Flexural Behaviour and Design of Hollow Flange Steel Beams

Flexural Behaviour and Design of

Hollow Flange Steel Beams

By

Tharmarajah ANAPAYAN

Faculty of Environmental and Engineering School of Urban Development

Queensland University of Technology

A THESIS SUBMITTED TO THE SCHOOL OF URBAN DEVELOPMENT QUEENSLAND UNIVERSITY OF TECHNOLOGY IN PARTIAL

FULFILLMENT OF REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

March 2010

Keywords

iii

KEYWORDS

LiteSteel beams, Hollow flange beams, Hollow flange steel beams, Lateral

distortional buckling, Lateral buckling tests, Section moment capacity tests, inelastic

reserve bending moments, Finite element analyses (FEA), Cold-formed steel

structures, Flexural members, Web stiffeners, Transverse web stiffeners.

Keywords

iv

Abstract

v

ABSTRACT

The LiteSteel Beam (LSB) is a new hollow flange channel section developed by

OneSteel Australian Tube Mills using a patented Dual Electric Resistance Welding

technique. The LSB has a unique geometry consisting of torsionally rigid rectangular

hollow flanges and a relatively slender web. It is commonly used as rafters, floor

joists and bearers and roof beams in residential, industrial and commercial buildings.

It is on average 40% lighter than traditional hot-rolled steel beams of equivalent

performance. The LSB flexural members are subjected to a relatively new Lateral

Distortional Buckling mode, which reduces the member moment capacity. Unlike the

commonly observed lateral torsional buckling of steel beams, lateral distortional

buckling of LSBs is characterised by simultaneous lateral deflection, twist and web

distortion.

Current member moment capacity design rules for lateral distortional buckling in

AS/NZS 4600 (SA, 2005) do not include the effect of section geometry of hollow

flange beams although its effect is considered to be important. Therefore detailed

experimental and finite element analyses (FEA) were carried out to investigate the

lateral distortional buckling behaviour of LSBs including the effect of section

geometry. The results showed that the current design rules in AS/NZS 4600 (SA,

2005) are over-conservative in the inelastic lateral buckling region. New improved

design rules were therefore developed for LSBs based on both FEA and experimental

results. A geometrical parameter (K) defined as the ratio of the flange torsional

rigidity to the major axis flexural rigidity of the web (GJf/EIxweb) was identified as

the critical parameter affecting the lateral distortional buckling of hollow flange

beams. The effect of section geometry was then included in the new design rules

using the new parameter (K). The new design rule developed by including this

parameter was found to be accurate in calculating the member moment capacities of

not only LSBs, but also other types of hollow flange steel beams such as Hollow

Flange Beams (HFBs), Monosymmetric Hollow Flange Beams (MHFBs) and

Rectangular Hollow Flange Beams (RHFBs).

The inelastic reserve bending capacity of LSBs has not been investigated yet

although the section moment capacity tests of LSBs in the past revealed that inelastic

reserve bending capacity is present in LSBs. However, the Australian and American

Abstract

vi

cold-formed steel design codes limit them to the first yield moment. Therefore both

experimental and FEA were carried out to investigate the section moment capacity

behaviour of LSBs. A comparison of the section moment capacity results from FEA,

experiments and current cold-formed steel design codes showed that compact and

non-compact LSB sections classified based on AS 4100 (SA, 1998) have some

inelastic reserve capacity while slender LSBs do not have any inelastic reserve

capacity beyond their first yield moment. It was found that Shifferaw and Schafer’s

(2008) proposed equations and Eurocode 3 Part 1.3 (ECS, 2006) design equations

can be used to include the inelastic bending capacities of compact and non-compact

LSBs in design. As a simple design approach, the section moment capacity of

compact LSB sections can be taken as 1.10 times their first yield moment while it is

the first yield moment for non-compact sections. For slender LSB sections, current

cold-formed steel codes can be used to predict their section moment capacities.

It was believed that the use of transverse web stiffeners could improve the lateral

distortional buckling moment capacities of LSBs. However, currently there are no

design equations to predict the elastic lateral distortional buckling and member

moment capacities of LSBs with web stiffeners under uniform moment conditions.

Therefore, a detailed study was conducted using FEA to simulate both experimental

and ideal conditions of LSB flexural members. It was shown that the use of 3 to 5

mm steel plate stiffeners welded or screwed to the inner faces of the top and bottom

flanges of LSBs at third span points and supports provided an optimum web stiffener

arrangement. Suitable design rules were developed to calculate the improved elastic

buckling and ultimate moment capacities of LSBs with these optimum web

stiffeners. A design rule using the geometrical parameter K was also developed to

improve the accuracy of ultimate moment capacity predictions.

This thesis presents the details and results of the experimental and numerical studies

of the section and member moment capacities of LSBs conducted in this research. It

includes the recommendations made regarding the accuracy of current design rules as

well as the new design rules for lateral distortional buckling. The new design rules

include the effects of section geometry of hollow flange steel beams. This thesis also

developed a method of using web stiffeners to reduce the lateral distortional buckling

effects, and associated design rules to calculate the improved moment capacities.

Publications

vii

PUBLICATIONS

Refereed International Conference Papers

1. Seo, J. K., Anapayan, T. and Mahendran, M. (2008) “Initial Imperfection

Characteristics of Mono-Symmetric LiteSteel Beams for Numerical

Studies”, proceedings of the 5th International Conference on Thin-Walled

Structures, Gold Coast, Australia, pp.451-460.

2. Anapayan, T. and Mahendran, M. (2009), “Improvements to the Design of

LiteSteel Beams Undergoing Lateral Distortional Buckling”, proceedings of

the 9th International Conference on Steel Concrete Composite

and Hybrid Structures, Leeds, UK, pp. 767-774.

QUT Conference Papers

1. Anapayan, T. and Mahendran, M. (2007) “Lateral Distortional Buckling

Behaviour of LiteSteel Beams”, BEE Postgraduate Research Conference on

Smart Systems: Technology, Systems and Innovation, Queensland

University of Technology, Brisbane, Australia.

2. Anapayan, T. and Mahendran, M. (2009) “Effect of Section Geometry on the

Lateral Distortional Buckling of LiteSteel Beams”, 3rd BEE Postgraduate

Research Conference on Smart Systems: Technology, Systems and

Innovation, Queensland University of Technology, Brisbane, Australia.

QUT Research Reports

1. Anapayan, T. and Mahendran, M. (2009a) “Lateral Buckling Tests of

LiteSteel Beams”, Research Report, Queensland University of Technology,

Brisbane, Australia.

Publications

viii

2. Anapayan, T. and Mahendran, M. (2009b) “Finite Element Models of

LiteSteel Beams Subject to Lateral Buckling Effects”, Research Report,

Queensland University of Technology, Brisbane, Australia.

3. Anapayan, T. and Mahendran, M. (2009c) “Parametric Studies and

Development of Design Rules for LiteSteel Beams Subject to Lateral

Buckling”, Research Report, Queensland University of Technology,


4. Anapayan, T. and Mahendran, M. (2009d) “Section Moment Capacity of

LiteSteel Beam”, Research Report, Queensland University of Technology,


5. Anapayan, T. and Mahendran, M. (2009e) “Effects of Web Stiffeners on the

Lateral Distortional Buckling Behaviour and Strength of LiteSteel Beams”,

Research Report, Queensland University of Technology, Brisbane, Australia.

Proposed International Journal Papers

1. Anapayan, T. and Mahendran, M. (2010a) “Lateral Buckling Tests of a New

Hollw Flange Channel Beam”, Journal of Construction Steel Research.

2. Anapayan, T. and Mahendran, M. (2010b) “Numerical Model of LiteSteel

beams Subject to Lateral Buckling”, Engineering Structures.

3. Anapayan, T. and Mahendran, M. (2010c) “Improved Design Rules for

LiteSteel Beams as Flexural Members including the Effects of Section

Geometry”, ASCE Journal of Structural Engineering.

4. Anapayan, T. and Mahendran, M. (2010d) “Section Moment Capacity of

LSBs”, ASCE Journal of Structural Engineering.

5. Anapayan, T. and Mahendran, M. (2010e) “Improvements of Lateral

Distortional Buckling Moment Capacity of LSBs by using Web Stiffeners”,

Thin-walled Structures.

Table of Contents

ix

TABLE OF CONTENTS

Keywords ………………………………………………………………………...iii

Abstract …………………………………………………………………………...v

Publications ………………………………………………………………………vii

Table of Contents …………………………………………………………………ix

List of Figures ……………………………………………………………………xv

List of Tables ……………………………………………………………………xxv

Statement of Original Authorship ………………………………………………xxxi

List of Symbols ………………………………………………………………..xxxiii

Acknowledgements …………………………………………………………….xxxv

CHAPTER 1

1.0 INTRODUCTION ...................................................................................... 1-1 1.1 Cold-Formed Steel Members ...................................................................... 1-1

1.2 Hollow Flange Steel Beams ........................................................................ 1-2

1.2.1 Hollow Flange Beams ........................................................................ 1-2

1.2.2 LiteSteel Beams ................................................................................. 1-4

1.3 Manufacturing Process of Hollow Flange Steel Beams .............................. 1-6

1.4 Applications of Hollow Flange Steel Beams .............................................. 1-7

1.5 Research Problem ....................................................................................... 1-9

1.6 Research Objectives and Specific Tasks ................................................... 1-12

1.7 Scope and Limitations ............................................................................... 1-14

1.8 Thesis Contents ......................................................................................... 1-14

CHAPTER 2

2.0 LITERATURE REVIEW ........................................................................... 2-1 2.1 Cold-Formed Steel Members ...................................................................... 2-1

2.2 Cold-Formed Steel Design Standards ......................................................... 2-5

2.3 Buckling Behaviour of Cold-Formed Steel Beams ..................................... 2-6

2.4 Buckling Behaviour of Hollow Flange Steel Beams .................................. 2-8

2.4.1 Local Buckling ................................................................................... 2-9

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x

2.4.2 Lateral Distortional Buckling ........................................................... 2-11

2.4.3 Lateral Torsional Buckling ............................................................... 2-13

2.5 Lateral Buckling Strength of Beams ......................................................... 2-13

2.5.1 Pre-Buckling Deflections ................................................................. 2-15

2.5.2 Post-Buckling Behaviour ................................................................. 2-15

2.5.3 Web Distortion ................................................................................. 2-16

2.5.4 Inelastic Behaviour ........................................................................... 2-17

2.5.5 Initial Geometric Imperfection and Twist ........................................ 2-18

2.5.6 Residual Stress ................................................................................. 2-20

2.5.7 Moment Distribution ........................................................................ 2-21

2.5.8 Load Height ...................................................................................... 2-23

2.5.9 Warping ............................................................................................ 2-24

2.6 Design Guidelines for Cold-Formed Hollow Flange Steel Beams ........... 2-25

2.6.1 Moment Capacity Based on AS 4100 (SA, 1998) ........................... 2-26

2.6.1.1 Section Moment Capacity ....................................................... 2-26

2.6.1.2 Member Moment Capacity ...................................................... 2-27

2.6.2 Moment Capacity Based on AS/NZS 4600 (SA, 2005) ................... 2-29

2.6.2.1 Section Moment Capacity ....................................................... 2-29

2.6.2.2 Member Moment Capacity ...................................................... 2-31

2.6.3 The Direct Strength Method ............................................................. 2-37

2.7 Hollow Flange Steel Beams with Web Stiffeners ..................................... 2-39

2.7.1 HFBs with Web Stiffeners ............................................................... 2-40

2.7.2 LSBs with Web Stiffeners ................................................................ 2-42

2.7.2.1 Stiffener Type and Configurations .......................................... 2-42

2.7.2.2 Design Methods ....................................................................... 2-43

2.7.2.3 Experimental Results of Kurniawan (2005) ............................ 2-45

2.7.2.4 Finite Element Analysis Results of Kurniawan (2005) ........... 2-45

2.8 Finite Element Analysis ............................................................................ 2-47

2.8.1 Finite Element Analyses of LSBs .................................................... 2-48

2.8.2 Finite Element Analyses of HFBs .................................................... 2-54

2.9 Experimental Investigation ....................................................................... 2-56

2.9.1 Tensile Coupon Tests ....................................................................... 2-56

2.9.2 Residual Stress Measurement ........................................................... 2-58

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xi

2.9.3 Initial Geometric Imperfection Measurement .................................. 2-59

2.9.4 Section Capacity Tests ..................................................................... 2-60

2.9.5 Lateral Buckling Tests ..................................................................... 2-60

2.9.6 Experimental Investigation of HFBs ................................................ 2-64

2.9.7 Experiments of other Cold-Formed Steel Beams ............................ 2-65

2.10 Literature Review Findings ....................................................................... 2-67

CHAPTER 3

3.0 MATERIAL PROPERTIES, RESIDUAL STRESSES AND GEOMETRIC

IMPERFECTIONS OF LSB SECTIONS ................................................... 3-1

3.1 Introduction ................................................................................................. 3-1

3.2 Tensile Coupon Tests to Determine the Mechanical Properties ................. 3-2

3.3 Residual Stress Measurements for LSB Sections ....................................... 3-7

3.3.1 Test Procedure .................................................................................... 3-7

3.3.2 Results ................................................................................................ 3-9

3.4 Initial Geometric Imperfection Measurements ......................................... 3-14

3.5 Conclusions ............................................................................................... 3-16

CHAPTER 4

4.0 LATERAL BUCKLING TESTS OF LSB SECTIONS ............................. 4-1

4.1 Introduction ................................................................................................. 4-1

4.2 Selection of Test Specimens ....................................................................... 4-2

4.3 Test Method ................................................................................................ 4-4

4.3.1 Support System .................................................................................. 4-7

4.3.1.1 Flange Twist Restraints ............................................................. 4-8

4.3.2 Loading System .................................................................................. 4-9

4.3.3 Measuring System ............................................................................ 4-11

4.3.4 Test Procedure .................................................................................. 4-13

4.4 Experimental Results and Discussions ..................................................... 4-15

4.5 Comparisons with Design Methods .......................................................... 4-21

4.6 Conclusions ............................................................................................... 4-29

Table of Contents

xii

CHAPTER 5

5.0 FINITE ELEMENT MODELLING OF LSBs SUBJECT TO LATERAL

BUCKLING EFFECTS ............................................................................... 5-1

5.1 Introduction ................................................................................................. 5-1

5.2 Model Description ....................................................................................... 5-1

5.2.1 Discretization of the Finite Element Mesh ......................................... 5-7

5.2.2 Material Model and Properties ........................................................... 5-8

5.2.3 Load and Boundary Conditions .......................................................... 5-9

5.2.3.1 Ideal Finite Element Model ....................................................... 5-9

5.2.3.2 Experimental Finite Element Model ....................................... 5-12

5.2.4 Initial Geometric Imperfections ....................................................... 5-16

5.2.5 Residual Stresses .............................................................................. 5-18

5.2.6 Analysis Methods ............................................................................. 5-21

5.3 Model Validation ....................................................................................... 5-22

5.3.1 Typical Buckling Modes of Ideal Finite Element Model ................. 5-23

5.3.2 Comparison of Elastic Buckling Moment Results ........................... 5-25

5.3.3 Comparison with Experimental Test Results ................................... 5-29

5.4 Conclusions ............................................................................................... 5-34

CHAPTER 6

6.0 PARAMETRIC STUDIES AND DESIGN RULE DEVELOPMENT ...... 6-1

6.1 Introduction ................................................................................................. 6-1

6.2 Parametric Study ......................................................................................... 6-2

6.3 Lateral Distortional Buckling Behaviour and Strength of LSBs ................. 6-4

6.3.1 Effects of Initial Geometric Imperfection Direction .......................... 6-4

6.3.2 Effects of Residual Stresses ............................................................... 6-7

6.4 Ultimate Moment Capacities of LSBs ...................................................... 6-12

6.5 Comparison of Member Moment Capacities of LSBs with AS/NZS 4600 (SA,

2005) Design Rules ................................................................................... 6-15

6.6 Proposed Design Rules for Member Moment Capacities of LSBs ........... 6-22

6.6.1 Calculation of Capacity Reduction Factor (Φ) ................................. 6-25

6.6.2 Moment Capacities of Hollow Flange Beams .................................. 6-30

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xiii

6.7 Effect of Section Geometry on the Lateral Distortional Buckling Moment

Capacities of LSBs .................................................................................... 6-36

6.8 Applicability of the Geometrical Parameter for Other Types of Hollow Flange

Steel Beams ............................................................................................... 6-55

6.9 Conclusions ............................................................................................... 6-62

CHAPTER 7

7.0 SECTION MOMENT CAPACITY OF LITESTEEL BEAM .................... 7-1

7.1 Introduction ................................................................................................. 7-1

7.2 Section Moment Capacity Tests of LSBs ................................................... 7-2

7.2.1 Test Set-Up and Procedure ................................................................ 7-3

7.2.2 Test Results and Discussion ............................................................... 7-6

7.2.3 Comparison of Ultimate Moment Capacities from Tests and Current

Design Rules ...................................................................................................... 7-11

7.3 Finite Element Modelling of LSBs to Determine their Section Moment

Capacities .................................................................................................. 7-16

7.3.1 Experimental Finite Element Model of LSBs .................................. 7-16

7.3.2 Finite Element Analyses of LSBs Subject to Local Buckling Effects7-27

7.4 Comparison of Ultimate Moment Capacities from FEA and Current Design

Rules .......................................................................................................... 7-31

7.5 Comparison of Ultimate Moment Capacities from FEA and Other Proposed

Design Rules ............................................................................................. 7-38

7.6 Discussion of Maximum Available Moment Capacity of LSBs and

Compressive Strain Limits ........................................................................ 7-41

7.7 Conclusions ............................................................................................... 7-49

CHAPTER 8

8.0 EFFECT OF WEB STIFFENERS ON THE LATERAL DISTORTIONAL

BUCKLING BEHAVIOUR AND STRENGTH OF LITESTEEL BEAMS8-1

8.1 Introduction ................................................................................................. 8-1

8.2 Elastic Buckling Analyses........................................................................... 8-4

Table of Contents

xiv

8.2.1 Finite Element Models ....................................................................... 8-6

8.2.2 Results .............................................................................................. 8-13

8.2.3 Determination of Optimum Spacing and Size of Web Stiffeners .... 8-17

8.3 Elastic Lateral Distortional Buckling of LSBs with Web Stiffeners ........ 8-23

8.4 Ultimate Member Moment Capacities of LSBs with Web Stiffeners ....... 8-31

8.5 Development of Design Rules ................................................................... 8-37

8.6 Conclusions ............................................................................................... 8-42

CHAPTER 9

9.0 Conclusions and Recommendations ............................................................ 9-1

9.1 Experimental Investigation of LSBs ........................................................... 9-3

9.2 Finite element Modelling of LSBs Subject to Lateral Buckling ................. 9-4

9.3 Parametric Studies and Design Rule Development ..................................... 9-5

9.4 Section Moment Capacity of LSBs ............................................................. 9-6

9.5 Effect of Web Stiffeners on the Lateral Distortional Buckling Moment Capacity

of LSBs ........................................................................................................ 9-6

9.6 Future Research ........................................................................................... 9-7

Appendix A…………………………………………………………………….A-1

Appendix B…………………………………………………………………….B-1

Appendix C…………………………………………………………………….C-1

Appendix D…………………………………………………………………….D-1

Appendix E…………………………………………………………………….E-1

References..........................................................................................................R-1

List of Figures

xv

LIST OF FIGURES

Figure 1.1: Cold-Formed Steel Structure .................................................................. 1-1

Figure 1.2: Cold-Formed Steel Cross-Sections ......................................................... 1-2

Figure 1.3: The Hollow Flange Beam ....................................................................... 1-3

Figure 1.4: Typical LSBs .......................................................................................... 1-4

Figure 1.5: HFS Manufacturing Process ................................................................... 1-6

Figure 1.6: Applications of LSBs ............................................................................. 1-7

Figure 1.6: Applications of LSBs ............................................................................. 1-8

Figure 1.7: Lateral Distortional Buckling of LSB .................................................. 1-10

Figure 1.8: HFB with Web Stiffener ....................................................................... 1-11

Figure 2.1: Various Shapes of Cold-Formed Steel Sections ..................................... 2-1

Figure 2.2: Different Types of Cold-Formed Steel Sections .................................... 2-2

Figure 2.3: Roll Forming Sequence for a Z-Section ………………………………2-3

Figure 2.4: Press Brake Dies ………………………………………………………2-3

Figure 2.5: Typical Stress-Strain Curves .................................................................. 2-4

Figure 2.6: Effects on Strain Hardening and Strain Ageing ..................................... 2-5

Figure 2.7: Different Buckling Modes of Z- Section ................................................ 2-7

Figure 2.8: Different Buckling Modes of Channel Section ...................................... 2-7

Figure 2.9: Flange Distortional and Lateral Distortional Buckling .......................... 2-8

Figure 2.10: HFB and LSB ....................................................................................... 2-9

Figure 2.11: Different Buckling Modes and Stresses of HFB Subject to Bending 2-10

Figure 2.12: Local Buckling Mode of LSB Sections .............................................. 2-11

Figure 2.13: Lateral Distortional Buckling Mode of LSB Sections ....................... 2-11

Figure 2.14: Elastic Lateral Distortional Buckling Moments ................................. 2-12

Figure 2.15: Lateral Torsional Buckling Mode of LSB Sections ........................... 2-13

Figure 2.16: Lateral Buckling Behaviour of Steel Beams ...................................... 2-14

Figure 2.17: Positive and Negative Imperfections of LSBs .................................... 2-19

Figure 2.18: Membrane and Flexural Residual Stresses ......................................... 2-20

Figure 2.19: Bending Moment Diagrams of Beams ............................................... 2-21

Figure 2.20: Effects of Moment Gradient ............................................................... 2-22

List of Figures

xvi

Figure 2.21: Warping Restraining Devices ............................................................. 2-25

Figure 2.22: Comparisons of Experimental and AS 4100 (1998) Predictions ........ 2-29

Figure 2.23: Stiffened Elements and Webs with Stress Gradient ........................... 2-31

Figure 2.24: Comparisons of Experiments and AS/NZS 4600 (1996) Predictions 2-33

Figure 2.25: Comparison of FEA Results with Avery et al.’s (1999b) Predictions 2-34

Figure 2.26: Comparisons of New Design Rules, FEA and Experiments (Φ=0.85) .. 2-

35

Figure 2.27: Comparisons of New Design Rules, FEA and Experiments (Φ=0.90) .. 2-

36

Figure 2.28: Stiffener Types .................................................................................... 2-40

Figure 2.29: Stiffener Configuration ....................................................................... 2-41

Figure 2.30: Special Stiffener Screw Fastened to HFB Flanges ............................. 2-42


Figure 2.32: Predicted Member Capacities of 250x60x2.0 LSB ............................ 2-44

Figure 2.33: FEA Models used by Mahaarachchi and Mahendran (2005c) ........... 2-46

Figure 2.34: Idealised Simply Supported Boundary Conditions ............................. 2-49

Figure 2.35: Ideal Finite Element Model ................................................................ 2-50

Figure 2.36: Experimental Finite Element Model ................................................... 2-51

Figure 2.37: Idealised Models of Residual Stresses for LSBs ................................ 2-52

Figure 2.38: Modified Ideal Finite Element Model (First Version) ........................ 2-53

Figure 2.39: Member Capacity Curves ................................................................... 2-53

Figure 2.40: Modified Ideal FE Model (Final Version) .......................................... 2-54

Figure 2.41: Finite Element Models of HFBs ......................................................... 2-55

Figure 2.42: Typical Stress-Strain Curves of the Base Steel used in LSB Sections ... 2-

57

Figure 2.43: Sectioning of LSBs ............................................................................. 2-59

Figure 2.44: Geometric Imperfection Test Set-up .................................................. 2-59

Figure 2.45: Section Capacity Test Set-up .............................................................. 2-60

Figure 2.46: Overall View of Test Rig .................................................................... 2-61

Figure 2.47: Support System ................................................................................... 2-61

Figure 2.48: Loading System .................................................................................. 2-62


Figure 2.50: Test Set-up of LSB with Stiffeners ..................................................... 2-63

List of Figures

xvii

Figure 2.51: Schematic Diagram for Lateral Buckling Tests of HFBs ................... 2-64

Figure 2.52: Schematic Diagram of Test Rig Including Support System ............... 2-65

Figure 2.53: Lateral Buckling Tsts of RHS Beams ................................................ 2-66

Figure 2.54: Test Arrangement for C- and Z- Section Beams ................................ 2-67

Figure 3.1: Tensile Test Coupons ............................................................................. 3-3

Figure 3.2: Tensile Test Arrangement ...................................................................... 3-4

Figure 3.3: Typical Stress-Strain Curves from Tensile Coupon Tests...................... 3-5

Figure 3.4: Strain Gauge Arrangement ..................................................................... 3-7

Figure 3.5: Sectioning Process of LSB ..................................................................... 3-8

Figure 3.6: Measured Released Strain along the Web Element ................................ 3-9

Figure 3.7: Measured Stresses along the Web Element of a 150x45x1.6 LSB ...... 3-10

Figure 3.8: Membrane Residual Stress Distribution ............................................... 3-11

Figure 3.9: Flexural Residual Stress Distribution ................................................... 3-11

Figure 3.10: Membrane Residual Stress Distribution for 150x45x1.6 LSB ........... 3-13

Figure 3.11: Geometric Imperfection Measurements ............................................. 3-14

Figure 3.12: Measured Imperfections of a 4 m Long 200x45x1.6 LSB Section .... 3-15

Figure 4.1: Experimental Results of Mahaarachchi and Mahendran (2005a) .......... 4-2

Figure 4.2: LSB Test Specimens ............................................................................... 4-3

Figure 4.3: Different Types of Test Methods ........................................................... 4-5

Figure 4.4: Overall View of Test Rig........................................................................ 4-6

Figure 4.5: Support System ....................................................................................... 4-7

Figure 4.6: Flange Twist at Failure of a 250x75x2.5 LSB with 3.5 m Span ............ 4-8

Figure 4.7: Flange Twist Restraint Arrangement of LSBs ....................................... 4-9

Figure 4.8: Loading System .................................................................................... 4-10

Figure 4.9: Data Logger and Load Cells ................................................................. 4-12

Figure 4.10: Wire Displacement Transducers (WDTs) .......................................... 4-12

Figure 4.11: Schematic Diagram of a Typical Test Specimen................................ 4-13

Figure 4.12: Schematic Diagram of Flange Twist Restraints ................................. 4-14

Figure 4.13: Typical Lateral Distortional Buckling Failure.................................... 4-15

List of Figures

xviii

Figure 4.14: A Closer View of Lateral Distortional Buckling Failure .................... 4-16

Figure 4.15: Local Web Buckling after Ultimate Failure ....................................... 4-16

Figure 4.16: Comparison of Flange Twist Condition at Failure ............................. 4-17

Figure 4.17: Shear Buckling Failure of 150x45x1.6 LSB with 1.2 m Span ........... 4-17

Figure 4.18: Moment vs Lateral Deflection Curves ................................................ 4-18

Figure 4.19: Comparison of Experimental Failure Moments with AS/NZS 4600 (SA,

2005) Predictions ..................................................................................................... 4-25

Figure 4.20: Typical Elastic Buckling Failure Mode from Finite Element Analysis 4-

28

Figure 5.1: Schematic Diagrams of Ideal and Experimental FE Models .................. 5-2

Figure 5.2: Actual and Idealised LSBs ...................................................................... 5-4

Figure 5.3: Typical Finite Element Mesh for LSB Models ....................................... 5-8

Figure 5.4: Stress-Strain Relationships ..................................................................... 5-9

Figure 5.5: Idealised Simply Supported Boundary Conditions ............................... 5-10

Figure 5.6: Boundary Conditions of the Ideal Finite Element Model of LSB ........ 5-11

Figure 5.7: Typical Loading Method for the Ideal Finite Element Model of LSB . 5-12

Figure 5.8: Loading and Boundary Conditions of the Experimental Finite Element

Model of LSB .......................................................................................................... 5-14

Figure 5.9: Loading Plate Twisting in the Experimental FE Model ....................... 5-15

Figure 5.10: Various Plate Elements in Experimental Finite Element Model ........ 5-16

Figure 5.11: Critical Buckling Mode from Elastic Buckling Analysis of Ideal Finite

Element Model ........................................................................................................ 5-17

Figure 5.12: Effect of Imperfection Direction Based on Nonlinear Analysis ......... 5-18

Figure 5.13: Residual Stress Distributions in LSB Sections ................................... 5-19

Figure 5.14: Typical Residual Stresses Distribution for LSB Sections .................. 5-20

Figure 5.15: Elastic Buckling Modes of 200x60x2.0 LSB ..................................... 5-23

Figure 5.15: Elastic Buckling Modes of 200x60x2.0 LSB ..................................... 5-24

Figure 5.16: Ultimate Failure Modes of 200x60x2.0 LSB ..................................... 5-25

Figure 5.17: Comparison of Elastic Buckling Moments ......................................... 5-28

Figure 5.18: Bending Moment vs Vertical Deflection at Mid-Span Curves for

150x45x1.6 LSB (3000 mm Span) .......................................................................... 5-30

List of Figures

xix


200x45x1.6 LSB (4000 mm Span) ......................................................................... 5-31


300x60x2.0 LSB (4000 mm Span) ......................................................................... 5-31

Figure 5.21: Bending Moment vs Lateral Deflection at Mid-Span Curves for

150x45x1.6 LSB (1800 mm Span) ......................................................................... 5-32


200x45x1.6 LSB (4000 mm Span) ......................................................................... 5-32


150x45x2.0 LSB (3000 mm Span) ......................................................................... 5-33

Figure 6.1: Positive and Negative Imperfections of LSBs ........................................ 6-5

Figure 6.2: Effects of Initial Geometric Imperfection Direction on the Ultimate

Moment Capacities of LSBs ..................................................................................... 6-7

Figure 6.3: Effects of Residual Stresses on the Ultimate Moment Capacities of

300x75x3.0 LSBs ...................................................................................................... 6-9

Figure 6.4: Comparison of Moment Capacities of 300x75x3.0 LSBs with and without

Residual Stresses ..................................................................................................... 6-10









Figure 6.9: Ultimate Moment Capacity Curves of LSBs ........................................ 6-13

Figure 6.10: Comparison of Moment Capacity Results from FEA with AS/NZS 4600

(SA, 2005) Design Curve ........................................................................................ 6-17

Figure 6.11: Comparison of FEA Moment Capacities with the Design Curve based

on Equations 6.7 (a) to (c) ....................................................................................... 6-23

List of Figures

xx

Figure 6.12: Comparison of Experimental Moment Capacities with the Design Curve

based on Equations 6.7 (a) to (c) ............................................................................. 6-24

Figure 6.13: Comparison of FEA and Experimental Moment Capacities with the

Design Curve based on Equations 6.7 (a) to (c) ...................................................... 6-24


on Equations 6.10 (a) to (c) ..................................................................................... 6-27


based on Equations 6.10 (a) to (c) ........................................................................... 6-28


based on Equations 6.11 (a) to (c) ........................................................................... 6-29


on Equations 6.11 (a) to (c) ..................................................................................... 6-30

Figure 6.18: Hollow Flange Beams ......................................................................... 6-30

Figure 6.19: Comparison of FEA Moment Capacities of HFBs from Avery et al.

(1999b) with Equations 6.7 (a) to (c) ...................................................................... 6-35

Figure 6.20: Non-Dimensional Member Moment Capacity versus Modified

Slenderness λd for LSBs .......................................................................................... 6-37

Figure 6.21: Non-Dimensional Member Moment Capacity versus Slenderness λ for

LSBs ........................................................................................................................ 6-38

Figure 6.22: Moment Capacity Design Curve for LSBs based on a Modified

Slenderness Parameter K1λ ..................................................................................... 6-39


Slenderness Parameter K2λd .................................................................................... 6-40


Slenderness Parameter Kλd ..................................................................................... 6-42

Figure 6.25: Comparison of Experimental Results with Equation 6.18 .................. 6-43

Figure 6.26: Moment Capacities of LSBs with Similar Values of GJf/EIxweb (Set 1) . 6-

44


45


46

List of Figures

xxi


(1999b) with Equation 6.18 .................................................................................... 6-47

Figure 6.30: Comparison of FEA Moment Capacities of Selected HFBs from Avery

et al. (1999b) with Equation 6.18 ............................................................................ 6-47

Figure 6.31: Moment Capacities of HFBs with Similar Values of GJf/EIxweb (Set 1) 6-

49

Figure 6.32: Moment Capacities of HFBs with Similar Values of GJf/EIxweb (Set 2) 6-

49

Figure 6.33: Moment Capacities of New LSBs with Different GJf/EIxweb Values . 6-51

Figure 6.34: Moment Capacities of Hollow Flange Steel Beams with GJf/EIxweb ≥

0.0811 ...................................................................................................................... 6-53

Figure 6.35: Moment Capacities of Hollow Flange Steel Beams with the Modified

Slenderness Parameter K as Defined in Equation 6.19 ........................................... 6-54

Figure 6.36: Moment Capacities of Hollow Flange Steel Beams with the Modified

Slenderness Parameter K as Defined in Equation 6.20 ........................................... 6-54

Figure 6.37: MHFB and RHFB Sections ................................................................ 6-55

Figure 6.38: Comparison of Moment Capacities of Hollow Flange Steel Beams with

Similar Values of GJf/EIxweb ................................................................................... 6-61

Figure 7.1: Schematic Diagram of the Test Set-Up .................................................. 7-3

Figure 7.2: Test Set-Up ............................................................................................. 7-5

Figure 7.2: Overall View of Test Set-up ................................................................... 7-5

Figure 7.3: Simply Supported Conditions at the End Supports ................................ 7-5

Figure 7.4: Load Application and Deflection Measurement ..................................... 7-6

Figure 7.5: Moment vs Vertical Deflection Curves of 150x45x2.0 LSB ................. 7-7



Figure 7.8: Plan View of Failed Specimen ............................................................... 7-9

Figure 7.9: Flange and Web Local Buckling ............................................................ 7-9

Figure 7.10: Flange Local Buckling ....................................................................... 7-10

Figure 7.11: Failure Mode of 300x60x2.0 LSB ...................................................... 7-13

Figure 7.12: Schematic Diagram of Experimental Finite Element Model.............. 7-17

List of Figures

xxii

Figure 7.13: Loading and Boundary Conditions of Experimental Finite Element

Model ...................................................................................................................... 7-18

Figure 7.14: Various Plate Elements in Experimental Finite Element Model ........ 7-19

Figure 7.15: Failure Modes from Finite Element Analyses of 150x45x2.0 LSB ... 7-20

Figure 7.16: Failure Modes from Finite Element Analyses of 300x60x2.0 LSB ... 7-21

Figure 7.17: Typical Buckling Mode after Failure from FEA ................................ 7-22

Figure 7.18: Bending Moment vs Vertical Deflection of 150x45x1.6 LSB ........... 7-23

Figure 7.19: Bending Moment vs Vertical Deflection of 200x45x1.6 LSB ........... 7-23

Figure 7.20: Stress-Strain Curves of 150x45x2.0 LSB ........................................... 7-26

Figure 7.20: Stress-Strain Curves of 150x45x2.0 LSB ........................................... 7-27

Figure 7.21: Stress Variation across the Cross-section of LSB from FEA ............. 7-29

Figure 7.21: Stress Variation across the Cross-section of LSB from FEA ............. 7-30

Figure 7.22: Strain Variation across the Cross-section of 150x45x3.0 LSB .......... 7-46

Figure 7.23: Strain along the Top Flange of 150x45x3.0 LSB ............................... 7-46

Figure 7.24: Strain Variation across the cross-section of 150x45x3.0 LSB as Fringe

Results ..................................................................................................................... 7-47

Figure 8.1: Lateral Distortional Buckling of LSBs ................................................... 8-1

Figure 8.1: Lateral Distortional Buckling of LSBs ................................................... 8-2

Figure 8.2: Use of Web Stiffeners in HFBs (Mahendran and Avery, 1997) ............. 8-2

Figure 8.3: Twist Restraint at the Supports ............................................................... 8-3

Figure 8.4: Types of Web Stiffeners Used by Avery and Mahendran (1997) and

Kurniawan (2005) ..................................................................................................... 8-5


Kurniawan (2005) ..................................................................................................... 8-6

Figure 8.5: Schematic Diagrams of Ideal and Experimental FE Models .................. 8-7

Figure 8.6: Experimental Finite Element Model of LSB with Web Stiffeners ......... 8-7

Figure 8.6: Experimental Finite Element Model of LSB with Web Stiffeners ......... 8-8

Figure 8.7: Experimental FE Model with Web Stiffeners and Flange Twist Restraints

................................................................................................................................... 8-9


................................................................................................................................. 8-10

List of Figures

xxiii

Figure 8.8: Ideal Finite Element Model with Full Twist Restraint at the Supports

(Including Flanges) and Web Stiffeners ................................................................. 8-10

Figure 8.9: Idealised Simply Supported Boundary Conditions .............................. 8-11

Figure 8.10: Loading Method of Ideal Finite Element Model ................................ 8-12

Figure 8.11: Elastic Lateral Distortional Buckling Failure Modes of LSBs with

Various Stiffener Arrangements ............................................................................. 8-15


Various Stiffener Arrangements ............................................................................. 8-16

Figure 8.12: LSBs with Web Stiffeners at Different Spacings ............................... 8-18

Figure 8.13: Elastic Lateral Buckling Modes of LSBs ........................................... 8-19

Figure 8.13: Elastic Lateral Buckling Modes of LSBs ........................................... 8-20

Figure 8.14: Elastic Lateral Distortional Buckling of LSB with Web Stiffener ..... 8-24

Figure 8.15: Modw/Mo versus Span for LSBs with Web Stiffeners ......................... 8-28

Figure 8.16: Modw/Mo versus Slenderness for LSBs with Web Stiffeners .............. 8-28

Figure 8.17: Comparison of Modw with Equation 8.2 .............................................. 8-29

Figure 8.18: Comparison of Modw with Equation 8.3 .............................................. 8-30

Figure 8.19: Lateral Buckling Mode of a 2 m Span 150x45x2.0 LSB from Non-linear

FEA ......................................................................................................................... 8-32

Figure 8.20: Ultimate Moments of LSBs with Web Stiffeners ............................... 8-36

Figure 8.21: Comparison of Ultimate Moments of LSBs with and without Web

Stiffeners ................................................................................................................. 8-36

Figure 8.22: Comparison of Ultimate Moments with Equation 8.4 ........................ 8-37





List of Figures

xxiv

List of Tables

xxv

LIST OF TABLES

Table 1.1: Geometry of HFB Sections ...................................................................... 1-3

Table 1.2: Mechanical Properties of LSBs ............................................................... 1-5

Table 1.3: LSB Section Dimensions ......................................................................... 1-5

Table 2.1: Avery et al.’s (1999b) Coefficients for Equation 2.32 ........................... 2-34

Table 2.2: Idealised Simply Supported Boundary Conditions ................................ 2-49

Table 2.3 Nonlinear Analysis Parameters ............................................................... 2-51

Table 2.4: Tensile Coupon Test Results ................................................................. 2-57

Table 3.1: Tensile Test Results ................................................................................. 3-5

Table 3.2: Comparison of Yield and Ultimate Stresses ............................................ 3-6

Table 3.3: Membrane Residual Stress of LSBs ...................................................... 3-13

Table 4.1: Details of Test Specimens ........................................................................ 4-4

Table 4.2: Lateral Buckling Test Results from this Study ...................................... 4-19

Table 4.3: Details and Results of Mahaarachchi and Mahendran’s (2005a) Lateral

Buckling Tests ......................................................................................................... 4-20

Table 4.4: Measured Properties and Capacities of LSBs Used in the Current Lateral

Buckling Tests ......................................................................................................... 4-23

Table 4.5: Measured Properties and Capacities of LSBs Used in the Lateral Buckling

Tests of Mahaarachchi and Mahendran (2005a) ..................................................... 4-24

Table 4.6: Comparison of Experimental Failure Moments of Mahaarachchi and

Mahendran (2005a) with AS/NZS 4600 (SA, 2005) Predictions ........................... 4-26

Table 4.7: Comparison of Experimental Failure Moments with AS/NZS 4600 (SA,

2005) Predictions .................................................................................................... 4-27

Table 4.8: Effect of Flange Twist Restraint from Finite Element Analysis ............ 4-28

Table 5.1: Nominal Properties of Available LSB Sections....................................... 5-3

Table 5.2: Elastic Section Modulus of Actual and Idealised LSBs .......................... 5-4

List of Tables

xxvi

Table 5.3: Elastic Lateral Buckling Moments of Actual and Idealised LSB Sections

................................................................................................................................. ..5-5

Table 5.4: Percentage Differences in Elastic Lateral Buckling Moments of Idealised

and Actual LSBs ........................................................................................................ 5-6


Table 5.6: Membrane Residual Stress Distribution of LSB Sections ..................... 5-20

Table 5.7: Comparison of Elastic Buckling Moments of LSB from FEA, Thin-Wall

and Pi and Trahair’s (1997) Equation ..................................................................... 5-26

Table 5.7 (Continued): Comparison of Elastic Buckling Moments of LSB from FEA,

Thin-Wall and Pi and Trahair’s (1997) Equation .................................................... 5-27

Table 5.8: Comparison of Experimental and FEA Ultimate Moment Capacities ... 5-30

Table 6.1: Nominal Dimensions of LSBs ................................................................. 6-3

Table 6.2: Effects of Initial Geometric Imperfection Direction on the Ultimate

Moment Capacities of LSBs ..................................................................................... 6-6

Table 6.3: Effects of Residual Stresses on the Ultimate Moment Capacities ........... 6-8

Table 6.4: Ultimate Moment Capacities of LSBs in kNm ...................................... 6-14

Table 6.5: Comparison of Moment Capacities from FEA and AS/NZS 4600 (SA,

2005) ........................................................................................................................ 6-19

Table 6.5 (continued): Comparison of Moment Capacities from FEA and AS/NZS

4600 (SA, 2005) ...................................................................................................... 6-20


4600 (SA, 2005) ...................................................................................................... 6-21


4600 (SA, 2005) ...................................................................................................... 6-22

Table 6.6: Capacity Reduction Factors for Eq.6.7 .................................................. 6-26

Table 6.7: Capacity Reduction Factors for Eq.6.10 ................................................ 6-27

Table 6.8: Capacity Reduction Factors for Eq.6.11 ................................................ 6-29

Table 6.9: Geometrical Dimensions of HFB Sections ............................................ 6-31

Table 6.10: Comparison of Avery et al.’s (1999b) FEA Results with Eq.6.7 ......... 6-32

Table 6.10 (continued): Comparison of Avery et al.’s (1999b) FEA Results with

Eq.6.7 ...................................................................................................................... 6-33

List of Tables

xxvii


Eq.6.7 ...................................................................................................................... 6-34

Table 6.11: Capacity Reduction factors for Eq.6.18 ............................................... 6-43

Table 6.12: Section Properties of LSBs including K .............................................. 6-44

Table 6.13: Section Properties of HFBs including K .............................................. 6-46

Table 6.14: Two New LSBs with Different GJf/EIxweb and K Values .................... 6-50

Table 6.15: FEA Moment Capacity Results of Two New LSBs ............................ 6-50

Table 6.16: Two New LSBs with Higher Values of GJf/EIxweb .............................. 6-52

Table 6.17: FEA Moment Capacity Results of Two New LSBs with Higher Values

of GJf/EIxweb ............................................................................................................ 6-52

Table 6.18: Dimensions of MHFB and RHFB Sections ......................................... 6-56

Table 6.19: Section Properties of MHFBs and RHFBs including K ...................... 6-56

Table 6.20: FEA Results of MHFB Sections without Residual Stresses ................ 6-57

Table 6.21: FEA Results of RHFB Sections without Residual Stresses ................. 6-58

Table 6.22: FEA Results of LSB Sections without Residual Stresses .................... 6-59

Table 7.1: Section Classification for LSBs ............................................................... 7-1

Table 7.2: Measured Dimensions of LSBs ............................................................... 7-2

Table 7.3: Measured Yield Stresses of LSBs ............................................................ 7-3

Table 7.4: Spans of Test Beams ................................................................................ 7-4

Table 7.5: Ultimate Moments of LSBs ................................................................... 7-10

Table 7.6: Section Moment Capacities from Tests and AS/NZS 4600 (SA, 2005) 7-13

Table 7.7: Measured Dimensions of LSBs used in Mahaarachchi and Mahendran’s

(2005b) Section Moment Capacity Tests ................................................................ 7-14

Table 7.8: Measured Yield Stresses of LSBs used in Mahaarachchi and Mahendran’s

(2005b) Section Moment Capacity Tests ................................................................ 7-14

Table 7.9: Section Moment Capacities from Mahaarachchi and Mahendran’s (2005b)

Tests and AS/NZS 4600 (SA, 2005) ....................................................................... 7-15

Table 7.10: Comparison of Experimental and FEA Ultimate Moment Capacities. 7-22

Table 7.11: Comparison of Mahaarachchi and Mahendran’s (2005b) Experimental

and FEA Ultimate Moment Capacities ................................................................... 7-24

Table 7.12: Ultimate Moments from the Ideal Finite Element Model .................... 7-28

Table 7.13: Ultimate Moment Capacities of LSBs from FEA ................................ 7-31

List of Tables

xxviii

Table 7.14: Compactness of LSBs Based on AS 4100 and AS/NZS 4600 ............. 7-33

Table 7.15: Section Moment Capacities of LSBs ................................................... 7-33

Table 7.16: Comparison of Ultimate Moment Capacities from FEA and Current

Design Rules ........................................................................................................... 7-35

Table 7.17: Comparison of Ultimate Moment Capacities from FEA and Eurocode 3

Part 1.3 (ECS, 2006 & 1996) .................................................................................. 7-37

Table 7.18: Comparison of Ultimate Moment Capacities from FEA and Shifferaw

and Schafer (2008) .................................................................................................. 7-40

Table 7.19: The Ratios of Mu/My and Section Compactness .................................. 7-41

Table 7.20: Dimensions and Properties of Non-Standard Compact LSBs ............. 7-42

Table 7.21: The Ratios of Mu/My of Some Non-Standard Compact LSBs ............. 7-42

Table 7.22: Comparison of Ultimate Moment Capacities from FEA and Eurocode 3

Part 1.3 (NSAI, 2006) for Non-Standard Compact LSBs ....................................... 7-44

Table 7.23: Comparison of Ultimate Moment Capacities from FEA and Shifferaw

and Schafer (2008) for Non-Standard Compact LSBs ............................................ 7-45

Table 7.24: Average and Maximum Membrane Strains of LSB Sections at Failure

............................................................................................................................. …7-48


Table 8.2: Elastic Lateral Distortional Buckling Moments of LSBs with Web

Stiffeners ................................................................................................................. 8-13

Table 8.3: Effect of Web Stiffener Arrangements on the Results of Mod from

Experimental Finite Element Models ...................................................................... 8-14

Table 8.4: Effect of Web Stiffener Spacing on the Elastic Distorional Buckling

Moments of LSBs in kNm ...................................................................................... 8-18

Table 8.5: Effect of Web Stiffener Sizes on the Elastic Lateral Distortional Buckling

Moments of LSBs .................................................................................................... 8-21

Table 8.5 (cont.): Effect of Web Stiffener Sizes on the Elastic Lateral Distortional

Buckling Moments of LSBs .................................................................................... 8-22

Table 8.5 (cont.): Effect of Web Stiffener Sizes on the Elastic Lateral Distortional

Buckling Moments of LSBs .................................................................................... 8-23

Table 8.6: Comparison of Elastic Buckling Moments ............................................ 8-25

Table 8.6 (continued): Comparison of Elastic Buckling Moments ......................... 8-26

List of Tables

xxix

Table 8.6 (continued): Comparison of Elastic Buckling Moments ......................... 8-27

Table 8.7: First Yield Moments of LSBs ................................................................ 8-29

Table 8.8: Comparison of Ultimate Moments with and without Web Stiffeners ... 8-33

Table 8.8 (continued): Comparison of Ultimate Moments with and without Web

Stiffeners (WS) ....................................................................................................... 8-34


Stiffeners (WS) ....................................................................................................... 8-35

Table 8.9: Section Properties of LSBs Including .................................................... 8-42

List of Tables

xxx

xxxi

STATEMENT OF ORIGINAL AUTHORSHIP

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the

best of my knowledge and belief, the thesis contains no material previously

published or written by another person except where due reference is made.

Tharmarajah Anapayan Signed: __________________________________________________ Date: __________________________________________________

xxxii

List of Symbols

xxxiii

LIST OF SYMBOLS

b = plate width

COV = Coefficient Of Variance

E = Young’s modulus

fcr = elastic critical buckling stress

fu = ultimate tensile strength

fyif = inner flange yield stress

fyof = outer flange yield stress

fyw = web yield stress

G = shear modulus

HFSB = Hollow Flange Steel Beam

HFB = Hollow Flange Beam

Ix = second moment of area about major axis

Iy = second moment of area about minor axis

Iw = warping constant

J = torsional constant

k = buckling co-efficient

K = geometrical parameter

L = span

λ = slenderness

λd = modified slenderness

λdw = modified slenderness with web stiffeners

LDB = Lateral Distortional Buckling

LSB = LiteSteel Beam

Mb = member moment capacity

Mc = critical moment

Mo = elatic lateral torsional buckling moment

Mod = elatic lateral distortional buckling moment

Modw = elatic lateral distortional buckling moment with web stiffeners

Mp = plastic moment

MPC = Multiple Point Constraint

List of Symbols

xxxiv

Ms = section moment capacity

My = first yield moment

υ = Poison’s ratio

Φ = capacity reduction factor

OATM = OneSteel Australian Tube Mills

S = plastic section modulus

SPC = Single Point Constraint

SSTM = Smorgon Steel Tube Mills

t = plate thickness

Z = full elastic section modulus

Zc = critical elastic section modulus

Ze = effective elastic section modulus

Acknowledgements

xxxv

ACKNOWLEDGEMENTS

The author wishes to express sincere gratitude to his supervisor, Professor Mahen

Mahendran for his patient guidance, invaluable expertise, rigorous discussions and

continuous support in many ways over the past three years. This study would not

have been success to this level without such assistance. The author would also like to

thank Dr. Jung Kwan Seo for his assistance and experience and friendship during his

postdoctoral study at QUT.

Author would like to thank QUT and OneSteel Australian Tube Mills (OATM) and

Australian Research Council (ARC) for providing financial support to this research.

The author would also like to thank Mr. Ross Dempsey, Manager - Research and

Testing, OneSteel Australian Tube Mills for his technical contributions, and his

overall support to the many different phases of this research project. Thanks also to

the School of Urban Development and the Faculty of Built Environment and

Engineering at QUT for providing the necessary facilities and technical support.

Many thanks to the structural laboratory staff members, particularly Mr. Arthur

Powell, Mr. Brian Pelin and Mr. Terry Beach for their assistance with operating the

equipment, fabrication and preparation of test set-up and specimens. Also many

thanks to staffs of high performance computing (HPC) and research support services

for providing necessary facilities and support with high performance computers and

relevant finite element packages. Special thank is given to Mr. Mark Barry for his

great help regarding HPC facilities.

The author wishes to thank Dr. John Papangelis for his assistance with THIN-WALL

program. Special thanks are given to senior postgraduate students, Dr. Yasintha

Bandulaheva and Mr. Win Kurniawan for their support during this research. It is also

important to thank fellow post-graduate students, Mr. Sivapathasunderam Jeyaragan,

Ms. Nirosha Dolamune Kankanamge, Mr. Poologanathan Keerthan, Mr.

Shanmuganathan Gunalan and Mr. Balachandren Baleshan for their support and

Acknowledgements

xxxvi

contribution to this research, and other postgraduate students for their friendship at

QUT.

Finally, the author wishes to express his sincere appreciation to his parents and sister,

particularly his mother, for their blessings, providing endless support and

encouragement and beliefs in his abilities.

Introduction

1-1

CHAPTER 1

1.0 INTRODUCTION

1.1 Cold-Formed Steel Members

Cold-formed steel members have been widely used in building applications for over

five decades. Their markets include the secondary cladding and purlin applications as

well as the primary applications as beams and columns of industrial, commercial and

housing systems. The reasons behind the growing popularity of these cold-formed

steel products include their ease of fabrication, high strength to weight ratio and

suitability for a wide range of applications. These advantages can result in more cost-

effective designs compared with hot-rolled steel members, especially in short-span

applications.

Figure 1.1: Cold-Formed Steel Structure

(www.structuretech.net)

Cold-formed steel members can be produced in a wide variety of section profiles, the

most commonly used of are the C- (channels) and the Z- sections. The thickness of

steel most frequently used for these structural members ranges from about 0.4 mm to

6.4 mm.

Introduction

1-2

Figure 1.2: Cold-Formed Steel Cross-Sections

Although these cold-formed steel members are considered to be more efficient than

hot-rolled steel members, they suffer from many complex buckling modes and their

interactions because they are usually slender sections that are either unsymmetric or

singly symmetric. Therefore an advanced cold-formed section, called the Hollow

Flange Steel Beams (HFSBs), was identified by cold-formed steel researchers,

manufacturers and designers as an alternative and improved section to replace the

conventional cold-formed C- and Z- sections and smaller hot-rolled I- and channel

sections (Dempsey, 1990 and Mahendran and Avery, 1997).

1.2 Hollow Flange Steel Beams

The Hollow Flange Steel beams (HFSB) are a new group of cold-formed steel

sections made of two torsionally rigid closed flanges and a slender web. Such

innovative sections have a unique geometry and light weight compared to traditional

hot-rolled steel members. They are also more efficient structurally than hot-rolled

steel members. Recently, two different types of HFSs such as Hollow Flange Beam

(HFB) and LiteSteel Beam (LSB) have been developed for use in the building and

construction industries. The first HFS manufactured by OneSteel Australian Tube

Mills (OATM) formerly known as Smorgon Steel Tube Mills (SSTM) during early

1990s is the HFB, which was also called as “DogBone”.

1.2.1 Hollow Flange Beams

The HFB is a unique cold-formed steel section developed for use as flexural

members. It was manufactured from a single strip of high strength steel (G450 steel

with a minimum guaranteed yield stress of 450 MPa) using electric resistance

Introduction

1-3

welding. The structural efficiency of the HFB due to the torsionally rigid closed

triangular flanges combined with economical fabrication process was the basis of

HFB development.

Figure 1.3: The Hollow Flange Beam

Table 1.1: Geometry of HFB Sections

Designation

Nominal Mass per m

Depth of Section D

Flange Width

B

NominalThick- -ness

t

Outside Bend

Radius Ro

Flange Flat

Width b

Web Depth

d

kg/m mm mm mm mm mm mm 45090HFB38 23.0 450 90 3.8 8.0 74.0 370 40090HFB38 21.5 400 90 3.8 8.0 74.0 320 35090HFB38 20.0 350 90 3.8 8.0 74.0 270

30090HFB38 18.5 300 90 3.8 8.0 74.0 220 30090HFB33 16.2 300 90 3.3 8.0 74.0 219 30090HFB28 13.8 300 90 2.8 8.0 74.0 218

25090HFB28 12.7 250 90 2.8 8.0 74.0 168 25090HFB23 10.5 250 90 2.3 8.0 74.0 168

20090HFB28 11.6 200 90 2.8 8.0 74.0 118 20090HFB23 9.6 200 90 2.3 8.0 74.0 118

(b) Isometric View (a) Cross-Sectional View

Introduction

1-4

Figures 1.3 (a) and (b) show the typical cross-section and an isometric view of HFB,

respectively while Table 1.1 presents the details of such HFBs. This doubly

symmetric member has been used as both compression and flexural members.

The HFBs when used as flexural members are subjected to a relatively new Lateral

Distortional Buckling (LDB) mode which reduces their moment capacity. This

caused the researchers to focus on this detrimental effect in the 1990s. It can be seen

in Table 1.1 that the flange width was 90 mm for all the HFBs and other flange

widths could not be manufactured using the existing equipment. The electric welding

process was also found to be somewhat expensive for the manufacturers. Therefore

the HFB production was discontinued in 1997.

1.2.2 LiteSteel Beams

The LiteSteel Beam (LSB) is the recently invented hollow flange steel beam

developed by OATM using a patented Dual Electric Resistance Welding (DERW)

technique. The LSB has a unique shape and manufacturing process which provides

an extremely efficient strength to weight ratio. It has potentially wide range of

applications in residential, commercial, and industrial construction, and is on average

40% lighter than traditional hot-rolled structural sections of equivalent bending

strength.

Figure 1.4: Typical LSBs

Introduction

1-5

Figure 1.4 shows the typical section of LSBs. The high strength steel material used

for LSBs is DuoSteel grade with a web yield stress of 380 MPa and a flange yield

stress of 450 MPa. Initially it is from a base steel with a yield stress fy of 380 MPa

and a tensile strength fu of 490 MPa. However, the cold-forming process improves

the yield stress and tensile strength of the LSB flanges to 450 MPa and 500 MPa,

respectively (not for web). The mechanical properties of steel used in the design of

LSBs are given in Table 1.2.

Table 1.2: Mechanical Properties of LSBs

Location Minimum

Yield Stress, fy (MPa)

Minimum Tensile Strength,

fu (MPa)

Minimum Elongation as a Proportion of Gauge Length

of So (%) Web 380 490 14

Flange 450 500 14 Currently there are 13 variations of the LSBs which range from a depth of 125 mm to

300 mm while the width of the hollow flange varies from 45 mm to 75 mm. The

thickness of steel used for the beams ranges from 1.6 mm to 3.0 mm. The LSB is

manufactured in standard lengths of 12 and 14.5 metres. Table 1.3 shows the section

dimensions for the range of commercially available LSB members.

Table 1.3: LSB Section Dimensions

20.07.93250 x 60 x 2.0 LSB

15.03.951.6 LSB15.04.87125 x 45 x 2.0 LSB15.04.271.6 LSB15.05.26150 x 45 x 2.0 LSB15.04.90200 x 45 x 1.6 LSB20.07.142.0 LSB20.08.81200 x 60 x 2.5 LSB

25.011.22.5 LSB25.013.3250 x 75 x 3.0 LSB20.08.71300 x 60 x 2.0 LSB25.012.12.5 LSB25.014.4300 x 75 x 3.0 LSBmmkg/mmm mm mm

d x bf x tFlange DepthMass

Designation

20.07.93250 x 60 x 2.0 LSB

15.03.951.6 LSB15.04.87125 x 45 x 2.0 LSB15.04.271.6 LSB15.05.26150 x 45 x 2.0 LSB15.04.90200 x 45 x 1.6 LSB20.07.142.0 LSB20.08.81200 x 60 x 2.5 LSB

25.011.22.5 LSB25.013.3250 x 75 x 3.0 LSB20.08.71300 x 60 x 2.0 LSB25.012.12.5 LSB25.014.4300 x 75 x 3.0 LSBmmkg/mmm mm mm

d x bf x tFlange DepthMass

Designation

Introduction

1-6

1.3 Manufacturing Process of Hollow Flange Steel Beams

Cold-formed members are usually manufactured by either roll forming or brake

pressing process. The HFSs are manufactured by roll forming from a single high

strength steel strip on a custom designed and built dual electric resistance welding

mill similar to those used for the manufacturing of circular, square, and rectangular

hollow sections. The process begins by feeding a large roll of sheet through a series

of flattening rollers. The steel is trimmed to appropriate width and the edges are

coiled over in a cold-formed process. This is followed by a complete penetration butt

weld along the length of the steel via a Dual Electric Resistance Welding (DERW)

process. This section is passed through another set of rollers which shape and size the

section and flanges to its final dimensions.

The HFS manufacturing process is illustrated in Figure 1.5. Cleaning and painting is

then performed prior to bundling and stacking. LSB is coated with the AZ+ alloy

coating system while HFB is coated with general water based paint. It provides a

coating thickness of 18-24 microns and protects up to twice the level provided by a

traditional steel tube primer and has resistance to scratching.

Figure 1.5: HFS Manufacturing Process

(http://www.litesteelbeam.com.au)

Introduction

1-7

1.4 Applications of Hollow Flange Steel Beams

Hollow flange steel beams are light weight and most economical cold-formed steel

sections. Even though different types of hollow flange steel beams have been

investigated by researchers in the past, the only such section that is currently

available is the LSB. It has found increasing popularity in residential, industrial and

commercial buildings not only due to their light weight and cost effectiveness, but

also due to their beneficial characteristics of including torsionally rigid flanges

combined with economical fabrication processes. The LSB sections can be used as

flexural members, truss members and studs in a range of building systems. They

have been used in both residential and commercial buildings. Some of the

applications of LSBs are illustrated in Figures 1.6 (a) to (e).

Figure 1.6: Applications of LSBs


(a) Residential Rafters (b) Floor Joists

(c) Roof Beams (d) Floor Bearers

Introduction

1-8

Figure 1.6: Applications of LSBs


The LSB is on average 40% lighter than traditional hot-rolled steel beams of

equivalent performance. This is because of the improved structural performance in

terms of load carrying capacity. The LSB can be lifted and carried like a timber beam

and can be easily worked to run services through or fix other materials to it.

The light weight of LSB provides it with a greater ease of constructability and on-site

versatility and limits the necessity of cranes and other heavy lifting equipment. The

beam material also ensures an ease of construction for the builder as standard power

tools can be used to cut, drill and install it. The connection attributes of LSB allowed

the builder to connect the floor bearers directly to the RHS posts and then fix the

floor joists to the bearers using Tek-screws and therefore off-site fabrication is not

required. The LSB is easy to weld like other structural steel beams if required. One

of the key benefits of LSB is its unique profile, with a thin flat web and two hollow

flanges at the top and bottom. The web is easily worked, allowing for cabling and

other services to be run through it.

LSB exhibits several practical advantages over conventional beams such as hot-

rolled steel and timber sections. Some of them are as follows;

(e) Purlins

Introduction

1-9

• LSB is 'termite proof' and non-combustible, i.e., steel is the preferred option

for the substructure.

• LSB has smaller deflection at cantilever end when compared with the

equivalent timber beam.

• Non shrinking and non creeping at ambient temperatures.

• LSB is more economical and durable than timber.

• Two men could easily handle and place the longest beam using simple lifting

aids.

• Ease of attachment of timber joists and balustrade posts.

• Ease of drilling holes on site to permit bolted assembly.

• Formwork unneeded.

• Economy in transportation and handling.

1.5 Research Problem

The use of thin-walled, cold-formed high strength steel products in the building

industry has significantly increased in recent years. This directed researchers to focus

in this area, particularly Hollow Flange Steel Beams (HFSBs), which are the newly

invented sections by OATM. The HFSs include HFBs and LSBs as described in the

last section. Since the HFB is currently not available in the industry, this research

was mainly focus on the LSBs and then the applicability of the outcomes (design

rules and recommendations) was investigated for HFBs.

The HFBs and LSBs when used as flexural members are subjected to a relatively

new Lateral Distortional Buckling (LDB) mode which reduces the member moment

capacity significantly for intermediate spans. Unlike the commonly observed lateral

torsional buckling of steel beams, the lateral distortional buckling of HFSs is

characterised by simultaneous lateral deflection, twist and cross sectional change due

to web distortion (see Figure 1.7). Although some research (Mahaarachchi and

Mahendran, 2005 a-e, Avery et al., 1999 a, b, and 2000 a, b) has been completed on

HFSs, the effect of hollow flanges and the relative rigidity between flanges and web

elements on the lateral distortional buckling behaviour is not fully understood.

Introduction

1-10

Figure 1.7: Lateral Distortional Buckling of LSB

The current design rules in AS/NZS 4600 (SA, 2005) for flexural members subject to

lateral distortional buckling were developed by Mahaarachchi and Mahendran

(2005d) based on the lower bound to the results from the numerical and experimental

studies of the currently available LSB sections (13 of them). The effect of LSB

section geometry was not considered despite the fact that the member geometry

influences the lateral distortional buckling behaviour. It is also necessary to develop

accurate design rules that are applicable to other hollow flange steel beams. The

critical geometric parameter that determines the lateral distortional buckling of LSBs

is unknown. In recent times, the manufacturing process of LSBs has been further

improved while a different steel grade has also been introduced. Such changes to

LSBs and their manufacturing process are likely to influence the lateral buckling

moment capacities of LSBs. Therefore it is necessary to verify the adequacy of

current design rules for the available LSBs. It is also important to fully understand

the effect of section geometry including the relative rigidity between hollow flange

and web elements on their lateral distortional buckling behaviour and to include a

suitable parameter in the relevant design rules.

Mahaarachchi and Mahendran (2005b) investigated the section moment capacity of

LSBs based on experiments. Their results showed that compact and non-compact

LSBs have inelastic reserve moment capacities although the current Australian and

American cold-formed steel codes limited it to the first yield moment. Their

experimental study did not investigate the possibility of including the available

Flange Lateral Deflection

Web Distortion

Section Twist

Tension

Compression

Introduction

1-11

inelastic bending capacity nor included the recently produced LSB sections. Also, a

finite element model to predict the section moment capacity of LSBs has not been

developed. Therefore it is necessary to investigate the presence of inelastic reserve

bending capacity of LSBs based on experimental and finite element analyses.

Past research by Avery and Mahendran (1997) and Mahendran and Avery (1997)

stated that the use of transverse web plate stiffeners effectively eliminated the

detrimental lateral distortional buckling of HFBs. Figure 1.8 shows the web stiffener

(connected to the flanges) arrangement developed by Avery and Mahendran (1997).

However, Kurniawan’s (2005) investigations on LSBs produced some conflicting

outcomes. His experimental studies based on quarter point loading showed that the

use of web stiffeners did not significantly improve the flexural moment capacity of

LSBs while his finite element analyses based on an ideal finite element model of

LSB with ideal support conditions and a uniform moment improved the lateral

buckling moment capacities. This contradiction should be investigated and the

optimum web stiffener configuration that improves the lateral distortional buckling

moment capacity of LSBs should be determined.

Figure 1.8: HFB with Web Stiffener

(Avery and Mahendran, 1997)

Simple design rules are not available to predict the elastic lateral buckling and

ultimate moments of LSBs with web stiffeners. Therefore it is important to

investigate the effect of web stiffeners on the lateral distortional buckling moment

capacities of LSBs and to develop suitable design rules to predict their elastic lateral

buckling and ultimate moment capacities.

Introduction

1-12

1.6 Research Objectives and Specific Tasks

The overall objective of this research is to investigate the member moment capacity

of LSBs with and without web stiffeners subject to lateral distortional and lateral

torsional buckling effects, and the section moment capacity of LSBs including their

inelastic reserve bending capacity so that safe and efficient design guidelines can be

developed for LSB flexural members in relation to their applications in the

construction industry.

Specific tasks are described next.

1. Conduct a series of lateral buckling tests on LSB sections subject to bending

and to compare the member moment capacity results with the predictions

from the current design rules in AS/NZS 4600 (SA, 2005). (Phase 1).

2. Develop suitable experimental and ideal finite element models of LSBs

subject to lateral distortional buckling and validate them using experimental

results and other numerical methods. (Phase 1).

3. Use the developed ideal finite element models with ideal support conditions

and a uniform moment in a parametric study to investigate the lateral

distortional and lateral torsional buckling modes of failures and the reduction

of member moment capacity due to lateral distortional buckling of LSBs, and

compare with predicted member moment capacities using the current

AS/NZS 4600 (SA, 2005) design rules. Modify and develop new design rules

if necessary. (Phase 1).

4. Investigate the effects of relevant geometrical parameters of LSBs such as

depth/thickness, slenderness of plate elements, torsional and flexural rigidity

of flanges and web using a parametric study based on the validated ideal

finite element model and determine the critical geometrical parameter that

influences the lateral distortional buckling capacity of LSBs. Develop new

design rules based on this geometrical parameter. (Phase 1).

Introduction

1-13

5. Verify the adequacy of the developed member moment capacity design rules

for lateral distortional buckling for other types of hollow flange steel beams

such as hollow flange beams. (Phase 1).

6. Conduct a series of section moment capacity tests to verify the existence of

inelastic bending capacity of LSBs. (Phase 2).

7. Develop suitable finite element models of LSBs to predict their section

moment capacity and validate them using experimental results. (Phase 2).

8. Use the validated finite element models to investigate the existence of

inelastic bending capacity and develop suitable design rules or

recommendations to calculate the section moment capacities of LSBs

including their inelastic bending reserve capacities. (Phase 2).

9. Develop suitable finite element models of LSBs with web stiffeners and

undertake a study to determine the optimum web stiffener configuration that

eliminates/reduces the detrimental lateral distortional buckling effects. (Phase

3).

10. Develop appropriate design rules to predict the elastic lateral distortional

buckling and ultimate member moment capacities of LSBs with the optimum

web stiffener configuration based on a detailed parametric study. (Phase 3).

11. Verify the applicability of the geometrical parameter found for unstiffened

LSBs to LSBs with web stiffeners. Develop or modify design equations based

on this geometrical parameter to predict the member moment capacities of

LSBs with web stiffeners. (Phase 3).

Ultimately, the accurate design rules developed for unstiffened and stiffened LSBs

will enhance their structural efficiency, mitigate lateral distortional buckling effects,

allow designers to make use of the increased lateral buckling capacities and the

Introduction

1-14

available inelastic reserve bending capacities, and increase their range of applications

in the construction industry.

1.7 Scope and Limitations

This research includes three major phases involving the following:

• Phase 1 – lateral distortional buckling of hollow flange steel beams and the

effects of their section geometry on lateral distortional buckling capacities

based on the experiments and finite element analyses,

• Phase 2 – section moment capacity of LSBs including the presence of their

inelastic reserve bending capacity based on the experimental and finite

element analyses and

• Phase 3 – effects of web stiffeners on the lateral distortional buckling

behaviour and capacity of LSBs based on finite element analyses.

The scope and limitations of this research based on the above three phases are as

follows:

1. Lateral distortional buckling behaviour of hollow flange steel beams subject

to a uniform bending moment with shear centre loading was considered.

However, the effects of moment gradient and load height were not

considered.

2. Elastic local buckling effects that may occur in thinner hollow flange steel

beams were not considered nor the interaction effects of local and lateral

buckling modes of failure.

3. Effects of transverse web stiffeners on the lateral distortional buckling

moment capacity of LSBs were considered under a uniform bending moment

with shear centre loading. However, the effects of moment gradient and load

height were not considered.

1.8 Thesis Contents

The outline of this thesis is as follows: Chapter 1 presents the introduction of thin-walled cold-formed steel members and

hollow flange steel beams such as Hollow Flange Beams (HFBs) and

Introduction

1-15

LiteSteel Beams (LSBs) and describes the manufacturing process,

mechanical properties, shapes and dimensions, and their applications in

the building industry. This chapter then provides the details of research

problems, objectives and research methods.

Chapter 2 presents the relevant literature review to successfully carry out this

research. It describes the buckling behaviour of cold-formed steel

members and hollow flanges sections. Further, it describes the past

research conducted on cold-formed steel members and LSBs subject to

lateral distortional buckling and the current design methods.

Chapter 3 presents the details of tensile coupon tests, residual stress measurements

and imperfection measurements of LSBs.

Chapter 4 presents the experimental investigation of LSBs subject to lateral

buckling. A comparison of the tests results with the current design rules

is also presented in this chapter.

Chapter 5 presents the details of the development and validation of both

experimental and ideal finite element models of LSBs subject to lateral

buckling.

Chapter 6 presents the details of a parametric study on the lateral buckling of LSBs,

effects of imperfections and residual stresses and the development of

design rules based on the ideal finite element model of LSBs subject to

lateral buckling. This chapter also includes the investigation on the

effects of section geometry of LSBs on the lateral distortional buckling

behaviour and the development suitable design rules based on a suitable

geometrical parameter.

Chapter 7 presents the details of the investigation of section moment capacity of

LSBs including inelastic bending reserve capacity based on experimental

and finite element analyses.

Introduction

1-16

Chapter 8 presents the details of the effect of web stiffeners on the lateral

distortional buckling moment capacity of LSBs based on finite element

analyses. Suitable design rules to calculate the elastic lateral buckling

and ultimate moment capacities of LSBs with web stiffeners are also

presented in this chapter.

Chapter 9 presents the significant findings from this research and the

recommendations for future research.

Literature Review

2-1

CHAPTER 2

2.0 LITERATURE REVIEW

2.1 Cold-Formed Steel Members

Cold-formed steel members are steel structural products that are made by bending

flat sheets of steel at ambient temperature into shapes which will support more than

the flat sheets themselves. Corrugated sheets, corrugated culverts, round grain bins,

retaining walls, rails, and other structures have been around for most of the 20th

century. Cold-formed steel for industrial and commercial buildings began about mid

20th century, and widespread usage of steel in residential buildings started in the last

three decades (Hancock et al., 2001). In recent years, higher strength materials and a

wider range of structural applications have caused a significant growth in cold-

formed steel structural members relative to the traditional heavier hot-rolled steel

structural members.

Figure 2.1: Various Shapes of Cold-Formed Steel Sections

(Yu, 2000) Figure 2.1 shows the cross sections of cold-formed steel sections with different

shapes used in the industry. Cold-formed structural steel members can be classified

into two major types such as individual structural framing members and panel and

decks. The usual shapes of the cold-formed steel used for individual structural

framing are channels (C- sections), Z- sections, angles, hat sections, I- sections, T-

sections and tubular members. Figure 2.2 (a) shows some cold-formed sections used

in structural framing. The major function of this type member is to carry load,

structural strength and stiffness are the main considerations in design. Such sections

can be used as primary framing members in buildings up to six stories in height. In

tall multi-story buildings the main framing is typically of heavy hot-rolled shapes

Literature Review

2-2

and the secondary elements may be of cold-formed steel members such as steel

joists, decks, or panels. In this case the heavy hot-rolled steel shapes and the cold-

formed steel sections supplement each other. Cold-formed sections are also used as

chord and web members of open web steel joists, space frames, arches and storage

racks.

Figure 2.2: Different Types of Cold-Formed Steel Sections

(Yu, 2000)

Another category of cold-formed steel sections is shown in Figure 2.2 (b). These

sections are generally used for roof decks, floor decks, wall panels, siding material

and bridge forms. Some deeper panels and decks are cold formed with web

stiffeners. Steel panels and decks not only provide structural strength to carry loads,

but they also provide a surface on which flooring, roofing, or concrete fill can be

applied. They can also provide space for electrical conduits, or they can be perforated

and combined with sound absorption material to form an acoustically conditioned

ceiling.

Cold-Formed Steel Manufacturing Process

Cold-formed members are manufactured by either roll forming or brake pressing

process. Roll forming consists of feeding a continuous steel strip through a series of

opposing rolls to progressively deform the steel plastically to form the desired shape.

Each pair of rolls produces a fixed amount of deformation in a sequence of type

shown in Figure 2.3. In this example, a Z- section is formed by first developing the

bends to form the lip stiffeners and then producing the bends to form the flanges.

(a) Structural Framing (b) Decks and Panels

Literature Review

2-3

Brake forming involves producing one complete fold at a time along the full length

of the section, using a machine called a press brake. Figure 2.4 illustrates the stages

of press braking processes. For sections with several folds, it is necessary to move

the steel plate in the press and to repeat braking operation several times.

Roll forming is the more popular process for producing large quantities of a given

shape. Hollow flange steel beams such as Hollow Flange Beams (HFBs) and

LiteSteel Beams (LSBs) are manufactured by roll forming process. The initial

tooling costs are high, but the subsequent labour content is low. Press braking is

normally used for low-volume production where a variety of shapes are required.

The main disadvantage of roll forming is the time it takes to change rolls for a

different size section.

