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RC BEAM-COLUMN JOINTS SEISMICALLY

RETROFITTED WITH SELECTIVE BEAM

WEAKENING AND LOCAL FRP STRENGTHENING

QINGKAI WANG

PhD

The Hong Kong Polytechnic University

2019

The Hong Kong Polytechnic University

Department of Civil and Environmental Engineering

RC BEAM-COLUMN JOINTS SEISMICALLY

RETROFITTED WITH SELECTIVE BEAM

WEAKENING AND LOCAL FRP STRENGTHENING

QINGKAI WANG

A Thesis Submitted in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

September 2018

I

CERTIFICATE OF ORIGINALITY

I hereby declare that this thesis is my own work and that, to the best of my

knowledge and belief, it reproduces no material previously published or written,

nor material that has been accepted for the award of any other degree or diploma,

except where due acknowledgement has been made in the text.

(Signed)

Qingkai WANG (Name of student)

II

III

ABSTRACT

The capacity design method has been widely accepted in the design of

reinforced concrete (RC) frames to ensure that they have sufficient plastic

deformation capacities when subjected to a seismic attack. The key of this

method is to make columns stronger than the connected beams at a joint, thus

realizing the so-called Strong-Column-Weak-Beam (SCWB) hierarchy.

However, many existing RC frames, especially those designed according to

outdated design codes; do not meet the SCWB principle. Inadequate

consideration of the contribution of a cast-in-place slab to the hogging moment

capacity of a beam is a common cause. Conventional seismic retrofit methods

have limited effects in enhancing the seismic safety of these RC frames. Against

the above background, this thesis presents a systematic study into a new seismic

retrofit method that involves beam weakening and FRP (fibre reinforced

polymer) strengthening (the BWFS method). This method was proposed by the

author’s research group.

In the experimental part of the study, three different beam weakening

techniques were studied by testing 9 full-scale RC beam-column joints (under a

combination of constant axial loading and cyclic lateral loading), including: (a)

the slab slit (SS) technique, in which a transverse slit is cut in the slab at each

beam end; (b) the beam web opening (BO) technique, in which an opening is

cut in the beam web; and (c) the beam section reduction (SR) technique, in

which a deep transverse groove is cut on the soffit of the beam near the joint.

IV

The test results show that (a) the SR technique reduces both the strength and

ductility of the specimen; (b) the BO technique leads to a ductile failure mode if

the opening size is sufficiently large; and (c) the SS technique has a small

negative effect on specimen ductility although it can effectively reduce the beam

strength. A combination of the SS and the BO techniques is shown to be an

efficient retrofit method in reducing the strength of a T-beam and enhancing the

ductility of the joint.

In the finite element (FE) analysis part, accurate three-dimensional (3D)

FE models for predicting the behaviour of T-beams with an opening and FRP

strengthening were developed. 3D FE models using either solid or shell

elements were both developed. The static analysis problem was regarded as a

dynamic problem and solved using the explicit centre-difference-method

(CDM). A few significant issues, such as the loading time, the damping scheme,

the computational time and the accuracy associated with the explicit dynamic

method, are discussed in depth. The 3D FE model built using shell elements was

then applied to study the issue of effective slab width of T-beams in a hogging

moment zone. Based on the results of parametric studies, new models for the

effective slab widths of T-beams of both interior and exterior beam-column

joints were proposed. Finally, 3D FE models for retrofitted RC joints were

developed and substantiated with test results; the verified FE model can be used

in further investigations of such seismically retrofitted RC beam-column joints.

V

ACKNOWLEDGEMENTS

My PhD research could not have been successfully completed with only

my own effort. I am so lucky and grateful to have received a great deal of help

and support from many people. In this regard, I would like to express my

sincere thanks to them.

First of all, I wish to express my sincere gratitude to my supervisor,

Professor Jin-Guang Teng, a distinguished scholar, for accepting me as one of

his PhD students six year ago; for his patience, motivation and wisdom. Prof.

Teng’s rigorous attitude towards academic research and creative and unique

insight into many academic problems have demonstrated the essential attributes

that a good researcher should possess. His enlightening guidance helped me

successfully finish my PhD research. The method and attitude for doing

research, which I have learned from him, will definitely benefit me in my future

career and life.

My sincere thanks also go to my two co-supervisors, Prof. Guang-ming

Chen of South China University of Technology (SCUT) and Prof. Shi-shun

Zhang of Huazhong University of Science and Technology (HUST). Prof.

Zhang provided me with much constructive advice on my experimental work.

The FE model for FRP-strengthened RC beams previously developed by Prof.

Chen during his PhD study (also supervised by Prof. J.G. Teng) served as the

foundation of my numerical studies presented in this thesis. Prof. Chen also

gave me a great deal of help with the revision of this PhD thesis, in addition to

VI

his enlightening guidance on my numerical research work. Their selfless

assistance helped me overcome many tough difficulties encountered in my PhD

study.

I would also like to express my thanks to both The Hong Kong Polytechnic

University and the Research Grants Council of the Hong Kong Special

Administrative Region for their financial support. Thanks are due to The Hong

Kong Polytechnic University for providing me with the research facilities.

I would like to give my special thanks to Mrs Anson for her great help with

the English languages aspects during the preparation of my thesis; my technical

writing skills have substantially improved as a result.

Special thanks also go to the technical staff of the laboratories of

Guangdong University of Technology (GDUT) and The Hong Kong Polytechnic

University (PolyU). They include but are not limited to Mr. Zhen-xiong Wang,

Mr. K.H. Wong, Mr. Y.H., Yiu and Mr. John Chan, who gave me valuable

assistance and advice for the my experimental work. Great thanks also go to

Prof. Li-juan Li, Dean of the School of Civil and Transportation Engineering for

making the laboratory facilities available for my experimental work.

Special thanks also go to many past and current members of Professor

Teng's research group: Dr. Guan Lin, Dr. Bing Fu, Dr. Yi-nan Yang, Dr.

Qiong-Guan Xiao, Dr. Bing Zhang, Dr. Jun-jie Zeng, Mr. Xue-fei Nie, Ms. Pang

Xie, Mr. Jie-kai Zhou, Mr. Pan Zhang and Prof. Jian-Guo Dai, not only for their

VII

discussions and constructive comments, but also for their encouragement during

difficult times of my PhD study.

Last but not least, I would link to thank my family and girlfriend for their

constant understanding, support and encouragement throughout my PhD study.

VIII

CONTENTS

CERTIFICATE OF ORIGINALITY ................................................................ I

ABSTRACT .................................................................................................... III

ACKNOWLEDGEMENTS ..............................................................................V

CONTENTS ..................................................................................................VIII

LIST OF NOTATIONS ............................................................................... XVIII

CHAPTER 1 INTRODUCTION ...................................................................... 1

1.1 BACKGROUD .............................................................................................. 1

1.1.1 Strong-Column-Weak-Beam Hierarchy in the Design of RC Frames .... 1

1.1.2 Problems in Realizing the Strong-Column-Weak-Beam Hierarchy ....... 2

1.1.3 Inadequacy of Existing Seismic Retrofit Methods ................................. 3

1.1.4 Effects of a Floor Slabs on the Flexural Capacity of a Supporting Beam

................................................................................................................ 4

1.1.5 Effects of Web Openings on the Performance of an RC Beam .............. 6

1.1.6 Effects of Slab Slits on the Performance of an RC frames ..................... 7

1.1.7 Proposed Seismic Retrofit Techniques ................................................... 7

IX

1.2 RESEARCH OBJECTIVES ........................................................................... 8

1.3 OUTLINE OF THE DISSERTATION ........................................................... 9

1.4 REFERENCES ............................................................................................. 12

CHAPTER 2 LITERATURE REVIEW ......................................................... 23

2.1 INTRODUCTION ........................................................................................ 23

2.2 POTENTIAL PROBLEMS AND SEISMIC RETROFIT OF RC FRAMES

WITH CAST-IN-PLACE SLABS ............................................................... 24

2.2.1 Potential Problems in RC Frames with Cast-in-place Slabs ................. 24

2.2.2 Seismic Retrofit of RC Frames ............................................................. 29

2.3 EXPERIMENTAL STUDIES AND FE MODELLING OF RC BEAMS

WITH WEB OPENINGS ............................................................................ 32

2.3.1 Experimental Studies of RC Beams with Web Openings ...................... 32

2.3.1 FE Studies of RC Beams with Web Openings ...................................... 39

2.4 EFFECTIVE SLAB WIDTH OF RC FRAMES .......................................... 42

2.5 CONCLUDING SUMMARY ...................................................................... 44

2.6 REFERENCES ............................................................................................. 46

CHAPTER 3 EXPERIMENTAL PREPARATION AND SPECIMEN

X

DETAILS .................................................................................. 67

3.1 INTRODUCTION ....................................................................................... 67

3.2 SPECIMEN DESIGN DETAILS ................................................................. 68

3.3 SPECIMEN CONSTRUCTION .................................................................. 72

3.4 MATERIAL PROPERTIES ......................................................................... 74

3.5 LOADING PROTOCOLS ........................................................................... 76

3.6 TEST SET-UP .............................................................................................. 77

3.7 INSTRUMENTATION ................................................................................ 78

3.7.1 Beam end Reaction Forces, Column Top Axial and Lateral loads ....... 79

3.7.2 Beam, Column and Joint Deformations ................................................ 79

3.7.3 Strain Gauges ........................................................................................ 85

3.8 REFERENCES ............................................................................................ 90

CHAPTER 4 EXPERIMENTAL STUDY OF THE SEISMIC

PERFORMANCE OF RETROFITTED RC

BEAM-COLUMN JOINTS................................................... 159

4.1 INTRODUCTION ..................................................................................... 159

4.2 FAILURE PROCESS AND FAILURE MODE ......................................... 159

XI

4.2.1 Cracking of Beams, Columns, Slabs and Joint Panel in the First Stage

Loading ............................................................................................... 161

4.2.2 Yield Point and Peak Point .................................................................. 164

4.2.3 Failure Mode ....................................................................................... 166

4.3 HYSTERETIC BEHAVIOUR AND ENVELOP CURVES ....................... 169

4.3.1 Hysteretic Behaviour ........................................................................... 169

4.3.2 Envelop Curves ................................................................................... 174

4.4 DEFORMATION BEHAVIOUR ............................................................... 175

4.4.1 Strains in the Steel Bars ....................................................................... 175

4.4.2 Rotations of Column Sections ............................................................. 183

4.4.3 Strain in Stirrups at Joint Region ........................................................ 184

4.4.4 Strains in FRP Jackets ......................................................................... 185

4.5 DISCUSSIONS .......................................................................................... 187

4.5.1 Specimen Ductility .............................................................................. 187

4.5.2 Energy Dissipation Capacity ............................................................... 191

4.5.3 Equivalent Viscous Damping Ratio ..................................................... 193

4.5.4 Stiffness Degradation .......................................................................... 196

XII

4.5.5 Plastic Hinge Lengths ......................................................................... 197

4.5.6 Deformation Components ................................................................... 201

4.6 CONCLUDING SUMMARY .................................................................... 206

4.7 REFERENCES .......................................................................................... 209

CHAPTER 5 THREE-DIMENSIONAL FINITE ELEMENT

MODELLING OF REINFORCED CONCRETE T-BEAMS

WITH A WEB OPENING WEAKENING AND

WITH/WITHOUT FRP SHEAR STRENGTHENING...... 299

5.1 INTRODUCTION ..................................................................................... 299

5.2 PROPOSED 3D FE MODELS .................................................................. 301

5.2.1 Modelling of concrete ......................................................................... 302

5.2.2 Definition of Damage Evolution ......................................................... 304

5.2.3 Modelling of Steel Reinforcement ...................................................... 305

5.2.4 Modelling of FRP Reinforcement. ...................................................... 306

5.2.5 Modelling of Bond Behaviour ............................................................ 307

5.2.6 Solution Strategy and Boundary Conditions ....................................... 311

5.3 VERIFICATION OF THE PROPOSED 3D FE MODEL ......................... 312

XIII

5.3.1 Specimen Dimensions and Material Properties ................................... 312

5.3.2 Load versus Displacement Curves ...................................................... 315

5.3.3 Crack patterns ...................................................................................... 317

5.3.4 Energy Release Behaviour .................................................................. 318

5.4 PARAMETRIC STUDIES ON THE EFFECTS OF KEY FACTORS ...... 320

5.4.1 Element Size ........................................................................................ 320

5.4.2 Loading Duration ................................................................................ 321

5.4.3 Damping Coefficient β ........................................................................ 322

5.4.4 Single versus Double Precision ........................................................... 326

5.4.5 FRP Confinement Effect...................................................................... 327

5.5 COMPARISON BETWEEN 2D AND 3D FE MODELS .......................... 328

5.6 SHEAR DEGRDATION OF CRACKED CONCRETE ............................ 320

5.6.1 The Effect of Number of Critical Cracks ............................................ 320

5.6.2 The Effect of Maximum Cracking Strain ............................................ 321

5.6.3 The Effect of Coefficient n of the Power Law .................................... 321

5.6.4 Comparison between the DP and DP+BC Models .............................. 321

XIV


5.8 REFERENCES .......................................................................................... 337

CHAPTER 6 3D SHELL MODELS FOR RC T-BEAMS .......................... 393

6.1 INTRODUCTION ..................................................................................... 393

6.2 PROPOSED 3D SHELL MODELS .......................................................... 394

6.2.1 3D Shell Models with a One-shell-layer Slab .................................... 395

6.2.2 3D Shell Models with a Two-shell-layer Slab .................................... 396

6.3 VERIFICATION OF 3D SHELL MODELS ............................................. 398

6.3.1 3D Shell models with One-shell-layer Slab ........................................ 399

6.3.2 3D Shell Models with a Two-shell-layer Slab .................................... 400

6.3.3 Efficiency and Accuracy of the 3D Shell Models ............................... 403

6.4 PARAMETRIC STUDIES ON THE EFFECTS OF SOME KEY FACTORS

.................................................................................................................. 404

6.4.1 Element Size ....................................................................................... 405

6.4.2 Loading Duration ................................................................................ 405

6.4.3 Damping Coefficient β ........................................................................ 406

6.5 3D SHELL MODELS FOR A T-BEAM WITH A WEB OPENING ......... 408

XV


6.7 REFERENCES ........................................................................................... 411

CHAPTER 7 EFFECTIVE SLAB WIDTH OF REINFORCED

CONCRETE FRAMES WITH SLAB UNDER TENSION 429

7.1 INTRODUCTION ...................................................................................... 429

7.2 PROPOSED SIMPLIFIED 3D FE MODELS ............................................ 432

7.2.1 FE Model Details and Boundary Conditions....................................... 432

7.2.2 Ultimate State in Simulation ............................................................... 434

7.2.3 Equations for Effective Slab Width ..................................................... 434

7.3 PARAMETRIC STUDIES ......................................................................... 435

7.3.1 Effect of Stress-strain Models of Steel ................................................ 435

7.3.2 Effect of Bond Slip Behavior .............................................................. 436

7.3.3 Effect of Yield Stress of Steel Bars ..................................................... 436

7.3.4 Effect of Concrete Strength ................................................................. 437

7.3.5 Effect of Beam Length, Width and Height .......................................... 437

7.3.6 Effect of Beam Reinforcement Ratio .................................................. 438

7.3.7 Effect of Column Width ...................................................................... 438

XVI

7.3.8 Effect of Slab Size. ............................................................................. 438

7.3.9 Effect of Slab Reinforcement Ratio .................................................... 439

7.3.10 Effect of Transverse Beam ................................................................ 439

7.4 DISCUSSIONS .......................................................................................... 440


7.6 REFERENCES .......................................................................................... 444

CHAPTER 8 THREE-DIMENSIONAL FINITE ELEMENT

MODELLING OF RETROFITTED RC

BEAM-COLUMN-SLAB JOINTS ....................................... 461

8.1 INTRODUCTION ..................................................................................... 461

8.2 THE PROPOSED 3D FE MODELS ......................................................... 462

8.3 PREDICTED RESULTS ............................................................................ 465

8.3.1 Load-displacement Curves.................................................................. 466

8.3.2 Crack Patterns ..................................................................................... 467

8.4 DISCUSSIONS .......................................................................................... 468

8.4.1 Confinement Effect of Joint Steel Stirrups ......................................... 468

8.4.2 Effect of Opening Shape ..................................................................... 469

XVII

8.4.3 Effect of Longitudinal Slit Length ...................................................... 470

8.4.4 Prediction Errors .................................................................................. 471


8.6 REFERENCES ........................................................................................... 472

CHAPTER 9 CONCLUSIONS ..................................................................... 509

9.1 INTRODUCTION ...................................................................................... 509

9.2 EXPERIMENTAL STUDIES OF RETROFITTED RC JONITS .............. 511

9.3 THREE-DIMENSIONAL FINITE ELEMENT MODELLING OF RC

T-BEAMS AND JOINTS .......................................................................... 514

9.3.1 3D Solid FE Model of T-beams with a Web Opening. ........................ 514

9.3.2 3D Shell FE Models for T-beams ........................................................ 517

9.3.3 Effective Slab Width of RC Frames .................................................... 519

9.3.4 3D FE Modelling of Retrofitted RC Joints ......................................... 521

9.4 FURTHER STUDIES ................................................................................. 522

9.5 REFERENCES ........................................................................................... 524

XVIII

LIST OF NOTATIONS

EDA : Dissipated energy of one hysteresis cycle;

effb : Effective slab width；

fb : Width of FRP;

jb : Joint width;

,rb ib : Length of right hand beam respective ith region measured by LVDTs;

wb : Beam web width;

D: Steel bar diameter;

ad : Maximum aggregate size;

td : Damage factor;

t bd : Vertical distance separating the two layers of 3D shell-2 models;

cE : Concrete elastic modulus;

fE : Elastic modulus of FRP sheet;

2sE : Modulus of hardening portion of steel stress-strain curve;

secE : Secant modulus of the compressive stress-strain curve of confined concrete;

totalE : Total energy for dynamic method;

IE : Internal energy;

XIX

VE : Energy dissipated by viscous effects;

FDE : Energy dissipated through frictional effects;

KEE : Kinetic energy;

WE : External work applied to the structural system；

'cf : Concrete cylinder strength;

'ccf : Compressive strength of confined concrete;

ckf : Concrete standard compressive strength;

cuf : Cube compressive strength of concrete;

yf : Bar yield stress;

uf : Bar ultimate stress;

frpf : Tensile strength of FRP sheet;

syf : Yield stress of slab bars;

cF : Column shear force;

b rightF : Beam tip load of the right hand T-beam;

b leftF : Beam tip load of the left hand T-beam;

1mF : Peak positive load of a hysteresis cycle;

2mF : Peak negative load of a hysteresis cycle;

XX

FG : Concrete fracture energy calculated by the area bounded by the strain-stress curve

of concrete under uni-axial tension;

fG : Interfacial fracture energy;

bh : Beam height;

jh : Joint height;

ch : Crack band width;

.rb ih : Respective distances between the top and bottom LVDTs for the right hand beam

respective ith region;

tc.1h : Respective distances between the right and left LVDTs for the upper column

respective ith region;

H : Distance between the inflection points of upper and lower columns;

cH : Length between the column-beam interface and the point of inflection.

c,elasticH : length of the column’s elastic region;

,e bI : Effective moment of inertia of cracked concrete area;

,g bI : Moment of inertia of the gross uncracked concrete area;

K : Stiffness matrix；

0l : Effective span of beam;

bL : Length of the beams;

XXI

,b elasticL : Length of beam elastic part;

bL : Length of a beam from the centre of the joint panel;

cL : Length of a column from the centre of the joint panel;

CM : Sum of column moment capacities;

BM : Sum of beam moment capacities;

topcM : Upper column moment capacity of a beam-column joint;

bottomcM : Lower column moment capacity a beam-column joint;

leftr bM : Left R-section beam moment capacity of a beam-column joint;

rightr bM : Left R-section beam moment capacity of a beam-column joint;

M : Mass matrix;

n : A parameter for power law damage model;

1n : A parameter controlling the transition from the elastic branch to the plastic branch;

ss : Slip between steel bar and concrete;

s : Slip between FRP sheet and concrete;

0s : Slip when the bond stress reached max ;

ns : Clear distance between neighbouring beams;

bs : Distance between slab bars;

S(y) :Area moment under the y point towards the neutral axis;

XXII

t : Slab thickness;

cV : Horizontal load acting on the end of the upper column;

tw : Crack opening displacement;

crw : Crack opening displacement at the point of concrete stress complete release;

tx : Line displacement along the X-direction of a node of the top slab shell layer of 3D

shell-2 models;

bx : Line displacement along the X-direction of a node of the bottom slab shell layer of

3D shell-2 models;

ty : Line displacement along the Y-direction of a node of the top slab shell layer of 3D

shell-2 models;

by : Line displacement along the Y-direction of a node of the bottom slab shell layer of

3D shell-2 models;

bz : Line displacement along the Z-direction of a node of the bottom slab shell layer of

3D shell-2 models;

tz : Line displacement along the Z-direction of a node of the top slab shell layer of 3D

shell-2 models;

s : Yield offset of steel stress-strain curve

: Mass-proportional damping coefficient;

c : Coefficient representing the initial tangent modulus of concrete;

: Stiffness-proportional damping coefficient;

XXIII

w : Width ratio factor;

: Axial compressive strain;

1 : Strain measured by the a inclined LVDTs for joint panel;

2 : Strain measured by the a inclined LVDTs for joint panel;

c : Strain corresponding to cylinder strength;

cc : Strain at compressive strength of confined concrete;

y : Bar yield strain;

u : Bar strain at ultimate strain

p : Strain at the peak stress of concrete stress-strain curve;

plt : Equivalent plastic strain;

h : hardening beginning strain;

s : Steel axial strain;

x : Strain in the joint width direction;

y : Strain in the joint height direction;

: Axial compressive stress;

p : Peak stress of concrete stress-strain curve;

bi : Stress in the i slab bar;

XXIV

maxb : Max stress of slab bars;

s : Steel axial stress;

0s : Yield stress of Ramberg Osgood (1943) model;

j : Angle of inclination of LVDT to the joint width direction;

,x t : Angular displacement along the X-direction of a node of the top slab shell layer of

3D shell-2 models;

,y t : Angular displacement along the Y-direction of a node of the top slab shell layer of

3D shell-2 models;

,z t : Angular displacement along the Z-direction of a node of the top slab shell layer of

3D shell-2 models;

,x b : Angular displacement along the X-direction of a node of the bottom slab shell layer

of 3D shell-2 models;

,y b : Angular displacement along the Y-direction of a node of the bottom slab shell layer


,z b : Angular displacement along the Z-direction of a node of the bottom slab shell layer


RBi : Average curvature of a beam ith region;

RBTi : Deformations measured by the LVDT located on the top surface of right hand

beam respective ith region;

XXV

RBBi :Deformations measured by the LVDT located on the bottom surface of right hand

beam respective ith region;

1TCR : Deformations measured by the LVDT located on the right hand surface of right

upper column respective ith region;

1TCL : Deformations measured by the LVDT located on the left hand surface of right

upper column respective ith region;

C : Flexural strength ratio;

: Coefficient of effective flange width to real flange width;

e sbys : A parameter accounting for slab bar strength effect on the effective slab width of

a T-beam at a exterior joint;

e cs : A parameter accounting for concrete compression strength effect on the effective

slab width of a T-beam at a exterior joint;

e fbs : A parameter accounting for flange bar spacing effect on the effective slab width

of a T-beam at a exterior joint;

e tbw : A parameter accounting for transverse beam width on the effective slab width of a

T-beam at a exterior joint;

i sbys : A parameter accounting for slab bar strength effect on the effective slab width of

a T-beam at a interior joint;

i bbys : A parameter accounting for slab bar strength effect on the effective slab width of


XXVI

i cs : A parameter accounting for concrete compression strength effect on the effective

slab width of a T-beam at a interior joint;

i bh : A parameter accounting for slab beam height effect on the effective slab width of a

T-beam at a interior joint;

i bw : A parameter accounting for beam width effect on the effective slab width of a

T-beam at a interior joint;

i fbs : A parameter accounting for flange bar spacing effect on the effective slab width of


i c : A parameter accounting for column weakening effect on the effective slab width of


e c : A parameter accounting for column weakening effect on the effective slab width of

a T-beam at a exterior joint;

elastic beam ， : Lateral displacement component due to beam elastic deformations；

,elastic column : Lateral displacement component due to column elastic deformations；

beam fixe end : Lateral displacement component due to tensile bar bond failure at

beam-column interface;

,hingebeam : Lateral displacement component due to beam hinge rotation;

,hingecol : Lateral displacement component due to column hinge rotation;

int jo : Lateral displacement component due to joint shear deformation;

1

CHAPTER 1

INTRODUCTION

1.1 BACKGROUND

1.1.1 Strong-Column-Weak-Beam Hierarchy in the Design of RC Frames

The capacity design method, which aims to ensure there is sufficient plastic

deformation capacity in reinforced concrete (RC) structures, was originally

proposed by Park and Paulay (1975) in New Zealand. The key principle of this

method is to make the beams framing into a joint, weaker than the columns. The

objective is to prevent failure by means of the storey sway mechanism (i.e.

when plastic hinges occur at the column ends), commonly known as the

strong-beam-weak-column (SBWC) mechanism. In general, the success of this

mechanism is ensured if the flexural strength ratio C , which is the sum of the

designed column moment capacities, CM , divided by

BM , the sum of

the designed beam moment capacities at a same joint, is greater than 1, as

express below:

C C BM M (1.1)

In China, this mechanism was first applied in 1989 using code GBJ11-89

for the seismic deign of buildings. The flexural strength ratio C was initially

2

set equal to 1.21 for RC frames of seismic grade 1. This value was later

increased to 1.7 in the recently updated edition of the code (GB-50011 2010).

In America, the SCWB mechanism was first adopted in the code ACI-318

1983: specifying the flexural strength ratio value to be 1.2. This value remains

the same today, but after 2002, ACI-318 (e.g. ACI-318 2014) requirements

replaced nominal values with design ones when calculating the moment

capacities of beams and columns.

The flexural strength ratio specified in Eurocode 8 (ENV 1998-1.3: 1995)

relates to the structural ductility class of the RC frame concerned. For instance,

the flexural strength ratio for RC frames of the highest ductility demand, is 1.35.

Those values are still the ones recommended today.

In New Zealand, initially, the flexural strength ratio was within the range

1.6 to 2.4 as specified in the design code NZS-3101(NZS-3101 1982). In 2006

(NZS-3101 2006), the value was modified, relating to two parameters and

, both of which are dynamic magnification factors ranging between 1.3 to 1.8

appropriate to the frame concerned.

1.1.2 Problems in Realizing the Strong-Column-Weak-Beam Hierarchy

The SCWB approach is widely adopted in the design of RC frame

structures subjected to seismic loading, because of its efficient

energy-dissipating capacity. However, many existing RC frames do not meet

3

SCWB criterion as most were designed using outdated codes, and have been

found to fail in adequately estimating the moment capacity of beams. A major

reason for this phenomenon lies in the neglecting, or downplaying, of the

contribution from the cast-in-place slab in tension, to the negative moment

capacity of the beam at the joint. This is a common characteristic of existing

Chinese codes concerned with the seismic design of RC structures (Lin et al.

2009; Zhang 2009). The fact that most RC frames failed during the 2008

Wenchuan earthquake (Chinese Academy of Building Research 2008) in

Sichuan Province, China, as a result of the storey sway mechanism (i.e. columns

fail before beams at a frame joint), has been attributed to the inadequacies of the

previous versions of the Chinese code (e.g., GB-50011 2002). Even though the

newly-revised version (GB-50011 2010), which came into practice in Dec. 2010,

takes into account the benefits of a cast-in-place slab to beam flexural capacity,

vague definition of the effective slab width contributing to the negative beam

moment capacity can also lead to an underestimation of slab contribution. Some

factors (e.g. in-fill walls, over-reinforced beam ends) can enhance the moment

capacities of beams and some (e.g. axial compression ratio, bi-directional

seismic action) can decrease the moment capacities of columns. Thus, the

specified flexural strength ratio in earlier codes could be inadequate and fail to

guarantee a beam sway failure mode.

1.1.3 Inadequacy of Existing Seismic Retrofit Methods

Seismic retrofit interventions are in great demand to enhance the seismic

4

safety of RC frames, but the SCWB mechanism condition can still be

compromised. To strengthen the columns is an easy and obvious option. Widely

used column strengthening techniques include: (1) concrete jacketing (e.g.

Thermou et al. 2007); (2) steel jacketing (e.g. Xiao and Wu 2003); and (3)

fibre-reinforced polymeric (FRP) jacketing (Teng et al. 2002). Cost-effective

seismic retrofitting is to enhance columns in both strength and ductility with

FRP jackets. However, this process can hardly change a storey sway failure

mechanism to a beam sway mechanism, especially when the columns are

non-circular in sectional shape because the strength improvement in this case

may not be sufficient to meet SCWB criterion. Even when the columns are, in

fact, sufficiently enhanced, failure may possibly occur at the beam-column

connections and the foundations, which are more complicated to be

reinforced/strengthened. Thus, a more effective seismic retrofit method is to

weaken the flexural capacities of T-shape beams, especially when under

negative bending.

1.1.4 Effects of a Floor Slabs on the Flexural Capacity of a Supporting

Beam

Numerous experimentally based research has conclusively shown that a

cast-in-place slab in tension significantly increases the negative flexural

capacity of a beam (e.g. Ehsani and Wight 1985; Durrani and Wight 1987;

Pantazopoulou and Moehle 1990; Pantazopoulou and French 2001; Zerbe and

Durrani 1990; Guimaraes et al 1992; Siao 1994; LaFave and Wight 1999; Shin

5

and LaFave 2004a, 2004b; and Canbolat and Wight 2008). The extent of the

contribution increases with increased lateral drift ratio (Durrani and Wight

1987). The contribution of a cast-in-place slab significantly reduce flexural

strength ratio (e.g. by 30%). Thus, the flexural strength ratios specified in many

existing design codes are insufficient for ensuring a beam sway mechanism.

Extensive FE studies carried out by Chinese scholars (e.g., Guan and Du

2005, Lin et al. 2009, Gao and Ma 2009, Tao 2010, Yang 2010, Chen 2010, Guo

2012) have investigated the effect of cast-in-place floor slabs on beam moment

capacity. In these studies, the RC frame structures simulated followed the

requirements specified in the Chinese code at the time when the studies were

carried out. Pushover analyses (Guan and Du 2005, Gao and Ma 2009, Yang

2010, Guo 2012) or elastic-plastic time history analyses (Lin et al. 2009, Tao

2010, Yang 2010, Chen 2010) were conducted to evaluate the effects of floor

slabs (Guan and Du 2005, Lin et al. 2009, Gao and Ma 2009, Tao 2010, Chen

2010, Guo 2012) and of different effective flange widths on the overall response

of RC frames. The simulation results indicated that cast-in place floor slabs

significantly improved the negative moment capacities of beams and frames,

designed according to Chinese seismic design code, all probably fail as a

non-ductile storey sway mechanism.

Various design codes take the effects of a cast-in-place slab in tension into

account by specifying an effective slab width in calculating the T-shaped beam

capacity (ACI-318 2008; NZS-3101 2006; Eurocode-8 2005). For instance, in

6

ACI-318 2008 and the New Zealand code NZS-3101 2006, the effective width

is specified as the smallest value of 1/4 of the beam span, 16 times the flange

thickness plus the beam web width, or the beam web plus half of the clear

distance between two adjacent beam webs.

1.1.5 Effects of Web Openings on the Performance of an RC Beam

Many studies since 1967 have explored the behaviour and design of a

rectangular or T-beam with circular or rectangular openings in the web for the

passage of utility ducts and pipes (e.g., Nasser et al. 1967; Mansur et al. 1999;

Pool and Lopes 1986; Hasnat and Akhtaruzzaman 1987; Kennedy and Abdalla

1992; Mansur 1998; Tan et al. 2001). These studies were concerned with

minimizing the effects of web openings in beams by surrounding the openings

with steel reinforcement strengthening. Their findings revealed that web

openings can significantly reduce the shear and flexural capacities of beams.

Five studies (Mansur et al. 1999; Abdalla et al. 2003; Maaddawy and

Sherif 2009; Pimanmas 2010, Nie 2018) studied the effects of creating an

opening in an existing beam and explored the efficiency of using bonded FRP as

a strengthening measure. The study of Mansur et al. (1999) and Nie (2018) were

concerned with T-beams. All these studies found that FRP strengthening, either

by the bonded FRP U jackets/full wraps (Abdalla et al. 2003; Maaddawy and

Sherif 2009, Nie 2018) or near-surface mounted FRP bars at corners (Pimanmas

2010), could offset the significant shear strength reduction caused by the

7

openings and effectively control shear

1.1.6 Effects of Slab Slits on the Performance of an RC frames

Zhang et al. (2011), Wang et al. (2012) and Zhang (2013) investigated the

effects of slab slits (referred to as SS hereafter) on the overall performance of

RC frames. Through numerical analysis of RC joints and frames, it was found

that the SS technique increased the probability of achieving an SCWB

mechanism (Zhang et al. 2011; Wang et al. 2012; Zhang 2013). In addition, this

retrofit measure performed better than FRP column strengthening. (Zhang

2013).

1.1.7 Proposed Seismic Retrofit Techniques

Local weakening or retrofitting for seismic design purposes is commonly

practised. For steel structures, a dog-bone design is adopted as a typical

weakening technique for new structures and, for existing structures, a seismic

retrofit to ensure the SCWB strength criterion (Popov et al. 1998). For RC

structures, local weakening by material removal to meet seismic concerns was

discussed in a preliminary and general manner at FEMA-356 (2000). Cutting

out the bottom longitudinal steel reinforcements, as a seismic retrofit technique

for exterior beam-column joints, was investigated by Pampanin (2006) and Kam

et al. (2009). However, cutting out bottom bars is insufficient to offset the

contribution of a cast-in-place slab in tension. Recently, a novel seismic retrofit

8

method called “beam-end weakening in combination with FRP strengthening”

method (BWFS method), was proposed by Teng et al. (2013). To weaken the

flexural capacities of the T-beams at a joint, particularly when the flange (i.e. the

cast-in-place slab) is in tension, is a primary measure.

To explore more suitable seismic retrofit techniques, three different

techniques as shown in Figure 1.1 proposed by Teng et al. (2013) were used for

this study: (a) separation of the slabs in the corner region from each supporting

beam by cutting a slit (including severing the steel bars crossing the slit)

between them (i.e. the slab slit (SS) technique); (b) drilling a web opening at the

beam end (i.e. the beam opening (BO) technique); (c) the beam section

reduction (SR) technique (e.g., creating a deep transverse groove (TG) on the

soffit of the beam near the joint). The latter two techniques are also combined

with associated strengthening measures (i.e., bonded FRP) to satisfy

serviceability and limit state requirements. The overall strategy of the proposed

method thus either applies the SS technique or the BWFS method when

retrofitting existing RC frames.

1.2 RESEARCH OBJECTIVES

This PhD research study aims to evaluate the effects of the retrofit

techniques proposed by Teng et al. (2013) on the achievement of ductile beam

sway mechanisms for RC frames subjected to seismic loading. The main

objectives of this study are as follows:

9

(1) To evaluate the efficiency of the three retrofit techniques by testing RC

beam-column joints under combined vertically constant axial loading and

horizontally cyclic loading. Although much research has concerned with the

local weakening of steel structures (i.e., dog-bone design ) in ensuring

SCWB hierarchy (Popov et al. 1998), few experiments relating to beam

local weakening, especially the newly proposed BWFS method, have been

conducted on RC joints,

(2) To develop advanced 3D finite element (FE) models for RC T-beams with a

web opening, with or without FRP strengthening, also for RC beam-column

joints retrofitted by the three proposed techniques.

(3) To conduct in-depth study on the issues of effective slab width of RC frames

based on extensive numerical investigations using a developed advanced FE

model.

1.3 OUTLINE OF THE DISSERTATION

The PhD dissertation is composed of the following chapters.

Chapter 1 presents the general background and research scope of the

present study.

Chapter 2 is an in-depth literature review of seismic

performance/behaviour of RC frames, RC beam-column joints with a

10

cast-in-place slab, beams with web openings. Both experimental studies and

numerical investigations were covered by the review.

In Chapter 3, details of the test program, and the process of preparing the

test specimens are presented. A total of 9, RC beam-column joints, divided into

two groups, one including a control RC joint and four retrofitted ones and

another one including a control RC joint and three retrofitted ones, were tested.

Test setup and boundary conditions for test specimens are presented. The layout

and functions of strain gauges and linear variable differential transformers

(LVDTs) are also described in detail. The retrofit procedure, including the

processes of creating web openings, transverse grooves and slab slits, and

applying FRP strengthening, is also introduced in detail.

Chapter 4 presents the experimental results of the 9 beam-column joints.

The failure process and failure modes, hysteretic behaviour are presented and

compared. Energy dissipation capacity, ductility, equivalent visual damping

ratio, stiffness degradation, plastic hinge length, beams and column deformation

contributions of the RC joints are discussed.

In Chapter 5, a 3D FE model using solid elements is developed for

T-beams with a web opening. A dynamic approach, using explicit time

integration method is adopted and extensive parametric studies are carried out to

investigate the effects of key factors of the dynamic approach (i.e. loading time

duration and damping schemes). The proposed 3D FE model is verified by the

11

RC specimens tested by Nie (2018), and its advantages over a 2D FE model are

discussed.

In Chapter 6, several 3D shell models (using shell elements for concrete) of

T-beams are proposed and assessed by the proposed 3D solid mode (using solid

elements for concrete) and test results. The predicted load-displacement curves

and crack patterns are used for in the assessment. The computational time is also

taken into consideration. The most efficient one is suggested based on accuracy

and computational efficiency.

In Chapter 7, the most effective 3D shell model is used in studying

effective slab width. Parameters including beam length, width, height, bar

reinforcement ratio, slab width, thickness, bar spacing, yield stress of steel bars,

transverse beam height, width, stirrup spacing and column width, are considered

in parametric studies. The effects of the bond-slip relationship between

longitudinal bars and concrete are also assessed. Two types of stress-strain

models for steel bars are used and assessed. Finally, two formulas are proposed

for the effective slab widths at the interior and exterior joints of RC frames,

respectively.

Chapter 8 presents a numerical investigation of RC beam-column joints.

Based on the 3D FE models developed in Chapter 5, 3D FE models for RC

joints retrofitted by the proposed techniques are proposed and accessed.

Load-displacement responses and crack patterns of the RC joints are compared

12

and discussed.

Finally, conclusions of this dissertation and an outline of the further

research needed are presented in Chapter 9.

1.4 REFERENCES

ABAQUS (2012). ABAQUS Analysis User's Manual (Version 6.12), Dassault

Systems SIMULIA Corporation, Providence, Rhode Island, USA.

ACI 318 (1983). Building Code Requirements for Structural Concrete and

Commentary (ACI 318-83), ACI Committee 318, American Concrete

Institute, Farmington Hills, MI.




Canbolat, B.B. and Wight, J.K. (2008). “Experimental investigation on seismic

behavior of eccentric reinforced concrete beam-column-slab connections”,

ACI Structural Journal, 105(2), 154-162.

Chen, X.B. (2010). The Effect of Floor Slab and Infill Walls on the Seismic

Behavior of Reinforced Concrete Frames, Master degree thesis: Fuzhou

University, China. (in Chinese)

13

Chinese Academy of Building Research (2008). Photo Collection of 2008

Wenchuan Earthquake Damage to Buildings, China Architectural &

Building Press, Beijing, China (in Chinese).

Durrani, A.J. and Wight, J.K. (1987). “Earthquake resistance of reinforced

concrete interior connections including a floor slab”, ACI Journal, 84(5),

400-406.

Ehsani, M.R. and Wight, J.K. (1985). “Effect of transverse beams and slab on

behavior of reinforced concrete beam to column connections”, ACI Journal,

82(2), 188-195.

Eurocode 8 (2004). Design of Structures for Earthquake Resistance – Part 1:

General Rules, Seismic Actions and Rules for Buildings (EN 1998-1: 2004),

CEN, Brussels.

FEMA-356 (2000). Pre-Standard and Commentary for the Seismic

Rehabilitation of Buildings. Federal Emergency Management Agency,

Washington, D.C.

Gao, Z.R. and Ma, Q.L. (2009). “Effect of cast-in-place slab on column-beam

strength ratio in frame structure”, Shanxi Architecture, 35(17), 1-3 (in

Chinese).

GBJ11-89 (1989). Code for Seismic Design of Buildings, Architectural &

14

Building Press, Beijing, China. (in Chinese)

GB-50011 (2001). Code for Seismic Design of Buildings, Architectural &




Guan, M.S. and Du, H.B. (2005). “Pushover analysis of effect of casting slab on

RC frame structure”, Earthquake engineering and engineering vibration,

5(5), 117-123. (in Chinese)

Guimaraes, G.N., Kreger, M.E. and Jirsa, J.O. (1992). “Evaluation of joint-shear

provisions for interior beam-column-slab connections using high-strength

materials”, ACI Structural Journal, 89(1), 89-98.

Guo, L. (2012). Research on RC Structures with Cast-in-Place Slab, Master

degree thesis: Beijing University of Technology, China. (in Chinese)

Hasnat, A., and Akhtanizzamam, A. A. (1987), "Beams with small rectangular

opening under torsion, bending, and shear", Journal of Structural

Engineering, 113(10), 2253-2270.

Hawileh, R., El-Maaddawy, T., and Naser, M. (2012). “Nonlinear finite element

modeling of concrete deep beams with openings strengthened with

externally-bonded composites”, Materials & Design (DOI:

15

10.1016/j.matdes.2012.06.004, 378-387).

Kennedy, J.B. and Abdalla, H. (1992). “Static response of prestressed girders

with openings”, Journal of Structural Engineering, ASCE, 118(2), 488-504.

LaFave, J.M. and Wight, J.K. (1999). “Reinforced concrete exterior wide

beam-column-slab connections subjected to lateral earthquake loading”,


Lin, X., Pan, P., Ye, L., Lu, X. and Zhao, S. (2009). “Analysis of the damage

mechanism of a typical RC frame in Wenchuan earthquake”, China Civil

Engineering Journal, 42(5), 13-20 (in Chinese).

Lu, X.Z., J.G. Teng, Ye, L.P. and Jiang, J.J. (2005). “Bond-slip models for FRP

sheets /plates bonded to concrete”, Engineering Structures, 27(6), 920-937.

Maaddawy, T. and Sherif, S. (2009). “FRP composite for shear strengthening of

reinforced concrete deep beams with openings”, Composite Structures,

89(1), 60-69.

Maaddawy, T. and El-Ariss, B. (2012). “Behavior of concrete beams with short

shear span and web opening strengthened in shear with CFRP composites”,

Journal of Composites for Construction, ASCE, 16(1), 47–59.

Mansur, M.A. (1998). “Effect of opening on the behaviour and strength of R/C

beams in shear”, Cement and Concrete Composites, 20(6), 477-486.

16

Mansur, M.A., Tan K.H. and Wei, W. (1999). “Effects of creating an opening in

existing beams”, ACI Structural Journal, 96(6), 899-906.

Nasser, K. W., Acavalos, A., and Daniel, H. R. (1967), "Behavior and design of

large openings in reinforced concrete beams", In Journal Proceedings (Vol.

64, No. 1, pp. 25-33).

Nie, X.F. (2018). Behavior and Modeling OF RC Beams with an

FRP-Strengthened Web Opening, Doctoral degree thesis: The Hong Kong

Polytechnic University.

NZS-3101 (1982). Code of Practice for the Design of Concrete Structures,

Standards New Zealand, Wellington, New Zealand.

NZS-3101 (2006). Concrete Structures Standard, Standards New Zealand,

Wellington, New Zealand.

OpenSees (2009). Open System for Earthquake Engineering Simulation, Pacific

Earthquake Engineering Research Center, University of California at

Berkeley, http://opensees.berkeley.edu.

Pantazopoulou, S.J. and Moehle J.P. (1990). “Identification of effect of slabs on

flexural behavior of beams”, Journal of Engineering Mechanics, ASCE,

116(1), 91-106.

Pantazopoulou, S.J. and French, C.W. (2001). “Slab participation in practical

17

earthquake design of reinforced concrete frames”, ACI Structural Journal,

98(4), 479-489.

Pimanmas, A. (2010). “Strengthening R/C beams with opening by externally

installed FRP rods: Behavior and analysis”, Composite Structures, 92(8),

1957-1976.

Pool, R. S., and Lopes, R. (1986), "Cyclically loaded concrete beams with web

openings", In Journal Proceedings, Vol. 83, No. 5, pp. 757-763.

Popov, E.P., Yang, T.S. and Chang, S.P. (1998). “Design of steel MRF

connections before and after 1994 Northridge earthquake”, Engineering

Structures, 20(12), 1030-1038.

Shin, M. and LaFave, J.M. (2004a). “Seismic performance of reinforced

concrete eccentric beam-column connections with floor slabs”, ACI

Structural Journal, 101(3), 403-412.

Shin, M. and LaFave, J.M. (2004b). “Reinforced concrete edge

beam-column-slab connections subjected to earthquake loading”, Magazine

of Concrete Research, 55(6), 273-291.

Siao, W.B. (1994). “Reinforced concrete column strength at beam/slab and

column intersection”, ACI Structural Journal, 91(1), 3-8.

Tan, K.H., Mansur, M.A., and Wei, W. (2001). “Design of reinforced concrete

18

beams with circular openings”, ACI Structural Journal, 98(3), 407-415.

Teng, J.G., Chen, J.F., Smith S.T. and Lam L. (2002). FRP-Strengthened RC

Structures, John Wiley and Sons Ltd, UK, November, 245pp.

Teng, J.G., Zhang, S.S., Jing, D.H., Nie, X.F. and Chen, G.M. (2013). “Seismic

retrofit of RC frames through beam-end weakening in conjunction with

FRP strengthening”, Proceedings of the 4th Asia-Pacific Conference on

FRP in Structures (APFIS 2013), 1-8.

Thermou, G.E., Pantazopoulou, S.J., Einashai, A.S. (2007). “Flexural behavior

of brittle RC members rehabilitated with concrete jacketing”, Journal of

Structural Engineering, ASCE, 133(10), 1373-1384.

Wang, X.G., Shan, M.Y., Ge, N. and Shu, Y.P. (2012). “Finite element analysis

of efficiency of slot-cutting around RC frame joint for ‘strong column and

weak beam’”, Journal of Earthquake Engineering and Engineering

Vibration, 32(1), 121-127 (in Chinese).

Xiao, Y. and Wu, H. (2003). “Retrofit of reinforced concrete columns using

partially stiffened steel jackets”, Journal of Structural Engineering, ASCE,

129(6), 725-732.

Yan, Y.L. (2010). A Study on Eurocodes about the RC Elements Design and

Comparison between Eurocodes and Chinese Codes, Master degree thesis:

19

China Academy of Building Research, China. (in Chinese).

Ye, L.P., Qu, Z., Ma, Q.L., Lin, X.C, Lu, X.Z. and Pan, P. (2008). “Study on

ensuring the strong column-weak beam mechanism for RC frames based on

the damage analysis in the Wenchuan earthquake”, Building Structure,

38(11), 52-67 (in Chinese).

Zerbe, H.E. and Durrani, A.J. (1990). “Seismic response of connections in

two-bay reinforced concrete frame subassemblies with a floor slab”, ACI


Zhang, J. (2013). Research on Efficiency of Slot-cutting around Frame Joint for

“Strong Column and Weak Beam" under Earthquake Action, Master degree

thesis: Hunan University, China (in Chinese).

Zhang, Y.P., Hao, Z.J., Shan, M.Y. and Ge, N. (2011). “Research on anti-seismic

performance for reinforced concrete frame joint with slot around”, Building

Science, 27(9), 7-11 (in Chinese).

20

(a) The slab slit (SS) technique

(b) The beam opening(BO) technique

lslsls

ls

sl l s

l s slSlitsColumn

Beam

ld ldComplete FRP wrap

NSM FRP strips

FRP U-jacket

Opening

ol

ho

FRP U-jacket

21

(c) The section reduction (SG) technique

Figure 1.1 The three seismic retrofit techniques (Teng et al. 2013)

ld ld

Gap

ol

ho

Gap

olFRP U-jacket

Remaining steel bars Remaining steel bars

ho

Groove Groove

22

23

CHAPTER 2

LITERATURE REVIEW

2.1 INTRODUCTION

This chapter presents literature review on relevant topic of the present PhD

research program. The chapter is organized as follows: experimental and finite

element (FE) studies on RC frames/joints with cast-in-place slabs are first

presented with the aim of pointing out the potential problems existing in those

reinforced concrete (RC) frames. Corresponding seismic retrofit methods for

those RC frames are then presented. As given in Chapter 1: creating a beam

opening is a proposed retrofit technique for RC frames with cast-in-place slabs,

hence existing experimental and numerical RC beam studies, in particular, RC

T-section beams (T-beams) with a web opening are reviewed. To better

seismically retrofit those RC frames, the contribution of cast-in-place slabs to a

beam negative moment capacity need to be quantitatively determined. Finally

effective slab width studies are reviewed, both experimentally and by means of

FE analysis

It should be noted that if not otherwise stated, all equations presented in

this chapter and the remainders of this thesis are written in terms of standard SI

unit designations (i.e. MPa and mm)

24

2.2 POTENTIAL PROBLEMS AND SEISMIC RETROFIT OF

RC FRAMES WITH CAST-IN-PLACE SLABS

The strong-column-weak-beam (SCWB) mechanism requires the flexural

strength ratio C larger than 1.0. However, the ignorance or underestimation of

the cast-in-place slabs contribution to the negative moment capacity of beams of

RC frames, designed according to previous seismic design codes, might lead to

storey sway failure when subjected to seismic loading. A review of existing

studies, including both experimental and numerical investigations, on the effect

of cast-in-place slabs on the capacity of RC beams and corresponding seismic

retrofit techniques are given below.

2.2.1 Potential Problems in RC Frames with Cast-in-place Slabs

In the early 1980s, researchers from the United States and Japan (Durrani

and Wight 1982, Otani et al. 1984, JTCC 1988) conducted experimental studies

on a full-scale 7-storey RC frame. Test results indicated that, the existence of

cast-in-place floor slabs in RC frames, when under lateral loading, could greatly

increase the flexural capacity of beams. Such an increase was, however, not

taken into consideration when designing the frame. This was because at that

time, such a consideration was not available in design provisions. Failure of the

test frame was consequently controlled by joints shear failure.

Qi and Pantazopoulou (1991) subjected a 1/4-scale single-story RC frame

with cast-in-place floor slabs to cyclic lateral loading. The test results indicated

25

that the contribution of cast-in-place slabs to the flexural capacity of the beams,

especially at the interior support, was considerable.

Ning et al. (2016) tested two spatial RC frames: a control specimen

without cast-in-place slabs and one with cast-in-place slabs. The specimens

were subjected to cyclic loading and tests results indicated that the existence of

cast-in-place slabs changed the failure mode of a RC frame from a typical

SCWB mechanism to the strong-beam-weak-column (SBWC) mechanism. The

slab reinforcement contribution to the negative moment capacity of the beams

was the main reason accounting for the occurrence of SBWC failure.

In addition to experimental RC frame tests, massive experimental research

on RC beam-column-slab subassemblies were conducted to assess the

cast-in-place slab effect on RC joints subjected to cyclic loading. Experimental

investigations of both exterior and interior RC joints with cast-in-slab are given

blow.

Ehsani and Wight (1985) conducted tests on six beam-column-slab

subassemblies, subjected to cyclic loading. Test variables including the ratio of

the column flexural capacity to those of the beam with slabs (i.e. flexural

strength ratio), the joint shear stress, and joint transverse reinforcement.

