Testing and Modeling of Reinforced Concrete Coupling Beams
Transcript of Testing and Modeling of Reinforced Concrete Coupling Beams
UNIVERSITY OF CALIFORNIA
Los Angeles
Testing and Modeling
of Reinforced Concrete
Coupling Beams
A dissertation submitted in partial satisfaction of the
requirements for the degree Doctor of Philosophy
in Civil Engineering
by
David Anthony Braithwaite Naish
2010
© Copyright by
David Anthony Braithwaite Naish
2010
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The dissertation of David Anthony Braithwaite Naish is approved.
__________________________________
Thomas Sabol
__________________________________
Ertugrul Taciroglu
__________________________________
Farzin Zareian
__________________________________
Jian Zhang
__________________________________
John Wallace, Committee Chair
University of California, Los Angeles
2010
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To my family
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Table of Contents
LIST OF FIGURES ........................................................................................................... vi LIST OF TABLES........................................................................................................... xvi LIST OF SYMBOLS ...................................................................................................... xvii ACKNOWLEDGEMENTS.............................................................................................. xx VITA............................................................................................................................... xxii ABSTRACT................................................................................................................... xxiv
CHAPTER 1 INTRODUCTION ...................................................................................... 1 1.1 Background......................................................................................................... 1 1.2 Objectives ......................................................................................................... 10 1.3 Organization...................................................................................................... 11
CHAPTER 2 LITERATURE REVIEW ......................................................................... 12 2.1 Conventionally Reinforced Coupling Beams ................................................... 12 2.2 Diagonally Reinforced Coupling Beams .......................................................... 14 2.3 Coupled Wall Behavior..................................................................................... 16
CHAPTER 3 EXPERIMENTAL PROGRAM ............................................................... 19 3.1 Beam Design..................................................................................................... 19 3.2 Material Properties............................................................................................ 29 3.3 Test Setup.......................................................................................................... 31 3.4 Loading Protocol............................................................................................... 32 3.5 Instrumentation ................................................................................................. 33
CHAPTER 4 EXPERIMENTAL RESULTS AND DISCUSSION ............................... 42 4.1 Detailing............................................................................................................ 46
4.1.1 Full Section vs. Diagonal Confinement .................................................... 46 4.1.2 Full vs. Half Confinement ........................................................................ 48
4.2 Slab Impact ....................................................................................................... 50 4.3 Frame Beam...................................................................................................... 56 4.4 Damage ............................................................................................................. 59
4.4.1 Damage at peak deformation .................................................................... 59 4.4.2 Residual damage at zero deformation....................................................... 68
4.5 Summary ........................................................................................................... 74
CHAPTER 5 SIMPLIFIED COMPONENT MODELING ............................................ 75 5.1 Effective Stiffness............................................................................................. 75 5.2 Slip/Extension Calculations .............................................................................. 83 5.3 Effect of Scale................................................................................................... 85 5.4 Load-Deformation Backbone Relations ........................................................... 88
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5.5 Application to Computer Modeling .................................................................. 92 5.5.1 Diagonally-reinforced coupling beams (2.0 < ln/h < 4.0) ......................... 93 5.5.2 Conventionally-reinforced coupling beams (3.0 < ln/h < 4.0) .................. 97 5.5.3 Extension to lower aspect ratios (1.0 < ln/h < 2.0).................................. 100
5.6 Nonlinear Component Modeling .................................................................... 102 5.6.1 Modeling overview ................................................................................. 102 5.6.2 Nonlinear modeling results ..................................................................... 105
5.7 Summary ......................................................................................................... 115
CHAPTER 6 FRAGILITY CURVES FOR COUPLING BEAMS.............................. 116 6.1 Sources of Data ............................................................................................... 116 6.2 Damage States................................................................................................. 117 6.3 Results............................................................................................................. 122 6.4 Modeling Parameters and Acceptance Criteria............................................... 127 6.5 Summary ......................................................................................................... 129
CHAPTER 7 SYSTEM MODELING .......................................................................... 130 7.1 Model Information .......................................................................................... 130
7.1.1 Baseline model........................................................................................ 130 7.1.2 Modified models ..................................................................................... 132 7.1.3 Loading ................................................................................................... 135
7.2 Nonlinear Analysis Results............................................................................. 136 7.2.1 Coupling beam rotations ......................................................................... 137 7.2.2 Inter-story drifts ...................................................................................... 143 7.2.3 Core wall shear ....................................................................................... 147
7.3 Summary ......................................................................................................... 148
CHAPTER 8 CONCLUSIONS..................................................................................... 149
APPENDIX A SUMMARY OF TEST RESULTS ...................................................... 154 APPENDIX B SLIP/EXTENSION CALCULATION EXAMPLE............................. 201 APPENDIX C PROCEDURE TO ESTIMATE ECIEFF................................................ 205 APPENDIX D MODELING PARAMETERS ............................................................. 207 APPENDIX E MATERIAL TESTING........................................................................ 212 APPENDIX F GROUND MOTION SELECTION METHODOLOGY ..................... 216 APPENDIX G LOAD-DEFORMATION BACKBONE DETERMINATION ........... 218
REFERENCES ............................................................................................................... 221
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List of Figures
Figure 1.1 Typical (a) plan and (b) elevation views of coupled corewall structure..... 3
Figure 1.2 Reactions to applied lateral load for system with (a) well-detailed coupling beams, and (b) poorly-detailed coupling beams ......................................... 4
Figure 1.3 Typical reinforcement pattern for conventional and diagonal reinforcement in coupling beams................................................................ 5
Figure 1.4 Confinement options provided in ACI 318: (a) ACI 318-05 Diagonal confinement; and (b) ACI 318-08 Full section confinement ...................... 8
Figure 1.5 Reinforcement congestion caused by using ACI 318-05 Diagonal confinement................................................................................................. 8
Figure 2.1 Distribution of energy dissipation in a core wall structure with (a) well-detailed coupling beams, and (b) poorly-detailed coupling beams........... 18
Figure 3.1 Test beam geometries (ln/h = 2.4) full section confinement: (a) CB24F, CB24F-RC, CB24F-PT, CB24F-1/2-PT elevation; (b) CB24F cross section; and (c) CB24F-RC, CB24F-PT, CB24F-1/2-PT cross section. (Dimensions are inches. 1in = 25.4mm) .................................................. 23
Figure 3.2 Slab geometry and reinforcement for CB24F-RC, CB24F-PT, and CB24F-1/2-PT: (a) Elevation view; and (b) plan view. (Dimensions are inches. 1in = 25.4mm)........................................................................................... 24
Figure 3.3 Slab geometry and PT reinforcement for CB24F-PT and CB24F-1/2-PT: (a) Plan view; and (b) photo of post-tensioning load application. (Dimensions are inches. 1in = 25.4mm) .................................................. 25
Figure 3.4 Test beam geometries (ln/h = 2.4) diagonal confinement (from left): (a) CB24D elevation; and (b) cross section, with diagonal bundle (Dimensions are inches. 1in = 25.4mm) .................................................. 26
Figure 3.5 Test beam geometries (ln/h = 3.33) full section confinement (from left): (a) CB33F elevation; and (b) cross-section (Dimensions are inches. 1in = 25.4mm) .................................................................................................... 26
Figure 3.6 Test beam geometries (ln/h = 3.33) diagonal confinement (from left): (a) CB33D elevation; and (b) cross-section, with diagonal bundle (Dimensions are inches. 1in = 25.4mm) .................................................. 27
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Figure 3.7 Test beam geometries (ln/h = 3.33) frame beam (from left): (a) FB33 elevation; and (b) cross-section. (Dimensions are inches. 1in = 25.4mm)................................................................................................................... 27
Figure 3.8 Photographs of test specimen construction: (a) CB24F beam construction; (b) CB24F-1/2-PT beam construction; (c) CB24D beam construction; (d) CB33F beam construction; (e) CB33D beam construction; (f) CB24F-RC beam and slab construction; and (g) CB24F-PT beam elevation ............. 28
Figure 3.9 Laboratory test setup................................................................................. 31
Figure 3.10 Loading protocol: (a) Load-controlled; and (b) Displacement-controlled. (1k = 4.45kN)............................................................................................ 33
Figure 3.11 Sensor layout for: (a) CB24F and CB24D, and (b) CB33F, CB33D, and FB33.......................................................................................................... 36
Figure 3.12 Sensor layout for (a) CB24F-RC, and (b) CB24F-PT and CB24F-1/2-PT37
Figure 3.13 Strain gauge layout for CB24F and CB33F. SG 12 and SG 14 are on horizontal crossties.................................................................................... 38
Figure 3.14 Strain gauge layout for CB24D and CB33D. SG 15 and SG 16 are located on horizontal crossties............................................................................... 39
Figure 3.15 Strain gauge layout for CB24F-RC, CB24F-PT, and CB24F-1/2-PT. SG 12 and SG 16 are located on horizontal crossties ..................................... 40
Figure 3.16 Strain gauge layout for FB33. SG 12 and SG 16 are located on horizontal crossties..................................................................................................... 41
Figure 4.1 Cyclic load-deformation: CB24F vs. CB24D (1in = 25.4mm)................. 47
Figure 4.2 Cyclic load-deformation: CB33F vs. CB33D (1in = 25.4mm)................. 48
Figure 4.3 Cyclic load-deformation: CB24F-PT vs. CB24F-1/2-PT (1in = 25.4mm)50
Figure 4.4 Moment curvature analysis summary (BIAX) for beam with and without slab (clockwise from top left): (a) Beam cross section with and without slab; (b) beam elevation with positive and negative moment capacities shown; (c) plot of Mn
- vs. curvature; and (d) plot of Mn+ vs. curvature.... 52
Figure 4.5 Cyclic load-deformation: CB24F vs. CB24F-RC (1in = 25.4mm)........... 53
Figure 4.6 Axial elongation vs. rotation: CB24F vs. CB24F-RC (1in = 25.4mm) .... 53
Figure 4.7 Cyclic load-deformation: CB24F-RC vs. CB24F-PT (1in = 25.4mm)..... 55
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Figure 4.8 Axial elongation vs. rotation: CB24F-PT vs. CB24F-RC (1in = 25.4mm)................................................................................................................... 55
Figure 4.9 Cyclic load-deformation: FB33 (1in = 25.4mm) ...................................... 57
Figure 4.10 Cyclic load-deformation: (a) FB33 vs. Xiao HB3-6L-T100, and (b) FB33 vs. Xiao HB4-6L-T100 ............................................................................. 58
Figure 4.11 (a) Deformation contributions for CB24F; and (b) Definition of different deformation types...................................................................................... 60
Figure 4.12 Deformation contributions for CB24F-1/2-PT ......................................... 61
Figure 4.13 CB24F damage photos: 0.75% - 10.0% rotation ...................................... 63
Figure 4.14 CB24D damage photos: 0.75% - 10.0% rotation...................................... 64
Figure 4.15 CB24F-PT damage photos: (a) 0.075% rotation; (b) 1% rotation; (c) 2% rotation; and (d) 3% rotation..................................................................... 65
Figure 4.16 CB24F-PT damage photos: (a) 4% rotation; (b) 6% rotation; (c) 8% rotation; and (d) 10% rotation................................................................... 66
Figure 4.17 CB24F-1/2-PT damage photos: (a) 0.075% rotation; (b) 1% rotation; (c) 2% rotation; and (d) 3% rotation .............................................................. 67
Figure 4.18 CB24F-1/2-PT damage photos: (a) 4% rotation; (b) 6% rotation; (c) 8% rotation; and (d) 10% rotation................................................................... 68
Figure 4.19 Residual (zero displacement) damage photos (CB24F) after cycles of rotations 1.0%-8.0%.................................................................................. 70
Figure 4.20 Residual (zero displacement) damage photos (CB24D) after cycles of rotations 1.0%-8.0%.................................................................................. 71
Figure 4.21 Residual (zero displacement) damage photos (CB24F-PT) after cycles of rotations 1.0%-8.0%.................................................................................. 72
Figure 4.22 Residual (zero displacement) damage photos (CB24F-1/2-PT) after cycles of rotations 1.0%-8.0% ............................................................................. 73
Figure 5.1 Effective stiffness plotted as a function of aspect ratio for various levels of displacement ductility (NZS 3101-1995). Included on the plot are test results at the corresponding ductility levels.............................................. 77
Figure 5.2 Effective secant stiffness values derived from test results: ln/h = 2.4 ...... 78
Figure 5.3 Yield rotation due to slip/extension for various aspect ratios and testing scales ......................................................................................................... 86
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Figure 5.4 Effective elastic stiffness as a function of gross section stiffness calculated for various aspect ratios and testing scales ............................................... 87
Figure 5.5 Determination of linearized backbone relation from test data.................. 89
Figure 5.6 Backbone load-deformation for full-scale beam models and ASCE 41-06 model (1/2-scale test results are dotted lines) ........................................... 89
Figure 5.7 Backbone load-deformation for full-scale beam models and ASCE 41-06 model modified to account for slip/extension deformations..................... 92
Figure 5.8 Modeling components: (a) Mn-hinge model; and (b) Vn-hinge model ..... 93
Figure 5.9 Cyclic load-deformation modeling results (ln/h = 2.4): CB24F vs. moment hinge model............................................................................................... 95
Figure 5.10 Cyclic load-deformation modeling results (ln/h = 2.4):CB24F vs. shear hinge model............................................................................................... 95
Figure 5.11 Cyclic load-deformation modeling results (ln/h = 3.33): CB33F vs. moment hinge model................................................................................. 96
Figure 5.12 Cyclic load-deformation modeling results (ln/h = 2.4): CB24F-RC vs. moment hinge model................................................................................. 97
Figure 5.13 Cyclic load-deformation modeling results (ln/h = 3.33): FB33 vs. moment hinge model............................................................................................... 98
Figure 5.14 Cyclic load-deformation modeling results (ln/h = 4.0): HB4-6L-T100 vs. moment hinge model................................................................................. 99
Figure 5.15 Cyclic load-deformation modeling results (ln/h = 1.17): CCB11 vs. moment hinge model............................................................................... 101
Figure 5.16 Definitions of parameters in elasto-plastic load-deformation relation ... 103
Figure 5.17 Modeling schematic (from left): (a) Typical beam cross-section; and (b) finite element discretization and loading. ............................................... 104
Figure 5.18 Total yield rotation for coupling beams at various aspect ratios and shear stress levels ............................................................................................. 106
Figure 5.19 Deformation contributions [%] at yield for various aspect ratios at (a) vn=6.0√f’c; and (b) vn=10.0√f’c .............................................................. 108
Figure 5.20 Beam chord rotation θu at onset of significant strength degradation for various aspect ratios and shear stresses .................................................. 109
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Figure 5.21 Beam lateral load, Vave, normalized with respect to beam shear strength from ACI, Vn ........................................................................................... 110
Figure 5.22 Load-deformation backbone relations comparing test results with the nonlinear model developed with VecTor5 and slip/extension for beams at aspect ratio 2.4 ........................................................................................ 112
Figure 5.23 Load-deformation backbone relation comparing test results with nonlinear VecTor5 and slip/extension model for beam at aspect ratio 1.17........... 113
Figure 6.1 Yield point determined from the Load-Deformation backbone relation, defined as the point at which stiffness changes substantially. ................ 118
Figure 6.2 Photo detailing DS1, in which there is light residual cracking evident (>1/16”)................................................................................................... 119
Figure 6.3 Photo detailing DS2, in which there is large residual cracking (>1/8”) and some light spalling of concrete ............................................................... 120
Figure 6.4 Determination of DS3, the onset of significant strength degradation due to severe damage to the concrete and reinforcement .................................. 121
Figure 6.5 Fragility curves for diagonally reinforced concrete coupling beams at high aspect ratio (2.0 < ln/h < 4.0) .................................................................. 125
Figure 6.6 Fragility curves for conventionally-reinforced concrete coupling beams with aspect ratio 2.0 < ln/h < 4.0 ............................................................. 126
Figure 6.7 Fragility curves for diagonally-reinforced concrete coupling beams with aspect ratio 1.0 < ln/h < 2.0 ..................................................................... 126
Figure 6.8 Fragility curves for conventionally-reinforced concrete coupling beams with aspect ratio 1.0 < ln/h < 2.0 ............................................................. 127
Figure 7.1 Coupling beam shear-displacement hinge backbone properties for baseline model....................................................................................................... 132
Figure 7.2 Coupling beam shear-displacement hinge backbone properties for Model 1................................................................................................................. 134
Figure 7.3 Coupling beam shear-displacement hinge backbone properties for Model 2................................................................................................................. 135
Figure 7.4 Perform 3D model (a) 3D view of structure; (b) North-South elevation view of structure; (c) East-West elevation view of structure; (d) plan view of structure; and (e) coupling beam locations in core wall of structure.. 137
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Figure 7.5 Coupling beam rotations (mean for 15 ground motions) for baseline model at MCE level. Dotted lines indicate mean ± one standard deviation. Vertical lines represent mean beam chord rotation at listed damage state................................................................................................................. 139
Figure 7.6 Coupling beam rotations (mean for 15 ground motions) for Model 1 at MCE level. Dotted lines indicate mean ± one standard deviation. Vertical lines represent mean beam chord rotation at listed damage state ........... 140
Figure 7.7 Coupling beam rotations (mean for 15 ground motions) for Model 2 at MCE level. Dotted lines indicate mean ± one standard deviation. Vertical lines represent mean beam chord rotation at listed damage state ........... 141
Figure 7.8 Coupling beam rotations (mean for 15 ground motions) at MCE level for all models (a) north-south side and (b) east-west side............................ 143
Figure 7.9 Inter-story drifts (mean for 15 ground motions) at MCE level for Baseline model. Dotted lines represent mean ± one standard deviation ............... 144
Figure 7.10 Inter-story drifts (mean for 15 ground motions) at MCE level for Model 1. Dotted lines represent mean ± one standard deviation ........................... 145
Figure 7.11 Inter-story drifts (mean for 15 ground motions) at MCE level for Model 2. Dotted lines represent mean ± one standard deviation ........................... 145
Figure 7.12 Inter-story drifts (mean for 15 ground motions) at MCE level for all models (a) north-south and (b) east-west................................................ 146
Figure 7.13 Core wall shear forces (mean for 15 ground motions) at MCE level for all models (a) north-south and (b) east-west................................................ 148
Figure A.1 Initial dimensions [in.] between sensor rods on sides A and B of specimen CB24F..................................................................................................... 155
Figure A.2 Actual displacement history of specimen CB24F................................... 155
Figure A.3 Cyclic load-deformation plot for CB24F................................................ 156
Figure A.4 Axial elongation for CB24F.................................................................... 156
Figure A.5 Deformation contributions for CB24F.................................................... 157
Figure A.6 Curvature profile for CB24F (a) positive loading cycles and (b) negative loading cycles.......................................................................................... 158
Figure A.7 Damage patterns at peak deformation CB24F front side (a) positive loading cycle, (b) negative loading cycle, (c) overall ............................. 159
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Figure A.8 Damage patterns at peak deformation CB24F back side (a) positive loading cycle, (b) negative loading cycle, (c) overall ............................. 160
Figure A.9 Initial dimensions [in.] between sensor rods on sides A and B of specimen CB24D .................................................................................................... 161
Figure A.10 Actual displacement history for specimen CB24D................................. 161
Figure A.11 Cyclic load-deformation relation for CB24D ......................................... 162
Figure A.12 Axial extension of CB24D...................................................................... 162
Figure A.13 Deformation contributions for CB24D ................................................... 163
Figure A.14 Curvature profiles for CB24D (a) positive loading cycles, and (b) negative loading cycles.......................................................................................... 164
Figure A.15 Damage patterns at peak deformation CB24D front side (a) positive loading cycle, (b) negative loading cycle, (c) overall ............................. 165
Figure A.16 Damage patterns at peak deformation CB24D back side (a) positive loading cycle, (b) negative loading cycle, (c) overall ............................. 166
Figure A.17 Initial dimensions [in.] between sensor rods on sides A and B of specimen CB24F-RC .............................................................................................. 167
Figure A.18 Actual displacement history for specimen CB24F-RC........................... 167
Figure A.19 Cyclic load-deformation plot for CB24F-RC ......................................... 168
Figure A.20 Axial extension of CB24F-RC................................................................ 168
Figure A.21 Deformation contributions for CB24F-RC ............................................. 169
Figure A.22 Curvature profiles CB24F-RC (a) positive loading cycles and (b) negative loading cycles.......................................................................................... 170
Figure A.23 Damage cracking patterns at peak deformations CB24F-RC (a) front side all cycles, (b) back side all cycles........................................................... 171
Figure A.24 CB24F-RC damage photos at peak rotation: 0.75%-4.0% beam chord rotation .................................................................................................... 172
Figure A.25 CB24F-RC damage photos at peak rotation: 6.0%-14.0% beam chord rotation .................................................................................................... 173
Figure A.26 CB24F-RC residual damage photos at zero rotation: after cycles at 0.75%-4.0% beam chord rotation ....................................................................... 174
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Figure A.27 CB24F-RC residual damage photos at zero rotation: after cycles at 6.0%-14.0% beam chord rotation ..................................................................... 175
Figure A.28 Initial dimensions [in.] between sensor rods on sides A and B of specimen CB24F-PT............................................................................................... 176
Figure A.29 Actual displacement history for specimen CB24F-PT ........................... 176
Figure A.30 Cyclic load-deformation relation for CB24F-PT.................................... 177
Figure A.31 Axial extension of CB24F-PT ................................................................ 177
Figure A.32 Load in prestressing tendons for CB24F-PT .......................................... 178
Figure A.33 Deformation contributions for CB24F-PT.............................................. 178
Figure A.34 Curvature profiles for CB24F-PT (a) positive loading cycles and (b) negative loading cycles ........................................................................... 179
Figure A.35 Damage cracking patterns at peak deformations CB24F-PT (a) front side all cycles, (b) back side all cycles........................................................... 180
Figure A.36 Initial dimensions [in.] between sensor rods on sides A and B of specimen CB24F-1/2-PT......................................................................................... 181
Figure A.37 Actual displacement history for specimen CB24F-1/2/PT ..................... 181
Figure A.38 Cyclic load-deformation plot for CB24F-1/2-PT ................................... 182
Figure A.39 Axial extension of CB24F-1/2-PT.......................................................... 182
Figure A.40 Load in prestressing tendons for CB24F-1/2-PT.................................... 183
Figure A.41 Deformation contributions for CB24-1/2-PT.......................................... 183
Figure A.42 Curvature profiles for CB24F-1/2-PT (a) positive loading cycles and (b) negative loading cycles ........................................................................... 184
Figure A.43 Damage cracking patterns at peak deformations CB24F-1/2-PT (a) front side all cycles, (b) back side all cycles ................................................... 185
Figure A.44 Initial dimensions [in.] between sensor rods on sides A and B of specimen CB33F..................................................................................................... 186
Figure A.45 Actual displacement history for specimen CB33F ................................. 186
Figure A.46 Cyclic load-deformation plot for CB33F................................................ 187
Figure A.47 Axial elongation of CB33F..................................................................... 187
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Figure A.48 Damage cracking patterns at peak deformations CB33F (a) front side all cycles, (b) back side all cycles................................................................ 188
Figure A.49 CB33F damage photos at peak rotation: 0.75%-6.0% beam chord rotation................................................................................................................. 189
Figure A.50 CB33F residual damage photos at zero rotation: after cycles at 1.0%-8.0% beam chord rotation ................................................................................ 190
Figure A.51 Initial dimensions [in.] between sensor rods on sides A and B of specimen CB33D .................................................................................................... 191
Figure A.52 Actual displacement history for specimen CB33D................................. 191
Figure A.53 Cyclic load-deformation plot for CB33D ............................................... 192
Figure A.54 Axial extension for CB33D .................................................................... 192
Figure A.55 Damage cracking patterns at peak deformations CB33D (a) front side all cycles, (b) back side all cycles................................................................ 193
Figure A.56 CB33D damage photos at peak rotation: 1.0%-6.0% beam chord rotation................................................................................................................. 194
Figure A.57 CB33D residual damage photos at zero rotation: after cycles at 1.0%-6.0% beam chord rotation ................................................................................ 195
Figure A.58 Initial dimensions [in.] between sensor rods on sides A and B of specimen FB33........................................................................................................ 196
Figure A.59 Actual displacement history for specimen FB33 .................................... 196
Figure A.60 Cyclic load-deformation plot of FB33.................................................... 197
Figure A.61 Axial extension of FB33 ......................................................................... 197
Figure A.62 Damage cracking patterns at peak deformations FB33 (a) front side all cycles, (b) back side all cycles................................................................ 198
Figure A.63 FB33 damage photos at peak rotation: 0.75%-6.0% beam chord rotation................................................................................................................. 199
Figure A.64 FB33 residual damage photos at zero rotation: after cycles at 1.0%-5.0% beam chord rotation ................................................................................ 200
Figure B.1 (a) Cross section of CB24F to be used for slip/ext calculation; and (b) definition of slip/extension crack and corresponding rotation................ 201
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Figure B.2 Elastic slip/extension moment-rotation hinge properties to be implemented in nonlinear model .................................................................................. 204
Figure B.3 Schematic of slip/extension springs in compound element .................... 204
Figure D.1 Schematic for Mn-hinge model, including elastic cross section, Mn-rotation hinges, and slip/extension hinges............................................................ 207
Figure D.2 Schematic for Vn-hinge model................................................................ 209
Figure E.1 Diagonal #7 bars; tested by twining laboratories; based on given fy, fu, and % elongation ........................................................................................... 212
Figure E.2 Concrete cylinders CB24F, CB24D, CB33F, CB33D; 6”x12” tested by twining laboratories; curve fit based on f’c ............................................. 213
Figure E.3 Concrete Cylinders CB24F-RC; 6”x12” tested by twining laboratories; 4”x8” tested by ucla; curve fit based on f’c............................................. 213
Figure E.4 Concrete Cylinders CB24F-PT; 6”x12” tested by twining laboratories; 4”x8” tested by ucla; curve fit based on f’c............................................. 214
Figure E.5 Concrete cylinders CB24F-1/2-PT; 6”x12” tested by twining laboratories; 4”x8” tested by ucla; curve fit based on f’c............................................. 214
Figure E.6 Concrete cylinder tests FB33; 6”x12” tested by twining laboratories; 4”x8” tested by ucla; curve fit based on f’c............................................. 215
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List of Tables
Table 2.1 Summary of results from Kwan and Zhao (2002) .................................... 14
Table 3.1 Test Matrix................................................................................................ 22
Table 3.2 Material Properties.................................................................................... 30
Table 4.1 Summary of predicted member strengths ................................................. 43
Table 4.2 Summary of experimental force results .................................................... 44
Table 4.3 Summary of experimental displacement results ....................................... 45
Table 4.4 Crack widths at peak rotation ................................................................... 62
Table 5.1 Effective stiffness values .......................................................................... 82
Table 5.2 Cyclic Degradation Parameters (Perform 3D)........................................ 100
Table 5.3 Geometric properties of beams used in nonlinear modeling procedure . 105
Table 5.4 Lower Bound Estimate ASCE 41-06 Modeling Parameters and Numerical Acceptance Criteria for Nonlinear Procedures - Diagonally-Reinforced Coupling Beams...................................................................................... 115
Table 6.1 Details of damage states for fragility relations ....................................... 122
Table 6.2 Summary of fragility function parameters for coupling beams .............. 123
Table 6.3 Limit/Damage State Comparisons (plastic hinge rotations) ................... 128
Table 6.4 ASCE 41-06 Modeling Parameters and Numerical Acceptance Criteria for Nonlinear Procedures - Diagonally-Reinforced Coupling Beams.......... 129
Table 7.1 Summary of varied coupling beam modeling parameters ...................... 136
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List of Symbols
Acw = cross-sectional area of concrete beam web
Ash = area of transverse reinforcement provided within given spacing, s
Avd = cross-sectional area of each diagonal group of bars
bw = width of beam web
d = depth of beam from extreme compression to extreme tension steel
db = diameter of rebar
Ec = modulus of elasticity of concrete
f’c = concrete compressive strength
fy = yield strength of reinforcement
fs = stress in steel reinforcement
fu = ultimate rupture strength of reinforcement
HL = height of lugs of steel reinforcement
h = beam depth
Ieff = effective section moment of inertia
Ig = gross section moment of inertia
K = stiffness of moment-rotation plot
Le = available elastic bond length
Lpy = available post-yield length
ld = development length of reinforcement
ln = clear span of beam
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ln/h = aspect ratio of beam
Mn = nominal moment capacity of beam
Mn+ = positive nominal moment capacity of beam
Mn- = negative nominal moment capacity of beam
Mpr = probable moment capacity of beam
My = yield moment of beam
SL = longitudinal spacing of lugs on steel reinforcement
s = longitudinal spacing of transverse reinforcement
ue = elastic bond stress
uf = frictional bond stress
uu = peak bond stress
V = beam shear
V@Mn = shear strength corresponding to nominal moment capacity
V(ACI) = shear strength based on ACI nominal shear strength eqn 21-9
Vave = average beam shear between yield and onset of strength degradation
Vmax = max shear force applied during test
Vn = nominal shear capacity of beam
Vr = residual capacity
Vy = yield strength of beam
vn = nominal shear stress of beam
x = depth of neutral axis
α = angle between diagonal bars and longitudinal axis of beam
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Δ = relative displacement of beam end
Δ@Vmax = relative displacement of beam end at peak load
Δu = relative displacement at onset of significant lateral strength degradation
Δy = relative displacement at yield
δext = deformation due to extension of reinforcement at beam-wall interface
δexty = deformation due to extension of reinforcement at beam-wall interface at yield
δs = deformation due to slip of reinforcement at beam-wall interface
δs1 = local slip at the peak bond stress
δtot = total deformation due to slip/extension
δtoty = total deformation due to slip/extension at yield
εu = ultimate (rupture) strain of reinforcement
εy = yield strain of reinforcement
θ = beam chord rotation defined as Δ/ln
θ@δtot = beam chord rotation due to slip/extension of reinforcement
θr = beam chord rotation at residual strength
θu = beam chord rotation at onset of significant lateral strength degradation
θx = maximum beam chord rotation
θy = beam chord rotation at yield
μ = displacement ductility defined as Δu/Δy
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Acknowledgements
I would first like to express my gratitude to my research advisor, Professor John W.
