (say, square or rectangle) and the aspect ratio (a =a/B) has to be surveyed. And for stiffened stub-columns, the critical local initial deflection mode along the length direction giving the lowest ductility should be first determined. The thickness of the plates is assumed as 20ram and the considered axial force, P, ranges from O.OPy to 0.SPy. Other pertinent parameters are given in Tables 1 and 2, where 9" * represents the optimum value of 9" obtained from elastic buckling theory ("DIN 4114" 1953).

Through parametric analyses, following conclusions are drawn for unstiffened stub-columns: (1) The effects of the cross-sectional shapes and column aspect ratios on the stub-column ductility are insignificant and the present empirical equation is based on the models with square sections and aspect ratios equal to 0.7; (2) Referring to the computed e,/ey versus R z and P/Py relations presented in Fig. 5, it is observed that the failure strain decreases as the increase of either R z or P/Py; (3) Considering the effects of axial loads, an equation of failure strain, eu/ey, versus flange width-thickness ratio, R z, is fitted as follows:

~;, 0 .108(1- P / Py )1.09 ~.~ : ( R f -0 .2 ) 3 " 2 ~ + 3 .58(1- P / py)O.839 < 20.0 (9)

The applicable range of this equation is R/= 0.2 -- 0.8, D/B = 0.75 --- 1.33, and P/Py = 0.0 --- 0.5. It should be noted that when the failure strain, 6u/ey (which is the average strain in the compressive flange), exceeds 20.0, the local maximum strain would be very large (say, 5% or larger) and the numerical analysis results would become unreliable. Thus, the upper bound of 6,,/6y is limited as 20.0 at present time although the consequent prediction will be on the safe side for some cases.

As for the stiffened stub-columns, the observations from the parametric analysis can be concluded as: (1) The critical local initial deflection mode along the length of stiffened stub-columns varies with different aspect ratios and the corresponding number of half-waves (m) is found as 2, 3, 4 and 5 for aspect ratios of 0.5, 0.7, 1.0 and 1.5, respectively; (2) The buckling mode of stub-columns has almost same shapes as the assumed initial deflection mode; (3) The influences of box cross-sectional shape and the aspect ratio on the ductility of stub-columns are not obvious and for simplification, they can be neglected in the design formulas of failure strain; (4) The effects of flange width-thickness ratio and stiffener's slenderness

~, ~0.1s ratio should be considered together and a combined parameter *-/,~s is introduced. Inversely

proportional relations of the failure strains to this combined parameter and the axial load are found (see - - 018

Fig. 6). On this basis, an equation of eu/~y v e r s u s R y e . s " , considering the effect of axial load, are fitted

as follows:

~'u _ 0"8(1 - P / Py )0"94 w

/ , • ~ - 0.18 ~'Y ~"'~f"s - 0 " 1 6 8 ) lzzs

+ 2.78(1 - P / P, )0.68 ~_~ 20.0 (10)

Here, R/ranges from 0.3 to 0.7, ~, is in a scope from 0.18 to 0.75, and P/Py is between 0.0 and 0.5. And

68 T. Usami et al.

Figure 5 Failure strains of unstiffened

stub-columns

Figure 6 Failure strains of stiffened

stub-columns

this equation is applicable to stiffened box stub-columns with a from 0.5 to 1.5 and B/D from 0.67 to 1.33. Moreover, it should be noted that this equation is fitted to give slightly smaller prediction of failure

strains for the cases with smaller -" = 0.~s K:% . This is for the reason that the numerical results of present

study are based on monotonically loading conditions and when applied to long columns with small - - 0 1 8 R / ~ " , they are found to yield larger ductility predictions compared with the cyclic experimental and

numerical results as presented later.

DUCTILITY OF ISOLATED PLATES UNDER UNIAXIAL COMPRESSION

For comparison, this paper also presents the failure strain equations based on isolated plates under uniaxial compression (Usami et al. 1995; Usami and Ge 1998). They are defined as follows:

Unstiffened olates: ~" 0.07 - - = + 1 .85 _< 20 .0 (11) " % (R: - 0 . 2 y "~

Stiffened nlates: e, 0.145 - - : + 1.19 < 20 .0 (12) 6, (x-, - 0.2) TM

Equation (11) is plotted in Fig. 5 and some computed results of stiffened plates (Usami and Ge 1998) are ~. . ~ ' - o 1 8 also plotted in Fig. 6, in the form of E / E y versus K/~s " �9 It is observed that the failure strains of

stub-columns subjected to compression and bending are larger than those of isolated plates under pure compression. When the axial load is so large as to approach the pure compression state two procedures will give similar predictions.

DUCTILITY OF SHORT CYLINDERS

The ductility of thin-walled steel short cylinders in compression and bending has been also investigated in a previous study (Gao et al. 1998a). Analytical models similar to those used for box stub-columns, which have been presented above, were employed for the cylinders. Main parameters controlling their behaviors are found to be the magnitude of the axial force and the radius-thickness ratio parameter, Rt, which is in the form of

Rt = 6y = ~/3(1 - v 2) oy d (13) o~, E 2t

Here d and t denote the diameter and thickness of the cylinder, respectively. And an empirical equation is proposed as follows:

Ductility Issues in Th&-Walled Steel Structures

~,, , O.120+4P/Py) 6--f = ( R t - 0.03) 1"4s (1 + P / P,)5 + 3.6(1 - P / P,) <__ 20.0

69

(14)

DUCTILITY EVALUATION PROCEDURE FOR THIN-WALLED STEEL STRUCTURES

By using the empirical equations of failure strains given above, a ductility evaluation procedure is proposed for practical structures composed of thin- walled steel beam-column members. It is applicable for the design of a new structure or evaluation of a existing structure, which are in the form of cantilever-typed columns or framing structures. The procedure involves the following steps: 1. Based on the general layout and loading condition

of the structure, establish the analytical model as Figure 7: Pushover analysis model shown in Fig. 7, by using beam elements, which facilitate the FEM modeling procedure but do not account for local buckling. Neither the residual stresses nor initial deflections are take into consideration. The material model defined in Fig. 4 is also utilized for the pushover analysis.

2. Carry out a planar pushover analysis. This procedure involves applying the constant vertical loads and incrementally increased lateral loads to represent the relative inertia forces which are generated at locations of sustained mass. An elastoplastic large displacement analysis is employed to account for the second-order effects.

3. The pushover analysis is terminated once the failure criterion is attained and this state is taken as the ultimate state of the structure, based on which the ductility capacity, 8u/By, of the structure can be determined.

Like the previous study (Usami et al. 1995), the failure of a structure composed of the thin-walled steel box members is assumed when the average strain over an effective failure length in the compressive flange (or in the maximum compressive meridional fiber for pipe sections) reaches its failure stain (Eqs. (9), (10) and (13)). The effective failure length, I,, of a box-sectioned member is assumed as the smaller one between 0.7 times of the flange width and the distance between two adjacent diaphragms (Usami et al. 1995). For pipe-sectioned structures, based on the observations in the previous studies (Gao et al. 1999a and 1999b), an empirical equation is proposed here to define the effective failure length:

1 -1)d (15) I t = 1.2(Rt--~.0s

The critical parts could be more than one place in a framing structure and all of them should be checked (see Fig. 7(b)). In a thin-walled steel structure, however, the excessive deformation tends to intensify in a local part and consequently the redistribution of the plastic stress becomes unexpected. Thus, once the failure criterion at any one of the critical parts is satisfied, the ultimate state of such a structure is though to be reached.

NUMERICAL EXAMPLES

To demonstrate its implementation, the proposed ductility evaluation procedure is applied to some cantilever-typed columns with box or pipe sections and a one-story rigid frame composed of box section

70 T. Usami et al.

TABLE 3

PARAMETERS OF CANTILEVER COLUMNS WITH UNSTIFFENED BOX SECTIONS

Specimen

UU1

UU6

UUll

U45-2513]

U45-4013]

U70-2513]

U70-4013]

h a B D t Material (mm) (mm) (mm) (mm) (mm) k RI P/PY Number

762 306 157 .5 120.0 4.51 0.362 0.664 0.2 I

1035 394 202.5 154.0 4.51 0.381 0.854 0.2 I

853 312 171 .5 127.0 10.5 0.406 0.297 0.2 I

485 278 144.0 108 .8 5 . 9 1 0.254 0.448 0.2 II

781 278 145.0 108 .8 5.91 0.404 0.451 0.2 II

786 434 222.0 167 .8 5.91 0.262 0.701 0.2 II

1217 434 222.9 167 .8 5 . 9 1 0.406 0.704 0.2 II

Notes: refer to Table 5 for details of material numbers.

TABLE 4 PARAMETERS OF CANTILEVER COLUMNS WITH STIFFENED BOX SECTIONS

Model

B1

B2

B3

B4

B5

B6

B7

B8

B14

B16

h (mm) 4311

7543

7559

10776

3264

5712

5551

5777

3403

5712

B t b, ts (mm) (mm) (mm) (mm) RI A, A u P/P,

1344 20 121 20 0.46 0.51 0.20 1.0 0.15

1344 20 121 20 0.46 0 .51 0.35 1.0 0.15

1344 20 121 20 0.46 0.28 0.35 0.5 0.15

1344 20 121 20 0.46 0.51 0.50 1.0 0.15

1023 20 105 20 0.35 0.21 0.20 0.5 0.15

1023 20 105 20 0.35 0.21 0.35 0.5 0.15

1023 20 179 20 0.35 0.23 0.35 1.0 0.15

1023 20 70 20 0.35 0.33 0.35 0.5 0.15

882 9 80 6 0.56 0.63 0.26 1.0 0.12

1023 20 105 20 0.35 0 .41 0.35 1.0 0.15

Notes" D = B; refer to Table 5 for details of material numbers.

Material Number

III

III

III

III

III

III

III

III

IV

III

members. Besides, the computed results are compared with those reported in previous studies (Usami 1996; Nishikawa, et al. 1996; Gao 1998; Gao et al. 1998b; Nishikawa et al. 1999), which are analyzed under cyclic lateral loading through experimental or numerical techniques. The ABAQUS program (1998) and a kind of beam element B21 are employed for the pushover analysis.

Cantilever Box Columns with and without Longitudinal Stiffeners

Recently extensive experimental study have been carried out to survey the behavior of steel cantilever box columns with and without stiffeners, which are subjected to cyclic lateral loading as well as a constant axial load (see Fig. 7(a)). A detailed summary of these studies has been reported in the literature (Usami 1996). According to this reference, the local buckling is observed to occur near the column base in the range of about 0.7B (B is the width of the flange) or between the transverse diaphragms, if any. And the mode shapes of the global and local buckling are found in the form of half sine-waves. These


observations are in good agreement with the effective length assumed above and the Material E buckling modes occurred in the Number (GPa) analyses of stub-columns. By I 197 introducing an index of ductility, II 216 8 95 / 8 y (895 is the top lateral displacement corresponding to III 206 95% of the maximum lateral load IV 206 after peak and 8y is the yield V 206 lateral displacement), empirical equations of ductility related to VI 206 some main parameters has been VII 206 developed for both the columns

VIII 206 with and without longitudinal stiffeners, which are defined as follows (Usami 1996):

T A B L E 5. M A T E R I A L PROPERTIES OF E X A M P L E S

v (MPa) E/gst g st /[~y

0.269 266 21.0 11.3

0.270 282 32.4 16.9

0.300 314 30 7.0

0.300 379 30 10.0

0.300 235 40 10.0

0.300 290 40 14.0

0.300 269 40 14.0

0.300 294 40 10.0

Notes: Refer to Tables 3, 4, 6, and 7. for material numbers

71

Unstiffened columns: 69s 0 .0670 - + 2 .60 (S = 1.09) (16) 8y [(1 + e / Py )RI ~~ ] 3"~

Stiffened columns: 89s 0 .0147 - - = + 4 .20 (S = 1.40) (17) 6y [(1 + P / Py )gf~~ 3"s

where S is the standard deviation; and ~ is the column slenderness ratio parameter given by t

_ ___2h 1 , / o , (18)

r ~: V E Here h is the column height and r is the radius of gyration of cross section. Equations (16) and (17) were fitted corresponding to the average curve for test data (i.e., the M curve plotted in Fig. 8 by the solid line) and the lower bound curves were also proposed as Eqs. (16) and (17) minus the standard deviation (S), as the M-S curve shown in Fig. 8 by the dashed line.

Several specimens in the form of both unstiffened and stiffened columns reported in the reference (Usami 1996) are adopted here to demonstrate the validity of the ductility evaluation method proposed in this paper. The parameters of the columns are presented in Tables 3, 4 and 5. The computed ductility estimations (8,,/8y) are presented in Fig. 8 compared with the empirical curves (Eqs. (16) and (17)), of which 89s/dy is denoted by 8u/dy for the accordance. For unstiffened columns (Fig. 8(a)), it is observed that the proposed method gives the ductility predictions very close to the lower bound curve (M-S Curve), which has been recommended for the practical use considering the required safety (Interim 1996). In Fig. 8(b) which is for stiffened columns, good agreement of computed results with the test curves is also observed. Aswell, the previous method proposed by Usami et al. (1995), where the failure strain equations based on isolated plates (Eqs. (11) and (12)) are used, is also applied to these examples and the obtained results are included in Fig. 8. It can been seen that the previous method underestimates the column ductility for most cases.

Cantilever Columns with Pipe Sections

The behavior of thin-walled steel cantilever-typed columns with pipe section has been investigated by some researchers (e.g., Nishikawa et al. 1996; Gao et al. 1998b). In the cyclic test on such columns by Nishikawa et al. (1996), the so-called elephant foot bulge mode was found to occur in the range of about

3.0\/R t (R is the radius of the pipe section) from the column base (Nakamura 1997). This range is

72

20

15

T. Usami et al.

Presen t ] O Previous I

( U s a m i et al. 1995) I I

, Cyclic Test ( U s a m i 1996) I M curve [

t M-S curve I

o . . . . . . . _.

i , i i i , i , . i

0 .2 0 .4 0 . 6 0 .8

(I+P/Py)R~ ~

(a) Unstiffened columns

20 ] �9 Presen t

.~ I O P r e v i o u s

15 i \ I ( U s a m i et al. 1995)

~ ]Cyclic Test ( U s a m i 1996)

~ . I 0 \~, 0 l . . . . . . . M-S . . . . .

5 - ~ ~ . . . . . . . . . . . . . . . . . . . .

0 , I , I , , , 0.] 0.2 0.3 01.4 01.5 01.6

(I+P/Py) �9 Rr" ~ o.s

(b) Stiffened columns

Figure 8 Ductility estimations of cantilever-typed columns with box sections

almost as same as the effective failure length assumed in this study (Eq. 15). Through numerical cyclic analyses, some researchers (Gao et al. 1998b) proposed an empirical equation for the ductility of cantilever-typed columns with pipe sections, which is given by

~9..__ff_5 _ . 0.24 (19) ~iy (1 + P / Py )2,3-~,3Rt

Nine such columns are investigated here, the parameters of which are presented in Table 6. The computed ductility estimations are plotted in the Fig. 9 by comparison with Eq. (19). It is found that all the points corresponding to the results of the present study lie in the vicinity of the equation curve. Thus, the applicability of the proposed method to steel columns with pipe sections is also verified.

One-story Rigid Frame

Although the behavior of thin-walled steel cantilever columns has been extensively investigated by researchers, available research findings on the thin-walled steel frames are too limited to supply sufficient information on the ductility evaluation (Nishikawa et al. 1999). The proposed method is expected to be a simple but efficient ductility evaluation tool for such structures.

A one-story rigid frame, which has been

Speci -men P1

P2

P5

P8a

P8b 1897 891

P8-15 4391 891

P10 3303 580

Pll 4391 891

P12 4391 891

Notes: see Table 5 for details of material properties

TABLE 6 P a r a m e t e r s o f C a n t i l e v e r P ipe C o l u m n s

h d t (mm) (mm) (mm) Rt ~ e/ev MaterialNumber

3403 891 9.00 0.110 0.26 0.12 VI

4391 891 7.32 0.115 0.30 0.15 V

4391 891 8.41 0.100 0.30 0.15 V

2598 891 11.2 0.075 0.18 0.15 V

12.6 0.067 0.13 0.15 V

11.2 0.075 0.30 0.15 V

20.0 0.031 0.37 0.09 VII

9.61 0.088 0.30 0.15 V

16.8 0.050 0.30 0.15 V

12 \ " � 9 P . . . . . t I

10 ~ " - - Empirical Curve I , {Gao et al. 1998b~l

8

4

2

0 t I i I t I i I

0 0.01 0.02 0.03 0.04 (l+P/Py)Rl'S~, ~

Figure 9: Ductility estimations of cantilever- typed columns with pipe sections


T A B L E 7 P A R A M E T E R S O F O N E - S T O R Y R I G I D F R A M E

B Element Plate (mm) Column Flange 600

Web 600 Beam Flange 600

Web 600

t b~ t~ a (mm) (mm) (mm) (mm) Rf 2~,

6 60 6 600 0.497 0.422 6 60 6 8 80 8 600 0.497 0.314 6 60 6

73

�9 - -Material No.

VIII

VIII

tested in a recent study (Nishikawa et al. 1999), is analyzed through proposed method. The general layout and some pertinent parameters of the frame are presented in Fig. 10 and Table 7. In the beam-column connection parts, all the panels of both beam and column sections are strengthened by doubling the thickness. The flame was tested under cyclic lateral loading with the constant vertical loads of P =0.12Py at the top of the frame.

The afore-mentioned method is applied to this structure, where it should be noted that in a flame system, the axial force of the columns varies with the change of lateral load. And this makes the trial and error method required for calculating the failure strains (see Eq. (7)). For this frame, the critical parts are the regions marked by (~), (~), (~), @, (~) and (~ in Fig. 10. And the place where the average compressive strain first reaches the corresponding failure strain is found at part (~). Figure 11 illustrates the normalized lateral force versus displacement curve from the pushover analysis compared by the normalized hysteretic curve from the cyclic test (Nishikawa et al. 1999). Both of the points corresponding to the maximum strength (6,.) and 95% of the maximum strength after peak on the test envelop curve (69s) are used to represent the

Figure 10 General layout of the frame

Figure 11 Force-displacement relation curve of the flame

ultimate state of the flame. The failure points are denoted by different marks in Fig. 11. It is observed that the computed ductility (ru/ry) of the proposed method is close to the 6~/~ from the cyclic test, whereas the ductility prediction based on previous method (Usami et al. 1995) is too conservative. In the light of safety required in practical design, the ductility capacity predicted by the proposed procedure is satisfactory.

74

CONCLUSIONS

T. Usami et al.

The ductility of thin-walled steel stub-columns with and without longitudinal stiffeners was investigated through extensive parametric analyses. The key parameters affecting the ductility of box stub-columns are found to be the flange width-thickness ratio, magnitude of the axial force, and the stiffener's slenderness ratio. The effects of the cross-sectional shape and the columns aspect ratio were also investigated and found insignificant for the ductility of stub-columns. On this basis, empirical equations for the ductility in terms of the failure strain were developed. Besides, empirical equations of failure strains proposed for isolated plates and short cylinders in previous studies (Usami et at. 1995; Gao et al. 1998a) were also presented in this paper.

Moreover, an evaluation procedure has been proposed to employ the ductility equations into the ductility estimation of practical steel structures composed of thin-walled box or pipe sections. A simplified pushover analysis was utilized and a failure criterion was defined. The procedure can be used to evaluate the ductility of thin-walled steel structures in the form of not only cantilever-typed columns but also framing structures. The proposed method was used to successfully evaluate the ductility of some cantilever-typed columns and a one-story rigid frame. By the comparison with ductility estimations obtained from cyclic tests or numerical analyses reported in the literature, the reliability of the proposed method was verified.

References

ABAQUS/Standard User's Manual. (1998). Ver. 5.7. "DIN 4114, Blatt2." (1953). Stahbau, Stabilitatsfalle (Knickung, Kippung, Beulung),

Berechnungsgrundlagen, Richtlinien, Berlin, Germany (in German). Fukumoto, Y., ed. (1997). Structural stability design- steel and composite structures. Elsevier Science

Ltd., Oxford. Galambos, T. V., ed. (1998). Guide to Stability Design Criteria for Metal Structures, 5th Ed., John Wiley

& Sons, Inc., New York. Gao, S. B., Usami, T., and Ge, H. B. (1998a). "Ductility of steel short cylinders in compression and

bending." a r. Engrg. Mech., ASCE, 124(2), 176-183. Gao, S. B., Usami, T., and Ge, H. B. (1998b). "Ductility evaluation of steel bridge piers with pipe

sections." a r. Engrg. Mech., ASCE, 124(3), 260-267. Nakamura, H. (1997). "Formulae for evaluating shear-bending buckling strength of steel piers with

circular cross section and applicability of the numerical buckling analysis method." Proc. of Nonlinear Numerical Analysis and Seismic Design of Steel Bridge Piers, JSCE, 37-42. (in Japanese)

Nishikawa, K., Yamamoto, S., Natori, T., Terao, O., Yasunami, H., and Terada, M. (1996). "An experimental study on improvement of seismic performance of existing steel bridge piers." 3'. of Struct. Engrg., 42A, 975-986 (in Japanese).

Nishikawa, K., Murakoshi, J., Takahashi, M., Okamoto, T., Ikeda, S., and Morishita, H. (1999) "Experimental study on strength and ductility of steel portal frame pier." a r. Struct. Engrg., JSCE, 45A, 235-244 (in Japanese).

Usami, T., ed. (1996). Interim guidelines and new technologies for seismic design of steel structures. Committee on New Technology for Steel Structures (CNTSS), JSCE (in Japanese).

Usami, T., Suzuki, M., Mamaghani, I. H. P., and Ge, H. B. (1995). "A proposal for check of ultimate earthquake resistance of partially concrete-filled steel bridge piers." Struct. Mech./Earthquake Engrg., JSCE, 508/I-31, 69-82 (in Japanese).

HIGH-PERFORMANCE STEEL STRUCTURES: RECENT RESEARCH

L.W. Lu, R. Sause and J.M. Ricles

Department of Civil and Environmental Engineering, Lehigh University Bethlehem, PA 18015-3176, USA

ABSTRACT

Much effort has been devoted in the recent years to the development of high-performance structures for civil and marine construction. Emphasis of this effort has been on the use of high-performance steels and innovative structural concepts to improve performance and reduce life-cycle cost. The paper first gives a summary of the properties of high-performance steels available in the market. This is followed by a description of research exploring application of such steels to I-girder bridges and critical elements in building structures within the framework of the current construction practice. Three innovative structural concepts are then presented: a post-tensioned connection for building frames resisting seismic forces, use of high performance dampers for dynamic response control, and unidirectional double hull structure for ships. Their potential applications are also discussed.

KEYWORDS

High-performance structure, high-performance steel, building, bridge, ship, weldability, fracture toughness, connection, seismic resistance.

INTRODUCTION

What is a high-performance structure? Presently, there is not a universally accepted answer to this question and different people are likely to provide different answers which will depend on the types of structures the individuals having in mind, the desired levels of performance and the performance of structures built according to the present practice. No attempt, therefore, will be made to define "high-performance." The following criteria are often used to judge the overall quality of a structure:

(1) (2) (3)

Performance under service load, Performance under overload, and Life-cycle cost.

75

76 L .W. Lu et al.

For a structure to be considered as a high-performance structure it should have one or more improvements related to these criteria. Different approaches may be adopted to achieve the desired improvements. This paper is concerned with (1) the use of high-performance materials and (2) the development of innovative structural concepts to enhance overall performance and to reduce life- cycle cost.

HIGH-PERFORMANCE STEELS AND THEIR PROPERTIES

A high-performance steel is defined as a steel that has the combined characteristics of high strength, good ductility, high toughness, good weldability and fabricability. These are the properties essential for successful construction of high-performance structures in a civil infrastructure system. For exposed structures, such as bridges and ships, good corrosion resistance is also necessary. From metallurgical composition and processing point of view, a yield strength above 450 MPa is considered as high strength. The fracture toughness, weldability and formability of the steels should be significantly better than those of the conventional steels. The key issue is the control of the amount of carbon and carbon equivalent (Lu, Dexter, Fisher, 1994).

The early attempts of using the traditional high strength steels in bridge and ship construction produced some unsatisfactory results. These steels were found difficult to fabricate due primarily to susceptibility to hydrogen cracking and the risk of brittle fracture associated with materials having inadequate fracture toughness. Other problems include: 1) welding defects other than hydrogen cracking, and 2) potential for stress-corrosion cracking. Many bridges fabricated in the 1960's and early 1970's from ASTM A514/A517 (690 MPa yield) steel suffered from hydrogen cracks which occurred during fabrication (Fisher, 1984). Many of these hydrogen cracks occurred in the longitudinal web/flange joint of welded built-up box sections used as tie girders in tied arch bridges (Anon, 1979, Fisher, Pense and Hausammann, 1982) as well as welded built-up plate girders. One example is the Gulf Outlet Bridge near New Orleans. Some bridges have also experienced hydrogen cracking in transverse groove welds, e.g. the 1-24 bridge over the Ohio River near Paducah, Kentucky (Fisher, 1984). Hydrogen cracking was also observed in the Navy's Seawolf submarine in the 120-S weld metal used with the 690 MPa yield strength HY-100 steel (Anon, 1991).

Hydrogen cracking is most effectively avoided by using steel and weld metal with microstructures that are not susceptible. It has been shown that susceptibility to hydrogen cracking increases significantly as the carbon content exceeds 0.1 percent (Graville, 1976). The susceptible microstructures are typically martensite. The new high-performance steels with low carbon contents are not susceptible to hydrogen cracking.

Microalloyed steels with low carbon content, high manganese levels and microalloy carbide and nitride formers have been available for sometime for use in construction of structures that require high strength, high fracture toughness, and good weldability. Over the past 15 years, low-carbon, age-hardenable steels have gained increasing usage in shipbuilding, heavy-vehicle manufacturing, and offshore structure construction because of their excellent weldability and fracture toughness. These steels have become known as High-Strength Low Alloy (HSLA) steels although their total alloy content is generally around four percent. Another method of increasing strength without increasing carbon and alloy content is controlled rolling combined with on-line accelerated cooling, i.e. thermo-mechanical controlled processing (TMCP).

These high-performance steels offer some clear benefits when compared with the traditional high strength steels (Bolliger, et al, 1988). Most are virtually immune to hydrogen cracking in the heat- affected zone (HAZ) of welds. This superior resistance to hydrogen cracking allows these steels to

High-Performance Steel Structures: Recent Research 77

be welded without the application of preheat in most situations. The low-carbon, fine-grained microstructure that results from typical processing yields a very favorable combination of high strength and high toughness. The excellent fabricability, strength and toughness make high- performance steel very attractive for use in many applications. For bridges, these advantages may allow consideration of lifting the onerous requirements for fabrication of fracture critical members (FCM). FCM are members subjected to tension which if fractured will cause failure of the structure.

The following are some examples of the currently produced high-performance steels:

Low Carbon Age-Hardening Nickel-Copper-Chromium-Molybdenum-Columbian and Nickel-Copper-Columbian Alloy Steels, ASTM A710.

High Yield Strength, Age-Hardening Alloy, Structural Steels (HSLA 80 and HSLA 100), MIL-S-24645A.

Structural Steel for Bridges, ASTM A709 Grade HPS 485W.

There are several copper-nickel high-performance steels for bridge construction under development at the ATLSS Center of Lehigh University (Gross, Stout, and Dawson, 1998).

APPLICATION OF HIGH-PERFORMANCE STEELS

A substantial number of studies have been carried out in the ATLSS Center and elsewhere to explore the use of high-performance steels in bridges, buildings, offshore structures, and ships. Brief descriptions of three of these studies are given below:

1-Girder Highway Bridges

The advantages of using high-performance steel in conventional I-girder highway bridges has been investigated by Sause and Fisher (1995). The investigation involved redesign of recently constructed highway bridges, using conventional steels with yield strengths of 250 MPa and 345 MPa, and using high-performance steels with yield strengths between 485 MPa and 825 MPa. The normalized weight of minimum weight girder cross-sections designed for each steel is plotted versus yield strength in Figure 1. The weight of the design using 345 MPa steel is taken as 100%, and the weight of the designs using other steels are normalized by the weight of the 345 MPa design. Three cases are considered: (1) design for strength and stability according to the AASHTO specifications without considering fatigue, indicated by the dashed line with circles; (2) design for strength and stability without considering fatigue and allowing the plastic moment to be used as the nominal bending strength of compact girder cross-sections, indicated by the dashed line with squares; and (3) design considering strength, stability, and fatigue indicated by the solid line with solid boxes. As seen in Figure 1, if fatigue is not considered, a higher steel yield strength usually results in a smaller weight per length. However, an exception occurs at 485 MPa because the AASHTO specifications permit the use of the plastic moment as the nominal bending strength of compact girder cross-sections only when the yield strength is no more than 485 MPa. As a result, higher strength steel girder cross-sections must be designed with the yield moment (yield stress at the extreme fiber) as the nominal bending resistance. This limitation in the design specifications is based on concern about the ductility of structural members fabricated from high-strength steel. Girders fabricated from high-performance steel may not require this limitation, although further study of this issue is needed. The dashed curve with squares in Figure 1 represents the case when the plastic moment is used as the nominal bending strength of all compact cross-sections. The solid

78

120

L. IV. Lu et al.

1 ................. 1 . . . . . . . . . . . . . . . t . . . . . I .... i . . . . . . . . . . . . . . . . .

lOO .s

-~ 80 --

6 0 , , I I ! ! I 40 60 80 1 O0 120

(275) (415) (550) (690) (830) Yield Stress, ksi (MPa)

Figure 1. Weight versus yield strength for minimum weight steel I-girder cross-sections

line shows that when fatigue of welds between transverse stiffeners and the web and flange plates is considered in design, potential decreases in weight with increasing yield strength end at a yield strength of 690 MPa, because of stress range limits for the details.

In addition to stability and fatigue, deflection under live load may also be a design constraint. The elastic live load deflection of I-girder bridges designed using high-performance steel was considered by Sause and Fisher (1995). AASHTO deflection criteria were applied to high- performance steel bridge designs to investigate whether these criteria are constraints on the use of high-performance steel. Live load deflections were calculated for minimum weight bridge designs developed for each yield strength level, and plotted versus yield strength level in Figure 2. The deflection limit is L/800. Figure 2 shows that the bridge designs at each strength level satisfy the deflection limit. However, the assumptions made in computing the live load deflections according to AASHTO may not be acceptable to many bridge engineers. With more conservative assumptions, the computed deflections for bridge designs at the highest yield strength levels may exceed the deflection limit.

75 I ............... 3

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Vleid Stress , ksi (MPa) Figure 2. Live load deflection versus yield strength for minimum weight bridge designs

Connections in Building Frames

The superior ductibility, toughness, and weldability of the high-performance steels make them ideal


material for critical elements in structural systems. Examples of critical elements, where such properties are required, include connecting plates in beam-to-column connections, connectors or connecting devices, shear links in eccentrically braced frames, tension members of structures in severe service environments, etc. These steels are also attractive for large and complex structures because of the possibility of requiring no pre- and post-weld treatment. A study of the use of A710 steel plates as the flange connecting plates in a beam-to-column web connection shown in Figure 3 was carried out. An identical connection, but with ASTM A572 (50) steel plates, was tested at Lehigh University. It failed prematurely due to fracture of one of the connecting plates. The fracture was predominately brittle in nature although there was evidence of several crack arrests which indicate some ductility in the region. The factors contributing to the fracture include: (1) a large amount of plastic strain imposed on the members, (2) strain concentrations at design details, and (3) orientation of the plates with the applied strain in the least fracture-resistant direction (the rolling direction was parallel to the fracture plane). A post-test examination showed that the defects in the weldments were no greater in size than might be found in typical structural welds. It is felt that this connection fractured in a brittle manner due to the large applied tensile strain which was concentrated at the design detail.

Figure 3. Beam-to-column connection test details with A710 steel.

For the A710 tests, the flange plates was orientated so that the applied strain was not in the direction of the least fracture resistant direction; the rolling direction of the plate was parallel to the applied tensile strain. The strain concentrations at the design details were difficult, if not impossible, to avoid in construction. The connection was, therefore, assembled as if in an actual construction environment. This specimen behaved in a very ductile manner and the ultimate load exceeded the calculated plastic limit load by about 20% (Lu and Fisher, 1990).

The ATLSS Center has developed a wedge and socket type joint for a beam-column connection in a

80 L.W. Lu et al.

Figure 4. ATLSS Connector

framed structure, as shown in Figure 4. This type of joint is designed for to be placed by a remote operator eliminating the need for an ironworker to make the connection (Viscomi, Michalerya and Lu, 1994). This concept could be used for many secondary bridge connections used for diaphragms and bracing. For ease of handling, it is necessary to make the wedge and socket as compact as possible. The socket is either welded directly to a column flange or to a plate which is then bolted to a column fange. A high-strength material that is easy to weld and cast is therefore required. Unfortunately, most of the available high-strength cast steels are not readily weldable. This makes the high- performance steels the ideal choice for the connector pieces. The ATLSS connectors, made with HSLA 80 steel, has been found, in laboratory testing, to perform well either as a shear connection or a part of a full or partial moment connection (Lu, et al 1995). These connectors have been successfully adopted in the construction of industrial plant structures.

Ultra High-Strength Structural Members

Work is in progress on several investigations on high-strength structural members made of high- performance steels with good weldability. One of these is a study on concrete filled tubular (CFT) columns subjected to axial compression or combined axial compression and bending moment (Varma, et al 1998). The tubes were made of A500 Grade 80 steel (550 MPa yield) and the concrete had a compressive strength of 110 MPa. The A500 Grade 80 material is similar to the HSLA 80 steel and can be readily cold formed and welded with one-sided welding (within certain plate thickness). The high-strength CFT members are ideal for use as columns in multi-story building frames.

Another study is on the local and lateral buckling behavior of flexural members made of HSLA 80 steel, whose stress-strain relationships are different from those of the conventional structural steels (Ricles, Sause and Green, 1998).

HIGH-PERFORMANCE STRUCTURES UTILIZING INNOVATIVE DESIGN CONCEPTS

Post-Tensioned Connections for Seismic-Resistant Building Structures

The common practice of designing a seismic-resistant steel structure is to utilize its ductility and


inelastic energy adsorpbtion capacity. The structure is allowed to yield and deform plastically under the design level of earthquake excitation. The individual members and connections must be able to sustain large plastic deformations, while maintaining their resistance. Large plastic deformations often result in excess damage to the structure, which may affect its function and safety.

To improve the seismic response of building frames a new type beam-to-column connection has been developed (Garlock, et al 1998). It is a post-tensioned (PT) connection and is similar to the unbonded PT connection developed previously for precast concrete construction. The connection can be easily incorporated into a conventional steel moment-resisting frame. One version of this connection is shown in Figure 5. It consists of bolted top and seat angles and post-tensioned high- strength strands running parallel to the beam and anchored outside the connection. The beam flanges are reinforced with cover plates in order to limit local beam yielding. Also, bearing plates are placed between the column flange and the beam flanges so that only the beam flanges and the cover plates are in contract with the column. The deformed configuration of the interior PT connection subjected to a pair of clockwise bending moments transmitted from the beams is shown in Figure 6. The moment-rotation (M-er) relationship of the connection is shown in Figure 7(a) and the load-deflection (P-A) relationship of a beam-column subassemblage containing such a connection is shown in Figure 7(b).

The behavior of the connection is characterized by a gap opening and closing at the beam-column interface. The moment to cause this separation is called the decompression moment. The connection initially behaves as a fully restrained connection; but after decompression it behaves as a partially restrained connection. The initial stiffness of the connection is the same as that of a fully restrained connection, where 0r remains equal to zero until the gap opens at decompression. The stiffness of the connection after decompression is associated with the flexrual stiffness of the tension angles and

Figure 5. Post-tensioned connection Figure 6. Deformed configuration of post-tensioned connection

82

M

L.W. Lu e t al.

A P

D-

A

(a) (b)

Figure 7. Moment-rotation and load-deflection relationships of post-tensioned connection

the axial stiffness of the post-tensioned strands. With continued loading, yielding will develop in the tension angles, which will cause further softening of the connection. Full yielding and strain hardening of the tension angles will follow. At a later stage, the compression angles will also yield. After yielding of the tension and compression angles, the M-0r curve becomes approximately linear because the connection's stiffness is provided primarily by the axial elastic stiffness of the PT strands. Upon unloading, the angles will dissipate energy until the gap between the beam flange and column face is closed (when er is again equal to zero). A complete reversal in moment will result in a similar behavior occurring in the opposite direction as shown in Figure 7. As long as the strands remain elastic and no significant yielding occurring in the beams, the post tension force is preserved and the connection and the subassemblage will remain self-centered upon unloading (Figure 7(b)). Accordingly, a frame constructed with PT connections will suffer to permanent sway or drift after a major earthquake event. Studies have shown that the behavior of the connection is controlled by the level of decompression moment, flexural strength and stiffness of the angles, and the elastic stiffness of the strands, while the amount of energy dissipation is related to the flexural strength of the angles. Further work on PT connections is currently in progress at Lehigh University.

Viscoelastic Dampers for Seismic Response Control of Building Structures

A major research was carried out at the ATLSS Center to study the effectiveness of viscoelastic (VE) dampers in reducing the earthquake response of building structures. The work included analytical modeling of the VE material and dampers, experimental investigation of the local and overall behavior of frames with VE dampers under simulated earthquake ground shaking and development of design methodologies (Kasai, et al 1993, Kasai and Fu, 1995 and Higgins and Kasai, 1998). The damping material was installed in V-braces of the frame, as shown in Figure 8.

A collaborative program which included shaking table tests three-story, single-bay steel frames with and without dampers, was conducted at the National Taiwan University of Science and Technology (Higgins, Chen and Chou, 1996). The test and analytical results demonstrate that a properly

High-Performance Steel Structures: Recent Research

Damper

\

(

yr/--

83

Figure 8. Frame with viscoelastic dampers

designed VE damped flame can perform in a damage free manner under major earthquakes generally considered in design codes. Recent work, however, has shown the undesirable effect of temperature sensitivity of the high damping VE material, especially when it is used in exposed structures (Fan, et al 1998). Material, such as natural rubber, whose damping and stiffness properties are almost unaffected by temperature changes, may be more desirable. Research on rubber damper is currently in progress at Lehigh University.

Double Hull Ship Structures

Another example of innovative design concept is the double hull structure for ships (Beach, 1990). Most of the ships in service today are of the conventional hull type, which is basically a single skin of steel plating stiffened by transverse web frames and longitudinal stiffeners. The double hull is fundamentally different from the conventional hull in that it has twin skins (or shells) of steel plating which are separated from each other and stiffened by longitudinal web plates or girders that span between transverse bulkheads. Other transverse components are eliminated, thus creating a simple unidirectional, longitudinal structure. Figure 9 compares the conventional hull and the new double hull.

The simple, unidirectional structure gives rise to several important advantages over the conventional hull. The redundant hull structure provides greater survivability for the ship when subjected to collision or grounding forces. The inner hull also serves as an additional barrier against leakages in case the outer hull is punctured. In the double hull the number of fatigue-critical details is significantly reduced because the longitudinal girders are not interrupted by stiffeners, brackets, or transverse frames between bulkheads. The long continuous welds may allow automatic welding and other advantages in producibility with consequent cost savings. It is envisioned that these ships will

84 L . W . Lu et al.

Figure 9. Conventional hull and double hull ship structures

be fabricated from high-performance steels which offer improved weldability, increased strength and toughness relative to conventional ship steels. The new design and materials should lead to safer and more affordable ships. Much of the research on double hull ships at Lehigh focused on hulls made of HSLA 80 steel (Pang, et al 1993, Dexter, et al 1996).

S U M M A R Y

Brief descriptions of some selected research to develop high-performance structures for civil and marine construction have been presented. Two approaches are adopted to achieve improvements in performance and life cycle cost. They are: (1) use of high-performance steels and (2) development of innovative design concepts. Potential applications of the developed technologies to bridges, buildings and ships have been discussed.

REFERENCES

Anon (1979). Welding Flaws Close Interstate Tied Arch Bridge, Engineering News Record, August 16.

Anon (1991). U.S. Navy Reports Welding Procedure Source of Cracks in First Seawolf Submarine, Welding Journal, 71(9), p. 5.

Beach, J.E. (1990). Advanced Surface Ship Hull Technology - Cluster B, ASNE Symposium on Destroyer, Cruiser and Frigate, 89-112.

Bolliger, W., Varughese, R., Kaufmann, E., Qin, W.F., Pense, A.W., and Stout, R.D. (1988). The Effect of Welding and Fabrication Operations on the Toughness of A710 Steel, in "Microalloyed HSLA Steels," ASM International, p. 277.


Dexter, R.J., Ricles, J.M., Lu, L.W., Pang, A.A., and Beach, J.E. (1996). Full-Scale Experiments and Analyses of Cellular Hull Sections in Compression, Joumal of Offshore Mechanics and Arctic Engineering, ASME, 118-3,232-237.

Fan, C.P., Lu, L.W., Sause, R., and Ricles, J.M. (1998). Research at Lehigh University on Use of Viscoelastic Material in Retrofitting Dynamically Load Structures. Proc. Third International Symposium on Civil Infrastructure Systems Research: Intelligent Renewal, Capri, Italy, 1997, World Scientific Publishing Co., 137-151.

Fisher, J.W., Pense, A.W., and Hausammann, H. (1982). Fatigue and Fracture Analysis of Defects in a Tied Arch Bridge, Proc. IABSE Colloquium on Fatigue of Steel and Concrete Structures, Lausanne, Switzerland.

Fisher, J.W. (1984). Fatigue and Fracture of Steel Bridges, Wiley Interscience, New York.

Garlock, M.M., Ricles, J.M., Sause, R., Peng, S.W., Zhao, C., and Lu, L.W. (1999). Post- Tensioned Seismically Resistant Connections for Steel Frames. To appear in Proc. 1998 Annual Technical Session, SSRC, Atlanta, GA.

Graville, B.A. (1976). Cold Cracking in Welds in HSLA Steels, Proc. International Conference on Welding of HSLA (Microalloyed) Structural Steels, ASM, Rome, Italy.

Gross, J.H., Stout, R.D., and Dawson, H.M. (1998). Copper-Nickel High-Performance 70W/100W Bridge Steels - Part II, Report No. 98-02, Center for Advanced Technology for Large Structural Systems, Lehigh University.

Higgins, C., Chen, S.J., and Chou, F.C. (1996). Testing and Analysis of a Steel Frame with Viscoelastic Dampers, Proc. Eleventh World Conference on Earthquake Engineering, Acapulco, Mexico, Paper No. 1961.

Higgins, C. and Kasai, K. (1998). Seismic Design, Analysis and Experiment of a Multi-Story Viscoelastically Damped Steel Frame, Special Issue on Passive Control, ISET Journal of Earthquake Technology, 35:4.

Kasai, K., Munshi, J.A., Lai, M.L., and Maison, B.F. (1993). Viscoelastic Damper Hysteretic Model: Theory, Experiment, and Application, Proc. ATC 17-1, Seminar on Seismic Isolation, Passive Energy Dissipation, and Active Control, Applied Technology Council, San Francisco, CA, 521-532.

Kasai, K. and Fu, Y. (1995). Seismic Analysis and Design Using Viscoelastic Dampers, Proc. Symposium on a New Direction in Seismic Design, Architectural Institute of Japan, Tokyo, Japan, 113-140.

Lu, L.W., Dexter, R.J., and Fisher, J.W. (1994). High-Performance Steels for Critical Civil Infrastructure Systems, Proc. Intemational Workshop on Civil Infrastructural Systems, Taipei, Taiwan, 283-298.

Lu, L.W., Viscomi, B.V., Fleischman, R.B., Lawrence, W.S., Rosa, A.M., and Garlock, R.B. (1995). Development and Experimental Investigation of New Types of Connections for Framed Structures Suited for Automated Construction, Proc. First European Conference in Steel Structures, Athens, Greece, 231-238.

86 L . W . Lu et al.

Pang, A.A., Tiberi, R., Lu, L.W., Ricles, J.M., and Dexter, R.J. (1993). Measured Imperfections and Their Effects on Strength of Component Plates of a Prototype Double Hull Structure, Journal of Ship Production, 11-1, 47-52.

Ricles, J.M., Sause, R., and Green, P.S. (1998). High-Strength Steel: Implications of Material and Geometric Characteristics on Inelastic Flexural Behavior, Engineering Structures 20:4-6, 325-335.

Sause, R. and Fisher, J.W. (1995). Application of High Performance Steel in Highway Bridges, Proc. ASM International Symposium on High Performance Steels for Structural Applications, Cleveland, OH.

Varma, A.H., Hull, B.K., Ricles, J.M., Sause, R., and Lu, L.W. (1998). Experimental Studies of High Strength CFT Beam-Columns, Proc. Fifth Pacific Structural Steel Conference, Seoul, Korea, 893-906.

Viscomi, B.V., Michalerya, W.D., and Lu, L.W. (1994). Automated Construction in the ATLSS Integrated Building Systems, Automation in Construction 3:1, 35-43.

A UNIFIED PRINCIPLE OF MULTIPLES FOR LATERAL DEFLECTION, BUCKLING AND VIBRATION

OF MULTI-STOREY, MULTI-BAY, SWAY FRAMES

W.P. Howson and F.W. Williams

Cardiff School of Engineering, Cardiff University, Cardiff CF24 3TB, UK

ABSTRACT

Cheap computing has rightly eliminated tedious hand methods from taught courses. Unfortunately, this has often unintentionally resulted in the elimination of procedures which give useful structural behavioural insights into whole families of structures using simplified models which give good approximate results. Such procedures help designers both to check that computer results are reasonable and also to gain insight from parametric studies with a manageable number of parameters. The present paper seeks to rescue one such procedure from relative obscurity and presents almost its only recent extension. It is the use of substitute one bay (or alternatively Grinter) frames to obtain results for unbraced and lightly braced multi-storey, multi-bay, sway frames which are exact when they are unbraced, have inextensible members and obey the Principle of Multiples and otherwise are good approximations. These results can incorporate cladding and are for : deflections caused by static lateral (e.g. wind) loading; critical buckling and; natural vibrations. These three types of problem are unified herein and, because adequate published results exist for the first two types, only (new) natural frequency results are given. These show that, for unbraced frames, the substitute frame gives acceptable accuracy for most purposes regardless of how closely the frame obeys the Principle of Multiples. They also apparently justify a new application, namely to analyse multi-storey, multi-bay frames with one or more braced bays by using a substitute frame which represents the bracing as cladding. Exceptionally, when all bays are cross-braced, the Principle of Multiples may be obeyed.

KEYWORDS

Wind load, buckling, vibration, substitute frame, Grinter, Principle of Multiples.

INTRODUCTION

Until the 1960's, the teaching and practice of structural engineering consisted mainly of understanding the underlying principles, then learning hand methods and practising their use extensively. Because hand solutions are tedious, engineers thought carefully about their initial design, both before analysing

87

88 W.P. Howson and F. IV. Williams

it and as the analysis gave intermediate results which were expected or otherwise. Experienced designers bought much of their experience by performing such calculations for each design they were responsible for, often including the analysis of several alternative structures before the design process was completed. Hence designers and their teachers were keen to develop insights which would enable them to choose a good design prior to computation commencing and/or to proceed from rejected designs to acceptable ones via as few analyses of trial designs as possible. The Principle of Multiples was one useful source of such insights.

The advent of computers and the rapid reduction in computing costs has required a different approach to the teaching of structures and to how designers obtain experience. The basic principles which need to be taught and understood have altered little but their implementation in computer programs has led to new (usually stiffness matrix based) methods being taught. Numerous hand methods are no longer taught because designers no longer require them and hand calculations are used only to confirm that students understand the underlying principles and how computer methods work. They are also used by designers to perform 'spot checks' and simple 'back of the envelope' type calculations to ensure that computer generated designs and/or analyses are reasonable, both to guard against data errors and an inappropriate choice of computer program. Therefore methods previously obtained to give insight, such as the Principle of Multiples, remain valuable instruments in the designer's armoury.

Computing is now so cheap that students and design engineers can build up experience very quickly by designing and/or analysing a large range of structures. However, it is tempting to overestimate the extent to which this enables insight and experience to be developed, because a complete structure usually involves so many design variables that, even if the required large number of computer runs could be afforded, designers would suffer from information overload unless structural insight can be used to categorise the results or to reduce their number. Here again, the Principle of Multiples can be of value.

In the authors' opinion, methods giving insight have often been removed from undergraduate courses, and hence largely lost to the profession, in the mistaken belief that their value has disappeared. Therefore, the Principle of Multiples is presented in this paper, starting with the static lateral (wind) load and critical buckling calculations with which it has historically been principally associated and then proceeding to examine its value in the vibration context.

THE PRINCIPLE OF MULTIPLES AND SUBSTITUTE FRAMES FOR LATERAL LOAD AND BUCKLING CALCULATIONS

The Principle of Multiples applies to unbraced, rigidly jointed, multi-bay, multi-storey plane frames and is exact on the basis of inextensible member theory. Since hand methods of analysis nearly all assume inextensible members this assumption is acceptable, although most computer programs use extensible theory both for convenience and to obtain additional accuracy. Numerous authors have dealt with the use of the Principle of Multiples and associated simple methods for lateral load and buckling calculations for such frames, e.g. Bolton (1976), Grinter (1936), Home and Merchant (1965), Home (1975), Lightfoot (1956, 1957, 1958, 1961), Naylor (1950), Williams (1977a, 1977b, 1979), Williams and Howson (1977), Williams and Butler (1988), Wood (1952,1974), Wood and Roberts (1975).

Figure 1 applies to lateral load calculations when F r 0, whereas it refers to buckling problems when F = 0 and W r 0. It is usual to perform the lateral load calculations with W = 0, but non-zero values of W can be used if the designer wishes to allow for the magnifying effect that vertical loads have on

Unified Principle of Multiples for Lateral Deflection, Buckling and Vibration

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, f , .

k3 k 3 2k 3

k4 2k 4 3 2k 4

8W

| 12k 1 8k2 [ 124 w ~f"~,~ 8F .IJ,

L

.5L

(d) (e)

Figure 1" Frames (a) - (d) comply with the Principle of Multiples and (e) is the corresponding Grinter frame

horizontal deflections caused by lateral loading. The Grinter frame of Figure 1 (e) is required later, but should be ignored for the time being. The Principle of Multiples proves that the frames of Figures 1 (a)-(d) share the same horizontal deflections for lateral loading problems and share the same critical value of W for buckling problems. The reasons are as follows.

In Figure 1, the k's are values of EI / g for the members, where EI is the flexural rigidity and g is length. Additionally, values are identical when the subscripts are identical, so that the frame of Figure 1 (a) is symmetrical. Note also that the vertical loading is symmetric. Therefore, considering buckling first, the frame must buckle with a symmetric or an anti-symmetric mode and it is easily proved that the anti-symmetric mode gives the lowest possible critical load. Hence any frame which is identical to frame (a) must have the same critical load Wc and the same deflected shape. Therefore any frame

90 W.P. Howson and F.W. Williams

obtained by superposing N (which need not be integer) such frames, in the sense implied by flames (b) and (c), must also share the critical load W c and the deflected shape of frame (a), even if the frames are all clamped together. Hence putting N=2 and N=4 gives the required proofs for frames (b) and (c), respectively. Moreover, frame (d) can be obtained by fastening together two frame (a)'s and a frame (b), which are situated side by side in the appropriate way. Since frames (a) and (b) share the same critical load Wc and sway with an anti-symmetric deflection pattem, the process of fastening them together to form frame (d) leaves the critical load We and the deflections unaltered.

The proof that frames (a)-(d) share the same deflections under lateral loading is essentially identical to the proof for buckling given above, if it is noted that the lateral loads of Figures (a)-(c) can be replaced by an anti-symmetric load pattern, which must cause an anti-symmetric deflection pattem, because the beams are assumed to be inextensible. For example, the F of Figure (a) can be replaced by F/2 at the left hand end of the top beam and F/2 at the right hand end of the top beam. Similarly, the 4F at the top left hand joint of Figure (d) can be replaced by loads of F/2, F, 3F/2 and F at the four top storey joints, etc.

Most multi-bay frames do not obey the Principle of Multiples, Home and Merchant (1965), Lightfoot (1956). However, a well established method exists for reducing multi-bay multi-storey frames to single bay multi-storey 'substitute' frames which can then be used to obtain approximate lateral loading or critical buckling results for the multi-bay case. The substitute frame has the same number of storeys and the same storey heights as the actual frame, but differs in that it has only one bay, is symmetric and carries symmetrical vertical loads. The required details of the substitute frame are found from the actual frame as follows: the substitute column k is equal to half the sum of the k's for all actual columns at the same storey level; the substitute beam k is equal to the sum of the k's for all beams at the same storey level; the horizontal loads at the nodes at both ends of a beam are equal to half of the sum of the horizontal loads at all actual nodes at that storey level; and the values of p for the substitute columns are equal to the sum of the axial forces in all actual columns at the same storey

level divided by the sum of the n2EI / g2 for all actual columns at the storey level. Hence p is equal

to the axial force in a substitute column divided by the value of its Euler load, n2EI / g2.

Applying the above rules to the flame of Figure l(d) gives the flame of Figure l(c), on which the forces 4F and 8F can be replaced by anti-symmetrical pairs of forces. Hence it can be deduced that when a frame obeys the Principle of Multiples the rules yield a substitute frame which gives exactly correct results for the actual frame, remembering that inextensible member theory is assumed.

The Grinter frame of Figure 1 (e) has been advocated as a means of obtaining approximate results for use in codes and has been known for a very long time, Grinter (1936), Wood (1974). Rules for obtaining it from the actual frame are identical to those given for the substitute frame above, except that, as can be seen by comparing Figures 1 (c) and (e), the vertical loads and column k's are twice as large and the beam k's are three times as large. Note that the rolling supports at the right-hand ends of the beams of the Grinter frame prevent rotation while leaving horizontal motion unrestrained. The Grinter frame has been favoured due to its computational simplicity, because computation only involves one node per storey and they are connected to form a chain, so that the stiffness matrix has the minimum possible bandwidth, i.e. it is tri-diagonal. Because of the symmetry, the substitute frame of Figure l(c) can be analysed by considering only half of the frame, which looks like the Grinter frame of Figure l(e) except that the beams are now of half length and are pinned to the rollers. Therefore, for buckling problems there is no horizontal force in the columns and so the stability functions n and o, Home and Merchant (1965), can be used to obtain an overall stiffness matrix with only one degree of freedom, the joint rotation, at each storey level. If the refinement of the n and o

Unified Pr&ciple of Multiples for Lateral Deflection, Buckl&g and Vibration 91

functions is not introduced, the problem is still very simple because it has only two degrees of freedom, rotation and horizontal deflection, at each storey level. Similarly, the lateral load calculations require only two degrees of freedom at each joint, or one degree of freedom if the Principle of Superposition is used to first apply the lateral loads with rotation prevented at the joints and then to calculate the clamping moments at the joints and apply them with reversed sign, so that the ensuing calculations are for a problem with no horizontal force in the columns and hence can again use the n and o functions. Thus the nodes form a chain and there is only one degree of freedom (rotation) at each storey level if the n and o functions are used (so that the overall stiffness matrix is tri-diagonal) and otherwise there are two degrees of freedom per node, i.e. rotation and horizontal displacement. Therefore, the substitute frame and the Grinter frame give identical results with identical computational effort.

The authors have always considered the substitute frame to be preferable to the Grinter frame because it gives much greater physical insight. In particular, as can now be seen, the substitute frame relates in an obvious way to the Principle of Multiples whereas the Grinter frame does not. For example, when an actual frame does not obey the Principle of Multiples, a 'feel' for the probable accuracy of results given by a substitute frame can be obtained by a quick estimate of how close an approximation to the actual frame can be obtained by applying the Principle of Multiples to the substitute frame.

The substitute frame has many fewer design variables than the actual frame. Therefore, parametric studies undertaken with the substitute frame can give the designer insights into the behaviour of the full range of possible actual frames, i.e. the full range of multi-storey multi-bays frames, with very small computational effort and without the designer overload referred to in the Introduction occurring.

S

EI

"37"

Figure 2: Simple system used to represent cladding for a Grinter frame, where the dashed line represents one storey (with flexural rigidity EI) of its column

A simple established way of allowing for cladding when using Grinter frames, Wood (1974), which can also be used for substitute frames, Williams (1979), is shown in Figure 2. The effect of the system shown is to resist relative horizontal movement of adjacent storeys with stiffness S, which is usually expressed in the dimensionless form

= Sg 3 / EI (1)


APPLICATION OF THE PRINCIPLE OF MULTIPLES TO VIBRATION PROBLEMS

The preceding sections have established a case for using substitute frames for wind load and buckling calculations, despite the availability of massive and cheap computer power, both for frames which do or do not obey the Principle of Multiples. Therefore, the focus is now changed to examine the extent and usefulness of the corresponding applications to vibration problems, to which very little attention has been given, Bolton (1978), Roberts and Wood (1981), Williams et. al. (1983). It is assumed that 'exact' member theory is used in the sense that the distributed mass of the members and the attached floors are incorporated when calculating the member stiffness matrices, which are therefore transcendental functions of both frequency and load per unit length. Hence, there are two restrictions to the application of the Principle of Multiples to vibration problems which were not there for the lateral load and buckling problems. These are that members sharing the same subscript on Figure 1 must have the same mass per unit length as well as the same value of k and that all bays must have identical spans. The second requirement occurs because, whereas a beam (because it is in contraflexure) contributes 6k to the overall stiffness matrix of the half substitute frame analysed for the lateral loading and buckling cases, in vibration problems the stiffness contributed depends both on k and on a dynamic stability function which is a transcendental function of both the beam span and the mass per unit length.

Therefore rules must be adopted to establish the values of g and la for the beams of the substitute frame. The rules adopted herein are that L is taken as the average value of the bay widths of the actual frame, so that EI can be calculated from the substitute beam k yielded by the rules given in the previous section, while l.t for the substitute beam is obtained by dividing its g into the total mass of all beams at the same storey level of the actual frame.

Arguments essentially identical to those in the previous section then show that frames (a)-(d) of Figure 1 have identical sway natural frequencies. It is impossible to devise a Grinter frame of the type shown in Figure l(e) which will share exactly the natural frequencies of the actual frame even when the actual frame obeys the Principle of Multiples. This is because the dynamic stability functions for a member built-in at its far end do not behave identically to those of a member which crosses an axis of anti-symmetry of the mode. Of course, this could be overcome by modifying the Grinter frame of Figure 1 (e) such that the right-hand ends of the beams, as well as being on rollers, are free to rotate. However, as well as this modified frame no longer strictly being a Grinter frame, it is essentially half of the substitute frame used previously with all flexural rigidities and loads doubled. Hence it shares exactly the computational advantages of the substitute frame without giving the insight advantages. Therefore, the authors consider that their preference for the substitute frame as opposed to the Grinter frame is additionally vindicated when vibration, as opposed to just lateral load and critical buckling problems, is considered.

RESULTS

To keep the description of the results concise, they were all obtained for variants of the four bay, eight storey frame with built-in foundations shown in Figure 3. This frame has sensible properties, as follows. Young's modulus (E) = 200 GN/m 2. The beams are all identical, with length 7.2m, second moment of area (I) 6,000cm 4, cross-sectional area (A) = 52.5cm 2 and mass per unit length (~t) = 3,500 kg/m, which includes an allowance for floor mass and the mass of live floor loading. The columns are all of length 4.0m and their other properties were identical for any chosen storey i (i = 1,2 ..... 8), such

Unified Principle of Multiples for Lateral Deflection, Buckling and Vibration

4 @ 7.2m

I

I

I e

I r

I f

I

f

I

f -

I

,,J I

I

,a I

I

@

93

Figure 3" The datum frame. The diagonals shown dashed are absent except for case 6 of Table 1

that Ii = 15,000 Yi cm4, Ai = 131.25~f~ cm2 and ~i = 100~]-kg/m, which is for columns with no

allowance for cladding, where

Yi = 1 + 0.35 ( 8 - i) (2)

Table 1 gives the first three natural frequencies of this structure as case 1, i.e. as the datum problem. The beams and columns were represented by Bernoulli-Euler member theory, with distributed mass allowed for exactly, Howson et. al. (1983). The results given by the substitute frame are compared with those given by the actual frame using inextensible and extensible member theory. Table 1 additionally contains the corresponding natural frequency results for all the remaining cases, each of which is a variant of the datum problem or of the substitute frame used to represent it, as briefly defined in the second column of the Table and more fully described as follows.

Cases 2 and 3 give altemative substitute frames for case 1, for instructional reasons. For case 2 the beams were analysed as massless and hmaped masses equal to half to the mass of the beam were added at each end of the beam. For case 3, the Grinter frame of Figure 1 (e) was used.

Case 4 used 'exact' theory, Howson et. al. (1983), to allow for the effect of axial force, as well as of distributed mass, on the flexure of the columns. The axial forces were obtained as if half of the mass of each beam had been lumped at its ends, both for the substitute and actual frames, but the beam

94

Case

10

W.P. Howson and F.W. Williams

TABLE 1

RESULTS FOR ALL PROBLEMS (HZ)

Description of Problem

Datum problem

Datum with the masses of beams of substitute frame lumped at their ends

Grinter frame results for datum

Datum and allow for effect of axial forces on column flexure

Datum with cladding added (~ = 5)

Datum with both central bay spans doubled

Datum with EI of second column from the left doubled

Datum with stiff cladding (g = 15)

for substitute frame, to represent structural bracing of one bay of actual frame

As case 5, except that g = 5 is represented exactly as in Figure 2, not by an equivalent diagonal

As case 8, except that g = 15 is represented exactly as in Figure 2, not by an equivalent diagonal

Substitute

0.234 0.756 1.437

0.234 0.758 1.446

0.234 0.754 1.428

0.208 0.708 1.374

0.557 1.504 2.518

0.172 0.565 1.095

0.240 0.783 1.510

0.901 2.363 3.054

0.557 1.504 2.519

0.901 2.364 3.054

Frame

Actual (EA--->oo)

0.234 0.753 1.432

as case 1

as case 1

0.207 0.705 1.369

0.557 1.503 2.515

0.172 0.563 1.086

0.239 0.779 1.505

0.901 2.362 3.004

as case 5

as case 8

Actual (True EA)

0.233 0.752 1.431

as case 1

as case 1

0.206 0.698 1.350

0.543 1.479 2.486

0.171 0.562 1.084

0.238 0.779 1.503

0.633 1.893 2.991

as case 5

as case 8

stiffnesses were still calculated using distributed mass. It should be noted that the axial forces were 22.6% of those which would have caused buckling of the substitute frame, i.e. the critical load factor for the substitute frame was 1/0.226 = 4.42.

Case 5 gives results when cladding (the mass of which was neglected) represented by bracing equivalent to ~ = 5 at every storey was added to the actual (i.e. multi-bay) datum problem. The

Unified Principle of Multiples for Lateral Deflection, Buckling and Vibration 95

authors deliberately chose software which does not have coding to represent the spring and rigid cranked beam system of Figure 2 when representing the substitute frame because such a feature is unlikely to be available to a designer seeking to use the substitute frame to undertake a parametric study. Instead the authors used approximately equivalent massless diagonal bracing in each of its bays. By assuming that the beams and columns were inextensible (which is reasonable because the cross-sectional area A of the beams and columns far exceeds that of the bracing) the bracing members for storey i were readily shown to have

A i = 1.263 yi ~ cm 2 (3)

for the substitute frame and one quarter of this value for each bay of the actual frame, for which all the diagonals were parallel to each other, so that the structure was not symmetric.

Cases 6 and 7 were included to show the effects of further deviation from the requirements of the Principle of Multiples. In case 6 the span of the two central bays was doubled, with the substitute beam length being taken as the average of the sum of the actual beam lengths. In case 7 the EI of the second column from the left was doubled at every storey level.

Case 8 was solved in order to see to what extent ~ (again modelled by diagonal bracing) could be used in the substitute frame to represent an actual frame which was braced only in the one bay indicated by the dashed lines on Figure 3. These diagonals and those of the substitute frame all have the value of A given by Eqn. 3.

Cases 9 and 10 are identical to cases 5 and 8 respectively, except that ~ for the substitute frames was modelled as shown in Figure 2, instead of by the equivalent diagonals of Eqn. 3.

SOME CONCLUSIONS FROM THE RESULTS OF TABLE 1

All cases of Table 1 (except cases 8 and 10 which are discussed later) demonstrate good agreement between the substitute frame results and those obtained for the actual frame when using extensible member theory, i.e. the true EA' s. This strongly suggests that the first three modes of the actual frame were sway dominated anti-symmetric ones, since these are the only modes which the substitute frame can find. The correctness of this conclusion was verified by calculating the natural frequency for the lowest non-sway (i.e. symmetric) mode and, in case 6, eliminating anti-symmetric modes between 0.562 Hz and 1.084 Hz for which the mode could be seen upon inspection to be a 'local' mode, i.e. one dominated by flexure of individual members with very little sway occurring.

By comparison with case 1, it can be seen from cases 2-6, respectively, that : the horizontal beam inertias are important but their transverse inertias have negligible effect; the Grinter frame results are very close to the substitute frame ones, so that the use of Grinter frames for structures which obey the Principle of Multiples may only cause very small errors; allowing for the flexural magnification due to axial forces of practical magnitudes causes significant reductions of the fundamental (12% in this case) and higher natural frequencies and these reductions can be calculated very accurately from the substitute frame; allowing for the stiffening effect of cladding can greatly increase the fundamental (by 133% in this case) and higher natural frequencies and again the substitute frame can be used to calculate these increases very accurately.


Note that none of the cases 1-5 of Table 1 obey the Principle of Multiples because the outer two of the five columns have twice the required properties, but that nevertheless the excellent agreement of the final two columns of results confirms that the inextensible assumption of the Principle of Multiples is extremely accurate. Cases 6 and 7 show that this agreement remains good for frames which depart more radically from obeying the Principle of Multiples.

The reason for the substitute frame results for cases 8 and 9 differing so much (by up to 42%) must principally be the extensibility of the beams and columns, because the actual frame with EA--->oo gave results almost identical to those of the substitute frame. Physical reasoning suggests that, because only one bay of the actual frame is braced, the extensions and contractions of beams and columns caused by the forces in the diagonal bracing will be largely confined to the beams in the braced bay and the two columns bounding the bay. This further suggests that the beams and columns of the substitute frame should not be treated as inextensible but should instead be given the EA values of an individual beam and column of the braced bay of the actual frame. When this was done the values of 0.901, 2.363 and 3.054 in Table 1 were replaced by 0.607, 1.861 and 2.777, i.e. the maximum difference of +42% from the 'full frame with actual EA' results was reduced to -7% for the third natural frequency and the fundamental was in error by only -4%. (Note that if the bracing of the actual frame is evenly distributed between the four bays, so that each bay has one quarter of the A, the substitute frame is unaltered but the actual frame results of 0.835, 2.248 and 2.992 are much closer to them, as would be expected because the columns of the actual frame will then change length very little.) Hence, the results of cases 8 and 10 lead to the tentative but important new result that the substitute frame method, with an appropriate value of ~ (modelled either via Figure 2 or the diagonals of Eqn. 3) and with appropriate values of EA, gives a useful indication of the natural frequencies for the sway modes of frames which have bracing in a minority of their bays.

Finally, comparison of the results of cases 9 and 10 with those of cases 5 and 8 justifies the use of the equivalent diagonal of Eqn. 3 when software incorporating the model of Figure 2 is not available.

FURTHER THOUGHTS

An unbraced frame is usually one of a set of similar frames which are parallel to it and are connected to it by beams perpendicular to it, e.g. to form a building of rectangular planform. The first author, together with a Master's student, have made a very promising preliminary investigation of predicting the sway modes of such structures which sway parallel to the frames, by applying the rules given earlier to obtain a substitute frame, but with the modification that all the frames are used when applying the rules, e.g. the substitute column k is equal to half of the sum of the k's for all actual columns at the same storey level, regardless of which frame the column lies in, etc. This concept is derived from the fact that floors can reasonably be regarded as being rigid in their own planes, so that all frames share the same horizontal displacements. Of course, such substitute frame results will be exact if the frames are identical to one another, are identically loaded and individually obey the Principle of Multiples. This is clearly true, because the frames would then deflect identically to one another even in the absence of floors and the substitute frame would share exactly the behaviour (i.e. lateral displacements, buckling load factor or natural frequencies) of the substitute frame yielded by one frame alone, since all the stiffnesses and loading of the latter substitute frame would be multiplied by the number of frames to give the former one.

Unified Principle of Multiples for Lateral Deflection, Buckl&g and Vibration

REFERENCES

97

Bolton A. (1976). A simple understanding of elastic critical loads. Struct. Engr. 54:6, 213-218. See also correspondence 54:11,457-462.

Bolton A. (1978). Natural frequencies of structures for designers. Struct. Engr. 56A:9, 245-253. See also correspondence 59A:3, 109-111.

Grinter L.E. (1936). Theory of Modern Steel Structures, Vol. 2. Macmillan, New York.

Home M.R. and Merchant W. (1965). The Stability of Frames. Pergamon Press, Oxford.

Home M.R. (1975). An approximate method for calculating the elastic critical loads of multi-storey plane frames. Struct. Engr. 53:6, 242-248.

Howson W.P., Banerjee J.R. and Williams F.W. (1983).Concise equations and program for exact eigensolutions of plane frames including member shear. Adv. Eng. Software, 5:3, 137-141.

Lightfoot E. (1956). The analysis for wind loading of rigid-jointed multi-storey building frames. Civil Engineering and Pubic Works Review. 51:601,757-759; 51:602, 887-889.

Lightfoot E. (1957). Substitute frames in the analysis of rigid jointed structures (Part 1). Civil Engineering and Public Works Review. 52:618, 1381-1383.

Lightfoot E. (1958). Substitute frames in the analysis of rigid jointed structures (Part 2). Civil Engineering and Public Works Review. 53:619, 70-72.

Lightfoot E. (1961). Moment Distribution. Spon, London.

Naylor N. (1950). Side-sway in symmetrical building frames. Struct. Engr. 28:4, 99-102.

Roberts E.H. and Wood R.H. (1981). A simplified method for evaluating the natural frequencies and corresponding modal shapes of multi-storey frames. Struct. Engr. 59B:1, 1-9. See also correspondence 59B:4, 64-65.

Williams F.W. (1977a). Simple design procedures for unbraced multi-storey frames. Proc. Inst. Civ. Engrs, Part 2 63, 475-479.

Williams F.W. (1977b). Buckling of multi-storey frames with non-uniform columns, using a pocket calculator program. Comput. Struct. 7:5, 631-637.

Williams F.W. and Howson W.P. (1977). Accuracy of critical loads obtained using substitute frames. Proc. Int. Conf. Stab. Steel Structs., Liege, 511-515.

Williams F.W. (1979). Consistent, exact, wind and stability calculations for substitute sway frames with cladding. Proc. Inst. Civ. Engrs. 67:2, 355-367.

Williams F.W., Bond M.D. and Fergusson L. (1983). Accuracy of natural frequencies given by substitute sway frames with cladding. Proc. Inst. Civ. Engrs. 2:75, 129-135.


Williams F.W. and Butler R. (1988). Simple calculations for wind deflections of multi-storey rigid sway frames. Proc. Instn. Cir. Engrs., Part 2 85, 551-565.

Wood R.H. (1952). Degree of fixity methods for certain sway problems. Struct. Engr. 30:7, 153- 162.

Wood R.H. (1974). Effective lengths of columns in multi-storey buildings. Struct. Engr. 52:7, 235- 244; 52:8, 295-302; 52:9, 341-346.

Wood R.H. and Roberts E.H. (1975). A graphical method of predicting sidesway in the design of multi-storey buildings. Proc. lnstn. Civ. Engrs., Part 2 59, 353-372.

Beams and Columns


THREE-DIMENSIONAL HYSTERETIC MODELING OF THIN-WALLED CIRCULAR STEEL COLUMNS

Lizhi Jiang and Yoshiaki Goto

Department of Civil Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, 466-8555, Japan

ABSTRACT

An empirical hysteretic model is presented to simulate the three-dimensional cyclic behavior of cantilever-type thin-walled circular steel columns subjected to seismic loading. This steel column is modeled into a rigid bar with multiple vertical springs at its base. Nonlinear hysteretic behavior of thin-walled columns is expressed by the springs. As the hysteretic model for the spring, we modify the Dafalias and Popov's bounding-line assumption in order to take into account the degradation caused by the local buckling. The material properties for the vertical spring are determined by using curve- fitting technique, based on the in-plane restoring force-displacement hysteretic relation at the top of the column obtained by FEM analysis. By properly increasing the number of springs, the homogeneity of thin-walled circular columns is maintained. Finally this model is used in three-dimensional earthquake response analysis.

KEYWORDS

Hysteretic model, Three-dimensional behavior, Local buckling effect, Steel, Thin-walled column, Empirical model, Earthquake response analysis

INTRODUCTION

In the three-dimensional earthquake response analysis for thin-walled steel columns used as elevated highway piers shown in Fig. 1, FEM analysis using shell elements is the only direct procedure that can consider both axial force and biaxial bending interaction and local buckling effect. However, it requires a large amount of computing. Herein, we propose a simple three-dimensional hysteretic model for thin-walled circular steel columns. To consider the three-dimensional interaction, Aktan And Pecknold (1974) developed a filament model. However, their model cannot consider the effect of the local buckling, since they adopt the bilinear relation for the hysteretic model of each filament. The model we propose herein is alike the filament model but uses fewer springs which simulate the three- dimensional interaction. As the hysteretic model for each spring, we modify the Dafalias and Popov (1976) bounding-line model in order to take into account the degradation caused by the local buckling. The force-displacement relationship for each spring is determined by using curve-fitting technique,

101

102 L. Jiang and Y. Goto

based on the in-plane restoring force-displacement hysteretic relation obtained by the FEM shell analysis. Liu et al (1999) also proposed an empirical hysteretic model ,but the application of this model is restricted to in-plane case. The validity of our model is examined by comparing with the results of the three-dimensional nonlinear dynamic response analysis using shell element.

Fig. l:Thin-walled steel columns of elevated highways in Japan

BOUNDING-LINE MODEL IN FORCE SPACE

In-plane Hysteretic Behavior of Thin-Walled Circular Steel Columns

From the elastic theory, Timoshenko and Gere (1961), the elastic buckling of columns with circular section is governed by two structural parameters R, and 3, .

R, = R. __.ay X/3( 1 - v 2) (1) t E

~ _ 2L 1 ~ - ~ . . . . (2)

r ~

where R and t are the radius and the thickness, respectively, of the thin-walled circular column; cry is

the yield stress of steel ; E is Young's modulus;v is Poisson's ratio; L is the height of column and r is the radius of gyration of cross section. In the plastic range, we assume that the hysteretic behavior of thin-walled circular steel columns is influenced by the axial load ratio P/Py (Py -Cry,, A and A is the

cross-sectional area) in addition to the two structural parameters R t and X. As a result of FEM

analysis, hysteretic behavior of thin-walled circular steel columns is classified into three types, depending on the value ofR t , as illustrated in Fig. 2 (a), (b) and (c).Herein, the material behavior of

steel is assumed to be represented by the three-surface cyclic plasticity model proposed by Goto et al (1998). The material constants used for the three-surface model is shown in Table 1.

Considering the sizes as well as the design loads of real columns, three parameters take the values as 0.1 ___ P/Py <_ 0.3,0.06 < R t < 0.12, and 0.2 < 3. < 0.5. These ranges for the three parameters indicate

that the hysteretic behavior of our concern corresponds to that shown in Fig. 2 (b). This hysteretic behavior is characterized by the gradual strength degradation with the increase of cyclic plastic deformation.

Three-Dimensional Hys tere t ic Mode l ing o f T h & - W a l l e d Circular Columns

TABLE 1 THREE-SURFACE MODEL PARAMETERS

Parameter E (Gpa) Cry (MPa) cru(MPa) v eyp" L/Cry [3

495.0 0.3 0.0183 0.581 SS400 (No.8) 206.0 289.6

103

t ~ H&i /E HmPo.

100 2 0.05 Note

Note: For details see Goto et al (1998).

Fig. 2: Classification of hysteretic behavior (P/Py = 0.1, A, = 0.2 )

Modi f i ed B o u n d i n g - L i n e M o d e l

Dafalias and Popov (1976) presented a bounding-line constitutive model to express the cyclic plasticity of metals. We modify this model to express the in-plane force-displacement relation of steel columns. As shown in Fig. 3, F and X e denote restoring force and plastic horizontal displacement at

the top of the column respectively. XX and YY are bounding-lines. In order to express the strength degradation under cyclic loading, the gradient K B of the bounding-lines are assumed to be negative,

being different from the original Dafalias-and-Popov model that adopts a positive gradient for the bounding-lines. The incremental force-displacement relation for the in-plane behavior of steel columns is expressed as follows, depending on whether the current state belongs to the elastic range or the plastic range.

(Elastic range) AF = K e ~ AX (3)

(Plastic range) AF = K e K e / ( K e + K e) �9 AX (4)

where K E is the elastic tangent stiffness and Kp is the plastic tangent stiffness. Based on the

bounding-line model, K p is give by

di Kp = K B + H ~ ~ (5)

6in --6 where K B is the slope of the bounding-line; H is the hardening shape parameter; 6 is the distance

from the current force state to the corresponding bound; 6in is the value of 6 at the initiation of each

loading process. In the elastic range represented by the straight lines OA and CD in Fig. 3, K e is zero;

when the force reaches the bounding-line BC , K e becomes the same as K B ;on the curves AB and DE,

K p is expressed by Eq. 5.

104 L. J iang and Y. Goto

x

Bounding-line

Y

A

>

" '

~in

Y

Fig. 3: Bounding-line model

E m p i r i c a l E q u a t i o n s to De te rmine Parameters

Five parameters are included in our hysteretic m o d e l : F e , K e , 6 i , , K B and H . Among them, two are

elastic parameters: F e is the elastic yielding force;K e is the elastic stiffness. The other three are

related to bounding-line model as mentioned in the previous sub-section. From Fig. 2 (b), it is observed tha tF E , K E , 6i, and K B all change with the increase of the plastic deformation. Thus, it is

assumed that these four parameters are extrapolated from their initial values F e , K e , 6i, andKB

following the same rule as

F e = f �9 F e (6)

K e = f . K e (7)

~,. = f . o ~ ~ (8)

K B = f ~ B (9)

where f = 1 - logo + W p / W e 1 C )" We =-2 "Fe ~ X e is the elastic work. W e is the accumulated plastic

work. C is an empirical function given by

C = 37.75 - 33 ~ )~ - 25 ~ P/Py - 125. R, (10)

The initial value of the four parameters: F e , K e ,(~in ,KB and H are given by

I 1o (O'y P (11) Fe - ~ ' ~ -~ - )

3EI 1 KE - L 3 ~ (1+ 5.85 ~ (R /L ) 2 ) (12)

K B / K E = (-0 .155" PIPe +0.1616)+( -0 .5085" P/Py -0 .1317) ~ ~, +(1 .06 . P/Py - 2.3) ~ R t (13)

6i, , /Fe = (2.7" P/Py + 0.48) + ( -0 .12" P/Py - 0.012)" X + (-22.967 ~ P/Py - 0.95)" R t (14)

Three-Dimensional Hysteretic Modeling of Thin-Walled Circular Columns 105

H = -5 .El0 8 �9 )~ + 3.E10 8 (15)

where A and I are the cross-sectional area and the second moment of inertia, respectively, of the steel

columns; P is the vertical dead load. The other variables have the same meaning as in Eq. 1 . F E

andK E are directly obtained from elastic theory;KB/KE, -~i,/FE and H are so determined by the

least square method that our model best fits the force-displacement relationship obtained by FEM analysis using shell elements under monotonic loading.

A comparison of the hysteretic loops between the present empirical hysteretic model and the FEM

shell model is shown in Fig. 4 for the steel column with P/Py = 0.1,R, = 0.07 and ~, = 0.5. The

present model will yield an acceptable result when applied to the practical design.

Fig. 4: Comparison between FEM model and empirical hysteretic model

MULTIPLE SPRING M O D E L F O R T H R E E - D I M E N S I O N A L ANALYSIS

Modeling of Thin-Walled Steel Columns

To express the three-dimensional hyteretic behavior, the steel column is modeled into a rigid bar with multiple vertical springs at its base, as illustrated in Fig. 5. At the column base, no horizontal relative displacement is assumed to occur.

Rigid body

Multiple springs

y -,,,, z <i- x

Y

os n of spring

Fig. 5: Modeling of steel column

Based on the three-dimensional modeling of steel columns, the following incremental force- displacement relation is obtained.

106

AFr =

AFz

R 2 n 7 � 9 c~ 0i)

R 2 n

�9 ( .~ k i �9 cos0/ �9 sin 0 i) 2v R "

- ~-. (~ ~i. cosO,)

L. Jiang and Y. Goto

R 2 n R "

�9 i.cosO/.sinOi - .cosOi L ~ T

n R2 (~~. ---oR ( ~ k i~ L 2 �9 k i �9 sin20i) L

- - - � 9 ki L �9 sinOi) ki

AX

where AF x ,AFr,AF z and AX,AY, AZ are force and displacement increments, respectively, at the

top of the columns; k i is the tangent stiffness for the ith spring ; 0 i is the angle that specifies the

location of the ith spring; n is the total number of springs.

The least number of springs that can have three-dimensional interactive effects is four. But this number of springs can not ensure the homogeneity. Fig. 6 (a) shows the non-homogeneous force-displacement

relations for the column model with four springs under horizontal force directions: 0~176 ~ and

45 ~ . However, if we increase the number of springs, the column comes to exhibit homogeneity as illustrated in Fig. 6 (b). The least number of springs that is required for homogeneity is 16.

Fig. 6: Homogeneity of multiple spring model

Constitutive Relation for Multiple Springs

The constitutive relation for the multiple springs is determined, based on the in-plane hysteretic model. From Eq.16, the in-plane force and displacement relation in the X direction is derived as

R 2 n l~i'X -" "-~" ( E ki " cOS20i)AX (17)

] [ . . a

By comparing Eq.17 with Eq.4, the multiple spring model parameters FE.,.pri,,g, Kespring, 6i,,.,.p,.i,,g,

KBvr,,g, and H.,r, ri,,g can be related as follows to the parameters of the in-plane hysteretic model.

L FEspring - F e �9 ( ~ , g" a) (18)

L 2 KEspring -. Ke . (._RT. g) (19)

L 6i,.,prZ,g = 6 i " (-R " g " a) (20)

L 2 K Bspring = K . �9 (--RT " g) (21)

Three-Dimensional Hysteretic Modeling of Thin-Walled Circular Columns 107

L 2 Hspring = H o (-R 5-o g ~ a) (22)

g = 1 / ' ~ cos 2 0 i and a = 0.87 .

/

where

THREE-DIMENSIONAL EARTHQUAKE RESPONSE ANALYSIS

Steel Column Model

In order to demonstrate the validity of the multiple spring model, a dynamic response analysis is

carried out under the N-S, E-W and U-D components of the Kobe earthquake ground acceleration

recorded by the Japan Meteorological Agency (JMA). Under the same ground acceleration, FEM

analysis using shell elements illustrated in Fig. 7 is also conducted to examine the accuracy of our

model. For the column material property, we adopts the three-surface model with the material

constants summarized in Table 1.

Fig. 7: FEM shell model

Earthquake Response

The results of the earthquake response analysis obtained by the empirical hysteretic model are shown in Figs. 8-10, in comparison with those obtained by the FEM shell model. Figure 8 illustrates the loci of the response sway displacement at the top of the column. Figure 9 shows the E-W component of the

Fig. 8: Loci of response sway displacement

108 L. Jiang and Y. Goto

sway displacement history, whereas Fig. 10 shows the hysteresis loops expressed in terms of the force- displacement relation. From Figs. 8-10; it is confirmed that the empirical hysteretic model can simulate the three-dimensional seismic behavior of the FEM shell model with an acceptable tolerance.

Fig. 9: Sway displacement history of the column (East-West)

Fig. 10: Comparison of hysteretic force-displacement relation (East-West)

SUMMARY AND CONCLUDING REMARKS

In view of the application to the practical design analysis, a three-dimensional hysteretic model for the thin-walled circular column is presented. This model is represented by a rigid bar with multiple vertical springs at its base. These multiple springs are used to consider both the axial force and biaxial bending interaction and the local buckling effect. The constitutive relation for each spring is determined by the curve-fitting technique, based on the in-plane hysteretic behavior of the FEM shell model. In order to examine the validity of the proposed hysteretic model, a three-dimensional earthquake response analysis is carried out for a steel column by using both the hysteretic model and the FEM shell model. As a result, it is confirmed that the proposed hysteretic model can simulate the three-dimensional seismic behavior of the FEM shell model with an acceptable tolerance.

References

Aktan A. E. and Pecknold A. (1974). R/C Column Earthquake Response in Two Dimensions. Journal of the Structural Division.ASCE. ST10, 1999-2015. Dafalias Y. E and Popov E. E (1976). Plastic Internal Variables Formalism of Cyclic Plasticity. Journal of Applied Mechanics. ASME. 43:12, 645-651. Goto Y. and Wang Q. Y. (1998). FEM Analysis for Hysteretic Behavior of Thin-Walled Columns. Journal of Structural Engineering. ASCE. 124:11, 1290-1301. Liu Q. Y. and Kasai A. (1999). Parameter Identification of Damage-based Hysteretic Model for Pipe- section Steel Bridge Piers. Journal of Structural Engineering. JSCE. 45A:3, 53-64. Shing-Sham L. and George T. W. (1984). Model for Inelastic Biaxial Bending of Concrete Members. Journal of Structural Engineering ASCE. 110:11, 2563-2584. Timoshenko S. P. and Gere J. M. (1961). Theory of Elastic Stability, McGraw-Hill Kogakusha, LTD.

LOCAL BUCKLING OF THIN-WALLED POLYGONAL COLUMNS SUBJECTED TO AXIAL COMPRESSION OR BENDING

J.G. Teng, S.T. Smith and L.Y. Ngok

Department of Civil and Structural Engineering The Hong Kong Polytechnic University, Hong Kong, P.R. China

ABSTRACT

Thin-walled polygonal section columns are a popular form of construction due mainly to aesthetic considerations. Limited literature exists, however, on the stability of the component plate elements of these columns. A finite strip model is used in this paper to investigate the local buckling behaviour and strength of these columns subject to either axial compression or uniform bending. Cross-sections of square, pentagonal, hexagonal, heptagonal and octagonal profiles are considered. Elastic local buckling coefficients are presented for a variety of plate width-to-thickness ratios. It is shown that the dimensionless buckling stress coefficient is influenced by two parameters: the nature of the applied loading and the number of sides of the section. The buckling stress coefficient is higher for bent sections than axially compressed ones, and this difference can be quite significant. Sections with an odd number of sides have an enhanced buckling capacity over those with an even number of sides, with pentagonal sections being the strongest under either axial compression or bending.

KEYWORDS

Buckling, stability, columns, finite strip method, local buckling, polygonal sections.

INTRODUCTION

Thin-walled polygonal section columns are a popular form of construction due mainly to aesthetic considerations. Common polygonal sections include square, pentagonal, hexagonal, heptagonal and octagonal shapes. These columns are generally subjected to axial and lateral loads. Limited literature exists on the stability of the component plate elements of these columns. This paper thus considers the elastic local buckling capacity of polygonal column sections subjected to axial compression or bending.

The local buckling of thin-walled columns of box sections has been quite extensively investigated. Few studies on local buckling in polygonal columns, however, are found in the literature. The local buckling of long polygonal tubes in combined bending and torsion was investigated by Wittrick and Curzon (1968) using an exact finite strip method. Bulson (1969) undertook a comprehensive test

109

110 J.G. Teng et al.

programme on thin walled columns with consisting of between four to forty sides. Only even numbers of sides were considered. The mode of collapse observed was elastic buckling of the fiat plate elements followed by plastic collapse of the junction between adjacent elements. Columns with more than eighteen sides were found to collapse in a manner similar to that of a circular tube.

Avent and Robinson (1976) conducted an elastic stability analysis of thin-walled regular polygonal columns by expanding nodal displacements into Fourier series. They derived buckling curves for axially loaded pin-ended columns with polygonal cross-sections. The buckling curves describe local plate buckling for short columns and Euler column buckling for longer ones. An increase in the local buckling capacity for sections with odd numbered panels was noted. The local buckling stress for sections with an even number of sides was found to match that of uniaxially compressed simply supported plates. As the number of sides is increased to 16, the critical local buckling load approaches that of a cylinder.

Kurt and Johnson (1978) considered imperfections in axially loaded pin-ended columns of polygonal sections. In their study they distorted the sides of a polygon by applying a midpoint lateral displacement and then utilised the same analytical solution technique as Avent and Robinson (1976). Similar buckling curves were produced to those of Advent and Robinson and it was found that as the number of sides increases, the polygon behaviour approaches that of a cylinder. In the Euler buckling range of response, initial imperfections decrease the predicted buckling strength. In the local buckling range of response, the buckling strength is increased by the imperfections.

More recently Koseko et al. (1983) undertook an experimental and theoretical study of the local buckling strength of thin walled steel members of octagonal cross-section. The finite strip method was employed for the theoretical work. Aoki et al. (1991) have since conducted experiments on columns varying from square to octagonal in cross-section. Residual stresses and geometric imperfections were measured and an empirical design formula was calibrated from the experimental results. Polygonal cross-sections appeared to be better than box sections in respect to the ultimate strength considerations. The most recent significant contribution to local buckling in polygonal section columns appears to have been made by Migita et al. (1992), who considered the interaction between local and overall buckling in polygonal section steel columns.

No study to date appears to have conducted a comparison between the buckling capacity of regular polygonal columns subjected to axial compression and that under bending. This in turn has prompted the current study. Five different thin-walled polygonal section forms are considered in this study. These are square, pentagonal, hexagonal, heptagonal and octagonal sections with four to eight sides respectively. These sections are shown in Figure 1 where their principal axes are indicated. It should be noted that while all sections are symmetric about the vertical or y axis, only sections with an even number of sides are symmetric about the horizontal or x axis. The width of a plate element is denoted as b while the thickness is represented by t. Thirteen different width-to-thickness ratios are considered for each section form and a summary of the dimensions of the sections investigated here are given in Table 1. The total cross-sectional area is kept constant for all five sections for each thickness. The number of sides of each section is denoted by n in Table 1.

STABILITY ANALYSIS

The program THIN-WALL (THIN-WALL 1996, Papangelis and Hancock 1995), developed by the University of Sydney, is used to study the local buckling of columns with polygonal sections subjected to axial compression or bending. The program is based on the well-established finite strip method of analysis (Cheung, 1976, Hancock 1978). In the finite strip method, thin walled sections are subdivided into longitudinal strips. The displacement functions, which are used to describe the displacement

Local Buckling of Thin-Walled Polygonal Columns 111

variation in the longitudinal direction, are assumed to be harmonic. Polynomial functions are used to describe the displacement variation in the transverse direction. The finite strip buckling analysis can be represented in matrix format as follows:

[K]{D}- A[G]{D} = O (1)

where [K] and [G] are the stiffness and stability matrices of the structure being investigated. The solution produces the eigenvalue, or the critical buckling load factor represented by t , and the eigenvector given in [D]. Substitution of the eigenvector into the assumed displacement functions gives the buckled shape of the plate assembly.

l y y y X X X X / / / / /J /

b Square Pentagon Hexagon Heptagon Octagon

Figure 1" Polygonal Shapes

TABLE 1" GEOMETRIC PROPERTIES OF POLYGONAL SECTIONS

t Section (mm) Square Pentagon Hexagon Heptagon Octagon

n = 4 n = 5 n = 6 n = 7 n = 8 b =126 mm b = 100.8 mm b = 84 mm b = 72 mm b = 63 mm b/t A b/t A b/t A b/t A b/t A

(mm2) (mm 2) (mm 2) (mm 2) (mm 2)

0.5 252 252 252 252 252 0.7 353 353 353 353 353 0.9 454 453 453 453 453 1.0 504 504 504 504 504 1.5 756 756 756 756 756 2.0 1008 1008 1008 1008 1008 2.5 1260 1260 1260 1260 1260 3.0 1512 1512 1512 1512 1512 4.0 2016 2016 2016 2016 2016 5.0 2520 2520 2520 2520 2520 6.0 3024 3024 3024 3024 3024 7.0 3528 3528 3528 3528 3528 8.0 4032 4032 4032 4032 4032

252.0 201.6 168.0 144.0 126.0 180.0 144.0 120.0 102.9 90.0 140.0 112.0 93.3 80.0 70.0 126.0 100.9 84.0 72.0 63.0 84.0 67.2 56.0 48.0 42.0 63.0 50.4 42.0 36.0 31.5 50.4 40.3 33.6 28.8 25.2 42.0 33.6 28.0 24.0 21.0 31.5 25.2 21.0 18.0 15.8 25.2 20.2 16.8 14.4 12.6 51.0 16.8 14.0 12.0 10.5 18.0 14.4 12.0 10.3 9.0 15.8 12.6 10.5 9.0 7.9


RESULTS

G e n e r a l

Figure 1 and Table 1 show the configurations and dimensions of the polygonal sections considered in this study. Two different loading scenarios were investigated: axial compression and bending. All results were generated for mild steel plate assemblies with an elastic modulus of 200,000 MPa and a Poisson's ratio of 0.3. The elastic local buckling capacity is given in terms of the dimensionless buckling stress coefficient kcr which is related to the critical stress ~ , through the following familiar expression (Bulson 1970, Trahair and Bradford 1998):

n-:E 1 O'cr =kcr 121"-v2'xz'/tx2~l j[b ) (2)

where E is the elastic modulus and v the Poisson's ratio. In the parametric study described below, the plate slenderness (width-to-thickness ratio) was varied from a minimum of about 10 to a maximum of 252. For each plate slenderness, a corresponding elastic local buckling capacity was determined.

4.4 -

.3 --

.~ 4 . 2 - -

0 m 4 . 1

�9 ~ 4 -

3.9--

3.8

~ ' A A A

# o i _r r

o o N

O &

, o square

.'. pentagon

[] hexagon

e heptagon

x octagon

I I I I

50 100 150 200

slenderness, b/t

Figure 2" Local Buckling Stress Coefficient versus Slenderness for Polygonal Sections Subjected to Axial Compression

t

250

Axial Compression

Figure 2 shows the elastic local buckling stress coefficient for the five polygonal sections investigated as a function of the plate slenderness or b/t ratio. Here the sections are subjected to uniform axial compression. From this figure it can be observed that the square, hexagonal and octagonal section columns have approximately the same buckling capacity. The heptagonal and pentagonal columns have a markedly increased buckling resistance. This increase can be attributed to the local buckling configuration of the individual plate elements. For a plate slenderness greater than about fifty, the buckling stress coefficient stays virtually constant. However, when the slenderness drops below about

Local Buckling of Thin- Walled Polygonal Columns 113

fifty, the buckling stress coefficient starts to reduce. The square, hexagonal and octagonal sections have a buckling stress coefficient of about four and this is consistent with that of a plate simply supported on all four edges and subjected to uniaxial compression (Bulson 1970).

Representative buckling modes are shown in Figure 3 for the five polygonal sections subjected to axial compression. The dotted lines in this figure represent typical buckled configurations while the continuous lines represent the original shapes. The square, hexagonal and octagonal sections all buckle in a similar manner with each plate element buckling in an opposite direction to the adjacent plate elements. These three sections have an even number of sides, that is, four, six and eight sides respectively. For sections with an odd number of sides, this alternating inward-outward bucking mode is incompatible with the number of sides. For these sections, two consecutive plate elements must buckle in the same direction, be that inward or outward or two half waves have to appear in one of the plate elements. These variations in buckling modes lead to a higher buckling capacity as seen for the pentagonal and heptagonal columns.

. . . . . . . . . . . . . " " - - " " f " - " i

i t ',\ / : ' )

Figure 3" Typical Local Buckling Modes under Axial Compression

6.5 f A A A A .8.

6

5.5 ^ , 0 0 0 0 0

[ ] [ ] [ ] [ ]

.~ex[l X .'<

x~ 4.5

4 I I

0 50 100 150 200

slenderness, b/t

Figure 4: Local Buckling Stress Coefficient versus Slenderness for Polygonal Sections Subjected to Bending

- 0

I I o(D

o square (x,y- axis)

-'- pentagon (y- axis)

[] hexagon (y- axis)

e heptagon (y- axis)

x octagon (x,y- axis) I I

250

Bending

Figure 4 shows the elastic local buckling stress coefficient for the five polygonal sections in bending as a function of the plate slenderness. For each section, a buckling analysis was carried out for bending about the x-axis in a positive and negative direction, and bending about the y-axis. The lowest of the


three buckling stresses is taken as the critical buckling stress. Figure 4 shows that the octagonal section has the lowest buckling resistance followed by the hexagonal, square, heptagonal and pentagonal section. The critical axis of bending for each section, which produces the lowest buckling coefficient is also reported in Figure 4. Figure 5 shows typical buckling modes of the five polygonal sections when subjected to bending in the weakest of the three directions.

. . . . J! . . . . . . Y

Figure 5: Typical Local Buckling Modes under Bending

Comparison between Axial Compression and Bending

The calculated buckling coefficient for bending is greater than the corresponding value for axial compression. This is as expected and is also reflected in the buckling modes of Figures 3 and 5. All sections subjected to bending experienced about a 25% increase in the buckling stress compared to axial compression.

For low b/t ratios the buckling capacity of all sections, subjected to axial compression or bending, reduces. This is believed to be due to membrane deformations in the plates which are not accounted for in classical theories for plate buckling.

CONCLUSIONS

The elastic local buckling capacity of polygonal sections has been investigated in this paper. Square, pentagonal, hexagonal, heptagonal and octagonal sections have been investigated, with elastic local buckling coefficients presented for a variety of plate width-to-thickness ratios. It has been shown that the dimensionless buckling stress coefficient is influenced by two parameters: the nature of the applied loading and the number of sides of the section. The buckling stress coefficient is higher for bent sections than axially compressed ones, and this difference can be quite significant. Sections with an odd number of sides have an enhanced buckling capacity over those with an even number of sides, with pentagonal sections being the strongest under either axial compression or bending. It should be noted that for sections subject to bending, the bending moment was applied in three different directions to find the weakest axis of bending. Further work should be carried out to establish if another axis of bending exists which leads to an even lower buckling stress.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the contribution to this work made by Mr K. K. Wong who carried out the initial calculations for the results presented here during his final year project supervised by the first author. The second author wishes to thank The Hong Kong Polytechnic University for providing him with a Postdoctoral Fellowship.

REFERENCES

Aoki, T., Migita, Y and Fukumoto, Y. (1991). Local Buckling Strength of Closed Polygonal Folded Section Columns. Journal of Constructional Steel Research 20, 259-270.

Local Buckling of Thin-Walled Polygonal Columns 115

Avent, R.R. and Robinson, J.H. (1976). Elastic Stability of Polygonal Folded Plate Columns. Journal of the Structural Division, ASCE 102(ST5), 1015-1029.

Bulson, P.S. (1969). The Strength of Thin Walled Tubes Formed from Flat Elements. International Journal of Mechanical Sciences 11, 613-620.

Bulson, P.S. (1970). The Stability of Flat Plates, Chatto & Windus, London, U.K. Cheung, Y.K. (1976). Finite Strip Method in Structural Analysis, Pergamon Press, Oxford, U.K. Hancock, G.J. (1978). Local, Distortional, and Lateral Buckling of I-Beams. Journal of the Structural

Division, ASCE 104(ST11), 1787-1798. Koseko, N., Aoki, T. and Fukumoto, Y. (1983). The Local Buckling Strength of the Octagonal Section

Steel Columns. Proceedings of Structural Engineering~Earthquake Engineering, JSCE 330, 27- 36.

Kurt, C.E. and Johnson, R.C. (1978). Cross Sectional Imperfections and Column Stability. Proceedings of the Structural Division, ASCE 104(ST12), 1869-1883.

Migita, Y., Aoki, T. and Fukumoto, Y. (1992). Local and Interaction Buckling of Polygonal Section Steel Columns. Journal of Structural Engineering, ASCE, 118(10), 2659-2676.

Papangelis, J.P. and Hancock, G.J. (1995). Computer Analysis of Thin-Walled Structural Members. Computers and Structures, 56(1), 157-176.

THIN-WALL (1996), Cross-Sectional Analysis and Finite Strip Buckling Analysis of Thin Walled Structures: Users Manual - Version 1.2, Centre for Advanced Structural Engineering, Department of Civil Engineering, The University of Sydney, Sydney, Australia.

Trahair, N.S. and Bradford, M.A. (1998). The Behaviour and Design of Steel Structures to AS 4100, Third Edition, E&FN Spon, London, U.K.

Wittrick, W.H. and Curzon, P.L.V. (1968). Local Buckling of Long Polygonal Tubes in Combined Bending and Torsion", International Journal of Mechanical Sciences 10, 849-857.


ULTIMATE LOAD CAPACITY OF COLUMNS STRENGTHENED UNDER PRELOAD

H. Unterweger

Department of Steel Structures, Technical University Graz Lessingstrasse 25, A- 8010 Graz, Austria

ABSTRACT

The ultimate load capacity of symmetric columns, which are strengthed under preload using steel plates, is presented. The analytic calculation model is an ideal column with geometric imperfections including second order effects. The decrease of the bending stiffness, due to the development of plastic zones in the cross section, is taken into account by a modification factor, based on comprehensive FE- calculations. By using modified buckling reduction factors it is possile to find out directly the extent of strengthening steel plates, depending on preload and type of cross section.

K E Y W O R D S

strengthen of steel columns, flexural buckling, strengthen steel plates, ultimate load capacity

I N T R O D U C T I O N

Due to increasing loads or changing of service conditions sometimes members of existing structures must be strengthen. Examples for primary compressed members are:

- columns of buildings

- chords of truss girders of old bridges, due to increased traffic loads

The strengthening design of the cross section of the member (base - section), using steel plates (welded or bolted to the member) has to consider the preload in the member due to at least permanent actions. In actual codes in Europe (e.g. Eurocode (1993), DIN (1990)) no procedures for strengthening of members are included.

117

118 H. Unterweger

For members without risk of stability failure the determination of the extent of the strengthening plates gives no problems (see also Fig. 2). But for members, whrere flexural buckling is relevant, the design procedure seems not clear at all. Publications in this field are very small and limited to the counties of Eastern Europe, summed up by Rebrov & Raboldt (1981). This paper presents the main results of a comprehensive study by the author (1996).

ASSUMTIONS AND EXTENT OF THE PRESENTATION

The presented results refer to columns with pinned ends on both sides (Figure 1). In the following only axial loading (constant axial force N) is taken into account. Additional limited bending moments can be also taken into account as shown in Unterweger (1996).

Figure 1 : System and type of cross sections of the presentation.

The base sections are universal rolled columns with H - shape. The strengthening plates are situated either on the outsides of the flanges (type 1) or on both sides of the web (type 2) and they have the same length as the columns. The result are also applicable to other section types like welded H - sections, hollow sections or channels, if they are symmetric to the buckling axis (see figure 1). The slenderness ratio of the individual parts of the cross section is limited in such way that local buckling is not possible until the plastic cross section resistance is reached (classified as class 1 and 2 in Eurocode).

CALCULATION PROCEDURE

Columns without buckling failure mode

For columns with small slenderness, e. g. short columns or columns which are supported by walls or bracings, the cross section resistance is relevant for design. The determination of the extent of the strengthening plates AA for a given axial force Nv+ AN, as shown in Figure 2, is very simple. If only the elastic resistance is taken into account the preload N v acts on the base section A 0 and only the additional load AN acts on the whole section A (Eqn. 1). That means that yielding of the base section limits the loading capacity. If the plastic resistance of the cross section is taken into account the whole section also acts for the preload N v (Eqn. 2). Therefore stress redistribution between base section and strengthening plates, due to plastification of the base section, is necessary (~,f, ~'m are partial safety factors).

Ultimate Load Capacity of Columns Strengthened under Preload 119

Figure 2 : Stress distribution due to axial force N and verification procedure for elastic (Eqn.1) and plastic cross section resistance (Eqn. 2).

With increasing preload the differencies between elastic and plastic resistance grow. At the theoretical border line case - yielding of the base section under preload - the strengthening plates are completely ineffective. Therefore the design procedure based on plastic resistance is simpler and much more economic, but leads to more or less plastifications of the base section.

The aim of this study is to show that also for slender columns plastification of the base section can be taken into account to exploit the full bearing capacity of the strengthening plates.

Columns under flexural buckling

In general for columns flexural buckling is the relevant failure mode for design, which will be discussed in the following. The partial factors are omitted (Tf, Tm).

Engineering solution

Considering the design procedure for columns, based on slenderness depending buckling reduction factors K: (e.g. European buckling curves a § d in Eurocode) the design procedure for determining the load capacity N R of a strengthen column follows Eqn. 3 (stress equation)

Nv AN + <fy N R = N v + A N ( 3 )

K: o . A o ir A - '

120 H. Unterweger

The first term considers the base section (A 0, v~ 0 for slenderness ratio 7~) under preload and the second term considers the strengthen section (A, 1< for slenderness ratio ~). The acting stresses are limited by the yield stress.

For short columns (~:-> 1,0) the verification procedure is equal to the elastic cross resistance (Eqn. 1). Therefore this procedure also seems uneconomic with increasing preload, because the capacity of the strengthening plates cannot be taken into account. This statement will be confirmed in the following.

Suggested solution

The extent of preload and strengthening is characterized by the preload ratio o~ - referred to the load capacity NR, 0 of the base section (Eqn. 4), and the strengthening ratio ~ (Eqn. 5, AA is the area of strengthening plates).

Nv Nv (~ - - (4)

NR, o Ao" too" fy

AA (5)

The simple analytic calculation model with the essential assumptions is shown in Figure 3. For determination of deformations and moments second order theory is used, including equivalent initial geometric bow imperfections (sine curve). This leads to increasing factors including the ideal elastic buckling load Nki,0 (base section) and Nki (strengthened section) respectively. The individual calculation steps are:

- calibration of the simple model in form of determining the initial bow imperfections e 0 to fulfil the ultimate load capacity of the base section according to the code buckling curves. Using the European buckling curves leads to Eqn. 6 for e 0, where a* is a constant depending on the relevant buckling curve (a* = 0,21 § 0,76 for curves a § d) and W 0 is the section modulus.

W o e 0 = a * . ( ~ 0 - 0 , 2 ) �9 A--o ( 6 )

- determination of deformation w v under preload N v of the base section.

- unloading of the strengthed section leads to the deformation w 0 - neglecting the residual stress distribution in the section

- Determination of the ultimate load N R of the strengthed section, with elastic cross resistance of the cross section.

Regarding a practical design procedure the resulting load capacity N R is expressed in form of a modified buckling reduction factor •* referred to the base section (Equ. 7 - 9). Doing this, the efficiency of the strengthening plates can be seen immediately.

N R = t c* .Ao . fy ( 7 )

(8)

1 / wo < K* = ~. 1 + ~ - • o~';Co) x ( l _ { x . ; C o . ~ o 2 ) . ~ 2 + ( 9 )

Ultimate Load Capacity of Columns Strengthened under Preload

The resulting load capacity N R depends on the following parameters:

- cross section parameters, slenderness and buckling reduction factor of the base section

(Ao, WO, I0 -> ~ 0 - > r'O) - cross section parameters, slenderness and buckling reduction factor of the strengthed section

(A,W, I - > ~ ) - preload ratio o~, strengthening ratio

121

Figure 3 : Calculation procedure for N R of the strengthed column under preload.

To determine the extent of the strengthening plates AA an iteration process is necessary, because W and depends on AA.

Reduction of load capacity due to welded strengthen plates

Due to the welding process of the strengthening plates residual stresses are introduced, which lead to a decrease of the buckling load capacity. Their quantities and distribution are in general hardly predictable, due to the high scatter of influence factors. Therefore the influence of the welding process on the load capacity is estimated in an engineering manner. Following the Europian buckling curves the effect of welding on the bearing capacity can be estimated in form of an additional geometric imperfection ev = 0,5. e 0 . Considering this effect in the analytic model (working with Wv* = w v + ev) leads to a maximum decrease of the load capacity of about 12 % for medium slenderness ratios, shown in Figure 4.

122 H. Unterweger

Figure 4 : Reduction fweld of the calculated ultimate load N R due to welding of the strengthening plates for careful execution.

Comparison of suggested and engineering solution

To evaluate the suggested load capacity (Equ. 7 - 9 -> NR, ex) with the engineering procedure (Equ. 3 -> NR, ing ) a comparison for practical columns is useful. In Figure 5 the increase of the load capacity using the suggested solution referred to the engineering procedure is shown for the two border line cases. On the one hand type 1- buckling about y- axis, which is the most effective case for the strengthening plates; and on the other hand type 2 - buckling about z- axis. In the first case the differencies in load capacity are given for all slenderness ratios, whereas in the latter case with increasing slenderness ratio the results are more and more equal. The differencies increase with growing preload ratio t~ and strengthening ratio ~. For the theoretical case of a preload ratio of t~ = 1,0 and small slenderness we get the highest difference AN = ~ . NR,ing, which is equal to the difference between elastic and plastic section resistance (Eqn. 1, 2). This example shows the economic advantages of the suggested solution.

Figure 5 : Increase of the ultimate load N R using the proposed solution compared to the engineering approach for two border line cases; a.) type 1, y - axis, b.) type 2, z- axis.

Ultimate Load Capacity of Columns Strengthened under Preload

Overestimation of load capacity due to plastification of the base section

123

The analytical model neglects the effect of plastification of the base section, which grows with increasing preload ratio o~ and also with higher material strength, because of increased plastic zones. To find out the extent of reduction of the load capacity comprehensive finite element calculations with ABAQUS (1996) were made. The web was modelled with shell elements and the flanges and strengthening plates with special beam elements, including progressive plastifications in thickness direction. The highest reduction of load capacity due to plastification are obtained for type 1 - buckling about y- axis. Fortunately the decrease of load capacity is very small, e.g. 2 - 5 % for a preload ratio ct = 0,5. From the results a simplified conservative procedure for practical use in form of a reduction factor fNR, plast (Figure 6) can be given.

Figure 6 : Reduction factor fNR,plast of the calculated ultimate load N R due to plastification of the base section.

Figure 7 : Load bearing capacity N R of strengthen columns using European rolled sections, expressed in form of a modified buckling factor •* a.) type 1 - y ; b.) type 2 - z.

124

Improved solution for practical design

H. Unterweger

For practical design a direct determination of the extent of strengthening plates AA, expressed by the strengthening ratio ~, is desirable. A comprehensive study shows that for every type of strengthened base section the ratios W / W 0 and ~ / 7~ 0 can be expressed in form of a linear relationship of ~. Intro- duction of these information in Eqn. 8 and 9 leads to equivalent buckling factors K:* depending on the slenderness of the base section, the preload ratio t~ and the strengthening ratio ~. In Figure 7 for type 1- buckling about y- axis and type 2 - buckling about z- axis, using European rolled sections, the load capacities in form of •* are given. The effectiveness of the strengthening plates is easy to survey.

In Figure 8 the suggested simple design procedure for direct determination of the extent of the strengthening plates, based on design charts for the global buckling reduction factors K:*, is shown.

In Unterweger (1996, 1998) the design charts for rolled European universal columns for type 1 and 2 are presented.

Figure 8 : Starting position and procedure of a practical design of column strengthening (partial factors omitted).

References

DIN 18800, Teil 1 und 2 (1990). Stahlbauten - Bemessung und Konstruktion. Deutsches Institut f'dr Normung.

Eurocode 3 (1993). Design of steel structures; Part 1.1: General rules. CEN.

Rebrov and Raboldt (1981). Zur Berechnung von Druckst~iben, die unter Belastung verst~kt werden. Informationen des VEB MLK 20.

Unterweger H. (1996). Berechnung von unter Belastung verstarkten stahlernen Druckstaben, unpublished.

Unterweger H. (1998). Druckbeanspruchbarkeit von unter Vorbelastung verst~kten Stiitzen. Stahlbau 68: 3, 196- 203.

CHAOTIC BELT PHENOMENA IN

NONLINEAR ELASTIC BEAM"

Zhang Nianmei 1 Yang Guitong 2 Xu Bingye ~

1 Department of Engineering Mechanics, Tsinghua

University, Beijing, China

2 Institute of Applied Mechanics, Taiyuan University

of Technology, Taiyuan, China

ABSTRACT

The chaotic motions of axial compressed nonlinear elastic beam subjected to transverse load

P 8P 0 ( f + coscot)sin rex = are studied in this paper. The constitutive equation of the beam is 1

threefold multinomial. The damping force in the system is nonlinear. Considering material and

geometric nonlinearity, nonlinear governing equation of the system is derived. By use of nonlinear

Galerkin method, differential dynamic system is set up. Melnikov method is used to analyze the

characters of the system. The results show that chaos may occur in the system when the load

parameters P0 and f satisfy some conditions. The zone of chaotic motion is belted. The route from

subharmonic bifurcation to chaos is analyzed in the paper. The critical conditions that chaos occurs

are determined.

125

126

KEYWORDS

Z. Nianmei et al.

chaos, bifurcation, heteroclinic orbit, periodic orbit, dynamic system, saddle

INTRODUCTION

The chaotic phenomena in solid mechanics fields bring more and more interesting. In 1988, F.C.Moon

analyzed the chaotic behaviors of beams experimentally first. Then he studied the dynamics response

of linear elastic beam subjected transverse periodic load. The chaotic motions of linear damping

beams have been studied by many scholars at home and abroad in resent years. The dynamic

behaviors of nonlinear damping beams subjected to transverse load P = S P o ( f + coscot)sin ~rx l

m a r e

studied in this paper. The critic conditions that chaos occurs in the system are determined by use of

Melnikov method. The results show that the chaotic areas may be limited ribbon zones.

BASIC EQUATIONS

The dynamic behavior of a simply supported nonlinearly elastic beam is studied. Two constant

compressive loads N are applied at its two ends. The length of the beam is l . The constitutive

relation of beam material satisfies:

o- =Eo~ +E16 2 ) (1)

where, E and E 1 are material constants.

We assume that deformation of the beam is still small deformation after buckling. The buckling

critical load of the beam is:

~2EI1A1

Here A I = 1 + 3E1602 . 11 stands for inertia moment, 11 = ~y2dA . A is the cross section area of the A

beam. 60 is the strain at neutral surface, it satisfies:

Chaotic Belt Phenomena in Nonlinear Elastic Beam 127

( 3~2El/a I ~'211 E1Eo 4 + E18o 3 + 1 - ~ 602 + 60 =0 (3)

AI 2 AI 2

The beam is subjected to transverse load P =6Po( f +coscot)sin ~r___x_x after buckling. Then the 1

governing equation of the system is:

c32M c32w 02w c3w Ow + N ~ + m = 8 P o ( f + coscot)sin ~rx _ 6 / . t ~ ~

a x ~ Ox ~ ~ T at a t (4)

where 8/~ is damping coefficient, m is the mass of unit length of the beam.

The boundary conditions of the system are:

w(o): wq)-o (5)

w"(0) = w"(l)=O (6)

Following formula can stand for the strain at the point which distancement from neutral plain is y:

O0 c = 60 - y c3 x (7)

where 0 is the rotating angle of cross section of the beam at x. It satisfies:

1 Ow sin 0 = (8)

1+~" 0 0 x

Submitting geometric relations and physical relations into eq.(4) and omitting the higher order items than three, follow formula can be obtained:

c, 4w_+c212 2 3 +6 w 4wl

128 Z . N i a n m e i et al.

V6(t~3W~ 2 {~2W q_,3~2W~ 2 04W 1

,,[ ,,sin"" =EI &(f + cosco -- / -- ~,

N m a2w + ~ ~ (9)

E11 E11 O t 2

- - X m W

Eq.(9) is turned into dimensionless form using dimensionless amounts x = - w = r =COot t ' 7 ' '

, cOO = '

coo ml4 "

According to boundary conditions (5) and (6), we suppose follow displacement mode:

n m w = c,o(v)sin rcx (10)

Applying Galerkin method to the dimensionless governing equation, differential dynamic system can

be obtained:

=-,~q,- p~' +,~o (," +,:os-,)- ~ ~] (11)

where

~-~(-~+~,~), p-,.,.,,(c,_- ~)

C 1 = A1 , C z = A1 , C4= 3ELI2 l+eo 2(1+ eo) 3 212(1+6o)312

P = ~ ISc~176 -~o - P~ , -N = N 12

E l l E l l EI1

If system (11) is not perturbed, g = 0. Then eq. (11) is integrable Hamilton system:

'r = v' (12)

Hamilton function means the total energy of kinetic energy and potential energy. The energy keeps

constant on the same orbit:

h = ~ + + = cons t (13) 2 2 4

Chaotic Belt Phenomena in Nonlinear Elastic Beam

The phase trajectory of undisturbed system may be determined by following formula:

129

d(p ~ ~(p4 (14) =+- 2h - aq92 2

The formula (14) shows that the phase trajectory has closed relation with the value of a, ft. The

dynamic response of the system in the case of stable post-buckling path (a > 0) and fl <0 are

studied in this paper.

DYNAMIC ANALYZATION

The unperturbed system have three balance states in real space. (0,0) is a center. (- ~]a/- fl,O) and

(~/a/- fl,O) are hyperbolic saddles. The heteroclinic orbits passing though two saddles are:

(15)

The Melnikov function ofheteroclinic orbits is as follows:

oo

-oo

: - ; ~0 + ~(,~ + ~ cos~,~0) (16)

here

~o : f ~ d ~ : . ; - s e t h ~ ~- a ~ - : - ; _0o 2 15r r

2 a - ~ ( o d r - 2 - - o o

(17)

(18)

(19)

130 Z. N i a n m e i et al.

The eq. (16) shows that constant load f and loading frequency ~ have great effect on the

conditions that there exists Smale horseshoe in the system.

1) When x/-a-f > 1, the conditions that Melnikov function has simple zero points is:

2o Poo ;to < _ - - < (20) 2fRa + 22 ,u 2 f & + 22

o r m

R 1 < P-~-~ <R 2 (21)

where

R 1 =

8a: 4~ 15fl

2 1+ ocosechI ol

R 2 =

8a: x/-a 15fl

m

The eq.(20) means that there exists a limited belted zone in P o - c o plane. When R 1 < ~P~ <R2,

there is Smale horseshoe in the Poincare map of the system.

1 2) When 0 < f < ---~, there exists sole nr*. It should satisfy:

4 a

- nr cosech zc~ = f 2

a. If 0 _< w _< w*, the critical condition chaos occurs is:

Chaotic Belt Phenomena in Nonl&ear Elastic Beam 131

r~ > R 1 (22) /.t

The above formula shows that chaotic area is half-infinite.

b. If nr > w*, the representative of critical condition that chaos occurs is the same as formula (21).

The above analyzation shows that the critical conditions there exists Smale horseshoe in the dynamic system have closed relations with loading way and frequency.

