PLASTIC HINGE LENGTH IN HIGH-RISE CONCRETE SHEAR WALLS by ALFREDO GUILLERMO BOHL ARBULU B.Sc, Universidad Peruana de Ciencias Aplicadas, Peru, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA November 2006 Â© Alfredo Guillermo Bohl Arbulu, 2006 11 ABSTRACT The fJexural displacement capacity of a concrete shear wall depends on the length of the plastic hinge. Typical building codes and several researchers recommend the use of an equivalent plastic hinge length at the base equal to half the wall length. However, the plastic hinge length is also influenced by parameters other than the wall length. There are currently no recommendations on what should be the plastic hinge length for parallel walls of different lengths in a high-rise building. A parametric study was conducted to investigate the parameters that affect the length of the plastic hinge in concrete walls. The walls were analyzed using program VecTor2. The analytical model was validated with tests results performed on wall specimens. The results obtained show that the inelastic curvatures vary linearly over the plastic hinge length. The shape of the strain profile in slender walls after cracking depends on the amount of reinforcement. Longer walls have larger plastic hinge lengths than shorter walls. Compressive axial loads reduce the plastic hinge length, tensile axial loads have the opposite effect. A simple shear model was proposed to estimate the increase in plastic hinge length when the shear stresses are high. Walls of different lengths interconnected by rigid slabs at various levels have different curvature distributions and plastic hinge lengths. The curvatures in the longer wall do not change whether it is alone or combined with a wall of shorter length. The shorter length wall is subjected to larger curvatures at the base when it is combined. A simple model was proposed to predict the maximum curvature in the shorter wall. iii TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS iii LIST OF TABLES vii LIST OF FIGURES xii LIST OF SYMBOLS xv ACKNOWLEDGEMENTS xx DEDICATION xxi CHAPTER 1: INTRODUCTION 1 1.1 Research significance 1 1.2 Objective of the thesis 3 1.3 Organization of the thesis 4 CHAPTER 2: LITERATURE REVIEW 5 2.1 Concept of plastic hinge length 5 2.2 Chan (1955) 10 2.3 Baker (1956) 12 2.4 Cohn and Petcu (1963) 12 2.5 Baker and Amarakone (1964) 13 2.6 Sawyer (1964) 16 2.7 Mattock (1964) 17 2.8 Corley (1966) 19 2.9 Mattock (1967) 20 2.10 ACI-ASCE Committee 428 (1968) 20 2.11 Priestly, Park and Potangaroa (1981) 21 2.12 Park, Priestly and Gill (1982) 22 2.13 Oesterle, Aristizabal-Ochoa, Shiu and Corley (1984) 23 2.14 Paulay(1986) 24 2.15 Zahn, Park and Priestly (1986) 24 2.16 Priestly and Park (1987) 25 2.17 Paulay and Priestly (1992) 26 2.18 Wallace and Moehle( 1992) 27 2.19 Moehle(1992) 27 2.20 Paulay and Priestly (1993) 27 2.21 Sasani and Der Kiureghian (2001) 27 2.22 Mendis(2001) 28 iv 2.23 Panagiotakos and Fardis (2001) 30 2.24 Thomsen and Wallace (2004) 32 2.25 Summary 34 CHAPTER 3: ANALYTICAL METHODS. 38 3.1 Introduction to program VecTor2 38 3.2 Theoretical bases of program VecTor2 38 3.3 Finite element formulation 38 3.4 Models for concrete in compression 39 3.4.1 Compression pre-peak response 39 3.4.2 Compression post-peak response 40 3.4.3 Compression softening 41 3.4.4 Confinement strength 42 3.5 Models for concrete in tension 42 3.5.1 Tension stiffening 43 3.5.2 Tension softening 44 3.5.3 Cracking criterion 45 3.6 Models for slip distortions in concrete 45 3.6.1 Stress-based model 46 3.6.2 Constant rotation lag model 47 3.7 Models for reinforcement 47 3.7.1 Stress-strain response 47 3.7.2 Dowel action 49 CHAPTER 4: COMPARISON OF ANALYTICAL PREDICTIONS WITH ISOLATED WALL TEST RESULTS 50 4.1 Scope of analysis 50 4.2 High-rise shear wall tested at the University of British Columbia 50 4.2.1 Description of the wall specimen 50 4.2.2 Instrumentation 51 4.2.3 Test procedure 52 4.2.4 Test results 52 4.2.5 Analytical model of the wall specimen 55 4.2.6 Analytical results 58 4.2.7 Comparison of experimental and analytical results 59 4.2.8 Equivalent plastic hinge length 62 4.2.9 Distribution of inelastic curvatures 66 4.2.10 Analysis of strain profiles 68 4.3 Rectangular shear wall tested at Clarkson University 79 4.3.1 Description of the wall specimen 79 4.3.2 Instrumentation 80 4.3.3 Test procedure 80 4.3.4 Test results 80 4.3.5 Analytical model of the wall specimen 80 4.3.6 Analytical results 83 4.3.7 Comparison of experimental and analytical results 83 CHAPTER 5: PARAMETRIC STUDY OF CONCRETE WALLS 89 5.1 Scope of analysis 89 5.2 Description of wall models 89 5.3 Analytical model of walls 90 5.4 Analytical results 94 5.5 Revision of analytical results 99 5.6 Influence of wall length 102 5.7 Influence of shear span 105 5.8 Influence of diagonal cracking 107 5.9 Influence of axial load U9 5.10 Summary 127 CHAPTER 6: SYSTEMS OF WALLS OF DIFFERENT LENGTHS CONNECTED TOGETHER B Y RIGID SLABS 128 6.1 Scope of analysis 128 6.2 Wall 1 combined with Wall 2 128 6.2.1 Description of wall system model 128 6.2.2 Analytical model of wall system 129 6.2.3 Analytical results 133 6.2.4 Discussion of analytical results 137 6.3 Wall 1 combined with Wall 3 144 6.3.1 Description of wall system model 144 6.3.2 Analytical model of wall system 145 6.3.3 Analytical results 150 6.3.4 Discussion of analytical results 151 6.4 Wall 1 combined with Column 1 153 6.4.1 Description of wall system model 153 6.4.2 Analytical model of wall system 154 6.4.3 Analytical results 158 vi 6.4.4 Discussion of analytical results 158 CHAPTER 7: CONCLUSIONS 160 REFERENCES 162 APPENDIX A: UBC WALL TEST 168 APPENDIX B: CLARKSON UNIVERSITY WALL TEST 184 APPENDIX C: CALCULATIONS FOR PARAMETRIC STUDY 186 Vl l L I S T O F T A B L E S Table 2.1 Summary of previous research done on plastic hinge length 37 Table 4.1 Material properties of UBC wall model 58 Table 4.2 UBC wall parameters 63 Table 4.3 Parameters required in plastic hinge length models 63 Table 4.4 Comparison of equivalent plastic hinge length models 64 Table 4.5 Material properties of UBC wall model with added reinforcement 75 Table 4.6 Material properties of specimen RW2 model 83 Table 4.7 Material properties of specimen RW2 model accounting for cover spalling 87 Table 5.1 Material properties of Wall 1 and 2 94 Table 5.2 Predicted and measured plastic hinge lengths for Wall 1 105 Table 5.3 Predicted and measured plastic hinge lengths for Wall 2 105 Table 5.4 Predicted and measured plastic hinge lengths for Wall 1 for a shear span of 35659mm 106 Table 5.5 Predicted and measured plastic hinge lengths for Wall 2 for a shear span of 35659mm 106 Table 5.6 Predicted and measured plastic hinge lengths for Wall 1 for a shear span of 27430mm 106 Table 5.7 Predicted and measured plastic hinge lengths for Wall 2 for a shear span of 27430mm 106 Table 5.8 Predicted and measured plastic hinge lengths for Wall 1 for a shear span of 19201mm 106 Table 5.9 Predicted and measured plastic hinge lengths for Wall 2 for a shear span of 19201mm 107 Table 5.10 Material properties of Wall 1 and 2 with a web thickness of 254mm 108 Table 5.11 Predicted and measured plastic hinge lengths for Wall 1 for a web thickness of 254mm and a shear span of 27430mm 110 Table 5.12 Predicted and measured plastic hinge lengths for Wall 2 for a web thickness of 254mm and a shear span of 27430mm 110 Table 5.13 Predicted and measured plastic hinge lengths for Wall 1 for a web thickness of 254mm and a shear span of 19201mm 110 Table 5.14 Predicted and measured plastic hinge lengths for Wall 2 for a web thickness of 254mm and a shear span of 19201mm I l l Table 5.15 Yield drifts and shear stresses of Wall 1 and 2 I l l Table 5.16 Predicted and measured plastic hinge lengths for Wall 1 for a shear span of 19201mm using shear model 117 Table 5.17 Predicted and measured plastic hinge lengths for Wall 1 for a web thickness of 254mm and a shear span of 27430mm using shear model 118 Table 5.18 Predicted and measured plastic hinge lengths for Wall 1 for a web thickness of 254mm and a shear span of 19201mm using shear model 118 Table 5.19 Yield curvature and yield moment of Wall 1 and 2 for different axial load ratios 122 Vl l l Table 5.20 Predicted and measured plastic hinge lengths for Wall 1 for a compressive axial load ratio of 0.3 122 Table 5.21 Predicted and measured plastic hinge lengths for Wall 2 for a compressive axial load ratio of 0.3 122 Table 5.22 Predicted and measured plastic hinge lengths for Wall 1 for a compressive axial load ratio of 0.2 122 Table 5.23 Predicted and measured plastic hinge lengths for Wall 2 for a compressive axial load ratio of 0.2 122 Table 5.24 Predicted and measured plastic hinge lengths for Wall 1 for no axial load 123 Table 5.25 Predicted and measured plastic hinge lengths for Wall 2 for no axial load 123 Table 5.26 Predicted and measured plastic hinge lengths for Wall 1 for a tensile axial load ratio of 0.02 123 Table 5.27 Predicted and measured plastic hinge lengths for Wall 2 for a tensile axial load ratio of 0.02 123 Table 5.28 Predicted and measured plastic hinge lengths for Wall 1 for a tensile axial load ratio of 0.05 123 Table 5.29 Predicted and measured plastic hinge lengths for Wall 2 for a tensile axial load ratio of 0.05 124 Table 5.30 Measured plastic hinge lengths at maximum drift 127 Table 6.1 Material properties in rectangular and triangular elements of Wall 1 and 2 132 Table 6.2 Material properties in truss elements of Wall 1 and 2 132 Table 6.3 Displacement components up to the fourth storey of Wall 1 and 2 combined for a drift of 2% 140 Table 6.4 Predicted and measured maximum curvature for Wall 2 combined 144 Table 6.5 Predicted and measured maximum curvature for Wall 3 combined 153 Table A. 1 List of zero readings 169 Table A.2 Mean values of zero readings 170 Table A.3 Experimental (pushing east) curvatures for a wall displacement of 105mm 171 Table A.4 Experimental (pushing west) curvatures for a wall displacement of 104mm 172 Table A.5 Experimental (pushing east) curvatures for a wall displacement of 132mm 173 Table A. 6 Experimental (pushing west) curvatures for a wall displacement of 138mm 174 Table A.7 Experimental (pushing east) curvatures for a wall displacement of 182mm 175 Table A.8 Experimental (pushing west) curvatures for a wall displacement of 187mm 176 Table A.9 Analytical curvatures for a wall displacement of 105mm 177 Table A. 10 Analytical curvatures for a wall displacement of 104mm 178 ix Table A. 11 Analytical curvatures for a wall displacement of 132mm 179 Table A. 12 Analytical curvatures for a wall displacement of 138mm 180 Table A. 13 Analytical curvatures for a wall displacement of 182mm 181 Table A. 14 Analytical curvatures for a wall displacement of 187mm 182 Table A. 15 Plane sections analysis to determine ultimate curvature 183 Table B . l Analytical strain profile at base for a drift of 1.5% 185 Table B.2 Analytical strain profile at base for a drift of 2% 185 Table B.3 Analytical strain profile accounting for cover spalling at base for a drift of 1.5% 185 Table B.4 Analytical strain profile accounting for cover spallingat base for a drift of 2% 185 Table C. 1 Curvatures up to the mid-height of Wall 1 for a drift of 2% 187 Table C.2 Interpolation of curvatures at storey heights of Wall 1 for a drift of 2% 189 Table C.3 Bending moments along the height of Wall 1 for a drift of 2% 190 Table C.4 Curvatures up to the mid-height of Wall 2 for a drift of 2% 191 Table C.5 Interpolation of curvatures at storey heights of Wall 2 for a drift of 2% 193 Table C.6 Bending moments along the height of Wall 2 for a drift of 2% 194 Table C.7 Average shear stress in Wall 1 195 Table C.8 Average shear stress in Wall 2 195 Table C.9 Average shear stress in Wall 1 for a shear span of 35659mm 195 Table C. 10 Average shear stress in Wall 2 for a shear span of 35659mm 195 Table C. 11 Average shear stress in Wall 1 for a shear span of 27430mm 195 Table C. 12 Average shear stress in Wall 2 for a shear span of 27430mm 196 Table C. 13 Average shear stress in Wall 1 for a shear span of 19201mm 196 Table C. 14 Average shear stress in Wall 2 for a shear span of 19201mm 196 Table C. 15 Average shear stress in Wall 1 for a web thickness of 254mm and a shear span of 27430mm 196 Table C. 16 Average shear stress in Wall 2 for a web thickness of 254mm and a shear span of 27430mm 196 Table C.17 Average shear stress in Wall 1 for a web thickness of 254mm and a shear span of 19201mm 197 Table C.l8 Average shear stress in Wall 2 for a web thickness of 254mm and a shear span of 19201mm 197 Table C. 19 Average shear stress in Wall 1 for a compressive axial load ratio of 0.3 197 Table C.20 Average shear stress in Wall 2 for a compressive axial load ratio of 0.3 197 Table C.21 Average shear stress in Wall 1 for a compressive axial load ratio of 0.2 197 Table C.22 Average shear stress in Wall 2 for a compressive axial load ratio of 0.2 198 Table C.23 Average shear stress in Wall 1 for no axial load 198 Table C.24 Average shear stress in Wall 2 for no axial load 198 Table C.25 Average shear stress in Wall 1 for a tensile axial load ratio of 0.02 198 Table C.26 Average shear stress in Wall 2 for a tensile axial load ratio of 0.02 198 Table C.27 Average shear stress in Wall 1 for a tensile axial load ratio of 0.05 199 Table C.28 Average shear stress in Wall 2 for a tensile axial load ratio of 0.05 199 Table C.29 Curvatures up to the mid-height of Wall 1 combined with Wall 2 for a drift of 2% 200 Table C.30 Interpolation of curvatures at storey heights of Wall 1 combined with Wall 2 for a drift of 2% 202 Table C.31 Displacement components at the first storey of Wall 1 combined with Wall 2 for a drift of 2% 203 Table C.32 Displacement components at the second storey of Wall 1 combined with Wall 2 for a drift of 2% 203 Table C.33 Displacement components at the third storey of Wall 1 combined with Wall 2 for a drift of 2% 204 Table C.34 Displacement components at the fourth storey of Wall 1 combined with Wall 2 for a drift of 2% 204 Table C.35 Slopes up to the mid-height of Wall 1 combined with Wall 2 for a drift of 2% 205 Table C.36 Bending moments along the height of Wall 1 combined with Wall 2 for a drift of 2% 206 Table C.37 Curvatures up to the mid-height of Wall 2 combined with Wall 1 for a drift of 2% 207 Table C.38 Interpolation of curvatures at storey heights of Wall 2 combined with Wall 1 for a drift of 2% 209 Table C.39 Displacement components at the first storey of Wall 2 combined with Wall 1 for a drift of 2% 210 Table C.40 Displacement components at the second storey of Wall 2 combined with Wall 1 for a drift of 2% 210 Table C.41 Displacement components at the third storey of Wall 2 combined with Wall 1 for a drift of 2% 211 Table C.42 Displacement components at the fourth storey of Wall 2 combined with Wall 1 for a drift of 2% 211 Table C.43 Slopes up to the mid-height of Wall 2 combined with Wall 1 for a drift of 2% 212 Table C.44 Bending moments along the height of Wall 2 combined with Wall 1 for a drift of 2% 213 Table C.45 Curvatures up to the mid-height of Wall 3 for a drift of 2% 214 Table C.46 Interpolation of curvatures at storey heights of Wall 3 for a drift of 2% 216 Table C.47 Bending moments along the height of Wall 3 for a drift of 2% 217 Table C.48 Curvatures up to the mid-height of Wall 1 combined with Wall 3 for a drift of 2% 218 Table C.49 Curvatures up to the mid-height of Wall 3 combined with Wall 1 for a drift of 2% 220 Table C.50 Interpolation of curvatures at storey heights of Wall 3 combined with Wall 1 for a drift of 2% 222 Table C.51 Bending moments along the height of Wall 3 combined with Wall 1 for a drift of 2% 223 Table C.52 Curvatures up to the mid-height of Column 1 for a drift of 2% 224 Table C.53 Curvatures up to the mid-height of Wall 1 combined with Column 1 for a drift of 2% 226 Table C.54 Curvatures up to the mid-height of Column 1 combined with Wall 1 for a drift of 2% 228 Xl l LIST OF FIGURES Figure 1.1 System of coupled walls subjected to lateral loading 2 Figure 1.2 System of walls of different lengths interconnected by rigid slabs 3 Figure 2.1 Displacement components of a reinforced concrete cantilever element 5 Figure 2.2 Displacement components beyond the yield displacement 6 Figure 2.3 Formulation of classical plastic hinge analysis 6 Figure 2.4 Bending moment-curvature relationship of different reinforced concrete elements 8 Figure 2.5 Curvature distribution along the length of a reinforced concrete cantilever element 9 Figure 2.6 Bending moment diagram for a reinforced concrete cantilever element 11 Figure 3.1 Concrete compression response 39 Figure 3.2 Concrete tension response 43 Figure 3.3 Reinforcement compression and tension response 48 Figure 4.1 Experimental curvatures (pushing east) for different wall displacement levels 54 Figure 4.2 Experimental curvatures (pushing west) for different wall displacement levels 54 Figure 4.3 Finite element model of UBC wall in FormWorks 57 Figure 4.4 Experimental (pushing east) and analytical curvatures for a wall displacement of 105mm.... 59 Figure 4.5 Experimental (pushing west) and analytical curvatures for a wall displacement of 104mm ...60 Figure 4.6 Experimental (pushing east) and analytical curvatures for a wall displacement of 132mm ....60 Figure 4.7 Experimental (pushing west) and analytical curvatures for a wall displacement of 138mm ...61 Figure 4.8 Experimental (pushing east) and analytical curvatures for a wall displacement of 182mm ....61 Figure 4.9 Experimental (pushing west) and analytical curvatures for a wall displacement of 187mm ...62 Figure 4.10 Comparison of equivalent plastic hinge length models 65 Figure 4.11 Linearly varying inelastic curvatures 67 Figure 4.12 Strain profile at construction joint for the case of no bending 69 Figure 4.13 Strain profile at 319mm from construction joint for the case of no bending 69 Figure 4.14 Strain profile at construction joint for a wall displacement of 12.8mm 70 Figure 4.15 Strain profile at 319mm from the construction joint for a wall displacement of 12.8mm 71 Figure 4.16 Strain profile at construction joint for a wall displacement of 2 5.4mm 72 Figure 4.17 Strain profile at 840mm from the construction joint for a wall displacement of 25.4mm 73 Figure 4.18 Strain profile at 840mm from the construction joint for a wall displacement of 48mm 74 Figure 4.19 Strain profile at construction joint for a wall displacement of 41.4mm 75 Figure 4.20 Strain profile at 840mm from the construction joint for a wall displacement of 41.4mm 76 Figure 4.21 Strain profile at 840mm from the construction joint for a wall displacement of 105mm 77 Figure 4.22 Strain profile at 840mm from the construction joint for a wall displacement of 281mm 78 Figure 4.23 Finite element model of specimen RW2 in FormWorks 82 Xl l l Figure 4.24 Strain profile at base for a drift of 1.5% 84 Figure 4.25 Strain profile at base for a drift of 2% 84 Figure 4.26 Area of concrete considered in the analysis to account for cover spalling 85 Figure 4.27 Finite element model accounting for cover spalling of specimen RW2 in FormWorks 86 Figure 4.28 Strain profile accounting for cover spalling at base for a drift of 1.5% 87 Figure 4.29 Strain profile accounting for cover spalling at base for a drift of 2% 88 Figure 5.1 Cross-section details of Wall 1 89 Figure 5.2 Cross-section details of Wall 2 .90 Figure 5.3 Finite element model of Wall 1 in FormWorks 92 Figure 5.4 Finite element model of Wall 2 in FormWorks 93 Figure 5.5 Strain profile of Wall 1 at base for a drift of 2% 95 Figure 5.6 Strain profile of Wall 1 at 1270mm from the base for a drift of 2% 95 Figure 5.7 Strain profile of Wall 2 at the base for a drift of 2% 96 Figure 5.8 Curvatures up to the mid-height of Wall 1 and 2 for a drift of 2% 97 Figure 5.9 Bending moments along the height of Wall 1 and 2 for a drift of 2% 98 Figure 5.10 Moment-curvature relationship of Wall 1 98 Figure 5.11 Moment-curvature relationship of Wall 2 99 Figure 5.12 Moment-curvature relationship of Wall 1 using Response-2000 constitutive models 101 Figure 5.13 Moment-curvature relationship of Wall 2 using Response-2000 constitutive models 101 Figure 5.14 Exterior and interior steel layers in zones in Wall 1 and 2 103 Figure 5.15 Steel strains up to the mid-height of Wall 1 and 2 for a drift of 2% 103 Figure 5.16 Moment-curvature relationships of Wall 1 and 2 104 Figure 5.17 Moment-curvature relationship of Wall 1 with a web thickness of 254mm 109 Figure 5.18 Moment-curvature relationship of Wall 2 with a web thickness of 254mm 109 Figure 5.19 Measured plastic hinge lengths vs. total drift for Wall 1 and 2 112 Figure 5.20 Measured plastic hinge lengths vs. plastic drift for Wall 1 and 2 113 Figure 5.21 Ratio of measured to predicted plastic hinge lengths vs. drift for Wall 1 for different shear stresses without accounting for shear 113 Figure 5.22 Ratio of measured to predicted plastic hinge lengths vs. drift for Wall 2 for different shear stresses without accounting for shear 114 Figure 5.23 Moment-curvature relationships of Wall 1 for different shear stresses 115 Figure 5.24 Ratio of measured to predicted plastic hinge lengths vs. drift for Wall 1 for different shear stresses using the shear model 119 Figure 5.25 Moment-curvature relationships of Wall 1 for different axial load ratios 121 Figure 5.26 Moment-curvature relationships of Wall 2 for different axial load ratios 121 Figure 5.27 Measured plastic hinge lengths vs. total drift for Wall 1 for different axial load ratios 124 xiv Figure 5.28 Measured plastic hinge lengths vs. total drift for Wall 2 for different axial load ratios 125 Figure 5.29 Ratio of measured to predicted plastic hinge lengths vs. drift for Wall 1 for different axial load ratios 126 Figure 5.30 Ratio of measured to predicted plastic hinge lengths vs. drift for Wall 2 for different axial load ratios 126 Figure 6.1 Model of Wall 1 and 2 combined 129 Figure 6.2 Finite element model of Wall 1 and 2 in FormWorks 131 Figure 6.3 Detail of elements representing the slabs at each storey in finite element model of Wall 1 and 2 132 Figure 6.4 Curvatures up to the mid-height of Wall 1 and 2 alone and combined for a drift of 2% 133 Figure 6.5 Bending moments along the height of Wall 1 and 2 alone and combined for a drift of 2%.. 134 Figure 6.6 Moment-curvature relationship of Wall 1 alone and combined 135 Figure 6.7 Moment-curvature relationship of Wall 2 alone and combined 135 Figure 6.8 Numerical scheme for curvature integration 136 Figure 6.9 Steel strains up to the mid-height of Wall 1 alone and combined for a drift of 2% 138 Figure 6.10 Steel strains up to the mid-height of Wall 2 alone and combined for a drift of 2% 138 Figure 6.11 Displacements at wall faces up to the fourth storey of Wall 1 and 2 combined for a drift of 2% 139 Figure 6.12 Displacements at wall faces up to the first storey of Wall 1 and 2 combined for a drift of 2% 140 Figure 6.13 Displacement components up to the fourth storey of Wall 1 and 2 combined for a drift of 2% 141 Figure 6.14 Slopes up to the mid-height of Wall 1 and 2 combined for a drift of 2% 142 Figure 6.15 Proposed model to determine maximum curvature in shorter wall 143 Figure 6.16 Cross-section details of Wall 3 145 Figure 6.17 Finite element model of Wall 3 in FormWorks 147 Figure 6.18 Finite element model of Wall 1 and 3 in FormWorks 149 Figure 6.19 Curvatures up to the mid-height of Wall 1 and 3 alone and combined for a drift of 2% 150 Figure 6.20 Moment-curvature relationship of Wall 3 alone and combined 151 Figure 6.21 Steel strains up to the mid-height of Wall 3 alone and combined for a drift of 2% 152 Figure 6.22 Cross-section details of Column 1 153 Figure 6.23 Finite element model of Column 1 in FormWorks 155 Figure 6.24 Finite element model of Wall 1 and Column 1 in FormWorks 157 Figure 6.25 Curvatures up to the mid-height of Wall 1 and Column 1 alone and combined for a drift of 2% 158 Figure 6.26 Steel strains up to the mid-height of Column 1 alone and combined for a drift of 2% 159 XV LIST OF SYMBOLS Ag: Gross area of cross-section Ac: Area of concrete As: Area of reinforcement Asc'. Area of compression reinforcement AsS: Area of tension reinforcement a: Maximum aggregate size as(. Zero-one variable in Equation 2.46 and 2.47 b: Width of member b^. Web thickness of wall C : Cohesion Cd- Compression softening strain softening factor, defined in Equation 3.10 Cs: Compression softening shear slip factor in Equation 3.9 c: Neutral axis depth at ultimate moment ca: Averaging factor, defined in Equation 3.5 c*: Coefficient used to reflect bar spacing in Equation 3.43 c,\ Coefficient that incorporates the influence of reinforcement bond characteristics, defined in Equation 3.19 d: Effective depth of member di. Bar diameter of the tension reinforcement dv: Lever arm of tensile force in the longitudinal reinforcement Ec: Concrete initial tangent stiffness Es: initial tangent stiffness or elastic modulus of reinforcement Esec: Concrete secant stiffness Est,: Strain hardening modulus F: Lateral force fe: Concrete cylinder uniaxial compressive strength fci. Average net concrete axial stress in the principal tensile direction f"x: Average concrete tensile stress due to tension stiffening / e *: Average concrete tensile stress due to tension softening fC2. Average net concrete axial stress in the principal compressive direction fÂ°2 'â€¢ Average concrete compressive stress contribution of unconfined concrete / c *: Average concrete compressive stress contribution of confined concrete fcc: Concrete cube strength fa.: Concrete cracking stress fen,'- Defined in Equation 3.24 fp: Peak concrete compressive stress fs: Reinforcement stress ft\ Tensile strength of steel f{. Concrete uniaxial tensile strength fu: Ultimate strength of reinforcement fy: Yield stress of the tension reinforcement fy: Yield stress of the compression reinforcement Gf. Energy required to form a complete crack of unit area Hâ€ž: Total height of wall h: Depth or height of the cross-section of member lz: Moment of inertia of the reinforcement ki. Factor that considers the influence of the tension reinforcement in Equation 2.12, 2.17 2.19 k2: Factor that considers the influence of the axial load, defined in Equation 2.13 k3: Factor that considers the influence of the concrete strength, defined in Equation 2.14 kc: Stiffness of the notional concrete foundation, defined in Equation 3.43 L: Length of member in the longitudinal direction L p : Plastic hinge length Lpy. Plastic hinge length in element"/" Lpconsi- Plastic hinge length for a constant inelastic curvature Lpcy: Plastic hinge length for cyclic loading LpHn'. Plastic hinge length for a linearly varying inelastic curvature Lp.meas'- Measured plastic hinge length LPtmon: Plastic hinge length for monotonic loading LpiPred. Predicted plastic hinge length Lr: Distance over which the crack is assumed to be uniformly distributed Ls: Distance from the section of maximum moment to the section of zero moment or shear span ls: Standard length in Equation 2.39 and 2.41 lâ€ž: Horizontal length of wall section M: Total bending moment M/. Moment due to flexure Mmax'. Maximum moment in the length of member Mâ€ž: Ultimate moment M v : Moment due to shear XVII My. Yield moment m: Bond parameter, defined in Equation 3.20 7VV: Axial compression due to shear n: Curve fitting parameter for stress-strain response of concrete in compression, defined in Equation 3.2 P: Axial load Pu: Axial compressive strength of member without any bending moment PER: Percentage ratio of volume of stirrups to volume of concrete core measured outside of stirrups Rm: Moment ratio, defined in Equation 2.32 RE: Strain ratio, defined in Equation 2.31 RFT: Defined in Equation 2.43 r. Ratio of the principal tensile strain to the principal compressive strain SPR: Defined in Equation 2.44 s: Crack spacing V: Shear force Vc: Shear force in concrete Vd- Dowel force Vdu'- Ultimate dowel force Vs: Shear force in longitudinal reinforcement Vz: Shear adjacent to a concentrated load or reaction at the section of maximum moment v: Shear stress v d: Local shear stress on the crack Va,max' Maximum local shear stress on the crack vco: Defined in Equation 3.32 vcr: Cracking shear stress vd: Shear resistance due to dowel action w: Average crack width wz: Uniformly distributed load at the section of maximum moment Zm: Slope of compression post-peak descending curve, defined in Equation 3.8 cc. Calibration parameter in Equation 2.10 a.i. Model parameter in Equation 2.39, 2.40 and 2.41 a2: Model parameter in Equation 2.39, 2.40 and 2.41 (J: Calibration parameter in Equation 2.10 Pd'. Softening parameter, defined in Equation 3.9 Pf. Strength enhancement factor, defined in Equation 3.14 A: Total displacement A/ Flexural displacement A p: Plastic or inelastic deformation As: Slip displacement Au: Ultimate displacement Aâ€ž: Shear displacement Ay: Yield displacement A6â€ž. Post-cracking rotation of the principal stress field A6e: Post-cracking rotation of the principal strain field 5S: Shear slip d]: Defined in Equation 3.29 Sci. Average net concrete axial strain in the principal tensile direction sC2. Average net concrete axial strain in the principal compressive direction See'. Concrete strain in the extreme compression fiber at yield curvature Ed,: Characteristic strain scr: Concrete cracking strain So,: Concrete strain in the extreme compression fiber at ultimate curvature e0: Concrete compressive strain corresponding \.ofc Bp-. Concrete compressive strain corresponding to fp es: Reinforcement strain Ssh'. Strain at the onset of strain hardening s,,: Reinforcement ultimate strain Exx: Total axial strain in the jc-direction Ey-. Yield strain Eyy: Total axial strain in the v-direction O: Internal angle of friction <f>: Total curvature 4>ma- Maximum curvature <f>max,i: Maximum curvature in element"/'" <j>p: Plastic or inelastic curvature <f>u: Ultimate curvature Yield curvature xix <j>y/. Yield curvature in element"/" ys: Shear slip strain yÂ°: Shear slip strain determined from the stress-based model y\: Shear slip strain determined from the constant rotation lag model Yxy-. Total shear strain X: Parameter that compares the stiffness of the concrete to the stiffness of the reinforcing bar, defined in Equation 3.42 &. Total rotation or slope 6C: Angle of the crack 6t: Total rotation or slope in element"/'" 6ic: Inclination of the principal stress field at cracking O1: Rotation lag 6â€ž: Angle between the normal to the crack surface and the longitudinal axis of the reinforcement 0P: Plastic or inelastic rotation or slope QpW- Plastic or inelastic rotation or slope in the length d/2 0U: Ultimate rotation or slope da. Inclination of the principal stress field p: Tension reinforcement ratio p'\ Compression reinforcement ratio Pt,: Reinforcement ratio at balanced ultimate strength condition in a member without compression reinforcement pâ€ž: Reinforcement ratio co: Tension reinforcement index, defined in Equation 2.15 a>': Compression reinforcement index, defined in Equation 2.16 mb: Tension reinforcement index for balanced ultimate strength condition, defined in Equation 2.24 Â£ Calibration parameter in Equation 2.10 Model error term in Equation 2.39, 2.40 and 2.41 \jf. Defined in Equation 3.30 A C K N O W L E D G E M E N T S xx I would like to thank my supervisor, Dr. Perry Adebar, for his support and guidance throughout this research. I also want to thank the Faculty of Graduate Studies, which granted me with a University of BC Graduate Fellowship. xxi To my parents, Alfredo and Berta 1 C H A P T E R 1: I N T R O D U C T I O N 1.1 Research significance The use of concrete shear walls to provide lateral strength and stiffness in high-rise buildings has become common practice. These walls require enough shear strength capacity and flexural displacement capacity to have an adequate seismic behaviour. The flexural displacement capacity depends on the compression strain capacity of concrete, the neutral axis depth and the length of the plastic hinge. The inelastic curvatures at the base of the wall are commonly assumed to be constant over a length known as the equivalent plastic hinge length. Different studies done in the past on concrete members have shown that the equivalent plastic hinge length is proportional to the member dimension (e.g.: beam depth, wall length). Building code requirements in ACI 318 2005 and CSA A23.3 2004 for determining confinement requirements at the ends of a concrete wall assume the plastic hinge length is equal to half of the wall length. Several empirical models have been developed to estimate the length of the plastic hinge in reinforced concrete members based on test results. A large number of these models were calibrated so that they will give the real total displacement or real total rotation at failure. These models consider different parameters and provide significantly different predictions. Most of the research done in the past has focused on the influence of the member dimension, the span of the member and the longitudinal reinforcement properties. The effect of the axial load ratio and the strain hardening has also been studied. It is still not clear which are the most relevant parameters that influence the length of the plastic hinge. Many of these models consider that the plastic hinge length is proportional to the member dimension only. This is due to the fact that the member dimension is commonly associated with the effect of diagonal cracking, which causes the length of the plastic hinge to increase. If the angle of the crack is considered to be 45Â°, it will extend to a distance from the support equal to the member dimension. Therefore, longer members have larger plastic hinge lengths. One of the objectives of this thesis will be to investigate in more depth how the member dimension and the plastic hinge length are related. The increase in the plastic hinge length due to the influence of diagonal cracking is difficult to quantify. A simple shear model will be developed to try to estimate the plastic hinge length when the shear stresses are high. 2 Previous investigations on plastic hinge length in concrete walls have been done for individual members. To the knowledge of the author, there are no previous studies done on concrete wall systems. It is not known how the interaction of the walls that are part of the same system will affect the length of the plastic hinge of these members. A system of coupled walls is connected together by very stiff beams (coupling beams). When a system of coupled walls with a high degree of coupling is subjected to lateral loading, the shear forces in the coupling beams induce high axial forces in the walls. Some walls are subjected to high axial tension, and others are subjected to high axial compression. The axial loads in the walls are expected to have a significant effect on the plastic hinge length. Walls subjected to compressive and tensile axial forces will be analyzed to study this effect. Figure 1.1 System of coupled walls subjected to lateral loading 7777777777777777777777Z77777Zy In high-rise buildings, it is common to have systems of walls of different lengths providing lateral resistance in the same direction of lateral loading. These walls are interconnected by rigid slabs at numerous floor levels. As a result, when this system is subjected to lateral loads, the displacement of all these walls is the same at the floor levels. 3 Figure 1.2 System of walls of different lengths interconnected by rigid slabs Displacements are equal at floor levels 77777777777777777777777, Along with the walls, the gravity columns that do not form part of the lateral force resisting system will also have the same displacement at the floor levels. This tends to increase curvature demand in these elements. The columns must be able to sustain that demand. An important objective of this thesis will be to analyze walls of different lengths connected together by rigid slabs to study the impact of the connection in the plastic hinge length. The effect on gravity columns will also be studied. 1.2 Objective of the thesis The main objective of this thesis is to investigate the parameters that affect the length of the plastic hinge in concrete walls, using static nonlinear finite element analysis. The parameters considered in this study were the wall length, the distance from the section of maximum moment to the section of zero moment, diagonal cracking due to shear, and the axial load. This parametric study was first performed for individual walls. Then, walls of different lengths interconnected by rigid slabs at numerous levels were analyzed to investigate how this affects the length of the plastic hinge. The analysis was performed using program VecTor2. VecTor2 is a computer program developed at the University of Toronto to perform nonlinear finite element analysis of reinforced concrete membrane structures, using the constitutive relationships of the Disturbed Stress Field Model. The program uses low-powered elements to model concrete structures. It considers a variety of effects to accurately predict the response, such as compression softening, tension stiffening, slip distortions and strain hardening. 4 Prior to the parametric study, analytical predictions made by program VecTor2 were verified with experimental results obtained from previous tests performed on cantilever walls to see how well the program predicts the response of these members. In particular, the curvature distributions along the height and the strain profiles along the wall section were examined and compared. 1.3 Organization of the thesis The thesis is divided in seven chapters. Chapter 1 is the introduction. It presents the objective of the thesis. Chapter 2 is the literature review. It first introduces the concept of plastic hinge length and the formulation of classical plastic hinge analysis. The results from 23 previous investigations on plastic hinge length and the models that were developed are presented. A summary of all these investigations, including the type of members studied and the parameters considered, is presented at the end of the chapter. Chapter 3 presents the analytical methods. It describes the theoretical bases of program VecTor2, as well as its finite element formulation and the description of the material models for concrete and steel. Only the material models used in this thesis are presented. Finally, a brief description of the pre-processor and post-processor for program VecTor2 is presented. Chapter 4 shows a comparison of analytical predictions from program VecTor2 with isolated wall test results. Two wall specimens were used for the comparison. The first is a high-rise shear wall tested at the University of British Columbia in 2000; the curvature distributions along the height were compared. The second is a rectangular shear wall tested at Clarkson University in 1995; the strain profiles along the wall section were compared. Chapter 5 presents a parametric study of concrete walls. The factors that affect the length of the plastic hinge were investigated. This study was performed for two individual walls. Chapter 6 presents the analysis of three systems of walls of different lengths interconnected by rigid slabs at numerous levels. The influence of the connection in the curvature distribution and the plastic hinge length of the walls were studied. Chapter 7 presents the conclusions of the thesis. 5 CHAPTER 2: LITERATURE REVIEW 2.1 Concept of plastic hinge length Consider a slender reinforced concrete cantilever element subjected to a lateral load at the top, like the one in Figure 2.1. The total lateral displacement at the top, A, is comprised of three components: the flexural displacement, A/, the shear displacement, A v , and the slip displacement, A :^ Figure 2.1 Displacement components of a reinforced concrete cantilever element â€”: t , J 1 f T \ 77777777. Therefore: A = A / + A V + A , (2.1) The shear and slip displacements are commonly not very important for high-rise buildings. In the classical formulation of plastic hinge analysis, it is considered that the total displacement of a reinforced concrete element after yielding has two components: the yield displacement, Ay, and the plastic deformation, Ap: Figure 2.2 Displacement components beyond the yield displacement A Ay Ap T~ 77777777. Therefore: (2.2) In practice, it is commonly assumed that the inelastic curvature in the plasticized region, although it has a certain variation, is constant over a length known as the plastic hinge length, Lp. This is shown in the Figure 2.3: Figure 2.3 Formulation of classical plastic hinge analysis Ay Ap r77777777. From Figure 2.3, the inelastic curvature, fa, is calculated from the total curvature, fa by: (2.3) Where fa is the yield curvature. The inelastic rotation, 0P, can be determined by integrating the inelastic curvatures: (2.4) Considering that the inelastic rotation is concentrated at the centroid of the inelastic curvatures, the inelastic displacement can be expressed as: f L \ L- â€” v 2 J (2.5) Combining Equation 2.2 and 2.5, the total displacement is: A = Aâ€ž+(*-4,)L. (2.6) The yield displacement can be determined by integrating the elastic curvatures, considering a linear variation as shown in Figure 2.3: faL2 (2.7) Then, the total displacement is: A ^ A = â€”â€” + f I ^ L- â€” v 2 y (2.8) This is the classical formulation of plastic hinge analysis, and it has been used over the years by many researchers to develop models for plastic hinge length. According to this formulation, the plastic hinge length is defined as the equivalent length over which the inelastic curvature is considered to be 8 constant. In reality, the inelastic curvature has a certain variation. Paulay and Priestly presented the actual curvature distribution of a prismatic reinforced concrete cantilever element, showing that the real spread of plasticity is longer that the equivalent plastic hinge length (Paulay and Priestly 1992: 139). Classical plastic hinge analysis is based on the assumption of an elasto-plastic behaviour and an equivalent plastic hinge length. The yield curvature used in Equation 2.8 can be determined considering that the moment-curvature relationship in the plastic hinge zone is elasto-plastic (Priestly and Park 1987: 71). This approximation depends on the actual shape of the moment-curvature relationship, which depends on the type of element. Figure 2.4 shows the difference between the equivalent yield curvature for a column and for a shear wall (Adebar and mraliim 2002: 408): The total displacement at the top can also be determined by calculating the curvature distribution along the length of the member, as shown in Figure 2.5 (Priestly and Park 1987: 71): 9 Figure 2.5 Curvature distribution along the length of a reinforced concrete cantilever element x theoretical curvature distribution 77777777. spread of plasticity tensile strain .penetration Then: L A = $<fi(x)xdx (2.9) o Equation 2.9 can be used to calculate the total displacement, considering a theoretical curvature distribution like the one in Figure 2.5. However, the actual displacement will be larger than the one calculated with this expression. This is due to the effect of tensile strain penetration and the spread of plasticity due to shear. Tensile strain penetration is the additional rotation in the plastic hinge zone due to the slippage of the longiradinal reinforcement. The extent of the tensile strain penetration depends on the development length of the bar (Paulay and Priestly 1992: 141). The spread of plasticity is caused by the presence of diagonal cracks due to high shear stresses, which produces higher steel strains than the ones due to pure flexure. For a crack angle of 45Â°, the influence of diagonal cracking is proportional to the depth of the member (Priestly and Park 1987: 71 - 72). The influence of tensile strain penetration and spread of plasticity can be taken into account in Equation 2.9 by considering a modified curvature distribution, shown in Figure 2.5. However, the most common approach in practice is to use plastic hinge analysis, in which the influence of tensile strain penetration and spread of plasticity is considered imphcitly in Equation 2.8 in the parameter Lp, in order to obtain a good estimate of the real displacement (Priestly and Park 1987: 72). Therefore, many of the models developed to determine the equivalent plastic hinge length in reinforced concrete elements were established on the basis that the plastic hinge length is proportional to the length of the member in the 10 longitoidinal direction, the depth of the member, and the longitudinal reinforcement properties; as it is presented in the following expression (Berry and Eberhard 2003: 13): Lp=aL + /3h + &ydb (2.10) - L: Length of the member in the longitudinal direction. h: Depth or height of the cross-section of the member. - fy: Yield stress of the tension reinforcement. db: Bar diameter of the tension reinforcement. In a more general sense, L in Equation 2.10 refers to the distance from the critical section to the point of contraflexure (point of zero moment). For the cantilever element shown in Figure 2.5, this distance is equal to the length of the member. When this is not the case, a different terminology will be used. Also, when referring to walls, the length in the longitudinal dimension (L) will be referred to as the height of the wall (Hw), and the depth of the cross-section (h) will be referred to as the wall length (/â€ž,). In Equation 2.10, the member length takes into account the curvature distribution along its length, the member depth considers the spread of plasticity, and the longitudinal reinforcement properties consider the effect of tensile strain penetration (Berry and Eberhard 2003: 13). Several researchers have used the form of Equation 2.10 or other similar to it and have calibrated them with experimental results to obtain the values for a, /?and Â£ Most of the studies done in plastic hinge length in reinforced concrete elements have been focused in the parameters presented in Equation 2.10. However, some researchers have also investigated the stress-strain properties of steel and the effect of axial compression, which have a significant influence in the length of the plastic hinge. In the case of walls, when subjected to some axial compression, the length over which the longitudinal reinforcement yields is reduced, and so does the plastic hinge length (Paulay and Uzumeri 1975: 596 - 597). In this chapter, the models developed by various researchers to determine the plastic hinge length for various kinds of concrete elements will be presented. 