FINITE ELEMENT STUDY OF RC SLAB IN PUNCHING SHEAR

FINITE ELEMENT STUDY OF RC SLAB

IN PUNCHING SHEAR

A. K. M. Jahangir Alam

DEPARTMENT OF CIVIL ENGINEERING BANGLADESH UNIVERSITY OF ENGINEERING

& TECHNOLOGY DHAKA, BANGLADESH

FINITE ELEMENT STUDY OF RC SLAB

IN PUNCHING SHEAR

by

A. K. M. Jahangir Alam

A thesis submitted to the Department of Civil Engineering,

Bangladesh University of Engineering & Technology, Dhaka, for fulfillment

of

DOCTOR OF PHILOSOPHY IN CIVIL ENGINEERING

OCTOBER 2014

FINITE ELEMENT STUDY OF RC SLAB IN PUNCHING SHEAR

Approve as to style and contents on the dated October 14, 2014 by: Professor Dr. Khan Mahmud Amanat Department of Civil Engineering, BUET

Supervisor Chairman

Professor Dr. A. M. M. Taufiqul Anwar Head, Department of Civil Engineering, BUET

Ex-Officio Member

Professor Dr. Sk. Sekender Ali Department of Civil Engineering, BUET

Member

Professor Dr. Ahsanul Kabir Department of Civil Engineering, BUET

Member

Professor Dr. Tahsin Reza Hossain Department of Civil Engineering, BUET

Member

Professor Dr. Jamilur Reza Choudhury Vice Chancellor University of Asia Pacific House # 53/1, Road 4/A, Dhanmondi Dhaka, Bangladesh

Member (External)

Professor Dr. Iftekhar Anam Department of Civil Engineering University of Asia Pacific House # 53/1, Road 4/A, Dhanmondi Dhaka, Bangladesh

Member (External)

To

My parents and wife Dr. Nasrin Sultana

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DECLARATION

Declared that except where specific references are made to other investigators, the

work embodied in this thesis is the result of the investigation carried out by the author

under the supervision of Dr. Khan Mahmud Amanat, Professor, Department of Civil

Engineering, Bangladesh University of Engineering & Technology, Dhaka,

Bangladesh. Neither this thesis nor any part of it has been submitted or is being

concurrently submitted elsewhere for any other purpose (except for publication).

____________________ October, 2014 A. K. M. Jahangir Alam

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ACKNOWLEDGMENT

The author wishes to express his deepest gratitude to Almighty Allah who is the

executive of everything and then his supervisor Dr. Khan Mahmud Amanat,

Professor, Department of Civil Engineering, Bangladesh University of Engineering &

Technology (BUET), for his constant guidance, continued encouragement, generous

help and unfailing enthusiasm at all the stages of this research work. His active

interest in this topic and valuable advice were the source of the author’s inspiration.

Sincere appreciation and gratitude are also expressed to Dr. A. M. M. Taufiqul

Anwar, Professor and Head, Department of Civil Engineering, BUET, for his

encouragement and cooperation. Thanks are also due to Dr. Sk. Sekender Ali,

Professor, Department of Civil Engineering, for his co-operation during his tenure as

Head, Department of Civil Engineering, BUET.

The author owes his thanks to the members of the Board of Post Graduate Studies of

the Department of Civil Engineering, BUET, and also to the members of the

Committee of Advanced Studies and Research for kindly approving the research

proposal and financing the experimental work reported in this thesis.

The author is grateful to Engr. Mohammad Mahfuzur Rahman, Sub-Divisional

Engineer (Civil), BUET, to support for software during analytical works. The author

is also grateful to Engr. M. M. Abdul Alim, Chief Engineer, BUET and other

colleagues for inspiration and support during the present study.

The author wishes to thank his family and friends who helped him with necessary

advice and information during the course of the study. Specially, author’s younger son

Md. Jarif Alam for his supports to produce sketch up model and elder son Md. Jahin

Alam for his help during preparation of this thesis.

Finally, the author likes to deliver his special thanks to his parents and beloved wife

Dr. Nasrin Sultana for their moral support to complete this work.

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ABSTRACT

Current design procedures and provisions for preventing punching shear

failure of reinforced concrete slabs, given in various codes of practice, are largely

based on studies of the behavior and strength of simply supported, small sized

conventional specimens extending to the nominal line of contra-flexure. Present day

codes of practice usually do not consider the effect of boundary restraint against

rotation. The contribution of flexural steel reinforcement is ignored by some of the

code provisions. Also, the effect of slab thickness and column size are not considered

for calculating punching shear capacity of slab in some codes.

A finite element study of punching shear behavior of reinforced concrete slab

is presented in this thesis. The numerical simulation is based on previous

experimental study of 15 reinforced concrete model slabs. Finite element analysis of

reinforced concrete slabs subjected to load producing punching shear is evaluated and

the validity of FE analysis has been verified through comparison with available

experimental data from other researchers as well. It has been shown that the load vs.

deflection diagrams and ultimate load capacities obtained from FE analysis closely

agree with the experimental results. Comparison of crack pattern of the slab also

shows good agreement between experiment and numerical prediction. It has been

shown that using appropriate method of solution and material model for numerical

simulation, significant benefit can be achieved employing finite element tools and

advanced computing facilities in obtaining safe and optimum solutions without doing

expensive and time-consuming laboratory tests.

Following the establishment of the validity and reliability of the FE modeling

scheme, a parametric study has been carried out to investigate the influence of the

flexural reinforcement on ultimate load capacity of slabs. Code-specified strength of

the specimen was calculated in accordance with the American, British, Canadian,

European, German and Australian codes. It has been observed from the study that

punching shear capacity may not be efficiently predicted in some of the codes.

The study is then extended for reinforced concrete multi-panel flat plates

subjected to punching shear. The study involves employment of a nonlinear material

model in finite element method of analysis based on past experimental investigations

which provides solution for realistic behavior of reinforced concrete slabs with

punching shear for concrete strengths, flexural reinforcement ratio, slab thickness and

( vii )

column size. It has been observed that depending on the degree of variation in these

parameters, the overall behavior of RC slab with punching shear changes

significantly.

A proposal for a reasonable estimate of punching shear capacity of flat plate

has been made in this thesis based on the findings of parametric study. The proposal

includes the effect of flexural reinforcement in addition to concrete strength in

calculating the punching capacity. The size effect of slab and column on punching

shear behavior of flat plate is also included in the proposal. The punching shear

capacity calculated using the proposed method is compared with the results of

nonlinear finite element analysis and values from different codes equations and have

been found to be in good agreement. It is expected that the findings of this study

would result in a more rational design of structural floor systems where concrete

punching phenomenon plays an important role.

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CONTENTS

Page No.

DECLARATION iv

ACKNOWLEDGMENT v

ABSTRACT vi

CONTENTS viii

LIST OF FIGURES xvi

LIST OF TABLES xxix

NOTATIONS xxx

CHAPTER-1 INTRODUCTION 1

1.1 GENERAL 1

1.2 LITERATURE REVIEW 3

1.3 CODE PROVISIONS FOR PUNCHING SHEAR 20

1.3.1 American (ACI 318-1) code 21

1.3.2 Australian (AS 3600-2009) code 21

1.3.3 Bangladesh (BNBC, 2006) code 22

1.3.4 British (BS 8110-97) code 22

1.3.5 Canadian CAN3-A23.3-M84 (1984) code 23

1.3.6 European Code (EC 2-1-1 (2003) and 23

CEB-FIP Model Code 90)

1.3.7 German (DIN 1045-1: 2008) code 24

1.4 MODELING GUIDELINE AND COMPLEXITIES 24

1.5 SCOPE AND OBJECTIVE 27

1.6 ORGANIZATION OF THE THESIS 29

CHAPTER-2 FE MODELING OF METHOD 31

2.1 INTRODUCTION 31

2.2 FE PROCEDURE 31

2.2.1 Global Formulation 32

2.2.1.1 Displacements 32

2.2.1.2 Strains and Stresses 32

2.2.1.3 Equilibrium 33

2.2.1.4 Principle of Virtual Displacements 33

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2.2.2 Discretization to Elements 34

2.2.2.1 Displacements 35

2.2.2.2 Strains and Stresses 35

2.2.2.3 Element Assembly 36

2.2.2.4 Virtual Strain Energy 36

2.2.2.5 Stiffness Matrix 38

2.2.3 Assembling the Load Vector 39

2.2.4 Equilibrium 39

2.3 SUITABLE ELEMENTS 40

2.3.1 Selection of Element 40

2.3.2 CHX60 Element 42

2.3.3 Displacements 43

2.3.4 Strains 43

2.3.5 Stresses 44

2.3.6 Shape functions 45

2.3.7 Element Stiffness Matrix 47

2.3.4 Numerical Solution of Element Integrals 48

2.4 INTEGRATION SCHEMES 49

2.5 STRAIN DISPLACEMENT RELATION 51

2.5.1 Equivalent Von Mises Strain 51

2.5.2 Principal Strains 51

2.5.3 Volumetric Strain 52

2.6 STRAIN MATRIX 52

2.7 STRESS-STRAIN RELATIONSHIP 54

2.8 REINFORCEMENT IN SOLID ELEMENT 54

CHAPTER-3 MATERIAL MODELLING 58

3.1 INTRODUCTION 58

3.2 BEHAVIOR OF CONCRETE IN COMPRESSION 59

3.2.1 Uniaxial Behavior 59

3.2.2 Biaxial Behavior 61

3.2.3 Triaxial Behavior 65

3.3 YIELD CRITEIA 68

3.3.1 The Tresca Yield Criterion 69

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3.3.2 The Von Mises yield criterion 70

3.3.3 The Mohr-Coulomb yield criterion 70

3.3.4 The Ducker-Prager yield criterion 72

3.4 CRACKING OF CONCRETE AND MODELLING 73

3.4.1 Smeared Cracking Model 73

3.4.1.1 Crack Initiation 75

3.4.1.2 Crack Stress-Strain Relation 76

3.4.2 Total Strain Cracking Model 77

3.4.2.1 Coaxial and Fixed Stress–Strain Concept 77

3.4.2.2 Lateral Expansion Effects due to Poisson’s Ratio 79

3.4.3 Tensile Behavior 81

3.4.3.1 Tension Softening Relations 81

3.4.3.2 Linear Tension Softening 83

3.4.3.3 Multi-linear Tension Softening 84

3.4.3.4 Nonlinear Tension Softening by Hordijk et al. 84

3.4.3.5 Brittle Cracking 85

3.4.3.6 Exponential Tension Softening 86

3.4.3.7 Constant Tension Softening 86

3.4.4 Shear Retention Relations 87

3.4.4.1 Full Shear Retention 87

3.4.4.2 Constant Shear Retention 87

3.4.5 Compressive Behavior 87

3.5 REINFORCEMENT 89

3.5.1 Bar Reinforcement 90

3.5.2 Reinforcement Modeling 91

3.5.2.1 Linear Elasticity 91

3.5.2.2 Von Mises Plasticity 91

3.5.2.3 Monti–Nuti Plasticity 91

3.5.2.4 Reinforcement Specials 92

CHAPTER-4 NONLINEAR SOLUTION TECHNIQUES 93

4.1 INTRODUCTION 93

4.2 BASIC NUMERICAL PROCESS FOR NONLINEAR PROBLEM 93

4.2.1 Method of Direct Iteration 94

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4.2.2 The Newton-Rapson Method 95

4.2.2.1 Regular Newton-Raphson. 95

4.2.2.2 Modified Newton-Raphson. 96

4.2.3 Linear Stiffness Iteration 98

4.2.4 Constant Stiffness Iteration 98

4.3 CONVERGENCE CRITERIA 98

4.3.1 Force Norm 100

4.3.2 Displacement Norm 100

4.3.3 Energy Norm 100

4.4 INCREAMENTAL PROCEDURE 101

4.4.1 Load and Displacement Control 101

4.4.2 Arc-Length Control 102

4.4.2.1 Spherical Path ARC length method 104

4.4.2.2 Updated Normal Plane Arc length method 105

4.4.2.3 Indirect Displacement Control 105

4.5 ITERATIVE SOLUTION METHODS 106

4.5.1 Conjugate Gradient 107

4.5.2 Generalized Minimal Residual 107

4.6 SOLUTION TERMINATION CRITERIA 107

4.6.1 Loading Based Termination 107

4.6.2 Result Based Termination 108

CHAPTER-5 EXPERIMENTAL DATA 110

5.1 INTRODUCTION 110

5.2 PREVIOUS EXPERIMENTAL PROGRAM BY ALAM (1997) 110

5.2.1 Specimen Details 110

5.2.2 Test Results 112

5.2.3 Comparison of Test Results with different code of predictions 116

5.3 TEST RESULTS OF OTHER RESEARCHERS 119

5.3.1 Bresler and Scordelis Beam 119

5.3.2 Toronto Beam 120

5.3.3 Kotsovos Beam 121

5.3.4 Slab Tested by Kuang and Morley 121

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CHAPTER-6 NUMERICAL EXAMPLES AND VALIDATION 123


6.2 MODELING OF TESTED SPECIMEN 123

6.3 ELEMENT SELECTION 124

6.4 MATERIAL MODEL OF CONCRETE 124

6.4.1 Compressive Behavior 124

6.4.2 Tensile Behavior 125

6.4.3 Shear Behavior 125

6.5 MODELING OF SLAB GEOMETRY 125

6.6 REINFORCEMENT MODEL 125

6.7 BOUNDARY CONDITION 126

6.8 LOADING 126

6.9 SOLUTION STRATEGY 126

6.10 RESULTS OF FE ANALYSIS 126

6.11 DISCUSSIONS ON FE ANALYSIS 132

6.11.1 Load-Deflection Behavior 132

6.11.2 Cracking 135

6.12 UPDATED FE MODEL 138

6.13 LOAD-DEFLECTION BEHAVIOR OF TESTED SLAB 138 USING UPDATED MODEL 6.14 COMPARISON OF TEST RESULTS AND ANALYSIS 141 WITH DIFFERENT CODE OF PREDICTIONS 6.15 PARAMETRIC STUDY 143

6.16 COMPARISON OF FE MODEL WITH TEST RESULTS 146 OF OTHER RESEARCHERS

6.16.1 Bresler and Scordelis Beam 146

6.16.2 Toronto Beam 147

6.16.3 Kotsovos Beam 148

6.16.4 Slab Tested by Kuang and Morley 149

6.17 SENSITIVITY OF MESH SIZE 151

6.18 VALIDATION OF FE ANALYSIS 153

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CHAPTER-7 FE MODELING AND STUDY ON 154

MULTIPANEL FLAT PLATES


7.2 BEHAVIORAL DIFFERENCE 154

7.3 FE MODELING OF MULTI PANEL FLAT PLATE 158

7.3.1 Flat Plate Building system 158

7.3.2 Boundary Condition and Loading 159

7.3.3 Material Model of Concrete in Slab 161

7.3.3.1 Compressive Behavior 161

7.3.3.2 Tensile Behavior 161

7.3.3.3 Shear Behavior 161

7.3.4 Reinforcement Modeling 161

7.4 FE MESHING OF MODEL 162

7.5 ANALYSIS PROCEDURE 164

7.6 PUNCHING TYPE OF FAILURE 164

CHAPTER-8 NUMERICAL EXAMPLES OF MULTI-PANEL 169 FLAT PLATES


8.2 DIFFERENT SLABS COSIDERED 169

8.3 RESULTS OBTAINED FROM ANALYSIS OF MODEL SLAB 171

8.3.1 Load-deflection behavior 171

8.3.2 Sensitivity of Tension Softening to Multi Panel Model Slab 176

8.3.3 Ultimate Failure Load of Multi Panel Flat Plate 181

8.3.4 Sensitivity of Flexural Steel into the Flat Plate 190

8.4 DISCUSSION ON RESULTS AND COMPARISON WITH CODES 193

8.4.1 Punching Shear Stress of Multi Panel Flat Plate 193

8.4.2 Non-Dimensional Punching Shear of Multi Panel Flat Plate 198

8.4.3 Effect of Concrete Strength 203

8.4.3.1 On 400mm x 400mm column 203

8.4.3.2 On 600mm x 600mm column 206

8.4.3.3 On 800mm x 800mm column 209

8.4.3.4 On Average thickness of Slab 211

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8.4.4 Effect of Flexural Reinforcement 214

8.4.4.1 On 400mm x 400mm column 214

8.4.4.2 On 600mm x 600mm column 217

8.4.4.3 On 800mm x 800mm column 219

8.4.4.4 On Average Thickness of Slab 221

8.4.5 Effect of Slab Thickness 226

8.4.5.1 On 400mm x 400mm column 226

8.4.5.2 On 600mm x 600mm column 229

8.4.5.3 On 800mm x 800mm column 231

8.4.5.4 On Average Column Size 234

8.4.6 Effect of Column Size 239

8.4.6.1 On 200mm thick slab 239



8.4.6.4 On Average Thickness of Slab 247

8.5 SUMMARY OF PRECEDING DISCUSSIONS 249

CHAPTER-9 SIMPLE PUNCHING SHEAR STRESS FORMULA 251 FOR MULTI PANEL FLAT PLATE 9.1 INTRODUCTION 251

9.2 BASIS OF PROPOSAL 251

9.2.1 Punching Shear Capacity of Slab 251

9.2.3 Relationship of Normalized Punching Shear with 253 Concrete Strength and Flexural Reinforcement 9.3 THE PROPOSED FORMULA 254

9.4 EFFECTIVENESS AND COMPARISON WITH CODE 255

9.4.1 Comparison with variable Concrete Strength 255

9.4.2 Application of Proposed Formula with variable Flexural 259 Reinforcement 9.4.3 Comparison with various codes of prediction 259

9.4.4 Comparison with test results 272

9.4.4.1 Comparison with Author’s Past Test Results. 272

9.4.4.2 Comparison with other’s Test Results. 272

9.5 CONCLUDING REMARKS 274

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CHAPTER-10 CONCLUSIONS AND RECOMMENDATIONS 275

10.1 CONCLUSIONS 275

10.2 RECOMMENDATIONS 278

REFFERENCE 279

APPENDIX-A DETAILS OF MODEL SLAB AN ANALYTICAL 294 RESULTS APPENDIX-B FAILURE LOAD AND PUNCHING SHEAR 298

STRESSES APPENDIX-C COMPARISON OF NORMALIZED PUNCHING 307 SHEAR OF MODEL SLAB

APPENDIX-D APPLICATION OF PROPOSED FORMULA FOR 314 PUNCHING SHEAR CAPACITY CALCULATION APPENDIX-E COMPARISON OF PROPOSED LOAD CARRYING 326 CAPACITY OF MODEL SLAB WITH VARIOUS

CODES APPENDIX-F COMPARISON OF PROPOSED FORMULA WITH 335 TEST RESULTS OF OTHER RESEARCHERS

APPENDIX- G LOAD DEFLECTION CURVE USING IDEAL 342 TENSION SOFTENING

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LIST OF FIGURES

Page No. Figure 2.1 Stress resultants for plate/shell 41

Figure 2.2 Assumption Regarding deformation of a plate/shell 41

Figure 2.3 Solid Isoparametric Elements 42

Figure 2.4 Displacements of solid element 43

Figure 2.5 Deformation of solid element 44

Figure 2.6 Cauchy stresses of solid element 45

Figure 2.7 Integration schemes in ζ direction for bricks 50

Figure 2.8 Example integration schemes ηξ x ηη x ηζ for brick 50

Figure 2.9 Embedded reinforcement in 3D solid concrete element 55

Figure 3.1 Uniaxial Stress-strain relationship of ordinary concrete 60

Figure 3.2 Complete stress-strain curve including post-peak response 61

Figure 3.3 Biaxial strength envelope of concrete 62

Figure 3.4 Biaxial Compressive Yield Surface for Concrete with 64 Low Confining Pressure

Figure 3.5 Triaxial stress-strain curve of concrete for different confining 66 pressures Figure 3.6 Axial stress-strain relationship for 51.8 MPa concrete 68

Figure 3.7 Axial stress-strain relationship for 77.46 MPa concrete 68

Figure 3.8 Tresca and Von Mises yield condition (in π-and rendulic plane) 69

Figure 3.9 Mohr-Coulomb and Drucker-Prager yield condition 71 (in π-and rendulic plane)

Figure 3.10 Multi-directional fixed crack model 74

Figure 3.11 Secant crack stiffness 76

Figure 3.12 Linear tension softening 81

Figure 3.13 Multi-linear tension softening 84

Figure 3.14 Hordijk tension softening 85

Figure 3.15 Brittle tension softening 85

Figure 3.16 Exponential tension softening 86

Figure 3.17 Ideal tension softening 86

Figure 3.18 Predefined compression behavior for Total Strain model 88

Figure 3.19 Reinforcement bar 90

( xvii )

Figure 4.1 Regular Newton-Raphson iteration 96

Figure 4.2 Modified Newton-Raphson iteration 97

Figure 4.3 Linear Stiffness iteration 98

Figure 4.4 Convergence Norm 99

Figure 4.5 Load and displacement control 101

Figure 4.6 Arc-length control 103

Figure 4.7 Result based termination criteria 109

Figure 5.1 Details of a typical model slab with reinforcement. 111

Figure 5.2 Typical cracking pattern on the top surface of a model slab 113

Figure 5.3 Typical cracking pattern on the bottom surface of a model slab 113

Figure 5.4 Deflection at slab center of all slabs under different loading. 115

Figure 5.5 Comparison of test results with different code of prediction at 117 h=80mm and b=245mm.

Figure 5.6 Comparison of test results with different code of prediction 117

at h=60mm and b=245mm.

Figure 5.7 Comparison of test results with different codes at same slab 118 thickness of h=80mm.

Figure 5.8 Comparison of test results with different codes at same slab 118 thickness of h=60mm.

Figure 5.9 Details of Bresler and Scordelis Beam. 120

Figure 5.10 Details of Toronto Beam. 120

Figure 5.11 Details of Kotsovos Beam. 121

Figure 5.12 Details of Kuang and Morley model slab. 122

Figure 6.1 Meshed model of a typical slab showing 127 ( a ) top surface, ( b ) bottom surface.

Figure 6.2 Embedded reinforcement in a typical slab model. 128

Figure 6.3 Deflected shape and stress contour shown on 129 ( a ) top surface ( b ) bottom surface of typical slab model.

Figure 6.4 Compressive stress-strain on top surfaces of typical slab. 130

Figure 6.5 Tensile stress-strain on bottom surfaces of typical slab. 131

Figure 6.6 Tensile stress-strain diagram of typical reinforcement. 131

Figure 6.7 Deflection contour of bottom surface of a typical slab. 132

Figure 6.8 Load-deflection curves of analyzed and tested model having 133 slab thickness = 80mm and width of edge beam = 245mm.

( xviii )

Figure 6.9 Load-deflection curves of analyzed and tested model having 133 slab thickness = 60mm and width of edge beam = 245mm.

Figure 6.10 Load-deflection curves of analyzed and tested model having 134 width of edge beam = 175mm.


Figure 6.12 Load-deflection curves of analyzed and tested model having 135 no edge beam.

Figure 6.13 Cracking pattern of a typical slab at bottom surface. 136

Figure 6.14 Cracking at bottom surface of SLAB6 showing 136 (a) experimental cracking pattern, (b) analytical cracking pattern.

Figure 6.15 Cracking at bottom surface of SLAB9 showing 137

(a) experimental cracking pattern, (b) analytical cracking pattern.





Figure 6.18 Load-deflection curves of analyzed and tested model having 139

slab thickness = 80mm and width of edge beam = 245mm.

Figure 6.19 Load-deflection curves of analyzed and tested model having 139 slab thickness = 60mm and width of edge beam = 245mm


Figure 6.21 Load-deflection curves of analyzed and tested model having 140 width of edge beam = 105mm .

Figure 6.22 Load-deflection curves of analyzed and tested model having 141 no edge beam.

Figure 6.23 Comparison of test results with different codes at same 142 slab thickness of h=80mm.

Figure 6.24 Comparison of test results with different codes at same 143 slab thickness of h=60mm.

( xix )

Figure 6.25 Normalized Punching Shear of Slab model having 144 slab thickness = 80mm and width of edge beam = 245mm (similar to SLAB-2).

Figure 6.26 Normalized Punching Shear of Slab model having 144

slab thickness = 60mm and width of edge beam = 245mm (similar to SLAB-6).





Figure 6.29 ( a ) Meshed Model (b) Deformed Shape of Bresler and 147 Scordelis Beam. Figure 6.30 Load-Deflection curve of Bresler and Scordelis Beam. 147

Figure 6.31 ( a ) Meshed Model (b) Deformed Shape of Toronto Beam. 148

Figure 6.32 Load-Deflection curve of Toronto Beam. 148

Figure 6.33 ( a ) Meshed Model (b) Deformed Shape of Kotsovos Beam. 149

Figure 6.34 Load-deflection curve of Kotsovos Beam. 149

Figure 6.35 Meshed model of Kuang and Morley model slab (top surface). 150

Figure 6.36 Meshed model of Kuang and Morley model slab (top surface). 150

Figure 6.37 Load-deflection curves of analyzed and tested model by 151 Kuang and Morley.

Figure 6.38 Meshed model of a slab having smaller size of mesh 152

(same model as shown in Figure 6.1). Figure 6.39 Load-deflection behaviors for various size of mesh 153

Figure 7.1 Perspective view of a typical building with flat Plate. 159

Figure 7.2 Typical geometry of multi panel model slab 160

Figure 7.3 Typical embedded reinforcement in the multi panel 162 model at central column.

Figure 7.4 Model slab after meshing. 163

Figure 7.5 Enlarged corner of meshed model. 163

Figure 7.6 Location of nodes points from central column along 165 center line of model MSLAB11-7.

Figure 7.7 Load-deflection curves of various nodes of model slab 165

MSLAB11-7 for 'cf =30 MPa and 0.50% flexural reinforcement.

( xx )

Figure 7.8 Deformed shape of a typical slab MSLAB11-7 before 166

failure load. Figure 7.9 Typical crack pattern at the bottom surface of slab 166

MSLAB11-7 before failure. Figure 7.10 Location of elements from central column along center 167

line of model MSLAB11-7. Figure 7.11 Stress-Strain curves for various element adjacent to central 168

column of model slab MSLAB11-7.

Figure 8.1 Load-deflection of slab MSLAB11 for 'cf =24 MPa 171

at a distance 320mm from the edge of the central column. Figure 8.2 Load-deflection of slab MSLAB12 for '

cf =40 MPa 172 at a distance 320mm from the edge of the central column.

Figure 8.3 Load-deflection of slab MSLAB13 for 'cf = 50 MPa 172

at a distance 288mm from the edge of the central column.

Figure 8.4 Load-deflection of slab MSLAB21 for 'cf = 30 MPa 173

at a distance 150mm from the edge of the central column. Figure 8.5 Load-deflection of slab MSLAB22 for '

cf = 24 MPa 173 at a distance 300mm from the edge of the central column.

Figure 8.6 Load-deflection of slab MSLAB23 for '








Figure 8.10 Load-deflection behaviors for ideal tension softening and 176

linear tension softening of model slab MSLAB11. Figure 8.11 Load-deflection behaviors for ideal tension softening 177

and linear tension softening of model slab MSLAB11, MSLAB12 and MSLAB13.

( xxi )

Figure 8.12 Load-deflection behaviors for ideal tension softening 178 and linear tension softening of model slab MSLAB21, MSLAB22 and MSLAB23.

Figure 8.13 Load-deflection behaviors for ideal tension softening 179

and linear tension softening of model slab MSLAB31, MSLAB32 and MSLAB33.

Figure 8.14 Ultimate punching failure loads of MSLAB11 for different '

cf . 181

Figure 8.15 Ultimate punching failure loads of MSLAB11 for 182 different flexural reinforcement ratio.

Figure 8.16 Ultimate punching failure loads of MSLAB12 for different '

cf . 182 Figure 8.17 Ultimate punching failure loads of MSLAB12 for 183

different flexural reinforcement ratio. Figure 8.18 Ultimate punching failure loads of MSLAB13 for different '












different flexural reinforcement ratio.

( xxii )

Figure 8.30 Ultimate punching failure loads of MSLAB33 for different 'cf . 189

Figure 8.31 Ultimate punching failure loads of MSLAB33 for 190

different flexural reinforcement ratio. Figure 8.32 Location of integration point for steel. 191 Figure 8.33 Stress-strain of Point-1 and Point-3 for steel. 192 Figure 8.34 Stress-strain of Point-2 for steel. 192 Figure 8.35 Punching shear stresses of MSLAB11 at various 193

compressive strength of concrete. Figure 8.36 Punching shear stresses of MSLAB12 at various 194








compressive strength of concrete. Figure 8.44 Non-dimensional stresses due to punching force 198

of MSLAB11 for various compressive strength of concrete. Figure 8.45 Non-dimensional stresses due to punching force 199



of MSLAB21 for various compressive strength of concrete.

( xxiii )

Figure 8.48 Non-dimensional stresses due to punching force 200 of MSLAB22 for various compressive strength of concrete.

Figure 8.49 Non-dimensional stresses due to punching force 201




of MSLAB33 for various compressive strength of concrete. Figure 8.53 Normalized punching shear strength at various compressive 204

strength of concrete of 200mm thick slab (400mm x 400mm column).

Figure 8.54 Normalized punching shear strength at various compressive 205
















( xxiv )


strength of concrete considering 400mm x 400mm column. Figure 8.63 Normalized punching shear strength at various compressive 213

strength of concrete considering 600mm x 600mm column. Figure 8.64 Normalized punching shear strength at various compressive 213

strength of concrete considering 800mm x 800mm column. Figure 8.65 Normalized punching shear of 200mm thick at various 215

reinforcement ratio (400mm x 400mm column). Figure 8.66 Normalized punching shear of 250mm thick at various 216





reinforcement ratio (600mm x 600mm column). Figure 8.71 Normalized punching shear of 200mm thick slab at various 220



reinforcement ratio (800mm x 800mm column). Figure 8.74 Average normalized punching shear strength at various 222

flexural reinforcement ratio for 'cf =24 MPa.

Figure 8.75 Average normalized punching shear strength at various 222






( xxv )



Figure 8.79 Normalized punching shear of model slabs having 0.5% flexural 227

reinforcement ratio (400mm x 400mm column). Figure 8.80 Normalized punching shear of model slabs having 1% flexural 227

reinforcement ratio (400mm x 400mm column). Figure 8.81 Normalized punching shear of model slabs having 1.5% flexural 228











reinforcement ratio (800mm x 800mm column). Figure 8.92 Average normalized punching shear of model slabs having 235

0.25% flexural reinforcement ratio. Figure 8.93 Average normalized punching shear of model slabs having 236

0.5% flexural reinforcement ratio.

( xxvi )

Figure 8.94 Average normalized punching shear of model slabs having 236 1% flexural reinforcement ratio.

Figure 8.95 Average normalized punching shear of model slabs having 237


2% flexural reinforcement ratio.

Figure 8.97 Normalized punching shear of model slabs having 0.5% flexural 240 reinforcement ratio (200mm thick slab).

Figure 8.98 Normalized punching shear of model slabs having 1% flexural 240

reinforcement ratio (200mm thick slab). Figure 8.99 Normalized punching shear of model slabs having 1.5% flexural 241

reinforcement ratio(200mm thick slab).

Figure 8.100 Normalized punching shear of model slabs having 2% flexural 241 reinforcement ratio (200mm thick slab).

Figure 8.101 Normalized punching shear of model slabs having 0.5% flexural 242 reinforcement ratio (250mm thick slab).

Figure 8.102 Normalized punching shear of model slabs having 1% flexural 243


reinforcement ratio (250mm thick slab). Figure 8.104 Normalized punching shear of model slabs having 2% flexural 244





reinforcement ratio (300mm thick slab). Figure 8.109 Average normalized punching shear of model slabs having 247

0.5% flexural reinforcement ratio.

( xxvii )

Figure 8.110 Average normalized punching shear of model slabs having 248 1% flexural reinforcement ratio.

Figure 8.111 Average normalized punching shear of model slabs having 248


2% flexural reinforcement ratio. Figure 9.1 Average normalized punching shear of all model slab for 253

variable compressive strength of concrete.

Figure 9.2 Average normalized punching shear of all model slab for variable 254 flexural reinforcement.

Figure 9.3 Application of proposed formula for variable strength of 256

concrete of ( a ) 0.25% , ( b ) 0.5% , ( c ) 1%, ( d ) 1.5% and ( e ) 2% flexural steel (400mm x 400mm column and 200mm thick slab).





Figure 9.6 Application of proposed formula for variable flexural 260

reinforcement of ( a ) 24 MPa, ( b ) 30 MPa, ( c ) 40 MPa, ( d ) 50 MPa and ( e ) 60 MPa (400mm x 400mm column and 200mm thick slab).





Figure 9.9 Punching shear load of obtained from FE analysis, calculated 263

by the proposed formula and various code of 200mm thick model slab having 400mm x 400mm column for ( a ) f’c = 24 MPa, ( b ) f’c = 30 MPa

( xxviii )

Figure 9.10 Punching shear load of obtained from FE analysis, calculated 264 by the proposed formula and various code of 250mm thick model slab having 400mm x 400mm column for

( a ) f’c = 24 MPa, ( b ) f’c = 30 MPa Figure 9.11 Punching shear load of obtained from FE analysis, calculated 265 by the proposed formula and various code of 300mm

thick model slab having 400mm x 400mm column for ( a ) f’c = 24 MPa, ( b ) f’c = 30 MPa Figure 9.12 Punching shear load of obtained from FE analysis, calculated 266

by the proposed formula and various code of 200mm thick model slab having 600mm x 600mm column for ( a ) f’c = 24 MPa, ( b ) f’c = 30 MPa

Figure 9.13 Punching shear load of obtained from FE analysis, calculated 267 by the proposed formula and various code of 250mm thick model slab having 600mm x 600mm column for ( a ) f’c = 24 MPa, ( b ) f’c = 30 MPa Figure 9.14 Punching shear load of obtained from FE analysis, calculated 268

by the proposed formula and various code of 300mm thick model slab having 600mm x 600mm column for

( a ) f’c = 24 MPa, ( b ) f’c = 30 MPa Figure 9.15 Punching shear load of obtained from FE analysis, calculated 269

by the proposed formula and various code of 200mm thick model slab having 800mm x 800mm column for ( a ) f’c = 24 MPa, ( b ) f’c = 30 MPa Figure 9.16 Punching shear load of obtained from FE analysis, calculated 270

by the proposed formula and various code of 250mm thick model slab having 800mm x 800mm column for ( a ) f’c = 24 MPa, ( b ) f’c = 30 MPa Figure 9.17 Punching shear load of obtained from FE analysis, calculated 271

by the proposed formula and various code of 300mm thick model slab having 800mm x 800mm column for

( a ) f’c = 24 MPa, ( b ) f’c = 30 MPa

( xxix )

LIST OF TABLES

Page No.

Table 5.1 Details of Reinforced Concrete Slab Specimens 111

Table 5.2 Test results, Non-dimensional and Normalized Punching 114 Shear Strength of Reinforced Concrete Slabs Table 8.1 Geometry of model slab 170

Table 8.2 Details of model slab MSLAB11 170

Table 8.3 Comparison of failure load using ideal and linear tension 180 softening.

Table 9.1 Average normalized punching shear capacity of all slabs 252

Table 9.2 Comparison of proposed formula with experimental result 273 of Alam (1997)

Table 9.3 Comparison of proposed formula with experimental result 273 of Kuang and Morley (1992)

( xxx )

NOTATIONS

B = Strain–displacement relation matrix

b = Width of edge beam in millimeter (mm)

ob = Perimeter at a distance 2d from the edge of column in millimeter.

c = Length or width of column or loaded area in millimeter

D = Stress–strain relation matrix

d = Effective depth (Distance from extreme compression fiber to centroid of

longitudinal tension reinforcements) in millimeter

dav = Average Effective depth ( Distance from extreme compression fiber to

centroid of longitudinal tension reinforcement ) in millimeter

E = Young’s modulus

Ec = Modulus of Elasticity of Concrete in MPa f = Vector of the nodal forces

'cf = Cylinder compressive strength of concrete at 28 days in MPa.

cuf = Uniaxial cube (compressive) strength of concrete in MPa

tf = Tensile Stress of concrete in MPa

yf = Yield Stress of Steel in MPa

G = Elastic shear modulus

Gf = Fracture Energy

g = Vector of the known body forces per unit volume

ge = Element body force per unit volume

h = Total thickness of slab in millimeter

J = Jacobian matrix

K = Stiffness matrix of the element

eK~ = Element local Cartesian coordinate system.

L = Differential operator defining a compatible strain field

N = Displacement interpolation matrix

N ′ = The shape functions

ne = The total number of elements

Pu = Ultimate load in Newton (N)

( xxxi )

r = Vector of the internal forces

St = Element surface, edge

T = Slab thickness in millimeter

Te = Element transformation matrix

TԐ = Strain transformation matrix

t = Traction forces vector on the boundary

te = Element tractions per unit area

u = Vector of the unknown nodal degrees of freedom

ue = Element nodal displacement vector

V = Total volume or domain of the model.

Vc = Punching shear strength provided by concrete in Newton (N)

Vn = Nominal Punching Shear Capacity in Newton (N)

Vp = Punching shear strength provided by concrete in Newton (N)

x = The approximation of any x

αs = 40 for interior column, 30 for edge column, 20 for corner column

β = Shear retention factor

βc = Ratio of long side to short side of concentrated load or reaction area

ρ = Flexural reinforcement ratio.

ρav = Average Reinforcement ratio in percentage

ε = Total Strain (mm/mm)

εcr = Crack Strain (mm/mm)

εe = Elastic Strain (mm/mm)

εcr,nn = Ultimate Crack Strain (mm/mm)

εp = Plastic Strain (mm/mm)

Ԑ0 = Initial strains vector

Ԑl = Local strain vector

σ = Total Stress in MPa

σ1, σ2, σ3 = Principal Stress in MPa

σ0 = Initial residual stresses vector

κ = hardening parameter

τ = Shearing stress in millimeter

( xxxii )

φ = Angle of internal friction

ϕ0 = The initial angle of internal friction

ρψ

776.7

3 '

+= cf

, for 'cf = 21 MPa to 48 MPa

)3.71(47.0 ρψ −= , for 'cf = above 48 MPa

υ = Poisson’s ratio

ξ, η and ζ = The intrinsic co-ordinates of any point within the element.

1

CHAPTER-1

INTRODUCTION

1.1 GENERAL

Because of architectural requirements, simplicity of construction and ease of use, RC

flat plate floors are popularly used in much low and medium height residential,

industrial and commercial structures. For the structural design of flat slab and

footings, punching shear is considered as one of the most critical parameters. This

punching shear failure is normally a brittle type failure of structural concrete which

occurs when shear stress around a slab column connection exceeds the shear capacity

of the slab, resulting in the column and the part of the slab punching through the slab

(Tan and Teng 2005). For the design of flat plates, structural concrete has to be

treated with caution, as they are susceptible to failure by punching shear of concrete

(Gardner 2005).

Such brittle type of punching shear failure, unlike flexural failure, occurs due to

crushing of concrete near to the support. The failure surface is similar to that of a

truncated cone or a pyramid around the column. The failure surface extends from the

bottom of the slab, at the support, diagonally upward to the top surface. The angle of

inclination with the horizontal face depends on the slab thickness and amount of

reinforcement in it. As punching shear failure is likely to occur suddenly with no

advance warning of distress, special attention should be given to those design

parameters which influence both punching shear strength and ductility of structural

members.

According to present state of research, punching shear capacity is related to concrete

strength, flexural reinforcement ratio, slab thickness, column sizes, edge restraint,

compressive membrane action of slab etc. Accounting the failure criterion and load-

rotation relationship of the slab, the punching shear strength of a flat plate may also

depend on the span of the slab, rather than on its thickness as proposed by some

researchers (Muttoni 2008). Mitchell et al. (2005) concluded that it is not clear

whether the punching strength is proportional to the square or cube root of the

2

concrete strength and that additional research is needed to enable the development of

design expressions for punching shear that are applicable to a wide range of concrete

strengths, especially high strength concrete. Flexural reinforcements embedded in the

flat plate play a significant role on punching shear capacity of slabs and a distinct

decrease in punching shear resistance with decreasing reinforcement ratio has been

observed by Dilger et al. (2005). The development of membrane (in-plane) forces

within slabs, under an applied load, significantly influence the slabs load carrying

capacity above the commonly adopted design value based solely on flexural behavior

(Bailey et al. 2008, Foster et al. 2004).

For the design of punching shear of flat plates, some of the present-day code

provisions usually specify the punching shear strength as a function of concrete

strength alone. For the design of punching shear, these code provisions rely mostly

on empirical methods derived from the test results on simply supported conventional

and thin slab specimens (Kuang and Morley 1992, Alam et al. 2009). Design codes

such as the American code (ACI 318-2011), Canadian Standard (CSA-A23.3-04

(R2010)) and Australian code (AS 3600-2009) do not reflect the influence of the

flexural reinforcement ratio on the punching capacity of slab-column connections.

Others codes like the British (BS 8110-1997) and European (Eurocode 2- 2003,

CEB-FIP-1990) consider the effect of flexural reinforcement on the punching shear

capacity of slabs. Some codes do not to take adequate account of the possible role of

specimen size and slab thickness (Lovrovich and McLean 1990; Mitchell et al.

2005).

In continuous slab, all panel edges cannot rotate freely, in contrast to its simply

supported counterpart. Investigations from multi-panel slabs will be more reasonable

than the results obtained by using isolated single span slab specimens. However,

multi-panel tests are time consuming, expensive and it is difficult to determine

experimentally the shear and moment applied to the individual slab-column

connections. An alternative to such expensive and difficult experimental procedure is

to perform the investigation by means of numerical finite element analysis.

3

Nonlinear finite analysis procedures are reliable and popular in recent years as

engineers attempt to more realistically model the behavior of structures subjected to

all types of loading. Computer simulation makes the accuracy for describing actual

behavior of structures, compare the behavior with laboratory experimenting methods,

prospects in the process of scientific research, and relation with experiment and

analysis methods. It is very important that before practical application finite element

analysis methods should be verified and validated comparing the analysis results

with reliable experiment data.

In this study, an advanced nonlinear finite element investigation of multi-panel flat

plate considering full scale with practical geometry has been carried out on the

behavior of punching shear characteristics of concrete slab in presence of flexural

reinforcement. At first stage, FE model has been used to simulate relevant

experiments carried out earlier (Alam et al. 2009). Good agreement has been

observed between numerical FE simulation and experiment, which establish the

validity of FE model. Later on, the same FE procedure has been used to analyze

multi-panel slab models and the results are presented in this study in an effort to

understand the actual punching shear behavior of slab systems.

1.2 LITERATURE REVIEW

A large number of investigators, on the basis of their experimental and analytical

studies on the punching shear behavior of slabs, have expressed their opinion against

the present punching shear provisions. They have shown that the code does not

usually provide an accurate prediction of the punching shear strength of reinforced

concrete slab for various end conditions, reinforcement ratio, span-to-depth ratio, etc.

Rankin and Long (1987) from their experiment recognized the importance of the

flexure and shear modes of punching failure to produce more consistent and

economic design procedure. They have drawn out that the punching strength of full

panel specimen is significantly greater than that of equivalent conventional slab, for

the effect of compressive membrane action in the full panel specimen gives

4

significantly better correlation with test result than present code method and other

procedures.

Punching shear tests of geometrically similar reinforced concrete slabs of different

sizes have been carried out by Bazant and Coa (1987). They have summarized that

the punching shear failure of slab without stirrup is not plastic but brittle. They have

found that larger the slab thickness, steeper the post-peak decline of the load

deflection diagram, thus the punching shear behavior of thin slab is closer to

plasticity and that of thick slab is closer to linear elastic fracture mechanics.

Regan and Jorabi (1988) have shown that analysis using current code provisions and

making separate calculations of full width shear strength and punching shear are

inappropriate. They proposed that design checks should be based on nominal shear

stresses obtained as the sum of stresses arising from two components of load bearing

action. The first is a symmetrical spreading of concentrated load and the second is

the spanning of the slab carrying the spread load between supports.

Gardner (1990) presents the result of an investigation relating punching shear to

concrete strength and steel ratio. He concluded that the shear capacity is proportional

to the cube root of concrete strength and steel ratio and that the ACI 318 (1983) and

CSA A23.3-M84 (1984) provision should be reviewed. He also opined that the shear

perimeter should be increased by using large columns and column capitals, if the

punching shear capacity is in doubt.

Results of an experimental investigation on the punching shear strength of reinforced

concrete slabs with varying span-to-depth ratio have been summarized by Lovrovich

and McLean (1990). They have reported that the ACI Building code does not

recognize span-to-depth ratio effects or the effects of restraining action at the support

when treating punching shear in reinforced concrete slabs. They observed that

punching shear strength were much greater than the values permitted by the ACI

Building code. This was especially true for those specimens with smaller span-to-

depth ratios. The higher strengths were a result of smaller span-to-depth ratios, in-

5

plane compressive forces caused by restraining action at the support and excellent

anchorage provided for the shear reinforcement.

McLean et al. (1990) concluded from their experimental work that punching shear

strengths are much higher than those predicted by ACI 318 (1983). They cited some

reasons for such higher strengths. The authors stated that the ACI Building code

allows only half of concrete contribution to the punching shear strength in a slab with

shear reinforcement than it allows in a slab without shear reinforcement. From the

test results, they have shown that the strength provided by the concrete is the same in

the specimens with and without shear reinforcement. The researchers stated that the

code recognizes only the shear reinforcement activated by the assumed 450 degree

failure surfaces. In the test specimens, the cracks were generally much flatter than

450 degree, thus activating substantially more shear reinforcement than is recognized

by the code. The study revealed that the relatively small span-to-thickness ratios of

the specimens resulted in different internal cracking patterns than those of thinner

slabs. This cracking was indicative of development of internal compression struts

similar to those observed in deep beams. They also argued with the code specified

upper limits on the punching shear strength in a slab, regardless of the amount of

shear reinforcement provided.

Kuang and Morley (1992) tested 12 restrained reinforced concrete slabs with

varying span-to-depth ratio, percentage of reinforcement, degree of edge restraint

and reported that the punching shear strengths are much higher than those predicted

by ACI 318 (1989) and BS 8110 (1985). They opined that no code specified method

predicts an enhancement in punching shear strength of restrained concrete slabs with

an increase in the degree edge restraint. In reality, they have suggested that there is a

definite enhancement in punching shear strength as the degree of edge restraint

increases. The codes do not give accurate predictions of the punching shear capacity

of restraint slab, and in view of the magnitude of the strength enhancement, the

authors have opined that it would evidently be beneficial if the effect of compressive

membrane action could be allowed for in the design codes.

6

Yamada et al. (1992) performed a research program for the determination of the

effect of shear reinforcement type and ratios on punching shear strength of

monolithic slab to column connections. Their experimental study showed that ACI

318-89 (1989) provisions for the computation of shear strength considering the

reinforcement contribution are justifiably conservative at low reinforcement ratios

(upto approximately 0.6 percent).They have also shown that the hat-shaped

reinforcement (this type of reinforcement did not conform to the requirements of

ACI 318-89) was not effective because of lack of proper anchorage and large

spacing. Double hooked reinforcement showed high effectiveness, which resulted in

a considerable increment of the punching shear resistance of the connections.

Loo and Guan (1997) presented in their research paper that nonlinear-layered finite

element method is capable of analyzing cracking and punching shear failure of

reinforced concrete flat plates with spandrel beams or torsion strips. Incorporating

the layered approach with transverse shear capacities, the procedure takes into

account the full interaction between the spandrel beam and adjoining slab. They

observed good correlation for punching shear strength, the collapse loads, load

deflection responses and crack patterns.

A model for predicting punching shear failures at interior slab-column connections

was developed by Hueste and Wight (1999) based on experimental results.

Experimental studies performed by researchers at various universities provided the

data used as the bases for the punching shear failure prediction model. Much of this

data was summarized by Pan and Moehle (1989), Megally and Ghali (1994), and Luo

and Durrani (1995). Their model had been incorporated into a new RC slab element

for the nonlinear analysis program, DRAIN-2DM, along with the desired unloading

behavior when a punch occurs. A four-story RC frame office building that

experienced punching shear failures during the Northridge earthquake was evaluated

using this new model and the occurrence of punching shear failures was successfully

post calculated for the ground motion recorded nearest the structure. The study

building was evaluated for three ground motions scaled to the same peak ground

acceleration. The building response varied for each record, but in general, it was

7

found that the inclusion of punching shear failures can modify the overall building

response in terms of drift, fundamental period, inelastic activity, and base shear

distribution. In the case of the study building, the presence of the stiffer moment-

resisting perimeter frames helped limit the magnitude of the effect that the punching

shear failures had on the overall structural response.

Mansur et al. (2000) represented an experimental study on a total of 14 restrained

ferrocement slabs. Test results revealed that the provision of end restraint leads to a

substantial enhancement in strength and stiffness of slabs, but the shape and location

of the critical punching shear perimeter remained unchanged. Both cracking and

punching shear loads increased with an individual increase in any of the test

parameters considered the thickness of the edge beam.

Xiao and Flaherty (2000) presented both experimental behavior and finite-element

analyses of reinforced concrete slab-column connections and concluded that the

numerical investigations provided good agreement between the predicted and the

measured test results of the ultimate load and associated deflection. They opined that

the single layered slab model provides the closest results compared with the

experimental tests.

Menetrey (2002) produced a synthesis of punching failure in reinforced concrete.

First, some recent experimental results are presented allowing one to show the

difference between flexural and punching failure. Second, the punching failure

mechanism is discussed based on results obtained with numerical simulations

demonstrating among others the influence of the concrete tensile strength. Then,

using these results, an analytical model is derived for punching load prediction. The

model allows a unified treatment of slabs with various types of reinforcement. He

concluded that the predicted failure load is successfully compared with the

experimental results available in database and also some special punching

experiments.

8

Hailgren and Bjerke (2002) investigated punching tests on two circular column

footings of reinforced concrete were simulated numerically. Their results show how

the failure mechanism differs from that of more slender slabs. A parametric study

also confirms that the punching shear strength of the analyzed slabs strongly depends

on the compressive strength of concrete. The results from their investigation could

preferably be used for the development of new design methods, or further

development of current design methods, in order to make them valid for slabs with

low shear-span to depth ratios also. Hereby, the new or further developed design

methods should be based on mechanical models rather than on empirically derived

formulas.

Hallgren (2002) concluded that current design methods and code formulas for the

assessment of the punching shear strength are normally based on tests on slabs with

relatively high slenderness, i.e., with high shear-span to depth ratios. Column

footings normally have low shear-span to depth ratios. Previous punching tests on

column footings indicate that the failure mechanism for punching of slabs with low

shear-span to depth ratios differs from that of slabs with high shear-span to depth

ratios. In this investigation, punching tests on two circular column footings of

reinforced concrete were simulated numerically. The results show how the failure

mechanism differs from that of more slender slabs. A parametric study also confirms

that the punching shear strength of the analyzed slabs strongly depends on the

compressive strength of concrete.

Salim and Sebastian (2003) presented the test results of an experimental study of the

ultimate punching load capacity of reinforced concrete slabs that are restrained by

means of incorporating hoop reinforcement. An upper bound-plastic solution for

predicting shear of laterally restrained slabs has been presented in which concrete is

assumed to be rigid plastic, with yielding controlled by a parabolic Mohr failure

criterion. They proposed a method that allows for the effect of compressive

membrane action and a membrane modified flexural theory of elasto-plasticity

developed by others is used to calculate the compressive membrane forces.

9

Islam (2004) found from his study that both span-to-depth ratio effects and the type

of support condition have significant influence on punching shear in concrete slab.

The author proposed to include the effect of steel percentage, support condition,

concrete strength and slab dimension in ACI-318 code.

Dilger et al. (2005) compared more than one thousand experiments on isolated

interior slab-column connections. The authors show a distinct decrease in punching

shear resistance with decreasing reinforcement ratio. It is therefore recommended

that the influence of the flexural reinforcement ratio should be added to the codes

based on ACI 318. The yield strength of the flexural reinforcement does not have to

be included in this formulation. The reinforcement ratio should be calculated in the

region where the punching cone occurs. A definition of the extent of this region

needs to be developed. They also delivered their comments that the influence of slab

thickness on the punching capacity is significant, but research on slabs with varying

thickness is very limited. There is no widely accepted formula to account for the slab

thickness in the punching load calculations.

Mitchell et al. (2005) concluded from comparison of test results and code

expressions that, as the effective depth increases the shear stress at punching failure

decreases. This size effect is significant and is an important feature to include in the

code design expressions. They commented that rectangularity of a column increases

the shear stress at punching failure. This geometric feature, expressed as the aspect

ratio of the long side to short side of a column, is an important feature to include in

all code design expressions.

Bailey et al. (2008), in their paper presented a comparison between a simple

analytical approach, based on rigid-plastic behavior with change of geometry, an

advanced finite element model (FEM), and fourteen tests on horizontally

unrestrained concrete slabs, which reached vertical displacements up to 10 times the

effective depth of the slab. Both analytical approaches predicted the membrane

behavior of the slabs, comprising compressive membrane action around the slab’s

perimeter and tensile membrane action in the central span region of the slab. The

10

simple approach produced good predictions of the load-displacement response

towards the end of the test, whereas the FEM produced reasonable predictions over

the full history of the test. They opined, the comparison of the simple approach

against the FEM and test results shows that it can safely be used in design for

predicting the load carrying capacity, due to membrane action, of concrete slabs

under large displacements.