ROUTE TO CHAOS

There is a set of periodic orbits circling the center (0,0)"

~Pk(Z')=+ - 0 + k 2 ) f l s n l+k2V, k

= _ v,k dn v,k l + k 2 l + k 2 l + k 2

(23)

The periods of the orbits are:

T 4~ l+k2 = K a

where K is first type Jocabi elliptic integration. Melnikov functions of subharmonic orbits are:

)) + COS ~7(2" + T O o

- - P o +

16a 2 __2~-- nk3(5 - k____:) (m,.)- Po 15p O+k F

fO~%m [ / ( K K ) n~l~ miseven Pz(m,n)= 2 ct rc '

- - ~ fll+k2 sh n = l and m is odd

here K ' = K~/1 - k 2

The threshold that odd order subharmonic bifurcation occurs is:

(24)

(25)

(26)

132 Z. Nianmei et al.

~0> 8k3etS/2 (5-k 2) sh(mZK' 15pz~O+k2) 5n \ 2K ) : Rm

When m ---> oo, that is k ---> 1, following formula can be obtained:

l imRm =R o : - ~ s h zw m---~oo 15/~rW

Comparing R1 ,R 2 with Ro, we know:

R 0 > R 1 , Ro > R 2

(27)

(28)

So the system will enter chaos status by limited odd order subharmonic bifurcation.

CONCLUSION

1 .Only when the undisturbed differential dynamic systems possess heteroclinic orbits, the chaotic belt

phenomena may occur after the system is perturbed.

2.The chaotic areas are affected by not only the ratio of constant load to the amplitude of periodic load but also loading frequency co. If the ratio of constant load to the amplitude of periodic load is

greater than 1, the chaotic area is belted at any loading frequency co. If the ratio is smaller than 1,

the area in which Smale horseshoe occurs is belted when nr > nr*. But the threshold is lower limit

only when 0 _< nr ___ m*

3.If the constant load equals to zero the area that chaos occurs is half infinite. 4.The system may enter chaos status by limited odd order subharmonic bifurcation.

REFERENCE

Moon F. C. (1988). Experiments on Chaotic Motions of A Forced Nonlinear Oscillator: Stranger

Attractors. J. Appl. Mech. 55, 190-196.

Panida Dinca Baran (1984). Mathematical Modes Used in Study The Chaotic Vibration of

Buckled Beams. Mechanics Research Communications 29:2, 189-196.

Zhang Nianmei and Yang G.T. (1996). Dynamic Subharmonic Bifurcation and Chaos of

Nonlinear Elastic Beam. J. Nonlinear Dynamic 3:2, 265-274.(in Chiness)

Frames and Trusses


INVESTIGATION OF ROTATIONAL CHARACTERISTICS OF COLUMN BASES OF

STEEL PORTAL FRAMES

T C H Liu and L J Morris

Manchester School of Engineering, Oxford Road, Manchester, M13 9PL, UK

ABSTRACT

Most of the portal frames are designed these days by the application of plastic analysis, with the normal assumption being made that the column bases are pinned. However, the couple produced by the compression action of the inner column flange and the tension in the holding down bolts will inevitably generate some moment resistance and rotational stiffness. Full-scale portal frame tests conducted during a previous research program had suggested that this moment can be as much as 20% of the moment of resistance of the column. The size of this moment of resistance is particularly important for the design of the tensile capacity of the holding down bolts and also the bearing resistance of the foundation. The present research program is aiming at defining this moment of resistance in simple design terms so that it could be included in the design of the frame. The investigation also included the study of the semi-rigid behaviour of the column base/foundation, which, to a certain extent, affects the overall loading capacity and stiffness of the portal frames. A series of column bases with various details were tested and were used to calibrate a finite element model which is able to simulate the action of the holding down bolts, the effect of the concrete foundation and the deformation of the base plate.

KEY WORDS

Column Base, Holding down bolts, Column flexibility, Portal Frame

INTRODUCTION

Steel portal frames, similar to most other structures, tend to be designed almost independent of the foundation condition, mainly because most practicing engineers cannot readily appreciate or quantify this interaction. While the design of column bases of most of the multi-storey frame structures is govemed by the large axial forces, column bases in portal frames are subjected to a relatively larger lateral shear (Bresler & Lin, 1959). Though there have been some studies recently, the interaction between the soil/foundation block/structural frame is probably the least understood aspect of the whole building. An on-going project was designed to investigate the effect of foundation to the overall

135

136 T.C.H. Liu and L.J. Morris

behaviour of steel portal frames following the series of full-scale tests. The research program had been divided into three phases, aiming to quantify the rotational and moment capacities of the column bases in order to check their effects on the overall frame behaviour and to recommend a suitable design for column base details. The first part, which is to be reported in this paper, was to look into the effect of various geometric parameters of the column bases such as the thicknesses of the base plates, column sizes, and size and length of holding down bolts. The study consists of a series of laboratory testing and computational modelling.

In the design of a typical portal frame, it is generally assumed that the bases are "pinned" for purposes of analysis, i.e. column does not transfer any moment to the foundation. A typical base consists of a base plate fillet welded to the end of the column member. The base is then attached to the concrete block by means of holding down bolts, anchored within the block. The normal detail for a "pinned" base is to locate two holding down bolts along the neutral axis of the column, one on either side of the web in an attempt to simulate a "pinned" base with the minimum cost. After completion of alignment the plate is grouted into position. In a previous research program, three three-dimensional full-scale pitched-roof portal frames of spans 12m, 12m and 25m respectively were tested. In additional to normal vertical load applied from the roof as in all the three frames, one of the columns in the second frame was also subjected to a horizontal load. In all cases, the columns, designed with "pinned bases", were built as mentioned above except that the concrete blocks were rest on floor. Table 1 shows the bending moment measured in the column just before the frames failed. Only the second frame failed with a plastic hinge formed near to the column head (Engel,1990; Liu, 1988).

Frame 1 Frame 2 Frame 3

TABLE 1 COLUMN BENDING MOMENT IN FULL-SCALE FRAMES

Column size Height Bending Moment near Bending Moment at (m) to column head (kNm) column base (kNm)

203x133x25UB 3.7 58 13.5 305x165x40UB 2.7 185 35 406xl 78x54UB 3.65 323 64

Though designed and constructed as "pinned", the bases had inevitably attracted some moments. Such moments might be about 20% of the column moment capacity (Liu, 1988) and have to be resisted by the coupled generated by the bearing compression of the base plates against the concrete blocks and the tension developed in the bolts. Since the bases were designed as "pinned", the size of the bolts were determined largely by the applied shear forces (Morris & Plum, 1995).

EXPERIMENTAL SET-UP

The objective of the isolated column base tests was primarily to calibrate the finite element model. The main feature in the set-up was to ensure that the numerical model was able to reveal a sufficiently accurate interaction between the column base plate and the concrete block. The column in the

TABLE 2 SUMMARY OF MATERIAL PROPERTIES Yield stress Modulus of Ultimate strength

(N/mm 2) Elasticity (N/mm 2) (N/mm 2)

Flange 348.20 187710 500.00 Web 401.00 189365 526.37

HD bolts Concrete

675.00 195200 feu=30N/mm 2 28500

845.00

Rotational Characteristics of Column Bases of Steel Portal Frames 137

Figure 1 Experimental set-up

arrangement was laid horizontal for the convenience of load application. It was loaded as a simple cantilever. The whole column base was 'rest' on a 500x1200x1500 concrete block. The whole set-up was geometrically symmetrical about the bottom of the concrete block as shown in Figure 1. A pair of one-metre long M24 holding-down bolts went through the two concrete blocks and held the two sides in position. The type of HD bolts used in the tests was of higher strength Grade 8.8 with an ultimate strength of 845N/mm 2. The material properties were shown in Table 2.

A well-established finite element package was previously developed (Liu, 1988) particularly for the analysis of the full-scale portal frame tests. It was also proved to be very successful for the analysis of various types of connections (Liu & Morris, 1991, 1991). In the finite element model, the steel columns were descretised into 8-noded shell elements and the concrete blocks were refined into 8- noded brick elements. The part of the concrete blocks beyond the tension flange of the columns was

Figure 2 F.E. mesh of the column base + concrete foundation block


Figure 3 Moment-rotation curves of the column

not modelled in order to reduce the problem size. Due to symmetry about the web plate, only half of the assembly was modelled. Link interface elements were placed in between the two components in order to determine whether or not they were in contact. The holding down bolts were modelled by line elements following the stress-strain characteristic which was obtained from a separated tension test. The bonding between the HD bolts and the concrete would quickly vanish after once or twice of loading and unloading. Therefore, it was assumed that the bolts were free to extend in tension from the beginning of the loading. Also, the pre-loads in the bolts, about 25kN, were ignored in the model. The base of the concrete block was assumed to be fixed. A point load was applied at a distance of 2m from the base plate. A typical mesh showing the deformation is shown in Figure 2. One of the crucial factors that can determine the accuracy the model is the effect of the base plate. Two thicknesses were used in the test, 12mm and 20mm representing two possible stiffnesses of the same column base.

Figure 4 Bolt forces vs. Applied Bending Moment


Figure 5 Effect of base plate thickness on bolt force

The moment-rotation characteristic and the bolt force vs. applied bending moment curves obtained from the F.E. models and the tests were plotted in Figures 3 and 4 respectively for the two different thicknesses of base plates. The comparison was excellent except that the F.E. models depicted a stiffer behaviour. This is mainly due to the in-accurate assessment of the compression stiffness of the concrete block. However, it is interested to note that, though there is a large difference in the stiffnesses between the two cases, the bolt forces do not differ a lot. The column base with a thicker base plate rotated about the toe of the base plate, i.e. about 220mm from the centroid. The bolt force would therefore be,

1 Mapp = 2.27M Pbolt - 2 0.22m app

where Mapp is the applied bending moment. This agrees very well with the results obtained for the 20mm case from the tests and F.E. modelling. For a more flexible base plate of thickness 12mm, the plate was able to bend and part of it was in contact with the concrete block. The prying action increased the forces in the bolts. However, after the bolts extended further, the prying action faded away and hence the bolt forces came back to a similar level as found with thicker plates. Since the centroids of the couple formed by the tension force in the bolts and the compression force by the reaction should normally be very close to the compression toe of the column, the bolt forces were fairly independent of the base plate thickness and bolt size. Figure 5 shows that the prying action increased the bolt forces by about 20% for thinner base plate.

MOMENT-ROTATION CHARACTERISTICS

Further computational analysis were carried out to examine the effect of various geometric parameters. The computational models were analysed upto a complete collapse, mainly due to bolt failure. Figure 6 shows the effect due to a variation of the base plate thickness. In general, a full range moment- rotation curve consists of four parts. The first part is the elastic regions where every component remains elastic. However, the behaviour is not linear, as a result of the moving centroid of the reaction from the concrete block. With high strength HD bolt, the elastic portion is followed by a static growth in moment of resistance due to an extensive flexural yielding in the base plate. Thereafter, the tensile


Figure 6 Effect of base plate thickness on rotational characteristics

membrance action of the base plate is able to support a further increase in the bolt force until the behaviour comes to a final stage where the bolts eventually fail.

The diameter of the holding down bolts affects directly the initial elastic rotational stiffness as shown in Figure 7. The moment carrying capacity at the second stage and the in-plane membrane stiffness in the third stage are basically not affected as they depend largely on the thickness of the base plate. Figure 8 summaries the effect on the elastic rotational stiffness due to the HD bolt diameter and the base plate thickness. In the range of consideration, the variations seem to be fairly linear. However, when the bolt diameter becomes very large, the stiffness of the column base with 12mm plate should approach a magnitude of about 12000kNrn/rad. When the base plate becomes very thick, the stiffness due to the pair of 24mm HD bolts would be 17000kNm/rad. Obviously, if similar analysis is extended

Figure 7 Effect of diameter of HD bolts on rotational characteristics


Figure 8 Factors affecting Rotational stiffness of column bases

to some smaller or weaker HD bolts, e.g. Grade 4.6, the ultimate moment capacity would be substantially reduced, as shown in Figure 7, by shortening the third stage.

FURTHER PARAMETRIC STUDY ON ROTATIONAL STIFFNESS

Apart from the study on various geometric factors, a series of parametric analyses was also carried out. The objective of this was to establish and quantify the effect of each of the components in the column base on its rotational flexibility of the column base. The components of interest include the stiffness of the concrete block, HD bolt and the base plate.

Five different cases were considered:

�9 Case 1: Rigid concrete block with infinitely rigid HD bolts;

�9 Case 2: No concrete block, but the base rotates about the toe with infinitely rigid HD bolts; �9 Case 3: Rigid concrete block with normal HD bolts; �9 Case 4: No concrete block, but the base rotates about the toe with normal HD rigid bolts; �9 Case 5: Normal concrete block and HD bolts

The F.E. analyses were carried out until the base plates had yielded extensively. For the cases where the thicknesses of base plates were 20mm, the average rotational stiffness upto 30kNm were noted; whereas in the cases of 12mm, the stiffnesses upto 15kNm were recorded. The results are tabulated in Tables 3 and 4.

Comparing cases 4 and 5, the flexibility due to compression of the concrete block is 23.27• .6

rad/kNm for the 20mm case and 16.87x 10 -6 rad/kNm when the thickness if 12ram. This is because the bearing area for the thinner plate is much larger than the thicker one. The difference between cases 2 and 3 shows that the flexibility due to extension of bolts are about 55x10 "6 rad/kNm from the two thickness cases. This agrees very well with the elastic flexibility obtained by simple calculation (58x 10 -6 rad/kNm) assuming all other components rigid. The flexibility due to base plate deformation is expected to dominate the difference between the two cases. The flexibility due to the bending of the 12mm plate together with the end-portion of the column is found to be 110.71x10 6 rad/kNm and that of the 20mm plate is only 34.68x 10 .6 rad/kNm.


TABLE 3 ROTATIONAL STIFFNESS FOR 20MM BASE PLATE

Rotation (x 10 "3 radian)

Case 1 0.805 Case 2 1.040 Case 3 2.110 Case 4 2.683 Case 5 3.381

(Test result = 7800 kNrn/rad I Single Bolt equivalent force (kN)

98.1 eccentricity (mm)

152.9

Stiffness (kNm/rad)

35104.0 68.2 220.0 28837.6 34.68 74.7 201.0 14221.4 70.32 68.2 220.0 11181.6 89.43 69.0 217.3 8873.3 112.70

Flexibility (~trad/kNm)

28.49

TABLE 4 ROTATIONAL STIFFNESS FOR 12MM BASE PLATE

Rotation (x 10 "3 radian)

Case 1 0.960 Case 2 1.660 Case 3 1.820 Case 4 2.458 Case 5 2.711

(Test result = 5300 kNm/rad' Single Bolt force (kN)

59.8

equivalent eccentricity (mm)

125.4

Stiffness (kNrn/rad)

15624.1

Flexibility (~trad/kNm)

64.00 34.1 220.0 9032.6 110.71 38.2 196.5 8248.8 121.23 33.9 221.4 6102.6 163.86 36.8 204.0 5533.2 180.73

CONCLUSION

In this paper, a few design parameters have been considered and their effects on the rotational stiffness been examined. They include the thickness of the base plate, bolt size and the stiffness of the concrete block. The contribution of flexibility by the concrete block is about 20% on the 20mm thick base plate whereas that on the 12mm thick base plate is only 8%. The stress distribution within the toe region is very complex. It requires further investigation. A factor, which is not considered here, is the reduced effective column section. The tensile stress transmitted from the bolts would diffuse gradually into the column. The effective stiffness of the column at the plate-column junction could probably be halved the normal value and thereby increases the rotational flexibility.

It is also not included in this part of the research the behaviour of the underlying soil. Any moment reversal could produce differential settlement causing possible rotation of the foundation block. This might lead to a reduction of the column base moment. While it is essential to quantify the possible stiffness and the moment capacity of the column base for their detail design, it is not recommendable to take this into account when designing the portal frame.

REFERENCES:

Bresler, B. & Lin, Y.Y. (1959), Design of steel structures, John Wiley & sons, N.Y.

Engel, P. (1990) The Testing and Analysis of Pitched Roof Portal Frames, Ph D Thesis, University of Salford.


Liu, T.C.H. (1988), Theoretical Modelling of Steel Portal Frame Behaviour, Ph D Thesis, University of Manchester.

Liu, T.C.H. & Morris, L.J., The development of a shear hinge and the effect on connection flexibility, Proc. Of the Asian-Pacific Conf on Computational Mechanics, Hong Kong, Dec 1991

Liu, T.C.H, & Morris, L.J., The effect of connection flexibility on portal frame behaviour, Int Workshop on connections in steel structures, AISC/Eurcom, Pittsburgh, April 1991

Morris, L.J. and Plum, D.R. (1995) Structural Steelwork Design to BS5950, Longman Scientific & Technical, 2 no Edition, U.K.


ULTIMATE STRENGTH OF SEMI-RIGID FRAMES UNDER NON-PROPORTIONAL LOADS

B.H.M. Chan, L.X.Fang and S.L. Chan

The Hong Kong Polytechnic University, Hunghom, Hong Kong SAR, PR China.

ABSTRACT

This paper presents a numerical procedure for practical design and elasto-plastic large deflection analysis of semi-rigid steel frames under non-proportional loads. Most structures are first under a set of vertical loads such as self-weight and live load before the application of the lateral loads due to wind or seismic forces. The response of a structure under this load sequence cannot be obtained by the principle of super-imposition of these two loading cases due to the non-linear structural behaviour. However, it is often treated in a non-linear analysis as proportional loads for simplicity, which contains a certain degree of uncertainty in accuracy. In this paper, the effects of load sequence are studied and a comparison is made between the case for a structure under proportional and non-proportional loads. It was found that the two results are considerably different that an accurate analysis should allow for this effect.

KEYWORDS

Steel frames, Semi-rigid Frames, Elasto-plastic analysis, Second-order inelastic analysis, Ultimate strength, Proportional and Non-proportional Loads

INTRODUCTION

Currently most of the second order inelastic analyses of steel-flamed structures are performed under the assumption of proportional extemal loads. However, real structures are often subjected to non-proportional loads. The objective of this paper is to study the load-deflection behaviour of steel flames under proportional and non-proportional loads. A numerical example with a portal flame of steel I-sections is analysed for this purpose using a geometric and material nonlinear finite element computer program, GM-NAF (Geometric & Material Non-linear Analysis of Frames).

145

146

PLASTIC HINGE M O D E L

B.H.M. Chan et al.

In the second-order inelastic analysis of steel frames, the first-yield moment, Mer, accounting for the residual stress, (Yres, Can be determined as,

Me r ._(O.y__O.res FIZ (1)

where cry is the specified yield strength, F is the axial load of the member, A is the cross-sectional area and Z is the elastic sectional modulus.

In this paper, the Section Assemblage Concept (Chan& Chui 1997) is adopted to determine the yielded zone, 2~, which is shown in Figure 1, for a section as,

F for ~ < d . 2Cryt 2

~ : (F-crytd) d for d d ~ + - - - - < ~ < - - + T 2Bcry 2 2 2

(2)

in which B is the overall breadth of the flanged section, d is the depth of the web, T and t are the thickness of flange and web, respectively.

(Yy

' . . . . . i . . . . . l o y

Stress Block

Figure 1" Stress distribution for wide-flange section under combined axial force and moment

Once the extent of the plastic region 2 ~ is known, the reduced moment capacity Mpr of the section under combined axial force and bending moment are obtained as follows,

Mp r =[BT(D_T)+I(d)2 _~//2/t]o.y for

Up r = I(O/2 _ ~br ~O.y for

d" ~<_--, 2

d d (3) - -<V<--+T 2 2

A spring is employed for simulation of yielding and the formation of plastic hinge. When no yielding occurs, the spring stiffness is infinite and, when a plastic hinge is formed, the spring stiffness is zero. The spring stiffness, ks, of sections between the first-yield and the fully plastic moments is then taken as,

Ultimate Strength of Semi-Rigid Frames under Non-Proportional Loads

6EI Mpr -M for Mer < M < Mpr ks= Z ]M-Mer]

(4)

147

where E is the elastic modulus of elasticity, I is the second moment of inertia and L is the member length and the strain hardening effect is ignored.

SEMI-RIGID CONNECTION MODEL

The exponential model as follows and proposed by Lui and Chen (1988) is used in this paper to demonstrate the moment-rotation behaviour of semi-rigid connections and is given by,

g c =nj~.lCj{1-expl-lOr]ll+kcf]Orl+g 0 . : 2ja)J

(5)

In equation (5), Mc is the moment applied at the connection; Or is the relative rotation corresponding to the moment Mc; Mo is the initial moment; kcf is the connection stiffness at the strain-hardening stage; a is the scaling factor; and Cj are the curve-fitting constants given in Lui and Chen (1988).

NUMERICAL PROCEDURE

For non-proportional loading, the Newton-Raphson method is used for the vertical loads in the first load sequence whilst the Minimum Residual Displacement method (Chan 1988) is used for lateral loads in the second load sequence. This arrangement is needed since the vertical loads can only be confined to the designed level by the Newton-Raphson method.

To detect the hysteretic behaviour of the plastic hinges and semi-rigid connections during loading stages, one can determine the sign of the incremental moment, AM, and then compared with that of the total moment, M. For the virgin loading path it can be sensed by

AM.M>0 (6)

while for the unloading path,

aM.M_<0. (7)

NUMERICAL EXAMPLE

The double-bay pitched-roof portal frame illustrated in Figure 2 is adopted for this example. Three types of semi-rigid beam-to-column connections are assumed. These connections are of the extended end plate (EEP), flush end plate (FEP) and top-and-seat angle (TSA) types. The out-of-plumbness for each column is taken as 1/200 (EC3 1993) and the residual stress pattern is considered according to E.C.C.S. (1983). To illustrate the effects of proportional loads (PL) and

148 B . H . M . Chan et al.

non-proportional loads (NPL) in the semi-rigid portal frame, the applications of the loading in PL and NPL cases for each type of semi-rigid connections are as follows:

1. PL: The vertical loads P and the lateral load H, which are described in Figure 2, are applied to the portal frame simultaneously with the same load factor (i.e. ~,v=~,h) until the frame collapses; and

2. NPL: The vertical loads P are firstly applied to the corresponding value of ~.v at which the frame collapses in the PL case (see Figure 3). Then, the lateral load H is applied until the frame collapses. This procedure is to ensure that the same vertical load level is maintained as a basis for comparison of the PL and NPL cases.

~,hH

~,vP LvP

.~ ~ 20 ~ 20 ~ ~

[. 4mL" 4m ..I. 4m ..I.. 4m ..I

E = 200 kNm -2 (Yy = 275 Nmm 2 P = 200 kN; H = 40 kN Column: 203x203x46UC Rat~er : 254x 146x31UB

Figure 2: Double-bay pitched-roof portal frame.

The load deflection curves for the semi-rigid frames are shown in Figures 3 and 4, and the maximum values of P, H with corresponding values of u are also given in parentheses in the figures. Note that, for the NPL cases, curves in Figure 3 are with respects to the first load sequences while Figure 4 for the second load sequences. Therefore, it is reminded that the curves for the NPL cases in Figure 4 do not start from the origin.

In Figure 4 it can be found that the ultimate lateral loads for the portal frame with EEL connections under PL and NPL are very close and the differences in ultimate lateral loads and displacements are within 5%. On the other hand, for the frame incorporated with TSA connections, the ultimate lateral load in the NPL case is about 14.6% greater than that in the PL case. However, for the frame with FEL connections, the behaviour in the PL and NPL cases is totally different. The ultimate lateral load in the NPL case is only 7.9% of the value in the PL case. It is because as observed in Figure 3 the stiffness of the portal frame with FEP commences to decrease from its elastic value, at which the prescribed vertical load level has been applied.

Ultimate Strength of Semi-Rigid Frames under Non-Proportional Loads

120

80

A z 60 Ik

40

20

24

22

2O

18

16

14

8

6

4

2

0

i (

a EEP-PL(P=107.27kN; u=5.724cm)

_- EEP-N PL(P--107.27kN; u=-3.063cm)

FEP-PL(P=100.52kN; u=8.177cm)

~, FEP-N PL(P=100.52kN; u=-4.112cm)

: TSA-PL(P=74.227kN; u=l 1.23cm)

- - - e- - - TSA-NPL(P=74.227kN; u=-3.419cm)

l l i , | , , ! ,

-6 -4 -2 0 2 4 6 8 10 12 14 16 18 u (cm)

100

Figure 3" Vertical loads P versus lateral displacement u.

149

-6 -4 -2 0 2 4 6 8 10 12 14 16 18 u (cm)

Figure 4" Lateral load H versus lateral displacement u.

150

CONCLUSIONS

B.H.M. Chan et al.

From the numerical example illustrated above shows that the ultimate strengths and the deformations of semi-rigid steel frames can be load-sequence dependent when both the geometric and material non-linearities are accounted for. Analysis based on proportional load approach can result in an under-estimation of the load-carrying capacity of structures.

REFERENCES

Chan, S.L. (1988). Geometric and Material Nonlinear Analysis of Beam-Columns and Frames using the Minimum Residual Displacement Method. Int. J. Num. Meth. in Engrg, 26, 267.

Chan, S.L. and Chui, P.P.T. (1997). A generalised design-based elastoplastic analysis of steel flames by section assemblage concept. Engrg. Struct., 19:8, 628.

EC3 (1993). Eurocode 3: Design of steel structures: Part 1.1 General rules and rules for buildings, European Committee for Standardization, Brussels.

ECCS (1983). Ultimate Limit State Calculation of Sway Frames with Rigid Joints, European Convention for Constructional Steelwork, Rotterdam.

Lui, E.M. and Chen, W.F. (1988). Behavior of braced and unbraced semi-rigid frames. Int. J. Solids. Struct., 24:9, 893.

S E C O N D - O R D E R PLASTIC A N A L Y S I S OF S T E E L F R A M E S

Peter Pui-Tak Chui ~ and Siu-Lai Chan 2

Ove Arup & Partners (Hong Kong) Ltd., HONG KONG 2 Dept. of Civil & Structural Engineering, The Hong Kong Polytechnic University, HONG KONG

ABSTRACT

A second-order refined-plastic-hinge method for determining the ultimate load-carrying capacity of steel frames is presented. Member imperfection and residual stress in hot-rolled I- and H-sections are considered. Second-order effect due to the geometrical nonlinearity is accounted for. In the present inelastic model, gradual degradation of section stiffness is allowed for simulating a more realistic and smooth transition from the elastic to fully plastic states. The developed model has been verified to be valid through a benchmark calibration frame.

INTRODUCTION

It has been long recognized that the second-order effects due to geometrical changes and inelastic material behaviour can dominate the load-carrying capacity of steel structures significantly, as shown in Fig. 1. However, the first-order elastic analysis is usually employed to estimate the member forces in conventional engineering design. In pace with the advent in computer technology, the sophisticated analysis is feasible. Recently, a refined method of analysis, which is called the Advanced Analysis, has been coded in the Australian limit states standard for structural steelwork (AS4100 1990). The basis of the Advanced Analysis is to consider initial imperfections and second- order effects so as to estimate the member forces and the overall structural behaviour accurately. This should result in more economical and safe selection of member size. The existing models for second-order plastic analysis can be broadly categorized into two types, namely the plastic-zone (Ziemian 1989) and the plastic-hinge (Gharpuray and Aristizabal-Ochoa 1989) models. In the plastic-zone method, the beam-column members are divided into many very fine fibres. Its results are generally considered as the exact solutions. However, it is much costly and, therefore, its solutions are usually used for calibrating of various plastic-hinge models. In the plastic-hinge method, a plastic hinge of zero-length is assumed to be lumped at a node. This eliminates the tedious integration process on the cross-section and permits the use of less elements per member. Therefore, it reduces computational time significantly. Although it can only predict approximately the strength and stiffness of a member, it is more suitable and practical in engineering design practice. In this paper, a refined-plastic-hinge model is proposed and studied.

151

152 P. P.- T. Chui and S.-L. Chan

FUNCTIONS OF YIELD SURFACES

In the present refined-plastic-hinge analysis, a function is employed to mathematically describe a limiting surface which is used to check whether or not the interaction point for axial-force and bending-moment lying outside this yield surface. As the name implies, a full-yield surface and an initial-yield surface are here used to define the ultimate strength surface and the initial yield surface respectively on the plane of normalized force diagram for a cross-section. The functions of these surfaces employed in this paper are defined as follows.

Full- Yield Surface

A full-yield surface is a strength surface of a section to control the combination of normalized axial force and moment. In other words, it represents the maximum plastic strength of the cross-section in the presence of axial force. Based on the British Standard BS5950 (1985), the Steel Construction

M / M p = 1 - 2 . 5 ( P / P y ) 2

M / M p = 1 . 1 2 5 ( 1 - P / P y ) when P / Py < 0.2 (1) when P / Py > 0.2

Institute (1988) has recommended a full-yield surface of hot-rolled 1-section for compact section bending about the strong axiS, as, in which M and P are moment and axial force acting on the section, Mp is the plastic moment capacity of the section under no axial force and Py is the pure crush load of the section.

Initial- Yield Surface

The European Convention for Constructional Steelwork (ECCS 1983) has provided a detailed and comprehensive information with regard to appropriate geometric imperfections, stress-strain relationship and residual stress for uses in the plastic zone analysis. The pattern of ECCS residual stress for hot-rolled I- and H-sections is shown in Fig. 2. The residual stress will result in the early yielding of a section and the initial-yield surface can be defined as,

Mer = Z e ( Oy - Ore s - P / A ) (2)

in which Mer is the reduced moment elastic capacity under axial force P, Ze is the elastic modulus, (Yy is the yield stress, Crre s is the residual stress and A is the cross-section area. In case of no residual stress and axial force, the M~r will become the usual maximum elastic moment (i.e. Mer = Zr Cry). As the normalized force point is within the initial yield surface, the member behaves elastically. The effect of residual stress on the moment-curvature relationship is illustrated in Fig. 3.

PROPOSED PLASTICITY METHOD

In the traditional plastic-zone (P-Z) method, beam-colunm members are divided into a large number of elements and sections are further subdivided into many fibres. The solutions by this method are generally considered as the exact solutions. However, the computation time required is much heavier and it is usually for research study, but not for practical design purpose. To simplify the inelastic analysis, a refined-plastic-hinge method is proposed because of its efficiency.

Second-Order Plastic Analysis of Steel Frames

Refined-Plastic-Hinge (R-P-H) Method

153

The proposed refined-plastic-hinge method is a plastic-hinge based inelastic analysis approach considering the stiffness degrading process of a cross-section under gradual yielding for the transition from the elastic to plastic states. In the proposed method, material yielding is allowed at nodal section only and can be represented by a pseudo-spring. The stiffness of the spring is dependent on the current force point on the thrust-moment plane. When the force point does not exceed the initial-yield surface, the section remains elastic and the spring stiffness is infinite. If the point reaches on the full-yield surface, the section will form a fully plastic hinge and the value of the spring stiffness will be zero. To avoid computer numerical difficulties, the limiting values of oo and zero will be assigned as 101~ and 10I~ respectively. When the force point lies between the surfaces, section will be in partial yielding and the function of the spring stiffness, t, is proposed to be given by,

t - 6EI IMpr-M I when Mer<M<Mpr (3) L I M - M rl

in which EI is the flexural rigidity, L is the element length, and Mer and Mpr are the reduced initial- and full-yield moments in the presence of axial force, P, shown in Fig. 4.

Movement Correction of Force Point

After a fully plastic hinge is formed at a section, correction of forces must be considered to insure the force point is not outside the maximum strength of section. As the axial force increases, the moment capacity will be reduced and hence the value of bending moment would decrease. If the force point is outside the full-yield surface, the point is assumed to shift orthogonally back onto the yield surface.

ELEMENT STIFFNESS

Assuming the section spring stiffness at the ends of an element to be t~ and t 2, an incremental form of element stiffness can be expressed (see Fig. 4) as,

/ AeMI/ tl -tl 0 0

AIM1[ = -t I 4EI/L +t 1 2EI/L 0

AiM: / 0 2EIIL 4EI/L +t 2 -t 2

AoM~) 0 0 -t~ t~

Ae01/ AiOl/

Ai02/ Ae 02)

(4)

in which the subscript "1" and "2" are referred to the node 1 and node 2, AeM and AiM are the incremental nodal moments at the junctions between the spring and the global node and between the beam and the spring and, Ae0 and Ai0 are the incremental nodal rotations corresponding to these moments. It is assumed that the loads are applied only at the global nodes and hence both AiM1 and AiM2 are equal to zero, we obtain,

154 P. P.-T. Chui and S.-L. Chan

Ai01) 1 A i02 ) = --~ -2EI/L 4EI/L +t 1 tAeO2)

(s)

in which 13 = (4EI/L+t0(4EI/L+t2) - 4(EI/L) 2. Eliminating the internal degrees of freedom by substituting the equation (5) into (4), the final incremental stiffness relationships for the element can be formulated as,

EA/L 0 0

0 tl -t12(K22 + t2)/13 tlt2K12/13 Ae01

0 tlt2K21/13 t2-t~(K11+t1)/~ t ao~ (6)

in which A is section area, AP is axial force increment and AL is axial deformation increment.

NUMERICAL EXAMPLE

The two-bay six-storey European calibration frame subjected to proportionally applied distributed gravity loads and concentrated lateral loads has been reported by Vogel (1985). The frame is assumed to have an initial out-of-plumb straightness and all the members are assumed to possess the ECCS residual stress distribution (ECCS 1983). The paths of load-deformation curves shown in Fig. 5 are primarily the same by the plastic-zone and the plastic-hinge analyses. The maximum capacity is reached at a load factor of 1.11 for Vogel's plastic-zone method (Vogel 1985), 1.12 for Vogel's plastic-hinge method (Vogel 1985), and 1.125 for the proposed refined-plastic hinge method. The maximum difference between these limit loads is less than 1.4%. This example shows the adequacy of the plastic hinge method for large deflection and inelastic analysis of steel frames.

The same frame has also been studied by the Cornell University inelastic program: the CU- STAND (Hsieh et al. 1989). The force diagrams of the frame with key values at specified locations and at the maximum load of the frame are plotted in Fig. 6. The ultimate load factors are 1.13 for the CU-STAND and 1.125 for the present study. The force distribution and the plastic hinge location obtained by the analyses are essentially similar. The CU-STAND hinge analysis detects a total of 19 plastic hinges while the present study detects 16 plastic hinges. The difference may be explained by the fact that the present limit load, which is less than that obtained by Hsieh et al. (1989), is not high enough to produce further fully plastic hinges at these three locations. Referring to the figure, the present bending moments at the three locations are very close to the fully plastic moment capacity of section just before structural collapse.

CONCLUSIONS

A plastic-hinge based approach for inelastic analysis of steel frames, the refined-plastic-hinge methods, is presented. The inelastic behaviour of a beam-column member can be simulated by a spring model allowing for degradable stiffness of sections between the elastic and plastic states. From the example, the inelastic behaviour of frame controls the ultimate load and should be

Second-Order Plastic Analysis of Steel Frames 155

considered. Generally speaking, based on the simplified numerical model employed, the proposed refined-plastic-hinge analysis is more suitable and practical in design practice when compared with the plastic-zone analysis.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge that the work described in this paper was substantially supported by a grant from the Research Grant Council of the Hong Kong Special Administration Region on the project "Static and Dynamic Analysis of Steel Structures (B-Q 193/97)". The support of the first author by Ove Arup and Partners(Hong Kong) Ltd. is also acknowledged.

REFERENCES

1. British Standard Institution (1985), BS5950: Part I." Structural Use of Steelwork in Building, BSI, London, England.

2. European Convention for Constructional Steelwork (1983), Ultimate Limit State Calculation of Sway Frames with Rigid Joints, ECCS, Technical Working Group 8.2, Systems, Publication No. 33.

3. Gharpuray, V. and Aristizabal-Ochoa, J.D. (1989), "Simplified Second-Order Elastic Plastic Analysis of Frames", J. of Computing in Civil Engng., 3:1, pp.47-59.

4. Standards Australia (1990), AS4100-1990 Steel Structures, Australian Institute of Steel Construction, Sydney, Australia.

5. Steel Construction Institute (1988), Introduction to Steelwork Design to BS5950: Part 1, SCI Publication No. 069, Berkshire, England.

6. Ziemian, R.D. (1989), Verification Study, School of Civil and Environmental Engng., Cornell Univ., Ithaca, N.Y.

7. Vogel, U. (1985), "Calibrating frames", Stahlbau, 54, October, pp.295-311. 8. Hsieh, S.H., Deierlein, G.G., McGuire, W. and Abel, J.F. (1989), "Technical manual for

CU-STAND", Structural Engineering Report No. 89-12, School of Civil and Environmental Engineering, Cornell University, Ithaca, N.Y., U.S.A.

156 P. P.-T. Chui and S.-L. Chan

8eooncl.Order Bmtk~

Unur Analym / (Flint-order Butlr

Bmtlr Bifurcation Load

Plastlo Umlt Load

Bastlc-Pl~Ic Analysis

8eaond-order Plutlc-hlnge Atolls

Aotual B e h ~ o u r oo%

Local and/or L~eml Torsional bucldlng 8eoond-order Plmtlc Zone

Generalised Displacement Fig. 1 General Analysis Types of Framed Structures

D

I i

I 0.5

0.5

I 0.5 B _1

~/~=os D / B < 12

03 ~ i ' ~ "-~ o a o~a I

3

03 ~/~=oa D/B > 12

Fig. 2 ECCS residual stress distribution for hot-rolled I-ssctlons

M/Mp ~ - - Idealized elastic-perfectly plasUc behaviour

or ~ r - - W'ithout residual s t resses

�9 IT ,"

, , . ~ / With residual st resses

/%, o-< : (Ty = y ie ld s t ress

o ~ +/+y Fig, 3 Moment-curvature relationship for I-ssctlon

with and without residual stresses

Section spring of stiffness, t2 Node1 ~ Node 2~.M2

P "'--2,.0,

Fig. 4 Internal forces of an element with end-ssction springs accounting for cross-ssction plsstlflcatlon employed by the present study

Second-Order Plastic Analysis of Steel Frames

1.2

1.0

0.9

0.8 ,,<: ,.z 0.7 0

,.~ 0.6

_9 0.5

0.4

0.3

0.2

0.1

0.0 0

Umiting load factor,)~ 1.11 1.12 1.125

_ ~ k N / m

-- IPESO0

2 LS?2 I = / -I '"'=~ ~' I

I ~L..L . . . . ~.l.~.,..~.,m / -~ F ;~-~ ~TM~ ~

/ ,, . L . . . . ~ ,L~M_ F~ ~IWN/m

/ -I "'= ~ I I E = 205 KN/mm < ~ ; " -"-'_~, .... ~_'-"

/ ~= ~ ~mr~ ~ ~ ~ ~' ~i / ~ = 1/450 (_P!astic zone) / 7 ~ ~/'~/7 7-/

= 1/300 (Plastic hinge) l< 2xe 112m _ _ l I

. . . . . �9 D Plastic zone (Vogel 1985)

. . . . . . 0 Plastic hinge (Vogel 1985) Refined-plastic hinge (this study) (5 6 (cm)

I I I I l I I 5 10 15 20 25 30 35

Fig. 5 Inelastic load-deflection behsvlour of Vogel six-storey frame

~ 81.4

255 I I 547 II ~ / f 142.7 . [2 ] L [~:6:3] [147~ [147.6] [147.6]

4O7 I I 879 II 4 I 146.5 [147.6]A 145.5 [147.5] [147.5] /

! ~ 8 . ; 152.4 [1

L/~ .4"/ 'q-__J~30.g 7 I 154.-g--- [230.3]/ 125.4P~.4]

/ / 112.5 [2~.s] $] / ~ [3o,.1] 7

~69] ~914] ~ ,j/W~.111.6 / [~2204.7 ] / ~10~:59]

(a) Axial force (kN) (b) Bending moment (kN-m) Values: Symbols: This study, k u =1.125 0 Plastic hinge location by CU-BTAND [CU-STAND, X u =1.1 3] ~ Plastic hinge location by this study

�9 Common plastic hinge location by CU-STAND and this study

Fig, 6 Comparslon of member forces of Vogel frame by Cornell studies and this study

157


STUDY ON THE BEHAVIOUR OF A NEW LIGHT-WEIGHT STEEL ROOF TRUSS

P. Makel~iinen and O. Kaitila

Laboratory of Steel Structures, Helsinki University of Technology, P.O.Box 2100, FIN-02015 HUT, Finland

ABSTRACT

The Rosette thin-walled steel truss system presents a new fully integrated prefabricated alternative to light-weight roof truss structures. The trusses will be built up on special industrial production lines from modified top hat sections used as top and bottom chords and channel sections used as webs which are jointed together with the Rosette press-joining technique to form a completed structure easy to transport and install. A single web section is used when sufficient and can be strengthened by double-nesting two separate sections or by using two or several lateral profiles where greater compressive axial forces are met.

A series of laboratory tests have been carried out in order to verify the Rosette truss system in practice. In addition to compression tests on individual sections of different lengths, tests have also been done on small structural assemblies, e.g. the eaves section, and on actual full-scale trusses of 10 metre span. Design calculations have been performed on selected roof truss geometries based on the test results, FE-analysis and on the Eurocode 3, U.S.(AISI) and Australian / New Zealand (AS) design codes.

KEYWORDS

Rosette-joint, truss testing, light-weight steel, roof truss, cold-formed steel, steel sheet joining.

INTRODUCTION

The Rosette-joining system is a completely new press-joining method for cold-formed steel structures. The joint is formed using the parent metal of the sections to be connected without the need for additional materials. Nor is there need for heating, which may cause damage to protective coatings. The Rosette technology was developed for fully automated, integrated processing of strip coil material directly into any kind of light-gauge steel frame components for structural applications, such as stud wall panels or roof trusses. The integrated production system makes prefabricated and dimensioned frame components and allows for just-in-time (JIT) assembly of frame panels or trusses without further measurements or jigs.

159

160 P. Mdkeldinen and O. Kaitila

This paper presents the first extended test programme performed on the ROSETTE light-weight steel roof truss system. Results of tests on individual members and full scale roof trusses are presented.

T H E R O S E T T E - J O I N T

The Rosette-joint is formed in pairs between prefabricated holes in one jointed part and collared holes in the other part. First, the collars are snapped into the holes. Then the Rosette tool heads penetrate the holes at the connection point, where the heads expand, and are then pulled back with hydraulic force. The expanded tool head crimps the collar against the hole. Torque is enhanced by multiple teeth in the joint perimeter. The joining process is illustrated in Figure 1 and the finished Rosette-joint is shown in Figure 2.

Figure 1: Rosette-joining process Figure 2: The Rosette-joint

D E S C R I P T I O N O F T H E R O S E T T E - R O O F T R U S S S Y S T E M

Rosette - trusses are assembled on special industrial production lines from modified hat sections used as top and bottom chords and channel sections used as webs, as portrayed in Figure 3, which are joined together with the Rosette press-joining technique to form a completed structure. The profiles are manufactured in two size groups using strip coil material of thicknesses from 1.0 to 1.5 mm. A single web section is used when sufficient, but it can be strengthened by double-nesting two separate sections and/or by using two or several lateral profiles where greater axial loads are met. At the present time, the application of the Rosette truss system is being examined in the 6 to 15 metre span range.

Figure 3: Cross-sections of the 89 mm Rosette chord and 38 mm web members

Study on the Behaviour o f a New Light-Weight Steel Roo f Truss

T E S T S O N I N D I V I D U A L M E M B E R S

161

Tests on Web Members

Axial compression tests were carried out on four differently arranged sets of 38 mm web sections of measured cross-sectional thickness 0.94 mm in order to verify their actual failure mode and load. The specimens in groups 1 to 3 were prepared for testing by casting each end in concrete, thus providing rigid end conditions. All specimens, including group 4, were placed firmly on solid smooth surfaces and the compressive force was applied axially on the gravitational centroid of the members.

The test results are summarized in Table 1. In test groups 1 and 2, they are quite consistent with analytical values determined according to Eurocode 3 , Part 1.3. Group 3 consists of two specimens of web members with two profiles freely nested one inside the other. The analytical compression capacity was obtained by simply multiplying the capacity of a single profile by two. The average maximum load from the tests was approximately three-fold the test value for a single profile. This high value is due to the greater torsion resistance of the nested profiles when compared to single profiles.

Test-group 4 differs from the first three groups in its overall arrangement and motives. The idea was to examine the way the joints connecting the web profile to the chord profiles perform under axial loading, and how much rotational support they give to the web profile that has been initially considered hinged at both ends. Each of the three test specimens consisted of a 1 060 mm long web profile element connected by Rosette-joints at each of its ends to a 400 mm long piece of chord profile. The length of the specimens was chosen great enough to prevent the failure of the joints before buckling occured. The distance between the midpoints of the joints was then 1 003 mm for all three specimens. The average maximum test load value was approximately 39 % larger than the analytical value calculated with an effective buckling length reduction factor of Kb = 1.0. The test load value corresponds to an analytical buckle half-wavelength of 780 mm (Kb = 0.78). This indicates that it would be safe to use an effective buckling length reduction factor ofKb = 0.9, as is quite usual practice in roof truss structures.

TABLE 1 38 MM WEB COMPRESSION TEST RESULTS

(T = TORSIONAL BUCKLING, F = FLEXURAL BUCKLING, D = DISTORTIONAL BUCKLING)

Total length after setup

mrn # 1 660 2 660 3 660

1061 1060 1060

Test Test pi~.e Group number

Theoretical Analytical Buckle Compression

Half-wavelength Capacity mm kN 330 33.44 330 33.44 330 33.44

Average: 530.5 530 530

25.56 25.56 25.56 Average:

1063 531.5 45.14 1061 530.5 45.14

Average: 1000 1000 1000

I000 I000 I000

9.27 9.27 9.27

Average:

Test

Result

kN 34.24 1.02 36.02 1.08 36.80 1.10 35.69 1.07 23.04 0.90 25.06 0.98 26.94 1.05 25.01 0.98

1.66 1.68 1.67 1.42 1.43 1.32

Ratio between test result

and analytical result

1.39

Failure Mode

T+D T+D T+D

T T T

F+T F+T

162

Tests on Chord Members

P. Mdkeldinen and O. Kaitila

Similar compression tests to those carried out on individual web profiles (test-groups 1 and 2) have been performed on chord profiles. The actual structure will include continuous chord members that are connected to web members at different intervals and laterally supported by braces at 600 mm intervals.

Test Group

#

1

TABLE 2 8 9 M M C H O R D C O M P R E S S I O N T E S T R E S U L T S

(TF = T O R S I O N A L - F L E X U R A L B U C K L I N G M O D E )

Test piece number

#

1

2 3

Total length after setup

mm

1258 1255 1255

Theoretical Buckle

Half-wavelength mm

629 627.5 627.5

4 1754 5 1751 6 1755

877 875.5 877.5

Analytical Test Ratio between Compression Result test result

Capacity and analytical kN kN result

52.68 47.28 0.90 52.68 46.92 0.89 52.68 49.85 0.95

Average: 48.02 0.91 St. deviation: 1.60

32.95 34.65 1.05 32.95 34.54 1.05 32.95 34.37 1.04

Average: I 34.52 1.05 St. deviation:

,, Failure Mode

TF TF TF

TF TF TF

It can be concluded that the design procedure used for the evaluation of the compression capacities is quite compatible with the test results. The analytical calculations and FE-analyses performed predicted a torsional-flexural buckling mode with a stronger deflection in the y-direction and the test results supported this prediction. Also the maximum loads observed in the tests comply with the analytical values to an acceptable degree.

TESTS ON F U L L - S C A L E T R U S S E S

General

Two full scale 10 metre span trusses have been tested according to the testing procedure described in Eurocode 3: Part 1.3 Appendix A4. The first truss passed the first phase of testing, i.e. the 'Acceptance Test', but failed during the load increase phase of the next test round, i.e. the 'Strength Test'. This failure was due to manufacturing difficulties and insufficient detail design of the truss (Kaitila 1998a). The information received from the first test was analysed and used to improve the details of the second truss while preserving the original basic geometry. The different phases and the results of the second truss test are given in the present chapter.

Test Set-Up

The test truss was manufactured from steel plate with cross-sectional wall thickness tobs = 0.95 mm (+ zinc coating), yield stressfy, obs = 368 N/mm ~, and modulus of elasticity E = 189 430 N/mm 2 (all values taken for steel in the direction of cold-forming).

The profiles used were a modified 89 mm chord and a new 29 mm web profile, as shown in Figure 4. The vertical web profiles on the supports were designed so that they rest against the bottom flange of the bottom chord and could thus directly transmit the load from the structure onto the support as compression, without the chord member having to support unneccessary shear force which would

Study on the Behaviour of a New Light-Weight Steel Roof Truss 163

cause strong distortion in the lower part of the chord member, as observed in the tests on eaves members.

Figure 4: The profiles used in the second truss test

The nominal geometry of the tested truss is outlined in Figure 5. The truss was symmetrical about its centre line with a top chord inclination of 18 degrees. The height at the support was approximately 490 mm, which gave the truss a total height of about 2100 mm. The top chords were connected to each other at mid-span using a short web member and specially manufactured jointing plates. The total mass of the actual truss was 75.5 kg.

Figure 5: Nominal geometry of the test truss with load cylinders

The truss was supported at the ends of the bottom chord with pinned supports. All horizontal displacements were prevented at the lefthand support and free in the plane of the structure at the righthand support. The support plates were long enough to allow for a sufficient support area for both web members at the support. The lateral supports were made at the top chord every 600 mm by simply bolting the top flange of the chord to the c 600 loading rig. The load cylinders were hinged in the plane of the structure but fixed in the plane perpendicular to that of the truss.

The dimensions of the actual truss differed quite little from the nominal values. The actual dimensions of the manufactured profiles differed from the nominal cross-sections by less than 5 %. The formation of the joints was done successfully this time without the problems that occurred in the manufacturing of the first test truss.

164

Outline o f Test Procedure

P. Mdikeldinen and O. Kaitila

The testing was performed according to the procedure described in Eurocode 3 Part 1.3 Appendix A4: Tests on Structures and Portions of Structures. This method includes three distinct phases, an 'Acceptance Test', a 'Strength Test' and a 'Prototype Failure Test'. The loading was applied at eighteen distinct points (nine on each side of the truss's midline) with c 600 mm space between them so, that at mid-point there was no load and thus the space between the two middle load cylinders was 1 200 mm.

Only symmetrical evenly distributed loading was considered in this test. The load was pumped into a hydrostatic pressure cylinder using a handpump and subsequently evenly divided between all 18 load cylinders. Each load cylinder had a 420 mm long loading pad which transmitted the load from the cylinder onto the structure. The loading pad is 80 mm wide which made it possible to place the 63 mm wide top chord profile centrally under the pad and leave a minimum space of approximately 8 mm for distortional or other deformation of the cross-section on both sides of the profile. Vertical deflections were measured with displacement bulbs at the mid-point and the quarter points of the bottom chord, and at the ends and the mid-point of the top chords. Horizontal displacement of the supports was also measured.

Computer Model o f the Test Truss

A STAAD III-analysis was performed for the design of the truss. The material values used for the model were:

�9 wall thickness t = 0.96 mm �9 yield strengthfy =fyb = 350 N/mm 2 �9 modulus of elasticity E = 210 000 N/mm 2

The connection (i.e. two joints) capacity used in the analysis was taken as Fc, conn = 10.8 kN.

Progression and Results o f t he Full Scale Truss Test

The second test truss successfully passed all phases of testing and the maximum load reached was 48.5 kN. The course of the test can be most simply explained with the aid of the diagram given in Figure 6 showing the deflection of the truss at mid-span measured from the bottom chord. The graph is complemented with numbers showing the different phases of testing.

Figure 6: Deflection at mid-span of the truss (see text for notes)

Study on the Behaviour of a New Light-Weight Steel Roof Truss 165

1. The test was begun at zero load and the load was steadily increased up to 25.16 kN, where it was held for one hour. The nonlinearities in the curve during load increase were caused by the movement in the joints due to production tolerances. Point 1 marks the beginning of the one hour period. During load increase or decrease, displacement values were taken at 5 second intervals. During the constant load phases, they were recorded every 30 seconds.

2. Point 2 marks the end of the one hour period. The maximum deflection at this stage was 11.28 mm or L / 850. The load was then gradually taken off.

3. The residual deflection after the 'Acceptance Test' phase was 1.74 mm (15 % of the maximum recorded). The allowable value is 20 %, so the truss passed this first phase successfully. The behaviour of the truss was very good during this first phase.

4. The test load was initially evaluated as 32.0 kN due to a miscalculation. Therefore a quick decision was made at the beginning of the one hour period of this second phase of testing, to increase the test load by 10 % up to 35.2 kN. Point 4 marks the small escalation caused by this mistake before the 10 % increase.

5. Point 5 shows the beginning of the one hour period of the 'Strength Test' phase at load 35.2 kN. 6. Point 6 marks the end of this one hour period. The maximum deflection recorded at this stage was

18.06 mm or L / 550. 7. Point 7 marks the residual deflection at mid-span after the removal of the load. This total residual

deflection was 4.51 mm, i.e. the deflection was decreased by 75 %, much more than the 20 % needed at this stage. No actual tear was observed, but a slight beginning of local deformations could be seen in the chord members in the area of the most heavily loaded joints, i.e. beginning shear deformations like the ones portrayed in Figure 7 were starting to appear, but in a much smaller scale than in the photographs.

Figure 7: Deformations at the left side support area of the top chord just before failure (left) and after failure (right)

The free edges of the top chord deformed into slight sine-shaped curves under loading, as expected. The deformation happened in such a way, that consecutive portions separated by web members were deformed in opposite directions, i.e. the first one towards the inside, the second one towards the outside etc. A similar deformation occurred in the bottom chord, although this part of the structure should primarily be under tensile stress. The effect of bending moment caused the deformation of the free edges of the bottom chord profiles. The individual web members did not show indication of insufficiency.

8. After the truss had satisfactorily passed the 'Strength Test'-phase, the last stage with loading up to failure was begun. During the increase of the load, the longer webs were considerably deformed in torsion and flexure. Nevertheless, the final failure did not occur directly due to this but to the joints in the first tension webs counting from outside, as expected from the computer analysis. The

166 P. Mgikelgiinen and O. Kaitila

failure load was 48.5 kN, although it can be argued that the load-bearing capacity of the truss was reached around a total load value of 46 kN, because of the strong torsional-flexural deformations of the longer web members.

CONCLUSIONS

This paper presents the general results of the first analysis including a test programme on the Rosette - steel roof truss system and individual members. The behaviour of the truss was linear and predictable throughout the testing procedure. The structure successfully passed the first and second stages of the Eurocode 3 testing procedure, 'Acceptance Test' and 'Strength Test', respectively. The manufacturing of the truss was carried out with a much better standard of quality than in the first test, where several imperfections caused the truss's early failure (Kaitila 1998a). The individual members acted well in this test. There was no significant plastic deformation before the last stages prior to failure.

The partial safety factors for the joints are considerably larger than those used for the members (t, = 1.25 compared with 1' = 1.1, respectively). Therefore it is not surprising that it is the joints that tend to become critical in the truss design. Furthermore, because the chord members did not cause any problems in this test, it might be concluded that the chord profile has unnecessary extra capacity and reasons for reducing the chord profile in size might exist. However, it is perhaps too early to draw such a conclusion, since the effects of this type of change need to be examined on the level of a complete structure.

The connection technique used to join together the top chords at mid-span should be studied and designed in a more efficient manner with an analysis extending to the effect of a suggested solution on the behaviour of the complete structure.

The truss passed the requirements set by the European design standard. Further optimization and more detailed design is needed for the application of the Rosette system to high-quantity production, but a strong confidence in the abilities of the system can be justified by this test.

ACKNOWLEDGEMENTS

The authors would like to acknowledge Mr. Kimmo J. Sahramaa (FUSA Tech Inc., Reston, VA, USA), the innovator of the Rosette-joint technology, and Mr. Juha Arola (Rosette Systems Ltd, Kauniainen, Finland) for the initiation and support of this research project.

REFERENCES

Kaitila O. (1998a). Design of Cold-Formed Steel Roof Trusses Using Rosette - Connections, Master's Thesis, Helsinki University of Technology, Espoo, Finland Kaitila O. (1998b). Second Full Scale Truss Test on a Rosette - Joined Roof Truss, Research Report TeRT-98-04, Helsinki University of Technology, Espoo, Finland Kesti J., Lu W., M~.kel/iinen P. (1998). Shear Tests for ROSETTE Connection, Research Report TeRT- 98-03, Helsinki University of Technology, Espoo, Finland M~kel~inen P., Kesti J., Kaitila O., Sahramaa K.J. (1998a). Study on Light-Gauge Roof Trusses with Rosette Connections, 14 th International Specialty Conference on Cold-Formed Steel Structures,

St.Louis, Missouri, USA M/J.kel~inen P., Kesti J., Kaitila O. (1998b). Advanced Method for Light-Weight Steel Truss Joining, Nordic Steel Construction Conference 98, Bergen, Norway

A PROPOSAL OF GENERALIZED PLASTIC HINGE MODEL FOR THE COLLAPSE BEHAVIOR OF STEEL FRAMES

GOVERNED BY LOCAL BUCKLING

Shojiro Motoyui and Takahiro Ohtsuka

Department of Built Environment, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama, 226-8502, Japan

ABSTRACT

It is necessary for evaluating true safety of structures to evaluate the safety by using an analytical method which can simulate the behavior to collapse. And a collapse of steel frames is due to local buckling, fracture, etc., we consider a collapse behavior caused by only elastoplastic local buckling in this paper. However, in present situation of computer performance, it is not realistic to analyze dynamically the whole frames with the finite element method which can express the influence of local buckling. Besides, as far as we know, none of the reports clarified the strength degradation behavior with local buckling considering the influence of applied axial force and bending moment equivalently. Then, we show a generalized plastic hinge model which is able to pursue the strength degradation behavior governed by local buckling to collapse, according to evaluating equivalently axial force and bending moment on N-M interaction relationships based on plasticity theory.

KEYWORDS

generalized plastic hinge model, local buckling, strength decrease behavior, collapse, plasticity theory, steel frames, numerical analysis, finite element method

INTRODUCTION

There are few studies on the response analysis of steel frames which have members with strength decrease governed by local buckling. The simplified model proposed in those studies, L. Meng et al.(1991), Yoda et al.(1991), Yamada and Akiyama(1996) are not clarified about evaluation axial force in local buckling, that is, those models don't evaluate equivalently axial force and bending moment for the influence of local buckling. Then, in order to propose a generalized plastic hinge model which can pursue the strength decrease behavior governed by local buckling based on plasticity theory, we clarify the following establishments according to the numerical results calculated with finite element method for simple structural model of steel member subjected to relatively high axial force. �9 Strength function which correspond to yield function in plasticity theory �9 Plastic potential which define a condition of plastic flow ~ Hardening and softening rule which define a movement of strength function of post yielding and

post buckling

167

168 S. Motoyui and T. Ohtsuka

MOVEMENT OF STRENGTH SURFACE

Analytical model

In this section, we consider previous establishments by the material and geometrical nonlinear analysis with finite element method. Table 1 shows measurement of model, and Table 2 shows material properties. Stress-strain relationship is elastic-perfectly plastic material. Analytical model shown in Fig. 1. We calculate in two kind of loading, one is that P,, and Pv are loading in the ratio of constant ( N/Q = 3,10,30 ), other is that P, is loading constantly ( N = 0.4Ny ) and P,, is loading variably.

TABLE 1 LIST OF MODEL: H-200X150X6X9

L b/tf d/t w A Aw I Ny M v (mm) (1) (1) (mm 2) (mm 2) (mm') (MN) (kN- m)

1000 8.3 33.3 3900 1200 3.1 x 107 1.5145 128.15

TABLE 2 MECHANICAL PROPERTIES

Cry E v G 6y (MPa) (GPa) (1) (GPa) (%)

388.34 206 0.3 79.2 0.1886

Strength function Figure 1: Analytical model

Axial displacement u and rotation angle at fixed end 0 are given by Eqn. 1, and axial force n, shear force q and fixed end moment m are given by Eqn. 2.

~-~/(L+8~)~+(8.)~-,~. o--8.1L (I) n = N cos 0 - Q sin 0, q = N sin 0 + Q cos 0, m = M (2)

where 8,,, 8v are horizontal and vertical displacement at free end, L is the member length, shown in Fig. 1. Then the relationship ~ and m for each loading pattern is shown in Fig. 2, in which ~=n/Ny ,Nyis the fully plastic axial force capacity, m=m/Mp,Mpis the full plastic moment. The initial full yield surface a~ in Fig. 1 is expressed as:

- ~ + ~1~1-1 - 0 Zone I (3)

= ~ = + 1 ~ 1 - 1 = 0 Zone II (4)

where N,~is the fully plastic force capacity of web, Mp:is the full plastic moment of flange, cr, r are constants obtained from N,~ and M p:.

Figure 2: ~ - ~ interaction curve

Generalized Plastic Hh~ge Mode l f o r the Collapse Behavior o f Steel Frames

Progress o f plastic displacement

169

Each displacement u, 0 are divided in terms of the elastic displacement and plastic one as follow:

u=u e +u p, 0=0 ~ +0 p (5)

where raising index e ,pare expressed elastic and plastic component respectively, and ue,oeare obtained as:

nL mL m u e = m 0 ~ = + ~ ( 6 )

EA ' 3 EI GAw L

where E, G , / , A and A, are the Young's modulus, shear modulus, moment of inertia, section area and web area respectively. However, shear deformation behaves elastically. Considering energy dimensional generalized plastic displacements ~p and Op are defined as Eqn.7, the relationship ~-~ and O~ is shown in Fig. 3. As shown in Fig. 3, the relationship ~-p and o~ is linear. Furthermore, diagramming the vector in Fig. 3 which cross at initial full yield surface in Fig. 2, the numerical results correspond to these vectors. Then, in the post buckling, the vectors of plastic displacement cross at initial full yield surface.

uP = NyU p, O p = MpO p (7)

Figure 3: Generalized plastic displacement m

Figure 4: S - ~- p relationship

Equivalent strength parameter

According to the plastic work ratiodWp which is defined by Eqn. 8, the relationship ~ TM and equivalent strength parameter S which is defined by Eqn. 9 is shown in Fig. 4.

dWp ~ n Au p + m A@ p (8 )

_ dw,,_Aw,, S - - - _ - - (9)

d-ffp A~-p

where auP,AO p are incremental plastic axial displacement and incremental plastic rotation angle at fixed end. Awp,A~-Pare incremental plastic work and incremental generalized plastic axial displacement respectively. As shown in Fig. 4, regardless of loading types, s of each loading type in Zone I are plotted in the same figure according to gP. Furthermore, as shown in Fig. 2, the points ( s =0.95,0.9,0.85,0.8) are plotted ( o A s . ) for each loading type and linked that points for each s , the strength surface is moving parallel to the initial full yield surface in Zone I.

Therefore, associate flow rule in plasticity theory can be applied in yielding and post buckling location. We assume that the shape of strength surface is equal to the full yield surface. And it is considered that plastic potential which define a condition of plastic flow is equal to strength surface.


GENERALIZED PLASTIC HINGE MODEL CONSIDERING LOCAL BUCKLING

Precondition

The development of plastic displacements conforms to associate flow rule. In this paper, we consider the strength surface for Zone I in Fig. 2 which is moving parallel to the initial full yield surface according to equivalent strength parameter. The relationship equivalent strength parameter and equivalent plastic displacement parameter is obtained from the results calculated with finite element method. And the relationship hysteresis characteristic under monotonic loading and that under cyclic loading is modeled by Kato and Akiyama (1973). However, structural member behaves without shear yielding and shear buckling.

Evaluate plastic and damage progress

Considering strength decrease governed by local buckling, the strength function for Zone I defined in Eqn. 3 for plus and minus ff is rewritten as follows:

~(~,m,~): I~1 +,lml- g : o (10)

Assuming associate flow rule, the incremental generalized plastic displacement vector A~P can be expressed as:

where aa~/~=~/lffl= v,8~/dm=~/Iml= ~, and a2p is energy dimensional incremental equivalent plastic displacement parameter, is condition OnA2p >__ 0, ~0_ 0, A2p~0 - 0 and A2pzx~o - o.

Then, we lead nodal displacement, nodal force and tangent stiffness matrix by using return mapping algorithm, M. Oritz and J.C.Simo (1992). Fig. 5 shows the properties of a element with plastic hinge at its two ends.

The displacement vector, its elastic vector and generalized plastic displacement vector at time t + At are 1+~' u ,'+~'u e and '+~'~P respectively. If we know plastic displacement vector'u p and equivalent plastic displacement parameter'2pat timet, elastic displacement vector'+~'ueand equivalent plastic parameter .... 2p are expressed as follows:

'+AtAp ='2p + A2p

(12)

(13)

where At is incremental time. Firstly, we try to obtain a trial force vector 'n~ according to freezing incremental plastic displacement during At.

Figure 5: Nodal displacement and nodal force

Genera l i z ed Plas t ic H i n g e M o d e l f o r the Col lapse Behav ior o f S t ee l F r a m e s

Elastic predictor Au p = O, trialgle t+Atl, l tl, ip , t .... N = K e tnal ue

AA v = 0 2,p = )tp = 'ri~t/l v

171

(14)

where elastic stiffness matrix K, is given as follows:

-~.. 0 o - k . . o o

kqq kqm 0 -kqq kqm kn n EA 12EI 1 EI (4 +y) kii 0 - kqm kii = T ' kqq-- L3 (l+r)' k,, = - - ~ L (1+7)

Ke= k.. 0 o SYM. kqq - kq~ k q,, 6E1 1 E1 (2- Z) 12E1 L J - L O+r) , r= ' L (1+7") GA.,

kii

If plastic displacement don't develop during At, 'ri~N and 'ri'S obtained from Eqn. 14 satisfy Eqn. 15. That is, when'ri"tN and 'ri"tg; don't satisfy Eqn.15, plastic displacement develop, we evaluate development of plastic displacement and correct trial force.

trial (~9( trial--l.l, trial~m, trial-~t_~ j <~ O

Plastic corrector '+~'ue='ri~'U ~ - - ~ P , Aid p = A2,pP-1t~}, ' +~ 'N = Ke(trialtl e -All p) (16)

'+~';tp , '+~'g ~('ri~ = t~,p + AA.p = 2p + A/].p )

(15)

These correct forces at time t + At should conform the strength function, therefore

'+A' q)('+A'~,'+*'~,'+~'S) = 0 (17)

This equation is nonlinear for A2~ so that we solve this by Newton method. To put it concretely, since the values of iteration step k are, Eqn. 17 can be expressed for node i and j at iteration step k + 1 as:

,ri~tn _k,~ (k) +k,,, (k) ' ~ vi (X')A)~'Pi Ny Vj (k) A~.pj tri~Zmi - k~i (~) ' (k)A/]'pi --~pk~J (k) l.t j (k)A2p j

(k+l) ~/ = +'C - Si ( .... l A pi + ( k) mApi ) Ny

,ri, t +k,~ (k) V~ (k)A2,v; -k,,,, (1,) n j N y -~y v j (k) AA.pj

Mp

,ri~t _ k o. (k) i u (k)AAp i _ k~ (k) m ) --CT--. i /-I ) ( k ) AA'vj Mv

(k+l) ~j Ny M p

(18) Considering to the first order term ofTaylor 's series ofEqn. 18 for SAip which is a variation OfAA, p

(k+l) ~, =(k) ~. _ a,; (k)6&,Zp, - a,j (k) 6A2pj (19)

(k+l) q) =(k) q)j _aii (k)gA2"pi -- air (k)gA2pj

where _ k~ +r: k~ + ((k)Api , k,~ v: ku k,, v: kq k,~ +v: k;~

Ny Mp Ny Mp Ny Mp Ny Mp

172 S. M o t o y u i a n d T. O h t s u k a

Equating Right-hand of Eqn. 19 with zero, then '"'8A2,,i and '"'"At,,, are obtained as follows:

where

/,,),s,,,x,,, = p , , In ~ _ p,~/',) aS, /',),SAX,,j = _p,, I,,)~ +p,/ , , ) r

(k+,) At,,i = (k)A2,,, + (k)6At ,,, (2 0 ) (k+,)/~,,j =(k)A~, N +(k)6A]~,,, j

_ ~i j a j i l~,jj p , = a,, , p , j_ , p . ,= , p , .= ~ i i l~, jj -- t~, O. ~ j i ~ i i ~ jj -- ~ ij ~ j i l~ H l~ jj -- ~ ij ~ j i ~ i i ~ jj -- ~ O. l~ j #

And elastic displacement vector and force vector of iteration step k + 1 are given in Eqn. 21, then we repeat that until accuracy reach a established value.

T a n g e n t s t i f f n e s s m a t r i x

(k+,) u" =(~)u" - (k) 6At, P -' { (k) (k)/xJ'v~ (k+l)N = K.(k+')u" (21)

We will have tangent stiffness matrix as follow. Rewriting elastic displacement as shown in Eqn. 16-a to the mention of rate, we have

du" = d f tri~ u " - k

then the rate of nodal force vector is expressed as follow:

trial v e 0

'ri:' " 0 (23) d N : K e d l l L e - ' g e d t r i a l u ; [ i

0

L {'o'o; /,,/M,

Beside, conforming to the rule as shown in Eqn. 24 during plastic flow.

a ~ = ~__a~ + ~___am + ~__a~ : o drd cGm OS

Then Eqn. 25 is given from Eqn. 23 and 24.

'ria'ue [V,/oNY 0

I 1 ( j ) t'Va'v~ : " d N Z - d S 2 __ i "Lei d - d Z ~ p i ~ . ~ l i / M p - d A ~ p j - H i d A ~ p i =0

' '~~ [~,,/M,

"~ [~/o N o

�9 dNj - dSj _ dA2v ~ v _ dA&z - Hj dA& z = 0

t(o-:-:-:-~/8-~)./ 1 c~ 'ri'u; v.j .,,

where dS={;;;}, g. =[~:7

(24)

(25)

Generalized Plastic Hinge Model for the Collapse Behavior of Steel Frames 173

Rearranging Eqn. 25, we obtain the following equations:

aii(k)6A~,pi +aO.(k)6A,~,pj = "Leidtriatll e , aji(k)6A~,p# +ajj(k)6A~,pj = "Lejdtrialll e (26)

t~.. IM, ta. IM, therefore, we can solve Eqn. 26 for dA2p~ and dA2pj :

[ ] [ ] dSXp, = fls, .L.,-fl,j "Les d "~ , dA2, : -fl,, "L., +fl,, "L u d""u (27) {s'.IM, J t ' . l M. {I'.IMpJ t~'.IM,

Substituting Eqn. 27 into Eqn. 23 and tangent stiffness matrix is given as follow:

where

[fljjVio/NY [Vi/oNY [-flJioi/NY 1 [ 0 0 aN= x. ~"%,-~ p..~,./M.. ~-p..../M,.L I -fl~vOINy .K, dt,~Q,u~l,61Mp_ .K, dt,~,u~ 0

t-n..~.lM, t P..s,.IM, J t~./M, :[".- - | +P,,".:, | .... "

/ . : { . . / u . o . . IM. o o o}'. : . : { o o o ".1". o . . IM.}"

(28)

Comparison the numerical results

Fig. 6 compares load-displacement curve subjected to static loading given by the proposed model and the finite element method in which 0p is an elastic rotation angle corresponding to M~. It can be seen that two solutions agree well regardless of loading types. What is more important is that the relationship ~ and ;tv using in Fig. 6 is the same one for each loading type.

Figure 6: Load-displacement curve (static)

In dynamic loading, using Newmark solution scheme and the Newmark's parameters/7 and 7" taken as 0.25 and 0.5, without considering effect of damping. A mass point m m =O.1046[MN.s2/m]is added to

the free end, and mass density p= 7.81xlO-9[N.s2/mm4]. Firstly, only Pvis loading at the almost static

rate until Pv is equal to 0.4Ny. Secondly, P~ keeps constant, P,, is cyclic loading as shown in Fig. 7 in

which Qpc = Mpc/L where Mvc is the full plastic moment in the present of axial force, P,, and Opc is the

elastic rotation angle corresponding to M v~, and T is the elastic first natural period of this structure. In

this situation, time increment At is 1.286 x 10-3[sec]. The vertical displacement and restoring force time


history are shown in Fig. 8 and Fig. 9. Fig. 10 shows the hysteresis characteristic under dynamic loading given by the proposed model and finite element method. Though external vertical force Pv is constant, the vertical restoring force N is variable, as shown in Fig. 9. Involving this, the hysteresis characteristic is not smooth like in static but waving, as shown in Fig. 10. As shown in Fig. 8,9 and 10, the results given by the proposed model correspond to the results given by finite element method.

Figure 7: Loading program

Figure 8: Vertical displacement

Figure 10: Load-displacement curve (dynamic)

Figure 9: Vertical force

CONCLUSIONS

We clarify the establishment which is to give the effect of local buckling based on plasticity theory according to the numerical results calculated with finite element method for simple structural model of steel member subjected to relatively high axial force ratio. Then according to these establishments, we propose a generalized plastic hinge model which takes local buckling into account, and we confirmed the proposed model can express the effect of local buckling by means of comparing with the results calculated with finite element method.

REFERENCES

Ohi K., Takahashi K. and Meng L.H. (1991). Multi-Spring Joint Model for Inelastic Behavior of Steel members with Local Buckling. Bulletin of Earthquake Resistant Structure Research Center, Institute of lndustrial Science, Univ. of Tokyo 24:March, 105-114 Yoda K., Kurobane Y., Ogawa K. and Imai K. (1991). Hysteretic Behavior and Earthquake Resistant Design of Single Story Building Frames with Thin-Walled Welded I-Sections. Journal of Struct. Constr. Engng, AIJ 424:June, 79-89 (in Japanese) Yamada S. and Akiyama H. (1996). Inelastic Response Analysis of Multi-Story Frames Based on the Realistic Behaviors of Members Proc. ICASS'96 1, 159-164 Kato B. and Akiyama H. (1973). Theoretical Prediction of the Load-Deflexion Relationship of Steel Members and Frames IABSE Symposium on Resistance and Ultimate Deformability of Structures Acted on by Well Defined Repeated Loads, 23-28 Oritz M. and Simo J.C. (!986). An Analysis of a New Class of Integration Algorithms for Elastoplastic Constitutive Relations. Int. J. Num. Mech. 23:3, 353-366

ADVANCED INELASTIC ANALYSIS OF SPATIAL STRUCTURES

J Y Richard Liew, H Chen and L K Tang

Department of Civil Engineering, National University of Singapore 10 Kent Ridge Crescent, Singapore 119260

ABSTRACT

This paper describes the methodology of an advanced analysis program for studying the large- displacement inelastic behaviour of steel frame structures. A brief review of the advanced inelastic analysis theory is provided, placing emphasis on a two-surface plastic hinge model for steel beam- columns, a thin-walled beam-column model for core-walls, and a four-parameter power model for semi-rigid connections. Numerical examples are provided to illustrate the acceptability of the use of the inelastic models in predicting the ultimate strength and inelastic behaviours of spatial frameworks.

INTRODUCTION

With the advancement of computer technology in the recent years, research works are currently in full swing to develop the advanced inelastic analysis methods and computer packages which can sufficiently represent the behavioural effects associated with member primary limit states such that the separated specification member capacity checks are not required. This paper presents the nonlinear inelastic models that can be used for analysing space frame structures within the context of advanced inelastic analysis. In the proposed approach, each steel framing member is modelled as one beam-column element. Plastic hinges are allowed to form at the element ends and within the element length. To allow for the gradual plastification effect, a two-surface model is adopted. The initial yield surface bounds the region of elastic sectional behaviour, while the plastic strength surface defines the state of full plastification of section. Smooth transition from the initial yield surface, as the force state moves to the plastic strength surface, is assumed. Core-walls provide a major part of the bending and torsional resistance in a building structure. They are modelled by thin-walled frame elements. The centre line of the core-wall is located on the shear centre axis. Any significant twisting action should be analysed to include both warping and torsional effects. Beam-to-column and beam-to-core-wall connections are modelled as rotational spring elements having the moment-rotation relationship described by the four-parameter power model. At last, the advanced analysis program is applied to investigate the collapse of a roof truss system, and perform nonlinear inelastic analysis of a core-braced frame with semi-rigid connections.

175

176 J . Y . R . L iew et al.

ADVANCED PLASTIC HINGE FORMULATION The basic feature of the proposed plastic hinge formulation is to use one beam-column element per member to model the nonlinear inelastic effects of steel beam-columns. The element stiffness matrix is derived from the virtual work equation based on the updated Lagrangian formulation. The elastic coupling effects between axial, flexural and torsional displacements are considered so that the proposed element can be used to predict the axial-torsional and lateral-torsional instabilities. By using the stability interpolation functions for the transverse displacements, the elastic flexural buckling loads of columns and frames can be predicted by modelling each physical member as one element. The member bowing effect and initial out-of-straightness are also considered so that the nonlinear behaviour of frame structures can be captured more accurately (Liew et al., 1999).

Material non-linear behaviour is considered by introducing plastic hinges at the element ends and within the element length if the sectional forces exceed the plastic criterion, which is expressed by an interaction function. If a plastic hinge is formed within the element length, the element is divided into two sub-elements at the plastic hinge location. The internal plastic hinge is modelled by an end hinge at one of the sub-element. The stiffness matrices for the two sub-elements are determined. The inelastic stiffness properties of the original element are obtained by static condensation of the "extra" node at the location of the internal plastic hinge. To allow for gradual plastification effect, the bounding surface theory in force space is adopted. Two interaction surfaces representing the state of the stress resultants on a section are employed (Liew and Tang, 1998). The yield surface bounds the region of elastic al behaviour, while the bounding surface defines the state of full plastification of the section. The bounding surface encloses the sectional force state and the yield surface at any stage during the plastic process. To avoid intersection of the surfaces, the yield and bounding surfaces are given the same shape. When the section is loaded, the force point travels through the elastic region and contacts the yield surface, which is given by

1-'y = f ( S - [ 3 / = f ( P-j31 QY-[32 Qz-~3 Mx_.~_..~4 My-J35 M z - [ 3 6 / _ l = 0 (1) ~ZySp ) ZyPy ' ZyQpy 'zyQp z ' ZyMpx ' ZyMpy ' ZyMpz

in which P, Qy, Qz, Mx, My, Mz are the sectional forces, Py, Qpy, Qpz, Mpx, Mpy, Mpz are the plastic capacities for each force component, j3 is the position vector of the yield surface's origo in the force space, and Zy is the yield surface size. The function Fy is defined that Fy = -1 corresponding to a

stress-free section, while Fy < 0 corresponds to a initial yielding or any subsequent yielding state. When the further loading takes place, the yield surface starts to translate so that the current force state remains on it during subsequent loading. For the advanced plastic hinge analysis, the plastic hardening parameter and transition parameter, which are specific for each force component, are crucial for the elasto-plastic behaviour of the element. They may be determined from experiments or numerical calibrations, and the details of such calibration work and further verification studies are demonstrated in Liew and Tang (1998).

MODELLING OF CORE-WALLS Core-walls are modelled by the thin-walled beam-column element for their proportional similarity to Vlasov's thin-walled beams and for their computational efficiency in the inelastic analysis (Liew et al., 1998). As shown in Fig. 1, the thin-walled beam-column element has an additional warping degree-of-freedom over the beam-column element at each end. The local coordinate is chosen: axis x lies on the shear centre axis, and y and z axes parallel to the principal y and ~, axes. Some force

and displacement components are referred to the shear centre, whereas the remaining ones are referred to the centroid of the section. However, before the element stiffness matrices are transformed into the global coordinate, it is necessary that all the forces and displacements are referred to a single point. The shear centre can be selected as the reference point. The detailed derivation for the elastic and geometric matrices of the thin-walled beam-column element is given

Advanced Inelastic Analysis of Spatial Structures 177

by Liew et al. (1997). Because the height-to-width ratio of core-walls is large and the axial force respective to the sectional area is small in practical building frames, material nonlinearity of core- walls is considered approximately, assuming that the plastic strength is controlled by the bending action only. The locations of the shear centre and the centroid of cross-section are assumed not to change due to the inelastic effects.

MODELLING OF SEMI-RIGID CONNECTIONS Beam-to-column connections can be modelled as rotational spring elements in the nonlinear analysis of semi-rigid frames (Hsieh, 1990; Chen et al., 1996). Many connection models have been proposed to describe the moment-rotation relationships of connections used in building steelworks (Liew et al., 1993). The present work adopts a four-parameter power model to represent the moment-rotation relationship of typical beam-to-column connections (Hsieh, 1990). The selection of this model is guided by its simplicity and robustness for representing the basic behaviour of typical connections, and for ease of implementation in the nonlinear inelastic analysis program. The four-parameter power model has the following form:

(Ke - K p ~

M - [ I + I ( K _Kp)O/Moln]/n+KpO (2)

in which I~ is the initial stiffness of connection, Kp is the strain-hardening stiffness of connection, M0 is a reference moment, and n is a shape parameter as shown in Fig. 2. The four-parameter model can easily encompass the more simple models. For examples, Eq. 2 becomes a linear model if I~ = Kp, a three-parameter power model if Kp=0, and a bilinear model when n is large.

In the structural design, it is unlikely that specific connection details will be known during the preliminary design until the structural members have been sized in the final design. Since connection flexibility will affect the structural response and therefore the required member sizes, there is a need to develop some means to account for connection behaviour during the analysis and design process before the final member sizes are selected. One solution is to use the standard connection reference curves which are based on the connection test database. An optimisation approach utilising the conjugate-gradient method is first used to find a set of parameters (M0, Ke, Kp, and n) which gives the best curve-fit to the experimental connection response data. The moment- rotation curves are then normalised with respect to the nominal connection capacity Mn, which equals to the moment at a rotation of 0.02 radian as shown in Fig. 2. The standard reference curve is calibrated by fitting a curve through the average of the normalised curves. The average values of M'=M/Mn, K'e=Ke/Mn, K'p=Kp/Mn and n in the standard reference curves for nine types of commonly used connections subjected to in-plane moment have been established (Hsieh, 1990). Then, for the analysis of the overall structure, only the connection type and nominal connection capacity would need to be defined without unnecessary concern over the final connection details. Based on the connection test database, a survey of the ratio of Mn/Mpb for different types of connections have been carried out, in which Mpb is the plastic bending capacity of beam where the semi-rigid connection is located. The standard reference curve parameters and values of Mn/Mpb for several types of connections are listed in table 1.

COLLAPSE ANALYSIS OF A ROOF TRUSS SYSTEM An accident took place when a roof truss system was assembled on site. Advanced analysis was carried out to investigate the cause of collapse. The roof truss system includes seven trusses connected by eight purlins at their top chords and its plan view is shown in Fig. 4. The span and height of each truss are L = 35.05m and h = 2.45m respectively, as shown in Fig. 5. All trusses are restrained from the displacement at the supports of bottom chord. The truss at axis 1 is laterally restrained at the mid-span of the top chord, while the other trusses are connected by purlins only.

178 J . Y . R . Liew et al.

The truss at axis 1 consists of initial out-of-straightness of double-curvature shape at the top chord, with maximum magnitude of (0.5L)/500 = L/1000 =.35 mm. The top chords of other trusses (from axes 2 to 7) consist of single-curvature initial out-of-straightness with a maximum magnitude L/500 = 70 mm at the mid-length. The lateral restraint and initial out-of-straightness of the top chords of all trusses are illustrated in Fig. 4. The supporting ends of all trusses are constrained from displacements in all directions and out-of-plane rotation, except that the rotational restraint of support A, whose position is shown in Fig. 4, is released to simulate a careless mistake made during the installation of the trusses.

The truss system is analysed for two loading conditions. Firstly the system is assumed to be subjected to only vertical load, so that the safety factor for the overall system under gravity can be evaluated. The vertical load at every truss includes (1) its self-weight, (2) eight concentrated load of 602.4N each on the connection with purlins to simulate the purlin weight, and (3) two concentrated load of 2530N each at mid-span of the truss, one at the top chord and the other at the bottom chord, to simulate the weight of Gusset plates and connections. This can be seen in Fig. 5. Subsequently the system is studied under full self-weight plus horizontal surged force created by the crane. A horizontal point load is applied at nodes B and C on the top chords of the truss at axis 7. Nodes B and C are located at nearly one third of the truss span, as shown in Figs. 4 and 5. This is to evaluate the horizontal surged forces required to cause the structural failure.

A separate analysis is also carried to evaluate the resistance of individual truss under two load situations: (1) gravity only, and (2) both the gravity and the horizontal surged force created by the crane. For the truss at axis 1, which has a lateral restraint at mid-span, its resistance is 1.49 times the gravity or 1.0 times the gravity plus a horizontal load, supplied at nodes B and C, of 29.5kN each. In contrast, for the truss at axis 2, without the lateral restraint, its capacity is only 0.38 times the total gravity. In other words, during the erection, the individual truss cannot resist its self-weight if lateral restraint is not provided. Since the restrained truss at axis 1 is required to provide the lateral restraint to the other six trusses by purlins, the maximum resistance of the truss is expected to be less than when it is acting alone.

When the gravity is applied progressively, the truss system collapse at the load factor 1.15. Fig. 6 shows the plots of applied load ratio versus lateral displacement at node B. The deformed shape of the truss system at collapse is shown in Fig. 7. This safety factor appears to be very small for the safe erection of steel structures. To investigate the effect of crane surge, the full self-weight of the structure is applied first, followed by two horizontal surged forces each at nodes B and C. Fig. 8 shows the horizontal load - displacement plots at node B for the truss at axis 1. The total maximum horizontal force that can be applied to cause the collapse of the overall truss system is 9.6 kN. The deformed shape of the trusses at collapse is shown in Fig. 9. This lateral load resistance is considered to be too small for practical viewpoint. Hence, a single point bracing at the mid-length of truss at axis 1 is not adequate in providing lateral restraint against normal impact load due to crane surge. The analysis concludes that more lateral restraints to the compression chord are necessary for safe erection of the roof trusses.

INELASTIC ANALYSIS OF SEMI-RIGID CORE-BRACED FRAMES Figures 10 &l 1 show a 24-storey core-braced frame with storey height h = 3.658 m and total height H = 87.792 m (Liew et al., 1998). Thickness of concrete core-walls is 0.254 m. Depth of concrete lintel beam is 1.219 m. A36 steel is used for all sections. Material properties of concrete are: modulus of elasticity Ec = 23,400 N/mm 2, and compressive strength f~ = 23.4 N/mm 2. The

structure is analysed for the most critical load combination of gravity loads and wind loads that act in the Y-direction. Core-walls are mainly subjected to the bending moment about the principle ~-


axis, which is parallel to the global X-axis. The bending moment about the principle ~-axis is

small. The plastic section modulus about the principle ~. axis of the channel-shaped core-wall section is Z = 2.549 m 3. In this example, the height-to-width ratio of core-walls is 24:1. It is assumed that the plastic resistance of core-walls is dominated by the plastic bending resistance about the principle 7.-axis, Mz = 0.8Z f" = 4.8x 10 4 kNm, only. The plastic resistance of core-walls has

been reduced to approximately account for the tensile cracking and axial force interaction effect.

In the nonlinear inelastic analysis, each steel column is modelled as one plastic hinge beam-column element, and each beam is modelled as four beam-column elements. Core-walls are modelled as thin-walled beam-column elements. Concrete lintel beams are rigidly connected to core-walls for resisting the lateral and torsional loads. All floors are assumed to be rigid in plane to account for the diaphragm action of concrete slabs. The gravity loads, which are equivalent to a uniform floor load of 4.8 kN/m 2, are applied as concentrated loads at the beam quarter points and at core-walls of every storey. The wind loads are simulated by applying the horizontal forces in the Y-direction at every frame joints of the front elevation, and are equivalent to a uniform pressure of 0.96 kN/m 2.

Firstly, inelastic analysis is performed on rigid core-braced frame. The loads are proportionally applied until the frame collapses at a load ratio of 1.787 when plastic hinges form at the bottom and the top of core-walls in the first storey. To study the lateral resistance capacity of core-walls, inelastic analysis is performed on core-braced frame with pin-connections. In this case, the whole building relies core-walls to provide the lateral resistance only. The limit load and initial lateral stiffness of the frame with pin connections are only 36% and 21% of those of the rigid frame. Similarly, to study the lateral resistance capacity of the pure steel frameworks, the elastic modulus and the compressive strength of concrete are assigned to be very small values. The frame collapses at a load ratio of 0.654, which is similar to that of the frame with pin-connections. It is noted that the inelastic lateral deflection behaviour of steel framework is more ductile than that of the frame with pin-connection. It can found that the building frame cannot only rely on core-walls or steel frameworks to provide the lateral resistance. Core-walls and steel frameworks must act together to withstand the external loads.

Semi-rigid construction is faster and cheaper than rigid construction. For high-rise building design, service wind drift is always the main concern. In order to reduce the number of moment connections in high-rise building construction, the use of core-braced frames with semi-rigid connections may provide optimum balance between the dual objectives of buildability and functionality (Chen et al., 1996). Different types of beam-to-column and beam-to-core-wall connections in the steel frameworks are assumed to study the connection effect on the inelastic limit loads and lateral deflections of the frame. The connection properties are given in table 1. The proposed semi-rigid formulation can model the torsional and both major- and minor-axis flexibility. However, in this analysis, only the relative rotations about the major-axis of beam section are allowed at the semi-rigid connections. This is due to two reasons: (1) at present there is little experimental information on the torsional and out-of-plane behaviours of semi-rigid connection, and (2) for typical framed structures with rigid floor, the torsional and out-of-plane effects of semi-rigid connections are not significant.

Inelastic analyses are performed on core-braced frames with 'DWA', 'TSAW' and 'EEP' connections. The inelastic limit loads and load - deflection curves are shown in Fig. 12. It can been seen from table 2 that if 'EEP' connections are adopted, the load and lateral stiffness can reach to 93% and 81% of those of the rigid frame. The limit load and inelastic stiffness of frame with 'DWA' connections are only a little higher than those of the frame with pin-connections. The limit load and inelastic behaviour of the frame with 'TSAW' connections are between those of the frame

180 J.Y.R. Liew et al.

with 'EEP' connections and the frame with 'DWA' connections. It can be concluded that if proper semi-rigid connections are used, the frame can be constructed much faster and cheaper than the rigid frame, at the same time satisfying the strength and serviceability limit states.

CONCLUSIONS The basic principles of the proposed advanced inelastic analysis program have been presented. Inelastic analysis has been applied to study the roof truss system and emphasises the importance of lateral brace to assure the system's stability, which is important for the safe erection of such structure. Inelastic analyses on core-braced frame with semi-rigid connections show that construction with proper selection of connections can satisfy limit states design and achieve fastrack construction. When properly formulated and executed, the advanced analysis can be used to assess the interdependence of member and system strength and stability, the actual failure mode and the maximum strength of the overall framework, and, hence, efficient and cost-effective design solutions can be obtained. This is in line with the modem design codes such as Eructed, which allows the use of advanced analysis for designing steel structures.

REFERENCES Chen, W.F., Goto, Y., and, Liew, J.Y.R. (1996), Stability Design of Semi-Rigid Frames, John

Wiley& Sons, NY. Hsieh, S.H. (1990), Analysis of three-dimensional steel frames with semi-rigid connections,

Structural Eng. Report 90-1, School of Civil and Environmental Eng., Comell University, NY. Liew, J.Y.R., White, D.W., and Chen, W.F. (1993), Limit-states design of semi-rigid frames using

advanced analysis: Part 1: Connection modelling and classification, J. Construct. Steel Res., 26, 1-27.

Liew, J.Y.R., Chen, H., Yu, C.H., Shanmugam, N.E., and Tang, L.K. (1997), Second-order inelastic analysis of three-dimensional core-braced frames, Research Report No: CE024/97, Dept. of Civil Eng., National University of Singapore.

Liew, J.Y.R., Chen, H., Yu, C.H., and Shanmugam, N.E. (1998), Advanced inelastic analysis of thin-walled core-braced frames, Proc. of the 2nd International Conference on Thin-Walled Structures, Dec. 2-4, 1998, Singapore.

Liew, J.Y.R., Chen, H., and Shanmugam, N.E. (1999), Stability functions for second-order inelastic analysis of space frames, Proc. of 4th International Conference on Steel and Aluminium Structures, June 20-23, 1999, Espoo, Finland.

Liew, J.Y.R., and Tang, L.K. (1998), Nonlinear refined plastic hinge analysis of space frame structures, Research Report No: CE029/99, Dept. of Civil Eng., National University of Singapore.

Table 1. Parameters and Mn/M values for connections under in-plane bending moment Mo' Ke' Kp' n Mn/Mpb

Connection type M0/Mn ~ n Kp/Mn

DWA 1.03 301 5.0 1.06 TSAW 0.94 363 6.9 1.11 0.4

EEP 0.97 309 5.5 1.20 1.0 DWA: Double web-angle connection TSAW: Top- and seat-angle connections with double web angles EEP: Extended end-plate connection without column stiffeners

At the beam framing about the major-axis

of column (see Fig. 3) 0.05

At the beam framing about the minor-axis

of column (see Fig. 3) 0.025

0.2 0.5


Fig. 2 Four-parameter power model

Fig. 1 Thin-walled beam-column element

Fig. 3 Beam-to-column connections

Fig. 4 Plan view of roof truss system Fig. 5 Elevation view of truss

Fig. 6 Load-lateral displacement curve under gravity load

Fig. 7 Deformed shape of roof truss system at collapse under gravity load

182 J . Y . R . L iew et al.

Fig. 8 Horizontal load-lateral displacement curve

Fig. 9 Deformed shape of roof truss system at collapse under the horizontal

surge forces

Fig. 10 Plan view of core-braced frame

Fig. 12 Top-storey load-deflection curves

Fig. 11 Elevation view of core-braced frame: (a) at axes 1, 2, 5, 6 (b) at axes 3, 4

Table 2. Comparison of limit loads and initial lateral stiffness

Connection types Pin

connection

Limit load 36%

Initial lateral stiffness

21%

DWA 40% 30% TSAW 65% 68%

EEP 93% 81% All % values are compared with the core-

braced frame with rigid connections

STABILITY ANALYSIS OF MULTISTORY FRAMEWORK

UNDER UNIFORMLY DISTRIBUTED LOAD

Chen Haojun and Wang Jiqing

Department of Construction Engineering, Changsha Communications University

45 Chiling Road, Changsha 410076 China

ABSTRACT

Problems of overall stability in a multistory framework become significant with the increase in its height. This paper presents the stability analysis to a one-bay multistory framework under uniformly distributed load by means of continuum model. Continuum model is a substituting column converted from multistory framework. So, the analysis to multistory frame, which is an indeterminate structure, is reduced to that to a determinate one. The formula of critical load is developed by Galerkin method. The effect of the axial compressive deformation of framework column is taken into consideration.

KEYWORDS

Multistory framework, overall stability, continuum model, uniformly distributed load, critical load, substituting column.

In the analysis to a multistory framework structure, one pays more attention to analysis to internal forces of a multistory framework at vertical and horizontal loads, than to analysis to overall stability. However, the problems of overall stability in a multistory framework become significant with the increase in height. Generally, the exact stability analysis of multistory frames can be solved by finite element method. This is an extremely complex procedure, even with the help of computer. The higher the structure is, the more complicated the problem is to handle. The critical load is usually obtained by determination of effective length factor of each framework column. In this paper, the framework structure will be taken as a whole for determination of the critical load. A critical load for

183

184 C. Haojun and IV. Jiqing

a one-bay multistory framework subjected to uniformly distributed load at floor level is developed by Galerkin method. The effect of axial compressive deformation on the critical load is taken into account in following analysis.

1. BASIC ASSUMPTIONS

During the analysis, following assumptions will be used. A). The material of the structure is homogeneous, isotropic and obeys Hook's law. B). The loads are applied statically and maintain their direction during buckling. C). The structure develops small deformation and the axial deformation in the beam is negligible when the axial framework buckles. D). All stories have the same height and the structure are at least four story high. E). The structure has a rectangular net work with elements attached by rigid joints to each other. F). The stiffness (El/l) of beams is the same. G). The inflection point is on the middle of the beam when the framework buckles.

2. SUBSTITUTING COLUMN

The continuum model of multistory framework is a substituting column converted from the framework. The substituting column is obtained from the original framework (Fig. 2.1a) in several steps. First, the UDL on the beam is transferred to the columns at floor levels (Fig. 2.1b) in the form of concentrated forces (the reactions on the beams). These concentrated forces are then distributed along story height (Fig. 2.1 c), in fact along the height of the framework. The beams are cut through at inflection points (Fig. 2.1 d) and finally the columns are added up into a single substitute cantilever

(Fig. 2.1 e).

P 46464+444

p 4~4~4+4~4

P

4 F 4 F 4 ~ 4 ~ 4 1 1

(D | @

l

4 4 t ~ !4 4 4~

4) t ~ q= q l ) m q:F 4q= qx4 ' I 144 4)

t 4) 4)

1 ,' l (~) (b) (c) (d) (e)

Fig. 2.1 Continuum Model

The bending stiffness of the substituting column is the sum of the bending stiffness of columns of the framework. The load on the substituting column equals the total load on the original framework. The distributed force along the height of substituting column is converted from the uniformly

Stability Analysis of Multistory Framework 185

distributed load at floor levels. The distributed moments along the substituting column are induced from deformation of the framework during buckling. In doing so, the framework is converted into a fixed-free column on which a distributed force and a distributed moment act. It should be noted that the difference of axial compressive deformation between two framework columns makes the framework have sway. This phenomenon is not shown in substituting column. Comparing actual column with substituting column, it is known that the restraint moment acting at floor level due to beam bending makes the column double-curvature between two beams for an actual framework. But for a substituting column, the restraint moment due to beam bending is distributed along the substituting column and does not make the substituting column double-curvature.

3. CRITICAL LOAD OF A ONE-BAY MULTISTORY FRAMEWORK

There is a one-bay multistory framework as shown in Fig. 3.1. The stiffness of beam of each floor level is Eblb except the top one of Eblb/2; and the stiffness of framework columns is Eclc. There are uniformly distributed loads at each floor level. According to preceding procedure, the substituting column is shown in Fig. 3. lb. When the framework buckles, it can be in equilibrium both in original configuration (undeformed configuration) and in slightly deformed configuration. Now, let us consider the equilibrium of framework in slightly deformed configuration.

EcI~/2

P I I I I I I I I I

Edj2

Edb I I I I I I I I I



E~b

EcIc /2 h

h

h

h ~y

I l l

ql I I t I

) ) m

) ~y

H=eh

(a) 0o)

Fig.3.1 Substituting Column

3.1 Distributed Moment When Framework Bends

The separated body for analysis may be taken as shown in Fig. 3.2 when the framework bends. It is cut at the middle point of beams (inflection points) and replaces with a shear force T. This shear force T can be obtained by the condition that the deformation at the middle point of beams (inflection

points) is equal to zero,

y, l TM (t/2) 3 - . - + ~ = 0 (3 .1)

2 3Ebl b

in which TM is the shear force in beams due to framework bending; l is the distance between the axes

of columns; Eb is the elastic modulus of beams; Ib is the moment of inertia of beam; Yb ~ is the first

186

derivative of framework.

C. Haojun and W. Jiqing

Eqn. 3.1 gives

12EbIb , TM = l 2 YM (3.2)

The distributed force along the framework column due to bending is

tM TM 12Ejb , (3.3) = - ~ - = h----~yM

Transfer of the shear forces at inflection points TM to the axis of the columns produces the concentrated moments acting on the column at floor level,

l 6Eblb , (3.4) M M = T M x - = ~ YM

2 l

Distribution of the concentrated moment M i along the column height leads to

mM -- MM 6Eblb ' (3.5) -----if--= hl YM

3.2 Consideration o f Axial Deformation o f Column

Shear force TM at beams makes the axial forces in two columns different. The axial force increases in TM in right column and decreases in left column. This variation of axial force causes an additional axial deformation in left and right columns. It is denoted by the sign AN. This deformation consists of two parts (Fig. 3.3). One (denoted by AN1) makes the beams bend and the other (denoted by Am) makes the columns bend. Hence,

or A~ = AN1 + AN2

Y~v = Y~vl + Y~v2 (3.6)

The bending moment at beam end due to AN1 is

12Eblb (3.7) MN1 = /2 AN1

Fig. 3.2 Separated Body When Buckling Fig. 3.3 Compressive Deformation

Letting 2AN1/I=y'N1, one obtains the distributed moment along column

Stabil i ty Analysis o f Mul t i s tory Framework

MN1 6Eblb , mN1 = ~ = - - ~ Y N 1

h lh

187

(3.8)

Variation of axial force in column due to AN~ is

TN1 = MN____A_I = _ 12Eblb l

Distribution of the force TN1 along the column leads to TNI 12EbI b ,

= - - ~ Y N 1 tN1 = h hl 2

y;,, (3.9)

(3.10)

According to Figs. 3.3 and 3.4, the compressive deformation distant to z from original O is

AN(Z)-- ~ ~(tM-~'tN1)d(dz E cAc

(3.11)

where A c is the cross-section area of column. Substitution of Eqns. 3.3 and 3.10 into Eqn. 3.11 leads to

AN(z ) = f ~ 12Eblb ( y~ -- y 'N,)d(dz EcI c (3.12)

Making use of Y'N =2AN/l, Ir=2Ac(l/2) 2, and differentiating twice, Eqn. 3.12 may be written in the form

,, 12Eblb YN = Eclrhl (Y~vl-YM) (3.13)

Integrating Eqn. 3.13 once and making use of the boundary condition, y"N(0)=0 and yN~(0)=yN2(0)=0, one obtains

" 1 2 E b l b ( Y N l _ y g 2 ) Y U - Eclrhl (3.14)

3.3 Equilibrium Differential Equation of Substituting Column

The Equilibrium differential equation is

Ec lcy" + ~ q ( y - rl)d ~ - ~m(~:)d~: = 0

It is known that m=2(mM+mN0, and making use of Eqns. 3.5 and 3.8, one obtains

(3.15)

m-- 12Eblb (Y~ - Y'~I) hl

(3.16)

Substituting Eqn. 3.16 into Eqn. 3.15, one obtains

Ec lc Y " + f q (Y - rl )d ~ - ~ 12 E b l b (Y " - Y 'N1 )dz = 0 hl (3.17)

188 C. Haojun and W. Jiqing

The bending deformation of the substituting column is

Y = YM + YJv2 (3.18)

3.4 Solution of Differential Equations

I

I

I

r i

Y

Z

Fig. 3.4 Coordinate for Calculation of Compressive Deformation

Y o Y o

Z Z Z

1 q

I

Fig. 3.5 Coordinate and Separated Body of Substituting Column

Combination of Eqns. (3.17), (3.13), (3.6) and (3.18) gives

Eclcy" + ~ q(y - ~7)dr - ~ 12Eblb (Y'M - Y N 1 ) d ~ = 0 hl (3.19a)

. 12EbIb , YN Eclrhl (Ym - Y~t ) =0 (3.19b)

Y~ = YN1 + Y~v2 (3.19c)

t p Y'= YM + YN2 (3.19d)

Arrangement of above equations and letting Kb=12Eblb/hl leads to the equilibrium differential equation

Eci~ ] " Kbq Eclcy + qz-x y

Eqn. (3.20) is solved by Galerkin method. Letting the approximate deflection curve be

(3.20)

7De y = 6 s i n ~

2H (3.21)

which satisfies the geometric and mechanic boundary conditions

Stability Analysis of Multistory Framework

y(O)= y'q): y"(O)= y"(l)=O

one obtains the Galerkin equation

f L ( y ) s i n az dz=O 2H

in which

Eclc ~ . Kbq L(y)= EclcY'V + qz- Kb - Kb ~cI~ ) y + qY ' -~l~ ~(y- rl)dr

Substituting Eqns. (3.24) and (3.21) into Eqn. (3.23) and making use of the integration

~ sin 2 az d z = H 2H 2

f z sin 2 nz dz = ( 1 I___]H 2 2H 4 + r e 2 )

f 7tz 7tz H c o s ~ s i n dz = 2H 2H n"

~ = 1 - cos ~ sin

2H 2H

H dz= m

one obtains

I ~r 14H ( r c l 2 ( 1 1--~]H2 ( Eclcl(rc ]2H 8Ec lc ~ -~ - &t ~ + rc 2 ) + 6 K b + K b ~ ) k,-~ ) -2

+~ - - - a + + a = o E~I, EcI, rc zc

Letting

F c =rc2EcI~//(2H) 2

F o =rc2Ej,.(2H) 2

and substituting these into Eqn. 3.25, one obtains

189

(3.22)

(3.23)

(3.24)

(3.25)

(qH)o. Fc + Kb + Kb Fc/Fo (3.26) = 0.279(1+ Kb/Fo)

Eqn. 3.26 is the critical load of the one-bay multistory framework according to Galerkin method by

190 C. Haojun and W. Jiqing

assuming the approximate deflection, y=Ssin('rrz/2H).

4. DISCUSSION

Consideration of Eqn. 3.26 leads to (1) If the stiffness of beams are equal to zero, that is, EbIb=0, then Kb=12Eblb/hl=0 and Eqn. 3.26

becomes

( q H ) c r = Fc _ :rt "2 E cI ~ 8.299Ec1~ 0.29----~ - 0.297 x-----------~" H -----5- = H 2 (4.1)

The exact value for Kb=0 is

7.837Ej~ ( q H ) c r = H 2

(4.2)

Comparison of Eqn. (4.1) with Eqn. (4.2) leads to the error less than 6%. (2) If the stiffness of beams tends to infinite, that is, EbIb---~ o0, then Kb-~ oo and Eqn. 3.27 becomes

(qH)~r = (F 0 + Fc)= 8.299(E~/~ + EcI r) 0.297 H 2

(4.3)

The exact value for Kb---> oo is

(qH)~ r = 7.837(Ecl c + Eclr)

H 2 (4.4)

(3) Eqn. (3.27) may be also written in the form

(qH)cr = Fc + K b + KbF ~ / F o

rO+X /Fo)

If it is taken that 3,=0.315, a more accurate value of the critical load may be obtained.

(4.5)

References

1 Zalka, K.A. and Armmer, G.S.T. (1992). Stability of Large Structure, Butterworth Heinmann Ltd. 2 Bao Shihua and Fang Ehua. (1994), Structural Design of Tall Building. Qinhua Publishing Housing,

Beijing. 3 Chen Haojun. (1996), Stability Analysis of Multistory Framework under Vertical Loading. Proceedings of International Conference on Advances in Steel Structures, 257-262. 4 Timoshenko, S.E and Gere, S.M. (1958), Theory of Elastic Stability, Chinese Science Publishing

House.

Space Structures


STUDIES ON THE M E T H O D S OF STABILITY F U N C T I O N AND FINITE E L E M E N T F O R S E C O N D - O R D E R

A N A L Y S I S OF F R A M E S T R U C T U R E S

S. L. CHAN and J.X. GU

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, CHINA

ABSTRACT

Imperfect beam-column element for second-order analysis of two- and three-dimensional frames is derived in this paper. Initial imperfection of element is restricted to a curvature in the form of a single sinusoidal half-wave. Force deformation equations and tangent stiffness matrix in Eulerian local coordinate system have been obtained using stability function method, as an extension of Oran's equations for straight element. Comparison is made between the present element and the cubic Hermit element by two numerical examples. The obtained results show accuracy and practicality of presented beam-column element.

KEYWORDS

Steel Frame, Structural analysis, Initial Imperfection, Finite element methods, Stability function method, Second-order analysis, Geometric nonlineality

INTRODUCTION

Second-order nonlinear analysis of steel frame has been studied extensively over the past few decades and is referred in modern design codes of practice such as the American Load and Resistance Factor Design (LRFD) specification (1986), the Australian Standard 4100 (1990) and the British Standard 5950 (1990). The finite element method and the method of stability function are the two main approaches.

The simplest and most typical stiffness matrix method of analysis is to extend the cubic Hermite element to the nonlinear case by inclusion of the geometric stiffness to the linear stiffness matrix to form the tangent stiffness matrix. This approach has been used by many researchers( for example, Barosum and Gallagher[ 1970], Meek and Tan[ 1984], and Chan and Kitipornchai[ 1987] ) and have been quite successful. However, the result by using a single element for each member was noted to

193

194 S.L. Chan and J.X. Gu

be somewhat different from the more accurate equilibrium curve obtained by using more elements. Using a single cubic element for each member has been demonstrated by So and Chan (1991) to contain an error of more than 20% for the simple case of a column with both ends pinned. The cause is due to the displacement function independent of the axial force, thus violating the equilibrium condition along the element. Albermani and Kitipornchai (1990) proposed an improved analysis technique to allow less elements to be used via the addition of some terms for large displacement effects. Izzuddin (1991) suggested the use of a higher order element for highly nonlinear analysis of frame. The element, although reported to be more accurate than the cubic Hermite element, does not consider the inter-dependence of the axial force and the element displacement. Recently, Chan and Zhou (1994, 1995) developed a pointwise equilibrating polynomial (PEP) element for slender frames. Their element includes initial imperfection and good results were obtained for second-order analysis using a single element to model each member.

As an exact solution of the beam-column, the method of Stability function has been widely studied (Livesley and Chandler [ 1956], Oran [ 1973], and Chen and Lui[ 1987]). The method develops the element matrix by solving the differential equilibrium equation of a beam-column under the action of axial load. Unlike the finite element approach, which assumes a displacement shape function. The accuracy of the analysis using stability function is affected only by the numerical truncating error. Although it has the disadvantage of inconsistency in stiffness expression and numerical problem when the axial force is close to zero, it enables only one beam-column element per member to capture the second-order effect. Satisfactory accuracy can generally be achieved without resorting to a fine discretization. Therefore, it can be used for analysis of structures accurately and economically. McConnel (1992) proposed force deformation equations for initially curved laterally loaded beam column, but his element was only for compression and the tangent stiffness matrix was not derived.

This paper presents an exact beam-column element allowing for second-order effect due to axial force and initial imperfection. Force deformation equations and tangent stiffness matrix in Eulerian local coordinate system have been obtained using the stability function method, as an extension of Oran's (1973) equations for straight element. Whilst all codes require the consideration of initial imperfection and the equivalent notational force is difficult to quantify, straight element may not be useful in practice. The correctness and effectiveness of presented beam-column element are demonstrated by several numerical examples.

ASSUMPTIONS

The present theory is based on the assumption of Timoshenko's beam-column. The cross-section of the element is doubly symmetric and the material is linearly elastic. The applied loads are conservative and nodal. Shear deformations and warping effects are neglected. Small strain but arbitrarily large deflections is considered. The initial shape of the element is assumed to in a half sine curve as follows,

�9 7 / ' x

V 0 - - V m o sm ~ (1) L

in which v0 is the lateral initial imperfection, Vmo is the magnitude of imperfection at mid-span, x is the distance along the element longitudinal axis and L is the element length.

Stability Function and Finite Element for Second-Order Analysis

FORCE D E F O R M A T I O N EQUATIONS

195

For a given axial compressive force, the equilibrium equation along the element length can be expressed as,

EI ~ = - dzV1 P(vo + VI) + M1 + M2 x- M~ (2) dx 2 L

in which EI is the usual flexural rigidity of the element, M1 and M2 are the nodal moments and vl is the lateral displacement induced by loads. Making use of the boundary conditions that when x=0 and x=L, v 1=0, we have,

[ j r ] M~ sin(c~-kx) L-x M2 sinkx x + q . :,rx Vl-- V s~n~z~ L --KL~ L 1-~q Vm~ (3)

Superimposing the deflection to the initial imperfection, we have the final offset of the element centroidal axis from the axis joining the two ends of the element as,

V -- VI + VO

L-x I Esin xl+• sin 'X P ' sin~) - ~ - sin~ L 1-q vm~

(4)

in which,

~ - ~ P PL 2 k = P �9 r q--Per ~ EI ' ' ----5---- (5 ,6 ,7)

Pcr is the buckling axial force parameter given by Pcr- 27 "2 EI

L 2

Differentiation Eqn. 3 with respect to x, and expressing the rotations at two ends as the nodal

d Vl . dvl rotations as, ~ x=0 = 01 ~ x=L = 02, we have,

Eli Vmo ] MI=--L- - C, Ol + c 2 0 2 +cOl---L--- ~ (8)

EII (VII Mz=- -L- - c281 + c l O 2 - c 0 (9)

Axial strain can be expressed in terms of the nodal shortening, u and the bowing due to initial imperfection and deflection as

u 1 r.vo [ ] = ~ + - - ;L 1 dv L 2 l_dx ] - -2] 'L ~X

dx (10)


Vmo - P = EA s = EA - bl (8~ + 82 )2_ bE (81- 82 )2_ bus-Z- (8~- 82)- bvv (11)

In Eqn. 8, 9 and 11, Cl, C2 and co are stability functions and bl, b2, bvs and bvv are curvature functions. They are required to be derived for the case of positive, zero and negative values of axial force parameter, q. in which, Cl, c2, bl, bE are correspondent to the terms by Oran (1973). The term co, can be expressed in terms of q, as,

za/r r - - for compression, q>0 (12)

c ~ ( 1 - q ) ( 1 - c o s r '

c 0 = 0, for no axial force, q=0 (13)

Co = - nqgs inhg/ , for tension, q<0 (14) (1 - q)(cosh ~ - 1)

bvs and bvv, can be expressed in terms of q, el, C2 and co as follows,

bvs= c l - c ~ + c2co (15) n"(1 - q)2 2(c~ + c2)(c~ - c2)

BVV = 2

z2q(2-)q) + 2Co )2 + C2Co )2 (16) 4(1 - q n'(1 - q 2(Cl + c2)(cl - c2

the axial force parameter, q, can be written as,

~2Fu )2 )2 VmO ] q = ~ - L ~ - b ~ ( O l + 0 2 -b2(0,-02 - b v s - - ~ ( O ~ - O 2 ) - b v v ( ~ ) 2 (17)

in which ~. is the slendemess ratio given by A= L/~I//A .

TANGENT STIFFNESS MATRIX

To complete the procedure for the Newton-Raphson type of incremental-iterative method, the tangent stiffness is required to formulate for the prediction of displacement increment subjected to an incremental force. Defining [Fi] and [ui] as the basic nodal variables at two ends of an element, we have,

[F] = [M,, M2, p]T (18)

[U] -" [01, 02, U] T (19)

The tangent stiffness equation for the incremental forces and displacements can then be written as,

Stability Function and Finite Element for Second-Order Analysis

[AF] =[ke][Au]

in which the element tangent stiffness matrix in local coordinate system is obtained from,

OFi OFi Oq kij = + ~ ~

Ouj c3q Ouj

Operating Eqn. 21, we have the following entries for the tangent stiffness matrix,

0 q _ Gl 0 q _ G2 0q 1 �9 o _ . . ' 2 ' 001 x2H 0 0 2 X H 0u LH

in which,

197

(20)

(21)

(22,23,24)

t t t VmO t V m 0 Gl = Cl 0~ + c2 02 + Co ~ ; G2 = c2'01 + Cl'02-co

L L (25, 26)

2 = 7 ~ + b 1 , ( 0 1 + 0 2 ) 2 , )2 , V m o / o , VmO 2 H ~ + bz (0l- 02 "+ bvs "-~\U1- 02) "+ b,,v (--L--) (27)

The resulting tangent stiffness matrix about a principal axis can be determined as,

EI [ke ] = ---L--

G12 G1G 2 GI C l + 2 H C2 + 2 H rt rt LH

G1G z G2 2 G 2 C 2 + C 1 +

rc2H ~:2H LH 2 G l G 2 71;

LH LH L2H _

(28)

Eqn.28 can be very easily extended to three-dimensional space by repeating the process for the other principal axis. The element-stiffness matrix, [ke] in Eqn. 28 will than be 6 by 6. The tangent stiffness in the local coordinate system can be evaluated as,

[kE] = [T][ke ][T ]T + [N] (29)

in which [T] is the internal to external transformation matrix relating the six independent internal force and moments to the external 12 forces and moments, [ke] is the element tangent stiffness matrix derived above and [N] is the matrix allowing for work done due to nodal displacements and initial stress (Ho and Chan, 1991). The complete element stiffness matrix in global coordinate system, [kG], can finally be determined as,

[kG] = [L][kE][L ]T (30)

in which [L] is the standard local to global transformation matrix (Gere and Weaver,1965).

198

NUMERICAL EXAMPLES

S.L. Chan and J.X. Gu

The derived element stiffness matrix is incorporated into the computer program NAF-NIDA (Chan, 1999). In smaller scope where the axial force is close to zero, the stability functions are expressed by interpolation in order to avoid numerical instability. The bucking behavior of column with both ends fixed is used to verify newly presented element. Fig. 1 shows the response of the column due to axial load and different value of initial imperfection. The load versus axial shortening curves for column obtained by the present single element and by 8 cubic straight elements are very close. The error arises since a smoothly curved member is replaced by eight segments.

4.0 0.001

rib3.0

1 present element o" ,-" 2.0 . . . . . . 8 cubic elements o o

o ~ P - - ~ ~[ " ~ P .~ 1.0

E=le7, 1=0.8333, A= I , L=100 (units: Ib, inch)

0.0 I t i

0.0 0.5 1.0 1.5 2.0

Axial shortening, u (in)

F i gu re 1. B u c k l i n g A n a l y s i s o f f i xed - f i xed C o l u m n

The second example is a 90 member hexagonal shallow dome, and its dimensions and properties are shown in Fig.2. Members with initial imperfections of various magnitudes and in the direction of the deflection caused by the external loads are assumed and the dome is analyzed. Their load deflection curves for these imperfections are plotted in Fig.2. In all cases, only a single element is used to model a member. It can be seen in the Figure that the cubic element over estimates considerably the buckling load of the structure than the presented element due to member under high axial force. Another observation gained from the analysis is that imperfection affects the buckling loads of the structure. When a member has a larger initial imperfection, the buckling load of the complete structure is reduced significantly. This observation cannot be found in bifurcation type of analysis.

CONCLUSION

Methods for analyzing large deflections and stability of frame structures in the past have been based on either the finite element approach or the stability function. The cubic finite element is inaccurate when a single element is used to model member under high axial load. The stability function is assumed straight in previous work. The exact stiffness matrix of an imperfect member under large

Stability Function and Finite Element for Second-Order Analysis 199

29.0~ 8 6 . 9 ~ ~ / ~ - - - . . ,58., .... ~ ~ ~ .

624

7200

Unit: mm �9 Loaded node

E = 1 . 9 5 e 5 N / m m 2 I y = 1 . 4 4 e 4 m m 4

G = 0 . 8 0 e 5 N / m m 2 I z = 1 . 4 4 e 4 m m 4

A = 1 4 2 . 3 m m 2 J = 2 . 8 9 e 4 1TIn'I 4

1500

Z "-" 1000 EL

-6 O ._1

500

Vmo/L =

-*- 0.0 cubic element - - 0.0 / '*- .001 present element

.005 J -*- .010"

I I I I

0 -10 -20 -30 -40 -50

D isp lacement , v (mm)

Figure 2. Buckling Analysis of Hexagonal Shallow Dome

axial force is derived in this paper and incorporated into a second-order analysis computer program NIDA for analysis of skeletal structures. The element is accurate even when the axial force is four times the Euler's buckling load, which refers to the extreme case of the buckling of a column with both ends fixed in direction and in rotation. Whilst all practical member possess initial imperfection, the derived element will be of great practical use.