2.2 Chan (1955) Chan performed tests on concrete columns to compare the assumption of plastic hinges concentrated at points with the real spread of plasticity. He also investigated the effect of lateral 11 confinement in the strain capacity of concrete. He reported the results obtained for 23 columns (Chan 1955: 121- 132). Consider the linear bending moment diagram for a reinforced concrete cantilever element subjected to a lateral load at a certain distance from the base, like the one in Figure 2.6: Figure 2.6 Bending moment diagram for a reinforced concrete cantilever element point of contrafiexure Chan considered the effect of strain hardening to define the length of the plastic hinge. For a linear bending moment diagram like the one shown in Figure 2.6, the length where the yield moment is exceeded is determined by (Chan 1955: 121 - 122): -*- = l y- (2.11) Ls: Distance from the section of maximum moment to the section of zero moment. My. Yield moment. Mâ€ž: Ultimate moment. Three types of specimens where tested: nine members with bent-in transverse reinforcement, seven members with spiral transverse reinforcement, and seven members with welded transverse reinforcement. The most important variables covered the following range (Chan 1955: 124): - Depth: 6". 12 - Width: 3 5/8 and 6". - Span length: 11 1/2, 12 and 52". Diameter of tension reinforcement: 1/2 and 5/8" bars. Diameter of compression reinforcement: 1/2 and 5/8" bars. Diameter of transverse reinforcement: 1/8, 3/16 and 1/4" bars. Eccentricity of axial load: 1/4 and 1/2". Concrete cube strength: Between 2.65 and 5.46 ksi. Each specimen was pin-ended and was loaded until failure. The axial load in the members with bent-in and spiral transverse reinforcement was held constant. The members with welded transverse reinforcement were subjected additionally to a transverse load at the midspan (Chan 1955: 125). The test results were used to develop stress-strain relationships for unconfined and confined concrete. Then, using these stress-strain curves, an elasto-plastic stress-strain curve for steel, a yield strain of 0.001, and an ultimate strain of 0.0035 for unconfined concrete; the yield moment and ultimate moment were calculated for each of the specimens tested. These values were then used in Equation 2.11 to determine the length of the plastic hinge. For members with low axial loads, the plastic hinge length did not vary significantly with the tension reinforcement ratio, and had a mean value of 0ALs. The plastic hinge length was greater for members with high axial loads, due to concrete spalling. The maximum value was approximately 0.7LS (Chan 1955: 129). 2.3 Baker (1956) Baker investigated the plastic deformations of hinges and members in concrete frames. He indicated that a safe estimate of the length of the plastic hinge in columns is between 0.5/7 and h (Baker 1956: 27). 2.4 Cohn and Petcu (1963) Cohn and Petcu performed tests on continuous concrete beams with two spans to investigate the factors that affect the rotational capacity of plastic hinges. The most important factor studied was the percentage of steel. They reported the results obtained for 10 beams (Cohn and Petcu 1963: 282 - 290). Two series of five concrete beams each were tested. The most important variables covered the following range (Cohn and Petcu 1963: 282 - 284): 13 Depth: 8cm. - Width: 15cm. Span length: 1.5m. Diameter of longitudinal reinforcement: 6, 8 and 10mm. Diameter of stirrups: 6mm. Average yield stress of reinforcement: Between 2550 and 3500 kg/cm2. Average tensile strength of reinforcement: Between 3920 and 4900 kg/cm2. Average concrete cube strength: 240 kg/cm2. Each beam was loaded with a concentrated load at a certain distance from the central support until failure. This distance was 40cm for one of the series, and 60cm for the other series (Cohn and Petcu 1963: 284). The test results were used to determine the bending moments at yielding and failure. These were calculated from the measured reactions in the beams using equilibrium equations, considering a linear variation of the bending moment diagram. Then, Equation 2.11 was used to determine the length of the plastic hinge at one side from the central support of the beams (Cohn and Petcu 1963: 284 - 285). The results obtained varied from 30 to 90% of the effective depth of the beam (Cohn and Petcu 1963: 290). 2.5 Baker and Amarakone (1964) Baker and Amarakone reported tests results on beams and columns that were performed to investigate how the following parameters influenced the moment-curvature relationship of these members: the strength of concrete and steel, percentage of steel, single loads and double loads, axial force, shear force, transverse reinforcement, and percentage of compression reinforcement. They reported the results obtained for 92 specimens (Baker and Amarakone 1964: 85 - 142). Baker and Amarakone proposed the following expression to determine the plastic hinge length for members with unconfined concrete (Baker and Amarakone 1964: 94): fr. \X (2.12) (2.13) k3 =0.9-/ Â« - 1 3 . 8 (2.14) 92 14 kj'. Factor that considers the influence of the tension reinforcement. It is equal to 0.7 for mild steel or 0.9 for cold-worked steel. k2: Factor that considers the influence of the axial load. k3: Factor that considers the influence of the concrete strength. It is equal to 0.6 for a concrete cube strength of 6 ksi (41.4 MPa) or 0.9 for a concrete cube strength of 2 ksi (13.8 MPa). Linear interpolation may be used between these values, arriving to Equation 2.14. d: Effective depth of the member. - P: Axial load. Pu: Axial compressive strength of the member without any bending moment. - fcc: Concrete cube strength, in MPa units. They indicated that for values of Lid and LJd commonly used in practice, the plastic hinge length is between OAd and 2 Ad. Three types of specimens where tested: 32 members with cold-worked steel, 30 members with mild steel, and 32 members subjected to bending and axial load (with both cold-worked and mild steel). The most important variables covered the following range (Baker and Amarakone 1964: 88 - 93): - Width: 6, 10 and 12". - Depth: 8 and 11". Concrete cylinder strength: Between 2.5 and 6.5 ksi. Yield stress of reinforcement: Between 37 and 85 ksi. Tension reinforcement index: Between 2.5 and 70.4%. Compression reinforcement index: Between 0.96 and 26.3%. Transverse reinforcement ratio: Between 0.05 and 1.26%. Axial load ratio (in this case, Plfrbd): Between 0.164 and 1.2. - Span length: 55, 80, 110 and 117". The tension and compression reinforcement index, co and co', respectively, are given by: co = co-p (2.15) (2.16) 15 p. Tension reinforcement ratio. - fc: Concrete cylinder uniaxial compressive strength. p'\ Compression reinforcement ratio. - fy\ Yield stress of the compression reinforcement. The test results were used to compare measured and predicted inelastic rotations. Since concrete behaves differently when it is confined by transverse reinforcement, the authors developed an expression to estimate the inelastic rotation for this case (Baker and Amarakone 1964: 97): (2.17) Ecu'. Concrete strain in the extreme compression fiber at ultimate curvature. ece: Concrete strain in the extreme compression fiber at yield curvature. Considering that the neutral axis depth at yielding and failure are equal, Equation 2.4 can be expressed as (Baker and Amarakone 1964: 91): ep=^-I^Lp (2.18) Where c is the neutral axis depth at ultimate moment. Therefore, combining Equation 2.17 and 2.18, the plastic hinge length for confined members is: (2.19) Although the parameter k2 is not included in Equation 2.19, the influence of the axial load is considered implicitly in the ratio eld. The test results showed that members with cold-worked steel had longer plastic hinge lengths. The ratio LJd did not have a significant effect on the plastic hinge lengths (Baker and Amarakone 1964: 116). A considerable scatter was observed for the inelastic rotation, because of the variation of the concrete strain at ultimate curvature. Baker and Amarakone indicated that the neutral axis depth has an important influence in the ultimate strain of confined concrete. Therefore, they plotted the experimental 16 plastic hinge lengths as a function of the ratio c/d, and found that there is a linear relation between these two parameters (Baker and Amarakone 1964: 95 - 97). 2.6 Sawyer (1964) Sawyer developed a design methodology for reinforced concrete frames based on a bilinear moment-curvature relationship (Sawyer 1964: 405 - 431). Some of the assumptions made to develop his method were the following: The maximum moment at any section is equal to the ultimate moment (Sawyer 1964: 409). The ratio M/Mâ€ž is equal to 0.85. This value was adopted based on previous test results obtained for beams (Sawyer 1964: 415). The plasticity spreads d/4 past the section in which the bending moment is equal to the yield moment (Sawyer 1964: 422). An expression to calculate the plastic hinge length can be derived from these assumptions (Mendis 2001: 190 - 191), considering a linear bending moment diagram like the one shown in Figure 2.6. Based on the second assumption, the length where the yield moment is exceeded is 0.154. The bilinear moment-curvature relationship developed by Sawyer is defined by the points (0, 0), (My, fa) and (Mu, fa). Based on this, the inelastic rotation at the section of maximum moment can be obtained by integrating the inelastic curvatures over the plastic region: Comparing this equation with Equation 2.4, it can be seen that they are both the same. Therefore: (2.20) 0.0754 (2.21) Including the spread of plasticity (third assumption), the plastic hinge length is given by: Lp =0.25^ + 0.0754 (2.22) 17 2.7 Mattock (1964) Mattock performed tests on simply supported beams subjected to a concentrated load at the midspan to investigate how their rotational capacity is influenced by the concrete strength, the effective depth, the distance from the section of maximum moment to the section of zero moment, and the amount and yield stress of the reinforcement. He reported the results obtained for 37 beams (Mattock 1964: 143 -181). The most important variables covered the following range (Mattock 1964: 145 - 149): Concrete cylinder strength: 4 and 6 ksi. Yield stress of tension reinforcement: 47 and 60 ksi. Yield stress of compression reinforcement: 50 and 70 ksi. Yield stress of stirrups: 50, 60 and 70 ksi. - Width: 6". - Effective depth: 10 and 20". - Span length: 55, 110 and 220". Tension reinforcement ratio: Between 1 and 3%. The test results were used to determine the spread of plasticity of the beam at each side of the midspan. Mattock indicated that a considerable spread of inelastic deformation occurred beyond a distance of half the effective depth, and that it depended on the distance from the section of maximum moment to the section of zero moment (equal to half of the beam length for these beams), the effective depth, and the amount of flexural reinforcement. He used the ratio of the total inelastic rotation in the length Ls, 0P, to the inelastic rotation in the length d/2, 6p^i, as a measure of the spread of plasticity (Mattock 1964: 158- 161). The total inelastic rotation in each beam was obtained from the plastic deformation at the midspan. The inelastic rotation in the length d/2 was obtained from the other measurements made at the midspan. Mattock plotted the values of 0p/0p,d/2 as a function of the parameters previously indicated. Based on his results, he developed the following expression to determine the spread of plasticity, using a least squares fit: 18 p,d/2 = 1+ 1.14. 0.411 d (2.23) Â®b =Pb f'c (2.24) Ls: Distance from the section of maximum moment to the section of zero moment, in meters. d: Effective depth of the member, in meters. cot. Tension reinforcement index for balanced ultimate strength condition. pb. Reinforcement ratio at balanced ultimate strength condition in a member without compression reinforcement. Mattock indicated that in Equation 2.23, the ratio of the difference between the tension and compression reinforcement index to the balanced tension reinforcement index is a measure of the strain hardening. As this ratio increases, the amount of strain hardening reduces, and so does the spread of plasticity. As he indicated, this is because the length of the plastic region is proportional to the difference between the yield moment and the ultimate moment, and this difference strongly depends on the amount of strain hardening in the tension reinforcement. This behaviour was also observed in the test results. The measured inelastic rotations for the tests beams were compared to the inelastic rotations obtained with Equation 2.23. The inelastic rotation in the length d/2 was calculated using compatibility and equilibrium equations (Mattock 1964: 163 - 164). The results showed that Equation 2.23 provides a lower bound for most of the beams tested. The plastic hinge length can be determined by (Mattock 1967: 521): (2.25) Therefore, combining Equation 2.23 and 2.25, the plastic hinge length is: 19 2.8 Corley (1966) Corley performed tests on simply supported beams subjected to a concentrated load at the midspan to investigate the effect of confinement of the concrete in compression and the effect of the size of the member in their rotational capacity. He reported the results obtained for 40 beams, which were an extension of those reported by Mattock in 1964. The size of the specimens in Corley's research extended the range covered by Mattock (Corley 1966: 121 - 146). The most important variables covered the following range (Corley 1966: 123 - 127): Concrete cylinder strength: 4 ksi. Yield stress of tension reinforcement: 60 ksi. Yield stress of compression reinforcement: 60 ksi. Yield stress of stirrups: 50 ksi. - Width: 3, 9 and 12". - Effective depth: 5, 10, 24 and 30". - Span length: 36, 72, 144, 165, 240 and 330". Tension reinforcement ratio: Between 1 and 3%. Transverse reinforcement ratio: Between 0.3 and 9%. The test results were used to determine the spread of plasticity of the beam at each side of the midspan. Corley used the ratio dplOp,di2 as a measure of the spread of plasticity, as Mattock did in 1964, and used the same procedure to detennine it from the test results. He also investigated the same parameters (Corley 1966: 139 - 141). These are all presented in Section 2.7. Corley plotted the values of dpIQpjiz as a function of the parameters indicated by Mattock, and found that the amount of flexural reinforcement did not have a significant influence; contrary to what Mattock determined. A pronounced scatter was found for the other variables. Therefore, Corley considered that using a least squares fit was not appropriate. Based on this, he suggested a simpler expression for the spread of plasticity: 0.064 Ls (2.27) 6 Where Ls and d are in meters. The measured inelastic rotations for the tests beams reported by Corley and Mattock were compared to the inelastic rotations obtained with Equation 2.27. The inelastic 20 rotation in the length d/2 was calculated using compatibility and equilibrium equations (Corley 1966: 143 - 144). The results showed that Equation 2.27 provides a lower bound for most of the beams tested. Combining Equation 2.25 and 2.27, the plastic hinge length is: Lp =0.5^ + 0.032 4d (2.28) 2.9 Mattock (1967) Mattock, in his discussion of Corley's paper, used the measured values of the total inelastic rotation and the inelastic rotation in the length d/2 that he and Corley determined, to calculate the plastic hinge length, using Equation 2.25 (Mattock 1967: 519 - 522). He plotted these values as a function of the distance from the section of maximum moment to the section of zero moment. Although the results showed a considerable scatter, Mattock proposed the following equation, which represents reasonably the measured data: 2.10 ACI-ASCE Committee 428 (1968) The ACI-ASCE Committee 428, on their progress report on code clauses for "Limit Design", proposed lower and upper bounds for the plastic hinge length in beams and frames (ACI-ASCE Committee 428 1968: 713 - 715). The plastic hinge length is between the following limits: L = 0.5d + 0.05L. (2.29) (2.30) In which: 0.004 -s, ce (2.31) s, cu -S. ce 21 RE: Strain ratio. Rm: Moment ratio. see\ Concrete strain in the extreme compression fiber at yield curvature; either calculated, or assumed between 0.001 and 0.002. So,: Concrete strain in the extreme compression fiber at ultimate curvature, neglecting effects of confinement, loading rate and strain gradients. It is assigned a value between 0.003 and 0.004. Mmax'. Maximum moment in the length of the member. The following expression was proposed to calculate the distance from the section of maximum moment to the section of zero moment for members subjected to uniformly distributed load: AM Ls=- . m a x (2.33) 4V2+-JwzMmaxRm Vz: Shear adjacent to a concentrated load or reaction at the section of maximum moment. wz: Uniformly distributed load at the section of maximum moment. 2.11 Priestly, Park and Potangaroa (1981) Priestly, Park and Potangaroa performed tests on spirally-confined concrete columns to study their behaviour under seismic loading. The effect of confinement reinforcement on the ductility of column plastic hinges was investigated. They reported the results obtained for five columns (Priestly, Park and Potangaroa 1981: 181 - 202). The five column specimens were of octagonal shape with the same dimensions: 600mm diameter, a height of 3.3m, and longitudinal reinforcement of 16-24mm bars equally spaced around the circle (Priestly, Park and Potangaroa 1981: 184 - 187). The most important variables covered the following range: Concrete cylinder strength: Between 26.6 and 32.9 MPa. - Axial load: Between 1920 and 6770 kN. - Axial load ratio (PlfAg): Between 0.237 and 0.737. Yield stress of longitudinal reinforcement: 303 and 307 MPa. Diameter of spiral reinforcement: 10, 12 and 16mm. Spiral reinforcement volumetric ratio: Between 0.75 and 2.61. 22 Yield stress of spiral reinforcement: Between 280 and 423 MPa. The columns were tested as pin-ended vertical members. A cyclic lateral load was applied to a heavily reinforced central stub located in the middle of the column span to produce a linear bending moment diagram above and below the point of application. A constant axial load was applied during the test. The tests continued until failure. The test results were used to compare experimental plastic hinge lengths with the values obtained using the expressions proposed by Baker and Amarakone (Equation 2.19) and Corley (Equation 2.28). The experimental plastic hinge lengths were obtained from the curvature distributions over the plastic region (Priestly, Park and Potangaroa 1981: 190 - 192). The experimental plastic hinge lengths were calculated by solving Equation 2.5 for L p ; setting <j> = </>u, where <f>u is ultimate curvature, which for the tests is the average curvature measured on either side of the central stub. An equivalent yield curvature and yield displacement was estimated from the test results. The plastic deformation was determined using Equation 2.2, subtracting the yield displacement from the measured total displacement. The results showed that the axial load level had very little influence on the experimental plastic hinge lengths. The models proposed by Baker and Amarakone and by Corley predict larger values than the ones measured. In order for the curvatures to be conservatively estimated, the plastic hinge length has to be mderestimated (Priestly, Park and Gill 1982: 942). Therefore, these models gave results that were not conservative. For the experimental results, the plastic hinge length had an average value of approximately 0.3/r 2.12 Park, Priestly and Gill (1982) Park Priestly and Gill performed tests on square-confined concrete columns to study their behaviour under seismic loading. This investigation was done in parallel with the experimental study carried by Priestly, Park and Potangaroa in 1981. The effect of confinement reinforcement on the ductility of column plastic hinges was also investigated here. They reported the results obtained for four columns (Priestly, Park and Gill 1982: 929 - 950). The four column specimens had the same dimensions: cross-sectional area of 550mm2, a height of 3.3m, longitudinal reinforcement of 12-24mm bars arranged symmetrically around the perimeter, and a longitudinal reinforcement ratio of 1.79% (Priestly, Park and Gill 1982: 932 - 934). The most important variables covered the following range: 23 Concrete cylinder strength: Between 23.1 and 41.4 MPa. - Axial load: Between 1815 and 4265 kN. Axial load ratio: Between 0.214 and 0.6. Yield stress of longitudinal reinforcement: 375 MPa. Diameter of stirrups: 10 and 12mm. Transverse reinforcement ratio: Between 1.5 and 3.5%. Yield stress of stirrups: Between 294 and 316 MPa. The test procedure used for the columns was very similar to the one used by Priestly, Park and Potangaroa in 1981 (Priestly, Park and Gill 1982: 937). It is presented in Section 2.11. The test results were used to compare experimental plastic hinge lengths with the values obtained using the expressions proposed by Baker and Amarakone (Equation 2.19) and Corley (Equation 2.28), as it was done by the authors in their research from 1981 (Priestly, Park and Gill 1982: 938 - 942). The same procedure was used to determine the experimental plastic hinge lengths. The results showed that the axial load level had very little influence on the experimental plastic hinge lengths; the mean value for these was 0A2h. The models proposed by Baker and Amarakone and by Corley predict larger values than the ones measured, so they were not conservative (as in the research in 1981). Corley's equation was more accurate than Baker and Amarakone's equation, because it does not consider the axial load, which is in agreement with the results of this investigation. On the other hand, Baker and Amarakone research suggested that the plastic hinge length depended on the axial load. For the experimental results, the plastic hinge length can be conservatively estimated as OAh. 2.13 Oesterle, Aristizabal-Ochoa, Shiu and Corley (1984) Oesterle, Aristizabal-Ochoa, Shiu and Corley conducted tests of isolated reinforced concrete structural walls subjected to inelastic load reversals to study their web crushing strength. In this study, they considered a plastic hinge length equal to the horizontal length of the wall section, lw, (Oesterle, Aristizabal-Ochoa, Shiu and Corley 1984: 233). 24 2.14 Paulay (1986) Paulay presented design procedures for ductile reinforced concrete walls for earthquake resistance. He indicated that the plastic hinge length was primarily a function of the wall length. Based on this, he suggested that the length of the plastic hinge is between 0.5/w and /w (Paulay 1986: 801 - 802). 2.15 Zahn, Park and Priestly (1986) Zahn, Park and Priestly performed tests on reinforced concrete bridge columns with different cross-sections subjected to combined axial load and bending to study their strength and ductility. As part of their work they tested the validity of an equation derived by Priestly and Park to calculate the plastic hinge length (Equation 2.36, presented in Section 2.16). This equation was also derived from tests on concrete bridge columns, and did not take into account the axial load, since it did not have a significant effect according to the results obtained. They reported the results obtained for 14 columns (Zahn, Park and Priestly 1986). Four types of section shapes were studied: two square sections with face loading, four square sections with diagonal loading, two octagonal sections with circular reinforcement, and six circular hollow sections. The length of all the test units was 3.9m, and the depth (or diameter) of all the cross-sections was 0.4m (Zahn, Park and Priestly 1986: 62 - 63). The most important variables covered the following range (Zahn, Park and Priestly 1986: 211 â€” 212): Internal diameter of hollow sections: 212, 252 and 292mm. Concrete cylinder strength: Between 27 and 40.1 MPa. Axial load ratio: Between 0.08 and 0.43. Yield stress of longitudinal reinforcement: Between 306, 337, 423 and 440 MPa. Longimdinal reinforcement ratio: Between 1.51 and 5.48%. Yield stress of stirrups: Between 318, 340 and 466 MPa. Transverse reinforcement ratio: Between 0.61 and 3.52%. The test procedure used for the columns was very similar to the one used by Priestly, Park and Potangaroa in 1981 (Zahn, Park and Priestly 1986: 66 - 67). It is presented in Section 2.11. 25 The test results were used to develop an expression to calculate the equivalent plastic hinge length. The experimental plastic hinge lengths were determined separately above and below the central stub, in both directions of loading. These were determined the same way as it was done by the authors in their research from 1981 (Zahn, Park and Priestly 1986: 71 - 72). The test observations showed that the inelastic curvatures spread over a longer portion of the column when the axial load was high, due to concrete spalling. Therefore, the authors concluded that the plastic hinge length is a function of this parameter. There was a tendency for the plastic hinge length to be smaller for axial loads ratios lower than 0.3. Based on this, the following expression was proposed to predict the plastic hinge length (Zahn, Park and Priestly 1986: 238 - 240): For circular hollow columns with one ring of reinforcement and no confinement, the following equation was recommended: 2.16 Priestly and Park (1987) Priestly and Park performed tests on concrete bridge columns with different cross-sections subjected to combined axial load and bending to study their strength and ductility. The influence of the following variables in the seismic behaviour of concrete bridge columns was investigated: the axial load, the amount and yield strength of the transverse reinforcement, and the aspect ratio (Priestly and Park 1987: 61-76). The section shapes tested were square (face and diagonally loaded), octagon with circular reinforcement (solid or hollow) and hollow square (Priestly and Park 1987: 61). In addition to these, smaller octagonal and rectangular sections were also tested (Priestly and Park 1987: 64). (2.34) Lp =0.064 +4.5rfA (2.35) The most important variables covered the following range (Priestly and Park 1987: 61): 26 - Depth: Between 400 and 750mm. Thickness of hollow sections: 120mm. Two different tests were performed, one for squat columns (with an aspect ratio of two) and one for slender columns (with an aspect ratio of four). Members with square and octagonal sections were tested as squat columns. The test procedure used for these was very similar to the one used by Priestly, Park and Potangaroa in 1981. It is presented in Section 2.11. The hollow-section columns were tested as cantilever elements, a constant axial load and a cyclic lateral load at the top were applied to these (Priestly and Park 1987: 64 - 66). The test results were used to develop an expression to calculate the equivalent plastic hinge length. The experimental plastic hinge lengths were determined the same way as it was done by the authors in their research from 1981. These values were correlated with the column parameters, arriving to the following expression (Priestly and Park 1987: 71 - 73): Lp=0.0SLs+6db (2.36) This equation was used to compute plastic hinge lengths for columns with different aspect ratios that were tested by other researchers (including the ones tested by the authors in 1981 and 1982), and were compared with the experimental plastic hinge lengths obtained in these tests. There was a good agreement between the experimental and predicted values for most cases. The average value for all tests was approximately 0.5h. The test results in this research also indicated that the plastic hinge length did not have any significant dependence on the axial load ratio, the longitudinal reinforcement ratio, and the yield stress of longitudinal reinforcement. 2.17 Paulay and Priestly (1992) Paulay and Priestly proposed the following expression to estimate the plastic hinge length (Priestly, Seible and Calvi 1996: 308 - 309): Lp = 0.08L, + 0.022/^, Â£ 0.044// , (2.37) Where fy is in MPa units. The authors indicated that for commonly used beam and column dimensions, Equation 2.37 gives plastic hinge lengths of approximately 0.5/2. They also indicated that the plastic region where special detailing requirements must be provided to have sufficient rotational capacity 27 is larger than the calculated equivalent plastic hinge length (Paulay and Priestly 1992: 141 - 142). For fy = 275 MPa, Equation 2.37 becomes the same as Equation 2.36. 2.18 Wallace and Moehle (1992) Wallace and Moehle presented an analytical procedure to determine the need of confined boundaries in concrete walls subjected to earthquake loading. They stated that the plastic hinge length is usually between 0.5/w and /â€ž, (Wallace and Moehle 1992: 1633 - 1634). 2.19 Moehle (1992) Moehle indicated that the equivalent plastic hinge length in reinforced concrete columns depends on the section depth, aspect ratio, bar diameter, and the axial and shear force. He stated that good correlation with experimental results may be obtained if a plastic hinge length equal to 0.5/i is used (Moehle 1992:411-412). 2.20 Paulay and Priestly (1993) Paulay and Priestly reported tests on ductile concrete walls of rectangular shape subjected to seismic loading to study out-of-plane buckling. In this study, they used the following expression to calculate the plastic hinge length (Paulay and Priestly 1993: 386 - 387): 4 =0.2/w+0.044tfw (2.38) Where Hw is the total height of the wall. The authors indicated that Equation 2.38 predicts conservatively the plastic hinge length, so that the curvature ductility demands are not underestimated; and that it gives a good approximation of the portion of the height of the wall over which out-of-plane buckling can occur. 2.21 Sasani and Der Kiureghian (2001) Sasani and Der Kiureghian developed probabilistic displacement capacity and demand models for reinforced concrete walls. They derived a model for the plastic hinge length in concrete walls, using the test results reported by Corley in 1966 (Section 2.8) and Mattock in 1967 (Section 2.9). From the 40 beams tested back then, they selected 29 of them with effective depths greater than 0.5m. This data was 28 used to estimate the parameters of the following two equations, which are normalized versions of Equation 2.28 and 2.29, respectively (Sasani and Der Kiureghian 2001: 220 - 221): f = c c 1 + a 2 ^ r + ^ L (2.39) d d% ^ = a l + a 2 ^ - + Â£L (2.40) ls: Standard length equal to lm (39.4"), inserted to make the model parameters dimensionless. Â«/, a2: Model parameters. Â£,L: Model error term. For the 29 beams selected, the mean values of ai and a2 for Corley's equation were 0.52 and 0.047, respectively; and for Mattock's equation they were 0.56 and 0.051. These results were used in Equation 2.39 and 2.40 to determine plastic hinge lengths as a function of the effective depth and of the distance from the section of maximum moment to the section of zero moment. The results obtained for both models overestimated the plastic hinge length for large values of the effective depth, which are representative of concrete walls. Therefore, the authors explored several different models, and finally arrived to the following expression: ,â€” y ^ = a , + a 2 ^ t ~ + ZL (2.41) d d The mean values of ai and a2 for this equation were now 0.427 and 0.077, respectively. These results were used in Equation 2.41 to determine plastic hinge lengths as a function of the effective depth and of the distance from the section of maximum moment to the section of zero moment. The results obtained showed that Equation 2.41 provides a better fit to the data, specially for large values of the effective depth. 2.22 Mendis (2001) Mendis performed tests on simply supported beams subjected to a concentrated load at the midspan to investigate how the plastic hinge length is influenced by the amount of tension, compression and transverse reinforcement, and shear and axial forces. He reported the results obtained for 13 beams (Mendis 2001: 189-195). 29 The most important variables covered the following range (Mendis 2001: 192 - 193): - Span length: 630, 750 and 938mm. - Width: 60mm. Depth: 164mm. - Axial load: 0, 50, 100 and 175 kN. Stirrup spacing: 30, 50, 75 and 150mm. Amount of tension reinforcement: 3Y12 and 4Y12 bars. Amount of compression reinforcement: 2Y12 and 3Y12 bars. Concrete cylinder strength: Between 37.4 and 57.9 MPa. The test results were used to compare experimental plastic hinge lengths with the values obtained using the expressions proposed by Baker and Amarakone (Equation 2.12 and 2.19), Sawyer (Equation 2.22), Mattock (Equation 2.26 and 2.29), Corley (Equation 2.28), ACI-ASCE Committee 428 (Equation 2.30), and Park, Priestly and Gill (who suggested a value of OAh). Most of the experimental results were between the ACI-ASCE bounds. The experimental results showed that the plastic hinge length increased with the length, the shear-span ratio (MlVh) and the longitudinal reinforcement ratio; and decreased with the transverse reinforcement ratio. For the specimens tested with axial loads, the plastic hinge length was approximately OAd, meaning that it was independent of the axial load (Mendis 2001: 193). Park, Priestly and Gill arrived to this same conclusion from their research in 1982 (see Section 2.12). Comparisons between the experimental and predicted results showed that the expression of Baker and Amarakone for unconfined concrete (Equation 2.12) overestimated the plastic hinge length by a small margin, while their equation for confined concrete (Equation 2.19) underestimated it. The equations proposed by Mattock and Corley significantly overestimated the plastic hinge length. Sawyer's equation also overestimated the plastic hinge length for many specimens. The value recommended by Park, Priestly and Gill gave good predictions for beams with axial loads. The test results were used to derive an equation to calculate the plastic hinge length for beams without axial loads (Mendis 2001: 193 - 194): Lâ€ž = 0.25d RFT02SPR05 PER 0.2 (2.42) RFT = bd xlOO SPR = ^-d 30 (2.43) (2.44) PER: Percentage ratio of volume of stirrups to volume of concrete core measured outside of stirrups. A s t : Area of tension reinforcement. Asc'. Area of compression reinforcement. b: Width of the member. For members with the same amount of tension and compression reinforcement (like symmetrical columns or walls), Equation 2.42 gives a plastic hinge length of zero, so it is not applicable for these cases. Mendis also reported measured plastic hinge lengths obtained from tests on high-strength concrete (up to 80 MPa) beams and columns with low axial load that were performed by other researchers, and compared these results with the ACI-ASCE bounds. Most of the experimental results were between these bounds, the author recommended their use to estimate the plastic hinge length for normal and high-strength concrete beams and columns with low axial loads. 2.23 Panagiotakos and Fardis (2001) Panagiotakos and Fardis reported tests on reinforced concrete members subjected to uniaxial bending, with and without axial loads, to derive expressions for deformations at yielding and failure, in terms of the member geometric and mechanical properties. These members are representative of beams, columns and walls. They reported the results obtained for 1012 specimens (Panagiotakos and Fardis 2001: 135-148). The experimental database used in this research was the following (Panagiotakos and Fardis 2001: 136-139): 266 specimens were representative of beams, they had unsymmetrical reinforcement and were not subjected to axial loads. All the specimens had rectangular cross-sections, except for two of them, which had T-sections. Beams with and without closely spaced stirrups were tested. 31 682 specimens were representative of columns, they had symmetrically reinforced square or rectangular cross-sections, subjected or not to axial load. Columns with and without closely spaced stirrups were tested. 61 specimens were representative of walls, with rectangular, barbelled or T-sections. Walls with and without confined boundaries were tested. 23 column specimens had diagonal reinforcement, combined or not with longitudinal bars. 824 specimens used hot-rolled steel, 129 specimens used heat-treated steel, and 59 specimens used brittle cold-worked steel. The most important variables covered the following range: Concrete cylinder strength: Between 15 and 120 MPa. Axial load ratio: Between 0 and 0.95. Shear-span ratio: Between 1 and 6.5. Diagonal reinforcement ratio: Between 0 and 1.125%. Hardening ratio (f/fy): Between 1.1 and 1.5 (where ft is the tensile strength of steel). Steel strain at peak stress: Between 4 and 15%. Most of the specimens were tested as simple or double cantilevers, and others were tested as simply supported beams with a concentrated load applied at the midspan. They were subjected to monotonic and cyclic loading. Most tests continued until failure (Panagiotakos and Fardis 2001: 136). The test results were used to compare measured and predicted total rotations. The results for 875 specimens subjected to monotonic or cyclic loading, for which the values of the rotations at failure were measured and where failure was due to bending, were used for the comparison. From the data used, 242 were monotonic tests and 633 were cyclic tests. Al l 61 walls specimens were used. There was some slippage of the longitudinal reinforcement for 703 of these tests, most of them for cyclic loading (Panagiotakos and Fardis 2001: 140). An approach used by the authors to determine the total rotation at failure was through plastic hinge analysis (Panagiotakos and Fardis 2001: 144 - 147). This was done by rearranging Equation 2.8. Setting L-Ls and <f> = fa, and dividing Equation 2.8 by Ls, an expression for the total rotation at failure is obtained: (2.45) 32 Equation 2.45 requires knowing the yield and ultimate curvature, and the plastic hinge length. The yield and ultimate curvature were determined from expressions based on basic principles of mechanics, since these expressions predicted the measured curvatures well on average, although with a large scatter. With expressions for the yield and ultimate curvature combined with Equation 2.45, the total rotations at failure for the 875 tests considered were used to arrive for expressions for the plastic hinge length that provided the best fit to this data. For this, the authors decided to use the same form of the expression proposed by Paulay and Priestly in 1992 (Equation 2.37), since the parameters included in this expression were the most significant. They developed the following expressions to determine the plastic hinge length: For cyclic loading: Lpcy=0.l2Ls+0.0Uasldbfy (2.46) For monotonic loading: h,^ =l-5LPi<y = 0.18L, + 0.02lasldbfy (2.47) Lpcy: Plastic hinge length for cyclic loading. Lp,mon' Plastic hinge length for monotonic loading. as{. Zero-one variable. It is equal to one if slippage of the longitudinal reinforcement is possible, and zero if it is not possible. - fy: Yield stress of the tension reinforcement, in MPa units. Equation 2.46 and 2.47 were used in conjunction with Equation 2.45 to calculate the total rotation at failure for the 875 tests considered. Equation 2.37 of Paulay and Priestly was also used with Equation 2.45 to calculate the total rotations at failure. These predictions were compared with the experimental results. The quality of the predictions was very similar for these three equations, meaning that the predictions of Equation 2.45 were not very sensitive to the expression used to calculate the plastic hinge length. A considerable scatter of the results was observed. 2.24 Thomsen and Wallace (2004) Thomsen and Wallace conducted tests of slender reinforced concrete walls with rectangular-shaped and T-shaped cross-sections with moderate amounts of transverse reinforcement in the boundaries 33 to evaluate the simplified displacement-based design approach in ACI 318 1999. The premises on which displacement-based design is founded on were verified with the experimental results. They reported the results obtained for four walls (Thomsen and Wallace 2004: 618 - 630). The four wall specimens included two with rectangular sections (named RW1 and RW2) and two with T-sections (named TW1 and TW2). All had an aspect ratio of three and were considered as four-storey walls. The walls were 4" thick, 48" long and 144" high. Additionally, the flanges in the T-shaped walls were 4" thick and 48" long, and floor slabs were provided every 36" over the height for these walls (Thomsen and Wallace 2004: 618-619). The most important variables covered the following range (Thomsen and Wallace 2004: 621 -622): Average concrete cylinder strength at first storey: 4.6, 4.9, 6.3 and 6.1 ksi. Average concrete strain at peak stress: 0.002. Concrete rupture strength: Between 13 and 14% of concrete cylinder strength. Diameter of boundary longitudinal reinforcement: #3 bars. Diameter of boundary transverse reinforcement: 3/16" smooth wire. Diameter of distributed horizontal and vertical web reinforcement: #2 bars. Average axial load ratio: 0.1. The wall specimens were tested as cantilever elements. They were subjected to cyclic lateral displacements applied at the top, and a constant compressive axial load (Thomsen and Wallace 2004: 622 - 623). The test results were used to verify the premises on which displacement-based design for slender concrete walls is based on (Thomsen and Wallace 2004: 626 - 629). Some of these premises are: The normal strain profile along the wall length at the critical section is linear. The yield curvature is equal to 0.003//w. The plastic hinge length is equal to 0.5/w. The experimental strain profiles were obtained from measurements made along the length of the wall. The analytical strain profiles were determined using a simplified version of Equation 2.8. Setting L = Hw, and considering that the plastic hinge length is very small compared to the height of the wall, Equation 2.8 becomes: 34 + {t-</>yypHw (2.48) Considering a yield curvature of 0.003//â€ž, and a plastic hinge length of 0.5/w, and solving Equation 2.48 for the total curvature, the following expression can be derived: f A H Y 2 ^ 0 . 