Specimens with cast-in-place slabs and transverse beams were compared with

those control examples without transverse beams and slab (referred as a plane

joint hereafter). Flexural strength ratios of 1.1, 1.5 and 2.0 were included in the

http://www.baidu.com/link?url=2hLGzDBkaOA9RnWMqxrKIlAHtI4diWPGstlg2cF4O0RsdepIZ-smUJ3EavQc2MHtpbERZgu4EB3MDkz-wrdpQRog1ic_3evPO43PvedmdXW

26

investigation. Only two longitudinal slab reinforcing bars, adjacent to the main

beam, were included in the calculation. All the slab longitudinal reinforcing bars

had yielded during test. Thus the flexural strength ratio was in fact lower than

anticipated. Specimens with flexural strength ratio equal to 1.1 failed by column

concrete crushing.

Durrani and Zerbe (1987) tested a total of six 3/4-scale joints under cyclic

lateral loading to study the effect of cast-in-place slabs on the behavior of

exterior joints. The test results showed that the cast-in-place floor slab had a

significant effect on the strength, stiffness and energy dissipation characteristics

of the joints. It was thus strongly suggested that the effect of cast-in-place floor

slabs be considered in the design of joints.

Durrani and Wight (1987) tested three interior RC beam-column-slab joints

with variables including joint shear stress level and an amount of joint hoop

reinforcement. All slab bars yielded at 4% storey drift ratio and the inclusion of

slab contribution to beam moment capacity in seismic design was

recommended.

Zerbe and Durrani (1990) tested two-bay frame subassemblies to study the

effect of slab on the behavior of beam-columns-slab connections. Each

subassembly was composed of two exterior joints and one interior joint. Three

subassemblies including a plane specimen C, specimen CTB with transverse

beams only and the remaining one CS1 consisting of both transverse beams and

27

slab were considered. Test results showed that continuous joints increased the

contribution of slab to the hogging moment of beams and introduced more shear

force in the joints than the individual joints did.

French (1991) collected test data of 20 beam-slab-column joints (13

interior joints and 7 exterior joints). The average underestimation of predicted

strength of interior joints when the cast-in-place floor slab was ignored was 25%

less than the test result, which is 17% for exterior joints.

A 1/4-scale single-story RC frame with cast-in-place floor slabs subjected

to cyclic lateral loading was tested by Qi and Pantazopoulou (1991). The test

results showed that the flexural capacity of the beams, especially at the interior

support greatly increased due to the existence of cast-in-place slabs.

Jiang et al. (1994) tested two specimens: the plane joint and one with a

cast-in-place slab. The test results showed that the cast-in-place floor slab

increased the beam negative flexural capacity by as much as 30%.

Shin and LaFave (2004a, b) tested four 2/3 scale RC edge

beam–column–slab subassemblies (i.e. two concentric and two eccentric

connections). The slab and transverse beam were only on one side of beams.

The test results revealed that the effective slab width, at peak storey shear force,

was actually wider than design recommendation.

Zhen et al. (2009) tested three groups of RC joints with different

28

reinforcement schemes under cyclic lateral loading. Each group was composed

of a plane joint and one/four/two joints with a cast-in-place floor slab. Test

results showed that the existence of a cast-in-place floor slab resulted in

specimen strengths, in group 1, 2 and 3, respectively about 1.6, 2.0 and 2.3 that

of those of the corresponding plane joints

Jiang et al. (2009) tested eight weak reinforced beam-column connections

and two strong ones under cyclic lateral loading. The experimental result

showed that due to the increased joint shear stress introduced by the

cast-in-place slab, the existence of the slab transformed the failure mode of a

weak joint from beam flexural failure to joint shear failure. The specimens with

a cast-in-place slab suffered more damage in the joint region than that suffered

by the corresponding plane specimens.

Li et al. (2012) tested five 3/4-scaled interior beam-column joints to

investigate the influence of floor slabs and column orientation on the seismic

performance of lightly-reinforced concrete beam-column joint subjected to

seismic loading. Test results show that the strengths of interior joints by about

11% to 27% increased by floor slabs

The effect on the seismic performance of RC frames with cast-in-place

floor slabs has been studies by many researchers (Guan and Du 2005, Lin et al.

2009, Gao and Ma 2009, Yang 2010, Chen 2010, and Guo 2012) using

numerical approaches as detailed in Table 2.1. All numerical results indicated

29

that cast-in place floor slabs could significantly increase the negative moment

capacity of the beams and thus lead to the SBWC mechanism in RC frames.

2.2.2 Seismic Retrofit of RC Frames

The literature review of experimental and FE analysis of RC frames/joints

with cast-in-place slabs given above indicate that the ignorance or

underestimation of the cast-in-place slabs contribution to a beam negative

moment strength would probably lead to a SBWC failure mode in existing RC

fames, subjected to seismic loading. To avoid such inductile failure mode,

seismic retrofitting of RC frames is in great need of further study.

As stated in Chapter 1, to strengthen the columns is an easy and obvious

option while cost-effective seismic retrofitting can enhance columns in both

strength and ductility with FRP jackets. This process however, changes a storey

sway failure mechanism to a beam sway mechanism, very little, especially when

the columns are non-circular in sectional shape because the strength

improvement may not be sufficient to meet SCWB criteria. Even when the

columns are, in fact, sufficiently enhanced, failure may possibly occur at the

beam-column connections and the foundations, the latter being more

complicated to reinforce. Thus, the benefit of a more effective seismic retrofit

method is that it weakens the flexural capacities of T-beams, especially when

under negative bending.

30

Recently, Teng et al. (2013) proposed a novel seismic retrofit method for

RC frames with cast-in-place slabs which violate the SCWB hierarchy. This

method is based on the concept of Beam-end Weakening in combination with

FRP Strengthening (referred to as the BWFS method hereafter for simplicity), to

implement the SCWB hierarchy. To weaken the flexural capacities of the

T-beams at a joint, particularly when the flange (i.e. the cast-in-place slab) is in

tension, is a primary measure. Three beam local weakening techniques were

also presented by Teng et al. (2013). Local weakening for seismic design

purposes or through retrofitting is more commonly practised. For steel

structures, a dog-bone design is adopted as a typical weakening technique for

new structures and, for existing structures, a seismic retrofit is aimed to ensure

the SCWB strength criteria (Popov et al. 1998). For RC structures, local

weakening by material removal to meet seismic concerns was discussed in a

preliminary and general manner at FEMA-356 (2000). Cutting the bottom

longitudinal steel reinforcements, as a seismic retrofit technique for exterior

beam-column joints, was investigated by Pampanin (2006) and Kam et al.

(2009). However, cutting out bottom bars is insufficient to offset the

contribution of a cast-in-place slab in tension.

As presented in Chapter 1, three different techniques are proposed: (a) the

slab slit (SS) technique; (b) the beam opening (BO) technique; (c) the beam

section reduction (SR) technique (e.g., creating a deep transverse groove (TG)

on the soffit of the beam near the joint). The effect of slits on RC joints and

31

frames has been studied through FE modelling by Zhang et al. (2011), Wang el

al.(2012) and Zhang (2013). The investigations were FE modelling based.

Conclusion was made by Zhang et al. (2011), Wang el al.(2012) and Zhang

(2013) that the SS technique can effectively change the failure mode from story

sway to beam sway mechanism. The BO technique has been recently

investigated by Nie (2018). Both the experimental test and the 2D FE

modellling of T-beams retrofitted by the BO technique were conducted. Test

results showed that the BO technique effectively decreased the T-beam negative

moment strength. The specimens with both BO weakening and FRP

strengthening had ductile failure modes. To the best of the candidate’s

knowledge, the last mentioned techniques are new and no relevant research

regarding its effectiveness and design methods is available.

More recently, Feng et al. (2017) proposed a novel method to improve the

seismic performance and progressive collapse resistance of RC frame structures

using kinked bar. The kinked bar has locally curved regions, which are usually

placed near the inflection points in beams. The curved region is gradually

straightened when subjected to tension force. The section where kinked bar

curved region is located has the lowest capacity and will firstly yield under

seismic loading. However, the section is of good ductile property as the kinked

bar can keep resisting force when straightened. The seismic performance and

progressive collapse resistance of RC frame structures will therefore be

improved. This novel method can be considered a new retrofit technique for

32

exiting RC frames, even though its original proposal was for new construction.

2.3 EXPERIMENTAL STUDIES AND FE MODELLING OF

RC BEAMS WITH WEB OPENINGS

2.3.1 Experimental Studies of RC Beams with Web Openings

Many studies have conducted experimental investigations regarding the

behavior and design of rectangular or T-beams with a rectangular or circular

opening for the passage of utility ducts and pipes since 1967 (e.g. Nasser et al.

1967; Mansur et al. 1985; Tan et al. 1996). Nasser et al. (1967) tested 10 beams

with openings. In the design of beams with openings, two groups of longitudinal

reinforcement were provided in each chord and several specimens were

reinforced with adequate stirrups and longitudinal bars in the chords. The beam

whose cross section was 41.7% (beam B-1) reduced and had an ultimate load 24%

lower than that of the control one. The existence of openings also lowered the

stiffness of beams.

Mansur et al. (1985) designed and tested 12 beams with openings under

concentrated load to verify the validity of the proposed design method for

strengthening beams. The failure mode of a beam with a web opening was chord

end concrete crushing. The existence of a opening of size 1200×180 mm2 and

800×220 mm2 decreased the ultimate load to less than 50% of that of the

control specimen. The application of diagonal bars as corner reinforcement

proved to be more effective in controlling crack propagation and thus the

33

increase of the ultimate loads of beams.

Tan et al. (1996) tested 15 T-beams with openings except for the control

beams to investigate the effect of opening size, location on continuous beams.

Two groups of specimens, namely group IT and group T were tested to

investigate the performance of a continuous T-beam under respectively negative

and positive loading. The result showed that the existence of openings decreased

the cracking and ultimate load as well as beam post-cracking stiffness. The

heights of all the openings were not larger than half of a T-beam height. The

openings were all located not closer than one-half the beam depth from the

supports or the concentrated loads. The test result of the IT group revealed that

enlarging the opening length could significantly decrease the post-cracking

stiffness and ultimate capacity of a T-beam and switch the failure mode from

flexural failure at solid section to a failure, due to hinge formation at the

opening corners. The location of opening had limited effect on the ultimate load

of beam. After testing the T group specimens, conclusions could be drawn that

the enlargement of the opening height could slightly lower the positive capacity

of a T-beam.

Conclusions can be drawn from the studies referred to above, that an

opening can significantly reduce the shear and flexural capacities of a beam. All

the above reviewed studies focused on the application of internal steel

reinforcement around the reserved opening to minimize that opening's effect on

its host beam. The effect of drilling an opening in an existing beam and the

34

application of external strengthening techniques are reviewed below.

Mansur et al. (1999) tested a series of T-beams to evaluate the effect of

creating beam web circular openings. The reinforcing stirrups crossing openings

were cut and no additional internal reinforcement around the openings was

provided. T-beams, subsequently repaired and strengthened by filling openings

with respective nonshrink construction grout and externally bonded FRP were

tested to evaluate their effects on recovering the T-beam strength degradation.

The result showed that the beneficial effect of grout was limited. The external

bonded FRP could not only fully help a T-beam regain its original stiffness and

ultimate load, but also enhance its ductility. The maximum crack width was also

well controlled.

Abdalla et al. (2003) conducted an experimental program of 10 RC beams

including five strengthened with bonded FRP around the openings, four tested

without FRP strengthening and a control solid beam. Parameters including

opening horizontal location, opening size, configuration and amount of FRP

were studied. The result revealed that those beams with openings weakening but

without FRP strengthening all suffered shear failure at the opening region under

very low loads. When the opening size was relatively small (), the beam

strengthened by bonded FRP could fully regain the original stiffness and

strength as a solid beam. As the opening size kept increasing, the failure mode

switched from mid-span flexure failure to shear failure at the opening region.

35

Allam (2005) conducted two series of test to investigate the efficiency of

strengthening schemes when web opening in shear zone of a beam on b. Internal

steel reinforcement, external steel plates and bonded CFRP sheets were all

applied in the test. The results indicated that the web opening decreased the

strength of beam from 167 kN to 105 kN when no strengthening scheme was

applied. Failure mode was also changed to shear failure at the opening zone

rather than flexural failure at the mid-span. When the beam with opening was

strengthened with steel reinforcement, external steel plates or bonded CFRP

sheets, its strength was obviously increased. The specimens strengthened with

external steel plates both outside and inside also failed due to flexural failure at

the mid span.

Maaddawy and Sherif (2009) studied the effect of FRP composites on

shear strengthening of RC deep beams with a group of 13 deep beams tested

under four-point bending. All the specimens were of 80×500 mm cross section

and a total length of 1200 mm. Maaddawy and Sherif (2009) concluded that the

effect of opening was primarily dependent on the degree of the interruption of

natural load path with variation of opening position and size. The strength gain

resulting from CEFP sheets was in a range of 35%-73% and the stiffness was

also upgraded.

Pimanmas (2010) examined the function of externally installed FRP robs

in shear strengthening RC beams with square or circular openings. A total

36

number of 13 beams were tested. The test result showed that an opening

significantly reduced the shear capacity of the beams and full depth FRP robs

was most effective in regaining strength and ductility. The flexural failure mode

was also restored with this kind of strengthening method.

Chin et al. (2012) conducted an experimental study on RC beams with

square openings (210×210 mm2). Six beams consisting of control beams,

un-strengthened and strengthened RC beams with square openings. FRP wrap

was employed as a strengthening technique. Experimental results showed that

the existence of openings significantly decreased the beam strength by about

70%. The CFRP wrap could increase the strength of the beam.

Maaddawyand and Ariss (2012) conducted an experimental study on a total

of 15 RC beams with openings, in addition to a solid beam and proposed

analytical formulation of shear resisting capacity for specimens to evaluate the

effect of CFRP composites on shear strengthening RC beams with opening. No

internal web reinforcement was provided to strengthen the opening region to

resemble the case of drilling an opening on a solid beam web. Variables

included the opening size and the amount of CFRP sheets. It is of note that for

FRP strengthened RC beams with web openings, vertical and horizontal FRP

sheets were arranged around the openings. Except that the bottom chords were

strengthened by CFRP wrapping, the top chords and the both sides of openings

were provided with U-shaped CFRP. The result indicated that the shear capacity

and stiffness of a beam were obviously degraded due to the existence of a web

37

openings and this weakening effect was effectively offset by the externally

boned CFRP sheets. The increase of shear capacity was not in the same scale as

the amount of CFRP applied. Increasing the opening size lowered the efficiency

of CFRP sheets as regards upgrading the shear resistance and stiffness of the

beam. The proposed analytical approach, used to predict the shear capacity of an

RC beam weakened by openings and shear strengthened by CFRP indicated a

good agreement with the test results.

Suresh and Prabhavathy (2015) tested 14 beams to investigate the

behaviour of steel fibre RC beams with opening strengthened by steel plates.

The results showed that the existence of web opening in shear zone decreased

the strength of beams by 55%-70%. Strengthening the beams with web opening

using steel plates considerably increased their strength and ultimate deflection.

Chin et al. (2016) investigated the behaviour of RC with a web opening,

with/without CFRP strengthening by testing 6 simply supported beams. The

result indicated that a web opening at the mid-span decreased the beam strength

to about 50% and externally bonded CFRP steel lead to 80-90% strength regain.

The experimental studies (Mansur et al. 1999; Abdalla et al. 2003; Yang et

al. 2006; Chin et al. 2012; Maaddawyand and Ariss (2012)) aimed to investigate

the effect of drilling a web opening in an existing beam. Design methods were

proposed for associated strengthening measures to regain the strength, reduced

due to the exiting of openings. Conclusions drawn suggest that the existence of

38

a web opening decreased the strength of RC beams and that strengthening with

bonded FRP reinforcement or steel plates could effectively enable the reduced

strength to be regained.

To investigate the proposed BO technique, Jing et al. (2013) tested two RC

T-beams weakened by rectangular opening: one beam (O-500×150) had a 150

mm (height) × 500 mm (length) opening but not FRP strengthening while the

other (FRP-500×220) had a 220 mm (height) × 500 mm (length) opening and

additional FRP strengthening. The result indicated that larger opening size lead

to more strength decrease and the bonded FRP reinforcement avoided

undesirable failure mode caused by the existence of web opening.

Recently, with the aim to reduce a T-beam strength while preserving its

ductility, Nie (2018) tested T-beams to investigate the effect of the BO

technique regarding decreasing the moment capacities of T-beams in his PhD

thesis. Except for a solid R-beam and T-beam, a total 12 T-beams with BO were

tested. Ten of the 12 T-beams were tested under negative loading and the

remaining two were tested under positive loading. Four of the ten specimens

were only weakened by opening and the remaining six were subjected to

additional FRP strengthening. Test results showed that the existence of an

opening effectively decreased T-beam strength. The specimens with additional

FRP strengthening had ductile failure modes. Nie (2018) also proposed a

method to predict the strength of RC T-beams with a web opening.

39

2.3.1 FE Studies of RC Beams with Web Openings

Several researchers (Madkour 2009; Pimanmas (2010); Chin et al. 2012;

Hawileh et al. 2012, Nie 2018) proposed FE models to simulate RC beams as

summarized in Table 2.3.

Madkour (2009) applied a damage-non-linear elastic model to conduct RC

beam 3D simulations. The isoperimetric quadratic element (20-node element),

elastic bar element (2-node element), and shell element were used for concrete,

reinforcing steel and FRP. The bond slip relation between FRP/steel

reinforcement and concrete was not mentioned by Madkour (2009). The

predicted crack patterns were poor.

Pimanmas (2010) examined the function of externally installed FRP bars in

shear strengthening of RC beams weakened by square or circular openings. In

addition to an experimental program, a 2D nonlinear FE analysis based on

smeared crack approach was conducted to investigate a preferable arrangement

of FRP robs round openings. A perfect bond between FRP robs and substrate

concrete was assumed by Pimanmas (2010). The predicted load-displacement

curves were comparable to the test results. The predicted crack patterns were

poor as the cracks were not clearly revealed.

Chin et al. (2012) conducted a numerical analysis of RC beams with large

square openings. 2D FE modelling experimental specimens with software

40

ATENA was carried on. The tensile behavior of concrete was modeled with a

combination of nonlinear fracture mechanics and crack band method, in which

the smeared crack concept was adopted. A displacement-controlled procedure

based on Newton-Raphson iterative method was applied in analysis. A Bond

slip relation between reinforcing steel and concrete was ignored and that

between CFRP and concrete was considered by applying the bond slip model

developed by Lu et al. (2005). Even though the predicted ultimate strength was

relatively close to the experimental result, the predicted stiffness was much

higher than that of test. Chin et al. (2012) assumed that the omission of bond

slip relation between reinforcing steel and concrete was believed to be the

cause.

Hawileh et al. (2012) carried out a 3D FE analysis on RC deep beams

tested by Maaddawy and Sherif (2009). A total of 12 specimens were modeled.

They included one solid beam, strengthened and un-strengthened beams

weakened by opening. Solid elements for concrete, multi-layer shell elements

for CFRP and link elements for steel reinforcement were applied. A special

interface element was used to represent the bond-slip behaviour between FRP

and substrate concrete. A model with perfect bond for the interface between FRP

and substrate concrete was also modelled for comparison. The models

incorporating bone-slip behavior showed great agreement with test results with

an average of 3.2% error for strengths and an average 14% error for ultimate

deflections. The omission of bond slip behavior led to higher predicted strengths

41

and stiffnesses.

Only Nie (2018) has conducted a 2D FE analysis on RC T-beams

weakened by opening using ABAQUS (2012). An explicit dynamic method was

applied to avoid numerical convergence problems. Several FE approaches, with

major difference in the applied models for concrete compression property and

damage under tension, were proposed to model a beam with web opening. The

damage plasticity (DP) approach with power law (PL) tension damage model

was recommended for beams with a flexural failure mode, while the brittle

cracking (BC) approach with secant modulus was recommended for beams with

a shear failure mode. The DP approach with PL tension damage model was most

suitable for the test T-beams with both opening weakening and associated FRP

strengthening, as they all had ductile failure mode, while the BC approach with

secant modulus was suggested for the T-beams weakened by a opening only.

The peak loads of T-beams were predicted with a relatively small error.

However, the 2D model generally overestimated specimen stiffness. Nie (2018)

stated that this may be due to the non-uniform distribution of longitudinal

compressive stresses in the concrete and the reinforcement of the flange along

the width direction, known as shear lag effect. Of interest is that the potential

out-of-plane deformation could not be captured by the 2D models. Nie (2018)

mentioned that a more accurate 3D FE model was needed to take the shear lag

effect into consideration. The cracks on the slab could not be directly presented

by the 2D models. Thus, 3D FE models of RC T-beam weakened by opening

42

should be proposed to better predict the behaviour of RC T-beams weakened by

opening.

2.4 EFFECTIVE SLAB WIDTH OF RC FRAMES

Extensive experimental studies (e.g. Jiang 1994; Bijan and Aalami 2001;

Huang et al. 2001) have indicated that the strains of steel bars in a cast-in-place

slab flange are not evenly distributed along the beam width direction. Instead,

the strain in a slab steel bar deceases with the increased distance between the

steel bar and the beam, due to the well-known shear lag effect. Thus, only steel

bars within a limited range of width away from the beam can reach their yield

stress at the failure of the beam (Wu et al. 2002; Wang et al. 2009; Zhen et al.

2009). In order to quantify the contribution of a cast-in-place slab to the beam

flexural capacity, an effective flange width ( effb ) has been proposed by previous

researchers (Wu et al. 2002; Wang et al. 2009). All the slab longitudinal steel

bars within the effective flange width are assumed to be equally strained in the

bending of the beam.

However, currently, a uniform method, to determine the effective width of

a slab, especially for an RC structure with cast-in-place slabs does not exist. The

effective widths of slabs stated in codes (GB 50010-2010, ACI318-05 &

FEMA-356, EC8, NZS-3101:2006), as indicated in Table 2.4, are also different.

Existing research in this area is experimentally based (Ehsani and Wigh 1982, T

Pantazopotrlou et al. 1988, Durrani and Zerbe 1990, French 1991, Li 1994,

43

Jiang et al. 1994, Wu et al. 2002, Wang et al. 2009, Zhen et al. 2009, Yang 2010,

Sun et al. 2010, Qi et al. 2010, He 2010, Ning et al. 2016). Table 2.5 presents a

summary of the suggested values of effective slab widths given by the above

researchers based on their experimental results. It is obvious that the effective

slab width beff is mainly related to beam width bw, beam height hb, effective span

of beam l0, slab thickness t, and clear distance between beams sn. Most

researchers determine the effective slab width when the storey drift rato ( ) is

equal to 1/50.

Some researchers, however, also proposed equations, based on their

simulation data, to determine the effective slab width for RC frames. Ning et al.

(2016) studied the effective slab width of an RC frame both experimentally and

by means of FE simulation. The FE model was first verified through

comparison with test result. Ning et al. (2016) then conducted parametric

studies on the axial compression ratio, concrete strength, reinforcement ratio of

slabs, thickness of slabs and dimension of the transverse beams. An equation

involving only the main beam width and height to predict the effective slab

width with a 95% guaranteed accuracy was proposed. However, Ning et al.

(2016) did not study the effect of beam length on beff, which is a parameter

considered in all design codes (e.g. GB 50010-2010, ACI318-05 & FEMA-356,

NZS-3101:2006) except for EC8. In addition, Ning et al. (2016) did not study

the effect of slab width on beff, a parameter also considered by all design codes

(e.g.GB 50010-2010, ACI318-05 & FEMA-356, EC8, NZS-3101:2006) except

44

EC8.

Many researchers (e.g. Castro et al. 2006, Nie and Tao 2012, Methee et al

2016, Galuppi and Royer-Carfagni 2016) have studied the effective slab width

of a composite structure under positive loading through FE modelling. Some,

such as Nie and Tao (2012), also studied the effective slab width of composite

structures under negative loading. By conducting parametric studies through FE

simulation on column dimensions, steel beam height, RC slab width and

thickness, transverse beam, and yield stress of the longitudinal reinforcement,

Nie and Tao (2012) found that the main factors influencing the negative

effective flange width, included the column dimensions, the steel beam height,

the flange width of the transverse beam, and the yield stress of the longitudinal

reinforcement. The slab width was found to be less influential.

2.5 CONCLUDING SUMMARY

This chapter has presented a literature review covering the potential

problems existing in RC frames and corresponding retrofit methods,

experimental and FE analysis on RC beams weakened by opening, effective slab

width of RC frames. The following conclusions can be drawn:

(1) Existing RC frames with cast-in-place slabs might fail by the

Strong-Beam-Weak-Column (SBWC) mechanism, rather than

Strong-Column-Weak-Beam (SCWB) mechanism, as the contribution from

cast-in-place slabs are ignored or underestimated.

45

(2) Three retrofit techniques, namely the slab slit (SS) technique, the beam

opening (BO) technique and the beam section reduction (SR) technique are

proposed based on the concept of Beam-end Weakening in combination with

FRP Strengthening, proposed by Teng et al. (2013). SS technique has only

been studied through FE modelling and no experimental tests and therefore

results are available. The BO technique has recently been investigated by

Nie (2018) through experimental tests on RC T-beams. The SR technique

for RC frames is new and no relevant research is available.

(3) The existence of web opening can decrease a RC beam shear and flexural

strengths. External bonded FRP is an efficient strengthening technique for a

RC beam with web opening. The BO technique for RC T-beams can

efficiently decrease the T-beam negative moment strength, proved by the

experiments tested by Nie (2018). The ductility of T-beams is good if

additional FRP strengthening is provided.

(4) Nie (2018) proposed 2D FE models for RC beams weakened by web

openings. Negative strength of T-beams with a web opening was well

predicted with a relatively small error. The 2D model, however, generally

overestimated the stiffness of specimens, possibly because shear lag and the

potential out-of-plane deformation could not be captured by the 2D models.

A more accurate 3D FE model for RC T-beams with a web opening is

needed.

46

(5) No uniform method to determine the effective width of a slab under tension

exists. Most research on effective slab width of RC frames is experimentally

based. Only Ning et al. (2016) conducted additional parametric studies

through FE modelling to determine the key factors affecting the effective

slab width of RC frames. Some factors, however, such as beam length and

slab width, were not considered by Ning et al. (2016) in their FE analysis. A

parametric study on the effective slab width of RC frames covering a wider

range of factors should be conducted.

2.6 REFERENCES












47

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48


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retrofit of RC frames through beam-end weakening in conjunction with

54

FRP strengthening”, Proceedings of the 4th Asia-Pacific Conference on

FRP in Structures (APFIS 2013), 1-8.







Vibration, 32(1), 121-127 (in Chinese).



129(6), 725-732.

Yan, Y.L. (2010). A Study on Eurocodes about the RC Elements Design and

Comparison between Eurocodes and Chinese Codes, Master degree thesis:

China Academy of Building Research, China. (in Chinese).

Ye, L.P., Qu, Z., Ma, Q.L., Lin, X.C, Lu, X.Z. and Pan, P. (2008). “Study on

ensuring the strong column-weak beam mechanism for RC frames based on

the damage analysis in the Wenchuan earthquake”, Building Structure,

38(11), 52-67 (in Chinese).

55






thesis: Hunan University, China (in Chinese).

Zhang, Y.P., Hao, Z.J., Shan, M.Y. and Ge, N. (2011). “Research on anti-seismic

performance for reinforced concrete frame joint with slot around”, Building

Science, 27(9), 7-11 (in Chinese).

56

Table 2.1 Numerical studies on the effect of cast-in-place floor slabs on the

behaviour of RC frames

Source Analysis type and FE models Software Guan and Du

(2005) Pushover analysis of a 3-storey-3-span RC frame.

SAP2000 (1998)

Lin et al. (2009

Elastic-plastic time history analysis of a 6-storey RC frame structure. Two 3D models for a plane frame and a frame with cast-in-place slabs was compared.

MSC.Marc (2005)

Gao and Ma (2009) Pushover analyses of two RC frames, with and without respective cast-in-place floor slabs.

SAP2000 (1998)

Yang (2010) Pushover analyses of five 6-storey 4x4-span RC frames with different slab widths.

SAP2000 (1998)

Chen (2010)

Elastic-plastic time history analyses of two 6-storey 6x3-span RC frames (one with floor slabs and one without floor slabs).

SAP2000 (1998)

Guo (2012)

Pushover analyses of established three FE models, one with cast-in-place slabs and slab reinforcement, one with cast-in-place slabs but without slab reinforcement and one without slabs.

SAP2000 (1998)

57

Table 2.2 Summary of experimental studies on RC beams with a web opening

Source

Beam dimensions Web opening

size Strengthening

method

Observed failure mode

Span/clear span Width Height Length×height

(mm×mm) Without strengthening With strengthening

(mm) (mm) (mm)

Nasser et al.

(1967) 3962.4/3657.6 228.6 457.2

1219.2× 190.5

Internal stirrups

and longitudinal

bars

No unstrengthened specimen

B-1:corner cracking

1219.2×203.2 B-2:shear failure away from the opening

Two 762×203.2 C-1:corner cracking; C2 and C3: flexural

failure away from the opening

762×203.2 D-1: diagonal tension in chords; D2-D4:

flexural failure away from the opening

Mansur et al.

(1985) 3300/3000 200 400

400 × 180

Internal stirrups

or combined use

of stirrups and

diagonal

reinforcement

No unstrengthened specimen Chord ends concrete crushing

600 × 180

800 × 180

1000 × 180

1200 × 180

800 × 140

800 × 220

58

Table 2.2 (cont.)

Tan et al. (1996) 3300/3000 200 500

Two 400 × 200

Internal stirrups

and longitudinal

bars

No unstrengthened specimen

Flexural failure failure away from the

opening

Two 600 × 200

IT2, IT7 :flexural failure failure away from the opening; IT5: local flexural failure at the chord ends; IT6: flexural failure in the

opening region

Two 800 × 200 Local flexural failure at the chord ends

Two 1000 ×

200 Local flexural failure at the chord ends

Four 400 × 200 Flexural failure away from the opening

600 × 120 Flexural failure away from the opening

600 × 200 TI,T3:Flexural failure failure away from the opening; T2: flexural failure in the

opening region

Mansur et al.

(1999) 2900/2600 200(c) 500

r=100 NA

Shear failure in the opening region Flexural failure away from the opening r=150 Bonded FRP

plates

r=200 NA

59

Table 2.2 (cont.)

Abdalla et al.

(2003) -/2000 100 250

100 × 100

Bonded FRP

sheets and wraps Shear failure in the opening region

Flexural failure away from the opening

200 × 100 Shear failure in the opening region



Allam (2005) 3200/3000 150 400 450 × 150

Internal steel

reinforcement

Shear failure in the opening region


External Steel

plates: outside Shear failure in the opening region

External Steel

plates: outside

and inside

Flexural failure away from the opening

Bonded FRP

sheets and

U-jackets


60

Table 2.2 (cont.)

Maaddawy and

Sherif (2009) 1200/1000 80 500

150 × 150 NA Two independent diagonal cracks in

the chords

No strengthened specimen

200 × 200 Bonded FRP

sheets and wraps

Two independent diagonal cracks in the

chords; 250 × 250 Relative rotation of three distinct

segments

Pimanmas

(2010) -/2100 400 160

r=150 Near-surface

mounted FRP

rods

Shear failure in the opening region C-rob2: Flexural failure away from the opening; The rest: shear failure in the

opening region 150 × 150

Chin et al. (2012) 2000/1800 300 120 210 × 210 Bonded FRP

sheets and wraps Shear failure in the opening region Shear failure in the opening region

210 × 210

61

Table 2.2 (cont.)

Maaddawy and

Ariss (2012) 2600/2400 85 400

200 × 200

Bonded FRP

sheets and

U-jackets


Shear failure in opening region after

debonding and rupture of FRP

350 × 200

500 × 120


500 × 160

500 × 200

Jing et al. (2013) 3300/3000 250 500 500 × 150 Bonded FRP

plate, U-jackets

and wraps

Shear failure in the opening region Shear failure in the opening region

500 × 220

Suresh and

Prabhavathy

(2015)

2200/2000 150 300

150 × 150

Steel fibers and

steel plates Shear failure in the opening region Shear failure in the opening region

200 × 150

250 × 150

300 × 150

62

Table 2.2 (cont.)

Chin et al. (2016) 2000/1800 120 300 800 × 140 Bonded FRP

sheets Shear failure in the opening region

Shear failure in the opening region after

debonding of FRP

Nie (2018) 3300/3000 250 500

700 × 300

Bonded FRP

plate, U-jackets

and wraps

Mixed local flexural and shear

failure at the chord ends Local flexural failure at the chord ends

800 × 280 Mixed local flexural and shear

failure at the chord ends Local flexural failure at the chord ends

600 × 220 Web chord shear failure Local flexural failure at the chord ends

700 × 200 Web chord shear failure Local flexural failure at the chord ends

600 × 280 No unstrengthened specimen Local flexural failure at the chord ends

700 × 260 No unstrengthened specimen Local flexural failure at the chord ends

63

Table 2.3 Summary of FE studies on RC beam with a web opening

Source Model

dimension

Flange width of one side ( mm)

Modelling of bond behaviour

Steel-to-concrete FRP-to-concrete

Madkour ( 2009 ) 3D NA Not mentioned Not mentioned Pimanmas (2010) 2D NA Perfect bond Perfect bond

Chin et al. (2012) 2D NA Perfect bond Bond-slip model

developed by Lu et al. (2005)

Hawileh et al. (2012)

3D NA Perfect bond Bond-slip relationship proposed by Xu and Needleman (1994)

Nie (2018) 2D 600 CEB-FIP (1993) Bond-slip relationship proposed by Lu et al.

(2005)

64

Table 2.4 Effective slab width definitions in negative moment of interior and exterior joints in different codes

Code Effective slab width of interior joints Effective slab width of exterior joints

GB 50010-2010 { , }eff o w n wb = min l / 3,b +s b +12t { , }eff o w n wb = min l / 3,b +s b +12t

ACI318-05 & FEMA-356 { 2 16 , }eff w o nb = b min l / , t s { 6 12 , }eff w o nb = b min l / , t s

EC8 8eff cb = b t 4eff cb = b t

NZS-3101:2006 1 21 2

1 2 2 3

{6 ,2 2 ,

}

eff w b Trans beam f NTB

b bn n

b b b b

b = b min h w bh hs s

h h h h

1 21 2

1 2 2 3

{6 ,2 , }b beff w b f NTB n n

b b b b

h hb = b min h b s sh h h h

Note: wb is the beam web width, ol is the effective span of beam, ns is the clear spacing between beams, t is the slab thickness, cb is the column width,

bh is the beam height, Trans beamw is web width of transverse beam, f NTBb is the distance at the critical section of the potential plastic region in the beam

between the web and a line drawn at 45⁰ from the intersection of a line drawn parallel to the web and touching the side of the column and the edge of the slab

65

Table 2.5 Suggested values of effective slab width by previous researchers

Time Researchers Value of beff Applicable conditions 1982 Ehsani and Wight 7bw Interior joints

1988 Pantazopotrlou et

al. bw+ 5hb

Interior joints under large deformation

1990 Durrani and Zerbe bw + 4 hb bw + 2 hb

Interior joints

1/ 33

Exterior joints

1/ 33

1991 French min{l0/4, bw + 16t, sn} Interior joints

1/ 50

1994 Li bw + 8t Interior joints 1994 Jiang et al. bw + 12t Interior joints

2002 Wu et al. bw + 12t Interior joints

1/ 50

2009 Wang et al. bw + 2t Interior joints

1/ 50

2009 Zhen et al.

min{ bw + 4hb, 0.4l0, sn}

Interior joints

1/ 50

min{ bw + 2 hb, 0.2 l0, sn}

Exterior joints

1/ 50

2010 Yang bw + (12-16)t Interior joints

2010 Sun

bw + min{max (l0/4, 2 hb), 1/2 sn }

Interior joints

1/ 50

bw + min{max (l0/5, 1.5 hb), 1/2 sn }

Exterior joints

1/ 50

66

Table 2.5 (cont.)

2010 Qi et al.

bw + min{ l0/4, 12t, sn } Interior joints

bw + min{l0/5, 8t, sn} Exterior joints

2010 He bw + 12t Interior joints

2016 Ning et al.

bw+ 6.4 hb Interior joints

bw + 5.4 hb Exterior joints

Note: beff: effective slab width; bw: beam width; hb: beam height; l0: effective span of

beam; t: slab thickness; sn: clear distance of beams.

1/ 50

1/ 50

1/ 50

1/ 50

1/ 50

67

CHAPTER 3

EXPERIMENTAL PREPARATION AND SPECIMEN

DETAILS

3.1 INTRODUCTION

The two groups of specimens shown in Table 3.1 were tested in order to

verify the effect of the proposed weakening techniques when realizing the

SCWB hierarchy. All the reinforced concrete (RC) joints of the two groups were

designed according to an earlier version of the Chinese code (GB-50011 2001)

to reflect the design philosophy of the current RC joint. The key design

principles are summarized below:

(1) For the control 3D beam-column joints, the flexural strength ratios of

column to R-beam (i.e. left( ) / ( )top bottom rightc c r b r bM M M M ) were more than 1.2

and those for the columns to T-section beams (T-beams) (i.e.

left( ) / ( )top bottom rightc c t b t bM M M M ) were less than 1.0.

(2) For retrofitted seismic resisting joints, the flexural strength ratios of columns

to T-beams (i.e. left/ top bottom rightc c t b t bM M M M ) were higher than 1.0.

(3) Flexural failure precedes failure due to shear.

68

(4) Joint shear failures should not occur before a beam or column failure.

(5) The column axial compression ratio was set to be 0.2.

(6) Openings or gaps are located at regions critical in flexure.

Except for the beam-column joint control specimens, the specimens of

each test group were weakened by one of web openings, transverse grooves on

the beam soffit, slits at the slab corner regions, or a combination of the above.

To avoid potential non-ductile shear failure due to the removal of concrete when

creating openings or transverse grooves, those weakened beams were

strengthened in shear using FRP U-jackets or combined FRP U-jackets and FRP

wrap.

In addition to the differences in the size as well as position and shape of

openings, grooves and slits, there are some other differences between the two

groups of specimens: (a) the flexural capacity of longitudinal beams of the

second test group was increased; (b) the joint region was provided with more

transverse reinforcement; (c) the loading protocols of the two series were

different. The second test group was designed based on the test results of the

first one to better fulfill the test objectives.

3.2 SPECIMEN DESIGN DETAILS

The second-degree earthquake-resisting rankings specified in the Chinese

69

code (GB-50011 2002) were adopted for the design and the corresponding

design strengths for both concrete and reinforcement. When naming these

specimens, the first letter of the labels, namely F and S, stands for first and

second group respectively. The letters O, G, S in the middle of the labels stand

for opening, groove, and slit, respectively. The number after these letters strands

for the dimensions of the opening, groove and slit. For the first test group,

16-mm-diameter deformed steel bars of 360 MPa design yield stress were used

for longitudinal bars in beams and columns. 8-mm-diameter plain steel bars of

300 MPa design yield stress were used for stirrups in beams, columns and slabs.

Joint regions were reinforced with 10-mm-diameter plain steel bars of 300 MPa

design yield stress. The elastic modulus of steel, sE was set to be 200kN/mm2.

C30 concrete with a target cube strength of 30.0 MPa, was adopted for the test

specimens. The beam shear span was 1400 mm and the inflection point of

column to the joint region was 1000 mm. The design details for the first test

group are listed in Table 3.1 and presented in Figure 3.1. Reinforcing and

geometrical details for all specimens are identical except for the stirrups and

slab longitudinal bars at the openings or cut slit positions. Eight column

longitudinal bars were uniformly distributed on each side of the columns. Four

16-mm-diameter bars were located in the tops of beams, close to the slab flange

and three were located in the lower part of the beams. A total of 16

8-mm-diameter bars were used to reinforce the slabs. Beams and column

stirrups were distributed at 80-mm centre-to-centre spacing along the length of

the beams to ensure that shear capacity of the beams is much higher than the

70

shear force corresponding to the moment capacity.

In the first test group, the specimen F-O-450-150 had two 450 mm long

and 150 mm high openings in the beam web adjacent to the slab bottom surface

and the beam-column connection. To avoid potential shear failure due to the

removal of concrete and stirrups, FRP U-jackets and FRP wraps were both used

to strengthen the weakened region. FRP fibre anchors were carefully designed

to guarantee that the U-jackets fail by FRP rupture.

Specimen F-G-50-200 had two 50 mm long, 200 mm deep grooves

adjacent to the beam-column connection on the beam soffits. To control crack

growth, FRP U-jackets were used to strengthen the regions around gaps. FRP

anchors were also used.

Specimen F-S-450-450 contained four 450-mm-long longitudinal slits and

four 450-mm-long transverse slits in the corner region to separate the slab from

the major and minor beams. Slab bars which crossed the slit locations were cut.

Specimen F-O-500-180 was weakened by including two 500 mm long and

180 mm high openings which were similarly strengthened against shear using

FRP U-jackets as in specimen F-O-450-150. Unlike the other weakened

specimens of this group, the openings in this specimen were formed by drilling

after concrete casting. The dimensions of these openings were determined after

the testing of other specimens in the first test group.

71

The actual materials properties, especially of concrete, of the first test

group were found to be higher than the expect values of concrete of 30 MPa

cylinder strength. The second test group was re-designed using 40 MPa cylinder

strength concrete. The beams in the second group were reinforced with six

20-mm-diameter deformed steel bars (instead of seven 16-mm-diameter steel

bars) with a 360 MPa design yield stress in the longitudinal direction. This

modification increased the beam strength to better realize column end failure of

the control specimen, which did not happen in the control specimen of the first

test group. To avoid premature joint failure in shear as observed when testing

specimen F-Control, the joint regions of the specimens in the second test group

were reinforced with 2 more 10-mm-diameter transverse reinforcing bars. The

design details of this second test group are given in Table 3.1.

Specimen S-G-50-200-100 contained grooves identical in size with those

of specimen F-G-50-200, but the grooves were located 100 mm distant from the

beam-column interface to avoid slippage of the beam bottom longitudinal bars

in the joint region. Specimen S-O-500-180 had slot-shaped openings, which

were different in shape from that of the first test group, as shown in Figure 3.2.

The creating of slot-shaped opening aims to weaken the T-beam like rectangular

one but with less concrete to be removed. Specimen S-O-500-180-S-300-300

was weakened with a combination of 500×180 mm2 web openings and 300

mm long slab slits.

72

3.3 SPECIMEN CONSTRUCTION

Because of the limited laboratory space in the HK Polytechnic University,

all specimens were constructed in the structural laboratory at the Guangdong

University of Technology. Specimen construction progress is illustrated in

Figure 3.3. The specimen formwork was constructed first. Due to the existence

of slab, the formwork was constructed vertically, in two phases. In the first

phase, formwork for the beam, slab, and bottom column parts was built. The

formwork for the second phase, for the top column frames, was erected after the

reinforcement cages had been placed in the lower formwork mould. The gaps in

the formwork, where wooden plates abutted were sealed with waterproof glue to

prevent mortar/water leakage during specimen casting. The formwork was

strong enough to avoid any obvious deformation during and after concrete

casting. To create openings, transverse grooves or slab slits, wooden boxes or

plates of the required size were placed at the appropriate locations in the

formwork prior to casting. Finished beam and column reinforcement cages were

placed in the formwork first. The placing of slab reinforcement followed.

Strain gauges were bonded to reinforcing bars and stirrups at critical

positions. To avoid damage during concrete casting, the strain gauges were

covered with waterproof and quakeproof glue. Each strain gauge wire was given

a unique label name to indicate the locations of the strain gauges

Mortar blocks with the required thickness were applied to guarantee the

73

thicknesses of concrete cover at different locations, as indicated in Table 3.2.

The specimens were cast when all the above processes had been completed.

C30 (the actual cylinder strength were about 40 MPa, which was not as

designed) and C40 commercial concrete with at least a 120-mm concrete slump

were used for the first and second batch specimen respectively because of the

high density of the steel reinforcements. All specimens were cast simultaneously

to guarantee concrete uniformity. At least 1 concrete cube and 3 cylinders were

prepared for each specimen.

After concrete casting, the specimens were thoroughly wetted three times

each day and the specimen surfaces covered with plastic sheeting to slow down

water evaporation. Two weeks after casting, the formwork was dismantled and

the curing of the whole specimens continued.

The smaller beam chord under the opening (referred to as the web chord

hereafter) was strengthened by FRP wrapping. The FRP splicing length was

greater than 150 mm to avoid failure of the splicing. As FRP U-jackets are

vulnerable to FRP debonding at the free end which lowers the strengthening

efficiency of the U-jackets, FRP fibre anchors were used to prevent that

debonding. The key external bonding processes of CFRP sheets to substrate

concrete and of the installation of FRP anchors in specimens are illustrated in

Figure 3.4. The radius of the rounded beam corners was 25 mm to allow for a

more effective FRP confinement. The surfaces of strengthened regions were

74

polished to remove external mortar before the strengthening. The FRP anchors

were prepared first. The drilled holes for FRP anchors were firstly filled with

Sikadur 330 and the straight part of FRP anchors, which was already saturated

with Sikadur 300, was then placed into the hole. After the anchors were installed,

CFRP sheets saturated with Sikadur 300 were bonded to the prepared beam

surfaces following a wet lay-up process, also saturated with Sikadur 330. Some

details of the FRP shear strengthening scheme are shown in Figure 3.5. The

anchor fan had a fan angle of approximately 36°. After saturation with Sikadur

300, the anchor was pressed/bonded to the surface of the corresponding FRP

sheet to anchor the FRP sheet.

The opening of specimen F-O-500-180 and the weakened specimens of the

second test group were created at least 28 days after concrete casting. The

procedure for drilling an opening in an existing RC structure is illustrated in

Figure 3.6. An opening was first created and then polished. The corners of the

critical beams were rounded and polished before application of the externally

bonded FRP.

3.4 MATERIAL PROPERTIES

The average concrete cylinder strength 'cf and corresponding strain o

were obtained by testing three concrete cylinders. According to GB/T

50081-2002, for C30 concrete, the loading rate is suggested to be in the range

0.3-0.5 MPa/s. When C30-C60 concrete is used, the loading rate is increased to

75

0.5-0.8 MPa/s. If the concrete grade is higher than C60, an 0.8-1.0 MPa/s

loading rate should be used. The cylinders were capped with plaster of 80 MPa

compressive strength before testing. The stress-strain curves for concrete

cylinders under uni-axial compression of the two groups are shown in Figure

3.8. The summarized data of the first and second test group are presented in

Table 3.3.

For each type of steel bar, at least 3 coupons were tested to obtain an

average yield stress yf and a corresponding yield strain y , an ultimate stress

uf and a corresponding ultimate strain u . Both strain gauges and the particle

image velocimetry (PIV) method were used to obtain steel deformations. The

deformation measured by strain gauges was used for the calculation of the

elastic modulus and the PIV method was used to obtain the hardening range of

the stress-strain curve, as most strain gauges fail at a strain of about 0.02.

However, it is of interest to note that the PIV method works beyond a strain of

0.2 and well into the large deformation range. For steel bars with an obvious

yield plateau, the initial elastic part of stress-strain curve was applied to

calculate elastic modulus. For those steel bars with no yield plateau, a “0.2%

off-set” criterion was adopted as the yielding point. The steel properties of the

two test groups are tabulated in Tables 3.4. The stress-strain curves obtained by

the PIV method are shown in Figures 3.8 for the two groups of specimens. Thus

it appears certain that the PIV method performs well in capturing the

stress-strain curves of steel bars over the complete range.

76

Seven coupons of CFRP sheet were tested to determine the material

properties according to ASTM-3039 (2008). The width and length of the coupon

test region were 25 mm and 250 mm respectively. The nominal thickness of

CFRP sheet provided by the manufacturer (i.e., 0.334 mm per ply) was used to

calculate the tensile strength and elastic modulus. The obtained tensile strength

and elastic modulus were 2820 MPa and 227 GPa respectively.

3.5 LOADING PROTOCOLS

Combined vertical axial load and lateral cyclic load were applied at the top

of the upper columns. The axial compression ratio was set to be 0.2. This value

fluctuated to a small extent during loading, due to the limitations of the test

equipment.

The cyclic load was quasi-statically applied at the top of the upper column

by an MTS machine. The displacement history applied was defined by the

storey drift ratio (ratio of the horizontal displacement to the story height, which

is 2400 mm) rather than the yield deformation as it is difficult to predict the

yield deformation of a T-beam, especially when it is weakened by an opening, a

transverse groove or slab slits. The loading protocol as illustrated in Figure 3.9(a)

is defined according to that used in Canbolat and Wight (2008). For each

inelastic cycle, two reverse cycles were applied in both the push and pull

directions as many researchers did (e.g. Shin and LaFave 2004, Canbolat and

Wight 2008, Park and Mosalam 2012). Between some inelastic cycles, a

77

reversed cycle of 0.5% storey drift ratio was inserted to quantify the stiffness

degradation (Canbolat and Wight 2008). Besides, in many seismic scenarios,

small cycles precede the main shock which can be represented by the inserted

low-level cycles. (ACI T1.1-01). The loading speed was slow to avoid

unnecessary additional inertia forces and to keep the axial load constant. The

loading sequence and protocol are depicted in Figure 3.9(a) and the applied

horizontal displacement magnitudes, durations and speeds are tabulated in Table

3.5. In the second test group, the first loading cycle was decreased to a 0.25%

storey drift ratio to better estimate specimen initial stiffness. There were no

inserted reversed cycles of 0.5% storey drift ratio as the inelastic cycles could be

used to assess stiffness degradation. Besides, the loading speed was very slow

and the loading was stopped at the peak displacement of each loading cycle for

recording. The test results of the first group showed that the inserted small

cycles did not have any significant effect, so they were not used in the second

group of tests. The loading protocol of the second test group is presented in

Figure 3.9(b) and tabulated in Table 3.5.

3.6 TEST SET-UP

As shown in Figure 3.10, a set of experimental devices had been designed

to provide the required boundary conditions. The hinges were used to represent

the inflection points of beams and columns. The beam ends were connected to

the supports using holes through the beam web. The foot of the lower column

was connected to a supporting hinge and the top of upper column was connected

78

to the MTS head.

The horizontal cyclic and axial loads were applied by a 50-ton MTS

actuator and a 320-ton hydraulic jack respectively. The jack was pulled and

pushed horizontally by the MTS and slid with the help of a sliding plate with a

maximum sliding displacement of 150 mm. Load cells measured the axial load

and the beam-end reaction forces.

To measure the external deformation of beams and columns, screw rods

were used to locate linear variable differential transformers (LVDTs). For the

first test group, the rod screws were placed at pre-defined locations of formwork

before concrete casting. However, some pre-located screw rods were damaged

during casting concrete. Thus, for the second test group, the rod screws rods

were placed and fixed at the designed positions through holes drilled after

concrete casting.

3.7 INSTRUMENTATION

During testing, external and internal instrumentation procedures were used

to monitor the following parameters: (1) beam-end reaction forces,

top-of-column axial and horizontal loads; (2) the deformations of beams,

columns and joints; (3) strains in the steel reinforcements and strengthening FRP.

Load cells, LVDTs and strain gauges were used to measure the load,

displacement and strain respectively.

79

3.7.1 Beam end Reaction Forces, Column Top Axial and Lateral loads

A 50-ton MTS actuator and a 320-ton jack applied the cyclic lateral load

and top-of-column axial load respectively. The horizontal load was recorded by

the MTS control system. The load cell connected to the hydraulic jack was used

to monitor axial load. The beam-end reaction forces were measured by the two

load cells above the beam-end supports. All load cells were calibrated before

testing.

