sherwood 1 and 2

One-Way Shear Behaviour of Large, Lightly-Reinforced Concrete Beams and Slabs

By:

Edward G. Sherwood

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Department of Civil Engineering University of Toronto

© Edward G. Sherwood (2008)

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ABSTRACT

One-Way Shear Behaviour of Large, Lightly-Reinforced Concrete Beams and Slabs Edward G. Sherwood, Department of Civil Engineering, University of Toronto Doctor of Philosophy, 2008

This research focuses on improving our understanding of the behaviour of large, lightly-

reinforced concrete beams and one-way slabs subjected to shear. Empirically-based shear design

methods, particularly those in the widely-used American Concrete Institute design code for concrete

structures (ACI-318) do not accurately predict the behaviour of these important structural elements,

and may produce unsafe designs in certain situations. Furthermore, the research community has not

reached consensus on the exact mechanisms of shear transfer in reinforced concrete. This has slowed

the replacement of empirically-based methods with rational methods based on modern theories of the

shear behaviour of reinforced concrete. Shear failures in reinforced concrete are brittle and sudden,

and typically occur with little or no warning. Furthermore, they are difficult to predict due to

complex failure mechanisms. It is critical, therefore, that shear design methods for reinforced

concrete be accurate, rational and theoretically sound.

An extensive experimental program consisting of load-testing thirty-seven large-scale

reinforced concrete beams and slabs has been performed. The results conclusively show that the ACI

shear design method can produce dangerously unsafe designs for thick concrete flexural elements

constructed without transverse reinforcement. However, safe predictions of the failure loads of small-

scale elements are produced. It is shown that the ACI design method does not account for the size-

effect in shear, in which the shear stress causing failure decreases as the beam depth increases.

Detailed measurements of flexural and shear stresses in the experimental specimens indicated

that aggregate interlock is the primary mechanism of shear transfer in slender, lightly-reinforced

concrete beams. It is also shown that the size-effect can be explained by reduced aggregate interlock

capacity in members with widely spaced cracks.

Digital three-dimensional topographical maps of the surfaces of failure shear cracks were

constructed by scanning the surfaces with a laser profilometer. It was shown that concrete made with

larger aggregate produced rougher cracks with a higher aggregate interlock capacity. The shear

strength of reinforced concrete is therefore directly related to the roughness of failure shear cracks,

and by extension the aggregate size, since larger aggregates produce cracks with larger asperities with

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improved aggregate-interlock capacity. Acoustic-emission monitoring techniques were employed to

characterize fracturing in large concrete beams.

Extensive studies on the ACI 318-05 requirements for crack control steel show that they do

not adequately prevent the formation of wide cracks, as they do not require a minimum bar diameter

for crack control reinforcement. It is shown that the ACI 318-05 requirements for crack control steel

were based partly on questionable interpretations of published experimental studies on crack widths

in large beams.

Various methods to eliminate the size effect in shear are explored, including the use of

stirrups or longitudinal reinforcement distributed over the beam height. Beam/slab width is shown to

have no effect on failure shear stress. It is concluded that the ACI shear design method should be

replaced with a rational, theoretically-sound shear design method. Modifications to Canadian shear

design methods are recommended.

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ACKNOWLEDGEMENTS

So many people have helped me out during my Ph.D. studies, that it is impossible to list them all in a single page of acknowledgements. I extend my sincerest gratitude to my Ph.D. supervisors –Professors Evan Bentz and Michael Collins. Their invaluable guidance and experience made the work described herein possible. I want to thank the dedicated and resourceful technical staff in the structures laboratory in the Department of Civil Engineering at the University of Toronto. Renzo Basset, John MacDonald, Joel Babbin, Giovanni Buzzeo and Al McClenaghan all helped make the experiments reported in this thesis run smoothly. Numerous graduate students have helped me during the construction and testing of my specimens. My sincerest thanks goes out to them all. I also wish to thank Professors Frank Vecchio and Constantin Christopoulos of the Department of Civil Engineering, and Mr. Gary Klein of Wiss, Janney, Elstner Associates for their thorough review of the thesis.

I want to offer my sincerest thanks to my wife, Toni, for her love, understanding and constant support during my graduate student career. And then, of course, there is my little son Colin, who makes everything worthwhile.

This thesis is dedicated in loving memory of my mom, Marjorie Sherwood.

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“To date, the majority of reinforced concrete beams which have been tested to failure range in depth from 10 to 15 in. Essentially, these are the beams on which all our design practices and

safety factors are based…How representative are the test results derived from such relatively small beams for the safety factors of large beams?”

Professor Gaspar Kani University of Toronto, 1967

Shear Behaviour of Large, Lightly-Reinforced Table of Contents Concrete Beams and One-Way Slabs

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TABLE OF CONTENTS

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

1.1 General............................................................................................................ 1

1.2 Inspiration for Current Study.......................................................................... 4

1.3 Objectives ..................................................................................................... 10

1.4 Organization of Thesis.................................................................................. 12

CHAPTER 2: BACKGROUND ......................................................................13

2.1 General.......................................................................................................... 13

2.2 Development of the ACI Shear Design Method........................................... 14

2.3 The Size Effect in Shear ............................................................................... 18

2.3.1 Current State of Experimental Data ................................................18 2.3.2 Leonhardt and Walther ....................................................................20 2.3.3 Kani .................................................................................................21 2.3.4 Shioya Tests ....................................................................................23 2.3.5 University of Toronto Tests ............................................................25

2.4 Design Methods Based on the MCFT .......................................................... 26

2.4.1 The Modified Compression Field Theory .......................................26 2.4.2 1994 CSA Methods .........................................................................32 2.4.3 2004 CSA Methods .........................................................................33 2.4.4 A Simplified Design Method based on the MCFT .........................36

2.5 Mechanisms of Shear Transfer and Failure .................................................. 36

2.5.1 Early Approaches ............................................................................37 2.5.2 Distribution of Shear Across Beam Depth ......................................37 2.5.3 Aggregate Interlock.........................................................................40 2.5.4 The a/d ratio.....................................................................................47

2.6 Concluding Remarks..................................................................................... 50

CHAPTER 3: WIDE BEAMS .........................................................................51

3.1 General.......................................................................................................... 51

3.2 Experimental Program –Beam AT-1 ............................................................ 55

3.2.1 Specimen Design and Construction ................................................55


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3.2.2 Material Properties ..........................................................................56 3.2.3 Experimental Setup ........................................................................59 3.2.4 Instrumentation................................................................................61

3.3 Experimental Results –Beam AT-1 .............................................................. 63

3.3.1 Load-Deflection Response ..............................................................64 3.3.2 Shear Strain Response .....................................................................69 3.3.3 Longitudinal Rebar Strain Response...............................................74

3.4 Discussion –Beam AT-1............................................................................... 81

3.4.1 Effect of Beam Width......................................................................81 3.4.2 The Effect of Beam Depth...............................................................84 3.4.3 Comparison with Bahen Centre Transfer Beams............................87

3.5 Concluding Remarks –Beam AT-1............................................................... 88

CHAPTER 4: ONE-WAY SLABS..................................................................90

4.1 General.......................................................................................................... 90

4.1.1 One-Way Slabs vs. Wide Beams.....................................................92

4.2 Experimental Program –AT-2 Series............................................................ 94

4.2.1 Specimen Design and Construction ................................................94 4.2.2 Material Properties ..........................................................................97 4.2.3 Test Setup –AT-2 Series..................................................................97 4.2.4 Instrumentation –AT-2 Series .......................................................103

4.3 Experimental Results –AT-2 Series............................................................ 107

4.3.1 General ..........................................................................................107 4.3.2 Load-Displacement Response –AT-2/3000 ..................................110 4.3.3 Longitudinal Steel Strain Response–AT-2/3000...........................113 4.3.4 Transverse Steel Strain Response–AT-3/3000..............................116 4.3.5 Load-Displacement Response -AT-2/250 and AT-2/1000 Series.120 4.3.6 Steel Strain Response –AT-2/250 and AT-2/1000 Series .............121 4.3.7 Failure Photos –AT-2 Series .........................................................129 4.3.8 Analysis of AT-2/250N.................................................................135

4.4 Experimental Program –AT-3 Series.......................................................... 140

4.4.1 Specimen Design and Construction ..............................................140 4.4.2 Instrumentation and Test Setup.....................................................142

4.5 Experimental Results –AT-3 Series............................................................ 144

4.5.1 Transverse Steel Strains ................................................................146 4.5.2 Failure Photos................................................................................147


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4.6 Discussion –AT Series................................................................................ 152

4.6.1 Beam/Slab Width ..........................................................................152 4.6.2 Shrinkage and Temperature Steel..................................................152 4.6.3 Effect of Slab Depth ......................................................................153 4.6.4 Final Remarks................................................................................154

CHAPTER 5: DEPTH AND AGGREGATE SIZE.................................... 159

5.1 General........................................................................................................ 159

5.2 Experimental Program ................................................................................ 162

5.2.1 Aggregate Size Series....................................................................166 5.2.2 Shear Reinforcement Series ..........................................................167 5.2.3 Longitudinal Reinforcement Series...............................................168 5.2.4 Experimental Setup .......................................................................169 5.2.5 Instrumentation..............................................................................171

5.3 Experimental Results –Aggregate Size Series............................................ 173

5.3.1 General ..........................................................................................173 5.3.2 The Effect of Depth.......................................................................176 5.3.3 ACI Code Predictions of Shear Strength.......................................180 5.3.4 The Effect of Aggregate Size ........................................................181 5.3.5 CSA Predictions of Shear Strength ...............................................184 5.3.6 High-Speed Photos of Failure .......................................................189

5.4 Shear Carried in the Compression Zone ..................................................... 193

5.4.1 General ..........................................................................................193 5.4.2 Specimen L-10N2..........................................................................193 5.4.3 Small Specimens ...........................................................................196 5.4.4 Is Shear Failure Caused by Crushing in the Compression Zone? .198 5.4.5 Analysis of a Concrete Tooth........................................................201

5.5 The Size Effect............................................................................................ 204

5.5.1 Crack Spacing and Widths ............................................................204 5.5.2 The Effective Crack Spacing Term...............................................210

5.6 Additional Topics ....................................................................................... 212

5.6.1 Specimen L-10H............................................................................212 5.6.2 Acoustic Emission Monitoring of L-50N2R.................................216 5.6.3 Crack Surface Roughness..............................................................221 5.6.4 Distribution of Strain in Longitudinal Steel ..................................228 5.6.5 Load Cycles –Specimen L-10N2 ..................................................229

5.7 Concluding Remarks................................................................................... 230


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CHAPTER 6: CONTROL OF CRACK WIDTHS .................................... 232

6.1 General........................................................................................................ 232

6.2 Crack Control in the ACI-318 Code........................................................... 234

6.2.1 Crack Control at the Level of the Tensile Steel ............................234 6.2.2 Skin Reinforcement .......................................................................238 6.2.3 Skin Reinforcement –CSA Code...................................................244

6.3 Skin Reinforcement Study .......................................................................... 245

6.3.1 General ..........................................................................................245 6.3.2 Experimental Program...................................................................245 6.3.3 Experimental Results.....................................................................248 6.3.4 Crack Widths as a Function of Steel Stress...................................251 6.3.5 Crack Widths at Maximum Service Load Steel Stress .................255 6.3.6 Predictions of Web Crack Width...................................................256 6.3.7 Another Look at Frantz and Breen ................................................261 6.3.8 Suggested Modifications to ACI Code..........................................266

6.4 Effect of Crack Control Steel on Shear Strength........................................ 268

6.4.1 General ..........................................................................................268 6.4.2 Experimental Program...................................................................269 6.4.3 Experimental Results.....................................................................272 6.4.4 Effect of Dowel Action and Aggregate Interlock .........................276 6.4.5 Code Estimates of the Shear Strength ...........................................278 6.4.6 Suggested Modifications to CSA Code.........................................283

6.5 Concluding Remarks................................................................................... 286

CHAPTER 7: LONGITUDINAL AND TRANSVERSE REINFORCEMENT ............................................................ 287

7.1 General........................................................................................................ 287

7.2 The Effect of Minimum Stirrups on Shear Strength................................... 288

7.2.1 General ..........................................................................................288 7.2.2 Experimental Behaviour................................................................289 7.2.3 High-Speed Photos ........................................................................296 7.2.4 Design Code Predictions ...............................................................298 7.2.5 Concluding Remarks .....................................................................301

7.3 The Effect of ρw on Shear Strength ............................................................ 303

7.3.1 General ..........................................................................................303 7.3.2 Experimental Behaviour................................................................304 7.3.3 Design Code Predictions ...............................................................308


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7.3.4 Acoustic Emission Monitoring......................................................312 7.3.5 Concluding Remarks .....................................................................313