Stress-Strain Relationship

There are two common types of typical stress-strain curves such as sharp yielding

type and gradual-yielding type. The sharp yielding type is a typical of stress-strain

curve of medium strength cold rolled steel which shows a linear region followed by a

distinct plateau then the strain hardening up to the ultimate tensile strength before

reaching the failure. On the other hand, high strength steel (i.e. G450) does not

exhibit a yield point with a yield plateau. Gradual-yielding occurs after the linear

region.

Ductility

Ductility is defined as the ability of a material to undergo sizable plastic deformation

without fracture. It reduces the harmful effects of stress concentrations and permits

cold-forming of a structural member without impairment of subsequent structural

Figure 2.3: Roll-Forming Sequence for a Z- Section

Figure 2.4: Press Brake Dies

Literature Review

2-4

behaviour. This is not only important for the cold-forming process, but also to avoid

catastrophic brittle behaviour in structural members. High strength steel has a lower

ductility as a consequence of higher yield stress as shown in Figure 2.5 (b). A

conventional measure of ductility is the percentage permanent elongation after

fracture in a 50 mm gauge length of a standard tension coupon. AS/NZS 4600 (SA,

2005) Clause 1.5.1.4 states the ductility criterion of cold-formed steel.

(a) Sharp Yielding (b) Gradual Yielding

Figure 2.5: Typical Stress-Strain Curves

(Yu, 2000)

Cold Work of Forming

The mechanical properties of sections made from sheet steel are affected by the cold

work of forming that takes place in the manufacturing process, specifically in the

regions of the bends. Hancock et al. (2001) summarized a study by Chajes et al.

(1963) of the effects of cold-work on the mechanical properties which take place

mainly in the bend region:

• An increase in the yield strength and ultimate tensile strength and decrease in

the ductility; which is dependent upon the amount of cold work.

• A Bauschinger effect; where difference occurred between the yield strength of

tension and compression.

• The larger the ratio of ultimate tensile strength to the yield strength, the larger

is the effect of strain hardening during cold work.

• Ageing of steel; which enhances the yield and ultimate tensile strengths,

decreases the ductility, and restores or partially restores the sharp yielding

characteristic.

Literature Review

2-5

Figure 2.6: Effects on Strain Hardening and Strain Ageing

(Hancock et al., 2001) The yield stress and the Young’s Modulus vary significantly depending on the

location in the cross section. The yield stress is higher in the rounded corners than in

the flats, but the Young’s Modulus is lower. The stress-strain relationship for the

corners is also different from that for the flats (Put et al., 1999).

Residual Stresses

Residual stresses occur as a result of manufacturing and fabricating processes.

Unlike in hot-rolled members which often have uniform membrane residual stresses,

the thin-walled members have uneven distribution and higher flexural stresses due to

the cold forming. The residual stress causes premature yielding than is expected if it

is neglected, and it reduces the member stiffness. Hancock et al. (2001) stated that

the increased residual stress is one of the factors that cause rapid fracture.

2.2 Cold-Formed Steel Design Standards

Design specifications for hot-rolled steel members can not be used for cold-formed

members as they have different material properties such as member thickness,

imperfections, residual stresses, stress-strain relationships as well as different

behaviour and different modes of failure. Cold-formed sections are thinner than hot-

rolled sections and are characterised by local instabilities while hot-rolled sections

rarely exhibit local buckling.

Current Australian cold-formed steel Standard AS/NZS 4600 (SA, 2005) was based

mainly on the latest AISI Specification. Standards Australia published AS 4100 (SA,

1998) for steel structures which is the latest version and has been most suitable for

Literature Review

2-6

hot-rolled members. British Standard BS 5950 Part 5 (BSI, 1998) specifies guidance

for the design of cold-formed structural steel works. Eurocode 3 (ECS, 1996) and

Canadian Standards are the other international standards providing design guidance

for cold-formed steel structures. The Direct Strength Method is the new design

method for cold-formed steel members and is adopted in the supplement to the North

American specification (AISI, 2004). It is also included in Section 7 of AS/NZS

4600 (SA, 2005). A Direct Strength Method Design Guide (CF06-1) was published

by AISI (2006).

2.3 Buckling Behaviour of Cold-Formed Steel Beams

Buckling is characterised by deformation of the plate elements or members under

compressive stresses. Beams when subject to bending action introduces tensile and

compressive stresses either side of the neutral axis where buckling is likely to occur

in the compressive stress region. Because of the open nature and the thin material

used in cold-formed steel, the failure mode under flexural action has higher

complexity than the hot-rolled members. Cross section instabilities in cold-formed

steel beams include local buckling, distortional buckling and lateral torsional

buckling.

Generally, lateral torsional buckling predominantly limits the cold-formed beam

design; hence requires lateral restraint provision. However, the improvement in the

shape has led to a new buckling mode, lateral distortional buckling.

The cross section deformations associated with each of the three buckling modes are

illustrated in Figure 2.7. Local buckling involves distortion of the cross section with

only rotation occurring at interior fold lines of the section. Distortional buckling

involves distortion of the cross section with rotation and translation occurring at

interior fold lines. Lateral torsional buckling excludes distortion of the cross section;

however, translation and rotation of the entire cross section occur. Local buckling

occurs in short span members while lateral torsional buckling occurs in long span

members and distortional buckling in beams with intermediate spans. Figures 2.8 (a)

to (d) show the different modes of buckling failure that occur in channel sections.

Literature Review

2-7

Figure 2.7: Different Buckling Modes of Z- Section

(Yu, 2006)

Figure 2.8: Different Buckling Modes of Channel Section Past investigations have revealed two distinctive distortional buckling modes that are

commonly observed in cold-formed steel members namely ‘flange distortional

buckling’ and ‘lateral distortional buckling’. Flange distortional buckling involves

rotation of a flange and lip about the flange/web junction of a C- section or Z-

section while lateral distortional buckling involves transverse bending of vertical

web (see Figures 2.9 (a) and (b)). Flange distortional buckling is most likely to occur

in the open thin-walled sections such as C- and Z- sections while lateral distortional

buckling is the most likely in hollow flange steel beams where the high torsional

rigidity of the tubular compression flange prevents it from twisting during lateral

displacement (Pi and Trahair, 1997). Clause 3.3.3.3 of AS/NZS 4600 (SA, 2005)

gives a comprehensive review of distortional buckling.

(a) Local (b) Distortional (c) Lateral Distortional (d) Lateral Torsional

Literature Review

2-8

Figure 2.9: Flange Distortional and Lateral Distortional Buckling

The flange distortional and lateral buckling of cold-formed C- and Z- section steel

members has been extensively investigated. Lau and Hancock (1987) presented

distortional buckling formulae for channel columns while Kwon and Hancock (1991)

proposed design equations for channel section columns undergoing local and

distortional buckling. Hancock et al. (1994) provided design strength curves for thin-

walled C- sections undergoing distortional buckling. Zhao et al. (1995) carried out

lateral buckling tests on rectangular hollow section beams and proposed design

formulations for member moment capacity while Pi and Trahair (1995) developed

lateral buckling strength equations for rectangular hollow sections. Rogers and

Schuster (1997) investigated the distortional buckling of cold-formed steel C-

sections in bending while Hancock (1997) provided a design method for distortional

buckling of C- section flexural members. Pi et al. (1998) investigated the lateral

buckling strength of channel section beams while Put et al. (1999) conducted lateral

buckling tests on channel beams. Put et al. (1998) carried out lateral buckling tests on

Z- beams while Pi et al. (1999) provided lateral buckling strength formula for Z

sections. Lecce and Rasmussen (2005) carried out experimental investigation on

distortional buckling of stainless steel channel sections and Yu and Schafer (2006)

investigated the distortional buckling behaviour of C- and Z- sections by a series of

experiments. It has been found that the cold-formed hollow flange steel beams

severely suffer from lateral distortional buckling due to torsionally rigid flanges with

slender web (Dempsey, 1990) as mentioned earlier.

2.4 Buckling Behaviour of Hollow Flange Steel Beams

Unlike hot-rolled heavy steel sections, structural behaviour of cold-formed Hollow

Flange Steel Beams (HFSBs) are mostly characterised by their high strength thinner

elements. In the design of cold-formed steel flexural members, the moment resisting

(a) Flange Distortional (b) Lateral Distortional

Literature Review

2-9

capacity and stiffness of the beam are the most important criteria. A brief review of

buckling behaviour and design aspects of HFSs such as Hollow Flange Beams

(HFBs) and LiteSteel Beams (LSBs) is presented in this section. Figures 2.10 (a) and

(b) show the typical cross sections of HFB and LSB, respectively.

Figure 2.10: HFB and LSB

2.4.1 Local Buckling

Since there are no free edges in HFSBs the propensity for local plate buckling is very

much reduced when compared with other cold-formed steel sections. However,

slender sections with short spans may locally buckle. The elastic critical stress (fcr)

for local buckling of a plate element in compression or bending is given by:

2

2

2

)1(12⎟⎠⎞

⎜⎝⎛

−=

btEkfcr υ

π (2.1)

Where k, b and t are the plate local buckling coefficient, plate width and plate

thickness, respectively while E is the Young’s modulus and υ is Poisson’s ratio. The

plate local buckling coefficient (k) depends upon the support conditions. Depending

on the restraint conditions along the longitudinal boundaries and the type of loading,

the plate local buckling coefficient (k) takes different values. A plate element is

defined as slender if the elastic critical local buckling stress (fcr) calculated using

Equation 2.1 is less than the material yield stress (fy). A slender section will buckle

locally before the squash load (Py) or the yield moment (My) is reached. Although

local buckling occurs at a stress level lower than the yield stress of steel, it does not

necessarily represent the failure of members. The failure is governed by post-

buckling strength which is generally much higher than the local buckling strength.

The theoretical analysis of post-buckling and failure of plates is extremely difficult,

and generally requires a computer analysis to achieve an accurate solution (Hancock

(a) HFB (b) LSB

Literature Review

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et al., 2001). If the elastic critical buckling stress exceeds the yield stress fy, the

compression element will buckle in the inelastic range (Yu, 2000).

The buckling behaviour of HFBs was investigated by Dempsey (1990) using a finite

strip buckling analysis program “BFINST6”. His buckling analysis has shown that

the buckling coefficients (k) are generally equal to or greater than 4.0 for flange

element and the web element, thus verifying that the flange and web elements are

adequately stiffened. Figure 2.11 shows the buckling stresses over a wide range of

half-wavelengths. Local buckling occurs in the top compression flange at a half-

wavelength of approximately the flat width of the compression element (Point A).

Both of the flange return and the compression portion of the web do not experience

local buckling because the stresses are lower and are not uniform and their flat width

to thickness ratio (b/t) is much smaller.

Figure 2.11: Different Buckling Modes and Stresses of HFB Subject to Bending

(Dempsey, 1990)

The buckling behaviour of LSBs was investigated by Mahaarachchi and Mahendran

(2005a-e) using both experimental and numerical analyses. The LSBs have small

width to thickness ratios when compared to many other cold-formed steel sections,

and gain increased rigidity by having no unstiffened elements (no free edges). This

considerably reduces the propensity to local plate buckling. However in the case of

slender LSB sections with intermediate spans, it has been noted that the sections

exhibited a local buckling in web element during load application, although these

buckles were not seen at the ultimate failure (Mahaarachchi and Mahendran, 2005a).

For very short spans, LSB sections exhibited local buckling failure, and a few of

them displayed weld failures which were caused by large local deformations

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(Mahaarachchi and Mahendran, 2005b). Figure 2.12 shows the typical local buckling

mode of LSBs obtained from finite element buckling analysis.

Figure 2.12: Local Buckling Mode of LSB Sections

2.4.2 Lateral Distortional Buckling

Lateral distortional buckling of hollow flange steel beams is characterised by

simultaneous lateral deflection, twist and cross section change due to web distortion.

The presence of a slender web with torsionally rigid flanges allows flange lateral

displacement and associated web distortion which can reduce the flexural torsional

buckling capacity. The cross-sectional distortion can cause significant strength

reductions, and is particularly severe in intermediate spans (Mahaarachchi and

Mahendran, 2005a). Figure 2.13 shows the typical lateral distortional buckling mode

of LSB sections.

Figure 2.13: Lateral Distortional Buckling Mode of LSB Sections

Lateral distortional buckling behaviour of HFBs has been investigated by many

researchers. Dempsey (1990) analysed the elastic lateral distortional buckling of

simply supported HFBs in uniform bending using a finite strip method incorporated

in the computer program THINWALL (Hancock and Papangelis, 1994). Heldt and

Tension

Compression

Tension

Compression

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Mahendran (1992) conducted investigations of lateral distortional buckling of HFBs

using both buckling analysis and experiments. Mahendran and Doan (1999) carried

out lateral distortional buckling tests on HFBs while Avery and Mahendran (1997)

and Mahendran and Avery (1997) investigated the use of web stiffeners to eliminate

the lateral distortional buckling of HFBs. Pi and Trahair (1997) developed a

nonlinear inelastic method to analyse the lateral distortional buckling behaviour of

HFBs and proposed simple formulations to determine the elastic lateral distortional

buckling moment.

Pi and Trahair (1997) stated that the survey of research information on HFBs

indicated that there is no simple formulation for predicting the effect of lateral

distortional buckling on the lateral buckling of HFBs. On this basis, they attempted

to find a simple but sufficiently accurate closed form solution for the effects of web

distortion on the elastic lateral buckling of simply supported HFBs in uniform

bending. They also attempted to develop an advanced theoretical method for

predicting the effects of stress-strain curve, residual stresses and geometrical

imperfections on the strengths of HFBs that fail by lateral-distortional buckling.

The equation for flexural torsional buckling moment resistance Mo (see Equation

2.20) was modified by Pi and Trahair (1997) by introducing an effective torsional

rigidity GJe (see Equation 2.22) in place of the nominal torsional rigidity (GJ) to

calculate the lateral distortional buckling moment resistance Mod (see Equation 2.21).

Figure 2.14: Elastic Lateral Distortional Buckling Moments

(Pi and Trahair, 1997)

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The elastic lateral distortional buckling moments, predicted by Thin-wall (MTW) and

obtained from Equation 2.21 (Mod) were compared with flexural torsional buckling

moment (Mo) for two HFB sections by Pi and Trahair (1997), and are shown in

Figure 2.14. It can be seen that the approximate values Mod are in close agreement

with the accurate Thin-wall values MTW, and also these lateral-distortional buckling

values are significantly lower than the flexural-torsional buckling moments Mo.

2.4.3 Lateral Torsional Buckling

Lateral torsional buckling is characterised by simultaneous cross-section twist and

lateral displacement of the compression flange. Very long span HFS members

exhibited lateral torsional buckling which is common to hot-rolled members. The

torsional stiffness of the flanges is proportional to the laterally unrestrained length of

the beam. As the length of the beam increases, the torsional stiffness of the flanges

decreases, and thus the beam tends to twist more and hence lateral torsional buckling.

Figures 2.15 (a) and (b) illustrate the lateral torsional buckling mode of LSBs.

Figure 2.15: Lateral Torsional Buckling Mode of LSB Sections

2.5 Lateral Buckling Strength of Beams

The lateral buckling strength of steel beams is governed chiefly by their elastic

lateral buckling resistance and the effects of yielding while the other factors such as

its pre-buckling and post- buckling behaviour and interactions with local and

distortional buckling may also be important. Elastic lateral buckling resistance is

significantly affected by restraints and the way in which the loading is distributed

through the structure. The elastic buckling resistance (see Equations 2.2 and 2.3)

(a) Cross Section (b) Isometric View

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decreases with the span and increases with the minor axis flexural rigidity (EIy), the

torsional rigidity (GJ) and the warping rigidity (EIw). For a simply supported beam in

uniform bending, the moment Mo at elastic buckling (Timoshenko and Gere, 1961) is

given by Equation 2.2.

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2

2

2

2

0 LEIGJ

LEI

M wy ππ (2.2)

Extensive research has shown that the bending moment distribution has a very

significant effect on the elastic buckling resistance (Trahair, 1993) and that uniform

bending is the worst case. Trahair (1995) illustrated the influence of elastic lateral

buckling on the strengths of beams and is shown in Figure 2.16, in which

o

sx

MM

=λ (2.3)

is a modified slenderness and Msx is the nominal major axis section capacity, as

governed by yielding and local buckling effects.

At low slenderness, the strengths of compact beams rise above the major axis section

capacity Msx = Mpx due to strain hardening effects. The strengths of intermediate

slenderness beams lie on a transition from the section capacity Msx to the elastic

buckling resistance Mo. At high slenderness, the beam strengths are close to the

elastic buckling strengths, but may rise above them due to pre-buckling and post-

buckling strengthening effects which are unaccounted for in the determination of

elastic lateral buckling moment.

Figure 2.16: Lateral Buckling Behaviour of Steel Beams

(Trahair, 1995)

Mo

Mo

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Many practical beams are of intermediate slenderness and they fail before the section

capacity Msx or the elastic buckling resistance Mo can be reached. This failure is due

to premature yielding resulting from initial imperfections and twist and residual

stresses. Web distortion also reduces the elastic buckling resistance. As stated in the

earlier sections, cold-formed hollow flange steel beams are more prone to lateral

distortional buckling. Some parameters which influence the beam strength are given

below.

2.5.1 Pre-Buckling Deflections

In the classical analysis of the lateral buckling of cold-formed steel members,

buckling is assumed to be independent of the pre-buckling deflections. This

assumption is valid only when there are small ratios of the minor axis flexural

stiffness and torsional stiffness to the major axis flexural stiffness. However, this

assumption may lead to inaccurate predictions of the buckling resistance when the

ratios are not small. The buckling resistance (Equation 2.2) obtained by including the

effects of the pre-buckling deflections may significantly exceed the theoretical

predictions which can be incorporated by a factor approximated by Pi and Trahair

(1992a, b).

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−⎟

⎠⎞

⎜⎝⎛ −

=

x

w

x

yo

EILEIGJ

EIEIM

M

2/11

122π

(2.4)

Past research on cold-formed channel sections (Pi et al., 1998), Z-sections (Pi et al.,

1999), rectangular hollow sections (Pi and Trahair, 1995) and HFBs (Pi and Trahair,

1997) incorporated the effects of pre-buckling deflections while the research on

LSBs (Mahaarachchi and Mahendran, 2005d) did not consider them.

2.5.2 Post-Buckling Behaviour

Beams subject to bending moment buckle in different modes and will remain that

mode shape into the post-buckling range until failure. Moments at failure may be

considerably higher than those at which buckling occurs. Imperfections increase

more in the post-buckling state than in the pre-buckling one. Significant

redistribution of stresses takes place and the flexural stiffness of the beams also

changed in the post-buckling state. Although significant increases in strength are

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only realised for very slender beams, this post-buckling behaviour causes the beam

strength to remain above the minor axis section capacity Msy, even when this is

greater than the elastic buckling resistance.

2.5.3 Web Distortion

While the flanges of hollow flange steel beams are very stiff torsionally, their webs

are comparatively flexible, and may allow web distortion effects to reduce their

resistances to lateral torsional buckling. The flanges of an I-section beam are not stiff

torsionally, and so web distortion does not become very significant unless the web is

particularly slender, or only the tension flange is restrained torsionally (Pi and

Trahair, 1997). The web flexibility allows significant flange displacements with only

small flange rotations. Web distortion introduces some effects that are not

encountered in lateral torsional buckling. First, web distortion may reduce the

effective torsional rigidity of the cross section. Second, the parallel flanges may have

different angles of twist rotation during buckling. Third, the symmetrical nature of

the hollow flange steel beams become unsymmetric during lateral distortional

buckling so that the centroid, shear centre and principal axes of the distorted cross

section are all different from those of undistorted cross section before buckling (Pi

and Trahair, 2000). Web distortion reduces the warping rigidities. The adoption of

reduced warping rigidity is important for the beams with small flange torsional

rigidities (Pi and Trahair, 2000) and this can be ignored for the hollow flange steel

beams as these have high flange torsional rigidities.

Past research identified that the lateral buckling strength of cold-formed channel

section beams (Pi et al., 1998) and Z- section beams (Pi et al., 1999) with web

distortion are lower than those without web distortion. However, the effects of web

distortion are small for these beams as they have low torsional rigidities.

Lateral buckling strengths of HFBs with web distortion are lower than those without

web distortion for beams of intermediate slenderness, but the effects of web

distortion on the strength of beams with very high and very low slenderness are small

(Pi and Trahair, 1997). Findings by Mahaarachchi and Mahendran (2005a) on LSBs

also support this fact.

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2.5.4 Inelastic Behaviour

In a short span beam, yielding occurs before the ultimate moment is reached, and

significant portions of the beams are inelastic when buckling commences. The

effective rigidities of these inelastic portions are reduced by yielding, and

consequently, the buckling moment is also reduced. In the inelastic range, the

buckling moment increases almost linearly with decreasing slenderness from the first

yield moment to the full plastic moment which is reached soon after the flanges are

fully yielded, when buckling is controlled by the strain hardening moduli. Residual

stresses and initial imperfections further reduce the inelastic buckling moment of a

beam. Theodore and Galambos (1963) presented a theoretical method for the

determination of the inelastic buckling strength of simply supported wide-flange

steel beams subjected to equal and opposite end moments which fail by lateral

buckling. The method they adopted was based on the determination of the reduction

in the lateral and torsional stiffness due to yielding. They considered the effect of

initial residual stresses as well. These theoretical derivations consisted of several

assumptions such as no initial imperfections, no cross section change during

buckling, etc.

Ma and Hughes (1996) investigated the lateral distortional buckling behaviour of

monosymmetric I-beams under distributed vertical loads by using an energy method.

They used nonlinear elastic theory to obtain the external work due to buckling and

developed a new formulation of total potential energy. Also they assumed that the

flanges buckle as rigid rectangular section beams and the web distorts as an elastic

plate during buckling. Further, they considered both cubic and 5th order polynomial

functions for web out-of-plane buckling deformation and concluded that 5th order

polynomial improved the accuracy of the results for the beams with uniformly

distributed vertical load.

Dekker and Kemp (1998) developed a simplified theoretical model for I-sections

undergoing lateral distortional buckling. Trahair and Hancock (2004) developed a

simple advanced method to design steel members against out-of-plane failure. They

included the effects of high moment, residual stresses and geometrical imperfections

on yielding. In this study, it was suggested that the nominal beam strength may be

obtained by making an inelastic lateral buckling analysis using inelastic moduli (EI =

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γIME, GI = γIMG) which are reduced below their elastic values to allow for the effects

of initial imperfections and twists and residual stresses. The design procedure to

calculate the beam nominal strength derived by Trahair and Hancock (2004) is given

next.

Inelastic lateral buckling moment MI under uniform moment is equal to the nominal

strength Mb and is given in Equation 2.5.

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2

2

2

2

LEIGJ

LEI

M wIMIM

yIMI

γπγγπ

(2.5)

Where

2

2.119.0 ⎟

⎠⎞

⎜⎝⎛−=

pxIM

MMγ while 0.1

2

≤⎟⎠⎞

⎜⎝⎛

pxMM (2.6)

Where, M is the bending moment at the cross section and Mpx is the major axis full

plastic moment. These formulations are suitable for hot-rolled compact I-section

beams but are not suitable for cold-formed steel beams as they have different initial

imperfections, residual stresses and cross sections.

Pi et al. (1998) investigated the inelastic lateral buckling behaviour of cold-formed

channel section beams while Pi et al. (1999) investigated that of cold-formed Z-

section beams. Pi and Trahair (1997) carried out a non-linear inelastic analysis of

HFBs and presented the effects of initial imperfections and twist, residual stresses,

moment distribution, load height and cross section.

2.5.5 Initial Geometric Imperfection and Twist

Geometric imperfections refer to deviation of a member from perfect geometry.

Imperfections of a member include bowing, warping and twisting as well as local

deviations. Local deviations are characterised by dents and regular undulations in the

plate. Imperfection and twist are generally caused by the fabrication process, storage,

transport, handling, installation and other factors. The initial imperfection and twist

may be in the positive direction or in the negative direction.

Pi et al. (1998) stated that the lateral buckling strengths of cold-formed channel

section beams with positive twist rotations were higher than those of the beams with

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negative twist rotations. This explains that the negative initial imperfection and twist

are more unfavourable to the lateral buckling strength and the positive initial

imperfection and twist are more desirable. Research on cold-formed Z- sections by Pi

et al. (1999) also supported this.

Figure 2.17: Positive and Negative Imperfections of LSBs

The magnitudes of the initial imperfection and twist vary randomly and should be

treated probabilistically (Schafer and Pekoz, 1998). However, many researchers

considered consistent magnitudes. Pi and Trahair (1997) found that the central initial

imperfection and twist Uso and Фso that are consistent with AS 4100 (SA, 1990) for

cold-formed hollow flange beams are given by Equations 2.6 (a) and (b).

1000Uso/L = 1000Фso (Mod/NyL) = -1.0 for λ ≥ 0.6 (2.6a)

1000Uso/L = 1000Фso (Mod/NyL) = -0.0001 for λ < 0.6 (2.6b)

where, Uso = initial imperfection

Фso = twist

Mod = elastic lateral distortional buckling moment (Equation 2.21)

λ = slenderness, od

px

MM

Ny = column elastic buckling load about the minor axis (Equation 2.7)

2

2

LEIN y

yπ

= (2.7)

Pi and Trahair (1997) investigated the effects of initial geometric imperfection and

twist of HFBs. They found that the strength increases as the initial imperfection

decreases, but the differences are small for beams with very low or very high

slenderness. This means the effects of initial geometric imperfections for beams with

(a) Positive Imperfection

(b) Negative Imperfection

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intermediate slenderness should be considered in the design as the lateral distortional

buckling is likely to occur.

Mahaarachchi and Mahendran (2005e) measured the initial imperfection and twist of

LSBs and concluded that the local plate imperfections were within the

manufacturer’s fabrication tolerance limit while the overall member imperfections

were less than the AS 4100 recommended limit of Span/1000 (SA, 1998). However,

it is necessary to measure the initial imperfections of the test beams prior to testing in

order to obtain accurate results.

2.5.6 Residual Stress

Residual stresses exist in the longitudinal and transverse directions. They vary

around the cross section and through thickness. The variation through thickness of

residual stress can be considered as the initial shear stress between layers within the

thickness of the plate. The longitudinal transverse residual stress includes membrane

and flexural (or bending) residual stresses. The membrane residual stress is constant

through thickness whereas the flexural residual stress is considered to vary through

thickness.

Figure 2.18: Membrane and Flexural Residual Stresses

(Schafer and Pekoz, 1998) For hot-rolled steel members, the residual stresses are mainly due to uneven cooling

after hot-rolling. Past researchers (Yang et al., 1952 and Tebedge et al. 1973) found

that the magnitude of maximum residual stress in hot-rolled steel sections made of

moderate strength steels is approximately equal to 30% of the yield stress of the

material and the residual stresses are assumed to be uniformly distributed through the

plate thickness, i.e. no bending residual stresses.

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The inelastic buckling moment varies markedly with both magnitude and the

distribution of the residual stresses. The moment at which inelastic buckling initiates

depends on the magnitude of the residual compressive stresses at the flange tips,

where yielding causes significant reductions in the effective rigidities. The flange tip

stresses are comparatively high in hot-rolled beams, especially those with high ratios

of flange to web area, and so inelastic buckling is initiated comparatively early in

these beams. The residual stresses in hot-rolled beams decrease away from the flange

tips, and so the extent of yielding increases and the effective rigidities steadily

decrease as the applied moment increases. Because of this, the inelastic buckling

moment decreases in an approximately linear fashion as the slenderness increases

(Trahair and Bradford, 1991).

For cold-formed steel members, the residual stresses are mainly caused by the cold-

forming process and thus are quite different from those of hot-rolled sections. In fact,

for cold-formed sections, flexural residual stresses are considered the most important

component and these stresses can be as high as 50% of the material yield stress

(Schafer and Pekoz, 1998). Pi et al. (1998) indicated that residual stresses reduced

the lateral buckling strength of cold-formed channel section beams particularly for

intermediate slenderness. Pi et al. (1999) also demonstrated the same fact for cold-

formed Z- sections and Pi and Trahair (1997) concluded this for HFBs.

Mahaarachchi and Mahendran (2005c) indicated the reduction on the lateral buckling

strength of LSBs was about 8%. Therefore, this effect can not be neglected.

2.5.7 Moment Distribution

Moment distribution along the span of the beam affects the lateral buckling strength

as this creates different yielding pattern throughout the span.

Figure 2.19: Bending Moment Diagrams of Beams

(a) (b) (c)

M M

βMM

-M

(d)

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Figure 2.19 (a) shows a uniform bending moment along the beam span caused by

equal and opposite end moments. This is the most severe case, for which yielding is

constant along the beam so that the resistance to lateral buckling is reduced

everywhere. Less severe case is the case of an unbraced beam with a load

concentrated in the centre (Figure 2.19 (b)), for which yielding is confined to a small

central portion of the beam, so that any reductions in the section properties are

limited to this region. Uniformly distributed load also has a similar effect. Even less

severe cases are those of beams with unequal end moments M and βM (Figure 2.19

(c)), where yielding is confined to small portions near the supports. The least critical

case is that of equal end moments that bend the beam in double curvature (Figure

2.19 (d)), for which the moment gradient is steepest and the regions of yielding are

most limited.

Put et al. (1999) plotted the member capacity curves for cold-formed channel beams

for different types of moment gradients (Figure 2.20). They also made a comparison

of the design methods of AS 4100 (SA, 1990) for hot-rolled beams and AS/NZS

4600 (SA, 1996) for cold-formed beams and suggested that the single design curve

of AS/NZS 4600 was unduly optimistic for near uniform bending, and unnecessarily

conservative for high moment gradients. The lateral distortional buckling strengths

increase with moment modification factor αm towards the linear elastic buckling

curve according to AS 4100 (SA, 1990) predictions. Further, they stated that AS

4100 (SA, 1990) is more accurate, because it makes allowance for the effects of the

moment distributions on the inelastic buckling resistance.

Figure 2.20: Effects of Moment Gradient

(Put et al., 1999)

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Pi and Trahair (1997) also provided member capacity curves for HFBs with different

types of moment distribution according to AS 4100 (SA, 1990) predictions.

However, the use of AS 4100 (SA, 1990) for cold-formed steel members is

questionable as cold-formed steel members have different material and mechanical

properties compared to hot-rolled steel members. This research is mainly concerned

with AS/NZS 4600 (SA, 2005) to develop design curves for LSBs and other cold-

formed hollow flange steel beams. It is possible to use appropriate moment

modification factors αm to modify the buckling moments. Recently, Kurniawan and

Mahendran (2009b) investigated the moment gradient effects on the lateral buckling

strength of LSBs. They found that the moment modification factor is approximately

equal to 1.0 for quarter point loading, which would be useful in the lateral buckling

tests of LSBs.

2.5.8 Load Height

In some cases a beam may have gravity loads that act at the top flange and move

laterally with the flange during buckling. These loads induce additional torques about

the beam axis, which decrease the resistance to buckling. Both elastic and inelastic

buckling resistances vary with the load height. The resistance to buckling is high

when the load acts below the shear centre axis, and it decreases significantly as the

point of application rises. Pi et al. (1998) investigated the effects of load height on

the lateral buckling strength of cold-formed channel sections and concluded that top

flange loading significantly reduces the strength while Pi et al. (1999) also indicated

the same fact for cold-formed Z-sections.

Trahair (1993) presented an equation to calculate the elastic lateral torsional buckling

moment of a simply supported beam with load height effects.

Pi and Trahair (1997) modified that equation to calculate the elastic lateral

distortional buckling moment of HFBs and is given below.

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛+=

od

yqm

od

yqmm

od

qd

MPy

MPy

MM ααα 4.04.01

2

(2.8)

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Where yq is the load height, Py is the transverse load and Mqd is the maximum elastic

buckling moment including the load height effects. Pi and Trahair (1997) concluded

that this equation provides more accurate predictions than when using the effective

length for top flange loading (i.e. effective length, Le = 1.4 L for top flange loading

based on AS 4100 (SA, 1998)). Recently, Kurniawan and Mahendran (2009a)

investigated the effects of load heights on the lateral buckling strength of LSBs.

2.5.9 Warping

Warping occurs when the twisting of a member results in the cross-section distorting

out-of-plane along the direction of the member’s longitudinal axis. Most of the cold-

formed members except closed tubular sections have cross-sections which tend to

warp when subjected to torsion. When a thin-walled member is restrained at any

particular cross-section, a complex distribution of longitudinal warping stresses is

developed. These stresses act in conjunction with those due to St. Venant torsion to

resist the applied torque.

In practical structures beam ends are connected by web cleats, web stiffeners and

additional web plates. These may induce flange end-restraining moments that oppose

the warping deformations and modify the elastic lateral buckling resistance of the

beam. It is important to determine the end warping restraint stiffness to predict the

elastic lateral distortional buckling resistance accurately. Pi and Trahair (2000)

investigated the effects of distortion and warping at beam support and proposed some

simple formulations to calculate the lateral distortional buckling moments which

allow for both torsional and warping rigidity reductions in uniform bending. They

also provided the warping restraint stiffnesses of some commonly used end-support-

conditions.

Ojalvo and Chambers (1977) investigated the effects of warping restraints on I-

beams theoretically. They considered both fixed and simply supported end conditions

in uniform bending and concluded that the fixed end boundary conditions produced

substantially high amount of buckling resistance, however, the fully fixed condition

is rarely practical. In addition, they theoretically analysed a simply supported beam

associated with warping free and fixed boundary conditions. From the analysis, they

stated that upgrading the simply supported end condition by restraining warping

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produces a significant improvement in the bucking moment. Further, they

recommended two types of warping restraining devices such as stiffener type (Figure

2.21 (a)) and tube type (Figure 2.21 (b)) welded on the flanges near to the support.

Figure 2.21: Warping Restraining Devices

(Ojalvo and Chambers, 1977)

Xiao et al. (2004) investigated the effects of warping stress on the lateral torsional

buckling of cold-formed zed-purlins. They developed an analytical model based on

energy method to analyse the elastic lateral torsional buckling of zed-purlins subject

to partial lateral restraint from the metal sheeting under a uniformly distributed uplift

load. They considered various loading positions and concluded that the influence of

warping stress is less when the load is acting at the web central line, hence increases

the critical load.

Different loading methods also induce warping restraints, specifically overhang

method loading increases the warping restraint. Mahaarachchi and Mahendran

(2005a) conducted lateral buckling tests of LSBs and reported that overhang method

loading over-predicted the failure moment by about 12%.

2.6 Design Guidelines for Cold-Formed Hollow Flange Steel Beams

Mahaarachchi and Mahendran (2005d) identified that the Australia/New Zealand

cold-formed steel design standard AS/NZS 4600 (SA, 1996) was most relevant for

the design of LSBs, as compared to the Australian hot-rolled steel design standard

AS 4100 (SA, 1998). The latter was deemed over-conservative in predicting the

ultimate member moment capacities. A new design rule based upon the work

undertaken by Mahaarachchi and Mahendran (2005d) was included on the latest

version of AS/NZS 4600 (SA, 2005). This section provides the design procedures

suitable for hollow flange steel beams such as HFBs and LSBs.

(a) Stiffeners Type (b) Tube Type

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2.6.1 Moment Capacity Based on AS 4100 (SA, 1998)

AS 4100 (SA, 1998) states the design procedures for local and lateral torsional

buckling failure modes of flexural members. Local buckling is governed by the

section moment capacity formula while lateral torsional buckling is governed by the

member moment capacity formula. There are no provisions made for flexural

members subjected to distortional buckling.

2.6.1.1 Section Moment Capacity

The section moment capacity of a beam is associated with yielding and/or local

buckling of the plate elements in the cross-section and is used where the compression

flange is fully laterally restrained. The nominal section moment capacity (Ms) as in

Clause 5.2.1 is given by Equation 2.9:

Ms = fy Ze (2.9)

Where, fy is the nominal yield stress. The effective section modulus (Ze) used to

calculate the section moment capacity (Ms) is governed by the compactness of the

section’s individual plate equations and is given by Equations 2.10 to 2.13:

For λe ≤ λep : (Compact Sections) Ze = Zc (2.10)

For λep < λe ≤ λey : (Non-Compact Sections) ( )⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛

−−

+= ZZZZ cepey

seye

λλλλ (2.11)

For λe > λey : (Slender Sections, Web elements) 2

⎟⎠⎞

⎜⎝⎛=

e

eyeZ

λλ (2.12)

For λe > λey : (Slender Sections, Flange elements) ⎟⎠⎞

⎜⎝⎛=

e

eyeZ

λλ (2.13)

where, λe = plate element slenderness

λey = plate element yield limit (Table 5.2 of AS 4100)

λep = plate element plasticity limit (Table 5.2 of AS 4100)

Zc = effective section modulus of the compact element


The element with the greatest ratio λe/λey is to be used for calculating the effective

section modulus (Ze). The plate element slenderness (λe) as in Clause 5.5.2 is given

by Equation 2.14:

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250

ye

ftb⎟⎠⎞

⎜⎝⎛=λ (2.14)

Where, b is the clear width of the element outstand from the face or between the

faces of the supporting plate element and t is the element thickness. The effective

section of the compact element (Zc) is given by Equation 2.15:

Zc = min [S, 1.5Z] (2.15)

Where, S is the plastic section modulus.

2.6.1.2 Member Moment Capacity

The member moment capacity of a flexural member is governed by the extent of

lateral restraint provided to the compression flange in order to prevent lateral

buckling. No specific provision are made in AS 4100 (SA, 1998) for lateral

distortional buckling. The nominal member moment capacity (Mb) without full

lateral restraint is specified in AS 4100 (SA, 1998), Clause 5.6.1 (see Equation 2.16)

while the nominal member moment capacity allowing for the lateral distortional

buckling is given by Pi and Trahair (1997) (see Equation 2.17).

Mb = αmαsMs ≤ Ms (2.16)

Mb = αmαsdMs ≤ Ms (2.17)

where, Ms = section moment capacity (Equation 2.9)

αm = moment distribution factor (=1 for constant moment)

αs = slenderness reduction factor

αsd = as above for distortional buckling

The slenderness reduction factors (αs and αsd) based on AS 4100 (SA, 1998) and Pi

and Trahair (1997), respectively, are given by Equations 2.18 and 2.19, respectively.

Bradford (1992) reported that “the relationship between distortional buckling

strength, yielding and elastic distortional buckling is the same as that between the

lateral buckling strength, yielding and elastic lateral buckling”. This implies that if

the elastic buckling moment (Mo) is replaced by the elastic buckling moment in the

distortional mode (Mod) then the method outlined by AS 4100 (SA, 1998) is

adequate for the design of hollow flange steel beams subject to lateral distortional

buckling.

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0.136.02

≤⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−+⎟

⎠⎞

⎜⎝⎛=

od

s

od

ss

MM

MMα (2.18)

0.18.26.02

≤⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−+⎟

⎠⎞

⎜⎝⎛=

od

s

od

ssd

MM

MMα (2.19)

The elastic lateral torsional buckling moment (Mo) of section with equal flanges for

simply supported beam under uniform bending as in AS 4100 (SA, 1998) Clause

5.6.1.1 is:

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2

2

2

2

0 LEIGJ

LEI

M wy ππ (2.20)

Pi and Trahair (1997) modified Equation 2.20 by introducing an effective torsional

rigidity (GJe) in place of the nominal torsional rigidity (GJ) to calculate the lateral

distortional buckling moment (Mod).

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+= 2

2

2

2

LEIGJ

LEIM w

ey

odππ (2.21)

where, EIy, EIw = minor axis flexural rigidity, warping rigidity

GJe = effective torsional rigidity

L = span

The effective torsional rigidity (GJe) is given by Equation 2.22:

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

23

2

23

91.02

91.02

dLEtGJ

dLEtGJ

GJf

f

e

π

π (2.22)

Where, GJf is the flange torsional rigidity and d1 is the clear web depth.

Comparisons between the variations of non-dimensional member capacity (Mu/Ms)

with slenderness (My/Mod)0.5 of experimental results, finite element analyses, AS

4100 (SA, 1998) and Pi and Trahair’s (1997) predictions made by Mahaarachchi and

Mahendran (2005d) are shown in Figure 2.22. With reference to Figure 2.22,

Mahaarachchi and Mahendran (2005d) concluded that both Pi and Trahair (1997)

and AS 4100 (SA, 1998) methods predicted the lower (conservative) bound of the

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experimental data. The design equation was particularly over-conservative in the low

slenderness region. Both methods tended to slightly underestimate the capacities at

the region of high slenderness, i e. conservative.

Figure 2.22: Comparisons of Experimental and AS 4100 (1998) Predictions

(Mahaarachchi and Mahendran, 2005d)

2.6.2 Moment Capacity Based on AS/NZS 4600 (SA, 2005)

The section moment capacity (Ms) is defined in Clause 3.3.2 of AS/NZS 4600 (SA,

2005) while the member moment capacity (Mb) is specified in Clause 3.3.3. AS/NZS

4600 (SA, 2005) covers the member moment capacity of cold-formed flexural

members for a number of buckling modes including lateral torsional and lateral

distortional buckling in Clauses 3.3.3.2 and 3.3.3.3, respectively.

2.6.2.1 Section Moment Capacity

The section moment capacity (Ms) is defined in Clause 3.3.2 of AS/NZS 4600 (SA,

2005) in a similar fashion to AS 4100. However, unlike AS 4100 (SA, 1998), the

effective section modulus (Ze) is based on the initiation of yielding in the extreme

compression fibre and therefore does not allow for the inelastic reserve capacity of

the section. The effects of local buckling are accounted for by using reduced widths

(be) of non-compact elements in compression for the calculation of the effective

section modulus. The plate element slenderness is a function of the applied stress (f*)

as shown in Equation 2.23:

Ms = fy Ze (2.23)

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Ef

tb

k

*052.1⎟⎠⎞

⎜⎝⎛=λ (2.24)

Where, λ, t = plate slenderness and thickness

k = plate buckling coefficient

b = flat width of element excluding radii

f* = design stress in the compression element

The procedure to find out the effective widths of uniformly compressed stiffened

elements for capacity calculations is given in Clause 2.2.1.2 of AS/NZS 4600 (SA,

2005).

For λ ≤ 0.673 : be = b (2.25a)

For λ > 0.673: be = ρb (2.25b)

ρ 0.1

22.01≤

⎟⎠⎞

⎜⎝⎛−

=λλ (2.26)

The procedure to find out the effective widths of stiffened elements with stress

gradient for capacity calculations is given in Clause 2.2.3.2 of AS/NZS 4600 (SA,

2005).

ψ−

=3

1e

ebb (2.27)

For ψ ≤ -0.236 : be2 = 2

eb (2.28a)

For ψ > -0.236 : be2 = be – be1 (2.28b)

Where

be = effective width from Equation 2.25 with f1* substituted for f* and

with k determined as follows;

k = 4 + 2(1- ψ)3 + 2(1- ψ)

ψ = *

*

1

2

ff

*

*

1

2

ff = web stresses calculated on the basis of the effective section

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Figure 2.23: Stiffened Elements and Webs with Stress Gradient

Mahaarachchi and Mahendran (2005d) calculated the section capacities of all

LiteSteel Beam sections using the design method in AS/NZS 4600 (SA, 1996)

method described above with local coefficients (k) equal to 4 and 24 for the

compression flange and web, respectively. The nominal yield stress (450 MPa) with

nominal dimensions was used to calculate the section capacity. They stated that the

AS/NZS 4600 (SA, 1996) section capacity method more accurately estimates the

reduction in capacity due to local buckling in non-compact and slender sections,

compared to the AS 4100 (SA, 1998) method. However, it did not permit the use of

inelastic reserve capacity and hence the section capacities were about 1.3 times the

yield moment capacity for LiteSteel beam sections. They concluded that the

adoption of AS/NZS 4600 (SA, 1996) predictions for section capacity design checks

of LSB members subject to pure bending moment was conservative.

2.6.2.2 Member Moment Capacity

Conventional hot-rolled universal and channel sections primarily exhibit lateral

torsional buckling in laterally unrestrained spans. This mode of buckling is

characterised by simultaneous cross-section twist and lateral displacement of the

compression flange. However, as mentioned previously, cold-formed flexural

members primarily fail by lateral distortional buckling, hence the methods presented

in AS 4100 (SA, 1998) are of little use with respect to the member moment capacity

of the hollow flange steel beams such as LSBs (Mahaarachchi and Mahendran,

2005a). Clause 3.3.3.3(b) of AS/NZS 4600 (SA, 1996) outlines the design rules for

members subject to bending under distortional buckling that involves transverse

b

(a) Actual Element

Stress f1* (Compression)

be

be

Stress f2* (Tension)

be

beStress f2* (Compression)

Stress f1* (Compression)

(b) Effective Element and Stress on Effective Element

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bending of a vertical web with lateral displacement of the compression flange while

the same clause from the latest version of AS/NZS 4600 (SA, 2005) outlines the

procedure developed by Mahaarachchi and Mahendran (2005d). Member moment

capacity Mb is given in Equation 2.29:

⎟⎠⎞

⎜⎝⎛=

ZZMM e

cb (2.29)

Where, Z is the full section modulus, Mc is the critical moment and Ze is the effective

section modulus. The critical moment Mc as in AS/NZS 4600 (SA, 1996) is given by

Equations 2.30a and b:

For λd < 1.414: ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

41

2d

yc MM λ (2.30a)

For λd ≥ 1.414: ⎟⎠⎞

⎜⎝⎛=

dyc MM 2

1λ

(2.30b)

Where, λd = member slenderness (Equation 2.31)


od

yd

MM

=λ (2.31)

The elastic lateral distortional buckling moment Mod can be determined using

Equations 2.21 and 2.22 or a buckling analysis program such as THINWALL. A

comparison of the AS/NZS 4600 (SA, 1996) design member moment capacities and

the experimental and FEA results of LSBs made by Mahaarachchi and Mahendran

(2005d) is shown in Figure 2.24.

Mahaarachchi and Mahendran (2005d) divided the ultimate moments from the

experiments by a moment modification factor of 1.09 to account for the non-uniform

moment distribution caused by the quarter-point loading. However, recent research

(Kurniawan and Mahendran, 2009b) on the effects of moment gradient on the lateral

distortional buckling of LSBs revealed that the moment distribution factor is closer

to 1.0 for quarter point loading. With reference to Figure 2.24, Mahaarachchi and

Mahendran (2005d) stated that the design rule of AS/NZS 4600 (SA, 1996) for

lateral distortional buckling was not suitable as it was quite conservative for beams

with low slenderness while being unconservative for intermediate slenderness

(inelastic buckling region). Further, they stated that the web distortion significantly

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reduced the ultimate moment capacities and are not accounted in AS/NZS 4600 (SA,

1996) predictions.

Figure 2.24: Comparisons of Experiments and AS/NZS 4600 (1996) Predictions


Alternative member capacity equations have been proposed by Trahair (1997). The

accuracy of these equations for the design of HFB flexural members was investigated

and lateral distortional buckling design curves were produced at QUT by Avery et al.

(1999b). Design curves for HFBs were derived using the finite element analysis

results of Avery et al. (1999a), which was verified against the lateral distortional

buckling test results of Mahendran and Doan (1999).

Trahair’s (1997) member capacity equation is given below:

sybobssnd

b MMMMMMc

babM ≥≥≤⎟⎟⎠

⎞⎜⎜⎝

⎛

+−

+= ;;1 2λ

(2.32)

Where, Mb is the member moment capacity, Ms is the section moment capacity and

Mo is the elastic lateral torsional buckling moment. The non-dimensional member

slenderness (λd) is given by:

od

sd M

M=λ (2.33)

The suitable coefficients (a, b, c and n) were established using the least square

method by Avery et al. (1999b). Values of a = 1.0, b = 0.0, c = 0.424, and n = 1.196

were found to minimise the total error for the Trahair’s (1997) design equations.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Member Slenderness

Mu/M

yExperimentalFEAAS 4600 (SA, 1996)

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However, this approach resulted in an unacceptable maximum unconservative error

of more than 10 percent for HFB sections. Therefore Avery et al. (1999b) derived

separate coefficient for each of the different thickness of the HFB sections. The

coefficients a, b, c and n for each thickness of HFBs proposed by Avery et al.

(1999b) are given in Table 2.1.

Table 2.1: Avery et al.’s (1999b) Coefficients for Equation 2.32

t = 3.8 mm t = 3.3 mm t = 2.8 mm t = 2.3 mm a 1.006 0.999 0.997 0.997 b 0.024 0.012 0.000 0.000 c 0.448 0.377 0.321 0.273 n 1.350 1.407 1.429 1.469

Even though this approach is more accurate for the HFB section range it is very

complicated and requires different design curves for each thickness of HFB.

Mahaarachchi and Mahendran (2005d) used the coefficients defined by Avery et al.

(1999b) to predict the moment capacities of LSBs and compared them with their

FEA results. They concluded that the predictions based on Avery et al.’s (1999b)

method were very similar to AS/NZS 4600 (SA, 1996) predictions and were not

suitable for LSB sections. It is quite conservative for beams of low slenderness while

being unconservative for beams of intermediate slenderness (inelastic buckling

region).

Figure 2.25: Comparison of FEA Results with Avery et al.’s (1999b) Predictions


In order to overcome these deficiencies Mahaarachchi and Mahendran (2005d)

proposed new design formulations which account for local, lateral distortional and

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mu/

Ms,

Mb/

Ms

125x45x1.6 LSB125x45x2.0 LSB150x45x1.6 LSB150x45x2.0 LSB200x45x1.6 LSB200x60x2.0 LSB200x60x2.5 LSB250x60x2.0 LSB250x75x2.5 LSB250x75x3.0 LSB300x60x2.0 LSB300x75x2.5 LSB300x75x3.0 LSBAvery et al.'s (1999b)

λd

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lateral torsional buckling modes of LSBs. They conducted statistical tests to

numerically model LSB buckling behaviour according to these three modes solving

for minimum total error for all sections of all spans. The intent was achieved by

minimising the square of the difference between the normalised analytical capacity

(Mu/My) and the normalised design capacity (Mb/My). The developed design rules

are given by Equations 2.34a, b and c. Figure 2.26 compares the predictions from

these design rules to experimental and finite element analysis results, which gave a

mean test to predicted ratio of 1.03. A capacity reduction factor (Φ) of 0.85 was

calculated based on the AISI procedure (AISI, 1996). The equations adequately

modelled the three buckling regions of LSBs given the limits of 0.63 and 1.59 for

member slenderness.

For λd ≤ 0.63: Mc = My (2.34a)

For 0.63 < λd < 1.59: ⎟⎠⎞

⎜⎝⎛=

dyc MM

λ63.0 (2.34b)

For λd ≥ 1.59: ⎟⎠⎞

⎜⎝⎛=

dyc MM 2

1λ

(2.34c)

Where, λd = member slenderness (Equation 2.31)


Figure 2.26: Comparisons of New Design Rules, FEA and Experiments (Φ=0.85)


0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.50 1.00 1.50 2.00 2.50

Mu/M

y, M

b/My

125x45x1.6 LSB

125x45x2.0 LSB150x45x1.6 LSB

150x45x2.0 LSB

200x45x1.6 LSB200x60x2.0 LSB

200x60x2.5 LSB250x60x2.0 LSB

250x75x2.5 LSB

250x75x3.0 LSB300x60x2.0 LSB

300x75x2.5 LSB

300x75x3.0 LSBAS4600

Equation 15Exp 125x45x1.6 LSB

Exp 125x45x2.0 LSB

Exp 150X45X16 LSBExp 150x45x2.0 LSB

Exp 200x45x1.6 LSB

Exp 200x60x2.5 LSBExp 250x75x2.5 LSB


Exp 300X60X20 LSB


λd

(1996)2.32

0.63 1.59

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A capacity reduction factor (Φ) of 0.90 is commonly associated with beams under

bending in accordance with AS/NZS 4600 (SA, 1996). Thus Mahaarachchi and

Mahendran (2005d) modified the above equations to account for the new capacity

reduction factor. The modified set of equations (2.35a, b and c) was reported to have

a mean test to predicted ratio of 1.10, and was recommended for inclusion into the

latest version of AS/NZS 4600 (SA, 2005). A graphical comparison between each of

the design methods is included in Figure 2.27.

For λd ≤ 0.59: Mc = My (2.35a)

For 0.59 < λd < 1.70: ⎟⎠⎞

⎜⎝⎛=

dyc MM

λ59.0 (2.35b)

For λd ≥ 1.70: ⎟⎠⎞

⎜⎝⎛=

dyc MM 2

1λ

(2.35c)

However, a thorough study of Mahaarachchi and Mahendran’s (2005c) finite element

analysis of LSBs revealed that the ideal model used to simulate LSBs under uniform

moment did not provide adequate twist restraint for the whole section including the

two rectangular flanges. Also, the application of boundary conditions and the type of

loading in this study was found to have some unacceptable stress distribution on the

LSB cross section at failure. Further, it was found that the member moment capacity

does not reach the section capacity as the length of the member decreases.

Figure 2.27: Comparisons of New Design Rules, FEA and Experiments (Φ=0.90)

(Mahaarachchi and Mahendran (2005d)

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

0.000 0.500 1.000 1.500 2.000 2.500

Mu/M

y, M

b/My

125x45x1.6 LSB125x45x2.0 LSB150x45x1.6 LSB150x45x2.0 LSB200x45x1.6 LSB200x60x2.0 LSB200x60x2.5 LSB250x60x2.0 LSB250x75x2.5 LSB250x75x3.0 LSB300x60x2.0 LSB300x75x2.5 LSB300x75x3.0 LSBEquation 15Equation 16

λd

2.32 2.33

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This research is mainly aimed at developing an accurate design curve using an

appropriate ideal finite element model including correct boundary conditions and

loading type. Also it is necessary to find out the appropriate geometrical parameters

which influence the lateral distortional buckling of LSBs and to include them in the

design procedures.

2.6.3 The Direct Strength Method

The Direct Strength Method (DSM) is a recently established method and was

formally adopted in the North American design specifications in 2004 as an

alterative to the traditional Effective Width Method. Accurate member elastic

stability is the fundamental idea behind the DSM. The Direct strength method is

predicted upon the idea that if an engineer determines all of the elastic instabilities

for the gross section (i.e., local, distortional and global buckling) and the load (or

moment) that causes the section to yield, then the strength can be directly determined

(Schafer, 2006). It is essentially an extension of the use of column curves for global

buckling, but with application to local and distortional buckling instabilities. Direct

strength method can be used with many advantages such as no effective width

calculation or iterations are required, gross sectional properties can be directly used,

explicit design method for distortional buckling, includes interaction of elements,

explores and includes all stability limit states and focus on the correct determination

of elastic buckling behaviour, instead of the empirical effective widths.

Elastic buckling analysis results are directly integrated into DSM. This provides a

general method of designing cold-formed steel members and creates the potential for

much broader extensions than the traditional specifications with limited applicability.

However, limitations of the direct strength method (as implemented in AISI 2004)

are given in the direct strength method design guide (CF06-1) and some of them are

presented here;

• overly conservative if very slender elements are used.

• shift in the neutral axis is ignored.

• empirical method calibrated only to work for cross-sections previously

investigated.

Section 7 of AS/NZS 4600 (SA, 2005) gives the direct strength method provisions

for cold-formed steel beams subject to bending with separate provisions for local

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buckling, lateral torsional buckling and distortional buckling. As stated in Clause

7.2.2.1 of AS/NZS 4600 (SA, 2005), the nominal member moment capacity (Mb)

shall be the minimum of the nominal member moment capacity (Mbe) for lateral

torsional buckling, the nominal member moment capacity (Mbl) for local buckling

and the nominal member moment capacity (Mbd) for distortional buckling.

Clause 7.2.2.2 of AS/NZS 4600 (SA, 2005) gives design procedure to determine the

nominal member moment capacity (Mbe) for lateral torsional buckling and is given in

Equations 2.36 (a), (b) and (c).

For Mo < 0.56 My: Mbe = Mo (2.36a)

For 2.78 My ≥ Mo ≥ 0.56 My: ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−=o

yybe

MMMM

36101

910 (2.36b)

For Mo > 2.78 My: Mbe = My (2.36c)

My= Zf fy (2.37)

Where, Mo is the elastic lateral torsional buckling moment, Zf is the full section

modulus and fy is the yield stress.

As in Clause 7.2.2.3 of AS/NZS 4600 (SA, 2005) the nominal member moment

capacity (Mbl) for local buckling is given in Equations 2.38 (a) and (b).

For λl ≤ 0.776: Mbl = Mbe (2.38a)

For λl > 0.776: bebe

ol

be

olbl M

MM

MMM

4.04.0

15.01 ⎟⎠⎞

⎜⎝⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−= (2.38b)

The non-dimensional member slenderness (λl) is given by:

ol

bel M

M=λ (2.39)

The elastic local buckling moment (Mol) is given by:

Mol = Zf fol (2.40)

Where fol is the elastic local buckling stress.

Clause 7.2.2.4 of AS/NZS 4600 (SA, 2005) provides the nominal member moment

capacity (Mbd) for distortional buckling and is given in Equations 2.41 (a) and (b).

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For λd ≤ 0.673: Mbd = My (2.41a)

For λd > 0.673: yy

od

y

odbd M

MM

MMM

5.05.0

22.01 ⎟⎠⎞

⎜⎝⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−= (2.41b)

The non-dimensional member slenderness (λd) is given by:

od

yd M

M=λ (2.42)

The elastic distortional buckling moment (Mod) is given by:

Mod = Zf fod (2.43)

where, fod is the elastic distortional buckling stress

It should be noted that the lateral distortional buckling is not included in these

provisions. Also, these design procedures are based on advanced buckling analyses

of cold-formed C- and Z- sections and the applicability of these predictions to hollow

flange steel beams such as HFBs and LSBs are questionable as their sectional

properties are different. However, their applicability should be investigated to

develop such type of design procedures for hollow flange steel beams.

2.7 Hollow Flange Steel Beams with Web Stiffeners

It is generally known that web stiffeners and batten plates increase the lateral

buckling strength of beams. This increase is considered to be due to the local

increment of both the torsional stiffness and the bending stiffness resulting from the

use of web stiffeners (Takabatake, 1988). Stiffeners have also been found to improve

the buckling capacity of members subject to distortional buckling as they act to

prevent distortion by coupling the rotational degrees of freedom of the top and

bottom flanges (Akay et al. 1977).

Avery and Mahendran (1997) investigated the lateral buckling capacity of Hollow

Flange Beams (HFBs) with web stiffeners using finite element analysis while

Mahendran and Avery (1997) carried out large scale experiments to validate the

results from finite element analysis and to improve the stiffener configuration. They

found that 5 mm web plate stiffeners welded to the flanges on both sides of the web

at third points of the span were adequate to eliminate the lateral distortional buckling

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of HFBs. However, research on LSBs with web stiffeners by Kurniawan (2005)

using finite element and experimental analyses stated that the use of two web

stiffeners were inadequate for LSBs. A review of these researches is presented in this

section.

2.7.1 HFBs with Web Stiffeners

Avery and Mahendran (1997) and Mahendran and Avery (1997) considered four

types of web stiffener configurations (Figures 2.28 and 2.29). They found that

longitudinal batten plate stiffeners located parallel to the web and attached to the

flanges were not effective enough in providing out of plane stiffness to prevent

relative rotation of the HFB flanges; thus web distortion was not prevented.

Figure 2.28: Stiffener Types


The cross stiffeners, box stiffeners and rectangular hollow section stiffeners only

provided a slightly higher strength increment than the transverse web stiffeners. This

was because the effect of a stiffener was mostly due to the constraints provided,

which were independent of the stiffener size. Their FEA revealed that a web stiffener

welded to flange only (Type D) was just as effective as a fully welded web stiffener

(Type A), while the web stiffener welded to the web only (Type E) had a slight

improvement (Figure 2.29). They stated that this was because the majority of the

strength increment was provided by tying together the rotational degrees of freedom

Plate, both sides. Welded to web and flanges

5 mm Plate, one side. Welded to web and flanges

Plate, both sides.

Plate, both sides.

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of the flanges, forcing the section to remain undistorted. This constraint is more

effective than a prevention of web distortion by welding the stiffener to the web only.

The effect of web stiffener thickness (5, 10, 15 and 20 mm) on the strength increment

was small. This is because the section properties of the stiffeners are less significant

than the nature of constraint they provided. Therefore they recommended that a plate

with a 5 mm (or larger) thickness can be used as a web stiffener. They also studied

the location of stiffeners at mid-span (1 stiffener), third points (2 stiffeners) and

quarter points (3 stiffeners) by using FEA. The use of three stiffeners or more did not

provide significantly greater strength than the strength obtained from two stiffeners.

They recommended using stiffeners at third points of the span which usually provide

an optimum compromise between the cost of fabrication and strength obtained.

Figure 2.29: Stiffener Configuration


In the experimental investigation, a screw fastened connection for the web stiffener

to the HFB flanges was developed as shown in Figure 2.30. The web stiffeners were

bent to fit the inclined flanges. These special stiffeners were compared with the

stiffener welded to the flanges and it was found that those stiffeners improve the

buckling capacity in a similar manner to those welded to flanges. This means

advantages in ease of installation and reduced cost. Furthermore, the screw fastening

did not introduce any geometrical imperfections and residual stresses. They

5 mm Plate, both sides, welded to web and flanges

RHS, both sides, welded to web and flanges

5 mm Plate, one side, welded to web and flanges

5 mm Plate, both sides, welded to web only

5 mm Plate, both sides, welded to flanges only

5 mm Plate, one side, welded to flanges only

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considered overhang loading method in the experimental investigations. Further

details of their experiments can be found in Section 2.9.6.

Figure 2.30: Special Stiffener Screw Fastened to HFB Flanges

(Mahendran and Avery, 1997)

2.7.2 LSBs with Web Stiffeners

Even though web stiffeners effectively eliminated the detrimental lateral distortional

buckling of HFBs the applicability of these results to LSB is questionable as the HFB

is a doubly symmetric section with two triangular hollow flanges whereas the LSB is

a monosymmetric channel section with two rectangular hollow flanges. Kurniawan

(2005) investigated the lateral distortional buckling behaviour of LSB sections with

web stiffeners. This section provides a summary of his study.

2.7.2.1 Stiffener Type and Configurations

Six types of stiffeners were considered with the recommendations of Avery and

Mahendran (1997) and Mahendran and Avery (1997) in order meet economical and

strength requirements. Figure 2.31 shows different types of stiffeners studied by

Kurniawan (2005). He used them at third points of the span as this provided an

optimum compromise between the cost and strength. He selected 250 x 60 x 2.0 LSB

with quarter point loading condition and a span of 3.5 m. Details of the experiments

can be found in Section 2.9.5.

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(Kurniawan, 2005)

2.7.2.2 Design Methods

Elastic buckling moments and member capacities of LSBs can be calculated using

appropriate design procedures.

Elastic Buckling Moment

Elastic lateral distortional buckling moment (Mod) of LSBs can be calculated using Pi

and Trahair’s (1997) formula (Equation 2.21) while the elastic lateral torsional

buckling moment (Mo) can be calculated using AS 4100 (SA, 1998) provision

(Equation 2.20). Both equations are derived for the beams without web stiffeners and

the applicability of these to the beams with web stiffeners is questionable as the

section properties of beams with web stiffeners are different to that of beams without

web stiffeners. Takabatake (1988) stated that the effects due to web stiffeners on the

second moment of area about the minor axis (Iy) can be neglected, but their effects

on the torsion constant (J) cannot be neglected. Further, it depends on the number of

stiffeners, size and the type of stiffeners. Kurniawan (2005) did not consider the

effects due to web stiffeners on the torsion constant.

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Member Moment Capacity

Member capacity (Mb) of LSBs undergoing lateral distortional buckling and lateral

torsional buckling can be calculated using appropriate clauses of AS/NZS 4600 (SA,

2005). Clause 3.3.3.3(b) of AS/NZS 4600 (SA, 2005) provides formulations to

determine Mb of beams undergoing lateral distortional buckling, which are presented

in the earlier section of this chapter (Equations 2.29 and 2.35a, b and c). Clause

3.3.3.2.1 of AS/NZS 4600 (SA, 2005) provides formulations to determine Mb of

beams undergoing lateral torsional buckling and is given by Equation 2.44 (This is

also given in Direct Strength Method as Equation 2.36).

For λb ≤ 0.60: Mc = My (2.44a)

For 0.60 < λb < 1.336: ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

3610111.1

2b

yc MM λ (2.44b)

For λb ≥ 1.336: ⎟⎠⎞

⎜⎝⎛=

byc MM 2

1λ

(2.44c)

where, λb = member slenderness (Equation 2.45)


o

yb M

M=λ (2.45)

Figure 2.32 shows the member capacities of 250x60x2.0LSB by Kurniawan (2005).

Figure 2.32: Predicted Member Capacities of 250x60x2.0 LSB

(Kurniawan, 2005)

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2.7.2.3 Experimental Results of Kurniawan (2005)

All the stiffener types were not able to eliminate the web distortion of the LSB

section during its lateral buckling failure. Even though expected capacity

improvement was approximately 58% the results from experiments showed a

maximum increase of only 11% with web stiffener and threaded rod stiffener.

Although all the stiffened LSBs failed in lateral distortional buckling mode, the

difference of top and bottom flange deflection was smaller than that for the

unstiffened LSBs. In addition, the use of stiffeners also improved the vertical

deflection. Stiffener Type D was found to have the greatest reduction of vertical

deflection by approximately 13%.

The connection type of the stiffener to the beams also was found to have some

effects on the capacity increase. Kurniawan (2005) stated that the TEK screw

fasteners were somewhat little loose due to the action of tying together the rotational

degrees of flange freedom. In particular, this occurred in the experiments using

stiffener types B, C, D and E because the screws were fastened at the flange corners

thus subjected to pull out action (tension). A screw has a lower resistance to tension

than to shear action. The web stiffener and the threaded rods were fastened and

bolted to the inner flanges, respectively hence the screws and bolts were subjected to

a shear action. He suspected this might be one of the reasons why other stiffener

types gave lower strength improvement than the web stiffener and threaded rods.

Ultimately, Kurniawan (2005) concluded that none of the proposed stiffener types

could effectively improve the LiteSteel beam bending strength while the lateral

buckling strengths of Hollow Flange beams, with similar section characteristic to

LSBs, were improved significantly by simply introducing two web stiffeners (Avery

and Mahendran, 1997). This led Kurniawan (2005) to undertake an advanced

computer analysis of finite element modelling in order to verify the experimental

investigations and to further observe the behaviour of stiffened LSBs.

2.7.2.4 Finite Element Analysis Results of Kurniawan (2005)

This investigation used both the experimental and ideal finite element models,

developed by Mahaarachchi and Mahendran (2005c). Details of this model can be

found in Section 2.8.1. The experimental model is a simulation of the actual test

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while the ideal model incorporated ideal constraints, nominal imperfections and a

uniform bending moment within the span. Residual stresses were not considered. The

analysis only considered stiffener Type A (web stiffener) as it was (except threaded

rods) found to be the best option. Threaded rods were not included due to the

complexity in modelling.

Elastic buckling analysis results showed that the elastic lateral distortional buckling

was almost eliminated when five or more web stiffeners were used. However, this is

not an efficient and practical solution.

The effects of section properties of web stiffeners are less significant than the nature

of the constraint they provided as suggested by Avery and Mahendran (1997).

Kurniawan (2005) also stated that there is a small strength increment (about 1%) for

the beam stiffened with 10 mm thickness of web stiffeners. A fully connected web

stiffener to the beam (ie. both flanges and web) was also found to have only a

marginal improvement compared to that connected to the flanges only, because the

majority improvement was provided by tying together the flanges.

Figure 2.33: FEA Models used by Mahaarachchi and Mahendran (2005c)

Kurniawan’s (2005) ideal finite element model gave a contradictory result in which

lateral distortional buckling was effectively reduced by just using two web stiffeners.

For 3 m span LSBs, the use of two web stiffeners enhanced the elastic buckling

capacity by 24%, three times higher than that predicted by experimental finite

element model.

Kurniawan (2005) undertook finite element analysis of HFBs with quarter point

loading to check whether the loading conditions might affect the number of web

stiffeners as Avery and Mahendran (1997) incorporated overhang method loading

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and concluded that two web stiffeners would adequately eliminate the lateral

distortional buckling. However, Kurniawan (2005) did not obtain strength increase

for HFB as much as Avery and Mahendran (1997) achieved. It should be noted that

Kurniawan (2005) considered only one HFB (30090HFB2.8) with 3 m span and

further investigation is required to conclude this fact.

Finally, Kurniawan (2005) used the same finite element model which was used by

Maharachchi and Mahendran (2005c). However, it was found that this finite element

model had some deficiencies such as inadequate lateral restraint, improper

application of end moments and loads and the direction of initial imperfections.

Experiments also found to be limited to only one LSB section. Therefore detailed

finite element analyses using both experimental model with quarter point loading and

ideal model of LSBs should be undertaken. Web plate stiffeners with 5 mm thickness

would be adequate. Further, large scale experimental analyses are also required to

validate the FEA results.

2.8 Finite Element Analysis

Finite element analysis (FEA) of cold-formed steel structures plays an increasingly

important role in engineering practice, as it is relatively inexpensive and time

efficient compared with physical experiments, especially when a parametric study of

cross-section geometries is involved. Furthermore, it is difficult to investigate the

effects of geometric imperfections and residual stresses of structural members

experimentally. Therefore, FEA is more economical than physical experiments,

provided the finite element model is accurate and the results could be validated with

sufficient experimental results.

The finite element analysis process involves three major phases such as pre-

processing, solution and post processing. The purpose of pre-processing is to develop

an appropriate finite element mesh, assign suitable material properties, and apply

boundary conditions in the form of restraints and loads. Governing equations are

assembled into a matrix form and are solved numerically in the “solution” phase.

The assembly process depends not only on the type of analysis (e.g. static or

dynamic), but also on the model’s element types and properties, material properties

and boundary conditions. Post-processing begins with a thorough check for problems

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that may have occurred during the solution stage. Most solvers provide a log file,

which should be searched for warning or error messages, and which will also provide

a quantitative measure of how well behaved the numerical procedures were during

solution.

2.8.1 Finite Element Analyses of LSBs

Mahaarachchi and Mahendran (2005c) conducted a numerical study of LSB sections

including elastic buckling and non-linear static analyses using a finite element

program called ABAQUS. They used MSC/PATRAN pre-processing facilities to

model the LSBs and ABAQUS (HKS, 2003) to analyse the model. They considered

both experimental and ideal finite element models. Experimental model was used to

validate the finite element models using experimental test results whereas the ideal

model was used to develop design curves (Figures 2.35 and 2.36). The cross-section

geometry of experimental models was based on measured dimensions. However,

they used nominal outside dimensions (OATM, 2008) to develop the ideal finite

element model instead of centreline dimensions. The geometric modelling process of

MSC/PATRAN is such that the specified section thickness is offset either side of

what is assumed to be a centreline dimension. This led to have somewhat higher

geometric area than actual. Also they ignored the corners of LSB sections in the

modelling for convenience as they verified that it did not affect the results

significantly.

Mahaarachchi and Mahendran (2005c) used shell elements (S4R5) to model the

LiteSteel beams. This element is a thin, shear flexible, isoparametric quadrilateral

shell with four nodes and five degrees of freedom per node, utilising reduced

integration and bilinear interpolation schemes. R3D4 rigid body elements and stiff

beam elements were used to create the ideal pinned member end restraints and

loading for both models. The R3D4 element is a rigid quadrilateral element with four

nodes and three translational degrees.

Using some convergence studies, Mahaarachchi and Mahendran (2005c) selected the

element sizes to be 5 mm in width and 10 mm in length in the longitudinal direction.

The support conditions were simulated to provide the required idealised simply

supported conditions where the following requirements were satisfied.

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• Simply supported in-plane – Both ends fixed against in-plane vertical

deflection, but unrestrained against in-plane rotation and one end fixed

against longitudinal horizontal displacement.

• Simply supported out-of-plane – Both ends fixed against out-of-plane

horizontal deflection and twist rotation, but unrestrained against minor axis

rotation and warping displacement.

Table 2.2: Idealised Simply Supported Boundary Conditions

T1 T2 T3 R1 R2 R3 One end Yes No No No Yes Yes Other end No No No No Yes Yes Mid span No Yes Yes Yes No No

Table 2.2 shows the boundary conditions of the simply supported beam. The presence

of symmetry allowed them to model only half the span which would reduce the

analysing time. T and R represent the translation and rotation, respectively and the

subscripts (1, 2, and 3) represent the direction. Field “Yes” means that it is free to

move in that direction. Figure 2.37 illustrates the global axes selected to input the

boundary conditions for the analysis.

Figure 2.34: Idealised Simply Supported Boundary Conditions

Figure 2.35 shows the load and boundary conditions of the ideal finite element model

developed by Mahaarachchi and Mahendran (2005c). An elastic strip width of 20

mm was included adjacent to the pinned end of all the sections in order to eliminate

the undesirable stress concentrations by concentrated loading. They used “rigid

beam” type Multiple Point Constraint (MPC) elements to spread the concentrated

moment evenly to the web and flanges at the shear centre of the cross section. Single

tie MPC and explicit MPCs with different degrees of freedom were applied to link

X, 1

Z, 3

Y, 2 Z, 3

X, 1

Y, 2

M

L/2

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the rigid beam elements and the elastic strip. Explicit MPC (UX, RY and RZ)

denoted that the degrees of freedom of X-translation (Code UX), Y-rotation (Code

RY) and Z-rotation (Code RZ) are linked.

Figure 2.35: Ideal Finite Element Model

(Mahaarachchi and Mahendran, 2005c)

Figures 2.36 (a) and (b) show the details of the experimental finite element model

developed by Mahaarachchi and Mahendran (2005c). Even though both overhang

method and quarter point loading were adopted in the experiments, only the quarter

point loading method was simulated. They used single point constraints and

concentrated nodal forces and applied loads as closely as possible. The experimental

specimens included a 70 mm width rigid plate at each support to prevent distortion

and twisting of the cross section. They modelled these stiffened plates as rigid body

using R3D4 elements. In ABAQUS (HKS, 2003) a rigid body is a collection of

nodes and elements whose motion is governed by the motion of a single node, known

as the rigid body reference node. Therefore they applied simply supported boundary

conditions to the node at the shear centre in order to provide an ideal pinned support.

They modelled the steel plate and bolts using rigid beam elements to simulate the

experimental set up.

Elastic modulus E of 200,000 MPa and Poisson’s ratio υ of 0.3 were adopted. They

used a nominal yield stress fy of 380 MPa for web and 450 MPa for flanges in the

ideal model while average measured yield stresses were used in the experimental

model.

X, 1

Z, 3

Y

20 mm Elastic

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Figure 2.36: Experimental Finite Element Model


Mahaarachchi and Mahendran (2005c) carried out both elastic buckling and

nonlinear static analyses. Elastic buckling analyses were used to obtain the

eigenvectors for the inclusion of geometric imperfections. Nonlinear static analysis

included the effects of large deformation, material yielding, imperfection and

residual stresses. Table 2.3 shows the parameters considered by Mahaarachchi and

Mahendran (2005c) in their nonlinear static finite element analysis. They explicitly

modelled the measured out-of-straightness and twist imperfections in the

experimental model while an assumed value of L/1000 was used in the ideal model.

They applied positive imperfections to the beams which would have given higher

lateral buckling strengths compared with negative imperfections (Pi et al., 1998).

Hence the negative imperfections should be considered in the future research of

LSBs.

Table 2.3 Nonlinear Analysis Parameters


Parameter Assigned Value Max. no. of load increments Between 50 and 250 Initial increment size 0.001 Min. increment size 1x10-7

No. of integration points per shell 9Automatic increment reduction Enabled Large displacements Enabled

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Figure 2.37 shows the measured residual stress distributions of LSBs by

Maaharachchi and Mahendran (2005e). The values shown in each figure are

expressed as a percentage of virgin plate’s nominal yield stress (fy) of 380 MPa. They

included the idealised residual stresses in both the ideal and the experimental models.

