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Seismic performance of buildings with permanent lateral demands Dupuis, Michael Robert Leo 2012

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    Seismic Performance of Buildings with Permanent Lateral Demands  by  Michael Robert Leo Dupuis B.Sc., Queen’s University, 2010   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 (Vancouver)  December 2012 © Michael Robert Leo Dupuis, 2012   ii  Abstract Architectural features and irregularities in the gravity system which apply permanent lateral demands to the seismic force resisting system are being incorporated in new buildings. There permanent lateral demands raised concerns within the Standing Committee on Earthquake Design due to the perceived potential for a seismic ratcheting effect to occur during seismic loading.  Nonlinear, parametric analyses were conducted in OpenSees to investigate the inelastic response of cantilevered and coupled shear wall buildings. The sensitivity of these buildings to permanent lateral demands was investigated across a domain of structural parameters including building height, building strength, and permanent lateral demands. Additional case studies considered the effect of vertical ground motions, subduction ground motions, coupling ratio, seismic demands, and investigated the behaviour of steel braced frame buildings.  The results demonstrate that a seismic ratcheting effect can develop and amplify inelastic deformation demands. The extent of ratcheting increases with the permanent lateral demands and is also highly dependent upon the hysteretic behaviour exhibited by the structural system. Systems with fat hysteresis - such as coupled shear walls and steel braced frames - demonstrate greater ratcheting than systems with flag-shaped hysteresis - such as cantilevered shear walls.  An irregularity class is proposed for the National Building Code of Canada which will limit the allowable permanent lateral demands when IEFaSa(0.2) ≥ 0.50. It is suggested to limit permanent lateral demands in coupled shear wall buildings and steel braced frame buildings to 10% of the yield strength required to resist earthquake loads. Cantilevered shear wall buildings may be subjected to larger permanent lateral demands corresponding to 40% of the yield strength required to resist earthquake loads. The more conservative limit of 10% is recommended for seismic force resisting systems, such as moment frames, which were not considered in this study. Within these limits, it is recommended that an amplification factor of 1.5 - accounting for the increased inelastic deformation demands - be applied to design deformations. The extent of ratcheting in buildings with permanent lateral demands exceeding the proposed limits is large and variable; therefore, the performance of such buildings should be validated with nonlinear dynamic analysis.    iii  Preface The OpenSees code for the analytical models used in this thesis was adopted from previous analytical models created by Mr. Best during an exploratory study. Mr. Best provided the original code for both single- and multiple-degree-of-freedom systems.  The OpenSees code for steel braced frame buildings was adapted from OpenSees models provided by Dr. Robert Tremblay and Dr. Carmen Izvernari from the University of Montreal.  From the original OpenSees code provided by Mr. Best, Dr. Tremblay, and Dr. Izvernari, significant additions and alterations were made. Seven additional material models were considered in the single- degree-of-freedom system and p-delta effects were modelled. The multiple-degree-of-freedom system was expanded to consider steel brace frame and coupled shear wall buildings, and fibre sections were implemented in both shear wall models.    iv  Table of Contents Abstract…………………………………………………………………………………….……………………………....….…ii Preface…………………………………………………………………………………………..………………...….…....….…iii List of Tables………………………………………..……………………………...………..………………...….…....….…viii List of Figures………………………………………………………………………………..….……………...….…....….….xi List of Symbols and Abbreviations …………………………………………………..………………...….……...….xxvi Acknowledgements…………………………………………………………………………..……………....…....….…xxviii Chapter 1 - Introduction ....................................................................................................................... 1 1.1 Impetus for Study .............................................................................................................................. 1 1.2 Objective of Thesis ............................................................................................................................ 3 1.3 Overview of Methodology ................................................................................................................. 3 1.4 Organization of Thesis ....................................................................................................................... 4 Chapter 2 - Background ........................................................................................................................ 6 2.1 SDOF System .................................................................................................................................... 6 2.1.1 Results and Conclusions ................................................................................................................. 7 2.1.2 Limitations and Recommendations ................................................................................................. 7 2.2 MDOF System................................................................................................................................... 8 2.2.1 Results and Conclusions ............................................................................................................... 10 2.2.2 Limitations and Recommendations ............................................................................................... 10 Chapter 3 - SDOF Study ..................................................................................................................... 12 3.1 Methodology ................................................................................................................................... 12 3.1.1 Model ........................................................................................................................................... 12 3.1.2 P-Δ Effects ................................................................................................................................... 14 3.1.3 Relative Strength Factor ................................................................................................................ 15 3.1.4 Applied Demand Ratio.................................................................................................................. 16 3.1.5 Ground Motions ............................................................................................................................ 17 3.1.6 Strength Allocation ....................................................................................................................... 18 3.1.7 Hysteretic Models ......................................................................................................................... 20 3.1.7.1 Clough ....................................................................................................................................... 21 3.1.7.2 Elastic Bi-Linear ........................................................................................................................ 24   v  3.1.7.3 Elastic Perfectly-Plastic.............................................................................................................. 26 3.1.7.4 Korchinski ................................................................................................................................. 27 3.2 Results and Discussion .................................................................................................................... 29 3.2.1 Influence of Hysteretic Behaviour ................................................................................................. 29 3.2.2 Inelastic Demand Amplifications .................................................................................................. 32 3.2.3 Effect of the Relative Strength Factor............................................................................................ 41 3.2.4 Variability .................................................................................................................................... 42 3.2.5 The Effect of Reverse Yield Strength ............................................................................................ 43 3.3 Conclusions ..................................................................................................................................... 49 Chapter 4 - Methodology .................................................................................................................... 54 4.1 Ground Motion Scaling.................................................................................................................... 57 4.2 SFRS Strength and Demand Relationships ....................................................................................... 60 4.3 Gravity System ................................................................................................................................ 62 4.4 Elastic Shear Wall ........................................................................................................................... 63 4.5 Plastic Hinge ................................................................................................................................... 63 4.6 Weight and Mass Distribution .......................................................................................................... 67 4.7 Gravity System Irregularities ........................................................................................................... 68 4.8 Initial Camber .................................................................................................................................. 71 4.9 Cantilevered Shear Walls ................................................................................................................. 71 4.9.1 Building Weights and Overturning Moments ................................................................................ 72 4.9.2 Effect of Axial Load in Shear Wall ............................................................................................... 73 4.9.3 Fibre Section ................................................................................................................................ 78 4.9.4 Pushover Response ....................................................................................................................... 82 4.9.5 Hysteretic Response ...................................................................................................................... 83 4.9.6 Ductility Capacity ......................................................................................................................... 84 4.10 Coupled Shear Walls ..................................................................................................................... 84 4.10.1 Building Weights and Overturning Moments............................................................................... 86 4.10.2 Coupling Beam Dimensions ........................................................................................................ 87 4.10.3 Coupling Beam Strength ............................................................................................................. 88 4.10.4 Coupling Beam Stiffness ............................................................................................................. 90 4.10.5 Coupling Beam Inelastic Response ............................................................................................. 91 4.10.6 Fibre Section............................................................................................................................... 92   vi  4.10.7 Pushover Response ..................................................................................................................... 94 4.10.8 Hysteretic Response .................................................................................................................... 98 4.10.9 Coupling Beam Response ......................................................................................................... 102 4.10.10 Ductility Capacity ................................................................................................................... 105 Chapter 5 - Results and Discussion ................................................................................................... 107 5.1 Cantilevered Shear Wall Buildings ................................................................................................ 108 5.1.1 Hysteretic Behaviour .................................................................................................................. 108 5.1.2 Maximum Base Curvature .......................................................................................................... 116 5.1.3 Maximum Roof Drift .................................................................................................................. 120 5.1.4 Residual Roof Drifts ................................................................................................................... 124 5.1.5 Discussion .................................................................................................................................. 129 5.2 Coupled Shear Wall Buildings ....................................................................................................... 134 5.2.1 Hysteretic Behaviour .................................................................................................................. 135 5.2.2 Maximum Base Curvature .......................................................................................................... 142 5.2.3 Maximum Roof Drift .................................................................................................................. 146 5.2.4 Residual Roof Drift ..................................................................................................................... 150 5.2.5 Discussion .................................................................................................................................. 155 Chapter 6 - Case Studies ................................................................................................................... 159 6.1 Gravity System Irregularity ............................................................................................................ 159 6.2 Strengthened Coupling Beams ....................................................................................................... 164 6.3 Seismic Zone ................................................................................................................................. 169 6.4 Subduction Ground Motions .......................................................................................................... 177 6.5 Vertical Ground Motions ............................................................................................................... 180 6.6 Steel Braced Frame Buildings ........................................................................................................ 186 6.6.1 SFRS Strength and Demand Relationships .................................................................................. 188 6.6.2 Building Design .......................................................................................................................... 190 6.6.3 Bracing Members ....................................................................................................................... 192 6.6.4 Beams and Columns ................................................................................................................... 192 6.6.5 Ductility Capacity ....................................................................................................................... 192 6.6.6 Extent of Seismic Ratcheting ...................................................................................................... 193 Chapter 7 - Recommendations .......................................................................................................... 206   vii  7.1 Current Study ................................................................................................................................ 206 7.2 Future Study .................................................................................................................................. 217 7.2.1 Axial Load in Shear Wall ............................................................................................................ 218 7.2.2 Shear Wall Section Geometry ..................................................................................................... 218 7.2.3 Coupling Beam Strength ............................................................................................................. 218 7.2.4 Refined Material Models............................................................................................................. 219 7.2.5 Shear Dominated Walls .............................................................................................................. 219 7.2.6 Additional SFRSs ....................................................................................................................... 220 7.2.7 Unique Mitigating Measures ....................................................................................................... 220 7.2.8 Seismic Hazard ........................................................................................................................... 221 Chapter 8 - Summary and Conclusions ............................................................................................ 222 References .......................................................................................................................................... 225 Appendices ......................................................................................................................................... 228 A.1 Ground Motions ............................................................................................................................ 228 A.2 Scaling of Ground Motions for MDOF Analyses ........................................................................... 234 A.3 Columns Inclinations .................................................................................................................... 239 A.4 Cantilevered Shear Wall Sections.................................................................................................. 248 A.5 Cantilevered Shear Wall Building Data ......................................................................................... 252 A.6 Coupled Shear Wall Sections ........................................................................................................ 255 A.7 Coupled Shear Wall Building Data................................................................................................ 258     viii  List of Tables Table 4.1: Height and floor plate dimensions of the six concrete wall building heights considered. ........ 57 Table 4.2: Concrete properties modelled for each building height. ......................................................... 57 Table 4.3: Plastic hinge lengths for different building heights. ............................................................... 64 Table 4.4: β for the roof drift demands in a 30 storey coupled shear wall building with each of the three types of structural irregularities. α = 0.2, Rα=0 = 4.0, subjected to the ten ground motions. ...................................................................................................................................... 69 Table 4.5: Building weights for the cantilevered shear wall models. ....................................................... 72 Table 4.6: Base overturning moments for the cantilevered wall models. ................................................. 72 Table 4.7: β for 5 storey cantilevered shear wall building with varying axial loads, Rα=0 = 4.0, α = 0.4. .............................................................................................................................................. 78 Table 4.8: Maximum Rα=0 considered for each cantilevered shear wall building height. .......................... 81 Table 4.9: Base curvature capacities of cantilevered wall sections. ......................................................... 84 Table 4.10: Building weights for the coupled shear wall models. ........................................................... 87 Table 4.11: Base overturning moments for the coupled wall models....................................................... 87 Table 4.12: Typical buildings in Vancouver with associated freestanding storeys (storeys above podia) and coupling ratios. Obtained from consultations with Bob Neville, P. Eng, and Svetlana Uranova, P. Eng, of Read Jones Christoffersen in Vancouver, May 2012. ...................... 89 Table 4.13: Base curvature capacities (rad/km) of the compression wall in the coupled shear wall systems. ..................................................................................................................................... 105 Table 6.1: Mean and maximum β for the maximum base curvature demands in a 30 storey coupled shear wall building with each of the three types of structural irregularities. α = 0.2, Rα=0 = 4.0, subjected to ten ground motions. ......................................................................................... 163 Table 6.2: Mean and maximum β for the maximum roof drift demands in a 30 storey coupled shear wall building with each of the three types of structural irregularities. α = 0.2, Rα=0 = 4.0, subjected to ten ground motions. ................................................................................................ 164 Table 6.3: Mean and maximum β for maximum base curvatures in the 30 storey coupled shear wall buildings with three considered coupling ratios for α = 0.2, Rα=0 = 4.0, subjected to ten ground motions. ......................................................................................................................... 167 Table 6.4: Mean and maximum β for maximum roof drifts in the 30 storey coupled shear wall buildings with three considered coupling ratios for α = 0.2, Rα=0 = 4.0, subjected to ten ground motions. ......................................................................................................................... 169   ix  Table 6.5: Mean and maximum β for maximum roof drifts in the 30 storey coupled shear wall building when located Vancouver and Victoria, α = 0.2, Rα=0 = 4.0, subjected to ten ground motions. .................................................................................................................................... 173 Table 6.6: Mean and maximum β for maximum roof drifts in the 10 storey coupled shear wall building when located in each of four considered Canadian cities, α = 0.2, Rα=0 = 2.0, subjected to ten ground motions. ................................................................................................ 176 Table 6.7: Mean roof drifts in the 10 storey coupled shear wall building when located in each of four considered Canadian cities, α = 0.0 and 0.2, Rα=0 = 2.0, subjected to ten ground motions. ... 176 Table 6.8: β for the maximum roof drift demands imposed in each of the four considered cases for each ground motion with and without vertical ground motion considered, α = 0.2, Rα=0 = 4.0. .... 184 Table 6.9: Maximum axial demands at the base of the exterior columns on the compression face of the 30 storey coupled shear wall building with Rα=0 = 4.0. .......................................................... 185 Table 6.10: Maximum axial demands at the base of the exterior columns on the tension face of the 30 storey coupled shear wall building with Rα=0 = 4.0. ............................................................... 186 Table 4.14: Building properties, MD: Moderately Ductile, LD: Limited Ductility, (Izvernari, Lacerte, & Tremblay, 2007). ...................................................................................................... 191 Table A.2.1: Ground motion scaling factors for cantilevered shear wall buildings. ............................... 237 Table A.2.2: Ground motion scaling factors for coupled shear wall buildings. ...................................... 238 Table A.3.1: Column inclination in the 5 storey cantilevered shear wall models. .................................. 240 Table A.3.2: Column inclination in the 10 storey cantilevered shear wall models. ................................ 240 Table A.3.3: Column inclination in the 20 storey cantilevered shear wall models. ................................ 