CHAPTER 8: CONCLUDING REMARKS ............................................... 315

8.1 General........................................................................................................ 315

8.2 Effect of Member Width............................................................................. 315

8.3 Effect of Member Depth and Aggregate Size............................................. 316

8.4 Control of Crack Widths............................................................................. 318

8.5 Effect of Longitudinal Reinforcement Ratio .............................................. 320

8.6 Where are all the Failures? ......................................................................... 321

8.7 Future Work................................................................................................ 326

REFERENCES……………………………………………………………...327

Appendix A: AT Series Experimental Data…………………………………333

Appendix B: L-Series Experimental Data………………………………….. 383

Appendix C: S-Series Experimental Data………………………………….. 465

Appendix D: Zurich Target Data………….................................................. 531

Appendix E: Concrete Mix Designs……...….............................................. 546

Shear Behaviour of Large, Lightly-Reinforced List of Figures Concrete Beams and One-Way Slabs

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LIST OF FIGURES

Figure 1-1: The Size Effect in Shear.................................................................................. 3

Figure 1-2: Cross-Section of Adel S. Sedra Lecture Theatre in the Bahen Centre ........... 5

Figure 1-3: Photos of Transfer Girder in Adel S. Sedra Lecture Theatre.......................... 5

Figure 1-4: Design of Transfer Girders, Adel S. Sedra Lecture Theatre........................... 6

Figure 1-5: Underground Liquid Natural Gas Storage Tanks Constructed in Japan......... 7

Figure 1-6: Typical Box Structure, Tokyo Underpass (Yoshida (2000)) .......................... 7

Figure 1-7: Typical Single-Cell Box Underground Structure for Toronto Subway .......... 8

Figure 1-8: Typical Hong Kong Mid-Rise Construction................................................... 9

Figure 1-9: Typical Hong Kong High-Rise Construction –External Transfer Plate ......... 9

Figure 1-10: Typical Hong Kong High-Rise Construction –Internal Transfer Plate ...... 10

Figure 2-1: One-Way and Two-Way Shear Failure in Slabs........................................... 14

Figure 2-2: Collapsed Roof of Air Force Warehouse...................................................... 16

Figure 2-3: Derivation of ACI 318 Equation (11-5)........................................................ 17

Figure 2-4: Summary of 60 Years of Shear Research on Members without Stirrups ..... 19

Figure 2-5: Kani’s Size Effect Tests ................................................................................ 22

Figure 2-6: Crack Diagrams of Kani’s Size Effect Tests Redrawn by MacGregor (1967)........................................................................................................................................... 23

Figure 2-7: Summary of Shioya et. al. (1989) and Shioya (1989) Tests ......................... 24

Figure 2-8: Relationships of the Modified Compression Field Theory........................... 27

Figure 2-9: Equilibrium Conditions and vci Relationship of the MCFT.......................... 28

Figure 2-10: Calculation of sx in the Application of the MCFT to Flexural Elements.... 29

Figure 2-11: Components of Shear Resistance in a Reinforced Concrete Beam ............ 37

Figure 2-12: Distribution of Shear Stress in a Cracked Reinforced Concrete Beam ...... 40

Figure 2-13: Model of Shear Failure as Suggested by Moe ............................................ 41

Figure 2-14: Measured and Theoretical Bond Forces Measured in Beams (Reproduced from Fenwick and Paulay (1968)) .................................................................................... 42

Figure 2-15: Measurement of Shear in the Compression Zone (Kani et. al. (1979)) ...... 43

Figure 2-16: Stress Conditions Above a Crack According to Tureyen and Frosch (2002)........................................................................................................................................... 44


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Figure 2-17: Failure of Reinforced Concrete Beams According to Bazant and Yu (2006a)........................................................................................................................................... 46

Figure 2-18: Model of Shear Failure by Gustafsson and Hillerborg (1988) ................... 46

Figure 2-19: The Compressive Force Path Concept (Kotsovos (1988)).......................... 46

Figure 2-20: Shear in Beam with no Shear Reinforcement According to Stratford and Burgoyne (2003) ............................................................................................................... 47

Figure 2-21: Beam Regions and Disturbed Regions........................................................ 48

Figure 2-22: Failure of Arch Action (Reproduced from Fenwick and Paulay (1968)) ... 48

Figure 2-23: Effect of a/d on Shear Strength (Adapted from Collins et. al. (2007) ........ 49

Figure 2-24: Shear Failure Modes in Reinforced Concrete Beams without Stirrups ...... 50

Figure 3-1: Shear Stress, vu, at which Stirrups are Required -ACI 318-05 ..................... 52

Figure 3-2: Design of Test Specimen AT-1..................................................................... 57

Figure 3-3: Formwork for Beam AT-1 ............................................................................ 58

Figure 3-4: Construction of Beam AT-1.......................................................................... 58

Figure 3-5: Test Setup of Beam AT-1 ............................................................................. 60

Figure 3-6: Instrumentation Layout -Beam AT-1............................................................ 62

Figure 3-7: Applied Load vs. Mid-Span Deflection -Beam AT-1................................... 64

Figure 3-8: Crack Patterns –Beam AT-1, South face ...................................................... 66

Figure 3-9: Crack Patterns –Beam AT-1, North Face ..................................................... 67

Figure 3-10: Failure Crack Pattern in the West End of Beam AT-1 ............................... 68

Figure 3-11: Measured Shear Strains in Zurich Target Grid –Beam AT-1 ..................... 70

Figure 3-12: Load-Deflection Curves, Beams AT-1, DB165 and DB180 ...................... 72

Figure 3-13: Shear Strains Measured on North Face –Beam AT-1, DB165 and DB180 74

Figure 3-14: Rebar Strain Profiles -Beam AT-1.............................................................. 76

Figure 3-15: Average Rebar Strains at Midspan, Quarterspans and Supports –AT-1..... 77

Figure 3-16: Effect of Beam Width on the Failure Shear Stress ..................................... 83

Figure 3-17: Failure Crack Surfaces -Beam AT-1........................................................... 83

Figure 3-18: Effect of Beam Depth on Failure Shear Stress of High-Strength Beams ... 85

Figure 4-1: One-Way Slab Supported on Beams on all Four Sides ................................ 91

Figure 4-2: One-Way Slab Supported on Beams on Two Sides...................................... 92

Figure 4-3: Design of AT-2 Series of Test Specimens .................................................... 95

Figure 4-4: AT-3 Series Formwork ................................................................................. 96


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Figure 4-5: AT-3/3000 Formwork ................................................................................... 96

Figure 4-7: Test Setup: AT-2/250N.............................................................................. 100

Figure 4-8: Test Setup: AT-2/1000W ........................................................................... 100

Figure 4-10: Test Setup: AT-2/3000............................................................................. 103

Figure 4-12: Reinforcement Strain Gauge Setup –AT-2 Series .................................... 106

Figure 4-13: Failure Crack Patterns -AT-2 Series ......................................................... 109

Figure 4-14: Initiation of Flexural Cracks at Locations of Shrinkage Reinforcement -AT-2/1000N........................................................................................................................... 110

Figure 4-15: Day 1 and Day 2 Load-Displacement Response -Specimen AT-2/3000.. 112

Figure 4-16: Load Measured by Load Cells as Percentage of Total Applied Load, and VCC Deflection as Percent of Other Deflections –Day 1............................................... 112

Figure 4-17: Load Measured by Load Cells as Percentage of Total Applied Load -Day 2......................................................................................................................................... 113

Figure 4-18: Strain Readings in Longitudinal Rebar Strain Gauges –AT-2/3000 ........ 114

Figure 4-19: Rate of Increase in Measured Strain as a Function of Applied Load (Day 2)......................................................................................................................................... 115

Figure 4-20: Strain Readings in Transverse Rebar Strain Gauges –AT-2/3000, Day 1 118

Figure 4-21: Strain Readings in Transverse Rebar Strain Gauges –AT-2/3000, Day 2 119

Figure 4-22: Average Strain Readings in Shrinkage Reinforcement -AT-2/3000 ........ 120

Figure 4-23: Load-Displacement Response –AT-2/250 Series ..................................... 122

Figure 4-24: Load-Displacement Response –AT-2/1000 Series ................................... 122

Figure 4-25: Load-Displacement Response -AT-2 Series ............................................. 122

Figure 4-26: Longitudinal Steel Strain Profiles -AT-2 Series ....................................... 123

Figure 4-27: Average Longitudinal Steel Strains -AT-2 Series..................................... 124

Figure 4-28: Ratios of Centreline Strains to Outer Strains –AT-2/1000 series and AT-1......................................................................................................................................... 127

Figure 4-29: Shear Stress vs. Average Shear Strain-AT-2 Series ................................. 128

Figure 4-30: Shrinkage and Temperature Reinforcement Strains–AT-2/1000N, AT-2/1000W and AT-2/3000 (Papp=854kN/m width) ........................................................... 128

Figure 4-32: Displacements, Shear Strains and Rebar Strains –AT-2/250W................ 132

Figure 4-33: High-Speed Digital Photos of Failure in Specimen AT-2/1000W ........... 133

Figure 4-34: High-Speed Digital Photos of Failure in Specimen AT-2/3000 ............... 134

Figure 4-35: Crack Patterns at Load Stages -Specimen AT-2/250N (West Face) ........ 136


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Figure 4-36: Photos of Failure in Specimen AT-2/250N............................................... 137

Figure 4-37: Analysis of AT-2/250N Assuming no Aggregate Interlock Action ......... 138

Figure 4-38: Specimen Design and Test Setup -AT-3 Series ........................................ 141

Figure 4-39: AT-3 Series Formwork ............................................................................. 141

Figure 4-40: Test Instrumentation -AT-3 Series............................................................ 143

Figure 4-41: Test Setup -AT-3 Series ............................................................................ 144

Figure 4-42: Load Deflection Curves -AT-3 Series ...................................................... 145

Figure 4-43: Transverse Steel Strains -AT-3 Specimens............................................... 146

Figure 4-44: High-Speed Digital Photos of Failure –East Face, AT-3/N1.................... 148

Figure 4-45: High-Speed Digital Photos of Failure – East Face, AT-3/N2................... 149

Figure 4-46: High-Speed Digital Photos of Failure – East Face, AT-3/T1 ................... 150

Figure 4-47: High-Speed Digital Photos of Failure – East Face, AT-3/T2 ................... 151

Figure 4-48: Effect of Beam or Slab Width on On-Way Shear Capacity...................... 156

Figure 4-49: Failure Crack Surface -AT-2/3000 ........................................................... 156

Figure 4-50: Effect of Shrinkage/Temperature Steel on Shear Strength of One-Way Slabs......................................................................................................................................... 157

Figure 4-51: Effect of Depth on Shear Strength of Wide Beams and Slabs.................. 158

Figure 4-52: Effect of sxe on Shear Strength of Wide Beams and Slabs........................ 158

Figure 5-1: Structure Using Thick One-Way Transfer Slab.......................................... 160

Figure 5-2: Idealization of Transfer Slab based on Unit Width Test Strips .................. 160

Figure 5-3: Main Reinforcement Details and Test Setup –Large Series ....................... 163

Figure 5-4: Main Reinforcement Details and Test Setup –Small Series ....................... 164

Figure 5-5: Formwork for Large Specimens ................................................................. 166

Figure 5-6: Photograph of L-10HS and S-10HS Following Shear Failure.................... 168

Figure 5-7: Photograph of S-Series Test Setup (East Face Shown) .............................. 169

Figure 5-8: Photographs of L-Series Test Setup............................................................ 170

Figure 5-9: Instrumentation Setup: L- and S- Series ...................................................... 172

Figure 5-10: Failure Crack Patterns in Small and Large Specimens without Stirrups .. 175

Figure 5-11: Typical Load Deflection Curves of Large and Small Specimens.............. 176

Figure 5-12: Effect of Depth on Failure Shear Stress.................................................... 177

Figure 5-13: Effect of Depth and Aggregate Size on Maximum Crack Width at Load Stage Prior to Failure (wmax) ........................................................................................... 178


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Figure 5-14: Effect of Depth and Aggregate Size on Deformations .............................. 179

Figure 5-15: Ability of CSA and ACI Codes to Predict Shear Failure in Large Specimens......................................................................................................................................... 180

Figure 5-16: Change in β due to Changes in the Maximum Aggregate Size ................ 182

Figure 5-17: Effect of ag,eff on Shear Deformations at a) Failure Load in Large Specimens and b) Constant Shear Stresses ....................................................................................... 183

Figure 5-18: Effect of Aggregate Size on the Shear Strength of Thick Slabs and Beams......................................................................................................................................... 187

Figure 5-19: Effect of Aggregate Size on the Shear Strength of Shallow Slabs and Beams......................................................................................................................................... 188

Figure 5-20: Progression of Shear Failure in Specimen L-40N1 .................................. 191

Figure 5-21: Progression of Shear Failure in Specimen S-20N1................................... 192