Figure 2.37: Idealised Models of Residual Stresses for LSBs

(Mahaarachchi and Mahendran, 2005e)

Kurniawan (2007) conducted a review of the appropriateness of the assumed

boundary conditions in the ideal finite element model developed by Maharaachchi

and Mahendran (2005c) and concluded that the existing model provided over-

conservative boundary conditions by releasing degrees of freedom beyond those

required. Kurniawan (2007) stated that Mahaarachchi and Mahendran (2005c)

considered a 20 mm elastic strip which was in addition to the considered span of the

member. Further, he found that they used external dimensions in their models. Shell

elements that were used in the LSB model, discretize a body by defining the

geometry at the reference surface which is the centreline of the body. Hence, their

finite element model using the nominal external dimensions as the centreline

dimensions simulated a larger LSB section. Kurniawan (2007) then modified this

ideal finite element model by including the required flange local twisting restraint

and eliminating the inaccuracies in Mahaarachchi and Mahendran’s (2005c) ideal

finite element model. Figure 2.38 illustrates the first modified version of the ideal

model developed by Kurniawan (2007).

(a) Flexural Residual Stresses (b) Membrane Residual Stresses

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Figure 2.38: Modified Ideal Finite Element Model (First Version)

(Kurniawan, 2007)

The accuracy of this model was reviewed by Parsons (2007a) and Kurniawan (2007)

and it was found that the elastic buckling moments from this model provide good

agreement with the results from Thin-Wall, but only for intermediate to long span

LSBs. However, this model underestimated the buckling strength, and its non-linear

ultimate strength was also found to be inaccurate for short to intermediate span

LSBs. The ultimate moment capacity did not reach the yield moment when the

member slenderness was reduced. The comparisons of the finite element analyses

results of Mahaarachchi and Mahendran (2005d) with that of Parsons (2007b) by

using Kurniawan’s (2007) first modified version confirmed this fact (Figure 2.39).

Figure 2.39: Member Capacity Curves

(Parsons, 2007b)

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In Figure 2.39, “JP” indicates Parsons’ (2007b) results and “DM” indicates

Mahaarachchi and Mahendran’s (2005d) results. It is clearly seen that the member

capacity does not reach the section capacity for low member slenderness values for

the ideal model of Kurniawan’s (2007) first modified version while that of

Mahaarachchi and Mahendran’s (2005d) model gives reasonable results because they

used equal and opposite end moments with warping fixed support conditions for

short span members and warping free conditions for long span members. However,

warping free support condition is to be used to develop design curves. Further, it

was found that the use of Multiple Point Restraint (MPC) elements for warping free

simulation was always associated with undesirable stress concentrations and thus

reduced the elastic buckling and the non-linear ultimate strengths of short to

intermediate span LSBs. Kurniawan (2007) therefore, further modified his first

version. The MPCs system was replaced with a simpler method of directly

restraining the degrees of freedom of the nodes at beam ends. Figure 2.40 shows the

loading and boundary conditions of the final version of the modified ideal model of

Kurniawan (2007).

Figure 2.40: Modified Ideal FE Model (Final Version)

(Kurniawan, 2007)

2.8.2 Finite Element Analyses of HFBs

The structural behaviour of cold-formed Hollow Flange Beams (HFBs) was studied

by Avery and Mahendran (1997), Avery et al. (1999a, 1999b and 2000) and Pi and

Trahair (1997) using finite element analyses.

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Flexural capacity of HFBs was investigated by Avery et al. (2000) using finite

element analyses. From their investigations, they discovered that the elastic lateral

distortional buckling moment and the ultimate capacities of HFBs can be accurately

predicted from their finite element analyses and therefore used them in the

development of design curves and suitable design procedures. Their study involved

two models, namely experimental and ideal models. The ABAQUS S4R5 shell

elements were employed to create the mesh and the R3D4 rigid body elements were

used to model the pinned end conditions. The loads and boundary conditions, as used

by Zhao et al. (1995) in the study of lateral buckling of cold-formed RHS beams,

were used in these models to provide ‘idealized’ simply supported boundary

conditions and a uniform bending moment was applied. However, they have not been

able to eliminate the warping restraints due to the overhang in the experimental

models. The models incorporated all the significant effects that might influence the

ultimate capacity of HFBs, including material inelasticity, local buckling, member

instability, web distortion, residual stresses and initial geometric imperfections. They

explicitly modelled the measured imperfections in the experimental models, with

magnitudes as measured by Mahendran and Doan (1999). A nominal imperfection

magnitude of L/1000 was used in the ideal model. Residual stresses were modelled

using Doan and Mahendran’s (1996) residual stress model which was based on the

measured residual stresses.

Figure 2.41: Finite Element Models of HFBs

(Avery et al., 2000)

Avery and Mahendran (1997) studied the lateral distortional buckling behaviour of

hollow flange beams with web stiffeners using finite element analysis. The finite

(b) Ideal HFB Model(a) Experimental HFB Model

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element analysis program, MSC/NASTRAN was used for this study and the

quadrilateral shell elements (QUAD4) were used in this finite element modelling.

Triangular shell elements (TRIA3) were used to model the stiffeners. Only half of

the beam was modelled by making use of the symmetry of geometry and loading

conditions about the centre plane of the span. They developed both ideal and

experimental models. The ideal model consisted of typical idealised simply

supported boundary conditions and overhang method loading provided that the

cantilever span was fully restrained against lateral buckling. Also, no warping

restraints were allowed at the support. The experimental model was similar to the

ideal model which simulated the actual experimental set up. However, warping

restraints induced by overhang length were not eliminated.

They compared both the elastic buckling and non-linear analysis results from

experimental and ideal finite element model and concluded that the difference is

insignificant except short spans. For short span members, an interactive buckling

mode between the main span and the cantilever was observed in the experimental

model. Further, they used their ideal finite element model for parametric studies

using elastic buckling analysis.

2.9 Experimental Investigation

Experimental methods are the base and a necessity for scientific research even

though they are very time-consuming and expensive. Experimental results can be

used to verify the numerical models that can then be used to expand the results to

enable a full understanding of the structural behaviour and the development of design

rules. Past experiments on LSBs and HFBs are summarised in this section.

Mahaarachchi and Mahendran (2005a, b, and e) carried out lateral buckling tests,

section capacity tests and tensile coupon tests on LSBs. They also measured the

initial imperfections and residual stresses for LSBs.

2.9.1 Tensile Coupon Tests

Mahaarachchi and Mahendran (2005e) tested 42 tensile coupons, taken form

different locations such as outside flange, inside flange and web of different LSBs.

Details of the test specimens were chosen in accordance with the recommendations

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of AS 1391 (SA, 1991). Strain gauges were attached to the tensile test coupons in

order to measure the strain at fracture including uniform elongation. Some coupons

had strain gauges on both sides and those measurements were used to obtain the

Young’s modulus of elasticity, and the stress-strain curves. Percentage elongations

and strains were also determined by measuring the longitudinal displacement of

sixteen 5 mm transverse gauge lines over the constant width section of the tensile

coupon.

The test results showed that the yield stresses exceeded the nominal flange yield

stress of 450 MPa and the web nominal yield stress of 380 MPa due to the heavy

cold-working involved in the making of LSB sections. Table 2.4 shows the average

measured yield stresses and ultimate stresses of the flange and the web of several

LSBs.

Table 2.4: Tensile Coupon Test Results


Location Yield Stress, fy (MPa)

Ultimate Stress, fu(MPa)

Flange

Outside Measured 516 568 Inside Measured 464 523

Average 490 546 Minimum (Dempsey 2001) 450 490

Web Measured 408 510 Minimum (Dempsey 2001) 380 500

Figure 2.42: Typical Stress-Strain Curves of the Base Steel used in LSB Sections


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Figure 2.42 shows the typical stress-strain curves of steel used in LSB sections. The

lack of yield plateau in the stress-strain curves of flange specimens indicating the

higher level of cold-working in the flanges. The web and flange yield stresses varied

depending on the thickness and LSB section (Mahaarachchi and Mahendran, 2005e).

2.9.2 Residual Stress Measurement

Generally, there are three groups of experimental methods used in residual stress

measurements such as destructive, semi-destructive and non-destructive methods.

Semi-destructive methods includes hole-drilling which is based on the fact that

drilling a hole releases the residual stresses, thus resulting in measurable

deformations on the surface adjacent to the hole. Non-destructive methods use

advanced techniques with high accuracy. One of the most frequently used non-

destructive methods is known as the X-ray diffraction technique. But it may not be

suitable for cold-formed steel. Other non-destructive techniques are the

electromagnetic method, neutron diffraction method, magnetic Barkhausen noise

method and the ultrasonic technique. The use of non-destructive methods is limited

because of the unavailability of highly specialised equipment.

The most commonly used destructive method is the sectioning method which is

considered to be the most reliable method of measuring residual stresses.

Mahaarachchi and Mahendran (2005e) used the sectioning method using electrical

strain gauges to measure the longitudinal and transverse residual stresses of LSBs.

However, since early tests showed the absence of transverse residual stresses, most

of the tests included only the longitudinal residual stress measurements. Electrical

strain gauges were used on both the inside and outside surfaces of the flanges. Due to

the difficulties of attaching the strain gauges on the inside corner, the measured

residual strains were extrapolated to the corners.

Figure 2.43 show some pictures of sectioning carried out by Mahaarachchi and

Mahendran (2005e) while the idealised residual stress distribution models for LSBs

are given in earlier section as Figure 2.37.

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Figure 2.43: Sectioning of LSBs


2.9.3 Initial Geometric Imperfection Measurement

Mahaarachchi and Mahendran (2005e) measured the initial imperfections of LSBs

using a Wild T05 theodolite and the imperfection measuring equipment specially

designed at QUT. The equipment consisted of levelled table with guided rails, a laser

sensor, travelator and a data log. They indicated that the measured thickness of

flanges was greater than the nominal value whereas the measured thickness of webs

was smaller than the nominal value. Figure 2.44 shows the test set-up for

imperfection measurement while Figure 2.50 shows the variation of measured

imperfections along the length of a 4 m span 200 x 60 x 2.0 LSB.

Figure 2.44: Geometric Imperfection Test Set-up


(a) Specimens with Strain Gauges

(b) Specimens Ready for Sectioning

(c) Sectioning of LSB using Band Saw

Travelator Table

Adjustable Legs (a) Measuring Table (b) Laser Sensor

Laser Sensor

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2.9.4 Section Capacity Tests

Mahaarachchi and Mahendran (2005b) carried out a total of 16 section capacity tests

for LSBs. Pairs of laterally restrained LSB sections with short spans were tested

using a four point bending test set-up as shown in Figure 2.45.

Tests results showed that the first yield of LSB sections occurred at about 0.75-0.82

of the theoretical first yield moment. Most of the specimens exhibited a flange local

buckling failure. Mahaarachchi and Mahendran (2005b) compared the test results

with predicted section capacities using AS 4100 (SA, 1998) and AS/NZS 4600 (SA,

1996). They found that all the experimental results exceeded the section moment

capacities predicted by both AS 4100 and AS/NZS 4600. AS 4100 predictions were

13% lower than the experimental results while AS/NZS 4600 predictions were lower

than the tested failure moments by 18%.

Figure 2.45: Section Capacity Test Set-up


2.9.5 Lateral Buckling Tests

Mahaarachchi and Mahendran (2005a) conducted a total of 48 simply supported

lateral buckling tests using a full scale test rig. They selected the test specimens from

the available 13 sections, and chose the beam span from 1200 mm to 4000 mm.

Figure 2.46 shows the overall view of the test rig used by Mahaarachchi and

Mahendran (2005a).

T-Shape Stiffeners

Plate Stiffeners

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Mahaarachchi and Mahendran (2005a)’s support system was based on the support

systems used by Zhao et al. (1995), Put et al. (1999) and Mahendran and Doan

(1999) with some modifications. The support conditions provided restraint against

in-plane and out-of-plane deflections and longitudinal twisting, and allowed major

and minor axis rotations. One of the support system was designed as a roller. Figure

2.47 illustrates the support system used.

Figure 2.46: Overall View of Test Rig

(Mahaarachchi and Mahendran, 2005a)

Figure 2.47: Support System


Mahaarachchi and Mahendran (2005a) reviewed the loading methods, the over-hang

loading method used by Zhao et al. (1995) and Mahendran and Doan (1999) and the

quarter point loading method used by Put et al. (1999), and proposed these two

Literature Review

2-62

methods with improved modifications. In the overhang method, the loads were

applied to the test beam at a distance of 1 m from the support in the upward direction

and this provided a uniform bending moment within the entire span whereas the

quarter point loading method provided a uniform bending moment only between the

points of load applications. Even though the overhang loading method is to be

preferred as it provides a uniform moment within the entire span, the overhang

component of the test beam provides a warping restraint to the test beam.

Mahaarachchi and Mahendran’s (2005a) test results showed that the overhang

loading method gave higher test capacity results by about 12% due to the effect of

warping restraints and hence they used the quarter point loading method. The loading

system was designed to prevent the possible restraints to the displacement and

rotations of the test beam using a special wheel system (Figure 2.48 (a)). Loads were

applied through the shear centre of the test beam to avoid any torsional and load

height effects to the test beam (Figure 2.48 (b)). The test results showed that the

lateral distortional buckling mode was most severe for intermediate spans. The

support conditions provided by Mahaarachchi and Mahendran (2005a) are likely to

allow local flange twist during the application of loading although the ideal simply

supported boundary conditions do not allow such twisting. Further experiments are

needed to identify the occurrence of such local flange twist. It is necessary to

restraint this twisting if it occurs during the experiments.

(a) Wheel System (b) Loading Arm

Figure 2.48: Loading System


Wheels

Smooth Tracks

Loading at Shear Centre

Literature Review

2-63

Kurniawan (2005) conducted an experimental investigation of LSBs with web

stiffeners. He tested a total of seven specimens of six stiffened 250 x 60 x 2.0 LSB

with different types of stiffeners (See Figure 2.49) and one unstiffened 250 x 60 x 2.0

LSB. The test set-up used by Kurniawan (2005) is shown in figure 2.50.


(Kurniawan, 2005)

Figure 2.50: Test Set-up of LSB with Stiffeners

(Kurniawan, 2005)

(a) Stiffener Type A (b) Stiffener Type B (d) Stiffener Type D (c) Stiffener Type C

(f) Stiffener Type F (e) Stiffener Type E

Additional Thin Plate

Literature Review

2-64

2.9.6 Experimental Investigation of HFBs

Mahendran and Avery (1997) conducted buckling experiments to investigate the

effects of different web stiffener configurations on the lateral buckling behaviour of

HFBs. The results of these experiments were also used to validate the finite element

model developed by Avery et al. (2000). The tests included ten 6 m long HFB

specimens, which were loaded to failure under a constant bending moment within a

span of 4.5 m as illustrated in Figure 2.51.

Figure 2.51: Schematic Diagram for Lateral Buckling Tests of HFBs

(Mahendran and Avery, 1997)

The experimental set-up used in this study consisted of a simply supported boundary

condition. Two load-controlled hydraulic jacks, located on the overhangs were used

to apply the loads and the web stiffeners at the support prevented any local bearing

failure of the bottom flange.

Mahendran and Doan (1999) conducted lateral buckling tests on HFBs. A purpose-

built test rig was used in this study to obtain the bending capacity of HFBs under

uniform bending moment. The test rig included a support system and a loading

system, which were attached to an external frame consisting of a main girder and

four columns as shown in the schematic diagram in Figure 2.52. The support system

was designed to ensure that the test beam had simply supported end conditions,

whereas the loading system was designed in such a way that no restraints were

induced as the beam deformed during loading.

0.55 m 0.55 m 4.5 m

Literature Review

2-65

Figure 2.52: Schematic Diagram of Test Rig Including Support System

(Mahendran and Doan, 1999)

Two vertical loads were applied at the end of two overhangs to produce a uniform

bending moment within the span of the specimen. The simply supported end

conditions of the span were simulated in a similar way to that of Zhao et al. (1995)

used for the Rectangular Hollow Sections (RHS) but were modified to suit the HFBs.

However, warping restraints induced by overhang of the beam could not be

eliminated in this system. The loading system included two hydraulic jacks instead of

gravity loading system used by Zhao et al. (1995). They were operated under load

control to ensure that the same load was applied and hence identical bending

moments were provided at the ends of the single span. Loading was applied at the

shear centre.

2.9.7 Experiments of other Cold-Formed Steel Beams

Zhao et al. (1995) conducted a series of lateral buckling tests of cold-formed RHS

beams to improve existing design rules for RHS beams. The section size used in the

testing program was 75 mm × 25 mm × 2.5 mm and the spans were varied from 2000

mm to 7000 mm. The loading system included a gravity load through the centroid of

the section and has simply supported end conditions. The layout of test setup is

shown schematically in Figure 2.53 (a).

Literature Review

2-66

Figure 2.53: Lateral Buckling Tsts of RHS Beams

(Zhao et al., 1995)

Zhao et al. (1995) stated that the support system used in their study (see Figure 2.53

(b)) was similar to that used by Trahair (1969) in his elastic lateral buckling tests of

aluminium I-beams and later by Papangelis and Trahair (1987) in their flexural

torsional buckling tests of arches.

Warping displacements were not prevented except by the adjacent cantilever lengths.

The restraint to warping provided by the cantilever lengths can be considered

minimal because significant warping does not occur in tubular sections, compared

with I-sections. Hollow flange steel beams such as LSBs and HFBs considered in

this research are open steel beams and hence they are expected to induce significant

amount of warping displacements compared to RHS beams. The loading system

included gravity loads being applied by suspending lead blocks on a platform which

was supported by hangers. Zhao et al. (1995) cited that the loading system used in

their study was similar to those used by Cherry (1960) and Hancock (1975), where

the vertical load applied acted through the centroid of the section and no restraints

were applied against out-of-plane movement at the loading point.

(a) Test Arrangement

(b) Support System

Literature Review

2-67

Figure 2.54: Test Arrangement for C- and Z- Section Beams

(Put et al., 1998 and 1999)

Put et al. (1998) conducted lateral buckling experiments on cold-formed lipped Z-

sections while Put et al. (1999) conducted lateral buckling tests on C- sections.

Although the support system was designed to achieve simple support end conditions

in these tests, they were different and complicated than the above mentioned loading

and support systems due to different geometric configurations of these section types

(see Figure 2.54). A gravity loading system was used for beam loading. This system

applied the vertical load in the loading drum. A low friction bearing system was used

to maintain vertical line of action and hence lateral buckling restraint effect was

eliminated. These test arrangements are not suitable for the buckling tests of LSBs

considered in this research program.

2.10 Literature Review Findings

Cold-formed steel members are increasingly used in the building industry due to

their light weight nature, ease of fabrication and structural and construction

efficiencies. The structural characteristics such as stress-strain relationships, initial

(b) Support System for Z- Section

(a) Test Arrangement for C- and Z-Section Beams

Literature Review

2-68

imperfections and residual stresses of cold-formed steel sections are different from

those of hot-rolled steel members mainly due to the manufacturing process which

involves cold-forming. The mechanical properties of cold-formed steel members are

affected by cold forming specifically in the regions of bends. The resulting changes

in material properties must be included in the design of cold-formed steel members.

Hollow flange steel beams such as HFBs and LSBs are some innovative cold-formed

steel members used in the building industry due to their superior structural and

construction efficiencies. However, the manufacturing of HFBs was discontinued in

1997 because of its relatively expensive electric welding process and lack of

equipment facilities.

Past research identified that hollow flange steel beams suffer from lateral distortional

buckling when used as flexural members. Lateral distortional buckling is

characterised by simultaneous lateral displacement, twist and web distortion which is

severe for intermediate spans and reduces the member moment capacity. AS 4100

(SA, 1998) does not provide any design procedure to determine the lateral

distortional buckling moment capacity of hollow flange steel beams while AS/NZS

4600 (SA, 1996) provides some design formulae. The equation given in AS 4100

(SA, 1998) to predict the lateral torsional buckling moment capacity was modified by

Pi and Trahair (1997) to predict the lateral distortional buckling moment capacity of

HFBs. However, it was not validated by experiments. Avery et al. (2000) then

modified Trahair’s (1997) equation to predict the lateral distortional buckling

moment capacity of HFBs by applying suitable coefficients for each thickness of

HFBs. But the applicability of these equations to predict the lateral distortional

buckling moment of LSBs is questionable.

The design formulae developed by Mahaarachchi and Mahendran (2005d) using

finite element analysis to predict the lateral distortional buckling moment capacity of

LSBs were found to be accurate and have been included in AS/NZS 4600 (SA,

2005). However, a thorough study of Mahaarachchi and Mahendran’s (2005c) finite

element analysis of LSBs revealed that the ideal model used to simulate LSBs under

uniform moment did not provide adequate twist restraint for the whole section

including the two rectangular flanges. Also, the application of boundary conditions

Literature Review

2-69

and the type of loading in this study found to have some unacceptable stress

distribution on the LSB cross section at failure. Therefore it is important to develop

an accurate ideal finite element model to predict the lateral distortional buckling

moment capacity of LSBs. Also, the existing design rules do not account for the

section geometry of LSBs and can only be used for the existing 13 LSB sections.

Therefore it is important to investigate the effect of section geometry on the lateral

distortional buckling moments of hollow flange steel beams and to develop a

comprehensive design procedure for all types of hollow flange steel beams (HFBs

and LSBs).

Mahaarachchi and Mahendran (2005a) carried out large scale experiments on LSBs

with quarter point and overhang type loading and found that the overhang loading

method increased the ultimate moments of LSBs due to warping. Therefore, they

used quarter point loading method and incorporated a moment modification factor of

1.09 to account for non-uniform bending moment along the span. However, this was

taken from AS 4100 (SA, 1998) which was developed for hot-rolled steel members

and the applicability of this to cold-formed steel members is questionable. Recent

research (Kurniawan and Mahendran, 2009b) on the effects of moment gradient on

the lateral distortional buckling capacity of LSBs revealed that the moment

distribution factor is closer to 1.0 for quarter point loading. Hence, this results can be

used in the experimental investigation.

Mahaarahchi Mahendran (2005b) carried out section moment capacity tests on LSBs

and their results revealed that some of the LSBs have inelastic reserve bending

capacity beyond their first yield moment. However, they did not develop finite

element models nor developed suitable design rules for LSBs to predict their section

moment capacities with inelastic reserve bending capacities.

Lateral buckling strengths of steel members are affected by initial imperfections,

residual stresses, moment distribution, web distortion, load height and pre-buckling

deformations and their effects on cold-formed hollow flange steel beams are

summarised in Section 2.5.

Web stiffeners were used with both HFBs and LSBs to eliminate the detrimental

effect of lateral distortional buckling. The results showed that the use of two web

Literature Review

2-70

stiffeners at third points of span were able to eliminate the lateral distortional

buckling of HFBs (Avery and Mahendran, 1997) while LSBs (Kurniawan, 2005)

required large number of web stiffeners. Overhang type loading method was used in

the experimental investigation of HFBs while quarter point loading was adopted for

LSBs. These different loading conditions and geometries of HFBs (doubly

symmetric with triangular hollow flanges) and LSBs (singly symmetric with

rectangular hollow flanges) would be some of the factors in determining the required

number of web stiffeners. It is important to investigate the effects of web stiffeners

on the lateral distortional buckling of LSBs.

Material Properties, Residual Stresses and Imperfections of LSBs

3-1

CHAPTER 3

3.0 MATERIAL PROPERTIES, RESIDUAL STRESSES AND GEOMETRIC

IMPERFECTIONS OF LSB SECTIONS

3.1 Introduction

The unique cold-forming and dual electric resistance welding process of LiteSteel

Beam (LSB) sections introduce considerable differences to the stress-strain curves,

residual stresses and initial geometric imperfections compared to the conventional

hot-rolled and cold-formed steel sections. Although Mahaarachchi and Mahendran

(2005e) measured the mechanical properties, residual stresses and initial geometric

imperfections of LSB sections the quality of the manufacturing process in relation to

cold-forming and electric resistance welding has been improved over the last three

years. Currently the LSB sections are made from a single strip of G60 galvanized

steel while the earlier LSB sections were manufactured from a single steel strip of TF

380 coil. The coating system has also been changed from EnviroKote water-based

primer paint protective coating system to AZ+ alloy coating system. Therefore it is

important to determine the accurate mechanical properties of these LSBs that can be

used in the finite element analyses of LSBs. The mechanical properties of LSBs were

measured using tensile coupon tests.

The residual stresses due to combined cold-forming and electrical resistance welding

process can have a significant effect on the behaviour and strength of LSB sections.

Unlike other cold-formed steel sections the LSB sections have both flexural and

membrane residual stresses due to cold-forming and welding processes, respectively.

Maharaachchi and Mahendran (2005e) measured the residual stresses of LSBs and

developed approximate residual stress models for both flexural and membrane

residual stresses. However, it was found that the sum of the membrane residual

forces was not equal to zero in their model. Further, their model had more

compressive stresses at the welding point despite the common belief that welding

creates tensile membrane residual stresses. In order to investigate these issues a

residual stress test was carried out for the web element of a 150x45x1.6 LSB section.


3-2

The LSB members exhibit a complicated post-buckling regime that is difficult to

predict. Today advanced computational modelling supplements experimental

investigations. Accuracy of computational models relies significantly on the various

inputs relating to material and section characteristics including section and member

imperfections used as input data. Mahaarachchi and Mahendran (2005e) measured

the initial geometric imperfections of a large number of LSBs and confirmed that

they were within the currently accepted fabrication tolerances. However, the initial

geometric imperfections were also measured for some LSBs in order to verify their

finding for the new LSBs.

This chapter presents the details of tensile coupon tests, residual stress and geometric

imperfection measurement tests of LSBs, and the results.

3.2 Tensile Coupon Tests to Determine the Mechanical Properties

The structural behaviour of LSBs depends on the mechanical properties of the steel

used, which are required in the finite element analyses of LSBs. Therefore tensile

tests of steel coupons taken from LSBs were conducted to determine the required

important mechanical properties based on the procedure specified in the Australian

Standard AS 1391 (SA, 2007).

Fifty four coupons were fabricated in the workshop at the Queensland University of

Technology. They were taken from the LSB sections used in the section and member

moment capacity tests. The coupons were cut in the longitudinal direction from

various locations of the beam, namely the outside flange, inside flange and web.

The coupon size and shape are important variables that can affect the behaviour of

tensile coupons. Previous research of G550 steel (Rogers and Hancock, 1997)

consisted of tensile testing of coupons with large grips. They have shown that uneven

gripping of these coupons might have occurred due to misalignment and improper

lubrication of the test machine grips, i.e. only one side of the grip portion of the

coupon was secured, resulting in either a slip of the test specimen as the load was

increased or eccentric tensile forces. This can cause a reduction in the true capacity

of the test specimens and influence their stress-strain behaviour. It is necessary to


3-3

design tensile coupons which limit the possibility of an eccentric connection between

the test machine grips and the test coupon. Therefore tensile coupon dimensions were

chosen in accordance with the recommendations of AS 1391 (SA, 2007), and are

shown in Figure 3.1. It is considered that the dimensions selected here do not unduly

affect the stress-strain behaviour of tensile coupons.

The coupons were immersed in a 1:1 diluted hydrochloric acid basin for about 45

minutes to remove the coating. The surface of the tensile coupons was then cleaned

with fine grade emery paper and then with an acetone solution. A 2 mm strain gauge

was attached to the tensile coupons using CN Cyanoacrylate adhesive and a 25 mm

gauge length extensometer was used in the middle of the specimen to measure the

strain more accurately.

Figure 3.1: Tensile Test Coupons The tensile coupon tests were carried out using the Tinius Olsen testing machine in

the QUT Structural Testing Laboratory. A displacement control method was used to

(b) Strain Gauge Arrangement

(a) Dimensions of Tensile Test Coupons

R20

15 mm

120 mm

25 mm

40 mm 40 mm


3-4

apply the tension load at a rate of 0.5 mm per minute. Special jaw systems that have

the ability to translate and rotate in-plane were attached to the test machine cross-

heads. This system of jaws minimised the presence of any end eccentricities due to

any misalignment of the grips and hence eliminated specimen twisting and bending

that usually occurs when the grips are tightened (see Figure 3.2). The tension force

was applied until a fracture occurred in the specimen. The measurements of load,

extensometer and strain gauge were recorded automatically at a fast rate (every

second) using a data acquisition system attached to a personal computer.

Figure 3.2: Tensile Test Arrangement

Test results derived based on the measured thicknesses are summarised in Table 3.1

while the typical stress-strain curves for the web and flange elements are given in

Figure 3.3. Test results show that the measured yield stresses exceed the nominal

flange yield stress of 450 MPa and the nominal web yield stress of 380 MPa. The

web and flange yield stresses varied depending on the thickness and LSB section.

Further details of tensile coupon tests including stress-strain curves can be found in

Appendix A.1.

(b) Extensometer

(a) Test Method


3-5

Figure 3.3: Typical Stress-Strain Curves from Tensile Coupon Tests

Table 3.1: Tensile Test Results

Test Specimen Location

Dimensions (mm) fy

(MPa) fu

(MPa)

Young’s Modulus

(GPa) fu/fy Thick

-ness Width

300x60x2.0 LSB

Outside Flange 2.22 12.03 557.7 592.9 193 1.06Inside Flange 2.02 12.03 496.3 534.2 210 1.08

Web 1.98 12.02 447.1 524.2 195 1.17

250x75x2.5 LSB


Web 2.54 12.03 446.0 515.4 208 1.16

200x45x1.6 LSB


Web 1.61 12.02 456.6 537.2 230 1.18

150x45x2.0 LSB


Web 1.97 12.01 437.1 516.4 208 1.18

150x45x1.6 LSB


Web 1.58 11.96 455.1 539.8 220 1.19

125x45x2.0 LSB


Web 1.94 11.89 444.4 532.3 227 1.20fy – Yield Stress, fu – Ultimate Stress

0.0

100.0

200.0

300.0

400.0

500.0

600.0

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00

% Strain

Stre

ss (N

/mm

2 )

Inside Flange

Outside Flange

Web


3-6

Table 3.1 presents the average values from three stress-strain curves for each

location. The ultimate stress (fu) was the maximum stress obtained from the stress-

strain curve while the yield stress (fy) was calculated based on the 0.2% proof stress.

These values are within 3% of the results from the manufacturers (OATM). The

measured yield and ultimate stresses are compared with the results from

Mahaarachchi and Mahendran (2005e) in Table 3.2. It is clearly seen that most of the

stresses for the new LSBs are higher than that of Mahaarachchi and Mahendran

(2005e). Although some of the results show about 20% increment when compared to

Mahaarachchi and Mahendran’s (2005e) results, on average the yield and ultimate

stresses were increased by about 8% and 3%, respectively. It should be noted that the

results for the new LSBs should be used in the numerical modelling of LSBs in this

research.

Table 3.2: Comparison of Yield and Ultimate Stresses

Test Specimen Location

Current Mahaarachchi and Mahendran (2005e) Current/MM

fy (MPa)

fu (MPa) fy (MPa) fu (Mpa) fy fu

300x60x2.0 LSB

Outside Flange 557.7 592.9 568 635 0.98 0.93 Inside Flange 496.3 534.2 492 557 1.01 0.96

Web 447.1 524.2 452 537 0.99 0.98

250x75x2.5 LSB


Web 446.0 515.4 420 531 1.06 0.97

200x45x1.6 LSB


Web 456.6 537.2 381 494 1.20 1.09

150x45x2.0 LSB


Web 437.1 516.4 373 507 1.17 1.02

150x45x1.6 LSB


Web 455.1 539.8 430 523 1.06 1.03

125x45x2.0 LSB


Web 444.4 532.3 377 496 1.18 1.07 Mean 1.08 1.03

fy – Yield Stress, fu – Ultimate Stress, MM – Mahaarachchi and Mahendran (2005e)


3-7

3.3 Residual Stress Measurements for LSB Sections

Residual stresses are important in the design of LSBs because the combined cold

roll-forming and dual electric resistance welding process used in the manufacturing

of LSBs is likely to cause higher initial residual stresses. These residual stresses

could reduce the bending moment capacities of LSBs. Therefore the use of a residual

stress model with accurate distribution and magnitudes of residual stresses is

important in any advanced numerical analysis of LSBs. Generally there are three

types of experimental methods used in residual stress measurements, namely,

destructive, semi-destructive and non-destructive methods. The most commonly used

destructive method is the sectioning method, which was used by Mahaarachchi and

Mahendran (2005e). They developed separate models for both flexural and

membrane residual stresses. However, the improvements to the LSB manufacturing

process over the last three years are likely to have modified the residual stress

distributions in LSBs. Also the membrane residual stress distribution developed by

Mahaarachchi and Mahendran (2005e) revealed some unbalanced forces in the LSB

cross-section. Therefore a residual stress test was carried out for the web element of a

150x45x1.6 LSB section.

3.3.1 Test Procedure

Ten strain gauges with 5 mm gauge length were attached to the inner and outer

surfaces of the web element of a 600 mm long 150x45x1.6 LSB section along the

longitudinal direction.

Figure 3.4: Strain Gauge Arrangement

(a) Specimen with Strain Gauges (b) Strain Gauge Locations

31 mm

22 mm

Neutral Axis28 mm

24 mm

5 mm

6.5 mm

0

1

2

3

4

Outer Surface Inner

Surface


3-8

The strain gauge locations are numbered as 0, 1, 2, 3 and 4 as shown in Figure 3.4.

The strain gauges were covered by a paper tape to avoid any damage during the

cutting process. They were connected to a data acquisition system which enabled the

recording of strains after each sectioning.

Figure 3.5: Sectioning Process of LSB

(a) First Transverse Cut (b) Specimen Ready for 2nd Transverse Cut

(c) Second Transverse Cut (d) Specimen Ready for Longitudinal Cut

(e) Longitudinal Cut (f) Small Pieces of LSB Specimen at the end


3-9

The specimen was first cut across the cross-section on either side of the installed

strain gauges using a band saw as shown in Figure 3.5 (a). A 100 mm specimen was

obtained after the first transverse cut on either side of strain gauges. The second

transverse cut was then carried out as shown in Figure 3.5 (c). Figure 3.5 (d) shows

the 50 mm specimen after two transverse cuts. The longitudinal sectioning was then

continued as shown in Figure 3.5 (e) to separate the strain gauges. Final strain gauge

readings were recorded after one hour and 24 hours of the completion of the

sectioning to allow for temperature effects. Figure 3.5 (f) shows the small pieces of

the LSB specimen at the end of sectioning.

3.3.2 Results

All the strains measured from the sectioning method are the strains released due to

sectioning. Final released strains were calculated from the difference between the

strain gauge reading before cutting and those obtained after 24 hours following

cutting. Figure 3.6 shows the measured released strains on both the outer and inner

surfaces of the web element. A positive value indicates tensile strain whereas a

negative value indicates compressive strain.

Figure 3.6: Measured Released Strain along the Web Element

-600

-400

-200

0

200

400

600

-60 -40 -20 0 20 40 60

Distance from the Web Neutral Axis (mm)

Stra

in (M

icro

Stra

in)

Outer Surface

Inner Surface


3-10

These released strains were converted to the released stresses using a Young’s

modulus value of 200,000 MPa. It can be assumed that all the residual stresses are

released during the sectioning process used in this research. Therefore the residual

stresses are equal to the calculated released stresses in magnitude but with an

opposite sign, i.e, released tensile stresses are equal to compressive residual stresses

and vice versa.

The residual stresses were then used to calculate the membrane and flexural residual

stresses using the following formulae.

Membrane residual stress, σm= (σi + σo)/2

Flexural residual stress, σf = (σi - σo)/2

Where

σi = calculated residual stress on the inner surface

σo= calculated residual stress on the outer surface

Figure 3.7 shows the calculated membrane and flexural residual stresses. The stresses

at the corners were extrapolated and further simplifications are made to determine

the membrane residual stress distribution shown in Figure 3.8.

Figure 3.7: Measured Stresses along the Web Element of a 150x45x1.6 LSB

-80

-60

-40

-20

0

20

40

60

80

-60 -40 -20 0 20 40 60

Distance from the Web Neutral Axis (mm)

Stre

ss (M

Pa)

Membrane Stress

Flexural Stress


3-11

The residual stresses are expressed as a ratio of the virgin plate’s yield stress of 380

MPa. The membrane residual stresses are then compared with those of Mahaarachchi

and Mahendran (2005e) in Figure 3.8.

Figure 3.8: Membrane Residual Stress Distribution

The new membrane residual stress along the web element shows a significant

reduction compared to those measured by Mahaarachchi and Mahendran (2005e).

However, the reduced values may not be used in numerical analyses as they were

derived from only one test. But the results confirm that there are compressive

membrane residual stresses at the web-flange junction, which is one of the main

objectives of this test.

Figure 3.9: Flexural Residual Stress Distribution

0.03fy

-0.23fy

0.11fy

-0.41fy

0.60fy

-0.41fy

-0.23fy

0.11fy 0.03fy

0.03fy

0.03fy

(a) Mahaarachchi and Mahendran (2005e)

-0.15fy

0.12fy

-0.15fy

(b) Current Research

0.24fy

0.24fy

0.24fy

1.07fy

0.41fy

0.8fy0.38fy

0.38fy 0.8fy

0.41fy

1.07fy

0.24fy

0.24fy

(a) Mahaarachchi and Mahendran (2005e) (b) Current Research

0.15fy


3-12

Figure 3.9 shows the comparison of flexural residual stresses between the current

research and that of Mahaarachchi and Mahendran (2005e). The residual stress along

the web element from this research reveals a noticeable reduction compared to those

measured by Mahaarachchi and Mahendran (2005e). Therefore it can be concluded

that both membrane and flexural residual stresses have been reduced in the new

LSBs due to possible improvements to the manufacturing process of LSBs.

However, the reduced value may not be used in numerical modelling as it was

obtained from only one test.

In order to determine the need to use accurate residual stresses in the numerical

analyses of LSBs, the individual effects of flexural and membrane residual stresses

on the flexural moment capacities of LSBs were thoroughly investigated by using

numerical analyses of LSBs. It was found that the flexural residual stresses play a

significant role in the reduction of moment capacities while the membrane residual

stresses have lesser effects. Further, the web membrane residual stresses were found

to have only a limited effect when compared to the flange membrane residual

stresses on the moment capacities of LSBs. More details can be found in Section

6.3.2 of Chapter 6 and Seo et al. (2008).

Based on the findings from the residual stress test and the numerical analyses of Seo

et al. (2008), it was decided not to use the reduced residual stress values obtained

from this research. Instead the flexural residual stress distribution of Mahaarachchi

and Mahendran (2005e) was retained while their membrane residual stress

distribution was modified for the web element and the left flange. The web

membrane residual stress values were changed from 0.60fy and -0.41fy to 0.50fy and -

0.50fy at the middle of the web and web-flange junction, respectively. The left flange

membrane residual stress value was slightly modified from 0.23fy to 0.2567fy for

150x45x1.6 LSB. Figure 3.10 shows the new membrane residual stress distribution

recommended for use in the numerical analyses of 150x45x1.6 LSB.


3-13

Figure 3.10: Membrane Residual Stress Distribution for 150x45x1.6 LSB

Table 3.3: Membrane Residual Stress of LSBs

LSB

Centreline Dimensions (mm) Membrane Residual Stress as a Ratio of fy

d

d1

bf

df

t

Left Flange

Right Flange

Web Top

MidWeb

Inside Flange

Left

Inside Flange Right

300x75x3.0 297.0 247.0 72.0 22.0 3.0 -0.2591 0.03 -0.50 0.50 0.11 0.03 300x75x2.5 297.5 247.5 72.5 22.5 2.5 -0.2556 0.03 -0.50 0.50 0.11 0.03 300x60x2.0 298.0 258.0 58.0 18.0 2.0 -0.2556 0.03 -0.50 0.50 0.11 0.03 250x75x3.0 247.0 197.0 72.0 22.0 3.0 -0.2591 0.03 -0.50 0.50 0.11 0.03 250x75x2.5 247.5 197.5 72.5 22.5 2.5 -0.2556 0.03 -0.50 0.50 0.11 0.03 250x60x2.0 248.0 208.0 58.0 18.0 2.0 -0.2556 0.03 -0.50 0.50 0.11 0.03 200x60x2.5 197.5 157.5 57.5 17.5 2.5 -0.2600 0.03 -0.50 0.50 0.11 0.03 200x60x2.0 198.0 158.0 58.0 18.0 2.0 -0.2567 0.03 -0.50 0.50 0.11 0.03 200x45x1.6 198.4 168.4 43.4 13.4 1.6 -0.2567 0.03 -0.50 0.50 0.11 0.03 150x45x2.0 148.0 118.0 43.0 13.0 2.0 -0.2615 0.03 -0.50 0.50 0.11 0.03 150x45x1.6 148.4 118.4 43.4 13.4 1.6 -0.2567 0.03 -0.50 0.50 0.11 0.03 125x45x2.0 123.0 93.0 43.0 13.0 2.0 -0.2615 0.03 -0.50 0.50 0.11 0.03 125x45x1.6 123.4 93.4 43.4 13.4 1.6 -0.2567 0.03 -0.50 0.50 0.11 0.03

Different values are used for other different LSB sections in order to make the net

membrane force in the cross-section to zero. Table 3.3 presents the proposed

membrane residual stress values for all the LSBs. It should be noted that no change

has been made to the flexural residual stresses proposed by Mahaarachchi and

Mahendran (2005e).

0.03fy

-0.2567fy

0.11fy

-0.50fy

0.50fy

-0.50fy

-0.2567fy

0.11fy 0.03fy

0.03fy

0.03fy


3-14

3.4 Initial Geometric Imperfection Measurements

Geometric imperfections of a member refer to deviation of a member from perfect

geometry and include bowing, warping, and twisting as well as local deviations. True

imperfection magnitude is important in investigating the structural behaviour of LSB

sections. In this research, attempts were made to determine the imperfection

magnitudes of LSB sections using measurements of the final profile of formed

sections.

The magnitudes of initial geometric imperfections were measured for some test

specimens using the imperfection measuring equipment specially designed and built

at the Queensland University of Technology (see Figure 3.11). The imperfection

measuring equipment included a levelled table with guided rails with an accuracy of

0.01 mm, a laser sensor, travelator to move the sensor and a data logger.

Measurements were taken along three lines in the longitudinal direction of the

specimen at 100 mm intervals. They were made to determine the initial crookedness

(lack of straightness) and t wist along the web and both flanges of the LSBs.

Figure 3.11: Geometric Imperfection Measurements

Travelator

Table

(a) Measuring Table (b) Laser Sensor


3-15

Typical imperfections measured along the length of a 4 m long 200x45x1.6 LSB are

shown in Figure 3.12. Global imperfections were measured at different locations of

the web and flange of the specimen and the average of those measurements are

presented in Figure 3.12 (a). Local imperfections were measured along the cross-

section of the web at quarter points and the mid span of the beam and are presented

in Figure 3.12 (b).

(a) Global Geometric Imperfections

(b) Local Geometric Imperfections along the Cross-Section

Figure 3.12: Measured Imperfections of a 4 m Long 200x45x1.6 LSB Section

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 500 1000 1500 2000 2500 3000 3500

Span Length (mm)

Impe

rfec

tions

(mm

)

Along the Web

Along the Flange

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-100 -50 0 50 100

Distance from Neutral Axis (mm)

Impe

rfec

tions

(mm

)

Quarter Point 1 Mid SpanQuarter Point 2


3-16

Measured imperfections of other LSB sections are presented in Appendix A.2. The

results show that the measured local plate imperfections are within the

manufacturer’s fabrication tolerance limit of depth or width/150 while the overall

member imperfections are less than the recommended limit of span/1000 (SA, 1998).

This demonstrates that the unique manufacturing process of LSB does not lead to

geometric imperfections that exceed the currently accepted fabrication tolerances.

Mahaarachchi and Mahendran (2005e) also measured these imperfections for a large

number of LSBs and confirmed this observation. The measured geometric

imperfection values and distribution can be used in the numerical modelling of LSBs

to improve its accuracy of simulating the structural behaviour of LSBs.

3.5 Conclusions

This chapter has presented the details of a series of tests to determine the mechanical

properties, residual stresses and initial geometric imperfections of the current LSB

sections manufactured using a unique cold-formed and dual electric resistance

welding process. A series of tensile tests was carried out based on coupons taken

from the web, outside flange and inside flange elements of six LSBs, and the

mechanical properties were compared with those of Mahaarachchi and Mahendran

(2005e). It was found that both the yield stress and ultimate tensile strength have

increased due to the improved manufacturing process over the last three years.

Residual stresses were measured along the web element of a 150x45x1.6 LSB

section, and it was found that the improved manufacturing process has reduced the

level of residual stresses in LSBs. The results also showed that there are compressive

membrane forces at the web-flange junction. The membrane residual stress

distribution was slightly modified so that the net membrane force in the LSB cross-

section is zero. Initial geometric imperfections of some LSB members showed that

the LSB imperfections are within the currently accepted fabrication tolerances.

Lateral Buckling Tests of LSB Sections

4-1

CHAPTER 4

4.0 LATERAL BUCKLING TESTS OF LSB SECTIONS

4.1 Introduction

Experimental investigations are important to fully understand the structural

behaviour of LiteSteel Beams (LSBs) subjected to flexural action and to calibrate the

numerical analyses of LSBs. Both elastic buckling and post-buckling behaviour of

LSBs can be investigated using lateral buckling tests. Mahaarachchi and Mahendran

(2005a) carried out a series of large scale lateral buckling tests of LSBs. However,

the quality of the LSB manufacturing process in relation to cold-forming and electric

resistance welding has improved over the last three years while the manufacturers are

currently using a different grade steel. It was found that the new LSB sections have

higher yield stresses and ultimate tensile strength when compared to those of the

earlier LSB sections (see Chapter 3). It is therefore believed that the lateral buckling

moment capacities of the currently available LSBs are higher than those of

Mahaarachchi and Mahendran (2005a).

Mahaarachchi and Mahendran (2005a) did not consider short span LSBs in their

lateral bucking tests. In their tests, both the top and bottom flanges of LSBs were

kept free at the supports which would have reduced the moment capacities due to the

occurrence of flange twist despite the fact that the idealised simply supported

boundary conditions do not allow such twisting. This effect can be minimised or

eliminated by plotting these results on the non-dimensional capacity curve using the

appropriate elastic lateral distortional buckling moments from numerical analyses.

Considering the new developments in LSB sections and their manufacturing and the

limitations in Mahaarachchi and Mahendran’s (2005a) series of tests as described

above, it was decided to undertake another series of lateral buckling test. In this

study a total of 12 tests was undertaken using the currently available LSBs. Some of

the tests considered the same spans used by Mahaarachchi and Mahendran (2005a)

for comparison purposes while other tests considered shorter spans subject to lateral

distortional buckling to obtain more points on the non-dimensional member moment


4-2

capacity curve. The LSBs with experimental failure moments below the AS/NZS

4600 (SA, 2005) member moment capacity curve were also chosen in this

experimental study (see Figure 4.1). The experimental results will be used to validate

the finite element analyses and to investigate the effects of section geometry on the

lateral distortional buckling moment capacities of LSBs, which is one of the major

objectives of this research. This chapter presents the details of this experimental

study and the results.

Figure 4.1: Experimental Results of Mahaarachchi and Mahendran (2005a)

4.2 Selection of Test Specimens

Six LSBs with different section geometries were considered in the experimental

study with the beam span ranging from 1200 mm to 4000 mm. The quarter point

loading method was considered for all the tests. Figure 4.1 shows the dimensionless

plot of Mahaarachchi and Mahendran’s (2005a) experimental failure moments with

the existing member capacity design curve from AS/NZS 4600 (SA, 2005) based on

measured LSB dimensions and thicknesses. They conducted some tests using the

overhang loading method, but used the quarter point loading for most of the tests.

This figure shows only the results from the quarter point loading tests of the

currently available 13 LSB sections although they tested some LSB sections, which

are not currently available.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2Slendreness, λd

Dim

ensi

onle

ss M

omen

t Cap

acity

, M u

/ My

AS/NZS 4600 (SA, 2005)300x75x3.0 LSB300x75x2.5 LSB300x60x2.0 LSB250x75x3.0 LSB250x75x2.5 LSB250x60x2.0 LSB200x60x2.5 LSB200x60x2.0 LSB200x45x1.6 LSB150x45x2.0 LSB150x45x1.6 LSB125x45x2.0 LSB125x45x1.6 LSBA

B

E F GH

C D


4-3

In Figure 4.1, points A and B correspond to 150x45x2.0 LSB and 150x45x1.6 LSB

with a span of 1.2 m, respectively. However, it was suspected that the shear buckling

between the support and the loading point might have reduced the capacity of these

beams even though Mahaarachchi and Mahendran (2005a) reported a lateral

distortional buckling failure. Current research considered a 150x45x2.0 LSB with 2

m span, and a 150x45x1.6 LSB with 1.8 m span, which are not likely to fail by shear

buckling based on preliminary finite element analyses. However, a 150x45x1.6 LSB

with 1.2 m span was also considered to check whether the beam fails by shear

buckling. Points C and D correspond to 250x75x2.5 LSB, points E and G correspond

to 200x45x1.6 LSB and points F and H correspond to 300x60x2.0 LSB with spans of

3 m and 4 m, respectively. Same tests were repeated with modified boundary

conditions as these failure moments are below the AS/NZS 4600 (SA, 2005) member

capacity curve. Full details of the test beam specimens are presented in Table 4.1

whereas Figure 4.2 illustrates the typical LSB test specimens.

Figure 4.2: LSB Test Specimens

Beam depth (d), flange width (bf) and the flange depth (df) were measured for each

test beam before testing while the thicknesses of LSB plate elements were carefully

measured using a micrometer. Accurate thickness of each plate element is important

to obtain the elastic lateral distortional buckling moment as a small change in

thickness would cause a significant change in the buckling capacities. The

riw ro

ro


4-4

thicknesses measured in the tensile coupon tests are likely to be more accurate as the

thickness of the coating in the test beam would affect the actual measurements. Table

4.1 presents the details of test specimens including the measured LSB dimensions

and the base metal thicknesses from tensile coupon tests. It can be seen that the outer

flange thickness (tof) is larger than the nominal thickness while the web thickness (tw)

is smaller than the nominal thickness due to the variation in cold-working within the

section. The measurements of small corners were not taken as it was difficult to

measure them, and it was decided to use the nominal corner dimensions provided by

the manufacturers, i.e. outer radius ro is equal to two times the thickness (2t) and the

inner radius riw is equal to 3 mm.

Table 4.1: Details of Test Specimens

No LSB Section Span

Thickness, t d df bf

Flange Twist

Restraint tof tif tw

1 250x75x2.5LSB 3500 2.90 2.60 2.54 251.0 75.0 25.5 No 2 300x60x2.0LSB 4000 2.22 2.02 1.98 302.0 60.0 20.5 No 3 200x45x1.6LSB 4000 1.79 1.66 1.61 201.0 45.0 14.8 No 4 300x60x2.0LSB 3000 2.22 2.02 1.98 299.0 60.0 20.0 No 5 200x45x1.6LSB 3000 1.79 1.66 1.61 201.0 45.0 14.9 Yes 6 150x45x1.6LSB 3000 1.75 1.62 1.58 150.0 46.0 15.1 Yes 7 150x45x2.0LSB 3000 2.22 2.05 1.96 150.0 45.0 15.0 Yes 8 200x45x1.6LSB 2000 1.79 1.66 1.61 200.0 45.0 14.9 Yes 9 150x45x2.0LSB 2000 2.22 2.02 1.97 151.0 45.0 14.9 Yes 10 150x45x1.6LSB 1800 1.77 1.63 1.58 150.0 46.0 14.6 Yes 11 125x45x2.0LSB 1200 2.16 1.97 1.94 125.0 45.0 14.6 Yes 12 150x45x1.6LSB 1200 1.77 1.63 1.58 150.5 45.5 14.6 Yes

Note: All dimensions are in mm. tof – outer flange thickness, tif – inner flange thickness, tw – web thickness.

4.3 Test Method

Experiments were conducted using a four-point bending arrangement to create a

uniform bending moment along the span between the loading points. The lateral

distortional buckling tests of LSBs can be conducted by two different testing

arrangements to produce a uniform bending moment. They are the overhang loading

method and the quarter point loading method. In the overhang loading method, loads

are applied on either side of the supports that will produce a uniform bending

moment throughout the span (see Figure 4.3 (a)). In the quarter point loading


4-5

method, loads are applied at quarter points within the span and a uniform bending

moment will be produced between the loading points (see Figure 4.3 (b)).

Figure 4.3: Different Types of Test Methods

Zhao et al. (1995) and Mahendran and Doan (1999) carried out lateral buckling tests

of cold-formed RHS beams and hollow flange beams, respectively, using the

overhang loading method. Put et al. (1998) used the quarter point loading method in

their investigation. Mahaarachchi and Mahendran (2005a) conducted lateral buckling

tests of LSBs using both the overhang and the quarter point loading methods in their

preliminary investigation in order to determine the most suitable method. They found

that the overhang loading system could cause undesirable warping effect due to the

overhang component of the test beam. Therefore they used the quarter point loading

system in most of their tests and considered a moment modification factor of 1.09 to

allow for the non-uniformity of bending moment within the span. Pokharel and

P P

Span, LOverhang

Bending Moment

Overhang

(a) Overhang Loading System

Span, L

P

L/4

P

L/4

Bending Moment

(b) Quarter Point Loading System


4-6

Mahendran (2006) also used the quarter point loading method to investigate the

bending moment capacities of LSBs with holes. Kurniawan and Mahendran (2009b)

found that the moment modification factor is much closer to 1.0 for LSBs under

quarter point loading. Hence the quarter point loading system was considered in this

research to study the lateral distortional buckling behaviour of LSBs.

A carefully designed special test rig was used to simulate a uniform moment between

the quarter points of LSB members. This test rig included special support conditions

that prevented the in-plane and out-of-plane deflections and twisting rotation without

restraining in-plane and out-of-plane rotations and warping displacements. Also it

was capable of applying the load through the shear centre of the mono-symmetric

LSB sections with no twisting and lateral restraints to the test beam.

Figure 4.4 shows the overall view of the test rig used in the lateral distortional

buckling tests that consisted of a support system and a loading system, which were

attached to an external frame consisting of two main beams and four columns. The

wheel system facilitates loading jacks to move laterally in both directions (along and

across the beam) without creating any restraints.

Figure 4.4: Overall View of Test Rig

Main Columns Main BeamsWheel System

Test Beam


4-7

4.3.1 Support System

The support system should ensure that the test beam is simply supported in-plane and

out-of-plane at the supports. The support system used in this experimental program

was developed by Mahaarachchi and Mahendran (2005a), which is similar to that

used by Zhao et al. (1995), Put et al. (1999) and Mahendran and Doan (1999), but

with some modifications. The support conditions provided fixity against in-plane

vertical deflections, out-of-plane deflections and twist rotations, but allowed major

and minor axis rotations. This means that the test beam could rotate freely about its

in-plane horizontal axis and vertical axis at the support, but did not twist.

Figure 4.5: Support System

Figure 4.5 shows the movable support at one end of the beam. The support of the

other end was the same except that the side bearing was prevented from rolling along

the running track by horizontal stops. The in-plane vertical movements and lateral

movements were prevented by the running tracks and side guides. The box-frame

Ball Bearing

LSB Section

Clamping Plate

Side Guide and Running Track

Box Frame


4-8

with the side bearing allowed the test beam to rotate about its major axis while the

top and bottom bearings allowed minor axis rotation and differential flange rotations

(about the minor axis) associated with warping displacement rotations.

The two supports were aligned to ensure that the vertical deflections remained in the

same plane. The test beam was connected to the support system by using 4 M10 bolts

and a 10 mm thick clamping plate. This plate was used to prevent web crippling and

twisting of the section at the supports.

4.3.1.1 Flange Twist Restraints

Past experimental research on LSBs (Mahaarachchi and Mahendran, 2005a, Pokharel

and Mahendran, 2006 and Kurniawan, 2007) considered the LSB flanges to be free

to twist despite the fact that the ideal simply supported boundary condition does not

allow flange twist. They provided steel plates connected to the full length of the web.

This does not prevent the isolated twisting of LSB flanges as observed during the

current experimental study (Figure 4.6). This could reduce the lateral buckling

moment capacity of LSBs. Therefore an attempt was made to prevent the flange twist

by welding a 6 mm thick plate stiffener between the inner face of the flanges (see

Figure 4.7). Since the weld at the inner face of the flanges was a small “tack” weld,

the effect of welding was considered to be negligible as it was generally used to hold

the plates in position and would not create any undesirable residual stresses.

Figure 4.6: Flange Twist at Failure of a 250x75x2.5 LSB with 3.5 m Span


4-9

Figure 4.7: Flange Twist Restraint Arrangement of LSBs

4.3.2 Loading System

A loading system was designed in order to apply two vertical loads at the quarter

points through the shear centre, which would produce a uniform bending moment

between the loading points. A gravity loading system was used by other researchers

in the past (Zhao et al., 1995 and Put et al., 1998) to investigate the lateral buckling

of simply supported beams. The gravity loads were applied by suspending a lead

block on a platform that was supported by hangers through the centroid of the

section. However, this method was considered tedious and labour intensive and

could not load the beam continuously. Mahendran and Doan (1999) used an

improved loading system in their lateral buckling tests of hollow flange beams

(HFB) where they applied the vertical loads using a hydraulic jack system operated

under load control. Mahaarachchi and Mahendran (2005a) indicated that the

hydraulic loading system used by Mahendran and Doan (1999) was also not the most

suitable method as it restrained the lateral movement of the test beam, did not allow

the continuation of loading into the post-ultimate load range and the whole loading

set-up was prone to damage. To eliminate the problems associated with this

hydraulic loading system, Mahaarachchi and Mahendran (2005a) improved and

Stiffener to eliminate flange twist


4-10

developed a new hydraulic loading system. In their system, two hydraulic rams were

connected to a specially designed wheel system on one end and to a load cell on the

other end. The load cell was then connected to a universal joint and then to the

loading point (shear centre) of the test beam. This system was operated under

displacement control with identical loads being applied at both loading points.

Figure 4.8: Loading System

(c) Loading at Shear Centre

Load position adjusting bolt

Load Cell

Universal Joint

Loading arm

(b) Hydraulic Pump

(a) Wheel System

P P

(d) Quarter Point Loading System

Support 1

Load 2

Load 1

Support 2


4-11

As Mahaarachchi and Mahendran (2005a) successfully used this improved hydraulic

loading system in their tests to investigate the lateral distortional buckling behaviour

of LSB sections, it was decided to use the same loading arrangement in this current

study. Figures 4.8 (a) and (b) show the wheel system and hydraulic pump,

respectively, used in this loading arrangement. In the lateral distortional buckling

tests, the loading system should not provide any restraint to the out-of-plane

movement of specimens at the loading points. The use of two sets of wheels that

allowed free translations longitudinally and transversely ensured that the loading arm

was always located vertically when the test beam deformed in-plane under the

applied loading. Figure 4.8 (c) shows the loading arm assembly with a universal

joint. The universal joint at a wheel system and at the connecting arm ensured that

the load was applied at the shear centre without applying a torque to the beam and

the load acted in the vertical plane when the beam deformed in-plane. Therefore all

six degrees of freedom were considered unrestrained at the loading positions of the

test beam. The loading arm was connected to the test beam at its centroid level using

3 M10 bolts. A steel plate of appropriate thickness was inserted between the loading

arm and the test beam web so that the loading arm was located at the shear centre.

This ensured the elimination of load height and torsional loading effects. Figure 4.8

(d) shows the overall view of the quarter point loading system.

4.3.3 Measuring System

There were two important parameters to be measured in these experiments, namely,

the applied load and the deflections. Two vertical loads at quarter points were

measured by two load cells of capacity 60 kN each. These load cells were attached to

the two loading arms as shown in Figure 4.9. The vertical deflections were measured

at mid-span and the bottom flange of two loading points using three wire

displacement transducers (WDT). The lateral deflections of top and bottom flanges at

mid-span were also measured using two wire displacement transducers (WDT) (see

Figure 4.10). These wire displacement transducers (WDT) were of the potentiometer

type.


4-12

Figure 4.9: Data Logger and Load Cells

Figure 4.10: Wire Displacement Transducers (WDTs)

All the load and displacement measurement units were connected to the data

acquisition (EDCAR) system which recorded the measurements automatically at

intervals of one second (s). The EDCAR unit included a HP3497A DATA

WDTs

Data logger Load cells


4-13

acquisition unit, a HP3498A extender and a PC. Calibration factors of the load cells

and wire displacement transducers (WDT) were determined and input to the EDCAR

unit before each test.

4.3.4 Test Procedure

Test specimens were cut 150 mm longer than their intended span since the

connection assembly needed an extra 75 mm at each support. Holes with 12 mm

diameter were drilled on the web at the loading and support positions to

accommodate M10 bolts as shown in Figure 4.11.

Note: all the dimensions are in mm.

Figure 4.11: Schematic Diagram of a Typical Test Specimen

Steel plate stiffeners of 6 mm thickness were welded between the inner faces of the

flanges at the middle of each support of the test beam using a tack weld as shown in

Figure 4.12. The widths of these plate stiffeners were 10 mm less than the flange

width in order to accommodate the web plates at the supports as shown in Figure

4.12 (b). Deflection measuring points were marked before the beam was positioned

and clamped to the test rig. The test beam was inserted within the box frame and the

clamping plates were bolted to the test beam (see Figure 4.5). These plates were used

to avoid web crippling and twisting of the section at the supports. The width of the

support plates (web plate) was 75 mm for the beams with the depth of 200 mm or

less, and 100 mm for the beams with the depth more than 200 mm. The depth of

these plates was chosen to cover the full web. The bolt spacing used for 75 mm width

support plates was 45 mm (vertical) x 45 mm (horizontal) while they were 160 mm

Support

Support plate P P

75

Load

Flange

Flange

L/2 L/4

Centreline

LoadL/4

Support

Web

45 45

20

120

45 45

20

120

Support plate Loading plate

45 50

45


4-14

(vertical) x 60 mm (horizontal) for 100 mm width support plates. The loading

devices were then bolted to the web at quarter points. The loading plates of 120 mm

x 20 mm x 10 mm thickness were used to connect the loading arm while appropriate

size nuts were used to set the loading point at the shear centre of LSBs. The support

frame was aligned to avoid any initial twisting while the loading jack and arm were

aligned in order to prevent any eccentricities. The jacks were connected in parallel to

ensure that equal vertical loads were applied at the shear centre of test beam. The

load cells and transducers were connected to the data acquisition system to record all

the measurements automatically. Each channel was individually checked to ensure

correct operation.

Note: all the dimensions are in mm.

Figure 4.12: Schematic Diagram of Flange Twist Restraints

The calibration factors of all the measuring devices were entered in the EDCAR unit.

A small load was applied first to allow the loading and support systems to settle on

wheels and bearings evenly. The measuring system was then initialized with zero

values. A trial load of 10% of the expected ultimate capacity was applied and

released in order to remove any slackness in the system and to ensure functionality.

The load was then applied gradually and smoothly using a manually operated

hydraulic pump (Figure 4.8 (b)) while the test data was recorded continuously at one

second intervals. The applied load started to drop off when the test beam buckled

out-of-plane. The loading was continued until the test beam failed by out-of-plane

buckling. The loading was also continued after failure in order to obtain the

(a) Front View (b) Cross Sectional View

6 mm Thick Plate Stiffeners

Tack weld

LSB

Web Plate


4-15

unloading characteristics of the test beam. A typical LSB specimen after failure is

shown in Figure 4.13.

Figure 4.13: Typical Lateral Distortional Buckling Failure

4.4 Experimental Results and Discussions

Twelve tests were conducted on LSB sections to investigate their lateral distortional

buckling behaviour and ultimate moment capacity. The first four tests did not include

the flange twist restraints while the remaining eight tests included them as shown in

Table 4.1. All the test beams except 150x45x1.6 LSB with a span of 1.2 m failed due

to lateral distortional buckling (Figure 4.14). Some local web buckling was observed

after the ultimate load was reached (Figure 4.15). Shear buckling was observed

between the loading points and the support for 150x45x1.6 LSB with a 1.2 m span.

Load Load


4-16

Figure 4.14: A Closer View of Lateral Distortional Buckling Failure

Figure 4.15: Local Web Buckling after Ultimate Failure

Figure 4.15 shows different views of local web buckling, which was observed just

after the ultimate failure. In the initial loading stages, the top and bottom flanges

slowly started to move laterally until its elastic buckling moment was reached. As the

applied moment reached its ultimate capacity, the lateral deflection of the top and


4-17

bottom flanges increased rapidly and the beam collapsed by lateral distortional

buckling modes as shown in Figure 4.14. At the beginning of the tests, lateral

deflections were negative but they gradually changed to positive. Flange twist was

not observed in test beams with plate stiffeners at the supports (see Figure 4.16).

Figure 4.16: Comparison of Flange Twist Condition at Failure

As expected, 150x45x1.6 LSB with 1.2 m span exhibited a shear buckling failure and

hence it did not reach the expected lateral distortional buckling capacity. Figure 4.17

shows the shear buckling failure of this test beam.

Figure 4.17: Shear Buckling Failure of 150x45x1.6 LSB with 1.2 m Span

Shear Buckling


4-18

During the test, both the applied quarter point loads (P) were measured and recorded

in kN using the EDCAR unit. The average of these two quarter point loads was then

used to calculate the applied uniform moment (M) using the following formula:

M = PL/4 (4.1)

Where, L is the span of the test beam.

(a) Moment vs Vertical Deflection of 3.5 m Span 250x75x2.5 LSB

(b) Moment vs Lateral Deflection of 3.5 m Span 250x75x2.5 LSB

Figure 4.18: Moment vs Lateral Deflection Curves

0

5

10

15

20

25

30

35

-10 10 30 50 70 90 110 130 150

Lateral Deflection at Mid Span (mm)

Mom

ent (

kNm

)

Top Flange (Tension)

Bottom Flange(Compression)

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90

Vertical Deflection (mm)

Mom

ent (

kNm

)

at Mid Span

Under the Load


4-19

Figures 4.18 (a) and (b) show the variation of applied moment vs vertical and lateral

deflections of a 250x75x2.5 LSB with 3.5 m span. All the other moment vs

deflection curves for the tested beams are given in Appendix B. Table 4.2

summarises the ultimate failure moments (Mu) and the type of failure for the tested

beams, and compares them with the corresponding results from Mahaarachchi and

Mahendran (2005a). It is clearly seen that all the test beams except 300x60x2.0 LSB

have higher lateral buckling moment capacities than that of Mahaarachchi and

Mahendran (2005a). This improvement is considered to be due to the improved

manufacturing process of LSBs and the resulting reduced residual stress effects, the

use of an higher strength steel and the improved simply supported boundary

conditions with flange twist restraints.

Table 4.2: Lateral Buckling Test Results from this Study

Test No LSB Sections Span

(mm)Mu

(kNm) Failure Mechanism Mu (MM) (kNm)

1 250x75x2.5LSB 3500 34.13 LDB, LB flange twist N/A 2 300x60x2.0LSB 4000 17.17 LDB, flange twist 16.94 3 200x45x1.6LSB 4000 5.92 LDB, flange twist 5.66 4 300x60x2.0LSB 3000 18.09 LDB, LB, flange twist 19.74 5 200x45x1.6LSB 3000 9.24 LDB 6.18 6 150x45x1.6LSB 3000 8.27 LDB 6.56 7 150x45x2.0LSB 3000 9.87 LDB 8.64 8 200x45x1.6LSB 2000 10.72 LDB, LB N/A 9 150x45x2.0LSB 2000 10.76 LDB 9.03 10 150x45x1.6LSB 1800 9.30 LDB N/A 11 125x45x2.0LSB 1200 10.83 LDB N/A 12 150x45x1.6LSB 1200 9.23 LDB, LB, Shear 8.02

Note: LDB – Lateral Distortional Buckling, LB – Local Buckling, N/A - Not Available, MM – Mahaarachchi and Mahendran (2005a).

The ultimate failure moments obtained in this study together with those from

Mahaarachchi and Mahendran’s (2005a) test will be used in the comparison with

predicted moment capacities from the current design methods. Although

Mahaarachchi and Mahendran’s (2005a) ultimate moments were slightly less than

those from this study, plotting them under a non-dimensional moment capacity curve

format would minimise or eliminate the effects of these differences.


4-20

Table 4.3: Details and Results of Mahaarachchi and Mahendran’s (2005a)

Lateral Buckling Tests

LSB Sections Span (mm)

Thickness, t (mm) d

(mm)bf

(mm)df

(mm) Mu

(kNm) Failure Mode tof tif tw

300x75x3.0LSB 4000 3.18 3.18 2.84 300 75.31 25.17 40.05 LDB 300x75x2.5LSB 4000 2.87 2.87 2.51 300 75.24 25.05 35.31 LDB 300x60x2.0LSB 4000 2.15 2.15 1.98 300 60.28 19.97 16.94 LDB 300x60x2.0LSB 3000 2.15 2.15 1.98 300 60.28 19.97 19.74 LDB, LB 250x75x3.0LSB 4000 3.08 3.08 2.77 250 76.35 25.22 33.35 LDB 250x75x2.5LSB 4000 2.79 2.79 2.48 250 75.98 24.92 28.37 LDB 250x75x2.5LSB 3000 2.79 2.79 2.48 250 75.98 24.92 29.85 LDB 250x60x2.0LSB 4000 2.09 2.09 1.96 250 60.47 20.12 17.28 LDB 250x60x2.0LSB 3000 2.09 2.09 1.96 250 60.47 20.12 18.25 LDB, LB 200x60x2.5LSB 4000 2.58 2.58 2.34 200 60.23 19.95 17.18 LTB 200x60x2.5LSB 3500 2.58 2.58 2.34 200 60.23 19.95 16.78 LTB 200x60x2.0LSB 4000 2.03 2.03 1.85 200 60.15 20.31 12.98 LTB 200x60x2.0LSB 3500 2.03 2.03 1.85 200 60.15 20.31 12.4 LDB 200x45x1.6LSB 4000 1.56 1.56 1.48 200 45.05 14.98 5.66 LTB 200x45x1.6LSB 3000 1.56 1.56 1.48 200 45.05 14.98 6.18 LDB 150x45x2.0LSB 3000 2.11 2.11 1.89 150 44.95 14.73 8.44 LDB 150x45x2.0LSB 2400 2.11 2.11 1.89 150 44.95 14.73 8.26 LDB 150x45x2.0LSB 2000 2.11 2.11 1.89 150 44.95 14.73 9.03 LDB 150x45x1.6LSB 3000 1.60 1.60 1.60 150 45.12 14.89 6.56 LDB 150x45x1.6LSB 2400 1.60 1.60 1.60 150 45.12 14.89 7.01 LDB 150x45x1.6LSB 2000 1.60 1.60 1.60 150 45.12 14.89 7.21 LDB 125x45x2.0LSB 3500 1.98 1.98 1.98 125 45.10 14.93 7.88 LTB 125x45x2.0LSB 2300 1.98 1.98 1.98 125 45.10 14.93 8.41 LTB 125x45x2.0LSB 2000 1.98 1.98 1.98 125 45.10 14.93 8.45 LDB 125x45x2.0LSB 1600 1.98 1.98 1.98 125 45.10 14.93 8.55 LDB 125x45x1.6LSB 3500 1.62 1.62 1.62 125 45.07 14.95 6.69 LTB 125x45x1.6LSB 2300 1.62 1.62 1.62 125 45.07 14.95 7.11 LTB 125x45x1.6LSB 2000 1.62 1.62 1.62 125 45.07 14.95 7.55 LDB 125x45x1.6LSB 1600 1.62 1.62 1.62 125 45.07 14.95 7.51 LDB

Note: LDB – Lateral Distortional Buckling, LTB – Lateral Torsional Buckling, LB – Local Buckling, tof – outer flange thickness, tof – inner flange thickness, tof – web thickness.


4-21

Table 4.3 shows the lateral buckling test details and the results from Mahaarachchi

and Mahendran (2005a). Although they conducted lateral buckling tests with both

the overhang and the quarter point loading methods, only the results from the latter

were considered due to several reasons including the undesirable warping effects

associated with the overhang loading method. Among their results from the quarter

point loading, the results for 150x45x2.0 LSB and 150x45x1.6 LSB sections with 1.2

m span (Points A and B in Figure 4.1) were not considered as their capacities were

reduced by premature shear buckling failures. Since Kurniawan and Mahendran

(2009b) showed that the moment modification factor for quarter point loading of

LSB flexural members can be taken as 1.0 the ultimate moment capacities from tests

will not be reduced as was done by Mahaarachchi and Mahendran (2005a).

4.5 Comparisons with Design Methods

The ultimate moment capacity results from this research and those from

Mahaarachchi and Mahendran (2005a) were plotted within the framework of non-

dimensional moment capacity curves and compared with the member moment

capacity predictions from the current design method given in AS/NZS 4600 (SA,

2005).

Clause 3.3.3.3 (b) of AS/NZS 4600 (SA, 2005) outlines the design rules for members

subject to bending under distortional buckling that involves transverse bending of a

vertical web with lateral displacements of the compression flange, which were

developed by Mahaarachchi and Mahendran (2005d). In this case, the nominal

member moment capacity Mb is given by Equation 4.2.

(4.2)

For hollow flange beams, it is appropriate to determine the effective section modulus

(Ze) at a stress corresponding to Mc/Z, where Mc is the critical moment as defined in

Equations 4.3 (a) to (d) and Z is the full elastic section modulus.

For λd ≤ 0.59: Mc = My (4.3a)

⎟⎠⎞

⎜⎝⎛=

ZMZM c

eb


4-22

For 0.59 < λd < 1.70: ⎟⎠⎞

⎜⎝⎛=

dyc MM

λ59.0 (4.3b)

For λd ≥ 1.70: ⎟⎠⎞

⎜⎝⎛= 2

1d

yc MMλ

(4.3c)

The non-dimensional slenderness λd is given by odyd MM /=λ ) (4.3d)

Where Mod is the elastic lateral distortional moment and My is the first yield moment.

Mod can be determined from an elastic buckling analysis program or by using

available buckling formulae while My is given by

My = Z fy (4.4)

where Z is the elastic section modulus and fy is the yield stress.

Calculations of Mod and My are quite important when plotting the experimental

points in the non-dimensional member capacity curve as a minor change in the

calculation could move the points to another location in the plot. Pi and Trahair’s

(1997) Mod equations (Eqs.2.21 and 2.22 in Chapter 2) are considered to be accurate

and have been used in the design capacity tables of LSBs. However, these equations

are valid only for a constant thickness throughout the cross section while the beams

used in the experimental study have different thicknesses. Hence Pi and Trahair’s

(1997) equations become unhelpful in this situation. The only option is to use a

software that allows for different thicknesses of plate elements in the cross section. A

well established finite strip analysis program Thin-Wall is considered to be capable

of using different thicknesses in obtaining the elastic lateral distortional buckling

moments Mod of tested sections and spans. It also has the capability to include the

corners of LSB cross-section. Being able to simulate the varying plate thickness and

corners makes Thin-Wall the most suitable in this case. Further, Thin-wall assumes

idealised simply supported boundary conditions at the supports while most of the

experiments in this research also considered the same. Thus Thin-Wall is more

suitable to obtain the Mod of tested LSBs in this research. Mahaarachchi and

Mahendran (2005a) also assumed the same despite the fact their test beams were not


4-23

fully prevented from flange twist. Hence their experimental Mod values can be

slightly less than those calculated using Thin-Wall. Since there were no other simple

options, Thin-Wall was used to calculate the elastic lateral distortional buckling

moments of all the tested beams, which included the measured dimensions including

thicknesses and nominal corners.