ACKNOWLEDGEMENT

The authors are thankful to the financial support by The Research Grant Council, Hong Kong SAR Government under the project "Analysis and design of steel frames allowing for beam warping and lateral-torsional buckling (B-Q233)"

REFERENCES

AISC, (1986). Load and Resistance Factor Design Specification for Structural Steel Buildings, AISC, Chicago, IL, U.S.A. A1-Bermani and Kitipornchai, S. (1990). Nonlinear analysis of thin-walled structures using least element/member, Journal of Structural Engineering, ASCE, 116:1, 215-234. Australian Standards (1990). AS4100-1990, Steel Structures, Standards Association of Australia, Sydney. Barosum, R.S., and Gallagher, R.H. (1970). Finite element analysis of torsional-flexural stability problems. International Journal for Numerical Methods in Engineering, 2, 335-52. BS5950 (1990). Structural Use of steelwork in building, British Standard Institutions.


Chan, S.L., et al. (1999). NAF-NIDA: Non-linear Integrated Design and Analysis of frames, User's Manual, 2nd Ed., The Hong Kong Polytechnic Univ., Hong Kong. Chan, S.L., and Kitipornchai, S. (1987). Geometric nonlinear analysis of asymmetric thin-walled beam-columns, Engrg. Struct., 9:4, 243-254. Chan, S.L., and Zhou, Z.H. (1995). Second-order elastic analysis of frames using single imperfect element per member. J. Struct. Engrg., ASCE, 121:6, 939-945. Gere, J.M., and Weaver, W.J. (1965). Analysis offramed structures, Van Nostrand Reinhold, New York. Ho, W.G.M., and Chan, S.L. (1991). Vibrational and bifurcation analysis of flexibly connected steel frames, J. Struct. Engrg., ASCE, 11'7:8, 2299-319. Izzuddin, B. A. (1991). Nonlinear dynamic analysis of framed structures, Ph.D. thesis, Imperial College, London, England. Liversley, R. K. and Chandler, D. B., Stability Functions for Sturctural Frameworks, Manchester University Press, Manchester, 1956. McConnel, R.K. (1992). Force deformation equations for initially curved laterally loaded beam column, J. Engrg. Mech., ASCE, 118:7, 1287-1302. Meek, J.L., and Tan, H.S. (1984). Geometrically nonlinear analysis of space frames by an incremental iterative technique, Computer Methods in Appl. Mech. and Engrg., 47, 261-282. Oran, C. (1973a). Tangent stiffness in plane frames. J. Struct. Div., ASCE, 99:ST6, 973-985. So, A.K.W., and Chan, S. L. (1991). Buckling and geometrically nonlinear analysis of frames using one element/member, J. Construct. Steel Research, 20, 271-289. Timoshenko, S.P., and Gere, J.M. (1961). Theory of elastic stability, 2nd Ed., McGraw-Hill Book Co., Inc., New York.

DYNAMIC STABILITY OF SINGLE LAYER RETICULATED DOME UNDER STEP LOAD

Ce Wang 1 and Shizhao Shen 2

1Department of Civil Engineering, Tsinghua University, Beijing, 100084 2Harbin University of Civil Engineering and Architecture, 150008

ABSTRACT

The present paper is concerned with dynamic stability of single layer reticulated domes. The updated Lagrangian formulation is employed to develop three dimensional beam elements nonlinear analysis which includes joints large displacements, large rotations and nonlinear material constitutive relation. Dynamic stability of latticed domes under step load are studied through various parameters such as span, rise-span ratio, elastic or elastic-plastic constitutive relation, including damping and without damping. The influence factors of material non-linearity, damping, initial geometry imperfection and initial static load for structure dynamic stability is analyzed. The simplified dynamic critical load calculating method is also suggested.

KEYWORDS: Dynamic Stability, Dynamic Stability Critical Load, Nonlinear Analysis, Reticulated Dome, Step Load, Updated Lagrangian Formulation

INTRODUCTION

Single layer reticulated dome is imperfection sensitive structure which may lose its stability under strong earthquake action and strong wind load. There are several methods that are adopted by numerous investigators to solve latticed dome static stability but few concerned with dynamic stability. Dynamic stability means structural stability under dynamic disturbance which is a research field closely related to stability theory and vibration theory. In the paper members of reticulated dome assumed as three dimensional beam element, the non-linearity of latticed domes include geometric non-linearity caused by joint large displacement, large rotation and nonlinear material constitutive relation. Nonlinear dynamic finite element method is the basis of latticed dome dynamic stability analysis. According to the continuum mechanics principle, the updated

201

202 C. Wang and S. Shen

Lagrangian formulation is employed to develop three dimensional beam element geometry nonlinear analysis which include joints large displacements and large rotations (Wang(1997)). The joint large rotation is modified because it doesn't accord with law of exchange so that Euler angle which describes rigid body motion around fixed point is used to simulate large joint rotation. In material nonlinear analysis the Mises yield criterion and Prandtl-Reuss flow rule are adopted to describe elastic-plastic constitutive relation. The Newmark integration combined with Newton-Raphon equilibrium iteration are used to solve structural nonlinear vibration equation that can improve the calculation precision and numerical stability.

The first task in structure dynamic stability analysis is to determine structure dynamic stability critical load which is very time consuming while structural geometry and material non-linearity considered in each numerical integration procedure. Trial calculations have to be employed in order to exactly judge dynamic critical load which makes the work more difficulty. The critical criterion which have theory basis and convenient in practical application is very important. The equation of motion approach is a famous method adopted by Budiansyk (1967). Structure vibration equations are numerically solved for various of the load parameters, thus obtaining the system responses. The load parameter at which there exists a large change in the response is called critical. Budiansyk criterion failed in some cases when structure dynamic response isn't sensitive to load changes. According to the concept of Liapunov stability, a motion is said to be stable if all of other neighbor motions stay close to it at all time; otherwise it said to be unstable. If structure tangent stiffness matrix is negative definite then structure transient response exponentially diverge. During Newmark integration procedure structure tangent stiffness matrix is triangle decomposed if there are negative values found in the diagonal elements the tangent stiffness matrix is negative definite. In the present paper the dual criterion is used to determine the critical load (Wang(1993)): if structure tangent stiffness matrix remains negative definite during several time steps and structure transient responses diverge then the load is called dynamic stability critical load.

Step load is the simplest dynamic load with constant amplitude at all time. Structure stability under step load represents its resistance to dynamic disturbance, it's also the basis for studying structure dynamic stability under strong earthquake. The are many factors which influence structure dynamic stability such as geometric parameters, material constitutive relation, damping, initial imperfection, initial static load, etc. The various influence factors including material non- linearity, damping, initial geometry imperfection are studied through a numerical example. By using parameter analysis method, structural dynamic stability critical load of various spans and rise-span ratio are calculated when material constitutive relation being elastic, elastic-plastic, including damping and without damping. Finally, a simplified dynamic critical load calculating method is suggested.

DYNAMIC STABILITY INFLUENCE FACTORS

Trial calculation is only valid method for structure dynamic stability critical load analysis. Increasing load step by step then calculating structure nonlinear dynamic response, structure vibration amplitude increased with the load. When structure vibration time history curve bifurcate and diverge, at the same time structure stiffness matrix is negative definite structure vibrate from stable state to unstable state, the load is called dynamic stability critical load. A 90 members dome is analyzed by various parameters which may influence structure dynamic

Dynamic Stability of Single Layer Reticulated Dome under Step Load 203

stability including material nonlinear, mass quantity, damping, half span load, impulsive load and initial geometry imperfection. The K6 type reticulated dome of 10 meters span which is

modeled as space frame with 1:10 rise to span ratio, shown in Fig. 1. The supports of the dome are

assumed to be pinned and restrained against translational motion. Vertical uniformly distributed pressure load is applied at the nodes symmetrically. The members are steel tube ~60mm • 3.5mm.

Fig. 1 Geometry of 90-member shallow dome

For static case, structure elastic stability load is 36.9 KN/m 2 solved by spherical constant arc-length

method. Assuming material is elastic perfect plastic with yield stresses 2.35e5 KN/m 2, the static

stability load reduced to 16.1 KN/m 2, the critical load reduced about 56% when material non-linearity

is included. Because elastic stability critical load of reticulated dome is high, the steel tubes became plastic before the structure reached its elastic critical load. Material non-linearity must be considered in reticulated dome stability analysis.

For dynamic case, step load distributed as static case is applied at the dome and uniformly distributed mass 500kg/m 2 lumped to the nodes. Structure fundamental period Tf=0.21s, time step At=0.005s,

damping matrix is neglected. Several levels of load are calculated, Fig. 2 shows node 3 displacement

history at elastic stage. Structure dynamic stability is sensitive to small load perturbation when

applied load is 14.70 KN/m 2 structure tangent stiffness matrix remains positive definite structure

vibrate stable, when the load reached 14.75 KN/m 2 structure tangent stiffness matrix became

negative definite at 0.39 seconds structure vibrate curve bifurcate, structures vibrate unstable

dynamic responses increase very fast. Assuming elastic perfect plastic material with yield stress 2.35e5 kN/m 2 structure dual nonlinear dynamic response is calculated again. When step load is 9.0KN/m 2 structure vibrate stable, until the load reached 9.1KN/m 2 structure lost stability (Fig. 3). If only material nonlinear is considered and geometric non-linearity neglected structure dynamic critical load is 11 KN/m 2 compare with only geometric non-linearity considered the critical load 14.75KN/m 2. According to the numerical example, material nonlinear influence is

large then geometric non-linearity. Structure dynamic stability load is smaller than that of static

stability load no matter material is elastic or elastic perfect plastic.

Damping influence is important in structure dynamic analysis which can largely reduces

vibration peak value and maintaining structure dynamic stability. The Rayleigh damping is used with damping ratio ~=0.05. Elastic dynamic stability critical load increase from 14.75KN/m 2 to

23KN/m 2, elastic-plastic dynamic stability load increase from 9.1 KN/m 2 to 15.5 KN/m 2, the

increase ratio is 56% and 70%, respectively. Node 3 elastic and elastic plastic time history with

damping is shown in Fig. 4.


Fig. 2 Node 3 elastic displacement history Fig. 3 Node 3 elastic-plastic displacement history

Fig. 4 Node 3 displacement history with damping Fig. 5 Node 3 elastic-plastic response

Fig. 6 Response with different load distribution Fig. 7 Node 3 response under impulsive load

Assuming two uniform distribution mass M=300kg/m 2, M=100kg/m 2, the other parameter is the same as previous analysis. The dynamic stability critical load is 9.5 KN/m 2 compare with 9.1KN/m 2 of M=500kg/m 2 (Fig. 5). It can be seen that the quantity of mass have less influence

in structure dynamic stability. Load distribution is also important in structure dynamic analysis assuming only half span applied uniform load with mass of M=300kg/m 2, elastic-plastic critical load is 11.8 KN/m 2 large then full span load distribution critical load of 9.1KN/m 2, but the total

load is half of full span load distribution. Increasing left half span load 50% and reduce right half span load 50% the total load remain the same, the critical load is 7.6KN/m 2 which reduce 16%

compare with full span load distribution (Fig. 6).

Loading time also should be considered, infinite loading time is step load, very short loading

Dynamic Stability of Single Layer Reticulated Dome under Step Load 205

time is impulsive load. There are two loading case calculated with time duration t o = 0.05 s and

t o = 0.025s, respectively. Structure elastic-plastic critical loads are 9.7 KN/m 2 and 14.4KN/m 2

while step loading case is 9.1KN/m 2. The critical load decreased when loading time increased,

under impulsive loading structure lost stability during free vibration state (Fig. 7).

Reticulated dome is imperfection sensitive structure with lower load bearing capacity than perfect

structure. The imperfection distribution and values are impossible to predict, here structure static

buckling modes are used to simulate initial geometry imperfection. Buckling mode is the tendency

of structure displacement at critical status, if imperfection mode is the same as buckling mode it will cause the worst influence to structure vibration. The first tenth static linear buckling modes are calculated from which choosing the detrimental imperfection mode. Assuming the maximum

imperfection value is 5cm for each buckling mode then calculating structure static linear buckling

load, the lowest buckling load with corresponding mode is chosen as imperfection mode. Without damping the elastic dynamic critical load and elastic-plastic critical load are 10KN/m 2 and 6KN/m 2,

the reduction ratio is 32% and 34% compare with perfect dome. Dynamic buckling mode is

different from perfect structure, the perfect dome buckling mode is symmetric large area collapse

on whole structure, but the imperfect structure buckling mode is part collapse near the maximum

imperfection point (Fig. 8).

Fig. 8 Maximum imperfection point time history and collapse mode

DYNAMIC STABILITY P A R A M E T E R ANALYSIS

In parameter analysis there are total 12 latticed domes fixed at the edge with spans 30m, 40m,

50m, 60m, each span have three rise to span ratio 1/10, 1/8, 1/6, respectively. The tube material

assumed as perfect elastic-plastic with yield stress 235KN/m 2. From previous analysis mass

quantity have few influences on stability so that the mass distribution is chosen constant 200kg/m 2 lumped at each joint. Rayleigh damping ratio is ~=0.05, time step At = 0.02. The

members for each span are tube 90 nos. q~140 • 5, 156 nos. q~159 • 5 ,240 nos. ~180 • 8, 342

nos. 00194 • 10, respectively. Structure static stability load is also calculated using load

incremental method. There are total 48 cases with different spans, elastic or elastic-plastic

constitutive relation, including or without damping, the results are shown in table 1.

206

Span Rise to (m) Span

1/10

3O 1/8

1/6

1/10

40 1/8

1/6

1/10

50 1/8

1/6

1/10

60 1/8

1/6

C. Wang and S. Shen

TABLE 1 STATIC AND DYNAMIC CRITICAL LOAD (KN/M 2)

Static Static Elastic Dynamic Elastic Plastic damp no damp

7.91 6.09 7.4 5.0

9.88 7.56 8.4 6.0

11.85 9.50 9.8 7.3

7.56 4.86 7.3 4.6

9.80 6.16 8.4 5.6

11.52 7.84 9.7 6.8

9.50 6.65 9.5 7.0

14.70 8.42 13.1 9.3

19.53 10.50 15.5 11.0

9.20 6.68 9.1 8.0

13.95 8.82 13.6 10.8

22.28 11.38 18.6 13.2

Plastic Dynamic damp no damp

5.4 3.4

6.0 4.2

6.5 5.0

4.6 2.8

6.0 3.4

6.8 4.0

5.8 4.2

7.5 5.0

10.0 6.0

5.8 5.0

7.3 5.2

10.5 6.2

It can be seen from table 1 that structure dynamic stability critical load is less than static critical load no matter material is elastic or elastic-plastic, elastic critical load is less than elastic-plastic critical load. If damping is included then the critical load increase 4 0 % ~ 50% when material is

elastic or elastic-plastic. The critical load increase with the rise to span ratio at the same span.

Structural dynamic stability is closely related to static stability, dynamic to static ratio is defined as

dynamic stability load divided by static stability load. Without damping elastic dynamic to static ratio is 0.56 "-~ 0.87, elastic-plastic dynamic to static ratio is 0.28 ~ 0.54. With damping elastic

dynamic to static ratio is 0.79 ~ 1.0, elastic-plastic dynamic to static ratio is 0.47 "~ 0.68. In order

to assess the reasonable range of dynamic to static ratio the follow simplify is proposed: without damping elastic dynamic to static ratio is 0.6, elastic-plastic dynamic to static ratio is 0.3; with damping elastic dynamic to static ratio is 0.8 and elastic-plastic dynamic to static ratio is 0.5.

The influences of initial geometry imperfection change with different imperfection mode and values. The imperfect 40m span dome with rise to span ratio 1/10, 1/8, 1/6 are analyzed, the imperfection mode is chosen from the first tenth static buckling mode as previous analysis with maximum imperfection values 4cm, 8cm, 12cm, respectively. The influence of material non-

linearity and damping are considered, the results shown in table 2. Structure static and dynamic stability loads decrease a lot with increasing imperfection value. With imperfection value 4cm

12cm, including damping, elastic dynamic critical load decrease about 35%'~ 62% , elastic-

plastic dynamic critical load decrease about 41%~- 59% compare with perfect dome. Without

damping, elastic dynamic critical load decrease about 2 1 % ~ 38%, elastic-plastic dynamic

critical load decrease about 13%~- 36%. Including damping the dynamic stability load is 0.4

0.6 times of perfect structure, 0.6 ~- 0.8 times without damping. With damping elastic imperfect

structure dynamic to static ratio decrease from 0.8 to 0.4, elastic-plastic dynamic to static ratio decrease from 0.6 to 0.4. Without damping elastic imperfect structure dynamic to static ratio

decrease from 0.6 to 0.25, elastic-plastic dynamic to static ratio decrease from 0.4 to 0.25.

Dynamic Stability of Single Layer Reticulated Dome under Step Load

TABLE 2 IMPERFECTION STATIC AND DYNAMIC CRITICAL LOAD (KN/M 2)

207

Rise to Imperfec Static Span :tion (cm) Elastic

0.0 7.56

1/10 4.0 4.62

8.0 3.10

12.0 2.88

0.0 9.80

1/8 4.0 7.94

8.0 5.11

12.0 3.60

0.0 11.52

1/6 4.0 11.70

8.0 9.60

12.0 6.48

Static Elastic Dynamic Plastic damp no damp

4.86 7.3 4.6

3.50 4.5 4.0

2.50 3.1 3.0

1.90 2.8 2.9

6.16 8.4 5.6

4.92 7.7 5.2

3.70 5.0 4.5

2.80 3.6 3.5

7.84 9.7 6.8

6.72 9.1 6.3

5.52 8.7 6.0

4.32 6.3 5.4

Plastic Dynamic damp no damp

4.6 2.8

3.0 2.7

2.5 2.1

1.9 1.8

6.0 3.4

4.0 3.2

3.5 3.0

2.8 2.5

6.8 4.0

6.0 3.9

5.0 3.8

4.0 3.5

In practical engineering application there are initial static loads before suddenly applied dynamic load. The 40m span dome with rise to span ratio 1/6 applied initial static load P0 = 2.0KN/m2,

P0 = 5-0KN/m2, P0 = 7.5KN/m2, other parameters is the same as previous analysis. First calculate

structure static nonlinear response under P0, second calculate structure dynamic response after

suddenly applied step load. The critical load is the sum of initial static load and dynamic load. When static load P0 increase the newly applied step load PD decrease, but the total load P

approach to structure static stability load Ps = 7.84KN/m2 (Fig. 9).

Fig. 9 Apex time history with initial static load

SIMPLIFY CALCULATION METHOD

Solving dynamic stability critical load is time consuming because it is dual nonlinear dynamic

analysis using many times trail calculation so that the simplify calculation method is needed for

practical engineering application. If structure dynamic critical load can be assumed among the

reasonable value then it can save much time during nonlinear FEM analysis. Referring to the

quasi-shell method in static stability analysis the simplified calculating method for structure

dynamic critical load is suggested as follows:


Pz~ = K~ x Kz x Ps (1)

Where K l is dynamic to static ratio, without damping elastic K 1 =0.6, elastic-plastic K 1 =0.3,

with damping elastic K 1 =0.8, elastic-plastic K 1 =0.5. K 2 is modified factor of imperfection,

with damping K 2 =0.4 ~ 0.6, without damping K 2 =0.6 ~- 0.8. Ps is static linear buckling load,

Ps: 0.8 • ~ ~ : (2)

Where R is radius of dome, E is modulus of elasticity, t is average equivalent member 2A -

thickness t = - ~ - , A is average member area, l is average member length, 6 is average

- 1

equivalent bending thickness 6 - (12~f3/) ~ , ] is average member moments of inertia.

CONCLUDING

Under step load structure dynamic stability critical load is less than static critical load no matter

material is elastic or elastic-plastic, elastic critical load is less than elastic-plastic critical load. The influence of material non-linearity is large than geometric non-linearity, when elastic-plastic is considered dynamic critical loads reduce 40%'~ 50%. Including damping structure dynamic critical loads increase 4 0 % ~ 50% compare with no damping case. Initial geometry imperfection

largely decrease structure dynamic stability load, the decrease ratio vary with different

imperfection mode and increasing with imperfection values. With damping imperfect structure dynamic stability critical load is only 0.4 ~ 0.6 times of perfect dome and 0.6 ~ 0.8 times

without damping. Initial static load also should be considered in structure dynamic stability analysis. The proposed simplify method can be used in assessing structure dynamic stability load in practical engineering application.

REFERENCES

Bathe K. J. and Bolourchi S. (1979). Large Displacement Analysis of Three Dimension Beam Structures. International Journal for Numerical Methods in Engineering Vol. 14, 961-986. Budiansky, B. (1967). Dynamic Buckling of Elastic Structures: Criteria and Estimates. Dynamic

Stability of Structures, Pergamon, New York. Simtses G. J. (1990). Dynamic Stability of Suddenly Loaded Structures. Springer-Verlag New

York Inc. Wang Ce and Shen Shizhao. (1993). Nonlinear Dynamic Response and Collapse Analysis of

Spatial Truss Structures. Symposium on nonlinear analysis and design for shell and spatial

structures. Tokyo. Wang Ce, Shen Shizhao and Chen Yunbo. (1996). Dynamic Stability of Reticulated Dome.

Proceedings of International Conference on Advances in Steel Structures. ICASS'96, Hong

Kong. Pergamon, Oxford, UK, Vol II, 1065-1070. Wang Ce. (1997). Dynamic Stability of Single Layer Reticulated Dome. PhD Dissertation.

Harbin University of Civil Engineering Civil Engineering and Architecture.

EXPERIMENTAL STUDY ON FULL-SIZED MODELS OF ARCHED CORRUGATED METAL ROOF

Liu Xiliangl, Zhang Yongl, Zhang Fuhai 2

1. Department of Civil Engineering, Tianjin University, Tianjin, 300072, China

2. Beijing Milky Way Metal Roof Forming Technology Institute, Beijing, 100021, China

ABSTRACT

Nine full-sized models of arched corrugated metal roof were tested to failure under static loads,

and four of these specimens were modeled using two different kinds of finite elements separately.

Through the description of the experimental processes and the analysis of the experimental results,

the load bearing performance and the failure model of this kind of structure could be seen clearly.

Based on theoretical and experimental results, some valuable conclusions were summarized and

some recommendations for further studies were proposed.

KEYWORDS

Arched Corrugated Metal Roof, full-sized model test, load bearing performance, thin-walled

structure, arch, shell, finite element method

1 INTROUDUCTION

Arched corrugated metal roof is an alternative to stressed skin diaphragm structures. It is composed

of a series of arched trough plates which are made of color-coated galvanized steel sheets

(thickness ranges from 0.6mm to 1.5mm) and coldly formed by special cold roll forming machine.

The steel sheets are firstly rolled into straight trough plates, and to obtain the desired curvature of

an arched roof, straight trough plates are cold-rolled again. With their lower sidings rolled out

209

210 L. Xiliang et al.

many tiny cross corrugations, they become curved, forming arched trough plates. Because arch

structure can translate the applied loads mainly into forces in the plane of its surface, so such

arched trough plate can be employed in larger span buildings (more than 30m) not only as an

accessory material to be used for simple coating, but also as load bearing skeleton. With the plate-

skeleton-combined structural style and the highly mechanized construction procedure, arched

corrugated metal roof possesses such advantages as strong spanning ability, light self-weight, fast

and easy construction, good waterproofing quality and attractive appearance etc. The combination

of these advantages certainly can result in cost saving. It is very suitable to be used in single layer

buildings, and if hoisting condition permits, it can also be used in multistory buildings. According

to the different sectional configurations of arched trough plates, this kind of structure can be

classified into several types. In China there are mainly three kinds of sectional configurations, so

there are three types of this kind of structure, which are respectively named MMR-118, MMR-178

and MMR-238. Figure l shows the outlines of their cross sections.

Figure 1

Design specifications and recommendations for cold-formed steel structural members are now

available in many countries, but none of the rules for the design and construction of arched

corrugated metal roof have been published all over the world till now. Even though it is a typical

kind of thin-walled steel structure, because of its peculiar characteristics, its performance under

load differs in several significant respects from that of ordinary cold-formed structural members.

As a result, design specifications for cold-formed steel structural members cannot possibly cover

the design features of this kind of structure completely, so it needs an appropriate design

specification. With no provision of certain design code, engineering accidents will be inevitable. In

the winter of 1996, a heavy fall of snow in the northeast of China caused more than 30000 m 2 of

this kind of roof to collapse.

According to former research work, there are mainly two kinds of mechanic models for this kind of

structure, namely arch and shell. However, for some reasons, none purely theoretical analysis on

this kind of structure can make satisfactory results [1 ], so experimental studies are essential here.

Nevertheless, just because of its special construction characteristics, it is almost impossible to

carry out scale model test, full-sized model tests are indispensable to the research of this kind of

structure.

After the engineering accidents mentioned above, the authors had carried out nine groups of large-

span model experiments on the spots of these accidents. Through these model tests the cause of

Study on Full-Sized Models of Arched Corrugated Metal Roof 211

these accidents and the load bearing performance of this kind of structure could be understood. By

comparing the theoretical results with the testing results, the great divergences between them could

be seen clearly. Aiming at reducing these divergences, some recommendations for further studies

are proposed.

2 OUTLINE OF EXPERIMENT

2.1 Model specimens

All of these tests were on-the-spot tests. The models studied here were the very structures that

survived from that heavy fall of snow. The steel plate used in these models had the yield strength

of 280Mpa and Young's modulus of 2.00 • 105 MPa. The sectional configurations of these trough

plates of these models were the same as that shown in fig.lc, namely trapezoid section. Five of

these models spaned 33m and the others spaned 22m. For the convenience of the application of

load, only one model was made up of six pieces of arched trough plates, the others were all made

up of four pieces. The cross section is shown in fig.2. In order to search for an effective measure to

raise the load bearing capacity of this kind of structure, three models were reinforced with tension

chords. The reinforcing pattern is shown in fig.3. The geometrical size and load patterns of these

models are described in tab.1. Because the width-to-span ratios of these models were very small,

their lateral rigidity was quite low. To avoid lateral buckling and something unwanted scaffolds

were placed under and by both sides of these models. The outlook of a model after being put in

order is shown in fig.4

Figure2: The cross section of models

Figure 3: The reinforcing pattern Figure 4: Testing ground

212

No.

1

2

3

4

5

6

7

8

9

L. Xiliang et al.

TABLE 1

Arch span

33(m)

33(m)

33(m)

33(m)

33(m)

22(m)

22(m)

22(m)

22(m)

Arch rise

6.6(m)

6.6(m)

6.6(m)

6.6(m)

6.6(m)

4.4(m)

4.4(m)

Plate thick,

1.25(mm)

1.25(mm)

1.25(mm)

1.25(mm)

1.25(mm)

1.00(mm)

1.00(mm)

Lateral

width

2440(mm)

2440(mm)

3660(mm)

2440(mm)

2440(mm)

2440(mm)

2440(mm)

Load pattern

Full span

Half span

Half span

Full span

Half span

Remarks

Local distributed load

Reinforced

Reinforced

Full span

Half span

Half span Triangular load distribution

Half span Reinforced

4.4(m)

4.4(m)

1.00(mm)

1.00(mm)

2440(mm)

2440(mm)

2.2 Loading method

As a kind of thin-walled structure, arched corrugated metal roof is very sensitive to concentrated

load which may cause local buckling of the structure at a relatively low load lever. In actual

engineering, large concentrated load should be avoided. To simulate the actual load-bearing pattern,

distributed loads were applied by using sandbags. From tab.1 we can see that No.3 model bore

local half-span distributed load, which means that only four out of the six trough plates bore half-

span distributed load, while the two edge trough plates were free from any external direct loads.

Tab.1 tells us that No.8 model bore triangularly distributed load. This loading pattern is to imitate

non-uniformly distributed snow load.

2.3 Observation method

Because this is a kind of flexible structure, its deformations are so large that any displacement

measuring instruments with conventional precision can not cover its deformation scope, therefore

levelling instruments were used to survey the vertical displacements, and theodolites were used to

measure the rotary angles of those surveying points. Through the values of these rotary angles, we

can figure out the horizontal displacements. 7V08 static electrical resistance strainometer was

employed to observe the distribution of strains in the models. The surveying points of

displacements and strains were arranged at such locations as two springs, L/8 section, L/4 section,

L/2 section, 3L/4 section and 7L/8 section.

Study on Full-Sized Models of Arched Corrugated Metal Roof

3 E X P E R I M E N T A L RESULTS

213

The ultimate load, maximum horizontal displacement (U) and its location, maximum vertical

displacement (V) and its location of each model are listed in tab.2

No.

1

2

3

4

5

6

7

8

9

TABLE 2

Ult imate load U Locat ion V Locat ion

0.87kN/m 2 38cm L/8 43cm L/2

0.56kN/m 2 52cm 3 L/4 57 cm 3 L/4

0.27kN/m 2 53cm L/4 54cm L/4

0.92kN/m 2 36cm L/8 42cm L/2

1.02kN/m 2 19cm L/4 23cm L/4

1.06kN/m 2 18cm L/8 27cm L/2

0.54kN/m 2 32cm 3 L/4 41 cm 3 L/4

1.02kN/m 2 31 cm 3L/4 39cm 3 L/4

1 l c m 1.28kN/m 2 L/4 16cm L/4

Studying the data got from electric resistance strainometer, it's hard to find the laws of the stress

distribution in these models' sections. Although the cross sections of the models and patterns of

external load were symmetric, the stresses in one section didn't show symmetry. The direction of

principal stress of a certain point changed form time to time with the load added. The tiny ripples in the

trough plates and the out door wind load may account for this to a certain extend. Certainly the stresses

measured couldn't reflect the laws of the distribution of the actual stresses, but as few of them exceeded

the yield point stress of the material, so they could qualitatively tell us it isn't strength that determines

this kind of structure's load bearing capacity. Though the width-to-thickness ratios of the trough plates

in these models are very large, local buckling models which is common for thin-walled members didn't

appear during these tests. This demonstrates clearly that the tiny ripples can strengthen the local

stability of the plates.

Both No.1 and No.6 models bore full-span uniformly distributed load, so their performances were

similar. When the load level wasn't high, their deformations were symmetric, as shown in fig.5. But

when the load was close to the ultimate load (shown in tab.2), a sudden change from symmetric

deformation to non-symmetric deformation happened, which caused the internal forces around L/8 in

this side to increase steeply. With a little more loads, the model lost its stability and buckled.

The failure model of this kind of structure under half-span distributed load was shown in fig. 6. I t ' s

easy to understand that the stability bearing capacity of this kind of structure under half-span load is


much lower than that under full-span load, while the stresses and displacements were much bigger. The

reason accounted for this was that the deviation between arch axis and pressure line in the half-span

load model was much larger than that in the full-span load model, so bending moments were prominent

here, which was very disadvantageous to any structures. According to the data provided by the local

meteorological department, after that fall of snow the basic snow load of the zone where the

accidents happened was about 0.521kN/m 2, and the gale also blow snow from windward side to

leeward side during the snow-fall. So the uneven half-span snow load was close to ultimate half

span load listed in tab.2. It's quite sure that half-span load pattern is the most dangerous load

pattern for this kind of structure.

Original shape

~--L / 8--4---L / 8-~L--L / 8---~L / 8--4--L / 8-~L--L / 8---4--L / 8--4---L / 8--*J

Original

I,--L. I B -'-4-- L I B--J---L I B--I--L I B --J'-- L I B - ' - ~ L I B---I---L I B---J'--L I B "-~

Figure5: Deformation Shape of Full Span Load Model Figure6: Deformation Shape of Half Span

Though there were two pieces of

trough plate free from direct

external load, the ultimate load of

No.3 model is not bigger than that

of No.2 model. This model test

indicates that as a kind of thin-

walled member with open cross

Load Model

I~'--L / 8--"#- L / 8--Jr--L / 8"-"IL- L / 8----i-- L / 8~L--L / 8 --J,"-'L / 8---l--- L / 8--4

Figure7: Deformation Shape of Reinforced Half-Span Load

Model

section, the trough plate's torsional rigidity was very small and its capacity of resisting torsional load

was poor. From this test we also can see that the coordination between two pieces of plates was bad,

and the lateral widths of other models had little effects on their load bearing capacity.

Fig.7 shows the deformation shape of the models reinforced with tension chords subjected to half

span load. Tab.2 tells that the reinforcing pattern shown in fig.3 has little effect on the load bearing

capacity of the structure under full span load, while under half span load the load bearing capacity

of the same structUre can be doubled. From fig.7 we can see that two chords restrain the 3L/4

section, where the largest deformation will take place without these chords. The tension chords can

make the distribution of the internal forces even more uniform.

4 COMPARISON OF EXPERIMENTAL AND T H E O R E T I C A L RESULTS

Because of the symmetry of the configuration and the load distribution along the longitudinal

direction of this kind of structure, it can be looked as a kind of arch structure and modeled with

thin-walled beam elements. The material constants, such as bending rigidity, axial rigidity, etc, are

Study on Full-Sized Models of Arched Corrugated Metal Roof

calculated according to the geometric size of unit width of its cross section.

215

To reflect such structure characteristics as thin wall, tiny ripples, doubly curved space

configuration, shell element is the most ideal one. The shell element used here is a kind of

generalized conforming quadrilateral flat shell element [2]. A piece of arched trough plate is

chosen as calculating model. Because the length-to-width ratio of the trough plate is very big, in

order to avoid deformed grid dividing, the size of shell element should be very small. So the

number of the shell elements in a piece of trough plate is great. This of course increases the amount

of calculation, while on the other hand this also can raise the calculating precision. In general, the

steel material used in this kind of structure is isotropic. But because of the ripples on the webs and

flanges of the trough plate, the webs and flanges will respond to load orthotropically. To analyses

the effect of the ripples an equivalent orthotropic fiat sheet is defined for the shell FEA model. The

material constants of the equivalent flat sheet can be acquired according to the equivalent condition [3].

The above experiments had indicated that it is global stability, not material strength, that control

the load bearing capacity of this kind of structure, so only geometric nonlinearity is considered in

this paper. For the same reason, local buckling isn't considered. To avoid the problem of material

nolinearity in theoretical analysis, yield criterion is adopted as the failure criterion. By the

programs based on above mentioned theory, specimen 1, 2, 6 and 7 had been calculated. The

ultimate loads of theoretical analysis and experiments and the errors of theoretical results

compared with experimental results are listed in tab.3.

TABLE 3

Experiment Arch model Error Model No. Shell model Error

1 0.87kN/m 2 2.17kN/m 2 149.4% 1.26kN/m 2 44.83%

2 0.56kN/m 2 1.06kN/m 2 89.29% 0.67kN/m 2 19.64%

6 1.02kN/m 2 5.76kN/m 2 464.7% 3.23kN/m 2 216.7% . . . . . . . . . . . .

1.89kN/m 2 0.54kN/m 2 250.0% 1.14kN/m 2 111.1%

As a kind of thin-walled steel structure, it is very sensitive to defects. Because the models used in

these experiments were the survivors of accidents, initial deformation and initial stress were

inevitable. In addition, all the tests were carried out outdoor, wind load will bring harmful effect on

the tests too. So from tab.3 we can see all the theoretical results are much higher than the

corresponding experimental results. But compared with half-span loading models, the errors of

full-span loading models are even larger, which indicates that this kind of structure under full-span

load is more sensitive to defects than that under half-span load.

It's obviously that the results calculated with shell FEA model are much closer to the experimental


results than that calculated with arch FEA model. This indicates that even though it's symmetric

along longitudinal direction, the arched trough plate, the structure's components have the property

of space load carrying because of its characteristics of thin wall and local ripple shape. The

construction of ripples on the plates certainly can strengthen the stiffness along longitudinal

direction, which makes the structure free from wavelike local buckling, but they weaken the

stiffness along span direction which is very disadvantageous for this kind of bearing structure.

Shell FEA model can reflect these factors to a certain extend.

From the analysis above, it's not difficult to find out that purely theoretical analysis on this kind of

structure has a distance from real application. Model test is indispensable here. But experimental

study requires testing of full-sized models, which are very expensive and the result is only

applicable for some special situations. So studying the relation between theoretical analysis and the

experimental results and finding out the appropriate calculating constants from experiments so as

to revise the calculation programs have great significance for the research of this kind of structure.

The authors of this paper are now preparing several groups of member tests in order to observe the

material constants of the arched corrugated trough plate. The material constants got from

experiments will be used in FEA.

5 CONCLUSION

Through the description of these full-sized model tests, the load bearing performance and the

failure model of arched corrugated metal roof are clear now. After pointing out that local buckling

and material strength are not the control factors to its load carrying capacity, two kinds of FEA

models were established for the its theoretical analysis. Though the theoretical results didn't agree

well with the test results, these deviations indicate that such structural characteristics of this kind

of structure as thin wall and local ripple shape have great effect on its load bearing performance.

To reduce the difference between theory analysis and experiment study, recommendations for

further research are proposed.

References

1. Zhang Yong, Liu Xiliang and Zhang Fuhai (1997) Experimental Study on Static Stability Bearing Capacity of Milky Way Arched Corrugated Metal Roof. J. of Building Structures, 18:6,

46-54 2. Zhang Fuhai, Zhang Yong and Liu Xiliang (1997) A Generalized Conforming Quadrilateral Flat Shell Element for Geometric Nonlinear Finite Element Analysis. J. of Building Structures

18:2, 66-71

3. Erdal Atrek, Arthur H.Nilson (1980) Nonlinear Analysis of Cold-Formed Steel Shear Diaphragms, J. of the Structural Division 3,693-710

QUASI-TENSEGRIC SYSTEMS AND ITS APPLICATIONS

Liu Yuxin 1 and Lti Zhitao 2

1Nanjing Architectural & Civil Engineering Institute; Nanjing 210009, China 2Southeast University, Nanjing 210018, China

ABSTRACT

Tensegric system is an optimum structural form in which the behavior of high strength in steel cable

can be utilized, but the reliability of this system is not very good because of the quasi-variable characteristics. Cable-nets are also an effective structure that could span large space. This paper

proposes a new concept of spatial structure in which we combine tensegrity with cable-nets to form a

quasi-tensegric system. So we can make use of the advantages of these two systems. A construction manner is developed. A quasi-tensegric system could be formed by the tensegric elements. This paper

divides the equilibrium state of quasi-tensegric system into two states: one is geometrical stable equilibrium state, the other is elastic state equilibrium state. A method is developed to calculate the form and internal forces in the geometrical stable equilibrium state and the convergence is provided. The results of calculating show that the method proposed has a good convergence and a high precision. Comparing incremental iterative method with dynamic relaxation method, the two methods are effective and reliable in engineering design.

KEYWORDS

Quasi-tensegric system, tensegrity, cable-nets, geometrical stability, equilibrium state, prestressed force, incremental iterative method, dynamic relaxation method

INTRODUCTION

Among reticulated structures composed of struts and cables, which require formfinding processes a specific class can be defined as funicular system's class (Liu and Motro,1995). Their stable shape is

directly related to a set of external actions. Two equilibrium states are defined. The first one which

doesn't take into account the member deformations corresponds to geometrical stable equilibrium state

(GSES), the second one is related to a computation of the equilibrium in the deformed shape under

217

218 L. Yuxin and L. Zhitao

extemal actions and is named the elastic state equilibrium state (ESES). A method for computing the coordinates for the GSES was obtained by using the theory of generalized inverse matrix (Liu and Lu et a1,1995). In order to determine the ESES leading to the value of node coordinates and internal forces, an alternate method was put forward. Computed results are compared with those obtained with a Newton Raphson method. We shall introduce briefly these main results in this paper. After giving the method of unstable systems, we discuss calculating procedure of cable-nets and simple tensegric system. And finally give the construction rule of quasi-tensegric systems.

INCREMENTAL ITERATIVE METHOD FOR UNSTABLE SYSTEM

Kinematic Relationship

Static and kinematic equations are established assuming classical hypothesis for reticulated structures with struts and cables. Assuming that free nodal displacements there are b members and n degreeS of freedom, the kinematic relationship can be expressed in matrix as follows

{e} = ([B] + [AB]){d} (1)

{e } is elastic deformation vector, [B] is the compatibility matrix and [AB] an increment of [B], {d} is the displacement vector in which boundary condition being included by deleting the corresponding

values. When II {d} II is very smaller, the second term can be neglected, then

[B] {d} = {e} (2)

For an unstable structure, there is no elastic deformation until the geometric stable equilibrium state and Eqn.2 become

[B] {d} = {0} (3)

Static Equilibrium Relationship

Static equilibrium equation can be derived from principle of virtual work. For a set of extemal actions

{f} and a virtual displacement { rid}, corresponding elongation { de } and internal forces {t} must satisfy

{f} r {d} = {t} r {e} (4)

Substituting Eqn.2 into Eqn.4 yields

({f}r _{t}r [B]){fd} = {0} (5)

It holds for arbitrary { 6at}, so that

Quasi-Tensegric Systems and Its Applications

[B] r {t}= { /} or [A]{t}={/}

The constitutive law can be expressed in matrix form as follows

{e} = [F]{t}

219

(6)

(7)

Where [F] is b-order diagonal matrix with

F, = (L / EA) , i=l ,b (8)

Stability Criterion and Convergence

When analyzing the form in geometric stable equilibrium state, we use the compatibility Eqn.3 instead

of static equilibrium equation Eqn.7. At equilibrium the total potential YI of the structure takes a local minimum value. The necessary and sufficient conditions for equilibrium are

oTI=0 (9)

521--I --- 0 (10)

Where 5 is a variational symbol related to the displacement space. The equilibrium is arbitrary or

stable according to the value of 6 2H. Condition expressed by Eqn.9 will be used in next section in order to choose a parameter leading to the stable equilibrium state. For the problem of GSES, we use

linear incremental method to solve the system of linear homogenous equation 3. That is to say an

iterative procedure will be used. As the incremental {d} is small, in each iterative step (say i-l), take the first order approximation, then

{d},_ 1 = { x ' } i _ l t (11)

Where t is a small parameter, {x'},_~ is the first order derivation of the displacement vector with regard

to t. If {d},_~ have been found out, [B]i. 1 can be calculated[2]. So we have

IN]i_ 1 { d I i -- {0) (12)

Based on the generalized inverse theory of matrix algebra, thesolution of Eqn.12) is

{d}, =([I]-[Bl+[B]{y},_l = [D],_I {a},_~ (13)

where { y} t-1 as well as {a} ,_~ are n and (n-r) dimension algebra vector, respectively. In this paper we

chose { 1 }-inverse, so { c~} i-1 is a (n-r) dimension vector. [D] ~-1 will be called as a displacement model matrix. Hence from i-1 step to i step, nodal position coordinate vector is


{X}, - - { X } ,-1 q- [D],_, {a},_l (14)

in which {o~} ,-1 is unknown. According to the criteria of energy, we can select

{a}~_ 1 -~b[D] T = i-1 {f} (15)

Here, it is clearly that ~b is a size parameter that controls the size of displacement increments in the gradient direction. Substituting Eqn. 15 to Eqn. 13 yields

{d}t = ~b[D]~_ 1 [D][, {f} (16)

Repeating the procedure from Eqn. 13 to Eqn. 16, we shall find out the nodal coordinates in GSES step

by step. Finally, the final space coordinates of nodes can be expressed as follows

oo

{x}= {X}o + ~--' {d}, (17) i=l

The convergence of Eqn. 17 depends on the extemal wind force {f}. This is the same behavior for any

unstable structure (Liu and Motro,1995).

STRUCTURAL ANALYSIS AND VECTORIAL SUBSPACES

Eqn.6 which describe the kinematic and static relations in a reticulated system connect two vectorial spaces, namely the node space R, and the member space Rb. External forces and displacements are

related to the first one, internal forces and elongation to the second one. If we call r the rank of the equilibrium matrix [A], a Gaussian elimination procedure on the augmented matrix [A:I], [/] being a diagonal unit matrix, leads to the transformation below (Pellegrino and Calladine,1986)

Gaussian transformation ~- ~- [A:I ] >[A:I] (18)

which can be put in the form

[ A ' I ] = Im (19)

Applying the transformation to static relationship Eqn.6, we get

Quasi- Tensegric Systems and Its Applications 221

The external force space {f} can be split in two subspaces: one is r-dimensioned and is a fitted external force subspace (forces can be equilibrated), the second one, of m-dimension with m--n-r corresponds to the forces which activate the mechanisms. Similar derivation (with [J] being a diagonal unit matrix) can be achieved on the compatibility equation 2 and lead to

The deformation space is composed of two subspaces, a "fitted" and a "non-fitted" subspace. Only the

former is compatible with the displacements. If we consider Eqns.19, 20 under an energy view, each

row of [It] and [lm], can be assimilated to displacements and each row of [Jr], [Js] to internal forces. In Eqn.20, when components of {d} belong to the mechanism subspace, corresponding values of {e} are

equal to zero and corresponding rows of [Br], which are related to the external forces are orthogonal to

the displacements. The m mechanism vectors are included in [lm] from Eqn.19 and the corresponding displacements can be computed by

{am } = [im IT {a} (21)

with {a} being composed of arbitrary constants combining elementary mechanisms.[D]=[lm] r is known as the displacement mode matrix. Similar analysis leads to the computation of any self-stress vector by

{t, } : [J~ ]r {/3} (22)

with {/3} being composed of arbitrary constants combining elementary self-stress states.[D]=[J,] r is known as the self-stress mode matrix. We note that for a structure that verifies compatibility condition,

the self-stress subspace is orthogonal to the elastic deformation one. If the structure is in equilibrium,

the mechanism displacement space is orthogonal to the external force space.

Governing Equations for GSES

Considering the compatibility equation, the matrix [B] is a bxn matrix, and generally this is not a square and even in this case its rank is not equal to n (=b); we can't use traditional procedures to solve it. We must introduce the Moore Penrose inverse and more precisely the { 1 }-inverse [B]-. With this, the general solution of the compatibility equation can be put in the form

{d} = [B]- {e} + ([I] - [B]-[B]){y} (23)

Where {7"} is a n-dimensional vector. The second term in Eqn.23 belongs to the mechanism displacement space and Eqn.23 is equivalent to

( [ I ] - [B]-[B]){y} = [D] {a} (24)


When {e}={0}, i.e. for a rigid displacement of the structure, the space coordinates can be found either

by general inverse method or by an iterative altemate method (Liu and Motro,1995), [D] is called as a

displacement model matrix.. Similar derivations give access to the intemal forces as

{t} = [A]- {f} + [S] {fl} (25)

Matrix [S] is called as a self-stress model matrix, {fl} is an unknown vector.

Elastic Stable Equilibrium State

In order to reach the ESES with a known GSES, it is necessary to determinate the values of vectors { a}

and {fl} in Eqns.24, 25. Assuming that {t}0 is the initial value of internal forces in GSES, we can obtain the elastic deformation vector {e} by combining Eqn.25 and 7 which describes the constitutive

law

{e} = [F]([A]- {f} - {t}0 ) + [F] [S] {/3} (26)

From the previous discussion, for an incompatible structure, the deformation space is orthogonal to the

self-stress space. This condition gives the needed equation for computing {fl}

[S] v {e} = [S] r [F]([A]- {f} - {t}0 ) + [S] r [F][S]{fl} = {0} (27)

Since the product [S] T [F][S] is not singular and the first term of Eqn.27 is known, we can find out {fl} ,and the relevant internal force {t} (Eqn.25) and {e} with the constitutive law (Eqn.7). The second term

can be equated by considering the orthogonality between the mechanism displacement and the non- fitted external forces, when structure is in equilibrium. Matrix [D] is a base of non-fitted external force

space, so we get with Eqn.23 and 24

[D] ~ {d} = [D] r [B]- {e} + [D] T [D] {a} = {0} (28)

where [D] r [D] is not singular, so we can find out { a} and simultaneously {d}. Then the position vector {x} is calculated with the GSES coordinates as first value. A new matrix [B] is derived and the

process is repeated until the increments become close to zero. Hence the ESES is found. The

convergence depends on the properties of the structure. If the rank is equal to the number of members

there is no self-stress state and the elastic deformation {e} is compatible with {d}: the process is

always convergent. When r<b, {e} doesn't fit with {d}. In this case, the external forces must be

divided into small increments, so that the fitting between {e } and {d} can be nearly satisfied.

NUMERICAL EXAMPLES

Unstable Cable

Quasi-Tensegric Systems and Its Applications 223

The first example is an unstable cable that has been researched by F. Baron and M.S. Vendatesan

(1971). The sectional area of the cable is 1.465 cm 2 and the stiffness is EA=12119.51kN. The initial

ESES is reached under two symmetric loads 17.8kN and an added 13.3kN local load applied to node 2.

Comparison with Baron's results is done for the GSES and the ESES. Coordinates of nodes are listed

in Table 1 and internal forces in Table 2. Baron's results are derived from a Newton-Raphson method.

TABLE 1

SPACE COORDINATES CALCULATED

Direction Node 1 Error(%) Node 2 Error(%)

31.1475 61.5544 xl (m) F. Baron

x2 (m) 13.9172 16.4714

xl(m) 31.1611 0.04 61.5217 -0.05 GSES

x2(m) 13.7810 -0.98 16.3073 -1.00

xl(m) 31.1540 0.02 61.5609 0.01 ESES

x2(m) 13.8517 -0.47 18.4969 0.15

TABLE 2

STRESES IN BARS (MPa)

Member No. Baron 1971 GSES Error(%) ESES Error(%)

1 367.908 371.255 0.91 365.276 -0.71

2 337.479 340.707 0.96 335.032 -0.73

3 383.088 386.694 0.94 381.266 -0.48

Our results are close to those of Baron. It can be noticed that the difference between GSES and ESES

is very small and could be neglected: a non-extension hypothesis could be acceptable.

Tensegrity System

A tensegric system (see Figure 1) is analyzed, in which an initial state is shown in Figure l a. The

calculating results by the method proposed in this paper are Shown in Figure lb and c. The initial nodal

6 i 2 X2

0.5m 0.5m

(a) Initial state

6 J

~ 1 x2

8,7 5,6

(b) Final State: vertical view (c) Final State: elevation

X 2

Figure 1" Tensegric element


coordinates are listed in Table 3. The nodal 5,6,7,8 are fixed, the nodal 1,2,3,4 are free. The final free nodal coordinates are listed in Table 4. In dynamic relaxation method (Liu, 1998), the coefficient

damping takes 500kg-m/s, the time incremental is 0.2s, nodal mass is 5000kg at each node. After

73695 times iteration get the final form. For incremental iterative method, the coefficient ~b =0.001, iterate 14166 times get the final results. Finally we reach the results which show in Table 4 and Table 5,

respectively. The two methods are effective for calculating tensegric system.

TABLE 3

THE INITIAL COORDINATES OF THE TENSEGRIC SYSTEM

Nodal No. 1 2 3 4

xl(m) -0.35355 -0.35355 -0.35355 0.35355

x2(m) 0.35355 0.35355 -0.35355 -0.35355

x3 (m) 0.97832 0.97832 0.97832 0.97832

5

0.5

0.5

0

6

-0.5

0.5

0

7 8

-0.5 0.5

-0.5 -0.5

0 0

TABLE 4

THE FINAL FREE COORDINATES OF THE TENSEGRIC SYSTEM

Methods

Nodal No.

x,(m)

x2(m)

x3(m)

Dynamic Relaxation (Liu, 1998)

1 2 3 4

0.0000 -0.50000 0.00000 0.50000

0.5000 0.00000-0.50000 0.00000

0.86603 0.86603 0.86603 0.86603

Incremental iterative

1 2 3 4

-0.00001 -0.50023 -0.00001 0.50021

0.50022 0.00000 -0.50022 -0.00000

0.86701 0.86701 0.86701 0.86701

TABLE 5

THE INITIAL INTERNAL FORCES OF THE TENSEGRIC SYSTEM

Methods Dynamic Relaxation Incremental iterative

Members 12,23,34,41 15,26,37,48 18,25,36,47 12,23,34,41 15,26,37,48 18,25,36,47

Stresses(kN) 0.50000 0.70711 -1.00000 0.49952 0.70673 -1.00000

THE CONSTRUCTION OF QUASI-TENSEGRIC SYSTEM

Cable net is a good structural system that has ability to span large space, but the stiffness is a big

problem. Tensegric system is a maximum economic form of structure, but its construction is very

complicate. This paper tries to combine the advantages of the two systems to form quasi-tensegric

systems.

According to the area to be covered, we at first design a cable-net, for example as shown in Figure 2a.

Of cause, in practical engineering the plan of structure may not be a rectangular. Then using the

tensegric element of Figure lb,c, we connect the tensegrity to cable-net one element by one element, as shown in Figure 2b. Finally we can form a double-layer quasi-tensegric spatial frame. In

construction, the key problems are the application of prestressed force technology and the connecting

Quasi-Tensegric Systems and Its Applications 225

form at nodes. These will be researched further in the future. And the whole structural analysis after integrating quasi-tensegric system is also important.

O Cable

Tensegric element Tensegric element

i I

Tensegric element

Tensegric element

Cable

Tensegric element

Cable

(a) Cable-net

Cable Cable Cable ' Cable

(b) Tensegric elements build on cable-net

Figure 2: Quasi-tensegric system

CONCLUSION

The results of our numerical examples assess the validity of the iterative alternate method we developed. As far as unstable system are concerned this method could be useful and specifically in the field of tensegric systems. Quasi-tensegric system is a new system that has the advantage of cable-net and tensegrity. It will be applied widely large span structure in practice.

ACKNOWLEDGMENT

This work was supported by National Natural Science Foundation of China (Project number 59508010 ). We express heartfelt thanks.

REFERENCES

Liu Y. (1998). Analysis of unstable systems and of tensegrity by dynamic relaxation. Chinese Journal of Spatial structures, 4:3, 26-30 Liu Y. and Motro R.(1995). Shape analysis and internal force in unstable structures. Journal of Southeast University, No.IA, 262-267

Liu Y., Lu Z., Han X., Jing J. (1995). Analysis for unstable cable-nets under static wind loads. Journal of Southeast University, No.IA, 531-535

Pellegrino S. and Calladine C.R.(1986). Matrix analysis of statically and kinematically indeterminate frameworks, lnt. J. Solids Structures, 22:4, 409-428

Baron F. and Vendtesan M.S. (1971). Nonlinear analysis of cable and truss structures. Journal of the structure Division, ASCE, 97:2, 679-710


Connections


THE DESIGN OF PINS IN STEEL STRUCTURES

R.Q. Bridge

School of Civic Engineering and Environment, University of Western Sydney, Nepean, Kingswood, NSW 2747, Australia

ABSTRACT

There has been a recent upsurge in the use of pins, particularly in architectural steel structures with visible tension and compression members. However, the rules for the design of pins vary quite considerably from code to code and this has caused some confusion amongst consulting structural engineers operating internationally. A comprehensive testing program has examined the influence of such parameters such as pin diameter, material properties of the pin, thickness of the loading plates, material properties of the loading plates and the distance of the pin to the edge of the loading plates. Modifications to current design procedures are proposed that take into account the different possible modes of failure.

KEYWORDS

Bearing, design, failure, pins, shear, steel structures, strength, tests

INTRODUCTION

As they have no head and are not threaded, pins cannot carry any axial forces and can only carry shear forces transverse across the pin. Despite this limitation, they are often used in structural applications by designers and architects for steel structures with visible tension and compression members, particularly in applications such as canopies, sporting stadiums, convention centres and bridges. In these cases, the pins are essentially subjected to static conditions and rotations are generally small.

The design procedures for pins can be found in most structural steel codes, standards and specifications. However, there is some disparity in the design values for three major of the major design conditions: shear of the pin; bearing on the pin; and bearing on the plies (plates) that load the pin. For instance, the Australian Standard AS4100-1998 has an apparently high design value for the strength of a ply (plate) in bearing and yet a low value for the strength of a pin in shear whereas Eurocode 3-1992 has a low value for plate bearing strength but a high value for pin shear strength. To explore this disparity, the behaviour of pins under load has been examined experimentally to determine the effects of the material and geometrical properties of both the pin and the loading plate on the

229

230 R.Q. Bridge

strength and mode of failure of the pin or plate. The results have been compared with design values from steel design standards and modifications to the design procedures have been proposed.

TEST PROGRAMME

The first series of test specimens consisted of a snug-fit single pin loaded in double shear by an interior plate between two exterior cover plates as shown in Figure 1. The lower half of the specimen was bolted and was designed to have a greater capacity than the pin. The specimens were tested to failure under load control in a 580 kN capacity tensile testing machine. Deformations of the interior pin plate relative to the exterior cover plates were measured. The main variables tested were the pin diameter df (10, 16 and 27mm), the interior plate thickness tp (3, 6, 10, 16 and 20mm) and the material properties of the pin. The pins were cut from two types of commercially available steel rod: black rod with a high ductility and low ratio of yield strength fyp to ultimate tensile strength fuf; and bright rod with a lower ductility. Typical stress-strain curves are show in Figure 2. The plate steel had a similar behaviour to the black pin. The distance from the pin to the edge of the plate in the direction of loading was kept generally within the limits of AS 4100-1998 for end plate tear-out to limit this mode of failure.

'.] t ~ - "1

100

50

95

T

Top interior test plate

Width = 100mm

Test pin

Cover plates t = 12mm

50

70 [ [ [ 1[ ~ Two M20 8.8 bolts

50

100

Bottom interior plate

All dimension in mm

Figure 1. Test specimen for double shear

600

The Design of Pins in Steel Structures

400

r~

\ 16mm bright pin m 200

16mm black pin

0.0 1.0 2.0 3.0 Strain %

231

Figure 2. Stress-strain curves for the two types of steel

Hayward and van Ommen (1992) conducted the tests. The geometrical properties, material properties, test results and modes of failure for the 18 test specimens are shown in Table 1. The primary mode of failure was either shearing of the pin (shear deformation of the pin generally being 25% of the pin diameter or greater) or large bearing deformations of the plate (63% of the pin diameter or greater). In some cases, large plate bearing deformations were observed prior to final shearing of the pin. In other cases, fracture of the plate occurred at the cross-section through the pin. These failures have been labelled as secondary modes of failure. Pin bearing failures were not observed.

TABLE 1 THE GEOMETRICAL AND MATERIAL PROPERTIES AND THE TEST RESULTS OF THE PIN TEST SPECIMENS

Test Pin No. df

Innl 1 10.06 2 10.04 3 10.06 4 16.13 5 16.14 6 16.13 7 26.95 8 26.95 9 26.95 10 9.97 11 10.09 12 10.00 13 15.97 14 15.97 15 15.97 16 26.90 17 26.90 18 26.90

Pin Pin Plate Plate Plate ~f ~f thick f~ ~p

MPa MPa mm MPa MPa 250 455 3.12 360 496 250 455 5.97 310 469 250 455 9.85 260 485 300 499 3.23 360 496 300 499 10.05 260 485 300 499 15.86 250 460 270 485 3.12 360 496 270 485 9.9 260 485 270 485 19.93 250 446 480 558 3.14 360 496 480 558 6.12 310 469 480 558 10.11 260 485 460 523 3.16 360 496 460 523 9.85 260 485 460 523 15.9 250 460 450 524 3.12 360 496 450 524 10.17 260 485 450 524 19.87 250 446

*Pin also sheared 25% of diameter

Max. Hole Load Elong. kN 53.6 54.0 54.3 97.0

150.8 146.5 113.0 346.0 344.0

53.6 56.8 56.4 92.5

137.0 131.0 110.0 352.0 350.0

Primary Failure

%

63.4 Pin shear 7.9 Pin shear 0.4 Pin shear

126.4 Plate bearing 7.5 Pin shear 2.3 Pin shear


72.3 Pin shear 8.9 Pin shear 1.5 Pin shear


78.7 Plate bearing 125.7 Plate bearing

5.7 Pin shear

Secondary Failure

Plate beating

Plate fracture Plate bearing

Plate bearing

Plate fracture Plate fracture*

232 R.Q. Bridge

Typical load-deformation curves are shown in Figure 3 for 10mm and 16mm diameter bright pins in three different thicknesses of plate. The 16mm diameter pin in 3mm plate (specimen 13) exhibited a primary plate bearing failure whereas the 10mm pin in 3mm plate (specimen 10 exhibited a secondary bearing failure. With plate bearing failures, hole elongations in excess of 60% of the hole diameter are attained. The other specimens shown in Figure 3 exhibited pin shear failures. Pin shear failures are associated with shear deformation through the pin itself of 25% of the pin diameter or more prior to failure, even for the pins manufactured from bright steel with a lower ductility than the black steel.

150

lOO

o ....- ~ f

50 -- "- . . . . �9

AI m

r

3mm piate, 16mm pin - 10mm plate, 16mm pin x 16mm plate, 16mm pin =-- 3mm plate, 10mm pin o-- 6mm plate, 10mm pin ,,- - 10mm plate, 10mm pin

0 0 5 10 15

Deformation (ram)

Figure 3. Typical load-deformation behaviour for pin shear and plate bearing failures

The second test series under deformation control was conducted by Sukkar (1998). This examined the effect of the shape of eye-bars (see Figure 4), typically used at the end of tension members, on end tear- out failures. Standards such as AS4100-1998, BS5950-1990 and Eurocode 3-1992 require an elongated end on the eye-bar (D3 > D2) whereas AISC-1993 permits a simpler circular eye bar end (D3 = D2). The eye-bar dimensions, material properties and test results are shown in Table 2.

D1

to

Figure 4. Typical shape of eye-bars at end of pinned tension members

The Des ign o f Pins in S tee l S t ruc tures

TABLE 2 DIMENSIONS AND MATERIAL PROPERTIES AND TEST RESULTS FOR EYE-BAR SPECIMENS

233

Test Head Pin Plate No. Type df tp

ITlln mnl

19 Elong. 20.00 5.0 20 Circ. 20.00 5.0 21 Elong. 20.00 6.0 22 Circ. 20.00 6.0 23 Elong. 20.00 8.0 24 Circ. 20.00 8.0 25 Elong. 27.00 5.0 26 Circ. 27.00 5.0 27 Elong. 27.00 6.0 28 Circ. 27.00 6.0 29 Elong. 27.00 8.0 30 Circ. 27.00 8.0

*Failure not reached. Instron

Eye-bar dimensions Pin Pin Plate Plate Max. D1 DE O3 fyf fuf fyp fup Load mm mm mm MPa MPa MPa MPa kN 22.5 15.0 22.5 730 870 280 440 46.9 22.5 15.0 15.0 730 870 280 440 44.4 22.5 15.0 22.5 730 870 280 440 52.6 22.5 15.0 15.0 730 870 280 440 47.0 22.5 15.0 22.5 730 870 280 440 53.6 22.5 15.0 15.0 730 870 280 440 52.7 30.0 20.0 30.0 730 870 280 440 51.4 30.0 20.0 20.0 730 870 280 440 53.0 30.0 20.0 30.0 730 870 280 440 54.4 30.0 20.0 20.0 730 870 280 440 54.0 30.0 20.0 30.0 730 870 280 440 62.2 30.0 20.0 20.0 730 870 280 440 63.1

6027 testing machine disabled by frame error

Failure mode

Tear-out Tear-out Tear-out Tear-out Deform* Deform* Tear-out Tear-out Tear-out Tear-out Deform* Deform*

under deformation control.

COMPARISON WITH DESIGN METHODS

The possible of modes of failure considered by most design codes are shown in Figure 5.

Figure 5. Modes of failures in pin connections.

The dimensions of eye-bars and the design strengths for the conditions of pin shear, pin bearing and plate bearing according to Australian, European, British and American practice are listed in Table 3.

TABLE 3 COMPARISON OF DESIGN STRENGTHS IN STEEL CODES AND SPECIFICATIONS

Steel code tp D2 D3 D4 Pin shear Pin bearing Plate bearing

AS4100-1998" _>0.25D2 20 .67D, 2 1.OD1 _> 1.ODI VI = 0.62fyiA I Vb = 1.4frrdrt p Vb = 3.2f, pdrt p

Eurocode 3-1992 n.a. _>0.75dp _> 1.1dp 21.1dp Vr = 0.60furAr Vb = 1.5f~rdrt p Vb = 1.5fypdrt p

BS5950-1990 !_>0.25D2 20.67D~ 2 1.OD~ _> 1.OD~ Vr = 0.60frrAr Vb = 1.2fedrtp Vb - 1.2fypdrt p

AISC-1993 20.12D~ 20.67D~ =l.OD2 n.a. Vt = 0.60feAr Vb = 1.4f~rdrt p Vb = 1.4fypdtt p

*In addition, AS4100-1998 requires Vb - f , paetp for plate tear-out where ae is the clear distance from the pin to the edge of the plate in the direction of loading.

234 R.Q. Bridge

In Table 3, Ay is the cross-sectional area of the pin, df is the diameter of the pin, frY is the yield stress of the steel in the pin, fuy is the ultimate tensile strength of the steel in the pin, tp is the thickness of the load-bearing plate, fyp is the yield stress of the steel in the plate, and f,p is the ultimate tensile strength of the steel in the plate. Most codes are similar with two major exceptions: Eurocode 3 uses the ultimate tensile strength of the pin in calculating the shear strength of the pin (similar to that for bolt strength in most steel codes); and AS4100-1998 uses the ultimate tensile strength of the plate (and a large factor of 3.2) in calculating the bearing strength of the plate. Therefore only AS4100-1998 and Eurocode 3-1992 are considered in the following comparisons of codes with the test strengths.

TABLE 4 COMPARISON OF TEST RESULTS WITH DESIGN VALUES PREDICTED BY AS4100-1998.

Test Max Vf Load/Vf Vb Load/Vb Vb Load/Vb Vb Load/Vb No. Load Pin Pin Pin§ Pin§ Plate Plate Tear-out Tear-out

, k N , k N , k N k N ~ , k N ,

1 53.6 24.6 2.18" 11.0 4.88 49.8 1.08 139.2 0.38 2 54.0 24.5 2.20* 21.0 2.57 90.0 0.60 251.9 0.21 3 54.3 24.6 2.20* 34.7 1.57 153.8 0.35 429.8 0.13 4 97.0 76.0 1.28" 21.9 4.43 82.7 1.17 139.3 0.70 5 150.8 76.1 1.98" 68.1 2.21 251.7 0.60 423.7 0.36 6 146.5 76.0 1.93" 107.4 1.36 376.6 0.39 634.2 0.23 7 113.0 191.0 0.59 31.8 3.56 133.5 0.85* 126.2 0.90 8 346.0 191.0 1.81" 100.9 3.43 414.1 0.84 391.4 0.88 9 344.0 191.0 1.80" 203.0 1.69 766.6 0.45 724.7 0.47 10 53.6 46.5 1.15" 21.0 2.55 49.7 1.08 140.2 0.38 11 56.8 47.6 1.19" 41.5 1.37 92.7 0.61 2 5 8 . 2 0.22 12 56.4 46.7 1.21" 67.9 0.83 156.9 0.36 i 441.3 0.13 13 92.5 114.3 0.81 32.5 2.85 80.1 1.15" 136.4 0.68 14 137.0 114.3 1.20" 101.3 1.35 244.1 0.56 415.7 0.33 15 131.0 114.3 1.15" 163.5 0.80 373.8 0.35 636.4 ! 0.21 16 110.0 317.1 0.35 52.9 2.08 133.2 0.83* 126.2 0.87 17 352.0 317.1 1.11" 172.4 2.04 424.6 0.83 402.2 0.88 18 350.0 317.1 1.10" 336.7 1.04 762.8 0.46 722.7 0.48 19 46.9 284.4 0.16 102.2 0.46 140.8 0.33 49.5 0.95* 20 44.4 284.4 0.16 102.2 0.43 140.8 0.32 33.0 1.35" 21 52.6 284.4 0.18 122.6 0.43 169.0 0.31 59.4 0.89* 22 47.0 284.4 0.17 122.6 0.38 169.0 0.28 39.6 1.19" 23 53.6 284.4 0.19 163.5 0.33 225.3 0.24 79.2 0.68* 24 52.7 284.4 0.19 163.5 0.32 225.3 0.23 52.8 1.00" 25 51.4 518.3 0.10 138.0 0.37 190.1 0.27 66.0 0.78* 26 53.0 518.3 0.10 138.0 0.38 190.1 0.28 44.0 1.20" 27 54.4 518.3 0.10 165.6 0.33 228.1 0.24 79.2 0.69* 28 54.0 518.3 0.10 165.6 0.33 228.1 0.24 52.8 1.02" 29 62.2 518.3 0.12 220.8 0.28 304.1 0.20 105.6 0.59* 30 63.1 518.3 0.12 220.8 0.29 1304.1 0.21 70.4 0.90*

* Asterisk indicates mode of failure predicted by the AS4100-1998 + Pin bearing ignored in predicting failure as none was observed in tests

The predicted failure modes from Table 3 compare well with actual failure modes in Table 1. However, for test specimens 1, 2, 3, 5, 6, 8, 9, 10, 1 l, 12, 14, 15 and 18 where the actual primary failure was by pin shear, the strength of the pin in shear predicted by AS4100-1998 was markedly lower than the test strengths, particularly for the ductile pins made from black steel rod. For the test specimens l, 4, 7, 8, 10, 13, 16 and 17 where the primary or secondary failure was by bearing of the

The Design o f Pins in Steel Structures 235

plate, the strength predicted by AS4100-1998 was close to the actual test strengths. The predicted bearing strengths for specimens 7, 16, and 17 appear a little high because the full bearing strength of the plate was not attained in the test due to premature fracture of the plate adjacent to the hole. For the eye-bars where plate tear-out was both the predicted and the actual mode of failure, AS4100-1998 provided a reasonable estimate of the test strength taking the edge distance a3 = D3. However, it is interesting to note that the elongation of the eye-bar as used by AS4100-1998, BS5950-1990 and Eurocode-1992 did little to improve the strength of the eye-bar and its use should be questioned.

When Eurocode 3-1992 is compared with the test results in a similar manner to that shown in Table 4 for AS4100-1998, it is found that the failure modes predicted by the code strengths do not compare well with actual test failure modes shown in Table 1. However, for the test specimens 1, 2, 3, 5, 6, 8, 9, 10, 11, 12, 14, 15 and 18 where the primary failure was by pin shear, the strength of the pin in shear predicted by Eurocode 3-1992 was close to the actual test strengths. For the test specimens 1, 4, 7, 8, 10, 13, 16 and 17 where the primary or secondary failure was by bearing of the plate, the strength predicted by Eurocode 3-1992 was significantly lower than the actual test strengths.

DESIGN RECOMMENDATIONS

From the comparisons with AS4100-1998 and Eurocode 3-1992, it was clear that the AS4100-1998 provided the best model for plate bearing strength based on the ultimate strength of the steel in the plate whereas Eurocode 3-1992 provided the best model for the pin shear strength, again based on the ultimate strength of the steel in the pin. It is therefore proposed that the strength Vf of a pin in shear should be given by

Vf = 0.62fqAf (1)

This is similar to the strength of a bolt in shear as given in AS4100-1998. The shear factor of 0.62 on the ultimate tensile strength is used to give the shear strength of the steel in the pin. In the tests, the mean value of this factor for the ductile black steel pins was 0.71 with a coefficient of variation of 0.08 with factors ranging from 0.74 for the 10mm diameter pins to 0.62 for the larger 27mm diameter pins. For the lower ductility bright steel pins, the mean value of the factor was 0.63 with a coefficient of variation of 0.03 with factors ranging from 0.65 for the 10mm diameter pins to 0.59 for the larger diameter 27mm pins. It is also proposed that the strength of the plate in bearing should be given by

v~ = 3.2Lpdjtp (2)

This is identical to the current requirements in AS4100-1998 for both pins and bolts. The bearing factor of 3.2 on the ultimate tensile strength is used to give the bearing strength of the steel in the plate. In the two tests that had primary bearing failures without plate fracture, the mean value of the factor was 3.74. In the other three bearing failure tests where premature plate fracture occurred, the mean value of this factor was still 2.67, a value close to 3.2.

It is proposed that a new serviceability condition for plate bearing be included in design codes. As shown in Figure 2 for the 3mm plate that failed in bearing, the bearing deformations of the plate at maximum load are very large and typically exceed 60% of the hole diameter. Using a proof load at 2% of the hole diameter as the basis to define the maximum service load Vs that can be sustained prior to the onset of large plate bearing deformations, a mean design value of bearing strength Vb~ for serviceability conditions has been determined as

Vbs = 1.6fypdftp (3)

236 R.Q. Bridge

The value of the factor 1.6 was derived from the eight tests that had primary and secondary bearing failures. It is close to the factors shown in Table 3 for plate bearing strengths based on the yield strength, indicating that this should be a serviceability condition, not a strength condition. Comparisons of the proposals with the test results are given in Table 5. Values are shown for both primary and secondary failure modes and indicate reasonable agreement over the test range.

TABLE 5. COMPARISON OF TEST RESULTS WITH DESIGN VALUES PREDICTED BY MODIFICATIONS TO CODES

Test Max Vf Load/Vf Load/Vb Service Vb~ VflVbs Load Pin Pin Plate Load Vs Plate kN kN kN

i i

1 53.6 44.8 1.20 1.08 20 18.1 1.11 2 54.0 44.7 1.21 3 54.3 44.8 1.21 4 97.0 126.4 1.17 30 30.0 1.00 5 150.8 ! 126.6 1.19 6 146.5 126.4 1.16 7 113.0 343.1 0.85 38 48.4 0.78 8 346.0 343.1 1.01 0.84 125 111.0 1.13 9 344.0 343.1 1.00 10 53.6 54.0 0.99 1.08 20 18.0 1.11 11 56.8 55.3 1.03 12 56.4 54.3 1.04 13 92.5 129.9 1.15 i 29 29.1 1.00 14 137.0 129.9 1.05 15 131.0 i 129.9 1.01 16 110.0 369.3 0.83 42 48.3 0.87 17 352.0 369.3 0.95 0.83 130 113.8 1.14 18 350.0 369.3 0.95

*Plate bearing (+pin shear) was a secondary failure mode in the tests.

Predicted Failure

Pin shear* Pin shear Pin shear Plate bearing Pin shear Pin shear Plate bearing Pin shear* Pin shear Plate bearing Pin shear Pin shear Plate bearing Pin shear Pin shear Plate bearing Pin shear *+ Pin shear

CONCLUSIONS

Tests have highlighted some deficiencies in current codes that are used to predict the strength of pins in plated structures. Modifications have been proposed that better model the modes of failure. Plate tear- out is an important consideration. A new serviceability condition is proposed. Bearing of the pin was not identified as a mode of failure and this aspect needs further examination.

R E F E R E N C E S

American Institute of Steel Construction AISC-1993, Load and resistance factor design specification for structural steel buildings - Second edition. American Institute of Steel Construction, Chicago.

Australian Standard AS4100-1990. Steel structures, Standards Australia, Sydney. British Standard BS5950-1990. Structural use of steel in buildings, British Standards Institution,

London. Eurocode 3-1992. ENV 1993-1-1 Design of steel structures- Partl.l: General rules and rules for

buildings, European Committee for Standardization, Brussels. Hayward, I.G.and Van Ommen, M.(1992). Pins in steel structures. B.E. Thesis, University of Sydney. Sukkar, T. (1998). Pins in steel Structures. B.E.Thesis, University of Western Sydney, Nepean.

FINITE ELEMENT MODELLING OF EIGHT-BOLT RECTANGULAR HOLLOW SECTION

BOLTED MOMENT END PLATE CONNECTIONS

A. T. Wheeler 1, M. J. Clarke 2 and G. J. Hancock 2

1 Department of Civic Engineering and Environment, The University of Western SydneymNepean, Kingswood, N.S.W., 2747, Australia

2 Department of Civil Engineering, The University of Sydney, Sydney, N.S.W. 2006, Australia

ABSTRACT

This paper describes the finite element modelling philosophy employed to analyse bolted moment end plate connections joining square and rectangular hollow sections which are subjected to pure bending. The ABAQUS finite element package (HKS, 1995) is used to simulate the experimental behaviour observed in tests performed at the University of Sydney. The parameters varied in both the experiments and the ABAQUS simulations include the end plate thickness, the section shape (square or rectangular), and the position of the bolts. The results obtained from the finite element analyses are evaluated and the appropriateness of the model assessed by comparing the numerically predicted ultimate loads and moment-rotation responses with those of the corresponding tests. Overall, it is concluded that the numerical analysis is effective in modelling the behaviour of the connections, although there are some failure modes observed experimentally which could not be directly reproduced in the finite element models.

KEYWORDS

Bolted connections, end plate connections, tubular sections, moment-rotation behaviour, ABAQUS.

INTRODUCTION

The increase in the use of rectangular hollow sections in mainstream structures coupled with the economics of prefabrication have highlighted the need for simple design methods that produce economical tubular connections. Although tubular connection design handbooks have been published recently (Syam and Chapman, 1996; AISC, 1997), the eight-bolt moment end plate connection described in this paper is one configuration for which a design model is not widely available. A suitable model is described in the companion paper by Wheeler et al. (1999). The eight-bolt connection described in this paper and depicted in Figure la represents one of two fundamental bolting arrangements studied by Wheeler (1998). The other bolting arrangement utilises four bolts, as shown in Figure lb. The eight-bolt detail described in this paper is superior to the four-bolt variant from the point of view of connection strength and stiffness, but is nevertheless more costly.

237

238 A.T. Wheeler et al.

Figure 1: Typical applications of bolted moment end plate connections using RHS

Realistic modelling of bolted end plate connections is highly complex because the problems are three-dimensional in nature, and involve the added complications of geometric and material nonlinearities, and contact/separation between various components (Bursi and Jaspart, 1997a, 1997b). Bursi and Jaspart (1997a, 1997b) also highlight the importance of correct element selection to obtain accurate solutions, and have endeavoured to establish benchmarks that can be used to calibrate finite element models.

This paper describes the finite element modelling philosophy employed to analyse eight-bolt moment end plate connections joining square and rectangular tubes subjected to pure bending. The ABAQUS finite element package (HKS, 1995) is used to simulate the behaviour observed in tests performed at the University of Sydney (Wheeler et al., 1995). The parameters varied in both the experiments and the ABAQUS simulations include the end plate thickness, the section shape (square or rectangular), and the position of the bolts. The results obtained from the finite element analyses are evaluated and the appropriateness of the model assessed by comparing the numerically predicted ultimate loads and moment-rotation responses with those of the corresponding tests.

DEVELOPMENT OF FINITE ELEMENT MODEL

Overview

The generation of a three-dimensional finite element model of the bolted tubular end plate connection was carried out using the PATRAN pre-processor (PDA Engineering, 1994). The connections were analysed using the ABAQUS finite element software package (HKS, 1995). The analysis incorporated the effects of both material and geometric nonlinearities.

The finite element model of a typical eight-bolt end plate connection is shown in Figure 2, with the vertical axis of symmetry along the beam length being utilised to reduce the size of the model. To aid the model verification process, the connection was divided into five individual sub-models, each of which represents a specific component of the connection. These components are labelled in Figure 2.

The model employed solid three dimensional brick elements for each of the components, with additional interface elements used to model the contact/separation between various surfaces. The material properties used for the various components of the model were determined from the engineering stress-strain curves obtained through tensile tests (Wheeler et al., 1995). It should be noted that the incorporation of material nonlinearity in an ABAQUS model requires the use of the true stress (Otrue)

versus the logarithmic plastic strain ( el pl ) relationship.

Finite Element Modell&g of Bolted Moment End Plate Connections 239

Beam Section

To model the beam section, eight-noded linear brick elements were utilised as shown in Figure 2. These elements are of type C3D8 in ABAQUS terminology. Two section sizes were employed in the tests: a square hollow section (SHS) of nominal dimensions 150x150• mm, and a rectangular hollow section (RHS) of nominal dimensions 200•215 mm.

Figure 2: Finite element model of eight-bolt connection

The tubular sections employed in the bolted end plate connections were manufactured using a cold- forming process. As a result, the material in the comers of the section was of higher strength than the material in the fiats. Consequently, different material properties were assigned to the comer and fiat regions of the sections in the finite element analysis. The relevant stress-strain curves are depicted in Figure 3.

Figure 3: Typical section material properties

End Plate

The general layout and the corresponding dimensions of the end plates are given in Figure 1 and Ta- ble 1 of the companion paper (Wheeler et al., 1999). For all tests, the edge distance (ae) to the centre of the bolt holes was 30 mm, and the diameter of the holes was 22 mm for M20 bolts.

In the finite element simulations, the end plate was modelled using eight-noded linear hybrid bricks, corresponding to element type C3D8H in ABAQUS. This element type was selected to prevent possible problems of volume strain locking, which can occur in the C3D8 linear elements (HKS, 1995). Following a convergence study, it was decided to use four elements through the thickness of the end plate for all analyses. A typical layout is shown in Figure 2.

The measured stress-strain relationships of the end plates follow the classic elastic-plastic-strain hardening pattem. Since the measured yield stresses (fy) and ultimate tensile strengths (fu) for the dif-


ferent end plate thicknesses (12 mm, 16 mm and 20 mm) were all within 3 percent, an average stress- strain relationship was used for all plate thicknesses. This average stress-strain curve was based on a yield stress of 351 MPa and an ultimate tensile strength of 492 MPa.

Bolts

In many cases, the ultimate strength of the connection was limited by tensile fracture of the bolts rather than end plate or section failure. Therefore, to simulate the connection behaviour accurately, each bolt was modelled as a separate entity using the nominal cross-sectional areas and measured material properties. Reflecting the experimental behaviour of a bolt in tension, in the finite element analyses the bolts were deemed to fracture when the strain reached 3 percent (Wheeler, 1998).

The interaction (i.e. contact and separation) between the bolt and the end plate was modelled using the INTER4 cubic interface elements (HKS, 1995). These assemblages were positioned between the underside of the bolt head and the end plate, and also between the bolt hole in the end plate and the bolt shank. The interface elements between the underside of the bolt head and the end plate were implemented as a "rough" interface to prevent slipping between the surfaces. The assemblages of interface elements between the bolt shank and the bolt hole modelled a frictionless interface to prevent the "penetration" of the bolt into the end plate at high rotations.

Weld

The connection between the tubular section and the end plate consisted of a combination butt-fillet weld. The weld was modelled as an individual component using eight noded linear brick elements (C3D8) and six noded linear triangular prism elements (C3D6) to encompass the butt and fillet portions, respectively. The specified nominal material properties of the weld metal (fy = 428 MPa, fu = 528 MPa) exceed those of the tubular section and the end plate.

Initial Stresses and End Plate Deformations

The cold-formed tubular sections used in the end plate connections contain residual stresses as a result of the manufacturing process. Welding the end plate to the tubular section induces residual stresses and bowing deformations in the end plate. Bolt pre-tensioning introduces further initial stresses in the connection. These heat induced distortions and the consequent initial stresses in the end plate may have a significant effect on the stiffness of the connection as the subsequent bolt pre- tensioning induces stresses into the end plate through the clamping action.

In the finite element analyses, the heat-induced deformations of the end plate were modelled by simply displacing the initial geometry as shown in Figure 4. The internal residual stress state resulting from welding was not modelled. An initial transverse displacement of 80, the magnitude of which depends on the end plate thickness and is based on measurements of test specimens, is applied to all four edges of the end plate with a linear variation to zero initial displacement at the flanges and webs.

Although initial end plate deformations were incorporated in the finite element model, verification studies have shown that the initial deformations have only a minor effect on the overall moment- rotation response for the eight-bolt connections (Wheeler, 1998).

End Plate Thickness tp (ram)

12

16

20

Initial Deformation 80 (mm)

2.0

1.0

0.75

Figure 4: Imposed initial end plate deformations

Finite Element Modelling of Bolted Moment End Plate Connections 241

Loading

To model the complete behaviour of the connection, the loading was carried out in five steps as shown in Figure 5. In the first two steps, displacements are applied to close the nominal gap between the solid elements and the appropriate rigid surfaces. In the third step, a concentrated load was applied to the end of the bolts to produce the pre-tension of 145 kN as specified in AS4100 for a friction grip connection employing high strength bolts (SA, 1998). In the fourth step, the bolts ends were fixed in their pre-tensioned position and equilibrium re-established. In the fifth and final step, the rigid end cap was rotated, thus applying a moment to the beam section and the connection.

Initial State Step 1 Close bolt rigid surfaces

Step 2 Close end plate rigid surfaces

Step 3 Pre-tension bolts (load P)

IF Step 4

Fix bolt ends in pre-tensioned position Rotate end cap

Figure 5: Schematic representation of loading procedure

S I M U L A T I O N OF C O N N E C T I O N S

The experimental and numerical results for the eight-bolt connections are given in Table 1. Graphical comparisons for all tests are presented in Wheeler (1998). When comparing the results, it should be noted that in the experimental study the tests were terminated when either a punching shear failure had occurred (Wheeler et al., 1999), when the load cells indicated a drop in bolt load, or when the section formed a plastic hinge. In the numerical analysis, the ultimate load was deemed to occur when the bolts reached their predefined fracture strain (3 percent) or when the section failed plastically. Consequently, punching shear failure was not considered in the finite element model.

The agreement between the experimental (Mcu) and numerical (ABAQUS) (mab) ultimate moments is excellent as indicated in Table 1, with the mean and standard deviation of the experimental-to- numerical ratio (mcu/Mab) being 0.96 and 0.07, respectively. Furthermore, if the tests that failed as a result of punching shear are ignored in the comparisons (Tests 2, 5, 8, 9, and 10), the mean and standard deviation are improved to 1.01 and 0.03, respectively. The comparison of experimental and numerical overall moment-rotation responses was generally good for the RHS (see Figures 6 and 7), but only fair for the SHS. The numerical predictions of the theoretical model (Mth) which considers yield line analysis, the stub tee analogy, beam section plasticity and punching shear (Wheeler et al., 1999), are also given in Table 1. With a mean theoretical-to-experimental ratio (Mcu/Mth) of 1.03 and a standard deviation of 0.05, the theoretical model is evidently very effective (Wheeler et al., 1999).


TABLE 1 COMPARISON OF EXPERIMENTAL, NUMERICAL AND THEORETICAL ULTIMATE MOMENTS

Test

1 (SHS) 2 (RHS) 3 (SHS) 4 (SHS) 5 (RHS) 6 (RHS) 7 (SHS) 8 (SHS) 9 (RHS) 10 (RHS)

Test Mcu

(kNm) 116.0 (S) 124.5 (P) 93.9 (B)

116.0 (S) 92.7 (P)

136.7 (S) 113.2 (B)

97.6 (P) 133.0 (P) 119.3 (P)

ABAQUS Mab (kNm) 110.8 131.5 95.7

111.9 114.7 137.4 115.1 105.6 136.0 133.3

(P) = Punching shear failure (S) = Section capacity failure (B) = Failure by yield line formation

and bolt fracture

Theoretical Mth

(~qrn) 116.3 (S) 116.8 (P) 92.8 (B)

116.3 (S) 87.6 (P)

128.4 (S) 116.3 (S) 104.9 (B) 123.2 (P) 110.0 (P)

Mean S.D.

Mcu/Mab

1.05 0.95 0.98 1.04 0.81 0.99 0.98 0.92 0.98 0.89 0.96 0.07

MeulMth

1.00 1.07 1.01 1.00 1.06 1.06 0.97 0.93 1.08 1.08 1.03 0.05

Figure 6: Effect of variation in end plate thickness for RHS connections

The numerical analyses demonstrate that the flexibility and strength of the connection depends on the flexibility of the end plate. This flexibility is a function of the thickness of the end plate and the position of the bolts relative to the section perimeter.

The effect of varying the end plate thickness is shown in Figure 6, in which the connection moment- rotation behaviour is presented for Tests 5, 2 and 6 which comprise end plate thicknesses of 12 mm, 16 mm and 20 mm, respectively. These three tests differ only in end plate thickness. A significant increase in the overall stiffness and strength is observed with an increase in the end plate thickness. In both the physical test and the ABAQUS model, the 20 mm end plate connection (Test 6) failed through the attainment of full plasticity in the beam section rather than the failure occurring in the connection itself. Conversely, the 16 mm and 12 mm end plate connections (Tests 2 and 5) failed through punching shear in the physical experiments, but are predicted to fail as a result of the bolts attaining their assumed fracture strain of 3 percent in the ABAQUS model. The ramifications of the inability of the ABAQUS model to consider the punching shear failure mode are particularly apparent for Test 5 (12 mm end plate) as indicated in Figure 6.

Finite Element Modelling of Bolted Moment End Plate Connections 243

Figure 7: Effect of bolt position on moment-rotation behaviour for RHS connections

The stiffening effect of the position of the bolts relative to the section perimeter is illustrated in Fig- ure 7. The three simulations presented in this figure have a constant end plate thickness of 16 mm, with the distance to the perimeter of the section (So) being varied. Increasing the value of So reduces the stiffness of the end plate, thus resulting in a more flexible moment-rotation response and lower ultimate strength (compare Tests 10 and 2 for which So = 45 mm and 35 mm, respectively).

As can be seen in Figures 6 and 7, the finite element analysis is reasonably effective in simulating the experimental moment-rotation response for the RHS connections. Generally the computed response is marginally stiffer than the experimentally measured one. However, the SHS connections (Tests 1, 3, 4, 7 and 8) are generally significantly stiffer in the finite element simulations than in the tests (Wheeler, 1998). It is believed that the additional stiffness in the SHS connections is associated with inadequate modelling of the bolts and their interaction with the end plate. The bolts in the SHS connection are positioned such that they restrain the comers of the section (i.e. the line of restraint between adjacent bolts passes through the comer of the section). Conversely, the positioning of the bolts in the RHS connections offers less restraint to the comers of the section, thus enabling a greater degree of flexibility within the end plate.

As can be seen in Figure 8, the yield mechanisms in the end plates vary depending on the shape of the beam section (SHS or RHS), which defines the positions of the bolts. For both the SHS and RHS, the pitch of the four bolts above and below the axis of bending is approximately constant. The distance between the bolts adjacent to the section webs varies according to the depth of the section. This distance was generally either 90 mm for the SHS or 170 mm for the RHS. The close proximity of the bolts in the SHS models causes high concentrations of stresses to form around the perimeter of the section and between the tensile bolts (Figure 8a). On the other hand, the additional spacing between the bolts in the RHS allows the formation of a horizontal yielded zone in the end plate at mid-depth (Figure 8b). These areas of high stress concentration observed in the finite element results are consistent with the yield line patterns observed experimentally and determined theoretically (Wheeler, 1998).

CONCLUSIONS

A numerical study of the behaviour of tubular bolted moment end plate connections has been described in this paper. The analyses were conducted using the commercially available finite element package ABAQUS. Brick elements were chosen to form the basis of the models used for this study as


Figure 8: Von Mises stresses (MPa) illustrating end plate yield line patterns

this type of element is easily adapted to model the interfaces between the connecting surface and the end plates and bolts.

Overall, the models simulated the behaviour of the eight-bolt connections well, with the mean and standard deviation of the ratio of the experimental and numerical ultimate moments being 0.96 and 0.07. Comparisons of the experimental and numerical moment-rotation responses of the connections were excellent for the eight-bolt connections comprising the RHS. The eight-bolt connections utilising the SHS were generally predicted to be stiffer than the corresponding test results. Although not fully investigated in this paper due to time constraints, it is thought that this additional stiffness may be due to the inadequate modelling of the bolts.

Although the predicted ultimate loads generally corresponded well with the experimental results, the numerical analyses did not specifically model the effects of punching shear (although the effects of shear yielding were of course modelled in the nonlinear material behaviour). The deformation and yielding patterns developed in the models correlated well with the experimental results and the yield line analyses developed in the corresponding theoretical models (Wheeler et al., 1999).

R E F E R E N C E S

AISC (1997). Hollow Structural Sections Connections Manual, American Institute of Steel Construction, Inc.

Bursi, O. S. and Jaspart, J. P. (1997a). Benchmarks for Finite Element Modelling of Bolted Steel Connections. Journal of Constructional Steel Research, Elsevier, 43:1, 17-42. Bursi, O. S. and Jaspart, J. P. (1997b). Calibration of a Finite Element Model for Isolated Bolted End Plate Steel Connec- tions. Journal of Constructional Steel Research, Elsevier, 44:3, 225-262.

HKS (1995). ABAQUS/Standard Users Manual, Version 5.5, Hibbitt, Karlsson and Sorensen, Inc.

PDA Engineering (1994). PATRAN 3, PDA Engineering, Costa Mesa, California.

SA (1998). AS 4100-1998: Steel Structures, Standards Australia, Sydney.

Syam, A. A. and Chapman, B. G., (1996). Design of Structural Steel Hollow Section Connections. Volume 1: Design Models, 1 st Edition, Australian Institute of Steel Construction, Sydney.

Wheeler, A. T. (1998). The Behaviour of Bolted Moment End Plate Connections in Rectangular Hollow Sections Sub- jected to Flexure. PhD Thesis, Department of Civil Engineering, The University of Sydney.

Wheeler, A. T., Clarke, M.J. and Hancock, G.J. (1995). Tests of Bolted Flange Plate Connections Joining Square and Rectangular Hollow Sections. Proceedings, Fourth Pacific Structural Steel Conference, Singapore, 97-104.

Wheeler A. T., Clarke M. J. and Hancock G. J. (1999). Analytical Model for Eight-Bolt Rectangular Hollow Section Bolted Moment End Plate Connections. Proceedings, Second International Conference on Advances in Steel Structures, Hong Kong, December.

F I N I T E E L E M E N T M O D E L L I N G O F D O U B L E B O L T E D C O N N E C T I O N S B E T W E E N C O L D - F O R M E D S T E E L S T R I P S U N D E R

S T A T I C S H E A R L O A D I N G

K.F.Chung 1 and K.H.Ip 2

t Department of Civil and Structural Engineering; 2 Department of Mechanical Engineering, the Hong Kong Polytechnic University, Hung Horn, Hong Kong.

ABSTRACT

In a complementary paper 1, it was reported that a finite element model with three- dimensional solid elements was successfully established to investigate the bearing failure o f bolted connections between cold-formed steel strips and hot rolled steel plates under static shear loading. Non-linear material geometrical and contact analysis was carried out to predict the load-extension curves o f bolted connections with cold-formed steel strips of high yield strength and low ductility. The predicted load-extension curves were found to follow closely the measured load-extension curves, and both the maximum load carrying capacities and the initial extensional stiffness were satisfactorily predicted

In this paper, the finite element model is further extended to examine the structural behaviour of bolted connections with two bolts, or double bolted connections between cold-formed steel strips and hot rolled steel plates under static shear loading. The effects o f strength degradation, hole clearance and bolt spacing on the load carrying capacity of double bolted connections are discussed. Comparison on the predicted load carrying capacity o f the finite element model with the bearing resistances given by the design rules from both BS5950: Part 5 2 and Eurocode 3: Part 1.3 3 is also presented.

KEYWORDS

Cold-formed steel, bearing failure, double bolted connections, high strength steel with low ductility.

INTRODUCTION

Galvanized cold-formed steel strips are commonly used in building construction, such as sections for secondary steel frames and purlins, and sheetings for roof cladding and floor decking. Cold-formed steel sections and sheetings are effective construction materials due to their high strength to weight ratio, high buildability during construction and also long-term durability against environmental attack. In building construction, cold-formed steel sections are usually bolted to hot rolled steel plates or members to form simple and moment connections.

With the development of material technology, high strength cold-formed steel products are available for building applications, but concern has been raised on the reduced ductility of the high strength steel (< 5%). Existing codified design rules 2-5 may not be necessarily

245

246 K.F. Chung and K.H. Ip

applicable for high strength low ductility steel, as the design rules are developed with low strength high ductility steel 6,7. Consequently, a close examination s on the resistance and the associated failure modes of bolted connections with high strength low ductility steel strip was carded out.

Three distinct failure modes were identified 1 from the finite element modelling, namely, (i) the bearing failure, (ii) the shear-out failure, and (iii) the net-section failure. Parametric runs 9 were also carded out to reveal the effects of geometrical and material properties on the resistances of different failure modes. It is found that while the existing design rules are sufficient for bolted connections with low strength steels, such as steel with yield strength at 280 N/mm 2 and 350 N/ram 2, they may not be conservative when applying to high strength low ductility steel.

In this paper, the finite element model is further extended to examine the structural behaviour of bolted connections with two bolts, i.e. double bolted connections between cold-formed steel strips and hot rolled steel plates under static shear loading as shown in Figure 1. The effects of strength degradation, hole clearance and bolt spacing on the load carrying capacity of typical double bolted connections are presented. The predicted load carrying capacity of the finite element model is also compared with the bearing resistances given by the design rules from both BS5950: Part 5 and Eurocode 3: Part 1.3; comparison with test data 10 is also presented.

FINITE ELEMENT MODELLING

The finite element package ANSYS (Verison 5.3) is used to predict the bearing behaviour in double bolted connections between cold-formed steel strips and hot rolled steel plates under static shear loading, and the following areas of interest are examined in detail:

a) Stress-strain curves

Two different stress-strain curves are proposed for the model as illustrated in Figure 2: �9 bi-linear elastro-plastic curve for low strength high ductility steel, designated as FEA-

Pr, �9 multi-linear elastro-plastic curve with strength degradation at large strain for high

strength low ductility steel, designated as FEA-pr.

b) Deformation Sequences

Due to the presence of clearance in bolt holes for easy installation, it is possible that the two bolts may not always come into contact with the cold-formed steel strips at the same time. The bolts may have a hole clearance of 1 mm to 2 mm typically. In order to examine the effect of hole clearance to the structural performance of the double bolted connection, three deformation sequences are considered as follows:

�9 Deformation sequence IA where Bolt 1 is always in direct contact with the cold- formed steel strip while Bolt 2 only comes into contact with the cold-formed steel strip aRer I mm (or 2 mm) extension.

�9 Deformation sequence IB which is similar to that of Deformation sequence 1,4 but with reverse order of bolts in contact, i.e. where Bolt 2 is always in direct contact with the cold-formed steel strip while Bolt 1 only comes into contact with the cold-formed steel strip after 1 mm (or 2 mm ) extension.

Finite Element Modelling of Double Bolted Connections 247

�9 Deformation sequence 11where both Bolts 1 and 2 always come into contact with the cold-formed steel strip together.

c) Bolt spacing

In BS5950: Part 5, the minimum bolt spacing Sp is recommended to be not less than 3 d, and the total load carrying capacity of a connection with multiple bolts may be obtained directly as the sum of the bearing resistances of all the bolts. No adverse interaction between bolts should be allowed for and this design method seems satisfactory for low strength high ductility steel. However, for high strength low ductility steel, it is necessary to investigate the minimum bolt spacing to avoid any adverse interaction of yield zones of the two bolts.

As the connection contains a plane of symmetry, the half model shown in Figure 3 is incorporated. The cold-formed steel strip, the hot rolled steel plate and the two bolt-washer assemblies are represented three-dimensionally by eight-node iso-parametic solid elements SOLID45, as they allow both geometric and material non-linearities. Contact between the various components is accomplished by employing contact elements CONTACT49. Shear load is applied to the finite element model by imposing incremental displacement to the end of the cold-formed steel strip, along the longitudinal direction of the model. Throughout the entire deformation range, the hot rolled steel plate and the root of the bolt are fixed in space. At present, only the bearing failure of double bolted connections is considered.

In typical fmite element models, there are over 3724 nodes, 2422 solid elements and 2022 contact elements. As the model is highly non-linear, the full Newton-Raphson procedure is employed to obtain solution after each displacement increment. For detail of the finite element model, see Reference 8.

RESULTS AND DISCUSSIONS

The load-extension curves for the double bolted connection with different stress-strain curves, deformation sequences and bolt spacings are presented in Figure 4. The von Mises stress distribution of the double bolted connections at various extensions are presented in Figure 5 while the deformed mesh of the double bolted connection is presented in Figure 6.

a) Stress-strain curves

With Sp = 3 d and Deformation sequence 11, the load carrying capacity of the connection is estimated to be 31.10 kN with material curve FEA-py, and 28.08 kN with material curve FEA-pr, as illustrated in Figure 4a. It is thus shown that the strength of the connection may be reduced by 10% when strength degradation is considered in high strength low ductility steel.

b) Deformation sequences

In Figure 4b, it is shown that the load-extension curves derived from both Deformation sequences IA and 1B follow each other fairly closely along the entire deformation range. By plotting the load-extension curve derived from Deformation sequence 11 on the same graph for direct comparison, it is shown that both the load carrying capacity and the extensional stiffness of the connection will be reduced approximately by half if only one bolt is in contact with the cold-formed steel strip. However, at 3 mm extension, the load


carrying capacity with Deformation sequences IA and IB are found to be 26.69 kN with 1 mm gap and 24.10 kN with a 2 mm gap, corresponding to a strength reduction of 0.95 and 0.85 respectively.

c) Bolt spacing

In Figure 4c, it is shown that with Deformation sequence 11, the load carrying capacity of the double bolted connection is found to be increased from 28.08 kN at Sp = 3 d or 36 mm to 31.82 kN at Sp = 4 d or 48 mm, i.e. an increase of 13% in strength. A close examination on the von Mises stress distribution of the cold-formed steel strip in Figure 5 reveals that under low applied loads, the yield zones in the cold-formed steel for both bolts are fairly localized around the bolt holes. However, under large applied load at 3 mm extension, it is evident that the yield zones of both bolts overlap, leading to significant reduction to the total load carrying capacity of the connection. Consequently, in bolted connections with high strength low ductility steel, it is recommended that the minimum bolt spacing should be 4 d.

COMPARISON WITH DESIGN RULES

In order to provide simple design rules in assessing the bearing resistance, Pb , of double bolted connections with high strength low ductility steel, a number of existing design rules are examined as follows:

Pb = (1.64 + 0.45 t ) t dpy = 2 . 5 t d p y = ( 4 - 0 . 1 d / t ) t dpy

from clause 8.2.5.2 of BS5950:Part 5 (A) from clause 8.4(4) with Table 8.4, EC3: Part 1.3 03) from page 133 & Table 4.12, Volume 1 of Reference 10 (C)

Substituting the numerical values of t = 0.99 mm, d = 12 mm and replacing py with f~ = 592 N/mm" (where py and f~ are the yield strength and the tensile strength respectively) into the above design rules, the beating resistances are summarized in Table 1 together with the f'mite element results. Based on the results from the present research project, it is shown that

a) Existing design rules tend to over-estimate the bearing resistances of bolted connections with high strength low ductility steel up to 30 % for both single and double bored connections when compared with test results.

b) The results from the finite element models are found to be conservative when compared with test results.

c) R is necessary to allow for adverse interaction of yield zones around boR holes indouble bored connections. At a boR spacing of 3 d, the reduction factor is estimated to be 27.13 / (2 x 14.43) or 0.94 based on test results, or 26.72 / (2 x 14.54) or 0.92 based on finite element results. Thus, a value of 0.90 is recommended for design purpose. Alternatively, the minimum boR spacing, Sp, for no adverse interaction should be increased and Sp = 4 d is recommended as appropriate.

CONCLUSIONS

A finite element model is presented to examine the structural performance of the bearing failure in double bolted connections between cold-formed steel strips and hot rolled steel plates under static shear loading. By incorporating bolt solid and contact elements, the model

Finite Element Modelling o f Double Bolted Connections 249

is demonstrated to able to capture non-linearities associated with geometry, material and contact (boundary) conditions. It is shown that existing design rules may not be applicable for high strength low ductility steel and new design rules are required to ensure structural adequacy. The bearing resistances of double bolted connections may be reduced by 10 % to 30 % due to strength degradation, hole clearance, and also adverse interaction of yield zones.

ACKNOWLEDGEMENT

The research project leading to the publication of this paper is supported by the Hong Kong Polytechnic University Research Committee (Project A/C code G-$565).

REFERENCES

1. Ip K.H. and Chung, K.F.: Failure modes of bolted cold-formed steel connections under static shear loading, Proceeding of the Second International Conference on Advances in Steel Structures, Hong Kong, December 1999.

2. BS5950: Structural use of steelwork in buildings: Part 5 Code of practice for the design of cold- formed sections, British Standards Institution, London, 1998.

3. Eurocode 3: Design of steel structures: Part 1.3: General rules - Supplementary rules for cold- formed thin gauge members and sheeting, ENV 1993-1-3, European Committee for Standardization.

4. Cold-formed steel structure code AS/NZ 4600: 1996, Standard Australia/Standards New Zealand, Sydney, 1996.

5. Toma, A.W., Sedlacek, G., and Weynand, K.: Connections in cold-formed steel, Thin Walled Structures, Vol. 16, pp219-237, 1993.

6. Holcomb, B.D., LaBoube, R.A., and Yu, W.W.: Tensile and bearing capacities of bolted connections, Final Summary Report, Civil Engineering Study 95-1, Cold Formed Steel Series, Centre for Cold Formed Steel Structures, Department of Civil Engineering, University of Missouri-Rolla, MO, U.S.A.

7. Rogers, C. A. and Hancock, G. J.: New bolted connection design formulae for G550 and G300 sheet steels less than 1.0 mm thick, Research Report No. R769, the Centre for Advanced Structural Engineering, University of Sydney, Sydney, Australia, 1998.

8. Chung, K.F. and Ip, K.H.: Finite element modelling of bolted connections between cold-formed steel strips and hot rolled steel plates under shear, Engineering Structures (to be published).

9. Chung K.F. and Ip, K.H.: Finite element modelling of cold-formed steel bolted connections, Proceedings of the Second European Conference on Steel Structures, Praha, May 1999, pp503 to 506.

10. Rogers, C. A.: Structural behaviour of thin sheet steels, Ph.D. dissertation, Department of Civil Engineering, the University of Sydney, Australia, 1998.

Table 1 Summary of bearing resistances - Design rules vs Finite element analysis

(A)

Single bolts e~(w0 14.77

03) 17.58 (C) 19.61 Finite element model (15.90 1.36) = 14.54

+Test value, Pr I 14. 43

Pr/Pb 0.977 0.821 0.736 0.992

I -

Double bolts Pb (ld~ Pr / Pb 29.54 0.918 35.16 0.772 39.21 0.692

(28.08-1.36) = 26.72 1.015

I 27.13 ] -

Note: * The model incorporates FEA-pr stress-strain curve, Deformation sequence 11 and Sp at 3d. A frictional force of 1.36 kN at zero extension is deducted from the load carrying capacity of the predicted resistance for direct comparison.

+ Averaged values from three test data in Table B55, Page 331 of Volume 2, Reference 10.


Double bolted connection : 100-G550-B2- 48 • 95 -M12 (page 38 of Reference 7).

Thickness, t = 0.99mm; Bolt diameter, d = 12mm, and Bolt spacing, Sp = 36mm.

Figure 1 Geometry of a double bolted connection

Figure 2 Proposed stress-strain curves for high strength low ductility cold-formed steel strips, FEA-pr and FEA-pr

Finite Element Modelling of Double Bolted Connections

35

30

25 Z �9 -~ 20 "o m 15 O -J 10

5

0

J

. oo ~ ; ~ ~ - ' - -

/

FEA-pr L . . . . FEA-py I

/ i 0 0.5 1 1.5 2 2.5 3

E x t e n s i o n ( m m )

Figure 4(a) Load-extension curves for double bol ted connect ions with different stress-strain curves (Bolts 1 and 2 in contact with CFS at the same time)

31.10kN 28.08kN

251

35- L

30 2 mm gap

25 Z

20 "O m 15 O

_! 10 i r ~ . . . . Bolt 1 first in contact 5 ~ Bolt 2 first in contact

Single bolt 0 i J

0 0.5 1 1.5 2 2.5 E x t e n s i o n ( m m )

Figure 4(b) Load-extension curves for double bolted connection with different deformation sequences

1 mm gap~ ~

FEA-pr

28.08kN 26.69kN 24.10kN

15.90kN

35

30

A 25 z

20 -o m 15 0 ,_1 10

5

0

j ] Sp=36m m . . . . Sp=48mm

, 1

0 0.5 1 1.5 2 2.5 E x t e n s i o n ( m m )

Load-extension curves for double bolted connection with different bolt spacing, Sp (Bolts 1 and 2 in contact with CFS at the same time)

o 31.82kN _ ~

. . . . . 28.08kN

Figure 4(c)

252

Figure 5

K.F. Chung and K.H. Ip

Distribution of von Mises stress of the double bolted connection

Figure 6 Deformed mesh of double bolted connection at 3mm extension (Failure mode - bearing failure of CFS strip)

ANALYTICAL MODEL FOR EIGHT-BOLT RECTANGULAR HOLLOW SECTION

BOLTED MOMENT END PLATE CONNECTIONS

A. T. Wheeler l, M. J. Clarke 2 and G. J. Hancock 2

~Department of Civic Engineering and Environment, The University of Western Sydney--Nepean Kingswood, N.S.W., 2747, Australia

2Department of Civil Engineering, The University of Sydney, Sydney, N.S.W., 2006, Australia

ABSTRACT

The increase in the use of rectangular hollow sections in mainstream structures has highlighted the need for simple design methods for the production of economical connections. This paper presents a new model for the determination of the serviceability limit moment and the ultimate moment capacity of bolted moment end plate connections utilising rectangular hollow sections and eight bolts positioned in an approximately equidistant sense around the perimeter of the section. The model considers the combined effects of prying action due to flexible end plates, and the formation of yield lines in the end plate. Failure modes involving plate yielding, bolt fracture, punching shear and beam section capacity are considered.

The model has been calibrated and validated using experimental data from an associated test program. The model constitutes a relatively simple method for predicting the serviceability limit moment and ultimate moment capacity of moment end plate connections utilising square and rectangular hollow sections and eight bolts.

KEYWORDS

Tubular, connections, moment end plate, structural design, prying, yield line.

INTRODUCTION

The use of moment end plate connections joining I-section members and their corresponding structural behaviour has been well documented (Murray, 1990). Contrastingly, research on end plate connections joining rectangular and square hollow sections has been limited and consequently few design models are available for routine use. Furthermore, documented studies have concentrated primarily upon pure tensile loading, or combined compression and bending, as in a column-to-column bolted flange splice connection (Packer et al., 1989; Kato and Mukai, 1991).

The eight-bolt moment end plate connection described in this paper and depicted in Figure I has a similar layout to that used by Kato and Mukai (1991), and represents one of two fundamental bolting arrangements studied by Wheeler (1998). The other bolting arrangement utilises four bolts, with the corresponding design model described in Wheeler et al. (1998).

253


Figure 1" Typical eight-bolt end plate application and layout

The theoretical model presented in this paper pertains to tubular eight-bolt end plate connections subjected to flexural loading. The model determines the yield moment of the connection using yield line analysis, and combines the yield line analysis with stub-tee analysis to predict the ultimate strength of the connection. Two additional failure modes observed in the experimental program, namely section capacity and punching shear, have also been included in the theoretical model. Full details of the derivation of the model are given in Wheeler (1998). The predictions of the model are compared with the results obtained from an associated experimental program (Wheeler et al., 1995).

EXPERIMENTAL PROGRAM

An experimental program in which ten eight-bolt connections were tested has been conducted at the University of Sydney (Wheeler et al., 1995). The connections were loaded in pure flexure by subjecting a beam, with a splice connection at mid-span, to four-point bending. As the sections were not susceptible to local buckling, the ultimate load of the specimen was limited to connection failure, which occurred due to tensile bolt fracture, excessive end plate deformations, section failure or punching shear failure. The experimental ultimate moment (Mcu) and the failure mode for each test are listed in Table 1. The end plate material properties of yield stress (fy) and ultimate tensile strength (fu), and the beam section dimensional details and measured ultimate moment capacity (Mus) are given in Table 2.

The parameters varied in the experimental program are also given in Table 1 and include the plate size (Wp, Dp), the plate thickness (tp), the section shape, and the positions of the bolts with respect to the section flange and web (So and g). The bolt and nut assemblies were M20 structural grade 8.8 (Grade

TABLE 1 END PLATE CONNECTION DETAILS AND TEST RESULTS

Specimen No.

Section Type SHS

2 RHS 3 SHS 4 SHS 5 RHS 6 RHS 7 SHS 8 SHS 9 RHS 10 RHS

Pla~ Dimensions(mm) Mcu Wp Dp So ~ (kNm)

16 280 280 35 30 116.0 16 230 330 35 15 124.5 12 280 280 35 30 93.9 20 280 280 35 30 116.0 12 230 330 35 15 92.7 20 230 330 35 15 136.7 16 260 260 25 35 113.2 16 300 300 45 25 97.6 16 210 310 25 20 133.0 16 250 350 45 10 119.3

Failure Mode*

Bolt Punching

Bolt Bolt

Punching Bolt Bolt

Punching Punching Punching

* Punching = Failure by section tearing away from plate at toe of weld (punching shear). Bolt = Failure by bolt fracture.

Analytical Model for Bolted Moment End Plate Connections

TABLE 2 END PLATE MATERIAL PROPERTIES AND BEAM SECTION DETAILS

255

End Plate Properties Beam Section Details tp (mm) fy (MPa) fu (MPa) Section Depth d (mm) Width b (mm) Thickness ts (mm) Mus (kNm)

12 354 499 SHS 151.0 150.9 9.0 119 16 349 482 RHS 199.5 101.5 9.1 138 20 351 496

8.8/T), with a measured yield strength and ultimate tensile strength of 195 kN and 230 kN, respectively. The connections were prefabricated using a combination fillet/butt weld joining the section to the end plate, with a nominal fillet leg length of 8 mm.

YIELD LINE ANALYSIS

The yield line analysis serves primarily to determine the failure mode of the end plate, with prying action of the bolts ignored. As a secondary function, the analysis provides an estimate of the yield moment of the connection (Mcy). To determine the critical yield line pattern, numerous plastic mechanisms were considered. Most of these entailed relatively complicated patterns and resulted in lengthy expressions for the collapse moment (Myl). The derivations of the collapse moments for the different mechanisms considered are given in Wheeler (1998). The three most critical end plate mechanisms are presented in Figure 2. For each test, the experimental yield moment (Mcy) and the corresponding calculated yield moments (My0 are presented in Table 3, with the critical mode highlighted. The yield mechanism termed "Mode 8" in fact corresponds to beam yield capacity, determined using the measured yield stress of the tubular section.

Figure 2: End plate yield line mechanisms

TABLE 3 THEORETICAL AND OBSERVED RESULTS FOR CONNECTION YIELD MOMENTS


It can be seen in Table 3 that the majority of the tests were govemed by section yielding (Mode 8). Additionally, the calculated yield moments for Modes 4 and 5 are virtually identical.

CUMULATIVE MODIFIED STUB-TEE METHOD

To consider both the combined effects of bolt prying and end plate yielding on the ultimate capacity of the connection, a modified version of the stub-tee analogy is employed. Stub-tee analogies have been used extensively to determine the strength of end plate connections in I-sections (Nair et al., 1974; Kennedy et al., 1981). Generally the stub-tee utilises a simple rigid plastic analysis of an analogous beam that represents the one-dimensional behaviour of the end plate, with yield lines parallel to the axis of bending only. However, in the eight-bolt tubular end plate connections bending occurs about two axes, with the yield lines not necessarily being parallel to either axis of bending. The model presented in this paper is consequently termed the "cumulative modified stub-tee method", and is based on the analysis of analogous beams in both orthogonal directions. The principle of superposition is then used to obtain the resultant connection behaviour.

Figure 3: Analogous beams for cumulative stub-tee model

Simple representations of the analogous beams used in the cumulative modified stub-tee method are shown in Figure 3. The beam referred to as "in-plane bending" models the effect of the bolts below the flange of the section, with plastic hinges forming at points 1, 2 and 3 as shown in Figure 3a. The beam referred to as "out-of-plane bending", models the effect of the bolts lying on either side of the section webs. In this case, plastic hinges are assumed to form at points 4 and 5 on both sides of the hollow section, as indicated in Figure 3b. To simplify the problem, the bolts above the neutral axis are assumed to have a negligible effect on connection strength and are ignored.

As defined by Kennedy et. al, (1981) the behaviour of the end plate may be defined as thick plate behaviour, intermediate plate behaviour and thin plate behaviour, depending on the thickness of the end plate (tp) and the magnitude of the applied load. In the cumulative stub-tee model, these categories are

Analytical Model for Bolted Moment End Plate Connections 257

identified by the position and number of yield lines. Thick plate behaviour occurs when the connection fails due to bolt fracture, with a yield line forming only at point 1. Intermediate plate behaviour occurs when the bolts fracture after the formation of yield lines at points 1, 2 and 4 (i.e. plastic mechanism 5). Thin plate behaviour corresponds to the formation of yield lines at points 1, 2, 3, 4 and 5 in the end plate (i.e. plastic mechanism 2), without deformation of the bolts.

To determine the moment capacity for the thick, intermediate and thin modes of behaviour, the analogous beams are analysed using statics as described by Wheeler (1998). The resulting capacities are given by Equations 1-3 following, in which it is assumed that the moment generated by the bending of the bolts is

m b = ~Td~b3fyb/32 (where db = bolt diameter, fyb = bolt yield stress), and Mip is the plastic moment for the

i th yield line. It is also assumed that the bolts below the flange reach their ultimate load, while those beside the webs of the section only reach a proportion (h) of their ultimate load based on their distance from the axis of rotation, h = (d- g)/(d +Soi).

/Mlp +2.B~ "(d +Soi +h.d)l . (d_ts ) (l) M Cthick = d

Mcint = (ap + Soi) +2. (ap+Soo) Jr d .(d-t~) (2)

Mlp +M2p Msp +M2p +M b M3p +M2p +2.M b / Mcthi" = d +2. Soo + ~:So: . (d-t ,) (3)

/

Since the yield lines invariably undergo significant rotations prior to the ultimate strength being reached, much of the material is stressed into the strain-hardening range. Consequently, the plastic moment Mip is defined in terms of a "design stress" (fp) rather than the yield stress (Packer et al., 1989).

1 2 fy + 2" fu (4) M i p = ---4" tp " f p " I i f P : 3

The stub-tee analogy assumes that the yield lines form in a linear fashion, transversely across the end plate. However, the yield line analysis for the eight bolt end plates indicates that such patterns rarely occur in practice. To compensate for this inconsistency, "equivalent lengths" (for in-plane and out-of- plane bending) are determined for the yield lines such that the total amount of internal work involved in the mechanism remains unchanged. The equivalent lengths of the yield lines used for the cumulative stub-tee analysis depend on the assumed plastic collapse mechanism. Furthermore, these yield line lengths represent the cumulative length of the x or y components of several yield lines. Full details are given in Wheeler (1998). The theoretical connection capacities based on the cumulative modified stub- tee method are listed in Table 4 (presented later).

PLASTIC SECTION CAPACITY

The plastic section capacity of the tubular member may also govem the ultimate moment that the connection can attain. For compact cross-sections, design specifications generally define the plastic section capacity as the yield stress (fy) times the plastic section modulus (S). Although appropriate for design, this method of calculating the section plastic capacity does not usually reflect the experimentally measured ultimate moment as the cold working of the section produces significant strain hardening of the material. A more accurate method to predict the experimental plastic section capacity is to use the design stress (fp) as defined in Equation 4, fumishing

M s = S.fp (5)


PUNCHING SHEAR

Punching shear failure (tearing of the end plate) occurs when the concentrated loads transferred from the section to the end plate exceed the shear capacity of the end plate over a localised region. To model punching shear failure, a simple approach is used in which it is assumed that shear failure planes are defined by the geometry of the connection. It is also assumed that the punching shear capacity of the end plate is not affected by any concomitant bending moment. The connection is considered to have failed in punching shear when the load in the tensile flange and adjacent regions of the section (Figure 4) exceed the shear capacity of a predefined "nominal shear length" of the end plate. The nominal shear length is the length around the perimeter of the section that is assumed to fail as a result of the section pulling out from the end plate. As shown in Figure 4, the nominal shear length is divided into two regions, corresponding to flange failure (/sf) and web failure (lsw).