Figure 5-22: Measurement of Shear Carried in Compression Zone of L-10N2............ 195

Figure 5-23: Calculation of Shear Carried in Compression Zone of L-10N2 According to Classic Theory of Mörsch ............................................................................................... 195

Figure 5-24: Measurement of Shear Carried in Compression Zone of S-40N1 ............ 197

Figure 5-25: Measurement of Shear Carried in Compression Zone of S-10N1 ............ 197

Figure 5-26: Measured Concrete Surface Strains –S-50N2........................................... 199

Figure 5-27: Measured Surface Strains –North End of S-40N2 .................................... 200

Figure 5-28: Analysis of Concrete Tooth in Specimen L-40N1.................................... 203

Figure 5-29: Crack Longitudinal Spacing in L-20N1 and S-20N1................................ 205

Figure 5-30: Crack Longitudinal Spacing in L-20N1 and S-20N1 (Fraction of d) ....... 205

Figure 5-31: Crack Longitudinal Spacing in L-20N1 and S-20N1 (Relative to Spacing at level of Reinforcement) .................................................................................................. 206

Figure 5-32: Average Crack Spacing (Sc) at Mid-Depth of Specimen.......................... 207

Figure 5-33: Aggregate Interlock at Cracks................................................................... 208

Figure 5-34: Maximum Crack Width in Large and Small Specimens........................... 209

Figure 5-35: The CSA 2004 Size Effect Term .............................................................. 211

Figure 5-36: High-Speed Photos of South-West Face of Specimen L-10H at Failure.. 214

Figure 5-37: Shear Stress-Strains at Quarterspans of Various Specimens .................... 215

Figure 5-38: Acoustic Emission Monitoring System: a) View of Data Acquisition Hardware, b) Data Acquisition Hardware and Test Setup (L-50N2), c) Schematic of AE Setup (Katsaga et. al. (2007)), and d) AE Sensor/Pulser ................................................ 217


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Figure 5-39: Located AE Events in L-50N2R ............................................................... 219

Figure 5-40: Average Number of AE Hits Per Second per Receiver ............................ 220

Figure 5-41: Crack Slip and Width in L-50N2R ........................................................... 220

Figure 5-43: Comparison of Failure Crack Surfaces in Normal and High-Strength Concrete .......................................................................................................................... 223

Figure 5-44: Scanned Failure Surfaces in Normal Strength and High-Strength Concrete with 3/8in. Maximum Aggregate Size ............................................................................ 224

Figure 5-45: Comparison of Failure Crack Surfaces in Normal-Strength Concrete ..... 225

Figure 5-46: Scanned Surfaces of Normal Strength Concrete Specimens .................... 226

Figure 5-47: Fracturing of Large Aggregates in L-50N1 .............................................. 227

Figure 5-48: Typical Crack Path in Specimens with 2in. Aggregate ............................ 227

Figure 5-49: Strains in Zurich Targets at Level of Longitudinal Steel.......................... 228

Figure 5-50: Measured Crack Widths and Slips in L-10N2 .......................................... 229

Figure 6-2: Rebar Spacing Requirements –Eq. 6-5 and Simplified Design Expressions......................................................................................................................................... 237

Figure 6-3: Side-Face Cracking in Large Beams (adapted from Frantz and Breen (1980))......................................................................................................................................... 238

Figure 6-4: Frantz and Breen (1980a) Skin Reinforcement Requirements .................... 239

Figure 6-5: ACI 318-02 Skin Reinforcement Requirements ......................................... 240

Figure 6-6: ACI 318-02 and ACI-318-05 Skin Reinforcement Requirements.............. 241

Figure 6-7: Effect of Skin Reinforcement According to Frosch (2002)........................ 242

Figure 6-8: ACI 318-05 Skin Reinforcement Requirements ......................................... 243

Figure 6-9: CSA Skin Reinforcement............................................................................ 244

Figure 6-10: Design of Skin Reinforcement .................................................................. 247

Figure 6-11: Crack Widths in Middle of Specimens at fs Ranging ............................... 250

Figure 6-12 Maximum Crack Widths in 48in. Wide Midspan Region, Measured Visually......................................................................................................................................... 252

Figure 6-13 Average Crack Width at Midheight of Midspan Region, Measured Visually......................................................................................................................................... 254

Figure 6-14 Average Crack Widths at Midheight Midspan Region, -Zurich Target Data......................................................................................................................................... 254

Figure 6-15: Expected Crack Widths at fs=0.67fy.......................................................... 255


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Figure 6-16: Widest Crack in Midspan Region and Crack Width Predictions by Eq. 6-8......................................................................................................................................... 258

Figure 6-17: Experimentally Determined Values of Ψs ................................................ 260

Figure 6-18: Effect of Bar Diameter on Crack Width in Web (Frantz and Breen, 1976)......................................................................................................................................... 262

Figure 6-19: Effect of Bar Size on Crack Magnification Ratio (Frantz and Breen, 1976)......................................................................................................................................... 264

Figure 6-20: Effect of Bar Size on Crack Extension into Web...................................... 264

Figure 6-21: Effect of ρsk on Crack Magnification Ratio .............................................. 266

Figure 6-22: Photographs of L-20D Cage Under Construction..................................... 270

Figure 6-23: Test Setup –Specimens S-20D1 and S-20D2............................................ 271

Figure 6-24: Load vs. Midspan Displacement Curves, L-20D, S-20D1, S-20D2......... 273

Figure 6-25: Failure Crack patterns in L-20D and L-20N1........................................... 274

Figure 6-26: Shear Stress vs. Shear Strain for Specimens L-20DR and L-20N2, Measured at Quarterspan ................................................................................................ 275

Figure 6-27: Analysis of Dowel Action in Specimen L-20D ........................................ 277

Figure 6-28: Size-Effect Factors for Members with Distributed Longitudinal Steel .... 280

Figure 6-29: Shear Strengths of Members with Crack Control Steel ............................. 282

Figure 6-30: CSA Size Effect Term for Members with Crack Control Steel................ 282

Figure 6-31: Definition of sd in Eq. 6-11 ....................................................................... 283

Figure 6-32: Shear Strengths of Members with Crack Control Steel Based on Eq. 6-11285

Figure 6-33: Size Effect Term for Members with Crack Control Steel Based on Eq. 6-11......................................................................................................................................... 285

Figure 7-1: L-Δ Curves of High-Strength Concrete Specimens with and without Stirrups......................................................................................................................................... 291

Figure 7-2: Photographs of Failed High-Strength Concrete Specimens........................ 292

Figure 7-3: Failure Crack Patterns –Specimens with and without Stirrups (d=1400mm)......................................................................................................................................... 293

Figure 7-4: Crack Spacing at d/2 from top of Member at Failure Loads ...................... 294

Figure 7-5: Maximum Measured Crack Widths at Load Stages.................................... 294

Figure 7-6: Measured Stirrup Stresses –East End of L-10HS ....................................... 295

Figure 7-7: Photos of L-10HS at Failure ....................................................................... 297

Figure 7-8: Relative Shear Strength of Members with and without Stirrups ................ 299


xviii

Figure 7-9: Load Deflection Curves and Failure Shear Stresses of L-20L, L-20LR, L-20N1 and L-20N2 ........................................................................................................... 306

Figure 7-10: Failure Crack Patterns (South Face): L-20L, L-20LR and L-20N1......... 307

Figure 7-11: Experimental and Predicted Impact of Changing ρw ................................ 310

Figure 7-12: Predictions of the Shear Strength of Very Lightly Reinforced Members. 311

Figure 7-13: AE Sensor Setup and Crack Patterns on North-East Face of L-20LR...... 314

Figure 7-14: Located AE Events in L-20LR.................................................................. 314

Figure 8-1: Ability of ACI 318-05 and CSA A23.3-04 Design Codes to Predict Beam Shear Failure in Members without Stirrups (adapted from Collins et. al. (2007)) ......... 322

Figure 8-2: Failure of le Viaduc de la Concorde, Laval, Quebec, September 30, 2006 323

Figure 8-3: Comparison of le Viaduc de la Concorde with L-Series Specimens (Chpt. 5)......................................................................................................................................... 325

xix

LIST OF TABLES

Table 2-1; Summary of Previous Experimental tests with d≥1000mm and a/d≥2.5 ....... 19

Table 3-1: 1971 and 2005 ACI Code Minimum Stirrup Requirements .......................... 52

Table 3-2: Experimental Results -AT-1........................................................................... 63

Table 3-3: Beam Width and Depth Series –Experimental Data ...................................... 72

Table 3-4: Shear Strain Data –AT-1, DB165 and DB180 ............................................... 73

Table 3-5: Experimental vs. Predicted Shear Capacities –Size Effect Series.................. 85

Table 3-6: Predicted Shear Capacities –Bahen Alternate Beam...................................... 87

Table 4-1: Concrete Material Properties -AT-2 Series .................................................... 97

Table 4-2: Steel Properties -AT-2 Series ......................................................................... 97

Table 4-3: Shrinkage Strains -AT-2/1000 Specimens ................................................... 104

Table 4-4: Shrinkage Strains -AT-2/3000...................................................................... 104

Table 4-5: As-Built Properties and Experimental Observations –AT-2 Series ............. 108

Table 4-6: Failure Shears –AT-2 Series......................................................................... 108

Table 4-7: Concrete Material Properties –AT-3 Series ................................................. 142

Table 4-8: Experimental Results -AT-3 Series .............................................................. 145

Table 4-9: Data for Narrow and Wide Beams Tested by Kani (1967) .......................... 155

Table 4-10: Predicted Failure Shears for Beam/Slab Width Series............................... 155

Table 5-1: Specimen Cast Dates, Test Dates and Major Variables Studied.................. 165

Table 5-2: Experimental Results -Aggregate Size and Stirrup Series ........................... 174

Table 6-1: Crack Width Data ......................................................................................... 251

Table 6-2: As-Built Properties and Experimental Results, L-20D and S-20D Series ... 273

Table 6-3: Summary of Experiments of Beams with Crack Control Reinforcement .... 281

Table 7-1: Experimental Data for Tests of Specimens with Minimum Stirrups ........... 299

Table 7-2: Experimental Results, L-20L and L-20LR................................................... 306

Table 7-3: Specimen Properties of Very-Lightly Reinforced Members........................ 311

Shear Behaviour of Large, Lightly-Reinforced Introduction Concrete Beams and One-Way Slabs

1

CHAPTER 1: INTRODUCTION

“I am among those who think that science has great beauty. A scientist in his laboratory is not only a technician: he is also a child placed before natural phenomena which impress him like a fairy tale.” -Marie Curie

This Chapter consists of a general introduction to the thesis, and presents an overall context into which the thesis can be placed. The inspiration for the current study is presented, and the overall goals of the thesis are discussed. Lastly, the organization of the thesis is presented.

1.1 General

It has long been a goal of the engineering profession to improve the quality of reinforced

concrete design procedures for shear. Unlike flexural failures, shear failures in reinforced

concrete structures are brittle and sudden. When they occur, they typically do so with

little or no warning. Furthermore, they tend to be less predictable than flexural failures,

due to considerably more complex failure mechanisms. Flexural design provisions are

based on the rational assumption that plane sections remain plane, and this assumption

has proven to be accurate over a wide range of reinforced concrete flexural elements.

However, the search continues for equally accurate shear design provisions, based on

equally rational assumptions.

Two leading reinforced concrete design codes were updated and reissued in 2004 and

2005: the American Concrete Institute’s “ACI-318-05 –Building Code Requirements for

Structural Concrete” (ACI Committee 318 (2005)) and the Canadian Standards

Association’s “A23.3-04 –Design of Concrete Structures” (CSA Committee A23.3

(2004)). Various updates and improvements in the two design codes represent the

culmination of intensive research efforts across North America. The shear design

provisions in the CSA code have been updated and modified based on ongoing research


2

into the shear behaviour of reinforced concrete. However, the corresponding design

provisions in the ACI code are still based on traditional empirical relationships developed

over 40 years ago. As such, there are considerable differences in various aspects of the

respective shear design methods. These differences include how each code accounts for

the effects of: a) member depth, b) maximum aggregate size, c) minimum stirrups, c)

reinforcement ratio, d) web width and e) crack control steel.

A particular aspect of the shear behaviour of reinforced concrete that is deserving of

additional attention is the effect of the maximum aggregate size on the shear response of

reinforced concrete sections. This is particularly true for reinforced concrete beams and

slabs constructed without stirrups, since aggregate interlock is generally, though by no

means universally, believed to be a dominant mechanism of shear transfer in these

element types. Increasing the size of the coarse aggregate produces rougher cracks that

are likely better able to transfer shear stresses. Likewise, reducing the maximum

aggregate size decreases the shear strength of a concrete section. In concrete elements

constructed with high-strength concrete, poor quality aggregate or light-weight aggregate,

the aggregate interlock capacity may be greatly reduced because coarse aggregate

particles will tend to fracture at cracks, resulting in smooth crack surfaces with a greatly

reduced aggregate interlock capacity. Self-consolidating concretes may also exhibit

reduced aggregate interlock capacity, since they are typically mixed with a smaller coarse

aggregate fraction and a smaller maximum aggregate size.