241 Table A.3.4: Column inclination in the 30 storey cantilevered shear wall models. ................................ 241 Table A.3.5: Column inclination in the 40 storey cantilevered shear wall models. ................................ 242 Table A.3.6: Column inclination in the 50 storey cantilevered shear wall models. ................................ 242 Table A.3.7: Column inclination in the 5 storey coupled shear wall models.......................................... 243 Table A.3.8: Column inclination in the 10 storey coupled shear wall models. ....................................... 243 Table A.3.9: Column inclination in the 20 storey coupled shear wall models. ....................................... 244 Table A.3.10: Column inclination in the 30 storey coupled shear wall models. ..................................... 244 Table A.3.11: Column inclination in the 40 storey coupled shear wall models. ..................................... 245 Table A.3.12: Column inclination in the 50 storey coupled shear wall models. ..................................... 245 Table A.3.13: Column inclination in the 8 storey steel braced frame models. ....................................... 246 Table A.3.14: Column inclination in the 12 storey steel braced frame models. ..................................... 246 Table A.3.15: Column inclination in the 16 storey steel braced frame models. ..................................... 247 Table A.5.1: Additional cantilevered shear wall data. ........................................................................... 252   x  Table A.5.2: Section properties and floor weights for the cantilevered shear wall models. .................... 252 Table A.5.3: Building weights for the cantilevered shear wall models. ................................................. 253 Table A.5.4: Base overturning moments for the cantilevered wall models. ........................................... 253 Table A.5.5: Overturning moment yield strength of the cantilevered shear wall buildings. ................... 254 Table A.7.1: Additional coupled shear wall data. ................................................................................. 258 Table A.7.2: Calculated section properties and floor weights for the coupled shear wall models. .......... 258 Table A.7.3: Building weights for the coupled shear wall models. ........................................................ 259 Table A.7.4: Base overturning moments for the coupled wall models. .................................................. 259 Table A.7.5: Overturning moment yield strength of the coupled shear wall buildings. .......................... 260    xi  List of Figures Figure 1.1: Inclined façade in a high-rise building. ................................................................................... 1 Figure 1.2: Schematic illustrating external (left) and internal (right) demands which result from inclined columns............................................................................................................................ 2 Figure 2.1: The idealized SDOF model (Best, Elwood, & Anderson, 2011). ............................................. 6 Figure 2.2: Clough hysteretic model used in the SDOF model with labelled tensile force yield plateau, Fyt, and compression force yield plateau, Fyc (Best, Elwood, & Anderson, 2011). .............. 7 Figure 2.3: Loading mechanisms modelled in exploratory study: fully inclined columns (left), inclined lobby (center), and eccentric floor spans (right) (Best, Elwood, & Anderson, 2011). ......... 9 Figure 2.4: Bending moment profiles for the three loading configurations modelled: fully inclined (left), inclined lobby (center), and eccentric floor spans (right) (Best, Elwood, & Anderson, 2011). ............................................................................................................................................ 9 Figure 3.1: SDOF system configuration. ................................................................................................ 14 Figure 3.2: Displacement-time histories of two identical SDOF systems with an otherwise identical ground motion applied in the forward and reverse directions. This analysis was done on a system with a period of 1.25 seconds, α = 0.4, R α=0 = 6.0, and the axial spring element was assigned Clough hysteretic model properties. ............................................................................... 17 Figure 3.3: Adjusted hysteretic models with (left) and without (right) reduced reverse yield strength (Best, Elwood, & Anderson, 2011). ............................................................................................. 19 Figure 3.4: Effect of a PLD on the required forward and reverse yield strengths of the SFRS. Original SFRS with α = 0.0 (left), SFRS with α = 0.4 and an unaltered reverse yield strength (center), and a system with α = 0.4 and a reduced reverse yield strength (right). ........................... 20 Figure 3.5: General backbone of the Clough model with partial unloading path indicated, © Mazzoni et al (2007), by permission. ........................................................................................... 21 Figure 3.6: Hysteretic behaviour of the Clough model with no unloading stiffness degradation, Clough, No Stiffness Degradation. ............................................................................................... 22 Figure 3.7: Hysteretic behaviour of the Clough model with mild unloading stiffness degradation, Clough, Mild Stiffness Degradation. ............................................................................................ 23 Figure 3.8: Hysteretic behaviour of the Clough model with moderate unloading stiffness degradation, Clough, Moderate Stiffness Degradation. ................................................................. 23 Figure 3.9: Hysteretic behaviour of the Clough model with large unloading stiffness degradation, Clough, Severe Stiffness Degradation. ......................................................................................... 24   xii  Figure 3.10: General backbone of the elastic bi-linear model, © Mazzoni et al (2007), by permission. .................................................................................................................................. 25 Figure 3.11: Hysteretic behaviour of the elastic bi-linear model, Elastic Bi-linear. ................................. 25 Figure 3.12: General backbone of the elastic perfectly-plastic model, © Mazzoni et al (2007), by permission. .................................................................................................................................. 26 Figure 3.13: Hysteretic behaviour of the elastic perfectly-plastic model, Elastic Perfectly Plastic........... 27 Figure 3.14: Hysteretic behaviour of Korchinski model, Modified Korchinski. ....................................... 28 Figure 3.15: Hysteretic behaviour of Korchinski model without hysteretic energy dissipation, Modified Korchinski, No Energy Dissipation. .............................................................................. 29 Figure 3.16: Effect of asymmetrical strengths to account for PLDs, corresponding to α = 0.8, on the inelastic behaviour of the Clough, No Stiffness Degradation hysteretic model. ............................. 30 Figure 3.17: Effect of asymmetrical strengths to account for PLDs, corresponding to α = 0.8, on hysteretic behaviour of the Elastic Perfectly Plastic hysteretic model. .......................................... 31 Figure 3.18: Force – Displacement hysteretic response of the SDOF systems with each of the four Clough hysteretic models when subjected to ground motion 2, T = 1.0 seconds, α = 0.4, R α=0 = 4.0. ........................................................................................................................................... 32 Figure 3.19: γ for the eight hysteretic models considered in the SDOF systems, α = 0.4, and R α=0 = 4.0. .............................................................................................................................................. 33 Figure 3.20: β produced by the eight hysteretic models considered in the SDOF systems, excluding P-∆ effects, α = 0.4, R α=0 = 4.0. ................................................................................................... 35 Figure 3.21: β produced by each of the eight hysteretic models considered in the SDOF systems, α = 0.4, R α=0 = 4.0. ......................................................................................................................... 36 Figure 3.22: Comparison of the SDOF and MDOF systems with the Clough, No Stiffness Degradation hysteretic model, with and without P-∆ effects, R α=0 = 4.0, α = 0.4. ......................... 37 Figure 3.23: β for the SDOF systems with the Clough, No Stiffness Degradation hysteretic model and varying α, R α=0 = 4.0. ............................................................................................................ 39 Figure 3.24: β for the SDOF systems with the Elastic Bi-Linear and Elastic Perfectly-Plastic hysteretic models and varying α, R α=0 = 4.0. The four collapses occurred for the Elastic Perfectly-Plastic hysteretic model regardless of PLDs. ................................................................. 40 Figure 3.25: β for the SDOF systems with the Modified Korchinski hysteretic model and varying α, R α=0 = 4.0. ................................................................................................................................... 41 Figure 3.26: β for the SDOF systems with the Clough, No Stiffness Degradation hysteretic model and varying R α=0, α = 0.4. ............................................................................................................ 42   xiii  Figure 3.27: β for the SDOF systems subjected to each of the ten ground motions with the Clough, No Stiffness Degradation hysteretic model, R α=0 = 4.0, α = 0.4. Also indicated are the mean, µ, and mean plus one standard deviation, µ + σ............................................................................ 43 Figure 3.28: Effect of a PLD on the Elastic Perfectly-Plastic hysteretic model for both reduced and unaltered reverse yield strengths. Original hysteretic model for α = 0.0 (left), α = 0.4 with an unaltered reverse yield strength (center), and α = 0.4 with a reduced reverse yield strength (right). ......................................................................................................................................... 44 Figure 3.29: β for the SDOF systems with the Elastic Perfectly-Plastic hysteretic model and two considered reverse yield strength philosophies, α = 0.4, R α=0 = 4.0. .............................................. 45 Figure 3.30: Hysteretic response of a SDOF system with the Elastic Perfectly-Plastic hysteretic model to ground motion 2, α = 0.0, R α=0 = 4.0, T = 1.0 seconds. .................................................. 46 Figure 3.31: Hysteretic response of a SDOF system with the Elastic Perfectly-Plastic hysteretic model to ground motion 2. The reverse yield strength has been reduced to account for the PLD, α = 0.4, R α=0 = 4.0, T = 1.0 seconds. ................................................................................... 47 Figure 3.32: Hysteretic response of a SDOF system with the Elastic Perfectly-Plastic hysteretic model to ground motion 2. The reverse yield strength has been unaltered despite the PLD, α = 0.4, R α=0 = 4.0, T = 1.0 seconds. ............................................................................................... 48 Figure 3.33: β for the SDOF systems with the Clough, No Stiffness Degradation hysteretic model and two considered reverse yield strength philosophies, α = 0.4, R α=0 = 4.0. ................................ 49 Figure 3.34: Effect of asymmetrical strengths to account for PLDs, corresponding to α = 0.8, on hysteretic behaviour of the Elastic Perfectly Plastic hysteretic model. .......................................... 50 Figure 3.35: Effect of a PLD, corresponding to α = 0.8, on the inelastic behaviour of the Clough, No Stiffness Degradation hysteretic model. .................................................................................. 51 Figure 3.36: Effect of PLD, corresponding to α = 0.8, on the hysteretic behaviour of the Clough, Moderate Stiffness Degradation hysteretic model. ....................................................................... 52 Figure 4.1: Structural configuration of 5 storey cantilevered concrete shear wall model. ........................ 55 Figure 4.2: Structural configuration of 5 storey coupled concrete shear wall model. ............................... 56 Figure 4.3: Acceleration response spectra for ten ground motions scaled to match the maximum elastic base overturning moment imposed by the Vancouver UHS on the 5storey cantilevered shear wall building. ..................................................................................................................... 59 Figure 4.4: Acceleration response spectra for ten ground motions scaled to match the maximum elastic base overturning moment imposed by the Vancouver UHS on the 50 storey cantilevered shear wall building. .................................................................................................. 60   xiv  Figure 4.5: Schematic of the general parametric relationship between the demands and capacities of the shear wall systems. The relative magnitudes shown correspond to a building with R α=0 = 2.0 and α = 0.5. ............................................................................................................................ 62 Figure 4.6: General backbone of the uniaxialMaterial Concrete01 material model, © Mazzoni et al (2007), by permission. ................................................................................................................. 65 Figure 4.7: General backbone of the uniaxialMaterial Hysteretic model, © Mazzoni et al (2007), by permission. .................................................................................................................................. 66 Figure 4.8: Hysteretic behaviour of longitudinal reinforcing steel when subjected to monotonous increasing cyclic displacements. .................................................................................................. 67 Figure 4.9: 10 storey cantilevered shear wall buildings with fully inclined columns corresponding to α = 0.2(left) and α = 0.8 (right) at Rα=0 = 2.0. ........................................................................... 70 Figure 4.10: 10 storey coupled shear wall buildings with fully inclined columns corresponding to α = 0.2(left) and α = 0.8 (right) at Rα=0 = 2.0. .................................................................................. 70 Figure 4.11: Overturning moment resistance mechanisms in a cantilevered shear wall, where M1 is the flexural moment developed at the base of the wall, © Composites, by permission (Park & Yun, 2011). ................................................................................................................................. 71 Figure 4.12: Base moment-curvature hysteretic response of a 5 storey cantilevered shear wall building with varying axial load in the shear wall, Rα=0 = 4.0, α = 0.0. Axial loads in the shear walls are, from top left to bottom right: zero, wall self-weight, wall self-weight plus 10% of floor weight, wall self-weight plus 20% of floor weight, wall self-weight plus 30% of floor weight, and wall self-weight plus 40% of floor weight. ........................................................ 75 Figure 4.13: Base moment-curvature hysteretic response of a 5 storey cantilevered shear wall building with varying axial load in the shear wall, Rα=0 = 4.0, α = 0.4. Axial loads in the shear walls are, from top left to bottom right: zero, wall self-weight, wall self-weight plus 10% of floor weight, wall self-weight plus 20% of floor weight, wall self-weight plus 30% of floor weight, and wall self-weight plus 40% of floor weight. ........................................................ 77 Figure 4.14: Schematic of a generalized cantilevered shear wall section with distributed and concentrated longitudinal reinforcing steel, unconfined concrete, and confined concrete. ............. 80 Figure 4.15: Base moment as a function of base curvature during pushover analyses of 5, 10, 20, and 30 storey cantilevered shear walls, Rα=0 = 2.0, α = 0.0............................................................ 82 Figure 4.16: Base moment-curvature hysteretic responses of 5, 10, 20, and 30 storey cantilevered shear walls, Rα=0= 2.0, α = 0.0. ..................................................................................................... 83   xv  Figure 4.17: Overturning moment resistance mechanism in a coupled shear wall, where M1 and M2 are flexural moments developed at the base of each wall, and T and C are axial forces developed by the coupling beams, © Composites, by permission (Park & Yun, 2011). ................. 85 Figure 4.18: Typical coupling beam cross-sections with horizontal (left) and diagonal (right) reinforcing schemes, © Canadian Journal of Civil Engineering, by permission (Chaallal & Gauthier, 2000)............................................................................................................................ 86 Figure 4.19: Coupling beam configuration in coupled wall model. ......................................................... 88 Figure 4.20: Hysteretic moment-curvature response of a coupling beam at mid height in a 5 storey building, Rα=0 = 2.0, α = 0.0. ........................................................................................................ 92 Figure 4.21: Schematic of a generalized coupled shear wall section with distributed and concentrated longitudinal reinforcing steel, unconfined concrete, and confined concrete. ............. 93 Figure 4.22: Overturning moment as a function of base curvature in the compression wall during pushover analyses of 5, 10, 20, and 30 storey coupled shear walls, Rα=0 = 2.0, α = 0.0. ................. 94 Figure 4.23: Overturning moment as a function of roof displacement during pushover analyses of 5, 10, 20, and 30 storey coupled shear walls, Rα=0 = 2.0, α = 0.0. ...................................................... 95 Figure 4.24: Overturning moment as a function of roof displacement for the 5 storey coupled shear wall model in each coupled wall and for the overall building, Rα=0 = 2.0, α = 0.0.......................... 96 Figure 4.25: Overturning moment as a function of roof displacement for the 30 storey coupled shear wall model in each coupled wall and for the overall building, Rα=0 = 2.0, α = 0.0................. 97 Figure 4.26: Coupling ratio as a function of roof displacement during the static pushover analyses of 5, 10, 20, 30, 40, and 50 storey coupled shear wall buildings designed for a coupling ratio of 0.7, Rα=0 = 2.0, α = 0.0. ............................................................................................................ 98 Figure 4.27: Base moment as a function of axial load in a shear wall of a 5 storey coupled shear wall building during a time-history analysis, Rα=0 = 2.0, α = 0.0. .................................................. 99 Figure 4.28: Base moment as a function of axial load in a shear wall of a 30 storey coupled shear wall building during a time history analysis, Rα=0 = 2.0, α = 0.0. ................................................ 100 Figure 4.29: Base moment as a function of curvature in the tension wall of a 5 storey coupled wall building during a time history analysis, Rα=0 = 2.0, α = 0.0. ........................................................ 101 Figure 4.30: Base moment as a function of curvature in the tension wall of a 30 storey coupled wall building during a time history analysis, Rα=0 = 2.0, α = 0.0. ........................................................ 102 Figure 4.31: Moment as a function of curvature for a third storey coupling beam in a 5 storey coupled shear wall building during a time-history analysis, Rα=0 = 2.0, α = 0.0. .......................... 103 Figure 4.32: Moment as a function of curvature for a fifth storey coupling beam in a 5 storey coupled shear wall building during a time-history analysis, Rα=0 = 2.0, α = 0.0. .......................... 104   xvi  Figure 5.1: Base moment versus curvature for a 5 storey cantilevered shear wall building with and without a PLD corresponding to α = 0.4, Rα=0 = 1.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .......................................................................................................... 109 Figure 5.2: Base moment versus curvature for a 5 storey cantilevered shear wall building with and without a PLD corresponding to α = 0.4, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .............. 110 Figure 5.3: Base moment versus curvature for a 5 storey cantilevered shear wall building with and without a PLD corresponding to α = 0.4, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .............. 110 Figure 5.4: Base moment versus curvature for a 5 storey cantilevered shear wall building with and without a PLD corresponding to α = 0.4, Rα=0 = 6.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .............. 111 Figure 5.5: Base moment versus curvature for a 10 storey cantilevered shear wall building with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .................................................. 112 Figure 5.6: Base moment versus curvature for a 10 storey cantilevered shear wall building with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The forward yield strength of this cantilevered wall not increased to account for the presence of the PLD. The initial and final resting states are identified with hollow and solid circles, respectively. .............................. 113 Figure 5.7: Base moment versus curvature for a 30 storey cantilevered shear wall building with varying PLDs, Rα=0 = 2.0, and subjected to ground motion 23. The initial and final resting states are identified with hollow and solid circles, respectively. .................................................. 114 Figure 5.8: Base moment versus curvature for a 50 storey cantilevered shear wall building with varying PLDs, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .................................................. 115 Figure 5.9: β for inelastic base curvatures in cantilevered shear wall buildings, Rα=0 = 2.0. ................... 117 Figure 5.10: β for inelastic base curvatures in cantilevered shear wall buildings, Rα=0 = 4.0. ................. 118 Figure 5.11: β for inelastic base curvatures in cantilevered shear wall buildings, Rα=0 = 6.0. ................. 119 Figure 5.12: β for inelastic base curvatures in cantilevered shear wall buildings subjected to each of the ten ground motions with the mean, µ, and mean plus one standard deviation, µ + σ, plotted, Rα=0 = 2.0, α = 0.6. ........................................................................................................ 120 Figure 5.13: β for roof drifts in cantilevered shear wall buildings, Rα=0 = 2.0. ....................................... 121 Figure 5.14: β for roof drifts in cantilevered shear wall buildings, Rα=0 = 4.0. ....................................... 122   xvii  Figure 5.15: β for roof drifts in cantilevered shear wall buildings, Rα=0 = 6.0. ....................................... 123 Figure 5.16: β for roof drifts in cantilevered shear buildings subjected to each of the ten ground motions with the mean, µ, and mean plus one standard deviation, µ + σ, plotted, Rα=0 = 2.0, α = 0.6. ......................................................................................................................................... 124 Figure 5.17: δ, the additional residual roof drift, expressed as a percentage of building height, produced in cantilevered shear wall buildings, Rα=0 = 2.0. .......................................................... 126 Figure 5.18: δ, the additional residual roof drift, expressed as a percentage of building height, produced in cantilevered shear wall buildings, Rα=0 = 4.0. .......................................................... 127 Figure 5.19: δ, the additional residual roof drift, expressed as a percentage of building height, produced in cantilevered shear wall buildings, Rα=0 = 6.0. .......................................................... 128 Figure 5.20: δ, the additional residual roof drift, expressed as a percentage of building height, produced in cantilevered shear wall buildings subjected to each of the ten ground motions with the mean, µ, and mean plus one standard deviation, µ + σ, plotted, Rα=0 = 2.0, α = 0.6........ 129 Figure 5.21: β for maximum roof drifts from each time-history analysis conducted on the cantilevered shear wall buildings. Each data point represents β from a single building and ground motion (630 data points). The data is organized by Rα=0 and α; no differentiation is made between buildings with different periods. .......................................................................... 