Figure 4: Punching shear failure regions

In Figure 4, s denotes the fillet weld leg length, dbh is the diameter of the bolt head, and it is assumed that the tubular section has an extemal comer radius of 2.5 times the wall thickness. Using the von Mises yield criterion, the moment capacity of the connection with respect to punching shear failure is given by

Mp s :~3.tp.(Isf.(d-ts)+Isw.(d-g)) (6)

The theoretical capacities of the connections tested in the experimental program with respect to the punching shear are shown in Table 4.

GENERALISED CONNECTION MODEL

The model described in this paper identifies three modes of failure, namely connection capacity (cumulative modified stub tee model), plastic section capacity, and punching shear. The computed capacities for each mode of failure are presented in Table 4, with the critical one highlighted.

Failure modes determined using the cumulative modified stub-tee model may be govemed by bolt capacity or end plate capacity. Bolt capacity (fracture of bolts) is associated with either thick or intermediate plate behaviour, while plate capacity occurs with thin plate behaviour and is independent of the bolt loads.

The results shown in Table 4 indicate that for the ten experimental tests carried out, four of these were limited in strength by punching shear and a further four were govemed by plastic section capacity. Only two tests were govemed by failure of the bolts according to the stub tee model. While the ultimate failure mode of the specimens was generally punching shear, bolt failure or section failure, substantial yielding in the end plates was observed in the experimental program. The failure criteria and failure loads for the standard SHS tests (Tests 1, 3, 4) and the RHS tests (Tests 2, 5, 6) are presented in Figures 4 and 5,

respectively.

Analytical Model for Bolted Moment End Plate Connections

TABLE 4 THEORETICAL AND OBSERVED ULTIMATE CONNECTION MOMENTS

259

Figure 5: Failure criteria for SHS connections (So = 35 mm)

Figure 6" Failure criteria for RHS connections (So = 35 mm)

260 A.T . Wheeler et al.

The failure criteria for the SHS (Figure 5) demonstrates that for the given end plate dimensions, an end plate thicker than 16 mm will result in plastic section failure, while an end plate thinner than 12 mm forms a mechanism (thin plate behaviour). Punching shear failure never governs for this configuration of SHS. In the case of the RHS (Figure 6), the depth-to-width aspect ratio results in punching shear failure becoming the dominant failure mode for end plate thicknesses in the range of 9 mm to 17 mm. Connections comprising end plates thicker than 17 mm will attain full plastic section capacity, while end plates thinner than 9 mm will fail as a result of a plastic mechanism forming in the end plate. The theoretical results depicted in Figures 4 and 5 are consistent with the experimental findings.

CONCLUSIONS

The analytical models presented in this paper constitute simple methods of predicting the ultimate strength of eight-bolt moment end plate connections joining square and rectangular hollow sections subjected to pure flexure. The model considers three failure modes which are end plate/bolt failure, plastic failure of the connecting beam section, and punching shear (tear out) failure. Plastic mechanism analysis comprising complex two-dimensional patterns of yield lines is employed for the investigation of end plate failure modes, and a modified version of stub tee analysis provides the means through which the effects of prying forces are incorporated in the model. The stub tee analysis is termed the "cumulative modified stub tee model" since it considers prying effects independently in the "in-plane" and "out-of- plane" bending directions for the end plate.

The experimental and analytical results indicate that for the SHS connections, plastic section capacity failure dominates, with end plate failure occurring only for the most flexible end plates. For the RHS connections, the failure mode is predominantly that of punching shear, with plastic section capacity limiting the strength for the thicker end plates. The model demonstrates excellent correlation with the test results and is effective in its consideration of all relevant failure modes that can occur.

R E F E R E N C E S

Kato, B. and Mukai, A. (1991). High Strength Bolted Flanges Joints of SHS Stainless Steel Columns. Proceedings International Conference on Steel and Aluminium Structures, Singapore, May 1991.

Kennedy, N. A., Vinnakota, S. and Sherbourne A. N. (1981). The Split-Tee Analogy in Bolted Splices and Beam-Column Connections. Joints in Structural Steelwork, John Wiley & Sons, London-Toronto, 1981.

Nair, R. S., Birkemoe, P. C. and Munse, W. H. (1974). High Strength Bolts Subject to Tension and Prying. Journal of the Structural Division, ASCE, 100:2, 351-372.

Murray, T. M. (1990). Design Guide for Extended End Plate Moment Connections, Steel Design Guide 4, American Institute of Steel Construction.

Packer, J. A., Bruno, L. and Birkemoe, P. C. (1989). Limit Analysis of Bolted RHS Flange Plate Joints. Journal of Structural Engineering, ASCE, 115:9, 2226-2241.

Wheeler, A. T., Clarke, M. J. and Hancock, G. J. (1995). Tests of Bolted Flange Plate Connections Joining Square and Rectangular Hollow Sections. Proceedings, Fourth Pacific Structural Steel Conference, Singapore, 97-104.

Wheeler A. T., Clarke M. J., Hancock G. J. and Murray, T. M. (1998). Design Model for Bolted Moment End Plate Connections Joining Rectangular Hollow Sections. Journal of Structural Engineering, ASCE, 124:2, 164-173.

Wheeler, A. T. (1998). The Behaviour of Bolted Moment End Plate Connections in Rectangular Hollow Sections Subjected to Flexure, PhD Thesis, Department of Civil Engineering, The University of Sydney.

PREDICTIONS OF ROTATION CAPACITY OF RHS BEAMS USING FINITE ELEMENT ANALYSIS

Tim Wilkinson and Gregory J. Hancock

Department of Civil Engineering, The University of Sydney, Sydney, NSW, 2006, Australia.

ABSTRACT

This paper describes finite element analysis of cold-formed RHS beams, to simulate a set of bending tests, and predict the rotation capacity of Class 1 and Class 2 beams. Introducing geometric imperfections into the model was essential to obtain rotation capacities that were close to the experimental results. A perfect specimen without imperfections achieved rotation capacities much higher than those observed. Introducing a bow-out imperfection, constant along the length of the beam, as was (approximately) measured experimentally, did not affect the numerical results significantly. To simulate the effect of the imperfections induced by welding the loading plates to the beams in the experiments, the amplitude of the bow-out imperfection was varied sinusoidally along the length of the beam. The magnitude of the imperfections had an unexpectedly large influence on the rotation capacity of the specimens. Larger imperfections were required on the more slender sections to simulate the experimental results.

KEYWORDS

Finite element analysis, beams, RHS, cold-formed steel, rotation capacity, local buckling.

INTRODUCTION

Wilkinson and Hancock (1997, 1998) describe tests on cold-formed RHS beams to examine the Class 1 flange and web slenderness limits. The sections represented a broad range of web and flange slenderness values, but it would have been desirable to test a much larger selection of specimens. A more extensive test program would have been expensive and time consuming. Finite element analysis provides a relatively inexpensive, and time efficient alternative to physical experiments

In order to model the plastic bending tests, the finite element program should include the effects of material and geometric non-linearity, residual stresses, imperfections, and local buckling. The program ABAQUS (Version 5.7-1) (Hibbit, Karlsson and Sorensen 1997), installed on Digital Alpha WorkStations in the Department of Civil Engineering, The University of Sydney, performed the numerical analysis.

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262 T. Wilkinson and G.J. Hancock

PHYSICAL MODEL AND FINITE ELEMENT MESH

A typical RHS has dimensions d, b, t, r e, referring to the depth, width, thickness and external corner radius. The depth refers to the larger of the dimensions of the rectangular shape. The rotation capacity, R, of a beam is defined only when the section can sustain its plastic moment, Mp. R is defined as R = K1/K p -1, where r,p=Mp/El is the plastic curvature, and K1 is the curvature (K) at which the moment drops back below the plastic moment.

Figure 1 shows the simplified testing arrangement for the RHS beams. The RHS were supplied by BHP Steel Structural and Pipeline Products, in either Grade C350 or Grade C450 (DuraGal). All beams were bent about the major axis, and most reached the plastic moment, Mp, and continued to deform plastically until a local buckle formed adjacent to the loading plate. A typical finite element mesh, replicating the test arrangement, is shown in Figure 2. The two relevant symmetry planes, at the mid-length of the beam, and through the minor principal axis of the RHS, have been used to reduce the size of the model.

Figure 1: Physical Model Figure 2: Typical Finite Element Mesh

E L E M E N T TYPE

The most appropriate element type to model the local buckling of the RHS was the shell element. The $4R5 element, defined as "4-node doubly curved general purpose shell, reduced integration with hourglass control, using five degrees of freedom per node" (Hibbit, Karlson, and Sorensen 1997), was used. The loading plates attached to the RHS beam were modelled as 3-dimensional brick elements, type C3D8 (8 node linear brick). The weld between the RHS and the loading plate was element type C3D6 (6 node linear triangular prism). The RHS was joined to the loading plates only by the weld elements. Details on the mesh refinement process have been omitted for brevity.

MATERIAL PROPERTIES

The cold-formed RHS have stress-strain curves that include gradual yielding, no distinct yield plateau, and strain hardening. There is variation of yield stress around the section, due to different amounts of work on the flat faces and corners during the production process, with higher yield stresses in the corners. Details of the material properties can be found in Wilkinson and Hancock (1997). The finite element model used three sets of material properties, as shown in Figure 3. Figure 4 compares the responses of a 150 x 50 x 3.0 C450 RHS from the experiment and for a geometrically perfect finite element. The post yielding moment in the ABAQUS was lower than in the experiment by approximately 3 %. The numerical model assumed the same material properties across the whole flange, web or corner, with discontinuity

Rotation Capacity of RHS Beams Us&g Finite Element Analys& 263

of properties at the junctions of the regions. In reality, there is a smooth increase of yield stress from the centre of a flat face, to the comer. The numerical model assigned the measured properties from the coupon cut from the centre of the face (which were the lowest across the face) to the entire face, resulting in a small underestimation of the moment. The slight error in predicted moment was not considered important, as the main aim of the analysis was to predict the rotation capacity. Figure 4 also shows that buckling occurred at much higher curvatures in the geometrically perfect model, compared to the experiment.

Figure 3: Different material properties around RHS Figure 4: Comparison of results from experiment, and perfect mesh

GEOMETRIC IMPERFECTIONS

The initial numerical analyses were performed on geometrically perfect specimens. It is known that imperfections must be included in a finite element model to simulate the true shape of the specimen and introduce some inherent instability into the model, in order to induce buckling.

Bow-out Imperfection

Measurement of the imperfections indicated that most RHS had an approximately constant "bow-out" along the length of each beam. For most cases, the web bulged outwards and the flange inwards. The magnitude of the bow was approximately d/500 (for the web), and -b/500 for the flange. Imperfection profiles are graphed in Wilkinson and Hancock (1997). However, the nature of the imperfection immediately adjacent to the loading plate was unknown, as it was not possible to measure the imperfections extremely close to the loading plate. The process of welding a flat plate to a web with a slight bow-out imperfection is certain to induce local imperfections close to the plate.

Figure 5 shows a typical mesh with the bow-out imperfection included. Figure 6 shows the moment curvature relationships obtained for a series of analyses on 150 x 50 • 3.0 C450 RHS with bow-out imperfections. The magnitude of the imperfection was either d/500 and -b/500 (approximately the magnitude of the measured imperfections), or d/75 and -b/75 (very much larger than the observed imperfections).

Compared to a specimen with no imperfection, the magnitude of the bow-out imperfection had a minor effect on the rotation capacity. In fact, the rotation capacity increased slightly as the imperfection increased. Even when the bow-out imperfections were included, the numerical results exceeded the observed rotation capacity by a significant amount. The conclusion is that the bow-out imperfection was not a suitable type of imperfection to include in the model.


Figure 5: Mesh incorporating "bow-out" Figure 6: Results for "bow-out" imperfection

Sinusoidal Varying Imperfection

It is more common to include imperfections that follow the buckled shape of a "perfect" specimen, such as by linear superposition of various eigenmodes. The approach taken was to vary the magnitude of the bow-out imperfection sinusoidally along the length of the specimen. The half wavelength of the imperfection is defined as L w. A typical specimen with the sinusoidally varying bow-out along the length is shown in Figure 7.

Figure 8 shows the results of selected analyses for a variety of imperfection wavelengths. The code "cont" in the legend to Figure 8 indicates the continuously varying imperfection. The specimen analysed was 150 x 50 x 3 RHS. A half wavelength of approximately d/2 (d is the depth of the RHS web) tended to yield the lowest rotation capacity and most closely matched the experimental behaviour (refer to the specimen with Lw = 70 mm). A half wavelength of d/2 was approximately equal to the half wavelength of the local buckle observed experimentally and in the ABAQUS simulations.

In Figure 8, the specimen with L w = 70 mm experienced a rapid drop in load after buckling, and had a buckled shape as shown in Figure 9 which matched the location of the local buckle in the experiments. A specimen with a slightly different imperfection profile, Lw = 60 mm, had a much flatter post buckling response, and the buckled shape include two local buckles, as shown in Figure 10. Both specimens buckle at approximately the same curvature.

To force one local buckle to form, and in the desired location, the imperfections were imposed only near the loading plate, as shown in Figure 11. Figure 8 includes the response of an additional specimen, with L w = 60 mm, but only the single imperfection. The curvature at which buckling initiated was barely unchanged, but the buckled shape changed, producing the desired shape of one buckle (Figure 9).

Imperfection Size

A variety of imperfection magnitudes was considered. The magnitude of the imperfections was varied from 6w = d/2000 to 6w = d/250, and 6f = -b/2000 to 6f = -b/250. Figure 12 shows the moment curvature graphs for a section with varying magnitudes of imperfection. It can be seen that increasing the imperfection size decreases the rotation capacity. For this example of a 150 x 50 x 3 RHS, applying an imperfection of 1/500 most closely matches the experimental response.

Rotation Capacity of RHS Beams Using Finite Element Analysis 265

Figure 7: Mesh with sinusoidal imperfection Figure 8: Results for sinusoidal imperfection

Figure 9: Specimen with one local buckle Figure 10: Specimen with two local buckles

Figure 11: Single imperfection Figure 12: Effect of imperfection magnitude


PREDICTIONS OF ROTATION CAPACITY

A large range of sections was then analysed. The sizes considered were either 150 • 150 (d/b = 1.0), 150 x 90 (d/b = 1.66), 150 x 75 (d/b = 2.0), 150 x 50 (d/b = 3.0), and 150 x 37.5 (d/b = 4.0), with a variety of thicknesses, and different imperfection sizes: d/250, d/500, d/lO00, d/1500, or d/2000 (for the web), and b/250, b/500, b/lO00, b/1500, or b/2000 (for the flange). The material properties assumed were those for specimen BS02 (see Wilkinson and Hancock 1997).

Figures 13 to 16 plot the relationship between web slenderness and rotation capacity for each aspect ratio considered and each imperfection size. The results are compared with the tests of Wilkinson and Hancock (1997, 1998), Hasan and Hancock (1988), and Zhao and Hancock (1991). It needs to be reinforced that the ABAQUS analyses were all performed on RHS with web depth d = 150 mm and material properties for specimen BS02 (Grade C450). The experimental results shown in comparison were from a variety of RHS with varying dimensions and material properties. Figure 17 compares the effect of aspect ratio with a given imperfection size. Some analyses were repeated using the material properties of a Grade C350 (Specimen BS 11) specimen, and the comparison between steel grades is shown in Figure 17. Note that the following figures use the AS 4100 definition of web slenderness (~.w), where Xw = (d- 2t)/rd'(fy/250).

Figure 13: Results for d/b = 1.0 Figure 14: Results for d/b = 1.66

Figure 15: Results for d/b = 2.0 Figure 16: Results for d/b = 3.0

Figure 17: Comparison of aspect ratio Figure 18: Comparison of steel grade

Rotation Capacity of RHS Beams Using Finite Element Analysis

Several observations can be made from the results:

267

Imperfection size had a lesser effect on the rotation capacity of the more slender sections (R < 1), and had a greater effect for stockier sections. For a given aspect ratio, the band of results encompassing the varying imperfection sizes widens as the slenderness decreases. This is an unexpected result of the study.

There is a clear non-linear trend between the web slenderness and rotation capacity for a given aspect ratio and imperfection size. The shape of the trend is similar regardless of aspect ratio and imperfection size (eg Figure 17). It may be possible to simplify the trend by a bi-linear relationship: a steep line for lower slenderness, and a line of less gradient at higher slenderness values. Sully (1996) found a similar bi-linear trend when comparing the critical local buckling strain of SHS (under pure compression and pure bending) to the plate slenderness.

No single line is a very good match for the experimental results. For example, the ABAQUS results for d/b = 3.0 and imperfection of 1/250 match the experimental results well when ~.w = 48, while in the range 58 < Xw < 65, the results for an imperfection of 1/500 provide a reasonable estimation of the experimental results, and for 75 < ~.w < 85 an imperfection of 1/2000 gives results closest to the experimental values. For d/b = 1.0, the ABAQUS results for an imperfection of 1/250 are close to the experimental results in the range 37 < Xw < 48, while in the range 25 < ~.w < 35, an imperfection of 1/2000 most accurately simulates the test results. This suggests considerable variability in the imperfections with changing aspect ratios and slenderness, and that as the slenderness increases, larger imperfections are required to simulate the experimental behaviour. There is no reason why the same magnitude of imperfections should be applicable to sections with a range of slenderness values. A possible explanation is that the true imperfections in the specimen were caused by the welding of the loading plate to the RHS. A thinner section was deformed more by a similar heat input, hence larger imperfections were induced. The sinusoidally varying bow-out imperfections simulated the effect of the imperfections caused by the weld, and hence greater imperfections were required as the slenderness increased.

Using the material properties of either Grade C350, or Grade C450 steel does not make a significant difference to the relationship between rotation capacity and web slenderness (Figure 18). However, both the Grade C350 and Grade C450 steel comes from the same virgin strip steel, and extra strength in the C450 specimens is obtained via the proprietary in-line galvanising process referred to as DuraGal. There is no reason to assume that the steel from a different supplier with different properties (eg a hot-formed steel) would produce the same relationship between rotation capacity and web slenderness.

For sections with d/b = 1.0, the values of R predicted by ABAQUS are consistently below the observed experimental values of Hasan and Hancock (1988) and Zhao and Hancock (1991), even when small imperfections are imposed. The ABAQUS simulations were performed with material properties taken from the specimens of Wilkinson and Hancock (1997), since the exact material properties from Hasan and Hancock, and Zhao and Hancock were unknown. Preliminary parametric studies showed that increasing strain hardening modulus increased the rotation capacity. If the strain hardening portion of the material properties assumed was different from tile "true" response of the sections of Hasan and Hancock, and Zhao and Hancock, the numerical simulations are likely to produce inaccurate results. In particular, Zhao and Hancock used Grade C450 specimens from a different supplier, Palmer Tube Mills Australia Pty Ltd, which were not in-line galvanised, so it is reasonable to assume that the material properties were different to those used in the ABAQUS simulations. The significance of material properties is a notable finding of the finite element study.

268

SUMMARY

T. Wilkinson and G.J. Hancock

This paper has described the finite element analysis of RHS beams. The finite element program ABAQUS was used for the analysis. The maximum loads predicted were slightly lower than those observed experimentally, since the numerical model assumed the same material properties across the whole flange, web or comer of the RHS In reality, the variation of material properties is gradual, with a smooth increase of yield stress from the centre of a flat face, to a maximum in the comer.

A perfect specimen without imperfections achieved rotation capacities much higher than those observed experimentally. Introducing a bow-out imperfection, constant along the length of the beam, as was (approximately) measured experimentally, did not affect the numerical results significantly. In order to simulate the effect of the imperfections induced by welding the loading plates to the beams in the experiments, the amplitude of the bow-out imperfection was varied sinusoidally along the length of the beam, and limited to be just near the loading plates. The size of the imperfections had an unexpectedly large influence on the rotation capacity of the specimens.

It is likely that the imperfection caused by welding the loading plates to the RHS was a major factor affecting the experientially observed behaviour. The sinusoidally varying imperfections in the ABAQUS model simulated the effects of the localised imperfections in the physical situation. Larger imperfections were required on the more slender sections to simulate the experimental results, since for the same type of welding, larger imperfections are induced in more slender sections.

REFERENCES

Hasan, S. W., and Hancock, G. J., (1988), "Plastic Bending Tests of Cold-Formed Rectangular Hollow Sections", Research Report, No R586, School of Civil and Mining Engineering, The University

o f Sydney, Sydney, Australia. (also published in Steel Construction, Journal of the Australian Institute of Steel Construction, Vol 23, No 4, November 1989, pp 2-19.)

Hibbit, Karlsson and Sorensen, (1997), "ABAQUS", Version 5.7, Users Manual, Pawtucket, RI, USA. Sully, R. M., (1996) "The Behaviour of Cold-Formed RHS and SHS Beam-Columns", PhD Thesis, School

of Civil and Mining Engineering, The University of Sydney, Sydney, Australia. Wilkinson, T. and Hancock, G. J., (1997), "Tests for the Compact Web Slenderness of Cold-Formed

Rectangular Hollow Sections", Research Report, No R744, Department of Civil Engineering, University of Sydney, Sydney, Australia.

Wilkinson T. and Hancock G. J., (1998), "Tests to examine the compact web slenderness of cold-formed RHS", Journal of Structural Engineering, American Society of Civil Engineers, Vol 124, No 10, October 1998, pp 1166-1174.

Zhao, X. L. & Hancock, G. J., (1991), "Tests to Determine Plate Slenderness Limits for Cold-Formed Rectangular Hollow Sections of Grade C450", Steel Construction, Journal of Australian Institute of Steel Construction, Vol 25, No 4, November 1991, pp 2-16.

ACKNOWLEDGEMENTS

This paper describes part of a research project is funded by CIDECT. The first author is funded by an Australian Postgraduate Award from the Commonwealth of Australia, supplemented by the Centre for Advanced Structural Engineering at The University of Sydney.

FAILURE MODES OF BOLTED COLD-FORMED STEEL CONNECTIONS

UNDER STATIC SHEAR LOADING

K. H. Ip 1 and K. F. Chung 2

1Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hum, Hong Kong

2Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung, Hum, Hong Kong

ABSTRACT

Three failure modes of bolted cold-formed steel (CFS) connections were predicted using a three-dimensional finite element (FE) model with material geometric and contact nonlinearity. The connections were under static shear loading up to 3 mm end extension, which is appropriate for the design of moment connections. The model can predict the propagation of yielding in the CFS strips, which characterizes the connection failure mode. Three distinct failure modes were observed from the simulation results, namely, (i) the bearing failure, (ii) the shear-out failure and (iii) the net-section failure. Through parametric runs, the effects of geometry and material properties on the failure modes were studied The results were also compared with the bearing resistances based on design rules in BS5950: Part 5.

KEYWORDS

Bolted connections, cold-formed steel, failure modes

INTRODUCTION

Galvanized cold-formed steel (CFS) sections can be found in various building applications, ranging from purlins and steel framing, to roof sheeting and floor decking. The advantages of using CFS sections are derived from their long-term durability together with high yield strengths and high buildability. In building construction, CFS sections are usually bolted to hot rolled steel (HRS) members to form shear and moment connections. With the development of material technology, high strength CFS sections are available for building applications. The established

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270 K.H. Ip and K.F. Chung

design codes 1-3, however, may be inappropriate for CFS sections with high yield strength but low ductility (< 5%). Consequently, a close examination on the resistance and the associated failure modes of bolted connections with high strength low ductility CFS strip is essential before the established design codes can be applied with confidence.

With the advent of computer hardware and software, numerical simulation has drawn the attention of researchers in many areas of engineering and science. In the field of solid mechanics, finite element (FE) method is perhaps the one having the greatest impact. The method is particularly useful in solving boundary value problems where large strains, nonlinear materials and contact surfaces are involved. Results from FE analysis provide a clear picture on the stress and the strain distributions in a structure, which is not easily obtained from physical tests. Besides, extensive parametric studies can be carried out to reveal the effects of geometrical and material properties on the performance of a structure.

The present study 4 concerns with finite element simulation on bolted connections between CFS strips and HRS plates under static shear loading. Emphasis is given to predict the possible failure modes, namely, (i) the bearing failure, (ii) the shear-out failure and (iii) the net-section failure aider calibration. Parametric runs will be carried out to reveal the effects of geometrical and material properties on the resistances of different failure modes. The results are then compared with design values to reveal the applicability of codified design rules 5.

FINITE ELEMENT ANALYSIS

The ANSYS (ver 4.3) finite element package is used to predict the load-extension curves of bolted connections between cold-formed steel strips and hot rolled steel plates under static shear loading. As the connection contains a plane of symmetry, the half model shown in Figure 1 is sufficient, where the edge distance Se and the specimen width W are indicated. The CFS strip, the HRS plate and the bolt-washer assembly are represented three-dimensionally by eight-node iso-parametic solid elements SOLID45, as they allow both geometric and material nonlinearities. Contact between the various components is accomplished by employing contact elements CONTACT49. The contact stiffness and the friction coefficient for all interfaces are assigned the values of 2 x 103 N/mm and 0.2, respectively. In a typical FE model, there are 1878 nodes, 1197 solid elements and 981 contact elements.

Plasticity in the CFS strip is considered by incorporating the von Mises yield criterion, the Prandtl-Reuss flow rule together with isotropic hardening rule. However, for simplicity, the bolt-washer assembly is linear elastic with Young's modulus, E, at 205 kN/mm 2 and Poisson's ratio v at 0.3.

Shear load is applied to the FE model by imposing incremental displacements to the end of the CFS strip, along the longitudinal direction of the specimen. Throughout the course of loading, the HRS plate and the root of the bolt are fixed in space. As the model is highly nonlinear, the full Newton-Raphson (N-R) procedure is employed to obtain the solution atter each displacement increment.

Failure M octes oj t~olted Cold-Formed Steel Connections 271

Figure 1 Finite element model of a bolted connection between CFS strip and HRS plate

Figure 2

True Strain (%)

Proposed stress-strain curves for cold-formed steel strips

RESULTS AND DISCUSSIONS

The FE model is first calibrated with the results from lap shear tests. Both G300 and

G550 cold-formed steel strips of different yield strengths py and thicknesses t are considered. Their material curves as deduced from standard coupon tests and they are presented in Figure 2. A negative slope is appended to each curve to simulate the effect of strength degradation at high tensile or compressive strains. The CFS strip is bolted to the HRS plate by a grade 8.8 bolt of 12mm diameter. Comparison between the predicted and the measured load-extension curves associated with bearing failure is given in Figure 3. Close agreement between the experimental and simulation results indicates the accuracy of the finite element model as well as the proposed material curves.


Figure 3 Theoretical and experimental load-extension curves for bolted connections with 12mm diameter bolts

By changing the dimensions of the CFS model, i.e. the edge distance Se and the specimen width W, three distinct failure modes are identified:

(i) Bearing failure

It prevails for strips having sufficiently large Se and W, as shown in Figure 4(a). The yield zone emerges from the bearing edge of the CFS strip owing to highly localized compressive stresses.

O0 Shear-out failure

It occurs when the edge distance Se of the specimen is small, as shown in Figures 4(b). Such failure is characterized by large shear stresses between the hole and the edge of the strip. Protrusion of the edge of the strip can be observed.

(iii) Net-section failure

It takes place for narrow specimens as shown in Figure 4(c). In contrast to bearing failure, the yield zone is developed from the tensile edges of the hole, accompanied by necking of the net-section.

The deformed meshes of each failure mode are also presented in Figure 5 for comparison.

Failure Modes of Bolted Cold-Formed Steel Connections 273

Figure 4 Failure modes of G550 CFS strip at 3ram extension (t - 1.60 m m with 12mm diameter bolts)


Figure 5 Deformed meshes of G550 CFS strip at 3mm extension (t = 1.60 mm with 12mm diameter bolts)

Failure Modes of Bolted Cold-Formed Steel Connections 275

A strength coefficient is established to compare the resistances of a bolted connections from finite element models to basic resistances of the connections, and the strength coefficient is defined as follows:

Res i s tan ce at 3ram Strength coeff icient = ( 1 )

t d U s

Through parametric runs, the effects of Se and W on the normalized resistance of the FE model are summarized in Figures 6 and 7 for the G300 and G550 strips, respectively. These figures also present the capacities of the connections based on the design formulae in Section 8.2 in BS5950: Part 5 [1 ]. A glance at these plots reveals that the FE predictions exhibit a similar trend with the design values. Maximum connection resistance is found to occur in the bearing mode. The results also demonstrate the independence of bearing resistance to Se and W when the bolt hole is sufficiently far from the sides of the strip. Inspection of Figure 6 shows that the design rules is conservative for predicting the resistance of G300 strips under net- section and bearing failures. In the FE model, transitions from the shear-out and the net-section failures to the bearing failure are found to occur at larger Se / d and W/d. In other words, sufficient distances, say Se / d > 4 and W / d > 5, should be provided for the CFS strip in order to secure the maximum connection resistance. Refer to Figure 7 for G550 strips, the design formulae are unsafe when applying to high strength steels.

Figure 6 Strength coefficient of bolted connection with G300 CFS strip ( t = 1.50 mm and d = 12 mm )

Figure 7 Strength coefficient of bolted connection with G550 CFS strip ( t = 1.60 mm and d = 12 mm )

276

C O N C L U S I O N S

K.H. Ip and K.F. Chung

A finite element model was employed to determine the resistance of bolted connections between cold-formed steel (CFS) strips and hot rolled steel (HRS) plates subject to static shear loading. By incorporating both solid and contact elements, the model is able to capture nonlinearities associated with geometry, materials and boundary conditions. T h e von Mises stress distributions in the CFS strips under different types of connection failure are also predicted. Results from parametric runs indicate that the existing design formulae are sufficient only for bolted connections with low strength steels, such as 280N/mm 2 and 350N/mm 2. However, the existing codified design rules may not to conservative when applying to high strength low ductility steel.

ACKNOWLEDGEMENT

The research project leading to the publication of this paper is supported by the Hong Kong Polytechnic University Research Committee (Project A/C code G-$565).

R E F E R E N C E S

1. BS5950: Structural use of steelwork in buildings: Part 5 Code of practice for the design of cold-formed sections, British Standards Institution, London, 1998.

2. Cold-formed steel structure code AS/NZ 4600: 1996, Standard Australia/Standards New Zealand, Sydney, 1996.

3. Eurocode 3: Design of steel structures: Part 1.3: General rules- Supplementary rules for cold-formed thin gauge members and sheeting, ENV 1993-1-3, European Committee for Standardization.

4. Chung, K.F. and Ip, K.H.: Finite element modelling of bolted connections between cold-formed steel strips and hot rolled steel plates under shear, Engineering Structures (to be published).

5. Chung~ K.F. and Ip, K.H.: Finite element modelling of cold-formed steel bolted connections, Proceedings of the Second European Conference on Steel Structures, Praha, May 1999, pp503 to 506.

DESIGN MOMENT RESISTANCE OF END PLATE CONNECTIONS*

Yongjiu Shi Jun Jing

Department of Civil Engineering, Tsinghua University, Beijing 100084, China

ABSTRACT

The end plate connection, either flush end plate or extended end plate, bolted with high strength friction fasteners, is one of the moment resistant connections recommended for steel portal frame design, and can be used for rafter to column connection or rafter splice. Current design rules specify that the tension force produced by the bending moment is triangularly distributed among the bolt rows in tension zone, if the end plate is stiff enough and its deformation is negligible. The engineering practice demonstrates that the end plate thickness usually varies from 12 to 36mm and its flexible deformation can not be neglected. In this paper, a finite element model is constructed to analyze connection behaviour under the applied bending moment and the model is verified by the available test results. The bolt tension force distribution and end plate deformation for connections with different configurations are compared. Finally, a modified design method is proposed.

KEY WORDS

Steel structures, End plate connection, High strength fastener, Portal frame design

INTRODUCTION

In design of steel portal frame, end plate connection is the most widely recommended economic moment-resistant joint with the advantage of fast erection and no field welding(Fig 1). The bolted end plate connection can be used for beam splice or beam to column connection and can be detailed as either flush or extended with or without stiffeners(CECS102:98, 1998). The moment resistance of end plate connections largely depends on the component behaviour in the tension zone, compression zone and shears zone, such as the bolt tension resistance, end plate yielding resistance and column web buckling resistance, etc. The traditional design guides(JGJ82-- 91,1992) suggest that the tension force produced by the bending moment is triangularly distributed among the bolt rows in tension zone under the assumption that both the flush and extended end plate is adequately stiff and its flexible deformation and prying force can be neglected(Fig. 2). The outermost row of bolts are assigned with the maximum

"Supported by National Natural Science Foundation of China

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278 Y. Shi and J. Jing

tension and the forces resisted by any row of bolts can be given by

Nti = Myi / E yi2 ( 1 )

It is required in Chinese code of practice that Nt~ should be limited to Na<~O.8P, where P is the bolt pretension.

f ( 3 ..... t i t Figure 1: End plate connections for portal frame

N t l .

v

Figure 2: Traditional design model

However, the end plate applied in the steel portal frame design may be just 20mm or less in thickness and the assumption described above may not be applicable. It is necessary to further investigate the design model that would be appropriate for connections in steel portal frame. In this paper, a finite element model is established to analyze the bolt force distribution for the beam to column connections. The contact pressure between end plate and column flange under different bending moment is also investigated. A revised design model is proposed for portal frame end plate connections.

ANALYTICAL MODELING

Traditionally, the T-stub or yielding line theory is used for analyzing the end plate deformation (Brown et al, 1996), and later, the 2D/3D finite element model was introduced(Sherbourne and Bahaari, 1994, Gebbeken et al, 1994). In this paper, a hybrid 2D/3D model was developed. The beam web and flange were modeled with plate element, while the end plate, bolt heads and nuts were represented by 3D block elements. A number of bar elements were adopted to simulate the bolt shank. The contact elements, which could resist compression but not tension, were used to simulate the interface between the end plate and the column flange. In establishing the finite element model, the bolt pretension were well simulated by temperature action, that is, a temperature stress were applied to the bolt shank, leading to the bolt to contract and subject to pretension. The established finite element model is shown in Fig. 3. The connection model is analyzed by loading increment method and the material properties are assumed remaining elastic.

Figure 3. Finite element model Figure 4: Tested connection and result comparison

To verify the finite element model, an end plate connection tested by Jenkins et al(1986) were analyzed again. The bolt tension force produced by the applied bending moment is compared in Fig. 4. It is noted that the finite element model simulates the tension force development very well, but gives higher value. Since calculated results are obtained in the elastic range of material properties, while partial plasticity

Design Moment Res&tance of End Plate Connections 279

may be developed under large bending moment during the tests, it is understandable that the calculated tension is larger than the measured tension. Both the experiment and calculation reveal that the tension force on the first row of bolts is well below that on the second row of bolts. The traditional design model(Fig 2) is inappropriate to the extended end plate connection.

PARAMETRIC STUDY

Based on the establishedmodel, some typical joints with flush or extended end plates(Fig. 5a) were investigated. The end plate thickness varies from t = 10mm to t = 40mm, and the high strength bolts are M20, Grade 8.8 with pretension P = 110kN. The extended part can be stiffened or unstiffened.

Figure 5: Bolt force versus applied bending moment


The end plate connection can fail with the following failure modes: (i)Fracture of bolts in tension zone; (ii)end plate yielding; and (iii)compression between the bolted plates diminishing. The moment resistance and failure modes may vary with the connection configuration. In this study, the compression C between the contact surfaces and the bolt tension T at the tension zone versus applied bending moment were obtained around each bolt. It should be noted that when there is no bending moment applied, the compression C and the bolt tension T is equal, that is C = T. It is also noted that with the increase of bending moment, the bolt tension T will increase, but compression C will reduce. The variations of the tension T and compression C with applied moment for different connections were shown in Fig. 5b-~5f.

It can be seen that for the flush end plate(Fig. 5b), the higher tension was developed in the first row of bolts even the applied moment is small and the connection fails with bolt fracture. The maximum moment resistance of the flush end plate connection is much less than that o f the corresponding extended end plate connection(Fig. 5d). Fig. 5c to 5e compares the effects of end plate thickness. When the thin plate is applied(Fig. 5c, t = 0.5d, where d is the bolt diameter), the end plate deformation and prying force is significant and connection fails with end plate yielding. When medium thickness of end plate is applied(Fig. 5d, t = d), the connection resistance largely depends on the tension resistance of the second row of bolts, where the maximum tension force is developed. However, if the extended part is stiffened(Fig. 5f), the maximum tension can be developed in both the first and the second rows of bolts and the higher moment resistance is achieved. When the thick plate is applied(Fig. 5e), the compression between the bored plates diminishes very quickly and the connection may fail by the separation of the bolted plates.

However, it is difficult or impossible to calculate the bolt tension T and/or compression C by a simple method. From the compression C and bolt tension T, the force Nt produced by bending moment and resisted by each bolt can be given by Nt = T - C. The distribution of Nt among the bolt rows are compared under the maximum bending moment(Table 1 and Fig. 6). It is noted that for the flush end plate connection(Fig. 6a), the tension force distribution is similar to traditional design model. For the extended end plate connection, the tension force between the beam flanges more or less distributed triangularly, but on the extended part, the tension force Ntl varies with the plate thickness and stiffening. The maximum Nt appears at the second row of bolts instead of the first row of bolts(Fig. 6b). Since the actual tension force on the extended part is far less than that calculated from the traditional design model(Fig. 2), the end plate yielding may more likely happen around the second row of bolts, rather than the first row of bolts as predicted by the current code of practice (CECS102:98, 1998). Even the extended part is stiffened(Fig. 6c), the tension force on the first row unlikely exceeds the force on the first row, unless the end plate is extremely thick.

Ntl

(a)Flush (b)Unstiffened (c)Stiffended (a)Unstiffened (b)Stiffened

Figure 6: Bolt force distributions Figure 7: Proposed design model

Design Moment Resistance of End Plate Connections

TABLE 1 BOLT TENSION UNDER MAXIMUM BENDING MOMENT

281

Extended End-plate

Unstiffened, t = 10mm

Unstiffened, t=20mm

Unstiffened, t=40mm

Stiffened, t=20mm

First row Ntl(kN)

26.3

97.5

105.3

137.9

Second row No(kN)

42.2

139.0

130.9

141.2

Ntl/Nt2

0.62

0.70

0.81

0.98

DE SIGN P R O P O S A L

From the above analysis, itis concluded that the traditional tension distribution model may not be applicable to the extended end plate connections. The necessary revision is proposed and recommended in this paper. Since the tension distribution Art for the flush end plate connection is close to the traditional design model, the force resisted by any row of bolts can still be calculated by Eq.(1), while for the extended end plate connections, the following design procedures were proposed"

(1)When the extended part unstiffened, the bolt force distribution can be assumed as Fig. 7a and the second row of bolts is supposed subject to the maximum tension, and tension force on any row in the tension zone can be given by

Nt2 = m YlY2~l 2 + Z Yi 2 +Y,,Y,-1 i = 2

Utl = Nt2~:l/~: 2

Nt~ = Nt2 Y,/Y2

where ~:1 = (ha + h2 - Ya )/ha and ~z = Y2/h2 y,----distance from the ith bolt row to the center of bolt group;

M--bending moment applied on the connection; n--number of bolt rows; m--number of bolt columns;

hi, h/--length of the extended part and the distance from the beam flange to the center of bolt group respectively.

(2) When the extended part stiffened, the bolt force distribution can be assumed as Fig. 7b and the tension force on the bolt rows adjacent to the beam flange in tension is equal. Tension force on any row in the tension zone can be given by

Nt2 "- m YlY2 + E Yi 2 +Y.Y,-1 (3a) i = 2

Ntl = Nt2 (3b)

N~ = Nt2 Yi/Y2 (3e)

(2a )

(2b)

(2c)

5O 5O u u L

i ,i i .-*-i4-

I I 1

.4. 4 - I I I I

i . i ~ .

I I

- - - I~- -~-

[ oo [

Figure 8.

.Io

I r

o

Example


The proposed design model by this paper is compared with the traditional design model, taking a typical extended end plate connection as an example(Fig 8). The applied bending moment is M = 160kNm, and the design results are listed in Table 2. It can be seen that because the maximum tension Art is assumed on the first row of bolts in the current code of practice, the larger flexible moment generated in the end plate happens on the extended part. Therefore, thicker end plate is required to prevent the connection failing in end plate yielding mode. However, if the bolt tension force is calculated by the proposed method where larger tension force appears on the second row of bolts instead of the first row, the moment generated in the end plate is reduced significantly. As the results, less thick end plate is required.

TABLE 2 DESIGN RESULTS

Bolt Tension(kN) First row(Nt0

Second row(Nt2)

Bolt Parameters

End Plate Thickness(mm)

Current Code

95.2

60.3

Grade 8.8, M22

27

This Paper

60.0

91.2

Grade 8.8, M22

22

CONCLUSION

A 3D finite element model was established in this paper to investigate the end plate connection behaviour. The analysis model is verified by the test results. Based on the comparison results obtained from analyzing some typical connections, it is concluded that the traditional design model may not be applicable to the end plate connection design. A revised model is proposed for end plate connection design and a formula is derived for evaluating the bolt tension distribution.

REFERENCE

Brown, D. G., Fewster, M. C., Hughes, A. F. and Owens G. W.(1996). A New Industry Standard for Moment Connection in Steelwork. The Structural Engineer 74:20, 335 - 342. CECS102:98 (1998). Technical Specification for Light Gauge Steel Structure of Low Rise Buildings with Portal Frames. Association of China Engineering Construction Standard, Beijing. Gebbeken, N., Rothert, H. and Binder B.(1994). On the Numerical Analysis of End-plate Connections. Journal of Constructional Steel Research 30, 177 - 196. Jenkins, W. M., Tong, C. S. and Prescott, A. T. (1986). Moment-transmitting Endplate Connections in Steel Construction, and a Proposed Basis for Flush Endplate Design. The Structural Engineer 64A:5, 121 - 132.

JGJ82--91 (1992). Specification for Design and Construction of High Strength Bolt in Steel Structures. Ministry of Construction, Beijing. Sherbourne, A. N. and Bahaari, M. R.(1994). 3d Simulation of End-plate Bolted Connections. Journal of Structural Engineering 120:11, 3122 - 3136.

THREADED BAR COMPRESSION STIFFENING FOR MOMENT CONNECTIONS

T. F. Nip 1 and J. O. Surtees 1

1 School of Civil Engineering, University of Leeds, Leeds LS2 9JT, UK.

ABSTRACT

A new means of providing local column reinforcement in the compression zone of high moment capacity end plate connections, using threaded bars, has been developed. Conventional welded plate stiffening is difficult to fabricate by automatic processes and is particularly expensive when used on site in structural upgrading schemes. In the new approach, a system of threaded bars is locked against inner flange faces of the column to transmit the horizontal compression force from incoming beams. Joint performance has been studied in a programme of tests on simple compression specimens and full scale beam/column/beam joints. The influences of concrete encasement and steel .sleeving on threaded rod capacity have also been investigated. In a typical connection using high strength threaded bar only, enhancements of column web bearing capacity in the order of 300% have been demonstrated. The validity of using simple compression tests to represent the compression zone of beam/column joints is examined by comparison with full scale connection tests. It is concluded that serviceability and ultimate conditions can be met fully by the proposed form of connection. Experience of fabricating and assembling the test specimens indicates that significant savings are possible in comparison with the true cost of providing welded stiffeners.

KEYWORDS

Steel frames, moment connections, bolted joints

INTRODUCTION

Beam-to-column end plate connections have been used extensively in multi-storey construction for resisting moments from wind and gravitational loading. UK and European design codes now provide the necessary 'continuous construction' background for member and connection design but magnitude of bending moment which may be transmitted by conventional end-plate connections is somewhat limited. This arises from the inherent incapacity of typical column flanges to resist normal force, even when reinforced locally by conventional welded stiffening. Demands for longer clear spans and lower

283

284 T.F. Nip and J.O. Surtees

floor construction depths have almost outstripped the capability of such connections. For instance, the maximum moment resistance of connections appropriate to a beam of depth 650mm is likely to be only 25% of the moment capacity of the beam.

Moment capacity may be increased by increasing either the bolt diameter or the total number of bolts in the tension cluster. The former of these has been explored at Leeds University, using backing angles to reinforce the column flange (Grogan and Surtees (1995) and Grogan and Surtees (1999)). The second approach entails increasing the number of bolt rows or the number of bolts per row. Several investig-ators have examined these options (Grundy et al (1980), Murray and Kukreti (1988) and Murray (1988)). The second option was explored without recourse to column stiffening but has been investigated more recently at Leeds University in a Science and Engineering Research Council (now EPSRC) supported project (Surtees and Yeung (1996)). In the latter investigation, which applied particularly to double-sided connections, local bending of the column flange was reduced by linking beam tension flange forces across the column via socketed couplers placed between opposing tension bolts. In tests on full-scale specimens, improvements in moment capacity up to 200% were observed, compared with less than 40% when using conventional welded reinforcement in the tension zone.

The work reviewed above has focused on tension zone stiffening and a principal feature has been the use of bolted forms of stiffening. Recently at Leeds University, the potential use of bolted compression zone stiffening was examined in depth and this paper presents details of tests on a particular form of stiffening element developed in that investigation, namely, the threaded bar stiffener.

USE OF THREADED BAR COMPRESSION ZONE STIFFENING

The term threaded bar is used to describe continuously threaded stock material readily available in various diameters and material grades at lengths up to 3m. It is usually cut to precise shorter lengths in the manufacturing process. In the present context, plain round bar with minimum threading to satisfy installation requirements is equally acceptable, though not necessarily cheaper than threaded bar.

Two forms of threaded bar compression stiffening element were used in the tests. The first consists of a short threaded bar with end nuts which fits between, rather than passes through, the column flanges. Blind flange holes may be used to locate the stiffener or, alternatively, thin punched retaining plates may be suspended from nearby end plate fixing bolts. The second form also has internal nuts but passes through the flanges to engage outer nuts and is therefore able to act as tension zone stiffening in the event of moment reversal.

Figure 1: Forms of threaded bar compression stiffening.

Threaded Bar Compression Stiffening for Moment Connections 285

This latter form must be positioned with appropriate installation clearance from beam flange surfaces. Both forms may be combined to provide compression zone stiffening in non-reversing or partially reversing moment connections. Figure l(a) shows a possible configuration for a non-reversing moment connection. Four threaded bar stiffeners are placed in line with the compression flange. On the tension side, a larger number of (off-line) stiffeners is necessary because of weaker participation of the web and presence of prying. In the full moment reversal connection shown in Figure 1 (b), tension zone requirements determine stiffener provisions at both flange levels.

TEST PROGRAMME

The primary objectives of the tests were (i) to verify the feasibility of threaded bar stiffening and (ii) to investigate the force distribution in the connection elements in order to establish design guidelines. To this end, the following aspects were studied in the tests: �9 threaded bar properties �9 bar configuration and size effects �9 influence of concrete encasement

Because of the large number of tests, all involving heavy sections, it was decided to confine testing mainly to compression zone specimens. Some tests on full connection specimens were, however, carried out for comparison purposes.

Compression tests

Compression zone check calculations invariably require that column web stiffening be provided. When this is done, the available compression zone capacity is usually much in excess of requirements and connection failure occurs elsewhere. In devising a new form of stiffening, the possibility of a more balanced provision for compression and tension zone failure might be allowed to advantage. However, most specimens tested in the present series had strengths and stiffnesses well in excess of what might be termed a minimum compression zone performance.

A full description of the isolated compression zone test specimens is given in Table 1. A threaded bar diameter of 24mm was used for most of the tests in recognition of the fabrication industry preference for M20 and M24 bolts. Larger sizes of threaded bar were tested both to examine their efficiency as concentrated compression stiffening and to detect potential assembly difficulties. In all cases except CB2 the threaded bar stiffening was prepared from plain material, leaving a small unthreaded central portion to accommodate ER strain gauges for direct force measurement. Test CB2 used commercial threaded bar. All bars were calibrated prior to testing and the characteristics of the two forms were compared for control purposes.

The compressive test load was restricted to 4110kN maximum by the capacity of the loading frame. The full connection tests described below used UB533x210x101kg (Grade 50) beam material throughout, corresponding to a nominal maximum compression flange force of 1730kN. Rather than represent the beam flange thickness correctly in the test rig and thereby limit the maximum test load, a 40x40x215 steel block and curtailed 25 thick end plate was used to load each side of the specimen. In this way, the true capacity of the stiffened column web was measured, rather than that of the beam flange. Dial gauges, LVDT deflection gauges and ER strain gauges were used to measure displacements and strain distributions in the test specimens. The spread of yield was monitored using heat applied resin coatings on the columns. A typical set up is shown in Figure 3.


TABLE 1

DESCRIPTION OF ISOLATED COMPRESSION ZONE SPECIMENS

Test Column description Size Length (mm)

CB 1 UC 254x254x73 700 CB2 UC 254x254x73 460

CB3 UC 254x254x73 460 CB4 UC 254x254x89 700 CB5 UC 305x305x118 700 CB6 UC 305x305x118 700 CB7 UC 305x305x118 700 CB8 UC 305x305x118 700 CB9 UC 305x305x118 700

CB 10 UC 305x305x 118 700 CC4 UC 305x305x118 700

UC 305x305x 118 700

Stiffener details Dimension Type (Fig. 2)

~24 HTS bars A M24 grade 8.8 A threaded bar

~24 HTS bars A ~24 HTS bars B ~24 HTS bars B ~24 HTS bars C ~24 HTS bars C ~30 HTS bars A(HP) ~t~45 mild steel D(HP) ~24 HTS bars B ~24 HTS bars A ~24 HTS bars A

Further reinforcement

25 mm thick backing plates

1.5" (0.25" thick ) HTS tubes Grade C60 concrete Grade C20 concrete

HP: with Hanging Plate

Figure 2: Threaded bar configurations in Table 1

Threaded Bar Compression Stiffen&g.for Moment Connections 287

Figure 3: Typical setup for isolated compression test

Cruciform tests

Two full size beam/column/beam connections, FB2 and FB4, were tested. The direct compression tests excluded interaction from nearby shear or tension zone forces and the whole-connection tests allowed the effect of these omissions to be studied. FB2 was an over-stiffened specimen with 8 threaded bars in the compression zone whereas FB4, with only 4 threaded bars placed directly against the incoming beam flanges, was marginally under-stiffened in relation to the tension capacity of the connection. Specimen FB2 was the whole-connection equivalent of CB5. Both specimens were constructed from UC305x305x118kg/m, UB533x210x101kg/m and 705x305x35 endplate (see Figure 4).

Figure 4: Test specimen FST4

288

RESULTS

T.F. Nip and J.O. Surtees

Table 2 summarises the isolated compression zone test results. Column web bearing capacity was increased substantially by the stiffening. Specimen CB5, which would be typical for beam depths of up to 600mm, was able to sustain a force equivalent to this order of beam plastic moment capacity using 8 M24 bars. The highest resistance recorded in the series was 4110kN, which would satisfy the required compression zone bearing capacity for all connections up to a beam depth of 900mm.

The general pattern of behaviour was similar in all the tests. Inter-crossing shear yield lines inclined at 45 ~ to the horizontal first occurred in the centre of the web panel. Heavy yielding then occurred at the flange/web junction near to the load and support points. As this spread into the web, the threaded bars absorbed an increasing proportion of the load and eventually buckled after substantial yielding. In the uncased Type A specimens (see Figure 2), a sidesway buckling mode occurred. In the remaining uncased specimens, stiffeners buckled without sidesway and were partially restrained at their ends.

Test

CB 1 CB2

TABLE 2

SUMMARY OF ISOLATED COMPRESSION ZONE TEST RESULTS

Nominal web bearing capacity F~w (~)

507 507

CB3 ,j 507 CB4 651

n

CB5 1063 | |

CB6 1063 1063

Failure load Fc Fc + Few (1~)

1575 1425 1425 2250 2780

Failure mode

2400 3200

3.107 21811 2.811 3.456 2.615 2.258 3.010 CB7

| l

CB8 1063 3308 3.112 ,, CB9 1063 2515 2.366

| |

CB 10 1063 4000 3.763 CC4 1063

1063 4110+ 3500 CC5

3.866 3.293

Web sidesway Web sidesway Web sidesway

Stiffeners yielding Stiffeners yielding Stiffeners yielding Stiffeners yielding

Web sidesway Stiffeners yielding

Web sidesway Concrete cracking Concrete cracking

+ capacity of loading rig reached before failure of specimen

The relationship between applied load and flange to flange displacement is shown in Figure 5. The compressibility of the connection was higher in the case of slender off-line stiffeners but was still well below an acceptable maximum value. Use of internal backing plates for the above case improved stiffness and strength significantly. In case of in-line stiffeners, concrete encasement prevented buckling and enabled them to develop their full yield capacity supplemented by the compression resistance of the concrete. The results in Table 2 show significant improvements in this respect.

It was established for in-line stiffeners that the total bearing resistance could be taken as the nominal web bearing capacity plus the compressive capacity of the stiffeners based on standard column design procedures with effective length equal to actual length and cross-section based on tensile stress area. For off-line stiffeners, the total bearing resistance is dependent on stiffener position and relative material strength.

Threaded Bar Compression Stiffening for Moment Connections 289

Figure 5" Deformation stiffeners in isolated compression zone tests (305x305xl 18kg/m UC)

Figure 6: Moment rotation relationship for whole-connection tests


Moment rotation curves for whole-connections FB2 and FB4 are presented in Figure 6. These display a rigid initial characteristic but are sufficiently ductile to develop 30 x 10 -3 rad rotation before failure. A summary of cruciform connection test results is shown in Table 3. A maximum applied moment of 1.2 Mp was recorded for both tests. The compressibility of the connections at compression flange level correlated well with corresponding values obtained from test CB5. Although FB4 is not fully equivalent to tests CB 1,CB2 or CB3, it was clear in the case of whole connections that sidesway buckling of the compression zone is totally inhibited by several factors. The carrying capacities obtained in the isolated compression tests understated the true capacities of whole connections

TABLE 3

SUMMARY OF WHOLE-CONNECTION TEST RESULTS

Test l[ Stiffening method

FB2 8 (4 off-line) ~24HTS bars

FB4 4 ~24 HTS bars

Failure moment + Mp Failure mode

1.229 Beam flange and web local buckling

1.2715 Beam flange and web local buckling

CONCLUSION

Threaded bar compression stiffening has been shown to be an effective and viable alternative to traditional welded plate stiffening. Tests have confirmed that bearing strengths much in excess of those required for current typical end plate connections are possible. Use of threaded bar as an effective form of tension stiffening has been considered incidentally in this paper because of its application to the whole-connection tests. The case for tension stiffening is strong and its eventual practical acceptance will undoubtedly increase the appeal of threaded bar compression stiffening.

ACKNOWLEDGEMENTS

The research described herein was funded by the Engineering and Physical Sciences Research Council and British Steel. Further support and advice was provided by the Steel Construction Institute and British Constructional Steelwork Association Ltd.

REFERENCES

Grogan W. and Surtees J. O. (1995) Column flange reinforcement in end plate connections using bolted backing angles. Nordic Steel Construction Conference, Malm6, Sweden, pp 87-94.

Grogan W. and Surtees J. O. (1999) Experimental behaviour of end plate connections reinforced with bolted backing angles. J. construct. Steel Research, vol. 50, pp71-96.

Grundy P., Thomas I. R. and Bennetts I.D. (1980) Beam-to-column moment connections. J. Struct. Div., Am Soc. Civ. Engnrs, pp313-330.

Murray T. M. and Kukreti A. R. (1988) Design of 8-bolt stiffened moment end plate. Engineering Journal, AISC, Vol.25, Pt. 2, pp45-53.

Murray T. M. (1988) Recent developments for the design of moment end-plate connections. J. Construct. Steel Research, Vol. 10, pp 133-162.

Surtees J.O. and Yeung K.W. (1996) A new form of high moment beam-to-column connection. EPSRC Final Report, Grant reference GR/J70758, University of Leeds.

Experimental Study of Steel I-Beam to CFT Column Connections

S.P. Chiew & C.W. Dai

School of Civil and Structural Engineering, Nanyang Technological University,

50 Nanyang Avenue, Singapore 639798.

ABSTRACT

This paper focused on the experimental study of the composite behavior of steel

universal beam to concrete-filled tube (CFT) column connections. Eight specimens were

designed and tested to failure, of which four specimens are simple beam-column

connections and the rest are rigid connections with different type of stiffening details. For

simple connections, the parameters investigated are the thickness and diameter of the

steel tube and the beam size. For the rigid connections, the stiffening details investigated

include cover plate, shear plate, extemal ring and re-bar respectively. Experimental

results showed that the simple connections have weaker ultimate strength, ductility and

stiffness, and their behaviors are influenced by the parameters investigated. All stiffening

details improved the composite behavior, but to different extent. The specimen with the

re-bar detail that is easy to fabricate and costs effective exhibited excellent behavior in

terms of ultimate strength, stiffness and ductility.

KEYWORDS: composite behavior, CFT column, re-bar stiffening detail.

1. INTRODUCTION

In the building construction industry, composite construction is gaining

widespread popularity in recent years. Its better structural performance and relatively

lower costs compared to conventional reinforced concrete construction makes it

especially attractive for high-rise building projects. In this connection, structural

291

292 S.P. Chiew and C.W. Dai

engineers have been experimenting with different types of composite columns in a hope

to produce the most aesthetically impressive and futuristic buildings. Undoubtedly, new

breakthroughs with this form of construction will usher a new and exciting era into the

building construction industry.

Basically, there are two types of composite columns: concrete-encased structural

steel section and concrete-filled tube (CFT) columns. CFT column has many advantages

over other types of column. Architecturally, CFT columns have many attractive features;

for example, the concrete filling has no visual effect on their external appearance. The

advantages from a structural point of view are, firstly, the triaxial confinement of the

concrete within the section, and secondly, the fire-resistance of the column which largely

depends on the residual capacity of the concrete core. During construction, the steel tube

will dispense with the need for formwork and prevents spilling of the concrete. Although

the CFT column is an economical form of composite construction, their uses to date have

been limited due to the lack of design information on the beam-to-column connections

and to the limited construction experience. While extensive data is available on CFT

column behavior under different loading conditions, relatively less work has been done

on the connections to these columns.

Experimental results on CFT column connections can vary significantly

depending on the tube shape and other connection requirements. Broadly speaking,

details can be generalized into two categories, i.e. connections with the beams attached to

the face of the steel tube only and connections that use elements embedded into or passed

through the concrete core. Connections to the face of the steel tube include welding the

beam directly to the tube surface, using fin-plate [ 1 ] or cover plate to connect the beam to

the tube and providing diaphragms or external tings [2,3] to stiffen the connections.

Connections with embedded or passed elements include through bolting beam end plates

and continuing structural steel shapes into and through the column [4,5].

This paper summarized an experimental investigation to study the connection

details to circular CFT columns. The objectives are: a) to investigate the effect of

different parameters on the composite behavior of the steel I-beam to CFT column ~J

connections, and hence, an strength and stiffness prediction for this type of connection

can be given; b) to compare the effect of different stiffening details on the composite

Experimental Study of Steel I-Beam to CFT Column Connections 293

behavior, so that a relatively good stiffening detail can be recommended and c) to provide

test results to verify the finite element model built for this kind of composite connection.

Experimental results showed that the simple connections have weaker ultimate strength,

ductility and stiffness, and their behaviors are influenced by the parameters investigated.

All stiffening details improved the composite behavior, but to different extent. The

specimen with the re-bar detail exhibited excellent behavior in terms of ultimate strength,

stiffness and ductility. This detail which is easy to fabricate and cost effective proved to

be the most promising of all.

2. EXPERMENTAL PROGRAM

2.1 Specimen Details

A total of eight specimens were tested in this study. All specimens are modeled to

�89 scale of the real size. Fig. 1 and Tables 1-2 show dimensions and details of the test

specimens. The materials used in the specimens are equivalent to BS4360 grade 43A

steel. Specimens consist of simple and rigid composite connections. For simple

connections, the parameters investigated are tube thickness (6.3mm and 8.0mm), tube

extemal diameter (219.1mm and 273mm) and beam size (203mm x 133mm x 31.3kg/m

and 254mm x 146mm x 31.25kg/m). For the rigid connections, the stiffening details

investigated include cover plate, shear plate, external ring and re-bar respectively.

Table 1. Details of Test Specimens

Specimen

UCN'I

Beam size (mm x mm x kg/m) .

203 x 133 x 31.3

Column size (mm x mm )

219.1 x 6.3

Stiffener type

No stiffener

Simple ucN-2 203 x 133 x 31.3 d~ 219.1 x 8.0 No stiffener

Connection UCN-3 203 x 133 x 31.3 d~ 273 x 6.3 No stiffener

UCN-4 254 x 146 x 31.25 qb 219.1 x 6.3 No stiffener , , ,

UCN-5' "203 x 133 x 31.3 d~ 219.1 x 6.3 Cover plate

Rigid UCN-6 203 x 133 x31.3 ~ 219.1 x 6.3 Shear plate

Connection UCN-7 203 x 133 x 31.3 (~ 219.1 x 6.3 External ring I

UCN-8 ' 203 x 133'x'31.3 d~ 219.1 x 613 Re-bar

294 S.P. Chiew and C. I41. Dai

Fig. 1 Overall Dimensions of Test Specimens

2.2 Load Application and Instrumentation

Monotonic static loads were applied as shown in Fig.2. Bonded strain gauges

were installed to observe the stress distribution in the flange, web and stiffeners (if

available). Also, 14 linear variable displacement transducers (LVDT) were used to

measure the vertical displacement, lateral displacement and the rotation of the 1-beam as

shown in Fig.3. In addition, two inclinometers were installed on the upper flange of the

steel 1-beam (near column) to measure the rotation of the steel I-beam. Loading was

terminated when the deformation was already excessive or when the composite

connection lost its ultimate capacity altogether.

Fig.2 Test Set-up Fig.3 Beam Rotation Measurement

Experimental Study of Steel I-Beam to CFT Column Connections

Table 2. Details of Rigid Connection

295

3. TEST RESULTS AND DISCUSIONS

3.1 Material Properties

Material tests were performed to determine the mechanic properties of steel and

concrete used in this experiment program. For concrete material, the 28 days cube


strength is 39.88 N/mm 2 and the cylinder strength is 34.4 N/mm 2. The properties of the

structural steel are summarized in table 3.

Coupon Thickness

(mm)

BF(203x133) 9.86

BW(203x133) 6.34

BF(254x146) 8.76

BW(254x146) 5.98

CL(219. lx6.3) 6.19

CL(219. lx8.0) 7.98

CL(273x6.3) 6.21

steel plate 10.47

Re-bar ~30

BF: beam flange

Table 3 Steel Mechanical Properties

Gy

(N/mm 2) 355.9

385.0

351.2

407.3

357.4 , ,

409.5

347.7

283.0

479.4

BW: beam web

6y E o. o ]o . Ob EIo.

(xl0 "6) (N/mm 2) (N/mm 2) (%) (N/mm 2) (%)

1725 206357 498.1 71.5 371.0 25.9

1867 206201 507.2 75.9 401.2 26.4

1709 205560 483.8 72.6 362.2 27.8

2030 200620 503.5 80.9 404.6 21.6

1716 208290 443.8 80.5 321.07 26.4

1962 208764 499.3 82.0 377.0 24.4

1665 208850 455.87 76.3 357.6 24.9

1363 207566 429.9 65.8 336.13 37.0

2601 184338 596.8 80.3 430.6 25.1

CL: column Cy: yield stress

Ou: ultimate stress Elo.: elongation

3.2 Load carrying capacity

The moment-rotation relationships of all specimens are shown in Fig.4 and Fig.5.

Table 4 shows the experimental and numerical analysis results. The yield load in table 4

was determined as the value at an intersection point between an initial tangential line

from the origin point and a tangential line with a 1/3 slope of the initial tangential line [6]

as shown in Fig.5. The yield loads of all specimens are compared with the numerical

analysis results. In order to evaluate the load carrying capacity of the connection, the

yield moment of the test result is also compared with the plastic moment capacity of the

steel I-beam.

For simple connections, except specimen UCN-2, all other specimens had a weak

load carrying capacity. This was reflected on the coefficient a - the value of ot is just

between 0.40-0.48. This means these connections can not even achieve half the beam' s

capacity. The specimen UCN-2 had a higher load carrying capacity and this illustrated

that the thickness of the steel tube is an important parameter on the load carrying

capacity. It is also found that the load carry capacity will decrease when the outside

Experimental Study of Steel I-Beam to CFT Column Connections E 297

Fig.4 Moment-Rotation relationship of Simple Connections

Fig.5 Moment-Rotation relationship of Rigid Connections

diameter of the steel tube increased by comparing specimens UCN-1 and UCN-3. Under

the same cross-section area, the selection of the higher and wider, but thinner steel I-

beam can improve the yield load of the connection about 42%, however, the value of the

coefficient c~ is almost the same (0.45 and 0.48 respectively). This means that the load

carrying capacity of the connection depends on the properties of the composite column,

the steel I-beam has lesser effect on it.

For rigid connections, all specimens have a higher load carrying capacity when

they are compared with the standard one (specimen UCN-1). This means the different


Table 4 Comparison of Numerical and Experimental Results

Specimen Test result

M~ (kN.m)

UCN-1 95.2

UCN-2 128.2

My, ( ~ . m ) FEA result

M . (kN.m)

My1/My2 1%

62.3 64.4 0.967 i37.95

87.8 0.968 137.95 90.7

55.3 UCN-3 96.1 54.8 0.991 137.95

UCN-4 140.4 88.5 91.7 ' 184.4

UCN-5 123.8 . .

UCN-6 118.8

UCN-7 138.8

UCN-8 229.4

81.9

88.8

78.8

0.965

0.962

0.887 78.8

90.8

137.95

137.95

137.95

o~ = M , , , / ~ ,

0.45

0.64

0.40

0.48

0.57

0.57

86.1 1.054 0.66

175.5 189.9 0.924 137.95 1.27 , .

Mu: ultimate moment Myl, My2: yield moment

Mp: plastic moment capacity of steel I-beam

Load 113 initial stiffness

Initial sty"

Yield Load

Def lect ion

Fig. 5 Determination of yield load

stiffeners are useful on improving the connection' s load carrying capacity, but their

effects are different. The yield load of specimen UCN-5 and UCN-6 are 1.26 times of that

of the specimen UCN-1. The external ring stiffener can enhance 46% of the yield load

and the degree can be higher if the external ring had a higher strength (the yield strength

is only 283N/mm 2 in this experimental project). The re-bar stiffener is the most effective

one in improving the load carrying capacity--the yield load is 2.82 times of the specimen

UCN-1.

The numerical analysis was carried out with MARC version K7.0 software


package--a general purpose finite element analysis program for nonlinear or linear stress

analysis in the static and dynamic regimes [7]. Three kinds of element are used in the

analysis: the 8-node doubly curved thick shell element is used to model the steel tube and

beam, the 8-node isoparametric three dimensional elements is used for the concrete and

the friction and gap link element is used to model the interaction between the steel tube

and the in-filled concrete. The Von Mises yield criterion and Buyukozturk yield criterion

are adopted for steel and concrete material respectively. The Newton-Raphson iterative

procedure is used in the analysis. The yield load was obtained by inputting actual material

properties into the program.

From table 4, it can be found the numerical results agree well with the test results

for all simple connections. For rigid connections, the prediction by numerical analysis is

acceptable, except specimen UCN-7, where the prediction is not so good--11% lower

than the test result. The good agreement between the numerical and test results shows that

it is feasible to use finite element analysis to predict the load carrying capacity of this

kind of composite connection.

3.3 Initial Stiffness and Ductility

Initial stiffness and ductility of each specimen is represented in table 5. They are

all compared with specimen UCN-1. For simple connections, from the table, it can be

found that the investigated parameters affect the initial stiffness and ductility in different

ways. When the thickness of the steel tube wall was increased, both the initial stiffness

and ductility of the connection were improved. The increase in the diameter of the steel

tube caused the increase of ductility, but it decreased the initial stiffness. The selection of

higher and wider but thinner beam (having the same cross-section area with UCN-1)

improved the initial stiffness of the connection, but it almost had no effect on ductility.

For rigid connections, the use of cover plate stiffener (UCN-5) improved the initial

stiffness of the connection, but it had no effect on ductility. Shear plate stiffener (UCN-6)

increased the ductility greatly, also had some useful on improving the initial stiffness.

The use of external ring stiffener (UCN-7) improved the ductility of the connection

obviously, but the initial stiffness decreased. The reason caused the reduction in initial

stiffness may be the use of lower strength steel plate (refer to table 3) in fabricating the


external ring. The connection that used re-bar as the stiffener (UCN-8) performed very

well in terms of initial stiffness and ductility. Compared with UCN-1, its initial stiffness

and ductility are 2.09 and 2.13 times respectively.

Table 5 Initial Stiffness and Ductility

Specime UCN- 1 UCN-2 UCN-3 UCN-4 UCN-5 UCN-6 UCN-7 UCN-8 Stiffness 1.0 1.30 0.73 1.52 1.31 1.11 0.78 2.09 Ductility 1.0 1.31 1.38 0.95 0.99 2.23 1.63 2.13

3.4 Failure Modes

Figure 6 shows the failure modes of some specimens. For all simple connections,

the failure began with the tube tearing at the beam flange attachment point on the tension

zone and then, with the increment in load, buckling appeared on the beam flange near

joint. For rigid connections, the failure modes of specimen UCN-5 and UCN-6 were

similar to those of the simple connections, but no buckling can be observed on

compression zone. The failure of the specimen UCN-7 began with the external ring

rupture at the beam flange attachment point, with the increment in load, the crack

developed through the external ring and finally the tube tore at the web attachment

position. Buckling can also be observed on the flange near the external ring (Fig.6 (c)).

The failure of the specimen UCN-8 began with the buckling near the end of the re-bar

(Fig.6 (d)). The twisting of the beam can be observed on the later stage (Fig.6 (e)). In

order to check the inside condition of the concrete core, specimen UCN-4 and UCN-8' s

steel tube skin around the joint were cut away after test. It was found that the concrete

core behind the connection exhibited no signs of crushing or distress even though it has a

lower compressive strength (Fig.6 (f)).

4. CONCLUSIONS

The following conclusions can be obtained from the tests:

1) Simple connections have weaker stiffness, ductility and strength. In this experiment,

their yield moment had only 40%-48% of the beam cross-section plastic moment

except UCN-2. The change of the thickness of the steel tube obviously influence the


Fig.6 Failure Modes

302 S.P. Chiew and C. HI. Dai

composite behavior of the connection.

2) With the same cross-section area, the selection of deeper and wider, but thinner steel

1-beam can improve the connection moment transfer capacity, but the ratio of the

yield moment to beam' s plastic moment capacity is almost the same.

3) Stiffeners have some effect on improving the connection' s stiffness, ductility and

strength. The degree of improvement on strength varies from 26% to 46% for shear

plate, cover plate and external ring.

4) Re-bar can greatly improve the composite behavior of the connection. The ultimate

strength of the connection with the re-bar stiffener is about 2.4 times of the

connection without stiffener. The main reasons are, firstly, the re-bar can improve the

boundary condition, eliminate the stress concentration point that appeared on the

column wall at the connections without stiffener; secondly, the re-bar can improve the

beam' s cross-section bending stiffness; and finally, the re-bar can move the plastic

hinge away from the column face.

5) Test results proved that the finite element models built up for the composite

connections are feasible to be used for the strength predictions.

ACKNOWLEDGEMENTS

The authors wish to Nanyang Technology University, Singapore for their

financial support in this project. The authors would also like to thank the technical staff

of the CSE Construction Lab and CSE Heavy Structures Lab for their assistance in

fabricating and testing all the specimens.

REFERENCES

1. Shakir-Khalil. H. (1992), "Full Scale Tests on Composite Connection" ASCE

Proceedings, Composite Construction of Steel and Concrete II, pp. 539-554.

2. Kato, B., Kimura, M., Ohta, H. and Mizutani, N. (1992), "Connection of Beam

Flange to Concrete-filled Tubular Columns" ASCE Proceedings, Composite

Construction of Steel and Concrete II, pp. 528-538.


3. Gang, H.G., Chung, I.Y., and Hong, S.G. (1998), "Performance of Concrete Filled

RHS Column-to-Beam Connections with Exterior Plate Diaphragm" Structural

Steel PSSC' 98 Vol.2, pp. 729-734.

4. Schneider, S.P. and Alostaz, Y.M. (1998), "Experimental Behavior of Connections

to Concrete-Filled Steel Tubes" . Journal of constructional steel research, Vol. 45,

No. 3, pp. 321-352, 1998.

5. Oh, Y.S., Shin, K.J. and Moon, T.S. (1998), "Test of Concrete-filled Box Column to

H-Beam Connections" . Structural Steel PSSC' 98 Vol.2, pp. 881-886.

6. Oh, Y.S., Shin, K.J., Lee, M.J. and Moon T.S. (1995), " A Study on the Bending

Behaviour of Connections for Empty and Concrete-Filled Box Steel Column and H-

Beam by Stiffened Triangular Plates" , Proceedings of 4 th pacific Structural Steel

Conference, V.2, Singapore, pp. 57-64.

7. MARC analysis research corporation, MARC manual Vol. A, 1995.


B E H A V I O U R OF T-END PLATE CONNECTION TO RHS PART I: EXPERIMENTAL INVESTIGATION

M. Saidani, M. R. Omair, and J. N. Karadelis

School of the Built Environment Coventry University, Priory Street, Coventry CV 1 5FB

ABSTRACT

This paper is concerned with investigating the behaviour of welded T-end plate connections to rectangular hollow sections when subjected to tension. A series of static tests were conducted to failure with varying parameters for the tube wall thickness and the cap plate thickness. The cleat plate thickness was kept constant for all tests. Stresses, strains and deflections at different locations in the connection were recorded and plotted against the applied load. Numerical modelling of the connection was undertaken using the finite element suite ANSYS as discussed in the companion paper.

KEYWORDS

Hollow sections, T-end plate connection, tests, tension, stresses, deformations, design.

INTRODUCTION

The excellent properties of structural hollow sections have long been known Cran (1977), CIDECT (1984), and Packer et al (1992). Connections made with hollow sections are often said to be complex and expensive. In reality they can be made simple and cost-effective. This is to add to their e,:cellent aesthetic appearance making them the ideal choice in many elegant structures.

Rectangular hollow section (RHS) members are often used as compressive members due to their good buckling stiffness. Often such members are also required, and indeed should be designed, to take also tensile forces. One of the simplest ways to connect tubular members is by cutting the ends and welding together. However, depending on joint configuration and number of members connected, this may result in complex and expensive connections. The alternative would be to connect the members together through some other means. Figure 1 shows types of end connection details for hollow tubes and which are used in practice. One of the most economic solutions is to weld a cap plate to the tube (CHS or RHS) and then weld on to it a cleat plate (Figure 2). The connection could be made entirely in the workshop, thus reducing labour work on site.

305

306 M. Saidani et al.

Figure 1: Type of end-connections

In the UK there is very little guidance on the design of welded T-end connections. Elsewhere, research work was mainly carded out by Kitipornchai and Traves (1986), Stevens and Kitipornchai (1990), and Granstrom (1979). The absence of design recommendation very often leads designers to specify uneconomical solutions.

Figure 2: Welded T-end plate connection

Research has shown that welded T-end connections subjected to uniform tension may fail in different ways. The failure mode is dictated by parameters such as:

�9 Tube wall thickness tw; �9 Cap plate thickness te; �9 Cleat plate thickness to; �9 Weld size and quality.

The possible resulting modes of failure are as follows:

�9 Tube yielding; �9 Local fracture in tube (in the region adjacent to weld); �9 Fracture of the weld; �9 Yielding of the cap plate; �9 Shear failure of the cap plate; �9 Yielding of the cleat plate.

Combination of more than one mode of failure is possible. In a truss environment (when the connection forms part of the truss assembly), there is also the possibility of the bolts failing.

Behaviour of T-End Plate Connections to R H S Part I 307

The problem being investigated in this research programme is to answer the fundamental question: how does the cap plate thickness influence the mode(s) of failure of the connection? In order to answer this question, an experimental programme was conducted on a series of specimens with varying parameters. In the companion paper, Karadelis et al (1999), a finite element model is developed, analysed and results are compared with the test results.

E X P E R I M E N T A L P R O G R A M M E

The experimental programme followed a similar procedure adopted by Stevens and Kitipornchai (1990) so that useful comparison could be made. The testing work included 8 specimens with varying tube wall and cap plate thickness. Each specimen was loaded in tension, taking all precautions to avoid any accidental eccentricity. Strains and deformations were also measured. The test arrangement for the strain gauges and LVDT's is shown in Figure 3. The programme of tests is summarised in Table 1. In order to keep the investigation manageable, only one cleat plate thickness was used and kept equal to 15mm for all specimens.

The test programme was devised to concentrate on the yielding of the tube wall and the deformation of the cap plate as these were found to be the main causes of failure. Strain gauges were located on the tube wall (four faces), the cap plate, and the cleat plate with the aim of closely monitoring strain (and stress) variations across the specimen. The LVDT's will give readings of the deformations and an indication of any in-plane and/or out-of-plane movements.

TABLE 1 TESTING PROGRAMME

Test No.

1 2 3 4 5 6 7 8

Cap plate thickness Cleat plate thickness Tube size Steel grade Comments (mm) (mm)

10 15 60x60x4.0 $275 test No.1 was 10 15 60x60x4.0 $275 repeated due 15 15 60x60x4.0 $275 to premature 20 15 60x60x4.0 $275 failure of the 30 15 60x60x4.0 $275 weld 10 15 80x80x4.0 $275 15 15 80x80x4.0 $275 20 15 80x80x4.0 $275

It is important that the strain gauges are kept far enough from welds in order to avoid any influence from the residual stresses on readings. The total length of the tube is 500mm again for the same raison. Strain gauges were placed on opposite sides so that in-plane and out-plane bending moments could be monitored and calculated. The general arrangement for the testing is shown in Figure 4. In total 12 stain gauges and 5 LVDT devises were used to monitor the joint behaviour and obtain the necessary information. In some locations (at some distance from the welds) rosette gauges were used with the aim of obtaining strains at different angles at a point. This was decided after the first two tests when it became evident that strains (stresses) were not uniaxial, but were in fact developing at an angle to the longitudinal axis of the tube.

British Steel was used for all the tests. Samples were cut out from each specimen and were tested in accordance with British Standards for testing in order to check the material properties (Young's modulus, yield strength, and ultimate tensile strength). Accurate material properties are important to


obtain since these are needed for accurate numerical modelling of the specimens as described in the companion paper, Karadelis et al (1999).

Finally, although two tube sizes were tested, the tube thickness was kept constant at 4mm, again with the aim of keeping the investigation manageable.

Figure 3: Specimen dimensions (all in mm)

Figure 4: Joint testing arrangement

THE SPECIMENS TESTING AND RESULTS

The DENISON machine with a capacity of 500kN was used for the testing of the joints. A tensile load is applied in increments of 10kN up to failure. The strains and deformations are recorded for

Behaviour of T-End Plate Connections to R H S Part I 309

each load increment into a computer logged to the testing machine. Using a simple spreadsheet program, stresses are calculated and various graphs are plotted.

Table 2 summarises the results from the specimen testing. For test No. 1, the failure was due to weld fracture. On close examination of the specimen it was discovered that weld penetration was not adequate. As a result, it was decided that welding should be done very carefully making sure it is evenly spread with sufficient material penetration. In the subsequent specimens, failure was mainly due to tube yielding. Yielding was also noticeable in the cap plate.

TABLE 2 SUMMARY OF TESTING RESULTS

Test No.

Cap plate thickness

Tube size Failure mode Full tension capacity Pc (kN) [theo]

First yield Py (kN) [exp]

Ultimate load Pu (kN) [exp]

10 60x60x4.0 Weld fracture 409.6 190 260 10 60x60x4.0 Tube yielding 409.6 230 313 15 60x60x4.0 Tube yielding 409.6 280 350 20 60x60x4.0 Tube yielding 409.6 280 385 30 60x60x4.0 Tube yielding 409.6 240 350 10 80x80x4.0 Local fracture 559.2 230 329

in tube 15 80x80x4.0 Tube yielding 559.2 320 450 20 80x80x4.0 Tube yielding 559.2 370 490

Pu/Py

1.37 1.36 1.25 1.38 1.46 1.43

1.40 1.32

In all the tests, the determination of the first yield point was proven very difficult to do accurately. The values shown in Table 2 were obtained by considering the load-axial deflection curve and taking the point where departure from the initial elastic path became measurable. The load-stress curves were also used to help with the determination of the first yield point. Figures 5 and 6 show typical load-deformation and load-stress curves for specimen No.2.

Using the strain gauge readings, in-plane and out-of-plane bending moments were calculated in order to check their relative significance. Again, typically, results for specimen No.2 are shown in Figures 7 and 8 respectively. For each specimen, the following graphs were plotted:

i. Strain vs. stress ii. Load vs. deformation

iii. Load vs. in-plane bending moment iv. Load vs. out-of-plane bending moment v. Load vs. axial force in the tube

Only samples of these graphs are shown for specimen 2.

As can be seen from Table 1, at the exception of specimen No.5, both the ultimate loads and observed first yield loads increase with the cap plate thickness. Similarly, with increased tube size, both the ultimate loads and observed first yield loads increased. Specimen No.5 was quite interesting. This particular test was repeated three times since there were some doubts about the results, and every time the results were almost the same. This seems to suggest that, as the cap plate thickness increases beyond 20mm (or may be 25mm), the ultimate capacity of the connection decreases. This could be explained by the fact that, as the cap plate becomes very thick, the tube would yield earlier resulting in a reduction in the joint capacity. Three particular modes of failure


were observed: (a) weld fracture; (b) tube yield failure; (c) local fracture of the tube wall at the vicinity of the weld.

Failure mode (a) would only occur if the weld were the weakest part of the joint. This could be avoided by carefully controlling the quality (and size) of the weld. Failure mode (b) is the more general one, occurring in almost all the tests. If the welds in the cap plate-tube connection and the cleat plate-tube connection are strong, then failure mode (c) may take place especially for connections with thinner plates. As can be seen from Figure 5, the axial deformation remains linear for load of up to 230kN (estimated first yield point). In this case, the failure load was 313kN. This gives a ratio of ultimate strength to yield strength of 1.36. In reality the ultimate load could be higher since the test was stopped as soon as the machine ceased taking any more loads. The stresses at mid-height of the tube (Figure 6) were linear up to about 120kN and thereafter non-linear. This was typical in all the specimens tested. Good agreement is obtained with the finite element modelling as shown in the companion paper, Karadelis et al (1999).

Load (kN) 35O

30O

250

2OO

150

100

50

0 0 10 20 30

O v e r a l l ax ia l d e f o r m a t i o n ( m m )

Figure 5: Load vs axial deformation (LVDT1)

Load (kN) 35O

3O0

25O

20O

150

100

5O

0 I . . . . i i

0 200 400 600 800

Stress in tube (N/mm 2)

Figure 6: Load vs axial stress in the tube (SG2/3)

35o ] Load (kN)

300 J ~ 250 f f

200 ~ 1 150

100

-50 0 50

Out-of-plane bending moment (kN.m)

-10 -5

350

300

250

200

150

100

50

0 0 5

In-plane bending moment (kN.m)

Figure 7" Load vs out-of-plane bending Figure 8: Load vs in-plane bending

It is also evident from Figures 6 (and in fact in other specimens), that extensive stress redistribution and strain hardening were taking place. Examination of Figures 7 and 8 show that the in-plane and

Behaviour of T-End Plate Connections to RHS Part I 311

out-of-plane bending moments were small and could therefore be ignored. As the load approaches the failure load, the deformations in the specimen become more important resulting in a sharp increase in in-plane and out-of-plane bending moments. Again, this was characteristic in all the joints tested.

CONCLUSIONS AND FUTURE WORK

The behaviour of welded T-end plate connections has been investigated through a series of tests. Numerical models have also been used to predict their behaviour. It was found that, apart from any weld defects, the mode of failure of the joint could be by generalised tube yielding or local fracture of the tube wall. It is suspected that as the cap plate gets thicker (more than 25mm), the capacity of the joint is reduced suggesting that joints with excessively thicker plates are less stronger than would normally be expected. The results also suggest that considerable stress re-distribution and strain hardening were taking place after the first yield More tests are under way for 'true' rectangular hollow section tubes (as compared to square). The effect of changing the orientation of the cap plate in relation to the tube will be examined. The finite element model will be further refined and benchmarked. The results will be used to produce design guideline for this type of connection, based on this work but also on previous published work.

A K N O W L E D G E M E N T S

The authors are very grateful for the generous support and contribution from British Steel, pipes and tubes (Corby, UK), especially to Mr Eddie Hole, sales manager, and Noel Yeomans, technical manager. Thanks also to Mr John Griffiths for his assistance is preparing and testing the specimens.

REFERENCES

Cran J.A. (1977). World wide applications of structural hollow sections, the sky's the limit. Symposium on tubular structures, Delft, The Netherlands, 23.1-23.14.

Comite International pour le Developpement et l'Etude de la Construction Tubulaire, British Steel, and the Commission of the European Communities (1984). Construction with hollow sections, Wellingborough, Northants, UK.

Packer J.A., Wardenier J., Kurobane Y., Dutta D., and Yeomans N. (1992). Design guide for rectangular hollow section (RHS) joints under predominantly static loading. Verlag TUV, Rheinland, Koln, Germany.

Kitipornchai S. and Traves W.H. (1989). Welded T-end connections for circular hollow tubes. Structural Engineering, ASCE 115:12, 3155-3170.

Stevens N.J. and Kitipornchai S. (1990). Limit analysis of welded tee end connections for hollow tubes. Structural Engineering, ASCE 116:9, 2309-2323.

Granstrom A. (1979). End plate connections for rectangular hollow sections. The Swedish Steel Construction Institute. Report 15:15.

Karadelis J.N., Saidani M., Omair M.R. (1999). Behaviour of end-plate connection to rectangular hollow section. Part II: Numerical modelling. ICASS99', Hong Kong, PRC.


THE BEHAVIOUR OF T-END PLATE CONNECTION TO SHS. PART II: A NUMERICAL MODEL

J N Karadelis, M Saidani, M Omair.

School of the Built Environment, Coventry University, Coventry, CV 1 5FB, UK.

ABSTRACT

The behaviour and performance of a family of structural connections made of square hollow sections (SHS) has been investigated in the laboratory and a series of data have been collected and presented in a graphical form. In parallel, a rigorous finite element model was developed capable of analysing the system of the SHS, the cap-plate, the cleat-plate and its surrounding weld. Evidence of non- linearity and deviation from the classical linear elastic theory led to a more complex numerical solution to fit more closely the experimental data. A specific methodology is presented, as it applies to analyses involving plasticity and large deflections (deformations). Test results obtained in the laboratory were compared with computed values from the finite element analysis and are presented graphically in the last pages of this paper. Satisfactory agreement was obtained between recorded and computed strains and displacements. The paper includes extensive discussion of the above results and the conclusions drawn from them. A brief account of directly related future research work is also given.

KEYWORDS

SHS, Non-linear, FE-Analysis, ANSYS, Stresses, Strains, Displacements.

INTRODUCTION

There is no doubt that the description of non-linear phenomena inevitably lead to non-linear equations which immediately render classical methods of mathematical analysis inapplicable. No method is yet known for finding the exact solution to a system of non-linear equations, such as the one shown below.

{Fe}=[ke(8,F)]{8 ~} (1)

Where: [ke(8,F)] = stiffness matrix of the element which is a function of {8} and {F}. {8 e} = displacement vector of the element. {F e} = load vector of the element.

313

314 J.N. Karadelis et al.

Non linear structural behaviour arises from a number of phenomena, which can be grouped into three main categories such as, changing status, geometric non-linearities and material non-linearities.

THEORY

Due to their nature, non-linear problems require special solution techniques. The established Newton-Raphson (N-R) method, Zienkiewicz O C et at (1994) is a series of successive linear approximations (iterations) with corrections and can be used in special algorithms to solve non-linear problems. The stiffness matrix [K] and the restoring force {F nr} vary with the applied load. Each linear approximation requires at least one iteration through the equation solver. The stiffness matrix [K] may be updated in every iteration, occasionally, or not at all. Accordingly, the method is called full, modified, or initial-stiffness (N-R) procedure.

GEOMETRIC NON-LINEARITIES (LARGE DISPLACEMENTS APPROACH).

If a structure undergoes large displacements as the load is applied incrementally, then the stiffness matrix will not be constant during the loading process. When a small tensile load is applied at the centre of the cap-plate welded around the periphery of a square hollow steel member (Figure 1, end of paper), the strain energy stored in the material is due to bending of the plate only. However, when the load becomes large enough to bow the plate significantly, the area of the plate around the line of application of the load will undergo further deformation to accommodate the additional strain. Therefore, the stiffness of the plate at this central region will increase. From the finite element point of view, geometric non-linearities are not difficult to deal with, provided that stresses do not affect the stiffness matrices significantly (that is, the stiffness matrices formed are not stress dependant).

In a typical case like the above, the load was divided into a series of sufficiently small increments (steps) and these were applied one at a time. ANSYS strongly recommends that for large displacements analyses the loads specified in the load steps should be 'stepped up' (as opposed to 'ramped on'). This means that the value of a particular load step will be reached during the first iteration and will be kept constant during the remaining iterations, until the end of the load step. This contributes to faster convergence. After each increment the deflections caused were calculated by using the linear version of the equation 1, above. That is, it was assumed that the stiffness matrix was constant during the application of each load increment. The initial stiffness matrix was used to generate the equations for the next increment and so on, until the process was completed. The original co-ordinates of the nodes were then shifted by an amount equal to the values of the displacements calculated. The new stiffness matrix for the deformed plate was re-calculated and the process was repeated until the total load was reached. The matrix notation of the incremental procedure, using the linear version of the equation 1 above, is shown below.

{AF, }= [K (i-1)]{ A ai} , V i ~ 9t N :i = 1,2,3,4,... (2)

(i= positive integer representing stage of incremental loading)

The initial Tangent Modulus, tE0, was taken from a non-linear stress-strain curve obtained from experimental observations. The initial stiffness matrix [K0], was then computed from the tangent modulus tE0 and the Poisson's Ratio, v. Note that v could also be defined as Tangent Poisson's ratio tv0 and can be obtained by evaluating the first derivative of the volumetric strain with respect to the axial strain curve. However, this was not found to be appropriate at this stage.

The Behaviour of T-End Plate Connections to SHS. Part H 315

It is important to keep the load increments small, so that the increments in displacement cause negligible changes in the stiffness matrix at each step. On the other hand, increments should be sufficiently large otherwise the cost of computer running may become excessive. These displacements were added together and gave the total displacement after the final increment. Stresses and strains corresponding to the above displacements were treated in the same manner.

ANSYS activates the large deflection analysis within the static analysis using the NLGEOM, ON option. It can be summarised as a three step process for each element.

1. Determination of the updated transformation matrix [Tn] for the element. 2. Extraction of the deformation displacement {Und}, from the total element displacement {Un}, in

order to compute the stresses and the restoring force {Fenr}. 3. After the rotational increments in {Au} are computed, node rotations must be updated

Any desired loading can be applied. However, the program must be allowed to iterate until the solution converges. Intermediate iterations are not in equilibrium and do not represent valid solutions. In order to obtain valid solutions at intermediate load levels, and/or to observe the structure under different loading configurations, multiple load steps may be desired.

A synoptical flow chart containing the main steps for large displacements analysis and demonstrating the presence of stress stiffening effects is shown below, in Figure 2. Simply stated, stress stiffening looks at the state of stress in a FE-model and calculates a stiffness matrix, [S], based upon it. Matrix [S] is then added to the usual (initial) matrix [K] and a new set of displacements are calculated.

Form

K

Calculate u0

and stresses.

Calculate S from stress state and add to K

V3

First iteration

Second and subsequent iterations.


Calculate displacements

and stresses using (K+Si) U i - F

Yes J

STOP

Second and subsequent iterations.

)

Figure: 2 The basic steps of a non-linear, large displacements analysis

MATERIAL NON-LINEARITIES

Material non-linearities occur when the stress is a non-linear function of the strain. The relationship is also path dependant, that is, the stress depends on the strain history as well as the strain itself. ANSYS accounts for several types of material non-linearities. Rate independent plasticity will be utilised here as it is characterised by irreversible straining, once a certain level of stress is reached.

Elasto-plastic behaviour. General approach.

ANSYS theory for elasto-plastic analysis, provides the user with three main elements: The yield criterion, the flow rule and the hardening rule.

The yield criterion determines the stress level at which yielding is initiated. For two and three dimensional stress systems this can be interpreted through the equivalent stress (~eq), MASE G E (1970). The flow rule determines the direction of plastic straining (ie: which direction the plastic strains flow) relative to x,y,z axes. Finally, the hardening rule describes the changes the yield surface undergoes with progressive yielding so that the various states of stress for subsequent yielding can be established. For an assumed perfectly plastic material the yield surface does not change during plastic deformation and therefore the initial yield condition remains the same. For a strain hardening material, however, plastic deformation is generally accompanied by changes in the yield surface. Two hardening rules are available and these are isotropic (work) hardening and kinematic hardening.

The Behaviour of T-End Plate Connections to SHS. Part H

Multilinear Isotropic Hardening

317

For materials with isotropic plastic behaviour, the assumption of isotropic hardening under loading conditions postulates that, as plastic strains develop, the yield surface simply increases in size and maintains its original shape. ANSYS, uses the von Mises yield criterion with the associated flow rule and isotropic (work) hardening. When the equivalent stress is equal to the current yield stress the material is assumed to undergo yielding. The yield criterion is known as the 'work hardening hypothesis' and assumes that the current yield surface depends only upon the amount of plastic work done.

The solutions of non-linear elastic and elasto-plastic materials are usually obtained by using the linear solution, modified with an incremental and iterative approach. The material is assumed to behave elastically before reaching yield as defined by Hooke's low. If the material is loaded beyond yielding, then additional plastic strains will occur. They will accumulate during the iteration process and after the removal of the load will leave a residual deformation.

In general:

~ --~n(ela) "q- iCn(pla) nt" ~ (3)

where: en = total strain for the current iteration. lgn(ela) = elastic strain for the current iteration. Agn(p la ) = additional plastic strain obtained from the same iteration. gn- l (p la) = total, previously obtained plastic strain.

Convergence is achieved when Agn(p la ) /gn(e la ) is less than a criterion value, Walz J E et al (1978). This means that very little additional plastic strain is accumulating and therefore the theoretical curve is very close to the actual one.

For the case of uniaxial tension, it is necessary to define the yield stress and the stress-strain gradients after yielding. For the uni-axial stress governing the SHS member alone, these are simply Crxx and exx. When Oxx becomes greater than the uniaxial yield stress then yielding takes place. The yield condition

G'xx > (5"yield (4)

is the well established von Mises yield criterion for one dimensional state of stress.

Typically, a set of flow equations (flow rule) can be derived from the yield criterion which imply that the plastic strains develop in a direction normal to the surface (associative rule). The associative flow rule for the von Mises yield criterion is a set of equations called the incremental Prandtl-Reuss flow equations MASE G E (1970). That is, the strain increment is split into elastic and plastic portions and the plastic strain increments are dependant on the deviator stresses. Hence, it is necessary to apply the total load on the structure in increments. These load steps need only start after the FE-model is loaded beyond the point of yielding. The size of the subsequent load steps depends on the problem. The load increments will continue until the total load has been reached or until plastic collapse of the structure has occurred.


As the load increases the plastic region spreads in the structure and the non-linear problem is approached using the full Newton-Raphson procedure. The stiffness used in the N-R iterations is the tangent stiffness and reflects the softening of the material due to plasticity. It should be noted that in general, the flatter the plastic region of the stress-strain curve, the more the iterations needed for convergence.

As plasticity is path dependant or a non-conservative phenomenon, it requires that in addition to multiple iterations per load step the loads be applied slowly, in increments, in order to characterise and model the actual load history. Therefore, the load history needs to be discretised into a number of load steps with the presence of convergence tests in each step. ANSYS recommends a practical rule for load increment sizes such as the corresponding additional plastic strain does not exceed the order of magnitude of the elastic strain. That is, the plasticity ratio:

Ae"(Pta) ___ 5 (5)

Cn(ela)

In order to achieve inequality 9, the following loading sequence was adopted:

Load step one was chosen so that to produce maximum stresses approximately equal to yield stress The yield stress was estimated from the experimental stress-strain curves and validated by performing a linear run with a unit load and by restricting the stresses to the critical stress of the material. This was found to be approximately equal to 200 kN. Successive load steps were chosen such as to produce additional plastic strain of the same magnitude as the elastic strain or less. This was achieved by applying additional load increments no larger that the load in step one, scaled by the ratio ET]E. Such as:

E T P~+, = .-E-- P~ V n ~ 9 ~ N (6)

where: E = Elastic slope Ev = Plastic slope with: Ev/E not less than 0.05

Table 1 below summarises the plasticity theory that characterises the elasto-plastic response of a certain type of materials. However, the basic steps characterising a non-linear elasto-plastic analysis are shown in a brief flow chart, in Figure 3.

Material Option

Multi-linear Isotropic Hardening

Yield Criterion

von Mises

Flow Rule

Associative (Prandtl-Reuss equations)

Hardening Rule

Isotropic

Material Response

Multilinear

Table: 1 Summary of the theories involved in a material with multi-linear isotropic hardening behaviour.

(

The Behaviour of T-End Plate Connections to SHS. Part H

( S T A R T )

Form K

Calculate u0~ (5"0, ~0

No.

(Linear elastic)

STOP -')

Load Step 1

\ . 7 5 / ~ Yes. (proceed with nonlinear analysis)

I Calculate "~ AUl, AG1 and AE:I

" and add to u0~ (Y0~ ~;0

Calculate displacements

and stresses and add to previous values.

Yield Criterion

Is ~ ~ ~ Load=Tgt.Load

�9 j J

~ Yes

( )

319

Incremental Procedure and repeated Iterations.

Figure: 3 The basic steps of a non-linear, elasto-plastic analysis

320

THE FINITE ELEMENT MODEL.

J.N. Karadelis et al.

The basic model consisted of the rectangular hollow section with the cap-plate fully integrated at one end and the cleat-plate on top (Figure: 1). As the structural member (tie) was symmetrical about a plane at right angles to its longitudinal axis, only half of the member was initially modelled. No modelling of the weld was present. Material properties such as a Young's Modulus value of 205000 N/mm 2 and a Poisson's Ratio of 0.27 were inserted in the program. Translational restraints were applied at the cut end and a negative pressure load of 1 kN at the cleat plate.

The correct choice of element is very important in finite element analysis. A 3D, 4 node tetrahedral structural solid element (SOLID 72) with three translational and three rotational degrees of freedom (DOF) per node was chosen. This is described by ANSYS as a general purpose element particularly suited to automatic meshing of irregular volumes. A linear elastic stress analysis was performed using the unit load and the results were normalised for other load values. These results are shown plotted on the same axes with the experimental results.

Utilising the experience obtained from the laboratory tests and the linear elastic analysis results, the model was divided into three substructures. The cap plate was allowed to undergo both, large displacements and material non-linearities, as previous experimental work with various cap-plate thicknesses demonstrated that when thin, the later undergoes excessive deflections under and near the beating of the cleat plate. The region of the SHS near the cap-plate was seen to develop excessive stress concentration. These stresses tended to exceed the yield stress of the material, hence non-linear material properties were attributed to it. Finally, the weld was treated linearly at this early stage of the investigation.

A non-linear (multi-linear, elasto-plastic) analysis featuring isotropic hardening effects was performed. The large displacements option was kept open to accommodate possible non-linear effects of the cap-plate. The results are shown plotted in the next pages:

RESULTS AND DISCUSSION

Figure: 4 shows the variation of strain with load just below the interface of the cap-plate and the hollow section. This is a region of high stress concentration as predicted by the finite element model. In order to save space, strains at positions 13 and 17 as well as 16 and 18 (Figure: 1, diagram of test specimen) were averaged and plotted in pairs. Plots from the linear and non-linear analyses were superimposed for ease of comparison. The same procedure is repeated for strains developing at positions 19 and 21 and also 20 and 22, measured 300 mm below the cap plate on the SHS and presented in Figure: 5. The finite element analysis results are plotted with them. It can be seen that the strain values predicted by the finite element model are in good agreement with the corresponding results obtained in the laboratory.

Figure: 6 shows the variation of displacements as measured at positions 5 and 8 and also 6 and 7. The agreement here cannot be considered as satisfactory as the one above. At the time of writing this paper the authors are investigating all the possibilities. It is anticipated that had the weld been modelled in a more rigorous manner, its contribution to the behaviour of the SHS connection would have been better represented. Another possibility currently under scrutiny is the reliability of the ageing DENISON machine on which some tests were carded out.

The Behaviour o f T-End Plate Connections to SHS. Part H

Figure: 4 Variation of Experimental and Calculated Average Strain with Load Strain Gauges: (13+17), (16+18)

321

Figure: 5 Variation of Experimental and Calculated Average Strain with Load. Strain Gauges: (19+21), (20+22)

Figure: 6 Variation of Average Displacement with Load. LVDTs: (5+8), (6+7)


CONCLUSIONS AND FUTURE WORK

The behaviour of the SHS connections has been modelled using finite element analysis techniques. Satisfactory agreement was obtained between the experimental and the calculated strains. The deformation of the cap-plate as noted in the laboratory was predicted accurately by the FE-model. Stress concentrations were also predicted to develop directly under the cap-plate indicating symptoms of weld yielding, possible separation from the parent metal and therefore the necessity to model the weld more accurately.

The finite element model is to be refined and calibrated. Efforts will be turned to the weld around the cap-plate and between the interface of the cap-plate and the cleat-plate. The improved finite element model will be used to carry out the appropriate parametric and sensitivity studies. When all parameters are investigated structural optimisation techniques will be used to optimise the connection and develop the appropriate design guides.

Figure: 1 SHS connection, FE-model and line diagram of the test specimen.

REFERENCES

Kohnke P (editor), (1998), ANSYS 5.4, Theory Manual. Canonsburg, PA USA.

Mase G E, 1970, Continuum Mechanics, McGraw-Hill book company.

Saidani M, Omair M R, Karadelis J N (1999), Behaviour of End Plate Connection to Rectangular Hollow Section, Part I, Experimental Investigation, ICASS99' Hong Kong, PRC.

Walz J E, Fulton R E, Cyrus N J (1978), Accuracy and Convergence of Finite Element Approximations., Proceedings, Second Conference on Matrix Methods in Structural Mechanics, Wright-Paterson Air Force Base, Ohio, USA.

Zienkiewicz O C, Taylor R L (1994), The Finite Element Method, Vol. 1-2, 4 th Ed., McGraw-Hill.

CYCLIC BEHAVIOUR OF BEAM-TO-COLUMN WELDED CONNECTIONS

Elena Mele 1, Luis Calado 2, Antonello De Luca I

1 Structural Analysis and Design Department (DAPS), University of Naples "Federico II", P.le Tecchio 80, 80125 Naples, Italy.

2 Civil Engineering Department (DECivil), Instituto Superior Tecnico, Av. Rovisco Pais, 1096 Lisboa Codex, Lisbon, Portugal.

ABSTRACT

In this paper the results of an experimental program devoted to the assessment of the cyclic behaviour of full scale, European type, beam-column subassemblages with welded connections are presented. Six tests (five cyclic and one monotonic) have been carded out on three different series of specimens, encompassing a total of eighteen tests. The tests have evidenced the effect of column size and panel zone design on the cyclic behaviour and failure modes of the connections, as well as the dependency of the moment capacity and of the maximum and cumulative plastic rotation of the joint upon the applied loading history.

KEYWORDS

welded connections, cyclic tests, loading histories, rotation capacity, panel zone, failure modes.

INTRODUCTION

The confidence of structural engineering in welded moment resisting frames (WMRFs) was strongly compromised by the performances observed in the earthquakes of Northridge (1994) and Hyogoken- nanbu (1995). Following these earthquakes, extensive unexpected brittle connection damage were detected in several frames, thus discovering the alarming problem of the high seismic vulnerability of the welded steel framed structures. The brittle modes of failure occurred at the beam-to-column joints have been defined "unexpected", since the WMRF connections were usually considered as the ones characterised by the more stable and ductile behaviour, giving rise to large rotational capacity and energy dissipation. It should be underlined though that, as reported by (Bertero et AI., 1994), almost all the types of failures occurred as a result of the Northridge seismic shaking, had been observed in past experimental tests carded out in U.S.A., as well as in Japan and Europe. However the experimental behaviour of the welded connections appears highly and perhaps randomly variable.

323

324 E. Mele et al.

Starting from these observations, significant research efforts have been undertaken in the United States (Mahin et AI., 1996; Malley, 1998), in Japan (Tanaka et AI., 1997; Nakashima et AI., 1998) and also in Europe (Mele et AI., 1997; Plumier et AI., 1998; Taucer et AI., 1998; Calado et AI., 1999), in order to enrich the experimental data base and to assess the major parameters affecting the cyclic behaviour of beam-to-column connections.

In this context, a wide experimental program has been carried out at the Material and Structures Test Laboratory of the Instituto Superior T6cnico of Lisbon on different types (both welded and bolted) of beam-to-column connections. The experimental tests have been performed on specimens representative of frame structure beam-to-column joints close to the ones typical of European design practice (beams less deep than the ones adopted in the current US design of SMRFs), with the aim of defining the effect of the column size and of the PZ design on the connection behaviour, varying the applied loading history. Some preliminary experimental results on the welded connections have been presented in (Mele et AI., 1997). In this paper a complete overview on the experimental program carded out on welded connections is reported. In particular the experimental results are presented through hysteresis loops obtained in the increasing amplitude tests; further, the failure modes of the specimens are described, and the major factors affecting the cyclic behaviour and the rotation capacity are assessed.

THE EXPERIMENTAL PROGRAM

Aim

The experimental program on welded beam-to-column connections presented in this paper was aimed at evaluating the effect of the column dimensiom and panel zone design on the cyclic behaviour, ultimate strength and deformation capacity of the welded connections, varying the applied loading history.

Specimen geometry

A total of 18 beam-to-column fully welded joints (3 series x 6 specimens) have been designed, fabricated and tested up to failure under different loading histories. The specimens, made of $235 JR steel, are T-shaped beam-column subassemblages, consisting of a 1000 mm long beam and a 1800 mm long column. In the three types of specimens, respectively appointed as BCC5, BCC6 and BCC8, the beam cross section is the same (IPE300), while the column cross section is varied, being respectively HE160B for the BCC5 series, HE200B for the BCC6 series, and HE240B for the BCC8 series. The section properties of beam and columns adopted in the three specimen types are reported in table 1.

Height (ram)

TABLE 1 BEAM AND COLUMN SECTION PROPERTIES

Beam Section

All specimens

IPE 300 300

BCC5

HE160B 160

Column Section

BCC6

HE200B 200

BCC8

HE240B 240

Width (mm) 150 160 200 240 t~ (mm) 7.1 9 8 10 tf(mm) 10.7 15 13 17 I (mm 4) 83356 x 103 24920 x 103 56960 x 103 112600 x 103

Wr (mm 3) 557 x 103 311X 10 3 570x 10 3

354 x 10 3 628 x 103 643 x 103 Wpl (mm 3) 938 x 10 3

1053 x 103

Cyclic Behaviour of Beam-To-Beam Welded Connections 325

Due to the relative cross-section dimensions of column and beam in the three series of connection specimens, the beam plastic modulus is respectively larger, approximately equal and smaller than the column plastic modulus for the BCC5, BCC6 and BCC8 series.

In all the specimens, the beam flanges have been connected to the column flange by means of complete joint penetration (CJP) groove welds, while fillet welds have been applied between both sides of the beam web and the column flange. The continuity of the connection through the column has been ensured by horizontal 10 mm thick plate stiffeners, fillet welded to the column web and flanges.

Material properties

The structural steel used for the specimens (beam, colunm, stiffener plates) is $235 JR type. The basic monotonic stress-strain curve and the mechanical properties of the specimen steel components have been determined through coupon tension tests. The average values of material properties (yield and ultimate stress) for the beam and column flanges and web are provided in table 2. In the same table are also provided the plastic and ultimate flexural capacities (Mp=Wpl x fy, Mu---Wpl x fy) of the beam and of the colunm, computed on the basis of the corresponding values of yield stress fly) and ultimate stress (fu) of the section flanges obtained from the tension tests.

fy (MPa) fu (MPa)

YR Mp (kNm) Mu (kNm)

TABLE 2 AVERAGE VALUES OF MATERIAL PROPERTIES AND DERIVED FLEXURAL CAPACITIES.

BCC5 BCC6 BCC8 B e a m

IPE300 flange web 274.8 305.5 404.6 412.6 1.47 1.35

166 234

Column HE160B

flange web 323.1 395.6 460.2 490.1 1.42 1.24

118 157

B e a m

IPE300 flange Web 278.6 304.9 398.8 411.4 1.43 1.35

169 242

Column HE200B

flange web 312.6 401.6 434.9 489.8 1.39 1.22

198 276

Beam IPE300

Flange web 292 300 445 450 1.53 1.50

183 280

Column HE240B

Flange web 300 309 457 469 1.52 1.52

316 482

Experimental set-up, instrumentation plan and loading histories

The test set-up, represented in figure 1, mainly consists in a foundation, a supporting girder, a reaction r.c. wall, a power jackscrew and a lateral frame. The power jackscrew (capacity 1000 kN, stroke + 400mm) is attached to a specific frame, pre-stressed against the reaction wall and designed to accommodate the screw backward movement. The specimen is connected to the supporting girder through two steel elements. The supporting girder is fastened to the reaction wall and to the foundation by means ofpre-stressed bars.

An automatic testing technique was developed to allow computerised control of the power jackscrew, of the displacement and of all the transducers used to monitor the specimens during the testing process. Specimens have been imtrtmaented with electrical displacement transducers (LVDTs), for carefully recording the various phenomena occurring during the tests. The same arrangement of LVDTs has been adopted for the three specimen types. The typical instrumentation set-up is provided in figure 2.

Each specimen type has been tested up to failure under several cyclic rotation histories. The complete set of loading histories is provided in table 3, where loadings are defined in terms of: applied beam tip displacement (d); applied beam tip displacement normalised to theoretical yield displacement dr (d/dy); interstory drift angle (d/H), i.e. d normalised to the distance between beam tip and column centreline H.

326 E. Mele et al.

Figure 1: Experimental set-up Figure 2: Specimen instrumentation

B BB C D E

m o n

TABLE 3 LOADING HISTORIES

BCC5 Ii BCC6

+ 75 + 7.5 + 7.5 + 75 + 7.5 + 7.5 _+ 75 _+ 7.5 _+ 7.5 _+ 75 _+ 7.5 _+ 7.5

Stepwise Increasing (ECCS) Stepwise Increasing (ECCS)

Monotonic Monotonic

ii Bcc8 I

_+37.5 +3.75 +3.75

I I Stepwise Increasing (ECCS)

+37.5 1___3.75 I_+3.75 Monotonic

EXPERIMENTAL RESULTS: GLOBAL BEHAVIOUR AND FAILURE MODES

In the following the experimental results obtained in the test program are provided. In particular the cyclic behaviour and the failure modes observed for the three sets of specimens are descn'bed, and the moment rotation hysteresis loops obtained in the stepwise increasing amplitude cyclic tests are provided. In the moment rotation hysteresis loops hereafter presented, the rotation values have been calculated both as the "unprocessed" total rotation given by the applied interstory drift angle d/H, and as the beam rotation ~ obtained through the measured LVTDs displacements at the beam cross sections. Correspondingly, in the M-d/H experimental curves the moment is evaluated at the centreline of the column, while in the Mb-tlh, curve the moment is evaluated at the column flee.

In figure 3 (a) the moment - total rotation (M-d/H) experimental curves resulting from the BCC5C, BCC6C and BCC8D tests (cyclic increasing stepwise amplitude) are plotted, while in figure 3 (b) both the corresponding moment - beam plastic rotation and the moment - panel zone rotation curves are plotted. The beam plastic rotation has been obtained through the measured displacements at the transducers 1 and 2 (see figure 2) by subtracting the contributions of the beam and column elastic rotations as well as of the panel zone distortion.

Cyclic Behaviour of Beam-To-Beam Welded Connections 327

Figure 3 (a): Moment-global rotation curves

Specimens BCC5

Figure 3 (b): Moment-beam plastic rotation and Moment-panel rotation curves

As can be derived from the curves reported in figure 3 (a) and (b), and as demonstrated also in the other tests carried out in the experimental program, the cyclic behaviour of the specimen BCC5 is characterised by a great regularity and stability of the hysteresis loops up to failure, with no deterioration of stif~ess and strength properties. The very last (18 th) cycle presents a sudden and sharp reduction of strength, corresponding to the collapse of the specimen, which occurred due to fracture initiated in the beam flange and propagated also in the web. During the test, significant distortion of the joint panel zone has been observed, while not remarkable plastic deformation in the beam occurred.

In table 4 a sunmm~ of the number of complete plastic cycles to collapse and the failure mode of the specimens is reported.

TABLE 4 NUMBER OF PLASTIC CYCLES AND FAILURE MODES OF B C C 5 SPECIMENS

,

,.= !.o. +o,. , . , , .r.-od. 1

..... . ..... [ 5 .... :48 Fracture of the beam flange near the weld. I

Crack on the beam flange d ~ to the weld, propagated in the beam web !

328

Specimens BCC6

E. Mele et al.

Throughout the test program, two different kinds of cyclic behaviour have been observed for the BCC6 specimens. In some cases (tests C and D) the behaviour of the specimens is close to the behaviour observed for the BCC5 type, with almost no deterioration of the mechanical properties up to the last cycle, during which the collapse occurred. On the contrary, for the other tests (A, B and BB) a gradual reduction of the peak moment at increasing number of cycles is evident. In these eases, starting from the very first plastic cycles, local buckling of the beam flanges occurred, and a well defined plastic hinge has formed in the beam. In the specimens BCC6 the contribution of the panel zone deformation has not been as significant as in the BCC5 specimen type. The collapse of the specimens BCC6A and BCC6B was due to fracture of the beam flange in the buckled zone. The specimens BCC6BB, BCC6C and BCC6D failed due to fracture in the beam flange along or close to the weld line. In table 5 a smmnm~ of the number of complete plastic cycles to collapse and the failure mode of the specimens is reported.

TABLE 5 NUMBER OF PLASTIC CYCLES AND FAILURE MODES OF BCC6 SPECIMENS

Specimens BCC8

The hysteresis loops obtained from the tests on the BCC8 specimens (except the one obtained in the C test) show a gradual reduction of the peak moment starting from the second cycle, where the maximum value of the applied moment has been usually registered. This deterioration of the flexural strength of the connection is related to occurrence and spreading of local buckling in the beam flanges and web. A well defined plastic hinge in the beam has formed in all the tested specimens. In the test C, where the specimen has been subjected to a constant amplitude rotation history, equal to 7.5% rad, an unstable behaviour of the specimen has been observed, with multiple buckling occurred in the beam flanges starting from the first plastic cycle, and a sudden failure occurred at the third plastic cycle due to the fracture in the beam flange along the weld. In the specimens BCC8 the panel zone deformation has not been remarkable, and the plastic deformation mainly took place in the beam. The collapse of the specimens BCC8A and BCC8D was due to fracture of the beam flange in the buckled zone. In the tests B, C and E the collapse of the specimens occurred dueto fracture in the beam, starting along the weld or very close to the weld line. In table 6 a sunmam3r of the number of complete plastic cycles to collapse and the failure mode of the specimens is reported.

TABLE 6 ~ E R OF PLASTIC CYCLES AND FAILURE MODES OF B C C 8 SPECIMENS

Cyclic Behaviour of Beam-To-Beam Welded Connections

COMPARISONS AND OBSERVATIONS

329

Panel zone and beam rotations.

The contribution of the total (elastic + plastic) panel zone deformation to the global rotation of the specimens has been, throughout the experimental program: remarkable (in average equal to the 80% of the total imposed rotation) in the BCC5 specimens, having the smallest column section (HE160B); less significant (in average equal to the 65% of the total imposed rotation) for the BCC6 specimens, with intermediate column section (HE200B); minor (40-50 % of the applied rotation) in the BCC8 specimens, characterised by the largest column section (HE240B). Consistently, the plastic rotations registered in the beam have been minor for the BCC5 specimens, comparable to the panel zone rotations in the BCC6 specimens, larger for the BCC8 specimens.

The values of the total rotation capacity, which, in the increasing amplitude test, reaches 0.064 rad for the BCC5 specimen, 0.053 rad for the BCC6 specimen and 0.046 rad (at maximum strength decrease not less than 90%) for the BCC8 specimen, correspond to low values of beam plastic rotations, respectively equal to 0.0057, 0.0175 and 0.0242 rad for the three specimens, thus confirming that large rotations can be experienced thanks to column web panel deformations.

Effect of column size on the cyclic behaviour and failure mode

The BCC5 specimens, even though able to experience high deformation levels, have shown brittle failure modes in all the cyclic tests, with hysteresis loops practically overlaid and no degradation of the flexural strength up to the very last cycle, where a sudden decay of the carrying capacity occurred due to fracture, generally developed in the proximity of the weld. On the contrary the BCC8 specimens have exhibited a typical ductile behaviour, with formation of a well defined plastic hinge in the beam starting from the first plastic cycles, and a gradual decrease of the peak moment at increasing number of cycles up to the collapse.

The BCC6 specimens displayed a behaviour sometimes closer to the BCC5 ones (tests BCC6C and BCC6D), sometimes to the BCC8 ones (tests BCC6A, BCC6B and BCC6BB), depending on the applied loading sequence. Also with regard to the final collapse of the specimens, in the former cases it involved fracture in the beam starting at or close to the weld location, while in the latter cases it was due to the cracking in the buckled zones of the beam flanges.

Effect of the loading history

The different cyclic histories applied to the specimens have evidenced the dependence of the plastic deformation capacity on the loading histories. The cmnulated plastic rotations computed on the basis of the test data, result in highly variable values for the BCC5 specimens (d/~l,c,m=0.65-0.27 rad), while the BCC8 specimens show, except test C, similar values for all the tests (d~pl.cum=0.48-0.55 rad).

In the tests B and BB, in which the BCC5 and BCC6 specimens have been subjected to the same loading history (constant amplitude rotation d/H=7,5%), the BCC5 specimens have shown the same failure mode and similar number of cycles to failure (test B: 5, test BB: 4). On the contrary the BCC6 specimens showed a different behaviour, since the BCC6B specimen experienced 11 plastic cycles and collapsed due to crack in the beam flange at the plastic hinge location, while the failure of the BCC6BB specimen occurred after 6 plastic cycles due to fracture in the beam flange along the weld, propagated also in the web. Similarly to the BCC5 specimens, the BCC8 specimens which have been subjected to the same loading history (BCCSB and BCCSE, constant cycle amplitude, d=37.5 ram) have shown the same collapse mode and close values of the number of plastic cycles (test B:16; test E:15).