As existing sources of good quality aggregate at quarries are exhausted, marginal sources

at these quarries will be increasingly exploited, particularly as increasingly stringent

environmental regulations inhibit the opening of new quarries. Furthermore, advances in

concrete mixing technologies have allowed concrete suppliers to achieve reasonably high

concrete compressive strengths despite using aggregate of marginal quality. Should

aggregate quality tend to decrease in the future, the aggregate interlock capacity of

concretes will also decrease, despite having otherwise adequate compressive cylinder

strengths. The importance of aggregate interlock must therefore be understood, and this


3

understanding must be reflected in structural design codes. An additional reason for the

importance of understanding aggregate interlock is that it has been proposed as a major

mechanism governing the “size effect in shear” (Figure 1-1).

Figure 1-1: The Size Effect in Shear

The size effect in shear is a phenomenon exhibited by slender reinforced concrete

members constructed without shear reinforcement in which the failure shear stress

decreases as the effective depth increases. However, the severity of the size effect is

neither universally known nor understood. There is still considerable debate as to how,

or indeed whether, design codes such as the ACI-318 code should account for the size

effect. While Shioya et. al. (1989) in Japan have conducted the most extensive series of

tests to study the size effect, there has been relatively little North American effort

directed towards a similar systematic study of the size effect in which such a large range

of depths has been studied. This lack of experimental data is a likely cause of the poor

understanding of the severity of the size effect. The ACI-318 code, for example, does not

account at all for the size effect. Equation (11-3) of the ACI-318 code predicts that at a

given concrete strength f’c the failure shear stress is constant for all effective depths.


4

1.2 Inspiration for Current Study

The work described in this thesis was inspired by a design situation discussed by Lubell

et. al. (2004). Figure 1-2 is a photograph showing one portion of the Bahen Centre, a

37,000m2 (400,000ft2) engineering building at the University of Toronto. Completed in

2002, it houses extensive offices, laboratories, teaching spaces and common areas. A

series of large lecture theatres are situated on the ground floor, and to provide adequate

sightlines, column loads from the upper stories must be transferred to more widely spaced

columns on the ground floor. This is a common design situation in large-scale

commercial, educational, residential or health-related structures, as architects commonly

desire open, column-free areas on lower levels to enable the creation of “feature spaces.”

A cross-section of the 278 seat Adel S. Sedra lecture theatre is shown in Figure 1-2, and

it can be seen that a series of large transfer girders has been provided to transfer column

loads from the upper eight stories to large columns at either side of the theatre. A

photograph of the girder in the middle of the theatre is shown in Figure 1-3. This beam

was designed using the 1994 CSA design code for concrete buildings (CSA Committee

A23.3 (1994)), and the design is shown in Figure 1-4. To safely transfer a column

service load of 9800kN, the beam has been heavily reinforced in both flexure and shear.

Because such a complex rebar cage is difficult and expensive to construct, the designer

could have chosen to modify the beam cross-sectional dimensions so as to reduce the

steel requirements. To maintain sightlines in the lecture theatre, however, only the beam

width can be modified. If the design was being carried out using ACI 318-05, an

alternative design meeting all requirements of the code could be that shown in Figure 1-4.

In this case, the beam width has been increased and the concrete strength doubled, such

that stirrups are no longer required. This modification has also allowed for a significant

reduction in flexural steel requirements. While this beam is heavier than the as-built

beam, such a design may in fact be less expensive to construct, particularly as

construction labour costs and steel material costs continue to increase, and as high-

strength concrete becomes more competitively priced.


5

Figure 1-2: Cross-Section of Adel S. Sedra Lecture Theatre in the Bahen Centre

Figure 1-3: Photos of Transfer Girder in Adel S. Sedra Lecture Theatre


6

Figure 1-4: Design of Transfer Girders, Adel S. Sedra Lecture Theatre

Interestingly, the ACI code predicts that increasing the beam width by 1.7m and

increasing the concrete strength to 70MPa, while at the same time reducing the

reinforcement ratio to 0.93%, is about equivalent to including fourteen legs of 15M

stirrups spaced at 300mm. The effective depth of 1700mm in the alternate beam is well

beyond the size of typical laboratory shear tests, and at that depth, the size effect can be

expected to dominate shear response. Indeed, it is 4.6 times deeper than the largest

slender beams in the database used to derive the applicable ACI shear design provisions.

A designer using the CSA A23.3-04 design code would find that the ACI 318-05

alternate beam is dangerously unsafe, and would be at risk of imminent shear failure.

Bearing in mind the brittleness of shear failures, the key question that the discrepancy

between the ACI and CSA codes raises is: “Which design code gives the more accurate

prediction of the failure shear stress of large, lightly-reinforced concrete beams and one-

way slabs?” Quite simply, is it safe to ignore the size effect in shear?


7

Answering this question will offer insight into the behaviour of not only the specific case

of the Bahen Centre transfer girders, but also the behaviour of all large concrete beams

and one-way slabs. For example, Figure 1-5 shows a typical cross-section of

underground liquid natural gas storage tanks constructed in Japan in which 9.8m thick

slabs without stirrups are used to resist hydrostatic uplift forces when the tanks are empty.

Figure 1-6 shows the cross-section of a box structure employed in an underpass in Tokyo

in which a 1.25m thick continuous one-way slab with stirrups is employed. Figure 1-7

shows the cross-section of a typical single-cell underground box structure used in a recent

extension to the Toronto subway in which a 1.4m thick one-way slab with stirrups is used.

Figure 1-5: Underground Liquid Natural Gas Storage Tanks Constructed in Japan

(Yoshida (2000))

Figure 1-6: Typical Box Structure, Tokyo Underpass (Yoshida (2000))


8

Figure 1-7: Typical Single-Cell Box Underground Structure for Toronto Subway (Collins and Kuchma (1999))

The construction practice used in the Bahen Centre, in which closely spaced loads from

upper stories are transferred to more widely spaced elements at lower levels, is by no

means rare. Examples of typical mid-rise and high-rise construction practices in Hong

Kong are illustrated in Figure 1-8 to Figure 1-10. In each of these structures, closely

spaced column or wall loads from upper stories are transferred to widely spaced elements

below to accommodate parking garages or ground level shopping and restaurant facilities.

Each of the transfer elements shown in the figures were constructed with stirrups.

Li et. al. (2006), for example, describe the design of an extensive residential development

in Hong Kong consisting of six 34 storey apartment blocks (one of which is shown in

Figure 1-9a)) constructed over an expansive parking garage. Each residential block

“…sits on a 2.7m thick transfer plate supported by columns and core walls. The transfer plate redistributes the loads from the shear wall structure above to widely spaced columns and core walls below. In general, a transfer plate can easily facilitate the architectural layout to provide column-free open space area at the lower stories. Because of such advantages, there is extensive use of transfer plates in high-rise buildings in areas where seismic hazards are not considered, for instance Hong Kong.”

The design considerations used in these structures are identical to those that resulted in

the transfer girders in the Bahen Centre –namely, the need for open spaces on lower

levels. These transfer elements in Hong Kong, however, all contain stirrups, and their

shear behaviour is not expected to be governed by the size effect.


9

Figure 1-8: Typical Hong Kong Mid-Rise Construction

(Li et. al. (2003))

Figure 1-9: Typical Hong Kong High-Rise Construction –External Transfer Plate

(Li et. al. (2006))

2.7m

a) High-Rise Development b) External Transfer Plate


10

Figure 1-10: Typical Hong Kong High-Rise Construction –Internal Transfer Plate (Su et. al. (2003))

1.3 Objectives

This thesis discusses the results of an extensive experimental program consisting of

thirty-seven tests on reinforced concrete beams and one-way slabs. The test specimens

ranged in heights from 280mm to 1.51 metres, in widths from 97mm to 3 metres, and in

mass from 117kg to 29 tonnes. The test specimens were designed to systematically study

the size effect in one-way shear in beams and slabs by studying the role played by


11

aggregate size, member width, member depth, concrete strength, minimum stirrups and

crack control steel. The thesis also presents some interesting results from a series of

collaborative studies carried out with the Rock Physics Group in the Department of Civil

Engineering at the University of Toronto in which acoustic emission monitoring was

employed on a number of large scale specimen to study concrete fracture and failure

mechanisms.

The overall objective of this thesis is to study the one-way shear behaviour of large,

lightly-reinforced concrete beams and slabs, particularly those constructed without shear

reinforcement. Specifically, the work investigates and attempts to quantify the following:

• The significance of the size effect in shear, including its causes, impact on design

choices, and how best to account for it in reinforced concrete design codes;

• The role played by interlocking of aggregate particles at cracks in reinforced

concrete in transferring shear stresses, and the importance of the surface

roughness of cracks in determining shear capacity;

• The effect of web width on the one-way shear capacity of beams and slabs;

• The effect of minimum stirrups on the one-way shear capacity of beams and slabs;

• The role played by crack control steel in determining shear behaviour, and

• The mechanisms of shear failure in lightly-reinforced reinforced concrete

members.

The work described herein will offer insight into the mechanisms of one-way shear

transfer in large, slender, lightly-reinforced concrete beams and slabs. It is hoped that the

experimental data and resulting analysis will support the development of rational,

theoretically-sound shear design provisions, the application of which will ensure

appropriate levels of safety for the types of large-scale structural elements shown in

Figures 1-4 to 1-7.


12

1.4 Organization of Thesis

This thesis is organized into 7 chapters and 5 appendices. Chapter 2 is a brief review of

the voluminous literature on one-way shear in reinforced concrete structures, focusing on

areas most relevant to the research reported in the thesis. Chapter 3 presents an

experimental program that investigates the effect of web width on the shear strength of

lightly-reinforced concrete beams. Chapter 4 describes an extension of the research

outlined in Chapter 3, in which a wide slab and equivalent slab design strips were loaded

to failure to assess the one-way shear behaviour of concrete slabs.

A major experimental program is outlined in Chapter 5, and its results are discussed in

Chapters 5, 6, and 7. The experimental program consists of tests on a series of 1.51m

high and equivalent small specimens constructed with varying aggregate sizes and

concrete strengths. Additional variables include the reinforcement ratio, ρw, and use of

minimum stirrups and horizontal crack control steel. Chapter 5 presents the results of the

aggregate size and concrete strength series, Chapter 6 presents the results of the tests

investigating the use of crack control steel, and Chapter 7 reports the results of tests on

members with stirrups and on a member with a very low reinforcement ratio. The large

members described in Chapter 5, 6 and 7 are referred to as the “L-“ series of specimens,

while the small members are referred to the “S-“ series of specimens. Additional results

from collaborative research conducted by the author and Tatyana Katsaga consisting of

acoustic emission monitoring of fracturing in several “L-“ series specimens are presented

in Chapter 5 and Chapter 7. Detailed experimental results are presented in Appendices A,

B and C for the AT, L- and S-series of experiments, respectively. Appendix D contains

readings of external zurich data targets for all of the experimental series, and Appendix E

contains concrete mix properties.

It is appropriate to note that the experiments described in Chapters 3 and 4 are the “AT”

series of tests and represent collaborative research conducted by the author and Adam

Lubell (Lubell (2006)).

Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs

13

CHAPTER 2: BACKGROUND

“One glance at a book and you hear the voice of another person, perhaps someone dead for 1,000 years. To read is to voyage through time.” –Carl Sagan

The purpose of this Chapter is to offer a brief presentation of the voluminous research that has been conducted on the shear behaviour of reinforced concrete. The emphasis is on research that relates directly to the findings reported in later Chapters. Major North American shear design provisions are described, as are previous studies on the size effect in shear. Mechanisms of shear transfer in reinforced concrete are described with a focus on aggregate interlock.

2.1 General

MacGregor and Bartlett (2000) note that when designing a reinforced concrete member,

flexure is usually considered first, with limits placed on the quantity of flexural

reinforcement so as to ensure a failure would occur gradually. The member is usually

then proportioned for shear such that the shear strength equals or exceeds the shear

required to cause flexural failure everywhere within the member. There is considerable

evidence that the size effect exists, though the ACI 318 code has yet to account for it in

its provisions. The purpose of a design code is to offer design methods to professionals

that, when used, will result in a structure with a minimum acceptable level of safety.

However, thick flexural members designed using the ACI 318 code are at risk of brittle

shear failure at or below the expected flexural capacity.

This Chapter will review the history of the ACI 318 shear design provisions and discuss

different approaches to the design and analysis of concrete sections subjected to shear.