0 0 1 ^ + 0.0015 â€” (2.49) Then, the strain profile along the length of the wall is given by multiplying this curvature by the distance to the neutral axis. The strain profiles were computed for different drift levels (A/Hw). The curvature was computed for each drift level, and the neutral axis depth associated to that curvature was obtained from a moment-curvature analysis. The sensitivity of the strain profile to the assumed yield curvature and plastic hinge length was then analyzed. This was done using the same procedure as before: consider different values for these parameters, plug them into Equation 2.48, consider a drift level, solve for the curvature, detennine the neutral axis depth, and calculate the strain profile. Variation of the yield curvature from 0.0025//w to 0.0035//â€ž, had a negligible impact on the strain profiles. However, the plastic hinge length had a very significant influence. The authors compared the experimental and analytical strain profiles, considering plastic hinge lengths of 0.33/w, 0.5/w and 0.67/w; and determined that plastic hinge lengths between 0.33/w and 0.5/w produced the best agreement between results. For the specimen RW2, the best agreement was found for 0.33/w. The experimental results also showed that the strain profiles are not linear, although the walls are slender. The greater difference between the experimental and analytical strain profiles was in the tension zone; the authors indicated that it was due to the influence of concrete cracking and slippage of the reinforcement. 2.25 Summary An extensive review of the models used to calculate the equivalent plastic hinge length has been presented. These models were developed for beams, columns and walls. Details on the tests performed to arrive at these expressions have been described. 35 In classical plastic hinge analysis, the plastic hinge length is defined as the equivalent length over which the inelastic curvature is considered to be constant. This definition has been used by most of researchers cited to derive their models. The influence of phenomena like tensile strain penetration and spread of plasticity is considered imphcitly through the plastic hinge length. The plastic hinge length is a function of several parameters. One of the most important is the depth of the member, as it has been included in most of the models presented. Other important parameters are the span of the member, the longitudinal reinforcement properties, the axial load ratio and the strain hardening. However, most researchers have not included the same parameters in their models, due to the characteristics of the specimens tested. Chan (1955), Cohn and Petcu (1963), Sawyer (1964) and the ACI-ASCE Committee 428 (1968) considered the effect of strain hardening. Mattock (1964) also considered this parameter along with the reinforcement properties. However, Corley (1966) found that these two parameters did not have a significant influence, based on additional test results. Mattock (1967) proposed a new expression that did not include these parameters. Sawyer (1964), Corley (1966), Mattock (1967) and Sasani and Der Kiureghian (2001) did not include the longitudinal reinforcement properties. On the other hand, Baker and Amarakone (1964), Zahn, Park and Priestly (1986), Priestly and Park (1987), Paulay and Priestly (1992), Mendis (2001) and Panagiotakos and Fardis (2001) considered that these were important parameters. Baker and Amarakone (1964) included the influence of the type of steel and the concrete strength. They also considered the effect of the axial load, which had a direct relation with the plastic hinge length according to the results of their research. Zahn, Park and Priestly (1986) also considered that the effect of the axial load and concrete strength was important. However, Priestly, Park and Potangaroa (1981), Park, Priestly and Gill (1982), Priestly and Park (1987), Mendis (2001) and Panagiotakos and Fardis (2001) conducted tests for members subjected to axial load and found no significant dependence between this parameter and the plastic hinge length. The ACI-ASCE Committee 428 (1968) proposed lower and upper bounds for the plastic hinge length, instead of a one equation. Many researchers arrived to the conclusion that a safe (lower bound) approximation for the plastic hinge length is 0.5h or 0.5/w. This is the value given in many concrete codes, including CSA A23.3. 36 Most of the research done in plastic hinge length has been more focused in beams and columns. Some models have been developed for walls, but have certain limitations. Oesterle, Aristizabal-Ochoa, Shiu and Corley (1984), Paulay (1986) and Wallace and Moehle (1992) suggested values for plastic hinge length between 0.5/w and lw. Paulay and Priestly (1993) proposed an equation for the plastic hinge length applicable to walls, which provides conservative predictions of the curvature ductility demand. Sasani and Der Kiureghian (2001) developed plastic hinge length models for reinforced concrete walls using the test results reported by Corley (1966) and Mattock (1967). However, these tests were performed on beams, which have a different behaviour compared to walls. Panagiotakos and Fardis (2001) derived expressions for the plastic hinge length for monotonic and cyclic loading using the results obtained for 875 specimens. Only 61 of these were walls, so these models may not be representative of this type of members. Thomsen and Wallace (2004) obtained an approximation of the plastic hinge length by comparing experimental and analytical strain profiles. The best agreement was found for values between 0.33/w and 0.5/w. Table 2.1 summarizes all the studies previously done on plastic hinge length presented in this chapter, including the type of members studied and the parameters considered by the authors to develop their models: 37 Table 2.1 Summary of previous research done on plastic hinge length Researchers reference Members studied Parameters considered Chan (1955) Columns Span, strain hardening Baker (1956) Beams and columns Depth Cohn and Petcu (1963) Beams Depth, strain hardening Baker and Amarakone (1964) Beams and columns Depth, span, reinforcement properties, axial load ratio Sawyer (1964) Beams and columns Depth, span, strain hardening Mattock (1964) Beams Depth, span, reinforcement properties, strain hardening Corley (1966) Beams Depth, span Mattock (1967) Beams Depth, span ACI-ASCE Committee 428 (1968) Beams and columns Depth, span, strain hardening Priestly, Park and Potangaroa (1981) Columns Depth Park, Priestly and Gill (1982) Columns Depth Oesterle, Aristizabal-Ochoa, Shiu and Corley (1984) Walls Depth Paulay (1986) Walls Depth Zahn, Park and Priestly (1986) Columns Span, reinforcement properties, axial load ratio Priestly and Park (1987) Columns Depth, span, reinforcement properties Paulay and Priestly (1992) Beams and columns Depth, span, reinforcement properties Wallace and Moehle (1992) Walls Depth Moehle (1992) Columns Depth Paulay and Priestly (1993) Walls Depth, span Sasani and Der Kiureghian (2001) Beams d > 0.5m Depth, span Mendis (2001) Beams and columns Depth, span, reinforcement properties Panagiotakos and Fardis (2001) Beams, columns and walls Span, reinforcement properties Thomsen and Wallace (2004) Walls Depth The models presented in this chapter will be used to determine the equivalent plastic hinge length for a test specimen representative of a shear wall in a high-rise building, in order to compare the results obtained. These results are presented in Chapter 4 (see Section 4.2.8). 38 CHAPTER 3: ANALYTICAL METHODS 3.1 Introduction to program VecTor2 Program VecTor2 will be the analysis tool used in this research. VecTor2 is a computer program developed to perform nonlinear finite element analysis of two-dimensional reinforced concrete membrane structures subjected to quasi-static loading. This program has been developed by researchers from the University of Toronto. In this chapter, a general overview of program VecTor2 will be presented, focusing basically on the models used in this research. The information presented in this chapter comes directly from the VecTor2 and FormWorks User's Manual (Wong and Vecchio 2002) and the VecTor Analysis Group website. Along with program VecTor2, program FormWorks is used as the pre-processor for the analysis, and program Augustus is used as the post-processor. 3.2 Theoretical bases of program VecTor2 The analysis was performed using the constitutive models of the Disturbed Stress Field Model (Vecchio 2000: 1070 - 1077), which is a refinement of the Modified Compression Field Theory (Vecchio and Collins 1986: 219- 231). Both analytical models can predict the behaviour of reinforced concrete elements subjected to in-plane normal and shear stresses, modeling cracked concrete as an orthotropic material with smeared, rotating cracks. Constitutive models for a variety of effects, such as compression softening and tension stiffening, are included to accurately predict the response (Wong and Vecchio 2002: 2). Additionally, the Disturbed Stress Field Model can consider different directions of the principal stresses and strains, and takes into account crack shear slip deformations (Wong and Vecchio 2002: 13). 3.3 Finite element formulation Program VecTor2 uses low-powered finite elements to model the structure. These include the 3-node constant strain triangle (with six degrees of freedom) and the 4-node plane stress rectangle (with eight degrees of freedom) to model concrete, and the 2-node truss bar element (with four degrees of freedom) to model discrete reinforcement. The reinforcement may be modeled as either smeared within concrete elements or as discrete bars. The stresses, strains and material properties are constant within each element (VecTor Analysis Group). 39 The finite element solution is based on a secant stiffness formulation that uses a total-load iterative procedure (VecTor Analysis Group). At each load step, the element stiffness matrices are calculated from the current stress-strain state and then assembled, and the nodal loads are calculated. The nodal displacements are determined and then used to calculate one strain tensor for each element, and then the principal strains are determined. These are used in the constitutive relationships for concrete and steel to calculate the stress tensor in each element. The secant moduli are then determined from the new stress-strain state, and compared to the secant moduli from the previous stress-strain state. If convergence is satisfactory, the analysis for that load step is completed and then continues to the next load step. If not, the analysis is repeated using the new stress-strain state. Usually, convergence is achieved after 10 to 30 iterations (Selby, Vecchio and Collins 1996: 306 - 307). This procedure continues until the specified force or the target displacement is reached, or until the structure becomes unstable. 3.4 Models for concrete in compression The compression response of concrete is described through nonlinear functions relating stresses and strains. Different models are used for the ascending and descending branches of the concrete response in uniaxial compression. These curves are then modified to account for second-order effects (Wong and Vecchio 2002: 45). Figure 3.1 Concrete compression response fc2 | post-peak response pre-peak response ^ 8 c 2 3.4.1 Compression pre-peak response Compression pre-peak response models compute the principal compressive stresses while the principal compressive strain is less than the strain corresponding to the peak compressive stress. The 40 model used for the ascending curve for concrete compressive stress is the one proposed by Popovics for normal-strength concrete. The stress-strain curve is described by (Wong and Vecchio 2002: 46 - 47): n â€” n (3.1) n = Ec-Et (3.2) sec , _ J p sec I (3.3) - fc2- Average net concrete axial stress in the principal compressive direction. sc2: Average net concrete axial strain in the principal compressive direction. sp: Concrete compressive strain corresponding tofp. - fp: Peak concrete compressive stress. n: Curve fitting parameter for stress-strain response of concrete in compression. Ec: Concrete initial tangent stiffness. Esec: Concrete secant stiffness. 3.4.2 Compression post-peak response Compression post-peak response models compute the principal compressive stresses while the principal compressive strain is greater than the strain corresponding to the peak compressive stress. The compressive stresses are computed as follows (Wong and Vecchio 2002: 52): fc2 = (1 (3.4) c a 0 < c a < l (3.5) ca: Averaging factor. fca2: Average concrete compressive stress contribution of unconfined concrete. fcb2: Average concrete compressive stress contribution of confined concrete. The stress contribution of unconfined concrete is determined using the Smith-Young model: fc2 ~ fp Exp 41 (3.6) The stress contribution of confined concrete is determined using the Modified Park-Kent model, in which the stresses decay linearly (Wong and Vecchio 2002: 53-54): [ v-zJp >0<fc<fp z = 0.5 3 +0.29k 145/;' -1000 - 0.002 + Jcl ,170, (3.7) (3.8) Zm: Slope of compression post-peak descending curve. - fc: Concrete cylinder uniaxial compressive strength, in MPa units. s0: Concrete compressive strain corresponding tofc. - fci. Average net concrete axial stress in the principal tensile direction. 3.4.3 Compression softening Compression softening is the reduction of compressive strength and stiffness of concrete due to transverse cracking and tensile straining. In VecTor2, compression softening is included by calculating a softening parameter, Bj, which varies between zero to one. The model used to determine this parameter is the Vecchio 1992-A (ecl/eC2-Â¥oTm). This factor is then applied to the uniaxial compressive strength and its corresponding strain to obtain the peak compressive strength and its corresponding strain, respectively; used in the compression response models previously described. The following expressions apply (Wong and Vecchio 2002: 58-61): 1 <1 Câ€ž = 1+c.c, [0 j v < 0.28 |fJ.35(r-0.28)08 ; A-> 0.28 r = <400 'c2 (3.9) (3.10) (3.11) (3.12) 42 'PdPieo',PdPiSo<eo<0 Â£c2 i Co < Ec2 < PdPl^o < 0 E0 ; ec2 <sa< J3d0,so < 0 (3.13) Cs: Compression softening shear slip factor. It is assigned a value of one i f shear slip is not considered, and 0.55 if it is considered. Cd'. Compression softening strain softening factor. r. Ratio of the principal tensile strain to the principal compressive strain. Scf. Average net concrete axial strain in the principal tensile direction. B(. Strength enhancement factor. The value of sp may be modified in some cases, as shown in Equation 3.13, so that compression response ascends for strains up to e0 and then descend. 3.4.4 Confinement strength Confinement increases the compressive strength and ductility of concrete. In VecTor2, the effect of confinement is included by calculating a strength enhancement factor, which is equal or greater than one. The model used to determine this parameter is the Kupfer-Richart model. This factor is then applied to the uniaxial compressive strength and its corresponding strain, to obtain the peak compressive strength and its corresponding strain, respectively; used in the compression response models previously described (Equations 3.12 and 3.13). For the case of biaxial compression, the strength enhancement factor for the direction of the largest compressive stress, fc2, is determined using the following expression (Wong and Vecchio 2002: 77 - 78): 2 *1 ; / c 2 < / c i < 0 (3.14) P, =1 + 0.92 (- f y -0.76 f-f } Jcl \ Jc J < fc J The strength enhancement factor for the direction offcl is determined by interchanging/; fovfc2 in Equation 3.14. 3.5 Models for concrete in tension The tension response of concrete is divided into uncracked and cracked response: 43 Figure 3.2 Concrete tension response i 1 > 8 c 1 8 c r Before cracking, the response is considered linear-elastic (Wong and Vecchio 2002: 64): fd = EcÂ£cl ; 0 < eel < ecr (3.15) ecr=~ (3.16) scr: Concrete cracking strain. - fcr: Concrete cracking stress. After cracking, VecTor2 does two calculations to detemiine the average concrete tensile stresses, one due to tension stiffening and the other due to tension softening. The larger of the two values is taken as the average post-cracking tensile stress (Wong and Vecchio 2002: 65): fel =Max(/;i,fc\) ; 0<scr<sc: (3.17) / c i : Average concrete tensile stress due to tension stiffening. / c , : Average concrete tensile stress due to tension softening. 3.5.1 Tension stiffening Tension stiffening is the presence of post-cracking tensile stresses between cracks in the vicinity of the reinforcement. In VecTor2, for discrete reinforcement elements, their tributary area in which the concrete within exhibits tension stiffening is delineated by a distance equal to 7.5 bar diameters from the 44 reinforcement element (Wong and Vecchio 2002: 64 - 65). Tension stiffening is included by decreasing gradually the average stress-strain response of concrete in tension. The Modified Bentz model is used for this purpose. This model is the same as the Bentz 1999 model when the steel is aligned with the x or y-axis (no skew steel). The concrete tensile stresses decay following these expressions (Wong and Vecchio 2002: 18): ^ (3-18) c,=2.2m (3.19) Â± = YA-*\cose\ (3.20) ct\ Coefficient that incorporates the influence of reinforcement bond characteristics. m: Bond parameter, in millimetres. 6â€ž: Angle between the normal to the crack surface and the longitudinal axis of the reinforcement. 3.5.2 Tension softening Tension softening is the presence of post-cracking tensile stresses in plain concrete. It is included by decreasing gradually the average stress-strain response of concrete in tension, as it is done for tension stiffening. The linear model, which does not consider residual tensile stresses, is used for this purpose. The concrete tensile stresses decrease linearly following these expressions (Wong and Vecchio 2002: 70 -72): 2Gf L~fcr ; l.lscr <sch <10f fcl - fcr Sch Â£cr J >0 (3.21) (3.22) eCh- Characteristic strain. Gf. Energy required to form a complete crack of unit area, it is assigned a value of 75 N/m. - L r : Distance over which the crack is assumed to be uniformly distributed, it is assigned a value of half the crack spacing. 45 3.5.3 Cracking criterion The concrete cracking stress usually decreases as the compressive stresses acting transversely increase, so it does not remain constant and may have a different value from the specified concrete uniaxial tensile strength,/",. The Mohr-Coulomb stress model is used to calculate the cracking stress. The following expressions apply (Wong and Vecchio 2002: 80 - 82): fcru 2C 1-Sin<& 2CosO CosO 1 + SirtO fcr fcru 1 + fc2 f'c) â€¢Olf^f^Kf, (3.23) (3.24) (3.25) C: Cohesion. - O: Internal angle of friction, it is assigned a value of 3 7Â°. 3.6 Models for slip distortions in concrete VecTor2 can take into account crack shear slip deformations, as it is formulated in the Disturbed Stress Field Model. When crack shear slip deformations are considered, the crack shear check requirement is eliminated (Wong and Vecchio 2002: 13). The model used to calculate the shear slip is the Hybrid-I model, which combines the Lai-Vecchio stress model (a stress-based model) and the constant rotation lag model (Wong and Vecchio 2002: 88). A hybrid model computes the shear slip strains using both the stress-based model and the constant rotation lag model, and takes the greater value: ys=Max(ras,rbs) (3.26) ys: Shear slip strain. y": Shear slip strain determined from the stress-based model. ybs: Shear slip strain determined from the constant rotation lag model. 46 3.6.1 Stress-based model The stress-based models relate the shear slip along the crack to the local shear stress along the crack. The shear slip strain is then computed from the shear slip with the following expression (Wong and Vecchio 2002: 85 - 86): s (3.27) Ss: Shear slip. s: Crack spacing. For the Lai-Vecchio stress model, the following expressions used are to calculate the shear slip (Wong and Vecchio 2002: 87): Ss=S;j-^<2w (3.28) " 1 - ^ s * = 0-5v a.m a x+v c o ^ 1.8W-08 +(0.234>v-0'707 -0.2)/ c c ( } W = (3.30) ^ci, max Va\max 24w 0.31 + -a + \6 V =IsÂ£. c o 30 (3.31) (3.32) / Â« ' = l - 2 / c (3.33) w. Average crack width, in millimetres. - v d: Local shear stress on the crack, in MPa units. : Maximum local shear stress on the crack, in MPa units. a: Maximum aggregate size, in millimetres. 47 3.6.2 Constant rotation lag model The constant rotation lag models relate the post-cracking rotation of the principal stress field to the post-cracking rotation of the principal strain field using a rotation lag (Wong and Vecchio 2002: 86): A#CT: Post-cracking rotation of the principal stress field. A0e: Post-cracking rotation of the principal strain field. 0l: Specified rotation lag. It is assigned a value of 10Â° for unreinforced elements, 7.5Â° for uniaxially reinforced elements and 5Â° for biaxially reinforced elements. The shear slip strain is then computed with the following expressions: 6â€ž. Inclination of the principal stress field. 0ic: Inclination of the principal stress field at cracking. Yxy-. Total shear strain. Ex*: Total axial strain in the ^ -direction. Syy\ Total axial strain in the v-direction. 3.7 Models for reinforcement The constitutive models for steel use nonlinear functions relating stresses and strains. Effect of strain hardening is considered (Wong and Vecchio 2002: 98). 3.7.1 Stress-strain response (3.34) 0â€ž=0lc+A0lT (3.35) (3.36) The model used for the monotonic stress-strain response of the reinforcement is the ductile steel reinforcement model. The response is trilinear, consisting of a linear-elastic response, a yield plateau, and a linear strain-hardening phase until rupture: 48 Figure 3.3 Reinforcement compression and tension response ^ ^ yield plateau elastic response strain hardening phase > 8 s The reinforcement stress, both in tension and compression, is given by the following expression (Wong and Vecchio 2002: 98 - 99): fy >Â£y< k| * fy + Esh(Â£s -Â£sh)> Ssh < \ss\ < â‚¬u 0 ' <\e} fu fy (3.37) (3.38) (3.39) 'sh fs: Reinforcement stress. Es: Initial tangent stiffness or elastic modulus of reinforcement. ss: Reinforcement strain. Sy-. Yield strain. ssh: Strain at the onset of strain hardening. Esh- Strain hardening modulus. Su\ Reinforcement ultimate strain. fu: Ultimate strength of reinforcement. 49 3.7.2 Dowel action Dowel action is the shear resistance offered by reinforcing bars crossing a crack. In VecTor2, dowel action models are used in conjunction with the stress-based slip distortion models for concrete. The shear resistance due to dowel action is calculated as a function of the shear slip at the crack. This shear resistance is then subtracted from the local shear stress on the crack, which reduces the shear slip. The model used to determine the shear resistance due to dowel action is the Tassios model. The following expressions are used to determine the shear resistance (Wong and Vecchio 2002: 103 - 104): V^EsI,X'S,iV^ (3.40) 'â€¢-lit (341) 64 if K = - ~ ^ (3.43) Vdu=\.21dl-\fcfy (3.44) v r f = ^ (3.45) Vd: Dowel force. Iz: Moment of inertia of the reinforcement. A: Parameter that compares the stiffness of the concrete to the stiffness of the reinforcing bar. VJu: Ultimate dowel force. kc: Stiffness of the notional concrete foundation. cb: Coefficient used to reflect bar spacing, it is assigned a value of 0.8. vd: Shear resistance due to dowel action. ps: Reinforcement ratio. As: Area of reinforcement. 50 CHAPTER 4: COMPARISON OF ANALYTICAL PREDICTIONS WITH ISOLATED W A L L TEST RESULTS 4.1 Scope of analysis Program VecTor2 was used to analyze isolated walls specimens that have been previously tested. The analytical predictions were compared with the test results available in order to see how well VecTor2 predicts the response of these members. In particular, the curvature distribution and the strain profile at critical sections were examined, since these results were then used to estimate plastic hinge lengths. The objective of this analysis was to test the validity of the results provided by VecTor2, so that the program can then be used to perform a parametric study of concrete walls. Also, the actual curvature distributions and strain profiles were investigated. Two wall specimens were analyzed. The first is a high-rise shear wall tested at the University of British Columbia in 2000 (Adebar, mrahim and Bryson 2004). The second is one of the rectangular shear walls tested at Clarkson University in 1995 (Thomsen and Wallace 1995); the wall analyzed is the rectangular specimen RW2 (see Section 2.24). 4.2 High-rise shear wall tested at the University of British Columbia A test was conducted on a large-scale model of a slender reinforced concrete cantilever shear wall from the core of a high-rise building at the University of British Columbia in 2000 by Adebar, mrahim and Bryson. The wall had an aspect ratio of 7.2, had a flanged cross-section, had a low amount of vertical reinforcement, and was subjected to a constant compressive axial load. The purpose of this test was to investigate the influence of cracking on the effective stiffness of the wall used for seismic analysis. Extensive concrete strains measurements were made at the faces of the wall over the cracked region (Adebar, mrahim and Bryson 2004: 1 - 2). 4.2.1 Description of the wall specimen The specimen was a 1/4 scale model of a typical wall from a high-rise building. It was 12200mm high and 1625mm long, with a flanged cross-section. The web was 1219mm long and 127mm thick, and the two flanges were 203mm long and 380mm thick (Adebar, mrahim and Bryson 2004: 4 - 5). The reinforcement was arranged symmetrically. The vertical reinforcement in the two flanges consisted of 5-10M reinforcing bars enclosed by 10M ties, spaced at 64mm in the lower 3m of the wall 51 and then spaced at 152mm over the remaining height of the wall. The vertical reinforcement in each flange was arranged in two layers, the exterior layer had three bars and the other layer had two bars. The clear cover of the ties was 6mm. The web had 10M reinforcing bars spaced at 305mm, vertically and horizontally. A total of four bars were used in the web. High-rise buildings typically have large perimeter walls below grade that are much larger than the tower walls. These are connected together by diaphragm action of the concrete slabs. Because of this, the section where the maximum bending moment occurs in the tower walls is usually at grade level, and there is no pullout of the vertical reinforcement from the foundation. To simulate this effect, the critical section was located at 426mm from the base, providing additional vertical reinforcement below a construction joint at this location (Adebar, mrahim and Bryson 2004: 5). The wall specimen was tested in a horizontal position because of the limited height of the laboratory. The base of the wall was post-tensioned to the floor to prevent movement during the test (Adebar, mrahim and Bryson 2004: 5 - 6). The average cylinder compressive strength was 49 MPa at the time of testing. The average yield strength of the reinforcing steel was 455 MPa, and the average ultimate strength was 650 MPa. The stress-strain response from four bar samples was measured, the yield plateau was very short (Adebar, mrahim and Bryson 2004: 6). 4.2.2 Instrumentation A cyclic lateral load was applied at the top of the wall by a hydraulic actuator. Dywidag bars were used to apply a uniformly distributed axial compression load. The wall displacement at the top was measured using linear variable differential transducers. Displacement transducers were also provided to measure movement of the base of the wall (Adebar, mrahim and Bryson 2004: 6 - 7). Twelve metal targets were glued to the concrete surface over the lower 5.08m of the wall height along each face to measure average concrete strains. Targets named TE1 to TE12 were located at the east face, and targets named TW1 to TW12 were located at the west face. The relative vertical displacements of the targets were measured with a large digital calliper. On each side of the wall, 10 targets located above the construction joint (TE3 to TE12 and TW3 to TW12) were used to measure strains over the cracked region. Targets TE3 and TW3 were located 59mm above the construction joint, and the rest were spaced at 500mm approximately. The two bottom targets located below the construction joint were more 52 closely spaced. Targets TE2 and TW2 were located 134mm below the construction joint, and targets TE1 and TW1 were located just above the base (Adebar, mrahim and Bryson 2004: 32). 4.2.3 Test procedure A constant compressive axial load of 0.\Agfc, equal to 1500 kN, was applied to the wall during the test. This load was applied through two hydraulic actuators located below the base of the wall that pulled the Dywidag bars. The cyclic lateral load at the top was applied at 11.76m from the base of the wall; that is, 11.33m from the construction joint. The wall was subjected to 13 displacement levels, which varied from 15 to 300mm. East was considered the positive direction of loading, and west the negative direction of loading. At each displacement level, four complete displacement cycles (zero, then maximum positive, then maximum negative, and finally zero) were performed (Adebar, Ibrahim and Bryson 2004: 6 - 7). 4.2.4 Test results The test results were used to determine curvature distributions along the height of the wall for different lateral displacement levels. The curvatures were determined from the axial strains at the faces of the wall, which were determined from the measured relative axial displacements of the targets. The lateral displacements were corrected so that they did not include the effect of the base rotation. Although the authors determined experimental curvatures for the wall, the curvatures presented in this section were recalculated from the original measured data, and differ from the results reported by the authors. During the test, some rotation at the base of the wall was measured. The total displacements included the lateral displacement component (wall displacement) and the rigid body motion of the wall due to base rotation (Adebar, mraruin and Bryson 2004: 8). Measurements from displacement transducers at the base were used to determine displacements due to base rotation to separate these two components. For each total displacement level, the corresponding wall displacement in each direction of loading was different (Adebar, Ibrahim and Bryson 2004: 27). The first cracking in the wall occurred at a wall displacement of 21mm (total displacement of 30mm) in the positive direction, and yielding of the vertical reinforcement at the construction joint occurred at a wall displacement of 46mm (total displacement of 60mm) in the positive direction (Adebar, mrahim and Bryson 2004: 9). The maximum wall displacement was 281mm (total displacement of 300mm) in the positive direction (Adebar, mraliim and Bryson 2004: 11). 53 The relative axial displacements of the targets in the compression and tension face of the wall were measured at the peak displacement of the first cycle for each wall displacement level. The difference between the vertical distance between targets at a particular displacement level and the initial vertical distance between targets (zero reading), divided by this initial distance, gives the average axial strain over that length at the faces of the wall for that displacement level. Then, the difference between the compression and tension strain, divided by the wall length between the two faces, gives the average curvature over the length between targets (Adebar, mrariim and Bryson 2004: 12 - 13). This procedure was used to calculate the curvature distribution at total displacement levels of 120, 150 and 200mm; for which the corresponding wall displacement levels were the following: Wall displacements of 105, 132 and 182mm, respectively; in the east (positive) direction of loading. Wall displacements of 104, 138 and 187mm, respectively; in the west (negative) direction of loading. A significant portion of the concrete cover fell off at a total displacement level of 300mm and some targets were lost, so the curvature distribution could not be determined for this displacement level (Adebar, mrahim and Bryson 2004: 14). Regarding the zero readings, several target measurements were made before and after the axial compression was applied. The average of all these readings was taken as the zero reading used in the calculations. Measurements that were significantly out of range were neglected to determine the zero readings. Figure 4.1 shows the experimental curvature distributions for the wall displacement levels mentioned in the east direction: 5 4 Figure 4.1 Experimental curvatures (pushing east) for different wall displacement levels 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 Curvature (rad/m) Figure 4.2 shows the experimental curvature distributions for the wall displacement levels mentioned in the west direction: Figure 4.2 Experimental curvatures (pushing west) for different wall displacement levels o 4 - , , , , 1 0.000 0.005 0.010 0.015 0.020 0.025 Curvature (rad/m) 55 As expected, the maximum curvature is located at the construction joint; that is, at approximately 426mm from the base. There were differences between the curvatures for both directions of loading; these become very significant near the construction joint for the higher displacement levels. This is because the wall was first pushed to the east and then to the west, so the crack pattern was not symmetrical (Adebar, mrahim and Bryson 2004: 10). An experimental moment-curvature relationship for the wall was developed by the authors, in order to estimate the yield curvature. The yield curvature was taken as the curvature prior to the yield plateau, which was approximately equal to 0.002 rad/m (Adebar, forahim and Bryson 2004: 15). 4.2.5 Analytical model of the wall specimen The prototype wall was modeled and analyzed using program VecTor2. The purpose of this analysis is to predict the curvature distribution along the height of the wall and compare these predictions with the experimental results. The strain profile along the length of the wall was also studied. Low-powered rectangular and triangular elements were used to model the concrete, with smeared steel to account for the presence of reinforcement. The constitutive models for concrete and steel described in Chapter 3 were used in the analysis. Three different material types were used to represent various regions of the wall in the finite element model: The first material type was used to represent the flanges in the lower 3m of the wall, in which the ties were spaced at 64mm. The second material type was used to represent the web of the wall. The third material type was used to represent the flanges over the remaining height of the wall, in which the ties were spaced at 152mm. As described in Section 4.2.1, the wall had a construction joint at 426mm from the base; and the maximum curvature occurs at this location, as shown in Figure 4.1 and 4.2. For this reason, the wall was modeled from the construction joint upwards. The horizontal and vertical displacements at the bottom in the analytical model were restrained. The finite element mesh was more refined near the base (critical section), in order to have a good representation of the strain profile along the length of the wall. The mesh was refined in such a way that the position of certain nodes coincides with the position of the targets. Meshes of 68x59, 68*87, 87*59 56 and 87x87 rectangular elements were used; 20 elements (21 nodes) were used in the transverse direction of the wall. This level of refinement was maintained up to approximately half the height of the wall. Up from this point, 16 elements (17 nodes) were used in the transverse direction, and then it was further reduced to eight elements (nine nodes) up to the top of the wall. The transitions were made using triangular elements. Al l nodes and elements were numbered in the horizontal (short) direction. The complete mesh consisted of 1487 nodes, 1360 rectangular elements and 60 triangular elements. The material properties used in the analysis were those reported in the description of the wall specimen, presented in Section 4.2.1. For the material properties that were not measured during the test, the values given by default in program VecTor2 were used. For the concrete properties, VecTor2 uses the following expressions to determine the tensile strength, the initial tangent elastic modulus and the cylinder strain atfc (Wong and Vecchio 2002: 146): / ; = 0 . 3 3 ^ (4.1) Ec = 5500VZ (4.2) sQ =1.8 + 0.0075/J (4.3) Where fc, f, and Ec are in MPa units, and e0 is in mrn/m. The maximum aggregate size given by default is 10mm. For the reinforcement properties, a modulus of elasticity of 200000 MPa was considered; while the strain hardening modulus and the strain at the onset of strain hardening were determined from the measured stress-strain curve. A monotonic lateral load was applied at the top of the wall. This load was applied in a displacement-control mode, in increments of 0.2mm. Although the actual wall was subjected to cyclic loading, applying a monotonic load seems reasonable, since the envelopes for the monotonic and cyclic response were almost the same. Additionally, the constant axial load of 1500 kN was applied; this load was equally distributed among all the nine nodes at the top. The self-weight of the wall was not considered. Figure 4.3 shows the finite element model of the wall specimen, created in the pre-processor FormWorks: Figure 4.3 Finite element model of UBC wall in FormWorks 1JWÂ» v M I U I I M I M â€¢ m i l i u m â€¢ M i n i m a * m i m m v i Table 4.1 shows the material properties in the different regions of the wall: 58 Table 4.1 Material properties of UBC wall model Concrete properties Material 1 Material 2 Material 3 Color Thickness (mm) 380 127 380 Cylinder compressive strength (MPa) 49 49 49 Reinforcement component properties jc-direction rÂ»direction jc-direction -^direction x-direction -^direction Reinforcement ratio (%) 0.831 0.65 0.259 0.259 0.346 0.65 Reinforcement diameter (mm) 10 10 10 10 10 10 Yield strength (MPa) 455 455 455 455 455 455 Ultimate strength (MPa) 650 650 650 650 650 650 Elastic modulus (MPa) 200000 200000 200000 200000 200000 200000 Strain hardening modulus (MPa) 4875 4875 4875 4875 4875 4875 Strain hardening strain (mm/m) 10 10 10 10 10 10 4.2.6 Analytical results The analytical predictions obtained from program VecTor2 were used to determine curvature distributions along the height of the wall for different lateral displacement levels. The curvatures were determined the same way as it was done for the experimental results in order to make comparisons, using the vertical nodal displacements at the faces of the wall obtained from the analysis. Since the model of the wall was fixed at the base, the total displacement and wall displacement are the same. As previously described, the position of certain nodes at the faces of the wall coincides with the position of the targets. The vertical nodal displacements at these particular nodes were used to calculate the curvatures. The difference in vertical displacements between two consecutive nodes at the face, divided by the vertical distance between these nodes, gives the average axial strain over that length. Then, the difference between the compression and tension strain, divided by the wall length between the two faces, gives the average curvature over the length between nodes. The curvature distribution was determined for the following wall displacement levels: Wall displacements of 105, 132 and 182mm; to be compared with experimental curvatures when the wall is pushed in the east direction. Wall displacements of 104, 138 and 187mm; to be compared with experimental curvatures when the wall is pushed in the west direction. 59 This way, the experimental and analytical average curvatures were calculated over the same lengths. The only exception is near the construction joint, due to the fact that for the analysis, the wall was modeled from the construction joint upwards. Therefore, the analytical curvature at this location was calculated for a shorter length than for the experimental curvature; this length goes from the location of the construction joint to the location of targets TE3 and TW3, located 59mm above the construction joint. 4.2.7 Comparison of experimental and analytical results Figure 4.4 to 4.9 show comparisons between the experimental and analytical curvature distributions for the wall displacement levels mentioned. Note that the vertical axis shows the distance from the construction joint, not the height of the wall: Figure 4.4 Experimental (pushing east) and analytical curvatures for a wall displacement of 105mm 4500 4000 3500 Â§ 3000 2500 2 Â« 2000 o o E 1500 1000 500 0 0. -500 qoo â€” * â€” Experimental (east) - - Â» - - Analyt ical Curvature (rad/m) Figure 4.5 Experimental (pushing west) and analytical curvatures for a wall displacement of 104mm Figure 4.6 Experimental (pushing east) and analytical curvatures for a wall displacement of 132mm -500 o.qoo o.oj â€”â€¢â€” Experimental (east) - -Â»- - Analytical Curvature (rad/m) Figure 4.7 Experimental (pushing west) and analytical curvatures for a wall displacement of 138mm 4500 â€”â€¢â€” Experimental (west) - -*- - Analytical -500 Curvature (rad/m) Figure 4.8 Experimental (pushing east) and analytical curvatures for a wall displacement of 182mm o.qoo -500 â€”â€¢â€” Experimental (east) --"--Analytical Curvature (rad/m) 62 Figure 4.9 Experimental (pushing west) and analytical curvatures for a wall displacement of 187mm There is a good agreement between the experimental and analytical curvatures prior to yielding for most wall displacement levels. For the higher wall displacement levels, there is a better agreement between the analytical curvatures and the experimental curvatures in the east direction, than with the experimental curvatures in the west direction. A reason for these discrepancies is that the model was constructed from the construction joint upwards and it was fixed at the base, so it is stiffer than the actual wall. Despite these differences, the analytical model seems to predict reasonably well the curvature distribution of the wall. 4.2.8 Equivalent plastic hinge length The models presented in Chapter 2 will be used to compute the equivalent plastic hinge length for the test wall, in order to compare the results obtained and see how accurate these models are. The relevant wall parameters required for these calculations are shown in Table 4.2: 63 Table 4.2 UBC wall parameters Parameter Symbol Value Height of wall (m) Ls â€” Hw 11.33 Horizontal length of wall (m) h = lw 1.625 Bar diameter of tension reinforcement (mm) dâ€ž 10 Yield stress of tension reinforcement (MPa) fy 455 Concrete cylinder compressive strength (MPa) fa 49 Axial load (kN) P 1500 Gross area of cross-section (mm2) 309093 Baker and Amarakone's equation for unconfined concrete was used (Equation 2.12). Regarding Mattock's equation (Equation 2.26), the second term in parenthesis was neglected, since the wall has the same amount of reinforcement in both flanges (p = p'). Regarding the bounds proposed by the ACI-ASCE Committee 428 (Equation 2.30), since the plastic hinge length is being calculated at failure, the maximum moment is equal to the ultimate moment (Rm = 1). In the equation proposed by Sasani and Der Kiureghian (Equation 2.41), the mean values of the model parameters were used, and the model error term was not included (& = 0). Panagiotakos and Fardis' equation for cyclic loading was used (Equation 2.46), and the second term in the equation was neglected, as it was considered that no significant slippage occurred (ast = 0). Other required parameters in the calculations are presented in Table 4.3: Table 4.3 Parameters required in plastic hinge length models Parameter Symbol Value Comments Effective depth (m) d 1.3 Calculated as 0.8/w, according toCSA A23.3 Area of reinforcement (mm2) As 1400 14-10M bars in the wall Area of concrete (mm2) Ac 307693 Axial compressive strength without bending moment (kN) Pu 15714 Calculated without using strength reduction factors Concrete cube strength (MPa) fee 58.8 Calculated with equation 3.33 Tension reinforcement factor (Baker and Amarakone's equation) k, 0.7 Mild steel Axial load factor (Baker and Amarakone's equation) k2 1.048 Calculated with equation 2.13 Concrete strength factor (Baker and Amarakone's equation) k3 0.411 Calculated with equation 2.14 Concrete strain in extreme compression fiber at yield curvature See 0.0015 Average of recommended values Concrete strain in extreme compression fiber at ultimate curvature &CU 0.0035 According to CSA A23.3 Strain ratio (ACI-ASCE equation) Rs 1.25 Calculated with equation 2.31 The equivalent plastic hinge length can also be determined experimentally from the test results available. Although the authors determined experimental plastic hinge lengths, the one presented in this 64 section was recalculated from the measured data, and differs from the results reported by the authors. In Equation 2.6, setting L = Hw, A = Aâ€ž and <f>= fa, this equation becomes: Au=Ay+(fa-fa)Lp f H â€” 2 (4.4) Solving Equation 4.4 for Lp: LP-HW-f A â€ž - A v l Hi-2 u y (4.5) As it was mentioned in Section 4.2.4, the maximum wall displacement was 281rnrn, the wall yield displacement was 46mm, and the yield curvature was 0.002 rad/m. Because some targets were lost at the ultimate displacement level, the ultimate curvature could not be determined experimentally. Therefore, a plane sections analysis of the wall was performed to predict the ultimate curvature. The analysis was done considering an elasto-plastic stress-strain curve for steel, the concrete cylinder strength, a concrete ultimate strain of 0.0035, a compressive axial load of 1500 kN, the CSA A23.3 stress block for concrete in compression, and strength reduction factors of unity. The estimated curvature capacity was 0.0236 rad/m. Using all these values in Equation 4.5, the equivalent plastic hinge length was lm or 0.62/* The predictions obtained are summarized in Table 4.4: Table 4.4 Comparison of equivalent plastic hinge length models Researchers reference Equation Lp(m) Baker and Amarakone (1964) Eq. 2.12 0.67 0.41 Sawyer (1964) Eq. 2.22 1.17 0.72 Mattock (1964) Eq. 2.26 2.19 1.35 Corley (1966) Eq. 2.28 0.97 0.60 Mattock (1967) Eq. 2.29 1.22 0.75 ACI-ASCE lower bound (1968) Eq. 2.30 0.83 0.51 ACI-ASCE upper bound (1968) 2.23 1.37 Zahn, Park and Priestly (1986) Eq. 2.34 0.64 0.40 Priestly and Park (1987) Eq. 2.36 0.97 0.59 Paulay and Priestly (1992) Eq. 2.37 1.01 0.62 Paulay and Priestly (1993) Eq. 2.38 2.34 1.44 Sasani and Der Kiureghian (2001) Eq. 2.41 0.75 0.46 Panagiotakos and Fardis (2001) Eq. 2.46 1.36 0.84 CSA A23.3 0.81 0.50 Experimental 1.00 0.62 65 Figure 4.10 shows these results graphically: Figure 4.10 Comparison of equivalent plastic hinge length models Rese.ncliei The results obtained using the models for plastic hinge length presented in Table 4.4 vary from 0.4/w to 1.44/w. The plastic hinge length given by the code is also presented. Most of these equations estimate a plastic hinge greater than the one given by the code; therefore, the code is conservative compared to most of these. The plastic hinge length obtained experimentally is also larger than the one predicted by the code. Comparing the experimental and predicted results from Figure 4.10, the equations of Mattock (1964), ACI-ASCE upper bound (1968) and Paulay and Priestly (1993) significantly overestimate the plastic hinge length. The equations of Sawyer (1964), Mattock (1967) and Panagiotakos and Fardis (2001) also overestimate it, but by a smaller margin. The equations of Baker and Amarakone (1964), ACI-ASCE lower bound (1968), Zahn, Park and Priestly (1986) and Sasani and Der Kiureghian (2001) underestimate the plastic hinge length. The best predictions are given by the equations of Corley (1966), Priestly and Park (1987) and Paulay and Priestly (1992). 