Muttani (2008) presented a mechanical explanation of the phenomenon of punching

shear in slabs without transverse reinforcement on the basis of the opening of a

critical shear crack. It leads to the formulation of a new failure criterion for punching

shear based on the rotation of a slab. This criterion correctly describes punching

shear failures observed in experimental testing, even in slabs with low reinforcement

ratios. Its application requires the knowledge of the load-rotation relationship of the

slab, for which a simple mechanical model is proposed. The resulting approach is

shown to give better results than current design codes, with a very low coefficient of

variation (COV). Parametric studies demonstrate that it correctly predicts several

aspects of punching shear previously observed in testing as size effect (decreasing

nominal shear strength with increasing size of the member). Accounting for the

proposed failure criterion and load-rotation relationship of the slab, the punching

shear strength of a flat slab is shown to depend on the span of the slab, rather than on

its thickness as often proposed.

Guandalini et al. (2009) presented results of a test series on the punching behavior of

slabs with varying flexural reinforcement ratios and without transverse reinforcement

are presented. The aim of the tests was to investigate the behavior of slabs failing in

punching shear with low reinforcement ratios. The size of the specimens and of the

aggregate was also varied to investigate its effect on punching shear. Measurements

at the concrete surface as well as through the thickness of the specimens allowed the

observation of phenomena related to the development of the internal critical shear

crack prior to punching. The results are compared with design codes and to the

critical shear crack theory. From that comparison, it is shown that the formulation of

ACI 318-08 (2008) can lead to less conservative estimates of the punching strength

for thick slabs and for lower reinforcement ratios than in the test results. Satisfactory

11

results are, on the other hand, obtained using Eurocode 2 and the critical shear crack

theory.

A total of 17 reinforced concrete footings were tested to investigate the punching

shear behavior of footings by Hegger et al. (2009). The test parameters investigated

are the shear span-depth ratio (a/d), concrete strength, and punching shear

reinforcement. The (a/d) ranged between 1.25 and 2.0, whereas the concrete strength

ranged between 20 and 40 MPa (2.9 and 5.8 ksi). To study the effect of soil-structure

interaction, five footings were realistically supported on sand. The remaining

specimens were supported on a column stub and a uniform surface load was applied.

The present experimental investigations indicated that the angle of the failure shear

crack is steeper in punching tests on compact footings than observed in tests on more

slender slabs. Furthermore, the (a/d) significantly affects the punching shear

capacity. Based on the test results, the ACI and Eurocode 2 provisions are critically

reviewed and improvements are proposed.

Ruiz et al. (2009) commented in their research paper that the traditional approach of

codes of practice for estimating the punching strength of shear-reinforced flat slabs is

based on the assumption that concrete carries a fraction of the applied load at

ultimate while the rest of the load is carried by the shear reinforcement. Concrete

contribution is usually estimated as a fraction of the punching strength of members

without shear reinforcement. The ratio between the concrete contribution for

members with and without shear reinforcement is usually assumed constant,

independent of the amount of shear reinforcement, flexural reinforcement ratio, and

bond conditions of the shear reinforcement. The limitations of such an approach are

discussed in this paper and a new theoretical model, based on the critical shear crack

theory, is presented to investigate the strength and ductility of shear-reinforced slabs.

The proposed approach is based on a physical model and overcomes most limitations

of current codes of practice

Yang et al. (2010) studied the punching shear behavior of slabs reinforced with high-

strength steel reinforcement and compared with that of slabs reinforced with

conventional steel reinforcement. The high-strength steel selected for this research

12

conforms to ASTM A1035-07. The influences of the flexural reinforcement ratio,

concentrating the reinforcement in the immediate column region, and using steel

fiber-reinforced concrete (SFRC) in the slab on the punching shear resistance, post-

cracking stiffness, strain distribution, and crack control were investigated. In

addition, the test results were compared with the predictions using various design

codes. The use of high-strength steel reinforcement and SFRC increased the

punching shear strength of slabs, and concentrating the top mat of flexural

reinforcement showed beneficial effects on post-cracking stiffness, strain

distribution, and crack control.

Gardner (2011) compared the punching shear provisions of ACI 318-08 (ACI

Committee 318 2008), BS 8110-97 (1997), DIN 1045-1 (2001), CEB-FIP MC90

(1993), EN1992-1-1 (2004), and Gardner (1996) for interior column slab connections

with and without moment transfer and edge and corner column slab connections with

published experimental data. The code equations cannot be directly compared due to

the different philosophies used in their derivations. Comparisons with experimental

data indicate that the equations of ACI 318-08 and Gardner (1996) for concentric

punching shear used the 5% probability value to determine the equation coefficients,

whereas CEB-FIP MC90, EN 1992-1-1, and DIN 1045-1 used the mean value

coefficients. ACI 318-08 and Gardner (1996) have satisfactory equation safety

indexes (3.40 and 3.14, respectively), whereas DIN 1045-1 and EN 1992-1-1 are

marginally less satisfactory at 2.8 and 3.0, respectively. Expected equation

coefficients derived using mean measured concrete strength—not code equation

coefficients—should be used to compare a prediction equation to a single result or

group of experimental results. The ACI 318-08 elastic eccentric shear interaction

method and the CEB-FIP MC90/EN 1992-1-1 plastic eccentric shear interaction

method are equally effective for interior column slab connections with moment

transfer and edge and corner column slab connections with the eccentricity towards

the slab. The BS 8110-97 equation is effective for interior column slab connections

with moment transfer.

Higashiyama et al. (2011) presented a design equation for the punching shear

capacity of steel fiber reinforced concrete (SFRC) slabs. The proposal is base on the

13

Japan Society of Civil Engineers (JSCE) standard specifications. Addition of steel

fibers into concrete improves mechanical behavior, ductility, and fatigue strength of

concrete. Previous studies have demonstrated the effectiveness of fiber reinforcement

in improving the shear behavior of reinforced concrete slabs. In this study, twelve

SFRC slabs using hooked-ends type steel fibers are tested with varying fiber dosage,

slab thickness, steel reinforcement ratio, and compressive strength. Furthermore, test

data conducted by earlier researchers are involved to verify the proposed design

equation. The proposed design equation addresses the fiber pull-out strength and the

critical shear perimeter changed by the fiber factor. Consequently, it is confirmed

that the proposed design equation can predict the punching shear capacity of SFRC

slabs with an applicable accuracy.

Mostafaei et al. (2011) have shown the use of externally post-tensioned fiber-

reinforced concrete decks in highway bridge structures is seen as available option in

the move toward the design and construction of high-performance structures.

However, with the thin unreinforced deck slabs that may result, punching shear is a

potential concern. An experimental program is described in which the punching

shear behavior of externally prestressed slabs is investigated, both with plain and

fiber-reinforced concrete specimens. Results indicate that significant improvements

in strength, ductility, energy absorption and non-brittleness of failure can be achieved

with fiber reinforcement. Nonlinear finite-element analysis procedures are used to

model the specimens, and reasonably accurate simulations of behavior are obtained.

Design code procedures are found to be unconservative in estimating the punching

shear strength of these elements, whereas a commonly used analytical model is found

to be overly conservative.

Rizk et al. (2011) commented that thick concrete plates are currently used for

offshore and nuclear containment concrete walls. In their research, five thick

concrete slabs with a total thickness of 300 to 400 mm (12 to 16 in.) were tested

under concentric punching loading. Four specimens had no shear reinforcement,

whereas the remaining one included T-headed shear reinforcement consisting of

vertical bars mechanically anchored at the top and bottom by welded anchor plates.

The main focus of this research was to investigate the influence of the size effect on

14

the punching shear strength of thick high-strength concrete plates. All tests without

shear reinforcement exhibited brittle shear failures. The addition of T-headed shear

reinforcement with a shear reinforcement ratio of approximately 0.68% by volume

changed the failure mode to ductile flexural failure. The test results revealed that

increasing the total thickness from 350 to 400 mm (14 to 16 in.) resulted in increased

punching capacity and at the same time resulted in a small increase in ductility

characteristics. An equation based on fracture mechanics principles is recommended

to account for the size effect factor. The proposed equation is verified using the test

results and is compared with the predictions of different design codes.

Trautwein et al. (2011) analyzed the punching strength of concrete flat slabs with

shear reinforcement that does not embrace flexural reinforcement. This paper also

reports the results of tests of slabs without shear reinforcement. Finally, they show

some comparisons of tests of similar slabs without shear reinforcement and slabs

with different types of shear reinforcement. The obtained results show that the use of

shear reinforcement elements without embracement in the flexural reinforcement

improves the punching strength of reinforced concrete flat slabs.

Choi and Kim (2012) tested three slab-column connections to investigate the moment

redistribution and punching shear resistance of flat plates under realistic loading and

boundary conditions. The test specimens were essentially identical except that they

had different reinforcement layouts within a span to impose different ratios of the

end span and mid span design moments to total static moment. The test results

showed that the different reinforcement layouts significantly and minutely influenced

the moment redistribution and the punching shear resistance, respectively. The

moment redistribution and punching shear resistance provisions in ACI 318 and EC2

were used to analyze the test results. New code recommendations for moment

redistribution limit and punching shear strength are proposed based on the novel

findings of this study.

The results of punching tests carried out at the Swiss Federal Institute of Technology

(ETH) in Zurich, Switzerland, on three full-scale reinforced concrete slab specimens

are presented and discussed by Heinzmann et al. (2012). Main focus is on the

15

punching failure modes obtained with different shear reinforcement arrangements

and their corresponding failure loads. One of the 350 mm (13.8 in.) thick specimens

had no shear reinforcement (Specimen SP1), whereas the other two were reinforced

with double-headed shear studs, which were placed locally around the column

(Specimen SP2) or over the entire slab (Specimen SP3). The three failure modes—

punching without shear reinforcement, punching outside the shear-reinforced zone,

and punching within the shear-reinforced zone due to concrete crushing were

observed. The experimental failure loads ranged from 43 to 85% of the computed

ultimate flexural load of the specimens. All three failure modes were correctly

predicted by the pertinent ACI 318-08 and EC2 Code provisions, as well as by the

critical shear crack theory (CSCT). Further, it is shown that the load-deflection

behavior of the slabs can be computed with a bending approach.

Lips et al. (2012) presented results of an extensive experimental campaign on 16 flat-

slab specimens with and without punching shear reinforcement. The tests aimed to

investigate the influence of a set of mechanical and geometrical parameters on the

punching shear strength and deformation capacity of flat slabs supported by interior

columns. All specimens had the same plan dimensions of 3.0 x 3.0 m (9.84 x 9.84 ft).

The investigated parameters were the column size (ranging between 130 and 520 mm

[approximately 5 and 20 in.]), the slab thickness (ranging between 250 and 400 mm

[approximately 10 and 16 in.]), the shear reinforcement system (studs and stirrups),

and the amount of punching shear reinforcement. Systematic measurements (such as

the load, the rotations of the slab, the vertical displacements, the change in slab

thickness, concrete strains, and strains in the shear reinforcement) allow for an

understanding of the behavior of the slab specimens, the activation of the shear

reinforcement, and the strains developed in the shear-critical region at failure.

Finally, the test results were investigated and compared with reference to design

codes (ACI 318-08 and EC2) and the mechanical model of the critical shear crack

theory (CSCT), obtaining a number of conclusions on their suitability.

Lontsoght et al. (2012) opined that current code provisions are based on shear tests

on heavily reinforced slender beams under point loads. The question remains if these

procedures are valid for wide beams and slabs under point loads close to the support.

16

To evaluate the shear capacity of reinforced concrete slabs and the associated

effective width, they executed a series of experiments is carried out on eight

continuous one-way slabs and twelve continuous slab strips loaded close to the

simple and continuous supports. Test results are compared to current code provisions

and methods to calculate the shear capacity from the literature. The influence of the

shear span to depth ratio, the size of the loading plate and the overall width of the

specimen are discussed. From these results follow that the behavior in shear of slabs

and beams is not identical. The effective slab width, used for calculating the beam

shear capacity, is recommended to be based on load spreading under 45° from the far

side of the loading plate towards the support.

Maya et al. (2012) concluded that the ultimate strength of reinforced concrete slabs is

frequently governed by the punching shear capacity, which may be increased with

addition of traditional fitments such as reinforcing steel, headed studs or shear heads.

In addition to these traditional methods of strengthening against punching, steel fibre

reinforcement has proved to be an effective and viable alternative. The addition of

fibres into the concrete improves not only the shear behavior but also the

deformation capacity of reinforced concrete slabs. This paper presents a mechanical

model for predicting the punching strength and behavior of concrete slabs reinforced

with steel fibres as well as conventional reinforcement. The proposed model is

validated against a wide number of available experimental data and its accuracy is

verified. On this basis, a simple design equation for the punching shear capacity of

steel fibre reinforced concrete (SFRC) slabs is proposed.

Hundreds of laboratory experiments have been conducted by Ospina et al. (2012) to

investigate the punching shear behavior of two-way reinforced concrete (RC) slabs at

interior supports. These experiments provide a mandatory frame of reference for the

development, calibration and evaluation of punching shear design provisions.

Unfortunately, because of the lack of dissemination and unavailability of some of the

references together with some level of arbitrariness by researchers and code

developers in selecting reference data, code provisions have been developed based

on a rather limited subset of the available test results. To overcome these limitations,

a task group was formed within ACI Committee 445 to gather, compile and post-

17

process the results from laboratory tests studying the concentric punching shear

behavior of two-way RC slabs without shear reinforcement at interior supports. The

development of the databank involved two stages: first, the creation of a "collected"

databank, where the characteristics of test specimens and test results were compiled

as faithfully as possible to what was reported by researchers. Secondly, the

development of a "selected" databank based on a series of Data Acceptance Criteria

(DAC), with the goal of endorsing a test result into an evaluation-level databank.

This paper describes the creation process and main features of the collected databank

and discusses several important aspects in databank development including the

selected platform being used to disseminate the information to users.

Peiris and Ghali (2012) presented a research paper on punching shear strength of

concrete. They concluded that ductility and the strength of flat plate connections with

their supporting columns are influenced by the concrete strength, the thickness of the

slab and the shear and the flexural reinforcements. The present paper concentrates on

the important effect of flexural reinforcement in the presence or the absence of shear

reinforcement.

Said et al. (2012) studied punching shear, both experimentally and analytically.

However, due to the number of parameters involved and the complexities in

modeling, current approaches used to estimate the punching shear capacity of

reinforced concrete (RC) slabs include mechanical models and design code

equations. Mechanical models are complex, while design code equations are

empirical. This study investigates the ability of artificial neural networks (ANN) to

predict the punching shear strength of concrete slabs. The parameters considered to

be the most significant in punching shear resistance of RC slabs were: concrete

strength, slab depth, shear span to depth ratio, column size to slab effective depth

ratio and flexure reinforcement ratio. Using a large and homogenous database from

existing experimental data reported in the literature, the ANN model is able to

predict the punching shear capacity of slabs more accurately than were the code

design equations.

18

Punching shear tests were conducted by Borges et al. (2013) on 13 reinforced

concrete flat plates with and without openings or/and shear reinforcement. The

openings (one or two) were adjacent to the shorter sides of rectangular supports and

had widths equal to those of the supports. The methods of calculating punching shear

strengths given in ACI 318-11 (2011) and MC90/EC2 (2003) are reviewed along

with some proposed formulations, and their predictions are compared with the test

results. Conclusions are drawn on the influence of plate depth on the unit shear

resistance from the concrete; the possibility of using straight projections of openings

onto control perimeters, rather than radial ones, to evaluate the effect of openings;

and the case for considering eccentricity in a pattern of openings as an influencing

factor and the detailing of shear reinforcement. It is shown that for the relatively

small openings considered, the provision of continuous bars adjacent to openings to

replace the areas of reinforcement seems to be an adequate approach to flexural

design. It is suggested that the shear stress used for the concrete in ACI 318-11

(2011) could be increased for the assessment of strengths at the edges of zones with

shear reinforcement (outer control perimeter).

Results from an experimental study aimed at investigating the behavior of full-scale

two-way flat slabs reinforced with glass fiber-reinforced polymer (GFRP) bars and

subjected to monotonically increase concentrated load is carried out by Dulude et al.

(2013). A total of 10 interior slab-column prototypes measuring 2.5 x 2.5 m (98 x 98

in.) were constructed and tested up to failure. The test parameters were: 1)

reinforcement type (GFRP and steel) and ratio (0.34 to 1.66%); 2) slab thickness

(200 and 350 mm [7.9 and 13.8 in.]); and 3) column dimensions (300 x 300 mm

[11.8 x 11.8 in.] and 450 x 450 mm [17.7 x 17.7 in.]). All test prototypes showed

punching shear failure and the crack patterns at failure were almost the same

regardless of reinforcement type or ratio. Besides, the GFRP-reinforced prototypes

showed lower punching capacity compared to that of the steel-reinforced ones when

the same reinforcement ratio was employed due to the lower modulus of GFRP bars

compared to steel. Predictions using different design guidelines were compared to

the experimental results obtained herein. The comparisons showed that the ACI

440.1R equation yielded very conservative predictions with an average Vtest/Vpred

equal to 2.10 ± 0.30.

19

Hawileh et al. (2013) developed a 3D nonlinear finite element (FE) model to predict

the punching shear of two-way reinforced concrete (RC) slabs. Six slabs casted with

regular and steel fiber reinforced concrete (SFRC) and reinforced with normal and

high strength steel reinforcement were modeled using ANSYS. The results were

validated with published experimental data. The computational model accounts for

the different nonlinear constitutive material laws by utilizing state-of-the-art

modeling methodologies. Concrete cracking and softening, effects of the embedded

steel fibers, steel yielding, and the bond-slip mechanism of the embedded steel

reinforcement were all taken into account. A good correlation was obtained between

the predicted and measured results at all stages of loading including failure of the

specimens. It is concluded that correlating models could be used as a reliable tool to

conduct parametric studies to evaluate the punching shear behavior of two-way slabs

cast with normal and SFRC concrete, reinforced with ordinary and high-strength

steel reinforcement.

Ruiz (2013) presented an extensive experimental campaign performed at the Ecole

Polytechnique Fédérale de Lausanne (EPFL) on the role of integrity reinforcement

by means of 20 slabs with dimensions of 1500 x 1500 x 125 mm (≈5 ft x 5 ft x 5 in.)

and various integrity reinforcement layouts. The performance and robustness of the

various solutions is investigated to obtain physical explanations and a consistent

design model for the load-carrying mechanisms and strength after punching failures.

Coronelli and Corti (2014) modeled for the nonlinear response of a flat slab

subjected to gravity and lateral cyclic loading. The model requires the definition of

the grid geometry and properties of point hinges in beam finite elements, and

modeling the nonlinear response in bending, torsion, and shear. The simulation is

carried out for experimental tests on a floor under gravity and lateral biaxial cyclic

loading of increasing amplitude. Pushover analyses have been performed under

gravity and horizontal loads in the two principal directions. Predictions are shown of

the global response and the connections of different column shapes and slab

reinforcement with the strength, drift capacity, and failure modes. The accuracy is

different in the two directions of loading due to the damage of the test slab for biaxial

20

cyclic loading. The results show the potential of the model for design and analysis of

existing flat slab structures.

Islam (2014) carried out a research program regarding to numerical modeling of

beam slab joint of flat plate. He obtained that the increase in ultimate load is more

prominent with lower slab thickness compared to higher slab thickness and it

decreases with increase of slab thickness. He also found that the increase in ultimate

load is more prominent with smaller size square column compared to larger size

square column. He opined that the support condition has a significant influence on

the punching shear strength of reinforced concrete flat plate.

The brief literature reviews shows that punching shear tests of the most researchers

are mainly based on simply supported and isolated slab specimens. The parameters

considered to be the most significant in punching shear resistance of reinforced

concrete slabs are concrete strength, slab depth, shear span to depth ratio, column

size to slab effective depth ratio and flexure reinforcement ratio etc. No code

specified method predicts an enhancement in punching shear strength of restrained

concrete slabs with an increase in the degree edge restraint. In reality, some

researchers have suggested that there is a definite enhancement in punching shear

strength as the degree of edge restraint increases. Ductility and the strength of flat

plate connections with their supporting columns are influenced by the concrete

strength, the thickness of the slab, the shear and the flexural reinforcements. The use

of high-strength steel reinforcement increases the punching shear strength of slabs,

and concentrating the top mat of flexural reinforcement showed beneficial effects on

post-cracking stiffness, strain distribution, and crack control. Some investigators

used three dimensional analyses as the aim was to understand flexural behavior on

punching shear behavior of RC flat slab.

1.3 CODE PROVISIONS FOR PUNCHING SHEAR

For the design of flat plates, flat slabs, bridge decks and column footings punching

shear strength of concrete in the vicinity of columns, concentrated loads or reactions

is one of the design criteria which governs the design. Thus, the critical shear section

for this type of shear should be located so as the perimeter of critical section is a

21

minimum, but need not approach closer than a certain distance from edge or corners

of columns, concentrated load or reaction areas. Different code provisions provide

the location of this critical section differently. But for all the codes, when this is

done, the shear strength is taken almost independent of the column size, slab depth,

span-to-depth ratio and edge restraint. It is to be noted that the nominal safety factor,

partial safety factors, reduction factors, etc. have been removed in the following code

equations.

1.3.1 American (ACI 318-11) code

According to ACI 318-11 code provision, the critical section for shear in slabs

subjected to bending in two directions follow the perimeter (b0) located at a distance

d/2 from the periphery of the concentrated load. According to this code, for non-

prestressed slabs and footing, nominal punching shear strength provided by concrete

shall be smallest of the following three equations,

Vc= (1 + 2 / βc) cf ′ b0d/6 ( 1.1a )

Vc= (1 + 0.5αsd / b0) cf ′ b0d/6 ( 1.1b )

Vc= 0.33 cf ′ b0d ( 1.1c )

Here,

βc= Ratio of long side to short side of concentrated load or reaction area.

αs = 40 for interior column, 30 for edge column, 20 for corner column

Vp= Punching shear strength provided by concrete in Newtons (N).

fc' = Uniaxial cylinder (compressive) strength of concrete in MPa

b0= Perimeter of critical section of slab or footing in millimeter (mm).

d = Effective depth (Distance from extreme compression fiber to centroid

of longitudinal tension reinforcement ) in millimeter (mm).

1.3.2 Australian (AS 3600-2009) code

According to AS 3600-2009 code provision, the critical section for shear in slabs

subjected to bending in two directions follow the perimeter (b0) located at a distance

0.5dav from the periphery of the concentrated load. According to this code, nominal

22

punching shear strength provided by concrete shall be calculated by the following

equation,

Vc= 0.34 cf ′ b0dav ( 1.2 )

Here,

Vc= Punching shear strength provided by concrete in Newtons (N).

fc' = Uniaxial cylinder (compressive) strength of concrete in MPa

b0 = Perimeter of critical section of slab or footing in millimeter (mm).

dav = Effective depth (Distance from extreme compression fiber to centroid


1.3.3 Bangladesh (BNBC, 2006) code

Provisions of punching shear strength of Bangladesh National Building Code

(BNBC, 2006) is akin to ACI 318-2011 and, thus, during the course of comparing

results with various code provisions reference will be made only to ACI 318-2011

code.

1.3.4 British (BS 8110-97) code

According to BS 8110-97 code the b0 is calculated at a distance of 1.5d from the

edge of column and the punching shear strength (Vp) of concrete is given by the

following equation,

Vp = 0.79 1003 ρ 3 /25cuf 4 /400 d [4(c + 3d)]d ( 1.3 )

Where,

ρ ≤ 3.0 percent, 400/d≥1.0 and fcu ≤ 40 MPa

Here,

Vp = Punching shear strength provided by concrete in Newtons (N).

fcu = Uniaxial cube (compressive) strength of concrete in MPa

c = Length or width or diameter of column or loaded area in millimeter.

d = Effective depth (Distance from extreme compression fiber to centroid


ρ = Reinforcement ratio.

23

1.3.5 Canadian (CAN3-A23.3-M84 (1984)) code

According to CAN3-A23.3-M84 (1984) Code, the critical section for punching shear

in slabs the perimeter (b0) located at a distance d/2 from the periphery of the

concentrated load. The punching shear strength provided by the concrete is given by

the following equation,

Vp = 0.4 cf ′ b0d (1.4)

Here,

Vp = Punching shear strength provided by concrete in Newtons (N).

cf ′ = Uniaxial cylinder (compressive) strength of concrete in MPa


d = Effective depth (Distance from extreme compression fiber to centroid of

longitudinal tension reinforcement) in millimeter (mm).

1.3.6 European Code (EC 2-1-1 (2003) and CEB-FIP Model Code 90)

According to CEB-FIP code, the critical section for punching shear follows the

perimeter (b0) located at a distance 2dav from the periphery of the concentrated load.

The punching shear strength is given by the following equation,

Vc = 0.18 [1+ av200/d ] [ 3 kav 100 cfρ ] b0dav ( 1.5 )

ρav is limited to a maximum value of 0.02.

Here,


fck= Characteristic cylinder strength of concrete in MPa





ρav = Reinforcement ratio.

24

1.3.7 German (DIN 1045-1: 2008) code

According to DIN 1045-1:2008 code the b0 is calculated at a distance of 1.5dav from

the edge of column and the punching shear strength of concrete is given by the

following equation,

Vc = 0.14 [1+ av200/d ] [ 3 kav 100 cfρ ] b0dav ( 1.6 )

Where b0 is the perimeter of the critical section located at a distance of 1.5dav from

the face of the column. The maximum value if ρav is the similar of 0.02 and 0.4 times

the ratio between the design concrete strength and the design steel strength.

Here,


fck= Characteristic cylinder strength of concrete in MPa





ρav = Reinforcement ratio.

1.4 MODELING GUIDELINE AND COMPLEXITIES

The response of a reinforced concrete structure is determined in part by the material

response of the plain concrete of which it is composed. Thus, analysis and prediction

of structural response to static or dynamic loading requires prediction of concrete

response to variable load histories. The fundamental characteristics of concrete

behavior are established through experimental testing of plain concrete specimens

subjected to specific, relatively simple load histories. Continuum mechanics provides

a framework for developing an analytical model that describes these fundamental

characteristics. Experimental data provide additional information for refinement and

calibration of the analytical model.

Concrete is a non-homogeneous composite and that the primary mechanism of

response is the development and propagation of discrete cracks, it is necessary to

consider the general framework of the model in establishing the experimental data

25

set. The response of plain concrete can be modeled at the scale of the coarse

aggregate with the model explicitly accounting for the response of the aggregate,

hydrated cement paste, the transition zone material as independent elements or as

components of a composite. However, while there may be available experimental

data that defines the response of aggregate and hydrated cement paste to general

loading, characterization of the transition zone must be accomplished indirectly.

Further, the random nature of the component material properties and distribution

adds complexity to models that are developed at this scale. In modeling the response

of a reinforced concrete structural element, it is reasonable to incorporate both the

microscopic response as well as the random nature of the concrete into a macro

model. The macro model describes the response of a body of concrete that is many

times the size of individual pieces of aggregate or of continuous zones of hydrated

cement paste. It is assumed that initially the concrete within the body is homogenous

and that the material response of the components is represented in the global

response of the concrete composite. For this investigation, plain concrete is idealized

as an initially homogenous material.

The idealization of concrete as a homogeneous body requires additional

consideration for the case of concrete subjected to moderate through severe loading.

At these load levels, the response of concrete is determined by the formation of

continuous crack systems. Some researchers have proposed models in which the

idealization of concrete as a continuum is abandoned in the vicinity of the crack, and

crack systems are modeled discretely. Development and calibration of such a model

requires experimental data defining the rate of crack propagation under variable

stress states and load histories. Currently, there are few data available characterizing

the concrete fracture process under multi-dimensional stress states. Additionally,

such a model requires special consideration within the framework of a finite element

program.

Other researchers have shown that it is possible to maintain the idealization of

concrete as a continuum in the presence of discrete cracks. In these models, the

material damage (evident in reduced material strength and stiffness) associated with

26

discrete cracking is distributed over a continuous volume of the material. Such

models include the fictitious-crack model, smeared-crack models and the crack-band

model. Modeling of concrete as a continuum results in a model that is compatible

with many existing computer codes as well as provides a basis for application of

existing continuum constitutive theory in developing models. For these reasons, in

this investigation concrete is considered to be a continuum.

Modeling concrete as an initially homogeneous material and assuming that the

discrete cracking is incorporated into a continuum model of concrete, it is necessary

that the experimental data set on which the analytical model will be developed and

calibrated be compiled from investigations that meet several criteria. The concrete

specimens must have critical zones that are sufficiently large that the concrete

composite in the vicinity is approximately a homogenous mixture. For load cases in

which the material response is determined by a global mechanism (e.g., micro

cracking) experimental measurement must define the deformation of the entire

concrete body to ensure that the deformation is representative of the composite. For

load cases in which the material response is determined by a local mechanism (e.g.,

formation of a continuous crack surface), it is necessary that experimental

measurement define the global deformation of the concrete body as well as the

deformation associated with the localized mechanism. This allows for appropriate

calibration of the continuum model.

The connections between the floor slab and column in a flat slab structure are

generally the most critical part as far as the strength is concerned because it is a

region where large moments and shear forces are concentrated. Despite an extensive

amount of experimental research work on shear strength of reinforced concrete slab,

there is still no single theory that can accurately predict the shear strength of a

reinforced concrete flat slab which consider the effect of concrete, flexural

reinforcement, column size and slab thickness simultaneously. With the advancement

in computing technology and numerical modeling of constitutive relationship of

reinforced concrete, many features have been implemented into the finite element

model to describe the behavior of reinforced concrete rationally.

27

Punching shear behavior is a 3-dimensional complex problem, in which tension,

compression, shear force, confinement action of concrete etc. are considerable factor

for numerical finite element analysis. Therefore, efficient use of the finite element

method for studying the behavior of reinforced concrete flat slabs and find out how

well it can predict the actual behavior is necessary to implement.

1.5 SCOPE AND OBJECTIVE

From the literature review and code provisions as stated earlier section, it is found

that present design rules for punching shear failure of reinforced concrete slabs,

given in various codes of practice, are largely based on studies of the behavior and

strength of simply-supported, conventional specimens extending to the nominal line

of contra-flexure. The code provisions rely mostly on empirical methods derived

from the test results on conventional and thin slab specimens. Test results from

simply supported slab specimens do not usually provide an accurate prediction of the

ultimate load capacity of a slab having lateral restraint. When the slab is restrained

against lateral deformation, this induces large restraining force within the slab and

between the supports, thus membrane forces are developed. The enhancement of

punching shear capacity can be attributed due to the presence of compressive

membrane action in the slab. The importance of compressive membrane stresses due

to edge restraint was not incorporated into the code formulations, which results in

conservative prediction.

Some of the present-day code provisions usually specify the punching shear strength

as a function of compressive strength of concrete alone. The parameters considered

to be the most significant in punching shear resistance of reinforced concrete slabs

are concrete strength, slab depth, shear span to depth ratio, column size to slab

effective depth ratio and flexure reinforcement ratio etc. are not properly

incorporated in some code provisions. Some codes do not acknowledge the possible

effect of flexural reinforcement on the punching shear behavior of reinforced

concrete slabs. Some codes do not take adequate account of the possible role of

specimen size and slab thickness.

28

From above discussion, there are scopes of work to investigate punching shear

provision of Flat Plate more accurately. Punching shear related parameters on multi-

panel Flat Plates will be more reasonable than the results obtained by using isolated

single span slab specimens in this regard. Continuity of slab and edge restraint will

be adjusted in multi-panel Flat Plate. Several parameters such as concrete strength,

flexural reinforcement, slab thickness, column size, span to depth ratio may be

applied to multi-panel Flat Plate.

However, multi-panel tests are time consuming, expensive and it is difficult to

determine experimentally the shears and moments applied to the individual slab-

column connections. An alternative to such expensive and difficult experimental

procedure is to perform the investigation by means of numerical finite element

analysis. Advanced nonlinear finite element investigation of multi-panel Flat Plate

considering full scale with practical geometry may be carried out on the behavior of

punching shear characteristics of concrete slab in presence of flexural reinforcement.

Thus, the objective of this study is to attempt to find out the punching shear features

of a multi-panel finite element model slab which is able to predict, with reasonable

accuracy, the ultimate punching load capacity and the correct mode of failure for a

large number of slabs which cover all factors affecting the behavior of reinforced

concrete slabs. The study may use three dimensional elements to study the behavior

of Flat Plates with different types of multi-panel Flat Plate model.

The objectives of this research work may be summarized as follows:

i) To develop a numerical finite element model including concrete nonlinearity

to compare the results of numerical analysis with test results.

ii) To simulate the actual punching shear capacity of slabs.

iii) To study the multi-panel flat plate on the punching shear strength of concrete.

iv) To find out the effect of concrete strength and flexural reinforcement ratio on

the punching shear strength of concrete.

v) To study the effect of slab thicknesses and column sizes on the punching

shear strength.

29

vi) To study the crack patterns for punching load.

vii) Parametric study on punching shear behavior.

viii) To formulate a guide line for more rational estimate of punching shear

capacity of slabs.

1.6 ORGANIZATION OF THE THESIS

The thesis covers mainly four parts, namely the background theory of finite element

analysis, the simulation of experimental investigations to validate finite element

model, the numerical analysis of multi-panel flat plate and the proposal for punching

shear capacity of RC slab.

After the introduction of punching shear and code provision in Chapter-1, Chapter-2

presents the theoretical background of FE modeling such as FE procedure, suitable

elements, integration scheme, stress-strain relationship and reinforcement in FE solid

elements. In Chapter-3, behavior of materials such as concrete and reinforcement in

FE are discussed. Nonlinear solution techniques including process of iteration,

convergence criteria, solution termination criteria are elaborated in Chapter-4.

Afterwards, Chapter-5 presents an overview of the test campaign that was used for

validation of FE model. Based on experimental investigation and numerical analysis,

an analytical model was developed. FE analysis of experimented works to validate

the analysis and parametric study based on experimental results presents in the

Chapter-6.

After the validation of the FE modeling technique in simulating the punching shear

behavior of flat plates and thus such modeling can be further applied to numerically

study the behavior of multi panel flat plate systems as an alternative to experiments

and presented in Chapter-7 and Chapter-8. In Chapter-7, various data for modeling

multi-panel flat plate and other input data are included. Results and discussions of FE

analysis of multi-panel flat plate are elaborated in Chapter-8.

30

According to the relationship of normalized punching shear with concrete strength,

flexural reinforcement and size effect of slab and column, an empirical equation for

calculating punching shear capacity is proposed in Chapter-9. The proposed equation

is verified by analyzed data and test results and comparison with various code of

prediction is also discussed in this chapter.

Finally, the thesis is finished with conclusions, followed by recommendation for

future research in Chapter-10.

31

CHAPTER-2

FINITE ELEMENT METHOD

2.1 INTRODUCTION

The finite element method is now firmly accepted as a most powerful general

technique for the numerical solution of variety of problems encountered in

engineering. For linear analysis, at least, the technique is widely employed as a

design tools. Similar acceptance for nonlinear situations is dependent on two major

factors. Firstly, in view of the increased numerical operations associated with

numerical problems, considerable computing power is required. Secondly, before the

finite element method can be used in design, the accuracy of any proposed solution

technique must be proven.

The development of improved element characteristics and more efficient nonlinear

solution algorithms and the experience gained in their application in engineering

problems have ensured that nonlinear finite element analysis can now be performed

with some confidence and the process is already economically acceptable for

selected industrial applications. Nonlinear analysis is necessary when,

o Designing high performance components.

o Establishing the causes of failure.

o Simulating true material behavior.

o Trying to gain a better understanding of physical phenomena.

Details of numerical analysis and procedure in accordance with TNO DIANA BV

(2010) have discussed in this chapter.

2.2 FE PROCEDURE

For linear elastic problems the system of equations to be solved is

Ku = f ( 2.1 )

Where, K is the system stiffness matrix, u is a vector of the unknown nodal degrees

of freedom such as displacements and rotations and f is the vector of the nodal

forces corresponding with the degrees of freedom u.

32

2.2.1 Global Formulation

When considering a general three-dimensional body, denoted by V, the problem is

identified by unknown displacements u and known body forces per unit volume g.

External forces in the form of concentrated forces and known tractions t are applied

to the part St of the boundary and are called the natural boundary conditions. The

displacements u are specified as known values u on the part Su of the boundary and

are called the essential boundary conditions. In the finite element method the body V

will be approximated as an assemblage of finite elements, which are connected by

nodal points on the element boundaries.

2.2.1.1 Displacements

In order to solve the problem the displacements u has to satisfy a continuity and

differentiability to the necessary degree. On the boundary Su the displacements must

satisfy the essential boundary condition,

( 2.2 )

The displacements of a particular point (x, y, z) are assumed to be continuous

functions expressed in terms of discretized variables at the nodal points and are

approximated as

( 2.3 )

where N is the displacement interpolation matrix and u is a vector of nodal point

variables such as components of displacements and rotations, and is denoted as the

vector of degrees of freedom. The interpolation matrix N comprises interpolation or

shape functions described in terms of independent variables, such as coordinates and

are locally defined for the individual elements.

2.2.1.2 Strains and Stresses

The strains at any point in the structure can be determined by,

( 2.4 )

where L is a differential operator defining a compatible strain field. Now the strain

field can be written as the derivative of the vector u as,

( 2.5 )

33

where the matrix B defines the strain–displacement relation for a particular point and

is called the differential matrix. Assuming linear elastic behavior, the relationship

between stresses and strains in a particular point can be written in the form,

( 2.6 )

where the matrix D is the stress–strain relation and is a function of material

properties like Young’s modulus E and Poisson’s ratio υ. The vector Ԑ0 denotes the

initial strains changes etc. and the vector σ0 contains the initial residual stresses.

2.2.1.3 Equilibrium

In a structural problem the governing equilibrium equations can be written as

( 2.7 )

where g is the vector of the known body forces per unit volume, with V as the total

volume or domain of the model. Vector t represents the known traction forces on the

boundary St such as surface, edge and point loads. For the derivation of the

equilibrium equations the stationarity condition of the total potential energy can be

used.

2.2.1.4 Principle of Virtual Displacements

A simpler way of introducing the equilibrium relationships of equation 2.7 can be

done by invoking the principle of virtual displacements. This principle states that an

elastic structure is in equilibrium under a given loading system if, for any virtual

displacement from a compatible state of deformation, the virtual work is equal to the

virtual strain energy. The virtual work equation can be written as

( 2.8 )

Where δԐ are the virtual strains which correspond to the virtual displacements δu,

Substituting (equation 2.3) and (equation 2.5) into (equation 2.8) gives,

( 2.9 )

34

where r is the vector of the internal forces corresponding to the vector of the nodal

degrees of freedom u. The principle of the virtual work states that (equation 2.9)

should be satisfied for any u so that,

( 2.10 )

These equations do not ensure that the equilibrium is satisfied at any point, but only

guarantee that the stresses satisfy equilibrium in a weighted average sense.

Substituting (equation 2.6) and (equation 2.5), the left hand side of (equation 2.10)

can be written as,

( 2.11 )

Combining the expression for r, in (equation 2.9) and (equation 2.11) we obtain

( 2.12 )

Where

( 2.13 )

is the system stiffness matrix, and f is the right hand side vector defined by

( 2.14 )

This provides a set of linear simultaneous equations which can be solved in a direct

or indirect way:

( 2.15 )

2.2.2 Discretization to Elements

In the Finite Element Method the solution domain V is divided into a finite number

of elements Ve, which are connected by nodal points at the inter-element boundaries.

In this way the solution domain is discretized and represented as a patch of elements.

35

The unknown displacements in each element are now approximated by continuous

functions expressed in terms of nodal variables. The functions over each finite

element are called interpolation or shape functions.

2.2.2.1 Displacements

In each element the displacements of an arbitrary point (x, y, z) can be measured in a

convenient local Cartesian coordinate system and are approximated by shape

functions and nodal variables

( 2.16 )

where N is the interpolation matrix with shape functions N(x, y, z) and ue the element

nodal displacement vector, expressed in local xyz axes. This element vector can be

composed from the nodal variables of the system degrees of freedom vector u by,

( 2.17 )

where Te is the element transformation matrix which transforms the corresponding

system degrees of freedom to the local element degrees of freedom, oriented in the

xyz coordinate system.

The rest is identical to global formulation as stated in the earlier section.

2.2.2.2 Strains and Stresses

Using the strain–displacement law for compatibility and assuming that the shape

functions N are known, the discrete form of the strain–displacement relation can be

written as

( 2.18 )

Likewise equation 2.6, for the entire domain, the relation between strains and

stresses, including initial strains and initial stresses, can be written for an element as

( 2.19 )

where D is the rigidity matrix representing the stress–strain law, usually derived from

Hooke’s law, varying from element to element. Often the matrix D is only defined in

a local element Cartesian (xl, yl, zl) coordinate system. In order to obtain the strains in

this system, it is necessary to apply a strain transformation

( 2.20 )

36

where TԐ is the strain transformation matrix. With equation 2.18 the local strain

vector Ԑl can now be related directly to the local element degrees of freedom vector

ue by

( 2.21 )

2.2.2.3 Element Assembly

During this process, also the constraints and the linear constraints (tyings) are

handled. The general tying equation for eccentric connection of three translations and

three rotations is

( 2.22 )

Where uxi denotes the x translation of the slave node, φyj the y rotation of the master

node, ∆x the eccentricity in x direction, etc.

2.2.2.4 Virtual Strain Energy

Now the structure has been idealized as an assemblage of elements, the integral form

of the virtual work equation 2.8 can be rewritten as a summation of the virtual work

done by the individual elements having volumes Ve and boundary surfaces Se like

( 2.23 )

where ne is the total number of elements, ge is the element body force per unit

volume and te are the element tractions per unit area acting along the element

boundary Se. For each element, its surface boundary Se can be separated in an

exterior part and an interior part having imaginary interfaces with adjacent elements.

The equation 2.23 is of fundamental importance for the displacement based Finite

Element Method and imposes some restrictions on the displacement functions. In the

37

‘Principle of Virtual Displacements’ finite element approximation, we will attempt to

ensure equilibrium, which for an element looks like

( 2.24 )

It can be proved that this theorem is only valid provided that all derivatives of u and

σ are finite through V. In general the stresses do not achieve continuity across the

element interfaces. However, if the shape functions are chosen such that the

displacements match at the nodes and the adjacent elements (i and j) have identical

displacements at their interface, then a continuity condition on the stresses in the

mean is met in the form of

( 2.25 )

where et is the contribution of applied external loads. This expression is another

approximation of satisfying equilibrium and therefore the equilibrium equation 2.24

is true within a single element and up to its surface boundary Se. Assuming that the

displacement functions satisfy the conditions equation 2.24, the integrations may

now be performed over the element volumes and surfaces. Substituting for the

element displacements and strains respectively equation 2.16 and 2.18, the virtual

work equation for an individual element can now be written as

( 2.26 )

The integral form of the element boundary tractions can be replaced by a

kinematically equivalent nodal force vector re corresponding with the element

degrees of freedom vector ue. Reordering and substituting re for the boundary

tractions, the virtual work equation can now be expressed in a form

( 2.27 )

As this relation is valid for any virtual displacement δue, the equilibrium equation

for an element can be written as

( 2.28 )

38

2.2.2.5 Stiffness Matrix

Equation 2.28 is valid for any stress–strain relation and in case of a linear elastic

behavior, substituting equation 2.19 for the stresses yields

( 2.29 )

Where

( 2.30 )

is the element stiffness matrix and

( 2.31 )

is the element contribution to the right hand side vector f .

Going back to equation 2.23 and using the piecewise approximation for the

displacements (equation 2.16) and the discrete strain–displacement relation (equation

2.18), the virtual work equation is now obtained by

( 2.32 )

Substitution of the stress–strain relation (equation 2.19) in case of linear elastic

behavior yields

( 2.33 )

With T for the transformation matrix (equation 2.30) is written as

( 2.34 )

The pre and post-multiplication with T transforms the element stiffness from local to

global coordinates.

39

2.2.3 Assembling the Load Vector

The load vector f is composed of the external nodal forces as specified in the input

file and of the assembly of the element loads. These element loads can be subdivided

into the following components,

1. Equivalent nodal forces due to thermal effects, effects resulting from

difference in concentration and initial strains. Summing these effects

results in an equivalent initial strain, which can be transformed to

nodal loads.

2. Equivalent nodal forces resulting from initial stresses.

3. Equivalent nodal forces resulting from loads on element boundaries.

4. Equivalent nodal forces resulting from acceleration effects (dead

weight).

Due to the LOADS command, the analysis calculates the above contributions per

element and after that, for each degree of freedom the contribution of the connected

elements are superposed and added to the external nodal point loads thus forming the

load vector f.

2.2.4 Equilibrium

Invoking the theorem of the virtual displacements, the equilibrium equations of

the element assemblage are

( 2.35 )

where the matrix K is the stiffness matrix of the element assemblage

( 2.36 )

and the vector f is the right hand side vector

( 2.37 )

W

2

2

T

T

e

t

p

T

w

e

s

T

t

t

d

A

r

With

2.3 SUIT

2.3.1 Selec

The selectio

The plate an

economy, bu

the nature o

plate elemen

The formula

without σ3 i

effect. Punc

shear forces

The formula

the thicknes

thickness of

distribution

Apart from

reinforcemen

TABLE EL

ction of Elem

n of element

nd shell elem

ut these elem

of whose be

nt and solid e

ation of plat

in the yield

ching failure

are concent

ation of plate

ss is linear

f plate is n

of strain.

m the diffe

nt (steel in z

EMENTS

ment

t type is alw

ments are v

ments are not

ehavior is th

element are

te/shell is ba

d criteria (Fi

e often occu

trated. The e

e/shell eleme

(Figure 2.2

not necessar

rences in

z-direction) a

40

ways related t

very attractiv

t suitable for

hree dimensi

as follows:

ased on the

igure 2.1). T

urs at locatio

ffect of σ3 c

ent assumes

2). The actu

rily linear,

the formul

and column

to the type o

ve on accoun

r the study o

ional. The m

two princip

This implies

on where lar

ould be sign

that the dist

ual distributi

and solid e

lations, it

in plate/shel

of problem to

nt of both s

of punching s

main differe

al stress (σ1

s that there

rge bending

nificant.

tribution of

ion of strain

element allo

is difficult

ll elements.

o be analyze

simplicity an

shear proble

ences betwee

and σ2), i.

is no triaxi

g moment an

strain throug

n through th

ows nonline

to simula

ed.

nd

em

en

e.

ial

nd

gh

he

ear

ate

F

F

P

r

e

I

h

Figure 2.1

Figure 2.2

Punching s

recommende

element. Thr

It has been

have shear

Stress res

Assumpti

shear behav

ed that the s

ree most com

recognized

locking pr

sultants for p

ion Regardin

vior is a

study of pun

mmonly used

that 8 noded

roblem. Wh

41

plate/shell

ng deformati

three dime

nching shear

d solid elem

d solid elem

hile the 32

ion of a plat

ensional pr

r problem us

ments are sho

ments produc

noded soli

te/shell

oblem. The

se three dim

wn in the Fi

ce very stiff

d element

erefore it

ensional sol

igure 2.3.

f response an

can be qui

is

lid

nd

ite

42

expensive to use, involving 96 degrees of freedom and a fairly high order of

integration for the element stiffness matrix. Therefore the 20 noded solid

isoparametric element is used to represent concrete. Each node has three degrees of

freedom. In order to cope with curved boundaries, this program uses an

isoparametric element.

( a ) 8 Noded

( b ) 20 Noded

( c ) 32 Noded

Figure 2.3 Solid Isoparametric Elements

2.3.2 CHX60 Element

The element CHX60 is a twenty-node isoparametric solid brick element (Figure

2.3b). It is based on quadratic interpolation and Gauss integration. The polynomials

for the translations uxyz can be expressed as,

43

( 2.38 )

Typically, a rectangular brick element approximates the following strain and stress

distribution over the element volume. The strain Ԑxx and stress σxx vary linearly in x

direction and quadratically in y and z direction. The strain Ԑyy and stress σyy vary

linearly in y direction and quadratically in x and z direction. The strain Ԑzz and stress

σzz vary linearly in z direction and quadratically in x and y direction.

2.3.3 Displacements

The basic variables in the nodes of solid elements are the translations ux, uy

and uz in the local element directions (Figure 2.4).

( 2.39 )

Figure 2.4 Displacements of solid element

2.3.4 Strains

The displacements in the nodes yield the deformations dux, duy and duz of an

infinitesimal part dx dy dz of the element Figure 2.5. From these deformations,

44

Figure 2.5 Deformation of solid element

the Green–Lagrange strains of equation,

( 2.40 )

With

( 2.41 )

These Green–Lagrange strains are derived for all integration points and may be

extrapolated to the nodes. The sign convention for strains is that an elongation yields

a positive strain.

2.3.5 Stresses

The program can calculate and output Cauchy stresses for all types of solid elements.

For some element types, it can determine and output generalized moments and forces

by integrating the Cauchy stresses in a user-specified thickness direction.

From the basic strains of equation (equation 2.41), the program derives the Cauchy

stresses of equation in (equation 2.42) the integration points.

45

( 2.42 )

Figure 2.6 shows these stresses on a unit cube in their positive direction. Note that

tension stress is positive.

Figure 2.6 Cauchy stresses of solid element

2.3.6 Shape functions

The shape functions can be defined by expressing the coordinates x = (x, y, z)T of an

arbitrary point within an element as functions of the parameters (ξ, η, ζ) and of the

global Cartesian coordinates of the element nodes xe as

( 2.43 )

where x is the approximation of x and N ′ denotes the shape functions. Definition of

the shape functions can be done in more than one way, so careful consideration is

necessary. Some considerations are described below in further detail. Above

definition of the shape function is almost identical to the definition of the

interpolation polynomial N for the displacements. Therefore it is very natural to

choose the same functions for the description of the geometry as for the

displacements. Thus for N ′ we write N. In this case we speak about isoparametric

elements.

Furthermore the condition must be satisfied that for instance for ξ = η = ζ = −1 the

global coordinates of node i must be reproduced. This should of course be the case

for all other corner nodes. For the mid-side nodes, usually the location of ξ, η or ζ =

0 is chosen. Thus if a mid-side node is not positioned exactly halfway between its

n

c

A

b

W

d

e

c

T

i

neighbor no

consequently

An example

been given b

Where i, j,

displacemen

elements of

coordinates

The shape fu

in curvilinea

odes, ξ = 0

y the integra

e of the shap

below for a l

k and l den

nts in global

f the vector

instead of th

function for 2

ar co-ordinat

0 does not

ation points a

pe functions

linear quadri

note the fou

X and Y dir

r ue. Substit

he displacem

20-noded so

te ξ, η and ζ

46

lie halfway

are not distri

s, or actually

ilateral elem

ur element n

rection respe

tution of u

ments.

olid element

.

y between ξ

ibuted symm

y the interpo

ment in two-d

nodes and u

ectively. Th

by x gives

are given by

ξ = −1 and

metrically.

olation polyn

dimensional

and v repr

herefore ui,j,k,

s the expres

y the follow

d ξ = 1, an

nomial N, h

space

( 2.44

( 2.45

resent e.g. th

,l and vi,j,k,l a

ssions for th

wing equation

( 2.46

( 2.47

( 2.48

nd

as

)

)

he

are

he

ns

)

)

)

w

T

e

i

n

g

2

N

t

where ξ, η a

The displac

expressed in

it should be

not to the ξ,

global co-or

2.3.7 Elem

Now the der

terms of the

and ζ are the

cement at an

n terms of the

noted that th

, η and ζ ax

rdinates is gi

ment Stiffnes

rivative of N

parametric c

intrinsic co-

ny point in

ese shape fu

he displacem

xis. Similarly

ven by :

ss Matrix

N with respe

coordinates

47

-ordinates of

nside the ele

unctions as fo

ments u, v an

y, the positio

ect to the glo

as

f any point w

ement, nam

ollows:

nd w are para

on at any po

obal coordin

within the ele

mely u, v an

allel to the x

oint within t

nates, can be

( 2.49

ement.

nd w, can b

( 2.50

, y and z, an

the element

( 2.51

e expressed

)

be

)

nd

in

)

in

48

( 2.52 )

Where the matrix J−1 denotes the inverse Jacobian matrix of the transformation from

the parametric to the Cartesian coordinates

( 2.53 )

Further we may write

( 2.54 )

Therefore the element stiffness matrix can be written as

( 2.55 )

With the above definition of the shape function, the B matrix can actually be

determined and the element stiffness matrix can be numerically solved, as holds for

all other element integrals. Therefore the import conclusion is that the element

integrals can be expressed in the nodal coordinates and the parametric coordinates on

the standard integration interval [−1, 1].