330 E. M e l e et al.

7. SUMMARY AND CONCLUSIVE REMARKS

In this paper an overview of a test program carried out on three series of welded connections has been provided, the global behaviour and the failure modes of the connections have been described, and the main differences in the performance of the three series of specimens have been emphasised. Some major aspects deriving from this preliminary analysis of the experimental data concern the effect of the loading history, of the column section and panel zone design on the cyclic behaviour, maximum rotation, energy dissipation and failure mode of the connections.

The quite high values of the maximum global rotations of the connections, especially if compared to the rotation capacities exhibited by US-type welded connections in past testing programs, can be related to the following aspects.

�9 According to the major findings reported by (Roeder & Foutch, 1996), the connection rotation capacity and ductility strongly decreases as the beam depth increases. Thus higher rotations are expected for beam-to-column connections usually adopted in Europe, where the depth of the beam section (db= 300 - 4 5 0 ram) is significantly less than the ones utilised in the US practice (db= 500- 1000 mm), due to the current adoption of perimeter frames configuration.

�9 Fully welded connections, as the ones adopted in the tested specimens, have already shown, in past experimental tests, higher rotation capacity than the BWWF connections (Tsai & Popov, 1995; Usami et AI., 1997).

�9 A significant contribution of panel zone deformation has been observed throughout the tests, suggesting the possibility of utilising the joint panel for providing energy dissipation and stable behaviour of the connections even at large number of cycles.

The design implications of this last aspect are currently being evaluated by the authors through the comparison with similar experimental data available in the inherent b~liography, through the evaluation of the provisions supplied by the seismic codes and through theoretical analyses.

REFERENCES

Bertero V.V., Anderson J.C., Krawinkler H. (1994). Performance of steel building structures during the Northridge earthquake. Ethq. Eng. Res. Center, Rep. UCB/EERC-94/09, University of California, Berkeley. Calado L., Mele E., De Luca A. (1999). Cyclic behaviour of steel semirigid beam-to-column connections, submitted for publication on: s Struct. Eng. ASCE. Mahin S.A., Hamburger R.O., Malley J.O. (1996). An integrated progrmn to improve the performance of welded steel frame buildings. Proc. 11 th WCEE, World Conf. Earthq. Eng., Elsevier Science Ltd., Paper No.ll14. MaUey J.O. (1998). SAC Steel Project: summary of Phase-I testing investigation results. Eng. Structs, 20:4-6, 300-309. Mele E., Calado L. Pucinotti R. (1997). Indagine sperimentale sul comportamento ciclico di alcuni collegamenti in acciaio. Proc. 8 ~h National Conf. Ethq. Engrg. ANIDIS, Taormina, Italy, 1031-1040. Nakashima M., Suita K., Morisako K., Maruoka Y. (1998). Tests on welded beam-column subassemblies. I: global behaviour. II detailed behaviour. J. Struct. Eng. ASCE, 124:11, 1236-1252. Plumier A. et AI. (1998). Resistance of steel connections to low-cycle fatigue. Proc.ll th ECEE, Balkema. Roeder C.W., Foutch D.A. (1996). Experimental results for seismic resistant steel moment frame connections. J. Struct. Eng. ASCE, 122:6, 581-588. Tanaka A., et AI. (1997). Seismic damage of steel beam-to-column connections - evaluation from statical aspects. Proc. STESSA '97, 2 "a Int. Conf. on Steel Structures in Seismic Areas, Kyoto, Japan, 856-865. Taucer F., Negro P., Colombo A. (1998). Cyclic testing of the steel frame. JRC ELSA Spec. Publ. No.L98.160, Dec. 1998 Tsai K.C., Popov E.P.(1995). Seismic steel beam-column moment connections. In: Metallurgy, Fracture Mechanics, Welding, Moment Connections and Frame Systems Behavior. Rep.SAC/BD-95/09, SAC Joint Venture, Sacramento, Cal. Usami T. et A1. (1997). Real scale model tests on flange fracture behaviour of beam adjacent to beam-to-column joint and the seismic resistance after repairing and strengthening. Proc. STESSA '97, 2 ~ Int. Conf. on Steel Structures in Seismic Areas, Kyoto, Japan, 955-962.

ADVANCED METHOD FOR MODELLING HYSTERETIC BEHAVIOUR OF SEMI-RIGID JOINTS

Y. Q. Ni, J. Y. Wang and J. M. Ko

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong

ABSTRACT

Modelling of the hysteretic behaviour at beam-to-column connections is an important issue for static and dynamic analysis of steel structures with semi-rigid joints. Some empirical models, such as the Ramberg-Osgood model, the Richard-Abbott model, and the Lui-Chen model, have been proposed for this purpose. To complete the interior and branch hysteresis curves, a set of empirical rules are indispensable to these models. However, some of the empirical hysteresis rules may conflict with the experimental observations. In this study, a mathematical hysteresis model-the Preisach model-is introduced to describe the hysteresis behaviour of steel semi-rigid connections. This phenomenological model can completely specify hysteresis curves without need of any empirical rules or additional conditions, and is really capable of representing hysteresis with non-local memory. A time-domain incremental method is presented to evaluate the transient dynamic response of flexibly jointed flames in terms of the Preisach model. A comparison of the analytical results with those by the Ramberg- Osgood model demonstrates the suitability of the Preisach model for this application.

KEYWORDS

Hysteresis, semi-rigid connection, Preisach model, phenomenological modelling, nonlinear dynamic response.

INTRODUCTION

The strength, stability, ductility and energy dissipation capacity of steel frames can be significantly influenced by the behaviour of beam-to-column connections. The semi-rigid characteristics of typical connections in steel frames have been widely recognized. To assess actual structural behaviour, it is necessary to incorporate the effect of connection flexibility in the analysis of these structures. This is particularly meaningful for the dynamic analysis because hysteretic damping at flexible connections may contribute significant energy dissipation. Some empirical models, such as the Lui-Chen exponential model (Lui and Chen 1986, Chui 1998), the Ramberg-Osgood model (Sveinsson and McNiven 1980, Chui and Chan 1996), and the Richard-Abbott model (Richard and Abbott 1975,

331

332 Y.Q. Ni et al.

Deierlein et al. 1990), have been widely used to represent the nonlinear moment-rotation behaviour of semi-rigid connections. These models are sufficient for describing the hysteresis loops under cyclic loading. However, for cases of transient loading or loading between variable limits, these models are incomplete and empirical rules have to be introduced to stipulate the paths of interior and branch hysteresis curves. It is intractable to encode these rules in computer program design and sometimes the hysteresis curves followed by the empirical rules conflict with the experimental observations.

The Bouc-Wen differential model has also been used to describe the hysteretic behaviour of semi-rigid steel connections subjected to dynamic loading (Mak 1995). This analytical model is mathematically tractable due to its complete capability in tracing transient hysteretic response. Given an arbitrary time history of the displacement, the hysteretic force can be completely specified by the Bouc-Wen model without need of empirical rules or additional conditions. However, it has been demonstrated that the differential-type models, including the Bouc-Wen model, can only represent the hysteresis with local memory (Ni et al. 1999). As a result, these models do not allow the crossing of minor loops which can arise in the measured hysteresis curves.

In the past decade, hysteresis phenomenon has been studied by mathematicians as a new branch of mathematics research (Macki et al. 1993). They explored the hysteretic nonlinearity in a purely mathematical form by introducing the concept of hysteresis operators. One of the mathematical hysteresis operators is the Preisach operator, or called Preisach model (Visintin 1994). The Preisach model formulates hysteretic constitutive relations in a conceptually simple and computationally elegant way. This model has several appealing features, including its ability to capture nonlocal memory, which make it capable of accurately modelling various hysteretic characteristics following a phenomenological approach. The Preisach model has been applied to describe the magnetomechanical hysteresis in ferromagnetic materials (Mayergoyz 1991), and the inelastic constitutive laws of ductile materials (Lubarda et al. 1993, Sumarac and Stosic 1996), piezoceramics and shape memory alloy (Hughes and Wen 1997, Song et al. 1999), and nonlinear vibration isolators (Wang et al. 1999).

The present study introduces the Preisach model to describe the moment-rotation hysteresis curves of semi-rigid joints and subsequently analyzes the transient dynamic response of flexibly connected steel frames. The Ramberg-Osgood model in conjunction with the empirical rules is first used to produce a set of hysteresis loops. By taking these numerical hysteresis loops as 'experimental' curves of semi- rigid joints, an identification technique is implemented to establish the corresponding representation in terms of the Preisach model. A time-domain incremental method is adopted to evaluate the transient dynamic response of a portal frame under seismic excitation and other dynamic loads. The hysteresis loop and dynamic response characteristics using the Preisach model are compared with those using the Ramberg-Osgood model to validate the suitability of thePreisach model.

HYSTERESIS MODELS

Experiments by many researchers have confirmed the hysteretic behaviour of typical beam-to.column connections under cyclic loading or dynamic excitation. The hysteretic behaviour provides energy absorption capacity that is beneficial for resistance against the earthquake, wind and other dynamic loads. The most direct indication of hysteretic behaviour is the hysteresis loops. The hereditary nature of hysteretic systems shows the multi-valuedness of the hysteretic force (joint moment) corresponding to one value of displacement (joint rotation) due to different past histories of deformation. Therefore, the hysteretic force depends not only on the instantaneous deformation but also the past history of deformation. The majority of existing models can only capture local history of the hysteresis. For most

Modell&g Hysteretic Behaviour of Semi-Rigid Jo&ts 333

practical hysteretic system, all the dominant extrema of the entire history leave their marks upon future states of the hysteresis. The models, which can capture nonlocal-memory hysteresis, usually give a finer fit for the experimental hysteresis curves. In this section, two hysteresis models, which will be used to represent the hysteretic behaviour of semi-rigid joints, are discussed.

Ramberg-Osgood Model

One of the hysteresis models commonly used to represent semi-rigid joints is the Ramberg-Osgood model (Jennings 1964). This model describes the hysteretic force-deflection skeleton (virgin) curve by a three-parameter polynomial as

F F 1,-1 u(F) = ~-(1 + A I- ~- ) (1)

where u and F represent the deflection and hysteretic force respectively; K, A and n are parameters controlling the curve shape. Eqn. 1 allows a smooth transition from the elastic to the plastic region and some freedom in the shape of the hysteresis. The ascending and descending branches of the hysteresis loops are described by the same basic equation as the skeleton curve but scaled by a factor of two, namely,

F-Fr F - F r in-1 u(F) = u r 3r (1 + A I ) (2) K 2K

where Ur and Fr are the deflection and force at the reference point of the curve.

Eqns. 1 and 2 are sufficient for describing the hysteresis loops under cyclic loading or repeated loading between fixed limits. However, for cases of transient loading or loading between variable limits, Eqns. 1 and 2 are incomplete because they give no indication of how the skeleton and branch curves can be linked together to give the response to other than cyclic loading. To complete the model, empirical hysteresis rules have to be introduced. Sveinsson and McNiven (1980) have given a detailed description of the empirical rules in thirteen phases by defining two types of skeletal curves and eleven types of branch curves (interior curves and bounding curves). As shown in Eqn. 2, since the current state (u, F) is only related to a specific previous state (Ur, Fr) at the reference point (reversal point) of the hysteresis curve, the Ramberg-Osgood model represents hysteresis with local memory.

Preisach Model

The Preisach model is constructed as a superposition of a continuous family of elementary rectangular loops, called relay hysteresis operators as shown in Figure 1. That is

r(t) = ~ /~(a, fl)G~[u](t)dc~lfl (3) S

where/.t(a,/5') is a weight function, called Preisaeh function, with support on a limiting triangle S of

the (a, fl)-plane with line a - fl being the hypotenuse and point (a o,/30 = - ao) being the vertex as

G~o[u]( t )

r

u(0 . , . . . _ . , , . . - -

Figure 1. Relay Hysteresis Operator

334 Y.Q. N i et al.

Figure 2. Input Sequence and Preisach Plane (Limiting Triangle) with Interface L(t)

shown in Figure 2. The triangle S in the half-plane a > fl is named Preisach plane. /.t(a, ,B) is equal to

zero outside S. G~/~ is the relay hysteresis operator with thresholds a >/3. It is a two-position relay

with only two values +1 or -1 corresponding to 'up' and 'down' positions respectively, i.e.,

+1 on [at, oo) G~p[u](t) = 1 on (-0% fl) (4)

The Preisach model can be interpreted as a spectral decomposition of a complicated hysteretic

constitutive law that has nonlocal memory, into the simplest hysteresis operators G ~ with local memory. Corresponding to an arbitrary input sequence u(t) shown in Figure 2, the triangle S can be

subdivided in two sets at any time instant t: S+(t) consisting of points (a, r ) for which the

corresponding Gaz-operators are in the 'up' position; and S-(t) consisting of points (a, r ) for which the

corresponding G,~z-operators are in the 'down' position. The interface L(t) between S+(t) and S-(t) is a

staircase line whose vertices have a and fl coordinates coinciding respectively with local maxima Mk

(k = 1, 2, ...) and minima mk (k = 1, 2, ...) of the input sequence at previous instants of time. The

nonlocal selective-memory is stored in this way. Thus, the output r(t) at any instant t can be

equivalently expressed as

r ( t ) = ~ ~ /.t ( et , f l ) d crd f l - ~ ~ /z ( et , f l ) d crd f l (5) s+(t) S-(t)

The Preisach function/.t(a, ,8) is usually determined by identification from experimental hysteresis loops. One of the advantages of the Preisach model is that it can be expressed in a real-time numerical simulation form. With the numerical formulation, the Preisach model is stated as

n(t)-I - + Z (rM . . . . --rM ..... . )+(rM.,u --rM ... . . . ) f o r fi(t)<O

r(t) = k=~ (6) n(t)-I + ~-](rM~ m~-rMk.m~_,)+(ru-ru. , ,_ ,) f o r fi(t)>_O

k=l

+r(t)

rat F ir:~ or:er

Figure 3. Determination of ra~ on First-Order Curve

Modelling Hysteretic Behaviour of Semi-Rigid Joints 335

where r- is the 'negative saturation' value of the output; ruk.m k is the value on the first-order curve

corresponding to the extrema {M k , m k } as shown in Figure 3.

The numerical implementation of the Preisach model is performed in the following steps. Firstly, the

Preisach plane (limiting triangle) is discretized into a squared mesh. A series of first-order transient

curves are entered that form the discrete sets of (a,/3, rap). The alternating series of dominant extrema

{Mk, mk} are then determined according to the time history of the input and updated at each new time

instant. Using {Mk, mk} and mesh value raft, all the terms in the parentheses in Eqn. 6 are computed by

numerical interpolation. The current output is then evaluated from Eqn. 6 with respect to monotonic

increasing and monotonic decreasing cases respectively.

TRANSIENT DYNAMIC RESPONSE OF A PORTAL FRAME

A one-storey portal frame with semi-rigid joints, as shown in Figure 4, is used to demonstrate the

dynamic response analysis results in terms of the Preisach model, and to compare the results with

those obtained by the Ramberg-Osgood model. The moment-rotation characteristics of the flexible connections are incorporated in the analysis as a spring point element. The equation of motion of the

structure can be expressed as

Figure 4. Schematic and Modelling of a Portal Frame

mii(t) + cft(t) + ku(t) = f (t) (7)

in which m, c and k are the mass, viscous damping coefficient and stiffness respectively; f(t) is an external excitation. Due to the nonlinearity of the semi-rigid connections, the stiffness k varies with

response and needs to be updated at each time step. On the assumption of linearly elastic columns and beams, the transient value of k can be expressed as

k = 12EIc 3EIc 1.0 Hc 3 ( 1 . 0 - - - ) H c 4EI~ K

- - 4 - c

H~ 1.0 + KcLb /6EI b

where Kc is the tangent stiffness of the connections obtained from the hysteresis loops.

(8)

In this study, the Ramberg-Osgood model with the parameters K = 1.26x 10 7 N-m/rad, A = 9.316x 10 9

and n = 5.5 is used to produce the 'experimental' hysteresis curves. The Preisach model is then

established by use of the 'experimental' first-order curve data. A time-domain incremental method is

presented to evaluate the transient dynamic response of the flexibly connected structure.

336 Y.Q. Ni et al.

Figures 5 and 6 illustrate the lateral displacement dynamic response of the frame and the moment- rotation hysteresis curves under the horizontal pulse excitationflt) = 50 kN for 0 < t < tcr " - 0.1S. Due to the hysteresis damping at connections, the dynamic response amplitude gradually attenuates with time. It is seen that both the predicted displacement response and the hysteresis loops in terms of the Preisach model agree well with those in terms of the Ramberg-Osgood model.

Usually, the local-memory hysteresis models cannot give a satisfactory representation of the minor hysteresis loops subjected to nonzero-mean excitation. Figures 7 and 8 show the displacement dynamic response and the corresponding hysteresis curves under a nonzero-mean cyclic loading. It is observed that, although the Ramberg-Osgood model and the Preisach model produce consistent displacement response, the steady-state minor hysteresis loops produced by the Ramberg-Osgood model have nearly zero enclosed area (the loading stiffness is almost identical to the unloading stiffness), showing obvious disagreement with actually observed minor loops. Contrarily, the Preisach model produces more reasonable minor hysteresis loops with a certain energy dissipation capability.

In order to verify the ability of the Preisach model to predict transient dynamic response, the dynamic response of the frame subjected to seismic excitation is analyzed. The ground acceleration excitation is the E1 Centro Earthquake with the peak acceleration value of 1.25g. Figure 9 shows the transient dynamic response of the structural lateral displacement in terms of the Ramberg-Osgood model and the Preisach model respectively. Figure 10 shows the corresponding moment-rotation hysteresis curves. The displacement dynamic response history obtained by the Preisach model coincides favourably with that obtained by the Ramberg-Osgood model. In particular, both the positive and negative response amplitudes predicted by the Preisach model are almost identical with the corresponding values predicted by the Ramberg-Osgood model. Also, the hysteresis loops arising from the two models match well with each other.

Figure 5. Lateral Displacement Response under Pulse Excitation ( t c r " - 0.1 S)

Figure 6. Trajectories of Hysteresis Loops under Pulse Excitation ( t c r = 0.1 S)

Modelling Hysteretic Behaviour of Semi-Rigid Joints 337

Figure 7. Nonzero-Mean Cyclic Loading and Corresponding Displacement Response

Figure 8. Trajectories of Hysteresis Loops under Nonzero-Mean Cyclic Loading

Figure 9. Lateral Displacement Response under Ground Seismic Excitation

Figure 10. Trajectories of Hysteresis Loops under Ground Seismic Excitation

338 Y.Q. Ni et al.

CONCLUDING REMARKS

This paper reports on the nonlinear dynamics of flexibly connected steel frames by using the Preisach model to represent the hysteretic behaviour of semi-rigid joints. The Preisach model possesses two salient attributes: (1) it is a phenomenological model and can describe transient hysteresis curves without needing any empirical rules; (2) it is a nonlocal-memory hysteresis model and is therefore capable of accurately depicting the minor loops and interior curves. The nonlinear dynamic response of a flexibly connected frame by using the Preisach model was analysed under various dynamic loads, and a good agreement with the results by the Ramberg-Osgood model was observed. This validated the suitability of the Preisach model for representing the hysteretic behaviour of semi-rigid joints.

ACKNOWLEDGEMENT

The funding support by The Hong Kong Polytechnic University to this research is gratefully

acknowledged.

References

Chui P.P.T. (1998). Geometric and Material Nonlinear Static and Dynamic Analysis of Steel Structures with Semi-Rigid Joints. Ph.D. Thesis, The Hong Kong Polytechnic University, Hong Kong.

Chui P.P.T. and Chan S.L. (1996). Transient Response of Moment-Resistant Steel Frames with Flexible and Hysteretic Joints. Journal of Constructional Steel Research 39, 221-243.

Deierlein G.G., Hsieh S.H. and Shen Y.J. (1990). Computer Aided Design of Steel Structures with Flexible Connections. Proc. 1990 National Steel Construction Conference, AISC, Chicago, USA.

Hughes D. and Wen J.T. (1997). Preisach Modeling of Piezoceramic and Shape Memory Alloy Hysteresis. Smart Materials and Structures 6, 287-300.

Jennings P.C. (1964). Periodic Response of a General Yielding Structure. ASCE Journal of the Engineering Mechanics Division 90, 131-163.

Lubarda V.A., Sumarac D. and Stosic S. (1993). Preisach Model and Hysteretic Behaviour of Ductile Materials. European Journal of Mechanics, A/Solids 12,445-470.

Lui E.M. and Chen W.F. (1986). Analysis and Behaviour of Flexibly-Jointed Frames. Engineering Structures 8, 107-118.

Macki, J.W., Nistri, P. and Zecca, P. (1993). Mathematical Models for Hysteresis. SlAM Reviews 35, 94-123. Mak W.H. (1995). System and Parameter Identification of Semi-Rigid Connections in Steel Structures.Ph.D.

Thesis, The Hong Kong Polytechnic University, Hong Kong. Mayergoyz I.D. (1991). Mathematical Models of Hysteresis, Springer-Verlag, New York, USA. Ni Y.Q., Ko J.M. and Wong C.W. (1999). Nonparametric Identification of Nonlinear Hysteretic Systems.ASCE

Journal of Engineering Mechanics 125, 206-215. Richard R.M. and Abbott B.J. (1975). Versatile Elastic-Plastic Stress-Strain Formula. ASCE Journal of the

Engineering Mechanics Division 101, 511-515. Song C.L., Brandon J.A. and Featherston C.A. (1999). Estimation of Local Hysteretic Properties for Pseudo-

Elastic Materials. Identification in Engineering Systems: Proceedings of the 2nd International Conference, Swansea, UK, 210-219.

Sumarac D. and Stosic S. (1996). The Preisach Model for the Cyclic Bending of Elasto-Plastic Beams. European Journal of Mechanics, A/Solids 15, 155-172.

Sveinsson B.I. and McNivwn H.D. (1980). General Applicability of a Nonlinear Model of a One Story Steel Frame. Report No. UCB/EERC-80/IO, University of California, Berkeley, California, USA.

Visintin A. (1994). Differential Models of Hysteresis, Springer-Verlag, Berlin, Germany. Wang J.Y., Ni Y.Q. and Ko J.M. (1999). Transient Dynamic Response of Preisach Hysteretic Systems. Proc.

International Workshop on Seismic Isolation, Energy Dissipation and Control of Structures, Guangzhou, China.

Cold-Formed Steel


BEHAVIOUR AND DESIGN OF COLD-FORMED CHANNEL COLUMNS

B. Young ~ and K.J.R. Rasmussen 2

i School of Civil and Structural Engineering, Nanyang Technological University, Singapore 639798 (Formerly, Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia)

2 Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia

ABSTRACT

The paper summarises recent research on cold-formed channel columns compressed between fixed ends and pinned ends. The research program formed the basis of the PhD thesis of the first author. The behaviour of channel columns was investigated experimentally and theoretically. Tests were performed over a range of lengths that involved pure local buckling, distortional buckling as well as overall flexural buckling and flexural-torsional buckling. Elastic and inelastic bifurcation analyses of locally buckled channel columns were used for the theoretical investigation. It is shown experimentally and theoretically that local buckling does not induce overall bending of fixed-ended singly symmetric columns, as it does of pin-ended singly symmetric columns.

The test strengths are compared with the design strengths predicted using the Australian/New Zealand, American and European specifications for cold-formed steel structures. It is shown that a fixed-ended channel column can be designed using an effective length of half of the column length and assuming the applied force acts at the centroid of the effective cross-section. Design recommendations for fixed- ended singly symmetric columns are proposed.

K E Y W O R D S

Bifurcation analysis, Buckling, Channel column, Cold-formed steel, Effective length, Finite strip method, Fixed-ended, Instability, Pin-ended, Steel structures, Structural design, Tests.

INTRODUCTION

The use of cold-formed steel structural members has increased rapidly in recent years. Cold-formed members can be used economically in domestic and small industrial building construction and other light structures. As compared to thicker hot-rolled members, cold-formed members provide enhanced strength to weight ratios and ease of construction. Cold-formed sections are normally thinner than hot-

341

342 B. Young and K.J.R. Rasmussen

rolled sections and have a different forming process. Consequently, the buckling and material behaviour can be quite different.

The boundary conditions for singly symmetric columns are important in the design of thin-walled columns. Local buckling of singly symmetric columns, such as channel sections, may cause overall bending of the column depending on whether the section is compressed between pinned or fixed ends. A uniformly compressed channel section undergoes a shift in the line of action of the intemal force when the section locally buckles. Rhodes and Harvey (1977) explained that the shift results from the asymmetric redistribution of longitudinal stress following the development of local buckling deformations, and leads to an eccentricity of the applied load in pin-ended channels. Hence, local buckling of pin-ended channel columns induces overall bending. Rasmussen and Hancock (1993) performed analytical studies and concluded that the phenomenon does not occur in fixed-ended channel columns, because the shift in the line of action of the internal force is balanced by a shift in the line of action of the external force. Consequently, local buckling does not induce overall bending. It follows that pin-ended and fixed-ended singly symmetric columns behave fundamentally different.

The different effects of local buckling on the behaviour of fixed-ended and pin-ended singly symmetric columns lead to inconsistencies in traditional design approaches. In the major codes of practice for cold-formed steel structures, full or partial rotational end restraint is accounted for solely by using effective lengths. Furthermore, the design strength of singly symmetric columns is reduced irrespective of the end support conditions to account for bending induced in pin-ended columns by the shift of the line of action of the internal force. This procedure is not rational for fixed-ended singly symmetric columns, which may remain straight after local buckling.

The purpose of this paper is to briefly summarise the tests, theoretical analyses and design analyses (Young 1997) performed at the University of Sydney on cold-formed steel channel columns compressed between fixed ends and pinned ends. The main emphasis of the research program was to obtain experimental evidence to demonstrate the differences in the behaviour and strength of fixed- ended and pin-ended singly symmetric columns resulting from local buckling. The research has been published recently in international journals and research reports; and reference is made to these publications for further details.

EXPERIMENTAL INVESTIGATION

The test program described in Young and Rasmussen (1998a and 1998b) provided experimental ultimate loads and failure modes for cold-formed plain and lipped channel columns compressed between fixed ends and pinned ends. All test specimens were brake-pressed from high strength zinc- coated Grade G450 steel sheets having a nominal yield stress of 450 MPa (Australian Standard, 1993). The test program comprised four different cross-section geometries, two series of plain channels and two series of lipped channels. The four test series were labelled P36, P48, L36 and L48 where "P" and "L" refer to "plain" and "lipped" channel respectively. The average values of measured cross-section dimensions of the test specimens are shown in Table 1 using the nomenclature defined in Fig. 1. The measured cross-section dimensions of each specimen are detailed in Young and Rasmussen (1998a and 1998b). The specimens were tested at various column lengths. The pin-ended specimens were tested using the same effective lengths as those for the fixed-ended specimens.

The base metal properties determined from coupon tests are also summarised in Table 1. The table

contains the measured Young's modulus (E) and the measured static 0.2% tensile proof stress (0"0.2). The coupon dimensions conformed to the Australian Standard AS1391 (1991) for the tensile testing of metals using 12.5 mm wide coupons and a gauge length of 50 mm. The stress-strain curves obtained

Behaviour and Design of Cold-Formed Channel Columns 343

from the coupon tests are detailed in Young and Rasmussen (1998a and 1998b). Residual stress measurements of the lipped channel specimens from Series L48 were obtained by Young and Rasmussen (1995b). The membrane and the flexural residual stresses were found to be less than 3% and 7% of the measured 0.2% tensile proof stress respectively. Hence, the residual stresses were deemed negligible compared with the 0.2% tensile proof stress. Local and overall geometric imperfections were measured prior to testing, and the imperfection profiles are detailed in Young and Rasmussen (1995a and 1995b).

Figure 1" Definition of symbols

TABLE 1 AVERAGE MEASURED SPECIMEN DIMENSIONS & MATERIAL PROPERTIES

P36 P48 L36

Specimen Dimensions Bl Bf Bw t t*

(mm) ( m m ) ( m m ) ( m m ) (mm) N/A 36.8 96.9 1.51 1.47 N/A 49.6 95.1 1.52 1.47 12.5 37.0 97.3 1.52 1.48

ri

(mm) 0.85 0.85 0.85

A (mm 2)

247

282 281

Material Properties E or02

(GPa) (MPa) 210 550 210 510 210 515

* Base metal thickness Note: 1 in. = 25.4 mm; 1 ksi = 6.89 MPa

A test rig shown in Fig. 2 was specifically designed and built for this test program. It consisted of a loading frame and two measurement frames. The measurement frames used spring systems and roller bearings to enable the frames to move longitudinally along the test specimen such that the deformation profiles could be obtained during testing. The pin-ended bearings were designed to allow rotations about the minor axis while restraining major axis rotations as well as twist rotations and warping. The fixed-ended bearings were designed to restrain both minor and major axis rotations as well as twist rotations and warping. Details of the test rig are given in Young and Rasmussen (1999a).

The load-deflection curves shown in Fig. 3 demonstrate that the shift in the line of action of the internal force caused by local buckling induces overall bending in a pin-ended channel but not in a fixed-ended channel. As a result of the different effects of local buckling, the strength of the fixed- ended specimen is higher than the strength of the pin-ended specimen, despite the fact that the specimens had the same effective length (ley) of 750 mm for Series L48. The effective length (ley) is assumed equal to half of the column length for the fixed-ended columns (ley= LF / 2) and equal to the column length for the pin-ended columns (ley = Lp), which includes the dimension of the pin-ended bearings. The fixed-ended column remained straight in both principal directions and no twisting of the cross-section was observed after local buckling until overall buckling occurred at ultimate. This

344 B. Young and K.J.R. Rasmussen

specimen exhibited nearly perfect bifurcation behaviour. In Fig. 3, the minor (u) and major (v) axis deflections and twist rotation (0) about the shear centre were measured at mid-length of the specimens. Further details of the different effects of local buckling on the behaviour of fixed-ended and pin-ended channels have been investigated by comparing strengths, load-shortening and load-deflection curves, as well as longitudinal profiles of buckling deformations, are given in Young and Rasmussen (1995a,b and 1999a) for all test specimens. The experimental and theoretical local buckling loads of the fixed- ended tests were also determined, as detailed in Young and Rasmussen (1998a and 1995b).

Figure 2: Failure of a fixed-ended Series L48 column at an effective length of 750mm

Figure 3: Load-Deflection curves for Series L48 at an effective length of 750mm

THEORETICAL ANALYSES

A technique for determining the overall flexural and flexural-torsional bifurcation loads of a locally buckled singly symmetric column is described in Young and Rasmussen (1997a). The overall bifurcation loads of locally buckled fixed-ended channel columns are determined using the theory presented by Rasmussen (1997). The theory is applicable to members of arbitrary cross-section shapes subjected to arbitrary types of loading. The members are assumed to be geometrically perfect in the overall sense but can include imperfections in the local mode.

The overall bifurcation analysis uses elastic and inelastic geometric non-linear finite strip local buckling analyses to determine the flexural and torsional tangent rigidities of the locally buckled section. These tangent rigidities are substituted into the flexural and flexural-torsional bifurcation equations to calculate the elastic and inelastic overall buckling loads. The elastic and inelastic tangent rigidities are obtained using the non-linear finite strip buckling analysis programs developed by Hancock (1985) and, Key and Hancock (1993) respectively.


The elastic and inelastic bifurcation loads obtained by Young and Rasmussen (1997b and 1998c) for flexural and flexural-torsional buckling are compared with tests on fixed-ended plain channel columns (Young and Rasmussen, 1998a) in Fig. 4 for Series P36. The bifurcation loads are shown as Ncr on the

vertical axis, non-dimensionalised with respect to the elastic local buckling load (Nz = A o'z), where crz is the elastic local buckling stress obtained using a finite strip buckling analysis (Hancock, 1978) and A is the full cross-sectional area. As shown algebraically in Young and Rasmussen (1997a), local buckling induces bending of a pin-ended singly symmetric column in the fundamental state but not of a fixed- ended singly symmetric column. Consequently, only fixed-ended singly symmetric columns exhibit bifurcation behaviour and only tests of fixed-ended columns are compared with bifurcation curves in Fig. 4.

Figure 4: Non-dimensionalised load (Ncr/Nz) vs column length (L) for fixed-ended P36 channel column (Wo/t = 0.02)

Figure 4 includes the flexural (F) and flexural-torsional (FT) bifurcation curves of both the distorted (locally buckled) and undistorted (non-locally buckled) cross-sections. The curves were obtained using a magnitude (Wo) of the local geometric imperfection (in the shape of the local buckling mode) of 2 % of the thickness (t). The test specimens failed in combined local (L) and flexural (F) buckling modes at short and intermediate lengths (L < 1500 mm), and in a pure flexural buckling mode at long lengths, as shown in Fig. 4. For the inelastic bifurcation analysis, the measured stress-strain curve was modelled using the Ramberg-Osgood expression (Ramberg and Osgood, 1943). In the expression, the parameter n = 8 was obtained for Series P36 using the measured stress-strain curve (Young and Rasmussen, 1998a), where the parameter n describes the shape of the stress-strain curve.

The variation of the inelastic bifurcation curves shown in Fig. 4 follows closely the test strengths, except at short lengths where the strength was govemed more by local buckling rather than overall instability. The test strengths are lower than the bifurcation curves, probably because of overall imperfections. The flexural buckling mode observed in the tests was accurately predicted by the elastic and inelastic bifurcation analysis for all column lengths. Further details of the elastic and inelastic bifurcation analyses are given in Young and Rasmussen (1997b and 1998c). It can be concluded from Fig. 4 that overall bifurcation analyses are useful for determining the critical buckling mode and the variation of the buckling load with column length.

346

DESIGN ANALYSES

B. Young and K.J.R. Rasmussen

The fixed-ended test strengths obtained by Young and Rasmussen (1998b) for Series L48 channels are compared in Fig. 5 with the unfactored design strengths predicted using the Australian/New Zealand Standard (AS/NZS 4600, 1996), European (1996) and American Iron and Steel Institute (AISI, 1996) specifications for cold-formed steel structures. Details of the calculation of the design strengths are given in Young and Rasmussen (1997c). The design strengths of the fixed-ended columns were calculated by assuming concentric loading (loading through the centroid of the effective cross-section). The ultimate loads of the tests are plotted against the effective length for minor axis flexural buckling (ley). The effective length is assumed equal to half of the column length for the fixed-ended columns (ley= LF / 2). The theoretical minor axis flexural buckling loads and flexural-torsional buckling loads of the undistorted cross-section as well as the experimental local buckling load are also shown in Fig. 5. These loads were determined in Young and Rasmussen (1995b).

Figure 5: Comparison of fixed-ended test strengths with design strengths for Series L48

The design strength predictions by the three specifications are conservative, as shown in Fig. 5. The failure modes observed in the Series L48 tests were combined local and distortional buckling modes at short lengths, combinations of these modes with the flexural-torsional buckling mode at intermediate lengths, and combined flexural and flexural-torsional buckling modes at long lengths. Flexural- torsional buckling failure modes were predicted by the three specifications for all column lengths which was in agreement with the failure modes observed in the tests, except at short lengths where neither flexural nor flexural-torsional buckling was found. The fact that the test strengths were conservatively predicted confirms the assumption that fixed-ended channel columns shall be designed by assuming loading through the effective centroid and an effective length of half of the column length. It also follows that the strength of a fixed-ended column is not reduced by the shift of the effective centroid. A comparison between the test strengths and the design strengths using the three specifications for both fixed-ended and pin-ended channel columns are given in Young and Rasmussen (1997c and 1998a,b) for all four test series.

Young and Rasmussen (1999b) presented a comparison between the experimentally measured shift of the effective centroid and the shift of the effective centroid predicted by AS/NZS 4600 and the AISI


Specification. It was concluded that effective width rules of the specifications accurately predict the direction and magnitude of the shift of the effective centroid for plain channels but not for lipped channels with slender flanges. Young and Rasmussen (1999b) proposed simple modifications to the current effective width rules that provide agreement between the measured and predicted shifts of the effective centroid for lipped channels. The modifications were shown to produce more accurate design strengths for lipped channel columns.

CONCLUSIONS

The research program of the PhD thesis of the first author has been summarised. The program was undertaken at the University of Sydney on cold-formed steel channel columns that involved tests, theoretical analyses and design analyses.

It was demonstrated experimentally and theoretically that the shift in the line of action of the internal force caused by local buckling deformations does not induce overall bending of fixed-ended singly symmetric columns as it does of pin-ended singly symmetric columns. For fixed-ended singly symmetric columns, the applied load always passes through the effective centroid of the cross-section. Hence, the effect of the shift in the line of action of the internal force due to local buckling should be ignored in the determination of the member strength of fixed-ended singly symmetric columns. It follows that for singly symmetric columns of the same effective length, the fixed-ended column strength is higher than the pin-ended column strength when the ultimate load exceeds the local buckling load.

A technique for determining overall flexural and flexural-torsional bifurcation loads of locally buckled singly symmetric columns has been presented and applied to fixed-ended channel sections. The analysis uses elastic or inelastic non-linear finite strip buckling analyses to determine the tangent rigidities of the locally buckled section. The tangent rigidities are substituted into the overall flexural and flexural-torsional bifurcation equations to produce the overall buckling loads. The comparison between elastic and inelastic bifurcation loads of plain channel section columns with tests indicated generally good agreement. It was concluded that overall bifurcation analyses are useful for determining the critical buckling mode and the variation of the buckling load with column length.

The test strengths were compared with design strengths obtained using the Australian/New Zealand (1996), European (1996) and American (1996) specifications for cold-formed steel structures. For fixed-ended columns, the design strength predictions by the three specifications were conservative for test Series L48. The design strengths were calculated assuming concentric loading through the effective centroid and an effective length of half of the column length. The overall failure modes predicted by the three specifications were in agreement with the failure modes observed in the tests at intermediate and long effective lengths but not at short effective lengths. On the basis of the comparison between test strengths and design strengths, it was recommended that fixed-ended columns failing by local and overall buckling shall be designed by assuming loading through the effective centroid (centroid of the effective cross-section) and using an effective length of half of the column length.

ACKNOWLEDGEMENTS

The first author wish to express his most sincere gratitude and appreciation to his PhD supervisor Assoc. Prof. Kim J. R. Rasmussen, for his invaluable guidance and support throughout the entire course of the candidature. The first author will always be greatly indebted to Assoc. Prof. Rasmussen for his helps.

348

REFERENCES

B. Young and K.J.R. Rasmussen

American Iron and Steel Institute (1996). Specification for the Design of Cold-Formed Steel Structural Members, AISI, Washington, DC.

Australian Standard (1991). Methods for Tensile Testing of Metals, AS 1391, Standards Association of Australia, Sydney, Australia.

Australian Standard (1993). Steel Sheet and Strip-- Hot-dipped zinc-coated or aluminium/zinc-coated, AS 1397, Standards Association of Australia, Sydney, Australia.

Australian/New Zealand Standard (1996). Cold-Formed Steel Structures, AS/NZS 4600:1996, Standards Australia, Sydney, Australia.

Eurocode 3, (1996). ENV 1993-1-3, Design of Steel Structures, Part 1.3: Supplementary Rules for Cold-Formed Thin Gauge Members and Sheeting. Draft February 1996, CEN, Brussels.

Hancock G.J. (1978). Local, Distortional and Lateral Buckling of I-Beams. Journal of the Structural Division, ASCE 104:11, 1787-1798.

Hancock G.J. (1985). Non-linear Analysis of Thin-walled I-Sections in Bending. Aspects of Analysis of Plate Structures, eds D.J. Dawe, R.W. Horsington, A.G. Kamtekar & G.H. Little, 251-268.

Key P.W. and Hancock G.J. (1993). A Finite Strip Method for the Elastic-Plastic Large Displacement Analysis of Thin-Walled and Cold-Formed Steel Sections. Thin-walled Structures 16, 3-29.

Ramberg W. and Osgood W.R. (1943). Description of Stress-Strain Curves by Three Parameters. Technical Note No. 902, National Advisory Committee for Aeronautics, Washington, D.C.

Rasmussen K.J.R. (1997). Bifurcation of Locally Buckled Members. Thin-Walled Structures 28:2, 117-154.

Rasmussen K.J.R. and Hancock G.J. (1993). The Flexural Behaviour of Fixed-ended Channel Section Columns. Thin-Walled Structures 17:1, 45-63.

Rhodes J. and Harvey J.M. (1977). Interaction Behaviour of Plain Channel Columns under Concentric or Eccentric Loading. Proceedings of the 2nd International Colloquium on the Stability of Steel Structures, ECCS, Liege, 439-444.

Young B. (1997). The Behaviour and Design of Cold-Formed Channel Columns, PhD Thesis Vol. 1 & 2, Department of Civil Engineering, University of Sydney, Australia.

Young B. and Rasmussen K.J.R. (1995a). Compression Tests of Fixed-ended and Pin-ended Cold-Formed Plain Channels. Research Report R714, School of Civil and Mining Engineering, University of Sydney, Australia.

Young B. and Rasmussen K.J.R. (1995b). Compression Tests of Fixed-ended and Pin-ended Cold-Formed Lipped Channels. Research Report R715, School of Civil and Mining Engineering, University of Sydney, Australia.

Young B. and Rasmussen K.J.R. (1997a). Bifurcation of Locally Buckled Channel Columns. Research Report R760, Department of Civil Engineering, University of Sydney, Australia.

Young B. and Rasmussen K.J.R. (1997b). Bifurcation of Singly Symmetric Columns. Thin-Walled Structures 28:2, 155-177.

Young B. and Rasmussen K.J.R. (1997c). Design of Cold-Formed Singly Symmetric Compression Members. Research Report R759, Department of Civil Engineering, University of Sydney, Australia.

Young B. and Rasmussen K.J.R. (1998a). Tests of Fixed-ended Plain Channel Columns. Journal of Structural Engineering, ASCE 124:2, 131-139.

Young B. and Rasmussen K.J.R. (1998b). Design of Lipped Channel Columns. Journal of Structural Engineering, ASCE 124:2, 140-148.

Young B. and Rasmussen K.J.R. (1998c). Inelastic Bifurcation Analysis of Locally Buckled Channel Columns. Proceedings of the 2nd International Conference on Thin-Walled Structures, Singapore, Elsevier Science, 409-416.

Young B. and Rasmussen K.J.R. (1999a). Behaviour of Cold-formed Singly Symmetric Columns. Thin-walled Structures 33:2, 83-102.

Young B. and Rasmussen K.J.R. (1999b). Shift of the Effective Centroid of Channel Columns. Journal of Structural Engineering, ASCE 125:5, 524-531.

SECTION MOMENT CAPACITY OF COLD-FORMED UNLIPPED CHANNELS

B. Young ~ and G.J. Hancock 2

1 School of Civil and Structural Engineering, Nanyang Technological University, Singapore 639798 (Formerly, Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia)


ABSTRACT

This paper describes a series of tests performed on cold-formed unlipped channels subjected to major axis bending. Traditionally, cold-formed steel members are thinner than hot-rolled steel members due to the limitations of the cold-formed technology. In the past, the typical thickness of cold-formed members is less than 3 mm. However, with the recent advancements made in the cold-forming technology, members of 12 mm and greater thickness can now be produced. Therefore, there is a possibility that thicker cold-formed members can be designed using hot-rolled steel structures standards. The objective of this test program is to determine the possibility of using the Australian Standard for hot-rolled steel structures in the design of thicker cold-formed unlipped channels subjected to major axis bending. In addition, the appropriateness of the section moment capacity design equations specified in the current Australian/New Zealand Standard and the American Specification for cold-formed steel structures for thicker cold-formed members is also investigated in this paper.

The test strengths are compared with the design strengths obtained using the Australian/New Zealand Standard and the American Specification for both the hot-rolled and cold-formed steel structures. The comparisons showed that the design strengths predicted by the hot-rolled and the cold-formed steel structures standards and specifications are conservative for thicker cold-formed channels.

KEYWORDS

Beam, Bending, Channel members, Cold-formed steel, Design strength, Experimental investigation, Hot-rolled steel, Moment capacity, Steel structures, Structural design, Test strength.

INTRODUCTION

Cold-formed steel structural members can be used very efficiently in many applications where conventional hot-rolled members proved to be uneconomic (Hancock, 1998). Hence, the use of cold-

349

350 B. Young and G.J. Hancock

formed steel structural members has increased rapidly. Up to the 1980s, the thickness of cold-formed members was l imited to 3 mm. This was due to the limitations o f the cold-forming technology in the

past. In the 1990s, cold-formed members o f 12 mm and greater thickness can be produced, and these

members are even thicker than some of the hot-rolled members. Therefore, the t h i cke r cold-formed

members may be used in place o f the thinner hot-rolled members in building construction.

The purpose of this paper is first to investigate the use of hot-rolled steel structures standards in the design of thicker cold-formed members. Therefore, a series of tests was conduced on cold-formed unlipped channels subjected to major axis bending (pure flexure in-plane bending upon the application of loads). The test results are compared with the design strengths predicted using the Australian Standard (AS 4100, 1998) for hot-rolled steel structures. The second purpose of this paper is to investigate the appropriateness of the section moment capacity design equations specified in the current cold-formed steel structures standards and specifications for thicker cold-formed members. The test strengths are compared with the design strengths predicted using the Australian/New Zealand Standard (AS/NZS 4600, 1996) and the American Iron and Steel Institute (AISI, 1996) Specification for cold-formed steel structures. Design recommendations are proposed for thicker cold-formed channel members in this paper. In addition, the paper also presents a comparison between the experimental results and the theoretical results of the cold-formed channel members. The theoretical elastic and plastic bending moments were calculated based on the measured material properties and the measured cross-section dimensions.


TABLE 1 MEASURED SPECIMEN DIMENSIONS FOR SERIES S 1

Specimen Web Flanges Thickness Radius Length d t ri L

75x40x4-a 75x40x4-b 100x50x4-a 100x50x4-b 125x65x4-a 125x65x4-b 200x75x5-a 200x75x5-b 250x90x6-a 250x90x6-b 300x90x6-a 300x90x6-b

(mm) 74.4 74.4 99.2 99.2 124.9 124.9 198.8 198.8 249.5 249.3 298.5 298.8

bi (mm) 40.3 40.2 50.3 50.4 65.5 65.5 75.9 75.9 90.1 90.0 91.2 91.2

(mm) 3.84 3.85 3.83 3.83 3.84 3.83 4.70 4.69 6.01 6.00 6.00 6.00

(mm) 3.9 3.9 4.1 4.1 3.9 3.9 4.2 4.2 7.9 7.9 8.4 8.4

(mm) 1268.0 1267.8 1269.9 1269.2 1269.2 1269.1 1272.4 1271.3 1269.2 1269.7 1269.8 1271.5

Note: 1 in. = 25.4 mm

Section Moment Capacity of Cold-Formed Unlipped Channels

TABLE 2 MEASURED SPECIMEN DIMENSIONS FOR SERIES 82

Specimen Web Flanges Thickness Radius Length d t ri L

80x40x4-a 80x40x4-b 140x50x4-a 140x50x4-b

150•215 150•215

(mm) 80.3 80.4

140.0 140.2 149.4 149.3

b: (mm) 39.7 39.6 49.9 50.1 75.6

75.5

(mm) 3.82 3.80

3.86 3.86 3.85 3.84

(mm) 4.0 4.0 4.0 4.0 4.0 4.0

(mm) 1202.0 1201.0 1251.0 1252.0 1050.0 1051.0

Note: 1 in. = 25.4 mm

351


Test Specimens

The tests were performed on unlipped channels cold-formed from structural steel coils. Two series of channels were tested, having nominal yield stresses of 450 MPa and 250 MPa for Series S1 and $2 respectively. The test specimens from the test Series S1 (called DuraGal) involve cold-forming of steel sections followed by in-line galvanising. This process considerably enhances the yield stress of the unformed material from 300 MPa to 450 MPa. The specimens were separated into two series of different nominal yield stress. The Series S 1 and $2 consisted of nine different section sizes, having the nominal overall depth of the webs (d) ranged from 75 mm to 300 mm, the nominal overall width of the flanges (by) ranged from 40 mm to 90 mm, and the nominal thicknesses (t) ranged from 4 mm to 6 mm. The length of the specimens was chosen, such that the section moment capacity could be obtained. Tables 1 and 2 show the measured specimen dimensions for the Series S1 and $2 respectively, using the nomenclature defined in Fig. 1. The specimens were labelled according to their cross-section dimensions. For example, the label "75•215 defines the specimen having nominal overall depth of the web of 75 mm, the overall flange width of 40 mm, and the thickness of 4 mm. The last letter "a" indicates that a pair of specimens ("a" and "b") was used in the test to provide symmetric loading for channel sections. The pair of specimens was cut from the same long specimen. Therefore, the cross-section dimensions and the material properties of the pair of specimens were nearly the same.

Material Properties

The material properties of all specimens were determined by tensile coupon tests. The coupons were taken from the centre of the web plate of the finished specimens belonging to the same batches as the bending tests. The coupon dimensions conformed to the Australian Standard AS 1391 (1991) for the tensile testing of metals using 12.5 mm wide coupons of gauge length 50 mm. The longitudinal coupons were also tested according to AS 1391 in a 300 kN capacity MTS displacement controlled testing machine using friction grips. A calibrated extensometer of 50 mm gauge length was used to measure the longitudinal strain. A data acquisition system was used to record the load and the gauge length extensions at regular intervals during the tests. The static load was obtained by pausing the applied straining for one minute near the 0.2% tensile proof stress and the ultimate tensile strength. This allowed the stress relaxation associated with plastic straining to take place. The material properties determined from the coupon tests are summarised in Table 3, namely the nominal and the

measured static 0.2% tensile proof stress (or02), the static tensile strength (O'u) and the elongation after

fracture (eu) based on a gauge length of 50 mm. The 0.2% proof stresses were used as the

corresponding yield stresses (fy).


T A B L E 3

NOMINAL AND MEASURED MATERIAL PROPERTIES

Test Series Specimen

dxbf•

75x40x4

Nominal

0"0.2 =f~ (MPa)

450

0"0. 2 =f~ (SPa)

450

Measured

0-u

(MPa)

525

(%) bl 20 S1 100x50x4 450 440 545 20

S1 125x65x4 450 405 510 23 S1 200x75x5 450 415 520 24 S1 250x90x6 450 445 530 21

300x90x6 450 435 535 23

80x40x4 250 280 370 35

140x50x4 250 290 380 39

S1

$2

$2

250 150x75x4 275 375 $2 37

Note: 1 ksi = 6.89 MPa

F igure 2: Schemat i c v iews o f bend ing test a r r a n g e m e n t

Section Moment Capacity of Cold-Formed Unlipped Channels 353

Figure 3" Bending test setup of specimens 200•215


Test Rig and Operation

The schematic views of the general test arrangement are shown in Figs 2a and 2b for the elevation and

sectional view respectively. Two channel specimens were used in the test to provide symmetric loading, and the specimens were bolted to the load transfer blocks at the two loading points and end supports. Hinge and roller supports were simulated by half rounds and Teflon pads. The simply supported specimens were loaded symmetrically at two points to the load transfer blocks within the span using a spreader beam. Half rounds and Teflon pads were also used at the loading points. In this testing arrangement, pure in-plane bending (no shear) of the specimens can be obtained between the two loading points without the presence of axial force. The distance between the two loading points was 480 mm for the Series S 1 and $2, and the distance from the support to the loading point was 350 mm for the Series S1. Two photographs of the test setup of specimens 200x75x5 are shown in Figs 3a

and 3b for the elevation and end view respectively.

A 2000 kN capacity DARTEC servo-controlled hydraulic testing machine was used to apply a

downwards force to the spreader beam. Displacement control was used to drive the hydraulic actuator at a constant speed of 0.8 mm/min and 0.6 mm/min for the Series S1 and $2 respectively. Three

displacement transducers were used to measure the vertical deflections and curvature of the specimens. A SPECTRA data acquisition system was used to record the load and the transducer readings at regular

intervals during the tests. The static load was recorded by pausing for one minute near the ultimate

load. This allowed the stress relaxation associated with plastic straining to take place.

TABLE 4 COMPARISON OF EXPERIMENTAL RESULTS WITH THEORETICAL RESULTS FOR SERIES S 1

Specimen dxbfxt

75x40x4 100•215 125•215 200•215 250•215 300•215

Experimental Ult. Moment per

Channel

M Exp

(kNm) 6.44 11.64 16.20

Theoretical Elastic

Me

(kNm) 5.50 9.53 14.80

Plastic

Mp

(kNm) 6.52 11.19 17.17

Comparison Elastic

M Exp

Me

Plastic

M Exp

Mp

1.17 0.99 1.22 1.04 1.09 0.94 1.06 0.90 40.48 38.05 45.10

79.90 77.96 93.10 1.02 0.86 92.89 98.77 119.47 0.94 0.78

Mean

COV Note: 1 in. = 25.4 mm; 1 kip = 4.45 kN 1.08 0.92

0.094 0.101

TABLE 5 COMPARISON OF EXPERIMENTAL RESULTS WITH THEORETICAL RESULTS FOR SERIES 82

Specimen dxbfxt

80x40x4 140x50x4


Channel

M Exp

Theoretical Elastic

Me

Plastic

Mp

Comparison Elastic

M Exp

Me

Plastic

M Exp

M r

(kNm) (kNm) (kNm) 5.51 3.72 4.43 1.48 1.24

10.11 14.50 150•215 16.11

Note: 1 in. = 25.4 mm; 1 kip = 4.45 kN

1.20 12.13 16.53

Mean

COV

1.43

1.13 0.97 14.28 1.35 1.14

0.141 0.128

Section M o m e n t Capacity o f Cold-Formed Unlipped Channels 355

Test Results

The experimental ultimate moments per channel (MExp) for bending about the major x-axis are given in

Tables 4 and 5 for the Series 1 (nominal yield stress of 450 MPa) and Series $2 (nominal yield stress

of 250 MPa) respectively. The moments were obtained using a quarter of the ultimate static applied load from the actuator multiplied by the lever arm (distance from the support to the loading point) of

the specimens. Out-of-plane bending was not observed in the tests.

C O M P A R I S O N OF E X P E R I M E N T A L R E S U L T S W I T H T H E O R E T I C A L R E S U L T S

The experimental ultimate moments per channel (MExp) obtained for the Series S 1 and $2 are compared

with the theoretical elastic (Me) and plastic (Mp) bending moments, as shown in Tables 4 and 5. The

elastic and plastic bending moments were calculated using the measured yield stress (fy), as listed in

Table 3, multiplied by the elastic (Zx) and plastic (Sx) section moduli of the full sections respectively

for bending about the major x-axis (Me =fy Zx and Mp = fy Sx). The elastic and plastic section moduli

were calculated based on the measured cross-section dimensions as detailed in Tables 1 and 2.

The theoretical elastic and plastic bending moments are generally conservative for the Series S 1 and

$2, except that the plastic bending moments are unconservative for the Series S1 having the mean

value of the experimental to theoretical bending moment (mExp/Mp) ratio of 0.92 and a coefficient of

variation (COV) of 0.101, as shown in Table 4.

TABLE 6

COMPARISON OF TEST STRENGTHS WITH DESIGN STRENGTHS FOR SERIES S 1

Specimen d x b f x t

Experimental

300x90x6

Ult. Moment per Channel

Ml,:xp

(kNm)

9~.,

12.7 16.1 20.5 19.5 18.7 18.7

75x40x4 6.44 100x50x4 11.64 125x65x4 16.20 200x75x5 40.48 250x90x6 79.90

92.89

AS 4100

Section

Non-compact Slender Slender Slender Slender Slender

Design

( M.,.x ) hot

AS/NZS 4600 & AISI

( M.,.x ) cold

Comparison AS 4100

M Exp

(M.,.x)hol

(kNm) 5.50 1.10 8.92 1.31 12.52 1.49 33.32 1.39 70.57 1.28 90.18 1.17

1.29

0.110 Note: 1 in. = 25.4 mm; 1 kip = 4.45 kN

(kNm) 5.83 8.87 10.85 29.21 62.62 79.10

Mean

COV

AS/NZS 4600 & AISI

M Exp

(M.,.x )c,,la

1.17 1.30

1.29

1.21

1.13 1.03

1.19 0.086

TABLE 7

COMPARISON OF TEST STRENGTHS WITH DESIGN STRENGTHS FOR SERIES 82

Specimen d x b f x t


Channel

M Exp

(kNm)

2~.,.

Design

80x40x4 5.51 10.0 140x50x4 14.50 12.9 150x75x4 16.11 19.6

Note: 1 in. = 25.4 mm; 1 kip = 4.45 kN

AS 4100

Section

Non-compact Non-compact

Slender

( M.,. x ) h,,~

(kNm) 4.23 10.72 10.95

AS/NZS 4600 & AISI

( M.,. x ) cold

(kNm)

Comparison AS 4100

M Exp

( M.,. x ) j,,,

3.72 1.30 10.11 1.35 12.26 1.47

Mean 1.37 COV 0.064

AS/NZS 4600 & AISI

M Exp

( M.,. x ) cord

1.48

1.43 1.31

1.41

0.062


COMPARISON OF TEST STRENGTHS WITH DESIGN STRENGTHS

The ultimate moments per channel obtained from the tests are compared with the section moment capacity (Msx) for bending about the major x-axis predicted using the AS 4100 for hot-rolled steel structures as well as using the AS/NZS 4600 and AISI Specification for cold-formed steel structures. Tables 6 and 7 show the comparison of the test strengths (mExp) with the unfactored design strengths (Msx)hot and (Msx)cota for hot-rolled and cold-formed steel structures standards respectively. The design strengths were calculated using the measured cross-section dimensions and the measured material properties. The values of the section slendemess ()~) calculated according to the AS 4100 are also given in Tables 6 and 7 for the Series S 1 and $2 respectively. The flanges of all channels were found to be the most slender element of the cross-sections.

The design strengths predicted by the hot-rolled and cold-formed steel structures standards are conservative for the Series S 1 and $2. The higher yield stress Series S 1 specimens are predicted less conservatively than the lower yield stress Series $2 specimens. The cold-formed steel structures standards are more accurate for predicting the section moment capacity for the Series S 1 having the mean value of the test strength to design strength (MExp / (Msx)cold) ratio of 1.19 and a coefficient of variation of 0.086, as shown in Table 6.

CONCLUSIONS AND DESIGN RECOMMENDATIONS

An experimental investigation of cold-formed unlipped channels subjected to major axis bending has been presented. The tests were conducted on channel members having plate thickness up to 6 mm. The test specimens have thicker plates than the traditional cold-formed thin gauge members. Two series of channels having nominal yield stresses of 450 MPa and 250 MPa were tested. The experimental results were compared with the theoretical elastic and plastic bending moments. It has been shown that the theoretical bending moments are generally conservative for all channels, except that the plastic bending moments are unconservative for channels having nominal yield stress of 450 MPa.

The test strengths were also compared with the design strengths obtained using the Australian Standard (AS 4100, 1998) for hot-rolled steel structures as well as using the Australian/New Zealand Standard (AS/NZS 4600, 1996) and the American Iron and Steel Institute (AISI, 1996) Specification for cold- formed steel structures. It is demonstrated that the design strengths predicted by the hot-rolled and the cold-formed steel structures standards and specifications are conservative for all tested channels. Therefore, it is recommended that the section moment capacity design equations specified in the AS 4100, AS/NZS 4600 and the AISI Specification can be used for cold-formed channel members having plate thickness up to 6 mm. The higher yield stress specimens are predicted less conservatively than the lower yield stress specimens.

REFERENCES



Australian Standard (1998). Steel Structures, AS 4100, Standards Association of Australia, Sydney, Australia.

Australian/New Zealand Standard (1996). Cold-Formed Steel Structures, AS/NZS 4600:1996, Standards Australia, Sydney, Australia.

Hancock, G.J., (1998). Design of Cold-Formed Steel Structures (To Australian/New Zealand Standard AS/NZS 4600:1996), 3rd Edition, Australian Institute of Steel Construction, Sydney, Australia.

WEB CRIPPLING TESTS OF HIGH STRENGTH COLD-FORMED CHANNELS

B. Young ~ and G.J. Hancock 2

School of Civil and Structural Engineering, Nanyang Technological University, Singapore 639798 (Formerly, Department of Civil Engineering, University of Sydney, Sydney, NSW 2006, Australia)


ABSTRACT

The paper presents a series of web crippling tests of high strength cold-formed unlipped channels subjected to the four ioading conditions specified in the Australian/New Zealand Standard (AS/NZS 4600, 1996) and the American Iron and Steel Institute (AISI, 1996) Specification for cold-formed steel structures. The four specified loading conditions are the End-One-Flange (EOF), Interior-One-Flange (IOF), End-Two-Flange (ETF) and Interior,Two-Flange (ITF) loading. The web slenderness values of the channel sections ranged from 15.3 to 45.

The test strengths are compared with the design strengths obtained using the AS/NZS 4600 and the AISI Specification. It is demonstrated that the design strengths predicted by the standard and the specification are generally unconservative for unlipped channels. Test strengths as low as 43% of the design strengths were obtained. For this reason, new web crippling design equations for unlipped channels are proposed in this paper. The proposed design equations are derived based on a simple plastic mechanism model, and the web crippling strength is obtained by dispersing the bearing load through the web. The proposed design equations are calibrated with the test results. It is shown that the web crippling strengths predicted by the proposed design equations are generally conservative for unlipped channels with web slenderness values of less than or equal to 45. The reliability of the current design rules and the proposed design equations used in the prediction of web crippling strength of cold-formed channels are evaluated using reliability analysis. The safety indices of the current design rules for different loading conditions are generally found to be lower than the target safety index specified in the AISI Specification, while the safety indices of the proposed design equations are higher than the target value.

KEYWORDS

Bearing capacity, Cold-formed channels, Design strength, High strength steel, Plastic mechanism model, Reliability analysis, Steel structures, Structural design, Test program, Test strength, Web crippling, Web slenderness.

357

358

INTRODUCTION

B. Young and G.J. Hancock

Web crippling is a form of localized buckling that occurs at points of transverse concentrated loading or supports. Cold-formed channels that are unstiffened against this type of loading are susceptible to structural failure caused by web crippling. The computation of the web crippling strength by means of theoretical analysis is quite a complex process as it involves a large number of variables. Hence, the current design rules found in most specifications for cold-formed steel structures are empirical in nature. The empirical design rules used in the Australia/New Zealand Standard (AS/NZS 4600, 1996) and the American Iron and Steel Institute (AISI, 1996) Specification for cold-formed steel structures were based on the experimental findings of Winter & Pian (1946), Zetlin (1955) and Hetrakul & Yu (1978) for sections with slender webs. The four loading conditions that are of prime interest are namely the End-One-Flange (EOF), Interior-One-Flange (IOF), End-Two-Flange (ETF) and Interior- Two-Flange (ITF) loading.

Although, according to Nash & Rhodes (1998), the computation of web crippling strength obtained using empirical methods is relatively rapid and safe within their range of application, this does not imply that empirical methods are without drawbacks. The equations, derived through empirical methods, are only applicable for a specific range and it may be difficult to ascertain the underlying engineering principles in parts of the complex equations. Therefore, there is a need to determine the appropriateness of the current design rules on the various types of steel members and to propose some design equations that are derived through a combination of both empirical and theoretical analyses.

In this paper, the appropriateness of the current design rules in the AS/NZS 4600 and the AISI Specification for unlipped channels subjected to web crippling is investigated. A series of tests was conducted under the four loading conditions specified in the AISI Specification. The web crippling test strengths are compared with the design strengths obtained using the AS/NZS 4600 and the AISI Specification. A set of equations to predict the web crippling strengths of unlipped channels with web slenderness (depth of the flat portion of the web to thickness ratio, h/t) values less than or equal to 45 is proposed. The proposed design equations are derived based on a simple plastic mechanism model, and these equations are calibrated with the test results. The proposed design equations are derived through a combination of theoretical and empirical analyses. Factors to account for the variation of the web slenderness of the channel sections have also been incorporated in the proposed design equations. In addition, the current design rules and the proposed design equations used in the prediction of web crippling strength are evaluated using reliability analysis. The safety indices of the current design rules and the proposed design equations are compared with the target safety index specified in the AISI Specification.

TEST PROGRAM

A series of tests was performed on cold-formed unlipped channels subjected to web crippling. The specimens were rolled from structural steel sheets having nominal yield stress of 450 MPa. The sections (called DuraGal) have in-line galvanising which increases the nominal yield stress from 300 MPa to 450 MPa when combined with roll-forming. The test specimens consisted of six different cross-section sizes, having a nominal thicknesses ranged from 4 mm to 6 mm, a nominal depth of the webs ranged from 75 mm to 300 mm, and a nominal flange widths ranged from 40 mm to 90 mm. The web slenderness (h/t): values ranged from 15.3 to 45.0, and these values were obtained using the measured cross-sectional dimensions. The specimens are considered to have stocky webs. The specimen lengths were determined according to the AS/NZS 4600 and the AISI Specification. Table 1 shows the nominal specimen dimensions, using the nomenclature defined in Fig. 1, where d is the overall depth of web, bf is the overall width of flange and t is the thickness of the channels. Young and

Web Crippling Tests of High Strength Cold-Formed Channels 359

Hancock (1998) also performed similar tests on cold-formed channels having different web slendemess and material properties. The web slenderness values ranged from 16.2 to 62.7, and had nominal yield

stress values of 250 MPa and 450 MPa.

The material properties of the test specimens were determined by tensile coupon tests. The coupons were taken from the centre of the web plate of the finished specimens. The tensile coupons were prepared and tested according to the Australian Standard AS1391 (1991) using 12.5 mm wide coupons of gauge length 50 mm. The static load was obtained by pausing the applied straining for one minute near the 0.2% tensile proof stresses and the ultimate tensile strength. This allowed the stress relaxation associated with plastic straining to take place. Table 1 summarises the material properties determined

from the coupon tests, namely the nominal and the measured static 0.2% tensile proof stress (o02), the

static tensile strength (ou) and the elongation after fracture (cu) based on a gauge length of 50 mm. The 0.2% proof stresses were used as the corresponding yield stresses.

The load or reaction forces were applied by means of bearing plates. The bearing plates were fabricated using high strength steel having a nominal yield stress of 690 MPa. All bearing plates were designed to act across the full flange widths of the channels excluding the rounded comer. The length of bearing (N) was chosen to be the full and half flange width of the channels. The channel specimens were tested using the four loading conditions according to the AISI Specification. These loading conditions are EOF, IOF, ETF and ITF as described earlier. Displacement control was used to drive the hydraulic actuator at a constant speed of 0.8 mm/min. The static load was recorded by pausing for one minute near the ultimate load. Details of the test set-up and test rig are given in Young and

Hancock (1998).

The experimental ultimate web crippling loads per web (PExp) a r e given in Tables 2 and 3. Two tests were repeated for 125x65x4 channel subjected to ITF loading condition, and the test results for the repeated tests are very close to their first test values, with a maximum difference of 1.5%. The small difference between the repeated tests demonstrated the reliability of the test results. For 75x40x4 channel (stockier web having h/t = 15.3) subjected to EOF loading condition, web crippling was not observed at ultimate load during testing, but specimens failed in overall twisting of the sections.

t_.

~ X

/ri

!-~ bs


TABLE 1 NOMINAL AND MEASURED MATERIAL PROPERTIES

Channel dx bfx t

(mm) 75x40x4 100x50x4 125x65x4 200x75x5

Nominal

0"0. 2 0"0. 2

(MPa) (MPa) 450 450 450 450

450 440 405 415

Measured O'u ~'u

(%) (MPa) 525 545 510 520

20 20 23 24

250x90x6 450 445 530 21 300x90x6 450 435 535 23

Note:lin. =25.4 mm; 1 ksi = 6.89MPa

COMPARISON OF TEST STRENGTHS WITH CURRENT DESIGN STRENGTHS

The web crippling loads per web obtained from the tests are compared with the nominal web crippling strengths predicted using the AS/NZS 4600 and the AISI Specification for cold-formed steel structures. Table 2 shows the comparison of the test strengths (PExp) with the unfactored design strengths (Pn). The current design strengths were calculated using the average measured cross-section dimensions and


the measured material properties as detailed in Table 1. A value of 203,000 MPa specified in the AISI Specification was used for the Young's modulus of elasticity (E) in calculating the design strength.

The current design strength (Pn) predicted by the standard and specification are unconservative, except that they closely predicted the web crippling strengths for the EOF loading condition in most of the cases. On average, the web crippling strength of a specimen subjected to either IOF or ETF loading condition was reached in the test at 67% and 66% of the value predicted by the specifications respectively, as shown in Table 2. For a specimen subjected to the ITF loading condition, the corresponding value is 56%. It is noteworthy that a test strength as low as 43% of the current design strength was obtained in the test for a certain specimen subjected to the ITF loading condition.

Figure 2: Mechanism model

PROPOSED DESIGN EQUATIONS

The nominal web crippling strength (P,,) of unlipped channels calculated according to the AS/NZS 4600 and AISI design rules are unconservative, as shown in Table 2, probably because they were calibrated for sections with more slender webs (h/t > 60). Hence, design equations for unlipped channels with stockier webs are proposed in this paper. It is assumed that the bearing load is applied eccentrically to the web due to the presence of the comer radii, which produces bending of the web out of its plane causing a plastic mechanism as shown in Fig. 2. A plastic mechanism model is used to establish design equations, which account for the eccentric loading of the web. This approach is similar to that used for square and rectangular hollow sections (SHS and RHS) by Zhao and Hancock (1992 and 1995) to determine the web crippling strengths for both interior and end bearing loads. The SHS and RHS tested by Zhao and Hancock (1992 and 1995) also had stockier webs than was intended for the AS/NZS 4600 and AISI web crippling equations.

The proposed equations for channel sections are summarised as:

Pp,,, = 1.44-0.0133 (1) r

where

Web Crippling Tests of High Strength Cold-Formed Channels

f yt 2 m p = 4

t r = r / + - -

2

t; N,,, = ed +

2

for Interior loading

for End loading

361

(2)

(3)

(4)

in which, Ppm is the web crippling strength predicted by using the plastic mechanism model, Mp is the plastic moment per unit length, r and r~ are the centreline and inside comer radii respectively, h is the depth of the flat portion of the web measured along the plane of the web, t is the thickness of the web, fy is the yield stress, d is the overall depth of the web and N is the length of the bearing. In Eqn. 4, Nm is the assumed mechanism length, as shown in Figs 3a and 3b for interior and end loading respectively. It is based on an assumption that the dispersion slope of the load through the corner and the web is 1:1 with correction factors i and e for interior and end loading respectively. The correction factors for interior loading are i = 1.3 and 1.4 for IOF and ITF respectively, and the correction factors for end loading are e = 1.0 and 0.6 for EOF and ETF respectively. Equation 1 also accounted for the web slenderness (h/t) of the channel sections, and the equation is calibrated with the test results.

Figure 3: Assumed plastic hinge position and mechanism length, Nm

COMPARISON OF TEST STRENGTHS WITH PROPOSED DESIGN STRENGTHS

The experimental ultimate web crippling loads per web (PExp) obtained from the tests are compared in Table 3 with the proposed design strengths (Ppm) using the plastic mechanism model. The proposed design strengths were calculated using the average measured cross-section dimensions and the measured material properties as detailed in Table 1.

TABLE 2 COMPARISON OF WEB CRIPPLING TEST STRENGTHS WITH CURRENT DESIGN STRENGTHS

tc' 7 B

a r, * %

c:

E

a 3 0 0

Note: 1 in, = 25.4 mm; 1 ksi = 6.89 MPa; 1 kip = 4.45 kN

TABLE 3 COMPARISON OF WEB CRIPPLING TEST STRENGTHS WITH PROPOSED DESIGN STRENGTHS

3 g G%

3 s % 3 ""s

% 9 % "rl 2 z 9 z

z

2 3

a

a r,

6 Note: 1 in. = 25.4 mrn; 1 ksi = 6.89 MPa; 1 kip = 4.45 kN


The proposed design strengths (Ppm) are generally conservative. The plastic mechanism model approach therefore appears to be suitable for unlipped channels with a web slenderness (h/t) value of less than or equal to 45.

TABLE 4

STATISTICAL PARAMETER FOR RELIABILITY ANALYSIS

Variables Material

(Tensile Yield Stress)

Fabrication (Mass)

Statistical Parameters

Mean Mm

COV VM

Mean F m

COV V F

Values 1.08

0.063 0.97 0.03i

RELIABILITY ANALYSIS

The safety index ([3) is a relative measure of the safety of the design. A lower target safety index of 2.5 for structural members is recommended as a lower limit for the AISI Specification. In general, if the safety index is greater than 2.5 (13 > 2.5), then the design is considered to be reliable.

The existing resistance (capacity) factor (q~) of 0.75 for web crippling strength of single unreinforced webs is given by the AS/NZS 4600 and the AISI Specification. This resistance (capacity) factor (q~ = 0.75) is used in the reliability analysis. A load combination of 1.25DL + 1.50LL is also used in the analysis, where DL is the dead load and LL is the live load. Accordingly, the safety index may be given as,

ln/MInFmPm / 0.691~

F = (5) 4V2M + V~ +CpV~ +0.212

The statistical parameters Mm, F m, V M and V F are mean values and coefficients of variation for material properties and fabrication variables respectively, and these values are obtained from BHP

Structural and Pipeline Products (1998), as shown in Table 4. The statistical parameters Pm and Vp are mean value and coefficient of variation for design equations, as shown in Tables 2 and 3 for current

design rules and proposed design equations respectively. The correction factor Cp is used to account for the influence due to a small number of tests (Pek6z and Hall 1988, and Tsai 1992), and the factor

Cp is given in Eqn. Fl . l -3 of the AISI Specification. The safety index in Eqn. (5) is detailed in Rogers

and Hancock (1996).

The safety indices (13) of the current design rules to predict the web crippling strengths for the four loading conditions are lower than the target safety index, except for the EOF loading condition as shown in Table 2. Safety indices as low as 0.48 were calculated for the ITF loading condition. However, this is not the case for the proposed design equations, the safety indices are higher than the target value for the four loading conditions as shown in Table 3. Therefore, the proposed design equations are much more reliable than the current design rules. The proposed design equations produce good limit state design when calibrated with the existing resistance (capacity) factors (~ - 0.75).

CONCLUSIONS

Web Crippling Tests of High Strength Cold-Formed Channels 365

A series of web crippling tests has been conducted to examine the appropriateness of the current design rules stipulated in the Australian/New Zealand Standard (AS/NZS 4600, 1996) and the American Iron and Steel Institute (AISI, 1996) Specification for cold-formed steel structures. Tests were performed on high strength cold-formed unlipped channels having nominal yield stress of 450 MPa, and the web slenderness values ranged from 15.3 to 45. The specimens were tested using the four loading conditions (EOF, IOF, ETF and ITF) according to the AISI Specification.

The test strengths were compared with the current design strengths obtained using AS/NZS 4600 and the AISI Specification. It is demonstrated that the current design strengths predicted by the standard and specification are unconservative for unlipped channels (single unreinforced webs), except that they closely predicted the web crippling strengths for the EOF loading condition in most of the cases. For a certain specimen subjected to ITF loading condition the test strength is only 43% of the current design strength predicted by the standard and specification. Since the design strengths obtained using the current design rules are generally unconservative for unlipped channels, therefore, a set of equations to predict the web crippling strengths have been proposed in this paper. The proposed design equations are derived based on a simple plastic mechanism model, and these equations are calibrated with the test results. It has been shown that the proposed design strengths are generally conservative for unlipped channels with web slenderness values of less than or equal to 45.

The reliability of the current design rules and the proposed design equations have been evaluated using reliability analysis. In general, the safety indices of the current design rules are lower than the target safety index of 2.5 as specified in the AISI Specification. Whereas the safety indices of the proposed design equations are higher than the target value. Therefore, it has shown that the proposed design equations are much more reliable than the current design rules for the prediction of web crippling strength of the tested channels. The proposed design equations are capable of producing reliable limit state designs when calibrated with the existing resistance (capacity) factors.

ACKNOWLEDGEMENTS

The authors are grateful to the Australian Research Council and BHP Structural and Pipeline Products for their support through an ARC Collaborative Research Grant. Test specimens were provided by BHP Steel.

REFERENCES



Australian/New Zealand Standard (1996). Standards Australia, Sydney, Australia.

Cold-Formed Steel Structures, AS/NZS 4600:1996,

BHP Structural and Pipeline Products (1998). Pipe, Tube and Structural Products - Mechanical Test Data. Somerton plant, NSW, Australia.


Hetrakul N. and Yu W.W. (1978). Structural Behavior of Beam Webs Subjected to Web Crippling and a Combination of Web Crippling and Bending. Final Report Civil Engineering Study 78-4, University of Missouri-Rolla, Mo, USA

Nash D. and Rhodes J. (1998). An Investigation of Web Crushing Behaviour in Thin-Wall Beams. Thin-Walled structures 32, 207-230.

Pektsz T.B. and Hall W.B. (1988). Probabilistic Evaluation of Test Results. Proceedings of the 9th International Specialty Conference on Cold-Formed Steel Structures, St. Louis, University of Missouri-Rolla, Mo, USA.

Rogers C.A. and Hancock G.J. (1996). Ductility of G550 Sheet Steels in Tension-Elongation Measurements and Perforated Tests. Research Report R735, Department of Civil Engineering, University of Sydney, Australia.

Tsai M. (1992). Reliability Models of Load Testing. PhD dissertation, Department of Aeronautical and Astronautical Engineering, University of Illinois at Urbana-Champaign.

Winter G. and Pian R.H.J. (1946). Crushing Strength of Thin Steel Webs. Cornell Bulletin 35, Part 1, Comell University, Ithaca, NY, USA.

Young B. and Hancock G.J. (1998). Web Crippling Behaviour of Cold-Formed Unlipped Channels. Proceedings of the 14th International Specialty Conference on Cold-Formed Steel Structures, St. Louis, University of Missouri-Rolla, Mo, USA, 127-150.

Zetlin L. (1955). Elastic Instability of Flat Plates Subjected to Partial Edge Loads. Journal of the Structural Division, ASCE 81:795, 1-24.

Zhao X.L. and Hancock G.J. (1992). Square and Rectangular Hollow Sections Subject to Combined Actions. Journal of Structural Engineering, ASCE 118:3, 648-668.

Zhao X.L. and Hancock G.J. (1995). Square and Rectangular Hollow Sections under Transverse End- Bearing Force. Journal of Structural Engineering, ASCE 121:9, 1323-1329.

LOCAL AND DISTORTIONAL BUCKLING OF PERFORATED STEEL WALL STUDS

Jyrki Kesti 1 and J. Michael Davies 2

~Laboratory of Steel Structures, Helsinki University of Technology, P.O. Box 2100, FIN-02015 HUT, Finland

2Manchester School of Engineering, University of Manchester, Manchester, M 13 9PL, UK

ABSTRACT

This paper considers the compression capacity of web-perforated steel wall studs. The web perforations decrease the local buckling strength of the web and the distortional buckling strength of the section. An analytical prediction of the compression capacity is described. Local and distortional buckling stresses are determined by replacing the perforated part of the web with plain plate of equivalent thickness. The effective area approach is used to consider local and distortional buckling. Comparison between the test results for short columns and the corresponding predictions shows that the method used gives reasonable results for web-perforated C-sections with or without web-stiffeners.

KEYWORDS

Cold-formed steel, wall stud, perforation, compression, local buckling, distortional buckling.

INTRODUCTION

Web-perforated steel wall studs are especially used in the Nordic countries as structural components in steel-framed housing. The slotted thermal stud offers a considerable improvement in thermal performance over a solid steel stud. The wall structure consists of web-perforated C-section studs with U-section tracks top and bottom and, for example, gypsum wallboards attached to the stud flanges. The sections investigated in this paper are shown in Figure 1. Both types of stud had six rows of slots with dimensions as shown in the Figure.

The perforations reduce the elastic local buckling stress of the web and also reduce the bending stiffness of the web which, in turn, results in decreased distortional buckling strength. The aim of this paper is to analyse the local and distortional buckling strength of the perforated steel stud. The local and distortional buckling modes are taken into account in design by using the effective area approach.

367

368 J. Kesti and J.M. Davies

Figure 1: Web-perforated C-section and web-stiffened C-section (Dimensions in mm)

ELASTIC BUCKLING STRESSES

Local Buckling Stress o f the Web o f a Perforated C-Section

The depth of the sections considered varied between 150 and 225 mm with a thickness between 1 and 2 mm. The overall depth of the perforations was 58 mm. Local buckling of the perforated region was studied using the elastic buckling analysis available in the NISA finite element software (1996). The analyses were carried out for both the isolated web element, which was assumed to be simply supported, and for the whole section, including the edge-stiffened flanges. The width of the flanges was 50 mm and the width of the stiffeners was 15 mm. Individual plate elements and the complete sections of 800 mm in length were modelled, including the perforations. A sufficient length was chosen so that the minimum local buckling stress could be achieved.

The elastic local buckling stress, l~rcr.perf, for simply supported perforated plate elements of different widths and thicknesses was determined using the finite element method (FEM). An analytical expression for the local buckling of a perforated plate may be achieved using a buckling factor of k = 4.0 and an equivalent thickness, tr, to~, for the whole plate. The equivalent thickness was determined in a manner similar to Salmi (1998):

/ O'cr ,perf. t r ,loc "--,tl

|1| O'cr ,entire t (1)

where O'cr,pe~ is the elastic buckling stress of the perforated plate and O'cr, entire is the elastic buckling stress of the entire plate. The elastic buckling stress of the equivalent plate with reduced thickness trtoc is thus the same as that of the perforated plate. The value for tr.toc was found to be in the range 0 .72t - 0.75t for the plates studied. Thus, for design purposes, the equivalent thickness value, tr.toc = 0.72t could be used for the whole range of sections.

Local buckling stresses for the whole of the perforated sections, including the flanges, were on average 75% higher than those of the simply supported perforated plates. This indicates that assuming the web part to be simply supported leads to quite conservative results and the contribution of the flanges to the local buckling of the web should generally be considered.

Distortional Buckling Stress

Because of the perforation of the web, the, transverse bending stiffness of the section is rather low and the section is sensitive to distortional buckling under compressive load. In the distortional mode of buckling, the edge-stiffened flange elements of the section tend to deform by rotation of the flange about the flange-web junction. The distortional buckling mode occurs at longer wavelengths than local buc.t-!ing. Numerical methods, such as the finite strip method (FSM), may be used to determine

Local and Distortional Buckling of Perforated Steel Wall Studs 369

the distortional buckling stress of the section. The Generalized Beam Theory (GBT) provides a particularly good tool with which to analyse distortional buckling in isolation and in combination with other modes. Some approximate manual methods have also been presented, namely the Eurocode 3 (1996) method, which is based on flexural buckling of the stiffener, and a more sophisticated model developed by Lau and Hancock (1987). The most recent method has been presented by Schafer and Pektiz (1999). Schafer's method was used in this study and it was modified to cover the perforated C- sections, as shown in Figure 2.

Figure 2: Notations for the perforated C-section and for the flange part alone

In the Schafer method, the closed-form prediction of the distortional buckling stress is based on the rotational restraint at the web/flange junction. The rotational stiffness may be expanded as a summation of the elastic and stress-dependent geometric stiffness terms with contributions from the flange and the web,

+k,)e-(k, (2)

where the subscript f indicates the flange and w the web. Buckling takes place when the elastic stiffness at the web/flange junction is eroded by the geometric effect, i.e.,

Q =0. (3)

Using (3) and writing the stress-dependent portion of the geometric stiffness explicitly,

ks = kcfe +kc~e--fcr,d ('kofg -at- k~c~g ) :0 . (4)

Therefore, the distortional buckling stress,f~r,a, is

k cge + k c~e f cr ,d -'~ "~

k c/g + k c,,e ' (5)

where the stiffness terms with the notations given in Figure 2 are:

1 ;4i 2 11 12 kc/e = EIx: (Xo: _hx: )2 + EIw: _Elx~ (xo: _hx: )2 + G1r (6) lyy

'[{ ! I 1 k#g (L ! A: (Xo/-h~zy( I~ ~ I ~ l + +Ix:+ (7) = _2yo(Xoi_h~ f 2 [, I~ [ Iy: ) h2x: +YoI Iyl

370

(1' k~c~g = s th~ ~,Lj 60

J. Kesti and J.M. Davies

(8)

in which L is the critical length for distortional buckling or the distance between restraints which limit rotation of the flange part of the section. The elastic stiffness term for the web is modified to take into account the more flexible perforated part of the web. Thus the perforated part of the web is replaced by a plate of equivalent thickness, tr, which has the same bending stiffness as the perforated web part. For the particular perforation type used in this study, the equivalent thickness was determined by means of the finite element method but, for simpler cases, this may be done by hand calculation. The elastic stiffness term for the web with a different thickness, tr, in the middle part may be expressed as:

1 = (9)

kc~e 6(l o2 3- b~+ 2b2w bwb~ l l2(l v2 )( b2~ w + h~ j+ EtZr- 2h~ bwb~h~ lj

The critical half-wave length for distortional buckling may now be expressed as:

11 TM

)2 l~y(x ~ hx)2 Lcr = ~r Ixe (xo - h x +I,o- - Iy

(10)

The manual calculation method for the distortional buckling of a C-section with reduced thickness in part of the web was verified by comparing its predictions with values given by GBT. C-sections with a height in the range 150 to 225 mm and a thickness in the range 1 to 2 mm were used for this comparison. The flange width was 50 mm and stiffener width was 15 mm. The reduced thickness value of 0.39t was used, corresponding to the studied perforation type. The mean ratio between the calculated and GBT value of 0.96 and the standard deviation value of 0.04 demonstrate a good performance for the proposed method.

Distortional buckling of the web-stiffened C-sections is much more complicated and is usually a combination of distortional buckling of the edge and web-stiffeners and should be thus determined by using the FSM or GBT.

DESIGN EXPRESSIONS

Current design recommendations do not include the design of perforated sections. In this study, the perforations were considered by using an equivalent thickness for the perforated part when determining the elastic local and distortional buckling stresses. In Eurocode 3, local buckling is taken into account by using effective widths for plane elements and distortional buckling is taken into account by reducing the thickness of the stiffeners. The reduction is based on the Eurocode column design curve ao with ct = 0.13.

Schafer and Pek6z (1999) proposed a new design method for considering local and distortional buckling in which local buckling and distortional buckling are seen as competitive buckling modes. Either one of them may be chosen to represent the buckling mode for each plane element. In order to properly integrate distortional buckling into the analysis, reduced post-buckling capacity in the distortional mode and the ability of the distortional mode to control the failure mechanism, even when

Local and Distortional Buckling of Perforated Steel Wall Studs 371

at a higher buckling stress than the local buckling, must be considered. Schafer and Pekoz therefore proposed a method where the critical buckling stress is defined for each plane element as:

L r "- min Lf cr,f , R d f ~r,a ] ( 11 )

The t e r m fcr, f is local buckling stress based on a buckling coefficient value of k = 4.0 and fcr, d is the distortional buckling stress. The reduction factor for the distortional buckling stress is as follows:

where

R d = 1 when ,;ta < 0.673 (12a)

1.17 - ~ + 0 . 3 when ,;ta> 0.673 (12b)

R d - Ad+l

2d = 4 f / f c r , d "

Finally local and distortional buckling are considered using an effective width approach in which the effective width of each plane element is determined using the well-known Winter reduction:

p = l

p = ( 1 - 0 . 2 2 / 2 ) / 2

2 = 4 f / f c r .

when 2<0.673 (13a)

when )1, > 0.673 (13b)

Schafer and Pek6z suggested that this reduction could be made for the entire member instead of each element if the buckling stresses were determined numerically. The above approach was examined for the strength capacity of laterally braced flexural members but, in this study, it has also been applied to compression members.

COMPARISON OF TEST RESULTS AND PREDICTED VALUES

Results from the compression testing of both section types are available. Salmi (1998) carried out some tests on perforated C- and web-stiffened C-sections. Kesti and Makel~iinen (1999) have also conducted tests on perforated web-stiffened C-sections. The dimensional notations are shown in Figure 3 and the measured dimensions, yield stresses and failure loads are given in Table 1 for web- stiffened C-sections and in Table 2 for C-sections. The quoted metal thickness value is the core thickness without the zinc layer. The TCJ- and TCS-Sections are from Salmi's test series.

Figure 3: Notations for section dimensions.

372 J. Kesti and J.M. Davies

TABLE 1 TEST DATA FOR WEB-STIFFENED C-SECTIONS

L h bl/b2 el/c2 el/e2 alia2 fl/f2 dl/d2 t Area Yield Failure A Stress load

~[rnrn] [nun] [mm] [nun] [mm] [mm] [mm] [nun] [mm] [mm 2] [N/mm 2] [kN] CC-1.2-W-1 800 173.7 49.8148.9116.2/16.4 22.8/24.8 9.3/9.2 22.4/22.4 13.1/10.7 1.15 301.3 386 64.4 CC-1.2-W-2 800 173.7 49.7/49.8 16.2/16.1 22.8/24.7 9.3/9.3 22.4/22.4 13.1/10.7 1.15 300.8 386 73.5 CC-1.5-W-1 800 174.1 49.9/50.2 16.2/16.8 23.6/22.9 8.8/8.3 23.3/22.5 10.0/10.9 1.47 377.2 380 96.2 CC-1.5-W-2 800 174.2 49.5/49.8 16.3/16.7 23.0/23.1 8.9/7.9 22.6/22.5 11.3/10.6 1.47 377.1 380 83.1

TCJ1 TCJ2 TCJ3 TCJ4 TCJ5

800 149.5 45.3/46.2 16.7/15.5 22.1/25.3 5.1/5.3 18.0/17.8 4.2/4.2 1.16 256.1 387 59.9 700 149.1 46.7/48.4 19.9/15.5 12.7/16.3 5.3/5.1 17.0/17.2 14.0/14.0 1.45 329.6 363 84.1 800 174.6 43.2/44.4 14.9/16.8 33.6/37.9 5.3/5.3 17.8/17.8 4.8/4.8 1.17 282.8 395 63.3 700 198.9 39.2/40.0 16.9/17.0 33.9/37.8 5.3/5.2 16.8/17.8 17.4/17.4 1.45 376.9 366 76.6 700 224.3 46.6/46.3 17.0/16.8 35.3/38.01 5.3/5.3 17.4/18.0 28.8/28.8 1.16 346.6 395 67.3

TCS1 TCS2 TCS3 TCS4 TCS5

TABLE 2 TEST DATA FOR C-SECTIONS

L h bl/b2 Cl/C2

[mm] [mm] [mm] [mm] 800 149.0 49.7/48.2 16.8/15.9 796 173.7 46.2/47.5 16.4/17.0 796 173.8 49.1/49.6 16.4/13.4 798 199.0 44.0/43.3 16.2/16.2 897 223.8 49.0/49.2 18.9/15.7

e t Area A Yield Failure Stress load

[mm] [mm] [mm 2] [N/mm 2] [kN] 45.5 1.16 257.1 388 52.5 57.9 1.17 284.1 392 55.3 57.9 1.95 476.6 356 108.3 70.5 1.45 378.1 366 74.5 82.9 1.16 346.6 395 57.3

In the analysis of the C-sections, the local buckling stress of the perforated section was determined by FEM in order to provide a more exact value for the effective width expressions. As mentioned above, the analytical method for the determination of the local buckling stress is quite conservative if the web is assumed to be simply supported without any contribution from the flanges. The effective width was determined by reducing the area of the whole web, t'hw. Nevertheless, the effective width can be, at most, the width of the entire web section. In the design of the web-stiffened section, the perforated web part was ignored and the plate element between the perforated part and plain part was assumed to be an unsupported element for the determination of the effective widths.

The elastic distortional buckling stresses for both the section types were determined using the Generalized Beam Theory taking into account the actual column length and the fixed-ended boundary conditions. The analysis was carded out using the computer program written by Davies and Jiang (1995). Manual calculation methods would require that the column is sufficiently long for the end boundary conditions to be insignificant. The perforated web part was replaced with an equivalent thickness of 0.39t corresponding to the same bending stiffness. The web-stiffeners of the TCJ-sections were relatively small and thus the web-stiffeners may buckle at a lower stress than the edge-stiffeners. Thus, the distortional buckling stress was determined separately for the modes which included either

buckling of the web-stiffeners or edge-stiffeners.

The predicted values were determined using both the EC3 method and Schafer's method. A comparison of the predicted and test values is given in Figure 4. The mean value for the capacity ratio Ntest/Np is 1.02 according to the EC3 method and 1.09 according to Schafer's method. The standard

deviations are 0.08 and 0.11 respectively.

Local and Distortional Buckling of Perforated Steel Wall Studs

1,50 1,40 1,30 1,20

z" 1,1o "~ 1,00 z 0,90

0,80 0,70 0,60 0,50

/ / - -~\ "it r \~ ,

"'- Schafer -4- EC3 d

r w ' - - - - - ' r

c . . )

c-,4 c ' ,4 ~ t . ~ , . - - , . _

r c _ ) ~ r c , . )

Figure 4: Comparison of test results and predicted values

373

THE DESIGN OF STUDS OF FULL LENGTH WHICH ARE RESTRAINED BY SHEETING

This study is mainly concerned with the local and distortional buckling of perforated wall studs. These buckling modes are taken into account in the design by using an effective area approach. Flexural buckling modes should also be considered in the design of studs of full length. Minor axis buckling is usually prevented by the sheeting connected to both flanges of the section. In any case, the screw connections have limited shear stiffness and the flexural buckling stress about the minor axis of the stud should be determined as the buckling stress of the compressed strut on an elastic foundation (H6glund, 1998). The web perforations decrease the flexural buckling capacity about the strong axis of the stud due to shear deformations. Allen (1969) presented a buckling load formula for sandwich structures with thick faces when the bending stiffness of the faces is significant compared to that of the whole structure. This sandwich theory was also applied to the perforated steel studs. The buckling load can be expressed as

1~ Ne/ N# Nel

Ncr ~d = Ne S,, S,, N e (14) �9 N, N #

1-~ S v Sv )

where Ne is the Euler load for the column ignoring the effect of shear deformation, Nef is the Euler buckling load of one flange part and Sv is the shear stiffness of the section. Flexural buckling may be taken into account by reducing the yield stress using the column curves given in Eurocode 3.

The elastic distortional buckling stress of the perforated stud is quite low when distortional buckling is free to develop. The sheathing screws offer considerable resistance to distortional buckling but the utilisation of this support requires that the sheeting retains its capacity and stiffness for the expected service life of the structure. The resistance of sheathing screws to distortional buckling may be taken into account by using a convenient buckling length in the rotational stiffness equations (6) - (8) or in the numerical analysis. However, the possibility of a failed screw connection in any location should be considered and a minimum buckling length of twice the screw pitch may conservatively be used in the design.

374

CONCLUSION

J. Kesti and J.M. Davies

An analytical prediction has been given for the local and distortional buckling capacity of perforated studs. The elastic local or distortional buckling strength may be determined by replacing the perforated web part with a plain plate of equivalent thickness. The effective cross-sectional area may be determined according to Eurocode 3 or by using the method suggested by Schafer. A comparison of test results for short perforated columns with the predicted values showed that the method used gives reasonable results for perforated C-sections with or without web-stiffeners. The analysis also showed that the local buckling stress of the perforated web of the C-section is conservative if it is determined assuming that the web is simply supported without any contribution from the flanges.

ACKNOWLEDGMENTS

This paper was prepared while the first author was on a one-year study leave at Manchester University. This leave was supported by The Academy of Finland. The facilities made available by the Manchester School of Engineering are gratefully acknowledged.

REFERENCES

Allen, H. (1969), Analysis and Design of Structural Sandwich Panels, Pergamon Press.

Davies, J. and Jiang, C. (1995). GBT - Computer program, public domain, University of Manchester.

Eurocode 3 (1996), CEN ENV 1993-1-3 Design of Steel Structures - Supplementary Rules for Cold Formed Thin Gauge Members and Sheeting, Brussels.

Htiglund, T. and Burstrand, H. (1998), Slotted Steel Studs to Reduce Thermal Bridges in Insulated Walls, Thin-Walled Structures, 32:1-3, 81-109.

Kesti, J. and Makelainen, P. (1999), Compression Behaviour of Perforated Steel Wall Studs, 4 th International Conference on Steel and Aluminium Structures ICSAS'99, Espoo, Finland, 123-130.

Lau, S. and Hancock, G. (1987), Distortional Buckling Formulas for Channel Columns, Journal of Structural Engineering, 113:5, 1063-1078.

NISA, Version 6.0 (1996), Users Manual, Engineering Mechanics Research Corporation (EMRC), Michigan.

Salmi, P. (1998), Uumasta termorei'itettyjen profiilien mitoituksesta, Ter~rakenteiden tutkimus ja kehitysp~iiv~it, 1998, Lappeenranta, Finland.

Schafer, B. and PekiSz, T. (1999). Local and Distortional Buckling of Cold-Formed Steel Members with Edge Stiffened Flanges, 4 th International Conference on Steel and Aluminium Structures ICSAS'99, Espoo, Finland, 89-97.

AN EXPERIMENTAL INVESTIGATION INTO COLD-FORMED CHANNEL SECTIONS IN

BENDING

V Enjily l, M H R Godley 2 and R G Beale 2

~Centre for Timber Technology and Construction, Building Research Establishment, Watford, WD2 7JR, UK

2Centre tbr Civil Engineering, Oxford Brookes Universky, Oxford, OX3 0BP, UK

ABSTRACT

The objective of this research was to investigate the post-buckling behaviour of cold-formed plain channel sections in bending.

26 cold-formed plain sections were tested with their unstiffened flanges in compression with a range of external flange width/thickness ratios (B/t) ranging from 5 to 94. The sections with B/t ratios less than 15 were able to carry the full plastic moment. Sections with higher B/t ratios developed trapezoidal yield lines with ultimate loads accurately predicted by the classical yield line theory of Murray.

24 channel sections were tested with their stiffened webs in compression with web width/thickness ratios (D/t) ranging from 18 to 186. Sections with D/t less than 60 carried the full plastic moment. Initial tests on specimens with D/t ratios between 60 and 100 failed with local web crushing. A modified loading procedure involving applying the load in the middle of the flanges was adopted which produced results with some sections with D/t ratios between 60 and 100 carrying the full plastic moment. All sections with D/t ratios in excess of 100 failed with a 'pitched roof' mechanism in the stiffened web. Murray' s theory was less able to predict collapse behaviour in this case but comparisons with the theory are given.

The experimental results are compared with BS5950 and design recommendations drawn up.

KEYWORDS

Cold-formed, steel, channels, bending, design

INTRODUCTION

Considerable research has been carried out, both experimentally and theoretically, into cold-formed channel sections in axial compression, for example, Rhodes and Harvey (1976), Stowell (1951),

375

376 V. Enjily et al.

Murray (1984) and Little (1982) but little research has been reported into bending, Rhodes (1982,1987), Enjily (1985). The objective of this research was to conduct a full experimental investigation of cold-formed plain channels in bending and to correlate the results with yield line theory.

UNSTIFFENED COMPONENTS (FLANGES) IN COMPRESSION

Experimental Technique

All specimens were tested in four point bending.

Load was applied by use of a screw jack and spreader beam so that the load shedding part of the experimental cycle could be followed. The end conditions were simply-supported. Two spans of 1000mm and 500mm were tested with a region of pure bending of 300mm and 200mm in the centre of each specimen. Measurements of central deflection were made by use of a dial gauge on the centre of the web which was in tension and by means of a cathetometer sighted on the web-flange junction. Load cells were placed under each support as shown in figure 1.

Figure 1: Experimental Layout and System of Loading

To ensure that the sections maintained shape, end-plates were welded onto the ends of each specimen. These end plates were made from 4mm thickness steel plate and extended 25mm beyond the outer fibres of the section. They also served to ensure that the end conditions were simply-supported (see figure 2).

A screw jack was used in order to apply displacement increments. The load was applied via two rollers nesting on the inside of the web. After each load increment was applied the system was allowed to stabilise. An initial set of experiments was performed on a channel with dimensions 100"50"1.6 and the stabilisation time was varied from between three and eight minutes in order to see if there was any significant creep. The resulting maximum loads only varied by 30N. Hence all experiments reported herein were tested at a stabilisation time of three minutes.

Experimental Investigation &to Cold-Formed Channel Sections in Bending 3 7 7

r i...:. --L r -

PLA___"_. Cl IO I$ - IECT I OI; l - I

i l ID PLATE

CUAI i I . [ x. tn rcT IO I t (

Figure 2" Details of end-plates

The experimental results are summarised in Table 1. Sample experimental curves are given in figure 3. Note the double 'kink' in the experimental curve of figure 3b. This occurred when one flange buckled before the second. In most cases failure occurred simultaneously in both flanges.

TABLE 1 TEST RESULTS FOR THE ULTIMATE LOADS OF CHANNEL SECTIONS WITH THEIR FLANGES IN COMPRESSION.

Experiment Section Size [Yiek [ Young's Reference (D*B*t) ! Slrer th ] Modulus

, ](N/Ill 12) ] (N/ram 2)

] M9 MI0 Mll

'MI2 _ MI3 MI4 Mi5 M16 Q~ Q2

.Q3 Q4

30* 8"i.6 ~ 232.5 45* 16"1.6 jr'232.5 " 60* 24"1.6 jr 232.5 75* 32"1.6 jr'232.5 90* 40"1.6 jr232.5

105" 48"1.6 jr232.5 120" 56"i.6 ~232.5 135" 64"1.6 4232.5 160" 80"!.6 Jr 183.(3 210"105"1.6 ~_183.(3 240"120"1.6 J183.G 270"135"1.6 /183.0

_Q5 3o0"150"1.6 T183.0 9

10 11 12

13 14 ,15 16

Q6 Q7

Q8 Q9 Q10

30* 8"1.6 4232.5 45* 16"i.6 ~232.5 60* 24"1.6 ]_232.5 75* 32"1.6 ~232.5 90* 40"1.6 [232.5

105" 48"1.6 42325 120" 56"1.6 ~232.5 135" 64"1.6 E232.5 160" 80"1.6 ~183.0 210"105"1.6 ~183.0 240"120"1.6 ~183.0 270"135"1.6 ~'83.0 300"150"1.6 UI.83.0

198700 198700 198700 198700 198700 198700 198700 198700 196000 196000 196000 196000 196000 198700 198700 198700 198700 198700 198700 198700 198700 196000 196000 196000 196000 196000

ExpeimentalIFuli-Plastic Experimental 1BS5950 l Experimental 1;pan l B/t Faih'e Load | Load (kN) Failur~ Load/] Failure Failure Load/ | imm) ratio (kN) t I Full Plastic 1 Load BS5950 [ t

i Load Failure Load -I 0.13 0.122 1.065 1. 0.122 1.065 ~ i000 4.50C 0.58 - 4 0.502 1.155 1.0.475 1222 I ~000 1. 9.50c 1.3(3 Jr I.!54 . 1.126 1.0.660 1.970 " J ~000 1. 14.50(3 1.89 Jr 2.079 0.909 1.1.153 1.639 "' j O00 1.19.50(3 2.19 Jr 3.275 0.669 ~ 1.635 1.339 .[ 000 1. 24.500 2.46 Jr 4.743 0.519 1.1.952 1.160 1. 000 ~29.500 2.71 4 6.684 0.418 1.2.132 1.271" ~ 000 1. 34.500 3.04 jr 8.497 0.358 1.2.259 1.346 J 000 ~39.500 3.80 Jr 10.495 0.362 1.2.355 1.614 1. 000 ~ 49.500 5.10 jr 18.166 0.281 1.2.689 1.897 ] 000 1. 65.125 5.40 Jr 23.773 0.227 ~ 2.908 1.857' 1. 000 1. 74.500 6.40 jr 30.133 0.212 1-3.140 2.038 .,. 1. 000 1-83"875 7.40 Jr36.,724 0.201 ].3.361 2.202 . ~ 000 1.93.250 0.32 jr 0.284 1.123 1.0.284 1.123 1. 500 [. 4.500 1.34 Jr 1.172 1.144 ~1.107 1.211 I 500 ~ 9.500( 2.90 Jr 2.693 1.077 ~1.339 1.884 ( 500 ~14.500 4.24 Jr 4.fl50 0.874 ~2.690 1.576 1. 500 ~ 19.500 5.02 Jr 7.641 0.657 1-3.816 1.316 I 500 1.24.500 5.82 Jr I1.068 0.526 [_4.556 1.278 1. 500 [_29.500 6.27 ].i5.i29 0.414 1-4.975 1.260 .... 1. 500 1.34.500 6.80 Jr 19.825 0.343 1-5.271 1.290 1. 500 1-39.500 8.70 ]-24.484 0.355 1-5.494 1.583 ~ 500 1-49.500

11.70 ]_42.388 0.276 [_6.273 1.865 ~ 500 1-65.125 13.10 Jr55.470 0.236 1-6.785 1.931 " I 500 ~74.500 11.90 ]_70.3"09 0.169 ~7.328 1.624 I 500 ~83.875 12.70 185.6"89 0.148 17.855 1.617 / 500 /9~ 250

Theoretical Analysis

To enable an understanding of the ultimate post yield behaviour of channels the simple yield line model described by Murray (1984) was used. Channels with flange/thickness ratios less than 15 were able to attain their full plastic moment capability. Longer flanges developed local buckles in the flanges which ultimately produced a series of yield lines as shown in Figure 4 In the analysis the compression elements were divided into a series of strips. Element equilibrium of each strip was used to derive values of the elemental force in each strip and their corresponding moments of resistance.


Figure 3a Figure 3b

Figure 3c Figure 3d

Figure 3: Typical experimental curves, flanges in compression

Figure 4: Side Elevation in the region of localised buckling

Experimental Investigation into Cold-Formed Channel Sections in Bending 379

These elemental values were then integrated to obtain the total moment of resistance of the compression flange. Full details of the procedure are found in Enjily (1985) and Enjily et al (1998). Figure 5 is a comparison between theory and experiment.

Figure 5: Comparison between theory and experiment (Specimen 14)

Discussion

In all cases the channels failed by forming a yield line mechanism in one, or both, of the unstiffened webs. The experimental results are summarised in Table 1. It can be seen that channel sections can carry the full plastic moment for flanges with B/t ratios less than 16.

The test results were compared with the maximum loads predicted by BS5950 Part 5(1987). The maximum loads predicted by BS5950 are also given in Table 1. The load factors against collapse calculated by BS5950 are seen to be very conservative giving results that vary from 1.065 to 2.2; the larger discrepancies occurring for the largest flange wide/thickness ratios.

Stiffened component (web) in compression

Experimental Technique

The initial arrangement of the test rig for specimens tested with the web in compression was identical to that for the case with the flanges in compression, with the exception that the cathetometer was used at mid-span. Initially, the load was applied through rollers sitting on the web. However, local crushing occurred under one or both rollers for experiments 1-8 and P6-P9 resulting in premature failure.

Owing to the thinness of the webs, and in order to overcome the bearing and tensile stresses exerted on the flanges by this loading, small steel plates were welded to the outer surface of the flanges (close to the free edge) at distances 300mm apart. Holes were drilled through the plates and flanges. Four long strip-plates were attached with plate hangers and bolted to the sections. 40mm bars were placed through the plate hangers to provide supports for a load-spreader beam. The screw jack was then placed on top of the spreader and reacted against an independently mounted beam. The force applied by the screw jack was then reacted through this improved loading arrangement into the flanges of the channel section putting it into bending. Figure 6 shows a diagram of the improved loading system.

This system was applied to specimens Y1-Y11.The results of the experiments with the webs in compression are summarised in Table 2.

Typical experimental curves are given in Figure 7. Local crushing was observed in most of the


experiments involving a span of 500mm leading to results significantly below the colTesponding curve for the 1000mm span

TABLE 2 TEST RESULTS FOR THE ULTIMATE LOADS OF CHANNEL SECTIONS WITH THE WEB IN COMPRESSION.

Experiment Section Size Yield Young's Experimental F u l l Experimental BS5950 Experimental Span D/t ratio Reference (D*B*t) Strength Modulus Failure Load Plastic Failure Load/ Fai lure Failure Load/ (mm)

(N/mm 2) (N/ram 2) (kN) Load (kN) Full Plastic Load BS5950 Load Failure Load

Y1 60* 24"1.6 210.0 199300 i.30 1.043 1.247 1.041 1.248 1000 36.500 Y2 75* 32"!.6 210.0 199300 2.20 1.877 !.172 1.830 1.202 I000 45.875 Y3 90* 40"1.6 210.0 199300 3.30 2.958 1.116 2.952 1.118 1000 55.250 Y4 105" 48"1.6 210.0 199300 4.80 4.284 i.120 4.257 1.127 1000 64.625 Y5 120" 56"1.6 210.0 199300 6.40 5.856 !.093 5.692 1.124 1000 74.000 Y6 135" 64"1.6 210.0 199300 8.20 7.674 !.068 7.209 1.138 1000 83.375 Y7 160" 80"1.6 210.0 1 9 9 3 0 0 12.10 12.043 !.005 10.470 1.158 1000 99.000 Y8 210"105"1.6 210.0 1 9 9 3 0 0 19.60 20.850 0.940 16.243 1.207 1 0 0 0 130.250 Y9 240* 120* 1.6 210.0 1 9 9 3 0 0 24.20 27.281 0.890 20.130 1.202 1 0 0 0 149.000 YIO 270"135"1.6 210.0 1 9 9 3 0 0 30.50 34.273 0.813 24.166 1.262 1 0 0 0 167.750 YI 1 300"150"1.6 210.0 1 9 9 3 0 0 33.90 41.704 0.793 28.305 1.198 1 0 0 0 186.500

1 30* 8"1.6 232.5 198700 0.26 0.285 0.913 0.285 0.913 500 17.750 2 45* 16"1.6 232.5 198700 i.39 1.172 1.181 1.169 1.189 500 27.125 3 60* 24"1.6 232.5 198700 3.05 2.693 1.132 2.689 1.134 500 36.500 4 75* 32"1.6 232.5 198700 5.48 4.850 1.130 4.842 1.132 500 45.875 5 90* 40"1.6 232.5 198700 8.41 7.642 1.101 7.620 1.104 500 55.250 6 105" 48"1.6 232.5 1 9 8 7 0 0 11.25 11.068 1.010 10.942 1.028 500 64.625 7 120" 56"1.6 232.5 1 9 8 7 0 0 11.68 15.129 0.772 14.537 0.803 500 74.000 8 135" 64"i.6 232.5 1 9 8 7 0 0 13.46 19.825 0.679 18.332 0.734 500 83.375

P6 160" 80"1.6 !83.0 1 9 6 0 0 0 18.10 24.488 0.739 21.841 0.829 500 99.000 P7 210"105"1.6 1 8 3 . 0 1 9 6 0 0 0 24.00 42.388 0.566 20.855 1.151 500 130.250 P8 240* 120 * 1.6 1 8 3 . 0 1 9 6 0 0 0 25.20 55.470 0.454 26.690 0.944 500 149.000 P9 270"135"1.6 1 8 3 . 0 1 9 6 0 0 0 27.90 70.310 0.397 33.180 0.841 500 167.750 P! 0 300* 150* 1.6 1 8 3 . 0 1 9 6 0 0 0 25.90 85.689 0.302 39.801 0.651 500 186.500

Theoretical Analysis

Discounting the results at 500mm spacing because of local crushing, sections with a web/thickness ratio less than 100 carried the full plastic moment. Sections with larger ratios failed by compression in the web forming a 'pitched roof' yield pattern. Using Murray's. theory and the mechanisms shown in figure 8 a theoretical prediction of the behaviour was made. Full details of the procedure are found in Enjily (1984). A typical comparison between theory and experiment is given in figure 9.

Discussion

From the experimental results at 1000mm it can be seen that for web/thickness ratios less than 100 that

Experimental Investigation into Cold-Formed Channel Sections in Bending 381

Figure 7: Typical experimental curves, web in compression

Figure 8" Theoretical model for beams with web in compression


Figure 9: Comparison of theory against experiment for specimen Y8

the channels were able to carry their full plastic moment. At 500mm span, a local crushing failure mode occurred before the full plastic moment was reached for web width/thickness ratios between 65 and 100. It is likely that if loading is such as to prevent local crushing that a design approach is to allow full plastic moments to be applied for ratios less than 100.

When moments of resistance are calculated by use of BS5950 (see Table 2) it can be seen that again BS5950 is conservative. However as the discrepancy does not exceed 26% the results from BS5950 are a good estimate of failure load.

For plain channels with their flanges (i.e. unstiffened elements) in compression the full plastic load can be used for flange/thickness ratios below 16. BS5950 is excessively conservative for flange/thickness ratios above 10.

For plain channels with their webs (i.e. stiffened element) in compression full plastic moment can probably be achieved for web/thickness ratios of up to 100. At ratios in excess of this figure BS5950 gives a good conservative prediction of performance.

REFERENCES

BS5950 Structural use of steelwork in building Part 5: Code of practice for design of cold formed sections BSI London 1987 Enjily V. (1985). The inelastic post-buckling behaviour of cold-formed sections, Ph. D. Thesis, Oxford Brookes University (formerly Oxford Polytechnic) Enjily V., Beale R.G. and Godley M.H.R. (1998) Inelastic Behaviour of Cold-Formed Channel Sections in Bending Proc. 2 "a Int. Co~f On Thin-walled Structures, Research & Development, Singapore, 1998, 197-204 Little G. H. (1982). Complete collapse analysis of steel columns, hTt. J. Mech. Sci. 24, 279-98 Murray N. W. (1984). Introduction to the theory of thin-walled structures, Clarendon Press, Oxford Rhodes J. and Harvey J.M. (1976) Plain channel sections in compression and bending beyond the ultimate load. Int. J. Mech. Sci. 18, 511-519 Rhodes J. (1982) The post-buckling behaviour of bending elements. Proc. Sixth Int. Speciality Conf. On Cold-Formed Steel Structures, St. Louis, 135-155 Rhodes J. (1987) Behaviour of Thin-Walled Channel Sections in Bending. Proc. Dynamics of Structures Congress '8 7, Orlando, 336-351

Composite Construction


FLEXURAL STRENGTH FOR NEGATIVE BENDING AND VERTICAL SHEAR STRENGTH

OF COMPOSITE STEEL SLAG-CONCRETE BEAMS

Qing-li Wang, Qing-liang Kang and Ping-zhou Cao

College of Civil Engineering, Hohai University, Nanjing, 210098, China

ABSTRACT

This paper is part of a summary on a series of tests and studies of 6 simply supported and 12 continuous composite steel slag-concrete beams. Using simple plastic theory and conversion of the steel member cross-section shape from " I " to rectangle, calculation formula of flexural strength of continuous composite beams for negative bending is obtained and this formula can provide accurate results no matter the cross-section neutral axis of the composite beam lies in the web or in the top flange of the steel member. Main factors affecting the strength of composite beams for vertical shear such as concrete slab, nominal shear span-ratio and force ratio, are discussed in this paper. It is necessary considering the effect of the concrete slab when calculating the strength of composite beams for vertical shear. Bending moment-ratio should be considered for continuous composite beams.

KEYWORDS

Composite steel slag-concrete beams, flexural strength for negative bending, vertical shear strength, conversion of cross-section, nominal shear span-ratio, force ratio.

INTRODUCTION

This paper is part of a summary on a series of tests and studies of 6 simply supported and 12 continuous composite steel slag-concrete beams. These tests indicate that at flexural failure around the interior prop of the continuous composite beams the concrete slab cracks and the stress in the main part of the steel member exceeds the yielding stress. The simple plastic theory is suitable to calculate the flexural strength of continuous composite beams for negative bending. Main factors affecting the strength of composite beams for vertical shear are approached. It is necessary considering the effect of concrete slab when calculating the strength of composite beams for vertical shear, and the bending moment-ratio must be considered for continuous composite beams. These are proved by the tests. Comparison of the composite steel slag-concrete beam with the composite steel common-concrete beam will be presented in another paper.

385

386 Q.-L. Wang et al.

FLEXURAL STRENGTH FOR NEGATIVE BENDING

The following method of calculating the flexural strength for negative bending, M'p, avoids the

complexities that arise in some other methods when the steel member cross-section is not geometrically symmetric about its centroidal axis as shown in Figure 1 (a). It can provide accurate results no matter the cross-section neutral axis of the composite beam lies in the web or in the top flange of the steel member. The main step of this is a conversion of the steel member cross-section shape from " I " to rectangle as shown in Figure 1 (a) which is the initial cross-section considered and (b) which is the conversed cross-section and during which following rules must be obeyed: (1) The relative position of the steel member center axis to the composite cross-section keeps

unchanged; (2) The steel member cross-section area keeps unchanged and (3) The steel member inertia moment about its center axis keeps unchanged.

Figure 1: Cross-section conversion of the steel member and stress distribution of the composite cross-section

New rectangle steel member cross-sectional dimensions are given by

ts = x/A.~ /O2Is )}

ds =~/12Is/As (1)

where d s and t s = the depth and breadth of the conversed rectangle cross-section respectively; I s =

the steel member inertia moment about its centroidal axis; A s = the steel member cross-section area.

At flexural failure, the whole of the concrete slab may be assumed to be cracked, and simple plastic theory is applicable, with all the steel at its design yield stress of frd for longitudinal reinforcement

and fsd for steel member respectively. The stresses are as shown in Figure 1 (c), and are separated

into two sets: those in Figure 1 (d) which correspond to the plastic moment of resistance of the

rectangle steel member alone, M ps, which is given by

M ps = f~a ts d2/4 (2)

and those in Figure 1 (e). The longitudinal force, F r , in Figure 1 (e) is

Flexural Strength for Negative Bending and Vertical Shear Strength

Fr =~rfr~

387

(3)

where A r = the cross-section area of longitudinal reinforcement within the effective breath of the concrete slab.

The flexural strength for negative bending is given by

M'p:Mm + Fr(d-ds /4+dt /2 -dr ) (4)

in which d , d r = the depths of the center axis of the steel member and the longitudinal reinforcement

below the top of the concrete slab respectively as shown in Figure 1 (a) and

Asf~a - Fr a, = (5)

2tsf sa

is the depth of tension zone of the rectangle steel member.

VERTICAL SHEAR STRENGTH

It is very difficult estimating the exact strength of composite beams to vertical shear theoretically for it is influenced by a lot of factors. In reinforced concrete beams, its vertical shear strength is taken into account even concrete cracks, for composite beam the strength of concrete slab for vertical shear should not be neglected too. If the cross-section area of concrete slab and force ratio are relative small and the steel member resists the main vertical shear, then it is feasible and convenient for calculation neglecting the effect of the concrete slab. Whereas a composite beam designed appropriately, the part of concrete slab should not be too small, the result would be too conservative if neglecting the effect of the concrete slab.

Main Influence Factors

In this paper vertical shear strength is derived based on test results with theoretical analysis, considering the main influence factors and the calculation model of the reinforced concrete beams. Tests by the author and others show: (1) The vertical shear strength increases as the cross-section area and the axial compressive stress of

the concrete slab increase. This is because concrete is not homogeneous material, which leads to the unusual shear stress distribution on cracked section of the concrete slab and very rough interface of crack, and there are friction and occlusive mechanism in the crack which will provide some vertical shear strength;

(2) Force ratio, ~ , could embody the contribution of the concrete slab and especially the longitudinal reinforcement inside it to the whole vertical shear strength of the composite beams, which is usually used in the negative moment region of continuous composite beams.