66 The experimental plastic hinge length is between the bounds proposed by the ACI-ASCE Committee 428. The lower bound gives a better prediction. The results provided by all these expressions are significantly different. This is because they have been derived from tests for different types of concrete members and consider different parameters. The last three equations in Table 4.4 (Equation 2.38, 2.41 and 2.46) where derived specifically for walls. These, however, are not providing good predictions compared to other equations. 4.2.9 Distribution of inelastic curvatures Figure 4.4 to 4.9 show the actual distribution of the inelastic curvatures. Since the elastic portion of the curvature was 0.002 rad/m, the inelastic curvatures can be visualized by shifting the vertical axis by this amount. The resulting curvatures suggest that the inelastic curvature over the plastic hinge length are not uniform, as it is commonly assumed; but have a linear variation over a distance equal to approximately the length of the wall (1625mm) measured from the construction joint, which is the critical section (Adebar, mrahim and Bryson 2004: 15 - 16). Both the experimental and analytical results are showing this trend. As seen in Chapter 2, in the classical formulation of plastic hinge analysis, the inelastic curvature is considered to be constant over the equivalent plastic hinge length. This equivalent length is commonly assumed to be 0.5h or 0.5/w, according to typical concrete codes and several researchers. However, the actual inelastic curvature has a certain variation. The test results and the analytical predictions from this study are suggesting that this variation may be considered linear over a length of I.Oh or 1.0/w; that is, twice the length normally considered. If we consider Lp to be the plastic hinge length for a linearly varying inelastic curvature, the resulting inelastic displacement at the top of a cantilever wall can be determined from the following formulation: Figure 4.11 Linearly varying inelastic curvatures 67 <|>P - t L P 4 The inelastic rotation can be determined by integrating the inelastic curvatures: 0â€ž <t>DL pLjp,lm (4.6) Considering that the inelastic rotation is concentrated at the centroid of the inelastic curvatures, the inelastic displacement can be expressed as: f r ^ Jp,lin (4.7) If we consider LPiCom, to be the plastic hinge length for a constant inelastic curvature, and setting Lp,iiâ€ž = 2Lpconsh Equation 4.7 can be expressed as: Hâ€ž â€” -2L â€ž ^ p,const (4.8) Compare this inelastic displacement with the one obtained for a uniform inelastic curvature: (4.9) 68 If the plastic hinge length is small compared to the height of the wall, both formulations give the same inelastic displacement. However, if this is not the case, the inelastic displacement is greater if a uniform inelastic curvature is considered. Although many researchers have considered a uniform inelastic curvature to develop their models for plastic hinge length, some have studied the actual curvature distribution in concrete elements. Paulay and Priestly plotted the actual curvature distribution of a prismatic reinforced concrete cantilever element. They showed that the actual extent of plasticity is approximately twice the equivalent plastic hinge length obtained considering a uniform inelastic curvature, and that the actual variation of the inelastic curvature can be reasonably considered as linear; as it was observed in this study (Paulay and Priestly 1992: 139). In the next chapter, a linearly varying inelastic curvature will be considered to perform a parametric study of concrete walls. 4.2.10 Analysis of strain profiles So far, the length of the plastic hinge has been obtained from the curvature distribution. The concept of curvature (strain gradient) is based on the hypothesis that plane sections remain plane after bending, which is a widely used engineering assumption. During the test, strain measurements were only made at the faces of the wall, and the curvatures were calculated assuming that the strains along the length of the wall had a linear variation. The analytical curvatures were also determined this way in order to compare results. Since this wall is very slender, it is expected than the actual variation of the strain profile will be close to linear. The average vertical strains in the elements along the length of the wall obtained from program VecTor2 were studied to verify if this assumption is true. First, we will consider the case where the wall is subjected to a constant compressive axial load only (before applying lateral displacements). It is expected that the strain profile in every cross-section will be constant. Figure 4.12 shows the strain profile at the construction joint: 69 Figure 4.12 Strain profile at construction joint for the case of no bending 0.00 -0.02 -0.04 -0.06 | -0.08 E J> -0.10 in -0.12 -0.14 -0.16 -0.18 200 400 600 800 1000 1200 1400 1600 Distance from left edge of wall (mm) Contrary to what it is expected, the strain profile is not constant. In order to investigate why, the strain profile at a certain distance from the construction joint is shown in Figure 4.13: Figure 4.13 Strain profile at 319mm from construction joint for the case of no bending 0.00 -0.02 -0.04 -0.06 | -0.08 E -0.10 -0.12 -0.14 -0.16 -0.18 200 400 600 800 1000 1200 1400 1600 Distance from left edge of wall (mm) 70 As we get further from the construction joint, the strain profile becomes almost constant. Therefore, the results are distorted near the construction joint due to the boundary conditions of the model. Figure 4.14 shows the strain profile for a wall displacement of 12.8mm at the construction joint, before cracking occurs: Figure 4.14 Strain profile at construction joint for a wall displacement of 12.8mm 0.1 -0.5 J â€” Distance from left edge of wall (mm) Prior to cracking, the strain profile is almost linear. There are some distortions near the base due to the boundary conditions, as seen previously. A least-squares fit was used to obtain a linear strain profile; the slope of this line is the curvature, in rad/m. Figure 4.15 shows the strain profile for this wall displacement level at 319mm from the construction joint: 71 Figure 4.15 Strain profile at 319mm from the construction joint for a wall displacement of 12.8mm 0.10 -0.40 -0.45 J Distance from left edge of wall (mm) At this distance from the construction joint, the strain profile has practically a perfect linear variation. Therefore, it can be concluded that the strain profiles remain linear before cracking. Figure 4.16 shows the strain profile for a wall displacement of 25.4mm at the construction joint, after cracking and prior to yielding: 72 Figure 4.16 Strain profile at construction joint for a wall displacement of 25.4mm 2.0 1.5 -1.0 J Distance from left edge of wall (mm) After cracking, the strain profile does not have a linear variation. Note that using a least-squares fit to determine the curvature gives a different result that if we use only the strains at the faces of the wall. Figure 4.17 shows the strain profile for this wall displacement level at 840mm from the construction joint: 73 Figure 4.17 Strain profile at 840mm from the construction joint for a wall displacement of 25.4mm 0.3 Distance from left edge of wail (mm) At this distance from the construction joint, the strain profile does not have a linear variation either. The distortions in this case are not due to the boundary conditions. Therefore, it can be concluded that after cracking, the strain profile does not remain linear, even though the wall is slender. Figure 4.18 shows the strain profile for a wall displacement of 48mm at 840mm from the construction joint, after yielding: 74 Figure 4.18 Strain profile at 840mm from the construction joint for a wall displacement of 48mm 2.5 -1.0 J Distance from left edge of wall (mm) As we get farther into the nonlinear range, the strain profile is less close to having a linear variation. Because these observations contradict the results that were expected, a new analysis was carried to study why the strain profiles do not have a linear variation after cracking. As it was previously described, the wall had a low amount of reinforcement. In order to investigate how the amount of reinforcement influences the shape of the strain profiles, a second finite element analysis of the wall specimen, with an increased amount of reinforcement, was performed using program VecTor2. A new model was constructed for this purpose. This model had the same characteristics as the previous one, described in Section 4.2.5, except for the amount of horizontal and vertical reinforcement. The same loads were also applied. The average vertical strains in the elements along the length of the wall obtained from this analysis were examined. Table 4.5 shows the material properties in the different regions of this new model of the wall: 75 Table 4.5 Material properties of UBC wall model with added reinforcement Concrete properties Material 1 Material 2 Material 3 Color Thickness (mm) 380 127 380 Cylinder compressive strength (MPa) 49 49 49 Reinforcement component properties ^-direction .y-direction .v-direction .redirection .v-direction .v-direction Reinforcement ratio (%) 1.5 2.0 0.5 0.5 1.5 2.0 Reinforcement diameter (mm) 10 10 10 10 10 10 Yield strength (MPa) 455 455 455 455 455 455 Ultimate strength (MPa) 650 650 650 650 650 650 Elastic modulus (MPa) 200000 200000 200000 200000 200000 200000 Strain hardening modulus (MPa) 4875 4875 4875 4875 4875 4875 Strain hardening strain (mm/m) 10 10 10 10 10 10 Figure 4.19 shows the strain profile for a wall displacement of 41.4mm at the construction joint for this new analysis, after cracking and prior to yielding: Figure 4.19 Strain profile at construction joint for a wall displacement of 41.4mm 1.5 -1.0 Distance from left edge of wall (mm) Comparing this figure with Figure 4.16, the variation of the strain profile is more close to linear. 76 Figure 4.20 shows the strain profile for this wall displacement level at 840mm from the construction joint: Figure 4.20 Strain profile at 840mm from the construction joint for a wall displacement of 41.4mm 1.2 -0.8 J Distance from left edge of wall (mm) At this distance from the construction joint, the strain profile is very close to linear. The distortions in Figure 4.19 are due to the boundary conditions. Therefore, it can be concluded that the amount of reinforcement in the wall has a very significant impact in the shape of the strain profile after cracking. With this amount of reinforcement, yielding occurs at a wall displacement of 70mm. Figure 4.21 shows the strain profile for a wall displacement of 105mm at 840mm from the construction joint, after yielding: 77 Figure 4.21 Strain profile at 840mm from the construction joint for a wall displacement of 105mm 2.5 -1.5 ' Distance from left edge of wall (mm) Comparing this figure with Figure 4.18, the variation of the strain profile is more close to linear after yielding at this distance from the construction joint. Figure 4.22 shows the strain profile for a wall displacement of 281mm at 840mm from the construction joint: 78 Figure 4.22 Strain profile at 840mm from the construction joint for a wall displacement of 281mm 20 -5 -1 Distance from left edge of wall (mm) As seen previously in Figure 4.18, as we get further into the nonlinear range, the strain profile is less close to having a linear variation. However, for this case, considering a linear variation is still a fair approximation. The four main conclusions made from this study can be summarized as the following: The boundary conditions of the model distort the shape of the strain profile. Before cracking, the strain profile remains linear. After cracking, the shape of the strain profile depends on the amount of reinforcement. As the amount of reinforcement is increased, the strain profile is closer to having a linear variation. The linearity of the strain profile degrades as the wall goes further into the nonlinear range. Therefore, considering that the strain profile is linear is not always true. It is important to take this into consideration when we design slender walls with low amounts of reinforcement. For the wall studied, there was no experimental information available regarding the axial strains along the cross-section, so a linear variation had to be assumed to determine the curvatures. Tests results showing the actual variation of axial strains along the cross-section of slender walls will be presented in the next section. 79 4.3 Rectangular shear wall tested at Clarkson University A series of tests were conducted at Clarkson University in 1995 on slender reinforced concrete walls with rectangular-shaped and T-shaped cross-sections with moderate amounts of transverse reinforcement by Thomsen and Wallace. The purpose of these tests was to evaluate the simplified displacement-based design approach in ACI 318 1999 (Thomsen and Wallace 2004: 618 - 630). A brief description of the tests and the results obtained has already been given in Section 2.24. These tests are of particular interest because extensive measurements of the concrete strains were made along the length of the wall at the base, so the actual variation of the axial strains was determined. Therefore, these test results can be compared with analytical predictions made with program VecTor2. The wall analyzed for these comparisons was the rectangular specimen RW2. The following sections will be focused on this specimen only; additional information needed for the analysis will be presented. 4.3.1 Description of the wall specimen The specimen RW2 was a 1/4 scaled model of a wall in an area of high seismicity. It had an aspect ratio of three and was considered as a four-storey wall. The wall was 102mm thick, 1219mm long and 3658mm high (Thomsen and Wallace 2004: 618 - 620). The reinforcement was arranged symmetrically. The boundary zones were 191mm long each. The vertical reinforcement in the two zones consisted of eight deformed #3 bars enclosed by 3/16" diameter smooth wire hoop and cross-ties spaced at 51mm. The vertical reinforcement in each zone was arranged in four layers (spaced at 51mm) of two bars each. The clear cover of the hoops was 9.5mm. The web had two deformed #2 bars in each layer spaced at 191mm, vertically and horizontally. A total of eight bars were used in the web. The wall had different concrete strengths at the base and at each of the four storeys at the time of testing. At the first storey, the compressive cylinder strength was 43.7 MPa and the rupture strength was 5.63 MPa. All three types of reinforcing steel used had different properties. The yield strength was 434, 448 and 434 MPa for the #3 bars, #2 bars and 3/16" smooth wires, respectively; and the ultimate strength was 641, 586 and 483 MPa, respectively. The modulus of elasticity was 200000 MPa for all three. The strain at the onset of strain hardening for the #3 bars was 16 mm/m (Thomsen and Wallace 1995: 148 -150). The stress-strain response of three types of reinforcing steel was measured (Thomsen and Wallace 1995: 182). 80 4.3.2 Instrumentation The wall was tested as a cantilever element. It was subjected to cyclic lateral displacements applied at the top by a hydraulic actuator. Additionally, a constant compressive axial load was applied at the top by hydraulic jacks (Thomsen and Wallace 2004: 622 - 623). Four wire potentiometers were used to measure lateral displacements at 914mm intervals along the height of the wall. Seven linear variable differential transducers were provided along the wall length to measure vertical displacements; these were placed vertically over a gage length of 229mm (Thomsen and Wallace 2004: 624). 4.3.3 Test procedure During the test, the constant axial load applied to the wall was on average 0.07Agfc, equal to 378 kN, calculated using the concrete cylinder strength of the first storey. However, the axial load had to be constantly adjusted, resulting in a considerable variation of it. The maximum axial load applied was 436 kN (Thomsen and Wallace 1995: 205). The lateral displacements applied to the wall consisted of 20 cycles. The drift for the 12 first cycles increased from 0.125 to 1.5%, and for the last eight cycles it increased from 1 to 2.5%. The drifts were increased every two cycles (Thomsen and Wallace 1995: 203). 4.3.4 Test results The test results were used to determine strain profiles along the length of the wall for different drifts levels. The axial strains were determined from the displacement readings measured with the seven transducers provided along the wall. These readings were divided by the gage length to obtain axial strains. Some of the readings at the surface were not reliable because of concrete spalling (Thomsen and Wallace 2004: 627). The experimental strain profiles were determined for the positive and negative direction of loading (Thomsen and Wallace 1995: 223). 4.3.5 Analytical model of the wall specimen The wall specimen was modeled and analyzed using program VecTor2. The purpose of this analysis is to predict the strain profile along the length of the wall and compare these predictions with the experimental results obtained by Thomsen and Wallace. Low-powered rectangular elements were used to 81 model the concrete, with smeared steel to account for the presence of reinforcement. The constitutive models for concrete and steel described in Chapter 3 were used in the analysis. Two different material types were used to represent various regions of the wall in the finite element model: The first material type was used to represent the confined boundaries of the wall. The second material type was used to represent the web of the wall. The analytical model was fixed at the bottom; both the horizontal and vertical displacements were restrained. The finite element mesh was refined in such a way that the position of certain nodes coincides with the position of the transducers. Meshes of 89x114, 102x114 and 114x114 rectangular elements were used; 12 elements (13 nodes) were used in the transverse direction of the wall. This level of refinement was used to model the whole wall. All nodes and elements were numbered in the horizontal (short) direction. The complete mesh consisted of429 nodes and 384 rectangular elements. The material properties used in the analysis were those reported in the description of the wall specimen, presented in Section 4.3.1. The concrete properties from the first storey were considered for the whole wall. The concrete tensile strength was taken as half of the rupture strength, that is, 2.8 MPa. For the material properties that were not measured during the test, the values given by default in program VecTor2 were used. The concrete initial tangent elastic modulus and the cylinder strain dXfc where determined with Equation 4.2 and 4.3, respectively. The maximum aggregate size was 10mm, given by default. For the reinforcement properties, the strain hardening modulus and the strain at the onset of strain hardening were determined from the measured stress-strain curves; except for the strain at the onset of strain hardening for the deformed #3 bar, which is given in Section 4.3.1. A monotonic lateral load was applied at the top of the wall. This load was applied in a displacement-control mode, in increments of 0.2mm. Additionally, a constant axial load of 436 kN was applied; this was the maximum axial load applied during the test. This load was equally distributed among all the 13 nodes at the top. The self-weight of the wall was not considered. Figure 4.23 shows the finite element model of the wall specimen, created in the pre-processor FormWorks: Table 4.6 shows the material properties in the different regions of the wall: 83 Table 4.6 Material properties of specimen RW2 model Concrete properties Material 1 Material 2 Color Thickness (mm) 101.6 101.6 Cylinder compressive strength (MPa) 43.7 43.7 Tensile strength (MPa) 2.8 2.8 Reinforcement component properties jc-direction ^-direction x-direction .y-direction Reinforcement ratio (%) 0.69 2.95 0.33 0.33 Reinforcement diameter (mm) 4.75 9.5 6.4 6.4 Yield strength (MPa) 434 434 448 448 Ultimate strength (MPa) 483 641 586 586 Elastic modulus (MPa) 200000 200000 200000 200000 Strain hardening modulus (MPa) 1119 5750 3391 3391 Strain hardening strain (mm/m) 2.2 16 2.3 2.3 4.3.6 Analytical results The analytical predictions obtained from program VecTor2 were used to determine strain profiles along the length of the wall for different drift levels. The axial strains were determined the same way as it was done for the experimental results in order to make comparisons, using the vertical nodal displacements along the wall length obtained from the analysis. As previously described, the position of certain nodes along the length of the wall coincides with the position of the transducers. The vertical nodal displacements at these particular nodes were used to calculate the axial strains. The difference in vertical displacements between two nodes in the same position along the wall length, divided by the vertical distance between these nodes, gives the average axial strain over that length at that position. This way, the experimental and analytical axial strains were calculated over the same length. The strain profile was determined for drift levels of 1.5 and 2%. 4.3.7 Comparison of experimental and analytical results Figure 4.24 and 4.25 show comparisons between the experimental strain profile, in both directions of loading, and the analytical strain profile; for the two drift levels mentioned: Figure 4.24 Strain profile at base for a drift of 1.5% 8 4 Both the experimental and the analytical results show that the strain profiles are not linear, although the walls are slender; which corifirms the results obtained for the UBC wall test. As it was 85 discussed previously, the nonlinearity of the strain profile is because this wall had a moderate amount of reinforcement. Although program VecTor2 predicts reasonably well the shape of the strain profiles along the length of the wall, there are differences between the experimental and analytical results, specially in the tension zone. There is a better agreement between the analytical and the experimental strain profiles in the positive direction of loading. The greater differences in the tension zone, as the authors indicated, are due to the influence of concrete cracking and slippage of the reinforcement (Thomsen and Wallace 2004: 628). The differences in the compression zone are due to the very significant damage and spalling of the concrete cover at the wall boundaries during the test (Thomsen and Wallace 2004: 626). The analytical predictions in the compression zone can be improved if the cover spalling at the wall boundary is considered in the finite element model. For this, only the concrete inside the core was modeled. The area of concrete in the wall boundary considered in the analysis was delimited by the longitudinal axis of the hoops. This is shown in Figure 4.26: Figure 4.26 Area of concrete considered in the analysis to account for cover spalling 167 mm 12 mm A second finite element analysis of the wall, considering cover spalling, was performed using program VecTor2. A new model was constructed for this purpose. This model had the same characteristics as the previous one, described in Section 4.3.5, except that a new material type was created for the confined boundary of the wall in the compression zone. This material type had a reduced thickness of 78mm to account for cover spalling at the sides, as shown in Figure 4.26. The nodes at the far right in the model were moved 12mm to the left to account for cover spalling at the face of the wall. The same loads were applied for this analysis. The vertical nodal displacements obtained were used to calculate the 86 axial strains along the length of the wall using the same procedure described in Section 4.3.6, for drift levels of 1.5 and 2%. Figure 4.27 shows the new finite element model of the wall specimen, created in the pre-processor FormWorks: Figure 4.27 Finite element model accounting for cover spalling of specimen RW2 in FormWorks Table 4.7 shows the material properties in the different regions of this new model of the wall: Table 4.7 Material properties of specimen RW2 model accounting for cover spalling Concrete properties Material 1 Material 2 Material 3 Color Thickness (mm) 101.6 101.6 77.75 Cylinder compressive strength (MPa) 43.7 43.7 43.7 Tensile strength (MPa) 2.8 2.8 2.8 Reinforcement component properties -v-direction -^direction jc-direction j-direction jc-direction j-direction Reinforcement ratio (%) 0.69 2.95 0.33 0.33 0.69 2.95 Reinforcement diameter (mm) 4.75 9.5 6.4 6.4 4.75 9.5 Yield strength (MPa) 434 434 448 448 434 434 Ultimate strength (MPa) 483 641 586 586 483 641 Elastic modulus (MPa) 200000 200000 200000 200000 200000 200000 Strain hardening modulus (MPa) 1119 5750 3391 3391 1119 5750 Strain hardening strain (mm/m) 2.2 16 2.3 2.3 2.2 16 Figure 4.28 and 4.29 show comparisons between the experimental strain profile, in both directions of loading, and the analytical strain profile; for the two drift levels mentioned: Figure 4.28 Strain profile accounting for cover spalling at base for a drift of 1.5% 30 25 Comparing these figures with Figure 4.24 and 4.25, there is an excellent agreement between the experimental and analytical strain profiles in the compression zone. Despite the discrepancies in the tension zone, the analytical model seems to predict reasonably well the strain profile of the wall. C H A P T E R 5: P A R A M E T R I C S T U D Y O F C O N C R E T E W A L L S 89 5.1 Scope of analysis Program VecTor2 was used to perform a parametric study of concrete walls. The factors that affect the plastic hinge length were investigated. The parameters considered in this study were the wall length, the distance from the section of maximum moment to the section of zero moment (shear span), the diagonal cracking and the axial load. 5.2 Description of wall models Two cantilever wall models were considered for this parametric study, which will be referred to as Wall 1 and Wall 2 throughout this chapter. Wall 1 was twice as long as Wall 2. The dimensions and material properties of these walls were typical of high-rise buildings. The two walls had a rectangular cross-section, and they were 54860mm high and 508mm thick. Wall 1 was 7620mm long. The boundary zones were 1219mm long, the vertical reinforcement in these consisted of 24-25M reinforcing bars enclosed by 15M ties spaced at 100mm. The clear cover of the ties was 40mm. The web had 15M reinforcing bars spaced at 150mm vertically and horizontally. The cross-section of Wall 1 is shown in Figure 5.1: Figure 5.1 Cross-section details of Wall 1 24-25M vertical with 15M ties @ 100mm 15M @ 150mm horizontal and vertical 24-25M vertical with 15M t i e s Â® 100mm 508 mm |5" 1219mm 1219 mm 7620 mm Wall 2 was 3810mm long. The boundary zones were 610mm long, the vertical reinforcement in these consisted of 12-25M reinforcing bars enclosed by 15M ties spaced at 100mm. The clear cover of the ties was 40mm. The web had 15M reinforcing bars spaced at 150mm vertically and horizontally. The cross-section of Wall 2 is shown in Figure 5.2: 90 Figure 5.2 Cross-section details of Wall 2 vertical with 15M @ 150mm horizontal 12-25M vertical with 15M ties @ 100mm a n d v e r t j c a l ! 5 M , j e s @ , 0 0 m m The concrete cylinder compressive strength of both walls was 40 MPa. The stress-strain curve of all the reinforcing bars was assumed to be the same as the one used for the UBC wall test, whose properties are shown in the bottom five rows of Table 4.1, except that the yield strength was taken as 400 MPa. Both walls were fixed at the base. They were subjected to a monotonically increasing lateral load at the top; as well as a constant compressive axial load of 0. lAgfc, equal to 15484 kN for Wall 1 and 7742 kN for Wall 2. Using these two wall models as the base cases, the parametric study was performed by changing the wall parameters and observing how this affects the length of the plastic hinge. 5.3 Analytical model of walls Wall 1 and 2 were modeled and analyzed using program VecTor2. Each wall was analyzed separately. Low-powered rectangular and triangular elements were used to model the concrete, with smeared steel to account for the presence of reinforcement. The constitutive models for concrete and steel described in Chapter 3 were used in the analysis. Two different material types were used to represent various regions of the two walls in the finite element model: The first material type was used to represent the confined boundaries of the wall. The second material type was used to represent the web of the wall. The analytical model was fixed at the bottom; both the horizontal and vertical displacements were restrained. The finite element mesh was more refined near the base (critical section) in both walls. Also, 91 the two walls had the same refinement in the vertical direction up to a half of the height, so that the curvatures could be calculated over the same average length. For Wall 1, meshes of 305x423 and 305x203 rectangular elements were used; 25 elements (26 nodes) were used in the transverse direction of the wall. This level of refinement was maintained up to half the height of the wall. Up from this point, 18 elements (19 nodes) were used in the transverse direction, then it was further reduced to 10 elements (11 nodes), and then to five elements (six nodes) up to the top of the wall. The transitions were made using triangular elements. All nodes and elements were numbered in the horizontal (short) direction. The complete mesh consisted of 2081 nodes, 1927 rectangular elements and 86 triangular elements. For Wall 2, meshes of 305x423, 305x203, 288x423 and 288x203 rectangular elements were used; 13 elements (14 nodes) were used in the transverse direction of the wall. This level of refinement was maintained up to half the height of the wall. Up from this point, eight elements (nine nodes) were used in the transverse direction, and then it was further reduced to five elements (six nodes) up to the top of the wall. The transitions were made using triangular elements. All nodes and elements were numbered in the horizontal (short) direction. The complete mesh consisted of 1276 nodes, 1139 rectangular elements and 30 triangular elements. The material properties used in the analysis were those reported in the description of the wall models, presented in Section 5.2. For the material properties not mentioned in that section, the values given by default in program VecTor2 were used. The concrete tensile strength, the initial tangent elastic modulus and the cylinder strain atfc where determined with Equation 4.1, 4.2 and 4.3, respectively. The maximum aggregate size was 10mm, given by default. For the reinforcement properties, these were the same as those for the UBC wall test, except for the yield strength, as it was previously mentioned. A monotonic lateral load was applied at the top of the walls. This load was applied in a displacement-control mode, in increments of 1mm. Additionally, a constant axial load of 15484 kN in Wall 1 and 7742 kN in Wall 2 was applied; these loads were equally distributed among the six nodes at the top of each wall. The self-weight of the walls was not considered. Figure 5.3 shows the finite element model of Wall 1, created in the pre-processor FormWorks: Figure 5.4 shows the finite element model of Wall 2, created in the pre-processor FormWorks: Table 5.1 shows the material properties i n the different regions o f the two walls: 94 Table 5.1 Material properties of Wall 1 and 2 Concrete properties Material 1 Material 2 Color Thickness (mm) 508 508 Cylinder compressive strength (MPa) 40 40 Reinforcement component properties jc-direction -^direction .v-direction -^direction Reinforcement ratio (%) 0.8 2.0 0.5 0.5 Reinforcement diameter (mm) 15 25 15 15 Yield strength (MPa) 400 400 400 400 Ultimate strength (MPa) 650 650 650 650 Elastic modulus (MPa) 200000 200000 200000 200000 Strain hardening modulus (MPa) 4875 4875 4875 4875 Strain hardening strain (mm/m) 10 10 10 10 5.4 Analytical results The analytical predictions obtained from program VecTor2 were used to determine curvature distributions along the height of the walls for different lateral displacement levels. The vertical nodal displacements at the faces of the walls were used to calculate the curvatures, as it was done for the UBC wall test, which is described in Section 4.2.6. The difference in vertical displacements between two nodes at the face, divided by the vertical distance between these nodes, gives the average axial strain over that length. Then, the difference between the compression and tension strain, divided by the wall length between the two faces, gives the average curvature over the length between nodes. The curvatures were calculated over average lengths of 847 or 1050mm; that is, between every two or three elements. The strain profile along the length of the two walls was investigated. For slender walls like the ones being studied, with the amount of reinforcement that they have, we expect the strain profile along the wall length to be close to linear, as seen in Section 4.2.10. The average vertical strains in the elements along the length of the walls obtained from program VecTor2 were studied to verify if this assumption is true. The ultimate drift for Wall 1 was 2%. Figure 5.5 shows the strain profile of Wall 1 for this drift at the base: Figure 5.5 Strain profile of Wall 1 at base for a drift of 2% 95 -10 Distance from left edge of wall (mm) The strain profile is not linear due to the boundary conditions, as seen in Section 4.2.10. Figure 5.6 shows the strain profile of Wall 1 for a 2% drift at 1270mm from the base: Figure 5.6 Strain profile of Wall 1 at 1270mm from the base for a drift of 2% 20 -10 Distance from left edge of wall (mm) 96 At this distance from the base, the strain profile very close to having a linear variation. Figure 5.7 shows the strain profile of Wall 2 for a 2% drift at the base: Distance from left edge of wall (mm) Because Wall 2 is more slender, the strain profile is linear even at the base. From these figures, it can be seen that considering that plane sections remain plane after bending is a reasonable assumption for these two walls. The curvatures were calculated from the strains at the wall faces considering a linear variation. Figure 5.8 shows the curvature distribution up to the mid-height of both walls for a drift of 2%: Figure 5.8 Curvatures up to the mid-height of Wall 1 and 2 for a drift of 2% 97 28 0 + â€” , râ€” , , , i 1 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Curvature (rad/m) As seen in Section 4.2.9, the inelastic curvatures can be well approximated as linearly varying. Wall 2 has larger elastic curvatures because it is more flexible. In order to determine the plastic hinge length from the curvature distribution, the yield curvature of the walls has to be estimated. Once the yield curvature is determined, the inelastic curvatures can be visualized by shifting the vertical axis by this amount, which defines the length of the plastic hinge for a linearly varying inelastic curvature. There are two ways to determine the yield curvature. One way is to calculate the curvature just above the base, using the procedure described in this section, for the yield displacement or drift (first yield of the longitudinal reinforcement). The yield drifts were determined by looking at the average vertical steel strains in the elements obtained from program VecTor2. For Wall 1, the steel started to yield at a drift of 0.61%, for which the curvature just above the base was 0.00045 rad/m. For Wall 2, the steel started to yield at a drift of 1.18%, for which the curvature just above the base was 0.0009 rad/m. A second way to estimate the yield curvature is from the moment-curvature relationships of the walls. To develop this relationship, the bending moment diagram along the height of the wall must be calculated at the same locations where the curvatures in Figure 5.8 were determined. For Wall 1 and 2, since they were subjected to a lateral load at the top, the bending moment diagram was linearly varying and could be easily determined. Figure 5.9 shows the bending moments along the height of both walls for adrift of 2%: Figure 5.9 Bending moments along the height of Wall 1 and 2 for a drift of 2% - W 1 --W2 20000 40000 60000 80000 Bending moment (kN.m) 100000 120000 Figure 5.10 shows the moment-curvature relationship of Wall 1: Figure 5.10 Moment-curvature relationship of Wall 1 120000 100000 f 80000 c i I 60000 O) c = c m 40000 20000 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Curvature (rad/m) 99 Figure 5.11 shows the moment-curvature relationship of Wall 2: Figure 5.11 Moment-curvature relationship of Wall 2 30000 25000 20000 o Â£ c â€¢D C 0) CO 15000 10000 5000 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Curvature (rad/m) From Figure 5.10 and 5.11, it can be seen that the yield curvature of 0.00045 rad/m for Wall 1 and 0.0009 rad/m for Wall 2 determined previously are approximately the same values estimated from the moment-curvature relationships. Both procedures are giving the same result, so these values were used to estimate the plastic hinge length. Similarly, the yield moment can be determined using these two procedures. The bending moment just above the base was calculated using the procedure described in this section for the yield drift. For Wall 1, this moment was 90680 kN.m; and for Wall 2, this moment was 22740 kN.m. These were approximately the same values estimated from the moment-curvature relationships shown in Figure 5.10 and 5.11. Therefore, these values were selected as the yield moments. 5.5 Revision of analytical results The moment-curvature relationships of Wall 1 and 2 were used to check the results obtained from program VecTor2, by comparing these with the moment-curvature relationships obtained from a sectional analysis. Program Response-2000 was used for this purpose. Response-2000 is a computer program developed to perform two-dimensional sectional analysis of concrete members. This program was also 100 developed at the University of Toronto, by Evan Bentz. More information about the program Response-2000 can be found in the Response-2000, Shell-2000, Triax-2000, Membrane-2000 User Manual (Bentz 2001). Response-2000 was used to perform a sectional analysis of Wall 1 and 2 and develop their moment-curvature relationships for pure bending. To compare the results obtained from VecTor2 and Response-2000, the material models used in both programs have to be the same. Response-2000 uses the Modified Compression Field Theory, and has specific constitutive models incorporated in it to account for effects like compression softening and tension stiffening. These models are different from the ones used in the VecTor2 analysis. Using different constitutive models can significantly change the response. Therefore, a new analysis for both walls was performed in VecTor2, using the same constitutive models as in Response-2000, in order to compare results. The new models used in this analysis are the following, the details of these models can be found in the VecTor2 and FormWorks User's Manual (Wong and Vecchio 2002: 45 - 88): Compression pre-peak response: Popovics for high-strength concrete. Compression post-peak response: Popovics for high-strength concrete. Compression softening: Vecchio-Collins 1986. Tension stiffening: Bentz 1999. Cracking criterion: Unixial cracking stress. Crack slip check: Vecchio-Collins 1986 (this check was not done before because it is not required when using the Disturbed Stress Field Model). Slip distortions: Not considered (since the Modified Compression Field Theory is being used). The moment-curvature relationship from the results of VecTor2 was determined using the same procedure described in Section 5.4. Figure 5.12 shows the moment-curvature relationship of Wall 1: Figure 5.12 Moment-curvature relationship ofWall 1 using Response-2000 constitutive models 120000 100000 â€¢g 80000 2 o 60000 E a> c â€¢o c m 40000 20000 â€¢ W1 VecTor2 â€” W1 Response 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 Curvature (rad/m) Figure 5.13 shows the moment-curvature relationship of Wall 2: Figure 5.13 Moment-curvature relationship of Wall 2 using Response-2000 constitutive models 30000 25000 W2 VecTor2 -W2 Response 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 Curvature (rad/m) 102 There is a good agreement between the results provided by VecTor2 and Response-2000. This analysis was only performed to check the VecTor2 predictions. The parametric study in this chapter was done using the constitutive models described in Chapter 3, since these models provided the best agreement with the experimental results from the tests described in Chapter 4. 