2.3.4 Numerical Solution of Element Integrals

The element integrals as described in the previous chapter can be solved analytically

or numerically, also called direct respectively numerical integration. Often it is not

feasible to solve the integrals analytically for an element in its most general

appearance. Therefore for the majority of the elements a numerical solution is

preferred. For both integration methods the interpolation polynomial N, which is

composed by a set of shape functions, has to be assumed firstly. These shape

functions express the approximated displacement field within the element in terms of

its nodal variables.

49

Principle of numerical integration: Numerical integration is based upon the

evaluation of the function to be integrated in a number of specific points, the so

called integration points. These function values in the integration points are then

weighted and summed to obtain the value of the integral. The weight function

depends on the method of integration. For finite element integration usually the

Gauss integration scheme is applied, as this method requires the least number of

integration points. Now the integration of a function f(x) can be rewritten numerically

as

( 2.56 )

Where ω ξi describes the weight function of the applied method for the specific

integration interval n, the number of integration points and xi the coordinate of the

integration point. In the case of the element stiffness matrix we rewrite equation

( 2.57 )

in which ωξ ωη and ωζ are the weight functions for each integration direction and

based on the standard interval and eK~ is expressed in the element local Cartesian

coordinate system. Above, the solution of the integral provides a simple algorithm to

determine the stiffness matrix.

2.4 INTEGRATION SCHEMES

As mentioned, the numerical integration is based upon the addition of weighted

function values as determined in the integration points. A minimum number of

integration points is required by the numerical integration method and depends on

the order of the interpolation polynomial. In order to integrate all of the terms in the

integrand a full integration scheme is necessary. Often however, especially in linear

elastic problems, some higher order cross terms of the polynomial can be ignored

safely, which reduces the required number of integration points and therefore is

called the reduced integration scheme.

It showed that the position of the Gauss integration points of the reduced scheme,

coincide with the optimal stress points, the so called Barlow stress points. It can be

50

shown that the error in the approximated strain and stress fields do not have the same

magnitude in the total integration area and that the approximation is at its best in the

Barlow points. In general it can be stated that the accuracy of the displacements is

not affected with a reduced integration scheme, whereas the stress and strain solution

is better in comparison with a full integration scheme.

In some cases (especially for coarse meshes) it is possible that zero-energy modes are

developed due to the reduced integration scheme, which causes a bad condition of

the total stiffness matrix and therefore a less accurate solution.

Integration Scheme for Bricks Element

Figure 2.7 shows the enumeration of the ζ planes for various ηζ . The principle of

enumeration of the integration points is illustrated in Figure 2.8.

Figure 2.7 Integration schemes in ζ direction for bricks

Figure 2.8 Example integration schemes ηξ x ηη x ηζ for brick

51

2.5 STRAIN DISPLACEMENT RELATION

To determine the strains of an element, the nodal results for this element are

transformed to the local Cartesian coordinate system of the element and placed in the

vector ue. The strain–displacement relation is defined by the matrix B as follows,

( 2.58 )

This relation is valid at any point within the element. But as the Finite Element

Method minimizes the error at the integration points, it is obvious that the strains will

be determined at these points. So for each integration point the expression

( 2.59 )

will be evaluated at the standard values for ξ, η and ζ .

2.5.1 Equivalent Von Mises Strain

Diana calculates the equivalent Von Mises strain according to

( 2.60 )

With the deviatoric strains:

( 2.61 )

The shear strains γ are defined as:

( 2.62 )

For some calculations the strains are placed in a strain matrix E which for the

general three-dimensional strain situation is given by

( 2.63 )

2.5.2 Principal Strains

Diana calculates the principal strains Ԑ1,2,3 as the roots of equation 2.64 ordered such

that Ԑ1 ≥ Ԑ2 ≥ Ԑ3.

F

Ԑ

a

2

D

2

F

f

w

c

w

o

For plane str

Ԑ3 is always

are the inpla

2.5.3 Volum

Diana calcul

2.6 STR

For the thre

form are giv

where Ԑx, Ԑy

components

written as

or simply ex

rain and axis

the out-of-p

ane principal

metric Strain

lates the volu

RAIN MATR

ee dimension

ven below:

y, Ԑz, are the

. Using the

xpressed as

symmetric e

plane strain,

l strains orde

n

umetric strai

RIX

nal element

normal strai

finite eleme

52

lements, how

while the fir

ered such tha

in Ԑvol by sum

, the strain-

in componen

ent idealizati

wever, the th

rst and secon

at Ԑ1 ≥ Ԑ2.

mmation of

-displacemen

nts and γxy, γ

ion, strain m

hird principa

nd principal

the principa

nt relationsh

γyz, γzx are th

matrix (equat

( 2.64

al strain

strains

al strains:

( 2.65

hips in matr

( 2.66

he shear stra

tion 2.66) ca

( 2.67

)

)

rix

)

ain

an

)

w

d

o

c

t

T

t

t

where [Bi] is

derivatives o

of the local c

co-ordinates

through the w

Thus

the inverse o

therefore the

s the 6x3 str

of the shape

co-ordinates

s is required

well known

of the Jacobi

e Cartesian d

rain matrix in

e functions. S

s of the elem

to obtain th

Jacobian ma

ian matrix w

derivatives a

53

n the equatio

Since the sh

ment (ξ, η, ζ),

he [B] matri

atrix which i

will be

are given by

ons 2.67 wh

hape function

, a transform

ix in the equ

is written as

ich contains

ns Ni are def

mation from l

uations 2.67.

:

( 2.68

s the Cartesia

fined in term

local to glob

. This is don

( 2.69

( 2.70

( 2.71

( 2.72

)

an

ms

bal

ne

9)

)

)

)

2

F

e

w

w

c

r

c

2

S

s

n

o

e

e

2.7 STR

For the line

expressed in

where [D] is

where E is

cracking an

reinforced co

crushing are

2.8 REIN

Steel bars a

specified loc

noded solid

only. The li

element as

element with

RESS-STRA

ar analysis

n the followi

s the elasticit

the Young

d crushing

oncrete struc

e taken into a

NFORCEM

are simulate

cations in the

element use

ine element

shown in F

h maximum

AIN RELAT

of uncracke

ing form :

ty matrix giv

's modulus

of concrete

ctures. All ch

account in a

MENT IN SO

ed by line e

e structure. A

ed in the pres

must lie pa

Figure 2.9. T

curvilinear c

54

TIONSHIP

ed concrete,

ven by :

of elasticity

e are the ma

hanges in m

new elastici

OLID ELEM

elements em

A three -nod

sent study. T

arallel to one

This line ele

co-ordinates

the stress-st

y, and v is

ajor sources

material prope

ity matrix.

MENT

mbedded in

ded line elem

This line elem

e of the cur

ement can b

s ξ = ±l, η =

train relation

the Poisson

s of nonline

erties due to

the concret

ment correspo

ment can car

rvilinear axi

be anywhere

±l and ζ = ±

nship may b

( 2.73

( 2.74

n's ratio. Th

earity in mo

cracking an

te element

onds to the 2

rry axial load

is of the sol

e in the sol

±1.

be

)

)

he

ost

nd

at

20

d

lid

lid

F

T

S

w

d

T

w

F

Figure 2.9

The displace

So that

where N is

displacemen

The virtual w

where dU =

As =

dl =

σl, Ԑl

For a horizo

Embedde

ement {u} of

s the shape

nt vector.

work of the l

= internal v

= cross-sec

line segm

l = the longit

ontal bar para

ed reinforcem

f any point o

e function

line element

virtual work

ctional area o

ment along th

tudinal stres

allel to the x

55

ment in 3D s

on the bar is

of concrete

t (steel bar) c

in the steel b

of steel bar;

he steel bar;

s and strain

x-axis,

solid concret

written as :

e element a

can be writte

bar;

and

along line se

te element

and {δ}e

en as :

egment, resp

( 2.75

( 2.76

is the nod

( 2.77

pectively.

( 2.78

)

)

dal

)

)

T

A

c

w

c

U

w

I

The equation

At any poin

curvilinear a

where X ′ ,

correspondin

Using the di

where ll, ml,

In terms of t

n becomes,

nt in the lin

axis. The loc

Y ′ and Z ′

ng displacem

isplacement

nl are the di

the shape fun

ne element,

cal strain in t

are local co

ments.

transformati

irection cosin

nction deriva

56

the local C

the steel bar

o-ordinates a

ion,

nes of the X

atives is writ

Cartesian axi

can be calcu

at a point, a

X' axis and ar

tten as

is X’ is tang

ulated as foll

and u ′ v ′ an

re written as

( 2.79

gential to th

lows:

( 2.80

nd w′ are th

( 2.81

( 2.82

( 2.83

)

he

)

he

)

)

)

T

w

s

A

w

T

T

a

The strain in

where B is t

strain in the

And the stiff

where Es is

The same ste

The final ex

adding the st

n steel can be

the nodal dis

steel bar is :

fness of the

the Young's

eps can be re

xpression fo

tiffness matr

e written as

splacement-s

:

embedded b

s modulus of

epeated for b

or the comp

rices for con

57

strain matrix

bar can be ex

f steel bar an

bars parallel

posite eleme

ncrete and ste

x. The relatio

xpressed as,

nd Js is the J

to y and z a

ent stiffness

eel together,

on between

acobian for

axis.

s is simply

, as follows:

( 2.84

the stress an

( 2.85

( 2.86

steel elemen

evaluated b

( 2.87

)

nd

5)

)

nt.

by

)

58

CHAPTER-3

MATERIAL MODELLING

3.1 INTRODUCTION

With the advancement of finite element technique and availability of high speed

computer, there has been a demand for refined and sophisticated model in order to

trace the response of RC structure in the nonlinear post-cracking and post-yield

range. The limitation of linear elastic, plastic and ultimate load theories necessitate

the nonlinear plastic analysis by a computational model such as finite element

method. Linear elastic theory can never predict the failure surface the true factor of

safety can never determine. To overcome this problem, plastic methods and ultimate

load theories have been developed. Since determination of the actual failure stress

distribution and collapse mechanism is not straightforward due to complex geometry

of structures and loading, this method can not be relied for important structures.

Moreover there is no consideration in these methods for several other factors such as,

inplane behavior, failure mechanism, of localized failure, effect of load history etc.

Although the use of finite element technique for RC structures is highly promising,

yet the task of modeling the material behavior was and remains a great challenge.

Ever since this method was applied to RC structure, concrete material modeling has

become a very active area of research. Although a great deal of progress has been

made by the researchers as seen in the state of art report of American Society of Civil

Engineers, ASCE (1982) and Christian and Hajime (1986), continuing research work

is still attempting to resolve some of the difficult issue. These include nonlinear

behavior, tension cracking, biaxial stiffening, strain softening, modeling of post

fracture behavior, interaction between the concrete and reinforcing bars etc. For the

development of a mathematical model, the typical experimental data is used for

concrete under uniaxial, biaxial and triaxial state of stress in order to simulate the

material behavior and to determine the various material constants in the

mathematical model.

3

I

f

s

i

b

s

W

b

t

c

o

i

c

m

c

p

3

A

s

s

s

T

d

m

A

p

t

s

o

3.2 BEH

It is a comm

for several

structures. I

index of its b

bond streng

strength of c

With the ev

behavior of

the effect o

conditions. T

of a biaxial

investigation

conditions,

modeling. H

concrete und

practice.

3.2.1 Unia

A common

section, is su

system is in

section of th

Therefore th

described by

may be calle

A typical re

presented in

the ultimate

small increm

of micro cra

HAVIOR OF

mon practice

requiremen

In fact the u

behavior in g

gth or even

concrete rule

ver increasi

concrete in

of fatigue on

The loading

or triaxial i

n regarding

no constitu

Hence, there

der biaxial a

axial Behavi

situation wi

ubjected to s

n equilibrium

he bar the to

he stress th

y = F/A, w

ed normal str

elationship

n Figure 3.1.

load, the cu

ments of stre

acks at the p

F CONCRE

e to determin

nts relating

uniaxial com

general. Be i

cyclic loadi

es.

ng awarene

the prototy

n concrete,

conditions

in nature, pa

g the behav

utive laws a

are need to

and triaxial c

ior

ith a simple

stress by opp

m and not ch

op part must

hroughout th

where A is t

ress or uniax

between str

After an ini

urve becomes

ss. The nonl

aste-aggrega

59

ETE IN COM

ne the uniax

to the desi

mpressive st

it the durabi

ing leading

ess amongst

ype structure

but once a

in real life s

articularly in

vior of con

are readily

have such f

conditions a

stress patter

posite forces

hanging with

t pull on the

he bar, acr

the area of

xial stress.

ress and str

tial linear po

s nonlinear,

linearity is p

ate interface

MPRESSIO

xial compres

ign and qua

trength of c

ility or tensil

to fatigue,

t the design

es, attempts

again mostly

structures are

n bridges an

ncrete has

available fo

facilities and

nd incorpora

rn is with un

s of magnitu

h time then

e bottom pa

ross any ho

the cross-se

rain for nor

ortion lasting

with large s

rimarily a fu

e. The ultima

ON

ssive strengt

ality contro

concrete is c

le strength, s

the uniaxial

ners regardin

have been m

y under uni

e seldom un

nd dams. Sin

been done

or analysis

d to study th

ate the resul

niform mate

ude F along

through eac

art with the s

orizontal sur

ection. This

rmal strengt

g up to abou

trains being

unction of th

ate stress is

th of concre

l of concre

considered a

shear strengt

l compressiv

ng the actu

made to stud

iaxial loadin

niaxial and a

nce not muc

e under suc

or numeric

he behavior

lts into desig

erial and cro

its axis. If th

ch transvers

same force F

rface, can b

type of stre

th concrete

ut 30 – 40%

registered f

he coalescenc

reached whe

ete

ete

an

th,

ve

ual

dy

ng

are

ch

ch

cal

of

gn

ss

he

sal

F.

be

ess

is

of

for

ce

en

60

a large crack network is formed within the concrete, consisting of the coalesced

micro cracks and the cracks in the cement paste matrix. The strain corresponding to

ultimate stress is usually around 0.003 for normal strength concrete. The stress-strain

behavior in tension is similar to that in compression.

The descending portion of the stress-strain curve, or in other words, the post-peak

response of the concrete, can be obtained by a displacement or a strain controlled

testing machine. In typical load controlled machines, a constant rate of load is

applied to the specimen. Thus any extra load beyond the ultimate capacity leads to a

catastrophic failure of the specimen. In a displacement controlled machine, small

increments of displacement are given to the specimen. Thus, the decreasing load

beyond the peak load can also be registered. The strain at failure is typically around

0.005 for normal strength concrete, as shown in Figure 3.2. The post peak behavior is

actually a function of the stiffness of the testing machine in relation to the stiffness of

the test specimen, and the rate of strain (Weiss, 2013). With increasing strength of

concrete, its brittleness also increases, and this is shown by a reduction in the strain

at failure.

Figure 3.1 Uniaxial stress-strain relationship for ordinary concrete (Weiss, 2013).

F

I

l

t

3

W

d

Figure 3.2

It is interest

linear stress-

the mismatc

3.2.2 Biax

When the st

direction, th

Complete st

ting to note

-strain relati

h and micro

xial Behavio

tress is zero

e stress is ca

tress-strain c

that althoug

ionships, the

cracking cr

or

only across

alled biaxial

61

curve includi

gh cement pa

e behavior fo

eated at the

s surfaces th

stress.

ing post-peak

aste and agg

or concrete is

interfacial tr

at are perpe

k response (

gregates indi

s nonlinear.

ransition zon

endicular to

(Weiss, 2013

ividually hav

This is due

ne.

one particul

3).

ve

to

lar

62

Kupfer et al. (1969) completed a series of tests to investigate the response of plain

concrete subjected to two-dimensional loading. Figure 3.3 shows the strength in the

principal directions compared with the uniaxial strength. One of the conclusions of

Kupfer et al (1969) is that the strength of concrete subjected to biaxial compression

may be up to 27% higher than the uniaxial compressive strength.

Figure 3.3 Biaxial strength envelope of concrete (Kufer and Hilsdorf, 1969)

63

Yin et al. (1989) completed a similar investigation. In these investigations concrete

plates (approximately 200 mm by 200 mm by 50 mm in dimension) were loaded to

failure at prescribed ratios of σ1 : σ2 with σ3 equal to zero. Loads were applied using

steel brushes to minimize stresses introduced through friction at the specimen

boundaries. The failure surfaces developed through these investigations are presented

in Figure 3.4. The result of the investigation conducted by Yin et al. (1989) show a

failure surface that is slightly stronger than that developed by Kupfer. The difference

in the failure surfaces may be due to a number of factors including load rate,

conditions of the specimens during testing, preparation of the specimens, properties

of the mixes or size effects. Yin et al. propose that the discrepancies are due in part to

differences in the type of coarse aggregate used in the two studies and in part to the

use by Kupfer of a slower rate of loading than is currently standard.

These two-dimensional failure surfaces are extended by data presented by Van Mier

(1986). Van Mier investigated the effect on the two-dimensional concrete failure

surface of applying low levels of confining pressure in the third dimension.

Two series of tests were completed in which concrete specimens were loaded at

prescribed ratios of σ1 : σ2 :σ3, with the stress in the one out-of-plane direction

maintained at 5 or 10 percent of one of the in-plane stresses. The results of these tests

show that a relatively small confining pressure in the out-of-plane reaction can

significantly increase the strength of concrete in the plane of the primary loading.

Biaxial compression showed similar micro cracks parallel to the free surfaces of the

specimen as uniaxial compression. At failure an additional 18-27 deg cracks on the

free surfaces of the specimen. Specimens subjected to combined tension and

compression behaved similarly to the specimens loaded in biaxial compression as

long as the applied tensile stress was less than 1/15 th of the compressive stress. The

strength of concrete under biaxial compression is larger than under uniaxial

compression. The large variation in water-cement ratio and cement content had no

significant effect on the biaxial strength. In the range of compression-tension and

biaxial tension, however, the relative strength decreases as the uniaxial strength

increases. The strength of concrete under biaxial tension is almost independent of the

64

stress ratiosigma1/sigma2 and equal to the uniaxial tensile strength. In the region of

biaxial compression the strains in the direction of the larger principal stress increase

in magnitude as the stress at failure increases in magnitude. For biaxial compression

Figure 3.4 Biaxial Compressive Yield Surface for Concrete with Low Confining Pressure [Data from Kupfer et al. (1969), Yin et al. (1989) and Van Mier (1986)]

and tension, the failure strains in the compressive stress decrease in magnitude as the

the simultaneously acting tensile stress increases. If the stresses increase beyond this

value the rate of volume reduction increases until at 80 to 90 percent of the ultimate a

point of inflection is reached. Further strain results in an increase in volume, it is

generally agreed that the inflection point coincides with the stress at which major

micro cracking of the concrete is initiated. Poisson’s ratio is constant beyond the

elastic limit and increases only at stresses beyond the point of inflection of the

volumetric strain relationship.

65

3.2.3 Triaxial Behavior

In the most general case, called triaxial stress, the stress is nonzero across every

surface element. According to Hooke’s law for isotropic solid materials, the strain

components and the related elastic stresses are defined below as mathematical

formulas. Hence, the triaxial entities are,

Using the Principle of Superposition, the total stain and stress in one direction become

For convenience, the stresses as function of strains may be defined in a matrix form

as indicated below. The stresses along the principal axes are

66

When a concrete structure undergoes a dynamic loading such an explosion or an

impact, its stress state is strongly heterogeneous and time-dependent. High triaxial

stress states occur, causing different damage modes which strongly depend on stress

state and loading path. The phenomena of brittle damage and irreversible strain such

as compaction need to be understood, and tests results at high levels of solicitations

and under various loading paths need to be performed.

In triaxial state of stress, the strength of concrete can increase considerably above the

uniaxial strength, in particular under hydrostatic stress conditions. Figure 3.5 shows

stress-strain curves from tests by Hobbs et al (1977) and Attard and Setunge (1996).

The tests were conducted under different confining pressures. All the stress-strain

curves basically followed a similar pattern. The initial tangent modulus was

approximately the same for all confining pressure, with linear portion of the

ascending curve extended with increasing confining pressure. The peak strength

increased with increasing confining pressure. The graph also shows that different

stress states can affect the ultimate strains of the test specimens.

Figure 3.5 Stress-strain curve of concrete for different confining pressures

67

Since the failure of concrete in a structure can occur differently under complex stress

states, the understanding of the behavior of concrete under multi axial stress states is

needed to develop the failure criteria for concrete. The compressive strength of

concrete is the principal property employed in the design of reinforced concrete

structures. After the pioneering investigation by Richart et al. (1920) on triaxial

behavior of concrete, many other researchers have also conducted studies on the

concrete behavior in multi axial compression and several failure criteria of concrete

have been proposed. More recently, the investigation has also been extended to

include high strength concrete. However, most of the triaxial tests were conducted on

equipment developed for rock testing and therefore the specimens were smaller (50

mm diameter) and the aspect ratio was kept at 2. With the advancement in servo-

hydraulic test equipment and digital technology, more complicated and sophisticated

triaxial tests can now be performed.

Tan (2005) performed experiment for triaxial stress of concrete having four groups

of concrete specimens termed as G15, G25, G50 and G80. The uniaxial compressive

strength of these groups was 10.35, 27.2, 51.8 and 77.46 MPa respectively. At least 3

specimens were tested to determine the uniaxial compressive strength for each group.

Typical stress-strain relationships of concrete under active confinement with various

confining stresses are compared in Figures 3.6 and 3.7.

Based on experimental results by Tan (2005), it can be observed that the axial stress

increases with increasing axial strain from the beginning up to the peak and then

drop. The peak stress level is related to the confinement level. The higher the

confinement, the higher the peak stress and peak strain the concrete can achieve. The

peak stresses and peak strains are calculated from the maximum load capacity and

the corresponding deformation respectively.

68

Figure 3.6 Axial stress-strain relationship for 51.8 MPa concrete (Tan, 2005).

Figure 3.7 Axial stress-strain relationship for 77.46 MPa concrete (Tan, 2005). 3.3 YIELD CRITEIA

The yield criterion determines the stress level at which plastic deformation begins

and can be written in the general form

f(αij) = K (κ) ( 3.1 )

69

Where f is some function and K a material parameter to be determined

experimentally. The term K may be a function of a hardening parameter κ.

On physical grounds, any yield criterion should be independent of the orientations of

the coordinate system employed and therefore it should be a function of the

following three stress invariants only,

J1= αii

J2= ½ αijαij

J3= 1/3αijαjkαki

( 3.2 )

3.3.1 The Tresca Yield Criterion

The yield condition of Tresca is a maximum shear stress condition which can be

expressed in the principal stress space ( σ1 ≥ σ2 ≥ σ3) as shown in Figure 3.8a.

( 3.3 )

with )(κσ the uniaxial yield strength as a function of the internal state variable ĸ.

The flow rule is in general given by the associated flow rule g ≡ f, which results for

the plastic strain rate vector in the principal strain space.

( 3.4 )

Figure 3.8 Tresca and Von Mises yield condition (in π-and rendulic plane)

70

3.3.2 The Von Mises yield criterion

The yield condition of Von Mises is a smooth approximation of the Tresca yield

condition: a circular cylinder in the principal stress space (Figure 3.8b). The yield

function of Von Mises is given by the square root formulation,

( 3.5 )

where )(κσ is the uniaxial yield strength as a function of the internal state variable

ĸ. The projection matrix P is given by

( 3.6 )

The flow rule is generally given by the associated flow rule g ≡ f, which results

for the plastic strain rate vector in

( 3.7 )

Figure 3.8 shows the geometric interpretation of the Von Mises yield surfaces to be

circular cylinder whose projection is a circle of radius. A physical meaning of the

constant κ can be obtained by considering the yielding of materials under simple

stress state. The case of pure shear (σ1 = - σ2 ,σ3 = 0) κ must be equal to the yield

shear stress. For most metals Von Mises’ law fits the experimental data more closely

than Tresca’s, but it frequently happens that the Tresca criterion is simpler to use in

theoretical applications.

3.3.3 The Mohr-Coulomb yield criterion

The yield condition of Mohr–Coulomb (Figure 3.9a) is an extension of the Tresca

yield condition to a pressure dependent behavior. The formulation of the yield

function can be expressed in the principal stress space ( σ1 ≥ σ2 ≥ σ3) as

( 3.8 )

71

With )(κc the cohesion as a function of the internal state variable ĸ and ϕ the angle

of internal friction which is also a function of the internal state variable. The initial

angle of internal friction is given by ϕ0. The flow rule is given by a general non-

associated flow rule g ≠ f but with the plastic potential given by

( 3.9 )

which results for the plastic strain rate vector

( 3.10 )

Figure 3.9 Mohr-Coulomb and Drucker-Prager yield condition (in π-and

rendulic plane)

Again as for the Tresca criterion, the complete yield surface is obtained by

considering all other stress combinations which can cause yielding (e.g. σ3≥σ1≥σ2).

In principal stress space this gives a conical yield whose normal section at any point

is an irregular hexagon. The conical, rather than cylindrical, nature of yielding

surface is a consequence of the fact that a hydrostatic stress does influence yielding

which is evident from the last term of Coulomb’s law. Mohr-Coulomb criterion is

applicable to concrete, rock and soil problems.

72

3.3.4 The Ducker-Prager yield criterion

The yield condition of Drucker–Prager is a smooth approximation of the Mohr–

Coulomb yield surface, which is a conical surface in the principal stress space

(Figure 3.9b). The formulation is given by,

( 3.11 )

with )(κc the cohesion as a function of the internal state variable ĸ. The projection

matrix is equal to the projection matrix of the Von Mises yield condition defined in

equation 3.6. The projection vector π is given by,

( 3.12 )

The scalar quantities αf, and β are given by

( 3.13 )

( 3.14 )

The angle of internal friction ϕ is also a function of the internal state variable. The

initial angle of internal friction is given by ϕ0. The flow rule is given by a general

non-associated flow rule g ≠ f , with the plastic potential given by,

( 3.15 )

with the scalar αg defined by the dilatancy angle Ѱ

( 3.16 )

Which results for the plastic strain rate vector in

( 3.17 )

with the scalar Ѱ defined by

( 3.18 )

73

3.4 CRACKING OF CONCRETE AND MODELLING

Concrete is a brittle and quasi brittle material. The constitutive behavior of quasi-

brittle material is characterized by tensile cracking and compressive crushing, and by

long-term effects like shrinkage and creep. The cracking can be modeled smear

cracking model with tension softening and shear retention and Total Strain crack

model describes the tensile and compressive behavior of a material with one stress-

strain relationship. The input for the Total Strain crack models comprises of Young's

modulus, Poisson's ratio, behavior of concrete in tension, shear and compression.

3.4.1 Smeared Cracking Model

The concept of a smeared crack model with strain decomposition was first proposed

by Litton (1974). Ever since it has been used by many other researchers, for instance

De Borst and Nauta (1985), Riggs and Powell (1986), Rots (1988).

The main concept of a smeared crack model is strain decomposition. The

fundamental feature of the decomposed crack model is the decomposition of the total

strain ε into an elastic strain εe and a crack strain εcr as,

( 3.19 )

This decomposition of the strain allows also for combining the decomposed crack

model with for instance a plastic behavior of the concrete in a transparent manner as

proposed by De Borst (1987).

The sub-decomposition of the crack strain εcr gives the possibility of modeling a

number of cracks that simultaneously occur. The basis feature of this multi-

directional fixed crack concept is that a stress si and strain eicr exists in the n-t

coordinate system that is aligned with each crack i (Figure 3.10).

If the vector which assembles the crack strain of each individual crack is denoted by

ecr, this yields,

( 3.20 )

with the crack strain for crack i given by ecr

( 3.21 )

74

Figure 3.10 Multi-directional fixed crack model For plane strain stress situation without loss of generality, the relation between the

global strain and the vector ecr is given by the transformation

( 3.22)

with N the assembled transformation matrix

( 3.23 )

with Ni the transformation matrix of crack i, which in the case of plane strain is

given by,

( 3.24 )

In a similar way it is possible to assemble a vector which contains the stress for each

Crack

( 3.25 )

with the crack stress for crack i given by,

( 3.26 )

The relation between the global stress and the vector scr can be derived as,

( 3.27 )

75

with the transformation matrix N given by equation 3.23. A basic assumption is that

the crack stresses are given as a function of the crack strains which results in the

general formulation,

( 3.28 )

Modeling of coupling effects between the different cracks is possible within this

general formulation but taking coupling into account would lead to an increasing and

unnecessary level of sophistication. For this reason, the crack stresses are solely

governed by the corresponding crack strains,

( 3.29 )

3.4.1.1 Crack Initiation

The constitutive model is complete if the criterion for crack initiation and the relation

between the crack stresses and the crack strains defined. The initiation of cracks is

governed by a tension cut-off criterion and a threshold angle between two

consecutive cracks. For successive initiation of the cracks the program for analysis

applies the following two criteria which must be satisfied simultaneously:

o The principal tensile stress violates the maximum stress condition.

o The angle between the existing crack and the principal tensile stress exceeds

the value of a threshold angle.

However, with these criteria it is possible that the tensile stress temporarily becomes

greater than three times the tensile strength while the threshold angle condition was

still not violated.

The principal tensile stress violates the maximum stress condition. The angle

between the existing crack and the principal tensile stress exceeds the value of a

threshold angle.

76

3.4.1.2 Crack Stress-Strain Relation

As defined in equation 3.29, the crack stresses in the n-t coordinate system of the

crack are determined by the corresponding crack strains,

( 3.30 )

A further simplification is made by ignoring the coupling between the normal stress crnnσ and the shear stress cr

nnτ . This results in the constitutive relation for crack i,

( 3.31 )

where the subscript i has been dropped for convenience. The Mode-I secant modulus

DI secant (Figure 3.11), is determined by use of the softening relation )( crnnnn

crnn f τσ =

according to:

( 3.32 )

In case of crack unloading, the secant stiffness remains constant. This means that

upon crack closing both the crack normal strain and the crack normal stress vanish.

Figure 3.11 Secant crack stiffness

The relation between the stress vector and the strain vector in the global coordinate

system can be derived starting from the strain decomposition equation 3.19, which

yields the relation for the global stress vector

( 3.33 )

77

Substitution of (equation 3.22) results in

( 3.34 )

With the notation crcrentcr eDS sec= for equation 3.31, and after substitution of the

relation between the stress vector in the global coordinate system and the stress

vector in the crack coordinate system (equation 3.27), the relation is written as

( 3.35)

which is written after some algebraic manipulations as

( 3.36 )

3.4.2 Total Strain Cracking Model

The constitutive model based on total strain is developed along the lines of the

Modified Compression Field Theory, originally proposed by Vecchio and Collins

(1986). The three-dimensional extension to this theory is proposed by Selby and

Vecchio (1993). Total Strain crack model describes the tensile and compressive

behavior of a material with one stress-strain relationship. These models can neither

be combined with other constitutive models, nor with ambient influence. This makes

the models very well suited for Serviceability Limit State (SLS) and Ultimate Limit

State (ULS) analyses, which are predominantly governed, by cracking or crushing of

the material. The input for the Total Strain crack models comprises two parts: (1) the

basic properties like the Young's modulus, Poisson's ratio, etc. and (2) the definition

of the behavior in tension, shear and compression.

3.4.2.1 Coaxial and Fixed Stress–Strain Concept

In total strain crack model, one commonly used approach is the coaxial stress-strain

concept, in which the stress-strain relationships are evaluated in the principal

directions of the strain vector. This approach, also known as the Rotating crack

model, is applied to the constitutive modeling of reinforced concrete during a long

period and has shown that the modeling approach is well suited for reinforced

concrete structures.

78

More appealing to the physical nature of cracking is the fixed stress-strain concept in

which the stress-strain relationships are evaluated in a fixed coordinate system,

which is fixed upon cracking. Both approaches are easily described in the same

framework where the crack directions are either fixed or continuously rotating with

the principal directions of the strain vector.

The basic concept of the total strain crack models is that the stress is evaluated in the

directions, which are given by the crack directions. The strain vector εxyz in the

element coordinate system xyz is updated with the strain increment ∆εxyz according

to,

( 3.37 )

which is transformed to the strain vector in the crack directions with the strain

transformation matrix T,

( 3.38 )

In a coaxial rotating concept the strain transformation matrix T depends on the

current strain vector, i.e.,

( 3.39 )

Where as in a fixed concept the strain transformation matrix is fixed upon cracking.

The behavior in compression is evaluated in a rotating coordinate system when the

material is not cracked, where in case of a fixed concept the compressive behavior is

evaluated in the fixed coordinate system determined by the crack directions.

The strain transformation matrix is determined by calculating the eigenvectors of the

strain tensor, e.g. with the Jacobi method. The strain tensor is given by

( 3.40 )

The eigen vectors are stored in the rotation matrix R which reads

79

( 3.41 )

with Cxn = Cosϕij the cosine between the i axis and the j axis. The strain

transformation matrix T is then calculated by substituting the appropriate values,

( 3.42 )

in a general three-dimensional stress situation. For the other stress situations the

appropriate sub-matrix should be taken. The constitutive model is then formulated in

the crack coordinate system which is generally given by

( 3.43 )

The updated stress vector in the element coordinate system is finally given by

( 3.44 )

The strain transformation matrix T is given by the current strain transformation

matrix in the coaxial rotating concept.

( 3.45 )

In a fixed concept, the strain transformation matrix T is given by the transformation

matrix at incipient cracking.

3.4.2.2 Lateral Expansion Effects due to Poisson’s Ratio

The Poisson effect of a material determines the lateral displacement of a specimen

subjected to a uniaxial tensile or compressive loading. If these displacements are

constrained a passive lateral confinement will act on the specimen. This effect is

considered important in a three-dimensional modeling of reinforced concrete

structures. In the work of Selby & Vecchio (1993) this effect is modeled through a

pre-strain concept in which the lateral expansion effects are accounted for with an

additional external loading on the structure. This implies that the computational flow

80

of the finite element engine is adapted to this method. The Poisson effect is taken

into account via the equivalent uniaxial strain concept. In case of linear-elastic

behavior the constitutive relationship in a three-dimensional stress–strain situation is

given by

( 3.46 )

( 3.47 )

This relationship is now expressed in terms of equivalent uniaxial strains as

( 3.48 )

with the equivalent uniaxial strain vector nstε defined by

( 3.49 )

Or

( 3.50 )

81

This concept is also applied to the nonlinear material model implemented in

numerical analysis. The stress vector in the principal coordinate system, is evaluated

in terms of the equivalent uniaxial strain vector, 123ε and not in terms of the principal

strain vector, nstε . The equivalent uniaxial strain vector is simply determined when

the principal strain vector and the (constant) Poisson’s ratio are known.

The tangent stiffness sub-matrix Dnst is slightly modified due to the equivalent

uniaxial strain concept. The matrix is given by

( 3.51 )

3.4.3 Tensile Behavior

The tensile behavior of reinforced concrete can be modeled using different

approaches, one resulting in a more complex description than the other. For the Total

Strain crack model, four softening functions based on fracture energy are

implemented, a linear softening curve, an exponential softening curve, the nonlinear

softening curve according to Reinhardt et al., and the nonlinear softening curve

according to Hordijk (1991), all related to a crack bandwidth as is usual in Smeared

crack models. Tensile behavior which is not directly related to the fracture energy

can also be modeled within the Total Strain concept. A constant tensile behavior, a

multi-linear behavior, and a brittle behavior are also implemented.

3.4.3.1 Tension Softening Relations

The relation between the crack stress crnnσ and the crack strain cr

nnε in the normal

direction can be written as a multiplicative relation,

( 3.52 )

in which ft is the tensile strength and crultnn.ε the ultimate crack strain. The general

function y (. . . ) represents the actual softening diagram. In numerical analysis both

the tensile strength and ultimate strain may be a function of temperature, (moisture)

concentration or maturity. Therefore the development of tensile strength and fracture

82

energy in time can be simulated. If the softening behavior on the constitutive level is

related to the Mode-I fracture energy IfG through an equivalent length or crack

bandwidth denoted as h , the following relation can be derived,

( 3.53 )

Thus the results become,

( 3.54 )

with the assumption that ft is a constant. Change from the variable crnnε to

( 3.55 )

and consequently d crnnε = cr

ultnn.ε dx results in the relation

( 3.56 )

where it is tacitly assumed that the ultimate crack strain crultnn.ε is finite. The final

expression for the ultimate crack strain is now given by,

( 3.57 )

with the factor α determined by the integral,

( 3.58 )

The factor crultnn.ε is assumed to be constant during the analysis and is considered to

be an element-related material property, which can be calculated from the material

properties, the tensile strength ft, the fracture energy IfG and the element area

represented by the equivalent length h.

3

I

s

F

T

w

T

3.4.3.2 Linea

In case of li

stress is give

Figure 3.12

The factor α

which result

The minimu

ar Tension S

inear tension

en by,

Linear ten

α (equation 3

ts in an ultim

um value of t

Softening

n softening a

nsion soften

.58) for the

mate crack st

the ultimate

83

as shown in

ning

ultimate cra

train

crack strain

Figure 3.12

ack strain is g

is then give

2, the relation

given by

en by

n of the crac

( 3.59

( 3.59a

( 3.59b

( 3.59c

ck

9 )

)

)

)

84

3.4.3.3 Multi-linear Tension Softening

Multi-linear behavior is completely defined by the user. If you define the behavior as

shown in Figure 3.13 then the initial slope should comply with basic equation, so

( 3.60 )

Figure 3.13 Multi-linear tension softening

3.4.3.4 Nonlinear Tension Softening by Hordijk et al.

Hordijk, Cornelissen & Reinhardt (1986) proposed an expression for the softening

behavior of concrete which also results in a crack stress equal to zero at a crack strain peak

nn.ε (Figure 3.14). The ultimate crack strain then reads,

( 3.61 )

The minimum value of the ultimate crack strain is then given by

( 3.62 )

and the reduced tensile strength reads

( 3.63 )

85

Figure 3.14 Hordijk tension softening

3.4.3.5 Brittle Cracking

Brittle behavior is characterized by the full reduction of the strength after the strength

criterion has been violated (Figure 3.15). This model involves a discontinuity. Before

the peak, there is only elastic strain. Beyond the peak, the stress drops to zero

immediately; the elastic strain vanishes and we have only crack strain. The sudden

stress drop, indicated by the dashed line in Figure 3.16.

Figure 3.15 Brittle tension softening

With tension-softening models, the ultimate strain is adapted to h, but with the brittle

cracking model the ultimate strain is fixed and always equal to ft/E so that a change

in h leads to a different energy Gf being consumed.

The issue is relevant especially for large scale unreinforced structures. Then, the

element dimensions and the crack band width may be large, so that the softening

diagram becomes very steep, brittle or even of the snap-back type. A solution may be

to refine the mesh and make sure that the ultimate strain of the softening diagram is

86

larger than ft/E. For reinforced structures, the issue is less relevant as the post-peak

input is based on tension stiffening considerations for distributed cracking rather than

fracture energy considerations for a single localized crack.

3.4.3.6 Exponential Tension Softening

Tension softening will occur exponentially as shown in Figure 3.16.

Figure 3.16 Exponential tension softening

3.4.3.7 Ideal Tension Softening

This is the ideal tension softening curve for concrete and is shown in Figure 3.17.

Value of tension for concrete is constant after peak. For reinforced concrete

structures, the post-peak is based on tension stiffening considerations by applying

reinforcement. No fracture energy based input is required for numerical analysis as

used in DIANA.

Figure 3.17 Ideal tension softening

87

3.4.4 Shear Retention Relations

Due to the cracking of the material the shear stiffness is usually reduced. This

reduction is generally known as shear retention. Diana offers two predefined

relations for shear retention: full shear retention and constant shear retention.

3.4.4.1 Full Shear Retention

In case of full shear retention the elastic shear modulus G is not reduced. The secant

crack shear stiffness is infinite if shear retention factor β=1. This implies that

( 3.64 )

3.4.4.2 Constant Shear Retention

In case of a reduced shear stiffness, the shear retention factor β is less or equal to

one, but greater than zero. The crack shear stiffness is then given by the general

relation,

( 3.65 )

In Total Strain Crack model, the modeling of the shear behavior is only necessary in

the fixed crack concept where the shear stiffness is usually reduced after cracking.

For the current implementation in Diana, only a constant shear stiffness reduction is

modeled, i.e.,

( 3.66 )

with β the shear retention factor, 0 ≤ β ≤ 1. For the rotating crack concept the shear

retention factor can be assumed equal to one. 3.4.5 Compressive Behavior

Concrete subjected to compressive stresses shows a pressure-dependent behavior,

i.e., the strength and ductility increase with increasing isotropic stress. Due to the

lateral confinement, the compressive stress–strain relationship is modified to

incorporate the effects of the increased isotropic stress.

Furthermore, it is assumed that the compressive behavior is influenced by lateral

cracking. To model the lateral confinement effect, the parameters of the compressive

88

stress-strain function, fcf and Ԑp, are determined with a failure function which gives

the compressive stress which causes failure as a function of the confining stresses in

the lateral directions.

If the material is cracked in the lateral direction, the parameters are reduced with the

factor βԐcr for the peak strain, and with the factor βσcr for the peak stress. It is tacitly

assumed that the base curve in compression is determined by the peak stress value,

( 3.67 )

The base function in compression, with the parameters fp and ɑp, is modeled with a

number of different predefined and user-defined curves.

The predefined hardening curves like (a) elastic, (b) ideal, (c) Thorenfeldt, (d) linear,

(e) multi-linear, (f) saturated type and (g) parabolic as shown in Figure 3.18

available.

Figure 3.18 Predefined compression behavior for Total Strain model

89

3.5 REINFORCEMENT

Embedded reinforcements add stiffness to the finite element model. Next to the

standard embedded reinforcements. The main characteristics of embedded

reinforcements are:

• Reinforcements are embedded in structural elements, the so-called mother

elements. Diana ignores the space occupied by an embedded reinforcement.

The mother element neither diminishes in stiffness, nor in weight. The

reinforcement does not contribute to the weight (mass) of the element.

• Standard reinforcements do not have degrees of freedom of their own. . In

standard reinforcements the strains in the reinforcements are computed from

the displacement field of the mother elements. This implies perfect bond

between the reinforcement and the surrounding material. However, with the

NOBOND input option can specify that the reinforcement is not bonded to

the embedding elements.

• Bond–slip reinforcements are only available as embedded lines in solid

elements. In this case the reinforcement bar is internally modeled as a truss or

beam elements, which are connected to the mother elements by line–solid

interface elements.

• In bond–slip reinforcements elastic or nonlinear bond–slip material behavior

may be defined for the line–solid interfaces in the bond–slip reinforcements.

• Bond–slip reinforcements may be applied for modeling slip of steel

reinforcement in concrete or for modeling interaction of pile foundations in

soil and rock.

90

3.5.1 Bar Reinforcement

Reinforcement bars may be embedded in various families of elements: beams, plane

stress, curved shell and solid. In finite element models with these elements, bar

reinforcements have the shape of a line.

Bars may also be embedded in plane strain and axisymmetric elements where they

have the shape of a point. The information in this section holds for the line-shaped

bars.

Figure 3.19 Reinforcement bar

Topology: The total length of the bar is considered to be divided in several particles

[Figure 3.19a]. By definition, a particle must be completely inside a structural

element. The so-called location points define the position of the particles in the finite

element model. Some location points are the intersections of the bar with the element

boundaries. Other location points are in-between these intersections, these points

define the curvature of the bar.

Usually, the location points are determined automatically by the program from input

of larger sections, this process is called preprocessing of reinforcement location. In

some cases it may be useful to specify the location points explicitly, which it call

element-by-element input.

Axes and variables: The program performs numerical integration of each particle of

a reinforcement bar separately, the isoparametric ξ axis is indicated in Figure 3.16a

for two particles. In this figure, integration points are marked with a small triangle ∆.

In each integration point the program determines an x) axis tangential to the bar

91

axis. The variables for a bar reinforcement are the strains Ԑxx and the stresses σxx

oriented in this x) axis (Figure 3.19b). The strains and stresses are coupled to the

degrees of freedom of the surrounding element.

Input data: The input data for bar reinforcements comprises the general material and

geometrical properties, the loading if appropriate and the specification of the location

of the bar.

3.5.2 Reinforcement Modeling

The reinforcement in a concrete structure can be modeled with the embedded

reinforcement types which are available in numerical FE analysis. The constitutive

behavior of the reinforcement can be modeled by an elasto-plastic material model

with hardening.

Furthermore, temperature influence on the Young’s modulus and thermal expansion

coefficient can be taken into account. Embedded reinforcements can be modeled as

bar or grid type. Four material models such as linear elasticity, two plasticity models

to model yielding of the reinforcement and special type material model are available.

3.5.2.1 Linear Elasticity

Embedded reinforcements in structural analysis require the input of Young’s

modulus E. Input of the thermal expansion coefficient α is only necessary in case of

temperature load.

3.5.2.2 Von Mises Plasticity

Von Mises plasticity and hardening models are available for embedded

reinforcements. Temperature influence on the plasticity is also possible.

3.5.2.3 Monti–Nuti Plasticity

The Monti–Nuti model is a special plasticity model for the cyclic behavior of steel. It

is available for embedded reinforcements. The model can be combined with different

hardening types.

92

3.5.2.4 Reinforcement Specials

Two special features can be applied for embedded reinforcements such as bonding

and corrosion influence.

Bonding Influence: For linear elasticity and plasticity it may require that the bar

reinforcement is not bonded to its mother elements. This option only applies for the

nonlinear calculation of the effects of post–tensioning.

Corrosion Influence: For embedded reinforcement it may specify corrosion

influence. It will require specifying corrosion influence time and corresponding

reduction factor of cross section data will require for this purpose.

93

CHAPTER-4

NONLINEAR SOLUTION TECHNIQUES

4.1 INTRODUCTION

In nonlinear Finite Element Analysis the relation between a force vector and

displacement vector is no longer linear. For several reasons, the relation becomes

nonlinear and the displacements often depend on the displacements at earlier stages,

e.g. in case of plastic material behavior. Just as with a linear analysis, it is required to

calculate a displacement vector that equilibrates the internal and external forces. In

the linear case, the solution vector could be calculated right away but in the nonlinear

case it cannot. To determine the state of equilibrium we not only make the problems

discrete in space (with finite elements) but also in time (with increments). To achieve

equilibrium at the end of the increment, it can use an iterative solution algorithm.

The combination of both is called an incremental-iterative solution procedure.

In this chapter we will consider a vector of displacement increments that must yield

an equilibrium between internal and external forces, and a stiffness matrix relating

internal forces to incremental displacements. In reality the physical meaning of items

in the ‘displacement’ vector can also be e.g. a velocity or a Lagrange multiplier. In

this chapter the physical meaning of what we call the displacement and force vector

and the stiffness matrix is irrelevant. Most often it represents a continuous system

that is approximated using the Principle of Virtual Work, Galerkin discretization or

another method.

4.2 BASIC NUMERICAL PROCESS FOR NONLINEAR PROBLEM

The use of finite element descretisation in a large class of nonlinear problems results

in a system of simultaneous equation of the form,

Hϕ + f = 0 ( 4.1 )

In which ϕ is the vector of the basic unknowns, f is the vector of applied `loads` and

H is the assembled `stiffness` matrix . For structural applications, the terms `load`

and `stiffness` are directly applicable, but for other situations the interpretation of

these quantities varies according to the physical problem under consideration.

94

If the coefficients of the matrix H depend on the unknowns ϕ or their derivatives, the

problem clearly becomes nonlinear. In this case, direct solution of equation system is

generally impossible and an iterative scheme must be adopted. Many options remain

open for iterative sequence to be employed. Some of the most generally applicable

materials available will now be outlined.

4.2.1 Method of Direct Iteration

In nonlinear Finite Element Analysis the relation between a force vector and

displacement vector is no longer linear. For several reasons, the relation becomes

nonlinear and the displacements often depend on the displacements at earlier stages,

e.g. in case of plastic material behavior. Just as with a linear analysis, we want to

calculate a displacement vector that equilibrates the internal and external forces. In

the linear case, the solution vector could be calculated right away but in the nonlinear

case it can not. To determine the state of equilibrium we not only make the problems

discrete in space (with finite elements) but also in time (with increments). To achieve

equilibrium at the end of the increment, we can use an iterative solution algorithm.

The combination of both is called an incremental-iterative solution procedure.

A vector of displacement increments that must yield an equilibrium between internal

and external forces, and a stiffness matrix relating internal forces to incremental

displacements. In reality the physical meaning of items in the `displacement' vector

can also be e.g. a velocity or a Lagrange multiplier. The physical meaning of what

we call the displacement and force vector and the stiffness matrix is irrelevant. Most

often it represents a continuous system that is approximated using the Principle of

Virtual Work, Galerkin discretization or another method.

In this approach successive solutions are performed, in each of the previous solution

for the unknowns ϕ is used to predict the current values of the coefficient matrix

H(ϕ) as,

ϕ = - [H(ϕ)]-1 f ( 4.2 )

95

then the iteration process yields the (r+1)th approximation to be,

ϕ(r+1) = - [H( ϕr )]-1 f ( 4.3 )

If the process is convergent then in the limit as r tends to infinity ϕr tend to the true

solution.

4.2.2 The Newton-Rapson Method

In numerical analysis, Newton's method (also known as the Newton–Raphson

method), named after Isaac Newton and Joseph Raphson, is perhaps the best known

method for finding successively better approximations to the zero. Newton's method

can often converge remarkably quickly, especially if the iteration begins "sufficiently

near" the desired value. Just how near "sufficiently near" needs to be, and just how

quickly "remarkably quickly" can be, depends on the problem.

Within the class of Newton-Raphson methods, generally two subclasses are

distinguished: the Regular and the Modified Newton-Raphson method. Both methods

use to determine the iterative increment of the displacement vector.

In a Newton-Raphson method, if u is the displacement, the stiffness matrix Ki

represents the tangential stiffness of the structure,

( 4.4 )

4.2.2.1 Regular Newton-Raphson.

In the Regular Newton-Raphson iteration the stiffness relation is evaluated every

iteration (Figure 4.1). This means that the prediction of is based on the last known or

predicted situation, even if this is not an equilibrium state.

F

T

w

i

A

i

c

M

i

i

d

4

F

i

d

s

s

t

b

Figure 4.1

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A disadvant

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4.2.2.2 Mo

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Regular N

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96

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97

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98

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99

specified maximum number of iterations have been reached or if the iteration

obviously leads to divergence. The detection of divergence is based on the same

norms as the detection of convergence. Figure 4.4 specifies the items used to set up

the various norms.

The choice of the proper norm and its convergence criterion depends on the type of

analysis. Using a lot of prescribed displacements generally makes the displacement

norm less useful. On the other hand, a structure that can expand freely will hardly

build up any internal forces and the force norm may be less useful. Always be sure

that the reference norm (the denominator in the ratios) has a reasonable value i.e., not

close to zero.

Figure 4.4 Convergence Norm

Experience shows that the convergence criterion for softening type behavior should

be stricter than the criterion that can be used in a hardening type analysis. If there is

any doubt about the criterion to be used, it is advisable to perform the analysis with

two distinct criteria and check the differences in results. If large differences occur, at

least the less strict norm was too large.

4

T

c

t

B

d

p

(

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4

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100

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101

minimize the norm, before the solution really converges to equilibrium. As with the

displacement norm, the energy norm also requires an additional iteration to detect

convergence.

4.4 INCREAMENTAL PROCEDURE

The incremental solution procedure consists of the load and displacement control and

the Arc-length method. The choice of method is depended on the results in the

current step of the increment. The initial choice of the step size for every increment

is an important factor in the incremental-iterative process. Therefore, two methods

are presented to determine step sizes and two methods to choose between loading

and unloading depending on the previous analysis results.

4.4.1 Load and Displacement Control

The iteration processes where the external load was increased at the start of the

increment, by directly increasing the external force vector fext. This is usually called

‘load control’ (Figure 4.5a). Another way to put an external load on a structure is to

prescribe certain displacements uc. This is called ‘displacement control’ (Figure

4.5b).

Figure 4.5 Load and displacement control

In case of displacement control the external force vector is not increased directly. To

get a proper first prediction of the displacements, the prescribed displacements must

be incorporated in the external force vector. This effective force can be calculated by

rewriting the displacement vector and splitting the displacement increment vector in

two parts: one referring to the unconstrained and an other referring to the constrained

102

displacements, respectively ∆uu and ∆uc. The stiffness matrix and force vector are

split likewise:

( 4.8 )

The unknown displacement increments ∆uu can be calculated from the first row in

equation 4.8.

( 4.9 )

Comparing displacement vector and equation 4.9 indicates that cuc uK Δ− 0 can be

regarded as the effective force vector, equivalent with the prescribed displacements.

In subsequent iterations, the iterative increments of the prescribed displacements are

zero and hence the effective force vector vanishes.

A similar effective force vector can be generated in case of influence of time on the

analysis e.g. prescribed temperature increments or viscoelastic material behavior. In

this case, the effective force vector contains the effect on the internal force vector

during the time increment if the displacements remain constant. The addition of this

effective force vector in the first prediction (zero iteration) will improve the

convergence of the iteration process significantly. In subsequent iterations, the time

does not change anymore and also this effective load vector will vanish. In real-life

analysis, the loading does not have to be restricted to load control, displacement

control or time increments, but they can be combined in any way. In that case the

‘real’ external load and the effective force vectors from prescribed displacement

increments and time influences must be used together.