The elastic section modulus Z of tested beams was also calculated using Thin-Wall

using the measured LSB dimensions and thicknesses. The first yield moment My was

then determined by using the measured yield stresses of tested beams. Corners of the

tested beams were not measured and it is not accurate to calculate Mod and My

without corners as the ultimate moment capacities of the tested beam had corners.

Therefore, as decided earlier, the nominal dimensions of corners were used rather

than obtaining those properties without corners. Table 4.4 gives the calculated

section properties, the elastic buckling and yield moment capacities, and the ultimate

moment capacities of the tests from this research while Table 4.5 gives the same for

the tests of Mahaarachchi and Mahendran (2005a).

Table 4.4: Measured Properties and Capacities of LSBs Used in the Current

Lateral Buckling Tests

LSB Section Span (mm)

fy (MPa)

Z (x103mm3)

My (kNm)

Mod (kNm) λd

Mu (kNm) Mu/My

300x60x2.0LSB 3000 557.7 101.70 56.72 22.82 1.58 18.09 0.32

300x60x2.0LSB 4000 557.7 103.10 57.50 18.68 1.75 17.17 0.30 250x75x2.5LSB 3500 552.2 119.00 65.71 47.83 1.17 34.13 0.52 200x45x1.6LSB 2000 536.9 39.60 21.26 11.54 1.36 10.72 0.50 200x45x1.6LSB 3000 536.9 39.95 21.45 8.54 1.58 9.24 0.43 200x45x1.6LSB 4000 536.9 39.95 21.45 6.87 1.77 5.92 0.28 150x45x2.0LSB 2000 537.6 32.58 17.52 14.33 1.11 10.76 0.61 150x45x2.0LSB 3000 537.6 32.46 17.45 10.54 1.29 9.87 0.57 150x45x1.6LSB 1200 557.8 26.79 14.94 16.65 0.95 9.29* 0.62 150x45x1.6LSB 1800 557.8 26.79 14.94 11.89 1.12 9.30 0.62 150x45x1.6LSB 3000 557.8 26.53 14.80 8.67 1.31 8.27 0.56 125x45x2.0LSB 1200 544.1 24.56 13.36 19.92 0.82 10.83 0.81

*Shear Failure.


4-24

Table 4.5: Measured Properties and Capacities of LSBs Used in the Lateral

Buckling Tests of Mahaarachchi and Mahendran (2005a)


fy (MPa)

Z (x103mm3)

My (kNm)

Mod (kNm) λd

Mu (kNm) Mu/My

300x75x3.0LSB 4000 528 173.90 91.82 52.11 1.33 40.05 0.44 300x75x2.5LSB 4000 511 157.90 80.69 45.00 1.34 35.31 0.44 300x60x2.0LSB 4000 568 104.00 59.07 18.95 1.77 16.94 0.29 300x60x2.0LSB 3000 568 104.00 59.07 23.53 1.58 19.74 0.33 250x75x3.0LSB 4000 506 132.80 67.20 51.46 1.14 33.35 0.50 250x75x2.5LSB 4000 525 120.90 63.47 44.54 1.19 28.37 0.45 250x75x2.5LSB 3000 525 120.90 63.47 53.77 1.09 29.85 0.47 250x60x2.0LSB 4000 580 79.12 45.89 18.63 1.57 17.28 0.38 250x60x2.0LSB 3000 580 79.12 45.89 22.66 1.42 18.25 0.40 200x60x2.5LSB 4000 496 70.34 34.89 22.90 1.23 17.18 0.49 200x60x2.5LSB 3500 496 70.34 34.89 25.34 1.17 16.78 0.48 200x60x2.0LSB 4000 473 56.17 26.57 17.68 1.23 12.98 0.49 200x60x2.0LSB 3500 473 56.17 26.57 19.28 1.17 12.40 0.47 200x45x1.6LSB 4000 478 36.14 17.27 6.29 1.66 5.66 0.33 200x45x1.6LSB 3000 478 36.14 17.27 7.73 1.49 6.18 0.36 150x45x2.0LSB 3000 498 32.01 15.94 10.26 1.25 8.44 0.53 150x45x2.0LSB 2400 498 32.01 15.94 12.22 1.14 8.26 0.52 150x45x2.0LSB 2000 498 32.01 15.94 14.00 1.07 9.03 0.57 150x45x1.6LSB 3000 540 25.12 13.56 8.19 1.29 6.56 0.48 150x45x1.6LSB 2400 540 25.12 13.56 9.61 1.19 7.01 0.52 150x45x1.6LSB 2000 540 25.12 13.56 10.89 1.12 7.21 0.53 125x45x2.0LSB 3500 503 23.73 11.94 8.87 1.16 7.88 0.66 125x45x2.0LSB 2300 503 23.73 11.94 12.58 0.97 8.41 0.70 125x45x2.0LSB 2000 503 23.73 11.94 14.01 0.92 8.45 0.71 125x45x2.0LSB 1600 503 23.73 11.94 16.50 0.85 8.55 0.72 125x45x1.6LSB 3500 549 19.71 10.82 7.33 1.22 6.69 0.62 125x45x1.6LSB 2300 549 19.71 10.82 10.09 1.04 7.11 0.66 125x45x1.6LSB 2000 549 19.71 10.82 11.09 0.99 7.55 0.70 125x45x1.6LSB 1600 549 19.71 10.82 12.85 0.92 7.51 0.69


4-25

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20

Slenderness, λd

Mu/M

y , M

b/My

AS/NZS 4600 (SA, 2005)EXP This ResearchEXP MM

Figure 4.19: Comparison of Experimental Failure Moments with AS/NZS 4600

(SA, 2005) Predictions

All the ultimate moment capacity and slenderness results were non-dimensionalised

for the purpose of comparison and are plotted in Figure 4.19. The test beam capacity

Mu and the AS/NZS 4600 (SA, 2005) moment capacities Mb are plotted as Mu/My

and Mb/My on the vertical axis whereas the non-dimensional member slenderness λd

(=(My/Mod)1/2) is plotted on the horizontal axis. The first yield moment My and λd

were calculated using Equations 4.4 and 4.3 (d), respectively. All of these values are

given in Tables 4.4 and 4.5.

Figure 4.19 shows the comparison of experimental results with the current AS/NZS

4600 (SA, 2005) design curve. Most of the experimental results from this research

were found to be higher than AS/NZS 4600 (SA, 2005) predictions. Table 4.6 shows

the ratio of Mu/My for Mahaarachchi and Mahendran’s (2005a) test results to the

predictions from AS/NZS 4600 (SA, 2005) design rules. It has a maximum value of

1.30 and a minimum value of 0.87. Only a few test results were less than the

predictions from the current design rule. Table 4.7 shows the ratio of Mu/My for the

test results from this research to the predictions from AS/NZS 4600 (SA, 2005). It

has a maximum value of 1.24 and a minimum value of 0.85. It is clearly seen that the

test results are on average about 13% higher than the predictions from the current


4-26

design rule. Overall, the current design curve in AS/NZS 4600 (SA, 2005) is quite

conservative compared to the test results from both this research and Mahaarachchi

and Mahendran (2005a).

Table 4.6: Comparison of Experimental Failure Moments of Mahaarachchi and

Mahendran (2005a) with AS/NZS 4600 (SA, 2005) Predictions

LSB Section Span (mm) λd

EXP Mu/My

AS/NZS 4600 (SA, 2005)

Ratio (EXP)/(AS/NZS 4600)

300x75x3.0LSB 4000 1.33 0.44 0.44 0.98 300x75x2.5LSB 4000 1.34 0.44 0.44 0.99 300x60x2.0LSB 4000 1.77 0.29 0.32 0.89* 300x60x2.0LSB 3000 1.58 0.33 0.37 0.90 250x75x3.0LSB 4000 1.14 0.50 0.52 0.96 250x75x2.5LSB 4000 1.19 0.45 0.49 0.90 250x75x2.5LSB 3000 1.09 0.47 0.54 0.87 250x60x2.0LSB 4000 1.57 0.38 0.38 1.00 250x60x2.0LSB 3000 1.42 0.40 0.41 0.96 200x60x2.5LSB 4000 1.23 0.49 0.48 1.03 200x60x2.5LSB 3500 1.17 0.48 0.50 0.96 200x60x2.0LSB 4000 1.23 0.49 0.48 1.02 200x60x2.0LSB 3500 1.17 0.47 0.50 0.93 200x45x1.6LSB 4000 1.66 0.33 0.36 0.92 200x45x1.6LSB 3000 1.49 0.36 0.39 0.91 150x45x2.0LSB 3000 1.25 0.53 0.47 1.12 150x45x2.0LSB 2400 1.14 0.52 0.52 1.00 150x45x2.0LSB 2000 1.07 0.57 0.55 1.02 150x45x1.6LSB 3000 1.29 0.48 0.46 1.05 150x45x1.6LSB 2400 1.19 0.52 0.50 1.04 150x45x1.6LSB 2000 1.12 0.53 0.53 1.01 125x45x2.0LSB 3500 1.16 0.66 0.51 1.30 125x45x2.0LSB 2300 0.97 0.70 0.61 1.16 125x45x2.0LSB 2000 0.92 0.71 0.64 1.11 125x45x2.0LSB 1600 0.85 0.72 0.69 1.03 125x45x1.6LSB 3500 1.22 0.62 0.49 1.27 125x45x1.6LSB 2300 1.04 0.66 0.57 1.15 125x45x1.6LSB 2000 0.99 0.70 0.60 1.17 125x45x1.6LSB 1600 0.92 0.69 0.64 1.08

Mean 1.03 COV 0.11

* Elastic region. Not considered in calculating Mean and COV.


4-27

Table 4.7: Comparison of Experimental Failure Moments with AS/NZS 4600

(SA, 2005) Predictions

LSB Section Span (mm) λd

EXP Mu/My

AS/NZS 4600 (SA, 2005)

Ratio (EXP)/(AS/NZS 4600)

300x60x2.0LSB 3000 1.58 0.32 0.37 0.85 300x60x2.0LSB 4000 1.75 0.30 0.32 0.92** 250x75x2.5LSB 3500 1.17 0.52 0.50 1.03 200x45x1.6LSB 2000 1.36 0.50 0.43 1.16 200x45x1.6LSB 3000 1.58 0.43 0.37 1.16 200x45x1.6LSB 4000 1.77 0.28 0.32 0.86** 150x45x2.0LSB 2000 1.11 0.61 0.53 1.15 150x45x2.0LSB 3000 1.29 0.57 0.46 1.23 150x45x1.6LSB 1200 0.95 0.62 0.62 1.00* 150x45x1.6LSB 1800 1.12 0.62 0.53 1.18 150x45x1.6LSB 3000 1.31 0.56 0.45 1.24 125x45x2.0LSB 1200 0.82 0.81 0.72 1.13

Mean 1.13 COV 0.11

* Not considered in calculating the Mean and COV (Shear Failure). ** Elastic region. Not considered in calculating Mean and COV. In this section, all the test ultimate moment capacities were compared with the

current design rule predictions in a non-dimensionalised format by using the

measured LSB dimensions, thicknesses and yield stresses and Thin-Wall software to

calculate the correct values of Mod and My required for non-dimensionalisation. This

approach allowed the comparison of all the test results in the same plot despite the

differences in thicknesses and yield stresses while also allowing for the effect of

corners to be included. Including the effects of corners and varying thickness and

yield stress was considered important when comparing with test results. However, in

some lateral buckling tests in this research and all of Mahaarachchi and Mahendran’s

(2005a) tests the twisting of flanges was not fully eliminated at the support. This

could have lead to slightly reduce Mod and thus lower values in the tests, but non-

dimensional plots would have eliminated this effect. Since the exact Mod values

could not be measured during tests they were calculated using Thin-Wall and used in

the calculations of non-dimensional slenderness (λd). However, Thin-Wall assumes

twist restraint of the entire cross-section at the support, and thus it would have given

higher Mod values for the tested beams without flange twist restraint. This implies


4-28

slightly lower λd values and hence higher predictions from AS/NZS 4600 (SA,

2005), leading to slightly lower ratios of (EXP) / (AS/NZS 4600) capacities. In

summary, the calculated ratios of ultimate moment capacities from experiments and

AS/NZS 4600 (SA, 2005) shown in Tables 4.6 and 4.7 and Figure 4.19 are likely to

be slightly higher if the effects of flange twist restraint were included via exact Mod

values. However, since the mean values of this ratio are already 1.03 and 1.13 in

Tables 4.6 and 4.7, it is concluded that the current design rule is quite conservative

compared to the experimental results from Mahaarachchi and Mahendran (2005a)

and this research.

Table 4.8: Effect of Flange Twist Restraint from Finite Element Analysis

LSB Sections with FTR/without FTR

Span (mm) Mod (kNm) Mu (kNm)

300x60x2.0 LSB 3000 1.11 1.08

200x45x1.6 LSB 4000 1.10 1.09 2000 1.08 1.06

Note: FTR – Flange Twist Restraint

Figure 4.20: Typical Elastic Buckling Failure Mode from Finite Element

Analysis

Table 4.8 shows the ratios of Mod and Mu for some test beams with and without

flange twist restraint (FTR) from preliminary finite element analysis while Figure

4.20 shows the typical elastic buckling failure modes of a test beam with and without

(a) with FTR (a) without FTR


4-29

flange twist restraint. The Mod values are on average about 10% higher for the beams

with flange twist restraint while the Mu values are on average about 8% higher.

However the non-dimensional slenderness λd (= (My/Mod)1/2) is reduced by only 5%

by considering the beams with flange twist restraint.

A capacity reduction factor (Φ) was calculated based on the test results in Tables 4.6

and 4.7 using the recommended AISI procedure (AISI, 2007). For an overall mean

and COV values of 1.05 and 0.11 based on 40 tests in Table 4.6 and 4.7 a capacity

reduction factor of. 0.91 was determined. This is greater than the recommended

capacity reduction factor of 0.90 in AS/NZS 4600 (SA, 2005). Hence, it confirms

that the current AS/NZS 4600 (SA, 2005) deign rule is conservative.

This study has also shown that despite improved manufacturing process and the use

of higher strength steel the same design curve can be used conservatively. It should

be noted that the current design method provided in AS/NZS 4600 (SA, 2005) was

developed by Mahaarachchi and Mahendran (2005d) based on the lower bound of

FEA and experimental results. Therefore it may be possible to improve the current

design curve. However, it can not be achieved by using only the test results as there

can be several shortcomings and limitations. The test results will be now used to

validate a finite element model of LSBs, which will be followed by a detailed

parametric study. Design curve will then be modified based on the finite element

analytical and experimental results.

4.6 Conclusions

This chapter has described the lateral buckling tests carried out to investigate the

lateral distortional buckling behaviour and member moment capacities of LSB

sections. A total of 12 tests were carried out with the beam span ranging from 1200

mm to 4000 mm, which included compact, non-compact and slender LSB sections.

The quarter point loading method was used and all the test beams failed in lateral

distortional buckling except 150x45x1.6 LSB with 1200 mm which exhibited a shear

buckling failure between the support and the loading positions. The test moment

capacity results from this research and Mahaarachchi and Mahendran (2005a) were


4-30

compared with the predictions from the current design rules in AS/NZS 4600 (SA,

2005). It was found that the test moment capacity results from this research were on

average about 13% higher than the AS/NZS 4600 (SA, 2005) predictions while those

of Mahaarachchi and Mahendran (2005a) were on average about 3% higher than the

predictions from AS/NZS 4600 (SA, 2005). The use of accurate Mod values for some

test beams without flange twist restraint would have given higher ratios of test

capacity to AS/NZS 4600 prediction. Therefore it is considered that the current

AS/NZS 4600 (SA, 2005) design rule for lateral distortional buckling is considered

to be conservative for LSBs. Further research using finite element analyses is likely

to develop improved design rules for LSBs.

Finite Element Modelling of LSBs

5-1

CHAPTER 5

5.0 FINITE ELEMENT MODELLING OF LSBs SUBJECT TO LATERAL

BUCKLING EFFECTS

5.1 Introduction

Finite Element Analysis (FEA) plays an important role in any research as it is

relatively inexpensive and time efficient compared with physical experiments. It is

particularly useful when a detailed parametric study into the effects of section

geometry is involved. Lateral distortional buckling behaviour of hollow flange steel

beams, particularly LSB sections, the effects of their geometry on lateral distortional

buckling capacity and the section moment capacity of LSBs were investigated using

finite element analyses in this research. Mahaarachchi and Mahendran (2005c)

developed the first finite element model of LSBs. The accuracy of this model was

improved by Kurniawan (2007) and Parsons (2007a) in relation to the boundary

conditions and dimensions considered by Mahaarachchi and Mahendran (2005c).

Kurniawan (2007) developed two modified versions of LSB finite element model to

investigate the moment distribution and load height effects on the moment capacities

of LSBs. The final version developed by Kurniawan (2007) was found to be the most

appropriate for use in this research on the lateral distortional buckling moment

capacities of LSBs. This chapter presents a detailed description of the finite element

model used in this research, which is capable of simulating the significant

behavioural effects of material inelasticity, buckling deformations including web

distortion, member instability, residual stresses and geometric imperfections. The

results from both elastic buckling and non-linear static analyses of LSBs are

compared with the finite strip analysis and experimental results, respectively. Details

of those comparisons are also presented in this chapter.

5.2 Model Description

Two types of finite element models were considered in this research, namely the

ideal finite element models and the experimental finite element models. The purpose

and the description of these models are given next.


5-2

• The ideal finite element models (Figure 5.1a) – These models incorporated

ideal constraints such as idealised simply supported boundary conditions and

a uniform bending moment throughout the span, nominal dimensions, yield

stresses, geometric imperfections and residual stresses. These idealised

conditions usually simulate the worst case, and hence they are commonly

adopted in the development of design curves as well as in the parametric

studies into the effects of section geometry of LSB on its member moment

capacity.

• The experimental finite element models (Figure 5.1b) – These models were

developed with the objective of simulating the actual test members’ physical

geometry, loads, constraints, mechanical properties, geometric imperfections

and residual stresses as closely as possible. They were used for the

comparison with experimental test results of LSBs subjected to quarter point

loads reported in Chapter 4. This comparison was intended to establish the

validity of the finite element model for explicit modelling of initial

geometric imperfections, residual stresses, lateral distortional buckling

deformations, and associated material yielding in non-linear static analyses.

Although this does not directly verify the suitability of the ideal finite

element model for its use in the development of design curves, this approach

is reasonably acceptable as the ideal conditions are simply a theoretical

assumption and are difficult to simulate in the real experiments.

Figure 5.1: Schematic Diagrams of Ideal and Experimental FE Models

Span/2

P

Span/4

(b) Experimental Model

Span/2

M

(a) Ideal Model Symmetric Plane


5-3

Both ideal and experimental finite element models did not consider the corners

although the actual LSB sections have corners as shown in Table 5.1. All the LSB

sections have an inner radius of 3 mm at the web-flange junction while the outer

radius equals to two times the thickness. However, including the corners in finite

element modelling would be cumbersome in relation to the application of mechanical

properties, geometric imperfections and residual stresses. Table 5.1 presents the

nominal dimensions and the yield stresses of both flange and web elements of LSBs.

Table 5.1: Nominal Properties of Available LSB Sections

LSB Section d d1 bf df t ro riw

fy Flange Web

mm mm mm mm mm mm mm MPa MPa

300x75x3.0 LSB 300 244 75 25 3.0 6.0 3.0 450 380 300x75x2.5 LSB 300 244 75 25 2.5 5.0 3.0 450 380 300x60x2.0 LSB 300 254 60 20 2.0 4.0 3.0 450 380 250x75x3.0 LSB 250 194 75 25 3.0 6.0 3.0 450 380 250x75x2.5 LSB 250 194 75 25 2.5 5.0 3.0 450 380 250x60x2.0 LSB 250 204 60 20 2.0 4.0 3.0 450 380 200x60x2.5 LSB 200 154 60 20 2.5 5.0 3.0 450 380 200x60x2.0 LSB 200 154 60 20 2.0 4.0 3.0 450 380 200x45x1.6 LSB 200 164 45 15 1.6 3.2 3.0 450 380 150x45x2.0 LSB 150 114 45 15 2.0 4.0 3.0 450 380 150x45x1.6 LSB 150 114 45 15 1.6 3.2 3.0 450 380 125x45x2.0 LSB 125 89 45 15 2.0 4.0 3.0 450 380 125x45x1.6 LSB 125 89 45 15 1.6 3.2 3.0 450 380

Note: d–depth, d1–clear web depth, bf – flange width, df – flange depth, t – thickness, ro – outer corner radius, riw – inner corner radius, fy – yield stress

In order to determine the level of approximations involved in using LSB sections

without corners in finite element analysis the effects of corners of LSB sections on

their elastic lateral distortional buckling moments and cross-sectional properties were

evaluated. Figures 5.2 (a) and (b) show the actual LSB with corners and the idealised

LSB without corners, respectively.


5-4

Figure 5.2: Actual and Idealised LSBs

Table 5.2: Elastic Section Modulus of Actual and Idealised LSBs

LSB Section Elastic Section Modulus

Z (x103 mm3) % difference (Idealised-Actual)/Actual

Actual Idealised 300x75x3.0 LSB 166.8 171.7 2.94% 300x75x2.5 LSB 140.6 144.0 2.42% 300x60x2.0 LSB 98.2 100.4 2.25% 250x75x3.0 LSB 129.5 133.5 3.09% 250x75x2.5 LSB 109.2 112.0 2.56% 250x60x2.0 LSB 76.2 78.0 2.35% 200x60x2.5 LSB 68.9 71.1 3.18% 200x60x2.0 LSB 55.9 57.3 2.50% 200x45x1.6 LSB 37.4 38.3 2.49% 150x45x2.0 LSB 30.8 31.9 3.40% 150x45x1.6 LSB 25.1 25.7 2.67% 125x45x2.0 LSB 23.9 24.8 3.60% 125x45x1.6 LSB 19.5 20.0 2.78%

Average 2.79%

Elastic section modulii of actual and idealised LSB sections which were calculated

using a well established finite strip analysis program, Thin-Wall, and the results are

compared in Table 5.2. It is seen that the idealised LSBs over-estimated the elastic

d1

ri

ro

ro ro

robf

df

d

df

t θ

(a) Actual LSB (b) Idealised


5-5

section modulus by 2.79% on average. Elastic lateral buckling moments were also

compared for both actual and idealised LSB sections using Thin-Wall. Table 5.3

presents the elastic lateral buckling moments of these LSB sections.

Table 5.3: Elastic Lateral Buckling Moments of Actual and Idealised LSB

Sections


1500 2000 3000 4000 Actual Ideal Actual Ideal Actual Ideal Actual Ideal

300x75x3.0 LSB 138.5 145.9 94.5 98.7 64.3 66.4 51.6 53.1 300x75x2.5 LSB 114.8 119.9 76.1 79.0 50.7 52.1 41.1 42.1 300x60x2.0 LSB 51.2 53.4 34.0 35.3 22.6 23.2 18.2 18.6 250x75x3.0 LSB 119.7 125.6 87.1 90.6 63.0 65.0 51.2 52.6 250x75x2.5 LSB 97.5 101.6 69.0 71.4 49.7 51.0 41.1 42.0 250x60x2.0 LSB 43.7 45.5 30.9 32.0 22.1 22.6 18.1 18.4 200x60x2.5 LSB 50.0 52.1 38.9 40.3 28.8 29.7 23.1 23.7 200x60x2.0 LSB 37.8 39.2 29.0 29.9 22.0 22.5 18.1 18.4 200x45x1.6 LSB 13.4 13.8 10.9 11.2 8.2 8.4 6.6 6.7 150x45x2.0 LSB 17.8 18.5 14.3 14.8 10.4 10.7 8.1 8.3 150x45x1.6 LSB 13.4 13.8 10.9 11.2 8.2 8.4 6.6 6.7 125x45x2.0 LSB 17.5 18.1 14.2 14.6 10.2 10.5 8.0 8.2 125x45x1.6 LSB 13.2 13.5 10.9 11.2 8.2 8.4 6.5 6.6

LSB Section 5000 6000 8000 10000 Actual Ideal Actual Ideal Actual Ideal Actual Ideal

300x75x3.0 LSB 43.6 44.7 37.7 38.7 29.6 30.3 24.2 24.8 300x75x2.5 LSB 35.3 36.0 30.9 31.5 24.7 25.2 20.5 20.8 300x60x2.0 LSB 15.5 15.8 13.5 13.7 10.7 10.9 8.8 8.9 250x75x3.0 LSB 43.2 44.3 37.3 38.2 29.1 29.7 23.7 24.2 250x75x2.5 LSB 35.3 36.0 30.8 31.4 24.4 24.9 20.1 20.5 250x60x2.0 LSB 15.4 15.7 13.4 13.6 10.5 10.7 8.6 8.8 200x60x2.5 LSB 19.2 19.7 16.4 16.8 12.6 12.9 10.2 10.4 200x60x2.0 LSB 15.3 15.6 13.2 13.4 10.3 10.5 8.4 8.6 200x45x1.6 LSB 5.6 5.7 4.6 4.7 3.5 3.6 2.9 2.9 150x45x2.0 LSB 6.6 6.8 5.6 5.7 4.3 4.4 3.4 3.5 150x45x1.6 LSB 5.4 5.5 4.6 4.7 3.5 3.6 2.9 2.9 125x45x2.0 LSB 6.5 6.7 5.5 5.6 4.1 4.2 3.3 3.4 125x45x1.6 LSB 5.3 5.4 4.5 4.6 3.5 3.5 2.8 2.8

Percentage differences of the elastic lateral buckling moments of actual and idealised

LSB sections are presented in Table 5.4. The use of idealised LSB section led to an


5-6

over-estimation of the elastic lateral buckling moments by 2.6% as shown in this

table. This percentage increase is similar to that obtained for elastic section modulii

of actual and idealised LSBs.

Table 5.4: Percentage Differences in Elastic Lateral Buckling Moments of

Idealised and Actual LSBs

LSB Section % (Idealised-Actual)/Actual

Average Span (mm) 1500 2000 3000 4000 5000 6000 8000 10000

300x75x3.0 LSB 5.3% 4.5% 3.4% 2.8% 2.6% 2.4% 2.3% 2.2% 3.2% 300x75x2.5 LSB 4.4% 3.9% 2.9% 2.4% 2.1% 1.9% 1.8% 1.7% 2.6% 300x60x2.0 LSB 4.4% 3.8% 2.8% 2.3% 2.0% 1.9% 1.8% 1.7% 2.6% 250x75x3.0 LSB 4.9% 4.0% 3.0% 2.7% 2.5% 2.4% 2.2% 2.2% 3.0% 250x75x2.5 LSB 4.2% 3.5% 2.6% 2.2% 2.0% 1.8% 1.8% 1.6% 2.5% 250x60x2.0 LSB 4.1% 3.4% 2.5% 2.1% 2.0% 1.8% 1.6% 1.6% 2.4% 200x60x2.5 LSB 4.4% 3.5% 2.9% 2.6% 2.5% 2.4% 2.3% 2.3% 2.9% 200x60x2.0 LSB 3.6% 2.9% 2.2% 1.9% 1.8% 1.8% 1.7% 1.7% 2.2% 200x45x1.6 LSB 3.1% 2.6% 2.1% 2.0% 2.0% 2.0% 2.0% 1.7% 2.2% 150x45x2.0 LSB 3.8% 3.3% 2.8% 2.6% 2.7% 2.7% 2.6% 2.6% 2.9% 150x45x1.6 LSB 3.1% 2.6% 2.1% 2.0% 1.8% 2.0% 1.7% 1.7% 2.1% 125x45x2.0 LSB 3.4% 3.1% 2.8% 2.8% 2.8% 2.6% 2.7% 2.4% 2.8% 125x45x1.6 LSB 2.8% 2.4% 2.1% 2.0% 1.9% 1.8% 1.7% 1.8% 2.1%

Average 4.0% 3.3% 2.6% 2.3% 2.2% 2.1% 2.0% 2.0% 2.6% Based on the results reported in Tables 5.2 and 5.4 on the effects of corners on the

elastic section modulus and the elastic lateral buckling moments of LSBs, it can be

concluded that the effect of corners is small and that it is adequate to use the

idealised LSB section in finite element modelling. It was found that the effect of

corners in the other sectional properties of LSBs such as torsional constant (J) and

the second moment of area (I) were similar to that of elastic section modulus (Z) and

hence the details are not provided. Most importantly, the moment capacity results

from the finite element analyses of idealised LSB sections will be non-

dimensionalised before they are used in the development of design rules and/or in

drawing important conclusions. This implies clearly that such small differences in

section properties and buckling moment capacities with the use of idealised LSB

sections without corners will not influence the final design rules or recommendations

of this research.


5-7

Both ideal and experimental finite element models were developed using MD

PATRAN (MSC Software, 2008) pre-processing facilities while they were analysed

using ABAQUS (HKS, 2007). MD/PATRAN (MSC Software, 2008) post-processing

facilities were then used to view the results from the ABAQUS analyses.

5.2.1 Discretization of the Finite Element Mesh

Shell elements are generally used to model thin-walled structures. ABAQUS (HKS,

2007) includes general purpose shell elements as well as elements that are

specifically formulated to analyse ‘thick’ and ‘thin’ shell problems. The general

purpose shell elements provide robust and accurate solutions in most applications

and have the capability of providing sufficient degrees of freedom. Therefore, local

buckling deformations and spread of plasticity could be explicitly modelled. The

shell element in ABAQUS (HKS, 2007) called S4R5 was used to develop the LSB

model. This element is thin, shear flexible, isometric quadrilateral shell with four

nodes and five degrees of freedom per node, utilizing reduced integration and

bilinear interpolation scheme.

One of the most important aspects of finite element modelling is to identify a suitable

mesh size for the accurate modelling of structural response. Finer meshes are

generally preferred to obtain accurate predictions although there is no general

guidelines on the required mesh density, which depends on the type of structure and

analysis involved (Ashraf et al., 2006). But finer meshes make the whole process

more expensive in terms of computational time. A compromise is therefore required

between the level of accuracy and the cost of a solution. Convergence study by

Mahaarachchi and Mahendran (2005c) indicated that a minimum mesh size density

comprising of 5 mm × 10 mm elements was required to represent accurate residual

stress distributions, spread of plasticity, and local buckling deformations of LSB

sections. Element widths equal to or less than 5 mm and a length of 10 mm were

selected as the suitable mesh size for the entire cross-section. Nine integration points

through the thickness of the elements were used to model the distribution of flexural

residual stresses in the LSB sections and the spread of plasticity through the

thickness of the shell elements. Kurniawan (2007) also used the same mesh size and


5-8

the number of integration points in his models of LSBs in order to investigate the

moment distribution and load height effects on the moment capacity of LSBs.

Figure 5.3: Typical Finite Element Mesh for LSB Models

Figure 5.3 shows the typical finite element mesh of the LSB used in this research.

The accuracy of the model and appropriateness of the finite element mesh density

and number of integration points was justified by the results of the verification

analyses presented in Section 5.3.3.

5.2.2 Material Model and Properties

The ABAQUS classical metal plasticity model was used in the analysis. This model

implements the von Mises yield surface to define isotropic yielding, and associated

plastic flow theory. This assumption is generally acceptable for most calculations

with metals. The ideal models included the nominal web and flange yield stresses of

380 and 450 MPa, respectively. These yield stresses are the minimum specified

values for the range of LSB sections (Dempsey, 2001). The yield stresses of web,

outside flange and inside flange were also measured using tensile coupon tests (see

Table 3.1 in Chapter 3 for details), and these measured yield stresses were used in the

experimental models of LSBs.

A perfect plasticity model based on simplified bilinear stress-strain curve with no

strain hardening was used for all the models. Isotropic hardening model that allows

strain hardening behaviour where yield stresses increase as plastic strain occurs was

Hollow Flanges

Web


5-9

not considered. This may be important when modelling sections subjected to

localised yielding involving strain hardening effects. However, since all the beams

modelled here were mainly subjected to lateral buckling effects, a simple elastic

perfect plastic model was assumed to be sufficient. Figure 5.4 shows the elastic

plastic material model used in the finite element model and the actual stress-strain

relationship of the steel. The elastic modulus E and Poisson’s ratio ν were taken as

200 000 MPa and 0.3, respectively, for both the ideal and experimental finite element

models.

Figure 5.4: Stress-Strain Relationships

5.2.3 Load and Boundary Conditions

The application of loads and boundary conditions of ideal and experimental finite

element models have similarities and contrasts. For simplicity, the boundary

conditions of these two models are described separately in the following sections.

5.2.3.1 Ideal Finite Element Model

“Idealised” simply supported boundary conditions with a uniform bending moment

throughout the span were considered to be the critical case for the development of

design curves. Therefore, the ideal finite element model of LSBs considered the

“Idealised” simply supported boundary conditions based on the following

requirements (Trahair, 1993, Zhao et al., 1995, Mahaarachchi and Mahendran,

2005a).

1. Simply supported in-plane - Both ends fixed against in-plane vertical deflection

but unrestrained against in-plane rotation, and one end fixed against

longitudinal horizontal displacement.


5-10

2. Simply supported out-of-plane - Both ends fixed against out-of-plane

horizontal deflection, and twist rotation, but unrestrained against minor axis

rotation and warping displacements of flanges.


T1 T2 T3 R1 R2 R3

One end Yes No No No Yes Yes

Other end No No No No Yes Yes

Mid span No Yes Yes Yes No No

Table 5.5 shows the boundary conditions of ideal simply supported beams. The

presence of symmetry allowed the use of only half the span, which would reduce the

analysis time. In Table 5.5, T and R represent the translation and rotation,

respectively and the subscripts (1, 2, and 3) represent the direction while field “Yes”

means that it is free to move in that direction. Figure 5.5 illustrates the global axes

selected to input the boundary conditions for the analysis.


Figure 5.6 shows the boundary conditions used in the ideal finite element model

considered in this research. The pin support at one end was modelled by using a

Single Point Constraint (SPC) of “234” applied to all the nodes at the end.

Symmetrical boundary condition of SPC “156” was applied to the mid-span of LSB

as only the half span was modelled due to the presence of symmetry conditions.

X, 1

Z, 3

Y, 2 Z, 3

X, 1

Y, 2

M

L/2


5-11

Figure 5.6: Boundary Conditions of the Ideal Finite Element Model of LSB

To simulate a uniform end moment across the section, linear forces were applied at

every node of the beam end, where the nodes above the middle of the web were

subject to tensile forces while the nodes below the middle of the web were subject to

compressive forces. The force at the middle of the web was zero and was linearly

increased within the cross section as shown in Figure 5.7. A tensile force of 1000 N

and a compressive force of 1000 N were applied at the nodes on the top and bottom

faces of LSB cross section. Figure 5.7 shows the applied loads on each node for

200x45x1.6 LSB section. The simulated moment due to the applied loads at each

node can be calculated by multiplying the load at each node by the distance of the

corresponding node to the middle of the web. The total moment is the arithmetic sum

of the above individual moments. Sample calculations can be found in Appendix C.1.

The loading and boundary conditions used in their ideal finite element model used in

SPC “234”

Symmetric Plane SPC “156”


5-12

this research are similar to that used by Kurniawan (2007) in his research on the

moment distribution effects of LSBs.

Figure 5.7: Typical Loading Method for the Ideal Finite Element Model of LSB

5.2.3.2 Experimental Finite Element Model

In the experimental study (Chapter 4), a quarter point loading was applied at the

shear centre with the “Idealised” simply supported boundary conditions as mentioned

in the above section. This was carefully simulated in the experimental finite element

(a) Front View (b) Isometric View

(c) Close-up View


5-13

model by applying the loads and accurate boundary conditions at the shear centre. A

single point constraint (SPC) and concentrated nodal forces were used in this model

to simulate the experimental boundary conditions and applied loads as closely as

possible. The presence of symmetry permitted modelling of only half the span.

Experimental specimens included a 75 mm width rigid plate at each support, which

was connected to the web of the LSB specimen by using four bolts to prevent

distortion and twisting of the cross-section. These stiffening plates were simulated

using thick shell elements and the web mechanical properties. Shell elements of 10

mm thickness were considered to be appropriate to predict the experimental failure

moments and load-deflection curves for most of the test beams. However, the use of

10 mm thick shell elements simulated the action of support plates for 150x45x2.0

LSB and 125x45x2.0 LSB resulted in highly rigid model, which over-predicted the

moment capacities. Hence 5 mm thick shell elements were used as support plates in

the experimental finite element models of those LSBs. Although past research

(Mahaarachchi and Mahendran, 2005c) considered Rigid Body R3D4 elements to

simulate the support plates, the preliminary finite element analysis showed that these

rigid body elements gave increased elastic buckling moments and ultimate failure

moments. Hence the R3D4 elements were not used in the current experimental finite

element models. Simply supported boundary conditions were applied to the node at

the shear centre in order to provide an ideal pinned support, which was connected to

the support plates using four rigid beam MPCs to simulate the bolt connections as

used in the experiments. In the experimental test set-up, a concentrated load was

applied at the shear centre at the quarter point of the span using a steel plate. The

steel plate was connected to the beam web by using three bolts along the beam

centreline. Same loading arrangement was implemented in the experimental finite

element model using a concentrated nodal load applied at the cross-section shear

centre while simulating the bolts using rigid beam MPCs as shown in Figure 5.8.

Thicker shell elements (10 mm) with the mechanical properties of web elements

were used to represent the loading plate.


5-14

Figure 5.8: Loading and Boundary Conditions of the Experimental Finite

Element Model of LSB

In the preliminary finite element analysis of experimental model the loading plate

twisted significantly and the LSB members deformed in the negative direction

although all the tested LSBs failed in the positive direction and the load plate

twisting was not possible. Therefore, a twist restraint of SPC “4” was provided at the

loading point together with the point load at the shear centre.

(a) Isometric View

Support at Shear Centre, (SPC 234)

Symmetric Plane


Support at Shear Centre

(b) Plan View

Loading Plate Twist Restraint (SPC 4)

Rigid Beam MPC


5-15

Figure 5.9: Loading Plate Twisting in the Experimental FE Model

Figure 5.9 shows the typical loading plate twisting observed in the experimental

finite element models of LSBs at ultimate failure. This is not possible and was

eliminated by restraining the degree of freedom “4” (rotation about the longitudinal

axis).

Steel stiffener plates of 6 mm thickness were welded to the inner surfaces of the

flanges at each support as flange twist restraints in the experimental testing except

the first four tests. These plates were modelled in the experimental finite element

model using the elastic perfect plastic material model and a yield stress of 300 MPa.

Welding process was not simulated as this was a “tack” weld and the effects are

negligible. Also the maximum bending moment occurred at mid-span while the

moment at the support was zero. Therefore a small change in the residual stress and

imperfection due to welding (if any) would not create any adverse effects on the

ultimate failure of the beam. Figure 5.10 identifies the various plate elements with

different mechanical material properties as defined in ABAQUS. Measured

dimensions and yield stresses were used for both the web and flange elements of

LSBs.

It should be noted that the first four tests did not include the stiffener plates

mentioned above and hence their experiment models did not include the flange twist

restraints.


5-16

Figure 5.10: Various Plate Elements in Experimental Finite Element Model

5.2.4 Initial Geometric Imperfections

The magnitude and direction of geometric imperfection are some of the important

parameters which reduce the buckling moment capacity of a beam. This should be

considered in finite element modelling as real beams are not perfectly straight. Based

on the geometric imperfection measurements reported in Mahaarachchi and

Mahendran (2005e) and Chapter 3, and the fabrication LSB tolerance limit, a value

of L/1000 was considered conservatively as the overall geometric imperfection in

both the ideal and experimental finite element models of LSBs. A value of depth or

width/150 was considered as the local plate imperfection. However, local plate

imperfection was not considered in the finite element models subject to lateral

buckling as there was no local buckling. The critical imperfection shape was

introduced by ABAQUS “*IMPERFECTION” option with the lateral distortional

buckling eigenvector obtained from an elastic buckling analysis, and therefore

included lateral displacement, twist rotation, and cross section distortion. Detail

coding to input the imperfection is given as follows.

Outside Flange

Inside Flange

Loading Plate

Support Plate

Flange Twist Restraint

Web

*Imperfection, File = file name, Step = 1

Mode Number, Imperfection value


5-17

For example, if a file with the name 300x75x3.0LSB3.0 (3000 mm span) is analysed

for elastic buckling analysis and the results show that the critical buckling mode is

“Mode 2”, then the imperfection coding is as follows.

Where, the imperfection magnitude is calculated to be 3000/1000 = 3.

Figure 5.11: Critical Buckling Mode from Elastic Buckling Analysis of Ideal

Finite Element Model

Figure 5.11 shows the critical buckling mode from elastic buckling analysis.

Preliminary investigations revealed that the imperfection direction such as “positive”

and “negative” influenced the ultimate moment capacity of LSBs subject to lateral

buckling. Figures 5.12 (a) and (b) show the non-linear analysis results for positive

and negative imperfection shapes.

Negative imperfection was implemented by simply replacing “3.0” by “-3.0” in the

imperfection code. For the beams subject to lateral buckling, negative imperfection

was found to be critical and was used in all the non-linear static analyses of ideal

LSBs. Detailed results and analyses of the effects of both positive and negative

imperfections will be discussed in the next chapter (Chapter 6). However, positive

imperfection was considered in the experimental finite element models as all the

observed experimental failure modes were similar to those with positive

imperfections.

*Imperfection, File = 300x75x3.0LSB3.0, Step = 1

2, 3.0

Deformed LSB

Tension

Compression

Midspan


5-18

Figure 5.12: Effect of Imperfection Direction Based on Nonlinear Analysis

5.2.5 Residual Stresses

The residual stress is an important parameter in the flexural strength of steel beams

as this can cause premature yielding, and reduce their bending strength. Both flexural

and membrane residual stresses were used in both ideal and experimental finite

element models. Figure 5.13 shows the residual stress distribution used in the

numerical modelling which include the flexural residual stress distribution used for

all the LSB sections and the membrane residual stress distribution for 200x45x1.6

LSB. Table 5.6 presents the values of membrane residual stresses for the available 13

LSB sections. Further details of how this table was developed are presented in

Chapter 3.




5-19

Figure 5.13: Residual Stress Distributions in LSB Sections

The residual stresses distributions shown in Figure 5.13 and Table 5.6 were modelled

using the ABAQUS *INITIAL CONDITIONS option, with TYPE = STRESS,

USER. The user defined initial stresses were created using the SIGINI Fortran user

subroutine. A subroutine defining the residual stress distribution for a beam section

is provided in Appendix C.2. This subroutine defines the local components of the

initial stress as a function of the global coordinates. The flexural residual stress is

also a function of the integration point number through the thickness. As the global

coordinates were used to define the local stress components, it was necessary to

allow for member imperfections in the calculations. Equations with the member

length as a variable and constant deformation factors obtained from the buckling

analysis were used to represent the imperfection of top and bottom flanges

approximately.

0.03fy

-0.2567fy

0.11fy

-0.50fy

0.50fy

-0.50fy

-0.2567fy

0.11fy 0.03fy

0.03fy

0.03fy

(b) Membrane Residual Stress for 200x45x1.6LSB

0.24fy

0.24fy

0.24fy

1.07fy

0.41fy

0.8fy 0.38fy

0.38fy 0.8fy

0.41fy

1.07fy

0.24fy

0.24fy

(a) Flexural Residual Stress


5-20

Table 5.6: Membrane Residual Stress Distribution of LSB Sections

LSB

Centreline Dimensions (mm) Membrane Residual Stress as a Ratio of fy

d

d1

bf

df

t

Left Flange

Right Flange

Web Top

MidWeb

Inside Flange

Left

Inside Flange Right

300x75x3.0 297.0 247.0 72.0 22.0 3.0 -0.2591 0.03 -0.50 0.50 0.11 0.03 300x75x2.5 297.5 247.5 72.5 22.5 2.5 -0.2556 0.03 -0.50 0.50 0.11 0.03 300x60x2.0 298.0 258.0 58.0 18.0 2.0 -0.2556 0.03 -0.50 0.50 0.11 0.03 250x75x3.0 247.0 197.0 72.0 22.0 3.0 -0.2591 0.03 -0.50 0.50 0.11 0.03 250x75x2.5 247.5 197.5 72.5 22.5 2.5 -0.2556 0.03 -0.50 0.50 0.11 0.03 250x60x2.0 248.0 208.0 58.0 18.0 2.0 -0.2556 0.03 -0.50 0.50 0.11 0.03 200x60x2.5 197.5 157.5 57.5 17.5 2.5 -0.2600 0.03 -0.50 0.50 0.11 0.03 200x60x2.0 198.0 158.0 58.0 18.0 2.0 -0.2567 0.03 -0.50 0.50 0.11 0.03 200x45x1.6 198.4 168.4 43.4 13.4 1.6 -0.2567 0.03 -0.50 0.50 0.11 0.03 150x45x2.0 148.0 118.0 43.0 13.0 2.0 -0.2615 0.03 -0.50 0.50 0.11 0.03 150x45x1.6 148.4 118.4 43.4 13.4 1.6 -0.2567 0.03 -0.50 0.50 0.11 0.03 125x45x2.0 123.0 93.0 43.0 13.0 2.0 -0.2615 0.03 -0.50 0.50 0.11 0.03 125x45x1.6 123.4 93.4 43.4 13.4 1.6 -0.2567 0.03 -0.50 0.50 0.11 0.03

Figure 5.14: Typical Residual Stresses Distribution for LSB Sections

In both the ideal and experimental finite element models of LSBs, the initial stresses

were applied in a *STATIC step with no loading and the standard model boundary

conditions to allow equilibration of the initial stress field before starting the response

history. The contours of residual stress after equilibration in a typical ideal finite

element model are shown in Figure 5.14. However, the equilibration of the initial

stress may require additional deformation to bring the model into equilibrium due to


5-21

the unbalanced stress. Past research (Yuan, 2004) in the finite element modelling of

hot-rolled I sections considered an additional “force field” in the *STATIC step to

reverse this extra initial deformations. This force field was the reaction forces

obtained from a preliminary finite element analysis with all the nodes fixed in the x,

y and z translation degrees of freedom. But this technique was found inappropriate

because the force field remains in the subsequent non-linear analysis step which

provides further restraint to the section. Nevertheless, this initial deformation effect

was considered to be insignificant in the analysis, thus no attempt was made in this

research to eliminate this. In summary, the application of residual stress in this

research was similar to that used by Kurniawan (2007), which was successfully

implemented for LSBs subject to lateral buckling.

5.2.6 Analysis Methods

Two methods of analysis, elastic buckling and nonlinear static analyses, were used in

this study. Elastic buckling analyses were carried out first and were used to obtain

the eigenvectors for the inclusion of geometric imperfections. Nonlinear static

analysis, including the effects of large deformation and material yielding, was

adopted to investigate the behaviour of LiteSteel beam sections up to failure.

ABAQUS uses the Newton-Raphson method to solve the non-linear equilibrium

equations. The RIKS method in ABAQUS was also included in the nonlinear

analysis. It is generally used to predict geometrically unstable nonlinear collapse of

structures. The RIKS method uses the load magnitude as an additional unknown and

solves simultaneously for loads and displacements. Therefore, another quantity

should be used to measure the progress of the solution. For this purpose, ABAQUS

uses variable arc-length constraint to trace the instability problems associated with

nonlinear buckling of beams. The parameters used for non-linear static analyses are

as follows:

• Typical maximum number of load increments = 100 (may vary),

• Initial increment size = 0.0001,

• Minimum increment size = 0.0000001,

• Automatic increment reduction enabled, and large displacements enabled.


5-22

The finite element models of LSBs were developed in MD/PATRAN and submitted

to ABAQUS (Version 6.7-1) for analysis. Following is a summary of the steps

involved in the finite element modelling of LSBs used in this research.

1. Define geometric surfaces for the flanges and the web.

2. Mesh those surfaces using shell S4R5 elements.

3. Define the support and mid-span boundary conditions and the loads.

(including the MPC rigid beam in the experimental model to simulate the

bolts).

4. Define and assign the mechanical properties for web and flanges (support,

load and stiffener plates for experimental model).

5. Define buckling analysis parameters and run bifurcation buckling analysis

using ABAQUS.

6. Obtain the critical buckling eigenvector and the required maximum

deformation factors for member equations to be included in the residual

stress input subroutine.

7. Prepare the residual stress input subroutine.

8. Define the non-linear static analysis parameters.

9. Using ABAQUS, run a non-linear static analysis, which consists of two

“load steps”:

Equilibration STATIC step – with the standard boundary

conditions, initial geometric imperfection and residual stress input

subroutine.

Ultimate load factor step – with the applied moment or load.

5.3 Model Validation

It is necessary to verify the accuracy of the developed finite element models prior to

their use in the development of member capacity curves and design

recommendations. This was achieved by conducting two series of comparisons. The

first series, presented in Section 5.3.2, involved comparison of the elastic lateral

distortional buckling moments obtained using the ideal finite element model with the

corresponding moment solutions obtained from the established finite strip analysis

program, Thin-Wall and the predictions from Pi and Trahair (1997). The second

series of comparisons, presented in Section 5.3.3, involved the use of the


5-23

experimental test results of LSB members subject to lateral distortional buckling

conducted in this research. Non-linear static analysis of the experimental finite

element model was used for the second series of comparisons in order to simulate the

experimental conditions as closely as possible. Deformation and stress contours of

the finite element analysis results were observed to assist with model verification.

Different types of possible buckling modes of LSBs were also carefully studied using

the analyses based on ideal finite element models.

5.3.1 Typical Buckling Modes of Ideal Finite Element Model

A series of elastic buckling analyses was conducted using the ideal finite element

model of LSBs developed in this research. The results showed that the LSB exhibited

three distinct buckling modes, namely local buckling for short span members, lateral

distortional buckling (LDB) for intermediate span members and lateral torsional

buckling (LTB) for long span members. It was confirmed that the LSBs with

intermediate spans commonly ranging from 1500 mm to 6000 mm (smaller sections

exhibited LDB at 750 mm and the LTB started at about 5000 mm) are prone to

lateral distortional buckling. It was found that the level of web distortion in lateral

distortional buckling varied as a function of beam slenderness, where increasing

beam slenderness reduced the web distortion and thus approached lateral torsional

buckling.

Figure 5.15: Elastic Buckling Modes of 200x60x2.0 LSB

(a) Local Buckling (500 mm)


5-24

Figure 5.15: Elastic Buckling Modes of 200x60x2.0 LSB

Figures 5.15 (a) to (d) show these three buckling modes of 200x60x2.0 LSB section

obtained from the elastic buckling analysis of its ideal finite element model. Figures

5.16 (a) and (b) show the ultimate failure modes of 200x60x2.0 LSB section obtained

from the non-linear static analysis of its ideal finite element model. For 500 mm

span, yielding occurred before local buckling at ultimate failure and the ultimate

capacity is its section moment capacity. More details of the finite element model of

(b) Lateral Distortional Buckling (4000 mm)

(c) Lateral Torsional Buckling (8000 mm)

(d) Close-up View (LTB)

No web Distortion


5-25

LSBs subject to local buckling will be discussed later in Chapter 7. When the beams

are subject to positive imperfection the lateral distortional and lateral torsional

buckling failure modes at ultimate failure were similar to those exhibited in elastic

buckling analyses while a failure mode as shown in Figure 5.16 (b) was observed for

the beams with a negative imperfection.

Figure 5.16: Ultimate Failure Modes of 200x60x2.0 LSB

5.3.2 Comparison of Elastic Buckling Moment Results

Elastic buckling moment results from the ideal finite element model were compared

with the corresponding elastic buckling solutions obtained from Thin-Wall and Pi

and Trahair’s (1997) equation for elastic lateral distortional buckling moment

(Eqs.2.21 and 2.22 in Chapter 2). Pi and Trahair’s (1997) equation has been verified

and adopted in the design capacity tables for LSBs. The comparison was intended to

verify the accuracy of the finite element type, mesh density, boundary conditions and

(a) Yielding (500 mm)

(b) LDB with Negative Imperfection (4000 mm)


5-26

loading method used in the finite element model for LSBs. The elastic buckling

moments from FEA, Thin-Wall and Pi and Trahair’s (1997) equation and the

percentage differences from the comparisons are summarised in Table 5.7.

Table 5.7: Comparison of Elastic Buckling Moments of LSB from FEA, Thin-

Wall and Pi and Trahair’s (1997) Equation

Span (mm)

300x75x3.0 LSB 300x75x2.5 LSB


% Difference Compared with FEA



FEA PT TW PT TW FEA PT TW PT TW 1000 183.8* 317.4 185.2 - 0.75% 107.8* 267.3 108.9 - 0.98% 1500 144.6 155.0 145.9 7.2% 0.93% 107.5* 127.6 108.9 - 1.34% 2000 97.9 102.8 98.7 5.0% 0.88% 78.4 82.4 79.0 5.16% 0.79% 3000 65.7 67.8 66.4 3.3% 1.13% 51.6 53.4 52.1 3.38% 0.93% 4000 52.4 53.7 53.1 2.6% 1.39% 41.6 42.7 42.1 2.74% 1.18% 6000 38.0 38.8 38.7 2.0% 1.71% 31.1 31.7 31.5 2.20% 1.55% 8000 29.7 30.2 30.3 1.7% 1.88% 24.7 25.2 25.2 1.95% 1.78% 10000 24.3 24.7 24.8 1.6% 2.02% 20.4 20.8 20.8 1.81% 1.91%

300x60x2.0 LSB 250x75x3.0 LSB 1000 44.0* 117.7 44.4 - 0.75% 223.0* 255.0 225.2 - 1.01% 1500 43.9* 56.5 44.4 - 1.02% 124.4 132.2 125.6 6.31% 1.01% 2000 35.0 36.7 35.3 4.75% 0.74% 89.7 93.6 90.6 4.28% 1.03% 3000 23.0 23.8 23.2 3.46% 0.96% 64.1 65.9 65.0 2.79% 1.31% 4000 18.4 18.9 18.6 2.90% 1.20% 51.8 52.9 52.6 2.25% 1.56% 6000 13.5 13.8 13.7 2.33% 1.63% 37.5 38.1 38.2 1.82% 1.84% 8000 10.7 10.9 10.9 2.06% 1.88% 29.1 29.6 29.7 1.63% 1.96% 10000 8.8 8.9 8.9 1.87% 1.95% 23.8 24.1 24.2 1.54% 2.02%

250x75x2.5 LSB 250x60x2.0 LSB 1000 130.9* 213.4 132.5 - 1.22% 52.9* 94.5 53.3 - 0.70% 1500 100.6 107.1 101.6 6.45% 0.95% 45.1 47.7 45.5 5.85% 0.80% 2000 70.8 73.9 71.42 4.36% 0.89% 31.7 33.0 32.0 4.29% 0.82% 3000 50.4 51.8 50.97 2.82% 1.11% 22.4 23.1 22.6 3.14% 1.12% 4000 41.4 42.4 41.99 2.32% 1.35% 18.2 18.7 18.4 2.64% 1.37% 6000 30.9 31.5 31.39 1.94% 1.68% 13.4 13.7 13.6 2.14% 1.72% 8000 24.4 24.8 24.86 1.81% 1.89% 10.5 10.7 10.7 1.86% 1.81% 10000 20.1 20.4 20.46 1.70% 1.94% 8.6 8.8 8.8 1.75% 1.97% * Subject to local buckling, TW – Thin-Wall, PT – Pi and Trahair’s (1997) Eq.


5-27

Table 5.7 (Continued): Comparison of Elastic Buckling Moments of LSB from

FEA, Thin-Wall and Pi and Trahair’s (1997) Equation

Span (mm)





FEA PT TW PT TW FEA PT TW PT TW 200x60x2.5 LSB 200x60x2.0 LSB

1000 86.0 93.1 86.9 8.23% 1.10% 67.3* 73.5 67.7 - 0.56% 1500 51.6 54.1 52.1 4.79% 1.07% 38.8 40.8 39.2 4.93% 0.90% 2000 39.8 41.2 40.3 3.46% 1.23% 29.6 30.6 29.9 3.55% 0.98% 3000 29.2 29.9 29.7 2.44% 1.58% 22.2 22.7 22.5 2.58% 1.31% 4000 23.3 23.7 23.7 2.05% 1.81% 18.1 18.5 18.4 2.22% 1.54% 6000 16.4 16.7 16.8 1.65% 2.01% 13.2 13.4 13.4 1.92% 1.90% 8000 12.6 12.8 12.9 1.53% 2.06% 10.3 10.5 10.5 1.80% 2.04% 10000 10.2 10.3 10.4 1.51% 2.16% 8.4 8.5 8.5 1.75% 2.11% 200x45x1.6 LSB 150x45x2.0 LSB 1000 25.4* 27.6 25.3 - -0.16% 26.7 28.3 27.1 5.82% 1.16% 1500 15.0 15.7 15.1 4.69% 0.87% 18.2 18.9 18.5 3.66% 1.37% 2000 11.4 11.8 11.5 3.79% 1.14% 14.5 14.9 14.8 2.81% 1.65% 3000 8.3 8.6 8.5 2.91% 1.46% 10.5 10.7 10.7 2.04% 1.91% 4000 6.7 6.8 6.8 2.53% 1.78% 8.2 8.3 8.3 1.62% 2.01% 6000 4.7 4.8 4.8 2.11% 2.07% 5.6 5.7 5.7 1.33% 2.12% 8000 3.7 3.7 3.7 1.94% 2.22% 4.3 4.3 4.4 1.34% 2.28% 10000 3.0 3.0 3.0 1.80% 2.23% 3.4 3.5 3.5 1.21% 2.22% 150x45x1.6 LSB 125x45x2.0 LSB 1000 20.4 21.6 20.6 6.06% 1.03% 24.7 26.0 25.1 5.16% 1.38% 1500 13.6 14.1 13.8 3.95% 1.10% 17.8 18.4 18.1 3.26% 1.57% 2000 11.0 11.4 11.2 3.16% 1.36% 14.4 14.7 14.6 2.50% 1.81% 3000 8.2 8.4 8.4 2.48% 1.71% 10.3 10.5 10.5 1.90% 2.13% 4000 6.6 6.7 6.7 2.14% 1.91% 8.0 8.1 8.2 1.63% 2.22% 6000 4.6 4.7 4.7 1.94% 2.15% 5.5 5.5 5.6 1.36% 2.23% 8000 3.5 3.6 3.6 1.81% 2.22% 4.1 4.2 4.2 1.33% 2.32% 10000 2.8 2.9 2.9 1.88% 2.39% 3.3 3.4 3.4 1.21% 2.25%

125x45x1.6 LSB 1000 18.5 19.5 18.7 5.24% 1.13% 1500 13.4 13.8 13.5 3.35% 1.27% 2000 11.0 11.3 11.2 2.75% 1.54% 3000 8.2 8.4 8.4 2.20% 1.83% 4000 6.5 6.6 6.6 2.07% 2.10% 6000 4.5 4.6 4.6 1.88% 2.24% 8000 3.4 3.5 3.5 1.80% 2.30% 10000 2.8 2.8 2.8 1.59% 2.13% * Subject to local buckling, TW – Thin-Wall, PT – Pi and Trahair’s (1997) Eq.


5-28

The comparison of elastic buckling moments from the three methods is also shown in

Figure 5.17, where the elastic buckling moments are plotted against the span. The

results from only four LSBs were plotted for clarity. The comparison shows that

FEA results agree well with the results from both Thin-Wall and Pi and Trahair’s

(1997) equation, where the average difference is about 1.5% and 2.9%, respectively.

While Pi and Trahair’s (1997) equation gives an approximate solution, the small

difference with Thin-Wall may be due to a very fine mesh density used in the finite

element model. Most of the bigger LSB sections exhibited local buckling at 1000

mm span as shown in Table 5.7. The local buckling moments from FEA agreed very

well with Thin-Wall results, where the percentage difference is less than 1% on

average. Pi and Trahair’s (1997) equation only provides solutions for lateral

distortional buckling, thus its short span results can not be compared with the FEA

results.

Figure 5.17: Comparison of Elastic Buckling Moments

Based on Table 5.7 and Figure 5.17, it can be confirmed that the ideal finite element

model developed in this research accurately predicts the elastic lateral distortional

buckling moments of all the LSB sections for a range of member slenderness. Since

the FEA results under-estimated the elastic buckling moments by 1.5% and 2.9% on

average when compared with predictions from Thin-Wall and Pi and Trahair’s

(1997) equation, respectively, this FEA model can be conservatively used in this

0

20

40

60

80

100

120

140

160

180

200

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Span (mm)

Elas

tic B

uckl

ing

Mom

ent,

(kN

m)

300x75x3.0 LSB - FEA

300x75x3.0 LSB - Pi and Trahair's Eq.

300x75x3.0 LSB - Thin-Wall











5-29

research for the development of design curves and parametric studies. Further, these

comparisons verify the suitability and accuracy of the element type, mesh density,

geometry, boundary conditions and the method used to generate the required uniform

bending moment distribution.

5.3.3 Comparison with Experimental Test Results

It is important to validate the finite element model for non-linear analyses prior to

using it to develop the member capacity curves for LSB sections subject to a uniform

bending moment. This was achieved by comparing the non-linear static analysis

results with the results obtained from experimental tests carried out in this research.

This comparison was intended to establish the validity of the shell element model for

explicit modelling of initial geometric imperfections and residual stresses, lateral

distortional and local buckling deformations, and the associated material yielding.

The accuracy of the residual stress models, local imperfection magnitudes, and the

finite element mesh density will also be established.

Table 5.8 compares the ultimate moment capacity results from experiments with the

non-linear static analyses using the experimental finite element model. A comparison

of FEA and experimental test results is also provided in the form of bending moment

versus vertical deflection curves in Figures 5.18 to 20 for different LSB sections.

These figures compare the measured experimental in-plane deflection to the

corresponding deflections predicted by the finite element analyses with residual

stresses and geometrical imperfections. The vertical deflection was taken at the

centre of the web at mid-span. As seen in these figures, bending moment versus

vertical deflection curves of finite element analyses agreed reasonably well with the

experimental curves. The reason for the observed small difference may be due to the

use of thicker shell elements for load and support plates in the finite element

analyses. Also, the vertical deflections from the experiments were not exactly

measured vertically due to the lateral deflection of beams during the experiments.

It should be noted that the first four tests did not consider the flange twist restraints

(FTR) while the corresponding experimental finite element models also did not


5-30

consider them. Since the test of 150x45x1.6 LSB with 1200 mm span failed by shear

buckling it was not modelled numerically for comparison purposes.

Table 5.8: Comparison of Experimental and FEA Ultimate Moment Capacities

Test No LSB Section Span

(mm)

Ultimate Moment Capacity (kNm) EXP/FEA

Experiment FEA 1 250x75x2.5 LSB* 3500 34.13 36.90 0.92 2 300x60x2.0 LSB* 4000 17.17 17.80 0.96 3 200x45x1.6 LSB* 4000 5.92 6.23 0.95 4 300x60x2.0 LSB* 3000 18.09 18.40 0.98 5 200x45x1.6 LSB 3000 9.24 8.92 1.04 6 150x45x1.6 LSB 3000 8.27 8.28 1.00 7 150x45x2.0 LSB 3000 9.87 10.50 0.94 8 200x45x1.6 LSB 2000 10.72 10.10 1.06 9 150x45x2.0 LSB 2000 10.76 11.20 0.96 10 150x45x1.6 LSB 1800 9.30 9.00 1.03 11 125x45x2.0 LSB 1200 10.83 10.40 1.04 Mean 0.99 COV 0.047

*Tests without Flange Twist Restraints.


150x45x1.6 LSB (3000 mm Span)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0 10 20 30 40 50 60 70


Mom

ent (

kNm

) EXP

FEA


5-31


200x45x1.6 LSB (4000 mm Span)


300x60x2.0 LSB (4000 mm Span)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 10 20 30 40 50 60 70


Mom

ent (

kNm

)

EXPFEA

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

0 10 20 30 40 50 60 70


Mom

ent (

kNm

)

EXPFEA


5-32


150x45x1.6 LSB (1800 mm Span)


200x45x1.6 LSB (4000 mm Span)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

-10 0 10 20 30 40 50 60 70 80 90

Lateral Deflection (mm)

Mom

ent (

kNm

)

EXP - Top Flange (Tension)

EXP - Bottom Flange (Compression)

FEA - Top Flange

FEA - Bottom Flange

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

-10 10 30 50 70 90 110 130 150


Mom

ent (

kNm

) EXP - Top Flange (Tension)


FEA - Top Flange

FEA - Bottom Flange


5-33


150x45x2.0 LSB (3000 mm Span)

Typical bending moment vs. lateral deflection curves are provided in Figures 5.21 to

23. These figures compare the measured experimental out-of-plane deflection at mid-

span for both the top and bottom flanges with the corresponding deflections

predicted by the finite element analyses. Appendix C.3 presents the remaining

moment versus deflection graphs for other LSB sections and spans. As seen in these

figures, the bending moment versus lateral deflection curves from finite element

analyses deviate slightly from the experimental curves. This could be due to the use

of twist restraint (SPC 4) at the loading point in finite element analyses while the

experimental lateral deflections could not have been measured exactly horizontal due

to the vertical deflection of test beams. The hydraulic jacks could also have imposed

a lateral restraint to the beam sections during testing. Even though the loading system

was designed to avoid any lateral restraints, there could have been friction in the

bearings. This was not measured and no attempts were made to include the friction

effects. However, it is considered that such lateral restraints had minimal effect on

the buckling moments of the tested LSB members.

0.0

2.0

4.0

6.0

8.0

10.0

12.0

-10 10 30 50 70 90 110 130 150

Lateral Deflection mm)

Mom

ent (

kNm

)



FEA - Top Flange

FEA - Bottom Flange


5-34

The comparisons provided in Table 5.8 and Figures 5.18 to 23 demonstrate that the

experimental finite element model predicts the ultimate failure moment accurately.

The mean ratio of the ultimate moment capacities from the finite element analyses

and experiments was 0.99 with a COV of 0.047. This result suggests that the

developed finite element model is accurate, considering the possible approximations

in the finite element models and limitations in the experimental measurements as

described above.

The finite element model provided reasonable comparisons with all the experimental

results. It is therefore reasonable to assume that the experimental comparisons

presented in this section establish the validity of the shell element model for explicit

modelling of initial geometric imperfections and residual stresses, lateral distortional

buckling deformations, and the associated material yielding. The suitability of the

residual stress model, geometric imperfection magnitudes, and the finite element

mesh density has also been verified.

5.4 Conclusions

This section has described the details of ideal and experimental finite element models

developed for the investigation into the behaviour and capacity of LSB flexural

members. The models accurately predicted both the elastic lateral distortional buckling

moments and the non-linear ultimate moment capacities of LiteSteel beam members

subject to pure bending. The models accounted for all the significant behavioural

effects including material inelasticity, lateral distortional buckling deformations,

member instability, web distortion, residual stresses, and geometric imperfections.

The models were validated by the comparison of elastic buckling and ultimate moment

capacity results with corresponding results from an established finite strip analysis

program Thin-Wall and Pi and Trahair’s (1997) equation, and experimental test

results, respectively. The validated model can be used for the development of design

curves for LSB flexural members.

Parametric Studies and Design Rule Development

6-1

CHAPTER 6

6.0 PARAMETRIC STUDIES AND DESIGN RULE DEVELOPMENT

6.1 Introduction

Chapter 4 provided the details of an experimental investigation into the lateral

distortional buckling behaviour of LiteSteel beams (LSBs) and the results while

Chapter 5 presented the details of finite element analyses of tested LSBs and the

results. Comparison of experimental results with the current design rules in AS/NZS

4600 (SA, 2005) for lateral distortional buckling showed that the experimental

results were on average about 13% higher than the predictions of the current design

rules while Mahaarachchi and Mahendran’s (2005a) experimental results were about

3% higher than the predictions of the current design rules. These comparisons

indicate that further improvements can be made to the current design rules for lateral

distortional buckling. Therefore a detailed parametric study was undertaken based on

the validated ideal finite element models of LSBs to improve the understanding and

knowledge of lateral distortional buckling behaviour of LSBs and to obtain

additional moment capacity results. It is important to understand the lateral

distortional buckling behaviour of LSBs as a function of their section geometry,

slenderness, geometric imperfections and residual stresses. The parametric study

included varying spans of the currently available 13 LSB sections manufactured by

OneSteel Australian Tube Mills. The results from the parametric study and

experimental investigation were then used to develop new/improved design rules.

This important chapter presents the details of parametric studies and the development

of design rules for lateral distortional buckling of LSBs.

Other types of hollow flange steel beams including the Hollow Flange Beam (HFB)

were also considered in this chapter to investigate the applicability of the developed

design rules to other hollow flange steel beams. Effects of geometric imperfections

and residual stresses on the member moment capacities of LSBs are also presented in

this chapter.


6-2

6.2 Parametric Study

The results presented in Chapter 5 confirmed that the experimental finite element

model of LSBs could accurately simulate the observed experimental behaviour of

LSBs in the lateral buckling tests whereas the ideal finite element model was

validated by a comparison of elastic lateral distortional buckling moments of LSBs

with the well established finite strip analysis program Thin-wall and Pi and Trahair’s

(1997) equations (Eq.2.21 and 2.22 in Chapter 2). In the parametric study the ideal

finite element model was used to analyse the lateral distortional buckling behaviour

of 13 LSBs with spans varying from 1 to 10 m and develop an extensive data base of

member moment capacities for the purpose of developing improved design rules.

This model accurately represents a simply supported LSB section subject to a

uniform bending moment, with idealised boundary conditions including no warping

restraints, rotational restraints, or cross-section distortion at the supports. Appropriate

initial geometric imperfections, residual stresses, buckling deformations, cross-

section distortion, material characteristics and spread of plasticity effects were

explicitly modelled. Nominal dimensions, thicknesses, yield stresses and material

properties provided by the LSB manufacturers were used in the numerical analyses.

The corners were not considered in the finite element analyses. However, the use of

non-dimensional moment capacity plots in the development of design rules was

considered to eliminate any effect due to the approximation of not including the

corners. The finite element models of LSBs were developed using MD/ PATRAN

(MSC, 2008) pre-processing facility and analysed using finite element solver

ABAQUS 6.7 (HKS, 2007) while MD/PATRAN (MSC, 2008) post-processing

facility was used to view the results from numerical analyses.

A significant amount of time and effort was required in creating the models in the

pre-processing phase, which included the geometry creation, mesh application and

the application of loads and boundary conditions. Therefore, PATRAN database

journal file containing instructions for the pre-processor was used to automatically

generate a model. Variables such as geometry, finite element mesh, loads, boundary

conditions, material properties and analysis input parameters were then automatically

created by rebuilding the journal file. This simple method was able to generate a

large number of models without creating each model separately. The created


6-3

ABAQUS input files were analysed using the bifurcation buckling solution sequence

to obtain the elastic buckling eigenvectors. The global geometric imperfections and

residual stresses were then included in the nonlinear analysis model, and the analysis

was continued using the nonlinear static solution sequence.

Table 6.1 presents the nominal dimensions of 13 LSBs considered in the parametric

study. The thickness range of LSB sections considered in this study was 1.6 mm to 3

mm, while the ranges of section depth and flange width were 125 mm to 300 mm and

45 mm to 75 mm, respectively. The hollow flange width to depth ratio is 3 for all the

LSB sections. Following nominal mechanical properties provided by the LSB

manufacturers were used:

Young’s modulus of elasticity = 200,000 MPa,

Poisson’s ratio = 0.3,

Flange yield stress = 450 MPa and

Web yield stress = 380 MPa

Table 6.1: Nominal Dimensions of LSBs

LSB Sections Depth

Clear Depth

of Web

Flange Width

Flange Depth

Thick--ness

d d1 bf df t (mm) (mm) (mm) (mm) (mm)

300x75x3.0 LSB 300 250 75 25 3.0 300x75x2.5 LSB 300 250 75 25 2.5 300x60x2.0 LSB 300 260 60 20 2.0 250x75x3.0 LSB 250 200 75 25 3.0 250x75x2.5 LSB 250 200 75 25 2.5 250x60x2.0 LSB 250 210 60 20 2.0 200x60x2.5 LSB 200 160 60 20 2.5 200x60x2.0 LSB 200 160 60 20 2.0 200x45x1.6 LSB 200 170 45 15 1.6 150x45x2.0 LSB 150 120 45 15 2.0 150x45x1.6 LSB 150 120 45 15 1.6 125x45x2.0 LSB 125 95 45 15 2.0 125x45x1.6 LSB 125 95 45 15 1.6

bf

d1

df

d


6-4

Based on AS/NZS 4600 (SA, 2005) design rules for local buckling, the web and

flange plate elements of all the LSB sections are fully effective if corners are

included. In this case, the effective section modulus is equal to the full section

modulus when the maximum compressive flange stress is taken as its yield stress.

However, if corners are not included as assumed in the finite element models, the

effective section moduli of five LSB sections are about 2% less than their full section

moduli. Local buckling could therefore occur in the case of these slender LSB

sections with short spans. However, the ultimate moment capacity results of such

cases were not considered in this research as it was focussed on lateral buckling only.

Appendix D.1 provides the sample calculations for the effective section moduli of

LSB sections.

6.3 Lateral Distortional Buckling Behaviour and Strength of LSBs

Due to the presence of torsionally rigid rectangular hollow flanges and a relatively

slender web, the dominant failure mode of LSBs is lateral distortional buckling. The

lateral distortional buckling behaviour of LSBs made of thin and high strength steel

is complicated and is dependent on a number of parameters including section

geometry. Initial geometric imperfection direction and residual stresses are also

considered critical for the lateral distortional buckling capacities of LSBs. The

effects of imperfection direction and residual stresses on the ultimate moment

capacities of LSBs are presented in this section while the effects of section geometry

will be presented later in this chapter.

6.3.1 Effects of Initial Geometric Imperfection Direction

Initial geometric imperfections are present in real beams and their magnitude and

direction influence the moment capacities of LSBs subject to lateral distortional

buckling. Past research has shown that the presence of initial geometric imperfection

reduces the capacity of LSBs. However, the effect of the direction of initial

geometric imperfection of LSBs on the moment capacity is not well understood.

Since LSBs are mono-symmetric sections their imperfection direction is likely to

have a significant effect on their moment capacity (Kurniawan and Mahendran,

2009b). Figure 6.1 shows the positive and negative imperfection directions of LSBs.


6-5

Figure 6.1: Positive and Negative Imperfections of LSBs

Both positive and negative initial geometric imperfections were included in the finite

element analyses and it was found that the negative imperfection was critical as the

ultimate moments were lower for the beams with negative imperfections than in the

case with positive imperfections. As stated in Chapter 5 a magnitude of L/1000 was

used as initial geometric imperfection in all the finite element models of LSBs. Table

6.2 presents the ultimate moment capacities of LSBs with positive and negative

imperfections. The residual stresses were not used in these models.