130 Figure 5.22: β for maximum roof drifts from each time-history analysis conducted on the cantilevered shear wall buildings. Each data point represents β from a single building and ground motion (630 data points). The data is organized by the period and α; no differentiation is made between Rα=0 = 2.0, 4.0, and 6.0. ............................................................ 132 Figure 5.23: Hysteretic behaviour and residual demands in four 5 storey cantilevered wall buildings with varying axial loads corresponding to wall self-weight (Wwall) alone, wall self-weight plus 10% of floor-weight (Wfloor), wall self-weight plus 20% of floor-weight, and wall self- weight plus 30% of floor-weight, Rα=0 = 4.0, α = 0.4. The initial and final resting states are identified with hollow and solid circles, respectively. ................................................................. 133 Figure 5.24: δ, the additional residual roof drift, expressed as a percentage of building height, produced from each time-history analysis conducted on the cantilevered shear wall buildings. Each data point represents δ from a single building and ground motion (1050 data points). The data is organized by the period and α; no differentiation is made between Rα=0 = 2.0, 4.0, and 6.0....................................................................................................................................... 134 Figure 5.25: Base moment versus curvature in the compression wall of a 5 storey coupled shear wall building with and without PLDs corresponding to α = 0.4, Rα=0 = 1.0, and subjected to   xviii  ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .................................................................................................................. 135 Figure 5.26: Base moment versus curvature in the compression wall of 5 storey coupled shear wall buildings with and without PLDs corresponding to α = 0.4, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .................................................................................................................. 136 Figure 5.27: Base moment versus curvature in the compression wall of 5 storey coupled shear wall buildings with and without PLDs corresponding to α = 0.4, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .................................................................................................................. 137 Figure 5.28: Base moment versus curvature in the compression wall of 5 storey coupled shear wall buildings with and without PLDs corresponding to α = 0.4, Rα=0 = 6.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .................................................................................................................. 138 Figure 5.29: Base moment versus curvature in the compression wall of 30 storey coupled shear wall buildings with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. ........................ 139 Figure 5.30: Base moment versus curvature in the compression wall of 50 storey coupled shear wall buildings with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. ........................ 141 Figure 5.31: Moment versus curvature in coupling beams on the fifth storeys of 10 storey coupled shear wall buildings with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .............. 142 Figure 5.32: β for inelastic base curvatures in the compression wall of coupled shear wall buildings, Rα=0 = 2.0. ................................................................................................................................. 143 Figure 5.33: β for inelastic base curvatures in the compression wall of coupled shear wall buildings, Rα=0 = 4.0. ................................................................................................................................. 144 Figure 5.34: β for inelastic base curvatures in the compression wall of coupled shear wall buildings, Rα=0 = 6.0. ................................................................................................................................. 145 Figure 5.35: β for inelastic base curvatures in the compression wall of coupled shear wall buildings subjected to each of the ten ground motions with the mean, µ, and mean plus one standard deviation, µ + σ, plotted, Rα=0 = 2.0, α = 0.2. .............................................................................. 146 Figure 5.36: β for roof drifts in coupled shear wall buildings, Rα=0 = 2.0. ............................................. 147 Figure 5.37: β for roof drifts in coupled shear wall buildings, Rα=0 = 4.0. ............................................. 148   xix  Figure 5.38: β for roof drifts in coupled shear wall buildings, Rα=0 = 6.0. ............................................. 149 Figure 5.39: β for roof drifts in coupled shear wall buildings subjected to each of the ten ground motions with the mean, µ, and mean plus one standard deviation, µ + σ, plotted, Rα=0 = 2.0, α = 0.2. ......................................................................................................................................... 150 Figure 5.40: δ, the additional residual roof drift, expressed as a percentage of building height, produced in coupled shear wall buildings, Rα=0 = 2.0. ................................................................. 151 Figure 5.41: δ, the additional residual roof drift, expressed as a percentage of building height, produced in coupled shear wall buildings, Rα=0 = 4.0. ................................................................. 152 Figure 5.42: δ, the additional residual roof drift, expressed as a percentage of building height, produced in coupled shear wall buildings, Rα=0 = 6.0. ................................................................. 153 Figure 5.43: δ, the additional residual roof drift, expressed as a percentage of building height, produced in coupled shear wall buildings subjected to each of the ten ground motions with the mean, µ, and mean plus one standard deviation, µ + σ, plotted, Rα=0 = 2.0, α = 0.2. .............. 154 Figure 5.44: β for the maximum roof drifts produced in the compression wall from each time- history analysis conducted on the coupled shear wall buildings. Each data point represents β from a single building and ground motion (1260 data points). The data is organized by the period and α; no differentiation is made between Rα=0 = 2.0, 4.0, and 6.0. .................................. 156 Figure 5.45: β for the maximum roof drifts produced in the compression wall from each time- history analysis conducted on the coupled shear wall buildings. Each data point represents β from a single building and ground motion (1260 data points). The data is organized by Rα=0 and α, with the relationship to the period ignored. ...................................................................... 157 Figure 5.46: δ, the additional residual roof drift, expressed as a percentage of building height, produced from each time-history analysis conducted on the coupled shear wall buildings. Each data point represents δ from a single building and ground motion (1260 data points). The data is organized by the period and α; no differentiation is made between Rα=0 = 2.0, 4.0, and 6.0....................................................................................................................................... 158 Figure 6.1: Three gravity system irregularities considered; fully inclined (left), inclined lobby (center), and eccentric floor spans (right) (Best, Elwood, & Anderson, 2011). ............................ 160 Figure 6.2: Bending moment distribution from each gravity system irregularity - fully inclined (left), inclined lobby (center), and eccentric floor spans - along the height of the shear wall in a 5 storey building (right) (Best, Elwood, & Anderson, 2011). ................................................... 160 Figure 6.3: Base moment versus curvature in the compression wall of three 30 storey coupled shear wall buildings with PLDs corresponding to α = 0.2, Rα=0 = 4.0, and subjected to ground motion 35. Each building has one of the three irregularities: fully inclined columns, inclined   xx  columns over the lobby, and eccentric floor spans. The initial and final resting states are identified with hollow and solid circles, respectively. ................................................................. 162 Figure 6.4: Base moment versus curvature in the compression wall of three 30 storey coupled shear wall buildings with PLDs corresponding to α = 0.2, Rα=0 = 4.0, and subjected to ground motion 164. Each building has one of the three irregularities: fully inclined columns, inclined columns over the lobby, and eccentric floor spans. The initial and final resting states are identified with hollow and solid circles, respectively. ................................................................. 162 Figure 6.5: Base moment versus curvature in the compression wall of three 30 storey coupled shear wall buildings with coupling ratios of 0.70, 0.77, and 0.84, α = 0.2, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .................................................................................................................. 166 Figure 6.6: Base moment versus curvature in the compression wall of three 30 storey coupled shear wall buildings with coupling ratios of 0.70, 0.77, and 0.84, α = 0.2, Rα=0 = 4.0, and subjected to ground motion 124. The initial and final resting states are identified with hollow and solid circles, respectively. .................................................................................................................. 166 Figure 6.7: Roof drift versus time in the 30 storey coupled shear wall buildings with three considered coupling ratios, α = 0.2, Rα=0 = 4.0, and subjected to ground motion 38. ................... 168 Figure 6.8: Roof drift versus time in the 30 storey coupled shear wall buildings with three considered coupling ratios, α = 0.2, Rα=0 = 4.0, and subjected to ground motion 59. ................... 168 Figure 6.9: Acceleration response spectra for ten ground motions scaled to match the maximum elastic base overturning moment imposed by the UHS in Vancouver, Victoria, Calgary, and Montreal on the archetype 30 storey cantilevered shear wall building (the 30 storey archetype building was not analyzed in Calgary and Montreal however the spectra are included for completeness and consistency). Each linearly scaled spectrum is indicated with a dashed line with the mean of the ten records in bold and the UHS indicated with a solid line. ........................................................................................................................................... 171 Figure 6.10: Base moment versus curvature in the compression wall of two 30 storey coupled shear wall buildings subjected to ground motion 2 linearly scaled to impose maximum elastic base overturning moments corresponding to the UHS in Vancouver and Victoria, α = 0.0, Rα=0 = 4.0. The initial and final resting states are identified with hollow and solid circles, respectively. .............................................................................................................................. 172 Figure 6.11: Base moment versus curvature in the compression wall of two 30 storey coupled shear wall buildings subjected to ground motion 2 linearly scaled to impose maximum elastic base overturning moments corresponding to the UHS in Vancouver and Victoria, α = 0.2, Rα=0 =   xxi  4.0. The initial and final resting states are identified with hollow and solid circles, respectively. .............................................................................................................................. 172 Figure 6.12: Base moment versus curvature in the compression wall of four 10 storey coupled shear wall buildings subjected to ground motion 2 linearly scaled to impose maximum elastic base overturning moments corresponding to the UHS in Vancouver and Victoria, α = 0.0, Rα=0 = 2.0. The initial and final resting states are identified with hollow and solid circles, respectively. .............................................................................................................................. 174 Figure 6.13: Base moment versus curvature in the compression wall of four 10 storey coupled shear wall buildings subjected to ground motion 2 linearly scaled to impose maximum elastic base overturning moments corresponding to the UHS in Vancouver and Victoria, α = 0.2, Rα=0 = 2.0. The initial and final resting states are identified with hollow and solid circles, respectively. .............................................................................................................................. 175 Figure 6.14: Subduction ground motion, Chile 2010, magnitude 8.8, Stn: Angol, ANGO, UCS (CESMD, 2012; Seismosignal, 2012)......................................................................................... 177 Figure 6.15: Subduction ground motion, Tohoku 2011, magnitude 9.0, Stn: Imaichi, TCG009, KNET. (CESMD, 2012; Seismosignal, 2012). ........................................................................... 178 Figure 6.16: Base moment versus curvature in the compression wall of a 30 storey coupled shear wall building with and without PLDs corresponding to α = 0.2 subjected to the 2010 Chile subduction ground motion, Rα=0 = 4.0. The initial and final resting states are identified with hollow and solid circles, respectively. ........................................................................................ 179 Figure 6.17: Base moment versus curvature in the compression wall of a 30 storey coupled shear wall building with and without PLDs corresponding to α = 0.2 subjected to the 2011 Tohoku subduction ground motion, Rα=0 = 4.0. The initial and final resting states are identified with hollow and solid circles, respectively. ........................................................................................ 180 Figure 6.18: Ground motion 549, Stn: USGS Bishop – LADWP South (PEER, 2012). ......................... 181 Figure 6.19: Ground motion 1, Stn: REHS Christchurch Resthaven (CESMD, 2012). .......................... 182 Figure 6.20: Base moment versus curvature in the compression wall of a 30 storey coupled shear wall building with and without PLDs corresponding to α = 0.2 and subjected to ground motion 549 with and without the vertical component, Rα=0 = 4.0. The initial and final resting states are identified with hollow and solid circles, respectively. .................................................. 183 Figure 6.21: Base moment versus curvature in the compression wall of a 30 storey coupled shear wall building with and without PLDs corresponding to α = 0.2 and subjected to ground motion 1 with and without the vertical component, Rα=0 = 4.0. The initial and final resting states are identified with hollow and solid circles, respectively. .................................................. 184   xxii  Figure 6.22: Structural configuration of 8 storey steel braced frame model with vertical columns (left) and with inclined columns, α = 0.4, Rα=0 = 2.0 (right) (Izvernari, Lacerte, & Tremblay, 2007). ........................................................................................................................................ 187 Figure 6.23: Structural configuration of 12 storey steel braced frame model with vertical columns (left) and with inclined columns, α = 0.4, Rα=0 = 2.0 (right) (Izvernari, Lacerte, & Tremblay, 2007). ........................................................................................................................................ 187 Figure 6.24: Structural configuration of 16 storey steel braced frame model with vertical columns (left) and with inclined columns, α = 0.4, Rα=0 = 2.0 (right) (Izvernari, Lacerte, & Tremblay, 2007). ........................................................................................................................................ 187 Figure 6.25: Schematic of the general parametric relationship between the demands and capacities of the steel braced frame systems. The relative magnitudes shown correspond to a building with Rα=0 = 2.0 and α = 0.5. ....................................................................................................... 190 Figure 6.26: Steel braced frame building overview: a) plan view of 8 and 12 storey buildings, b) braced frame elevation for the 8 storey building, © Izvernari, Lacerte, & Tremblay (2007), by permission. ........................................................................................................................... 191 Figure 6.27: Base shear versus first storey drift in the 8 storey concentrically braced frame building with and without PLDs corresponding to α = 0.4, Rα=0 = 1.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. ... 194 Figure 6.28: Base shear versus first storey drift in the 8 storey concentrically braced frame building with and without PLDs corresponding to α = 0.4, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. ... 195 Figure 6.29: Base shear versus first storey drift in the 8 storey concentrically braced frame building with and without PLDs corresponding to α = 0.4, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. ... 195 Figure 6.30: Base shear versus first storey drift in the 8 storey concentrically braced frame building with and without PLDs corresponding to α = 0.4, Rα=0 = 6.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. ... 196 Figure 6.31: Base shear versus first storey drift in the 8 storey concentrically braced frame building with varying PLDs, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. ...................................... 197 Figure 6.32: Base shear versus first storey drift in the 8 storey concentrically braced frame building with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. ...................................... 198   xxiii  Figure 6.33: Base shear versus first storey drift in the 8 storey concentrically braced frame building with varying PLDs, Rα=0 = 6.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. ...................................... 199 Figure 6.34: Base shear versus first storey drift in the 12 storey concentrically braced frame building with varying PLDs, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .............................. 200 Figure 6.35: Base shear versus first storey drift in the 12 storey concentrically braced frame building with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .............................. 201 Figure 6.36: Base shear versus first storey drift in the 12 storey concentrically braced frame building with varying PLDs, Rα=0 = 6.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .............................. 202 Figure 6.37: Base shear versus first storey drift in the 16 storey concentrically braced frame building with varying PLDs, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .............................. 203 Figure 6.38: Base shear versus first storey drift in the 16 storey concentrically braced frame building with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .............................. 204 Figure 6.39: Base shear versus first storey drift in the 16 storey concentrically braced frame building with varying PLDs, Rα=0 = 6.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. .............................. 205 Figure 7.1: β in cantilevered shear wall buildings at each α for all considered Rα=0, periods, and ground motions. ......................................................................................................................... 208 Figure 7.2: β in coupled shear wall buildings at each α for all considered Rα=0, periods, and ground motions. .................................................................................................................................... 208 Figure 7.3: Deformation amplifications in cantilevered and coupled shear wall buildings corresponding to the mean β (solid line) for the worst performing building (considering all periods and Rα=0) and the maximum β (dashed line) for all ground motions and buildings considered. ................................................................................................................................ 209 Figure 7.4: γ corresponding to the mean γ for the worst performing cantilevered shear wall building (considering all periods) as a function of α for different Rα=0. ..................................................... 211 Figure 7.5: γ corresponding to the mean γ for the worst performing cantilevered shear wall building (considering all Rα=0) as a function of α for different periods. ..................................................... 211   xxiv  Figure 7.6: γ corresponding to the mean γ for the worst performing coupled shear wall building (considering all periods) as a function of α for different Rα=0. ..................................................... 212 Figure 7.7: γ corresponding to the mean γ for the worst performing coupled shear wall building (considering all Rα=0) as a function of α for different periods. ..................................................... 212 Figure 7.8: Prescriptive γ proposed for design as well as γ corresponding to the mean γ for the worst performing cantilevered and coupled shear wall building (considering all Rα=0 and periods). ...... 214 Figure 7.9: Proposed additions to Table 4.1.86 in the NBCC to account for the new structural irregularity of PLFs on the SFRS. .............................................................................................. 215 Figure 7.10: Proposed additions to Section 4.1.8.10 in the NBCC to account for the new structural irregularity of PLFs on the SFRS. .............................................................................................. 216 Figure A.1.1: Ground motion 2, Stn: USGS 1095 Taft Lincoln School (Best, Elwood, & Anderson, 2011; Seismosignal, 2012). ........................................................................................................ 228 Figure A.1.2: Ground motion 16, Stn: 9102 Dayhook TR (Best, Elwood, & Anderson, 2011; Seismosignal, 2012)................................................................................................................... 229 Figure A.1.3: Ground motion 23, Stn: ENEL 99999 Calitri (Best, Elwood, & Anderson, 2011; Seismosignal, 2012)................................................................................................................... 229 Figure A.1.4: Ground motion 26, Stn: 6098 Site 2 (Best, Elwood, & Anderson, 2011; Seismosignal, 2012). ........................................................................................................................................ 230 Figure A.1.5: Ground motion 35, Stn: CDMG 57007 Corralitos (Best, Elwood, & Anderson, 2011; Seismosignal, 2012)................................................................................................................... 230 Figure A.1.6: Ground motion 38, Stn: CDMG 57217 Coyote lake Dam (SW Abut) (Best, Elwood, & Anderson, 2011; Seismosignal, 2012). ................................................................................... 231 Figure A.1.7: Ground motion 59, Stn: CDMG 89156 Petrolia (Best, Elwood, & Anderson, 2011; Seismosignal, 2012)................................................................................................................... 231 Figure A.1.8: Ground motion 124, Stn: USGS 5108 Santa Susana Ground (Best, Elwood, & Anderson, 2011; Seismosignal, 2012). ....................................................................................... 232 Figure A.1.9: Ground motion 133, Stn: CUE 99999 Nishi-Akashi (Best, Elwood, & Anderson, 2011; Seismosignal, 2012). ........................................................................................................ 232 Figure A.1.10: Ground motion 164, Stn: Ichinoseki (IWT010) NS (Best, Elwood, & Anderson, 2011; Seismosignal, 2012). ........................................................................................................ 233 Figure A.2.1: Uniform hazard spectrum for Vancouver, British Columbia (NBCC, 2010). ................... 235 Figure A.4.1: Concrete core section for 5 storey cantilevered shear wall model, units in mm (Yathon, 2010). ......................................................................................................................... 248   xxv  Figure A.4.2: Concrete core section for 10 storey cantilevered shear wall model, units in mm (Yathon, 2010). ......................................................................................................................... 248 Figure A.4.3: Concrete core section for 20 storey cantilevered shear wall model, units in mm (Yathon, 2010). ......................................................................................................................... 249 Figure A.4.4: Concrete core section for 30 storey cantilevered shear wall model, units in mm (Yathon, 2010). ......................................................................................................................... 