4.4.2 Arc-Length Control

The arc-length method was originally introduced by Riks (1972, 1779) and Wempner

(1971) with later modifications being made by the Criesfield (1981, 1983), Ramm

(1981, 1982), Schweizerhof and Wriggers (1986), Forde and Stiemer (1987),

Gierlinski and Graves-Smith (1985), Belleni and Chulya(1987). The arc-length

method is suitable for nonlinear static equilibrium solution of unstable problems.

Application of arc-length method involves the tracing of complex path in the load-

103

displacement response into the buckling/post buckling regimes. The arc-length

method uses the explicit spherical iteration to maintain the orthogonality between

arc-length radius and orthogonal directions as described by Forde and Stiemer

(1987). It is assumed that all load magnitudes are controlled by a single parameter

(i.e. the total load factor). Unsmooth or discontinuous load-displacement response in

the cases often seen in contact analyses and elastic-perfectly plastic analyses can not

be traced effectively by the arc-length solution method. Mathematically, the arc-

length method can be viewed as the trace of a single equilibrium in a space spanned

by the nodal displacement variables and the total load factor. Therefore, all options

of Newton-Raphson methods are still the basic method for arc-length solution.

In an ordinary iteration process the predictions for the displacement increments can

become very large. This is the case especially if the load-displacement curve is

almost horizontal. If a fixed load increment is prescribed, this results in very large

predictions for the displacements. The problem can be overcome with the use of an

arc-length method. Using the arc-length method the snap-through behavior of Figure

4.6a can be analyzed, just as displacement control could. Here, however, it is

possible to define a system of loads that could not be substituted by prescribed

displacements.

Figure 4.6 Arc-length control

Moreover, the arc-length method is also capable of passing snap-back behavior

[Figure 4.6b], where displacements control fails. The Arc-length method constrains

the norm of the incremental displacements to a prescribed value. This is done by

104

simultaneously adapting the size of the increment. Note that the size is adapted

within the iteration process and is not fixed at the moment the increment starts.

Arc-length methods may only be used in combination with nodal or element loads,

not with prescribed non-zero displacements or with time steps. Arc-length methods

fail if the loading is dominantly nonconservative. Arc-length methods adapt the

loading during iterations in one load step. For adaptive loading in consecutive load

steps use the adaptive loading options.

In Arc length control the external force vector multiplies a unit load and can change

every iteration. Thus the results,

( 4.10 )

The solution δui is now split in two parts;

( 4.11 )

and

( 4.12)

The total iterative increment is then derived from,

( 4.13 )

FE analysis offers a quadratic and a linearized constraint, leading to the Spherical

Path Arc-length method and the Updated Normal Plane method.

4.4.2.1 Spherical Path ARC length method

In the spherical constraint, the constraint equation is,

( 4.14 )

Where ∆l is the required arc length. Thus,

( 4.15 )

105

with

( 4.16 )

4.4.2.2 Updated Normal Plane Arc length method

The second constraint is a linearized constraint. If spherical equation as shown earlier

is matched for ∆ui-1, then the constraint equation for ∆ui = ∆ui−1 + δui can be written

as,

( 4.17 )

where the quadratic term in δui is ignored. Thus,

( 4.18 )

Geometrically this constraint means that the iterative increment must be

perpendicular to the total increment at the previous iteration. The solution is

projected on the plane, normal to the previous solution, hence the method is referred

to as the Updated Normal Plane method.

4.4.2.3 Indirect Displacement Control

In the previous description of the constraint equations all displacements were

gathered together. For global nonlinear behavior this is adequate, but for local

collapse mechanisms the method can perform better if only a part of the

displacements is considered. The constraint equations can remain the same, but

instead of using the vectors δu and ∆u vectors δv and ∆v are considered, defined by,

( 4.19 )

In the extreme case that only one item in v is non-zero, the arc length is defined as

the displacement of the corresponding degree of freedom. A constant arc length

106

during the analysis will result in this case in equal displacement increments for this

degree of freedom. Because the loading is defined as an external force, this type of

control is called Indirect Displacement control. A variant of Indirect Displacement

control is Crack Mouth Opening Displacement control, usually called CMOD. This

can be used, just as in experiments, to control the increase in crack width et cetera. In

case of CMOD control, a vector is formed with new ‘degrees of freedom’ that can

e.g. represent the difference in displacements on opposite nodes on a crack plane.

( 4.20 )

As long as the displacement increments per step remain relatively small, the

difference between the Spherical Path method and the Updated Normal Plane method

are small. More important than the choice between these two methods is the choice

of the value for the arc length l.

4.5 ITERATIVE SOLUTION METHODS

The direct solution method has some drawbacks. The most important one is that the

background storage requirements can be extremely high for large three dimensional

problems. Another disadvantage is that the Gauss decomposition without pivoting

(i.e., interchanging rows and columns) is not numerically stable if the stiffness matrix

is not positive definite. For these reasons two iterative methods are available in

Diana as alternatives for the direct solution method. The common idea of all iterative

methods for solving the linear system of equations Ku = f is to generate a sequence

of approximations ui to the solution vector u via the recursion

( 4.21 )

in which Q is the preconditioning matrix or the preconditioner. In some way, Q

should resemble the inverse of the stiffness matrix K. Two algorithms to compute the

iteration parameters have been implemented in Diana, the Conjugate Gradient

107

method (CG) and the Generalized Minimal Residual algorithm (GMRES). Diana

uses CG for symmetric matrices and GMRES for nonsymmetric matrices.

4.5.1 Conjugate Gradient

The Conjugate Gradient method introduced by Hestenes & Stiefel (1952), is

currently the most popular and probably the best iterative method for systems with a

symmetric positive definite stiffness matrix, for example all linear elastic problems.

The CG algorithm generates γi such that all residuals ri are perpendicular. By making

clever use of the symmetry of K it is possible to orthogonalize the residual ri against

all previous residuals by making use of only the residuals of the two previous

iterations. This method is used in the current FE analysis of this thesis.

4.5.2 Generalized Minimal Residual

The GMRES method introduced by Saad & Schultz (1986), converges even if the

stiffness matrix is not positive definite, for instance if the stiffness matrix is

nonsymmetric. The iteration parameters are computed by orthogonalizing the

residual explicitly against all previous residuals. To be able to do this, all residuals

must be stored. Moreover, the number of computations per iteration increases since

the orthogonalization process becomes more expensive every iteration. Therefore the

iteration is restarted after a fixed number of residuals have been added to the basis.

4.6 SOLUTION TERMINATION CRITERIA

It can be specified a solution termination criterion to stop the execution of the load or

time steps in the current execution. Solution termination is based on loading, or

based on resulting strain or stress.

4.6.1 Loading Based Termination

Criteria based on loading are useful in combination with Arc-length methods or

adaptive loading. If a loading based termination criterion is specified then the

analysis takes the number of steps from the size of step command as a maximum

number. The stop criterion will stop the execution of the current execution command

108

block. The analysis will continue with the next block (if any). The following types of

load based termination criteria may be assigned;

( i ) Total load based termination : Solution will be terminated if total load is

reached.

( ii ) Increment load based termination : Execution will stop when the incremental

load is less than assigned incremental load. This type termination criteria is

used in the present analysis.

( iii ) Sign change based termination : Solution will stop when the sign of the load

vector changes. This criterion is for use in combination with automatic

loading–unloading or loading–unloading based on negative pivots. Sign for

automatic loading–unloading for the incremental step sizes similar to

spherical path arc-length control. This method may only be used in

combination with arc-length control.

4.6.2 Result Based Termination

With a termination criterion based on analysis results, Diana will stop the execution

of the steps in the current execution block if the value of certain strains or stresses

exceeds a specified extreme. The following types of results based termination criteria

may be assigned;

( i ) Selection of based termination : Selection of certain parts of the model for

which the criterion for resulting strain or stress must be applied. Otherwise the

analysis applies the specified results criterion on all elements of the model.

( ii ) Strain based termination : Execution will stop based on the total Green–

Lagrange strain as shown in Figure 4.7a.

109

Figure 4.7 Result based termination criteria.

( iii ) Stress based termination : Execution will stop based on total Cauchy stress as

shown in Figure 4.7b.

110

CHAPTER-5

EXPERIMENTAL DATA

5.1 INTRODUCTION

An experimental program was carried out on the author’s post graduate study which

was comprised of a planned series of tests on restrained as well as unrestrained slabs,

variation of flexural reinforcement and slab thickness (Alam, 1997). The

experimental study describes punching shear tests conducted on reinforced concrete

slabs with their edges restrained as well as unrestrained. Edge restraint was provided,

by means of edge beams of various dimensions, to mimic the behavior of continuous

slabs. A total of 15 model slabs were tested in an effort to ascertain the influence of

the degree of boundary restraint, percentage of steel reinforcement, and slab

thickness of the slab models on their structural behavior and punching load-carrying

capacity. The cracking pattern and load-deflection behavior of the slabs tested was

also monitored closely. The test program was carried out to provide basic

information on the real punching behavior of restrained slabs subjected to

concentrated loading and may also be usefully applied in the assessment of existing

structures with laterally restrained slab construction.

Experimental data of other researchers are briefly discussed in this chapter. Some

analysis, discussion of test result, comparison of test result are also added.

5.2 PREVIOUS EXPERIMENTAL PROGRAM BY ALAM (1997)

5.2.1 Specimen Details

A total of 15 square reinforced concrete slab specimens was constructed and tested

by Alam (1997). Twelve of these slabs had edge restraints in the form of edge beam,

whereas the other three samples were plain normal slabs having no edge beams.

Width of edge beam, slab thickness and reinforcement ratio were test variable

elements for different samples having one or more than one variability. Details of the

slab samples are given in Table 5.1 and typical plan and sectional details of slabs

with edge beam are shown on the Figure 5.1.

111

Table 5.1 - Details of Reinforced Concrete Slab Specimens (Alam, 1997)

Slab Sample

Width of edge beam ( b )

Slab thick-ness ( h )

Reinfor-cement

ratio ( ρ )

Main bars in each

direction

Extra top bars in each

direction

Edge beam reinforce-

ment

mm mm % No.-mm φ no.-mm φ no.-mm φ SLAB1 245 80 0.5 15- 6 15- 6 4-16 SLAB2 245 80 1.0 30-6 30-6 4-16 SLAB3 245 80 1.5 16-10 16-10 4-16 SLAB4 245 60 0.5 11-6 11-6 4-16 SLAB5 245 60 1.0 22-6 22-6 4-16 SLAB6 245 60 1.5 33-6 33-6 4-16 SLAB7 175 80 1.0 30-6 30-6 4-16 SLAB8 175 60 0.5 11-6 11-6 4-16 SLAB9 175 60 1.0 22-6 22-6 4-16 SLAB10 105 80 1.0 30-6 30-6 4-16 SLAB11 105 60 0.5 11-6 11-6 4-16 SLAB12 105 60 1.0 22-6 22-6 4-16 SLAB13 0 80 1.0 30-6 30-6 *3-16 SLAB14 0 60 0.5 11-6 11-6 *3-16 SLAB15 0 60 1.0 22-6 22-6 *3-16 *These reinforcements were provided at the extended bottom section of slab. All stirrups for edge beam were 6 mm φ @ 88 mm c/c., span=1200 mm

1200

mm

105/

175/

245

mm

60/8

0 m

m

120

mm

Loading block of size 120mm X 120 mm

6mm Ø @ 88mm c/c

4 - 16mm Ø (for edge beam)

6/10 mm Ø bar

SECTION X-X

PLAN VIEW

105/175/245 mm

1200mm105/

175/

245

mm

X

105/175/245 mm

X

Figure 5.1 Details of a typical model slab with reinforcement (Alam, 1997).

112

The concrete used in the specimens consisted of ordinary Portland cement, natural

sand and crushed stone aggregate with maximum size 10 mm. The water cement

ratio for concrete was 0.45. Both 6 mm and 10 mm diameter steel bars having

average yield strength of 421 MPa were used in the slab panels and stirrup of edge

beams. Flexural reinforcement in the edge beams were provided by 16 mm diameter

steel bars with average yield strength 414 MPa.

5.2.2 Test Results

All the models underwent punching type of failure with their inherent brittle

characteristics and failed in a punching shear mode. Most of the slab samples failed

at a load much higher than those predicted by the codes. The cracking pattern of the

top surface of all the slabs were very much localized and approximately had a size of

average 120mm x 120mm as shown in Figure 5.2. The cracking patterns at the

bottom surface of slabs having low percentage of reinforcement were more severe

than those having higher percentages of steel. It has been noticed that the surface

area of cracked zone for the slabs having wider edge beams were more than those

slabs having smaller edge beams. It has also been observed for all the samples that

the deflection at support was negligible, pointing out to the fact that support fixity

was ensured, albeit approximately, during the testing of the models. A typical crack

pattern after failure on the bottom surface of slab model is shown on Figure 5.3.

Test results obtained from this study have been analyzed and shown in the Table 5.2.

It has been found that ultimate punching shear capacity and behavior of slab samples

are dependent on restraining action of slab edges, flexural reinforcement ratio, slab

thickness and span-to-depth ratio of the slab.

Analysis of test result as shown in Table 5.2, where non-dimensional punching shear

strength Vu / fc'b0d, [where, d=effective depth of slab, b0= 4 x (120+d)] and

normalized punching shear strength Vu / f 'c b0d of each specimen have been given.

There is a general trend to increase the load carrying capacity of slabs with the

increase of width of edge beam as well as flexural reinforcement of slab.

113

Normalized punching shear strength of all slabs are higher that of ACI code formula

(Vu / f 'c b0d = 0.33) and Canadian code formula (Vu / f 'c b0d = 0.4).

Figure 5.2 Typical cracking pattern on the top surface of a model slab (Alam, 1997)

Figure 5.3 Typical cracking pattern on the bottom surface of a model slab (Alam, 1997).

114

Table 5.2 Test results,Non-dimensional and Normalized Punching Shear Strength of Reinforced Concrete Slabs(Alam, 1997)

Slab

Sample

Slab

thick-

ness

(h)

Reinfor-

cement

ratio ( ρ )

Width

of Edge

Beam

( b )

Experimental

Failure

Load

(Vu)

Cylinder

Strength

(fc')

Non-

Dimensiona

l Strength

(Vu / fc'b0d)

Normalized

Punching Shear

Strength

(Vu / f 'c b0d)

mm % mm kN MPa

SLAB1 80 0.5 245 225.16 38.51 0.1099 0.6820

SLAB2 80 1.0 245 242.09 37.42 0.1216 0.7439

SLAB3 80 1.5 245 142.95 28.19 0.0953 0.5061

SLAB4 60 0.5 245 138.12 38.24 0.1062 0.6569

SLAB5 60 1.0 245 147.59 36.60 0.1186 0.7175

SLAB6 60 1.5 245 130.51 41.95 0.0915 0.5927

SLAB7 80 1.0 175 181.64 32.45 0.1052 0.5994

SLAB8 60 0.5 175 133.27 41.30 0.0949 0.6099

SLAB9 60 1.0 175 115.51 33.14 0.1025 0.5902

SLAB10 80 1.0 105 188.89 37.45 0.0948 0.5802

SLAB11 60 0.5 105 112.88 40.43 0.0821 0.5221

SLAB12 60 1.0 105 115.73 37.04 0.0919 0.5593

SLAB13 80 1.0 0 171.96 37.72 0.0857 0.5263

SLAB14 60 0.5 0 84.73 34.71 0.0718 0.4230

SLAB15 60 1.0 0 91.76 33.03 0.0817 0.4696

Load deflection curve for all slabs is shown if Figure 5.4. It may be recalled that

complete load-deflection curves of the entire slab tested could not be traced due to

limitation of available instruments.

It appears that, deflection for both 80mm and 60mm thick slabs are very close for all

types of reinforcement ratio. Deflection of 80mm thick slabs is smaller than that of

60mm thick slab for same loading. Deflection is also very close in same thickness of

slab with different reinforcement ratio. Although the higher the reinforcement, the

smaller the deflection was observed for same loading as shown in these figures. The

value of deflection decreased, in general, as the slab thickness increases. Again, the

heavily reinforced slabs, on the whole, underwent lesser deflections and showed

slightly higher stiffness. It is, however, clear from figures that central slab deflections

were smaller for the slabs restrained by edge beams. The value of deflection

115

decreased, in general, as the degree of edge restraint increased. In general, for

smaller span to depth ratio of sample, the slab deflections at center were smaller than

those of higher span to depth ratio.

Figure 5.4 Test deflectionat slab center of all slabs under different loading

(Alam 1997, Alam et al. 2009a and 2009b). During the tests, the development of cracking and the width of cracks were carefully

observed and monitored at various load increments. Cracking on the underside of the

slabs developed as a series of cracks radiating from the centrally loaded area. As the

load increased, the widths of the cracks increased as expected. For lower level of

reinforcement (ρ=0.5 percent), numbers of cracks were small and more spalling

occurred than others. For higher level of flexural reinforcement (ρ=1.5 percent),

cracks were concentrated in the middle portion of the slab.

0

50

100

150

200

250

0 2 4 6 8 10 12 14 16 18

App

lied

Loa

d (

kN )

Deflection at Slab Center (mm)

Load Deflection Curve for all Samples

SLAB1SLAB2SLAB3SLAB4SLAB5SLAB6SLAB7SLAB8SLAB9SLAB10SLAB11SLAB12SLAB13SLAB14SLAB15

116

Cracking pattern were fine and large in number in case of strongly restrained slabs,

for moderately restrained slabs such cracks were found to be wider and fewer in

number. In case of strongly restrained slabs, due to the presence of in-plane forces,

the width of the cracks was less and consequently the total energy due to punching

was distributed among a large number of fine cracks (Alam 1997, Kuang and Morley

1992). On the other hand, in slab having lesser amount of lateral restraint, initially

produced cracks could widen and thereby, the total energy was distributed to lesser

but wider cracks.

5.2.3 Comparison of Test Results with different code of predictions A comparison of the experimental failure loads and the punching shear strength

predicted by various codes is shown in Figures 5.5 to 5.8. It is once again noted that

the nominal safety factor, partial safety factors, reduction factors, etc. have been

removed in this exercise and the actual strength of concrete of individual slabs has

been considered while plotting the graphs. In general, it was observed that

experimental punching failure load of most of the slabs was higher than punching

load carrying capacity calculated by the American code (ACI 318-2011), Australian

code (AS 3600-2009), British code (BS 8110-85), Canadian Standard (CSA-A23.3:

2004), European CEB-FIP –1990 code and German code (DIN 1045-1 : 2008). CEB-

FIP-1990 code is comparatively close to the experimental punching failure load. The

American code (ACI 318-2011) and Australian code (AS 3600-2009) are very close

to each other in all slabs.

Figure 5.5 and 5.6, represents the experimental failure load and ultimate punching

load carrying capacity of slabs having h=80mm of SLAB1, SLAB2, SLAB3 and

h=60mm of SLAB4, SLAB5, SLAB6 with same width of edge beam (b=245mm). It

is evident from these figures that the punching load carrying capacity having higher

the slab thickness is higher than smaller one. It is also shown that, experimental

punching load carrying capacity of 1% flexural reinforcement is higher than all

others of slabs of same thickness. Punching shear strength capacity with 1.5%

flexural reinforcement as calculated by British, CEB-FIP-1990, Canadian and

German codes are close to the experimental load carrying capacity. In Figure 5.7, it

117

is observed that punching shear capacity of slabs having 1.0 percent reinforcement

and 80mm thickness, load carrying capacity in accordance with British and Canadian

are very close. This tendency is also evident in Figure 5.8 for 1% flexural

reinforcement.

Figure 5.5 Comparison of test results with different code of prediction

at h=80mm and b=245mm.

Figure 5.6 Comparison of test results with different code of prediction at h=60mm and b=245mm.

118

Figure 5.7 Comparison of test results with different codes at same slab thickness

of h=80mm.


of h=60mm.

It appears that for slab samples having 0.5 percent reinforcement, load carrying

capacity predicted by the European and Canadian codes were close and in SLAB14,

this is very close to experimental punching capacity. German code is very

conservative than all other codes in 0.5% flexural reinforcement.

119

From the above discussion, it can be concluded that the present codes may not be

capable of predicting the punching shear strength of reinforced concrete slabs

satisfactorily taking into account the effect of edge restraint. For all the slabs tested

having more than 0.5% flexural reinforcement, the prediction of ACI 318-11 and AS

3600-09 are most conservative. On the other hand, although European CEB-FIP

codes are very much on the conservative side, its prediction of punching failure load

is better and more economical than the others.

5.3 TEST RESULTS OF OTHER RESEARCHERS

The FE analysis modeling has been used to simulate other independent experimental

investigations presented by Bresler and Scordelis (1963), Vecchio and Shim (2004),

Kotsovos (1984) and Kuang and Morley (1992).

It is to be noted that, several experimental data for various beams tested by by

Bresler and Scordelis (1963), Vecchio and Shim (2004), Kotsovos (1984) with and

without shear reinforcement are available. The current research is related to shear

failure condition. Thus, experimental data of beam without shear reinforcement have

been chosen for current analysis. Experimental model and data for both beam and

slab model are explained in the following sections.

5.3.1 Bresler and Scordelis Beam

The classic series of beam tests conducted by Bresler and Scordelis (1963) some 50

years ago to investigate the behavior of reinforced concrete in shear, is commonly

regarded as a benchmark against which finite element analysis models can be

calibrated. The 12 beams tested by Bresler and Scordelis (1963) consisted of four

series of three beams; each series differed in amount of longitudinal reinforcement,

amount of shear reinforcement, span length, cross-section dimensions, and concrete

strength. The typical cross-section details of beam tested by Bresler and Scordelis

(1963) for which FE analysis was developed is shown in Figure 5.9. Material

behavior of concrete and steel used for experimental works such as 'cf = 22.6 MPa,

yf = 555 MPa are used.

120

2- #9 (645 sq. mm each)

LONGITUDINAL SECTION OF BRESLER AND SCORDELIS BEAM

a

a

3660 mm 220220

556

2- #9 (645 sq. mm each)

556

Section a-a310

4 - #9

461

Load

Figure 5.9 Details of Bresler and Scordelis Beam.

5.3.2 Toronto Beam

The 12 Toronto beams named as tested in the University of Toronto and presented by

Vecchio and Shim (2004) were nominally identical to the Bresler–Scordelis (1663)

beams in terms of cross-section dimensions and reinforcement provided. Cross-

section details of a typical beam are shown in Figure 5.10. Material behavior of

concrete and steel used for experimental works are used as 'cf = 22.6 MPa, yf = 545

MPa for M25 rod and yf = 436 MPa for M30 rod.

2- M30 (700 sq. mm each)

LONGITUDINAL SECTION OF TORONTO BEAM

b

b3660 mm 220220

552

2- M25 (500 sq. mm each)

552

Section b-b305

2-M30

457

Load

2-M25

Figure 5.10 Details of Toronto Beam.

121

5.3.3 Kotsovos Beam

FE analysis of Kotsovos (1984) reinforced concrete beam subjected to two-point

loading as shown in Figure 5.11. Material behavior of concrete and steel used for

experimental works such as 'cf = 38MPa, yf = 502MPa are used.

As= 84.83 sq. mm

LONGITUDINAL SECTION OF KOTSOVOS BEAM

c

c918 mm 4141

102 10

2

Section c-c51

As (84.83 sq-mm)

90

Load Load204 204

Figure 5.11 Details of Kotsovos Beam.

5.3.4 Slab Tested by Kuang and Morley

Kuang and Morley (1992) tested 12 restrained reinforced concrete slabs with

varying span-to-depth ratio, percentage of reinforcement, degree of edge restraint

Three slabs denoted S1, S2 and S3 were modeled having same slab thickness of

60mm and same width of edge beam 280mm. Flexural reinforcement ratio for S1,

S2, and S3 of 0.30%, 1.0% and 1.6% respectively are used. Cube strength of

concrete ( cuf ) 48.7 MPa, 33.8 MPa and 41.2 MPa for S1, S2, and S3 respectively

are used. Average yield strength of steel 386 MPa for slab panel and 515 MPa for

edge beam are used. 120mm x 120mm loading block were used at geometric center

of slab. Typical geometry of model slab is shown in Figure 5.12.

122

4 - 12 Ø (for edge beam)

4/6 mm Ø bar 6mm Ø @ 110mm c/c

120 m

mX1 X1

PLAN VIEW

SECTION X1-X1

280 m

m12

00 m

m

1200 mm280 mm

60 m

m

280 mm

280 m

m

300 mm

Figure 5.12 Details of Kuang and Morley model slab.

123

CHAPTER-6

NUMERICAL EXAMPLES AND VALIDATION

6.1 INTRODUCTION

A numerical simulation for experimental punching shear behaviors of reinforced

concrete slabs has been developed based on author’s previous experimental studies

(Alam 1997) in this chapter. The numerical simulations using for reinforced concrete

slabs were based on the total strain crack model approach (Vecchio and

Collins 1986, Selby and Vecchio 1993). This study involves the development of a

nonlinear strategy which implements solution for a realistic description of the

deflection and crack process related to punching shear of RC slabs for several type of

slab thickness, edge restraints and reinforcement ratio. The simulation would

investigate experimental behavior of structural concrete slabs under different

loadings and it is thought that the findings would form the basis of further numerical

investigation on the punching shear behavior of RC slabs.

The FE analysis modeling is simulated with test result of other researchers. In this

case also, very good agreement has been obtained between the FE analysis and the

experimental data. A parametric study has also conducted by using exactly similar

geometry and material properties of each slab except flexural reinforcement of all

model slabs also presented in this chapter.

6.2 MODELING OF TESTED SPECIMEN

The modeling of a concrete structure, which has to be analyzed, can be divided into

three major parts: (i) the modeling of the geometry of structure, selection of

appropriate element and proper finite element meshing. (ii) the modeling of the

physical behavior of the materials which are applied in the structure, for instance

concrete and reinforcing steel, (iii) the modeling of the structural effects which

influence the behavior of the structure, for instance large displacements.

The finite element software DIANA 8.1 developed by TNO DIANA BV (2003) was

used to develop finite element model of concrete slabs at the preliminary stage of this

124

thesis work. In this thesis, it was decided to focus on modeling both the load

deflection characteristics of the slabs and cracking. Stress-strain behavior of a typical

slab model, which was not included in the experiment, is also discussed. After

availability of DIANA 9.4 developed by TNO DIANA BV (2010), FE model is

slightly revised to obtain better results as discussed in Section 6.12 and this FE

model is used next analytical works.

6.3 ELEMENT SELECTION

The elements adopted were twenty-node isoparametric solid brick element (elements

CHX60). The element CHX60 is based on quadratic interpolation and Gauss

integration. The basic variables in the nodes of this element are the translations in the

local element directions. In this rectangular brick element approximates the strain εxx

and stress σxx vary linearly in x direction and quadratically in y and z direction over

the element volume. Distribution of strain εyy and εzz, stress σyy and σzz follow

similar approximation.

Gaussian integration scheme 3×3×3 was used which yields optimal stress points. The

Green-Lagrange strains are derived for all integration points and may be extrapolated

to the nodes. The most important feature of this element is that it can represent both

linear and nonlinear behavior of the concrete. For the linear stage, the concrete is

assumed to be an isotropic material up to cracking. For the nonlinear part, the

concrete may undergo plasticity and/or creep.

6.4 MATERIAL MODEL OF CONCRETE

The input for the total strain crack models comprises two parts: (1) the basic

properties like the Young's modulus, Poisson's ratio, tensile and compressive

strength, etc and (2) the definition of the behavior in tension, shear, and compression.

6.4.1 Compressive Behavior

The pre-defined curves are the ideal curve and the brittle curve, and the linear and

exponential softening curves based on the compressive fracture energy. Other

available hardening-softening curves in compression are the parabolic, parabolic-

125

exponential, and the hardening curve according to Thorenfeldt et al. (1987). The pre-

defined Thorenfeldt curve is used in the present study.

6.4.2 Tensile Behavior

For the total strain crack model, four softening functions based on fracture energy are

used. Those are linear softening curve, exponential softening curve, nonlinear

softening curve according to Reinhardt (1984), and nonlinear softening curve

according to Hordijk (1991). Ideal tensile behavior, multi-linear behavior and brittle

behavior can also implemented in analysis.

The nonlinear tension softening curve according to Hordijk (1991) is used in the

present study.

6.4.3 Shear Behavior

A constant shear retention factor= 0.2 was considered for the reduction of shear

stiffness of concrete due to cracking.

6.5 MODELING OF SLAB GEOMETRY

The full-scale geometry of all slabs was modeled by using the finite element program

and meshed model of a typical slab is shown in Figure 6.1. The slab and edge beam

mesh were completely built with solid elements CHX20.

6.6 REINFORCEMENT MODEL

The reinforcement mesh in a concrete slab was modeled with the bar reinforcement

embedded in the solid element. In finite element mesh, bar reinforcements have the

shape of a line, which represents actual size and location of reinforcement in the

concrete slab and beam. Thus in the present study, reinforcements are used in a

discrete manner exactly as they appeared in the actual test specimens. The

constitutive behavior of the reinforcement can be modeled by an elastoplastic

material model with hardening. Tension softening of the concrete and perfect bond

between the bar reinforcement and the surrounding concrete material was assumed.

126

This was considered reasonable since welded mesh reinforcement was used in the

tests. Typical reinforcement in finite element model is shown in Figure 6.2.

The steel reinforcement behaves elastically up to the Von Mises yield stress of 421

MPa for slab and 414 MPa for edge beam.

6.7 BOUNDARY CONDITION

The edge beams of the slab were vertically restrained, as in the experimental set-up.

The beams were simply supported along four sides. One corner had all transnational

degrees of freedom fixed, while diagonally opposite of that corner was fixed with

two degrees of freedom so as to prevent the slab from moving and rotating in its own

plane.

6.8 LOADING

Loading was applied within at 120 mm x 120mm area of central portion of slab

model at the top surface to simulate actual experimental loading. A load step of 2 kN

applied for nonlinear analysis.

6.9 SOLUTION STRATEGY

A commonly used modified Newton–Raphson solving strategy was adopted,

incorporating the iteration based on conjugate gradient method with arc-length

control. Force norm convergence criteria was used in this study. The line search

algorithm for automatically scaling the incremental displacements in the iteration

process was also included to improve the convergence rate and the efficiency of

analyses. Second order plasticity equation solver solved physical nonlinearity with

total strain cracking. Reinforcement was evaluated in the interface elements.

Accuracy checked by the norms of residual vector. Incremental loading based

solution termination criteria was used to stop the execution.

6.10 RESULTS OF FE ANALYSIS

The aim of this study is to compare results of finite element analysis with the actual

experimental investigation. Thus, analysis results were accumulated in the same

manner as experimental test results. Typical deflected shape and stress contour of

slab model is shown in Figure 6.3. Experimental failure on top surface of slab model

127

was very localized which is represented analytically by stress contour on top surface

as shown in Figure 6.3 (a).

( a )

( b )

Figure 6.1 Meshed model of a typical slab showing ( a ) top surface, ( b ) bottom surface (Alam and Amanat 2012a).

128

Figure 6.2 Embedded reinforcement in a typical slab model (Alam and Amanat

2012a). Compressive stress developed on top surface and tensile stress developed on bottom

surface at the central region of slab as shown in Figure 6.3. Maximum compressive

stresses produced on top surface, which are concentrated around and within loading

block. But higher value of tensile stress developed outside of the loading block as

shown in Figure 6.3(b) and indicative to failure surface at that portion. This

analytical stress concentration at bottom surface as shown in Figures 6.3 is analogous

to experimental failure surface as well as cracking pattern of slabs.

Stress-strain curve of concrete is shown in Figure 6.4 and 6.5. In Figure 6.4, related

stress and strain are calculated at the integration point positioned on the top face of

slab. In Figure 6.4, element ET1 is located at the central top surface of slab model,

where as element ET2 and ET3 are 90mm and 150mm apart from slab center

respectively (Figure 6.1a). Concrete failed by compression at central region on top

surfaces earlier than apart from center as shown in Figure 6.4. In Figure 6.5, related

stress and strain are calculated at the integration point positioned on the bottom face

of slab. In this figure, element EB1 is located at the central bottom surface of slab

129

( a )

( b )

Figure 6.3 Deflected shape and stress contour shown on ( a ) top surface ( b ) bottom surface of typical slab model (Alam and Amanat 2012a).

130

model, where as element EB1 and EB2 are 90mm and 150mm apart from slab center

respectively (Figure 6.1b). Higher strain at same tensile stress obtained on central

region on bottom surface in compare to others. Higher the strain, higher the crack

width produced there. Stress-strain curve of typical embedded reinforcement at

central zone of slab is also shown in Figure 6.6. It is to note that reinforcements at

bottom surface of slab remain elastic and tensile strength is much lesser than yield

strength of steel, thus failure of steel does not occur here. Deflection contour for a

specific applied load is also shown in Figure 6.7. Deflection of slab decreases

gradually from center toward the edge beam as shown in Figures 6.7.

Figure 6.4 Compressive stress-strain on top surfaces of typical slab (Alam and

Amanat 2012a).

-35

-30

-25

-20

-15

-10

-5

0-0.0025 -0.0020 -0.0015 -0.0010 -0.0005 0.0000

Com

pres

sive

Str

ess (

MPa

)

Strain (mm/mm)

Element ET1Element ET2Element ET3

131

Figure 6.5 Tensile stress-strain on bottom surfaces of typical slab (Alam and

Amanat 2012a).

Figure 6.6 Tensile stress-strain diagram of typical reinforcement (Alam and

Amanat 2012a).

0.00

0.50

1.00

1.50

2.00

2.50

0.000 0.001 0.001 0.002 0.002 0.003 0.003 0.004

Ten

sile

Str

ess (

MPa

)

Strain (mm/mm)

Element EB1Element EB2Element EB3

0

10

20

30

40

50

60

0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030

Stre

ss (M

Pa)

Strain (mm/mm)

SLAB-7, 0.50% Rod

132

Figure 6.7 Deflection contour of bottom surface of a typical slab (Alam and

Amanat 2012a).

6.11 DISCUSSIONS ON FE ANALYSIS

6.11.1 Load-Deflection Behavior

It appeared that, deflection for both 80mm and 60mm thick slabs are very close for

all types of reinforcement ratio. Deflection of 80mm thick slabs is smaller than that

of 60mm thick slab for same loading. Deflection is also very close for slabs of same

thickness but with different reinforcement ratio. A little higher deflection was

observed for slabs with less reinforcement at same level of loading.

Load deflection curve for both experiment and FE analysis of all slabs are shown in

Figures 6.8 to 6.12. Due to instrumental limitation, complete experimental load-

deflection curves upto failure load could not be traced in the experiment. A

horizontal line in all load-deflection curves drawn in Figures 6.8 to 6.12 showing

experimental failure load. It is clear from Figures 6.8 to 6.12 that analytical load

deflection behavior of all model slabs are reasonably matched with experimental

result. In case of same width of edge beam, variation of deflection occurred due to

the variation of slab thickness and reinforcement ratio. Deflections of slabs without

edge beam are higher than all other slabs with edge beam as shown in Figure 6.12. It

is obvious that flexural reinforcement play a significant role in the behavior of RC

slab subjected to punching force.

133

Figure 6.8 Load-deflection curves of analyzed and tested model having slab thickness = 80mm and width of edge beam = 245mm (Alam and Amanat 2012a).

Figure 6.9 Load-deflection curves of analyzed and tested model having

slab thickness = 60mm and width of edge beam = 245mm (Alam and Amanat 2012a).

0

50

100

150

200

250

300

0 10 20 30 40 50 60

Loa

d (k

N)

Deflection (mm)

Analysis (SLAB-1)Test (SLAB-1)Analysis (SLAB-2)Test (SLAB-2)Analysis (SLAB-3)Test (SLAB-3)

SLAB-1

SLAB-2

SLAB-3

Exp. Failure Load of SLAB-1



0

50

100

150

200

250

300

0 10 20 30 40 50 60

Loa

d (k

N)

Deflection (mm)

Analysis (SLAB-4)Test (SLAB-4)Analysis (SLAB-5)Test (SLAB-5)Analysis (SLAB-6)Test (SLAB-6) SLAB-5

SLAB-6

SLAB-4


Exp. Failure Load of SLAB-5Exp. Failure Load of SLAB-6

134


width of edge beam = 175mm (Alam and Amanat 2012a).


width of edge beam = 105mm (Alam and Amanat 2012a).

0

50

100

150

200

250

300

0 10 20 30 40 50 60

Loa

d (k

N)

Deflection (mm)


SLAB-7

SLAB-8

SLAB-9




0

50

100

150

200

250

300

0 10 20 30 40 50 60

Loa

d (k

N)

Deflection (mm)


SLAB-10

SLAB-11

SLAB-12




135

Figure 6.12 Load-deflection curves of analyzed and tested model having no edge

beam (Alam and Amanat 2012a). The value of deflection decreased, in general, as the slab thickness increases. Again,

the heavily reinforced slabs, on the whole, underwent lesser deflections and showed

slightly higher stiffness. Higher reinforcement increases tensile strength capacity at

extreme fibre of slab, which causes lesser deflection.

The tensile strength of concrete is an important property because the slab will crack

when the tensile stress in the extreme fibre is exceeded. Due to increase of load,

crack width and depth will also increase which results increase of slab deflection.

Higher the slab thickness, due to increase of effective depth of concrete, tensile

strength at extreme fibre will be lower for slab loading. Due to higher section

modulus of 80mm thick slab, deflection of 80mm thick slab is smaller than that of

60mm slab.

6.11.2 Cracking

Figure 6.13 shows the crack pattern at bottom surface of finite element model of a

typical slab for applied load of 180 kN, where uniaxial principal strain characteristics

is used. Cracks at the bottom surface are propagated toward edge beam and major

0

50

100

150

200

250

300

0 20 40 60 80 100 120

Loa

d (k

N)

Deflection (mm)


SLAB-13

SLAB-14

SLAB-15




136

cracking area is concentrated to central region of slab. The major cracking produced

a circular bounded area in both analysis and experiment.

The trend and area of cracking is also similar as shown in Figures 6.15 to 6.17, where

similar grid line inserted on experimental cracked slab to compare with analytical

cracking area. Cracking area is smaller in case of strongly restrained slabs. For

comparatively smaller restrained slab such area is increased accordingly as shown in

Figures 6.14 to 6.17.

Figure 6.13 Cracking pattern of a typical slab at bottom surface (Alam and

Amanat 2012a).

( a )

( b )

Figure 6.14 Cracking at bottom surface of SLAB6 showing (a) experimental cracking pattern, (b) analytical cracking pattern (Alam and Amanat 2012a).

137

( a )

( b )

Figure 6.15 Cracking at bottom surface of SLAB9 showing (a) experimental cracking pattern, (b) analytical cracking pattern (Alam and Amanat 2012a).

( a )

( b )

Figure 6.16 Cracking at bottom surface of SLAB10 showing (a) experimental cracking pattern, (b) analytical cracking pattern.

( a )

( b )

Figure 6.17 Cracking at bottom surface of SLAB15 showing (a) experimental cracking pattern, (b) analytical cracking pattern.

138

6.12 UPDATED FE MODEL

After availability of DIANA 9.4 version developed by TNO DIANA BV (2010),

some additional features of this software have been explored. Thus, there were

scopes to update the FE model of slab as mentioned above and following

modification are applied.

In the FE model, the punching behavior of the slab as well as the detailed stress

condition and failure modes is studied around the central portion of slab. For this

reason, nonlinear material behavior for all FE elements around middle half central

portion the slab was applied. To make the FE modeling and analysis numerically

efficient and less time consuming, linear load deflection response by using elastic

material properties such as modulus of elasticity and Poisson’s ratio were applied to

other elements of the model slab.

The pre-defined Thorenfeldt compressive curve (Figure 3.18c) and ideal tension

softening curve (Section 3.4.3.7) are used in this study. In this curve value for

fracture energy is not necessary in input data.

A constant shear retention factor= 0.01 is considered for the reduction of shear

stiffness of concrete due to cracking. The Von Mises yield stress of 421 MPa for slab

and for edge beam 414 MPa is used for steel. Gaussian integration scheme 2×2×2 is

used which yields optimal stress points.

6.13 LOAD-DEFLECTION BEHAVIOR OF TESTED SLAB USING

UPDATED MODEL

Load deflection curve for both experiment and updated FE analysis of all slabs are

shown in Figures 6.18 to 6.22. It is shown from all those figures that experimental

and analytical load-deflection behaviors of all model slabs are reasonably matched.

Ultimate failure loads for model slabs can be comfortably traced from those curves.

Failure load of FE analysis are very close to experimental failure load. Similar trend

of load deflection behavior of numerical analysis and experimental data indicate to

have similar nature of other parameters for structural designing of slab. Thus, this

updated FE model is used in analysis of test results conducted by other researchers,

parametric study of author’s tested slab and multi-panel flat stab as discussed in next

sections and chapters.

F

F

Figure 6.18

Figure 6.19

0

20

40

60

80

100

120

140

160

180

200

0

Loa

d (k

N)

Load-defleslab thicknAmanat 20

Load-defslab thickAmanat 2

2

AnalysiTest (SLAnalysiTest (SLAnalysiTest (SL

ection curvesess = 80mm

012b, 2013a)

flection curvkness = 60m2013a).

4 6Defle

is (SLAB-4)LAB-4)is (SLAB-5)LAB-5)is (SLAB-6)LAB-6)

s of analyzedm and width o)

ves of analyzmm and width

8 10ection (mm)

SL

Exp. Failure Lo

d and tested of edge beam

zed and testeh of edge bea

12 14)

LAB-5

SLAB-

Expoad of SLAB-6

model havinm = 245mm

d model havam = 245mm

4 16

-6

SLAB-4

Exp. Failure L

p. Failure Load of

ng (Alam and

ving m (Alam and

18 20

4

Load of SLAB-4

f SLAB-5

d

F

F

Figure 6.20

Figure 6.21

0

50

100

150

200

250

300

0

Loa

d (k

N)

Load-defwidth of

Load-defwidth of

5

flection curvedge beam =

flection curvedge beam =

10

SLA

Exp. Failure L

ves of analyz= 175mm (A

ves of analyz= 105mm (A

15Deflection (mm

B-7 SLA

Load of SLAB-7

Exp. Failu

zed and testeAlam and Am

zed and testeAlam and Am

20m)

SLAB-8

AB-9

Ex

7

ure Load of SLA

d model havmanat 2013a)

d model havmanat 2013a)

25

Analysis (SLATest (SLAB-7Analysis (SLATest (SLAB-8Analysis (SLATest (SLAB-9

xp. Failure Load

AB-9

ving ).

ving ).

30

AB-7)7)AB-8))

AB-9)9)

d of SLAB-8

F

6

A

s

b

h

s

I

p

w

6

a

v

Figure 6.22

6.14 COMDIFFE

A compariso

shear capaci

be noted tha

have been re

slabs has bee

In Figure 6

percent rein

with British

6.24. It appe

as shown in

very close to

Load-defbeam (Al

MPARISON ERENT CO

on of the ex

ity predicted

at the nomin

emoved in th

en considere

6.23, it is o

nforcement a

h and Canad

ears that for

n Figure 6.24

o experiment

flection curvlam and Am

OF TEST RODE OF PR

xperimental

d by various

nal safety fa

his exercise

ed while plot

bserved tha

and 80mm t

dian are very

slab sample

4, load carry

tal punching

ves of analyzmanat 2013a)

RESULTS AREDICTION

failure loads

codes is sho

ctor, partial

and the actu

tting the grap

at punching

thickness, lo

y close. Thi

es without ed

ying capacit

g capacity.

zed and teste.

AND ANALNS

s, analytical

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safety facto

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phs.

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dge restraint

ty predicted

d model hav

LYSIS WIT

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of concrete

city of slab

g capacity i

is also evid

t (SLAB14 a

by the Can

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TH

the punchin

d 6.24. It is

n factors, et

e of individu

bs having 1

in accordanc

dent in Figur

and SLAB15

nadian code

e

ng

to

tc.

ual

.0

ce

re

5)

is

142


of h=80mm.

In general, it was observed that experimental and analytical punching failure load of

most of the slabs was higher than punching load carrying capacity calculated by the

American (ACI 318-2011), Australian (AS 3600-2009), British (BS 8110-97),

Canadian (CSA-A23.3-04 (R2010)) and German (DIN 1045-1 : 2008) codes. BS

8110-97 and Canadian (CSA-A23.3-04 (R2010)) codes are comparatively close to

the experimental and analytical punching failure load. The American (ACI 318-

2011) and Australian (AS 3600-2009) codes are very close to each other in all slabs.

From the above discussion, it can be concluded that some of the present codes are

not sufficiently capable for predicting the punching shear strength of reinforced

concrete slabs. For all the slabs tested having more than 0.5% flexural reinforcement,

the prediction of American and Australian codes are most conservative. On the other

hand, although British code (which considered effect of reinforcement) predictions

are on the conservative side, its prediction of punching failure load is better than the

others.

0

50

100

150

200

250

SLAB7 (ρ=1.0%) SLAB10 (ρ=1.0%) SLAB13 (ρ=1.0%)

Loa

d in

kN

h=80

Exp. Failure LoadACI 318AS 3600BS 8110CAN3-A23.3DIN 1045-1Analysis

143


of h=60mm.

6.15 PARAMETRIC STUDY

Load verses deflection diagram and failure load during testing has been reasonably

simulated using finite element model. A parametric study has been conducted by

using exactly similar geometry and material properties of each slab except flexural

reinforcement of all model slabs. Load-displacement response and punching failure

load for various reinforcement of each model slab monitored from the analysis. For

each model having various reinforcement ratios, different ultimate failure load

obtained. For comparison purpose, 245mm width of edge beam and 80mm thick slab

(similar to SLAB-2), 245mm width of edge beam and 60mm thick slab (similar to

SLAB-6), 175mm width of edge beam and 60mm thick slab (similar to SLAB-9),

105mm width of edge beam and 60mm thick slab (similar to SLAB-12) have been

chosen.

The normalized punching shear strength in accordance with ACI and Canadian code

formula ( dbfV c 0' ) [where, d=effective depth of slab, b0= 4 x (width of loading

block + d)] of various slab, have been analyzed in this study. Normalized punching

0

20

40

60

80

100

120

140

160

SLAB8 (ρ=0.5%)

SLAB9 (ρ=1.0%)

SLAB11 (ρ=0.5%)

SLAB12 (ρ=1.0%)

SLAB14 (ρ=0.5%)

SLAB15 (ρ=1.0%)

Loa

d in

kN

h=60mm Exp. Failure LoadACI 318AS 3600BS 8110CAN3-A23.3DIN 1045-1Analysis

s

e

F

F

shear strengt

edge restrain

Figure 6.25

Figure 6.26

0.

0.

0.

0.

0.

0.

0.

1.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ths are plott

nt and slab th

Normaliz80mm an

Normaliz60mm an

.30

.40

.50

.60

.70

.80

.90

.00

20

30

40

50

60

70

80

90

00

20

ted for differ

hickness as s

zed Punchingnd width of e


30

30

rent reinforc

shown in Fig

g Shear of Sedge beam =


40f 'c (MPa

40

f 'c (MPa

cement ratio

gures 6.25 to

lab model h= 245mm (sim


50a)

2% Flexural 1.5% Flexura1% Flexural R0.5% Flexural

50

a)

2% Flexural R1.5% Flexural1% Flexural R0.5% Flexural

of specimen

o 6.28.

aving slab thmilar to SLA


60

Reinforcementl Reinforcemen

Reinforcementl Reinforcemen

60

Reinforcement Reinforcemen

Reinforcement Reinforcemen

n having sam

hickness = AB-2).

hickness = AB-6).

70

tnt

nt

70

nt

nt

me

F

F

Figure 6.27

Figure 6.28

0.

0.

0.

0.

0.

0.

0.

1.

Normaliz60mm an

Normaliz60mm an

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

30

40

50

60

70

80

90

00

20



30

30

2% Flexural R1.5% Flexura1% Flexural R0.5% Flexura



40

f 'c (MP

2%1.51%0.5

40

f 'c (MPa

Reinforcementl ReinforcemenReinforcementl Reinforcemen



50

Pa)

% Flexural Rein5% Flexural Re% Flexural Rein5% Flexural Re

50

a)

nt

nt



60

nforcementeinforcementnforcementeinforcement

60

hickness = AB-9).

hickness = AB-12).

70

70

146

Normalized punching shear strengths for all slabs are higher than those of ACI code

( dbfV c 0' = 0.33) and Canadian code ( dbfV c 0

' = 0.4 ) as shown in all those

figures. The normalized punching load-carrying capacity of the all slab panels

increased with the increase of steel reinforcement ratio from 0.5 percent to 2.0

percent and decreasing tendency with increase of compressive strength of concrete

( 'cf ) as shown in Figures 6.25 to 6.28.

6.16 COMPARISON OF FE MODEL WITH TEST RESULTS OF OTHER RESEARCHERS

The same FE analysis model has been used to simulate other independent

experimental investigations presented by Bresler and Scordelis (1963), Vecchio and

Shim (2004), Kotsovos (1984) and Kuang and Morley (1992). Detail geometry,

material properties, support condition and loading are shown in the previous chapter

(Section 5.3). FE model and comparison with test results for both beam and slab

model are explained in the following section.

6.16.1 Bresler and Scordelis Beam

The typical meshed model of beam tested by Bresler and Scordelis (1963) for which

FE analysis was developed is shown in Figure 6.29. Similar material behavior of

concrete and steel used for experimental works as well as similar analysis procedure

is used in the FE analysis.

Load-deflection curves of analyzed and tested model of Bresler and Scordelis beam

is shown in Figure 6.30. Load-deflection curve of experimental works and FE

analysis are almost matched. Failure load of tested beam and analysis was 320 kN

and 316 kN respectively. In both cases, very good agreement has been found

between the FE analysis and the experimental data.

147

( a )

( b )

Figure 6.29 ( a ) Meshed Model (b) Deformed Shape of Bresler and Scordelis

Beam.

Figure 6.30 Load-Deflection curve of Bresler and Scordelis Beam.

6.16.2 Toronto Beam

The Meshed model of Toronto beam is similar to Bresler-Scordelis beam and shown

in Figure 6.31. Load-deflection curves of analyzed and tested model by Toronto

beam is Figure 6.32. Load-deflection curve of experimental works and FE analysis

are almost matched. Failure load of tested beam and analysis was 360 kN and 341

kN respectively. Good agreement has been obtained between the FE analysis and the

experimental data in both load-deflection curve and failure load.

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7

Loa

d (k

N)

Deflection (mm)

FE AnalysisTest (Bresler and Scordelis Beam)

148

( a )

( b )

Figure 6.31 ( a ) Meshed Model (b) Deformed Shape of Toronto Beam.

Figure 6.32 Load-Deflection curve of Toronto Beam.

6.16.3 Kotsovos Beam

FE meshing of Kotsovos (1984) reinforced concrete beam subjected to two-point

loading as shown in Figure 6.33. Load-deflection curves of analyzed and tested

model of Kotsovos beam is shown in Figure 6.34. Load-deflection curve of

experimental works and FE analysis are almost matched. Failure load of tested beam

and analysis was 36 kN and 35.74 kN respectively and almost same. In both

defection and failure behavior, very good agreement has been obtained between the

FE analysis and the experimental data.

0

50

100

150

200

250

300

350

400

0 2 4 6 8

Loa

d (k

N)

Deflection (mm)

FE AnalysisTest (Toronto Beam)

149

( a ) ( b )

Figure 6.33 ( a ) Meshed Model (b) Deformed Shape of Kotsovos Beam.

Figure 6.34 Load-deflection curve of Kotsovos Beam.

6.16.4 Slab Tested by Kuang and Morley

Typical geometry and meshed model of Kuang and Morley (1992) slab is shown in

Figure 6.35. Deformed shape of this model is shown in Figure 6.36.

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5

Loa

d (k

N)

Deflection (mm)

FE AnalysisTest (Kotsovos Beam)

150

Figure 6.35 Meshed model of Kuang and Morley model slab (top surface).

Figure 6.36 Meshed model of Kuang and Morley model slab (top surface).

151

Load-deflection curves of analyzed and tested model by Kuang and Morley (1992) is

shown in Figure 6.37. Load-deflection curves of experimental works and FE analysis

are almost matched. Experimental failure load of S1, S2 and S3 were 101 kN, 118

kN and 149 kN respectively. Where as in FE analysis of those models are 103.8 kN,

116.5 kN and 145.5 kN respectively. In this case also, very good agreement has been

obtained between the FE analysis and the experimental data.

Figure 6.37 Load-deflection curves of analyzed and tested model by Kuang and

Morley.

6.17 SENSITIVITY OF MESH SIZE

Physical scale and size effects influence the failure of structures and structural

components. This can be especially true when failure is due to brittle, quasi- brittle,

or ductile fracture. When simulating ductile fracture using the finite element method,

mesh size effects are also encountered. A common approach for analyzing the

response of hull structures due to grounding and impact, for example, is to eliminate

elements or to allow elements to split when a critical strain to failure is achieved.