Two-way shear, characterized by shear failures consisting of a truncated cone of

concrete (Figure 2-1) is not considered, though it is certainly worthy of future study. For

this thesis, only one-way (or “beam action”) shear is considered, in which failure is

characterized by a uniform failure surface across the member width.


14

Figure 2-1: One-Way and Two-Way Shear Failure in Slabs

2.2 Development of the ACI Shear Design Method

In 1899 the Swiss engineer Ritter, and in 1902 the

German engineer Mörsch, published papers in which

they outline a 45o truss analogy for the shear design of

reinforced concrete members with web reinforcement.

The model is an elegant simplification of the highly

indeterminate system of internal stresses in a cracked

beam in which shear is visualized to be transferred

through the cracked web by a field of diagonal

compression in the concrete and tension in transverse

reinforcement.

The 45o truss model allowed designers to calculate tensile stresses in longitudinal steel

and stirrups and compressive stresses in the uncracked compression zone and inclined

struts. To produce the expression shown below for the shear strength of a concrete

section, it was assumed in the model that shear cracks formed at an angle, θ, of 45o:

sbfA

jdbVv

w

vv

w== (2-1)


15

Where: V = shear force at a section Av = area of stirrups bw = beam width fv = stress in stirrups jd = flexural lever arm s = stirrup spacing Reflecting the design philosophy at the time, fv was taken by to be the safe working stress

in the stirrups. While Mörsch knew from observation that failure shear cracks did not

necessarily form at 45o, he saw no way to calculate the angle of what he termed

secondary inclined cracks.

The 45o truss model entered use in various design methods and still forms the basis for

the ACI expression for the shear resistance provided by stirrups. (The current ACI

expression has simplified the equation by replacing the term jd with d.) As its use

became more widespread, however, it was criticized for being overly conservative. In

particular, the model assumes that only transverse reinforcement is effective at carrying

shear, thereby predicting that a section without stirrups or bent-up bars will have no shear

strength whatsoever. Clearly this is not the case. Extensive research efforts were

undertaken in order to ascertain the so-called “concrete contribution” to shear resistance,

which was eventually set at an empirically derived safe working shear stress of Vc/bwd =

vc = 0.03f’c. For the first time, the shear resistance of a reinforced concrete section was

divided up into two components: a concrete contribution (Vc) and a web reinforcement

contribution (Vs) predicted by the 45o truss model:

sc VVV += (2-2)

This method was used to design numerous concrete structures in the post-war

construction boom of the 1950’s and early 1960’s. In 1955, however, a considerable

portion of the roof of the Wilkins Air Force Warehouse in Selby, Ohio collapsed

(Anderson (1957)). The collapsed portions of the beams supporting the roof had been

designed without stirrups, assuming that they could safely resist a working shear stress of

0.6MPa (90psi = 0.03 x 3000psi specified concrete strength). However, shear failure

occurred at a shear stress of approximately 0.5MPa (70psi), corresponding to about 80%

of the safe service load on the roof. It therefore became apparent that unsafe designs

could result from what had previously been considered to be a safe, conservative method.


16

Other design codes at the time used similar empirical expressions to calculate the

concrete contribution to shear strength. As one example, the 1966 CSA S6 bridge design

code (CSA Committee S6 (1966)) permitted a working shear stress in the concrete of

1.1(f’c)0.5 before stirrups were required. For the Air Force Warehouse beams, this

corresponds to a working stress of 1.1(f’c)0.5 = 1.1(3000)0.5=60psi, which is 86% of the

failure shear stress.

Figure 2-2: Collapsed Roof of Air Force Warehouse

As a result of the warehouse collapse, extensive research was undertaken to derive a

better expression for Vc. In 1962, these efforts resulted in what was believed to be a

simple, conservative expression for the failure shear based on a curve-fit through 194

experimental data points as shown in Figure 2-3 (ACI Committee 326 (1962)). This

well-known expression (Equation 2-3) entered design use through incorporation into the

1963 American Concrete Institute Design Code (ACI Committee 318 (1963)), and has

remained essentially unchanged since that time:

'c

w'c

w

c f5.3MVdρ

2500f1.9db

V≤+= (psi units) (2-3)

'c

w'c

w

c f29.0MVdρ

17f71

dbV

≤+= (MPa units)


17

A considerable innovation in Equation (2-3) was the use of the parameter M/(ρwVd). The

parameter was chosen because the stress, fs, in the longitudinal flexural steel at shear

failure is directly proportional to this parameter, and it was observed that the shear stress

at failure decreased as fs increased. Low values of the term 1000ρwVd/M(f’c)0.5 in Figure

2-3 represent sections with small reinforcement ratios and/or subjected to high moment in

relation to the shear. Equation 2-3 is currently Equation (11-5) of the 2005 version of the

ACI 318 design code.

Figure 2-3: Derivation of ACI 318 Equation (11-5) (Reproduced from ACI Committee 326 (1962))

Noting that the range of practical values of 1000ρwVd/M(f’c)0.5 tends to be to the left in

Figure 2-3, and that the predicted influence of steel stress is small, the developers of the

1963 ACI 318 Code included a simplified version of Equation (2-3) as shown below:

dbf2V w'cc = (psi units) (2-4)

dbf0.167V w'cc = (MPa units)

Equation (2-4) is currently Equation (11-3) of the 2005 version of the ACI 318 design

code, and sees far greater use in practical design situations than does equation 11-5 due to

its simplicity. In applying Equations (11-3) and (11-5), the ACI 318-05 code limits

(f’c)0.5 to 100psi (8.3MPa).


18

2.3 The Size Effect in Shear

2.3.1 Current State of Experimental Data

A review of Equations (11-3) and (11-5) reveals that they do not account for the size

effect in shear (Figure 1-1). Equation (11-3), for example, predicts that only the concrete

strength need be considered in design, and that the member depth, M/Vd ratio and ρw can

be safely neglected.

The largest slender beam (a/d>2.5) in the database used to derive Equation (11-5) of the

ACI 318 code had an effective depth, d, of 14.75 in. (375mm), and the average depth for

all of the beams was 13.4 in. (340mm). Given the limited size range of the beams studied,

it is not surprising that the resulting design expression did not account for the size effect

in shear. Equations (11-3) and (11-5), while conservative for shallow members, can be

unconservative for thick members constructed without web reinforcement. The difficulty

in applying empirical design equations to situations outside the scope of the dataset used

to derive the equations is apparent when applying the ACI one-way shear design

expressions to thick, slender flexural members without stirrups.

In a comprehensive review of the previous sixty years of research into the shear

behaviour of reinforced concrete flexural members, Collins et. al. (2007) assembled an

extensive database of 1849 shear tests as summarized in Figure 2-4. This figure, which

excludes nineteen tests published in 2007 and reported in Chapters 5 and 7 of this thesis,

indicates that only about 1.2% of the tests reported in the literature consisted of slender

beams (a/d>2.5) with a very large effective depth, which for the purposes of this figure is

defined as 1000mm (40in.) (see Table 2-1). Furthermore, 84% of the shear failures

occurred in beams with an effective depth less than 16in. (406mm). It is apparent, then,

that despite the continued interest in shear research since the ACI 318 shear provisions

were finalized in 1963, there has been relatively little effort directed at investigating the

size effect in shear. It is thus not surprising the size effect, and how (or indeed, whether)

to account for it in design codes, remains a controversial subject.


19

Table 2-1; Summary of Previous Experimental tests with d≥1000mm and a/d≥2.5

bw d ρw a/d ag f'cKani (1967) 3042 154 1095 2.70 2.50 19 26.4 1.65

3043 154 1092 2.71 3.00 19 27.0 1.143044 152 1097 2.72 3.98 19 29.5 1.053045 155 1092 2.70 5.00 19 28.3 1.023046 155 1097 2.70 6.99 19 26.7 1.063047 155 1095 2.69 8.00 19 26.7 1.01

Kani (1969) 3061 154 1091 0.80 3.10 19 27.4 0.67

Bhal (1968) B4 240 1200 1.26 3.00 30 25.2 0.77

Niwa et. al. (1987) 1 600 2000 0.28 3.00 25 27.1 0.392 600 2000 0.14 3.00 25 26.2 0.373 300 1000 0.14 3.00 25 24.6 0.41

Shioya et. al. (1989) S3: No. 4 500 1000 0.40 3.00 10 27.2 0.53S3: No. 5 500 1000 0.40 3.00 25 21.9 0.65S3: No. 6 1000 2000 0.40 3.00 25 28.5 0.47S3: No. 7 1500 3000 0.41 3.00 25 24.3 0.43S4: No. 6 500 1000 0.40 3.00 5 28.2 0.42

Kawano and A4A 600 2000 1.20 3.00 40 22.2 0.65Watanabe (1997) A4B 600 2000 1.20 3.00 40 23.1 0.58

Yoshida (2000) YB2000/0 300 1890 0.74 2.86 9.5 33.6 0.47

ShenCao (2001) SB2003/0 300 1925 0.36 2.81 9.5 30.8 0.43SB2012/0 300 1845 1.52 2.93 9.5 27.5 0.85

Higgins et. al. (2004) 37T 355.6 1151 0.74 2.91 19 31.8 0.64

Average 0.71COV 46%

Specimen PropertiesSpecimenResearcher vexp/vACI (11-3)

Figure 2-4: Summary of 60 Years of Shear Research on Members without Stirrups

(Collins, Bentz and Sherwood (2007))


20

The following sections will provide a summary of some of the major previous efforts

directed at studying the size effect in shear, including some of those summarized in Table

2-1

2.3.2 Leonhardt and Walther

Leonhardt and Walther noted the lack of experimental data at the time on deep members

without stirrups in a series of articles published in German in 1961 and 1962, and

translated into English in 1962 (Leonhardt and Walther (1962)), when they wrote:

“(w)ith regard to the many shear tests which have been carried out in recent years, more particularly in the United States, it is a striking feature that the beams investigated nearly always had an effective depth of about 30cm and were 2-3m in length. The question arises as to whether the results of these laboratory tests are valid also for larger structures. This question is all the more important because in some countries empirical formulae have been based on these tests. It is therefore necessary to check whether the laws of similarity are validly applicable to shear failure tests.”

In one of the earliest attempts to quantify the size effect in shear, Leonhardt and Walther

(1962) tested two series of beams without stirrups in which the effective depth was varied,

while a/d was kept constant at 3.0. One series (D-series) consisted of beams with

effective depths of 70mm, 140mm, 210mm and 280mm with the same number of

reinforcing bars and scaled bar diameters so as to achieve “complete similarity” between

specimens. Both the reinforcement ratio and number of bars were kept constant, and as a

result, the so-called “bond quality” decreased as the depth increased. A second series (C-

series) consisted of beams with identically scaled concrete cross-sections with depths of

150mm, 300mm, 450mm and 600mm. This series was constructed with identical bar

diameters and reinforcement ratios, resulting in a larger number of bars as the depths

increased. The “bond quality” was thus kept similar for the C-series.

The authors found that the size effect was more severe for the D-series in which the bond

quality decreased as the depth increased than it was for the C-series. They concluded that

in beams with constant bond quality, the shear strength is fairly independent of beam size.


21

2.3.3 Kani

Kani (1967) echoed Leonhardt’s critical view of the unsatisfactory state of experimental

data on the shear strength of beams without stirrups when he stated that:

“(t)o date (1966), the majority of reinforced concrete beams which have been tested to failure range in depth from 10 to 15 in. Essentially, these are the beams on which all our design practices and safety factors are based…How representative are the test results derived from such relatively small beams for the safety factors of large beams?”

In Kani’s solution to what he termed the “riddle of shear failure” (Kani (1964)) he

predicted that

“…(a)ll other factors being equal, the safety factor decreases as the depth of the beam increases.”

This prediction was investigated in the 1967 paper in a classic study on the size effect in

shear. In the 1967 paper, Kani references earlier studies on the effect of depth, and notes

that these studies have tended to find that the influence of depth does not extend to

effective depths beyond a critical threshold, and that the concept of a critical threshold is

at odds with his “rational theory.” Rüsch et. al. (1962), for example, are quoted as saying,

“(i)t seems that for beams tested under uniformly distributed load, a change in depth beyond a critical value does not have any influence on the load-carrying capacity. This critical beam depth is 15 to 20 cm (6 to 8 in).”

Forsell (1954) is also quoted as suggesting that the critical depth for point-loaded beams

is 30 to 40 cm (12 to 16 in.).

Kani (1967) tested a large number of reinforced concrete beams without stirrups in which

the a/d ratio and effective depth, d, was systematically varied in order to assess the effect

of beam depth. The results are summarized in Figure 2-5, where it can be clearly seen

that as the effective depth increased, the failure shear stress decreased at a/d values

exceeding about 2.0.