249 Figure A.4.5: Concrete core section for 40 storey cantilevered shear wall model, units in mm (Yathon, 2010). ......................................................................................................................... 250 Figure A.4.6: Concrete core section for 50 storey cantilevered shear wall model, units in mm (Yathon, 2010). ......................................................................................................................... 251 Figure A.6.1: Concrete core section for 5 storey coupled shear wall model, units in mm (Yathon, 2010). ........................................................................................................................................ 255 Figure A.6.2: Concrete core section for 10 storey coupled shear wall model, units in mm (Yathon, 2010). ........................................................................................................................................ 255 Figure A.6.3: Concrete core section for 20 storey coupled shear wall model, units in mm (Yathon, 2010). ........................................................................................................................................ 256 Figure A.6.4: Concrete core section for 30 storey coupled shear wall model, units in mm (Yathon, 2010). ........................................................................................................................................ 256 Figure A.6.5: Concrete core section for 40 storey coupled shear wall model, units in mm (Yathon, 2010). ........................................................................................................................................ 257 Figure A.6.6: Concrete core section for 50 storey coupled shear wall model, units in mm (Yathon, 2010). ........................................................................................................................................ 257     xxvi  List of Symbols and Abbreviations Acronym/Symbol Expanded α Applied Demand Ratio β Relative Amplification Factor γ Inelastic Amplification Factor ∆ Generic displacement quantity CR Coupling Ratio DOF Degree-of-Freedom Dst Static Displacement resulting from the permanent lateral demands E Elastic Modulus Ec Concrete Modulus of Elasticity FE max Maximum elastic seismic force demand FPLD Magnitude of the permanent lateral demand Fy Base Yield Strength G Modulus of Rigidity I Moment of Inertia IE Effective Moment of Inertia IG Un-cracked Moment of Inertia K Stiffness L1 Length of the spring in the single-degree-of-freedom system L2 Length of the beam-column in the single-degree-of-freedom system MDOF Multiple Degree of Freedom NBCC National Building Code of Canada P Synthetic P-∆ force applied to SDOF system PLD Permanent Eccentric Load R Relative strength factor SCED Standing Committee on Earthquake Design Sd Spectral Displacement SDOF Single Degree of Freedom SFRS Seismic Force Resisting System T Fundamental Period ȕ Ground Acceleration   xxvii  UBC Uniform Building Code UHS Uniform Hazard Spectrum w Distributed Load     xxviii  Acknowledgments First and foremost I would like to thank my advisors, Dr. Ken Elwood and Dr. Don Anderson, who provided the guidance, vision, and wisdom which kept this study focused and on track. Thanks to Tyler Best for laying the foundation of this project with his preliminary work. Thanks to Majid and Andrew –the steel braced frame analyses would not have been completed without these two men. Sincere thanks to Frank Mckenna and the OpenSees team of developers for the development of OpenSees and supporting documentation. Their constant effort towards maintaining the online help forums is thoroughly appreciated and this research project would not have achieved the outcomes it did without these tools. Thanks to everyone in the office for the support and distractions which actually made coming to the office an enjoyable experience; Majid, Alex, Xavier, Michael, Osmar, Jeff, Carla, and Ehsan all provided significant help. Thanks to my family, both near and far for all the support: my mom, sister, aunt, and grandmother in Calgary, as well as Bob and Marietta in Burnaby.    1  Chapter 1 - Introduction  1.1 IMPETUS FOR STUDY  Architects are driven to produce innovative and aesthetically appealing buildings. In the past this desire to build unique buildings was often curtailed by the limited accuracy of structural analysis – it was too much work and too expensive. This is quickly changing and relatively recent advances in state of the art finite element design software are allowing for the construction of radical and fascinating buildings.  Radical new buildings, often in seismically active areas such as the CCTV building in Beijing shown in Figure 1.1 include architectural features such as inclined façades. These inclined façades may be used over several storeys, or throughout the entire height of a building. Other increasingly common architectural features include eccentric floor spans, cantilevered floor spans, and columns with a lateral offset at each floor level.  Figure 1.1: Inclined façade in a high-rise building. The structural system in buildings is often idealized as two distinct systems: a seismic force resisting system (SFRS) which resists lateral demands, and a gravity system which supports the weight of the building. Building codes required that the gravity system be detailed to maintain its strength through lateral displacements, however, the force demands on each system have been assumed to behave independently. Each of the architectural features listed above introduce an irregularities to the gravity   2  system. As a result of these irregularities, there is an interaction between the gravity system and the SFRS which is not present in regular buildings. Because of this new interaction, permanent lateral demands (PLDs) are imposed on the SFRS in buildings. This interaction between the two previously independent structural systems is illustrated in Figure 1.2. In this case, transformation of gravity demands in a five storey shear wall building to PLDs acting on the SFRS, occur as a result of inclined columns.                Figure 1.2: Schematic illustrating external (left) and internal (right) demands which result from inclined columns. Engineers account for PLDs in the design of the SFRS by increasing strength. However, when strong ground motions occur, almost all structures are designed to become inelastic and exhibit ductility (Bernal, 1998; Tremblay, Léger, & Tu, 2001). This behaviour is desired because it allows for increased energy dissipation, decreases the required strength of the SFRS, and provides ample warning of failure (Bernal, 1998; Jennings & Husid, 1968; Paulay & Priestley, 1992). Although buildings are often designed to exhibit inelastic behaviour, PLDs have raised concerns, due to the perceived potential for a ratcheting effect to occur during seismic loading.     3  The effects of PLDs on the inelastic behaviour of buildings have thus far been ignored. It had been hypothesized, that the presence of PLDs could soften the SFRS and reduce its resistance to seismic demands in a manner similar to P-Δ effects. However, unlike P-Δ effects, PLDs maintain a constant orientation; the plastic hinge would be softened in one direction and stiffened in the other. A concern of the engineering community in Canada was that this would result in a seismic ratcheting effect causing buildings with PLDs to displace laterally to an extent not recognized in the building codes.  Despite their increasing prevalence, and poorly understood inelastic behaviour; there are currently no design codes which account for the effects of PLDs on SFRSs (NBCC, 2010). While these PLDs are induced by various irregularities in the gravity system, the implications of their effects are much broader. The phenomenon is not one of inclined columns, or even inclined facades, but rather SFRSs which support PLDs. As architectural trends continue to evolve, other yet to be conceived irregularities may also impose the PLDs on SFRSs. Beyond the scope of design, PLDs may eventually need to be addressed in the seismic evaluation of buildings after significant ground motions because PLDs could result from differential settling of a foundation due to liquefaction.  1.2 OBJECTIVE OF THESIS  It is necessary to quantify the additional inelastic displacement demands created by PLDs on buildings. The primary goal of this work is to parametrically define the scope of influence of PLDs for different structural systems and recommend provisions for the National Building Code of Canada (NBCC). To accomplish this, models were developed to represent reinforced concrete shear walls in both the cantilevered and coupled directions. Additional consideration was given to steel braced frame buildings. Design guidelines are to be formulated which will allow for architectural features such as inclined facades and stepped columns to be implemented while satisfying performance objectives established in the NBCC.  1.3 OVERVIEW OF METHODOLOGY  Two broad categories of analytical models were developed and implemented into the Open System for Earthquake Engineering Simulation (OpenSees) for nonlinear parametric analysis. The ‘workspace’ tool, available online through the Network for Earthquake Engineering Simulation (NEEShub) was used   4  extensively to handle the large computational demands of the study, especially for the complex MDOF systems (Kisseberth, 2012; NEEShub, 2012).  The first category of models, simple single-degree-of-freedom (SDOF) systems, was used to roughly identify the nature of the behaviour resulting from PLDs. This was done to investigate how sensitive the ratcheting response is to the hysteretic behaviour exhibited by the SFRS. The SDOF system was relatively simple, with few input parameters. Therefore, it was an ideal platform to explore the sensitivity of the inelastic displacement demands to the hysteretic behaviour of the SFRS.  Once a better understanding of the ratcheting phenomenon causing was developed, the study evolved to MDOF systems. From observations of the SDOF systems, models were developed which incorporated appropriate hysteretic models. Two types of buildings were considered: cantilevered shear wall and coupled shear wall, with an additional case study investigating steel braced frames. Concrete shear wall buildings consisting of five to fifty stories were considered in the study as well as eight, twelve, and sixteen storey steel braced frame buildings. The PLD imposed on building and the strength of the SFRS was each parametrically varied across a wide domain.  Each shear wall building was subjected to ten crustal ground motions were selected from the PEER Strong Motion Database (PEER 2012). These ground motions, all recorded on site class C soil conditions, are consistent with those used in the British Columbia schools retrofit project (Bebamzadeh et al 2012). Building metrics are compared between otherwise identical buildings with and without PLDs to quantify the effect on inelastic performance.  The shear wall buildings are used to identify the severity of displacement amplifications for each SFRS under a range of loading configurations and to recommend codes to account for PLDs on SFRSs. Case studies are then completed to investigate the influence of factors not explicitly considered in the study, such as vertical ground motion, subduction ground motions, coupling ratio, type of gravity system irregularity. A final case study considers the performance of steel braced frame buildings.  1.4 ORGANIZATION OF THESIS  This thesis is organized into eight chapters with seven appendices. Chapter 2 outlines the previous study by Best, Elwood, and Anderson (2011). The main findings of this study and the limitations will be   5  discussed to form context for the approach taken in this study. Chapter 3 introduces the SDOF systems implemented in OpenSees and the eight different hysteretic models which were considered. The behaviour of each hysteretic model is discussed as well as how the findings from the SDOF study shaped the scope and direction of the MDOF study. Chapter 4 describes the cantilevered shear wall building models and the coupled shear wall building models which, together, were the main focus of this study. The hysteretic behaviour of these buildings is explored and the potential effects of different modelling assumptions are considered. Chapter 5 presents the results from the MDOF systems. The inelastic displacement demands for the buildings under various conditions are discussed and performance trends are identified. Six case studies are included in Chapter 6, which were used to investigate the influence of different conditions not considered in the main body of this study. Such conditions include vertical and subduction ground motions, coupling ratio, type of gravity system irregularity, seismic demands, and the behaviour of SFRSs other than shear walls – steel braced frames. Chapter 7 develops a rational design procedure to account for PLDs in buildings along with the study’s limitations and recommendations for future research. Finally, conclusions from this work are given in Chapter 8.   6  Chapter 2 - Background  An initial study: “Inclined Columns and how they Influence the Dynamic Response of Structures” was undertaken by Best, Elwood, and Anderson (2011). This study was initiated by the Standing Committee on Earthquake Design (SCED), an organization of seismologists, engineers, researchers, and private sector representatives from across Canada. The inelastic response of buildings with PLDs was investigated in order to determine if PLDs significantly affected the inelastic behaviour of buildings.  2.1 SDOF SYSTEM  Modelling began with simple SDOF systems with an axial spring element modelling an idealized SFRS. These systems each consisted of a lumped mass pinned the end of a spring element and a simple damper providing 5% damping, as shown in Figure 2.1. The mass was constrained in the vertical degree-of- freedom and the end of the spring element was pinned at the end opposite the mass.  Figure 2.1: The idealized SDOF model (Best, Elwood, & Anderson, 2011). Through adjusting the magnitude of the lumped mass and keeping the stiffness of the spring element constant, the period of this simple model was varied between 0.5 and 5.0 seconds. The magnitude of the FDL (Figure 2.1) was manipulated to model different magnitudes of PLDs on the idealized SFRS (Best, Elwood, & Anderson, 2011).  Constant stiffness, k, was assigned to the axial spring element and inelastic behaviour was modelled with the Clough hysteretic model, shown in Figure 2.2. The strength of this axial spring element was modified, to represent SFRSs with different strengths (Best, Elwood, & Anderson, 2011). Ten crustal ground motions were considered.   7   Figure 2.2: Clough hysteretic model used in the SDOF model with labelled tensile force yield plateau, Fyt, and compression force yield plateau, Fyc (Best, Elwood, & Anderson, 2011).  2.1.1 Results and Conclusions The results of the analyses conducted on the SDOF systems considered demonstrated that displacement amplifications could result from PLDs. The study found that the inelastic displacement demands increased as the strength of the SFRS decreased relative to seismic demands. This was attributed to increased cycles achieving yield, and thus increased ratcheting of the SFRS. As one would expect, it was demonstrated that as the magnitude of the PLD increased, so too did the observed inelastic displacement demands.  The inelastic displacement demands were also variable across the range considered SDOF system periods. However, the SDOF system did not produce a consistent correlation between fundamental period and the increases in inelastic displacement demands induced by PLDs.  2.1.2 Limitations and Recommendations There were several limitations identified in the study and to address these, recommendations were put forward for future studies by Best, Elwood, and Anderson (2011). The study considered a single hysteretic model (Clough), which was used to represent the inelastic behaviour of the SFRS. The conclusions drawn were the result of the interaction of this hysteretic model and the PLDs. The sensitivity of the inelastic displacement demands to changes in the hysteretic models was poorly understood.   8  Although, the SDOF system is a highly simplified representation of SFRSs, there are several hysteretic models that can be implemented. The interaction between each of these hysteretic models and the PLDs could differ from the Clough model and result in different inelastic displacement demands. It was therefore encouraged that future work utilize different hysteretic models with PLDs and to explore the sensitivity of the ratcheting phenomenon to the inelastic behaviour of SFRSs (Best, Elwood, & Anderson, 2011).  A second major limitation of the study was that it didn’t include P-∆ effects in the SDOF analyses (Best, Elwood, & Anderson, 2011). P-∆ effects promote instability and would tend to promote increased inelastic displacements. Furthermore, it was supposed that P-∆ effects and PLDs could have a compounding, producing a combined inelastic displacement demands more severe than the linear sum of their independent influences. P-∆ effects become more significant as lateral displacements increase, therefore, they have more significant effects systems with large periods. It was hypothesized that the inclusion of P-∆ effects in the SDOF system may have produced results demonstrating a correlation between the inelastic displacement demands and the period.  2.2 MDOF SYSTEM  A building type was modelled in the MDOF system; cantilevered shear wall buildings. The building model consisted of a single shear wall up the height of the building with a single line of exterior columns on either side. Building inelasticity was idealized as purely flexural and was lumped within a plastic hinge at the base of the shear wall. This plastic hinge was modelled with displacement-based beam-column elements with a idealized Clough hysteretic model. The building above, both the gravity system and the SFRS were modelled as fully elastic. The weight and mass of the wall was lumped at each storey level. Elastic truss elements were used to model floor slabs and columns. Buildings with 5, 10, 20, 30, 40, and 50 storeys were modelled, each with floor weights adjusted such that periods of 0.5, 1.0, 2.0, 3.0, 4.0, and 5.0 seconds were achieved, respectively. Each storey height was 3.5m with the exception of the first storey which was given a height of 4.5m to simulate an atrium. The irregularity in the gravity system (inclined columns, inclined columns over the first storey, or eccentric floors spans) was varied to achieve different magnitudes of PLDs in terms of base overturning moment demands. As was done with the SDOF systems, the strength of the SFRS was adjusted to achieve different strengths relative to the seismic force demands imposed by the ten ground motions, which were the same ground motions used in the SDOF system, only scaled to the Vancouver Uniform Hazard Spectrum (UHS) (Best, Elwood, &   9  Anderson, 2011). The irregularities in the gravity system considered are illustrated in Figure 2.3 (Best, Elwood, & Anderson, 2011). Although the three irregularities appear very different, they impose similar demands on the SFRS. Each irregularity was modified to impose the same PLDs (base overturning moments) at the base of the shear wall as indicated in Figure 2.4 (Best, Elwood, & Anderson, 2011).  Figure 2.3: Loading mechanisms modelled in exploratory study: fully inclined columns (left), inclined lobby (center), and eccentric floor spans (right) (Best, Elwood, & Anderson, 2011).  Figure 2.4: Bending moment profiles for the three loading configurations modelled: fully inclined (left), inclined lobby (center), and eccentric floor spans (right) (Best, Elwood, & Anderson, 2011).    10  2.2.1 Results and Conclusions As with the SDOF system, it was found that the inelastic displacement demands increased as the strength of the SFRS decreased relative to the seismic demands. This was again attributed to the increased yielding cycles and thus an increased ratcheting of the building. The PLD was also again found to be positively correlated with increased inelastic displacement demands, which was expected, given that PLDs were the impetus of the study.  Contrary to the SDOF systems, the building period was found to be loosely correlated to the displacement amplifications observed in the cantilevered shear wall building (Best, Elwood, & Anderson, 2011). The taller the building (larger periods), the greater the relative increases in inelastic displacement demands observed. As previously explained, this discrepancy was partially attributed to P-∆ effects on the MDOF system which were excluded from the SDOF system (Best, Elwood, & Anderson, 2011).  Although each irregularity was scaled to impose the same base overturning moment demands, the distribution of the moment demands along the height of the shear wall were dependent upon the specific irregularity in the gravity system. Because inelastic behaviour was considered throughout the plastic hinge region, these variations could change the inelastic response of the buildings. However, it was found that the different irregularities in the gravity system (inclined columns, inclined columns over the first storey, or eccentric floors spans) produced very similar inelastic displacement demands. Inclination over the lobby alone promoted the smallest inelastic displacement demands while eccentric floor spans and fully inclined columns both resulted in larger demands. The difference between the eccentric floor spans and fully inclined columns was found to be minimal. The fully inclined column irregularity was also identified as that likely to be most prevalent in new buildings (Best, Elwood, & Anderson, 2011). For these two reasons, the fully inclined column irregularity was used for the main body of the analyses.  2.2.2 Limitations and Recommendations The MDOF system in was limited to cantilevered shear walls. The behaviour of both concrete shear walls and steel braced frame models was not considered. As was the case with the SDOF systems, the cantilevered shear wall buildings relied upon the use of the Clough hysteretic model to represent inelastic behaviour (Best, Elwood, & Anderson, 2011). In the SDOF systems, the Clough hysteretic model was used to represent idealized global behaviour of the SFRS. This differed from the cantilevered shear wall buildings where the Clough hysteretic model was used to model flexural inelasticity in the plastic hinge.   11  Specific recommendations made at the conclusion of the study included refining inelastic models used within the plastic hinge. The Clough hysteretic model considers flexural behaviour independently of the axial behaviour. It was therefore a poor representation of cantilevered shear walls, where axial load interacts with flexural demands and significantly influences the hysteretic behaviour of SFRS.   12  Chapter 3 - SDOF Study  Inelastic time-history analyses were completed on the SDOF systems in OpenSees. These were done to develop a better understanding of the interaction between PLDs and the hysteretic behaviour exhibited by the SFRS. The SDOF systems were relatively simple compared to the MDOF systems that were later developed. Therefore, they represented an ideal platform to develop a better understanding of the fundamental nature of the seismic ratcheting effect.  Significant inelastic displacement demands were observed in the original study by Best, Elwood, and Anderson (2011), both for the SDOF and MDOF systems. However, nothing was known about how the assumed hysteretic model (Clough) had affected these amplifications. The period, relative strength of the SFRS, and magnitude of the PLDs had all been widely varied to investigate the nature of the seismic ratcheting phenomenon. However, the hysteretic model had not undergone similar variation to investigate the sensitivity of the resulting ratcheting effect to changes in hysteretic behaviour. To address this, eight hysteretic models are implemented in the SDOF systems to investigate the relationship between the ratcheting phenomenon and the hysteretic behaviour of the SFRS.  This chapter presents the SDOF systems and the domain of inputs through which they were parametrically varied. Using the results of time-history analyses, the influence of the hysteretic model on the ratcheting effect and the resulting inelastic displacement demands is explored. The consequences of the results of the SDOF systems on the modelling approach taken in the MDOF systems models are also discussed.  3.1 METHODOLOGY  In the SDOF analysis, there was an opportunity to conduct extensive parametric analyses due to the simplicity of the model, and thus low computational requirements. A wide domain of input values was considered for the magnitude of the PLD, the relative strength of the SRFS, the period of the systems.  3.1.1 Model The idealized SFRS in the OpenSees model consisted of a simple spring element. This element was assigned a constant length, L1, of one meter and a constant cross-sectional area, A, of one square meter.   