However, an important complication arises because of the observed mesh size

0

20

40

60

80

100

120

140

160

0 2 4 6 8 10 12 14 16 18

Loa

d (k

N)

Deflection (mm)

Analysis (S3)Test (S3, 1.60% Rod)Analysis (S2)Test (S2, 1.00% Rod)Analysis (S1)Test (S1, 0.30% Rod)

152

sensitivity whereby strain to failure generally increases with finer finite element

meshes.

The concrete model in plain concrete zone is adjusted according to the fracture

energy balance in terms of the element size in computation. In order to examine the

mesh sensitivity of finite element analysis, two different meshes as shown in Figure

6.1 and 6.38 are used.

Figure 6.38 Meshed model of a slab having smaller size of mesh (same model as

shown in Figure 6.1). To compare analytical results for both sizes of mesh, load-deflection curve is drawn

and shown in Figure 6.39. As shown in Figure 6.39, load deflection behavior of

model having smaller element (Figure 6.38) is almost similar to that of meshed

model as shown in Figure 6.1.

In the present case for punching shear capacity analysis, it can be seen that the effect

of mesh size is negligible and the computed shear behavior is successfully common.

To make the FE modeling and analysis numerically efficient and less time

consuming, meshed model with smaller number of element were used in the present

study. The proposed model offers the stable convergence of the punching shear

capacity of reinforced concrete flat plate.

153

Figure 6.39 Load-deflection behaviors for various size of mesh

6.18 VALIDATION OF FE ANALYSIS

Such very good matching between the FE modeling and several experimental data

establishes the validity of the FE modeling technique in simulating the punching

shear behavior of flat plates and thus such modeling can be further applied to

numerically study the behavior of multi panel flat plate systems as an alternative to

experiments. Effect of edge restraint and flexural reinforcement were obtained from

above study, which is used in multi panel flat plate effectively.

It is found from parametric study that, the normalized punching shear carrying

capacity of RC slab has a decreasing tendency with increase of compressive strength

of concrete ( 'cf ). Increase of punching shear capacity of the all slab panels with the

increase of flexural reinforcement ratio from 0.5 percent to 2.0 percent is also

investigated from parametric study.

0

50

100

150

200

250

300

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00

Loa

d (k

N)

Deflection (mm)

Mesh as Figure 6.1 (SLAB-2)Mesh as Figure 6.38 (SLAB-2)

154

CHAPTER-7

FE MODELING AND STUDY ON MULTIPANEL FLAT PLATES

7.1 INTRODUCTION

According to present state of research, punching shear capacity is related to concrete

strength, punching perimeter of slab, slab thickness, reinforcement ratio, edge

restraint etc. Continuity of flat plate or effect of edge restraint on punching shear

behavior of slab is investigated. Application of continuity of flat plate to multi-panel

flat plates will be more reasonable than the results obtained by using isolated single

span slab specimens. However, multi-panel tests are time consuming, expensive and

it is difficult to determine experimentally the shears and moments applied to the

individual slab-column connections. An alternative to such expensive and difficult

experimental procedure is to perform the investigation by means of numerical finite

element analysis.

In this study, an advanced nonlinear finite element investigation of multi-panel flat

plate considering full scale with practical geometry has been carried out on the

behavior of punching shear characteristics of concrete slab in presence of flexural

reinforcement. FE model has been developed to simulate relevant experiments

carried out in the previous chapter. Good agreement has been observed between

numerical FE simulation and experiment, which establish the validity of FE model.

Same FE procedure has been used to analyze multi-panel slab models and the results

are presented in this study in an effort to understand the actual punching shear

behavior of slab systems.

7.2 BEHAVIORAL DIFFERENCE

Test results from simply supported slab specimens do not usually provide an accurate

prediction of the ultimate load capacity of a slab having lateral restraint (Kaung and

Morley 1992). When the slab is restrained against lateral deformation, this induces

large restraining force within the slab and between the supports, thus membrane

forces are developed (Selim and Sebastian 2003). The enhancement of punching

shear capacity can be attributed due to the presence of in-plane compressive

155

membrane action in the slab (Kaung and Morley 1992; McLean et al. 1990; Selim

and Sebastian 2003). The importance of compressive membrane stresses due to edge

restraint was not incorporated into the code formulations, which results in

conservative prediction.

The present exercise reveals that the edge restraint has a significant effect on the

ultimate punching load of reinforced concrete slabs, resulting in a significant

increase of punching shear resistance in the slabs and effectively enhancing the load-

carrying capacity of the member subjected to punching load. The enhancement in the

punching load carrying capacity of slabs due to edge continuity may be attributed to

the possible influence of in-plane restraint. This may be due to the lateral slab

expansion and possible restraint against outward movement by the edge beams.

Continuous slabs deflect less than similar simply supported slabs under the action of

load. This helps the slabs having edge continuity to sustain more punching load.

Enhancement of punching shear capacity due to restraint action was also advocated

by other researchers like Salim and Sebastian (2003), Kuang and Morley (1992),

Lovrovich and McLean (1990), McLean, et al. (1990) and Rankin and Long (1987).

Similar to the findings of other researchers, compressive membrane forces were, in

fact, developed in the slabs due to edge restraints.

Kuang and Morley (1992) tested a total twelve slabs which were supported and

restrained on all four sides by edge beams. Different degrees of edge restraint were

provided by different sizes of edge beam. They observed that a restrained slab with

low percentage of steel failed in punching shear mode when subjected to

concentrated loading. This indicated lateral restraint may also change the mode of

failure because the membrane forces developed enhances the shear and flexural

capacity of the slab and at the same time reduces the ductility of the slab. It is

apparent that from the above test results that the restraint can considerably enhance

the load carrying capacity of slab, but reduce the ductility of the slab. However, the

degree of the enhancement in strength due to the membrane action is difficult to

quantify since it depends on the in-plane restraint provided by the surrounding

156

structure. Others (Aoki and Seki 1971, Tong and Batchelor 1971) observed that

restraint will enhance punching shear strength of slab in all cases.

The enhancement of load carrying capacity can also be attributed to the presence of

in-plane compressive membrane action in the slab (Fenwick and Dickson 1989).

Membrane action generally occurs after cracking of the concrete or yielding of the

reinforcement, and has been found to result in substantial enhancement in the load

carrying capacity of restrained concrete slabs (Kuang and Morley 1992). These arise

from the coupling of in-plane and bending deformations in flexural cracked

reinforced concrete members and the restraint in the in-plane deformation provided

by the surrounding structure and the boundaries (Fenwick and Dickson 1989).

The restraining effects of the slab produces a higher load carrying capacity for the

spandrel beam in flat plate slab when compared with an isolated beam. This increase

in capacity is a result of the slab restraints on both the elongation and the rotation of

the edge beam (Loo and Falamaki 1992). Apart from this, the rotation of spandrel

beam also produces a vertical displacement at the beam slab interface. The vertical

displacement will be restrained by the vertical stiffness of slab. The restraining

effects of the slab on the rotation of beam also increases the strength of beam (Loo

and Falamaki 1992), thereby reducing the moment of the column-slab junction and

indirectly increasing the punching shear capacity.

In a normal simply supported reinforced concrete design, the neutral axis is located

closer to the compression face of the member, and so strain of the middle depth of

the slab is tensile over the full length, indicating expansion. Conventionally, this

length change is ignored. In practice, this expansion results in a compressive force

that enhances the performance of the member by reducing the magnitude of the

tensile force required in the reinforcement for a given load (Fenwick and Dickson

1989). Thus, slab deflection at column junction in the presence of edge beam also

affects the punching shear capacity.

157

Taylor and Hayes (1965) carried out a series of tests on the effect of edge restraint.

The slabs were divided into three groups depending on the amount of tension

reinforcement which was zero, 1.57% and 3.14%. The restraint was imposed by a

heavy welded steel frame which surrounded the slabs, i.e. the edges of slab were

restrained against lateral movement. All slabs without reinforcement were tested in

the restrained condition. For pairs of slab with reinforcement, one of each pair was

tested in the simply supported condition and the other in the restrained condition.

The test results indicated that for slab with low percentage of reinforcement, the

restraint significantly increased the ultimate load up to 60%. This group of slabs

exhibited high ductility and were more likely to fail in flexural mode. Tile ductile

behavior allowed compressive membrane forces to fully develop. The flexural

capacity was thus significantly increased as observed by other researchers (Roberts

1969). However, for slabs with high reinforcement ratio, the enhancement by

restraint was less significant and in some cases there was virtually no increase in

strength. Punching shear failure is critical for this group of slabs and the slabs

suddenly rupture. It is possible that the slab fails before the membrane action has

developed.

Hewitt and Batchelor (1975) stated that restraining forces at the slab boundaries can

result from compressive membrane (arch) action as well as from “fixed boundary

action”. The compressive membrane action gives a net in-plane force at the slab

boundaries, while fixed boundary action is due to moment restraint with no net in-

plane force at the slab boundary. Compressive membrane forces can be induced in a

cracked concrete slab but, unlike fixed boundary moments, cannot occur in a slab

that is uncracked or made from a material having the same stress-strain relationships

in compression and tension. Thus, compressive membrane action can occur in a

cracked unreinforced concrete slab, whereas fixed boundary action in a cracked slab

requires the provision of tension reinforcement at the boundary. A restrained

reinforced slab loaded to its punching load goes through the following stages: fixed

boundary action, cracking, compressive membrane action superimposed on fixed

boundary action, and finally punching shear failure.

158

The area of the slab beyond the line of contra flexure and external frames enhances

the capacity of slabs due to the action of in-plane compression forces. The portion of

the slab beyond the line of contraflexure acts as a tension ring which reacts against

compressive forces induced in the inner portion of the slab. Hewitt and Batchelor

(1975) wrote that the first model which took fixed boundary action into account, was

the model developed by Kinnunen and Nylander (1960).

Tests carried out by Csagoly (1979) for the Ontaria Ministry of Transportation and

Communications also indicated the increase in punching resistance due to membrane

action. Very high factors of safety against punching of slabs designed by

conventional methods were found. While Hewitt and Batchelor (1975) made a clear

distinction between the compressive membrane action and the fixed boundary action,

Bakker (2008) and Wei (2008) name the existence of any restraining force

“compressive membrane action”.

Effect of continuity can be effectively used in multi panel flat. In multi panel flat

plate, lateral deformation is resisted by the continuity of slab which induces large

restraining force within the flat plate, thus membrane forces are developed. The

enhancement of punching shear capacity in multi panel is occurred due to the

presence of membrane action in compare to simply supported slab.

7.3 FE MODELING OF MULTI PANEL FLAT PLATE

7.3.1 Flat plate Building system

Finite element model of the multi-panel full-scaled reinforced concrete is obtained

from a typical building structure with flat plate floor system as shown in Figure 7.1.

Multi panel flat plate is ideal for short span floors which are subjected to uniformly

distributed loads, and is used extensively in residential buildings as well as in certain

areas of commercial buildings. Flat plate can also be used for cantilever balconies as

permanent formwork. The design of multi panel flat plate concrete buildings

composed of flat plates and columns is considered. The analysis to calculate the

displacements and the internal forces, due to gravity load on flat plate buildings, has

to be preceded by defining the structural idealization model. FE analysis of the multi-

159

panel full-scaled reinforced concrete flat plate is considered in this thesis. Geometry

of a typical model is shown in Figure 7.2. Variable column size ( a x b ) of 400mm x

400mm, 600mm x 600mm and 800mm x 800mm are used for analysis. Similarly

variable slab thicknesses ( T ) of 200mm, 250mm and 300mm are used.

7.3.2 Boundary Condition and Loading

In this thesis, multi panel flat of a floor is simulated. Thus boundary condition and

support is idealized as actual building system. In current model, all columns are

vertically restrained at bottom ends and horizontally restrained both at top and

bottom ends. Uniformly distributed load was applied on the top surface of slab to

simulate actual behavior of practical slabs.

Figure 7.1 Perspective view of a typical building with flat plate.

160

6000 6000

6000

6000

1500

1500

1500

Slab Thickness = T (mm)

1500

X2

1500

1500

Section X2-X26000 60001500 1500

1500 1500

Plan View

X2a

b

aa a

Figure 7.2 Typical geometry of multi panel model slab.

161

7.3.3 Material Model of Concrete in Slab

The nonlinear finite element program DIANA 9.4 (2010) is used in this study. The

total strain approach with fixed smeared cracking (i.e. the crack direction is fixed

after crack initiation) is used in this study. For this approach, compression and

tension stress–strain curve are used.

7.3.3.1 Compressive Behavior

The available hardening-softening curves in compression are the parabolic, the

parabolic-exponential, and the hardening curve according to Thorenfeldt et al.

(1987). The pre-defined Thorenfeldt compressive constant curve (Figure 3.18c) is

used in the present study. Cylinder compressive strength of concrete at 28 days age

( 'cf ) is considered as ideal properties of concrete. Relationship of compressive

strength of concrete with Young’s modulus ( '4730 cc fE = ) and Poisson’s ratio

for concrete = 0.15 are used in this study.

7.3.3.2 Tensile Behavior

The pre-defined nonlinear linear tension-softening curve (Section 3.4.3.2) and ideal

softening curve (Section 3.4.3.7) are used in the present study. Relationship of

compressive strength of concrete with tensile strength ( '333.0 ct ff = ) is

used in this study. Ultimate value of plastic strain 0.02 (William et. al 1985, Amanat

1997) is used for linear tension softening in the FE analysis of this study.

7.3.3.3 Shear Behavior

The modeling of the shear behavior is only necessary in the fixed crack concept

where the shear stiffness is usually reduced after cracking. A constant shear retention

factor= 0.01 was considered for the reduction of shear stiffness of concrete due to

cracking.

7.3.4 Reinforcement Modeling

The reinforcement mesh in a concrete slab was modeled with the bar reinforcement

embedded in the solid element. In finite element mesh, bar reinforcements have the

shape of a line, which represents actual size and location of reinforcement in the

162

concrete slab and beam. Thus in the present study, reinforcements are used in a

discrete manner exactly as they appeared in the actual slab. The constitutive behavior

of the reinforcement modeled by an elastoplastic material model with hardening.

Tension softening of the concrete and perfect bond between the bar reinforcement

and the surrounding concrete material was assumed. This was considered reasonable

since welded mesh reinforcement was used in the tests. Typical reinforcement in

finite element model is shown in Figure 7.3.

Figure 7.3 Typical embedded reinforcement in the multi panel model at central

column.

The Von Mises yield stress of 421 MPa Young’s modulus of 200000 MPa and

Poisson’s ratio = 0.30 for steel reinforcement is used in this study. Similar types of

nonlinear parameters were also used in FE analysis of slab by Bailey et al. (2008).

7.4 FE MESHING OF MODEL

The twenty-node isoparametric solid brick element (elements CHX60) was adopted

for this study. Gaussian 2x2x2 integration scheme was used which yields optimal

stress points. The typical full model and enlarged portion of same model after

meshing are shown in Figures 7.4 and 7.5.

F

F

Figure 7.4

Figure 7.5

Mo

Enlarge

odel slab afte

ed corner of

er meshing (A

meshed mod

Alam and A

del (Alam an

Amanat 2014

nd Amanat 2

).

2014).

164

7.5 ANALYSIS PROCEDURE

A commonly used modified Newton–Raphson solving strategy was adopted,

incorporating the iteration based on conjugate gradient method with arc-length

control. Force norm convergence criteria was used in this study. The line search

algorithm for automatically scaling the incremental displacements in the iteration

process was also included to improve the convergence rate and the efficiency of

analyses. Second order plasticity equation solver solved physical nonlinearity with

total strain cracking. Reinforcement was evaluated in the interface elements.

Accuracy checked by the norms of residual vector. Incremental loading based

solution termination criteria was used to stop the execution.

7.6 PUNCHING TYPE OF FAILURE

All the models underwent punching type of failure with their inherent brittle

characteristics and failed in a punching shear mode. Most of the slab samples failed

at a load much higher than those predicted by the codes. The cracking patterns of the

top surface of all the slabs were very much localized. Punching type of failure is

confirmed by load-defection analysis and cracking pattern of a typical model slab

MSLAB11-7 ( 'cf =30 MPa and 0.50% flexural reinforcement) as discussed below.

To compare load deflection behavior of model MSLAB11-7 for various nodes

adjacent to central column is shown in Figure 7.6. In this figure node B00, B80,

B160, B320, B480 and B640 are located at bottom surface of slab at a distance 0mm,

80mm, 160mm, 320mm, 480mm and 640mm respectively from edge of middle

column toward the edge column along center line. Similarly nodes T00, T80, T160,

T320, T480 and T640 are located at top surface of slab. Load-deflection curve of

those nodes are shown in Figure 7.7. Deflections of node located same section of

slab such as B00 and T00, B80 and T80, B160 and T160, B320 and T320, B480 and

T480, B640 and T640 are almost matched. Similar deflect of top and bottom fiber at

any load is indicating no differential horizontal movement in same section of slab.

No differential horizontal movement of top and bottom chords at same section of

slab during failure load indicates that failure due to bending moment is not occurred

for model slab in this study.

165

It is clear from Figure 7.7 that punching type brittle failure occurs at and around

80mm from the edge of column. Deflected shape of this typical model slab before

failure load is shown in Figure 7.8. Punching type deflected shape before failure

adjacent to central column is observed as shown in Figure 7.8. Later on, serious shear

cracks at the bottom surface of slab around the central column before failure are

visible as shown in Figure 7.9.

8080160

B00B80B160B320B640

160

T00T80T160T320T640

Column

Slab

160160

B480

T480

Figure 7.6 Location of nodes points from central column along center line of

model MSLAB11-7.

Figure 7.7 Load-deflection curves of various nodes of model slab MSLAB11-7

for 'cf =30 MPa and 0.50% flexural reinforcement (Alam and Amanat

2014).

0

200

400

600

800

1000

1200

1400

1600

1800

0 10 20 30 40 50 60

Loa

d (k

N)

Deflection (mm)

Node B640Node B480Node B320Node B160Node B80Node B00Node T640Node T480Node T320Node T160Node T80Node T00

F

F

F

b

8

l

Figure 7.8

Figure 7.9

For analysis

bottom leve

8910 along

located adja

Deformed(Alam an

Typical before fai

s of stress-str

el of slab ar

center line o

acent to colu

d shape of nd Amanat 2

crack patterilure (Alam

rain of elem

re considere

of middle co

umn. Size of

a typical s014).

rn at the band Amanat

ents adjacen

d. Location

olumn is sho

f each eleme

slab MSLAB

ottom surfat 2014).

nt to central

n of those e

own in Figu

ent is 160mm

B11-7before

ace of slab

column, thre

elements 89

ure 7.10. Ele

m, thus elem

e failure loa

MSLAB11

ee elements

04, 8907 an

ement 8910

ment 8907 an

ad

-7

at

nd

is

nd

167

8904 are located at a distance 160mm and 320mm respectively from the edge of

column. Stress-strain curves for element 8904, 8907 and 8910 are shown in Figure

7.11. Stress-strain for each element is taken from integration point of bottom level.

Compressive stress is obtained as expected. According to Figure 7.11 element 8907

fails with brittle type of failure which is an indication of punching failure.

From above discussion and figures, punching failure of slab at middle column is

confirmed and ultimate failure load is obtained from load-deflection curve of slab

adjacent to middle column.

160

Central Column

Top Level of Slab

200

400

160160160

891089078904

y

xz

d

Figure 7.10 Location of elements from central column along center line of model MSLAB11-7.

168

Figure 7.11 Stress-Strain curves for various elements adjacent to central column

of model slab MSLAB11-7.

-35

-30

-25

-20

-15

-10

-5

0-0.0015 -0.0010 -0.0005 0.0000

Stre

ss, σ

z (M

Pa)

Strain (mm/mm)

Element 8904Element 8907Element 8910

169

CHAPTER-8

NUMERICAL EXAMPLES OF MULTI-PANEL FLAT PLATES

8.1 INTRODUCTION

In the FE model, the punching behavior of the slab as well as the detailed stress

condition and failure modes is studied around the central column. For this reason

nonlinear material behavior for all slab elements around the central column upto

1/4th of the adjacent span was studied. To make the FE modeling and analysis

numerically efficient and less time consuming, linear load deflection response by

using elastic material properties such as modulus of elasticity and Poisson’s ratio

were applied to other elements of the model away from central column.

Results of FE analyses of each model obtained from this study shows that ultimate

punching shear load and behavior of slab samples are dependent on flexural

reinforcement ratio, compressive strength of concrete, slab thickness, column size

which are discussed in detail in the following paragraphs. In this research work, slab

deflections are also studied to evaluate the actual punching shear behavior of slabs.

8.2 DIFFERENT SLABS COSIDERED

Multi-panel full-scaled reinforced concrete flat plate is modeled in present thesis.

Column size and slab thicknesses of model are summarized in Table 8.1.

The model consists of four equal panels, each of 6000mm square with nine square

columns of size 400mm x 400mm, 600mm x 600mm and 800mm x 800mm. The slab

is extended 1500 mm outward from all columns to simulate continuous action

beyond the column lines. All columns are extended by 1500mm from both top and

bottom surface of slab. Minimum slab thickness criteria in accordance with ACI code

is used to select slab thickness of studied flat plate. Variation of slab thickness such

as slab thickness of 200mm, 250mm and 300mm are used to investigate the effect of

punching shear of flat plate. Clear cover for flat plate from rod center of 25mm is

used.

170

A total 225 model slabs with variation of compressive strength of concrete ( 'cf ) and

percentage of flexural reinforcement are analyzed in this study. Compressive strength

of 24, 30, 40, 50 and 60 MPa for concrete are considered for analysis. Percentage of

flexural reinforcements having 0.25%, 0.5%, 1%, 1.5% and 2% for each compressive

strength of concrete are used. Details input data of a typical model is shown in Table

8.2. Similar data for other models are shown in the Appendix.

Table 8.1 Geometry of model slab

Model Group Slab Thickness Column size T (mm) a (mm) b (mm)

MSLAB11 200 400 400 MSLAB12 250 400 400 MSLAB13 300 400 400 MSLAB21 200 600 600 MSLAB22 250 600 600 MSLAB23 300 600 600 MSLAB31 200 800 800 MSLAB32 250 800 800 MSLAB33 300 800 800

Table 8.2 Details of model slab MSLAB11

Model

Slab thickness

Column Size

'cf ft Ec

ρ

(mm) (mm x mm) (MPa) (MPa) (MPa) (%)

MSLAB11-1 200 400 x 400 24 1.630 23180 0.25 MSLAB11-2 200 400 x 400 30 1.820 25920 0.25 MSLAB11-3 200 400 x 400 40 2.106 29940 0.25 MSLAB11-4 200 400 x 400 50 2.350 33470 0.25 MSLAB11-5 200 400 x 400 60 2.579 36670 0.25MSLAB11-6 200 400 x 400 24 1.630 23180 0.50MSLAB11-7 200 400 x 400 30 1.820 25920 0.50 MSLAB11-8 200 400 x 400 40 2.106 29940 0.50 MSLAB11-9 200 400 x 400 50 2.350 33470 0.50 MSLAB11-10 200 400 x 400 60 2.579 36670 0.50 MSLAB11-11 200 400 x 400 24 1.630 23180 1.00MSLAB11-12 200 400 x 400 30 1.820 25920 1.00MSLAB11-13 200 400 x 400 40 2.106 29940 1.00 MSLAB11-14 200 400 x 400 50 2.350 33470 1.00 MSLAB11-15 200 400 x 400 60 2.579 36670 1.00 MSLAB11-16 200 400 x 400 24 1.630 23180 1.50 MSLAB11-17 200 400 x 400 30 1.820 25920 1.50 MSLAB11-18 200 400 x 400 40 2.106 29940 1.50 MSLAB11-19 200 400 x 400 50 2.350 33470 1.50 MSLAB11-20 200 400 x 400 60 2.579 36670 1.50 MSLAB11-21 200 400 x 400 24 1.630 23180 2.00 MSLAB11-22 200 400 x 400 30 1.820 25920 2.00 MSLAB11-23 200 400 x 400 40 2.106 29940 2.00 MSLAB11-24 200 400 x 400 50 2.350 33470 2.00 MSLAB11-25 200 400 x 400 60 2.579 36670 2.00

171

8.3 RESULTS OBTAINED FROM ANALYSIS OF MODEL SLAB

8.3.1 Load-deflection behavior

Slab deflection behavior of each geometry of model slabs considering ideal tension

softening are shown in Figures 8.1 to 8.9. Similar load-deflection curves of other

models are included in the Appendix. Shortening of column for each load is deducted

from slab deflection to calculate actual slab deflection. Reaction of central column

for each load step is considered as punching load.

It is found that, for same deflection, load carrying capacities of slabs having 2%

flexural reinforcement are higher than 0.25% flexural reinforcement. Similarly, from

Figures 8.1 to 8.9, it is observed that slabs having higher compressive strength of

concrete ( 'cf ) sustain higher punching shear load carrying capacities than lower '

cf

at same deflection.


cf =24 MPa at a distance 320mm from the edge of the central column.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 5 10 15 20 25 30

Loa

d (k

N)

Deflection (mm)

2.00% Flexural Steel1.50% Flexural Steel1.00% Flexural Steel0.50% Flexural Steel0.25% Flexural Steel

172


cf =40 MPa at a distance 320mm from the edge of the central column.


cf = 50 MPa at a distance 288mm from the edge of the central column.

0

500

1000

1500

2000

2500

3000

0 5 10 15 20

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

3500

0 2 4 6 8 10 12

Loa

d (k

N)

Deflection (mm)


173





0

500

1000

1500

2000

2500

0 2 4 6 8 10 12 14

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

0 5 10 15 20 25 30

Loa

d (k

N)

Deflection (mm)


174





0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 2 4 6 8 10

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

0 5 10 15 20 25 30

Loa

d (k

N)

Deflection (mm)


175





0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 2 4 6 8 10 12

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

3500

0 2 4 6 8

Loa

d (k

N)

Deflection (mm)


176

Value of deflection is decreased in general with the increase of reinforcement ratio

and compressive strength of concrete. The heavily reinforced slabs, on the whole,

showed slightly higher stiffness and underwent lesser deflections. Higher

reinforcement and compressive strength of concrete increase tensile strength capacity

at extreme fibre of slab, causes lesser deflection. Similar trend of load deflection

behavior of numerical analysis indicates to have similar nature of other parameters

for structural designing of slab.

8.3.2 Sensitivity of Tension Softening to Multi Panel Model Slab

Tensile behavior of concrete using ideal tension softening and linear tension

softening are compared for each geometric group of model having several concrete

strength and flexural reinforcement ratios. Some model slabs are analyzed using

linear tension softening of concrete (Section 7.3.3.2) as well. All other parameters

and analysis procedure are remained exactly same as earlier models. Load-deflection

behavior of same model considering both ideal tension softening and linear tension

softening are compared and shown in Figures 8.10 to 8.12.

MSLAB11 ( 'cf = 30 MPa, ρ = 0.5%)

MSLAB11 ( '

cf = 30 MPa, ρ = 1%)

Figure 8.10 Load-deflection behaviors for ideal tension softening and linear tension softening of model slab MSLAB11.

0

500

1000

1500

2000

0 2 4 6 8 10

Loa

d (k

N)

Deflection (mm)

Ideal Tension SofteningLinear Tension Softening

0

500

1000

1500

2000

0 10 20 30

Loa

d (k

N)

Deflection (mm)

Ideal Tension softeningLinear Tension Softening

177

MSLAB11 ( 'cf = 30 MPa, ρ = 1.5%)

MSLAB11 ( '

cf = 30 MPa, ρ = 2%)

MSLAB12 ( 'cf = 30 MPa, ρ = 1%)

MSLAB12 ( '

cf = 30 MPa, ρ = 1.5%)

MSLAB13 ( 'cf = 30 MPa, ρ = 1%)

MSLAB13 ( '

cf = 50 MPa, ρ = 1.5%)

Figure 8.11 Load-deflection behaviors for ideal tension softening and linear

tension softening of model slab MSLAB11, MSLAB12 and MSLAB13.

0

500

1000

1500

2000

0 10 20 30

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

0 10 20 30

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

0 5 10 15 20

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

0 5 10 15 20

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

0 5 10 15 20

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

3500

0 5 10 15

Loa

d (k

N)

Deflection (mm)


178

MSLAB21 ( 'cf = 30 MPa, ρ = 1%) MSLAB21 ( '

cf = 40 MPa, ρ = 1.5%)


cf = 50 MPa, ρ = 2%)

MSLAB23 ( 'cf = 30 MPa, ρ = 1%)

MSLAB23 ( '

cf = 40 MPa, ρ = 1%)

Figure 8.12 Load-deflection behaviors for ideal tension softening and linear tension softening of model slab MSLAB21, MSLAB22 and MSLAB23.

0

500

1000

1500

2000

2500

0 5 10 15

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

0 5 10

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

0 10 20 30

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

0 2 4 6 8

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

3500

0 5 10 15

Loa

d (k

N)

Deflection (mm)


179

MSLAB31 ( 'cf = 24 MPa, ρ = 0.5%)

MSLAB31 ( '

cf = 40 MPa, ρ = 0.5%)


cf = 50 MPa, ρ = 1.5%)


cf = 30 MPa, ρ = 1%) Figure 8.13 Load-deflection behaviors for ideal tension softening and linear

tension softening of model slab MSLAB31, MSLAB32 and MSLAB33.

0

500

1000

1500

2000

2500

0 10 20 30

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

0 10 20 30

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

0 5 10 15 20

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

3500

0 2 4 6 8

Loa

d (k

N)

Deflection (mm)


0

500

1000

1500

2000

2500

3000

3500

4000

0 2 4 6

Loa

d (k

N)

Deflection (mm)


180

From Figures 8.10 to 8.13, it is found that load-deflection of all model groups with

various type of concrete strength and flexural reinforcement ratios are very close. For

comparison purpose, failure load using both tensile behaviors are tabulated in Table

8.3.

Table 8.3 Comparison of failure load using ideal and linear tension softening.

Model Slab thickness

Column Size

'cf ρ

Failure Load by

Ideal Tension

(V1)

Failure Load by Linear Tension

(V2)

V2/V1

(mm) (mm x mm) (MPa) (%) (kN) (kN)

MSLAB11

200 400 x 400 30 0.50 1552.80 1506.90 0.97 200 400 x 400 30 1.00 1683.00 1576.80 0.94

200 400 x 400 30 1.50 1750.60 1651.60 0.94

200 400 x 400 30 2.00 1818.40 1729.20 0.95

MSLAB12 250 400 x 400 30 1.00 2047.00 1870.00 0.91

250 400 x 400 30 1.50 2117.00 1971.00 0.93

MSLAB13 300 400 x 400 30 1.00 2382.00 2169.60 0.91

300 400 x 400 50 1.50 3229.00 2974.00 0.92

MSLAB21 200 600 x 600 30 1.00 2146.00 1918.00 0.89

250 600 x 600 40 1.50 2514.00 2246.00 0.89

MSLAB22 250 600 x 600 30 1.00 2310.00 2036.00 0.88

250 600 x 600 50 2.00 3169.00 2827.50 0.89

MSLAB23 300 600 x 600 30 1.00 2834.00 2648.00 0.93

300 600 x 600 40 1.00 3273.00 2984.00 0.91

MSLAB31 200 800 x 800 24 0.50 1987.00 1814.00 0.91

200 800 x 800 40 0.50 2238.00 2080.00 0.93

MSLAB32 250 800 x 800 30 1.00 2849.00 2469.00 0.87

250 800 x 800 50 1.50 3648.00 3176.00 0.87

MSLAB33 300 800 x 800 24 2.00 3240.00 2989.00 0.92 300 800 x 800 30 1.00 3380.00 3096.00 0.92

Average 0.92

Failure load for linear softening material are slightly smaller than ideal tension

softening of concrete. As shown in Table 8.3, it is found that failure load using linear

tension softening is average 0.92 times of ideal tension softening concrete behavior.

Thus, smaller value of ultimate failure load using linear tension softening of all

model slabs are accepted for next analysis considering 92% of ideal softening of

concrete and shown in next sections.

8

U

U

s

v

A

i

h

U

o

i

f

F

8.3.3 Ultim

Ultimate fai

Ultimate pu

strength of c

various flexu

According to

increase of f

higher upto

Ultimate pun

of 0.25%, 0

increment ra

for all flexur

Figure 8.14

5

10

15

20

25

Failu

re L

oad

in k

N

mate Failur

ilure load of

nching shea

concrete ( f

ural reinforc

o those figur

flexural rein

1% flexural

nching shea

0.5%, 1%, 1

ate is higher

ral reinforce

Ultimate

0

500

000

500

000

500

24

0.25%0.50%1.00%1.50%2.00%

re Load of M

f all model s

ar failure of '

cf ) of 24 MP

cement ratios

res punching

nforcement r

reinforceme

ar failure of m

1.5% and 2%

when comp

ment ratios

punching fa

30

% Flexural Steel% Flexural Steel% Flexural Steel% Flexural Steel% Flexural Steel

Multi Panel

slabs are sho

model slabs

Pa, 30 MPa

s such as 0.2

g shear failu

atio. Rate of

ent than that

model slabs

% are also

pressive stren

as shown in

ailure loads o

0 4

f'c (MPa)

Flat Plate

own graphic

s are plotted

a, 40 MPa, 5

25%, 0.5%, 1

ure load for e

f increment

of above thi

for each fle

increased w

ngth of conc

those figure

of MSLAB1

0

)

cally in Figu

d for various

50 MPa and

1%, a.5% an

each 'cf is i

of punching

is in all case

exural reinfo

with increase

crete is more

es.

1 for differe

50

ures 14 to 3

s compressiv

d 60 MPa an

nd 1.5%.

ncreased wi

g shear load

es.

orcement rat

e of 'cf . Th

e than 30 MP

ent '

cf .

60

1.

ve

nd

ith

is

tio

his

Pa

F

F

Figure 8.15

Figure 8.16

5

10

15

20

25

Failu

re L

oad

in k

N

5

10

15

20

25

30

35

40

Failu

re L

oad

in k

N

Ultimate reinforce

Ultimate

0

500

000

500

000

500

0.25

f'c=24f'c=30f'c=40f'c=50f'c=60

0

500

000

500

000

500

000

500

000

24

0.25%

0.50%

1.00%

1.50%

2.00%

punching fament ratio.

punching fa

0.50

4 MPa0 MPa0 MPa0 MPa0 MPa

30

% Flexural Steel

% Flexural Steel

% Flexural Steel

% Flexural Steel

% Flexural Steel

ailure loads o

ailure loads o

0 1.0

Flexural R

0 4

f'c (MPa

of MSLAB1

of MSLAB1

00 1

Reinforcemen

0

a)

1 for differe

2 for differe

.50

nt Ratio (%)

50

ent flexural

ent 'cf .

2.00

60

F

F

Figure 8.17

Figure 8.18

5

10

15

20

25

30

35

40

Failu

re L

oad

in k

N

5

10

15

20

25

30

35

40

45

Failu

re L

oad

in k

N

Ultimate reinforce

Ultimate

0

500

000

500

000

500

000

500

000

0.25

f'c=24f'c=30f'c=40f'c=50f'c=60

0

500

000

500

000

500

000

500

000

500

24

0.25%0.50%1.00%1.50%2.00%


punching fa

0.50


30


ailure loads o

ailure loads o

0 1.0

Flexural R

0 4f'c (MPa

of MSLAB1

of MSLAB1

00 1

Reinforcemen

0a)

2 for differe

3 for differe

.50

nt Ratio (%)

50

ent flexural

ent '

cf .

2.00

60

F

F

Figure 8.19

Figure 8.20

5

10

15

20

25

30

35

40

45

Failu

re L

oad

in k

N

5

10

15

20

25

30

35

Failu

re L

oad

in k

N

Ultimate reinforce

Ultimate

0

500

000

500

000

500

000

500

000

500

0.25

f'c=24f'c=30f'c=40f'c=50f'c=60

0

500

000

500

000

500

000

500

24

0.25%0.50%1.00%1.50%2.00%


punching fa

0.50


30


ailure loads o

ailure loads o

0 1.0

Flexural R

0 4

f'c (MPa

of MSLAB1

of MSLAB2

00 1

Reinforcemen

0

a)

3 for differe

21 for differe

.50

nt Ratio (%)

50

ent flexural

ent '

cf .

2.00

60

F

F

Figure 8.21

Figure 8.22

5

10

15

20

25

30

35

Failu

re L

oad

in k

N

5

10

15

20

25

30

35

40

45

Failu

re L

oad

in k

N

Ultimate reinforce

Ultimate

0

500

000

500

000

500

000

500

0.25

f'c=24f'c=30f'c=40f'c=50f'c=60

0

500

000

500

000

500

000

500

000

500

24

0.25%

0.50%

1.00%

1.50%

2.00%


punching fa

0.50


30

% Flexural Steel

% Flexural Steel

% Flexural Steel

% Flexural Steel

% Flexural Steel

ailure loads o

ailure loads o

0 1.0

Flexural R

0 4

f'c (MPa

of MSLAB2

of MSLAB2

00 1

Reinforcemen

0

a)

21 for differe

22 for differe

.50

nt Ratio (%)

50

ent flexural

ent '

cf .

2.00

60

F

F

Figure 8.23

Figure 8.24

5

10

15

20

25

30

35

40

45

Failu

re L

oad

in k

N

10

20

30

40

50

Failu

re L

oad

in k

N

Ultimate reinforce

Ultimate

0

500

000

500

000

500

000

500

000

500

0.25

f'c=24f'c=30f'c=40f'c=50f'c=60

0

000

000

000

000

000

24

0.25%0.50%1.00%1.50%2.00%


punching fa

0.50


30


ailure loads o

ailure loads o

0 1.0

Flexural R

0 4f'c (MPa

of MSLAB2

of MSLAB2

00 1

Reinforcemen

0a)

22 for differe

23 for differe

.50

nt Ratio (%)

50

ent flexural

ent 'cf .

2.00

60

F

F

Figure 8.25

Figure 8.26

10

20

30

40

50

Failu

re L

oad

in k

N

5

10

15

20

25

30

35

40

Failu

re L

oad

in k

N

Ultimate reinforce

Ultimate

0

000

000

000

000

000

0.25

f'c=24f'c=30f'c=40f'c=50f'c=60

0

500

000

500

000

500

000

500

000

24

0.25%0.50%1.00%1.50%2.00%


punching fa

0.50


30


ailure loads o

ailure loads o

0 1.0

Flexural R

0 4

f'c (MPa

of MSLAB2

of MSLAB3

00 1

Reinforcemen

0

a)

23 for differe

1 for differe

.50

nt Ratio (%)

50

ent flexural

ent '

cf .

2.00

60

F

F

Figure 8.27

Figure 8.28

5

10

15

20

25

30

35

40

Failu

re L

oad

in k

N

5

10

15

20

25

30

35

40

45

Failu

re L

oad

in k

N

Ultimate reinforce

Ultimate

0

500

000

500

000

500

000

500

000

0.25

f'c=24f'c=30f'c=40f'c=50f'c=60

0

500

000

500

000

500

000

500

000

500

24

0.25%

0.50%

1.00%

1.50%

2.00%


punching fa

0.50


30

% Flexural Steel

% Flexural Steel

% Flexural Steel

% Flexural Steel

% Flexural Steel

ailure loads o

ailure loads o

0 1.0

Flexural R

0 4

f'c (MPa

of MSLAB3

of MSLAB3

00 1

Reinforcemen

0

a)

1 for differe

2 for differe

.50

nt Ratio (%)

50

ent flexural

ent 'cf .

2.00

60

F

F

Figure 8.29

Figure 8.30

5

10

15

20

25

30

35

40

45

Failu

re L

oad

in k

N

10

20

30

40

50

Failu

re L

oad

in k

N

Ultimate reinforce

Ultimate

0

500

000

500

000

500

000

500

000

500

0.25

f'c=24f'c=30f'c=40f'c=50f'c=60

0

000

000

000

000

000

24

0.25%0.50%1.00%1.50%2.00%


punching fa

0.50


30


ailure loads o

ailure loads o

0 1.0

Flexural R

0 4f'c (MPa

of MSLAB3

of MSLAB3

00 1

Reinforcemen

0a)

2 for differe

3 for differe

.50

nt Ratio (%)

50

ent flexural

ent '

cf .

2.00

60

F

8

I

e

s

p

a

T

i

F

a

r

c

Figure 8.31

8.3.4 Sens

In FE mod

element. Ba

size and loc

present stu

appeared in

To assess

integration

Figure 8.32

a distance 1

respectively

column as b

10

20

30

40

50

Failu

re L

oad

in k

N

Ultimate reinforce

sitivity of Fl

del, steel is

ar reinforce

cation of re

dy, reinfor

n the actual

stress-stra

points suc

2 are consid

120mm from

y. “Point-3”

bottom rod.

0

000

000

000

000

000

0.25

f'c=24f'c=30f'c=40f'c=50f'c=60


lexural Stee

s used as th

ements hav

einforcemen

cements ar

slab.

in behavio

ch as “Poin

dered for an

m the edge

” is located

.

0.50


ailure loads o

el into the Fl

he bar rein

e the shape

nt in the co

re used in a

or of steel

nt-1”, “Poi

nalysis. “Po

e of central

d at a distan

0 1.0

Flexural R

of MSLAB3

lat plate

nforcement

e of a line,

oncrete slab

a discrete m

l for mod

nt-2” and

oint-1” and

column as

nce 2600m

00 1

Reinforcemen

3 for differe

embedded

which repr

b and beam

manner exa

el MSLAB

“Point-3”

“Point-2” a

top rod and

mm from ed

.50

nt Ratio (%)

ent flexural

in the sol

resents actu

. Thus in th

actly as the

B11-7, thre

as shown

are located

d bottom ro

ge of centr

2.00

id

ual

he

ey

ee

in

at

od

ral

191

6000 600060

0060

001500

1500

1500

Slab Thickness = 200 mm

1500

X3

200

1500

1500

Section X3-X3

400

6000 60001500 1500

400 400

1500 1500

Plan View of MSLAB11

X3

400

400

400

400

400

400

400

400

400

400

400

400

400

400

400

400

400

400

x

z

POINT -3POINT -2

POINT -1

Enlarge -A

Enlarge-A

4500 4500

POINT -3POINT -2

POINT -1

Figure 8.32 Location of integration point for steel.

Stress-strain curves of steel into the model flat plate at integration Point-1, Point-2

and Point-3 are shown in Figures 8.33 and 8.34. As shown in Figure 8.33, higher

value of stress and strain is obtained at Point-1 than that of Point-3. Stress-strain

behaviors of steel in both curves are elastic and tensile. Both curves are laid on same

line and steel do not reach to yield. Maximum value of stress is reached to 343 MPa

near column (Point-1) and 103 MPa at middle span (Point-3) of slab, which are

81.47% and 24.45% of yield stress of steel.

192

Compressive and elastic stress-strain behavior of steel at Point-2 is obtained as

shown in Figure 8.34 and the maximum stress does not reach to yield. Maximum

value of stress is reached to 250 MPa which is 60% of yield stress of steel.

Figure 8.33 Stress-strain of Point-1 and Point-3 for steel.

Figure 8.34 Stress-strain of Point-2 for steel.

0

50

100

150

200

250

300

350

400

0.000 0.001 0.002 0.003

Stre

ss, σ

z(M

Pa)

Strain (mm/mm)

Point-1Point-3

-400

-350

-300

-250

-200

-150

-100

-50

0-0.003 -0.002 -0.001 0.000

Stre

ss, σ

z(M

Pa)

Strain (mm/mm)

Point-2

193

8.4 DISCUSSION ON RESULTS AND COMPARISON WITH CODES

A total 225 model slabs with variation of compressive strength of concrete ( 'cf ) and

percentage of flexural reinforcement are analyzed in this thesis. Each of those model

slabs is individual and analytical result of each model is discussed in earlier sections.

In this section, various effects on punching shear of model slabs will be discussed

and compared in some groups with each other and also will be compared according

to various code of prediction. Grouping of model will be dependent on concrete

strength, flexural reinforcement ratio, slab thickness and column size. Normalized

punching shear according to ACI and Canadian code are also compared in this

section.

8.4.1 Punching Shear Stress of Multi Panel Flat Plate

The punching shear stress ( dbV o ) at a distance d/2 from edge of column [where, V

= punching failure load, d=effective depth of slab, b0= 4 x (side of column + d)] of

various model slabs, have been shown in Figures 8.35 to 8.43. Punching shear stress

for different flexural reinforcement ratios are increased with the increase of

compressive strength of concrete. Higher the flexural reinforcement ratio, the higher

punching shear stresses are obtained as shown in those figures.

Figure 8.35 Punching shear stresses of MSLAB11 at various compressive strength

of concrete.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

20 30 40 50 60

Punc

hing

She

ar S

tres

s (M

Pa)

f 'c (MPa)


194


of concrete.


of concrete.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

20 30 40 50 60

Punc

hibg

She

ar S

tres

s (M

Pa)

f 'c (MPa)


0.00

1.00

2.00

3.00

4.00

5.00

6.00

20 30 40 50 60

Punc

hibg

She

ar S

tres

s (M

Pa)

f 'c (MPa)


195


of concrete.


of concrete.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

20 30 40 50 60

Punc

hibg

She

ar S

tres

s (M

Pa)

f 'c (MPa)


0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

20 30 40 50 60

Punc

hibg

She

ar S

tres

s (M

Pa)

f 'c (MPa)


196


of concrete.


of concrete.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

20 30 40 50 60

Punc

hibg

She

ar S

tres

s (M

Pa)

f 'c (MPa)


0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

20 30 40 50 60

Punc

hibg

She

ar S

tres

s (M

Pa)

f 'c (MPa)


197


of concrete.


of concrete.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

20 30 40 50 60

Punc

hibg

She

ar S

tres

s (M

Pa)

f 'c (MPa)


0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

20 30 40 50 60

Punc

hibg

She

ar S

tres

s (M

Pa)

f 'c (MPa)


198

It is also found from Figures 35 to 43 that, punching shear stress is higher for higher

level of flexural reinforcement ratio. Although, the effect flexural reinforcement

toward punching shear stress is smaller for higher thicknesses of slabs like 250mm

and 300mm.

8.4.2 Non-Dimensional Punching Shear of Multi Panel Flat Plate

The non-dimensional shear due to punching load ( dbfV c 0' ) at a distance d/2 from

edge of column [where, V = punching failure load, d=effective depth of slab, b0= [4

x (side of column + d)] of various model slabs, have been shown in Figures 8.44 and

8.52.

The non-dimensional stress due to punching failure load of the all slab panels

decreases with increase of compressive strength of concrete as shown in all those

figures. Thus, contribution of concrete strength for punching shear load decreases

with the increase of concrete strength.

Figure 8.44 Non-dimensional stresses due to punching force of MSLAB11 for

various compressive strength of concrete.

0.00

0.10

0.20

0.30

20 30 40 50 60

Non

-Dim

ensi

onal

Str

ess f

or P

unch

ing

Loa

d

f 'c (MPa)


199





0.00

0.05

0.10

0.15

0.20

20 30 40 50 60

Non

-Dim

ensi

onal

Str

ess f

or P

unch

ing

Loa

d

f 'c (MPa)


0.00

0.05

0.10

0.15

0.20

20 30 40 50 60

Non

-Dim

ensi

onal

Str

ess f

or P

unch

ing

Loa

d

f 'c (MPa)


200





0.00

0.05

0.10

0.15

0.20

20 30 40 50 60

Non

-Dim

ensi

onal

Str

ess f

or P

unch

ing

Loa

d

f 'c (MPa)


0.00

0.05

0.10

0.15

0.20

20 30 40 50 60

Non

-Dim

ensi

onal

Str

ess f

or P

unch

ing

Loa

d

f 'c (MPa)


201





0.00

0.05

0.10

0.15

0.20

20 30 40 50 60

Non

-Dim

ensi

onal

Str

ess f

or P

unch

ing

Loa

d

f 'c (MPa)


0.00

0.05

0.10

0.15

0.20

20 30 40 50 60

Non

-Dim

ensi

onal

Str

ess f

or P

unch

ing

Loa

d

f 'c (MPa)


202





0.00

0.05

0.10

0.15

0.20

20 30 40 50 60

Non

-Dim

ensi

onal

Str

ess f

or P

unch

ing

Loa

d

f 'c (MPa)


0.00

0.05

0.10

0.15

0.20

20 30 40 50 60

Non

-Dim

ensi

onal

Str

ess f

or P

unch

ing

Loa

d

f 'c (MPa)


203

It is also found from Figures 44 to 52 that, non dimensional punching shear stress is

higher for higher level of flexural reinforcement ratio. Although, the effect flexural

reinforcement toward punching shear stress is smaller for higher thicknesses of slabs

like 250mm and 300mm.

8.4.3 Effect of Concrete Strength

The FE results of multi-panel slab models are compared according to concrete

strength in this section. Figures 8.53 to 8.64 are included with three group of model

slab. The groups are based on similar column sizes such as 400mm x 400mm,

600mm x 600mm and 800mm x 800mm. Effect of concrete strength on punching

shear capacity are also found on the basis of average normalized punching shear

strengths of various thicknesses of slab. Details of data are included in the Appendix.

8.4.3.1 On 400mm x 400mm column

Normalized punching shear strength ( dbfV c 0' ) of total 75 model slab having

400mm x 400mm column size are analyzed. Normalized punching shear strength of

all model slabs having 400mm x 400mm column are higher than ACI

( dbfV c 0'33.0= ) and Canadian ( dbfV c 0

'40.0= ) code. Average value of this

parameter for all slabs for this column is 0.60 which is 81% and 50% higher than

calculated by ACI and Canadian code respectively. Average normalized punching

shear strengths of 200mm thick slab is 0.67, 250mm thick slab is 0.60 and 300mm

thick slab is 0.54 with standard deviations of 6.83%, 3.97% and 3.02% respectively.

Thus, punching shear capacity of smaller thick slab is higher than those of higher

thick slab. Similarly minimum average normalized punching shear strength for

0.25%, 0.5%, 1%, 1.5% and 2% flexural reinforcement are 0.53, 0.56, 0.59, 0.62 and

0.63 respectively. There is an increasing tendency of punching shear capacity as

flexural reinforcement of slab is increased for 400mm x 400mm column. But this

tendency is very small for flexural reinforcement ratio is above 1.5%.

Normalized punching shear strength of model slab having slab thicknesses of

200mm, 250mm and 300mm are shown in Figure 8.53 to 8.55. According to Figure

8

d

F

p

A

h

a

n

T

d

c

l

F

8.53, the nor

decreases w

For, concret

punching loa

According t

having 250m

around 40 M

normalized p

The normal

decrease wit

concrete stre

load carrying

Figure 8.53

0

0

0

0

0

0

0

0

0

0

1

rmalized pu

with increase

e strength of

ad carrying i

o Figure 8.5

mm thick dec

MPa. For,

punching loa

ized punchi

th increase o

ength of 30 M

g is observe

Normalizof concre

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

unching shea

s of compre

f 50 MPa to

is observed.

54, the norm

crease with i

concrete str

ad carrying i

ing shear ca

of compressi

MPa to 60 M

d as shown i

zed punchingete of 200mm

30

ar capacity o

essive streng

60 MPa, ver

malized punc

increase of c

rength of 40

is observed.

apacity of a

ive strength

MPa, increas

in Figure 8.5

g shear strenm thick slab

40

f 'c (MPa

2%1.1%0.0.

of all slab pa

gth of concr

ry small or n

hing shear c

compressive

0 MPa to 6

all slab pan

of concrete u

sing tendency

55.

ngth at variou(400mm x 4

50

a)

% Flexural Rei5% Flexural R

% Flexural Rei5% Flexural R25% Flexural

anels having

rete upto aro

no decrease

capacity of a

e strength of

60 MPa, no

nels having

upto around

y of normali

us compress400mm colum

60

inforcementReinforcementinforcement

ReinforcementReinforcement

200mm thic

ound 50 MP

of normalize

all slab pane

concrete up

o decrease

300mm thic

d 30 MPa. Fo

ized punchin

sive strength mn).

0

t

ck

Pa.

ed

els

to

of

ck

or,

ng

F

F

Figure 8.54

Figure 8.55

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

Normalizof concre

Normaliz of concre

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20


zed punchingte of 300mm

30

30



40

f 'c (MPa

40

f 'c (MPa

21100



50

a)

2% Flexural 1.5% Flexur1% Flexural 0.5% Flexur0.25% Flexu

50

a)

2% Flexural Re.5% Flexural R% Flexural Re

0.5% Flexural R0.25% Flexural



6

Reinforcemenal ReinforcemeReinforcemenal Reinforceme

ural Reinforcem

60

einforcementReinforcementeinforcementReinforcementl Reinforcemen

sive strength mn).

sive strength mn).

60

ntent

ntent

ment

0

t

tnt

206

Thus, contribution of concrete strength for punching shear capacity decreases with

the increase of concrete strength for smaller thicknesses of slab up 40 MPa of

concrete strength. According to Figures 8.53 to 8.55, normalized punching shear

strengths are increased with increase of flexural reinforcement ratios of slab.

Although, very small or no increase of this parameter are obtained for 300mm thick

slab having 1.5% to 2% reinforcement.