22

Figure 2-5: Kani’s Size Effect Tests

While both providing convincing evidence that the size effect in shear is considerably

more severe than that suggested by Leonhardt and Walther, and finding no evidence for a

critical depth beyond which the size effect is not critical, Kani’s tests may have had one

important unintended consequence. Kani chose a high reinforcement ratio of about 2.8%

(close to the balanced reinforcement ratio) for all of the size effect beams in the 1967

paper. Because such a high reinforcement ratio was used, when ACI Equations (11-3)

and (11-5) are employed to predict the shear strengths of the members, it can be seen that

generally safe predictions result. Equation (11-5), in particular, is effective at capturing

the effect of a/d at the largest depth.

It is therefore instructive to add to the above figure a little-known test by Kani (1969) in

which a 42.9in. deep beam without stirrups, at an a/d ratio of 3.1 and a reduced

reinforcement ratio of 0.80% failed at a value of β = V/bwd(f'c)0.5 of 1.33. Both Equations

(11-3) and (11-5) of the ACI 318 code predict β = 2.0 for this beam, resulting in ratios of


23

vexp/vpred of 0.67 for both equations. Without an understanding of the effect of ρw on

shear strength, it is easy to study Kani’s size effect study and conclude that, if the size

effect does exist, the ACI shear provisions safely account for it.

Brock (1967) criticized the results shown in Figure 2-5 by noting the slenderness of the

cross-sections of the largest members, suggesting that the ratio of depth:width is of

primary importance. The fact that the largest members had a series of lateral supports to

account for the slenderness was not noted by Brock. The lack of an effect of beam width

at a constant depth found in a companion series of tests reported in the paper was

apparently not persuasive.

MacGregor (1967) agreed with Kani’s basic finding that the shear strength decreases as

the depth increases. In the context of research results presented in Chapter 5 (Figure

5-10), it is interesting to note that MacGregor was struck by the similarity of crack

patterns in members of varying depths at a/d=4, as shown in Figure 2-12.

Figure 2-6: Crack Diagrams of Kani’s Size Effect Tests Redrawn by MacGregor (1967)

2.3.4 Shioya Tests

Perhaps the most extensive series of tests on the size effect in shear were conducted by

Shioya et. al. (1989) and Shioya (1989), the main results of which are summarized in


24

Figure 2-7. These tests were carried out with the aim of assessing the shear strength of

the large slabs used in the underground LNG tanks shown in Figure 1-5.

The authors note that reinforced concrete structures have been “gradually increasing in

size as a result of advances made in materials and…improvement in design and

construction techniques” and, in language that is similar to that of Leonhardt’s and Kani’s,

that it is “…difficult to estimate the accurate shear strength of large reinforced concrete

structures” due to a lack of experimental data.

Figure 2-7: Summary of Shioya et. al. (1989) and Shioya (1989) Tests

In these tests, a series of reinforced concrete beams were loaded to shear failure, with

depths ranging from 4in to 118in. (100mm to 3000mm) and maximum aggregate sizes

from 1mm to 25mm. All test specimens were uniformly loaded across their entire spans,

and the span-to-depth ratio, L/d, was kept constant at 12. The results shown in Figure 2-7

clearly show the reduction in shear strength as the effective depth increases, and they also

show an increase in shear strength as the maximum aggregate size increases. The

inability of ACI Equation 11-3 to accurately predict the shear strength of the largest

members is clear. The size effect in shear was attributed by the authors to a combined

action of a size effect in concrete flexural tensile strength and “shear transfer across

cracked surfaces.”


25

These tests have been recently criticized (Brown et. al. (2006)), in part because the

largest specimen would not have been able to support its own weight. However, the tests

were conducted to assess the shear strength of foundation slabs such as that shown in

Figure 1-5 where the design load is hydrostatic uplift. It is therefore irrelevant that it

could not support its own weight, since self-weight is not a design load for structural

elements of this type (Bentz (2007)). It is also worth noting that the smaller specimens

contained extra reinforcing bars at midspan to prevent flexural failures and were cut off

within the shear span. It is known that bar cutoffs can reduce shear strength, and

correction factors to account for the bar cutoffs are presented in Shioya (1989) Most

importantly, beams with the two largest depths did not contain extra midspan

reinforcement, and hence there were no bar cutoffs.

The specimens tested by Shioya et. al. were loaded under uniform load at an L/d ratio of

12. For a beam tested under uniform load, an equivalent a/d ratio can be determined, as

suggested by Kani (1963), by calculating equivalent points loads at L/4 from the

centrelines of the supports. The Shioya tests therefore have equivalent a/d ratios of

12/4=3.

2.3.5 University of Toronto Tests

Extensive tests have been carried out at the University of Toronto on the size effect in

shear (Collins and Kuchma (1999), Yoshida (2000), ShanCao (2001), Angelakos et. al.

(2002)) A total of twenty three specimens reported in these references were constructed

without stirrups and with an effective depth greater than 890mm. These specimens have

consistently shown that there exists a size effect, and that the ACI code can not account

for it. In particular, the shear strengths of high strength concrete specimens reported by

Angelakos et. al. were poorly predicted by the ACI code. These tests will be discussed in

subsequent chapters when comparing them to the tests conducted by the author.


26

2.4 Design Methods Based on the MCFT

2.4.1 The Modified Compression Field Theory

A considerable contribution to the advancement of shear design methods was the

development of the “general method” of shear design (AASHTO (1994), CSA

Committee A23.3 (1994), Collins et. al. (1996)) based on the Modified Compression

Field Theory (Vecchio and Collins (1986)). Shear design methods based on the MCFT

have a theoretical base and are not derived by empirical curve fits to data from beam tests.

As such, MCFT-based shear provisions are able to predict the behaviour of reinforced

concrete elements in shear where no experimental data is available. In the context of the

research presented in this thesis, advantages of MCFT-based design methods include the

ability to predict the size effect in shear and the effect of aggregate size.

A significant strength of the original Compression Field Theory (Mitchell and Collins

(1974), Collins (1978)) was the use of a variable angle truss model in which the angle of

inclination, θ, of diagonal compressive stresses, f2, was found based on strains in the web

using Mohr’s circle. Further studies (Vecchio and Collins (1986), Bhide and Collins

(1989)) on reinforced concrete biaxial elements were directed at studying the relationship

between diagonal compressive stresses, f2, and diagonal compressive strains, ε2, and the

relationship between coexisting diagonal tensile stresses, f1, and tensile strains, ε1. This

extensive research resulted in a suite of equilibrium, compatability and stress-strain

relationships which collectively form the Modified Compression Field Theory (Vecchio

and Collins (1986), Collins and Mitchell (1991)). See Figure 2-8. The size effect in

shear can be predicted by the MCFT using Equations 9, 10 and 15 in Figure 2-8.

In the MCFT, it is assumed that cracks are aligned in the principal (1-2) directions. The

crack width, w, is calculated assuming that small tensile strains in concrete between

cracks can be ignored, so that crack width is the product of the crack spacing in the

principal tensile direction, sθ, and the principal tensile strain, ε1. The term sθ is predicted


27

to be a function of the ability of the reinforcement in the x and z-directions to control

crack spacings in the x and z-directions, respectively (Equation 10). The terms sx and sz

in equation 10 are the crack spacings that would occur if a member were subjected to

pure longitudinal tension and pure transverse tension, respectively, and are functions of

the crack control characteristics of the reinforcing bars.

Figure 2-8: Relationships of the Modified Compression Field Theory

(Bentz et. al. (2006))

In analyzing a reinforced concrete element using the MCFT, the capacity of cracks to

transmit shear stresses due to interlocking of aggregate particles may limit the shear

strength as indicated by Equation (2-5) below, which is a combination of Equations 9 and

15 in Figure 2-8:

1624

31.0

18.0

1

'

++

≤

g

cci

as

fv

θε (2-5)

Equation (2-5) predicts the aggregate interlock shear capacity at a crack, vci, and was

derived using the experimental data of Walraven (1981). Walraven found that the

aggregate interlock capacity depended upon the concrete strength, f’c, crack width, w, and

maximum aggregate size, ag. Walraven found that as cracks increased in width or as the


28

aggregate size was reduced, the aggregate interlock capacity decreased. Any action that

thus serves to increase the crack width in Equation (2-5) by either increasing the crack

spacing, sθ, or the principal tensile strain, ε1, will reduce the aggregate interlock capacity.

This is shown further in Figure 2-9, where the equilibrium conditions on average and at a

crack are illustrated. On average, there is a tensile stress f1 in the principal direction, with

average stresses in the steel fsx and fsz. At a crack, equilibrium requires that the stresses

locally increase in the steel, and it may further require that a shear stress vci act on the

crack. The vci term is limited by Equation (2-5), which is illustrated in Figure 2-9 for

different maximum aggregate sizes. Failure may occur due to sliding on the crack if the

aggregate interlock capacity is exceeded, and this is particularly the case if the steel is

yielding in one or both directions, or if there is steel in only one direction.

Figure 2-9: Equilibrium Conditions and vci Relationship of the MCFT

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1Crack Width, w (mm)

ag=0mmag=10mm

ag=20mm

v v vci

v

v

v v

fzfz

fz

fx fx

fx

fsz

fsx

fszcr

fsxcr

f1

θ θ

Crack Width, w = sθε1 (mm)


29

Expressions to estimate sx and sz in the equation for sθ are provided by Collins and

Mitchell (1991). For elements without reinforcement in the z-direction (for example

beams and slabs without stirrups), only a single crack would occur if the element were

subjected to pure transverse tension, hence sz equals infinity, and Equation 10 in Figure

2-8 reduces to:

sθ = sx/sinθ (2-6)

The term sx can be calculated based on crack spacing expressions in the 1978 CEB-FIP

code (CEB (1978)), and the expression for sx in members with deformed bars is given

below (see Figure 2-10):

sx = 2(cx + sxb/10) + 0.1db/ρx (2-7)

Figure 2-10: Calculation of sx in the Application of the MCFT to Flexural Elements

Equation (2-7) indicates that crack spacing at the location of maximum crack width under

uniform strain depends primarily upon cx, which is the maximum distance to reinforcing

bars that are able to control crack spacing. Evidence for this phenomenon was found by

Shioya (1989) who determined that, on average, the horizontal crack spacing at mid-

height of the size effect tests was about 0.5d for all depths tested. That is, as the distance

from mid-height to the longitudinal tensile steel increased, the crack spacing increased

roughly in proportion to cx.


30

The 1978 CEB-FIP code recognized that significant variation in crack spacings occur in

reinforced concrete elements by defining a characteristic crack width which is expected

to be exceeded by only 5% of cracks. This characteristic crack width is 1.7 times the

average crack width. By assuming an average horizontal crack spacing of 0.5d at

midheight for members without stirrups, it can be expected that the characteristic crack

spacing is equal to 1.7 x 0.5d = 0.85d. To simplify the calculation of sx for the purposes

of design, Collins et. al. (1996) suggest using sx=0.9d for elements without stirrups. For

elements with longitudinal web reinforcement of a sufficient area, sx can be set equal to

the vertical spacing between layers. To be effective, each layer must have an area of steel

equal to 0.003bwsx. The term sθ can be taken as 300mm (12 in.) in members with stirrups

in recognition of the enhanced crack control characteristics provided by well-detailed

stirrups (Collins and Mitchell (1991)).

In developing design expressions based on the MCFT, the following expression was

derived for the shear strength of a reinforced concrete section (Collins and Mitchell

(1991)):

θcotjdsfA

+θcotjdbf=V+V=V vvw1sc (2-8)

In reinforced concrete with deformed bars subjected to monotonic, short-term loads in

which the principal tensile strain exceeds the cracking strain, the principal tensile stress,

f1, in equation 2-8 is found by:

1

cr1 ε500+1

f=f (2-9)

The average and local stresses shown in Figure 2-9 equilibrate the same external forces

and are thus equivalent. Hence, it can be shown that f1 in equation (2-8) must be limited

to the aggregate interlock capacity of the crack such that:

)(tan1 vyvci ffvf ρθ +≤ (2-10)


31

where it is assumed that stirrups are yielding at the crack, fy is the stirrup yield stress, fv is

the average stirrup stress between cracks and ρv is the stirrup reinforcement ratio and is

equal to Av/bws. In Equation (2-10), it can be seen that if stirrups are highly stressed on

average or if there is a small quantity of stirrups (or indeed no stirrups) then the shear

strength can be limited by the aggregate interlock capacity, and failure can occur by

slipping on the crack.

Recalling Equation (2-5) in which vci is inversely related to ε1 and sθ, it can now be seen

how the MCFT predicts the size effect in shear. Since sx=0.9d, doubling the depth is

predicted to double the horizontal crack spacing at mid-height. The crack spacing in the

principal direction, sθ will also increase, based on Equation (2-6), resulting in increased

crack widths. These increased crack widths result in reduced aggregate interlock

capacity, which limits the principal tensile stress that can be sustained in the web. This

reduced capacity to transfer tension in the web precipitates failure at a reduced shear

stress. Fundamentally, the size effect is predicted to be a result of reduced crack control

characteristics of the longitudinal reinforcement as the depth is increased.