5.6 Influence of wall length The first parameter studied was the wall length. As it was discussed in Chapter 2, many models developed for plastic hinge length consider that it is proportional to the wall length only, so it is an important parameter to be considered. The results obtained for Wall 1 and 2 will be used to analyze how the plastic hinge length is influenced by the wall length. As seen in Section 5.4, the yield curvature and yield moment for Wall 1 were 0.00045 rad/m and 90680 kN.m, respectively; and for Wall 2 they were 0.0009 rad/m and 22740 kN.m. The yield curvature was used to measure the plastic hinge length for a drift of 2% from the curvature distribution in Figure 5.8. For Wall 1, the plastic hinge length was 8.39m or 1.10/w; and for Wall 2, it was 6.21m or 1.63/â€ž,. Therefore, predicting the plastic hinge length as a function of the wall length only does not work for these walls. A different approach to predict the plastic hinge length is by using moments; that is, the yield moment determined from the moment-curvature relationship defines the length of the plastic hinge. For a linear bending moment diagram like the one shown in Figure 2.6, the length where the yield moment is exceeded can be determined from Equation 2.11, repeated here for convenience: Lp My - ^ = 1 yâ€” (5.1) Ls M m a x From Figure 5.9, the maximum moment in Wall 1 was 105633 kN.m, and for Wall 2 it was 25406 kN.m. Using Equation 5.1, the plastic hinge length for Wall 1 was 7.77m or 1.02/w; and for Wall 2, it was 5.76m or 1.51/w. This flexural prediction matches well with the measured plastic hinge lengths. The spread of yielding was larger in Wall 1. To explain why a longer wall has a larger plastic hinge length, the tensile steel strains in the wall were observed. The average vertical steel strains in the elements along the height of the walls obtained from program VecTor2 were studied for this. The steel strains were observed in the exterior and interior steel layer in the boundary zones, shown in Figure 5.14: Figure 5.14 Exterior and interior steel layers in zones in Wall 1 and 2 exterior steel layer in zone Interior steel layer in zone exterior steel layer In zone 103 interior steel layer In zone yt \ t 1 5 C > < 5 C 1 I I Wall 1 Wall 2 Figure 5.15 shows the steel strains up to the mid-height of both walls for a drift of 2%: Figure 5.15 Steel strains up to the mid-height of Wall 1 and 2 for a drift of 2% â€” W1 ext â€” â€” -W1 int â€¢ W2ext â€” W2int 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 Strain The steel strains exceed the yield strain (0.002 for these walls) at a larger distance from the base in Wall 1, so the spread of yielding was larger in this wall. The reason why the steel strains are larger in Wall 1 can be explained from the curvature distribution in Figure 5.8. At this drift, the curvatures at the base are approximately the same. For the same curvature, the steel strains will be larger in the longer wall; and therefore, it will have a larger spread of yielding. The reason why a longer wall has a larger plastic hinge length can also be explained by comparing the moment-curvature relationships of both walls. The plastic hinge length increases as the 104 difference between the yield and ultimate moment increases, which is also reflected in Equation 5.1. By plotting the moment-curvature relationships together, it can be observed that the slope of the post-yielding phase is bigger in Wall 1; therefore, it has a larger plastic hinge length. This is shown in Figure 5.16: Figure 5.16 Moment-curvature relationships of Wall 1 and 2 120000 i , 0-) , râ€” , , , , -I 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Curvature (rad/m) It can be concluded that longer walls have larger plastic hinge lengths because there is a bigger difference between the yield and ultimate moment for these. However, predicting the plastic hinge length as a function of the wall length only does not provide good results. Better predictions can be made by using a flexural prediction, like the one shown in Equation 5.1. The measured (from curvature distribution) and predicted (using Equation 5.1) plastic hinge lengths were deterrnined for different drift levels, using the same procedure described in Section 5.4 and this section. The results are shown in Table 5.2 and 5.3: 105 Table 5.2 Predicted and measured plastic hinge lengths for Wall 1 Drift (%) Mmax (kN.m) A/y(kN.m) Z,(m) Lp(m) Wl Predicted Wl Measured 2.0 105632.9 90680.0 54.860 7.77 8.39 1.8 104908.8 90680.0 54.860 7.44 8.08 1.6 104316.3 90680.0 54.860 7.17 7.40 1.4 103345.3 90680.0 54.860 6.72 7.18 1.2 102281.0 90680.0 54.860 6.22 6.53 Table 5.3 Predicted and measured plastic hinge lengths for Wall 2 Drift (%) Mmax (kN.m) My(kN.m) Ls(m) I 0(m) W2 Predicted W2 Measured 2.0 25405.7 22740.0 54.860 5.76 6.21 1.8 25136.9 22740.0 54.860 5.23 5.54 1.6 24769.3 â€¢ 22740.0 54.860 4.49 4.56 1.4 24050.6 22740.0 54.860 2.99 3.12 1.2 23079.6 22740.0 54.860 0.81 1.02 The flexural prediction provides good results for all drift levels. 5.7 Influence of shear span The second parameter studied was the shear span. This parameter has also been included in many models for plastic hinge length presented in Chapter 2, so it has a sigmficant importance. So far, the analysis for Wall 1 and 2 had been performed for a shear span equal to the wall height. To study the influence of the shear span, three new analyses for each wall were performed in VecTor2 for the following shear spans: - 35659mm (2/3 of wall height). - 27430mm (1/2 of wall height). - 19201mm (1/3 of wall height). The models used for these analyses had the same characteristics as the previous ones, described in Section 5.3, except that the lateral load was now applied at these heights measured from the base. The results from the analysis were used to determine the curvatures and bending moments for different drift levels, using the same procedure described in Section 5.4 and 5.6. Table 5.4 to 5.9 show the measured (from curvature distribution) and predicted (using Equation 5.1) plastic hinge lengths for different drift levels. These drifts were measured at the location of the lateral load (A/L,): Table 5.4 Predicted and measured plastic hinge lengths for Wall 1 for a shear span of 35659mm Drift (%) Mm* (kN.m) My (kN.m) Ls(m) Zâ€ž(m) Wl Predicted Wl Measured 1.6 106852.2 90680.0 35.659 5.40 6.06 1.4 106338.7 90680.0 35.659 5.25 5.72 1.2 105650.5 90680.0 35.659 5.05 5.74 1.0 103803.3 90680.0 35.659 4.51 5.01 Table 5.5 Predicted and measured plastic hinge lengths for Wall 2 for a shear span of 35659mm Drift (%) Mmax (kN.m) My (kN.m) L,(m) ID(m) W2 Predicted W2 Measured 1.6 25941.9 22740.0 35.659 4.40 4.74 1.4 25774.3 22740.0 35.659 4.20 4.29 1.2 25275.1 22740.0 35.659 3.58 3.61 1.0 24494.2 22740.0 35.659 2.55 2.61 Table 5.6 Predicted and measured plastic hinge lengths for Wall 1 for a shear span of 27430mm Drift (%) Mmax (kN.m) My (kN.m) Ls(m) Lâ€ž (m) Wl Predicted Wl Measured 1.6 108123.6 90680.0 27.430 4.43 5.55 1.4 107983.7 90680.0 27.430 4.40 5.37 1.2 106905.7 90680.0 27.430 4.16 5.31 1.0 105764.6 90680.0 27.430 3.91 4.61 0.8 104069.4 90680.0 27.430 3.53 4.00 Table 5.7 Predicted and measured plastic hinge lengths for Wall 2 for a shear span of 27430mm Drift (%) Mmax (kN.m) My (kN.m) Ls(m) Lâ€ž(m) W2 Predicted W2 Measured 1.6 26585.2 22740.0 27.430 3.97 4.28 1.4 26288.9 22740.0 27.430 3.70 3.98 1.2 26025.6 22740.0 27.430 3.46 3.50 1.0 25603.2 22740.0 27.430 3.07 3.06 0.8 24610.2 22740.0 27.430 2.08 2.16 Table 5.8 Predicted and measured plastic hinge lengths for Wall 1 for a shear span of 19201mm Drift (%) Mmax (kN.m) My (kN.m) Ls(m) ID(m) Wl Predicted Wl Measured 1.2 108372.4 90680.0 19.201 3.13 4.45 1.0 107362.4 90680.0 19.201 2.98 4.31 0.8 106567.5 90680.0 19.201 2.86 3.40 0.6 104616.6 90680.0 19.201 2.56 3.01 107 Table 5.9 Predicted and measured plastic hinge lengths for Wall 2 for a shear span of 19201mm Drift (%) Mmax (kN.m) My (kN.m) Ls(m) Lâ€ž(m) W 2 Predicted W 2 Measured 1.2 26766.2 22740.0 19.201 2.89 3.16 1.0 26282.3 22740.0 19.201 2.59 2.80 0.8 25911.7 22740.0 19.201 2.35 2.47 0.6 24840.3 22740.0 19.201 1.62 1.63 It can be observed that as the shear span gets smaller, the measured plastic hinge length becomes larger than the predicted one; specially for Wall 1 at high drifts. The reason for this is that the shear stress becomes larger as the shear span decreases, because the lateral force required to produce the same maximum moment at the base becomes larger. This produces more diagonal cracking, which increases the tensile forces in the longitudinal reinforcement, and so, the length of the plastic hinge. Therefore, a pure flexural prediction does not work when shear stresses are high. However, for low shear stresses, the prediction is good, independent of the shear span. The influence of the shear span is strongly related to the influence of diagonal cracking, which will be studied in the next section. 5.8 Influence of diagonal cracking The third parameter studied was the diagonal cracking. As it was discussed in Chapter 2, the reason why many of the models developed for plastic hinge length consider that it is proportional to the wall length is because the longer it is, more significant is the influence of diagonal cracking. Therefore, diagonal cracking was related to the wall length. In this study, diagonal cracking was related to the magnitude of the shear stress. The results of all the previous analyses made for Wall 1 and 2 were used to study the influence of diagonal cracking. Additionally, two new analyses for each wall were performed in VecTor2, this time reducing the thickness of the web, so that the shear stresses become higher (that is, they were now flanged walls). These new analyses were performed for the following shear spans: - 27430mm (1/2 of wall height). - 19201mm (1/3 of wall height). The models used for these analyses had the same characteristics as the previous ones, described in Section 5.3, with the lateral load being applied at these heights measured from the base. The only difference was that the web thickness was reduced from 508mm to 254mm. The results from the analysis were used to determine the curvatures and bending moments for different drift levels, using the same procedure described in Section 5.4 and 5.6. 108 Table 5.10 shows the material properties in the different regions of these new models of the two walls: Table 5.10 Material properties of Wall 1 and 2 with a web thickness of 254mm Concrete properties Material 1 Material 2 Color Thickness (mm) 508 254 Cylinder compressive strength (MPa) 40 40 Reinforcement component properties x-direction j>-direction x-direction -^direction Reinforcement ratio (%) 0.8 2.0 0.5 0.5 Reinforcement diameter 15 25 15 15 (mm) Yield strength (MPa) 400 400 400 400 Ultimate strength (MPa) 650 650 650 650 Elastic modulus (MPa) 200000 200000 200000 200000 Strain hardening modulus (MPa) 4875 4875 4875 4875 Strain hardening strain (mm/m) 10 10 10 10 Since the walls in these new analyses had a different web thickness, the moment-curvature relationship was also different. The yield curvature and yield moment were determined using the same procedures described in Section 5.4. Figure 5.17 shows the moment-curvature relationship for Wall 1 with a web thickness of 254mm: 109 Figure 5.17 Moment-curvature relationship of Wall 1 with a web thickness of 254mm 100000 90000 80000 70000 f 1. 60000 ** C i o 50000 E a i f 40000 c 4) CO 30000 20000 10000 0.0000 0.0005 0.0010 0.0015 Curvature (rad/m) 0.0020 0.0025 Figure 5.18 shows the moment-curvature relationship for Wall 2 with a web thickness of 254mm: Figure 5.18 Moment-curvature relationship of Wall 2 with a web thickness of 254mm 30000 25000 â€¢g" 20000 o 15000 E o i c -a c S 10000 5000 0 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 Curvature (rad/m) 110 The yield curvature of both walls is approximately the same, and yield moment has decreased. The yield curvature and yield moment for Wall 1 were now 0.00045 rad/m and 84700 kN.m, respectively; and for Wall 2 they were now 0.0009 rad/m and 21420 kN.m. Table 5.11 to 5.14 show the measured (from curvature distribution) and predicted (using Equation 5.1) plastic hinge lengths for different drift levels. Again, these drifts were measured at the location of the lateral load (A/Ls): Table 5.11 Predicted and measured plastic hinge lengths for Wall 1 for a web thickness of 254mm and a Drift (%) M*Â»r (kN.m) My (kN.m) Ls(m) Ln(m) Wl Predicted Wl Measured 1.8 100292.3 84700.0 27.430 4.26 6.26 1.6 99837.0 84700.0 27.430 4.16 5.83 1.4 97971.7 84700.0 27.430 3.72 5.06 1.2 98202.1 84700.0 27.430 3.77 4.72 1.0 97801.7 84700.0 27.430 3.67 4.35 0.8 96663.3 84700.0 27.430 3.39 3.89 Table 5.12 Predicted and measured plastic hinge lengths for Wall 2 for a web thickness of 254mm and a Drift (%) Mmax (kN.m) My(kN.m) Ls(m) Lâ€ž(m) W2 Predicted W2 Measured 1.8 25112.2 21420.0 27.430 4.03 4.42 1.6 24815.9 21420.0 27.430 3.75 4.13 1.4 24431.9 21420.0 27.430 3.38 3.72 1.2 24149.4 21420.0 27.430 3.10 3.30 1.0 23716.0 21420.0 27.430 2.66 2.79 0.8 23161.9 21420.0 27.430 2.06 2.06 Table 5.13 Predicted and measured plastic hinge lengths for Wall 1 for a web thickness of 254mm and a Drift (%) Afmar (kN.m) M y(kN.m) Ls(m) ID(m) Wl Predicted Wl Measured 1.8 101277.6 84700.0 19.201 3.14 5.86 1.6 101004.9 84700.0 19.201 3.10 5.57 1.4 100235.0 84700.0 19.201 2.98 4.98 1.2 99442.0 84700.0 19.201 2.85 4.63 1.0 99117.5 84700.0 19.201 2.79 4.00 0.8 97243.5 84700.0 19.201 2.48 3.55 0.6 96679.0 84700.0 19.201 2.38 3.34 I l l Table 5.14 Predicted and measured plastic hinge lengths for Wall 2 for a web thickness of 254mm and a Drift (%) Mmax (kN.m) My(kN.m) Ls(m) Lp(m) W2 Predicted W2 Measured 1.8 25796.5 21420.0 19.201 3.26 3.68 1.6 25500.8 21420.0 19.201 3.07 3.47 1.4 25376.0 21420.0 19.201 2.99 3.49 1.2 24945.9 21420.0 19.201 2.71 3.16 1.0 24441.0 21420.0 19.201 2.37 2.81 0.8 24108.8 21420.0 19.201 2.14 2.37 0.6 23296.6 21420.0 19.201 1.55 1.63 It is clear from these results that as the shear span decreases and the shear stresses become higher, the plastic hinge length is much larger than the one determined from the pure flexural prediction. To make a better prediction, we have to account for the diagonal cracking. A measure of the diagonal cracking in the walls is the average shear stress, determined by: v = -V 0.Slwbw (5.2) v: Shear stress. V: Shear force. bw\ Web thickness of wall. Although the lateral force acting on the walls keeps increasing until they reach their capacity, its increase after yielding is not very large. Therefore, the average shear stress in the walls after yielding remained approximately constant. Table 5.15 shows the average shear stress after yielding in Wall 1 and 2 calculated with Equation 5.2 for the six analyses presented so far in this chapter. The yield drifts (first yield of the longitudinal reinforcement) are also shown; these were determined by looking at the average vertical steel strains in the elements obtained from program VecTor2: Table 5.15 Yield drifts and shear stresses of Wall 1 and 2 Ls (mm) bw (mm) v(MPa) A A (%) Wall 1 Wall 2 Wall 1 Wall 2 54860.0 508.0 0.60 0.30 0.61 1.18 35659.0 508.0 0.95 0.45 0.43 0.79 27430.0 508.0 1.25 0.60 0.35 0.63 19201.0 508.0 1.80 0.85 0.28 0.47 27430.0 254.0 2.30 1.15 0.37 0.65 19201.0 254.0 3.35 1.65 0.30 0.49 112 The results obtained so far show that the plastic hinge length increases as the lateral displacement or drift increases. This trend is plotted in Figure 5.19, which shows the measured plastic hinge lengths for Wall 1 and 2 as a function of the total drift for the six analyses presented so far in this chapter. The continuous lines show the results for Wall 1, and the discontinuous lines show the results for Wall 2: Figure 5.19 Measured plastic hinge lengths vs. total drift for Wall 1 and 2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Total drift (%) Figure 5.20 shows the measured plastic hinge lengths for Wall 1 and 2 as a function of the plastic drift. The plastic drift was calculated as the difference between the total drift and the yield drift: 113 The quality of the predictions was evaluated by plotting the ratio of the measured to predicted plastic hinge length for different drift levels and shear stresses. This is shown in Figure 5.21 and 5.22: Figure 5.21 Ratio of measured to predicted plastic hinge lengths vs. drift for Wall 1 for different shear stresses without accounting for shear â€”â€¢â€”v = 0.60 MPa â€¢-â€¢â€”v = 0.95 MPa v= 1.25 MPa -*â€”v= 1.80 MPa - * â€” v = 2.30 MPa --â€¢ v = 3.35 MPa 1.0 1.2 1.4 1.6 Drift (%) 1.8 2.0 2.2 114 Figure 5.22 Ratio of measured to predicted plastic hinge lengths vs. drift for Wall 2 for different shear stresses without accounting for shear 2.0 1.5 0.5 0.0 -â€¢â€”v = 0.30 MPa -Â«--v = 0.45 MPa - v = 0.60 MPa â€”*â€”v = 0.85 MPa -*--v= 1.15 MPa ........ v= 1.65 MPa 0.4 0.6 0.8 1.0 1.2 1.4 Drift (%) 1.6 1.8 2.0 2.2 The predictions for Wall 2 are good for all cases because the shear stresses are not very high, since this wall is more slender than Wall 1. For Wall 1, it can be seen that the measured plastic hinge lengths are very large for high shear stresses. As seen in Section 2.1, diagonal cracking changes the curvature distribution along the height (see Figure 2.5). This has an influence in the moment-curvature relationship. Figure 5.23 shows the moment-curvature relationship for Wall 1 for low and high shear stresses. These relations were developed for the same web thickness but different shear spans (see Table 5.15): 115 Figure 5.23 Moment-curvature relationships of Wall 1 for different shear stresses 120000 v = 0.60 MPa v=1.80 MPa 20000 4-0-+ , . , , . 1 0.0000 0.0010 0.0020 0.0030 0.0040 Curvature (rad/m) This difference in the moment-curvature relationship is the reason why the pure flexural prediction, which is based on the moment-curvature relationship developed for low shear stresses (pure bending), is not providing good results for Wall 1 for high shear stresses. In order to improve the predictions for Wall 1, a very simple shear model was developed. As it was mentioned previously, high shear stresses produce more diagonal cracking, which increases the tensile forces in the longitudinal reinforcement. Therefore, we must take into account these additional tensile forces. A simple way to determine the additional tension due to shear (shift) is using the following expression: ^=^cotee+vccotee (5.3) Nv: Axial compression due to shear. Vs: Shear force in the longitudinal reinforcement. 6C: Angle of the crack. Vc: Shear force in the concrete. 116 It was consider that the total shear force was carried by the reinforcement (V = Vs and Vc = 0), because it gives a smaller shift (which is more conservative, since this gives a smaller plastic hinge length). Also, it was assumed that 6C = 45Â°. Therefore, the additional tension due to shear is: N V â€” = - (5-4) 2 2 The additional moment due to shear is then given by: Mv=^dv (5.5) - M v : Moment due to shear. a\: Lever arm of the tensile force in the longitudinal reinforcement. For these walls, it was considered that: rfv=0.S7w (5.6) So, for high shear stresses, the bending moment diagram is shifted by an amount equal to Mv. The total bending moment at any section is given by: M=Mf+Mv (5.7) Where Mf is the moment due to flexure. This simple model was used to predict the plastic hinge length in Wall 1 for high shear stresses. In typical concrete codes, the following expression is proposed for the cracking shear stress: . v ( T=0 .17VZ (5.8) Where v c r and fc are in MPa units. For a concrete cylinder strength of 40 MPa, used for Wall 1 and 2, the cracking shear stress is 1.08 MPa. However, Figure 5.21 for Wall 1 shows that for shear stresses up to 1.25 MPa, the pure flexural prediction provides good results. Also, Figure 5.22 for Wall 2 shows that the pure flexural prediction provides good results in all cases, for which the highest shear stress is 1.65 MPa. Therefore, the shear model should be used for shear stresses that are higher than these. 117 Figure 5.21 suggests that the influence of diagonal cracking in the plastic hinge length is significant for shear stresses higher or equal than 1.8 MPa; this value is equal to 0.29 times the square root of/V Based on this, the cracking shear stress should be: v â€ž = 0 . 3 V / c The following expression was proposed to account for shear stresses: (5.9) Mâ€ž = 0 ; v < vâ€ž v , ; v>v c 1 2 * (5.10) Where vâ€ž. is calculated with Equation 5.9. The plastic hinge length in Wall 1 was predicted for different drift levels and for shear stresses of 1.8, 2.3 and 3.35 MPa, using this model. For the linear bending moment diagram of Wall 1, since it is now shifted by M v , the yield moment is located at a larger distance from the base, so the plastic hinge length is longer. The moment due to shear was first calculated with Equation 5.5, and then added to the maximum bending moment at the base. Then, this moment was applied in Equation 5.1 to determine the plastic hinge length. The results are shown in Table 5.16 to 5.18: Table 5.16 Predicted and measured plastic hinge lengths for Wall 1 for a shear span of 19201mm using shear model Drift (%) F(kN) lw (mm) bw (mm) v(MPa) dv (mm) Mv (kN.m) 1.2 5644.1 7620.0 508.0 1.82 6096.0 17203.2 1.0 5591.5 7620.0 508.0 1.81 6096.0 17042.9 0.8 5550.1 7620.0 508.0 1.79 6096.0 16916.7 0.6 5448.5 7620.0 508.0 1.76 6096.0 16607.0 Drift (%) M/(kN.m) Mmax (kN.m) My (kN.m) Ls(m) Z,â€ž(m) Wl Predicted Wl Measured 1.2 108372.4 125575.6 90680.0 19.201 5.34 4.45 1.0 107362.4 124405.3 90680.0 19.201 5.21 4.31 0.8 106567.5 123484.2 90680.0 19.201 5.10 3.40 0.6 104616.6 121223.7 90680.0 19.201 4.84 3.01 Table 5.17 Predicted and measured plastic hinge lengths for Wall 1 for a web thickness of 254mm and a shear span of 27430mm using shear model Drift (%) K(kN) lw (mm) bw (mm) v(MPa) dv (mm) Mv (kN.m) 1.8 3656.3 7620.0 254.0 2.36 6096.0 11144.4 1.6 3639.7 7620.0 254.0 2.35 6096.0 11093.8 1.4 3571.7 7620.0 254.0 2.31 6096.0 10886.5 1.2 3580.1 7620.0 254.0 2.31 6096.0 10912.1 1.0 3565.5 7620.0 254.0 2.30 6096.0 10867.6 0.8 3524.0 7620.0 254.0 2.28 6096.0 10741.2 Drift (%) M/CkN.m) Mmax (kN.m) My(kN.m) Ls(m) Lv [m) Wl Predicted Wl Measured 1.8 100292.3 111436.7 84700.0 27.430 6.58 6.26 1.6 99837.0 110930.8 84700.0 27.430 6.49 5.83 1.4 97971.7 108858.3 84700.0 27.430 6.09 5.06 1.2 98202.1 109114.3 84700.0 27.430 6.14 4.72 1.0 97801.7 108669.3 84700.0 27.430 6.05 4.35 0.8 96663.3 107404.5 84700.0 27.430 5.80 3.89 Table 5.18 Predicted and measured plastic hinge lengths for Wall 1 for a web thickness of 254mm and a shear span of 19201mm using shear model Drift (%) V(kN) lw (mm) bw (mm) v(MPa) dv (mm) Mv (kN.m) 1.8 5274.6 7620.0 254.0 3.41 6096.0 16077.0 1.6 5260.4 7620.0 254.0 3.40 6096.0 16033.7 1.4 5220.3 7620.0 254.0 3.37 6096.0 15911.5 1.2 5179.0 7620.0 254.0 3.34 6096.0 15785.6 1.0 5162.1 7620.0 254.0 3.33 6096.0 15734.1 0.8 5064.5 7620.0 254.0 3.27 6096.0 15436.6 0.6 5035.1 7620.0 254.0 3.25 6096.0 15347.0 Drift (%) M/(kN.m) M ^ k N . m ) My(kN.m) M m ) M m ) Wl Predicted Wl Measured 1.8 101277.6 117354.6 84700.0 19.201 5.34 5.86 1.6 101004.9 117038.6 84700.0 19.201 5.31 5.57 1.4 100235.0 116146.5 84700.0 19.201 5.20 4.98 1.2 99442.0 115227.6 84700.0 19.201 5.09 4.63 1.0 99117.5 114851.6 84700.0 19.201 5.04 4.00 0.8 97243.5 112680.1 84700.0 19.201 4.77 3.55 0.6 96679.0 112025.9 84700.0 19.201 4.68 3.34 The quality of these predictions was evaluated by plotting the ratio of the measured to predicted plastic hinge length. This is shown in Figure 5.24: 119 Figure 5.24 Ratio of measured to predicted plastic hinge lengths vs. drift for Wall 1 for different shear stresses using the shear model 2.0 1.5 =! 1.0 0.5 0.0 v = 0.60 MPa - -Â» - -v = 0.95 MPa v= 1.25 MPa - * - - v = 1 . 8 0 MPa - * â€” v = 2.30 MPa â€¢ â€¢ â€¢ â€¢ v = 3.35 MPa 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Drift (%) Although the predictions are now very large for low drifts (which is conservative), they are good for high drifts, which are of more interest. Considering the simplicity of the model used, the results are satisfactory. The reason why this shear model does not work for low drifts is because at this point, there are not many diagonal cracks formed in the wall, so their influence is still not significant. It can be concluded that the diagonal cracking in walls has a significant influence in the plastic hinge length. For high shear stresses, the actual plastic hinge length is larger than the one predicted using a pure flexural prediction. The effect of shear needs to be included to estimate the length of the plastic hinge. 5.9 Influence of axial load The fourth parameter studied was the axial load. As seen in Chapter 2, the axial load was also considered in some of the models developed for plastic hinge length. Different conclusions were made regarding the influence of axial load. Some researchers concluded that it has a significant influence in the plastic hinge length, while others did not find any significant dependence. Studying the influence of axial load is of particular importance for coupled walls, which are subjected to high tensile and compressive axial loads. Previous studies presented in Chapter 2 were done 120 for compressive axial loads only, not tensile axial loads. In this study, the influence of both tensile and compressive axial loads was investigated. The first analysis made for Wall 1 and 2, for a shear span equal to the wall height and a web thickness of 508mm, described in Section 5.3, were performed for a compressive axial load ratio (P/fcAg) of 0.1. To study the influence of compressive axial loads, three new analyses for each wall were performed in VecTor2 for the following axial load ratios: - 0.3 (46452 kN for Wall 1 and 23226 kN for Wall 2). - 0.2 (30968 kN for Wall 1 and 15484 kN for Wall 2). Zero. To study the influence of tensile axial loads, two additional analyses for each wall were performed. The tensile axial load applied cannot exceed the pure tension capacity of the walls (fyAs\ equal to 15200 kN for Wall 1 and 7680 kN for Wall 2. The following axial load ratios were selected: - 0.02 (3800 kN for Wall 1 and 1920 kN for Wall 2), which is equal to 1/4 of the pure tension capacity. - 0.05 (7600 kN for Wall 1 and 3840 kN for Wall 2), which is equal to 1/2 of the pure tension capacity. The models used for these analyses had the same characteristics as the previous ones, described in Section 5.3. The only difference was the magnitude of the constant axial loads; these loads were also equally distributed among all the six nodes at the top of each wall. The results from the analysis were used to determine the curvatures and bending moments for different drift levels, using the same procedure described in Section 5.4 and 5.6. Since the axial load was different for all these analyses, the moment-curvature relationship was also different in all cases. The yield curvature and yield moment were also different; these were determined using the same procedures described in Section 5.4. Figure 5.25 shows the moment-curvature relationship for Wall 1 for the six axial load ratios considered (the negative sign indicates compression): 121 Figure 5.25 Moment-curvature relationships of Wall 1 for different axial load ratios 180000 160000 140000 120000 = 100000 o Â£ O) 00 40000 20000 T P " * -â€¢ P/fcAg = -0.30 â€¢ PIT cAg = -0.20 Â» P/fcAg = -0.10 Â« P/fcAg = 0.00 - PffcAg = 0.02 â€¢ P/fcAg = 0.05 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Curvature (rad/m) Figure 5.26 shows the moment-curvature relationship for Wall 2 for the six axial load ratios considered: Figure 5.26 Moment-curvature relationships of Wall 2 for different axial load ratios 45000 40000 35000 â€¢g" 30000 z | 25000 o a 20000 c c m 15000 10000 5000 0 â€¢ P/fcAg = -0.30 â€¢ PffcAg = -0.20 â€¢ P/fcAg = -0.10 Â« P/fcAg = 0.00 - P/fcAg = 0.02 â€¢ P/fcAg = 0.05 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Curvature (rad/m) 122 Table 5.19 shows the yield curvature and yield moment for each of the cases analyzed: Table 5.19 Yield curvature and yield moment of Wall 1 and 2 for different axial load ratios P'fA P(kN) tp\, (rad/m) My (kN.m) Wall 1 Wall 2 Wall 1 Wall 2 Wall 1 Wall 2 -0.30 -46452.0 -23226.0 0.00062 0.00120 157840.0 39480.0 -0.20 -30968.0 -15484.0 0.00053 0.00104 127380.0 31800.0 -0.10 -15484.0 -7742.0 0.00045 0.00090 90680.0 22740.0 0.00 0.0 0.0 0.00037 0.00075 47910.0 12150.0 0.02 3800.0 1920.0 0.00035 0.00070 36270.0 9300.0 0.05 7600.0 3840.0 0.00032 0.00064 23900.0 6040.0 Table 5.20 to 5.29 show the measured (from curvature distribution) and predicted (using Equation 5.1) plastic hinge lengths for different drift levels. The results for Wall 1 and 2 for a compressive axial load ratio of 0.1 were presented in Table 5.2 and 5.3, respectively: Table 5.20 Predicted and measured plastic hinge lengths for Wall 1 for a compressive axial load ratio of 0.3 Drift (%) Mmax (kN.m) A^OcN.m) Ls(m) Lp(m) Wl Predicted Wl Measured 1.2 170170.2 157840.0 54.860 3.98 4.28 1.0 168431.2 157840.0 54.860 3.45 3.43 0.8 160306.4 157840.0 54.860 0.84 0.78 Table 5.21 Predicted and measured plastic hinge lengths for Wall 2 for a compressive axial load ratio of 0.3 Drift (%) Mmax (kN.m) My (kN.m) Ls (m) ID(m) W2 Predicted W2 Measured 2.0 42522.0 39480.0 54.860 3.92 3.94 1.8 41781.4 39480.0 54.860 3.02 3.03 1.6 40722.6 39480.0 54.860 1.67 1.30 Table 5.22 Predicted and measured plastic hinge lengths for Wall 1 for a compressive axial load ratio of 0.2 Drift (%) M Â« a x (kN.m) My (kN.m) Ls(m) LD (m) Wl Predicted Wl Measured 1.4 141950.3 127380.0 54.860 5.63 5.93 1.2 141741.8 127380.0 54.860 5.56 5.64 1.0 139481.6 127380.0 54.860 4.76 4.71 0.8 134066.9 127380.0 54.860 2.74 2.61 Table 5.23 Predicted and measured plastic hinge lengths for Wall 2 for a compressive axial load ratio of 0.2 Drift (%) Mmax (kN.m) My (kN.m) Mm) Z,â€ž(m) W2 Predicted W2 Measured 2.0 35472.5 31800.0 54.860 5.68 5.38 1.8 34781.2 31800.0 54.860 4.70 4.50 1.6 33826.7 31800.0 54.860 3.29 3.17 1.4 32614.3 31800.0 54.860 1.37 1.23 123 Table 5.24 Predicted and measured plastic hinge lengths for Wall 1 for no axial load Drift (%) MÂ«ax (kN.m) My (kN.m) Ls(m) Ln(m) Wl Predicted Wl Measured 2.0 57690.8 47910.0 54.860 9.30 10.10 1.8 56730.7 47910.0 54.860 8.53 9.32 1.6 55929.8 47910.0 54.860 7.87 8.59 1.4 55145.3 47910.0 54.860 7.20 7.87 1.2 54278.5 47910.0 54.860 6.44 6.97 Table 5.25 Predicted and measured plastic hinge lengths for Wall 2 for no axial load Drift (%) Mmax (kN.m) My (kN.m) Ls(m) Lâ€ž (m) W2 Predicted W2 Measured 2.0 13720.5 12150.0 54.860 6.28 6.24 1.8 13391.3 12150.0 54.860 5.09 5.78 1.6 13314.5 12150.0 54.860 4.80 4.99 1.4 12957.9 12150.0 54.860 3.42 4.00 1.2 12683.6 12150.0 54.860 2.31 2.21 5.26 Predicted and measured plastic hinge lengths for Wall 1 for a tensile axial load ratio Drift (%) Mmax (kN.m) My (kN.m) Ls(m) ID(m) Wl Predicted Wl Measured 2.0 44381.7 36270.0 54.860 10.03 10.01 1.8 43778.3 36270.0 54.860 9.41 9.73 1.6 42906.0 36270.0 54.860 8.48 8.86 1.4 42116.0 36270.0 54.860 7.61 8.06 1.2 41260.2 36270.0 54.860 6.64 6.84 5.27 Predicted and measured plastic hinge lengths for Wall 2 for a tensile axial load ratio Drift (%) Mmax (kN.m) My (kN.m) Ls(m) LB(m) W2 Predicted W2 Measured 2.0 10461.8 9300.0 54.860 6.09 6.18 1.8 10275.3 9300.0 54.860 5.21 5.65 1.6 10209.4 9300.0 54.860 4.89 5.05 1.4 9918.7 9300.0 54.860 3.42 3.82 1.2 9627.9 9300.0 54.860 1.87 2.17 .28 Predicted and measured plastic hinge lengths for Wall 1 for a tensile axial load ratio Drift (%) Mmax (kN.m) My(kN.m) L (m) Zâ€ž(m) J-'s V 1 1 1 / Wl Predicted Wl Measured 2.0 30381.5 23900.0 54.860 11.70 11.19 1.8 29778.0 23900.0 54.860 10.83 10.29 1.6 29031.9 23900.0 54.860 9.70 9.17 1.4 28423.0 23900.0 54.860 8.73 8.35 1.2 27792.1 23900.0 54.860 7.68 7.03 124 Table 5.29 Predicted and measured plastic hinge lengths for Wall 2 for a tensile axial load ratio of 0.05 Drift (%) Mmax (kN.m) A/>,(kN.m) Is(m) LD(m) W2 Predicted W2 Measured 2.0 6906.9 6040.0 54.860 6.89 6.04 1.8 6868.5 6040.0 54.860 6.62 5.87 1.6 6665.5 6040.0 54.860 5.15 5.14 1.4 6616.1 6040.0 54.860 4.78 3.79 1.2 6292.4 6040.0 54.860 2.20 1.18 It can be observed that the flexural prediction provides good results for the different axial load ratios and drift levels. Figure 5.27 shows the measured plastic hinge lengths for Wall 1 as a function of the total drift: Figure 5.27 Measured plastic hinge lengths vs. total drift for Wall 1 for different axial load ratios - P/fcAg = -0.30 - PffcAg = -0.20 P/fcAg = -0.10 - P/fcAg = 0.00 P/fcAg = 0.02 - P/fcAg = 0.05 Figure 5.28 shows the measured plastic hinge lengths for Wall 2 as a function of the total drift: These plots show that the plastic hinge length of the wall reduces when it is subjected to compressive axial forces, and increases when it is subjected to tensile axial forces. This is because the compression load reduces the length in which the inelastic curvatures occur, and so, the length of the plastic hinge. The tension load has the opposite effect. The quality of the predictions was evaluated by plotting the ratio of the measured to predicted plastic hinge length for different drift levels and axial loads. This is shown in Figure 5.29 and 5.30: 126 Figure 5.29 Ratio of measured to predicted plastic hinge lengths vs. drift for Wall 1 for different axial load ratios 2.0 1.5 5 =! 1.0 0.5 0.0 â€” Â» â€” P/fcAg = -0.30 - â€” - P/fcAg = -0.20 -â€¢-A--- P/fcAg = -0.10 P/fcAg = 0.00 * P/fcAg = 0.02 PffcAg = 0.05 0.6 0.8 1.0 1.2 1.4 1.6 Drift (%) 1.8 2.0 2.2 Figure 5.30 Ratio of measured to predicted plastic hinge lengths vs. drift for Wall 2 for different axial load ratios 2.0 1.5 =i 1.0 0.5 0.0 - â€¢ â€” P/fcAg = -0.30 - â€¢ â€¢ - - P f f c A g = -0.20 â€¢â€¢Â»â€¢ â€¢â€¢ PffcAg = -0.10 - * â€” PffcAg = 0.00 PffcAg = 0.02 PffcAg = 0.05 0.6 0.8 1.0 1.2 1.4 1.6 Drift (%) 1.8 2.0 2.2 127 Most of the predictions shown in these figures are good. Some of the predictions for Wall 2 are not very good for the lowest drifts. This is because the wall at these drifts has just started yielding, the plastic hinge length is very small, and the ratio of measured to predicted becomes either too large or too small. For large drifts, all the predictions are good. It can be concluded that the length of the plastic hinge reduces with the addition of compression and increases with the addition of tension. The flexural prediction can be used to estimate the plastic hinge length in walls subjected to compressive and tensile axial loads. 5.10 Summary Table 5.30 shows the measured plastic hinge lengths at maximum drift as a function of the parameters studied for the 22 analyses presented in this chapter. These are measured plastic hinge lengths considering a linearly varying inelastic curvature: Table 5.30 Measured plastic hinge lengths at maximum drift Wall L, (m) v(MPa) Lp(m) Lpllw Wall 1 54.860 0.60 -0.10 8.39 1.10 0.15 Wall 2 54.860 0.30 -0.10 6.21 1.63 0.11 Wall 1 35.659 0.95 -0.10 6.06 0.80 0.17 Wall 2 35.659 0.45 -0.10 4.74 1.25 0.13 Wall 1 27.430 1.25 -0.10 5.55 0.73 0.20 Wall 2 27.430 0.60 -0.10 4.28 1.12 0.16 Wall 1 19.201 1.80 -0.10 4.45 0.58 0.23 Wall 2 19.201 0.85 -0.10 3.16 0.83 0.16 Wall 1 27.430 2.30 -0.10 6.26 0.82 0.23 Wall 2 27.430 1.15 -0.10 4.42 1.16 0.16 Wall 1 19.201 3.35 -0.10 5.86 0.77 0.30 Wall 2 19.201 1.65 -0.10 3.68 0.97 0.19 Wall 1 54.860 1.00 -0.30 4.28 0.56 0.08 Wall 2 54.860 0.50 -0.30 3.94 1.04 0.07 Wall 1 54.860 0.80 -0.20 5.93 0.78 0.11 Wall 2 54.860 0.40 -0.20 5.38 1.41 0.10 Wall 1 54.860 0.32 0.00 10.10 1.33 0.18 Wall 2 54.860 0.16 0.00 6.24 1.64 0.11 Wall 1 54.860 0.25 0.02 10.01 1.31 0.18 Wall 2 54.860 0.12 0.02 6.18 1.62 0.11 Wall 1 54.860 0.17 0.05 11.19 1.47 0.20 Wall 2 54.860 0.08 0.05 6.04 1.59 0.11 CHAPTER 6: SYSTEMS OF WALLS OF DIFFERENT LENGTHS CONNECTED TOGETHER BY RIGID SLABS 128 6.1 Scope of analysis Program VecTor2 was used to investigate plastic hinge lengths in a system of walls of different lengths interconnected by rigid slabs at various levels. The parametric study in Chapter 5 was performed for Wall 1 and 2 separately. A new analysis for these two walls was performed, being under the same conditions as the ones described in Section 5.2, except that they were now combined together, to see how this influences the length of the plastic hinge. Additional analyses with walls of other lengths were also performed. In high-rise buildings, it is common to have parallel walls of different lengths providing lateral resistance. These walls, as well as the gravity columns, are interconnected by rigid slabs at numerous floor levels. As a result, the lateral displacement of all these elements is the same at these levels. To study how the plastic hinge length in walls of different lengths is influenced when they are connected together, three new analyses were performed in VecTor2 for the following wall systems: The first system consisted of Wall 1 and 2 combined together. Wall 1 was two times longer than Wall 2. The second system consisted of Wall 1 combined together with a wall that was four times shorter. This wall will be referred to as Wall 3. The third system consisted of Wall 1 combined together with a column that was eight times shorter. This column will be referred to as Column 1. 6.2 Wall 1 combined with Wall 2 The description of Wall 1 and 2 and their individual analytical models have already been given in Section 5.2 and 5.3, respectively. The description of the analytical model for these two walls combined together and the results obtained will be presented in the following sections. 6.2.1 Description of wall system model The model used to analyze Wall 1 and 2 combined is presented in Figure 6.1. The slabs were provided every 2743mm, resulting in a 20 storey building. The thickness of the slabs was 203mm. 129 Figure 6.1 Model of Wall 1 and 2 combined 54860 mm W1 W2 f / / / / / / / / / / / / / / / / / / / / / , 2743 mm 6.2.2 Analytical model of wall system Wall 1 and 2 were modeled and analyzed together using program VecTor2. The finite element mesh used for each wall, as well as the boundary conditions, material properties and loads, were the same as the ones described in Section 5.3. The slabs interconnecting the walls at each floor level were modeled using truss bar elements. This type of element in VecTor2 is typically used to model the reinforcing steel as discrete bars (see Section 3.3), so reinforcement properties have to be assigned to it. These elements were provided with a very high strength and stiffness, so that they do not yield and do not deform axially, in order to force the lateral displacements in both walls to be the same. A new material type was created to represent this region of the wall system. 