Wallace, for his guidance and support during my graduate studies. Thanks are also
extended to the members of the doctoral committee, Professors Ertugrul Taciroglu,
Thomas Sabol, Jian Zhang, and Farzin Zareian for their invaluable advice and insightful
comments.
This research was conducted in close collaboration with practicing engineers from
Magnusson Klemencic Associates (MKA), Inc., in Seattle, Washington. Particularly, I
would like to thank our collaborators Andy Fry and Ron Klemencic, whose practical
input was essential to the completion of the project. Thanks also go to Dr. Brian Morgen
of MKA for his advice during the testing phase. Thanks are extended also to Paul
Briennen of PCS, for his input during the testing phase.
The research has been funded by the Charles Pankow Foundation, with significant
in-kind support provided by Webcor Concrete; this support is gratefully acknowledged.
As well, material contributions from Catalina Pacific Concrete, SureLock, and Hanson
Pacific are appreciated. Linas Vitkas at Twining Laboratories is thanked for his
assistance with material testing.
I would particularly like to thank Senior Development Engineer Steve Keowen
for his invaluable help with strain gauge installation, specimen construction, placement,
and cleanup. Assistant Development Engineer Dr. Alberto Salamanca was absolutely
essential to the completion of this project, and his assistance with specimen testing, data
xxi
acquisition, and post processing is greatly appreciated. Thanks also go to Senior
Development Engineer Harold Kasper for his help with material testing. IT Manager
Steve Kang is thanked for his help during the testing phase.
Thanks are extended to laboratory assistants Joy Park, Nolan Lenahan, and
Cameron Sanford for their help in test preparation and completion. Special thanks are
given to Dr. Anne Lemnitzer, Sarah Taylor-Lange, and Dr. Derek Skolnik for their
assistance during the testing phase, and to Aysegul Gogus, Marisol Salas, and Zeynep
Tuna, whose model was used as the basis for the system modeling studies, for their
assistance and advice during the modeling phase. I would particularly like to thank Alicia
Kinoshita for her wise advice and support throughout my graduate studies.
Finally, I would like to especially thank my family for their continued support
now and throughout my education.
xxii
VITA
March 2, 1984 Born Van Nuys, CA
2006 B.S., Civil and Environmental Engineering
University of California, Los Angeles
Los Angeles, CA
2004-2006 Engineering Aide
Dept. of Civil and Environmental Engineering
Los Angeles, CA
2008 M.S., Civil and Environmental Engineering
University of California, Los Angeles
Los Angeles, CA
2006-2010 Graduate Student Researcher
Dept. of Civil and Environmental Engineering
2008-2010 Teaching Assistant
Dept. of Civil and Environmental Engineering
Los Angeles, CA
xxiii
PUBLICATIONS AND PRESENTATIONS
Naish, D., Wallace, J., Testing and Modeling of Reinforced Concrete Coupling Beams.
9th US National and 10th Canadian Conference on Earthquake Engineering, Toronto, Canada, (accepted for publication).
Naish, D., Wallace, J., Fry, J. A., Klemencic, R., Modeling of Reinforced Concrete
Coupling Beams. 7th International Conference on Urban Earthquake Engineering & 5th International Conference on Earthquake Engineering Proceedings, Tokyo, Japan, March 3-5, 2010.
Naish, D., Wallace, J., Fry, J.A., Klemencic, R., Experimental Evaluation and Analytical
Modeling of ACI 318-05/08 Reinforced Concrete Coupling Beams Subjected to Reversed Cyclic Loading, Report No. UCLA-SGEL 2009/06, August 25, 2009.
Naish, D., Wallace, J., Klemencic, R., Fry, J.A., Testing and Modeling of Reinforced
Concrete Coupling Beams. Oral Presentation, 6th Annual Meeting of the Network for Earthquake Engineering Simulation Consortium, Inc. (NEESinc), Portland, OR, June 18, 2008.
Naish, D., Wallace, J., Klemencic, R., Fry, J.A., Testing and Modeling of RC Link
Beams. Oral Presentation, American Concrete Institute Spring Convention, Los Angeles, CA, April 2, 2008.
xxiv
ABSTRACT OF THE DISSERTATION
Testing and Modeling
of Reinforced Concrete
Coupling Beams
by
David Anthony Braithwaite Naish
Doctor of Philosophy in Civil Engineering
University of California, Los Angeles, 2010
Professor John Wallace, Chair
An efficient structural system for tall building construction to resist
earthquake loads consists of reinforced concrete shear walls connected by
coupling beams. Construction of coupling beams that satisfy the strength and
detailing requirements set forth in ACI 318-05 for diagonally reinforced coupling
beams is cumbersome and costly; therefore, ACI 318-08 provides a new detailing
option which aims to improve the constructability while maintaining adequate
xxv
strength and ductility. Eight half-scale specimens were tested to compare the
performance of beams constructed utilizing new and old detailing options, to
compare beams with diagonal reinforcement to beams with straight bars at higher
aspect ratios, and to assess the impact of reinforced and post-tensioned slabs. Test
results indicate that the new detailing approach provides equal, if not improved
behavior as compared to the alternative detailing approach and that including a
slab had only a modest impact on strength, stiffness, ductility, and observed
damage.
Understanding of the load-deformation characteristics is essential to
modeling the overall system response to seismic loading. Modeling studies are
performed to evaluate the effectiveness of current modeling approaches with
respect to key performance parameters, including effective elastic stiffness,
ductility, and residual strength. As well, most of the experimental work that has
been performed on coupling beams has been on specimens at less than full-scale.
The impact of this scaling effect on the full-scale models is presented and shown
to be potentially significant on the expected ductility of the member. A brief
summary of simplified modeling techniques using commercially available
software is presented to provide practical applications for design engineers.
Results indicate that these simple modeling approaches reasonably capture
measured force versus deformation behavior.
1
Chapter 1 Introduction
This chapter presents a brief summary of information regarding background and
motivation for this research. It lays out the specific objectives of the testing program, and
the basis of these objectives with respect to desired outcomes from a design engineer’s
viewpoint. An overview of the dissertation is also presented.
1.1 Background
Tall building construction is common in metropolitan areas and it has become
increasingly important to provide methods of construction that improve both seismic
performance and constructability. Reinforced concrete core walls, with coupling beams
above openings to accommodate doorways, are an efficient lateral-force-resisting system
for tall buildings (Fig. 1.1). When subjected to strong shaking, coupling beams act as
fuses and typically undergo large inelastic rotations to dissipate energy. When properly
detailed, the coupling beam links the behavior of the two independent shear walls into a
coupled system. Careful design of the beams is essential to achieve the desired degree of
2
coupling and level of energy dissipation. The degree of coupling of the system directly
impacts the system reaction to lateral forces (Fig. 1.2) (Harries et al., 2000).
Various testing programs have been carried out to assess the load – deformation
behavior of coupling beams [(Paulay, 1971), (Paulay and Binney, 1974), (Barney et al.,
1980), (Tassios et. al., 1996), (Xiao et. al., 1999), (Galano and Vignoli, 2000), (Kwan and
Zhao, 2001), (Fortney, 2005)]. Primary test variables in these studies were the ratio of the
beam clear span to the beam total depth (commonly referred to as the beam aspect ratio)
and the arrangement of the beam reinforcement. In a majority of these studies, the load –
deformation behavior of low-aspect ratio beams (1.0 to 1.5) constructed with beam top
and bottom longitudinal reinforcement were compared with beams constructed with
diagonal reinforcement. Concrete compressive strengths for most tests were around 4 ksi
(~25 to 30 MPa). Although these tests provided valuable information, they do not address
issues for current tall building construction, where beam aspect ratios are typically
between 2.0 and 3.5 and concrete strengths are in the range of 6 to 8 ksi (~40 to 55 MPa).
In addition, in none of the prior studies was a slab included as part of the test specimen;
whereas the slab might restrain axial elongations and impact stiffness, strength, and
deformation capacity [(Klemencic et. al., 2006), (Kang and Wallace, 2005), (Kang and
Wallace, 2006)]. Slabs commonly exist and use of post-tensioned slabs is common for
current construction.
3
Post-tensioned Floor Slab
Concrete Wall PierConcrete Coupling Beam
A A
(a) Post-tensioned Floor Slab
Concrete Wall PierConcrete Coupling Beam
A A
(a)
Concrete Floor Slab
RC Coupling Beam
Concrete Wall Pier
Wall OpeningA-A
(b)
Concrete Floor Slab
RC Coupling Beam
Concrete Wall Pier
Wall OpeningA-A
(b)
Figure 1.1 Typical (a) plan and (b) elevation views of coupled corewall structure
4
M M M MT
T
C
C CTVbase
Vbase
Lateral Load
Lateral Load
(a) (b)
M M M MT
T
C
C CTVbase
Vbase
Lateral Load
Lateral Load
(a) (b)
Figure 1.2 Reactions to applied lateral load for system with (a) well-detailed coupling
beams, and (b) poorly-detailed coupling beams
Use of diagonal reinforcement in coupling beams with aspect ratio (clear length to
total depth) less than four was introduced into ACI 318-99. Two groups of diagonal bars
are placed such that they intersect at the center of the beam, and are at the top and bottom
of the beam depth at the beam-wall interface (see Fig. 1.3). These two groups of diagonal
bars, and the concrete they encase, are commonly assumed to form a truss, with one
group serving as the tension member and the other as the compression member. To
enhance the compressive strength and deformation capacity of the diagonal truss
members as well as to suppress buckling of the diagonal bars, use of transverse
reinforcement around the diagonal bar groups is required (ACI 318-99). The quantity of
transverse reinforcement required is the same as that used for columns, which is
5
substantially more than that used in most of the prior coupling beam test programs.
Nominal transverse reinforcement also is required around the entire beam cross section.
Conventionally Reinforced Concrete Coupling Beam
Diagonally Reinforced Concrete Coupling Beam
Conventionally Reinforced Concrete Coupling Beam
Diagonally Reinforced Concrete Coupling Beam
Figure 1.3 Typical reinforcement pattern for conventional and diagonal reinforcement
in coupling beams
Providing transverse reinforcement around the diagonal bar bundles as detailed in
ACI 318-05 S21.7.7 (Fig. 1.4(a)) presents significant difficulties with regards to
constructability. Specifically, placing hoops around the diagonal bundles is difficult
where the diagonal groups intersect at beam mid-span, particularly for shallow beams
(aspect ratio greater than 2.5), for which the intersection of the bars is much longer. As
well, it is very difficult to place hoops around the diagonal bundles at the beam-wall
interface, particularly for deep beams (aspect ratio less than 2.0) due to interference with
6
the wall boundary vertical reinforcement (Fig. 1.5). To overcome these construction
difficulties, ties or hoops are placed around the entire intersection region, rather than each
bundle individually; however, it is unclear if the modified detailing meets the intent of
the code and the coupling beam performance is acceptable.
To combat these issues, ACI 318-08 S21.9.7 introduced an alternative detailing
option, where transverse reinforcement is placed around the beam cross section to
provide confinement and suppress buckling, and no transverse reinforcement is provided
directly around the diagonal bar bundles (Fig. 1.4(b)). Use of this detailing option avoids
the problems noted where the diagonal bars intersect and at the beam-wall interface,
reducing the construction time for a typical floor by a day or two (ENR, 2007). While the
volumetric ratio of steel used for this detail may increase, the overall cost is lower due to
the reduced construction time.
Procedures for the design of diagonally-reinforced concrete coupling beams are
given in ACI 318-08 S21.9.7. Specifically, coupling beams with aspect ratio less than 2.0
and expected shear stress greater than '4 psicf must be reinforced with diagonally-
placed bars. The strength of beams with diagonal reinforcement is determined by ACI
318-08 Equation 21-9, reproduced here (Eqn. 1.1):
2 sin 10 'n vd y c cwV A f f Aα= ≤ (Eq. 1.1)
7
where Avd is the area of a bundle of diagonal reinforcement, fy is the yield strength of the
reinforcement, and α is the angle of inclination of the diagonal bars. Thus the shear
strength is based solely on the vertical component of the area of the diagonal
reinforcement. As discussed above, there are two options for confinement of the diagonal
bars, named “diagonal confinement” and “full section confinement” in this text,
corresponding to S21.9.7.4(c) and S21.9.7.4(d), respectively. The diagonal confinement
option requires that transverse reinforcement satisfying S21.6.4 be placed around each
diagonal bundle, with nominal longitudinal and transverse reinforcement required around
the entire beam cross-section. The full section confinement option requires that transverse
reinforcement satisfying S21.6.4 be placed around the entire cross-section. In either case,
the area of transverse reinforcement required, Ash, is governed by ACI 318-08 equations
21-4 and 21-5, reproduced here as equations 1.2 and 1.3:
'
0.3 1gc csh
yt ch
As b fAf A
⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦ (Eq. 1.2)
'
0.09 c csh
yt
s b fAf
= (Eq. 1.3)
where s is the spacing of the transverse reinforcement longitudinally, bc is the cross-
section dimension to the outside edges of the transverse reinforcement, Ag is the gross
area of concrete, and Ach is the area measured to the outside edges of the transverse
reinforcement.
8
SECTION
Spacing measured perpendicular to the axis of the diagonal bars not to exceed 14 in., typical
*
*
(a)
SECTION
Spacing measured perpendicular to the axis of the diagonal bars not to exceed 14 in., typical
*
*
(a)Alternate consecutive crosstie 90-deg hooks, both horizontally and vertically, typical
Spacing not to exceed 8 in., typical
SECTION
*Spacing not to exceed 8 in., typical
* *
(b)Alternate consecutive crosstie 90-deg hooks, both horizontally and vertically, typical
Spacing not to exceed 8 in., typical
SECTION
*Spacing not to exceed 8 in., typical
* *
(b)
Figure 1.4 Confinement options provided in ACI 318: (a) ACI 318-05 Diagonal
confinement; and (b) ACI 318-08 Full section confinement
ln
h
Symmetry
α
Reinforcement Congestion
Wall Boundary Reinforcementln
h
Symmetry
α
Reinforcement Congestion
Wall Boundary Reinforcement
Figure 1.5 Reinforcement congestion caused by using ACI 318-05 Diagonal
confinement
9
In beams with aspect ratio (ln/h) approaching four, the angle of inclination (α) of
the diagonal reinforcement is often very small (~10°), making placement of the diagonal
reinforcement more difficult, as the diagonal bars are more likely to be obstructed by
transverse reinforcement. Given this fact, design engineers prefer to use straight
(longitudinal) flexural reinforcement, if the shear demand and required displacement
ductility are low ( )'4 psicf< .
Nonlinear modeling of coupling beams has received increased attention as the use
of performance-based design for tall core wall buildings has become more common
(Wallace, 2007). Modeling parameters for diagonally-reinforced coupling beams were
introduced into Table 6-17 of FEMA 273 (1997); given the limited test data available,
only one row of modeling parameters is provided, and these parameters remain
unchanged in FEMA 356 (2000) and ASCE 41-06 (2007). Of particular interest is the
selection of the effective secant bending stiffness at yield c effE I and the allowable plastic
rotation prior to significant lateral strength degradation. The value used for coupling
beam bending stiffness has a significant impact on degree of coupling [(Coull, 1974),
(Harries, 2001)]. Understanding of the behavior of the coupling beams is essential for
understanding of the behavior of the system as a whole.
10
1.2 Objectives
While several research programs have been undertaken to test and model reinforced
concrete coupling beams, there are several critical gaps in the research related to current
construction practices. With this in mind, a research program was developed to address
key issues, including aspect ratio, residual strength, concrete compressive strength, slab
impact, and detailing of transverse reinforcement. Specifically the following objectives
were deemed particularly important for study.
1) To test beams with aspect ratios more representative of those used in
current tall building construction, namely beams with aspect ratio
greater than 2.0.
2) To test the specimens completely to failure, specifically to determine
residual strength and total plastic rotation capacities.
3) To test beams with material properties that correspond to those used by
practicing designers in current construction.
4) To determine the impact of the slab (both conventional reinforced
concrete and post-tensioned) on the performance of the coupling beam.
5) To investigate and compare the performance of the new detailing
provisions in ACI 318-08 for diagonally reinforced coupling beams to
that of the old detailing provisions from ACI 318-05.
6) To provide simple procedures for implementation of design parameters
in nonlinear modeling by practicing engineers.
11
1.3 Organization
The dissertation is organized into eight chapters. This introduction provided in Chapter 1
is followed by a review of relevant research in Chapter 2. Chapter 3 summarizes all of the
test specimen design parameters as well as all the testing protocols and procedures.
Chapter 4 is a presentation and discussion of all the relevant test results. Chapter 5
provides an overview of current modeling techniques and the development of simplified
modeling techniques for design engineers. A procedure for developing fragility relations
for coupling beams is introduced in Chapter 6 and Chapter 7 provides an overview of
analytical studies to assess the impact of variation of modeling parameters on the overall
system response. Conclusions to the research project are provided in Chapter 8.
12
Chapter 2 Literature Review
This chapter presents summaries of some of the relevant work that has been previously
done on coupling beams and coupled walls. Testing programs for both diagonally and
conventionally reinforced coupling beams as well as modeling studies for effective
elastic stiffness and load-deformation relations are investigated.
2.1 Conventionally Reinforced Coupling Beams
Conventionally reinforced concrete coupling beams are fairly deep beams that link shear
walls in a core wall system. They are reinforced with longitudinal flexural reinforcement.
Paulay (1971) conducted tests at the University of Canterbury in the early 1970s,
investigating the behavior of short and relatively deep concrete beams in shear walls.
Barney et al. (1980) investigated coupling beams subjected to load reversals.
They tested beams at aspect ratios 2.5 and 5.0, and observed displacement ductility
values of 7.8 and 10.0, respectively. Xiao et. al. (1999) investigated the behavior of high
strength concrete coupling beams with conventional longitudinal reinforcement subjected
to cyclic loading protocols. The main purpose of the testing program was to investigate
13
differences between flexural reinforcement configurations, by concentrating flexural
reinforcement at the beam edges and by distributing flexural reinforcement vertically
along the beam depth. The tests were conducted on 1:2 scale beams, with aspect ratio 3
and 4. The concrete compressive strength was 10 ksi and the steel yield strength was
around 70 ksi. The study found that conventionally reinforced beams with aspect ratio 3
achieved maximum chord rotation of 3.6%, corresponding to displacement ductility of
6.0, with a shear stress of '4.8 cf ; beams with aspect ratio 4 achieved maximum chord
rotation of 4.6%, corresponding to a displacement ductility of 6.2, with a shear stress of
'3.7 cf .
Kwan and Zhao (2002) investigated deep coupling beams subjected to cyclic
loading. They tested five 1:2-scale conventionally-reinforced coupling beams with aspect
ratio between 1 and 2, specifically 1.17, 1.40, 1.75, and 2.00. The average concrete
compressive strength was 5.5 ksi and the average yield strength of the reinforcement was
75 ksi. The results are provided in Table 2.1. They can be summarized as follows.
Generally speaking, displacement ductility increased with increasing aspect ratio and
with decreasing shear stress.
14
Table 2.1 Summary of results from Kwan and Zhao (2002)
ID ln/h Transverse reinforcement spacing [in.]
vn ( )' psicf θu [%] μ
CCB1 1.17 3.0 9.25 5.7 4.0
CCB12 1.17 2.0 8.95 4.3 4.3
CCB2 1.40 3.0 7.77 4.3 5.0
CCB3 1.75 3.0 7.16 3.6 5.0
CCB4 2.00 3.0 6.15 5.1 6.0
2.2 Diagonally Reinforced Coupling Beams
Diagonal reinforcement was introduced as a potential alternative to conventional
longitudinal reinforcement in coupling beams by Paulay and Binney in 1974. The
purpose of providing diagonal reinforcement was to improve performance of coupling
beams with respect to sliding shear failures at high shear stress levels. The idea behind
placing reinforcement diagonally was to enable the beam to act as a cross bracing with
equal diagonal tension and compression capacity. In other words, the diagonal bars can
act as a truss to resist the lateral loads, with one group of bars in compression and the
other in tension. These beams were found to have excellent ductility and energy
dissipation properties. However, because the diagonal bars are placed in compression in
each loading cycle, stability of these diagonal bars is a major issue, and providing
15
adequate transverse reinforcement to protect against buckling of the diagonal bars is the
main detailing provision (Paulay and Binney, 1974).