Review of the above equations reveals that the principal tensile stress, ε1, also affects

both the aggregate interlock capacity (by increasing crack widths) and the sustainable

value of the average tensile stress (Equation (2-9)). The principal tensile stress can be

calculated as shown below:

θcot)εε(+ε=ε 22xx1 (2-11)

Any action that serves to increase the longitudinal strain in the web, εx, is thus predicted

to reduce shear capacity, and these actions can include axial tension, increased moment-

to-shear ratio, reduced longitudinal reinforcement ratio or the use of reinforcement with a

reduced elastic modulus (such as fibre-reinforced polymer (FRP) bars). The effect of εx

on shear capacity has been referred to as the “strain-effect” (Bentz et. al. (2006)).


32

2.4.2 1994 CSA Methods

The General Method of Shear Design developed by Collins and Mitchell (1991) and

Collins et. al. (1996) was implemented in the 1994 version of the CSA A23.3 concrete

design code. The basic expressions for the method are shown below:

θcots

dfAφ+dbfλβφ3.1=V+=VV vyvc

vw'ccsgcgrg (2-12)

The β term models the ability of a cracked web to transfer tension, and is a function of

the average tension stress in the concrete (Equation (2-9)), but is limited by the aggregate

interlock shear capacity (Equations (2-10) and (2-5)). In deriving expressions for β, it is

assumed that fv=fy in Equation (2-11). The principal tensile strain is found using

Equation (2-11). The longitudinal strain in the web, εx, is assumed to be equal to the

strain in the longitudinal steel and for non-prestressed members not subjected to axial

load, it can be found as:

ss

vffx AE

d/M+θcotV5.0=ε (2-13)

As there is an interactive relationship between θ, εx and β, values of θ and β are provided

in tables for different combinations of εx and the index value vf/λφcf’c. Two tables are

provided –one for members with stirrups, and one for members without stirrups. These

tables are not reproduced here for the sake of brevity, but it is interesting to note that

most cells in the table for members with stirrups are governed by the crack-slip limitation,

while all cells in the table for members without stirrups are governed by crack slip. Thus,

for most cells β is a function of both ε1 and sθ, and can be found by the following:

16+asε24

+31.0

18.0=β

g

θ1 (2-14)

The General Method of Shear Design is also implemented in the AASHTO-LRFD design

code for bridges (AASHTO (1994, 2004)).


33

The simplified method of shear design in the 1994 CSA A23.3 design code calculates the

concrete contribution to shear strength using Equation (11-3) of the ACI code as shown

below:

dbfλφ.=V w'

ccc 20 (MPa units) (2-15)

The coefficient was changed from 0.167 to 0.2 to reflect the lower material safety factor

φc in the CSA code. Equation (2-15) only applies to members with a minimum quantity

of stirrups. For members with no stirrups, or less stirrups than the minimum:

dbfλφdbfλφ+d

=V w'

ccw'

ccc 1.01000

260≥ (2-16)

Equation (2-16) accounts for the size effect in shear and reduces to Equation (2-15) for

d=300mm. To use Equations (2-15) and (2-16) to predict experimental results, φc should

be set equal to 1 and the equations should be multiplied by 0.167/0.2.

2.4.3 2004 CSA Methods

The recently developed Simplified Modified

Compression Field Theory (Bentz et. al. (2006)) is

a simplification of the Modified Compression Field

Theory designed for “back-of-the-envelope”

calculations. The 2004 CSA A23.3 general shear

design method is based on the SMCFT, and

represents a simplification of the general method in

the AASHTO-LRFD and the 1994 CSA Standards.

Simple expressions have been developed for β, the

crack angle, θ and the longitudinal strain in the

web, εx, thereby eliminating the need for iteration.

One of the major assumptions in the development of SMCFT was that aggregate

interlock governs shear failure of members without stirrups.


34

The SMCFT employs the following relationship to determine the shear resistance of a

concrete section:

θ+β=+= cotdsfA

dbfVVV vyv

vw'csc (2-17)

The term β in Equation (2-17) is a function of 1) the longitudinal strain at the mid-depth

of the web, εx, 2) the crack spacing at the mid-depth of the web and 3) the maximum

coarse aggregate size, ag. It is calculated using an expression that consists of a strain

effect term and a size effect term:

m)effect ter (sizem)effect ter(strain =)s+(1000

1300)1500ε+(1

0.40=β

xex (2-18)

The longitudinal strain at the mid-depth of a beam web is conservatively assumed to be

equal to one-half the strain in the longitudinal tensile reinforcing steel. For sections that

are neither prestressed nor subjected to axial loads, εx is calculated by:

ss

fvfx A2E

V/dMε

+= (2-19)

The effect of the crack spacing at the beam mid-depth is accounted for by use of the

crack spacing parameter, sx. This crack spacing parameter is equal to the smaller of

either the flexural lever arm (dv=0.9d or 0.72h, whichever is greater) or the maximum

distance between layers of longitudinal crack control steel distributed along the height of

the web. To be effective, the area of the crack control steel in a particular layer must be

greater than 0.003bwsx.

The term sxe is referred to as an “equivalent crack spacing factor” and was first developed

(Collins and Mitchell (1991)) to model the effects of different maximum aggregate size

on the shear strength of concrete sections by modifying the crack spacing parameter. For

concrete sections with less than the minimum quantity of transverse reinforcement and


35

constructed with a maximum aggregate size of 19mm (3/4in.), sxe is equal to sx. For

concrete with a maximum aggregate size other than 19mm, sxe is calculated as follows:

xg

xxe 0.85s≥

a+1635s

=s (2-20)

The 0.85sx limit on sxe is based on research presented in Chapter 5. To account for

aggregate fracturing at high concrete strengths, an effective maximum aggregate size,

ag,eff, is calculated by linearly reducing ag to zero as f’c increases from 60 to 70MPa. The

term ag is equal to zero if f’c is greater than 70MPa (Angelakos et. al. (2001)). Further,

the square root of the concrete strength is limited to a maximum of 8MPa.

Since specimens with transverse reinforcement do not exhibit any significant size effect,

sxe is set equal to 300mm for specimens with at least the minimum quantity of stirrups as

per Equation (2-21). This has the effect of setting the size effect term to 1.

'c

w

yv f0.06=sbfA

(2-21)

The angle of inclination of the cracks at the beam mid-depth, θ, is calculated by the

following equation:

°≤++°= 75)2500s)(0.887000ε(29θ xex (2-22)

A Note about Subscripts

The SMCFT uses the terms sx and sxe when referring to the crack spacing and effective

crack spacing. The implementation of the SMCFT in the 2004 CSA A23.3 code uses the

terms sz and sze, and changes the 16 in Equation (2-20) to a 15. This change is intended

to reflect the fact that the CSA A23.1 code that applies to concrete aggregate is metric

and specifies metric aggregate sizes. That is, sze in the CSA code reduces to sz for

ag=20mm, while in Equation (2-20), which represents the definitive form of the effective

crack spacing parameter as used in the SMCFT, sxe reduces to sx for ag=3/4in. (19mm).


36

In this thesis, Equation (2-20) is used for all analyses of beam strengths with hard

conversions of aggregate sizes, but the terms sz, sze and sx, sxe are used interchangeably.

2.4.4 A Simplified Design Method based on the MCFT

Lubell et. al. (2004) have proposed a simplified expression for the concrete contribution

to shear strength based on the MCFT:

dbf+s1000

208=V w

'c

xec (MPa units) (2-23)

dbf+s38

100=V w

'c

xec (psi units)

The term 208/(1000+sxe) is derived from Equation (2-18) using an assumed value for εx

of 0.833x10-3 and is intended to be a useful simplification for design. It is a slightly

modified version of an expression presented by Collins and Kuchma (1999), and was

shown by Lubell et. al. to accurately account for size and aggregate effects.

2.5 Mechanisms of Shear Transfer and Failure

The fundamental mechanisms by which flexural elements transfer shear are illustrated in

the simple free-body diagrams in Figure 2-11 (MacGregor and Bartlett (2004)). A

member without transverse reinforcement transfers vertical shear, Vc, through a

combination of shear in the compression zone, Vcz, a vertical force in the longitudinal

steel due to dowel effects, Vd, and the vertical component of aggregate interlock stresses,

va, integrated over the surface of the crack. These three components -the force in the

compression zone, the force due to aggregate interlock and the dowel force- collectively

form the concrete contribution to shear resistance, Vc. The relative proportions of each of

these components acting at a concrete section have been the subject of research over the

years and remain a matter of some debate. Factors which can affect the relative

proportions include: the depth of the compression zone, span-to-depth ratio, crack width,

crack roughness, ρw, number and layout of bars, cover thickness, concrete strength,


37

reinforcement modulus, among others. While some shear can be transferred by residual

tension across the crack, this component is likely small relative to the other components,

particularly for wider cracks. The horizontal component of the integrated aggregate

interlock stresses is resisted by an increased tensile force in the longitudinal steel, T.

These same components of the vertical shear force act in a member with stirrups (Figure

2-11b). The presence of stirrups provides an additional vertical force totaling Vs, the

steel contribution to shear resistance. In deriving the steel contribution, it is typically

assumed the stirrups are yielding at the crack (i.e. fv=fy).

Figure 2-11: Components of Shear Resistance in a Reinforced Concrete Beam

2.5.1 Early Approaches

Early attempts at developing rational theories of the shear strength of reinforced concrete

without stirrups tended to neglect the role played by aggregate interlock (Zwoyer and

Siess (1954), Moretto (1955), Moody et. al. (1954), Hanson (1958), Bresler and Pister

(1958), Walther (1962)). Both implicit or explicit in these early theories is the

assumption that all the vertical shear force in cracked concrete sections without shear

reinforcement is carried by the uncracked concrete compression zone.

2.5.2 Distribution of Shear Across Beam Depth

These early approaches tended to disregard the classic work reported by Mörsch (1909).

Mörsch determined that the vertical shear stress distribution could be calculated at a

section if adjacent flexural stresses were known. The shear stress distribution in a

cracked member derived by him consisted of a parabolic distribution above the neutral

Avfy


38

axis in the uncracked compression zone, increasing from zero at the top of the beam to a

maximum at the neutral axis, and a constant shear stress below the neutral axis.

To illustrate, consider the cracked reinforced concrete beam without stirrups of width bw

shown in Figure 2-12a). The section a-b-c-d between two cracks is shown in Figure

2-12b), along with the horizontal forces due to flexural stresses and vertical forces due to

shear acting on the section. Because the moment is higher on the right than it is on the

left, the compressive force in the concrete and tensile force in the steel are slightly larger

on the right than they are on the left. For simplicity, it is assumed that the flexural lever

arm jd does not change over the distance dx.

Consider now the section a-b-e-f in the compression zone of a-b-c-d, as shown in Figure

2-12c), where the horizontal face e-f is located a distance z from the neutral axis. Sides

a-e and b-f are subjected to trapezoidal horizontal stress blocks, resulting in horizontal

compressive forces Cae and Cbf. Because Cbf is slightly larger than Cae, a horizontal force

of (Cbf - Cae) must act on the horizontal plane e-f (Jourawski (1856)). The compression

force Cae can be found by:

wcecaae b)zkd)(f+f(21

=C , where cace fkdz

=f (2-24)

Hence, wcaae b)zkd)(kdz

+1(f21

=C w2

ca b)kd)()kdz

(1(f21

= (2-25)

Similarly, w2

cbbf b)kd)()kdz

(1(f21

=C (2-26)

By considering the entire flexural stress blocks at a and b, the moments at a and b can be

calculated as:

jd)kd(bf21

=M wcaa and jd)kd(bf21

=M wcbb (2-27)

By substituting the relationships in (2-27) into (2-26) and (2-25), the following

relationships for Cae and Cbf are determined, based on the moments at the sections:

))kdz

(1(jd

M=C 2a

ae (2-28)


39

))kdz

(1(jd

M=C 2b

bf (2-29)

The horizontal shear force on the face e-f induces a complementary shear stress of:

dxb

CC=v

w

aebf

ef (2-30)

))kdz

(1(jd)dx(b

MM= 2

w

ab

))kdz

(1(jdb

V= 2

w (2-31)

Equation (2-31) defines a parabola, and it can be seen that the shear stress in the

compression zone increases from 0 at z=kd (the top of the beam) to a maximum of

V/bwjd at z=0 (at the neutral axis). Horizontal equilibrium of the small element i-j-k-e

requires that the horizontal shear stress on e-f equal the vertical shear stress at z from the

neutral axis. Hence, the vertical shear stress distribution is parabolic in the compression

zone.