3.7.2 Beam, Column and Joint Deformations

Similar to Kam (2010), a beam was divided into the three regions of elastic

region, plastic hinge region and bond failure region. The deformations of each

region under loading were measured by LVDTs located on the beam surface.

The arrangement and labelling of the LVDTs are shown in Figure 3.12. The

lateral displacement drift , of a beam-column joint was decomposed into

components expressed in the following equation:

, , , intdrift drift beam drift column drift jo (3.1)

, , ,hinge+drift beam elastic beam beam fixe end beam (3.2)

, , ,hingedrift column elastic column col (3.3)

where ，elastic beam and ,elastic column are the top-of-column lateral displacement

80

components due to beam and column elastic deformations respectively.

beam fixe end is caused by a tensile bar bond failure at the beam-column

interface. ,hingebeam and ,hingecol are the components resulting from the

rotations of left and right beams, upper and lower columns plastic hinges

respectively, and int jo is caused by the joint shear deformation.

3.7.2.1 Beam deformation

The average curvature RBi of a beam ith region was determined using the

following equation:

. ,

= RBTi RBBiRBi

rb i rb ih b

(3.4)

where RBTi and RBTi are the total deformations measured by the top and

bottom LVDTs located on the respective ith region, .rb ih and ,rb ib are the

respective distances between the top and bottom LVDTs and the length of the

measured region, as shown in Figures 3.12.

For the specimens of the first test group, instrumented with 14 LVDTs, the

top-of-column lateral displacement components drift,beam , originating from the

beam drift beam , were calculated as follows:

81

2drift,beam beamb

HL

(3.5)

1 1,1

.1

1 2 1 2,2 ,1

.2

1 1,1 ,2 ,1

.1

( 0.5 )

( ) ( ) ( 0.5 )

( 0.5 )

RBT RBBbeam b rb

rb

RBT RBT RBB RBBb rb rb

rb

LBT LBBb lb rb rb

lb

L bh

L b bh

L b b bh

(3.6)

where H is the distance between the upper and lower column inflection points ,

bL is the length of the beams. The first part of Equation (3.6) is the beam drift

component resulting from bar slippage. The second part accounts for plastic

hinge deformation and the third part represents the deformation in the elastic

region.

Due to the limited number of LVDTs, only two LVDTs were used to

measure the plastic hinge deformation of the left hand beam. The sum of the

bond failure deformation and the plastic hinge deformation were measured by

only one pair of LVDTs.

According to Kam (2010), the lateral drift component originating from

deformation of the beam elastic region can be determined using the following

basic structural mechanics equation:

3

, ,

,

2( )3 0.2 2b b elastic b b elastic

elastic,beamc e b c w b b

V L qV L HE I E b h L

(3.7)

82

where bV is the shear force in the beam, measured by the load cell at the

beam-end support, ,b elasticL is the length of the elastic part of the beam, wb and

bh are beam web width and beam height respectively, ,c e bE I is the effective

flexural stiffness of the prismatic element taking into account the concrete

cracking effect. The effective moment of inertia, ,e bI , given by equation (3.6),

allows for the possible double counting of elastic deformations involved in the

measurement of average curvatures across the beam plastic hinge region; q is

the T-beam shape factor calculated using equation (3.7) according to mechanics

of materials theory:

, ,0.5e b g bI I (3.8)

2

,

( )bb

g b

A S yq dAI b

(3.9)

where ,g bI is the moment of inertia of the gross uncracked concrete area, Ab is

the total area of that concrete, S(y) is the area moment under the y point towards

the neutral axis and b is the section width (i.e. bw for the beam web and bf for

the beam flange).

3.7.2.2 Column deformation

Due to the limited number of LVDTs, only deformations in the column

hinge region were measured. The average curvature 1TC was obtained using

83

following equation:

1 11

tc.1 tc,1

= TCR TCLTC h b

(3.10)

where 1TCR and 1TCL are the average deformations measured by the right

hand and left hand LVDTs respectively, installed on the upper column, tc.1h

and tc,1b are the distances between the two respective LVDTs and the length of

the region measured , as shown in Figure 3.13 (a).

The lateral drift component drift,column caused by the upper column hinge

deformation ,hinge,col top was determined using the following equation:

1 1 1 1ct,1( 0.5 ) ( 0.5 )TCR TCL BCR BCL

col,hinge,top c c bt,1tc.1 bc.1

H b H bh h

(3.11)

where cH is the length between the column-beam interface and the point of

inflection.

Similar to beams, the elastic deformation of the upper column is calculated

using Equation 3.12:

3

c, c,

,c3 0.2c elastic c elastic

elastic,column,topc e c c c

V H qV HE I E b h

(3.12)

where cV , obtained from the MTS data, is the horizontal load acting on the end

84

of the upper column; c,elasticH is the length of the column’s elastic region, q

(=1.2) is the rectangular section shape factor calculated using Equation (3.10)

and cb and ch are column width and height respectively. The effective

moment of inertia, ,e cI , is given by Equation 3.13:

,c ,c0.7e gI I (3.13)

3.7.2.3 Joint shear distortion

If the joint deformation can be measured by two LVDTs based on the plane

strain cycle, then

1 cos(2 ) sin(2 )2 2 2

x y x y jj j

(3.14)

2 cos(2 ) sin(2 )2 2 2

x y x y jj j

(3.15)

( ) tan(2 )j x y j (3.16)

11

jL

(3.17)

22

jL

(3.18)

2 2j j jL h b (3.19)

85

where 1 and 2 are the strains measured by the two respective inclined

LVDTs; x and y are the strains in the joint width and height directions

respectively, j is the angle of inclination of LVDT to the joint width direction;

j is the joint shear distortion; jL is the length of joint core along the diagonal

line and jh and jb are joint height and width respectively .

From Equations 3.15-3.20, the joint distortion j can be given by the

following equation:

2 21 1

2j j jj j

h bh b

(3.20)

The ,hingecol can be determined using following equation:

, int ( )cdrift jo j c b c

b

HH h hL

(3.21)

Due to the existence of the floor slab, the joint shear distortion cannot be

directly measured. Therefore, the deformation of a joint was monitored using

the interior strain gauges mounted on the transverse stirrups at the joint.

3.7.3 Strain Gauges

Due to the limited number of data logger channels, strain gauges were

situated on reinforcing bars only at the critical beams, slabs and columns

86

positions, the joint transverse reinforcement, and the FRP. Strain gauges were

bonded on the exterior faces of the reinforcing bars. The arrangement and

labelling of the strain gauges on steel reinforcement are shown in Figures 3.13

to 3.21 and those on FRP reinforcement are in Figure 3.22. The meanings of the

labels are also illustrated in these figures.

3.7.3.1 Strain gauges on beam reinforcement

Those strain gauge with labels starting with TB (i.e. Beam Top) or BB (i.e.

Beam Bottom), were placed on reinforcing bars at the critical beam sections (i.e.

the section of maximum moment) and at weakened sections (i.e. those sections

weakened by the inclusion of an opening or gap). The purpose of the layout of

strain gauges chosen along the longitudinal direction was to measure the strain

distribution and the development of plastic hinges. The two specimens with

openings include more strain gauges along the longitudinal bars, covering the

complete lengths of openings so as to examine the effect of openings on strain

distribution.

In the first test group, the two strain gauges with labels starting with TS (i.e.

Transverse beam Stirrup), were symmetrically located on two stirrups of the two

transverse beams so as to monitor the symmetrical behavioural characteristics of

the specimens and the torsion in the transverse beams. The strain gauges were

not deployed in the second test group as they were found to be of little function.

3.7.3.2 Strain gauges on slab reinforcement

87

The strain gauges with labels starting with TL (i.e. Top Longitudinal) or BL

(i.e. Bottom Longitudinal), were positioned parallel to the strain gauges on the

main longitudinal beam bars in the transverse direction. In this way, the shear

lag effect and the contribution of slab bars were evaluated. Three strain gauges

with labels starting with TT (i.e. Transverse Top), among the first test group,

were located on the slab transverse bars, with the aim of examining the effects

of longitudinal slits on those bars. After testing the first group of specimens,

these strain gauges were found to be of little value as the strain levels reached in

the transverse bars were very low. Consequently strain gauges were not used in

the second test group.

In the second test group, more strain gauges, instead, were placed on the

longitudinal bars in the right hand slab of specimen S-O-500-180 to estimate the

effects of a large opening on the deformation of the slab. The strain gauges

installed on the slab reinforcement for specimen S-O-500-180-S-300-30, with

slab slits separating the slab from beams, were located identically to those of the

other three specimens in the same group.

3.7.3.3 Strain gauges on column reinforcement

The strain gauge labels on the reinforcements of the top and bottom

columns began with the letters TC (i.e. Top Column) and BC (i.e. Bottom

Column) respectively. The function of the strain gauges was to measure the

plastic hinge length and the strain levels in the steel bars in columns. More

88

strain gauges were used for the columns of the control specimens than for the

other specimens tested because the control ones had longer column plastic

hinges owing to the strong-beam-weak-column (SBWC) mechanism.

3.7.3.4 Strain gauges on joint transverse reinforcement

The strain gauges mounted on the joint stirrups were used to evaluate the

shear deformation of the joints, to assess the strain levels in the stirrups and

hence find whether a joint panel is sufficiently confined.

3.7.3.5 Strain gauges on FRP jackets

The strain gauges mounted on the FRP were located in areas of potentially

large tensile strains. According to an FEM prediction using ABAQUS, the FRP

U Jackets would experience maximum tensile strain in those areas of wide crack

propagation, whereas FRP wraps experience maximum tensile strain in areas

where concrete bulges under compression. The strain gauge labels had initial

letters U or W, to indicate FRP U-jackets or FRP Wraps. In the first test group,

strain gauges were only arranged on the FRP of the right beam. Thus, only U

and W were used to name these strain gauges. In the second test group, strain

gauges were pasted on both left and right beams and R and L were placed in

front of U and W to name these strain gauges.

3.7.3.6 The coordinate system

For the easy analysis of the test results, a Cartesion coordinate has to be

89

defined. The directions of the coordinates are shown in Figures 3.13 to 3.21.

The X-axis is in the longitudinal direction of the main beams and the Y-axis is

parallel to the transverse beams. The Z-axis follows the longitudinal direction of

the columns. The origin of the coordinate system is located at the geometrical

centre of the joint regions.

3.7.3.7 Sign of top-of-column lateral displacement

When the specimens experienced a “push” action, the lateral displacement

was in the positive X direction and most of the slab to the right hand side of the

column was, therefore, under compression. When the lateral displacement was

in the opposite direction, the slab to the right side of column was in tension.


This chapter has presented the details of the experimental program. Two

test groups were designed and prepared. The following conclusions can be

drawn:

(1) The specimens were designed according to an old version of the Chinese

seismic design code (GB-50011 2001 Code for Seismic Design of

Buildings). The expected failure mode for control specimens was column

end failure. In terms of those retrofitted by the three proposed techniques,

the expected failure model was beam end flexural failure.

90

(2) The control specimen of the second test group had stronger beams and joint

region than the control specimen F-Control to better realize column end

failure.

(3) The loading protocols were determined based on the storey drift ratio as the

yield displacement is hard to predict for specimens with a cast-in-place slab.

3.9 REFERENCES

ACI T1. 1–01. (2001). Acceptance Criteria for Moment Frames Based on

Structural Testing.

Canbolat, B. B., and Wight, J. K. (2008). "Experimental investigation on

seismic behavior of eccentric reinforced concrete beam-column-slab

connections", ACI structural journal, 105(2), 154.



Kam, W. Y. (2010). Selective weakening and post-tensioning for the seismic

retrofit of non-ductile RC frames, Doctoral degree thesis: The Hong Kong

Polytechnic University, University of Canterbury.

Park, S., and Mosalam, K. M. (2012). "Experimental investigation of nonductile

RC corner beam-column joints with floor slabs", Journal of Structural

Engineering, 139(1), 1-14.

91

Shin, M., and LaFave, J. M. (2004). "Seismic performance of reinforced

concrete eccentric beam-column connections with floor slabs", Structural

Journal, 101(3), 403-412.

92

(a) Elevation of specimen F-Control and F-S-450-450

(b) Elevation of specimen F-O-450-150

Openings

93

(c) Elevation of specimen F-O-500-180

(d) Elevation of specimen F-G-50-200

Grooves

Openings

94

(e) Elevation of specimen S-Control

(f) Elevation of specimen S-G-50-200-100

Grooves

95

(g) Elevation of specimens S-O-500-180 and S-O-500-180-S-300-300

96

(h) Plan view of specimens F-control, F-G-50-200, F-O-450-150 and F-O-500-180

97

(i) Plan view of specimen F-S-450-450

98

(j) Plan view of specimens S-Control, S-G-50-200-100 and S-O-500-180

99

(k) Plan view of specimen S-O-500-180-S-300-300

Figure 3.1 Specimen reinforcement details

100

(a) 50×200 mm2 groove, 100-mm away from beam-column interface of

specimen S-G-50-200-100

(a) Slot-shape opening of specimen S-O-500-180

Figure 3.2 The new groove position and opening shape

100 mm

180 mm

500 mm

101

(a) Specimens formworks (b) Steel cages

(c) Placing steel cages into formworks (d) Specimens ready for casting

(e) Casting specimens (f) Concrete cubes and cylinders

Figure 3.3 Specimens construction process

Wooden boxes creating openings

102

(a) Drilling holes for FRP anchors

(b) Applying primer-Sikadur 330 for anchors

103

(c) Installation of FRP anchors with straight part saturated by Sikadur 300

(d) Applying primer-Sikadur 330 for CFRP sheets

104

(e) Saturating CFRP sheet with Sikadur 300

(f) Attaching CFRP sheet to the specimen surface

(g) Saturating FRP anchor fan with Sikadur 300

105

(h) Pressing FRP anchor fan on the surface of CFRP sheet

(i) Curing CFRP U-jacket, CFRP wrap and FRP anchors

Figure 3.4 Installing FRP anchors and externally bonding CFRP sheets to

substrate concrete

106

CFRP Sheet

200

10090

80

(a) Details of FRP anchor

(b) Details of FRP shear strengthening scheme of specimen F-O-450-150

(c) Details of FRP shear strengthening scheme of specimen F-O-500-180

107

(d) Details of FRP shear strengthening scheme of specimen F-G-50-200

(e) Details of FRP shear strengthening scheme of specimens S-O-500-180

and S-O-500-180-S-300-300

108

(f) Details of FRP shear strengthening scheme of specimen S-G-50-200-100

(g) FRP anchor components

Figure 3.5 Details of FRP shear strengthening scheme (all units in mm)

109

(a) Creating opening

(b) Grinding the inner surfaces of opening and rounding the corners of

chord and beam

110

(c) Roughing surfaces around opening

(d) Opening ready for shear strengthening

Figure 3.6 Procedure of creating an opening in existing specimen

111

(a) 66-days-old concrete of the first test group

(b) 116-days-old concrete of the first test group

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000 2500 3000

Stre

ss (M

Pa)

Strain (με)

Cylinder-1

Cylinder-2

Cylinder-3

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000 2500 3000

Stre

ss (M

Pa)

Strain (με)

Cylinder-1

Cylinder-2

Cylinder-3

112

(c) Concrete of F-O-500-180

(d) 33-days-old concrete of the second test group

0

5

10

15

20

25

30

35

40

0 500 1000 1500 2000 2500 3000 3500 4000

Stre

ss (M

Pa)

Strain (με)

Cylinder-1

Cylinder-2

Cylinder-3

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000 2500 3000

Stre

ss (M

Pa)

Strain (με)

Cylinder-1

Cylinder-2

Cylinder-3

113

(e) 73-days-old concrete the second test group

(f) 108-days-old concrete the second test group

Figure 3.7 Concrete stress-strain curves under uni-axial compression test

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000 2500 3000 3500 4000

Stre

ss (M

Pa)

Strain (με)

Cylinder-1

Cylinder-2

Cylinder-3

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000 2500 3000 3500

Stre

ss (M

Pa)

Strain (με)

Cylinder-1

Cylinder-2

Cylinder-3

114

(a) D8 plain bars of the first test group

(b) D16 deformed bars of the first test group

0

50

100

150

200

250

300

350

400

450

500

0 50000 100000 150000 200000 250000 300000

Stre

ss (M

Pa)

Strain (με)

HPB335-D8-1

HPB335-D8-2

HPB335-D8-3

0

100

200

300

400

500

600

700

0 50000 100000 150000 200000 250000

Stre

ss (M

Pa)

Strain (με)

HRB400-D16-1

HRB400-D16-2

HRB400-D16-3

115

(c) D8 plain bars of the second test group

(d) D10 plain bars of the second test group

0

100

200

300

400

500

600

0 50000 100000 150000 200000 250000 300000

Stre

ss (M

Pa)

Strain (με)

HPB335-D8-1

HPB335-D8-2

HPB335-D8-3

0

100

200

300

400

500

600

0 50000 100000 150000 200000 250000 300000

Stre

ss (M

Pa)

Strain (με)

HPB335-D10-1

HPB335-D10-2

HPB335-D10-3

116

(e) D16 deformed bars of the second test group

(f) D20 deformed bars of the second test group

Figure 3.8 Stress-stain curves of steel bars obtained by PIV method

0

100

200

300

400

500

600

700

0 50000 100000 150000 200000 250000 300000

Stre

ss (M

Pa)

Strain (με)

HRB400-D16-1

HRB400-D16-2

HRB400-D16-3

0

100

200

300

400

500

600

700

0 50000 100000 150000 200000 250000 300000

Stre

ss (M

Pa)

Strain (με)

HRB400-D20-1

HRB400-D20-2

HRB400-D20-3

117

(a) The first test group

(b) The second test group

Figure 3.9 Loading protocol for two test groups

118

(a) Loading and measuring devices

(b) Reaction wall

Figure 3.10 Experimental setup

119

Figure 3.11 Deformation disassembling of specimens

120

(a) LVDTs arrangement for specimens F-Control, F-G-50-200, F-S-450-450

Labeling guide:

T: LVDT at the column Top

M: LVDT at the Middle

T/BCL/R: LVDT on the Top/Bottom

Column Left/ Right side

R/LBT/B: LVDT on the Right/Left

Beam Top/Bottom side

121

(b) LVDTs arrangement for specimens F-O-450-150 and F-O-500-180

122

(c) LVDTs arrangement for specimens of the second test group

Figure 3.12 LVDTs arrangement

123

(a) Strain gauges distribution from front view and sectional view

Labeling guide: TC: Top Column bars BC: Bottom Column bars TB: Top Beam bars BB: Bottom Beam bars CS: Connection Stirrups

Labeling guide: TL: Top Longitudinal bars on slab BL: Bottom Longitudinal bars on slab

X

Z

124

(b) Strain gauges distribution from top view

Labeling guide: TS: Transverse beam Stirrups TT: Top Transverse bars on slab

Push Direction

Pull Direction

X

Y

125

(c) Strain gauges distribution on different bars

Figure 3. 13 The labels and distribution of strain gauges on steel reinforcement of specimen F-Control (all units in mm)

126




X

Z

127



X

Y

Push Direction Pull Direction

128


Figure 3. 14 The labels and distribution of strain gauges on steel reinforcement of specimen F-G-50-200 (all units in mm)

129




X

Z

130



X

Y


131


Figure 3. 15 The labels and distribution of strain gauges on steel reinforcement of specimen F-S-450-450 (all units in mm)

132




X

Z

133



X

Y


134


Figure 3. 16 The labels and distribution of strain gauges on steel reinforcement of specimen F-O-450-150 (all units in mm)

135




X

Z

136



X

Y


137


Figure 3. 17 The labels and distribution of strain gauges on steel reinforcement of specimen F-O-500-180 (all units in mm)

138


Labeling guide: TC: Top Column bars BC: Bottom Column bars CC: Central Column bars TB: Top Beam bars BB: Bottom Beam bars CS: Connection Stirrups


X

Z

139


X

Y


140


Figure 3.18 The labels and distribution of strain gauges on steel reinforcement of specimen S-Control (all units in mm)

141



X

Z


142


X

Y


143


Figure 3.19 The labels and distribution of strain gauges on steel reinforcement of specimen S-G-50-200-100 (all units in mm)

144



X

Z


145


X

Y


146


Figure 3.20 The labels and distribution of strain gauges on steel reinforcement of specimen S-O-500-180 (all units in mm)

147



X

Z


148


X

Y


149


Figure 3.21 The labels and distribution of strain gauges on steel reinforcement of specimen S-O-500-180-S-300-300 (all units in mm)

150

(a) Strain gauges on the FRP of specimen F-O-450-150

(b) Strain gauges on the FRP of specimen F-G-50-200

(c) Strain gauges on the FRP of specimen F-O-500-180

151

(d) Strain gauges on the FRP of specimen S-G-50-200-100

(e) Strain gauges on the FRP of specimens S-O-500-180 and

S-O-500-180-S-300-300

Figure 3.22 Strain gauges on FRP (all units in mm)

152

Table 3.1 Specimens details of the two test groups

Test Unit

Beam width

× height

(mm×mm)

Column

width ×

height

(mm×mm)

Slab width

× height

(mm×mm)

Top

reinforceme

nt in beam

Bottom

reinforceme

nt in beam

Joint

transverse

reinforceme

nt

Top &

Bottom

reinforceme

nt in slab

Reinforcem

ent in

column

Stirrups Weakening

method

FRP

Strengtheni

ng method

First test

group

F-Control

400×200 320×320 650×100 4C16 3C16 B10@80 B8@150 8C16 B8@80

Control

specimen /

F-O-450-150 450 mm × 150

mm opening

FRP

U-jacket &

FRP wrap

F-G-50-200 50 mm × 200

mm groove

FRP

U-jacket

F-S-450-450 450-450 slits /

F-O-500-180 500 mm × 180

mm opening

FRP

U-jacket &

FRP wrap

153

Table 3.1 (cont.)

Second

test group

S-Control

400×200 320×320 650×100 3C20 3C20 B10@60 B8@150 8C16 B8@80

Control

specimen /

S-G-50-200-1

00

50 mm × 200

mm groove

100 mm away

from column

face

Two-layer

FRP

U-jacket

S-O-500-180

500 mm × 180

mm

slot-shaped

opening

FRP

U-jacket &

FRP wrap

S-O-500-180-

S-300-300

500 mm × 180

mm opening

and 300-300

slits

FRP

U-jacket &

FRP wrap

154

Table 3.2 Cover concrete thickness

Items

Longitudinal bars to each

sides of column

Longitudinal bars to

top/bottom side of beam


left/right side of beam


top/bottom side of slab

Cover concrete thickness (mm)

40 40 30 15

Table 3.3 Strengths of concrete cylinders of the two test groups

Test group

Days or Specimen

Items Cylinder-1 Cylinder-2 Cylinder-3 Average

First 66

Strength (MPa)

39.3 38.2 40.5 39.3 166 40.3 41.2 41.5 41

F-O-500-180 39.3 38.2 40.5 39.3

Second 33 36.9 40.9 35.2 37.7 73 38.6 40.5 39 39.3 108 40.9 38.2 40.3 39.8

155

Table 3.4 Steel reinforcement properties of the two test groups

Test group

Type Number Weight

(g) Length (mm)

Density (kg/m3)

Area (mm2)

Original Gauge length

Lo (mm)

Ultimate Gauge length

Lu (mm)

Elongation after

Fracture

Yield Stress (MPa)

Average (MPa)

E (MPa) by Strain

Gages

Average (MPa)

First

D8 1 216.5 555 0.00785 49.69 40 51 27.50% 348

340 2.03E+05

2.00E+05 2 217.6 554 0.00785 50.04 40 50 25.00% 334 2.08E+05 3 217.1 552 0.00785 50.1 40 48 20.00% 338 1.88E+05

D10 1 298.2 482 0.00785 78.81 50 65 30.00% /

364.5 /

2.11E+05 2 308.9 499 0.00785 78.86 50 65.5 31.00% 369 2.10E+05 3 311 501 0.00785 79.08 50 65 30.00% 360 2.12E+05

D16 1 851.4 554 0.00785 195.77 80 93 16.25% 491

483.3 2.07E+05

2.00E+05 2 846.5 552 0.00785 195.35 80 97.5 21.88% 483 1.96E+05 3 847 556 0.00785 194.06 80 94.8 18.50% 476 1.97E+05

156

Table 3.4 (cont.)

Second

D8 1 163 403 0.00785 51.52 40 50.7 26.75% 352.3

348.55 2.04E+05

2.04E+05 2 168 402 0.00785 53.24 40 50.7 26.75% 353.8 2.04E+05 3 162 400 0.00785 51.59 40 50.6 26.50% 344.8 2.03E+05

D10 1 254 403 0.00785 80.29 50 65.8 31.60% 354.5

353.5 2.11E+05

2.10E+05 2 252 400 0.00785 80.25 50 64.3 28.60% 350.7 2.08E+05 3 257 401 0.00785 81.64 50 / / 352.5 2.07E+05

D16 1 633 405 0.00785 199.1 80 / / 470.6

477.7 2.00E+05

1.99E+05 2 615 400 0.00785 195.86 80 100.7 25.88% 435.5 1.99E+05 3 619 402 0.00785 196.15 80 101.3 26.63% 484.8 1.98E+05

D20 1 970 401 0.00785 308.15 100 / / 460.3

453.6 1.96E+05

1.96E+05 2 979 405 0.00785 307.93 100 / / 441.2 1.96E+05 3 981 406 0.00785 307.8 100 / / 446.9 1.96E+05

157

Table 3.5 Loading schemes of the two test groups

`

Drift ratio

Direction 0.25% 0.50% 1% 1.50% 2% 2.50% 3% 4% 5%

Push(mm) 6 12 24 36 48 60 72 96 120

Pull (mm)

First

Cyclic times

/ 1 2 2 2 2 2 2 2

Speed (mm/s)

0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4

Second

Cyclic times

1 1 2 2 2 2 2 2 2

Speed (mm/s)

0.1 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4

158

159

CHAPTER 4

EXPERIMENTAL STUDY OF THE SEISMIC

PERFORMANCE OF RETROFITTED RC

BEAM-COLUMN JOINTS

4.1 INTRODUCTION

The experimental results for the two test groups introduced in Chapter 3

will be presented in this Chapter. The general test observations including the

failure process and failure mode, hysteretic performance and envelope curves,

and interior deformation of the test specimens are presented. Also included for

discussion to further illustrate the effect of the proposed retrofit techniques on

the seismic performance of the specimens are ductility properties, energy

dissipation characteristics, equivalent viscous damping, stiffness degradation,

plastic hinge lengths and deformation components. Conclusions are then drawn

and presented.

4.2 FAILURE PROCESS AND FAILURE MODE

The two test group were to investigate the effect of the three proposed

retrofit techniques, the slab slit (SS) technique, the beam opening (BO)

technique and the beam section reduction (SR) technique (e.g., creating a deep

transverse groove (TG)), on the performance of RC beam-column-slab joints

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(RC 3D joints) subjected to a combination of horizontal cyclic and axial loads.

The failure process/failure modes (including crack propagation process, steel

yielding) are presented below. Among the five specimens of the first group, the

control specimen unexpectedly failed due to shear failure near the supports at

the beam ends. Thus, the beam ends of remaining specimens in the first test

group were all strengthened, in this regard, with FRP sheets, to prevent this

unwanted failure mode from interfering with the purposes of the study. The

failed control specimen beam ends were strengthened by recasting the beam

ends with fresh concrete and strengthened by FRP sheets as shown in Figure 4.1.

The strengthened control specimen was retested until the lateral storey drift ratio

reached 5%. In the first test group, the BO technique was investigated by testing

specimens F-O-450-150 and F-O-500-180, the TG and SS techniques by

specimens F-G-50-200 and F-S-450-450 respectively.

The specimens of the second test group were designed based on the

experimental results/observations of the first test group. As the joint region of

the specimens in the first test group was not strong enough to ensure that the

failure of the beams and/or columns occurs first, the joint region of the second

test group was provided with 50% more transverse stirrups. The

moment-capacity of T-beams was also enhanced by increasing the area of

longitudinal bars. With these enhancements, the design principles mentioned in

the former chapter can be better followed.

In the second test group, except for one control specimen, the other three

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were retrofitted using one of the proposed techniques or a combination. For

specimen S-O-500-180, a new slot-shaped opening was tried. The slot-shaped

opening was 500 mm in length and 180 mm in height. The obvious advantage of

the slot-shaped opening over the rectangular is its smaller open area. For

specimen S-G-50-200-100, 50x200mm transverse grooves were placed 100 mm

distant from the beam-column interface to avoid beam bottom bars slipping

failure. For the third retrofitted specimen S-O-500-180-S-300-300, a

combination of the WB and SS techniques was applied for retrofitting purposes.

The experimental results are given below.

As mentioned in the previous chapter, during the testing, an axial load was

first applied at the end of the upper column first. When the axial load ratio

reached 0.2, the horizontal load was applied. Failure process and crack

behaviour evolution with the increase of displacement, are summarized in Table

4.1. Selected photographs taken during the testing are presented in Figures 4.2

to 4.10 to better indicate the cracking behaviour and failure mode as well. It

should be noted that the blue and red lines on the specimens identify the cracks

which arose when the specimen had undergone particular numbers of loading

cycles.

4.2.1 Cracking of Beams, Columns, Slabs and Joint Panel in the First Stage

Loading

Drift ratios when cracks first appeared on the beams, columns and joint

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panel of specimens are tabulated in Table 4.1. The crack distributions of

different stages are shown in Figures 4.2 to 4.10. Except for specimen F-Control,

whose flexural cracks initiating were first observed at the bottom of the beam

when the axial compression ratio reached 0.2, due to axial deformation of the

column, specimens of the first test group got beam flexural cracks during the

first cycle of 0.5% drift ratio. Figure 4.2(a) shows the cracks on beams of

specimen F-Control at 0.5% drift ratio. Shear cracks on the joint panel and

flexural cracks on the bottom column of specimen F-Control happened at 1.0%

drift ratio, as shown in Figure 4.2(b). For specimen F-G-50-200, by the end of

the cycle of 0.5% drift ratio, many flexural cracks had developed at the bottom

of the beam, with fewer at the top of slab, as shown in Figure 4.3(a). A crack

also showed on the bottom. At 1.0% drift ratio, cracks appeared on the joint

region of specimen F-G-50-200 as shown in Figure 4.3(b). As shown in Figure

4.4(a), flexural cracks arose on the slab and beams around the corners of the

beam openings of specimen F-O-450-150 at 0.5% drift ratio. Flexural cracks

emerged in the bottom column but fewer in the top column at 1.0% drift ratio.

Shear cracks arose on the specimen F-O-450-150 joint panel at 1.5% drift ratio,

as indicated in Figure 4.4(c). For specimen F-O-500-180, some flexural cracks

arose from the beam bottom surface and extended towards the FRP anchors

during the cycle 1 of 0.5% drift ratio as indicated in Figure 4.5(a). Cracks in the

slab emerged from the top corners of the openings. Meanwhile, one flexural

crack showed on the lower column. During the following two cycles of 1.0%

drift ratio, new cracks, close to the transverse beam, appeared on the specimen

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F-O-500-180 bottom column and joint panel, as shown in Figure 4.5(b). As

shown in Figure 4.6(a), many flexural cracks arose on the specimen

F-S-450-450 beam and some on its lower column and the slab during cycle 1.

The cracks on the slab emerged from the end of transverse slits and extended to

the slab edge.

In the second test group, the right and left beam tip loads were released to

zero after applying an axial load of 0.2 axial compression ratio. The drift ratio of

the initial cycle was decreased to 0.25%, rather than 0.5% in the first test group.

For control specimen S-Control, some flexural cracks arose from the beam

bottom surface during cycle 1 of 0.25% drift ratio, shown in Figure 4.7(a). A

flexural crack showed on its bottom column and one on its left slab. At 1.0%

drift ratio, several shear cracks showed on the joint panel around the transverse

beam, as indicated in Figure 4.7(b). As shown in Figure 4.8(a), a flexural crack

arose on the top surface of the specimen S-G-50-200-100 left slab during cycle

1 of 0.25% drift ratio. Flexural cracks showed on the bottom column of

specimen S-G-50-200-100 at 0.5% drift ratio. At the following drift ratio, shear

cracks arose on the joint panel as presented in Figure 4.8(c). As shown in

Figures 4.9(a) and 4.10(a), flexural cracks on the slab and beams of specimens

S-O-500-180 and S-O-500-180-S-300-300 arose during the second cycle of 0.5%

drift ratio. At the 1.0% drift ratio, several flexural and shear cracks arose on the

columns and joint panel respectively of the two specimens as presented in

Figures 4.9(b) and 4.10(b).

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4.2.2 Yield Point and Peak Point

As indicated in Table 4.1, for the first test group, specimens F-Control and

F-O-500-180 reached their strengths at 2.0% drift ratio, Specimens F-G-50-200

and F-S-450-450 at 1.5% and specimens F-O-450-150 at 2.5% drift ratio. For

the second test group, all specimens reached their strength at 2.0% drift ratio.

All specimens of the two test groups, except for specimen S-G-50-200-100, had

all their beam and slab bars yielded before they reached their peak loads.

Specimen S-G-50-200-100 had all its slab bars yielded at 4.0% drift ratio. This

is probably because the positions of strain gauges on its slab bars were not

transversely parallel to the beam groove region where beam flexural

deformation was concentrated. Except for specimens F-G-50-200 and

S-G-50-200-100, in which joint stirrups did not yield, all specimens had a port

of or all joint stirrups yielded during test. Specimens F-Control, F-O-450-50,

S-Control and S-O-500-180 also have obviously column bars obviously yielded.

For specimen F-Control, shear cracks arising near the beam pinned region

were wide at 2.0% drift ratio as indicated in Figure 4.2(c). The joint panel also

had many shear cracks at this drift ratio. For specimen F-G-50-200, when the

drift ratio was increased to 1.5%, cracks due to beam bottom bars slippage were

clear as indicated in Figure 4.3(c). Specimen F-O-450-150 got many cracks

arose on the bottom column due to the pull force provided by the web chords at

2.5% drift ratio, as shown in Figure 4.4.(c). The main beam longitudinal bars of

specimen F-O-500-180 yielded under negative bending at the drift ratio of 1.5%.

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A joint stirrup yielded, as indicated by readings of strain gauges at this drift

ratio. At the 2.0% drift ratio, more cracks appeared near the FRP anchors due to

the pull-out action of the FRP anchors as shown in Figure 4.5(c). The beam

longitudinal bars of specimen F-S-450-450 yielded under negative bending

when the drift ratio was increased to 1.5%. At the drift ratio of 2.0%, two

middle stirrups in the joint region yielded. The slab cracks initiating from the

transverse slit ends were widely open and the number of cracks on the joints

panel were obviously increased, as indicated in Figure 4.6(c). At a drift ratio of

2.0%, many cracks arose around the specimen S-Control transverse beam as

shown in Figure 4.7(c). As the drift ratio grew to 2.5%, some joint stirrups close

to its bottom column yielded. The specimen S-G-50-200-100 beam longitudinal

bars yielded under negative bending at 1.5% drift ratio. At the drift ratio of

2.0%, many cracks arose around its transverse beam and bottom column as

shown in Figure 4.8(c). The specimen S-O-500-180 beam top bars and slab bars

yielded under negative bending at 1.5% drift ratio. As the drift ratio was

increased to 2.0%, several cracks originating from the anchor positions showed

on the top and bottom surfaces of slabs as shown in Figure 4.9(c). Many cracks

showed on the top and bottom columns at this drift ratio. At the drift ratio of

2.5%, two specimen S-O-500-180 joint middle stirrups yielded. The top beam

bars and the uncut slab bars of specimen S-0-500-180-S-300-300 yielded at the

1.0% drift ratio. Afterwards, at the drift ratio of 2.0%, many cracks arose on the

top, bottom columns and around the transverse beam.

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4.2.3 Failure Mode

For specimen F-Control, due to shear failure caused by the very large

openings originating within the beam pinned region as shown in Figure 4.2(d),

the test was stopped at a drift ratio of 4.0% during push loading. At the 4.0%

drift ratio, the beams of this specimen had entered their yield state. The failed

control specimen beam ends was then strengthened as shown in Figure 4.1. Old

concrete was removed and replaced with new concrete of 40 MPa cylinder

strength. To avoid shear failure, steel bars were used to strengthen the pinned

region. A wooden mould was built to protect the original shape of the specimen.

The strengthened control specimen was then further tested starting from the drift

ratio reached at the test stoppage point. The strengthened control specimen

finally failed by joint shear failure as shown in Figure 4.2(e), rather than column

end failure. This unexpected failure mode was due to actual strengths of the

steel reinforcement and concrete involved were not as designed. The

deformation of specimen S-Control was concentrated at the bottom column

hinge region at the later drift ratio. As shown in Figure 4.7(e), the bottom

column concrete at the column-beam intersection was crushed. Specimen

S-Control failed because the bottom column bars buckled and were unable to

sustain the axial load. The buckling of bottom column bars is clearly shown in

Figure 4.7(e).

The failure mode of specimen F-G-50-200 is shown in Figure 4.3(d).The

beam bottom bars slippage in the joint region was the cause of the failure.

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Deformation of specimen S-G-50-200-100 was also clearly concentrated at the

beam hinges formed at the beam groove region. Concrete was pulled out when

the beam bottom bars slipped as shown in Figure 4.8(e). The failure mode of

specimen S-G-50-200-100 is similar to that of specimen F-G-50-200, even

though the grooves were placed 100 mm away from the beam-column interface,

with the aim to avoid beam bottom bars slippage.

In F-S-450-450, a large opening appeared at the beam-to-column interface

and also on the slab, starting from the transverse beam as shown in Figure

4.6(d).The specimen F-S-450-450 achieved the designed failure mode as 75%

slab longitudinal bars were removed.

For specimen F-O-450-150, damage and deformation were concentrated at

the joint region as shown in Figures 4.4-(e) due to joint shear failure. The test

result indicated that the 450x150 mm2 opening size was not sufficiently large to

enable a switch from joint shear failure mode to flexural failure. Thus, a larger

opening size was designed for another specimen F-O-500-180. As shown in

Figure 4.5(d), the existence of a four hinges mechanism (four hinges formed at

the ends of the two chords) was obvious in this specimen. For specimen

S-O-500-180, bottom column concrete at the column-beam intersection was

crushed and the column bars buckled when the specimen failed, as shown in

Figure 4.9(d). Even though the opening size of specimen S-O-500-180 is the

same to that of specimen F-O-500-180, the beams of specimen S-O-500-180

were stronger as they were provided with more longitudinal steel reinforcement.

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The joint region of specimen S-O-500-180 was also provided with more

transverse steel reinforcement. Thus the failure of this specimen happened on

the bottom column. When slab slits were provided to weaken the T-beam

moment capacity in addition to 500×180 mm2 slot-shaped openings, the

specimen S-O-500-180-S-300-300 failed in a ductile manner, as indicated in

Figure 4.10(d). The concrete at the top part of the T-beams of specimen

S-O-500-180-S-300-300 crashed when the specimen failed. The deformation of

main beams obviously followed the four hinges mechanism.

The specimen test results of the two test groups, as presented above,

demonstrated that three proposed retrofit techniques have great effects on the

seismic performance of 3D RC joints if they are well designed. The existence of

a transverse groove can greatly lower the moment-capacity of a T-beam.

However, the failure mode was non-ductile because the main beam bottom

longitudinal bars slipped through the joint region even though the groove was

not placed close to beam-column interface. The slab slitting method can

effectively lower the moment capacity of a T-beam as it removes the

contribution of the cut slab longitudinal bars. Meanwhile, the failure mode is

still ductile. The beam opening size should be well designed (e.g. increasing the

opening size) to lead to an obvious four-hinges mechanism. The slab slits can

help a specimen, not sufficiently weakened by the beam opening technique, to

realize a failure following the four-hinges mechanism.

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4.3 HYSTERETIC BEHAVIOUR AND ENVELOP CURVES

4.3.1 Hysteretic Behaviour

Hysteresis curves of axial compression load are shown in Figure 4.11. The

axial compression load was kept almost unchanged with a relatively small

variation due to limitation of loading equipment. The variation were as big

when lateral drift ratio was close to 5% for specimens S-Control and

S-O-500-180 as at the drift ratio, the two specimens failed due to bottom

column bars buckling and capacity for axial loading dropped obviously.

The hysteretic properties of the beams and the upper column of the

specimens are shown in Figures 4.12 and 4.13. The beam tip loads (the loads at

the beam end support) of the two T-beams were recorded by the load cells

installed in the beam end supports. The figures indicate that the hysteretic

performances of the two T-beams were almost the same. The beam tip loads

were used to calculate the column shear force using Equation (4.1):

(( ) ) /c b right b left b drift cF F F L N d L / 2 (4.1)

where cF is the column shear force and b rightF and b leftF are the beam tip

loads of the right and left T-beams respectively; bL and cL represent the

lengths of beams and columns from the centre of the joint panel respectively;

driftd is the lateral displacement.

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The absolute load value recorded by the MTS machine was higher than

that of the column shear force as the former includes additional friction forces,

as demonstrated in Figure 4.14. As shown in these figures, the effect of friction

was quite significant during the pull action, varying between different test

specimens. This effect could not be accurately measured. The predicted column

strengths shown in Table 4.2 were based on cross-sectional analysis using actual

material properties. Second-order effect was incorporated when calculating the

predicted column strengths. The calculated column shear force was obtained

using Equation (4.1). In Table 4.2, the weakening ratio represents the weakening

degree when comparing the sum of the absolute values of the measured T-beam

negative and positive strengths to that of the control specimens.

The specimen F-control had a gradual yielding load-displacement curve

due to the existence of a wide slab and reached its strength at the drift ratio of

2.0%. The strengths of the left and right T-beams were close to each other as

indicated in Figure 4.12(a).The positive and negative yield loads of the T-beams

were 87.2 kN and 156.6 kN respectively, close to those derived from sectional

analysis. These values were obtained before the beam end of the control

specimen was strengthened. The strengthened specimen was tested at drift ratios

of 4.0% and above. As shown in Figure 4.13(a), the retrofitted specimens had

relatively higher strengths in the push direction, which might be the result from

a regain of strength after being retrofitted. In the pull action, the strength was

lower due to the damage caused during the push action. Pinching behaviour was

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obvious and the reloading stiffness decreased with increased lateral drift ratio.

Specimen F-G-50-200 had a relatively short yielding process and reached

its strength at 1.5% drift ratio. Obviously different from the other specimens, the

specimen retrofitted by TG technique failed quickly after reaching its strength

as shown in Figures 4.12(b) and 4.13(b). According to damage propagation

observations, the slippage of bottom beam bars in the joint region was the major

cause of this non-ductile failure mode. The measured strength of the T-beams

under positive and negative loadings was 87.4 kN and 85.6 kN respectively.

Though the sum of these values was close to that of the R-beam, such a

non-ductile failure mode is not permitted for seismic design. Hence, the TG

retrofit method should be redesigned to prevent bottom beam bar slipping

failure.

Specimen F-O-450-150 also displayed gradual yielding and reached its

strength at 2.5% drift ratio. The strength measured in the push action indicated

in the right hand part of Figure 4.13(c) was a little higher than those due to pull

action as shown in the left hand part of the figure. The T-beams reached their

negative strengths at 4.0% drift ratio, which is much later than the control case.

The obtained strength of the T-beams in the push action was 88.5 kN and 138.5

kN under positive and negative loadings respectively. Their sum was only 7.0%

lower than that of the control specimen F-Control, which was far from the

requirement to offset the contribution from slab. Thus, the designed opening

size of specimen F-O-450-150 was not big enough to ensure a beam flexural

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failure mode.

This opening size of specimen F-O-500-180 was designed after testing the

specimen F-O-450-150. The specimen F-O-500-180 reached its strength at 2.0%

drift ratio as shown in Figure 4.13(d). The T-beams also obtained their strength

at 2.0% drift ratio, which was quicker than the specimen F-O-450-150 as joint

deformation of the specimen F-O-450-150 was the major contribution to lateral

drift. The measured positive and negative strength of the T-beams was 88.2 kN

and 115.7 kN respectively and the weakening ratio was 16.4%, which was more

than twice that of the specimen F-O-450-150. Even though the weakening ratio

was only half of that of the R-beam, the specimen F-O-500-180 still failed in a

ductile manner.

Specimen F-S-450-450 had a quicker yielding process because 75% of the

longitudinal slab bars had been cut. The specimen reached its strength at 1.5%

drift ratio and had a relative fast post-peak ascending range as indicated in

Figure 4.13(e). The T-beams reached their yield and strength at 1.5% drift ratio,

also earlier than those of control specimen did because of the removal of 75%

slab longitudinal bars. The measured positive and negative strength of the

T-beams were 83.1 kN and 109.4 kN respectively, and the weakening ratio was

21.0%. Even though the strength was still higher than those of the R-beam, the

specimen retrofitted by the SS method was able to perform in a ductile manner

under cyclic loading. In this sense, the method was a good option for retrofitting

a beam-column joint.

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Due to this MTS machine problem, the test on specimen S-control was

stopped several times during the test. The specimen failed due to hinge

formation in the bottom column as indicated in Figure 4.7(e). The test was

stopped at the first cycle of the 5% drift ratio as the column bars buckled. The

specimen reached its strength at 2.0% drift ratio and its two T-beams at 2.5%

drift ratio. The measured negative and positive strength of the T-beams was

144.4 kN and 112.8 kN respectively and the yield load from sectional analysis

was 156.4 kN and 119.6 kN, which meant the T-beams did not reached their

strength during testing. As indicated in Figure 4.13(f), the beam tip load

hysteresis curves were similar except for the final hoop, which was affected

because the bottom column bars buckled leading to a sharp decrease of axial

load.

The grooves of Specimen S-G-50-200-100 were 200 mm deep and 50 mm

wide and placed 100 mm away from the beam-column intersection. The

specimen reached its strength at 2.0% drift ratio. The measured maximum

negative and positive beam tip loads were 111.0 kN and 107.3 kN respectively.

The weakening ratio was 15.1 %. The ductility of the specimen was still poor

due to the occurrence of bottom beam bars slippage in the joint region, even

though the grooves were placed 100 mm away from the beam-column

intersection. The negative beam tip load has a sudden drop as demonstrated in

Figure 4.12(g) when the bottom beam bars began slipping in the joint region.

When the specimen S-O-500-180 was under pull action in the last cycle, of

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the 5% drift ratio, the column bars finally buckled because they lacked concrete

cover protection. The specimen and its beams reached their strength at 2.5%

drift ratio. The measured maximum negative and positive beam tip loads were

120.4kN and 119.3 kN respectively. The weakening ratio was 6.8%. The beam

tip loads obtained were similar except for the final hoop during which the

bottom column bars buckled

The opening dimensions of the specimen S-O-500-180-S-450-450 were the

same as those of specimen S-O-500-180. The specimen reached its strength at

2.0% drift ratio but the T-beams reached their negative strength at 4.0% drift

ratio, slower than for the other specimens. The measured maximum negative

and positive beam tip loads were 111.7 kN and 102.6 kN respectively, which are

lower than the measured maximum negative and positive beam tip loads of

specimen F-O-500-180, 120.4 kN and 119.3 kN respectively. The hysteresis

curves for the two T-beams were similar and of good ductile property as

indicated in Figure 4.12(i).

4.3.2 Envelop Curves

Envelop curves of calculated column shear forces and beam tip loads of

specimens are shown in Figures 4.15 and 4.16. It is obvious that the specimens

retrofitted by the proposed techniques had lower stiffness and strength than

those of control specimens. Among the first test group, the specimen

F-G-50-200 had the lowest post-cracking stiffness, in addition to the lowest

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strength. Among the second test group, the specimen S-O-500-180-S-300-300

had the lowest stiffness. As shown in Figure 4.16(a), both the BO and SS

techniques decreased the stiffness of T-beams under both negative and positive

loading. The TG technique only had obvious effect on the stiffness and strength

of a T-beam when it was under negative loading. When the grooves were placed

100 mm away from the beam-column interface, the effect of the TG technique

on T-beam stiffness became weaker, as demonstrated in Figure 4.16(b). The

positive strength of specimens F-O-450-150, F-O-500-180 and S-O-500-180

was very close to that of control specimens, which meant that the positive

strength of the T-beams was slightly affected by the existence of BO. The SS

technique deceased both negative and positive strength of T-beams.

4.4 DEFORMATION BEHAVIOUR

Many strain gauges were mounted on reinforcing steel bars to measure the

interior deformation of the specimens under cyclic loading. The labels and

major functions of the strain gauges were illustrated in the previous Chapter 3.

Axis directions and the distribution of strain gauges were also presented in the

previous chapter. Except for some gauges broken before and during testing, the

remaining strain gauges were used to measure the deformation of steel bars at

the peak displacement for each loading drift ratio.

4.4.1 Strains in the Steel Bars

4.4.1.1 X Direction

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Along the X axis, the strain gauges measured the strain distribution in the

steel bars of the main beam and slab. When a specimen was under a push or a

pull action, the right T-beam experienced a hogging moment or sagging moment

respectively. It should be noted that in Figure 4.17, the strains were normalized

to the yield strains of corresponding steel bars. In these figures, the black dashed

lines indicate the yielding of steel bars.

Figure 4.17(a) shows the strain variation along the X axis in one top

reinforcing bar of the T-beam of the control specimen. The test data for the

retrofitted control specimen were not available because the wiring had all been

cut. When the specimen was under a push action/pull action, the top steel bar

was in compression/in tension respectively. In general, the stain gauges closest

to the column recorded large strains as they were located at the section of

highest moment. Some strain gauges measured tensile strains in the top steel

bars when a specimen was under push action, as indicated by the left hand

figure of Figure 4.17(a). This could be because the plastic extension of a steel

bar under tension could not be offset by the compression strain.

For specimen F-G-50-200, the greatest deformation occurred near the

transverse groove as indicated in Figure 4.17(b). It should be noted that the

grooves were 50 mm wide and located close to the beam-column interface.

Obviously, the plastic hinge length of the T-beam was shortened as the beam

deformation was mostly concentrated near the hinge. The plastic extension

deformation of the top beam bars was concentrated where the transverse groove

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was created. The obtained tension strains were still higher even when the beam

top part was under compression, which was because tension plastic strain was

unrecoverable. Strains measured by strain gauge TB11 dropped after the 2.0%

drift ratio, possibly due to slippage of the beam bottom bars.

When a T-beam was weakened by a 450x150 mm2 opening, the

deformation of the beam was much different from the one without opening, as

indicated in Figure 4.17(c). The right hand side of Figure 4.17(c) displays the

elevated strains of a beam top longitudinal bar with the increased drift ratio

when the specimen was under pull action. In this situation, the middle three

strain gauges measured smaller strains and the two close to the corner of the

opening measured higher strains. Strain gauge TB11 failed at 2.5% drift ratio.

Thus unreasonable tensile strain was obtained by this stain gauge when the

specimen was under push action at 2.5% drift ratio as indicated in the left hand

picture of Figure 4.17(c).

As shown in Figure 4.17(d), a beam top steel bar of the specimen

F-O-500-180 deformed similarly to that of specimen F-O-450-150. As the

opening size of specimen F-O-500-180 was larger, the four hinges mechanism

was more obvious, as demonstrated by the experimental observation that, at the

same drift ratio, the hinges at the chord ends rotated more severely. This

phenomenon became more obvious as the lateral drift ratio increased. As

indicated in the right hand picture of Figure 4.17(d), the strain gauge TB14,

which was closer to one corner of the opening, measured a greater steel

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deformation. Strain gauge TB11 measured increasing strain, before the 1.5%

drift ratio, during pushing action. When the drift ratio was increased to a higher

level, the strain gauge measured a descending deformation trend, perhaps

because the effect of the four hinges mechanism overwhelmed the plane

sectional deformation.