Since the 1970s, several testing programs have been undertaken to investigate the
behavior of coupling beams with diagonal reinforcement. Barney et al. (1980) also tested
diagonally reinforced beams at aspect ratios 2.5 and 5.0. These beams exhibited
displacement ductility values of 9.0 and 10.2, respectively. Tassios et al. (1996)
investigated coupling beams with several layouts of flexural reinforcement, and found
that diagonally reinforced beams tended to perform better than beams with conventional
reinforcement. Specifically, the diagonally reinforced beams had concrete compressive
strengths of 4 ksi and yield strength of the reinforcement of 73 ksi. One beam was tested
with aspect ratio 1 and reached 8.2% rotation prior to significant (>15%) strength
reduction, corresponding to displacement ductility of 5.6, with a maximum shear stress of
'10 cf . Similarly, a beam with aspect ratio 1.66 was tested and achieved maximum
chord rotation of 8.8%, displacement ductility of 5.2, and shear stress of '10 cf .
Galano and Vignoli (2000) performed similar studies on short coupling beams
with different reinforcement layouts. Two different configurations were tested with
diagonal reinforcements, the main difference between the two being the detailing of the
transverse reinforcement around the diagonal bars. The beams were tested after a time of
between 4 and 5 years to simulate the conditions under which the beams would be
subjected in actual usage. The concrete compressive strengths of the beam specimens
ranged from 5.8 ksi to 7.8 ksi. The beams were all aspect ratio 1.5, approximately 1:2
16
scale. The beams exhibited displacement ductility values of 7.0 and 5.0 corresponding to
volumetric steel ratios of 0.0039 and 0.0031, respectively, indicating that higher steel
ratios can lead to larger plastic rotation capacities.
Kwan and Zhao (2002) also investigated a diagonally reinforced beam for
comparison with the conventionally reinforced beams tested. The beam had aspect ratio
1.17, with the same material properties as the other tests. The specimen reached a
maximum chord rotation of 5.4% (displacement ductility of 4.0) and a maximum shear
stress of '9.8 cf . As well, they found that the energy dissipation characteristics of the
diagonally reinforced beam were much better than those of the conventionally-reinforced
beams.
Fortney (2005) tested a number of different types of coupling beams including
one diagonally-reinforced coupling beam, which had aspect ratio 2.56. The steel strength
was 62.5ksi for yield, and 100ksi for ultimate, and the concrete compressive strength was
5.5ksi. The diagonally-reinforced specimen reached a maximum chord rotation of 5.8%
and a maximum shear stress of '13.6 cf , while having a design shear stress '8.0n cv f= .
2.3 Coupled Wall Behavior
Several studies have investigated the behavior of coupled wall systems, where coupling
beams are used to link shear walls to add system strength and stiffness. However, of
interest in this study is the impact of coupling beam behavior on overall system behavior.
17
Harries et al. (2000) conducted several studies on the design and behavior of
coupling beams and their impact on the behavior of coupled wall systems. Well-detailed
coupling beams above the second floor of multi-story buildings generally develop plastic
hinges simultaneously, with similar end rotations over the entire height of the structure.
This mechanism allows energy dissipation to be distributed in the coupling beams over
the building height, rather than primarily focused in the wall piers at the base of the
building (Fig. 2.1). Ideally, the mechanism through which energy is dissipated should
involve plastic hinges first in most of the beams and then at the base of the walls. The
variables that are used to achieve this performance are strength, stiffness, ductility, and
energy dissipation capacity.
Paulay (1980) performed studies on design of coupled wall systems, and
investigated the importance of understanding coupling beam behavior. Paulay introduced
the idea of using diagonal reinforcement to help prevent sliding shear failures in squat
beams (ln/h < 4). This use of diagonal reinforcement is essential to provide adequate
ductility in the coupling beams to ensure that the majority of the energy is dissipated in
the coupling beams.
18
Lateral Load
Lateral Load
(a) (b)
Lateral Load
Lateral Load
(a) (b)
Figure 2.1 Distribution of energy dissipation in a core wall structure with (a) well-
detailed coupling beams, and (b) poorly-detailed coupling beams
19
Chapter 3 Experimental Program
This chapter provides details of the beam prototype designs and the resulting test
specimen design. As well, a discussion of the procedures employed to characterize the
mechanical properties of the structural materials used in the test specimens is provided.
Test methods and test protocols are described.
3.1 Beam Design
The test beam prototypes were based on two common tall building configurations for
residential and office construction. Typical wall openings and story heights produce
coupling beams with aspect ratios of approximately 2.4 for residential buildings and 3.33
for office buildings. A coupling beam with cross-section dimensions of 24" 30"x and
24" 36"x reinforced with two bundles of 8-#11 diagonal bars is common for residential
and office construction, respectively. The nominal shear strengths of the residential and
office beams, determined using ACI 318-08 equation 21-9:
2 sin 10 'n vd y c cwV A f f Aα= ≤ (Eq. 3.1)
20
are 7.3 'c cvf A and 4.8 'c cvf A , for aspect ratios of 2.4 (α=15.7°) and 3.33 (α=12.3°),
respectively, for Grade 60 reinforcement, where α represents the degree of inclination of
the diagonal bars with respect to the longitudinal axis of the beam. Due to geometric and
strength constraints of an existing structural steel reaction frame, tests were conducted on
one-half scale replicas of the prototype beams. Thus the test specimens were either
12" 15"x or 12" 18"x with two bundles of 6-#7 diagonal bars (Figs. 3.1-3.5), for the
residential and office beams, respectively. For aspect ratio 3.33, a 12" 18"x specimen
with 3-#6 top and bottom longitudinal reinforcement (referred to as “frame beam”) was
also tested (Fig. 3.8). The maximum shear stress expected for the frame beam, based on
reaching prM at the beam-wall interface at the beam ends, was 3.6 'cf . This limit was
selected based on input from practicing engineers; at higher shear stresses, use of
diagonal reinforcement is common.
As stated previously, the configuration of the transverse reinforcement was a
primary variable of the test program. Beams with transverse reinforcement provided
around the bundles of diagonal bars (referred to as “diagonal confinement”) were
designed according to ACI 318-05 S21.7.7.4, whereas beams with transverse
reinforcement provided around the entire beam cross section (referred to as “full section
confinement”) were designed according to ACI 318-08 S21.9.7.4(d). Volumetric ratios of
transverse reinforcement and the ratios of bar spacing to bar diameter ( )/ bs d for the one-
half scale test beams were selected to be similar to the prototype beams. Due to
21
maximum spacing limits, the volumetric ratios of transverse reinforcement provided in
both the prototype and test beams exceed that calculated using the requirement for
columns (ACI 318-08 21.6.4.4); therefore, even though the provided transverse
reinforcement exceeds the minimum required, the tests are representative of beams
designed to satisfy minimum code requirements. The test beam geometries and
reinforcement configurations are summarized in Table 3.1 and Figures 3.1-3.8.
Three test specimens with aspect ratio of 2.4 were constructed with 4”-thick slabs.
One specimen (CB24F-RC) included a slab reinforced with #3 bars @12” spacing, on the
top and bottom in the transverse direction, and on the top only in the longitudinal
direction, without post-tensioning strands (Fig. 3.3). Two specimens (CB24F-PT and
CB24F-1/2-PT) contained a similar reinforced-concrete slab, but also were reinforced
with 3/8” 7-wire strands post-tensioned to apply 150 psi to the slab in the longitudinal
direction (Figs. 3.3-3.4). Specimen notation is given in Table 3.1.
22
Table 3.1 Test Matrix
Transverse Reinforcement
Beam ln/h type α[°]
Full Section Diagonals
AshactAshreq x
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
AshactAshreq y
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
Description
CB24F #3 @ 3" N.A. 1.34 (1.25)1
1.24 (1.09)1
Full section confinement ACI 318-08
CB24D #2 @ 2.5" #3 @ 2.5" 1.92 2.44 Diagonal
confinement ACI 318-05
CB24F-RC #3 @ 3" N.A. 1.34 (1.25)1
1.24 (1.09)1
Full section conf.
w/ RC slab ACI 318-08
CB24F-PT #3 @ 3" N.A. 1.34 (1.25)1
1.24 (1.09)1
Full section conf.
w/ PT slab ACI 318-08
CB24F-1/2-PT
2.4 residential 15.7
#3 @ 6" N.A. 0.67 (0.63)1
0.62 (0.55)1
Full section conf.
(reduced) w/ PT slab
ACI 318-08
CB33F #3 @ 3" N.A. 1.34 (1.25)1
1.26 (1.06)1
Full section confinement ACI 318-08
CB33D
12.3
#2 @ 2.5" #3 @ 2.5" 1.92 2.44 Diagonal
confinement ACI 318-05
FB33
3.33 office
0 #3 @ 3” N.A. - - Frame beam
with 6-#6 straight bars
1Full scale prototypes
23
Section A-ASection A-A Section A-ASection A-A
(a)
(b) (c)
Section A-ASection A-A Section A-ASection A-A
(a)
(b) (c)
Figure 3.1 Test beam geometries (ln/h = 2.4) full section confinement: (a) CB24F,
CB24F-RC, CB24F-PT, CB24F-1/2-PT elevation; (b) CB24F cross section;
and (c) CB24F-RC, CB24F-PT, CB24F-1/2-PT cross section. (Dimensions
are inches. 1in = 25.4mm)
24
(a)(a)
(b)(b)
Figure 3.2 Slab geometry and reinforcement for CB24F-RC, CB24F-PT, and CB24F-
1/2-PT: (a) Elevation view; and (b) plan view. (Dimensions are inches. 1in
= 25.4mm)
25
(a)(a)
(b)(b)
Figure 3.3 Slab geometry and PT reinforcement for CB24F-PT and CB24F-1/2-PT: (a)
Plan view; and (b) photo of post-tensioning load application. (Dimensions
are inches. 1in = 25.4mm)
26
Section B-BSection B-B
Figure 3.4 Test beam geometries (ln/h = 2.4) diagonal confinement (from left): (a)
CB24D elevation; and (b) cross section, with diagonal bundle (Dimensions
are inches. 1in = 25.4mm)
Section C-CSection C-C
Figure 3.5 Test beam geometries (ln/h = 3.33) full section confinement (from left): (a)
CB33F elevation; and (b) cross-section (Dimensions are inches. 1in =
25.4mm)
27
Section D-DSection D-D
Figure 3.6 Test beam geometries (ln/h = 3.33) diagonal confinement (from left): (a)
CB33D elevation; and (b) cross-section, with diagonal bundle (Dimensions
are inches. 1in = 25.4mm)
Section E-ESection E-ESection E-ESection E-E
Figure 3.7 Test beam geometries (ln/h = 3.33) frame beam (from left): (a) FB33
elevation; and (b) cross-section. (Dimensions are inches. 1in = 25.4mm)
28
(a)(a)
(b)(b)
(c)(c)
(d)(d)
(e)(e)
(f)(f)
(g)(g)
Figure 3.8 Photographs of test specimen construction: (a) CB24F beam construction;
(b) CB24F-1/2-PT beam construction; (c) CB24D beam construction; (d)
CB33F beam construction; (e) CB33D beam construction; (f) CB24F-RC
beam and slab construction; and (g) CB24F-PT beam elevation
29
3.2 Material Properties
Material samples were taken and tested in order to determine representative properties for
both concrete compressive strength and steel tensile strengths. Concrete cylinders were
tested to determine f’c for each test specimen on the day of testing. Concrete cylinders
were tested both in the UCLA material testing laboratory as well as at Twining Testing
Labs in Long Beach, CA, in order to provide redundancy, and to help avoid errors in the
material testing process. Rebar coupons were tested in order to determine yield and
ultimate tensile strengths for steel in the coupling beam specimens. Rebar in each
specimen was taken from the same batch to ensure consistency from test to test. These
material properties are summarized in Table 3.2.
30
Table 3.2 Material Properties
Beam f’c[psi] fy[psi] fu[psi]
CB24F 6850
CB24D 6850
CB24F-RC 7305
CB24F-PT 7242
CB24F-1/2-PT 6990
CB33F 6850
CB33D 6850
FB33 6000
70000 90000
31
3.3 Test Setup
All beam specimens were tested in the UCLA Structural/Earthquake Engineering
Research Laboratory. The setup shown in Figure 3.9, where the test specimen was placed
in a vertical position with end blocks simulating wall boundary zones at each beam end,
was used for all tests. The top and bottom surfaces of the end blocks were grouted and
post-tensioned to the steel reaction frame (top) and to the laboratory strong floor (bottom)
to minimize slip between the surfaces as well as to provide for fixed end conditions. Two
vertical hydraulic actuators on each side of the beam specimen were used to ensure zero
rotation at the top of the specimen, while maintaining constant (zero) axial force in the
beam.
Figure 3.9 Laboratory test setup
32
The lateral load was applied via a horizontal actuator, with the line of action of
the actuator force passing through the mid-span (mid-height) of the test specimen to
achieve zero moment at the beam mid-span. To prevent out-of-plane rotation or twisting,
a sliding truss system was attached between the steel reaction frame and the reinforced
concrete reaction wall.
3.4 Loading Protocol
The testing procedure included load-controlled and displacement-controlled cycles (Fig.
3.10). Load-control was performed at 0.125, 0.25, 0.50, and 0.75Vy, where
2y y nV M l= to ensure that the load-displacement behavior prior to yield was captured.
Based on use of nominal material properties, Vy was estimated as 120 and 100 kips for
the residential and office beams, respectively.
Beyond 0.75Vy, displacement-control was used in increments of percent chord
rotation (θ), defined as the relative lateral displacement over the clear span of the beam
(Δ) divided by the beam clear span (ln). The applied chord rotation excluded any
contributions due to translations (sliding) and rigid rotation of the bottom support block,
as these deformations were measured during the test and excluded in real-time. Three
cycles were applied at each load increment for load-controlled testing, and three cycles
were applied in displacement-control at each increment of chord rotation up to 3%, which
is approximately the allowable collapse prevention (CP) limit state for ASCE 41-06. Two
cycles were applied at each increment of chord rotation exceeding 3%.
33
-100
-50
0
50
100
Late
ral L
oad
[k]
ln/h = 2.4ln/h = 3.33 (a)
-100
-50
0
50
100
Late
ral L
oad
[k]
ln/h = 2.4ln/h = 3.33 (a)
-12
-8
-4
0
4
8
12
Rota
tion
[%]
(b)-12
-8
-4
0
4
8
12
Rota
tion
[%]
(b)
Figure 3.10 Loading protocol: (a) Load-controlled; and (b) Displacement-controlled.
(1k = 4.45kN)
3.5 Instrumentation
Each of the test specimens was heavily instrumented. Linear Variable Differential
Transformers (LVDTs) were placed on the specimen to measure key deformation
quantities; Figures 3.11 and 3.12 show the sensor layouts for the different test specimens.
34
Wire Potentiometers (WPs) were used to measure large displacements (>5”), used to
determine beam chord rotations. As well, strain gauges were placed on diagonal,
transverse, and longitudinal reinforcement, as well as on the surface of the concrete slab
(Fig. 3.13-3.16). LVDTs were TransTek models 0242-0000, 0243-0000, and 0244-0000
for different strokes of 0.5”, 1”, and 2” respectively. SGs were Texas Measurements
model YEFLA-5-005LE. WPs were UniMeasure model P1010-20 with stroke of 20”.
Longitudinal LVDTs (#1-12) measured flexural deformations, diagonal LVDTs
(#13-24) measured shear deformations, longitudinal LVDTs (#54-57) at the beam-wall
interface measured slip/extension deformations, transverse LVDTs (#50-53) at the beam-
wall interface measured any sliding of the beam with respect to the wall, longitudinal
LVDTs (#40-41) spanning the full length of the beam measured axial elongation of the
beam, and all other LVDTs (#30-33 and AC-1,2) were used to measure the relative tip
displacement of the beam. The relative tip displacement was calculated using the
following relationship between sensor data:
32 33
1 2 30 31 32 332 2 n
AC AC DC DC DC DC lL −
⎛ ⎞+ + −⎛ ⎞ ⎛ ⎞Δ= − − ×⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
(Eq. 3.2)
where AC1 and AC2 represent the data from AC-LVDT 1 and 2 respectively; DC30,
DC31, DC32, and DC33 represent the displacement data from DC-LVDTs 30, 31, 32,
and 33 respectively; and L32-33 represents the horizontal distance between DC-LVDTs 32
and 33. The first term in the equation is the absolute transverse displacement of the beam
35
tip (center of the cross-section width). The second term in the equation is the absolute
transverse displacement of the base of the beam (center of the cross-section width). The
third term is the tip displacement due to rigid rotation of the beam.
Data from several different sensors were used to calculate values plotted in all
results. Individual sensor data are available from the authors. Eventually, the data will be
uploaded to the Network for Earthquake Engineering Simulation (NEES) data repository.
Data also will be stored on a data server at UCLA.
36
(a)(a)
(b)(b)
Figure 3.11 Sensor layout for: (a) CB24F and CB24D, and (b) CB33F, CB33D, and
FB33
37
(a)(a)
(b)(b)
Figure 3.12 Sensor layout for (a) CB24F-RC, and (b) CB24F-PT and CB24F-1/2-PT
38
Figure 3.13 Strain gauge layout for CB24F and CB33F. SG 12 and SG 14 are on
horizontal crossties
39
Figure 3.14 Strain gauge layout for CB24D and CB33D. SG 15 and SG 16 are located
on horizontal crossties
40
Figure 3.15 Strain gauge layout for CB24F-RC, CB24F-PT, and CB24F-1/2-PT. SG 12
and SG 16 are located on horizontal crossties
41
Figure 3.16 Strain gauge layout for FB33. SG 12 and SG 16 are located on horizontal
crossties
42
Chapter 4 Experimental Results and Discussion
Results from the tests are presented and discussed. Overall load-displacement relations
are compared to assess the impact of providing full section confinement as opposed to
confinement around the diagonals for both residential- and office-use beams. The role of
transverse reinforcement is examined by comparing load-displacement relations for the
beams, including one beam with only one-half of the code-required transverse
reinforcement. Other comparisons are made that examine the effect of the floor slab (both
reinforced concrete (RC) and post-tensioned reinforced concrete (PT)) on the beam load-
deformation response, including the effective elastic bending stiffness at yield as well as
the influence of scale on the test results. Table 4.1 summarizes the calculated strengths,
and Tables 4.2-3 summarize the actual strengths and deformations of each test specimen
at major points.
43
Table 4.1 Summary of predicted member strengths
Beam Mn+ [in-k] Mn
- [in-k] V@Mn [k]@
'n
c cv
V Mf A
Vn(ACI)[k] ( )'
n
c cv
V ACIf A
CB24F 2850 2850 158.3 10.65 136.3 9.15
CB24D 2850 2850 158.3 10.65 136.3 9.15
CB24F-RC 2890 (3550)1
2890 (3350)1
160.6 (191.7)1
10.45 (12.50)1 136.3 8.87
CB24F-PT 3160 (3960)1
3160 (3625)1
175.6 (210.7)1
11.45 (13.75)1 136.3 8.90
CB24F-1/2-PT
3145 (3940)1
3145 (3610)1
174.7 (209.7)1
11.61 (13.90)1 136.3 9.06
CB33F 3615 3615 120.5 6.77 107.8 6.03
CB33D 3615 3615 120.5 6.77 107.8 6.03
FB33 1450 1450 48.3 2.89 - -
1Calculations that consider the impact of the slab [1 in-k = 113 mm-kN, 1 in = 25.4 mm, 1 k = 4.45 kN]
44
Table 4.2 Summary of experimental force results
Beam Vy [k] '
y
c cv
Vf A
Vave [k] 'ave
c cv
Vf A
Vmax [k] max
'c cv
Vf A
CB24F 121.3 8.14 154.9 10.40 171.0 11.48
CB24D 128.8 8.65 150.7 10.12 159.2 10.69
CB24F-RC 147.2 9.57 181.0 11.77 190.8 12.41
CB24F-PT 163.2 10.65 198.9 12.98 211.8 13.82
CB24F-1/2-PT 158.1 10.51 182.4 12.12 189.6 12.60
CB33F 107.7 6.03 118.3 6.62 124.0 6.94
CB33D 95.94 5.37 114.7 6.42 120.6 6.75
FB33 47.86 2.86 56.3 3.37 58.1 3.48
Note: Vave is defined as the average shear force resisted by the beam between the yield point and the onset of significant lateral strength degradation. [1 in-k = 113 mm-kN, 1 in = 25.4 mm, 1 k = 4.45 kN]
45
Table 4.3 Summary of experimental displacement results
Beam Δy [in] θy [%] Ieffy/Ig [%] Δ@Vmax [in] Δu [in] θu [%] μ
CB24F 0.360 1.00 10.8 1.08 3.42 9.50 9.50
CB24D 0.363 1.01 11.4 2.16 3.15 8.75 8.66
CB24F-RC 0.362 1.00 13.1 2.16 3.69 10.25 10.25
CB24F-PT 0.361 1.00 14.5 2.16 3.24 9.00 9.00
CB24F-1/2-PT 0.365 1.01 14.0 1.08 2.97 8.25 8.17
CB33F 0.600 1.00 15.4 1.80 5.40 9.00 9.00
CB33D 0.601 1.00 13.7 3.60 5.25 8.75 8.75
FB33 0.306 0.51 13.8 1.20 3.00 5.00 9.80
Note: Δu is the displacement at which significant lateral strength degradation occurs (0.8Vave). [1 in-k = 113 mm-kN, 1 in = 25.4 mm, 1 k = 4.45 kN]
46
4.1 Detailing
This section briefly provides a discussion of results based on the comparison between
different detailing configurations. Namely, there are comparisons of the ACI 318-05 and
ACI 318-08 detailing provisions, as well as comparisons of the ACI 318-08 full section
confinement provision to possible reductions in the amount of transverse necessary.
4.1.1 Full Section vs. Diagonal Confinement
Figure 4.1 is a plot of the load-deformation response of CB24F and CB24D, and is
representative of the general behavior of all specimens tested. The yield load for both
beams occurred at approximately 1% beam chord rotation, and significant strength
degradation began at approximately 8% total beam chord rotation. Strength and
deformation characteristics for all beams are summarized in Tables 4.1-4.3.
Load-deformation responses of CB24F and CB24D are very similar over the full
range of applied rotations (Fig. 4.1). Notably, both beams achieve large rotation (~8%)
without significant degradation in the lateral load carrying capacity, and the beams
achieve shear strengths of 1.25 and 1.17 times the ACI nominal strength (Table 4.1-4.3).
The shear strength of CB24D degraded rapidly at around 8% rotation, whereas CB24F
degraded more gradually, maintaining a residual shear capacity of ~80% of Vave at
rotations of 10%. Vave is defined as the average shear force resisted by the beam between
the yield point and the onset of significant lateral strength degradation.
47
Figure 4.2 is a plot of load vs. rotation relations for the 3.33 aspect ratio beams
with full section confinement (CB33F) vs. diagonal confinement (CB33D). Similar to the
2.4 aspect ratio beams, Figure 4.2 reveals that the beams have similar strength (Table
4.1), stiffness, deformation, and damage (Table 4.4) characteristics.
The test results presented in Figures 4.1-4.2 indicate that the full section
confinement option of ACI 318-08 provides equivalent, if not improved performance,
compared to confinement around the diagonals per ACI 318-05.
-4.32 -2.16 0 2.16 4.32Relative Displacement [in]
-200
-100
0
100
200
Late
ral L
oad
[k]
-12 -6 0 6 12Beam Chord Rotation [%]
-890
-445
0
445
890
Late
ral L
oad
[kN
]
CB24FCB24D
Vn (ACI)
Vn (ACI)
Figure 4.1 Cyclic load-deformation: CB24F vs. CB24D (1in = 25.4mm)
48
-6 -3 0 3 6Relative Displacement [in]
-150
-100
-50
0
50
100
150
Late
ral L
oad
[k]
-10 -5 0 5 10Beam Chord Rotation [%]
-670
-335
0
335
670
Late
ral L
oad
[kN
]CB33FCB33D
Vn (ACI)
Vn (ACI)
*
**
* Stroke of controlling sensor exceeded
** Stroke of LVDT exceeded
Figure 4.2 Cyclic load-deformation: CB33F vs. CB33D (1in = 25.4mm)
4.1.2 Full vs. Half Confinement
The transverse reinforcement used for CB24F-1/2-PT was one-half that used for CB24F-
PT to assess the impact of using less than the code-required transverse reinforcement
given that the requirements of ACI 318-08 S21.6.4 are based on column requirements.
Figure 4.3 plots load-deformation responses and reveals similar loading and unloading
relations up to 3% total rotation, which approximately corresponds to the Collapse
Prevention limit state per ASCE 41-06. At higher rotations (θ ≥ 4%), modest strength
degradation is observed for CB24F-1/2-PT, whereas the strength of CB24F-PT continues
49
to increase slightly; however, both beams achieve rotations of ~8% before significant
lateral strength degradation (< 0.8Vave).
The results indicate that the one-half scale coupling beams tested with ACI 318-
08 detailing are generally capable of achieving total rotations exceeding 8%, whereas
ASCE 41 limits plastic rotation to 3% without strength degradation and 5% with 20%
strength degradation. The potential influence of scale on the test results is discussed later
(Section 5.3). The test results indicate that there is little difference in load-deformation
response between CB24F-PT and CB24F-1/2-PT; therefore, the potential to reduce the
quantity of required transverse reinforcement exists, but requires further study since only
one beam test was conducted. A discussion of crack patterns and deformation
characteristics is provided in §4.4.