The block of concrete h-g-c-d at the bottom of section a-b-c-d as shown in Figure 2-12d)

can be considered so as to determine the distribution of vertical shear stress in the tension

zone, vtz. The difference in tensile force in the steel, ΔT, must be resisted by horizontal

shear stresses on the face h-g, which are equal to ΔT/bwdx. It can be shown that:

jdVdx

=jdMΔ

=TΔ

Hence,

jdbV

=)jd(dxb

Vdx=

dxbTΔ

www

Rotational equilibrium requires that the horizontal shear stress on face h-g equal the

vertical shear stresses of faces h-d and g-c. Since the horizontal shear stress on face h-g

is independent of the distance between this face and the steel, the shear stress in the

tension zone, vtz, is constant within the tension zone. The shear vtz is equal to the

maximum shear stress at the neutral axis from Equation (2-30). The complete vertical


40

shear stress distribution as determined by Mörsch is shown in Figure 2-12f).

Considerations of rotational equilibrium of elements 1, 2 and 3 in the tension zone

indicate that vertical shear in the tension zone is transferred across cracks. While the

shear stress v=V/bwjd was later simplified in design codes to V/bwd, it is important to note

that it has a very real physical significance, and was found by Mörsch based simply on

the requirements of equilibrium.

Figure 2-12: Distribution of Shear Stress in a Cracked Reinforced Concrete Beam

2.5.3 Aggregate Interlock

As shear research progressed in the 1960’s it was gradually realized that aggregate

interlock did play a significant role in shear behaviour (Moe (1962), Fenwick and Paulay

(1964, 1968), MacGregor (1964), Taylor (1970), MacGregor and Walters (1967), Kani et.

al. (1979)). These researchers realized that in order for the stress in the longitudinal

tensile steel to change along the span, shear stresses had to be transferred across cracks


41

by aggregate interlock action. Two of these early references are discussions of Kani

(1964), in which he outlined his solution to the riddle of shear failure, which, while a

useful tool to conceptualize shear failure, does not consider aggregate interlock.

Moe (1962) suggested the shear stress distribution shown in Figure 2-13, and compared it

to the distribution according to the classical theory of Mörsch. He considered a portion

of a beam between two adjacent flexural cracks as a vertical cantilever fixed at the

beam’s neutral axis. Kani (1964) later called this a concrete tooth. Because the moment

is higher on the left side of the tooth shown in Figure 2-13, it is pulled towards the left by

the longitudinal reinforcement. The resulting bending of the tooth is resisted “…by shear

forces, Vr, which are transferred by interlocking of grains across the bending cracks.”

Essentially, these Vr forces constitute a force couple that counteracts the moment cause

by ΔT. As the cracks widen, Moe suggested that Vr decreases, causing bending stresses

to develop at the root of the concrete tooth. Moe concluded that “At a certain value of

the crack width the stresses in the cantilever become high enough to cause failure.” This

failure is in the form of an inclined crack caused by the tensile bending stress at the root

of the tooth, as shown in Figure 2-13. Moe also suggested that as the Vr forces broke

down, the shear force carried in the compression zone may increase to compensate.

Figure 2-13: Model of Shear Failure as Suggested by Moe

Fenwick and Paulay (1968) were the first to quantify the importance of what they termed

aggregate interlock. Through direct measurements on cracked beams and subassemblies,

they were able to conclude that at least 60%, and possibly as much as 75%, of the vertical


42

shear is carried by aggregate interlock across the flexural cracks. The authors suggest

that it is unlikely that more than 20% is carried by dowel action, with the remaining

portion carried in the compression zone. In Figure 2-14, at a theoretical bond force of 6.5

kips, about 75% of the total shear was carried by aggregate interlock action in beam FA4.

Figure 2-14: Measured and Theoretical Bond Forces Measured in Beams (Reproduced

from Fenwick and Paulay (1968))

A later study by Taylor (1970) “…supported the view that has been growing the past two

decades that shear force carried by beams is not just confined to the compression zone.”

He found that 20-40% of the shear was transferred in the compression zone, 33-50% due

to aggregate interlock, and 15-25% by dowel action. It was suggested that near failure

dowel action breaks down, with shear transferred to aggregate interlock. “The aggregate

interlock mechanism was probably the next to fail, causing an abrupt and sometimes

explosive failure of the compressive zone.” Thus failure was attributed to breakdown in

aggregate interlock. Factors such as aggregate quality and strong concrete in relation to

the aggregate strength are identified as affecting aggregate interlock capacity and shear

strength. The size effect is attributed to reduced aggregate interlock capacity in large

beams due not to wider cracks, but rather to not scaling aggregate size in relation to the

depth, which results in proportionally smoother cracks.

Kani et. al. (1979) report a major study investigating shear transfer mechanisms. A 48in.

deep beam without stirrups was loaded over several months and cycled repeatedly. Both


43

prior to loading and as the test progressed numerous external gauge points, many in the

form of strain rosettes, were installed so as to map in significant detail the directions of

principal strains in the member. Of interest were the authors’ findings that 19%, 32%

and 17% of the vertical shear at three adjacent cracks was transferred in the compression

zone, with the remainder transferred by aggregate interlock and dowel action. See Figure

2-15. The authors note that “…the measurements indicate that some 50 to 60 percent of

the vertical shear force may be transferred through aggregate interlock.”

Figure 2-15: Measurement of Shear in the Compression Zone (Kani et. al. (1979))

The mechanism of shear failure proposed in Kani’s original solution to the riddle of shear

failure (Kani (1964)) involved the flexural failure of concrete teeth as shown in Figure

2-13. Kani’s original mechanism did not include aggregate interlock shear stresses on the

side faces of the tooth, hence the full moment resistance of a tooth was provided by

flexural stresses at the base of the tooth. Kani et. al. (1979) report that measurement of

flexural stresses at the base of concrete teeth indicated that they accounted for only 40%

of the required tooth moment resistance, with the remainder provided by aggregate

interlock and dowel action. Fenwick and Paulay (1968) found that 20% of the moment

resistance of the tooth was provided by flexural stresses at the root of the tooth.

Gergely (1969) is quoted by ASCE-ACI Committee 426 (1973) as confirming that

between 20-40% of shear is carried in the compression zone, 33-50% due to aggregate


44

interlock, and 15-25% by dowel action. Krefeld and Thurston (1966) found a similar

proportion of the shear to be carried by dowel action, and ASCE-ACI Committee 426

(1973) quote Parmelee (1961) and Baumann (1968) as also finding a similar proportion

of the shear to be carried by dowel action.

Different Perspectives

Despite the classic studies described above, it is still a commonly held assumption in

many models of reinforced concrete that vertical shear is transferred solely through the

uncracked compression zone. Some examples are as follows.

Tureyen and Frosch (2002) found a relationship between the neutral axis depth, c, and

shear strength when analyzing members with different reinforcement ratios and moduli.

They proposed the following expression for Vc in terms of c, in which the coefficient of 5

was derived based on an empirical curve fit to beam data. The neutral axis depth is found

based on the elastic properties of the section.

cbf5=V w'cc (2-32)

Equation (2-32) does not account for the size effect or the strain effect caused by M/Vd.

A behavioural model was developed to justify the equation in which it was assumed that

all of the shear was carried in the uncracked compression zone (Figure 2-16). The

authors felt that this shear distribution was a “reasonable approximation,” as direct

measurement of the distribution has “not been possible to date” and is “debatable.”

Figure 2-16: Stress Conditions Above a Crack According to Tureyen and Frosch (2002)


45

Bazant and Yu (2005a,b) review the extensive work on the size effect conducted by

Bazant, and describe a series of finite element studies on small and large beams (Figure

2-17). In these studies it was found that shear failure in slender beams is governed by

concrete crushing at the tip of the critical crack. Aggregate interlock shear stresses are

not considered, and shear is transferred to the support by direct strut action.

Gustafsson and Hillerborg (1988) conducted a finite element analysis of shear failure as

shown in Figure 2-18 using the fictitious crack approach. Dowel and aggregate effects

were neglected, and the two sides of the crack were considered to be in contact only for

widths less than a critical width. Several crack locations were considered, and failure

was assumed to occur by concrete crushing above the crack. Ultimate shear strength

corresponded to the crack location resulting in the lowest shear strength.

Kotsovos (1988) developed a model for shear behaviour termed the Compressive Force

Path Concept (Figure 2-19). The model assumes shear failure occurs by excessive tensile

stresses perpendicular to the compressive path. These can occur due to changes in the

direction of the force path requiring a tensile resultant (T in Figure 2-19a)), high tensile

stresses at the tip of cracks (t2), and dilation in the vertical direction due to varying

intensity of the compressive stress field (t1). It should be noted that T represents a tensile

force that must be developed by unrealistically high tensile stresses in cracked concrete.

Furthermore, the assumed stress conditions in the compression zone indicate that all of

the shear is carried above the neutral axis.

Stratford and Burgoyne (2003) have analyzed FRP-reinforced beams without stirrups

assuming shear is transferred as shown in Figure 2-20. Shear is assumed to be transferred

by beam action to the right of the failure crack (i.e. varying intensity of the

compression/tension forces), and by arch action between the crack and the support (i.e.

varying lever-arm between constant compression/tension forces). Beam action is

accomplished by bending of concrete teeth similar to Kani’s model (Kani (1964)), but

flexural resistance of the teeth is provided solely by flexural stresses at the root of the

tooth. Aggregate interlock is not considered.


46

Figure 2-17: Failure of Reinforced Concrete Beams According to Bazant and Yu (2006a)

Figure 2-18: Model of Shear Failure by Gustafsson and Hillerborg (1988)

Figure 2-19: The Compressive Force Path Concept (Kotsovos (1988)) a) Compressive Force Path b) Stress Conditions in Compression Zone


47

Figure 2-20: Shear in Beam with no Shear Reinforcement According to Stratford and

Burgoyne (2003)

2.5.4 The a/d ratio

In discussing one-way shear in reinforced concrete, a distinction must be made between

behaviour in beam regions (B-regions) and disturbed region (D-regions). In regions of

members away from discontinuities, load is transferred by beam action, in which the

assumption that plane-sections-remain-plane is accurate. Beam action in reinforced

concrete consists of a change in the compressive and tensile flexural forces at a constant

lever arm. In regions of members within about a member depth from a discontinuity,

load is transferred primarily by arch action, in which the lever arm between constant

flexural forces changes.

Fenwick and Paulay (1968) showed that when beam action broke down in members with

small a/d ratios, redistribution of stresses could occur and engage arch action, which was

termed a secondary shear transfer mechanism. At larger a/d ratios arch action can not be

engaged due to geometric incompatibility, with failure occurring due to either tension in

the top of the compression zone due to the large eccentricity of the thrust line, or crushing

due to a reduction in the size of the compression zone. Kani (1964) also discusses arch


48

action, referring to it as “secondary strut” action, and notes that it is not a reliable

mechanism in members with larger a/d ratios.

Figure 2-21: Beam Regions and Disturbed Regions

Figure 2-22: Failure of Arch Action (Reproduced from Fenwick and Paulay (1968))

Considerably higher loads can be reached in members where secondary strut action can

occur. This is generally observed in beams where the shear span-to-depth ratio, a/d, is

less than about 2.5 (see Figure 2-5). In members with a/d<2.5, strut-and-tie methods can

be applied to determine the expected shear capacity, but this is beyond the scope of this

thesis. Figure 2-23 is adapted from Collins et. al. (2007), and it can be seen that taking

the higher of the shear strengths predicted by 2004 CSA code strut-and-tie provisions and

sectional models accurately predicts the variation in Vc in beams with varying a/d.


49

Figure 2-23: Effect of a/d on Shear Strength (Adapted from Collins et. al. (2007)

Figure 2-24 is reproduced from ACI-ASCE Committee 426 (1973) in which a valuable

discussion is provided of shear transfer mechanisms in reinforced concrete. The different

forms of shear failure are shown. Failure in deep members (a/d<2.5) is characterized by

1) crushing of the strut, 2) tension in the top face of the strut, 3) anchorage failure or 4)

splitting at the level of the steel. Failure in slender members (a/d>2.5) is characterized by

a sudden formation of an inclined flexure-shear crack, in which both beam action can no

longer be maintained, and strut action can not be engaged.


50

Figure 2-24: Shear Failure Modes in Reinforced Concrete Beams without Stirrups

2.6 Concluding Remarks

Analysis of the literature has shown that there is evidence that the size effect in shear

exists and is significant. Yet there is debate as to how or whether to account for it in the

ACI 318 code. There is also evidence that aggregate interlock is the dominant

mechanism of shear transfer in slender flexural elements without stirrups. The results of

the classic tests that established the importance of aggregate interlock, however, have

been neglected in some modern theories of the shear behaviour of reinforced concrete.

There is clearly a need for further investigation of the size effect, with particular

emphasis on the role off aggregate interlock. The experiments described in the following

chapters have been designed to address these issues, with the ultimate goal of improving

the generality, accuracy and safety of shear design methods

sherwood 1 and 2

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