13  These values meant that axial stiffness, k, and the elastic modulus, E, were synonymous (Equation 1). An advantageous extension of this was that force equaled stress and displacement equaled strain. Therefore, although both terms in each synonymous pair are used in this section, they refer to identical quantities.   ݇ = ܣܧ ܮ = 1 ∗ ܧ1 = ܧ (1)  The restrained end of the spring element was pinned and the opposite end was assigned a roller support which prevented movement in the vertical-direction but allowed for horizontal motion. A constant lateral force, representing the PLDs from an irregularity in the gravity system, was applied to free end of the spring element (Figure 3.1) and maintained throughout the time-history analysis. Mass was assigned to this free end to achieve fundamental periods, T, between 0.5 and 5.0 seconds at 0.25 second intervals. The spring element was analogous to a SFRS in which strains were analogous to displacement demands.  ‘Synthetic’ P-Δ effects were modelled to maintain consistency with the MDOF systems, which included P-∆ effects. In order to apply P-Δ effects to the SDOF systems, Element 2 (Figure 3.1); an elastic beam column, was added extending from Node 2 to Node 3. This element was supported by a horizontal roller a vertical roller at Node 3 as indicated in Figure 3.1. A vertical force, P was then applied Node 3, which was used to simulate the self-weight of the buildings represented by the idealized SDOF systems.  Stiffness proportional damping, resulting in a damping ratio of 5%, was implemented with the “Rayleigh” command in OpenSees. This command allowed for damping to be applied to the spring element representing the SFRS and not the elastic beam-column element which was used to apply synthetic P-Δ forces. The assumption of pure, stiffness proportional, Rayleigh damping is a simplification of real structural damping, however, it is appropriate for a study comparing relative changes in inelastic displacement demands.     14                      Figure 3.1: SDOF system configuration.  3.1.2 P-Δ Effects As discussed in section 2.1.2, one of the limitations of the original study by Best, Elwood, and Anderson (2011) was that it did not include P-Δ effects in the SDOF systems. This made it difficult to compare the results of the SDOF systems to the results of the MDOF systems, which included P-Δ effects. To remedy this problem, synthetic P-Δ forces were modelled in the SDOF systems. The SDOF model configuration (Figure 3.1) was incompatible with true P-Δ effects; therefore, synthetic P-Δ effects were added.  An elastic beam-column, with extremely high axial stiffness was added to the model extending from Node 2 to Node 3. This element was supported with a vertical roller at Node 3. Node 2 and Node 3 were vertically aligned at the beginning of each analysis and a vertical load, P, representing the self-weight of Element 1 Element 2   PLD Node 1 Node 2 Node 3 ȕ ȕ P     15  an equivalent cantilevered shear wall building with period, T, was then applied acting upwards at Node 3. In this way, an additional horizontal force was applied to Node 2 when it displaced laterally relative to Node 3. This lateral force created additional demands in the SDOF system which analogous to true P-Δ effects.  Element 2 was specified to be extremely long, causing the rotation of Element 2 from vertical to be relatively small and proportional to the lateral displacement of Node 2. Knowing this, the lateral force component on Node 2 from the vertical load, P, could be taken as proportional to the lateral displacement.  The rotation of Element 2 about Node 3 was small; therefore, the synthetic P-Δ force acting at Node 2 was proportional to the ratio of the vertical load, P, and the length of the elastic beam-column, L2. The resulting P-Δ force was equal to this ratio multiplied by the lateral displacement of Node 2, ∆, from its original position as shown in Equation 2.   ܨ௉ି௱ = ܲ ∗ ∆ܮଶ  (2)  Determining the appropriate magnitude for the ratio of P/L2 was challenging. It was not clear what magnitude the higher order effects should have since they were being added to a simple SDOF system. The solution was to link each SDOF system with a corresponding cantilevered shear wall building with a corresponding fundamental period. The ratio of P/L2 was modified to scale the relative strength of P-Δ effects on the axial spring element to be proportional to relative strength of P-Δ effects on the shear wall in the cantilevered shear wall buildings, which will be discussed further in section 4.9.  3.1.3 Relative Strength Factor To determine the influence that the extent of yielding had on the inelastic displacement demands, a range of yield strengths were considered in the axial spring element. The relative strength factor, R α=0, was defined as the ratio of the maximum elastic force demands, FE max, to the yield strength, Fy, of the SFRS. The relative strength factor, given in Equation 3, was assigned magnitudes of 1.0, 2.0, 4.0, and 6.0. R α=0 is related to, but should not be confused with, relative strength factors specified in the building codes. Rather, R α=0 is a term devised to simply relate strength of the SFRS in a building with vertical columns to the elastic demands. The yield strengths of the SFRSs were selected to achieve each desired relative strength factor, R α=0.   16    ܴఈୀ଴ = ܨா	௠௔௫ܨ௬  (3)  The yield strength of the spring element, needed to achieve the desired R α=0 was determined from the elastic response of the SDOF, with spectral displacement, Sd, and stiffness, k, as shown in Equation 4. In this way, the strength of the structural system was adjusted to achieve each desired R α=0.   ܨ௬ = ܨா	௠௔௫ܴఈୀ଴ = ܵௗܴ݇ఈୀ଴ (4)  3.1.4 Applied Demand Ratio To simulate the demands on the SFRS from irregularities in the gravity system, PLD that was applied to the SDOF system at Node 2 (Figure 3.1). A range of PLD magnitudes were considered, representative of various column inclinations, or alternatively greater floor span eccentricity in a real building. One may recall from section 3.1.3 that the yield strength, Fy, was scaled to achieve the various R α=0. This R α=0 was determined from the ratio of Fy of the spring element and FE max developed by a specific ground motion. Similarly, it is convenient to express the PLD as a function of Fy,, of the axial spring element in the SDOF systems. Therefore, the PLD applied was expressed as an applied demand ratio, α, where α is defined as the ratio of the PLD, FPLD, to the yield strength, Fy (Equation 5).   ߙ = ܨ௉ா௅ ܨ௬  (5)  Since, the yield strength, Fy, was earlier defined as the maximum elastic force, FE max, divided by the desired relative strength factor, R α=0, then the above relationship can be used to determine the PLD, FPLD (Equation 6). For the SDOF systems, α was given the range of values ranging from 0 to 0.8, resulting in PLDs up to 80% of the yield strength of the SFRS with no PLD.   ܨ௉ா௅ = ߙܨ௬ = ߙܨா௠௔௫ܴఈୀ଴  (6)    17  3.1.5 Ground Motions Ten crustal ground motions were selected from the PEER Strong Motion Database (PEER, 2012). These ground motions, from site class C soil conditions, are consistent with those used in the British Columbia schools retrofit project (Bebamzadeh et al, 2012). Relative inelastic demands between systems with and without various PLDs were the sole concern of the SDOF component of the study; therefore the original un-scaled ground motions were used.  Ground motions are inherently random and asymmetric, both in terms of energy distribution and force application (Moustafa & Takewaki, 2010). They often exert more energy in one direction than the other. This can result in greater yielding of inelastic structures in one direction. Typically, the greater the asymmetry of a ground motion, the greater the unbalanced response is, which can then result in large inelastic displacement demands (Moustafa & Takewaki, 2010). Since the PLD applied to the system is by nature asymmetric, the relative orientation of the ground motion and PLD is important. The importance of the orientation of the ground motion and the PLD is shown in Figure 3.2.  Figure 3.2: Displacement-time histories of two identical SDOF systems with an otherwise identical ground motion applied in the forward and reverse directions. This analysis was done on a system with a period of 1.25 seconds, α = 0.4, R α=0 = 6.0, and the axial spring element was assigned Clough hysteretic model properties.   18  The asymmetry of ground motions is highly variable and it was important that be considered in the analyses. This was true not only for consistency, but also to ensure that the results reported the maximum amplifications which were induced by the PLD. To account for the asymmetry of each ground motion, the ground motions were applied in both directions and the direction which caused the largest inelastic displacement demands was used.  3.1.6 Strength Allocation When designing SFRS subjected to PLDs, engineers account for the presence of the PLD by asymmetrically adjusting the strength of the SFRS. If possible, an engineer will strengthen the SFRS in the direction of the PLD to account for the additional demands. In a flexure dominated shear wall, this is accomplished through adding longitudinal reinforcing steel to one side of the shear wall. Other similar techniques would be applied, if possible, in different types of SFRSs.  Engineers are afforded certain flexibility the specific way in which they account for the presence of the PLDs; for the purpose of this parametric study it was necessary to assume one logical approach. This approach assumed that the SFRS system would be strengthened by the magnitude of the PLD in one direction and weakened by the same amount in the opposite direction. Therefore the desired yield strength in the forward direction, Fy+, which is to be taken as the direction in which the PLD acts, is expressed in Equation 7.   ܨ௬ା = ܨ௬(1 + ߙ) = ܨா௠௔௫ܴఈୀ଴ (1 + ߙ) (7)  In the direction opposite that of the PLD, a strengthening effect occurs; because of the PLD, the SFRS will experience less force demand in this reverse direction. Engineers have a choice between accounting for the presence of the PLD by reducing the reverse yield strength of the SFRS (Equation 8), or by simply ignoring this strengthening effect and maintaining the original yield strength (Equation 9). Although this could be considered an inefficient use of materials, there may be factors which dictate minimum reverse yield strength such as large axial core loads, minimum steel requirements, or SFRS which have inherently symmetrical strengths. The relationship between the original and adjusted strengths is shown for a simple hysteretic model in Figure 3.3.    19   ܨ௬ି = ܨ௬(1 − ߙ) = ܨா௠௔௫ܴఈୀ଴ (1 − ߙ) (8)   ܨ௬ି = ܨ௬ = ܨா௠௔௫ܴఈୀ଴  (9)   Figure 3.3: Adjusted hysteretic models with (left) and without (right) reduced reverse yield strength (Best, Elwood, & Anderson, 2011). The effect that PLDs have on the yield strengths and hysteretic behaviour of the SFRS is indicated in Figure 3.4 below. In this figure, three Clough hysteretic models are compared. The first is representative of a SFRS with no PLDs. A second model represents a SFRS where a large PLD (α = 0.4) has been applied and the engineer has opted to reduce the reverse yield strength to account for the strengthening effect. Finally, a third Clough hysteretic model represents a SFRS with an identical PLD applied, where the engineer has not reduced the reverse yield strength. One should note that unlike α, the R α=0 of the SFRS does not affect the shape of the hysteretic behaviour, but simply scales the entire model to greater or lesser yield strengths. To account for the potential for both approaches to be taken, both design philosophies were considered in the SDOF system analyses.    20   Figure 3.4: Effect of a PLD on the required forward and reverse yield strengths of the SFRS. Original SFRS with α = 0.0 (left), SFRS with α = 0.4 and an unaltered reverse yield strength (center), and a system with α = 0.4 and a reduced reverse yield strength (right).  3.1.7 Hysteretic Models Eight different hysteretic models were considered in the SDOF systems. These included a Clough model; uniaxialMaterial Hysteretic, elastic perfectly-plastic model; uniaxialMaterial ElasticPP, elastic bi-linear model; uniaxialMaterial ElasticBilin, and a flag shaped model developed specifically to model cantilevered shear walls by Korchinski (2007); uniaxialMaterial ADK (OpenSees, 2012). Four variations of the Clough model were implemented each with different unloading stiffness degradation. Single variations of both the elastic perfectly-plastic model and the elastic bi-linear model were used (Mazzoni et al, 2007). Finally, two variations of a modified Korchinski model were used; one with hysteretic energy dissipation and one without.   21  3.1.7.1 Clough The hysteretic model used to develop the four Clough models was the ‘uniaxialMaterial Hysteretic’ model available in the library of models for OpenSees users. The input arguments allowed for the definition of either a simple bi-linear or tri-linear hysteretic backbone with no pinching of the unloading path. Hysteretic strength degradation was not modeled. For simplicity, only the degree to which the unloading stiffness degraded was varied between the four Clough models used. The general form of the Clough material model used in this study can be seen in Figure 3.5.   Figure 3.5: General backbone of the Clough model with partial unloading path indicated, © Mazzoni et al (2007), by permission. As one can see from Figure 3.5, the degradation of the Clough model was dictated by an input parameter $beta, where the unloading stiffness was given by the relationship shown in Equation 10, where μ was the relative strength factor of the system.   ܭ௨௡௟௢௔ௗ௜௡௚ = ߤି$௕௘௧௔ܭ௜௡௜௧௜௔௟ (10)  This model was defined to have a perfectly plastic yield plateau and unloaded at the same stiffness as the elastic portion of the curve when $beta was set to zero. When the model returned to zero force/stress, the hysteretic path returned to the previous point of maximum displacement. This then repeated in the opposite direction. The first variation of the Clough model used was specified to have zero unloading degradation. The hysteretic energy dissipation in this variation was the largest because of the large area   22  enclosed within each hysteretic loop. The hysteretic shape for the first variation of the Clough model is given in Figure 3.6.  Figure 3.6: Hysteretic behaviour of the Clough model with no unloading stiffness degradation, Clough, No Stiffness Degradation. The three additional variations of the Clough model were assigned incrementally greater unloading degradation. The parameter $beta was modified to 0.5, 0.75, and finally 1.0 shown in Figure 3.7, Figure 3.8, and Figure 3.9, respectively. In general, as the unloading slope degrades, the hysteretic energy dissipation of each hysteretic model decreases, because of decreased area within each hysteretic loop (Lestuzzi, Belmouden, & Trueb, 2007).   23   Figure 3.7: Hysteretic behaviour of the Clough model with mild unloading stiffness degradation, Clough, Mild Stiffness Degradation.  Figure 3.8: Hysteretic behaviour of the Clough model with moderate unloading stiffness degradation, Clough, Moderate Stiffness Degradation.   24   Figure 3.9: Hysteretic behaviour of the Clough model with large unloading stiffness degradation, Clough, Severe Stiffness Degradation.  3.1.7.2 Elastic Bi-Linear The elastic bi-linear hysteretic model had the same backbone definition as the Clough models presented in section 3.1.7.1; that is, a bi-linear, elastic perfectly-plastic backbone curve. However, whereas the Clough models unloaded along a new path when the loading direction was reversed, the elastic bi-linear model unloaded directly along the original backbone. This model does not dissipate any energy through hysteretic cyclic loading as no area is enclosed by the loading and unloading paths (Lestuzzi, Belmouden, & Trueb, 2007).  This model allowed for the definition of both the pre- and post-yield moduli in the forward and reverse direction. The transition point between the pre- and post-yield was defined by a strain coordinate in both the forward and reverse direction. The generic shape of the material backbone is indicated in Figure 3.10.    25   Figure 3.10: General backbone of the elastic bi-linear model, © Mazzoni et al (2007), by permission. Figure 3.11 below shows the hysteretic behaviour of this model, although it appears identical to the backbone curve presented in Figure 3.10, it in fact depicts many cycles of loading. The unloading path along the backbone so each new cycle is hidden behind the initial backbone.  Figure 3.11: Hysteretic behaviour of the elastic bi-linear model, Elastic Bi-linear.    26  3.1.7.3 Elastic Perfectly-Plastic The elastic perfectly plastic model was very similar to the Clough and elastic bi-linear models presented in sections 3.1.7.1 and 3.1.7.2. The backbone definition was identical and it differed only in its unloading characteristics. This model unloaded at the initial loading stiffness similar to the original Clough model however, unlike the Clough model, the unloading transitioned to reverse loading without a change in slope. This model produced the largest hysteretic loops out of all of the hysteretic models implemented in the SDOF system. For this reason, it was effective at dissipating energy (Lestuzzi, Belmouden, & Trueb, 2007).  This material accepts four arguments which define the elastic perfectly-plastic backbone curve. The material modulus, $E, is defined, as well as both the forward and reverse strains at which plastic behaviour begins, $eypsyP and $epsyN, respectively. Finally, there is an optional argument to specify an initial non-zero strain offset, which was set to zero. The general form of the backbone for this inelastic material model is shown in Figure 3.12 and the resulting cyclic behaviour hysteretic is given in Figure 3.13.   Figure 3.12: General backbone of the elastic perfectly-plastic model, © Mazzoni et al (2007), by permission.   27   Figure 3.13: Hysteretic behaviour of the elastic perfectly-plastic model, Elastic Perfectly Plastic.  3.1.7.4 Korchinski This hysteretic model was implemented using the OpenSees framework and referenced by the OpenSees executable as a dynamically linked library upon execution. The modified Korchinski model, ‘uniaxialMaterial ADK’, used in this project was originally developed to represent high-rise cantilevered concrete shear walls by Korchinski (2007). It was developed from experimental data gathered from a large scale shear wall test specimen and allowed for a nonlinear analysis of shear walls accounting for cracking and axial load to be conducted in a SDOF analysis (Korchinski, 2007). The Korchinski model is effective at representing the behaviour of high-rise concrete shear walls before and after cracking (Korchinski, 2007).  Within the framework of the hysteretic model, the transition curve from elastic to inelastic behaviour is dictated, in part, by an input parameter. This parameter is used to ensure a smooth transition from elastic to inelastic behaviour (Korchinski, 2007). The value of this parameter was previously manually modified to achieve a smooth transition curve. This had been practical given the original scope of use, however, given the parametric nature of the analyses conducted in this research; this manual entry was no longer feasible. It was necessary to use a constant value of this parameter for the entire range of the input   28  parameter values. The result was that the backbone curve of the material model was slightly irregular as shown in both Figure 3.14 and Figure 3.15. Furthermore, the exact shape of this irregularity was slightly variable between analyses. This was acceptable, as it was determined during the original development of the material model that the inelastic displacement demands were not sensitive to this transition parameter (Korchinski, 2007).  Two variations of this model were implemented, identical to each other with the exception of the shear at which cracks close. For the first variation, shown in Figure 3.14, cracks closed at 30% of the shear stress at flexural capacity. In the second variation, given in Figure 3.15, cracks closed at 60% of the shear stress at flexural capacity (Korchinski, 2007). Since this was the same stress at which the cracks opened, the second variation loads along the same path on which it previously unloaded.   Figure 3.14: Hysteretic behaviour of Korchinski model, Modified Korchinski.   29   Figure 3.15: Hysteretic behaviour of Korchinski model without hysteretic energy dissipation, Modified Korchinski, No Energy Dissipation.  3.2 RESULTS AND DISCUSSION  In the subsequent sections, the influence of the considered parameters on the response of the SDOF systems with PLDs will be described. Specific cases for R α=0, α, and particular hysteretic models are used as examples. General trends across the parametric domain are discussed in the sub-section entitled “Inelastic Demand Amplifications”. Unless explicitly stated, the results presented in this section are from models where P-∆ effects were considered and the reverse yield strength of the system was reduced to account for the strengthening effect of the PLD.  3.2.1 Influence of Hysteretic Behaviour The PLDs resulted in very different inelastic displacement demands depending on which of the eight hysteretic models was used. The Elastic Perfectly-Plastic and Elastic Bi-Linear hysteretic models showed no effect from the application of PLDs however others, such as the Clough, No Stiffness Degradation model demonstrated significant ratcheting. After considering the definition of each hysteretic model, it   30  was realized that the PLD altered the hysteretic shape of some models (Clough (Figure 3.16) and Modified Korchinski) while having no effect on other models (elastic perfectly plastic and elastic bi- linear (Figure 3.17)). Shifting the strengths of the SFRS about the zero-force axis caused asymmetrical stiffness in some models, once yielding occurred during cyclic loading (Figure 3.16).   Figure 3.16: Effect of asymmetrical strengths to account for PLDs, corresponding to α = 0.8, on the inelastic behaviour of the Clough, No Stiffness Degradation hysteretic model.    31   Figure 3.17: Effect of asymmetrical strengths to account for PLDs, corresponding to α = 0.8, on hysteretic behaviour of the Elastic Perfectly Plastic hysteretic model. The effect of the asymmetric stiffness on the hysteretic response of the SDOF systems can be best illustrated by looking at the response of otherwise identical SDOF systems models to an identical ground motion with each of the four Clough hysteretic models considered. The hysteric models with more asymmetrical stiffness (Clough, No Stiffness Degradation) displayed more seismic ratcheting and larger inelastic displacement demands as indicated in Figure 3.18.   32   Figure 3.18: Force – Displacement hysteretic response of the SDOF systems with each of the four Clough hysteretic models when subjected to ground motion 2, T = 1.0 seconds, α = 0.4, R α=0 = 4.0. Similar to the Clough models with unloading stiffness degradation, it was found that the two Korchinski models exhibited relatively small inelastic displacement demands as a result of PLDs. Similar to the Elastic Bi-Linear hysteretic model, the elastic stiffness in each direction of the Korchinski hysteretic model was not influenced by the PLD. The Korchinski models were still affected to a small extent because the PLD created uneven ‘flag’ loops in each direction. As a consequence, mild to moderate inelastic displacement demands were produced from each of the two Korchinski models.  3.2.2 Inelastic Demand Amplifications In order to quantify the displacement amplifications in the SDOF system resulting from the PLD, two amplification factors were devised. The first amplification factor, the inelastic amplification factor, γ, captured the combined influence of relative strength factor and the PLDs on the inelastic displacements of the SDOF systems. This factor was determined from the ratio of the inelastic displacement demands of a   33  SDOF system with PLDs, ∆(R α=0,α) to the displacement demands in an equivalent elastic system without PLDs, ∆(R α=0=1,α=0).   ߛ = ߂(ܴఈୀ଴,ߙ) ߂(ܴఈୀ଴ = 1,ߙ = 0) (11)   Figure 3.19: γ for the eight hysteretic models considered in the SDOF systems, α = 0.4, and R α=0 = 4.0. Although the inelastic amplification factor, γ, is commonly used to examine the effects of inelasticity on displacement demands, for this study, it was more appropriate to quantify the inelastic displacement demands created by the PLDs alone. Therefore, the maximum inelastic displacement demands were compared with and without PLDs. A parameter – the relative amplification factor, β, is introduced here to relate performance metrics in a building with PLDs corresponding to α, ∆(R α=0,α), to an equivalent building without PLDs, ∆(R α=0,α=0):    34   ߚ = ߂(ܴఈୀ଴,ߙ) ߂(ܴఈୀ଴,ߙ = 0) (12)  A β from each of the ten ground motions for each SDOF system was calculated and the mean response was determined. Analyses where a ‘collapse’ (defined in this study as non-convergence at large displacement demands) occurred were excluded from the calculation of β. In these collapse cases, the inelastic displacement demands increased became infinitely large due to the onset of structural instability. Collapse cases are extremely important because were indicative of very large lateral displacements leading to instability from P-∆ effects. Collapses, where they occurred, are presented along with the for each SDOF system configuration.  