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


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


Los Angeles, CA

2004-2006 Engineering Aide

Dept. of Civil and Environmental Engineering

Los Angeles, CA

2008 M.S., Civil and Environmental Engineering


Los Angeles, CA

2006-2010 Graduate Student Researcher


2008-2010 Teaching Assistant


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


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


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


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


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]


-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]


-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


-200

-100

0

100

200

Late

ral L

oad

[k]


-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)


-0.4

0

0.4

0.8

1.2

Axia

l elo

ngat

ion

[in]


-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


-200

-100

0

100

200

Late

ral L

oad

[k]


-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)


-0.4

0

0.4

0.8

1.2

Axi

al elo

ngat

ion

[in]


-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.


-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)


-1.5

-1

-0.5

0

0.5

1

1.5

V/V

nFB33Xiao ln/h=3(a)


-1.5

-1

-0.5

0

0.5

1

1.5

V/V

n

FB33Xiao ln/h=4(b)


-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%


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%


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






Figure 4.19 Residual (zero displacement) damage photos (CB24F) after cycles of

rotations 1.0%-8.0%

71







Figure 4.20 Residual (zero displacement) damage photos (CB24D) after cycles of

rotations 1.0%-8.0%

72







Figure 4.21 Residual (zero displacement) damage photos (CB24F-PT) after cycles of

rotations 1.0%-8.0%

73







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


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


-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).


-150

-100

-50

0

50

100

150

Late

ral L

oad

[k]

-660

-330

0

330

660

Late

ral L

oad

[kN

]


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.


-75

-50

-25

0

25

50

75

Late

ral L

oad

[k]

-330

-165

0

165

330

Late

ral L

oad

[kN

]


(FB33)


-75

-50

-25

0

25

50

75

Late

ral L

oad

[k]

-330

-165

0

165

330

Late

ral L

oad

[kN

]


(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.


-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

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

0.2

0.4

0.6

0.8

1

Prob

abili

ty o

f dam

age

stat

e oc

curr

ing


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


Figure 6.7 Fragility curves for diagonally-reinforced concrete coupling beams with


127


0

0.2

0.4

0.6

0.8

1

Prob

abili

ty o

f dam

age

stat

e oc

curr

ing


Figure 6.8 Fragility curves for conventionally-reinforced concrete coupling beams with


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

0.25

0.5

0.75

1

1.25

1.5Sh

ear F

orce

[V/V

yexp

]

(θy, Vyexp)


(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

0.25

0.5

0.75

1

1.25

1.5

Shea

r For

ce [V

/Vye

xp]

(θy, Vyexp)


(θ10, Vrexp)


0

0.25

0.5

0.75

1

1.25

1.5

Shea

r For

ce [V

/Vye

xp]

(θy, Vyexp)


(θ10, Vrexp)

Figure 7.2 Coupling beam shear-displacement hinge backbone properties for Model 1

135


0

0.25

0.5

0.75

1

1.25

1.5

Shea

r For

ce [V

/Vye

xp]

(θy, Vyexp)


(θ10, Vrexp)


0

0.25

0.5

0.75

1

1.25

1.5

Shea

r For

ce [V

/Vye

xp]

(θy, Vyexp)


(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


(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


(b)

-0.03 -0.02 -0.01 0 0.01 0.02 0.03Rotation (rad)

0

10

20

30

40

Floo

r Lev

el


(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

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

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

10

20

30

40

Floo

r Lev

el

BaselineModel 1Model 2 (a)


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


(a)

-12000 -8000 -4000 0 4000 8000 12000Shear Force [k]

0

10

20

30

40

Floo

r Lev

el


(a)

148

-12000 -8000 -4000 0 4000 8000 12000Shear Force [k]

0

10

20

30

40

Floo

r Lev

el


(b)

-12000 -8000 -4000 0 4000 8000 12000Shear Force [k]

0

10

20

30

40

Floo

r Lev

el


(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


-200

-100

0

100

200

Late

ral L

oad

[k]


Figure A.3 Cyclic load-deformation plot for CB24F


-0.4

0

0.4

0.8

1.2

Axi

al e

long

atio

n [in

]


-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%


-0.00125-0.001

-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


161

A.2 CB24D


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


-200

-100

0

100

200

Late

ral L

oad

[k]


Figure A.11 Cyclic load-deformation relation for CB24D


-0.4

0

0.4

0.8

1.2

Axi

al e

long

atio

n [in

]


-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


-0.0015-0.0012-0.0009-0.0006-0.0003

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%


-0.00125-0.001

-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


166

(a)(a)

(b)(b)

(c)(c)

Figure A.16 Damage patterns at peak deformation CB24D back side (a) positive loading


167

A.3 CB24F-RC


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


-200

-100

0

100

200

Late

ral L

oad

[k]


Figure A.19 Cyclic load-deformation plot for CB24F-RC


-0.4

0

0.4

0.8

1.2

Axi

al e

long

atio

n [in

]


-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


-0.003

-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%


-0.0008

-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


Beam

SlabAfter Rotation = 0.0075

Beam


Beam


Beam


Beam


Beam


Beam

Figure A.26 CB24F-RC residual damage photos at zero rotation: after cycles at 0.75%-

4.0% beam chord rotation

175


Beam


Beam


Beam


Beam


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


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


-200

-100

0

100

200

Late

ral L

oad

[k]


Figure A.30 Cyclic load-deformation relation for CB24F-PT


0

0.4

0.8

1.2

Axi

al e

long

atio

n [in

]


-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


-0.0008

-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%


-0.0005

-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


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


-200

-100

0

100

200

Late

ral L

oad

[k]


Figure A.38 Cyclic load-deformation plot for CB24F-1/2-PT


-0.4

0

0.4

0.8

Axi

al e

long

atio

n [in

]


-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


-0.0012

-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%


-0.0016

-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


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


-150

-100

-50

0

50

100

150

Late

ral L

oad

[k]


Figure A.46 Cyclic load-deformation plot for CB33F


0

0.4

0.8

1.2

Axi

al e

long

atio

n [in

]


-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


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








Figure A.50 CB33F residual damage photos at zero rotation: after cycles at 1.0%-8.0%

beam chord rotation

191

A.7 CB33D


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


-150

-100

-50

0

50

100

150

Late

ral L

oad

[k]


Figure A.53 Cyclic load-deformation plot for CB33D


0

0.2

0.4

0.6

0.8

1

Axi

al e

long

atio

n [in

]


-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


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







Figure A.57 CB33D residual damage photos at zero rotation: after cycles at 1.0%-6.0%

beam chord rotation

196

A.8 FB33


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


-80

-40

0

40

80

Late

ral L

oad

[k]


Figure A.60 Cyclic load-deformation plot of FB33


0

0.2

0.4

0.6

0.8

1

Axi

al e

long

atio

n [in

]


-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


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







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


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


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


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]


Figure E.4 Concrete Cylinders CB24F-PT; 6”x12” tested by twining laboratories;


0 0.002 0.004 0.006 0.008 0.01

ε [in/in]

0

2

4

6

8

σ [k

si]


Figure E.5 Concrete cylinders CB24F-1/2-PT; 6”x12” tested by twining laboratories;


215

0 0.002 0.004 0.006 0.008 0.01ε [in/in]

0

2

4

6

8

σ [k

si]


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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Documents

Testing and Modeling of Reinforced Concrete Coupling Beams