Tension

Compression


Tension

Compression


6-6

Table 6.2: Effects of Initial Geometric Imperfection Direction on the Ultimate

Moment Capacities of LSBs

Span (mm)

300x75x3.0 LSB Span (mm)

300x60x2.0 LSB Pos IMP

Neg IMP

Neg/Pos IMP

Pos IMP

Neg IMP

Neg/Pos IMP

1500 78.38 69.79 0.89 1750 35.74 31.20 0.87 3000 58.69 53.56 0.91 3000 21.59 20.25 0.94 4000 49.87 46.41 0.93 4000 17.87 17.05 0.95 6000 38.41 35.55 0.93 6000 13.84 12.91 0.93 8000 31.14 28.99 0.93 8000 11.16 10.43 0.94 10000 26.48 24.34 0.92 10000 10.09 8.92 0.88

Span (mm)

200x60x2.5 LSB Span (mm)


Neg IMP

Neg/Pos IMP

Pos IMP

Neg IMP

Neg/Pos IMP

1500 32.37 28.92 0.89 1500 13.20 11.43 0.87 3000 26.49 24.41 0.92 3000 8.18 7.69 0.94 4000 22.51 21.20 0.94 4000 6.80 6.36 0.93 6000 17.05 16.04 0.94 6000 5.17 4.70 0.91 8000 13.72 12.83 0.94 8000 4.35 3.88 0.89 10000 11.70 10.87 0.93

Span (mm)


Neg IMP

Neg/Pos IMP

1000 11.73 10.60 0.90 2000 9.77 8.94 0.91 4000 6.64 6.28 0.95 6000 4.98 4.72 0.95 8000 4.09 3.79 0.93

Neg IMP – Negative Imperfection, Pos IMP – Positive Imperfection

Table 6.2 presents the ultimate moment capacities (in kNm) of five LSB sections

with positive and negative geometric imperfections and the ratios of ultimate

moment capacities of LSBs with negative and positive imperfections. The ultimate

moment capacities are also plotted against their span in Figure 6.2. It can be seen that

the ultimate moment capacities of LSBs with negative imperfections are always

lower than that of LSBs with positive imperfections. The ratios of ultimate moment

capacities with negative and positive imperfections given in Table 6.2 show that the

effect of negative imperfection is higher for small spans compared with large spans

while it is smaller for intermediate spans. However, this effect appears to increase

again for very large spans.


6-7

Figure 6.2: Effects of Initial Geometric Imperfection Direction on the Ultimate

Moment Capacities of LSBs

6.3.2 Effects of Residual Stresses

Cold-forming and electric resistance welding processes used in the manufacturing of

LSBs lead to both flexural and membrane type residual stresses (Mahaarachchi and

Mahendran, 2005e). Flexural residual stresses are caused by the cold-forming

process while the membrane residual stresses are due to the welding process. The

flexural and membrane residual stress distributions used in the numerical analyses

are presented in Chapter 5. Table 6.3 presents the ultimate moment capacities of five

LSB sections obtained from finite element analyses with flexural residual stress only,

membrane residual stress only and both flexural and membrane residual stresses. It

should be noted that the critical negative initial imperfection was used in all the finite

element analyses.

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Span (mm)

Ulti

mat

e M

omen

t Cap

aciti

es (k

Nm

)

300x75x3.0 LSB - Pos IMP

300x75x3.0 LSB - Neg IMP










6-8

Table 6.3: Effects of Residual Stresses on the Ultimate Moment Capacities

300x75x3.0 LSB Span (mm) w.o. RS

with RS Effect of RS F M F + M F/w.o RS M/w.o RS (F + M)/w.o RS

1500 69.79 62.87 67.04 60.84 0.90 0.96 0.87 3000 53.56 49.15 50.10 46.29 0.92 0.94 0.86 4000 46.41 43.30 43.42 40.92 0.93 0.94 0.88 6000 35.55 34.24 33.88 32.45 0.96 0.95 0.91 8000 28.99 28.15 27.44 26.60 0.97 0.95 0.92 10000 24.34 23.74 23.14 22.55 0.98 0.95 0.93

300x60x2.0 LSB Span (mm) w.o. RS with RS Effect of RS

F M F + M F/w.o RS M/w.o RS (F + M)/w.o RS 1750 31.20 27.48 28.82 25.62 0.88 0.92 0.82 3000 20.25 19.32 18.70 17.77 0.95 0.92 0.88 4000 17.05 16.32 15.70 14.98 0.96 0.92 0.88 6000 12.91 12.60 11.98 11.57 0.98 0.93 0.90 8000 10.43 10.23 9.63 9.44 0.98 0.92 0.90 10000 8.92 8.74 8.32 8.13 0.98 0.93 0.91

200x60x2.5 LSB Span (mm) w.o RS with RS Effect of RS

F M F + M F/w.o RS M/w.o RS (F + M)/w.o RS 1500 28.92 27.02 27.85 26.25 0.93 0.96 0.91 3000 24.41 22.69 22.92 21.68 0.93 0.94 0.89 4000 21.20 19.84 19.96 18.95 0.94 0.94 0.89 6000 16.04 15.38 15.32 14.73 0.96 0.96 0.92 8000 12.83 12.47 12.29 11.94 0.97 0.96 0.93 10000 10.87 10.57 10.45 10.16 0.97 0.96 0.93

200x45x1.6 LSB Span (mm) w.o RS with RS Effect of RS

F M F + M F/w.o RS M/w.o RS (F + M)/w.o RS 1500 11.43 10.49 10.74 9.85 0.92 0.94 0.86 3000 7.69 7.34 7.19 6.85 0.96 0.94 0.89 4000 6.36 6.16 5.96 5.76 0.97 0.94 0.91 6000 4.70 4.61 4.44 4.33 0.98 0.94 0.92 8000 3.88 3.80 3.68 3.60 0.98 0.95 0.93

150x45x1.6 LSB Span (mm) w.o RS

with RS Effect of RS F M F + M F/w.o RS M/w.o RS (F + M)/w.o RS

1000 10.60 9.70 10.13 9.44 0.92 0.96 0.89 2000 8.94 8.24 8.41 7.87 0.92 0.94 0.88 4000 6.28 6.01 5.98 5.71 0.96 0.95 0.91 6000 4.72 4.55 4.49 4.35 0.96 0.95 0.92 8000 3.79 3.72 3.65 3.55 0.98 0.96 0.94

w.o. RS – without Residual Stress, F – Flexural RS, M – Membrane RS


6-9

Effects of flexural residual stresses, membrane residual stresses and both residual

stresses are presented in the last three columns of Table 6.3. It can be seen that the

flexural residual stresses significantly reduced the moment capacities of LSBs in the

case of intermediate spans while their effect is not significant in the case of large

spans. In contrast, the effect of membrane residual stresses is low for LSBs with

intermediate spans whereas it is significant in the case of large spans. The effect of

combined flexural and membrane residual stresses is found to be almost the addition

of the individual effects of flexural and membrane residual stresses. Since both

flexural and membrane residual stresses are present in LSBs, they should be

considered in the analysis and design of LSBs. Overall effect of residual stresses is

higher for LSBs with intermediate spans than with large spans.

Figure 6.3: Effects of Residual Stresses on the Ultimate Moment Capacities of

300x75x3.0 LSBs

Figure 6.3 shows the variation of moment capacities of 300x75x3.0 LSBs without

residual stresses, with flexural residual stresses, membrane residual stresses and both

residual stresses as a function of span. The moment capacity curves of other LSBs

are presented in Appendix D.2.

20.0

25.0

30.0

35.0

40.0

45.0

50.0

55.0

60.0

65.0

70.0

75.0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

Span (mm)

Ulti

mat

e M

omen

t Cap

aciti

es (k

Nm

)

300x75x3.0 LSB - without RS

300x75x3.0 LSB - with Flexural RS

300x75x3.0 LSB - with Membrane RS

300x75x3.0 LSB - with Flexural + Membrane RS


6-10

The moment capacity results shown in Table 6.3 and Figure 6.3 are now plotted in a

non-dimensionalised format of moment capacity (Mu/My) vs member slenderness

(λd), where Mu is the ultimate moment, My is the first yield moment and λd =

(My/Mod)1/2.

Figure 6.4: Comparison of Moment Capacities of 300x75x3.0 LSBs with and

without Residual Stresses



0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50

Slenderness, λd

Dim

ensi

onle

ss M

omen

t Cap

acity

, Mu/M

y


300x75x3.0 LSB - with RS

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50

Slenderness, λd

Dim

ensi

onle

ss M

omen

t Cap

acity

, Mu/M

y




6-11





0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50

Slenderness, λd

Dim

ensi

onle

ss M

omen

t Cap

acity

, Mu/M

y



0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50

Slenderness, λd

Dim

ensi

onle

ss M

omen

t Cap

acity

, Mu/M

y




6-12



Figures 6.4 to 6.8 compare the ultimate moments of five LSBs with and without

residual stresses. A similar pattern can be observed in all of them where the residual

stress effect is significant for LSBs with intermediate slenderness and this effect is

reduced with increasing slenderness. For LSBs with intermediate slenderness (from

0.50 to 1.10), there is about 16% reduction due to residual stresses. For LSBs with

high slenderness (from 1.11 to 1.70), it is about 10% while for those with very high

slenderness (above 1.71), it is about 8%. The results also showed that the effect of

residual stress is significant for slender LSBs in comparison with compact LSBs.

Here, 300x60x2.0 LSB and 200x45x1.6 LSB are slender beams based on AS/NZS

4600 design rules (see Table D.2 of Appendix D.1).

6.4 Ultimate Moment Capacities of LSBs

The ultimate moment capacities of 13 LSBs shown in Table 6.1 with varying spans

were obtained by using the ideal finite element model with the critical negative

geometric imperfections and residual stresses. Figure 6.9 and Table 6.4 present the

ultimate moment capacity results of LSBs subject to lateral distortional and lateral

torsional buckling as a function of span.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50

Slenderness, λd

Dim

ensi

onle

ss M

omen

t Cap

acity

, Mu/M

y150x45x1.6 LSB - without RS



6-13

Figure 6.9: Ultimate Moment Capacity Curves of LSBs

In Figure 6.9 the ultimate moment capacities of LSBs subject to lateral distortional

and lateral torsional buckling are shown. The ultimate moment capacities of LSBs

subject to local buckling effects were not considered in this research as it is focused

on lateral buckling effects. The next chapter will provide the design and details of

LSBs subject to local buckling effects including the section capacity tests, finite

element modelling and design.

0

10

20

30

40

50

60

70

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

Span (mm)

Mom

ent C

apac

ity, M

u (k

Nm

)300x75x3.0 LSB300x75x2.5 LSB300x60x2.0 LSB250x75x3.0 LSB250x75x2.5 LSB250x60x2.0 LSB200x60x2.5 LSB200x60x2.0 LSB200x45x1.6 LSB150x45x2.0 LSB150x45x1.6 LSB125x45x2.0 LSB125x45x1.6 LSB


6-14

Table 6.4: Ultimate Moment Capacities of LSBs in kNm Span (mm) 300x75x3.0LSB Span (mm) 300x75x2.5LSB Span (mm) 300x60x2.0LSB

1500 60.84 1750 47.26 1750 25.62 2000 54.52 2000 44.41 2000 23.45 3000 46.29 3000 36.91 3000 17.77 4000 40.92 4000 32.62 4000 14.98 6000 32.45 6000 26.43 6000 11.57 8000 26.60 8000 22.02 8000 9.43 10000 22.55 10000 18.81 10000 8.13

Span (mm) 250x75x3.0LSB Span (mm) 250x75x2.5LSB Span (mm) 250x60x2.0LSB

1250 53.63 1500 41.82 1500 24.28 1500 51.49 2000 38.39 2000 20.91 2000 48.24 3000 34.50 3000 17.29 3000 43.59 4000 31.72 4000 15.04 4000 39.41 6000 26.24 6000 11.82 6000 31.97 8000 22.07 8000 9.65 8000 26.40

10000 18.92 10000 8.28 10000 22.49

Span (mm) 200x60x2.5LSB Span (mm) 200x60x2.0LSB Span (mm) 200x45x1.6LSB 1000 28.45 1250 21.26 1250 10.84 1250 27.08 1500 20.08 1500 9.85 1500 26.25 2000 18.54 2000 8.43 2000 24.41 3000 16.64 3000 6.85 3000 21.68 4000 14.75 4000 5.76 4000 18.95 6000 11.79 6000 4.33 6000 14.73 8000 9.71 8000 3.60 8000 11.94

10000 8.29 10000 3.37 10000 10.16

Span (mm) 150x45x2.0LSB 150x45x1.6LSB 125x45x2.0LSB 125x45x1.6LSB

750 13.53 10.10 10.81 8.50 1000 12.13 9.44 10.58 8.21 1250 11.63 8.87 10.37 7.93 1500 11.16 8.47 10.14 7.82 2000 10.23 7.87 9.46 7.38 3000 8.53 6.71 8.13 6.47 4000 7.13 5.71 6.93 5.61 6000 5.36 4.35 5.26 4.31 8000 4.37 3.55 4.27 3.51 10000 4.00 3.20 3.78 3.07


6-15

As seen from the results of LSBs with intermediate slenderness, the moment capacity

is reduced below the first yield moment due to the interaction of yielding and

buckling effects. This inelastic lateral distortional buckling capacity is influenced by

residual stress distributions and initial geometric imperfections. For LSBs with high

slenderness, the ultimate moment capacity can be predicted approximately by the

elastic lateral distortional buckling moment Mod. This indicates that the effects of

yielding, residual stresses and initial geometric imperfections are very small for

slender beams. Figure 6.9 shows that the moment capacity curves of LSBs with

identical flange properties merge with increasing span. For example,

• Group 1 – 300x75x3.0 LSB and 250x75x3.0 LSB

• Group 2 – 300x75x2.5 LSB and 250x75x2.5 LSB

• Group 3 – 300x60x20 LSB, 250x60x2.0 and 200x60x2.0 LSB

It was also observed that for LSBs with very high slenderness, the ultimate moment

capacity exceeded the elastic lateral distortional buckling moment. This is due to the

effects of pre-buckling deflections, which transform a straight beam into a “negative

arch” and thus increases its moment capacity (Trahair, 1993), in particular for small

beams such as the 150 and 125 LSBs with spans more than 8000 mm. However, this

effect can be ignored as it is unlikely for very slender beams to be used without any

lateral restraint.

6.5 Comparison of Member Moment Capacities of LSBs with AS/NZS 4600

(SA, 2005) Design Rules

In this section, the ultimate moment capacities of LSBs obtained from finite element

analyses are compared with the predictions from the current design rules in AS/NZS

4600 (SA, 2005). Clause 3.3.3.3(b) of AS/NZS 4600 (SA, 2005) presents the design

rules for members subject to bending under distortional buckling that involves

transverse bending of a vertical web with lateral displacement of the compression

flange. The member moment capacity, Mb, is given by Equation 6.1:

⎟⎠⎞

⎜⎝⎛=

ZZMM e

cb (6.1)


6-16

where

Z = full section modulus

Mc = critical moment

Ze = effective section modulus

For LSBs, it is appropriate to determine the effective section modulus at a stress

corresponding to Mc/Z, where Mc is the critical moment as defined in Equation 6.2.

For λd ≤ 0.59: Mc = My (6.2a)

For 0.59 < λd < 1.70: ⎟⎠⎞

⎜⎝⎛=

dyc MM

λ59.0 (6.2b)

For λd ≥ 1.70: ⎟⎠⎞

⎜⎝⎛= 2

1d

yc MMλ

(6.2c)

where

λd = member slenderness (Equation 6.3)


od

yd

MM

=λ (6.3)

The elastic lateral distortional buckling moment Mod can be calculated using Pi and

Trahair’s (1997) equations as provided in Equations 6.4 and 6.5 or an elastic

buckling analysis program such as Thin-Wall or CUFSM.

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+= 2

2

2

2

LEIGJ

LEIM w

ey

odππ (6.4)

where

EIy = minor axis flexural rigidity

EIw = warping rigidity

GJe = effective torsional rigidity

L = span

The effective torsional rigidity (GJe) is given by Equation 6.5:


6-17

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

23

2

23

91.02

91.02

dLEtGJ

dLEtGJ

GJf

f

e

π

π (6.5)

where

GJf = flange torsional rigidity

t = nominal thickness

d1 = clear depth of the web

Figure 6.10: Comparison of Moment Capacity Results from FEA with AS/NZS

4600 (SA, 2005) Design Curve

Figure 6.10 compares the member moment capacities from finite element analyses

with the AS/NZS 4600 (SA, 2005) design curve based on the above equations. The

ultimate moment capacities (Mu) and the elastic lateral distortional buckling

moments (Mod) were obtained from finite element analyses while the first yield

moments (My) were obtained by using Equation 6.6 where the elastic section

modulus (Z) was calculated as shown in Appendix D.1.

My = Z fy (6.6)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40

Slenderness, λd

Mu/M

y, M

b/My

AS 4600 (2005)300x75x3.0 LSB300x75x2.5 LSB300x60x2.0 LSB250x75x3.0 LSB250x75x2.5 LSB250x60x2.0 LSB200x60x2.5 LSB200x60x2.0 LSB200x45x1.6 LSB150x45x2.0 LSB150x45x1.6 LSB125x45x2.0 LSB125x45x1.6 LSB

0.59 1.70


6-18

As stated in the earlier chapters the corners of LSBs were ignored in the calculation

of section properties (Z) since the finite element models also did not include the

corners. The nominal flange yield stress of 450 MPa was used to calculate the first

yield moment My.

Sample calculations of the effective section moduli of LSB sections based on

AS/NZS 4600 (SA, 2005) are presented in Appendix D.1. They show that some LSB

sections without corners are likely to exhibit local buckling effects as their Ze values

are about 2% less than their Z values when the maximum compressive stress is taken

as its yield stress (450 MPa). However, only the FEA moment capacities of LSB

members subject to lateral distortional and lateral torsional buckling are considered

in this research in the comparison with design rules and in developing the new design

rules. Therefore there is no need to allow for any local buckling effects as a result of

the reduced Z values of some LSB sections.

Figure 6.10 clearly shows that almost all the finite element analysis data points are

above the current design curve for intermediate slenderness (inelastic lateral buckling

region). Experimental study (Chapter 4) also showed that the moment capacities of

LSBs were higher than the predictions from the current design rule in the inelastic

region. Table 6.5 compares the member moment capacities from finite element

analyses and AS/NZS 4600 (SA, 2005) for each LSB and span. The mean and COV

values of the ratio of member moment capacities from FEA and AS/NZS 4600 (SA,

2005) are 1.08 and 0.088 for LSBs within the inelastic lateral buckling region (0.59 <

λd < 1.70). For these calculated mean and COV values, a capacity reduction factor

(Φ) of 0.96 was determined using the recommended AISI procedure (AISI, 2007).

This is greater than the recommended capacity reduction factor of 0.90 in AS/NZS

4600 (SA, 2005) for flexural members and hence confirms that the current AS/NZS

4600 (SA, 2005) design rule is conservative in the inelastic lateral buckling region.


6-19

Table 6.5: Comparison of Moment Capacities from FEA and AS/NZS 4600 (SA,

2005)


Mod (kNm)

My (kNm) λd

FEA Mu

(kNm)

Mu / My Ratio FEA/(AS/NZS

4600) FEA AS/NZS 4600

300x75x3.0LSB

1500 144.55 77.24 0.73 60.84 0.79 0.81 0.98 2000 97.87 77.24 0.89 54.52 0.71 0.66 1.06 3000 65.69 77.24 1.08 46.29 0.60 0.54 1.10 4000 52.37 77.24 1.21 40.92 0.53 0.49 1.09 6000 38.00 77.24 1.43 32.45 0.42 0.41 1.02 8000 29.71 77.24 1.61 26.60 0.34 0.37 0.94 10000 24.29 77.24 1.78 22.55 0.29 0.31 0.93*

300x75x2.5LSB

1750 94.02 64.79 0.83 47.26 0.73 0.71 1.03 2000 78.39 64.79 0.91 44.41 0.69 0.65 1.06 3000 51.62 64.79 1.12 36.91 0.57 0.53 1.08 4000 41.59 64.79 1.25 32.62 0.50 0.47 1.07 6000 31.05 64.79 1.44 26.43 0.41 0.41 1.00 8000 24.72 64.79 1.62 22.02 0.34 0.36 0.93 10000 20.43 64.79 1.78 18.81 0.29 0.32 0.92*

300x60x2.0LSB

1750 41.99 45.17 1.04 25.62 0.57 0.57 1.00 2000 35.04 45.17 1.14 23.45 0.52 0.52 1.00 3000 22.99 45.17 1.40 17.77 0.39 0.42 0.93 4000 18.36 45.17 1.57 14.98 0.33 0.38 0.88 6000 13.50 45.17 1.83 11.57 0.26 0.30 0.86* 8000 10.65 45.17 2.06 9.43 0.21 0.24 0.89* 10000 8.76 45.17 2.27 8.13 0.18 0.19 0.93*

250x75x3.0LSB

1250 160.82 60.06 0.61 53.63 0.89 0.97 0.92 1500 124.35 60.06 0.69 51.49 0.86 0.85 1.01 2000 89.72 60.06 0.82 48.24 0.80 0.72 1.11 3000 64.12 60.06 0.97 43.59 0.73 0.61 1.19 4000 51.78 60.06 1.08 39.41 0.66 0.55 1.20 6000 37.46 60.06 1.27 31.97 0.53 0.47 1.14 8000 29.14 60.06 1.44 26.40 0.44 0.41 1.07 10000 23.75 60.06 1.59 22.49 0.37 0.37 1.01

250x75x2.5LSB

1500 100.64 50.38 0.71 41.82 0.83 0.83 1.00 2000 70.79 50.38 0.84 38.39 0.76 0.70 1.09 3000 50.41 50.38 1.00 34.50 0.68 0.59 1.16 4000 41.43 50.38 1.10 31.72 0.63 0.54 1.18 6000 30.87 50.38 1.28 26.24 0.52 0.46 1.13 8000 24.40 50.38 1.44 22.07 0.44 0.41 1.07 10000 20.07 50.38 1.58 18.92 0.38 0.37 1.01

*outside the inelastic lateral buckling region (elastic buckling region)


6-20

Table 6.5 (continued): Comparison of Moment Capacities from FEA and

AS/NZS 4600 (SA, 2005)

LSB Sections

Span (mm)

Mod (kNm)

My (kNm) λd

FEA Mu

(kNm)


4600) FEA AS/NZS 4600

250x60x2.0 LSB

1500 45.10 35.10 0.88 24.28 0.69 0.67 1.03 2000 31.69 35.10 1.05 20.91 0.60 0.56 1.06 3000 22.36 35.10 1.25 17.29 0.49 0.47 1.05 4000 18.19 35.10 1.39 15.04 0.43 0.42 1.01 6000 13.37 35.10 1.62 11.82 0.34 0.36 0.92 8000 10.50 35.10 1.83 9.65 0.27 0.30 0.92* 10000 8.60 35.10 2.02 8.28 0.24 0.24 0.96*

200x60x2.5 LSB

1000 85.99 31.98 0.61 28.45 0.89 0.97 0.92 1250 63.39 31.98 0.71 27.08 0.85 0.83 1.02 1500 51.59 31.98 0.79 26.25 0.82 0.75 1.10 2000 39.80 31.98 0.90 24.41 0.76 0.66 1.16 3000 29.19 31.98 1.05 21.68 0.68 0.56 1.20 4000 23.26 31.98 1.17 18.95 0.59 0.50 1.18 6000 16.42 31.98 1.40 14.73 0.46 0.42 1.09 8000 12.60 31.98 1.59 11.94 0.37 0.37 1.01 10000 10.19 31.98 1.77 10.16 0.32 0.32 1.00*

200x60x2.0 LSB

1250 48.68 25.79 0.73 21.26 0.82 0.81 1.02 1500 38.84 25.79 0.81 20.08 0.78 0.72 1.08 2000 29.57 25.79 0.93 18.54 0.72 0.63 1.14 3000 22.16 25.79 1.08 16.64 0.65 0.55 1.18 4000 18.13 25.79 1.19 14.75 0.57 0.49 1.16 6000 13.19 25.79 1.40 11.79 0.46 0.42 1.08 8000 10.27 25.79 1.58 9.71 0.38 0.37 1.01 10000 8.37 25.79 1.75 8.29 0.32 0.32 0.99*

200x45x1.6 LSB

1250 18.65 17.23 0.96 10.84 0.63 0.61 1.02 1500 14.96 17.23 1.07 9.85 0.57 0.55 1.04 2000 11.37 17.23 1.23 8.43 0.49 0.48 1.02 3000 8.33 17.23 1.44 6.85 0.40 0.41 0.97 4000 6.67 17.23 1.61 5.76 0.33 0.37 0.91 6000 4.74 17.23 1.91 4.33 0.25 0.28 0.91* 8000 3.65 17.23 2.17 3.60 0.21 0.21 0.98* 10000 2.96 17.23 2.41 3.37 0.20 0.17 1.14*



6-21


AS/NZS 4600 (SA, 2005)

LSB Sections

Span (mm)

Mod (kNm)

My (kNm) λd

FEA Mu

(kNm)


4600) FEA AS/NZS 4600

150x45x2.0 LSB

750 38.84 14.35 0.61 13.53 0.94 0.97 0.97 1000 26.74 14.35 0.73 12.13 0.85 0.81 1.05 1250 21.31 14.35 0.82 11.63 0.81 0.72 1.13 1500 18.21 14.35 0.89 11.16 0.78 0.66 1.17 2000 14.52 14.35 0.99 10.23 0.71 0.59 1.20 3000 10.48 14.35 1.17 8.53 0.59 0.50 1.18 4000 8.17 14.35 1.33 7.13 0.50 0.45 1.12 6000 5.62 14.35 1.60 5.36 0.37 0.37 1.01 8000 4.26 14.35 1.83 4.37 0.30 0.30 1.02* 10000 3.43 14.35 2.05 4.00 0.28 0.24 1.17*

150x45x1.6 LSB

750 30.57 11.58 0.62 10.10 0.87 0.96 0.91 1000 20.38 11.58 0.75 9.44 0.82 0.78 1.04 1250 15.96 11.58 0.85 8.87 0.77 0.69 1.11 1500 13.61 11.58 0.92 8.47 0.73 0.64 1.14 2000 11.02 11.58 1.02 7.87 0.68 0.58 1.18 3000 8.24 11.58 1.19 6.71 0.58 0.50 1.16 4000 6.56 11.58 1.33 5.71 0.49 0.44 1.11 6000 4.60 11.58 1.59 4.35 0.38 0.37 1.01 8000 3.52 11.58 1.81 3.55 0.31 0.30 1.01* 10000 2.84 11.58 2.02 3.20 0.28 0.25 1.12*

125x45x2.0 LSB

750 33.65 11.15 0.58 10.81 0.97 1.00 0.97* 1000 24.72 11.15 0.67 10.58 0.95 0.88 1.08 1250 20.47 11.15 0.74 10.37 0.93 0.80 1.16 1500 17.82 11.15 0.79 10.14 0.91 0.75 1.22 2000 14.35 11.15 0.88 9.46 0.85 0.67 1.27 3000 10.31 11.15 1.04 8.13 0.73 0.57 1.29 4000 7.99 11.15 1.18 6.93 0.62 0.50 1.24 6000 5.47 11.15 1.43 5.26 0.47 0.41 1.14 8000 4.14 11.15 1.64 4.27 0.38 0.36 1.07 10000 3.33 11.15 1.83 3.78 0.34 0.30 1.14*



6-22


AS/NZS 4600 (SA, 2005)

LSB Sections

Span (mm)

Mod (kNm)

My (kNm) λd

FEA Mu

(kNm)


4600) FEA AS/NZS 4600

125x45x1.6 LSB

750 25.97 9.00 0.59 8.50 0.94 1.00 0.94* 1000 18.53 9.00 0.70 8.21 0.91 0.85 1.08 1250 15.25 9.00 0.77 7.93 0.88 0.77 1.15 1500 13.37 9.00 0.82 7.82 0.87 0.72 1.21 2000 11.01 9.00 0.90 7.38 0.82 0.65 1.26 3000 8.20 9.00 1.05 6.47 0.72 0.56 1.28 4000 6.47 9.00 1.18 5.61 0.62 0.50 1.25 6000 4.50 9.00 1.41 4.31 0.48 0.42 1.15 8000 3.43 9.00 1.62 3.51 0.39 0.36 1.07 10000 2.77 9.00 1.80 3.07 0.34 0.31 1.11*

Mean 1.08 COV 0.088

*outside the inelastic lateral buckling region (elastic buckling region); not considered

in the calculation of Mean and COV values.

6.6 Proposed Design Rules for Member Moment Capacities of LSBs

The comparison of FEA and experimental member moment capacity results with the

predictions from the current design rules in AS/NZS 4600 (SA, 2005) in the last

section and Chapter 4, respectively, showed that the current design rule is

conservative in the inelastic buckling region while it is adequate in the elastic

buckling region. Therefore the member moment capacity results from experiments

and finite element analyses were combined and used to improve the current design

equations. Experimental and finite element analyses reveal the presence of at least

three buckling modes for LSB flexural members, namely, local, lateral distortional

and lateral torsional buckling. Current design rules consider three distinct regions

such as local buckling/yielding, inelastic and elastic buckling regions, which

correspond to the above buckling modes. Since the current design rule accurately

predicts the moment capacities of LSBs in the elastic buckling region (mostly subject

to lateral torsional buckling), a new design rule was developed for the inelastic

lateral distortional buckling region. The new design equation was established by

solving for minimum total error for all the 13 LSB sections and spans considered


6-23

here. This was achieved by minimising the square of the difference between the non-

dimensionalised moment capacity results (Mu/My) from FEA and experiments and

that predicted by the new equation (Mb/My). The new design rule for the inelastic

buckling region is given by Equation 6.7(b) and Figure 6.11 compares the design

curve based on this equation with the current AS/NZS 4600 (SA, 2005) design curve

and FEA results. Figures 6.12 and 6.13 show the comparison of the design curve

based on Eqs.6.7 (a) to (c) with experimental results and a combination of

experimental and FEA results, respectively.

For λd ≤ 0.54: Mc = My (6.7a)

For 0.54 < λd < 1.74: Mc = My (0.28 2dλ – 1.20 λd + 1.57) (6.7b)

For λd ≥ 1.74: ⎟⎟⎠

⎞⎜⎜⎝

⎛= 2

1

d

yc MMλ

(6.7c)

Figure 6.11: Comparison of FEA Moment Capacities with the Design Curve

based on Equations 6.7 (a) to (c)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/M

y, M

b/My

AS/NZS 4600 (2005)Equation 6.7300x75x3.0 LSB300x75x2.5 LSB300x60x2.0 LSB250x75x3.0 LSB250x75x2.5 LSB250x60x2.0 LSB200x60x2.5 LSB200x60x2.0 LSB200x45x1.6 LSB150x45x2.0 LSB150x45x1.6 LSB125x45x2.0 LSB125x45x1.6 LSB

0 5 1 7


6-24

Figure 6.12: Comparison of Experimental Moment Capacities with the Design

Curve based on Equations 6.7 (a) to (c)

Figure 6.13: Comparison of FEA and Experimental Moment Capacities with the

Design Curve based on Equations 6.7 (a) to (c)

The mean and COV values of the ratios of ultimate member moment capacities from

FEA, experiments and FEA and experiments, and Eq.6.7 (b) were calculated, ie.

FEA / Eq.6.7 (b), Test / Eq.6.7 (b) and (FEA + Test) / Eq.6.7 (b), and are presented

in Table 6.6. The corresponding capacity reduction factors (Φ) were also determined

using the AISI procedure (AISI, 2007), and are included in Table 6.6. The AISI

procedure is described next.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40

Slenderness, λd

Mb/M

y, M

u/My

Equation 6.7EXP MM - 300x75x3.0LSBEXP MM - 300x75x2.5LSBEXP MM - 300x60x2.0LSBEXP MM - 250x75x3.0LSBEXP MM - 250x75x2.5LSBEXP MM - 250x60x2.0LSBEXP MM - 200x60x2.5LSBEXP MM - 200x60x2.0LSBEXP MM - 200x45x1.6LSBEXP MM - 150x45x2.0LSBEXP MM - 150x45x1.6LSBEXP MM - 125x45x2.0LSBEXP MM - 125x45x1.6LSBThis Research - 300x60x2.0LSBThis Research - 250x75x2.5LSBThis Research - 200x45x1.6LSBThis Research - 150x45x2.0LSBThis Research - 150x45x1.6LSBThis Research - 125x45x2.0LSB

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/

My,

Mb/

My

Eq.6.7EXP This ResearchEXP MMFEA


6-25

6.6.1 Calculation of Capacity Reduction Factor (Φ)

The American cold-formed steel structures standard (AISI, 2007) recommends a

statistical model to determine the capacity reduction factor for limit states design in

clause F1.1. This model accounts for the variations in material, fabrication and the

loading effects. The capacity reduction factor Φ is given by Equation 6.8.

2222 qppfmo VVCVV

mmm ePFMC +++−= βφφ (6.8)

where,

Cф = Calibration coefficient, 1.52

Mm = Mean value of material factor, 1.1

Fm = Mean value of fabrication factor, 1.0

Pm = Mean value of the tested to predicted load ratio

βo = Target reliability index, 2.5

Vm = Coefficient of variation of material factor, 0.1

Vf = Coefficient of variation of fabrication factor, 0.05

Cp = Correction factor depending on the number of tests, ⎟⎠⎞

⎜⎝⎛

−⎟⎠⎞

⎜⎝⎛ +

211

mm

n

Vp = Coefficient of variation of tested to predicted load ratio, but not less than 6.5%

Vq = Coefficient of variation of load effect, 0.21

n = number of tests

m = degrees of freedom, n-1

Using the values of common parameters given above, Equation 6.8 leads to Equation

6.9.

20566.05.2672.1 ppVC

meP +−=φ (6.9)

Vp, Pm and Cp values have to be determined from experiments or analyses. The use

of FEA provides a large number of moment capacity results (about 110 results in this

research) and hence finite element analyses are increasingly used for the

development of design rules. However, experimental results are also needed to

demonstrate that the developed design rules are accurate. In this research the


6-26

developed design rules were calibrated using FEA and experimental moment

capacity results separately and in combination as shown in Table 6.6.

Table 6.6: Capacity Reduction Factors for Eq.6.7

Results Mean, Pm COV, Vp Φ FEA / Eq.6.7 (b) 1.02 0.066 0.92 EXP / Eq.6.7 (b) 0.98 0.105 0.86

(FEA + EXP) / Eq.6.7 (b) 1.01 0.080 0.90

As seen in Table 6.6, a capacity reduction factor of 0.92 was obtained for FEA

results, which is greater than the recommended factor of 0.90 in AS/NZS 4600 (SA,

2005). However, it was 0.86 for experiments, which is less than the recommended

value. This is because of comparatively low mean and high COV values of the ratios

of experimental to predicted moment capacities. As shown in Figure 6.12, most of

the experimental data points of Mahaarachchi and Mahendran (2005a) were below

the developed design curves which caused the reduction of the mean value. This may

be due to the approximate elastic lateral distortional buckling moments (Mod) used in

plotting the data points. The Mod value was calculated using Thin-Wall for the tests

of Mahaarachchi and Mahendran (2005a). Thin-Wall assumes ideal supported

conditions (i.e. no flange twist) although local flange twist was not fully eliminated

in their tests. The use of accurate Mod values for these tests would have given higher

ratios of test/predicted moment capacities and hence a greater capacity reduction

factor as discussed in Chapter 4. Nevertheless, the capacity reduction factor was 0.90

when both finite element analysis and experimental results were considered, and this

is considered adequate. Therefore it is recommended that the developed design

equation (Eq.6.7 (b)) can be used in the design of LSBs with a capacity reduction

factor of 0.90 within the guidelines of AS/NZS 4600 (SA, 2005).

Although Equation 6.7 (b) was developed for LSBs subject to lateral buckling with

an acceptable capacity reduction factor of 0.90 by considering both finite element

analysis and experimental results, attempts were also made to develop design rules

by considering FEA results and experimental results separately. They were also

developed with an acceptable capacity reduction factor of 0.9. Equation 6.10 was


6-27

developed by considering FEA moment capacity results only. Figure 6.14 compares

the FEA results with the developed design equation. The mean, COV and capacity

reduction factor for the moment capacity ratios of FEA / Eq.6.10 (b), EXP / Eq.6.10

(b) and (FEA + EXP) / Eq.6.10 (b) are given in Table 6.7.

For λd ≤ 0.54: Mc = My (6.10a)

For 0.54 < λd < 1.74: Mc = My (0.244 2dλ – 1.114 λd + 1.530) (6.10b)

For λd ≥ 1.74: ⎟⎟⎠

⎞⎜⎜⎝

⎛= 2

1

d

yc MMλ

(6.10c)





(FEA + EXP) / Eq.6.10 (b) 1.00 0.080 0.89

A comparison of experimental results with Equation 6.10 is provided in Figure 6.15.

As shown in Table 6.7 and Figure 6.15 the new design equation is unconservative

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/

My,

Mb/

My

AS/NZS 4600 (2005)

Equation 6.10

FEA


6-28

when compared with experimental results. Hence a lower capacity reduction factor

of 0.84 has to be used. However, an acceptable capacity reduction factor of 0.89 was

obtained when both FEA and experimental results were compared with this equation.



When only the experimental results were considered, the following equations were

developed.

For λd ≤ 0.54: Mc = My (6.11a)

For 0.54 < λd < 1.74: Mc = My (0.351 2dλ – 1.359 λd + 1.631) (6.11b)

For λd ≥ 1.74: ⎟⎟⎠

⎞⎜⎜⎝

⎛= 2

1

d

yc MMλ

(6.11c)

Figure 6.16 compares the experimental results with the above design equation based

on experimental results only. The mean, COV and capacity reduction factor for the

moment capacity ratios of FEA / Eq.6.11 (b), EXP / Eq.6.11 (b) and (FEA + EXP) /

Eq.6.11 (b) are given in Table 6.8.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/

My,

Mb/

My

Equation 6.10

EXP This Research

EXP MM


6-29




A comparison of FEA results with Equation 6.11 is provided in Figure 6.17. As

shown in Table 6.8, Figures 6.16 and 6.17, Eq.6.11 is very conservative when

compared with FEA results only with a capacity reduction factor of 0.95 while it was

0.94 when compared with both FEA and experimental results. Therefore, the use of

Equation 6.11 is considered to be over-conservative and is not recommended in this

research. As discussed earlier, the elastic lateral distortional buckling moments (Mod)

of Mahaarachchi and Mahendran’s (2005a) experimental data points were

approximate and the use of accurate Mod values was expected to give more

reasonable values of capacity reduction factors with Equations 6.7 and 6.10.

Therefore, Equations 6.7 (a) to (c) are recommended for the design of LSB flexural

members while Equations 6.10 (a) to (c) can also be used although they are slightly

unconservative. Equations 6.7 (a) to (c) will be used in the following section to

investigate their suitability to other types of hollow flange steel beams such as

Hollow Flange Beams (HFB).


(FEA + EXP) / Eq.6.11 (b) 1.05 0.082 0.94

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/M

y, M

b/My

Equation 6.11

EXP This Research

EXP MM


6-30



6.6.2 Moment Capacities of Hollow Flange Beams

Hollow Flange Beam (HFB) shown in Figure 6.18 is the first hollow flange section

developed by the LSB manufacturers. Table 6.9 shows the nominal dimensions of

HFBs while their nominal flange and web yield stresses are 550 MPa and 475 MPa,

respectively. Although the HFB is currently not available in the building industry it

is considered in this research to investigate the applicability of the developed lateral

distortional buckling design rules for other hollow flange steel beams.

Figure 6.18: Hollow Flange Beams

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/M

y, M

b/My

Equation 6.11

FEA

(b) Isometric View(a) Cross-Sectional View


6-31

Table 6.9: Geometrical Dimensions of HFB Sections

Designation D B t Ro b d mm mm mm mm mm mm

45090HFB38 450 90 3.8 8 74 370 40090HFB38 400 90 3.8 8 74 320 35090HFB38 350 90 3.8 8 74 270

30090HFB38 300 90 3.8 8 74 220 30090HFB33 300 90 3.3 8 74 219 30090HFB28 300 90 2.8 8 74 218

25090HFB28 250 90 2.8 8 74 168 25090HFB23 250 90 2.3 8 74 168

20090HFB28 200 90 2.8 8 74 118 20090HFB23 200 90 2.3 8 74 118

Avery et al. (1999a, 1999b and 2000) investigated the lateral buckling behaviour of

HFB sections shown in Table 6.9 except 20090HFB28 using finite element analyses,

and developed suitable design procedures. Their FEA results of ultimate moment Mu

and elastic lateral distortional buckling moment Mod are given in Table 6.10. The

first yield moments My of these HFBs with corners were calculated by using their Z

values obtained from Thin-Wall. Since Thin-Wall includes HFBs and LSBs with

corners among its standard sections, it was used here instead of using the basic

principles as for LSBs without corners. Preliminary calculations showed that the

HFB sections with their thickness less than or equal to 2.8 mm are likely to have

local buckling effects in their flanges in the case of short spans. However, comparing

the first yield moments and the ultimate moments, and the chosen spans chosen for

each HFB in Table 6.10, it is clear that Avery et al. (1999a, b, 2000) has only

considered the FEA moment capacities of HFB members subject to lateral

distortional and lateral torsional buckling. Therefore there is no need to allow for any

local buckling effects as a result of the reduced Z values of these HFB sections. The

ultimate moments are compared with the predictions from the design rule developed

for LSBs (Equation 6.7) in Table 6.10 and Figure 6.19. Only the moment capacities

in the inelastic lateral buckling region were considered in the calculation of mean and

COV of the ratios of member moment capacities from FEA and Eq.6.7 (b). The mean

and COV values of this ratio were 1.09 and 0.047 that gave a capacity reduction

factor of 0.98 based on the AISI (2007) procedure. This indicates that Eq.6.7 (b) is

very conservative for HFBs.


6-32

Table 6.10: Comparison of Avery et al.’s (1999b) FEA Results with Eq.6.7

HFB Section Span (mm)

Mod (kNm)

My (kNm) λd

FEA Mu (kNm)

Mu/My Ratio FEA/Eq.6.7 FEA Eq.6.7

45090HFB38

1500 194.76 207.13 1.03 141.15 0.68 0.63 1.08 2000 125.56 207.13 1.28 107.39 0.52 0.49 1.06 2500 94.87 207.13 1.48 86.77 0.42 0.41 1.03 3000 78.45 207.13 1.62 74.17 0.36 0.36 1.00 4000 60.91 207.13 1.84 58.48 0.28 0.29 0.96* 5000 50.81 207.13 2.02 49.64 0.24 0.25 0.98* 6000 43.76 207.13 2.18 43.29 0.21 0.21 0.99* 8000 34.24 207.13 2.46 34.62 0.17 0.17 1.01*

40090HFB38

1500 179.13 175.29 0.99 126.01 0.72 0.66 1.09 2000 119.08 175.29 1.21 99.62 0.57 0.53 1.08 2500 92.48 175.29 1.38 82.94 0.47 0.45 1.05 3000 78.87 175.29 1.49 72.36 0.41 0.40 1.02 4000 61.37 175.29 1.69 58.22 0.33 0.34 0.97 5000 51.35 175.29 1.85 49.67 0.28 0.29 0.97* 6000 44.20 175.29 1.99 43.36 0.25 0.25 0.98* 8000 34.49 175.29 2.25 34.61 0.20 0.20 1.00* 11000 26.89 175.29 2.55 26.89 0.15 0.15 1.00*

35090HFB38

1000 316.85 145.26 0.68 132.96 0.92 0.89 1.03 1500 164.76 145.26 0.94 111.38 0.77 0.69 1.11 2000 114.08 145.26 1.13 92.34 0.64 0.57 1.11 2500 91.25 145.26 1.26 78.93 0.54 0.50 1.08 3000 78.09 145.26 1.36 70.20 0.48 0.45 1.06 4000 62.22 145.26 1.53 58.28 0.40 0.39 1.03 5000 52.07 145.26 1.67 49.91 0.34 0.35 0.99 6000 44.73 145.26 1.80 43.55 0.30 0.31 0.97* 8000 34.76 145.26 2.04 34.69 0.24 0.24 1.00* 11000 25.89 145.26 2.37 26.84 0.18 0.18 1.04*

30090HFB38

1000 278.84 116.99 0.65 112.11 0.96 0.91 1.05 1500 152.79 116.99 0.88 97.23 0.83 0.73 1.13 2000 111.22 116.99 1.03 85.15 0.73 0.63 1.15 2500 91.51 116.99 1.13 76.22 0.65 0.57 1.14 3000 79.29 116.99 1.21 69.26 0.59 0.53 1.13 4000 63.47 116.99 1.36 58.43 0.50 0.46 1.09 5000 52.98 116.99 1.49 50.23 0.43 0.41 1.06 6000 45.36 116.99 1.61 43.84 0.37 0.36 1.03 8000 35.07 116.99 1.83 34.83 0.30 0.30 0.99* 11000 26.02 116.99 2.12 26.72 0.23 0.22 1.03* 14000 20.63 116.99 2.38 22.20 0.19 0.18 1.08*

*outside the inelastic lateral buckling region (elastic lateral buckling region)


6-33


Eq.6.7

HFB Sections

Span (mm)

Mod (kNm)

My (kNm) λd

FEA Mu (kNm)

Mu/My Ratio FEA/Eq.6.7FEA Eq.6.7

30090HFB33

1000 243.53 102.19 0.65 93.49 0.91 0.91 1.01 1500 130.01 102.19 0.89 81.33 0.80 0.73 1.10 2000 93.09 102.19 1.05 71.25 0.70 0.62 1.12 2500 76.41 102.19 1.16 63.40 0.62 0.56 1.11 3000 66.54 102.19 1.24 57.88 0.57 0.51 1.10 4000 54.05 102.19 1.38 49.78 0.49 0.45 1.08 5000 45.67 102.19 1.50 43.26 0.42 0.40 1.05 6000 39.46 102.19 1.61 38.10 0.37 0.36 1.02 8000 30.84 102.19 1.82 30.59 0.30 0.30 0.99* 11000 23.06 102.19 2.11 23.60 0.23 0.23 1.02* 14000 18.36 102.19 2.36 19.65 0.19 0.18 1.07*

30090HFB28

1500 107.93 87.18 0.90 67.22 0.77 0.72 1.07 2000 75.39 87.18 1.08 58.13 0.67 0.60 1.11 2500 61.66 87.18 1.19 51.74 0.59 0.54 1.10 3000 54.49 87.18 1.26 47.36 0.54 0.50 1.09 4000 44.08 87.18 1.41 40.66 0.47 0.44 1.07 5000 37.82 87.18 1.52 35.81 0.41 0.39 1.04 6000 33.06 87.18 1.62 31.89 0.37 0.36 1.02 8000 26.25 87.18 1.82 26.00 0.30 0.30 0.99* 11000 19.86 87.18 2.10 20.34 0.23 0.23 1.02* 14000 15.90 87.18 2.34 16.99 0.19 0.18 1.07*

25090HFB28

1500 99.68 67.60 0.82 56.56 0.84 0.77 1.08 2000 74.60 67.60 0.95 52.36 0.77 0.68 1.14 2500 63.05 67.60 1.04 49.12 0.73 0.63 1.16 3000 55.86 67.60 1.10 46.03 0.68 0.59 1.16 4000 46.18 67.60 1.21 40.84 0.60 0.53 1.14 5000 39.33 67.60 1.31 36.32 0.54 0.48 1.12 6000 34.14 67.60 1.41 32.51 0.48 0.44 1.10 8000 26.80 67.60 1.59 26.38 0.39 0.37 1.05 11000 20.10 67.60 1.83 20.51 0.30 0.30 1.02* 14000 16.02 67.60 2.05 16.97 0.25 0.24 1.06* 18000 12.58 67.60 2.32 14.34 0.21 0.19 1.14*

*outside the inelastic lateral buckling region (elastic lateral buckling region); not considered in the calculation of Mean and COV values.


6-34


Eq.6.7

HFB Sections

Span (mm)

Mod (kNm)

My (kNm) λd

FEA Mu (kNm)

Mu/My Ratio FEA/Eq.6.7 FEA Eq.6.7

25090HFB23

1500 78.58 55.94 0.84 44.48 0.80 0.76 1.05 2000 57.03 55.94 0.99 40.57 0.73 0.66 1.11 2500 47.79 55.94 1.08 37.75 0.67 0.60 1.13 3000 42.51 55.94 1.15 35.24 0.63 0.56 1.12 4000 35.83 55.94 1.25 31.67 0.57 0.51 1.12 5000 31.11 55.94 1.34 28.64 0.51 0.46 1.10 6000 27.40 55.94 1.43 25.85 0.46 0.43 1.08 8000 21.93 55.94 1.60 21.54 0.39 0.37 1.05 11000 16.70 55.94 1.83 16.97 0.30 0.30 1.02* 14000 13.40 55.94 2.04 14.16 0.25 0.24 1.06* 18000 10.57 55.94 2.30 11.99 0.21 0.19 1.13*

20090HFB23

1500 74.00 41.05 0.74 36.19 0.88 0.83 1.06 2000 59.11 41.05 0.83 35.01 0.85 0.76 1.12 2500 51.49 41.05 0.89 34.08 0.83 0.72 1.15 3000 46.31 41.05 0.94 33.26 0.81 0.69 1.18 4000 38.75 41.05 1.03 31.16 0.76 0.63 1.20 5000 33.16 41.05 1.11 28.74 0.70 0.58 1.20 6000 28.83 41.05 1.19 26.25 0.64 0.54 1.19 8000 22.67 41.05 1.35 21.85 0.53 0.46 1.15 11000 17.02 41.05 1.55 17.23 0.42 0.38 1.10 14000 13.56 41.05 1.74 14.21 0.35 0.33 1.05* 18000 10.65 41.05 1.96 11.84 0.29 0.26 1.11* 25000 7.80 41.05 2.29 9.20 0.22 0.19 1.18*

Mean 1.09 COV 0.047

*outside the inelastic lateral buckling region (elastic lateral buckling region); not considered in the calculation of Mean and COV values.


6-35


(1999b) with Equations 6.7 (a) to (c)

As seen in Table 6.10 and Figure 6.19, Equation 6.7 (b) is conservative in the

inelastic lateral buckling region. However, Avery et al. (1999b) did not include the

membrane residual stresses in their finite element analyses of HFBs despite the fact

that the manufacturing process of HFBs also involved an electric welding process

similar to that used for LSBs. This welding process would have created some

membrane residual stresses in HFBs, which would have reduced their moment

capacities. The effects of membrane residual stress of LSBs were investigated in this

research and are presented in Table 6.3. It was found that on average the membrane

residual stress reduced the lateral buckling moment capacities of LSBs by about 6%.

It is believed that the effect of welding process in the making of HFBs is similar to

that of LSBs and hence the effects of membrane residual stresses on the moment

capacities of HFBs could also be similar to that of LSBs. Therefore, the mean FEA to

predicted moment capacity ratio is likely to be reduced by about 6% if the membrane

residual stress is considered in the analysis of HFBs. The capacity reduction factor

will also be reduced from 0.98. Therefore, no attempts were made to modify Eq.6.7

(b) or to develop a new design rule to predict the moment capacities of HFBs subject

to lateral distortional buckling.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/M

y, M

b/My

Equation 6.745090HFB3840090HFB3835090HFB3830090HFB3830090HFB3330090HFB2825090HFB2825090HFB2320090HFB23


6-36

6.7 Effect of Section Geometry on the Lateral Distortional Buckling Moment

Capacities of LSBs

Lateral distortional buckling is a complex phenomenon which involves not only

lateral deflection and twist but also web distortion. This is not commonly observed in

conventional hot-rolled I-sections or other types of beams unless the flanges are

restrained torsionally or the web is particularly slender (Pi and Trahair, 1997). Cold-

formed hollow flange steel beams such as LSBs and HFBs have torsionally rigid

flanges and relatively slender webs and are thus subjected to the more detrimental

lateral distortional buckling and associated reduction of inelastic lateral buckling

moment capacities. Pi and Trahair (1997) developed a closed form solution to

calculate the elastic lateral distortional buckling moments of beams by introducing an

effective torsional rigidity term (GJe) in place of the torsional rigidity term (GJ) in

the well known equation for lateral torsional buckling moment, Mo (Equation 6.12).

The effective torsional rigidity term is given by Equation 6.5.

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+= 2

2

2

2

LEIGJ

LEIM wy

oππ (6.12)

where



GJ = torsional rigidity

L = span

When plotting the non-dimensional member capacity curves for beams subject to

lateral distortional buckling, a modified member slenderness parameter

{λd=(My/Mod)1/2} is used instead of λ {=(My/Mo)1/2}. This procedure was used in the

earlier sections of this chapter. However, a closer look at the finite element analysis

data points for LSBs plotted in the non-dimensional moment capacity versus

modified slenderness λd format as shown in Figure 6.20 reveals that the points are

scattered to some extent. Although suitable design rules in the form of Eqs.6.7 (a) to

(c) have been developed based on these data points through a process of minimising

the total error, it underestimates the member moment capacities of some LSB


6-37

sections (compact sections) while overestimating them for other LSB sections

(slender sections) as shown in Figures 6.11 and 6.13. If the member capacity

equations are developed based on less scattered data, this shortcoming will be

eliminated and their accuracy will be equally good for all the LSB sections.

Following sections discuss the use of other modified member slenderness parameters

to achieve this.

Figure 6.20: Non-Dimensional Member Moment Capacity versus Modified

Slenderness λd for LSBs

The FEA member moment capacities of LSBs are plotted in the non-dimensional

member capacity versus slenderness λ format in Figure 6.21 where the slenderness λ

was based on lateral torsional buckling moment Mo )(o

y

MM

= . This approach does not

include the effect of web distortion observed with lateral distortional buckling and

hence leads to more scattered data as shown in Figure 6.21. The elastic lateral

torsional buckling moments (Mo) were calculated based on Eq.6.12 and are presented

in Appendix D.3. Comparison of data points in Figures 6.20 and 6.21 clearly

demonstrates this. The use of a modified member slenderness parameter, λd

)(od

y

MM

= , in Figure 6.20 considers the effects due to web distortion and hence

reduces the scatter among the data points. However, further improvements are

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/

My


6-38

possible through the introduction of a geometrical parameter with λ or λd to address

the effect of web distortion in LSBs and HFBs.

Figure 6.21: Non-Dimensional Member Moment Capacity versus Slenderness λ

for LSBs

Trahair (1995a) introduced a geometrical parameter Et3L2 / GJfd1 in the equation for

effective torsional rigidity GJe (Eq.6.5) and stated that it was a measure of the

relative magnitude of the flexural rigidity of the web in comparison with the torsional

rigidity of the flanges in the investigation of elastic lateral distortional buckling of

hollow flange beams. An attempt was therefore made to determine a geometrical

parameter K1 in terms of Et3L2 / GJfd1 that can be used to modify the slenderness

parameter as K1λ instead of λd in order to reduce the scatter of FEA data points.

When the structural parameter K1 defined by Equation 6.13 was developed and used,

it was found that the scatter of data points was reduced as seen in Figure 6.22.

4/1

1

231

35.03.0

1

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

dGJLEt

K

f

(6.13)

where K1 is a factor determined based on several trial and error attempts to reduce

the scatter of data points. The member moment capacity design rules in this case are

given by the following equations.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λ (MY/Mo)1/2

Mu/M

y


6-39

For K1λ ≤ 0.39: Mc = My (6.14a)

For K1λ > 0.39: Mc = My (0.29(K1λ)2 – 1.23K1λ + 1.44) (6.14b)


Slenderness Parameter K1λ

For the design rules developed (Eq.6.14b) the mean and COV of the ratio of FEA to

predicted moment capacities were 1.00 and 0.069 with a capacity reduction factor of

0.90. However, a comparison of FEA data points in Figures 6.20 and 6.22 reveals

that the scatter has not been reduced much when λd was replaced by K1λ. Therefore

an attempt was made to plot FEA data points in the Mu/My vs K2λd format, where

od

yd M

M=λ and K2 is a geometrical parameter defined in terms of Et3L2/GJfd1 by

Equation 6.15.

15.0

1

232

5.05.0

1

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

dGJLEt

K

f

(6.15)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Modified Slenderness, K1λ

Mu/M

y, M

b/My

FEA

Equation 6.14


6-40


Slenderness Parameter K2λd

New design rules were developed for the FEA data points plotted in the new format.

For these design rules given by Eq.6.16, the mean and COV of the ratio of FEA to

predicted moment capacities were 1.00 and 0.048 with a capacity reduction factor of

0.90.

For K2λd ≤ 0.55: Mc = My (6.16a)

For K2λd > 0.55: Mc = My (0.347(K2λd)2 – 1.48K2λd + 1.71) (6.16b)

Comparison of Figures 6.20, 6.22 and 6.23 show that the plot in the Mu/My vs K2λd

format has the least scatter of data points among them. However, it is to be noted that

the parameter Et3L2/GJfd1 has already been included in the Mod equation via GJe

equation inod

yd M

M=λ . This implies that the parameter Et3L2/GJfd1 has been used

twice by considering K2λd to include the effects of web distortion in LSBs. This does

not appear to be appropriate. The following equations clearly explain this fact.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Modified Slenderness, K2λd

Mu/

My,

Mb/M

yFEA

Equation 6.16


6-41

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+= 2

2

2

2

LEIGJ

LEIM w

ey

odππ ,

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

23

2

23

91.02

91.02

dLEtGJ

dLEtGJ

GJf

f

e

π

π,

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

23

2

23

91.02

91.02

dGJLEtd

LEt

GJ

f

e

π

π

Apart from this, the calculation of K2 for each span is not a simple task for designers.

Therefore, from a design point of view, the use of Equation 6.16 with K2λd was not

considered suitable. Several other parameters such as depth/thickness,

width/thickness, depth/width, slenderness of plate elements, torsional rigidity and

flexural rigidity were considered to determine a simple geometrical parameter which

would reduce the scatter of FEA data points of LSBs. Finally it was found that the

use of a new K parameter defined as a function of the ratio of torsional rigidity of the

flanges to the major axis flexural rigidity of web (GJf/EIxweb) considerably reduced

the scatter of FEA data points. The new parameter K is defined by Equation 6.17.

Figure 6.24 shows the FEA data points plotted in the non-dimensional moment

capacity (Mu/My) versus modified slenderness (Kλd) format.

xweb

f

EIGJ

K+

=85.0

1 (6.17)

where

GJf = torsional rigidity of the flange

EIxweb = major axis flexural rigidity of the web

Divide by GJf


6-42


Slenderness Parameter Kλd

Based on the FEA moment capacity results plotted in Figure 6.24, new design rules

were developed as given by Equation 6.18.

For Kλd ≤ 0.52: Mc = My (6.18a)

For Kλd > 0.52: Mc = My (0.199(Kλd)2 – 1.013Kλd + 1.475) (6.18b)

Comparison of Figures 6.20, 6.23 and 6.24 reveal that the moment capacity plot in

the new Mu/My versus Kλd format in Figure 6.24 has little scatter among the data

points. Therefore Equations 6.18 (a) and (b) are considered to be accurate and

recommended as alternative improved equations to Eqs.6.7 (a) to (c). Since the

horizontal axis was changed to the modified slenderness (Kλd) the elastic buckling

region as provided in Equation 6.7 (c) does not have any meaning. Hence there are

only two regions in Equations 6.18 (a) and (b), namely, the local buckling/yielding

region (Kλd ≤ 0.52) and the lateral buckling region (Kλd > 0.52).

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Modified Slenderness, Kλd

Mu/M

y , M

b/My

300x75x3.0 LSB300x75x2.5 LSB300x60x2.0 LSB250x75x3.0 LSB250x75x2.5 LSB250x60x2.0 LSB200x60x2.5 LSB200x60x2.0 LSB200x45x1.6 LSB150x45x2.0 LSB150x45x1.6 LSB125x45x2.0 LSB125x45x1.6 LSBEquation 6.116.18


6-43

Figure 6.25: Comparison of Experimental Results with Equation 6.18

Figure 6.25 compares the experimental moment capacity results with Equation 6.18

based on the new geometrical parameter K. To plot the experimental data points,

torsional rigidity of the flange and the major axis flexural rigidity of the web were

calculated based on the measured dimensions of the tested beams and the parameter

K was calculated as shown in Appendix D.4. The mean, COV and capacity reduction

factor for the ratios of FEA / Eq.6.18 (b), EXP / Eq.6.18 (b) and (FEA + EXP) /

Eq.6.18 (b) are given in Table 6.11. The capacity reduction factor obtained in the

case of FEA and FEA+EXP were 0.90 and 0.89 and thus confirm the adequacy of the

new design rules. However, it was only 0.83 when only the experimental moment

capacity results were considered. As explained in Section 4.5 of Chapter 4, the use of

accurate Mod values for the tests of Mahaarachchi and Mahendran (2005a) will

eliminate the approximation in the evaluation of test results and thus increase the

capacity reduction factor.

Table 6.11: Capacity Reduction factors for Eq.6.18


(FEA + EXP) / Eq.6.18 (b) 0.99 0.064 0.89

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40


Mb/M

y, M

u/My

Equation 6.11MM EXP - 300x75x3.0LSBMM EXP - 300x75x2.5LSBMM EXP - 300x60x2.0LSBMM EXP - 250x75x3.0LSBMM EXP - 250x75x2.5LSBMM EXP - 250x60x2.0LSBMM EXP - 200x60x2.5LSBMM EXP - 200x60x2.0LSBMM EXP - 200x45x1.6LSBMM EXP - 150x45x2.0LSBMM EXP - 150x45x1.6LSBMM EXP - 125x45x2.0LSBMM EXP - 125x45x1.6LSBThis Research - 300x60x2.0LSBThis Research - 250x75x2.5LSBThis Research - 200x45x1.6LSBThis Research - 150x45x2.0LSBThis Research - 150x45x1.6LSBThis Research - 125x45x2.0LSB

8


6-44

Table 6.12: Section Properties of LSBs including K

LSB Section Ixweb (103mm4)

EIxweb (106Nmm2)

Jf (103 mm4)

GJf (106Nmm2) GJf/EIxweb K

300x75x3.0 LSB 3906 781 250 160 12812 0.0164 1.0224 300x75x2.5 LSB 3255 651 042 140 11204 0.0172 1.0192 300x60x2.0 LSB 2929 585 867 57 4589 0.0078 1.0655 250x75x3.0 LSB 2000 400 000 160 12812 0.0320 0.9718 250x75x2.5 LSB 1667 333 333 140 11204 0.0336 0.9677 250x60x2.0 LSB 1544 308 700 57 4589 0.0149 1.0289 200x60x2.5 LSB 853 170 667 68 5400 0.0316 0.9729 200x60x2.0 LSB 683 136 533 57 4589 0.0336 0.9677 200x45x1.6 LSB 655 131 013 19 1524 0.0116 1.0440 150x45x2.0 LSB 288 57 600 22 1786 0.0310 0.9746 150x45x1.6 LSB 230 46 080 19 1524 0.0331 0.9691 125x45x2.0 LSB 143 28 579 22 1786 0.0625 0.9091 125x45x1.6 LSB 114 22 863 19 1524 0.0667 0.9024

Table 6.12 presents the values of torsional rigidity of flange (GJf) and major axis

flexural rigidity of web (EIxweb) for all the 13 LSB sections considered in this

research. It also includes the values of the important geometrical parameter K used in

the development of Eqs.6.18 (a) and (b). As seen in Table 6.12, each LSB section has

unique values of GJf/EIxweb and K. Figures 6.26 to 6.28 show the plots of FEA

moment capacities in the non-dimensional moment capacity format of Mu/My versus

slenderness λd in an attempt to study the effect of K and GJf/EIxweb on the moment

capacity curves of LSBs.

Figure 6.26: Moment Capacities of LSBs with Similar Values of GJf/EIxweb

(Set 1)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/M

y, M

b/My

Equation 6.7300x75x3.0 LSB300x75x2.5 LSB


6-45


(Set 2)

Figure 6.26 shows the FEA moment capacity data points of 300x75x3.0 LSB and

300x75x2.5 LSB, which have similar GJf/EIxweb values of 0.0164 and 0.0172 (see

Table 6.12). It is clearly seen that the data points follow the same trend. Similarly in

Figure 6.27, the FEA moment capacity data points of six LSBs with different

dimensions but with similar values of GJf/EIxweb (about 0.03 as seen in Table 6.12)

are plotted, which show the same trend. Figure 6.28 is another example where two

LSBs (125x45x2.0 LSB and 125x45x1.6 LSB) with similar values of GJf/EIxweb

(about 0.06) follow the same trend. Hence it is concluded that as demonstrated by

Figures 6.26 to 28 the chosen parameters GJf/EIxweb and K are appropriate in

reducing the scatter of FEA data points of LSBs. Another important finding was that

LSBs with high values of GJf/EIxweb plotted above the design curve and those with

low values plotted below the design curve (Figure 6.28). In other words, non-

dimensional moment capacity ratios (Mu/My) increase with the parameter GJf/EIxweb

for a given slenderness. As shown earlier, this scatter can be significantly reduced by

using the K factor which includes the parameter GJf/EIxweb.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/M

y, M

b/My

Equation 6.7250x75x3.0 LSB250x75x2.5 LSB200x60x2.5 LSB200x60x2.0 LSB150x45x2.0 LSB150x45x1.6 LSB


6-46


(Set 3)

An attempt was then made to verify the applicability of Equation 6.18 based on the

new K factor for HFB sections. For this purpose, the torsional rigidity of triangular

hollow flanges and the major axis flexural rigidity of web were calculated as for

LSBs. However, the corners of HFBs can not be ignored as they are large compared

to the corners in LSBs. Therefore, Thin-wall was used to calculate the torsional

constant of triangular hollow flange (Jf) with corners. Since Avery et al.’s (1999b)

finite element analyses also considered the corners of HFBs, the same configuration

of HFBs was used in the calculation of parameter K. Table 6.13 shows the values of

torsional rigidity of flange, major axis flexural rigidity of web and the parameter K

for HFBs with corners.

Table 6.13: Section Properties of HFBs including K

HFB Sections Ixweb (103 mm4)

EIxweb (106 Nmm2)

Jf (103 mm4)

GJf (106 Nmm2) GJF/EIxweb K

45090HFB38 16040 3208023 387.0 30960 0.0097 1.0546 40090HFB38 10377 2075307 387.0 30960 0.0149 1.0287 35090HFB38 6233 1246590 387.0 30960 0.0248 0.9925 30090HFB38 3372 674373 387.0 30960 0.0459 0.9396 30090HFB33 2888 577690 340.7 27256 0.0472 0.9370 30090HFB28 2417 483477 293.2 23456 0.0485 0.9344 25090HFB28 1106 221276 293.2 23456 0.1060 0.8506 25090HFB23 909 181763 240.4 19232 0.1058 0.8509 20090HFB23 315 62983 240.4 19232 0.3054 0.7130

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/M

y, M

b/M

yEquation 6.7

300x60x2.0 LSB

200x45x1.6 LSB

125x45x2.0 LSB

125x45x1.6 LSB


6-47


(1999b) with Equation 6.18

Figure 6.29 shows the comparison of Avery et al.’s (1999b) ultimate moment

capacities of HFBs from their finite element analyses with Equations 6.18 (a) and

(b). It shows that the moment capacity data points are more scattered than in the case

of LSBs (compare with Figure 6.24). A closer look at Figure 6.29 shows that the data

points of 20090HFB23, 25090HFB23 and 25090HFB28 caused this scatter. Hence

Figure 6.30 was plotted without considering these three HFBs.

Figure 6.30: Comparison of FEA Moment Capacities of Selected HFBs from

Avery et al. (1999b) with Equation 6.18

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60


Mu/M

y, M

b/My

Equation 6.1145090HFB38 40090HFB3835090HFB3830090HFB3830090HFB3330090HFB2825090HFB2825090HFB2320090HFB23

6.18

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60


Mu/M

y, M

b/My

Equation 6.11

45090HFB38

40090HFB38

35090HFB38

30090HFB38

30090HFB33

30090HFB28

6.18


6-48

As seen in Figure 6.30, the FEA moment capacity data points of HFBs have only a

small scatter when the data points of 20090HFB23, 25090HFB23 and 25090HFB28

were excluded. It was found that the values of GJf/EIxweb for these three HFBs were

much higher than those of the remaining HFBs as shown in Table 6.13. This ratio

was about 0.106 for 25090HFB28 and 25090HFB23 and 0.305 for 20090HFB23

while the remaining HFBs have a ratio less than 0.0485. The ratios of GJf/EIxweb for

LSBs varied from 0.0078 to 0.0667 and the moment capacity results of LSBs agree

well with Equation 6.18 as seen in Figure 6.24.

Based on the results of LSBs and HFBs, it can be observed that the moment capacity

results of LSBs and HFBs with GJf/EIxweb values in the range of 0.0078 and 0.0667

agree well with Equations 6.18 (a) and (b) while those with a ratio of 0.1058 and

higher do not agree. This indicates that there is a need to define suitable lower and

upper limits of this ratio when Equation 6.18 can be used for LSBs and HFBs. The

upper limit is expected to be between 0.0667 and 0.1058 based on the above

observations. However, in order to determine the lower limit, further finite element

analyses are needed. Finite element analyses showed that the level of web distortion

is very small for the beams with high values of GJf/EIxweb. For example, 125x45x2.0

LSB (0.0625) and 125x451.6 LSB (0.0667) have high values of GJf/EIxweb among the

13 LSBs and their finite element analysis data points were seen to plot well above the

Mu/My versus λd design curve defined by Eq.6.7 as seen in Figure 6.11 or 6.28. They

are small sections with smaller web depth and the tendency to fail by web distortion

is low. Similarly, 20090HFB23 (0.3054), 25090HFB23 (0.1058) and 25090HFB28

(0.1060) have high values of GJf/EIxweb among the nine HFBs and their FEA moment

capacity data points were also seen to plot well above the design curve defined by

Eq.6.7 as seen in Figure 6.19. These beams are also beams with a smaller web depth

and web distortion is less likely to occur. The plot of HFB data points with similar

values of GJf/EIxweb were compared in Figures 6.31 and 6.32 to validate the

applicability of this parameter. These figures show that HFBs with similar values of

GJf/EIxweb follow the same trend in Mu/My versus λd plots where 30090HFB38,

30090HFB33 and 30090HFB28 have GJf/EIxweb values of 0.0459, 0.0472 and 0.0485,

respectively, while those of 25090HFB28 and 25090HFB23 are about 0.106.

Therefore it can be concluded that the use of parameter GJf/EIxweb is appropriate in


6-49

the investigation into the effects of section geometry of hollow flange steel beams

subject to lateral distortional buckling.

Figure 6.31: Moment Capacities of HFBs with Similar Values of GJf/EIxweb

(Set 1)

Figure 6.32: Moment Capacities of HFBs with Similar Values of GJf/EIxweb

(Set 2)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/

My,

Mb/

My

Equation 6.7

25090HFB28

25090HFB23

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/

My,

Mb/

My

Equation 6.7

30090HFB38

30090HFB33

30090HFB28


6-50

Having established the appropriateness of the use of parameter GJf/EIxweb and K in

including the effects of section geometry, further studies were undertaken using

finite element analyses of two new non-standard LSB sections with different values

of GJf/EIxweb. Table 6.14 shows their dimensions, GJf/EIxweb and K values. The

GJf/EIxweb value of 300x45x3.6 LSB is 0.0021, which is much less than that of the

currently available LSBs (0.0078) while the other LSB section chosen, 135x50x1.6

LSB, has a GJf/EIxweb value of 0.0811, which is between 0.0667 and 0.1058. Table

6.15 presents the finite element analysis results of these two LSBs subject to lateral

distortional and lateral torsional buckling effects only. Appendix D.1 provides the

details of their section moduli.

Table 6.14: Two New LSBs with Different GJf/EIxweb and K Values

Table 6.15: FEA Moment Capacity Results of Two New LSBs

LSB

Sections Span Mod (kNm) My (kNm) λd Mu (kNm) Mu/My

300x45x3.6 LSB

750 134.95 65.61 0.70 51.36 0.78 1000 84.90 65.61 0.88 43.84 0.67 1500 46.76 65.61 1.18 31.77 0.48 2000 32.41 65.61 1.42 24.14 0.37 3000 20.46 65.61 1.79 16.20 0.25 4000 15.09 65.61 2.08 12.28 0.19 5000 11.99 65.61 2.34 9.84 0.15

135x50x1.6 LSB

1000 24.69 10.70 0.66 9.92 0.93 1250 19.81 10.70 0.73 9.65 0.90 1500 17.22 10.70 0.79 9.50 0.89 2000 14.31 10.70 0.86 9.11 0.85 4000 8.84 10.70 1.10 7.36 0.69 6000 6.27 10.70 1.31 5.84 0.55 8000 4.82 10.70 1.49 4.77 0.45 10000 3.90 10.70 1.66 4.08 0.38

LSB Sections

Depth Clear Web depth

Flange Width

Flange Depth

Thick--ness

GJF/EIxweb K d d1 bf df T (mm) (mm) (mm) (mm) (mm)

300x45x3.6 LSB 300 270 45 15 3.6 0.0021 1.1163

135x50x1.6 LSB 135 101 50 17 1.6 0.0811 0.8812


6-51

Figure 6.33: Moment Capacities of New LSBs with Different GJf/EIxweb Values Figure 6.33 shows a comparison of the FEA moment capacity data points of the new

LSBs as plotted in the non-dimensional member moment capacity Mu/My versus

modified slenderness Kλd format. As seen in this figure, the FEA moment capacities

of LSBs with GJf/EIxweb values in the range of 0.0021 and 0.0811 agree well with the

design curve based on Equation 6.18. Based on the comparisons in Figure 6.29, the

moment capacities of HFBs with a GJf/EIxweb value of 0.1058 did not comply with

Equation 6.18. Further, it is obvious that the value of K decreases with increasing

values of GJf/EIxweb and this is the reason why the FEA data points of HFBs with

higher values of GJf/EIxweb plot below the developed design curve. Therefore, it was

decided to define a lower limit for the values of K instead of an upper limit for

GJf/EIxweb. For lower values of GJf/EIxweb, the K values become higher which will

bring the FEA data points towards right in the Mu/My versus Kλd plots as for

300x45x3.6 LSB in Figure 6.33, i e. conservative. Therefore an upper limit may not

be essential for K.

An attempt was made to plot the FEA data points of all the HFBs with the GJf/EIxweb

values of greater than 0.0811 in the Mu/My versus λd format to observe the variations

in the plot. For this purpose two more non-standard LSB sections with higher values

of GJf/EIxweb were considered in the finite element analyses. They are 125x45x1.8

LSB and 125x47x2.4 LSB with GJf/EIxweb values of 0.1123 and 0.2065, respectively.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40


Mu/M

y, M

b/My

Equation 6.18Available LSBs300x45x3.6LSB135x50x1.6LSB


6-52

The dimensions and the FEA results of these LSBs are given in Tables 6.16 and 6.17,

respectively. Figure 6.34 shows the moment capacities of these HFBs and LSBs with

GJf/EIxweb values greater than 0.0811.

Table 6.16: Two New LSBs with Higher Values of GJf/EIxweb

Table 6.17: FEA Moment Capacity Results of Two New LSBs with Higher

Values of GJf/EIxweb

LSB Sections

Depth

Clear Depth

of Web

Flange Width

Flange Depth

Thickness t

GJF / EIxweb K d d1 bf df web flange

(mm) (mm) (mm) (mm) (mm) (mm)

125x45x1.8 LSB 125 89 45 18 1.8 1.8 0.1123 0.8438

125x47x2.4 LSB 125 85 47 20 1.8 2.4 0.2065 0.7666

LSB Sections Span Mod (kNm) My (kNm) λd Mu (kNm) Mu/My

125x45x1.8 LSB

750 31.88 9.93 0.56 9.67 0.97 2000 14.81 9.93 0.82 8.93 0.90 3000 11.11 9.93 0.95 8.12 0.82 4000 8.80 9.93 1.06 7.41 0.75 6000 6.14 9.93 1.27 5.87 0.59 10000 3.79 9.93 1.62 3.91 0.39

125x47x2.4 LSB

1000 32.19 13.10 0.64 13.36 1.02 3000 15.82 13.10 0.91 11.49 0.88 4000 12.67 13.10 1.02 10.33 0.79 6000 8.92 13.10 1.21 8.26 0.63 10000 5.52 13.10 1.54 5.73 0.44


6-53

Figure 6.34: Moment Capacities of Hollow Flange Steel Beams with

GJf/EIxweb ≥ 0.0811

As seen in Figure 6.34, the FEA moment capacity data points of different hollow

flange steel beams with different values of GJf/EIxweb followed a similar trend

irrespective of the GJf/EIxweb values (when GJf/EIxweb values are greater than 0.0811).