~=Arfry/(Asfy ) (6)

where fry = the yielding stress of the longitudinal reinforcement and fy = the yielding stress of

the steel member. The effect of the concrete slab enhances as force ratio increases mainly due to the effect of pin to concrete slab and restriction to crack of the longitudinal reinforcement;

388

(3) Nominal shear span-ratio, 2' ,

Q.-L. Wang et al.

M ,~ '= ~ ( 7 )

Vh'

in which M , V = the bending moment and vertical shear on the composite cross-section respectively, h '= the whole depth of the composite beam; There is decrease trend of the vertical shear strength of the composite beams as 2' increases when 2' < 4;

(4) The vertical shear strength increases as the transverse reinforcement ratio and the tension yielding stress of the transverse reinforcement increase and

(5) Bending moment ratio, m, must be considered for continuous composite beams.

m (8)

where M- = the negative moment of a point of inflection; M § = the positive moment of a point of inflection.

Vertical Shear o f Strength

Although there are not effective compositive actions on the prop cross-section of simply supported composite beams and the interior prop cross-section of continuous composite beam, functions of the steel member could be added to that of the concrete slab. The following formulas imitate that of the reinforced concrete beams.

For simply supported composite beams subjected to concentrated loads at midspan the vertical shear strength, V,, is provided by the steel member and the concrete slab together

V. = V c + V~ (9)

where Vs = the vertical shear strength of the steel member alone, which is given by

V s =dwtwfy /V~ (10)

where d , , t , = the depth and breadth of the web of the steel member respectively; V c = the vertical

shear strength of the concrete slab alone, which is given by

o/a ) Vc = -j-~+ b f ~ + P,~ f r~ bc hc (11)

wherebc, h c = the effective breadth and depth of the concrete slab; fc = the axial compressive

strength of concrete; p,, = the transverse reinforcement ratio; f~v = the yielding stress of the

transverse reinforcement; a and b = the coefficients decided by tests, a = 0.2 and b = 1.5.

For continuous composite beams subjected to concentrated loads at midspan moment-ratio must be considered and then

Flexural Strength for Negative Bending and Vertical Shear Strength 389

V, = Vc + V s (12) l + m

Figure 2 shows the comparisons of V, with V~st and V~est with V s. The averages of Vu/Vtest and

V,~s,/Vs are 0.935 and 1.401 for simply supported beams (1--6), 0.978 and 1.3 for continuous beams

(7--12) respectively.

Figure 2: Comparisons of V u with V, est and V~est with V s

CONCLUSIONS

(1) Simple plastic theory is suitable for calculating the flexural strength for negative bending of continuous composite steel slag-concrete beams with compact steel member cross-section. Calculation formula presented in this paper can provide accurate solutions even the neutral axis of the composite cross-section lies in the top flange of the steel member.

(2) Vertical shear strength of composite beam increases as the cross-section area and the axial compressive stress of the concrete slab and force ratio increase. There is decrease trend of the vertical shear strength of composite beams as nominal shear span-ratio increases when 2'< 4. Bending moment ratio must be considered for continuous composite beams.

(3) Results of calculation with formula about vertical shear strength of composite beams have good accordance with that of test, concrete slab can provides 28.6%V u and 23.1%Vu for simply

supported and continuous beams respectively.

REFERENCES

1. GBJ10---89, Reinforced Concrete Structure Design Code, Construction Industry Publishing Company, Beijing, China, 1989.

2. GBJ17--89, Steel Structure Design Code, Construction Industry Publishing Company, Beijing, China, 1990.

3. JBJ12m82, Light Reinforced Concrete Structure Design Rule, Construction Industry Publishing Company, Beijing, China, 1982.

4. Johnson, R. P. (1984). Composite Structures of Steel and Concrete, Volume 1: Beams, Columns, Frames and Applications in Building, Granada, London, England

5. Qing-li Wang. Practical Study and Theoretical Analysis on Mechanical Performance and Deformation Behavior of Continuous Composite Beams, Ph.D. Dissertation, Northeastern University, Shenyang, China, July 1998.


CONCRETE-FILLED STEEL TUBES AS COUPLING BEAMS FOR RC SHEAR WALLS

J.G. Teng 1, J.F. Chen 2 and Y.C. Lee 1

1 Department of Civil and Structural Engineering Hong Kong Polytechnic University, Hong Kong, China

2 Built Environment Research Unit, School of Engineering and the Built Environment, Wolverhampton University, Wulfruna Street,

Wolverhampton WV 1 1SB UK

ABSTRACT

Coupling beams in a reinforced concrete coupled shear wall structure are generally designed to provide a ductile energy dissipating mechanism during seismic attacks. This paper explores the use of concrete-filled rectangular tubes (CFRTs) as coupling beams and describes an experimental investigation into this form of construction to study their load carrying capacity, ductility and energy absorption characteristics. Results from six tests on simplified CFRT coupling beam models subject to static and cyclic loads are presented. These results demonstrate that CFRT beams have good ductility and a good energy absorption capacity. They are therefore suitable as coupling beams for shear walls particularly if the effect of local buckling is minimised by the use of steel plates of an appropriate thickness.

KEYWORDS

Coupling beams, concrete-filled steel tubes, shear walls, tall buildings, seismic design, ductility.

INTRODUCTION

Reinforced concrete (RC) coupled shear walls are commonly found in high-rise buildings. For buildings subject to seismic attacks, properly designed coupled walls offer excellent ductility through inelastic deformations in the coupling beams, which can dissipate a great amount of seismic energy. It is thus essential that the coupling beams be designed to possess sufficient ductility.

The traditional way of constructing a ductile RC coupling beam is to use a large amount of steel reinforcement, particularly diagonal reinforcement (e.g. Pauley and Binney, 1974, Park and Paulay, 1975). However, diagonal reinforcement is only effective for coupling beams with span-to-depth ratios less than two. For larger span-to-depth ratios, the inclination angle of diagonal bars becomes too small for them to contribute effectively to shear resistance (Shiu et al, 1978). However, deep coupling beams

391


are often not desirable because their depths may interfere with clear floor height. Furthermore, with the increased use of high strength concrete, it is more difficult to achieve ductility in RC beams as the section size reduces and the brittleness of the concrete increases. Therefore, the exploration of alternative coupling beam forms offering good ductility is worthwhile.

As an alternative to RC coupling beams, Harries et al. (1993) and Shahrooz et al. (1993) studied the use of steel I-beams as coupling beams. As a structural material, steel is much stronger and much more ductile than concrete. However, steel beams may suffer from inelastic lateral buckling and local buckling which limit their ductility. Although local buckling may be prevented by the proper use of lateral stiffeners (Harris et al., 1993), such stiffening is labour intensive and may lead to uneconomic designs.

More recently, steel coupling beams encased in normally reinforced concrete have been studied (Liang and Han, 1995; Wang and Sang, 1995; Gong et al., 1997). These studies show that the encasement of concrete leads to increases in stiffness and strength which should be properly considered in design and that the concrete is likely to spall during cyclic deformations.

This paper explores the use of concrete-filled rectangular tubes (CFRTs) as coupling beams and describes an experimental investigation into this form of construction to study their load carrying capacity, ductility and energy absorption characteristics. Extensive recent research has been carried out on the behaviour of concrete filled steel tubes, particularly as columns (e.g. Ge and Usami, 1992; Shams and Saadeghvaziri, 1997; Uy, 1998). In such tubes, the concrete infill prevents the inward buckling of the tube wall while the steel tube confines the concrete and constrains it from spalling. The combination of steel and concrete in such a manner makes the best use of the properties of both materials and leads to excellent ductility. To the authors' best knowledge, CFRTs have not previously been used as coupling beams, although their use in buildings and other structures, particularly as columns, has been extensive. Apart from ductility considerations, CFRT beams are much simpler to construct than RC beams because both the placement of complicated reinforcement and temporary formwork are eliminated. Compared with steel coupling beams, CFRT beams are more economic due to significant savings in steel.

." ~hear ~lall ' L .A dShe~'.wall , ".

'I L/2 ~1

w.- I

L I l l l l l l l l l l l Shear orco

Bending moment

a) Prototype structure

IIIIllllllLll

Model structure

Figure 1: Modelling of coupling beams

SPECIMEN DESIGN AND PREPARATION

Modelling of Coupling Beams

During an earthquake, the coupling beams provide an important energy dissipation mechanism in a coupled wall structure through inelastic deformations. These beams are subject to large shear forces

Concrete-Filled Steel Tubes as Coupling Beams for RC Shear Walls 393

and bending moments, with the effect of axial forces being small. The shear force and bending moment distributions in a coupling beam with the point of contraflexure at the mid-span are shown in Figure 1 a. These force distributions can be modelled by a cantilever beam under a point load at its free end (Figure l b). This cantilever beam system was thus used in the present study to simulate the behaviour of a coupling beam under seismic loading. The effect of embedment was not considered and the wall was assumed to provide a rigid support to the beam. In practical applications, a sufficient embedment length should be used to prevent premature failures in the embedment zones. An existing approach for designing the concrete embedment for steel coupling beams (Marcakis and Mitchell, 1980; Harries et al., 1993) can be used for designing the concrete embedment for CFRT coupling beams.

Design of Specimens

Eight cantilever beams were tested in this study, consisting of two control rectangular hollow section (RHS) tubes and six CFRTs. All tubes had a wall thickness of 2 mm, with a cross-sectional height of 200 mm and width of 150 mm. The variable for the CFRTs was the concrete strength, designed to cube strengths of 40, 60 or 90 MPa (referred to as Grade 40, Grade 60 and Grade 90 concrete respectively in the paper). The eight specimens were divided into two series, each consisting of one RHS tube and three CFRTs filled with concrete of different grades. The two series of specimens were tested under static loads and cyclic loads respectively.

Preparation of Specimens

The fabrication of the RHS tubes was by cold-bending and welding. Two channels were first made from steel sheets using a bending machine. Subsequently, the two channels, with their edges facing each other, were welded together to form a RHS tube with a welding seam at the mid-height of each web. Two types of steels with slightly different properties were used (Table 1). These properties were determined by tensile tests using samples from the same plates used for fabricating the RHS tubes.

TABLE 1 SPECIMEN DETATILS

Specimen Steel properties, MPa Concrete properties, MPa Yield stress Ultimate

stress Young' s modulus

Compressive Strength, 28th day

Compressive strength,

day of beam test

Splitting tensile strength, 28 th days

RHSs 290 441 194,000 N/A N/A N/A TG40s 290 441 194,000 45.3 45.2 3.03 TG60s 290 441 194,000 87.6 84.3 4.57 TG90s 290 441 194,000 92.5 94.1 4.68 RHSc 290 365 216,000 N/A N/A N/A TG40c 290 441 194,000 41.3 45.1 3.36 TG60c 290 441 194,000 87.6 90.62 4.57 TG90c 290 365 216,000 112.0 109.5 6.51

Test type

Static Static Static Static Cyclic Cyclic Cyclic Cyclic

The fabricated RHS tubes were then filled with fresh concrete. For each of the specimens, six 100x100x100 mm 3 concrete cubes and three concrete cylinders with a diameter of 100mm and a height of 200mm were cast to test their compressive and splitting tensile strengths. Measured concrete properties are shown in Table 1. The actual concrete strength for Grade 60 (Specimens TG60s and TG60c) was as high as that for Grade 90 (Specimens TG90s and TG90c) probably due to mixing problems. While this was undesirable, the specimens were still suitable for the present study and are still referred to using their intended concrete grades (ie TG60 and TG90) in this paper.


In order to prevent the concrete core from being pushed out when a CFRT specimen was loaded, two 6 mm thick steel plates were welded to the ends of each CFRT beam when the concrete age was 28 days. This simulated the antisymmetric condition at the point of contraflexure in a full coupling beam.

EXPERIMENTAL SET-UP

The experimental set-up for static loading tests is shown in Figure 2. Beam specimens were clamped between two large angle plates, which were in turn fixed on the floor by four high strength bolts. The embedment length of the beams was 440mm. Loads were applied at 460 mm from the fixed end. The span to depth ratio of the beams was 460/200=2.3 which was the smallest value possible because of restrictions of the pre-installed anchor plates on the strong floor. A hydraulic jack was fixed onto the floor to load the beam horizontally for convenience. Displacements at the loading position, the mid- span and near the fixed end were measured by electronic displacement transducers. Furthermore, a number of strain gauges were installed near the fixed end (Figure 2). Two displacement transducers were also used to measure the translation and rotation of the fixed end support. The effect of small support movements has been removed in the values of displacements presented in this paper.

For cyclic loading tests, two hydraulic jacks were used. Because of this arrangement, the displacement transducer at the loading point was moved to the tip of the beam. The positions of other transducers were the same as in the static tests. The deflection at the loading position was inferred from the measured values at the tip in an approximate manner assuming either the beam deformed elastically or rigid-plastically with a plastic hinge at the fixed support. Details are given in Lee (1998). No strain measurement was undertaken in the cyclic tests.

L I -HSS~

q

-~C) I

-~C)

'Dlsptacenent Tronsducers

'='I

45 Degree ~ S • R o s e t t

-- __~Str~In G~uge

a) Plan b) Section A-A Figure 2: Experimental set-up for static loading test

TEST PROCEDURE

Static Loading Test

In static loading tests, the specimens were monotonically loaded until failure. The strains and displacements were recorded at different load levels, from which load-deflection curves were plotted. These curves were used to determine the values of the 'yield load' Py and the corresponding deflection at the loading position dy (Lee, 1998). Based on observations during the experiments, the 'yield load' was defined as the load when local buckling of the compression flange occurred and corresponds to a strong change in slope of the load deflection curve. This yield load Py and the deflection dy were later used to control the load/displacement levels in the cyclic loading tests.


Cyclic Loading Test

The loading sequence used in the cyclic loading tests is shown in Figure 3. Load control was used before the yield load was reached. Two cycles of reversed cyclic loading were carried out at a load level of P=0.8Py. Three additional cycles were then carried out at P = Py. Thereafter, deflection control at multiples of dy was used. Three complete cycles were carried out at each selected value of deflection until the specimen failed. The loads or displacements were carefully controlled during cyclic tests, nevertheless, some small deviations from the intended values still existed. Displacements were monitored and recorded throughout the test.

Figure 3: Loading history for cyclic tests Figure 4: Load-deflection curves

under static loading

STATIC TEST RESULTS

Figure 4 shows the load-deflection curves of the loading point for all four static test specimens. The rapidly descending load-deflection curve after buckling of the RHS tube indicates that its load carrying capacity was reduced quickly, exhibiting very limited ductility. The ultimate strengths of the concrete filled tubes are almost triple that of the corresponding RHS tube. The extended plateaux in the load- deflection curves alter yielding show that CFRT beams are very ductile. These effects of the concrete infill are well known. The ductile behaviour of the CFRT beams was terminated by tensile rupture of the tension flange which occurred significantly earlier in Specimen TG90s than in the other two CFRT beams. Specimens TG40s and TG60s showed similar ductility, though they were filled with concrete of rather different strengths. The effect of the concrete strength on ductility is thus believed to be small.

Although local buckling of the steel tube was observed in all tests, the final failure modes were different for RHS and CFRT specimens (Figure 5). The local buckling of the compression flange near the fixed end occurred at a load of approximately 40 kN, leading to immediate collapse of Specimen RHSs. Shear buckling occurred on both webs at the same load. No crack was found on the tensile flange of Specimen RHSs.

For the three CFRT beams, outward local buckling was observed on the compression flanges at a load of approximately 80kN. Shear buckling occurred later on the webs at about 100kN. Clearly, the concrete infill constrained the plate to buckle only away from it, which led to a higher buckling strength, as has been shown by many authors (eg Wright, 1993; Smith et al., 1999). Strain readings showed that the tensile flange had yielded and the compression flange was close to yielding when local buckling occurred. Fracture cracks were found on the tension flanges of CFRT specimens at final failure, indicating the full use of the steel strength. The final failure of CFRT members was by rupture of steel of the tension flange and is referred to as a flexural failure.


Figure 5: Static loading test: buckling of the compression flange

Table 2 shows the experimental ultimate loads for all the static test specimens. The calculated ultimate flexural failure loads according to the approach in BS 8110 (1985) for reinforced concrete beams and using the ultimate stress of steel are also listed for comparison. Clearly, experimental observations are in good agreement with theoretical predictions for CFRT specimens, with discrepancies within 3%. These calculations did not consider local buckling effects, so the calculated ultimate flexural failure load of Specimen RHSs of 93.78kN is more than double the value actually achieved during the test (42.51kN). The chief contribution of the concrete infill is thus to provide constraint to the steel tube. The ultimate strength of CFRT beams increases with the concrete strength. However, this increase is small. Table 2 shows that the concrete strength for TG60s and TG90s is almost twice that for TG40s, but the increase in the experimental ultimate load is only less than 3% while the theoretical increase is less than 6%.

TABLE 2 STATIC ULTIMATE LOADS

Specimen fcu, MPa Test ultimate load, Predicted ultimate Test / Prediction kN load, kN

RHSs N/A 42.51 93.78 0.453 TG40s 45.2 117.14 114.2 1.026 TG60s 84.3 120.24 119.55 1.006 TG90s 94.1 119.57 120.38 0.993

CYCLIC TEST RESULTS

Test Observations and Failure Modes

Local buckling was observed on both flanges of all the cyclic test specimens. During load reversal, a buckled flange was straightened again under tension. The compression-tension cyclic stresses caused degradation in both steel and concrete, so that the maximum load reached in a cyclic test is considerably lower than that in the corresponding static test.

For Specimen RHSc, local buckling was observed in both flanges. No crack developed in the flanges, indicating that the steel tensile strength was not fully utilised. By contrast, cracks developed in both flanges of TG40c and TG60c, and in one of the flanges of TG90c at final failure. Figure 6 shows one


of the flanges for each of the three cyclic test specimens after final failure. All the CFRT specimens failed after 14 to 15 loading cycles.

Figure 6: Failure mode under cyclic loading

Hysteretic Responses

The hysteretic load-deflection responses of the loading position from all four cyclic tests are shown in Figure 7. The load carrying capacity of Specimen RHSc (Figure 7a) was quickly reduced from about 40kN in the first few cycles, to less than 20kN at the 8 th cycle and to less than 10kN at the 14 th cycle, confirming the lack of ductility as observed in the static loading test. Because the areas surrounded by the hysteresis loops represent the amount of energy absorbed by the test specimen, the energy absorption capacity of RHS tubes is thus very limited and reduces quickly under large cyclic deformations.

As observed in the static tests, the ultimate strength of CFRT beams is significantly higher than their hollow counterparts. While the differences in the load carrying capacity in the plastic range are not large between the three CFRT cyclic test specimens, it is worth noting that TG90c, which had the highest concrete strength (Table 1), showed the lowest load carrying capacity.

Compared with the results from the static loading tests, the maximum load carrying capacities of CFRT beams under cyclic loading are about 20-30% lower, with the difference between the two TG40 specimens being the smallest and that between the two TG90 specimens the largest. This indicates that a CFRT beam with a lower strength concrete behaves better than one filled with concrete of a higher strength.

The CFRT beams exhibited strength and stiffness degradations under reversed cyclic loading and pinching is seen for all of them (Figure 7). The main reason is believed to be the degradation of concrete in strength and stiffness when subject to reversed cyclic loading which leads to shear cracks in both directions. Slipping between the steel tube and the concrete may also have been a significant factor. The slipping behaviour may be improved by using shear connectors such as those used by Shakir-Khalil et al. (1993).

Overall, the hysteretic responses of these beams are good and are better than normal reinforced concrete beams, but are not as good as deep RC beams with proper diagonal reinforcement (Park and


Paulay, 1975). Significant improvements to the cyclic behaviour of these beams should be achievable by using thicker steel plates so that the effect of local buckling is minimised.

Figure 7: Hysteretic load-deflection responses at loading position

CONCLUSIONS

This paper has explored the use of concrete filled steel tubes as coupling beams for reinforced concrete coupled shear wall structures. Six concrete filled rectangular steel tubes and two rectangular hollow steel tubes have been tested under static and cyclic loadings. The mutual constraints of the steel tube and the concrete infill lead to higher strength and good ductility. The strength and ductility of these beams are insensitive to concrete strength, but cyclic degradation seems to increase with concrete strength. The use of high strength concrete thus seems to be undesirable. The hysteretic responses of these beams under cyclic loads show that they have a good energy absorption capacity. Therefore, these beams are suitable as coupling beams, particularly if local buckling is minimised by using relatively thick steel plates and slipping between the steel and concrete is reduced using some form of shear connectors. Further research is required to better understand this form of coupling beams.

Concrete-Filled Steel Tubes as Coupling Beams for RC Shear Walls

ACKNOLWEDGEMENTS

The authors are grateful to Dr. Y.L. Wong for helpful discussions on the subject.

399

R E F E R E N C E S

BS 8110 (1985). Structural Use of Concrete. British Standards Institution, London. Gong B., Shahrooz B.M. and Gillum A.J. (1997). Seismic Behaviour and Design Of Composite

Coupling Beams. Proc. of the Engineering Foundation Conference 1997, ASCE, New York, NY, USA, 258-271.

Ge, H.B. and Usami, T. (1992). Strength of Concrete-Filled Thin-Walled Steel Box Columns: Experiment. Journal of Structural Engineering, ASCE, 118:11, 3036-3051.

Harries K.A., Mitchell D., Cool W.D. and Redwood R.G. (1993). Seismic Response of Steel Beams Coupling Concrete Walls. Journal of Structural Engineering, ASCE, 119:12, 3611-3629.

Lee Y.C. (1998). Concrete Filled Steel Tubes as Coupling Beams for Concrete Shear Walls, BEng Dissertation, Dept of Civil & Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China.

Liang, Q. and Han, X. (1995). The Behaviour of Stiffening Beams and Lintel Beams under Cyclic Loading. Journal of South China University of Technology (Natural Science), 23:1, 26-33.

Marcakis, K. and Mitchell, D. (1980). Precast Concrete Connections with Embedded Steel Members", PCI Journal, 25:4, 88-116.

Park, R. and Paulay, T. (1975). Reinforced Concrete Structures, John Wiley and Sons, New York, N.Y.

Paulay T. and Binney J.R. (1974). Diagonally Reinforced Coupling Beams of Shear Walls. Shear in Reinforced Concrete: Publication No. 42, ACI, Detroit, Mich., 579-598.

Shahrooz B.M., Remmetter M.E. and Qin F. (1993). Seismic Design and Performance of Composite Coupled Walls. Journal of Structural Engineering, ASCE, 119:11,3291-3309.

Shakir-Khalill, H. and Hassan, N.K.A. (1993) Push Out Resistance of Concrete-Filled Tubes. Tubular structures VI, (ed by Grundy, Holgate & Wong), Balkema, Rotterdam.

Shams, M. and Saadeghvaziri, M.A. (1997). State of the Art of Concrete-Filled Steel Tubular Columns. A CI Structural Journal, 94:5, 558-571.

Shiu K.N., Barney G.B., Fiorato A.E. and Corley W.G. (1978) Reversed Load Tests of Reinforced Concrete Coupling Beams. Proc., Central American Conference on Earthquake Engineering, 239- 249.

Smith, S.T., Bradford, M.A. and Oehlers, D.J. (1999). Elastic Buckling of Unilaterally Constrained Rectangular Plates In Pure Shear. Engineering Structures, 21,443-453.

Uy, B. (1998). Concrete Filled Fabricated Steel Box Columns for Multistorey Buildings: Behaviour and Design. Progress in Structural Engineering and materials, 1:2, 150-158.

Wang, Z. and Sang, W. (1995). Beating Behaviour and Calculation Method of Steel Reinforced Concrete Coupling Beams. Journal of South China University of Technology (Natural Science), 23:1, 35-43.

Wright, H. (1993). Buckling of Plates in Contact with a Rigid Medium. The Structural Engineer, 71:2, 209-215.


Experimental Study of High Strength Concrete Filled Circular Steel Columns

Y. C. Wang Manchester School of Engineering, University of Manchester, Manchester M13 9PL UK

ABSTRACT

In places where usable floor space is at a premium, it is desirable to use the most structurally efficient load bearing columns. In concrete filled steel tubes, the beneficial interaction between the steel casing and the concrete core gives a load carrying system that is highly efficient. When High Strength Concrete (HSC) is used, the column load bearing performance is further improved.

This paper presents the results of a series of parametric experimental study on HSC filled circular steel columns under axial compression. The parameters examined in these tests are: concrete grade, steel grade, column slenderness and steel contribution factor.

The objectives of these tests are threefold:

1. To experimentally investigate the performance of HSC filled steel tubular columns; 2. To assess whether the design rules for normal strength concrete (NSC) filled columns

can be extrapolated to HSC filled columns, and 3. To examine the structural load bearing efficiency by changing different design

parameters.

From the results of this experimental study, the following main findings have been obtained:

1. It is conservative to extrapolate the design method for NSC filled steel columns to HSC filled ones;

2. The advantage of HSC in resisting compressive load can be effectively utilised in HSC filled columns, even for slender columns where HSC does not offer much improved rigidity to resist flexural buckling;

3. The improved column strength due to concrete confinement effect is noticeable only for short columns;

4. The confinement effect may be appreciably reduced by a small eccentricity, and 5. The ductility of HSC filled columns is similar to that of NSC filled columns.

401

402 Y. C. Wang

1. Introduction

In places where usable floor space is at a premium, it is desirable to use the most structurally efficient load beating columns. Concrete filled hollow steel columns are more structurally efficient in resisting compressive loads than either bare steel columns or reinforced concrete columns. They also have a number of other advantages including rapid construction, enhanced concrete strength and ductility due to the confinement effect and inherent high fire resistance. Normal strength concrete (NSC) filled columns are now being increasingly used in the construction of multi-storey and high rise buildings and design recommendations for this type of construction are now firmly established [1,2]. NSC is assumed to have the maximum cube strength of about 60N/mm 2.

Using high strength concrete (HSC) can further improve the structural load bearing efficiency of concrete filled columns and improve their durabilit3'. However, before HSC filled columns can be used with confidence and improved economy, their superior load carrying capacity should be confirmed and suitable design guidelines developed.

HSC filled steel tubes have been investigated by a number of researchers. For example, O'Shea and Bridge [3] concentrated on local buckling of thin walled tubes filled with HSC. Cai & Gu [4] studied the confinement effect on HSC in short columns.

This paper reports the results of a series of tests on HSC (C100) and NSC (C40) filled circular hollow section (CHS) steel columns. The objectives of these tests were threefold:

(1) To assess whether the design rules for NSC filled columns can be extrapolated to cater for HSC filled columns;

(2) To experimentally study the performance of HSC filled columns, in particular, the confinement effect on the column strength and ductility, and

(3) To examine the structural load bearing efficiency by changing different design parameters.

2. Test programme

2.1 Test parameters

This series of tests were carried out to examine the influence of a number of design parameters on column performance. In total, 2 pairs of 12 columns were made and tested. Table 1 gives the values of test parameters for each pair of columns.

2.2 Test set up

All columns were cast in December 1996 and tests were carried out about six months after casting. For each column, three concrete cubes of 100 mm and two concrete prisms of 90 mm square and 300 mm in height were cast, to be tested on the column loading day. For each concrete mix, three concrete cubes of 100 mm were cast. These cubes were tested after 28 days for quality control.

Four strain gauges were attached to the external surface of the steel tube at two opposing sides at each column mid-height. Two strain gauges at each side measure the horizontal and longitudinal strains in the steel respectively. For the shortest columns of 500 mm, a vibrating strain gauge was cast in the column centre to measure the concrete axial main.

Study of High Strength Concrete Filled Circular Steel Columns 403

Column tests were carried out in the BRE axial test machine with the maximum capacity of 5000 kN. Each column was simply supported about one axis and rotationally restrained in the perpendicular direction using roller supports. The end support increased the column length by about 80 mm at each end. Therefore, the total column length (L) should be the column specimen length (L0) plus 160 mm.

All columns were intended to be subjected to compression axial load and were checked to be so by human eyes only. Also each column had some initial imperfections and the end supports had to be adjusted for each test. Inevitably, the column could not be aligned perfectly nor in the central position. This led to a small eccentricity, and bending moment in each column. The amount of this equivalent eccentricity will be evaluated for calculating the column strength.

Each column was loaded incrementally until it reached its strength when it could not sustain the applied load. The test was continued to study the column response during unloading at increasing deformation until the column eventually found a stable position.

3. High strength concrete mechanical properties

For each column, three cube tests and two prism tests were carried out to determine various properties of concrete. During each prism test, the concrete strain was measured and the complete concrete stress-strain relationship up to the maximum stress was established. Results of the compressive strength, the corresponding strain and the Young's modulus are given in Table 2. The Young's modulus was obtained by using the proposed stress-strain relationship from Cla)~ton [5]. It is observed that the stiffness of HSC is only slightly higher (about 25%) than that of NSC, and also that concrete strain at prism strength is almost independent of the concrete grade.

4. Test observation and results

When high strength concrete fails in compression, the failure mode is brittle, this was observed during each prism test when HSC failure was accompanied by a noisy bang. In contrast, HSC filled steel columns failed in a ductile manner, similar to NSC. This was demonstrated by the ability of HSC filled columns to deform under decreasing loads and to find a stable position after reaching the peak strength.

Different failure modes were observed for different columns. For short columns (Lo/D=3), the failure mode was clearly local due to extensive concrete crushing and steel yielding. NSC filled columns exhibited very ductile behaviour, with column failure due to splitting of the cold rolled steel tube at the welding edge. HSC filled short columns also behaved in a ductile way. Confinement effect was observed by the fact that the failure strain in HSC was several times higher than the prism crush strain. Nevertheless, the extent of concrete confinement in HSC was lower than in NSC filled columns and no steel tube splitting occurred.

Global buckling was the dominant failure mode for the longest columns (Lo/D=25). Due to inevitable eccentricity induced bending effect, global buckling was not very clearly demonstrated in most columns. However, for the two columns that had very little bending moment, column failure was indicated by a sudden large lateral movement.

404 Y.C. Wang

All columns w~th the intermediate length (Lo/D= 15) failed in a mixed mode, both axial strain and lateral deflection increased at steady but faster rates until peak applied load.

Table 3 presents results for all columns, including the column eccentricity e. To verify the accuracy of the design method and to check the effectiveness of the confinement effect, it was necessary to evaluate the column eccentricity. This value is calculated from the two axial strain readings in the steel tube using the following equation:

A6.EI e - [1]

N.D where Ae = the difference in longitudinal strain recorded by the two strain gauges El =composite section flexural stiffiaess N =applied load in the column D =column cross-section diameter

Equation (1) is based on elastic analysis, therefore, the value of eccentricity was obtained from the average of the few earlier load increments.

In table 3, all design strengths were calculated taking into account the eccentricity and by setting the partial safety factors for steel and concrete to 1.0. Also, the short term concrete modulus of elasticity in Table 3 was used for each column.

5. Analysis of test results

The test results have been analysed by a comparison against the predictions of various design methods for concrete filled columns. From this comparison, a number of conclusions may be drawn. This paper presents some of the more important ones.

5.1 Accuracy of current design rules for HSC filled columns

The current design rules for concrete filled columns have been derived from test results on NSC. From the comparative results in Table 3, it may be concluded that these design rules give quite accurate predictions for NSC filled columns (TI&T2, T5&T6, T9&TI0). Furthermore, it seems that these design rules may be extended to HSC filled steel columns, as indicated by the overall accuracy in Table 3. Indeed, the current design rules give conservative results for HSC filled columns, thus they are acceptable for safety.

Nevertheless, for HSC filled steel cohmms, Table 3 suggests that the accuracy of the NSC-based design rules depends on the column slenderness. While the code predictions are quite accurate for short columns, discrepancy between predicted and test results increases at higher column slenderness. Figure 1 presents the results in Table 3. It is clear that as the slenderness of HSC filled steel columns increase, both BS 5400 Part 5 [2] and Eurocode 4 Part 1.1 [2] predict lower column strength. Whilst this means that both design methods are safe to use for HSC filled steel columns, it also suggests that it is possible to use a higher cohann buckling curve for HSC filled steel columns for improved column efficiency. However, this can only be confirmed atter more extensive experimental studies.

Study of High Strength Concrete Filled Circular Steel Columns

5.2 Effect of confinement on concrete

405

Strength It is now well reax~gnised that when concrete is under tri-axial compression, both its load carrying capacity and ductility are increased. Concrete confinement can be obtained through placing hoop reinforcement or using steel casing. For concrete filled columns, although increase in the concrete strength is at the expense of a reduction in the steel strength, the overall effect is a net increase in the column strength.

This confinement effect diminishes for slender columns. Although BS 5400 Part 5 [1] gives a limiting length of L/D=25, realistically, the confinement effect is noticeable only for columns of

m

L/D not greater than 5. In Eur~xxte 4 Part 1.1 [2], the limiting column slenderness is at 2 =0.5. Nevertheless, in places where the column foot print is large, a L/D ratio of less than 5 is realistic. Thus, it is beneficial to explore the enhancement due to concrete confinement.

However, the effect of confinement is greatly reduced by bending in the column. To illustrate the effect of concrete confinement, only Eurocxxle 4 Part .1.1 [2] is used in this paper. Results are given in Table 4 for L0/D=3. Without bending moment, the squash load of a column can be increased by up to 20% due to enhancement. However, with an eccentricity to diameter (e/D) ratio of only 3%, column strength increase due to the confinement effect is reduced by about 30%. For columns in simple construction, BS 5950 Part 1 [6] gives a nominal eccentricit)" of 100mm plus D/2 for beam reactions. For medium rise buildings, this end bending moment can give a significant eccentricity to the overall column axial load, which may completely remove the enhancement due to confinement. For example, for a 10 storey building with 300 mm diameter columns, the column eccentricity (e/D) to the overall axial load of the bottom floor column is about 8%. Therefore, to make use of the enhancement in design, an accurate assessment of the column ~ t r i c i t y should be carried out.

Duetili~ ~ One of the main concerns with using HSC is its lack of ductility and its brittle and explosive failure. However, in the author's tests, no HSC filled steel column suffered from this failure mode and all columns performed in a ductile manner.

The ductilit3, of a column is rather difficult to quantify. The unloading slope of the column may give some indication. Figure 2 plots the load-axial strain relationship for tests T5-T8, two of which used HSC and the other two NSC. In this figure, the applied load is norrnalised with regard to the column test strength. The unloading slope seems to be comparable between NSC and HSC filled columns. However, while the two NSC filled column curves are almost identical, there is a great variability in the behaviour of the two HSC filled columns.

On the other hand, if the column ductility is measured by the maximum concrete strain reached at the peak column strength, the enhanced strain due to the conflnernent effect may be predicted using the equation obtained by Mander et al [7]:

8cc I Cr cc |1 - 1 + 5 - [ 2 ]

s \ 0"~

406 Y. C. Wang

Table 5 gives a comparison between test results and predictions using equation (4). Since the confinement effect is negligible for slender columns, the comparison was carried out for short columns (I.o/D=3) only. In addition, the theoretical value of the concrete strength enhancement factor (t~/(rck) has been calculated using recommendations in Eurocode 4 Part 1.1 [2].

Table 5 only indicates a broad agreement between the predictions of equation (4) and test results. Nevertheless, it suggests that the confinement effect can significantly increase the concrete ductility and that equation (4) gives conservative results.

Table 5:

Test ID

T9

T10

TII

T12

T17

T18

T27

T28

Increased concrete strain due to confmement effect

t~cc/cck according to EC4 test

1.774 (1.544) 15.8

1.796 (1.56) 13.2

1.230 (1.162) 2.03

1.227 (1.213) 2.36

1.60(1.458) 4.0

1.571 (1.469) 6.12

1.442 (1.18) 5.07

1.412 (1.288) 4.5

2

0.17

0.17

0.21

0.21

0.20

0.21

0.18

0.19

NB: Values in brackets include the effect of eccentricity.

model [7]

4.87 (3.72)

4.98 (3.8)

2.15(1.81)

2.14 (2.07)

4.0 (3.29)

3.86 (3.35)

3.21 (1.9)

3.06(2.44)

5.3 Effect of high strength steel

One of the original objectives of this series of tests was to examine the effectiveness of using high strength materials, including both high strength steel and high strength concrete. The effect of using HSC has already been discussed in 5.1. It seems that despite only a modest increase in HSC modulus of elasticity., column test strength increases in line with increase in the column squash load regardless of the column slenderness.

However, unless the column is short, using high strength steel only gives a small increase in the column strength. Tests T13-TI8 are directly comparable to Test T23-T28, the only difference being that $355 steel was used in the former and S275 steel was used in the latter. Table 6 gives increases in the column strength due to high strength steel. Clearly, the benefit of using high strength steel diminishes at higher column slenderness.

Table 6: Comparison between results for different grades of steel L/D=25 L/D=15 L/D=5 1.075 1.386 1.368

6. Conclusion

This paper has presented the results of a series of compression tests on NSC and HSC filled circular steel columns. From an analysis of the test results, the following conclusions may be drawn:


(1) Using HSC can significantly increase the strength of concrete filled columns. This conclusion

applies to a wide range of tested column slenderness (2, = 0.2-1.4 ). (2) Since the modulus of elasticit3' of HSC is only slightly higher than that of NSC, it follows that

a higher column buckling curve may be used in design calculations for HSC filled steel columns. However, a large number of tests should be carried out for confirmation. In the meantime, the design rules for NSC filled steel columns may conservatively be used for HSC filled columns.

(3) Using high strength steel is far less effective tlwu using HSC in increasing the column strength. (4) The benefits of concrete confinement in increasing the concrete strength and ductility are

realised for short columns only. Furthermore, the increase in concrete strength can be reduced by a small ~ t r i c i t y . Therefore, in order to reliably use the beneficial effect of confining concrete, accurate assessment of the column eccentricity should be made in design calculations.

Acknowledgments

The tests reported in this paper were carried out by the author at the Building Research Establishment and he acknowledges the technical support of various BRE staff members. He also thanks Mr. Nigel Clayton of BRE for the concrete prism tests.

References

1. Design of composite bridges: use of BS 5400: Part 5:1979 for Department of Transport structures, Department of Transport, London, December 1987

2. Eurocode 4: Design of composite steel and concrete structure, Part 1.1: General rules and rules for buildings, British Standards Institution, London, 1994

3. O'Shea M D and Bridge R Q, "Circular thin walled concrete filled steel tubes", Proceedings of the 4 th Pacific Structural Steel Conference, Vol. 3: Steel-concrete composite structures, pp. 53- 60, 1995

4. Cai, S H and Gu W P, "Behaviour and ultimate strength of steel tube confined high strength concrete columns", Proceedings of 4 th International S3~posium on Utilization of high strength/high performance concrete, pp. 827-833, Paris 1996

5. BS 5950: Structural use of steelwork in buildings, Part 1: Code of practice for design in simple and continuous construction: hot rolled sections, British Standards Institution, London, 1990

6. Clayton N, "High grade concrete - stress-strain behaviour", BRE Client Report CR44/97, Building Research Establishment, 1997

7. Mander J B, Priestley M J N and Park R, "Theoretical stress-strain model for confined concrete", Journal of Structural Engineering, Vol. 114, No. 8, pp. 1804-1826, American Society of Civil Engineering, 1988

Y.C. Wang

TI,T2 168.3 5.0 T3,T4 168.3 5.0 T5,T6 168.3 5.0 T7,T8 168.3 5.0 T9,TI0 168.3 5.0 TI1,T12 168.3 5.0 T13,T14 168.3 10.0 T15,T16 168.3 10.0 T17,T18 168.3 10.0 T23,T24 168.3 10.0 T25,T26 168.3 10.0 T27,T28 168.3 10.0

Table 2: Measured Material

Lo (mm) Steel $355

8rade

Test ID

4200 4200 S355 C100 2500 $355 C40 2500 $355 C100 500 $355 C40 500 S355 CI00 4200 $355 C100 2500 $355 C100 500 $355 C100 4200 S275 CI00 2500 S275 C100 500 $275 C100

Concrete 8rade C40

TI 438

Steel yield stress (Nlmm 2)

52

Properties Cube strength (Nlmrn 2)

40.8

T2 438 51.8 T3 438 123.5 T4 438 121.2 T5 438 47.5 T6 438 47.8 T7 438 116.0 T8 438 115.3 T9 438 46 T10 438 44.7 T l l 438 115.3 T12 438 113.8 TI3 i480 120.7

Cylinder strength (N/mm 2)

41.3 106.3 91.0 39.0 39.0 97.8

408

Table 1: Test parameters Test ID D(mm) t(mm)

Young's modulus (N/mm 2) 41000

6max

(mm/m)

2.3

45000 , 2.4 52500 12.83

53000 '2 .05 39500 t2 .2 42000 2.05 50500 2.45

101.0 54000 2.8 37.3 41000 2.05 36.5 41500 2.15

53500 2.48 52500 2.78 49000 2.4 50500 2.25 49500 2.48

99.5 100.0 92.7

T14 480 119.5 82.3 TI5 480 113.8 93.3 T16 480 114.2 90.5 52500 2.20 TI 7 480 126.0 87.8 52000 2.95 TI8 480 120.0 9 0 . 8 49000 2.23

50000 2.78 T23 330.5 116.8 , 98.8

T24 330.5 118.7 98.5 50500 2.78

T25 330.5 113.6 89.3 53500 2.23 T26 330.5 116.3 95.8 52500 I 2.48 T27 330.5 116.0 91.8 152000 ! 2.38

T28 330.5 114.2 97.0 ~ ' i 50500 2.58


Table 3: Comparison between design strength and test results Test lD ~ e(mm) Test comparison between design calculations

and test results load BS 5400 Part 5 [ 1] Euroexxte 4 Part 1.1

t21 (kN) (kN) _pred/test (kN~.__pred/test

T1 1.16 3 900 , 964 1.071 961 1.068 , _ . . . _ . . _ _ . _ . _

T2 1.14 5 950 ! 963 0.993 932 0.981 T3 1.---'~43 2.5 1550 ~ 0.754 1153 0.744 T4 1.35 5 1400 ~ 1124 0.803 ! 1053 0.752 T5 0.7 4 1300 i 1382 1.063 ~ 1448 1.114 T6 0.7 2 1445 1465 1 . 0 1 4 1513 1.047 T7 0.85 5 2330 1858 0.791 2007 0.854 T8 0.85 2.5 2450 2004 0.818 2197 0.897 T9 0.17 5 2360 2002 0.848 1891 0.801 T10 0.17 5 2360 1988 0.842 1879 0.796 T l l 0.21 5 3250 i 2784 0.857 2944 0.906 T12 0.21 1 3250 ~ 0.950 32/2 1.007

v ~

T13 1.36 4.5 1900 ~ 0.827 1481 0.779 T14 1.33 1.5 2400 1661 0.692 1623 0.676 T15 0.83 2 3350 2855 0.852 2957 0.883 T16 ~ 0.2 3650 3032 0.831 3076 0.843 T17 ~ 4 4550 4326 0.951 3943 0.867 T18 0.21 3 4550 4386 0.964 4099 0.901 T23 1.25 6 1800 1429 0.794 ~ 0.738 T24 1.24 0.5 2200 1605 0.729 11627 ! 0.739

!T25 0.74 3 2600 2333 0.897 "2465 ! 0.948

L T26 0.75 3.5 2450 2314 0.945 : 1.021

I T , 0.1 lO 2,0 2 60 10. 64 , T28 0.19 5 3400 3206 0.943 3 - ~ 10.952

Table 4: Effect of column squash load increase due to confinement effect Test iD With bending moment Without bending moment T9 1.145 1.207 T10 1.148 1.210

. . . . .

T1 i 1.066 1.094 . . . . .

TI2 1.087 1.092 ,,

TI7 1.109 1.142 T18 1.113 1.137 T27 1.056 1.137 T28 1.091 1.130

1.2

1

0.8 e a

t! G

P ti

g! n.

u)

0.6

.- U

0.4

0.2

0

0

ts

0

n Grade C40 [ ;r concrete

- 00

=: -

- BS5400 joEC4 1 K

9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Nondimensional slenderness

Fig 1: Comparison between predicted and test results


A

Z v

"o a 0 . J

C40

C100

I o I t I t I

0 -2000 -4000 -6000 -8000 - 10000

vertical micro strain

I

-12000

Fig 2 �9 L o a d - s t r a i n curves , L = 2 . 5 m


STRENGTH AND DUCTILITY OF HOLLOW CIRCULAR STEEL COLUMNS FILLED WITH FIBRE REINFORCED CONCRETE

G. Campione, N. Scibilia, G. Zingone

Dipartimento di Ingegneria Strutturale e Geotecnica Universitb, di Palermo, 1-90128, ITALY

ABSTRACT

The focus of the present investigation is the study of the behaviour of hollow circular steel cross- sections filled with fibre reinforced concrete (FRC), subjected to monotonic loads. Using the same volume percentages of fibres, the influence of different types of fibres (steel, polyolefin) on the behaviour of the columns was investigated. Results of fibre reinforced composite columns were compared with those of columns filled with plain concrete, showing the advantages of using FRC compared to plain concrete, in terms of both strength and ductility. A simplified analytical model to predict load-deformation curves for composite members in compression is proposed, and the comparison between experimental and analytical results has shown good agreement. Finally, a comparison is made between the bearing capacity of circular hollow steel columns filled with plain concrete evaluated according to recent European and International codes and that evaluated using the proposed model.

KEYWORDS

Circular steel columns, fibre reinforced concrete, composite members, strength, ductility, active confinement.

INTRODUCTION

In the design of tubular structures, after determining the shell plate thickness satisfying tensile stress, the stability of the shell should be checked for compressive stresses against buckling. A thin-walled cylinder shell subjected to compression may fail either due to the instability of the shell, involving bending of the axis, or due to local instability, as shown in Figure. 1, and also depends on the ratio of the thickness to the radius of the shell wall and on the length of the columns. Failure of this type of structure is due to the formation of characteristic wrinkles or bulges, circular or lobed in shape. To mitigate or prevent this type of failure it is common to encase or fill steel profiles with concrete. The coupling of concrete and steel shapes makes it possible to obtain structural elements which, compared to the single constituent elements, ensure high performance in terms of both resistance and ductility, Cosenza & Pecce (1993).

413

414 G. Campione et al.

Modern codes like the Eurocode relating to composite steel-concrete structures contemplate the use of composite steel-concrete columns made using W shapes or thin-walled tubular profiles with a circular, rectangular or square section, as shown in Figure 2. Concrete, inside or outside the profile, exerts beneficial confinement action against phenomena of local or overall instability and markedly increases the resisting and dissipative capacity of steel columns, Schneider (1998). A further advantage is the increase in fire resistance, Lie (1994), Frassen et al. (1998).

Figure 1: Overall and local instability

Dwelling on the case of tubular steel columns, we can observe that, filled with concrete, they present deeply different behaviour from hollow ones and much better performances. There is not only a higher bearing capacity, but also greater safety against flexural instability, which makes it possible to construct particularly slender columns. The concrete is highly confined and hence is more ductile and has a greater bearing capacity, Shakir et al. (1994). The coupling of several tubular sections makes it possible to face very high stresses, greatly increasing the critical load values, both against local and overall buckling of composite columns.

Figure 2: Typical composite members

Recent applications of the structural elements concern arch road bridges in which the deck is supported by hollow steel columns filled with concrete. These very slender elements, thanks to the combined use of the two materials, can face the high stresses induced by external loads, with the evident advantage of constituting carpentry during the construction of the structure.

Recent theoretical and experimental studies have shown that if traditional reinforcement made up of bars and stirrups is added inside tubes, or these are filled with fibre reinforced concrete (FRC), their

Strength and Ductility of Hollow Circular Steel Columns 415

bearing capacity and ductility increase, Campione et. al (1999). In the last few decades interest in the field of composite materials and especially in fibre reinforced concrete has led to the development of new types of fibres (carbon, polyolefin, kevlar, steel) with different shapes (hooked, crimped, deformed). Due to the bridging capacity of the fibres across the cracks, FRC behaves better than either plain concrete or plain concrete confined with traditional reinforcement in terms of energy absorption capacity, and sometimes strength, especially when a high volume fraction of fibres (1-2 %) is used, Campione et al. (1999).


In the present section there are briefly mentioned experimental results discussed in detail in a previous investigation by the authors, Campione et al. (1998). The experimental research involved the casting of different types of composite members: steel columns, steel columns filled with normal strength concrete (NSC) and steel columns filled with fibre reinforced concrete (FRC). Different types of fibres (polyolefin straight, hooked and crimped steel fibres) were added to the fresh concrete at a dosage of 2% by volume. The fibres had the characteristics shown in Table 1.

TABLE 1 CHARACTERISTICS OF THE FIBRES

Type of fibres

Shape Diameter equiv.

, (ram)

Length. Lf (mm)

Tensile strength f't (MPa)

Modulus of Weight elasticity density Ef (MPa) (kg/ m 3)

Poyolefin 0.80 25 375 12000 Hooked steel ~ "-" 0.50 35 1115 207000 Crimped steel ~ 1.00 50 1037 207000

900 7860 7860

The columns had a circular cross-section welded along their length; the yielding stress was 206 MPa and the ultimate stress 324 MPa; the internal diameter was 120 mm and the thickness 3.5 mm, the length of the entire columns was 1000 mm.

Figure 3: Composite columns tested in compression

The columns were tested in uniaxial compression utilising a universal testing machine operating in displacement control (Figure 3). A load cell and several LVDT's connected to a data acquisition system were used to record the load P and the vertical deformation 8 of the columns. Monotonic tests were carried out on 100x200 mm cylinders in concrete and FRC both in indirect split tension and in compression to characterise the materials as shown in Figure 4. It is interesting to observe that adding fibres to the matrices the behaviour of the latter changes significantly, especially in


the sottening branch, in terms of both energy absorption capacity and residual strength. More details on the strength and strain values and cyclic response of the materials are given in a previous paper, Campione et al. (1998). Table 2 gives the most rapresentative experimental results of the compression and indirect split tension tests, particularly the maximum compressive strength f c, and corresponding strain e0, and maximum tensile strength ft.

TABLE 2 EXPERIMENTAL RESULTS FOR COMPRESSION AND INDIRECT TENSION TESTS

Types of fibres

Matrix

E;o

0.0032

f~ (A/IPa) 25.20

(*)f, (MPa)

1.64 Hooked steel 0.0061 27.45 3.56

Polyolefin 0.0034 29.34 2.42 Crimped steel 35.40 0.0064 2.72 (*) ft=2P/0td h)

Figure 4: Monotonic tests in compression of FRC with 2% fibres

Figure 5 gives load-deformation (P-8) curves in compression for steel pipes and composite members in the case of montonic loads. Experimental results have shown that columns filled with FRC exhibit higher strength than those filled with plain concrete. The maximum strength of composite members filled with FRC is 20 % higher than that recorded for steel pipes filled with FRC. After the peak load was reached, failure was due to the crushing of concrete and to local and global buckling of the steel pipes. At this point, the peak load and also the stiffness decreased. By contrast, the addition of fibres ensured better softening behaviour and more available ductility.

Figure 5" Load vs. deformation curves for composite columns with 2% fibres

Strength and Ductility of Hollow Circular Steel Columns

STRENGTH OF COMPOSITE COLUMNS SUBJECTED TO COMPRESSIVE LOADS

417

Several European and international codes give design rules and simplified formulae to predict the bearing capacity of steel columns filled with plain concrete. These are able to take into account the strength of the materials, and the buckling problems of the composite members. When columns are subjected to axial forces the properties which must be taken into account in the design of members include strength of the constituent materials, local instability, and the capacity to transfer internal stresses between the steel pipe and the concrete core.

For composite members having a transverse circular cross-section, EC4 allows one to neglect local buckling problems when the ratio between the diameter d and the thickness t obeys the relationship d/t<90 e 2, where e 2=235/fy, and fy is the yield stress of the steel. According to EC4, the axial plastic force of the transverse cross-section of a steel column filled with plain concrete can be obtained, as a first approximation, as the sum of the contributions of the concrete core and the steel section, also taking into account the increase in strength due to the confinement in the concrete core and the reduction in steel stress due to the biaxial state of stress in the steel pipes; this is done by introducing the coefficients rl, which depend on the slenderness of the composite members. The ultimate axial load capacity of the columns is obtained by multiplying the axial plastic force by the reduction coefficient ~,

which depends on the slenderness ,i.

~ , . . _ _ _ . . . . . . _ . _ . ~

TTT

~~- ~" t ~'II,"L'"ll

T ... . . . . . l~s ,t

Figure 6: Distribution of tensions in the steel pipes

It is interesting to observe that EC4 and LRFD give very conservative values of the maximum beating capacity for short columns. In fact, in composite members loaded concentrically the concrete core and steel tube are subjected to a combined state of stresses (Figure 6). For this reason many researchers, like in Pecce (1993), have modelled the behaviour of composite members subjected to compression considering a triaxial state of stresses in the concrete core and a biaxial state in the steel pipes and imposing the compatibility for each loading step in terms of longitudinal and lateral strains. Unfortunately, few experimental data are available for plain concrete and fibre reinforced concrete subjected to triaxial stresses, and so it is difficult to calibrate the parameters of an analytical model of concrete core subjected to a multiply stresses.

In the present investigation the authors refer to a monoaxial state of stresses for concrete, in which the confinement effect due to the lateral pressure f'~ of the steel pipes, varying at each loading step, is taken into account. For steel pipes a biaxial state of stresses is assumed and the relationship between longitudinal and circumferential stresses is that proposed by Von Mises up to failure condition:

2 2


For steel in uniaxial tension an elasto-plastic stress-strain relationship was assumed, having conventional yielding stress of 300 MPa. When the yielding condition occured in steel pipe due to biaxial stresses a linear reduction of longitudinal stress ~s,I up to 2 % strain was assumed, in accordance with experimental data (Figure 7-b). It was also assumed that circumferential stress ~s,t in steel pipes, evaluated using Eqn. 1, increased up to the yielding value and consequently that in the concrete core the confinement effect was maximum (Figure 7-a). The longitudinal stress in the concrete core was obtained utilising the Mander et al. (1988) model in which the lateral pressure fl is assumed variable.

The Mander et al. (1988) stress-strain curve for concrete adopted is:

~c t3~ . - - - r ~ cc c~r = (2)

r

r - I + ~

The r coefficient is related with the initial tangent modulus Er and with the secant modulus Eso~=o~dec~ as follow:c

E~ r = ~ ( 3 )

The maximum strength in the concrete core ~cr is:

C~cc = f~"(--1.254 + 2 . 2 5 4 " 1 1 + ~ - - ~ 7.94.f/ 2 - ~ '1 (4)

L' L )

The lateral pressure ft on the concrete core is:

t ft' = 2.o, . , . - d (5)

The maximum longitudinal strain is:

E - 11] e~=eo" I+ tf~' (4)

Figure 7 shows the longitudinal stress-strain relationship for plain concrete (f'c=25.2 MPa) and steel obtained assuming different values of maximum longitudinal stresses.

To determine the load-deformation curves (P-8) of the columns the compatibility of the lateral and longitudinal deformations was assumed and an equilibrium equation was utilised in which the stresses were calculated through constitutive relationships given before.

The vertical loads of the composite members were obtained by considering of the steel pipes and concrete core to strength, without any reduction in the transverse cross-section of the concrete core, because of the effective confinement reached in concrete inside steel pipes.

Strength and Ductility of Hollow Circular Steel Columns 419

Figure 7: Analytical longitudinal stress-strain curves : a) concrete; b) steel

Figure 8 shows experimental and analytical results obtained with the model proposed. It is interesting to observe that the model permits one to obtain a good level of approximation in terms of maximum bearing capacity and corresponding deformation, but overestimates initial stiffness. This is due perhaps to the fact that initial deformations are affected by boundary test effects that the model does not take into account.

Figure 8: Comparison between analytical and experimental results for composite members

In the case of FRC the buckling effects are reduced by the presence of fibres and failure is due essentially to plasticization of materials ensuring a good prevision of experimental results with the analytical model based on these concepts. In the case of steel pipes filled with plain concrete, the buckling effects reduce the bearing capacity of the columns in the softening branch and the analytical model is not able to predict these effects.

TABLE 3 ULTIMATE LOAD OF COMPOSITE MEMBERS: EXPERIMENTAL AND ANALYTICAL VALUES

Columns filled with

Plain concrete FRC with hooked steel fibres

FRC filled with polyolefin fibres FRC filled with crimped fibres

g g NpI, R kc Per g me~ee Nsp

0.303 0.977 678 0.640 552 876 889 0.308 0.976 700 0.650 568 966 976 0.311 0.975 718 0.660 581 1030 980 0.322 0.972 778 0.670 623 1250 1055


Table 3 gives the maximum bearing capacity evaluated with EC4, LRFD and the model proposed. For the latter the strength of the columns is penalised to take into account the effect due to the slenderness

of the columns through the ~ coefficient proposed by EC4. In Table 3, ~, and ~ are the slenderness as evaluated according to EC4 and LRFD, ~ NplR and P= the bearing capacity of the composite columns according to EC4 and LRFD respectively and ~ Ar the bearing capacity evaluated according to the method proposed penalising the axial plastic force with the ~ coefficient, and finally N~, the experimental values.

CONCLUSIONS

The strength of circular columns filled with plain concrete and FRC depends on the mechanical and geometrical properties of the steel columns (type of steel, thickness, diameter) and also on the strength of the concrete and type and volume percentage of fibres. Experimental data from tests on materials (compression and tension tests) and from compressive tests on steel columns filled with plain concrete and FRC have shown the advantages of using FRC rather than plain concrete. Simplified analytical model to predict the load-deformation curves in compression of composite members is proposed. It is based on monoaxial state of stresses for concrete, in which the confinement effect due to the lateral pressure f'l of the steel pipes, is variable at each loading step and steel pipe is in biaxial state of stresses. From the comparison between the experimental and analytical results obtained using the simplified method good agreement is found in term of maximum strength and maximum displacements, but different values are recorded in the softening branch, because of the instability phenomena that the model do not take into account.

REFERENCES

Campione G. Scibilia N. & Zingone G. (1998) Comportamento ciclico in compressione di colonne composte in acciaio e calcestruzzo fibrorinforzato. 3 ~ Workshop Italiano sulle Strutture Composte, Ancona, 29-30 Ottobre. Campione G, Mindess S, Scibilia N., &.Zingone G, (1999). Compressive behaviour of circular steel columns filled with fibre reinforced concrete: experimental investigation and comparison with EC4 code. Costruzioni Metalliche 5. Cosenza E., Pecce M. (1993). Resistenza a compressione di colonne con sezione di acciaio riempita di calcestruzzo. Atti delle Giornate Italiane delle Costruzioni in Acciaio, Viareggio 24-27 ottobre, 290- 303. Eurocode 4 - Common Unified Rules for Composite Steel and Concrete Structures, European Committee for Standardization (CEN), ENV 1994-1-1. Frassen J.M., Talamona D., Kruppa J., Cajot L.G. (1998). Stability of steel columns in case of fire: experimental evaluation. Journal of Structural Engineering ASCE, 124:2, 158-163. Lie T.T. (1994). Fire resistance of Circular Steel Columns Filled with Bar-Reinforced Concrete. Journal of Structural Engineering ASCE 120: 5, 1489-1509. Manual of Steel Construction:Load and resistance factor design (LRFD) (1994) 2nd Ed., Am. Inst. of Steel Construction, Chicago, IL 1. Mander J. B., Priestley, M.J.N. & Park, R. (1988). Theoretical stress-strain model for confined concrete, Journal of Structural Eng#leering ASCE 114:8, 1804-1825. Pecce, M.(1993). La modellazione del comportamento in condizioni ultime di colonne composte con sezione circolare in acciaio riempita di calcestruzzo e sottoposte a sforzo normale centrato. 1 ~ Workshop Italiano sulle Strutture Composte, Trento, 17-18 maggio. Schneider S.P. (1998). Axially loaded concrete-filled steel tubes. Journal of Structural Engineering, ASCE, 124: 10, 1125-1138. Shakir-Khail H. (1994). Experimental study of concrete-filled rectangular hollow section columns. Structural Engineering Rewiew, 6: 2,. 85-96.

AXIAL COMPRESSIVE STRENGTH OF STEEL AND COMPOSITE COLUMNS FABRICATED WITH HIGH STENGTH STEEL PLATE

B. Uy 1

ISchool of Civil & Environmental Engineering, The University of New South Wales, Sydney, NSW, 2052, AUSTRALIA

ABSTRACT

The design of tall buildings has recently provided many challenges to structural engineers. One such challenge is to minimise the cross-sectional dimensions of columns to ensure greater floor space in a building is attainable. This has both an economic and aesthetics benefit in buildings, which require structural engineering solutions. The use of high strength steel in tall buildings has the ability to achieve these benefits as the material provides a higher strength to cross-section ratio. However as the strength of the steel is increased the buckling characteristics become more dominant with slenderness limits for both local and global buckling becoming more significant. To arrest the problems associated with buckling of high strength steel, concrete filling and encasement can be utilised as it has the affect of changing the buckling mode which increases the strength and stiffness of the member. This paper describes an experimental program undertaken for both encased and concrete filled composite columns, which were designed to be stocky in nature and thus fail by strength alone. The columns were designed to consider the strength in axial compression and were fabricated from high strength steel plate. In addition to the encased and concrete filled columns, unencased columns and hollow columns were also fabricated and tested to act as calibration specimens. A model for the axial strength was suggested and this is shown to compare well with the test results. Finally aspects of further research are addressed in this paper.

KEYWORDS

Columns, composite construction, high strength steel, steel structures, tall buildings

INTRODUCTION

The design of tall building gravity load systems is influenced heavily by the ability to resist axial force with the smallest cross-sectional sizes available. Recent developments in the quality of high strength steel have seen it become extremely attractive for the design of tall buildings in Australia and future landmark buildings are earmarked to utilise high strength steel in Japan. The benefits of the use of

421

422 B. Uy

high strength steel can be utilised in a braced frame where the external spandrel frame is used to resist gravity loads alone. High strength steel is most efficient when it is allowed to develop its' full yield stress. Thus high strength steel is efficient when local and overall buckling can be eliminated in a column design. This paper will summarise the previous applications of high strength steel in tall buildings in major cities throughout the world. The summary includes a description of the column type and grade of steel used to illustrate the methods in which high strength steel is being used.

Based on the previous applications, a detailed experimental program was conducted which reflected current and future uses of high strength steel in composite columns. The experiments were based on the behaviour of high strength steel columns in pure compression and consisted of both bare steel sections and composite sections. These experiments will be described here and a numerical model for the calculation of the pure compressive strength will be presented and shown to provide a conservative estimate of the column cross-section strength.

COMPLETED AND PLANNED PROJECTS

Previous projects, which have been completed and planned, are summarised herein. This list will identify the type of projects and the potential benefits achieved from the use of high strength steel. In particular, this table reflects tall building projects in Australia where high strength steel has been used. In the design of Star City, the major benefits derived from the use of high strength steel were in providing additional car space in the basement levels. This was a mandatory requirement for the project by the Sydney City Council. The use of high strength steel in the other Australian buildings was justified in reducing column sizes and thus providing additional floor area and car park spaces in the building. The Shimizu Super High Rise (SHR), which is a proposed project in Tokyo, Japan, will use high strength steel in box columns for the exterior spandrel frame. Figure 1 also illustrates the cross-section geometries utilised for each of these projects.

TABLE 1 PROJECTS UTILISING HIGH STRENGTH STEEL

Building

Grosvenor Place

City

Sydney Perth

Year Number of Completed Storeys 1988 50

Central Park 1989 50 300 Latrobe St. Melbourne 1990 30

20 Star City Shimizu SHR

Sydney 1997 Tokyo Proposed 120

Column Type Encased Encased Encased Encased Filled

Steel Grade (MPa) 690 690 690 690 600

Figure 1: High strength steel composite cross-sections

Axial Compressive Strength of Steel and Composite Columns

PREVIOUS RESEARCH

423

Previous research into high strength structural steel for columns has been mainly carried out in the regions where it has been applied in practice and this includes research in both Australia and Japan. Firstly Rosier and Croll (1987) considered the benefits of high strength quenched and tempered steel being applied in structures such as bridges, buildings and silos. This study included consideration of the economics of the material over conventional mild structural steel and showed the significant advantages that could be derived from its use.

Rasmussen and Hancock (1992 and 1995) conducted tests on both high strength steel fabricated I- sections and box sections. These tests established local buckling slenderness limits for these high strength steel sections. Furthermore, slender columns were tested and the behaviour of these was compared with the slender column curves of the existing Australian Standard AS 4100-1990 (Standards Australia 1990). It was found that providing the local buckling slenderness limits were adhered to, then the slender column behaviour could be described using this standard developed specifically for mild structural steel.

Hagiwara et al. (1995) and Mochizuki et al. (1995) considered the behaviour of high strength structural steel for the application in super high rise buildings in Japan. These studies considered the reliability inspection and the welding process for heavy gauge steel plate. These studies are pertinent to the application of the use of high strength steel in projects such as the Shimizu Super High Rise in Tokyo, Japan.

Uy (1996) considered the behaviour of concrete filled steel box columns filled with concrete. These studies considered the advantages derived from filling the sections with concrete to increase the local buckling stresses. Furthermore, the members were considered under combined bending and compression to assess the strength of short columns. The results of these columns were compared with columns designed with normal strength structural steel, to show the reduced cross-sectional dimensions able to be achieved. Furthermore, comparisons of the cross-sectional ductility were made and showed that composite members composed of high strength structural steel still had a large degree of reserve of strength after peak loading conditions.

EXPERIMENTS

This section outlines the test program undertaken which included column tests and numerous material property tests. The test set-up for the columns will be described and the results will then be presented. A general review and description of the failure modes will then be provided.

Column Tests

The test program consisted of eight columns, of which four columns were fabricated I sections and four were fabricated box sections. The columns and their pertinent dimensions are shown in Figure 2

Figure 2: Column cross-sections

424 B. Uy

Tensile Coupon Tests

To determine the stress-strain characteristics of the steel plate in tension a series of tensile coupons were produced from the virgin steel plate and tested in an Instron uniaxial testing machine. Pertinent data for these test coupons are provided in Table 2. Four tests were conducted with a mean value for yield stress of 784 MPa being established. Whilst high strength steel is not considered to have a defined strain hardening region, the tests revealed an increase in stress after yielding and the mean ultimate stress of the material in tension was determined to be 817 MPa.

TABLE 2 TENSILE COUPON TESTS

Specimen Number Yield Stress, cry (MPa)

765.3

Ultimate Stress Cru (MPa)

809.7 2 781.3 816.8 3 796.4 808.9 4 793.7 830.9

Mean 784.2 816.6 Standard Deviation 14.2 10.2

Compressive Stub Column Tests

The stress-strain characteristics can vary in tension and compression, which can be due to the effects of inelastic local buckling and/or residual stresses. To try and identify these differences a series of stub column tests were undertaken. These stub column tests were able to establish a clear reduction in the mean yield stress of at least 30 MPa, which could be attributed to the effects of inelastic local buckling. However, the ultimate stress of the stub column tests was virtually identical to the tensile coupon tests with a mean value of 817 MPa being attained in both.

TABLE 3 STUB COLUMN TESTS

Specimen Number

Average Standard Deviation

Yield Stress, oy (MPa)

757.3 757.3 750.0 754.9

4.2

Ultimate Stress Cru (MPa)

833.0 839.1 779.0 817.0 33.1

Column Test Set-Up

Eight columns were tested in pure compression in an Amsler 5,000 kN capacity compression testing facility. The column test set-up is illustrated in Figure 3 which shows the general characteristics of the testing machine platens and the instrumentation used in the testing. The test set-up highlights the end conditions, which were provided to ensure a uniform loading surface to the column. Using steel plates with recessed edge, the plates were filled with a very stiff plaster and a small preload was applied until the plaster cured and reached an appropriate stiffness. In addition to this strain gauges were used on

Axial Compressive Strength of Steel and Composite Columns 425

the steel surfaces, which was useful in tracing the load-strain characteristics for both the determination of yielding and local buckling of the steel plates. Furthermore, linear varying displacement transducers (LVDT's) were used to measure the load-axial shortening characteristics which was also useful in the determination of yielding and ultimate loading of the column members.

Figure 3: Photograph of column test set-up

Load-Deflection Results

The axial load - axial shortening of each column was recorded and these were useful in being able to ascertain the point at which yielding took place and the point of ultimate failure, which was usually characterised by concrete crushing and softening. Figure 4 illustrates typical load-deflection results for the columns tested. For both the fabricated I section and box section columns, one can see that the composite sections have a larger stiffness, as well as achieving a larger ultimate capacity. Furthermore, the apparent ductility of both the bare still sections and the composite sections is shown to be quite adequate with a significant post peak reserve of strength being displayed.

Figure 4: Load-deflection results

426 B. Uy

Load-Strain Results

The load strain results were used to identify yielding of the steel sections in compression, and the strain gauges also proved useful in identifying the onset of inelastic local buckling. Figure 5 shows a set of typical load-strain results for the columns tested. The fabricated I section columns were designed so that the plates were compact for local buckling, however inelastic local buckling was still evident after the ultimate load was reached and this is defined by the erratic behaviour of the strain gauges as shown below. The fabricated box columns were also designed with compact plates, however, inelastic local buckling was more controlled and greater stress redistribution capability was noted by the smooth nature of the strain gauges after the peak load was reached as shown below.

Figure 5: Load-strain results

Failure Modes

All columns were tested in pure compression and thus failure was essentially primary compression. Failure was initiated by compressive yield since most plate sections were compact. Once yielding began, concrete crushing and inelastic local plate buckling of the plate elements usually followed this. Whilst the failure mode was primary compressive each of the members displayed a significant reserve of strength and thus highlighted a ductile failure plateau. Figure 6 shows the failure modes for six of the columns tested in this paper. This photograph highlights the local buckling modes for both the bare steel and composite sections as well as highlighting the concrete crushing for the composite sections.

Figure 6: Failure modes of columns

Axial Compressive Strength of Steel and Composite Columns

COMPRESSIVE STRENGTH MODEL

427

All the columns tested in this paper were tested in pure compression and in order to predict the strength of these members an axial compressive strength model is proposed which considers both the steel and concrete contributions to axial strength. The model shown in Equation 1 was used to provide a prediction of the axial strength of each of the columns

N u = Nuc + Nus (1)

where Nu is the ultimate axial strength of the composite column, Nuc=fc.Ac is the concrete contribution to axial strength and Nus=fy.As is the steel contribution to axial strength.

COMPARISONS

Table 4 summarises the specimen names, pertinent geometric and material properties and results of the tests. The strength of each of the tests, Nu.test was determined from the peaks of the load-deflection graphs, whilst the theoretical value for ultimate load Nu.theory was determined using Equation 1. The ratio of the test results to theoretical results is also calculated in Table 4 for each specimen.

The theoretical results generally provide a conservative estimate of strength for the steel section columns. Furthermore, the composite section columns are also well represented by the model. There is a slight value of non-conservatism in the strength determination of the composite columns, which could be due to the assumed maximum concrete stress. Eurocode 4 (British Standards Institution 1994) suggests a maximum stress equivalent to the cylinder stress be used and this generally accounts for some confinement effect. The maximum stress utilised herein was equal to the cylinder strength and thus illustrates that some confinement may be present. However, it may be necessary to impose a factor to account for creep in the composite columns and thus a value less than the cylinder strength may need to be applied for design. The mean value of the ratio of strength shows that the model overestimates the strength of the columns by 1% and there is a 6 % standard deviation associated with the model. The model is therefore shown to be quite acceptable for use in design.

TABLE 4 STRENGTH COMPARISONS

Test Name

HSSI1 HSSI2

1 2 3 HSCI1 4 HSCI2 5 HSSH1 6 HSSH2 7 HSCB1 8 HSCB2

me (mm 2)

0 0

9,500 9,500

0 0

10,000 10,000

NA-NOT APPLICABLE

AS (mm 2) 1,500 1,500 1,500 1,500 2,100 2,100 2,100 2,100

f~ (MPa)

NA NA

fy (MPa)

750

Nu.test (kN) 1,163 1,140

Nu.theory (kY) 1,125 1,125

Nu.test/ Nu.theory

1.03 1.01 750

50 750 1,408 1,600 0.88 50 750 1,590 1,600 0.99

NA 750 1,644 1,575 1.04 NA 750 1,561 1,575 0.99 50 750 1,940 2,075 0.93 50 750 2,132 2,075 1.03

Mean Standard Deviation

0.99 0.06

428 B. Uy

CONCLUSIONS

This paper has described the advantages of the use of high strength steel in tall buildings. A brief overview of projects to utilise these structural forms has been provided and the reasons for their use have been given. In particular the use of high strength steel plate in composite column forms was identified as a potentially useful application. An extensive experimental program was conducted to consider the axial compressive behaviour of both high strength steel columns and high strength steel composite columns. A compressive strength model was then proposed and found to be conservative in its prediction of the axial compressive strength of all the columns tested. Further research is however necessary to consider the behaviour of these columns in combined bending and compression. Furthermore the effects of interaction buckling due to local and global buckling also need to be considered as part of future research in this area.

ACKNOWLEDGEMENTS

This project was sponsored by an Australian Research Council Grant and supported in kind by Bisalloy Steels at Unanderra. This assistance is gratefully acknowledged.

REFERENCES

British Standards Institution (1994) Eurocode 4, ENV 1994-1-1 1994. Design of composite steel and concrete structures, Part 1.1, General Rules and Rules for Buildings.

Hagiwara, Y., Kadono, A., Suzuki, T. Kubodera, I. Fukasawa, T. and Tanuma, Y. (1995) Application of HT780 high strength steel plate to structural member of super high rise building: Part 2 Reliability inspection of the structure. Proceedings of the Fifth East Asia - Pacific Conference on Structural Engineering and Construction, Building for the 21st Century, Gold Coast, pp. 2289-2294.

Mochuziki, H., Yamashita, T., Kanaya, K., and Fukasawa, T. (1995) Application of HT780 high strength steel plate to structural member of super high rise building: Part 1 Development of high strength steel with heavy gauge and welding process. Proceedings of the Fifth East Asia - Pacific Conference on Structural Engineering and Construction, Building for the 21st Century, Gold Coast, pp. 2283-2288.

Rasmussen, K.J.R., and Hancock, G.J. (1992) Plate slenderness limits for high strength steel sections. Journal of Constructional Steel Research, 23, pp. 73-96.

Rasmussen, K.J.R., and Hancock, G.J. (1995) Tests of high strength steel columns. Journal of Constructional Steel Research, 34, pp. 27-52.

Rosier, G.A. and Croll, J.E. (1987) High strength quenched and tempered steels in structures. Seminar Papers of Association of Consulting Structural Engineers of New South Wales, Steel in Structures, Sydney.

Standards Australia (1990) Australian Standard, Steel Structures, AS4100-1990, Sydney, Australia.

Uy, B. (1996) Behaviour and design of high strength steel-concrete filled box columns. Proceedings of the International Conference on Advances in Steel Structures, Hong Kong, pp. 455-460.

CONCRETE FILLED COLD-FORMED C450 RHS COLUMNS SUBJECTED TO CYCLIC AXIAL LOADING

X. L. Zhao, R. H. Grzebieta, P. Wong and C. Lee

Department of Civil Engineering, Monash University, Clayton, VIC 3168, Australia

ABSTRACT

This paper describes a series of static and cyclic axial compression tests on empty and concrete-filled Rectangular Hollow Sections (RHS). Columns were made from cold formed C450 (450 MPa nominal yield stress) RHS with two different plate slenderness ratios. Two loading protocols were applied, namely Cyclic-Direct (full axial displacement applied and load oscillated) and Cyclic-Incremental (load oscillated at several accumulating axial displacement increments). First-cycle buckling loads were noted in the tests and compared with design loads predicted using various national standards and CIDECT. The paper further demonstrates that concrete filling increases the post-peak-load residual strength and reduces the rate of residual strength loss per cycle for thinner RHS columns subjected to overload cyclic situations. The authors also suggest that the AISC Seismic Provisions (1997) may be conservative with respect to compact behaviour for the specified width-to-wall thickness ratio limits.

KEYWORDS

Buckling, Bracing, Cold-Formed, Columns, Cyclic Loading, Concrete-Filling, Steel Hollow Sections

INTRODUCTION

Cold-formed tubular sections are widely used in steel structures. Cold-formed rectangular steel tubular braces have recently become popular in seismic regions, especially for high rise structures (Liu and Goel (1988)). Tests were performed in New Zealand (Walpole (1995)) and in USA (Jain et al. (1980), Sherman and Sully (1994)) on cold-formed RHS empty members. The test results showed that the capacity of cold- formed tubular members subjected to cyclic axial compression and tension reduced significantly due to local buckling in the sections. The magnitude of the local buckles also increased under repeated loading. Tests performed at Monash University have also demonstrated that the capacity of empty cold-formed tubular members reduces significantly when subjected to cyclic bending moments (Grzebieta et al. (1997)). Therefore it is necessary to study in detail the behaviour of cold-formed tubular sections during cyclic loading, and to investigate the possibility of improving the earthquake resistance of tubular members. It was suggested (Walpole (1995)) that the width-to-thickness limit for RHS under compression given in NZS3404 (1992) needs to be reduced for cold-formed sections under earthquake generated forces. In fact, a much lower limit of width-to-thickness ratio is given in AISC Seismic Provisions for Structural Steel Buildings (AISC (1997)) for RHS bracing members.

429

430 X.L. Zhao et al.

Most of the steel hollow sections manufactured in Australia are in Grade C350 and C450 (minimum yield stress of 350 and 450 MPa). The width-to-thickness (H/t) limits are lower for C450 RHS than those of C350 RHS (Zhao and Hancock (1991)), where H is the overall width or depth (whichever is larger) of an RHS and t is the thickness of the RHS. It is the C450 RHS that will be studied in this paper. The calculated (H/t) limit using AISC (1997) is 16.7 for C450 empty RHS under cyclic axial load. It can be demonstrated that most of the C450 sections manufactured in Australia, which have a H/t ratio ranging from 12.5 to 75 (AISC (1992), Tubemakers (1994)), do not comply with these limits.

The technique of filling tubes with concrete increases the ductility and prevents or delays local buckling of tubular sections under cyclic loading (Nakai et al. 1994, Ge & Usami 1994, 1996, Ricles 1995, Haijar and Gourley 1997, Zhao and Grzebieta 1999). Most of the tubular sections in these studies were either hot-rolled or welded box sections. Little research was performed on concrete-filled cold-formed RHS columns under cyclic axial loading (Liu & Goel 1988). This paper forms part of a research project on Tubular Structures under Large Amplitude Dynamic Loading currently running at Monash University, Australia. Only the case of empty and concrete-filled cold-formed RHS under cyclic axial loading is reported in this paper.

The paper describes a series of static and cyclic axial compression tests on empty and concrete-filled columns made from cold-formed C450 RHS profiles. The two loading protocols used, the Cyclic-Direct procedure and the Cyclic-Incremental procedure are outlined. The first-cycle buckling loads noted in the tests are then compared with design loads predicted using various national standards and CIDECT formulae. The paper demonstrates that concrete filling increases the post-peak-load residual strength and reduces the rate of residual strength loss per cycle for thinner RHS columns subjected to overload cyclic situations. It also proposes that the AISC Seismic Provisions (1997) specify width-to-wall thickness ratio limits that may be conservative in regards to full effective sections under cyclic loading.

WIDTH-TO-THICKNESS RATIOS

Width-to-thickness ratio limits are given in various codes (SSRC (1991), Rondal et al (1996)) to prevent local buckling of RHS under static compression. The concept of a form factor (a ratio of effective area to gross area) is used in AS4100 to indicate the effect of local buckling. Local buckling occurs if the form factor is less than 1.0. A width to thickness ratio limit is also given in AISC Seismic Provisions for Structural Steel Buildings (AISC (1997)) for RHS bracing members in concentrically braced frames. A summary of width-to-thickness ratio limits for different countries is presented in Table 1 where the following four width-to-thickness (H/t) limits are given:

1. Empty RHS under static load (loaded to failure) 2. Empty RHS bracing members under cyclic load (repeated loading) 3. Concrete-filled RHS under static load (loaded to failure) 4. Concrete-filled RHS bracing members under cyclic load (repeated loading)

Table 1 shows that the width-to-thickness ratio limits increase for concrete-filled RHS.

For static loaded structures, the limit for concrete-filled RHS given in Eurocode 4 is about 13% higher than that for empty RHS given in Eurocode 3. A theoretical analysis determining the width-to-thickness limit for concrete-filled RHS under static loading was carried out by Uy et al. (1998) and Wright (1995). Depending on the boundary conditions, Uy et al. (1998) showed that the limit can increase by either 27% or 53% whereas Wright (1995) found it increased by 22% or 68% compared with that given in AS4100- 1998.

In the case of cyclic loaded structures, the limit (18.0) for concrete-filled RHS members in the USA is about 7.8% higher than that (16.7) for empty RHS members.

Concrete Filled Cold-Formed C450 RHS Columns

TABLE 1

235 WIDTH-TO-THICKNESS RATIO LIMITS ( ~ = ~ , ~y in MPa)

~Jy

431

Sections, Loading Empty RHS, Static loading

Empty RHS, Cyclic loading Concrete- filled RHS, Static load

Concrete- filled RHS, Cyclic load

Country/ Region

Australia

Reference

AS4100 (SAA (1998))

H/t limits

3+40.2e New Zealand NZS 3404 (NZS (1992)) 3+40.2e Canada CAN/CSA-S16.1-M89 (CSA (1989)) 3+37.6e Japan AIJ (1990) 3+47.8e UK BS5950 Part 1 (BSI (1990)) 3+42.2e Europe Eurocode 3 Part 1.1 (EC3 (1992)) 3+42.0e USA AISC-LRFD (AISC (1993)) 3+40.8e USA

Europe

Research Australia

Research Australia

Research UK

AISC Seismic Provisions (AISC (1997))

Eurocode 4 (1992), Rondal et al. (1996) Uy et al. (1998) assuming Simply- Supported boundary condition in Finite Strip Analysis Uy et al. (1998) assuming Fixed boundary condition in Finite Strip Analysis Wright (1995) assuming Simply- Supported boundary condition

Wright (1995) assuming Fixed boundary condition

AISC Seismic Provisions (1997)

Research in UK

USA

3+18.9e

52e

50 for ~y : 300 MPa 60 for Cyy = 300 MPa 50 for Cyy - 275 MPa 69 for (Yy - - 275 MPa 3+20.8e

H/t limit (C450RHS) 32.1 32.1 30.2 37.5 33.5 33.4 32.5 16.7

37.6

40.8

49.0

39.1

53.9

18.0

When comparing cyclic loaded members to static loaded members Table 1 shows that the limits decrease for the cyclic loading case. The limit (16.7) for empty RHS bracing members under cyclic load is about half of that (30.2 to 37.5) for empty RHS under static load. The limit (18.0) for concrete-filled RHS bracing members under cyclic load is about 33% to 48% of that (37.6 to 53.9) for static load.


Materials

Cold-formed RHS were supplied by BHP Steel- Structural Pipeline and Products (Newcastle, NSW). The nominal dimensions, width-to-thickness ratio and measured material properties are shown in Table 2 where a cross-section number (S1 or $2) is given. Three tensile coupons were extracted from the flat surfaces of each tube size. The tensile coupon tests were performed according to the Australian Standard


AS1391 (SAA (1991)). The 0.2% proof stress was adopted as the yield stress for the cold-formed steel tubes. Six concrete cylinders with a diameter of 100 mm and a height of 200 mm were tested to determine the compression strength of the concrete filler. The average compression strength was found to be 44.4 MPa.

TABLE 2

CROSS-SECTION SIZES AND MATERIAL PROPERTIES

Cross- RHS Size Ratio Ratio section H x B x t (H-2t)/t H/t number (mm) S 1 100 x 50 x 2 48 50 $2 100 x 50 x 4 23 25

Form factor (SAA (1998))

Yield stress (MPa)

Ultimate tensile strength (MPa)

Comply with AISC (1997)

0.746 429 501 No 1.0 481 533 No

Test Specimens

Twelve tests were performed as listed in Table 3 and Table 4. The specimen label system used in this paper is: the first two letters (S 1 or $2) refers to the cross-section number defined in Table 2, the second part of the label (H or CF) refers to the filler material (Hollow or Concrete Filled), the third part of the label (SC, CD or CI) refers to the loading scheme (Static Compression, Cyclic-Direct or Cyclic- Incremental) as defined in the section on "Test Procedures".

TABLE 3

FIRST CYCLE PEAK LOADS (IN KN) FOR EMPTY RHS

Specimen Label S 1HSC

Measured First cycle peak load P1 (kN) 158

S1HCD 158 S1HCI 173 S2HSC 360 S2HCD 351 S2HCI 322 Mean -- COV

Pke=0.5 Pke=0.65 Pke=0.7

158 131 121 158 131 121 158 131 121 352 265 237 352 265 237 352 265 237

-_

Pk~.5]Pl Pke~.65/Pl Pke~.V/Pl

1.0 0.829 0.766 1.0 0.829 0.766

0.913 0.766 0.699 0.978 0.736 0.658 1.003 0.755 0.675 1.093 0.823 0.736 0.998 0.790 0.717 0.058 0.053 0.065

TABLE 4

FIRST CYCLE PEAK LOADS (IN KN) FOR CONCRETE-FILLED RHS

Specimen Label S 1CFSC S 1CFCD S 1CFCI S2CFSC S2CFCD S2CFCI Mean COV

Tested AISC Eurocode 4 P l (kN) (1993) 311 292 273 275 292 273 274 292 273 441 431 425 421 431 425 397 431 425

AU CIDECT

294 292 294 292 294 292 437 428 437 428 437 428

PAISC/ PEC4/ Pl

0.939 1.062 1.066 0.977 1.024 1.086 1.026 0.056

PAIJ Pl /el 0.878 0.945 0.993 1.069 0.996 1.073 0.964 0.991 1.010 1.038 1.071 1.101 0.985 1.036 0.064 0.056

PCIDECT /P I

0.939 1.062 1.066 0.971 1.017 1.078 1.022 0.056

Concrete Filled Cold-Formed C450 RHS Columns 433

The end-to-end length of all specimens was 2500 mm. Each end was welded to a 20 mm thick steel plate which in turn was bolted to end supports. The column end conditions were treated as fixed-fixed. The corresponding effective length factor (ke) is 0.5 in an idealised condition, 0.65 according to Galambos (1998) and 0.7 as defined in AS4100-1998. The modified slenderness ratio (~,n) is about 96 for columns with S 1 RHS and 116 for columns with $2 RHS based on AS4100-1998. The maximum measured initial mid-span lateral deflection was within L/5000 where L is the column length.

Test Set Up

A test rig was built on the N. W. Murray strong floor in the Civil Engineering Laboratory, Monash University, as shown in Figure 1. A 1000 kN capacity Instron Performance Reckoner actuator was used together with an Instron 8500 controller to apply the cyclic axial loading. Two spring pots were used to measure the axial deformation and the mid-span lateral deformation.

Figure 1 Test Set Up

Test Procedures

Different researchers used different loading histories. For example, a loading history was defined by Liu and Goel (1998) in terms of a yield deformation Ay, where Ay was the axial deformation in tension of the specimen corresponding to the nominal yield stress. Five cycles were applied at each axial deflection increment of 5Ay, 10Ay and 15Ay. A similar loading history was defined by Walpole (1995), where one cycle was applied at each axial deflection increment of 2Ay, 5Ay, 10Ay, 15Ay, 20Ay and 25Ay. In the tests by Sherman and Sully (1994), the applied axial displacement was between 5 mm in compression and 5 mm in tension, or 10 mm in compression and 5 mm in tension with the number of cycles varying from 18 to 40.

Constant axial displacement ranges were used in the current test program. Ten cycles were applied for each axial displacement range. Two types of loading schemes were used. One is called Cyclic-Direct as shown in Figure 2 (a) where the cyclic load was directly applied when the axial displacement reached 10 mm. The other loading scheme is called Cyclic-Incremental as shown in Figure 2(b) where the cyclic load was applied at several accumulating axial displacement increments.

Test Results

Inward folding mechanism with cracks was observed for the empty S 1 RHS under cyclic loading as shown in Figure 3 (a), whereas an outward folding mechanism was observed for the filled S 1 RHS as shown in Figure 3 (b). Cracks at tube corners were observed in S 1CFCI but not in S 1CFCD. No folding


mechanism was observed for the empty and filled $2 RHS as shown in Figure 3 (c). The peak load obtained in each test is listed in Tables 3 and 4. The peak load for the S 1 RHS increased by about 58% due to the concrete filing. The increase in peak load for the $2 RHS was about 23%. The load versus axial deflection curves are compared in Figure 4 for hollow and filled sections, where both upper bound (peak load in first cycle of each increment) and lower bound (peak load in last cycle of each increment) curves are given. It seems that the concrete filling increases the residual load capacity of RHS to cyclic load especially for thinner sections. No local failure was observed for $2 RHS in spite of the fact that their H/t ratio was larger than the limit specified in AISC (1997), ie. the code may be too conservative.

Figure 2 Loading Schemes

Figure 3 Failure modes

Figure 4 Load versus Axial Deflection Curves

Concrete Filled Cold-Formed C450 RHS Columns

FIRST CYCLE BUCKLING LOAD AND RESIDUAL STRENGTHS

435

The first cycle buckling load (ie the peak load Pl) for empty sections is presented in Table 3. The nominal column capacity predicted by AS4100-1998 using different values of effective length are compared with test values. It seems that the use of an effective length factor of 0.5 gives the best agreement whereas using the recommended factor of 0.7 provides a conservative value.

The first cycle buckling load for the concrete-filled RHS is presented in Table 4. Nominal capacities determined using AISC (1993), Eurocode 4, AIJ (Matsui et al. 1997) and CIDECT (Bergmann et al 1995) formulae are compared with the experimental values. All the design formulae give good predictions.

The residual strength (P/PI) for the CD (Cyclic-Ddirect loading) test series is plotted in Figure 5 (a) against the number of cycles. It can be seen that the residual strength reduces more rapidly for thinner empty S 1 sections than for thicker $2 sections. Furthermore, concrete filling induces more increase in residual strength for thinner sections than for the thicker sections.

The residual strength for the CI (Cyclic-Incremental loading) test series was also plotted against the number of cycles. A typical graph is shown in Figure 5 (b) for specimen S1CHCI where A1 is the first cycle axial deflection when the load peaks at Pl and A2 to A6 are the axial displacement at different increments. It can be seen that the rate of reduction of residual load per cycle is about the same at different axial deflection increments. This can also be seen in Figure 2 (b).

Figure 5 Residual Strength versus Number of Cycles

CONCLUSIONS

1. The effective length factor of 0.7 recommended in AS4100-1998 for a fully clamped empty RHS columns subjected to axial compression may be too conservative. An effective length factor of 0.5 is more appropriate.

2. Concrete filling reduces the rate of reduction in residual load (per cycle) for RHS subjected to cyclic loading especially for thinner sections. Concrete filling induces more increase in residual strength for thinner sections than for thicker sections.

3. No local failure mechanism was observed for all the $2 RHS profiles tested, which have a H/t ratio larger than the limit specified in AISC (1997), ie. the core may be too conservative.

4. The predicted first cycle buckling load for concrete-filled RHS members using AISC, Eurocode 4, AIJ and CIDECT formulae is in good agreement with tested values.

436

REFERENCES

X.L. Zhao et al.

AIJ (1990). Standard for Limit State Design of Steel Structures. Architectural Inst. of Japan, Tokyo. AISC (1992). Design Capacity Tables for Structural Steel Hollow Sections. AISC, Sydney. AISC (1993). LRFD Specification for Structural Steel Buildings. AISC, Chicago, Illinois. AISC (1997). Seismic Provisions for Structural Steel Buildings. AISC, Chicago, Illinois. Bergmann, R. et al. (1995). Design Guide for Concrete-Filled Hollow Section Columns under Static and Seismic Loading, CIDECT, TOV- Verlag GmbH, K61n. BSI (1990). Structural use of Steelwork in Building, BS5950, Part 1, British Standards Inst., London. CSA (1989). Steel Structures for Buildings. CAN/CSA-S 16.1-M89, Rexdale, Ontario. Eurocode 3 (1992). Design of Steel Structures, Part 1.1. ENV 1993-1-1. Eurocode 4 (1992). Design of Composite Steel and Concrete Structures, Part 1.1. ENV 1994-1-1. Galambos, T.V. (1998). Guide to Stability Design Criteria for Metal Structures, John Wiley & Sons, NY. Ge, H.B. and Usami, T. (1994). Strength Analysis of Concrete-Filled Thin-Walled Steel Box Columns, J. Construct. Steel Research, 30, 259-281. Ge, H.B. and Usami, T. (1996). Cyclic Tests of Concrete Filled Steel Box Columns." J. Struct. Engrg., ASCE, 122(10), 1169-1177. Grzebieta, R.H., Zhao, X.L. and F. Purza (1997). Multiple Low Cycle Fatigue of SHS Tubes subjected to Gross Pure Bending, Proc., SDSS'97, Nagoya, Japan, 847-854. Haijar, J.F. and Gourley, B.C. (1997). A Cyclic Nonlinear Model for Concrete-Filled Tubes - I: Formulation. J. Struct. Engrg., ASCE, 123(6), 736-744. Jain, A.K., Subhash, C., Goel, M. and Hanson, R.D. (1980). Hysteretic Cycles of Axially Loaded Steel Members. J. Struct. Engrg., ASCE, 106(ST8), 1777-1795. Liu, Z.Y. and Goel, S. (1988). Cyclic Load Behaviour of Concrete-Filled Tubular Braces. J. Struct. Engrg., ASCE, 114(7), 1488-1506. Matsui, C., Mitani, I., Kawano, A. and Tsuda, K. (1997). AIJ Design Method for Concrete Filled Steel Tubular Structures. ASCCS Seminar, September, Innsbruck, 93-116. Nakai, H. et al. (1994). Experimental Study on Ultimate Strength and Ductility of Concrete-Filled Thin- Walled Steel Box Columns under Seismic Load. J. Struct. Engrg., JSCE, 40A, 1401-1412. NZS 3404 (1992). Steel Structures Standard. Standards Association of New Zealand, Wellington. Ricles, J.M. (1995). Seismic Performance of CFT Columns-to-WF Beam Moment Connections. Proc., 3rd International Workshop on Connections in Steel Structures, Trento, Italy. Rondal, J., Wurker, K.G., Dutta, D., Wardenier, J. and Yeomans, N. (1996). Structural Stability of Hollow Sections, Verlag TUV Rheinland GmbH, Ktiln, Germany. SAA (1991). Methods for Tensile Testing of Metals. AS 139 l, Standards Assoc. Australia, Sydney. SAA (1998). Steel Structures. AS4100-1998, Standards Association of Australia, Sydney. Sherman, D. and Sully, R. M.(1994). Tubular Bracing Member Under Cyclic Loading. Proc., 4th Pacific Structural Steel Conference, Singapore. SSRC (1991). Stability of Metal Structures - A World View. Structural Stability Research Council. Tubemakers (1994). Design Capacity Tables for DuraGal Steel Hollow Sections. Structural Products Division of Tubemakers of Australia Limited, Newcastle, NSW, Australia. Uy, B., Wright, H.D. and Diedricks, A.A. (1998). Local Buckling of Cold-Formed Steel Sections Filled with Concrete. Proc., 2 no Int. Conf. Thin-Walled Structures, Singapore, 367-374. Walpole, W.P. (1995). Behaviour of Cold-Formed Steel RHS Members Under Cyclic Loading. Proc., Technical Conf. National Society for Earthquake Engrg., Waikanae, New Zealand, 44-50. Wright, H. D. (1995). Local Stability of Filled and Encased Steel Sections", J. Struct. Engrg., ASCE, 121(10), 1382-1388 Zhao, X.L. and Hancock, G.J. (1991). Tests to Determine Plate Slenderness Limits for Cold-Formed Rectangular Hollow Sections of Grade C450." Steel Construction, AISC, 25(4), 2-16. Zhao, X.L. and Grzebieta, R.H. (1999). Void-Filled SHS Beams subjected to Large Deformation Cyclic Bending. J. Struct. Engrg., ASCE, 125(9).

R E S E A R C H ON THE HYSTERETIC B E H A V I O R OF HIGH S T R E N G T H C O N C R E T E F I L L E D S T E E L T U B U L A R M E M B E R S U N D E R C O M P R E S S I O N AND B E N D I N G

Zhan Wang ~ and Yonghui Zhen 2

Department of Civil Engineering, Shantou University, Daxue Road, Shantou City, Guangdong Province, P.R.China

2 Department of Civil Engineering, Harbin University of Civil Engineering and Architecture(HUAE), Haihe Road, Harbin City, Heilongjiang Province, P.R.China

ABSTRACT

In this paper, force-displacement hysteretic loops of high strength concrete filled steel tubular members under compression and bending are calculated by using finite element method according to the steel constitutive model which is suitable for multiaxial cyclic loading and modified bounding surface model which is suitable for multiaxial cyclic compression of concrete. The test for hysteretic behavior of high strength concrete filled steel tubular members have been done. On the basis of theoretical analysis and experiment study, the force-displacement hysteretic loop characteristic of high strength concrete filled steel tubular members under lateral loading are discussed.

KEYWORDS

Concrete filled steel tube, high strength concrete, hysteretic loops, aseismatic behavior

1.1NTRODUCTION

Recently, high strength concrete (HSC) is being spread and applied in engineering areas. The trend of increment will be obvious year by year. The weakness of HSC is its high brittlement. Its failure, especially in complex stress state, will be controlled by brittlement ,and its reliability will be lowered. The members composed of steel tube and HSC (HCFST), as an ideal high strength and high ductility members, is the best way of the application of HSC to reality. On the basis of the research of HCFST force-displacement hysteretic loop under compression and bending and determining the model of hysteretic loop, we can analyze the HCFST elasto-plastic earthquake reaction by using shear building model. For the research of mechanism of HCFST member under compression and bending, the aseismatic behavior and the determination of force-displacement hysteretic loop model, the theoretical calculation of force-displacement hysteretic loop is very important. In this paper, by making use of finite element method and some necessary tests, hysteretic behavior of HCFST under compression and

437

438 Z. Wang and Y. Zhen

bending is studied. It not only has important practical value on HCFST aseismatic design, but also has theoretical significance on further research of HCFST. Illt21

2.THE CALCULATION OF FORCE-DISPLACEMENT HYSTERETIC LOOP

2.1. Calculating hypothesises and test method

HCFST under compression and bending is belonged to three dimension problem. Three dimension finite element should be used to resolve it. In this paper author resort to three dimension twenty nodes iso-parametric element which has so high precision in each element that it has being widely used on resolving three dimension problems.

There are some hypothesises on HCFST analysis by using finite element method, t31 a) Horizontal section hypothesis b) Constitutive relationship of steel is two linear random strengthen model. c) Constitutive relationship of core concrete is modified bounding surface model.

There are two loading ways in calculation. One is loading with force, which is used for vertical load. The other is loading with displacement, which is used for horizontal load. It mainly accords to the following. a) If loading with force, we will have great trouble near peak value and find no ways to calculate

decent part. b) With respect to horizontal section hypothesis, we know the resultant force of imposed load and

that section is still plane, but we don't know how the imposed forces distributed. In this case, lateral load P and axial load N are not able to turned into equivalent joint force on member joint. It is hardly carried out even loading with force in elastic part on this question.

2.Z Calculating method

The target of discussed in this paper can be regarded as a part of a frame column between inflection point and fixed end with lateral deflection, which stands for real work conditions of the column. The member under compression and bending is only discussed here, that is, a constant axial force is applied on a cantilever column first, lateral force is increased continuously later. In this case, we research the relationship between lateral force and lateral displacement.

There are two methods on calculating force-displacement hysteretic loop, i.e model column and data analysis method. As there is bigger error in model column method, data analysis is used here.

3.TEST RESEARCH

3.1. General features of test

In order to make a further research on behavior of HCFST under compression and bending and check the accuracy of theory, we carried out some experiments of force-displacement hysteretic loop. There are basements which is twenty millimeters made of steel on up and bottom of test specimens. The column is meld with stiffening rib of twenty-five millimeter. So the rigidity of each side of column is big enough. Loading sensors, displacement sensors and strain gauges are connected through the IMP(Isolated Measurement Pods) to the computer. Test data are able to gathered automaticly. The interval time are 1500 milliseconds with continuous gathering control. According to supervising data curve, P - 6 hysteretic loops are drawn

Research on the Hysteretic Behavior of High Strength Concrete 439

3.Z Experiment result

Figure 1 is the analysis between experiment and calculating result in this paper. These two are basely identical. The difference is mainly raised by the following, a) The rigidity of theoretical curve is higher because bending deflection is considered only except for

shear deflection on theoretical calculation. b) The rigidity of real curve is lower because of crack and frictional force between each link of

loading equipment.

In addition to this, experiment result from others are gathered here. Figure 2 and figure 3 show this compare. The parameters of member in figure 2 are as following: �9 133• seamless steel tube

which is 1260 millimeters long, the yield strength of steel is347.7 N/mm 2 , the cubic strength of

concrete is 70.2 N/mm 2 . The subjected axial loads are 385 KN and 865 KN. The parameters of

member in figure 3 are as following: �9 108 • 5 seamless steel tube which is 1100 millimeters long, the

yield strength of steel is 327.8 N/mm 2 , the cubic strength of concrete is 33.8 N/mm 2 . The subjected

axial loads are 20 KN and 270 KN.

In the figure 1-~3, the left is experiment result and the right is calculating result.

TABLE 1 FEATURES OF TEST SPECIMENS

Number of specimen D x t x L f~ f~,, r tll .~(~') Loading way

Z1-20 108x4.5x 1250 312.4 77.1 1.07 200 Cyclic

Z1-30 108 x 4.5 x 1250 312.4 77.1 1.07 300 Cyclic

Z2-20 114 x 6.0 x 1250 319.3 77.1 1.04 200 Cyclic

Z2-30 114 x 6.0 x 1250 319.3 77.1 1.04 300 Cyclic

Z3-20 l14x6.0x 1450 319.3 77.1 1.36 200 Cyclic

Z3-30 l14x6.0x 1450 319.3 77.1 1.36 300 Cyclic

Note[l]- r = 2Cry [11

rcfc,


Figure 1. The comparison between theoretical calculation and the test result

Research on the Hysteretic Behavior of High Strength Concrete 441

Figure 2. The comparison between theoretical calculation and the test result in reference 4

Figure 3. The comparison between theoretical calculation and the test result in reference 5

THE FORCE-DISPLACEMENT HYSTERETIC LOOP CHARACTERISTIC

We can find some characteristics of force-displacement hysteretic loop of high strength concrete filled steel tubular members from the theoretical analysis and experiment research result. a) The shape of hysteretic loop is closed to that of steel member under the condition of without local

buckle. And it is also analogous to the loop of general concrete filled steel tube member.


b) No matter how the parameters change, the hysteretic loop has great plumpness and no pinched or reduced phenomenon appear.

4.CONCLUSION

The calculation method in this paper has its new characteristics on how to select constitutive relationship and construct the model of finite element. On the basis of theoretical analysis and experiment study, the characteristic of force-displacement hysteretic loop of HCFST under compression and bending are discussed. From above, I think the following should be further studied: a) The basic property of polygon HCFST should be studied by making use of programme in this

paper. b) The property of the member of eccentric compression should be studied by making use of

calculating method in this paper. c) The lateral force resisting property of short column of HCFST should be studied considering of

shear deflection.

REFERENCE

l.Shantong Zhong.(1994). Concrete filled steel tubular structures. Heilongjiang science and technology press,. 2.Linhai Han.(1996) Mechanics of concrete filled steel tubular. Dalian science and engineering university press. 3.Yonghui Zhen.(1998). The hysteretic behavior studies of high strength concrete filled steel tububular members subjected to compression and bending. Master thesis of HUAE. 4.Weibo Yan.(1998). Theoretical analysis and experimental research for the hysteretic behaviors of high strength concrete filled steel tubular beam-columns. Master thesis of HUAE. 5.Yongqing Tu.(1994). The hystersis behavior studies of concrete filled steel tubular membersw subjected to compression and bending. Doctor thesis of HUAE.

DESIGN OF COMPOSITE COLUMNS OF ARBITRARY CROSS- SECTION SUBJECT TO BIAXIAL BENDING

S. F. Chen 1, j. G. Teng 2 and S. L. Chan 2

1 Department of Civil Engineering, Zhejiang University, Hangzhou 310027, China 2 Department of Civil and Structural Engineering,

The Hong Kong Polytechnic University, Hong Kong, China

ABSTRACT

In this paper, an iterative Quasi-Newton procedure based on the Regula-Falsi numerical scheme is proposed for the rapid design of short concrete-encased composite columns of arbitrary cross-section subjected to biaxial bending. The stress resultants of the concrete are evaluated by integrating the concrete stress-strain curve over the compression zone, while the stress resultants of the encased structural steel and the steel reinforcing bars are obtained using the fiber element method. A particularly important feature of the present method is the use of the plastic centroidal axes of the cross-section as the reference axes of loading in the iterative solution process. This ensures the convergence of the solution process for all cross-sectional conditions. Numerical examples are presented to demonstrate the validity, accuracy and capacity of the proposed method.

KEYWORDS

Composite Columns, Arbitrary Cross-Sections, Irregular Cross-Sections, Structural Design, Biaxial Bending

INTRODUCTION

Composite steel-concrete construction has been widely used in many structures such as buildings and bridges. The concrete-encased composite column is one of the common composite structural elements. Many studies have examined the behaviour and strength of biaxially loaded composite columns of doubly-symmetric cross-sections. Several researchers, including Johnson and Smith (1980), Lachance (1982) and Roik and Bergmann (1984) proposed simple methods for the analysis and design of rectangular composite columns under biaxial loading. E1-Tawil et al. (1995) developed an iterative computer method for biaxial bending of encased composite columns using the fiber element method and generated numerical results to evaluate the uni- and biaxial bending strengths of composite columns predicted by ACI-318 (1992) and AISC-LRFD (1993) provisions. Munoz and Hsu (1997) proposed a generalized interaction equation for the analysis and design of biaxially loaded square and

443

444 S.F. Chen et al.

rectangular columns. They compared their results with test results of many columns and predictions of the current ACI (1992) and AISC (Manual 1986) design methods.

In the design of building comer columns, comer walls, and core walls, irregular cross-sections or regular cross-sections with asymmetrically placed structural steel and/or steel reinforcement are often used to suit irregular plan layouts and/or eccentric load requirements. Little work has been carried out on composite columns of such irregular sections. Design rules for these columns also do not exist in design codes such as ACI-318 (1992) and EuroCode 4 (1994). Roik and Bergmann (1990) appears to have presented the only study on the design of rectangular composite section with an asymmetrically placed steel section. They proposed an approximate design method, which is a simple modification of the design approach for composite columns with a doubly symmetrical cross-section given in Eurocode 4 (1984). Only sections with the structural steel mono-symmetrically placed were considered. Rotter (1985) presented a moment-curvature analysis of arbitrary sections subject to axial load and biaxial bending using Green's theorem in integration which can be used to analyze composite steel-concrete columns. The emphasis of his study is on predicting the moment-curvature relationship rather than the design of such sections.

This paper provides a brief description of a general iterative computer method for the rapid design and analysis of arbitrarily shaped concrete-encased composite columns with arbitrarily distributed structural steel and steel reinforcement subjected to biaxial bending. A detailed presentation of the method is given in Chen et al. (1999). The method employs the iterative Quasi-Newton procedure within the Regula-Falsi numerical scheme. The stress resultants of the concrete are evaluated by integrating the concrete stress-strain curve over the compression zone, while those of structural steel and reinforcement are obtained using the fiber element method, in which the steel sections are discretized into small areas (fibers). Numerical examples are presented to demonstrate the validity, accuracy and capability of the proposed method.

REFERENCE LOADING AXES

For any cross-section under biaxial loading, the exact location of the neutral axis is determined by two parameters: the orientation On and the depth dn (Figure 1). The Quasi-Newton method has been adopted for the solution of 0 n and dn and found to be effective in many studies (e.g. Brondum-Nielsen, 1985; Yen, 1991). When dealing with irregular cross-sections, especially those with the arrangement of structural steel and reinforcement being strongly eccentric, convergence of the iterative process cannot be guaranteed. Indeed, when such a column is subjected to an axial load with magnitude approaching the axial load capacity under pure compression, the origin of the loading axes may fall outside the iso- load contour if this origin is located at the geometric centroid of the cross-section as usual (Yau et al., 1993). As a result, the inclination of the resultant bending moment resistance O~m may change in a range less than 2re when On varies from 0 to 2~r, resulting in non-uniqueness or non-existence of the solution of On (Yau et al., 1993). In this paper, this difficulty is overcome by using the plastic centroid as the origin of the reference loading axes. By taking the plastic centroidal axes as the reference loading axes, the existence and uniqueness of the solution of On are always ensured and the convergence of the iterative solution process is guaranteed.

For an arbitrary composite cross-section, the plastic centroid may be determined as follows (Roik and Bergmann, 1990)

XcAcf~c/rc +XsA~L/r, +XrArfy/7"r YcAcfc~/rc +Y~A~L/r~ +YrArfy/7"r = , r.c= Acfcc/rc+A~L/r~+Arfy/rr (1) Ypc AcLc/rc+AsL/r~+A~fy/rr

Design of Composite Columns of Arbitrary Cross-Section 445

where Ac, Ar and As are the total areas of concrete, reinforcing bars and structural steel, respectively; fcc, fy and fi are the respective specified strengths according to design codes such as Eurocode 4 (1994); Yc, Yr and 7~ are the corresponding partial safety factors, Xc, Yc, Xr, Yr. Xs and Ys are the respective centroid coordinates in the global XCY system.

v~ T"_~-. \ vmax \dn ~''-1~- \ 1

(a) (b) (c) Figure 1: Arbitrarily shaped cross-section: (a) Cross-section consisting of several regions; (b) Strain

distribution; (c) Stress block for concrete in compression

CROSS-SECTIONAL DESIGN

Basic Assumptions

The proposed design method is based on the following basic assumptions:

(1) Plane sections before deformation remain plane after deformation. Consequently, the strain at any point of the cross-section is proportional to its perpendicular distance from the neutral axis.

(2) The cross-section reaches its failure limit state when the strain of the extreme fiber of the concrete in compression attains the maximum strain gcu.

(3) The stress-strain relationship of concrete in compression is represented by a parabola and then a horizontal line:

( c f~22/(~<~0)and crc=fic(CO<_C <-Ccu) (2) Crc=fcc 2~oo-Co J

The structural steel and the steel reinforcing bars are assumed to be elastic-perfectly plastic. (4) Tensile strength of concrete is neglected.

Stress Resultants in the Cross-section

The cross-section may assume any shape with multiple openings. The entire section may be divided into several regions if necessary (Figure 1 a). For convenience of calculation, each region is treated as the superposition of a solid section occupying the entire area of the region and a number of negative sub-sections representing the openings (Figure 1 a). Consequently, the entire section consists of ns solid subsections (i.e. regions completed filled with structural materials) and no negative subsections (i.e.


openings). All subsections may be arbitrarily polygonal. The xoy coordinate system represents the reference loading axes with its orgin as determined by Eqn. 1. The uov system is related to the xoy system by a rotational transformation with the u-axis being parallel to the neutral axis.

For the ith subsection (either solid or negative), the coordinates of the jth vertex in the uov coordinate system are related to those in the xoy system by

uy=-xjcOS On+yjsin O., vj=yscOS On-Xjsin O. (3)

where On is the orientation of the neutral axis.

In order to evaluate the stress resultants of the concrete, the intersection points of a subsection with the neutral axis are first determined. These intersection points together with the vertices of the subsection above the neutral axis form the compression zone of the subsection (Figure 1). The stress resultants of the concrete can then be found by integrating the concrete stress-strain curve (Figures l b and 1 c) over this polygonal compression zone as given in Eqn. 4, assuming that the vertices of the zone are numbered sequentially (either clockwisely or anti-clockwisely)

It Nzci =]Pzcil = ICrc dud~ IJ=l uj o

n c Uj+l v(u) n c Uj+l v(u)

, M u c i = P i Z I I [-ere (~ + vn)]dud~' Mvc i = P i E f f~ (4) j=l uj 0 j=l uj 0

where nc is the total number of vertices of the compression zone; P (u)=v(u)-v. is the linear equation of the boundary line, with v. being the v-coordinate of the neutral axis; pF1 when P~ci >0, and p ~ -1 when Pzci<O; Unc+l--Ul and Vnc+l--V 1.

The total stress resultants contributed by the concrete of the whole cross-section are then given by a summation over all subsections

n s +n o n s +n o n s +n o

Nzc = ZciNzci, Muc= ZciMuci, Mvc= ZciMvci (5) i=1 i=1 i=1

in which c~ l for a solid subsection and cu -1 for a negative subsection. The bending moments of the concrete about the x- and y-axes can be easily obtained by coordinate transformation

Mxc=MuccOS O.-Mvcsin 0~, Myc=Mucsin O.+MvccOS O. (6)

The fiber element method (Mirza and Skrabek, 1991) is used to calculate the stress resultants carried by the structural steel and the steel reinforcement. The steel section is subdivided into small areas referred to as fiber elements and the reinforcing bars are treated as individual fibers. Both the structural steel and the reinforcing bars are assumed to be elastic-perfectly plastic. The stress resultants of the whole composite cross-section, axial force and bending moments about the x- and y- axes, can then be written as

mr ms

N z =Nzc+~-'~croAo+ff'o',jAsj j=l j=l

mr ms

M x = Mxc - ~" (O'rj -Crcj)ArjYrj - ~ (o',j -Crcj)Asjysj (7) j=l j=l

mr ms

My = My c + Z ( O'rj -- O'cj ) m r j X rj + Z ( O'sj - - O'cj ) A s j X sj

j=l j=l

Design of Composite Columns of Arbitrary Cross-Section 447

where mr and ms are the numbers of reinforcing bars and steel fibers into which the structural steel is discretized, Crrj, Crsj and ere are the stresses of the reinforcement, the structural steel and the concrete at the center of thejth bar or steel fiber. If vj---Vn <0, Crcj =0.

Iterative Solution Procedure

For a given composite cross-section subjected to an axial load Nzd at o and bending moments Mxd and Myd about the x- and y-axes respectively (Figure 1), the depth and orientation of the neutral axis dn and On can be determined by the following iterative procedure:

(1) Initial values of On and d, are first specified, for which the axial force Nz is calculated using the first expression of Eqn. 7.

(2) The calculated axial force Nz is compared to the design value Nza and iteratively adjusted using the following equation until Nz is equal to Nzd with a given tolerance:

dn '-dn dn, k = d n + (Nzd - Nz) (8)

N~'-Nz

in which Nz' and Nz are the axial force capacities calculated with the neutral axis depths dn' and dn respectively, with Nz' being greater than the design value Nzd and Nz being smaller than Nzd.

(3) The bending moments Mx and My are found using the second and third expressions of Eqn. 7, and then the angle Ctm=arctg(My/Mx) is determined.

(4) The value of am is compared to the design value ama=arctg(Myd/Mxd) and iteratively adjusted using the following equation:

O,'-On On, k = O n + (Ctmd - - a m ) (9)

a m t--a m

in which am' and am are the inclination of the resultant bending moment calculated with the neutral axis orientations 0,,' and 0,, respectively, with am' being greater than the design value Ctnd, and am being smaller than Ctmd. Steps (1) to (3) are repeated until O~m and amd are identical within a given tolerance.

(5) If the task is to check the adequacy of an existing design, the calculated resultant bending moment Mr is compared to the design value Mra~(Mxd2+My2) 1/2. If Mr >--Mrd, the cross-section is adequate, otherwise the structural steel and/or reinforcing bars should be increased and/or the section enlarged.

(6) If the task is to design a section, all structural parameters except the reinforcing bars are first specified. Only steel bars of identical properties and size may be used, and the bar diameter can be determined iteratively starting from an initial assumed value. For any value of the bar diameter, the corresponding resultant bending moment capacity can be found and the bar diameter is iteratively adjusted using the following equation:

~bk=~b + r (Mrd_Mr) (10) M r ' - M r

where Mr' and Mr are the resultant bending moments resisted by the section with bar diameters ~b' and r respectively, with Mr' being greater than the design value Mrd and Mr being smaller than Mrd. The required bar diameter is found when Mr is equal to Mra within a given tolerance.

448

NUMERICAL EXAMPLES

S.F. Chen et al.

A computer program has been developed based on the method presented in this paper. Two numerical examples are presented below to demonstrate the validity, accuracy and capability of the proposed method. Further numerical examples can be found in Chen et al. (1999). Tolerances used for the axial load, the inclination of resultant bending moment and the resultant bending moment are 10 -5, 10 -4 and 10 -5 respectively. The larger tolerance for the inclination of resultant moment was used as it was found to converge more slowly than the other two parameters.

Rectangular Columns with Asymmetrically Placed Steel H-Section

Two rectangular cross-sections with asymmetrically placed structural steel are shown in Figure 2. The maximum load-carrying capacities of the cross-sections under uniaxial loading at different eccentricities were determined in tests by Roik and Bergmann (1990). The cube strength of concrete and yield stresses of structural steel and reinforcing bars are listed in Table 1.

Figure 2: Rectangular cross-sections with asymmetrically placed structural steel

TABLE 1 MATERIAL PROPERTIES (ROIK AND BERGMANN, 1990)

Specimen Cross- Section

V l l , V12, V13 V1

V21, V22, V23 V2

Lk fsflange fs.web A,test _ (N/ram 2 ) (N/mm ~) (N/mm 2 ) fN/mm ~ )

37.4 206 220 420

37.4 255 239 420

TABLE 2 LOAD CARRYING CAPACITIES AND COMPARISON WITH TEST RESULTS

Specimen er(mm) Nz.comp(kN) Nz.test(KN) (Nz.comp-Nz.test)/Nz.test

VII 0 3608 3617 -0.25%

V 12 -40 2654 2825 -6.05% V13 100 1937 1800 7.61% V21 0 2880 2654 8.52% V22 -40 2107 1998 5.46% V23 100 2036 1706 19.34%

Design o f Composite Columns o f Arbitrary Cross-Section 449

In order to compare the present results with the test results, all material partial safety factors were taken to be unity in the analysis. The stress-strain curve from Eurocode 4 (1994) for concrete was used, that is, fcc = 0.85fck, c0 = 0.002 and ecu=0.0035 in Eqn. 2. Table 2 gives the computed load- carrying capacities of the six specimens and their comparison with the test results, where er is the load eccentricity in the Y-direction with reference to the geometric centroid of the cross-section (Figure 2). It is seen that the computed values agree closely with the test results. This demonstrates the accuracy and validity of the proposed numerical method.

Column of Asymmetric Polygonal Cross-Section

A composite column cross-section, as shown in Figure 3, is subject to the design loads as follows: Nzd = 4320.5 kN, Mxd = 950.15 kNm, Mrd = -577.49 kNm

in which X and Y are the geometric centroidal axes of the cross-section. The size and layout of the encased structural steel and the distribution of the steel reinforcing bars are shown in Figure 3. The reinforcing bars are assumed to have the same diameter of 12 mm. The task here is to check if the cross-section has an adequate capacity to carry the given design loads or to determine the required bar diameter if this section is inadequate.