Figure 4.17(e) presents the top beam bar deformation of specimen

F-S-450-450. At lower drift ratios, strain gauge TB11 measured the highest

deformation. In later loading cycles, strain gauge TB12 gave the highest strains,

because it was located near a large flexural crack.

Figure 4.17(f) shows the strain distribution along the X axis of a beam top

reinforcing bar of the specimen S-Control. At first the strain gauges closer to the

column indicated more deformation, as they were located at the section of

highest moment. As the lateral drift ratio grew larger, concrete cracking lead to

strain redistribution and thus, some part of the steel bar was the most strained

even though not the closest to the column.

For specimen S-G-50-200-100, the highest deformation occurred at the

groove position as indicated by the red dashed lines in Figure 4.17(g). The

grooves were located 100 mm away from the beam-to-column interface. The

plastic extension deformation of the top beam bars was also concentrated where

the transverse groove was created, similar to Specimen F-G-50-200. Strain

gauge TB12 became detached from the steel bar after the 2.5% drift ratio. Thus,

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even under push action, the strain gauge still measured a large tensile strain,

which was nearly constant during the later drift ratios.

The beam deformation of specimen S-O-500-180 was very different from

the specimen S-control and the specimen S-G-50-200-100, even though the

opening size was not sufficiently large to lead to ductile failure mode. The left

hand picture of Figure 4.17(h) displays the strain distribution of a top

longitudinal T-beam bar. As mentioned above, when a specimen is under push

action, the right hand beam is subjected to negative bending. The strain gauges

TB11 and TB15 on a top longitudinal bar, were placed near the left and right

hand sides of the opening respectively and their data revealed that the two ends

of the strong chord were deformed to a greater extent than the middle part,

similar to the specimen F-O-450-150 and F-O-500-180 cases.

The T-beam deformation of specimen S-O-500-180-S-300-300 showed no

obvious difference to that of specimen S-O-500-180. The deformation measured

by strain gauge TB14 during push action was larger than that by TB15, which

might be due to the existing of slab slits, as slits separated slabs and main beams

in the longitudinal direction.

4.4.1.2 Y Direction

Along the Y direction, the strain distribution was measured to reveal the

slab contribution to T-beam flexural capacity. Except for the cut slab bars

passing the slab slits, all slab bars of specimens yielded during test.

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For the control specimen F-Control, slab steel bars farther away from main

beam in the transverse direction were recorded less deformation before 1.0%

lateral drift ratio, which was due to shear lag effect. This effect was not

obviously presented by strain gauges data at higher drift ratios. A possible

reason was cracks propagating along the transverse direction were not parallel

to the Y direction. Another possible reason was strain gauges were not longer

tightly attached to slab bars after several cyclic loading cycles. Almost all steel

bars yielded when drift ratio reached 2.0%, as indicated in Figure 4.18(a). The

strain gauge TL11 might have become detached from the steel bar after the 2.0%

drift ratio as it remained almost constant as drift ratio was increased. Strain

gauge TL21 measured a large deformation, possibly because it bridged a large

crack.

For specimen F-G-50-200, all slab top bars yielded at the 1.0% drift ratio,

as indicated in Figure 4.18(b). The strain gauges TL11 and TL21 might be

detached from the steel bars after 1.0% and 2.0% drift ratios respectively thus

the recorded strains by them were strange afterwards.

The shear lag effect on slab for specimen F-O-450-150 was obvious before

2.0% lateral drift ratio as slab steel bars farther away from main beam in the

transverse direction were recorded less deformation, indicated in Figure 4.18(c).

All slab top bars yielded at 2.5% drift ratio and strain gauge TL11 also failed at

this drift ratio.

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As shown in Figure 4.18(d), the slab top bars of specimen F-O-500-180

yielded at 1.5% drift ratio, less than that in specimen F-O-450-150, as the

specimen F-O-500-180 chord end hinges rotated more severely.

As strain gauge TL41 of specimen F-S-450-450 broke before the test,

unlike the other specimens, the strain distribution on the left hand side of slab

was plotted in Figure 4.18(e) instead. As expected, the cut slab bars experienced

very low strain levels under conditions of both positive and negative loading.

For the control specimen S-control, there was shear lag effect in the slab

transverse direction and almost all the steel bars yielded when the lateral drift

ratio reached 2.5%, as indicated in Figure 4.18(f),

For specimen S-G-50-200-100, the shear lag effect is also obvious in the

right hand picture of Figure 4.18(g).

Figures 4.18(h) and (i) show the strain distribution in the slabs of

specimens S-O-500-180 and S-O-500-180-S-300-300 in the transverse direction.

For the specimen weakened by openings only, the strain distribution of the top

slab bars in the slab transverse direction indicated the shear lag effect at a low

lateral drift ratio (lower than 1.0%). Crack propagation with increased drift ratio

might result in some strain gauges capturing much higher deformations (more

than 2 times the yield strain). For the specimen weakened by additional slab

slits, the cut slab bars contributed marginally to the beam moment capacity at

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the monitored section.

4.4.1.3 Z Direction

Along the Z axis direction, only data from strain gauges on column

longitudinal bars were plotted. As measurements of strain in the two column

bars were nearly symmetrical about the z axis. The strain distribution in one

column bar is given in Figure 4.19.

Except for the specimens F-G-50-200, F-O-500-180, S-G-50-200-100,

whose column longitudinal bars kept elastic until the end of the test as indicated

in Figures 4.19(b).(e) and (g), all specimens had yielding longitudinal bars in

the upper column or bottom column during test.

As shown in Figure 4.19(a), the bars in the upper column of specimen

F-Control stayed elastic during the test and those in the lower column, reached

yield load after a 2.5% drift ratio. It was clear that strain gauges closest to the

joint region measured more deformation. As shown in Figure 4.19(c), the

specimen F-O-450-150 bottom column bars compression yielding occurred after

2.5% drift ratio as strain gauge BC32 recorded large compression deformation.

As shown in Figure 4.19(e), due to the lack of surrounding slab protection, steel

bars in the upper column of specimen F-S-450-450 was recorded more

deformation. Strain gauge TC31 recorded yielding of the corresponding steel

bar in the upper column at 5% drift ratio. As shown in Figure 4.19(f), the upper

column bars stayed elastic during test and the lower bottom column bars

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reached their yield load after a 2.0% drift ratio. At peak displacement of the 4.0%

drift ratio cycle, large deformation existed in the bottom column bars and when

the drift ratio reached 5%, the column bottom bars buckled. The specimen could

not bear any axial loading and thus, the strain variation after ultimate loading is

not plotted in Figure 4.19(f). The openings size of S-O-500-180 was not big

enough to switch the column flexural failure to a four hinges formation failure.

Thus, the bottom column bars yielded after a 2.5% drift ratio and at the second

cycle of the final cycle group, some bottom column bars buckled after spalling

of the concrete cover. Though specimen S-O-500-180-S-300-300 went through

WBSC failure, the monitored bottom column bar reached its compression

yielding strain as demonstrated in Figure 4.19(i).

4.4.2 Rotations of Column Sections

The strain levels of bottom column eight column bars close to the joint

region were plotted in a 3D form (as indicated in Figure 4.20) to show column

section deformation under pull action. The column section deformation of

specimen F-O-50-180 is not presented because most of the relevant strain

gauges broke when creating the openings. As indicated by these 3D figures, the

plane section assumption was basically obeyed at low lateral drift ratios. When

the drift ratio increased, cracks propagated and concrete gradually spalled,

which led to strain redistribution and disturbance of the plane section

assumption. In addition, plastic deformation of the steel bars was irreversible,

which contributed to the breaking of the plane section assumption at high drift

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ratios. For specimens F-S-450-450, S-O-500-180-S-300-300, the column

deformation basically followed the plane section assumption, even at the last

cycle as indicated in Figure 4.20(h), because they failed in WBSC mode. As

indicated in Figures 4.20(b)-(d), the column section deformation of specimens

F-G-50-200, F-O-450-150, F-S-450-450 reached its peak rotation at 2.0%, 2.5%

and 2.5% lateral drift ratio respectively.

4.4.3 Strain in Stirrups at Joint Region

The interior deformation of the joint was measured by the strain gauges

mounted on the joint stirrups. Strain gauge CS1 was located on the uppermost

stirrup and CS4 on the lowermost. As indicated in Figure 4.21, except for the

specimen F-G-50-200, all specimens had one or more yielding joint stirrups

when the drift ratio reached 2.5%. Generally, the control specimen suffered a

severest joint shear deformation. However, CS1 of the specimen F-S-450-450

measured larger strains than the control specimen did. This might be due to the

lack of surrounding slab protection. The strain gauge CS3 of specimen

F-O-500-180 recorded higher deformations than CS4, which might be due to the

removal of beam web concrete.

Though two strain gauges were used to record the deformation of one

stirrup in the specimens' joints region of the second test group, only data from

one strain gauge for each monitored stirrup was plotted. As indicated in Figure

4.22, except for the stirrup monitored by strain gauge CS42, specimen

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S-Control joint stirrups kept elastic during testing. The strain gauge CS42

measured higher deformation as the stirrup it monitored was closest to the

bottom column among the four monitored stirrups. For specimen

S-G-50-200-100, all joint stirrups were elastic during testing as indicated in

Figure 4.22. The specimen S-O-500-180, whose joint middle two stirrups had

more deformation than those of the other specimens, as indicated in Figures

4.22(b) and (c).The reason might due to the middle two lacking the protection of

beam web concrete, which was removed to create openings. The existence of

SSs resulted in smaller joint shear deformation, as indicated by comparing

specimen S-O-500-180-S-300-300 joint stirrups strain level with that of

specimen S-O-500-180.

4.4.4 Strains in FRP Jackets

The strain gauge arrangements on FRP are indicated in Figure 3.22 of

former chapter. The strain gauges on the left hand beam behaved similarly to

those on the right. As shown in Figures 4.23-4.28, all FRP sheets stayed elastic

until the end of the test. Figure 4.25(b) presents the strain distribution of the

FRP sheet at the two ends of the web chord of specimen F-O-500-180. When

the specimen was under pull action, the right hand T-beam was under negative

bending. The bottom surface of the left end and the top surface of the other end

of the chord were under compression. At this moment, RW5 and RW6 obtained

the highest strains, caused by concrete expansion. Similarly, when the specimen

was under push action, strain gauges LW5 and LW6 obtained similar results.

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This phenomenon was not obvious when the T-beam was under positive

bending.

For specimen S-G-50-200-100, as indicated in Figure 4.26, the strain gauge

(RU1) close to the column face and the upper corner of the groove measured the

largest strain after a 3.0% drift ratio, because the cracks propagating from the

corner became larger with the increased drift ratio. Of the strain gauges RU4,5

and 6, on the other sides of the groove, the one closer to the bottom of the

T-beams measured the largest deformation. For the specimens with beam

openings, as shown in Figures 4.27 and 4.28, strain gauges on the U-jackets

displayed lower strain levels, which meant the U-jackets were strong enough.

For strain gauges on the FRP wrap, RW1, RW3, RW5 and RW7 were mounted

on the corners of the web chord. RW4 and RW8 were mounted on the bottom

surface of the chord. When the right hand beam was under negative loading,

RW4 and RW3 measured the largest deformation at the left hand end of the

chord. Because at this scenario, the beam deformed following the four hinges

mechanism. The bottom concrete of the chord left end was under compression

and bulged. When the right hand beam was under positive loading, RW4 and

RW3 still measured the largest deformations as the four hinges mechanism was

not obvious and cracks propagated from the bottom surface. At the other end of

the chord, when the right hand beam was under negative moment (i.e. the

specimen was under pull action), RW5 measured the largest deformation as the

concrete of the top side bulged. When the beam was under positive moment,

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RW5 measured a relatively larger deformation as that part was still under

compression.

4.5 DISCUSSIONS

The original aim of retrofitting RC beam column joints is to avoid the story

sway failure mechanism. The flexural strength ratios of columns to beams with

cast-in-place slab should be increased to more than 1.0. The failure mode should

be ductile. Apart from the beam strength and the specimen failure mode, the

specimen ductility, energy dissipation capacity, equivalent viscous damping

ratio hyst and peak-to-peak stiffness are also important factors which can be

used to assess the seismic performance of these specimens. A retrofitted

specimen should have better ductility and energy dissipation capacity than the

original one which might fail due to storey sway mechanism. The peak-to-peak

stiffness should not be obviously decreased as according to seismic design

codes (e.g. GB-50011 2001), a reliable RC structure should be able to stand

obvious deformation (e.g. 2% drift ratio). The energy dissipation capacity is a

direct index and hyst is a more effective index to assess the seismic

performance of specimens. Thus, these factors are discussed below.

4.5.1 Specimen Ductility

Yielding of the tested specimens was not easily detected due to the

existence of a wide slab. According to Hu (2005), there are four methods for

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defining yield deformation: (1) as the deformation where yielding first occurs in

the system (see Figure 4.29(a)); (2) as the elastic limit of an equivalent

elastic-perfectly plastic curve with the same elastic stiffness and strength as

those of the test curve (see Figure 4.29(b)); (3) as the elastic limit of an

equivalent elastic-perfectly plastic curve which provides an equal area to that of

the test envelope curve before the peak load (strength), with the real test

strength being taken as the equivalent strength (see Figure 4.29(c)); and (4) as

the elastic limit of an equivalent elastic-perfectly plastic curve with a reduced

stiffness which is equal to the secant stiffness at either the first yielding or at a

certain value (e.g. 75%) of the strength, whichever is less (see Figure 4.29(d)).

Among the four definitions, the first is inappropriate for the present study

because first yielding is vague and thus hard to recognize. Of the last three

methods, the second is suitable for materials with an obvious yielding plateau,

such as hot-rolled steel. The third definition is hard to apply. The final definition

provides the most appropriate and general way and has been claimed the most

suitable for various structures such as concrete, masonry, steel as well as timber

structures (Park 1989).

As for the definition of the ultimate state of a structural member, possible

definitions for the ultimate deformation are: (1) as that corresponding to a

particular limit value for the material ultimate strain (e.g. the attainment of a

specified concrete ultimate compressive strain in the case of reinforced concrete

structures (see Figure 4.30(a)); (2) as that corresponding to the ultimate load of

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a test load-displacement curve (see Figure 4.30(b)); (3) as the value where the

load capacity of a test specimen has undergone a small reduction, for example, a

10% to 30% reduction (see Figure 4.30(c)) ; (4) as the value when the material

fractures or elements buckles; for example, the fracture of transverse reinforcing

steel or the buckling of longitudinal reinforcing steel in the case of reinforced

concrete; see Figure 4.30(d).

Definition (1) is inappropriate as it is evident that the maximum available

deformation does not necessarily correspond to a specified extreme concrete

compressive strain. It should be recognized that most structures have some

deformation capacity after achieving their ultimate load without significant

reduction in load capacity. It is reasonable to include this extra deformation

capacity in defining the ultimate deformation. Hence, definition (2) is

conservative. Park (1989) recommended defining the ultimate deformation

using the criteria (3) and (4) together, whichever occurs first.

The ductility parameter is commonly defined by the following

equation:

u

y

(4.2)

where and are the ultimate and yield displacements respectively.

As indicated in Figure 4.29, the definition (4) for the yield deformation

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was applied to determine the yield displacements of specimens and 75% of

strength was used to calculate the secant stiffness. And the definition (3), for

ultimate state, was applied and a 20% reduction of load after the peak load was

chosen as the ultimate state. The results are summarized in Table 4.3.

The positive parts of the envelope curves, as shown in Figure 4.13, were

used to determine the yield and ultimate displacements of tested specimens

except for the specimen F-Control. Specimen F-Control failed at 4.0% lateral

drift ratio group due to shear failure at the beam pinned position. It was then

strengthened with steel reinforcement and FRP sheets. The test was carried out

about 2 months after being retrofitted. Therefore, the strength of the specimen

was slightly increased at the first cycle loading owing to crack closing and

concrete strength increase.

In terms of the yield deformation, the control specimen F-Control had the

smallest yield displacement among the first group as the stiffness degradation

ratio due to retrofitting is higher than that of strength. As for ductility, the

control specimen had good ductility as the beam yielded first before joint shear

failure. The specimen F-G-50-200 was the poorest as indicated in Figure 4.31. It

is obvious from the envelope curve shown in Figure 4.13(b) that specimen

F-G-50-200 strength falls quickly after peak stress. The slits on the slab can

effectively lower the strength specimen F-S-450-450. The ductility was

decreased by the existence of slab slits as the control specimen did not failed by

column end failure. The BO techniques can reduce strength with ductility

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remaining almost unchanged (the ductilities of specimens F-O-450-150 and

F-O-500-180, which are 4.4 and 4.1 respectively, is close to that of specimen

F-control, which is 4.4).

Unlike the first test group, specimen S-G-50-200-100 of the second test

group had the smallest yield displacement among the second group. That's

because specimen S-G-50-200-100 yield load was also obviously lowered but

the stiffness was less affected by the TG technique than in the first test group as

the grooves were placed 100 mm away from the beam-column interface. The

other three specimens had close yield displacements as both strength and

stiffness were lowered by the BO technique and the combination of the BO and

SS techniques. As shown in Figure 4.31, the ductility of specimen S-Control

was obvious lower than that of specimen F-Control, which was mainly because

the specimen S-Control failed by column end failure. The ductility of specimen

S-G-50-200-100 was much higher than that of specimen F-G-50-200 as the

grooves were placed 100 mm distant from the column face, which postponed

the slippage of bottom longitudinal bars. The specimen S-O-500-180 had

slightly higher ductility than the specimen S-Control. The combined use of the

BO and SS techniques in specimen S-O-500-180-S-300-300 obviously reduced

specimen strength but increased ductility.

4.5.2 Energy Dissipation Capacity

As indicated in Figure 4.32, dissipated energy EDA by each cycle is

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defined by the area enclosed by the cycle. Energy dissipation curves for each

individual cycle for each specimen are plotted in Figure 4.33. It is obvious that

almost all the first hysteresis cycle of each drift ratio dissipate more energy than

the second one. That's because crack propagation decreases the stiffness and

strength of a specimen. Except for specimen F-G-50-200 which has poor

ductility and fails quickly after the 2.0% lateral drift ratio, the remaining four

specimens in the first test group possessed increasing energy dissipation

capacity with increased drift ratio. Column bars buckling lead to the first cycle

at the 5% drift ratio of the control specimen S-Control, dissipating much more

energy than the first cycle of the 4.0% drift ratio. That also resulted in a

consumption of a bit more energy in the second cycle the 4.0% drift ratio. than

in the first cycle. Except for specimen S-G-50-200-100, which has poor ductility

and failed quickly after the 2.0% lateral drift ratio, the drift ratio at the peak load,

the remaining three specimens of the second test group displayed increasing

energy dissipation capacity with increased lateral drift ratio.

As shown in Figure 4.34(a), specimen F-G-50-200 dissipates more energy

than the other four before 2.5% lateral drift ratio, as the specimen suffered more

damage at first cycle owing to the existence of grooves. After failure (lower than

80% of the peak load), less energy is dissipated as demonstrated in Figure

4.33(a). Specimen F-S-450-450 dissipates the least amount of cumulative

energy at first but becomes the one consuming the most energy when reaching

the 3.0% drift ratio. Thus, specimen F-S-450-450 has the best energy dissipation

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capacity in the group after entering yielding stage. Among the remaining three

specimens, the control specimen F-Control dissipates more energy than the

other two do on account of its higher strength, but similar hysteresis curves.

As shown in Figure 4.34(b), specimen S-G-50-200-100 dissipates more

energy than the other three specimens before 3.0% lateral drift ratio, because it

suffered more damage. After failure, the specimen dissipates less energy as

demonstrated in Figure 4.33(b). Among the remaining three specimens, the

control specimen S-control dissipates more energy than the other two on

account of its higher strength.

4.5.3 Equivalent Viscous Damping Ratio

Equivalent viscous damping ratio hyst represents the dissipation due to

the nonlinear (hysteretic) behaviour (Blandon and Priestley 2005), which is a

more effective factor to assess the seismic performance of specimen. Figure

4.35 shows the equivalent damping ratio hyst (referred to as EDR hereafter)

development with increasing lateral drift ratio. The EDR was obtained using

Jacobsen's (1930) approach, dividing dissipated energy by the stored elastic

energy, as indicated in Figure 4.32 and expressed in Equation (4.3).

1 1 2 2

12

EDhyst

m m m m

AF F

(4.3)

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where EDA is the energy dissipated by a cycle; 1mF and 1m are the positive

peak load and its corresponding displacement of the cycle; and 2mF are 2m

the negative peak load and its corresponding displacement of the cycle.

As shown in Figure 4.35(a), the EDR of specimen F-G-50-200 is the

second lowest at first, but quickly exceeds those of the remaining specimens.

The reason for this phenomenon is that specimen F-G-50-200 was the least

ductile and failed quickly after peak load, which lead to its EDR increasing

most quickly after the 1.0% drift ratio. Specimen F-S-450-450 possesses the

second highest EDR after the 1.5% drift ratio, because of its better energy

dissipation capacity and lower stored elastic energy. The remaining specimens

display similar EDR curves. Obvious discrepancies are evident as indicated in

Figure 4.35(a), probably due to the different mechanisms of energy dissipation

of different weakening scheme.

Figure 4.35(b) shows the EDR variation with increased lateral drift ratio

for the second test group. As shown in Figure 4.35(b), the EDRs of all

specimens in the second test group show a growing trend except for a drop at

the 1.5% drift ratio, which is not obvious in the first test group. This might be

because the specimen cracking of the second test group, which occurred during

the 1.0% drift ratio, was more serious than for the first test group. After the 1.5%

drift ratio, continuous yielding of steel bars and concrete cracking caused the

damping ratio to increase.

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The EDR of specimen S-G-50-200-100 was the lowest at first but quickly

exceeded those of the other specimens in the second test group. As specimen

S-G-50-200-100 is the least ductile, it fails quickly after yielding, which leads to

its EDR becoming the highest after the 1.0% drift ratio. Specimens

S-O-500-180 and S-O-500-180-S-300-300 have similar EDR curves versus

lateral drift ratio. The EDR of the control specimen greatly increases at 4.0%,

when the plastic region of the lower column suffered serious damage.

As indicated in Figures 4.35(a) and (b), the EDRs of the second cycle are

lower than those of the first, because less energy is dissipated during the second

cycle but the stored elastic energy remains almost unchanged, as indicated by

the close PTPSs of the first and the second cycles as shown in Figure 4.36 in

next section. The EDR of specimen F-G-50-200 at the second cycle of the 2.0 %

drift ratio is higher than that of the first cycle. This may be because beam

bottom longitudinal bars slipped significantly at this drift ratio, also indicated by

the obvious PTPS degradation between the two cycles as shown in Figure

4.36(a). Even though the specimen F-G-50-200 EDR curve of first cycles keeps

increasing after the 2.0% drift ratio, that of the second cycles descends after the

2.0 % drift ratio due to serious slipping problem brought about by the first

cycles. For the control specimen F-Control and the specimens with openings

F-O-450-150 and F-O-500-180, the distance between the EDRs of first and

second cycles is noticeable at first, but gradually becomes smaller and finally

close to zero at 5% drift ratio. The distance between the two EDRs of first and

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second cycles of specimen F-S-450-450, however, is almost constant with

increased drift ratio.

The bottom column concrete of the control specimen S-Control in the

second test group was seriously damaged during the second cycle of the 4.0%

lateral drift ratio, leading to the EDR of the second cycle suddenly higher than

that of the first cycle. The EDRs of second cycles for the second test group

increase with the increasing drift ratio between 1.0% and 1.5% drift ratios,

unlike the case with the first cycles. This might also be due to the serious

cracking which occurred in the first cycle of the 1.0% drift ratio.

4.5.4 Stiffness Degradation

Figure 4.32 illustrates the definition of peak-to-peak stiffness Keff

(referred to as PTPS hereafter). Keff is defined as the slope of the line linking the

peak-to-peak force-displacement points at each imposed displacement cycle. It

can be taken as an average secant stiffness for both positive and negative

displacement peaks as illustrated by the following equation:

1 2eff

1 2

m m

m m

F FK

(4.4)

Figure 4.36 shows the PTPS variation with increasing lateral drift ratio. As

indicated in Figure 4.36(a), the specimen S-G-50-200-100 has the closest initial

PTPS to that of the specimen F-control. In contrast, the BO and SS techniques

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effectively decreased the initial PTPS of specimens. That means the TG

technique has less effect on specimen initial PTPS than other retrofit techniques.

The control specimen and specimen F-G-50-200 have the highest PTPS at first.

However, the PTPS of the specimen F-G-50-200 descends at a faster rate than

the other four and becomes the lowest after the 1.0% drift ratio. Specimen

F-S-450-450 has a slightly higher PTPS than specimen F-O-500-180 at first but

becomes lower after the 1.5% drift ratio. The control specimen F-Control, the

specimens F-O-450-150 and F-o-500-180 have parallel descending PTPS curves

in relation to lateral drift ratio.

Figure 4.36(b) shows PTPS variation with the increased lateral drift ratio

of the second test group. It is obvious that the control specimen has the highest

PTPS at first. The specimen S-G-50-200-100 has a slightly higher PTPS than

for the specimen S-O-500-180. However, the PTPS of the specimen

S-G-50-200-100 decreases at the faster speed than the others and has the lowest

value after the 2.0% drift ratio, when the peak load has been reached. The

control specimen S-Control and specimen S-O-500-180 share similar PTPS

curves. Specimen S-O-500-180-S-300-300 has the lowest PTPS first. Its PTPS

degrades the most slowly and gets much closer to those of the control specimen

S-Control and specimen S-O-500-180 at the 4.0% drift ratio.

4.5.5 Plastic Hinge Lengths

According to Hines et al. (2004), the equivalent plastic hinge length is a

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length of a structural member over which the plastic curvature can be assumed

constant. A physical plastic hinge, within which the plasticity region actually

spreads, does not include the strain penetration. By assuming that the plastic

curvature is linearly distributed over the physical plastic hinge region, Hines et

al.(2004) proposed the following equation to relate the length of the physical

plastic hinge to that of the equivalent plastic hinge:

/ 2p ppl ypL L L (4.5)

where pL is the length of the equivalent plastic zone, pplL is the length of the

physical plastic zone and ypL is the plastic hinge length due to yield

penetration into the base. Figure 4.37 clearly presents the plastic hinge

composition.

Zhao (2012) investigated plastic hinges in RC structures through

experiments and finite element (FE) modelling. According to Zhao (2012), the

plastic hinge of an RC column consists of three physical zones: the bar yielding

zone, concrete crushing zone, and curvature concentration zone. The bar

yielding zone was defined as the region in which the reinforcing steel in tension

had reached or exceeded its yield stress. The strain concentration zone, whose

length was less than or equal to that of the bar yielding zone was the region

where most of the plastic curvature was concentrated. The concrete crushing

zone was defined as the region in which compressive strains greater than 0.002,

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the strain at the peak concrete strength or 0.006, the strain at complete concrete

crushing. Through FE modelling and experimental studies, Zhao (2012) stated

that the lengths of the bar yielding zone and the concrete crushing zone served

as upper and lower bounds of the physical plastic zones respectively.

Jiang et al. (2014) applied a digital image correlation (DIC) method to

continuously record variation of the strain field on the external column face.

Jiang et al. (2014) focused on the curvature concentration zone, which is similar

to the strain concentration zone studied by Zhao (2012), as it has a closer

relationship with the curvature plastic hinge. The curvature concentration zone

was measured with the DIC method and determined using the dividing point

concept. The dividing point is a point on one side of which curvatures keep on

increasing, while on the other side of the point, they remain almost constant as

the applied displacement increases. The dividing point defines the edge of the

curvature concentration zone. Jiang et al. (2014) also mentioned that the lengths

of curvature concentration zones of the tested columns were consistent with the

lengths of bar yielding zones. In addition, Jiang et al. (2014) applied the

relationship proposed by Hines et al. (2004) to relate the physical plastic zone to

the equivalent plastic zone.

In the author's tests, only strain gauges were available for use to determine

the length of the physical plastic region. The plastic hinge length accounting for

yield penetration into the base could not be estimated. The bar yielding region

was taken as the physical plastic region and the strain concentration zone was

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determined by the dividing point method. The strain distribution of a

longitudinal bar under tension is more reliable in determining a bar yielding

region. The lengths of yielding and strain concentration zones measured by

strain gauges through dividing point are summarized in Table 4.4.

As shown in Figure 4.17(a), at 3.0% drift ratio, the x-coordinate value of

the beam bar yielding region of specimen F-control is about 360 mm and half of

the column width is 160 mm , which means the bar yielding region length is

about 200 mm (i.e. 360-160=200mm). This value is smaller than that of

specimen S-Control mainly because specimen F-control failed at 4.0% lateral

drift ratio. At that drift ratio, the length of the yielding zone is not fully

developed.

For specimen F-G-50-200, the strain concentration zone was located at the

groove region. As indicated in Figure 4.17(b), the x-coordinate values of the

beam bar yielding region and the strain concentration zone are about 350 mm

and 280 mm respectively. Thus the lengths of the bar yielding region and the

strain concentration zone are about 190 mm and 120 mm respectively (i.e.

350-160=190, 280-160=120). It can be conclude that the transverse groove

shortened the plastic hinge length.

As for specimen F-S-450-450, the x-coordinate value of the beam bar

yielding region was 470 mm as indicated in Figure 4.12(e), which means that

the length of the bar yielding region was 310 mm. The strain concentration zone

201

length was hard to determine as the number of strain gauges was insufficient.

The length of the strain concentration zone is more than 120 mm as indicated in

Figure 4.17(e).

As shown in Figure 4.17(f), the x-coordinate values of the beam bar

yielding region and the strain concentration zone of specimen S-control are

about 520 mm and about 270 mm respectively, which means the lengths of the

bar yielding region and the strain concentration zone are about 360 mm and 110

mm respectively.

For specimen S-G-50-200-100, the length of the bar yielding region is

about 310mm. It is obvious that the strain concentration zone is located within

the groove region within the red dashed line as shown in Figure 4.36(b). The

strain concentration zone length was hard to determine as the number of strain

gauges was insufficient.

The specimens with beam openings had several plastic regions and thus,

are not included in the discussion.

4.5.6 Deformation Components

LVDTs were applied to measure the deformation of T-beams and columns

based on the plane cross-section assumption. Due to the existence of transverse

beams and slab, the joint deformation could not be measured. The arrangement

of LVDTs was described in Chapter 3 and how the deformations of beams and

202

columns can be obtained was also explained in Chapter 3.

4.5.6.1 Beam drift versus beam tip load hysteresis curves

Beam tip load versus beam displacement hysteresis curves were plotted to

reveal the beam behaviour under cyclic loading. As mentioned previously, the

beam tip loads of the left and right hand beams were directly measured by load

cells placed on the beam supports. The beam drift was calculated using LVDTs

data and equations in Chapter 3. The elastic range deformations of beams and

columns were calculated using the Equations (3.5) and (3.10) respectively. The

plastic range deformations of beams and columns were determined using

LVDTs data and Equations (3.4) and (3.9) respectively. In Figures 4.38, a

positive displacement or load means the left or right beam is sustaining a

sagging moment where the compression is in the slab. When a specimen was

under push action, the left hand beam sustained a hogging moment and the right

hand beam a sagging moment.

Figure 4.38, giving curves of beam tip load versus calculated beam

displacement, does not include specimens with openings. That's because the

deformation of beams with openings could not be accurately measured by the

LVDTs as the plane cross-section assumption was not satisfied at larger lateral

drifts. The testing of specimen F-Control stopped at the first cycle of 4.0% drift

ratio. The deformations of beams and columns of the strengthened specimen

F-Control were not recorded by LVDTs.

203

As indicated in Figure 4.38, the left and right hand beams of specimens

showed similar behaviour under cyclic loading. In the first test group, two sets

of LVDTs, one for beam end rotation and the other for plastic concentration,

were used to measure the plastic deformation of the left hand beam but only one

set was used on the right hand beam due to insufficient LVDTs available. The

LVDTs measured length for the specimens F-control, F-S-450-450 and

F-G-50-200 plastic region was 360mm, which was equal to the effective height

of the main beam. The length for the beam end rotation was 120 mm, which was

the shortest length possible allowing for the placing of LVDTs. If the beam bar

bond-slipping phenomenon was obvious, more plastic deformation in the left

hand beam of the first test group would be measured by LVDTs than for the

right hand one as only the left beam was equipped with LVDTs for the beam end

rotation. As indicated in Figures 4.38(b) and (c), more deformations in the left

hand beams of specimens F-G-50-200 and F-S-450-450 were measured in the

later cycles than in the right hand beams. However, this phenomenon was not

obvious on the specimen F-Control. The reason for this was the existence of

grooves on beams and slab slits, which weakened the bond between concrete

and steel bars at the beam-column interface. Thus, at larger lateral drift ratios,

beam end rotation resulting from beam bar bond slipping was more obvious.

The second test group had only one set of LVDTs available to measure the

deformation of the plastic region, whose length for measurement was 360 mm

for the S-Control specimen and 250 mm for the S-G-50-200-100. As the test on

204

the specimen S-Control was twice stopped due to an MTS problem, the

calculated beam deformation were much in error as indicated in Figure 4.38(d).

The red line showed a huge error during the last cycle owing to the bottom

column bars buckling. The beam drift versus beam tip load curve for the

specimen S-G-50-200-100 is shown in Figure 4.38(e). As indicated in Figure

4.38(e), the two beams show similar behaviour under cyclic loading.

4.5.6.2 Deformation contributions of beams and columns to the lateral drift

The deformation contributions of beams and columns to the lateral drift of

the first and second test groups are presented in Figure 4.39. The column plastic

hinge lengths for LVDTs measurement were all equal to 280 mm. As the

deformation of those beams with openings could not be measured by LVDTs,

their deformation contributions could not be determined. The LVDT data

relating to the column plastic zones of specimen F-Control were lost, thus the

columns contributions in the case of the F-Control specimen is also unknown.

As a result, in Figures 4.39(a) and (b), the plotted contribution ratios of beams

and columns are the sums of the left and right hand beams and the top and

bottom columns respectively. Only the first cycle of each drift ratio was

included. Shown in Figure 4.39(a) are the variation curves of contribution ratio

of beam to the lateral drift versus the drift ratio. It can be seen that except for the

two control specimens F-Control and S-Control case, which have a decreasing

contribution ratio, the contribution of beam has a tendency to increase. The

descending scope of specimen S-Control is more obvious than that of the

205

specimen F-Control. The beams contribution ratio for specimen S-Control was

lower than for specimen F-Control after the 1.0% drift ratio. This implies the

beams of specimen S-Control were stronger and thus the deformations of joint

panel and columns dominated the specimen's deformation. The beams

contribution ratio for specimen S-G-50-200-100 shows an ascending trend with

increased drift ratio and the rate of ascent increases after the 2.0% drift ratio,

where the specimen reached its peak load and beam bottom bars slipping

became more severe.

Except for the specimens S-Control and S-O-500-180, whose column

contribution ratios drop during the middle cycles and increase after a 3.0% drift

ratio as the two specimens all suffered column bottom bars buckling at the final

drift ratio, the contribution ratios of columns for all the available specimens

show a descending trend with increased drift ratio, as shown in Figure 4.39(b).

It should be due to the slab slits existence, the LVDT sets had error in measuring

the column plastic deformation at a higher drift ratio. That is why the column

contribution ratio of specimen F-S-450-450 became close to 0% at the 4.0%

drift ratio. The ratio was accurate before the 1.5% drift ratio after a check of the

obtained data. The decreasing column deformation contribution ratio means the

deformations of specimens were gradually dominated by the deformations of

beams as the drift ratio increased. The column contribution ratio for specimens

F-G-50-200 and S-G-50-200-50 decreases quickly with increased drift ratio and

becomes close to zero at the final drift ratio, because the slipping of beam bars

206

dominates the specimen's deformation. The greater column contribution ratio for

the control specimen S-Control than for specimen S-O-500-180 reveals that the

existence of beam openings lessens the column deformation contribution ratio.

In addition, larger web openings weaken the column deformation contribution,

indicated by comparing the column contribution ratio of specimen F-450-150

and F-500-180. The existence of slab slits further lessens the column

deformation contribution ratio as indicated in Figure 4.39(b).


This chapter has presented the results of two test groups of RC

beam-column joints are presented in this chapter. The effects of the proposed

retrofit techniques on the performance of RC beam-column joints (RC 3D joints)

have been explored. Based on the test results and discussion above in this

chapter, the following conclusions can be drawn:

(1)The existence of a transverse groove (TG) can lower the moment-capacity of

a T-beam greatly. However, the failure mode was non-ductile because the

main beam bottom longitudinal bars slipped in the joint region, even though

the groove was not placed close to the beam-column interface. The slab

slitting (SS) method can effectively lower the moment capacity of a T-beam

as it removes the contribution of the cut slab longitudinal bars. Meanwhile,

the failure mode is beam end flexural failure. The beam opening (BO) size

should be well designed (e.g. increasing the opening size)to lead to an

207

obvious four-hinges mechanism if the BO technique is applied as a retrofit

measure. The slab slits can help a specimen, not sufficiently weakened by

the BO technique, to realize the four-hinges mechanism. Slot-shaped

openings have similar effect on the seismic behaviour of a RC beam-column

joint as the rectangular one does.

(2) As the horizontal load recorded by the MTS machine included friction force,

which varied during test and among different specimens, the calculated

column shear force was used for discussions instead. The specimens

retrofitted by the proposed techniques had lower stiffness and strength than

the control specimens. Both the BO and SS techniques decrease the stiffness

of T-beams under both negative and positive loading. The TG technique

only has obvious effect on the stiffness and strength of a T-beam when it

was under negative loading. When the groove is not placed close to the

beam-column interface, the effect of the TG technique on T-beam stiffness

becomes weaker. The positive strength of T-beams is slightly affected by the

existence of BO. The SS technique decreases both negative and positive

strength of T-beams.

(3) When the control specimen has good ductility, the TG technique leads to

very poor ductility. The BO technique keeps the specimen ductility almost

unchanged. The SS technique decreases the ductility of a specimen. When

the control specimen has a poor ductility, the TG technique has a small

effect on specimen ductility but leads to a smaller yield displacement than

208

that of control specimen. The BO technique slightly increases the specimen

ductility. When the combination of the BO and SS techniques is applied,

specimen ductility can be increased in addition to effectively decreasing the

moment-capacity of T-beams.

(4) The Specimen retrofitted by the SS technique had better energy dissipation

capacity than those by the BO and the TG technique. The specimens

retrofitted the TG technique had best energy dissipation capacity at first but

worse energy dissipation capacity after they reached peak load.

(5) Equivalent viscous damping ratios (EDRs) of specimens basically increased

with increasing drift ratio. The EDRs of those specimens weakened by TGs

were almost the highest before they failed. The EDR of the specimen

F-S-450-450 was the second highest among the first test group. The BO

technique has slight effect on a specimen EDR. Generally, the EDRs of the

second cycle of each drift ratio are lower than those of the first cycle.

(6) Unlike the BO, SS methods and their combination, the TG method has less

effect on the initial PTPS of specimens. In addition, the PTPS of the

specimens weakened by TGs, reduced the most quickly with increasing drift

ratio, as those specimens had poor ductility and failed soon after peak load

had been reached. The specimen retrofitted by a combination of the BO and

SS techniques has the lowest speed of PPTS descending.

209

(7) The TG technique shortens plastic hinges length of T-beams as the

deformation of T-beams is concentrated on the groove region.

(8) Except for two control specimens F-Control and S-Control, beams’

deformation contribution to the lateral drift ratio increased with the

increasing lateral drift ratio. Thus the contribution of columns deformation

was decreased with the increasing lateral drift ratio.

4.7 REFERENCES

Blandon, C. A., & Priestley, M. J. N. (2005), "Equivalent viscous damping

equations for direct displacement based design", Journal of earthquake

Engineering, 9(sup2), 257-278.

Hines, E. M., Restrepo, J. I., and Seible, F. (2004), "Force-displacement

characterization of well-confined bridge piers", ACI Structural

Journal, 101(4), 537-548.

Jacobsen, L. S. (1930), "Steady forced vibration as influenced by

damping", Trans. ASME-APM, 52(15), 169-181.

Jiang, C., Wu, Y. F., & Wu, G. (2014), "Plastic hinge length of FRP-confined

square RC columns", Journal of Composites for Construction, 18(4),

04014003.

Zhao, X. (2012). Investigation of plastic hinges in reinforced concrete (RC)

210

structures by finite element method and experimental study, Doctoral

dissertation: City University of Hong Kong

211

(a) Removing the cracked concrete and clean the exposed surface with clean

water

(b) Shear strengthening the opening region with steel bars

(c) Building up wooden temple for concrete casting;

212

(d) Recasting specimen

Figure 4.1 The control specimen strengthening process

(a) Flexural cracks appeared on beam at 0.5% drift ratio

(b) Cracks first appeared on bottom column and joint panel at 1% drift ratio

213

(c) Cracks distribution at peak load of 2% drift ratio

(d) Cracks distribution at first failure

214

(e) Cracks distribution at 5% after being strengthening

Figure 4.2 Failure mode and cracks distribution of Specimen F-Control

215

(a) Flexural cracks appeared on beam, slab and column at 0.5% drift ratio

(b) Cracks appeared on joint panel at 1% drift ratio

(c) Cracks distribution at peak load of 1.5% drift ratio

216

(d) Cracks distribution after failure at 5 % drift ratio

Figure 4.3 Failure mode and cracks distribution of specimen F-G-50-200

(a) Flexural cracks first appeared on beam and slab at 0.5% drift ratio

Bottom beam bars slippage

217

(b) Cracks appeared on column at 1% drift ratio

(c) Cracks appeared on joint panel at 1.5% drift ratio

(d) Cracks distribution at peak load of 2.5 % drift ratio

218

(e) Cracks distribution after failure at 5 % drift ratio

Figure 4.4 Failure mode and cracks distribution of specimen F-O-450-150


219


(c) Cracks distribution at peak load of 2 % drift ratio

Local flexural rotation

220

(d) Cracks distribution and deformation shape after failure at 5 % drift ratio

Figure 4.5 Failure mode and cracks distribution of specimen F-O-500-180


221


(b) Cracks appeared on joint panel at 1.5% drift ratio

222


223

(d) Cracks distribution after failure at 5 % drift ratio

Figure 4.6 Failure mode and cracks distribution of specimen F-S-450-450

(a) Flexural cracks appeared on beam at 0.25% drift ratio

(b) Cracks appeared on slab and column at 0.5% drift ratio

224

(c) Cracks appeared on joint panel at 1% drift ratio

(d) Cracks distribution at peak load of 2 % drift ratio

225

(e) Column bottom bars buckled at 5% drift ratio

Figure 4.7 Failure mode and crack distributions of specimen S-Control

226

(a) A flexural crack appeared on slab at 0.25% drift ratio

(b) Cracks appeared on bottom column at 0.5% drift ratio

(c) Cracks appeared on joint panel at 1% drift ratio

227

(d) Cracks distribution at peak load of 2 % drift ratio

(e) Cracks distribution and failure mode at 5 % drift ratio

Figure 4.8 Failure mode and cracks distribution of specimen S-G-50-200-100

Bottom beam bars slippage

228

(a) Cracks appeared on beam, slab and column at 0.5% drift ratio


229


(d) Cracks distribution and failure mode at 5 % drift ratio

Figure 4.9 Failure mode and cracks distribution of specimen S-O-500-180

(a) Cracks appeared on beam and slab at 0.5% drift ratio

230

(b) Cracks appeared on column and joint panel 1% drift ratio


231

(d) Cracks distribution and failure mode at 5 % drift ratio

Figure 4.10 Failure mode and crack distribution of specimen S-O-500-180


232

(a) Specimen F-Control

(b) Specimen F-G-50-200

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

200

400

600

800

1000

1200

Lateral Drift(%)

Axi

al L

oad(

kN)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144

Lateral Displacement (mm)

J-Control-Axial Load-Strengthened J-Control-Axial Load

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

200

400

600

800

1000

1200

Lateral Drift (%)

Axi

al L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Axial Load

233

(c) Specimen F-O-450-150

(d) Specimen F-O-500-180

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

200

400

600

800

1000

1200

Lateral Drift(%)

Axi

al L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Axial load

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

200

400

600

800

1000

1200

1400

Lateral Drift(%)

Axi

al L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Axial load

234

(e) Specimen F-S-450-450

(f) Specimen S-Control

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

200

400

600

800

1000

1200

Lateral Drift (m%)

Axi

al L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Axial load

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

100

200

300

400

500

600

700

800

900

1000

Lateral Drift (%)

Axi

al L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Axial load-1 Axial load-2 Axial load-3

235

(g) Specimen S-G-50-200-100

(h) Specimen S-O-500-180

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

100

200

300

400

500

600

700

800

900

1000

Lateral Drift (m%)

Axi

al L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Axial load

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

100

200

300

400

500

600

700

800

900

1000

Lateral Drift (%)

Axi

al L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Axial load

236

(i) Specimen S-O-500-180-S-300-300

Figure 4.11 Hysteresis curves of the top-of-column axial loads


-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

100

200

300

400

500

600

700

800

900

1000

Lateral Drift (m%)

Axi

al L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Axial load

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-200

-150

-100

-50

0

50

100

150

Lateral Drift(%)

Bea

m T

ip L

oad

Load

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-Control-Left-Strengthened F-Control-Right-Strengthened F-Control-Left F-Control-Right

237



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-100

-80

-60

-40

-20

0

20

40

60

80

100

Lateral Drift(%)

Bea

m T

ip L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-G-50-200-Left F-G-50-200-Right

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-O-450-150-Left F-O-450-150-Right

238



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-O-500-180-Left F-O-500-180-Right

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-S-450-450-Left F-S-450-450-Right

239



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144 Lateral Displacement (mm)

S-Control-Left-1 S-Control-Left-2 S-Control-Left-3 S-Control-Right-1 S-Control-Right-2 S-Control-Right-3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Bea

m T

ip L

oad

Load

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


S-G-50-200-100-Left S-G-50-200-100-Right

240


(i) Specimen S-O-500-180-S-300-300

Figure 4.12 Hysteresis curves of the beam tip loads

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


S-O-500-180-Left S-O-500-180-Right

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


S-O-500-180-S-300-300-Left S-O-500-180-S-300-300-Right

241



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Col

umn

Shea

r For

ce (k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Envelop Curve F-Control F-Control-Strengthened Yield point Peak point Failure point

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Col

umn

Shea

r For

ce(k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Envelop Curve F-G-50-200 Yield point Peak point Failure point

0.75Hpeak

Hpeak

δy=-21.8 mm

0.80Hpeak

δu=-96.2 mm

0.75Hpeak

Hpeak 0.80Hpeak

δu=51.4 mm δy=24.3 mm

242



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Col

umn

Shea

r For

ce(k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Envelop Curve F-O-450-150 Yield point Peak point Failure point

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Col

umn

Shea

r For

ce(k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Envelop Curve F-O-500-180 Yield point Peak point Failure point

δu=111.1 mm δy=25.5mm

0.75Hpeak

Hpeak

0.80Hpeak

δu=101.1 mm δy=24.9 mm

0.75Hpeak

Hpeak

0.80Hpeak

243



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Col

umn

Shea

r For

ce(k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Envelop Curve F-S-450-450 Yield point Peak point Failure point

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Col

umn

Shea

r For

ce(k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Envelop Curve S-Control Yield point Peak point Failure point

δu=72.8 mm δy=23.7 mm

0.75Hpeak

Hpeak

0.80Hpeak

δu=86.9 δy=28.6 mm

0.75Hpeak

Hpeak

0.80Hpeak

244



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Col

umn

Shea

r For

ce(k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Envelop Curve S-G-50-200-100 Yield point Peak point Failure point

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Col

umn

Shea

r For

ce(k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Envelop Curve S-O-500-180 Yield point Peak point Failure point

δu=61.2 mm δy=20.5 mm

0.75Hpeak

Hpeak

0.80Hpeak

δu=106.00mm

δy=29.88mm

0.75Hpeak

Hpeak

0.80Hpeak

245

(i) Specimen S-O-500-180-S-300-300

Figure 4.13 Column shear force versus top-of-column lateral drift ratio hysteresis

and envelop curves


-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Col

umn

Shea

r For

ce(k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


Envelop Curve S-O-500-180-S-300-300 Yield point Peak point Failure point

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-200

-150

-100

-50

0

50

100

150

Lateral Drift (%)

Col

umn

Shea

r For

ce (k

N)

F-Control-MTS-Retrofit F-Control-Calculated with Beam Tip Loads-Strengthened F-Control-MTS F-Control-Calculated with Beam Tip Loads

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144 Lateral Displacement (mm)

δu=102.8 mm δy=28.6 mm

0.75Hpeak

Hpeak

0.80Hpeak

246



-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift(%)

Col

mun

She

ar F

orce

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-G-50-200-MTS F-G-50-200-Calculated with Beam Tip Loads

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Displacement(mm)

Col

umn

Shea

r For

ce (k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-O-450-150-MTS F-O-450-150-Calculated with Beam Tip Loads

247



Figure 4.14 Comparison between lateral load applied by MTS and calculated

column shear force

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift(%)

Col

umn

Shea

r For

ce (k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


J-O-500-180-MTS J-O-4500-180-Calculated with Beam Tip Loads

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-150

-100

-50

0

50

100

150

Lateral Drift (%)

Col

umn

Shea

r For

ce (k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-S-450-450-MTS F-S-450-450-Calculated with Beam Tip Loads

248


(a) The second test group

Figure 4.15 Envelop curves of column shear force versus lateral drift ratio

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-200

-150

-100

-50

0

50

100

150

Lateral Drift (%)

Col

umn

Shea

r For

ce (k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-Control F-G-50-200 F-O-450-150 F-O-500-180 F-S-450-450

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-200

-150

-100

-50

0

50

100

150

200

Lateral Drift (%)

Col

umn

Shea

r For

ce (k

N)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


S-Control S-G-50-200-100 S-O-500-180 S-O-500-180-S-300-300

249



Figure 4.16 Envelop curves of beam tip loads versus lateral drift ratio

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-200

-150

-100

-50

0

50

100

150

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-Control F-G-50-200 F-S-450-450 F-O-450-150 F-O-500-180

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-200

-150

-100

-50

0

50

100

150

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144



250

(a) Specimen F-control


200 250 300 350 400 450-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05 Push-TB11,TB12,TB13

X Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3%

200 250 300 350 400 4500

0.2

0.4

0.6

0.8

1

1.2

1.4 Pull-TB11,TB12,TB13

X Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3%

200 300 400 500 600 700-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4 Push-TB11,TB12,TB13,TB14

X Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

200 300 400 500 600 7000

0.5

1

1.5

2

2.5

3

3.5 Pull-TB11,TB12,TB13,TB14

X Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

251



200 400 600 800 1000-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5 Push-TB11,TB12,TB13,TB14,TB15

X Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

200 400 600 800 10000

0.5

1

1.5

2

2.5 Pull-TB11,TB12,TB13,TB14,TB15

X Direction (mm) N

orm

aliz

ed S

train

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

200 300 400 500 600 700-1

-0.5

0

0.5

1

1.5

2 Push-TB11,TB12,TB13,TB14

X Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

200 300 400 500 600 7000

0.5

1

1.5

2

2.5 Pull-TB11,TB12,TB13,TB14

X Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

252


(f) Specimen S-control

200 300 400 500 600 700-1

-0.5

0

0.5

1


X Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

200 300 400 500 600 7000

0.5

1

1.5

2

2.5

3

3.5

4 Pull-TB11,TB12,TB13,TB14

X Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

200 300 400 500 600-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4


X Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4%

200 300 400 500 6000

0.5

1

1.5

2

2.5


X Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4%

253



200 300 400 500 600-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 Push-TB11,TB12,TB13,TB14

X Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

200 300 400 500 600-1

-0.5

0

0.5

1

1.5


X Direction (mm) N

orm

aliz

ed S

train

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

200 300 400 500 600 700-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 Push-TB11,TB12,TB13,TB14,TB15

X Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

200 300 400 500 600 7000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 Pull-TB11,TB12,TB13,TB14,TB15

X Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

254

(i) Specimen S-O-500-180-S-300-300

Figure 4.17 Strain variation along the X direction at the peak displacement of

each cycle

(a) Specimen F-control

200 300 400 500 600 700-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3 Push-TB11,TB12,TB13,TB14,TB15

X Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

200 300 400 500 600 700-1

0

1

2

3

4

5

6 Pull-TB11,TB12,TB13,TB14,TB15

X Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-200 -300 -400 -500 -600 -700-0.5

0

0.5

1

1.5

2 Push-TL11,TL21,TL31,TL41

Y Direction (mm)

Nor

mai

lzed

Stra

in

0.5% 1% 1.5% 2% 2.5% 3%

-200 -300 -400 -500 -600 -7000

0.5

1

1.5

2

2.5

3

3.5 Pull-TL11,TL21,TL31,TL41

Y Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3%

255



-200 -300 -400 -500 -600 -700-0.5

0

0.5

1

1.5

2

2.5


Y Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-200 -300 -400 -500 -600 -7000.5

1

1.5

2

2.5

3

3.5

4 Pull-TL11,TL21,TL31,TL41

Y Direction (mm) N

orm

lized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-200 -300 -400 -500 -600 -700-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5 Push-TL11,TL21,TL31,TL41

Y Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-200 -300 -400 -500 -600 -7000

1

2

3

4

5


Y Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

256



-200 -300 -400 -500 -600 -700-0.4

-0.2

0

0.2

0.4

0.6

0.8


Y Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4%

-200 -300 -400 -500 -600 -7000.5

1

1.5

2

2.5

3

3.5


Y Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4%

-200 -300 -400 -500 -600 -700-1

0

1

2

3

4

5


Y Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-200 -300 -400 -500 -600 -700-0.5

0

0.5

1

1.5

2

2.5

3


Y Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

257

(f) Specimen S-control


-200 -300 -400 -500 -600 -700-1

0

1

2

3

4

5


Y Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4%

-200 -300 -400 -500 -600 -7000

1

2

3

4

5

6

7


Y Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4%

-200 -300 -400 -500 -600 -700-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4


Y Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-200 -300 -400 -500 -600 -7000

0.5

1

1.5

2

2.5

3


Y Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

258


(i) Specimen S-O-500-180-S-300-300

Figure 4.18 Strain variation along the Y direction at the peak displacement of

each cycle

-200 -300 -400 -500 -600 -700-0.5

0

0.5

1

1.5

2

2.5

3

3.5


Y Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-200 -300 -400 -500 -600 -7000

1

2

3

4

5


Y Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-200 -300 -400 -500 -600 -700-0.2

0

0.2

0.4

0.6

0.8

1


Y Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-200 -300 -400 -500 -600 -700-0.2

0

0.2

0.4

0.6

0.8

1


Y Direction (mm)

Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

259



-1 -0.5 0 0.5 1 1.5-600

-400

-200

0

200

400

600

800 Push-TC31-TC35,BC31-BC34

Normalized Strain

Z D

irect

ion

(mm

)

0.5% 1% 1.5% 2% 2.5% 3%

-1 -0.5 0 0.5-600

-400

-200

0

200

400

600

800 Pull-TC31-TC35,BC31-BC34

Normalized Strain

Z D

irect

ion

(mm

)

0.5% 1% 1.5% 2% 2.5% 3%

-0.4 -0.2 0 0.2 0.4-600

-400

-200

0

200

400

600


Normalized Strain

Z D

irect

ion

(mm

)

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-0.8 -0.6 -0.4 -0.2 0 0.2-600

-400

-200

0

200

400

600


Normalized Strain

Z D

irect

ion

(mm

)

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

260



-3 -2 -1 0 1-600

-400

-200

0

200

400

600


Normalized Strain

Z D

irect

ion

(mm

)

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-6 -4 -2 0 2-600

-400

-200

0

200

400

600


Normalized Strain

Z D

irect

ion

(mm

)

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-0.5 0 0.5 1-600

-400

-200

0

200

400

600

800 Push-TC32,TC34,TC35,BC32,BC34

Normalized Strain

Z D

irect

ion

(mm

)

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-1 -0.5 0 0.5-600

-400

-200

0

200

400

600

800 Pull-TC32,TC34,TC35,BC32,BC34

Normalized Strain

Z D

irect

ion

(mm

)

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

261



-1.5 -1 -0.5 0 0.5 1-600

-400

-200

0

200

400

600


Normalized Strain

Z D

irect

ion

(mm

)

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-1 -0.5 0 0.5 1-600

-400

-200

0

200

400

600


Normalized Strain Z

Dire

ctio

n (m

m)

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-6 -4 -2 0 2-600

-400

-200

0

200

400

600 Push-TC71-TC75,CC7,BC71-BC74

Normalized Strain

Z D

irect

ion

(mm

)

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4%

-2 0 2 4 6 8-600

-400

-200

0

200

400

600 Pull-TC71-TC75,CC7,BC71-BC74

Normalized Strain

Z D

irect

ion

(mm

)

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4%

262



-1 -0.5 0 0.5-400

-300

-200

-100

0

100

200

300


Normalized Strain

Z D

irect

ion

(mm

)

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-0.5 0 0.5 1-400

-300

-200

-100

0

100

200

300


Normalized Strain

Z D

irect

ion

(mm

)

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-4 -3 -2 -1 0 1-400

-300

-200

-100

0

100

200

300

400 Push-TC71-TC72,CC7,BC71,BC73

Normalized Strain

Z D

irect

ion

(mm

)

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-2 0 2 4 6-400

-300

-200

-100

0

100

200

300

400 Pull-TC71-TC72,CC7,BC71,BC73

Normalized Strain

Z D

irect

ion

(mm

)

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

263

(i) Specimen S-O-500-180-S-300-300

Figure 4.19 Strain variation along the Z direction at the peak displacement of

each cycle


-1.5 -1 -0.5 0 0.5 1-400

-300

-200

-100

0

100

200

300


Normalized Strain

Z D

irect

ion

(mm

)

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-1 -0.5 0 0.5 1-400

-300

-200

-100

0

100

200

300


Normalized Strain Z

Dire

ctio

n (m

m)

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-150-100

-500

50100

150

-200

-100

0

100

200-1

-0.5

0

0.5

1

1.5

X Direction (mm) Y Direction (mm)

Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3%

264



-150-100

-500

50100

150

-200

-100

0

100

200-1.5

-1

-0.5

0

0.5

1


Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-150-100

-500

50100

150

-200

-100

0

100

200-1.5

-1

-0.5

0

0.5

1


Nor

mal

ized

Stra

in

0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

265

(d )Specimen F-S-450-450

(e) Specimen S-Control

-150-100

-500

50100

150

-200

-100

0

100

200-1

-0.5

0

0.5

1


Nor

mal

ized

Stra

in 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-120-100

-80-60

-40-20

0

-200

-100

0

100

200-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4


Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2%

266

(f) Specimen S-G-50-200-100

(g) Specimen S-O-500-180

-150-100

-500

50100

150

-200

-100

0

100

200-2

-1.5

-1

-0.5

0

0.5

1


Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5%

-150-100

-500

50100

150

-200

-100

0

100

200-8

-6

-4

-2

0

2

4


Nor

mal

ized

Stra

in

0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4%

267

(h) Specimen S-O-500-180-S-300-300

Figure 4.20 Rotations of bottom column critical section under push action

(a) CS1 strain gauge

-150-100

-500

50100

150

-200

-100

0

100

200-1.5

-1

-0.5

0

0.5

1

1.5


Nor

mal

ized

Stra

in 0.25% 0.5% 1% 1.5% 2% 2.5% 3% 4% 5%

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-Control-CS1 F-G-50-200-CS1 F-S-450-450-CS1

268

(b) CS2 strain gauge

(c) CS3 strain gauge

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-1

0

1

2

3

4

5

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-Control-CS2 F-S-450-450-CS2

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-G-50-200-CS3 F-S-450-450-CS3

269

(d) CS4 strain gauge

Figure 4.21 Strain level of joint stirrups at the peak displacement of each cycle of

the first test group

(a) CS12 strain gauge

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

1

2

3

4

5

6

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


F-Control-CS4 F-O-450-150-CS4 F-G-50-200-CS4 F-S-450-450-CS4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


S-Control-CS12 S-G-50-200-100-CS12 S-O-500-180-CS12

270

(b) CS22 strain gauge

(c) CS32 strain gauge

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144


S-Control-CS22 S-G-50-200-100-CS22 S-O-500-180-CS22 S-O-500-180-S-300-300-CS22

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-0.5

0

0.5

1

1.5

2

2.5

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144



271

(d) CS42 strain gauge

Figure 4.22 Strain level of joint stirrups at the peak displacement of each cycle of

the second test group

Figure 4.23 Strain level of FRP U-jacket at the peak displacement of each cycle of

specimen F-G-50-200

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-0.5

0

0.5

1

1.5

2

2.5

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -120 -96 -72 -48 -24 0 24 48 72 96 120 144



-6 -4 -2 0 2 4 6-0.02

0

0.02

0.04

0.06

0.08

0.1

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


U1 U2 U3 U4 U5

272

(a) Strain gauges mounted on the FRP U-jacket

(b) Strain gauges mounted on the FRP wrap

Figure 4.24 Strain level of FRP at the peak displacement of each cycle of

specimen F-O-450-150

-6 -4 -2 0 2 4 6-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


U1 U2 U3

-6 -4 -2 0 2 4 6-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


U1 U4 U5

-6 -4 -2 0 2 4 6-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


W1 W2 W3

273




specimen F-O-500-180

-6 -4 -2 0 2 4 6-0.05

0

0.05

0.1

0.15

0.2

Lateral Drift (%)

Nom

aliz

ed S

train

-144-96 -48 0 48 96 144


U1 U2 U3

-6 -4 -2 0 2 4 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Lateral Drift (%)

Nom

aliz

ed S

train

-144-96 -48 0 48 96 144


U4 U5 U6

-6 -4 -2 0 2 4 60

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Lateral Drift (%)

Nom

aliz

ed S

train

-144-96 -48 0 48 96 144


U7 U8 U9

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

0.3

0.4

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


RW1 RW2 RW3 RW4 RW5

-6 -4 -2 0 2 4 6

0

0.2

0.4

0.6

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


RW6 RW7 RW8 RW9 RW10

-6 -4 -2 0 2 4 6

0

0.1

0.2

0.3

0.4

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


LW1 LW2 LW3 LW4 LW5

-6 -4 -2 0 2 4 6

0

0.1

0.2

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


LW6 LW7 LW8 LW9 LW10

274

Figure 4.26 Strain level of FRP U-jacket at the peak displacement of each cycle of

specimen S-G-50-200-100


-6 -4 -2 0 2 4 6-0.05

0

0.05

0.1

0.15

0.2

Lateral Drift (%)

Nom

aliz

ed S

train

-144-96 -48 0 48 96 144 Lateral Displacement (mm)

RU1 RU2 RU3

-6 -4 -2 0 2 4 6-0.05

0

0.05

0.1

0.15

0.2

Lateral Drift (%)

Nom

aliz

ed S

train


RU4 RU5 RU6

-6 -4 -2 0 2 4 6-0.05

0

0.05

0.1

0.15

0.2

Lateral Drift (%) N

omal

ized

Stra

in


LU1 LU2 LU3

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


RU1 RU2 RU3

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


RU3 RU4 RU5

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


LU1 LU2 LU3

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


LU3 LU4 LU5

275



specimen S-O-500-180

-6 -4 -2 0 2 4 6-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Lateral Drift (%)

Nom

aliz

ed S

train


RW1 RW2 RW3 RW4

-6 -4 -2 0 2 4 6-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Lateral Drift (%)

Nom

aliz

ed S

train


RU9 RU10

-6 -4 -2 0 2 4 6-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Lateral Drift (%)

Nom

aliz

ed S

train


RW5 RW6 RW7 RW8

-6 -4 -2 0 2 4 6-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Lateral Drift (%)

Nom

aliz

ed S

train


LW1 LW2 LW3 LW4

-6 -4 -2 0 2 4 6-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Lateral Drift (%)

Nom

aliz

ed S

train


LU9 LU10

-6 -4 -2 0 2 4 6-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Lateral Drift (%)

Nom

aliz

ed S

train


LW5 LW6 LW7 LW8

276


-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

0.3

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


RU1 RU2 RU3

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

0.3

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


RU3 RU4 RU5

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

0.3

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


LU1 LU2 LU3

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

0.3

Lateral Drift (%)

Nom

aliz

ed S

train

-144 -96 -48 0 48 96 144


LU3 LU4 LU5

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

0.3

0.4

0.5

Lateral Drift (%)

Nom

aliz

ed S

train


RW1 RW2 RW3 RW4

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

0.3

0.4

0.5

Lateral Drift (%)

Nom

aliz

ed S

train


RU9 RU10

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

0.3

0.4

0.5

Lateral Drift (%)

Nom

aliz

ed S

train


RW5 RW6 RW7 RW8

277

(B) Strain gauges mounted on the FRP wrap


specimen S-O-500-180-S-300-300

(a) Definition 1 for yield deformation

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

0.3

0.4

0.5

Lateral Drift (%)

Nom

aliz

ed S

train

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

0.3

0.4

0.5

Lateral Drift (%)

Nom

aliz

ed S

train

-6 -4 -2 0 2 4 6-0.1

0

0.1

0.2

0.3

0.4

0.5

Lateral Drift (%)

Nom

aliz

ed S

train


LW1 LW2 LW3 LW4


LU9 LU10


LW5 LW6 LW7 LW8

278

(b) Definition 2 for yield deformation

(c) Definition 3 for yield deformation

279

(d) Definition 4 for yield deformation

Figure 4.29 Definitions for the yield deformation (Hu 2005)

(a) Definition 1 for the ultimate state

280

(b) Definition 2 for the ultimate state

(c) Definition 3 for the ultimate state

281

(d) Definition 4 for the ultimate state

Figure 4.30 Definitions for the ultimate state (Hu 2005)

Figure 4.31 Specimens ductility

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

μ δ

282

Figure 4.32 Energy dissipated, equivalent viscous damping ratio and effective

secant stiffness definitions for hysteresis curves

F

△

A2

A1

AED

△

1

△

2

F1

F2

Keff

Fm2

△

m2

Fm1

△

m1

1 21 2 1

3 41 1 2

1 1 2 2

( )2

( )2

12

d i ED

EDhyst

m m m m

F FA

F FA

E A A

AF F

F3

F4

283



Figure 4.33 Dissipated energy of each hysteresis cycle

0 1 2 3 4 5 60

2

4

6

8

10

12

14

Lateral Drift(%)

Dis

sipa

ted

Ener

gy o

f Eac

h H

yste

resi

s C

ycle

(kN

·m)


0 1 2 3 4 5 60

5

10

15

20

25

Lateral Drift (%)

Dis

sipa

ted

Ener

gy o

f Eac

h H

yste

resi

s C

ycle

(kN

·m)


284



Figure 4.34 Development of cumulative dissipated energy

0 1 2 3 4 5 60

10

20

30

40

50

60

70

80

Lateral Drift(%)

Cum

ulat

ive

Dis

sipa

ted

Ener

gy (k

N·m

)


0 1 2 3 4 5 60

10

20

30

40

50

60

70

80

90

Lateral Drift (%)

Cum

ulat

ive

Dis

sipa

ted

Ener

gy (k

N·m

)


285



Figure 4.35 Equivalent viscous damping ratio development

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Lateral Drift (%)

Equ

ival

ent V

isco

us D

ampi

ng R

atio

ζhy

st


0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Lateral Drift (%)

Equ

ival

ent V

isco

us D

ampi

ng R

atio

ζ


286



Figure 4.36 Evolution of the stiffness degradation

0 1 2 3 4 5 60

1000

2000

3000

4000

5000

6000

7000

Lateral Drift (%)

Pea

k-to

-pea

k St

iffne

ss K

eff (

kN/m

)


0 1 2 3 4 5 60

1000

2000

3000

4000

5000

6000

7000

8000

Lateral Drift (%)

Pea

k-to

-pea

k St

iffne

ss K

eff (

kN/m

)


287

Figure 4.37 Components of plastic hinge


-50 -40 -30 -20 -10 0 10 20 30 40 50-200

-150

-100

-50

0

50

100

150

Beam Tip Displacement (mm)

Bea

m T

ip L

oad(

kN)

Left beam Right beam

288


(c) Specimen F-S-450-450

-80 -60 -40 -20 0 20 40 60 80-100

-80

-60

-40

-20

0

20

40

60

80

100

Beam Tip Displacement (mm)

Bea

m T

ip L

oad(

kN)


-80 -60 -40 -20 0 20 40 60 80-150

-100

-50

0

50

100

Beam Tip Displacment (mm)

Bea

m ti

p lo

ad(k

N)


289

(d) Specimen S-Control

(e) Specimen S-G-50-200-100

Figure 4.38 Hysteresis curves of beam tip load versus calculated beam tip

displacement

-50 -40 -30 -20 -10 0 10 20 30 40 50-200

-150

-100

-50

0

50

100

150

Beam Displacement (mm)

Bea

m T

ip L

oad

(kN

)


-80 -60 -40 -20 0 20 40 60 80-150

-100

-50

0

50

100

150

Beam Tip Displacment (mm)

Bea

m T

ip L

oad(

kN)


290

(a) Contribution of beam deformation

(b) Contribution of column deformation

Figure 4.39 Deformation contributions of beams and columns to the lateral drift

ratio

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0 1 2 3 4 5 6

Con

trib

utio

n of

bea

m d

efor

mat

ion

to th

e la

tera

l dri

ft

Lateral drift ratio (%)

F-Control F-S-450-450 F-G-50-200 S-Control S-G-50-200-100

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

0 1 2 3 4 5 6

Con

trib

utio

n of

col

umn

defo

rmat

ion

to th

e la

tera

l dri

ft

Lateral drift ratio (%)

F-O-450-150 F-O-500-180 F-S-450-450 F-G-50-200 S-Control S-O-500-180 S-G-50-200-100 S-O-500-180-S-300-300

291

Table 4.1 Failure process and mode

Specimen

Drift ratio when beam

concrete first

cracked (%)

Drift ratio when

column concrete

first cracked

(%)

Drift ratio when joint

concrete first

cracked (%)

Drift ratio at peak load (%)

Beam Steel bars

yielding

Slab longitudinal

bars yielding

Joint stirrup

yielding

Column Steel bars yielding

Failure mode

F-Control

During applying 970 kN

axial load.

1.0 1.0 2.0 Yielded at 1.5% drift

ratio

Yielded at 1.5% drift

ratio

One stirrup yielded at 1.5% and all yielded

at 3.0% drift ratio

Steel bars at bottom column

yielded at 2.5% drift ratio under

tension

Before being retrofitted, shear failure

at the beam end pin region at 4% drift ratio. After being retrofitted, shear failure at the joint

panel

292

Table 4.2 (cont.)

F-G-50-200 0.5 0.5 1.0 1.5

Beam bottom bars yielded at 1.0% drift ratio under

compression

Some close to the beam yielded at 1.0% drift

ratio and all yielded at 1.5% drift

ratio

No yielding No yielding Beam bottom bars

slipping in the joints region

F-O-450-150 0.5 1.0 1.5 2.5 Yielded at 1.5% drift

ratio



ratio

Stirrups yielded at 2.0% ratio



compression

Shear failure at the joint pane

293

Table 4.3 (cont.)

F-O-500-180 0.5 0.5 1.0 2.0 Yielded at 1.5% drift

ratio



ratio

One stirrup yielded at 1.5% and all yielded

at 4.0% drift ratio

No yielding Local flexural failure at

the chord ends

F-S-450-450 0.5 0.5 1.5 1.5 Yielded at 1.5% drift

ratio

The uncut slab bars yielded at 1.0% drift

ratio

Two stirrups

yielded at 2.0% drift

ratio and all yielded at 3% drift

ratio

Steel bars at top column yielded at 5.0% drift ratio under

compression

Beam flexural failure

294

Table 4.4 (cont.)

S-Control 0.25 0.5 1.0 2.0 Yielded at 1.5% drift

ratio


ratio and all yielded at 2% drift

ratio

Some stirrups

yielded at 2.5% drift

ratio



compression

Bottom column bars buckling

S-G-50-200-100 0.25 0.5 1.0 2.0 Yielded at 1.5% drift

ratio



ratio

No yielding No yielding Beam bottom bars

slipping in the joints region

295

Table 4.5 (cont.)

S-O-500-180 0.5 0.5 1.0 2.0 Yielded at 1.0% drift

ratio

All yielded at 1.0%

drift ratio

Two middle stirrups in

joint yielded at 2.5% ratio



compression

Bottom column bars buckling

S-O-500-180-S-300-300 0.5 1.0 1.0 2.0 Yielded at 1.0% drift

ratio

50% slab bars yielded

at 2.0% drift ratio

One middle stirrup in

joint yielded at 2.5% ratio and two

middle ones yielded at 4.0% drift



compression

Local flexural failure at the chord ends

296

Table 4.2 Strengths and weakening ratios of test and prediction

Test group

Items

Beam Column Strength (kN)

Weakening ratio

Calculated column shear force (kN)

Predicted column shear

force (kN) - + Total

First

Cross-sectional analysis

R-section 95.6 72.2 167.8 -30.7% 101.6 150.1 T-section 154.2 87.8 242 0.0% 146.5 150.1

F-Control 156.6 87.2 243.8 0.0% 130.2 150.1 F-G-50-200 85.6 87.4 173 -29.0% 96.9 150.1 F-O-450-150 138.3 88.5 226.8 -7.0% 120.2 150.1 F-O-500-180 115.7 88.2 203.9 -16.4% 190.4 150.1 F-S-450-450 109.4 83.1 192.5 -21.0% 110.9 150.1

Second

Cross-sectional analysis

R-section 97.8 97.8 195.6 -29.1% 118.2 150.8 T-section 156.4 119.6 276 0.0% 166.7 150.8

S-Control 144.4 112.8 257.2 0.0% 147.0 150.8 S-G-50-200-100 111 107.3 218.3 -15.1% 122.0 150.8

S-O-500-180 120.4 119.3 239.7 -6.8% 132.5 150.8 S-O-500-180-S-300-300 111.7 102.6 214.3 -16.7% 118.6 150.8

297

Table 4.3 Specimen ductility

Specimen Calculated peak column strength

(kN)

Displacement at 75% peak load (mm)

Yield displacement

(mm)

Displacement at failure

(mm) Ductility

F-Control 128.5 16.3 21.8 96.2 4.4 F-G-50-200 96.9 18.2 24.3 51.4 2.1 F-O-450-150 120.2 19.1 25.5 111.1 4.4 F-O-500-180 109.4 18.7 24.9 101.3 4.1 F-S-450-450 110.9 17.7 23.7 72.8 3.1

S-Control 147.0 21.5 28.6 86.9 3.0 S-G-50-200-100 122.0 15.4 20.5 61.2 3.0

S-O-500-180 132.5 22.4 29.9 94.3 3.2 S-O-500-180-S-300-300 118.6 21.5 28.6 102.7 3.6

298

Table 4.4 The lengths of yielding and strain concentration zones

Items F-Control F-G-50-200 F-S-450-450 S-Control S-G-50-200-100 Yielding zone length (mm) 200 190 310 360 310 Strain concentration zone

length (mm) /* 120

More than 120

110 /**

Note:*: This value could not be determined as the pre-failure of the specimen F-Control.

**: This value could not be determined as limited number of strain gauges.

299

CHAPTER 5

THREE-DIMENSIONAL FINITE ELEMENT

MODELLING OF REINFORCED CONCRETE

T-BEAMS WITH A WEB OPENING WEAKENING

AND WITH/WITHOUT FRP SHEAR

STRENGTHENING

5.1 INTRODUCTION

A beam sway (BS) mechanism (i.e. one with plastic hinges at beam ends)

is the most preferred failure mode for a reinforced concrete (RC) frame

subjected to seismic attack, because of its efficient energy-dissipation capacity.

To ensure the BS mechanism failure and enhance the seismic safety of existing

RC frames, appropriate seismic retrofitting is needed for those RC structures

designed according to the out-dated and inadequate design codes. One technique

is to include web openings at beam ends, the consequential reduced shear

capacity can be compensated by shear strengthening (e.g. using FRP). This

approach has been recently proposed by the authors’ research group based on

the previous studies on the behaviour of RC beams containing web openings for

other purposes, e.g. openings for the passing of the pipes or wires of ventilation

and air conditioning systems (Mansur et al. 1999; Abdalla et al. 2003;

300

Maaddawy and Sherif 2009; Madkour 2009; Pimanmas 2010; Chin et al. 2012;

Hawileh et al. 2012; Maaddawy and Ariss 2012). These studies showed that

web openings can significantly reduce beam flexural capacities. Recently, this

proposed technique was validated through a series of tests by Nie et al. (2018)

and Nie (2018) on full scale T-beams. A few recent studies (Madkour 2009;

Pimanmas 2010; Chin et al. 2012; Hawileh et al. 2012, Nie 2018) used

two-dimensional (2D) (Pimanmas 2010; Chin et al. 2012; Nie 2018) and

three-dimensional (3D) (Madkour 2009; Hawileh et al. 2012) finite element (FE)

models to simulate RC beams with web openings. The last three of those studies

(Chin et al. 2012; Hawileh et al. 2012, Nie 2018) also modelled bond slip

behaviour between the shear strengthening FRP and the beam concrete. Among

the five studies, only Nie (2018) proposed a 2D FE models concerned

specifically with RC T-beams weakened by web openings. However, Nie’s

(2018) 2D models are subjected to an obvious limit: it can't predict the

non-uniform deformations/stresses in the transverse direction, especially those

in the flange of the T-beam, necessitating the development of a more

accurate/powerful 3-D FE model for RC beams with and without opening in the

present study.

To gain an in-depth understanding of the potential of this retrofit technique,

a 3D finite element (FE) model, using the general purpose FE programme

ABAQUS (2012) for RC T-beams with web openings, was proposed and

substantiated with test results. The applicability of using an explicit dynamic

301

method, which is not subject to numerical convergence problems, to the

simulation of RC T-beams with web openings under static load, is also

examined and described below. In particular, the effect of some key parameters

including loading scheme, duration and damping ratio, among others, were

evaluated. The results of the 3D FE model are then compared with the

predictions of the 2D FE models used in Nie (2018) to show the advantages of

the proposed 3-D FE model.

5.2 PROPOSED 3D FE MODEL

The proposed 3D FE models are based on the 2D FE model of Chen et al

(2011), which did overcome the convergence problem and provided accurate

predictions on the test results of RC beam, flexurally strengthened by FRP

sheets and failed by intermediate crack induced debonding (IC debonding)

under monotonic loading. The 2D FE model, which is based on the

smeared-crack approach, employed the crack band model of Bazant and Planas

(1998) to overcome the mesh sensitivity problem. According to Chen et al.

(2011), to precisely predict flexural cracks and the associate FRP debonding

failure, an accurate modelling of concrete cracking behaviour (especially the

post-cracking behaviour of concrete), accurate bond-slip modelling of bond

behaviour between steel reinforcement and concrete and that between FRP and

substrate concrete are needed (Chen et al. 2011). In the proposed 3D FE model,

all these key components are considered and included, as will be detailed below.

Taking advantage of the symmetry of the specimen and the loading pattern, only

302

a half width of beam was modelled in the proposed FE model. The 3D model is

meshed with 20 mm elements, which is small enough to obtain reliable results

according to a separate convergence study.

5.2.1 Modelling of concrete

The concrete is represented by the 3D solid elements C3D8R in ABAQUS

(2012). The accurate representation of cracked concrete behaviour is essential

for the accurately modelling of the cracking behaviour of an RC beam. There

are three built-in models of concrete in ABAQUS (2012). These consist of the

smeared concrete cracking in ABAQUS/Standard, the brittle cracking (i.e. BC)

model in ABAQUS/Explicit and the concrete damaged plasticity (i.e. DP)

model in both ABAQUS/Standard and ABAQUS/Explicit. The brittle cracking

model assumes linear behaviour in compression, which is not appropriate when

compression failure dominates the failure mode. Therefore, in the proposed

model, the concrete damaged plasticity model of ABAQUS (2012) was adopted

to model concrete cracking. For modelling compressive behaviour of concrete,

the constitutive equation proposed by Saenz (1964), for the state of uni-axial

compression, was adopted following Chen et al. (2011).

2 1 / 2 / /p p p p

(5.1)

where and are axial compressive stress and strain respectively; p

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and p are the peak stress and its corresponding strain; c is the coefficient

representing the initial tangent modulus of concrete. In the study, c was set

to be equal to the elastic modulus of concrete Ec. The latter three parameters

( p , p and ) were given by the average values of three cylinder tests, as

given in Tables 5.1 and 5.2. When there are no available cylinder test results p

and Ec were set equal to 0.002 and 4730 'cf respectively.

According to Malm (2009), using a stress-strain relationship for tensile

behaviour of cracked concrete introduces considerable mesh sensitivity for the

parts of an RC specimen without steel reinforcement. In those places it is more

appropriate to use fracture energy or stress-crack opening laws for modelling

the tensile behaviour cracked concrete. The tension-softening relationship

proposed by Hordijk (1991), as shown in Equation (5.2) was adopted following

Chen et al. (2011) for its good performance (Malm, 2009).

t

2cr 2

3 w-cw -c3t t t

1 1t cr cr

w w= 1+ c e - 1+c ef w w

(5.2)

Fcr

t

Gw = 5.14f (5.3)

where t is the tensile stress normal to the crack, tw is crack opening

displacement; crw is crack opening displacement at the point of concrete stress

304

complete release; FG is the concrete fracture energy calculated by the area

bounded by the strain-stress curve of concrete under uni-axial tension; 1c and

2c are constants determined from tensile tests of the concrete, which can be set

equal to 3.0, 6.93, respectively; for normal concrete. tf and FG can be

estimated from the cylinder compressive strength based on the equations in

CEB-FIP (1993), if no specific test data are provided for determining these two

parameters.

2

' 3c

tf - 8f = 1.410

Mpa (5.4)

0.7'

20.0469 0.5 2610

cF a a

fG d d

(5.5)

where ad represents the maximum aggregate size, which is assumed as 20 mm

if no test data is available.

5.2.2 Definition of Damage Evolution

In the proposed 3D FE model, a damage evolution curve is employed to

describe the progressive degradation of concrete when it enters the softening

phase. The damage model used in Chen el al. (2011) was adopted for simulation

purposes as it takes into account the element size effect. In the damage model,

the unloading path is assumed to return to the original point and therefore the

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equivalent plastic strain plt is set equal to zero. Thus, the damage model is a

elastic (ELA) damage model. The damage factor dt , expressed in terms of

crack-opening displacement tw for cracked concrete as below:

( ) /

tt

t c t c

wdw h E

(5.6)

where ch is crack band width which is defined as the characteristic crack

length of an element in ABAQUS. In the present study, it is equal to the element

size with a reduced integration scheme as recommend in Rot's (1988).

5.2.3 Modelling of Steel Reinforcement

The steel reinforcements in the 3D FE model are modelled with B31 beam

elements. A beam element is more suitable than a truss element for representing

steel reinforcement as it can resist local moments. Two types of steel bar, type

one bar and type two bar are assumed, as indicated in Figure 5.1. The

stress-strain curve of type 1 bar possesses an obvious yielding platform and a

hardening range; whereas, that of Type 2 features a gradual yielding process. To

better capture type 2 bar behaviour under monotonic loading, a model proposed

by Ramberg Osgood (1943) was applied. The equation proposed by Ramberg

Osgood is:

306

1

0

= ( )ns ss s

soE

(5.7)

where s is the axial strain and s is the axial stress; s is the yield offset

and 0s is the yield stress. n1 is a parameter controlling the transition from the

elastic branch to the plastic branch. The hardening ratio of the material is

decreased by increasing “n1”. Commonly used value for n1 is 5 or greater. The

first part of the equation at the right hand side represents elastic strain and the

second part, plastic strain. 0.2% is a generally used value for the yield offset s

if the yield stress is hard to determine. This is also the value suggested in the

Chinese code (GB/T 228.1-2010) and the British standard (BS EN ISO

6892-1:2009). Slab steel bars arranged in the transverse direction are assumed

to experience marginal slip during the loading. Thus these steel bars are

perfectly embedded in concrete. For those steel bars in the longitudinal direction,

interfacial cohesive elements are employed to simulate the bond-slip behaviour

between the longitudinal steel bars and concrete, as will be detailed in

Sub-section two .

5.2.4 Modelling of FRP Reinforcement.

In the proposed 3D FE model, the CFRP reinforcement is represented by

S4R 4-node shell elements and treated as a linear elastic brittle material with an

elastic modulus of 227.380 GPa. Only the fibre direction is provided with

stiffness. The Poisson's ratio of the FRP is set as small as possible and as it can't

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be set to be equal to 0, value of 0.001 is used. The FRP reinforcement ruptures

when it reaches its tensile strength of 2820 MPa. It should be noted that the

values of elastic modulus and tensile strength for FRP were calculated based on

the nominal thickness of CFRP sheet coupons according to ASTM-3039(2008).

Bond-slip behaviour between FRP and concrete surface is represented by

cohesive elements, as detailed below.

5.2.5 Modelling of Bond Behaviour

In the proposed 3D FE model, the bond behaviour between steel

reinforcement and concrete and between FRP and concrete is modelled by

employing cohesive elements COH3D8, which are built using the separate

nodes at the same locations of the two adjacent elements, representing the

concrete and the reinforcements (either FRP or steel reinforcement) respectively.

The shear force between the two elements connected via the cohesive element is

equal to the interfacial shear stress (i.e. bond stress) multiplied by the bonded

area represented by the section area of the cohesive element.

For bond-slip behaviour of the steel-concrete interface, the CEB-FIP(1993)

bond-slip model, as expressed by the following Equations (5.8-5.10), were

adopted to define the cohesive elements, following Chen et al. (2012).

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11

2

3 2

( )

max s

max 1 s 2s

sm

s s

s

s s sax m f 3

sf 3

ax 2 s

s

0 s ss

s s s=

s s s s ss s

s > s

s

(5.8)

max max2.0 , 0.15s s sck ff deformed bar (5.9)

max 0.3s sf ckf plain bar (5.10)

where s is the local shear bond stress between steel bars and concrete; ss is

the slip; maxs is the local shear bond strength; s

f is the failure bond stress;

ckf is the concrete standard compressive strength; =0.4 for deformed bars

and 0.5 for plain bars; , for deformed bars;

for plain bars.

No relative displacement is allowed for in the direction normal to the steel

reinforcement and concrete interface, following Chen et al. (2012).

As one B31 element used for modelling the steel bars has only two nodes,

a 3-D prism cohesive element modelling approach using the cohesive element

COH3D8 was proposed to connect the corresponding nodes of concrete and

steel bar, as shown in Figure 5.2. In this modelling approach, the first two nodes,

Nodes 1 and 2 of the cohesive element are connected to one concrete element

node, which means the two nodes are located at the same position. The third

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node, Node 3 is connected to a bar element node. The forth node, Node 4, is

used to create an area for the cohesive section. All the degrees of freedom

(DOFs) of node 4 are set equal to those of Node 3 to secure an unchanged area.

The area of cohesive element is determined by the positions of sectional nodes

and the thickness of cohesive elements and the shear force between the two

elements connected via the cohesive element is equal to the interfacial shear

stress (i.e. bond stress) multiplied by the bonded area represented by the section

area of the cohesive element. Thus, if the sectional area created by the four

nodes is not equal to the circumference of a bar, the shear stress along the

steel-concrete interface direction should be adjusted to keep the shear force

unchanged.

To verify the reliability of the proposed 3D prism cohesive element

modelling approach, a prism cohesive element was built and modelled using

ABAQUS (2012). a assumed circumference of a deformed bar equal to 10 mm

was applied. As the area of the prism cohesive element is only half that of a

rectangular section corresponding to a cuboid cohesive element, which is

usually used, the shear stress was multiplied by 2. Figure 5.3 indicates that the

3D prism cohesive model can accurately reproduce the input data, thus

demonstrating that the proposed model is reliable.

The bond behaviour between FRP and concrete was modelled using

rectangular section cohesive elements. The simplified bond-slip model

developed by Lu et al (2005) was adopted to define the bond behaviour of the

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FRP-concrete interface. The model can be expressed by following equation:

0

0

0

0

( 1)

max

m

ss

ax

0 s s=

ss

e s s

(5.11)

0.0195o w ts f (5.12)

max 1 w tf (5.13)

2 /1 /

f ww

f w

b bb b

(5.14)

0.550.395( )t cuf f (5.15)

max 0

123

fGs

(5.16)

20.308f w tG f (5.17)

where is the local shear bond stress between FRP sheet and concrete; max

is the local bond strength; s is the slip; s0 is the slip when the bond stress

reached max ; w is the width ratio factor; fb is the width of FRP; cb is the

width of beam; cuf is the cube compressive strength of concrete; fG is the

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interfacial fracture energy; 1 =1.5 and / 1f wb b for the case of presence

study.

In the direction normal to the interface, it was assumed there is no relative

displacement between FRP and concrete. That is because the effect of normal

stress on IC de-bonding is insignificant (Chen et al 2011) and besides, IC

de-bonding generally propagates parallel to the FRP- to-concrete interface (e.g.,

Brena et al. 2003; Matthys 2000).

5.2.6 Solution Strategy and Boundary Conditions

Convergence issue is a big problem to be overcome when using a 3D FE

model to simulate structural failure associated with concrete cracking, especially

when bond behaviour is incorporated in the FE model (Chen et al. 2015). The

existence of a web opening further worsens the convergence problem. Thus, the

dynamic explicit approach, which adopts the central difference integration

method, was used to avoid the convergence issue. The validity and accuracy of

the dynamic explicit method have been proved by Chen el al. (2015). In the

proposed a 3D FE model, the densities of materials are set equal to the actual

values, of 2.5e-9 ton/mm3 for concrete and cohesive elements, 7.25e-9 ton/mm3

for steel elements and 1.75e-9 ton/mm3 for FRP elements. The

stiffness-proportional damping coefficient β was used to damp out the dynamic

effect associated with high mode and a value of 0.000002 was adopted based on

parameter study. To minimize the dynamic effect, a displacement-control load is

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applied following the smooth loading scheme shown in Figure 5.5, as suggested

by Chen et al. (2015). The loading duration was set equal to 50 times T0, the

natural period of vibration of simulated specimen. The parameter values were

determined based on a careful parametric study, as will be described below.

The boundary conditions of the 3D models are illustrated in Figure 5.4.

The displacement-control load was applied centrally at the top loading point.

The left hand supporting points are restrained in both the Y and Z directions and

the right hand supporting points are restrained in the Y direction only. None of

the nodes on the section of symmetry are allowed for any displacement in the X

direction.

5.3 VERIFICATION OF THE PROPOSED 3D FE MODEL

5.3.1 Specimen Dimensions and Material Properties

The results for the specimens tested by Nie (2018) were employed to verify

the proposed 3D FE models. Those specimens include T-beams with slab flange

widths satisfying the requirements of the Chinese design code GB-50011 (2010).

The behaviour of T-beams with web openings, with or without FRP

strengthening, was investigated when subjected to monotonic loading. Sufficient

details of the specimens are available in Nie (2018).

5.3.1.1 Specimens' dimensions

Two groups of full-scale RC beams, one including a rectangular beam

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(R-beam) and 7 T-beams, the other 6 T-beams, had been tested by Nie (2018).

The aims of their test program had been to estimate the performance of the

proposed retrofit technique of creating a web opening in existing RC T-beams to

decrease their flexural strength along with associated FRP shear strengthening

measures. The first test group included two control specimens, consisting of a

solid rectangular beam CB-Rec and a solid T-beam CB-T. The remaining

specimens were all weakened with web openings. In the first test group, two

opening sizes (length × depth = 700 mm ×300 mm and 800 mm ×280 mm) were

studied. For each opening size, three specimens were tested, one with a web

opening only, one with a web opening and associated FRP shear strengthening,

and third one also with a web opening and associated FRP strengthening, but

tested under a positive loading condition only (i.e. the slab was in compression).

For the second test group, four opening sizes (length × height=600mm ×

220mm, 700 mm × 200 mm, 600mm × 280 mm and 700mm × 260 mm

respectively) were studied by testing six T-beams. Two of the six T-beams were

weakened by a web opening (opening sizes of length × height=600mm ×

220mm, 700 mm × 200 mm) only. The remaining four specimens have

additional FRP strengthening and their openings are of different opening sizes.

The names of specimens indicated their opening size, bending direction and

having/having not FRP strengthening. FRP strengthening. details of the main

variables of the specimens are listed in Table 5.1.

The length of all specimens was 3500 mm with a clear span of 3300 mm.

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The width of the beam web was 250 mm. Except for the rectangular beam

CB-Rec, all T-beams had two 600-mm-wide and 100-mm-deep flanges on each

side of the beam, leading to a total slab width of 1450 mm. Web openings were

located 250 mm from specimen centres, to simulate the real situation of web

openings adjacent to columns of 500-mm width along the main beam direction.

Web openings were placed adjacent to the flange. The control specimens (i.e.

CB-Rec and CB-T) failed due to concrete compression crushing. For specimens

weakened by openings but without FRP strengthening (i.e.O-700-300-N,

O-800-280-N, O-600-220-N and O-700-200-N), the two have larger web

opening ( specimens O-700-300-N, O-800-280-N) failed by local flexural or

mixed flexural an shear failure at the ends of web and flange chords (i.e. the

chords in the bound of beam web and flange). Specimens O-600-220-N and

O-700-200-N failed due to diagonal cracks developing in span of web chord.

Those with additional FRP strengthening, failed by local flexural rotation at the

two ends of the web chord and flange chord.

5.3.1.2 Material properties

Commercial concrete was used and its strength was obtained by averaging

the strengths of three concrete cylinders. The concrete strength of each

specimen is listed in Table 5.1. Longitudinally placed deformed steel bars of

20-mm diameter (D20 bars) were used for beam tension reinforcements. Four

D20 bars were placed at the top side (i.e. the side with slab flanges) and three

D20 bars at the bottom side. Plain steel bars of 8-mm diameter (D8 bars) were

315

used as steel stirrups and slab reinforcements. The spacings for steel stirrups and

slab longitudinal bars were 100 mm and 200 mm respectively. There were two

layers of D8 longitudinal slab reinforcing bars in the slab. For the first test

group, the yield load and ultimate stresses of the D8 steel bars were 307 MPa

and 447 MPa, respectively; and 475 MPa and 625 MPa, respectively for the

D20 steel bars. The yield and ultimate stresses of D8 steel bars were 349 MPa

and 526 MPa respectively for the second test group and those for the D20 bars

were 434 MPa and 559 MPa respectively. The properties of type 1 bar are

tabulated in Table 5.3 based on coupon tests. Only the D8 bar of the first group

is type 2 bar. To better describe the stress-strain curves obtained from tests, the

average stress 0.2% obtained at a yield offset of 0.2%, is applied to 0s and

n1 is set equal to 9.0, which is determined by the obtained peak stress and the

corresponding strain of the D8 steel bars in the first test group. The elastic

modulus for all steel bars was assumed to be 200 GPa. Except for D8 bar of the

first test group, which is a type 2 bar, the material properties of the type 1 bars

are tabulated in Table 5.3. CFRP, which was used in the U-shaped, or wrapping,

form, was of 0.334mm fabric design thickness, 2820 MPa tensile strength and

227 GPa elastic modulus. The bond stress and corresponding slip of Lu etal.'s

(2005) model for specimens with FRP strengthening were tabulated in Table

5.4.

5.3.2 Load versus Displacement Curves

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Figure 5.6 shows the predicted curves for all specimens. The predicted

curves for the control specimens match well with the test results (Figure 5.6(a)).

For those specimens with openings with and without FRP strengthening, the FE

model predicts peak load (load capacity) quite accurately, but higher

post-cracking stiffness (Figures 5.6(b), 5.6(c), 5.6(e) and 5.6(f)) for specimens

without FRP strengthening. This is because shear damage is underestimated by

the proposed 3D models. This is confirmed by the fact that for those models

provided with additional FRP strengthening, there were better predictions of

post-cracking stiffness. Except for the two specimens loaded in positive bending

as indicated in Figure 5.6(d). Almost all cracking loads were slightly

overestimated by the 3D FE Models. For the post-yielding range (i.e. the range

after a beam yields) load-displacement response, the behaviour of the specimens

with web openings but without FRP strengthening was not well predicted by the

3D mode: the predicted load-displacement curve in the post-yielding range

show higher stiffnesses than those in the test curves. This might also because

shear damage is underestimated by the proposed 3D models. In Figure 5.6(c)

and (e), the FE curves for models F-800-200-N and F-600-220-N stop earlier

than the test results. This is because local debonding at the FRP-concrete

interface at the opening’s corner near the loading point leads to great local

vibration and a sudden drop in the predicted curves.

The test and predicted load capacities are tabulated in Table 5.5. For the

CB-R specimen who had a obvious yielding load, its yield load is tabulated as

317

specimen predicted load. For specimens with web openings but without FRP

strengthening, the peak loads are tabulated. For models with both web openings

and FRP strengthening, the loads at 33 mm displacement, 2% of beam effective

length, are tabulated. It can be seen from Table 5.4 that the predicted load

capacities are much closer to the test results except for the two specimens

loaded in positive bending. The average of the ratios (including the two

specimens subject to positive bending) of predicted load to test result is 1.03,

with a standard deviation of 0.09.

5.3.3 Crack patterns

The predicted crack patterns are shown in Figures 5.7(a)-5.7(n). In these

Figures, the FRP had been removed to better demonstrate the cracks

propagating in and near the web chords. Except for the beam CB-Rec, the main

test cracks are indicated by red dashed lines, as shown in Figure 5.7(b)-(n).

Figure 5.7(a), predicts many cracks concentrated in the middle of the

specimen at the ultimate state for the R-beam, which is similar to those of the

test result shown in Nie (2018). As for the CB-T beam, the 3D FE model

predicts good crack pattern predictions in terms of the number and distribution

of the major cracks in the beam web and slab flange. For the specimens with

web openings, the major cracks are also well predicted by the 3D FE models.

For those specimens with the larger web openings (e.g. O-700-300-N), the

predicted major cracks are located near the ends of the chords and become

318

wider as the four hinges mechanism (i.e. local flexural rotation at the ends of

web and flange chords) becomes more obvious, which is close to test

observation. In comparison, for the specimens with smaller web openings (e.g.

O-600-220-N), additional flexural cracks and shear cracks are predicted to

appear near mid-span of the beams and on the web chord respectively,

consistent with test results. When FRP is provided to control shear crack

propagation (e.g. F-O-600-220-N), more flexural cracks are predicted near

specimen mid-spans, which is consistent with test results. For those specimens

tested in positive bending, many flexural cracks are predicted in the beam webs,

in addition to those near the ends of chords. This is also similar to the test

results.

5.3.4 Energy Release Behaviour

Energy analyses predicted by a 3D FE model can be obtained and used to

better understand and evaluate the reliability of predicted results. As presented

ABAQUS (2012) manual, the total energy totalE is given as:

total I V FD KE WE E E E E E (5.18)

where IE is internal energy, VE is energy dissipated by viscous effects, FDE

is total energy dissipated through frictional effects, KEE is kinetic energy and

WE is external work applied to the structural system. For a structure in a stable

319

state, WE is equal to IE . Therefore, when making a quasi-static analysis for

which the static structural response dominates the structural response, the

difference between WE and IE ought to be very small. VE is relatively

small compared to IE . That is because viscous forces aim to damp out local

dynamic effects. A viscous force which is too large leads to obviously higher

predicted load capacity of the structure and an unreliable result. FDE is equal

to 0 in this study as no friction force is applied in the proposed FE model. For a

quasi-static analysis, the dynamic energy of any particular analysis also

accounts for a small portion of the internal energy (normally below 1%).

Figure 5.8 shows the curves of the ratios of external work, viscous energy,

kinetic energy to the internal energy for specimen CB-T. As indicated in Figure

5.8, the external work to internal energy ratio is nearly equal to 100% except at

the very beginning. The energy due to viscous effects is negligible (0.01%) and

decreasing at first but suddenly increases to a value of about 10% upon the

specimen cracking, due to kinetic energy increase; then followed by a gradually

decrease to the value of below 1%. The kinetic energy to internal energy ratio is

relatively small (1%) at first, gradually decreasing as displacement increases,

except that, at about a 3-mm displacement, specimen cracking causes a sudden

dynamic energy increase (to a value of about 2%), then it quickly decreases to a

value below 0.1% , as indicated in Figure 5.8. The above results imply that the

predicted structural responses are essentially static except for a few discrete

320

moments, such as at the beginning of loading and at the appearance of

significant cracks.

5.4 PARAMETRIC STUDIES ON THE EFFECTS OF KEY

FACTORS

This section concerns the choice of the values for the key factors, including

element size, loading duration, damping coefficient β and computation precision

setting as they relate to the accuracy of the 3D FE analysis. In addition, as the

web chords of some T-beams with web openings were strengthened by FRP

wrapping, the confinement effect provided by FRP wrapping was also explored

ensure the validity of the 3D FE analysis. It should be noted that in the

following parametric studies, when the value of one parameter is changed, those

of the rest parameters were kept unchanged and equal to those in the reference

case specified in Subsection 5.2.6.

5.4.1 Element Size

Two control specimens, and specimens O-700-300-N, F-700-300-N were

used for a convergence study of element size. In Figure 5.9, the label ELE4050

means the element sizes are either 40 mm or 50 mm, which are dependent on

the dimensions of specimens, spaces between stirrups and bars, etc. ELE20 and

ELE10, indicate that all element sizes are 20 mm or 10 mm respectively. As

shown in Figure 5.9, the element size of 20 mm is small enough for

reliable/accurate results to be obtained as the numerical results obtained using

321

the element size of 20 mm are very close to those using element size of 10 mm.

The predicted load-displacement curve for F-700-300-N-ELE10 stops at about

30 mm as from then on, FRP-concrete local interface debonding causes a

sudden drop of load, bringing about a large kinetic energy increase. However,

the obtained numerical results are enough to indicate that a 20 mm element size

is small enough for the proposed 3-D FE models.

5.4.2 Loading Duration

As explicit centre difference method (CDM) is used as time integration

method to solve the 3D FE models, the loading duration should be long enough

to minimize the dynamic effect and inertia forces effects. However, longer

loading durations lead to longer computational times and larger accumulation

errors (Chen et al. 2015). Thus, the loading duration effect was investigated to

determine a suitable value of loading time with both accuracy and

computational time kept in mind.

The FE models for CB-T, O-700-300-N and F-700-300-N mentioned above

were used to study the loading duration effect. The natural period T0 of a 3D

model was first determined by ABAQUS (2012). Then loading durations

ranging from 12.5T0 to 200T0 were applied for specimens CB-T and

O-700-300-N first. Loading durations ranging from 25T0 to 100T0 were applied

to specimen F-700-300-N for further confirmation. As shown in Figure 5.10 (a),

the predicted loading-displacement curve for specimen CB-T with a 50 T0

322

loading duration is close to those with longer loading durations. Figure 5.10 (b)

indicates that shorter loading durations lead to higher cracking and peak loads.

For specimen O-700-300-N, the 50 T0 loading duration leads to a slightly higher

peak load than for longer loading durations, but the difference is less than 5%,

which is acceptable. Based on the above discussions, a loading duration of 50 T0

was chosen in this study.