The results from the SDOF system with each of the eight hysteretic models used to represent inelastic behaviour are presented (Figure 3.20), with the Clough, No Stiffness Degradation model corresponding to the original hysteretic model used by Best, Elwood, and Anderson (2011). Since the analysis results presented in Figure 3.20 were obtained without considering P-∆ effects, the models can be compared directly to the results from the study by Best, Elwood, and Anderson (2011).   35   Figure 3.20: β produced by the eight hysteretic models considered in the SDOF systems, excluding P-∆ effects, α = 0.4, R α=0 = 4.0. As one can see in Figure 3.20, the hysteretic model considered significantly impacted the β exhibited by the SDOF systems. The Clough, No Stiffness Degradation model used by Best, Elwood, and Anderson (2011) produces the largest amplifications while the other Clough models showed decreasing levels of amplifications as their unloading stiffness decreased. Both variations of the Korchinski model showed relatively mild amplifications through the entire period range considered. Finally, the Elastic Bilinear and the Elastic Perfectly-Plastic models both returned β of exactly 1.0 – corresponding to no change in the inelastic displacement demands as a result of the PLDs. It was initially surprising that the Elastic Bi- Linear and Elastic Perfectly-Plastic models produced β of exactly 1.0, however, it was found that this was the case regardless of R α=0, α, and T. This indicated that the PLD had no impact on the inelastic displacements of these models. The reason for this result, and ultimately the primary source of variations in β between the hysteretic models, was the effect of PLDs on the hysteretic behaviour discussed in section 3.2.1.   36  The PLD did not impact the hysteretic shape of both the Elastic Bi-Linear and Elastic Perfectly-Plastic models; therefore their inelastic displacements were identical regardless of the PLD applied. Other hysteretic models were affected to varying degrees; resulting in varying degrees of inelastic displacement amplifications as is seen throughout section.  While Figure 3.20 was useful to bring context to the results produced by Best, Elwood, and Anderson (2011), it was important to include P-∆ effects in order for comparison to MDOF systems to be appropriate. The β for the same parametric conditions considered in Figure 3.20 with the addition of P-∆ effects are given in Figure 3.21.  Figure 3.21: β produced by each of the eight hysteretic models considered in the SDOF systems, α = 0.4, R α=0 = 4.0. The consideration of P-∆ effects significantly increased the β for the Clough, No Stiffness Degradation model. The difference is particularly pronounced in the higher period ranges, where the β increased from approximately 1.15 in Figure 3.20 to 1.7 in Figure 3.21. Other hysteretic models considered were affected to lesser extents, with mild increases in the amplifications seen across the entire period range of the   37  analyses. However, the Elastic Bilinear and the Elastic perfectly Plastic hysteretic models were not affected by P-Δ effects and again produced β = 1.0, for the entire period range considered. Although four collapses are indicated along the Elastic Perfectly-Plastic curve, these were a result of the high R α=0 of the system and occurred regardless of the PLDs.  One main concern presented by Best, Elwood, and Anderson (2011) was the inconsistency between the β which resulted from the SDOF systems the corresponding MDOF systems. In the MDOF systems, larger β were observed at higher periods, a trend not observed in the results from the SDOF systems. At the time, it was hypothesized that this discrepancy was due to the exclusion of P-∆ effects in the SDOF systems. The results of the analyses conducted in the SDOF systems with P-Δ effects for the Clough, No Stiffness Degradation model confirmed this hypothesis. The β are now more consistent with the MDOF results from the exploratory study as shown in Figure 3.22.  Figure 3.22: Comparison of the SDOF and MDOF systems with the Clough, No Stiffness Degradation hysteretic model, with and without P-∆ effects, R α=0 = 4.0, α = 0.4.    38  In addition, to making the SDOF and MDOF systems more consistent, the inclusion of P-Δ effects had another significant effect on the behaviours observed. There was now the potential for the SDOF system to become unstable and to ultimately collapse if the displacements became too great. These collapses are extremely significant because they demonstrated that the lateral displacements were significant enough that the system became unstable. Analysis cases which resulted in collapse are indicated along with the number of collapses which occurred out of the ten ground motions considered for each data point. The mean β for the original Clough, No Stiffness Degradation hysteretic model, with R α=0 = 4.0 and varying α are shown in Figure 3.23. Note that for large α, β becomes extremely large at higher fundamental periods.  The β produced by PLDs corresponding to α = 0.8 are less than those produced by PLDs corresponding to α = 0.6 at fundamental periods between of 3.25 and 4.00 seconds. This is due to the collapses which occurred as a result of PLDs corresponding to α = 0.8 in this period range. The simultaneous exclusion of these severe collapse cases from the determination of β for α = 0.8, and inclusion when coming short of collapse at the more moderate α = 0.6 resulted in the discontinuity in this period range.    39   Figure 3.23: β for the SDOF systems with the Clough, No Stiffness Degradation hysteretic model and varying α, R α=0 = 4.0. The high β for the SDOF systems with large PLDs indicates that there were large inelastic displacement demands when the Clough, No Stiffness Degradation hysteretic model was considered. This was also true for the other Clough hysteretic models, although to decreasing extents for the variations with significant degradation in the unloading stiffness. The large β from this model were not exhibited other hysteretic models considered. As mentioned, the Elastic Bi-Linear and the Elastic Perfectly-Plastic hysteretic models produced β = 1.0 (Figure 3.24) for all considered α, R α=0, and T.   40   Figure 3.24: β for the SDOF systems with the Elastic Bi-Linear and Elastic Perfectly-Plastic hysteretic models and varying α, R α=0 = 4.0. The four collapses occurred for the Elastic Perfectly-Plastic hysteretic model regardless of PLDs. Although not as dramatic the behaviour of the Elastic Bi-Linear and the Elastic Perfectly-Plastic hysteretic models, the Modified Korchinski and the Modified Korchinski, No Energy Dissipation hysteretic models also showed relatively small β as a result of PLDs. These minimal amplifications are presented in Figure 3.25 for the Korchinski hysteretic model.   41   Figure 3.25: β for the SDOF systems with the Modified Korchinski hysteretic model and varying α, R α=0 = 4.0.  3.2.3 Effect of the Relative Strength Factor Increasing R α=0 of the SDOF systems resulted in increased inelastic displacement demands as seen in Figure 3.26. Increasing R α=0 of the system had no direct influence on the asymmetry in stiffness of the hysteretic model. Large R α=0 values had the effect of scaling down the entire hysteretic loop relative to the seismic demands, which facilitated yielding during a greater number of cycles. The increased propensity to yield associated with higher R α=0 caused large inelastic displacement demands. Clough hysteretic models had loading stiffness dependant on the previous maximum displacement in each direction.   42   Figure 3.26: β for the SDOF systems with the Clough, No Stiffness Degradation hysteretic model and varying R α=0, α = 0.4.  3.2.4 Variability While mean β were the primary concern, it was important to quantify the variability of β between individual ground motions. Mean β was useful for quantifying the domain of influence for PLDs and the magnitude of their influence - it helped to characterize the nature of the ratcheting phenomenon In order for this characterization to be useful in establishing design limits, it needs to be demonstrated that the domain of influence is consistent for all ground motions. Without quantifying variability, it is conceivable that specific ground motions cause massive β while others cause no amplifications whatsoever. In this case, it would be very difficult to safely design buildings with PLDs to withstand these specifically dangerous ground motions. If β from each ground motion to varies significantly, it will be more difficult to define effective design limits and even more difficult to justify them.    43  A PLD resulting in α = 0.4 was applied to the SDOF systems as shown in Figure 3.27. Two interesting conclusions can be drawn from the scatter in β. First, some ground motions are prone to producing large β across the entire period range considered while others show consistently show minimal displacements - well below the mean. Secondly, β produced from each ground motion are variable across the domain of considered periods. This variability was greater for ground motions which caused relatively large β. Finally, the overall variability between ground motions increased with the period of the system with one or two ground motions producing relatively large β at periods greater than 3.0 seconds.  Figure 3.27: β for the SDOF systems subjected to each of the ten ground motions with the Clough, No Stiffness Degradation hysteretic model, R α=0 = 4.0, α = 0.4. Also indicated are the mean, µ, and mean plus one standard deviation, µ + σ.  3.2.5 The Effect of Reverse Yield Strength To accommodate PLDs, engineers have two logical alternatives; reduce the strength of the SFRS in the reverse direction thus utilizing the strengthening influence of the PLD, or to leave the strength in the   44  reverse direction unaltered. The effect of a PLD on the required forward yield strength and the alternatives for reverse yield strength are indicated in Figure 3.28 for the Elastic Perfectly-Plastic hysteretic model.  Figure 3.28: Effect of a PLD on the Elastic Perfectly-Plastic hysteretic model for both reduced and unaltered reverse yield strengths. Original hysteretic model for α = 0.0 (left), α = 0.4 with an unaltered reverse yield strength (center), and α = 0.4 with a reduced reverse yield strength (right). When the reverse yield strength of the SFRS was reduced, this Elastic Perfectly-Plastic hysteretic model maintained its original shape. For this approach, the only effect was that the model reoriented about a force equal to the PLD and its corresponding static displacement. As discussed in section 3.2.2 such a shift had no effect on the inelastic displacement demands of the SDOF systems because the hysteretic model maintained its original symmetric stiffness. However, when the reverse yield strength was not reduced to accommodate the PLDs, such as in the ‘α = 0.4, Unaltered’ case, inelastic displacements increased and β > 1.0 (Figure 3.29).   45   Figure 3.29: β for the SDOF systems with the Elastic Perfectly-Plastic hysteretic model and two considered reverse yield strength philosophies, α = 0.4, R α=0 = 4.0. These results can be explained by examining the hysteretic response of the three cases identified in Figure 3.28 to the same ground motion at, R α=0 = 4.0, and T = 1.0 seconds, given in Figure 3.30, Figure 3.31, and Figure 3.32.   46   Figure 3.30: Hysteretic response of a SDOF system with the Elastic Perfectly-Plastic hysteretic model to ground motion 2, α = 0.0, R α=0 = 4.0, T = 1.0 seconds.   47   Figure 3.31: Hysteretic response of a SDOF system with the Elastic Perfectly-Plastic hysteretic model to ground motion 2. The reverse yield strength has been reduced to account for the PLD, α = 0.4, R α=0 = 4.0, T = 1.0 seconds. The hysteretic behaviour in Figure 3.30 and Figure 3.31 is the exact same, simply oriented about different initial states. The response in Figure 3.30 began with an initial state at the origin (zero displacement, zero force) while the response in Figure 3.31 began with an initial resting state at the force of the PLD and corresponding static displacement. If this static displacement of the building was accounted for with an initial building camber, as was done in this study, PLDs have no effect on the inelastic displacement demands on the SFRS.  The same was not true if the reverse yield strength was not reduced to account for the PLD (Figure 3.32). As one can see, this third case allowed little yielding in the reverse direction because of the strengthening effect of the PLD. As a result, it rarely yielded in reverse and had difficulty recovering the forward inelastic displacements which developed.   48   Figure 3.32: Hysteretic response of a SDOF system with the Elastic Perfectly-Plastic hysteretic model to ground motion 2. The reverse yield strength has been unaltered despite the PLD, α = 0.4, R α=0 = 4.0, T = 1.0 seconds. Larger inelastic displacement demands were produced in the SDOF systems when the reverse yield strength of the hysteretic model was not reduced. The effect of the reverse yield strength on the Clough, No Stiffness Degradation hysteretic model is shown in Figure 3.33, which shows that β was consistently greater when the reverse yield strength was left unaltered. Similar to the Elastic Perfectly-Plastic case, the when the reverse yield strength was not reduced, the SDOF system had difficulty recovering the inelastic displacements which occurred in the direction. As a result, there was a greater propensity to ratchet during cyclic loading.   49   Figure 3.33: β for the SDOF systems with the Clough, No Stiffness Degradation hysteretic model and two considered reverse yield strength philosophies, α = 0.4, R α=0 = 4.0.  3.3 CONCLUSIONS  The hysteretic model considered has an enormous impact on the seismic ratcheting effect. Depending on which hysteretic model was used, the SDOF systems responded very differently to PLDs. When the reverse yield strengths were reduced, the Elastic Bi-Linear and Elastic Perfectly-Plastic hysteretic models were not influenced by the presence of PLDs (β = 1.0 for all R α=0, α, and T). Other hysteretic models, such as the Clough and Korchinski models were affected, but to very different degrees.  It was discovered that PLDs only promote seismic ratcheting if the shape of the hysteretic model is affected by the application of PLDs. In Figure 3.34, the lower curve represents the original hysteretic model. When the forward and reverse yield strengths are adjusted to account for the PLD, the hysteretic   50  model is shifted upwards. Because of the PLD and the resulting static displacement, the initial resting state is also shifted upwards and to the right. The relative position of the initial resting state within the shifted hysteretic model is unchanged, as is the shape of the hysteretic model itself. As a result, the inelastic displacements not affected by the PLD, provided that an initial static displacement (camber) is provided to account for the PLD.   Figure 3.34: Effect of asymmetrical strengths to account for PLDs, corresponding to α = 0.8, on hysteretic behaviour of the Elastic Perfectly Plastic hysteretic model. Importantly, if the reverse yield strength of these hysteretic models – which are not changed by the PLD - was not reduced, then the PLD would amplify the inelastic displacements.  In such a case, the SFRS would have an increased effective strength in the reverse direction and thus be more prone to yielding in one direction than the other (asymmetrical strengths).    51  The mechanism responsible seismic ratcheting in the SDOF systems can be better understood by considering the effect of PLDs on the shape of the different Clough hysteretic models. The Clough, No Stiffness Degradation hysteretic model (Figure 3.35) exhibited the largest β of all eight hysteretic models.   Figure 3.35: Effect of a PLD, corresponding to α = 0.8, on the inelastic behaviour of the Clough, No Stiffness Degradation hysteretic model. As with the Elastic Perfectly-Plastic and Elastic Bi-Linear hysteretic models, the upwards shift has no effect on the relative position of the initial resting state to the yield plateaus. However, the shape of the Clough hysteretic model is influenced by its position relative to the zero-stress (force) axis. The shift upwards creates a reduced effective stiffness in the forward direction which results in asymmetrical stiffness in the inelastic behaviour. It is this asymmetrical stiffness which results in seismic ratcheting and promoted the development of large inelastic displacement demands. When the reverse yield strength was not reduced the hysteretic models had asymmetrical strength as well, this combination lead to even larger inelastic displacement demands.    52  The significance of effective asymmetric stiffness for the ratcheting effect is confirmed by considering the Clough models with reduced unloading stiffness. In these other Clough hysteretic models, the unloading stiffness is smaller and therefore less stiffness asymmetry is produced by the PLD (Figure 3.36).  Figure 3.36: Effect of PLD, corresponding to α = 0.8, on the hysteretic behaviour of the Clough, Moderate Stiffness Degradation hysteretic model. From the SDOF system analyses considering different hysteretic models, it is clear that the mechanism driving the ratcheting effect is the asymmetry in hysteretic properties which can result from PLDs. If the reverse yield strength is not reduced, or if the shape of the hysteretic model is influenced by the PLDs, the asymmetrical properties promote seismic ratcheting in the direction of the PLDs. The effects of R α=0 and T on the SDOF systems are less consistent. The seismic ratcheting becomes more severe with increased PLDs and promoted collapse for some cases.  A simplified lumped plasticity hysteretic model, as used by Best, Elwood, and Anderson (2011), is not appropriate for the MDOF systems. The seismic ratcheting phenomenon is very sensitive to the inelastic behaviour of the SFRS system, and therefore a more comprehensive model is required, which accurately captures the axial-flexural interaction.   53  The seismic ratcheting effect is more significant in SFRS with unaltered reverse yield strength and asymmetrical strength. Since this is an approach which may be adopted by some engineers, it was decided that this conservative case would be considered in the MDOF systems.   54  Chapter 4 - Methodology  At the conclusion of the SDOF study, it was clear that comprehensive MDOF models were required to properly characterize and quantify the nature of the risk posed by PLDs. The SDOF results were dependent upon the assumed hysteretic model and inelastic displacement amplifications were highly variable depending on the modelling approach.  Two concrete shear wall MDOF systems are outlined in this chapter, cantilevered concrete shear wall buildings and coupled concrete shear wall buildings. The main focus of this study was the behaviour of these two SFRSs; however, a third SFRS - steel braced frames - was also considered and presented as a case study in section 6.6. The buildings were modelled in 2 dimensions, with 2 translational and 1 rotational degrees of freedom. The buildings consisted of a single interior concrete shear wall system with a single line of concrete columns running up each side of the building as shown in Figure 4.1 and Figure 4.2. The columns and shear wall were connected by an elastic floor plate which extended the width of the building between the exterior columns before being cantilevered out 0.6 meters on either side of the building from the center of the columns. This floor plate - modelled with truss elements - constrained the lateral deflections of the columns to that of the shear wall and also transferred the PLDs resulting from inclined columns to the shear wall. The floor plate varied in size depending on the height of the building, and its dimensions are included in Table 4.1.    55   Figure 4.1: Structural configuration of 5 storey cantilevered concrete shear wall model. The coupled wall model, shown in Figure 4.2 for the 5 storey building, was more complicated, as it had two coupled walls connected by coupling beams at each floor level. As with the cantilevered model, the floor elements were pinned to nodes located at the centroid of each wall along its height. Infinitely stiff and strong rigid links were used to connect the centroidal nodes to the line of nodes along the interior of each wall. Coupling beams then connected the interior line of nodes on each wall.   56   Figure 4.2: Structural configuration of 5 storey coupled concrete shear wall model.  In the SDOF system, periods between 0.5 seconds and 5.0 seconds were considered at every 0.25 second period increment. This meant that the system was analyzed at 19 different periods which resulted in a large number of analyses. Such a large number of analyses were impractical for the MDOF systems due to vastly increased computational demands. Therefore, 6 fundamental periods were considered in the concrete shear wall buildings; 0.5, 1.0, 2.0, 3.0, 4.0, and 5.0 seconds corresponded to 5, 10, 20, 30, 40, and 50 storey buildings, respectively. For each building height analyzed, an assumed concrete cross section was used for all relative strength factors and PLDs. The size of this concrete core increased as the height of the building increased (Yathon, 2010). The concrete core sections used for cantilevered building models are given in Appendix A.4 and the concrete core sections used for coupled building models are given in Appendix A.6 (Yathon, 2010). A typical storey height of 3.5 meters was assumed in all buildings and a first storey height of 4.5 meters was assumed. The heights of each of the six MDOF system buildings are indicated in Table 4.1.    57  Table 4.1: Height and floor plate dimensions of the six concrete wall building heights considered. Number of Storeys, N Building Height, h (m) Floor Plate Width (m) 5 18.5 27 10 36 27 20 71 27 30 106 27 40 141 30 50 176 37  In addition to the building dimensions, the concrete properties considered varied with the height of the building. The taller the building, the stronger the concrete used, and the higher it’s associated modulus, Ec, as indicated in Table 4.2. Table 4.2: Concrete properties modelled for each building height. Number of Storeys, N Concrete Strength, f’c (MPa) Concrete Modulus of Elasticity, Ec (MPa) 5 25 22500 10 30 24648 20 35 26622 30 40 28460 40 45 30187 50 50 31820  4.1 GROUND MOTION SCALING  For the concrete building models, it was convenient to scale each of the ground motions so that they each imposed same elastic base moment demands on a given model. This allowed for the same fibre section model to be used, regardless of the particular ground motion.    58  In order to determine the elastic base moment demands which would be representative of the hazard in Vancouver, a modal analysis, considering the first ten modes, was performed for each MDOF model. The Square Root Sum of Squares (SRSS) method was used to combine the modal responses to the NBCC 2010 UHS for Vancouver at a 2% in 50 year probability of exceedance (NBCC, 2010).  Once the elastic overturning moment demands were determined from the Vancouver UHS, the ground motions were linearly scaled to match the elastic overturning moment demands at each building height. Elastic dynamic analyses were conducted on each building model with each un-scaled ground motion. With the elastic overturning moment demands resulting from each un-scaled ground motion, (ME max)dynamic and the elastic overturning moment demands from the Vancouver UHS, (M E max))SRSS, a scaling factor was determined for each ground motion. The calculation of this scaling factor is shown in Equation 13. As a result of this approach, each ground motions was representative of the seismic hazard in Vancouver. The scaling factors applied to each ground motion for each shear wall model are given in Appendix A.2.   ݈ܵܿܽ݅݊݃	ܨܽܿݐ݋ݎ = 	 ൫ܯா೘ೌೣ൯ௗ௬௡௔௠௜௖ ൫ܯா೘ೌೣ൯ௌோௌௌ  (13)  The resulting acceleration spectra for the ten ground motions scaled for the 5 and 50 storey cantilevered buildings models are given in Figure 4.3 and Figure 4.4, respectively. For the 5 storey cantilevered shear wall building, the scaled ground motions match the Vancouver UHS spectrum very closely at the fundamental period of 0.5 seconds, as expected. For the 50 storey buildings, however, the spectra resulting from the scaled ground motions fall well below the Vancouver UHS spectrum at 5.0 seconds. This is because there were significant contributions from the higher modes to the seismic demands in the 50 storey buildings and by scaling the ground motions to the Vancouver UHS with the SRSS method, these contributions from higher modes were included in the scaling process.   