130 Since the slabs are very stiff, the strains in the transverse direction of the wall at the floor levels are very small. To simulate that effect in VecTor2, the rectangular elements located at each storey level were also provided with very stiff properties. These elements had a height equal to the slab thickness (203mm). These were only provided up to half the height of the wall, where the mesh was more refined. A new material type was created to represent this region of the wall system. Al l nodes and elements were numbered in the horizontal (short) direction of the wall system. The complete mesh consisted of 3357 nodes, 3066 rectangular elements, 116 triangular elements and 20 truss elements. A monotonic lateral load was applied at the top of the Wall 1. This load was applied in a displacement-control mode, in increments of 1mm. Additionally, a constant axial load of 15484 kN in Wall 1 and 7742 kN in Wall 2 was applied; these loads were equally distributed among the six nodes at the top of each wall. The self-weight of the walls was not considered. Figure 6.2 shows the finite element model of Wall 1 and 2, created in the pre-processor FormWorks: Figure 6.2 Finite element model of Wall 1 and 2 in FormWorks Figure 6.3 shows a more detailed view of the elements representing the slabs at each storey in finite element model: 132 Figure 6.3 Detail of elements representing the slabs at each storey in finite element model of Wall 1 and 2 Table 6.1 and 6.2 show the material properties in the different regions of the wall system: Table 6.1 Material properties in rectangular and triangular elements of Wall 1 and 2 Concrete properties Material 1 Material 2 Material 3 Color Thickness (mm) 508 508 2000 Cylinder compressive strength (MPa) 40 40 40 Reinforcement component properties jc-direction -^direction x-direction j>-direction x-direction jÂ»-direction Reinforcement ratio (%) 0.8 2.0 0.5 0.5 0.5 0.5 Reinforcement diameter (mm) 15 25 15 15 15 15 Yield strength (MPa) 400 400 400 400 400 400 Ultimate strength (MPa) 650 650 650 650 650 650 Elastic modulus (MPa) 200000 200000 200000 200000 200000 200000 Strain hardening modulus (MPa) 4875 4875 4875 4875 4875 4875 Strain hardening strain (mm/m) 10 10 10 10 10 10 Table 6.2 Material properties in truss elements of Wall 1 and 2 Reinforcement properties Material 4 Color Cross-sectional area (mm2) 2500000 Reinforcement diameter (mm) 55 Yield strength (MPa) 4000 Ultimate strength (MPa) 4000 Elastic modulus (MPa) 1000000 Strain hardening modulus (MPa) 1 Strain hardening strain (mm/m) 80 133 6.2.3 Analytical results The results from the analysis were used to determine the curvatures, using the same procedure described in Section 5.4 and 5.6. The ultimate drift for the wall system was 2%. Figure 6.4 shows the curvature distribution up to the mid-height of both walls for this drift, when they are alone and combined: Figure 6.4 Curvatures up to the mid-height of Wall 1 and 2 alone and combined for a drift of 2% 28 0-1 , , ^ , =2 1 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 Curvature (rad/m) Several observations can be made for this figure. Wall 1 is pulling Wall 2 along, resulting in equal curvatures in the elastic range. After yielding, a more complex phenomenon is occurring. The curvature distribution in Wall 1 remains approximately the same, while the curvatures in Wall 2 at the base have increased considerably. The curvature distribution will be examined in more detail in the next section. The influence of the interconnection in the moment-curvature relationship of the walls was analyzed. To develop this relationship, the bending moment diagram along the height of the wall must be calculated. The truss elements representing the slabs forced Wall 1 and 2 to displace the same at the storey levels, resulting in high axial forces in these elements. Therefore, both walls were now subjected to a system of lateral forces at each storey level, and Wall 1 is additionally subjected to the lateral load 134 acting at the top. The axial forces in the truss elements obtained from program VecTor2 were used to deterrnine the system of lateral forces. Then, the bending moment diagram was calculated. Figure 6.5 shows the bending moments along the height of both walls for a drift of 2%: Figure 6.5 Bending moments along the height of Wall 1 and 2 alone and combined for a drift of 2% Bending moment (kN.m) Figure 6.6 shows the moment-curvature relationship of Wall 1: Figure 6.6 Moment-curvature relationship of Wall 1 alone and combined 120000 100000 80000 o E o> c 'â€¢5 c 0 CO 60000 40000 20000 â€¢ W1A â€¢ W1C 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 Curvature (rad/m) Figure 6.7 shows the moment-curvature relationship of Wall 2: Figure 6.7 Moment-curvature relationship of Wall 2 alone and combined 30000 25000 1 20000 z 5 15000 E o> c = c m 10000 5000 W2A â€¢ W2C 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 Curvature (rad/m) 136 It can be seen that the moment-curvature relationship does not change when the walls are connected, so this is not causing the differences in the curvature distributions shown in Figure 6.4. Therefore, as seen in Section 5.4, the yield curvature and yield moment for Wall 1 were 0.00045 rad/m and 90680 kN.m, respectively; and for Wall 2 they were 0.0009 rad/m and 22740 kN.m. The deflected shapes and slopes of both walls were also investigated. The total displacement at each storey, as well as its flexural and shear displacement components, were determined. The total horizontal displacements at any level or location were obtained directly from program VecTor2. The flexural displacement at a particular storey was calculated by mtegrating the curvatures along the height up to that level, using the second moment-area theorem: Ay =^(f>{x)xdx (6.1) This integral was solved with the following numerical scheme: Figure 6.8 Numerical scheme for curvature integration End V ij)i-i (hi - \ -Axi Then: <t>i-x + Ax, Ax, (6.2) The shear displacement at a particular storey was then calculated as the difference between the total and flexural displacement: A V = A - A (6.3) 137 The slopes were calculated at the same location as the curvatures, by integrating the curvatures along the height using the first moment-area theorem: (6.4) Using the numerical scheme from Figure 6.8: 2 J (6.5) 6.2.4 Discussion of analytical results The curvature distribution in Figure 6.4 shows that the curvatures in Wall 1 remain almost the same when it is connected with Wall 2, and its plastic hinge length has increased slightly. In Wall 2, the curvatures have increased considerably at the base, and its plastic hinge has been reduced. Using the yield curvature to measure the plastic hinge length for a drift of 2% from the curvature distribution when the walls are connected together, it was equal to 9.60m for Wall 1 and 4.64m for Wall 2. The change in the plastic hinge lengths can also be observed from the distribution of tensile steel strains in the wall. The average vertical steel strains in the elements along the height of the walls obtained from program VecTor2 were studied for this. The steel strains were observed in the exterior and interior steel layer in the boundary zones, which were shown in Figure 5.14. Figure 6.9 shows the steel strains up to the mid-height of Wall 1, alone and combined, for a drift of 2%: 138 Figure 6.9 Steel strains up to the mid-height of Wall 1 alone and combined for a drift of 2% 28 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 Figure 6.10 shows the steel strains up to the mid-height of Wall 2, alone and combined, for a drift of 2%: Figure 6.10 Steel strains up to the mid-height of Wall 2 alone and combined for a drift of 2% 28 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 Strain 139 For Wall 1, the steel strains in both layers are approximately the same, but they exceed the yield strain (0.002 for these walls) at a slightly larger distance from the base when it is combined. For Wall 2, the steel strains are larger for the combined case, and they exceed the yield strain at a slightly smaller distance from the base. All these observations are revealing that the curvature distributions and plastic hinge lengths in both walls are different, even when they are combined together by rigid slabs. In order to understand this phenomenon, the deflected shapes and slopes of both walls were studied. Figure 6.11 shows the total lateral displacements at the faces of both walls (right face for Wall 1 and left face for Wall 2) up to the fourth storey when they are combined, for a drift of 2%; these were obtained directly from program VecTor2: Figure 6.11 Displacements at wall faces up to the fourth storey of Wall 1 and 2 combined for a drift of 2% 12 10 at Â«> I <â€”W1C â€” -W2C 60 80 100 Displacement (mm) 140 160 Although the lateral displacements in both walls are equal at the storey levels, these are different between these levels. So, the deflected shapes are not strictly the same. The bigger differences between the displacements in both walls are located at the bottom. Figure 6.12 shows a more detailed view of the lateral displacements at the faces of both walls up to the first storey when they are combined, for a drift of 2%: 140 Figure 6.12 Displacements at wall faces up to the first storey of Wall 1 and 2 combined for a drift of 2% 3.0 0 2 4 6 8 10 12 14 16 18 20 Displacement (mm) The displacements are not the same between storey levels. Wall 1 has larger displacements than Wall 2. It can also be seen that Wall 2 ends up having bigger slopes. The variation of the slope in Wall 2 is larger than in Wall 1, which is the reason why the curvatures are larger in this wall for the combined case. Table 6.3 shows the displacement components up to the fourth storey of both walls when they are combined for a drift of 2%: Table 6.3 Displacement components up to the fourth storey of Wall 1 and 2 combined for a drift of 2% Storey Height Wall 1 Wall 2 (m) Af (mm) Av (mm) A (mm) A/ (mm) Av (mm) A (mm) 0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1 2.74 8.02 10.59 18.61 10.38 8.23 18.61 2 5.49 34.02 18.84 52.87 38.88 13.99 52.87 3 8.23 70.91 23.46 94.37 74.26 20.11 94.37 4 10.97 112.96 27.48 140.44 114.45 25.99 140.44 Figure 6.13 shows these results graphically: 141 Figure 6.13 Displacement components up to the fourth storey of Wall 1 and 2 combined for a drift of 2% 12 Displacement (mm) The total displacements are the same at the storey heights, but the flexural displacement is larger in Wall 2 at the lower levels. Wall 1 has higher shear displacements at the lower levels. Therefore, the displacements in Figure 6.12 in this wall up to the first storey are mostly due to shear. Although Wall 1 has larger total displacements at the bottom, Wall 2 has larger flexural displacements. The difference in shear displacements is causing the slopes to be different, as shown in Figure 6.14: 142 Figure 6.14 Slopes up to the mid-height of Wall 1 and 2 combined for a drift of 2% 28 0.000 0.005 0.010 0.015 0.020 0.025 Slope (rad) The slopes in Wall 2 are larger at the lower levels; this was also deduced by looking at Figure 6.12. The inelastic rotation in Wall 2 is concentrated in a shorter height than in Wall 1. It can be concluded that the curvature distribution of Wall 1 and 2 when they are connected are different because of two reasons. First, the displacements between storey levels are different in both walls, so the deflected shapes are not the same. Second, the shear displacements at the lower levels are larger in Wall 1 than in Wall 2 when they are connected; resulting in lower flexural displacements, lower slopes, and lower curvatures for Wall 1. Since the curvatures are not the same, the plastic hinge lengths are also different. Another important conclusion is that when walls of different lengths are combined together, the curvature demand for the shorter wall is much larger than when it is alone. This wall has to be able to sustain that demand. In order to predict the maximum curvature in Wall 2 for the combined case, a very simple model was developed, based on the observations made from this study. It is shown in figure 6.15: The following assumptions were made regarding this model: The inelastic curvatures are linearly varying. - The plastic hinge length and the maximum curvature in Wall 1 (longer wall) do not change when it is combined with Wall 2 (shorter wall). - The rotations or slopes in both walls at a distance equal to the plastic hinge length of Wall 1 from the section of maximum curvature are the same. Based on these assumptions, the maximum curvature in Wall 2 can be estimated. From Figure 6.15, the total rotation in Wall 1 is determined by integrating the curvatures: ox = max,l (6.6) The total rotation in Wall 2 is determined similarly: L P,2 (6.7) Since we are considering that the rotations are the same: 0X =02 (6.8) 144 Replacing Equation 6.6 and 6.7 in Equation 6.8, and solving for ^ .j, we finally get: max.] 'P,I 'P.2J (6.9) The measured values of the parameters in Equation 6.9 were used to predict the maximum curvature in Wall 2, and then it was compared with the measured value, obtained from Figure 6.4. The maximum curvature, yield curvature and plastic hinge length for Wall 1 when it is alone are already known. The yield curvature of Wall 2 is also known. The plastic hinge length used for Wall 2 was the one measured in the combined case, that is, 4.64m. The result is shown in Table 6.4: Table 6.4 Predicted and measured maximum curvature for Wall 2 combined Drift (%) LP,J (M) LP,2 (m) faj (rad/m) 0yi2 (rad/m) 4>max.i (rad/m) <t>max.2 (rad/m) Predicted Measured 2.0 8.39 4.64 0.00045 0.00090 0.00326 0.00472 0.00456 The proposed model provides a good prediction of the maximum curvature in Wall 2. 6.3 Wal l 1 combined with Wal l 3 Wall 1 was then combined with a wall that was four times shorter. This wall will be referred to as Wall 3 throughout this chapter. Wall 3 was first analyzed individually, and then it was connected together with Wall 1 to compare the results obtained. The description of the analytical model for Wall 3 alone and combined together with Wall 1, and the results obtained, will be presented in the following sections. 6.3.1 Description of wall system model Wall 3 was first analyzed alone as a cantilever element. It had a rectangular cross-section, and it was 54860mm high, 508mm thick and 1905mm long. The boundary zones were 305mm long, the vertical reinforcement in these consisted of 6-25M reinforcing bars enclosed by 15M ties spaced at 100mm. The clear cover of the ties was 40mm. The web had 15M reinforcing bars spaced at 150mm vertically and horizontally. The cross-section of Wall 3 is shown in Figure 6.16: 145 Figure 6.16 Cross-section details of Wall 3 6-25M vertical with 15M @ 150mm horizontal 6-25M vertical with 15M ties @ 100mm and vertical 15M ties @ 100mm The material properties of Wall 3 were the same ones used for Wall 1 and 2, described in Section 5.2. The wall was fixed at the base. It was subjected to a monotonically increasing lateral load at the top; as well as a constant compressive axial load of 0.2Agfc, equal to 7742 kN. This is the same axial load used for Wall 2. Smaller walls usually have a larger tributary area, so the axial load ratio was increased for Wall 3. The model used to analyze walls 1 and 3 combined was the same as the one presented in Figure 6.1, except that Wall 2 was replaced by Wall 3. The slabs were provided every 2743mm, resulting in a 20 storey building. The thickness of the slabs was 203mm. 6.3.2 Analytical model of wall system Wall 3 was first modeled and analyzed individually using program VecTor2. The model was very similar to the one created for Wall 1 and 2, described in Section 5.3. Low-powered rectangular and triangular elements were used to model the concrete, with smeared steel to account for the presence of reinforcement. The constitutive models for concrete and steel described in Chapter 3 were used in the analysis. The same two material types from Wall 1 and 2 were used to represent the confined boundaries and the web of Wall 3 in the finite element model. The analytical model was fixed at the bottom; both the horizontal and vertical displacements were restrained. The finite element mesh was more refined near the base (critical section). Wall 3 had the same refinement as Wall 1 in the vertical direction up to a half of the height, so that the curvatures could be calculated over the same average length. Meshes of 305x423, 305x203, 324x423 and 324x203 rectangular elements were used; six elements (seven nodes) were used in the transverse direction of the wall. This level of refinement was 146 mamtained up to half the height of the wall. Up from this point, five elements (six nodes) were used in the transverse direction. The transitions were made using triangular elements. Al l nodes and elements were numbered in the horizontal (short) direction. The complete mesh consisted of 857 nodes, 717 rectangular elements and seven triangular elements. The material properties used in the analysis were the same ones used for Wall 1 and 2, presented in Section 5.3. A monotonic lateral load was applied at the top of the wall. This load was applied in a displacement-control mode, in increments of 1mm. Additionally, a constant axial load of 7742 kN was applied; this load was equally distributed among the six nodes at the top of the wall. The self-weight of the wall was not considered. Figure 6.17 shows the finite element model of Wall 3, created in the pre-processor FormWorks: 147 Figure 6.17 Finite element model of Wall 3 in FormWorks 1000. The material properties in the different regions of the wall were the same ones used for Wall 1 and 2 alone, presented in Table 5.1. 148 Wall 1 and 3 were then modeled and analyzed together using program VecTor2. The finite element mesh used for each wall, as well as the boundary conditions, material properties and loads, were the same as the ones described in Section 5.3 and this section. As it was done for the model of Wall 1 and 2 combined together, described in Section 6.2.2, the slabs interconnecting the walls at each floor level were modeled using truss bar elements with very stiff properties. The rectangular elements located at each storey level, which had a height equal to the slab thickness, were also provided with very stiff properties. The same material types were used for these elements. All nodes and elements were numbered in the horizontal (short) direction of the wall system. The complete mesh consisted of 2938 nodes, 2644 rectangular elements, 93 triangular elements and 20 truss elements. A monotonic lateral load was applied at the top of the Wall 1. This load was applied in a displacement-control mode, in increments of 1mm. Additionally, a constant axial load of 15484 kN in Wall 1 and 7742 kN in Wall 3 was applied; these loads were equally distributed among the six nodes at the top of each wall. The self-weight of the walls was not considered. Figure 6.18 shows the finite element model of Wall 1 and 3, created in the pre-processor FormWorks: 149 Figure 6.18 Finite element model of Wall 1 and 3 in FormWorks The detail of the elements representing the slabs at each storey level in the finite element model was the same as the one shown in Figure 6.3. The material properties in the different regions of the wall system were the same ones used for Wall 1 and 2 combined, presented in Table 6.1 and 6.2. 150 6.3.3 Analytical results The results from the analysis were used to detennine the curvatures, using the same procedure described in Section 5.4 and 5.6. The ultimate drift for the wall system was 2%. Figure 6.19 shows the curvature distribution up to the mid-height of both walls for this drift, when they are alone and combined: Figure 6.19 Curvatures up to the mid-height of Wall 1 and 3 alone and combined for a drift of 2% 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070 Curvature (rad/m) The same phenomenon observed for Wall 1 and 2, presented in Section 6.2.3, is seen here. The curvature distribution in Wall 1 remains approximately the same, while the curvatures in Wall 3 at the base have increased considerably. The moment-curvature relationship of Wall 3 was also determined, using the procedure described in Section 5.4 when it is alone, and the procedure described in Section 6.2.3 when it is combined. It is presented in Figure 6.20: 151 Figure 6.20 Moment-curvature relationship of Wall 3 alone and combined 10000 9000 8000 7000 z a. 6000 c o 5000 E 4000 3000 2000 1000 W3A â€¢ W3C 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070 Curvature (rad/m) It can be seen that the moment-curvature relationship does not change when the walls are connected. The yield curvature for Wall 3 is 0.00207 rad/m. 6.3.4 Discussion of analytical results The curvature distribution in Figure 6.19 shows that the curvatures in Wall 1 remain almost the same when it is connected with Wall 3, and its plastic hinge length has increased slightly. In Wall 3, the curvatures have increased considerably at the base. This is the same phenomenon observed for Wall 1 and 2 combined together. However, when Wall 3 is alone, the curvatures at 2% drift do not reach the yield curvature. Therefore, it is not yielding, so no plastic hinge is formed. When combined with Wall 1, the curvatures in Wall 3 increase, causing it to yield and develop a plastic hinge. This can also be observed from the distribution of tensile steel strains in the wall. The average vertical steel strains in the elements along the height of the walls obtained from program VecTor2 were studied for this. The steel strains were observed in the exterior steel layer in the boundary zones. Figure 6.21 shows the steel strains up to the mid-height of Wall 3, alone and combined, for a drift of 2%: 152 Figure 6.21 Steel strains up to the mid-height of Wall 3 alone and combined for a drift of 2% 28 24 20 2> Ol x 12 8 4 0 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 Strain The steel strains do not reach the yield strain when Wall 3 is alone, but they do when it is combined. Using the yield curvature to measure the plastic hinge length for a drift of 2% from the curvature distribution when the walls are connected together, it was equal to 9.58m for Wall 1 and 2.06m for Wall 3. In conclusion, the curvature distribution and plastic hinge length of Wall 1 and 3 when they are connected are different, as it was observed for Wall 1 and 2 connected, for the same reasons explained in Section 6.2.4. Also, the curvature demand for the shorter wall is much larger when it is combined than when it is alone, and it has to be able to sustain that demand. The maximum curvature in Wall 3 was predicted using the model presented in Figure 6.15. The measured values of the parameters in Equation 6.9 were used in the prediction, and then it was compared with the measured value, obtained from Figure 6.19. The maximum curvature, yield curvature and plastic hinge length for Wall 1 when it is alone are already known. The yield curvature of Wall 3 is also known. The plastic hinge length used for Wall 3 was the one measured in the combined case, that is, 2.06m. The result is shown in Table 6.5: 153 Table 6.5 Predicted and measured maximum curvature for Wall 3 combined Drift (%) LP,i (m) Lp.2 (m) <t>yj (rad/m) <j>y,2 (rad/m) </>maU (rad/m) <t>mm.2 (rad/m) Predicted Measured 2.0 8.39 2.06 0.00045 0.00207 0.00326 0.00530 0.00581 The proposed model provides a good prediction of the maximum curvature in Wall 3. 6.4 Wall 1 combined with Column 1 Wall 1 was finally combined with a column that was eight times shorter. This column will be referred to as Column 1 throughout this chapter. Column 1 was first analyzed individually, and then it was connected together with Wall 1 to compare the results obtained. The description of the analytical model for Column 1 alone and combined together with Wall 1, and the results obtained, will be presented in the following sections. 6.4.1 Description of wall system model Column 1 was first analyzed alone as a cantilever element. It had a rectangular cross-section, and it was 54860mm high, 508mm thick and 953mm long. The vertical reinforcement consisted of 20-25M reinforcing bars enclosed by 15M ties spaced at 100mm. The clear cover of the ties was 40mm. The cross-section of Column 1 is shown in Figure 6.22: Figure 6.22 Cross-section details of Column 1 20-25M vertical with 15M t iesÂ® 100mm J u c c 1 o __aâ€”rJ 953 mm The material properties of Column 1 were the same ones used for Wall 1 and 2, described in Section 5.2. The wall was fixed at the base. It was subjected to a monotonically increasing lateral load at the top; as well as a constant compressive axial load of 0AAgfc, equal to 7742 kN. This is the same axial load used for Wall 2 and 3. Gravity columns typically have a larger tributary area compared to walls, so the axial load ratio was increased for Column 1. 154 The model used to analyze Wall 1 and Column 1 combined was the same as the one presented in Figure 6.1, except that Wall 2 was replaced by Column 1. The slabs were provided every 2743mm, resulting in a 20 storey building. The thickness of the slabs was 203mm. 6.4.2 Analytical model of wall system Column 1 was first modeled and analyzed individually using program VecTor2. The model was very similar to the one created for Wall 1 and 2, described in Section 5.3. Low-powered rectangular and triangular elements were used to model the concrete, with smeared steel to account for the presence of reinforcement. The constitutive models for concrete and steel described in Chapter 3 were used in the analysis. Only one material type was used to represent the whole column in the finite element model, it was the same one used to represent the confined boundaries of Wall 1 and 2. The analytical model was fixed at the bottom; both the horizontal and vertical displacements were restrained. The finite element mesh was more refined near the base (critical section). Column 1 had the same refinement as Wall 1 in the vertical direction up to a half of the height, so that the curvatures could be calculated over the same average length. Meshes of 318x423 and 318x203 rectangular elements were used; three elements (four nodes) were used in the transverse direction of the wall. This level of refinement was maintained up to half the height of the wall. Up from this point, two elements (three nodes) were used in the transverse direction, and then it was further reduced to one element (two nodes) up to the top of the wall. The transitions were made using triangular elements. Al l nodes and elements were numbered in the horizontal (short) direction. The complete mesh consisted of 345 nodes, 242 rectangular elements and eight triangular elements. The material properties used in the analysis were the same ones used for Wall 1 and 2, presented in Section 5.3. A monotonic lateral load was applied at the top of the column. This load was applied in a displacement-control mode, in increments of 1mm. Additionally, a constant axial load of 7742 kN was applied; this load was equally distributed in the two nodes at the top of the column. The self-weight of the column was not considered. Figure 6.23 shows the finite element model of Column 1, created in the pre-processor FormWorks: 155 Figure 6.23 Finite element model of Column 1 in FormWorks 1 non s . The material properties of the column were the same ones used for the material 1 in Wall 1 and 2 alone, presented in Table 5.1. 156 Wall 1 and Column 1 were then modeled and analyzed together using program VecTor2. The finite element mesh used for each of them, as well as the boundary conditions, material properties and loads, were the same as the ones described in Section 5.3 and this section. As it was done for the model of Wall 1 and 2 combined together, described in Section 6.2.2, the slabs intercormecting the wall and the column at each floor level were modeled using truss bar elements with very stiff properties. The rectangular elements located at each storey level, which had a height equal to the slab thickness, were also provided with very stiff properties. The same material types were used for these elements. All nodes and elements were numbered in the horizontal (short) direction of the wall system. The complete mesh consisted of 2426 nodes, 2169 rectangular elements, 94 triangular elements and 20 truss elements. A monotonic lateral load was applied at the top of the Wall 1. This load was applied in a displacement-control mode, in increments of 1mm. Additionally, a constant axial load of 15484 kN in Wall 1 and 7742 kN in Column 1 was applied; these loads were equally distributed among the six nodes at the top of the wall and the two nodes at the top of the column. The self-weight of the wall and the column was not considered. Figure 6.24 shows the finite element model of Wall 1 and Column 1, created in the pre-processor FormWorks: 157 Figure 6.24 Finite element model of Wall 1 and Column 1 in FormWorks The detail of the elements representing the slabs at each storey level in the finite element model was the same as the one shown in Figure 6.3. The material properties in the different regions of the wall system were the same ones used for Wall 1 and 2 combined, presented in Table 6.1 and 6.2. 158 6.4.3 Analytical results The results from the analysis were used to determine the curvatures, using the same procedure described in Section 5.4 and 5.6. The ultimate drift for the wall system was 2%. Figure 6.25 shows the curvature distribution up to the mid-height of the wall and the column for this drift, when they are alone and combined: Figure 6.25 Curvatures up to the mid-height of Wall 1 and Column 1 alone and combined for a drift of 2% 28 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 Curvature (rad/m) The same phenomenon observed for the previous wall systems, presented in Section 6.2.3 and 6.3.3, is seen here. The curvature distribution in Wall 1 remains approximately the same, while the curvatures in Column 1 at the base have increased considerably. 6.4.4 Discussion of analytical results The curvature distribution in Figure 6.25 shows that the curvatures in Wall 1 remain almost the same when it is connected with Column 1, and its plastic hinge length has increased slightly. In Column 1, the curvatures have increased considerably at the base. This is the same phenomenon observed for Wall 1 and 2 and Wall 1 and 3 combined together. However, when Column 1 is alone, the curvatures at 2% drift do not reach the curvature at cracking. The column remains uncracked, so no plastic hinge is formed. 159 This is due to the high axial load applied to it. When combined with Wall 1, the curvatures in Column 1 increase, causing it to crack, but they are not large enough to make the column yield and develop a plastic hinge. This can also be observed from the distribution of tensile steel strains in the wall. The average vertical steel strains in the elements along the height of the walls obtained from program VecTor2 were studied for this. The steel strains were observed in the exterior steel layer. Figure 6.26 shows the steel strains up to the mid-height of Column 1, alone and combined, for a drift of 2%: Figure 6.26 Steel strains up to the mid-height of Column 1 alone and combined for a drift of 2% I ' I ": â€”Â£0â€” " M I â€¢ â€¢ 1 : I f 1 â€¢ i 1 ft : * I] I D 1 Â° I 4 1 I k \ I Â£. Q \ 1 O V *\ -0.0010 -0.0005 0.0000 0.0005 0.0010 0.0015 0.0020 Strain The steel strains are of compression when Column 1 is alone, but they are of tension when it is combined. The tensile strains in the combined case do not reach the yield strain. In conclusion, as seen for the previous wall systems in Sections 6.2.4 and 6.3.4, the curvature distributions of Wall 1 and Column 1 when they are connected are different. Also, the curvature demand for the column is much larger when it is combined than when it is alone, and it has to be able to sustain that demand. It is important to consider this effect in the design of gravity columns in high-rise buildings. 160 C H A P T E R 7: C O N C L U S I O N S The plastic hinge length is a function of several parameters. Past research has focused primarily on the influence of member depth, member span and longitudinal reinforcement properties. Some studies have also considered the influence of axial load ratio and strain hardening. Many researchers arrived to the conclusion that a good lower bound approximation for the plastic hinge length is 0.5h or 0.5/w, which is the value given in many concrete codes, including CSA A23.3. Several empirical models have been developed to predict the length of the plastic hinge. These expressions provide very different results because they have been derived from tests for different types of concrete members and consider different parameters. Past research has focused mainly on beams and columns, not walls. Program VecTor2 is a powerful analysis tool that provides good predictions of the response of reinforced concrete members. Analytical predictions of curvatures distributions and strain profiles obtained from VecTor2 have been verified with experimental results from tests performed on wall specimens. The curvature distribution along the height of walls, determined experimentally and analytically, has been investigated. The results indicate that the distribution of inelastic curvatures can be well approximated as linearly varying. The concept of curvature is based on the hypothesis that plane sections remain plane after bending, which is commonly assumed for slender concrete members. The strain profile along the length of walls has also been investigated to verify this assumption. The results indicate that before cracking, the strain profile remains linear. However, after cracking, the shape of the strain profile depends on the amount of reinforcement. As the amount of reinforcement increases, the strain profile is closer to have a linear variation. Also, the hnearity of the strain profile degrades as the wall goes further into the nonlinear range. The effect of the length of the wall in the plastic hinge length is more related to the magnitude of the steel strains than to the effect of diagonal cracking. Longer walls have larger plastic hinge lengths because they have larger steel strains than shorter walls. There is a bigger difference between the yield and ultimate moment for longer walls, which increases the slope of the post-yielding phase of the moment-curvature relationship. 161 For low shear stresses, a pure flexural prediction can be used to estimate the plastic hinge length. When the shear stresses are high, the actual plastic hinge length is longer due to the effect of diagonal cracking. The effect of shear has to be included to estimate the length of the plastic hinge for these cases. A simple shear model has been proposed, which provides reasonable results for high drifts. The length of the plastic hinge reduces with the addition of axial compression, and increases with the addition of axial tension. In a system of two walls of different lengths interconnected by rigid slabs at numerous floor levels, the curvature distributions along the height of the walls are different because of two reasons. First, the displacements between storey levels are different in both walls, specially at the lower levels, so the deflected shapes are not the strictly the same. Second, the shear displacements at the lower levels are larger in the longer wall; resulting in lower flexural displacements, lower slopes, and lower curvatures for this wall. Since the curvatures are not the same, the plastic hinge lengths are also different. When two walls of different lengths are combined together, the curvature demand of the shorter wall is much larger than when it is alone for the same drift level. The curvature distribution in the longer wall remains the same. The shorter wall has to be designed to sustain that larger curvature demand. A simple model to predict the maximum curvature in the shorter wall has been proposed. This effect is of particular importance in the design of gravity columns in high-rise buildings, which may be subjected to tensile strains and also have to sustain larger curvature demands, even i f they do not provide lateral resistance. 162 R E F E R E N C E S ACI-ASCE Committee 428, 1968. "Progress Report on Code Clauses for Limit Design". ACI Journal, 65(9): 713-715. Adebar, P., 2005. "High-Rise Concrete Wall Buildings: Utilizing Unconfined Concrete for Seismic Resistance". CONMAT'05: Third International Conference on Construction Materials: Performance, Innovations and Structural Implications, Vancouver, B.C., 13 pp. Adebar, P., and mrahim, A.M.M., 2002. "Simple Nonlinear Flexural Stiffness Model for Concrete Structural Walls". Earthquake Spectra, 18(3): 407-426. Adebar, P., and mrahim, A.M.M., 2004. "Effective Flexural Stiffness for Linear Seismic Analysis of Concrete Walls". Canadian Journal of Civil Engineering, 31: 597 - 607. Adebar, P., mrahirn, A.M.M., and Bryson, M. , 2004. "Test of a High-Rise Shear Wall: Effective Stiffness for Seismic Analysis". ACI Structural Journal. In press (to be published in 2007). Adebar, P., Mutrie, J., and DeVall, R., 2005. "Ductility of Concrete Walls: The Canadian Seismic Design Provisions 1984 to 2004". Canadian Journal of Civil Engineering, 32: 1124 - 1137. Baker, A.L.L., 1956. "The Ultimate Load Theory Applied to the Design of Reinforced and Prestressed Concrete Frames". Concrete Publications Ltd, London. Baker, A.L.L., and Amarakone, A.M.N., 1964. "Inelastic Hyperstatic Frames Analysis". Proceedings of the International Symposium on the Flexural Mechanics of Reinforced Concrete, ASCE-ACI, Miami: 85 - 142. Bentz, E.C., 2000. "Sectional Analysis of Reinforced Concrete Members". Ph.D. Thesis, Department of Civil Engineering, University of Toronto. Bentz, E.C., 2001. "Response-2000, Shell-2000, Triax-2000, Membrane-2000 User Manual". Department of Civil Engineering, University of Toronto. Bentz, E.C., 2005. "Explaining the Riddle of Tension Stiffening Models for Shear Panel Experiments". Journal of Structural Engineering, ASCE, 131(9): 1422 - 1425. 163 Berry, M.P., and Eberhard, M.O., 2003. "Performance Models for Flexural Damage in Reinforced Concrete Columns". Pacific Earthquake Engineering Research Center Report 2003/18, University of California, Berkeley, California. Berry, M.P., and Eberhard, M.O., 2005. "Practical Performance Models for Bar Buckling". Journal of Structural Engineering, ASCE, 131(7): 1060 - 1070. Bryson, M. , 2000. "Testing of the Lateral Stiffness of a Slender Concrete Shear Wall". Department of Civil Engineering, University of British Columbia, Vancouver, B.C. Cement Association of Canada, 2006. "Concrete Design Handbook: 3 r d. Edition". Ottawa, Ontario. Chan, W.W.L., 1955. "The Ultimate Strength and Deformation of Plastic Hinges in Reinforced Concrete Frameworks". Magazine of Concrete Research, 7(21): 121 - 132. Cohn, M.Z., and Petcu, V.A., 1963. "Moment Redistribution and Rotation Capacity of Plastic Hinges in Redundant Reinforced Concrete Beams". Indian Concrete Journal, 37(8): 282 - 290. Corley, W.G., 1966. "Rotational Capacity of Reinforced Concrete Beams". Journal of the Structural Division, ASCE, 92(ST5): 121 - 146. David, S., 2004. "Plastic Hinge Length in Isolated Walls". Department of Civil Engineering, University of British Columbia, Vancouver, B.C. Duong, K.V., 2006. "Seismic Behaviour of a Shear-Critical Reinforced Concrete Frame: An Experimental and Numerical Investigation". M.A.Sc. Thesis, Department of Civil Engineering, University of Toronto. Federal Emergency Management Agency, 1997. "NEHRP Guidelines for the Seismic Rehabilitation of Buildings, FEMA 273". Washington, D.C. Federal Emergency Management Agency, 1998. "Evaluation of Earthquake Damaged Concrete and Masonry Wall Buildings: Basic Procedures Manual, FEMA 306". Washington, D.C. 164 Ibrahim, A.M.M., 2000. "Linear and Nonlinear Flexural Stiffness Models for Concrete Walls in High-Rise Buildings". Ph.D. Thesis, Department of Civil Engineering, University of British Columbia, Vancouver, B.C. Institution of Civil Engineers, 1964. "Ultimate Load Design of Concrete Structures". Research Report, William Clowes & Sons Ltd, London. Kowalsky, M.J., 2001. "RC Structural Walls Designed According to UBC and Displacement-Based Methods". Journal of Structural Engineering, ASCE, 127(5): 506 - 516. Lehman, D.E., and Moehle, J.P, 2000. "Seismic Performance of Well-Confined Concrete Bridge Columns". Pacific Earthquake Engineering Research Center Report 1998/01, University of California, Berkeley, Cahfornia. Mattock, A.H., 1964. "Rotational Capacity of Hinging Regions in Reinforced Concrete Beams". Proceedings of the International Symposium on the Flexural Mechanics of Reinforced Concrete, ASCE-ACI, Miami: 143 - 181. Mattock, A.H., 1967. "Discussion of 'Rotational Capacity of Reinforced Concrete Beams', by W.G. Corley". Journal of the Structural Division, ASCE, 93(ST2): 519 - 522. Mendis, P., 2001. "Plastic Hinge Lengths of Normal and High-Strength Concrete in Flexure". Advances in Structural Engineering, 4(4): 189 - 195. Moehle, J.P., 1992. "Displacement-Based Design of RC Structures Subjected to Earthquakes". Earthquake Spectra, 8(3): 403 - 428. Montoya, E., Vecchio, F.J., and Sheikh, S.A., 2001. "Compression Field Modeling of Confined Concrete". Structural Engineering and Mechanics, 12(3): 231 - 248. Oesterle, R.G., Aristizabal-Ochoa, J.D., Shiu K.N., and Corley, W.G., 1984. "Web Crushing of Reinforced Concrete Structural Walls". ACI Journal, 81(3): 231 - 241. Panagiotakos, T.B., and Fardis, M.N., 2001. "Deformations of Reinforced Concrete Members at Yielding and Ultimate". ACI Structural Journal, 98(2): 135 - 148. 165 Park, R., and Paulay, T., 1975. "Reinforced Concrete Structures". John Wiley & Sons, New York. Park, R., Priestly, M.J.N., and Gill, W.D., 1982. "Ductility of Square-Confined Concrete Columns". Journal of the Structural Division, ASCE, 108(ST4): 929 - 950. Paulay, T., 1986. "The Design of Ductile Reinforced Concrete Structural Walls for Earthquake Resistance". Earthquake Spectra, 2(4): 783 - 823. Paulay, T., 2001. "Seismic Response of Structural Walls: Recent Developments". Canadian Journal of Civil Engineering, 28: 922 - 937. Paulay, T., and Priestly, M.J.N., 1992. "Seismic Design of Reinforced Concrete and Masonry Buildings". John Wiley & Sons, New York. Paulay, T., and Priestly, M.J.N., 1993. "Stability of Ductile Structural Walls". ACI Structural Journal, 90(4): 385-392. Paulay, T., and Uzumeri, S.M., 1975. "A Critical Review of the Seismic Design Provisions for Ductile Shear Walls of the Canadian Code and Commentary". Canadian Journal of Civil Engineering, 2: 592 -601. Priestly, M.J.N., and Park, R., 1987. "Strength and Ductility of Concrete Bridge Columns Under Seismic Loading". ACI Structural Journal, 84(1): 61 - 76. Priestly, M.J.N., Park, R., and Potangaroa, R.T., 1981. "Ductility and Spirally-Confined Concrete Columns". Journal of the Structural Division, ASCE, 107(ST1): 181-202. Priestly, M.J.N., Seible, F., and Calvi, G.M., 1996. "Seismic Design and Retrofit of Bridges". John Wiley & Sons, New York. Sasani, M. , and Der Kiureghian, A., 2001. "Seismic Fragility of RC Structural Walls: Displacement Approach". Journal of Structural Engineering, ASCE, 127(2): 219 - 228. Sawyer, H.A., 1964. "Design of Concrete Frames for Two Failure Stages". Proceedings of the International Symposium on the Flexural Mechanics of Reinforced Concrete, ASCE-ACI, Miami: 405 -431. 166 Selby, R.G., Vecchio, F.J., and Collins, M.P., 1996. "Analysis of Reinforced Concrete Members Subject to Shear and Axial Compression". ACI Structural Journal, 93(3): 306 - 315. Thomsen, J.H., and Wallace, J.W., 1995. "Displacement-Based Design of Reinforced Concrete Structural Walls: An Experimental Investigation of Walls with Rectangular and T-Shaped Cross-Sections". Report No. CU/CEE-95-06, Department of Civil and Environmental Engineering, Clarkson University, Postdam, New York. Thomsen, J.H., and Wallace, J.W., 2004. "Displacement-Based Design of Slender Reinforced Concrete Structural Walls: Experimental Verification". Journal of Structural Engineering, ASCE, 130(4): 618 â€” 630. Vecchio, F.J., 1989. "Nonlinear Finite Element Analysis of Reinforced Concrete Membranes". ACI Structural Journal, 86(1): 26 - 35. Vecchio, F.J., 2000. "Disturbed Stress Field Model for Reinforced Concrete: Formulation". Journal of Structural Engineering, ASCE, 126(9): 1070 - 1077. Vecchio, F.J., and Collins, M.P., 1986. "The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear". ACI Journal, 83(2): 219 - 231. Vecchio, F.J., and Shim, W., 2004. "Experimental and Analytical Reexamination of Classic Concrete Beam Tests". Journal of Structural Engineering, ASCE, 130(3): 460 - 469. VecTor Analysis Group Website, University of Toronto: www.civ.utoronto.ca/vector/ Wallace, J.W., 1998. "A Designer's Guide to Displacement-Based Design of RC Structural Walls". Department of Civil Engineering, University of California, Los Angeles, California. Wallace, J.W., and Moehle, J.P., 1992. "Ductility and Detailing Requirements of Bearing Wall Buildings". Journal of Structural Engineering, ASCE, 118(6): 1625 - 1644. Wong, P.S., and Vecchio, F.J., 2002. "VecTor2 and FormWorks User's Manual". Department of Civil Engineering, University of Toronto. 