The stress-strain curve of Eqn. 2 for concrete (Eurocode 4, 1994) is used in the calculation, with fcc=O.85fck/Yc, c0=0.002 and 6cu=0.0035. The specified strengths and safety factors are taken as follows:

fck=30 N/mm2,fs=355N/mm2,fy=460 N/mm2; yc=l.5, ?~=1.1 and yr=l.15

(1) Checking the adequacy of the pre-defined cross- section

Coordinates of the plastic centroid o of the predefined cross-section (Figure 3):

Xpt =-25.962 mm, Ypt =-27.694 mm Design loads with reference to the plastic centroidal axis system xoy (Figure 3): Nza = 4320.5 kN, area = 150.739 ~ Mra-=951.966 kNm Calculated load carrying capacity of the cross- section:

Nz = 4320.5 kN, am = 150.739 ~ Mr=774.575 kNm As Mr<Mra, the pre-defined cross-section is inadequate.

Figure 3: Cross-section of a column

(2) Deisgn of the required bar diameter

The program was instructed to find the required bar diameter for the corner column to achieve an adeqaute resistance for the design loads. The required bar diameter was found to be ~eq=20mm, with which the cross-section has a load carrying capacity of Nz=4320.5kN, Ctm=150.739 ~ and Mr = 951.972 kNm.

CONCLUSIONS

An iterative numerical method for the rapid design of short biaxially loaded composite columns of arbitrary cross-section has been presented in this paper. In the proposed method, the plastic centroidal axes of the cross-section are taken as the reference loading axes, which ensures the uniqueness and


convergence of the solution of the neutral axis orientation for all cross-sectional conditions. Six composite column specimens of rectangular cross-section with asymmetrically placed structural steel were first analyzed. The results agree well with the test results. The bending moment capacity of a composite column of asymmetric polygonal section is then checked and the required bar diameter is successfully designed. Quick convergence was observed in all cases studied herein. The proposed method has thus been shown to be effective and accurate, and is directly applicable in practical design.

ACKNOWLEDGMENTS

The work presented in this paper is the result of a collaborative effort between the Department of Civil Engineering of Zhejiang University and the Department of Civil and Structural Engineering of The Hong Kong Polytechnic University. The authors wish to thank The Hong Kong Polytechnic University for its financial support provided through the Area of Excellence Scheme and Dr. Y. L. Wong for helpful discussions.

REFERENCES

ACI Committee 318 (1992). Building Code Requirements for Structural Concrete (ACI 318-92) and commentary (A C1318R-92) , ACI, Detroit, MI.

AISC-LRFD (1993). Load and Resistance Factor Design Specification for Structural Steel Buildings, AISC, Chicago, IL.

Brondum-Nielsen T. (1985). Ultimate Flexural Capacity of Cracked Polygonal Concrete Sections under Biaxial Bending. ACI Struct. J., 82:6, 863-870.

Chen S. F., Teng J. G. and Chan S. L. (1999). Biaxial Bending Design of RC and Composite Columns of Arbitrary Cross-Section. To be published.

E1-Tawil S., Sanz-Picon C. F. and Deierlein G. G. (1995). Evaluation of ACI 318 and AISC (LRFD) Strength Provision for Composite Beam-Colunms. J. Construct. Steel Res., 34, 103-123.

Eurocode 4, (1994). Eurocode 4: Design of Composite Steel and Concrete Structures, Commission of the European Communities, British Standards Institution, London.

Johnson R. P. and Smith D. G. E. (1980). A Simple Design Method for Composite Columns. Struct. Engr., 58A:3, 85-93.

Lachance L. (1982). Ultimate Strength of Biaxially Loaded Composite Sections. J. Struct. Div., ASCE, 108:10, 2313-2329.

Manual of Steel Construction (1986). Load and Resistance Factor Design, 1st Ed., Amer. Inst. Steel Construct., Chicago, Ill.

Mirza S. A. and Skrabek B. W. (1991). Reliability of Short Composite Beam-Clolumn Strength Interaction. J. Struct. Engrg., ASCE, 117:8, 2320-2339.

Munoz P. R. and Hsu C. T. T. (1997). Biaxially Loaded Concrete-Encased Composite Columns: Design Equation. J. Struct. Engrg., ASCE, 123:12, 1576-1585.

Roik K. and Bergmann R. (1984). Composite Columns-Design and Examples for Construction. Composite and Mixed Constructions, Proc. U. S.-Japan Joint Seminar, ASCE, Univ. Washington, 267-278.

Roik K. and Bergrnann R. (1990). Design Method for Composite Columns with Unsymmetrical Cross- Sections. J. Construct. Steel Res., 15, 153-168.

Rotter J. M. (1985). Rapid Exact Inelastic Biaxial Anlysis. J. Struct. Engrg., ASCE, 111:12, 2659- 2674.

Yau C. Y., Chan S. L. and So A. K. W. (1993). Biaxial Bending Design of Arbitrarily Shaped Reinforced Concrete Column. ACI Struct. J., 90:3, 269-278.

Yen J. R. (1991). Quasi-Newton Method for Reinforced Concrete Column Analysis and Design. J. Struct. Engrg., ASCE, 117:3, 657-666.

EFFECTS OF LOADING CONDITIONS ON BEHAVIOUR OF

SEMI-RIGID BEAM-TO-COLUMN COMPOSITE CONNECTIONS

Y. L. Wong, J. Y. Wang and S. L. Chan

Department of Civil and Structural Engineering,

The Hong Kong Polytechnic University, Hong Kong

ABSTRACT

In the design and analysis of moment resistance framed structures, the internal joints are generally

modeled as two separate connections with independent characteristics. However, in the case of

composite frames, the interaction between two connections located on both sides of the column can be

significant, especially under the action of anti-symmetrical moments.

In this paper, the experimental results of four composite beam-column internal joints are presented.

The specimens were constructed using either flush endplate connections or partial depth endplate

connections. Three different loading conditions were considered, namely, (1) hogging moment

developed at connection of one side of the joint; (2) sagging moment developed at connection of one

side of the joint; (3) hogging moment developed at one connection and sagging moments developed at

the other connection simultaneously. Test results indicate that the connection subjected to loading case

(1) or (2) has the moment capacity over 20% higher than the corresponding value developed under the

loading case (3).

Finally, the experimental results are compared with the calculations of an advanced component-based

model. It is demonstrated that the analytical results were in good agreement with the experimental

results.

KEYWORD

Composite, End-plate connection, Loading condition, Unbalanced moment, Component-based model.

451

452 Y.L. Wong et al.

INTRODUCTION

In design practice, the moment-rotation properties of connections located at two sides of internal joints

have been modeled as two separate connections with independent. The influences of loading

conditions are often ignored. However, in the case of composite frames under lateral loads, the

interaction between connections located in interior joints, especially under the action of unbalanced

moments may be significant. Previous experimental study of composite frames (Leon et al 1987)

indicated that the positive moment capacity of an exterior connection was about 25% higher than that

of the interior connection, while their capacities under negative moments were comparable. This is

mainly because the improvement in strength and stiffness of composite structures, as compared with

the steel counterparts, is achieved through the composite action of the slab. However, the linkage

between the concrete slab and the column has been considered as a weak point and some researchers

(Zandonini 1989, Kato and Tagawa 1984) emphasized the importance of interaction between column

and slab. However, up to date, systematic study on this aspect is not available.

The objective of this study is to experimentally quantify the interaction between connections under

unbalanced moments. Theoretical analysis using a component-based model was also carried out. The

contents of this paper are arranged as follows. In the first section, an experimental programme was

presented. In the next section, a component-based model is introduced. The test results and the

verification of the analytical results are presented in the third section. Finally, conclusions are drawn at

the end of the paper.

SPECIMEN DESCRIPTION AND TEST ARRANGEMENT

Four specimens (SPFM1, SPPM2, SPMM3 and SPMM4) were constructed and tested. Connection

types of flush endplate and partial depth endplate were selected (see Figure 1), where 'L' and 'R'

denote the left and right connections respectively. All test specimens comprised two 305x127UB37

steel beams. The conventional metal decking floor system comprising a concrete slab supported by a

profiled steel decking was chosen. The concrete slab was 1200mm width and 130mm overall depth

filled with normal weight concrete with design strength of 40 N/mm 2. The steel decking was 1.0mm

BONDEK II. Headed connectors of 19mm diameter and 100mm pre-welded length were adopted with

Figure 1 Steel connection details

Effects of Loading Conditions on Behaviour of Beam-Column Connections 453

one connector per trough. Longitudinal reinforcement was 10T10, and the corresponding ratio was

0.86% of the concrete area above the ribs of the decking. All bolts used in connections were 20mm

diameter of grade 8.8 and tightened to 180 Nm by a hand torque wrench to provide the comparability

and consistency of the specimen.

Figure 2 Test set-up

Figure 3 Different loading conditions and beam-end supports

The general arrangement of the test set-up including a reaction flame and a loading system is

illustrated in Figure 2. The column of a test specimen is hinge-connected to a ground-beam at the


bottom, while the top of the column is connected with the actuator by low-friction swivel hinge

bearings, which allow free rotation of the actuator as the specimen deforms. Load cells were fixed at

the beam far ends through rigid bars to record the reaction forces for calculating the connection

moments. The beam length from the column centerline to the beam-end support is 2.0m. The overall

height of the column measured from the bottom hinge to the center of the actuator is 2.165m. The

connection rotation is defined as the change in angle of steel beam relative to the column centerline.

The lateral load was applied to the specimen at the column head by a computer-controlled double-

acting hydraulic actuator with a maximum capacity of + 500kN and an available stroke up to 250mm.

Different loading conditions (see Figure 3) were considered, namely, (1) only hogging moment

achieved in connection (Case 1); (2) only sagging moment achieved in connection (Case 2); (3)

hogging and sagging moments achieved at both connections simultaneously (Case 3).

COMPONENT BASED MODEL OF COMPOSITE JOINT

Figure 4 shows the modeling of an interior composite joint. The rigid bars on the left and right sides

represent the boundary between connections and composite beams. The central rigid body coincides

with the centroidal axis of the column. The rigid body and bars are connected by a continuum of

nonlinear distributed springs in axial compression or tension to simulate the behaviour of connection

components. Four types of elements are used to represent the connection components, including: (1)

elements corresponding to beam flanges with endplate attached; (2) elements corresponding to beam

webs with endplate attached; (3) elements of reinforcement; (4) elements of concrete slab. The

idealized configuration for the tension and compression regions in concrete slab is illustrated in Figure

5. The constitutive rule and detailed derivation of properties for those elements may refer to Wang

(1999).

Rigid body d bar

,,lement

�9 Y~

am 'ange 1 !! !

(a) Connection model (undeformed) (b) Connection model (deformed)

Figure 4 Modeling of composite joint


Figure 5 Idealized configuration of slab

Figure 6 Force equilibrium in connections

The discrete bodies of left and right connections are shown in Figure 6. For the sake of easy and clear

expression, it is assumed that the left connection is under sagging moment and the right one is under

hogging moment. According to the force equilibrium, we have,

nsR n sL n con

E f i + F r + F; = N. ; Z f + Z f + Fr = NL (1) i=1 i=1 i=1

nsR nr

~"~fi(YoR - Y i ) + ~ fi(YoR - Yi) + M*~ = M R (2) i=1 i=1

nsL nson

~-~fi(YoL - Y i ) + ~-'~f (YoL - Y i ) + Mr = ML (3) i=1 i=1

456 Y.L . W o n g et al.

where nsL and n,R are the number of elements in the left and right steel connections respectively; rt r

and n co . are element numbers of reinforcing bars and concrete slab respectively; YoL and Yon are

neutral line positions in left and right connections respectively; N z and N R are axial forces in left and

right beams. F* indicates the force transferred from the opposite connection, and the additional

moment then achieved is denoted by M*. Those values are calculated as follows,

n r

Fr =Fr = ~ ~ f i (4) i=I

nr M; = ~--'~f/(YoL - Y i ) (5)

i=l b; ....

F~ = be9" + b----~ 2 f (6)

M;= b; beff q- b~ Ei:I f / ( Y o . - Yi ) (7)

in which, bef f and b~ are the effective widths of concrete slab corresponding to the contacted column

flange and fill-in concrete and the opposite side respectively.

The force f / o f each element can be directly obtained from the constitutive rule of elements.

f i = f i ( d i ) (8)

whereas the element deformation d i in the left and right connections is expressed as"

di =(Y0L --Yi)OL (i = 1, 2 . . . . . nsL ) & ( i = 1, 2 . . . . . nco. )

di =(YOR -- Yi)On ( i = 1, 2 . . . . . nsR )

(9)

(10)

Special attention should be paid to the reinforcement elements, where d i is calculated as:

di = OR(Yon -- Y i ) + OL (YoL -- Y i ) ( i = 1, 2 . . . . . rt r ) (11)

Since the nonlinear constitutive relations of connection elements, a numerical procedure with iteration

is needed to determine the moment-rotation curve of connection(s).

EVALUATION OF EXPERIMENTAL AND MODELING RESULTS

Table 1 lists the experimental ultimate moments of connections with various configurations and under

different loading conditions. As expected, flush endplate connections achieve higher resistance than

partial depth endplate connections. It is worth to note that the connection resistance without concerning

the interaction (Case 1 and 2 for specimens SPMM3 and SPMM4 respectively) was over 20% higher

than that encountered by the influence from the opposite connection (Case 3 for specimens SPFM 1 and

SPPM2).


T A B L E 1

C O M P A R I S O N OF U L T I M A T E M O M E N T S U N D E R D I F F E R E N T L O A D I N G C O N D I T I O N S

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Loading Specimen Mult ~ Flush Partial Depth ( a )/( b ) \ ~ ~ (a) (b) (c)

I l m l

Case 1 SPMM3 ( i ) -239.01 -172.85 1.38

Case 2 SPMM4 ( ii ) 227.88 171.23 1.33

Case 3 SPFM1L/SPPM2L ( iii ) 175.03 128.89 1.36

Case 3 SPFM1R/SPPM2R (iv) -171.69 -142.11 1.21

( i)/( iv ) ( v ) 1.39 1.22 -

( ii )/( iii ) ( vi ) 1.30 1.33 -


Figure 7 Moment-rotation curves of specimens

The complete moment-rotation curves of test specimens obtained experimentally and analytically are

illustrated in Figure 7. In general, good agreement between the experimental and theoretical results has

been achieved, at least up to the maximum moment levels. It is also evident that the proposed model

can simulate the interaction effect existing between composite connections located at two sides of the

column. The dashed line in Figure 7(a) to (d) represents the moment-rotation performance of a

connection with the absence of the opposite connection.

CONCLUSION

Experimental work and rational analysis based on an advanced component-based model indicate that

the interaction effects between composite connections of interior joints are significant when they are

subjected to the unbalanced moments. This effect should be considered in the design of unbraced

composite frames.

Reference

Kato B. and Tagawa Y. (1984). Strength of composite beams under seismic loading, Composite and

Mixed Construction (edited by Roeder, C. W.), Proc. of the US/Japan joint seminar, ASCE, 42-49.

Leon R. T., Ammerman D., Lin J. and McCauley R. (1987). Semi-rigid composite steel frames,

Engineering Journal AISC, 4th quarter, 147-155.

Wang J. Y. (1999). Nonlinear analysis of semi-rigid composite joints under lateral loading m

Experimental and theoretical study, Ph. D thesis, the Hong Kong Polytechnic University.

Zandonini R. (1989). Semi-rigid composite joints, Structural Connections: Stability and Strength

(edited by Narayanan, R.), Elsevier Applied Science, 63-120.

S T E E L - C O N C R E T E C O M P O S I T E

C O N S T R U C T I O N W I T H P R E C A S T C O N C R E T E H O L L O W C O R E F L O O R

D. Lam l, K. S. Elliott 2 and D. A. Nethercot 2

1School of Civil Engineering, University of Leeds, Leeds, LS2 9JT, UK

2 School of Civil Engineering, University of Nottingham, Nottingham, NG7 2RD, UK

ABSTRACT

Precast concrete hollow core floor units (hcu) are widely used in all types of multi-storey steel framed buildings where they bear onto the top flanges of universal beams. In the current UK market, hcu's account for about 50% of all floors used in steel framed buildings, with an annual production of about 4 million square metres worth s The steel beam is normally designed in bending in isolation from the concrete slab and no account is taken of the composite beam action available with the precast units. A programme of combined experimental and numerical studies has been conducted, with the aim of deciding on a suitable approach for the design of composite steel beams that utilise precast concrete hollow core slabs. The results show that the precast slabs may be used compositely with the steel beams in order to increase both flexural strength and stiffness at virtually no extra cost, except for the headed shear studs. For typical geometry and serial sizes, the composite beams were found to be twice as strong and three times as stiff as the equivalent isolated steel beam. The failure mode was ductile, and may be controlled by the correct use of small quantities of tie steel and insitu infill concrete placed between the precast units.

KEYWORDS

Composite, concrete, hollow core, precast, push-off, shear studs, steel, structural design

INTRODUCTION

An effective way to improve structural efficiency is to utilise the favourable structural properties of the basic components and to combine them in a manner that leads to maximum performance in a safe and cost effective way. Composite action between steel beams and concrete slabs through the use of shear connectors is responsible for a considerable increase in the load-carrying capacity and stiffness of the steel beams, which when utilised in design, can result in significant savings in steel weight and / or in

459

460 D. L a m e t al.

construction depth. These economies have largely accounted for the dominance of composite steel frame construction in the commercial building sector in the UK in recent years. Composite construction of steel frames with profiled steel decking to support floor slabs is now common in multi- storey construction, but the use of precast prestressed concrete hollow core units (hcu' s) in conjunction with the steel frame to provide composite action is relatively new. Figure 1 shows the composite beam with precast hollow core slabs.

Figure 1: Composite beam with precast hollow core slabs

Hcu's are now the most widely used type of precast floor in Europe; annual production is about 20 million m 2, representing 40 to 60 per cent of the precast flooring market. This success is largely due to the highly efficient design and production methods, choice of unit depth and structural efficiency. The design of dry cast hcu originated in the United States in the late 1940s following the development of the high strength strand that could be reliably pre-tensioned over distances of 100m to 150m. This coincided with advancements in zero slump (hence the term 'dry') concrete production, which inevitably led to factory made hcu. Units have longitudinal voids and are produced on a long prestressing bed either by slip form or extrusion and are then saw cut to length. The degree of prestress and the depth of unit are the two main design parameters. The depth ranges from 150 to 400mm, with the performance limited to a maximum span / depth ratio of around 50, although 35 is more usual for normal office loading conditions.

Although hcu's are widely used in all types of multi-storey buildings and account for approximately 50% of all floors used in steel framed buildings, the steel beam is normally designed in bending in isolation from the hcu' s slab and no composite beam action is considered in design with the hcu' s. The main reason for this is the uncertainty over the ability of the hcu's to satisfactorily transfer the shear and compressive forces. Whilst the potential to generate worthwhile composite action is not in doubt, little research into this problem has previously been undertaken. Recent work by the authors (1) is aimed at rectifying this by carrying out a systematic study into the behaviour of the composite beam in bending.

To study the flexural behaviour of the hcu slabs and steel beam composite construction, the major issues that were addressed were: (a) the compression behaviour of the hcu slabs, and (b) the transfer of the horizontal shear forces between the steel beam and the concrete slab. To achieve this, full scale bending tests were supplemented by: (a) horizontal eccentric compression tests and (b) horizontal push-off tests, as shown in Figure 2. In this study, 3 no. of full scale 6.0m long composite beam tests were supplemented by 12 shear stud push-off tests and 5 no. composite slab compression tests. In addition to the experimental work described, analytical studies using the finite element technique were

S tee l - Concrete Composite Construction with Concrete Hollow Core Floor 461

employed to carry out parametric studies. This paper concentrates on presenting the full scale bending tests and the numerical simulation. Suitable references are also made to the supporting scientific work.

Figure 2: Simplification of testing regime for [a] full scale bending test [b] isolation eccentric compression slab tests [c] isolated push-off tests

TEST SET UP

Three full scale bending tests comprised a 356 x 171 x 51 serial size $275 UB loaded in 4-point bending over a 5.7 m simply supported span as shown in Figure 3, with 150 mm deep x 1200 mm wide hcu's connected through 125 mm high x 19 mm diameter 'TRW Nelson' headed studs at 150 mm spacing(s) along the full length of the beam, giving 11 studs between the support and load positions. A 6.0m nominal length universal beam, with a 150ram thick hcu will be capable to carry a general office floor of 6.0m x 16.0m space free from columns. The characteristic cube strength for the hcu's is taken as 50 N/mm 2. The specimens were simply supported over a span of 5.7 metres and loaded by two point loads spaced symmetrically at 1.5 metres from each end support. All three specimens were similarly constructed, with the differences being the transverse reinforcement and insitu joint. Web stiffeners were used to eliminate local failure due to web buckling or flange yielding at the loading position.

Figure 3: Plan view of the beam test

462 D. L a m et al.

The slabs were placed directly on to the UB with a minimum bearing of 50 mm. The gap between the ends of the hcu' s was carefully monitored during placing to ensure a 65 mm gap width was maintained throughout. The tops of four cores per hcu, i.e. 2 nd, 4 th, 8 th and 10 th core, were left open for a length of 500mm to allow the placing of transverse reinforcement, giving an average bar spacing of 300 mm. Figure 4 shows the specimen before the insitu infill was cast.

Figure 4: Test specimen before casting of the insitu infill

Following the horizontal compression tests ~2) and push-off tests ~3), it was decided that transverse reinforcement ofT8 and T16 bars should be used for the full scale bending tests. T16 bars were used in test CB 1 to prevent tensile splitting and to confine the concrete slab from splitting failure, while T8 bars were used in test CB2 to allow tensile splitting to take place in a controlled manner. Insitu concrete with the design cube strength of 25 N/mm 2 was placed into the longitudinal and transverse joints and opened cores and compacted using a 25mm diameter vibrating poker to form the composite slab.

In addition, specimen CB3 with debonded joints between the insitu and precast concrete was tested to observe the effect of a debonded insitu joint due to shrinkage. Two sheets of polythene were cast between the insitu concrete infill and the hcu to ensure a proper separation between the insitu infill and hcu, so that bonding and aggregate interlocking between the insitu infill and hcu could not be achieved, see Figure 5. Transverse reinforcement of T8 bars was chosen for this test; identical to the arrangement of Test CB2.

Figure 5: Composite beam with debonded insitu joint


TEST R E S U L T S

The results of the bending tests are given in Table 1 and Figure 6, where the increases in moment capacity and flexural stiffness of the composite beam compared to the bare steel UB are apparent. The elastic neutral axis of the composite section normally lies close to the interface between the steel and the concrete. As the bending moment increased, the bottom flange of the steel beam yielded and the neutral axis moved towards the compression zone, causing tensile cracking at the underside of the slab. When the stress at the outer surface of the concrete slab reached approximately 0.67fcu, spoiling of the concrete began and the ultimate strength of the section was then fully mobilized. As the curvature of the section is further increased, the load carried remains approximately constant and crushing of the slab occurs. Failure of the shear connectors may also occur, which would reduce the load carrying capacity of the composite section. No slippage between the slab and the end of the UB occurred for loads in the working load range. However, slip does have a considerable influence on the development of the ultimate moment capacity.

TABLE 1 BEAM TEST RESULTS

Test

Reference

Tie-steel area

Insitu cube

Max. test moment,

ratio (%)

strength (N/mm 2)

MR (kNm)

Mid-span Ratio Initial deflection of flexural

at MR MR / stiffness, (mm) MR(steel) Ki

(kNm/mm)

CB1-T16 0.45 32.0 496 32

CB2-T8 0.11 25.5 470 35

CB3-T8* 0.11 26.5 345 27

356•215 UB - - 245 51 *included polythene at insitu-precast interface.

2.02

1.92

1.42

1.00

End slip at max.

test moment

(mm)

25.5 0.4

25.4 2.6

18.9 5.9

7.7

Note: Span/360 = 15.8ram(Deflection limit, BS5950.)

Figure 6" Applied moment vs. vertical mid-span deflection curves of bending tests

464 D. Lam et al.

The sudden reduction in strength in tests CB 1 & 2 was due to the fracture of the shear studs at one end of the beam. The span / deflection ratio when this occurred was about 165: 1, i.e. much larger than the allowable limit of 360:1 used in limit state design. Failure of test CB3 was due to concrete failure around the shear studs.

In test CB 1, the first crack was observed at an applied moment of 342 kNm. This moment of 342 kNm is about 0.69 times the ultimate strength of test CB 1 and may conveniently be taken as the working load. This caused the neutral axis to move towards the compression zone, which in turn resulted in tension and cracking in the soffit of the precast slab. As the load was further increased, yielding of the steel section and cracking in the underside of the hcu extended over the full length of the slab, with a gradual reduction in stiffness. At the applied moment reached 490 kNm, i.e. twice the ultimate capacity of the bare steel beam, the sudden fracture of several shear studs precipitated a rapid reduction of load. No yielding or bond failure was not detected in the T16 transverse reinforcement - stresses were less than 20% of the yield stress at failure.

In test CB2, the deformation was linear up to 245 kNm. Reduction in stiffness continued with yielding in the steel section and extended cracking in the hcu. A maximum load plateau was reached at 470 kNm with continuous deflection. Yielding of the transverse reinforcement occurred as the load reached the maximum. Tensile splitting occurred in the top surface of the concrete slab due to yielding of the transverse bars, causing concrete failure around the shear studs and a gradual reduction in load carrying capacity. There was fracture of some of the shear studs and failure at the interface by the crushing of concrete around the headed studs.

The main difference caused by the introduction of the pre-cracked joint in test CB3 was the position of the neutral axis from the start of the test, which was located 20 mm below the steel concrete interface, indicating a reduced effective breadth of concrete slab caused by the pre-cracked joint. Although deformation remained linear up to 150 kNm, the position of the neutral axis moved from 20 mm to 58 mm below the steel concrete interface which suggested further reduction of the effective concrete section. At the ultimate moment of 345 kNm, i.e. some 35 % less than in the previous tests, the transverse reinforcement was fully yielded, leading to further tensile splitting of the slab.

The stiffness of the pre-cracked specimen CB3 was approximately 0.71 of that in the former tests, indicating a reduced effective breadth when the interface bond is destroyed. A similar result was found in the eccentric compression tests (z), where the resistance of the pre-cracked specimen was 0.72 times that in the normally bonded specimens, also indicating a reduced concrete section. The ultimate moment in Test CB3 was, by chance, equal to the final post-fracture resistance in tests CB 1 & 2, indicating that in spite of the different modes of failure equilibrium is reached at the same level.

NUMERICAL SIMULATION

A numerical model based on the finite element method has been developed using ABAQUS (4). A two- dimensional model of the composite steel-concrete beam is shown in Figure 7. The model is set up to the same dimensions as the full scale bending test specimens. Although a 2-D model has its limitations when dealing with a 3-D structure, (the 2-D model used precluded the 3 rd dimensional effect where certain failure mechanisms might be critical), it is extremely useful when the modelling is admissible on account of economy (computational time, input/output), ready visualization and the relative ease with which parametric studies may be conducted. Three types of elements were used: 4-node plane stress elements to model the steel beam, 8-node concrete elements for the concrete slab and spring element for the shear studs. The shear studs were modelled as non-linear springs using the actual load - slip behaviour of the shear studs obtained from the experimental load vs. slip curve of the push-off tests (3). Each node of the steel element is connected to the node of the concrete element at the interface


by the spring element, i.e. at 150c/c. The test parameters, including the material properties used are identical to the ones used for the full scale bending tests.

Figure 7: Finite element mesh of composite beam model

Figure 8: Moment-deflection curves for beam tests and FE analyses

The results of the FE analyses are shown in Figure 8. These show close agreement in terms of failure moment and moment-deflection characteristics for the three composite beams analysed herein. The percentage differences between the test and FE results for CB 1, CB2 and CB3 were 1%, 2% and 7% respectively. The FE analysis accounted for all major material features and was able to model concrete cracking and crushing as well as shear stud failure, although the post failure conditions could not be followed. The results showed that the 2-D model is suitable for the analysis and can be used to carry out parametric studies (5) on composite beams.

466

CONCLUSIONS

D. Lam e t al.

Based on the authors' own extensive experimental and numerical study, it has been shown that precast slabs may be used compositely with steel beams in order to increase both flexural strength and stiffness at virtually no extra cost, except for the headed shear studs. For typical geometry and serial sizes, the composite beams were found to be twice as strong and three times as stiff as the equivalent isolated steel section. The failure mode was ductile, and may be controlled by the correct use of small quantities of tie steel and insitu infill concrete placed between the precast units. A simple 2-D numerical model has been developed and can be effectively predicted moment and deflection of the composite beam with precast hollow core slabs.

REFERENCES

1. Lam, D., 'Composite Steel Beams Using Precast Concrete Hollow Core Floor Slabs', Ph.D. Thesis, School of Civil Engineering, University of Nottingham, March, 1998.

2. Lam, D., Elliott, K. S. and Nethercot, D. A., 'Experiments on Composite Steel Beams with Precast Concrete Hollow Core Floor Slabs', submitted for publication.

3. Lam, D., Elliott, K. S. and Nethercot, D. A., 'Push-off tests on shear studs with hollow-cored floor slabs', The Structural Engineer, Vol. 76, No. 9, 1998, pp 167-174.

4. ABAQUS user manual, Version 5.3.1 (1994), Hibbitt, Karlsson & Sorensen, inc., 1080 Main Street, Pawtucket, RI 02860-4847, USA.

5. Lam, D., Elliott, K. S. and Nethercot, D. A., 'Parametric Study on Composite Steel Beams with Precast Concrete Hollow Core Floor Slabs', submitted for publication.

ACKNOWLEDGEMENTS

The authors acknowledge the support provided by the EPSRC, Bison Floors Ltd., UK and the skilled assistance provided by the technical staff in the Civil Engineering Laboratory at Nottingham University.

Testing and Numerical Modelling of Bi-Steel Plate Subject to Push-Out Loading

Simon K. Clubley Robert Y. Xiao

Department of Civil & Environmental Engineering University of Southampton, UK

ABSTRACT

Bi-Steel panels are a newly patented composite construction system developed by British Steel Plc. They comprise of steel plates permanently coupled by a matrix of transverse friction welded rods. The shear strength and deformation capacity of the Bi-Steel unit subject to push out load is discussed in this paper. Numerical modelling by the use of finite element analysis has been conducted on Bi-Steel plates with and without in-filled concrete. The results of non-linear analysis are compared with experimental data. Both material and geometrical non-linearity were considered in the computing analysis. A design model has been suggested for deformation calculation due to shear action.

1. INTRODUCTION

Performance of a composite steel and concrete structure is dependent upon the efficient interaction and effective transfer of shear between these two materials. Traditional shear connector design includes provision of adequate resistance to section uplift in addition to longitudinal slip. There are a number of types of shear connectors being used in design such as welded shear studs and mechanical fixed shear connectors. It is important to note that ultimate strength design of shear connectors assumes individual studs has sufficient ductility to redistribute load over the array so that all fail as a group under shear actions. In the absence of heavy concentrated loads, connectors are spaced uniformly between supports and sections of maximum bending moment.

Bi-Steel adopts the high-speed friction weld process to attach shear rods to both steel panels as shown in Figure 1. The design philosophy of Bi-Steel is systems modular based. It is envisaged that minimal work for shear connections should be carried out on site, with prefabrication and preparation essentially carried out in workshops. Concrete will be poured once panels are delivered and positioned on site. In situ panels may be connected together by either bolting or welding. Large hydrostatic pressures during pumping can be sustained due to the density of shear connectors. This system has extremely strong potential to be used for many different types of structure.

In collaboration with British Steel an extensive laboratory test programme was undertaken at Southampton to establish the ultimate shear strength of Bi-Steel specimens subject to push out load. This will be compared with traditional shear connector push-out testing. Research is available by Moy, Xiao & Lillestone (1), Oehlers & Sved (2), Kalfas, Pavlidis & Galoussis (3), Uy & Bradford (4) & (5) and Schuurman & Stark (6). The performance of shear studs in high and normal strength concrete was examined in the work of An & Cederwall (7). Prediction of expected shear stud strength with associated design formulae will be proposed as for the standard composite construction suggested by Oehlers & Johnson (8). Research conducted into double skin composite panels using profiled sheeting in Hossain & Wright (9) by numerical modelling and laboratory test data will also be examined.

467

468 S.K. Clubley and R.Y. Xiao

Matrix voids and material imperfections account for the small difference in repeatable trials of push-out tests. Generally, scatter is accounted for by assuming the lowest recorded or derived failure load. Mechanical shear connectors only exhibit very limited shear/slip capacity when used in composite construction. Analysis in Oehlers & Sved (2) and Kemp & Trinchero (10) suggests development of yield at failure loads disappears as flexural strength reduces.

Figure 1: Pre-fabricated Bi-Steel section.

This paper presents the application of the finite element analysis program Ansys and associated theory to model experimental behaviour of Bi-Steel panels including a wide range of variable geometric specifications. Subsequent evaluation will suggest theory not readily available from data collected in a laboratory test programme.

2. TEST PROGRAMME

The initial laboratory test programme commissioned by British Steel consisted of fifteen specimens. Details of all tests can be found in the confidential report to British Steel. Due to the nature of the contract, not all results of testing will be released here. There are a further twenty large specimens being planned for testing. They will be fully published separately in the future. Primary objectives of the test programme were to examine the strength and stiffness of shear connector studs when the concrete is subject to a shear action relative to the steel plates. The design of test specimens will be briefly introduced here. Plate spacing within the Bi-Steel unit was kept constant at 200mm during which five sizes of plate thickness were investigated. Shear connector size remained constant at 25mm diameter and was arranged in a regular grid matrix of 200mm in both directions.

Before commencing the test programme representative steel coupons and concrete cubes were tested from each unit to establish key material properties. Prior to concrete casting, two weldable strain gauges were spot welded to each stud. Spot welding of strain gauges is a new technique for attachment to steel rods. To simulate constraint provided by a larger panel, six 16mm diameter threaded steel studs were located in the mould and the concrete was cast. The moulds were stripped after one day and together with the cubes both left in water to cure. The 100mm cubes were removed and tested at intervals to evaluate if the concrete had achieved the required strength. Following Bi-Steel unit removal from the water a drying period of 24 hours elapsed before 50mm

Testing and Modellling of Bi-Steel Plate Subject to Push-Out Loading 469

square nuts and washers were placed on the studs and tightened. Protruding wires from the strain gauges were connected to a data logger. In addition, nine displacement transducers were placed around the unit to measure steel/concrete slip and displacement of the steel panels at the end of the studs. Before commencing specimen test each potentiometer was zeroed and calibrated in conjunction with the 1000KN capacity jack and spot-welded strain gauges. These gauges are pre- welded on shear connector rods and post welded on steel plates. A single push out load was applied by a hydraulic jack as shown in Figure 2 of the test set up.

Figure 2: Test set-up for the specimen.

Table 1 gives some typical test results for different thickness Bi-Steel plates. The load-strain relationships of steel plates are shown in Figure 3.

Figure 3: Load-strain relationship for varying plate thickness.


Table 1: Summary of typical tested specimens.

Plate Thickness

(mm)

Failure Load (KN)

Maximum Slip (mm)

Failure Mode

12 900 1.95 Brittle 10 870 4.70 Ductile 8 850 3.40 Ductile 6 820 5.63 Ductile 5 770 6.00 Ductile

3. NON-LINEAR NUMERICAL MODELLING

A comprehensive numerical programme was conducted on the tested specimens. The initial stages of numerical modelling have been concentrated on the definition of the geometric model, associated constraints and the constituent material properties. Subsequent selection of finite element types in the computer program will determine the mathematical model applied to achieve solution convergence. This in turn will govern the level of physical behavioural model accuracy. Both material and geometric non-linearity were considered. The data obtained from the laboratory is limited and does not detail behaviour at varying geometric specification. A wide range of parameters was selected to examine their influence on the shear strength of Bi-Steel plates. Numerical analysis has enabled all geometric and material properties to be varied. From this research, conclusions about Bi-Steel behaviour and corresponding theoretical modelling are suggested.

Figure 4: Meshed geometric model prior to solution.

Geometric non-linearity will occur if the model is allowed to experience large strain displacement. Following initial geometric construction the model requires a mesh of nodes and elements as shown in Figure 4. Symmetry has been used to provide efficient computing. Meshing parameters and subsequent specification can prove detrimental to the success of the solution. The level of mesh refinement will largely govern solution accuracy and the cost of associated computation time. Bi- Steel is a complicated three-dimensional problem and requires careful use of isoparametric elements to construct curved surfaces. Within the Ansys program Solid45 and Solid65 were chosen to represent steel and concrete entities respectively. Solid45 has eight nodes each with three degrees of


freedom; design is primarily that of three-dimension consideration. Rate dependent and independent behaviour is permitted through load application with tolerance for stress stiffening, large deflections and large strain capability. Tetrahedral shape definition is to be avoided in favour of prismatic assignment. Solid65 is capable of cracking in three orthogonal directions in tension, crushing in compression, plastic deformation and creep. The element is defined by eight nodes each having three degrees of freedom. All models were computed on Silicon Graphics Octane machines and solved on a mainframe Silicon Graphics Power Challenge 2000 tandem CPU array.

3.1 Model l ing without use of a contact element.

Initial computing simulation without a gap element between steel and concrete materials involves the use of entity merge commands in areas considered to be key load paths. The merge will represent full interaction at steel and concrete interface. The increase in recorded data is representative of over stiffening. For this reason it was considered appropriate to invoke area merge commands between contact surfaces of the shear connector and concrete only. Entity interface at these two points would ensure load path transference between the concrete core and steel skin across the whole plate length. The standard non-contact modelling may be considered as the benchmark for subsequent gap element as shown in Figure 5. Loading is assumed uniform over area, eccentricity is zero and an arbitrary friction constant is defined. The material specification used was typical of the material used in the laboratory tests.

Figure 5: Standard contact Bi-Steel model (concrete omitted for clarity).

Initial data analysis indicated the presence of over stiffness and the subsequent tendency to produce failure loads generally in excess of what could be reasonably expected under laboratory conditions. Considering the models individually, the most accurate models are those with reduced plate thickness, which typically experience ductile deformation. Large plate sizes appear accurate during early to mid stages of load application. All of the aforementioned modelling errors are a result of the unrealistically high surface interaction invoked at the shear connector. The result summary graph, Figure 6, indicates a key relationship between plate and connector spacing which was also later noted with increased accuracy in the contact element model study. It would appear that the greatest governing factor defining failure strength is plate spacing. Connector pitch adjusts failure load marginally, producing a group of lines clustered and collecting in a similar fashion around a locus. A change in the mode of failure appears between plate spacing one hundred and two hundred millimetres, indicating the possibility of plastic hinge formation and the corresponding relocation at varying model geometry.


Figure 6: Plate spacing, connector spacing and plate thickness influence on shear strength.

From the regression analysis in Figure 6 the following equations define failure load as a function of plate spacing. In Figure 6 the regression trendline is indicated in bold.

�9 6mm steel plates

�9 8mm steel plates F = 0.02D 2 - 10D + 390 (1)

F = 0.009D 2 - 4.6D + 77 (2)

Where: F = Failure load per shear connector (KN) D = Plate spacing (mm)

3.2 Model l ing with a contact element.

Following the benchmark analysis of the standard model the next stage of accuracy progression is the introduction of a gap element. Upon initial consideration of software documentation and element libraries it was decided that Contac52 would represent the interface between steel and concrete surfaces and provide a more realistic load path for transmission of normal and shear force. Contac52 allows numerous parameter definitions, most importantly values for normal and tangential stiffness, Kn & Ks. The gap element may be judged analogous to a spring assembly. Consequently, incorrect stiffness definition will produce either unrealistic elastic 'bounce' during load application or too large an inertia to movement caused by shear action. The introduction of tangential stiffness, Ks, is a key step regarding the modelling of energy dissipation during shear action, characteristic of a classical push-out test. The ability to successfully transfer energy and maintain the desired load path propagation is defined by modelling accuracy of physical chemical bonds and corresponding friction force generated. Upon examination of the relative slip versus load graph produced by numerical analysis shown in Figure 7, it has been noted that large geometry models slip considerable less in longitudinal direction UX than smaller specification models. Behaviour appears not subject to proportionality with respect to their global size. This performance is curious due to the expected deflection of large span beams according to classical structural


mechanics. A possible explanation for this may be due to the fact that shear connectors can be divided into two categories, either rigid or flexible. The corresponding classification provides for alternate failure mechanisms. Rigid connectors tend to exhibit higher stress concentrations in the concrete surrounding them resulting in crushing. Flexible connectors are generally more consistent in failure behaviour. Therefore, smaller size equals increased rigidity, which implies in the case of Bi-Steel, extended crushing appears locally to shear connectors. Relative slip is possible through and past crushed concrete zone with applied force not transferred to steel plates fully due to reduced surface area. Consequently, relative slip of concrete versus steel increases. In contradiction, large span shear connector equals increased flexibility producing reduced concentrated local crushing. Traction force and corresponding friction force increases, which implies passage through crushed matrix reduces.

Figure 7: Relative slip versus load comparison for varied spacing of 6mm thick plate.

When plate spacing is increased to three hundred millimetres or greater, all regions of maximum stress intensity are generated at weld interface with considerable local problems of concrete crushing. In addition, the nearest load path is the adjacent steel plate resulting in large UZ wave displacement particularly evident at large connector spacing. The concrete core has now ceased to become an effective load path. Subsequently, smaller plate sizes are capable of displaying ductile deformation before ultimate load as indicated at the weld interface. Therefore, the curve reduction noted is smaller, but large plate thickness inhibit ductile behaviour due to increased local stiffness which in turn promotes earlier failure at the weld interface with regression of a plastic hinge into the shear connector. Connector spacing of one hundred millimetres or less is very unlikely to deform in a UZ wave shape even for small plate thickness at high loads. This factor allows the whole plate surface area to remain in contact with the concrete surface with retention of the chemical interface bond. Increased shear action is necessary to remove the resistance to shear force.


4. MATHEMATICAL MODELLING OF PLATE DEFLECTED SHAPE

A mathematical model is required to support the Bi-Steel design process. This must compare favourably with previous experimental data and in addition correlate with numerical modelling already conducted. The solution to the problem is sought with the application of the Laplace equation.

4.1 Formulation of deflected shape by the Laplace equation.

Consider that plate displacement surface between shear connectors is governed by the Laplace Equation 3. This assumption comes from the plate measurements recorded in the laboratory tests.

Testing and Modellling of Bi-Steel Plate Subject to Push-Out Loading

Hence the UZ deformation shape of the plate is described by:

~ u(x, y) 4A I 1 1 = ~ e -y sinx + - e -3y sin3x + ...... ~r 3

(5)

475

4.2 Validation against the test data.

The value of the interaction displacement is defined by the UZ displacement at the first shear connector, weld perimeter. Currently, this is obtained from numerical modelling analysis in the absence of concise 'real' world physical data. It was found that the equation was very accurate at the considered point of contraflexure but accuracy reduced moderately at peak/trough values. Typically accuracy error moved between 0.1% and 40% for displacement predictions at midpoint and peak/troughs respectively. Quantitatively, the discrepancy in each case is only several hundredths of a millimetre. Measurement this small would be extremely difficult to record consistently in the laboratory during specimen loading. Table 2 indicates the accuracy of Equation 5.

Table 2: UZ deformation shape error at midpoint between shear connectors.

Plate Size 6 8 10 12 14 16 (mm)

Numerical 2.77 2.89 5.23 1.94 0.83 0.34 Modelling (ram)

Equation (5) 2.59 2.99 5.00 1.89 0.82 0.36 Answer (ram)

% Error 6.94 3.34 4.60 2.65 1.22 5.56 Difference

It is shown that greatest error is achieved consistently on plate sizes that promote ductile wave displacements of the Bi-Steel plate between the shear connectors. However, co-ordinates of peak/trough displacement at one third, two thirds distance between the shear connectors display a consistent error difference of approximately 30% regardless of the steel plate thickness. Further evaluation indicates that higher load conditions produce increased displacement stability, while lower load predictions become more difficult to make accurately. Generally, the Laplace equation was consistently more accurate at the one-third point than at two-thirds distance between shear connector one and two. Error difference experienced between the two positions was typically in the region of 20%.

5. CONCLUSIONS

The following conclusions can be drawn based on the testing, numerical modelling and mathematical analysis for Bi-Steel plates.

1. From the recent testing it has revealed that Bi-Steel rods and plates have significant shear strength. The shear strength is greatly affected by several important parameters. These include plate spacing, rod spacing and rod diameter.

2. From load-deformation relationships it can be seen that Bi-Steel plates have high ductility and deformation capacity. For very thick plates (> 14mm), the failure can be brittle if Bi-Steel rod numbers are small. The failure will be initiated by the shear failure of local welds.


3. Graphical plots from numerical analysis show plate thickness and plate spacing govern stress distribution at local weld perimeter. Rod spacing will largely determine the out of plane UZ plate deformation shape.

4. Preliminary design formulae for shear strength of Bi-Steel plate have been proposed. These include the consideration of plate spacing and rod diameter. A general equation is being developed for design purposes.

5. The deformation shape of Bi-Steel plates has been established through the derivation of the Laplace equation. Validation against the test results has proved Equation (5) is accurate. This will be very useful for the serviceability design of Bi-Steel plate.

Further testing, numerical simulation and design procedures are being conducted. These research results will be published in stages according to the plan.

Acknowledgements:

This research was jointly funded by British Steel Plc and a CASE studentship from the Engineering and Physical Science Research Council (EPSRC).

References:

1. MOY S. S. J., XIAO R. Y. and LILLESTONE D. Tests for British Steel on the shear strength of the studs used in the Bi-Steel system. University of Southampton - Department of Civil & Environmental Engineering, 199.8, May. 2. OEHLERS D. J. and SVED G. Composite beams with limited slip capacity shear connectors. Journal of Structural Engineering, 1995, Volume 121, June, 932-938. 3. KALFAS C., PAVLIDIS P. and GALOUSSIS E. Inelastic behaviour of shear connection by a method based on FEM. Journal of Constructional Steel Research, 1997, Volume 44, No. 1-2, 107- 114. 4. UY B. and BRADFORD M. A. Local buckling of thin steel plates in composite construction: Experiment and theory. Proceedings of the Institution of Civil Engineers - Structures and Buildings, 1995, Volume 110, November, 426-440. 5. UY B. and BRADFORD M. A. Elastic local buckling of steel plates in composite steel - concrete members. Journal of Engineering Structures, 1996, Volume 18, No. 3, 193-200. 6. SCHUURMAN R. G. and STARK J. W. B. Longitudinal shear resistance of composite slabs. Proceedings of the Engineering Foundation Conference, 1997, 89-103. 7. AN L. and CEDERWALL K. Push-out tests on studs in high strength and normal strength concrete. Journal of Constructional Steel Research, 1996, Volume 36, No.l, 15-29. 8. OEHLERS D. J. and JOHNSON R. P. The strength of stud shear connections in composite beams. The Structural Engineer, 1987, Volume 65B, No.2, June, 44-48. 9. ANWAR HOSSAIN K. M. and WRIGHT H. D. Performance of profiled concrete shear panels. Journal of Structural Engineering- ASCE, 1998, Volume 124, 368-381. 10. KEMP A. R. and TRINCHERO P. E. Horizontal shear failures around connectors used with steel decking. Proceedings of the Engineering Foundation Conference, 1997, 104-118. 11. CLUBLEY S. K., XIAO R. Y. and MOY S. S. J., Computational structural analysis and testing of Bi-Steel p l a t e s - Six and Twelve month progress report for British Steel. University of Southampton - Department of Civil & Environmental Engineering, 1999, June, 145 & 270 pages.

RECTANGULAR TWO-WAY RC SLABS BONDED WITH A STEEL PLATE

J. W. Zhangl, J.G. Teng 2 and Y.L. Wong 2

1 Department of Structural Engineering, Southeast University, Nanjing, China. 2 Department of Civil and Structural Engineering

The Hong Kong Polytechnic University, Hong Kong, China.

ABSTRACT

External bonding of steel plates has been widely used for retrofitting RC structures. Many studies have been carried out on RC beams bonded with steel plates, but little research exists on two-way RC slabs strengthened using this technique. This paper is therefore concerned with the strength of rectangular two-way RC slabs bonded with steel plates subject to a central patch load. Experimental results on square RC slabs bonded with square steel plates are first summarized. A yield line analysis of rectangular two-way plated RC slabs is then presented based on experimental observations of the formation of yield lines. Finally, a design procedure based on the yield line analysis is proposed for practical use, which incorporates an empirical modification factor based on the experimental results.

KEYWORDS: Slabs, Steel Plates, Yield Line Analysis, Strengthening, Bonding, Adhesive

INTRODUCTION

Among the many strengthening techniques available, the method of plate bonding has been an attractive one in recent years, due to its simplicity and speed of application and minimum increases in structural self-weight and size. Steel plates and fibre-reinforced plastic (FRP) plates have both been used in plate bonding, depending on the requirement of a particular situation. Steel plates have been used very widely to strengthen RC beams and also slabs.

Two recent cases of plate bonding to slabs are reported in Civil Engineers Australia (1995) and Godfrey and Sharkey (1996), and both used steel plates. Although a great deal of research has been carried out in recent years on this method of strengthening for RC beams, the only previous study on two-way slabs is that by Erki and Hefferman (1995) which reported some tests on small two-way slabs bonded with FRP sheets to enhance the punching shear failure load.

This paper is therefore concerned with the strength of rectangular two-way RC slabs bonded with steel plates subject to a central patch load. High patch loads requiting strengthening of structures often arise in practice. Examples include local loads due to the installation of a piece of heavy equipment and column loads on floor slabs due to the removal or addition of columns. Experimental results on square

477

478 J.W. Zhang et al.

RC slabs bonded with square steel plates are first summarized. A yield line analysis of rectangular two- way plated RC slabs is then presented based on experimental observations of the formation of yield lines. Finally, a design procedure based on the yield line analysis is proposed for practical use, which incorporates an empirical modification factor based on the experimental results. It should be remarked that if a single steel plate is too big for convenient handling in construction, a number of orthogonally placed steel strips may be used instead to achieve the same amount of external reinforcement. The work presented here is equally applicable to such slabs.

EXPERIMENTS ON SQUARE SLABS

Experimental Results

A total of five square RC slabs bonded with square steel plates were tested by the authors (Zhang et al., 1999). Only a brief summary of the experimental results in relation to the yield line analysis to be

TABLE 1 PROPERTIES OF MATERIALS

Materials

Concrete

Rebar

Type

1800x 1800x70 (mm)

Mild steel d76.5,@ 150mm centres, in both directions, average concrete cover for the two directions =16.5mm

Steel plate Mild steel

Adhesive ET epoxy resin

*Assumed values

Elastic Compressive Yield Ultimate modulus strength stress tensile stress (N/mm 2 ) (Nlmm z ) (N/mm z ) (N/mm 2 )

- 26.4 - -

200000*

200000*

5960

340 431

- 335 417

94 - 11

TABLE 2 EXPERIMENTAL RESULTS AND COMPARISON WITH YIELD LINE ANALYSIS

Specimen

SB1 (control)

Dimensions of steel plate (in mm) and plate-to-slab

area ratio (%)

No plate

Initial cracking

loadPcr (kY) and relative

increase Ycr against

SB1 (%)

21 (0.00)

Experimental

ultimate load Pe (kN) and relative

increase Ye against

SBI(%)

55.0 (0.00)

Theoretical ultimate

load Pu (1~)

45.2

SB2 500•215 (8.65) 40 (90.5) 67.5 (22.7) 56.0

SB3 500•215 (8.65) 40 (90.5) 65.0 (18.2) 56.0

SB4 850•215 (25.0) 60 (186) 85.0 (54.5) 75.7

SB5 1400• 1400• 1 100 (376) 165 (200) 200 (67.8)

Pu

ee

0.82

0.83

0.86

0.92

1.21

presented in this paper is given here. The slabs all had the same dimensions of 1800 mm x 1800 mm x 70 mm. They were simply supported with a span of 1700 mm between supports and subject to a

Rectangular Two-Way RC Slabs Bonded with a Steel Plate 479

central patch load over an area of 150 mm x 150 mm. Details of the material properties and dimensions of the bonded steel plates are given in Tables 1 and 2 respectively. Table 2 also gives the initial cracking loads and ultimate loads of the test slabs.

These experiments showed that bonding of steel plates to the soffit of slabs can greatly increase both the cracking load and the ultimate load of two-way RC slabs. Debonding of the steel plate from the slab is unlikely as in all four tests on plated slabs, no debonding failure was found. This is contrary to the case of plated beams. In this sense, the plate bonding method is more suited for slabs than for beams. Failure of the plated slabs was by the formation of yield lines and the failure mode was ductile. The final cracking patterns and hence yield line patterns of slabs SB2, SB3 and SB4 are similar. Most of the cracks were on the soffit of the slab. At the edges of the steel plate, an abrupt change in stiffness and strength occurs. As a result, main cracks occurred around the plate perimeter. In addition, in each zone between the corner of the steel plate and that of the slab, there were four or five main diagonal cracks. The final cracking patterns of slab SB4 are shown in Figure 1. Slab SB5 which was bonded with a large steel plate had a different yield line pattern.

Figure 1 Final Cracking Patterns of Slab SB4 Figure 2 Triangular Yield Line Pattern

for a Square Plated Slab

TABLE 3 ACCURACY OF THE TRIANGULAR YIELD LINE PATTERN

Slabs k 2 ( k 3 )

SB1 0.0882

SB2 0.294

SB3 0.294 SB4 0.5 SB5

k 1

0.267

0.207

0.207 0.146

Pe (kN)

55

67.5

65 85

Pu (kN)

45.2

56.0

56.0 75.7

Error (%)

-17.8

-17.0

-13.8 -10.9

0.824 0.052 165 200 21.2

Yield Line Analysis

Based on the experimentally observed yield line pattems, yield line analyses (Johansen, 1962; Jones and Wood, 1967; Kong and Evans, 1987) were carried out for the test slabs (Zhang et al., 1999). In particular, three of four different yield line pattems explored were suitable for slabs with cracking patterns similar to those shown in Figure 1, with the fourth being suitable for slab SB5. The differences in these three yield line patterns lie in the assumption of yield line patterns in the zones between the comers of the steel plate and the slab.

480 J. IV. Z h a n g et al.

Zhang et al. (1999) compared the accuracy of these three yield line patterns in predicting the test results. The triangular pattern (Figure 2) was identified by Zhang et al. (1999) as the most suitable for use in design as long as the width ratio between the steel plate and the slab k2 (or k3 denoting the width ratio between the patch load and the slab for the RC control slab SB 1) is not too large (Table 3). Although the triangular pattern is not as accurate as the most complicated pattern assumed in Zhang et al. (1999), it is simpler than the latter and conservative for all applicable slabs (SB 1-SB4).

YIELD LINE ANALYSIS OF RECTANGULAR SLABS

Yie ld L i n e Pat tern

The above comparisons between yield line analyses and tests show that the yield line method can predict the ultimate strength of plated slabs closely and is suitable for design use. In order for the method to be more generally applicable, a yield line analysis of rectangular slabs adopting the triangular yield line pattern is briefly presented in this section. A more detailed description of this analysis can be found in Teng et al. (1999). In this analysis, the rectangular slab is assumed to have an aspect ratio of less than 2 and to be symmetrically bonded with a steel plate and subjected to a central patch load. In order for the triangular yield line pattern to be the critical pattern, it is further assumed that the longer direction of the slab coincides with that of the steel plate. Only isotropic steel reinforcement with equal sagging and hogging yield moments is considered. Due to symmetry, only a quarter of a rectangular plated RC slab is considered, as shown in Figure 3. This quarter of the plated slab is then divided into four rigid regions A, B, C and D by yield lines. The negative yield line on the top surface of the slab is assumed to be inclined to the slab edges at an angle of 45 degrees.

aJ2

s

o~ ~ A bd2

t 'q -.>_

~5 C ~

Figure 3 Yield Line Pattern for a Rectangular Plated Slab

Vir tua l W o r k E q u a t i o n

Before proceeding to the detailed derivation, it is necessary to define some notation. The width and

length of the slab are a~ a n d a 2 respectively, while those of the steel plate are b~ and b 2 . The location

of the negative yield line is determined by k~ = s / a~. The steel plate-to-slab width and length ratios

are denoted by k 2 and k 4 respectively. That is, k 2 = b 1 / a~ and k 4 = b 2 / a z . The aspect ratio of the

slab a 2 / a~ is denoted by k s .


Based on geometrical compatibility requirements, the rotations of the rigid regions A, B, and C

around their own rotational axes are respectively found to be 0 A =2/(a~-b~), 0 B = I / H and

0 c = 2/(a 2 -b2) . Here H is the perpendicular distance from the plate comer to the adjacent negative yield line, and can be found using simple geometric relations as

H = a+,~[(l - k 2 - 2k, ) + ks O - k4 )l/ 4 . (t)

Based on the virtual work principle, the expressions of extemal virtual workE e and intemal virtual

work Eg can be found without difficulty:

Ee = Pu x 1.0 = P. (2)

- 2 k 1 1 - 2 k 1 8k t Ei = 4mp 5 + ~ + ] (3)

k l - k 2 k s 0 - k 4 ) O - k z - 2 k , ) + k s O - k 4 )

where m.,, denotes the yield moment per unit length of the unplated part of the slab. Combination of

Eqns 2 and 3 then yields the desired ultimate load Pu as

I - 2 k 1 1 - 2 k 1 8k l P. = 4mp 5 + ~ + ] (4) kl-k+ ks( l -k4) O-k2-2kl)+ksO-k4)

Minimization of Pu with respect to k t and ignoring the contribution of the hogging moment in this minimization leads to

1 k I - - ~ ( 1 - k 2 ,-l--O-k4)k 5 -420-k2) ( l - k4 )ks ) (5)

which can be substituted into Eqn. 4 to find the minimum value of Pu.

Eqns 4 and 5 include square plated slabs as a special case. For the case of k 5 = 1 and k 2 = k 4 which corresponds to the case of a square slab bonded symmetrically with a square plate, Eqns 4 and 5 reduce to Eqns 6 and 7 below:

where

1 - k 1 - k 2+2kl z ] P,, =8mp ( l _ k _ k z X l _ k 2 ) (6)

k, : 2 - ' f 2 ( l -k2 ) (7) 2

DESIGN PROCEDURE

The main task in designing the steel plate for strengthening a given RC slab is to determine the size of the required steel plate provided the thickness of the steel plate required to prevent failure in the

482 J.W. Zhang et al. central plated part is determined by a suitable method. A discussion of this issue is given in Teng et

al. (1999). To this end, the strength P,c of the existing RC slab needs to be found first, using

Puc =4mpT (8)

where

I k s - T ~ 1-T~ 4T~] r = +

k 5 - k 3 T 2 (9)

T~ = 1 - 2k 3 + k 5 - T 2 (10)

T 2 = ~/2(1-k3)(k 5 -k3 ) (11)

Eqn. 8 is obtained from Eqn. 4 by replacing k 2 with k 3 and k 4 with k3/k 5 respectively, with k 3 =

c/a~. For convenient use in design, the strengths of unplated slabs are tabulated in Table 4 in a

dimensionless manner.

TABLE 4

DIMENSIONLESS STRENGTHS euc / mp OF UNPLATED RC SLABS

k3

0.05 0.10 0.15 0.20 0.25 0.30

k 5 1.0 1.2 1.4 1.6 1.8 2.0

10.36 10.48 10.83 11.30 11.86 12.47 10.83 10.92 11.27 11.77 12.35 12.99 11.35 11.41 11.76 12.28 12.89 13.57 11.94 11.95 12.31 12.85 13.50 14.21 12.61 12.56 12.92 13.49 14.18 14.94 13.37 13.26 13.62 14.22 14.95 15.76

Once the existing strength Puc is known and the required strength Pa of the strengthened slab is

specified, the strengthening ratio required fl is calculated as

f l = Pd (12) Puc

For design use, the strength of the plated slab can be predicted by the following modified form of Eqn. 4:

4mplk -2kl 1-2k ~ 8kl )1 + ~ + - (13) Pd--" Ifig 51-k2 k50-k4) O-k2-2kl)-~t-k50-k4

where the term ~ is introduced to account for the differences between the test results and the yield

line analysis. For square slabs, the value of ~t can be found using the following empirical formula

obtained by curve-fitting the results of Table 3:

= 1.63k 3 - 1.18k 2 + 0.37k 2 + 0.80 (14)


For rectangular plated slabs, ~ is expected to be a function of both k 2 and k 4. As experimental results are only available for square plated slabs, a conservative approach for rectangular slabs is to

adopt the greater of k 2 and k 4 for k 2 in Eqn. 14 to find the value of ~ . Combination of Eqns 8, 12,

13 and 14 results in the following expression for /3 for a given steel plate size chosen to strengthen the slab

1 Iks -D 1-D 4D 1 13 = ~-T-~ 1 - k 2 + (1 - k4)k-------~ + ~ D 1

(15)

in which

D =(1-k2)+O-k4)k 5 -D, (16)

D, = 4 2 0 - k 2 Xl-k4)k5 (17)

For convenience in design use, the values of fl can be tabulated. An example is shown in Table 5.

TABLE 5 STRENGTHENING RATIO fl OF PLATED RC SLABS

k5

1.0 1.2 1.4 1.6 1.8 2.0

k 2 ( k 4 =0.60, k 3 =0.10)

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

1.727 1.769 1.818 1.876 1.946 2.032 2.140 2.187 2.251 1.660 1.710 1.768 1.838 1.922 2.025 2.155 2.228 2.323 1.602 1.659 1.726 1.806 1.901 2.019 2.166 2.259 2.379 1.556 1.619 1.693 1.780 1.886 2.014 2.175 2.285 2.424 1.520 1.588 1.668 1.762 1.875 2.013 2.185 2.307 2.46'1 1.493 1.565 1.650 1.749 1.868 2.014 2.195 2.327 2.493

Such tables enable a simple determination of fl once the dimensions of the slab and the plate are

known, or the determination of the steel plate size to arrive at a given value of ft.

Although Eqns 13 and 15 may be applicable to slabs with a very large steel plate as was used in the square test slab SB5 since a modification factor ~ has been included, it is necessary at this point of the time to restrict the application of these formulas to slabs with a bonded steel plate satisfying the following conditions:

0.3 < k 2 < 0.7 (18)

0.3 _< k 4 < 0.7 (19)

The lower bound is to ensure that punching shear failure does not become critical while the upper bound is to prevent a yield line pattern different from that assumed in Figure 3 to become critical. The above limitations allow plates up to half the size of the slab to be used in strengthening, which are already capable of producing a large strength increase. These limits are to be further justified in future work.

484 J.W. Zhang et al.

CONCLUSIONS

This paper has been concemed with the strength of two-way rectangular RC slabs bonded with a steel plate subject to a central patch load. Based on experimental observations of the formation of yield lines in square plated slabs and the previous success of the yield line method for these slabs, a yield line analysis of rectangular two-way plated RC slabs has been presented. Finally, a design procedure based on the yield line analysis has been proposed for practical use, which incorporates an empirical modification factor based on the experimental results. Experiments on rectangular plated slabs are required to validate the proposed design method.

ACKNOWLEDGEMENTS

The work described here forms part of the project "Strengthening of Reinforced Concrete Slabs by Plate Bonding" carried out in the Department of Civil and Structural Engineering, The Hong Kong Polytechnic University in collaboration with the Department of Structural Engineering, Southeast University. The authors wish to thank Professors Z.T. Lu of Southeast University and J.M. Ko of The Hong Kong Polytechnic University for their support to this collaborative project and The Hong Kong Polytechnic University for the financial support provided through a central research grant (G- $567). The authors would also like to thank Messrs Zheng Xian-Yuan and Mr Chen Mao-Lin for their assistance in the experimental work.

REFERENCES

Civil Engineers Australia. (1995). Epoxy Adhesive Used to Bond Steel Plates. Civil Engineers Australia Sep., 55. Erki M.A. and Hefferman P.J. (1995). Reinforced Concrete Slabs Externally Strengthened with Fibre-Reinforced Plastic Materials. Non-Metallic (FRP) Reinforcement for Concrete Structures, L. Taerwe ed., E & FN Spon, 509-516. Kong, F.K. and Evans, H.R.(1987). Reinforced and Prestressed Concrete, 3rd Edition, Chapman and Hall, London. Godfrey J. and Sharkey P. (1996). Plate Bonding to Strengthen Hall Floor. Construction Repair July/August, 39-40. Johansen K. W. (1962). YieM Line Theory, Cement and Concrete Association, London. Jones L.L. and Wood R.H. (1967). Yield Line Analysis of Slabs, American Publishing Company, Inc., USA. Teng J.G., Zhang J.W. and Wong Y.L. (1999). Strength of Two-Way Rectangular RC Slabs Externally Bonded with a Steel Plate. To Be Published. Zhang J.W., Teng J.G., Wong Y.L. and Lu Z.T. (1999). Behaviour of Two-way RC Slabs Externally Bonded with a Steel Plate. To Be Published.

Bridges


Structural Performance Measurements and Design Parameter Validation for Tsing Ma Suspension Bridge

C.K. Lau 1 W.P. Mak I K.Y. Wong 1 Deputy Director Chief Engineer Senior Engineer

W.Y. Chan' K.L. Man' K.F. Wong 2 Engineer Engineer System Analyst

'Highways Department, The Government ofHong Kong Special Administrative Region 2E&M Section, Tsing Ma Management Limited

ABSTRACT

An intensive bridge monitoring system has been installed on the Tsing Ma Suspension Bridge for monitoring the structural performance and evaluating the health (safety) conditions of the bridge. After more than two years of operation, the system has collected and archived a substantial amount of data. Numerous works for data processing, analysis and interpretation have been carried out to assess the structural performance of the bridge as well as to validate the various bridge design parameters. This paper presents the findings from the measured results including: wind, temperature, traffic and bridge responses. The measured results are then used to compare with the design parameters of the bridge for evaluating the current loading and health conditions.

KEYWORDS

bridge monitoring system, measurement, design parameters, load monitoring, response monitoring

INTRODUCTION

The Tsing Ma Suspension Bridge, which forms a key part of the most essential strategic transportation network (road and railway) linking the Hong Kong International Airport to the urban areas of Hong Kong, was opened to traffic on 22 May 1997. Figure 1 shows the location of Tsing Ma Bridge in the Tsing Ma Control Area. In order to monitor and evaluate the structural health and performance of the bridge, Highways Department of the Government of the Hong Kong Special Administrative Region has devised and implemented an intensive structural monitoring system on the bridge, named as Wind And Structural Health Monitoring System (WASHMS) [1 and 2]. The system collects environmental and applied loads information (such as wind, temperature, seismic and traffic) and structural response records of the bridge (such as vibrations, displacements and strains) in a continuos manner from

487

488 C.K. Lau et al.

approximately 350 sensors installed at different locations of the bridge. According to the nature of the signals to be collected, these sensors are divided into seven groups, namely, anemometer, temperature measurement assembly, accelerometer, strain gauge, level sensing system, displacement transducer and weigh-in-motion system. Figure 2 shows the layout of the sensory system in Tsing Ma Bridge.

WIND LOAD MONITORING

The Tsing Ma Suspension Bridge was designed to resist a maximum steady wind speed of 50 m/s plus a maximum fluctuating wind speed of 80 m/s (3-second gust). Such design parameters are based on a wind return period of 1 in 120 years. A summary of the design wind speeds and corresponding horizontal wind loaded lengths for Tsing Ma Suspension Bridge is given in Table 1.

TABLE 1 STRUCTURAL DESIGN WIND SPEEDS FOR BRIDGE DECK

WITH AND WITHOUT HIGHWAY AND RAILWAY LIVE LOADS (120 YEARS RETURN PERIOD)

Design Wind Speeds for Deck without Highway and Railway Live Load Design Wind Speeds for Deck with combined Highway and Railway Live Load Design Wind Speeds for Deck with Railway Live Load only

Hourly Mean Wind Speed at

Deck Level 50m/s

25 m/s

28 m/s

Maximum 3-second Wind Speed Horizontal Wind Loaded Length

20morless 100m 600m 1000m 80m/s 72m/s 65 m/s 63 m/s

2000m 60 m/s

44m/s 38 m/s 34 m/s 33 m/s 33 m/s

50 m/s 43 m/s 39 m/s 38 m/s 37 m/s

The above information was derived from the wind data obtained from an observation station at Waglan Island which is located some 5 kilometres to the south-east of Hong Kong Island. As the topographical conditions of the bridge site and that of Waglan Island are different, it is necessary to verify the above wind design parameters. The bridge deck was designed for aerodynamic stability according to the parameters given in Table 2.

TABLE 2 DESIGN CRITICAL WIND SPEEDS FOR AERODYNAMIC STABILITY CHECK

Angle of Incidence Design Critical Wind Speeds (One-minute Mean Wind Speed) Bridge Deck Alone Bridge Deck with Traffic and Trains

-5 ~ 31 m/sec 25 m/sec -2.5 ~ 50 m/sec 40 m/sec

0 o 74 m/sec 50 m/sec 2.5 ~ 50 m/sec 40 m/sec 5 ~ 31 m/sec 25 m/sec

(Note : Wind speeds in 1 in 200 years return period were used in the design in deriving the critical wind speeds for checking the bridge's aerodynamic stability)

Wind measurement data for Tsing Ma Suspension Bridge are given by 6 No. anemometers respectively installed at the tower top of the bridge (2 No.), mid-span at deck level (2 No.) and Ma Wan side-span deck level (2 No.). The measured wind data are used to derive (a) wind rose diagrams

Structural Performance Measurements for Tsing Ma Suspension Bridge 489

of hourly mean wind speeds and 3-second gust wind speeds - showing wind speeds, wind directions and frequencies of occurrence and (b) wind structure - including 3-second gust wind speed and hourly mean wind speed, gust factors, turbulence intensities and wind spectrum. Some typical wind monitoring results for the bridge obtained in the past two years are given in Figure 3 to Figure 6.

T E M P E R A T U R E M O N I T O R I N G

The structural design for temperature load of the bridge is based on two parameters, namely effective bridge temperature and differential temperature. Effective bridge temperatures are functions of total solar radiation and values of shade air temperature, whilst differential temperature is a function of total solar radiation and extreme ranges of shade air temperature. The parameters for the design of steel and composite bridges are based on theoretical approaches and experimental data adopted for the design of concrete bridges under Hong Kong climatic conditions[3]. It is thus necessary to verify the original design parameters by measuring the temperature distribution of the bridge. A total of 115 temperature sensors are installed at various locations of the bridge deck to measure respectively the ambient air temperature, structural steel temperature including steel sections, main suspension cables, steel cladding and asphalt temperature. The bridge is designed for temperature values with a 120-year return period and the corresponding designed maximum and minimum coincident effective bridge temperature are given in Table 3.

TABLE 3 DESIGNED MAXIMUM/MINIMUM EFFECTIVE TEMPERATURE

FOR TSING MA SUSPENSION BRIDGE

Design Max Effective Temp (oc) Deck 46 Main Cables 50 Suspenders 50

Design Min Effective Temp (oc) -2 -2 -2

Towers 36 2

Figure 7 is a summary of the annual temperature variation on Tsing Ma Suspension Bridge during the past two years. Six different curves are presented in the plot, namely (i) mean effective temperature, (ii) maximum effective temperature, (iii) minimum effective temperature, (iv) mean ambient temperature, (v) maximum ambient temperature and (vi) minimum ambient temperature. The lowest mean ambient temperature was recorded in December and a value of 13.5 ~ was recorded. The highest mean ambient temperature was recorded in August and a value of 34.5 ~ was recorded. It is noted that during the period of low temperature, i.e., at night, values of effective temperature and ambient temperature run very closely to each other as there is no solar energy gained by the structure. However, during the hottest period of the measurement, i.e., during noon time and in the summer season, values of effective temperature are about 8 to 12 ~ higher than the recorded ambient temperature. The highest effective temperature measured on the bridge was 44.5 ~ which was recorded in July 1998. This value is close to the design threshold limit of 46 ~ An interesting point to note however concerning the temperature in the year 1998 is that it is the warmest year since 1884 (according to the information from Hong Kong Observatory).

D I S P L A C E M E N T M O N I T O R I N G

Longitudinal movement of the bridges at the expansion joint is mainly affected by the temperature of the structure. The designed maximum longitudinal movement value for Tsing Ma Suspension Bridge

490 C.K. Lau et al.

is + 835mm. Figure 8 and 9 illustrate the longitudinal movement and range of longitudinal movement of Tsing Ma Suspension Bridge at the expansion joint at the Tsing Yi Abutment. It is noted that the trend of the movement, when converted into effective temperature of the bridge, can be readily represented by a straight line. The gradient of the fitted straight line for Tsing Ma Suspension Bridge is 24.4 mm per ~ while the range of movement as revealed from Figure 9 is between i 450mm only. This also illustrates that the longitudinal movement of the bridge is well within the allowable tolerance. Vertical displacement of the bridge deck of Tsing Ma Suspension Bridge is monitored by means of a level sensing system which, by means of a pair of fluid conduits, measures the change of fluid pressure at various locations of the deck and derives the corresponding reference datum. Vertical displacement of the deck is basically a function of temperature and live load. Figure 10 illustrates the measured vertical movement of the bridge deck at mid-span in the past two years. It can be revealed from the figure that the maximum downward deflection of the bridge deck due to combined traffic and temperature effect is within a range of 1.4m, which is well within the designed values of 6m (respectively 4.7 m for live load and 1.3 m for temperature effect).

TENSILE LOAD MONITORING ON SUSPENDERS

Figure 11 illustrates the measured load on the suspenders of Tsing Ma Suspension Bridge derived by field measurement. The field measurement was conducted in 1998 and a total of 2 x 95 = 190 No. suspenders were measured. Portable accelerometers were used to measure the ambient vibration of the suspenders. The time series data were then converted to frequency domain (spectrum analysis) to give the natural frequencies of the suspenders under ambient condition. These frequencies were then used to derive the tensile force of the individual suspender during the period of measurement. (Note : Tension in a taut string/wire is a function of material/sectional properties and fundamental frequency)

According to the design information, the self-weight of an 18 metres deck unit for the bridge (Dead Load + Superimposed Dead Load) is about 516 tons. This weight is to be taken by two groups of suspenders (south and north) during the erection stage, i.e., about 258 tons per group. The increase of the suspenders' tensile force since erection represents the presence of other superimposed dead load, including railway slab, servicing and live load on the bridge. However, the result of the field measurement indicated that all the tensile load now taken by the suspenders are within the Serviceability Limit State for material strength, i.e., 448 tons per group of suspenders. The typical range of tensile load for the suspender is between 300 tons and 400 tons. The corresponding current maximum tensile load taken by each 76mm diameter hanger strand is 75 - 100 tons.

VEHICULAR TRAFFIC LOAD MONITORING

The intensity of vehicular traffic loads on long-span bridges is govemed by the effects of groups of vehicles in traffic jams and is derived by statistical simulation from local vehicular characteristic data and future prediction of change. The required loaded length used in the design is based on number of daily traffic jams, locations of the jams, duration and distribution of vehicular types and traffic flow during the jams. The design HA Lane Factor for Tsing Ma Suspension Bridge is 3.6. In order to assess the validity of the above design parameter, it is necessary to have a corresponding traffic load

monitoring for the bridge.

Weigh-in-motion sensors are provided on the Lantau Fixed Crossing so that vehicular number, axle- weights, speeds and type of vehicles crossing the Tsing Ma Suspension Bridge can be measured. The measured data are then used to formulate a database to derive the percentage of goods vehicles and to

Structural Performance Measurements for Tsing Ma Suspension Bridge 491

compare with the HA Lane Factor used in traffic load design. Figure 12 illustrates the monthly daily average percentage of goods vehicles crossing the bridge. It is noted that the current percentage of goods vehicles using the bridge is about 34% of the total vehicle. This value is well below the design percentage of 60%.

RAILWAY LOAD MONITORING

Railway traffic load is one of the most important parameters affecting the structural design of the bridge. Loads due to railway traffic on Tsing Ma Suspension Bridge are monitored by means of strain gauges installed at waybeams which support the railway trackform. The monitoring works include the conversion of the recorded waybeam strains into bogie load data and train load data and subsequent derivation of the train weight, passing rate and rainflow counts for fatigue life estimation. Figures 13 illustrates the vertical acceleration record on Tsing Ma Bridge during the past two years of operation. It can been denoted that the average value of vertical acceleration is in the range of 100 to 150 mm/sec 2. The results also reveal that the derived maximum accelerations (on a monthly basis) run closely with the desirable operational upper limit for train running but are still well below the allowable maximum value. Figure 14 is a train load monitoring plot showing the load configuration of a typical 7-car train (i.e., 14 No. bogies) passing the bridge. It can been seen from the derived bogie loads that they are all within the designed envelop (tare load and crush load) and is in agreement with the designed load pattern for the designed train.

STRAIN/STRESS MONITORING OF VARIOUS STRUCTURAL MEMBERS

Strain gauges are installed at a number of critical locations on the bridge to measure the change in strain of the structural members under different loading conditions. The instrumented locations include chord members of the longitudinal trusses, cross-frame chord members, plan bracing members, deck trough and rocker bearings at Ma Wan Tower. The measured strains are recorded and then used to derive axial, shear and bending stresses of the members and the corresponding loading effects. Figure 15 illustrates the measured strain results of the chord members on the outer longitudinal truss at Chainage 24662.5. Figure 16 shows the load monitoring results for the outer rocker bearing at Ma Wan Abutment during the past two years. The measured stress values are used to compare with the designed values of the members at both the Serviceability Limit State and Ultimate Limit State, whilst the measured strain can be used to establish the rainflow counts for fatigue life estimate. Again, it can be revealed that the current stress levels of the critical structural members are well below the designed limits.

DYNAMIC RESPONSE M O N I T O R I N G

The dynamic characteristics of a structure can be represented by its mode shapes and frequencies. Accelerometers are installed at various strategic locations of the bridge deck and main cables so that their dynamic characteristics or response under vibration can be measured and monitored. Table 4 shows the measured results of the first four frequencies and their corresponding mode shapes of Tsing Ma Suspension Bridge. The corresponding values computed by the designer of the bridge and others are also illustrated for comparison. It could be revealed that the measured results are in general higher than those derived in the design and the measured values obtained during bridge construction. This implies that the stiffness of the as-built bridge-deck is stiffer than that of the designed values and that during construction stage.

492 C.K. Lau et al.

TABLE 4

COMPARISON OF COMPUTED AND MEASURED FREQUENCIES

OF TSING MA SUSPENSION BRIDGE

Type and Order of Mode Shape

Lateral Mode 1st

2nd 3rd 4th

Vertical Mode 1st

2nd 3rd 4th

Computed MMHK 1

(Designer)

0.065 Hz 0.164 Hz

0.112 Hz 0.141 Hz

_ _ -

_ _ -

Computed FNp2

(Checker)

0.064 Hz 0.149 Hz 0.266 Hz 0.455 Hz

0.112 Hz 0.133 Hz 0.179 Hz 0.233 Hz

Measured HKPU 3

( without paving )

0.069 Hz 0.164 Hz 0.214 Hz 0.226 Hz

0.113 Hz 0.139 Hz 0.184 Hz 0.241 Hz

Rotational Mode 1st 0.259 Hz 0.235 Hz

2nd 0.276 Hz 0.268 Hz 3rd --- 0.409 Hz 4th --- 0.533 Hz

MMHK1 _ Mott MacDonald Hong Kong Limited HKPU 3 - Hong Kong Polytechnic University HyD 5 - Highways Department

0.267 Hz 0.320 Hz

Measured THU 4

(with paving )

0.069 Hz 0.161 Hz 0.242 Hz 0.246 Hz

0.114 Hz 0.137 Hz 0.183 Hz 0.240 Hz

0.265 Hz 0.320 Hz 0.485 Hz 0.591 Hz

Measured HyD 5

(as-built)

0.070 Hz 0.170 Hz 0.254 Hz 0.301 Hz

0.114 Hz 0.133 Hz 0.187 Hz 0.249 Hz

0.270 Hz 0.324 Hz 0.486 Hz 0.587 Hz

FNP 2 - Flint & Neill Partnership. THU4- Tsinghua University

C O N C L U S I O N

The measured results on w i n d , traffic and temperature loads indicate that the loads acting on the

bridge are far less than the design load values. The measured/der ived results on bridge responses

indicate that the current stresses and displacements at critical locations are far below the design

response values. It is therefore concluded that the bridges are currently under healthy condition.

A C K N O W L E D G E M E N T

The authors wish to express their thanks to Director of Highways, Mr. K.S. Leung, for permission to

publish this paper. Any opinions expressed or conclusions reached in the text are entirely those of the

authors.

References

1. Lau, C. K. and Wong, K.Y., "Design, Construction and Monitoring of the Three Key Cable-Supported Bridges in Hong Kong", Proceedings of the Fourth International Conference on Structures in the New Millennium", 3-5 September 1997 in Hong Kong, A.A. Balkema, Rotterdam, Netherlands.

2. Lantau Fixed Crossing Project Management Office, Highways Department, "Structural Health Monitoring System", Highway Contract No. HY/93/09 - Electrical and Mechanical Services in Lantau Fixed Crossing, The Hong Kong Government, 1993.

3. Highways Department, "Structures Design Manual", The Government of Hong Kong Special Administrative Region, 1997.

Structural Perform

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a Suspension Bridge

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WIND CHARATERISTICS AND RESPONSE OF TSING MA BRIDGE DURING TYPHOON VICTOR

L.D.Zhu 1, Y.L.Xu 1, K.Y.Wong 2 and K.W.Y.Chan 2

1 Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

2 Lantau Fixed Crossing Office, Highways Department, Hong Kong, China

ABSTRACT

On 2 August 1997, Typhoon Victor just crossed over the Tsing Ma Bridge in Hong Kong. The Wind and Structural Health Monitoring System (WASHMS) installed on the Bridge timely recorded both wind and structural response time-histories of seven hours duration. The recorded wind and structural response data are analysed in this paper for evaluating wind characteristics and acceleration response of the Bridge. The result shows that during Typhoon Victor, both mean and turbulent wind characteristics varied considerably due to the change of wind direction and the upwind terrain. Larger turbulence intensities and gust factors are obtained during Typhoon Victor, compared with those due to seasonal trade wind. It is also confirmed that the wind excitation mechanism of the Bridge in the lateral direction is different from that in the vertical direction or the rotation. The alongwind acceleration response of the Bridge is approximately proportional to mean wind speed square while the vertical acceleration and torsional angular acceleration are almost proportional to mean wind speed cubic. Furthermore, the natural frequencies identified from the acceleration response spectra are consistent with those obtained from the ambient vibration measurement or the numerical analysis carried out before.

KEYWORDS

Typhoon Victor, Suspension bridge, Wind characteristics, Acceleration response, Natural frequency.

INTRODUCTION

With the increase of span length of modem suspension bridges, the prediction of bridge response to strong winds becomes more and more important for the bridge constructed within a wind-prone area. To this end, some analytical methods, computational fluid dynamics technique, and wind tunnel test technique have been developed in the past two or three decades. To verify these analytical and numerical methods as well as wind tunnel tests, the field measurements of wind characteristics and bridge response play an important role. However, field measurement data, especially those during severe storms such as typhoons, are very limited up to now.

497

498 L.D. Zhu et al.

Figure 1: The moving path of Typhoon Victor Figure 2: Schematic diagram of the topography ofHong Kong

On 2 Aug. 1997, about three months after the opening of the Tsing Ma Bridge in Hong Kong, Typhoon Victor just crossed over the Bridge and made landfall over the western part of the New Territories. The WASHMS installed on the Bridge by the Highways Department of Hong Kong Special Administrative Region timely recorded wind speed and bridge response time-histories of seven hours duration (Lau et al, 1998). These recorded wind and structural response data are analysed in this paper to evaluate wind characteristics and acceleration response of the Bridge and to provide a basis for the verification of the currently used analytical or numerical or experimental methods at a late stage. Before the presentation and discussion of the measured wind characteristics and bridge responses, a brief introduction of Typhoon Victor, the Bridge and its surroundings, and the measurement instrumentation is given first.

TYPHOON VICTOR

Tropical depression Victor originated in the middle of the South China Sea on 31 July 1997 and its intensity continuously increased afterwards (Lee et al, 1998). The tropical depression Victor first moved northwesterly for 12 hours and then had a sudden turn to near north and remained in almost the same direction during its passage over Hong Kong (see Figure 1). The tropical depression Victor became a real typhoon when it entered the region of 250km south of Hong Kong at 8:00 on 2 August 1997 HKT (Hong Kong Time). At 19:00 HKT on 2 August, the centre of Typhoon Victor moved into the region about 8km east of Cheung Chau Island (see Fig. 2). The lowest air pressure measured on Cheung Chau Island at sea level was 972hPa. Typhoon Victor then crossed over the Tsing Ma Bridge at 20:05 and made landfall over the western part of the New Territories. Victor crossed over the whole Hong Kong within 2 hours at the average translational speed about 25km per hour. After leaving Hong Kong, Typhoon Victor continued moving in the north until it decayed on 3 August in the Southeast of China. The measured highest 10 minute mean wind speed in the wall area of the Typhoon during its passage over Hong Kong was about 110km per hour (30.6m/s) at a 500 m height above the ground, just 2 hours after its landfall (Lee et al, 1998).

TSING MA BRIDGE AND TOPOGRAPHY

The Tsing Ma Bridge in Hong Kong is a suspension bridge with an overall length of 2160m and a main span of 1377m, carrying a dual three-lane highways on the upper level of the bridge deck and two railway tracks and two carriageways on the lower level within the bridge deck (see Figs.3 and 5). The

Wind Response to Tsing Ma Bridge During Typhoon Victor 499

alignment of bridge deck deviates from the east-west axis for about 17 ~ in anticlockwise. The bridge deck is 41m wide and 7.643m high (see Fig. 5). The two bridge towers of 206 m high are made of prestressed concrete. The east bridge tower sits on the Northwest shoreline of Tsing Yi Island, called the Tsing Yi tower while the west bridge tower sits on Ma Wan Island, called the Ma Wan tower.

Hong Kong is situated in the coastal area of South China. Not only there are many islands in Hong Kong, but also there are many mountains covering most areas of the territory. The topography of Hong Kong thus varies from place to place (see Fig. 2). The local topography surrounding the Tsing Ma Bridge within the dashed circle of 5kin in radius is a typical example. The bridge is embraced by sea, islands, and mountains of 200 to 500m high. If taking the bridge as a centre, the whole surrounding area may be roughly classified into seven types of regions (I to VII), bounded by seven lines R1, R2, R3, R4, R5, R6 and R7 as shown in Fig. 2. The TsinYi Island adjacent to the Bridge is in Region I and VII. The top levels of Tsing Yi Island are 218m in the north (Region I) and 334m in the south (Region VII). The Ma Wan Island adjacent to the Bridge is in Regions III and IV. The top level of Ma Wan Island is 69m only.

Figure 3: Elevation ofTsing Ma Bridge Figure 4: Locations of anemometers and accelerometers

Figure 5: Positions of sensors on cross section of bridge deck

INSTRUMENTATION AND DATA ANALYSIS

There are altogether seven different type sensors installed for WASHMS, including, amongst others, six anemometers and 24 uni-axial servo type accelerometers (Lau, et al, 1998). Two digital ultrasonic anemometers (AneU), called Gill Wind Master Ultrasonic Anemometer, were installed on the north side and south side, respectively, of the bridge deck at the mid-span. They are specified as WITJN01 and WITJS01 in Figs 4 and 5. Each ultrasonic anemometer can measure three components of wind velocity simultaneously. Two analogue mechanical anemometers (AneM) were located at two sides of the bridge deck near the middle of the Ma Wan side span, specified as WITBN01 at the north side and WITBS01 at

500 L.D. Zhu et al.

the south side in Figs. 4 and 5. Each analogue mechanical anemometer consists of a horizontal component, called RM YOUNG 05106 Horizontal Anemometer, and a vertical component, named as RM YOUNG 27106 Vertical Anemometer. Another two analogue mechanical anemometers (AneM) of horizontal component only were arranged at the level of 11 m above the top of each bridge tower. They are specified as WITPT01 for the Tsing Yi tower and WITET01 for the Ma Wan tower in Fig. 4.

The servo accelerometers are of the brand Allied Signal Aerospace Q-Flex QA700. Three different types of arrangement for acceleration measurement were used in the system, namely AccT, AccB and AccU as indicated in Fig. 4, representatively representing Tri-axial measurement (three uni-axial accelerometers assembled orthogonally to each other), Bi-axial measurement (two uni-axial accelerometers assembled perpendicularly to each other) and Uni-axial measurement (by using only one accelerometer to give signal in one prescribed direction). A total of 12 uni-axial accelerometers were located at the four sections of the bridge deck. At each section, there are two accelerometers measuring acceleration in the vertical direction and one accelerometer measuring acceleration in the lateral direction, as shown in Fig.5. A set of three uni-axial accelerometers is located at Ma Wan Anchorage for seismic load measurement.

The measurements of wind speed and bridge acceleration response were carried out during the passage of Typhoon Victor over Hong Kong. The sample frequencies were set as 2.56Hz for recording wind speed and 25.6Hz for recording acceleration response, respectively. The recording duration was 7 hours for each channel, from 17:00 to 24:00 on 2 Aug. 1997. Thus, the data number of each time history is 64512 for wind speed and 645120 for bridge acceleration response.

By using MATLAB as a platform, some programs were developed to analyse the measured data to obtain mean wind speed, turbulent intensity, integral scale of turbulence, friction velocity, gust factor, wind spectrum, acceleration standard deviation response, and response spectrum of acceleration. The variations of these parameters during the passage of Typhoon Victor were also studied. Each sample (time history) of seven hours duration was evenly divided into seven segments. The segment smooth method and the hamming window were applied in the spectral analysis. For wind spectral analysis, one- hour segment was further divided into 11 sub-segments of 10-minute duration, with an overlapped length of 5 minutes between two neighbouring sub-segments. The 1536 data points in the 10-minute sub-segment were zero-padded to 2048 points to meet the requirement of Fast Fourier Transformation (FFT). The frequency resolution in the wind spectral analysis was thus 0.00175Hz. The spectral analysis of acceleration response was based on one-hour segment, the data point number for FFT and the overlapped length were selected as 8192 and 2.6 minutes, respectively. As a result, one-hour segment of acceleration response could be divided into 21 sub-segments which are averaged and smoothed in the spectral analysis, and the frequency resolution was 0.004375 Hz.

MAIN RESULTS: WIND CHARACTERISTICS

The wind characteristics around the Tsing Ma Bridge during the passage of Typhoon Victor varied with time due to the change of wind direction and upwind terrain to the Bridge. The wind characteristics also varied with position due to the size of the Bridge and the nature of a typhoon called the heterogeneity of typhoon planetary boundary layer (TPBL).

Mean Wind

Figures 6 and 7 show variations of 10-minute-averaged mean wind direction and mean wind speed in the horizontal plane with time, respectively. The mean wind was found to be nearly horizontal during the passage of Typhoon Victor at the site of the Tsing Ma Bridge. Figure 6 shows that there is a sudden change of wind direction from north-east to south-west within 20 minutes from 19:50 to 20:10. Correspondingly, the mean wind speed is very small during this period. This is because during this period, Typhoon Victor's eye just crossed over the Bridge. It is also seen from Fig. 6 that the mean wind


to the Bridge blew from north-east within Region I from 17:00 to 19:50, and from south-west within Region V from 21:00 to 22:00, and from south-west within Region VI from 22:00 to 24:00, respectively. The angles between the mean wind measured at the top of the tower and the longitudinal axis of the bridge were about 20 ~ , 34 ~ and 52 ~ , correspondingly. During the period of 20:10 to 21:00, the mean wind direction of Typhoon Victor at the site of Tsing Ma Bridge was unstable and varied from Region II to Region IV due to the landfall on the mountain areas.

The maximum 10-minute mean wind speeds were measured as 12.9m/s at the deck level and 16rn/s at the tower-top level before the Typhoon Victor crossed the Bridge. After the crossing, they became, respectively, l l .7m/s and 14.2m/s during 20:10 to 21:00, 18.5m/s and 21.1m/s during 21:00 to 22:00, and 17.4rn/s and 23.3rn/s during 22:00 to 24:00. The maximum hourly mean wind speeds before the crossing were 10.1m/s at the deck level and 13.9m/s at the tower-top level. After the crossing, they became, respectively, 7.9m/s and 9.3m/s in the duration of 20:00 to 21:00, 14.9m/s and 17.8m/s in the duration of 21:00 to 22:00, and 15.7m/s and 21.2m/s in the duration of 22:00 to 24:00. The highest 10- minute and one-hour mean wind speeds at the tower-top level occurred between 22:00 and 23:00.

Figure 6: Variation of 10min mean wind direction at WITPT01

Figure 7: Variation of 10min mean wind speed at WITPT01

Figure 8: Variation of gust factor with gust duration at WlTJS01

The mean wind speed profile during a typhoon is not well known yet. It also cannot be exactly explored this time, for only two level wind speeds were available. However, by fitting two level mean wind speeds to the power law mean wind profile, it was found that the mean value of exponent for the power law was 0.324 when wind blew from north-east within Region I and 0.199 when wind blew from south- west within Region V. These values are very close to those specified in the Hong Kong Wind Code (1983) for the build up terrain and general terrain respectively.

Turbulence Intensity and Gust Factor

Compared with seasonal trade winds, the turbulence intensity of wind due to a typhoon is relatively higher. The measured largest hourly mean longitudinal turbulence intensity was about 33% at the deck level and 27% at the top level. The measured largest hourly mean lateral turbulence intensity was about 27% at the deck level and 23% at the top level. The corresponding value for vertical turbulence intensity was about 25% at the deck level.

With respect to the gust factor based on the hourly mean wind speed, it is found that for a given longitudinal turbulence intensity, the factor is approximately proportional to the logarithm of gust duration (see Fig. 8). The factor is also almost proportional to longitudinal turbulence intensity for a given gust duration. Thus, by best fitting the measured data, the following empirical formula is obtained for the estimation of the gust factor during Typhoon Victor.

G(T, Iu) : 1 - 0.5377(Iu) 1~ In(T/3600) (1)

where, T is the gust duration in second; I u is the longitudinal turbulence intensity.

502

Wind Auto Spectra

L.D. Zhu e t al.

Friction velocity u, was estimated through the horizontal shear stress (Tieleman and Mullins 1980) at

the deck level. The friction velocities were found to be 1.23m/s, 1.09m/s and 0.86m/s for Regions I, V

and VI, respectively. The reduced auto spectra of three components of fluctuating wind (nSu/U, 2 ,

nS v/u 2 , nS w/u, 2 ) varied strongly with the time due to the change of wind direction and upwind terrain

and also with the height of the anemometer position. However, the slopes of all auto spectra were

approximately equal to -5/3 in the reduced frequency range f > 0.3 for nSu/u 2 and f > 0.6 - 0.9 for

nSv/u , 2 and nS w/u, 2 . The reduced frequency f is equal to nz/U(z), where U(z) is the mean speed at

height z and n is the frequency in Hertz. The auto spectra can be fitted using non-linear least square method with the following objective function.

nSa/u. 2 = a f / 0 +bfl/m) 5m/3 (2)

where the subscript a of S can be u,v, or w; and a, b, m are the parameters to be fitted. The results showed that these parameters scatter in a wide range, depending on the upwind terrain and the position of anemometers. Figures 9 to 11 display the longitudinal, lateral, and vertical wind spectra measured from WITJS01 during the period of 22:00 to 23:00 on 2 August 1997. The corresponding fitted curves together with the von Karman spectra, Kaimal spectra and Simiu spectra are also plotted in these figures (Morfiadakis et al 1995, Kaimal et al. 1972, Simiu & Scanlan 1996). It is seen that the spectra using Eq. 2 can fit the measured spectral data well in both low and high frequency regions. The von Karman spectra, using measured integral scales, fit the measured spectral data better than Kaimal and Simiu spectra, especially in the low frequency region.

Figure 9" nS u/u 2 at WITJS01 Figure 10: nS v/u2. at WITJS01 Figure 11" nS w/u 2 at WITJS01

MAIN RESULTS: ACCELERATION RESPONSE

Standard Deviation and Peak Factor

Due to the limitation of space, only the acceleration responses of the Bridge at the mid-span after the crossing of Typhoon Victor are presented. Figure 12 illustrates the time histories of lateral, vertical and torsional acceleration responses during the period of 22:00 to 23:00. Figure 13 shows variations of the

aL aV aT standard deviations (0-60,0-60,0-60) of lateral, vertical and torsional accelerations with time in 10-minute

aL aV and aT interval. The maximum values of 0-60,0-60 0-60 were found to be 0.588cm/s 2 , 3.082 cm/s 2 and

0.0010 rad/s 2 , occurring at about 21:35, 23:35 and 22:45, respectively. Figure 14 shows the relationship

of O"60aL, (5.60aV , 0-60aT with the 10 minutes mean wind speed (U6o). It can be seen that the vertical and

aV and aT increase almost proportionally to the cube of the mean wind torsional acceleration responses c60 0-60


- - aL increases almost proportionally to the square of the speed U60 while the lateral acceleration response ~60

mean wind speed U60. The maximum values of the one-hour standard deviation acceleration responses

aL aV and aT d u r i n g 22:00"~23:00 were 0.468 cm/s 2 2.050cm/s 2 and 0.0007 rad/s 2 respectively. CY360 , l~Y360 I~Y360 , ,

The maximum peak accelerations during the period of 22:00 to 23:00 were 1.95 cm/s 2 , 9.78 cm/s 2 and

0.0034 rad/s 2 . In terms of one-hour duration, the averages of the measured peak factors for seven hour

duration were 4.86, 4.29 and 5.73, respectively, for lateral, vertical and torsional acceleration responses, but in terms of 10-minute duration they became 3.71, 3.04 and 3.51, respectively.

Probability analysis of acceleration peak factor was also performed. It was found that the peak factor distributions comply with the Gaussian distribution approximately.

Figure 13" Variation of standard deviation of acceleration

Figure 12: Responses of lateral, vertical and torsional accelerations at the mid span (22:00 to 23:00)

Figure 14: Acceleration response via mean wind speed

Figure 15" Acceleration spectra of the Bridge at mid-span (22:00 to 23:00)

Acceleration Response Spectra

Figure 15 illustrates the auto spectra of the lateral, vertical and torsional acceleration responses at the mid-span of the Bridge for the period of 22:00 to 23:00. The first two peak frequencies identified from these spectra were 0.0688Hz (0.068Hz)[0.069Hz] and 0.2656Hz (0.285Hz)[0.297Hz] for lateral

504 L.D. Zhu et al.

acceleration, and 0.2656Hz (0.271Hz)[0.267Hz] and 0.4844Hz (0.475Hz) for torsional acceleration. The first three peak frequencies identified from the vertical spectrum were 0.1375Hz (0.137Hz) [0.139Hz], 0.1813Hz (0.189Hz)[0.184Hz] and 0.325 Hz (0.325Hz)[0.327Hz]. Compared with the numbers in the above parenthesises and square brackets that were obtained by Xu et al (1997) from the eigenvalue analysis and the ambient vibration measurement respectively, one can see that three sets of the natural frequency results are very close. The relative difference of the lateral frequencies is less than 11% and that of the torsional frequencies is less than 2% whilst that of the vertical frequencies is less than 4%. Furthermore, these frequencies were found to remain almost constant during the passage of Typhoon Victor in spite of the variation of the wind speed, wind direction, and upwind terrain.

CONCLUSIONS

The recorded wind and structural response data were analysed in this paper for evaluating wind characteristics and acceleration response of the Bridge. The results show that during Typhoon Victor, both mean and turbulent wind characteristics varied considerably due to the change of wind direction and the upwind terrain. Turbulence intensities and gust factors measured during Typhoon Victor were higher than those due to seasonal trade winds. An empirical formula for gust factor as a function of turbulence intensity and time duration was also provided based on the measured results. It was confirmed that the wind excitation mechanism of the Bridge in the lateral direction was different from that in the vertical direction or the rotation. The alongwind acceleration response of the Bridge was approximately proportional to mean wind speed square while the vertical acceleration and torsional angular acceleration were almost proportional to mean wind speed cubic. Furthermore, the natural frequencies identified from the acceleration response spectra were consistent with those obtained from the ambient vibration measurement or the numerical analysis carried out before.

ACKNOWLEDGMENTS

The work described in this paper was supported by both the Hong Kong Polytechnic University through a studentship to the first writer and the Research Grants Council of Hong Kong through a grant to the second writer (Project No. PolyU 5027/98E). Any opinions and conclusions presented in this paper are entirely those of the writers.

REFERENCES

Code of Practice on Wind Effects: Hong Kong-1983, Building Department, Hong Kong Kaimal J.C., Wyngaard J.C., Izumi Y. and Cote R. (1972), Spectral characteristics of surface-layer turbulence, J. Royal Meteorol. Soc., Vol. 98:132-148. Lau, C.K., Wong K.Y. and Chan K.W.Y.(1998), Preliminary mornitoring results of Tsing Ma Bridge, The 14 th National Conference on Bridge Engineering, Shanghai, Vol.2:730-740 Li P.W., Poon H.T. and Lai S.T. (1998), Observational study of Typhoon Victor (9712) during its passage over Hong Kong, 12 th Guangdong-HongKong-Macau Seminar on Hazardous Weather, Hong Kong (in Chinese). Morfiadakis E.E., Glinou G.L. and Koulouvari M.J. (1995), The suitability of the von Karman spectrum for the structure of turbulence in a complex terrain wind farm, J. Wind Eng. Ind. Aerodyn. Vol 62:237- 257 Simiu E.& Scanlan R.H. (1996), Wind effects on structures, John Wiley & Sons, INC, New York. Tieleman H.W., Mullins S.E. (1980), The structure of moderately strong winds at a Mid-Atlantic coastal site (below 75m), Proceedings of Fifth International Conference on Wind Engineering, Pergamon Press, Oxford, Vol 1:145-159 Xu Y.L., Ko J.M. and Zhang W.S. (1997), Vibration studies of Tsing Ma Suspension Bridge, J. Bridge Eng. ASCE, Vol 2:149-156.

STRUCTURAL PERFORMANCE MEASUREMENT AND DESIGN PARAMETER VALIDATION FOR KAP SHUI MUN BRIDGE

C.K. Lau ~ W.P. Mak ~ K.Y. Wong ~ Deputy Director Chief Engineer Senior Engineer

K.L. M a n I W.Y. Chan ~ K.F. Wong z Engineer Engineer System Analyst

tHighways Department, The Government of the Hong Kong Special Administrative Region /E&M Section, Tsing Ma Management Limited (The Operator of Tsing Ma Control Area)

ABSTRACT

A structural health monitoring system has been operating on the Kap Shui Mun (cable-stayed) Bridge, as part of a strategic road and rail Link in Hong Kong, in order to monitoring and evaluating structural performance of the bridge structures. Valuable field measurement data of the bridge responses under various environmental and traffic conditions are being obtained to establish a baseline reference. This paper briefly describes the system of data acquisition and presents some findings in the evaluation of the structural health conditions of the bridge with reference to the design values.

KEYWORDS

cable-stayed bridge, structural health monitoring, wind, temperature, traffic load, bridge response

INTRODUCTION

Kap Shui Mun Bridge (KSMB), forming part of the Lantau Link in Hong Kong, is the first road and rail cable-stayed bridge ever built in Hong Kong. Since its opening in May 1997, this strategic crossing has been serving as the only road and rail link to Lantau Island and the Hong Kong International Airport (Figure 1). The 430m main span of the double-deck cable-stayed bridge has adopted an innovative steel-concrete composite design in its main span and coupled with concrete side spans at the two ends (Figure 2). As a pioneer in Hong Kong to monitor structural performance of bridges by field measurement, Highways Department (HyD) has installed a permanent On-structure Instrumentation System (OSIS) on the bridge. This OSIS is used to collected the measured information for structural health monitoring and design data validation works.

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506 C.K. Lau et al.

OVERVIEW OF THE ON-STRUCTURE INSTRUMENTATION SYSTEM

The OSIS for KSMB is comprised of a total of 270 sensors of different types permanently installed on the bridge (Figure 2), such as anemometers, accelerometers, level sensing system, strain gauges, temperature sensors, displacement transducers and weigh-in-motion sensors. Streams of data signal. are continuously transmitted from these sensory systems to the controlling computer system (CLFC) located in the Bridge Monitoring Room at 1 ~t Floor of the Tsing Yi Administration Building. In the transmission network, there are acquisition computers which are housed inside air-conditioned cabinets of the two Outstation Units located inside the lower deck of the bridge. Analogue and digital signals are collected into these acquisition computers via 313 numbers of signal channels and undergo signal conversion with appropriate signal conditioning and amplification devices before transmitting via a token-ring network over fiber optic cables to CLFC in Bridge Monitoring Room. The measured data are then processed by tailor-made programmes, operating in Unix platform, for real-time display of strategic data as well as subsequent data analysis and archiving.

WIND LOAD MONITORING

The cable-stayed bridge was designed to withstand an hourly mean wind speed of 50 rn/s and a maximum 3-second gust wind speed of 80 m/s. The design wind speeds under different live load conditions on deck structure are given in Table 1 below.

TABLE 1 DESIGN WIND SPEEDS FOR BRIDGE STRUCTURES

Live Load Conditions on Deck Structure

Without Highway and Railway Live Loads With Combined Highway & Railway Live Loads With Railway Live Load

Mean Hourly Wind Speed

at Deck Level

50 m/s 25 m/s 28 m/s

Max. Wind Gust Speed (rru's) Horizontal Wind loaded length <_20m 100m 600m 1000m

80 72 65 63 44 38 34 33 50 43 39 38

Min. Gust Wind Speed

50m/s 25 m/s 28 m/s

The above wind design parameters were derived from available statistical wind records obtained at Waglan Island which is situated about 30 kilometers from the bridge site. Different topography would have influence on the wind parameters. A database established from the wind measurement data is used to verify the design wind parameters. Two anemometers of ultrasonic type are installed at deck level of around 59mPD and positioned at mid-span on outrigger trusses on both sides of the deck. Wind rose diagram is derived from the wind measurement, showing wind speeds, wind directions and frequencies of occurrence, as shown in Figure 3 for a two-year period since the opening of the bridge. The measurement record illustrates a relatively mild condition of wind speed in the past two years.

TEMPERATURE MONITORING

The design temperature data specified for KSMB was generally adopted from Structural Design Manual [3]. These design values were derived by theoretical approach based on experimental data previously used in deriving the temperature data for design of concrete bridges under Hong Kong climatic conditions. With temperature measurement in place, the range of design temperature parameters specific for this steel composite bridge can be verified. Temperature sensors are installed in various locations of the bridge. Temperatures are recorded in structural steel beams at top and

Structural Performance Measurement for Kap Shui Mun Bridge 507

bottom deck levels, steel cladding on fascia structures, concrete tower legs, road asphalt on upper carriageways. Air temperatures above, inside and below the deck are also recorded. Differential temperature within the bridge as induced by solar radiation and re-radiation can then be identified. The theoretical effective temperature, with appropriate weighting of cross-sectional areas, can also be verified by calculation. The design range of effective bridge temperature for different structural components are tabulated in Table 2 below for information.

TABLE 2 DESIGN EFFECTIVE BRIDGE TEMPERATURE

Structural Components Design Effective Bridge Teml

Steel Composite Main Span *

Concrete Side Span * 36 ~

Stay Cables 50 ~

Concrete Towers 36 ~ Note : * Thickness of deck surfacing is 100mm.

perature for 120-year return period Maximum Minimum

40 ~ 0 ~

0~ -2 ~

-1 ~

Figure 4 illustrates the variation of temperature recorded in bridge-deck over a 12-month period. Both shade air temperature and calculated effective bridge temperatures exhibit a typical seasonal variation. During the period, the maximum and minimum effective bridge temperatures of 36 ~ and 8 ~ were recorded respectively. It demonstrates that temperature variation in bridge-deck has remained within the design temperature range of 40 ~ and 0 ~ (for a 120-year return period).

Since the alignment of the bridge orientates in a North-East to South-West direction at 55 ~ from the North, it is anticipated that the solar radiation along the locus path from sunrise to sunset will induce a thermal gradient across the hollow section of the twin tower legs. Platinum resistance thermometers installed at 48m above deck level are used to monitor the temperature of concrete tower legs. Typical variations of temperature gradient across the hollow section during a hot summer period of 72 hours are shown in Figures 5 and 6. The temperatures on east face of tower leg build up quicker in the morning while west face is still under shade. The temperatures on west face begin catching up after mid-day when the sun is high up. There is also a gradual reversal of temperature gradient in late afternoon between outside (sensor H) and inside face (sensor A) of concrete tower wall as the shade temperature inside tower legs remain reasonably constant throughout the day. The difference in temperatures between the two faces of wall has reached a maximum of 5 ~ which is still within the permissible range. The corresponding ambient temperature is also plotted for comparison. Incidentally, there was a shower in the morning on the first day which caused the temperature drop.

VEHICULAR TRAFFIC LOAD MONITORING

In the design of highway live loads on bridges, various combinations of vehicular loading patterns are explored to encompass the most adverse configuration of vehicular traffic loads. The vehicular traffic loads and associated load factors are derived by statistical simulation on the basis of projected future growth of traffic volume and the proportion of heavy goods vehicle population amongst other classes of vehicles. The distribution of different classes of vehicles in traffic jams is in fact the governing factor in deriving the vehicular traffic loads. The loaded length used in design also depends on frequency of daily traffic jams, locations of traffic jams, duration and distribution of vehicular types as well as traffic flow during traffic jams. For Lantau Link, a weigh-in-motion (WIM) sensory system is installed on both bounds of the carriageways near the Lantau Toll Plaza. The WIM system (bending

508 C.K. Lau et al.

plate type) is used to record vehicle count, to measure axle weight and travelling speed and to identify type of vehicles crossing the bridges. A database is being established from these statistical records to verify the population of heavy goods vehicles. Correlation can then be made with the design HA lane factor of 3.6 adopted in highway live load design. Figure 7 is a monthly average statistics of daily vehicle count and proportion of different types of goods vehicles in traffic population. Population of goods vehicle is demonstrated to be around 34%, which is well below the design value of 60%.

DYNAMIC RESPONSE MONITORING

As a means to monitor the global dynamic characteristics of the bridge, accelerometers are installed on the deck to record its ambient vibration status. From accelerometer data, natural frequencies are derived. Mode shapes and modal damping values of the deck are subsequently determined. Since there are only three uni-axial accelerometers permanently installed at mid-span section of the deck, additional measurement points were established on the deck by setting up portable measuring equipment at four sections of the main span to obtain additional deck vibration data over short periods of time. The measured and computed frequencies of the deck are in Table 3 below.

TABLE 3 COMPARISON OF COMPUTED AND MEASURED NATURAL FREQUENCIES

Mode No.

Computed Frequencies (Hz)

Consultant[ HyD

1 0.378 0.41 2 0.509 0.58 3 0.645 0.93 4 0.955 1.51 5 1.199 1.74 6 1.296 2.81 7 1.483 2.88

1 0.383 0.49 2 0.876 1.15 3 1.035 2.45 4 1.298 3.06 5 N/A 3.39

1 0.676 0.77

2 1.137 1.62

3 1.442 2.18 4 N/A 2.69

5 N / A 3.35

Measured Type of Mode Shape Frequencies (Hz)

University HyD ! (A) For Deck Vertical-dominant Modes

0.39 0.39 Vertical 1 i i

0.66 N / A Towers i i

1.07 0.68 Vertical 2 i i

1.54 1.05 Vertical 3 i |

1.81 N/A Towers 2.71 ' 1.35 ' Vertical 4 (Towers in phase) 3.08 1.53 . Vertical 5 (Towers out of phase)

(B) For I)eck Lateral-dominant Modes 0.49 0.41 Lateral 1

i i ,

1.25 0.90 Lateral 2, Towers, Torsion 2 | |

2.12 N/A Torsion 2, Lateral 2, Towers i |

2.93 1.31 Ma Wan Side-span Lateral, Tower ! i

3.20 N/A (C) For l)eck Torsion-dominant Modes

0.83 0.81 Torsion 1 I I

1.39 1.18 Torsion 2 i !

1.90 1.77 Torsion 3 I I

2.56 N/A Torsion 4 m j

3.39 N/A : Torsion 5

In above table, the natural frequencies previously computed by Consultant by means of computer model simulation are presented for comparison. With the limited number of measurement points on the deck, the mode shapes of towers and side spans, coupling modes of deck, towers and stay cables cannot be identified. A university was once commissioned to conduct a field measurement of ambient vibration of the cable-stayed bridge with a total of 34 measurement points (18 nos. along the deck, 4

Structural Performance Measurement for Kap Shui Mun Bridge 509

nos. at each tower and 8 nos. in stay cables). Their corresponding measured values are also tabulated for comparison. However, the two measurement results do not provide sufficient agreement to ascertain the correctness of the measured natural frequencies. Further measurement would be required to verify the values. In structural design of this road and rail bridge, there are certain criteria stipulated by railway operation to ensure acceptable levels of comfort for users of the railway. Some of these are more stringent than those required for highway traffic. One of them is the limiting of vertical acceleration of the deck. The measured vertical acceleration of the deck has demonstrated that the deck dynamic response is far below the design normal maximum value of 0.05g and only occasionally approaches the threshold of desirable upper limit of 0.03g as shown in Figure 8.

STRAIN AND STRESS MONITORING

As the bridge is constantly responding to the ever-changing environmental loading and fluctuating live load in addition to static permanent loads, various structural components of the bridge are subject to varying strain and stress conditions. Monitoring of strain in certain critical structural components by strain gauges and level sensing system can provide a health histories of the bridge and establish baseline references for damage detection/assessment. During June 1999, Typhoon Maggie approached Hong Kong and there was a change in direction of her path. Records of structural responses in some of the instrumented structural components are extracted for a 60-minute period as presented in Figure 9 to 11. It is noticed that the change of wind incidences has induced a corresponding changes in bridge responses, causing an observable reduction in both the vertical oscillation of the deck and variation in strain/stress levels recorded in the sensory systems. Nevertheless, the strain/stress levels were well below permissible design values.

CONCLUSION

The structural health monitoring system enable continuous collection of field measurement data, reflecting different bridge responses under various types of environmental and applied loads. A health history with baseline references can be established over times. Current structural health monitoring work shows that the bridge is in a healthy condition.

ACKNOWLEDGEMENT

The authors express their thanks to Director of Highways, Mr. K.S. Leung, for permission to publish this paper. Any opinions expressed or conclusions reached in the text are entirely those of the authors.

REFERENCES

1. Lau, C.K. and Wong, K.Y., "Design, Construction and Monitoring of the Three Key Cable-supported Bridges in Hong Kong", Proceedings of" the Fourth International Conference on Structures in the New Millennium", 3-5 September 1997 in Hong Kong, A.A. Balkema, Rotterdam, Netherlands.

2. Lantau Fixed Crossing Project Management Office, Highways Department, "Structural Health Monitoring System", Highway Contract No. HY/93/09 - Electrical and Mechanical Services in Lantau Fixed Crossing, The Hong Kong Government, 1993.

3. Highwaysn Department, "Structures Design Manual", The Government of Hong Kong Special Administrative Region, 1997.

510 C

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Structural Perform

ance Measurem

ent for Kap Shui M

un Bridge

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512 C

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FREE AND FORCED VIBRATION OF LARGE-DIAMETER SAGGED CABLES TAKING INTO ACCOUNT

BENDING STIFFNESS

Y. Q. Ni, J. M. Ko and G. Zheng

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong

ABSTRACT

In this paper, a finite element formulation for vibration analysis of large-diameter structural cables taking into account bending stiffness is developed. This formulation is suitable for suspended and inclined cables with sag-to-span ratio not limited to being small. The proposed method provides a tool for accurate evaluation of natural frequencies and mode shapes of structural cables. A numerical verification is made by analyzing with the proposed method the modal properties of a set of cables with different degrees of bending stiffness and sag extensibility and comparing them with the results available in the literature. The modal behaviour and dynamic response of the main cables of the Tsing Ma Bridge in free cable stage are also predicted and compared with the measurement results.

KEYWORDS

Sagged cable, bending stiffness, finite element method, modal property, transient dynamic response, three-dimensional analysis.

INTRODUCTION

Advance in modem construction technology has resulted in increasing application of large-diameter structural cables in long-span cable-supported bridges. The Tsing Ma Bridge, a suspension bridge with the main span of 1377m, has been built recently in Hong Kong. As a result of carrying both road and rail traffic, the Tsing Ma Bridge has the most heavily loaded cables in the world. The cable section of the Tsing Ma Bridge is about 1.1m in diameter after compacting. The Akashi Kaikyo Bridge in Japan, which is the world's longest suspension bridge with the main span of 1990m, also has the main cables of about 1.1m diameter. It is well known that the existing theory for cable analysis is developed on the assumption that the cable is perfectly flexible and only capable of developing uniform normal stress over the cross-section. For the large-diameter structural cables, however, the effect of the bending stiffness should be not negligible in performing accurate dynamic analyses.

513

514 Y.Q. Ni et al.

Stay cables are the most crucial elements in cable-stayed bridges. Changes in cable forces through degradation or other factors affect internal force distributions in the deck and towers and influence bridge alignment, and are therefore important in assessing the structural condition. The dynamic method has been applied to quantitative measurement of cable tension forces (Takahashi et al. 1983; Okamura 1986; Kroneberger-Stanton and Hartsough 1992; Casas 1994). In most of these applications, the cables were idealized as taut strings by ignoring the sag and bending stiffness effects. This idealization simplifies the analysis but may introduce unacceptable errors in tension force evaluation (Casas 1994; Mehrabi and Tabatabai 1998). Recent research efforts in this subject have been devoted to the development of accurate analytical models to relate the modal properties to cable tension by considering bending stiffness and/or sag extensibility (Zui et al. 1996; Yen et al. 1997; Mehrabi and Tabatabai 1998; Russell and Lardner 1998). Zui et al. (1996) derived empiric formulas for estimating cable tension from measured frequencies, which took into account the effects of bending stiffness and cable sag. Yen et al. (1997) proposed a bridge cable force measurement scheme that considered sag extensibility, bending stiffness, end conditions, and intermediate springs and/or dampers. Mehrabi and Tabatabai (1998) developed a general finite difference formulation for free vibration of a fiat-sag horizontal cable accounting for sag extensibility and bending stiffness. Russell and Lardner (1998) related the tension to natural frequencies of an inclined cable through formulating a curvature equation

which took into account the sag extensibility.

This paper presents a new finite element formulation for free and forced vibration of cables. This formulation takes into account the combined effects of all important parameters involved, such as sag extensibility, bending stiffness, end conditions, cable inclination, and lumped stiffness and mass. The formulation is first developed for the pure cable without considering bending stiffness. Then the additional contribution of the flexural, torsional and shear rigidities to stiffness matrix is derived by reference to a curvilinear coordinate system. Analytical results using the proposed formulation are verified with available results in the literature and compared with experimentally measured modal data

of bridge cables.