50
-5 -2.5 0 2.5 5Relative Displacement [in]
-200
-100
0
100
200
Late
ral L
oad
[k]
-14 -7 0 7 14Beam Chord Rotation [%]
-980
-490
0
490
980
Late
ral L
oad
[kN
]
CB24F-PTCB24F-1/2-PT Vn (ACI)
Vn (ACI)
Figure 4.3 Cyclic load-deformation: CB24F-PT vs. CB24F-1/2-PT (1in = 25.4mm)
4.2 Slab Impact
Four beams with aspect ratio of 2.4 were tested to systematically assess the impact of a
slab on the load-deformation responses. CB24F did not include a slab, whereas CB24F-
RC included an RC slab, and CB24F-PT and CB24F-1/2-PT included PT slabs (with 150
psi of prestress). Comparing the load-displacement responses of CB24F vs. CB24F-RC,
Figure 4.5 reveals that the slab increases shear strength by 17% (155 k to 181 k);
however, this strength increase can be taken into account by considering the increase in
nominal moment strength due to the presence of the slab, i.e. slab concrete in
compression at the beam-wall interface at one end, and slab tension reinforcement at the
beam-wall interface at the other end (Figure 4.4 and Table 4.1). For example,
51
consideration of the slab produces increases of approximately 20% in the nominal
moment capacities, which also provide similar increases in beam shear (since yielding of
diagonal reinforcement limits the shear forces on the beams). The results indicate that the
higher test shear strength observed is primarily due to the increase in nominal moment
capacity when a slab is present.
The presence of a slab, and in particular, a post-tensioned slab, might impact the
load-deformation behavior by restraining the axial growth along the member length.
Figure 4.6 plots the axial growth of CB24F vs. CB24F-RC and reveals that the axial
growth is very similar for the two tests. Both beams grow approximately one inch over
the course of the test, with relatively large cracks observed at the beam-wall interface.
Strength degradation for CB24F is noted at 8%, due to the buckling and eventual fracture
of the diagonal bars, leading to axial shortening, whereas the axial extension in CB24F-
RC remains stable over the entire test due to the presence of the slab.
52
0 0.00216 0.00432Curvature [in-1]
0
1000
2000
3000
4000
Mom
ent [
in-k
]
No SlabSlab
Mn+ Mn
-
0 0.0005 0.001 0.0015 0.002 0.0025Curvature [in-1]
No SlabSlab
Mn+ (slab) = 3550 in-k
Mn+(no slab) = 2850 in-k
Mn- (slab) = 3350 in-k
Mn-(no slab) = 2850 in-k
0 0.00216 0.00432Curvature [in-1]
0
1000
2000
3000
4000
Mom
ent [
in-k
]
No SlabSlab
Mn+ Mn
-
0 0.0005 0.001 0.0015 0.002 0.0025Curvature [in-1]
No SlabSlab
Mn+ (slab) = 3550 in-k
Mn+(no slab) = 2850 in-k
Mn- (slab) = 3350 in-k
Mn-(no slab) = 2850 in-k
Figure 4.4 Moment curvature analysis summary (BIAX) for beam with and without
slab (clockwise from top left): (a) Beam cross section with and without slab;
(b) beam elevation with positive and negative moment capacities shown; (c)
plot of Mn- vs. curvature; and (d) plot of Mn
+ vs. curvature
53
-5 -2.5 0 2.5 5Relative Displacement [in]
-200
-100
0
100
200
Late
ral L
oad
[k]
-14 -7 0 7 14Beam Chord Rotation [%]
-980
-490
0
490
980
Late
ral L
oad
[kN
]
CB24FCB24F-RC Vn (ACI)
Vn (ACI)
Figure 4.5 Cyclic load-deformation: CB24F vs. CB24F-RC (1in = 25.4mm)
-5 -2.5 0 2.5 5Relative Displacement [in]
-0.4
0
0.4
0.8
1.2
Axia
l elo
ngat
ion
[in]
-14 -7 0 7 14Beam Chord Rotation [%]
-1
0
1
2
3
Axi
al elo
ngat
ion
[cm
]
CB24FCB24F-RC
Figure 4.6 Axial elongation vs. rotation: CB24F vs. CB24F-RC (1in = 25.4mm)
54
Load-deformation responses for CB24F-RC vs. CB24F-PT are compared in
Figure 4.7 and display similar overall behavior, with CB24F-PT experiencing higher
shear forces (13.0 'c cwf A ) than CB24F-RC (11.8 'c cwf A ). This increase in strength is
primarily due to the axial force applied to the specimen by the tensioned strands, which
provided approximately 150 psi stress to the slab and increased the nominal moment
strength (Table 4.1). Between 8% and 10% rotations, strength degradation is more
pronounced for CB24F-PT than CB24F-RC, with 30% reduction for CB24F-PT vs. 10%
for CB24F-RC, possibly due to the presence of pre-compression.
A plot of axial elongation of CB24F-RC vs. CB24F-PT, (Fig. 4.8), indicates that
the PT slab with 150 psi prestress grows 30-40% less than the RC slab. As well, the PT
slab, like the RC slab in CB24F-RC, helps to maintain the axial integrity of the beam for
rotations exceeding 6%.
55
-5 -2.5 0 2.5 5Relative Displacement [in]
-200
-100
0
100
200
Late
ral L
oad
[k]
-14 -7 0 7 14Beam Chord Rotation [%]
-980
-490
0
490
980
Late
ral L
oad
[kN
]CB24F-RCCB24F-PT Vn (ACI)
Vn (ACI)
Figure 4.7 Cyclic load-deformation: CB24F-RC vs. CB24F-PT (1in = 25.4mm)
-5 -2.5 0 2.5 5Relative Displacement [in]
-0.4
0
0.4
0.8
1.2
Axi
al elo
ngat
ion
[in]
-14 -7 0 7 14Beam Chord Rotation [%]
-1
0
1
2
3
Axi
al elo
ngat
ion
[cm
]
CB24F-RCCB24F-PT
Figure 4.8 Axial elongation vs. rotation: CB24F-PT vs. CB24F-RC (1in = 25.4mm)
56
4.3 Frame Beam
FB33 was tested to assess the impact of providing straight bars as flexural reinforcement
instead of diagonal bars in beams with relatively low shear stress demand (< 4.0 'cf ).
A plot of load vs. deformation for FB33 (Fig. 4.9) indicates that plastic rotations greater
than 4% can be reached prior to strength degradation. These results correspond well with
prior test results (Xiao et. al., 1999) on similarly sized beams. Specimen HB3-6L-T100 at
aspect ratio 3.0 achieved maximum shear stresses of about 4.7 'cf and plastic chord
rotations greater than 3.5% (Fig. 4.10(a)). Specimen HB4-6L-T100 at aspect ratio 4.0
reached maximum shear stresses of about 3.7 'cf and plastic chord rotations greater
than 4.5% (Fig. 4.10(b)).
Compared with CB33F and CB33D (Fig. 4.2), FB33 experiences pinching in the
load-deformation plot, indicating that less energy is dissipated. As well, the beams with
diagonal reinforcement exhibited higher ductility, reaching plastic rotations exceeding
7% prior to strength degradation. However, for beams that are expected to experience
shear forces less than 5.0 'c cwf A , frame beams with straight bars can provide
significant ductility (θp > 4%), and are much easier to construct than diagonally-
reinforced beams. Therefore, adding a shear stress limit of 5.0 'cf for conventionally-
reinforced coupling beams with aspect ratio between 2 and 4 to ACI 318-08 21.9.7 might
be prudent. At a minimum, ACI 318 should add commentary to note the significant
57
difference in deformation capacity between diagonally- and longitudinally-reinforced
coupling beams.
-5 -2.5 0 2.5 5Relative Displacement [in]
-80
-40
0
40
80La
tera
l Loa
d [k
]-8 -4 0 4 8
Beam Chord Rotation [%]
-356
-178
0
178
356
Late
ral L
oad
[kN
]
Figure 4.9 Cyclic load-deformation: FB33 (1in = 25.4mm)
58
-8 -6 -4 -2 0 2 4 6 8Beam Chord Rotation [%]
-1.5
-1
-0.5
0
0.5
1
1.5
V/V
nFB33Xiao ln/h=3(a)
-8 -6 -4 -2 0 2 4 6 8Beam Chord Rotation [%]
-1.5
-1
-0.5
0
0.5
1
1.5
V/V
nFB33Xiao ln/h=3(a)
-8 -6 -4 -2 0 2 4 6 8Beam Chord Rotation [%]
-1.5
-1
-0.5
0
0.5
1
1.5
V/V
n
FB33Xiao ln/h=4(b)
-8 -6 -4 -2 0 2 4 6 8Beam Chord Rotation [%]
-1.5
-1
-0.5
0
0.5
1
1.5
V/V
n
FB33Xiao ln/h=4(b)
Figure 4.10 Cyclic load-deformation: (a) FB33 vs. Xiao HB3-6L-T100, and (b) FB33 vs.
Xiao HB4-6L-T100
59
4.4 Damage
The following sections provide quantitative and qualitative descriptions of damage of the
test specimens. Crack widths are detailed and photographs are provided for damage at
peak deformations as well as residual damage at zero deformation following large
deformation cycles.
4.4.1 Damage at peak deformation
All of the test specimens exhibited similar damage states and deformation characteristics.
Each specimen had hairline diagonal cracking (<1/64”) at beam chord rotations less than
1%, and only specimens not detailed with full section confinement experienced large
shear cracks (>1/8”) at 6% rotation. However, each beam exhibited fairly large flexural
and slip/extension cracking (>1/4”) prior to 3% rotation at the beam-wall interface.
Figure 4.11 is a plot of the relative contributions of shear, flexure, and
slip/extension deformations (and a figure showing the definition of these deformations) to
the overall deformation of CB24F, and is representative of the behavior of all beams
tested, except for the beam with one-half of the required transverse reinforcement
CB24F-1/2-PT. This plot shows that shear deformations account for less than 20% of the
total beam chord rotation (at peak value), while flexure and slip/extension each account
for approximately 40% of beam chord rotation at low rotations (<1%). At high rotations
(>3%), slip/extension accounts for nearly 80% of measured peak beam chord rotation.
60
Lateral strength degradation began with the buckling of the diagonal reinforcement,
followed by the fracture of both the diagonal rebar and the hoops/crossties at the beam-
wall interface. Figure 4.12 plots the relative contributions of shear, flexure, and
slip/extension deformations to the overall deformation of CB24F-1/2-PT. The plot shows
that shear deformations represented closer to 40% of the total deformations, while
flexural deformations represented around 15-20%. This is due to the fact that there is less
transverse steel around the entire cross-section to resist diagonal cracking.
0 0.01 0.02 0.03 0.04Beam Chord Rotation [rad.]
0
20
40
60
80
100
% C
ontr
ibut
ion
FlexureSlip/Ext.Shear
Flexure
Shear
Slip/Extension
Flexure
Shear
Slip/Extension
Figure 4.11 (a) Deformation contributions for CB24F; and (b) Definition of different
deformation types
61
0 0.01 0.02 0.03 0.04Rotation [% drift]
0
20
40
60
80
100
% C
ontri
butio
n
FlexureSlipShear
Figure 4.12 Deformation contributions for CB24F-1/2-PT
Figures 4.13 and 4.14 are photos of CB24F and CB24D at the peak of every
displacement stage between 0.075% and 10% total rotations, respectively, and reveal that
maximum diagonal crack widths for CB24F were less than 0.02” and flexural crack
widths of 0.08 and 0.125” were measured at 3 and 6% rotations (Table 4.4). In general,
diagonal crack widths for CB24D were larger than for CB24F, possibly due to the
reduced transverse reinforcement around the full section. The results indicate beams
detailed with full section confinement might require fewer repairs than beams detailed
with diagonal confinement following an earthquake.
Diagonal crack widths for CB24F-1/2-PT (Figs. 4.17-18) are much larger than
those observed for CB24F-PT (Figs. 4.15-16), especially for rotations exceeding 6%. At
4% rotation, 1/16” diagonal cracks were noted in CB24F-1/2-PT, whereas diagonal
cracks were still hairline in CB24F-PT. Beyond 4% rotation, for CB24F-1/2-PT, spalling
62
of cover concrete was noted, with 1/4” diagonal cracks noted at 6% rotation; buckling
and fracture of reinforcement, and crushing of the core concrete were noted for rotations
between 8 and 10%. In contrast, minimal damage was observed for CB24F-PT (Figs.
4.17-18), with hairline diagonal cracks and flexural crack widths of less than 1/4”, with
most of the rotation due to rebar slip/pullout at the beam-wall interface (approximately
1/2” at 6% rotation). Crack widths for all specimens at peak deformations of 1%, 3%, and
6% rotations are summarized in Table 4.4. More photos of damage for all specimens are
provided in Appendix A.
Table 4.4 Crack widths at peak rotation
1% 3% 6% Beam
Slip/ext Flexure Shear Slip/ext Flexure Shear Slip/ext Flexure Shear
CB24F 0.125 0.065 hairline 0.400 0.080 hairline 0.750 0.125 0.015
CB24D 0.125 0.095 hairline 0.375 0.125 0.016 0.500 0.250 0.125
CB24F-RC 0.095 0.045 hairline 0.500 0.125 0.016 0.500 0.375 0.065
CB24F-PT 0.065 0.030 hairline 0.250 0.190 hairline 0.500 0.250 hairline
CB24F-1/2-PT 0.065 0.015 hairline 0.375 0.190 0.031 0.625 0.375 0.250
CB33F 0.125 0.065 hairline 0.315 0.065 hairline 0.500 0.250 0.015
CB33F 0.125 0.065 hairline 0.250 0.125 0.016 0.500 0.190 0.125
FB33 0.060 0.030 hairline 0.250 0.250 0.125 - - -
Note: All measurements in inches. [1 in = 25.4 mm]
63
Rotation = 0.0075
Rotation = 0.01
Rotation = 0.02
Rotation = 0.03
Rotation = 0.04
Rotation = 0.06
Rotation = 0.08
Rotation = 0.10
Figure 4.13 CB24F damage photos: 0.75% - 10.0% rotation
64
Rotation = 0.0075
Rotation = 0.01
Rotation = 0.02
Rotation = 0.03
Rotation = 0.04
Rotation = 0.06
Rotation = 0.08
Rotation = 0.10
Figure 4.14 CB24D damage photos: 0.75% - 10.0% rotation
65
Rotation = 0.0075
Rotation = 0.01
Rotation = 0.02
Rotation = 0.03
Figure 4.15 CB24F-PT damage photos: (a) 0.075% rotation; (b) 1% rotation; (c) 2%
rotation; and (d) 3% rotation
66
Rotation = 0.04
Rotation = 0.06
Rotation = 0.08
Rotation = 0.10
Figure 4.16 CB24F-PT damage photos: (a) 4% rotation; (b) 6% rotation; (c) 8% rotation;
and (d) 10% rotation
67
Rotation = 0.0075
Rotation = 0.01
Rotation = 0.02
Rotation = 0.03
Figure 4.17 CB24F-1/2-PT damage photos: (a) 0.075% rotation; (b) 1% rotation; (c) 2%
rotation; and (d) 3% rotation
68
Rotation = 0.04
Rotation = 0.06
Rotation = 0.08
Rotation = 0.10
Figure 4.18 CB24F-1/2-PT damage photos: (a) 4% rotation; (b) 6% rotation; (c) 8%
rotation; and (d) 10% rotation
4.4.2 Residual damage at zero deformation
The degree of damage at zero applied chord rotation is of interest since it is a better
measure of potential repair costs versus measured crack widths for peak loads. Pictures
showing the residual damage of each beam after each rotation level are also shown in
Figures 4.19-22 and Appendix A. Understanding of expected residual damage levels is
69
important for design and consulting engineers as this is the damage that is likely to be
seen once an earthquake has stopped shaking. The information obtained based on residual
damage patterns can be used to develop fragility relations, which will be discussed in
Chapter 6. These fragility relations can be used to help identify expected damage levels
and subsequent repair procedures in specific components in a building following a
seismic event.
Prior to 2% beam chord rotation, all beam specimens showed similar residual
damage, with small cracks (< 1/16”) focused at the beam-wall interface. At 3% rotation,
CB24F and CB24D both showed light spalling at the ends of the beam, with residual
cracks exceeding 1/16”. Test specimens with slabs began to show cracking in the slab,
and CB24F-1/2-PT began to have spalling at the beam ends. After 6% rotation, CB24F
and CB24D showed significant spalling at beam ends, and had fairly large residual
cracking (> 1/8”). Beams with slabs showed large cracks in the slab, and CB24F-1/2-PT
showed significant spalling of concrete along the beam length, and in the slab. Strength
loss due to fracture of diagonal reinforcement and crushing of concrete followed after 8%
rotation for most test specimens, with CB24F-RC reaching 10% rotation prior to
significant strength loss.
70
After Rotation = 0.01
After Rotation = 0.02
After Rotation = 0.03
After Rotation = 0.04
After Rotation = 0.06
After Rotation = 0.08
Figure 4.19 Residual (zero displacement) damage photos (CB24F) after cycles of
rotations 1.0%-8.0%
71
After Rotation = 0.01
After Rotation = 0.02
After Rotation = 0.03
After Rotation = 0.04
After Rotation = 0.06
After Rotation = 0.08
Figure 4.20 Residual (zero displacement) damage photos (CB24D) after cycles of
rotations 1.0%-8.0%
72
After Rotation = 0.01
After Rotation = 0.02
After Rotation = 0.03
After Rotation = 0.04
After Rotation = 0.06
After Rotation = 0.08
Figure 4.21 Residual (zero displacement) damage photos (CB24F-PT) after cycles of
rotations 1.0%-8.0%
73
After Rotation = 0.01
After Rotation = 0.02
After Rotation = 0.03
After Rotation = 0.04
After Rotation = 0.06
After Rotation = 0.08
Figure 4.22 Residual (zero displacement) damage photos (CB24F-1/2-PT) after cycles
of rotations 1.0%-8.0%
74
4.5 Summary
This chapter presented the results of tests on eight coupling beams subjected to reversed
cyclic loading. The results indicate that the new detailing provision for diagonally-
reinforced coupling beams in ACI 318-08 §21.9.7.4(d) yields performance that is at least
equivalent to the performance of beams detailed according to the old detailing provision
ACI 318-05 §21.7.7.4(c). Including a reinforced concrete slab on the test specimen was
found to provide a nominal increase in shear strength, corresponding to the increase in
flexural capacity of approximately 20%. Investigating damage in the test specimens
showed that slip/extension deformations accounted for a large portion of the overall
rotation (~50%), with flexural and shear deformations accounting for similar amounts.
Coupling beams with conventional reinforcement were shown to perform well, reaching
total chord rotations of 5.0%, corresponding to displacement ductility of 9.8, prior to
strength degradation. For beams with low shear demand ( )'5.0 cf< and relatively high
aspect ratio (ln/h > 3.0) diagonal bar placement can be particularly difficult; therefore,
conventional reinforcement is a useful alternative.
75
Chapter 5 Simplified Component Modeling
Typical modeling procedures for coupling beams are discussed and results generated with
models are compared to test results. Specifically, models for effective secant stiffness at
yield are presented to provide a direct comparison between typical parameters used by
engineers and values obtained via testing. As well, the impact of scaling test specimens is
investigated to allow test results to be applied to full-scale models. Based on these
studies, backbone relations are fit to all test results and modified to represent the behavior
of the beam at full-scale. These backbone relations can be used directly in computer
software; representative model load-deformation results are compared for one of the
beams tested.
5.1 Effective Stiffness
Elastic analysis approaches require estimation of the effective elastic bending and shear
stiffness values. In the Federal Emergency Management Agency’s Prestandard for the
Seismic Rehabilitation of Buildings (FEMA 356), stiffness values of
0.5 c gE I and 0.4 c cwE A are recommended for bending and shear, respectively. ASCE 41-
76
06 including Supplement #1 incorporates a lower value for effective stiffness of 0.3 c gE I ,
with a mean value obtained from tests of 0.2 c gE I (Elwood et. al., 2007). The New
Zealand Concrete Structures Standard Part 1 – The Design of Concrete Structures (NZS-
3101 1995) includes an equation to estimate the effective bending stiffness that depends
on the expected ductility demand as:
2( / )
c gc eff
n
A E IE I
B C h l×
=+ ×
(Eq. 5.1)
where A, B, and C vary with ductility [A=1.0 and 0.40; B=1.7 and 1.7; C=1.3 and 2.7; for
ductility=1.25 and 6.0]. For beams with aspect ratio ln/h = 2.4, Equation 5.1 yields a
beam with effective elastic stiffness of around fifty percent of the gross section
stiffness, 0.5 c gE I , whereas for a ductility ratio of 6, the effective (secant) stiffness drops
to eighteen percent of the gross section properties, 0.18 c gE I . All of these values are
summarized and compared with the test results in Figure 5.1. The test results are average
effective secant stiffness values for the test specimens at aspect ratios 2.4 and 3.33, at
different ductility levels (defined as displacement normalized by yield displacement). The
test results indicate a much lower effective stiffness than that predicted by the NZS
relation (Ieff/Ig = ~0.05-0.12 for the test results vs. ~0.2 to 0.5 for NZS).
77
1 2 3 4Ln/h
0
0.2
0.4
0.6
I eff/
I g
μ=1.25μ=3.0μ=4.5μ=6.0
Figure 5.1 Effective stiffness plotted as a function of aspect ratio for various levels of
displacement ductility (NZS 3101-1995). Included on the plot are test results
at the corresponding ductility levels.
Figure 5.2 plots the secant stiffness normalized with respect to the concrete gross
section stiffness versus the chord rotation. Secant stiffness is calculated assuming fixed
end conditions according to: 3
12n
c effV lE I ×
=×Δ
. The initial stiffness of each residential beam
is approximately 0.25 c gE I , with an effective stiffness at the yield rotation (~1.0%
rotation) of 0.12 c gE I . Effective secant stiffness values corresponding to ASCE 41-06
limit states are approximately 0.15 c gE I at Immediate Occupancy (~0.6% rotation),
0.075 c gE I at Life Safety (~1.8% rotation), and 0.05 c gE I at Collapse Prevention (~3%
rotation). The effective stiffness ratio ( eff gI I ) does not vary significantly for the three
different configurations (Fig. 5.2), i.e. beam without slab (CB24F, CB24D), beam with
78
RC slab (CB24F-RC), and beam with PT slab (CB24F-PT, CB24F-1/2-PT). The initial
stiffness ratio for the beams with slabs is moderately higher (~25%) for rotations up to
about 2%; however, after significant flexural cracks form at the slab-wall interface,
generally at ~3% rotation, the stiffness ratio is nearly the same for all three test
configurations.
0 2 4 6Beam Chord Rotation [%]
0
0.1
0.2
0.3
I eff
/Ig
CB24F-PTCB24F-RCCB24F
Figure 5.2 Effective secant stiffness values derived from test results: ln/h = 2.4
The low secant stiffness ratios ( eff gI I ) relative to recommended values (Table
5.1) might imply that significant damage (cracking, concrete spalling) is required to
achieve these ratios. However, photos of beam damage, Figures 4.13-14 for the beams
without slabs, and Figures 4.15-18 for the beams with slabs, do not show significant
spalling and diagonal crack widths are limited to 1/32” even at 6% total rotation (Table
79
4.4); damage is concentrated at the beam-wall interface in the form of slip/extension
cracks. The photos also indicate that the quantity of beam transverse reinforcement is
sufficient to keep crack widths small for peak shear stresses as large as10.5 to 13.8 'cf .
The larger diagonal crack widths observed for CB24F-1/2-PT, with only one-half the
required transverse reinforcement, indicate that the quantity of transverse reinforcement
provided in CB24F, CB24F-RC, and CB24F-PT could likely be reduced moderately
without compromising deformation capacity. Current modeling of the load-deformation
response of coupling beams tends to focus on shear behavior (NZS 3101:2006); however,
for the 2.4 and 3.33 aspect ratio beams tested, flexural and slip/extension deformations at
and adjacent to the beam-wall interface generally accounted for more than 85% of the
total rotation.
It is important to note that axial deformations were not restrained in any way in
the tests; whereas redistribution of shear between walls might lead to axial compression
in coupling beams. This was not considered in the tests. However, two specimens,
CB24F-PT and CB24F-1/2-PT, were constructed with 150 psi post-tensioned strands as a
means to provide some restraint on axial growth. This was seen in Figure 4.8, which
plotted axial growth of CB24F-PT over the full-range of applied rotations, and showed
that the post-tensioning did restrain the axial deformations in the member. The impact of
this axial restraint on the effective stiffness is evident in Figure 5.2, which shows
moderate, not substantial, difference in stiffness (~0.15EcIg vs. 0.12EcIg at yield). A
recent study by Bower (2008) investigated the effect of axial restraint on stiffness and
ductility of diagonally-reinforced coupling beams using finite element analysis. The
80
results of this work indicate that the impact of axial restraint on the initial stiffness can be
substantial (as much as 100% difference); however, the impact on the system is minimal.
Of the various approaches noted above for estimating the effective flexural
stiffness at yield, i.e. FEMA 356 ( )0.5 c gE I , ASCE 41-06 ( )0.3 c gE I , and NZS-3101
1995 for low ductility ( )0.5 c gE I , only ASCE 41-06 (2007) addresses the impact of
slip/extension on the effective stiffness at yield [it is noted that median effective stiffness
reported by Elwood et al (2007) is actually 0.2 c gE I at low axial load, the value of
0.3 c gE I is used as a compromise to address issues associated with deformation
compatibility checks for gravity columns].