167 Zahn, F.A., Park, R., and Priestly, M.J.N., 1986. "Design of Reinforced Concrete Bridge Columns for Strength and Ductility". Research Report 86-7, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand. APPENDIX A: UBC W A L L TEST 169 Table A.1 List of zero readings Target Reading 1 (mm) Reading 2 (mm) Reading 3 (mm) Reading 4 (mm) Reading 5 (mm) Reading 6 (mm) First Second TW1 TW2 288.37 289.53 289.71 289.55 289.57 289.63 TW2 TW3 194.74 194.08 193.95 194.01 193.83 193.84 TW3 TW4 521.09 521.15 521.12 521.04 521.00 521.03 TW4 TW5 521.06 521.08 521.08 521.04 521.04 521.04 TW5 TW6 520.93 520.91 520.77 520.70 520.65 520.61 TW6 TW7 521.51 521.33 521.42 521.28 521.00 521.23 TW7 TW8 520.11 521.20 520.10 520.03 521.30 519.83 TW8 TW9 520.40 520.82 520.42 520.34 520.05 520.28 TW9 TW10 520.62 520.52 520.56 520.49 520.48 520.44 TW10 TW11 520.52 520.50 520.44 521.74 521.00 521.20 TW11 TW12 520.59 520.40 520.42 520.78 521.00 521.82 TE1 TE2 291.37 294.36 294.78 294.73 294.80 294.82 TE2 TE3 191.04 191.61 191.24 191.98 190.92 191.12 TE3 TE4 521.10 521.10 521.28 521.30 521.09 521.00 TE4 TE5 521.06 520.89 521.07 520.90 520.95 520.89 TE5 TE6 521.25 521.07 521.26 521.02 521.04 520.97 TE6 TE7 521.06 521.12 521.08 521.06 521.09 521.03 TE7 TE8 521.03 521.46 521.16 520.99 520.95 521.08 TE8 TE9 521.06 521.07 521.34 520.98 521.05 521.26 TE9 TE10 521.33 521.16 521.14 521.02 521.11 521.16 TE10 TE11 521.16 521.24 521.13 521.05 521.10 521.22 TE11 TE12 521.33 521.28 521.15 521.25 521.41 521.07 Marked cells show measurements out of range, these were neglected. Table A.2 Mean values of zero readings Target Reading 1 (mm) Reading 2 (mm) Reading 3 (mm) Reading 4 (mm) Reading 5 (mm) Reading 6 (mm) Mean value (mm) First Second TW1 TW2 289.53 289.71 289.55 289.57 289.63 289.60 TW2 TW3 194.08 193.95 194.01 193.83 193.84 193.94 TW3 TW4 521.09 521.15 521.12 521.04 521.00 521.03 521.07 TW4 TW5 521.06 521.08 521.08 521.04 521.04 521.04 521.06 TW5 TW6 520.93 520.91 520.77 520.70 520.65 520.61 520.76 TW6 TW7 521.51 521.33 521.42 521.28 521.23 521.35 TW7 TW8 520.11 520.10 520.03 519.83 520.02 TW8 TW9 520.40 520.42 520.34 520.28 520.36 TW9 TW10 520.62 520.52 520.56 520.49 520.48 520.44 520.52 TW10 TW11 520.52 520.50 520.44 521.00 521.20 520.73 TW11 TW12 520.59 520.40 520.42 520.78 521.00 520.64 TE1 TE2 294.36 294.78 294.73 294.80 294.82 294.70 TE2 TE3 191.04 191.61 191.24 190.92 191.12 191.19 TE3 TE4 521.10 521.10 521.28 521.30 521.09 521.00 521.15 TE4 TE5 521.06 520.89 521.07 520.90 520.95 520.89 520.96 TE5 TE6 521.25 521.07 521.26 521.02 521.04 520.97 521.10 TE6 TE7 521.06 521.12 521.08 521.06 521.09 521.03 521.07 TE7 TE8 521.03 521.16 520.99 520.95 521.08 521.04 TE8 TE9 521.06 521.07 521.34 520.98 521.05 521.26 521.13 TE9 TE10 521.33 521.16 521.14 521.02 521.11 521.16 521.15 TE10 TE11 521.16 521.24 521.13 521.05 521.10 521.22 521.15 TE11 TE12 521.33 521.28 521.15 521.25 521.41 521.07 521.25 Table A.3 Experimental (pushing east) curvatures for a wall displacement of 105mm Target Reading (mm) Zero reading (mm) Cumulative (mm) Height (mm) Strain Curvature (rad/m) First Second TW1 TW2 290.24 289.60 289.60 146.07 0.0022 0.00228 TW2 TW3 196.22 193.94 483.54 388.43 0.0117 0.00821 TW3 TW4 524.50 521.07 1004.61 745.27 0.0066 0.00476 TW4 TW5 523.21 521.06 1525.67 1266.32 0.0041 0.00305 TW5 TW6 521.96 520.76 2046.43 1787.29 0.0023 0.00190 TW6 TW7 521.92 521.35 2567.78 2308.37 0.0011 0.00106 TW7 TW8 520.78 520.02 3087.80 2829.24 0.0015 0.00129 TW8 TW9 520.52 520.36 3608.16 3349.88 0.0003 0.00029 TW9 TW10 520.75 520.52 4128.68 3870.67 0.0004 0.00038 TW10 TW11 521.16 520.73 4649.41 4391.55 0.0008 0.00069 TW11 TW12 520.97 520.64 5170.05 4912.50 0.0006 0.00072 Top of wall - - - 11760.00 0.0000 0.00000 TE1 TE2 294.26 294.70 294.70 146.07 -0.0015 0.00228 TE2 TE3 190.88 191.19 485.88 388.43 -0.0016 0.00821 TE3 TE4 520.54 521.15 1007.03 745.27 -0.0012 0.00476 TE4 TE5 520.53 520.96 1527.99 1266.32 -0.0008 0.00305 TE5 TE6 520.69 521.10 2049.09 1787.29 -0.0008 0.00190 TE6 TE7 520.74 521.07 2570.16 2308.37 -0.0006 0.00106 TE7 TE8 520.71 521.04 3091.21 2829.24 -0.0006 0.00129 TE8 TE9 521.04 521.13 3612.33 3349.88 -0.0002 0.00029 TE9 TE10 521.06 521.15 4133.49 3870.67 -0.0002 0.00038 TE10 TE11 520.99 521.15 4654.64 4391.55 -0.0003 0.00069 TE11 TE12 520.97 521.25 5175.88 4912.50 -0.0005 0.00072 Top of wall - - - 11760.00 0.0000 0.00000 Wall length: 1625 mm Table A.4 Experimental (pushing west) curvatures for a wall displacement of 104mm Target Reading Zero reading (mm) Cumulative (mm) Height (mm) Strain Curvature (rad/m) First Second (mm) TW1 TW2 288.76 289.60 289.60 146.07 -0.0029 0.00433 TW2 TW3 193.67 193.94 483.54 388.43 -0.0014 0.01050 TW3 TW4 520.59 521.07 1004.61 745.27 -0.0009 0.00364 TW4 TW5 520.65 521.06 1525.67 1266.32 -0.0008 0.00259 TW5 TW6 520.38 520.76 2046.43 1787.29 -0.0007 0.00149 TW6 TW7 521.02 521.35 2567.78 2308.37 -0.0006 0.00052 TW7 TW8 519.50 520.02 3087.80 2829.24 -0.0010 0.00155 TW8 TW9 520.11 520.36 3608.16 3349.88 -0.0005 0.00039 TW9 TW10 520.28 520.52 4128.68 3870.67 -0.0005 0.00031 TW10 TW11 520.87 520.73 4649.41 4391.55 0.0003 0.00010 TW11 TW12 520.58 520.64 5170.05 4912.50 -0.0001 0.00006 Tope )f wall - - - 11760.00 0.0000 0.00000 TE1 TE2 295.92 294.70 294.70 146.07 0.0041 0.00433 TE2 TE3 194.18 191.19 485.88 388.43 0.0157 0.01050 TE3 TE4 523.75 521.15 1007.03 745.27 0.0050 0.00364 TE4 TE5 522.75 520.96 1527.99 1266.32 0.0034 0.00259 TE5 TE6 521.98 521.10 2049.09 1787.29 0.0017 0.00149 TE6 TE7 521.18 521.07 2570.16 2308.37 0.0002 0.00052 TE7 TE8 521.84 521.04 3091.21 2829.24 0.0015 0.00155 TE8 TE9 521.21 521.13 3612.33 3349.88 0.0002 0.00039 TE9 TE10 521.18 521.15 4133.49 3870.67 0.0001 0.00031 TE10 TE11 521.20 521.15 4654.64 4391.55 0.0001 0.00010 TE11 TE12 521.24 521.25 5175.88 4912.50 0.0000 0.00006 Top of wall - - - 11760.00 0.0000 0.00000 Wall length: 1625 mm Table A.5 Experimental (pushing east) curvatures for a wall displacement of 132mm Target Reading (mm) Zero reading (mm) Cumulative (mm) Height (mm) Strain Curvature (rad/m) First Second TW1 TW2 288.10 289.60 289.60 146.07 -0.0052 0.00214 TW2 TW3 196.79 193.94 483.54 388.43 0.0147 0.01028 TW3 TW4 526.00 521.07 1004.61 745.27 0.0095 0.00659 TW4 TW5 524.08 521.06 1525.67 1266.32 0.0058 0.00413 TW5 TW6 522.34 520.76 2046.43 1787.29 0.0030 0.00235 TW6 TW7 522.00 521.35 2567.78 2308.37 0.0012 0.00114 TW7 TW8 520.86 520.02 3087.80 2829.24 0.0016 0.00138 TW8 TW9 520.45 520.36 3608.16 3349.88 0.0002 0.00048 TW9 TW10 520.84 520.52 4128.68 3870.67 0.0006 0.00073 TW10 TW11 521.14 520.73 4649.41 4391.55 0.0008 0.00077 TW11 TW12 520.86 520.64 5170.05 4912.50 0.0004 0.00060 Top c )f wall - - - 11760.00 0.0000 0.00000 TE1 TE2 294.20 294.70 294.70 146.07 -0.0017 0.00214 TE2 TE3 190.80 191.19 485.88 388.43 -0.0020 0.01028 TE3 TE4 520.49 521.15 1007.03 745.27 -0.0013 0.00659 TE4 TE5 520.49 520.96 1527.99 1266.32 -0.0009 0.00413 TE5 TE6 520.69 521.10 2049.09 1787.29 -0.0008 0.00235 TE6 TE7 520.75 521.07 2570.16 2308.37 -0.0006 0.00114 TE7 TE8 520.72 521.04 3091.21 2829.24 -0.0006 0.00138 TE8 TE9 520.81 521.13 3612.33 3349.88 -0.0006 0.00048 TE9 TE10 520.86 521.15 4133.49 3870.67 -0.0006 0.00073 TE10 TE11 520.91 521.15 4654.64 4391.55 -0.0005 0.00077 TE11 TE12 520.96 521.25 5175.88 4912.50 -0.0006 0.00060 Top of wall - - - 11760.00 0.0000 0.00000 Wall length: 1625 mm Table A.6 Experimental (pushing west) curvatures for a wall displacement of 138mm Target Reading (mm) Zero reading (mm) Cumulative (mm) Height (mm) Strain Curvature (rad/m) First Second TW1 TW2 288.95 289.60 289.60 146.07 -0.0022 0.00307 TW2 TW3 193.70 193.94 483.54 388.43 -0.0012 0.01472 TW3 TW4 520.47 521.07 1004.61 745.27 -0.0012 0.00557 TW4 TW5 520.74 521.06 1525.67 1266.32 -0.0006 0.00375 TW5 TW6 520.35 520.76 2046.43 1787.29 -0.0008 0.00191 TW6 TW7 520.98 521.35 2567.78 2308.37 -0.0007 0.00048 TW7 TW8 519.61 520.02 3087.80 2829.24 -0.0008 0.00172 TW8 TW9 520.12 520.36 3608.16 3349.88 -0.0005 0.00055 TW9 TW10 520.44 520.52 4128.68 3870.67 -0.0002 0.00012 TW10 TW11 521.11 520.73 4649.41 4391.55 0.0007 0.00041 TW11 TW12 520.65 520.64 5170.05 4912.50 0.0000 0.00013 Top c )f wall - - - 11760.00 0.0000 0.00000 TE1 TE2 295.51 294.70 294.70 146.07 0.0028 0.00307 TE2 TE3 195.52 191.19 485.88 388.43 0.0227 0.01472 TE3 TE4 525.26 521.15 1007.03 745.27 0.0079 0.00557 TE4 TE5 523.82 520.96 1527.99 1266.32 0.0055 0.00375 TE5 TE6 522.31 521.10 2049.09 1787.29 0.0023 0.00191 TE6 TE7 521.11 521.07 2570.16 2308.37 0.0001 0.00048 TE7 TE8 522.09 521.04 3091.21 2829.24 0.0020 0.00172 TE8 TE9 521.35 521.13 3612.33 3349.88 0.0004 0.00055 TE9 TE10 521.18 521.15 4133.49 3870.67 0.0001 0.00012 TE10 TE11 521.18 521.15 4654.64 4391.55 0.0001 0.00041 TE11 TE12 521.15 521.25 5175.88 4912.50 -0.0002 0.00013 Top of wall - - - 11760.00 0.0000 0.00000 Wall length: 1625 mm Table A.7 Experimental (pushing east) curvatures for a wall displacement of 182mm Target Reading Zero reading (mm) Cumulative (mm) Height (mm) Strain Curvature (rad/m) First Second (mm) TW1 TW2 289.65 289.60 289.60 146.07 0.0002 0.00151 TW2 TW3 198.00 193.94 483.54 388.43 0.0209 0.01402 TW3 TW4 528.79 521.07 1004.61 745.27 0.0148 0.01021 TW4 TW5 525.59 521.06 1525.67 1266.32 0.0087 0.00600 TW5 TW6 523.16 520.76 2046.43 1787.29 0.0046 0.00340 TW6 TW7 522.12 521.35 2567.78 2308.37 0.0015 0.00132 TW7 TW8 520.98 520.02 3087.80 2829.24 0.0019 0.00157 TW8 TW9 520.45 520.36 3608.16 3349.88 0.0002 0.00053 TW9 TW10 521.09 520.52 4128.68 3870.67 0.0011 0.00108 TW10 TW11 521.15 520.73 4649.41 4391.55 0.0008 0.00080 TW11 TW12 520.87 520.64 5170.05 4912.50 0.0004 0.00071 Tope )f wall - - - 11760.00 0.0000 0.00000 TEl TE2 294.03 294.70 294.70 146.07 -0.0023 0.00151 TE2 TE3 190.83 191.19 485.88 388.43 -0.0019 0.01402 TE3 TE4 520.22 521.15 1007.03 745.27 -0.0018 0.01021 TE4 TE5 520.41 520.96 1527.99 1266.32 -0.0011 0.00600 TE5 TE6 520.62 521.10 2049.09 1787.29 -0.0009 0.00340 TE6 TE7 520.72 521.07 2570.16 2308.37 -0.0007 0.00132 TE7 TE8 520.68 521.04 3091.21 2829.24 -0.0007 0.00157 TE8 TE9 520.77 521.13 3612.33 3349.88 -0.0007 0.00053 TE9 TE10 520.81 521.15 4133.49 3870.67 -0.0007 0.00108 TE10 TE11 520.89 521.15 4654.64 4391.55 -0.0005 0.00080 TE11 TE12 520.88 521.25 5175.88 4912.50 -0.0007 0.00071 Top of wall - - - 11760.00 0.0000 1 0.00000 Wall length: 1625 mm Table A.8 Experimental (pushing west) curvatures for a wall displacement of 187mm Target Reading (mm) Zero reading (mm) Cumulative (mm) Height (mm) Strain Curvature (rad/m) First Second TW1 TW2 288.86 289.60 289.60 146.07 -0.0025 0.00341 TW2 TW3 193.53 193.94 483.54 388.43 -0.0021 0.02253 TW3 TW4 520.33 521.07 1004.61 745.27 -0.0014 0.00808 TW4 TW5 520.52 521.06 1525.67 1266.32 -0.0010 0.00577 TW5 TW6 520.29 520.76 2046.43 1787.29 -0.0009 0.00278 TW6 TW7 520.95 521.35 2567.78 2308.37 -0.0008 0.00125 TW7 TW8 519.67 520.02 3087.80 2829.24 -0.0007 0.00159 TW8 TW9 520.07 520.36 3608.16 3349.88 -0.0006 0.00061 TW9 TW10 520.28 520.52 4128.68 3870.67 -0.0005 0.00034 TW10 TW11 520.86 520.73 4649.41 4391.55 0.0002 0.00008 TW11 TW12 520.96 520.64 5170.05 4912.50 0.0006 0.00034 Top of wall - - - 11760.00 0.0000 0.00000 TE1 TE2 295.58 294.70 294.70 146.07 0.0030 0.00341 TE2 TE3 197.78 191.19 485.88 388.43 0.0345 0.02253 TE3 TE4 527.25 521.15 1007.03 745.27 0.0117 0.00808 TE4 TE5 525.31 520.96 1527.99 1266.32 0.0083 0.00577 TE5 TE6 522.98 521.10 2049.09 1787.29 0.0036 0.00278 TE6 TE7 521.73 521.07 2570.16 2308.37 0.0013 0.00125 TE7 TE8 522.04 521.04 3091.21 2829.24 0.0019 0.00159 TE8 TE9 521.35 521.13 3612.33 3349.88 0.0004 0.00061 TE9 TE10 521.20 521.15 4133.49 3870.67 0.0001 0.00034 TE10 TE11 521.21 521.15 4654.64 4391.55 0.0001 0.00008 TE11 TE12 521.28 521.25 5175.88 4912.50 0.0001 0.00034 Top of wall - - - 11760.00 0.0000 0.00000 Wall length: 1625 mm Table A.9 Analytical curvatures for a wall displacement of 105mm Node Node 1 y- Node 2 y- Node 2 y-coord (mm) Difference Height (mm) Strain Curvature First Second displ (mm) displ (mm) (mm) (rad/m) 1 22 0.00 0.78 58.71 58.71 29.36 0.0134 0.00985 22 148 0.78 6.87 579.63 520.92 319.17 0.0117 0.00822 148 274 6.87 9.51 1100.55 520.92 840.09 0.0051 0.00375 274 400 9.51 10.27 1621.47 520.92 1361.01 0.0015 0.00139 400 526 10.27 10.90 2142.39 520.92 1881.93 0.0012 0.00118 526 652 10.90 11.35 2663.31 520.92 2402.85 0.0009 0.00091 652 778 11.35 11.62 3184.23 520.92 2923.77 0.0005 0.00066 778 904 11.62 11.75 3705.15 520.92 3444.69 0.0002 0.00046 904 1030 11.75 11.86 4226.07 520.92 3965.61 0.0002 0.00041 1030 1156 11.86 11.91 4746.99 520.92 4486.53 0.0001 0.00033 Top of wall - - - _ 11330.00 0.0000 0.00000 21 42 0.00 -0.16 58.71 58.71 29.36 -0.0027 0.00985 42 168 -0.16 -1.03 579.63 520.92 319.17 -0.0017 0.00822 168 294 -1.03 -1.57 1100.55 520.92 840.09 -0.0010 0.00375 294 420 -1.57 -1.98 1621.47 520.92 1361.01 -0.0008 0.00139 420 546 -1.98 -2.34 2142.39 520.92 1881.93 -0.0007 0.00118 546 672 -2.34 -2.67 2663.31 520.92 2402.85 -0.0006 0.00091 672 798 -2.67 -2.96 3184.23 520.92 2923.77 -0.0006 0.00066 798 924 -2.96 -3.22 3705.15 520.92 3444.69 -0.0005 0.00046 924 1050 -3.22 -3.46 4226.07 520.92 3965.61 -0.0005 0.00041 1050 1176 -3.46 -3.68 4746.99 520.92 4486.53 -0.0004 0.00033 Top of wall - - - - 11330.00 0.0000 0.00000 Wall length: 1625 mm Table A.10 Analytical curvatures for a wall displacement of 104mm Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 1 22 0.00 0.78 58.71 58.71 29.36 0.0133 0.00981 22 148 0.78 6.82 579.63 520.92 319.17 0.0116 0.00816 148 274 6.82 9.37 1100.55 520.92 840.09 0.0049 0.00365 274 400 9.37 10.13 1621.47 520.92 1361.01 0.0015 0.00139 400 526 10.13 10.77 2142.39 520.92 1881.93 0.0012 0.00118 526 652 10.77 11.21 2663.31 520.92 2402.85 0.0009 0.00091 652 778 11.21 11.49 3184.23 520.92 2923.77 0.0005 0.00067 778 904 11.49 11.61 3705.15 520.92 3444.69 0.0002 0.00046 904 1030 11.61 11.72 4226.07 520.92 3965.61 0.0002 0.00041 1030 1156 11.72 11.77 4746.99 520.92 4486.53 0.0001 0.00033 Top of wall - - - - 11330.00 0.0000 0.00000 21 42 0.00 -0.16 58.71 58.71 29.36 -0.0026 0.00981 42 168 -0.16 -1.02 579.63 520.92 319.17 -0.0017 0.00816 168 294 -1.02 -1.56 1100.55 520.92 840.09 -0.0010 0.00365 294 420 -1.56 -1.97 1621.47 520.92 1361.01 -0.0008 0.00139 420 546 -1.97 -2.33 2142.39 520.92 1881.93 -0.0007 0.00118 546 672 -2.33 -2.66 2663.31 520.92 2402.85 -0.0006 0.00091 672 798 -2.66 -2.95 3184.23 520.92 2923.77 -0.0006 0.00067 798 924 -2.95 -3.21 3705.15 520.92 3444.69 -0.0005 0.00046 924 1050 -3.21 -3.45 4226.07 520.92 3965.61 -0.0005 0.00041 1050 1176 -3.45 -3.67 4746.99 520.92 4486.53 -0.0004 0.00033 Top of wall - - - - 11330.00 0.0000 0.00000 Wall length: 1625 mm Table A.11 Analytical curvatures for a wall displacement of 132mm Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 1 22 0.00 0.93 58.71 58.71 29.36 0.0158 0.01171 22 148 0.93 8.10 579.63 520.92 319.17 0.0138 0.00967 148 274 8.10 12.41 1100.55 520.92 840.09 0.0083 0.00580 274 400 12.41 13.80 1621.47 520.92 1361.01 0.0027 0.00217 400 526 13.80 14.47 2142.39 520.92 1881.93 0.0013 0.00123 526 652 14.47 15.01 2663.31 520.92 2402.85 0.0010 0.00103 652 778 15.01 15.28 3184.23 520.92 2923.77 0.0005 0.00067 778 904 15.28 15.47 3705.15 520.92 3444.69 0.0004 0.00054 904 1030 15.47 15.58 4226.07 520.92 3965.61 0.0002 0.00042 1030 1156 15.58 15.63 4746.99 520.92 4486.53 0.0001 0.00033 Top of wall - - - - 11330.00 0.0000 0.00000 21 42 0.00 -0.19 58.71 58.71 29.36 -0.0032 0.01171 42 168 -0.19 -1.20 579.63 520.92 319.17 -0.0020 0.00967 168 294 -1.20 -1.81 1100.55 520.92 840.09 -0.0012 0.00580 294 420 -1.81 -2.25 1621.47 520.92 1361.01 -0.0009 0.00217 420 546 -2.25 -2.63 2142.39 520.92 1881.93 -0.0007 0.00123 546 672 -2.63 -2.96 2663.31 520.92 2402.85 -0.0006 0.00103 672 798 -2.96 -3.26 3184.23 520.92 2923.77 -0.0006 0.00067 798 924 -3.26 -3.53 3705.15 520.92 3444.69 -0.0005 0.00054 924 1050 -3.53 -3.77 4226.07 520.92 3965.61 -0.0005 0.00042 1050 1176 -3.77 -4.00 4746.99 520.92 4486.53 -0.0004 0.00033 Top of wall - - - - 11330.00 0.0000 0.00000 Wall length: 1625 mm Table A.12 Analytical curvatures for a wall displacement of 138mm Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 1 22 0.00 0.97 58.71 58.71 29.36 0.0165 0.01221 22 148 0.97 8.56 579.63 520.92 319.17 0.0146 0.01021 148 274 8.56 13.17 1100.55 520.92 840.09 0.0089 0.00617 274 400 13.17 14.62 1621.47 520.92 1361.01 0.0028 0.00225 400 526 14.62 15.29 2142.39 520.92 1881.93 0.0013 0.00124 526 652 15.29 15.84 2663.31 520.92 2402.85 0.0010 0.00104 652 778 15.84 16.11 3184.23 520.92 2923.77 0.0005 0.00067 778 904 16.11 16.29 3705.15 520.92 3444.69 0.0004 0.00054 904 1030 16.29 16.41 4226.07 520.92 3965.61 0.0002 0.00042 1030 1156 16.41 16.46 4746.99 520.92 4486.53 0.0001 0.00033 Top of wall - - - - 11330.00 0.0000 0.00000 21 42 0.00 -0.20 58.71 58.71 29.36 -0.0034 0.01221 42 168 -0.20 -1.25 579.63 520.92 319.17 -0.0020 0.01021 168 294 -1.25 -1.86 1100.55 520.92 840.09 -0.0012 0.00617 294 420 -1.86 -2.30 1621.47 520.92 1361.01 -0.0009 0.00225 420 546 -2.30 -2.68 2142.39 520.92 1881.93 -0.0007 0.00124 546 672 -2.68 -3.02 2663.31 520.92 2402.85 -0.0006 0.00104 672 798 -3.02 -3.32 3184.23 520.92 2923.77 -0.0006 0.00067 798 924 -3.32 -3.58 3705.15 520.92 3444.69 -0.0005 0.00054 924 1050 -3.58 -3.83 4226.07 520.92 3965.61 -0.0005 0.00042 1050 1176 -3.83 -4.06 4746.99 520.92 4486.53 -0.0004 0.00033 Top of wall - - - - 11330.00 0.0000 0.00000 Wall length: 1625 mm Table A.13 Analytical curvatures for a wall displacement of 182mm Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 1 22 0.00 1.14 58.71 58.71 29.36 0.0193 0.01458 22 148 1.14 10.31 579.63 520.92 319.17 0.0176 0.01235 148 274 10.31 17.78 1100.55 520.92 840.09 0.0143 0.00965 274 400 17.78 20.51 1621.47 520.92 1361.01 0.0052 0.00380 400 526 20.51 21.23 2142.39 520.92 1881.93 0.0014 0.00132 526 652 21.23 21.82 2663.31 520.92 2402.85 0.0011 0.00111 652 778 21.82 22.18 3184.23 520.92 2923.77 0.0007 0.00078 778 904 22.18 22.37 3705.15 520.92 3444.69 0.0004 0.00056 904 1030 22.37 22.49 4226.07 520.92 3965.61 0.0002 0.00044 1030 1156 22.49 22.55 4746.99 520.92 4486.53 0.0001 0.00034 Top of wall - - - - 11330.00 0.0000 0.00000 21 42 0.00 -0.26 58.71 58.71 29.36 -0.0044 0.01458 42 168 -0.26 -1.54 579.63 520.92 319.17 -0.0025 0.01235 168 294 -1.54 -2.24 1100.55 520.92 840.09 -0.0013 0.00965 294 420 -2.24 -2.72 1621.47 520.92 1361.01 -0.0009 0.00380 420 546 -2.72 -3.12 2142.39 520.92 1881.93 -0.0008 0.00132 546 672 -3.12 -3.46 2663.31 520.92 2402.85 -0.0007 0.00111 672 798 -3.46 -3.77 3184.23 520.92 2923.77 -0.0006 0.00078 798 924 -3.77 -4.05 3705.15 520.92 3444.69 -0.0005 0.00056 924 1050 -4.05 -4.30 4226.07 520.92 3965.61 -0.0005 0.00044 1050 1176 -4.30 -4.53 4746.99 520.92 4486.53 -0.0004 0.00034 Top of wall - - - - 11330.00 0.0000 0.00000 Wall length: 1625 mm Table A.14 Analytical curvatures for a wall displacement of 187mm Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 1 22 0.00 1.16 58.71 58.71 29.36 0.0197 0.01487 22 148 1.16 10.52 579.63 520.92 319.17 0.0180 0.01260 148 274 10.52 18.15 1100.55 520.92 840.09 0.0147 0.00986 274 400 18.15 21.17 1621.47 520.92 1361.01 0.0058 0.00414 400 526 21.17 21.91 2142.39 520.92 1881.93 0.0014 0.00134 526 652 21.91 22.50 2663.31 520.92 2402.85 0.0011 0.00112 652 778 22.50 22.86 3184.23 520.92 2923.77 0.0007 0.00079 778 904 22.86 23.06 3705.15 520.92 3444.69 0.0004 0.00056 904 1030 23.06 23.18 4226.07 520.92 3965.61 0.0002 0.00044 1030 1156 23.18 23.24 4746.99 520.92 4486.53 0.0001 0.00035 Top of wall - - - - 11330.00 0.0000 0.00000 21 42 0.00 -0.26 58.71 58.71 29.36 -0.0045 0.01487 42 168 -0.26 -1.57 579.63 520.92 319.17 -0.0025 0.01260 168 294 -1.57 -2.28 1100.55 520.92 840.09 -0.0014 0.00986 294 420 -2.28 -2.77 1621.47 520.92 1361.01 -0.0009 0.00414 420 546 -2.77 -3.17 2142.39 520.92 1881.93 -0.0008 0.00134 546 672 -3.17 -3.51 2663.31 520.92 2402.85 -0.0007 0.00112 672 798 -3.51 -3.82 3184.23 520.92 2923.77 -0.0006 0.00079 798 924 -3.82 -4.10 3705.15 520.92 3444.69 -0.0005 0.00056 924 1050 -4.10 -4.35 4226.07 520.92 3965.61 -0.0005 0.00044 1050 1176 -4.35 -4.58 4746.99 520.92 4486.53 -0.0004 0.00035 Top of wall - - - - 11330.00 0.0000 0.00000 Wall length: 1625 mm 183 to â€¢ X Table A.15 Plane sections analysis to determine ultimate curvature C T tw -> y to fa -*â€¢ >â€¢ Concrete properties Wall dimensions / c ( M P a ) 49.0 If! (mm) 203.0 0.0035 tfl (mm) 380.0 1.00 1/2 (mm) 203.0 Reinforcement properties tf2 (mm) 380.0 / w (mm) 1219.0 / y ( M P a ) 455.0 tw (mm) 127.0 Es (MPa) 200000.0 Axial load 1.00 P ( k N ) -1500.0 Stress block factors Yield strain Section centroid Forces in concrete 0.777 0.0023 x (mm) 812.50 c (mm) 148.23 Pi 0.848 a (mm) 125.62 -1816.31 Mc (kN.m) 1361.67 Forces in reinforcement Layer x (mm) A, (mm2) e, / f ( M P a ) P, (kN) V (mm) Ms (kN.m) 1 21.00 300.0 -0.0030 -455.00 -125.09 -791.50 99.01 2 182.00 200.0 0.0008 159.49 31.90 -630.50 -20.11 3 355.00 100.0 0.0049 455.00 45.50 -457.50 -20.82 4 660.00 100.0 0.0121 455.00 45.50 -152.50 -6.94 5 965.00 100.0 0.0193 455.00 45.50 152.50 6.94 6 1270.00 100.0 0.0265 455.00 45.50 457.50 20.82 7 1443.00 200.0 0.0306 455.00 91.00 630.50 57.38 8 1604.00 300.0 0.0344 455.00 136.50 791.50 108.04 Total 316.31 Total 244.31 Results Prim -1500.00 M r (kN.m) 1605.98 <l>u (rad/m) 0.0236 APPENDIX B: CLARKSON UNIVERSITY W A L L TEST Table B.l Analytical strain profile at base for a drift of 1.5% Node Node 1 y- Node 2 y-displ (mm) Distance along wall length (mm) Distance between nodes (mm) Strain (mm/m) First Second displ (mm) 1 27 0.00 5.71 0.00 228.60 24.96 3 29 0.00 5.07 177.80 228.60 22.20 5 31 0.00 4.92 381.00 228.60 21.52 7 33 0.00 3.26 609.60 228.60 14.27 9 35 0.00 1.59 838.20 228.60 6.93 11 37 0.00 0.27 1041.40 228.60 1.18 13 39 0.00 -1.48 1219.20 228.60 -6.47 Table B.2 Analytical strain profile at base for a drift of 2% Node Node 1 y-displ (mm) Node 2 y-displ (mm) Distance along wall length (mm) Distance between nodes (mm) Strain (mm/m) First Second 1 27 0.00 6.56 0.00 228.60 28.67 3 29 0.00 5.60 177.80 228.60 24.49 5 31 0.00 5.14 381.00 228.60 22.48 7 33 0.00 3.64 609.60 228.60 15.93 9 35 0.00 2.39 838.20 228.60 10.45 11 37 0.00 0.39 1041.40 228.60 1.69 13 39 0.00 -2.15 1219.20 228.60 -9.41 Table B.3 Analytical strain profile accounting for cover spalling at base for a drift of 1.5% Node Node 1 y-displ (mm) Node 2 y-displ (mm) Distance along wall length (mm) Distance between nodes (mm) Strain (mm/m) First Second 1 27 0.00 5.49 0.00 228.60 24.02 3 29 0.00 4.52 177.80 228.60 19.77 5 31 0.00 3.18 381.00 228.60 13.89 7 33 0.00 3.03 609.60 228.60 13.23 9 35 0.00 1.48 838.20 228.60 6.48 11 37 0.00 -0.23 1041.40 228.60 -0.99 13 39 0.00 -1.76 1219.20 228.60 -7.69 Table B.4 Analytical strain profile accounting for cover spalling at base for a drift of 2% Node Node 1 y-displ (mm) Node 2 y-displ (mm) Distance along wall length (mm) Distance between nodes (mm) Strain (mm/m) First Second 1 27 0.00 6.40 0.00 228.60 27.98 3 29 0.00 5.18 177.80 228.60 22.66 5 31 0.00 3.41 381.00 228.60 14.90 7 33 0.00 3.47 609.60 228.60 15.18 9 35 0.00 2.26 838.20 228.60 9.88 11 37 0.00 -0.35 1041.40 228.60 -1.54 13 39 0.00 -2.33 1219.20 228.60 -10.20 APPENDIX C: CALCULATIONS FOR PARAMETRIC STUDY Table C.l Curvatures up to the mid-height of Wall 1 for a drift of 2% Node Node 1 y- Node 2 y- Node 2 y- Difference Height (mm) Strain Curvature (rad/m) First Second displ (mm) displ (mm) coord (mm) (mm) 1 53 0.00 15.39 846.60 846.60 423.30 0.0182 0.00326 53 105 15.39 30.70 1693.20 846.60 1269.90 0.0181 0.00320 105 183 30.70 49.08 2743.00 1049.80 2218.10 0.0175 0.00271 183 235 49.08 62.44 3589.60 846.60 3166.30 0.0158 0.00237 235 287 62.44 74.59 4436.20 846.60 4012.90 0.0143 0.00214 287 365 74.59 87.95 5486.00 1049.80 4961.10 0.0127 0.00190 365 417 87.95 94.19 6332.60 846.60 5909.30 0.0074 0.00117 417 469 94.19 97.11 7179.20 846.60 6755.90 0.0035 0.00064 469 547 97.11 99.87 8229.00 1049.80 7704.10 0.0026 0.00052 547 599 99.87 101.58 9075.60 846.60 8652.30 0.0020 0.00042 599 651 101.58 103.17 9922.20 846.60 9498.90 0.0019 0.00040 651 729 103.17 105.05 10972.00 1049.80 10447.10 0.0018 0.00038 729 781 105.05 106.53 11818.60 846.60 11395.30 0.0018 0.00037 781 833 106.53 107.97 12665.20 846.60 12241.90 0.0017 0.00036 833 911 107.97 109.70 13715.00 1049.80 13190.10 0.0016 0.00034 911 963 109.70 111.03 14561.60 846.60 14138.30 0.0016 0.00033 963 1015 111.03 112.30 15408.20 846.60 14984.90 0.0015 0.00032 1015 1093 112.30 113.79 16458.00 1049.80 15933.10 0.0014 0.00030 1093 1145 113.79 114.92 17304.60 846.60 16881.30 0.0013 0.00029 1145 1197 114.92 115.99 18151.20 846.60 17727.90 0.0013 0.00027 1197 1275 115.99 117.22 19201.00 1049.80 18676.10 0.0012 0.00026 1275 1327 117.22 118.14 20047.60 846.60 19624.30 0.0011 0.00024 1327 1379 118.14 118.99 20894.20 846.60 20470.90 0.0010 0.00023 1379 1457 118.99 119.96 21944.00 1049.80 21419.10 0.0009 0.00022 1457 1509 119.96 120.67 22790.60 846.60 22367.30 0.0008 0.00020 1509 1561 120.67 121.31 23637.20 846.60 23213.90 0.0008 0.00019 1561 1639 121.31 122.03 24687.00 1049.80 24162.10 0.0007 0.00017 1639 1691 122.03 122.55 25533.60 846.60 25110.30 0.0006 0.00016 1691 1743 122.55 123.01 26380.20 846.60 25956.90 0.0005 0.00015 1743 1821 123.01 123.52 27430.00 1049.80 26905.10 0.0005 0.00014 26 78 0.00 -5.66 846.60 846.60 423.30 -0.0067 0.00326 78 130 -5.66 -10.98 1693.20 846.60 1269.90 -0.0063 0.00320 130 208 -10.98 -14.31 2743.00 1049.80 2218.10 -0.0032 0.00271 208 260 -14.31 -16.24 3589.60 846.60 3166.30 -0.0023 0.00237 260 312 -16.24 -17.93 4436.20 846.60 4012.90 -0.0020 0.00214 312 390 -17.93 -19.76 5486.00 1049.80 4961.10 -0.0017 0.00190 390 442 -19.76 -21.07 6332.60 846.60 5909.30 -0.0015 0.00117 442 494 -21.07 -22.26 7179.20 846.60 6755.90 -0.0014 0.00064 494 572 -22.26 -23.63 8229.00 1049.80 7704.10 -0.0013 0.00052 572 624 -23.63 -24.65 9075.60 846.60 8652.30 -0.0012 0.00042 624 676 -24.65 -25.63 9922.20 846.60 9498.90 -0.0012 0.00040 676 754 -25.63 -26.78 10972.00 1049.80 10447.10 -0.0011 0.00038 754 806 -26.78 -27.67 11818.60 846.60 11395.30 -0.0011 0.00037 806 858 -27.67 -28.53 12665.20 846.60 12241.90 -0.0010 0.00036 858 936 -28.53 -29.56 13715.00 1049.80 13190.10 -0.0010 0.00034 936 988 -29.56 -30.36 14561.60 846.60 14138.30 -0.0009 0.00033 988 1040 -30.36 -31.13 15408.20 846.60 14984.90 -0.0009 0.00032 1040 1118 -31.13 -32.06 16458.00 1049.80 15933.10 -0.0009 0.00030 188 Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 1118 1170 -32.06 -32.78 17304.60 846.60 16881.30 -0.0009 0.00029 1170 1222 -32.78 -33.48 18151.20 846.60 17727.90 -0.0008 0.00027 1222 1300 -33.48 -34.31 19201.00 1049.80 18676.10 -0.0008 0.00026 1300 1352 -34.31 -34.96 20047.60 846.60 19624.30 -0.0008 0.00024 1352 1404 -34.96 -35.59 20894.20 846.60 20470.90 -0.0007 0.00023 1404 1482 -35.59 -36.34 21944.00 1049.80 21419.10 -0.0007 0.00022 1482 1534 -36.34 -36.93 22790.60 846.60 22367.30 -0.0007 0.00020 1534 1586 -36.93 -37.49 23637.20 846.60 23213.90 -0.0007 0.00019 1586 1664 -37.49 -38.16 24687.00 1049.80 24162.10 -0.0006 0.00017 1664 1716 -38.16 -38.68 25533.60 846.60 25110.30 -0.0006 0.00016 1716 1768 -38.68 -39.18 26380.20 846.60 25956.90 -0.0006 0.00015 1768 1846 -39.18 -39.78 27430.00 1049.80 26905.10 -0.0006 0.00014 Wall length: 7620 mm Table C.2 Interpolation of curvatures at storey heights of Wall 1 for a drift of 2% Storey Height (mm) Curvature (rad/m) 423.30 0.00326 1269.90 0.00320 2218.10 0.00271 1 2743.00 0.00252 3166.30 0.00237 4012.90 0.00214 4961.10 0.00190 2 5486.00 0.00150 5909.30 0.00117 6755.90 0.00064 7704.10 0.00052 3 8229.00 0.00047 8652.30 0.00042 9498.90 0.00040 10447.10 0.00038 4 10972.00 0.00037 11395.30 0.00037 12241.90 0.00036 13190.10 0.00034 5 13715.00 0.00034 14138.30 0.00033 14984.90 0.00032 15933.10 0.00030 6 16458.00 0.00029 16881.30 0.00029 17727.90 0.00027 18676.10 0.00026 7 19201.00 0.00025 19624.30 0.00024 20470.90 0.00023 21419.10 0.00022 8 21944.00 0.00021 22367.30 0.00020 23213.90 0.00019 24162.10 0.00017 9 24687.00 0.00017 25110.30 0.00016 25956.90 0.00015 26905.10 0.00014 10 27430.00 0.00013 Table C.3 Bending moments along the height of Wall 1 for a drift of 2% Storey Height (m) Force (kN) Shear (kN) Moment incr. (kN.m) Moment (kN.m) 0 0.00 0.0 1925.5 815.1 105632.9 0.42 0.0 1925.5 1630.1 104817.9 1.27 0.0 1925.5 1825.8 103187.7 2.22 0.0 1925.5 1010.7 101362.0 1 2.74 0.0 1925.5 815.1 100351.3 3.17 0.0 1925.5 1630.1 99536.2 4.01 0.0 1925.5 1825.8 97906.1 4.96 0.0 1925.5 1010.7 96080.3 2 5.49 0.0 1925.5 815.1 95069.6 5.91 0.0 1925.5 1630.1 94254.6 6.76 0.0 1925.5 1825.8 92624.4 7.70 0.0 1925.5 1010.7 90798.7 3 8.23 0.0 1925.5 815.1 89788.0 8.65 0.0 1925.5 1630.1 88972.9 9.50 0.0 1925.5 1825.8 87342.8 10.45 0.0 1925.5 1010.7 85517.0 4 10.97 0.0 1925.5 815.1 84506.3 11.40 0.0 1925.5 1630.1 83691.3 12.24 0.0 1925.5 1825.8 82061.2 13.19 0.0 1925.5 1010.7 80235.4 5 13.72 0.0 1925.5 815.1 79224.7 14.14 0.0 1925.5 1630.1 78409.6 14.98 0.0 1925.5 1825.8 76779.5 15.93 0.0 1925.5 1010.7 74953.7 6 16.46 0.0 1925.5 815.1 73943.1 16.88 0.0 1925.5 1630.1 73128.0 17.73 0.0 1925.5 1825.8 71497.9 18.68 0.0 1925.5 1010.7 69672.1 7 19.20 0.0 1925.5 815.1 68661.4 19.62 0.0 1925.5 1630.1 67846.3 20.47 0.0 1925.5 1825.8 66216.2 21.42 0.0 1925.5 1010.7 64390.5 8 21.94 0.0 1925.5 815.1 63379.8 22.37 0.0 1925.5 1630.1 62564.7 23.21 0.0 1925.5 1825.8 60934.6 24.16 0.0 1925.5 1010.7 59108.8 9 24.69 0.0 1925.5 815.1 58098.1 25.11 0.0 1925.5 1630.1 57283.0 25.96 0.0 1925.5 1825.8 55652.9 26.91 0.0 1925.5 1010.7 53827.2 10 27.43 0.0 1925.5 5281.6 52816.5 11 30.17 0.0 1925.5 5281.6 47534.8 12 32.92 0.0 1925.5 5281.6 42253.2 13 35.66 0.0 1925.5 5281.6 36971.5 14 38.40 0.0 1925.5 5281.6 31689.9 15 41.15 0.0 1925.5 5281.6 26408.2 16 43.89 0.0 1925.5 5281.6 21126.6 17 46.63 0.0 1925.5 5281.6 15844.9 18 49.37 0.0 1925.5 5281.6 10563.3 19 52.12 0.0 1925.5 5281.6 5281.6 20 54.86 1925.5 1925.5 0.0 0.0 191 Table C.4 Curvatures up to the mid-height of Wall 2 for a drift of 2% Node Node 1 y- Node 2 y- Node 2 y- Difference Height (mm) Strain Curvature (rad/m) First Second displ (mm) displ (mm) coord (mm) (mm) 1 29 0.00 7.52 846.60 846.60 423.30 0.0089 0.00300 29 57 7.52 14.01 1693.20 846.60 1269.90 0.0077 0.00257 57 99 14.01 20.58 2743.00 1049.80 2218.10 0.0063 0.00214 99 127 20.58 24.25 3589.60 846.60 3166.30 0.0043 0.00157 127 155 24.25 27.94 4436.20 846.60 4012.90 0.0044 0.00153 155 197 27.94 30.86 5486.00 1049.80 4961.10 0.0028 0.00109 197 225 30.86 32.74 6332.60 846.60 5909.30 0.0022 0.00091 225 253 32.74 34.56 7179.20 846.60 6755.90 0.0021 0.00088 253 295 34.56 36.55 8229.00 1049.80 7704.10 0.0019 0.00080 295 323 36.55 38.09 9075.60 846.60 8652.30 0.0018 0.00076 323 351 38.09 39.58 9922.20 846.60 9498.90 0.0018 0.00074 351 393 39.58 41.34 10972.00 1049.80 10447.10 0.0017 0.00071 393 421 41.34 42.69 11818.60 846.60 11395.30 0.0016 0.00068 421 449 42.69 43.98 12665.20 846.60 12241.90 0.0015 0.00065 449 491 43.98 45.49 13715.00 1049.80 13190.10 0.0014 0.00062 491 519 45.49 46.63 14561.60 846.60 14138.30 0.0014 0.00059 519 547 46.63 47.71 15408.20 846.60 14984.90 0.0013 0.00057 547 589 47.71 48.96 16458.00 1049.80 15933.10 0.0012 0.00054 589 ' 617 48.96 49.90 17304.60 846.60 16881.30 0.0011 0.00051 617 645 49.90 50.77 18151.20 846.60 17727.90 0.0010 0.00048 645 687 50.77 51.76 19201.00 1049.80 18676.10 0.0009 0.00045 687 715 51.76 52.49 20047.60 846.60 19624.30 0.0009 0.00042 715 743 52.49 53.16 20894.20 846.60 20470.90 0.0008 0.00040 743 785 53.16 53.92 21944.00 1049.80 21419.10 0.0007 0.00037 785 813 53.92 54.47 22790.60 846.60 22367.30 0.0007 0.00034 813 841 54.47 54.98 23637.20 846.60 23213.90 0.0006 0.00032 841 883 54.98 55.54 24687.00 1049.80 24162.10 0.0005 0.00030 883 911 55.54 55.94 25533.60 846.60 25110.30 0.0005 0.00028 911 939 55.94 56.30 26380.20 846.60 25956.90 0.0004 0.00026 939 981 56.30 56.71 27430.00 1049.80 26905.10 0.0004 0.00024 14 42 0.00 -2.16 846.60 846.60 423.30 -0.0025 0.00300 42 70 -2.16 -3.96 1693.20 846.60 1269.90 -0.0021 0.00257 70 112 -3.96 -5.93 2743.00 1049.80 2218.10 -0.0019 0.00214 112 140 -5.93 -7.31 3589.60 846.60 3166.30 -0.0016 0.00157 140 168 -7.31 -8.57 4436.20 846.60 4012.90 -0.0015 0.00153 168 210 -8.57 -10.00 5486.00 1049.80 4961.10 -0.0014 0.00109 210 238 -10.00 -11.07 6332.60 846.60 5909.30 -0.0013 0.00091 238 266 -11.07 -12.08 7179.20 846.60 6755.90 -0.0012 0.00088 266 308 -12.08 -13.28 8229.00 1049.80 7704.10 -0.0011 0.00080 308 336 -13.28 -14.21 9075.60 846.60 8652.30 -0.0011 0.00076 336 364 -14.21 -15.10 9922.20 846.60 9498.90 -0.0011 0.00074 364 406 -15.10 -16.18 10972.00 1049.80 10447.10 -0.0010 0.00071 406 434 -16.18 -17.01 11818.60 846.60 11395.30 -0.0010 0.00068 434 462 -17.01 -17.82 12665.20 846.60 12241.90 -0.0010 0.00065 462 504 -17.82 -18.81 13715.00 1049.80 13190.10 -0.0009 0.00062 504 532 -18.81 -19.58 14561.60 846.60 14138.30 -0.0009 0.00059 532 560 -19.58 -20.32 15408.20 846.60 14984.90 -0.0009 0.00057 560 602 -20.32 -21.22 16458.00 1049.80 15933.10 -0.0009 0.00054 192 Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 602 630 -21.22 -21.91 17304.60 846.60 16881.30 -0.0008 0.00051 630 658 -21.91 -22.59 18151.20 846.60 17727.90 -0.0008 0.00048 658 700 -22.59 -23.40 19201.00 1049.80 18676.10 -0.0008 0.00045 700 728 -23.40 -24.02 20047.60 846.60 19624.30 -0.0007 0.00042 728 756 -24.02 -24.62 20894.20 846.60 20470.90 -0.0007 0.00040 756 798 -24.62 -25.35 21944.00 1049.80 21419.10 -0.0007 0.00037 798 826 -25.35 -25.90 22790.60 846.60 22367.30 -0.0007 0.00034 826 854 -25.90 -26.44 23637.20 846.60 23213.90 -0.0006 0.00032 854 896 -26.44 -27.09 24687.00 1049.80 24162.10 -0.0006 0.00030 896 924 -27.09 -27.58 25533.60 846.60 25110.30 -0.0006 0.00028 924 952 -27.58 -28.06 26380.20 846.60 25956.90 -0.0006 0.00026 952 994 -28.06 -28.63 27430.00 1049.80 26905.10 -0.0005 0.00024 Wall length: 3810 Table C.5 Interpolation of curvatures at storey heights of Wall 2 for a drift of 2% Storey Height (mm) Curvature (rad/m) 423.30 0.00300 1269.90 0.00257 2218.10 0.00214 1 2743.00 0.00182 3166.30 0.00157 4012.90 0.00153 4961.10 0.00109 2 5486.00 0.00099 5909.30 0.00091 6755.90 0.00088 7704.10 0.00080 3 8229.00 0.00078 8652.30 0.00076 9498.90 0.00074 10447.10 0.00071 4 10972.00 0.00069 11395.30 0.00068 12241.90 0.00065 13190.10 0.00062 5 13715.00 0.00061 14138.30 0.00059 14984.90 0.00057 15933.10 0.00054 6 16458.00 0.00052 16881.30 0.00051 17727.90 0.00048 18676.10 0.00045 7 19201.00 0.00043 19624.30 0.00042 20470.90 0.00040 21419.10 0.00037 8 21944.00 0.00036 22367.30 0.00034 23213.90 0.00032 24162.10 0.00030 9 24687.00 0.00029 25110.30 0.00028 25956.90 0.00026 26905.10 0.00024 10 27430.00 0.00024 Table C.6 Bending moments along the height of Wall 2 for a drift of 2 % Storey Height (m) Force (kN) Shear (kN) Moment incr. (kN.m) Moment (kN.m) 0 0.00 0.0 463.1 196.0 25405.7 0.42 0.0 463.1 392.1 25209.6 1.27 0.0 463.1 439.1 24817.6 2.22 0.0 463.1 243.1 24378.5 1 2.74 0.0 463.1 196.0 24135.4 3.17 0.0 463.1 392.1 23939.4 4.01 0.0 463.1 439.1 23547.3 4.96 0.0 463.1 243.1 23108.2 2 5.49 0.0 463.1 196.0 22865.1 5.91 0.0 463.1 392.1 22669.1 6.76 0.0 463.1 439.1 22277.0 7.70 0.0 463.1 243.1 21837.9 3 8.23 0.0 463.1 196.0 21594.8 8.65 0.0 463.1 392.1 21398.8 9.50 0.0 463.1 439.1 21006.7 10.45 0.0 463.1 243.1 20567.6 4 10.97 0.0 463.1 196.0 20324.5 11.40 0.0 463.1 392.1 20128.5 12.24 0.0 463.1 439.1 19736.4 13.19 0.0 463.1 243.1 19297.3 5 13.72 0.0 463.1 196.0 19054.2 14.14 0.0 463.1 392.1 18858.2 14.98 0.0 463.1 439.1 18466.2 15.93 0.0 463.1 243.1 18027.0 6 16.46 0.0 463.1 196.0 17784.0 16.88 0.0 463.1 392.1 17587.9 17.73 0.0 463.1 439.1 17195.9 18.68 0.0 463.1 243.1 16756.8 7 19.20 0.0 463.1 196.0 16513.7 19.62 0.0 463.1 392.1 16317.7 20.47 0.0 463.1 439.1 15925.6 21.42 0.0 463.1 243.1 15486.5 8 21.94 0.0 463.1 196.0 15243.4 22.37 0.0 463.1 392.1 15047.4 23.21 0.0 463.1 439.1 14655.3 24.16 0.0 463.1 243.1 14216.2 9 24.69 0.0 463.1 196.0 13973.1 25.11 0.0 463.1 392.1 13777.1 25.96 0.0 463.1 439.1 13385.0 26.91 0.0 463.1 243.1 12945.9 10 27.43 0.0 463.1 1270.3 12702.8 11 30.17 0.0 463.1 1270.3 11432.5 12 32.92 0.0 463.1 1270.3 10162.3 13 35.66 0.0 463.1 1270.3 8892.0 14 38.40 0.0 463.1 1270.3 7621.7 15 41.15 0.0 463.1 1270.3 6351.4 16 43.89 0.0 463.1 1270.3 5081.1 17 46.63 0.0 463.1 1270.3 3810.8 18 49.37 0.0 463.1 1270.3 2540.6 19 52.12 0.0 463.1 1270.3 1270.3 20 54.86 463.1 463.1 0.0 0.0 Table C.7 Average shear stress in Wall 1 Drift (%) Yield drift (%) Plastic drift (%) F(kN) lw (mm) bw (mm) v(MPa) 2.0 0.61 1.39 1925.5 7620.0 508.0 0.62 1.8 0.61 1.19 1912.3 7620.0 508.0 0.62 1.6 0.61 0.99 1901.5 7620.0 508.0 0.61 1.4 0.61 0.79 1883.8 7620.0 508.0 0.61 1.2 0.61 0.59 1864.4 7620.0 508.0 0.60 Table C.8 Average shear stress in Wall 2 Drift (%) Yield drift (%) Plastic drift (%) F(kN) lv (mm) bw (mm) v(MPa) 2.0 1.18 0.82 463.1 3810.0 508.0 0.30 1.8 1.18 0.62 458.2 3810.0 508.0 0.30 1.6 1.18 0.42 451.5 3810.0 508.0 0.29 1.4 1.18 0.22 438.4 3810.0 508.0 0.28 1.2 1.18 0.02 420.7 3810.0 508.0 0.27 Table C.9 Average shear stress in Wall 1 for a shear span of 35659mm Drift (%) Yield drift (%) Plastic drift (%) F(kN) lw (mm) bw (mm) v(MPa) 1.6 0.43 1.17 2996.5 7620.0 508.0 0.97 1.4 0.43 0.97 2982.1 7620.0 508.0 0.96 1.2 0.43 0.77 2962.8 7620.0 508.0 0.96 1.0 0.43 0.57 2911.0 7620.0 508.0 0.94 Table C.10 Average shear stress in Wall 2 for a shear span of 35659mm Drift (%) Yield drift (%) Plastic drift (%) F(kN) K (mm) bw (mm) v(MPa) 1.6 0.79 0.81 727.5 3810.0 508.0 0.47 1.4 0.79 0.61 722.8 3810.0 508.0 0.47 1.2 0.79 0.41 708.8 3810.0 508.0 0.46 1.0 0.79 0.21 686.9 3810.0 508.0 0.44 Table C.l l Average shear stress in Wall 1 for a shear span of 27430mm Drift (%) Yield drift (%) Plastic drift (%) F(kN) lw (mm) bw (mm) v(MPa) 1.6 0.35 1.25 3941.8 7620.0 508.0 1.27 1.4 0.35 1.05 3936.7 7620.0 508.0 1.27 1.2 0.35 0.85 3897.4 7620.0 508.0 1.26 1.0 0.35 0.65 3855.8 7620.0 508.0 1.25 0.8 0.35 0.45 3794.0 7620.0 508.0 1.23 Table C.12 Average shear stress in Wall 2 for a shear span of 27430mm Drift (%) Yield drift (%) Plastic drift (%) F(kN) lv (mm) bw (mm) v(MPa) 1.6 0.63 0.97 969.2 3810.0 508.0 0.63 1.4 0.63 0.77 958.4 3810.0 508.0 0.62 1.2 0.63 0.57 948.8 3810.0 508.0 0.61 1.0 0.63 0.37 933.4 3810.0 508.0 0.60 0.8 0.63 0.17 897.2 3810.0 508.0 0.58 Table C.13 Average shear stress in Wall 1 for a shear span of 19201mm Drift (%) Yield drift (%) Plastic drift (%) F(kN) L (mm) bw (mm) v(MPa) 1.2 0.28 0.92 5644.1 7620.0 508.0 1.82 1.0 0.28 0.72 5591.5 7620.0 508.0 1.81 0.8 0.28 0.52 5550.1 7620.0 508.0 1.79 0.6 0.28 0.32 5448.5 7620.0 508.0 1.76 Table C.14 Average shear stress in Wall 2 for a shear span of 19201mm Drift (%) Yield drift (%) Plastic drift (%) F(kN) K (mm) bâ€ž (mm) v(MPa) 1.2 0.47 0.73 1394.0 3810.0 508.0 0.90 1.0 0.47 0.53 1368.8 3810.0 508.0 0.88 0.8 0.47 0.33 1349.5 3810.0 508.0 0.87 0.6 0.47 0.13 1293.7 3810.0 508.0 0.84 Table C.15 Average shear stress in Wall 1 for a web thickness of 254mm and a shear span of 27430mm Drift (%) Yield drift (%) Plastic drift (%) F(kN) K (mm) bw (mm) v(MPa) 1.8 0.37 1.43 3656.3 7620.0 254.0 2.36 1.6 0.37 1.23 3639.7 7620.0 254.0 2.35 1.4 0.37 1.03 3571.7 7620.0 254.0 2.31 1.2 0.37 0.83 3580.1 7620.0 254.0 2.31 1.0 0.37 0.63 3565.5 7620.0 254.0 2.30 0.8 0.37 0.43 3524.0 7620.0 254.0 2.28 Table C.16 Average shear stress in Wall 2 for a web thickness of 254mm and a shear span of 27430mm Drift (%) Yield drift (%) Plastic drift (%) F(kN) lw (mm) bw (mm) v(MPa) 1.8 0.65 1.15 915.5 3810.0 254.0 1.18 1.6 0.65 0.95 904.7 3810.0 254.0 1.17 1.4 0.65 0.75 890.7 3810.0 254.0 1.15 1.2 0.65 0.55 880.4 3810.0 254.0 1.14 1.0 0.65 0.35 864.6 3810.0 254.0 1.12 0.8 0.65 0.15 844.4 3810.0 254.0 1.09 Table C.17 Average shear stress in Wall 1 for a web thickness of 254mm and a shear span of 19201mm Drift (%) Yield drift (%) Plastic drift (%) V (kN) 4 (mm) bw (mm) v(MPa) 1.8 0.30 1.50 5274.6 7620.0 254.0 3.41 1.6 0.30 1.30 5260.4 7620.0 254.0 3.40 1.4 0.30 1.10 5220.3 7620.0 254.0 3.37 1.2 0.30 0.90 5179.0 7620.0 254.0 3.34 1.0 0.30 0.70 5162.1 7620.0 254.0 3.33 0.8 0.30 0.50 5064.5 7620.0 254.0 3.27 0.6 0.30 0.30 5035.1 7620.0 254.0 3.25 Table C.18 Average shear stress in Wall 2 for a web thickness of 254mm and a shear span of 19201mm Drift (%) Yield drift (%) Plastic drift (%) F(kN) lw (mm) bw (mm) v(MPa) 1.8 0.49 1.31 1343.5 3810.0 254.0 1.74 1.6 0.49 1.11 1328.1 3810.0 254.0 1.72 1.4 0.49 0.91 1321.6 3810.0 254.0 1.71 1.2 0.49 0.71 1299.2 3810.0 254.0 1.68 1.0 0.49 0.51 1272.9 3810.0 254.0 1.64 0.8 0.49 0.31 1255.6 3810.0 254.0 1.62 0.6 0.49 0.11 1213.3 3810.0 254.0 1.57 Table C.19 Average shear stress in Wall 1 for a compressive axial load ratio of 0.3 Drift (%) Yield drift (%) Plastic drift (%) F(kN) ly, (mm) bv (mm) v(MPa) 1.2 0.78 0.42 3101.9 7620.0 508.0 1.00 1.0 0.78 0.22 3070.2 7620.0 508.0 0.99 0.8 0.78 0.02 2922.1 7620.0 508.0 0.94 Table C.20 Average shear stress in Wall 2 for a compressive axial load ratio of 0.3 Drift (%) Yield drift (%) Plastic drift (%) F(kN) lv (mm) bw (mm) v(MPa) 2.0 1.51 0.49 775.1 3810.0 508.0 0.50 1.8 1.51 0.29 761.6 3810.0 508.0 0.49 1.6 1.51 0.09 742.3 3810.0 508.0 0.48 Table C.21 Average shear stress in Wall 1 for a compressive axial load ratio of 0.2 Drift (%) Yield drift (%) Plastic drift (%) F(kN) lw (mm) bw (mm) v(MPa) 1.4 0.69 0.71 2587.5 7620.0 508.0 0.84 1.2 0.69 0.51 2583.7 7620.0 508.0 0.83 1.0 0.69 0.31 2542.5 7620.0 508.0 0.82 0.8 0.69 0.11 2443.8 7620.0 508.0 0.79 Table C.22 Average shear stress in Wall 2 for a compressive axial load ratio of 0.2 Drift (%) Yield drift (%) Plastic drift (%) K(kN) lw (mm) bw (mm) v(MPa) 2.0 1.34 0.66 646.6 3810.0 508.0 0.42 1.8 1.34 0.46 634.0 3810.0 508.0 0.41 1.6 1.34 0.26 616.6 3810.0 508.0 0.40 1.4 1.34 0.06 594.5 3810.0 508.0 0.38 Table C.23 Average shear stress in Wall 1 for no axial load Drift (%) Yield drift (%) Plastic drift (%) F(kN) lw (mm) bw (mm) v(MPa) 2.0 0.55 1.45 1051.6 7620.0 508.0 0.34 1.8 0.55 1.25 1034.1 7620.0 508.0 0.33 1.6 0.55 1.05 1019.5 7620.0 508.0 0.33 1.4 0.55 0.85 1005.2 7620.0 508.0 0.32 1.2 0.55 0.65 989.4 7620.0 508.0 0.32 Table C.24 Average shear stress in Wall 2 for no axial load Drift (%) Yield drift (%) Plastic drift (%) F(kN) lw (mm) bv (mm) v(MPa) 2.0 1.08 0.92 250.1 3810.0 508.0 0.16 1.8 1.08 0.72 244.1 3810.0 508.0 0.16 1.6 1.08 0.52 242.7 3810.0 508.0 0.16 1.4 1.08 0.32 236.2 3810.0 508.0 0.15 1.2 1.08 0.12 231.2 3810.0 508.0 0.15 Table C.25 Average shear stress in Wall 1 for a tensile axial load ratio of 0.02 Drift (%) Yield drift (%) Plastic drift (%) F(kN) lw (mm) bw (mm) v(MPa) 2.0 0.55 1.45 809.0 7620.0 508.0 0.26 1.8 0.55 1.25 798.0 7620.0 508.0 0.26 1.6 0.55 1.05 782.1 7620.0 508.0 0.25 1.4 0.55 0.85 767.7 7620.0 508.0 0.25 1.2 0.55 0.65 752.1 7620.0 508.0 0.24 Table C.26 Average shear stress in Wall 2 for a tensile axial load ratio of 0.02 Drift (%) Yield drift (%) Plastic drift (%) F(kN) lw (mm) bw (mm) v(MPa) 2.0 1.11 0.89 190.7 3810.0 508.0 0.12 1.8 1.11 0.69 187.3 3810.0 508.0 0.12 1.6 1.11 0.49 186.1 3810.0 508.0 0.12 1.4 1.11 0.29 180.8 3810.0 508.0 0.12 1.2 1.11 0.09 175.5 3810.0 508.0 0.11 Table C.27 Average shear stress in Wall 1 for a tensile axial load ratio of 0.05 Drift (%) Yield drift (%) Plastic drift (%) F ( k N ) lw (mm) bw (mm) v(MPa) 2.0 0.58 1.42 553.8 7620.0 508.0 0.18 1.8 0.58 1.22 542.8 7620.0 508.0 0.18 1.6 0.58 1.02 529.2 7620.0 508.0 0.17 1.4 0.58 0.82 518.1 7620.0 508.0 0.17 1.2 0.58 0.62 506.6 7620.0 508.0 0.16 Table C.28 Average shear stress in Wall 2 for a tensile axial load ratio of 0.05 Drift (%) Yield drift (%) Plastic drift (%) F ( k N ) lw (mm) bw (mm) v(MPa) 2.0 1.17 0.83 125.9 3810.0 508.0 0.08 1.8 1.17 0.63 125.2 3810.0 508.0 0.08 1.6 1.17 0.43 121.5 3810.0 508.0 0.08 1.4 1.17 0.23 120.6 3810.0 508.0 0.08 1.2 1.17 0.03 114.7 3810.0 508.0 0.07 Table C.29 Curvatures up to the mid-height of Wall 1 combined with Wall 2 for a drift of 2% Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 1 81 0.00 15.30 846.60 846.60 423.30 0.0181 0.00317 81 161 15.30 30.51 1693.20 846.60 1269.90 0.0180 0.00311 161 281 30.51 47.37 2743.00 1049.80 2218.10 0.0161 0.00248 281 361 47.37 59.96 3589.60 846.60 3166.30 0.0149 0.00229 361 441 59.96 71.56 4436.20 846.60 4012.90 0.0137 0.00209 441 561 71.56 82.29 5486.00 1049.80 4961.10 0.0102 0.00156 561 641 82.29 88.74 6332.60 846.60 5909.30 0.0076 0.00123 641 721 88.74 94.21 7179.20 846.60 6755.90 0.0065 0.00106 721 841 94.21 98.66 8229.00 1049.80 7704.10 0.0042 0.00073 841 921 98.66 101.03 9075.60 846.60 8652.30 0.0028 0.00055 921 1001 101.03 102.93 9922.20 846.60 9498.90 0.0022 0.00046 1001 1121 102.93 104.52 10972.00 1049.80 10447.10 0.0015 0.00034 1121 1201 104.52 106.27 11818.60 846.60 11395.30 0.0021 0.00043 1201 1281 106.27 107.81 12665.20 846.60 12241.90 0.0018 0.00038 1281 1401 107.81 109.28 13715.00 1049.80 13190.10 0.0014 0.00031 1401 1481 109.28 110.77 14561.60 846.60 14138.30 0.0018 0.00037 1481 1561 110.77 112.19 15408.20 846.60 14984.90 0.0017 0.00035 1561 1681 112.19 113.52 16458.00 1049.80 15933.10 0.0013 0.00028 1681 1761 113.52 114.82 17304.60 846.60 16881.30 0.0015 0.00033 1761 1841 114.82 116.04 18151.20 846.60 17727.90 0.0014 0.00031 1841 1961 116.04 117.17 19201.00 1049.80 18676.10 0.0011 0.00024 1961 2041 117.17 118.26 20047.60 846.60 19624.30 0.0013 0.00028 2041 2121 118.26 119.27 20894.20 846.60 20470.90 0.0012 0.00026 2121 2241 119.27 120.18 21944.00 1049.80 21419.10 0.0009 0.00020 2241 2321 120.18 121.06 22790.60 846.60 22367.30 0.0010 0.00023 2321 2401 121.06 121.85 23637.20 846.60 23213.90 0.0009 0.00021 2401 2521 121.85 122.56 24687.00 1049.80 24162.10 0.0007 0.00017 2521 2601 122.56 123.21 25533.60 846.60 25110.30 0.0008 0.00019 2601 2681 123.21 123.79 26380.20 846.60 25956.90 0.0007 0.00017 2681 2801 123.79 124.32 27430.00 1049.80 26905.10 0.0005 0.00014 26 106 0.00 -5.17 846.60 846.60 423.30 -0.0061 0.00317 106 186 -5.17 -10.01 1693.20 846.60 1269.90 -0.0057 0.00311 186 306 -10.01 -12.98 2743.00 1049.80 2218.10 -0.0028 0.00248 306 386 -12.98 -15.18 3589.60 846.60 3166.30 -0.0026 0.00229 386 466 -15.18 -17.04 4436.20 846.60 4012.90 -0.0022 0.00209 466 586 -17.04 -18.81 5486.00 1049.80 4961.10 -0.0017 0.00156 586 666 -18.81 -20.30 6332.60 846.60 5909.30 -0.0018 0.00123 666 746 -20.30 -21.65 7179.20 846.60 6755.90 -0.0016 0.00106 746 866 -21.65 -23.00 8229.00 1049.80 7704.10 -0.0013 0.00073 866 946 -23.00 -24.18 9075.60 846.60 8652.30 -0.0014 0.00055 946 1026 -24.18 -25.26 9922.20 846.60 9498.90 -0.0013 0.00046 1026 1146 -25.26 -26.38 10972.00 1049.80 10447.10 -0.0011 0.00034 1146 1226 -26.38 -27.38 11818.60 846.60 11395.30 -0.0012 0.00043 1226 1306 -27.38 -28.32 12665.20 846.60 12241.90 -0.0011 0.00038 1306 1426 -28.32 -29.30 13715.00 1049.80 13190.10 -0.0009 0.00031 1426 1506 -29.30 -30.19 14561.60 846.60 14138.30 -0.0011 0.00037 1506 1586 -30.19 -31.03 15408.20 846.60 14984.90 -0.0010 0.00035 1586 1706 -31.03 -31.90 16458.00 1049.80 15933.10 -0.0008 0.00028 201 Node Node 1 y- Node 2 y- Node 2 y- Difference (mm) Height (mm) Curvature (rad/m) First Second displ (mm) displ (mm) coord (mm) Strain 1706 1786 -31.90 -32.70 17304.60 846.60 16881.30 -0.0009 0.00033 1786 1866 -32.70 -33.45 18151.20 846.60 17727.90 -0.0009 0.00031 1866 1986 -33.45 -34.24 19201.00 1049.80 18676.10 -0.0007 0.00024 1986 2066 -34.24 -34.95 20047.60 846.60 19624.30 -0.0008 0.00028 2066 2146 -34.95 -35.62 20894.20 846.60 20470.90 -0.0008 0.00026 2146 2266 -35.62 -36.32 21944.00 1049.80 21419.10 -0.0007 0.00020 2266 2346 -36.32 -36.96 22790.60 846.60 22367.30 -0.0007 0.00023 2346 2426 -36.96 -37.56 23637.20 846.60 23213.90 -0.0007 0.00021 2426 2546 -37.56 -38.18 24687.00 1049.80 24162.10 -0.0006 0.00017 2546 2626 -38.18 -38.74 25533.60 846.60 25110.30 -0.0007 0.00019 2626 2706 -38.74 -39.28 26380.20 846.60 25956.90 -0.0006 0.00017 2706 2826 -39.28 -39.84 27430.00 1049.80 26905.10 -0.0005 0.00014 Wall length: 7620 Table C.30 Interpolation of curvatures at storey heights of Wall 1 combined with Wall 2 for a drift of 2% Storey Height (mm) Curvature (rad/m) 423.30 0.00317 1269.90 0.00311 2218.10 0.00248 1 2743.00 0.00238 3166.30 0.00229 4012.90 0.00209 4961.10 0.00156 2 5486.00 0.00138 5909.30 0.00123 6755.90 0.00106 7704.10 0.00073 3 8229.00 0.00063 8652.30 0.00055 9498.90 0.00046 10447.10 0.00034 4 10972.00 0.00039 11395.30 0.00043 12241.90 0.00038 13190.10 0.00031 5 13715.00 0.00034 14138.30 0.00037 14984.90 0.00035 15933.10 0.00028 6 16458.00 0.00030 16881.30 0.00033 17727.90 0.00031 18676.10 0.00024 7 19201.00 0.00026 19624.30 0.00028 20470.90 0.00026 21419.10 0.00020 8 21944.00 0.00022 22367.30 0.00023 23213.90 0.00021 24162.10 0.00017 9 24687.00 0.00018 25110.30 0.00019 25956.90 0.00017 26905.10 0.00014 10 27430.00 0.00011 203 Table C.31 Displacement components at the first storey of Wall 1 combined with Wall 2 for a drift of 2% Storey 1 Height (m) 2.74 Height (m) Curvature (rad/m) Ax (m) x(m) Flexural displ (mm) 0.42 0.00317 2.32 1.27 0.00311 0.85 1.47 5.04 2.22 0.00248 0.95 0.52 2.65 2.74 0.00238 0.52 0.00 0.33 Total 8.02 Total displacement (mm) 18.61 Shear displacement (mm) 10.59 Table C.32 Displacement components at the second storey of Wall 1 combined with Wall 2 for a drift of 2% Storey 2 Height (m) 5.49 Height (m) Curvature (rad/m) Ax (m) x(m) Flexural displ (mm) 0.42 0.00317 5.06 1.27 0.00311 0.85 4.22 12.33 2.22 0.00248 0.95 3.27 9.91 2.74 0.00238 0.52 2.74 3.83 3.17 0.00229 0.42 2.32 2.50 4.01 0.00209 0.85 1.47 3.52 4.96 0.00156 0.95 0.52 1.73 5.49 0.00138 0.52 0.00 0.20 Total 34.02 Total displacement (mm) 52.87 Shear displacement (mm) 18.84 Table C.33 Displacement components at the third storey of Wall 1 combined with Wall 2 for a drift of 2% Storey 3 Height (m) 8.23 Height (m) Curvature (rad/m) Ax (m) x(m) Flexural displ (mm) 0.42 0.00317 7.81 1.27 0.00311 0.85 6.96 19.63 2.22 0.00248 0.95 6.01 17.18 2.74 0.00238 0.52 5.49 7.32 3.17 0.00229 0.42 5.06 5.21 4.01 0.00209 0.85 4.22 8.60 4.96 0.00156 0.95 3.27 6.47 5.49 0.00138 0.52 2.74 2.32 5.91 0.00123 0.42 2.32 1.40 6.76 0.00106 0.85 1.47 1.84 7.70 0.00073 0.95 0.52 0.84 8.23 0.00063 0.52 0.00 0.09 Total 70.91 Total displacement (mm) 94.37 Shear displacement (mm) 23.46 Table C.34 Displacement components at the fourth storey of Wall 1 combined with Wall 2 for a drift of 2% Storey 4 Height (m) 10.97 Height (m) Curvature (rad/m) Ax (m) x(m) Flexural displ (mm) 0.42 0.00317 10.55 1.27 0.00311 0.85 9.70 26.92 2.22 0.00248 0.95 8.75 24.44 2.74 0.00238 0.52 8.23 10.82 3.17 0.00229 0.42 7.81 7.92 4.01 0.00209 0.85 6.96 13.69 4.96 0.00156 0.95 6.01 11.22 5.49 0.00138 0.52 5.49 4.44 5.91 0.00123 0.42 5.06 2.91 6.76 0.00106 0.85 4.22 4.49 7.70 0.00073 0.95 3.27 3.16 