It is observed that normalized punching shear strength ( dbfV c 0' ) of total 75

model slab having column size of 600mm x 600mm are higher than those of ACI (

dbfV c 0'33.0= ) and Canadian ( dbfV c 0


parameter for all slabs for this column is 0.55 which is 67% and 38% higher than

calculated by ACI and Canadian code respectively. Average normalized punching

shear strengths of 200mm thick slab is 0.62, 250mm thick slab is 0.53 and 300mm

thick slab is 0.50 with standard deviations of 7.94%, 3.76% and 3.1% respectively.

Thus, punching shear capacity of smaller thick slab is higher than those of higher

thick slab. Similarly minimum average normalized punching shear strength for

0.25%, 0.5%, 1%, 1.5% and 2% flexural reinforcement are 0.46, 0.50, 0.54, 0.57 and

0.59 respectively. There is an increasing tendency of punching shear capacity as

flexural reinforcement of slab is increased for 600mm x 600mm column.


200mm, 250mm and 300mm are shown in Figure 8.56 to 8.58. As shown in the

Figure 8.56, the normalized punching shear capacity of all slab panels having

200mm thick decreases with increases of compressive strength of concrete.

According to Figure 8.57, the normalized punching shear capacity of all slab panels

having 250mm thick decrease with increase of compressive strength of concrete upto

around 40 MPa. For, concrete strength of 40 MPa to 60 MPa, no decrease of

normalized punching load carrying is observed. Punching shear capacity is slightly

increased from 50 MPa concrete and reinforcement ratio from 0.5 and higher.

T

d

c

l

s

f

d

C

A

w

i

F

The normal

decrease wit

concrete stre

load carryin

strength for

for smaller t

decreasing te

Contribution

According to

with increas

increase of t

Figure 8.56

0.

0.

0.

0.

0.

0.

0.

0.

0.

ized punchi

th increase o

ength of 40 M

ng is observe

punching sh

thicknesses o

endency is o

n of concrete

o Figures 8.5

se of flexura

this paramete

Normalizof concre

0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

20

ing shear ca

of compressi

MPa to 60 M

ed as shown

hear capacity

of slab. For

observed upt

e has a posit

56 to 8.58, n

al reinforcem

er are obtain


30

apacity of a

ive strength

MPa, increas

n in Figure

y decreases w

higher thick

to 40 MPa of

tive tendency

normalized p

ment ratios

ned for 300m


40

f 'c (MPa

all slab pan

of concrete u

sing tendency

8.58. Thus,

with the incr

k slab such a

f concrete st

y above 40 M

punching she

of slab. Alt

mm thick slab


50

a)

2% Flexural R1.5% Flexural1% Flexural R0.5% Flexural0.25% Flexur

nels having

upto around

y of normali

, contributio

rease of con

as 250mmm

trength.

MPa concret

ear strengths

though, very

b.


60

Reinforcementl ReinforcementReinforcementl Reinforcemental Reinforcemen

300mm thic

d 40 MPa. Fo

ized punchin

on of concre

ncrete streng

or above, th

te in this cas

s are increase

y small or n

sive strength mn).

0

t

tnt

ck

or,

ng

ete

gth

his

se.

ed

no

F

F

Figure 8.57

Figure 8.58

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Normalizof concre

Normalizof concre

0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20



30

30



40

f 'c (MPa

21100

40

f 'c (MPa



50

a)

2% Flexural Rei1.5% Flexural R1% Flexural Rei0.5% Flexural R0.25% Flexural R

50

a)

2% Flexural Re1.5% Flexural R1% Flexural Re0.5% Flexural R0.25% Flexural



6

inforcementReinforcementinforcement

ReinforcementReinforcement

60

einforcementReinforcementeinforcementReinforcementl Reinforcement

sive strength mn).

sive strength mn).

0

0

t

209


Normalized punching shear strength ( dbfV c 0' ) of total 75 model slab

having 800mm x 800mm column are higher than those of ACI (

dbfV c 0'33.0= ) and Canadian ( dbfV c 0


parameter for all slabs for this column is 0.50 which is 51% and 25% higher

than calculated by ACI and Canadian code respectively. Average normalized

punching shear strengths of 200mm thick slab is 0.53, 250mm thick slab is

0.49 and 300mm thick slab is 0.46 with standard deviations of 9.05%, 5.31%

and 4.15% respectively. Thus, punching shear capacity of smaller thick slab is

higher than those of higher thick slab. Similarly minimum average normalized

punching shear strength for 0.25%, 0.5%, 1%, 1.5% and 2% flexural

reinforcement are 0.39, 0.43, 0.48, 0.52 and 0.54 respectively. There is an

increasing tendency of punching shear capacity as flexural reinforcement of

slab is increased for 800mm x 800mm column.


200mm, 250mm and 300mm are shown in Figure 8.59 to 8.61. As shown in

the Figure 8.59 to 8.61, the normalized punching shear capacity of all slab

panels having 200mm, 250mm and 300mm thick decreases with increases of

compressive strength of concrete.

For 250mm and 300mm thick slab such decrements are very slow rate in

compare to 200mm thick slab. Thus, contribution of concrete strength for

punching shear capacity decreases with the increase of concrete strength. For

higher thick slab such as 250mmm or above, this decreasing tendency is very

small.

F

F

Figure 8.59

Figure 8.60

0

0

0

0

0

0

0

0

0

0

1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Normalizof concre

Normalizof concre

.00

.10

.20

.30

.40

.50

.60

.70

.80

.90

.00

20

00

10

20

30

40

50

60

70

80

90

00

20



30

30



40

f 'c (MPa

2% F1.5%1% F0.5%0.25

40

f 'c (MPa

21100



50

a)

Flexural Reinfo% Flexural ReinFlexural Reinfo

% Flexural Rein% Flexural Re

50

a)

2% Flexural R1.5% Flexural 1% Flexural R0.5% Flexural 0.25% Flexura



60

forcementnforcementforcementnforcementeinforcement

6

einforcementReinforcemeneinforcementReinforcemen

al Reinforceme

sive strength mn).

sive strength mn).

0

60

nt

ntnt

F

8

I

s

a

t

A

p

s

p

T

t

A

p

Figure 8.61

8.4.3.4 On

In earlier se

shown. In th

analyzed on

thicknesses o

According t

panels havin

strength of c

punching lo

Thus, contri

the increase

According t

panels havin

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

1.

Normalizof concre

n Average th

ctions, norm

his section,

n the basis o

of slab as sh

to Figure 8.

ng 400mm

concrete upt

ad carrying

ibution of co

of concrete

to Figure 8.

ng 600mm

00

10

20

30

40

50

60

70

80

90

00

20


ickness of Sl

malized punc

effect of c

of average n

hown in Figu

62, the norm

x 400mm c

to around 40

is observed

oncrete stren

strength and

63, the norm

x 600mm c

30


lab

ching shear s

oncrete stre

normalized

ures 8.62 to 8

malized pun

column decr

0 MPa. Very

d from conc

ngth for pun

d after 40 MP

malized pun

column decr

40

f 'c (MPa

21100


strength of s

ngth to pun

punching sh

8.64.

nching shear

reases with

y small or n

crete strength

nching shear

Pa, it is very

nching shear

reases with

50

a)



same thickne

nching shear

hear strengt

r capacity o

increase of

no increase o

h of 40 MP

r capacity d

y small.

r capacity o

increase of

60


sive strength mn).

ess of slab a

r capacity a

ths of variou

of the all sla

f compressiv

of normalize

Pa to 60 MP

decreases wi

of the all sla

f compressiv

0

t

tnt

are

are

us

ab

ve

ed

Pa.

ith

ab

ve

s

p

T

t

A

p

s

K

t

i

F

strength of c

punching lo

Thus, contri

the increase

According t

panels havin

strength of c

Kinnunen an

to the crush

influences th

Figure 8.62

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

concrete upt

ad carrying

ibution of co

of concrete

to Figure 8.

ng 800mm

concrete from

nd Nylander

hing of con

he shear stre

Normalizof concre

20

to around 50

is observed

oncrete stren

strength and

64, the norm

x 800mm c

m 24 MPa to

r (1960) mod

ncrete. This

ngth of reinf

zed punchingete consideri

30

0 MPa. Very

d from conc

ngth for pun

d after 50 MP

malized pun

column decr

o 60 MPa.

del assumed

implies tha

forced concr

g shear strenng 400mm x

40

f 'c (MPa

2%1.51%0.50.2

y small or n

crete strength

nching shear

Pa, it is very

nching shear

reases with

d that the pu

at compress

rete slab.

ngth at varioux 400mm co

50

a)

% Flexural Rein5% Flexural Re% Flexural Rein5% Flexural Re25% Flexural R

no increase o

h of 50 MP

r capacity d

y small.

r capacity o

increase of

unching failu

sive strength

us compresslumn.

nforcementeinforcementnforcementeinforcementReinforcement

of normalize

Pa to 60 MP

decreases wi

of the all sla

f compressiv

ure occurs du

h of concre

sive strength

60

ed

Pa.

ith

ab

ve

ue

ete

F

F

Figure 8.63

Figure 8.64

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Normalizof concre

Normalizof concre

20

0

0

0

0

0

0

0

0

0

0

0

20



30

30



40

f 'c (MPa

21100

40

f 'c (MPa



50

a)


50

a)

2% Flexural R1.5% Flexural1% Flexural R0.5% Flexural0.25% Flexura

us compresslumn.

us compresslumn.


Reinforcementl ReinforcemenReinforcementl Reinforcemenal Reinforceme

sive strength

sive strength

60

t

tnt

60

nt

ntent

214

Test results compared by Mitchell, Cook and Dilger (2005) to the square root and

cube root of the concrete strength. The two functions were normalized to give a value

of1.0 at a concrete strength of 30 MPa. For each of the tests, the normalized shear

ratio is taken as the failure load divided by the failure load for the case with a

concrete compressive strength of 30 MPa, the cube root function appears to fit the

data for high strength concrete in a more conservative manner. However, Mitchell,

Cook and Dilger (2005) concluded that it is not clear whether the punching strength

is proportional to the square or cube root of the concrete strength and that additional

research is needed to enable the development of design expressions for punching

shear that are applicable to a wide range of concrete strengths, especially high

strength concrete.

Islam (2004) found from his study the effect of concrete strength on punching shear

in concrete slab. The author proposed to include the effect of concrete strength in

ACI-318 code.

It may be summarized that punching shear capacity is decreased with increase of

compressive strength of concrete upto certain limit. This decreasing tendency is

higher for smaller column size than that of higher sized column.

8.4.4 Effect of Flexural Reinforcement

The FE results of multi-panel slab models are compared according to flexural

reinforcement of slab in this section. Figures 8.65 to 8.78 are included with three

group of model slab. The groups are based on similar column sizes such as 400mm x

400mm, 600mm x 600mm and 800mm x 800mm. Effect of flexural reinforcement on

punching shear capacity are also shown on the basis of average normalized punching

shear strengths of various thicknesses of slab.


The normalized punching shear strengths of various slabs are plotted against

percentage of flexural reinforcement and shown in Figure 8.65 to 8.67. It has found

that having same concrete strength, normalized punching shear strength is increased

w

r

r

A

p

M

a

c

f

F

a

c

c

s

F

with additio

rate of incre

ratio than th

According t

punching str

MPa and 30

and 60 MPa

close curves

flexural rein

For higher th

all strength

capacity wi

concrete stre

small in thos

Figure 8.65

0.40

0.50

0.60

0.70

0.80

0.90

1.00

on of flexura

ease of punc

e above of th

to Figure 8

rength were

0 MPa with c

a concrete. F

s indicate tha

nforcement o

hicknesses o

of concrete

ill increase

ength for hig

se cases.

Normalizratio (400

0

0

0

0

0

0

0

0.00%

al reinforcem

hing load ca

his ratio.

8.65, For 20

e obtained in

compare to h

For such hig

at effect of c

of slab is very

of slab such

e as shown

with increa

gher thickne

zed punching0mm x 400m

0.50%

f'c=24 MPaf'c=30 MPaf'c=40 MPaf'c=50 MPaf'c=60 MPa

ment ratio fr

arrying capa

00mm thick

n case of lo

higher concr

gher concrete

concrete stre

y small.

as 250mm a

in Figure 8

ase of flexu

sses of slab

g shear of 20mm column)

1.00%

Reinforcem

from 0.25%

acity is highe

k slab, high

ower strength

rete strength

e strength cu

ength above

and 300mm,

8.66 and 8.6

ural reinforc

toward flexu

00mm thick .

1.50%

ment in %

to 2% perce

er upto 1% r

her values o

h of concret

such as 40

urves are ve

e 30 MPa co

, curves are

67. Thus, pu

cement, but

ural reinforc

at various re

2.00%

ent. Althoug

reinforceme

of normalize

te such as 2

MPa, 50 MP

ry close. Th

oncrete towar

very close f

unching she

influence

cement is ver

einforcement

2.50%

gh

ent

ed

24

Pa

his

rd

for

ear

of

ry

t

F

F

Figure 8.66

Figure 8.67

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Normalizratio (400

Normalizratio (400

0

0

0

0

0

0

0

0.00%

0

0

0

0

0

0

0

0

0.00%



0.50%


0.50%




1.00%

Reinforcem

1.00%

Reinforcem

50mm thick .

00mm thick .

1.50%

ment in %

1.50%

ment in %

at various re

at various re

2.00%

2.00%

einforcement

einforcement

2.50%

2.50%

t

t

8

T

p

t

w

r

r

A

s

a

a

M

F

o

t

v

F

8.4.4.2 On

The normal

percentage o

that having

with additio

rate of incre

ratio than th

According t

strength wer

and 40 MPa

are very clo

MPa concret

For 250 mm

obtained up

thick slab as

very close fo

Figure 8.68

0.40

0.50

0.60

0.70

0.80

0.90

1.00

600mm x 60

lized punch

of flexural re

same concre

on of flexura

ease of punc

e above of th

o Figure 8.6

re obtained i

a. For 50 MP

ose. This clo

te toward fle

m thick slab

to 1% flexu

s well as all p

or all strengt

Normalizratio (600

0

0

0

0

0

0

0

0.00%

00mm colum

hing shear

einforcemen

ete strength,

al reinforcem

hing load ca

his ratio.

68, For 200m

in case of low

Pa and 60 M

ose curves in

exural reinfo

as shown i

ural reinforc

percentages

th of concret


0.50%


mn

strengths of

nt and shown

normalized

ment ratio fr

arrying capa

mm thick sla

wer strength

Pa concrete,

ndicate that

orcement of

in Figure 8.

cement. For

of reinforce

te as shown


1.00%

Reinforcem

f various s

n in Figure

d punching s

from 0.25%

acity is highe

ab, variation

h of concrete

, such higher

effect of co

slab is very

69, influenc

above 1%

ment of 300

in Figure 8.6

00mm thick .

1.50%

ment in %

slabs are pl

8.68 to 8.70

hear strengt

to 2% perce

er upto 1% r

n of normali

e such as 24

r concrete st

oncrete stren

small.

ce of concre

flexural ste

0mm thick sl

69 and 8.70.

at various re

2.00%

lotted again

0. It has foun

th is increase

ent. Althoug

reinforceme

ized punchin

MPa, 30 MP

trength curv

ngth above 4

ete strength

eel of 250m

lab, curves a

einforcement

2.50%

nst

nd

ed

gh

ent

ng

Pa

es

40

is

mm

are

t

F

F

Figure 8.69

Figure 8.70

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Normalizratio (600

Normalizratio (600

0

0

0

0

0

0

0

0.00%

fffff

0

0

0

0

0

0

0

0

0.00%



0.50%


0.50%




1.00%

Reinforcem

1.00%

Reinforcem

50mm thick .

00mm thick .

1.50%

ment in %

1.50%

ment in %

at various re

at various re

2.00%

2.00%

einforcement

einforcement

2.50%

2.50%

t

t

219

Thus, punching shear capacity will increase with increase of flexural reinforcement,

but influence of concrete strength for higher thicknesses of slab toward flexural

reinforcement is very small in those cases.


The normalized punching shear strengths of various slabs are plotted against

percentage of flexural reinforcement and shown in Figure 8.71 to 8.73. It has found

that having same concrete strength, normalized punching shear strength is increased

with addition of flexural reinforcement ratio from 0.25% to 2% percent. Although

rate of increase of punching load carrying capacity is higher upto 1% reinforcement

ratio than the above of this ratio. Thus, punching load-carrying capacity of the all

slab panels increased with the increase of steel reinforcement.

According to Figure 8.71, For 200mm thick slab, variation of normalized punching

strength were obtained in case of lower strength of concrete such as 24 MPa, 30 MPa

and 40 MPa. For 50 MPa and 60 MPa concrete, such higher concrete strength curves

are very close. This close curves indicate that effect of concrete strength above 40

MPa concrete toward flexural reinforcement of slab is very small.

For 250 mm thick slab as shown in Figure 8.72, influence of concrete strength is

obtained upto 1.5% flexural reinforcement. For above 1.5% flexural steel of 250mm

thick slab as well as all percentages of reinforcement of 300mm thick slab, curves are

very close for all strength of concrete as shown in Figure 8.72 and 8.73.

Thus, punching shear capacity will increase with increase of flexural reinforcement,

but influence of concrete strength for higher thicknesses of slab toward flexural

reinforcement is very small in those cases.

F

F

Figure 8.71

Figure 8.72

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Normalizreinforce

Normalizreinforce

0

0

0

0

0

0

0

0

0.00%

fffff

0

0

0

0

0

0

0

0

0.00%

fffff

zed punchingment ratio (8


0.50%


0.50%


g shear of 20800mm x 80


1.00%

Reinforcem

1.00%

Reinforcem

00mm thick 00mm colum


1.50%

ment in %

1.50%

ment in %

slab at variomn).

slab at variomn).

2.00%

2.00%

ous

ous

2.50%

2.50%

F

8

I

p

a

t

i

I

p

f

c

i

m

s

Figure 8.73

8.4.4.4 On

In earlier se

presented. In

analyzed on

thicknesses

in Figures 8

It has found

punching sh

from 0.25%

capacity is h

increment o

multi-panel

shown in Fig

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Normalizreinforce

n Average Th

ctions, norm

n this sectio

n the basis o

of slab and p

.74 to 8.78.

d from table

hear strength

% to 2% per

higher upto

f punching

flat plate is

gures 8.74 to

0

0

0

0

0

0

0

0

0.00%


hickness of S

malized punc

on, effect of

of average n

plotted again

and graphs

h is increase

rcent. Althou

1% reinforc

shear capac

almost simil

o 8.78.

0.50%



Slab

ching shear s

concrete str

normalized

nst percentag

s that having

ed with add

ugh rate of

ement ratio

city due to p

lar to all typ

1.00%

Reinforcem


strength of s

rength to pun

punching sh

ge of flexura

g same conc

dition of flex

f increase of

than the abo

presence of

e of column

1.50%

ment in %

slab at variomn).

same thickne

nching shea

hear strengt

al reinforcem

crete strength

xural reinfo

f punching

ove of this r

flexural rein

ns and concre

2.00%

ous

ess of slab a

ar capacity a

ths of variou

ment as show

h, normalize

orcement rat

load carryin

ratio. Trend

nforcement

ete strength

2.50%

are

are

us

wn

ed

tio

ng

of

of

as

F

F

Figure 8.74

Figure 8.75

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Average reinforce

Average reinforce

0

0

0

0

0

0

0

0

0

0

0

0.00%

0

0

0

0

0

0

0

0

0

0

0

0.00%

normalized ment ratio fo


0.50%

400 mm x 40600 mm x 60800 mm x 80ACI CodeCanadian Co

0.50%

400 mm x 400600 mm x 600800 mm x 800ACI CodeCanadian Cod

punching shor '

cf =24 M

punching shor '

cf =30 M

1.00%

Reinforcem

00mm Column00mm Column00mm Column

ode

1.00%

Reinforcem


de

hear strengthMPa.

hear strengthMPa.

1.50%

ment in %

nnn

1.50%

ment in %

h at various f

h at various f

2.00%

2.00%

flexural

flexural

2.50%

2.50%

F

F

Figure 8.76

Figure 8.77

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Average reinforce

Average reinforce

0

0

0

0

0

0

0

0

0

0

0

0.00%

0

0

0

0

0

0

0

0

0

0

0

0.00%



0.50%


0.50%


punching shor '

cf =40 M

punching shor '

cf =50 M

1.00%

Reinforcem


de

1.00%

Reinforcem


de

hear strengthMPa.

hear strengthMPa.

1.50%

ment in %

1.50%

ment in %

h at various f

h at various f

2.00%

2.00%

flexural

flexural

2.50%

2.50%

F

D

d

H

s

b

G

s

r

S

t

t

c

o

o

i

Figure 8.78

Dilger et al

distinct decr

He added, a

seems to lea

bar spacing

Gardner (20

shear capaci

reinforcemen

Significant y

the effective

transferred t

conclude tha

on the shear

of flexural r

include the e

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Average reinforce

l. (2005) stu

rease in pun

a concentrat

ad to a smal

does not lea

005) noted th

ity, the beha

nt should ne

yielding of f

e area resist

through the

at the width

r capacity of

einforcemen

effect of stee

0

0

0

0

0

0

0

0

0

0

0

0.00%


udied over o

nching shear

tion of flexu

l increase in

d to a reduct

hat while inc

avior of the

ever be less t

flexural rein

ing the shea

portion of

and hence th

f the connect

nt on punchin

el percentage

0.50%


punching shor '

cf =60 M

one thousan

r resistance

ural reinforc

n the punchi

tion in the bo

creasing the

connection

than 0.5% an

nforcement p

ar. If it is as

the depth o

he depth of

tion. Islam (

ng shear in c

e in ACI-318

1.00%

Reinforcem


de

hear strengthMPa.

nd test resul

with decrea

cement in th

ing shear str

ond strength

flexural ste

becomes m

nd will rarely

produces lar

ssumed that

of slab that

the crack ha

2004) found

concrete slab

8 code.

1.50%

ment in %

h at various f

lts and conc

asing reinfor

he vicinity o

rength only i

h along the b

eel increases

ore brittle a

y exceed 2%

rge crack, w

t little or no

is cracked,

ave a signific

d from his st

b. The autho

2.00%

flexural

cluded that,

rcement rati

of the colum

if the reduce

bars.

the punchin

and practical

% in real slab

which decrea

shear can b

it is easy

cant influenc

tudy the effe

or proposed

2.50%

a

io.

mn

ed

ng

lly

bs.

se

be

to

ce

ect

to

225

Percentage of flexural reinforcement is often used as an index for the dowel effect.

Shear strength is expected to increase with increasing flexural reinforcement ratios

and increasing concrete strength. However, according to the work on dowel action in

reinforced concrete beam (Baumann et al, 1970), the rate of increase of shear

strength decreases at higher concretes strengths and flexural reinforcement ratios.

Kinnunen and Nylander (1960) tested a number of slabs with ring reinforcement in

which steel ratios was equal to those in other tests with two way reinforcement. By

comparisons, they concluded that dowel action carries about 30% of the total shear.

However, Criswell (1974) concluded that this effect is not important.

The failure modes of the dowel mechanism defined by Vintzeleou and Tassios(1986)

might explain the reason for the contradiction noted above. They stated that there are

two possible failures model of dowel mechanisms:

(1) yield of the steel bar and concrete crushing under the dowel.

(2) concrete splitting.

Guandalini et al. (2009) investigated the punching strength of slabs with low

reinforcement ratios. The scope of their research was slabs with low reinforcement

ratios, because there was not much data available for slabs with low reinforcement ratios

failing in shear, as researchers tried to avoid flexural failures, and because the code

provisions differ significantly. The results were recorded as load-deflection curves which

show unexpectedly low strengths for slabs with low reinforcement ratios. They reported

that, for thick slabs with low reinforcement ratios, ACI 318-08 is less conservative.

The values given by Eurocode 2 are in better correlation with the experimental

results. Guandalini et al. (2009) concluded that future research is needed to investigate

this observation and that special attention should be given to the cases in which the code

provisions significantly underestimate the punching shear strength.

However, concentration of flexural reinforcement in the column region(critical

perimeter) is to be encouraged because it improves the behavior of the slab in the

service load range. Concentration increases the stiffness of the slab, increases the

load for the first yielding of the flexural reinforcement, and consequently results in

smaller maximum crack widths for a given loading.

226

Due to increase of applied load, cracking of concrete propagates at the tension zone

of concrete, which decrease the effective depth of slab for resisting the shear. If it is

assumed that little or no shear can be transferred through the portion of the depth of

slab that is cracked, it is easy to conclude that the width and hence the depth of the

crack have a significant influence on the shear capacity of the connection. With

present of flexural reinforcement, this propagation crack will be reduced, thus the

load carrying capacity increased.

8.4.5 Effect of Slab Thickness

Effect of slab thickness is obtained in the FE results of multi-panel slab models and

discussed in this section. Figures 8.79 to 8.91 are included with three group of model

slab. The groups are based on similar column sizes such as 400mm x 400mm,

600mm x 600mm and 800mm x 800mm. Effect of slab thickness on punching shear

capacity are also shown on the basis of average normalized punching shear strengths

of various flexural reinforcement ratios.


The normalized punching shear strengths of various thicknesses of slab are plotted

against compressive strength of concrete having of same percentage of flexural

reinforcement and shown in Figure 8.79 to 8.82. It is clearly obtained the influence

of slab thickness to punching shear capacity of multi-panel flat plate from all those

figures. The smaller the slab thickness the higher punching shear capacity was

obtained.

According to Figures 8.79 to 8.82, the smaller thick slab such as 200mm thick,

punching shear capacity decreases significantly with increase of concrete strength

upto 50 MPa of concrete strength. For higher thick slab such as 250mm and 300mm,

this decrement is very small upto 40 MPa concrete and 1% flexural reinforcement.

For such higher thick slab, punching shear capacity has a slightly increasing

tendency for above 1% reinforcement and above 40 MPa concrete strength.

F

F

Figure 8.79

Figure 8.80

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Normalizreinforce

Normalizreinforce

0

0

0

0

0

0

0

0

0

0

0

20

0

0

0

0

0

0

0

0

0

0

0

20



30

30

g shear of m400mm x 40


40

f 'c (MPa

40f 'c (MPa

model slabs h00mm colum


50

a)

Slab ThicknSlab ThicknSlab ThicknCanadian CACI Code

50a)

Slab ThicknesSlab ThicknesSlab ThicknesCanadian CodACI Code

aving 0.5% mn).

aving 1% flemn).

60

ness 200mm ness 250mmness 300mm

Code

60

ss 200mm ss 250mmss 300mmde

flexural

exural

F

F

Figure 8.81

Figure 8.82

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Normalizreinforce

Normalizreinforce

0

0

0

0

0

0

0

0

0

0

0

20

00

10

20

30

40

50

60

70

80

90

00

20



30

30



40f 'c (MPa

40f 'c (MPa



50a)


50a)

Slab ThiSlab ThiSlab ThiCanadiaACI Cod

aving 1.5% mn).

aving 2% flemn).

60


60

ickness 200mmickness 250mmickness 300mm

an Codede

flexural

exural

m mm

8

T

t

s

c

p

p

A

p

F

M

i

s

F

8.4.5.2 On

The normali

thicknesses

same percen

clearly obtai

panel flat p

punching sh

According t

punching sh

For higher t

MPa concre

increasing t

strength.

Figure 8.83

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n 600mm x 6

ized punchin

of slab are

ntage of flex

ined the infl

plate from a

ear capacity

to Figures 8

hear capacity

thick slab su

ete. For such

tendency for

Normalizreinforce

20

600mm colum

ng shear stre

plotted aga

xural reinfor

luence of sla

all those figu

y was obtaine

8.83 to 8.86

y decreases

uch as 250m

h higher thi

r above 0.5


30

mn

engths of 60

ainst compre

rcement and

ab thickness

ures. The sm

ed.

6, the small

significantly

mm and 300m

ick slab, pu

5% reinforc


40f 'c (MPa

00mm x 600

essive streng

d shown in F

to punching

maller the s

er thick sla

y with incre

mm, this dec

unching shea

ement and


50a)


0mm column

gth of concr

Figure 8.83

g shear capa

slab thickne

ab such as 2

ease of conc

crement is sh

ar capacity h

above 40 M

aving 0.5% mn).

60


n and variou

rete having

to 8.86. It

acity of mult

ess the high

200mm thic

crete strengt

hown upto 3

has a slight

MPa concre

flexural

us

of

is

ti-

her

ck,

th.

30

tly

ete

F

F

Figure 8.84

Figure 8.85

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2

Normalizreinforce

Normalizreinforce

20

20



30

30



40f 'c (MPa)

40f 'c (MPa



50)

Slab ThicknSlab ThicknSlab ThicknCanadian CoACI Code

50a)

Slab ThickneSlab ThickneSlab ThickneCanadian CoACI Code

aving 1% flemn).

aving 1.5% mn).

60

ness 200mm ness 250mmness 300mmode

60

ess 200mm ess 250mmess 300mmode

exural

flexural

F

8

T

t

s

A

c

c

t

f

Figure 8.86

8.4.5.3 On

The normali

thicknesses

same percen

According t

capacity of

concrete stre

thickness ab

flexural rein

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalizreinforce

n 800mm x 8

ized punchin

of slab are

ntage of flexu

to Figures 8

all thick sla

ength. Varia

bove 0.25%

nforcement a

0

1

2

3

4

5

6

7

8

9

1

20


00mm colum

ng shear stre

plotted aga

ural reinforc

8.87, for 0.2

ab are almo

ation of punc

flexural rein

above 0.50%

30


mn

engths of 80

ainst compre

cement and s

25% of flex

ost matched

ching shear c

nforcement a

%, this decrem

40f 'c (MPa


00mm x 800

essive streng

shown in Fig

xural reinfo

although de

capacity is o

as shown in

ment is show

50a)

Slab ThicSlab ThicSlab ThicCanadianACI Cod

aving 2% flemn).

0mm column

gth of concr

gure 8.87 to

orcement, pu

ecreases wit

obtained with

Figures 8.8

wn upto 40 M

60

ckness 200mm ckness 250mmckness 300mmn Codee

exural

n and variou

rete having

8.91.

unching she

th increase

h various sla

8 to 8.91. F

MPa concrete

us

of

ear

of

ab

or

e.

F

F

Figure 8.87

Figure 8.88

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Normalizreinforce

Normalizreinforce

0

0

0

0

0

0

0

0

0

0

0

20

00

10

20

30

40

50

60

70

80

90

00

20



30

30



40f 'c (MPa

40f 'c (MPa



50a)


50a)


aving 0.25%mn).

aving 0.5% mn).

60


60


% flexural

flexural

F

F

Figure 8.89

Figure 8.90

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Normalizreinforce

Normalizreinforce

0

0

0

0

0

0

0

0

0

0

0

20

0

0

0

0

0

0

0

0

0

0

0

20



30

30



40f 'c (MPa

40f 'c (MPa



50a)

Slab ThickneSlab ThickneSlab ThickneCanadian CoACI Code

50a)


aving 1% flemn).

aving 1.5% mn).

60

ess 200mm ess 250mmess 300mmode

60


Code

exural

flexural

0

F

8

T

h

c

v

t

t

t

T

p

f

I

c

o

t

Figure 8.91

8.4.5.4 On

The average

higher than

column size

value for 20

than 300mm

thick slab is

thick slab.

The average

plotted agai

flexural rein

It has found

capacity is d

of punching

thickness of

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Normalizreinforce

n Average Co

e normalized

250mm thic

of 400mm x

00mm thick

m thick slab.

s 9.43% hig

e normalized

inst compre

nforcement a

d from figur

decreased wi

g shear capa

f slabs.

0

0

0

0

0

0

0

0

0

0

0

20


olumn Size

d punching

ck slab and

x 400mm. S

slab is 17.5%

For 800mm

gher than 25

d punching s

essive streng

and shown in

res that havi

ith increase

city with in

30


shear capac

22% higher

imilarly, for

% higher tha

m x 800mm

50mm thick

shear strengt

gth of conc

n Figure 8.92

ing same fle

of slab thick

ncrease of co

40f 'c (MPa


city of 200m

than 300mm

r 600mm x 6

an 250mm th

column valu

slab and 13

ths of variou

crete having

2 to 8.96.

exural reinfo

kness. Altho

oncrete stren

50a)


aving 2% flemn).

mm thick sl

m thick slab

600mm colum

hick slab an

ue, this valu

3.7% higher

us thickness

g of same p

orcement, pu

ough, decrea

ngth is highe

60


Code

exural

lab is 10.75

b having sam

mn value, th

nd 24% high

ue for 200m

r than 300m

ses of slab a

percentage

unching she

asing tendenc

er for small

%

me

his

her

mm

mm

are

of

ear

cy

ler

A

w

r

O

T

h

M

p

(

f

p

r

b

i

F

According to

which indic

reinforcemen

On the other

This means

having smal

Muttoni (20

predicts seve

(decreasing

for the prop

punching sh

rather than o

both span-to

influence on

Figure 8.92

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

o Figure 8.9

cate, effect

nt.

r hand, all cu

, the effect

ler flexural r

008) perform

eral aspects

nominal she

posed failur

hear strength

on its thickne

o-depth ratio

n punching sh

Average flexural r

20

2 and 8.93,

of slab thic

urves of 300

of slab thi

reinforcemen

med parame

of punching

ear strength

re criterion

h of a flat pl

ess as often

o effects and

hear in conc

normalized reinforcemen

30

three curves

ckness in sm

0mm thick sl

ckness towa

nt ratio is ve

etric study

g shear previ

with increas

and load-ro

late is show

proposed. Is

d the type o

crete slab.

punching shnt ratio.

40

f 'c (MPa

s of various t

maller for 0

labs are alm

ard punchin

ery small.

and demon

ously observ

sing size of t

otation relat

wn to depend

slam (2004)

of support c

hear of mode

50

a)

Slab ThickneSlab ThickneSlab ThickneACI CodeCanadian Cod

thicknesses

0.25% and 0

most flattenin

ng shear cap

nstrated tha

ved in testing

the member

tionship of

d on the spa

found from

condition ha

el slabs havin

60

ess 200mm ess 250mmess 300mm

de

are very clo

0.5% flexur

ng than other

pacity of sla

at it correct

g assize effe

r). Accountin

the slab, th

an of the sla

his study th

ave significa

ng 0.25%

se

ral

rs.

ab

tly

ect

ng

he

ab,

hat

ant

F

F

Figure 8.93

Figure 8.94

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Average flexural r

Average flexural r

20

20



30

30



40

f 'c (MPa

40

f 'c (MPa

hear of mode

hear of mode

50

a)

Slab ThicknSlab ThicknSlab ThicknACI CodeCanadian Co

50

a)

Slab ThicknSlab ThicknSlab ThicknACI CodeCanadian Co

el slabs havin

el slabs havin

60


ode

60


ode

ng 0.5%

ng 1%

0

F

F

Figure 8.95

Figure 8.96

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Average flexural r

Average flexural r

20

20



30

30



40

f 'c (MPa

40

f 'c (MPa

hear of mode

hear of mode

50

a)

Slab ThicknSlab ThicknSlab ThicknACI CodeCanadian C

50

a)

Slab ThickSlab ThickSlab ThickACI CodeCanadian C

el slabs havin

el slabs havin

60


Code

60

kness 200mm kness 250mmkness 300mm

Code

ng 1.5%

ng 2%

0

0

238

According to Bazant and Cao (1987), the larger the slab thickness, the steeper the

post peak decline of the load-deflection diagram. Thus, the punching shear behavior

of thin slabs is closer to plasticity and that of thick slabs is close to linear elastic

fracture mechanics. This independently confirms the applicability of the size-effect

law, since this law predicts exactly such kind of behavior.

Elstner and Hognestad (1956) questioned the extrapolation of observations on thick

footing slabs to flat plate floors from a theoretical point of view, since lower thickness-to

span ratios and higher moment-to-shear ratios are more associated with floor slabs than

with footings.

Collins and Kuchma (1999) investigated the importance of the size effect on beams,

slabs and footings and concluded that the size effect has to be taken into account and that

high-strength concrete members display a more significant size effect. They pointed out

that the shear stress at failure decreases, both as the member depth increases and as the

maximum aggregate size decreases. According to Collins and Kuchma (1999), the size

effect had to be studied especially in slabs and footings, as these members can be both

very thick and very lightly reinforced.

Mitchell, Cook and Dilger (2005) stated that it is difficult to gather experimental data

solely on the size effect, as many reported experiments varied other parameters together

with the thickness. For example, the reinforcement ratio was changed together with the

slab thickness to keep the ratio of flexural capacity to shear capacity constant.

Mitchell, Cook and Dilger (2005) gathered information of tests where only the size was

varied. It is clear from the data that there is a size effect for slabs thicker than about 200

mm (8 in). The data also show a size effect, even for slabs with a thickness smaller than

200 mm (8 in). Tests with varying maximum aggregate size are not included. As can be

seen, the shear stress at punching failure decreases as the effective depth increases.

According to Mitchell, Cook and Dilger (2005) the size effect is significant, but the

available data are scarce.

239

According to Sundquist (2005), no good analysis method has been presented to date that

can really explain the size effect. A model developed by Hallgren (1996) was cited,

based on fracture mechanics that incorporated the aggregate size.

Guandalini et al. (2009) performed series of 11 punching tests on flat plates. The

tests are useful to complement available punching test series performed in the past,

as the tests presented in this paper systematically explore the domain of slabs with

low flexural reinforcement ratios. The tests have confirmed that, due to size effect,

the punching strength decreases with increasing slab thickness. At the same time, the

deformation at failure decreases.

Borges et al. (2013) concluded that there is some indication of a size effect on

punching resistance, which can be related to the slab effective depth, even though

some differences could have been expected due to the different arrangements of the

supports.

It can be concluded that having same flexural reinforcement, punching shear capacity

is decreased with increase of slab thickness. Decreasing tendency of punching shear

capacity due to slab thickness with smaller strength of concrete is higher. For higher

slab thickness, decreasing tendency of punching shear capacity is very small. The

contribution of slab thickness may be included in the presently recognized codes.

8.4.6 Effect of Column Size

The FE results of multi-panel slab models are compared according to column size as

shown in Figures 8.97 to 8.108. The models are grouped on the basis of similar slab

thicknesses of 200mm, 250mm and 300mm.

8.4.6.1 On 200mm thick slab

The normalized punching shear strengths of various size of columns of all model

slabs having 200mm thick are plotted against compressive strength of concrete and

shown in Figure 8.97 to 8.100. It is clearly obtained the influence of column size to

punching shear capacity of multi-panel flat plate from all those figures. The smaller

the column size the higher punching shear capacity was obtained.

A

r

s

F

F

According t

reinforcemen

shear capaci

Figure 8.97

Figure 8.98

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

to Figures 8

nt are almo

ity for 400m

Normalizreinforce

Normalizreinforce

0

0

0

0

0

0

0

0

0

0

0

20

0

0

0

0

0

0

0

0

0

0

0

20

8.97, variatio

ost regular.

mm x 400mm



30

30

on of punch

Above 0.5

m and 600mm

g shear of m200mm thick


40f 'c (MPa

40f 'c (MPa

hing shear ca

% flexural

m x 600mm c

model slabs hk slab).


50a)

400600800CanAC

5a)

400m600m800mCanadACI C

apacity for

reinforceme

columns are

aving 0.5%

aving 1% fle

0

0mm x 400mm 0mm x 600mm 0mm x 800mm nadian CodeI Code

0

mm x 400mm Comm x 600mm Comm x 800mm Co

dian CodeCode

0.5% flexur

ent, punchin

very close.

flexural

exural

60

ColumnColumnColumn

60

olumnolumnolumn

ral

ng

F

F

Figure 8.99

Figure 8.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Normalizreinforce

0 Normalizreinforce

0

0

0

0

0

0

0

0

0

0

0

20

0

0

0

0

0

0

0

0

0

0

0

20

zed punchingment ratio(2


30

30



40f 'c (MPa

40f 'c (MPa



50a)

406080CA

50a)

468CA

aving 1.5%

aving 2% fle

0

00mm x 400mm00mm x 600mm00mm x 800mm

Canadian CodeACI Code

0

400mm x 400mm600mm x 600mm800mm x 800mmCanadian CodeACI Code

flexural

exural

60

m Columnm Columnm Column

60

m Columnm Columnm Column

8

T

s

s

p

t

A

r

c

2

F

8.4.6.2 On

The normali

slabs having

shown in Fig

punching sh

the column s

According t

reinforcemen

capacity for

24MPa to 40

Figure 8.10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

250mm thic

ized punchin

g 250mm thi

gures 8.101

hear capacity

size the high

to Figure 8.

nt is almost

r 600mm x 6

0 MPa concr

1 Normalizreinforce

20

ck slab

ng shear str

ick are plott

to 8.104. It i

y of multi-pa

her punching

101, variatio

regular. Ab

600mm and

rete.


30

rengths of v

ted against c

is clearly ob

anel flat plat

g shear capac

on of punch

bove 0.5% fl

800mm x 8


40

f 'c (MPa

various size

compressive

btained the in

te from all t

city was obta

hing shear ca

lexural reinfo

800mm colu


5

a)

400mm x 600mm x 800mm x Canadian ACI Code

of columns

e strength of

nfluence of c

those figures

ained.

apacity for

forcement, pu

umns are ve

aving 0.5%

0

400mm Colum600mm Colum800mm ColumCode

e

of all mod

f concrete an

column size

s. The small

0.5% flexur

unching she

ry close fro

flexural

60

mnmnmn

del

nd

to

ler

ral

ear

om

F

F

Figure 8.10

Figure 8.10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 Normalizreinforce

3 Normalizreinforce

0

4

6

8

20

0

1

2

3

4

5

6

7

8

9

1

20



30

30



40

f 'c (MPa

40f 'c (MPa



5

a)

400m600m800mCanACI

5a)

400m600m800mCanaACI

aving 1% fle

aving 1.5%

50

mm x 400mm mm x 600mm mm x 800mm

nadian CodeI Code

50

mm x 400mm Cmm x 600mm Cmm x 800mm Cadian CodeCode

exural

flexural

60

ColumnColumnColumn

60

ColumnColumnColumn

F

8

T

s

s

p

t

A

r

s

f

Figure 8.10

8.4.6.3 On

The normali

slabs having

shown in Fig

punching sh

the column s

According t

reinforcemen

shear capaci

from 24MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

4 Normalizreinforce

300mm thic

ized punchin

g 300mm thi

gures 8.105

hear capacity

size the high

o Figures 8.

nt are almo

ity for 600m

a to 40 MPa

0

1

2

3

4

5

6

7

8

9

1

20


ck slab

ng shear str

ick are plott

to 8.108. It i

y of multi-pa

her punching

.105, variatio

ost regular.

mm x 600mm

concrete.

30


rengths of v

ted against c

is clearly ob

anel flat plat

g shear capac

ons of punch

Above 0.5

m and 800m

40

f 'c (MPa


various size

compressive

btained the in

te from all t

city was obta

hing shear c

% flexural

mm x 800mm

50

a)

400mm x 40600mm x 60800mm x 80Canadian CoACI Code

aving 2% fle

of columns

e strength of

nfluence of c

those figures

ained.

capacity for

reinforceme

m columns a

0


ode

exural

of all mod

f concrete an

column size

s. The small

0.5% flexur

ent, punchin

are very clo

60

del

nd

to

ler

ral

ng

se

F

F

Figure 8.10

Figure 8.10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5 Normalizreinforce

6 Normalizreinforce

0

1

2

3

4

5

6

7

8

9

1

20

0

1

2

3

4

5

6

7

8

9

1

20



30

30



40

f 'c (MPa

40

f 'c (MPa



5

a)

400m600m800mCanaACI

5

a)


aving 0.5%

aving 1% fle

50

mm x 400mm Cmm x 600mm Cmm x 800mm Cadian CodeCode

50

mm x 400mm Cmm x 600mm Cmm x 800mm C

dian CodeCode

flexural

exural

60

ColumnColumnColumn

60

ColumnColumnColumn

F

F

Figure 8.10

Figure 8.10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

7 Normalizreinforce

8 Normalizreinforce

20

0

1

2

3

4

5

6

7

8

9

1

20



30

30



40

f 'c (MPa

40

f 'c (MPa



50

a)


50

a)

400mm600mm800mmCanadACI C

aving 1.5%

aving 2% fle

0

mm x 400mm Cmm x 600mm Cmm x 800mm C

dian CodeCode

0

m x 400mm Cm x 600mm Cm x 800mm Cdian CodeCode

flexural

exural

60

ColumnColumnColumn

60

olumnolumnolumn

8

T

6

o

T

m

F

r

M

c

a

F

8.4.6.4 On

The average

600mm x 60

of 800mm x

The average

model slabs

Figure 8.109

reinforcemen

Moe (1961)

column base

and 3.0d, wh

Figure 8.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Average Th

e normalized

00mm colum

x 800mm col

e normalized

thick are pl

9 to 8.112.

nt, punching

) assumed a

ed on test d

here d is the

9 Average flexural r

0

0

0

0

0

0

0

0

0

0

0

20

ickness of Sl

d punching s

mn are 20%

lumn.

d punching s

lotted agains

It has foun

g shear capac

a linear vari

data when th

slab thickne


30

lab

shear capaci

and 10% hig

shear streng

st compressi

nd from tho

city is decrea

iation in sh

he side lengt

ess.


40f 'c (MPa

ity of slab f

gher respect

gths of vario

ive strength

se figures th

ased with in

ear strength

th of loaded

hear of mode

50a)


for 400mm x

tively in com

ous size of c

of concrete

hat having

ncrease of co

h with side

d area was b

el slabs havin

60


de

x 400mm an

mpare to tho

columns of a

and shown

same flexur

lumn size.

dimension

between 0.75

ng 0.50%

nd

se

all

in

ral

of

5d

F

F

Figure 8.11

Figure 8.11

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00



0

0

0

0

0

0

0

0

0

0

0

20

0

0

0

0

0

0

0

0

0

0

0

20



30

30



40f 'c (MPa

40

f 'c (MPa

hear of mode

hear of mode

50a)

400 mm x600 mm x800 mm xACI CodeCanadian

50

a)

400 mm x 600 mm x 800 mm x ACI CodeCanadian C

el slabs havin

el slabs havin

60

x 400mm Columx 600mm Columx 800mm ColumeCode

60

400mm Colum600mm Colum800mm Colum

Code

ng 1%

ng 1.5%

mnmnmn

mnmnmn

F

R

T

t

a

b

t

r

p

r

8

A

a

s

n

s

Figure 8.11

Regan (1986

The test resu

that it excee

about 0.75d)

below that p

than 0.75d, t

resulting in

practice to p

rather than in

8.5 SUMM

Analysis of

analysis of m

strength, co

numerical F

slab panel d

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0


6) tested five

ults confirm

eds 0.75d. W

), the slab fa

predicted by

the length o

an increase

provide drop

ncreasing th

MARY OF

f punching

multi-panel

lumn sizes

FE analysis,

decreases wi

00

10

20

30

40

50

60

70

80

90

00

20


e slabs wher

med the linea

When the loa

ailed in local

y tile linear

of critical sec

e in shear st

p panels or c

he column siz

PRECEDIN

shear behav

RC flat pla

and flexural

the normali

th increase

30


re the loaded

ar relationshi

aded area is

l crushing an

relationship

ction becom

trength of sl

capitals to in

ze.

NG DISCUS

vior of rein

ates for seve

l reinforcem

zed load-car

of compress

40

f 'c (MPa

hear of mode

d area is the

ip for tile lo

very small

nd therefore

p. If the loa

me greater as

lab. Therefo

ncrease the p

SSIONS

nforced con

eral types of

ment ratio ha

rrying capac

sive strength

50

a)


el slabs havin

only signifi

oaded dimen

(side dimen

the strength

aded dimens

the loaded a

ore it is very

punching sh

ncrete slab b

f slab thickn

as been dev

city of the a

h of concrete

60


de

ng 2%

icant variabl

nsion provide

nsion less tha

h of slab is f

sion is great

area increas

y common

hear resistanc

based on F

ness, concre

veloped. Fro

all multi-pan

e upto aroun

le.

ed

an

far

ter

es

in

ce

FE

ete

om

nel

nd

250

48 MPa. Very small or no increase of normalized punching load carrying is observed

above the concrete strength of 48 MPa. It has found that punching shear capacity is

increased with addition of flexural reinforcement ratio. Although rate of increase of

punching load carrying capacity is higher upto 1% reinforcement ratio than the above

of this ratio. Size of column for flat plate is important for determining punching shear

load carrying capacity. It has been found that for slabs having same size and

reinforcement, the punching shear capacity decreases with a corresponding increase

in the column size. It has found that having same flexural reinforcement, punching

shear capacity is decreased with increase of slab thickness.

251

CHAPTER-9

SIMPLE PUNCHING SHEAR STRESS FORMULA FOR

MULTI PANEL FLAT PLATE

9.1 INTRODUCTION

From the analysis of all 225 model slabs, it is established that punching shear

capacity of multi-panel flat plate is dependent on compressive strength of concrete,

flexural reinforcement, column size and slab thickness. From analysis and

discussion, it is established that punching shear capacity is decreased with increase of

compressive strength of concrete upto certain limit and increase with increase of

flexural reinforcement. Decreasing tendency of punching shear capacity with

increase of slab thickness and column size has also investigated from earlier

chapters. According to the relationship of normalized punching shear with concrete

strength, flexural reinforcement and size effect of slab and column, an empirical

equation for calculating punching shear capacity is proposed in this chapter. The

proposed equation is verified by analyzed data and test results and comparison with

various code of prediction is also discussed in this chapter.

9.2 BASIS OF PROPOSAL

9.2.1 Punching Shear Capacity of Slab

Summary of analysis of 225 model slab are tabulated in Table 9.1 as average value

of normalized punching shear. In this table average normalized punching shear of

200mm, 250mm and 300mm thick slab are used. According to this table, average

normalized punching shear strength of 400x400 mm2 is 0.66 with 4.78% standard

deviation, 600x 600 mm2 is 0.60 with 5.12% standard deviation and 800x 800 mm2 is

0.54 with 6.10% standard deviation. For all model slabs irrespective of column size,

slab thickness, flexural reinforcement and concrete strength, the average normalized

punching shear strength is 0.60.

According to table 9.1, normalized punching shear strength of all model slabs is

higher than ACI and Canadian building code requirements. According to American,

Australian, Bangladesh and Canadian, punching shear capacity is function of square

252

root of concrete strength. British, European and German code recognize the effect of

flexural reinforcement in addition to concrete strength. In this study, effect of

concrete strength, flexural reinforcement, slab thickness and column size to punching

shear capacity is clearly identified.

Table 9.1 Average normalized punching shear capacity of all slabs

Flexural Reinforcement

( ρ )

f'c

(MPa)

Normalized Punching Shear, dbfV c 0'

400 x 400 mm2

Column

600 x 600 mm2

Column

800 x 800 mm2

Column Average ACI

Code Canadian

Code

0.25% 24 0.58 0.52 0.45 0.52 0.33 0.4 0.25% 30 0.55 0.49 0.42 0.49 0.33 0.4 0.25% 40 0.54 0.47 0.42 0.48 0.33 0.4 0.25% 50 0.53 0.46 0.4 0.46 0.33 0.4 0.25% 60 0.53 0.46 0.39 0.46 0.33 0.4 0.50% 24 0.61 0.56 0.51 0.56 0.33 0.4 0.50% 30 0.58 0.53 0.48 0.53 0.33 0.4

0.50% 40 0.56 0.50 0.45 0.50 0.33 0.4 0.50% 50 0.56 0.50 0.44 0.50 0.33 0.4 0.50% 60 0.56 0.51 0.43 0.50 0.33 0.4

1.00% 24 0.65 0.58 0.54 0.59 0.33 0.4 1.00% 30 0.62 0.56 0.53 0.57 0.33 0.4 1.00% 40 0.59 0.55 0.51 0.55 0.33 0.4

1.00% 50 0.59 0.54 0.49 0.54 0.33 0.4 1.00% 60 0.61 0.56 0.48 0.55 0.33 0.4 1.50% 24 0.67 0.60 0.56 0.61 0.33 0.4

1.50% 30 0.64 0.58 0.55 0.59 0.33 0.4 1.50% 40 0.62 0.57 0.53 0.57 0.33 0.4 1.50% 50 0.62 0.57 0.52 0.57 0.33 0.4

1.50% 60 0.62 0.58 0.52 0.57 0.33 0.4 2.00% 24 0.69 0.62 0.58 0.63 0.33 0.4 2.00% 30 0.65 0.60 0.56 0.60 0.33 0.4 2.00% 40 0.63 0.59 0.55 0.59 0.33 0.4 2.00% 50 0.63 0.59 0.54 0.59 0.33 0.4 2.00% 60 0.64 0.60 0.54 0.59 0.33 0.4

Average 0.60 0.55 0.50 0.55 0.33 0.4 Standard

Deviation (%) 4.39% 4.72% 5.59% 4.86%

9

A

9

V

r

F

w

o

s

t

w

c

t

w

F

9.2.3 RelaFlex

Average nor

9.1 and 9.2.