The contribution of slip/extension to the yield rotation is estimated for the beams
tested using the approach recommended by Alsiwat and Saatcioglu (1992), where the
crack width that develops at the beam-wall interface depends on bar slip and bar
extension (strain). Using a coupling beam effective stiffness derived from a moment-
curvature analysis of the beam cross-section at the beam-wall interface ( )~ 0.5 c gE I and
the slip/extension model noted above, the effective stiffness at yield reduces to 0.12 c gE I ,
which is consistent with the effective stiffness at the yield rotation (approximately 1.0%
for all beams) derived for the tests (Fig. 5.2). Additional details of the slip/extension
calculations are included in Section 5.2 and Appendix B.
Table 5.1 provides a summary of the effective stiffness and yield rotation for each
of the different models discussed above. Based on these results, use of the model detailed
in ASCE 41-06 Supplement #1 is recommended, i.e., use a moment-curvature analysis to
81
define the secant stiffness at the yield point and include a slip/extension spring.
Alternatively, as noted in ASCE 41-06 (2007), the effective bending stiffness can be
defined to provide an equivalent stiffness that combines both curvature and slip
deformations (~ 0.12 c gE I for the test beams). Use of a value of 0.15 to 0.20 c gE I is
suggested given that the test program is limited and does not address the potential
stiffening impact of coupling beam axial load due to redistribution of forces from tension
to compression walls. The impact of variation of coupling beam stiffness on system level
responses is addressed later via a sensitivity study for a 42-story building.
82
Table 5.1 Effective stiffness values
EcIeff [% EcIg] θy [% drift]
Test Results 14.0 (12.5)1
0.70 (1.00)1
FEMA 356 50.0 0.23
ASCE 41 30.0 0.39
ASCE 41 S1, w/slip hinge 16.5 (13.0)1
0.75 (0.95)1
NZS-3101 95 (μ=1) 50.0 0.23
1 1/2-scale test results
83
5.2 Slip/Extension Calculations
As stated in the previous section, slip and extension of the flexural reinforcement
contributes a large portion (40-50%) of the beam deformations prior to yield. The
approach developed by Alsiwat and Saatcioglu (1992) is used to model this contribution
to the yield rotation. The calculations are provided here for the case of determining the
slip/extension rotations at the yield point of the flexural reinforcement. If adequate
embedment of the flexural reinforcement is provided, then the slip contribution is
negligible. However, this calculation is also included for the sake of completeness. The
parameters that are required for this calculation are db, A, ld, f’c, fy, My and fs. All
dimensions are in mm and MPa (1 mm = 0.0397 in, 1 MPa = 0.145 ksi).
Preliminary calculations:
[ ]4y b
ed
f du MPa
l×
=×
(Eq. 5.2)
[ ]4
s be
e
f dL mmu×
=×
(Eq. 5.3)
20 [ ]4 30
b cu
d fu MPa′⎛ ⎞= − ×⎜ ⎟
⎝ ⎠ (Eq. 5.4)
130 [ ]s
c
mmf
δ =′
(Eq. 5.5)
84
In these calculations, ue represents the elastic bond stress, Le represents the elastic
region length, uu represents the peak bond stress, and δs1 represents the local slip at the
peak bond stress.
Calculations for fs = fy:
2.5
1 [ ]es s
u
u mmu
δ δ⎛ ⎞
= ×⎜ ⎟⎝ ⎠
(Eq. 5.6)
1.25 [ ]2
eexty y
L mmδ ε= × × (Eq. 5.7)
[ ]toty s exty mmδ δ δ= + (Eq. 5.8)
@ [ ]totytoty rad
d xδ
δθ =
− (Eq. 5.9)
@
[ ]y
toty
MK mm MPa
δθ= − (Eq. 5.10)
In these calculations, δs represents the slip of the reinforcement, δexty represents
the extension of the bar due to accumulation of strain along its length at yield, δtoty
represents the total displacement of the bar at the beam-wall interface at yield, θ@δtoty
represents the angle of the crack that opens at the beam-wall interface due to the
slip/extension of the bar at yield, and K represents the stiffness of the corresponding
moment-rotation hinge that can be implemented in a structural model.
As well, the slip/extension model can be used to modify post-yield behavior.
85
Calculations for εs = εu:
'
5.5 0.07 [ ]27.6
cLf
L
fSu MPaH
⎛ ⎞= − × ×⎜ ⎟⎝ ⎠
(Eq. 5.11)
[ ]4
s bpy
f
f dL mmu
Δ ×=
× (Eq. 5.12)
( )
[ ]2
y u pyext exty
Lmm
ε εδ δ
+ ×= + (Eq. 5.13)
[ ]tot s ext mmδ δ δ= + (Eq. 5.14)
@ [ ]tottot rad
d xδδθ =−
(Eq. 5.15)
In these calculations, uf represents the frictional bond stress, SL and HL are the
spacing and height of the lugs on the reinforcement respectively, and Lpy is the post-yield
length. The end result is the rotation due to slip/extension of the reinforcement.
5.3 Effect of Scale
As previously stated, the tests were conducted at one-half scale; therefore, it is important
to understand the potential impact of scale on the effective yield stiffness as well as the
overall load-deformation behavior. Slip/extension deformations are not directly scalable
by linear or square scaling of dimensions and bar sizes. The relative contribution of
flexural deformations (curvature) and slip/extension to the yield rotation of the test beams
at full scale (i.e. prototype beams) is assessed using the same approach as noted in the
86
previous paragraph for the one-half scale beams. The study is extended to consider
coupling beam aspect ratios beyond those tested, by varying the beam length. Results are
reported in Figure 5.3, where the effective yield rotation is plotted against beam aspect
ratio (ln/h) for various scale factors.
1 2 3 4ln/h
0.002
0.003
0.004
0.005
0.006
Slip
Rot
atio
n [ra
d]
1/2-scale2/3-scale3/4-scaleFull-scale
Figure 5.3 Yield rotation due to slip/extension for various aspect ratios and testing
scales
For a given scale factor, variation of the aspect ratio has only a moderate impact
on the slip rotation, producing roughly a 15 to 20% increase from aspect ratios of 1.0 to
3.0. However, for a given aspect ratio, slip rotation at yield is significantly impacted by
scale, with a 35 to 40% reduction for beams at one-half versus full scale. The effective
bending stiffness at yield for the one-half scale tests of 0.12 c gE I increases to 0.14 c gE I
87
for the full-scale prototypes due to the reduction in the relative contribution of slip
rotation. Based on these results, we recommend use of an effective yield stiffness value
of 0.15 to 0.20c g c gE I E I for full-scale coupling beams. Figure 5.4 provides a summary
of calculated values of effective yield stiffness for coupling beams with aspect ratios
2.0 4.0nl h≤ ≤ , for both full-scale and half-scale beams (for comparison purposes).
Specific examples of the implementation of these calculations are provided in Appendix
C.
2 2.4 2.8 3.2 3.6 4ln/h
0.05
0.1
0.15
0.2
0.25
Eff
ectiv
e St
iffne
ss [I
eff/
I g]
Full-scale1/2-scale
Figure 5.4 Effective elastic stiffness as a function of gross section stiffness calculated
for various aspect ratios and testing scales
88
5.4 Load-Deformation Backbone Relations
In this section, backbone relations are derived both from test results and using ASCE 41
to provide a simple yet accurate method for estimating the overall load-deformation
behavior of coupling beams. Linearized backbone relations for normalized shear strength
versus rotation are plotted in Figure 5.6 as dotted lines for the three configurations of
beams tested, i.e. beams with no slab (CB24F, CB24D, CB33F, CB33D), beam with RC
slab (CB24F-RC), and beams with PT slab (CB24F-PT and CB24F-1/2-PT). These
backbone relations are determined as shown in Figure 5.5, which plots the peaks of the
load-deformation curves for CB24F and CB24D. The backbone relations that are
modified to represent full-scale beams are also plotted in Figure 5.6, as discussed in the
prior subsection. For configurations with multiple tests, an average relation is plotted.
The results for all seven tests are very consistent, with a yield rotation of
approximately 1.0%, initiation of shear strength degradation at 8.0% rotation, and the
residual shear strength reached at 12.0% rotation. Backbone relations modified to
represent full-scale beams indicate that the total rotations at yield, strength degradation,
and residual strength are reduced to 0.70%, 6.0%, and 9.0%, respectively (from 1.0%,
8.0%, and 12.0%). The impact of slab on shear strength also is apparent in Figure 5.6,
with the ratios of ave nV V being approximately 1.1 (no slab), 1.3 (RC slab), and 1.4 (PT
slab).
89
0 2 4 6 8 10 12 14Beam Chord Rotation [%]
0
0.2
0.4
0.6
0.8
1
1.2
1.4
V/V
ncod
e
CB24FCB24DLinear Backbone
Figure 5.5 Determination of linearized backbone relation from test data
0 2 4 6 8 10 12 14Rotation [% drift]
00.2
0.40.6
0.81
1.2
1.41.6
V/V
ncod
e
PT SlabRC SlabNo SlabASCE 41-06
Figure 5.6 Backbone load-deformation for full-scale beam models and ASCE 41-06
model (1/2-scale test results are dotted lines)
90
ASCE 41-06 with Supplement #1 modeling parameters also are plotted on Figure
5.6 and indicate that the test beams are more flexible at yield and that they attain
substantially higher deformation capacity prior to lateral strength degradation than the
ASCE 41 backbone relation. The elastic stiffness of the ASCE 41 relation is based on a
bending stiffness of 0.3 c gE I , or about double that derived for full-scale beams from the
test data. The plastic rotation capacity given by ASCE 41-06 Table 6-18 is limited to 3%,
whereas the backbone relations for the full-scale beams derived from the test data yield at
approximately 0.7% rotation and reach 6.0% rotation prior to strength degradation, or a
plastic rotation of 5.3%. Therefore, relative to ASCE 41-06, the relations derived for the
full-scale beams have a lower effective yield stiffness (0.14EcIg/0.3EcIg = 0.47) and
substantially greater deformation capacity (5.3%/3.0% = 1.77).
The tests also reveal that a residual strength equal to 0.3Vn can be maintained to
very large rotations (10 to 12%) compared to the ASCE 41-06 residual strength ratio of
0.8 at a plastic rotation value of 5.0%. Therefore, it is reasonable to use a plastic rotation
value of 5.0% with no strength degradation, with moderate residual strength (0.3Vn) up to
a plastic rotation of 7.0%. It is noted that the ASCE 41-06 relation applies to all
diagonally-reinforced coupling beams, including beams with aspect ratios significantly
less than the values of 2.4 and 3.33 investigated in this test program. Results presented in
Fig. 5.6 apply for the beam aspect ratios tested (2.4 and 3.33), as well as to beams
between these ratios. It is reasonable to assume these values can be extrapolated modestly
to apply to beams with 2.0 4.0nl h≤ ≤ .
91
As an exercise to evaluate the potential for developing load-deformation
backbone relations for beams at any aspect ratio, the ASCE 41-06 backbone relation can
be modified to account for the effect of slip/extension deformations on both effective
elastic stiffness and plastic rotation capacities. In the previous section, the slip/extension
model was used to modify the elastic stiffness, the final result being an effective stiffness
of 0.15EcIg for beams with aspect ratio of 2.4. The slip/extension model can then be used
to modify the post-yield behavior defined by the ASCE relation. Equations 5.11 to 5.14
can be used to determine the ultimate rotation due to slip/extension, which can be added
to the rotation determined for the flexural model (ASCE 41 relation), to give the overall
plastic rotation capacity of the beam. The final result is presented in Figure 5.7, showing
the modified ASCE relation, which provides a very good approximation for the load-
deformation backbone of the full-scale beams. While this may not be the most realistic
modeling procedure, it clearly indicates the impact of the slip/extension hinge on both the
elastic and post-yield behaviors of beams with aspect ratios between 2 and 4.
92
0 2 4 6 8 10 12 14Beam Chord Rotation [%]
00.2
0.40.60.8
11.2
1.41.6
V/V
ncod
e
PT SlabRC SlabNo SlabASCE 41 mod for slip/ext
Figure 5.7 Backbone load-deformation for full-scale beam models and ASCE 41-06
model modified to account for slip/extension deformations
5.5 Application to Computer Modeling
Based on the backbone and effective stiffness relations discussed above, nonlinear
modeling approaches commonly used by practicing engineers were investigated to assess
how well they were able to represent the measured test results. Two models were
considered, one utilizing a rotational spring at the ends of the beam to account for both
nonlinear flexural and shear deformations (Mn hinge) and one utilizing a nonlinear shear
spring at beam mid-span to account for both shear and shear deformations (Vn hinge).
Both models were subjected to the same loading protocol used in the tests (Fig. 3.10).
93
5.5.1 Diagonally-reinforced coupling beams (2.0 < ln/h < 4.0)
The Mn-hinge model (Fig. 5.8(a)) consists of an elastic beam cross-section with EcIeff =
0.5EcIg, elastic-rotation springs (hinges) at each beam-end to simulate the effects of
slip/extension deformations, and rigid plastic rotational springs (hinges) at each beam-
end to simulate the effects of nonlinear deformations. The stiffness of the slip/extension
hinges were defined using the Alsiwat and Saatcioglu (1992) model discussed above,
whereas the nonlinear flexural hinges are modeled using the backbone relations derived
from test results (Fig. 5.6, excluding the elastic portion). The Vn-hinge model (Fig.
5.8(b)) also consists of an elastic beam cross-section and slip/extension hinges. However,
instead of using flexural hinges at the beam ends, a shear force versus displacement hinge
(spring) is used at the beam mid-span to simulate the effects of nonlinear deformations.
The shear hinge properties are defined using the backbone relations derived from the test
results (Fig. 5.6).
Mn-Rotation Springs
Slip/Ext. Springs
(a)Mn-Rotation Springs
Slip/Ext. Springs
(a) Vn-Displacement Hinge
Slip/Ext. Springs
(b)Vn-Displacement Hinge
Slip/Ext. Springs
(b)
Figure 5.8 Modeling components: (a) Mn-hinge model; and (b) Vn-hinge model
Figure 5.9 and Figure 5.10 show cyclic load-deformation plots for the two models
and the test results for CB24F. Both models accurately capture the overall load-
displacement response of the member; however, the Mn-hinge model (Fig. 5.9) captures
94
the unloading characteristics better than the Vn-hinge model (Fig. 5.10), due to the fact
that unloading stiffness modeling parameters, which help to adjust the slope of the
unloading curve, are available for the flexural hinges in the commercial computer
program used, but not for the shear hinges. While the current version (4.0.3) of the
program does not have these unloading stiffness parameters for the shear hinges, it is
noted that the next version is expected to incorporate these parameters. As noted
previously, for the beam test aspect ratios (2.4 and 3.33), flexural and slip/extension
deformations account for approximately 80-85% of total deformation whereas shear
deformations generally account for only l5-20% of total deformation. Therefore, in both
models, the flexural and shear hinges are used to represent flexural deformations,
whereas shear deformations are not considered. Therefore, depending on the computer
program used, modeling studies similar to those presented here should be conducted to
calibrate available model parameters with test results. Specifically, these models were
created using CSI Perform 3D (Computers and Structures, 2006), as it is the common
program used by design engineers in nonlinear modeling of structural systems. The
parameters used in each model are summarized in detail in Appendix D.
95
-0.12 -0.06 0 0.06 0.12Beam Chord Rotation [rad]
-200
-100
0
100
200
Late
ral L
oad
[k]
-890
-445
0
445
890
Late
ral L
oad
[kN
]
Test (CB24F)Mn Hinge
Figure 5.9 Cyclic load-deformation modeling results (ln/h = 2.4): CB24F vs. moment
hinge model
-0.12 -0.06 0 0.06 0.12Beam Chord Rotation [rad]
-200
-100
0
100
200
Late
ral L
oad
[k]
-890
-445
0
445
890
Late
ral L
oad
[kN
]
Test (CB24F)Vn Hinge
Figure 5.10 Cyclic load-deformation modeling results (ln/h = 2.4):CB24F vs. shear hinge
model
96
Beams at aspect ratio 3.0nl h ≥ exhibit predominately flexural behavior, and
therefore only the Mn-hinge model is presented for this case. The properties of the
slip/extension hinge, plastic moment-rotation hinge, and elastic concrete cross-section are
determined in the same way as for the 2.4 aspect ratio beam, but for the CB33F cross-
section. Much like for the residential beam, the Mn-hinge provides a good approximation
of the overall load-deformation response of the member, when compared with test results
(Fig. 5.11).
-0.1 -0.05 0 0.05 0.1Beam Chord Rotation [rad]
-150
-100
-50
0
50
100
150
Late
ral L
oad
[k]
-660
-330
0
330
660
Late
ral L
oad
[kN
]
Test (CB33F)Mn Hinge
Figure 5.11 Cyclic load-deformation modeling results (ln/h = 3.33): CB33F vs. moment
hinge model
Based on the analysis and test results, the main impact that the slab has on the
behavior of the coupling beams is to provide an increase in capacity by approximately
20%. This impact can easily be implemented into the computer model, by simply
97
increasing the capacity of the moment-rotation hinge by 20%. The results of this model
compared to the test results for the coupling beam with reinforced concrete slab (Fig.
5.12) show that the model can once again accurately capture the nonlinear behavior of the
coupling beam.
-0.12 -0.08 -0.04 0 0.04 0.08 0.12Beam Chord Rotation [rad]
-200
-100
0
100
200
Late
ral L
oad
[k]
-890
-445
0
445
890
Late
ral L
oad
[kN
]
Test (CB24F-RC)Mn Hinge
Figure 5.12 Cyclic load-deformation modeling results (ln/h = 2.4): CB24F-RC vs.
moment hinge model
5.5.2 Conventionally-reinforced coupling beams (3.0 < ln/h < 4.0)
In the testing program, in addition to the diagonally-reinforced coupling beams tested,
one conventionally-reinforced beam was also tested. Brief modeling studies were
conducted and compared with those of previous tests on similar beam specimens. The
98
model is composed similarly to the Mn-hinge model for beams CB33F and CB33D, with
changes made only to the shear strength (different cross-section), plastic rotation capacity
(different ASCE 41 parameters and different slip/extension parameters) and the energy
dissipation factors (straight bars instead of diagonal bars, the values for which are listed
in Table 5.2). The load-deformation response of the model is plotted against that of FB33
in Figure 5.13, and shows good correspondence with test results. The model can simulate
the pinching of the load-deformation plot, with some simple modification of the energy
dissipation parameters.
-0.08 -0.04 0 0.04 0.08Beam Chord Rotation [rad]
-75
-50
-25
0
25
50
75
Late
ral L
oad
[k]
-330
-165
0
165
330
Late
ral L
oad
[kN
]
Test (CB33F)Mn Hinge
(FB33)
-0.08 -0.04 0 0.04 0.08Beam Chord Rotation [rad]
-75
-50
-25
0
25
50
75
Late
ral L
oad
[k]
-330
-165
0
165
330
Late
ral L
oad
[kN
]
Test (CB33F)Mn Hinge
(FB33)
Figure 5.13 Cyclic load-deformation modeling results (ln/h = 3.33): FB33 vs. moment
hinge model
This model is also constructed for a test beam of similar geometry, tested by Xiao
et al., with aspect ratio ln/h=4.0 (Fig. 5.14). The model accurately captures the elastic
behavior of the beam, and follows very closely the unloading characteristics in the
99
nonlinear range. Also, by using the slip/extension spring for both elastic and inelastic
deformations, the model is able to reasonably capture the strength degradation at high
rotations. For beams with aspect ratio greater than 3.0, placement of diagonal
reinforcement can become difficult, and the potential gain from using diagonal as
opposed to longitudinal reinforcement may not justify its use. Thus, beams with low
shear stress requirements ( 5 cf ′≤ ) can be designed with longitudinal reinforcement with
only minor sacrifices to ductility as compared to diagonal reinforcement.
-6 -4 -2 0 2 4 6Beam Chord Rotation [%]
-50-40-30-20-10
01020304050
Late
ral L
oad
[k]
-200
-100
0
100
200
Late
ral L
oad
[kN
]
modelXiao ln/h=4
-6 -4 -2 0 2 4 6Beam Chord Rotation [%]
-50-40-30-20-10
01020304050
Late
ral L
oad
[k]
-200
-100
0
100
200
Late
ral L
oad
[kN
]
modelXiao ln/h=4
Figure 5.14 Cyclic load-deformation modeling results (ln/h = 4.0): HB4-6L-T100 vs.
moment hinge model
100
Table 5.2 Cyclic Degradation Parameters (Perform 3D)
Energy Factor Model
Y U L R X Unloading Stiffness Factor
Mn-hinge 0.50 0.45 0.40 0.35 0.35 0.50
Vn-hinge 0.50 0.45 0.40 0.35 0.35 --
Frame beam 0.50 0.40 0.35 0.17 0.17 0.75
5.5.3 Extension to lower aspect ratios (1.0 < ln/h < 2.0)
The modeling studies performed in this research were aimed at beams with aspect ratios
between 2.0 and 4.0, and while these beams make up the majority of coupling beam
geometries in current tall building construction, lower aspect ratio beams (ln/h < 2.0) need
to be studied. Most of the prior research on coupling beams has been conducted on test
beams with these lower aspect ratios.
The Mn-hinge model was applied to a specimen (ln/h = 1.17) tested by Kwan et al.
(Fig. 5.15). The model is able to capture the unloading/energy dissipation characteristics
of the beam, using the same parameters as for higher aspect ratio coupling beams.
However, the model does not represent the overall load-deformation behavior of the test
results well. The predicted load-deformation behavior in the elastic range is much stiffer
than that of the test specimen, due to the fact that for such a low aspect ratio,
slip/extension deformations and flexural deformations are smaller. Therefore, for such a
low aspect ratio, the shear deformations have a major effect on the overall behavior of the
101
model. Studies on shear-flexure interaction (e.g. Massone et al., 2009) for coupling
beams should be performed to investigate the relationship between shear deformations,
aspect ratio, and shear stress. This will help to better understand and model the elastic
behavior of such low aspect ratio beams.
-8 -6 -4 -2 0 2 4 6 8Beam Chord Rotation [%]
-80
-60
-40
-20
0
20
40
60
80
Late
ral L
oad
[k]
-350
-175
0
175
350
Late
ral L
oad
[kN
]
modelKwan ln/h=1.17
Figure 5.15 Cyclic load-deformation modeling results (ln/h = 1.17): CCB11 vs. moment
hinge model
102
5.6 Nonlinear Component Modeling
This section provides an overview of a nonlinear modeling procedure to develop load-
deformation backbone curves for coupling beam components of any aspect ratio. A
summary of the results of one such study and the potential application to simplified
modeling techniques is provided. Limitations of this study and recommendations for
future work also are discussed.
5.6.1 Modeling overview
A study was performed to assess the impact of various factors on nonlinear modeling of
coupling beams. The goal of the study was to provide reasonable modeling parameters
based on nonlinear fiber modeling, to be easily implemented in commercial software by
practicing engineers. While the study is not exhaustive, it does provide a reasonable
framework for developing simplified nonlinear load-deformation backbone curves.
Specifically, beams at aspect ratios between 1.0 and 4.0 at aspect increments of
0.25 were considered at various levels of shear stress ranging from
'6.0 to 10.0 [ ]n cv f ksi= . Each beam was loaded monotonically to failure, and an elasto-
plastic relation was fit to the resulting load-deformation behavior for each beam as
indicated in Appendix G, to provide the parameters θy, θu, θr, θx, Vave, and Vr, as shown in
Fig. 5.16.
103
θy θu θr θx
Vr
Vave
θy θu θr θx
Vr
Vave
Figure 5.16 Definitions of parameters in elasto-plastic load-deformation relation
VecTor5 is a program developed at the University of Toronto to perform
nonlinear sectional analysis of two-dimensional frame structures. It is based on the
Modified Compression Field Theory (MCFT) and the Disturbed Stress Field Model
(DSFM), both developed at University of Toronto (Vecchio and Collins, 1986; Vecchio,
2001). It uses a rotating, smeared crack approach based on a total load, secant stiffness
formulation. MCFT is typically used in the development of models to incorporate shear-
flexure interaction, which is especially important for members at low aspect ratio (ln/h <
2.0), where shear deformations can contribute a substantial amount to the overall
deformation of a member (Massone et. al. 2009). The theoretical basis for VecTor5 is
discussed in Guner (2008).
VecTor5 was used to model the nonlinear load-deformation behavior of the
beams according to the general outline discussed above, first accounting for both shear
and flexural deformations, and then accounting for flexural deformations only. In this
104
way, the relative contributions of shear and flexural deformations were determined.
Because VecTor5 does not account for slip of the reinforcement, the slip model discussed
earlier was again utilized to model the deformations due to slip/extension. As in the
slip/extension deformation study discussed earlier, one beam cross-section was used, with
different steel areas to account for different levels of shear stress, and different beam
spans to account for different aspect ratios. A schematic of the beam configurations
considered and summary of the parameters used are shown in Figure 5.17 and Table 5.3.
The model is loaded monotonically in displacement increments to failure. The right end
of the beam is not restrained axially. The model uses the default constitutive properties
for concrete and steel as defined by Guner (2008).
24”
30”
3 node
Ln
8-Bar Bundle Typ. [db]
5”
#5 x
@4” o.c.
V
24”
30”
3 node
Ln
8-Bar Bundle Typ. [db]
5”
#5 x
@4” o.c.
#5 x
@4” o.c.
V
Figure 5.17 Modeling schematic (from left): (a) Typical beam cross-section; and (b)
finite element discretization and loading.