Fig. 5.11 plotted the dynamic energy versus displacement curves on

logarithmic scales. It is obvious that the spacings between two neighbouring

dynamic energy-displacement curves are almost the same. That is because the

loading duration was double. At initial and final states, the dynamic energy is

close to zero, because the loading speed of the smooth loading scheme is zero at

the beginning and the end of loading duration. It can be found from numerical

results not presented here that cracking of concrete and local debonding of the

FRP-concrete interface lead to some local fluctuations of the dynamic

energy-displacement curves. It is of interest to note that the each of the main

local fluctuations of load-displacement curves shown in Figure 5.10 occurs at

the same displacement as that in the dynamic energy-displacement curves,

suggesting that the local fluctuations in the load-displacement response are

associated with the local dynamic effects, which will be damped out by the

damping system of the model.

5.4.3 Damping Coefficient β

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Rayleigh damping of ABAQUS (2012) was adopted to define material

viscous damping. The viscous damping matrix C is expressed as:

C M K (5.19)

where M is the mass matrix and K is the stiffness matrix, and are

mass-proportional and stiffness-proportional damping coefficients respectively.

Thus the damping ratio for the jth model of system can be expressed as :

2 2j

jj

(5.20)

where j is the circular frequency corresponding to the jth model. The

relationship between damping ratio and circular frequency is illustrated in

Figure 5.12. If parameter is equal to zero, the damping ratio is proportional

to the stiffness matrix. Then damping ratio is a linearly increasing function of

circular frequency. Under this circumstance, a higher vibration mode is assigned

with higher viscous damping. If parameter is equal to zero, viscous

damping is proportional to mass, which leads to higher viscous damping at

lower modes of vibration. When both parameters are larger than zero, the

damping ratio will not be lowest at the beginning. In Figures 5.13 to 5.15, the

first five modes of vibration of the models CB-Rec, CB-T and O-700-300-N are

presented. The vibration amplitudes are 100-times enlarged to better present the

results. The corresponding frequencies are tabulated in Table 5.4. As indicated

324

in Figures 5.13 to 5.15, the first mode of vibration is the expected deformation

shape. Therefore, to damp out the dynamic effect of higher vibration modes and

minimize dynamic effect associated with the lower mode (e.g. first mode), only

is needed.

To determine the value of to be used in the proposed FE models, CB-T,

O-700-300-N and F-700-300-N were used for a parameter study. The values of

are from zero to 5e-5 and the predicted load-displacement curves are shown

in Figure 5.16. As indicated in Figure 5.16, when is equal to zero or below

5e-7, the predicted curves drop before yield load. When is higher than 1e-5,

the FE models predict higher post-cracking stiffness and load capacity due to

the effect of viscous force associated with damping (Chen et al. 2015).

Furthermore, when explicit CDM is used, the stable time increment largely

decreases due to the increased (ABAQUS 2012), which leads to greatly

increased computational times. Therefore, the with a value between 5e-7

and 1e-5 should be an reasonable value for the proposed FE model. This

conclusion is further confirmed when checking predicted kinetic energy as

shown in Figure 5.17. It is obvious that when is below 5e-7, the predicted

kinetic energy is much higher. When is above 1e-5 (e.g. 5e-5), the predicted

kinetic energy is also higher than for those with between 5e-7 and 1e-5. To

further narrow the range of , the FE models of CB-T, O-700-300-N and

F-700-300-N with equal to 1e-6, 2e-6 and 4e-6 were studied and the results

325

are shown in Figure 5.18; for all three values of , similar load-displacement

and kinetic energy-displacement curves were predicted except that for CB-T,

unexpected kinetic energy appears at a displacement around 28 mm when is

equal to 1e-6. As a result, = 2e-6 was chosen for all the specimens for the

proposed 3D FE models.

To find the reason for the early failure of FE models with very small (e.g.

below 5e-7), the predicted crack patterns for model O-700-300-N, with =0,

was studied as shown in Figure 5.19. The positions of a, b, c and d are

illustrated in Figure 5.16(b). Points a and b occur immediately before and after a

local peak load of 81.9 kN; and c and d are the lowest (-17.1kN) and highest

(107.9 kN) points respectively after points a and b. As shown in Figure 5.19 (a),

some cracks emerge at the corners of the opening at moment a. At the

displacement of moment b, corner cracks keep growing and a new flexural

crack propagates on the slab. As the displacement increases to moment c,

numerical local cracks are predicted as indicated by the large number cracked

elements and at d, cracks elements cover the whole specimen. These

strange/doubtful phenomena were actually not observed in tests (Nie 2018),

which can be explained as follows. When is artificially/ideally set to zero,

the local vibrations associated with the initiation and propagation of critical

cracks, which are high vibration mode, can't be efficiently damped out, leading

to the quick and unreal appearance of a large number of cracks, first next to the

critical local crack and then propagate to the whole beams; as a result, global

326

dynamic effect appears and the kinetic energy becomes doubtfully large (Figure

5.17(b)). As a result, the related numerical results should be unreliable.

To summarize, in the proposed 3-D FE model, the stiffness-proportional

Rayleigh damping (characterized by stiffness-proportional damping coefficient

) is favoured and should be small enough to avoid the unnecessary viscous

force associated with damping, which may cause the over-shooting of the

predicted load. On the other hand, the damping cannot be such small that the

local dynamic effects (local vibration) associated with the cracking of concrete

cannot be efficiently damped out.

5.4.4 Single versus Double Precision

Even though simulation with single precision requires less computational

time, simulation with double precision is suggested for FE simulation for

improved accuracy, especially when the number of increments is very large,

which is the case of explicit CDM. To investigate the computation precision

effect on the 3D FE modelling, the FE models for CB-Rec, CB-T and

O-700-300-N were used. The predicted load-displacement curves are shown in

Figure 5.20. As indicated in Figure 5.20, the FE models calculated using single

precision predict similar load-displacement curves to those calculated with

double precision, except that the model CB-Rec calculated with single precision

fails earlier than that with double precision. When checking the kinetic energy

of these FE models, it was found that single precision calculation predict a

327

slightly higher kinetic energy than those calculated using double precision,

which is more obvious at the beginning and the end of loading, as shown in

Figure 5.21. Figure 5.22 shows the predicted crack patterns for the above three

specimens based on single precision calculation. It is obvious that at the

yielding displacement, the FE models simulated with single precision predict

crack patterns well. As the displacement increases, numerous numerical local

cracks appear, making the crack pattern deviated from that crack pattern

observed in test. Similar phenomena exist with the other two specimens with

less difference between the test observed crack pattern and numerical crack

patterns. The inaccuracy (or errors) in predicting the crack patterns can be

attributed to the error accumulation nature of to the explicit integration method.

The issues can be well solved when the calculations were carried out with

double precision mode. Thus, to obtain reliable 3D FE predictions, calculations

with double precision were adopted for the FE simulations in the present study

if not otherwise specified.

5.4.5 FRP Confinement Effect

To avoid non-ductile shear failure caused by the removal of beam concrete

with opening, FRP is applied to strengthen the beam in the vicinity of the

opening where potential shear failure might occur. The existence of FRP can

both increase the shear capacity of a weakened region and at the same time,

improve the compressive strength of the concrete, especially the web chord

confined with FRP wraps where FRP confinement effect should not be ignored.

328

Yu et al.'s (2009a, b) modified Plastic-damage model for a concrete column

with FRP confinement has been proved to be very accurate for modelling the

mechanical properties of FRP-confined concrete under compression. Therefore,

that model was adopted for modelling the concrete of the web chord

strengthened by FRP wrapping, as shown in Figure 5.23.

Figure 5.24 shows the typical numerical results with and without

considering FRP confinement. It is obvious that the inclusion of the

confinement effect on the web chord leads to more accurate predicted load

carrying capacity and post-peak load-displacement response. When the opening

is smaller and the web chord bigger (e.g. 700 × 200 mm2 opening), the

difference between the two FE models with and without the confinement effect,

is bigger, which is understandable because the member section affected by the

FRP is larger for the chord with larger cross section.

5.5 COMPARISON BETWEEN 2D AND 3D FE MODELS

2D FE Modelling using the dynamic method has been proposed by Chen et

al. (2011, 2015) and used by several researchers (Fu 2016, Nie 2018) and the

predicted results match well with test results for RC rectangular beams.

However, when the slab flange is wide, 2D models may do not perform well as

they do on rectangular beams because the uniform distribution of stress/strain

along the width direction cannot be satisfied in this circumstance.

The differences in modelling performance between 2D and 3D FE models

329

for T-beams will be examined in this section. As the ELA tension damage model

is used by the author, for comparison convenience, the 2-D modelling results

predicted by the damage plasticity (DP) approach with ELA tension damage

model are used for comparison purpose, even though the results predicted by DP

approach with ELA damage model is not better than those with power law (PL)

tension damage, as stated by Nie (2018). The 2-D FE models were meshed with

10-mm elements. CPS4R elements and T2D2 elements were used to represent

concrete and steel reinforcement respectively. 2D cohesive elements were

applied to model the bond-slip behaviour of the FRP-concrete and the

steel-concrete interfaces.

Figure 5.25 shows the results predicted by the 2D and 3D FE models for

the specimens CB-Rec, CB-T, O-700-300-N, O-700-200-N, F-700-300-N and

F-700-200-N. The 2D CB-T FE model overestimates the post-cracking stiffness

as demonstrated in Figure 5.25(a), though it predicts a yield load which is as

close to the test result as the 3D model. This is because the non-uniform flange

deformation of the T-beam in the direction perpendicular to the longitudinal axis

of the beam cannot be captured by a 2D FE model. Moreover, the 3D FE model

performs better than the 2D in predicting the CB-T beam post-yielding response.

The 2D FE models also behave worse than the 3D FE model when modelling a

T-beam with web opening weakening but no FRP strengthening against shear.

As shown in Figure 5.25(b), when the opening size is large (e.g. 700 × 300

mm2), the 2D FE model predicts a appreciably higher post-cracking stiffness

330

and lower peak load than the test results, whereas the predictions of the 3D FE

model are much closer to the result in terms of both post-cracking stiffness and

peak load. When the size of the opening is smaller (e.g. 700 × 200 mm2), the 2D

FE model predicts a much lower peak load than the test result, whereas the

prediction of the 3D FE model match well with the test result. When modelling

T-beams with web opening weakening and FRP strengthening, the 2D FE

models predict peak load higher than both test results and the 3D FE model

predictions, especially for the specimen with smaller opening (e.g. 700 × 200

mm2). Post-cracking stiffness are also overestimated by 2D FE models. When

the opening is smaller (e.g. 700×200 mm2), the 2D predicts a hardening

load-displacement curve which is deviated from both test result.

The predicted crack patterns of the specimens CB-R, O-700-200-N,

F-700-200-N of 2D and 3D models are compared in Figure 5.26, which shows

that the 2D and 3D models predicted similar crack patterns, both in the beam

web and slab, though the 2D models were unable to clearly show the cracks

propagating on the slab.

5.6 SHEAR DEGRADATION OF CRACKED CONCRETE

As mentioned earlier, the proposed 3D FE models predicted poorly the

post-yielding load-displacement curves for T-beams with a web opening but

without FRP strengthening, and a possible reason is that the DP model

underestimated the effect of shear resistance degradation of cracked concrete.

331

To investigate the effect of shear degradation, a new model using a combination

of the DP model and the BC model (namely the brittle crack model (ABAQUS

2012)) to represent concrete (i.e. a DP+BC model) was adopted, following Chen

et al. (2012). The model is illustrated in Figure 5.27. The concrete above the

opening was modelled with the DP model to capture the compression damage

associated with concrete compression failure and the remaining concrete was

modelled with the BC model considering that this concrete was mainly

subjected to tensile damage associated with shear tension failure. It should be

noted that the BC model assumes the concrete to be elastic under compression.

When using the BC model, the postcracking shear resistance is defined using a

power law as indicated in Equations 5.21 and 5.22:

( )ckc nnG e G (5.21)

( ) (1 )ck

ck nnnnn

max

eee

(5.22)

where G is the shear modulus of uncracked concrete; cG is the postcracking

shear modulus; cknne is the cracking strain; and maxe is the maximum cracking

strain corresponding to the complete loss of aggregate interlock.

As shown in Figure 5.28, as the coefficient n increases, the shear modulus

G decreases at a faster speed with a larger cracking strain. To define the failure

state, a number of critical cracks with the minimum and maximum values to be

332

1 and 3, respectively, is also specified. Therefore, three parameters, namely, the

number of critical cracks, the maximum cracking strain and the coefficient n,

may affect the concrete shear degradation behaviour in the BC model. The

specimen O-700-300-N was simulated with the proposed model to examine the

effects of the three parameters. The following reference parameters were

adopted in the simulation if not otherwise specified: maxe =0.0075, n=5 and the

number of critical cracks = 2.

5.6.1 The Effect of Number of Critical Cracks

As indicated in Figure 5.29, a larger number of critical cracks leads to a

more gradual descending post-peak load-displacement response and a higher

strength. The predicted strength increases appreciably when the number of

critical cracks increases from 1 to 2. When the number of critical cracks is equal

to 1, the DP+BC model predicts a critical crack happening at a displacement of

around 24 mm, leading to a sudden increase in the predicted kinetic energy.

5.6.2 The Effect of Maximum Cracking Strain

Maximum cracking strains equal to 0.005, 0.006 and 0.0075 were adopted

in the simulation for comparison. As indicated in Figure 5.30, a lower maximum

cracking strain leads to a lower predicted strength and softer post-peak response.

This is because a lower cracking strain results in a faster shear modulus

degradation (Figure 5.28).

333

5.6.3 The Effect of Coefficient of the Power Law

As shown in Figure 5.28, a larger n value results in a faster degradation

speed of the shear resistance. Figure 5.31 shows that when n increases from 1 to

4, the predicted strength becomes lower and the post-peak load-displacement

response becomes softer. When n is larger than 4, the effect of n is marginal,

except that when n = 7, the DP+BC model predicts a premature failure.

5.6.4 Comparison between the DP and DP+BC Models

In this section, the results predicted with the DP+BC model are compared

with those predicted with the proposed DP model presented in section 5.3. The

following values were used for the three parameters in the DP+BC model:

number of critical cracks = 2; maximum cracking strain = 0.006; and n = 5 for

the BC part, since these values provide the best predicted load-displacement

curve compared with the test curve as shown in Figure 5.29. As indicated in

Figure 5.32, the two models predict similar load-displacement curves, with the

DP+BC model predicting a post-peak load-displacement curve slightly closer to

the experimental one. The slightly better performance of the DP+BC model is

due to the fact that the DP+BC model can better consider the effect of shear

degradation of cracked concrete. However, the DP+BC model predicts larger

kinetic energy when the applied displacement is larger than about 12 mm, which

indicates that the DP+BC model may predict more cracks than the DP model.

334

The predicted crack patterns by the two FE models are compared in Figure

5.33. It can be seen that the predicted cracks propagating in the web chord are

similar although the DP+BC model predicts slightly more flexural cracks at the

mid-span of the specimen.

The above comparison results show that the DP model can provide

predictions comparable to the DP+BC model for RC T-beams with a web

opening but without FRP strengthening, but it is easier to create an FE model

using the former model; as a result, the DP model will be used in the remainder

of this PhD research if not otherwise specified.


Three-dimensional (3D) finite element (FE) modelling T-beams with web

opening was conducted using ABAQUS (2012) in this chapter. To avoid

convergence problems, a dynamic explicit approach which adopts the central

difference integration method was used. In this study, a 3D prism cohesive

model was proposed for the steel-concrete interface. The effect of the element

size, loading duration, damping ratio, and calculation precision which affect

dynamic explicit approach prediction accuracy, were studied. The confinement

effect provided by the web chord FRP was also investigated. Finally, the

predicted results of the 3D models were compared with the 2D results predicted

by the damage plasticity (DP) approach with elastic (ELA) tension damage

model, which was proposed by Nie (2018). The following conclusions can be

335

drawn based on the achieved modelling results:

(1) The proposed 3D FE models simulated well, the behaviour of the specimens

tested by Nie (2018) except for the two under positive loading. The

yield/peak loads and test crack patterns of all specimens were accurately

predicted, though post-yielding curves of T-beams with web opening were

not well captured. A possible reason is shear damage is underestimated by

the proposed models.

(2) The proposed 3D prism cohesive model is verified as reliable for 3D FE

analysis purposes when used for the steel-concrete interface and it was

successfully applied in the proposed 3D FE models for T-beams.

(3) For a quasi-static analysis, the dynamic energy of any particular analysis

should account for a small portion of the internal energy (normally below

1%), except for a few discrete moments, such as at the beginning of loading

and at the appearance of significant cracks.

(4) Element size, loading duration and stiffness damping coefficient were

determined based on convergence studies. A 20-mm element size, 50T0 (T0

is the FE model natural period) and equal to 2e-6 were found

appropriate for the proposed 3D models. Only the value of is needed to

define Rayleigh damping, since the first mode of vibration is the expected

deformation shape.

336

(5) Modelling with double precision computation is suggested for 3D FE

simulation. Single precision modelling may cost less in computational time,

but more errors accumulate as the number of increments increases, which

was obviously indicated by the predicted crack patterns.

(6) Yu et al.'s (2010a, b) modified plastic-damage model for a concrete column

confined with FRP was applied for the concrete in web chords with FRP

wrapping. When the confinement effect of the FRP wrapping is simulated,

more accurate results are obtained, especially when modelling a T-beam

with a smaller opening and a bigger web chord.

(7) The proposed 3D FE models are better predictors of reality than the 2D FE

models proposed by Nie (2018), in the modelling of T-beams, especially

those weakened by web openings and with no FRP shear strengthening. The

2D FE models overestimated the post-cracking stiffness of T-beams, which

was better predicted by the proposed 3D FE models.

(8) A DP+BC model, which is capable of modeling the shear degradation effect

of cracked concrete, was examined and the predictions were compared with

those from the DP model. The effects of three parameters, namely, the

number of critical cracks, the maximum cracking strain and the coefficient

of power law, of the BC model, were investigated through parametric

studies. The results indicated that the DP+BC model predicted the

load-displacement curves and crack patterns similar to those predicted with

337

the DP model; the DP+BC model predicted slightly better the post-peak

load-displacement response. The DP model is thus recommended for

modelling RC T-beams with a web opening but without FRP strengthening

since it is more convenient to create an FE model using the DP model.

5.8 REFERENCES



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346

Figure 5.1 Two kinds of Stress-strain models for steel bar

Figure 5.2 The proposed cohesive element model

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

50

100

150

200

250

300

350

400

450

500

550

Strain

Sre

ss (M

Pa)

Type 1 Rebar Type 2 Rebar

Bar element node

Nodes 1 and 2

Node 3 Node 4

Yielding point

0.2%

Hardening point

Ultimate point

0.2%

Es

347

Figure 5.3 Predicted stress-slip curve of proposed cohesive model versus input

data

(a) CB-Rec

0 0.5 1 1.50

2

4

6

8

10

12

Slip (mm)

She

ar s

tres

s (M

Pa)

Input Prediciton

' 30cf MPa

Deformed bar of 10 mm circumference

348

(b) CB-T

(c) F-700-300-N

Figure 5.4 Boundary conditions of 3D FE models

Figure 5.5 Smooth loading scheme

Load

X-symmetric section

Restrained in Y direction Restrained in both Z and Y direction

Y direction

Loading time

Dis

plac

emen

t

349

(a) CB-Rec and CB-T

(b) O-700-300-N and F-700-300-N

0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

CB-Rec-FE CB-T-FE CB-Rec-Test CB-T-Test

0 10 20 30 40 50 60 70 800

50

100

150

200

250

Displacement (mm)

Loa

d (k

N)

O-700-300-N-FE F-700-300-N-FE O-700-300-N-Test F-700-300-N-Test

Element size=20 mm =2e-6

Loading duration=50T0

Element size=20 mm =2e-6 Loading duration=50T0

350

(c) O-800-280-N and F-800-280-N

(d) F-700-300-P and F-800-280-P

0 10 20 30 40 50 60 70 800

50

100

150

200

250

Displacement (mm)

Loa

d (k

N)

O-800-280-N-FE F-800-280-N-FE O-800-280-N-Test F-800-280-N-Test

0 10 20 30 40 50 60 700

50

100

150

200

250

300

350

Displacement (mm)

Loa

d (k

N)

F-700-300-P-FE F-800-280-P-FE F-700-300-P-Test F-800-280-P-Test

Element size=20 mm =2e-6 Loading duration =50T0


351

(e) O-600-220-N, F-600-220-N and F-600-280-N

(f) O-700-200-N, F-700-200-N and F-700-260-N

Figure 5.6 Comparisons of predicted load-displacement curves and test results

0 10 20 30 40 50 60 700

50

100

150

200

250

300

350

400

450

Displacement (mm)

Loa

d (k

N)

O-600-220-N-FE F-600-220-N-FE F-600-280-N-FE O-600-220-N-Test F-600-220-N-Test F-600-280-N-Test

0 10 20 30 40 50 60 70 800

50

100

150

200

250

300

350

400

Displacement (mm)

Loa

d (k

N)

O-700-200-N-FE F-700-200-N-FE F-700-260-N-FE O-700-200-N-Test F-700-200-N-Test F-700-260-N-Test


Element size=20 mm =2e-6 Loading time =50T0

352

(a) CB-Rec

(b) CB-T

(c) O-700-300-N

Test crack pattern indicated by red dashed lines


353

(d) O-800-280-N

(e) O-600-220-N

(f) O-700-200-N




354

(g) F-700-300-N

(h) F-800-280-N

(i) F-700-300-P



355

(j) F-800-280-P

(k) F-600-220-N

(l) F-700-200-N




356

(m) F-600-280-N

(n) F-700-260-N

Figure 5.7 Predicted crack patterns

Figure 5.8 Energy relations of model CB-T

0 10 20 30 40 50 6010-8

10-6

10-4

10-2

100

102

104

Displacement (mm)

Per

cent

(%)

External work/Internal energy Energy by viscous effects/Internal energy Kinetic energy/Internal energy



357

(a) CB-Rec

(b) CB-T

0 10 20 30 40 50 600

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

Test CB-T-ELE4050 CB-T-ELE20 CB-T-ELE10

0 10 20 30 40 50 600

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

Test CB-T-ELE4050 CB-T-ELE20 CB-T-ELE10

=2e-6 Loading duration =50T0

=2e-6 Loading duration =50T0

358

(c) O-700-300-N

(c) F-700-300-N

Figure 5.9 Convergence study on element size

0 10 20 30 40 50 60 700

50

100

150

200

250

Displacement (mm)

Loa

d (k

N)

Test T-700-300-N-ELE4050 T-700-300-N-ELE20 T-700-300-N-ELE10

0 10 20 30 40 50 60 70 800

50

100

150

200

250

300

Displacement (mm)

Loa

d (k

N)

Test F-700-300-N-ELE4050 F-700-300-N-ELE20 F-700-300-N-ELE10

=2e-6 Loading duration=50T0


359

(a) CB-T

(b) O-700-300-N

0 10 20 30 40 50 600

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

Test CB-T-12.5T0

CB-T-25T0

CB-T-50T0

CB-T-100T0

CB-T-200T0

0 10 20 30 40 50 60 700

20

40

60

80

100

120

140

160

180

200

Displacement (mm)

Loa

d (k

N)

Test O-700-300-N-12.5T0

O-700-300-N-25T0

O-700-300-N-50T0

O-700-300-N-100T0

O-700-300-N-200T0



360

(c) F-700-300-N

Figure 5.10 The loading duration effect on the predicted load-displacement

curves

(a)CB-T

0 10 20 30 40 50 60 70 800

50

100

150

200

250

Displacement (mm)

Loa

d (k

N)

Test F-700-300-N-25T0

F-700-300-N-50T0

F-700-300-N-100T0

0 10 20 30 40 50 6010-8

10-6

10-4

10-2

100

102

104

106

Displacement (mm)

Kin

etic

Ene

rgy

(kN

. mm

)

CB-T-12.5T0

CB-T-25T0

CB-T-50T0

CB-T-100T0

CB-T-200T0


361

(b) O-700-300-N

(c) F-700-300-N

Figure 5.11The loading duration effect on the dynamic energy

0 10 20 30 40 50 60 7010-8

10-6

10-4

10-2

100

102

104

106

Displacement (mm)

Kin

etic

Ene

rgy

(kN

. mm

)

O-700-300-N-12.5T0

O-700-300-N-25T0

O-700-300-N-50T0

O-700-300-N-100T0

O-700-300-N-200T0

0 10 20 30 40 50 60 70 80

10-6

10-4

10-2

100

102

104

Displacement (mm)

Kin

etic

Ene

rgy

(kN

*mm

)

F-700-300-N-25T0

F-700-300-N-50T0

F-700-300-N-100T0

362

Figure 5.12 Relation between Rayleigh damping ratio and frequency

(a) 1st mode of vibration

(b) 2nd mode of vibration

363

(c) 3rd mode of vibration

(d) 4th mode of vibration

(e) 5th mode of vibration

Figure 5.13 First 5 modes of vibration of model CB-Rec

364




365



Figure 5.14 First 5 modes of vibration of model CB-T


366




367


Figure 5.15 First 5 modes of vibration of model O-700-300-N

0 10 20 30 40 50 60

0

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

Test CB-T-No damping CB-T-Beta=1e-8 CB-T-Beta=5e-8 CB-T-Beta=1e-7 CB-T-Beta=5e-7 CB-T-Beta=1e-6 CB-T-Beta=5e-6 CB-T-Beta=1e-5 CB-T-Beta=5e-5

Element size=20 mm Loading duration=50T0

368

000(a) CB-T

(b) O-700-300-N

Figure 5.16 The damping ratio beta effect on the predicted load-displacement

curves

(a) CB-T

0 10 20 30 40 50 60 70-50

0

50

100

150

200

Displacement (mm)

Loa

d (k

N)

Test O-700-300-N-No damping O-700-300-N-Beta=1e-8 O-700-300-N-Beta=5e-8 O-700-300-N-Beta=1e-7 O-700-300-N-Beta=5e-7 O-700-300-N-Beta=1e-6 O-700-300-N-Beta=5e-6 O-700-300-N-Beta=1e-5 O-700-300-N-Beta=5e-5

0 10 20 30 40 50 6010-8

10-6

10-4

10-2

100

102

104

106

Displacement (mm)

Kin

etic

Ene

rgy

(kN

. mm

)

CB-T-No damping CB-T-Beta=1e-8 CB-T-Beta=5e-8 CB-T-Beta=1e-7 CB-T-Beta=5e-7 CB-T-Beta=1e-6 CB-T-Beta=5e-6 CB-T-Beta=1e-5 CB-T-Beta=5e-5


a b c

d

369

(b) O-700-300-N

Figure 5.17 The damping ratio beta effect on the dynamic energy

(a) CB-T

0 10 20 30 40 50 60 7010-8

10-6

10-4

10-2

100

102

104

106

Displacement (mm)

Kin

etic

Ene

rgy

(kN

. mm

)

O-700-300-N-No damping O-700-300-N-Beta=1e-8 O-700-300-N-Beta=5e-8 O-700-300-N-Beta=1e-7 O-700-300-N-Beta=5e-7 O-700-300-N-Beta=1e-6 O-700-300-N-Beta=5e-6 O-700-300-N-Beta=1e-5 O-700-300-N-Beta=5e-5

0 10 20 30 40 50 600

200

400

600

Loa

d (k

N)

Displacement (mm)

0 10 20 30 40 50 600

2.5

5

7.5x 104

Kin

etic

Ene

rgy

(kN

. mm

)

0 10 20 30 40 50 600

2.5

5

7.5x 104

0 10 20 30 40 50 600

2.5

5

7.5x 104

Test CB-T-Beta=1e-6 CB-T-Beta=2e-6 CB-T-Beta=4e-6


370

(b) O-700-300-N

(c) F-700-300-N

Figure 5.18 The damping ratio beta effect on the predicted load-displacement

curves and dynamic energy

0 10 20 30 40 50 60 700

100

200

Loa

d (k

N)

Displacement (mm)

0 10 20 30 40 50 60 700

2500

5000

7500

10000

Kin

etic

Ene

rgy

(kN

. mm

)

0 10 20 30 40 50 60 700

2500

5000

7500

10000

0 10 20 30 40 50 60 700

2500

5000

7500

10000

Test O-700-300-N-Beta=1e-6 O-700-300-N-Beta=2e-6 O-700-300-N-Beta=4e-6

0 10 20 30 40 50 60 70 800

50

100

150

200

250

Loa

d (k

N)

Displacement (mm)

0 10 20 30 40 50 60 70 800

0.4

0.8

1.2

1.6

2x 104

Kin

etic

Ene

rgy

(kN

. mm

)

0 10 20 30 40 50 60 70 800

0.4

0.8

1.2

1.6

2x 104

0 10 20 30 40 50 60 70 800

0.4

0.8

1.2

1.6

2x 104

Test F-700-300-N-Beta=1e-6 F-700-300-N-Beta=2e-6 F-700-300-N-Beta=4e-6



371

(a) Crack pattern at moment a

(b) Crack pattern at moment b

(c) Crack pattern at moment c

372

(d) Crack pattern at moment d

Figure 5.19 Crack patterns of model O-700-300-N with damping ratio beta equal

to zero at four specified displacements

Figure 5.20 The computation precision effect on the predicted load-displacement

curves

0 10 20 30 40 50 60 70 80 900

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

CB-Rec-Single CB-Rec-Double CB-T-Single CB-T-Double O-700-300-N-Single O-700-300-N-Double

=2e-6 Loading duration=50T0 Element size=20 mm

a b

373

Figure 5.21 The computation precision effect on the dynamic energy

(a) Yielding state of CB-Rec

0 10 20 30 40 50 60 70 80 9010-10

10-8

10-6

10-4

10-2

100

102

104

106

Displacement (mm)

Kin

etic

Ene

rgy

(kN

. mm

)

CB-R-Single CB-R-Double CB-T-Single CB-T-Double O-700-300-N-Single O-700-300-N-Double0 3

10-3

100

104

Single precision

Double precision

374

(b) Moment a of CB-Rec

Single precision

Single precision

Double precision

375

(c) Moment b of CB-Rec

(d) Ultimate state of CB-Rec

Single precision

Double precision

Double precision

376

(e)Yielding state of CB-T

Single precision

Double precision

377

(f) Ultimate state of CB-T

Single precision

Double precision

378

(g) Peak load state of O-700-300

Single precision

Double precision

379

(h) Ultimate state of O-700-300

Figure 5.22 Crack patterns of 3D FE models simulated with single and double

precision

Single precision

Double precision

380

Figure 5.23 The part of FE model considering FRP confinement effect

Figure 5.24 FRP confinement effect (CE) on load-displacement curves

0 10 20 30 40 50 60 70 800

50

100

150

200

250

300

350

400

Displacement (mm)

Loa

d (k

N)

F-700-300-N-Test F-700-200-N-Test F-700-300-N-FE-With-CE F-700-300-N-FE-Without-CE F-700-200-N-FE-With-CE F-700-200-N-FE-Without-CE


Element size=20 mm

Web chord considering FRP confinement effect

381

(a) CB-Rec and CB-T

(b) O-700-300-N and O-700-200-N

0 5 10 15 20 25 30 35 40 450

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

CB-Rec-Test CB-Rec-2D CB-Rec-3D CB-T-Test CB-T-2D CB-T-3D

0 10 20 30 40 50 600

50

100

150

200

250

300

350

Displacement (mm)

Loa

d (k

N)

O-700-300-N-Test O-700-300-N-2D O-700-300-N-3D O-700-200-N-Test O-700-200-N-2D O-700-200-N-3D

382

(c) F-700-300-N and F-700-200-N

Figure 5.25 Predicted load versus displacement curves of 2D and 3D models

(a) CB-T

0 10 20 30 40 50 600

50

100

150

200

250

300

350

400

450

500

Displacement (mm)

Loa

d (k

N)

F-700-300-N-Test F-700-300-N-2D F-700-300-N-3D F-700-200-N-Test F-700-200-N-2D F-700-200-N-3D

2D

3D

383

(b) O-700-200-N

(c) F-O-700-200-N

Figure 5.26 Comparison between crack patterns predicted with 2D and 3D FE

models

2D

3D

2D

3D

384

Figure 5.27 Implementation of the BC+DP model in the FE model

Figure 5.28 Shear modulus degradation with different values of coefficient n

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02

ρ=G

c/G

ennck

n=1

n=5

n=2

n=3

n=4

n=6

n=7

DP part

BC part

emaxck

385

Figure 5.29 The effect of number of critical cracks

Figure 5.30 The effect of maximum cracking strain

0.1

1

10

100

1000

10000

100000

1000000

10000000

0

25

50

75

100

125

150

175

200

0 5 10 15 20 25 30 35 40 45

Loa

d (k

N)

Inte

rnal

Ene

rgy

(N·m

m)

Displacement (mm)

Test

O-700-300-N-DP+BC-N=5-0.0075-1-TZ

O-700-300-N-DP+BC-N=5-0.0075-2-TZ

O-700-300-N-DP+BC-N=5-0.0075-3-TZ

0.1

1

10

100

1000

10000

100000

1000000

0

25

50

75

100

125

150

175

200

0 5 10 15 20 25 30 35 40 45

Kin

etic

ene

rgy

(N·m

m)

Loa

d (k

N)

Displacement (mm)

Test

O-700-300-N-DP+BC-N=5-0.005-2-TZ

O-700-300-N-DP+BC-N=5-0.006-2-TZ

O-700-300-N-DP+BC-N=5-0.0075-2-TZ

386

Figure 5.31 The effect of coefficient of power law

Figure 5.32 Comparison of load-displacement curves predicted with the DP and

the DP+BC models

0

25

50

75

100

125

150

175

200

225

0 10 20 30 40

Loa

d(kN

)

Displacement(mm)

Test O-700-300-N-DP+BC-N=1-0.006-2-TZ O-700-300-N-DP+BC-N=2-0.006-2-TZ O-700-300-N-DP+BC-N=3-0.006-2-TZ O-700-300-N-DP+BC-N=4-0.006-2-TZ O-700-300-N-DP+BC-N=5-0.006-2-TZ O-700-300-N-DP+BC-N=6-0.006-2-TZ O-700-300-N-DP+BC-N=7-0.006-2-TZ

0.1

1

10

100

1000

10000

100000

0

25

50

75

100

125

150

175

200

0 5 10 15 20 25 30 35 40 45

Loa

d (k

N)

Kin

etic

Ene

rgy

(N·m

m)

Displacement (mm)

Test O-700-300-N-DP O-700-300-N-DP+BC-N=5-0.006-2-TZ

387

(a) O-700-300-N-DP

(b) O-700-300-N-DP+BC-N=5-0.006-2-TZ

Figure 5.33 Comparison of crack patterns predicted with the DP and the DP+BC

models

Table 5.1 Specimen dimensions and test method

Source Specimen Beam dimensions Slab dimensions Opening size

Bending direction l0/s, mm bw, mm hb/ h'b, mm bslab, mm t, mm

Fabrication of web opening / Length/ Height, mm

Nie (2018) Batch 1

CB-Rec

3300/1650 250 500/460

NA Negative(a)

CB-T Negative O-700×300-N

1400 1000

Pre-formed/700/300 Negative

F-700×300-N Negative F-700×300-P Positive(b)

O-800×280-N Pre-formed/800/280

Negative F-800×280-N Negative F-800×280-P Positive

Nie (2018) Batch 2

O-600×220-N

3300/1650 250 500/460 1400 1000

Post-cut/600/220 Negative

F-600×220-N Negative O-700×200-N

Pre-formed/700/200 Negative

F-700×200-N Negative F-600×280-N Post-cut/ 600/280 Negative F-700×260-N Pre-formed/ 700/260 Negative

Note: l0: effective span of beam; s: shear span of beam. bw: beam width; hb: beam height; h'b: beam effective height; bslab: slab width; t: slab thickness; (a) The beam flange was in tension; (b) The beam flange was in compression.

389

Table 5.2 Specimen material properties

Source Specimen

Concrete Steel reinforcement CFRP

reinforcement

fc’ (MPa)

Ec (GPa) c

Tension bars (deformed

bars) Number/D/ fy/fu, Mpa

Compression bars

(deformed bars)

Number/D/ fy/fu, Mpa

Stirrups (plain bars) D/spacing/ fv/fv, Mpa

Slab bars (plain bars) Number/D/ fv/fv, Mpa

Es, Gpa

tf ffrp,

Mpa Ef , Gpa

Nie (2018) Batch 1

CB-Rec 42.5 24.9 0.0026

4/20/ 475/625

3/20/ 475/625

8/100/ 307/447

Two layers 6/8/

307/447 200

NA CB-T 55.2 25.1 0.0033 O-700×300-N 42.5 24.9 0.0026 F-700×300-N 41 23.3 0.0028

0.334 2820 227 F-700×300-P 44.1 / / O-800×280-N 42.5 24.9 0.0026 NA F-800×280-N 41 23.3 0.0028

0.334 2820 227 F-800×280-P 44.1 / /

Nie (2018)

O-600×220-N 40.3 / /

4/20/ 434/559

3/20/ 475/625

8/ 100/ 349/526

Two layers 6/8/

200 NA

F-600×220-N 0.334 2820 227

390

Batch 2 O-700×200-N 36.2 349/526 NA F-700×200-N 39.6

0.334 2820 227 F-600×280-N 42

F-700×260-N

Note: f'c: concrete cylinder compressive strength; Ec: concrete elastic modulus. c : concrete compressive strain at concrete cylinder strength; D: steel bar

diameter (in mm unit); fy: bar yield stress; fu: bar ultimate stress; Es :steel bar elastic modulus; tf : FRP sheet thickness; ffrp: tensile strength of FRP sheet; Ef:

elastic modulus of FRP sheet.

391

Table 5.3 Steel properties of type 1 bar of the two test groups

Source Nie (2018)

Batch 1 Nie (2018)

Batch 2 D/mm 20 20 8

Es (MPa) 200000 200000 200000 fy (MPa) 475 434 349

h 0.03 0.028 0.013

Es2 (MPa) 2680 2450 4490 fu (MPa) 625 559 526

Note: D: steel bar diameter; Es :steel bar elastic modulus; fy: deformed bar yield stress;

h : hardening beginning strain; Es2: modulus of hardening curve; fu: deformed bar

strength.

Table 5.4 Peak bond stress and corresponding slip of Lu et al.'s (2005)model

Source Specimen fc’ (MPa)

Peak bond stress (MPa)

so (mm)

Nie (2018) Batch 1

F-700×300-N 41.00 3.68 0.0478 F-700×300-P 44.10 3.83 0.0498 F-800×280-N 41.00 3.68 0.0478 F-800×280-P 44.10 3.83 0.0498

Nie (2018) Batch 2

F-600×220-N 40.30 3.64 0.0474 F-700×200-N 39.60 3.61 0.0469 F-600×280-N

42.00 3.73 0.0484 F-700×260-N

392

Table 5.5 Test and predicted loads

Specimen Test result (kN)

3D FE models (kN) 2D FE models (kN)

Prediction Prediction/test Prediction Prediction/test

CB-Rec 340 335.6 0.99 333.5 0.98 CB-T 510 513.0 1.01 510.5 1.00

O-700×300-N 182 189.2 1.04 165.2 0.91 O-800×280-N 181 189.5 1.05 158.9 0.88 O-600×220-N 316 310.9 0.98 243.8 0.77 O-700×200-N 300 300.5 1.00 236.5 0.79 F-700×300-N 207 215.2 1.04 216.8 1.05 F-800×280-N 219 219.9 1.00 221.5 1.01 F-600×280-N 260 258.8 1.00 279.7 1.08 F-700×260-N 270 269.2 1.00 296 1.10 F-600×220-N 407 385.0 0.95 407.4 1.00 F-700×200-N 410 386.5 0.94 406.4 0.99 F-700×300-p 228 284.3 1.25 277.1 1.22 F-800×280-P 215 262.8 1.22 249.6 1.16 Average = 1.03 0.99

STD = 0.09 0.12

393

CHAPTER 6

3D SHELL MODELS FOR RC T-BEAMS

6.1 INTRODUCTION

The proposed 3D FE model presented in chapter 5 is, referred to hereafter

as a 3D solid model, given that 3D solid elements are used to represent concrete.

As revealed in the previous chapter, this model has performed well in modelling

the load-displacement curve and crack patterns of RC T-section beams

(T-beams). The 3D solid model, however, is not economical of computational

time as the number of elements is greater than that of the 2D FE model

proposed by Nie (2018), although it does provide more accurate predictions. To

reduce the FE modelling computational time of the dynamic explicit method,

reducing element numbers is a good method when the loading time duration and

the incremental time step length are fixed. Thus, a 3D shell FE model (referred

to as 3D shell model), in which the 3D solid elements are replaced by 3D S4R

shell elements for modeling concrete, was proposed for modelling T-section

beams (T-beams). Compared to the 2D FE model, the 3D shell model can

simulate the non-uniform deformation in the transverse direction of a T-beam.

Compared with the 3D solid model, the construction of a 3D shell model is

easier. Thus, several 3D shell models are proposed based on different

considerations/assumptions and compared in this chapter, with the aim of

394

finding the most suitable 3D shell models which can be a substitute of 3D solid

model for modelling T-beams.

6.2 PROPOSED 3D SHELL MODELS

The 3D shell models are proposed based on the 3D solid model presented

in chapter 5. The major difference between 3D shell models and 3D solid

models is the type of concrete element as mentioned earlier. In the construction

of a 3D shell model, concrete of the beam web part was represented by 3D S4R

shell elements available in ABAQUS (2012). B31 beam elements were used to

model steel bars. Bond-slip behaviour between steel bars and concrete are

considered using the proposed 3D prism cohesive model. As for the slab

component, if there is only one layer of longitudinal bars, the slab shell layer is

located at the height of the longitudinal bars, as shown in Figure 6.1(a). If two

layers of longitudinal reinforcing bars exist, which is a usual case, the slab is

modelled by one shell layer as shown in Figure 6.1(a) or two shell layers as

shown in Figure 6.1(b). The effect of considering bond-slip between slab

longitudinal steel bars and concrete in the FE model needs to be investigated ,

so slab bars are either modeled by the discrete bar method which models bars

with beam element B31, or the smeared bar method in which an embedded layer

of steel within the concrete layer is defined to simulate the bars. As a result, a

total of seven 3D shell models, including 3 models with one layer of shell for

slab modeling and 4 models with two layers of shell for slab modelling, are

proposed and discussed next, as shown in Table 6.1. In Table 6.1, except for the

395

T-3D-Solid model proposed in chapter 5, all remaining models use shell

elements to represent concrete. In the seven models, only T-2D-Shell-1-SB

model uses the smeared bar method to model slab bars, the rest six 3D shell

models use the discrete bar method to model slab bars, with or without

considering bond-slip behavior between bars and concrete. The models not

considering bond-slip behaviour between slab bars and concrete have “NBS” in

their names. For those models with two slab shell layers, if the deformation of

the two layers follows the plain sectional assumption such as an integral shell

layer, they have “R” in their names.

6.2.1 3D Shell Models with a One-shell-layer Slab

The positions of the shell layers of web and slab in the FE models with

one-shell-layer slabs (referred to hereafter as 3D 1-shell models) are shown in

Figure 6.1(a). Making using of the symmetry condition, only a half the T-section

beam is modeled to reduce the calculation time. As a result, the thickness of the

shell layer of beam web is half that of the beam web. In the beam web, the

stirrups or longitudinal bars at the same position (same height) are modelled

with only one bar meshed with the B31 beam elements with the section area and

bond area considered. The bond-slip behaviour of the steel-concrete interface is

modelled using the proposed 3D prism cohesive model detailed in Chapter 5.

For the slab, as there is only one shell layer with a thickness equal to that of the

slab, that layer is assumed located at the middle of the slab. It should be noted

that the slab bars of the T-3D-Shell-1-SB model are smeared in the slab shell, so

396

that the area of the steel layer area is equal to the sum of the complete slab bar

area. One smeared steel layer is used to model the slab longitudinal and

transverse bars as they had a same bar spacing but were orthogonal. Bond-slip

behaviour of the slab steel-concrete interface is ignored (i.e. there is no slip

between the bar and surrounding concrete) in the T-3D-Shell-1-SB model.

For the T-3D-Shell-1 and T-3D-Shell-1-NBS models, slab bars are

modelled with beam elements B31. The slab bars sited in the same position are

modelled with only one bar of the same sectional area to their sectional area

sum. For the T-3D-Shell-1 model, the bond-slip behaviour between slab

longitudinal bars and concrete are modelled using the proposed triangular

cohesive model. The T-3D-Shell-1-NBS model does not incorporate bond-slip

behaviour between slab longitudinal bars and concrete (i.e. there is no slip

between the bar and surrounding concrete). For both T-3D-Shell-1-SB and

T-3D-Shell-1-NBS models, the transverse bars are connected directly to the

concrete elements, hence ignoring the bond-slip behaviour of the steel-concrete

interface.

6.2.2 3D Shell Models with a Two-shell-layer Slab

The positions of the shell layers representing the web and slab of a 3D FE

model in the case of a two-shell-layer slab (referred to, below, as a 3D 2-shell

model) are as shown in Figure 6.1(b). The beam web in 3D 2-shell models is

represented in the same way as for 3D 1-shell models. The two shell layers of

http://www.baidu.com/link?url=kknmU5pXmwJaI--LpU3lQ2VtPcnol5X9x-fe6ffEtu9K-wnCrWWr3k6re8VqZA3te3o7fQXL23p-TarMSKCQspViRjgF69Y-ETLy2UwIM3_

397

the beam slab are located at the respective heights of the two slab bar layers.

The thickness of each shell layer is half the thickness of the real beam slab. The

offset values for each shell layer were defined the positions of each layers of

slab relative to the position of bars. A discrete bar method was applied to model

both the longitudinal and the transverse slab bars. For both T-3D-Shell-2 and

T-3D-Shell-2-NBS models, no direct interaction condition is applied to the two

slab layers. For the T-3D-Shell-2-R and T-3D-Shell-2-NBS-R models, direct

interaction is applied to the two slab layers using equation restraints in

ABAQUS (2012). To ensure the two slab layers deform in the same way as the

one layer, the following equations are applied to any two nodes, with one node

in each of the two slab layers but located in the same position along both the X

and Z directions:

`

, ,

, ,

, ,

,

,

000

000

x t x b

y t y b

z t z b

t b

t b z t t b

t b x t t b

y yx x dz z d

(6.1)

where ,x t , ,y t , ,z t , tx , ty and tz are the angular and line displacements

of the node of the top layer and ,x b , ,y b , ,z b , bx , by and bz are the

angular and line displacements of the corresponding node of the bottom layer

and t bd is the vertical distance separating the two layers. It should be noted

398

that the above restraints are proposed based on the assumption that the

deformation of the nodes in the layer following a plane section.

In all 3D 2-shell models, no bond-slip behaviour is allowed at the interface

between slab transverse bars and concrete. For the T-3D-Shell-2 and

T-3D-Shell-2-R models, the proposed 3D prism cohesive model introduced in

the last chapter will be used for modelling of bond-slip behaviour and applied in

the longitudinal direction. The bond-slip is not considered (i.e. there is no slip

between the bar and surrounding concrete) in the T-3D-Shell-2-NBS and

T-3D-Shell-2-NBS-R models.

6.3 VERIFICATION OF 3D SHELL MODELS

The specimen CB-T described and tested by Nie (2018) was modeled

herein to verify the proposed 3D shell models. To save time, only half of the

specimen was modelled taking advantage of symmetry. The boundary

conditions of a 3D shell model are illustrated in Figure 6.2. Next, the predicted

load-displacement curves and crack patterns using the shell models were

compared with test results and those of the 3D solid model to assess the

accuracy/efficiency of the proposed 3D shell model, based on which the most

suitable 3D shell model will be determined. The effect of bond-slip behaviour

between steel bars and concrete in the longitudinal direction will also be

evaluated and discussed. The comparisons between the best 3D shell model with

2D models in terms of the load-displacement curve will also be conducted. The

399

element size was chosen to 20 mm, the loading time duration was 50T0, the

damping coefficient β was 2e-6 according to the convergence studies presented

later.

6.3.1 3D Shell models with One-shell-layer Slab

The predicted load-displacement curves are shown in Figure 6.3. As

indicated in figure, the 3D-Solid model of the T-beam (referred to hereafter as a

T-3D-Solid model) predicts a slightly more accurate load-displacement curve

than those of the 3D 1-shell models. The 3D 1-shell models predict an

appreciably higher post-cracking stiffness and yield load than those of the

T-3D-Solid model, although both the shell model and the solid model produce

quite similar results for post-yielding loading displacement responses. Among

those 3D 1-shell models, the two which ignore the bond-slip behaviour of

longitudinal bars in the slab, predict a slightly higher post-cracking stiffness and

yield load than those predicted by the 3D-Shell-1 model, which incorporate the

modeling of the bond-slip model. Based on the above, it is apparent that the

considering bond-slip behaviour of bars in the longitudinal direction results in a

more accurate predicted load-displacement curve, especially for the

post-cracking stiffness and yield load of the T-beams.

The predicted crack patterns of the 3D 1-shell models are shown in Figure

6.4, from which it is seen that the web crack and slab crack patterns appear and

connected at the web-flange interface. The test crack pattern is indicated by

400

red-dashed lines, indicating that the beam web cracks are well predicted by the

3D 1-shell models. For cracks propagated in the slab, however, the prediction of

T-beam 3D-Shell-1 model (referred to hereafter as a T-3D-Shell-1 model) is the

most successful. The other two models predict more densely spaced cracking

near the centre of the specimen slab, this can be attributed to that they ignore the

bond-slip behaviour of the longitudinal bars in slab. Thus, the

T-3D-Shell-1model, which considers the bond-slip behavior of the longitudinal

bars, is the best of the 3D 1-shell models for simulating the behaviour of RC

T-beams.

6.3.2 3D Shell Models with a Two-shell-layer Slab

The load displacement curves predicted by predicted 3D 2-shell models are

shown in Figure 6.5. Again, the T-3D-Solid model also predicts a better/accurate

load-displacement curves than those of the 3D 2-shell models. Similar to the 3D

1-shell model, the 3D 2-shell models without considering the bond-slip behavior

of longitudinal bars in slab (i.e. T-3D-Shell-2-NBS and T-3D-Shell-2-NBS-R)

predict appreciably higher post-cracking stiffnesses and yield loads than those

of the T-3D-Solid model, whereas the 3D 2-shell models (T-3D-Shell-2 and

T-3D-Shell-2-R) which ignore that bond-slip behaviour, predict obviously better

results (closer to test results and the predictions of T-3D-Solid model) in terms

of post-cracking stiffnesses and yield loads, thus demonstrating that the

inclusion of the bond-slip behaviour in the 3D 2-shell models leads to a more

accurate predicted load-displacement curve.

401

The predicted cracks patterns using the 3D 2-shell model are shown in

Figure 6.6 with the test observed crack patters marked by red-dashed lines. It

should be noted that in Fig. 6.6., the web, top and bottom slab crack patterns are

placed, in the figure, at the top, middle and bottom respectively to demonstrate a

whole image of the crack pattern. It is obvious that the 3D-Shell-2 and

3D-Shell-2-R models of the T-beam (referred to, below, as T-3D-Shell-2 model

and the T-3D-Shell-2-R modes) predict better crack patterns regarding the

number and positions of cracks than the two models not considering the

bond-slip behavior of the longitudinal bars in the slab. Comparing Figures 6.6(a)

and (c) or (b) and (d), it is seen that the plane sectional restraint condition

applied for the two slab layers has a significant effect on the predicted crack

patter: it causes the cracks on the two shell layers to appear and propagate in the

same position, whereas in the two shell models (3D-Shell-2 and

3D-Shell-2-NBS) without including such restraint of plan section, even the main

crack may appear at obviously different locations. Comparing the predicted

load-displacement responses, it can be seen that the effect of the plan section

restrain is marginal. In summary, applying the plan section restraint leads to a

more accurate predictions, especially for crack patters.