Very smoo

reinforcemen

Figure 9.1, t

with increas

or no incre

strength of a

that having

with additio

carrying cap

this ratio. A

with increas

Figure 9.1

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

ationship of ural Reinfo

rmalized pun

Effect of sla

oth curve o

nt to norma

the normaliz

se of compre

ase of norm

above 48 MP

same concre

on of flexura

pacity is hig

Above 1% f

e flexural re

Average nocompressiv

0

0

0

0

0

0

0

0

0

0

0

20

Normalizedorcement

nching shea

ab thickness

of the effe

alized punc

zed punchin

essive streng

malized pun

Pa to 60 MP

ete strength,

al reinforcem

gher upto 1%

flexural rein

einforcement

ormalized puve strength o

30

d Punching

ar strength o

s and column

ect of conc

hing shear

ng shear cap

gth of concr

nching load

Pa. Similarly

normalized

ment ratio.

% flexural re

nforcement,

t ratio is alm

unching sheaof concrete.

40

f 'c (MPa

g Shear with

of all model

n size is not

crete streng

strength is

acity of the

rete upto aro

carrying is

y, as shown i

d punching s

Rate of inc

einforcemen

increase of

most linear.

ar of all mod

50

a)


h Concrete

slabs is sho

considered

gth as well

produced.

all slab pan

ound 48 MP

observed f

in Figure 9.2

hear strengt

rement of p

nt ratio than

punching s

el slab for v

einforcementReinforcementeinforcementReinforcement

Reinforcement

Strength an

own in Figu

in the figure

l as flexur

As shown

nels decreas

Pa. Very sma

from concre

2, it has foun

th is increase

punching loa

the above

shear capaci

ariable

60

nd

ure

es.

ral

in

es

all

ete

nd

ed

ad

of

ity

F

9

I

d

t

s

e

A

p

c

c

s

s

N

Figure 9.2

9.3 THE

Increase of

decreasing t

thickness an

shear with

effect of slab

According t

parameter o

curves as s

calculation

shear capac

square interi

Nominal Pun

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Average noflexural rei

E PROPOSE

punching s

tendency of

nd column s

crushing str

b and colum

to the data

f punching

tated in the

is establishe

city has bee

ior columns.

nching Shea

0

0

0

0

0

0

0

0

0

0

0

0.00%

ormalized puinforcement.

ED FORMU

shear capaci

the same wi

size are inv

rength of co

mn have been

obtained fr

shear streng

e earlier cha

ed. The foll

en proposed

. Safety facto

ar Capacity,

0.50%

f'c = 24 MPaf'c = 30 MPaf'c = 40 MPaf'c = 50 MPaf'c = 60 MPa

unching shea.

ULA

ity with inc

ith increase

vestigated. R

oncrete, flex

n established

from FE an

gth, curves s

apters, a pr

lowing emp

. The propo

or is not incl

Vn )1( −= ψ

1.00%

Reinforcem

aaaaa

ar of all mod

crease of fle

of crushing

Relationship

xural reinfor

in this study

alysis, relat

shown in Fi

roposal for

irical formu

osed formul

luded in the

3 31)(1( + ρ

1.50%

ment in %

el slab for v

exural reinfo

strength of

of normali

rcement of

y.

tionship bet

igure 9.1, 9.

punching s

ula to calcul

la will be a

proposed fo

fdc

c)5.00++

2.00%

ariable

forcement an

concrete, sla

zed punchin

slab and siz

ween variou

.2 and simil

hear capaci

late punchin

applicable f

ormula.

dbfc 0'

2.50%

nd

ab

ng

ze

us

lar

ity

ng

for

255

Here,

Vn = Nominal Punching Shear Capacity in Newton (N).

ρψ

776.7

3 '

+= cf

, for 'cf = 21 MPa to 48 MPa

)3.71(47.0 ρψ −= , for 'cf = above 48 MPa

0.15.0310≤

++

dcc

ρ= Flexural reinforcement ratio. c= Side of column in millimeter (mm).

d= Effective depth of slab in millimeter (mm). '

cf = Cylinder compressive strength of concrete at 28 days in MPa.

ob = Perimeter at a distance 2d from the edge of column in millimeter.

The normalized punching shear capacity using proposed formula is compared with

nonlinear analysis and shown in the following sections.

9.4 EFFECTIVENESS AND COMPARISON WITH CODE

Proposed formula for calculating normalized punching shear capacity with variable

crushing strength of concrete as well as flexural reinforcement is compare with all

FE model slab used in this study. Comparison of model slab having column size

400mm x 400mm, 600mm x 600mm and 800mm x 800 with 200mm, 250mm and

300mm thick slab respectively are shown in the following sections. Related curves

for comparison with all other model slab are shown in the Appendix. Ratios of

ultimate failure load by proposed formula and FE analysis for all 225 model slabs is

found an average value of around 1.00 with 6.31% standard deviation.

9.4.1 Comparison with variable Concrete Strength

Normalized punching shear strength with variable concrete strength calculated by

proposed formula and FE model slabs of 200mm thick slab (400mm x 400mm

column), 250mm thick slab (600mm x 600mm column) and 300mm thick slab

(800mm x 800mm column) are shown in Figure 9.3 to 9.5. According to those

figures proposed formula is almost matched with FE analysis.

FFigure 9.3

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( a )

( c )

Application( a ) 0.25%(400mm x

0 30f 'c

FP

0 30f 'c

FP

n of propose% , ( b ) 0.5%

400mm colu

40 50c (MPa)

FE AnalysisProposed Formu

40 50c (MPa)


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( e ) ed formula fo

% , ( c ) 1%, (umn and 200

60

ula

60

ula

30 40f 'c (M

FE AnPropo

for variable s( d ) 1.5% an0mm thick sl

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

50 6MPa)

nalysisosed Formula

( b )

( d )

strength of cond ( e ) 2% flab).

30 40f 'c (M

FE APropo

30 40

f 'c (M

FE APropo

60

)

oncrete of flexural steel

50 6MPa)

Analysisosed Formula

50 6

MPa)


l

60

60

F

Figure 9.4

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

2

( a )

( c )


0 30f 'c

FP

20 30

f


600mm colu

40 50c (MPa)


40 50

f 'c (MPa)

FE AnalysisProposed Form

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( e ) ed formula fo

% , ( c ) 1%, (umn and 250

60

ula

60

mula

30 40f 'c (M

FE AnPropos


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

50 6MPa)

nalysissed Formula

( b )

( d )


30 40

f 'c (M

FP

30 40f 'c (M

FEPr

60


50 6

MPa)

FE AnalysisProposed Formula

50 6MPa)

E Analysisroposed Formul

l

60

a

60

la

F

Figure 9.5

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

2

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

2

(

( c


20 30f 'c

FE Pro

0 30f 'c

a )

c )


800mm colu

40 50c (MPa)

Analysisoposed Formula

40 50c (MPa)


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( e )

ed formula fo% , ( c ) 1%, (

umn and 300

60

60

mula

30 40f 'c (M

FE AnaPropos


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0 50MPa)

alysised Formula

( b

( d


30 40f 'c (M

FE AProp

30 40f 'c (M

FEPr

60

b )

)


0 50 6MPa)

Analysisposed Formula

0 50 6MPa)


l

60

60

la

259

9.4.2 Application of Proposed Formula with variable Flexural Reinforcement

Comparison of normalized punching shear strength calculated by proposed formula

and FE analysis of model slabs of 200mm thick slab (400mm x 400mm column),

250mm thick slab (600mm x 600mm column) and 300mm thick slab on (800mm x

800mm column) with variable flexural reinforcement are shown in Figures 9.6 to 9.8.

Application of same models as shown in Figures 9.3 to 9.5 are used in Figures 9.6 to

9.8 and found to be very close. Slight variation is found for higher flexural

reinforcement such as 1.5% and 2%. Size effect of slab such as variation of slab

thickness and column size of proposed formula is also working properly as shown in

those figures.

9.4.3 Comparison with Various Codes of Prediction

Ultimate punching shear load carrying capacity using proposed formula is compared

with ACI, Canadian, European and British code as shown in Figures 9.9 to 9.17. FE

analytical failure load is also included in those figures. ACI and Canadian code do

not consider the effect of flexural reinforcement. Eurocode and British code consider

the effect of flexural reinforcement on punching shear capacity and code formula are

difference than those of ACI and Canadian code, thus ultimate punching load

capacity is plotted against flexural reinforcement ratio in those figures. The British

code formula has a limitation for maximum cube compressive strength of 40 MPa.

So, concrete strength of 24 MPa and 30 MPa of each column size and each thickness

of slab have to include in those figures.

According to all those Figures 9.9 to 9.17, the tendency of increment the punching

shear capacity of flat plate due to the presence of flexural reinforcement in analysis,

proposed formula, Eurocode 2 and British code are almost similar. In all cases

proposed formula is also almost matched with FE analysis. In some cases, analytical

punching shear capacity is slightly higher than the proposed formula. Thus, the

proposed formula is on safe side in those cases as well.

F

Figure 9.6

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

( a

( c

Application( a ) 24 MP(400mm x

0% 1.0Rein

0% 1.0Reinfo

a )

c )

n of proposePa, ( b ) 30 M400mm colu

00% 2.0nforcement in

FE Analysis

Proposed Fo

0% 2.0orcement in %


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( e )

ed formula foMPa, ( c ) 40umn and 200

00%%

s

ormula

00%

mula

1.00%Reinforc

FE AProp

) for variable f MPa, ( d ) 5

0mm thick sl

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

2.00%cement in %


( b )

( d )

flexural reinf50 MPa and lab).

1.00%Reinforcem

FE Pro

1.00%Reinforce

FP

)

forcement of( e ) 60 MPa

2.00%ment in %


2.00%ement in %

FE AnalysisProposed Formul

f a

la

FFigure 9.7

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

( a

( c


0% 1.0

Rein

FE APropo

% 1.0Reinfo

FP

a )

c )

n of proposedPa, ( b ) 30 M600mm colu

00% 2.0

nforcement in


0% 2.0orcement in %


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( e )

d formula foMPa, ( c ) 40umn and 250

00%

%

00%

la

1.00%Reinforc

FE AnPropos

) or variable fl MPa, ( d ) 5

0mm thick sl

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

2.00%cement in %

nalysissed Formula

( b )

( d )

lexural reinfo50 MPa and lab).

1.00%

Reinforce

FE AnPropo

1.00%

Reinfor

FE AnPropo

%

)

forcement of ( e ) 60 MPa

2.00%

ment in %

nalysissed Formula

2.00%

rcement in %

nalysisosed Formula

f a

%

%

F

Figure 9.8

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00

( a

( c


0% 1.00Rein

0% 1.00Reinfo

FE Pro

a )

c )

n of proposedPa, ( b ) 30 M800mm colu

0% 2.0nforcement in


0% 2.0rcement in %


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( e )

d formula foMPa, ( c ) 40umn and 300

00%%

mula

00%

1.00%Reinforc

FE AProp

) or variable fl MPa, ( d ) 5

0mm thick sl

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

2.00%cement in %


( b )

( d )

lexural reinfo50 MPa and lab).

1.00%Reinforce

FE AnPropos

1.00%Reinfor

FE Analy

Proposed

%

)


2.00%ment in %

nalysissed Formula

2.00%rcement in %

ysis

d Formula

f a

%

263

( a )

( b )

Figure 9.9 Punching shear load of obtained from FE analysis, calculated by the proposed formula and various code of 200mm thick model slab having 400mm x 400mm column for ( a ) f’c = 24 MPa, ( b ) f’c = 30 MPa

0

500

1000

1500

2000

2500

0.00% 0.50% 1.00% 1.50% 2.00%Reinforcement in %

FE AnalysisProposed Formula Eurocode 2 British code ACI code Canadian code

Ulti

mat

e Lo

ad (k

N)

0

500

1000

1500

2000

2500



Ulti

mat

e Lo

ad (k

N)

264

( a )

( b )


0

500

1000

1500

2000

2500

3000



Ulti

mat

e Lo

ad (k

N)

0

500

1000

1500

2000

2500

3000

0.00% 0.50% 1.00% 1.50% 2.00%

Reinforcement in %


Ulti

mat

e Lo

ad (k

N)

265

( a )

( b )


0

500

1000

1500

2000

2500

3000

3500



Ulti

mat

e Lo

ad (k

N)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0.00% 0.50% 1.00% 1.50% 2.00%

Reinforcement in %


Ulti

mat

e Lo

ad (k

N)

266

( a )

( b )


0

500

1000

1500

2000

2500

3000

3500

4000



Ulti

mat

e Lo

ad (k

N)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0.00% 0.50% 1.00% 1.50% 2.00%

Reinforcement in %


Ulti

mat

e Lo

ad (k

N)

267

( a )

( b )


0

500

1000

1500

2000

2500

3000

3500

4000



Ulti

mat

e Lo

ad (k

N)

0

500

1000

1500

2000

2500

3000

3500

4000

0.00% 0.50% 1.00% 1.50% 2.00%

Reinforcement in %


Ulti

mat

e Lo

ad (k

N)

268

( a )

( b )


0

500

1000

1500

2000

2500

3000

3500

4000



Ulti

mat

e Lo

ad (k

N)

0

500

1000

1500

2000

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Present design rules for punching shear failure of reinforced concrete slabs, given in

various codes of practice, are largely based on studies of the behavior and strength of

simply-supported, conventional specimens extending to the nominal line of

contraflexure (Kuang and Morley 1992, Alam et al. 2009). The code provisions rely

mostly on empirical methods derived from the test results on conventional (Salim

and Sebastian 2003) and thin slab specimens (Lovrovich and McLean 1990). In a

continuous slab, all panel edges cannot rotate freely, in contrast to its simply

supported counterpart. As a result, punching shear failure load calculated by these

codes are smaller than the punching shear behavior of continuous multi panel flat

plate.

9.4.4 Comparison with Test Results

9.4.4.1 Comparison with Test Results of Alam (1997).

The punching shear capacity using proposed formula is compared with previous

experimental works of Alam (1997) as shown in Table 9.2. The experimental

program carried out by Alam (1997) was comprised of a planned series of tests on

restrained as well as unrestrained slabs, variation of flexural reinforcement and slab

thickness. Edge restraint was provided by means of edge beams of various

dimensions to mimic the behavior of continuous slabs.

As shown in Table 9.2, experimental failure load due to punching shear of highly

restraint slabs are very close to calculated punching shear failure load according to

proposed formula. Highly restrained tested slab are very similar to multi panel flat

plate. Thus proposed formula is matched with experimental works carried out by the

author.

9.4.4.2 Comparison with other Test Results.

The punching shear capacity using proposed formula is compared with test result of

other researchers. Kuang and Morley (1992) carried out an experimental program

regarding punching shear capacity of slab with variation of restraint of slabs. They

concluded that the enhancement of punching shear capacity can be attributed due to

the presence of in-plane membrane action imposed by edge restraint in the slab. The

273

punching shear capacity using proposed formula is compared with experimental

works of Kuang and Morley (1992) and shown in Table 9.3.

Table 9.2 Comparison of proposed formula with experimental result of Alam (1997).

Model t d c ρ f'c

Failure Load calculated by

Proposed Formula

Experimental Failure Load Vtest/ Vn

(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) SLAB1 80 70 120 0.005 38.51 223.88 225.16 1.006 SLAB2 80 70 120 0.010 37.42 238.32 242.09 1.016 SLAB3 80 70 120 0.015 28.19 230.49 142.95 0.620 SLAB4 60 50 120 0.005 38.24 135.06 138.12 1.023 SLAB5 60 50 120 0.010 36.60 141.47 147.59 1.043 SLAB6 60 50 120 0.015 41.95 166.17 130.51 0.785 SLAB7 80 70 120 0.010 32.45 228.76 181.64 0.794 SLAB8 60 50 120 0.005 41.30 145.62 133.27 0.915 SLAB9 60 50 120 0.010 33.14 134.62 115.51 0.858

SLAB10 80 70 120 0.010 37.45 223.91 188.89 0.844 SLAB11 60 50 120 0.005 40.43 144.85 112.88 0.779 SLAB12 60 50 120 0.010 37.04 151.88 115.73 0.762 SLAB13 80 70 120 0.010 37.72 238.85 171.96 0.720 SLAB14 60 50 120 0.005 34.71 128.68 84.73 0.658 SLAB15 60 50 120 0.010 33.03 134.39 91.76 0.683

Table 9.3 Comparison of proposed formula with experimental result of Kuang and Morley (1992)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) S1-C03 60 49 120 0.003 38.96 134.62 101 0.750 S1-C10 60 49 120 0.010 27.04 134.73 118 0.876 S1-C16 60 49 120 0.160 32.96 246.77 149 0.604 S2-C03 40 31 120 0.003 38.48 71.98 49 0.681 S2-C10 40 31 120 0.010 36.64 77.95 70 0.898 S2-C16 40 31 120 0.016 34.08 86.92 68 0.782 S1-B10 60 49 120 0.010 36.72 147.60 116 0.786 S2-B03 40 31 120 0.003 40.64 76.88 42 0.546 S2-B10 40 31 120 0.010 47.60 88.85 69 0.777 S1-A10 60 49 120 0.010 37.20 138.95 99 0.712 S2-A03 40 31 120 0.003 38.24 75.74 43 0.568 S2-A10 40 31 120 0.010 48.24 89.65 63 0.703

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As shown in Table 9.3, experimental punching shear failure load carried out by

Kuang and Morley (1992) of highly restraint slabs are also very close to calculated

punching shear failure load according to proposed formula. Thus proposed formula is

also matched with experimental works in this case.

Comparison of the punching shear capacity using proposed formula with test result

of other researchers like Elstner and Hognestad (1956), Moe (1961), Mowrer and

Vanderbilt (1967), Kinnunen et al. (1978), Regan and Zakaria (1979), Rankin and

Long (1987), Gardner (1990), Marzouk and Hussein (1991), Tomaszewicz (1993),

Hallgren (1996), Ramdane (1996), Kevin (2000), Sundquist and Kinnunen (2004),

Birkle and Dilger (2008), Marzouk and Hussein (2007) and Marzouk and Rizk

(2009) are included in the Appendix. The average ratio of experimental failure load

obtained by those researchers to predicted failure load calculated by the proposed

formula is within the range of 0.45 to 0.71 with 4.63% to 13.92% standard deviation.

All these experimental works to obtain punching shear capacity of flat plate were

carried out of simply supported slabs. Test results of simply supported slab

specimens will not be similar to the load capacity of slabs having lateral restraint.

Thus, punching shear failure load obtained by these researchers are smaller than that

calculated by multi panel flat plate proposed formula. It may be mentioned that past

experimental results of Alam (1997) for specimens having simple supports and no

edge restraints also have shown similar response.

9.5 CONCLUDING REMARKS

From the above discussion, it may be concluded that the proposal for estimating

punching shear capacity made in this study predicts the capacity more reasonably

taking into account the effect of flexural steel and effect of column size and slab

thickness which some well practiced codes do not account for. Thus, the proposed

equation may be accepted by codes after applying some appropriate factor of safety

on it.

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CHAPTER-10

CONCLUSIONS AND RECOMMENDATIONS

10.1 CONCLUSIONS

Analysis of punching shear behavior of reinforced concrete slab based on

experimental works and nonlinear finite element investigation of same experimental

works has been carried out. Finite element analysis of previous experimental works

along with comprehensive parametric study has been performed. After this study on

previous experimental works, the validation of material model and FE procedure has

been established. The same FE procedure has been used to analyze multi-panel flat

plate considering full-scale with practical geometry is carried out on the behavior of

punching shear characteristics of concrete slab. A nonlinear solution technique for a

realistic modeling of the punching shear behavior of multi-panel RC flat plates for

different slab thickness, concrete strength, column sizes and flexural reinforcement

ratio has been used. Finally an empirical equation for calculating punching shear

capacity of flat plate is proposed. The following conclusions may be derived from

this study;

( i ) Nonlinear finite element analysis for punching shear behavior of reinforced

concrete slabs can effectively be used to simulate the actual behavior of

reinforced concrete slab under punching load and provide a virtual testing

scheme of structures to explore their behavior under different loadings and

other effects under different conditions.

( ii ) Using appropriate simulation technique and material model for numerical

simulation, it can be demonstrated that significant benefits can be achieved

using finite element tools and advanced computing facilities to obtain safe

and optimum solutions without the need for expensive and time-consuming

laboratory testing.

( iii ) Punching shear strength observed from punching tests conducted on the

restrained reinforced concrete slabs has been found to be higher than the

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predictions of present-day design provisions. Present code provision

underestimates the punching shear capacity of slabs as the code expressions

are based on tests conducted on simply-supported slabs with their edges

unrestrained. The strength enhancement in such case increases with the

degree of edge restraint.

( iv ) The level of flexural reinforcement has a positive effect on the ultimate

punching shear capacity of the reinforced concrete slabs. Although British,

European CEB-FIP and German codes recognize the influence of percentages

of steel, American, Australian and Canadian codes completely ignore the

possible influence of the amount of flexural reinforcement in formulating the

equations of punching shear capacity of slabs. The provision of all these

codes may thus be reviewed to accommodate the influence of flexural steel

more rationally.

( v ) Load deflection behavior of experimental works and FE analysis has been

reasonably predicted using finite element model. In finite element simulation,

the load-displacement response can be predicted over the full load range of

the analysis. Punching failure load can also be predicted by FE analysis.

( vi ) Cracking is an important phenomenon on punching shear behavior of

reinforcement concrete slab. Such cracking areas and pattern of cracking can

be effectively simulated by finite element analysis.

( vii ) For lower level of reinforcement ratios (ρ = 0.50% and 1.0%), some cracking of

the slab is predicted in the immediate vicinity of the column, but punching occurs

before yielding of the entire slab reinforcement. For higher reinforcement ratios

( ρ = 1.5%), punching occurs before any yielding of the reinforcement takes place

and hence a brittle failure is experienced.

( viii ) The magnitude of punching load carrying capacity of slabs increases with the

increase of compressive strength of concrete and the normalized load-

carrying capacity of the all multi-panel slab panel decreases with increase of

277

compressive strength of concrete upto around 48 MPa. Very small or no

increase of normalized punching load carrying is observed above the concrete

strength of 48 MPa.

( ix ) Decreasing tendency of normalized punching shear capacity due to increasing

strength of concrete is higher for smaller column size than that of higher sized

column.

( x ) From the analysis of multi-panel flat plate, it has been found that punching

shear capacity increases with the addition of flexural top reinforcement ratio

from 0.25% to 2.00% percent. However, the rate of increase of load carrying

capacity in punching shear continues to be higher upto 1% reinforcement

ratio after which the effect becomes insignificant.

( xi ) Size of column for flat plate is important for determining punching shear load

carrying capacity. It has been found that for slabs with same size and

reinforcement, the normalized punching shear capacity decreases with the

increase of column size.

( xii ) It has found that for same flexural reinforcement, normalized punching shear

capacity decreases with increase of slab thickness. However, the effect of slab

thickness on normalized punching shear capacity of slab with smaller flexural

reinforcement ratio is very small. Decreasing tendency of punching shear

capacity due to slab thickness with smaller strength of concrete is higher.

( xiii ) The contribution of slab thickness and column dimension may be included in

the presently recognized codes.

The punching shear capacity using proposed formula has been found to agree well

with the results of nonlinear finite element analysis. The load carrying capacities

predicted by the codes are much smaller and over-conservative due to exclusion of

the effect of concrete strength and flexural steel efficiently. The inclusion of size

effect such as slab thickness and column size is also incorporated in the proposed

formula. The proposed empirical equation can be used for estimating the punching

278

capacity of slabs more reasonably and after applying appropriate safety factor, the

proposal may be incorporated in codes so that designer may use the proposed

formula for calculating punching shear capacity of slab for optimum design of

building structure.

10.2 RECOMMENDATIONS

The proposal for calculating punching shear capacity of flat plate with interior square

column is recommended in this study. This formula can predict the punching shear

capacity more reasonably, taking into account the effect of flexural steel and size

effect which some well practiced codes do not consider. The proposed equation may

be verified by nonlinear finite element analysis and experimental works with the

variation of concrete strengths, flexural reinforcement ratios, column shape and size,

slab thickness etc. Additional research work may be carried out to calculate punching

shear capacity for multi-panel flat plate with rectangular column, combined effect of

punching shear and unbalanced moment to verify the proposed equation.

279

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294

APPENDIX-A DETAILS OF MODEL SLAB AN ANALYTICAL RESULTS

Table A1 Details of model slab MSLAB12

Model Slab

thickness Column

Size '

cf ft Ec

ρ

(mm) (mm x mm) (MPa) (MPa) (MPa) (%) MSLAB12-1 250 400 x 400 24 1.630 23180 0.25 MSLAB12-2 250 400 x 400 30 1.820 25920 0.25 MSLAB12-3 250 400 x 400 40 2.106 29940 0.25 MSLAB12-4 250 400 x 400 50 2.350 33470 0.25 MSLAB12-5 250 400 x 400 60 2.579 36670 0.25 MSLAB12-6 250 400 x 400 24 1.630 23180 0.50 MSLAB12-7 250 400 x 400 30 1.820 25920 0.50 MSLAB12-8 250 400 x 400 40 2.106 29940 0.50 MSLAB12-9 250 400 x 400 50 2.350 33470 0.50

MSLAB12-10 250 400 x 400 60 2.579 36670 0.50 MSLAB12-11 250 400 x 400 24 1.630 23180 1.00 MSLAB12-12 250 400 x 400 30 1.820 25920 1.00 MSLAB12-13 250 400 x 400 40 2.106 29940 1.00 MSLAB12-14 250 400 x 400 50 2.350 33470 1.00 MSLAB12-15 250 400 x 400 60 2.579 36670 1.00 MSLAB12-16 250 400 x 400 24 1.630 23180 1.50 MSLAB12-17 250 400 x 400 30 1.820 25920 1.50 MSLAB12-18 250 400 x 400 40 2.106 29940 1.50 MSLAB12-19 250 400 x 400 50 2.350 33470 1.50 MSLAB12-20 250 400 x 400 60 2.579 36670 1.50 MSLAB12-21 250 400 x 400 24 1.630 23180 2.00 MSLAB12-22 250 400 x 400 30 1.820 25920 2.00 MSLAB12-23 250 400 x 400 40 2.106 29940 2.00 MSLAB12-24 250 400 x 400 50 2.350 33470 2.00 MSLAB12-25 250 400 x 400 60 2.579 36670 2.00


Model Slab

thickness Column

Size '

cf ft Ec

ρ



295


Model Slab

thickness Column

Size '

cf ft Ec

ρ




Model Slab

thickness Column

Size '

cf ft Ec

ρ



296


Model Slab

thickness Column

Size '

cf ft Ec

ρ




Model Slab

thickness Column

Size '

cf ft Ec

ρ



297


Model Slab

thickness Column

Size '

cf ft Ec

ρ




Model Slab

thickness Column

Size '

cf ft Ec

ρ



T

F

Table B1

Flexural

Rod

( ρ ) (M

0.25% 24

0.25% 30

0.25% 40

0.25% 50

0.25% 60

0.50% 24

0.50% 30

0.50% 40

0.50% 50

0.50% 60

1.00% 24

1.00% 30

1.00% 40

1.00% 50

1.00% 60

1.50% 24

1.50% 30

1.50% 40

1.50% 50

1.50% 60

2.00% 24

2.00% 30

2.00% 40

2.00% 50

2.00% 60

FAILURE L

Punching

f'c

Failu

Load

V

MPa) (N)

4.00 1291.0.00 1347.0.00 1468.0.00 1590.0.00 1762.4.00 1359.0.00 1428.0.00 1539.0.00 1692.0.00 1846.4.00 1485.0.00 1548.0.00 1644.0.00 1792.0.00 1983.4.00 1561.0.00 1610.0.00 1730.0.00 1864.0.00 2064.4.00 1618.0.00 1672.0.00 1775.0.00 1948.0.00 2128.

A

LOAD AND

g shear stress

ure

d

Side of

Column

a

) (mm)

.68 400

.25 400

.04 400

.13 400

.35 400

.94 400

.58 400

.99 400

.80 400

.81 400

.43 400

.73 400

.96 400

.44 400

.89 400

.06 400

.55 400

.52 400

.29 400

.30 400

.10 400

.93 400

.42 400

.93 400

.88 400

APPENDIX

D PUNCHIN

ses of model

Effective

depth of

slab

d

4

(mm)

175

175

175

175

175

175

175

175

175

175

175

175

175

175

175

175

175

175

175

175

175

175

175

175

175

X-B

NG SHEAR

l slab MSLA

b0 =

4x(a+d)

Punc

She

Str

V/b

(mm) (MP

2300 3.22300 3.32300 3.62300 3.92300 4.32300 3.32300 3.52300 3.82300 4.22300 4.52300 3.62300 3.82300 4.02300 4.42300 4.92300 3.82300 4.02300 4.32300 4.62300 5.2300 4.02300 4.2300 4.42300 4.82300 5.2

R STRESSE

AB11 ching

ear

ess

bod

Non-

dimens

onal

Stres

V/f'c b

Pa)

21 0.133

35 0.111

65 0.091

95 0.079

38 0.073

38 0.140

55 0.118

83 0.095

21 0.084

59 0.076

69 0.153

85 0.128

09 0.102

45 0.089

93 0.082

88 0.161

00 0.133

30 0.107

63 0.092

13 0.085

02 0.167

16 0.138

41 0.110

84 0.096

29 0.088

ES

-

si-

l

s

od

Normal-

ized

Punchin

g Shear

37 0.66

16 0.61

12 0.58

90 0.56

30 0.57

08 0.69

83 0.65

57 0.60

41 0.59

65 0.59

38 0.75

83 0.70

22 0.65

91 0.63

21 0.64

16 0.79

34 0.73

75 0.68

26 0.66

55 0.66

75 0.82

85 0.76

03 0.70

68 0.68

82 0.68

T

Table B2

Flexural

Rod

( ρ ) (M

0.25% 240.25% 300.25% 400.25% 500.25% 600.50% 240.50% 300.50% 400.50% 500.50% 601.00% 241.00% 301.00% 401.00% 501.00% 601.50% 241.50% 301.50% 401.50% 501.50% 602.00% 242.00% 302.00% 402.00% 502.00% 60

Punching

f'c

Failu

Load

V

MPa) (N)

4.00 158350.00 171480.00 190990.00 215640.00 236664.00 164430.00 179280.00 200000.00 225650.00 246464.00 173840.00 188340.00 209850.00 239450.00 263804.00 181310.00 194780.00 217210.00 247660.00 274234.00 186900.00 201110.00 225860.00 253400.00 28501

g shear stress

ure

d

Side of

Column

a

) (mm)

504 400 880 400 920 400 480 400 608 400 316 400 896 400 080 400 576 400 680 400 432 400 424 400 520 400 576 400 008 400 136 400 824 400 120 400 640 400 336 400 072 400 120 400 600 400 048 400 160 400

ses of model

Effective

depth of

slab

d

4

(mm)

225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225

l slab MSLA

b0 =

4x(a+d)

Punc

She

Str

V/b

(mm) (MP

2500 2.82500 3.02500 3.42500 3.82500 4.22500 2.92500 3.2500 3.52500 4.02500 4.32500 3.02500 3.32500 3.72500 4.22500 4.62500 3.22500 3.42500 3.82500 4.42500 4.82500 3.32500 3.52500 4.02500 4.52500 5.0

AB12 ching

ear

ess

bod

Non-

dimens

onal

Stres

V/f'c b

Pa)

82 0.11705 0.10140 0.08483 0.07621 0.07092 0.12119 0.10656 0.08801 0.08038 0.07309 0.12835 0.11173 0.09326 0.08569 0.07822 0.13446 0.11586 0.09640 0.08888 0.08132 0.13858 0.11902 0.10050 0.09007 0.084

-

si-

l

s

od

Normal-

ized

Punching

Shear

73 0.57 16 0.56 49 0.54 67 0.54 01 0.54 18 0.60 62 0.58 89 0.56 02 0.57 30 0.57 88 0.63 16 0.61 33 0.59 51 0.60 82 0.61 43 0.66 54 0.63 65 0.61 81 0.62 13 0.63 84 0.68 92 0.65 04 0.63 01 0.64 44 0.65

T

Table B3

Flexural

Rod

( ρ ) (M

0.25% 240.25% 300.25% 400.25% 500.25% 600.50% 240.50% 300.50% 400.50% 500.50% 601.00% 241.00% 301.00% 401.00% 501.00% 601.50% 241.50% 301.50% 401.50% 501.50% 602.00% 242.00% 302.00% 402.00% 502.00% 60

Punching

f'c

Failu

Load

V

MPa) (N)

4.00 188210.00 200680.00 232270.00 258960.00 283804.00 193230.00 211740.00 242140.00 270880.00 302124.00 204530.00 219140.00 252080.00 288020.00 326114.00 206000.00 222910.00 260760.00 297090.00 335264.00 207950.00 224990.00 262930.00 300430.00 33708

g shear stress

ure

d

Side of

Column

a

) (mm)

136 400 888 400 724 400 616 400 016 400 368 400 472 400 440 400 848 400 280 400 344 400 440 400 892 400 244 400 124 400 064 400 160 400 648 400 956 400 664 400 568 400 952 400 360 400 352 400 880 400

ses of model

Effective

depth of

slab

d

4

(mm)

275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275

l slab MSLA

b0 =

4x(a+d)

Punc

She

Str

V/b

(mm) (MP

2700 2.52700 2.72700 3.2700 3.42700 3.82700 2.62700 2.82700 3.22700 3.62700 4.02700 2.72700 2.92700 3.42700 3.82700 4.32700 2.72700 3.02700 3.52700 4.02700 4.52700 2.82700 3.02700 3.52700 4.02700 4.5

AB13 ching

ear

ess

bod

Non-

dimens

onal

Stres

V/f'c b

Pa)

53 0.10570 0.09013 0.07849 0.06982 0.06360 0.10885 0.09526 0.08165 0.07307 0.06775 0.11495 0.09840 0.08488 0.07739 0.07377 0.11500 0.10051 0.08700 0.08052 0.07580 0.11603 0.10154 0.08805 0.08054 0.075

-

si-

l

s

od

Normal-

ized

Punching

Shear

56 0.52 01 0.49 82 0.49 98 0.49 37 0.49 84 0.53 51 0.52 15 0.52 30 0.52 78 0.53 48 0.56 84 0.54 49 0.54 76 0.55 32 0.57 56 0.57 01 0.55 78 0.56 00 0.57 53 0.58 67 0.57 10 0.55 85 0.56 09 0.57 57 0.59

T

Table B4

Flexural

Rod

( ρ ) (M

0.25% 240.25% 300.25% 400.25% 500.25% 600.50% 240.50% 300.50% 400.50% 500.50% 601.00% 241.00% 301.00% 401.00% 501.00% 601.50% 241.50% 301.50% 401.50% 501.50% 602.00% 242.00% 302.00% 402.00% 502.00% 60

Punching

f'c

Failu

Load

V

MPa) (N)

4.00 150510.00 152260.00 170380.00 185470.00 198164.00 167250.00 175810.00 188600.00 206810.00 225764.00 181700.00 197430.00 218500.00 234870.00 254384.00 190160.00 208380.00 231280.00 247110.00 272044.00 197520.00 217300.00 237360.00 255940.00 27738

g shear stress

ure

d

Side of

Column

a

) (mm)

120 600 600 600 840 600 720 600 680 600 560 600 120 600 000 600 160 600 680 600 000 600 320 600 000 600 760 600 800 600 640 600 800 600 880 600 120 600 440 600 240 600 040 600 600 600 440 600 800 600

ses of model

Effective

depth of

slab

d

4

(mm)

175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175

l slab MSLA

b0 =

4x(a+d)

Punc

She

Str

V/b

(mm) (MP

3100 2.73100 2.83100 3.3100 3.43100 3.63100 3.03100 3.23100 3.43100 3.83100 4.3100 3.33100 3.63100 4.03100 4.33100 4.63100 3.53100 3.83100 4.23100 4.53100 5.03100 3.63100 4.03100 4.33100 4.73100 5.

AB21 ching

ear

ess

bod

Non-

dimens

onal

Stres

V/f'c b

Pa)

77 0.11581 0.09314 0.07842 0.06865 0.06008 0.12824 0.10848 0.08681 0.07616 0.06935 0.13964 0.12103 0.10033 0.08669 0.07851 0.14684 0.12826 0.10656 0.09101 0.08364 0.15101 0.13338 0.10972 0.09411 0.085

-

si-

l

s

od

Normal-

ized

Punching

Shear

56 0.57 36 0.51 85 0.50 84 0.48 09 0.47 85 0.63 80 0.59 69 0.55 62 0.54 94 0.54 96 0.68 13 0.66 07 0.64 66 0.61 82 0.61 61 0.72 80 0.70 66 0.67 11 0.64 36 0.65 17 0.74 35 0.73 94 0.69 44 0.67 52 0.66

T

Table B5

Flexural

Rod

( ρ ) (M

0.25% 240.25% 300.25% 400.25% 500.25% 600.50% 240.50% 300.50% 400.50% 500.50% 601.00% 241.00% 301.00% 401.00% 501.00% 601.50% 241.50% 301.50% 401.50% 501.50% 602.00% 242.00% 302.00% 402.00% 502.00% 60

Punching

f'c

Failu

Load

V

MPa) (N)

4.00 188010.00 197780.00 210320.00 242420.00 266724.00 198280.00 209990.00 224140.00 259400.00 291824.00 200300.00 212590.00 238830.00 275410.00 308184.00 207160.00 221030.00 251960.00 287220.00 320304.00 209290.00 228730.00 259450.00 291550.00 33212

g shear stress

ure

d

Side of

Column

a

) (mm)

112 600 816 600 212 600 200 600 264 600 876 600 900 600 488 600 032 600 240 600 024 600 936 600 320 600 112 600 816 600 656 600 300 600 604 600 240 600 072 600 908 600 304 600 584 600 572 600 200 600

ses of model

Effective

depth of

slab

d

4

(mm)

225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225

l slab MSLA

b0 =

4x(a+d)

Punc

She

Str

V/b

(mm) (MP

3300 2.53300 2.63300 2.83300 3.23300 3.53300 2.63300 2.83300 3.03300 3.43300 3.93300 2.73300 2.83300 3.23300 3.73300 4.3300 2.73300 2.93300 3.33300 3.83300 4.33300 2.83300 3.03300 3.43300 3.93300 4.4

AB22 ching

ear

ess

bod

Non-

dimens

onal

Stres

V/f'c b

Pa)

53 0.10566 0.08883 0.07026 0.06559 0.05967 0.11183 0.09402 0.07549 0.06993 0.06570 0.11286 0.09522 0.08071 0.07415 0.06979 0.11698 0.09939 0.08487 0.07731 0.07182 0.11708 0.10249 0.08793 0.07847 0.074

-

si-

l

s

od

Normal-

ized

Punching

Shear

55 0.52 88 0.49 08 0.45 53 0.46 99 0.46 13 0.55 43 0.52 55 0.48 99 0.49 55 0.51 24 0.55 54 0.52 04 0.51 42 0.52 92 0.54 63 0.57 92 0.54 48 0.54 74 0.55 19 0.56 74 0.58 27 0.56 74 0.55 85 0.56 45 0.58

T

Table B6

Flexural

Rod

( ρ ) (M

0.25% 240.25% 300.25% 400.25% 500.25% 600.50% 240.50% 300.50% 400.50% 500.50% 601.00% 241.00% 301.00% 401.00% 501.00% 601.50% 241.50% 301.50% 401.50% 501.50% 602.00% 242.00% 302.00% 402.00% 502.00% 60

Punching

f'c

Failu

Load

V

MPa) (N)

4.00 224990.00 246740.00 271760.00 298680.00 345694.00 231760.00 251980.00 280430.00 317180.00 359084.00 239800.00 260720.00 301020.00 335280.00 391184.00 244410.00 267510.00 311420.00 359860.00 405904.00 250570.00 274590.00 318130.00 364340.00 41584

g shear stress

ure

d

Side of

Column

a

) (mm)

952 600 440 600 680 600 872 600 992 600 664 600 880 600 344 600 884 600 852 600 072 600 280 600 240 600 848 600 840 600 164 600 176 600 200 600 672 600 040 600 712 600 924 600 360 600 476 600 400 600

ses of model

Effective

depth of

slab

d

4

(mm)

275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275

l slab MSLA

b0 =

4x(a+d)

Punc

She

Str

V/b

(mm) (MP

3500 2.33500 2.53500 2.83500 3.3500 3.53500 2.43500 2.63500 2.93500 3.33500 3.73500 2.43500 2.73500 3.3500 3.43500 4.03500 2.53500 2.73500 3.23500 3.73500 4.23500 2.63500 2.83500 3.33500 3.73500 4.3

AB23 ching

ear

ess

bod

Non-

dimens

onal

Stres

V/f'c b

Pa)

34 0.09756 0.08582 0.07010 0.06259 0.05941 0.10062 0.08791 0.07230 0.06573 0.06249 0.10371 0.09013 0.07848 0.06906 0.06754 0.10578 0.09224 0.08074 0.07422 0.07060 0.10885 0.09531 0.08279 0.07532 0.072

-

si-

l

s

od

Normal-

ized

Punching

Shear

74 0.48 55 0.47 06 0.45 21 0.44 99 0.46 03 0.49 73 0.48 28 0.46 59 0.47 22 0.48 38 0.51 03 0.49 82 0.49 97 0.49 77 0.52 58 0.52 26 0.51 09 0.51 48 0.53 03 0.54 85 0.53 51 0.52 26 0.52 57 0.54 20 0.56

T

Table B7

Flexural

Rod

( ρ ) (M

0.25% 240.25% 300.25% 400.25% 500.25% 600.50% 240.50% 300.50% 400.50% 500.50% 601.00% 241.00% 301.00% 401.00% 501.00% 601.50% 241.50% 301.50% 401.50% 501.50% 602.00% 242.00% 302.00% 402.00% 502.00% 60

Punching

f'c

Failu

Load

V

MPa) (N)

4.00 151050.00 158560.00 180180.00 189630.00 201524.00 182870.00 192090.00 205950.00 215940.00 235984.00 198830.00 218810.00 243020.00 252220.00 267694.00 208250.00 228340.00 253520.00 270230.00 302154.00 223430.00 237240.00 265860.00 287740.00 31780

g shear stress

ure

d

Side of

Column

a

) (mm)

548 800 620 800 820 800 396 800 260 800 776 800 960 800 512 800 424 800 892 800 304 800 128 800 272 800 272 800 924 800 512 800 440 800 244 800 316 800 556 800 312 800 404 800 616 800 484 800 048 800

ses of model

Effective

depth of

slab

d

4

(mm)

175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175 175

l slab MSLA

b0 =

4x(a+d)

Punc

She

Str

V/b

(mm) (MP

3900 2.23900 2.33900 2.63900 2.73900 2.93900 2.63900 2.83900 3.03900 3.3900 3.43900 2.93900 3.23900 3.53900 3.73900 3.93900 3.03900 3.33900 3.73900 3.93900 4.43900 3.23900 3.43900 3.93900 4.23900 4.6

AB31 ching

ear

ess

bod

Non-

dimens

onal

Stres

V/f'c b

Pa)

21 0.09232 0.07764 0.06678 0.05595 0.04968 0.11181 0.09302 0.07516 0.06346 0.05791 0.12121 0.10656 0.08970 0.07392 0.06505 0.12735 0.11171 0.09296 0.07943 0.07327 0.13648 0.11590 0.09722 0.08466 0.077

-

si-

l

s

od

Normal-

ized

Punching

Shear

22 0.45 74 0.42 60 0.42 56 0.39 92 0.38 16 0.55 38 0.51 54 0.48 33 0.45 76 0.45 14 0.59 69 0.59 90 0.56 39 0.52 54 0.51 71 0.62 15 0.61 29 0.59 92 0.56 38 0.57 64 0.67 59 0.63 74 0.62 43 0.60 76 0.60

T

Table B8

Flexural

Rod

( ρ ) (M

0.25% 240.25% 300.25% 400.25% 500.25% 600.50% 240.50% 300.50% 400.50% 500.50% 601.00% 241.00% 301.00% 401.00% 501.00% 601.50% 241.50% 301.50% 401.50% 501.50% 602.00% 242.00% 302.00% 402.00% 502.00% 60

Punching

f'c

Failu

Load

V

MPa) (N)

4.00 201600.00 212400.00 243650.00 265600.00 282804.00 231280.00 243930.00 266840.00 288400.00 307054.00 243210.00 262130.00 291780.00 310410.00 332304.00 246210.00 268910.00 306230.00 335620.00 365274.00 251040.00 272660.00 308770.00 339130.00 37444

g shear stress

ure

d

Side of

Column

a

) (mm)

088 800 096 800 528 800 040 800 080 800 880 800 380 800 460 800 016 800 592 800 112 800 356 800 872 800 172 800 040 800 104 800 160 800 312 800 252 800 768 800 404 800 696 800 704 800 304 800 400 800

ses of model

Effective

depth of

slab

d

4

(mm)

225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225 225

l slab MSLA

b0 =

4x(a+d)

Punc

She

Str

V/b

(mm) (MP

4100 2.4100 2.34100 2.64100 2.84100 3.04100 2.54100 2.64100 2.84100 3.4100 3.34100 2.64100 2.84100 3.4100 3.34100 3.64100 2.64100 2.94100 3.34100 3.64100 3.94100 2.74100 2.94100 3.34100 3.64100 4.0

AB32 ching

ear

ess

bod

Non-

dimens

onal

Stres

V/f'c b

Pa)

19 0.09130 0.07664 0.06688 0.05707 0.05151 0.10464 0.08889 0.07213 0.06233 0.05564 0.10984 0.09416 0.07936 0.06760 0.06067 0.11192 0.09732 0.08364 0.07296 0.06672 0.11396 0.09835 0.08368 0.07306 0.067

-

si-

l

s

od

Normal-

ized

Punching

Shear

11 0.45 68 0.42 60 0.42 76 0.41 11 0.40 45 0.51 81 0.48 23 0.46 25 0.44 55 0.43 99 0.54 47 0.52 91 0.50 73 0.48 00 0.47 12 0.54 72 0.53 30 0.52 28 0.51 60 0.51 34 0.56 85 0.54 37 0.53 35 0.52 76 0.52

T

Table B9

Flexural

Rod

( ρ ) (M

0.25% 240.25% 300.25% 400.25% 500.25% 600.50% 240.50% 300.50% 400.50% 500.50% 601.00% 241.00% 301.00% 401.00% 501.00% 601.50% 241.50% 301.50% 401.50% 501.50% 602.00% 242.00% 302.00% 402.00% 502.00% 60

Punching

f'c

Failu

Load

V

MPa) (N)

4.00 257910.00 276110.00 308440.00 337070.00 366604.00 266500.00 288360.00 315900.00 347520.00 376834.00 289520.00 311020.00 346470.00 393390.00 423564.00 294190.00 324140.00 366890.00 412890.00 444174.00 298110.00 330790.00 376830.00 422830.00 45484

g shear stress

ure

d

Side of

Column

a

) (mm)

128 800 196 800 484 800 788 800 016 800 056 800 648 800 004 800 208 800 320 800 240 800 244 800 720 800 920 800 680 800 976 800 436 800 960 800 960 800 760 800 168 800 952 800 320 800 320 800 480 800

ses of model

Effective

depth of

slab

d

4

(mm)

275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275 275

l slab MSLA

b0 =

4x(a+d)

Punc

She

Str

V/b

(mm) (MP

4300 2.4300 2.34300 2.64300 2.84300 3.4300 2.24300 2.44300 2.64300 2.94300 3.4300 2.44300 2.64300 2.94300 3.34300 3.54300 2.44300 2.74300 3.4300 3.44300 3.74300 2.54300 2.84300 3.4300 3.54300 3.8

AB33 ching

ear

ess

bod

Non-

dimens

onal

Stres

V/f'c b

Pa)

18 0.09034 0.07761 0.06585 0.05710 0.05125 0.09344 0.08167 0.06694 0.05819 0.05345 0.10263 0.08793 0.07333 0.06658 0.05949 0.10374 0.09110 0.07749 0.06976 0.06252 0.10580 0.09319 0.07958 0.07185 0.064

-

si-

l

s

od

Normal-

ized

Punching

Shear

09 0.45 78 0.43 52 0.41 70 0.40 17 0.40 39 0.46 13 0.45 68 0.42 88 0.42 31 0.41 20 0.50 77 0.48 32 0.46 65 0.47 97 0.46 37 0.51 14 0.50 76 0.49 98 0.49 26 0.48 50 0.51 32 0.51 97 0.50 15 0.51 41 0.50

307

APPENDIX-C

COMPARISON OF NORMALIZED PUNCHING SHEAR OF MODEL SLAB

Table C1: Normalized Punching Shear of 200mm thick Flat Plate


( ρ )

f'c

(MPa)


400mm x 400mm Column



Average ACI Code

Canadian Code

0.25% 24 0.66 0.57 0.45 0.56 0.33 0.4 0.25% 30 0.61 0.51 0.42 0.51 0.33 0.4 0.25% 40 0.58 0.5 0.42 0.50 0.33 0.4 0.25% 50 0.56 0.48 0.39 0.48 0.33 0.4 0.25% 60 0.57 0.47 0.38 0.47 0.33 0.4 0.50% 24 0.69 0.63 0.55 0.62 0.33 0.4 0.50% 30 0.65 0.59 0.51 0.58 0.33 0.4 0.50% 40 0.6 0.55 0.48 0.54 0.33 0.40.50% 50 0.59 0.54 0.45 0.53 0.33 0.4 0.50% 60 0.59 0.54 0.45 0.53 0.33 0.4 1.00% 24 0.75 0.68 0.59 0.67 0.33 0.41.00% 30 0.7 0.66 0.59 0.65 0.33 0.4 1.00% 40 0.65 0.64 0.56 0.62 0.33 0.4 1.00% 50 0.63 0.61 0.52 0.59 0.33 0.4 1.00% 60 0.64 0.61 0.51 0.59 0.33 0.4 1.50% 24 0.79 0.72 0.62 0.71 0.33 0.4 1.50% 30 0.73 0.70 0.61 0.68 0.33 0.4 1.50% 40 0.68 0.67 0.59 0.65 0.33 0.4 1.50% 50 0.66 0.64 0.56 0.62 0.33 0.4 1.50% 60 0.66 0.65 0.57 0.63 0.33 0.4 2.00% 24 0.82 0.74 0.67 0.74 0.33 0.4 2.00% 30 0.76 0.73 0.63 0.71 0.33 0.4 2.00% 40 0.7 0.69 0.62 0.67 0.33 0.4 2.00% 50 0.68 0.67 0.60 0.65 0.33 0.4 2.00% 60 0.68 0.66 0.60 0.65 0.33 0.4


Deviation (%) 6.83% 7.94% 8.30% 7.53%

308



( ρ )

f'c

(MPa)





Average ACI Code

Canadian Code

0.25% 24 0.57 0.52 0.45 0.51 0.33 0.4 0.25% 30 0.56 0.49 0.42 0.49 0.33 0.4 0.25% 40 0.54 0.45 0.42 0.47 0.33 0.40.25% 50 0.54 0.46 0.41 0.47 0.33 0.4 0.25% 60 0.54 0.46 0.4 0.47 0.33 0.4 0.50% 24 0.6 0.55 0.51 0.55 0.33 0.4 0.50% 30 0.58 0.52 0.48 0.53 0.33 0.4 0.50% 40 0.56 0.48 0.46 0.50 0.33 0.4 0.50% 50 0.57 0.49 0.44 0.50 0.33 0.4 0.50% 60 0.57 0.51 0.43 0.50 0.33 0.4 1.00% 24 0.63 0.55 0.54 0.57 0.33 0.4 1.00% 30 0.61 0.52 0.52 0.55 0.33 0.4 1.00% 40 0.59 0.51 0.5 0.53 0.33 0.4 1.00% 50 0.6 0.52 0.48 0.53 0.33 0.4 1.00% 60 0.61 0.54 0.47 0.54 0.33 0.4 1.50% 24 0.66 0.57 0.54 0.59 0.33 0.4 1.50% 30 0.63 0.54 0.53 0.57 0.33 0.4 1.50% 40 0.61 0.54 0.52 0.56 0.33 0.4 1.50% 50 0.62 0.55 0.51 0.56 0.33 0.4 1.50% 60 0.63 0.56 0.51 0.57 0.33 0.4 2.00% 24 0.68 0.58 0.56 0.61 0.33 0.42.00% 30 0.65 0.56 0.54 0.58 0.33 0.4 2.00% 40 0.63 0.55 0.53 0.57 0.33 0.4 2.00% 50 0.64 0.56 0.52 0.57 0.33 0.42.00% 60 0.65 0.58 0.52 0.58 0.33 0.4


Deviation (%) 3.97% 3.76% 4.74% 4.05%

309



( ρ )

f'c

(MPa)





Average ACI Code

Canadian Code

0.25% 24 0.52 0.48 0.45 0.48 0.33 0.4 0.25% 30 0.49 0.47 0.43 0.46 0.33 0.4 0.25% 40 0.49 0.45 0.41 0.45 0.33 0.40.25% 50 0.49 0.44 0.40 0.44 0.33 0.4 0.25% 60 0.49 0.46 0.40 0.45 0.33 0.4 0.50% 24 0.53 0.49 0.46 0.49 0.33 0.4 0.50% 30 0.52 0.48 0.45 0.48 0.33 0.4 0.50% 40 0.52 0.46 0.42 0.47 0.33 0.4 0.50% 50 0.52 0.47 0.42 0.47 0.33 0.4 0.50% 60 0.53 0.48 0.41 0.47 0.33 0.4 1.00% 24 0.56 0.51 0.50 0.52 0.33 0.4 1.00% 30 0.54 0.49 0.48 0.50 0.33 0.4 1.00% 40 0.54 0.49 0.46 0.50 0.33 0.4 1.00% 50 0.55 0.49 0.47 0.50 0.33 0.4 1.00% 60 0.57 0.52 0.46 0.52 0.33 0.4 1.50% 24 0.57 0.52 0.51 0.53 0.33 0.4 1.50% 30 0.55 0.51 0.50 0.52 0.33 0.4 1.50% 40 0.56 0.51 0.49 0.52 0.33 0.4 1.50% 50 0.57 0.53 0.49 0.53 0.33 0.4 1.50% 60 0.58 0.54 0.48 0.53 0.33 0.4 2.00% 24 0.57 0.53 0.51 0.54 0.33 0.42.00% 30 0.55 0.52 0.51 0.53 0.33 0.4 2.00% 40 0.56 0.52 0.50 0.53 0.33 0.4 2.00% 50 0.57 0.54 0.51 0.54 0.33 0.42.00% 60 0.59 0.56 0.50 0.55 0.33 0.4


Deviation (%) 3.02% 3.10% 3.82% 3.19%

310

Table C4 Normalized punching shear of slabs having column size 400mm x 400mm


( ρ )

f'c

(MPa)


Slab Thickness

200mm

Slab Thickness

250mm

Slab Thickness

300mm

Average ACI Code

Canadian Code

0.25% 24 0.66 0.57 0.52 0.58 0.33 0.4 0.25% 30 0.61 0.56 0.49 0.55 0.33 0.4 0.25% 40 0.58 0.54 0.49 0.54 0.33 0.4 0.25% 50 0.56 0.54 0.49 0.53 0.33 0.4 0.25% 60 0.57 0.54 0.49 0.53 0.33 0.4 0.50% 24 0.69 0.6 0.53 0.61 0.33 0.4 0.50% 30 0.65 0.58 0.52 0.58 0.33 0.4 0.50% 40 0.6 0.56 0.52 0.56 0.33 0.4 0.50% 50 0.59 0.57 0.52 0.56 0.33 0.4 0.50% 60 0.59 0.57 0.53 0.56 0.33 0.4 1.00% 24 0.75 0.63 0.56 0.65 0.33 0.4 1.00% 30 0.7 0.61 0.54 0.62 0.33 0.4 1.00% 40 0.65 0.59 0.54 0.59 0.33 0.4 1.00% 50 0.63 0.6 0.55 0.59 0.33 0.41.00% 60 0.64 0.61 0.57 0.61 0.33 0.4 1.50% 24 0.79 0.66 0.57 0.67 0.33 0.4 1.50% 30 0.73 0.63 0.55 0.64 0.33 0.41.50% 40 0.68 0.61 0.56 0.62 0.33 0.4 1.50% 50 0.66 0.62 0.57 0.62 0.33 0.4 1.50% 60 0.66 0.63 0.58 0.62 0.33 0.4 2.00% 24 0.82 0.68 0.57 0.69 0.33 0.4 2.00% 30 0.76 0.65 0.55 0.65 0.33 0.4 2.00% 40 0.7 0.63 0.56 0.63 0.33 0.4 2.00% 50 0.68 0.64 0.57 0.63 0.33 0.4 2.00% 60 0.68 0.65 0.59 0.64 0.33 0.4


Deviation (%) 6.83% 3.97% 3.02% 4.34%

311



( ρ )

f'c

(MPa)


Slab Thickness

200mm

Slab Thickness

250mm

Slab Thickness

300mm

Average ACI Code

Canadian Code

0.25% 24 0.57 0.52 0.48 0.52 0.33 0.40.25% 30 0.51 0.49 0.47 0.49 0.33 0.4 0.25% 40 0.5 0.45 0.45 0.47 0.33 0.4 0.25% 50 0.48 0.46 0.44 0.46 0.33 0.4 0.25% 60 0.47 0.46 0.46 0.46 0.33 0.4 0.50% 24 0.63 0.55 0.49 0.56 0.33 0.4 0.50% 30 0.59 0.52 0.48 0.53 0.33 0.4 0.50% 40 0.55 0.48 0.46 0.50 0.33 0.4 0.50% 50 0.54 0.49 0.47 0.50 0.33 0.4 0.50% 60 0.54 0.51 0.48 0.51 0.33 0.4 1.00% 24 0.68 0.55 0.51 0.58 0.33 0.4 1.00% 30 0.66 0.52 0.49 0.56 0.33 0.4 1.00% 40 0.64 0.51 0.49 0.55 0.33 0.4 1.00% 50 0.61 0.52 0.49 0.54 0.33 0.4 1.00% 60 0.61 0.54 0.52 0.56 0.33 0.4 1.50% 24 0.72 0.57 0.52 0.60 0.33 0.4 1.50% 30 0.7 0.54 0.51 0.58 0.33 0.4 1.50% 40 0.67 0.54 0.51 0.57 0.33 0.4 1.50% 50 0.64 0.55 0.53 0.57 0.33 0.41.50% 60 0.65 0.56 0.54 0.58 0.33 0.4 2.00% 24 0.74 0.58 0.53 0.62 0.33 0.4 2.00% 30 0.73 0.56 0.52 0.60 0.33 0.42.00% 40 0.69 0.55 0.52 0.59 0.33 0.4 2.00% 50 0.67 0.56 0.54 0.59 0.33 0.4 2.00% 60 0.66 0.58 0.56 0.60 0.33 0.4


Deviation (%) 7.94% 3.76% 3.10% 4.75%

312



( ρ )

f'c

(MPa)


Slab Thickness

200mm

Slab Thickness

250mm

Slab Thickness

300mm

Average ACI Code

Canadian Code

0.25% 24 0.45 0.45 0.45 0.45 0.33 0.4 0.25% 30 0.42 0.42 0.43 0.42 0.33 0.4 0.25% 40 0.42 0.42 0.41 0.42 0.33 0.40.25% 50 0.39 0.41 0.40 0.40 0.33 0.4 0.25% 60 0.38 0.40 0.40 0.39 0.33 0.4 0.50% 24 0.55 0.51 0.46 0.51 0.33 0.40.50% 30 0.51 0.48 0.45 0.48 0.33 0.4 0.50% 40 0.48 0.46 0.42 0.45 0.33 0.4 0.50% 50 0.45 0.44 0.42 0.44 0.33 0.4 0.50% 60 0.45 0.43 0.41 0.43 0.33 0.4 1.00% 24 0.59 0.54 0.50 0.54 0.33 0.4 1.00% 30 0.59 0.52 0.48 0.53 0.33 0.4 1.00% 40 0.56 0.50 0.46 0.51 0.33 0.4 1.00% 50 0.52 0.48 0.47 0.49 0.33 0.4 1.00% 60 0.51 0.47 0.46 0.48 0.33 0.4 1.50% 24 0.62 0.54 0.51 0.56 0.33 0.4 1.50% 30 0.61 0.53 0.50 0.55 0.33 0.4 1.50% 40 0.59 0.52 0.49 0.53 0.33 0.4 1.50% 50 0.56 0.51 0.49 0.52 0.33 0.4 1.50% 60 0.57 0.51 0.48 0.52 0.33 0.4 2.00% 24 0.67 0.56 0.51 0.58 0.33 0.42.00% 30 0.63 0.54 0.51 0.56 0.33 0.4 2.00% 40 0.62 0.53 0.50 0.55 0.33 0.4 2.00% 50 0.60 0.52 0.51 0.54 0.33 0.42.00% 60 0.60 0.52 0.50 0.54 0.33 0.4


Deviation (%) 8.30% 4.74% 3.82% 5.57%

313

Table C7 Average normalized punching shear capacity of all slabs grouping to same concrete strength.