As seen in Figure 6.34, Eq.6.7 (b) is very conservative for these sections and hence

the use of Eq.6.18 (b) based on the modified slenderness Kλd will be useful. For the

applicability of Eqs.6.18 (a) and (b) it is therefore reasonable to define a single K

value for the hollow flange steel beams with the values of GJf/EIxweb greater than

0.0811. Based on this, a value of 0.8812 was defined as the lower limit for K based

on the limiting GJf/EIxweb value of 0.0811. Equation 6.19 defines the K value with its

lower limit. The FEA moment capacities of the above mentioned hollow flange steel

beams were plotted in the Mu/My versus Kλd format with the same K value of 0.88 in

Figure 6.35.

xweb

f

EIGJ

K+

=85.0

1 ≥ 0.88 (6.19)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/

My,

Mb/

My

Equation 6.7125x47x2.4 LSB (0.2065)125x45x1.8 LSB (0.1123)135x50x1.6 LSB (0.0811)25090HFB28 (0.1060)25090HFB23 (0.1058)20090HFB23 (0.3054)


6-54

Figure 6.35: Moment Capacities of Hollow Flange Steel Beams with the

Modified Slenderness Parameter K as Defined in Equation 6.19

As seen in Figure 6.35, Equation 6.18 with a lower limit of 0.88 for K gives

reasonable approximations for both LSBs and HFBs. However, some of the FEA

data points of HFBs with high slenderness were below the design curve predicted by

Eq.6.18. Therefore the lower limit of K was increased to 0.90 as shown next.

xweb

f

EIGJ

K+

=85.0

1 ≥ 0.90 (6.20)

Figure 6.36: Moment Capacities of Hollow Flange Steel Beams with the

Modified Slenderness Parameter K as Defined in Equation 6.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40


Mu/

My,

Mb/M

yEquation 6.18Available LSBs125x45x1.8 LSB135x50x1.6 LSB125x47x2.4 LSB25090HFB2825090HFB2320090HFB23

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40


Mu/M

y, M

b/M

y

Equation 6.18Available LSBs125x45x1.8 LSB135x50x1.6 LSB125x47x2.4 LSB25090HFB2825090HFB2320090HFB23


6-55

Figure 6.36 shows that Equation 6.18 with a lower limit for K of 0.90 as defined in

Eq. 6.20 is accurate for all the hollow flange steel beams such as HFBs and LSBs.

Hence it is recommended that Equations 6.18 and 6.20 can be used to accurately

predict the member moment capacities of hollow flange steel beams with varying

section geometries.

6.8 Applicability of the Geometrical Parameter for Other Types of Hollow

Flange Steel Beams

Although hollow flange beams and LiteSteel beams are the common shapes

introduced by the industry in recent times, other types of hollow flange steel beams

such as Monosymmetric Hollow Flange Beam (MHFB) and Rectangular Hollow

Flange Beam (RHFB) shown in Figure 6.37 are likely to be introduced by the

industry in the future. Three MHFBs and four RHFBs with different GJf/EIxweb

values were therefore considered in this research to investigate the applicability of

the developed design rules for lateral distortional buckling. Their dimensions are

given in Table 6.18. The dimensions of RHFB sections are similar to those of LSBs.

The section slenderness characteristics and the yield stresses of RHFBs and MHFBs

are similar to that of LSBs and HFBs used in this research.

Figure 6.37: MHFB and RHFB Sections

(a) MHFB

bf

df

D

bf

(b) RHFB

D

df


6-56

Table 6.18: Dimensions of MHFB and RHFB Sections

Hollow Flange Steel Beams

Depth Clear Depth of Web

Flange Width

Flange Depth Thickness

D d1 bf df t MHFB Sections (mm) (mm) (mm) (mm) (mm) 26585MHFB36 265 215 85 25 3.6 18079MHFB32 180 130 79 25 3.2 21090MHFB38 210 138 90 36 3.8 RHFB Sections

300x60x2.0 RHFB 300 260 60 20 2.0 250x75x2.5 RHFB 250 200 75 25 2.5 200x45x1.6 RHFB 200 170 45 15 1.6 125x45x2.0 RHFB 125 95 45 15 2.0

The corners were not considered in the analyses of MHFBs and RHFBs as for LSBs.

The torsional constant of flange Jf and the major axis flexural rigidity of the web

EIxweb, and the two important geometrical parameters GJf/EIxweb and K were calculated

for these sections. The relevant calculations are presented in Appendix D.5 while

Table 6.19 presents the results.

Table 6.19: Section Properties of MHFBs and RHFBs including K

MHFB Sections Iweb (103 mm4)

EIxweb (106 Nmm2)

Jf (103 mm4)

GJf (106 Nmm2) GJf/EIxweb K

26585MHFB36 2982 596303 58 4653 0.0078 1.0657 18079MHFB32 586 117173 49 3943 0.0337 0.9676 21090MHFB38 832 166445 138 11072 0.0665 0.9026 RHFB Sections

300x60x2.0 RHFB 2929 585867 57 4589 0.0078 1.0655 250x75x2.5 RHFB 1667 333333 140 11204 0.0336 0.9677 200x45x1.6 RHFB 655 131013 19 1524 0.0116 1.0440 125x45x2.0 RHFB 143 28579 22 1786 0.0625 0.9091

Both elastic and nonlinear finite element analyses were carried out for the beams

shown in Table 6.18. Nonlinear finite element analyses were undertaken using the

critical negative imperfection of L/1000, but without any residual stresses as their

residual stresses are not known. Flange and web yield stresses of 550 MPa and 475

MPa, respectively, were considered for MHFBs as they were the nominal yield

stresses of HFBs while the LSB flange and web yield stresses of 450 MPa and 380


6-57

MPa, respectively were considered in the finite element analyses of RHFBs. It was

observed that the ultimate moments were critical with negative imperfections for

singly symmetrical sections such as LSBs and MHFBs while both positive and

negative imperfections gave the same results for doubly symmetric sections such as

HFBs and RHFBs.

Tables 6.20 and 6.21 present the ultimate moments, elastic lateral buckling moments

and the first yield moments of MHFBs and RHFBs, respectively. The ultimate FEA

moments of all the available 13 LSBs without residual stresses are provided in Table

6.22. The FEA moment capacity results given in these tables are used to plot Figures

6.38 (a) to (d) for different types of hollow flange steel beams but with similar values

of GJf/EIxweb.

Table 6.20: FEA Results of MHFB Sections without Residual Stresses

MHFB

Sections Span (mm)

Mod (kNm)

My (kNm) λd

Mu (kNm) Mu/My

26585MHFB36

1200 217.06 101.53 0.68 91.81 0.90 1500 152.72 101.53 0.82 86.92 0.86 2000 101.25 101.53 1.00 73.09 0.72 3000 61.25 101.53 1.29 53.06 0.52 4000 44.44 101.53 1.51 40.54 0.40 6000 28.97 101.53 1.87 27.54 0.27 8000 21.57 101.53 2.17 21.70 0.21

18079MHFB32

1200 110.03 51.75 0.69 48.94 0.95 1500 84.32 51.75 0.78 47.69 0.92 2000 61.91 51.75 0.91 44.78 0.87 3000 41.24 51.75 1.12 35.80 0.69 4000 31.05 51.75 1.29 29.33 0.57 6000 20.78 51.75 1.58 20.87 0.40 8000 15.61 51.75 1.82 16.71 0.32

21090MHFB38

1200 219.61 80.52 0.61 77.96 0.97 1500 169.78 80.52 0.69 75.44 0.94 2000 127.27 80.52 0.80 73.90 0.92 3000 87.52 80.52 0.96 68.53 0.85 4000 66.99 80.52 1.10 59.91 0.74 6000 45.48 80.52 1.33 44.55 0.55 8000 34.35 80.52 1.53 35.28 0.44 10000 27.58 80.52 1.71 28.29 0.35


6-58

Table 6.21: FEA Results of RHFB Sections without Residual Stresses

RHFB Sections

Span (mm)

Mod (kNm)

My (kNm) λd

Mu (kNm) Mu/My

300x60x2.0 RHFB

1300 42.08 45.18 1.04 31.71 0.70 1500 33.53 45.18 1.16 27.58 0.61 2000 22.80 45.18 1.41 20.52 0.45 2500 18.19 45.18 1.58 16.83 0.37 3000 15.69 45.18 1.70 15.09 0.33 4000 12.80 45.18 1.88 12.71 0.28 6000 9.57 45.18 2.17 9.90 0.22 8000 7.59 45.18 2.44 7.83 0.17

250x75x2.5 RHFB

1200 99.30 50.4 0.71 48.50 0.96 1500 71.25 50.4 0.84 45.07 0.89 2000 51.13 50.4 0.99 39.19 0.78 3000 37.41 50.4 1.16 33.02 0.66 4000 31.11 50.4 1.27 28.90 0.57 6000 23.39 50.4 1.47 23.12 0.46

200x45x1.6 RHFB

800 24.05 17.23 0.85 14.99 0.87 1000 16.80 17.23 1.01 12.68 0.74 1500 10.14 17.23 1.30 8.97 0.52 2000 7.89 17.23 1.48 7.36 0.43 3000 5.95 17.23 1.70 5.86 0.34 4000 4.81 17.23 1.89 4.90 0.28 6000 3.44 17.23 2.24 3.68 0.21

125x45x2.0 RHFB

500 44.74 11.15 0.50 11.36 1.02 750 25.04 11.15 0.67 11.27 1.01 1000 18.66 11.15 0.77 10.87 0.98 1250 15.63 11.15 0.84 10.51 0.94 1500 13.70 11.15 0.90 10.22 0.92 2000 11.12 11.15 1.00 9.40 0.84 3000 8.05 11.15 1.18 7.65 0.69 4000 6.25 11.15 1.34 6.29 0.56 6000 4.29 11.15 1.61 4.62 0.41 8000 3.25 11.15 1.85 3.60 0.32


6-59

Table 6.22: FEA Results of LSB Sections without Residual Stresses


1500 69.79 1750 N/A 1750 31.20 2000 63.70 2000 52.38 2000 27.89 3000 53.56 3000 42.50 3000 20.25 4000 46.41 4000 37.03 4000 17.05 6000 35.55 6000 29.29 6000 12.91 8000 28.99 8000 24.05 8000 10.43 10000 24.34 10000 20.48 10000 8.93

Span (mm) 250x75x3.0LSB Span (mm) 250x75x2.5LSB Span (mm) 250x60x2.0LSB1250 58.28 1500 47.20 1500 28.22 1500 56.88 2000 44.14 2000 23.96 2000 54.28 3000 39.60 3000 19.62 3000 48.80 4000 35.70 4000 16.80 4000 44.06 6000 29.12 6000 12.95 6000 35.13 8000 24.02 8000 10.45 8000 28.81 10000 20.40 10000 8.92 10000 24.26


1000 30.65 1250 23.87 1250 12.96 1250 30.05 1500 22.80 1500 11.43 1500 28.92 2000 21.38 2000 9.76 2000 27.50 3000 18.83 3000 7.69 3000 24.41 4000 16.52 4000 6.36 4000 21.20 6000 12.85 6000 4.70 6000 16.04 8000 10.42 8000 3.88 8000 12.83 10000 8.88 10000 3.63 10000 10.87

Span (mm) 150x45x2.0LSB 150x45x1.6LSB 125x45x2.0LSB 125x45x1.6LSB

750 14.63 11.13 11.33 9.04 1000 13.40 10.60 11.18 8.84 1250 12.96 10.10 11.05 8.71 1500 12.40 9.60 10.89 8.55 2000 11.46 8.94 10.47 8.21 3000 9.50 7.48 9.12 7.28 4000 7.83 6.28 7.69 6.18 6000 5.76 4.72 5.68 4.68 8000 4.63 3.79 4.53 3.74 10000 4.20 3.39 3.99 3.22


6-60

(a) Sections with a GJf/EIxweb Value of 0.0078

(b) Sections with GJf/EIxweb Values of 0.0116-0.0149

Figure 6.38: Comparison of Moment Capacities of Hollow Flange Steel Beams

with Similar Values of GJf/EIxweb

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/

My,

Mb/

My

Equation 6.7

300x60x2.0 LSB (0.0078)

26585MHFB36 (0.0078)

3006020RHFB (0.0078)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

MU/M

y, M

b/M

y

Equation 6.7250x60x2.0 LSB (0.0149)200x45x1.6 LSB (0.0116)200x45x1.6 RHFB (0.0116)


6-61

(c) Sections with GJf/EIxweb Values of 0.0331-0.0337

(d) Sections with GJf/EIxweb Values of 0.0625-0.0667

Figure 6.38: Comparison of Moment Capacities of Hollow Flange Steel Beams

with Similar Values of GJf/EIxweb

As seen in these figures, the FEA moment capacity data points of different hollow

flange steel beams follow the same trend provided they have similar GJf/EIxweb (or K)

values. However, in this case, the ultimate moment capacities of these beams were

obtained without considering residual stresses. If the effect of residual stress

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/M

y, M

b/My

Equation 6.7

18079MHFB32 (0.0337)

250x75x2.5 LSB (0.0336)

200x60x2.0 LSB (0.0336)

150x45x1.6 LSB (0.0331)

250x75x2.5 RHFB (0.0336)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd

Mu/M

y, M

b/My

Equation 6.7125x45x2.0 LSB (0.0625)125x45x1.6 LSB (0.0667)21090MHFB38 (0.0665)125x45x2.0 RHFB (0.0625)


6-62

variation of the hollow flange steel beams is similar, then it can be concluded that the

use of Equation 6.18 with parameter K as defined in Eq.6.20 is suitable for all the

hollow flange steel beams such as LSB, HFB, MHFB and RHFB.

6.9 Conclusions

This chapter has presented the details of a detailed parametric study into the lateral

distortional buckling behaviour of hollow flange steel beams such as LSBs, HFBs

and their variations (MHFBs and RHFBs)). The effects of initial geometric

imperfections and residual stresses were presented first for LSBs. The comparison of

ultimate moment capacities of LSBs from finite element analyses and experimental

studies with the current design rules in AS/NZS 4600 (SA, 2005) showed that the

current design rule was conservative by about 8% in the inelastic buckling region.

New improved design rules were developed for the LSBs based on both FEA and

experimental results, which can be used to predict the moment capacities of LSBs

with a capacity reduction factor of 0.90. The applicability of the developed design

rule was investigated for HFBs and it was found that the design rule developed for

monosymmetric LSBs was very conservative as HFB is a doubly symmetric section.

A geometrical parameter defined as the ratio of flange torsional rigidity to the major

axis flexural rigidity of the web (GJf/EIxweb) was found to be a critical parameter that

reduced the scatter in the FEA data points of hollow flange steel beams in the non-

dimensionalised moment capacity plots based on Mu/My versus λd. New design rules

were developed by using a modified slenderness parameter Kλd where K was

determined as a function of GJf/EIxweb. The new design rules based on the modified

slenderness parameter Kλd were found to be accurate in calculating the moment

capacities of not only LSBs and HFBs but also other types of hollow flange steel

beams such as MHFBs and RHFBs if their residual stress variations are similar to

that of LSBs and HFBs. The developed design rules in this research can be used in

the design of hollow flange steel beams subject to uniform bending while appropriate

moment modification factors developed by Kurniawan and Mahendran (2009b) can

be used for other types of loadings.

Section Moment Capacity of LSB

7-1

CHAPTER 7

7.0 SECTION MOMENT CAPACITY OF LITESTEEL BEAM

7.1 Introduction

The LiteSteel Beams (LSBs) with intermediate and long spans are subjected to

lateral distortional and lateral torsional buckling, respectively, while short span LSBs

exhibit local buckling. Earlier chapters provided the details of experiments and finite

element analyses of LSBs subject to lateral buckling effects and the development of

accurate member moment capacity design rules. In the developed design rules, the

section moment capacity of LSBs was limited to the first yield moment. However,

Mahaarachchi and Mahendran’s (2005b) section moment capacity tests revealed that

the moment capacities of compact and non-compact LSB sections could be higher

than their first yield moments. Therefore an attempt was made to investigate the

inelastic reserve capacity of compact and non-compact LSBs. Section moment

capacity tests were carried out on selected compact, non-compact and slender LSB

sections while finite element analyses were conducted for all 13 LSB sections. Table

7.1 presents the section classification for LSBs with corners, which was determined

in accordance with AS 4100 (SA, 1998). The relevant calculations are given in

Appendix E.1. In Table 7.1, S denotes slender sections while NC and C represent

non-compact and compact sections.

Table 7.1: Section Classification for LSBs

Designation Compactness Web Flange Overall

300x75x3.0 LSB NC C NC 300x75x2.5 LSB S NC S 300x60x2.0 LSB S NC S 250x75x3.0 LSB C C C 250x75x2.5 LSB NC NC NC 250x60x2.0 LSB S NC S 200x60x2.5 LSB C C C 200x60x2.0 LSB NC NC NC 200x45x1.6 LSB S NC S 150x45x2.0 LSB C C C 150x45x1.6 LSB NC NC NC 125x45x2.0 LSB C C C 125x45x1.6 LSB C NC NC


7-2

7.2 Section Moment Capacity Tests of LSBs

Although Mahaarachchi and Mahendran (2005b) conducted section moment capacity

tests on all the available 13 LSBs, further tests were carried out in this research as the

quality of the LSB manufacturing process in relation to cold-forming and electric

resistance welding has improved over the last three years. It was also intended to

verify the test results of Mahaarachchi and Mahendran (2005b), in particular to

verify the presence of any inelastic reserve capacity for LSBs in bending.

For each LSB test beam, the following dimensions, beam depth (d), flange width (bf)

and the flange depth (df) and the thicknesses of LSB plate elements, were carefully

measured using a vernier calliper and a micrometer. Accurate thickness of each plate

element is important to obtain the elastic lateral distortional buckling moment as a

small change in thickness will cause significant changes to the buckling capacities of

LSBs. The LSB plate thicknesses were also accurately measured in the tensile

coupon tests after removing the coating. Table 7.2 presents the details of test

specimens including the measured LSB dimensions and the base metal thicknesses

from tensile coupon tests. The measurements of small corners were not taken as it

was difficult to measure them. Hence the nominal corner dimensions provided by the

manufacturers were used, i.e. the outer radius ro is equal to twice the thickness (2t)

and the inner radius riw is equal to 3 mm. It should be noted that the flange yield

stresses of 300x75x3.0 LSB and 250x60x2.0 LSB were based on the tensile test

results provided by the LSB manufacturers. Table 7.3 presents the measured yield

stresses of outer flange, inner flange and web elements of LSBs.

Table 7.2: Measured Dimensions of LSBs

Test No. LSB Sections

Thickness (mm) d (mm)

bf (mm)

df (mm) tof tif tw

1 150x45x1.6 LSB 1.77 1.63 1.58 150.1 45.2 14.8 2 200x45x1.6 LSB 1.79 1.66 1.61 200.0 45.5 15.2 3 150x45x2.0 LSB 2.22 2.02 1.97 150.1 45.4 14.8 4 250x75x2.5 LSB 2.90 2.60 2.54 251.1 75.0 25.5 5 300x75x3.0 LSB 3.22 3.13 3.09 299.0 74.6 24.8 6 250x60x2.0 LSB 2.18 2.02 1.95 250.1 60.4 20.4 7 300x60x2.0 LSB 2.22 2.02 1.98 300.1 60.0 19.8

Note: tof – outer flange thickness, tif – inner flange thickness, tw – web thickness.


7-3

Table 7.3: Measured Yield Stresses of LSBs

Test No. LSB Sections fyof (MPa) fyif (MPa) fyw (MPa) 1 150x45x1.6 LSB 557.8 487.5 455.1 2 200x45x1.6 LSB 536.9 491.3 456.6 3 150x45x2.0 LSB 537.6 491.8 437.1 4 250x75x2.5 LSB 552.2 502.2 446.0 5 300x75x3.0 LSB 497.8* 481.5* 440.1* 6 250x60x2.0 LSB 523.0* 473.0* 429.9* 7 300x60x2.0 LSB 557.7 496.3 447.1

Note: fyof – outer flange yield stress, fyif – inner flange yield stress, fyw – web yield stress. * from LSB manufacturers (OATM).

7.2.1 Test Set-Up and Procedure

The section moment capacities of LSBs were determined based on four-point

bending tests of a pair of LSBs connected back to back with web plate stiffeners at

the loading and support locations. This allowed the use of a symmetric and

convenient test set-up and loading arrangement. Figure 7.1 shows the schematic

diagram of the four point bending test arrangement used in this study.

Figure 7.1: Schematic Diagram of the Test Set-Up

Spreader beam

Loading

Load Cell Rollers

Load transfer plate

Roller bearingsTransducersSupport

T-Stiffeners

T-Stiffeners

a b a

LSB


7-4

Table 7.4: Spans of Test Beams

Test No. LSB Sections a

(mm) b

(mm) Span (mm)

Total Length of Test Beam (mm)

1 150x45x1.6 LSB 575 575 1725 1820 2 200x45x1.6 LSB 750 750 2250 2345 3 150x45x2.0 LSB 575 450 1600 1695 4 250x75x2.5 LSB 925 925 2775 2870 5 300x75x3.0 LSB 1100 1100 3300 3395 6 250x60x2.0 LSB 925 600 2450 2545 7 300x60x2.0 LSB 1100 500 2700 2795

Test beam dimensions “a” and “b” were selected so that lateral buckling is

eliminated in the tests. Maharachchi and Mahendran (2005b) stated that shear

buckling is likely to occur in the test section between the loading and support

locations if “a” is too small. Therefore the dimension “a” was chosen to be equal to

or greater than dimension “b” although the critical segment where a uniform bending

moment occurs is segment “b”. However, the top compression flange of the test

beam was laterally restrained by placing large LSB members on either side as shown

in Figure 7.2. Frictionless bolts were placed between the lateral restraints (large

LSB) and the compression flange of the test beam to resist any lateral movement

while allowing the test beam to move vertically. Table 7.4 gives the spans and total

lengths of test beams.

T – shaped steel plate stiffeners were used to connect the LSBs back to back and to

support and transfer the loads. Steel plates with the same height as the LSB web

element were also attached to the test beam on both sides of the web to avoid any

relative movement between LSBs. All the plates and T-stiffeners were connected to

the web of the test specimens by using 18 mm diameter bolts with a vertical spacing

of 100 mm symmetrically from the centreline and a horizontal spacing of 45 mm.

However, the vertical bolt spacing was limited to 45 mm for 200x45x1.6 LSB,

150x45x2.0 LSB and 150x45x1.6 LSB due to their smaller web depths. Figure 7.2

shows the overall view of the test set-up.


7-5

Figure 7.2: Test Set-Up

Figure 7.2: Overall View of Test Set-up Test specimens were supported on half rounds placed upon ball bearing as shown in

Figure 7.3. The bottom surfaces of the half rounds and alloy balls were machine

ground and polished to a high degree of smoothness, and smooth ball bearing

surfaces were lubricated to further facilitate the sliding of the half rounds on the ball

bearing when the beam deflected under load. The ends of the test beam were free to

rotate upon the half rounds. Thus it was considered that simply supported conditions

were simulated accurately at the end supports.

Figure 7.3: Simply Supported Conditions at the End Supports

Support

T-Shaped Stiffeners

Lateral Restraint

Spreader Beam

Load Cell

Hydraulic Ram

Back to Back Test Beam


7-6

The simply supported LSB specimens were tested by loading them symmetrically at

two points through a spreader beam that was loaded centrally by a hydraulic ram and

an ‘Enerpac” electrical hydraulic pump. This four-point bending arrangement

provided a central region of uniform bending moment and zero shear force. A 500

kN “Phillips” load cell connected to the EDCAR data acquisition system was used to

measure the applied load. During the tests, the vertical deflections were measured

using displacement transducers located at the top and bottom flanges of the specimen

at mid-span and loading points. The EDCAR data acquisition system was used to

record the applied load and all the deflections until the failure of specimen. Figure

7.4 shows the details of load application and deflection measurement technique used

in the experiments.

Figure 7.4: Load Application and Deflection Measurement

7.2.2 Test Results and Discussion

The applied bending moment was calculated by multiplying the measured applied

load with the distance between the support and the loading point. Typical moment

versus deflection curves are shown in Figures 7.5 to 7.7. The deflections measured at

the loading points and the mid-spans are denoted as “Load-point” and “Mid-span” in

these figures. Other curves are presented in Appendix E.2.

Displacement Transducers

“Enerpac” Hydraulic Ram

Loading arm


7-7

0

2

4

6

8

10

12

14

16

18

20

22

0 5 10 15 20 25 30 35 40Deflection (mm)

Mom

ent (

kNm

)Load-point

Mid-span

Figure 7.5: Moment vs Vertical Deflection Curves of 150x45x2.0 LSB


0

10

20

30

40

50

60

70

80

90

100

0 5 10 15 20 25 30 35 40 45Deflection (mm)

Mom

ent (

kNm

)

Load-point

Mid-span


7-8


All the specimens failed by local buckling of the top compression flange at mid-span

near the maximum load that was followed by a rapid unloading and increased

deflection. Local web buckling was also observed soon after flange local buckling.

Elastic buckling was not observed in any test as the flange elements of all the LSBs

were either compact or non-compact as shown in Table 7.1. Figures 7.5 to 7.7 show

the moment versus deflection curves for compact, non-compact and slender LSBs,

respectively. For the compact section (150x45x2.0 LSB), the vertical deflection

increased linearly with moment until the ultimate moment, which was followed by a

long horizontal plateau as shown in Figure 7.5. This was as expected for a compact

section. For non-compact section (300x75x3.0 LSB), the plateau was not significant

as shown in Figure 7.6 while for slender section (200x45x1.6 LSB), the load dropped

suddenly with increasing deflection after failure as shown in Figure 7.7. The good

agreement with the expected moment versus deflection curves for compact, non-

compact and slender LSB sections confirms the accuracy of the experimental

investigation in relation to loading method and measurements. Typical local buckling

failures observed in the tests are shown in Figures 7.8 to 7.10. Local buckling failure

occurred within the mid-span of the test beam in most cases.

0

2

4

6

8

10

12

14

16

18

20

22

0 5 10 15 20 25 30

Deflection (mm)

Mom

ent (

kNm

)

Load-point

Mid-span


7-9

Figure 7.8: Plan View of Failed Specimen

Figure 7.9: Flange and Web Local Buckling

T - Stiffeners

Frictionless Bolts Lateral Restraint


7-10

Figure 7.10: Flange Local Buckling There was no sudden unloading associated with lateral deflection while no specimen

failed due to insufficient material ductility. Although the failure modes of tested

specimens were similar, there were some differences in the way the failure occurred.

For compact LSB sections, large flange deformations and yielding occurred at

moments closer to the failure moment. For non-compact sections, yielding and large

flange deformations appeared to occur earlier while for slender sections, local web

buckling occurred, which was followed by large flange deformations and yielding.

There was no welding failure in this series of tests although Mahaarachchi and

Mahendran (2005b) reported a weld failure in the test of 200x45x1.6 LSB. This

confirms that the welding strength of the new LSB sections is structurally adequate.

The ultimate moments of tested specimens are given in Table 7.5 while the next

section compares these results with the section moment capacities predicted by the

current design rules.

Table 7.5: Ultimate Moments of LSBs

Test No. LSB Sections Ultimate Moment Mu (kNm)

1 150x45x1.6 LSB 16.18 2 200x45x1.6 LSB 20.88 3 150x45x2.0 LSB 20.20 4 250x75x2.5 LSB 70.68 5 300x75x3.0 LSB 93.00 6 250x60x2.0 LSB 42.12 7 300x60x2.0 LSB 53.36

T - Stiffeners

Web Plate with M18 Bolts


7-11

7.2.3 Comparison of Ultimate Moment Capacities from Tests and Current

Design Rules

The section moment capacity (Ms) of cold-formed steel section is usually based on

the initiation of yielding in the extreme compression fibre in the Australian cold-

formed steel standard AS/NZS 4600 (SA, 2005). The inelastic reserve capacity is

allowed subject to restrictive conditions as discussed in Section 7.4. Effects of local

buckling are accounted for by using the effective widths (be) of slender elements in

compression in the calculation of effective section modulus (Ze). The plate element

slenderness is a function of the applied stress (f *) as shown in Equation 7.2. This

accounts for the reduction in the strength due to local buckling effects with

increasing member slenderness. Clause 3.3.2 of AS/NZS 4600 (SA, 2005) provides

the equation for section moment capacity (Ms) as given in Equation 7.1, where fy is

the yield stress.

Ms = fy Ze (7.1)

Ef

tb

k

*052.1⎟⎠⎞

⎜⎝⎛=λ (7.2)

where

λ = plate slenderness

k = plate buckling coefficient

b = flat width of element excluding radii

t = thickness of the uniformly compressed stiffened elements

f* = design stress in the compression element

The procedure to determine the effective widths of uniformly compressed stiffened

elements for capacity calculations is given in Clause 2.2.1.2 of AS/NZS 4600 (SA,

2005).

For λ ≤ 0.673 : be = b (7.3a)

For λ > 0.673: be = ρb (7.3b)


7-12

ρ 0.1

22.01≤

⎟⎠⎞

⎜⎝⎛−

=λλ (7.4)

The section moment capacities of tested LSBs were calculated based on the AS/NZS

4600 (SA, 2005) design method described above and without considering any

inelastic reserve capacity. It is noted that the design provisions in NAS (AISI, 2004)

are identical to those of AS/NZS 4600 and hence the comparisons and findings are

the same for both codes. It was found that the effective width is equal to the actual

width for all the elements of 13 LSBs sections when their corners are included. The

full section modulus (Z) can therefore be used to calculate the section moment

capacity. Therefore the section moment capacity Ms is equal to the first yield

moment My for all the currently available 13 LSBs when their corners are included.

Sample effective width calculations are given in Appendix D.1. The elastic section

modulus (Z) of tested beams was calculated by using Thin-Wall based on their

measured dimensions. Although the corners of LSBs were not measured, the nominal

corners were used in the calculations. Measured outer flange yield stress was

considered as fy in Equation 7.1. Table 7.6 presents the section moment capacities

(Ms) based on AS/NZS 4600 (SA 2005), ultimate moments from experiments (Mu)

and their ratios (Mu / Ms). It must be noted that in this case Ms is equal to My for all

LSBs. It can be seen that the ratio of Mu/Ms (or Mu/My) for 150x45x2.0 LSB is 1.15,

which is a compact section, while it is about 1.0 for slender sections except

300x60x2.0 LSB. For non-compact sections this ratio is 1.08 on average. This

confirms that there is some inelastic reserve moment capacity for compact and non-

compact sections while slender sections do not have it. The reason why this ratio is

less than unity for 300x60x2.0 LSB is not known. However, this is the most slender

section with very deep web element among the available 13 LSBs. It is likely the

prediction of AS/NZS 4600 (SA, 2005) is high in this case of slender LSB. Figure

7.11 shows the failure mode of 300x60x2.0 LSB which exhibits flange and web local

buckling due to the presence of a slender web element.


7-13

Table 7.6: Section Moment Capacities from Tests and AS/NZS 4600 (SA, 2005)

Test No LSB Sections

Compact-

-ness

Elastic Section

Modulus Z(103 mm3)

Section Capacity, Ms AS/NZS 4600

(kNm)

Ultimate Moment

Mu (kNm)

Mu/Ms

1 150x45x1.6 LSB NC 26.71 14.90 16.18 1.09 2 200x45x1.6 LSB S 40.15 21.56 20.88 0.97 3 150x45x2.0 LSB C 32.75 17.61 20.20 1.15 4 250x75x2.5 LSB NC 120.10 66.32 70.68 1.07 5 300x75x3.0 LSB NC 174.70 86.97 93.00 1.07 6 250x60x2.0 LSB S 80.12 41.90 42.12 1.01 7 300x60x2.0 LSB S 102.80 57.33 53.36 0.93

Figure 7.11: Failure Mode of 300x60x2.0 LSB

Section moment capacity test results from Mahaarachchi and Mahendran (2005b)

were also considered in this research as the current tests did not include all the

available LSBs. Measured dimensions and yield stresses of Mahaarachchi and

Mahendran (2005b) were used to calculate the elastic section modulus (Z) and the

section moment capacities of LSBs. As decided earlier, nominal corners were

included in the calculations. Tables 7.7 to 7.9 give the details of the tests conducted

by Mahaarachchi and Mahendran (2005b) and the results.


7-14

Table 7.7: Measured Dimensions of LSBs used in Mahaarachchi and

Mahendran’s (2005b) Section Moment Capacity Tests

LSB Thickness, t (mm) d

(mm) bf

(mm)df

(mm) Z

(103mm3) tof tif tw 300x75x3.0 LSB 3.18 3.18 2.84 300 75.31 25.17 173.90 300x75x2.5 LSB 2.87 2.87 2.51 300 75.24 25.05 157.90 300x60x2.0 LSB 2.15 2.15 1.98 300 60.28 19.97 104.00 250x75x3.0 LSB 3.08 3.08 2.77 250 76.35 25.22 132.80 250x75x2.5 LSB 2.79 2.79 2.48 250 75.98 24.92 120.90 250x60x2.0 LSB 2.09 2.09 1.96 250 60.47 20.12 79.12 200x60x2.5 LSB 2.58 2.58 2.34 200 60.23 19.95 70.34 200x60x2.0 LSB 2.03 2.03 1.85 200 60.15 20.31 56.17 200x45x1.6 LSB 1.56 1.56 1.48 200 45.05 14.98 36.14 150x45x2.0 LSB 2.11 2.11 1.89 150 44.95 14.73 32.01 150x45x1.6 LSB 1.60 1.60 1.60 150 45.12 14.89 25.12 125x45x2.0 LSB 1.98 1.98 1.98 125 45.1 14.93 23.73 125x45x1.6 LSB 1.62 1.62 1.62 125 45.07 14.95 19.71

Note: tof – outer flange thickness, tif – inner flange thickness, tw – web thickness, Z – elastic section modulus.

Table 7.8: Measured Yield Stresses of LSBs used in Mahaarachchi and

Mahendran’s (2005b) Section Moment Capacity Tests

LSB Sections fyof (MPa) fyif (MPa) fyw (MPa) 300x75x3.0 LSB 528 438 431 300x75x2.5 LSB 511 457 434 300x60x2.0 LSB 568 492 452 250x75x3.0 LSB 506 459 406 250x75x2.5 LSB 525 478 420 250x60x2.0 LSB 580 502 448 200x60x2.5 LSB 496 465 388 200x60x2.0 LSB 473 439 386 200x45x1.6 LSB 478 442 381 150x45x2.0 LSB 498 451 373 150x45x1.6 LSB 540 483 430 125x45x2.0 LSB 503 455 377 125x45x1.6 LSB 549 478 431

Note: fyof – outer flange yield stress, fyif – inner flange yield stress, fyw – web yield stress.


7-15

Table 7.9: Section Moment Capacities from Mahaarachchi and Mahendran’s

(2005b) Tests and AS/NZS 4600 (SA, 2005)

LSB Sections Compactness Ms (kNm) Mu (kNm) Mu/Ms 300x75x3.0 LSB NC 91.82 103.90 1.13 300x75x2.5 LSB S 80.69 85.80 1.06 300x60x2.0 LSB S 59.07 52.40 0.89 250x75x3.0 LSB C 67.20 77.89 1.16 250x75x2.5 LSB NC 63.47 71.49 1.13 250x60x2.0 LSB S 45.89 47.33 1.03 200x60x2.5 LSB C 34.89 52.47 1.50 200x60x2.0 LSB NC 26.57 31.80 1.20 200x45x1.6 LSB S 17.27 17.36 1.00 150x45x2.0 LSB C 15.94 19.63 1.23 150x45x1.6 LSB NC 13.56 14.94 1.10 125x45x2.0 LSB C 11.94 14.38 1.20 125x45x1.6 LSB NC 10.82 12.95 1.20

As seen in Table 7.9, the Mu/Ms ratios of compact sections are 1.20 on average

except 200x60x2.5 LSB, which has a value of 1.50 while non-compact sections have

a ratio of 1.15 on average. Slender sections have a ratio of 1.0 on average. It can be

seen that the ratio of Mu/Ms for 300x60x2.0 LSB from Mahaarachchi and

Mahendran’s (2005b) test was 0.89, which compares well with the ratio of 0.93 from

the experiments of this research. This confirms the lower section moment capacity

ratio observed with the slender 300x60x2.0 LSB section. The Mu/Ms ratio is below 1

for 300x60x2.0 LSB, which indicates that it could not reach the first yield moment.

This is as predicted by AS 4100 (SA, 1998) for this slender section according to its

classification. However, AS/NZS 4600 (SA, 2005) predicted that 300x60x2.0 LSB

section will reach its first yield moment and hence Ms is equal to My and the Mu/Ms

ratio becomes less than 1.0 for 300x60x2.0 LSB. This implies that AS/NZS 4600

(SA, 2005) is unconservative in predicting the section moment capacities of some

slender LSBs. However, experimental results alone are not sufficient to confirm this.

Based on the section moment capacity test results from Mahaarachchi and

Mahendran (2005b) and this research, it is concluded that only compact and non-

compact LSB sections have inelastic reserve capacity. Since it is not accurate to

develop inelastic reserve capacity design rules based on experimental results alone,

numerical studies were also conducted for all the available 13 LSBs. The following


7-16

section provides the details of finite element analyses of LSBs to determine their

section moment capacities.

7.3 Finite Element Modelling of LSBs to Determine their Section Moment

Capacities

Although two types of finite element models, ideal and experimental finite element

models, were considered in the investigation of lateral buckling capacities of LSBs as

described in Chapter 5, only the experimental finite element model was considered in

the investigation of the section moment capacities of LSBs. This experimental model

included the actual experimental conditions with measured dimensions and yield

stresses. The results from the experimental finite element models were compared

with test results to validate the finite element models in relation to the element type,

mesh size, initial geometrical imperfections, residual stress, local buckling

deformation and material yielding. Following the validation of the model, idealised

simply supported boundary conditions, nominal dimensions and yield stresses were

applied to this model and analyses were conducted using this model in order to

develop suitable design rules. Details of the experimental finite element models used

this study are described next.

7.3.1 Experimental Finite Element Model of LSBs

A total of seven section moment capacity tests were carried out in this research and

all of them were modelled using MSC/ PATRAN (MSC Software, 2008) pre-

processing facilities while ABAQUS (HKS, 2007) was used to analyse the models.

MSC/PATRAN (MSC Software, 2008) post-processing facilities were then used to

view the results of ABAQUS analyses. The shell element in ABAQUS (HKS, 2007)

called S4R5 was used to develop the LSB model as in the previous models of LSBs

subject to lateral buckling described in Chapter 5. A mesh size of 5 mm x 10 mm

was selected to be appropriate, i e. 5 mm along the cross section and 10 mm along

the longitudinal direction.


7-17

Figure 7.12: Schematic Diagram of Experimental Finite Element Model

Figure 7.12 shows the schematic diagram of the experimental finite element model.

In the experimental study, two LSBs were connected back to back and the load was

applied through the spreader beam at mid-span (see Figures 7.1 and 7.2). This type of

arrangement was considered to eliminate the twisting of test beams, which is

considered to be equivalent to the shear centre loading. Such back to back beam

testing is commonly used in section moment capacity tests. However, it is not

necessary to model the actual experimental set-up including two LSBs connected

back to back and the spreader beam. A simplified model of single LSB loaded at its

shear centre was considered to be appropriate as it will simulate the actual

experimental conditions. Therefore, the experimental finite element model shown in

Figure 7.13 was considered to be appropriate in the validation of numerical analyses.

Only half the span was modelled due to the symmetrical nature of loading and

boundary conditions of the test set-up. The material model, mechanical properties

and boundary conditions were the same as for the experimental finite element models

of LSBs described in Chapter 5.

Span/2

P

a b / 2


7-18

Figure 7.13: Loading and Boundary Conditions of Experimental Finite Element

Model

Idealised simply supported boundary conditions and a point load were applied at the

shear centre as shown in Figures 7.13 (a) and (b). In the experiments, only the top

flange (compression) was laterally restrained as shown in Figure 7.2. This was

simulated in the finite element model by using the boundary condition of SPC 345 at

(a) Isometric View

Support at Shear Centre, (SPC 234)

Symmetric Plane (SPC 156)


(b) Plan View

Support at Shear Centre Loading at Shear Centre

Lateral Restraints (SPC 345)

(c) Lateral Restraints of Flanges


7-19

all the nodes on the outer face of the flanges where lateral deflection (SPC 3),

twisting (SPC 4) and minor axis rotation (SPC 5) were locked as shown in Figure

7.13 (c). Although the experiments included lateral restraints in the top flange only,

the finite element models included such restraints for both top and bottom flanges as

the preliminary finite element analyses revealed some lateral displacements of

bottom flange (tension flange) due to numerical instability.

The web side plates that were used to connect the LSBs and T-Stiffeners were

modelled as rigid body by using R3D4 elements. In ABAQUS (HKS, 2007) a rigid

body is a collection of nodes and elements whose motion is governed by the motion

of a single node, known as the rigid body reference node. The motion of the rigid

body can be prescribed by applying boundary conditions at the rigid body reference

node. So the simply supported boundary conditions and the load were applied on the

corresponding rigid body reference nodes at the shear centre. Figure 7.14 identifies

the various plate elements with different mechanical material properties as defined in

ABAQUS. The elastic perfect plastic material model with measured yield stresses

was considered in this study.

Figure 7.14: Various Plate Elements in Experimental Finite Element Model

A geometrical imperfection was included in the finite element model. An

imperfection magnitude of depth or width/150 as described by the manufacturers was

considered to be appropriate as the local plate imperfection. Global geometrical

imperfection was not included as the LSBs were subject to only local buckling

Web

Outside Flange

Inside Flange

Steel Stiffeners as Loading and Support Plates


7-20

effects. Usually, the initial buckling mode from the elastic buckling analyses is

considered to be critical and this buckling mode is used to apply the initial geometric

imperfections. However, most of the initial elastic buckling modes of LSBs from the

finite element models exhibited a local buckling failure between the support and the

load while higher modes revealed a local buckling failure at mid-span as seen in

Figure 7.15 (a). Therefore, the initial geometric imperfections were applied based on

the higher modes of the elastic buckling analyses for those beams. Both membrane

and flexural residual stresses were also included in the finite element analyses. These

residual stress distributions were the same as used in the LSB models subject to

lateral buckling described in Chapter 5. In the SIGINI Fortran user subroutine (see

Appendix C), the lateral deflections at the top and bottom flanges were set to zero as

there were no lateral displacements.

Figure 7.15: Failure Modes from Finite Element Analyses of 150x45x2.0 LSB

(a) Elastic Buckling Analyses

(b) Non-linear Static Analyses


7-21

Figure 7.16: Failure Modes from Finite Element Analyses of 300x60x2.0 LSB

Both elastic and non-linear static analyses were carried out for the developed LSB

models. Figures 7.15 and 7.16 show the failure modes of 150x45x2.0 LSB and

300x60x2.0 LSB, respectively, from both elastic and non-linear static analyses. The

failure shape obtained from non-linear static analysis was similar to that exhibited in

the experiment. It can be seen that the ultimate failure mode of 300x60x2.0 LSB

exhibits a web local buckling (Figure 7.16 (b)) while that of 150x45x2.0 LSB

exhibits only yielding (Figure 7.15 (b)). This agreed with the experimental failure

modes of these slender (300x60x2.0 LSB) and compact (150x45x2.0 LSB) sections.

As described above, significant level of web local buckling could be the reason for

the reduced ratio of Mu/Ms for the slender 300x60x2.0 LSB section. It should be

noted that the local buckling failure in these beams occurred at mid-span within the

loading points during the experiments and the corresponding finite element model

was able to simulate the local buckling failure and the location where it occurred.

Further, Figure 7.17 shows the typical post-ultimate failure mode obtained from

(b) Non-linear Static Analysis

(a) Elastic Buckling Analysis


7-22

finite element analyses. Flange and web local buckling with yielding as seen in this

figure agreed well with the failure mode in the experiments (Figure 7.9). Thus

Figures 7.15 to 7.17 confirm that the developed finite element model accurately

predicts the failure modes of LSBs.

Figure 7.17: Typical Buckling Mode after Failure from FEA

Table 7.10 compares the ultimate moment capacity results from the non-linear finite

element analyses and experiments undertaken in this research. A comparison of FEA

and experimental test results is also provided in the form of bending moment versus

vertical deflection curves in Figures 7.18 and 7.19 for different LSB sections. Other

curves are presented in Appendix E.3. These figures compare the measured

experimental vertical deflections at loading point and mid-span with the

corresponding deflections predicted by FEA. The good agreement between the

results from experiments and finite element analyses indicates that the developed

finite element model is accurate.

Table 7.10: Comparison of Experimental and FEA Ultimate Moment Capacities

LSB Sections Ultimate Moments (kNm) Experiments FEA EXP/FEA

300x75x3.0 LSB 93.00 93.4 1.00 300x60x2.0 LSB 53.36 53.0 1.01 250x75x2.5 LSB 70.68 68.0 1.04 250x60x2.0 LSB 42.12 40.5 1.04 200x45x1.6 LSB 20.88 20.6 1.01 150x45x2.0 LSB 20.20 20.1 1.00 150x45x1.6 LSB 16.18 15.6 1.04

Mean 1.02 COV 0.018


7-23

Figure 7.18: Bending Moment vs Vertical Deflection of 150x45x1.6 LSB

Figure 7.19: Bending Moment vs Vertical Deflection of 200x45x1.6 LSB

The comparisons provided in Table 7.10 and Figures 7.18 and 7.19 demonstrate that

the experimental finite element model predicts the ultimate failure moments of LSBs

accurately. The mean ratio of the ultimate moment capacities from experiments and

finite element analyses was 1.02 with a COV of 0.018. This result suggests that the

model is accurate, considering the possible sources of error caused by unavoidable

differences between the experimental test and finite element model.

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25 30

Deflection (mm)

Mom

ent (

kNm

)

Load-point:ExpMid-span:ExpLoad-point:FEAMid-span:FEA

0

2

4

6

8

10

12

14

16

18

20

22

0 5 10 15 20 25 30

Deflection (mm)

Mom

ent (

kNm

)



7-24

An attempt was made to numerically model the section capacity tests carried out by

Mahaarachchi and Mahendran (2005b) based on the experimental finite element

model developed here. The measured dimensions and yield stresses in Tables 7.7 and

7.8 were used in these finite element models. Table 7.11 compares the ultimate

moment capacities from experiments and finite element analyses.

Table 7.11: Comparison of Mahaarachchi and Mahendran’s (2005b)

Experimental and FEA Ultimate Moment Capacities

LSB Sections Ultimate Moments (kNm) EXP/FEA EXP FEA

300x75x3.0 LSB 103.90 89.5 1.16 300x75x2.5 LSB 85.80 80.7 1.06 300x60x2.0 LSB 52.40 54.8 0.96 250x75x3.0 LSB 77.89 70.2 1.11 250x75x2.5 LSB 71.49 65.8 1.09 250x60x2.0 LSB 47.33 43.7 1.08 200x60x2.5 LSB 52.47 37.5 1.40* 200x60x2.0 LSB 31.80 28.4 1.12 200x45x1.6 LSB 17.36 16.9 1.03 150x45x2.0 LSB 19.63 17.0 1.15 150x45x1.6 LSB 14.94 14.1 1.06 125x45x2.0 LSB 14.38 13.4 1.07 125x45x1.6 LSB 12.95 11.7 1.11

Mean 1.08 COV 0.051

*Not considered in calculating Mean and COV. As seen in Table 7.11, most of the FEA ultimate moments are less than the

experimental failure moments. But it does not mean that the finite element model

was inadequate in predicting the ultimate moments of Mahaarachchi and

Mahendran’s (2005b) tests as the same model accurately predicted the ultimate

moments of the tests carried out in this research. The main difference between the

two series of tests is in relation to the yield stresses of test specimens. Therefore, it is

suspected that the measured yield stresses provided in Mahaarachchi and Mahendran

(2005b) were not accurate for their section moment capacity test specimens. Also it

appears that the ratio of the outer flange yield stress to web yield stress (fyof/fyw) has

a significant influence on their ultimate moments. When the yield stresses of web,

inner flange and outer flange were taken as 450, 460 and 528 MPa in the finite

element analyses instead of the reported values in Table 7.8 (431, 438 and 528 MPa),


7-25

the ultimate moment of 300x75x3.0 LSB was 91.6 kNm (increased from 89.5 kNm).

Such small changes to even the yield stresses of web and inside flange elements

appear to lead to increased ultimate moments. Hence it is possible that the difference

in FEA and experimental ultimate moments might have been caused by differences

in the yield stresses used in the analyses. It is noted that the mean and COV of the

ratio of ultimate moments from experiments and FEA are 1.08 and 0.051,

respectively, even with the yield stresses reported in Table 7.8.

It is therefore reasonable to assume that the experimental comparisons presented in

this section establish the validity of the shell element model for explicit modelling of

initial geometric imperfections and residual stresses, local buckling deformations,

and associated material yielding of LSBs. The suitability of the residual stress model,

geometric imperfection magnitudes, and the finite element mesh density has also

been verified.

Finite element analyses reported so far included an elastic perfect plastic material

model. This could have lead to the under-estimation of the ultimate moment

capacities of compact LSB sections. Therefore the measured stress-strain

relationships for outside and inside flanges and web elements shown in Figures 7.20

(a) to (c) were used in the non-linear analyses of one of the compact sections,

150x45x2.0 LSB. Measured stress-strain relationships using tensile coupon tests

were simplified as shown in Figures 7.20 (a) to (c), and the true stresses and strains

were also plotted in these figures. Relevant calculations are presented in Appendix

E.4. The ultimate moment capacity from this analysis is 19.0 kNm, which is only 1%

higher than the corresponding value of 18.8 kNm obtained using an elastic perfect

plastic model. This is possibly due to the smaller strain hardening modulus of flange

elements as seen in Figures 7.20 (a) and (b). This result confirms that it is adequate

to use an elastic perfect plastic material model for LSBs.


7-26

Figure 7.20: Stress-Strain Curves of 150x45x2.0 LSB

(a) Outer Flange

0.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450

Strain

Stre

ss, M

Pa

EngineeringTrue

(b) Inner Flange

0

100

200

300

400

500

600

700

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Strain

Stre

ss, M

Pa

EngineeringTrue


7-27

Figure 7.20: Stress-Strain Curves of 150x45x2.0 LSB

It is necessary to obtain the section moment capacities of all the available 13 LSBs

with nominal dimensions and yield stresses in order to investigate the inelastic

reserve capacity of each LSB. This can be achieved by using the non-linear analyses

based on the validated finite element model. Next section provides these details.

7.3.2 Finite Element Analyses of LSBs Subject to Local Buckling Effects

It was believed that the ideal finite element model used to develop the design curves

for LSBs subject to lateral buckling as provided in Chapter 5 can be used with

reduced spans to obtain the section moment capacities of LSBs. Therefore, an

attempt was made to develop the ideal finite element models of LSBs with nominal

dimensions and yield stresses excluding corners as for the ideal models of LSBs

subject to lateral buckling. However, in this case, the top and bottom flanges were

laterally restrained by using SPC 345 as for the finite element models used in this

chapter to resist any lateral buckling deformations.

Preliminary finite element analyses without residual stresses revealed that the ratios

of ultimate moment (Mu) from FEA to the section moment capacity Ms based on

AS/NZS 4600 (SA, 2005) did not exceed 1.06 even for compact LSBs although they

were about 1.15 from the experiments. If the residual stresses were included, the

0

100

200

300

400

500

600

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Strain

Stre

ss, M

Pa

EngineeringTrue

(c) Web


7-28

ultimate moment ratios would have been less than 1.06. Table 7.12 presents the

ultimate moments from the ideal finite element models without residual stresses for

LSBs with a span of 500 mm. The section moment capacity Ms was calculated by

using AS/NZS 4600 (SA, 2005) without corners as shown in Appendix D.1. Here,

250x75x2.5 LSB is a non-compact section and 200x45x1.6 LSB is a slender section

while 150x45x2.0 LSB is a compact section based on AS 4100 (SA, 1998).

Table 7.12: Ultimate Moments from the Ideal Finite Element Model

As seen in Table 7.12, the ratios of Mu/Ms from the ideal finite element model for

250x75x2.5 LSB and 200x45x1.6 LSB agreed reasonably well with those from

experiments. However, the ratio of Mu/Ms from the ideal finite element model was

only 1.06 while it was 1.15 from the experiment for the compact LSB section. Based

on this, it was concluded that the ideal finite element model developed was not able

to predict the section moment capacity of LSBs. Several attempts were made to

investigate the reason for this and to create an appropriate finite element model.

Finally, it was found that the method used to create a uniform bending moment along

the span based on linearly varying tension and compression nodal forces has not

allowed the section to exceed its yield moment.

The longitudinal stresses across the cross-section of LSBs are shown in Figures 7.21

(a) to (c). These figures also show the stress variations when experimental finite

element models were used with nominal dimensions and yield stresses for the three

LSBs chosen. The stress variations across the section of 150x45x2.0 LSB based on

the experimental finite element model with nominal dimensions and yield stresses

were considered to be appropriate as this shows that most of the web and flange

sections away from the neural axis have yielded, i e. showing section plastification.

However, the stress variation from the ideal finite element model reveals that most of

the web element has not yielded. The stress variation of 250x75x2.5 LSBs also

LSB Sections Ms (kNm) Mu (kNm) Ideal FE Model

FEA Mu/Ms

EXP Mu/Ms

250x75x2.5 LSB 49.30 52.21 1.06 1.07 200x45x1.6 LSB 17.20 17.39 1.01 0.97 150x45x2.0 LSB 14.35 15.19 1.06 1.15


7-29

confirms this. For 200x45x1.6 LSB, the stress variation is similar in both ideal and

experimental models as this is a slender section without any inelastic reserve moment

capacity. Figure 7.21(d) shows the longitudinal stress variation along the length of

150x45x2.0 LSB at ultimate failure based on the experimental finite element model

with nominal dimensions. It can be seen that the stresses are similar along the

longitudinal axis at midspan between the loading points (right side of loading plate in

Figure 7.21 (d)). This also confirms the absence of any torsional moment in LSB

flanges and thus also the accuracy of applying the loads through the shear centre of

LSBs. The stress variations of other LSBs are similar to that of Figure 7.21 (d).

Figure 7.21: Stress Variation across the Cross-section of LSB from FEA

-80

-60

-40

-20

0

20

40

60

80

-500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350 400 450 500

Stress, (MPa)

Dis

tanc

e ac

ross

Sec

tion,

(mm

)

Ideal Model

EXP Model (nominal dimensionsand yield stresses)

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

120

140

-500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350 400 450 500

Stress, (MPa)

Dis

tanc

e ac

ross

the

Sec

tion,

(mm

)

Ideal Model

EXP Model (nominal dimensionsand yield stresses)

(b) 250x75x2.5 LSB

(a) 150x45x2.0 LSB


7-30

Figure 7.21: Stress Variation across the Cross-section of LSB from FEA

Based on Table 7.12 and Figures 7.21 (a) to (c), it was decided to use the

experimental finite element model with nominal dimensions and yield stresses to

obtain the section moment capacities of LSBs. All the available 13 LSBs were

analysed using this experimental finite element model with nominal dimensions and

yield stresses. Initial geometric imperfections and residual stresses were also

included. For the tested LSBs, the experimental finite element models were simply

modified by replacing the measured dimensions and yield stresses with nominal

dimensions and yield stresses. Elastic perfect plastic material model with nominal

yield stress was used as the effect of including strain hardening gave only a very

small increase to the moment capacity (<1%). The ultimate moments and the

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

120

-500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350 400 450 500

Stress, (MPa)

Dis

tanc

e ac

ross

the

Sec

tion,

(mm

)

Ideal Model

EXP Model (with nominaldimensions and yield stresses)

(c) 200x45x1.6 LSB

(d) Longitudinal Stress Variation along the Length of 150x45x2.0 LSB


7-31

dimensions “a” and “b” of the LSB models used in the analyses are presented in

Table 7.13. Here, “a” is the distance between the support and loading point while “b”

is the distance between the loads (see Figures 7.1 and 7.12). The effect of residual

stresses is also presented in this table and was found to be very small with an average

reduction of 2%. It is important to compare the ultimate moments obtained from

finite element analyses with the predictions from the current design rules. Next

section provides these details.

Table 7.13: Ultimate Moment Capacities of LSBs from FEA

LSB Sections a (mm)

b (mm)

Span (mm)

Mu without RS(kNm)

Mu with RS (kNm)

Effect of RS

300x75x3.0 LSB 1100 1100 3300 82.9 81.1 0.98300x75x2.5 LSB 1100 1100 3300 68.3 66.7 0.98300x60x2.0 LSB 1100 500 2700 45.6 44.1 0.97250x75x3.0 LSB 920 920 2760 66.3 65.5 0.99250x75x2.5 LSB 920 920 2760 55.1 54.0 0.98250x60x2.0 LSB 920 600 2440 36.7 35.8 0.98200x60x2.5 LSB 750 740 2240 35.5 35.1 0.99200x60x2.0 LSB 750 740 2240 28.2 27.7 0.98200x45x1.6 LSB 750 740 2240 17.9 17.5 0.98150x45x2.0 LSB 570 460 1600 16.0 15.8 0.99150x45x1.6 LSB 570 460 1600 12.7 12.5 0.98125x45x2.0 LSB 500 500 1500 12.7 12.6 0.99125x45x1.6 LSB 500 500 1500 10.1 9.97 0.99

Note: RS – Residual Stress.

7.4 Comparison of Ultimate Moment Capacities from FEA and Current

Design Rules

The procedure to calculate the section moment capacity of steel sections is provided

in AS 4100 (SA, 1998), Eurocode 3 Part 1.3 (ECS, 1996 & 2006) and AS/NZS 4600

(SA, 2005). The AS/NZS 4600 (SA, 2005) procedure is given in Section 7.2.3 of this

chapter, which limits the section moment capacity to the first yield moment.

The nominal section moment capacity (Ms) is defined in AS 4100 (SA, 1998) as

follows:

Ms = fy Ze (7.5)

Where, fy = nominal yield stress

Ze = effective section modulus


7-32

The effective section modulus (Ze) allows for the effects of local buckling. The

section moment capacity (Ms) of a section is governed by the compactness of its

plate elements and is given by Equations 7.6 (a) to (d):

For λe ≤ λep : (Compact Sections) Ze = Zc (7.6a)

For λep < λe ≤ λey: (Non-Compact Sections) ( )⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛

−−

+= ZZZZ cepey

seye

λλλλ (7.6b)

For λe > λey (Slender Sections, Web elements) 2

⎟⎠⎞

⎜⎝⎛=

e

eye ZZ

λλ (7.6c)

For λe > λey (Slender Sections, Flange elements) ⎟⎠⎞

⎜⎝⎛=

e

eye ZZ

λλ (7.6d)

Where, λe = plate element slenderness λey = plate element yield limit (Table 5.2 of AS 4100)

λep = plate element plasticity limit (Table 5.2 of AS 4100)

Zc = effective section modulus of the compact element


The element with the greatest ratio λe/λey is to be used in calculating the effective

section modulus (Ze). The plate element slenderness (λe) is given by Equation 7.7:

250

ye

ftb⎟⎠⎞

⎜⎝⎛=λ (7.7)

where, b is the clear width of the element outstand from the face or between the faces

of the supporting plate element and t is the thickness.

The effective section modulus of the compact element (Zc) is given by Equation 7.8:

Zc= min [S, 1.5Z] (7.8)

where S is the plastic section modulus.

The section moment capacities and the plate slenderness values of LSBs were

calculated based on the above AS 4100 (SA, 1998) procedure. The corners were not

included and the centreline dimensions were used as assumed in finite element

analyses. Sample calculations are presented in Appendix E.5. Based on AS/NZS

4600 (SA, 2005) it was found that some of the LSBs have ineffective horizontal


7-33

flange elements (ie. slender elements) when corners were not included. Hence their

section moment capacities are slightly less than their first yield moments. Table 7.14

gives the section compactness of LSBs based on AS/NZS 4600 (SA, 2005) and AS

4100 (SA, 1998). Based on AS/NZS 4600 (SA, 2005), there are only five slender

LSBs as indicated by “bold S” while the slenderness values of the other three LSBs

denoted as “S” are very close to the limiting value of 0.673 (refer Appendix D.1 for

further details).

Table 7.14: Compactness of LSBs Based on AS 4100 and AS/NZS 4600

LSB Sections Section Compactness AS 4100 AS/NZS 4600

300x75x3.0 LSB NC C 300x75x2.5 LSB S S 300x60x2.0 LSB S S 250x75x3.0 LSB NC C 250x75x2.5 LSB NC S 250x60x2.0 LSB S S 200x60x2.5 LSB NC C 200x60x2.0 LSB NC S 200x45x1.6 LSB S S 150x45x2.0 LSB C C 150x45x1.6 LSB NC S 125x45x2.0 LSB C C 125x45x1.6 LSB NC S

Table 7.15: Section Moment Capacities of LSBs

LSB Sections My (kNm)

Ms (kNm) Ms/My AS 4100 AS/NZS 4600 AS 4100 AS/NZS 4600

300x75x3.0 LSB 77.24 81.97 77.24 1.06 1.00 300x75x2.5 LSB 64.79 63.30 63.47 0.98 0.98 300x60x2.0 LSB 45.17 40.71 44.31 0.90 0.98 250x75x3.0 LSB 60.06 68.60 60.06 1.14 1.00 250x75x2.5 LSB 50.38 51.39 49.30 1.02 0.98 250x60x2.0 LSB 35.10 33.80 34.39 0.96 0.98 200x60x2.5 LSB 31.98 37.30 31.98 1.17 1.00 200x60x2.0 LSB 25.79 26.31 25.24 1.02 0.98 200x45x1.6 LSB 17.23 16.48 17.20 0.96 1.00 150x45x2.0 LSB 14.35 16.96 14.35 1.18 1.00 150x45x1.6 LSB 11.58 12.35 11.56 1.07 1.00 125x45x2.0 LSB 11.15 13.21 11.15 1.18 1.00 125x45x1.6 LSB 9.00 9.60 8.98 1.07 1.00


7-34

Table 7.15 gives the first yield moment My, section moment capacity Ms based on

AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 2005), and the ratios of Ms/My. As seen

in this table, the section moment capacity predictions of AS 4100 (SA, 1998) for

compact and non-compact LSB sections are more than the first yield moments (i e.,

Ms/My >1). However, these ratios are less than one for slender sections as expected.

Based on AS/NZS 4600 (SA, 2005) predictions, these ratios are unity for compact

sections, but they are 0.98 for slender sections.

Table 7.16 compares the FEA ultimate moment capacities with the section moment

capacity predictions of AS 4100 (SA, 1998) and AS/NZS 4600 (SA, 2005). It can be

seen that the predictions of AS/NZS 4600 (SA, 2005) are conservative by about 10%

for compact and non-compact sections. This is because in the AS/NZS 4600 (SA,

2005) calculations the inelastic reserve capacity was not used. Clause 3.3.2.3 of

AS/NZS 4600 (SA, 2005) states that the inelastic reserve capacity may be used

subject to four conditions. Two of them are: the effect of cold-forming is not

included in determining the yield stress (fy); the ratio of the depth of the compressed

portion of the web to its thickness does not exceed the slenderness ratio λ1 defined as

1.11/(fy/E)1/2. Currently available LSBs do not satisfy both these conditions. In

relation to the first condition, if an increased yield stress (fya) based on Clause 1.5.1.2

has already been used to include the effect of cold-forming, the inelastic reserve

capacity cannot be used. However, Clause 1.5.1.2 refers to strength increase resulting

from cold-forming in relation to cold-working of the corners (bends) of cold-formed

sections. The section moment capacities of LSBs are currently not based on the basic

yield strength of 380 MPa (yield strength of parent steel plate). Instead it is based on

a higher flange yield stress of 450 MPa that includes the benefit of significant cold-

working of hollow flange elements, and not due to that of corners as stated in Clause

1.5.1.2. The rectangular and square hollow sections (RHS and SHS) are

manufactured using a very similar method to that of LSBs, and their inelastic reserve

bending capacities are calculated using AS 4100 (SA, 1998) based on the increased

yield stress enhanced by the cold-working of their flange elements. Hence it is

possible to use the available inelastic reserve capacity of LSBs although it does not

satisfy the first condition of Clause 3.3.2.3. In relation to the second condition

relating to web slenderness, Clause 3.3.2.3 appears to be quite restrictive as λ1 value

for LSB sections is only about 23.4 (when a yield stress of 450 MPa is used). Other


7-35

design codes such as Eurocode 3 Part 1.3 (ECS, 2006) does not have such limits. In

summary although the compact and non-compact LSB sections have inelastic reserve

bending capacities as shown by both experiments and FEA, it is not possible to take

advantage of them using the current AS/NZS 4600 (SA, 2005) design rules.

In the case of some slender sections, AS/NZS 4600 (SA, 2005) predictions are

reasonable, but appear to be slightly less than the ultimate moments from FEA. The

AS 4100 (SA, 1998) predictions are less than both FEA and AS/NZS 4600 (SA,

2005) moment capacities for slender sections, implying that AS 4100 (SA, 1998)

design rules are more conservative for slender sections. However, the predictions of

AS 4100 (SA, 1998) are higher than the moment capacities from FEA for some of

the non-compact and compact sections. In general, AS 4100 (SA, 1998) design rules

appear to predict the available inelastic reserve capacity of non-compact and compact

LSBs reasonably well, considering the observation that FEA predictions are less than

the corresponding experimental capacities.

Table 7.16: Comparison of Ultimate Moment Capacities from FEA and Current

Design Rules

LSB Sections Ms (kNm) Mu (kNm)FEA

(FEA Mu) / (Ms AS 4100)

(FEA Mu) / (Ms AS/NZS 4600)AS 4100 AS/NZS 4600


From the above comparisons of section moment capacities of LSBs, AS 4100 (SA,

1998) design rules are more suited for predicting the inelastic reserve bending


7-36

capacity of LSBs. However, in principle, they cannot be used for LSBs as they are

cold-formed sections. Hence the European cold-formed steel structures standard

Eurocode 3 Part 1.3 (ECS, 2006 & 1996) was used to calculate the section moment

capacities of LSBs as they do not include a conservative limit for web slenderness.

The section moment capacity of a cold-formed steel member is defined in Eurocode

3 Part 1.3 (ECS, 2006) as follows,

If eleff WW < , 0, / MybeffRdc fWM γ= (7.9)

If eleff WW = ,

( ) 00

_

max

_

, /14 MeeelplelybRdc WWWfM γλλ ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −−+= 0/ Mybpl fW γ≤ (7.10)

where Weff, Wel and Wpl are the effective section modulus, gross elastic section

modulus and the plastic section modulus, respectively.

max

_

λ is taken as the slenderness of the element which correspond to the largest value

of 0

__

ee λλ . The plate element slenderness, p

_

λ is defined in EN 1993-1-5.

For double supported plane elements pe

__

λλ = and ( )ψλ +−+= 3055.025.05.00

_

e

For outstand elements pe

__

λλ = and 673.00

_

=eλ .

For stiffened elements pe

__

λλ = and 65.00

_

=eλ .

Table 7.17 presents the section moment capacities (Ms) of LSBs determined using

the above rules. Relevant calculations are presented in Appendix E.6.


7-37

Table 7.17: Comparison of Ultimate Moment Capacities from FEA and

Eurocode 3 Part 1.3 (ECS, 2006 & 1996)

LSB Sections Compact-ness

Ms (kNm) EC3 Pt. 1.3

Mu (kNm) FEA

Mu FEA / Ms EC3 Pt. 1.3

300x75x3.0 LSB NC 81.71 81.1 0.99 300x75x2.5 LSB S 61.50 66.7 1.08 300x60x2.0 LSB S 38.74 44.1 1.14 250x75x3.0 LSB NC 63.52 65.5 1.03 250x75x2.5 LSB NC 49.30 54.0 1.10 250x60x2.0 LSB S 32.68 35.8 1.10 200x60x2.5 LSB NC 34.71 35.1 1.01 200x60x2.0 LSB NC 25.24 27.7 1.10 200x45x1.6 LSB S 16.20 17.5 1.08 150x45x2.0 LSB C 16.17 15.8 0.98 150x45x1.6 LSB NC 11.57 12.5 1.08 125x45x2.0 LSB C 12.58 12.6 1.00 125x45x1.6 LSB NC 8.98 9.97 1.11

Mean 1.06 COV 0.049

The comparison of ultimate moments from FEA and Eurocode 3 Part 1.3 (ECS,

2006) in Table 7.17 shows that the section moments capacities of compact and some

of the non-compact LSBs are predicted well by Eurocode 3 Part 1.3 (ECS, 2006).

However, it is conservative for some non-compact LSBs such as 250x75x2.5 LSB,

200x60x2.0 LSB, 150x45x1.6 LSB and 125x45x1.6 LSB. The horizontal flange

elements of these LSBs were found to be slender as shown in Table E.7 of Appendix

E.6. This is the reason for the underestimation of the section moment capacities of

these LSBs. Further, Eurocode 3 Part 1.3 (ECS, 2006) was conservative for slender

sections. This implies that Eurocode 3 Part 1.3 (ECS, 2006) design rules are more

conservative in the case of slender and some non-compact sections. The FEA

ultimate moments of some compact and non-compact sections indicated by bold

letters in Table 7.17 agreed well with Eurocode 3 Part 1.3 design rules with a mean

of 1.00 and COV of 0.020.

In the above calculations and discussions, a higher yield stress of 450 MPa was used

for LSB outer flanges. However, Eurocode 3 Part 1.3 (ECS, 2006) design rules only

allow the basic yield strength, fyb, to be used. This is similar to the AS/NZS 4600

(SA, 2005) design rues which do not allow the use of an enhanced yield stress due to

cold-forming when calculating the inelastic reserve bending capacity. However,


7-38

based on the reasons discussed earlier by comparing LSBs with RHS and SHS

sections, it is proposed that Eurocode 3 Part 1.3 (ECS, 2006) design rules are used to

predict the inelastic reserve capacities of compact and non-compact LSB sections

using the enhanced yield stress of flanges (450 MPa).

7.5 Comparison of Ultimate Moment Capacities from FEA and Other

Proposed Design Rules

The inelastic reserve bending strength of cold-formed steel sections was first

investigated by Reck et. al (1975). Their test results showed that cold-formed steel

sections did not achieve higher inelastic bending capacities like the hot-rolled

sections due to the inability of cold-formed sections to sustain high compressive

strains. The ratio of the compressive strain to yield strain (Cy) was found to be a

function of the compressive flange’s width to thickness ratio (b/t). Yener and Pekoz

(1983, 1985) developed design rules to determine the inelastic bending capacity

based on the recommended ratio of compressive strain to yield strain (Cy) as a

function of the b/t ratio of compression elements. These design rules are adopted in

the NAS (AISI, 2004) and AS/NZS 4600 (SA, 2005). Recently Shifferaw and

Schafer (2008) investigated the inelastic bending capacity of conventional open cold-

formed steel members such as C and Z-section beams and proposed suitable design

rules under the direct strength method format. They state that inelastic reserve

bending capacity is available in cold-formed steel beams.

AS/NZS 4600 (SA, 2005) and NAS (AISI, 2004) design rules allow the calculation

of inelastic bending capacity based on the calculated maximum compressive strains.

However, in the last section, the inelastic reserve bending capacities of LSBs were

not calculated based on these rules, as they did not meet the two conditions including

the limit on web slenderness. In their report, Shifferaw and Schafer (2008) state that

the presence of reduced inelastic bending capacity in cold-formed steel beams in

comparison to hot-rolled steel beams is due to higher web to flange area,

unsymmetric sections resulting in first yield occurring in the tension flange and the

inability of cold-formed steel sections to sustain high compressive strains. However,

LSBs despite being cold-formed, do not have the above shortcomings as they are not


7-39

the conventional open cold-formed sections. The presence of rectangular hollow

flanges eliminates the above problems and hence appears to lead to higher inelastic

bending capacities in the case of compact and non-compact LSB sections. Therefore

it is feasible to use the available inelastic bending capacity equations.

Shifferaw and Schafer (2008) developed suitable design equations based on a

comprehensive finite element analysis study and available experimental results of

cold-formed steel sections. Their first set of equations was based on the average

membrane compressive strains (Cy). They developed separate equations for Cy as a

function of slenderness for local and distortional buckling cases based on the back-

calculated strain values corresponding to the ultimate moments obtained from FEA.

Their equation in the case of local buckling is given next.

If lyl λλ < , 544.0

⎟⎠⎞

⎜⎝⎛=

l

lyyC

λλ (7.11)

where, 776.0=lyλ and crlyl MM=λ . Here, Mcrl is the elastic local buckling

moment.

They then developed the following equation for the inelastic bending capacity (Mn)

that lie between the yield moment (My) and plastic moment (Mp) capacities.

42.2

11 ⎟⎠⎞

⎜⎝⎛−=

−−

yyp

yn

CMMMM

for Cy >1 (7.12)

Shifferaw and Schafer (2008) also gave relevant equations for the inelastic bending

capacity as a function of slenderness using the direct strength method (DSM) format

by combining Eqs. 7.11 and 7.12.

If lyl λλ < , ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−−+=

32.1

1ly

lypyn MMMM

λλ (7.13)

The DSM based design equation above was used to predict the inelastic reserve

capacity of compact and non-compact LSBs, and the results are shown in Table 7.18


7-40

and compared with FEA ultimate moments. This comparison provides mean and

COV values of 1.03 and 0.006 for compact and non-compact LSB sections. This

indicates a good agreement with the proposed equations of Shifferaw and Schafer

(2008). In comparison with Eurocode 3 Part 1.3 (ECS, 2006) design equations, these

equations predicted the section moment capacities of all the compact and non-

compact LSBs more accurately. Therefore it is recommended Shifferaw and

Schafer’s (2008) design equations are used to predict the inelastic reserve capacities

of compact and non-compact LSBs while the section moment capacities of slender

LSBs can be predicted by AS/NZS 4600 (SA, 2005), NAS (AISI, 2004) and

Eurocode 3 Part 1.3 (ECS, 2006).


Shifferaw and Schafer (2008)

LSB Sections Mcrl (kNm)

My (kNm)

Mp (kNm) λl

Shifferaw & Schafer

Mnl (kNm)

FEA Mu (kNm)

FEA Mu /

Mnl

300x75x3.0 LSB 163.14 77.24 91.40 0.688 79.32 81.1 1.02 300x75x2.5 LSB 95.99 64.79 76.71 0.822 S 66.7 S 300x60x2.0 LSB 39.35 45.17 53.75 1.071 S 44.1 S 250x75x3.0 LSB 191.2 60.06 71.01 0.560 63.88 65.5 1.03 250x75x2.5 LSB 114.71 50.38 59.63 0.663 52.12 54.0 1.04 250x60x2.0 LSB 46.50 35.10 41.58 0.869 S 35.8 S 200x60x2.5 LSB 109.09 31.98 37.80 0.541 34.18 35.1 1.03 200x60x2.0 LSB 57.24 25.79 30.53 0.671 26.62 27.7 1.04 200x45x1.6 LSB 22.12 17.23 20.43 0.883 S 17.5 S 150x45x2.0 LSB 54.29 14.35 16.96 0.514 15.44 15.8 1.02 150x45x1.6 LSB 28.52 11.58 13.70 0.637 12.07 12.5 1.04 125x45x2.0 LSB 63.92 11.15 13.20 0.418 12.30 12.6 1.02 125x45x1.6 LSB 33.71 9.00 10.68 0.517 9.70 9.97 1.03

Mean 1.03 COV 0.006

Note: S – Slender section.