To further investigate the effect of applying the plan sectional restraint,

several models, tabulated in Table 6.2 were studied. The differences between the

models lie in the density of the nodes applied with plan sectional restraints. As

the element size of the T-3D-Shell-2-R model is 20 mm, the restraint distances

402

(distance between nodes with the plan section restraint) in both X and Z

directions was chosen as 20 mm in the T-3D-Shell-2-R. For the

T-3D-Shell-2-R-X100 and T-3D-Shell-2-R-X200 models, the restraint distances

in the X direction are 100 and 200 mm respectively but the restraint distance in

the Z direction still remains as 20 mm. Similarly, the T-3D-Shell-2-R-Z100 and

T-3D-Shell-2-R-Z200 models have the restraint distances of 100 and 200 mm in

the Z direction with the restraint distance along the X direction, remaining

unchanged at 20 mm.

The predicted load-displacement curves of the models listed in Table 6.2

are shown in Figure 6.7. As indicated in Figure 6.7, decreasing the density of

plane sectional restraints by increasing the restraint distances along the X or Z

directions (i.e. transverse and longitudinal directions) has a minor effect on the

predicted load-displacement curves. This is to be expected, as the T-3D-Shell-2

model, in which the plane section restrain is applied, predicts a

load-displacement curve, close to that of the T-3D-Shell-2-R model as discussed

above. The predicted crack patterns are shown in Figure 6.8. It is found that

increasing the restraint distance in the Z direction, results in predicted cracks in

the two slab layers, appearing and propagating at different position along the Z

direction. This situation is more obvious with increasing restraint distance in the

Z direction. Decreasing restraint distance along the X direction has much less

effect on predicted crack patterns than a decrease in restraint distance along the

Z direction, Hence, increasing restraint distance along the X direction is the

403

more suitable modelling option in term of the accuracy of predicting the

cracking behavior of the T-section beams.

6.3.3 Efficiency and Accuracy of the 3D Shell Models

The predicted results reveal that the 3D-Shell-1 and 3D-Shell-2-R models

to be the best 3D 1-shell model and 3D 2-shell model respectively for modelling

a T-beam. As indicated in Figure 6.9, the two shell models predict close

load-displacement curves, both are close to the prediction of the 3D-solids

model. The comparisons of predicted crack patterns shown in Figure 6.4(b) and

6.6(c), indicate that both models satisfactorily predict crack patterns in beam

web and slab. However, the T-3D-Shell-2-R model is less economical of

computational time than the T-3D-Shell-1 model and, even less economical than

the T-3D-Solid model, as demonstrated in Table 6.3 which lists the

computational time. It should be noted that the same computer was used for all

computational times given in Table 6.3. The computational cost for the

T-3D-Shell-1 model was about 64.5% of that for the T-3D-Solid model, whereas

the computational cost for the T-3D-Shell-2-R model was more than twice

(about 2.22 times) that of the T-3D-Solid model. Of further interest is that the

decreasing plane sectional restraint density between the two slabs can reduce the

computational time cost when using 3D 2-shell models, as also illustrated in

Table 6.3. Decreasing restraint density in the X direction, is more effective in

reducing computational time than decreasing along the Z direction.

404

To compare the performance of 3D-shell model and 2D FE model, the

load-displacement curve predicted by the T-3D-Shell-1 model is compared with

the 2D FE model of Nie (2018) in Figure 6.10. The T-3D-Shell-1 model, similar

to the T-3D-Solid model presented in chapter 5, is better than the 2D FE model

in predicting post-cracking stiffness as this model is able to capture non-uniform

deformation in the transverse direction.

Thus, in terms of accuracy, modeling convenience (i.e. ease of building the

FE model) and computational efficiency (assessed by computational time), the

T-3D-Shell-1 model can be considered a good substitute for the T-3D-Solid

model, and the T-3D-Shell-2-R with an appropriate restraint distance (e.g.

T-3D-Shell-2-R-X200), although more complicated in building FE model

compared with the T-3D-Shell-1 model, can also be a suitable choice, especially

when the locations (along the beam height) of the bars in slab needs to be

considered.

6.4 PARAMETRIC STUDIES ON THE EFFECTS OF SOME

KEY FACTORS

Of the 3D shell models, the T-3D-Shell-1 model has been found the most

suitable 3D FE model substitute for the T-beam. According to Chen et al. (2015)

and the results presented in the previous chapter (Chapter 5), accurate choices of

element size, loading time duration and damping coefficient β of Rayleigh

damping, are important for dynamic FE modelling if reliable results are to be

405

obtained. Thus a parametric study was carried out in a similar way as described

in the previous chapter, to investigate the effects of element size, loading time

duration and the damping coefficient β on the numerical results predicted by the

T-3D-Shell-1 model, with the objective of finding most suitable values for them.

As in the previous chapter, in the parametric study, when one parameter is

changed, the rest parameters will be kept unchanged.

6.4.1 Element Size

A convergence study for different element sizes was carried and the

predicted load-displacement curves are shown in Figure 6.11, in which

T-3D-Shell-1-ELE40 and T-3D-Shell-1-ELE20 stand for FE models meshed

with element sizes 40 mm and 20 mm respectively, and T-3D-Shell-1-ELE4050

meshed mostly with 50 mm elements in the Z and X directions and 40 mm in

the Y direction. As indicated in Figure 6.11, the three models predict similar

load-displacement curves. As the FE model with 20 mm elements provides

predictions almost as accurate as the model with the 10 mm elements, the

former, i.e. FE model meshed element size of 20 mm, was chosen for the

T-3D-Shell-1 model, for modelling T-beams in this study.

6.4.2 Loading Duration

The natural period T0 of the T-3D-Shell-1model was first predicted using

ABAQUS (2012). Loading time spans of 12.5T0 , 25T0, 50T0, 100T0 and 200T0

406

were then applied to the T-3D-Shell-1 model. The predicted load-displacement

curves are shown in Figure 6.12. It is evident that load time spans of 12.5 T0 and

25 T0 are not long enough for the T-3D-Shell-1 model to reach stability. When

the load time span is increased to 50 T0, the predicted load-displacement curve

is much closer to those with longer long loading time. The predicted kinetic

energy-displacement curve plotted in a logarithmic form is shown in Figure

6.13(a), which shows that even though kinetic energy increases as load time

duration decreases, kinetic energy is within a reasonable range except at the

beginning of loading (as indicated in Figure 6.13(b)) and without any

unexpected sudden increase. Sudden increases of kinetic energy, featured by the

local peaks of the kinetic energy, caused by concrete cracking are efficiently

damped out with the increase of displacement. Given this, a load time span

equal to 50 T0 was chosen for the T-3D-Shell-1 model.

6.4.3 Damping Coefficient β

As stated in chapter 5, only the stiffness-proportional damping is needed in

Rayleigh damping, so only the coefficient β value needs to be determined for FE

modelling using a dynamic explicit method. The T-3D-Shell-1 model with a

coefficient β value ranging from 0 to 5e-5 was studied. The predicted

load-displacement curves predicted with various values of β are shown in Figure

6.14. As illustrated in Figure 6.14, when β is equal to, or below, 1e-7, the

predicted load-displacement curve deviates significantly from the test response

before it reaches the yield state. When β is larger than 1e-5, the T-3D-Shell-1

407

model predicts higher cracking and yield loads, due to the over-shooting effect

on load arising from viscous force associated with the velocity. Thus, the

suggested value for β is between 5e-7 and 1e-5. Predicted kinetic

energy-displacement curves in logarithmic form are shown in Figure 6.15. It is

obvious that when β is not larger than 1e-7, the predicted kinetic energy shows a

sudden increase after concrete cracking and cannot not damped down as

displacement increases. When β is larger than 1e-5, the kinetic energy is also

slightly higher than those corresponding to a β value ranging from 5e-7 to 1e-5,

which is still reasonable. To further narrow the range for β, the T-3D-Shell-1

model with β equal to 1e-6, 2e-6 and 4e-6 was studied. It was found that the

predicted load-displacement curves for the three damping ratio were quite close.

The predicted kinetic energy-displacement curves were also within a reasonable

range. A value for β of 2e-6 was chosen because the T-3D-Shell-1 model

predicts a more stable result than with this value of β as the predicted curves are

sufficiently stable when compared with those with β equal to 4e-6. It is of

interest to note that local peaks of kinetic energy at the two significant moments

corresponding to concrete cracking and steel bars yielding, regardless of the

specific values of β.

To study the effect of insufficient β on predicted crack patterns, the

predicted crack patterns of the T-3D-Shell-1 model with β equal to 1e-8 at

different displacements (marked as moments “a”, “b”, “c” and “d” in Figure

6.14) are presented in Figure 6.17, in addition those of T-3D-Shell-1 model with

http://www.baidu.com/link?url=gkNKygcuKULE6LahxspToE8hSBuPyzBVp1R7ZHab0lhNaoqwZaVEJo8eLGBMjtBl1NiUorP8UxrWEAwltl3re9WPxfpxQ8gw5uUgsbULDMkwPrjCeKxaKYoV0vjVtECw

408

β equal to 2e-6 for comparison. The displacements at moments “a” to “d” are as

indicated in Figure 6.14. Moments “a” and “b” are occurring immediately

before and after appearance of a new crack (at the displacement of 3.28 mm, as

predicted by the model with β equal to 2e-6). Moments “c” and “d” are those at

the lowest and highest points (at the displacement of 5.2 mm and 43.28 mm) of

the load-displacement curve, respectively. As shown in Figure 6.17(a), at the

moment “a”, three flexural cracks propagate in the beam web and slab. When

reaching the displacement of moment “b”, for the model with insufficient β ,

many unexpected large strains and associated cracks arise on the web and slab,

due to local vibration caused by crack initiation and propagation and which has

not been effectively damped out due the low value of damping associated with

the very low value of β. When displacement increases to that of moment “c”,

additional large strains continue to emerge until, finally, at displacement of

moment “d”, the predicted crack pattern is spread over the FE model of

insufficient β , which is totally unreal/unreliable. Thus, β must be large enough

(e.g. >1e-7 for the studied case) to damp out dynamic effects associated with

local vibration caused by local cracking, so as to obtain reliable predicted crack

patterns. However, it should not too large (e.g. <5e-5 for the studied case) to

avoid viscous force leading to the over-shooting (overestimation) of load.

6.5 3D SHELL MODELS FOR A T-BEAM WITH A WEB

OPENING

The above sections have shown that compared with 3D solid model. The

409

most suitable 3D 1-shell models (i.e. the 3D-Shell-1 model) well predicts the

cracking and load-displacement behavior of a T-beam, and reduces

computational time as well. Next, the performance of the 3D-Shell-1model will

be further verified by modelling of T-beams weakened by web openings. The

O-700-300-N specimen tested by Nie (2018) was chosen for the analysis. A

convergence study related to element size was first conducted. As shown in

Figure 6.18, even when the element size is 20 mm, the predictions of the

O-700-300-N specimen still cannot converge, suggesting that the element size

needs to be smaller. The 3D-Shell-1 model of O-700-300-N based on 10 mm

elements, however, costs more computational time than that the 3D Solid model

meshed with 20 mm elements. Given that the 3D Solid model is more accurate

than the 3D-Shell-1 model, the latter proves not to be a satisfactory 3D Solid

model substitute when modelling a T-beam with web opening weakening. This

is partially because when the slab is modelled with only one shell layer, that

layer lacks the ability to simulate cracks propagating in slab originated from the

right bottom corner of the opening in the web. As shown in Figure 6.19, the

cracks emerging from the right corner of the opening near the slab are critical

and extend to the bottom of the beam. As shell elements was used instead of

solid element to simulation slab, the progressive propagation of these cracks

within the slab could be effectively simulated by the shell elements.

Theoretically, this problem can be solved by modeling the slab using muti-layer

shells; it can be envisaged that this will lead to complicated restraint conditions

between the different shell layers which will greatly increase the computational

410

cost as discussed above, making this modeling approach much less appealing.


The 3D Solid model proposed in Chapter 5 proved to be both reliable and

accurate when modelling a T-section beam (T-beam), but is not economical in

computational time compared with the 2D FE model proposed by Nie (2018).

Thus, representation of the beam with shell elements was alternatively proposed,

by which the number of elements can be greatly reduced. Several shell element

models have been built and used for simulation studies. A T-beam tested by Nie

(2018) was modeled using these shell element models and the 3D T-beam solid

model presented in chapter 5 was used as the control model for comparison

purpose. The following conclusions are drawn based on the numerical results,

comparisons and discussion presented in this chapter:

(1) The 3D-Solid model for T-beams predicts the most accurate results in terms

of the load-displacement curve and crack patterns.

(2) Those 3D 1-shell and 2-shell models incorporating the bond-slip relationship

between slab longitudinal bars and concrete, predict better

load-displacement curves and crack patterns than those which do not

consider the bond slip behaviour.

(3) In the 3D 2-shell model, the plane section restraints applied between the two

shells representing slab have only slight effect on the predicted

411

load-displacement curves. The crack patterns, however, are better predicted

when plan section restraints are applied. Decreasing the density of restraints

in the X direction has less effect on the predicted crack patterns than

decrease of restraint density in the Z direction.

(4) Both 3D-Shell-1 and 3D-Shell-2-R models work well as models for the

T-beam. As the 3D-Shell-1 model uses less computational time than the

3D-Shell-2-R mode, the former is the better substitute for 3D solid model in

modeling T-beam.

(5) A 20 mm element size, a time span of 50T0 and a damping coefficient β

equal to 2e-6, are suggested for the FE 3D-Shell-1 model of the T-beam.

(6) An element size of 20 mm is not small enough for the 3D-Shell-1 model to

obtain convergence when it used to model a T-beam weakened by web

openings. Thus, the 3D-Shell-1 model is not a good substitute for the 3D

solid model when modelling such a beam. A 3D shell model with muti-layer

shell slab might be a solution to the convergence problem, but it is not an

efficient method as it not only requires more computational time but gives

less accurate predictions than the 3D solid model.

6.7 REFERENCES



412

Chen, G.M., Teng, J.G., Chen, J.F. and Xiao, Q.G. (2015). “Finite element

modeling of debonding failures in FRP-strengthened RC beams: A dynamic

approach”, Computers and Structures, 158, 167–183.




413

(a) model with a one-shell-layer slab

(b) model with a two-shell-layer slab

Figure 6.1 Shell layer positions of 3D shell models

Figure 6.2 Boundary conditions of 3D 1-shell models

x

y

x

y

z

y

414

Figure 6.3 Predicted load-displacement curves of 3D FE 1-shell models versus

test result and that of the 3D solid model

(a) T-3D-Shell-1-SB

(b) T-3D-Shell-1

0 10 20 30 40 50 600

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

Test result T-3D-Solid T-3D-Shell-1-SB T-3D-Shell-1 T-3D-Shell-1-NBS

Element size=20 mm Loading duration=50T0 Damping coefficient β=2e-6

415

(c) T-3D-Shell-1-NBS

Figure 6.4 Predicted crack patterns of 3D 1-shell models

Figure 6.5 Predicted load-displacement curves of 3D 2-shell models versus test

result and that of 3D solid model

0 10 20 30 40 50 600

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

Test result T-3D-Solid T-3D-Shell-2 T-3D-Shell-2-NBS T-3D-Shell-2-R T-3D-Shell-2-NBS-R


416

(a) T-3D-Shell-2

(b) T-3D-Shell-2-NBS

(c) T-3D-Shell-2-R

(d) T-3D-Shell-2-NBS-R

Figure 6.6 Predicted crack patterns of 3D 2-shell models

417

(a) Different restraint density along X direction

(b) Different restraint density along Z direction

Figure 6.7 The effect of plane sectional restraint density between the two slab

shell layers on the predicted load-displacement curves

0 10 20 30 40 50 600

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

Test result T-3D-Shell-2-R T-3D-Shell-2-R-X100 T-3D-Shell-2-R-X200

0 10 20 30 40 50 600

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

Test result T-3D-Shell-2-R T-3D-Shell-2-R-Z100 T-3D-Shell-2-R-Z200



418

(a) T-3D-Shell-2-R-X100

(b) T-3D-Shell-2-R-X200

(c) T-3D-Shell-2-R-Z100

419

(d) T-3D-Shell-Dy-2-R-Z200

Figure 6.8 The effect of plane sectional restraint density between the two slab

shell layers on the predicted crack patterns

Figure 6.9 Comparison between predicted and test load-displacement curves.

0 10 20 30 40 50 600

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

Test result T-3D-Solid T-3D-Shell-1 T-3D-Shell-2-R


420

Figure 6.10 Comparison of predicted load-displacement curves between the 2D

and 3D-Shell-1 models

Figure 6.11 Effect of element size on the predicted load-displacement curves

0 5 10 15 20 25 30 35 400

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

CB-T-Test CB-T-2D CB-T-3D-Shell-1-BS

0 10 20 30 40 50 600

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

Test T-3D-Shell-1-ELE4050 T-3D-Shell-1-ELE20 T-3D-Shell-1-ELE10

Loading duration=50T0 Damping coefficient β=2e-6

421

Figure 6.12 Effect of loading time on the predicted T-3D-Shell-1 model

load-displacement curves

(a) Curves of kinetic energy versus displacement

0 10 20 30 40 50 600

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

Test T-3D-Shell-1-12.5T0

T-3D-Shell-1-25T0

T-3D-Shell-1-50T0

T-3D-Shell-1-100T0

T-3D-Shell-1-200T0

0 10 20 30 40 50 6010-8

10-6

10-4

10-2

100

102

104

106

Displacement (mm)

Kin

etic

Ene

rgy

(kN

. mm

)

3D-Shell-1-BS-12.5T 3D-Shell-1-BS-25T 3D-Shell-1-BS-50T 3D-Shell-1-BS-100T 3D-Shell-1-BS-200T

Element size=20 mm Damping coefficient β=2e-6

422

(b) Curves of kinetic energy to internal energy ratio versus displacement

Figure 6.13 Effect of loading time on the predicted kinetic energy

Figure 6.14 The effect of β value on the predicted load-displacement curves

0 10 20 30 40 50 6010-5

10-4

10-3

10-2

10-1

100

101

102

103

Displacement (mm)

Kin

etic

Ene

rgy/

Inte

rnal

Ene

rgy

(%)

3D-Shell-1-BS-12.5T 3D-Shell-1-BS-25T 3D-Shell-1-BS-50T 3D-Shell-1-BS-100T 3D-Shell-1-BS-200T

0 10 20 30 40 50 600

100

200

300

400

500

600

Displacement (mm)

Loa

d (k

N)

Test 3D-Shell-1-BS-No damping 3D-Shell-1-BS-Beta=1e-8 3D-Shell-1-BS-Beta=5e-8 3D-Shell-1-BS-Beta=1e-7 3D-Shell-1-BS-Beta=5e-7 3D-Shell-1-BS-Beta=1e-6 3D-Shell-1-BS-Beta=5e-6 3D-Shell-1-BS-Beta=1e-5 3D-Shell-1-BS-Beta=5e-5

a b c d

Element size=20 mm

Loading duration =50T0

1%

423

(a) Curves of kinetic energy versus displacement

(b) Curves of kinetic energy to internal energy ratio versus displacement

Figure 6.15 The effect of β value on predicted kinetic energy

0 10 20 30 40 50 6010-8

10-6

10-4

10-2

100

102

104

106

Displacement (mm)

Kin

etic

Ene

rgy

(kN

. mm

)

3D-Shell-1-BS-No damping 3D-Shell-1-BS-Beta=1e-8 3D-Shell-1-BS-Beta=5e-8 3D-Shell-1-BS-Beta=1e-7 3D-Shell-1-BS-Beta=5e-7 3D-Shell-1-BS-Beta=1e-6 3D-Shell-1-BS-Beta=5e-6 3D-Shell-1-BS-Beta=1e-5 3D-Shell-1-BS-Beta=5e-5

0 10 20 30 40 50 6010-6

10-4

10-2

100

102

104

Displacement (mm)

Kin

etic

Ene

rgy/

Inte

rnal

Ene

rgy

(%)

3D-Shell-1-BS-No damping 3D-Shell-1-BS-Beta=1e-8 3D-Shell-1-BS-Beta=5e-8 3D-Shell-1-BS-Beta=1e-7 3D-Shell-1-BS-Beta=5e-7 3D-Shell-1-BS-Beta=1e-6 3D-Shell-1-BS-Beta=5e-6 3D-Shell-1-BS-Beta=1e-5 3D-Shell-1-BS-Beta=5e-5

1%

424

Figure 6.16 The effect of β value on load-displacement curves and kinetic energy

curves

(a)Moment a

0 10 20 30 40 50 600

200

400

600

Loa

d (k

N)

Displacement (mm)

0 10 20 30 40 50 600

4000

8000

12000

Kin

etic

Ene

rgy

(kN

. mm

)

0 10 20 30 40 50 600

4000

8000

12000

0 10 20 30 40 50 600

4000

8000

12000

Test 3D-Shell-1-BS=1e-6 3D-Shell-1-BS=2e-6 3D-Shell-1-BS=4e-6


=1e-8

=2e-6

425

(b) Moment b

(c) Moment c

=1e-8

=2e-6

=2e-6

=1e-8

426

(d) Moment d

Figure 6.17 Predicted crack patterns with β =1e-8 and 2e-6 at critical moments

0 10 20 30 40 50 60 70 80

0

50

100

150

200

250

300

350

Displacement (mm)

Loa

d (k

N)

Test O-700-300-N-Shell-1-ELE4050 O-700-300-N-Shell-1-ELE20 O-700-300-N-Shell-1-ELE10

Loading duration=50T0

Damping coefficient β=2e-6

=1e-8

=2e-6

427

Figure 6.18 Convergence study on element size for modeling O-700-300-N

specimen

Figure 6.19 Predicted crack pattern of 3D shell-1 model for specimen

O-700-300-N

428

Table 6.1 Model names and details

Model name Concrete element

type

Layer number

Bond-slip between slab bars and

concrete Other issues

3D-Solid Solid / √

/

3D-Shell-1-SB

Shell

1 Smeared bars

3D-Shell-1 √ 3D-Shell-1-NBS ×

3D-Shell-2

2

√ 3D-Shell-2-NBS ×

3D-Shell-2-R √ The deformation of two slab layers are restrained 3D-Shell-2-NBS-R ×

Table 6.2 Model names and restraint frequency

Model name Plane sectional Restraint

distance along X direction (mm)

Plane sectional Restraint distance along Z direction (mm)

T-3D-Shell-2-R 20 20 T-3D-Shell-2-R-X100 100 20 T-3D-Shell-2-R-X200 200 20 T-3D-Shell-2-R-Z100 20 100 T-3D-Shell-2-R-Z200 20 200

Table 6.3 Total time cost by FE models

Model Total time T-3D-Shell-1 15:07:02

T-3D-Shell-2-R 51:53:57 T-3D-Solid 23:26:40

T-3D-Shell-2-R-X100 17:22:04 T-3D-Shell-2-R-X200 16:17:23 T-3D-Shell-2-R-Z100 30:28:47 T-3D-Shell-2-R-Z200 27:9:14

429

CHAPTER 7

EFFECTIVE SLAB WIDTH OF REINFORCED

CONCRETE FRAMES WITH SLAB UNDER

TENSION

7.1 INTRODUCTION

As previously indicated in Chapter 2, the longitudinal bars within the

effective slab width contribute greatly to the negative bending strength of a

T-section beam (T-beam). However, currently, a uniform method, to determine

the effective width of a slab under tension, especially for an RC structure with

cast-in-place slabs does not exist. Existing research in this area is

experimentally based (Ehsani and Wigh 1982, Durrani and Zerbe 1987, T

Pantazopotrlou et al. 1988, French 1991, Li 1994, Jiang et al. 1994, Wu et al.

2002, Wang et al. 2009, Zhen et al. 2009, Yang 2010, Sun et al. 2010, Qi et al.

2010, He 2010). Table 2.1 is a summary of the suggested values of effective

slab widths given by these researchers, based on their experimental results. It is

obvious that the effective slab width beff is mainly related to beam width bw,

beam height hb, effective span of beam l0, slab thickness t, and clear distance

between beams sn. Most researchers determine the effective slab width when the

storey drift ratio is equal to 1/50. Some researchers, however, have used

430

equations, based on their simulation data, to determine the effective slab widths.

Ning et al. (2016) studied the effective slab width of an RC frame both

experimentally and by means of FE simulation. Ning et al. (2016) conducted

parametric studies on the axial compression ratio, concrete strength,

reinforcement ratio of slabs, thickness of slabs and dimension of the transverse

beams. An equation involving only the main beam width and height to predict

the effective slab width with a 95% guaranteed accuracy was then proposed.

Many researchers have studied the effective slab width of a composite

structure under positive loading (under which the slab is under compression). A

few, such as Nie and Tao (2012), also studied the effective slab width of

composite structures under negative loading (under which the slab is under

compression). By conducting parametric studies through FE simulation on

column dimensions, steel beam height, RC slab width and thickness, transverse

beam, and yield stress of the longitudinal reinforcement, Nie and Tao (2012)

found that the main factors which influence the negative effective flange width,

included the column dimensions, the steel beam height, the flange width of the

transverse beam, and the yield stress of the longitudinal reinforcement. The slab

width was found to be less influential.

In Table 7.1, results of two papers (Ning et al., 2016 and Nie and Tao,

2012) giving the effective slab width under negative bending, based on FE

simulation results are compared. Ning et al. (2016) studied the effective slab

width of an RC frame and Nie and Tao (2012) studied the effective slab width

431

of a composite structure at ultimate state under both positive and negative

bending. Ning et al. (2016) did not study the effect of beam length on beff, which

is a parameter considered in all design codes except EC8. Nie and Tao (2012)

found the beam length was less influential. In addition, Ning et al. (2016) did

not study the effect of slab width on beff, a parameter also considered by all

design codes except EC8. The equations used by Nie and Tao (2012) and Ning

et al. (2016) for calculating the effective slab width under negative bending use

the stress in slab bars to calculate the effective width. There is a slight

difference in that Nie and Tao (2012), in their calculations, used yield stress

instead of maximum stress in slab bars, which was used in Ning et al. (2016)'s

equation.

By comparing the five different codes (GB 50010-2010, ACI318-05 &

FEMA-356, EC8, NZS-3101:2006) as indicated in Table 2.2, it is apparent that

the effective slab width of an interior joint is wider than that of an exterior joint

under the same conditions. Only Eurocode 8 takes column width into account

when determining the effective slab width. The application of this code is the

simplest. In contrast, the equation suggested in NZS-3101:2006 is the most

complicated of all. The equations suggested in the Chinese code GB

50010-2010, ACI318-05 and FEMA-356 are similar. The difference in these

equations in the codes for effective slab widths further strengthen the need to

clarify which are the key factors affecting effective slab width of a RC joint.

In this chapter, a parametric study on the effective slab width for RC

432

structures with slab under tension is conducted via FE modelling to find the key

factors affecting the effective slab width. Simple equations for slab effective

width at both exterior and interior joints are then proposed, based on the results

of the parametric study. It is to be noted that the modelling relates only to weak

beam and strong column joints which fail due to beam flexural failures.

7.2 PROPOSED SIMPLIFIED 3D FE MODELS

7.2.1 FE Model Details and Boundary Conditions

The 3D shell models, with only one shell slab layer, as proposed in chapter

6, were used for the parameter study. In the models, to guarantee all the slab

longitudinal bars are under tension when the beam enters the yielding stage, the

thickness of this layer must be small relative to the depth of the main beam.

Slabs under negative bending were the cases studied and described in this

chapter. The control model and the others simulated are listed in Table 7.2. As

shown, the main parameters studied are the main beam dimensions of beam

width, height and length, the beam reinforcement ratio, the slab width and

thickness, the slab longitudinal bar distance, the column width and transverse

beam dimensions and stirrup spacing. The effect of bond-slip between concrete

and bars, yield stress of bars, and the concrete strength on the predicted

effective slab width was also studied. The frame of the control model are shown

in Figure 7.1. The control models are shown in Figure 7.2. Only one half of a

specimen was modelled and a concentrated load was applied to simulate the

433

situation (the distribution of moment) of an RC structure under lateral force. A

frame with following details was chosen to reflect the commonly accepted

parameters in practice, as detailed next. The main beam span is 4000 mm and

the height and width are 400 mm and 200 mm respectively. The width of the

slab flange at one side is 3000 mm. The column width and height are both 400

mm. Concrete with cylinder strength of 40 MPa was assumed for slab, beam

and column. Deformed bars with a yield stress of 400 MPa and plain bars with a

yield stress of 300 MPa respectively constitute the main beam and slab

longitudinal bars. The stirrups in the main beam consist of 8 mm plain bars with

a yield stress of 300 MPa at a 100 mm spacing. The slab bars are reinforced

with two layers 8 mm plain bars at a 200 mm spacing. The height and width of

the transverse beams are 300 mm and 150 mm. Two deformed bars comprise

the upper and lower longitudinal bars. The stirrups are also 8 mm plain bars

with a yield stress of 300 MPa at a 100 mm spacing.

The boundary conditions of the control models are illustrated in Figure 7.2.

To save computational time, only half of a specimen was modelled. The exterior

edge of the slab and the transverse beam are symmetrically restrained in the X

direction as the joints belong to the RC multi-bay frame in Figure 7.1. The

column was modelled by restraining all the degrees of freedom of element

nodes within the column section, similar to Nie and Tao (2012), as the column

is assumed to be elastic and sufficiently strong to take the loading. For an

exterior joint, only one beam under negative loading was modelled. For an

434

interior joint, two beams, under respective positive and negative loading, were

modelled.

7.2.2 Ultimate State in Simulation

As suggested in the Chinese deign code (GB50010-2010), a 1/50 drift ratio

is acceptable as representing the ultimate state for most RC Frames. In this

chapter, drift ratio is defined by dividing applied displacement by the length of

beam, which is 4000 mm for control model and a drift ratio of 1/50 was

assumed as the ultimate state during the FE simulations. If the peak load is

reached, however, before that ultimate state assumption, that peak load is used

for the calculation of effective slab width.

7.2.3 Equations for Effective Slab Width

Nie and Tao (2012) and Ning et al. (2016) calculated the effective slab

width effb of the FE models, using the predicted stress of the slab bars at the

same section. However, when the flange is wide, the plane section assumption

may be violated. Thus, in this chapter, the effective slab width has been

back-calculated using the peak moment obtained for the critical section. With

this peak moment value, the effective slab width can be calculated based on the

plane section assumption. The equations used in Teng et al. (2002) were

adopted herein. The part accounting for FRP contribution was removed.

435

10 ( )cu sieff w bi

c fr i

f Ak bx b bd

(7.1)

11

( ) ( ) ( ) ( )2

nfcu si

peak 2 eff w bi si si siic fr i

hf Ah h hM k bx - k x b b - A - d2 d 2 2

(7.2)

In the above equation, the slab bars are treated as a layer of steel with a

thickness of si

fr i

Ad

mm. The concrete tension stress is not included as it's

negligible. The compressive zone height x and the effective slab width effb

can be obtained using Equations 7.1 and 7.2.

7.3 PARAMETRIC STUDIES

The exterior joint and interior joint models were used in the following

parametric study and the summarized results are shown in Tables 7.3 and 7.4.

The calculated effective flange width is based on Equations 7.1 and 7.2. It

should be noted that in the parametric studies presented next, when one

parameter is changing, the rest parameters (conditions) are kept the same as

those in the reference beam.

7.3.1 Effect of Stress-strain Models of Steel

Elastic-perfect plastic and nonlinear stress-strain curves related to the

hardening range shown in Figure 7.3 were used in two models to investigate the

steel stress-strain effect on the predicted effective slab width. As indicated in

436

Tables 7.3 and 7.4, the steel stress-strain model effect is is small. To simplify

the parametric study, the elastic-perfect plastic model was used in the remaining

analyses.

7.3.2 Effect of Bond Slip Behaviour

The bond slip effect between concrete and bars was investigated. The

bond-slip behaviour between steel bars and concrete surface is represented by

COH3D8 cohesive elements and CEB-FIP(1993) bond-slip model was used.

The proposed 3D prism cohesive model was used in simulation to connect steel

bars and concrete. The elements of steel bars for models E-BS-N and I-BS-N,

were directly connected to the concrete nodes. For models E-C and I-C,

bond-slip behaviour was assumed for longitudinal bars in the beam and slab and

stirrups in the beams. As shown in Tables 7.3 and 7.4, the bond slip effect on

the predicted effective slab width of interior joint is obvious. The inclusion of

the bond-slip behaviour also leads to more accurate predictions as demonstrated

in chapter 6. Hence bond slip effect was considered in the remaining analyses.

7.3.3 Effect of Yield Stress of Steel Bars

The strength effect of steel bars was investigated by changing yield stress

of longitudinal bars in both main beam and slab bar. Beam bars of 300-600 MPa

yield stress and slab bars of 200-400 MPa yield stress were assumed in the

simulations. As shown in Tables 7.3 and 7.4, for both interior and exterior joints,

437

increasing the bar yield stress of main beam longitudinal bars and slab bars

leads to less predicted effective slab width of both exterior and interior joints.

The effect of slab bar yield stress is more obvious than that of main beam

longitudinal bars. The effect of the yield stress of main beam longitudinal bars

is insignificant for exterior joints.

7.3.4 Effect of Concrete Strength

To investigate the concrete strength effect on predicted effective slab width,

concrete of a cylinder strength of 20, 30, 40, 50, 60 MPa was assumed. As

shown in Tables 7.3 and 7.4, an increase in concrete strength results in a greater

effective slab width, except for model E-CS-60, exterior joint with concrete

strength equal to 60MP. It is possible that increased concrete strength leads to

increased beam shear strength and thus the shear deformation along the

transverse direction is weakened. Thus the deformation of the critical section is

more uniform. The effect of concrete strength is more obvious for interior joints

than for exterior joints.

7.3.5 Effect of Beam Length, Width and Height

The effects of beam heights of 300 mm to 800 mm, beam widths of 100

mm to 300 mm, half beam lengths of 1000 mm to 3000 mm were all examined.

The study on beam length effect was only considered for exterior joints as the

results of exterior joints have indicated that the effect of the beam length is

438

insignificant, which is probably because the effective slab width is more

strongly affected by the critical section properties and boundary conditions. As

shown in Tables 7.3 and 7.4，the effect of beam height and width is more

significant for interior joints than for exterior joints. Increasing the beam height/

width leads to wider predicted effective slab width of interior joints.

7.3.6 Effect of Beam Reinforcement Ratio

In this study, the longitudinal reinforcement ratio of main beam is changed

through changing the diameter of the bars. The cohesive element area is also

modified accordingly. The area of the upper bars is always equal to that of the

lower bars. As shown in Tables 7.3 and 7.4, the beam longitudinal

reinforcement ratio of one side is from 0.5% to 1.5%. The effect of beam

longitudinal reinforcement is insignificant for both exterior and interior joints.

7.3.7 Effect of Column Width

Only column width has been taken into consideration in this study, due to

limitations of the proposed models. The column section was assumed to be

square. Column widths of 300 to 600 mm were included in the analyses. As

shown in Tables 7.3 and 7.4, increasing column width has almost no effect for

both exterior and interior joints. A possible reason is that, the proposed models

assume the column to be elastic and sufficiently strong

7.3.8 Effect of Slab Size.

439

The effects of flange widths of 1600 mm, 2000 mm, 3000 mm and

thickness of 80 mm, 100 mm 120 mm, 140 mm were studied and the effect of

flange thickness was found to be small. A possible reason is that the flange

thickness is relatively small compared to the beam height. The change of flange

thickness slightly changed the position of slab layer in the model. Increasing the

flange thickness slightly increases the effective slab width in exterior joints. As

shown in Tables 7.3, an increase in the flange width leads to a almost the same

predicted effective slab width of exterior joints, which indicates that the

effective slab width will keep constant if the flange width is wider than it. Thus

the effect of flange width was not included into the parametric study for interior

joints.

7.3.9 Effect of Slab Reinforcement Ratio

The slab reinforcement ratio effect was investigated by changing the slab

bar spacing as, in practice, changing slab bar spacing is more usual than

changing the diameter. As indicated in Tables 7.3 and 7.4, slab bar spacing has

a great effect on the predicted effective slab width. A decrease in spacing leads

to a decrease in effective slab width. This may be because increasing bar density

restricts the crack propagation in the transverse direction. Besides, the tensile

stress of slab concrete was not considered in the calculation when using

Equations 7.1 and 7.2.

7.3.10 Effect of Transverse Beam

440

To study the effect of a transverse beam on predicted effective slab width,

the transverse beam height, width and also stirrups related to beam torsional

stiffness, were investigated. As indicated in Tables 7.3 and 7.4, transverse beam

width only, has an obvious effect on predicted effective slab width at exterior

joints. Increasing the transverse beam width increases predicted effective slab

width at exterior joints. For interior joints, the effect is marginal.

7.4 DISCUSSIONS

According to the FE modelling results, the key factors influencing the

effective slab width at exterior joints includes concrete strength, slab bar yield

stress, spacing and transverse beam width. For interior joints, the factors of

concrete strength, beam height, width, longitudinal bar yield stress, slab bar

yield stress and spacing are of great effect. As the column is assumed to be

elastic, its effects are insignificant. The effect of slab thickness is very limited,

even though it is included in many proposed equations for effective slab width

(French 1991, Li 1994, Jiang et al. 1994, Wu et al. 2002, Wang et al. 2009,

Yang 2010, Qi et al. 2010, He 2010, GB 50010-2010, ACI318-05 & FEMA-356,

EC8, NZS-3101:2006).

The predicted effective slab width at interior joints is greater than that for

exterior joints. This is confirmed by equations proposed in the literature

(Durrani and Zerbe 1990, Zhen et al. 2009, Sun et al. 2010, Qi et al. 2010) for

exterior and interior joints. The reason relates to this fact is that an interior joint

441

slab under hogging moment is provided with additional restraint by the slab

subject to a sagging moment.

To generate a simple equation to calculate effective width, two equations

(7.3) and (7.4) were proposed for exterior joints and interior joints respectively.

Only the above key factors are included in the proposed equations. As shown in

Figure 7.4 and 7.5, approximately linear relationship exists between each of

these key factors and the effective flange width, with the only exception being

the flange width for exterior joints.

'

6000( 0.0011 0.783)

(0.0033 0.2755)(0.0013 0.1482)(0.002 0.1167)

eff w e sbys e cs e fbs e tbw

e sbys sy

e cs c

e fbs fr

e tbw tb

b bf

fd

w

(7.3)

where e sbys , e cs , e fbs , e tbw account for the effects of slab bar yield

stress syf , concrete strength 'cf , flange bar spacing frd and transverse beam

width tbw of exterior joints.

'

6000( 0.0021 1.4089)

(0.0087 0.4025)(0.0004 0.6245)(0.0013 0.5231)( 0.0003 0.8884)

(0.0025 0.2672)

eff w i sbys i cs i bh i bw i bbys i fbs

i sbys sy

i cs c

i bh b

i bw b

i bbys by

i fbs fb

b bf

fhw

fd

(7.4)

442

where i srs , i cs , i bh , i bw , i fw , i frp account for the effects of slab bar

strength syf , concrete strength 'cf , beam height bh , beam width bw , beam bar

yield stress byf and flange bar spacing fbd at interior joints. The coefficients

accounting for the effects of key factors were calculated independently and their

inter-effects were not considered. This is similar to Nie and Tao (2012), who

generated a equation for calculating effective slab with of composite frame

system by using three coefficients 0 , tr and r to represent the effects

resulting from different key factors. In the equation for 0 , the contribution

from column dimension and steel beam height to 0 , were also independently

considered. Coefficients tr and r , which respectively presents the effect of

transverse beam and bar yield stress, were also independently considered.

As the column is assumed to be elastic, to accurately predict the effective

slab width, a weakening coefficient i c and e c for interior and exterior

joints should be included. i c and e c equal to 0.5 are suggested for interior

and exterior joints respectively.


Many researchers (Ehsani and Wigh 1982, T Pantazopotrlou et al. 1988,

Durrani and Zerbe 1990, French 1991, Li 1994, Jiang et al. 1994, Wu et al.

2002, Wang et al. 2009, Zhen et al. 2009, Yang 2010, Qi et al. 2010, He 2010)

have recognised the need to identify the effective slab width of an RC frame and

443

numerous experimental studies have been conducted. Parametric studies, based

on the experimental approach, have been carried out but these studies have been

limited to study the effect of a few significant factors, although various different

equations have been proposed in the codes of different countries (GB

50010-2010, ACI318-05 & FEMA-356, EC8, NZS-3101:2006). Thus,

parametric studies covering a wide range factors have been conducted in this

chapter using the 3D 1-shell model proposed in Chapter 6.

Based on the FE modelling results and subsequent discussions presented

above, the following conclusions can be drawn:

(1) The key factors affecting the predicted effective slab width at both exterior

and interior joints include concrete strength, slab bar spacing, yield stress of

steel.

(2) The transverse beam width only has an obvious effect on the effective slab

width for an exterior joint. The effects of beam bar yield stress, beam height

and width are more obvious for an interior joint than for an exterior joint.

(3) The effective slab width for an interior joint is much wider than that for an

exterior joint with the same properties.

(4) Based on numerical results, two simple equations for predicting the effective

slab width were proposed for interior and exterior joints respectively. Only

key factors are included in the equations. A weakening coefficient

444

accounting for effect of column should be included to more accurately

predict the slab effective width.

7.6 REFERENCES




Durrani, A.J. and Zerbe, H.E. (1987). “Seismic resistance of R/C exterior

connections with floor slab”, Journal of Structural Engineering, ASCE,

113(8), 1850-1864.

Ehsani, M.R. and Wight, J.K. (1982). Behavior of External Reinforced Concrete

Beam to Column Connections Subjected to Earthquake Type Loading, Univ.

of Michigan, Ann Arbor, Mich.



CEN, Brussels.

French, C.W. (1991). Effect of Floor Slab on Behavior of Slab-Beam-Column

Connections, Design of Beam-Column Joints for Seismic Resistance.



445

He, J. (2010). Study on Influence of Cast-in-situ Slab on RC Frame Structure

Achieving “Strong Column Weak Beam”, Master degree thesis: Huazhong

University of Science and Technology, China. (in Chinese)

Jiang, Y.S., Chen, Z.F., Zhou, X.P. et al. (1994). “Seismic studies of frame

joints with cast-in-place slab”, Journal of Building Structures, 15(6), 11-16.

(in Chinese)

Nie, J. G., and Tao, M. X. (2012),"Slab spatial composite effect in composite

frame systems. I: Effective width for ultimate loading capacity",

Engineering structures, 38, 171-184.

Ning, N., Qu, W., and Ma, Z. J. (2016), "Design recommendations for achieving

“strong column-weak beam” in RC frames", Engineering Structures, 126,

343-352.



Qi, C.H. et al. (2010). “Effect of floor slabs on the mechanical properties of

reinforced concrete frame structures”, Journal of Catastrophology, 25(S0),

105-110. (in Chinese)

Sun, Y. (2010). The effect of Slabs on Strong Column-Weak Beam Mechanism

of RC Frame Structures, Master degree thesis: Harbin Institute of

446

Technology, China. (in Chinese)

Teng, J. G., Chen, J. F., Smith, S. T., and Lam, L. (2002), FRP: strengthened

RC structures. Frontiers in Physics, 266.

Wang, S.G., Hang, X.L. and Ji, J. (2009). “The effect of slabs on the failure

mode of reinforced concrete frame structures”, Journal of Civil,

Architectural & Environmental Engineering, 31(1), 66-71.

Wu, Y., Lei, J.C., Y, H. et al. (2002). “Discussion on the problem that slab

reinforcement getting involved in the negative moment capacity of beam

end”, Journal of Chongqing Jianzhu University, 24(3), 33-37. (in Chinese)

Yang, Z.L. (2010). Research on “Strong Column Weak Beam” Yield

Mechanism Factors of Reinforced Concrete Frame Structure, Master

degree thesis: Qingdao Technological University, China. (in Chinese)

Zhen, S.J., Jiang, L.X., Zhang, W.P. and Gu, X.L. (2009). “Experimental

research and analysis of effective flange width of beam end section in

cast-in-site concrete frames”, Structural Engineers, 25(2), 134-140. (in

Chinese)

447

Figure 7.1 Frame and joints of the FE models

(a) Exterior joint

Symmetric section

Load

Symmetric section

Constraints of all

DOFs

448

(b) Interior joint

Figure 7.2 Control FE models of exterior and interior joints

Figure 7.3 Two stress-strain models for steel bars

Load

Load

Load

Symmetric section

Constraints of all

DOFs

Symmetric section

449

(a) Effect of slab bar yield stress

(b) Effect of concrete strength

y = -0.0011x + 0.783

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

0 100 200 300 400 500

Eff

ectiv

e sl

ab w

idth

/ Pra

tical

wid

th

Yield strength (MPa)

Slab rebar yield stress

线性 (Slab rebar yield stress)

y = 0.0033x + 0.2755

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

35.0%

40.0%

45.0%

50.0%

0 10 20 30 40 50 60 70

Eff

ectiv

e sl

ab w

idth

/ Pra

tical

wid

th

Concrete strength (MPa)

Concrete strength

线性 (Concrete strength)

450

(d) Effect of flange bar spacing

(e) Effect of transverse beam width

Figure 7.4 Key factors effecting the predicted slab width of exterior joints

y = 0.0013x + 0.1482

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

0 50 100 150 200 250 300 350

Eff

ectiv

e sl

ab w

idth

/ Pra

tical

wid

th

Flange rebar distance (mm)

Flange rebar distance

线性 (Flange rebar distance)

y = 0.002x + 0.1167

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

0 50 100 150 200 250

Eff

ectiv

e sl

ab w

idth

/ Pra

tical

wid

th

Transverse beam width (mm)

Transverse beam width

线性 (Transverse beam width)

451

(a) Effect of beam bar yield stress

(b) Effect of slab bar yield stress

y = -0.0003x + 0.8884

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

90.0%

0 100 200 300 400 500 600 700

Eff

ectiv

e fla

nge

wid

th/ P

ratic

al w

idth


Beam bar yield stress

线性 (Beam bar yield stress)

y = -0.0021x + 1.4089

0.0%

20.0%

40.0%

60.0%

80.0%

100.0%

120.0%

0 100 200 300 400 500

Eff

ectiv

e fla

nge

wid

th/ P

ratic

al w

idth


Slab rebar yield stress

线性 (Slab rebar yield stress)

452

(c) Effect of concrete strength

(d) Effect of beam height

y = 0.0087x + 0.4025

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

90.0%

100.0%

0 10 20 30 40 50 60 70

Eff

ectiv

e fla

nge

wid

th/ P

ratic

al w

idth

Concrete strength (MPa)

Concrete stength

线性 (Concrete stength)

y = 0.0004x + 0.6245

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

90.0%

100.0%

0 100 200 300 400 500 600 700 800 900

Eff

ectiv

e fla

nge

wid

th/ P

ratic

al w

idth

Beam height (mm)

Beam height

线性 (Beam height)

453

(e) Effect of beam width

(f) Effect of flange bar spacing

Figure 7.5 Key factors affecting the predicted slab width of interior joints

y = 0.0013x + 0.5231

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

90.0%

100.0%

0 50 100 150 200 250 300 350

Eff

ectiv

e fla

nge

wid

th/ P

ratic

al w

idth

Beam width (mm)

Beam width

线性 (Beam width)

y = 0.0025x + 0.2762

0.0%

20.0%

40.0%

60.0%

80.0%

100.0%

120.0%

0 50 100 150 200 250 300 350

Eff

ectiv

e fla

nge

wid

th/ P

ratic

al w

idth

Flange rebar spacing (mm)

Flange rebar spacing

线性 (Flange rebar spacing)

454

Table 7.1 Some existing parametric studies on the effective slab width

Time Researchers software Parameters Model Equation used to

calculate effb Proposed effb

2016 Ning et al. Abaqus Axial compression ratio, concrete strength,

reinforcement ratio of slabs, thickness of the slabs and the stiffness of the transverse beams

3D solid element models 1

max

n

bi bi

eff wb

sb b

at 2% drift ratio

6.4eff w bb b h

2012 Nie and Tao MSC.Marc

(2005)

Column dimension, steel beam height, RC slab width and thickness, transverse beam flange

width, and yielding strength of the longitudinal reinforcement

3D nonlinear shell-solid elaborate finite element model

1

sy

n

bi bi

eff w

sb b

f

at 2% drift ratio

eff w fb b b

Note: beff: effective slab width; bw: beam width; hb: beam height; bi : the i slab bar stress; maxb : max stress of slab bars; bs : distance

between slab bars; syf : yield stress of slab bars; : coefficient; fb : the flange width.

455

Table 7.2 Parameters investigated in studying slab effective width Parameters Values/considerations

Stress-strain curves Elastic-perfect plastic or nonlinear

stress-strain curve considering hardening range

Bond-slip effect With or without including the

bond-slip relationship Yielding stress of bars fy (MPa) 200 300 400 500 600

Concrete strength fc (MPa) 20 30 40 50 60 Column width (mm) 300 400 500 600

Beam

Length (mm) 2000 3000 4000 5000 6000 Width (mm) 100 150 200 250,300 Height (mm) 300 400 500 600 700 800 Longitudinal

reinforcement ratio (%) 0.5 0.75 1 1.25 1.5 ( through changing

area)

Flange

Width (mm) 2000 2500 3000 Thickness (mm) 80 100 120 140

Longitudinal reinforcement distance

(mm) 100 160 200 240 300

Transverse beam

Height (mm) 200 300 400

Width (mm) 100 150 200

Stirrup spacing (mm) 60 100 200

456

Table 7.3 Predicted effective flange width of exterior joints

Model name Parameter Parameter values Sectional strength (Kn·m)

Calculated flange

width of one side

(mm)

Calculated flange width of one side/flange

real width

E-C Stress-strain

curves

Elastic-perfect plastic model

226.5 1283 42.8%

E SSC-N Nonlinear stress-strain

model considering hardening range

233.4 1362 45.4%

E-BS-N

Bond-slip

Tied to concrete node 230.1 1324 44.1%

E-C Bonded to concrete node

through cohesive element

226.5 1283 42.8%

E-BBYS-300 Beam bar yielding

stress

300 201.9 1294 43.1% E-C 400 226.5 1283 42.8%

E-BBYS-500 500 252.0 1282 42.7% E-BBYS-600 600 272.6 1225 40.8% E-SBYS-200 Slab bars

yielding stress

200 212.7 1692 56.4% E-SBYS-300 300 226.5 1283 42.8% E-SBYS-400 400 232.3 1012 33.7%

E-CS-20 Concrete strength (MPa)

20 190.7 953 31.8% E-CS-30 30 212.0 1153 38.4%

E-C 40 226.5 1283 42.8% E-CS-50 50 238.0 1385 46.2% E-CS-60 60 234.6 1330 44.3%

E-BH-300

Beam height (mm)

300 157.6 1313 43.8% E-C 400 226.5 1283 42.8%

E-BH-500 500 298.1 1291 43.0% E-BH-600 600 372.4 1314 43.8% E-BH-700 700 444.2 1315 43.8% E-BH-800 800 517.4 1322 44.1%

457

Table 7.3 ( Cont.)