59   Figure 4.3: Acceleration response spectra for ten ground motions scaled to match the maximum elastic base overturning moment imposed by the Vancouver UHS on the 5storey cantilevered shear wall building.    60   Figure 4.4: Acceleration response spectra for ten ground motions scaled to match the maximum elastic base overturning moment imposed by the Vancouver UHS on the 50 storey cantilevered shear wall building.  4.2 SFRS STRENGTH AND DEMAND RELATIONSHIPS  For the purposes of conducting a parametric study with a broad range of considered wall strengths and PLD demands, it was necessary to develop consistent definitions for the wall strengths and demands imposed. Similar to the approach taken for the SDOF study, the analyses were parametrically varied using the relative strength factor, R α=0, and the applied demand ratio, α. While the SDOF systems dealt with axial force in a spring, the shear wall buildings were designed in terms of the moments at the base of the shear wall. Through the use of these two easily defined parameters, R α=0 and α, the relationships between the dead load demands; MPLD, elastic overturning moment demand; ME max, and the yield moment strength of the wall in each direction; My+ and My-, were easily defined. The magnitudes of α considered were 0.0, 0.05, 0.1, 0.2, 0.4, 0.6, and 0.8. R α=0 of 1.0, 2.0, 3.0 and 4.0 were considered.   61   R α=0 is related to, but should not be confused with force reduction factors specified in the building codes. Rather, R α=0 is a term devised to simply relate strength of the SFRS in a building with vertical columns to the elastic demands imposed on it by the seismic loading. This relationship is shown in Equation 14.   ܯ௬ = ܯ	ா	௠௔௫ܴఈୀ଴  (14)  Once the base yield strength of the wall was calculated, based on the desired Rα=0, the dead moment demands at the base of the shear wall were determined from α. This dead moment demand is a ratio of the base yield strength at the bottom of the wall, and as a consequence, it is dependent Rα=0 in addition to α (Equation 15). As a consequence of the dependence of MPLD on both α and R α=0, the column inclination in each analysis was increased by increasing α, and decreased by increasing R α=0.   ܯ௉௅஽ = ߙܯ௬ = ߙ	ܯ	ா	௠௔௫ܴఈୀ଴  (15)  To account for the PLDs, MPLD, imposed by the inclined columns, the yield strength of the wall was increased in the forward (positive) direction and unchanged in the reverse (negative) direction. This final adjustment is shown in Equation 16 through Equation 18. The resulting moment capacities of the shear wall system are shown in Figure 4.5 in comparison to the seismic and dead load demands on the system.   ܯ௬ା = ܯ௬ + ܯ௉ா௅  (16)   ܯ௬ା = ܯ	ா	௠௔௫(1 + ߙ)ܴఈୀ଴  (17)   ܯ௬ି = ܯ	ா	௠௔௫ܴఈୀ଴  (18)    62   Figure 4.5: Schematic of the general parametric relationship between the demands and capacities of the shear wall systems. The relative magnitudes shown correspond to a building with R α=0 = 2.0 and α = 0.5.  4.3 GRAVITY SYSTEM  Columns and floor plates were used to transfer PLDs to the concrete shear wall from the gravity system. The columns and floor plates were modelled as a pure gravity system - the columns and floor plates supported axial forces but did not transfer shear or flexural forces. Columns and floor plates were modelled as truss elements. Pinned connections were considered between columns and floor plate elements and the bases of the exterior columns were pinned. This was an appropriate approach because when lateral deflections are imposed on the gravity system, cracking and spaling typically occur at the connections between floor plates and columns. As a result, plastic hinges with negligible flexural resistance form at these connections while the SFRS is still within the elastic range (ATC, 2010).   63   Axial behaviour in these truss elements was assumed to remain elastic. This is a reasonable assumption because, although some cracking and spaling occurs at the connections between the floor plates and columns, this is accounted for and does not compromise axial load capacity (ATC, 2010).  4.4 ELASTIC SHEAR WALL  Two different approaches were taken to model the shear wall properties, depending on the elevation of the wall. A plastic hinge was assumed to form in the region above the fixed base of each shear wall. The inelastic flexural response of the wall was considered to occur entirely within this region. Above the plastic hinge, the walls were assumed to remain elastic throughout the duration of the analyses. In this elastic range the wall between each storey was modelled with an elastic beam-column element. The flexural stiffness of these elements was assumed to be 0.8EcIg, to account for the influence of some cracking, and the axial stiffness was assumed to be EcAg (ASCE, 2007). Where Ec is the elastic modulus of concrete, Ig is the moment of inertia of the gross concrete section, and Ag is the gross area of the concrete section.  4.5 PLASTIC HINGE A plastic hinge formed at the base of the concrete shear walls when R > 1.0 and the inelastic behaviour of the shear wall was assumed to occur within this plastic hinge region. It was assumed that the shear walls were governed by flexure and shear deformations were ignored. This simplification was appropriate given that when properly detailed, shear walls are designed to experience flexural yielding before shear failure. As an extension of this simplification, the effects of permanent shear demands on the plastic hinge were not considered directly. Given that shear demands can amplify during the nonlinear dynamic response, it is possible that the presence of the PLD could promote shear failure in the plastic hinge. This effect was not considered in this study.  The length of the plastic hinge region was dependent on the length of the shear wall. This has been found to be an appropriate assumption in previous studies on shear wall behaviour (Bohl, 2006). The plastic hinge length was taken to be lesser of half of the wall length and the first storey height. It was assumed that yielding would not progress into the second storey because of the presence of the second storey floor   64  plate connection. This is because the floor plate provides containment to the hinging mechanism and prevents yielding above. The plastic hinge lengths for the various building heights are given in Table 4.3. Table 4.3: Plastic hinge lengths for different building heights. Number of Storeys, N Cantilevered Shear Wall Plastic Hinge Length (m) Coupled Shear Wall Plastic Hinge Length (m) 5 2.3 1.15 10 2.75 1.5 20 3.75 2.0 30 4.5 2.25 40 4.5 2.875 50 4.5 3.375  An important lesson was learned in the SDOF systems analyses. It was discovered that the hysteretic model representing inelasticity in the SFRS had a large impact on the extent of seismic ratcheting. The effect of the PLD on the loading and unloading stiffness of the SDOF system seemed to be the main driver of seismic ratcheting. For this reason, a different, more comprehensive approach was taken to that used previously by Best, Elwood, and Anderson (2011). Rather than lump inelastic behaviour into a single model, a complex fibre model was used to represent individual material components of the shear wall throughout the plastic hinge.  Three material models were considered within the plastic hinge for both the cantilevered and coupled shear wall models. The uniaxialMaterial Concrete01 material model, shown in Figure 4.6 was used as the template to represent both confined and unconfined concrete.    65    Figure 4.6: General backbone of the uniaxialMaterial Concrete01 material model, © Mazzoni et al (2007), by permission. This material model assumed zero tensile strength – representative of cracked concrete – which was appropriate given that concrete typically cracks early during significant ground motion. Both concrete models were assigned the same compressive strength, f’c, or $fpcu, as indicated in Table 4.2 above. It was assumed that the strengthening effect of confinement on the concrete was negligible. The strain at this maximum strength, epsc0, was set to be 0.002 for all unconfined and confined concrete strengths. When pushed beyond the maximum compressive strength, the concretes were modeled according to Mander’s Model (Mander, Priestley, & Park, 1988). The unconfined concrete degraded to zero ultimate capacity at 350% of the strain at maximum strength (0.007). The confined concrete retained more strength after exceeding its capacity and maintained 80% of its maximum strength at 1000% of the strain at maximum stress (0.02).  The uniaxialMaterial Hysteretic material model was used for the longitudinal reinforcing steel in the cantilevered shear walls. As one can see in Figure 4.7, this material model is a somewhat simplified representation of true steel behaviour. This is because it ignores both Bauschinger effects and strain hardening. Young’s Modulus of 200GPa and strength of 400MPa were assumed for all steel in all of the buildings modelled regardless of the height of the building. The exception to this were buildings where fully elastic behaviour was desired (Rα=0 = 1.0). For these cases, the steel was defined to have enormous strength to insure there was no yielding of the steel in wall.    66   Figure 4.7: General backbone of the uniaxialMaterial Hysteretic model, © Mazzoni et al (2007), by permission. The backbone of the hysteric steel behaviour was modelled as elastic perfectly-plastic with elastic response transitioning to plastic yielding at a strain of 0.002 and associated stress of 400MPa. Unloading occurred at the same stiffness as the original loading, however the stiffness of the material degraded proportionally to the maximum historic strains as shown in Figure 4.8.   67   Figure 4.8: Hysteretic behaviour of longitudinal reinforcing steel when subjected to monotonous increasing cyclic displacements.  4.6 WEIGHT AND MASS DISTRIBUTION  There were two sources of mass in the shear wall buildings: the self-weight of the concrete shear wall and the floor weight. The self-weight of the shear wall was proportional to the area of the cross-section; however, the floor weight was not proportional to the area of the floor plate. Rather, the floor weight was specified so that each shear wall building had a fundamental period of 0.1N, where N is the number of storeys.  It was convenient to apply weight and mass independently to the building model. The self-weight of the wall and the mass corresponding to the floor weight were lumped together at each storey. This lumped mass was applied at each storey node along the height of the shear wall. For cantilevered wall buildings,   68  this resulted in a single lumped mass at each floor, for coupled wall buildings, the mass at each storey was split into two halves and lumped along the height of each coupled wall.  The weight distribution of the model was critical for two reasons. The weight supported by the irregularities in the gravity system has a direct impact on the magnitude of the PLDs. Conversely, the weight supported by the shear walls has a strengthening effect and influences the hysteretic properties exhibited. In practice, the ratio of weight supported by shear walls is variable. This ratio depends on the tributary areas, floor plate geometry, and column spacing among other contributing factors. The strengthening effect of this axial load on the concrete shear walls cannot be ignored because it creates inherent flexural strength. When large, this axial load can cause significant inherent flexural strength which creates an upper limit to the Rα=0 which can be achieved. Since large Rα=0 were found to promote the greatest inelastic displacement demands in the SDOF systems, it was important that the buildings be able to represent the largest Rα=0 which might be achieved in practice. The self-weight of the wall alone represented a realistic and conservative case for the weight which would be supported by shear walls in practice. This weight was lumped along the shear wall height, with one lumped weight per storey in the cantilevered shear walls and two in the coupled shear walls. The floor weight was applied to the nodes along the exterior columns, with each side of the building supporting half.  4.7 GRAVITY SYSTEM IRREGULARITIES  There are multiple irregularities in the gravity system which can create PLDs on the SFRS. Three main irregularities were considered in this study: fully inclined columns, inclined columns over the lobby, and eccentric floor spans. The PLDs from each of the three loading mechanisms are very similar and the seismic responses were found to very consistent by Best, Elwood, and Anderson (2011).  The consistency between the inelastic responses of the three considered gravity irregularities was investigated in a further case study, which is included in section 6.1. In this case study, it was found that fully inclined columns yield un-conservative results relative to the eccentric floor span irregularity. Eccentric floor spans tend to promote larger roof drifts and base curvatures than do fully inclined columns when each irregularity is scaled to impose the same PLDs. To determine if this additional risk is significant, the eccentric floor span irregularity was considered. A single coupled shear wall building, which demonstrated large inelastic displacement demands when subjected to the inclined column irregularity, was instead considered with an eccentric floor span irregularity. It was found that the   69  demands were increased, as expected, however these additional demands were moderate and no additional collapses were instigated by this irregularity in the gravity system. β determined when this building was subjected to each of the three loading mechanisms are given in Table 4.4. The fully inclined column case was the impetus for this study and hypothesized to be the most widespread irregularity in practice. Therefore, the fully inclined column irregularity was used for the study, despite yielding mildly un-conservative relative to the eccentric floor span irregularity. Table 4.4: β for the roof drift demands in a 30 storey coupled shear wall building with each of the three types of structural irregularities. α = 0.2, Rα=0 = 4.0, subjected to the ten ground motions. Type of Irregularity β Fully Inclined 1.60 Inclined Lobby 1.10 Eccentric Floor Spans 1.83  The fully inclined column irregularity is illustrated for the 10 storey cantilevered wall model in Figure 4.9 and the 10 storey coupled wall model in Figure 4.10. A proportion of the self-weight of the building was carried by the exterior columns, which, when inclined, transferred a lateral force to the shear wall through the floor plate at each storey. For both types of wall models, the magnitude of the PLD was varied by altering the inclination of the columns on both sides of the building, with larger inclinations corresponding to increased PLDs.    70   Figure 4.9: 10 storey cantilevered shear wall buildings with fully inclined columns corresponding to α = 0.2(left) and α = 0.8 (right) at Rα=0 = 2.0.   Figure 4.10: 10 storey coupled shear wall buildings with fully inclined columns corresponding to α = 0.2(left) and α = 0.8 (right) at Rα=0 = 2.0.   71  4.8 INITIAL CAMBER  As a result of PLDs, the shear walls displaced laterally during static conditions. It was important to consider how these lateral displacements would be accounted for in practice. Concrete shear walls are constructed floor by floor. After each storey is poured and attains acceptable strength, the second storey follows. Despite the lateral displacements imposed by the PLDs, the shear wall would still be constructed vertically, floor after floor, resulting in a vertical shear wall despite the presence of a PLD. This is analogous to the way in which beams are often cambered to account for permanent dead loads. The shear walls were therefore modelled with an initial camber and when subjected to PLDs, displaced laterally to a vertical position typical of an unloaded shear wall.  4.9 CANTILEVERED SHEAR WALLS  Although there were many similarities between the cantilevered and coupled shear wall buildings, there were also fundamental distinctions. A schematic of a cantilevered shear wall system is shown in Figure 4.11 where M1 is the flexural resistance at the base of the wall. The modeling approaches taken for the cantilevered model are discussed in this section.  Figure 4.11: Overturning moment resistance mechanisms in a cantilevered shear wall, where M1 is the flexural moment developed at the base of the wall, © Composites, by permission (Park & Yun, 2011).    72  4.9.1 Building Weights and Overturning Moments The weights of the cantilevered shear wall models are shown in Table 4.5. The wall self-weight was dependent upon the cross section of the shear wall while the additional floor weight per storey was adjusted to achieve the desired fundamental building period. Because the cantilevered shear wall considered were relatively flexible compared to each corresponding coupled shear wall building, the floor weights per storey were small, relative to the coupled shear wall buildings. Table 4.5: Building weights for the cantilevered shear wall models. Number of Storeys, N Weight of Wall Per Storey (kN) Floor Weight Per Storey (kN) Total Weight Per Typical Storey (kN) Total Building Weight (kN) 5 665 4182 4847 23 999 10 836 3432 4268 42 390 20 1693 2591 4284 85 088 30 2580 1856 4436 132 189 40 3781 1862 5643 224 375 50 5248 3109 8357 416 014  Once the floor weights of the building were determined, the associated overturning moments, shown in Table 4.6, were calculated from the Vancouver UHS. This was done using the SRSS modal combinations method with the first ten modes considered, and first five modes for 5 storey buildings. Table 4.6: Base overturning moments for the cantilevered wall models. Number of Storeys, N Overturning Moment (kNm) 5 162 307 10 270 797 20 549 980 30 957 037 40 1 541 424 50 3 348 235     73  4.9.2 Effect of Axial Load in Shear Wall In mid- and especially high-rise concrete construction, the concrete shear walls generally support a portion of the floor weight in addition to their own self weight. As a consequence, the amount of axial load in shear walls is variable, and highly dependent upon the specific floor plan geometry and column spacing. In order to investigate the influence of the assumed axial load on the hysteretic behaviour of the cantilevered shear walls, dynamic analyses were conducted with a range of considered axial loads.  For the bulk of the analyses conducted, the axial load in the cantilevered shear walls resulted from the self-weight of the wall alone. All of the floor weight was supported by the columns. This small yet realistic axial load was considered because it allowed for large Rα=0 which were shown to promote seismic ratcheting by Best, Elwood, and Anderson (2011). In this respect, it was conservative to consider relatively small axial loads; however axial loads also have significant effects on the hysteretic behaviour of cantilevered shear walls. Although the same wall strength would be achieved, the source of this strength (reinforcing steel or axial load) had a significant influence on hysteretic behaviour exhibited. It was not clear how the different potential hysteretic behaviours would influence the seismic ratcheting, therefore this influence required investigation to insure that the most conservative approach could be considered.  When the axial load in the shear wall was exactly zero, the moment curvature behaviour at the base of the wall was found to be very similar to the Clough hysteretic model. This was logical because the flexural strength was derived from the longitudinal reinforcing steel the wall flanges and the inelastic behaviour of the longitudinal reinforcing steel was modelled a Clough hysteretic model. The Clough hysteretic model was used in the plastic hinge region by Best, Elwood, and Anderson (2011) and as a consequence similar hysteretic responses were exhibited between the two sets of models.  Although possible, it is extremely unlikely that a shear wall would have zero axial load. Shear walls will almost always support their own self weight, regardless of the floor plate and column configuration around them. Therefore, the assumed model, with the shear wall supporting its self-weight, represents a realistic lower bound. As the axial load increases, the area of longitudinal reinforcing steel is decreased and as a result, the behaviour of the wall becomes less akin to the Clough hysteretic model and increasingly flag-shaped, as is seen in Figure 4.12.  Although the shape of the hysteresis is highly dependent on the assumed axial load, the maximum inelastic curvatures, presented in Figure 4.12 for a 5 storey cantilevered shear wall building, do not   74  change significantly. The maximum curvatures observed at the base of the shear wall in this building were consistently about 2 rad/km, regardless of the axial load.   75   Figure 4.12: Base moment-curvature hysteretic response of a 5 storey cantilevered shear wall building with varying axial load in the shear wall, Rα=0 = 4.0, α = 0.0. Axial loads in the shear walls are, from top left to bottom right: zero, wall self-weight, wall self-weight plus 10% of floor weight, wall self-weight plus 20% of floor weight, wall self-weight plus 30% of floor weight, and wall self-weight plus 40% of floor weight.   76  The extent to which PLDs promotes seismic ratcheting in each of the models is critically important. Therefore, the relationship between the axial load and the resulting inelastic amplifications needed to be understood. In the results from the SDOF systems, it was clear that the Clough hysteretic model promoted the greatest extent of seismic ratcheting. Therefore, as the axial load increased, and the hysteretic behaviour transitioned from Clough to pinching behaviour, it was anticipated that the inelastic displacement demands would decrease. This was desirable because the assumption of low axial load would again provide a conservative representation of the range of behaviour which might be expected in practice. Therefore, this conservative behaviour needed to be confirmed in the MDOF cantilevered walls.  Because the wall with zero axial loads produced similar hysteretic behaviour to that modelled by best, Elwood, and Anderson (2011) it serves as an effective baseline for comparison. The hysteretic moment- curvature responses of the 5 storey cantilevered shear wall model with varying axial loads, Rα=0 = 4.0, and α = 0.4 are shown in Figure 4.13. There is a great deal of seismic ratcheting in cases with small axial loads. This ratcheting and the associated inelastic displacements consistently decrease as the axial loads in the walls increase. In walls with large axial load, pinching behaviour effectively inhibits seismic ratcheting from occurring.    77   Figure 4.13: Base moment-curvature hysteretic response of a 5 storey cantilevered shear wall building with varying axial load in the shear wall, Rα=0 = 4.0, α = 0.4. Axial loads in the shear walls are, from top left to bottom right: zero, wall self-weight, wall self-weight plus 10% of floor weight, wall self-weight plus 20% of floor weight, wall self-weight plus 30% of floor weight, and wall self-weight plus 40% of floor weight.   78  These results indicated that the assumed lower bound case (self-weight of wall alone) for axial load was indeed the critical governing case, producing the largest inelastic displacements when under the influence of a PLD. The axial load was found to provide a restorative pinching effect which mitigated the ratcheting phenomenon. The relative amplification factors, β, observed for the inelastic roof displacements of the 5 storey models with varying axial loads and α = 0.4 are given in Table 4.7. As suspected from the behaviour shown in Figure 4.13, the observed amplifications decreased as the axial load in the shear walls was increased. Therefore, the relatively low axial load resulting from the wall self-weight alone provided the worst-case scenario for cantilevered shear wall performance. It was therefore reasonable to assume that the model with low axial loads produced results conservative relative to buildings with greater axial loads in the cantilevered walls. Table 4.7: β for 5 storey cantilevered shear wall building with varying axial loads, Rα=0 = 4.0, α = 0.4. Axial Load β 0 1.77 WW 1.48 10% FW + WW 1.29 20% FW + WW 1.28 30% FW + WW 1.34 40% FW + WW 1.42  The results produced by Best, Elwood, and Anderson (2011) indicated a great propensity for the seismic ratcheting phenomenon to occur. However, the cantilevered shear wall buildings considered used a simple Clough hysteretic model which did not exhibit any pinching behaviour. After examining the effects of axial load on hysteretic behaviour in buildings with PLDs, it was clear that pinching behaviour mitigated the ratcheting phenomenon. As a result, the models developed by Best, Elwood, and Anderson (2011) were particularly susceptible to seismic ratcheting. Therefore, although large inelastic displacement demands were observed, the cantilevered shear wall buildings considered did not exhibit hysteretic behaviour which would be expected in practice.  