7-41

7.6 Discussion of Maximum Available Moment Capacity of LSBs and

Compressive Strain Limits

Table 7.19 provides the ratios of ultimate FEA moments to the first yield moments of

LSBs without corners. It is quite easy to calculate the actual section moment capacity

of each LSB using the ratios of Mu/My. Table 7.19 also provides the section

compactness of LSBs based on AS 4100 (SA, 1998) in both cases of with and

without corners.

Table 7.19: The Ratios of Mu/My and Section Compactness

LSB Sections Mu/My Section Compactness

Without Corners With Corners 300x75x3.0 LSB 1.05 NC NC 300x75x2.5 LSB 1.03 S S 300x60x2.0 LSB 0.98 S S 250x75x3.0 LSB 1.09 NC C 250x75x2.5 LSB 1.07 NC NC 250x60x2.0 LSB 1.02 S S 200x60x2.5 LSB 1.10 NC C 200x60x2.0 LSB 1.07 NC NC 200x45x1.6 LSB 1.02 S S 150x45x2.0 LSB 1.10 C C 150x45x1.6 LSB 1.08 NC NC 125x45x2.0 LSB 1.13 C C 125x45x1.6 LSB 1.11 NC NC

As seen in Table 7.19, compact and non-compact sections exhibited some amount of

inelastic reserve capacity, i e. about 9% on average with a maximum of 13%, while

slender sections do not have any inelastic reserve capacity. However, the ratio of

Ms/My has the values of about 1.18 for compact sections (Table 7.15) such as

200x60x2.5 LSB, 150x45x2.0 LSB and 120x45x2.0 LSB although the FEA results

give only about 1.10 to 1.13. It was found that the shape factor (S/Z) for all the

available LSBs without corners was 1.18 and the relevant calculations are provided

in Appendix E.5. Hence the achievable maximum inelastic reserve moment capacity

is 18% of My, which is only for compact sections. Since there are only two compact

sections in the available 13 LSBs, some non-standard LSBs were created and

analysed using finite element analyses. For this purpose, some slender and non-

compact LSBs were converted to compact sections by increasing their thicknesses


7-42

and the details of these non-standard LSBs are presented in Table 7.20. The section

moduli of these beams can be calculated by using Thin-Wall or the procedure shown

in Appendix D.1. The flange and web yield stresses were taken as 450 MPa and 380

MPa as for standard LSBs available in the industry.

Table 7.20: Dimensions and Properties of Non-Standard Compact LSBs

LSB Sections Depth

Clear Depth

of Web

Flange Width

Flange Depth Thickness Section

Modulus

D d1 bf df T Z (mm) (mm) (mm) (mm) (mm) (103 mm3)

300x75x3.9 LSB 300 250 75 25 3.9 220.53 250x75x3.3 LSB 250 200 75 25 3.3 146.23 200x45x3.0 LSB 200 170 45 15 3.0 69.64 150x45x3.0 LSB 150 120 45 15 3.0 46.77

Table 7.21: The Ratios of Mu/My of Some Non-Standard Compact LSBs

LSB Sections My (kNm) Mp (kNm)

FEA Mu (kNm) Mu/My

300x75x3.9 LSB 99.24 117.28 107.0 1.08 250x75x3.3 LSB 65.80 77.76 72.5 1.10 200x45x3.0 LSB 31.34 37.08 34.1 1.09 150x45x3.0 LSB 21.05 24.79 23.8 1.13

Table 7.21 presents the first yield moment, the ultimate moment capacity from finite

element analyses and the ratios of Mu/My for these non-standard LSBs. As seen in

this table, the ratios of Mu/My are in the range of 1.08 to 1.13 for these compact LSB

sections. This may be due to the inability of finite element analysis to simulate the

true inelastic bending capacity of steel beams. Mahaarachchi and Mahendran’s

(2005b) test results have shown the presence of full plastic moment capacity for

compact LSB sections, ie. many LSBs reached about 1.20 My, which is the plastic

moment capacity for LSBs with corners. Past research (Greiner, 2001) has shown

that finite element modelling based on mechanical properties derived from tensile

coupon tests is unable to capture the full plastic moment capacity of steel beams. The

section moment capacity equations based on Eurocode 3 Part 1.3 (ECS, 2006) and

Shifferaw and Schafer (2008) were able to predict the section moment capacities of


7-43

compact and non-compact LSB sections based on FEA. However, this may be due to

the development of these equations based on FEA results. It is possible for compact

cold-formed steel sections such as LSBs to develop their full plastic moment

capacities (Mp). Mahaarachchi and Mahendran’s (2005b) test results show that

compact LSB sections are capable of reaching their full plastic moment capacities in

addition to the use of an enhanced flange yield stress of 450 MPa due to cold-

working. However, as a safer conservative approach, it is recommended to use the

results of Mu/My provided in Table 7.19 based on FEA results or Equations 7.9 and

7.10 based on Eurocode 3 Part 1.3 (ECS, 2006) or Equation 7.13 based on Shifferaw

and Schafer (2008). Hence the maximum amount of inelastic reserve capacity for

compact LSB sections is 1.13 My.

Although FEA was not able to predict the plastic moment capacities of four non-

standard compact LSBs (Table 7.21), it was believed that FEA would predict the full

plastic moment capacities of very compact LSBs with thicker plate elements.

Therefore two such sections, 150x45x4.0 LSB and 150x45x5.0 LSB, were analysed.

However, the ratios of Mu/My for these highly compact LSBs were found to be 1.12

in comparison to their Mp/My ratio of 1.17. It was then found that the use of different

yield stresses for flange and web elements also influenced the ultimate moment

capacities of LSBs obtained from FEA. Hence when the same LSB sections

(150x45x4.0 and 150x45x5.0 LSB) were analysed using the same flange and web

yield stress of 450 MPa, FEA predicted their plastic moment capacities. However,

FEA was not able to predict the plastic moment capacity of 150x45x2.0 LSB

although it is classified as compact based on AS 4100 (SA, 1998). Appendix E.7

provides the details of FEA and the results.

An attempt was then made to obtain the section moment capacities of conventional

hot-rolled I- and C-sections. For this purpose, 150UB14.0 and 150PFC17.7 sections

were modelled without corners using the same flange and web yield stress of 320

MPa as shown in their design capacity tables (AISC, 1994). It was found that FEA

was able to predict the full plastic moment capacity of the doubly symmetric

150UB14.0 section, but not that of the monosymmetric 150PFC17.7. The ratio of

Mu/My was only 1.14 for the mono-symmetric hot- rolled PFC section in comparison

to its Mp/My ratio of 1.19 (see Appendix E.7). This contradicts the moment capacities


7-44

given in AISC (1994), which recommends the full plastic moment capacity for

150PFC17.7.

Based on the results in Appendix E.7, it can be concluded that conventional finite

element analyses may not able to predict the full plastic moment capacities of

compact mono-symmetric steel sections unless they are made of very thick plate

elements (with small b/t ratios). Further experiments are needed to confirm these

observations.


Eurocode 3 Part 1.3 (NSAI, 2006) for Non-Standard Compact LSBs

Non-Standard LSB Sections

Ms (kNm) EC3 Part 1.3

Mu (kNm) FEA

Mu FEA / Ms EC3 Part 1.3

300x75x3.9 LSB 117.28 107.0 0.91 250x75x3.3 LSB 73.74 72.5 0.98 200x45x3.0 LSB 37.08 34.1 0.92 150x45 3.0 LSB 24.79 23.8 0.96

A comparison of FEA ultimate moments with the predictions from Eurocode 3 Part

1.3 (ECS, 2006) is presented in Table 7.22. As seen in Table 7.22, Eurocode 3 Part

1.3 (ECS, 2006) was not be able predict the section moment capacities of non-

standard compact LSBs from FEA except 250x75x3.3 LSB despite the fact it well

predicted the section moment capacities of standard compact and non-compact LSBs.

It was found that Eurocode 3 Part 1.3 (ECS, 2006) design rules predicted the full

plastic moment capacities for those non-standard LSBs except 250x75x3.3 LSB as

shown in Appendix E.6. This implies that the current Eurocode 3 Part 1.3 (ECS,

2006) design rules allow the full plastic moment capacities for highly compact cold-

formed sections including LSBs. The FEA do not predict the full plastic moment

capacities for these sections and hence resulted in the disagreement between their

results in Table 7.22.

A comparison of FEA ultimate moments with the predictions of Shifferaw and

Schafer (2008) is presented in Table 7.23. As seen in this table, FEA ultimate

moments agreed well with the predictions from Shifferaw and Schafer’s (2008)

design rules for non-standard compact LSBs. Unlike Eurocode 3 Part 1.3 (ECS,


7-45

2006) design rules, Shifferaw and Schafer’s equation does not predict the full plastic

moment capacities and hence appear to be conservative for these very compact

sections. Therefore, it is recommended to use Shifferaw and Schafer’s (2008) design

rule (Equation 7.13) to predict the section moment capacities of compact and non-

compact LSBs.


Shifferaw and Schafer (2008) for Non-Standard Compact LSBs

The above discussion appears to indicate that compact LiteSteel beams with

torsionally rigid flanges and no free edges have inelastic reserve moment capacity

despite the fact they are cold-formed sections. This inelastic reserve capacity can be

calculated based on Shifferaw and Schafer’s and Eurocode 3 part 1.3 (ECS, 2006)

design equations. Some current cold-formed steel codes such as AS/NZS 4600 (SA,

2005) and NAS 2007 (AISI, 2007) have restrictions based on slenderness as there is

a concern about the excessive strains at failure that may also lead to fracture in the

section. It is believed that the maximum longitudinal strain at failure should not

exceed three times the yield strain to avoid material fracture (εmax < 3εy). Also, the

inelastic reserve moment that could be achieved will also depend on the maximum

strain at failure. Therefore, an attempt was made to obtain the membrane strain

variation across the LSB cross-sections including the maximum strain. The strain

variations across 150x45x3.0 LSB at the ultimate moment are shown in Figure 7.22.

Strains of the nodes at the edge of the cross section at mid-span were used in plotting

this figure. However, it was found that the strain varied along the flange element as

shown in Figure 7.23.

Non-Standard LSBs

Mcrl (kNm)

My (kNm)

Mp (kNm) λl

Shifferaw & Schafer

Mnl (kNm)

FEA Mu (kNm)

FEA Mu /

Mnl 300x75x3.9 LSB 347.09 99.24 117.28 0.535 106.25 107.0 1.01 250x75x3.3 LSB 251.38 65.80 77.76 0.512 70.86 72.5 1.02 200x45x3.0 LSB 135.27 31.34 37.08 0.481 34.02 34.1 1.00 150x45x3.0 LSB 170.12 21.05 24.79 0.352 23.47 23.8 1.01


7-46

Figure 7.22: Strain Variation across the Cross-section of 150x45x3.0 LSB

Figure 7.23: Strain along the Top Flange of 150x45x3.0 LSB

As seen in Figure 7.23, the strain values increased from 6.25 x 10-3 to 17.5 x 10-3.

Therefore, the maximum strain (εmax) on this LSB at failure is 17.5 x 10-3 at the outer

most corner. The yield strain (εy) of LSB can be calculated as follows.

-80

-60

-40

-20

0

20

40

60

80

-7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

Strain (10-3)

Dis

tanc

e ac

ross

Sec

tion,

(mm

)


7-47

Yield strain, εy = yield stress / Young’s modulus

= 450 / 200000

εy = 2.25 x 10-3

therefore, 3εy = 6.75 x 10-3.

In this case, εmax > 3εy for 150x45x3.0 LSB, hence material fracture may occur.

Figure 7.24 shows the membrane strain variation across the section and along one

element at mid-span as a fringe result from finite element analyses. Maximum

longitudinal strains for all the available 13 LSBs and 4 non-standard LSBs were

obtained at ultimate failure from FEA. Table 7.24 presents the maximum membrane

strains and Mu/My ratios of these LSBs. The average membrane strains along the

horizontal flange elements are also included in Table 7.24.

Figure 7.24: Strain Variation across the cross-section of 150x45x3.0 LSB as

Fringe Results


7-48

Table 7.24: Average and Maximum Membrane Strains of LSB Sections at Failure

Available LSBs Section Compact--ness

Mu/My Maximum

Strain, εmax, (10-3)

Average Strain, (10-3)

Cy (εmax / εy)

300x75x3.0 LSB NC 1.05 3.49 3.34 1.55 300x75x2.5 LSB S 1.03 3.32 3.31 1.48 300x60x2.0 LSB S 0.98 3.27 3.19 1.45 250x75x3.0 LSB NC 1.09 3.71 3.69 1.65 250x75x2.5 LSB NC 1.07 3.41 3.40 1.52 250x60x2.0 LSB S 1.02 3.25 3.12 1.44 200x60x2.5 LSB NC 1.10 3.89 3.87 1.73 200x60x2.0 LSB NC 1.07 3.44 3.43 1.53 200x45x1.6 LSB S 1.02 2.92 2.91 1.30 150x45x2.0 LSB C 1.10 4.17 4.01 1.85 150x45x1.6 LSB NC 1.08 3.53 3.52 1.57 125x45x2.0 LSB C 1.13 6.75 5.87 3.00 125x45x1.6 LSB NC 1.11 4.53 4.11 2.01

Non-Standard LSBs 300x75x3.9 LSB C 1.08 3.81 3.79 1.69 250x75x3.3 LSB C 1.10 3.97 3.95 1.76 200x45x3.0 LSB C 1.09 4.06 4.00 1.80 150x45x3.0 LSB C 1.13 17.52 11.58 7.79

As seen in Table 7.24, the results for 125x45x2.0 LSB and 150x45x3.0 LSB show

that their maximum compressive strains at the ultimate failure are greater than 3εy

(6.75 x 10-3). This indicates that these compact LSBs may fail by material fracture

before they reach their higher inelastic moment capacities. However, Mahaarachchi

and Mahendran’s (2005b) tests showed that there was no material fracture during the

tests of many compact sections, which reached their full plastic moment capacities

(1.20 My). Also the FEA results showed that the compression flanges yielded first

and that the maximum compressive strains were higher than the maximum tensile

strains in all cases (13 available and 4 non-standard compact LSBs). Experiments

also revealed that the failure was due to inelastic buckling of compression flanges.

Table 7.24 results show that higher Mu/My ratios are achieved for the most compact

sections. Since the slenderness values of compression plate elements of these

sections are small, they are able to reach higher compressive strains at failure and

hence higher moment capacities. Further experiments and numerical modelling are


7-49

required before full plastic moment capacities are adopted for compact LSB sections.

It is recommended that the section moment capacity equation (Equation 7.13) based

on Shifferaw and Schafer (2008) is used. Eurocode 3 Part 1.3 design rules can also

be used for this purpose.

If a simplified design approach is needed, it is recommended that the section moment

capacity of the currently available compact LSBs is taken as 1.10My. Since the

inelastic reserve capacity of non-compact sections is in most cases about 5% of My, it

can be neglected and the first yield moment (My) can be taken as their section

moment capacity.

7.7 Conclusions

This chapter has presented the details of an experimental investigation, finite element

analyses and a parametric study on the section moment capacities of LSBs. Four

point bending tests were carried out for seven LSBs. Experimental ultimate moment

capacities from this research and Mahaarachchi and Mahendran (2005b) were

compared with those predicted by the current design rules for section moment

capacity based on AS/NZS 4600 (SA, 2005). Appropriate finite element models were

developed and validated using the experimental results. The validated finite element

models with nominal dimensions and yield stresses were used to obtain the section

moment capacities of all the available 13 LSBs. A comparison of the section moment

capacity results from finite element analyses, experiments and design codes showed

that compact and non-compact LSBs based on AS 4100 (SA, 1998) have some

inelastic reserve capacity while slender LSBs do not have any inelastic reserve

capacity beyond their first yield moment. This chapter has presented the section

moment capacities of LSBs based on experiments, finite element analyses and the

current steel design codes and has made some suitable recommendations.

Although the currently available LSBs exceed the slenderness limits and other

conditions of AS/NZS 4600 (SA, 2005) and NAS (AISI, 2007) for inelastic reserve

bending capacity, considerable inelastic bending capacities exist for LSBs as evident

from experiments and finite element analyses of LSBs and should be included in

design. For this purpose, it is recommended that the inelastic bending capacity


7-50

equations developed by Shifferaw and Schafer (2008) are used to predict these

capacities for non-compact and compact LSBs. Eurocode 3 Part 1.3 (ECS, 2006)

design equations can also be used. As a simple design approach, it is also possible to

use the ultimate moment capacity of compact LSB sections as 1.10 times their first

yield moment while it is the first yield moment for non-compact sections. For slender

LSB sections, current cold-formed steel codes can be used to predict their section

moment capacities.

Effect of Web Stiffeners on the Lateral Buckling of LSBs

8-1

CHAPTER 8

8.0 EFFECT OF WEB STIFFENERS ON THE LATERAL DISTORTIONAL

BUCKLING BEHAVIOUR AND STRENGTH OF LITESTEEL BEAMS

8.1 Introduction

The LiteSteel Beams (LSBs) with intermediate and long spans are subjected to

lateral distortional and lateral torsional buckling, respectively. Lateral distortional

buckling occurs due to the presence of torsionally rigid rectangular flanges and a

relatively slender web. Simultaneous lateral displacement, section twist and web

distortion occur during this lateral distortional buckling of LSB as seen in Figure 8.1.

Lateral distortional buckling significantly reduces the flexural moment capacity of

LSBs with intermediate spans as shown in Chapter 6. Such moment capacity

reduction can be eliminated if the observed web distortion in LSBs is eliminated or

reduced. Past research (Avery and Mahendran, 1997, Mahendran and Avery, 1997)

has shown that the use of web stiffeners reduces web distortion and hence improves

the flexural moment capacity of hollow flange steel beams such as Hollow Flange

Beams (HFB) (see Figure 8.2). Avery and Mahendran (1997) stated that web

stiffeners act to prevent web distortion by coupling the rotational degrees of freedom

of the top and bottom flanges of HFBs. Hence they found that simple plate stiffeners

welded or screw fastened to only the top and bottom flanges were able to reduce the

web distortion and thus improve the lateral buckling moment capacities of HFBs.

Figure 8.1: Lateral Distortional Buckling of LSBs

(a) Experiments


8-2

Figure 8.1: Lateral Distortional Buckling of LSBs

(a) Welded Web Stiffeners (b) Screw-fixed Web Stiffeners

Figure 8.2: Use of Web Stiffeners in HFBs (Mahendran and Avery, 1997)

Although Avery and Mahendran (1997) and Mahendran and Avery (1997) showed

that the use of web stiffeners significantly improved the flexural moment capacity of

HFBs using both large scale experiments and finite element analyses, Kurniawan’s

(2005) investigations on LSBs produced some conflicting outcomes. His

experimental studies based on quarter point loading showed that the use of web

stiffeners did not significantly improve the flexural moment capacity of LSBs while

his finite element analyses based on an ideal finite element model of LSB with ideal

support conditions and a uniform moment gave improved buckling moment

capacities. It is unlikely that this conflicting outcome was caused by the difference in

moment distributions, in which case, the experimental studies should have given a

Web Distortion Lateral Displacement

Section Twist

(b) FEA


8-3

higher elastic buckling capacity. Instead it may be due to the lack of flange twist

restraints at the supports in his lateral buckling experiments (see Figure 8.3 (a)). The

use of web side plates alone was unable to provide the required flange twist restraint

(FTR).

Figure 8.3: Twist Restraint at the Supports

As described in Chapter 4, this leads to local flange twists at the supports as shown in

Figure 8.3 (b) and thus does not produce the ideal simply supported boundary

conditions in which the entire section has full twist restraint. Transverse web

stiffeners were provided at the supports for most of the lateral buckling tests

described in Chapter 4 in order to provide full twist restraint at the supports as shown

in Figure 8.3 (c). The difference between the support conditions in Kurniawan’s

(2005) experimental and finite element analyses might have caused the moment

capacity differences observed by him. Further, Kurniawan’s (2005) finite element

analyses were limited to elastic buckling analyses while his experimental study was

also limited to one LSB section. Therefore a thorough investigation is required to

investigate the effect of web stiffeners on the lateral distortional buckling and

ultimate strength behaviour of LSBs. For this purpose the validated finite element

models of LSBs developed in Chapter 5 were used by including the required web

stiffeners at the supports and appropriate locations within the span. It is important to

investigate the reasons for the conflicting outcomes of Kurnaiawan (2005) and then

to determine the most suitable and cost-effective type, size and spacing of the

(a) Support without FTR (b) Local Flange Twist (c) Support with FTR


8-4

required web stiffeners that will provide the improved lateral distortional buckling

capacities for LSBs. This was first undertaken using a series of elastic buckling

analyses. Both elastic and non-linear lateral buckling analyses were then undertaken

for the chosen web stiffener arrangement and suitable design rules were also

developed. This chapter presents the details of this investigation and the results.

8.2 Elastic Buckling Analyses

Avery and Mahendran (1997) and Mahendran and Avery (1997) found that the use of

5 mm thick steel plate stiffeners screwed or welded to the top and bottom flanges of

HFBs (Hollow Flange Beams) at third points within the span (Figure 8.2) was the

most optimum arrangement to improve the lateral buckling moment capacities based

on their experimental and finite element analyses. Kurniawan (2005) also found that

the use of 5 mm steel plate stiffeners at third points within the span improved the

lateral buckling moment capacity of LSBs based on his finite element analyses. They

considered various types of web stiffeners such as angle sections, threaded rod

fasteners, square hollow sections, LSBs and rectangular hollow sections (see Figure

8.4) and concluded that steel plates screwed or welded to the hollow flanges

provided the most simple and cost-effective web stiffener arrangement. Further they

reported that the use of other types of web stiffeners and arrangements did not

increase the lateral buckling moment capacities of hollow flange steel beams with

compared to their cost. Therefore 5 mm steel plate web stiffeners were considered in

this research. However, the use of this plate web stiffener with LSBs as used in the

experiments of Kurniawan (2005) must be investigated first since he stated that the

use of web stiffeners did not significantly improve the buckling moment capacity of

LSB. As mentioned earlier in this chapter, the use of additional web stiffeners at the

supports to provide the required flange twist restraint is likely to eliminate this

problem.


8-5

(i) Avery and Mahendran (1997)


Kurniawan (2005)


8-6

(ii) Kurniawan (2005) Figure 8.4: Types of Web Stiffeners Used by Avery and Mahendran (1997) and

Kurniawan (2005)

8.2.1 Finite Element Models

Two types of finite element models were used in this research, namely, ideal and

experimental finite element models as shown in Figure 8.5. Ideal models of LSBs

were based on ideal simply support conditions and a uniform moment. Ideal simply

supported boundary conditions were implemented by fixing the vertical and lateral

deflections and twist of the section at the supports. Experimental finite element

models were used to simulate the LSBs as used in the lateral buckling experiments

with quarter point loading. Chapter 5 provides the details of these models of LSBs

(a) 5 mm Plate Screwed to Flanges Only (b) 65x65x5 EA Screwed

to Flanges only

(d) 125x45x2.0 LSB Screwed to Flanges only

(c) 125x45x2.0 LSB Screwed to Flanges only

(e) 50x50x2.5 SHS Screwed to Flanges only

(f) M16 Threaded Rod Fastened Flanges only


8-7

without web stiffeners that have been validated using the experimental results in

Chapter 4. These validated models were modified by including the required web

stiffeners in this research. Nominal dimensions of LSBs were used in the analyses.

Steel plates with 5 mm thickness and a yield stress of 300 MPa were considered at

the supports and one third points of the beam span. The plate stiffeners at the

supports provided the required flange twist restraint. Figure 8.6 shows the

experimental finite element model of LSB with web stiffeners.

Figure 8.5: Schematic Diagrams of Ideal and Experimental FE Models

Figure 8.6: Experimental Finite Element Model of LSB with Web Stiffeners

Span/2

P

Span/4

(b) Experimental Model

Span/2

M

(a) Ideal Model Symmetric Plane

(a) Finite Element Mesh (5 mm x 10 mm)


8-8

Figure 8.6: Experimental Finite Element Model of LSB with Web Stiffeners

Shell elements of 5 mm width and 10 mm length were used as shown in Figure 8.6

(a). Figure 8.6 (b) shows the various plates used in the experimental finite element

model with web stiffeners at supports and suitable locations within the span. It also

includes the usual web side plates used at the supports. Figure 8.6 (c) shows the cross

sectional view of LSB with web stiffeners, which includes the support and mid-span

boundary conditions and loading. The loading and boundary conditions were the

same as used in the experimental finite element models of LSBs described in Chapter

5. In the finite element models of stiffened LSBs, the web stiffeners were connected

to the inner flange surface by a process of “equivalencing” the nodes of the web

stiffener plate and the nodes of the inner surface of the flange so that the web

(b) Various Plate Elements

Loading Plate

Web Side Plate

Web stiffener providing flange twist restraint Web Stiffener

(c) Boundary Conditions and Loading

Support Boundary Condition, SPC 234

MPC

Mid-Span Boundary Condition, SPC 156

Web Stiffener

5 mm Gap


8-9

stiffener plate and the flange can act as an integral member. The welding process was

not modelled as it was decided to recommend a “tack” weld and the effects of this

welding on the residual stresses were considered to be negligible. A 5 mm gap was

provided between the stiffener and the web element as it is not practical to provide

the stiffener next to the web due to the corners present in LSBs.

Elastic buckling analyses were undertaken with varying arrangements of web

stiffeners to investigate the need for web stiffeners at the supports, ie. no web

stiffeners, web stiffeners at the supports providing flange twist restraint, web

stiffeners at third points within the span and web stiffeners at the supports and third

span points. Figures 8.7 (a) to (d) provide their details.


(b) Web Stiffener at the Supports Providing Flange Twist Restraint

(a) No Web Stiffeners


8-10

Figure 8.7: Experimental FE Model with Web Stiffeners and Flange Twist

Restraints

Figure 8.8: Ideal Finite Element Model with Full Twist Restraint at the

Supports (Including Flanges) and Web Stiffeners

(d) Web Stiffeners at the Supports and Third Span Points

(c) Web Stiffener at third points within the Span (No Flange Twist Restraint at the Support)

Symmetric Plane, SPC 156 SPC 234

Web Stiffener


8-11

Figure 8.8 shows the typical ideal finite element model of LSBs with web stiffeners.

It was used in the parametric study after resolving the conflicting outcomes from

Kurniawan (2005). Hence flange twist restraints at the supports were not modelled

explicitly using web stiffeners as shown in Figures 8.7 (b) and (d). Instead they were

included in the models via idealised simply supported boundary conditions which

provide full twist restraint to the entire section at the supports as shown in Figure 8.8.

The idealised boundary conditions at the support and the boundary condition of

symmetric plane are presented in Table 8.1. The presence of symmetry allowed the

use of only half the span, which reduced the analysis time. In Table 8.1, T and R

represent the translation and rotation, respectively and the subscripts (1, 2, and 3)

represent the direction while field “Yes” means that it is free to move in that

direction. Figure 8.9 illustrates the global axes selected to input the boundary

conditions for the analysis. The section twist was restrained by fixing the X axis

rotation (SPC 4) while the vertical and lateral displacements were also fixed at all the

nodes of the end-span of LSB as shown in Figure 8.8. Therefore, additional web

stiffener plates are not needed at the support as the idealised simply supported

boundary condition provides the required flange twist restraint and eliminate the

local flange twist (Figure 8.3 (b)).


T1 T2 T3 R1 R2 R3

One end Yes No No No Yes Yes

Other end No No No No Yes Yes

Mid span No Yes Yes Yes No No


X, 1

Z, 3

Y, 2 Z, 3

X, 1

Y, 2

M

L/2


8-12

It should be noted that in the actual experiments, simply supported boundary

conditions were applied at the shear centre through web side plates which have no

connection to the flanges and hence those flanges were free to twist. Therefore,

experimental models require a flange twist restraint at the support.

To simulate a uniform end moment across the section, linear forces were applied at

every node of the beam end, where the nodes above the middle of the web were

subject to tensile forces while the nodes below the middle of the web were subject to

compressive forces. The force at the middle of the web was zero and was linearly

increased within the cross section as shown in Figure 8.10. A tensile force of 1000 N

and a compressive force of 1000 N were applied at the nodes on the top and bottom

faces of LSB cross section. This loading method was the same as that used in the

finite element models of LSBs without web stiffeners, as described in Chapter 5.

Figure 8.10: Loading Method of Ideal Finite Element Model


8-13

8.2.2 Results

Table 8.2 presents the elastic lateral distortional buckling moments of LSBs with

varying web stiffener arrangements from both experimental and ideal finite element

models described in the previous section. It includes the elastic lateral distortional

buckling moments (Mod) from the ideal finite element model (as given in Chapter 6),

the elastic torsional buckling moment Mo calculated using Eq.8.1, and the values of

Mod with various arrangements of web stiffeners as obtained from the experimental

finite element models.

The elastic lateral torsional buckling moment Mo can be calculated by using the

following equation.

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+= 2

2

2

2

LEIGJ

LEIM wy

oππ (8.1)

where



GJ = torsional rigidity

L = span

Table 8.2: Elastic Lateral Distortional Buckling Moments of LSBs with Web

Stiffeners

LSB Sections

Span (mm)

Mo (kNm) Eq. 8.1

Mod (kNm) Ideal

Model

Mod (kNm) Experimental Model

No WS (a)

WSs (b)

WSTP (c)

WS (d)

300x60x2.0 LSB

3000 33.95 22.99 22.05 23.80 25.02 29.06 4000 24.66 18.36 17.55 19.46 19.66 22.81

200x45x1.6 LSB

3000 10.68 8.33 8.14 8.95 9.00 10.18 4000 7.89 6.67 6.43 7.05 6.89 7.63

150x45x2.0 LSB

2000 18.35 14.52 12.15 14.30 13.01 15.81 3000 11.96 10.48 9.01 10.42 9.39 10.93

WSs – Web stiffeners at the supports providing flange twist restraint, WSTP – Web stiffeners at third points within the span, WS – Web stiffeners at the supports and at third span points.


8-14

Table 8.3: Effect of Web Stiffener Arrangements on the Results of Mod from

Experimental Finite Element Models


WSs / No WS

WSTP / No WS

WS / WSs

WS / No WS

300x60x2.0 LSB

3000 1.08 1.13 1.22 1.32 4000 1.11 1.12 1.17 1.30

200x45x1.6 LSB

3000 1.10 1.11 1.14 1.25 4000 1.10 1.07 1.08 1.19

150x45x2.0 LSB

2000 1.18 1.07 1.11 1.30 3000 1.16 1.04 1.05 1.21

WSs – Web stiffeners at the supports providing flange twist restraint, WSTP – Web stiffeners at third points within the span, WS – Web stiffeners at the supports and at third span points. As seen in Tables 8.2 and 8.3, the elastic lateral distortional buckling moments (Mod)

were increased by 19 to 32% for the LSBs and spans considered here when web

stiffeners were used at the third points within the span and the supports. However,

when the web stiffeners were used only at third span points of span without any

stiffeners at the supports (without flange twist restraint at the support), the

improvement to elastic lateral distortional buckling moment was not significant. This

difference is small for 200x45x1.6 LSB with 4 m span and 150x45x2.0 LSB with 3

m span because these LSBs exhibit lateral torsional buckling with very small web

distortion for these spans. The results in the Tables 8.2 and 8.3 clearly demonstrate

the need to use web stiffeners at both the supports an third span points.

Figures 8.11 (a) to (d) show the elastic lateral distortional buckling modes obtained

for 200x45x1.6 LSB section from finite element analyses based on the experimental

finite element model with various configurations of web stiffeners. A comparison of

Figures 8.11 (a) and (b) clearly demonstrates that the use of web stiffeners at the

supports significantly reduced the local flange twist at the support. Comparison of

Figures 8.11 (a) and (c) shows that web distortion was reduced when web stiffeners

were used at third span points. Although the use of web stiffeners at third span points

reduced the web distortion the use of web stiffeners at the supports is also important

as this further improved the moment capacities by avoiding local flange twist as

shown in Figure 8.11 (d).


8-15


Various Stiffener Arrangements

(a) No Web Stiffeners

Web Distortion

(b) Web Stiffeners at the Supports


8-16


Various Stiffener Arrangements

(c) Web Stiffeners at third Span Points

(d) Web Stiffeners at the Supports and third Span Points

Less Web Distortion


8-17

Based on these finite element elastic buckling analyses (Tables 8.2 and 8.3 and

Figure 8.11), it is concluded that the use of web stiffeners at every third point within

the span can effectively improve the lateral distortional buckling moment capacity of

LSBs provided web stiffeners are also used at the supports. This simulates the

idealised simply supported boundary conditions with full twist restraint. These

results also provide the explanation why Kurniawan’s (2005) experimental and

numerical analyses gave conflicting outcomes in relation to the buckling capacity

improvements due to web stiffeners.

Having confirmed the effectiveness of using web stiffeners in improving the lateral

distortional buckling moment capacities of LSBs, it is now necessary to investigate

the optimum size and spacing of the required plate web stiffeners. For this purpose, a

series of elastic buckling analyses was conducted using the ideal finite element

model to investigate the lateral distortional buckling moment capacities of LSBs as a

function of web stiffener thickness and spacing. The following section provides the

details of these analyses and the results.

8.2.3 Determination of Optimum Spacing and Size of Web Stiffeners

It was decided to use steel plates welded to the inner faces of top and bottom flanges

as web stiffeners. However, the location or the number and spacing of web stiffeners,

which effectively improve the lateral distortional buckling moment capacity, has to

be determined. It is obvious that the moment capacity will increase with increasing

number of web stiffeners. However, the cost will also increase with it. Therefore a

series of elastic buckling analyses was undertaken for LSBs with 5 mm thick plate

stiffeners at varying spacings of span/2, span/3 and span/4 as shown in Figures 8.12

(a) to (c) using the ideal finite element model and the results are presented in Table

8.4.

It should be noted that there is no need of steel plates at the support for these ideal

finite element models as the idealised simply supported boundary conditions provide

the required restraints against flange twist as mentioned earlier in this chapter.


8-18

Figure 8.12: LSBs with Web Stiffeners at Different Spacings

Table 8.4: Effect of Web Stiffener Spacing on the Elastic Distorional Buckling

Moments of LSBs in kNm

LSB Sections

Span (mm)

Web Stiffener Spacing Ratio

Span/2 Span/3 Span/4 (Span/3) / (Span/2)

(Span/4) / (Span/3)

300x75x3.0 LSB

2000 125.44 132.28 135.71 1.05 1.03 4000 58.59 61.24 62.61 1.05 1.02 6000 39.69 40.69 41.35 1.03 1.02

200x60x2.5 LSB

2000 48.03 50.75 52.06 1.06 1.03 4000 24.32 24.96 25.40 1.03 1.02 6000 16.67 16.82 16.97 1.01 1.01

150x45x2.0 LSB

2000 16.02 16.73 17.01 1.04 1.02 4000 8.31 8.40 8.49 1.01 1.01 6000 5.65 5.67 5.69 1.00 1.00

(a) Span/2

(c) Span/4

(b) Span/3

Web Stiffeners (5 mm thick)

Web Stiffeners (5 mm thick)


8-19

As seen in Table 8.4, the elastic lateral distortional buckling moments (Mod) increase

with decreased stiffener spacing. The ratio of Mod values for span/3 and span/2 was

about 1.05 for intermediate spans while they were about 1.02 for span/4 and span/3.

This indicates that the degree of improvement to Mod is not significant when the web

stiffener spacing was reduced from span/3 to span/4. An additional web stiffener thus

only provides about 2% increase in Mod. Therefore span/3 was considered to be the

optimum web stiffener spacing based on both member capacity and cost. Avery and

Mahendran (1997) also made a similar recommendation based on their elastic

buckling studies of HFBs with web stiffeners.

Figure 8.13: Elastic Lateral Buckling Modes of LSBs

(a) Span/2

(b) Span/3


8-20

Figure 8.13: Elastic Lateral Buckling Modes of LSBs

Figures 8.13 (a) to (c) show the elastic lateral distortional buckling modes of a

300x75x3.0 LSB with 4 m span with various web stiffener spacings. They show that

the level of web distortion was decreased with increasing number of web stiffeners.

By considering both the cost and the capacity improvement, web stiffener spacing of

span/3 was considered to be adequate. Table 8.4 clearly demonstrates this as the

maximum improvement is only about 3% when the web stiffeners are used at a

spacing of span/4 when compared to that of span/3.

For the chosen web stiffener spacing, it is important to investigate the effects of

different sizes (thicknesses) of web stiffeners and to determine the optimum size.

Four thicknesses of 3 mm, 4 mm, 5 mm and 10 mm were considered in this

investigation. Table 8.5 presents the elastic lateral distortional buckling moments

(Mod) of various web stiffener sizes for all the available 13 LSBs. The buckling

moment capacities increase with increasing thickness of web stiffeners for

intermediate spans while this increment is very small for long spans for which web

distortion is small. Buckling moment capacity improvement was about 1.4 % for

intermediate spans when 5 mm web stiffeners were replaced with 10 mm web

stiffeners. Therefore, the use of 10 mm web stiffeners cannot be justified.

(c) Span/4


8-21

Table 8.5: Effect of Web Stiffener Sizes on the Elastic Lateral Distortional

Buckling Moments of LSBs

LSB Sections

Span (mm)

Modw (kNm) % Increase

3 mm 4 mm 5 mm 10 mm4 mm

vs 3 mm

5 mm vs

4 mm

10 mm vs

5 mm

300x75x3.0 LSB

2000 131.13 131.80 132.28 133.77 0.51 0.36 1.13 3000 82.53 82.84 83.06 83.76 0.38 0.27 0.84 4000 60.97 61.13 61.24 61.62 0.26 0.18 0.62 6000 40.60 40.65 40.69 40.83 0.12 0.10 0.34 8000 30.66 30.68 30.70 30.76 0.07 0.07 0.20 10000 24.71 24.71 24.72 24.76 0.00 0.04 0.16

300x75x2.5 LSB

2500 85.34 85.72 85.98 86.88 0.45 0.30 1.05 3000 69.88 70.15 70.34 70.97 0.39 0.27 0.90 4000 51.70 51.84 51.94 52.29 0.27 0.19 0.67 6000 34.46 34.51 34.54 34.68 0.15 0.09 0.41 8000 26.05 26.07 26.08 26.15 0.08 0.04 0.27 10000 21.02 21.03 21.03 21.07 0.05 0.00 0.19

300x60x2.0 LSB

2500 36.86 37.00 37.11 37.56 0.38 0.30 1.21 3000 30.09 30.19 30.26 30.58 0.33 0.23 1.06 4000 22.18 22.23 22.27 22.45 0.23 0.18 0.81 6000 14.74 14.76 14.77 14.85 0.14 0.07 0.54 8000 11.13 11.14 11.14 11.18 0.09 0.00 0.36 10000 8.97 8.97 8.98 9.00 0.00 0.11 0.22

250x75x3.0 LSB

1250 214.23 215.51 216.41 219.27 0.60 0.42 1.32 2000 121.35 121.94 122.35 123.66 0.49 0.34 1.07 3000 78.41 78.66 78.84 79.43 0.32 0.23 0.75 4000 58.54 58.66 58.75 59.05 0.20 0.15 0.51 6000 39.31 39.35 39.37 39.48 0.10 0.05 0.28 8000 29.78 29.80 29.81 29.86 0.07 0.03 0.17 10000 24.02 24.03 24.03 24.06 0.04 0.00 0.12

250x75x2.5 LSB

2000 102.56 103.05 103.40 104.54 0.48 0.34 1.10 3000 66.43 66.66 66.81 67.34 0.35 0.23 0.79 4000 49.65 49.76 49.84 50.13 0.22 0.16 0.58 6000 33.36 33.40 33.42 33.53 0.12 0.06 0.33 8000 25.31 25.33 25.34 25.39 0.08 0.04 0.20 10000 20.46 20.47 20.47 20.50 0.05 0.00 0.15

250x60x2.0 LSB

2000 44.36 44.55 44.69 45.25 0.43 0.31 1.25 3000 28.56 28.64 28.70 28.97 0.28 0.21 0.94 4000 21.29 21.33 21.37 21.51 0.19 0.19 0.66 6000 14.28 14.30 14.31 14.36 0.14 0.07 0.35 8000 10.82 10.83 10.83 10.86 0.09 0.00 0.28 10000 8.74 8.74 8.74 8.76 0.00 0.00 0.23


8-22

Table 8.5 (cont.): Effect of Web Stiffener Sizes on the Elastic Lateral

Distortional Buckling Moments of LSBs

LSB Sections

Span (mm)



vs 3 mm

5 mm vs

4 mm

10 mm vs

5 mm

200x60x2.5 LSB

1000 113.70 114.33 114.81 116.63 0.55 0.42 1.59 2000 50.43 50.61 50.75 51.27 0.36 0.28 1.02 3000 33.14 33.21 33.27 33.49 0.21 0.18 0.66 4000 24.91 24.94 24.96 25.08 0.12 0.08 0.48 6000 16.81 16.81 16.82 16.86 0.00 0.06 0.24 8000 12.73 12.73 12.74 12.76 0.00 0.08 0.16 10000 10.25 10.25 10.25 10.27 0.00 0.00 0.20

200x60x2.0 LSB

1500 56.33 56.59 56.79 57.59 0.46 0.35 1.41 2000 41.11 41.27 41.38 41.85 0.39 0.27 1.14 3000 27.09 27.16 27.21 27.41 0.26 0.18 0.74 4000 20.38 20.41 20.43 20.54 0.15 0.10 0.54 6000 13.78 13.79 13.79 13.84 0.07 0.00 0.36 8000 10.47 10.48 10.48 10.50 0.10 0.00 0.19 10000 8.46 8.46 8.46 8.48 0.00 0.00 0.24

200x45x1.6 LSB

1500 20.33 20.42 20.49 20.89 0.44 0.34 1.95 2000 14.73 14.78 14.82 15.05 0.34 0.27 1.55 3000 9.65 9.67 9.69 9.79 0.21 0.21 1.03 4000 7.24 7.25 7.26 7.31 0.14 0.14 0.69 6000 4.88 4.89 4.89 4.91 0.20 0.00 0.41 8000 3.70 3.70 3.70 3.72 0.00 0.00 0.54 10000 2.98 2.98 2.98 3.00 0.00 0.00 0.67

150x45x2.0 LSB

750 50.70 50.99 51.22 52.34 0.57 0.45 2.19 1000 35.41 35.59 35.74 36.44 0.51 0.42 1.96 1500 22.47 22.57 22.61 22.93 0.45 0.18 1.42 2000 16.65 16.69 16.73 16.90 0.24 0.24 1.02 3000 11.10 11.12 11.13 11.20 0.18 0.09 0.63 4000 8.39 8.39 8.40 8.44 0.00 0.12 0.48 6000 5.66 5.67 5.67 5.68 0.18 0.00 0.18 8000 4.28 4.28 4.28 4.29 0.00 0.00 0.23 10000 3.38 3.38 3.38 3.39 0.00 0.00 0.30

150x45x1.6 LSB

1000 28.80 28.94 29.07 29.71 0.49 0.45 2.20 1500 18.35 18.42 18.47 18.77 0.38 0.27 1.62 2000 13.63 13.67 13.70 13.86 0.29 0.22 1.17 3000 9.11 9.12 9.14 9.21 0.11 0.22 0.77 4000 6.89 6.90 6.91 6.94 0.15 0.14 0.43 6000 4.68 4.68 4.68 4.70 0.00 0.00 0.43 8000 3.55 3.55 3.55 3.56 0.00 0.00 0.28 10000 2.86 2.86 2.86 2.86 0.00 0.00 0.00


8-23

Table 8.5 (cont.): Effect of Web Stiffener Sizes on the Elastic Lateral

Distortional Buckling Moments of LSBs

LSB Sections

Span (mm)



vs 3 mm

5 mm vs

4 mm

10 mm vs

5 mm

125x45x2.0 LSB

750 45.78 46.04 46.25 47.25 0.57 0.46 2.16 1000 32.84 33.00 33.13 33.72 0.49 0.39 1.78 2000 15.98 16.02 16.04 16.18 0.25 0.12 0.87 3000 10.74 10.75 10.76 10.82 0.09 0.09 0.56 4000 8.14 8.14 8.15 8.18 0.00 0.12 0.37 6000 5.50 5.50 5.50 5.51 0.00 0.00 0.18 8000 4.15 4.15 4.15 4.16 0.00 0.00 0.24 10000 3.33 3.33 3.33 3.34 0.00 0.00 0.30

125x45x1.6 LSB

1000 26.74 26.88 26.99 27.53 0.52 0.41 2.00 2000 13.11 13.14 13.16 13.29 0.23 0.15 0.99 3000 8.82 8.83 8.84 8.90 0.11 0.11 0.68 4000 6.70 6.70 6.71 6.74 0.00 0.15 0.45 6000 4.55 4.55 4.56 4.57 0.00 0.22 0.22 8000 3.45 3.45 3.45 3.46 0.00 0.00 0.29 10000 2.77 2.77 2.77 2.78 0.00 0.00 0.36

Based on these elastic buckling analyses of LSBs with web stiffeners, it was decided

to use 5 mm thick plate web stiffeners, welded to the inner faces of top and bottom

flanges at third points within the span and supports as the optimum web stiffener

configuration. It was then decided to obtain the elastic lateral distortional buckling

moments and the nonlinear ultimate moments of all the 13 LSBs using the ideal

finite element model shown in Figure 8.8. The following section presents the details

of the elastic buckling analyses of LSBs with web stiffeners.

8.3 Elastic Lateral Distortional Buckling of LSBs with Web Stiffeners

The ideal finite element model of LSBs with the optimum web stiffener arrangement

was considered in these elastic buckling analyses. As explained earlier, web

stiffeners were not explicitly modelled at the supports, instead they were simulated

via idealised simply supported conditions that provided full twist restraint. The

lateral distortional buckling mode obtained from these analyses revealed reduced

web distortion when compared to that of LSBs without web stiffeners. Figure 8.14


8-24

shows the elastic lateral distortional buckling mode of 1500 mm span 150x45x2.0

LSB with web stiffeners. It shows that the web distortion is small.

Figure 8.14: Elastic Lateral Distortional Buckling of LSB with Web Stiffener Some LSBs exhibited a local buckling failure mode in the case of some intermediate

spans despite the fact they exhibited a lateral distortional buckling mode without web

stiffeners. This demonstrates that lateral distortional buckling is delayed for some

LSBs by using web stiffeners. Table 8.6 presents the elastic lateral torsional buckling

moments (Mo), elastic lateral distortional buckling moments without web stiffeners

(Mod) and the elastic lateral distortional buckling moments (Modw) of LSBs with web

stiffeners. It also compares the ratios of these buckling moments.

Web Stiffener


8-25

Table 8.6: Comparison of Elastic Buckling Moments

LSB Sections

Span (mm)

Mo (kNm)

Mod (kNm)

Modw (kNm) Mo/Mod Modw/Mod Modw/Mo

300x75x3.0 LSB

1500 219.9 144.55 LB 1.52 - - 2000 150.3 97.87 132.28 1.54 1.35 0.88 3000 92.62 65.69 83.06 1.41 1.26 0.90 4000 67.35 52.37 61.24 1.29 1.17 0.91 6000 43.87 38.00 40.69 1.15 1.07 0.93 8000 32.63 29.71 30.70 1.10 1.03 0.94 10000 26.00 24.29 24.72 1.07 1.02 0.95

300x75x2.5 LSB

1750 153.1 94.02 LB 1.63 - - 2000 129.0 78.39 LB 1.65 - - 2500 98.32 63.31 85.98 1.55 1.36 0.87 3000 79.63 51.62 70.34 1.54 1.36 0.88 4000 57.96 41.59 51.94 1.39 1.25 0.90 6000 37.77 31.05 34.54 1.22 1.11 0.91 8000 28.10 24.72 26.08 1.14 1.06 0.93 10000 22.39 20.43 21.03 1.10 1.03 0.94

300x60x2.0 LSB

1750 65.70 41.99 LB 1.56 - - 2000 55.25 35.04 LB 1.58 - - 2500 42.00 28.26 37.11 1.49 1.31 0.88 3000 33.95 22.99 30.26 1.48 1.32 0.89 4000 24.66 18.36 22.27 1.34 1.21 0.90 6000 16.05 13.50 14.77 1.19 1.09 0.92 8000 11.93 10.65 11.14 1.12 1.05 0.93 10000 9.51 8.76 8.98 1.09 1.03 0.94

250x75x3.0 LSB

1250 252.9 160.82 216.41 1.57 1.35 0.86 1500 198.4 124.35 171.77 1.60 1.38 0.87 2000 138.9 89.72 122.35 1.55 1.36 0.88 3000 87.64 64.12 78.84 1.37 1.23 0.90 4000 64.37 51.78 58.75 1.24 1.13 0.91 6000 42.25 37.46 39.37 1.13 1.05 0.93 8000 31.52 29.14 29.81 1.08 1.02 0.95 10000 25.15 23.75 24.03 1.06 1.01 0.96

250x75x2.5 LSB

1500 170.28 100.64 LB 1.69 - - 2000 119.43 70.79 103.40 1.69 1.46 0.87 3000 75.44 50.41 66.81 1.50 1.33 0.89 4000 55.44 41.43 49.84 1.34 1.20 0.90 6000 36.40 30.87 33.42 1.18 1.08 0.92 8000 27.16 24.40 25.34 1.11 1.04 0.93 10000 21.67 20.07 20.47 1.08 1.02 0.94

Mod – elastic lateral distortional buckling moment without web stiffeners from FEA. Modw – elastic lateral distortional buckling moment with web stiffeners from FEA. Mo – elastic lateral torsional buckling moment from Eq. 8.1. LB – Local Buckling.


8-26

Table 8.6 (continued): Comparison of Elastic Buckling Moments

LSB Sections

Span (mm)

Mo (kNm)

Mod (kNm)


250x60x2.0 LSB

1500 73.19 45.10 LB 1.62 - - 2000 51.13 31.69 44.69 1.61 1.41 0.87 3000 32.18 22.36 28.7 1.44 1.28 0.89 4000 23.61 18.19 21.37 1.30 1.17 0.91 6000 15.49 13.37 14.31 1.16 1.07 0.92 8000 11.55 10.50 10.83 1.10 1.03 0.94 10000 9.21 8.60 8.74 1.07 1.02 0.95

200x60x2.5 LSB

1000 133.89 85.99 114.81 1.56 1.34 0.86 1250 99.63 63.39 86.73 1.57 1.37 0.87 1500 79.43 51.59 69.86 1.54 1.35 0.88 2000 56.77 39.80 50.75 1.43 1.28 0.89 3000 36.45 29.19 33.27 1.25 1.14 0.91 4000 26.96 23.26 24.96 1.16 1.07 0.93 6000 17.79 16.42 16.82 1.08 1.02 0.95 8000 13.30 12.60 12.74 1.06 1.01 0.96 10000 10.62 10.19 10.25 1.04 1.01 0.97

200x60x2.0 LSB

1250 82.75 48.68 LB 1.70 - - 1500 66.05 38.84 56.79 1.70 1.46 0.86 2000 47.27 29.57 41.38 1.60 1.40 0.88 3000 30.39 22.16 27.21 1.37 1.23 0.90 4000 22.49 18.13 20.43 1.24 1.13 0.91 6000 14.85 13.19 13.79 1.13 1.05 0.93 8000 11.10 10.27 10.48 1.08 1.02 0.94 10000 8.86 8.37 8.46 1.06 1.01 0.95

200x45x1.6 LSB

1250 29.40 18.65 LB 1.58 - - 1500 23.39 14.96 20.49 1.56 1.37 0.88 2000 16.67 11.37 14.82 1.47 1.30 0.89 3000 10.68 8.33 9.69 1.28 1.16 0.91 4000 7.89 6.67 7.26 1.18 1.09 0.92 6000 5.20 4.74 4.89 1.10 1.03 0.94 8000 3.89 3.65 3.70 1.06 1.01 0.95 10000 3.10 2.96 2.98 1.05 1.01 0.96

150x45x2.0 LSB

750 59.52 38.84 51.22 1.53 1.32 0.86 1000 40.79 26.74 35.74 1.53 1.34 0.88 1250 31.10 21.31 27.63 1.46 1.30 0.89 1500 25.20 18.21 22.61 1.38 1.24 0.90 2000 18.35 14.52 16.73 1.26 1.15 0.91 3000 11.96 10.48 11.13 1.14 1.06 0.93 4000 8.90 8.17 8.40 1.09 1.03 0.94 6000 5.90 5.62 5.67 1.05 1.01 0.96 8000 4.41 4.26 4.28 1.04 1.00 0.97 10000 3.53 3.43 3.38 1.03 0.99 0.96


8-27

Table 8.6 (continued): Comparison of Elastic Buckling Moments

LSB Sections

Span (mm)

Mo (kNm)

Mod (kNm)


150x45x1.6 LSB

750 49.50 30.57 LB 1.62 - - 1000 34.00 20.38 29.07 1.67 1.43 0.86 1250 25.96 15.96 22.53 1.63 1.41 0.87 1500 21.05 13.61 18.47 1.55 1.36 0.88 2000 15.34 11.02 13.70 1.39 1.24 0.89 3000 10.01 8.24 9.14 1.21 1.11 0.91 4000 7.45 6.56 6.91 1.14 1.05 0.93 6000 4.94 4.60 4.68 1.07 1.02 0.95 8000 3.70 3.52 3.55 1.05 1.01 0.96 10000 2.95 2.84 2.86 1.04 1.01 0.97

125x45x2.0 LSB

750 53.69 33.65 46.25 1.60 1.37 0.86 1000 37.70 24.72 33.13 1.53 1.34 0.88 1250 29.16 20.47 26 1.42 1.27 0.89 1500 23.83 17.82 21.48 1.34 1.21 0.90 2000 17.52 14.35 16.04 1.22 1.12 0.92 3000 11.51 10.31 10.76 1.12 1.04 0.93 4000 8.59 7.99 8.15 1.08 1.02 0.95 6000 5.70 5.47 5.50 1.04 1.01 0.96 8000 4.27 4.14 4.15 1.03 1.00 0.97 10000 3.41 3.33 3.33 1.02 1.00 0.98

125x45x1.6 LSB

750 44.75 25.97 LB 1.72 - - 1000 31.48 18.53 26.99 1.70 1.46 0.86 1250 24.37 15.25 21.25 1.60 1.39 0.87 1500 19.93 13.37 17.58 1.49 1.31 0.88 2000 14.67 11.01 13.16 1.33 1.20 0.90 3000 9.64 8.20 8.84 1.18 1.08 0.92 4000 7.19 6.47 6.71 1.11 1.04 0.93 6000 4.78 4.50 4.56 1.06 1.01 0.95 8000 3.58 3.43 3.45 1.04 1.01 0.96 10000 2.86 2.77 2.77 1.03 1.00 0.97

Mod – elastic lateral distortional buckling moment without web stiffeners from FEA. Modw – elastic lateral distortional buckling moment with web stiffeners from FEA. Mo – elastic lateral torsional buckling moment from Eq. 8.1. LB – Local Buckling. As seen in Table 8.6, the use of web stiffeners improved the elastic lateral

distortional buckling moment capacities. The ratio of Modw/Mod reduces with

increasing span while the ratio of Modw/Mo increases. The elastic lateral distortional

buckling moment of stiffnened LSBs (Modw) approaches the elastic lateral torsional

buckling moment (Mo) with increasing spans. The ratio of Modw/Mo of 0.97 reveals

this fact. It is important to develop a relationship between Mo and Modw in order to


8-28

calculate Modw without undertaking any finite element analyses. For this purpose, the

ratio of Modw/Mo was plotted against span in Figure 8.15 while Figure 8.16 shows the

variation of this ratio with non-dimensional slenderness λ = (My/Mo)1/2 .

Figure 8.15: Modw/Mo versus Span for LSBs with Web Stiffeners

Figure 8.16: Modw/Mo versus Slenderness for LSBs with Web Stiffeners

The first yield moment My was calculated for LSBs without corners as described in

Appendix D.1 and the values are presented in Chapter 6. However, Table 8.7

presents the first yield moments (My) of all the available 13 LSBs.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

Span, (mm)

Mod

w/M

o

300x75x3.0 LSB300x75x2.5 LSB300x60x2.0 LSB250x75x3.0 LSB250x75x2.5 LSB250x60x2.0 LSB200x60x2.5 LSB200x60x2.0 LSB200x45x1.6 LSB150x45x2.0 LSB150x45x1.6 LSB125x45x2.0 LSB125x45x1.6 LSB

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40

Slenderness (λ)

Mod

w/M

o

300x75x3.0 LSB300x75x2.5 LSB300x60x2.0 LSB250x75x3.0 LSB250x75x2.5 LSB250x60x2.0 LSB200x60x2.5 LSB200x60x2.0 LSB200x45x1.6 LSB150x45x2.0 LSB150x45x1.6 LSB125x45x2.0 LSB125x45x1.6 LSB


8-29

Table 8.7: First Yield Moments of LSBs

LSB Sections My (kNm)

300x75x3.0 LSB 77.24 300x75x2.5 LSB 64.79 300x60x2.0 LSB 45.17 250x75x3.0 LSB 60.06 250x75x2.5 LSB 50.38 250x60x2.0 LSB 35.10 200x60x2.5 LSB 31.98 200x60x2.0 LSB 25.79 200x45x1.6 LSB 17.23 150x45x2.0 LSB 14.35 150x45x1.6 LSB 11.58 125x45x2.0 LSB 11.15 125x45x1.6 LSB 9.00

Based on the variation of FEA data points in Figure 8.16, two possible equations

were developed. Equation 8.2 was a linear equation while Equation 8.3 was a second

order polynomial equation and the relevant curves are shown in Figures 8.17 and

8.18.

Figure 8.17: Comparison of Modw with Equation 8.2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Slenderness (λ)

Mod

w/M

o

FEAEquation 8.2


8-30

Figure 8.18: Comparison of Modw with Equation 8.3

Figures 8.17 and 8.18 present the comparison of FEA elastic lateral distortional

buckling moments of LSBs with web stiffeners (Modw) with the developed design

equations as shown next.

Modw/Mo = 0.065 λ + 0.84 (8.2)

Modw/Mo = - 0.30 λ2 + 0.14 λ + 0.80 (8.3)

where, λ = (My/Mo)1/2

The ratios of FEA to predicted buckling moment ratios were obtained and the mean

and COV values were calculated for both equations. Equation 8.2 has a mean FEA to

predicted value of 1.00 and a COV of 0.017 while those for Equation 8.3 are 1.00

and 0.015. This indicates that the developed equations are accurate to predict the

lateral distortional buckling moments of LSBs with the chosen web stiffener

configuration in this research. Although Equation 8.2 is considered to be a simple

equation, it over-estimates the buckling moments at higher slenderness values (i e, λ

> 2). Therefore, it is recommended to avoid using Equation 8.2 for LSBs with high

slenderness values while Equation 8.3 is reasonable for any slenderness values.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Slenderness (λ)

Mod

w/M

o

FEAEquation 8.3


8-31

8.4 Ultimate Member Moment Capacities of LSBs with Web Stiffeners

It is now necessary to predict the ultimate moment capacities of stiffened LSBs

subjected to lateral buckling. For this purpose, non-linear finite element analyses

were carried out for the available 13 LSBs with web stiffeners. Ideal finite element

models used in the investigation of elastic lateral distortional buckling of LSBs with

the optimum web stiffener configuration of 5 mm stiffeners at third span points was

adopted in the nonlinear analyses. Geometrical imperfection and both flexural and

membrane residual stresses were included in this study. The imperfection values,

direction and the residual stress distribution are the same as that used in the

investigation of lateral distortional buckling of LSBs described in Chapter 5.

Negative imperfection was found to be critical and this was used in this study. A

value of span/1000 was considered as the initial geometric imperfection. The lateral

buckling failure mode of a 2 m span 150x45x2.0 LSB obtained from the non-linear

finite element analysis is given in Figure 8.19 (a) while Figure 8.19 (b) shows the

non-linear failure mode of the same LSB without web stiffeners. These figures show

that web distortion has been significantly reduced by the use of web stiffeners. The

ultimate moment capacities from FEA with and without web stiffeners are presented

and compared in Table 8.8. The ultimate moment capacities in the case of LSBs with

web stiffeners are about the same even when smaller stiffener thicknesses of 3 mm

and 4 mm are used. The reduction in ultimate capacities is likely to be less than 1%

in all cases. Therefore, it is considered that the findings in this section including the

design rules are equally applicable to stiffener thicknesses in the range of 3 to 5 mm.


8-32

Figure 8.19: Lateral Buckling Mode of a 2 m Span 150x45x2.0 LSB from

Non-linear FEA

(a) With Web Stiffeners

Very Small Web Distortion

(b) Without Web Stiffeners

Large Web Distortion


8-33

Table 8.8: Comparison of Ultimate Moments with and without Web Stiffeners


Mu (kNm) % increase without WS with WS

300x75x3.0 LSB

2000 54.52 65.37 19.91 3000 46.29 54.76 18.30 4000 40.92 46.29 13.12 6000 32.45 34.60 6.62 8000 26.60 27.56 3.59 10000 22.55 23.02 2.10

300x75x2.5 LSB

3000 36.91 46.19 25.16 4000 32.62 39.05 19.71 6000 26.43 29.17 10.36 8000 22.02 23.33 5.95 10000 18.81 19.52 3.80

300x60x2.0 LSB

3000 17.77 22.62 27.32 4000 14.98 17.98 20.00 6000 11.57 12.81 10.71 8000 9.43 10.02 6.24

250x75x3.0 LSB

1250 53.63 60.51 12.82 1500 51.49 59.30 15.16 2000 48.24 56.42 16.96 3000 43.59 49.54 13.65 4000 39.41 43.03 9.20 6000 31.97 33.18 3.78 8000 26.40 27.05 2.46 10000 22.49 22.77 1.24

250x75x2.5 LSB

2000 38.39 47.20 22.95 3000 34.50 41.36 19.89 4000 31.72 36.17 14.04 6000 26.24 28.10 7.07 8000 22.07 22.81 3.36 10000 18.92 19.29 1.96

250x60x2.0 LSB

2000 20.91 26.53 26.92 3000 17.29 21.07 21.86 4000 15.04 17.29 14.97 6000 11.82 12.62 6.80 8000 9.65 10.05 4.17 10000 8.28 8.52 2.91

WS – Web Stiffeners


8-34


Stiffeners (WS)

LSB Sections Span (mm) Mu (kNm) % increase without WS with WS

200x60x2.5 LSB

1000 28.45 31.89 12.11 1250 27.08 31.30 15.57 1500 26.25 30.29 15.39 2000 24.41 27.97 14.60 3000 21.68 23.64 9.04 4000 18.95 20.01 5.64 6000 14.73 15.03 2.02 8000 11.94 12.12 1.49 10000 10.16 10.27 1.17

200x60x2.0 LSB

1500 20.08 24.46 21.83 2000 18.54 22.74 22.68 3000 16.64 19.19 15.30 4000 14.75 16.23 10.04 6000 11.79 12.26 4.02 8000 9.71 9.89 1.83

200x45x1.6 LSB

1500 9.85 12.37 25.55 2000 8.43 10.49 24.56 3000 6.85 7.88 15.11 4000 5.76 6.26 8.55 6000 4.33 4.51 4.21

150x45x2.0 LSB

1000 12.13 13.83 14.01 1250 11.63 13.23 13.75 1500 11.16 12.56 12.54 2000 10.23 11.16 9.12 3000 8.53 8.93 4.69 4000 7.13 7.33 2.80 6000 5.36 5.40 0.62 8000 4.37 4.40 0.76 10000 4.00 4.00 0.00

150x45x1.6 LSB

1000 9.44 11.16 18.25 1250 8.87 10.70 20.60 1500 8.47 10.13 19.61 2000 7.87 9.10 15.61 3000 6.71 7.28 8.42 4000 5.71 5.98 4.65 6000 4.35 4.45 2.29 8000 3.55 3.62 1.87 10000 3.20 3.24 1.35



8-35


Stiffeners (WS)


Mu (kNm) % increase without WS with WS

125x45x2.0 LSB

750 10.81 11.49 6.27 1000 10.58 11.46 8.37 1250 10.37 11.23 8.29 1500 10.14 10.81 6.68 2000 9.46 9.82 3.86 3000 8.13 8.31 2.24 4000 6.93 7.04 1.50 6000 5.26 5.29 0.50 8000 4.27 4.27 0.00 10000 3.78 3.78 0.00

125x45x1.6 LSB

1000 8.21 9.17 11.71 1250 7.93 9.07 14.43 1500 7.82 8.78 12.29 2000 7.38 8.11 9.86 3000 6.47 6.78 4.82 4000 5.61 5.74 2.31 6000 4.31 4.37 1.20 8000 3.51 3.53 0.74 10000 3.07 3.09 0.85


As seen in Table 8.8, the increase in the non-linear lateral buckling moment capacity

of LSBs is high for intermediate spans (up to 27%) while it is small for long spans.

This is as expected since lateral torsional buckling is the dominant buckling mode for

long span members. Thus web distortion is reduced for long spans and hence the

increase in the non-linear moment capacity is also small. Figure 8.20 shows the

variation of ultimate moment capacities of LSBs with web stiffeners while Figure

8.21 shows a comparison of ultimate moment capacities of some LSBs with and

without web stiffeners. The increase in the ultimate moments of LSBs due to the use

of web stiffeners with its span is evident in Figure 8.21. As mentioned above the

moment capacity improvement is high for LSBs with intermediate spans.


8-36

Figure 8.20: Ultimate Moments of LSBs with Web Stiffeners

Figure 8.21: Comparison of Ultimate Moments of LSBs with and without Web

Stiffeners

0

10

20

30

40

50

60

70

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Span (mm)

Ulti

mat

e La

tera

l Buc

klin

g M

omen

ts (k

Nm

) 300x75x3.0 LSB300x75x2.5 LSB300x60x2.0 LSB250x75x3.0 LSB250x75x2.5 LSB250x60x2.0 LSB200x60x2.5 LSB200x60x2.0 LSB200x45x1.6 LSB150x45x2.0 LSB150x45x1.6 LSB125x45x2.0 LSB125x45x1.6 LSB

0

10

20

30

40

50

60

70

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Span (mm)

Ulti

mat

e La

tera

l Buc

klin

g M

omen

ts (k

Nm

)

300x75x3.0 LSB with WS

300x75x3.0 LSB without WS








8-37

8.5 Development of Design Rules

The ultimate moments from FEA were non-dimensionalised and compared with the

design curve for the LSBs without web stiffeners developed in Chapter 6. Figure

8.22 shows the comparison of these FEA results with Equation 8.4 developed in

Chapter 6 as Equation 6.7.

For λd ≤ 0.54: Mc = My (8.4a)

For 0.54 < λd < 1.74: Mc = My (0.28 2dλ – 1.20 λd + 1.57) (8.4b)

For λd ≥ 1.74: ⎟⎟⎠

⎞⎜⎜⎝

⎛= 2

1

d

yc MMλ

(8.4c)

where, od

yd

MM

=λ

Figure 8.22: Comparison of Ultimate Moments with Equation 8.4

Figure 8.22 compares the ultimate moments with Equation 8.4 where the slenderness

λdw is given by the following equation.

odw

ydw

MM

=λ

Here, the elastic lateral distortional buckling moments of LSBs with web stiffeners

Modw were used instead of Mod.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd, λdw

Mu/M

y, M

b/My

Equation 8.4

FEA


8-38

As seen in Figure 8.23, it can be concluded that Equation 8.4 is conservative in

predicting the ultimate moments of LSBs with web stiffeners as most of the FEA

data points are above the design curve. The ratios of the ultimate moments from FEA

and Eq. 8.4 were calculated and the mean of this ratio was found to be 1.06 and the

COV was 0.053. The capacity reduction factor Φ was calculated using the

recommended AISI procedure (AISI, 2007). It was 0.96 compared with the

recommended capacity reduction factor of 0.90 for cold-formed steel members in

bending. Based on the mean and the capacity reduction factor values, it is concluded

that Equation 8.4 is conservative. It should be noted that only the ultimate moments

in the inelastic region were considered in these calculations (i e. 0.54 < λdw < 1.74).

Since the FEA data points are well above the design curve based on Equation 8.4 in

the intermediate slenderness region, it was considered that the use of equations

developed for the back to back LSBs may be more suitable. The design equations for

the ultimate moment capacities of back to back LSBs were developed by Jeyaragan

and Mahendran (2009) and are given next.

For λd ≤ 0.65: Mc = My (8.5a)

For 0.65 < λd < 1.80: Mc = My (0.28 2dλ – 1.29 λd + 1.73) (8.5b)

For λd ≥ 1.80: ⎟⎟⎠

⎞⎜⎜⎝

⎛= 2

1

d

yc MMλ

(8.5c)


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λd, λdw

Mu/M

y, M

b/My

FEA

Equation 8.5


8-39

Figure 8.23 shows that most of the FEA data points are below the design curve based

on Equation 8.5. The mean of the ratio of ultimate moment capacities from FEA and

this equation was found to be 0.97 and the associated COV was 0.049 with a capacity

reduction factor of 0.88. This indicates that Equation 8.5 is also not suitable to

predict the ultimate moment capacities of stiffened LSBs subject to lateral buckling.

Therefore a new design equation (Eq. 8.6) was developed by solving for minimum

total error for the available FEA ultimate moments (92 results).

For λdw ≤ 0.60: Mc = My (8.6a)

For 0.60 < λdw < 1.70: Mc = My (0.29 2dwλ – 1.26 λdw + 1.65) (8.6b)

For λdw ≥ 1.70: ⎟⎟⎠

⎞⎜⎜⎝

⎛= 2

1

dw

yc MMλ

(8.6c)


The mean and COV of the ratio of ultimate moment capacities from FEA and

Equation 8.6 were calculated to be 1.02 and 0.050, respectively for the inelastic

region. The capacity reduction factor in this case was found to be 0.92, which is

slightly higher than the recommended value of 0.90. Since no experiments have been

carried out in this study, it is appropriate to accept a slightly higher capacity

reduction factor of 0.92.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, λdw

Mu/M

y, M

b/My

Equation 8.6

FEA


8-40

A geometrical parameter was found to reduce the scatter of the FEA data points of

the ultimate moment capcacities of LSBs without web stiffeners as described in

Chapter 6. An attempt was also made to determine the applicability of the same

geometrical parameter in the investigation of LSBs with web stiffeners. Figure 8.25

shows the comparison of the FEA ultimate moments of LSBs with web stiffeners and

the new design equation with geometrical parameter as presented in Chapter 6 (Eq.

6.18).

For Kλdw ≤ 0.52: Mc = My (8.7a)

For Kλdw > 0.52: Mc = My (0.199(Kλdw)2 – 1.013Kλdw + 1.475) (8.7b)

xweb

f

EIGJ

K+

=85.0

1 (8.8)

where

GJf = torsional rigidity of the flange



0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, Kλdw

Mu/M

y, M

b/My

FEAEquation 8.7


8-41

As seen in Figure 8.25, Equation 8.7 is conservative at the beginning of the inelastic

region. The mean and COV of the ultimate moment capacities from FEA and this

equation were found to be 1.03 and 0.038, respectively, with a capacity reduction

factor of 0.93. Therefore, a new design equation was developed by solving for

minimum total error for the available FEA ultimate moments as shown next.

For Kλdw ≤ 0.58: Mc = My (8.8a)

For Kλdw > 0.58: Mc = My (0.212(Kλdw)2 – 1.09Kλdw + 1.56) (8.8b)


Equation 8.8 predicted the ultimate moment capacities accurately as reflected by the

mean and COV of the ratio of ultimate moment capacities from FEA and Equation

8.8, which were 1.02 and 0.043, respectively. The capacity reduction factor was

found to be 0.92, which is slightly greater than the recommended value of 0.90.

Calculation of geometrical parameter K is presented in Appendix D.4 and the section

properties of LSBs including the major axis flexural rigidity of web and the torsional

rigidity of flanges are given in Table 6.12 of Chapter 6. However, the same table is

reproduced here as Table 8.9 for the sake of completeness.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Slenderness, Kλdw

Mu/M

y, M

b/My

FEAEquation 8.8


8-42

Table 8.9: Section Properties of LSBs Including

LSB Section Ixweb (103mm4)

EIxweb (106Nmm2)

Jf (103 mm4)

GJf (106Nmm2) GJf/EIxweb K

300x75x3.0 LSB 3906 781 250 160 12812 0.0164 1.0224 300x75x2.5 LSB 3255 651 042 140 11204 0.0172 1.0192 300x60x2.0 LSB 2929 585 867 57 4589 0.0078 1.0655 250x75x3.0 LSB 2000 400 000 160 12812 0.0320 0.9718 250x75x2.5 LSB 1667 333 333 140 11204 0.0336 0.9677 250x60x2.0 LSB 1544 308 700 57 4589 0.0149 1.0289 200x60x2.5 LSB 853 170 667 68 5400 0.0316 0.9729 200x60x2.0 LSB 683 136 533 57 4589 0.0336 0.9677 200x45x1.6 LSB 655 131 013 19 1524 0.0116 1.0440 150x45x2.0 LSB 288 57 600 22 1786 0.0310 0.9746 150x45x1.6 LSB 230 46 080 19 1524 0.0331 0.9691 125x45x2.0 LSB 143 28 579 22 1786 0.0625 0.9091 125x45x1.6 LSB 114 22 863 19 1524 0.0667 0.9024

8.6 Conclusions

This chapter has presented the details of an investigation on the effects of web

stiffeners on the lateral distortional buckling moment behaviour and capacity of

LSBs. Various types of web stiffener configurations including their size and spacing

were considered using a series of elastic buckling analyses. It was found that 5 mm

thick steel plates welded to the inner surfaces of the top and bottom flanges at the

beam supports and at third points within the span considerably improved the lateral

distortional buckling moment capacities of LSBs. It was found that this improvement

was achieved when web stiffeners are also used at the supports, thus providing flange

twist restraints to the entire section including its flanges.

The use of web stiffeners reduced the level of web distortion considerably and thus

allowed the LSB members to achieve at least 85% of lateral torsional buckling

capacity for short and intermediate spans, but reached about 97% of lateral torsional

buckling capacity for long spans. The web stiffeners can also be screw-fixed instead

of welding to the inner faces of top and bottom flanges of LSBs. Thinner web

stiffeners (3 mm or 4 mm) can also be considered to be equally effective for thinner

and smaller LSBs. Suitable equations were developed to calculate the elastic lateral

distortional buckling moments of LSBs with the above mentioned web stiffener

configurations.


8-43

The ultimate moment capacities of LSBs with web stiffeners were compared with the

developed design rules for single and back to back LSBs without web stiffeners.

Since they were not suitable, a new design rule was developed to accurately predict

the ultimate moment capacities of LSBs with web stiffeners subject to lateral

buckling.

It was found that the use of a geometrical parameter K significantly reduced the

scatter in the FEA data points where this parameter K was the same as that used in

the lateral buckling investigation of LSBs without web stiffeners. A new design rule

with the geometrical parameter K was also developed to accurately predict the

ultimate moment capacities of LSBs with web stiffeners.

Conclusions and Recommendations

9-1

CHAPTER 9

9.0 Conclusions and Recommendations

This thesis has described a detailed investigation into the flexural behaviour of

LiteSteel Beams (LSBs) based on experimental and finite element analyses. This

investigation included three phases. In the first phase the member moment capacity

of LSBs and other types of hollow flange sections such as Hollow Flange Beams

(HFBs), Monosymmetric Hollow Flange Beams (MHFBs) and Rectangular Hollow

Flange Beams (RHFBs) subject to lateral distortional buckling was investigated

while in the second phase, the section moment capacity of LSBs subject to local

buckling effects including the inelastic reserve moment capacity was investigated. In

the third phase the use of web stiffeners was investigated in order to improve the

lateral distortional buckling moment capacities of LSBs.