FINITE ELEMENT FORMULATION

Three-Node Curved Element of Pure Cable

Without losing generality, the cable static equilibrium profile is assumed in the x-y plane as shown in Figure 1. This initial (static) configuration is defined by x(s) and y(s), here s denotes the arc length coordinate. Let L, E, A and m be the cable length, modulus of elasticity, cross-sectional area, and mass per unit length respectively. In static equilibrium state, the cable is subjected to dead loads (cable self weight and lumped masses) and the cable tension is H(s). The cable is then subjected to the action of

dynamic external forces px(s, t), py(s, t), and pz(s, t). The dynamic configuration of the cable is described by the displacement responses u(s, t), v(s, t), and w(s, t) measured from the position of static

equilibrium in the x-, y- and z-directions respectively. Let U = {u(s, t) v(s, t) w(s, t)} r and P = {px(s, t) p~(s, 0 pz(s, O} T.

By using the Lagrangian strain measure, the extensional strain in the cable due to dynamic loads,

ignoring bending stiffness, can be expressed as

~ eO+el (dx Ou dY Ov 1 ~s 2 -~s z --~s . . . . ds Os+--'ds Os)+2"[( ) +( ) +( )2] (1)

The finite element formulation is derived from the Hamilton's principle

Free and Forced Vibration of Large-Diameter Sagged Cables 515

OU 51 = 5~ I~ . . . . . . . . OUrc3t OUc3t V,. EA2 ~2 . . . . . H(s) e]dsdt + ~ ~SU r (q + P c Ot )dsdt = 0 (2)

where V,. is the elastic strain energy stored in the initial state; q is the dead load vector existent in the initial state; c = diag[cx Cy ez] is the viscous damping coefficient matrix.

An isoparametric curved element with three-nodes is introduced to describe the cable. As shown in Figure 1, the shape functions in the natural coordinate system are given by

1 ( 1 _ ~ 2 ~2 _1 1 2 N 1 = �89 - ~- ) , N 2 = 1 - ' N3 -2 (1 + ~ ) - 5 (1-~, ) (3)

and the coordinates and the displacement functions are expressed as

x = Z Nixi, Y = Z NiYi (4)

U = Z N i u i , v = Z N i v i , w = Z N i w i (5)

By defining nodal displacement vector r}~ }r {5}={{8}[ {8} r {8}3 ={uljvljwlju2jv2jw2ju3jv3jWaj (6)

Eqn. 5 can be expressed as T T U={u v w} r =[N,I N2I N3I ] {{5} r {5}2 r {8}3 } =[N]{8} (7)

Substituting Eqns. 4 to 7 into Eqn. 1 yields

e o =[Bo]{d}=[{Bol } {B02 } {B03}]{6}, e t =[Bt]{d}=[{BI1 } {Bt2 } {B13}]{6 } (8a, b)

, , 1 {u'N; v'N; w'N;} (9a, b) {B0i } = -55-1{x'N[ y N i 0} , {Bli } = --2j 2

where J = ds/d~ and the prime denotes the derivative with respect to ~.

After substituting Eqns. 7 to 9 into Eqn. 2, integrating Eqn. 2 by parts, and considering the static equilibrium equation in the initial state, the governing motion equation for the elementj is derived as

[M/]{~jI+[Cs]{fJSI+[Koj + Kls({Uj})+ K2j({UjI{ujIT)]{Uj} = {Pj.} (10)

in which,

[Mj] =mJ~+][N]r[N]d~, , [Cj] =JI+([N]r[c][N]d~ ( l la , b)

{Pj} =J~+~[N] r{PId~ , [K0j] = EAJ~+][Bo]r[Bo]d{ +I~+~H[N']r[N']d~ (11c, d)

[ K l j ] = EAJ~+~([Bt]T[Bo]+ 2[Bo][Bt])d~ , , [ K 2 j ] = 2EAJ~+~[Bt]T[Bt]d~ ( l le , f)

(a) 1 (xlj, Yv)

y tj

_= x ~ 3 (x3j, y3j) ,r 7

z

(b)

~=-1 ~=0 ~=+1

1 2 ~' 3

Figure 1. Three-Node Curved Cable Element: (a) Physical Coordinate; (b) Natural Coordinate

516 Y . Q . N i e t al.

The global equation of the cable is then obtained through assembling the element mass matrix, damping matrix, stiffness matrix, and nodal load vector by the standard assembly procedure. It is noted that in Eqn. 10 the stiffness matrix includes linear stiffness [K0], cubic stiffness [K1] and quadratic stiffness [/(2]. The present study only addresses linear problem of cable dynamics by discarding the nonlinear stiffness terms. The nonlinear dynamic analyses of cables refer to Ni et al. (1999a, b).

Cable Taking into Account Bending Stiffness

The additional stiffness matrix stemming from flexural rigidities of the cable is derived with respect to the same curved element as shown in Figure 1. However, a new local coordinate system in terms of tangential and normal axes is introduced to relate displacements with stress resultants. As shown in Figures 2 and 3, the nodal displacements at any node i are expressed as

{ t ~ } i - - {U i V i W i ~si ~ti Ozi }T (12)

where ui is the in-plane displacement in tangential direction; v; is the displacement in transverse

direction; wi is the displacement in z-direction; Osi is the total rotation in tangential direction; 0ti is the

angle of twist; and Ozi is the total rotation of transverse bending. Similar to Eqn. 7, the displacement vector is expressed with isoparametric interpolation functions as

{U}={u v w O~ 0 t O z } r = [ N l I N 2 I N3I]{{b'}~ {b'}~ {8}~}r=[N]{b "} (13)

The strain vector is written as

{ s } - {x" z K" t a Yv, Yw~ } (14)

where Ir and x~ are the in-plane and out-of-plane curvature changes respectively; a is the cross-

sectional torsion change; Yvs and Yws are the shear strains. They are expressed as

aO z 1 ~u a~s ~t K" z - d - - - . , K" t = (15a, b)

as R as as R

ao, Os ~ aw ct . . . . as R ' Yvs = -~s - O z , Yws +0s (15c, d,e)

in which R is the curvature radius of the element. It is noted that R is not a constant in the case of sagged cables. It is calculated using the formulae

R [1 (dd_~Yx) ]3/2 dx "d2y) = + 2 /( _--z--w- (16)

The strain-displacement relation can be obtained from Eqns. 13 and 15 as

{s} = [B ] {J ' } =[ [B1] [B:z] [B3]]{J'} (17)

i~si S

\ Oz~ / 5 / o,,

I ~-X

Iz

Msi s

Mzi R rds I",

~ X

Figure 2. Displacements at Node i Figure 3. Stress-Resultants at Node i

in which,

Free and Forced Vibration of Large-Diameter Sagged Cables

I N; 0 0 0 0 1 0 0 - R . N ' - J . N i

, 0 o 0 [Bi ] "- ~ R - ; 0 0 0

L0 o R.N~ R J . N i 0

and the stress-strain relationship is given by

R . N ;

0

0

- R J . N i

0

517

(18)

{o-} = {M z m t T V z V s }T =diag[Eiz E1 s GJ flGA flGA]{g} = [D]{e'} (19)

The additional element stiffness matrix due to flexural rigidities is obtained from Eqns. 17 and 19 as

tXa I : JI _ [Sl [OISld (20)

The additional stiffness matrix given in Eqn. 20 is derived by referring to the local coordinate system.

It should be transformed into the element stiffness relation in the global x-y-z coordinate system before

performing assembly to obtain overall stiffness matrix. Similarly, the element stiffness matrix given in

Eqn. 11, with 9• dimension, should be expanded as an 18x 18 matrix to cater for the rotation degrees

of freedom.

NU MERICAL VERIFICATION

The proposed formulation has been encoded into a versatile finite element program. In this section, a

numerical verification is conducted through comparing the computed results by the present method

with the analytical results available in Mehrabi and Tabatabai (1998). They used the string equation

and a finite difference formulation to calculate the frequencies of the first two in-plane modes of four

suspended cables with a same length (100m) but different sag-extensibility (~2) and bending-stiffness

(~) parameters. Table 1 shows the parameters of the four cables (the definition of the parameters refers

to the reference). Cable 1 (~2 = 0.79, ~ = 605.5) has a moderate sag and a low bending stiffness; Cable

2 (L2 = 50.70, ~ = 302.7) has a large sag and an average bending stiffness; Cable 3 (~2 _ 1.41, ~ =

50.5) has a moderate sag and a high bending stiffness; Cable 4 (~2 = 50.70, ~ = 50.5) has a large sag

and a high bending stiffness. Modal properties of the four cables are analysed by using the proposed

finite element formulation. The static profiles of the cables are assumed as parabolas. Sixty equi-length

Cable No. ~2

1 0.79 605.5 2 50.70 302.7 3 1.41 50.5 4 50.70 50.5

TABLE 1 Cable Parameters

m (kg/m) g (N/kg) L (m) H (106N) E (Pa) A (m 2) J (m 4)

400.0 9.8 100.0 2.90360 1.5988e+10 7.8507e-03 4.9535e-06 400.0 9.8 100.0 0.72590 1.7186e+10 7.6110e-03 4.6097e-06 400.0 9.8 100.0 26.13254 2.0826e+13 7.8633e-03 4.9204e-06 400.0 9.8 100.0 0.72590 4.7834e+08 2.7345e-01 5.9506e-03

TABLE 2 Comparison of Computed Frequencies of In-Plane Modes (Hz)

Cable String equation No. ~2 ~ 1st mode 2nd mode

1 0.79 6 0 5 . 5 0.426 0.852 2 50.70 302 .7 0.213 0.426 3 1.41 50.5 1.278 2.556 4 50.70 50.5 0.213 0.426

Finite difference method 1 st mode 2nd mode

0.440 0.428 1.399 0.447

0.853 0.464 2.679 0.464

Present method 1 st mode 2nd mode

0.441 0.854 0.421 0.460 1.400 2.682 0.438 0.461

518 Y.Q. Ni et al.

elements are used in the computation. Table 2 presents a comparison of the computed natural frequencies obtained by the string equation, the finite difference formulation and the present method. It is observed that for all the four cases the results by the present method match well with those by the finite difference formulation (both the methods take into account sag extensibility and bending stiffness). It is also seen that the computed natural frequencies using the taut string equation (ignoring

sag extensibility and bending stiffness) are quite different from those calculated with the present method and finite difference formulation, indicating a considerable influence of the sag extensibility

and bending stiffness.

CASE STUDY: TSING MA BRIDGE CABLES

The modal properties of the Tsing Ma Bridge (Figure 4) in different construction stages have been measured through ambient vibration survey (Ko and Ni 1998). One stage under measurement is the free cable stage. In this stage, only the tower-cable system was erected but none of deck segments has been hoisted into position. The modal parameters of the main span cable and the Tsing Yi side span cable in the free cable stage are calculated and compared with the measurement results. The cable length and sag are 1397.8m and 112.5m for the suspended main span cable and 329.1m and 5.7m for the inclined Tsing Yi side cable. The horizontal component of the tension force is 122642 kN for both the cables. The main span cable is partitioned into 77 elements and the Tsing Yi side span cable is partitioned into 17 elements in the computation. The analyses are conducted by assuming the cable supports to be pinned ends and fixed ends respectively. It is seen in Tables 3 and 4 that the computed natural frequencies of both in-plane and out-of-plane modes of the two cables agree favorably with the

measurement results.

The dynamic responses of the two cables under in-plane and out-of-plane excitations are then analyzed using the present method. For the damped response analysis, the damping is assumed in the Rayleigh

damping form [C] = cz[M] + [3[K] with the coefficients cz = 0.05 and 1~ = 0.01. Figure 5 shows the predicted lateral dynamic response of the damped Tsing Yi side span cable at the cable midspan when an out-of-plane pulse excitation is laterally exerted at the same position. The pulse excitation is F(t) =

500 kN for 0 < t < tcr = 0.5s. Figure 6 illustrates the predicted vertical dynamic response of the damped Tsing Yi side span cable at the cable midspan when an in-plane harmonic excitation is vertically

applied at the same position. The harmonic excitation is F(t) = F0.cos2nfi with F0 = 500 kN and f = 0.03569 Hz. It is observed that the damped dynamic response rapidly attains the steady state after several cycles, having the response frequency identical to the exciting frequency. Figure 7 shows the

Figure 4. Elevation of Tsing Ma Bridge

Free and Forced Vibration of Large-Diameter Sagged Cables 519

TABLE 3 Natural Frequencies of Main Span Cable in Free Cable Stage (Hz)

Mode Out-of-plane modes In'plane modes No. 1 st 2nd 3rd 1 st 2nd 3rd

Computed: pinned ends 0 . 0 5 2 2 0 . 1 0 4 0 0 .1 5 5 7 0 .1008 0 . 1 4 7 1 0.2081 Computed: fixed ends 0 . 0 5 2 8 0 . 1 0 5 2 0 .1 5 7 8 0 .1 0 2 0 0 . 1 4 8 8 0.2091

Measured 0.0530 0 . 1 0 5 0 0 . 1 5 6 0 0 .1 0 2 0 0 . 1 4 3 0 0.2070

TABLE 4 Natural Frequencies of Tsing Yi Side Span Cable in Free Cable Stage (Hz)

Mode No.

Computed: pinned ends Computed: fixed ends

Measured

Out-of-plane modes 1st 2nd 3rd

0.2352 0 . 4 6 9 6 0.7154 0.2450 0 . 4 9 4 6 0.7534 0.2360 0 . 4 7 7 0 0.7400

In-plane modes 1st 2nd 3rd

0.3527 0 . 4 6 9 3 0.7216 0.3569 0 . 4 9 4 3 0.7593 0.3430 0 . 4 7 8 0 0.7310

Figure 5. Damped Response of Cable Midspan under Lateral Pulse Excitation (tcr = 0.5s)

Figure 6. Damped Response of Cable Midspan under Vertical Harmonic Excitation

Figure 7. Undamped Response of Cable Midspan under Vertical Harmonic Excitation

520 Y.Q. Ni et al.

vertical dynamic response of the undamped main span cable at the cable midspan when a vertical

harmonic excitation is exerted at the same position. The harmonic excitation is F(t) = F0-cos2nfi with

F0 = 1000 kN and f = 0.05 Hz. Both the responses with and without consideration of bending stiffness

are given. The difference of response amplitudes between the two sequences is not significant, while

the transient responses at a same instant may be distinct from each other due to the phase shift.

CONCLUDING REMARKS

This paper reports on the development of a finite element formulation for free and forced vibration

analysis of structural cables taking into account both sag extensibility and bending stiffness. The

predicted results by the proposed formulation agree favorably with the analytical results available in

the literature and with the measurement results of real bridge cables. The numerical simulations show that the cable bending stiffness contributes a considerable effect on the natural frequencies when the

tension force is relatively small, and affects higher modes more significantly than lower modes. The

proposed method will be used to provide the training data required for developing a multi-layer neural

network for identifying the cable tension from measured multi-mode frequencies. By interchanging the

input and output roles in the training of the network, a functional mapping for the inverse relation can

be directly established using the neural network which then serves as a tension force identifier.

ACKNOWLEDGEMENTS

This study was supported by The Hong Kong Polytechnic University under grants G-YW29 and G-

V785. These supports are gratefully acknowledged.

References

Casas J.R. (1994). A Combined Method for Measuring Cable Forces: The Cable-Stayed Alamillo Bridge, Spain. Structural Engineering International 4:4, 235-240.

Ko J.M. and Ni Y.Q. (1998). Tsing Ma Suspension Bridge: Ambient Vibration Survey Campaigns 1994-1996. Preprint, The Hong Kong Polytechnic University.

Kroneberger-Stanton K.J. and Hartsough B.R. (1992). A Monitor for Indirect Measurement of Cable Vibration Frequency and Tension. Transactions of the ASAE 35:1, 341-346.

Mehrabi A.B. and Tabatabai H. (1998). Unified Finite Difference Formulation for Free Vibration of Cables. ASCE Journal of Structural Engineering 124:11, 1313-1322.

Ni Y.Q., Lou W.J. and Ko J.M. (1999a). Nonlinear Transient Dynamic Response of a Suspended Cable. Submitted to Journal of Sound and Vibration.

Ni Y.Q., Zheng G. and Ko J.M. (1999b). Nonlinear Steady-State Dynamic Response of Three-Dimensional Cables. Intermediate Progress Report No. DG1999-03C, The Hong Kong Polytechnic University.

Okamura H. (1986). Measuring Submarine Optical Cable Tension from Cable Vibration. Bulletin of JSME 29:248, 548-555.

Russell J.C. and Lardner T.J. (1998). Experimental Determination of Frequencies and Tension for Elastic Cables. ASCE Journal of Engineering Mechanics 124:10, 1067-1072.

Takahashi M., Tabata S., Hara H., Shimada T. and Ohashi Y. (1983). Tension Measurement by Microtremor- Induced Vibration Method and Development of Tension Meter. IHI Engineering Review 16:1, 1-6.

Yen W.-H.P., Mehrabi A.B. and Tabatabai H. (1997). Evaluation of Stay Cable Tension Using a Non- Destructive Vibration Technique. Building to Last Structures Congress." Proceedings of the 15th Structures Congress, ASCE, Vol. I, 503-507.

Zui H., Shinke T. and Namita Y. (1996). Practical Formulas for Estimation of Cable Tension by Vibration Method. ASCE Journal of Structural Engineering 122:6, 651-656.

STABILITY ANALYSIS OF CURVED CABLE-STAYED BRIDGES

Yang-Cheng Wang I , Hung-Shan Shu ! and John Ermopoulos 2

l Department of Civil Engineering, Chinese Military Academy, Taiwan, ROC P.O. Box 90602-6, Feng-Shan, 83000, Taiwan, ROC

2 Department of Civil Engineering, National Technical University of Athens 42 Patission Street, 10682--Athens, Greece

ABSTRACT

The objective of this study is to investigate the stability behaviour of curved cable-stayed bridges. In recent days, cable-stayed bridges become more popular due to their pleasant aesthetic and their long span length. When the span length increases, cable-stayed bridges become more flexible than the conventional continuous bridges and therefore, their stability analysis is essential. In this study, a curved cable-stayed bridge with a variety of geometric parameters including the radius of the curved bridge deck is investigated. In order to study the stability effects of the curved cable-stayed bridges, a three-dimensional finite element model is used in which the eigen-buckling analysis is applied to find the minimum critical loads. The numerical results first indicate that as the radius of the bridge deck increases the fundamental critical load decreases. Furthermore, as the radius of the curved bridge deck becomes greater than 500m, the fundamental critical loads are not significantly decreased and they are approaching to those of the bridge with straight bridge deck. The comparison of the results between the curved bridges with various radiuses and that of a straight bridge deck determines the curvature effects on stability analysis. In order to make the results useful, they are non-dimensionalized and presented in graphical form, for various values of the parameters that are interested in the problem.

KEYWORDS

Stability Analysis, Curved, Cable-Stayed Bridges, Bridges, Buckling

INTRODUCTION

Cable-stayed bridges have been known since the beginning of the 18th Century (Leonhardt, 1982 and Chang et al., 1981), but they have been widely used only in the last 50 years (O'Connor 1971, Troitsky 1988). The span length of cable-stayed bridges increases (Ito 1998 and Wang 1999a) due to the use of computer technology and the high strength material; some of them have curved decks due to the pleasant aesthetic and the functional reasons (Menn 1998, and Ito 1998). These structures

521

522 Y.-C. Wang et al.

utilize their material well since all of their components are mainly axially loaded (Wang 1999b). The geometric nonlinearity induced by the pylon, the deck and the cables' arrangement influences the analysis results (Ermopoulos et al. 1992, Troitsky 1988, and Xanthakos 1994), especially for the curved cable-stayed bridges. Generally, this influence is small, but if the pylons and the deck are flexible, and cables' slope is small, then this influence becomes significant and stability analysis may be necessary. In this paper an elastic stability analysis of a cable-stayed bridge with two pylons and curved deck is performed. The considered loads include a uniform load along the entire span and a concentrated moving load. A nonlinear finite element program and the Jocobi eigen-solver technique are used to determine the critical loads and their corresponding buckling mode shapes. The results are presented in graphical form for a wide range of the parameters of the problem.

GEOMETRY AND LOADING

The geometry, the notation, and the loading of the curved cable-stayed bridges structural model are presented in Figure 1. The bridge is symmetric and is composed of three major elements: (a) the bridge deck with various radiuses ranging from 250m to infinity, i.e., straight roadway, (b) the two pylons and (c) the cables. Two cases of the bridge span lengths are considered. In Case I (Figure 1) the projective length of the bridge remains constant no matter what is the radius, and in Case II (Figure 2) the total curved bridge length remains 460m; the bridge deck has a constant cross-section along the whole span. It is supported at the ends of the both side spans by rollers while at the intersection points with the pylons is attached with a pinned connection. The pylons are fixed at their bases; they have a constant cross-section and their intersection with the deck lies on the one third of their total height from the supports. The projective distance between the two pylons is Ll=220m; the projective distance of the side spans is L 2 =120m each, for both cases. The height H

of the pylon above the deck varies between 0.165 x L and 0.542x L. These limiting values

correspond to the top cable's slope of 20 ~ and 50 ~ , respectively. The ratio Ip/I b (where Iv is the

moment of inertia of the pylon and I b is the moment of inertia of the bridge deck) varies between

0.25 and 4. In order to take the cables' arrangement into account in buckling analysis, the distance d is introduced as shown in Figure 1. The ratio d/H varies between 0.2 (harp-system) and 0.95 (fan- system). The cables are of constant cross-section, they support the deck every 20m and are attached to the pylons by hinges.

Figure 1 (a) Side view and (b) Plan View of the Curved Cable-Stayed Bridge Case I: with variable total curved length

Stability Analysis of Curved Cable-Stayed Bridges 523

Figure (2) Plane View of the Curved Cable-Stayed Bridge Case II" with constant total curved length

Element

Deck

Pylon

Cable

TABLE 1 ELEMENT'S PROPERTIES

Area (m 2 )

0.300

0.100

Moment of Inertia (m 4 )

0.200

0.050 0.300 0.200 0.500 0.005

0.800 . -

Table 1 shows the area and the moment of inertia used in different elements. The Young's modulus (E) is taken to be 21 x 10 6 t/m 2 for the deck and the pylons, and 17 x 10 6 t/m 2 for the cables. The applied loading is consisted of a uniformly distributed load (q) along the deck, and a moving concentrated load (P) at a distance (e) from the left deck's support. Two values of the q/P ratio are

considered, i.e. 0 and 0.07(m -! ). During the critical load search this ratio remains constant for a given set of geometric parameters. The total number of finite elements used in the whole structure was 96.

FINITE ELEMENT MODEL AND IDEALIZATION

Numerical methods such as finite difference and finite element methods are powerful tools in recent days (Bathes 1982). In this study finite element method is used.

Finite Element Model

Two different types of three-dimensional element such as beam and spar have modeled the curved cable-stayed bridge. Forty-six beam elements model the bridge deck; fifteen beam elements model each pylon; and twenty spar elements which can only resist tension forces, model the stayed cables. Each beam element consists of six degrees of freedom, i.e. translation in x-, y- and z-direction and rotation about x-, y- and z-axis. Each spar element consists of three degrees of freedom, i.e. translation in the three directions.

Boundary condition is one of the most important factors in buckling analysis. The bases of pylons are considered as fixed; the end of side span is simply supported; and the connection between the pylon and the bridge deck is coupled in both vertical and lateral directions.


Idealization

An exact formulation and finite element analysis were made within the limitations of the following assumptions: 1 .Members are initially straight and piecewise prismatic. 2.The material behavior is linearly elastic and the moduli of elasticity E in tension and compression

are equal. 3.Statically concentrated and uniformly distributed loads only apply on the structure. The loading is

proportional to each other thus the load state increases in a manner such that the ratios of the forces to one another remain constant.

4.No local buckling is considered. 5.The effect of residual stress is assumed negligibly small.

NUMERICAL RESULTS AND DISCUSSION

Based on the finite element model and the eigen-buckling analysis procedure (Ermopoulos et al. 1992, Vlahinos et al. 1993), the critical loads for various sets of geometric parameters are calculated. The fundamental critical load and its corresponding mode shape are found. In all cases the anti- symmetric modes' critical load was the lowest while the second mode is always symmetric. Figure 3 shows the undeformed and the first three buckling mode shapes for a set of geometric parameters (for radius R=300m, Ip / I b = 4, H/L=0.262, d/H=0.6 and the deck's dead load only).

Figure 3. Buckling Mode Shapes of the Curved Cable-Stayed Bridges

Figure 4 shows several curves of critical loads Pcr versus the distance e from the left deck support

(load eccentricity), for H/L=0.262, d/H=0.20 (harp-type) and d/H=0.95 (fan-type) represented in (a) and (b), respectively with the uniform load q=0. The solid lines correspond to Ip / I b = 4 and the

dashed lines correspond to I p /I b = 1.


Figure 4 indicates that the ratio of Ip / I b is one of the most important factors for the minimum

critical loads of this type of structure. The harp-type bridge (d/H=0.2) represented in (a) has the ratio of H/L=0.262 and the fan-type bridge (d/H=0.95) represented in (b) has H/L=0.126. Figure 4(a) shows that the minimum critical loads occurs around the mid-span and are almost the same for the curved-deck bridges with Ip / I b = 1.0. When the ratio becomes Ip / Ib= 4, the fundamental

critical loads increase for the curved-deck bridge with radius less than 500m. Figure 4(b) first shows that fan-type bridge has lower fundamental critical loads than harp-type, and if the radius decreases the fundamental critical load decreases for all ratios of Ip / I b . Based on Figure 4, the

ratio of Ip / I b , the cable arrangement, and the radius of the curved bridge deck play the most

important role for buckling analysis of this type of structures.

Figure 4 Minimum Critical Loads versus Eccentricity of the Concentrated Load for Various Radiuses

Figure 5 represents the fundamental critical load for the same bridge but subjected only to its dead load. It becomes obviously that the fundamental critical loads are almost the same when the radius is greater than 500 m for the harp-type bridge (d/H=0.2) represented in (a). For fan-type bridge (d/H=0.95) represented in Figure (b), if the radius decreases, the fundamental critical loads decrease.

Figure 6 represents several curves of fundamental critical loads. It is for dead and the moving concentrated load with ratio q/P=0.07 applied at the midpoint of the middle span versus the Ip /I b

ratio. Two cases are shown in Figure 6; the bridge with H/L = 0.262 is represented in (a), and with H/L=0.165 is represented in (b) for various values of radius. It can be seem that the minimum critical load of the straight-deck bridges decreases when the ratio of Ip /I b increases, which means

that the flexural interaction between the pylons and the bridge deck decrease. If the radius of the curved bridge deck is less than 300m, the minimum critical load increases when the ratio of Ip /I b

increases, which is different from those bridges having large radius.


Figure 5 Minimum Critical Loads versus Eccentricity of Concentrated and dead Loads for Radiuses

Figure 6 Fundamental Critical Loads versus Ratio Ip /I b for Various Radiuses

Figure 7 shows the radius effects on minimum critical load for various ratios of d/H and Ip /I b .

Coupling parameters of d/H and radius of curved-deck, the minimum critical loads of curved-deck bridges are significantly different from those of straight-deck bridges. Figure 7(a) shows that a curved-deck bridge with H/L=0.262 having the ratios of d/H=0.4 and Ip/Ib=4.0 has the optimum critical load when the radius is less than 500 m. If the radius is.

greater than 500 m, the bridge with d/H=0.2 gives the optimum critical load. For H/L=0.165 (Figure 7b), there are different sets of geometric parameters and different radius for this optimum.


Figure 7 Fundamental Critical Loads versus the Ratio of d/H for Radiuses

Regarding Case II (as represented in Figure 2), the stability behavior of both cases is similar but the minimum critical loads are greater than those of Case I. Figure 8 shows four curves to compare the minimum critical loads for the optimum design parameters represented in Figure 7(a).

Figure 8 Comparison of the Fundamental Critical Load of Case I and II CONCCLUDING REMARKS

In a common sense if a bridge's span length increases, the bridge becomes more flexible and then the critical load decreases but this study shows that this kind of sense is not suitable to apply to the. curved cable-stayed bridges. Due to the axial components of cable reactions (Wang 1999), the curved bridge deck has less axial forces acting on the bridge deck than the straight bridge deck has. For the geometric parameters considered, the minimum critical load significantly increases when the radius of the curved-deck bridges less than 300 m. On the other hand, if the radius is greater


than 500 m, he characteristics of the minimum critical loads are similar to those of straight-deck bridge even though the curved-deck bridges have higher minimum critical loads.

Reference

F. Leonhardt (1982), Briiken/Bridges. Architectural Press, London. Fu-Kuei Chang and E. Cohen (1981), Long-span bridges: state of art, Journal of structural Division, ASCE. C. O'Connor (1971), Design of Bridge Superstructures, John Wiley, New York. M.S. Troitsky (1988), Cable-Stayed Bridges: An Approach to Modem Bridge Design, 2 nd Edition, Van Nostrand Reinhold, New York. Manabu Ito (1999), The Cable-Stayed Meiko Grand Bridges, Nagoya, Structural Engineering Intemational (SEI), IABSE, Vol. 8, No.3, pp.168-171. Christian Menn (1999), Functional Shaping of Piers and Pylons, Structural Engineering International, IABSE, Vol.8, No.4, pp.249-251. Manabu Ito (1999), Wind Effects Improve Tower Shape, Structural Engineering International, IABSE, Vol.8, No.4, pp.256-257. Yang-Cheng Wang (1999a), Kao-Pin Hsi Cable-Stayed Bridge, Taiwan, China, Structural Engineering International, Journal of IABSE, Vol.9, No.2, pp.94-95. Yang-Cheng Wang (1999b), Number Effects of Cable-Stayed-Bridges on Buckling Analysis, Journal of Bridge Engineering, ASCE, Vol.4, No.4. Yang-Cheng Wang (1999c), Effects of Cable Stiffness on a Cable-Stayed Bridge, Structural Engineering and Mechanics, Vol.8, No.l, pp.27-38. John CH. Ermopoulos, Andreas S. Vlahinos and Yang-Cheng Wang (1992), Stability Analysis of Cable-Stayed Bridges, Computers and Structures, Vol.44, No.5, pp. 1083-1089. Andreas S. Vlahinos, John CH. Ermopoulos and Yang-Cheng Wang (1993), Buckling Analysis of Steel Arch Bridges, Journal of Constructional Steel Research, Vol. 33, No.2, pp.100-108. Klaus-Jurgen Bathe (1982), Finite Element Procedures in Engineering Analysis, Prentice-Hall, Inc. Petros P. Xanthakos (1994), Theory and Design of Bridges, John Wiley & Sons, Inc. New York, USA. Anthony N. Kounadis (1989), AYNAMIKH Tf2N ZYNEXf2N EAAZTIK~N ZYZTHMAT~N, EKDOZEIZ ZYMEQN, (in Greek).

EXPERT SYSTEM OF FLEXIBLE PARAMETRIC STUDY ON CABLE-STAYED

BRIDGES WITH MACHINE LEARNING

Bi Zhou ~ and Masaaki Hoshino 2

1, 2 Dept. of Transportation Engineering College of Science and Technology, Nihon University

(24-1, Narashinodai 7, Funabashi, Chiba 274-8501, Japan)

ABSTRACT

The development of practical expert systems is mostly concentrated on how to acquire experiential knowledge from domain experts successfully. However, frequently, the acquiring progress is difficult and the representation is incomplete. Furthermore, the experiential knowledge may be entirely lacking when the design situation changes or technology comes new. The present study is to develop a cable- stayed bridge expert system of how knowledge in the cable-stayed bridges may be generated from hypothetical designs with machine learning for the parametric study processed as flexible as possible.

KEYWORDS

cable-stayed bridge, structural design, multiple regression analysis, expert system, machine learning, object-oriented method

INTRODUCTION

Modern structures such as cable-stayed bridges involve a relatively new knowledge that may be entirely lacking when the design situation changes or technology comes new. Formalised knowledge and knowledge evolving procedures are difficult to acquire, store and represent. In view of expert systems, the knowledge obtained from experts or documentary materials (such as guidelines, books or papers) usually only contains general explanations about possible configurations with few recommendations which play a conceptual control or value-restricted role in selecting candidate designs.

By introducing the concepts of static knowledge and dynamic knowledge, this paper presents an exploration for the expert systems of how to generate the domain knowledge from hypothetical designs with the change of the design situation and apply it to the knowledge evolution with the ability of learning. The candidate related knowledge (CRK), that is regarded as having influence on the design situation, is used to supplement the relative knowledge constantly and is concentrated on hypothetical

529

530 B. Zhou and M. Hoshino

designs, which also may be decomposed into the hypothetical designs from recorded designs as well as from candidate designs recommended by the system or specified by the designer. In this system, a learning, representing and storing method of Object-Oriented Multiple Regression Model (OOMRM) (Zhou and Hoshino 1998; 1999) is introduced to explain and represent the design situation. This paper focuses on the process of the system realisation and not on the explanation of the multiple regression method (Balakrishnan et al. 1965; Eric and John 1977; Hald 1952; Harald 1966; Hwang et al. 1994; John 1967; Robb 1980; Samuel 1962; Warren 1976; William and Douglas 1980), the object-oriented programming technologies (Booch 1994; Rumbaugh et al. 1991), OOMRM (Zhou and Hoshino 1998) and the parametric studies of the cable-stayed bridges (Agrawal 1997; Hegab 1989; Krishna et al. 1985).

SIMPLE CONCEPT OF OOMRM

The Object-Oriented Multiple Regression Model (OOMRM) (Zhou and Hoshino 1998; 1999) is a method of knowledge engineering that integrates the object-oriented programming (OOP) and the multiple regression analysis for explaining and predicting (inference mechanism) the comple x engineering designs with the ability of adequacy and learning of the design situation. The method deals CRK with an adding-MRM-overriding process through a number of temporary views for the design situation to build up the knowledge base more completely and accurately, which can bring together knowledge from different domains about the design situation, by the process of trial-and-use. It allows evolution of the knowledge representation and storage with the change of CRK or with the change of design situations. Both the numerical and qualitative knowledge are represented as STATE, RECOMMEND and RULE, which we will discuss later, and the refined knowledge is organised by OOP.

The MRM general class, which is used to create different objects as instances for different purposes and to apply the adding-MRM-overriding process, involves following items:

LIll

CRK:

Condition:

Explanation:

Evaluation:

Prediction:

Source:

. . . .

Condition: Girder continuous at tower 0.45>=Span_ratio>=0.35..

-

~ n : Girder continuous at towel ~ - E ~ MRM (13, P . . . . . ) O. 40>=Span_ratio>=O.~ 35. f : ~ v a l u a t i o n : Mcc, SSn, Sn...

~xplanation: M R M (]3, P .. . . . ) [ Prediction: yo - ~' < y0 ": Y0 +

Mcc, SSR, SR... [ Source: design, expert, site ...

Prediction: Y0 -~ ' < Yo ": Y0 § ~'

Source: design, expert, site ... m

Figure 1" MRM general class and its derived objects of cable_ tension temporary views

(1) object name which describes the substance of the situation; (2) CRK which is relative to the object; (3) given conditions which should be satisfied; (4) possible actions which may be performed:

(a) explanation of the situation which describes the relationship between CRK (from MRM); Co) evaluation of the situation which indicates the degree of strength and validity of the actions

(from MRM); (c) prediction of the situation which is based on stored knowledge (from MRM);

Expert System of Flexible Parametric Study on Cable-Stayed Bridges

(5) source which the knowledge comes from.

531

Figure 1 shows the representation of MRM (the MRM general class) and its derived objects of cable_ tension temporary views. The properties of Explanation, Evaluation and Prediction are not explicitly stored as static attribute values, and may be defined by functional expression or dynamic data established by MRM that having the attributes of CRK, Condition and Source as arguments.

FLEXIBLE PARAMETRIC STUDY ON CABLE-STAYED BRIDGES

The present system focuses the design objectives on three parts for the preliminary design of the cable- stayed bridges:

(1) the variations of topology arrangement of the side span length, the middle span length and the tower height above deck;

(2) the variations of cable arrangement with respect to the spacing of girder-cable connections and the unsupported spacing in middle span;

(3) the variations of cross section of girder, cable and tower.

The parametric studies on the cable-stayed bridges have been reported on several papers (Agrawal 1997; Hegab 1989; Krishna et al. 1985). However, a common feature of these papers is that the range of the parameters is restricted in detailed values for widely interpolating by programming reuse and the variety of the parameters are fixed in several situations for applying to different design situations. This study is tried to make the parameters representation space as flexible as possible by the introduction of the derived design parameters that could adjust the design parameters each other with the preliminarily stored knowledge, and represent functional explanations according to the design situation.

side_length I~ mid length JA side_length ~]

(span ratio ) 1 (span ratio ) "[" (span ratio ) "1 - ~ - (t . . . . _ g _ s t i f f n ~ -

///// I \ \ ~ ~ [~nsupp~ / / / / X 2 / / I \ \ ~ ~

I [cable number~4] @ [lenght side x cable ratio s] . . . . . . . . "__ ," I ~ (caote~gsnlyness)

J [cable number~4] @ [(lenght mid x cable ratio m] I_ r'

Figure 2: General arrangement: some parametric properties and derived parametric properties

Table 1 shows some design properties used in the system for describing the candidate designs. A candidate design is one of the hypothetical designs that is temporarily expected to explain and represent a certain design situation by the design properties according to the experience. The table contains two groups of properties: parametric properties and derived parametric properties. The parametric properties (specified by the designer or suggested by the system) describe the candidate design, while the derived parametric properties describe the relationships between the parametric properties. To make the system more flexible, the parametric properties are restricted within the interpolation range of the derived parametric properties (interpolation properties), not within the range of the parametric properties themselves. The range of the interpolation properties is designed flexibly that it may be modified or extended when new knowledge is learned. E.g., at the beginning, we would learn the knowledge for the span_ratio range between 0.35~0.4; while the training examples increase, the range would be extended to 0.35~0.45. The derived parametric properties are purposely defined in the form of ratios, which aim to be perceived easily by the designer to select and compare between the candidate designs. Figure 2 shows some parametric properties and derived parametric properties in the

532

general arrangement.

B. Zhou and M. Hoshino

TABLE 1 DESIGN PROPERTIES

Properties

Side span length Middle span length Middle span unsupported length Tower height above desk Girder inertia moment Tower inertia moment Girder area

Total cable cross-sectional area Tower area Cable number

] Acronym Parametric Properties

side_length mid_length unsupportlength tower_height girder_inetria tower inetria girder_area cable area t o w e r a r e a

cable nwnber

Restriction

span_ratio span_ratio unsupportratio tower ratio a cable_stiffness, tower_stiffness tower_stiffness cable_stiffness, tower_stiffness cable_sttffness tower_stiffness 4-100

Derived Parametric Properties

I Acronym span_ratio unsupportratio tower_ratio cable_g_stiffness tower..g_stiffness

Properties Side span to main span ratio Middle span unsupported spacing to total span ratio Tower height above desk to total span ratio Cable to girder stiffness Tower to ~irder stiffness

Interpolation Range

INTERPOLATION PROPERTIES

TABLE 2 PRODUCTIVE PROPERTIES

Properties I Acronym Productive Properties

Maximum girder moment at middle span ]girder m max Minimum girder moment at support I girder .m m in Maximum cable tension [cable__tension Tower base moment [tower_m Maximum girder deflection Igirder deflection Tower tip deflection I tower~deflection

Derived Productive Properties Conversion weight of girder Weight of cable Conversion weight of tower

girdercweigh cableweight tower cweight

I Expected Range

OBJECTWE PROPERTIES

OBJECTWE PROPERTIES

Instead of dealing with the complete results from the structural analysis program, only six important productive properties and three derived productive properties are stored for evaluating the candidate design. The productive properties describe the structural behaviour due to the specified design in terms of stress, deflection and weight. They are sufficient for evaluating the behaviour in the preliminary design stage. Both the productive properties and the derived productive properties are expected to be within the range of the objective properties (given specifications or design constraints).

DEVELOPING STATIC KNOWLEDGE AND DYNAMIC KNOWLEDGE

As mentioned before, systematic and general knowledge of the cable-stayed bridges is hard to find for a variety of design situations. Fragmental design recommendations can be abstracted from experts or limited descriptions in documentary materials, which usually play a conceptual control or value- restricted role in the design process; here, we call it static knowledge. In the present system, the static knowledge is obtained from guidelines, books and papers (Agrawal 1997; Carl et al. 1992; Troitsky 1988; Hegab 1989; Hunt et al. 1997; Krishna et al. 1985; Starossek 1996; Xanbakos 1993), and is represented as STATE, RECOMMEND and RULE.

Expert System of Flexible Parametric Study on Cable-Stayed Bridges 533

(1) STATE that cable_tension decreases rapidly with the increase of cable_number. (Agrawal, 1997)

(2) RECOMMEND unsupportlength 20-30% larger than supportlength. (Troitsky 1988, pp.181) (3) IF the mid_length is in the range of 140-150m,

THEN RECOMMEND supportlength of 20m. (Troitsky 1988, pp.181)

However, the static knowledge, which is represented as the pure abstract statement of the general recommendation (1), is difficult for designers to make accurate and convincing decisions in the practical designs. Eventually, the detailed numerical design specification and evaluation should be mainly depended on the designers' experience and heuristic judgement, i.e. on subjective decisions. Similarly, the other two value-restricted recommendations (2), (3), which are abstracted from past experience of existing design comparisons, can not be easily adapted to the variations of upcoming design situations. Therefore, only the static knowledge may be insufficient in explaining and representing the design situation for practical designs and satisfying the improvement for future designs.

In contrast to the world static, if the knowledge is stored in the form of an organised database of evaluated designs with the design properties and the productive properties, and is processed and represented by OOMRM, the recommendations can be updated and re-represented at any time and given functional expressions, in cases of the representation of the design situation is not complete; CRK that is relative to the design situation is changed; and adjustment is necessary to match the change of the design situation. Accordingly, we introduce the concept of dynamic knowledge to remedy the defect of the static knowledge by means of continuous acquiring and improving the knowledge with the adding-MRM-overriding method for forming the functional expressions. E.g., omitting the conditions in the rule, the dynamic knowledge of cable_tension and its derived objects of the value-restricted temporary views can be represented as follows (training in a particular design situation that having the total cable area per plane kept constant for all the candidate designs).

(4) THEN RECOMMAND the influence on cable_tension IS cable_tension[span_ratio, cable_number, cable_area]

AND The prediction of cable_tension IS Y o - ~P < cable _ tension ~spanratio, cable_nu mber, cabl e_area ], a ] < Y o + ~P.

(5) THEN RECOMMAND the influence on cable_tension IS span_ratio(-0.8995) > cable_area(0.7587) > cable_number(O.lO02)

AND The prediction of cable_tension IS 84.7976 <= cable_tension[[0.40,20,O.500]',95] <= 106.0321 (95% prediction interval) (training the system with 36 set of examples).

(6) THEN RECOMMAND the influence on cable_tension IS span_ratio(-0.9230) > cable__area(0.7603) > cable_number(O.1272)


(7) THEN RECOMMAND the influence on cable_tension IS span_ratio(-0.8233) > cable_area(0.5799) > cable_number(O.lO08)



The coefficients in the parentheses indicate the influence of the parameters on cable-tension varying one parameter with others held constant (Zhou and Hoshino 1999). The dynamic knowledge is regarded as the lower hierarchy of the static knowledge, that the static knowledge represents the abstraction of the dynamic knowledge, while the dynamic knowledge represents the value-unrestricted situation in the functional expression. Often the dynamic knowledge can be translated into the static knowledge represented as the abstract form to play a conceptual control or value-restricted role in the design process, usually at the sacrifice of the value-unrestricted and the numerical prediction effects. Accordingly, in the above training situation, the dynamic knowledge (5), (6) and (7) can be translated into the following abstract rule.

(8) THEN RECOMMAND the influence on cable_tension IS span_ratio > cable_area > cable_number.

Different from many hierarchical knowledge classifications that have many relative hierarchies (Reich and Fenves 1995; Kushida et al. 1997), both the static knowledge and the dynamic knowledge are processed, organised and represented with the relationship between CRK (Zhou and Hoshino 1998; 1999). As a simply example, in investigating the effect of cable stiffness on the behaviour of the structure, instead of facing enormous raw data obtained from structural analysis software arranged by the relational order or internally organised by the hierarchical classification tree, we can just link the name of cable_area to an object of a predefined general class that explains the design situation and predicts its productive properties within its CRK established by OOMRM.

SYSTEM GENERAL STRUCTURE

Figure 3 illustrates the architecture and the flow of the general system. The static knowledge, which is learned by the designer from documentary materials or experts who are in the fields of application, has conceptual or value-restricted influence on the candidate design or the hypothetical design. Influenced by the static knowledge or specified by the designer, sometimes by a heuristic selection, the parametric properties of the candidate design are specified as a temporary view for the design situation. If the derived parametric properties of the specified parametric properties are within the range of the pre- stored interpolation properties, the specified design is then submitted to OOMRM to give the explanation of the design situation and give the prediction as the productive properties. The explanation is describing the situation of the specified design, and the prediction is submitted to the evaluation decision process for evaluating. If the derived parametric properties are exceeding the interpolation properties, the specified design is then submitted to the traditional structural analysis program to give the productive properties for evaluating.

As a matter of fact, every engineering design may be regarded as an estimation or prediction of a certain specified design situation, which we call it the design situation temporary view, including explanation and problem solving. Because, no matter how many times the design has been confirmed in the past, it is always subject to the future confirmation by different design situations, different design methods or different practical uses. It is useful to explain and predict what will happen when changes are performed on the any of the structural parameters. The evaluation decision is simply made from the objective properties using the IF-THEN rules. If the productive properties are within the range of the objective properties, the specified parametric properties of the candidate design are regarded as the acceptable design and are then stored into the dynamic knowledge as improved knowledge, sometimes being accompanied with the change of the interpolation properties.

However, the productive properties that are generated from the specified design usually exceed the expected range of the objective properties, and should be adapted. Frequently, for a complex structure, several alternatives are always available for consideration. Comparing the productive properties with

Expert System of Flexible Parametric Study on Cable-Stayed Bridges 535

the objective properties, sometimes redesigns should be carried out to converge the productive properties on the objective properties. The back propagation method (Guo and Xiao 1991), which is a structural back propagation optimum method, is integrated into the system for providing possible design properties according to the objective properties by fixing some design properties within the range in advance. The redesign is a process of giving recommendations that may be adopted by the designer. If the designer adopts the recommendations, the design properties will be within the interpolation range in Table 1. The process of the redesign iterates until the candidate design satisfies the desired requirements. Because of the adjustable ability of the cable-stayed bridge in later designs and erection stages, usually the designer selects a partial set of the recommendations for redesigning the candidate design and the final design selection is mostly based on the subject decision. Finally, the acceptable design for the dynamic knowledge can be translated into the static knowledge and both of them can be used for future candidate designs.

Figure 3: Architecture and flow of the system

CONCLUSIONS AND FUTURE DIRECTIONS

The flexible parametric study in the present system includes two meanings: the flexible range of the design properties and the flexible representation of the knowledge. The flexible representation of both static and dynamic knowledge based on STATE, RECOMMEND and RULE with conceptual and functional expressions is clearer and more convenient for designers to make decisions than the representation of only numerical or only qualitative knowledge. Especially, when the knowledge is vague or the design situation is changed, it is difficult for designers to make accurate and convincing decisions.

In contrast to the restricted and narrowed parametric knowledge in the cable-stayed bridges, the system intends to acquire, store and represent the knowledge that is relevant to the design situation incrementally and continually, and adapts it to the change of the design situation. The system is mainly based on the traditional structural analysis program for the knowledge extension, on MRM for the knowledge analysis and on OOP for the knowledge engineering. This method can be used very efficiently for the knowledge acquisition, storage and representation to the expert systems and very


economically for the optimum designs using past experience.

In consideration of the variety of the cable-stayed systems, future directions should be turned to integrate three additional groups of properties into the system: construction properties describing the variety of erection methods; under construction productive properties describing the control situation during the erection phases; and the final cost properties.

ACKNOWLEDGEMENTS

The first writer should greatly appreciate the financial support for this research from Kameda Gumi Co., Ltd., Japan.

REFERENCE

1. Agrawal T. P. (1997). "Cable-Stayed Bridges- Parametric Study". J. Struct. Engrg., ASCE, 2:2, 61-67. 2. Balakrishnan A., George V. D. and Lotfi Z. (1965). Probability, Random Variables, and Stochastic

Processes. Mcgraw-Hill Book Comp. 3. Booch G. (1994). Object-Oriented Analysis and Design with Applications. 2nd ed. Addison Wesley

Longman, Inc. CA. 4. Carl C. et al. (1992). Guidelines for Design of Cable-Stayed Bridges. ASCE Committee on Cable-Stayed

Bridges. 5. Eric H. A. and John E. J. (1977). Statistical Methods for Social Scientists. Academic Press Inc., New York. 6. Guo W. F. and Xiao R. C. (1991). Liner and Non-Liner Bridge Structural Analysis Program System.

Shanghai Institute of Urban Construction. 7. Hald A. (1952). Statistical Theory with Engineering Application. John Wiley & Sons, New York. 8. Harald C. (1966). Mathematical Methods of Statistics. Overseas Publications, Ltd. Tokyo. 9. Hegab H. I. A. (1989). "Parametric Investigation of Cable-Stayed Bridges". J. Struct. Engrg., ASCE, 114:8,

1917-1928. 10. Hunt I. (1997). "Initial Thought on the Design Cable-Stayed Bridge". Proc. Instn. Civ. Engrs Structures &

Bridges. 1997, 112, May, 218-226. 11. Hwang J. N., Lay S. R., Martin R. D. and Schimert, J. (1994). "Regression Modelling in Back-Propagation

and Projection Pursuit Learning". Translations on Neural Networks, IEEE, 5:3, 342-353. 12. John R. W. (1967). Prediction Analysis. D. Van Nostrand Company Inc., Toronto. 13. Krishna P., Arys A. S. and Agrawal T. P. (1985). "Effect of Cables Stiffness on Cable-Stayed Bridges". J.

Struct. Engrg., ASCE, 111:9, 2008-2020. 14. Kushida M., Miyamoto A. and Kinoshita K. (1997). "Development of Concrete Bridge Rating Prototype

Expert System with Machine Learning". J. Comput.. Engrg., ASCE, 11:4, 238-247. 15. Reich Y. and Fenves S. J. (1995). "System that Learns to Design Cable-Stayed Bridges". J. Struct. Engrg.

12:7, 1090-1100. 16. Robb J. M. (1980).Aspects of Multivariate Statistical Theory. John Wiley & Sons, New York. 17. Rumbaugh J., Blaha M., Premerlani W., Eddy F. and Lorensen W. (1991). Object Oriented Modelling and

Design. General Electric Research and Development Centre Schenectady, New York. 18. Troitsky M. S. (1988). Cable-Stayed Bridges: an Approach to Modern Bridge Design. Second Ed., Van

Nostrand Reinhold, New York, N.Y. 19. Samuel S. W. (1962). Mathematical Statistics. John Wiley & Sons, New York. 20. Starossek U. (1996). "Cable-Stayed Bridge Concept for Longer Spans". J. Bridge Engrg., ASCE, 1:3,99-

103 21. Warren G. (1976). Statistical Forecasting. John Wiley & Sons, New York. 22. Xanbakos P. P. (1993). Theory and Design of Bridges. John Wiley & Sons, Inc. New York, N.Y. 23. William W. H. and Douglas C. M. (1980). Probability and Statistics in Engineering and Management

Science. John Wiley & Sons, New York. 24. Zhou B. and Hoshino M. (1998). "OOMRM: Object-Oriented Multiple Regression for Complex

Engineering Designs". Advances in Engrg. Computational Tech. Civil-Comp Ltd., UK. 249-255. 25. Zhou B. and Hoshino M. (1999). "Knowledge-Based Multiple Regression Model for Complex Engineering

Designs". J. Struct. Engrg., JSCE, 45A. 501-510.

PARAMETER STUDIES OF MOVING FORCE IDENTIFICATION IN LABORATORY

Tommy H. T. CHAN, Ling YU, S. S. LAW, and T. H. YUNG

Department of Civil and Structural Engineering The Hong Kong Polytechnic University

Hunghom, Kowloon, HONG KONG

ABSTRACT

The parameters of both vehicle and bridge play an important role in moving force identification. This paper aims to investigate the effect of various parameters on Time Domain Method (TDM) and Frequency-Time Domain Method (FTDM). For this purpose, a steel bridge model and a vehicle model were constructed in laboratory. Bending moment and acceleration responses of the bridge were simultaneously measured when the model vehicle moved across the bridge at different speeds. The moving forces were identified using the TDM and FTDM and rebuilt responses were calculated from the identified forces for comparison of identification accuracy. Assessment results show that both the TDM and FTDM are effective and acceptable with higher accuracy but the TDM is better than the FTDM. Further work includes enhancement of the two methods and merging them into a Moving Force Identification System (MFIS).

KEYWORDS

Moving Force Identification, Bridge-Vehicle Interaction, Bending Moment, Acceleration Response, Measurement, System Identification

INTRODUCTION

Force identification or force reconstruction from dynamic responses of bridges is an important inverse problem. Many methods have been presented for its prediction in recent years (Fryba 1972, Moses 1984, Hoshiya and Maruyama 1987, Brigges and Tse 1992,). Stevens (1987) gave an excellent survey of the literature on the force identification problem as well as an overview. However, some of the above mentioned methods measure only static axle loads. O'Connor and Chan (1988) suggested an advanced force identification method - Interpretive Method I (IMI) to interpret the force history, which is an advancement of the weight-in-motion methods mentioned above and is able to measure the dynamic axle forces of multi-axle system. Based on system identification theory, the authors have developed another two moving force identification methods, namely Time Domain Method (Law, Chan and Zeng 1997) and Frequency-Time Domain Method (Law, Chan and Zeng) respectively.

537

538 T.H.T. Chan et al.

Recently, a new method similar to IMI, called Interpretive Method (IMII), has also been published (Chan, Law and Yuan 1999). The preliminary and comparative studies showed that all these four methods could identify moving forces with acceptable accuracy. However, each method has its merits, limitations and disadvantages. They should be improved and strengthened for practical application in field tests. In order to enhance the four methods and merge them into a Moving Force Identification System (MFIS), the effects of various parameters on two of the methods, namely the TDM and FTDM had been critically investigated in laboratory. The parameters include mode numbers of bridge, sampling frequencies, vehicle speeds, computational time, sensor numbers and locations. Acceptable results were obtained and some observations had been made in this paper.

BRIEF DESCRIPTION OF THEORY

Referring to Figure 1, the bridge deck is considered as a simply supported beam with a span length L and constant flexural stiffness El, constant mass per unit length p , and viscous proportional damping

C, and the effects of shear deformation and rotary inertia are not taken into account (Bernoulli-Euler beam). The force P moves from left to fight at a speed c.

l< x=ct 1 >1 L

Figure 1. Moving force on a steel beam bridge

An equation of motion in term of the modal displacement q, (t) can be given as

2 ~n(t)+Z~,conO(t)+coZqn(t)=--~ZPn(t ) (n = 1,2,...~)

Where

(1)

n2n-2 I ~ C nTact co" = L ------7- ' (" = 2pco, ' pn ( t )= P ( t ) s i n ( ~ ) (2)

are the modal frequency of the nth mode, the damping ratio of the nth mode and modal force respectively. If the time-varying force P(t) is known, the equation (1) can be solved to yield q, (t)

then the dynamic deflection v(x,t) can be found from the q,(t)and the nth mode shape

function O, (x). This is called the forward problem. The moving force identification is an inverse

problem, in which the unknown time-varying force P(t) could be identified based on measuring the displacements, bending moments or accelerations of practical structures. Two methods are developed for the purpose.

Time Domain Method (TDM)

As mentioned above, the equation (1) can be solved in time domain by the convolution integral and the dynamic deflection v(x,t) of the beam at point x and time t can be obtained as

2 sinnntzt t v(x't)= ~ L f~e-~"~ sinco"(t- r)sinnntZr p(r) (3)

Where co', = co. ~/1 - ( 2 , therefore, the bending moment of the beam at point x and time t is

02V(x,t) ~2EI1r2n 2 f~ n~cr m(x,t) = -El = e (4) OX2 n=l pL3co' sinnntztL t--r176 L P(r)dr

Parameter Studies of Moving Force Identification in Laboratory 539

The acceleration at the point x and time t is

[ ncizzt t n~rcr'dr] (5) a(x, t) = i;,(x,t) = ~ 2_7 ~ , (x) P(t)sin(----~-) + ~ h', ( t - r)P(r)sin( L ) ,=I pL

Where

J~, (t) = 1. e-r176 n )2 _ O.)n'2 ]sin CO'nt + [-- 2~,(O,(O', ]COS CO'nt} (6) (O n

Assuming that both the time-varying force P(t) and the bending moment m(x,t)or the acceleration

a(x,t) are step functions in a small time interval At, equation (4) or (5) can be rewritten in discrete

terms and rearranged into a set of equations as follows

BN• PNB • = RN• (7) Where, P is the time series vector of time-varying force P(t), R is the time series vector of the measured response of the bridge deck at the point x, such as the bending moment m(x,t) or

acceleration a(x,t). The system matrix B is associated with the system of bridge deck and the force.

The subscripts N B = L / cAt and N are the numbers of sample points for the force P(t) and measured

response R respectively when the force goes through the whole bridge deck.

Frequency-Time Domain Method (FTDM)

Equation (1) can also be solved in the frequency domain. Performing the Fourier Transform for

Equations (1) and v(v,t) = s ap, (x)q n (t), the Fourier Transform of the dynamic deflection v(x,t) is n=l

2 V ( x, (O ) = ~.~ --7-j- O , ( x ) H , ( (O ) P ( (O ) ( 8 )

n=l /a,.

Where H,((O) and P((O)are the Fourier Transform of q,(t)and P(t) respectively. Similarly, the

relationships between bending moment or acceleration and dynamic deflection can also be used to execute the corresponding Fourier Transform. Finally, a set of N-order simultaneously equations can be established in the frequency domain. The force P((O)consisted of the real and imaginary parts can

be found by solving the N-order linear equations. The time history of the time-varying force P(t) can then be obtained by performing the inverse Fourier Transformation. From the procedures mentioned above, initially, the governing equations are formulated in the frequency domain. However, the solution is obtained in the time domain. Therefore this method is named frequency-time domain method.

The above procedure is derived for a single force identification in TDM and FTDM methods. They can be modified for multi-force identification using the linear superposition principle.

EXPERIMENTAL DESIGN

The model car and model bridge deck were constructed in the laboratory. An Axle-Spacing-to-Span- Ratio (ASSR) is defined as the ratio of the axle spacing between two consecutive axles of a vehicle to the bridge span length. Here, the ASSR was set to be 0.15. The model car had two axles at a spacing of 0.55 m and it ran on four rubber wheels. The static mass of the whole vehicle was 12.1 kg in which the mass of rear wheel was 3.825 kg. The model bridge deck consisted of a main beam, a leading beam and a trailing beam as shown in Figure 2. The leading beam was used to achieve acquired constant speed of vehicle when the model car approached the bridge. The trailing beam was for the slowing down of the car. The main beam with a span of 3.678 m long and 101 mm x 25 mm uniform

cross-section, was simply supported. It was made from a solid rectangular mild steel bar with a


density of 7335 kg / m 3 and a flexural stiffness EI = 29.97kN/m 2 . The first three theoretical natural

frequencies of the main beam bridge was calculated as f~ = 4.5 Hz, f2 = 18.6 Hz, and f3 = 40.5 Hz.

Figure 2. Experimental setup for moving force identification

The U-shape aluminum track was glued to the upper surface of the main beam as a guide for the car. The model car was pulled along the guide by a string wound around the drive wheel of an electric motor. The rotational speed of motor could be adjusted. Seven photoelectric sensors were mounted on the beams to measure and check the uniformity of moving speed of the car. Seven equally spaced strain gauges and three equally spaced accelerometers were mounted at the lower surface of the main beam to measure the response. Bending moment calibration was carried out before actual testing program by adding masses at the middle of the main beam. In addition, a 14-channel type recorder was employed to record the response signals. Where Channels 1 to 7 were for logging the bending moment response signals from the strain gauges. Channels 8 to 10 were for the accelerations from the accelerometers. The channel 11 was connected to the entry trigger. In the meantime, the response signals from Channels 1 to 7 and Channel 11 were also recorded in the hard disk of personal computer for easy analysis. The software Global Lab from the Data Translation was used for data acquisition and analysis in the laboratory test. Before exporting the measured data in ASCII format for identification calculation, the Bessel IIR digital filters with lowpass characteristics was implemented as cascaded second order systems. The Nyquist fraction value was chosen to be 0.05.

PARAMETER STUDIES

For practical reason, one parameter was studied at a time. The examination procedure was to examine one parameter in each case to isolate the case with the highest accuracy for the corresponding parameter and then another parameter was examined. The parameters, such as the mode number, the sampling frequency, the speed of vehicle, the computational time, the sensor and sensor locations were considered as variables to examine their effect on the accuracy of force identification. There are two ways to check this kind of effects. One is that the identified results are checked directly by comparing the identified forces with the true forces. However, because the true forces are unknown, it is difficult to proceed. The other way is that the identified results are checked indirectly by comparing the measured responses (bending moments, displacements or accelerations) with the rebuilt responses calculated from the identified forces. The accuracy is quantitatively defined as Equation (9), called a Relative Percentage Error (RPE).

RPE = EJftn, e - ~'dentl x 100% (9) El/.el

Equation (9) is also used to calculate the relative percentage errors between the measured and rebuilt responses from identified forces instead of comparing the identified forces with the true forces directly. The measured response (R ....... d) and rebuilt response (Rreb,,izt)are herein substituted for the

true force (ftr,,e)and identified force (fide,,)in equation (9) respectively. In the present parameter

studies, most of results were from the comparison of the relative percentage errors between the measured and rebuilt response only for the bending moment response. Regarding the results associated with the accelerations, they will be reported separately.

Parameter Studies of Moving Force Identification in Laboratory

Effects of Mode Number

541

For comparing the effects of different Mode Number (MN) on identified results in the TDM and FTDM, it was assumed that the sampling frequency (f~) and vehicle speed (c) were constant, and the

case of fs = 250Hz, c = 15 Units (1.52322 m/s) was chosen. The data at all the seven measurement

stations for bending moments were employed to identify the moving forces. The mode number was varied from MN=3 to MN=10. The identified forces were calculated first, and then the rebuilt responses from the identified forces were then computed accordingly. The Relative Percentage Error (RPE) for both the TDM and FTDM are shown in Figure 3.

Figure 3. Effects of mode number

For the TDM, the RPE values at the middle measurement stations are always less than the ones at the two end measurement stations. This is associated with the signal noise ratio of various measurement stations because there are bigger responses at the middle stations than those at the two end stations. It is found that when the mode number is equal to or bigger than four, the relative percentage errors are reduced dramatically. This means the TDM is effective if the required mode number is achieved or exceeded, otherwise, the TDM will be failed. The minimum RPE value case is of MN=5, the maximum RPE value case is of the biggest mode number involved (MN=I 0). This also shows that the case MN=5 is the most optimal case in this kind of comparisons. Similar conclusions are drawn for the FTDM. However, the biggest difference from the TDM is that the RPE is independent of the increment MN after MN=5.

By comparing the identification accuracy by the TDM and FTDM in Figure 3, it can be seen that the results are very close to each other when the MN is equal to 5 and 6 respectively, especially at the middle measurement stations. However, it can be seen from the identified forces in Figure 4 that the FTDM is worse than the TDM because it has components with higher frequency noise.

Figure 4. Identified forces (MN=5, fs = 250Hz, c = 15 Units )

Effects of Sampling Frequency

The sampling frequency fs should be high enough so that there is sufficient accuracy in the discrete

integration in equation (4) and (5) [Law et al 1997]. In the present study, the data was acquired at the sampling frequency 1000 Hz per channel for all the cases. This sampling frequency was higher than the practical demand because only a few of lower frequency modes were usually used in the moving force identification. Therefore, the sequential data acquired at 1000 Hz was sampled again in a few intervals in order to obtain a new sequential data at a lower sampling frequency. Here, a new


sequential data at the sampling frequency of 333,250, and 200 Hz would be obtained by sampling the data again every third, fourth and fifth point respectively. For the case of the vehicle running at 15 Units, the bending moment data was acquired at the different frequencies of 200, 250 and 333 Hz respectively. The RPE results between the rebuilt and measured bending moment responses are calculated and listed in Table 1 for both the TDM and FTDM.

TABLE 1 EFFECTS OF SAMPLING FREQUENCY (c = 15 Units)

S t a . T O M F T D M

No. MN=3 MN=4 MN=5 MN=3 MN=4 MN=5 I II III I II III I II III I II III I II III I II III

1 13.8 783. 412. 6.44 6.05 6.11 8.86 5.32 3.75 188. 1541 1724 6.35 16.1 445. 3.38 4.39 2005 2 7.15 609. 244. 2.81 2.74 2.69 3.34 2.61 2.40 167. 1618 1636 2.68 10.3 419. 2.51 2.16 1370 3 6.50 358. 185. 2.74 2.10 1.95 2.87 2.10 1.94 163. 1679 1647 2.09 8.50 427. 2.01 2.14 1230 4 3.61 216. 216. 3.15 2.96 2.80 3.72 2.71 2.12 165. 1686 1678 2.69 8.05 433. 2.09 2.22 1321 5 6.27 359. 189. 3.16 2.74 2.53 3.58 2.68 2.44 164. 1676 1657 2.61 8.62 430. 2.42 2.13 1238 6 7.41 614. 245. 4.74 4.58 4.32 5.42 4.31 3.45 169. 1615 1654 4.22 10.0 424. 2.89 2.70 1387 7 17.3 780. 420. 6.61 5.84 5.94 9.36 5.19 4.05 187. 1514 1717[ 5.89 16.0 446. 3.92 4.30 2095

Case I, II, and III is for 200,250 and 333 Hz respectively.

For completely comparing the effects of different sampling frequency, the effect of mode number on identification accuracy is also incorporated in the study. It is found that the higher the sampling frequency is, the lower the RPE values are for all the measurement stations in the TDM. This shows that the higher sampling frequency is better than the lower sampling frequency, and the TDM method has higher identification accuracy if the response is acquired at a higher sampling frequency. In Table 1, it is shown that the FTDM method is failed when the sampling frequency fs = 333 Hz and mode

number MN=3 because the RPE values are too big to accept for all the measurement stations. However, The FTDM method is still effective for the case in which the mode number is bigger than 3, f~ = 200 Hzandf, = 250 Hz respectively. By comparing the RPE values at a lower sampling

frequency f~ = 200 Hz with that at f , = 250 Hz, it is found that the identification accuracy at

fs = 200 Hz are higher than one at f , = 250 Hz. It shows that the identified results are acceptable

and useful if more mode number and suitable sampling frequency is determined in FTDM method.

Effects o f Various Vehicle Speeds

In this section, some limitations on identified methods TDM and FTDM should be considered firstly. In particular, necessary RAM memory and CPU speed of personal computer are required for both the TDM and FTDM. Otherwise, they will take very long execution time due to the bigger system coefficient matrix B in equation (7), or they cannot execute at all due to inefficient memory. As the mode number, the sampling frequency and bridge span length had not been changed for this case, a change of the vehicle speed would mean a change of the sampling point number, namely change of dimensions of matrix B in equation (7). Therefore, in order to make TDM and FTDM effective and to analyze the effects of various vehicle speeds on the identified results, the case of MN=4 and fs = 200 Hz was selected. When the test was carried out, the three vehicle speeds were set

manually to 5 Units (0.71224 m/s), 10 Units (1.08686 m/s), and 15 Units (1.52322 m/s) respectively. After acquiring the data, the speed of vehicle was calculated and the uniformity of speed was checked. If the speed was stable, the experiment was repeated five times for each speed case to check whether or not the properties of the structure and the measurement system had changed. If no significant change was found, the recorded data was accepted. The RPE values between the rebuilt and measured bending moment responses are calculated and listed in Table 2. It shows that the TDM is effective for all the three various vehicle speeds. The RPE values tend to reduce for each measurement station as the vehicle speed increases. But, the RPE values are close to each other in the case of 10 Units and 15

Parameter Studies of Moving Force Identification in Laboratory 543

Units. It shows that the identification accuracy for the faster vehicle speed is higher than that at slower vehicle speed. However, the FTDM is not effective in the case of lower vehicle speed 5 Units, but the identified results are getting better and better as the vehicle speed increases. Fortunately, the identified result is acceptable in the case of 15 Units in the FTDM.

TABLE 2 EFFECTS OF VEHCILE SPEEDS (MN=4, fs = 200Hz )

Stat ion TDM FDTM No. 5 Units 10 Units 15 Units 5 Units 10 Units 15 Units

1 6.81 5.40 6.45 1045.69 101.53 6.35 2 5.54 2.49 2.81 708.18 46.57 2.68 3 5.88 3.03 2.74 621.36 24.83 2.09 4 8.67 3.01 3.15 562.90 45.11 2.69 5 4.50 2.76 3.16 586.98 23.98 2.61 6 4.66 3.93 4.74 647.38 44.70 4.22 7 6.56 7.94 6.61 965.75 94.26 5.89

Effects of Various Measured Station Number

This section estimates the effects of measurement station number ( N t ) on the identified accuracy. The

N t was set to 2, 3, 4, 5 respectively while the other parameters MN=5, f , = 250 Hz, c = 15 Units

were not changed. The RPE values between the rebuilt and measured responses are given in Table 3. The results in Table 3 show that the TDM is required to have at least three measurement stations to get the two correct moving forces for the front and rear wheel axles respectively. But the FTDM should have at least one more measurement station, i.e. 4, to get the same moving forces. However, the errors are increased obviously when the measurement station number is equal to 5 for the FTDM.

TABLE 3 EFFECTS OF MEASUREMENT STATIONS

TDM Station No. 2 3 4

1 (L/a) * * * 2 (2L/8) * * 1.50 3 (3L/8) 2003.03 2.15 2.08 4 (4L/8) * 2.27 * 5 (5L/8) 2029.36 2.48 2.04 6 (6L/8) * * 2.38 7 (7L/8) * * * Asterisk * indicates the station is not chose.

5

1.91 2.21 2.62 2.39 2.82

FTDM 2 3 4 5

* * 2 . 67 27.06 1192.66 86.92 2.35 14.82

* 115.42 * 27.95 1198.49 87.52 2.49 14.73

* * 2.93 26.62

Comparison of computational time

The computational time consists of three periods, i.e., i) forming the system coefficient matrix B in equation (7), ii) identifying forces by solving the equation and iii) reproducing the responses. The

above parts are same for the TDM and FTDM. The case described here is of MN=5, f , = 250 Hz,

c = 15 Units, N t = 7 by using a Pentium II 266 MHz CPU, 64M RAM computer. The total sampling

points are 700 for bending moment response at each measurement station and the total sampling points are 604 for each wheel axle force in the time domain. Therefore, the dimensions of matrix B are (7 x 700, 2 • 604). The execution time recorded is listed in Table 4 for the comparison on each period

of the TDM and FTDM in details. It shows that the FTDM takes much longer than the TDM method in forming the coefficient matrix B. The execution time in other two parts is almost the same for the two methods. The TDM takes shorter time than the FTDM from the point of view of the total execution time.


TABLE 4 COMPARISON OF COMPUTATION TIME (in Second)

PERIOD TDM Forming coefficient matrix B

Identifying forces Rebuilding responses

Total

FTDM 332.69 1059.57 1837.97 1834.07 55.04 53.99

2225.7 2947.63

CONCLUSIONS

Parameter studies on moving force identification in laboratory test have been carried out in this paper. These parameters include the mode numbers, the sampling frequencies, the vehicle speeds, the computational time, the sensor numbers and locations. The study suggests the following conclusions: (1) A minimal necessary mode number is required for both the TDM and FTDM. It should be equal to or bigger than 4. If first five modes are determined to identify the moving forces, the identification accuracy is the highest in the cases studied. (2) The TDM has higher identification accuracy when the higher sampling frequency is employed. However the FTDM is failed if adopting the higher sampling frequency and the lower mode number. (3) The faster car speed is of benefit to both the TDM and FTDM, but FTDM method is not suitable for the slower car speed case. (4) At least three and four measurement stations are required to identify the two wheel axle forces for the TDM and FTDM respectively. (5) The TDM takes shorter time than the FTDM. (6) Both the TDM and FTDM can effectively identify moving forces in time domain and frequency domain respectively, and can be accepted as a practical application method with higher identification accuracy. (7) From the point of view of all the parameter effects on the identification accuracy, the TDM is the best identification method. It should be firstly recommended as a practical method to be incorporated in the future developed Moving Force Identification System (MFIS).

ACKNOWLEDGMENT The present project is supported by the Hong Kong Research Grants Council.

REFERENCES 1. Briggs J.C. and Tse M.K. (1992). Impact force Identification using Extracted Modal Parameters and

Pattern Matching. Int. J. Impact Engineering 12:3, 361-372. 2. Chan T.H.T. and O'Connor C. (1990). Wheel Loads from Highway Bridge Strains: Field Studies.

Journal of Structural Engineering 116:7, 1751-1771. 3. Chan T.H.T. Law S.S. Yung T.H. and Yuan X.R. (1999). An Interpretive Method for Moving Force

Identification. Journal of Sound and Vibration 219:3, 503-524. 4. Fryba L. (1972). Vibration of Solids and Structure under Moving Loads, Noordhoff International

Publishing, Prague. 5. Hoshiya M. and Maruyama O. (1987), Identification of Running Load and beam system. Journal of

Engineering Mechanics ASCE, 113, 813-824. 6. Law S.S. Chan T.H.T. and Zeng Q.H. (1997). Moving Force Identification: A Time Domain

Method, Journal of Sound and Vibration, 201:1, 1-22. 7. Law S.S. Chan T.H.T. and Zeng Q.H. Moving Force Identification-Frequency and Time Domain

Analysis, Journal of Dynamic System, Measurement and Control (accepted for publication) 8. Moses F. (1984). Weigh-In-Motion System using Instrumented Bridge, Journal of Transportation

Engineering ASCE, 105(TE3), 233-249. 9. O'Connor C. and Chan T.H.T. (1988). Dynamic Wheel Loads from Bridge Strains, Journal of

Structural Engineering, 114:8, 1703-1723. 10.Stevens K. K. (1987). Force Identification Problems-An Overview, Proceeding of SEM Spring

Conference on Experimental Mechanics, 838-844.

SEISMIC ANALYSIS OF ISOLATED STEEL HIGHWAY BRIDGE

Xiao-Song LI 1 and Yoshiaki GOTO 2

1 Research Associate, 2 Professor

Dept. of Civil Engineering, Nagoya Institute of Technology

Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan

ABSTRACT

Seismic isolators with dissipation devices have been widely used for highway bridges in Japan,

because they may effectively absorb energy and reduce inertia force induced by earthquake. The main

factors that influence the response of the isolated bridge are initial stiffness and yield force of the

isolator. These quantities should be appropriately designed. Besides, introduction of the isolators leads

to an interaction between the bridge pier and the isolator and increases the computational difficulty

due to the nonlinearity that occurs in both the pier and the isolator. The purpose of this paper is to

investigate the seismic response of the isolated bridges subjected to ground motions, where we

examine how the behaviors of the bridge pier are influenced by the initial stiffness and yield force of

the isolator and the elongation of natural period of the bridge. Then, the applicability of the

'Displacement Conservation Principle' for predicting the maximum responses of the piers of the

isolated steel bridges is numerically examined. The numerical results show that the application of the

'Displacement Conservation Principle' may be reasonably safe and accurate for the practical design of

isolated steel piers.

KEYWORDS

seismic isolation design, nonlinear dynamic analysis, steel highway bridge

545

546

INTRODUCTION

X.-S. Li and Y. Goto

After the great earthquake happened in Kobe, in 1995, seismic isolators with dissipation devices have

been widely used for highway bridges in Japan. Due to the significant increase of the natural period,

the isolators may effectively absorb energy and reduce inertia force induced by earthquake. The main

factors that influence the response of the isolated bridge are initial stiffness and yield force of the

isolator. These quantities should be appropriately designed. Besides, introduction of the isolators leads

to an interaction between the bridge pier and the isolator and increases the computational difficulty

due to the nonlinearity that occurs in both the pier and the isolator. The purpose of this paper is to

investigate the seismic response of the isolated bridges subjected to ground motions, where we

examine how the behaviors of the bridge pier are influenced by the initial stiffness and yield force of

the isolator and the elongation of natural period of the bridge. Then, the applicability of the

'Displacement Conservation Principle' for predicting the maximum responses of the piers of the

isolated steel bridges is numerically examined. The numerical results show that the application of the

'Displacement Conservation Principle' may be reasonably safe and accurate for the practical design of

isolated steel piers.

ANALYTICAL MODEL

A typical isolated bridge is used as an analytical model, as shown in Fig.1. The total weight of the

bridge M=1067ton consists of the weight of the deck Mb=0.95M and the weight of the pier Mp=0.05M.

For the height of piers, two values are considered, that is, H=13m for Model-l, and H=11m for

Model-2. The corresponding fundamental natural periods for the two piers are Tl=0.705s and

T2=0.549s.

Fig.l: Analytical Model

A lead-rubber bearing (LRB) is assumed as an isolator with dissipation device that has bilinear yield

stiffness as shown in Fig.2. In Fig.2, Qy and Uby are the yield force and yield displacement,

Seismic Analysis of Isolated Steel Highway Bridge 547

respectively. K~,~ and Kb2 are the elastic stiffness and post-yield stiffness with a relation of Kbz=KbJ6.5;

Kr~ is the equivalent stiffness and UBe=0.7Ub .... which are suggested by the 'Manual of Menshin

(isolation and dissipation) Design of Highway Bridges' (1992).

Fig.2: Hysteresis Behavior of Isolator

ANALYTICAL METHOD

A numerical method that considers both geometrical and material nonlinearity is used to carry out the

dynamic analysis (Li and Goto, 1998). The post-yield modulus of material is assumed to be Ep=E/100.

The effect of damping is considered by a mass-proportional damping matrix. The damping coefficient

is set to h=0.01 for elasto-plastic analysis and h--0.05 for elastic analysis.

Two standard ground accelerations suggested by Japan Road Association are used for the analysis.

One is Type 2 at Soil Group II (hard soil site), the other is Type 2 at Soil Group Ill (soft soil site).

Both acceleration waves are illustrated in Fig.3. The time interval adopted in the numerical integration

is 0.01s.

Fig.3: Ground Accelerations

In order to investigate the effect of the initial stiffness and yield force of the isolator, the calculation is

carried out by changing Qy/Py and I~I/K p from 0.2 to 0.9 that is the possible range in practical design,

where Py=2( cr y-Mg/A)/(HB) and Kp=3EI/H 3 are the yield force and elastic stiffness of the pier.

548

NUMERICAL RESULTS

X.-S. Li and Y. Goto

Typical responses of an isolated bridge are shown in Fig.4. It can be seen from Fig.4 (a) that the

displacement of the pier is much smaller than that of the deck due to the isolator. Furthermore, the pier

is damaged little and the energy induced by seismic wave is almost dissipated in the isolator, as

illustrated in Fig.4 (b). In the following, the effects of the initial stiffness Kb~ and yield force Qy of the

isolator and the elongation of natural period of the bridge are investigated. Then, the applicability of

the 'Displacement Conservation Principle' for predicting the maximum responses of the piers of the

isolated steel bridges is numerically examined.

Fig.4: Responses of Model-1 with Qy/Py=0.6 and Kbl/I~ =0.5 Subjected to Wave Type 2-111

Effect of K~l and Qy on Pier and Isolator

With the designated yield force ratios Qy/Py=0.3, 0.5 and 0.7, the maximum response displacements of

the pier and the isolator are obtained by changing the initial stiffness ratio Kbl/Kp from 0.2 to 0.9. The

relations of ductility factors//p and/1 b of the pier and the isolator vs. the initial stiffness ratio Kb~/K p

of the isolator are shown in Fig.5 for Model-1 and Model-2 subjected to waves Type 2-II and Type

2-lit. In this figure, the ductility factors for piers and isolators are defined as /1 p=Upmax/Upy (with solid

line) and /1 b=Ubmax/Uby (with dotted line), where Upmax=maximum displacement of pier, Upy=yield

displacement of pier, and Ubmax=maximum displacement of isolator and Uby=yield displacement of

isolator. It should be noted that the/1 p may be considered as the maximum response or the ductility

factor of the pier, while/.t b denotes only the ductility factor of the isolator because the Uby varies with

Qy/Py or Kb~/K p. Similarly, the relations that vary with the yield force ratio Qy/Py are shown in Fig.6 for

the designated initial stiffness ratios Kbl/Kp=0.3, 0.5 and 0.7.

From Fig.5, it can be seen that the both ductility factors ~t p and /~ b of the pier and the isolator

increase with the increase of the initial stiffness ratio Kbl/K. p of the isolator for all cases. However,


some influences caused by the wave types can be found as follows. The maximum responses of the

pier subjected to wave Type 2-11" almost linearly increase as Kbl/Kp increases from 0.2 to 0.9 as

shown in Fig.5(a), while those subjected to wave Type 2-m" increase a little shapely when Kbl/I~ >0.6.

Furthermore, the values of It p for Model-2 with a smaller ratio of Qy/Py=0.3 become greater than

those with Qy/Py=0.5 and 0.7 when Kbl/Kp~0.6, as shown in Fig.5(c). The relations between the

ductility factor It b of the isolator and Kbl/Kp exhibit a different tendency depending on the value of

Qy/Py. That is, It b with Qy/Py=0.3 exhibits a large increase, while that with Qy/Py=0.7 shows just a

small increase.

Fig.5: Effect of Initial Stiffness Ratio Kbl/Kp of Isolator on Ductility Factors It p and It b

Fig.6: Effect of Yield Force Ratio Qy/Py of Isolator on Ductility Factors It p and It b

550 X.-S. Li and Y. Goto

Figure 6 shows that the ductility factor/1 p of the pier increases with the increase of yield force ratio

Qy/Py for most values of Kbl/Kp, while the ductility factor/1 b of the isolator decreases. A different

tendency, however, is observed for Model-2 with Kb~/Kp=0.7 as shown in Fig.6(c). In this case, the

maximum value of /.t p is obtained for the yield force ratio of Qy/Py=0.2 that is the smallest one. This

phenomenon is identical with that observed in Fig.5(c) and is also identical with those obtained for

concrete piers by Kawashima & Shoji (1998). It can be concluded that an isolator with a higer

stiffness and a smaller yield force may result in much damage of the pier when subjected to wave

Type 2- Ill.

Fig.7: Relations between Energy Ratio Ep/Ef and Period Ratio Tp/Tf

Effect of Elongation of Natural Period

The main difference between isolated bridges and conventional bridges is characterized by a

significant increase in the natural period of isolated bridges due to the introduction of the isolator that

may also effectively absorb energy. As a result, the inertia force induced by earthquake is considerably

reduced in isolated bridges. In order to find an appropriate period range for the isolated bridge, an

energy ratio Ep/Ef is used to evaluate the damage of the pier, where Ep = 4Ppdup and Ef = ~Pfduf

are the energy absorbed in the pier with the isolator and that without the isolator, respectively. And, a

period ratio Tp/Tf is used as the abscissa, where Tp = 2nx/m/K e and Tf - -2nx/m/K p are the

fundamental periods of the isolated bridge and the conventional one, respectively. K e is the equivalent

stiffness of the isolated bridge and defined as Ke=KBKp/(KB+Kp), where KB is the equivalent stiffness

of the isolator as shown in Fig.2. Figure.7 shows the relations between the energy ratio Ep/Ef and the

period ratio Tp/Tf for Model-1 and Model-2. It can been seen from Fig.7 (a) that a general tendency is

that the energy ratio Ep/Ef decreases as the period ratio Tp/Tf increases, except for a few points. The

three points over 0.4 are corresponding to the cases: (Qy/Py=0.3, Kbl/Kp=0.8, 0.9) and (Qy/Py=0.2,

Kbl/Kp=0.7) for Model-2 subjected to wave Type 2-Ill. If both Qy/Py and Kbl/Kp are limited to the


range from 0.3 to 0.7 that are favorable in the practical design, the maximum value of Ep/Ef will be

less than 0.3 as shown in Fig.7 (b). This result implies that the range from 0.3 to 0.7 is appropriate for

both Qy/Vy and Kbl/Kp of the isolator. The corresponding period ratio Tp/Tf is between 2.1 and 3.7.

Furthermore, by noting that Ep/Ef approaches zero when Tp/Tf > 3.0, the range from 2.1 to 3.0 of the

period ratio Tp/Tf may be more appropriate. This is based on the fact that too much elongation of the

natural period may lead to an undesirable large displacement of the deck.

Application of 'Property of Displacement Conversation'

As previously described, the interaction between the pier and the isolator results in the computational

difficulty due to the nonlinearity that occurs in both the pier and the isolator. To avoid the difficulty,

some simple methods have been proposed for predicting the dynamic response of the pier, such as the

method based on the 'Energy Conservation Principle'. For the conventional bridge pier that has a

shorter fundamental period, the method has been verified applicable and widely used in practical

design. For the isolated bridge that has a longer fundamental period, however, it overestimates or

underestimates the response of the pier. Here, based on the 'Displacement Conservation Principle' that

states 'the maximum elasto-plastic deformation of a system with a long fundamental period is

approximately equal to the maximum elastic deformation of the same system', the maximum

responses of steel piers are predicted by elastic dynamic analysis. In this method, the maximum elastic

response displacements of both the pier and the isolator are first dynamically calculated for the

isolated bridge with the elastic stiffness Kp of the pier and the equivalent stiffness KB of the isolator.

Then, the maximum lateral force of the pier can be obtained by equating the elastic response

displacement to the static elasto-plastic response displacement.

Fig.8: Comparison of Maximum Responses of Piers Obtained

by Simple Method and Analytical Method

The comparison of the maximum responses of the piers obtained by the simple method and by the

elasto-plastic dynamic method are shown in Fig.8 (a), where RF and ~ are the maximum lateral

552 X.-S. Li and Y. Goto

force and displacement of the pier, respectively. The subscripts e and p denote the elastic dynamic

response and the elasto-plastic dynamic response. It is observed that the maximum lateral force ratio

RFJRFp is closer to 1.0 when Tp/Tf is smaller than 3.0, whilst the maximum lateral displacement ratio

d d d p is in the range from 0.77 to 1.86. Similarly, if both Qy/Py and Kbl/K p are limited to the range

from 0.3 to 0.7, RFe/RF p and ~ o/~ p will take the values between 1.0 to 1.81. The approximate results

obtained by the 'Displacement Conservation Principle ' may be considered to be reasonably safe and

accurate for practical design of the isolated steel piers.

SUMMARY AND CONCLUDING REMARKS

From the above analysis, the following conclusions are obtained.

The response of an isolated bridge is greatly influenced by the initial stiffness and yield force of the

isolator. The both ductility factors for the pier and the isolator increase with the increase of the initial

stiffness ratio Kb~/I< v of the isolator. The maximum response displacement of the pier increases with

the increase of yield force ratio Qy/Py for most values of Kb~/l ~ , while the ductility factor r b of the

isolator decreases. An isolator with a higher stiffness and a smaller yield force may result in much

damage of the pier when subjected to wave Type 2-m'.

As a general tendency, the energy ratio Ep/Ef decreases as the period ratio Tp/Tf increases. Within the

range from 0.3 to 0.7 for both Qy/Py and Kb~/Kp of the isolator, the energy ratio Ep/Ef will be less than

0.3. Furthermore, by noting that Ep/Ef approaches zero when Tp/Tf > 3.0, the range from 2.1 to 3.0 of

the period ratio Tp/Tf may be more appropriate.

The applicability of the 'Displacement Conservation Principle' for predicting the maximum responses

of steel piers of the isolated bridges is numerically confirmed. The numerical results show that both

RF~RFp and d d ~ p will take the values between 1.0 and 1.81 when Qy/Py and Kbl/Kp are limited to

the range from 0.3 to 0.7. The approximate results may be considered to be reasonably safe and

accurate for practical design of the isolated steel piers.

REFERENCE

Kawashima, K. and Shoji, G. (1998). Interaction of Hysteretic Behavior between Isolator/Damper and

Pier in an Isolated Bridge, J. Struct. Engrg. JSCE. Vol.44A, pp.733-741.

Li, X.S. and Goto, Y. (1998). A three-dimensional nonlinear seismic analysis of frames considering

panel zone deformation, Struct. Eng.~Earthquake Eng., JSCE, Vol.15, No.2, pp.201-213, 1998, Oct.

Ministry of Construction (1992). Manual of Menshin Design of Highway Bridges, Civil Engineering

Research Center, Tokyo, Japan (In Japanese)

Shear analysis for asphalt concrete deck pavement of steel bridges

Xudong Zha 1 and Qiuming Xiao 2

1Department of Highway and Bridge Engineering, Changsha Communications University,

Changsha, 410076, PRC

2Department of Highway and Bridge Engineering, Changsha Communications University,

Changsha, 410076, PRC

ABSTRACT

This paper is based on the elastic layered theory of rigid support. The horizontal shear stresses between

the bridge deck pavement of asphalt concrete and steel box girders, the maximum principal stresses

and the maximum shear stresses at the bottom of the deck pavement layer has been analyzed under the

normal motion and the emergency break of motor vehicles. Based on this result, the shear resistance index of binder course has been put forward so as to supply the binder course design of steel bridge deck pavement with a scientific basis.

KEYWORDS

steel bridge, bridge deck pavement, asphaltconcrete, horizontal shoving, binder course, elastic layer, principal stress, shear stress, shear strength, shear resistance index

PREFACE

In the late 70s, China started building steel bridges with orthotropic plate decks, and the relative

decking technology research has been done for 20 years. From Mafang Bridge to recently completed

Xiling Bridge, Humen Bridge and Qingma Bridge, the deck pavement technology has been

continuously improved and developed. However, up to now, bridge deck pavement engineering of steel bridges in China has not achieved a complete success.

553

554 X. Zha and Q. Xiao

There are three main reasons for the distresses of bridge deck pavement of Xiling Yangzi Bridge,

Chongqing Highway Scientific Research Institute (1998). The first is that the deck pavement system was not perfect, the second the qualification of construction teams could not meet the requirement for

the works, and the third the influence of heavy truck vehicles used for the Three Gorges was not fully

estimated. The local temperature conditions was not completely occupied during the designing of

Humen Bridge, and the worse thing was that the deck pavement thickness and structure were changed

at the time of construction, thus reducing the material quality, and the thermal stability distress

appeared due to the poor construction control. Although Qingma Bridge built in accordance with the

English bridge deck pavement improved the softening point of hard asphalt to 70~ the deck failed

to pass the test of high temperature in Hong Kong. In the meantime, the use of Eliminator waterproof

adhesive agent left the hidden trouble of air bubbles.

Now, China is building a lot of steel bridges with orthotropic plate decks. The bridges under

construction are Jiangyin Yangzi Bridge, Xiamen Haicang Bridge, Nangjing Yangzi Bridge II,

Chongqing Egongyan Yangzi Bridge, and etc. The bridge deck pavement of asphalt concrete was

employed in most of bridge construction. Therefore, the bridge deck pavement technology should be

urgently solved for steel box girders. The paper has put forward the shear resistance index for asphalt

concrete bridge deck pavement through the theoretical analysis as far as the horizontal shoving

problem between asphalt concrete of the bridge deck pavement and the steel sheets are concerned. This

has provided the design of binder course for bridge deck pavement and program alternatives with a scientific basis.

BASIC THEORY

Due to the strength and rigidity of asphalt concrete is far from those of the steel box girders, the geometric sizes of the wheel load is much less than the length and width of the steel box girders. In the

meantime, under the wheel load, the reducing speed of the horizontal distribution of the asphalt concrete stress is rapid. Therefore, the asphalt concrete deck pavement of steel bridges can be regarded as a elastic layer, while the steel box girders can be simplified as rigid support. Thus, the theory of

elastic layers on the rigid support can be used for the stress analysis of deck pavement, while the motor

vehicle load simplified as the uniformly distributed loads of double vertical and horizontal circles,

I T M ~1~ ~1 T M ~1~" x ~1~ ~1

q ~ o ~ / y

Asphalt concrete deck | pavment E, h, ~t 1 r / / / Binder course

/ / , < \ V / , , K \ \ ~z Steel girder

Figure 1: Mechanical model of bridge deck pavement analysis

Shear Analysis for Asphalt Concrete Deck Pavement of Steel Bridges 555

Highway Planning and Design Institute of Ministry of Communications (1997). The mechanical model

is seen in Figure 1, x showing the longitudinal direction and y the horizontal one.

SHEAR STRESS ANALYSIS

Nowadays there exits a trend to apply SMA material for bridge deck pavement of asphalt concrete in

China. The pavement will be built in two layers. The size of the aggregate for the upper layer is

2-5mm larger than that for the lower layer, the general thickness of the pavement 6cm and the

modulus value of SMA measured in laboratory 1400---1600MPa. Therefore, in Figure 1, when the

analysis is carried out, the layer thickness h = 6cm, the modulus E = 1500MPa, the Poisson's ratio ~t =

0.25. The load of motor vehicles will be the standard axle load of 100kN, and the radius 5 = 10.65cm,

the vertical load p = 0.7MPa. The horizontal load will be determined in accordance with the friction

coefficient f between wheel and deck pavement, i.e.:

q = f . p (1)

Where: q is the horizontal load (MPa); p the vertical load (MPa); f the friction coefficient,

0.2 represents normal motion of motor vehicles, and 0.5 accidental emergency break, Fang Fusen

(1993).

According to the elastic-layered theory and the above data chosen, this paper has analyzed the

maximum principal stress cr 1 at the bottom of the pavement (z = 6cm), the maximum shear stress Xmax

and the horizontal shear stress Xxz. The different stress distributions will be computed under normal

motion of motor vehicles and emergency conditions, as shown in Figure 2--Figure 7. The maximum

and minimum value of various kinds of stress will be seen in Table 1. From Figure 2-Figure 7, the

maximum and minimum value of various stress at the bottom of the deck pavement layer will always

appear on x-direction in the center of the singular wheel. The relevant profile distributions in x-

direction will be seen in Figure 8-Figure 10.

TABLE 1

MAXIMUM AND MINIMUM VALUE OF VARIOUS KINDS OF STRESS (MPa)

Items

Maximum value Minimum value

f= 0.2

Gl "~m~x "l~xz

0.048 0.267 0.068 -0.215 0.000 -0.189

f=0 .5

ffl "Cmax "l;xz

0.078 0.355 0.000 -0.153 0.000 -0.283

From the above figures and tables, the maximum principal stress is mostly under compressive

condition. The maximum tension stress under both conditions will often be less than 0.1MPa, while the

cleavage strength of SMA between 1.2-1.6MPa. This shows that the fatigue cracking will not

generally appear in the paving layer. The maximum value of the maximum shear stress will be less

than 0.40MPa. Since SMA is of excellent stability at high temperature. The test shows that the shear

strength is between 1.0-1.4MPa. Therefore, the fatigue shear distress will not normally occur inside


Figure 2: Distribution of principal stress Cl (f = 0.2)

Figure 3: Distribution of maximum shear stress Xm~x (f = 0.2)

Figure 4: Distribution of horizontal shear stress Xx~ (f= 0.2)

Shear Analysis for Asphalt Concrete Deck Pavement of Steel Bridges 557

Figure 5: Distribution of principal stress c~ (f= 0.5)

Figure 6: Distribution of maximum shear stress Xm~x (f = 0.5)

Figure 7: Distribution of horizontal shear stress x= (f= 0.5)


Figure 8: Distribution of principal stress t~ l at center of singular wheel in x-direction

Figure 9: Distribution of maximum shear stress "lTma x at center of singular wheel in x-direction

Figure 10: Distribution of horizontal shear stress Xxz at center of singular wheel in x-direction

Shear Analys& for Asphalt Concrete Deck Pavement of Steel Bridges

the paving layer.

559

As to the horizontal shear stress between the paving layer and steel sheets, the maximum absolute

value will appear at the place of 9.585cm (0.95) from the center of singular wheel along the driving direction. The relevant value of the normal motion and emergency break of motor vehicles will

separately be -0.189MPa and -0.283MPa. Under normal driving condition, the area of reverse

direction for the horizontal shear stress will happen at the bottom of the front edge of singular wheel

(Figure 10). The maximum value is 0.068MPa. Through analysis, the larger shear stress will produce

between the deck pavement layer at 0.96 from the center of singular wheel and the steel sheet under

wheel loads, and this will result in horizontal shoving. Therefore, the design of the binder course

should meet the requirement of the shear action. Because the deck pavement layer will bear the

repeated load of motor vehicles, with the consideration of overloads and heavy loads, the shear

resistance strength of the binder course should be of enough safety. The shear resistance index is as

follows, Fang Fusen (1993):

1; 1;a < 1;R = - ( 2 )

K,

In which 1;a is the computed horizontal shear stress value (MPa) ; 1;R is the allowable horizontal shear

stress (MPa); 1; is the shear resistance strength (MPa); I~ the structural coefficient of the shear

resistance strength, this is related to the acting times of axle loads.

K~ should be determined in line with the traffic conditions, the application conditions and the

importance of the steel bridge through the fatigue shear tests. I~ will be 2-3 at the time of normal

driving, while I~ will be 1.5-2 when in emergency. From it, the shear resistance strength index of the binder course will be 0.4-0.6MPa.

CONCLUSION

This paper analyzes the maximum principal stresses, the maximum shear stresses and the horizontal

shear stresses in the deck pavement layer of asphalt concrete for steel bridges under the load of motor

vehicles. The analyzing result shows that the fatigue tension cracks and fatigue shear distress within

the paving layer will not generally happen. However, the greater horizontal shear stresses will take

place between the deck pavement and the steel box girders. This is the main cause for the horizontal

shoving of the deck pavement of steel bridges. The maximum absolute value will appear at 0.98 from

the center of singular wheel. Therefore, in order to prevent larger horizontal shoving of the deck

pavement, the shear resistance in the binder course should be controlled. In consideration of the

material, the thickness of the deck pavement of asphalt concrete for steel bridges and the traffic and

application conditions in China, it is suggested that the shear strength of the binder course between the

deck pavement and steel sheets at high temperature season should not be less than 0.4-0.6MPa.


REFERENCES

Chongqing Highway Scientific Research Institute (1998). Design Programs and Testing Guidelines for the Bridge Deck Pavement of Xiamen Hiachang Steel Bridge and Approach Bank, Chongqing Highway Scientific Research Institute, PRC.

Fang Fusen (1993). Pavement engineering, People Transportation Publishing House, PRC.

Highway Planning and Design Institute of Ministry of Communications (1997). Specifications for Design of Highway Asphalt Pavement (JTJO14-97), People Transportation Publishing House, PRC.

Zhu Zhaohong, Wang Binggang and Guo Dazhi (1988). Mechanical computation for pavement, People Transportation Publishing House, PRC.

INDEX OF C O N T R I B U T O R S

Volumes I and II

Alikhail, M.M. 849 Amano, M. 563 Ansourian, P. 679 Asta, A.D. 931,939

Bailey, C.G. 1055 Bao, S.H. 897 Beale, R.G. 375 Bernuzzi, C. 991 Bian, C.Y. 1005 Bingye, X. 125 Bradford, M.A. 571 Bridge, R. 639 Bridge, R.Q. 229

Calado, L. 323 Campione, G. 413 Cano, V. 1141 Cao, P.-Z. 385 Carroll, C. 775 Chan, B.H.M. 145 Chan, C.-M. 1081 Chan, F. 655 Chan, K.W.Y. 497 Chan, S.L. 145, 193, 443, 451,791, 1125, 1167 Chan, S.-L. 151,823 Chan, T.H.T. 537, 1109 Chan, W.Y. 487, 505 Chang, K.L. 775 Chau, K.T. 1125 Chen, H. 175 Chen, J. 839 Chen, J.F. 391 Chen, S.F. 443 Chiew, S.P. 291 Choi, C.K. 1117 Chu, A.Y.T. 791 Chui, P.P.-T. 151,823 Chung, K.F. 245, 269 Clandening, K. 775 Clarke, M.J. 237, 253 Clubley, S.K. 467 Combescure, A. 713

Dai, C.W. Dasui, W. Davies, J.M. De Luca, A. Deqing, G. Dezi, L. Dou, Y.

Elliott, K.S. Enjily, V. Ermopoulos, J.

Fang, L.X. Fuhai, Z.

Ge, H.B. Ghojel, J. Godley, M.H.R. Goto, Y. Grundy, P. Grzebieta, R.H. Gu, J.X. Guitong, Y. Gupta, L.M. Gusic, G.

Han, L.-H. Hancock, G.J. Haojun, C. Hara, T. Hautala, K.T. He, J.L. Hensman, J.S. Ho, G.W.M. Hong, T. Hoshino, M. Howson, W.P. Huang, K.S. Hudramovych, V.S.

291 731 367 323

999, 1013 931,939

831

459 375 521

145 209

63, 563 1047 375

101,545, 705, 1021 983, 1047

429 193 125 921 613

1039 25, 237, 253, 261,349, 357

183 881 6O5

1031 3

747 697 529 87

631 721

I2 Index of Contributors

Ip, K.H. 245, 269 Luo, Y.F. Luong, M.P.

631 663

Jian, Z. Jiang, L. Jiang, S.C. Jing, J. Jiqing, W. Jonaidi, M. Jullien, J.F.

Kaitila, O. Kang, Q.-L. Karadelis, J.N. Kasai, A. Kawanishi, N. Kesti, J. Ko, J.M. Koss, L. Kumar, A.N.

Lam, D. Lam, P.H. Lam, S.S.E. Lau, C.K. Law, S.S. Lee, C. Lee, Y.C. Lee, Y.-Y. Lennon, T. Leoni, G. Li, G.Q. Li, Q.Z. Li, S. Li, X.-S. Li, Y. Li, Y.Y. Li, Z.X. Liew, J.Y.R. Limam, A. Lin, H. Liu, K. Liu, L. Liu, P. Liu, T.C.H. Liu, X. Lu, L.W. Lu, X. Lui, T.H. Luo, Y.

731 101 Mak, W.P.

1031 M/ikel/iinen, P. 277 Man, K.L. 183 Mashiri, F.R. 679 Mele, E.

613, 1 1 4 1 Mizutani, A. Moore, D.B. Morris, L.J. Motoyui, S. Mtiller, A.

159 385

305,313 1157 705 367

331,513, 1109 849

1133

459 739 579

487, 505 537 429 391 905

1055 931,939

1031 631

965, 975 545 809 913

1109 175 613

1089 965

1089 1081 135 831

75 783 579 799

Nethercot, D.A. Ng, C.-F. Ngok, L.Y. Ni, Y.Q. Nianmei, Z. Nip, T.F. Nuttall, H.

Obata, M. Oehlers, D.J. Ohga, M. Ohtsuka, T. Omair, M. Omair, M.R. Osterrieder, P. Outinen, J. Owens, G.

Pan, Q. Panzeri, N. Patterson, N.L. Pircher, M. Poggi, C.

Qinghua, H.

Rahimian, A. Rasmussen, K.J.R. Rees, S.J.W. Ricles, J.M. Romero, E.M. Ronghe, G.N.

487, 505 159, 1063 487, 505

983 323

1021 1055

135 167

1073

3, 459 905 109

331,513, 1109 125 283 747

1021 571 881 167 313 305

1073 1063 1187

1151 621

1047 639 621

587

755, 857 341 775

75 755 92!

Index of Contributors I3

Rotter, J.M. 39

Saidani, M. 305, 313 Sause, R. 75 Schmidt, H. 597, 605 Scibilia, N. 413 Scott, D. 747 Shan, L. 1099 Shen, J. 783 Shen, S. 201 Shen, S.Z. 51 Shen, Z. 799 Shen, Z.-Y. 13 Shi, W. 783 Shi, Y. 277, 975 Shigematsu, T. 881 Shu, H.-S. 521 Smith, S.T. 109, 571 Song, C.Y. 687 Song, M.K. 1117 Surtees, J.O. 283

Tang, L.K. 175 Teng, J.G. 109, 391,443, 477, 647, 655, 687, 697 Toledano, M. 947 Tong, L. 799 Toribio, J. 947, 955

Unterweger, H. 117 Usami, T. 63, 563, 1157 Uy, B. 421

Wei, X.X. Werner, F. Wheeler, A.T. Wilkinson, T. Williams, F.W. Winterstetter, T.A. Wong, K.F. Wong, K.Y. Wong, M.B. Wong, P. Wong, Y.L.

Xianzhong, Z. Xiao, Q. Xiao, R.Y. Xie, X. Xiliang, L. Xu, Y.L. Xu, Z.G.

Yam, L.H. Yan, H. Yangji, C. Yi, S.C. Yiyi, C. Yong, Y. Yong, Z. Yoshida, S. Young, B. Yu, L. Yue, Y. Yung, T.H. Yuxin, L.

1125 1073

237, 253 261

87 597

487, 505 487, 497, 505

1047 429

451,477, 1167

731 553 467

1151 209, 587

497, 815, 873 1005

913 1099 731 897 731

1013 209 671

341,349, 357 537 587 537 217

Valiente, A. 955 Vincent, Y. 1141

Wang, C. 201,975 Wang, J.Y. 331,451, 1167 Wang, L. 889 Wang, L.Y. 873 Wang, Q. 889 Wang, Q.-L. 385 Wang, R.L. 1005 Wang, Y. 799 Wang, Y.C. 401, 1175 Wang, Y.-C. 521 Wang, Z. 437 Wang, Z.-M. 1081

Zandonini, R. Zha, X. Zhai, Y. Zhang, C. Zhang, J.W. Zhang, K. Zhang, W.S. Zhao, X. Zhao, X.L. Zhao, X.-L. Zhao, Y. Zhen, Y. Zheng, G. Zheng, Y. Zhitao, L.

991 553

1089 705 477 965 815 831

429, 849, 983 1047 647 437 513 63

217

14 Index of Contributors

Zhou, B. Zhu, L.D. Zhu, X.

529 Zingone, G. 497 Zuyan, S.

1151

413 731

KEYWORD INDEX

Volumes I and II

ABAQUS acceleration response active confinement adaptive h-refinement ADAS devices adhesive airport airy stress function aluminium alloy sections anisotropic plate ANSYS arbitrary cross-sections arch arched corrugated metal roof ASB aseismatic behavior asphalt concrete austenitic stainless steel austeno-ferritic steel axial compression axial load axisymmetric axisymmetric shell

barrel vault beam(s) beam-column beam-to-column steel joints bearing bearing capacity bearing failure bending bending moment bending stiffness biaxial bending bifurcation bifurcation analysis bifurcation buckling binder course body force bolted connections bolted joints bonding box section

237 497, 537

413 1117 775 477 731

1125 799 579 313 443 209 209

1055 437 553 605

1141 597

621,639 713 671

1099 261,349

1039 991 229 357 245

349, 375 537 513 443

126, 687, 721 341 671 553

1125 237, 269

283 477

63

BP algorithm 1167 bracing 429 bridge(s) 75, 521 bridge deck pavement 553 bridge monitoring system 487 bridge response 505 bridge tower section 1109 bridge-vehicle interaction 537 buckling 87, 109, 341,429, 521, 571, 613, 621,

647, 655, 679, 687 buckling modes 1117 building 75

cable 873 cable-beam 731 cable-nets 217 cable-stayed bridge(s) 505, 521,529 calculation 721 CFT column 291 channel column 341 channel members 349 channels 375 chaos 126 circular inclusion 1125 circular line load 631 circular steel columns 413 circumferential weld imperfection 639 class hierarchy 975 CLS element 1117 coal storage 1099 code 1031 cold-formed 25, 375, 429 cold-formed channels 357 cold-formed steel 159, 245, 261,269, 341,

349, 367 collapse 167 collapse mechanism 621 column(s) 109, 421,429, 1047 columns base 135 column flexibility 135 combined loading 597 compartment fires 39 complete load-deflection analysis 51 complicated structural member 1109

I5

16 Keyword Index

component-based model composite composite beam composite behavior composite columns composite construction composite foundation composite members composite slab composite steel slag-concrete beams composite structure(s) concrete concrete-filled steel tube(s) concrete-filling conical shells connection(s) consideration of constraints constraint continuum model conversion of cross-section coupling beams cracked cylindrical bars creep critical load(s) CTOD cumulative damage model curved cyclic behaviour cyclic loading cyclic tests cylindrical arc-length method cylindrical shell(s) cylindrical steel shells

451 39, 451,459, 1047, 1167

965 291

443, 747 3, 421

1151 413 747 385

755, 931,939 459

391,437, 1039 429 621

3, 75, 253, 1055, 1167 1073 1099

183 385 391 947 931

183, 631, 1117 1133

13 521 991

63, 429, 563 639 587

605, 679, 705 597

D.O.F. 1117 damage index 991 damage tolerance 955 damper 857 deep mixing method 1151 deformation(s) 305, 721 design 229, 305, 375, 655, 1031 design formula 51 design parameters 487 design strength 349, 357 diamond buckling 705 discrete optimization 1081 displacements 313 distortional buckling 367 double bolted connections 245 drilling 1117 ductile fracture 1021

ductility 63, 391, 413, 563, 991 dynamic characteristics 809 dynamic postbuckling 713 dynamic relaxation method 217 dynamic stability 201 dynamic stability critical load 201 dynamic system 126

earthquake response analysis 101 effective length 341 effective stress concentration factor 999 eigenvalue 679 elastic layer 553 elasticity 1125 elastic-plastic problem 671 elasto-plastic analysis 145 elasto-plastic response analysis 1157 elasto-plastique analysis 1141 electronic information 1187 elephant-foot buckling 705 elevated temperatures 605, 1047, 1063 empirical model 101 end plate connection(s) 237, 277, 451 energy absorption 755, 775 energy absorption device 857 energy release rate 947 engineering education 1187 engineering safe design 955 equilibrium state 217 experiment(s) 647, 721 experiment study 965 experimental device 1141 experimental investigation 349 expert system 529 external pressure 621

failure 29 failure modes 269, 323 fatigue 965 fatigue crack 1005 fatigue life curve 999 fatigue reliability 975 fatigue resistance (FR) line 991 fatigue strength 983, 999 FE-analysis 313 fibre reinforced concrete 413 fibre resistance 1039 finite element(s) 713, 1109 finite element analysis 261 finite element buckling analysis 1117

Keyword Index I7

finite element method(s) 167, 193, 209, 513, 671, 1005

finite strip method 109, 341,579 fire 39, 1055 fire protection 1047 fire resistance 1031, 1047 fire temperatures 1063 fire-resistance 1031 fixed-ended 341 flexible shear connection 931,939 flexural buckling 117 flexural strength for negative bending 385 flexure-torsion coupled 897 floor systems 39 fluctuating wind 831 folded plate structures 1117 force ratio 385 fracture criteria 947, 955 fracture toughness 75 frame(s) 3, 63 frame-shear wall 889 frangible roof design 671 free vibration problem 881 frequency 889 full-scale test 731 full-sized model test 209 functional 889 furnace 1047

high temperature properties holding down bolts hollow core hollow sections hoop stress horizontal shoving horizontal vibration hot-rolled steel human response hybrid optimization hysteresis hysteretic loops hysteretic model

imperfections incremental iterative method infinite elastic plane initial deflection initial imperfection m-plane bending instability interaction internal pressure ~rregular cross-sections ISO834 fire curve

1063 135 459 305

1125 553 897 349 849

1081 331, 1167

437 101

647, 687 217

1125 63

193, 579 983 341 597 671 443

1047

galloping 873 generalized plastic hinge model 167 genetic algorithms 1081 geometric nonlineality 193 geometrical stability 217 glass curtain walls 799 global optimal solution 1089 grinter 87

Hankel singular value approach 913 heteroclinic orbit 126 hexahedron element 1013 high performance steel 75 high performance structure 75 high rise 857 high rise building 783 high strength concrete 437, 1047 high strength fastener 277 high strength steel 357, 421,947 high strength steel with low ductility 245

J-integral joints junctions

large deflections large deformation large steel silo large-scale test lateral torsional buckling lightweight floor lightweight steel limit load limit state line spring element load bearing performance load monitoring loading condition loading histories local buckling local buckling effect lock mitre gate longitudinal rupture

1133 3

655

39 671 663

1055 1073 849 159 51

721 1013 209 487 451 323

109, 167, 261,367, 1073 101

1005 1133

I8 Keyword Index

long-term analysis low-cycle fatigue low-rise structure

machine learning material material properties maximum response displacement measurement measurement system mechanical properties membrane effects metallurgical behaviour Mexico city misaligned welded joint modal damping modal damping ratio modal property mode interaction mode shape mode switching model order determination moment capacity moment connections moment end plate moment-resisting frames moment-rotation behaviour monotonic loading Monte-Carlo-Simulation moving force identification moving rivulet multiple regression analysis multiply cross section multi-storey multi-storey framework multi-variable

natural frequency neural network nominal shear span-ratio non axisymetric imperfections nonlinear nonlinear analysis nonlinear behaviour nonlinear dynamic analysis nonlinear dynamic response nonlinear response nonlinear stability numerical analysis numerical simulation

931 991 857

529 1141 1063 1157

487, 537 613

1063 39

1141 755 999 849 815 513 697 849 697 913 349 283 253 991 237 563

1073 537 873 529

1109 39

183 1089

497, 849, 881 1167 385 713

313, 613, 731 201,687, 705, 775

939 545 331

39 631

167, 1021 731, 1141

object 975 objective function 1099 object-oriented method 529 object-oriented technique 975 oil storage tank 671 open sections 25 optimal analysis 1099 optimization 857 outriggers 747 overall stability 183

paddy rice 663 panel zone 323 parametric study 815 perforation compression 367 performance based design 775 performance index optimization 913 periodic orbit 126 phenomenological modelling 331 pin-ended 341 pins 229 pipe section 63 pipelines 1133 plastic bifurcation 705 plastic mechanism model 357 plasticity 39, 721, 1021 plasticity theory 167 polygonal sections 109 portal frame design 277 portal frames 135 post-buckling 39, 571,639, 687, 697 post-buckling analysis 579 post-buckling equilibrium path 587 precast 459 preflex 965 preisach model 331 pressure 613, 1133 prestress 731 prestressed 965 prestressed force 217 prestressed steel 921 prestressing 931,939 prestressing steel wires 955 principal stress 553 principle of multiples 87 prism element 1013 probabilistic fracture mechanics 1005 professional practice 1187 proportional and non-proportional loads 145 prying 253 Pseudo excitation method 815

Keyword Index I9

push-off 459 pushover analysis 63

quality control 739 quasi-tensegric system 217

rain 873 random vibration 831 R-curve 1133 re-bar stiffening detail 291 recticulated dome 201 rehabilitation 921 reinforcement ring 1125 reliability 1005 reliability analysis 357 research 1031 residual microstress 721 residual response displacement 1157 residual stress(es) 63, 639, 1141 resonance response 873 response analysis 831 response monitoring 487 restraint 39 reticulated domes 51 reticulated saddle shells 51 reticulated shallow shells 51 reticulated shells 51 reticulated vaults 51 retrofit 571 rheology of rice 663 RHS 261 ringbeams 655 rivet 1125 roof truss 159 Rosette-joint 159 rotation capacity 261, 323 rupture 1133

saddle 126 safety 1133 sagged cable 513 second-order analysis 193 second-order inelastic analysis 145 section properties 1109 seismic design 391,755, 775, 857, 991 seismic isolation design 545 seismic resistance 75 seismic response(s) 13, 809, 815 seismic response analysis 1157

self-stressing force 921 semi-continuous frames 991 semi-rigid connection 331, 815 semi-rigid frames 145 settlement 679, 1151 shaking table test 13 shear 229 shear buckling 563, 679 shear coefficient 1109 shear resistance index 553 shear strength 553, 563 shear stress 553 shear studs 459 shear walls 391 shell(s) 209, 613, 647, 655, 687, 697, 721 shell buckling 597, 605 shell structure 881 shells of revolution 697 ship 75 ship lock 1005 SHS 313 silo with internal ties 663 silos 639, 647, 655 single curvature cable-suspended roof 831 singularity detection 1089 site measurement 783 slabs 477 slim floor 1055 'Slimdek' 1055 slipping cables 931,939 soft soil 1151 soil-structure interaction 1157 space frame 631 stability 109, 647, 655, 1151 stability analysis 51,521 stability design 605 stability function method 193 standards 25 static 965 static strength 983 static tests 799 steady-state 1063 steel 101,375, 459 steel bridge 553 steel frame(s) 145, 167, 193, 283, 815, 1073 steel framed structures 13 steel highway bridge 545 steel hollow section(s) 429, 849, 983 steel jacket 975 steel oil tank 1151 steel plates 477, 571 steel portal frame 1081

I 10 Keyword Index

steel sheet joining 159 steel structure(s) 3, 25, 229, 277, 341,349, 357,

421,731, 1031, 1063 steel-concrete 965 steelwork construction 1187 steelwork erection 739 step load 201 storage 663 strains 313 strength 229, 413 strength decrease behavior 167 strengthen of steel columns 117 strengthen steel plates 117 strengthening 477, 921 stress intensity factor 947, 1013 stresses 305, 313 structural analysis 193, 671,739 structural design 3, 25, 253, 341,349, 357, 443,

459, 529 structural fire design 1063 structural health monitoring 505 structural integrity 955 structural modelling 739 structural optimization 1089 structural stability 663 structural steelwork 739 stub-column 63 subcrust 1151 substitute frame 87 substituting column 183 super-convergent patch recovery 1117 supervision 739 surface crack 1013 survival rate 999 suspension bridge 497 system identification 537

TADAS devices 775 tall building(s) 391,421,747, 755, 889 tall building structure 897 tall-pier aqueduct 809 tank nuclear reactor 1141 tanks 639, 655 tapered beam 1073 technology limit 1099 temperature 505 temperature field 1039 T-end plate connection 305 tendon 921 tensegrity 217 tension 305

test program 357 test strength 349, 357 testing 1047 testing frame 921 tests 229, 305, 341 theoretical study 873 theory of flow 721 thermal behaviour 1141 thermal buckling 39 thermal expansion 39 thickness imperfection 613 thin plate systems 913 thin shells 713 thin spherical joint 631 thin-walled column 101 thin-walled member 881 thin-walled sections 983 thin-walled shell structure 639 thin-walled steel structure 63 thin-walled structure 209 three-dimensional analysis 513 three-dimensional behavior 101 three-dimensional degenerated shell element 587 time history analysis 775 torsion 597 torsional rigidity 1109 toughness 1133 traffic load 505 transfer matrix method 881 transient dynamic response 513 transient-state 1063 transition junctions 647 transverse 809 transversely stiffening 831 triaxial apparatus 663 truss 1089 truss testing 159 T-section 655 tubular 253 tubular sections 25, 237 typhoon victor 497

ultimate capacity 939 ultimate load capacity 117 ultimate strength 145 unbalanced moment 451 uniformly distributed load 183 unilateral buckling 571 unstiffened plate 563 updated langrangian formulation 201

Keyword Index I l l

variation vertical load vertical shear strength vibration vibration characteristics vibration mode shape vibration test data viscous damper von Karman's model

889 897 385

87, 889 783 881 913 755 571

web slenderness weldability welded connections welded joint welding wind wind characteristics wind load wind-reducing random response (the) worst-case fatigue notch factor

357 75

323 1013

1133, 1141 505,873

497 87

831 999

WA wall stud web crippling web plates of girders

1089 367 357 587

yield line yield line analysis

253 477


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