8.23 0.00063 0.52 2.74 1.07 8.65 0.00055 0.42 2.32 0.63 9.50 0.00046 0.85 1.47 0.81 10.45 0.00034 0.95 0.52 0.38 10.97 0.00039 0.52 0.00 0.05 Total 112.96 Total displacement (mm) 140.44 Shear displacement (mm) 27.48 Table C.35 Slopes up to the mid-height of Wall 1 combined with Wall 2 for a drift of 2% Height (m) Curvature (rad/m) Ax (m) Slope incr. (rad) Slope (rad) 0.42 0.00317 1.27 0.00311 0.85 0.0027 0.0027 2.22 0.00248 0.95 0.0026 0.0053 3.17 0.00229 0.95 0.0023 0.0076 4.01 0.00209 0.85 0.0019 0.0094 4.96 0.00156 0.95 0.0017 0.0112 5.91 0.00123 0.95 0.0013 0.0125 6.76 0.00106 0.85 0.0010 0.0134 7.70 0.00073 0.95 0.0008 0.0143 8.65 0.00055 0.95 0.0006 0.0149 9.50 0.00046 0.85 0.0004 0.0153 10.45 0.00034 0.95 0.0004 0.0157 11.40 0.00043 0.95 0.0004 0.0161 12.24 0.00038 0.85 0.0003 0.0164 13.19 0.00031 0.95 0.0003 0.0167 14.14 0.00037 0.95 0.0003 0.0171 14.98 0.00035 0.85 0.0003 0.0174 15.93 0.00028 0.95 0.0003 0.0177 16.88 0.00033 0.95 0.0003 0.0179 17.73 0.00031 0.85 0.0003 0.0182 18.68 0.00024 0.95 0.0003 0.0185 19.62 0.00028 0.95 0.0002 0.0187 20.47 0.00026 0.85 0.0002 0.0189 21.42 0.00020 0.95 0.0002 0.0192 22.37 0.00023 0.95 0.0002 0.0194 23.21 0.00021 0.85 0.0002 0.0196 24.16 0.00017 0.95 0.0002 0.0197 25.11 0.00019 0.95 0.0002 0.0199 25.96 0.00017 0.85 0.0002 0.0201 26.91 0.00014 0.95 0.0001 0.0202 Table C.36 Bending moments along the height of Wall 1 combined with Wall 2 for a drift of 2% Storey Height (m) Force (kN) Shear (kN) Moment incr. (kN.m) Moment (kN.m) 0 0.00 0.0 1726.0 730.6 106307.1 0.42 0.0 1726.0 1461.2 105576.5 1.27 0.0 1726.0 1636.6 104115.3 2.22 0.0 1726.0 906.0 102478.7 1 2.74 -44.9 1726.0 740.1 101572.7 3.17 0.0 1770.9 1499.2 100832.6 4.01 0.0 1770.9 1679.2 99333.4 4.96 0.0 1770.9 929.5 97654.2 2 5.49 321.0 1770.9 681.7 96724.6 5.91 0.0 1449.9 1227.5 96043.0 6.76 0.0 1449.9 1374.8 94815.5 7.70 0.0 1449.9 761.1 93440.7 3 8.23 -90.4 1449.9 632.9 92679.6 8.65 0.0 1540.3 1304.0 92046.8 9.50 0.0 1540.3 1460.5 90742.7 10.45 0.0 1540.3 808.5 89282.2 4 10.97 -252.7 1540.3 705.5 88473.7 11.40 0.0 1793.0 1518.0 87768.2 12.24 0.0 1793.0 1700.1 86250.3 13.19 0.0 1793.0 941.1 84550.2 5 13.72 -166.0 1793.0 794.1 83609.0 14.14 0.0 1959.0 1658.5 82814.9 14.98 0.0 1959.0 1857.5 81156.4 15.93 0.0 1959.0 1028.3 79298.9 6 16.46 -69.8 1959.0 844.0 78270.6 16.88 0.0 2028.8 1717.6 77426.6 17.73 0.0 2028.8 1923.7 75709.0 18.68 0.0 2028.8 1064.9 73785.3 7 19.20 -14.2 2028.8 861.8 72720.4 19.62 0.0 2043.0 1729.6 71858.6 20.47 0.0 2043.0 1937.2 70129.0 21.42 0.0 2043.0 1072.4 68191.8 8 21.94 30.5 2043.0 858.3 67119.4 22.37 0.0 2012.5 1703.8 66261.1 23.21 0.0 2012.5 1908.3 64557.3 24.16 0.0 2012.5 1056.4 62649.1 9 24.69 67.4 2012.5 837.6 61592.7 25.11 0.0 1945.1 1646.7 60755.1 25.96 0.0 1945.1 1844.3 59108.3 26.91 0.0 1945.1 1021.0 57264.0 10 27.43 112.5 1945.1 5181.1 56243.0 11 30.17 -42.1 1832.6 5084.6 51061.9 12 32.92 -89.7 1874.7 5265.3 45977.3 13 35.66 -89.4 1964.4 5511.0 40712.0 14 38.40 -52.8 2053.8 5706.0 35201.1 15 41.15 -22.6 2106.6 5809.4 29495.1 16 43.89 -11.9 2129.2 5856.7 23685.7 17 46.63 -26.6 2141.1 5909.5 17829.0 18 49.37 -48.3 2167.7 6012.2 11919.4 19 52.12 124.9 2216.0 5907.2 5907.2 20 54.86 2091.1 2091.1 0.0 0.0 Table C.37 Curvatures up to the mid-height of Wall 2 combined with Wall 1 for a drift of 2% Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 27 107 0.00 11.38 846.60 846.60 423.30 0.0134 0.00456 107 187 11.38 21.65 1693.20 846.60 1269.90 0.0121 0.00388 187 307 21.65 31.63 2743.00 1049.80 2218.10 0.0095 0.00296 307 387 31.63 36.59 3589.60 846.60 3166.30 0.0059 0.00198 387 467 36.59 39.24 4436.20 846.60 4012.90 0.0031 0.00121 467 587 39.24 41.02 5486.00 1049.80 4961.10 0.0017 0.00074 587 667 41.02 42.80 6332.60 846.60 5909.30 0.0021 0.00086 667 747 42.80 44.34 7179.20 846.60 6755.90 0.0018 0.00076 747 867 44.34 45.78 8229.00 1049.80 7704.10 0.0014 0.00059 867 947 45.78 47.08 9075.60 846.60 8652.30 0.0015 0.00065 947 1027 47.08 48.26 9922.20 846.60 9498.90 0.0014 0.00060 1027 1147 48.26 49.35 10972.00 1049.80 10447.10 0.0010 0.00046 1147 1227 49.35 50.25 11818.60 846.60 11395.30 0.0011 0.00049 1227 1307 50.25 51.07 12665.20 846.60 12241.90 0.0010 0.00045 1307 1427 51.07 51.83 13715.00 1049.80 13190.10 0.0007 0.00035 1427 1507 51.83 52.45 14561.60 846.60 14138.30 0.0007 0.00037 1507 1587 52.45 53.03 15408.20 846.60 14984.90 0.0007 0.00035 1587 1707 53.03 53.56 16458.00 1049.80 15933.10 0.0005 0.00028 1707 1787 53.56 54.03 17304.60 846.60 16881.30 0.0005 0.00031 1787 1867 54.03 54.45 18151.20 846.60 17727.90 0.0005 0.00029 1867 1987 54.45 54.85 19201.00 1049.80 18676.10 0.0004 0.00023 1987 2067 54.85 55.20 20047.60 846.60 19624.30 0.0004 0.00025 2067 2147 55.20 55.51 20894.20 846.60 20470.90 0.0004 0.00024 2147 2267 55.51 55.81 21944.00 1049.80 21419.10 0.0003 0.00019 2267 2347 55.81 56.06 22790.60 846.60 22367.30 0.0003 0.00021 2347 2427 56.06 56.28 23637.20 846.60 23213.90 0.0003 0.00019 2427 2547 56.28 56.47 24687.00 1049.80 24162.10 0.0002 0.00015 2547 2627 56.47 56.64 25533.60 846.60 25110.30 0.0002 0.00017 2627 2707 56.64 56.79 26380.20 846.60 25956.90 0.0002 0.00016 2707 2827 56.79 56.91 27430.00 1049.80 26905.10 0.0001 0.00013 40 120 0.00 -3.32 846.60 846.60 423.30 -0.0039 0.00456 120 200 -3.32 -5.55 1693.20 846.60 1269.90 -0.0026 0.00388 200 320 -5.55 -7.42 2743.00 1049.80 2218.10 -0.0018 0.00296 320 400 -7.42 -8.84 3589.60 846.60 3166.30 -0.0017 0.00198 400 480 -8.84 -10.08 4436.20 846.60 4012.90 -0.0015 0.00121 480 600 -10.08 -11.28 5486.00 1049.80 4961.10 -0.0011 0.00074 600 680 -11.28 -12.28 6332.60 846.60 5909.30 -0.0012 0.00086 680 760 -12.28 -13.19 7179.20 846.60 6755.90 -0.0011 0.00076 760 880 -13.19 -14.10 8229.00 1049.80 7704.10 -0.0009 0.00059 880 960 -14.10 -14.90 9075.60 846.60 8652.30 -0.0009 0.00065 960 1040 -14.90 -15.64 9922.20 846.60 9498.90 -0.0009 0.00060 1040 1160 -15.64 -16.39 10972.00 1049.80 10447.10 -0.0007 0.00046 1160 1240 -16.39 -17.05 11818.60 846.60 11395.30 -0.0008 0.00049 1240 1320 -17.05 -17.69 12665.20 846.60 12241.90 -0.0007 0.00045 1320 1440 -17.69 -18.34 13715.00 1049.80 13190.10 -0.0006 0.00035 1440 1520 -18.34 -18.92 14561.60 846.60 14138.30 -0.0007 0.00037 1520 1600 -18.92 -19.48 15408.20 846.60 14984.90 -0.0007 0.00035 1600 1720 -19.48 -20.06 16458.00 1049.80 15933.10 -0.0006 0.00028 208 Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 1720 1800 -20.06 -20.58 17304.60 846.60 16881.30 -0.0006 0.00031 1800 1880 -20.58 -21.08 18151.20 846.60 17727.90 -0.0006 0.00029 1880 2000 -21.08 -21.61 19201.00 1049.80 18676.10 -0.0005 0.00023 2000 2080 -21.61 -22.08 20047.60 846.60 19624.30 -0.0006 0.00025 2080 2160 -22.08 -22.53 20894.20 846.60 20470.90 -0.0005 0.00024 2160 2280 -22.53 -23.00 21944.00 1049.80 21419.10 -0.0004 0.00019 2280 2360 -23.00 -23.42 22790.60 846.60 22367.30 -0.0005 0.00021 2360 2440 -23.42 -23.82 23637.20 846.60 23213.90 -0.0005 0.00019 2440 2560 -23.82 -24.24 24687.00 1049.80 24162.10 -0.0004 0.00015 2560 2640 -24.24 -24.62 25533.60 846.60 25110.30 -0.0004 0.00017 2640 2720 -24.62 -24.98 26380.20 846.60 25956.90 -0.0004 0.00016 2720 2840 -24.98 -25.36 27430.00 1049.80 26905.10 -0.0004 0.00013 Wall length: 3810 mm Table C.38 Interpolation of curvatures at storey heights of Wall 2 combined with Wall 1 for a drift of 2% Storey Height (mm) Curvature (rad/m) 423.30 0.00456 1269.90 0.00388 2218.10 0.00296 1 2743.00 0.00242 3166.30 0.00198 4012.90 0.00121 4961.10 0.00074 2 5486.00 0.00081 5909.30 0.00086 6755.90 0.00076 7704.10 0.00059 3 8229.00 0.00062 8652.30 0.00065 9498.90 0.00060 10447.10 0.00046 4 10972.00 0.00047 11395.30 0.00049 12241.90 0.00045 13190.10 0.00035 5 13715.00 0.00036 14138.30 0.00037 14984.90 0.00035 15933.10 0.00028 6 16458.00 0.00029 16881.30 0.00031 17727.90 0.00029 18676.10 0.00023 7 19201.00 0.00024 19624.30 0.00025 20470.90 0.00024 21419.10 0.00019 8 21944.00 0.00020 22367.30 0.00021 23213.90 0.00019 24162.10 0.00015 9 24687.00 0.00016 25110.30 0.00017 25956.90 0.00016 26905.10 0.00013 10 27430.00 0.00011 Table C.39 Displacement components at the first storey of Wall 2 combined with Wall 1 for a drift of 2% Storey 1 Height (m) 2.74 Height (m) Curvature (rad/m) Ax (m) x(m) Flexural displ (mm) 0.42 0.00456 2.32 1.27 0.00388 0.85 1.47 6.77 2.22 0.00296 0.95 0.52 3.24 2.74 0.00242 0.52 0.00 0.37 Total 10.38 Total displacement (mm) 18.61 Shear displacement (mm) 8.23 Table C.40 Displacement components at the second storey of Wall 2 combined with Wall 1 for a drift of 2% Storey 2 Height (m) 5.49 Height (m) Curvature (rad/m) Ax (m) x(m) Flexural displ (mm) 0.42 0.00456 5.06 1.27 0.00388 0.85 4.22 16.56 2.22 0.00296 0.95 3.27 12.13 2.74 0.00242 0.52 2.74 4.24 3.17 0.00198 0.42 2.32 2.36 4.01 0.00121 0.85 1.47 2.56 4.96 0.00074 0.95 0.52 0.92 5.49 0.00081 0.52 0.00 0.11 Total 38.88 Total displacement (mm) 52.87 Shear displacement (mm) 13.99 Table C.41 Displacement components at the third storey of Wall 2 combined with Wall 1 for a drift of 2% Storey 3 Height (m) 8.23 Height (m) Curvature (rad/m) Ax (m) x(m) Flexural displ (mm) 0.42 0.00456 7.81 1.27 0.00388 0.85 6.96 26.35 2.22 0.00296 0.95 6.01 21.03 2.74 0.00242 0.52 5.49 8.12 3.17 0.00198 0.42 5.06 4.91 4.01 0.00121 0.85 4.22 6.26 4.96 0.00074 0.95 3.27 3.46 5.49 0.00081 0.52 2.74 1.23 5.91 0.00086 0.42 2.32 0.90 6.76 0.00076 0.85 1.47 1.30 7.70 0.00059 0.95 0.52 0.64 8.23 0.00062 0.52 0.00 0.08 Total 74.26 Total displacement (mm) 94.37 Shear displacement (mm) 20.11 Table C.42 Displacement components at the fourth storey of Wall 2 combined with Wall 1 for a drift of 2% Storey 4 Height (m) 10.97 Height (m) Curvature (rad/m) Ax (m) x(m) Flexural displ (mm) 0.42 0.00456 10.55 1.27 0.00388 0.85 9.70 36.14 2.22 0.00296 0.95 8.75 29.92 2.74 0.00242 0.52 8.23 11.99 3.17 0.00198 0.42 7.81 7.46 4.01 0.00121 0.85 6.96 9.95 4.96 0.00074 0.95 6.01 5.99 5.49 0.00081 0.52 5.49 2.34 5.91 0.00086 0.42 5.06 1.87 6.76 0.00076 0.85 4.22 3.19 7.70 0.00059 0.95 3.27 2.39 8.23 0.00062 0.52 2.74 0.95 8.65 0.00065 0.42 2.32 0.68 9.50 0.00060 0.85 1.47 1.00 10.45 0.00046 0.95 0.52 0.50 10.97 0.00047 0.52 0.00 0.06 Total 114.45 Total displacement (mm) 140.44 Shear displacement (mm) 25.99 Table C.43 Slopes up to the mid-height of Wall 2 combined with Wall 1 for a drift of 2 % Height (m) Curvature (rad/m) Ax (m) Slope incr. (rad) Slope (rad) 0.42 0.00456 1.27 0.00388 0.85 0.0036 0.0036 2.22 0.00296 0.95 0.0032 0.0068 3.17 0.00198 0.95 0.0023 0.0092 4.01 0.00121 0.85 0.0013 0.0105 4.96 0.00074 0.95 0.0009 0.0114 5.91 0.00086 0.95 0.0008 0.0122 6.76 0.00076 0.85 0.0007 0.0129 7.70 0.00059 0.95 0.0006 0.0135 8.65 0.00065 0.95 0.0006 0.0141 9.50 0.00060 0.85 0.0005 0.0146 10.45 0.00046 0.95 0.0005 0.0151 11.40 0.00049 0.95 0.0004 0.0156 12.24 0.00045 0.85 0.0004 0.0160 13.19 0.00035 0.95 0.0004 0.0164 14.14 0.00037 0.95 0.0003 0.0167 14.98 0.00035 0.85 0.0003 0.0170 15.93 0.00028 0.95 0.0003 0.0173 16.88 0.00031 0.95 0.0003 0.0176 17.73 0.00029 0.85 0.0003 0.0178 18.68 0.00023 0.95 0.0002 0.0181 19.62 0.00025 0.95 0.0002 0.0183 20.47 0.00024 0.85 0.0002 0.0185 21.42 0.00019 0.95 0.0002 0.0187 22.37 0.00021 0.95 0.0002 0.0189 23.21 0.00019 0.85 0.0002 0.0191 24.16 0.00015 0.95 0.0002 0.0192 25.11 0.00017 0.95 0.0002 0.0194 25.96 0.00016 0.85 0.0001 0.0195 26.91 0.00013 0.95 0.0001 0.0197 Table C.44 Bending moments along the height of Wall 2 combined with Wall 1 for a drift of 2% Storey Height (m) Force (kN) Shear (kN) Moment incr. (kN.m) Moment (kN.m) 0 0.00 0.0 693.4 293.5 26421.1 0.42 0.0 693.4 587.0 26127.6 1.27 0.0 693.4 657.5 25540.6 2.22 0.0 693.4 364.0 24883.1 1 2.74 44.9 693.4 284.0 24519.1 3.17 0.0 648.5 549.0 24235.1 4.01 0.0 648.5 614.9 23686.1 4.96 0.0 648.5 340.4 23071.2 2 5.49 -321.0 648.5 342.4 22730.8 5.91 0.0 969.5 820.8 22388.4 6.76 0.0 969.5 919.3 21567.6 7.70 0.0 969.5 508.9 20648.3 3 8.23 90.4 969.5 391.3 20139.4 8.65 0.0 879.1 744.2 19748.2 9.50 0.0 879.1 833.6 19003.9 10.45 0.0 879.1 461.4 18170.3 4 10.97 252.7 879.1 318.6 17708.9 11.40 0.0 626.4 530.3 17390.3 12.24 0.0 626.4 594.0 16860.0 13.19 0.0 626.4 328.8 16266.0 5 13.72 166.0 626.4 230.0 15937.2 14.14 0.0 460.4 389.8 15707.2 14.98 0.0 460.4 436.6 15317.4 15.93 0.0 460.4 241.7 14880.9 6 16.46 69.8 460.4 180.1 14639.2 16.88 0.0 390.6 330.7 14459.1 17.73 0.0 390.6 370.4 14128.4 18.68 0.0 390.6 205.0 13758.0 7 19.20 14.2 390.6 162.3 13553.0 19.62 0.0 376.4 318.7 13390.7 20.47 0.0 376.4 356.9 13072.0 21.42 0.0 376.4 197.6 12715.1 8 21.94 -30.5 376.4 165.8 12517.5 22.37 0.0 406.9 344.5 12351.7 23.21 0.0 406.9 385.8 12007.3 24.16 0.0 406.9 213.6 11621.4 9 24.69 -67.4 406.9 186.5 11407.9 25.11 0.0 474.3 401.5 11221.4 25.96 0.0 474.3 449.7 10819.8 26.91 0.0 474.3 249.0 10370.1 10 27.43 -112.5 474.3 1455.3 10121.1 11 30.17 42.1 586.8 1551.9 8665.8 12 32.92 89.7 544.7 1371.1 7114.0 13 35.66 89.4 455.0 1125.5 5742.9 14 38.40 52.8 365.6 930.4 4617.4 15 41.15 22.6 312.8 827.0 3687.0 16 43.89 11.9 290.2 779.7 2860.0 17 46.63 26.6 278.3 726.9 2080.3 18 49.37 48.3 251.7 624.2 1353.4 19 52.12 -124.9 203.4 729.2 729.2 20 54.86 328.3 328.3 0.0 0.0 214 Table C.45 Curvatures up to the mid-height of Wall 3 for a drift of 2% Node Node 1 y- Node 2 y- Node 2 y- Difference Height (mm) Strain Curvature First Second displ (mm) displ (mm) coord (mm) (mm) (rad/m) 1 15 0.00 1.21 846.60 846.60 423.30 0.0014 0.00148 15 29 1.21 2.39 1693.20 846.60 1269.90 0.0014 0.00143 29 50 2.39 3.76 2743.00 1049.80 2218.10 0.0013 0.00137 50 64 3.76 4.80 3589.60 846.60 3166.30 0.0012 0.00130 64 78 4.80 5.76 4436.20 846.60 4012.90 0.0011 0.00124 78 99 5.76 6.86 5486.00 1049.80 4961.10 0.0011 0.00118 99 113 6.86 7.69 6332.60 846.60 5909.30 0.0010 0.00112 113 127 7.69 8.45 7179.20 846.60 6755.90 0.0009 0.00107 127 148 8.45 9.35 8229.00 1049.80 7704.10 0.0009 0.00103 148 162 9.35 10.03 9075.60 846.60 8652.30 0.0008 0.00098 162 176 10.03 10.66 9922.20 846.60 9498.90 0.0008 0.00095 176 197 10.66 11.41 10972.00 1049.80 10447.10 0.0007 0.00092 197 211 11.41 11.96 11818.60 846.60 11395.30 0.0006 0.00086 211 225 11.96 12.47 12665.20 846.60 12241.90 0.0006 0.00083 225 246 12.47 13.06 13715.00 1049.80 13190.10 0.0006 0.00079 246 260 13.06 13.47 14561.60 846.60 14138.30 0.0005 0.00074 260 274 13.47 13.84 15408.20 846.60 14984.90 0.0004 0.00070 274 295 13.84 14.25 16458.00 1049.80 15933.10 0.0004 0.00066 295 309 14.25 14.55 17304.60 846.60 16881.30 0.0003 0.00062 309 323 14.55 14.82 18151.20 846.60 17727.90 0.0003 0.00060 323 344 14.82 15.14 19201.00 1049.80 18676.10 0.0003 0.00059 344 358 15.14 15.37 20047.60 846.60 19624.30 0.0003 0.00055 358 372 15.37 15.58 20894.20 846.60 20470.90 0.0002 0.00053 372 393 15.58 15.83 21944.00 1049.80 21419.10 0.0002 0.00052 393 407 15.83 15.98 22790.60 846.60 22367.30 0.0002 0.00047 407 421 15.98 16.11 23637.20 846.60 23213.90 0.0002 0.00046 421 442 16.11 16.26 24687.00 1049.80 24162.10 0.0001 0.00044 442 456 16.26 16.37 25533.60 846.60 25110.30 0.0001 0.00042 456 470 16.37 16.46 26380.20 846.60 25956.90 0.0001 0.00041 470 491 16.46 16.56 27430.00 1049.80 26905.10 0.0001 0.00039 7 21 0.00 -1.18 846.60 846.60 423.30 -0.0014 0.00148 21 35 -1.18 -2.30 1693.20 846.60 1269.90 -0.0013 0.00143 35 56 -2.30 -3.67 2743.00 1049.80 2218.10 -0.0013 0.00137 56 70 -3.67 -4.73 3589.60 846.60 3166.30 -0.0013 0.00130 70 84 -4.73 -5.77 4436.20 846.60 4012.90 -0.0012 0.00124 84 105 -5.77 -7.03 5486.00 1049.80 4961.10 -0.0012 0.00118 105 119 -7.03 -8.01 6332.60 846.60 5909.30 -0.0012 0.00112 119 133 -8.01 -8.96 7179.20 846.60 6755.90 -0.0011 0.00107 133 154 -8.96 -10.13 8229.00 1049.80 7704.10 -0.0011 0.00103 154 168 -10.13 -11.03 9075.60 846.60 8652.30 -0.0011 0.00098 168 182 -11.03 -11.93 9922.20 846.60 9498.90 -0.0011 0.00095 182 203 -11.93 -13.01 10972.00 1049.80 10447.10 -0.0010 0.00092 203 217 -13.01 -13.86 11818.60 846.60 11395.30 -0.0010 0.00086 217 231 -13.86 -14.68 12665.20 846.60 12241.90 -0.0010 0.00083 231 252 -14.68 -15.68 13715.00 1049.80 13190.10 -0.0010 0.00079 252 266 -15.68 -16.46 14561.60 846.60 14138.30 -0.0009 0.00074 266 280 -16.46 -17.21 15408.20 846.60 14984.90 -0.0009 0.00070 280 301 -17.21 -18.12 16458.00 1049.80 15933.10 -0.0009 0.00066 215 Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 301 315 -18.12 -18.83 17304.60 846.60 16881.30 -0.0008 0.00062 315 329 -18.83 -19.53 18151.20 846.60 17727.90 -0.0008 0.00060 329 350 -19.53 -20.37 19201.00 1049.80 18676.10 -0.0008 0.00059 350 364 -20.37 -21.04 20047.60 846.60 19624.30 -0.0008 0.00055 364 378 -21.04 -21.69 20894.20 846.60 20470.90 -0.0008 0.00053 378 399 -21.69 -22.47 21944.00 1049.80 21419.10 -0.0008 0.00052 399 413 -22.47 -23.09 22790.60 846.60 22367.30 -0.0007 0.00047 413 427 -23.09 -23.69 23637.20 846.60 23213.90 -0.0007 0.00046 427 448 -23.69 -24.42 24687.00 1049.80 24162.10 -0.0007 0.00044 448 462 -24.42 -24.99 25533.60 846.60 25110.30 -0.0007 0.00042 462 476 -24.99 -25.55 26380.20 846.60 25956.90 -0.0007 0.00041 476 497 -25.55 -26.24 27430.00 1049.80 26905.10 -0.0007 0.00039 Wall length: 1905 mm Table C.46 Interpolation of curvatures at storey heights of Wall 3 for a drift of 2% Storey Height (mm) Curvature (rad/m) 423.30 0.00148 1269.90 0.00143 2218.10 0.00137 1 2743.00 0.00133 3166.30 0.00130 4012.90 0.00124 4961.10 0.00118 2 5486.00 0.00115 5909.30 0.00112 6755.90 0.00107 7704.10 0.00103 3 8229.00 0.00100 8652.30 0.00098 9498.90 0.00095 10447.10 0.00092 4 10972.00 0.00089 11395.30 0.00086 12241.90 0.00083 13190.10 0.00079 5 13715.00 0.00076 14138.30 0.00074 14984.90 0.00070 15933.10 0.00066 6 16458.00 0.00064 16881.30 0.00062 17727.90 0.00060 18676.10 0.00059 7 19201.00 0.00057 19624.30 0.00055 20470.90 0.00053 21419.10 0.00052 8 21944.00 0.00049 22367.30 0.00047 23213.90 0.00046 24162.10 0.00044 9 24687.00 0.00043 25110.30 0.00042 25956.90 0.00041 26905.10 0.00039 10 27430.00 0.00039 Table C.47 Bending moments along the height of Wall 3 for a drift of 2% Storey Height (m) Force (kN) Shear (kN) Moment incr. (kN.m) Moment (kN.m) 0 0.00 0.0 135.4 57.3 7428.0 0.42 0.0 135.4 114.6 7370.7 1.27 0.0 135.4 128.4 7256.1 2.22 0.0 135.4 71.1 7127.7 1 2.74 0.0 135.4 57.3 7056.6 3.17 0.0 135.4 114.6 6999.3 4.01 0.0 135.4 128.4 6884.7 4.96 0.0 135.4 71.1 6756.3 2 5.49 0.0 135.4 57.3 6685.2 5.91 0.0 135.4 114.6 6627.9 6.76 0.0 135.4 128.4 6513.3 7.70 0.0 135.4 71.1 6384.9 3 8.23 0.0 135.4 57.3 6313.8 8.65 0.0 135.4 1 14.6 6256.5 9.50 0.0 135.4 128.4 6141.9 10.45 0.0 !35.4 71.1 6013.5 4 10.97 0.0 135.4 57.3 5942.4 11.40 0.0 135.4 114.6 5885.1 12.24 0.0 135.4 128.4 5770.5 13.19 0.0 135.4 71.1 5642.1 5 13.72 0.0 135.4 57.3 5571.0 14.14 0.0 135.4 114.6 5513.7 14.98 0.0 135.4 128.4 5399.1 15.93 0.0 135.4 71.1 5270.7 6 16.46 0.0 135.4 57.3 5199.6 16.88 0.0 135.4 114.6 5142.3 17.73 0.0 135.4 128.4 5027.7 18.68 0.0 135.4 71.1 4899.3 7 19.20 0.0 135.4 57.3 4828.2 19.62 0.0 135.4 114.6 4770.9 20.47 0.0 135.4 128.4 4656.3 21.42 0.0 135.4 71.1 4527.9 8 21.94 0.0 135.4 57.3 4456.8 22.37 0.0 135.4 114.6 4399.5 23.21 0.0 135.4 128.4 4284.9 24.16 0.0 135.4 71.1 4156.5 9 24.69 0.0 135.4 57.3 4085.4 25.11 0.0 135.4 114.6 4028.1 25.96 0.0 135.4 128.4 3913.5 26.91 0.0 135.4 71.1 3785.1 10 27.43 0.0 135.4 371.4 3714.0 11 30.17 0.0 135.4 371.4 3342.6 12 32.92 0.0 135.4 371.4 2971.2 13 35.66 0.0 135.4 371.4 2599.8 14 38.40 0.0 135.4 371.4 2228.4 15 41.15 0.0 135.4 371.4 1857.0 16 43.89 0.0 135.4 371.4 1485.6 17 46.63 0.0 135.4 371.4 1114.2 18 49.37 0.0 135.4 371.4 742.8 19 52.12 0.0 135.4 371.4 371.4 20 54.86 135.4 135.4 0.0 0.0 Table C.48 Curvatures up to the mid-height of Wall 1 combined with Wall 3 for a drift of 2% Node Node 1 y- Node 2 y- Node 2 y- Difference Height (mm) Strain Curvature (rad/m) First Second displ (mm) displ (mm) coord (mm) (mm) 1 67 0.00 15.02 846.60 846.60 423.30 0.0177 0.00310 67 133 15.02 29.99 1693.20 846.60 1269.90 0.0177 0.00307 133 232 29.99 47.07 2743.00 1049.80 2218.10 0.0163 0.00251 232 298 47.07 60.20 3589.60 846.60 3166.30 0.0155 0.00238 298 364 60.20 71.74 4436.20 846.60 4012.90 0.0136 0.00209 364 463 71.74 82.30 5486.00 1049.80 4961.10 0.0101 0.00155 463 529 82.30 89.14 6332.60 846.60 5909.30 0.0081 0.00130 529 595 89.14 95.15 7179.20 846.60 6755.90 0.0071 0.00114 595 694 95.15 99.06 8229.00 1049.80 7704.10 0.0037 0.00066 694 760 99.06 101.27 9075.60 846.60 8652.30 0.0026 0.00053 760 826 101.27 103.17 9922.20 846.60 9498.90 0.0022 0.00046 826 925 103.17 104.79 10972.00 1049.80 10447.10 0.0015 0.00034 925 991 104.79 106.54 11818.60 846.60 11395.30 0.0021 0.00042 991 1057 106.54 108.08 12665.20 846.60 12241.90 0.0018 0.00038 1057 1156 108.08 109.54 13715.00 1049.80 13190.10 0.0014 0.00030 1156 1222 109.54 111.02 14561.60 846.60 14138.30 0.0017 0.00036 1222 1288 111.02 112.43 15408.20 846.60 14984.90 0.0017 0.00035 1288 1387 112.43 113.74 16458.00 1049.80 15933.10 0.0013 0.00027 1387 1453 113.74 115.03 17304.60 846.60 16881.30 0.0015 0.00032 1453 1519 115.03 116.24 18151.20 846.60 17727.90 0.0014 0.00030 1519 1618 116.24 117.35 19201.00 1049.80 18676.10 0.0011 0.00024 1618 1684 117.35 118.43 20047.60 846.60 19624.30 0.0013 0.00028 1684 1750 118.43 119.42 20894.20 846.60 20470.90 0.0012 0.00026 1750 1849 119.42 120.33 21944.00 1049.80 21419.10 0.0009 0.00020 1849 1915 120.33 121.19 22790.60 846.60 22367.30 0.0010 0.00023 1915 1981 121.19 121.97 23637.20 846.60 23213.90 0.0009 0.00021 1981 2080 121.97 122.68 24687.00 1049.80 24162.10 0.0007 0.00017 2080 2146 122.68 123.34 25533.60 846.60 25110.30 0.0008 0.00019 2146 2212 123.34 123.92 26380.20 846.60 25956.90 0.0007 0.00017 2212 2311 123.92 124.44 27430.00 1049.80 26905.10 0.0005 0.00013 26 92 0.00 -4.98 846.60 846.60 423.30 -0.0059 0.00310 92 158 -4.98 -9.82 1693.20 846.60 1269.90 -0.0057 0.00307 158 257 -9.82 -12.86 2743.00 1049.80 2218.10 -0.0029 0.00251 257 323 -12.86 -15.11 3589.60 846.60 3166.30 -0.0027 0.00238 323 389 -15.11 -17.04 4436.20 846.60 4012.90 -0.0023 0.00209 389 488 -17.04 -18.86 5486.00 1049.80 4961.10 -0.0017 0.00155 488 554 -18.86 -20.40 6332.60 846.60 5909.30 -0.0018 0.00130 554 620 -20.40 -21.77 7179.20 846.60 6755.90 -0.0016 0.00114 620 719 -21.77 -23.13 8229.00 1049.80 7704.10 -0.0013 0.00066 719 785 -23.13 -24.30 9075.60 846.60 8652.30 -0.0014 0.00053 785 851 -24.30 -25.37 9922.20 846.60 9498.90 -0.0013 0.00046 851 950 -25.37 -26.47 10972.00 1049.80 10447.10 -0.0010 0.00034 950 1016 -26.47 -27.46 11818.60 846.60 11395.30 -0.0012 0.00042 1016 1082 -27.46 -28.37 12665.20 846.60 12241.90 -0.0011 0.00038 1082 1181 -28.37 -29.34 13715.00 1049.80 13190.10 -0.0009 0.00030 1181 1247 -29.34 -30.21 14561.60 846.60 14138.30 -0.0010 0.00036 1247 1313 -30.21 -31.03 15408.20 846.60 14984.90 -0.0010 0.00035 1313 1412 -31.03 -31.90 16458.00 1049.80 15933.10 -0.0008 0.00027 219 Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 1412 1478 -31.90 -32.68 17304.60 846.60 16881.30 -0.0009 0.00032 1478 1544 -32.68 -33.43 18151.20 846.60 17727.90 -0.0009 0.00030 1544 1643 -33.43 -34.21 19201.00 1049.80 18676.10 -0.0007 0.00024 1643 1709 -34.21 -34.91 20047.60 846.60 19624.30 -0.0008 0.00028 1709 1775 -34.91 -35.58 20894.20 846.60 20470.90 -0.0008 0.00026 1775 1874 -35.58 -36.29 21944.00 1049.80 21419.10 -0.0007 0.00020 1874 1940 -36.29 -36.92 22790.60 846.60 22367.30 -0.0007 0.00023 1940 2006 -36.92 -37.52 23637.20 846.60 23213.90 -0.0007 0.00021 2006 2105 -37.52 -38.14 24687.00 1049.80 24162.10 -0.0006 0.00017 2105 2171 -38.14 -38.71 25533.60 846.60 25110.30 -0.0007 0.00019 2171 2237 -38.71 -39.24 26380.20 846.60 25956.90 -0.0006 0.00017 2237 2336 -39.24 -39.80 27430.00 1049.80 26905.10 -0.0005 0.00013 Wall length: 7620 mm Table C.49 Curvatures up to the mid-height of Wall 3 combined with Wall 1 for a drift of 2% ode Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 27 93 0.00 6.60 846.60 846.60 423.30 0.0078 0.00581 93 159 6.60 11.04 1693.20 846.60 1269.90 0.0052 0.00399 159 258 11.04 12.83 2743.00 1049.80 2218.10 0.0017 0.00169 258 324 12.83 14.30 3589.60 846.60 3166.30 0.0017 0.00170 324 390 14.30 15.56 4436.20 846.60 4012.90 0.0015 0.00149 390 489 15.56 16.65 5486.00 1049.80 4961.10 0.0010 0.00108 489 555 16.65 17.37 6332.60 846.60 5909.30 0.0009 0.00101 555 621 17.37 17.95 7179.20 846.60 6755.90 0.0007 0.00087 621 720 17.95 18.36 8229.00 1049.80 7704.10 0.0004 0.00060 720 786 18.36 18.62 9075.60 846.60 8652.30 0.0003 0.00059 786 852 18.62 18.82 9922.20 846.60 9498.90 0.0002 0.00052 852 951 18.82 18.96 10972.00 1049.80 10447.10 0.0001 0.00040 951 1017 18.96 19.07 11818.60 846.60 11395.30 0.0001 0.00042 1017 1083 19.07 19.15 12665.20 846.60 12241.90 0.0001 0.00039 1083 1182 19.15 19.22 13715.00 1049.80 13190.10 0.0001 0.00032 1182 1248 19.22 19.27 14561.60 846.60 14138.30 0.0001 0.00035 1248 1314 19.27 19.31 15408.20 846.60 14984.90 0.0000 0.00033 1314 1413 19.31 19.33 16458.00 1049.80 15933.10 0.0000 0.00027 1413 1479 19.33 19.33 17304.60 846.60 16881.30 0.0000 0.00029 1479 1545 19.33 19.32 18151.20 846.60 17727.90 0.0000 0.00028 1545 1644 19.32 19.30 19201.00 1049.80 18676.10 0.0000 0.00023 1644 1710 19.30 19.27 20047.60 846.60 19624.30 0.0000 0.00025 1710 1776 19.27 19.23 20894.20 846.60 20470.90 0.0000 0.00023 1776 1875 19.23 19.17 21944.00 1049.80 21419.10 -0.0001 0.00019 1875 1941 19.17 19.10 22790.60 846.60 22367.30 -0.0001 0.00021 1941 2007 19.10 19.03 23637.20 846.60 23213.90 -0.0001 0.00019 2007 2106 19.03 18.94 24687.00 1049.80 24162.10 -0.0001 0.00016 2106 2172 18.94 18.85 25533.60 846.60 25110.30 -0.0001 0.00017 2172 2238 18.85 18.75 26380.20 846.60 25956.90 -0.0001 0.00016 2238 2337 18.75 18.63 27430.00 1049.80 26905.10 -0.0001 0.00013 33 99 0.00 -2.77 846.60 846.60 423.30 -0.0033 0.00581 99 165 -2.77 -4.76 1693.20 846.60 1269.90 -0.0024 0.00399 165 264 -4.76 -6.36 2743.00 1049.80 2218.10 -0.0015 0.00169 264 330 -6.36 -7.64 3589.60 846.60 3166.30 -0.0015 0.00170 330 396 -7.64 -8.77 4436.20 846.60 4012.90 -0.0013 0.00149 396 495 -8.77 -9.85 5486.00 1049.80 4961.10 -0.0010 0.00108 495 561 -9.85 -10.76 6332.60 846.60 5909.30 -0.0011 0.00101 561 627 . -10.76 -11.59 7179.20 846.60 6755.90 -0.0010 0.00087 627 726 -11.59 -12.39 8229.00 1049.80 7704.10 -0.0008 0.00060 726 792 -12.39 -13.08 9075.60 846.60 8652.30 -0.0008 0.00059 792 858 -13.08 -13.72 9922.20 846.60 9498.90 -0.0008 0.00052 858 957 -13.72 -14.36 10972.00 1049.80 10447.10 -0.0006 0.00040 957 1023 -14.36 -14.93 11818.60 846.60 11395.30 -0.0007 0.00042 1023 1089 -14.93 -15.48 12665.20 846.60 12241.90 -0.0006 0.00039 1089 1188 -15.48 -16.05 13715.00 1049.80 13190.10 -0.0005 0.00032 1188 1254 -16.05 -16.56 14561.60 846.60 14138.30 -0.0006 0.00035 1254 1320 -16.56 -17.06 15408.20 846.60 14984.90 -0.0006 0.00033 1320 1419 -17.06 -17.57 16458.00 1049.80 15933.10 -0.0005 0.00027 221 Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 1419 1485 -17.57 -18.04 17304.60 846.60 16881.30 -0.0006 0.00029 1485 1551 -18.04 -18.50 18151.20 846.60 17727.90 -0.0005 0.00028 1551 1650 -18.50 -18.98 19201.00 1049.80 18676.10 -0.0005 0.00023 1650 1716 -18.98 -19.41 20047.60 846.60 19624.30 -0.0005 0.00025 1716 1782 -19.41 -19.83 20894.20 846.60 20470.90 -0.0005 0.00023 1782 1881 -19.83 -20.27 21944.00 1049.80 21419.10 -0.0004 0.00019 1881 1947 -20.27 -20.67 22790.60 846.60 22367.30 -0.0005 0.00021 1947 2013 -20.67 -21.05 23637.20 846.60 23213.90 -0.0005 0.00019 2013 2112 -21.05 -21.46 24687.00 1049.80 24162.10 -0.0004 0.00016 2112 2178 -21.46 -21.83 25533.60 846.60 25110.30 -0.0004 0.00017 2178 2244 -21.83 -22.19 26380.20 846.60 25956.90 -0.0004 0.00016 2244 2343 -22.19 -22.57 27430.00 1049.80 26905.10 -0.0004 0.00013 Wall length: 1905 mm Table C.50 Interpolation of curvatures at storey heights of Wall 3 combined with Wall 1 for a drift of 2% Storey Height (mm) Curvature (rad/m) 423.30 0.00581 1269.90 0.00399 2218.10 0.00169 1 2743.00 0.00170 3166.30 0.00170 4012.90 0.00149 4961.10 0.00108 2 5486.00 0.00104 5909.30 0.00101 6755.90 0.00087 7704.10 0.00060 3 8229.00 0.00059 8652.30 0.00059 9498.90 0.00052 10447.10 0.00040 4 10972.00 0.00041 11395.30 0.00042 12241.90 0.00039 13190.10 0.00032 5 13715.00 0.00034 14138.30 0.00035 14984.90 0.00033 15933.10 0.00027 6 16458.00 0.00028 16881.30 0.00029 17727.90 0.00028 18676.10 0.00023 7 19201.00 0.00024 19624.30 0.00025 20470.90 0.00023 21419.10 0.00019 8 21944.00 0.00020 22367.30 0.00021 23213.90 0.00019 24162.10 0.00016 9 24687.00 0.00017 25110.30 0.00017 25956.90 0.00016 26905.10 0.00013 10 27430.00 0.00011 Table C.51 Bending moments along the height of Wall 3 combined with Wall 1 for a drift of 2% Storey Height (m) Force (kN) Shear (kN) Moment incr. (kN.m) Moment (kN.m) 0 0.00 0.0 632.3 267.7 9621.2 0.42 0.0 632.3 535.3 9353.6 1.27 0.0 632.3 599.5 8818.2 2.22 0.0 632.3 331.9 8218.7 1 2.74 70.3 632.3 252.8 7886.8 3.17 0.0 562.0 475.8 7634.0 4.01 0.0 562.0 532.9 7158.2 4.96 0.0 562.0 295.0 6625.4 2 5.49 58.8 562.0 225.4 6330.4 5.91 0.0 503.2 426.0 6104.9 6.76 0.0 503.2 477.1 5678.9 7.70 0.0 503.2 264.1 5201.8 3 8.23 155.3 503.2 180.1 4937.6 8.65 0.0 347.9 294.5 4757.5 9.50 0.0 347.9 329.9 4463.0 10.45 0.0 347.9 182.6 4133.1 4 10.97 114.7 347.9 123.0 3950.5 11.40 0.0 233.2 197.4 3827.5 12.24 0.0 233.2 221.1 3630.1 13.19 0.0 233.2 122.4 3408.9 5 13.72 62.5 233.2 85.5 3286.5 14.14 0.0 170.7 144.5 3201.1 14.98 0.0 170.7 161.9 3056.5 15.93 0.0 170.7 89.6 2894.7 6 16.46 19.8 170.7 68.1 2805.1 16.88 0.0 150.9 127.8 2737.0 17.73 0.0 150.9 143.1 2609.3 18.68 0.0 150.9 79.2 2466.2 7 19.20 9.7 150.9 61.8 2387.0 19.62 0.0 141.2 119.5 2325.1 20.47 0.0 141.2 133.9 2205.6 21.42 0.0 141.2 74.1 2071.7 8 21.94 15.1 141.2 56.6 1997.6 22.37 0.0 126.1 106.8 1941.0 23.21 0.0 126.1 119.6 1834.3 24.16 0.0 126.1 66.2 1714.7 9 24.69 24.5 126.1 48.2 1648.5 25.11 0.0 101.6 86.0 1600.3 25.96 0.0 101.6 96.3 1514.3 26.91 0.0 101.6 53.3 1418.0 10 27.43 -30.9 101.6 321.1 1364.6 11 30.17 51.3 132.5 293.1 1043.6 12 32.92 27.9 81.2 184.5 750.5 13 35.66 13.9 53.3 127.1 566.0 14 38.40 4.9 39.4 101.4 438.9 15 41.15 2.5 34.5 91.2 337.5 16 43.89 -3.0 32.0 91.9 246.3 17 46.63 12.8 35.0 78.4 154.4 18 49.37 55.0 22.2 -14.5 76.0 19 52.12 -131.6 -32.8 90.5 90.5 20 54.86 98.8 98.8 0.0 0.0 224 Table C.52 Curvatures up to the mid-height of Column 1 for a drift of 2% Node Node 1 y- Node 2 y- Node 2 y- Difference Height (mm) Strain Curvature First Second displ (mm) displ (mm) coord (mm) (mm) (rad/m) 1 9 0.00 0.01 846.60 846.60 423.30 0.0000 0.00113 9 17 0.01 0.00 1693.20 846.60 1269.90 0.0000 0.00109 17 29 0.00 0.00 2743.00 1049.80 2218.10 0.0000 0.00109 29 37 0.00 -0.02 3589.60 846.60 3166.30 0.0000 0.00105 37 45 -0.02 -0.05 4436.20 846.60 4012.90 0.0000 0.00104 45 57 -0.05 -0.08 5486.00 1049.80 4961.10 0.0000 0.00103 57 65 -0.08 -0.13 6332.60 846.60 5909.30 0.0000 0.00099 65 73 -0.13 -0.17 7179.20 846.60 6755.90 -0.0001 0.00098 73 85 -0.17 -0.24 8229.00 1049.80 7704.10 -0.0001 0.00097 85 93 -0.24 -0.30 9075.60 846.60 8652.30 -0.0001 0.00094 93 101 -0.30 -0.37 9922.20 846.60 9498.90 -0.0001 0.00092 101 113 -0.37 -0.46 10972.00 1049.80 10447.10 -0.0001 0.00092 113 121 -0.46 -0.54 11818.60 846.60 11395.30 -0.0001 0.00088 121 129 -0.54 -0.63 12665.20 846.60 12241.90 -0.0001 0.00087 129 141 -0.63 -0.75 13715.00 1049.80 13190.10 -0.0001 0.00086 141 149 -0.75 -0.86 14561.60 846.60 14138.30 -0.0001 0.00082 149 157 -0.86 -0.97 15408.20 846.60 14984.90 -0.0001 0.00081 157 169 -0.97 -1.11 16458.00 1049.80 15933.10 -0.0001 0.00080 169 177 -1.11 -1.24 17304.60 846.60 16881.30 -0.0002 0.00077 177 185 -1.24 -1.38 18151.20 846.60 17727.90 -0.0002 0.00075 185 197 -1.38 -1.55 19201.00 1049.80 18676.10 -0.0002 0.00074 197 205 -1.55 -1.70 20047.60 846.60 19624.30 -0.0002 0.00071 205 213 -1.70 -1.86 20894.20 846.60 20470.90 -0.0002 0.00069 213 225 -1.86 -2.06 21944.00 1049.80 21419.10 -0.0002 0.00069 225 233 -2.06 -2.23 22790.60 846.60 22367.30 -0.0002 0.00066 233 241 -2.23 -2.41 23637.20 846.60 23213.90 -0.0002 0.00064 241 253 -2.41 -2.64 24687.00 1049.80 24162.10 -0.0002 0.00063 253 261 -2.64 -2.83 25533.60 846.60 25110.30 -0.0002 0.00060 261 269 -2.83 -3.03 26380.20 846.60 25956.90 -0.0002 0.00059 269 281 -3.03 -3.28 27430.00 1049.80 26905.10 -0.0002 0.00058 4 12 0.00 -0.90 846.60 846.60 423.30 -0.0011 0.00113 12 20 -0.90 -1.79 1693.20 846.60 1269.90 -0.0010 0.00109 20 32 -1.79 -2.89 2743.00 1049.80 2218.10 -0.0010 0.00109 32 40 -2.89 -3.76 3589.60 846.60 3166.30 -0.0010 0.00105 40 48 -3.76 -4.62 4436.20 846.60 4012.90 -0.0010 0.00104 48 60 -4.62 -5.69 5486.00 1049.80 4961.10 -0.0010 0.00103 60 68 -5.69 -6.53 6332.60 846.60 5909.30 -0.0010 0.00099 68 76 -6.53 -7.37 7179.20 846.60 6755.90 -0.0010 0.00098 76 88 -7.37 -8.40 8229.00 1049.80 7704.10 -0.0010 0.00097 88 96 -8.40 -9.22 9075.60 846.60 8652.30 -0.0010 0.00094 96 104 -9.22 -10.04 9922.20 846.60 9498.90 -0.0010 0.00092 104 116 -10.04 -11.04 10972.00 1049.80 10447.10 -0.0010 0.00092 116 124 -11.04 -11.84 11818.60 846.60 11395.30 -0.0009 0.00088 124 132 -11.84 -12.63 12665.20 846.60 12241.90 -0.0009 0.00087 132 144 -12.63 -13.60 13715.00 1049.80 13190.10 -0.0009 0.00086 144 152 -13.60 -14.37 14561.60 846.60 14138.30 -0.0009 0.00082 152 160 -14.37 -15.14 15408.20 846.60 14984.90 -0.0009 0.00081 160 172 -15.14 -16.08 16458.00 1049.80 15933.10 -0.0009 0.00080 225 Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 172 180 -16.08 -16.83 17304.60 846.60 16881.30 -0.0009 0.00077 180 188 -16.83 -17.57 18151.20 846.60 17727.90 -0.0009 0.00075 188 200 -17.57 -18.48 19201.00 1049.80 18676.10 -0.0009 0.00074 200 208 -18.48 -19.21 20047.60 846.60 19624.30 -0.0009 0.00071 208 216 -19.21 -19.92 20894.20 846.60 20470.90 -0.0008 0.00069 216 228 -19.92 -20.81 21944.00 1049.80 21419.10 -0.0008 0.00069 228 236 -20.81 -21.51 22790.60 846.60 22367.30 -0.0008 0.00066 236 244 -21.51 -22.20 23637.20 846.60 23213.90 -0.0008 0.00064 244 256 -22.20 -23.06 24687.00 1049.80 24162.10 -0.0008 0.00063 256 264 -23.06 -23.74 25533.60 846.60 25110.30 -0.0008 0.00060 264 272 -23.74 -24.41 26380.20 846.60 25956.90 -0.0008 0.00059 272 284 -24.41 -25.24 27430.00 1049.80 26905.10 -0.0008 0.00058 Col. length: 952.5 mm Table C.53 Curvatures up to the mid-height of Wall 1 combined with Column 1 for a drift of 2% Node Node 1 y- Node 2 y- Node 2 y- Difference Height (mm) Strain Curvature (rad/m) First Second displ (mm) displ (mm) coord (mm) (mm) 1 61 0.00 15.45 846.60 846.60 423.30 0.0182 0.00316 61 121 15.45 30.84 1693.20 846.60 1269.90 0.0182 0.00315 121 211 30.84 47.94 2743.00 1049.80 2218.10 0.0163 0.00253 211 271 47.94 60.94 3589.60 846.60 3166.30 0.0154 0.00237 271 331 60.94 72.75 4436.20 846.60 4012.90 0.0140 0.00213 331 421 72.75 83.68 5486.00 1049.80 4961.10 0.0104 0.00160 421 481 83.68 91.00 6332.60 846.60 5909.30 0.0086 0.00137 481 541 91.00 97.35 7179.20 846.60 6755.90 0.0075 0.00120 541 631 97.35 100.38 8229.00 1049.80 7704.10 0.0029 0.00055 631 691 100.38 102.43 9075.60 846.60 8652.30 0.0024 0.00050 691 751 102.43 104.19 9922.20 846.60 9498.90 0.0021 0.00044 751 841 104.19 105.72 10972.00 1049.80 10447.10 0.0015 0.00033 841 901 105.72 107.38 11818.60 846.60 11395.30 0.0020 0.00041 901 961 107.38 108.89 12665.20 846.60 12241.90 0.0018 0.00037 961 1051 108.89 110.33 13715.00 1049.80 13190.10 0.0014 0.00030 1051 1111 110.33 111.76 14561.60 846.60 14138.30 0.0017 0.00035 1111 1171 111.76 113.12 15408.20 846.60 14984.90 0.0016 0.00033 1171 1261 113.12 114.38 16458.00 1049.80 15933.10 0.0012 0.00026 1261 1321 114.38 115.62 17304.60 846.60 16881.30 0.0015 0.00031 1321 1381 115.62 116.77 18151.20 846.60 17727.90 0.0014 0.00029 1381 1471 116.77 117.82 19201.00 1049.80 18676.10 0.0010 0.00023 1471 1531 117.82 118.84 20047.60 846.60 19624.30 0.0012 0.00026 1531 1591 118.84 119.78 20894.20 846.60 20470.90 0.0011 0.00025 1591 1681 119.78 120.63 21944.00 1049.80 21419.10 0.0008 0.00019 1681 1741 120.63 121.44 22790.60 846.60 22367.30 0.0010 0.00022 1741 1801 121.44 122.16 23637.20 846.60 23213.90 0.0009 0.00020 1801 1891 122.16 122.82 24687.00 1049.80 24162.10 0.0006 0.00016 1891 1951 122.82 123.41 25533.60 846.60 25110.30 0.0007 0.00018 1951 2011 123.41 123.95 26380.20 846.60 25956.90 0.0006 0.00016 2011 2101 123.95 124.42 27430.00 1049.80 26905.10 0.0005 0.00013 26 86 0.00 -4.96 846.60 846.60 423.30 -0.0059 0.00316 86 146 -4.96 -9.92 1693.20 846.60 1269.90 -0.0059 0.00315 146 236 -9.92 -13.06 2743.00 1049.80 2218.10 -0.0030 0.00253 236 296 -13.06 -15.36 3589.60 846.60 3166.30 -0.0027 0.00237 296 356 -15.36 -17.29 4436.20 846.60 4012.90 -0.0023 0.00213 356 446 -17.29 -19.12 5486.00 1049.80 4961.10 -0.0017 0.00160 446 506 -19.12 -20.67 6332.60 846.60 5909.30 -0.0018 0.00137 506 566 -20.67 -22.02 7179.20 846.60 6755.90 -0.0016 0.00120 566 656 -22.02 -23.36 8229.00 1049.80 7704.10 -0.0013 0.00055 656 716 -23.36 -24.52 9075.60 846.60 8652.30 -0.0014 0.00050 716 776 -24.52 -25.57 9922.20 846.60 9498.90 -0.0012 0.00044 776 866 -25.57 -26.64 10972.00 1049.80 10447.10 -0.0010 0.00033 866 926 -26.64 -27.60 11818.60 846.60 11395.30 -0.0011 0.00041 926 986 -27.60 -28.50 12665.20 846.60 12241.90 -0.0011 0.00037 986 1076 -28.50 -29.45 13715.00 1049.80 13190.10 -0.0009 0.00030 1076 1136 -29.45 -30.30 14561.60 846.60 14138.30 -0.0010 0.00035 1136 1196 -30.30 -31.10 15408.20 846.60 14984.90 -0.0009 0.00033 1196 1286 -31.10 -31.95 16458.00 1049.80 15933.10 -0.0008 0.00026 227 Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 1286 1346 -31.95 -32.71 17304.60 846.60 16881.30 -0.0009 0.00031 1346 1406 -32.71 -33.43 18151.20 846.60 17727.90 -0.0009 0.00029 1406 1496 -33.43 -34.19 19201.00 1049.80 18676.10 -0.0007 0.00023 1496 1556 -34.19 -34.88 20047.60 846.60 19624.30 -0.0008 0.00026 1556 1616 -34.88 -35.53 20894.20 846.60 20470.90 -0.0008 0.00025 1616 1706 -35.53 -36.21 21944.00 1049.80 21419.10 -0.0006 0.00019 1706 1766 -36.21 -36.82 22790.60 846.60 22367.30 -0.0007 0.00022 1766 1826 -36.82 -37.40 23637.20 846.60 23213.90 -0.0007 0.00020 1826 1916 -37.40 -38.00 24687.00 1049.80 24162.10 -0.0006 0.00016 1916 1976 -38.00 -38.55 25533.60 846.60 25110.30 -0.0006 0.00018 1976 2036 -38.55 -39.06 26380.20 846.60 25956.90 -0.0006 0.00016 2036 2126 -39.06 -39.60 27430.00 1049.80 26905.10 -0.0005 0.00013 Wall length: 7620 mm Table C.54 Curvatures up to the mid-height of Column 1 combined with Wall 1 for a drift of 2% I First â€¢fade Second Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) 27 87 0.00 1.91 846.60 846.60 423.30 0.0023 0.00550 87 147 1.91 3.31 1693.20 846.60 1269.90 0.0017 0.00404 147 237 3.31 4.16 2743.00 1049.80 2218.10 0.0008 0.00238 237 297 4.16 4.37 3589.60 846.60 3166.30 0.0002 0.00164 297 357 4.37 4.49 4436.20 846.60 4012.90 0.0001 0.00141 357 447 4.49 4.52 5486.00 1049.80 4961.10 0.0000 0.00102 447 507 4.52 4.47 6332.60 846.60 5909.30 -0.0001 0.00097 507 567 4.47 4.37 7179.20 846.60 6755.90 -0.0001 0.00083 567 657 4.37 4.19 8229.00 1049.80 7704.10 -0.0002 0.00059 657 717 4.19 3.98 9075.60 846.60 8652.30 -0.0002 0.00055 717 777 3.98 3.75 9922.20 846.60 9498.90 -0.0003 0.00050 777 867 3.75 3.48 10972.00 1049.80 10447.10 -0.0003 0.00039 867 927 3.48 3.21 11818.60 846.60 11395.30 -0.0003 0.00040 927 987 3.21 2.93 12665.20 846.60 12241.90 -0.0003 0.00038 987 1077 2.93 2.63 13715.00 1049.80 13190.10 -0.0003 0.00031 1077 1137 2.63 2.33 14561.60 846.60 14138.30 -0.0004 0.00033 1137 1197 2.33 2.03 15408.20 846.60 14984.90 -0.0004 0.00032 1197 1287 2.03 1.69 16458.00 1049.80 15933.10 -0.0003 0.00026 1287 1347 1.69 1.38 17304.60 846.60 16881.30 -0.0004 0.00028 1347 1407 1.38 1.05 18151.20 846.60 17727.90 -0.0004 0.00027 1407 1497 1.05 0.70 19201.00 1049.80 18676.10 -0.0003 0.00022 1497 1557 0.70 0.37 20047.60 846.60 19624.30 -0.0004 0.00023 1557 1617 0.37 0.03 20894.20 846.60 20470.90 -0.0004 0.00022 1617 1707 0.03 -0.34 21944.00 1049.80 21419.10 -0.0004 0.00018 1707 1767 -0.34 -0.69 22790.60 846.60 22367.30 -0.0004 0.00019 1767 1827 -0.69 -1.05 23637.20 846.60 23213.90 -0.0004 0.00018 1827 1917 -1.05 -1.44 24687.00 1049.80 24162.10 -0.0004 0.00015 1917 1977 -1.44 -1.80 25533.60 846.60 25110.30 -0.0004 0.00016 1977 2037 -1.80 -2.17 26380.20 846.60 25956.90 -0.0004 0.00016 2037 2127 -2.17 -2.56 27430.00 1049.80 26905.10 -0.0004 0.00014 30 90 0.00 -2.53 846.60 846.60 423.30 -0.0030 0.00550 90 150 -2.53 -4.38 1693.20 846.60 1269.90 -0.0022 0.00404 150 240 -4.38 -5.91 2743.00 1049.80 2218.10 -0.0015 0.00238 240 300 -5.91 -7.03 3589.60 846.60 3166.30 -0.0013 0.00164 300 360 -7.03 -8.04 4436.20 846.60 4012.90 -0.0012 0.00141 360 450 -8.04 -9.03 5486.00 1049.80 4961.10 -0.0009 0.00102 450 510 -9.03 -9.86 6332.60 846.60 5909.30 -0.0010 0.00097 510 570 -9.86 -10.63 7179.20 846.60 6755.90 -0.0009 0.00083 570 660 -10.63 -11.39 8229.00 1049.80 7704.10 -0.0007 0.00059 660 720 -11.39 -12.05 9075.60 846.60 8652.30 -0.0008 0.00055 720 780 -12.05 -12.68 9922.20 846.60 9498.90 -0.0007 0.00050 780 870 -12.68 -13.34 10972.00 1049.80 10447.10 -0.0006 0.00039 870 930 -13.34 -13.93 11818.60 846.60 11395.30 -0.0007 0.00040 930 990 -13.93 -14.51 12665.20 846.60 12241.90 -0.0007 0.00038 990 1080 -14.51 -15.13 13715.00 1049.80 13190.10 -0.0006 0.00031 1080 1140 -15.13 -15.69 14561.60 846.60 14138.30 -0.0007 0.00033 1140 1200 -15.69 -16.25 15408.20 846.60 14984.90 -0.0007 0.00032 1200 1290 -16.25 -16.84 16458.00 1049.80 15933.10 -0.0006 0.00026 229 Node Node 1 y-displ (mm) Node 2 y-displ (mm) Node 2 y-coord (mm) Difference (mm) Height (mm) Strain Curvature (rad/m) First Second 1290 1350 -16.84 -17.38 17304.60 846.60 16881.30 -0.0006 0.00028 1350 1410 -17.38 -17.92 18151.20 846.60 17727.90 -0.0006 0.00027 1410 1500 -17.92 -18.49 19201.00 1049.80 18676.10 -0.0005 0.00022 1500 1560 -18.49 -19.01 20047.60 846.60 19624.30 -0.0006 0.00023 1560 1620 -19.01 -19.53 20894.20 846.60 20470.90 -0.0006 0.00022 1620 1710 -19.53 -20.08 21944.00 1049.80 21419.10 -0.0005 0.00018 1710 1770 -20.08 -20.59 22790.60 846.60 22367.30 -0.0006 0.00019 1770 1830 -20.59 -21.09 23637.20 846.60 23213.90 -0.0006 0.00018 1830 1920 -21.09 -21.63 24687.00 1049.80 24162.10 -0.0005 0.00015 1920 1980 -21.63 -22.12 25533.60 846.60 25110.30 -0.0006 0.00016 1980 2040 -22.12 -22.61 26380.20 846.60 25956.90 -0.0006 0.00016 2040 2130 -22.61 -23.14 27430.00 1049.80 26905.10 -0.0005 0.00014 Col. length: 952.5 mm UBC Civil Engineering Grad Students, 2004/2005 (Contains some students who graduated in 2003/04. Many students did not have their photo taken) LEGEND: SPECIALIZATION GROUP ID CM Construction Management EF Environmental Fluid Mechanics EQ Earthquake Engineering G Geotechnical Engineering GE Environmental Geotechnics H Hydrotechnical Engineering M Materials Engineering PC Pollution Control S Structural Engineering T Transportation Engineering Reem Hameed Abdul-Hafidh MEng C M Yapo Alle-Ando M A S c E F AH Amini Asalemi PhD G Parmeshwaree Bahadoorsingh PhD P C Armin Bebamzadeh M A S c S Chrisopher Borstad M A S c EF Alireza Biparva M A S c M Christian Brumpton MEng P C Cynthia Evelyn Bluteau M A S c E F Kelly Lynn Bush M A S c P C Alfredo Guillemo Bohl M A S c S ,00 i ^ ^r^h^o Jess ica Erin Campbell M A S c G
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Plastic hinge length in high-rise concrete shear walls Bohl, Alfredo Guillermo
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Title | Plastic hinge length in high-rise concrete shear walls |
Creator |
Bohl, Alfredo Guillermo |
Publisher | University of British Columbia |
Date Issued | 2006 |
Description | The flexural displacement capacity of a concrete shear wall depends on the length of the plastic hinge. Typical building codes and several researchers recommend the use of an equivalent plastic hinge length at the base equal to half the wall length. However, the plastic hinge length is also influenced by parameters other than the wall length. There are currently no recommendations on what should be the plastic hinge length for parallel walls of different lengths in a high-rise building. A parametric study was conducted to investigate the parameters that affect the length of the plastic hinge in concrete walls. The walls were analyzed using program VecTor2. The analytical model was validated with tests results performed on wall specimens. The results obtained show that the inelastic curvatures vary linearly over the plastic hinge length. The shape of the strain profile in slender walls after cracking depends on the amount of reinforcement. Longer walls have larger plastic hinge lengths than shorter walls. Compressive axial loads reduce the plastic hinge length, tensile axial loads have the opposite effect. A simple shear model was proposed to estimate the increase in plastic hinge length when the shear stresses are high. Walls of different lengths interconnected by rigid slabs at various levels have different curvature distributions and plastic hinge lengths. The curvatures in the longer wall do not change whether it is alone or combined with a wall of shorter length. The shorter length wall is subjected to larger curvatures at the base when it is combined. A simple model was proposed to predict the maximum curvature in the shorter wall. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-02-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0063296 |
URI | http://hdl.handle.net/2429/31861 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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