The LSB flexural members are subjected to a relatively new lateral distortional

buckling mode, which reduces their member moment capacities. A detailed

investigation into the flexural behaviour of LSBs and their member moment

capacities was undertaken in the first phase of this research using experimental and

finite element analyses. It included 12 lateral buckling tests of LSBs using a quarter

point loading arrangement, finite element modelling of tested LSBs, and a detailed

parametric study to develop suitable design rules. Numerical studies in this phase

entail the development of two finite element models, namely, experimental and ideal

models to simulate the flexural behaviour of LSBs including their lateral buckling

characteristics. A general purpose finite element analysis program ABAQUS Version

6.7 (HKS, 2007) and MSC PATRAN (PATRAN, 2008) were used in this study.

Experimental finite element models were used to simulate the tested LSBs and to

validate the models by a comparison of experimental and finite element analysis

results while ideal finite element models were used to develop member moment

capacity data under uniform moment conditions that were used to propose suitable

design rules. Chapters 4, 5 and 6 presented the details of Phase one of this research

project and the results.


9-2

In Phase two of this research, the section moment capacity of LSBs was investigated

based on experiments and finite element analyses. It included four point bending

tests of seven LSBs, numerical simulations and a detailed parametric study of section

moment capacity of LSBs including inelastic reserve capacity of LSBs. Suitable

experimental finite element models were developed and their results were compared

with experimental results for validation purposes. The validated model was then used

in the parametric study. The results from the parametric study and experiments were

used to review the available design rules for section moment capacity including the

presence of inelastic reserve bending capacity. Chapter 7 presented the details of

Phase 2 of this research project and the results.

Phase three of this research included an investigation on the effects of web stiffeners

on the lateral distortional buckling moment behaviour and capacity of LSBs. This

included finite element analyses of LSBs with different configurations of web

stiffeners in order to develop an optimum web stiffener configuration, and a detailed

parametric study to develop suitable design rules with optimum web stiffener

configuration. Chapter 8 presented the details of Phase 3 of this research project and

the results.

Chapter 3 of this thesis presented the details of tensile coupon tests and residual

stress and geometric imperfection measurements of LSBs used in the experimental

study. Chapter 4 presented the details of lateral buckling tests of LSBs. It includes a

comparison of the experimental results with the member moment capacity

predictions from AS/NZS 4600 (SA, 2005). Chapter 5 presented the details of the

finite element analyses of LSBs subject to lateral buckling including validation of

finite element models. The details of the detailed parametric study of LSBs subject to

lateral distortional buckling and the development of design rules are presented in

Chapter 6. The effect of section geometry and the applicability of developed design

rules for other types of hollow flange sections are also included in this chapter.

Chapter 7 presented the details of section moment capacity of LSBs including the

inelastic reserve moment capacity based on experiments and finite element analyses.

Chapter 8 presented the details of the effect of web stiffeners on the lateral

distortional buckling strength of LSBs. It includes the details of the optimum web


9-3

stiffener configuration to improve the lateral distortional buckling moment capacity

of LSBs and suitable design rules.

The most important outcomes obtained from this research are as follows.

Significantly improved understanding and knowledge of the flexural

behaviour of LSBs including their section and member moment capacities.

Development of accurate member capacity design rules for LSBs and other

hollow flange sections based on AS/NZS 4600 (SA, 2005). Some of them

included the effect of section geometry on the lateral distortional buckling

moment capacity using a geometrical parameter K (

xweb

f

EIGJ

+=

85.0

1 ).

Where, (GJf) is the flange torsional rigidity and (EIxweb) is the major axis

flexural rigidity of web. The new design rules can be used to accurately

predict the member moment capacity of LSBs with a capacity reduction

factor of (Φ) 0.90.

Assessment of the current design rules for the section moment capacity of

LSBs.

Confirmation of the presence of inelastic reserve moment capacity for

compact LSBs and suitable design rules to include it in design.

Development of an optimum web stiffener configuration to improve the

lateral distortional buckling moment capacity of LSBs and associated design

rules to predict the improved elastic buckling and ultimate member moment

capacities of LSBs with optimum web stiffeners.

Following important conclusions and recommendations have been drawn based on

the specific topics investigated in this research.

9.1 Experimental Investigation of LSBs

The measured initial geometric imperfections were well below the fabrication

tolerance limits for flexural members.

Residual stresses of LSBs were measured and the current membrane residual

stress distributions of LSBs proposed by Mahaarachchi and Mahendran

(2005e) were improved.


9-4

The mechanical properties of the tested LSB specimens were determined

using standard tensile coupon tests based on AS 1391 (SA, 2007).

It was found that flange twist restraints are needed at the supports for LSBs in

order to reach their theoretical ultimate lateral distortional buckling moment

capacities.

The failure mode of LSB flexural members was governed by lateral

distortional buckling for intermediate spans and the level of web distortion

was reduced for long span members.

Experimental lateral buckling moment capacity results from this research

were on average about 13% higher than the predictions of AS/NZS 4600 (SA,

2005) while those of Mahaarachchi and Mahendran (2005a) were on average

about 3% higher than the predictions of AS/NZS 4600 (SA, 2005). The use of

accurate Mod values for some test beams without flange twist restraint would

have given higher ratios of test moment capacity to AS/NZS 4600 (SA, 2005)

prediction.

9.2 Finite element Modelling of LSBs Subject to Lateral Buckling

The approximation of LSBs’ round corners with right angle corners in the

finite element modelling has negligible effect on the section properties as

well as the elastic lateral buckling moments.

Finite element analyses using the experimental finite element models

developed in this research well predicted the ultimate moments and load-

deflection curves obtained from experiments.

Ideal finite element models developed in this research were able to predict the

elastic lateral buckling moments of LSBs with the moments from a well

established finite strip analysis program Thin-Wall and the elastic lateral

distortional buckling moments from Pi and Trahair’s (1997) equation with an

average deviation of 1.5% and 2.9%, respectively.

The developed finite element model was able to capture both the elastic and

non-linear ultimate strength behaviour of LSBs.


9-5

9.3 Parametric Studies and Design Rule Development

Negative geometric imperfection was found to be critical in finite element

analyses of LSBs, and was considered in the parametric study.

Residual stresses significantly reduced the lateral distortional buckling

moment capacity of LSBs with a maximum reduction of 16% for

intermediate spans while it was about 10% for long spans.

The effect of flexural residual stresses was higher than the effect of

membrane residual stress in the case of intermediate spans while the effect of

membrane residual stress was higher than the effect of flexural residual stress

in the case of large spans.

The comparison of ultimate member moment capacities of LSBs from finite

element analyses and experimental studies with the current design rules in

AS/NZS 4600 (SA, 2005) showed that the current design rule was

conservative by about 8% in the inelastic lateral buckling region.

New improved design rules were developed for LSBs based on both FEA and

experimental results, which can be used to predict the member moment

capacities of LSBs with a capacity reduction factor (Φ) of 0.90.

The applicability of the developed design rule was investigated for HFBs and

it was found that the design rule developed for monosymmetric LSBs was

very conservative as HFB is a doubly symmetric section.

A geometrical parameter defined as the ratio of flange torsional rigidity to the

major axis flexural rigidity of the web (GJf/EIxweb) was found to be a critical

parameter that reduced the scatter in the FEA data points of hollow flange

sections in the non-dimensionalised moment capacity plots based on Mu/My

versus λd.

New design rules were developed by using a modified slenderness parameter

Kλd where K was determined as a function of GJf/EIxweb. The new design

rules based on the modified slenderness parameter Kλd were found to be

accurate in calculating the moment capacities of not only LSBs and HFBs but

also other types of hollow flange sections such as MHFBs and RHFBs if their

residual stress variations are similar to that of LSBs and HFBs.


9-6

9.4 Section Moment Capacity of LSBs

Appropriate finite element models were developed to predict the section

moment capacity of LSBs and validated using the experimental results.

A comparison of the section moment capacity results from finite element

analyses, experiments and current design codes showed that compact and

non-compact LSBs based on AS 4100 (SA, 1998) have some inelastic reserve

bending capacity while slender LSBs do not have any inelastic reserve

bending capacity beyond their first yield moment.

It is recommended that the inelastic bending capacity equations developed by

Shifferaw and Schafer (2008) can be used to predict the inelastic bending

capacities of compact and non-compact LSBs.

Eurocode 3 Part 1.3 (ECS, 2006) design equations can also be used to predict

the section moment capacity of LSBs.

As a simple approach, the ultimate moment capacity of compact LSB

sections can be taken as 1.10 times their first yield moment while it is the first

yield moment for non-compact sections. For slender LSB sections, current

cold-formed steel codes can be used to predict their section moment

capacities.

9.5 Effect of Web Stiffeners on the Lateral Distortional Buckling Moment

Capacity of LSBs

Elastic buckling finite element analyses showed that 3 to 5 mm thick steel

plates welded to the inner surfaces of the top and bottom flanges at the beam

supports and at third points within the span considerably improved the lateral

distortional buckling moment capacities of LSBs.

It was found that this improvement was achieved when web stiffeners are

also used at the supports, thus providing flange twist restraints to the entire

LSB section including its flanges.

The use of web stiffeners reduced the level of web distortion considerably

and thus allowed the LSB flexural members to achieve at least 85% of lateral

torsional buckling capacity for short and intermediate spans, but reached

about 97% of lateral torsional buckling capacity for long spans.


9-7

The web stiffeners can also be screw-fixed instead of welding to the inner

faces of top and bottom flanges of LSBs.

Suitable design equations were developed to calculate the elastic lateral

distortional buckling moments of LSBs with the recommended optimum web

stiffener configurations.

It was found that the developed design rules for single and back to back LSBs

without web stiffeners were not suitable to predict the lateral buckling

moments of LSBs with web stiffeners.

A new design rule was developed to accurately predict the ultimate moment

capacities of LSBs with web stiffeners subject to lateral buckling. This can be

used with a capacity reduction factor (Φ) of 0.90.

The use of a geometrical parameter K significantly reduced the scatter in the

FEA data points where this parameter K was the same as that used in the

lateral buckling investigation of LSBs without web stiffeners.

A new design rule with the geometrical parameter K was also developed to

more accurately predict the ultimate moment capacities of LSBs with web

stiffeners.

The developed design equations for LSBs with 5 mm thick steel plate

stiffeners welded to the inner faces of the flanges of LSBs at third span points

and supports is recommended to be equally applicable when thinner web

stiffeners (3 mm or 4 mm) are used for thinner and smaller LSBs.

9.6 Future Research

It is recommended that the following research projects are undertaken in the future to

advance the knowledge in this field.

Experimental measurements of residual stresses in other types of hollow

flange sections such as MHFBs and RHFBs. Measured residual stresses can

then be used in the finite element analyses of these sections to confirm the

adequacy of the developed design rules in Phase one of this research.

Effects of support conditions on the flexural behaviour and strength of LSBs.

Further section moment capacity tests of compact LSBs to more accurately

predict the inelastic reserve capacity of compact LSBs.


9-8

Section moment capacity of other types of hollow flange sections such as

MHFBs and RHFBs including their inelastic reserve bending capacities.

Lateral buckling tests of LSBs and other types of hollow flange sections with

web stiffeners.

Lateral buckling behaviour of LSBs with web stiffeners under non-uniform

moment distributions and transverse loading.

Appendix A

A-1

APPENDIX A

A.1: Stress-Strain Curves for LSB Section Material

Figure A.1: Stress-Strain Curve – 2.23 mm Outside Flange of 300x60x2.0 LSB

Figure A.2: Stress-Strain Curve – 2.01 mm Inside Flange of 300x60x2.0 LSB

0.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50

% Strain

Stre

ss (N

/mm

2 )

Strain GaugeExtensometerE=200GPa0.2% Proof Stress

557.7

0.0

100.0

200.0

300.0

400.0

500.0

600.0

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50

% Strain

Stre

ss (N

/mm

2 )

Extensometer

E=220GPa

0.2% Proof Stress

496.3

Appendix A

A-2

Figure A.3: Stress-Strain Curve – 1.98 mm Web of 300x60x2.0 LSB

Figure A.4: Stress-Strain Curves – 250x75x2.5 LSB

0.0

100.0

200.0

300.0

400.0

500.0

600.0

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

% Strain

Stre

ss (N

/mm

2 )

WebOutside FlangeE=210GPaInside Flange

0.0

100.0

200.0

300.0

400.0

500.0

600.0

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00

% Strain

Stre

ss (N

/mm

2 ) Extensometer

E=190GPa

0.2% Proof Stress

447.1

Appendix A

A-3



0.0

100.0

200.0

300.0

400.0

500.0

600.0

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00

% Strain

Stre

ss (N

/mm

2 )

WebOutside FlangeInside FlangeE=200GPa

0.0

100.0

200.0

300.0

400.0

500.0

600.0

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00

% Strain

Stre

ss (N

/mm

2 )

Inside FlangeOutside FlangeWebE=215GPa

Appendix A

A-4



0.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

Strain (%)

Stre

ss (M

Pa) Outside flange

Inside flangeWebE=200GPa

0

100

200

300

400

500

600

700

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00

Strain (%)

Stre

ss (M

Pa)

Outside flange

Inside flange

Web

E=200GPa

Appendix A

A-5

A.2: Measured Global Geometric Imperfections of LSBs

Figure A.9: Global Geometric Imperfection along the Web for 250x75x2.0 LSB

Figure A.10: Global Geometric Imperfection for 150x45x1.6 LSB

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 500 1000 1500 2000 2500 3000 3500

Span Length (mm)

Impe

rfec

tion

(mm

)

Along 1Along 2Along 3Along 4Along 5

2

1

4

5

3Neutral Axis

-1.8

-1.5

-1.2

-0.9

-0.6

-0.3

0.0

0.3

0.6

0.9

1.2

1.5

1.8

0 500 1000 1500 2000 2500 3000 3500

Span Length (mm)

Impe

rfec

tions

(mm

)

Flange - Average

Web - Average

Web Bottom

Web Middle

Web Top

Flange Location 1

Flange Location 2

1 2

Appendix B

B-1

APPENDIX B:

B.1: Moment vs Deflection Curves for the Tested LSB Specimens

Figure B.1: Moment vs Vertical Deflection of 300x60x2.0 LSB

Figure B.2: Moment vs Lateral Deflection of 300x60x2.0 LSB

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

0 10 20 30 40 50 60 70 80


Mom

ent (

kNm

)

3 m span, at Mid Span

3 m Span, under the Load

4 m Span, at MidSpan


0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

-10 10 30 50 70 90 110 130 150


Mom

ent (

kNm

)

3 m Span, at Top Flange (Tension)

3 m Span, at Bottom Flange (Compression)

4 m Span, at Top Flange (Tension)

4 m Span, at Bottom Flange (Compression)

Appendix B

B-2



0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 10 20 30 40 50 60 70 80


Mom

ent (

kNm

)

2 m Span, MidSpan






0.0

2.0

4.0

6.0

8.0

10.0

12.0

-10 10 30 50 70 90 110 130 150


Mom

ent (

kNm

)

2 m Span, Top Flange (Tension)

2 m Span, Bottom Flange (Compression)


3 m Span, Bottom Flange (Compression)


4 m, Bottom Flange (Compression)

Appendix B

B-3



0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 10 20 30 40 50 60 70 80 90 100


Mom

ent (

kNm

)

2.0 m Span, at MidSpan2.0 m Span, under the Load3.0 m Span, at MidSpan3.0 m Span, under the Load

0.0

2.0

4.0

6.0

8.0

10.0

12.0

-10 10 30 50 70 90 110 130 150

Lateral Deflection at Mid Span(mm)

Mom

ent (

kNm

)

2.0 m Span, at Top Flange (Tension)

2.0 m Span, at Bottom Flange (Compression)



Appendix B

B-4



0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0 10 20 30 40 50 60


Mom

ent (

kNm

)

1.2 m span, at MidSpan

1.2 m Span, under the Load

1.8 m Span, at MidSpan


3.0 m Span, at MidSpan


0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

-10 10 30 50 70 90 110 130


Mom

ent (

kNm

)







Appendix B

B-5

Figure B.9: Moment vs Vertical Deflection of 1.2 m Span 125x45x2.0 LSB

Figure B.10: Moment vs Lateral Deflection of 1.2 m Span 125x45x2.0 LSB

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 5 10 15 20 25 30 35


Mom

ent (

kNm

)

MidSpanunder the Load

0

2

4

6

8

10

12

-2 0 2 4 6 8 10 12 14 16 18


Mom

ent (

kNm

)

Top Flange (Tension)

Bottom Flange (Compression)

Appendix C

C-1

APPENDIX C:

C.1: Sample Calculation of Application of End Moment in the Ideal FE Model

of LSB

Nodal loads were applied at each node to create a uniform end moment across the

section of LSB. Sample calculations for an ideal model of 200x45x1.6 LSB are given

next.

Step 1

Find the linear equation of load across the cross section.

Note: Centreline dimensions are used

Figure C.1: Load Application Details

Table C.1: Applied Load Across the Cross Section

Load Distance Along Y Direction 1000 0

0 99.2 -1000 198.4

The equation is

Load = -10.08064516x + 1000 (C.1)

Where, x is the distance along Y direction.

This equation was used as a “spatial function” in MD/PATRAN and applied to all

the nodes at the end support of LSB.

99.2 mm

1000 N

-1000 N

Centreline

0 mm

198.4 mm

Y

Appendix C

C-2

Figure C.2: Applied Load Vs Cross Section Height

Step 2

Calculate the applied moment.

Find the load information from the input file to the elastic buckling analysis.

Figure C.3: Calculation of Applied Moment

Note: It is important to have the finite element mesh so that there is a node at the

middle of the web. Then the loads will be created symmetrically about the centreline

of the web.

Moment at web middle due to the force couple (F, -F)

= F * Lever arm (anti-clockwise moment)

Due to the symmetrical loading, the moment can be written as follows;

Moment = 2*F*(Lever arm/2)

Lever arm/2 for the force F = 99.2-Y1

y = -10.08064516x + 1000

-1500

-1000

-500

0

500

1000

1500

0 50 100 150 200 250

Distance Along Y Direction

Load

F

Lever arm

-F

YY1

Appendix C

C-3

Table C.2 shows the calculation of applied moment for a 200x45x1.6 LSB ideal FE

model. Here, the node number and the corresponding loads are extracted from the

input file of the elastic buckling analysis. Distance along the Y direction can be

calculated using the equation we obtained before.

For example,

Node 1

Load = 864.92 N

From equation, Load = -10.08064516*Distance along Y direction + 1000

Hence, Distance along Y direction = (1000-864.92)/10.08064516

= 13.40 mm

Lever arm/2 = 99.2-13.4

= 85.80 mm

Applied Moment = 2 * load * (lever arm/2)

= 2 * 864.92 * 85.8 * 10-6 kNm

= 0.1484 kNm

Table C.2: Calculation of Applied Moment for 200x45x1.6 LSB

Node No Load (N) Distance along

Y Direction (mm) Lever arm/2

(mm) Applied Moment

(kNm) 1 864.92 13.40 85.80 0.1484 2 864.92 13.40 85.80 0.1484 3 864.92 13.40 85.80 0.1484 4 864.92 13.40 85.80 0.1484 5 864.92 13.40 85.80 0.1484 6 864.92 13.40 85.80 0.1484 7 864.92 13.40 85.80 0.1484 8 864.92 13.40 85.80 0.1484 9 864.92 13.40 85.80 0.1484 10 864.92 13.40 85.80 0.1484

1512 909.95 8.93 90.27 0.1643 1513 954.97 4.47 94.73 0.1809 1514 1000.00 0.00 99.20 0.1984

Appendix C

C-4

Table C.2 (continued): Calculation of Applied Moment for 200x45x1.6 LSB

Node No Load (N) Distance along

Y Direction (mm) Lever arm/2

(mm) Applied Moment

(kNm) 2116 1000.00 0.00 99.20 0.1984 2117 1000.00 0.00 99.20 0.1984 2118 1000.00 0.00 99.20 0.1984 2119 1000.00 0.00 99.20 0.1984 2120 1000.00 0.00 99.20 0.1984 2121 1000.00 0.00 99.20 0.1984 2122 1000.00 0.00 99.20 0.1984 2123 1000.00 0.00 99.20 0.1984 2124 1000.00 0.00 99.20 0.1984 3626 954.97 4.47 94.73 0.1809 3627 909.95 8.93 90.27 0.1643 4230 814.04 18.45 80.75 0.1315 4231 763.17 23.49 75.71 0.1156 4232 712.29 28.54 70.66 0.1007 4233 661.41 33.59 65.61 0.0868 4234 610.53 38.64 60.56 0.0740 4235 559.66 43.68 55.52 0.0621 4236 508.78 48.73 50.47 0.0514 4237 457.90 53.78 45.42 0.0416 4238 407.02 58.82 40.38 0.0329 4239 356.15 63.87 35.33 0.0252 4240 305.27 68.92 30.28 0.0185 4241 254.39 73.96 25.24 0.0128 4242 203.51 79.01 20.19 0.0082 4243 152.64 84.06 15.14 0.0046 4244 101.76 89.11 10.09 0.0021 4245 50.88 94.15 5.05 0.0005 4246 0.00 99.20 0.00 0.0000

Total Moment 4.9269

Total Moment applied = 4.9269 kNm

Step 3

Application of this applied moment.

The critical buckling load factor obtained from the elastic buckling analysis and the

maximum load factor obtained from the non-linear static analysis should be

multiplied by this applied moment to determine the actual values.

Appendix C

C-5

C.2: Residual Stress Subroutine

This is the residual stress subroutine for an ideal model of 200x45x1.6 LSB with 3 m

span. SUBROUTINE SIGINI(SIGMA,COORDS,NTENS,NCRDS,NOEL,NPT,LAYER,KSPT) C INCLUDE 'ABA_PARAM.INC' C REAL X,Y,Z,nipt,ipt,Fy,IMPBUF,IMPBLF,IMPTLF,IMPTUF,sigmaout,MEMB DIMENSION SIGMA(NTENS), COORDS(NCRDS) C X=COORDS(1) Y=COORDS(2) Z=COORDS(3) midspan=1500. IMP=-3.0 TUF=0.455 TLF=0.478 BUF=0.984 BLF=1.01 BF=43.4 nipt=9. Fy=380. C IF(KSPT.EQ.1.) THEN ipt=1. ENDIF IF(KSPT.EQ.2.) THEN ipt=2. ENDIF IF(KSPT.EQ.3.) THEN ipt=3. ENDIF IF(KSPT.EQ.4.) THEN ipt=4. ENDIF IF(KSPT.EQ.5.) THEN ipt=5. ENDIF IF(KSPT.EQ.6.) THEN ipt=6. ENDIF IF(KSPT.EQ.7.) THEN ipt=7. ENDIF IF(KSPT.EQ.8.) THEN ipt=8. ENDIF IF(KSPT.EQ.9.) THEN ipt=9. ENDIF C IF(X.LE.(midspan)) THEN IMPTUF=Z-(IMP*TUF)*X/midspan IMPTLF=Z-(IMP*TLF)*X/midspan IMPBUF=Z-(IMP*BUF)*X/midspan IMPBLF=Z-(IMP*BLF)*X/midspan

Note: TUF, TLF, BUF and

BLF are the lateral deflections

at mid-span of the critical

elastic buckling mode after

bifurcation buckling analysis.

TUFTLF

BUFBLF

Appendix C

C-6

ENDIF C C FLEXURAL RESIDUAL STRESS IF((NOEL.GE.9151.).AND.(NOEL.LE.10500.)) THEN sigmaout=(0.24*Fy+0.83*Fy*IMPTUF/BF)*(1.-2.*(nipt-ipt)/(nipt-1.)) ELSEIF((NOEL.GE.1801.).AND.(NOEL.LE.3150.)) THEN sigmaout=(0.24*Fy+0.83*Fy*IMPBLF/BF)*(1.-2.*(nipt-ipt)/(nipt-1.)) ELSEIF((NOEL.GE.10951.).AND.(NOEL.LE.12300.)) THEN sigmaout=(0.38*Fy+0.42*Fy*IMPTLF/BF)*(1.-2.*(nipt-ipt)/(nipt-1.)) ELSEIF((NOEL.GE.1.).AND.(NOEL.LE.1350.)) THEN sigmaout=(0.38*Fy+0.42*Fy*IMPBUF/BF)*(1.-2.*(nipt-ipt)/(nipt-1.)) ELSEIF(((NOEL.GE.10501.).AND.(NOEL.LE.10950.)).OR. & ((NOEL.GE.1351.).AND.(NOEL.LE.1800.))) THEN sigmaout=(0.41*Fy)*(1.-2.*(nipt-ipt)/(nipt-1.)) ELSEIF(((NOEL.GE.8701.).AND.(NOEL.LE.9150.)).OR. & ((NOEL.GE.3151.).AND.(NOEL.LE.3600.))) THEN sigmaout=(0.24*Fy)*(1.-2.*(nipt-ipt)/(nipt-1.)) ELSEIF((NOEL.GE.3601.).AND.(NOEL.LE.8700.)) THEN sigmaout=(0.24*Fy)*(1.-2.*(nipt-ipt)/(nipt-1.)) ENDIF C C MEMBRANE RESIDUAL STRESS IF((NOEL.GE.3601.).AND.(NOEL.LE.8700.)) THEN IF((Y.GE.13.4).AND.(Y.LE.99.2)) THEN MEMB=(0.01166*Y-0.65618)*Fy ELSEIF((Y.GE.99.2).AND.(Y.LE.185)) THEN MEMB=(-0.01166*Y+1.65618)*Fy ENDIF ELSEIF((NOEL.GE.10951.).AND.(NOEL.LE.12300.)) THEN MEMB=0.11*Fy-0.08*Fy*IMPTLF/BF ELSEIF((NOEL.GE.1.).AND.(NOEL.LE.1350.)) THEN MEMB=0.11*Fy-0.08*Fy*IMPBUF/BF ELSEIF(((NOEL.GE.10501.).AND.(NOEL.LE.10950.)).OR. & ((NOEL.GE.1351.).AND.(NOEL.LE.1800.))) THEN MEMB=0.03*Fy ELSEIF(((NOEL.GE.8701.).AND.(NOEL.LE.9150.)).OR. & ((NOEL.GE.3151.).AND.(NOEL.LE.3600.))) THEN MEMB=-0.2567*Fy ENDIF SIGMA(1)=sigmaout+MEMB c SIGMA(2)=0 SIGMA(3)=0 C RETURN END

Appendix C

C-7

C.3: Comparison of Bending Moment vs Deflection Curves from Experiments

and Finite Element Analyses

Figure C.4: Bending Moment vs Vertical Deflection Curves at Mid-Span for

250x75x2.5 LSB (3500 mm Span)

Figure C.5: Bending Moment vs Lateral Deflection Curves at Mid-Span for

250x75x2.5 LSB (3500 mm Span)

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

-10 0 10 20 30 40 50 60 70


Mom

ent (

kNm

)

EXP

FEA

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

-10 10 30 50 70 90


Mom

ent (

kNm

)


EXP - Bottom Flange(Compression)

FEA - Top Flange

FEA - Bottom Flange

Appendix C

C-8


300x60x2.0 LSB (4000 mm Span)


300x60x2.0 LSB (3000 mm Span)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

0 5 10 15 20 25 30


Mom

ent (

kNm

)

EXP

FEA

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

-10 10 30 50 70 90 110 130 150


Mom

ent (

kNm

) EXP - Top Flange (Tension)


FEA - Top Flange

FEA - Bottom Flange

Appendix C

C-9


300x60x2.0 LSB (3000 mm Span)


200x45x1.6 LSB (3000 mm Span)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0 5 10 15 20 25 30 35 40 45 50


Mom

ent (

kNm

)

EXP

FEA

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

-10 10 30 50 70 90


Mom

ent (

kNm

)


EXP - Bottom Flange(Compression)

FEA - Top Flange

FEA - Bottom Flange

Appendix C

C-10


200x45x1.6 LSB (3000 mm Span)


200x45x1.6 LSB (2000 mm Span)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 5 10 15 20 25


Mom

ent (

kNm

) EXP

FEA

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

-10 10 30 50 70 90 110 130


Mom

ent (

kNm

)



FEA - Top Flange

FEA - Bottom Flange

Appendix C

C-11


200x45x1.6 LSB (2000 mm Span)


150x45x2.0 LSB (3000 mm Span)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 10 20 30 40 50 60 70 80 90


Mom

ent (

kNm

)

EXP

FEA

0.0

2.0

4.0

6.0

8.0

10.0

12.0

-10 0 10 20 30 40 50


Mom

ent (

kNm

)

EXP - Top Flange (Tension)EXP - Bottom Flange (Compression)FEA - Top FlangeFEA - Bottom Flange

Appendix C

C-12


150x45x2.0 LSB (2000 mm Span)


150x45x2.0 LSB (2000 mm Span)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 5 10 15 20 25 30 35 40 45 50


Mom

ent (

kNm

)

EXPFEA

0

2

4

6

8

10

12

14

-10 0 10 20 30 40 50 60 70 80 90


Mom

ent (

kNm

)

EXP - Top Flange (Tension)EXP - Bottom Flange (Compression)FEA - Top FlangeFEA - Bottom Flange

Appendix C

C-13


150x45x1.6 LSB (3000 mm Span)


150x45x1.6 LSB (1800 mm Span)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0 5 10 15 20 25 30 35 40 45 50


Mom

ent (

kNm

)

EXP

FEA

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

-10 0 10 20 30 40 50 60


Mom

ent (

kNm

)

Experiment- Top Flange (Tension)

Experiment- Bottom Flange (Compression)

FEA - Top Flange

FEA - Bottom Flange

Appendix C

C-14


125x45x2.0 LSB (1200 mm Span)


125x45x2.0 LSB (1200 mm Span)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 5 10 15 20 25 30 35


Mom

ent (

kNm

)

EXPFEA

0.0

2.0

4.0

6.0

8.0

10.0

12.0

-2 0 2 4 6 8 10 12 14 16 18


Mom

ent (

kNm

)



FEA - Top Flange

FEA - Bottom Flange

Appendix D

D-1

APPENDIX D:

D.1: Sample Calculations of Moment Capacities

Sample Calculations Based on AS/NZS 4600 (SA, 2005) Design Rules

300x75x3.0 LSB section – based on nominal dimensions and yield stresses without

corners. Centreline dimensions are used.

Section Moment Capacity

Nominal section capacity (Ms) based on initiation of yield Clause 3.3.2

Ms = Ze fy

Assume f* = fy in the top fibre of the section

Assuming full depth to be effective

Web

tw = 3 mm

d1 = 250 + 3 = 253 mm

d1 / tw = 253/3 = 84.33 < 200 Clause 2.1.3.4a

t = 3 mm

d = 300 mm

d1 = 250 mm

df = 25 mm

bf = 75 mm

fyf = 450 MPa

fyw = 380 MPa

bf

df

d d1

Appendix D

D-2

Horizontal Flange Element

Both longitudinal edges connected to other stiffened elements

b = 75-3 = 72

t = 3

b / t = 72/3 = 24 < 500 Clause 2.1.3.1(b)

Vertical Flange Element

b = 25-3 = 22

t = 3

b / t = 22/3 = 7.33 < 500 Clause 2.1.3.1(b)

Effective widths of elements

Horizontal Flange Element

K = 4 b = 72 t = 3 f* = 450 MPa

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

Ef

tb

k

*052.1λ Clause 2.2.1.2 (4)

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

200000450

372

4052.1λ

673.0599.0 <=λ

be = b = 72 mm, fully effective


f2 / (148.5-22) = 450/148.5

f2 = 383.33 MPa

f1 = 450 MPa

*1

*2

f

f=ψ Clause 2.2.3.2 (5)

450

33.383=ψ = 0.852

k = 4 + 2(1-ψ)3 + 2(1 – ψ) Clause 2.2.3.2 (4)

k = 4.30

f2

450

22

148.5

Appendix D

D-3

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

Ef

tb

k

*052.1λ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

200000450

322

30.4052.1λ

673.0176.0 <=λ

ψ > - 0.236

be1 = be / (3- ψ) Clause 2.2.3.2 (1)

be1 = 22 / (3-0.852) = 10.24 mm

be2 = be – be1 = 22 – 10.24 = 11.76 mm Clause 2.2.3.2 (3)

Fully effective

Web

f2* = -383.33 MPa

f1* = 383.33 MPa

*1

*2

f

f=ψ

33.38333.383−

=ψ = -1

k = 4 + 2(1-ψ)3 + 2(1 – ψ) Clause 2.2.3.2 (4)

k = 24

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

Ef

tb

k

*052.1λ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

20000033.383

3253

24052.1λ

673.0793.0 >=λ

be = ρb

λλρ

22.01−=

793.0

793.022.01−

=ρ

911.0=ρ

be = 253 x 0.911 = 230.5

Appendix D

D-4

be1 = be / (3- ψ) Clause 2.2.3.2 (1)

be1 = 230.5 / (3+1) = 57.62 mm

Since ψ < -0.236 be2 = be/2 = 230.5/2 = 115.25

be1 + be2 > compression portion of the web

Web is fully effective

My = Zfy

where Z is the full section modulus

Z = 171.7 x 103 mm3 (from Thin-wall)

My = 171.7 x 103 x 450 x 10-6 = 77.27 kNm

Since 300x75x3.0 LSB is fully effective, its effective section modulus Ze is equal to

its full section modulus Z. Therefore the section moment capacity Ms is equal to the

first yield moment My of this LSB. However, some other LSBs are not fully effective

as their flanges are not fully effective when the corners are not considered in the

calculations. Tables D.1 and D.2 give the details of calculations leading to their

effective widths.

Table D.1: Width to Thickness Ratio of LSBs

LSB Sections d (mm)

d1 (mm)

bf (mm)

df (mm)

t (mm)

Width to Thickness Ratio (b/t)

Web Horizontal

Flange Element

Vertical Flange

Element 300x75x3.0 LSB 300 250 75 25 3.0 84.33 24.00 7.33 300x75x2.5 LSB 300 250 75 25 2.5 101.00 29.00 9.00 300x60x2.0 LSB 300 260 60 20 2.0 131.00 29.00 9.00 250x75x3.0 LSB 250 200 75 25 3.0 67.67 24.00 7.33 250x75x2.5 LSB 250 200 75 25 2.5 81.00 29.00 9.00 250x60x2.0 LSB 250 210 60 20 2.0 106.00 29.00 9.00 200x60x2.5 LSB 200 160 60 20 2.5 65.00 23.00 7.00 200x60x2.0 LSB 200 160 60 20 2.0 81.00 29.00 9.00 200x45x1.6 LSB 200 170 45 15 1.6 107.25 27.13 8.38 150x45x2.0 LSB 150 120 45 15 2.0 61.00 21.50 6.50 150x45x1.6 LSB 150 120 45 15 1.6 76.00 27.13 8.38 125x45x2.0 LSB 125 95 45 15 2.0 48.50 21.50 6.50 125x45x1.6 LSB 125 95 45 15 1.6 60.38 27.13 8.38

Appendix D

D-5

Table D.2: Effective Width of Horizontal Flange Element in LSBs

LSB Sections Horizontal Flange Element Is

λ > 0.673? Comments K B λ be 300x75x3.0 LSB 4 72.0 0.5988 72.00 No Compact 300x75x2.5 LSB 4 72.5 0.7236 69.73 Yes Slender 300x60x2.0 LSB 4 58.0 0.7236 55.79 Yes Slender 250x75x3.0 LSB 4 72.0 0.5988 72.00 No Compact 250x75x2.5 LSB 4 72.5 0.7236 69.73 Yes Slender 250x60x2.0 LSB 4 58.0 0.7236 55.79 Yes Slender 200x60x2.5 LSB 4 57.5 0.5739 57.50 No Compact 200x60x2.0 LSB 4 58.0 0.7236 55.79 Yes Slender 200x45x1.6 LSB 4 43.4 0.6768 43.28 Yes Slender 150x45x2.0 LSB 4 43.0 0.5364 43.00 No Compact 150x45x1.6 LSB 4 43.4 0.6768 43.28 Yes Slender 125x45x2.0 LSB 4 43.0 0.5364 43.00 No Compact 125x45x1.6 LSB 4 43.4 0.6768 43.28 Yes Slender

As seen in Table D.2, the flange slenderness values (λ) of eight LSB sections are

greater than 0.673 and hence these sections are considered to be slender according to

AS/NZS 4600 (SA, 2005). However, the flange slenderness values of three LSBs

(200x45x1.6 LSB, 150x45x1.6 LSB and 125x45x1.6 LSB) are closer to the limiting

value of 0.673. Hence only five LSBs marked by bold italic letters in the table

should be considered as slender. The effective width be was calculated using the

procedure mentioned above (be = ρb).

Table D.3: Effective Width of Vertical Flange Element in LSBs

LSB Sections Compression Flange Element (Vertical)

f1 f2 ψ K b λ be be1 be2 be1+be2

300x75x3.0 LSB 450 383.33 0.852 4.303 22.0 0.176 22.0 10.24 11.76 22.0 300x75x2.5 LSB 450 381.93 0.849 4.309 22.5 0.216 22.5 10.46 12.04 22.5 300x60x2.0 LSB 450 395.64 0.879 4.245 18.0 0.218 18.0 8.49 9.51 18.0 250x75x3.0 LSB 450 369.84 0.822 4.368 22.0 0.175 22.0 10.10 11.90 22.0 250x75x2.5 LSB 450 368.18 0.818 4.376 22.5 0.215 22.5 10.31 12.19 22.5 250x60x2.0 LSB 450 384.68 0.855 4.296 18.0 0.217 18.0 8.39 9.61 18.0 200x60x2.5 LSB 450 370.25 0.823 4.366 17.5 0.167 17.5 8.04 9.46 17.5 200x60x2.0 LSB 450 368.18 0.818 4.376 18.0 0.215 18.0 8.25 9.75 18.0 200x45x1.6 LSB 450 389.21 0.865 4.275 13.4 0.202 13.4 6.28 7.12 13.4 150x45x2.0 LSB 450 370.95 0.824 4.362 13.0 0.155 13.0 5.98 7.02 13.0 150x45x1.6 LSB 450 368.73 0.819 4.373 13.4 0.200 13.4 6.15 7.25 13.4 125x45x2.0 LSB 450 354.88 0.789 4.442 13.0 0.154 13.0 5.88 7.12 13.0 125x45x1.6 LSB 450 352.27 0.783 4.455 13.4 0.198 13.4 6.04 7.36 13.4

Appendix D

D-6

As seen in Table D.3, be1+be2 = be for all 13 LSBs. Therefore, the vertical flange

elements are all fully effective.

Table D.4: Effective Width of Web Element in LSBs

LSB Sections Web Element

f1 f2 ψ K b λ be be1 be2 be1+be2 be/2 300x75x3.0 LSB 383.33 -383.33 -1 24 253.0 0.793 230.6 57.64 115.28 172.9 115.3 300x75x2.5 LSB 381.93 -381.93 -1 24 252.5 0.948 204.6 51.14 102.29 153.4 102.3 300x60x2.0 LSB 395.64 -395.64 -1 24 262.0 1.251 172.6 43.15 86.29 129.4 86.3 250x75x3.0 LSB 369.84 -369.84 -1 24 203.0 0.625 203.0 50.75 101.50 152.3 101.5 250x75x2.5 LSB 368.18 -368.18 -1 24 202.5 0.746 191.4 47.84 95.68 143.5 95.7 250x60x2.0 LSB 384.68 -384.68 -1 24 212.0 0.998 165.6 41.39 82.78 124.2 82.8 200x60x2.5 LSB 370.25 -370.25 -1 24 162.5 0.601 162.5 40.63 81.25 121.9 81.3 200x60x2.0 LSB 368.18 -368.18 -1 24 162.0 0.746 153.1 38.27 76.54 114.8 76.5 200x45x1.6 LSB 389.21 -389.21 -1 24 171.6 1.016 132.3 33.08 66.16 99.2 66.2 150x45x2.0 LSB 370.95 -370.95 -1 24 122.0 0.564 122.0 30.50 61.00 91.5 61.0 150x45x1.6 LSB 368.73 -368.73 -1 24 121.6 0.701 119.0 29.76 59.52 89.3 59.5 125x45x2.0 LSB 354.88 -354.88 -1 24 97.0 0.439 97.0 24.25 48.50 72.8 48.5 125x45x1.6 LSB 352.27 -352.27 -1 24 96.6 0.544 96.6 24.15 48.30 72.5 48.3

As seen in Table D.4, be1+be2 > be/2 (compression portion of web) for all 13 LSBs.

Therefore, the web elements are all fully effective.

In summary, when the LSBs are considered without corners (as used in finite

element modelling), their horizontal flange elements are not fully effective for five

LSBs as shown in Table D.2. Therefore the effective section moduli of these five

LSBs are less than their full section moduli. However, it must be noted that the

flange elements of LSBs are fully effective if their corners are included as shown in

the LSB manufacturer’s manuals (OATM, 2008). The procedure to calculate the

effective section moduli of these LSBs is given next.

Appendix D

D-7

Calculation of Elastic Section Moduli of LSBs

The elastic section moduli of all the available 13 LSBs are calculated by using the

basic principles. The effective widths of the horizontal compression flange elements

calculated in Table D.2 are used here. Tables D.5 and D.6 provide the details of the

effective second moment of area calculations for 300x75x3.0 LSB and 300x75x2.5

LSB.

Table D.5: Calculation of Effective Second Moment of Area of 300x75x3.0 LSB

Element No

b (mm)

t (mm)

A (mm2)

y (mm)

Ay (mm3)

(y-y1)2 (mm2)

A(y-y1)2 (mm4)

I (mm2)

1 253.0 3 759 148.50 112711.5 0.00 0.00 4048569.252 22.0 3 66 11.00 726.0 18906.25 1247812.50 2662.00 3 22.0 3 66 286.00 18876.0 18906.25 1247812.50 2662.00 4 22.0 3 66 11.00 726.0 18906.25 1247812.50 2662.00 5 22.0 3 66 286.00 18876.0 18906.25 1247812.50 2662.00 6 72.0 3 216 22.00 4752.0 16002.25 3456486.00 162.00 7 72.0 3 216 0.00 0.0 22052.25 4763286.00 162.00 8 72.0 3 216 275.00 59400.0 16002.25 3456486.00 162.00 9 72.0 3 216 297.00 64152.0 22052.25 4763286.00 162.00 Total 1887 280219.5 21430794.00 4059865.25

Centroid, y1 = ΣAy / ΣA = 280219.5 / 1887 = 148.50 mm

bf

df

d d1

9

5

1

4

7

6

8

3

2 y = 0

Appendix D

D-8

Effective second moment of area about the major axis, Ie = 21.43 * 106 + 4.060 * 106

= 25.49 * 106 mm4

Effective elastic section modulus, Ze = Ie / y = 25.49 * 106 / (297-148.50)

= 171.65 *103 mm3

This is very close to the value obtained from Thin-wall (171.7*103 mm3). This

confirms the accuracy of calculations used here.

Here, y is the distance from the neutral axis to the extreme fibre of compression

flange (top flange).

For 300x75x3.0 LSB, My = Ms = Zfy = Zefy = 171.65 * 103 * 450 * 10-6 = 77.24 kNm

Since 300x75x2.5 LSB section is not fully effective, Ze is not equal to Z.

Table D.6: Calculation of Second Moment of Area of 300x75x2.5 LSB

Element No

b (mm)

t (mm)

A (mm2)

y (mm)

Ay (mm3)

(y-y1)2

(mm2) A(y-y1)2

(mm4) I

(mm2) 1 252.50 2.5 631 148.75 93898.44 1.47 929.65 3353844.4 2 22.50 2.5 56 11.25 632.81 18574.00 1044787.24 2373.05 3 22.50 2.5 56 286.25 16101.56 19241.45 1082331.56 2373.05 4 22.50 2.5 56 11.25 632.81 18574.00 1044787.24 2373.05 5 22.50 2.5 56 286.25 16101.56 19241.45 1082331.56 2373.05 6 72.50 2.5 181 22.50 4078.13 15634.11 2833682.96 94.40 7 72.50 2.5 181 0.00 0.00 21767.00 3945269.29 94.40 8 69.73 2.5 174 275.00 47941.61 16246.96 2832382.69 90.80 9 69.73 2.5 174 297.50 51864.10 22489.07 3920589.14 90.80 Total 1567.4 231251.0 17787091.34 3363706.99

Centroid, y1 = 147.54 mm

Effective second moment of area about major axis, Ie = 21.151 * 106 mm4

Effective elastic section modulus, Ze = 21.151 * 106 / (297.50-147.54)

= 141.04 *103 mm3

Full section modulus Z can be calculated by replacing the compression flange

elements (elements 8 and 9) by the actual widths (equal to 72.5 mm) or by using

Thin-Wall. Z for 300x75x2.5 was calculated to be 144.0 *103 mm3.

For 300x75x2.5 LSB,

Ms = Zefy = 141.04 * 103 * 450 * 10-6 = 63.47 kNm

My = Zfy = 144.00 * 103 * 450 * 10-6 = 64.80 kNm

Appendix D

D-9

Based on this, Ze = 0.98Z

This means, there is only 2% reduction due to the ineffective plate element widths.

Table D.7 provides the comparison of Ze and Z for all 13 LSBs.

Table D.7: Comparison of Z and Ze for LSBs

LSB Sections Z (103 mm3)

Ze (103 mm3)

% Reduction (1-(Ze/Z))*100

Ms (kNm)

My (kNm)


Effective Widths and Section Moduli of the New Non-standard LSBs

Four non-standard LSBs were considered in the parametric study of this research.

They are

1. 300x45x3.6 LSB – Ze = Z = 145.79 x103 mm3

2. 135x50x1.6 LSB – Ze = 22.92 x103 mm3 and Z = 23.78 x103 mm3

3. 125x45x1.8 LSB – Ze = Z = 22.07 x103 mm3

4. 127x45x2.4 LSB – Ze = Z = 29.10 x103 mm3

The effective widths of these LSBs were calculated by using the same AS/NZS 4600

(SA, 2005) procedure used with the 13 standard LSBs and it was found that the

effective widths are equal to the actual widths except in the case of 135x50x1.6

LSBs. The flange flat element of this LSB has a slenderness value of 0.755 (>0.673)

and hence its effective section modulus will be less than the full section modulus.

Appendix D

D-10

D.2: Effects of Residual Stresses on the Ultimate Moment Capacities of LSBs

Subject to Lateral Distortional Buckling

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

22.0

24.0

26.0

28.0

30.0

32.0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

Span (mm)

Ulti

mat

e M

omen

t Cap

aciti

es (k

Nm

)





Figure D.1: Effects of Residual Stresses for 300x60x2.0 LSB

8.0

10.0

12.0

14.0

16.0

18.0

20.0

22.0

24.0

26.0

28.0

30.0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

Span (mm)

Ulti

mat

e M

omen

t Cap

aciti

es (k

Nm

)






Appendix D

D-11

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

1000 2000 3000 4000 5000 6000 7000 8000 9000

Span (mm)

Ulti

mat

e M

omen

t Cap

aciti

es (k

Nm

)






3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Span (mm)

Ulti

mat

e M

omen

t Cap

aciti

es (k

Nm

)






Appendix D

D-12

D.3: Elastic Lateral Torsional Buckling Moments of LSBs

Elastic lateral torsional buckling moments of LSBs were calculated using Eq.6.12.

Table D.8: Elastic Lateral Torsional Buckling Moments of LSBs

LSB Sections Span (mm) Mo

(kNm) LSB

Sections Span (mm)

Mo (kNm)

300x75x3.0 LSB

1500 219.9

300x75x2.5 LSB

1750 153.1 2000 150.3 2000 129.0 3000 92.62 3000 79.63 4000 67.35 4000 57.96 6000 43.87 6000 37.77 8000 32.63 8000 28.10 10000 26.00 10000 22.39

300x60x2.0 LSB

1750 65.70

250x75x3.0 LSB

1250 252.9 2000 55.25 1500 198.4 3000 33.95 2000 138.9 4000 24.66 3000 87.64 6000 16.05 4000 64.37 8000 11.93 6000 42.25 10000 9.51 8000 31.52

10000 25.15

250x75x2.5 LSB

1500 170.28

250x60x2.0 LSB

1500 73.19 2000 119.43 2000 51.13 3000 75.44 3000 32.18 4000 55.44 4000 23.61 6000 36.40 6000 15.49 8000 27.16 8000 11.55 10000 21.67 10000 9.21

200x60x2.5 LSB

1000 133.89

200x60x2.0 LSB

1250 82.75 1250 99.63 1500 66.05 1500 79.43 2000 47.27 2000 56.77 3000 30.39 3000 36.45 4000 22.49 4000 26.96 6000 14.85 6000 17.79 8000 11.10 8000 13.30 10000 8.86 10000 10.62

Appendix D

D-13

Table D.8 (continued): Elastic Lateral Torsional Buckling Moments of LSBs

LSB Sections Span (mm) Mo

(kNm) LSB

Sections Span (mm)

Mo (kNm)

200x45x1.6 LSB

1250 29.40

150x45x2.0 LSB

750 59.52 1500 23.39 1000 40.79 2000 16.67 1250 31.10 3000 10.68 1500 25.20 4000 7.89 2000 18.35 6000 5.20 3000 11.96 8000 3.89 4000 8.90 10000 3.10 6000 5.90

8000 4.41 10000 3.53

150x45x1.6 LSB

750 49.50

125x45x2.0 LSB

750 53.69 1000 34.00 1000 37.70 1250 25.96 1250 29.16 1500 21.05 1500 23.83 2000 15.34 2000 17.52 3000 10.01 3000 11.51 4000 7.45 4000 8.59 6000 4.94 6000 5.70 8000 3.70 8000 4.27 10000 2.95 10000 3.41

125x45x1.6 LSB

750 44.75 1000 31.48 1250 24.37 1500 19.93 2000 14.67 3000 9.64 4000 7.19 6000 4.78 8000 3.58 10000 2.86

Appendix D

D-14

D.4: Calculation of Geometrical Parameter K for LSB Sections

The geometrical parameter K for any beam is defined as follows;

xweb

f

EIGJ

K+

=85.0

1

where, GJf = torsional rigidity of the flange


Consider a 300x75x3.0 LSB cross-section without corners Calculation of torsional rigidity of flange, GJf

From the basic principles, Torsional Constant, Jf = ⎟⎠⎞

⎜⎝⎛

tSAm

24

where,

Am = area enclosed by the median line of the cross section.

S = perimeter along the median line of the section.

Jf = ttdtb

tdtb/)}(){(2

)}(*){(*4 2

−+−−−

Jf = 3/)}325()375{(2

)}325(*)375{(*4 2

−+−−−

Jf = 160.15 * 103 mm4 Shear modulus of elasticity, G = 80 000 MPa

d

b

t

b = 75 mm d = 25 mm t = 3.0 mm

Appendix D

D-15

Torsional rigidity of flange, GJf = 80 000 * 160.15 * 103

= 12812 * 106 Nmm2

Calculation of major axis flexural rigidity of web, EIxweb

From basic principles, Ixweb = 12

31td

= 12250*3 3

= 3906.25 * 103 mm4

Elastic modulus, E = 200 000 MPa

Flexural rigidity of web, EIxweb = 200 000 * 3906.25 * 103

= 781 250 * 106 Nmm2

Hence xweb

f

EIGJ = 6

6

10*78125010*12812

= 0.0164

Therefore,

xweb

f

EIGJ85.0

1K+

=

0164.085.0

1+

=K

K = 1.0224

d1 = 250 mm

t

Appendix D

D-16

D.5: Calculation of Geometrical Parameter K for Monosymmetric Hollow

Flange Beams (MHFBs) with Triangular Flanges

Consider a 26585MHFB36 cross-section without corners Calculation of torsional rigidity of flange, GJf

From the basic principles, Torsional Constant, Jf = ⎟⎠⎞

⎜⎝⎛

tSAm

24

where,

Am = area enclosed by the median line of the cross section.

S = perimeter along the median line of the section.

Jf = [ ] ttatdtb

tdtb

/)()()(

)(*)(21*4

2

−+−+−

⎥⎦⎤

⎢⎣⎡ −−

Jf = [ ] 6.3/)6.360.88()6.325()6.385(

)6.325(*)6.385(21*4

2

−+−+−

⎥⎦⎤

⎢⎣⎡ −−

Jf = 58.168 * 103 mm4

Shear modulus of elasticity, G = 80 000 MPa Torsional rigidity of flange, GJf = 80 000 * 58.168 * 103

= 4653.4 * 106 Nmm2

b = 85 mm d = 25 mm t = 3.6 mm a = 88.60 mm

d

b

t

a

Appendix D

D-17

Calculation of major axis flexural rigidity of web, EIxweb The calculation method for the major axis flexural rigidity is the same for any hollow

flange sections as they all have a rectangular web element.

Therefore, depth of the web, d1 = 215 mm, t = 3.6 mm

Ixweb = 12

31td

= 12

215*6.3 3

= 2981.5125 * 103 mm4

Elastic modulus, E = 200 000 MPa

Flexural rigidity of web, EIxweb = 200 000 * 2981.5125* 103

= 596 302.5 * 106 Nmm2

Hence, xweb

f

EIGJ

= 6

6

10*5.59630210*4.4653

= 0.0078

Therefore,

xweb

f

EIGJ85.0

1K+

=

0078.085.0

1+

=K

K = 1.0657

Note: Calculation of GJf and EIxweb for RHFB is similar to that of LSBs as the flange

and web dimensions are the same.

Appendix E

E-1

APPENDIX E: E.1: Section Compactness of LSBs with Corners Based on AS 4100 (SA, 1998)

Top Flange Element Web Element Overall Compact-

-ness LSB Sections d (mm)

d1 (mm)

bf (mm)

df (mm)

t (mm)

ro (mm)

riw (mm)

fy (MPa) b λef Compact-

-ness b λew Compact--ness Flange Web

300x75x3.0 LSB 300 244 75 25 3.0 6.0 3.0 450 380 63.0 28.17 C 244 100.3 NC NC 300x75x2.5 LSB 300 244 75 25 2.5 5.0 3.0 450 380 67.0 35.96 NC 244 120.3 S S 300x60x2.0 LSB 300 254 60 20 2.0 4.0 3.0 450 380 53.0 35.55 NC 254 156.6 S S 250x75x3.0 LSB 250 194 75 25 3.0 6.0 3.0 450 380 66.0 29.52 C 194 79.73 C C 250x75x2.5 LSB 250 194 75 25 2.5 5.0 3.0 450 380 67.0 35.96 NC 194 95.67 NC NC 250x60x2.0 LSB 250 204 60 20 2.0 4.0 3.0 450 380 53.0 35.55 NC 204 125.8 S S 200x60x2.5 LSB 200 154 60 20 2.5 5.0 3.0 450 380 52.0 27.91 C 154 75.95 C C 200x60x2.0 LSB 200 154 60 20 2.0 4.0 3.0 450 380 53.0 35.55 NC 154 94.93 NC NC 200x45x1.6 LSB 200 164 45 15 1.6 3.2 3.0 450 380 38.8 32.53 NC 164 126.4 S S 150x45x2.0 LSB 150 114 45 15 2.0 4.0 3.0 450 380 38.0 25.49 C 114 70.27 C C 150x45x1.6 LSB 150 114 45 15 1.6 3.2 3.0 450 380 38.8 32.53 NC 114 87.84 NC NC 125x45x2.0 LSB 125 89 45 15 2.0 4.0 3.0 450 380 38.0 25.49 C 89 54.86 C C 125x45x1.6 LSB 125 89 45 15 1.6 3.2 3.0 450 380 38.8 32.53 NC 89 68.58 C NC

Note:

• S – Slender, NC- Non Compact, C- Compact.

• Flanges are assumed to be lightly welded cold-formed elements. 250

yftb⎟⎠⎞

⎜⎝⎛=λ

• For Flange - λep = 30, λey = 40, for web - λep = 82, λey = 115

Appendix E

E-2

E.2: Moment versus Vertical Deflection Curves of Section Capacity Tests

0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40

Deflection (mm)

Mom

ent (

kNm

) Load-pointMid-span

Figure E.1: Moment vs Vertical Deflection Curves of 300x60x2.0 LSB

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

0 5 10 15 20 25 30 35 40

Deflection (mm)

Mom

ent (

kNm

)

Load-point

Mid-span


Appendix E

E-3

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25 30 35

Deflection (mm)

Mom

ent (

kNm

)

Mid-span


0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25 30

Deflection (mm)

Mom

ent (

kNm

)

Load-pointMid-span


Appendix E

E-4

E.3: Comparison of Bending Moment versus Vertical Deflection Curves from

Experiments and FEA

0

10

20

30

40

50

60

70

80

90

100

0 5 10 15 20 25 30 35 40 45

Deflection (mm)

Mom

ent (

kNm

)



0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40

Deflection (mm)

Mom

ent (

kNm

)



Appendix E

E-5

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

0 5 10 15 20 25 30 35 40

Deflection (mm)

Mom

ent (

kNm

)



0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25 30 35

Deflection (mm)

Mom

ent (

kNm

) Mid-span:ExpLoad-point:FEAMid-span:FEA


Appendix E

E-6

0

2

4

6

8

10

12

14

16

18

20

22

0 5 10 15 20 25 30 35 40

Deflection (mm)

Mom

ent (

kNm

)



Appendix E

E-7

E.4: Calculation of True Stress-Strain Curves for 150x45x2.0 LSBs Figures E.10 (a) to (c) show the measured stress-strain curves for 150x45x2.0 LSBs

from tensile coupon tests.

Figure E.10: Measured Stress-Strain Curves of 150x45x2.0 LSB

(a) Outer Flange

0.0

100.0

200.0

300.0

400.0

500.0

600.0

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50

% Strain

Stre

ss (M

Pa)

Strain GaugeExtenso MeterTangent0.2% proof stress

(b) Inner Flange

0.0

100.0

200.0

300.0

400.0

500.0

600.0

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00

% Strain

Stre

ss (M

Pa)

Strain GaugeExtenso MeterTangent0.2% Proof Stress

Appendix E

E-8

Figure E.10: Measured Stress-Strain Curves of 150x45x2.0 LSB Based on the above figures, some important stress and strain values (engineering) are

considered in plotting the true stress-strain curve. Table E.1 shows these values

Table E.1: Measured (Engineering) Stress and Strain Values of 150x45x2.0 LSB

Outer Flange Inner Flange Web Stress Strain Stress Strain Stress Strain

0.0 0.00000 0.0 0.00000 0 0.00000 539.3 0.00423 500.5 0.00443 402 0.00255 568.0 0.00649 514.0 0.00552 432 0.00360 581.0 0.00935 526.0 0.00802 441 0.00480 586.0 0.01406 533.0 0.01241 443 0.00472 587.0 0.04250 536.0 0.02165 446 0.01360

536.0 0.09000 452 0.01651 468 0.02522 488 0.03982 508 0.06091 515 0.08138

These values are converted to true stress, strain values by using following equations.

σtrue = σeng (1 + εeng)

εtrue = ln (1 + εeng)

0.0

100.0

200.0

300.0

400.0

500.0

600.0

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

% Strain

Stre

ss (M

Pa)

Strain GaugeExtenso MeterTangent0.2% Proof Stress

(c) Web

Appendix E

E-9

Table E.2 shows the true stress and strain values which were used in the finite

element analyses.

Table E.2: True Stress, Strain Values of 150x45x2.0 LSB

Outer Flange Inner Flange Web

Stress Strain Stress Strain Stress Strain 0.0 0.00000 0.0 0.00000 0.0 0.00000

541.6 0.00422 502.7 0.00442 403.0 0.00254 571.7 0.00647 516.8 0.00550 433.6 0.00360 586.4 0.00931 530.2 0.00799 443.1 0.00479 594.2 0.01396 539.6 0.01234 445.1 0.00471 611.9 0.04162 547.6 0.02142 452.1 0.01350 584.2 0.08618 459.5 0.01637

479.8 0.02490 507.4 0.03904 538.9 0.05913 556.9 0.07824

The following yield stresses are calculated based on the measured stress-strain

curves.

Outer flange = 539.3 MPa

Inner flange = 500.5 MPa

Web = 443.0 MPa

Elastic section modulus of 150x45x2.0 LSB was found to be 31.89 x 103 mm3. This

was calculated with nominal dimensions without corners.

Therefore, the first yield moment, My = fy x Z = 539.3 x 31.89 x 103 x 10-6

= 17.20 kNm

Since this LSB is compact, section moment capacity Ms = My = 17.20 kNm

Appendix E

E-10

E.5: Section Compactness and Section Moment Capacity of LSBs

Based on AS 4100 (SA, 1998) Appendix E.1 provided the plate slenderness and the section compactness of LSBs

with corners. This section provides the section compactness and the section moment

capacities of LSBs without corners. Centreline dimensions are used in all the

calculations.

Consider 300x75x3.0 LSB

Plate Slenderness, 250

ye

ftb⎟⎠⎞

⎜⎝⎛=λ

For Flange

250y

eff

tb⎟⎠⎞

⎜⎝⎛=λ

250450

372

⎟⎠⎞

⎜⎝⎛=efλ

= 32.20

Assuming lightly welded cold-formed steel

λep = 30, λey = 40

λep < λef < λey

flange is non-compact

bf

df

d d1

t = 3 mm

d = 300 mm

d1 = 250 mm

df = 25 mm

bf = 75 mm

fyf = 450 MPa

fyw = 380 MPa

Appendix E

E-11

for Web

250y

ewf

tb⎟⎠⎞

⎜⎝⎛=λ

250380

3253

⎟⎠⎞

⎜⎝⎛=ewλ

= 103.97

Assuming lightly welded cold-formed steel

λep = 82, λey = 115

λep < λef < λey

web is non-compact

Overall section is non-compact.

Full elastic section modulus Z = 171.65 * 103 mm3 (Thin-wall or Appendix D.1)

Plastic section modulus (S)

S = ((d1 + t)2/4) * t + 4 * t * (df - t)(d-df)/2 + t (bf –t)(d - t) + t (bf – t) (d1 + t)

= (2532/4) * 3 + 4 * 3 * (25-3)(300-25)/2 + 3(75-3)(300-3) + 3(75-3)(250 + 3)

= 203.11 * 103 mm3

Zc = min (1.5Z, S)

Zc = min (1.5 * 171.65 * 103, 203.11 * 103)

Zc = 203.11 * 103 mm3

Since the LSB is non-compact,

( )⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛

−−

+= ZZZZ cepey

seye

λλλλ

( ) 33 10*65.17111.20382115

97.10311510*65.171 ⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛

−−

+=eZ

Ze = 182.16 * 103 mm3

therefore, Ms = fyZe = 450 * 182.16 * 103 * 10-6

Ms = 81.97 kNm.

Similarly, the effective section modulus and the section moment capacities of other

LSBs were calculated based on the above procedure and the results are presented in

Table E.3.

Appendix E

E-12

Table E.3: Nominal Dimensions and Yield Stress of LSBs

LSB Sections d (mm)

d1 (mm)

bf (mm)

df (mm)

t (mm)

fy (MPa) Flange Web

300x75x3.0 LSB 300 250 75 25 3.0 450 380 300x75x2.5 LSB 300 250 75 25 2.5 450 380 300x60x2.0 LSB 300 260 60 20 2.0 450 380 250x75x3.0 LSB 250 200 75 25 3.0 450 380 250x75x2.5 LSB 250 200 75 25 2.5 450 380 250x60x2.0 LSB 250 210 60 20 2.0 450 380 200x60x2.5 LSB 200 160 60 20 2.5 450 380 200x60x2.0 LSB 200 160 60 20 2.0 450 380 200x45x1.6 LSB 200 170 45 15 1.6 450 380 150x45x2.0 LSB 150 120 45 15 2.0 450 380 150x45x1.6 LSB 150 120 45 15 1.6 450 380 125x45x2.0 LSB 125 95 45 15 2.0 450 380 125x45x1.6 LSB 125 95 45 15 1.6 450 380

Table E.4: Slenderness and Compactness of LSBs

LSB Sections

Horizontal Flange Element Web

b λef Compact-

-ness b λew Compact--ness

300x75x3.0 LSB 72.0 32.20 NC 253.0 104.0 NC 300x75x2.5 LSB 72.5 38.91 NC 252.5 124.5 S 300x60x2.0 LSB 58.0 38.91 NC 262.0 161.5 S 250x75x3.0 LSB 72.0 32.20 NC 203.0 83.43 NC 250x75x2.5 LSB 72.5 38.91 NC 202.5 99.86 NC 250x60x2.0 LSB 58.0 38.91 NC 212.0 130.7 S 200x60x2.5 LSB 57.5 30.86 NC 162.5 80.14 C 200x60x2.0 LSB 58.0 38.91 NC 162.0 99.86 NC 200x45x1.6 LSB 43.4 36.39 NC 171.6 132.2 S 150x45x2.0 LSB 43.0 28.85 C 122.0 75.21 C 150x45x1.6 LSB 43.4 36.39 NC 121.6 93.70 NC 125x45x2.0 LSB 43.0 28.85 C 97.0 59.79 C 125x45x1.6 LSB 43.4 36.39 NC 96.6 74.44 C

When calculating the effective section modulus of LSBs with slender web and non-

compact flange, the section modulus of web element is factored by the ratio of (λey/

λew)2.

When calculating the effective section modulus of LSBs with non-compact web and

flanges, the effective section modulus values were calculated by using both λew and

λef values separately and the lower section modulus was selected. Table E.5 presents

Appendix E

E-13

the details of full section modulus Z, plastic section modulus S, critical section

modulus Zc and the effective section modulus Ze. The section moment capacity Ms is

also given in this table.

Table E.5: Section Moduli and Section Moment Capacities of LSBs

LSB Sections Overall

Compact--ness

Z (103 mm3)

S (103 mm3)

S/Z Zc

(103 mm3) Ze

(103 mm3) Ms

(kNm)

300x75x3.0 LSB NC 171.65 203.11 1.18 203.11 182.16 81.97 300x75x2.5 LSB S 143.98 170.47 1.18 170.47 140.67 63.30 300x60x2.0 LSB S 100.38 119.44 1.19 119.44 90.46 40.71 250x75x3.0 LSB NC 133.47 157.81 1.18 157.81 152.45 68.60 250x75x2.5 LSB NC 111.96 132.50 1.18 132.50 114.20 51.39 250x60x2.0 LSB S 78.00 92.39 1.18 92.39 75.10 33.80 200x60x2.5 LSB NC 71.07 84.00 1.18 84.00 82.89 37.30 200x60x2.0 LSB NC 57.32 67.84 1.18 67.84 58.47 26.31 200x45x1.6 LSB S 38.29 45.40 1.19 45.40 36.63 16.48 150x45x2.0 LSB C 31.89 37.68 1.18 37.68 37.68 16.96 150x45x1.6 LSB NC 25.74 30.45 1.18 30.45 27.44 12.35 125x45x2.0 LSB C 24.77 29.34 1.18 29.34 29.34 13.21 125x45x1.6 LSB NC 19.99 23.73 1.19 23.73 21.34 9.60

Table E.6: Section Moduli and Shape Factors of Non-Standard LSBs

LSB Sections Z (103 mm3)

S (103 mm3) S/Z

300x75x3.9 LSB 220.53 260.62 1.18 250x75x3.3 LSB 146.23 172.80 1.18 200x45x3.0 LSB 69.64 82.39 1.18 150x45x3.0 LSB 46.77 55.09 1.18

It can be noted that the shape factor (S/Z) for all the 13 LSBs and non-standard LSBs

are about 1.18.

Appendix E

E-14

E.6: Section Moment Capacity of LSBs Based on

Eurocode 3 Part 1.3 (ECS, 1996 & 2006)

The procedure to calculate the slenderness of each plate element of 300x75x3.0 LSB

is presented here. It should be noted that centreline dimensions are used.

Consider 300x75x3.0 LSB

Eurocode 3 Part 1.3 (ECS, 1996) was used

For Horizontal flange element

1=ψ 4=σk Uniformly compressed stiffened element (Table 4.1)

p

_

λ ( ) 5.0

2

2112

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −⎟⎟⎠

⎞⎜⎜⎝

⎛=

σπνEk

ft

b ybp ( ) 673.0599.04200000

4503.01120.3

725.0

2

2

<=⎥⎦

⎤⎢⎣

⎡××

×−⎟⎠⎞

⎜⎝⎛=

π


f2 / (148.5-22) = 450/148.5

f2 = 383.33 MPa

f1 = 450 MPa

*1

*2

f

f=ψ Table 4.1

450

33.383=ψ = 0.852

bf

df

d d1

t = 3 mm

d = 300 mm

d1 = 250 mm

df = 25 mm

bf = 75 mm

fyf = 450 MPa

fyw = 380 MPa

f2

450

22

148.5

Appendix E

E-15

ψ+

=05.1

2.8K Table 4.1

K = 4.31

p

_

λ ( ) 5.0

2

2112

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −⎟⎟⎠

⎞⎜⎜⎝

⎛=

σπνEk

ft

b ybp ( ) 673.0599.04200000

4503.01120.3

225.0

2

2

<=⎥⎦

⎤⎢⎣

⎡××

×−⎟⎠⎞

⎜⎝⎛=

π

673.0176.0 <=λ

For Web element

σ1 = σ2 = 383.33 MPa

11

2 −==σσ

ψ 9.23=σk (Table 4.1)

p

_

λ ( ) 5.0

2

2112

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −⎟⎟⎠

⎞⎜⎜⎝

⎛=

σπνEk

ft

b ybp ( ) 673.0794.09.23200000

33.3833.01120.3

2535.0

2

2

>=⎥⎦

⎤⎢⎣

⎡××

×−⎟⎠⎞

⎜⎝⎛=

π

It should be noted that these slenderness values are the same as obtained based on

AS/NZS 4600 (SA, 2005) procedures as shown in Appendix D.

Now the section moment capacity of LSBs can be calculated based on Cl 6.1.4.1 of

Eurocode 3 Part 1.3 (ECS, 2006).

If the effective section modulus Weff is less than the gross elastic section modulus

Wel

M c.Rd = Weff fyb / γMO (E.1)

If the effective section modulus Weff is equal to the gross elastic section modulus Wel

(E.2)

λeo = 0.65 (stiffened element)

( ) MOeo

eelplelybRdc WWWfM γ

λλ /1*4* max

. ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−+=

Appendix E

E-16

Table E.7 presents the slenderness values of compression horizontal flange element

of all the available 13 LSBs and 4 non-standard LSBs and their section moment

capacities. It should be noted that the plate elements are considered to be slender if λp

> 0.673.

Table E.7: Slenderness of Compression Horizontal Flange Element and the

Section Moment Capacities of LSBs

LSB Sections λp Compactness Ms (kNm) 300x75x3.0 LSB 0.599 Compact 81.71 300x75x2.5 LSB 0.723 Slender 61.50 300x60x2.0 LSB 0.723 Slender 38.74 250x75x3.0 LSB 0.599 Compact 63.52 250x75x2.5 LSB 0.723 Slender 49.30 250x60x2.0 LSB 0.723 Slender 32.68 200x60x2.5 LSB 0.574 Compact 34.71 200x60x2.0 LSB 0.723 Slender 25.24 200x45x1.6 LSB 0.677 Slender 16.20 150x45x2.0 LSB 0.536 Compact 16.17 150x45x1.6 LSB 0.677 Slender 11.57 125x45x2.0 LSB 0.536 Compact 12.58 125x45x1.6 LSB 0.677 Slender 8.98

Non-Standard LSBs 300x75x3.9 LSB 0.455 Compact 117.28 250x75x3.3 LSB 0.542 Compact 73.74 200x45x3.0 LSB 0.349 Compact 37.08 150x45 3.0 LSB 0.349 Compact 24.79

The section moment capacities of these LSBs were calculated based on these

slenderness values and Eq.E.1 (slender sections) and E.2 (compact sections).

It should be noted that for 300x75x3.3 LSB, 200x45x3.0 LSB and 150x45x3.0 LSB

the section moment capacity Ms is equal to their plastic moments.

For example,

300x75x3.3 LSB, S = 117.28 * 1000 / 450 = 260.62 * 103 mm3 (see Table E.6).

Appendix E

E-17

E.7: Predicting the Plastic Moment Capacity of Compact Steel Beams Using

Finite Element Analyses

It was found that finite element analyses did not predict the full plastic moment

capacity of compact LSBs although experimental results showed that some compact

LSBs reached their full plastic moment capacities. Therefore an attempt was made to

investigate this by using finite element analyses (FEA). Table E.8 shows the details

of this investigation and the results.

Table E.8: Ultimate Moments of LSBs from FEA

LSB Sections

My (kNm)

Mp (kNm) Mp/My

fyf = 450 MPa, fyw = 380 MPa fyf = fyw= 450 MPa

Mu (kNm) Mu/My

Mu (kNm) Mu/My

150x45x2.0 LSB 14.35 16.96 1.18 15.8 1.10 16.2 1.13

150x45x3.0 LSB 21.05 24.79 1.18 23.8 1.13 24.5 1.16

150x45x4.0 LSB 27.42 32.19 1.17 30.8 1.12 31.8 1.16

150x45x5.0 LSB 33.48 39.16 1.17 37.4 1.12 38.7 1.16

Table E.8 shows the first yield moment (My) and the plastic moment (Mp) of four

compact LSB sections as well as their ultimate moments (Mu) from FEA. It was

found that using the same yield stress for web and flange elements of LSBs in FEA

gave improved ultimate moment capacities. It can be seen that Mu/My ratio increased

from 1.10 to 1.13 when the thickness was increased from 2 mm to 3 mm while the

ratio was similar (1.12) for the LSBs with 3 to 5 mm thicknesses when the nominal

flange (450 MPa) and web (380 MPa) yield stresses were used. However, when the

flange and web yield stresses were taken to be the same at 450 MPa, higher Mu/My

ratios of 1.16 were obtained, i e. very close to their Mp/My ratios of 1.17. However,

the Mu/My ratio was 1.13 for 150x45x2.0 LSB section in comparison to its Mp/My

ratio of 1.18. These results in Table E.7 indicate that FEA are not able to predict the

full plastic moment capacities when a smaller yield stress was used for the web

element. They also show that FEA are only able to predict the full plastic moment

capacities of thicker and very compact sections.

Appendix E

E-18

An attempt was then made to obtain the section moment capacities of conventional

hot-rolled I- and C-sections. For this purpose, 150UB14.0 and 150PFC17.7 sections

were modelled without corners using the same flange and web yield stress of 320

MPa as shown in their design capacity tables (AISC, 1994). Table E.9 shows the

details of these sections and the results.

Table E.9: Ultimate Moments of 150UB14.0 and 150PFC17.7 from FEA

Hot-Rolled Sections My (kNm) Mp (kNm) Mp/My Mu (kNm) Mu/My

150UB14.0 27.39 32.2 1.18 32.5 1.19 150PFC17.7 34.85 41.51 1.19 39.7 1.14

As seen in Table E.9, FEA was able to predict the full plastic moment capacity of

compact hot-rolled I-sections (150UB14.0). However, the ratio of Mu/My was only

1.14 for the mono-symmetric hot- rolled PFC section in comparison to its Mp/My

ratio of 1.19. This contradicts the moment capacities given in AISC (1994), which

recommends the full plastic moment capacity for 150PFC17.7.

Based on the results in Tables E.8 and E.9, it can be concluded that conventional

finite element analyses may not able to predict the full plastic moment capacities of

compact mono-symmetric steel sections unless they are made of very thick plate

elements (with small b/t ratios). Further experiments are needed to confirm these

observations.

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R-1

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Flexural Behaviour and Design of Hollow Flange Steel Beams

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