Flexural Reinforcement f'c

Normalized Punching Shear

( ρ ) (MPa)

400 mm x 400mm Column



Average

0.25% 24 0.58 0.52 0.45 0.52

0.50% 24 0.55 0.49 0.42 0.49

1.00% 24 0.54 0.47 0.42 0.48

1.50% 24 0.53 0.46 0.40 0.46

2.00% 24 0.53 0.46 0.39 0.46

0.25% 30 0.61 0.56 0.51 0.56

0.50% 30 0.58 0.53 0.48 0.53

1.00% 30 0.56 0.50 0.45 0.50

1.50% 30 0.56 0.50 0.44 0.50

2.00% 30 0.56 0.51 0.43 0.50

0.25% 40 0.65 0.58 0.54 0.59

0.50% 40 0.62 0.56 0.53 0.57

1.00% 40 0.59 0.55 0.51 0.55

1.50% 40 0.59 0.54 0.49 0.54

2.00% 40 0.61 0.56 0.48 0.55

0.25% 50 0.67 0.60 0.56 0.61

0.50% 50 0.64 0.58 0.55 0.59

1.00% 50 0.62 0.57 0.53 0.57

1.50% 50 0.62 0.57 0.52 0.57

2.00% 50 0.62 0.58 0.52 0.57

0.25% 60 0.69 0.62 0.58 0.63

0.50% 60 0.65 0.60 0.56 0.60

1.00% 60 0.63 0.59 0.55 0.59

1.50% 60 0.63 0.59 0.54 0.59

2.00% 60 0.64 0.60 0.54 0.59

Average 0.60 0.55 0.50 0.55

Standard Deviation (%)

4.34% 4.75% 5.57% 4.86%

dbfV c 0'

FFigure D1

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

APPLICPUNCHI

( a

( c

Application( a ) 0.25%(MSLAB12

0 30f 'c

30 4f 'c

ACATION OFING SHEAR

a )

c )


2, 400mm x

40 50c (MPa)


40 50(MPa)

FE AnalysisProposed For

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

APPENDIXF PROPOSER CAPACI

( e )


400mm colu

60

la

60

rmula

30 40f 'c (M

X-D ED FORMUITY CALCU

) or variable s( d ) 1.5% anumn and 250

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

50 6MPa)


ULA FOR ULATION

( b )

( d )

strength of cond ( e ) 2% f0mm thick s

30 40

f 'c (M

30 40f 'c (M

0

ula

)

oncrete of flexural steelslab).

0 50 6

MPa)


50 6MPa)


l

60

la

60

ula

F

Figure D2

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( a

( c


0 30 4f 'c

0 30 4f 'c

a )

c )


3, 400mm x

40 50(MPa)


40 50(MPa)

FE AnalysisProposed Fo

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( e )


400mm colu

60

mula

60

sormula

30 40f 'c (M


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

50 6MPa)


( b )

( d )

strength of cond ( e ) 2% f00mm thick

30 40f 'c (M

FP

30 40f 'c (M

0

mula

)


50 6MPa)


50 6MPa)


l

60

ula

60

mula

F

Figure D3

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( a

( c


30 4f 'c

30 4f 'c

a )

c )


1, 600mm x

40 50(MPa)


40 50(MPa)


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( e )


600mm colu

60

mula

60

rmula

30 40f 'c (M

FP


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

50 6MPa)


( b )

( d )


30 40f 'c (M

F

P

30 40f 'c (M

FP

0

ula

)


50 6MPa)

E Analysis

roposed Formul

50 6MPa)


l

60

la

60

ula

F

Figure D4

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( a

( c


30 4f 'c

0 30 4f 'c

a )

c )


3, 600mm x

40 50(MPa)

FE Analysis

Proposed Form

40 50(MPa)

FE Analysis

Proposed For

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( e )


600mm colu

60

mula

60

rmula

30 40f 'c (M

F

P


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

50 6MPa)

FE Analysis

Proposed Formu

( b )

( d )


30 40f 'c (M

FE

Pr

30 40f 'c (M

F

P

0

ula

)


50 6MPa)

E Analysis

roposed Formula

50 6MPa)

FE Analysis

Proposed Formu

l

60

a

60

ula

F

Figure D5

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( a

( c


0 30f 'c

0 30f 'c

a )

c )


1, 800mm x

40 50(MPa)


40 50(MPa)

FE AnalysisProposed Fo

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( e )


800mm colu

60

mula

60

sormula

30 40f 'c (M

FP


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

50 6MPa)


( b )

( d )


30 40

f 'c (M

30 40

f 'c (M

FP

0

ula

)


50 6

MPa)


50 6

MPa)


l

60

mula

60

la

F

Figure D6

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( a

( c


0 30 4f 'c

0 30f 'c

a )

c )


2, 800mm x

40 50(MPa)


40 50(MPa)

FE Analysis

Proposed For

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

( e )


800mm colu

60

ula

60

rmula

30 40

f 'c (M


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20

50 6

MPa)


( b )

( d )


30 40

f 'c (M

FP

30 40f 'c (M

0

ula

)


50 6

MPa)

FE AnalysisProposed Formula

50 6MPa)

FE Analysis

Proposed Formu

l

60

a

60

ula

F

Figure D7

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( a

( c

Application( a ) 24 MP(MSLAB12

% 1.00%Reinf

% 1.00%Reinforc

a )

c )

n of proposePa, ( b ) 30 M2, 400mm x

2.00%forcement in %

FE Analysis

Proposed Form

2.00%cement in %


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( e )

ed formula foMPa, ( c ) 40

400mm colu

3.00%%

mula

3.00%

mula

1.00%Reinforc

F

P

) or variable f MPa, ( d ) 5umn and 250

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

2.00% 3cement in %

FE Analysis

Proposed Formula

( b )

( d )

flexural reinf50 MPa and 0mm thick s

1.00%Reinforcem

F

P

1.00%Reinfor

3.00%

)


slab).

2.00%ment in %

FE Analysis

Proposed Formula

2.00%cement in %

FE Analysis

Proposed Formula

f a

3.00%

3.00%

a

F

Figure D8

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( a

( c


% 1.00%Reinf

FE Analy

Proposed

% 1.00%Reinforc

F

P

a )

c )


2.00%forcement in %

ysis

d Formula

2.00%cement in %

FE Analysis

Proposed Formu

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( e )

ed formula foMPa, ( c ) 40

400mm colu

3.00%%

3.00%

ula

1.00%Reinforc

FE APropo

) or variable f MPa, ( d ) 5umn and 300

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

2.00% 3cement in %


( b )

( d )

flexural reinf50 MPa and 0mm thick s

1.00%Reinforcem

FE

Pro

1.00%Reinfor

FE A

Propo

3.00%

)


slab).

2.00%ment in %

Analysis

oposed Formula

2.00%cement in %

Analysis

osed Formula

f a

3.00%

3.00%

F

Figure D9

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( a

( c


% 1.00%Reinf

FEPr

% 1.00%Reinforc

a )

c )


2.00%forcement in %

E Analysisroposed Formula

2.00%cement in %


0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( e )

ed formula foMPa, ( c ) 40x 600mm col

3.00%%

a

3.00%

rmula

1.00%

Reinforc

F

P

) or variable f MPa, ( d ) 5lumn and 20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

2.00% 3

cement in %

FE Analysis

Proposed Formu

( b )

( d )

flexural reinf50 MPa and 00mm thick

1.00%Reinforcem

FE AnaPropose

1.00%

Reinfor

F

P

3.00%

la

)

forcement of( e ) 60 MPaslab).

2.00%ment in %

alysised Formula

2.00%

cement in %

FE Analysis

Proposed Formu

f a

3.00%

3.00%

ula

F

Figure D10

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( a

( c

Applicatio( a ) 24 MP(MSLAB23

% 1.00%Reinf

F

P

% 1.00%

Reinforc

F

P

a )

c )

on of proposPa, ( b ) 30 M3, 600mm x

2.00%forcement in %

FE Analysis

Proposed Formu

2.00%

cement in %

FE Analysis

Proposed Formu

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( e )

sed formula MPa, ( c ) 40x 600mm col

3.00%%

ula

3.00%

ula

1.00%

Reinforc

FE An

Propo

) for variable MPa, ( d ) 5lumn and 30

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

2.00% 3

cement in %

nalysis

sed Formula

( b )

( d )

flexural rein50 MPa and 00mm thick

1.00%Reinforcem

FE A

Prop

1.00%

Reinforcem

FE

Pro

3.00%

)

nforcement o( e ) 60 MPaslab).

2.00%ment in %

Analysis

osed Formula

2.00%

ment in %

Analysis

posed Formula

of a

3.00%

3.00%

F

Figure D11

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( a

( c


% 1.00%Reinf

% 1.00%Reinforc

a )

c )

on of proposePa, ( b ) 30 M1, 800mm x

2.00%forcement in %

FE Analysis

Proposed For

2.00%cement in %

FE Analysis

Proposed Fo

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( e )

ed formula fMPa, ( c ) 40x 800mm col

3.00%%

rmula

3.00%

s

ormula

1.00%

Reinforc

F

P

) for variable f MPa, ( d ) 5lumn and 20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

2.00% 3

cement in %

FE Analysis

Proposed Formu

( b )

( d )


1.00%Reinforcem

FE

Pr

1.00%

Reinfor

FE

Pr

3.00%

ula

)


2.00%ment in %

E Analysis

roposed Formul

2.00%

cement in %

E Analysis

roposed Formula

of a

3.00%

la

3.00%

a

F

Figure D12

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( a

( c


% 1.00%Reinf

% 1.00%

Reinforc

a )

c )

on of proposePa, ( b ) 30 M2, 800mm x

2.00%forcement in %

FE Analysis

Proposed Form

2.00%

cement in %

FE Analysis

Proposed For

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

( e )

ed formula fMPa, ( c ) 40x 800mm col

3.00%%

mula

3.00%

rmula

1.00%

Reinforcemen

FE

Pro

) for variable f MPa, ( d ) 5lumn and 25

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00%

2.00% 3

nt in %

E Analysis

oposed Formula

( b )

( d )


1.00%Reinforcem

F

P

1.00%Reinfor

F

P

3.00%

)


2.00%ment in %

FE Analysis

Proposed Formula

2.00%cement in %

FE Analysis

Proposed Formul

of a

3.00%

a

3.00%

la

326

APPENDIX-E

COMPARISON OF PROPOSED LOAD CARRYING CAPACITY OF MODEL SLAB WITH VARIOUS CODES

Table E1: Ultimate Load Carrying Capacity of MSLAB11

Rod f'c Analytical

Failure Load (N)

Proposed Formula

(N)

Eurocode 2-2003 and CEB-FIP

Model Code 90 (N)

British (BS 8110-97) code (N)

ACI code (N)

Canadian Code (N)

0.25% 24.00 1291680 1251423 520784 421215 656622 788736 0.25% 30.00 1347248 1335655 560956 453706 734126 881833 0.25% 40.00 1468044 1439372 617352 847696 1018253 0.25% 50.00 1590128 1544460 664973 947753 1138442 0.25% 60.00 1762352 1691872 706597 1038211 1247101 0.50% 24.00 1359944 1308597 655995 530575 656622 788736 0.50% 30.00 1428576 1399183 706597 571502 734126 881833 0.50% 40.00 1539988 1512087 777636 847696 1018253 0.50% 50.00 1692800 1617826 837620 947753 1138442 0.50% 60.00 1846808 1772239 890050 1038211 1247101 1.00% 24.00 1485432 1393612 826311 668328 656622 788736 1.00% 30.00 1548728 1494857 890050 719881 734126 881833 1.00% 40.00 1644960 1623574 979533 847696 1018253 1.00% 50.00 1792436 1731905 1055092 947753 1138442 1.00% 60.00 1983888 1897206 1121134 1038211 1247101 1.50% 24.00 1561056 1462414 945762 764941 656622 788736 1.50% 30.00 1610552 1573010 1018716 823947 734126 881833 1.50% 40.00 1730520 1715811 1121134 847696 1018253 1.50% 50.00 1864288 1830339 1207615 947753 1138442 1.50% 60.00 2064296 2005036 1283205 1038211 1247101 2.00% 24.00 1618096 1522399 1040846 841846 656622 788736 2.00% 30.00 1672928 1641504 1121134 906784 734126 881833 2.00% 40.00 1775416 1797219 1233849 847696 1018253 2.00% 50.00 1948928 1921861 1329025 947753 11384422.00% 60.00 2128880 2105293 1412214 1038211 1247101

327


Rod f'c Analytical

Failure Load (N)

Proposed Formula

(N)


Model Code 90 (N)


ACI code (N)

Canadian Code (N)

0.25% 24.00 1583504 1608972 743041 591056 917640 1102270 0.25% 30.00 1714880 1717271 800357 636649 1025953 1232376 0.25% 40.00 1909920 1850621 880823 1184668 1423025 0.25% 50.00 2156480 1985735 948767 1324499 1590990 0.25% 60.00 2366608 2175263 1008154 1450916 1742843 0.50% 24.00 1644316 1682482 935957 744512 917640 1102270 0.50% 30.00 1792896 1798949 1008154 801942 1025953 1232376 0.50% 40.00 2000080 1944112 1109511 1184668 1423025 0.50% 50.00 2256576 2080061 1195095 1324499 1590990 0.50% 60.00 2464680 2278593 1269901 1450916 1742843 1.00% 24.00 1738432 1791787 1178959 937810 917640 1102270 1.00% 30.00 1883424 1921959 1269901 1010150 1025953 1232376 1.00% 40.00 2098520 2087452 1397573 1184668 1423025 1.00% 50.00 2394576 2226735 1505378 1324499 1590990 1.00% 60.00 2638008 2439265 1599605 1450916 1742843 1.50% 24.00 1813136 1880246 1349389 1073379 917640 1102270 1.50% 30.00 1947824 2022441 1453477 1156177 1025953 1232376 1.50% 40.00 2172120 2206043 1599605 1184668 1423025 1.50% 50.00 2476640 2353294 1722994 1324499 1590990 1.50% 60.00 2742336 2577904 1830844 1450916 1742843 2.00% 24.00 1869072 1957370 1485053 1181293 917640 1102270 2.00% 30.00 2011120 2110506 1599605 1272415 1025953 1232376 2.00% 40.00 2258600 2310710 1760425 1184668 1423025 2.00% 50.00 2534048 2470964 1896219 1324499 1590990 2.00% 60.00 2850160 2706805 2014911 1450916 1742843

328


Rod f'c Analytical

Failure Load (N)

Proposed Formula

(N)


Model Code 90 (N)


ACI code (N)

Canadian Code (N)

0.25% 24.00 1882136 1966521 999333 782923 1211285 1454997 0.25% 30.00 2006888 2098887 1076418 843316 1354258 1626736 0.25% 40.00 2322724 2261871 1184638 1563762 1878393 0.25% 50.00 2589616 2427009 1276018 1748339 2100107 0.25% 60.00 2838016 2658655 1355889 1915210 2300552 0.50% 24.00 1932368 2056366 1258789 986193 1211285 1454997 0.50% 30.00 2117472 2198716 1355889 1062266 1354258 1626736 0.50% 40.00 2421440 2376137 1492206 1563762 1878393 0.50% 50.00 2708848 2542297 1607310 1748339 2100107 0.50% 60.00 3021280 2784947 1707918 1915210 2300552 1.00% 24.00 2045344 2189962 1585609 1242239 1211285 1454997 1.00% 30.00 2191440 2349061 1707918 1338062 1354258 1626736 1.00% 40.00 2520892 2551330 1879627 1563762 1878393 1.00% 50.00 2880244 2721564 2024616 1748339 2100107 1.00% 60.00 3261124 2981324 2151345 1915210 2300552 1.50% 24.00 2060064 2298079 1814824 1421816 1211285 1454997 1.50% 30.00 2229160 2471873 1954814 1531491 1354258 1626736 1.50% 40.00 2607648 2696275 2151345 1563762 1878393 1.50% 50.00 2970956 2876248 2317294 1748339 2100107 1.50% 60.00 3352664 3150772 2462342 1915210 2300552 2.00% 24.00 2079568 2392341 1997280 1564761 1211285 1454997 2.00% 30.00 2249952 2579507 2151345 1685462 1354258 1626736 2.00% 40.00 2629360 2824202 2367634 1563762 1878393 2.00% 50.00 3004352 3020067 2550267 1748339 2100107 2.00% 60.00 3370880 3308317 2709898 1915210 2300552

329


Rod f'c Analytical

Failure Load (N)

Proposed Formula

(N)


Model Code 90 (N)


ACI code (N)

Canadian Code (N)

0.25% 24.00 1505120 1496800 615472 512288 885013 1063079 0.25% 30.00 1522600 1597549 662948 551805 989474 1188558 0.25% 40.00 1703840 1721602 729598 1142547 1372429 0.25% 50.00 1854720 1847296 785878 1277406 1534422 0.25% 60.00 1981680 2023611 835069 1399328 1680875 0.50% 24.00 1672560 1565184 775267 645294 885013 1063079 0.50% 30.00 1758120 1673532 835069 695070 989474 1188558 0.50% 40.00 1886000 1808574 919024 1142547 1372429 0.50% 50.00 2068160 1935046 989915 1277406 1534422 0.50% 60.00 2257680 2119737 1051878 1399328 1680875 1.00% 24.00 1817000 1666869 976549 812831 885013 1063079 1.00% 30.00 1974320 1787966 1051878 875531 989474 1188558 1.00% 40.00 2185000 1941922 1157630 1142547 1372429 1.00% 50.00 2348760 2071494 1246926 1277406 1534422 1.00% 60.00 2543800 2269208 1324977 1399328 1680875 1.50% 24.00 1901640 1749162 1117719 930334 885013 1063079 1.50% 30.00 2083800 1881443 1203937 1002097 989474 1188558 1.50% 40.00 2312880 2052245 1324977 1142547 1372429 1.50% 50.00 2471120 2189230 1427182 1277406 1534422 1.50% 60.00 2720440 2398181 1516515 1399328 1680875 2.00% 24.00 1975240 1820908 1230091 1023867 885013 1063079 2.00% 30.00 2173040 1963368 1324977 1102845 989474 1188558 2.00% 40.00 2373600 2149615 1458186 1142547 1372429 2.00% 50.00 2559440 2298696 1570666 1277406 1534422 2.00% 60.00 2773800 2518095 1668980 1399328 1680875

330


Rod f'c Analytical

Failure Load (N)

Proposed Formula

(N)


Model Code 90 (N)


ACI code (N)

Canadian Code (N)

0.25% 24.00 1880112 1924457 857355 701020 1211285 1454997 0.25% 30.00 1977816 2053991 923489 755095 1354258 1626736 0.25% 40.00 2103212 2213488 1016334 1563762 1878393 0.25% 50.00 2424200 2375095 1094731 1748339 2100107 0.25% 60.00 2667264 2601786 1163255 1915210 2300552 0.50% 24.00 1982876 2012380 1079950 883026 1211285 1454997 0.50% 30.00 2099900 2151685 1163255 951140 1354258 1626736 0.50% 40.00 2241488 2325310 1280205 1563762 1878393 0.50% 50.00 2594032 2487917 1378956 1748339 2100107 0.50% 60.00 2918240 2725376 1465270 1915210 2300552 1.00% 24.00 2003024 2143118 1360338 1112286 1211285 1454997 1.00% 30.00 2125936 2298814 1465270 1198085 1354258 1626736 1.00% 40.00 2388320 2496757 1612584 1563762 1878393 1.00% 50.00 2754112 2663349 1736974 1748339 2100107 1.00% 60.00 3081816 2917553 1845698 1915210 2300552 1.50% 24.00 2071656 2248922 1556987 1273078 1211285 1454997 1.50% 30.00 2210300 2418998 1677089 1371279 1354258 1626736 1.50% 40.00 2519604 2638600 1845698 1563762 1878393 1.50% 50.00 2872240 2814724 1988071 1748339 2100107 1.50% 60.00 3203072 3083375 2112512 1915210 2300552 2.00% 24.00 2092908 2341168 1713522 1401069 1211285 1454997 2.00% 30.00 2287304 2524330 1845698 1509143 1354258 1626736 2.00% 40.00 2594584 2763791 2031259 1563762 1878393 2.00% 50.00 2915572 2955466 2187945 1748339 2100107 2.00% 60.00 3321200 3237551 2324897 1915210 2300552

331


Rod f'c Analytical

Failure Load (N)

Proposed Formula

(N)


Model Code 90 (N)


ACI code (N)

Canadian Code (N)

0.25% 24.00 2249952 2352114 1132577 782923 1570184 1886107 0.25% 30.00 2467440 2510434 1219941 843316 1755519 2108732 0.25% 40.00 2717680 2705375 1342590 2027099 2434954 0.25% 50.00 2986872 2902893 1446153 2266366 2722361 0.25% 60.00 3456992 3179960 1536674 2482679 2982197 0.50% 24.00 2317664 2459575 1426628 986193 1570184 1886107 0.50% 30.00 2519880 2629837 1536674 1062266 1755519 2108732 0.50% 40.00 2804344 2842046 1691166 2027099 2434954 0.50% 50.00 3171884 3040787 1821618 2266366 2722361 0.50% 60.00 3590852 3331015 1935641 2482679 2982197 1.00% 24.00 2398072 2619366 1797023 1242239 1570184 1886107 1.00% 30.00 2607280 2809661 1935641 1338062 1755519 2108732 1.00% 40.00 3010240 3051591 2130244 2027099 2434954 1.00% 50.00 3352848 3255205 2294565 2266366 2722361 1.00% 60.00 3911840 3565898 2438191 2482679 2982197 1.50% 24.00 2444164 2748683 2056800 1421816 1570184 1886107 1.50% 30.00 2675176 2956554 2215456 1531491 1755519 2108732 1.50% 40.00 3114200 3224956 2438191 2027099 2434954 1.50% 50.00 3598672 3440218 2626266 2266366 2722361 1.50% 60.00 4059040 3768570 2790655 2482679 2982197 2.00% 24.00 2505712 2861427 2263584 1564761 1570184 1886107 2.00% 30.00 2745924 3085292 2438191 1685462 1755519 2108732 2.00% 40.00 3181360 3377967 2683319 2027099 2434954 2.00% 50.00 3643476 3612237 2890303 2266366 2722361 2.00% 60.00 4158400 3957007 3071218 2482679 2982197

332


Rod f'c Analytical

Failure Load (N)

Proposed Formula

(N)


Model Code 90 (N)


ACI code (N)

Canadian Code (N)

0.25% 24.00 1510548 1742177 710160 603362 1113403 1337421 0.25% 30.00 1585620 1859442 764940 649904 1244823 1495283 0.25% 40.00 1801820 2003832 841844 1437397 1726604 0.25% 50.00 1896396 2150131 906782 1607059 1930402 0.25% 60.00 2015260 2355351 963541 1760445 2114649 0.50% 24.00 1828776 1821772 894539 760013 1113403 1337421 0.50% 30.00 1920960 1947882 963541 818638 1244823 1495283 0.50% 40.00 2059512 2105062 1060412 1437397 1726604 0.50% 50.00 2159424 2252267 1142209 1607059 1930402 0.50% 60.00 2359892 2467235 1213705 1760445 2114649 1.00% 24.00 1988304 1940126 1126788 957335 1113403 1337421 1.00% 30.00 2188128 2081075 1213705 1031181 1244823 1495283 1.00% 40.00 2430272 2260270 1335727 1437397 1726604 1.00% 50.00 2522272 2411083 1438761 1607059 1930402 1.00% 60.00 2676924 2641209 1528819 1760445 2114649 1.50% 24.00 2082512 2035910 1289676 1095727 1113403 1337421 1.50% 30.00 2283440 2189876 1389158 1180248 1244823 1495283 1.50% 40.00 2535244 2388678 1528819 1437397 1726604 1.50% 50.00 2702316 2548120 1646748 1607059 1930402 1.50% 60.00 3021556 2791325 1749824 1760445 2114649 2.00% 24.00 2234312 2119418 1419336 1205888 1113403 1337421 2.00% 30.00 2372404 2285232 1528819 1298906 1244823 1495283 2.00% 40.00 2658616 2502011 1682522 1437397 1726604 2.00% 50.00 2877484 2675531 1812307 1607059 1930402 2.00% 60.00 3178048 2930898 1925746 1760445 2114649

333


Rod f'c Analytical

Failure Load (N)

Proposed Formula

(N)


Model Code 90 (N)


ACI code (N)

Canadian Code (N)

0.25% 24.00 2016088 2239942 971669 810984 1504930 1807723 0.25% 30.00 2124096 2390711 1046621 873541 1682563 2021096 0.25% 40.00 2436528 2576355 1151845 1942856 2333761 0.25% 50.00 2656040 2764454 1240695 2172179 2609224 0.25% 60.00 2828080 3028308 1318355 2379503 2858262 0.50% 24.00 2312880 2342278 1223944 1021540 1504930 1807723 0.50% 30.00 2439380 2504420 1318355 1100339 1682563 2021096 0.50% 40.00 2668460 2706508 1450899 1942856 2333761 0.50% 50.00 2884016 2895772 1562817 2172179 2609224 0.50% 60.00 3070592 3172159 1660640 2379503 2858262 1.00% 24.00 2432112 2494448 1541716 1286762 1504930 1807723 1.00% 30.00 2621356 2675668 1660640 1386020 1682563 2021096 1.00% 40.00 2917872 2906061 1827595 1942856 2333761 1.00% 50.00 3104172 3099964 1968571 2172179 2609224 1.00% 60.00 3323040 3395840 2091792 2379503 2858262 1.50% 24.00 2462104 2617598 1764586 1472776 1504930 1807723 1.50% 30.00 2689160 2815555 1900701 1586382 1682563 2021096 1.50% 40.00 3062312 3071158 2091792 1942856 2333761 1.50% 50.00 3356252 3276154 2253147 2172179 2609224 1.50% 60.00 3652768 3588847 2394180 2379503 2858262 2.00% 24.00 2510404 2724966 1941992 1620844 1504930 1807723 2.00% 30.00 2726696 2938155 2091792 1745872 1682563 2021096 2.00% 40.00 3087704 3216871 2302094 1942856 2333761 2.00% 50.00 3391304 3439969 2479671 2172179 2609224 2.00% 60.00 3744400 3768297 2634883 2379503 2858262

334


Rod f'c Analytical

Failure Load (N)

Proposed Formula

(N)


Model Code 90 (N)


ACI code (N)

Canadian Code (N)

0.25% 24.00 2579128 2737706 1265821 1038571 1929083 2317217 0.25% 30.00 2761196 2921980 1363463 1118684 2156781 2590728 0.25% 40.00 3084484 3148879 1500541 2490436 2991515 0.25% 50.00 3370788 3378778 1616289 2784392 3344615 0.25% 60.00 3666016 3701265 1717459 3050149 3663842 0.50% 24.00 2665056 2862785 1594466 1308216 1929083 2317217 0.50% 30.00 2883648 3060957 1717459 1409128 2156781 2590728 0.50% 40.00 3159004 3307955 1890127 2490436 2991515 0.50% 50.00 3475208 3539277 2035926 2784392 3344615 0.50% 60.00 3768320 3877083 2163363 3050149 3663842 1.00% 24.00 2895240 3048770 2008438 1647868 1929083 2317217 1.00% 30.00 3110244 3270261 2163363 1774980 2156781 2590728 1.00% 40.00 3464720 3551852 2380861 2490436 2991515 1.00% 50.00 3933920 3788845 2564514 2784392 3344615 1.00% 60.00 4235680 4150471 2725037 3050149 3663842 1.50% 24.00 2941976 3199286 2298776 1886083 1929083 2317217 1.50% 30.00 3241436 3441234 2476098 2031570 2156781 2590728 1.50% 40.00 3668960 3753637 2725037 2490436 2991515 1.50% 50.00 4128960 4004188 2935239 2784392 3344615 1.50% 60.00 4441760 4386368 3118967 3050149 3663842 2.00% 24.00 2981168 3330514 2529888 2075703 1929083 2317217 2.00% 30.00 3307952 3591078 2725037 2235817 2156781 2590728 2.00% 40.00 3768320 3931732 2999003 2490436 2991515 2.00% 50.00 4228320 4204406 3230338 2784392 3344615 2.00% 60.00 4548480 4605697 3432538 3050149 3663842

335

APPENDIX-F COMPARISON OF PROPOSED FORMULA WITH

TEST RESULTS OF OTHER RESEARCHERS Table F1 Comparison of Proposed Formula with Test Result by Elstner and Hognestad (1956)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) A-1b 152 118 254 0.0116 25.20 710.17 365 0.514 A-1c 152 118 254 0.0116 29.00 742.64 356 0.479 A1-d 152 118 254 0.0116 36.80 797.50 351 0.440 A-1e 152 118 254 0.0116 20.30 554.34 356 0.642 A2-b 152 114 254 0.0250 19.50 591.38 400 0.676 A-2c 152 114 254 0.0250 37.40 863.62 467 0.541 A-7b 152 114 254 0.0250 27.90 783.37 512 0.654 A-3b 152 114 254 0.0374 22.60 779.78 445 0.571 A-3c 152 114 254 0.0374 26.50 760.96 534 0.702 A-3d 152 114 254 0.0374 34.50 868.26 547 0.630 A‐4 152 118 356 0.0118 26.10 917.07 400 0.436 A‐5 152 114 356 0.0250 27.80 999.26 534 0.534 A‐6 152 114 356 0.0374 25.00 1034.26 498 0.482 A‐13 152 114 356 0.0055 26.20 710.26 236 0.332 B‐1 152 114 254 0.0048 14.20 404.90 178 0.440 B‐2 152 114 254 0.0048 47.60 740.47 200 0.270 B‐4 152 114 254 0.0101 47.70 802.31 334 0.416 B‐9 152 114 254 0.0200 43.90 871.16 505 0.580 B ‐14 152 114 254 0.0302 50.50 993.65 578 0.582

Average 0.522 Standard deviation (%) 11.65%

Table F2 Comparison of Proposed Formula with Test Result by Moe (1961)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) R-1 152 114 305 0.0138 27.50 812.15 394 0.485 R-2 152 114 152 0.0138 26.50 509.44 312 0.612

S1-60 152 114 254 0.0106 23.20 653.89 390 0.596 S2-60 152 114 254 0.0103 22.00 543.28 356 0.655 S3-60 152 114 254 0.0113 23.80 571.69 334 0.584 S1-70 152 114 254 0.0106 24.40 664.78 393 0.591 S2-70 152 114 254 0.0102 25.30 669.75 379 0.566 S4-70 152 114 254 0.0113 35.10 749.35 374 0.499 S4A-70 152 114 254 0.0113 20.40 529.28 312 0.589 S5-60 152 114 203 0.0106 22.10 470.71 343 0.729 S5-70 152 114 203 0.0106 24.20 571.11 379 0.664


336

Table F3 Comparison of Proposed Formula with Test Result by Mowrer and Vanderbilt (1967)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) M1A 152 114 305 0.015 23.00 773.78 433 0.560

M3-1-0 70 51 152 0.011 21.10 156.97 79 0.503 M3-l-0a 70 51 152 0.022 18.00 162.02 99 0.611 M-4-1-0 70 51 203 0.011 15.50 141.97 93 0.655 M-4-2-0 70 50 203 0.022 27.20 205.13 133 0.648 M-5-1-0 70 51 254 0.011 23.30 243.79 109 0.447 M-5-2-0 70 51 254 0.022 22.90 265.57 152 0.572 M-6-1-0 70 51 305 0.011 23.00 283.34 114 0.402 M-6-2-0 70 50 305 0.022 26.40 283.57 159 0.561 M-7-1-0 70 51 356 0.011 27.70 304.10 139 0.457 M-7-2-0 70 51 356 0.022 25.00 365.46 184 0.503 M-8-1-0 70 51 406 0.011 24.90 373.32 145 0.388 M-8-2-0 70 50 406 0.022 24.60 399.18 185 0.463 M-3-1-2 70 50 152 0.011 27.00 146.09 102 0.698 M-2-1-0 70 50 102 0.011 28.50 112.94 86 0.761 M-2-2-0 70 50 102 0.022 24.90 133.62 102 0.763 M-3-l-0b 70 50 152 0.022 53.80 226.11 172 0.761 M-3-l-4a 70 50 152 0.011 21.10 153.13 99 0.646 M-3-l-4b 70 50 152 0.011 20.00 125.73 112 0.891


Table F4 Comparison of Proposed Formula with Test Result by Kinnunen et al. (1978)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) 1 120 100 125 0.0080 35.70 388.03 216 0.557 3 119 99 125 0.0081 28.60 359.02 194 0.540 5 220 200 250 0.0080 30.30 1480.66 603 0.407 6 219 199 250 0.0080 28.60 1282.04 600 0.468

13 118 98 125 0.0035 33.30 315.72 145 0.459 14 119 99 125 0.0034 31.40 343.47 148 0.431 17 220 200 250 0.0034 31.70 1397.61 489 0.350 18 217 197 250 0.0035 30.20 1351.68 444 0.328


337

Table F5 Comparison of Proposed Formula with Test Result by Regan and Zakaria (1979)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) SS2 100 77 200 0.0120 23.30 289.87 176 0.607 SS4 100 77 200 0.0092 33.40 335.82 194 0.578 SS6 100 79 200 0.0075 21.70 324.63 165 0.508 SS7 100 79 200 0.0080 31.20 365.76 186 0.509 SS8 250 200 250 0.0098 36.30 1595.19 825 0.517 SS9 160 128 160 0.0098 34.50 594.25 390 0.656 SS10 160 128 160 0.0098 35.70 604.49 365 0.604 SS11 80 64 80 0.0098 34.50 161.00 117 0.727 SS12 80 64 80 0.0098 35.70 162.58 105 0.646 SS13 80 64 80 0.0098 37.80 165.21 105 0.636


Table F6 Comparison of Proposed Formula with Test Result by Rankin and Long (1987)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) 1 52 41 100 0.0042 31.50 82.26 36 0.438 2 52 41 100 0.0056 31.50 93.04 49 0.527 3 52 41 100 0.0069 31.50 94.81 57 0.601 4 52 41 100 0.0082 36.20 100.35 56 0.558 5 52 41 100 0.0088 36.20 94.29 57 0.605 6 52 41 100 0.0103 36.20 96.03 66 0.687 7 52 41 100 0.0116 30.40 99.24 71 0.715 8 52 41 100 0.0129 30.40 100.58 71 0.706 9 52 41 100 0.0145 30.40 102.15 79 0.773

10 52 41 100 0.0052 30.60 82.43 44 0.534 11 52 41 100 0.008 30.60 85.80 55 0.641 12 52 41 100 0.0111 30.60 98.91 67 0.677 13 52 41 100 0.006 35.30 96.63 49 0.507 14 52 41 100 0.0069 35.30 97.91 52 0.531 15 52 41 100 0.0199 35.30 104.76 85 0.811 1A 58 47 100 0.0044 29.40 95.31 45 0.472 2A 58 47 100 0.0069 29.40 111.04 66 0.594 3A 58 47 100 0.0129 29.40 118.95 90 0.757 4A 58 47 100 0.0199 31.70 129.67 97 0.748 IB 46 35 100 0.0042 39.60 75.39 29 0.385 2B 46 35 100 0.0069 39.60 82.52 38 0.460 3B 46 35 100 0.0129 39.60 88.91 57 0.641 4B 46 35 100 0.0199 31.70 88.68 73 0.823 1C 65 54 100 0.0042 28.30 112.17 63 0.562 2C 65 54 100 0.0069 33.50 127.23 88 0.692 3C 65 54 100 0.0129 33.50 149.04 124 0.832 4C 65 54 100 0.0199 28.30 150.36 126 0.838


338

Table F7 Comparison of Proposed Formula with Test Result by Gardner (1990)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) 8 101 76 102 0.0205 24.10 232.63 129 0.555 9 101 76 102 0.0205 22.60 227.46 136 0.598

10 101 76 102 0.0205 24.60 205.79 129 0.627 11 152 113 152 0.0214 22.60 440.23 311 0.706 12 152 113 203 0.0214 24.80 624.25 357 0.572 13 153 122 203 0.0066 24.80 602.59 271 0.450 14 102 73 152 0.0501 25.00 335.89 202 0.601 15 102 81 152 0.0147 25.00 273.52 160 0.585 16 102 86 152 0.0045 23.20 251.25 107 0.426 17 102 81 102 0.0147 25.50 247.93 121 0.488 19 152 123 203 0.0047 22.10 571.83 271 0.474 20 152 113 203 0.0214 15.10 521.22 278 0.533 21 153 122 203 0.0066 16.10 418.77 230 0.549 23 102 81 152 0.0147 14.50 208.31 108 0.518 25 153 122 203 0.0066 52.10 736.06 306 0.416 26 102 73 203 0.0501 52.10 538.43 323 0.600 27 102 81 152 0.0147 52.10 389.69 243 0.624 28 102 86 152 0.0045 52.10 376.51 148 0.393


Table F8 Comparison of Proposed Formula with Test Result by Marzouk and Hussein (1991)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) NS1 120 95 150 0.0147 42.00 453.77 320 0.705 HS1 120 95 150 0.0049 67.00 439.37 178 0.405 HS2 120 95 150 0.0084 70.00 470.81 249 0.529 HS7 120 95 150 0.0119 74.00 563.09 356 0.632 HS3 120 95 150 0.0147 69.00 560.40 356 0.635 HS4 120 90 150 0.0237 66.00 518.10 418 0.807 NS2 150 120 150 0.0094 30.00 540.59 396 0.733 HS5 150 125 150 0.0064 68.00 669.06 365 0.546 HS6 150 120 150 0.0094 70.00 740.34 489 0.661 HS8 150 120 150 0.0111 69.00 750.05 436 0.581 HS9 150 120 150 0.0161 74.00 736.18 543 0.738 HS10 150 120 150 0.0233 80.00 809.97 645 0.796 HS11 90 70 150 0.0095 70.00 316.90 196 0.618 HS12 90 70 150 0.0152 75.00 388.57 258 0.664 HS13 90 70 150 0.0187 68.00 353.96 267 0.754 HS14 120 95 220 0.0147 72.00 664.75 498 0.749 HS15 120 95 300 0.0147 71.00 831.26 560 0.674

Average 0.660Standard deviation (%) 10.46%

339

Table F9 Comparison of Proposed Formula with Test Result by Tomaszewicz (1993)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) 65-1-1 300 275 200 0.0149 64.00 3035.26 2050 0.675 95-1-1 300 275 200 0.0149 84.00 3477.33 2250 0.647

115-1-1 300 275 200 0.0149 112.00 3107.94 2450 0.788 95-1-3 300 275 200 0.0255 90.00 3423.49 2400 0.701 65-2-1 225 200 150 0.0175 70.00 1598.05 1200 0.751

95-2-1D 225 200 150 0.0175 88.00 1957.66 1100 0.562 95-2-1 225 200 150 0.0175 87.00 1946.51 1300 0.668

115-2-1 225 200 150 0.0175 119.00 1740.62 1400 0.804 95-2-3 225 200 150 0.0262 90.00 1847.44 1450 0.785

95-2-3D 225 200 150 0.0262 80.00 1800.07 1250 0.694 95-2-3D+ 225 200 150 0.0262 98.00 2234.82 1450 0.649 115-2-3 225 200 150 0.0262 108.00 2346.07 1550 0.661 95-3-1 113 88 100 0.0184 85.00 396.80 330 0.832


Table F10 Comparison of Proposed Formula with Test Result by Hallgren (1996)

Model t d c ρ f'c Failure Load calculated by

Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) HSC0 225 200 250 0.008 90.00 1871.29 965 0.516 HSC1 225 200 250 0.008 91.00 1873.36 1021 0.545 HSC2 225 194 250 0.0082 86.00 2148.75 889 0.414 HSC4 225 200 250 0.0119 92.00 2001.98 1041 0.520 HSC6 225 201 250 0.006 109.00 1831.90 960 0.524 HSC9 225 202 250 0.0033 84.00 1708.67 565 0.331

N/HSC8 225 198 250 0.008 95.00 2319.65 944 0.407 Average 0.465

Standard deviation (%) 8.12%

340

Table F11 Comparison of Proposed Formula with Test Result by Ramdane (1996)

Model t d c ρ f'c


Proposed Formula

Experimental Failure

Load Vtest/ Vn

(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) 1 125 98 150 0.0058 88.20 593.83 224 0.377 2 125 98 150 0.0058 56.20 452.22 212 0.469 3 125 98 150 0.0058 26.90 374.55 169 0.451 4 125 98 150 0.0058 58.70 456.05 233 0.511 6 125 98 150 0.0058 101.80 637.97 233 0.365

12 125 98 150 0.0128 60.40 536.53 319 0.595 13 125 98 150 0.0128 43.40 468.50 297 0.634 14 125 98 150 0.0128 60.80 508.96 341 0.670 16 125 98 150 0.0128 98.40 552.58 362 0.655 21 125 98 150 0.0128 41.90 446.87 286 0.640 22 125 98 150 0.0128 84.20 541.13 405 0.748 23 125 100 150 0.0087 56.40 487.46 341 0.700 25 125 100 150 0.0127 32.90 444.69 244 0.549 26 125 100 150 0.0127 37.60 434.97 294 0.676 27 125 102 150 0.0103 33.70 411.98 227 0.551

Average 0.573Standard deviation (%) 11.78%

Table F12 Comparison of Proposed Formula with Test Result by Kevin (2000)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) Vn (kN) Vtest(kN) P100 135 100 200 0.0097 39.40 543.25 330 0.607 P150 190 150 200 0.009 39.40 942.34 583 0.619 P200 240 200 200 0.0083 39.40 1422.85 904 0.635 P300 345 300 200 0.0076 39.40 2512.78 1381 0.550 P400 450 400 300 0.0076 39.40 4690.52 2224 0.474 P500 550 500 300 0.0076 39.40 7046.72 2681 0.380


Table F13 Comparison of Proposed Formula with Test Result by Sundquist and Kinnunen (2004)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) C1 120 100 250 0.008 24.00 460.04 270 0.587 C2 120 100 250 0.008 24.40 463.85 250 0.539 D1 145 125 150 0.0064 27.20 536.06 265 0.494


341

Table F14 Comparison of Proposed Formula with Test Result by Birkle and Dilger (2008)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) 1 160 124 250 0.0154 33.10 848.64 483 0.569 7 230 190 300 0.013 33.50 1670.22 825 0.494

10 300 260 350 0.011 31.00 2458.11 1046 0.426 Average 0.496

Standard deviation (%) 7.18% Table F15 Comparison of Proposed Formula with Test Result by Marzouk and Hussein (2007)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) NSC1 200 158 250 0.0217 35.00 1182.59 678 0.573 HSC1 200 138 250 0.0248 68.50 1309.48 788 0.602 HSC2 200 128 250 0.0268 70.00 1211.23 801 0.661 HSC3 200 158 250 0.0167 66.70 1443.11 802 0.556 HSC4 200 158 250 0.0113 61.20 1408.67 811 0.576 HSC5 150 113 250 0.0188 70.00 949.97 480 0.505 NSC2 200 163 250 0.0052 33.00 1090.94 479 0.439 NSC3 150 105 250 0.004 34.00 596.71 228 0.382


Table F16 Comparison of Proposed Formula with Test Result by Marzouk and Rizk (2009)

Model t d c ρ f'c


Proposed Formula


(mm) (mm) (mm) MPa Vn (kN) Vtest(kN) NS1 150 105 250 0.0045 44.70 635.12 219 0.345 NS2 200 153 250 0.0055 50.20 1131.40 491 0.434 NS3 250 183 250 0.0035 35.00 1266.30 438 0.346 HS1 250 183 250 0.0035 70.00 1460.75 574 0.393 NS4 300 218 250 0.0073 40.00 1796.86 882 0.491 HS2 300 218 250 0.0073 64.70 2181.27 1023 0.469 HS3 300 220 250 0.0043 76.00 2289.47 886 0.387 HS4 350 268 400 0.0113 75.00 3209.62 1721 0.536 HS6 350 263 400 0.0144 65.40 3195.70 2090 0.654 NS4 400 313 400 0.0157 40.00 3404.10 2234 0.656 HS7 400 313 400 0.0157 60.00 3975.95 2513 0.632


342

APPENDIX-G

LOAD DEFLECTION CURVE USING

IDEAL TENSION SOFTENING

Figure G.1 Load-deflection curves of slab MSLAB11 for '

cf =24 MPa.


cf =30 MPa.

0

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cf =40 MPa.


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cf =60 MPa.


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cf =50 MPa.


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Figure G.13 Load-deflection curves of slab MSLAB13for '

cf =40 MPa.


cf =50 MPa.

0

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cf =30 MPa.


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cf =50 MPa.


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cf =24 MPa.


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cf =40 MPa.


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cf =60 MPa.


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cf =30 MPa.


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cf =50 MPa.


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cf =40 MPa.


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Deflection (mm)


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Deflection (mm)


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cf =60 MPa.


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cf =30 MPa.


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Deflection (mm)


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Deflection (mm)


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cf =50 MPa.


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Deflection (mm)


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Deflection (mm)


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cf =24 MPa.


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Loa

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Deflection (mm)


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Deflection (mm)


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cf =40 MPa.


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Deflection (mm)


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Deflection (mm)


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Deflection (mm)


FINITE ELEMENT STUDY OF RC SLAB IN PUNCHING SHEAR

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Transcript of FINITE ELEMENT STUDY OF RC SLAB IN PUNCHING SHEAR