E-BL-1000

Beam length (mm)

1000 224.4 1259 42.0% E-BL-1500 1500 224.2 1258 41.9%

E-C 2000 226.5 1283 42.8% E-BL-2500 2500 225.7 1274 42.5% E-BL-3000 3000 226.9 1287 42.9% E-BW-100

Beam width(mm)

100 202.9 1113 37.1% E-BW-150 150 216.4 1206 40.2%

E-C 200 226.5 1283 42.8% E-BW-250 250 230.3 1300 43.3% E-BW-300 300 232.1 1303 43.4%

E-BTRR-0.5 Beam tension reinforcement

ratio (%)

0.5 171.5 1240 41.3% E-BTRR-0.75 0.75 198.0 1250 41.7%

E-C 1 226.5 1283 42.8% E-BTRR-1.25 1.25 249.5 1253 41.8% E-BTRR-1.5 1.5 278.0 1286 42.9% E-CW-300

Column width (mm)

300 223.5 1249 41.6% E-C 400 226.5 1283 42.8%

E-CW-500 500 231.0 1334 44.5% E-CW-600 600 231.5 1340 44.7% E-FW-1500

Flange width (mm)

1600 222.7 1240 77.50% E-FW-2000 2000 227.3 1292 64.60%

E-C 3000 226.5 1283 42.8% E-FRD-100

Flange bar distance(mm)

100 259.7 836 27.9% E-FRD-160 160 234.1 1096 36.5%

E-C 200 226.5 1283 42.8% E-FRD-240 240 216.5 1404 46.8% E-FRD-300 300 210.3 1652 55.1%

E-FH-80 Flange height

(mm)

80 226.4 1240 41.3% E-C 100 226.5 1283 42.8%

E-FH-120 120 225.1 1312 43.7% E-FH-140 140 225.1 1361 45.4%

E-TBH-200 Transverse beam height

(mm)

200 217.9 1186 39.5% E-C 300 226.5 1283 42.8%

E-TBH-400 400 236.6 1399 46.6% E-TBW-100 Transverse 100 196.1 946 31.5%

458

E-C beam width (mm)

150 226.5 1283 42.8% E-TBW-200 200 249.8 1553 51.8%

Table 7.4 Predicted effective flange width of interior joints

Model

name Parameter Parameter values

Sectional

strength

(Kn*m)

Calculated

flange

width of

one side

(mm)

Calculated

flange width

of one

side/flange

real width

I-C

Stress-strain

curves

Elastic-perfect plastic

model 307.0 2274 75.8%

I SSC-N

Nonlinear stress-strain

model considering

hardening range

311.2 2331 77.7%

I-BS-N

Bond-slip

Tied to concrete node 348.3 2860 95.3%

I-C

Bonded to concrete node

through cohesive

element

307.0 2274 75.8%

I-BBYS-300

Beam bar

yielding stress

300 287.9 2362 78.7%

I-C 400 307.0 2274 75.8%

I-BBYS-500 500 325.3 2177 72.6%

I-BBYS-600 600 342.3 2064 68.8%

I-SBYS-200 Slab bars

yielding stress

200 303.8 3347 100.0%

I-SBYS-300 300 307.0 2274 75.8%

I-SBYS-400 400 310.5 1741 58.0%

I-CS-20

Concrete

strength

(MPa)

20 241.7 1683 56.1%

I-CS-30 30 280.4 2047 68.2%

I-C 40 307.0 2274 75.8%

I-CS-50 50 333.8 2525 84.2%

I-CS-60 60 358.2 2756 91.9%

I-BH-300

Beam height

(mm)

300 203.5 2208 73.6%

I-C 400 307.0 2274 75.8%

I-BH-500 500 430.8 2500 83.3%

I-BH-600 600 555.5 2632 87.7%

I-BH-700 700 673.2 2672 89.1%

I-BH-800 800 799.9 2747 91.6%

459

Table 7.4（cont.）

I-BW-100

Beam width(mm)

100 254.0 1908 63.6% I-BW-150 150 293.9 2253 75.1%

I-C 200 307.0 2274 75.8% I-BW-250 250 337.5 2570 85.7% I-BW-300 300 355.6 2719 90.6%

I-BTRR-0.5 Beam tension reinforcement

ratio (%)

0.5 260.9 2343 78.1% I-BTRR-0.75 0.75 284.9 2321 77.4%

I-C 1 307.0 2274 75.8% I-BTRR-1.25 1.25 329.3 2230 74.3% I-BTRR-1.5 1.5 359.3 2289 76.3% I-CW-300

Column width (mm)

300 309.5 2301 76.7% I-C 400 307.0 2274 75.8%

I-CW-500 500 308.1 2289 76.3% I-CW-600 600 313.2 2358 78.6% I-FRD-100

Flange bar distance(mm)

100 359.1 1513 50.4% I-FRD-160 160 331.9 2095 69.8%

I-C 200 307.0 2274 75.8% I-FRD-240 240 304.1 2682 89.4% I-FRD-300 300 292.4 3123 100.0%

I-FH-80 Flange height

(mm)

80 303.0 2135 71.2% I-C 100 307.0 2274 75.8%

I-FH-120 120 308.0 2287 76.2% I-FH-140 140 308.2 2290 76.3%

I-TBH-200 Transverse beam height

(mm)

200 324.2 2510 83.7% I-C 300 307.0 2274 75.8%

I-TBH-400 400 310.8 2325 77.5% I-TBW-100 Transverse

beam width (mm)

100 319.8 2449 81.6% I-C 150 307.0 2274 75.8%

I-TBW-200 200 322.8 2490 83.0%

460

461

CHAPTER 8

THREE-DIMENSIONAL FINITE ELEMENT

MODELLING OF RETROFITTED RC

BEAM-COLUMN-SLAB JOINTS

8.1 INTRODUCTION

An experimental study of RC beam-column joints, retrofitted by the

proposed techniques has been presented in Chapters 3 and 4. The 3D FE

analysis of T-beams with a web opening has been presented in Chapter 5. The

FE modelling of RC beam-column joints has previously been conducted by a

few researchers. For example, Li et al.(2009) used DIANA software to model

test specimens and carried out parametric studies. The FE modelling of RC

beam-column joints weakened by openings however, has not been previously

studied. The effect of slits on RC joints and frames has been studied through FE

modelling by Zhang et al. (2011), Wang el al. (2012) and Zhang (2013). The

investigations were based, simply, on FE modelling not substantiated by

experimental data. It is noted that RC Joints with transverse grooves (TGs) had

not as yet, been examined either experimentally or by FE modelling. Thus, to

fill the gap in the existing knowledge on FE analysis of RC beam-column joints

with weakened beams, 3D FE modelling of RC beam-column joints retrofitted

462

by the proposed techniques has been investigated by this study and will be

presented in this chapter.

8.2 THE PROPOSED 3D FE MODELS

The proposed FE models for retrofitted RC beam-column joints are based

on the T-beam models using solid elements described in Chapter 5. Similar to

the 3D models in Chapter 5, the element size chosen for the RC joint models

was 20 mm. The concrete was modelled by the 3D solid elements C3D8R in

ABAQUS (2012) and the constitutive equation proposed by Saenz (1964) for

the state of uni-axial compression was adopted. p and Ec were determined

based on test results. If test results were not available, they were set equal to

0.002 and 4730 'cf respectively as in Chapter 5. In terms of tension-softening

curves, the constitutive equation curve proposed by Hordijk (1991) was adopted.

For tension damage, the power low model used in Nie (2018), presented in

Equation 8.1 was adopted.

1 (1 )ntt

cr

wdw

(8.1)

where n is equal to 5 following Chen et al. (2012); is crack opening

displacement; is crack opening displacement at the complete release of

concrete stress.

According to the FE analysis of T-beams with weakening openings and

463

FRP strengthening described in Chapter 5, the FRP confinement effect should

not be ignored. Thus, Yu el al.'s (2009a, b) modified Plastic-damage model was

adopted for the concrete of the web chord, which was strengthened by FRP

wrapping.

As concrete in the joint region concrete was reinforced by the steel stirrups,

the stress-strain model for confined concrete proposed by Mander et al. (1988),

as expressed in Equation (8.2), was employed for the concrete in the joint

region.

'

ccc r

f xrfr -1+ x

(8.2)

where 'ccf is compressive strength of confined concrete; cc is the compressive

strain of concrete at ultimate state; c

cc

x

in which c is compressive

concrete strain. The cc and r can be obtained using the following

equations:

'

'1 5 1cccc co

co

ff

(8.3)

c

c sec

ErE E

(8.4)

in which and are unconfined concrete strength and corresponding 'cof co

464

strain, respectively; is equal to 0.002 if test data is not available, cE is

equal to '4730 cof and secE is equal to '

cc

cc

f

.

The steel bars were modelled with the B31 beam elements and an

elastic-perfect plastic model was applied to represent their properties if a yield

platform was available, as in Chapter 5. Otherwise, the model proposed by

Ramberg Osgood (1943) was applied. The bond-slip behaviour between steel

bar and concrete was represented by COH3D8 cohesive elements and the

CEB-FIP(1993) bond-slip model was used. The 3D prism cohesive model

proposed in Chapter 5 was used to simulate bond behaviour between steel bars

and concrete.

FRP used to wrap the concrete chord was modelled with the S4R 4-node

shell elements and treated as a linear elastic brittle material with elastic modulus

of 227.380 GPa based on test material properties. Only in the fibre direction was

defined with stiffness. The Poisson's ratio of the FRP was set as 0.001. The FRP

reinforcement fails when it reaches its tensile rupture strength of 2820 MPa,

obtained from coupons test. The simplified bond-slip model Lu et al. (2005)

was adopted to define the bond behavior of FRP and concrete.

The dynamic explicit approach was used to avoid convergence problems,

as for the FE models discussed in Chapters 5 and 6. When building the 3D

model, the densities of materials were set equal to the actual values: 2.5e-9

co

465

ton/mm3 for concrete and cohesive elements, 7.25e-9 ton/mm3 for steel

elements and 1.75e-9 ton/mm3 for FRP elements. The stiffness-proportional

damping coefficient β was set equal to 0.000002 based on the parametric study

shown in Chapter 5.

The boundary conditions are shown in Figure 8.1. To save computational

time, only half of a specimen was modelled. Nodes on the section of symmetry

were permitted no displacement in the X direction. Coupling constraints were

applied to the five reference points (referred to as RPs below).

As in experimental test, axial load was applied first and then lateral load ,

in FE modelling, Loads were applied in two steps. In the first step, an axial load

equal to 485 kN was applied to the column top through RP 5. In this step, for

the first test group, the linear displacements of RPs 1-3 were restrained in the X ,

Y and Z directions. For the second test group, Y displacements of the RPs 1 and

3 were not restrained as, during testing, the beam tip loads of the second test

group were released to zero after applying the axial load. In the second test

group, a horizontal load was applied through RP4, until the specimens failed.

During this modelling procedure, the RPs1-3 restraints were kept unchanged for

the first test group. For the second test group, restraints on the vertical

displacements of RPs 1 and 3 were provided. The loading time of the first step

was 0.1s and 50T0 (T0 is the natural period of a 3D model) for the second step.

8.3 PREDICTED RESULTS

466

8.3.1 Load-displacement Curves

The predicted curves of lateral loads, beam tip loads, and initial stiffness

values versus lateral drift ratios or displacements are shown in Figures 8.2 to 8.4.

In these figures, the drift ratio values are defined by dividing lateral

displacement by storey height (2400 mm). The peak lateral loads are given in

Table 8.1. The envelope curves are used for comparison herein, as the

specimens were tested under cyclic loading. In Figure 8.2, the 3D models

predict satisfactory results for all specimens except for the two with TGs

(F-G-50-200 and S-G-50-200-100). For the specimens F-G-50-200 and

S-G-50-200-100, the FE model predicts slightly higher post-cracking stiffness

and peak lateral load. The descending portion after peak load is not well

captured either. This can be attributed to the severe slippage of the steel bars in

beam bottom due to the existence of transverse grooves (placed close to or 100

mm away from the beam-column interface) subjected to cyclic loading. As the

3D model was subjected to a monotonic load, the cyclic loading effect was not

simulated. The FE model predicts a slightly lower peak lateral load for

specimen F-S-450-450, but the descending portion is well captured. For

specimens F-S-450-450, S-O-500-180 and S-O-500-180-S-300-300, the FE

models predict slightly lower peak lateral loads.

In Figure 8.3, curves of the left and right beam tip loads versus lateral drift

ratio are presented. The left and right beams are under respective negative and

positive loading. It is seen that the positive beam tip load have all been well

467

predicted, whereas, the FE model predicts negative loads higher than test values

for F-G-50-200, S-Control and S-G-50-200. The beam tip loads after peak

points are generally overestimated by the 3D FE models, which might because

that the FE models do not consider the cumulative damage arising from cyclic

loading. The residual deformations of beams, columns and joints are thus

underestimated at a given load. This phenomenon is more obvious for those

specimens did not fail by beam flexural failure.

In Figure 8.4, the first cycles of test load-displacement curves are shown

for to substantiate the proposed models. The FE model predicts a slightly higher

initial stiffness than the test value. The specimen S-450-450 showed a 2mm

initial displacement, which is probably due to ‘mechanical slack’, tiny gaps

existing between testing equipment components.

8.3.2 Crack Patterns

The comparisons between numerical and test crack patterns and test cracks

are shown in Figures 8.5 and 8.6. Test observed crack propagation at the

ultimate state when specimens were subject to push action, are highlighted in

black in Figure 8.5 as a reference to verify the accuracy of the FE models in

predicting crack patterns. As shown in Figure 8.6(a), the cracks in the slab of

the specimen F-Control are well captured. Main cracks in the joints region, are

also correctly predicted. For specimens F-G-50-200 and S-G-50-200-100, the

predicted cracks caused by beam bar bars slipping in the joints region closely

468

match with the test results, even though those test cracks were more severe as

cyclic loading was applied. For specimen F-S-450-450, the slab and beam

flexural cracks are also precisely predicted. For specimens F-O-450-150 and

F-O-500-180, the slab cracks caused by the four hinges mechanism (i.e. local

flexural rotation at the two ends of web and flange chords) are also well

captured. Specimen S-Control failed due to hinge formation in bottom column,

which is also predicted by FE model as shown in Figure 8.6(f). The cracks on

the top column, beams and slab are also well predicted. For specimen

S-O-500-180, the cracks in the beam web above the opening are well predicted.

The cracks arising from the anchor within the right hand side slab are also

predicted. For specimen S-O-500-180-S-300-300, the cracks originated from the

ends of the longitudinal slits within the right hand side slab are well captured

with, the same can be said for those cracks caused by four hinges mechanism.

8.4 DISCUSSIONS

8.4.1 Confinement Effect of Joint Steel Stirrups

The comparison between the models with and without a consideration of

the joint stirrups confinement effect is shown in Figure 8.7. As indicated in the

Figure 8.7, the differences between the two models for the first test group are

smaller than for the second, as heavier joint stirrups were provided for the

second test group. The difference became more obvious with increasing

displacement, this is because the joint deformations became more obvious in the

469

later range. Stirrups confinement had more obvious effect on the specimens

F-Control, S-Control as the joint deformations were more severe in these

specimens. In general, it can be seen that the inclusion of the confinement effect

for joint concrete results in more accurate predicted peak loads, except for the

two specimens with TGs.

8.4.2 Effect of Opening Shape

A slot-shaped opening has been tested on Specimens S-O-500-180 and

S-O-500-180-S-300-300 and this opening shape led to the four hinges type

deformation in its weakening T-beam as the rectangular one did on the beams of

Specimen F-O-500-180. To get a in-depth comparison between the two types of

openings, a 3D FE model S-O-500-180-R with rectangular-shaped beam

openings was built and studied. The FE mode S-O-500-180-R has the same

properties as the FE model S-O-800-180 except for the opening shape. The

predicted load-displacement curves are presented in Figure 8.8. It is obvious

that the two types of beam opening predicted almost the same

load-displacement curves. The predicted crack patterns by FE model

S-O-500-180-R is shown in Figure 8.9. The comparison between Figures 8.9

and 8.6(h) indicates that the two 3D models also predicted very close crack

patterns, except for some insignificant cracks (e.g. the cracks on the beam web

above the slot-shaped opening). Thus slot-shaped opening is a better option for

the BO technique as it removes less beam web concrete as the rectangular one

470

does.

8.4.3 Effect of Longitudinal Slit Length

When applying the SS technique to retrofit RC beam-column joints

F-S-450-450 and S-O-500-180-S-300-300, both longitudinal and transverse slits

were created. The effect of the transverse slits is noticeable as the longitudinal

slab bars crossing the slits were cut. The effects of longitudinal slits on the test

specimens were not clear. Thus two additional 3D FE models F-S-450-200 and

F-S-450-0 were built to study the effect of longitudinal slit length. Similar to FE

model F-S-450-450, both the models F-S-450-200 and F-S-450-0 had 450 mm

long transverse slits, while the model F-450-200 had 200 mm long longitudinal

slits and model F-S-450-0 had no longitudinal slit. The predicted

load-displacement curves of the FE models with longitudinal slit of different

size were presented in Figure 8.10. It is obvious that the length of longitudinal

slits has marginal effect on the predicted load-displacement curves. The model

with shorter longitudinal slits predicted a slight higher peak load. The three FE

models with longitudinal slits different in length also predicted very close crack

patterns as shown in Figure 8.10(a), (b) and Figure 8.6(e). The reason

accounting for above observation might be that the cut transverse slab bars

crossing the longitudinal slits had little effect on the moment capacity of T-beam

along the longitudinal direction. Both transverse and longitudinal slits are

created when applying the SS technique as in a real situation, a seismic load

may come from any direction.

471

8.4.4 Prediction Errors

As shown in Table 8.1, it is clear that the predicted lateral loads are more

accurate than the predicted beam tip loads. This is because the predicted peak

lateral loads happened at a 2% drift ratio, but the predicted peak beam tip loads

usually occurred at a later drift ratio as demonstrated in Figures 8.3 and 8.8.

This might be because the cumulative damage due to the cyclic loading is not

simulated in the FE analysis, which can also be the reason why the FE model’s

predictions on the specimens with TGs are not as good as those without TGs.

Also in these cases, the mechanical slack in the testing equipment may play a

role in the difference. The predicted positive beam tip loads are closer to test

results than the negative loads. This may be because some specimens failing by

joint shear failure, column hinge formation or beam bottom bar slippage, so that

these beams did not reach their negative strengths during testing. Besides, the

retrofitted techniques have more effect on beam’s negative strength, as observed

and discussed in Chapter 4. Cyclic loading might further increase this effect. As

the FE models do not simulate the cumulative damage due to cyclic loading,

higher negative loads are predicted, as clearly demonstrated in Figure 8.3.


3D FE models for RC joints retrofitted by the proposed techniques have

proposed and assessed in this Chapter. The following conclusions can be drawn

based on the predicted results and discussions:

472

(1) The proposed 3D FE models perform well in predicting load-displacement

curves and crack patterns of retrofitted RC joints. The predictions for

specimens with transverse grooves are not as good as the others because the

beam bottom bars slippage was more noticeable under cyclic loading.

(2) The predicted initial stiffness is slightly greater than in the test value, which

can be attributed to the mechanical slack of test equipment arising from the

small gap between the mechanical components and the corresponding test

equipment.

(3) To obtain more reliable predictions, the joint stirrup confinement effect

should be considered in the 3D FE analysis. The model proposed by Mander

et al. (1988) can be applied. The consideration of the joint stirrup

confinement effect has more effect in the specimens F-Control and

S-Control, which failed by joint shear failure and bottom column hinge

formation, respectively.

(4) The ignorance of cyclic loading effect leads to a slightly higher predicted

negative beam tip loads. The main reason is that cumulative damage caused

by cyclic loading during testing was not simulated in FE analysis.

8.6 REFERENCES



473

CEB-FIP. (1993). CEB-FIP Model Code 90, Thomas Telford, London.

Chen, G.M., Chen, J.F. and Teng, J.G. (2012). “On the finite element modelling

of RC beams shear-strengthened with FRP”, Construction and Building

Materials, 32, 13-26.

Hordijk, D.A. (1991). Local Approach to Fatigue of Concrete, PhD thesis, Delft

University of Technology.



Li, B., Tran, C. T. N., and Pan, T. C. (2009)." Experimental and numerical

investigations on the seismic behavior of lightly reinforced concrete

beam-column joints", Journal of structural engineering, 135(9), 1007-1018.

Mander, J. B., Priestley, M. J., and Park, R. (1988). "Theoretical stress-strain

model for confined concrete", Journal of structural engineering, 114(8),

1804-1826.






474

Ramberg, W., and Osgood, W. R. (1943)," Description of stress-strain curves by

three parameters".

Rots, J. G. (1988). Computational Modeling of Concrete Fracture, Ph.D. thesis,

Delft University of Technology.

Saenz, L. P. (1964). “Discussion of ‘Equation for the stress-strain curve of

concrete’ by P. Desayi and S. Krishan”, ACI Journal, 61(9), 1229-1235.




Vibration, 32(1), 121-127. (in Chinese)

Yu, T. T. J. G., Teng, J. G., Wong, Y. L., and Dong, S. L. (2010a), "Finite


model", Engineering Structures, 32(3), 665-679.

Yu, T., Teng, J. G., Wong, Y. L., and Dong, S. L. (2010b). "Finite element

modeling of confined concrete-II: Plastic-damage model", Engineering

structures, 32(3), 680-691.



Thesis: Hunan University, China. (in Chinese)

475

Zhang, Y.P., Hao, Z.J., Shan, M.Y. and Ge, N. (2011). “Research on

anti-seismic performance for reinforced concrete frame joint with slot

around”, Building Science, 27(9), 7-11. (in Chinese)

476

Figure 8.1 3D FE Model Control specimen

(a) F-Control

0 1 2 3 4 5 60

20

40

60

80

100

120

140

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


F-Control-Test F-Control-FE

Coupling restraint Coupling restraint

Horizontal Load

Axial Load

RP5

RP4

RP1 RP2 RP3

Coupling restraint

477

(b) F-G-50-200

(c) F-S-450-450

0 1 2 30

20

40

60

80

100

120

Lateral Drift(%)

Lat

eral

Loa

d (k

N)

0 24 48 72


F-G-50-200-Test F-G-50-200-FE

0 1 2 3 4 5 60

20

40

60

80

100

120

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


F-S-450-450-Test F-S-450-450-FE

478

(d) F-O-450-150

(e) F-O-500-180

0 1 2 3 4 5 60

20

40

60

80

100

120

140

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


F-O-450-150 Test F-O-450-150-FE

0 1 2 3 4 5 60

20

40

60

80

100

120

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


F-O-500-180 Test F-O-500-180-FE

479

(f) S-Control

(g) S-G-50-200-100

0 1 2 3 4 50

20

40

60

80

100

120

140

160

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120


S-Control-Test S-Control-FE

0 1 2 3 40

20

40

60

80

100

120

140

Lateral Drift(%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 Lateral Displacement (mm)

S-G-50-200-100-Test S-G-50-200-100-FE

480

(h) S-O-500-180

(i) S-O-500-180-S-300-300

Figure 8.2 Predicted lateral load versus test result

0 1 2 3 4 5 60

20

40

60

80

100

120

140

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


S-O-500-180-Test S-O-500-180-FE

0 1 2 3 4 5 60

20

40

60

80

100

120

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


S-O-500-180-S-300-300-Test S-O-500-180-S-300-300-FE

481

(a) F-Control

(b) F-G-50-200

0 1 2 3 4 5 6-200

-150

-100

-50

0

50

100

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

0 24 48 72 96 120 144



0 1 2 3-150

-100

-50

0

50

100

Lateral Drift(%)

Bea

m T

ip L

oad

(kN

)

0 24 48 72


F-G-50-200-Test F-G-50-200-FE

482

(c) F-S-450-450

(d) F-O-450-150

0 1 2 3 4 5 6-150

-100

-50

0

50

100

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

0 24 48 72 96 120 144


F-S-450-450-Test F-S-450-450-FE

0 1 2 3 4 5 6-200

-150

-100

-50

0

50

100

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

0 24 48 72 96 120 144


F-O-450-150-Test F-O-450-150-FE

483

(e) F-O-500-180

(f) S-Control

0 1 2 3 4 5 6-150

-100

-50

0

50

100

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

0 24 48 72 96 120 144


F-O-500-180-Test F-O-500-180-FE

0 1 2 3 4 5 6-200

-150

-100

-50

0

50

100

150

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

0 24 48 72 96 120 144



484

(g) S-G-50-200-100

(h) S-O-500-180

0 1 2 3-200

-150

-100

-50

0

50

100

150

Lateral Drift(%)

Bea

m T

ip L

oad

(kN

)

0 24 48 72


S-G-50-100-200-Test S-O-50-100-200-FE

0 1 2 3 4 5 6-200

-150

-100

-50

0

50

100

150

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

0 24 48 72 96 120 144


S-O-500-180-Test S-O-500-180-FE

485

(i) S-O-500-180-S-300-300

Figure 8.3 Predicted beam-tip load versus test result


0 1 2 3 4 5 6-200

-150

-100

-50

0

50

100

150

Lateral Drift (%)

Bea

m T

ip L

oad

(kN

)

0 24 48 72 96 120 144


S-O-500-180-S-300-300-Test S-O-500-180-S-300-300-FE

0 5 10 150

50

100


Lat

eral

Loa

d (k

N)


0 2 4 6 8 10 120

20

40

60

80


Lat

eral

Loa

d (k

N)

F-G-50-200-Test S-G-50-200-FE

0 5 10 150

20

40

60

80


Lat

eral

Loa

d (k

N)

F-S-450-450-Test F-S-450-450-FE

0 2 4 6 8 10 120

20

40

60

80


Lat

eral

Loa

d (k

N)

F-O-450-150Test F-O-450-150-FE

0 2 4 6 8 10 120

20

40

60

80


Lat

eral

Loa

d (k

N)

F-O-500-180Test F-O-500-180-FE

486


Figure 8.4 Predicted initial stiffness versus test result

0 1 2 3 4 5 6 70

20

40

60

80

100


Lat

eral

Loa

d (k

N)


0 1 2 3 4 5 6 70

10

20

30

40

50

60


Lat

eral

Loa

d (k

N)

S-G-50-200-100-Test S-G-50-200-100-FE

0 1 2 3 4 5 6 70

10

20

30

40

50

60


Lat

eral

Loa

d (k

N)

S-O-500-180-Test S-O-500-180-FE

0 1 2 3 4 5 6 70

10

20

30

40

50

60

Top-of-column Displacement (mm) L

ater

al L

oad

(kN

)

S-O-500-180-S-300-300-Test S-O-500-180-S-300-300-FE

487


488


489

(c) Specimen F-S-450-450

490


491

(e) Specimen F-O-500-180

492


493


494


495

(i) Specimen S-O-500-180-S-300-300

Figure 8.5 Specimen crack patterns under push action

496

(a) F-Control

(b) F-G-50-200

497

(c) F-S-450-450

(d) F-O-450-150

498

(e) F-O-500-180

(f) S-Control

499

(g) S-G-50-200-100

(h) S-O-500-180

500

(i) S-O-500-180-S-300-300

Figure 8.6 Predicted crack patterns of 3D FE models

(a) F-Control

0 1 2 3 4 5 60

50

100

150

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


F-Control-Test F-Control-FE F-Control-FE-NJC

501

(b) F-G-50-200

(c) F-S-450-450

0 1 2 30

20

40

60

80

100

120

Lateral Drift(%)

Lat

eral

Loa

d (k

N)

0 24 48 72


F-G-50-200-Test F-G-50-200-FE-NJC F-G-50-200-FE

0 1 2 3 4 5 60

20

40

60

80

100

120

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


F-S-450-450-Test F-S-450-450-FE F-S-450-450-NJC-FE

502

(d) F-O-450-150

(e) F-O-500-180

0 1 2 3 4 5 60

50

100

150

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


F-O-450-150-Test F-O-450-150-FE F-O-450-150-NJC-FE

0 1 2 3 4 5 60

20

40

60

80

100

120

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


F-O-500-180-Test F-O-500-180-FE F-O-500-180-NJC-FE

503

(f) S-Control

(g) S-G-50-200-100

0 1 2 3 4 50

20

40

60

80

100

120

140

160

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120


S-Control-Test S-Control-FE S-Control-FE-NJC

0 1 2 3 40

50

100

150

Lateral Drift(%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96


S-G-50-100-200-Test S-G-50-100-200-FE S-G-50-100-200-FE-NJC

504

(h) S-O-500-180

(i) S-O-500-180-S-300-300

Figure 8.7 The effect of joints stirrup confinement

0 1 2 3 4 5 60

50

100

150

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


S-O-500-180-Test S-O-500-180-FE S-O-500-180-NJC

0 1 2 3 4 5 60

20

40

60

80

100

120

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


S-O-500-180-S-300-300-Test S-O-500-180-S-300-300-FE S-O-500-180-S-300-300-FE-NJC

505

Figure 8.8 The effect of opening shape

Figure 8.9 Crack pattern predicted by FE model S-O-500-180-R

0 1 2 3 4 5 60

20

40

60

80

100

120

140

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


S-O-500-180-Test S-O-500-180-FE S-O-500-180-R-FE

506

Figure 8.10 Effect of longitudinal slit length

(a) F-S-450-200

0 1 2 3 4 5 60

20

40

60

80

100

120

Lateral Drift (%)

Lat

eral

Loa

d (k

N)

0 24 48 72 96 120 144


F-S-450-450-Test F-S-450-0-FE F-S-450-200-FE F-S-450-450-FE

507

(b) F-S-450-0

Figure 8.11 Crack pattern predicted by FE models F-S-450-200 and

F-S-450-0

508

Table 8.1 Predicted loads versus test results

Specimen Beam tip load (kN)

Top of column load (kN) Positive Negative

Prediction Test Error (%) Prediction Test Error (%) Prediction Test Error (%)

First test

group

F-Control 87.8 87.2 0.7% -161.1 -156.6 2.9% 133.6 130.2 2.6% F-G-50-200 88.6 87.4 1.4% -109.8 -85.6 28.3% 105.2 96.9 8.6% F-S-450-450 79 83.1 -4.9% -116.8 -109.4 6.8% 104.4 110.9 -5.9% F-O-450-150 86.0 88.5 -2.8% -153.7 -138.3 11.1% 120.5 120.2 0.2% F-O-500-180 86.4 88.2 -2.0% -128.7 -115.7 11.2% 109.6 109.4 0.2%

Second test

group

S-Control 119.4 112.8 5.9% -160.7 -144.4 11.3% 153.1 147 4.1% S-G-50-200-100 119.1 107.3 11.0% -135.6 -111.0 22.2% 125.8 122 3.1%

S-O-500-180 112.9 119.3 -5.4% -132.4 -120.4 10.0% 126.8 135.6 -6.5% S-O-500-180-S-300-300 99 102.6 -3.5% -122.5 -111.7 9.7% 110.4 118.6 -6.9%

509

CHAPTER 9

CONCLUSIONS

9.1 INTRODUCTION

This thesis has presented a systematic study into a new seismic retrofit

approach called “beam weakening in combination with FRP (fibre reinforced

polymer) strengthening” (BWFS) method through both experimental study and

numerical investigation. The study has been focused on exploring the effects of

the proposed retrofitted techniques on the seismic performance of RC

beam-column joints via experimental study, and developing advanced

three-dimensional (3D) finite element (FE) models capable of accurately

modelling of T-section beams (T-beam) with a web opening (with or without

FRP strengthening) and the retrofitted 3-D RC joints. The developed 3-D FE

model was also used to study the issue of effective slab width with slab under

tension.

In the first stage of this study, an experimental study had been carried out

to explore the most effective techniques of the BWFS of retrofitting RC

beam-column joints, in the purpose to find a suitable approach for achieving

ductile beam sway mechanisms. Three different beam weakening techniques

were studied by testing 9 full-scale RC joints (under combined vertically

510

constant axial loading and horizontally cyclic loading), including: (a) the slab

slit (SS) technique; (b) the beam web opening (BO) technique; (c) the beam

section reduction (SR) technique, in which a deep transverse groove (TG) was

cut on the soffit of the beam near the joint. The latter two techniques had been

carried out in associated strengthening measures (i.e., strengthening the

weakening area in shear using FRP) to satisfy both serviceability and

load-carrying capacity requirements.

The study was then focused on developing of 3D FE solid models (using

solid elements for modelling concrete) capable of accurately modelling of

T-beams with opening, with and without FRP strengthening in terms of the

load-displacement response and cracking behaviour (cracking patterns). The 3D

shell models (using shell elements for modelling concrete) for T-beams has also

been proposed and accessed. An explicit dynamic approach was adopted in

obtaining the solution in which the structural problem is regarded as dynamic

problem and solved using explicit centre-difference-method (CDM). Effects of

several significant parameters, such as loading duration, damping schemes,

computational time and accuracy associated with the explicit dynamic method

have been assessed and discussed in-depth in this thesis.

The final part of the thesis has been concerned with the application of

developed advanced 3D solid FE model and 3D shell FE model. First, the most

suitable 3D shell FE model was used to study the issues of effective slab width

of T-beams in hogging moment zone (and slab in tension). A wide range of key

511

parameters influencing effective slab width have been investigated through

parametric studies and new models of effective slab width were proposed for

both interior and exterior joints. Finally, the 3D FE solid model was used for

numerical simulations of the retrofitted RC joints.

9.2 EXPERIMENTAL STUDIES OF RETROFITTED RC

JOINTS

To investigate the effectiveness of the proposed retrofit techniques in

transferring a storey sway failure mechanism to a beam sway mechanism, two

test groups, including a total of 9 specimens, were tested under cyclic loading.

Except for two control specimens, which was designed to violate the SCWB

mechanism (i.e. failing by storey sway mechanism), two specimens retrofitted

by creating 450×150 mm2 and 500×180 mm2 size openings were tested. It was

found that the specimens with the larger opening achieved the beam sway

mechanism. To reduce the opening area, an RC joint retrofitted by slot-shaped

openings was then tested. A 50×200 mm2 TG was made in the soffit of the

beam next to the joint. The first specimen so weakened, adjacent to the

beam-column interface, failed due to slipping of the bars in beam bottom within

the joint region. Another specimen of the same size, weakened by TGs placed

100 mm away from the beam-column interface, was also tested. The failure

mode was similar to the first. A specimen weakened by 450 mm long slab slits

along both the longitudinal and transverse directions was also tested. The steel

bars crossing the slits had all been cut. Finally, a combination of the SS and BO

512

techniques was investigated. Based on the test results and discussions presented

in Chapter 4, the following conclusions can be drawn:

(1) The existence of a transverse groove (TG) can lower the moment-capacity of

a T-beam greatly. However, the failure mode was non-ductile because the

main beam bottom longitudinal bars slipped in the joint region, even though

the groove was not placed close to the beam-column interface. The slab

slitting (SS) method can effectively lower the moment capacity of a T-beam

as it removes the contribution of the cut slab longitudinal bars. Meanwhile,

the failure mode is beam end flexural failure. The beam opening (BO) size

should be well designed (e.g. increasing the opening size)to lead to an

obvious four-hinges mechanism if the BO technique is applied as a retrofit

measure. The slab slits can help a specimen, not sufficiently weakened by

the BO technique, to realize the four-hinges mechanism. Slot-shaped

openings have similar effect on the seismic behaviour of a RC beam-column

joint as the rectangular one does.

(2) As the horizontal load recorded by the MTS machine included friction force,

which varied during test and among different specimens, the calculated

column shear force was used for discussions instead. The specimens

retrofitted by the proposed techniques had lower stiffnesses and strengths

than control specimens. Both the BO and SS techniques decrease the

stiffness of T-beams under both negative and positive loading. The TG

technique only has obvious effect on the stiffness and strength of a T-beam

513

when it was under negative loading. When the groove is not placed close to

the beam-column interface, the effect of the TG technique on T-beam

stiffness becomes weaker. The positive strength of T-beams is slightly

affected by the existence of BO. The SS technique deceases both negative

and positive strength of T-beams.

(3) When the control specimen has good ductility, the TG technique leads to

very poor ductility. The BO technique keeps the specimen ductility almost

unchanged. The SS technique decreases the ductility of a specimen. When

the control specimen has a poor ductility, the TG technique has a small

effect on specimen ductility but leads to a smaller yield displacement than

that of control specimen. The BO technique slightly increases the specimen

ductility. When the combination of the BO and SS techniques is applied,

specimen ductility can be increased in addition to effectively decreasing the

moment-capacity of T-beams.

(4) The Specimen retrofitted by the SS technique had better energy dissipation

capacity than those by the BO and the TG technique. The specimens

retrofitted the TG technique had best energy dissipation capacity at first but

worse energy dissipation capacity after they reached peak load.

(5) Equivalent viscous damping ratios (EDRs) of specimens basically increased

with increasing drift ratio. The EDRs of those specimens weakened by TGs

were almost the highest before they failed. The EDR of the specimen

514

F-S-450-450 was the second highest among the first test group. The BO

technique has slight effect on a specimen EDR. Generally, the EDRs of the

second cycle of each drift ratio are lower than those of the first cycle.

(6) Unlike the BO, SS methods and their combination, the TG method has less

effect on the initial PTPS of specimens. In addition, the PTPS of the

specimens weakened by TGs, reduced the most quickly with increasing drift

ratio, as those specimens had poor ductility and failed soon after peak load

had been reached. The specimen retrofitted by a combination of the BO and

SS techniques has the lowest speed of PPTS descending.

(7) The TG technique shortens plastic hinges length of T-beams as the

deformation of T-beams is concentrated on the groove region.

(8) Except for two control specimens F-Control and S-Control, beams’

deformation contribution to the lateral drift ratio increased with the

increasing lateral drift ratio. Thus the contribution of columns deformation

was decreased with the increasing lateral drift ratio

9.3 THREE-DIMENSIONAL FINITE ELEMENT

MODELLING OF RC T-BEAMS AND JOINTS

9.3.1 3D Solid FE Model of T-beams with a Web Opening.

3D FE models for T-beams retrofitted by BO techniques were developed

and verified in Chapter 5. The RC T-beams tested by Nie (2018), with web

515

openings, with or without FRP strengthening, were used for assessing the

accuracy of the 3D FE model. To avoid convergence problems, an explicit

dynamic approach which uses explicit centre-difference-method (CDM), was

adopted to obtain the numerical results. In this study, a 3D Prism cohesive

model was proposed for modelling the steel-concrete interface. The effects of a

number of significant parameters affecting the accuracy of the explicit dynamic

approach, including the element size, loading time, damping ratio, calculation

time and calculation precision, were studied. The confinement effect provided

by FRP on the web chord was also investigated. Finally, the predicted results of

the 3D FE solid model were compared with the predicted results of the 2D FE

models proposed by Nie (2018). The following conclusions can be drawn based

on the results and discussions presented in Chapter 5:

(1) The proposed 3D FE models predicted well, the behaviour of the beam

specimens tested by Nie (2018), except for the two beams under positive

loading. Furthermore, the yield/peak loads and test crack patterns of all

specimens were also accurately predicted, though the post-yielding curves of

T-beams with web opening were not well captured. A possible reason is that

the damage in shear is underestimated by the proposed models.

(2) The proposed 3D prism cohesive element modelling method was shown to

be effective and reliable for modelling the steel-concrete interface in the 3D

solid FE model.

516

(3) For a quasi-static analysis, the dynamic energy should be a small portion of

the internal energy (normally below 1%), except for a few discrete moments

when local dynamics events occur, such as at the beginning of loading, at

the appearance of significant cracks, and at yielding of the main tensions

bars.

(4) Element size, loading time and stiffness damping coefficient were

determined based on convergence studies. A 20-mm element size, 50T0 (T0

is the FE model natural period) and value of 2e-6 were found

appropriate for the proposed 3D FE models. Only the value of is needed

to define the Rayleigh damping, since the vibrations associated with high

modes are expected to damped out in the adopted explicit dynamic method.

(5) The double precision computation was suggested for numerical simulation

using the 3D solid FE. Single precision computation costs less

computational time, but leads to more accumulation errors and unreal (or

even problematic) crack patterns in the later stage of simulation.

(6) When the confinement effect of the FRP wrapping was simulated using Yu

et al.'s (2009a, b) modified plastic-damage model for FRP-confined concrete,

more accurate results can be obtained, especially for a T-beam with a

smaller opening and a bigger web chord.

(7) The proposed 3D solid FE models are better than the 2D FE model proposed

517

by Nie (2018) in the modelling of T-beams, especially for those weakened

by web openings but without FRP shear strengthening. The 2D FE models

usually overestimate the post-cracking stiffness of T-beams, which is better

predicted by the proposed 3D FE models.

(8) A DP+BC model, which is capable of modeling the shear degradation effect

of cracked concrete, was examined and the predictions were compared with

those from the DP model. The effects of three parameters, namely, the

number of critical cracks, the maximum cracking strain and the coefficient

of power law, of the BC model, were investigated through parametric

studies. The results indicated that the DP+BC model predicted the

load-displacement curves and crack patterns similar to those predicted with

the DP model; the DP+BC model predicted slightly better the post-peak

load-displacement response. The DP model is thus recommended for

modelling RC T-beams with a web opening but without FRP strengthening

since it is more convenient to create an FE model using the DP model.

9.3.2 3D Shell FE Models for T-beams

The 3D solid model proposed in Chapter 5 proved to be both reliable and

accurate when modelling a T-beam, but was not as economical as the 2D FE

model of Nie (2018) in terms of computational time. A decrease in the number

of elements proved to be a satisfactory option when the dynamic explicit

method is used. Based on the above considerations, several 3D shell FE models,

518

in which concrete was modelled by shell elements (referred as 3D shell models)

were presented in Chapter 6. Different 3D FE models were proposed and

assessed, which included two 3D shell models with slab modelling by only one

shell layer (i.e. 3D 1-shell models) and a few 3D FE models with slab modelled

using two shell layers (i.e. 3D 2-shell models). For both types of shell models,

the effects of the bond-slip behaviour between longitudinal bars and concrete in

slab were investigated. For the 3D 2-shell models, effects of the plane section

restraint applied to the two slab shell layers were investigated. The T-beams

tested by Nie (2018) and the 3D solid model presented in Chapter 5 were used

for comparison purpose in the above assessment. The following conclusions are

drawn based on the numerical results and discussion presented in Chapter 6:

(1) The 3D-Solid models for T-beams are most accurate in terms of the

load-displacement curve and crack patterns.

(2) The 3D 1-shell and 2-shell models incorporating the bond-slip relationship

between slab longitudinal bars and concrete, predict better

load-displacement curves and crack patterns than those not considering the

bond slip behaviour.

(3) In the 3-D 2-shell model, the plane section restraints applied between the

two shells representing slab have only slight effect on the predicted

load-displacement curves. The crack patterns, however, are better predicted

when plan section restraints are applied. Decreasing the density of restraints

519

in the X direction has less effect on the predicted crack patterns than

decrease of restraint density in the Z direction, which is the direction of

longitudinal bars.

(4) Both 3D-Shell-1 and 3D-Shell-2-R models (i.e., models with plane section

restraints applied between the two shells) work well as models for the

T-beam. As the 3D-Shell-1 model costs less computational time than the

3D-Shell-2-R mode, the former is the better substitute for 3D solid FE

model in modelling T-beam.

(5) A 20 mm element size, a loading time of 50T0 and a damping coefficient β

of 2e-6, are suggested for the FE 3D-Shell-1 model for modeling T-beams.

(6) An element size of 20 mm is not small enough for the 3D-Shell-1 model to

obtain mesh size convergence when it was used to model a T-beam

weakened by web openings. Thus, the 3D-Shell-1 model is not a good

substitute for the 3D solid FE model for modelling such beams. It not only

requires more computational time but provides less accurate predictions than

the 3D solid FE model.

9.3.3 Effective Slab Width of RC Frames

Using the 3D 1-shell model developed in Chapter 6, parametric studies

were carried out to investigate the effects of a wide range of factors on the

effective slab width of T-section RC beam with slab in tension, with following

520

factors investigated: beam length, width, height, bar reinforcement ratio, slab

width, thickness, bar spacing, yield stress of steel bars, transverse beam height,

width stirrup spacing, column width. The effects of the bond-slip relationship

between longitudinal bars and concrete were also investigated. Two types of

stress-strain model for steel bars were studied and described parametrically.

Based on the numerical results and discussions presented in Chapter 7, the

following conclusions can be drawn:

(1) The key factors affecting the predicted effective slab width at both exterior

and interior joints include concrete strength, slab bar spacing, yield stress of

steel.

(2) The transverse beam width only has an obvious effect on the effective slab

width of an exterior joint. The effects of beam bar yield stress, beam height

and width are more obvious for an interior joint than for an exterior joint.

(3) The effective slab width for interior joints is wider than that for interior

joints.

(4) Based on the numerical modelling results, two simple equations for

predicting effective slab width were proposed for interior and exterior joints.

Only key factors were included in the equations. With the objective of

providing more accurate predictions, a weakening coefficient accounting for

column effects was included.

521

9.3.4 3D FE Modelling of Retrofitted RC Joints

3D FE models developed for RC joints retrofitted by the proposed

techniques was presented in Chapter 8, which were based on the verified 3D

solid models described in Chapter 5. The 3D FE models for RC joints were

subjected to monotonic loading, rather than the cyclic loading in the tests. The

envelope curves derived from these test results were then used for comparison

purposes. The following conclusions can be drawn based on the predicted

results:

(1) The proposed 3D FE models perform well in predicting load-displacement

curves and crack patterns of the retrofitted RC joints. The predictions for

specimens with transverse grooves are not as good as the others because bars

slippage at beam bottom is noticeable due to cyclic loading, which was not

considered in the 3D FE models.

(2) The predicted initial stiffness are slightly larger than those of the test values

because of the mechanical slack existing in the testing equipments. Such

slack was caused by very small failures of the mechanical components of the

test equipments used to fit together precisely.

(3) The confinement effect of stirrups should be considered in the 3D FE model

to obtain more reliable/accurate predictions. In this study the model

proposed by Mander et al. (1988) was applied.

522

(4) The confinement effect of stirrups has a larger effect on the specimens

F-Control and S-Control, in which joint shear failure occurred and bottom

column hinge formed respectively.

(5) The ignorance of the effect of cyclic loading leads to slightly higher negative

beam tip loads. This is mainly because the cumulative damage caused by

cyclic loading during testing is not simulated in FE analyses.

9.4 FURTHER STUDIES

During this study, the effectiveness of the proposed retrofit techniques for

RC frames were investigated through testing RC joints subject to applied cyclic

loading. Due to the limit of time, money and test equipment, only 9 specimens

were tested. To produce comprehensive retrofitting guidelines, more serials of

tests should be conducted. Firstly, due to the limit of test facilities, the applied

axial load was constant. However, in a real situation, the axial load might

increase with the increasing drift ratio, which could decrease the specimen

ductility. Thus, if possible, a serial of test with axial load keeping changing

should be conducted. Secondly, the specimens retrofitted by the TG technique

had poor seismic performance due to the bottom beam bar slippage within the

joint region. Thus, another modification, like filling the groove with a material

with lower compressive strength, should be done for this technique. Finally, to

better verify the effectiveness of the proposed retrofit techniques in realising the

SCWB mechanism, experimental studies on retrofitted RC frames subjected to

523

seismic loading should be further explored. These RC frames should be

designed according to previous editions of the seismic design codes. When

subjected to seismic loading, the control specimen should fail by storey sway

mechanism. After applying the proposed retrofit techniques, a beam sway

mechanism should be realised.

The proposed 3D FE models representing retrofitted RC joints considered

monotonic loading only, which needs to be improved to consider the effect of

cyclic loading. The 3D models for the RC Joints retrofitted by the TG technique,

overestimated the peak load, the corresponding displacement at the peak load

and the post-cracking stiffness. This may be due to the incapability to model the

effects of cyclic loading. Thus, a more advanced FE model of retrofitted RC

joints, subjected to cyclic loading, should be developed in the future. The

Software OpenSEES (2009) can be used for this purpose. An appropriate FE

modelling of retrofitted RC frames subject to seismic loading should then be

conducted. With a reliable FE model, parametric studies on the opening sizes,

locations, and various combinations of the three techniques should be done,

with the aim to produce comprehensive retrofitting guidelines. Besides, the

effect of axial load applied on the column top should be investigated with the

verified FE model.

Parametric studies related to effective slab width of RC frames have been

conducted and two simple equations proposed for interior and exterior joints.

The effect of BO or SS on the effective slab width has not be investigated. As

524

both the BO and SS techniques might actually be used in practice for the

retrofitting of existing RC frames, the effects of BO and SS on the effective slab

width cannot be ignored. The effect of the SS on the effective slab width is

obvious as slab bars crossing the slits are cut. The BO technique might change

the failure mode from beam flexural failure to four hinges failure. The existence

of web openings has a significant effect on the effective slab width in composite

steel beams and this has been confirmed by Alsarraf and El Din (2016). Thus

the effect of BO on the effective slab width of a RC frame should also be

investigated in the future.

9.5 REFERENCES



Alsarraf, M. A., and El Din, H. S. (2016). "Effects of Web Openings on the

Effective Slab Widths in Composite Steel Beams", International Journal

of Engineering and Technology, 8(1), 6.

Mander, J. B., Priestley, M. J., and Park, R. (1988). "Theoretical stress-strain

model for confined concrete", Journal of structural engineering, 114(8),

1804-1826.



525





Yu, T. T. J. G., Teng, J. G., Wong, Y. L., and Dong, S. L. (2010), "Finite


model", Engineering Structures, 32(3), 665-679

526

527

APPENDIX A FRP ANCHOR DESIGN

As shown in Figure A.1, there are three typical adhesive anchor failure

modes, which are concrete cone failure, combined failure and anchor failure

(Kim and Smith 2010).

The anchor resistance force ( uN ) is given as below:

min( , , )u cc cb arN N N N (A-1)

1.5cc 9.68 ' ef cN h f (cone failure ) (A-2)

04.62 ( ' 20 ) cb ef cN d h f MPa (combined failure) (A-3)

09.07 ( ' 20 ) cb ef cN d h f MPa (combined failure ) (A-4)

0.59ar FRP FRP FRPN w t f (anchor failure ) (A-5)

where efh = effective embedment depth of the anchor (mm); 'cf =concrete

cylinder compressive strength (MPa); 0d =diameter of the anchor hole (mm);

FRPw and FRPt = width (mm) and thickness (mm), respectively, of the fiber

sheet used in construction of the FRP anchor; and FRPf = flat coupon tensile

rupture FRP strength (MPa).

The expected failure mode is FRP rupturing before anchor failure. Thus the

design anchor should satisfy the following equation:

FRP uF N (A-6)

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where FRPF is the resistance force of one FRP stripe (N).

0.334 ,5022

2 0

7

82FRP u

t mmw mmG GPa

f MPa

' 35cf MPa

47090 FRP uF wt N

From equation（A-5) and（A-6）： FRPw >84.7mm; FRPw =90mm

From equation（A-2) and（A-6）： efh > 87.7; 令 efh ≈90mm

From equation（A-4) and（A-6）： 0d >18.4mm；令 0d ≈19mm

The diameter of the anchor holes should not be too big to void small space

between anchor holes. 0d =12 mm is applied into anchor design. The failure

force is then calculated as below:

09.07 9.07 3.14 12 90 30758.2 cb efN d h N

The FRP effective rupture stress is about 60% of u . The design anchor

meets the requirement for the expected failure mode.

The free end of anchor will be fanned out at an angle about 36°

(2arctan(25/80)) and attached to the FRP sheet to avoid the FRP sheet

debonding from the U jacket end.

529

Figure A.1 Typical adhesive anchor failure modes (Kim and Smith 2010)

Figure A.2 The detail of FRP sheet anchor

150 450 150450

180

25 5050 2525505025

80

36°

Figure A.3 The layout of FRP Sheet anchors of specimen F-O-450-150

530

200

50 100200

50100

5050

80

36°

2525

Figure A.4 The layout of FRP Sheet anchors of specimen F-G-50-200

REFERENCES

Kim SJ, Smith ST. (2010) "Pullout strength models for FRP anchors in

uncracked concrete", Journal of Composites for Construction, 14(4),

406–414.

Koutas, L., and Triantafillou, T. C. (2012), "Use of anchors in shear

strengthening of reinforced concrete T-beams with FRP", Journal of

Composites for Construction, 17(1), 101-107.

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