105
Table 5.3 Geometric properties of beams used in nonlinear modeling procedure
db [in.] ln/h ln [in.]
vn=6[√f’c] vn=7[√f’c] vn=8[√f’c] vn=9[√f’c] vn=10[√f’c] 1.00 30.0 0.86 0.93 0.99 1.05 1.11 1.25 37.5 0.93 1.01 1.08 1.14 1.21 1.50 45.0 1.01 1.09 1.16 1.23 1.30 1.75 52.5 1.07 1.16 1.24 1.32 1.39 2.00 60.0 1.14 1.23 1.32 1.40 1.47 2.25 67.5 1.20 1.30 1.39 1.47 1.55 2.40 72.0 1.24 1.34 1.43 1.52 1.60 2.50 75.0 1.26 1.36 1.46 1.55 1.63 2.75 82.5 1.32 1.43 1.52 1.62 1.70 3.00 90.0 1.38 1.49 1.59 1.69 1.78 3.25 97.5 1.43 1.54 1.65 1.75 1.85 3.50 105.0 1.48 1.60 1.71 1.81 1.91 3.75 112.5 1.53 1.65 1.77 1.88 1.98 4.00 120.0 1.58 1.71 1.83 1.94 2.04
5.6.2 Nonlinear modeling results
Figure 5.18 plots total beam chord rotation (including flexural, shear, and slip/ext
deformations) at yield for different aspect ratios and levels of shear stress. Yield
deformations follow a generally linear increasing trend. The differences in yield rotation
between different levels of shear stress are consistent over the range of aspect ratios.
Therefore it is reasonable to assume that a linear interpolation can be used between the
shear stresses plotted to determine the yield rotation at a given aspect ratio.
106
1 2 3 4ln/h
0.4
0.6
0.8
1
1.2
1.4
1.6
Beam
Cho
rd R
otat
ion
[%] vn=6√f'c
vn=7√f'c
vn=8√f'c
vn=9√f'c
vn=10√f'c
Figure 5.18 Total yield rotation for coupling beams at various aspect ratios and shear
stress levels
The above result was based on the VecTor5 and slip/extension model including
the impact of shear deformations; however, the models also can be created that neglect
shear deformations. The difference between the two results (with and without shear
deformations) represents the contribution of shear deformations to the total deformation.
Figure 5.19 plots the percent contributions of the different deformation components
(shear, flexure, and slip/extension) for various aspect ratios at shear stresses of
'6.0 [ ]n cv f ksi= and '10.0 [ ]n cv f ksi= . As discussed, the results are fairly consistent
between the different levels of shear stresses, and therefore, only the outer bounds are
plotted. Results can be interpolated between shear stresses if necessary. Slip/extension
deformations account for approximately the same percentage (~40%) of overall beam
107
chord rotation for all aspect ratios. For aspect ratios ln/h < 2.3, at a constant shear stress,
shear deformations (~35%) are more significant than flexural deformations (~25%); for
aspect ratios ln/h < 2.3, at a constant shear stress level, shear deformations (~20-25%)
account for less of the total rotation than do flexural deformations (~35-40%). At aspect
ratio ln/h = 2.4, shear and flexural deformations are very comparable (~30% each), with
slip/extension deformations contributing 40%. This differs slightly from results observed
during testing, where shear deformations at yield contributed approximately 20%,
flexural deformations contributed approximately 30%, and slip/extension deformations
contributed approximately 50%. This difference in shear deformations can be explained
by the fact that the amount of transverse confinement steel in the test specimens was
slightly higher than that in the model, due to spacing and scaling issues, a fact which
would lead to reduced shear deformations compared with the prototype model.
108
1 2 3 4Ln/h
0
10
20
30
40
50
% C
ontri
butio
n [θ
y]
Slip/ExtShearFlexure (a)
1 2 3 4Ln/h
0
10
20
30
40
50
% C
ontri
butio
n [θ
y]
Slip/ExtShearFlexure (a)
1 2 3 4Ln/h
0
10
20
30
40
50
% C
ontri
butio
n [θ
y]
Slip/ExtShearFlexure (b)
1 2 3 4Ln/h
0
10
20
30
40
50
% C
ontri
butio
n [θ
y]
Slip/ExtShearFlexure (b)
Figure 5.19 Deformation contributions [%] at yield for various aspect ratios at (a)
vn=6.0√f’c; and (b) vn=10.0√f’c
Figure 5.20 plots θu, defined as the beam chord rotation at significant strength
degradation (< 0.8Vave) or total plastic rotation capacity, for various aspect ratios and
shear stresses. For aspect ratios ln/h ≥ 1.5, the plot again shows a generally linear
109
increasing trend with increasing aspect ratio and decreasing shear stress level. There is
however, a jump in plastic rotation capacity at very low aspect ratios. The resolution on
the geometries tested is fairly large, so the exact point at which this jump occurs is not
obvious. Further studies could be completed to investigate the impact of different cross-
section geometries on this plastic rotation capacity.
1 2 3 4ln/h
3
4
5
6
7
Beam
Cho
rd R
otat
ion
[%] vn=6√f'c
vn=7√f'c
vn=8√f'c
vn=9√f'c
vn=10√f'c
Figure 5.20 Beam chord rotation θu at onset of significant strength degradation for
various aspect ratios and shear stresses
Figure 5.21 plots beam shear strength determined from the nonlinear section
analysis, Vave, normalized with respect to ACI-calculated shear strength, Vn. For beams
with aspect ratio ln/h ≥ 1.5, the maximum capacity of the beam increases slightly with
increasing aspect ratio and decreasing shear stress. All beams are able to hold larger loads
110
than those predicted by ACI; at any given aspect ratio, the capacity vs. Vn(ACI) increases
by approximately 15-20% from a shear stress of '10.0 [ ]n cv f ksi= to '6.0 [ ]n cv f ksi= .
At lower shear stress levels, the amount of steel in the section is less, yielding a lesser
impact on the overall strength of the member, defined by the ACI equation for shear
strength of a coupling beam (Eq. 3.1), which is directly proportional to the steel area.
1 2 3 4ln/h
0
0.5
1
1.5
2
Vav
e/V
n
vn=6√f'c
vn=7√f'c
vn=8√f'c
vn=9√f'c
vn=10√f'c
Figure 5.21 Beam lateral load, Vave, normalized with respect to beam shear strength from
ACI, Vn
The results from Figures 5.18-21 can be used to develop overall elasto-plastic
load-deformation curves of the form in Figure 5.16. One such example is shown in Figure
5.22, for the full-scale prototypes of the beam specimens with aspect ratio 2.4 tested in
this research program. The model does not represent the overall load-deformation
111
perfectly. It underestimates both the elastic stiffness, EcIeff, and the overall plastic rotation
capacity, θu, while overestimating the strength of the section. This example indicates
some of the limitations of VecTor5 in representing the behavior of coupling beams at
higher aspect ratios. First, VecTor5 does not consider slip of the flexural reinforcement.
While this can be separately modeled, there is an impact to not having it directly included
in the analysis. Therefore, if slip/extension was directly included in the model, VecTor5
may be able to better predict the overall plastic rotation capacity. Secondly, VecTor5
does not allow for diagonal reinforcement. So in this model, all flexural reinforcement is
modeled as longitudinal, a fact which has substantial impact on the cyclic energy
dissipation characteristics and degradation at high levels of rotation.
112
0 2 4 6 8 10 12 14Beam Chord Rotation [%]
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
V/V
ncod
e
PT SlabRC SlabNo SlabVecTor5 w/ slip/ext
Figure 5.22 Load-deformation backbone relations comparing test results with the
nonlinear model developed with VecTor5 and slip/extension for beams at
aspect ratio 2.4
Figure 5.23 plots the load-deformation backbone for a beam tested by Kwan
(2002) at aspect ratio 1.17, compared to that predicted using VecTor5. In this case, the
VecTor5 model predicts the plastic rotation capacity fairly well, which is expected, as
shear damage is expected to play a much larger role in a beam at such a low aspect ratio,
and VecTor5 accounts for this shear failure mode. The elastic stiffness of the member is
slightly overestimated by the model.
113
0 2 4 6 8 10 12Beam Chord Rotation [%]
0
0.5
1
1.5
2
V/V
ncod
e
Kwan ln/h=1.17VecTor5 w slip/ext
Figure 5.23 Load-deformation backbone relation comparing test results with nonlinear
VecTor5 and slip/extension model for beam at aspect ratio 1.17
The general conclusions that can be drawn from this modeling study are as
follows. For beams with aspect ratios lower than 2.25, shear deformations are particularly
important, and should be considered in the analysis. For beams at higher aspect ratio, ln/h
> 2.25, flexural deformations are more important to the analysis, and shear deformations
are less likely to contribute substantially to the overall deformation of the member.
Simplified nonlinear deformation backbone relations can be developed using modeling
procedures that account for interaction of flexural and shear components of deformation;
however, slip/extension deformations generally must be incorporated separately. These
procedures are particularly effective in modeling members at low aspect ratios, where
shear deformations contribute substantially to the overall deformation capacity of the
member. They are less effective in members at higher aspect ratios, where shear plays
114
less of a role. These nonlinear load-deformation backbone relations can be easily
implemented in structural modeling approaches commonly used in commercial software
such as Perform 3D (Computers and Structures, 2006), to accurately represent the
behavior of coupling beams. Further study regarding direct incorporation of
slip/extension of flexural reinforcement and including diagonal steel placement should be
performed to help provide a better overall understanding and prediction of the behavior
of diagonally-reinforced coupling beams.
Based solely on this modeling study, Table 5.4 provides lower bound estimates
for the modeling parameters and numerical acceptance criteria in ASCE 41-06 for
diagonally-reinforced coupling beams. These values can be directly implemented in
nonlinear models using software such as Perform 3D to represent the behavior of
coupling beams accurately and include the effect of slip/extension on the plastic rotation
capacity. Used in conjunction with an effective elastic stiffness of 0.20EcIg, these
parameters account for the impact of slip/extension deformations on the overall load-
deformation behavior of the member.
115
Table 5.4 Lower Bound Estimate ASCE 41-06 Modeling Parameters and Numerical
Acceptance Criteria for Nonlinear Procedures - Diagonally-Reinforced
Coupling Beams
Conditions Plastic Hinge Rotation (radians)
Residual Strength
Ratio
Acceptable Plastic Hinge Rotation (radians)
ln/h 'w w c
Vt l f
a b c IO LS CP
≤ 2.0 ≤ 6.0 0.037 0.057 0.30 0.007 0.020 0.037
≤ 2.0 ≥ 8.0 0.034 0.054 0.30 0.006 0.018 0.034
≥ 3.0 ≤ 6.0 0.043 0.063 0.30 0.009 0.022 0.043
≥ 3.0 ≥ 8.0 0.037 0.057 0.30 0.007 0.020 0.037
5.7 Summary
This chapter presented a summary of component modeling procedures including
considerations of effective elastic stiffness, slip/extension deformations, residual strength,
and plastic rotation capacity. A value of 0.20EcIg is recommended for coupling beam
effective elastic stiffness for conservatism. In the next chapter, fragility functions will be
investigated in coordination with this modeling study to evaluate the potential to further
modify the parameters in Table 5.4 to more accurately represent the load-deformation
characteristics of diagonally-reinforced coupling beams.
116
Chapter 6 Fragility Curves for Coupling Beams
In this chapter, an overview of a procedure for development of fragility curves for
diagonally and conventionally reinforced concrete coupling beams is provided. Fragility
curves are especially important to performance based earthquake engineering, as they
link engineering design parameters such as beam chord rotation to specified damage
states such as yielding, concrete spalling, and rebar buckling.
6.1 Sources of Data
The collection of data for coupling beam tests is not as extensive as that for columns,
shear walls, or joints. Given the lack of data, it was not possible to create various bins to
address design parameters, such as variations in transverse steel reinforcement ratio,
maximum shear stress, and aspect ratio. Available test data were collected and organized
first simply by aspect ratio; the potential to sort data by maximum shear stress was
considered, but not implemented.
As noted in Chapters 1 and 2, a majority of coupling beam tests have been
performed at low aspect ratios (ln/h < 2.0), with detailing that did not conform to modern
117
codes (ACI 318-99). Therefore, the first set of fragility was developed for beams with
relatively higher aspect ratio (2.0 < ln/h < 4.0), on diagonally reinforced coupling beams.
Even within this range of aspect ratios, there are relatively few tests of diagonally-
reinforced beams besides the ones reported in this study. The only relevant study
identified was conducted at University of Cincinnati by Fortney (2005).
Two sets of fragility relations were developed for conventionally-reinforced
coupling beams, one for higher aspect ratios (3.0 < ln/h < 4.0) and one for lower aspect
ratios (1.0 < ln/h < 2.0). Test data were taken from Naish et al. (2009), Xiao et al. (1999),
Tassios et al. (1996), Galano and Vignoli (2000), Kwan et al. (2002), and Paulay (1971).
The final set of fragility curves was developed for diagonally-reinforced coupling beams
at low aspect ratios (1.0 < ln/h < 2.0). Data were taken from tests performed by Tassios et
al. (1996), Kwan et al. (2002), Galano and Vignoli (2000), and Paulay (1974).
6.2 Damage States
The definition of damage states is essential to the development of fragility curves. Four
total damage states were defined for the scope of this project including yield: (1) Yielding
of the test specimen (either diagonal or longitudinal bars), (2) DS1-Minor damage, (3)
DS2-Major damage I, and (4) DS3-Major damage II. These damage states were
determined for each test specimen based on investigation of load-deformation relations as
well as photographs and descriptions of damage provided in the papers/reports by
authors.
118
The first important point considered was yielding of the test specimen. More
specifically, this was defined as the point at which the effective stiffness of the load-
deformation plot changed substantially. This was chosen over simply defining the first
point at which any flexural reinforcement yielded due to difficulties in defining first yield
due to the potential for noise variations and inaccuracies in the strain gage data. As well,
the impact of one bar or a layer of bars yielding is far less important than the impact of a
major change in beam stiffness. An example of the determination of yield point is shown
in Figure 6.1.
θyθy
Figure 6.1 Yield point determined from the Load-Deformation backbone relation,
defined as the point at which stiffness changes substantially.
Damage state 1 (DS1) was defined as minor damage of the test specimen.
Specifically, minor damage was defined as damage that could be easily repaired using
119
methods such as epoxy injection of residual cracks. Beams were determined to have
suffered minor damage if they had residual cracks less than 1/16” wide, most of which
occurred at the beam-wall interface, with some limited flexural cracking also occurring.
This was based mostly on investigation of damage photographs from test specimens.
Photos of this damage state shown in Figure 6.2.
Figure 6.2 Photo detailing DS1, in which there is light residual cracking evident
(>1/16”)
Damage state 2 (DS2) was defined as major damage (I) of the test specimen.
Major damage (I) was defined as damage that would require repair in the form of
substantial epoxy injection of residual cracks (both in the beam and slab) as well as
120
replacement of spalled concrete. A specimen experienced DS2 if it had residual cracks
greater than 1/8” and minor spalling of concrete (generally at the beam-wall interface).
Photos of this damage state are shown in Figure 6.3.
Figure 6.3 Photo detailing DS2, in which there is large residual cracking (>1/8”) and
some light spalling of concrete
Damage state 3 (DS3) was defined as major damage (II) of the test specimen.
Major damage (II) was defined as very substantial damage that would require significant
repair. A beam was said to suffer DS3 if the member had significant strength degradation
(<0.8Vn), buckling and/or fracture of the diagonal or flexural reinforcement, and crushing
of the concrete. To provide repair, it would be required to chip away all damaged
concrete, attach mechanical couplers to any reinforcement still embedded in the walls,
replace any damaged or fractured reinforcement, and replace the damaged concrete. This
damage state is shown in Figure 6.4.
121
0.8 Vn
θu
0.8 Vn
θu
Figure 6.4 Determination of DS3, the onset of significant strength degradation due to
severe damage to the concrete and reinforcement
A summary of the definitions of the different damage states and procedures for
repair for each is provided in Table 6.1.
122
Table 6.1 Details of damage states for fragility relations
Damage State Definition of damage Repair procedures
Yield -Substantial change in stiffness of load-deformation plot
none
DS1-Minor damage -Residual cracks greater than 1/16”
-Epoxy injection of cracks (200”-240” in length)
DS2-Major damage (I)
-Residual cracks greater than 1/8” -Minor spalling of concrete
-Epoxy injection of cracks in beam (600”-720”) and slab (300”) -Replacement of spalled concrete
DS3-Major damage (II)
-Significant strength degradation (<0.8Vn) -Buckling/fracture of reinforcement -Crushing of concrete
-Chip away damaged concrete -Attach mechanical couplers to remaining bars -Replace damaged/fractured reinforcement -Replace damaged concrete
6.3 Results
The results of the investigation are presented here in the form of fragility curves. Fragility
curves are defined as cumulative density functions plotted against the desired response
quantity. In the case of this study, beam chord rotation (θ), defined as the relative
displacement (Δ) normalized by beam span (ln), was the desired response quantity or
Engineering Demand Parameter (EDP). The data were fit with lognormal distributions,
123
which are commonly used and found to fit the data reasonably well. The parameters θm
and s.d. are defined as the mean rotation of the distribution and logarithmic standard
deviation for a given damage level, respectively. The results for all studies are provided
in Table 6.2.
Table 6.2 Summary of fragility function parameters for coupling beams
Beam Category Damage State θm [%] s.d.[%]
Yielding 0.84 0.39
DS1 1.79 0.38
DS2 3.52 0.44
Diagonally-reinforced
1.0 < ln/h < 2.0
DS3 5.43 0.95
Yielding 0.97 0.26
DS1 2.03 0.39
DS2 3.94 0.35
Diagonally-reinforced
2.0 < ln/h < 4.0
DS3 6.02 1.00
Yielding 0.85 0.25
DS1 1.37 0.21
DS2 2.64 0.33
Conventionally-reinforced
1.0 < ln/h < 2.0
DS3 4.28 0.74
Yielding 0.72 0.20
DS1 1.37 0.21
DS2 2.64 0.33
Conventionally-reinforced
2.0 < ln/h < 4.0
DS3 4.07 0.75
124
Figure 6.5 shows fragility curves for higher aspect ratio diagonally reinforced
beams, 2.0 < ln/h < 4.0. The graph can be read as follows. For a given beam chord
rotation, say 2.0%, the probability of a beam reaching DS1 and requiring minor repair is
approximately 50%; for a chord rotation of 5.5%, the probability of reaching DS3 and
requiring major repair is approximately 30%. Based on these results, the following
information can be obtained. The mean rotation at which yielding occurs is
approximately 0.97%, with a standard deviation of 0.26%. The mean rotation at which
DS1 is reached is 2.03% with a standard deviation of 0.39%. The mean rotation at which
DS2 is reached is 3.94%, with a standard deviation of 0.35%. And the mean rotation at
which DS3 is reached is 6.03%, with a standard deviation of 1.01%.
125
0 2 4 6 8 10Beam Chord Rotation [%]
0
0.2
0.4
0.6
0.8
1
Prob
abili
ty o
f dam
age
stat
e oc
curr
ing
YieldDS1-Minor repairDS2-Major repair 1DS3-Major repair 2
Figure 6.5 Fragility curves for diagonally reinforced concrete coupling beams at high
aspect ratio (2.0 < ln/h < 4.0)
Similar conclusions can be drawn from plots of the fragility relations for the three
damage states and yielding both for conventionally reinforced coupling beams at high
aspect ratios, 2.0 < ln/h < 4.0 (Figure 6.6), and for diagonally and conventionally
reinforced beams at low aspect ratios, 1.0 < ln/h < 2.0 (Figs 6.7-6.8). These fragility
functions can be used in performance based earthquake engineering applications to help
define damage states as a function of the engineering demand parameter, beam chord
rotation.
126
0 1 2 3 4 5 6Beam Chord Rotation [%]
0
0.2
0.4
0.6
0.8
1
Prob
abili
ty o
f dam
age
stat
e oc
curr
ing
YieldDS1-Minor repairDS2-Major repair 1DS3-Major repair 2
Figure 6.6 Fragility curves for conventionally-reinforced concrete coupling beams with
aspect ratio 2.0 < ln/h < 4.0
0 2 4 6 8 10Beam Chord Rotation [%]
0
0.2
0.4
0.6
0.8
1
Prob
abili
ty o
f dam
age
stat
e oc
curr
ing
YieldDS1-Minor repairDS2-Major repair 1DS3-Major repair 2
Figure 6.7 Fragility curves for diagonally-reinforced concrete coupling beams with
aspect ratio 1.0 < ln/h < 2.0
127
0 1 2 3 4 5 6Beam Chord Rotation [%]
0
0.2
0.4
0.6
0.8
1
Prob
abili
ty o
f dam
age
stat
e oc
curr
ing
YieldDS1-Minor repairDS2-Major repair 1DS3-Major repair 2
Figure 6.8 Fragility curves for conventionally-reinforced concrete coupling beams with
aspect ratio 1.0 < ln/h < 2.0
6.4 Modeling Parameters and Acceptance Criteria
As stated previously, DS3 represents substantial damage and strength loss of the member.
Thus DS3 is analogous to the collapse prevention (CP) limit state in ASCE 41. Table 6.3
provides a comparison of the rotation values for CP and the mean rotation values for
DS3. The CP values for conventionally-reinforced coupling beams are organized based
on shear demand and conforming transverse reinforcement. However, the CP values for
diagonally-reinforced beams are independent of aspect ratio, shear stress, or transverse
reinforcement. As well, there is a significant difference between the CP and DS3 values,
most notably for the diagonally-reinforced beams. This indicates the potential to provide
128
more rows of values for the diagonally-reinforced beams. Values for conventionally-
reinforced beams are fairly thorough, with distinctions made based on shear demand and
conforming reinforcement, and therefore could be left intact.
Table 6.3 Limit/Damage State Comparisons (plastic hinge rotations)
Coupling Beams CP [rad] DS3 [rad]
1.0 < ln/h < 2.0 0.035 Conventionally-reinforced 2.0 < ln/h < 4.0
0.020-0.0251 0.034
1.0 < ln/h < 2.0 0.046 Diagonally-reinforced 2.0 < ln/h < 4.0
0.030 0.050
1represents a range of values depending on shear stress demand
Based on the results and discussions in Chapters 5 and 6, further modifications to
the modeling parameters for diagonally-reinforced coupling beams in ASCE 41-06 are
shown in Table 6.4. These values are based on the median values of the fragility curves,
consistent with current ASCE 41 values (Elwood et al., 2007). The rotation values are
plastic deformation values, i.e. neglecting elastic deformations. These values are meant to
provide a best estimate of the deformation and residual strength capacities based on test
results shown in Figures 6.5-6.8, for the purposes of nonlinear modeling of diagonally-
reinforced coupling beams. Linear interpolation is permitted between values listed in the
table.
129
Table 6.4 ASCE 41-06 Modeling Parameters and Numerical Acceptance Criteria for
Nonlinear Procedures - Diagonally-Reinforced Coupling Beams
Conditions Plastic Hinge Rotation (radians)
Residual Strength
Ratio
Acceptable Plastic Hinge Rotation (radians)
ln/h 'w w c
Vt l f
a b c IO LS CP
≤ 2.0 ≤ 6.0 0.045 0.065 0.30 0.007 0.020 0.045
≤ 2.0 ≥ 8.0 0.035 0.055 0.30 0.006 0.018 0.035
≥ 3.0 ≤ 6.0 0.050 0.070 0.30 0.009 0.022 0.050
≥ 3.0 ≥ 8.0 0.045 0.065 0.30 0.007 0.020 0.045
6.5 Summary
This chapter presented a procedure for developing fragility relations for coupling beams.
In this study, the bins of data were organized based on reinforcement (diagonal vs.
conventional) and aspect ratio (low vs. high). A potentially useful future study would be
to further divide test data into bins based on level of shear stress. Further, the nonlinear
modeling studies of Chapter 5 and the fragility studies from this Chapter were used to
develop modeling parameters and acceptance criteria.
130
Chapter 7 System Modeling
This chapter briefly presents a parametric study of the impact of coupling beam strength
and stiffness parameters on the system behavior, where coupling beam chord rotation is
used as the engineering demand parameter to measure performance.
7.1 Model Information
This section defines the modeling parameters used for this study, as well as the loading
history. Finally, a description of the model changes and the impact of the model changes
on beam chord rotation are assessed.
7.1.1 Baseline model
The model used here was developed as part of a study to investigate and compare the
performance of buildings designed using code-based approaches vs. buildings designed
using performance-based approaches, a study which comprised the master’s research of
Zeynep Tuna (Tuna, 2009). The 42-story building (referred to as 2A) was designed by
131
Englekirk Partners Inc. (EPI). The coupling beams were designed based on results from
this research (Naish et al, 2009).
The building was designed prescriptively according to IBC 2006, which
incorporates ASCE 7-05 and ACI 318-08. A modal response spectrum analysis was used
with 5% damping. The design is a dual system with a core wall and four-bay special
moment frames along each side of the building. The specifics of the design are described
by Tuna (2009). Of interest in this study was the impact of the coupling beam design and
modeling parameters on system behavior.
The building lateral force resisting system was modeled in Perform 3D. The
coupling beams were modeled using the Vn-hinge model described in §5.5. Specifically,
the following properties were used:
Baseline Model:
0.2c eff c gE I E I=
( )exp 2 1.17 siny s yV A f α= × × ×
exp exp1.33u yV V= ×
exp exp0.25r uV V= ×
The lower flexural stiffness (EcIeff) was used in place of slip/extension hinges, as
discussed in §5.1. The ultimate strength of the model considers overstrength due to the
presence of the slab. Figure 7.1 plots the load-deformation backbone curve used to define
132
the shear-displacement hinge properties. The same cyclic energy dissipation factors as
described in Appendix D were used in this model. This design was considered to have the
most reasonable assumptions, and was therefore the baseline for this study.
