sherwood 1 and 2
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Sherwood 1 and 2TRANSCRIPT
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
Shear Behaviour of Large, Lightly-Reinforced Table of Contents Concrete Beams and One-Way Slabs
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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
Shear Behaviour of Large, Lightly-Reinforced Table of Contents Concrete Beams and One-Way Slabs
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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
Shear Behaviour of Large, Lightly-Reinforced Table of Contents Concrete Beams and One-Way Slabs
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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
Shear Behaviour of Large, Lightly-Reinforced List of Figures Concrete Beams and One-Way Slabs
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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
Shear Behaviour of Large, Lightly-Reinforced List of Figures Concrete Beams and One-Way Slabs
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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
Shear Behaviour of Large, Lightly-Reinforced List of Figures Concrete Beams and One-Way Slabs
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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
Shear Behaviour of Large, Lightly-Reinforced List of Figures Concrete Beams and One-Way Slabs
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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
Shear Behaviour of Large, Lightly-Reinforced List of Figures Concrete Beams and One-Way Slabs
xvii
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
Shear Behaviour of Large, Lightly-Reinforced List of Figures Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Introduction Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Introduction Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Introduction Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Introduction Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Introduction Concrete Beams and One-Way Slabs
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?
Shear Behaviour of Large, Lightly-Reinforced Introduction Concrete Beams and One-Way Slabs
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))
Shear Behaviour of Large, Lightly-Reinforced Introduction Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Introduction Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Introduction Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Introduction Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Introduction Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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)
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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)
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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).
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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))
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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.”
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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)
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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)
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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)).
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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)).
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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).
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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,
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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)
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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)
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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.
Shear Behaviour of Large, Lightly-Reinforced Background Concrete Beams and One-Way Slabs
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