0 2 4 6 8 10 12 14 16Beam Chord Rotation [%]
0
0.25
0.5
0.75
1
1.25
1.5Sh
ear F
orce
[V/V
yexp
]
(θy, Vyexp)
(θ2, Vuexp) (θ6, Vuexp)
(θ10, Vrexp)
0 2 4 6 8 10 12 14 16Beam Chord Rotation [%]
0
0.25
0.5
0.75
1
1.25
1.5Sh
ear F
orce
[V/V
yexp
]
(θy, Vyexp)
(θ2, Vuexp) (θ6, Vuexp)
(θ10, Vrexp)
Figure 7.1 Coupling beam shear-displacement hinge backbone properties for baseline
model
7.1.2 Modified models
Two parameters, stiffness and strength, were deemed particularly important for study.
Wide ranges of stiffness values have been used to define the elastic behavior of coupling
beams in both linear and nonlinear models. Common methods for determining flexural
stiffness values (EcIeff) were discussed thoroughly in §5.1, and values range from 0.15EcIg
133
to 0.5EcIg. With such a range of values being commonly used, it is important to consider
the impact of stiffness variation on the system behavior (i.e. EDPs). Typically, larger
stiffness values directly correspond to higher degrees of coupling as discussed in
Chapters 1 and 2. Therefore large differences in stiffness can have a large impact on the
behavior of the system. This fact is already understood by engineers, and therefore, this
study focused on looking at a smaller range of stiffness values, 0.15EcIg to 0.25EcIg.
Prescriptive methods to design for coupling beam strength are based solely on the
amount of diagonal reinforcement in the member. They do not consider the impact of the
slab (concrete and reinforcement), which can increase the flexural strength by around
20%, as discussed in S4.2. Therefore, the second parameter for study was the shear
strength.
Two models were constructed, in addition to the baseline model, with the
following characteristics:
Model 1: Model 2:
0.15c eff c gE I E I= 0.25c eff c gE I E I=
( )exp 2 siny s yV A f α= × × × ( )exp 2 1.17 siny s yV A f α= × × ×
exp exp1.15u yV V= × exp exp1.33u yV V= ×
exp exp0.25r uV V= × exp exp0.25r uV V= ×
134
The two models should provide a bound on variations of EDPs, since one has low
stiffness and strength, whereas the other has slightly high values for each parameter.
Again the models were constructed using the Vn-hinge model, with modified elastic
stiffness properties in place of the slip/extension hinge. Figures 7.2-7.3 display the shear-
deformation hinge properties for both models.
0 2 4 6 8 10 12 14 16Beam Chord Rotation [%]
0
0.25
0.5
0.75
1
1.25
1.5
Shea
r For
ce [V
/Vye
xp]
(θy, Vyexp)
(θ2, Vuexp) (θ6, Vuexp)
(θ10, Vrexp)
0 2 4 6 8 10 12 14 16Beam Chord Rotation [%]
0
0.25
0.5
0.75
1
1.25
1.5
Shea
r For
ce [V
/Vye
xp]
(θy, Vyexp)
(θ2, Vuexp) (θ6, Vuexp)
(θ10, Vrexp)
Figure 7.2 Coupling beam shear-displacement hinge backbone properties for Model 1
135
0 2 4 6 8 10 12 14 16Beam Chord Rotation [%]
0
0.25
0.5
0.75
1
1.25
1.5
Shea
r For
ce [V
/Vye
xp]
(θy, Vyexp)
(θ2, Vuexp) (θ6, Vuexp)
(θ10, Vrexp)
0 2 4 6 8 10 12 14 16Beam Chord Rotation [%]
0
0.25
0.5
0.75
1
1.25
1.5
Shea
r For
ce [V
/Vye
xp]
(θy, Vyexp)
(θ2, Vuexp) (θ6, Vuexp)
(θ10, Vrexp)
Figure 7.3 Coupling beam shear-displacement hinge backbone properties for Model 2
7.1.3 Loading
Nonlinear response history analyses were performed for five hazard levels consisting of
15 pairs of ground motions in the study by Tuna (2009). However, only one of these
hazard levels, MCE, was used for this study. The MCE hazard level is defined as the
Maximum Considered Earthquake, with a probability of 2% in 50 years, corresponding to
a return period of 2475 years. The earthquake ground motion was applied after a gravity
load of P=1.0D+0.25L. The ground motion selection methodology was defined by Tuna
(2009) and is reproduced in Appendix F. A summary of the modeling parameters for this
study is provided in Table 7.1.
136
Table 7.1 Summary of varied coupling beam modeling parameters
Model Stiffness Peak Strength Hazard Level
Baseline 0.20EcIg 1.33Vy MCE
1 0.15EcIg 1.15Vy MCE
2 0.25EcIg 1.33Vy MCE
7.2 Nonlinear Analysis Results
The building was analyzed in Perform 3D, for the 15 ground motions at the MCE level.
The results for the three different models are presented here. Coupling beam rotations
along the height of the structure were used as the primary performance metric. However,
considerations of core wall shear force and displacements also were considered to more
thoroughly investigate the impact of the parametric variations. Figure 7.4 provides
several views of the model implemented in Perform 3D, and shows the locations of the
coupling beams in the core wall.
137
(a)(a)
(b)
Elev B-B
(b)(b)
Elev B-B
(c)
Elev C-C
(c)(c)
Elev C-C
C
CB B
(d)
C
CB B
C
CB B
(d)
East CBsWest CBs
North CB
South CB
(e)
East CBsWest CBs
North CB
South CB
(e)
Figure 7.4 Perform 3D model (a) 3D view of structure; (b) North-South elevation view
of structure; (c) East-West elevation view of structure; (d) plan view of
structure; and (e) coupling beam locations in core wall of structure
7.2.1 Coupling beam rotations
Coupling beam rotations for the baseline model are provided in Figure 7.5. The plot
shows the mean rotations for coupling beams on the north-south sides and the east-west
138
sides of the core wall. The dotted lines represent the mean rotation values +/- one
standard deviation. The maximum mean values of coupling beam chord rotations are in
the east-west beams and at MCE level reach approximately 1.5%. If mean plus one
standard deviation is considered, the maximum rotation is approximately 3.0%. North-
south coupling beams reach mean maximum chord rotations of approximately 0.8%. The
mean plus standard deviation rotation for the north-south beams is approximately 1.5%.
The MCE is the maximum considered earthquake; therefore, these rotations are the
maximum demands that are expected for design and analysis considerations. Recall that
the Collapse Prevention limit state from ASCE 41-06 for coupling beams is given as
3.0%, which is not reached even with the consideration of the mean plus one std dev.
This is also well below the provided limit based on testing results of 6.0%. Based on
fragility relations defined in Chapter 6, the coupling beams in this structure will not
require any substantial repair in a major earthquake, and are unlikely to require anything
but minor repair in the form of epoxy injection of large residual cracking at the beam-
wall interface; approximately 10% of the EW beams are likely to require this minor
repair in the MCE event.
139
-0.03 -0.02 -0.01 0 0.01 0.02 0.03Rotation [rad]
0
10
20
30
40
Floo
r Lev
elN-SE-W
DS1
DS1
Figure 7.5 Coupling beam rotations (mean for 15 ground motions) for baseline model
at MCE level. Dotted lines indicate mean ± one standard deviation. Vertical
lines represent mean beam chord rotation at listed damage state
Coupling beam rotations for Model 1 (model that does not consider the impact of
the slab on the strength and stiffness) are plotted in Figure 7.6. The plot shows the mean
rotations for coupling beams on the north-south sides and the east-west sides of the core
wall. The dotted lines represent the mean rotation values +/- one standard deviation. The
maximum rotations both in the north-south and east-west beams are approximately 2.0%.
Consideration of mean plus standard deviation gives a maximum beam chord rotation of
approximately 3.5% for beams on the north-south sides of the building. This value
exceeds the Collapse Prevention limit state of 3.0%, but is still significantly less than the
maximum chord rotation limit imposed on the model based on test results. Based on the
fragility relations defined in the previous chapter, there is an approximately 50% chance
140
that coupling beams on the north-south side of the building are likely to require minor
repair in the MCE event. There is a small potential (approximately 10%) for major
damage in some of the north-south coupling beams in the event of MCE shaking.
-0.04 -0.02 0 0.02 0.04Rotation [rad]
0
10
20
30
40
Floo
r Lev
el
N-SE-W
DS1 DS1
DS2
DS2
Figure 7.6 Coupling beam rotations (mean for 15 ground motions) for Model 1 at MCE
level. Dotted lines indicate mean ± one standard deviation. Vertical lines
represent mean beam chord rotation at listed damage state
Figure 7.7 plots coupling beam rotations for Model 2 (model that considers higher
effective elastic stiffness). The plot shows the mean rotations for coupling beams on the
north-south sides and the east-west sides of the core wall. The dotted lines represent the
mean rotation values +/- one standard deviation. The maximum bean chord rotation is
approximately 1.5% and occurs in the east-west beams. However, the behavior of both
the north-south beams and the east-west beams are very similar. The mean plus one
141
standard deviation chord rotation is approximately 2.5%, which is less than the Collapse
Prevention limit state in ASCE 41. There is a small (~10%) chance that these beams will
require minor repair in an MCE event. These beams are unlikely to require major repair
even in the event of major shaking.
-0.03 -0.02 -0.01 0 0.01 0.02 0.03Rotation [rad]
0
10
20
30
40
Floo
r Lev
el
N-SE-W
DS1
DS1
Figure 7.7 Coupling beam rotations (mean for 15 ground motions) for Model 2 at MCE
level. Dotted lines indicate mean ± one standard deviation. Vertical lines
represent mean beam chord rotation at listed damage state
Figure 7.8 is a plot of the mean coupling beam rotations for all the models on (a)
north-south and (b) east-west sides of the building, to provide a direct comparison
between the different models. For the east-west beams, Model 2 and the Baseline model
have very similar coupling beam rotations; while Model 1 has higher rotations in the
coupling beams, especially in mid-height beams, with rotations reaching 2.0% vs. 1.5%
142
for the other two models. However, rotations in the north-south beams are different; the
Baseline model has substantially lower rotations (~1.0%) than either other model. Model
1 has the highest rotations, exceeding 2.0%, and Model 2 has rotations in excess of 1.5%.
Model 1 is expected to have the highest rotations as it has a lower elastic stiffness and,
more significantly, lower peak strength, meaning that yielding occurs sooner and that
increased shaking to the building causes increased plastic deformations. More interesting
is the difference between the Baseline model and Model 2, which has a higher elastic
stiffness and the same peak strength compared to the Baseline model. Due to their higher
stiffness, the coupling beams in Model 2 yielded prior to those in the Baseline model, and
thus plastic deformations were larger for Model 2 relative to the baseline model.
Therefore, the parameters used for coupling beam strength and stiffness will have a
substantial impact on coupling beam rotation.
-0.03 -0.02 -0.01 0 0.01 0.02 0.03Rotation (rad)
0
10
20
30
40
Floo
r Lev
el
BaselineModel 1Model 2
(a)
-0.03 -0.02 -0.01 0 0.01 0.02 0.03Rotation (rad)
0
10
20
30
40
Floo
r Lev
el
BaselineModel 1Model 2
(a)
143
-0.03 -0.02 -0.01 0 0.01 0.02 0.03Rotation (rad)
0
10
20
30
40
Floo
r Lev
el
BaselineModel 1Model 2
(b)
-0.03 -0.02 -0.01 0 0.01 0.02 0.03Rotation (rad)
0
10
20
30
40
Floo
r Lev
el
BaselineModel 1Model 2
(b)
Figure 7.8 Coupling beam rotations (mean for 15 ground motions) at MCE level for all
models (a) north-south side and (b) east-west side
7.2.2 Inter-story drifts
Plotted in Figures 7.9-7.11 are inter-story drifts for the Baseline model, Model 1, and
Model 2 for the MCE. The results are the mean for 15 ground motions, and the dotted
lines represent mean +/- one standard deviation. The maximum drift for the baseline
model is approximately 1.5% in the east-west direction. The maximum drift for the other
two models is approximately 1.4%, both in the east-west direction. Figure 7.12, which is
a plot of the mean inter-story drifts for all models, shows that the models all have very
similar patterns of inter-story drift. In the east-west direction, the baseline model has
higher drifts (~1.5%) compared with Models 1 and 2 (~1.4%). However, in the north-
144
south direction, Models 1 and 2 have higher drifts (~1.3%) compared to the baseline
model (1.1%). Therefore, the parameters used for coupling beam strength and stiffness do
not have a substantial impact on inter-story drift. This is likely because the wall behavior
has the biggest impact on lateral displacements, and wall strain demands are relatively
low (Tuna, 2009).
-0.02 -0.01 0 0.01 0.02Inter-story Drift
0
10
20
30
40
Floo
r Lev
el
E-WN-S
Figure 7.9 Inter-story drifts (mean for 15 ground motions) at MCE level for Baseline
model. Dotted lines represent mean ± one standard deviation
145
-0.02 -0.01 0 0.01 0.02Inter-story Drift
0
10
20
30
40
Floo
r Lev
elE-WN-S
Figure 7.10 Inter-story drifts (mean for 15 ground motions) at MCE level for Model 1.
Dotted lines represent mean ± one standard deviation
-0.02 -0.01 0 0.01 0.02Inter-story Drift
0
10
20
30
40
Floo
r Lev
el
E-WN-S
Figure 7.11 Inter-story drifts (mean for 15 ground motions) at MCE level for Model 2.
Dotted lines represent mean ± one standard deviation
146
-0.02 -0.01 0 0.01 0.02Inter-story Drift
0
10
20
30
40
Floo
r Lev
el
BaselineModel 1Model 2 (a)
-0.02 -0.01 0 0.01 0.02Inter-story Drift
0
10
20
30
40
Floo
r Lev
el
BaselineModel 1Model 2 (a)
-0.03 -0.02 -0.01 0 0.01 0.02 0.03Inter-story Drift
0
10
20
30
40
Floo
r Lev
el
BaselineModel 1Model 2 (b)
-0.03 -0.02 -0.01 0 0.01 0.02 0.03Inter-story Drift
0
10
20
30
40
Floo
r Lev
el
BaselineModel 1Model 2 (b)
Figure 7.12 Inter-story drifts (mean for 15 ground motions) at MCE level for all models
(a) north-south and (b) east-west
147
7.2.3 Core wall shear
Plotted in Figure 7.13 are the mean core wall shear forces over the building height for all
three models subjected to 15 ground motions. The results for each model are almost
identical; however in the north-south direction, Model 2 has slightly higher shear forces
in the core walls (~7.5% higher) than the Baseline model and Model 1. Therefore,
variations in coupling beam strength and stiffness parameters do not greatly impact the
core wall shear forces.
-12000 -8000 -4000 0 4000 8000 12000Shear Force [k]
0
10
20
30
40
Floo
r Lev
el
BaselineModel 1Model 2
(a)
-12000 -8000 -4000 0 4000 8000 12000Shear Force [k]
0
10
20
30
40
Floo
r Lev
el
BaselineModel 1Model 2
(a)
148
-12000 -8000 -4000 0 4000 8000 12000Shear Force [k]
0
10
20
30
40
Floo
r Lev
el
BaselineModel 1Model 2
(b)
-12000 -8000 -4000 0 4000 8000 12000Shear Force [k]
0
10
20
30
40
Floo
r Lev
el
BaselineModel 1Model 2
(b)
Figure 7.13 Core wall shear forces (mean for 15 ground motions) at MCE level for all
models (a) north-south and (b) east-west
7.3 Summary
This chapter presented a summary of a parametric study performed to determine the
sensitivity of different Engineering Demand Parameters to variations in stiffness and
strength of coupling beams. Three models, with varying strength and stiffness
parameters, of a 42-story building were loaded with 15 ground motions at MCE level.
The results indicate that coupling beam chord rotations were most sensitive to changes in
stiffness and strength, while inter-story drift and shear wall force were relatively
unaffected by the changes to the coupling beam designs.
149
Chapter 8 Conclusions
Eight coupling beam specimens with ln/h ratios of 2.4 and 3.33, and varying geometries
and reinforcement layouts, were tested under reversed cyclic loading and double
curvature bending. Modeling studies were also performed to investigate the applicability
of easily implemented modeling parameters that can accurately capture the load-
deformation behavior of coupling beams. Fragility relations were developed for both
diagonally- and conventionally-reinforced coupling beams at both low and high aspect
ratios. System-level studies were performed to investigate the impact of stiffness and
strength variations on different Engineering Demand Parameters. The following
conclusions can be drawn from the results.
1) Beams detailed according to the new provision in ACI 318-08, which allows for
full section confinement, have performance, in terms of strength and ductility, that
is better than beams detailed according to the old provision in ACI 318-05, which
requires confinement of the diagonal bar groups.
150
2) Including a reinforced concrete slab increases the beam shear strength
approximately 15-20%, whereas adding post-tensioning increases the beam shear
strength an additional 10%. However, the strength increase was directly related to
the increase in beam moment strength, as the beam shear force was limited by
flexural yielding.
3) Beams detailed to satisfy 1/2*Ash perform well at chord rotations θ < 3.0%.
However, at very large rotations (θ > 6.0%), the beams experienced greater levels
of damage (i.e. more spalling of cover concrete and substantially larger shear
cracks > 1/4”) compared with beams detailed to satisfy Ash. The results indicate
that the amount of transverse reinforcement required could be modestly reduced
for the beam aspect ratios tested, especially for beams with lower ductility
requirements (θ < 3.0%.). However, further study is necessary to determine if less
transverse reinforcement could be used for rotations exceeding 3%, or for beams
with lower aspect ratios (ln/h < 2).
4) Effective elastic stiffness values for test beams are determined to be ~15% of the
gross section stiffness, values that are much less than FEMA 356 and ASCE 41
prescribed values of 50% and 30%, respectively. Designers should therefore
utilize the slip/extension hinge model detailed in Supplement 1 to ASCE 41 to
better approximate the elastic stiffness of the coupling beam. As a rule of thumb it
is recommended to use a value of 0.20EcIg for coupling beam elastic stiffness.
151
5) Most damage experienced by coupling beams with aspect ratio ranging from 2.4
to 3.33 is concentrated at the beam-wall interface in the form of slip/extension of
diagonal reinforcement, even when axial load is applied to the beam via post-
tensioning. Beams not detailed with full section confinement experience more
damage at large rotations (θ > 6.0%).
6) ACI 318-08 implies equivalence between diagonally-reinforced and “frame
beams” for aspect ratios between 2.0 and 4.0. However, frame beams typically
achieve maximum plastic chord rotations of 3.5 to 4.0%, for cases where the
expected shear stresses are 4.0 to 5.0 'cf , or about one-half the values for
diagonally-reinforced coupling beams tested. Changes to ACI 318 code should be
considered to reduce the shear stress allowed for frame beams ( )e.g., 5.0 'cf ,
or to the ACI commentary to identify this significant difference in performance.
7) While flexural and shear deformation contributions are equivalent regardless of
the scale of the specimen, deformations due to slip and extension of the flexural
reinforcement at the beam-wall interface must be modified to account for the
scale of the given specimen. Thus the behavior of beam specimens tested at less
than full scale must be modified to account for the scale at which the test was
conducted.
152
8) Simple nonlinear models, either moment-hinge or shear-hinge, accurately
represent the load-deformation behavior of test beams. The flexural hinge model
better matches the test results in the unloading and reloading range, due to the
specific modeling parameters available in the computer software used (unloading
stiffness modeling parameters), although both models produce acceptable results
up to 3% total rotation for beams with ln/h between 2.0 and 4.0. Therefore,
depending on the computer program used, the influence of modeling parameters
on the load versus deformation responses should be compared with test results to
ensure that they adequately represent observed behavior. [Note that it is likely that
the unloading stiffness parameters will be included for the shear-hinge in the next
version of Perform 3D, so this discrepancy will no longer be an issue.]
9) Studies of shear-flexure interaction show that flexural deformations are
particularly important in beams with aspect ratios exceeding 2.25, while shear
deformations are most important in beams with aspect ratios less than 2.25. Using
a model that incorporates shear-flexure interaction can be used in addition to a
model incorporating slip/extension to obtain a reasonable lower-bound estimate
for the overall load-deformation behavior of a coupling beam.
10) Fragility relations can be particularly useful for engineers to evaluate the potential
damage and subsequent repair of coupling beams based on the expected beam
153
chord rotation in a given earthquake. They can also be used in coordination with
nonlinear modeling techniques to develop simplified modeling parameters and
acceptance criteria to further aid engineers in design. With this in mind,
recommendations for modifications to the nonlinear modeling parameters in
ASCE 41 for diagonally-reinforced coupling beams are provided. Specifically, it
is recommended that additional rows be added to account for different aspect
ratios and shear stresses.
11) Variations in coupling beam stiffness and strength can have impacts on the
behavior of the system as a whole. However, changes to these parameters have
the most impact directly on coupling beam rotation demands. There is little
impact from these parametric variations on both inter-story drift and core wall
shear forces.
154
Appendix A Summary of Test Results
Appendix A presents a summary of test results for each test specimen. Specifically, load-
deformation, axial elongation, deformation contributions, and damage patterns and
photographs are provided for all test specimens.
155
A.1 CB24F
Figure A.1 Initial dimensions [in.] between sensor rods on sides A and B of specimen
CB24F
-5-4-3-2-1012345
Rel.
Disp
. [in
.]
-12
-8
-4
0
4
8
12
Rota
tion
[%]
Figure A.2 Actual displacement history of specimen CB24F
156
-4.32 -2.16 0 2.16 4.32Relative Displacement [in]
-200
-100
0
100
200
Late
ral L
oad
[k]
-12 -6 0 6 12Beam Chord Rotation [%]
Figure A.3 Cyclic load-deformation plot for CB24F
-5 -2.5 0 2.5 5Relative Displacement [in]
-0.4
0
0.4
0.8
1.2
Axi
al e
long
atio
n [in
]
-14 -7 0 7 14Beam Chord Rotation [%]
-1
0
1
2
3
Axi
al e
long
atio
n [c
m]
CB24F
Figure A.4 Axial elongation for CB24F
157
0 0.01 0.02 0.03 0.04Beam Chord Rotation [rad.]
0
20
40
60
80
100
% C
ontri
butio
n
FlexureSlip/Ext.Shear
Figure A.5 Deformation contributions for CB24F
158
0 6 12 18 24 30 36Position along beam length [in.]
-0.0006
-0.00045
-0.0003
-0.00015
0
0.00015
0.0003
0.00045
0.0006
Curv
atur
e [1/
in.]
1.00%1.50%2.00%3.00%
0.11%0.29%0.50%0.75%
0 6 12 18 24 30 36Position along beam length [in.]
-0.00125-0.001
-0.00075-0.0005
-0.000250
0.000250.0005
0.000750.001
0.00125
Curv
atur
e [1/
in.]
1.00%1.50%2.00%3.00%
0.11%0.29%0.50%0.75%
Figure A.6 Curvature profile for CB24F (a) positive loading cycles and (b) negative
loading cycles
159
(a)(a)
(b)(b)
(c)(c)
Figure A.7 Damage patterns at peak deformation CB24F front side (a) positive loading
cycle, (b) negative loading cycle, (c) overall
160
(a)(a)
(b)(b)
(c)(c)
Figure A.8 Damage patterns at peak deformation CB24F back side (a) positive loading
cycle, (b) negative loading cycle, (c) overall
161
A.2 CB24D
Figure A.9 Initial dimensions [in.] between sensor rods on sides A and B of specimen
CB24D
-5-4-3-2-1012345
Rel.
Disp
. [in
.]
-12
-8
-4
0
4
8
12R
otat
ion
[%]
Figure A.10 Actual displacement history for specimen CB24D
162
-4.32 -2.16 0 2.16 4.32Relative Displacement [in]
-200
-100
0
100
200
Late
ral L
oad
[k]
-12 -6 0 6 12Beam Chord Rotation [%]
Figure A.11 Cyclic load-deformation relation for CB24D
-5 -2.5 0 2.5 5Relative Displacement [in]
-0.4
0
0.4
0.8
1.2
Axi
al e
long
atio
n [in
]
-14 -7 0 7 14Beam Chord Rotation [%]
-1
0
1
2
3
Axi
al e
long
atio
n [c
m]
Figure A.12 Axial extension of CB24D
163
0 0.01 0.02 0.03 0.04Rotation [% drift]
0
20
40
60
80
100
% C
ontri
butio
n
FlexureSlipShear
Figure A.13 Deformation contributions for CB24D
164
0 6 12 18 24 30 36Position along beam length [in.]
-0.0015-0.0012-0.0009-0.0006-0.0003
00.00030.00060.00090.00120.0015
Curv
atur
e [1/
in.]
1.00%1.50%2.00%3.00%
0.11%0.29%0.50%0.75%
0 6 12 18 24 30 36Position along beam length [in.]
-0.00125-0.001
-0.00075-0.0005
-0.000250
0.000250.0005
0.000750.001
0.00125
Curv
atur
e [1/
in.]
1.00%1.50%2.00%3.00%
0.11%0.29%0.50%0.75%
Figure A.14 Curvature profiles for CB24D (a) positive loading cycles, and (b) negative
loading cycles
165
(a)(a)
(b)(b)
(c)(c)
Figure A.15 Damage patterns at peak deformation CB24D front side (a) positive loading
cycle, (b) negative loading cycle, (c) overall
166
(a)(a)
(b)(b)
(c)(c)
Figure A.16 Damage patterns at peak deformation CB24D back side (a) positive loading
cycle, (b) negative loading cycle, (c) overall
167
A.3 CB24F-RC
Figure A.17 Initial dimensions [in.] between sensor rods on sides A and B of specimen
CB24F-RC
-6-5-4-3-2-10123456
Rel.
Disp
. [in
.]
-16-12-8-40481216
Rot
atio
n [%
]
Figure A.18 Actual displacement history for specimen CB24F-RC
168
-4.32 -2.16 0 2.16 4.32Relative Displacement [in]
-200
-100
0
100
200
Late
ral L
oad
[k]
-12 -6 0 6 12Beam Chord Rotation [%]
Figure A.19 Cyclic load-deformation plot for CB24F-RC
-5 -2.5 0 2.5 5Relative Displacement [in]
-0.4
0
0.4
0.8
1.2
Axi
al e
long
atio
n [in
]
-14 -7 0 7 14Beam Chord Rotation [%]
-1
0
1
2
3
Axi
al e
long
atio
n [c
m]
Figure A.20 Axial extension of CB24F-RC
169
0 0.01 0.02 0.03 0.04Rotation [% drift]
0
20
40
60
80
100
% C
ontr
ibut
ion
FlexureSlipShear
Figure A.21 Deformation contributions for CB24F-RC
170
0 6 12 18 24 30 36Position along beam length [in.]
-0.003
-0.00225
-0.0015
-0.00075
0
0.00075
0.0015
0.00225
0.003
Curv
atur
e [1/
in.]
1.00%1.50%2.00%3.00%
0.11%0.29%0.50%0.75%
0 6 12 18 24 30 36Position along beam length [in.]
-0.0008
-0.0006
-0.0004
-0.0002
0
0.0002
0.0004
0.0006
0.0008
Curv
atur
e [1/
in.]
1.00%1.50%2.00%3.00%
0.11%0.29%0.50%0.75%
Figure A.22 Curvature profiles CB24F-RC (a) positive loading cycles and (b) negative
loading cycles
171
(a)(a)
(b)(b)
Figure A.23 Damage cracking patterns at peak deformations CB24F-RC (a) front side all
cycles, (b) back side all cycles.
172
Rotation = 0.0075Slab
Beam
Rotation = 0.0075
Beam
SlabRotation = 0.0075
Beam
SlabRotation = 0.01
Beam
SlabRotation = 0.015
Beam
SlabRotation = 0.02
Beam
SlabRotation = 0.03
Beam
SlabRotation = 0.04
Beam
Figure A.24 CB24F-RC damage photos at peak rotation: 0.75%-4.0% beam chord
rotation
173
SlabRotation = 0.06
Beam
SlabRotation = 0.08
Beam
SlabRotation = 0.10
Beam
SlabRotation = 0.12
Beam
SlabRotation = 0.14
Beam
Figure A.25 CB24F-RC damage photos at peak rotation: 6.0%-14.0% beam chord
rotation
174
After Rotation = 0.0075Slab
Beam
After Rotation = 0.0075
Beam
SlabAfter Rotation = 0.0075
Beam
SlabAfter Rotation = 0.01
Beam
SlabAfter Rotation = 0.015
Beam
SlabAfter Rotation = 0.02
Beam
SlabAfter Rotation = 0.03
Beam
SlabAfter Rotation = 0.04
Beam
Figure A.26 CB24F-RC residual damage photos at zero rotation: after cycles at 0.75%-
4.0% beam chord rotation
175
SlabAfter Rotation = 0.06
Beam
SlabAfter Rotation = 0.08
Beam
SlabAfter Rotation = 0.10
Beam
SlabAfter Rotation = 0.12
Beam
SlabAfter Rotation = 0.14
Beam
Figure A.27 CB24F-RC residual damage photos at zero rotation: after cycles at 6.0%-
14.0% beam chord rotation
176
A.4 CB24F-PT
Figure A.28 Initial dimensions [in.] between sensor rods on sides A and B of specimen
CB24F-PT
-5-4-3-2-1012345
Rel.
Disp
. [in
.]
-12
-8
-4
0
4
8
12R
otat
ion
[%]
Figure A.29 Actual displacement history for specimen CB24F-PT
177
-4.32 -2.16 0 2.16 4.32Relative Displacement [in]
-200
-100
0
100
200
Late
ral L
oad
[k]
-12 -6 0 6 12Beam Chord Rotation [%]
Figure A.30 Cyclic load-deformation relation for CB24F-PT
-5 -2.5 0 2.5 5Relative Displacement [in]
0
0.4
0.8
1.2
Axi
al e
long
atio
n [in
]
-14 -7 0 7 14Beam Chord Rotation [%]
-1
0
1
2
3
Axi
al e
long
atio
n [c
m]
Figure A.31 Axial extension of CB24F-PT
178
-16-15-14-13-12-11-10-9-8-7-6
Load
in te
ndon
[k]
-16-15-14-13-12-11-10-9-8-7-6
Load
in te
ndon
[k]
-16-15-14-13-12-11-10-9-8-7-6
Load
in te
ndon
[k]
Figure A.32 Load in prestressing tendons for CB24F-PT
0 0.01 0.02 0.03 0.04Rotation [% drift]
0
20
40
60
80
100
% C
ontri
butio
n
FlexureSlipShear
Figure A.33 Deformation contributions for CB24F-PT
179
0 6 12 18 24 30 36Position along beam length [in.]
-0.0008
-0.0006
-0.0004
-0.0002
0
0.0002
0.0004
0.0006
0.0008
Curv
atur
e [1/
in.]
1.00%1.50%2.00%3.00%
0.11%0.29%0.50%0.75%
0 6 12 18 24 30 36Position along beam length [in.]
-0.0005
-0.000375
-0.00025
-0.000125
0
0.000125
0.00025
0.000375
0.0005
Curv
atur
e [1/
in.]
1.00%1.50%2.00%3.00%
0.11%0.29%0.50%0.75%
Figure A.34 Curvature profiles for CB24F-PT (a) positive loading cycles and (b)
negative loading cycles
180
(a)(a)
(b)(b)
Figure A.35 Damage cracking patterns at peak deformations CB24F-PT (a) front side all
cycles, (b) back side all cycles
181
A.5 CB24F-1/2-PT
Figure A.36 Initial dimensions [in.] between sensor rods on sides A and B of specimen
CB24F-1/2-PT
-4
-3
-2
-10
1
2
3
4
Rel.
Disp
. [in
.]
-10-8-6-4-20246810
Rot
atio
n [%
]
Figure A.37 Actual displacement history for specimen CB24F-1/2/PT
182
-4.32 -2.16 0 2.16 4.32Relative Displacement [in]
-200
-100
0
100
200
Late
ral L
oad
[k]
-12 -6 0 6 12Beam Chord Rotation [%]
Figure A.38 Cyclic load-deformation plot for CB24F-1/2-PT
-5 -2.5 0 2.5 5Relative Displacement [in]
-0.4
0
0.4
0.8
Axi
al e
long
atio
n [in
]
-14 -7 0 7 14Beam Chord Rotation [%]
-1
0
1
2
3
Axi
al e
long
atio
n [c
m]
Figure A.39 Axial extension of CB24F-1/2-PT
183
-13
-12
-11
-10
-9
-8
-7
-6
Load
in te
ndon
[k]
-13
-12
-11
-10
-9
-8
-7
-6
Load
in te
ndon
[k]
-13
-12
-11
-10
-9
-8
-7
-6
Load
in te
ndon
[k]
Figure A.40 Load in prestressing tendons for CB24F-1/2-PT
0 0.01 0.02 0.03 0.04Rotation [% drift]
0
20
40
60
80
100
% C
ontr
ibut
ion
FlexureSlipShear
Figure A.41 Deformation contributions for CB24-1/2-PT
184
0 6 12 18 24 30 36Position along beam length [in.]
-0.0012
-0.0009
-0.0006
-0.0003
0
0.0003
0.0006
0.0009
0.0012
Curv
atur
e [1/
in.]
1.00%1.50%2.00%3.00%
0.11%0.29%0.50%0.75%
0 6 12 18 24 30 36Position along beam length [in.]
-0.0016
-0.0012
-0.0008
-0.0004
0
0.0004
0.0008
0.0012
0.0016
Curv
atur
e [1/
in.]
1.00%1.50%2.00%3.00%
0.11%0.29%0.50%0.75%
Figure A.42 Curvature profiles for CB24F-1/2-PT (a) positive loading cycles and (b)
negative loading cycles
185
(a)(a)
(b)(b)
Figure A.43 Damage cracking patterns at peak deformations CB24F-1/2-PT (a) front side
all cycles, (b) back side all cycles
186
A.6 CB33F
Figure A.44 Initial dimensions [in.] between sensor rods on sides A and B of specimen
CB33F
-6-5-4-3-2-10123456
Rel.
Disp
. [in
.]
-10-8-6-4-20246810
Rot
atio
n [%
]
Figure A.45 Actual displacement history for specimen CB33F
187
-6 -3 0 3 6Relative Displacement [in]
-150
-100
-50
0
50
100
150
Late
ral L
oad
[k]
-10 -5 0 5 10Beam Chord Rotation [%]
Figure A.46 Cyclic load-deformation plot for CB33F
-5 -2.5 0 2.5 5Relative Displacement [in]
0
0.4
0.8
1.2
Axi
al e
long
atio
n [in
]
-14 -7 0 7 14Beam Chord Rotation [%]
-1
0
1
2
3
Axi
al e
long
atio
n [c
m]
Figure A.47 Axial elongation of CB33F
188
(a)(a)
(b)(b)
Figure A.48 Damage cracking patterns at peak deformations CB33F (a) front side all
cycles, (b) back side all cycles
189
Rotation = 0.0075
Rotation = 0.01
Rotation = 0.015
Rotation = 0.02
Rotation = 0.03
Rotation = 0.04
Rotation = 0.06
Figure A.49 CB33F damage photos at peak rotation: 0.75%-6.0% beam chord rotation
190
After Rotation = 0.01
After Rotation = 0.015
After Rotation = 0.02
After Rotation = 0.03
After Rotation = 0.04
After Rotation = 0.06
After Rotation = 0.08
Figure A.50 CB33F residual damage photos at zero rotation: after cycles at 1.0%-8.0%
beam chord rotation
191
A.7 CB33D
Figure A.51 Initial dimensions [in.] between sensor rods on sides A and B of specimen
CB33D
-5-4-3-2-1012345
Rel.
Disp
. [in
.]
-8
-6
-4
-20
2
4
6
8R
otat
ion
[%]
Figure A.52 Actual displacement history for specimen CB33D
192
-6 -3 0 3 6Relative Displacement [in]
-150
-100
-50
0
50
100
150
Late
ral L
oad
[k]
-10 -5 0 5 10Beam Chord Rotation [%]
Figure A.53 Cyclic load-deformation plot for CB33D
-5 -2.5 0 2.5 5Relative Displacement [in]
0
0.2
0.4
0.6
0.8
1
Axi
al e
long
atio
n [in
]
-14 -7 0 7 14Beam Chord Rotation [%]
-1
0
1
2
3
Axi
al e
long
atio
n [c
m]
Figure A.54 Axial extension for CB33D
193
(a)(a)
(b)(b)
Figure A.55 Damage cracking patterns at peak deformations CB33D (a) front side all
cycles, (b) back side all cycles
194
Rotation = 0.01
Rotation = 0.015
Rotation = 0.02
Rotation = 0.03
Rotation = 0.04
Rotation = 0.06
Figure A.56 CB33D damage photos at peak rotation: 1.0%-6.0% beam chord rotation
195
After Rotation = 0.01
After Rotation = 0.015
After Rotation = 0.02
After Rotation = 0.03
After Rotation = 0.04
After Rotation = 0.06
Figure A.57 CB33D residual damage photos at zero rotation: after cycles at 1.0%-6.0%
beam chord rotation
196
A.8 FB33
Figure A.58 Initial dimensions [in.] between sensor rods on sides A and B of specimen
FB33
-5-4-3-2-1012345
Rel.
Disp
. [in
.]
-8
-6
-4
-20
2
4
6
8R
otat
ion
[%]
Figure A.59 Actual displacement history for specimen FB33
197
-5 -2.5 0 2.5 5Relative Displacement [in]
-80
-40
0
40
80
Late
ral L
oad
[k]
-8 -4 0 4 8Beam Chord Rotation [%]
Figure A.60 Cyclic load-deformation plot of FB33
-5 -2.5 0 2.5 5Relative Displacement [in]
0
0.2
0.4
0.6
0.8
1
Axi
al e
long
atio
n [in
]
-14 -7 0 7 14Beam Chord Rotation [%]
-1
0
1
2
3
Axi
al e
long
atio
n [c
m]
Figure A.61 Axial extension of FB33
198
(a)(a)
(b)(b)
Figure A.62 Damage cracking patterns at peak deformations FB33 (a) front side all
cycles, (b) back side all cycles
199
Rotation = 0.0075
Rotation = 0.01
Rotation = 0.015
Rotation = 0.02
Rotation = 0.03
Rotation = 0.04
Rotation = 0.05
Rotation = 0.06
Figure A.63 FB33 damage photos at peak rotation: 0.75%-6.0% beam chord rotation
200
After Rotation = 0.01
After Rotation = 0.015
After Rotation = 0.02
After Rotation = 0.03
After Rotation = 0.04
After Rotation = 0.05
Figure A.64 FB33 residual damage photos at zero rotation: after cycles at 1.0%-5.0%
beam chord rotation
201
Appendix B Slip/Extension Calculation Example
Problem Statement
For the cross-section of CB24F, determine the rotation, θ, due to slip/extension of
flexural reinforcement at the beam-wall interface prior to yielding.
(a) (b)
θd
x
δtot
θd
x
δtot
Figure B.1 (a) Cross section of CB24F to be used for slip/ext calculation; and (b)
definition of slip/extension crack and corresponding rotation
202
Given:
2 2
'
7 / 8" 22.23
0.6 387
6.8 46.970 482.7
33" 83812.625"
@
2200 ( )
5"
b
b
c
y
d
s y
y
d mm
A in mm
f ksi MPaf ksi MPa
L mmd
f f
M in k from M analysis
x
φ
= =
= =
= == =
= ==
=
= − −
=
Calculations: Determine @ totM δθ−
(482.7 ) (22.23 ) 3.24 4 (838 )y b
ed
f d MPa mmu Mpal mm× ×
= = =× ×
22.23 1.744 4 3.2
s b se s
e
f d f mmL fu MPa× ×
= = = ×× ×
' 46.89(20 ) (20 22.23/ 4) 18.14 30 30
b cu
d fu MPa= − × = − × =
1 '
30 0.80sc
mmf
δ = =
203
@ fs = fy
'
''
'2.5
1
@
1.74 840
3.194
( ) 0.0104
1.25 / 2 1.05
0.0104 1.05 0.0417"
0.0417" 0.0054712.625" 5"
2200
/ 2200 / 0.0054 40
tot
e e y
y be
e
es s
u
ext y e
tot s ext
tot
y
y
L L f mm
f du MPa
L
u mmu
L mm
mm mm
d xM in k
K M
δ
δ δ
δ ε
δ δ δ
δθ
θ
= = × =
×= =
×
= × =
= × × =
= + = + =
= = =− −
= −
= = = 2200
Result
This M-θ relation represents the deformation characteristics of the beam in the elastic
region due to slip/extension of the flexural reinforcement. It can be implemented as an
M-θ hinge in a model to modify the elastic stiffness of the member.
204
-0.008 -0.004 0 0.004 0.008Rotation (rad)
-4000
-2000
0
2000
4000
Mom
ent [
in-k
]
Figure B.2 Elastic slip/extension moment-rotation hinge properties to be implemented
in nonlinear model
Slip/Extension springsSlip/Extension springs
Figure B.3 Schematic of slip/extension springs in compound element
205
Appendix C Procedure to Estimate EcIeff
Problem Statement
Determine an estimate for the effective elastic stiffness (EcIeff) as a function of the gross
section stiffness (EcIg), considering the influence of slip/extension deformations.
Calculations
1) Calculate θslip@yield, by following the procedure in Appendix B.
2) Calculate θflex@yield, by the following:
a. @
miny
ACIy
M
VV V
⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠
b. Use EIeff for flexure = 0.5EIg
c. 2 2
@12 ( .) 6
y n y nflex
eff g
V l V lyield
EI for flex EIθ
× ×= =
× ×
3) Calculate θtot:
a. tot slip flexθ θ θ≅ +
4) Calculate EcIeff:
206
a. 2
12eff y n
g g tot
EI V lEI EI θ
×=
× ×
Results
This value of EcIeff as a function of EcIg can be directly input to a model (modifying the
flexural stiffness) in place of a slip/extension hinge model. This has the same impact as
the slip hinge, but would ease computation time. This procedure is practical only for
members that have relatively low shear deformations, and is shown mainly as an exercise
to illustrate a potential alternative method to implement the decreased stiffness due to the
slip/extension deformations (rather than using a slip/extension hinge).
207
Appendix D Modeling Parameters
Summarized below are the parameters used in modeling of diagonally reinforced
coupling beams using CSI Perform 3D. Specifically, these parameters are for CB24F.
Mn-Hinge Model
The Moment-hinge model consists of an elastic RC cross-section, Mn-θ hinges, and
Slip/Extension hinges. The properties listed are for CB24F.
Mn-Rotation Springs
Slip/Ext. Springs
Figure D.1 Schematic for Mn-hinge model, including elastic cross section, Mn-rotation
hinges, and slip/extension hinges
The elastic RC cross-section has the following properties:
Cross Section: Beam, Reinforced Concrete Section
Shape and Dimensions
Section Shape: Rectangle
B: 12 [in] D: 15 [in]
208
Section Stiffness
Axial Area: 180 [in2]
Shear Area: 0 (Shear area = 0 means no shear deformation)
Material Stiffness
Young’s Modulus: 1800 [ksi] Poisson’s Ratio: 0.2 Shear Modulus: 692
The Slip/Extension Hinges have the following properties:
Inelastic: Semi-Rigid Moment Connection
Basic F-D Relationship
K0: 402200 [k-in/rad2]
FU: 3100 [in-k]
DX: 0.14 [rad]
The Mn-θ hinges have the following properties:
Inelastic: Moment Hinge, Rotation Type
Basic F-D Relationship
FY: 2350 [in-k] DU: 0.075 [rad]
FU: 2500 [in-k] DX: 0.130 [rad]
Strength Loss
DL: 0.08 FR/FU: 0.3
DR: 0.1 Interaction Factor: 0.25
Cyclic Degradation
209
Point Energy Factor
Y 0.5
U 0.45
L 0.4
R 0.35
X 0.35
Unloading Stiffness Factor: 0.5
Alternatively, similar results can be obtained by modifying the cross-section properties
such that Young’s Modulus = 0.15EcIg = 540 [ksi], and eliminating the slip/ext hinge
altogether.
Vn-Hinge Model
The Shear-hinge model consists of an elastic RC cross-section, Vn-δ hinges, and
Slip/Extension hinges. The properties listed are for CB24F.
Vn-Displacement Hinge
Slip/Ext. Springs
Figure D.2 Schematic for Vn-hinge model
The elastic RC cross-section has the following properties:
210
Cross Section: Beam, Reinforced Concrete Section
Shape and Dimensions
Section Shape: Rectangle
B: 12 [in] D: 15 [in]
Section Stiffness
Axial Area: 180 [in2]
Shear Area: 0 (Shear area = 0 means no shear deformation)
Material Stiffness
Young’s Modulus: 1800 [ksi] Poisson’s Ratio: 0.2 Shear Modulus: 692
The Slip/Extension Hinges have the following properties:
Inelastic: Semi-Rigid Moment Connection
Basic F-D Relationship
K0: 402200 [k-in/rad2]
FU: 3100 [in-k]
DX: 0.14 [rad]
211
The Vn-δ hinges have the following properties:
Inelastic: Shear Hinge, Displacement Type
Basic F-D Relationship
FY: 130 [k] DU: 2.7 [in]
FU: 136 [k] DX: 4.7 [in]
Strength Loss
DL: 2.88 [in] FR/FU: 0.3
DR: 3.31 [in] Interaction Factor: 0.25
Cyclic Degradation
Point Energy Factor
Y 0.5
U 0.45
L 0.4
R 0.35
X 0.35
Alternatively, similar results can be obtained by modifying the cross-section properties
such that Young’s Modulus = 0.15EcIg = 540 [ksi], and eliminating the slip/ext hinge
altogether.
212
Appendix E Material Testing
0 0.04 0.08 0.12 0.16 0.2ε [in/in]
0
20
40
60
80
100
σ [k
si]
bar1bar2bar3
Figure E.1 Diagonal #7 bars; tested by twining laboratories; based on given fy, fu, and %
elongation
213
0 0.002 0.004 0.006 0.008 0.01ε [in/in]
0
2
4
6
8
σ [k
si]
cyl1cyl2cyl3cyl4cyl5cyl6
Figure E.2 Concrete cylinders CB24F, CB24D, CB33F, CB33D; 6”x12” tested by
twining laboratories; curve fit based on f’c
0 0.002 0.004 0.006 0.008 0.01ε [in/in]
0
2
4
6
8
10
σ [k
si]
twining1twining2twining3twining4ucla1ucla2ucla3
Figure E.3 Concrete Cylinders CB24F-RC; 6”x12” tested by twining laboratories;
4”x8” tested by ucla; curve fit based on f’c
214
0 0.002 0.004 0.006 0.008 0.01
ε [in/in]
0
2
4
6
8
10
σ [k
si]
twining1twining2twining3twining4ucla1ucla2ucla3
Figure E.4 Concrete Cylinders CB24F-PT; 6”x12” tested by twining laboratories;
4”x8” tested by ucla; curve fit based on f’c
0 0.002 0.004 0.006 0.008 0.01
ε [in/in]
0
2
4
6
8
σ [k
si]
twining1twining2twining3twining4ucla1ucla2ucla3
Figure E.5 Concrete cylinders CB24F-1/2-PT; 6”x12” tested by twining laboratories;
4”x8” tested by ucla; curve fit based on f’c
215
0 0.002 0.004 0.006 0.008 0.01ε [in/in]
0
2
4
6
8
σ [k
si]
twining1twining2twining3twining4ucla1ucla2ucla3
Figure E.6 Concrete cylinder tests FB33; 6”x12” tested by twining laboratories; 4”x8”
tested by ucla; curve fit based on f’c
216
Appendix F Ground Motion Selection
Methodology
(Tuna, 2009)
F.1 Ground Motion Selection and Scaling Assumptions
• Tmin & Tmax at 0.5sec & 10.0 sec.
• Maximum acceptable scale factor = 4.0
• No restriction on magnitude
• Rmin & Rmax at 0.0 and 70.0 Km
• Min and max shear wave velocity = 200.0 and 700.0 m/s
• Low pass filter frequency lower than 0.1 Hz
• Used a subset of NGA database (no aftershocks & etc.)
• Diversify motions from various events as much as possible
F.2 Procedure
1. Target spectrum obtained from Marshal for 5% damping
2. A subset of NGA database is used to identify motion.
3. Records are ranked according to the error between target spectrum and geometric mean
217
of ground motion pairs.
4. A weight function of 10% for periods between 0.5 to 3 seconds, 60% from 3 seconds
to 7 seconds, and 30% from 7 seconds to 10 seconds is used.
5. From each earthquake not more than 2 records were selected.
6. The records are filtered using 8-node filter and down-sampled with dt=0.04 sec.
218
Appendix G Load-Deformation Backbone
Determination
In this appendix, a brief overview of a standardized process to develop elasto-plastic
load-deformation backbone curves is presented in a series of figures.
1) Determine the load-deformation backbone relation based on the test data by plotting
the peak of each loading increment.
2) Using two points on the load-deformation plot, θ1 and θ2, determine the average shear
strength over this yield plateau. θ1 and θ2 are defined as a rotation after yield and a
219
rotation before strength degradation, respectively. Vave is defined as the average shear
strength over the yield plateau.
Vave
θ1 θ2
Vave
θ1 θ2
3) The initial slope of the idealized backbone curve (red) is based on the secant slope at
2/3 of Vave. The point at which this line crosses Vave defines the yield point of the beam. θy
is defined as the beam chord rotation at yield.
Vave
2/3 Vave
θy θ1 θ2
Vave
2/3 Vave
θy θ1 θ2
220
4) Define the ultimate rotation θu as the rotation corresponding to onset of significant
lateral strength degradation (<0.8Vave). The plastic rotation capacity is defined as the
rotation between the θy and θu.
Vave
2/3 Vave
0.8 Vave
θy θ1 θ2 θu
Vave
2/3 Vave
0.8 Vave
θy θ1 θ2 θu
5) Define θr as the rotation corresponding to the residual capacity of the member
(0.3Vave). θr is assumed as 1.15θu. This rotation governs the degradation of the member.
The residual capacity can be maintained to θx defined as the maximum beam chord
rotation.
Vave
2/3 Vave
0.8 Vave
θy θ1 θ2 θu
0.3 Vave
θr θx
Vave
2/3 Vave
0.8 Vave
θy θ1 θ2 θu
0.3 Vave
θr θx
221
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