4.9.3 Fibre Section Although the Clough hysteretic model considered by Best, Elwood, and Anderson (2011) was easy to implement and computationally efficient, it had limitations. The idealized model does not account for the effects of migrating neutral axis location during loading and unloading on the behaviour (Chaallal &   79  Gauthier, 2000). More importantly, the model does not consider axial-flexure interaction (Chaallal & Gauthier, 2000). In light of the significant influence of hysteretic behaviour observed in the SDOF systems, these limitations, necessitated the use of a comprehensive fibre section within the plastic hinge at the base of the shear walls.  A fibre section consisting of nine steel, eleven unconfined concrete, and eleven confined concrete material fibres was considered in the cantilevered shear wall within the plastic hinge. This fibre model was implemented in displacement based beam-column elements which extended the height of the plastic hinges given in Table 4.3. The distinction between unconfined and confined concrete was made based on an assumed clear cover of 20mm over 20M ties (NBCC, 2010). Concrete enclosed within the 20M ties was assumed to be confined, and the remaining concrete area was considered as unconfined. This is shown in a generalized schematic of the cantilevered shear wall section in Figure 4.14. In the fibre section, the area and properties of each unconfined concrete and confined concrete fibre was dependent on the shear wall section of the building, which was dependant only on building height.    80   Figure 4.14: Schematic of a generalized cantilevered shear wall section with distributed and concentrated longitudinal reinforcing steel, unconfined concrete, and confined concrete. The strength of each shear wall was variable depending on of the number of storeys, Rα=0, and α. For each building, the area of the concentrated longitudinal steel in the flanges was adjusted in order to achieve different strengths. In each fibre section, there were five steel fibres evenly distributed in the web which corresponded to minimum required distributed steel (NBCC, 2010). There were also two steel fibres in each flange which modelled the combination of concentrated longitudinal steel as well as the distributed steel that would be in the flanges. For large Rα=0, the area of the four steel fibres in the flanges was   81  reduced to achieve lower strength in both loading directions. These four steel fibres in the flanges had a minimum area corresponding to the minimum distributed steel requirements in the NBCC (NBCC, 2010).  Reducing the area of steel in the flanges was not always effective at reducing strength; a significant amount of wall strength was derived simply from axial load in the wall. Depending the height of the building, it was not always possible to achieve high Rα=0. The maximum attainable Rα=0 for each cantilevered shear wall building height are given in Table 4.8. Table 4.8: Maximum Rα=0 considered for each cantilevered shear wall building height. Number of Storeys, N Maximum Rα=0 5 6.0 10 4.0 20 2.0 30 2.0 40 2.0 50 2.0  To account for PLDs, engineers would ideally add strength to a wall in the forward direction and decrease strength in the reverse direction by adjusting the longitudinal steel areas. However, it was also not always possible to reduce the yield strength in the reverse direction because of inherent strength from the axial load and minimum steel requirements. This was especially true for tall buildings which had much greater axial loads and large concrete sections.  Engineers would likely take advantage of the strengthening effect of the PLD in the reverse direction by removing steel. However, if the yield strength in the reverse direction was governed by axial load and minimum steel, an engineer would be unable to make this adjustment. It was consistently observed in the results from the SDOF analyses more significant ratcheting occurred when the strength of the wall in the reverse direction was not reduced. To maintain a conservative modelling approach, this limiting case was assumed for all buildings. To account for PLDs, the steel area in the required flange and the opposite flange was left unaltered.    82  4.9.4 Pushover Response Prior to conducting the parametric study, it was necessary to validate the behaviour of the cantilevered shear wall buildings and ensure that they exhibited the desired yield strengths in each direction. The buildings were subjected to a displacement controlled pushover analysis with a triangular load distribution. This section considers the pushover response of the 5, 10, 20, and 30 storey cantilevered shear wall buildings with Rα=0 = 2.0 and α = 0.0 (Figure 4.15). The elastic overturning moment demands on the 5 storey building were 162 000kNm and the shear wall had a base moment capacity of 81 000kNm. The elastic overturning demands on the 30 storey building were 957 000kNm and the shear wall a base moment capacity of 478 500kNm. As expected, distinct tri-linear behaviour was exhibited, with decreases in the effective stiffness at the onset of cracking and yielding of the tensile reinforcing steel. Furthermore, the stresses and strains in the individual fibres corresponded with this progressive yielding.   Figure 4.15: Base moment as a function of base curvature during pushover analyses of 5, 10, 20, and 30 storey cantilevered shear walls, Rα=0 = 2.0, α = 0.0.   83  4.9.5 Hysteretic Response Static pushover analyses were used achieve the desired yield strengths in each direction (forward and reverse). Once the buildings were confirmed to have the correct yield strengths, the cantilevered shear wall buildings were subjected to dynamic time-history analyses. Figure 4.16 shows the hysteretic response, of the same 5, 10, 20, and 30 storey buildings discussed in section 4.9.4 to ground motion 2. The hysteretic responses of the 40 and 50 storey buildings are again omitted for to preserve the visual clarity of the other results. The 40 and 50 storey cantilevered shear wall buildings were found to exhibit the smallest base curvatures and highest flexural yield strengths.  Figure 4.16: Base moment-curvature hysteretic responses of 5, 10, 20, and 30 storey cantilevered shear walls, Rα=0= 2.0, α = 0.0.    84  4.9.6 Ductility Capacity The curvature capacity of shear walls is governed by the crushing strain of concrete which is typically between 0.003 and 0.004 (White, 2004). A value of 0.0035 was considered - consistent with the NBCC (2010). From this assumed limit, the curvature capacities of the different sections were determined from static pushover analyses. From these pushover analyses, it was found that the curvature capacity –beyond which significant damage could be expected - of the cantilevered shear walls was not significantly influenced by Rα=0 or α. Therefore, a single curvature capacity is given for each building height (Table 4.9). Table 4.9: Base curvature capacities of cantilevered wall sections. Number of Storeys, N Base Curvature Capacity (rad/km) 5 1.0 10 0.8 20 0.6 30 0.5 40 0.4 50 0.3  Global drift limits prevent instability and ensure post-earthquake access to buildings (Tremblay, Léger, & Tu, 2001). Inter-storey drifts limits have been established at 2.5% for normal occupancy buildings and 1% for post-disaster buildings by the NBCC (2010). The tall buildings initiative established global acceptance criteria which included a limit of maximum transient inter-storey drift of 3% and a residual inter-storey drift limit of 1% (PEER, 2010).  4.10 COUPLED SHEAR WALLS  The modeling considerations of the coupled wall shear wall buildings are discussed in this section. The fibre model used for the plastic hinge region is presented along with the assumed coupling beam   85  properties. The effect of these coupling beam properties on the seismic response of the system is discussed and the overall behaviour of the coupled wall system is analyzed.  Coupled shear walls are inherently more complex than cantilevered shear walls. In cantilevered shear wall buildings, resistance to overturning moments is provided by the flexural capacity at the base of the cantilevered shear wall. In coupled shear wall buildings, resistance to the overturning moment is derived from two resistance mechanisms. One of these two mechanisms is the flexural capacity at the base of the two coupled walls, labeled M1 and M2, in Figure 4.17. In addition, to this flexural capacity, a second resistance mechanism is provided through coupling (axial) forces, labeled T and C, generated in the shear walls. These axial forces are developed by the coupling beams which connect the two shear walls at each floor level.    Figure 4.17: Overturning moment resistance mechanism in a coupled shear wall, where M1 and M2 are flexural moments developed at the base of each wall, and T and C are axial forces developed by the coupling beams, © Composites, by permission (Park & Yun, 2011). There are two common approaches to reinforce coupling beams which are used in practice. Engineers may use either horizontal reinforcement or diagonal reinforcement (or a mixture of both), each of these reinforcing schemes are shown in Figure 4.18. The behaviour of coupled shear wall buildings is highly dependent upon the properties of the coupling beams, in terms of their stiffness, yield strength, and   86  displacement ductility capacity. Prior to significant yielding, it is the stiffness of coupling beams which governs the axial forces which they generate in the shear walls. After significant yielding has occurred, the yield strength and ductility of the coupling beams are important. In order for coupling beams to contribute to the overturning moment resistance of the building, they must maintain sufficient strength and generate axial forces in the coupled shear walls, even once significant yielding has occurred (Chaallal & Gauthier, 2000).  Figure 4.18: Typical coupling beam cross-sections with horizontal (left) and diagonal (right) reinforcing schemes, © Canadian Journal of Civil Engineering, by permission (Chaallal & Gauthier, 2000).  4.10.1 Building Weights and Overturning Moments For simplicity and consistency, the same concrete core wall sections were used for the coupled shear wall buildings that were used for the cantilevered shear wall buildings (Yathon, 2010). The wall sections were simply considered in perpendicular orientations corresponding to coupled and cantilevered behaviour. The cross-sectional areas of the walls were identical for both building types; therefore, the self-weight of the shear walls was also consistent, as shown in Table 4.10. Coupled shear walls are inherently stiff relative to cantilevered shear walls (Chaallal & Gauthier, 2000). Therefore, larger floor weights were required to achieve the same fundamental periods of 0.5, 1.0, 2.0, 3.0, 4.0, and 5.0 seconds for the 5, 10, 20, 30, 40, and 50 storey buildings, respectively.    87  Table 4.10: Building weights for the coupled shear wall models. Number of Storeys, N Weight of Wall Per Storey (kN) Floor Weight Per Storey (kN) Total Weight Per Storey (kN) Total Building Weight (kN) 5 665 6075 6740 33 461 10 836 8077 8913 88 835 20 1693 8870 10 563 210 662 30 2580 8142 10 722 320 762 40 3781 10 643 14 424 575 583 50 5248 13 445 18693 932 776  As a result of the relatively large floor weights, the coupled shear wall buildings were subjected to greater elastic overturning moment demands from the Vancouver UHS than were the cantilevered shear wall buildings. The elastic overturning moment demands of the six coupled shear wall models are given in Table 4.11. Table 4.11: Base overturning moments for the coupled wall models. Number of Storeys, N Overturning Moment (kNm) 5 242 300 10 580 021 20 1 302 274 30 2 219801 40 3 666 898 50 7 050 505   4.10.2 Coupling Beam Dimensions The coupling beam dimensions considered were taken from a study done on coupled shear walls: ‘Seismic Demands in High-Rise Concrete Walls’ by White (2004). In this study a constant coupling beam length of 2.44 meters (8.0 feet) was used with a depth of 1.67 meters (5.5 feet). The width of the coupling beams was taken to be equal to the width of the shear wall web to which they were connected. For simplicity and consistency, the coupling beam lengths and depths were kept constant across the entire   88  range of the parametric analysis (all T, Rα=0, and α). The widths of the coupling beams were matched to the thickness of the coupled walls to which they were connected.  The connection between each coupling beam and the centroid of shear wall was modelled with a rigid- beam element (Figure 4.19.). This allowed for stiff wall properties to be considered at each end of the relatively flexible coupling beam. The ratio of the rigid-beam length to the depth of the coupling beams was approximately 1, therefore, the rigid  beam length did not need to be reduced to correctly model the stiffness, as is required if the ratio is high (Moustafa & Takewaki, 2010).  Figure 4.19: Coupling beam configuration in coupled wall model.  4.10.3 Coupling Beam Strength A properly designed coupling beam will yield in bending before it yields in shear (White, 2004). Due to the geometry of coupling beams, shear demands are limited by the bending capacity. Therefore, it was only necessary to model the bending capacity of the coupling beams in order to achieve the desired yield strengths.  The shear forces which are developed in the coupling beams create axial forces in the two coupled shear walls. A compressive force is generated in one wall and a tensile force is generated in the other. Together,   89  these two coupling forces contribute to the overturning moment resistance of the SFRS (White, 2004). The ratio of the total overturning moment resistance of the SFRS that is provided by the coupling forces in the walls is known as the coupling ratio. The coupling ratio is calculated as indicated in Equation 19, where the T is the axial force generated in each wall when all the coupling beams have yielded in the same direction. L is the length between the centroids of the two walls, and M1 and M2 are the yield moments at the base of the two coupled shear walls (Park & Yun, 2011).   ܥܴ = ܶܮ ܶܮ + ܯଵ + ܯଶ (19)  This calculation is made assuming that both walls have fully yielded, and that all the coupling beams have fully yielded along the height of the building. Such a situation of full building yield is theoretical, as it is unrealistic that every coupling beam and both shear walls would yield, especially in high-rise buildings. Therefore, the coupling ratio of a coupled shear wall building may vary to some extent in practice. Table 4.12 shows the coupling ratios occupational use for various mid- and high rise coupled shear wall buildings in Vancouver. The office buildings have an average coupling ratio of approximately 0.79 while the residential buildings have a lower average coupling ratio closer to 0.69. This difference is reflective of the different architectural demands associated with the two occupational uses - office buildings tend to have greater storey heights and can therefore accommodate deeper, stronger coupling beams. Table 4.12: Typical buildings in Vancouver with associated freestanding storeys (storeys above podia) and coupling ratios. Obtained from consultations with Bob Neville, P. Eng, and Svetlana Uranova, P. Eng, of Read Jones Christoffersen in Vancouver, May 2012. Occupancy Storeys, N Coupling Ratio Office 15 0.74 Office 31 0.80 Office 35 0.80 Office 25 0.82 Residential 12 0.58 Residential 12 0.61 Residential 10 0.63 Residential 10 0.72 Residential 9 0.73 Residential 12 0.75 Residential 6 0.78   90   As recommended in the ‘Tall Building Report’ conducted by the Applied Technology Council, a coupling ratio of 0.7 was used in the coupled shear wall buildings. This coupling ratio, CR, was applied in Equation 20 to calculate the required flexural strength, of the coupling beams in each building (ATC, 2010). Where My is the flexural yield strength of each coupling beam, ME is the elastic overturning moment resistance of the entire building, LCB is the length of each coupling beam, N is the number of storeys in the building, and L is the distance between the centroids of the walls. At each considered building height, the yield strength of the coupling beams was only influenced by Rα=0 of the SFRS.   ܯ௬ = ܥܴ ∗ ܯா ∗ ܮ஼஻2 ∗ ܰ ∗ ܴఈୀ଴ ∗ ܮ (20)  When the coupling beams fully yield, they transfer a coupling force to each of the coupled shear walls - compressive in one wall and tensile in the other. The magnitude of this maximum coupling force in each wall, Tmax, is given in Equation 21.   ௠ܶ௔௫ = 2 ∗ ܰ ∗ ܯ௬ܮ௖௕ = 	 ܥܴ ∗ ܯாܴ ∗ ܮ  (21)   4.10.4 Coupling Beam Stiffness The stiffness of the coupling beams was considered independently of yield strength in this study. The stiffness of the coupling beams was dictated by their dimensions, which varied only with building height. In previous studies on coupled shear walls, it was found that using an effective flexural stiffness of 0.15ECIG produced a good match between test results and analytical models, where IG is the moment of inertia of the gross section (ATC, 2010). The same approach was considered in this study.  The same study found that the relationship between the shear modulus, GC, and the concrete modulus, EC, varies depending on the ratio of the length, LCB, to the height, hCB, of the coupling beams. This ranged from GC = 0.1EC at LCB/hCB = 1.4 to GC = 0.4EC at LCB/hCB = 2.0 (ATC, 2010). In the current study LCB/hCB = 1.45 producing a shear modulus of GC = 0.127EC, and thus shear stiffness of 0.127ECAG.    91  In order to model the stiffening effect of the floor slabs on the coupling beams, the axial stiffness of the coupling beams was amplified to 100AGEC, where AG was the gross cross-sectional area of each coupling beam. The floor slab would be relatively stiff and control the axial demands imposed on the coupling beams (ATC, 2010).  4.10.5 Coupling Beam Inelastic Response The shear and axial properties of the coupling beams were defined as perfectly elastic, with inelasticity occurring only in flexure. The Clough hysteretic model used to model this inelastic flexural behaviour. Although this model is a simplification of true behaviour; it was an appropriate representation (White, 2004). The Clough hysteretic model does not consider axial-flexural interaction. Axial loads can develop due to the constraining effect of the floor slab; however, these axial compressive loads are unreliable and are typically ignored in design. The Clough model is shown in Figure 4.20 below, which depicts the inelastic hysteretic response of a mid-height coupling beam.    92   Figure 4.20: Hysteretic moment-curvature response of a coupling beam at mid height in a 5 storey building, Rα=0 = 2.0, α = 0.0.  4.10.6 Fibre Section As discussed in section 4.9.3, the Clough hysteretic model was not appropriate for modeling the flexural behaviour at the base of the cantilevered shear wall because it ignored axial-flexural interaction. Because of the coupling forces in the walls, axial loads change significantly during the hysteretic response of coupled shear walls. Therefore, it was especially important that the inelastic model capture the effects that these axial variations would have on the flexural response. The use of a relatively complex fibre section to represent behaviour in the plastic hinge region was necessary. This fibre section was implemented in displacement based beam-column elements which extended the height of the plastic hinge regions given in Table 4.3.    93  Three material models were considered in each coupled shear wall fibre section; longitudinal reinforcing steel, unconfined concrete, and confined concrete; these are identified in section 4.5. Mirrored fibre sections were implemented in each wall of the six coupled shear wall buildings. These mirrored fibre sections were identical with the exception of the concentrated steel areas in the ends of the walls. The fibre section in each wall was comprised of nine steel fibres, eleven unconfined concrete fibres, and eleven confined concrete fibres.   Figure 4.21: Schematic of a generalized coupled shear wall section with distributed and concentrated longitudinal reinforcing steel, unconfined concrete, and confined concrete. As with the cantilevered shear wall buildings, the area of concentrated steel in each of the two flanges was adjusted to achieve the desired strengths in each direction. These strengths ultimately depended on Rα=0 and α.    94  4.10.7 Pushover Response As with the cantilevered shear wall buildings, it was necessary to investigate the behaviour of the coupled shear walls during both a static pushover analysis and during a dynamic earthquake analysis. In this section, the observed inelastic behaviour of the coupled shear walls is discussed. In every coupled wall shear wall building, the flexural strength of the coupling beams was selected to achieve a coupling ratio of 0.7. The response of 5, 10, 20, and 30 storey coupled shear wall buildings during displacement-controlled static pushover analyses are presented in Figure 4.22 and Figure 4.23. The behaviour in the compression wall – so called because it was loaded in compression by the coupling force – is considered rather than the tension wall because it exhibited more reliable behaviour.  Figure 4.22: Overturning moment as a function of base curvature in the compression wall during pushover analyses of 5, 10, 20, and 30 storey coupled shear walls, Rα=0 = 2.0, α = 0.0.   95   Figure 4.23: Overturning moment as a function of roof displacement during pushover analyses of 5, 10, 20, and 30 storey coupled shear walls, Rα=0 = 2.0, α = 0.0. As with the cantilevered shear walls, tri-linear behaviour was observed when considering the moment curvature response at the base of the compression shear wall. Transitions in stiffness occurred corresponding to cracking of concrete and yielding of the tensile reinforcing steel. These transitions were less distinct than for the cantilevered shear walls because the coupling beams yielded progressively throughout the analysis.  The overturning moment resistance in the coupled shear walls was derived from both flexural resistance at the base of each wall and the coupling forces in the walls. The moment at the base of each coupled shear wall in the 5 and 30 storey buildings is shown in Figure 4.24 and Figure 4.25 as a function of the roof displacement. Also plotted is the total overturning moment resistance of the building from the combination of the flexural and coupling effects.   96   Figure 4.24: Overturning moment as a function of roof displacement for the 5 storey coupled shear wall model in each coupled wall and for the overall building, Rα=0 = 2.0, α = 0.0.   97   Figure 4.25: Overturning moment as a function of roof displacement for the 30 storey coupled shear wall model in each coupled wall and for the overall building, Rα=0 = 2.0, α = 0.0. Although the coupled shear walls were designed for a coupling ratio of 0.7, the coupling ratio exhibited during the building response was variable, depending upon the relative stiffness and strengths of the walls and coupling beams. Progressive yielding of different building components during the static pushover resulted in the varying effective coupling ratio that can be seen in Figure 4.26.   98   Figure 4.26: Coupling ratio as a function of roof displacement during the static pushover analyses of 5, 10, 20, 30, 40, and 50 storey coupled shear wall buildings designed for a coupling ratio of 0.7, Rα=0 = 2.0, α = 0.0.  4.10.8 Hysteretic Response In this section, the hysteretic behaviour of the coupled walls when subjected to seismic loading is analyzed. Figure 4.27 and Figure 4.28 show the curvatures at the base of one of the coupled shear walls as a function of the axial load for a 5 and 30 storey building, respectively. These buildings were each designed for Rα=0 = 2.0 and no PLD. The axial dead load in each 5 and 30 storey wall from the self- weight of the building was -1 544kN and -38 246kN, respectively.  The 5 and 30 storey walls then experienced coupling forces with maximum magnitudes of approximately 30 000kN and 170 000kN, respectively. When the walls were in axial compression, they developed high base moments. However when in tension, the corresponding base moments were much smaller. This can   99  be explained by considering the stiffness of the shear walls. When in compression, there was a significant contribution from concrete to the overall section stiffness causing greater base moments than in tension for the corresponding roof displacements in the opposite direction.  A second factor contributing to greater compression stiffness of the shear walls was the C- shape of each wall. The large concrete flange on the outside of each wall contributes significantly to the stiffness of the wall when in compression. Due to the geometry of coupled shear walls, this concrete flange experiences compressive flexural strains when the wall was put into compression.  Figure 4.27: Base moment as a function of axial load in a shear wall of a 5 storey coupled shear wall building during a time-history analysis, Rα=0 = 2.0, α = 0.0.   100   Figure 4.28: Base moment as a function of axial load in a shear wall of a 30 storey coupled shear wall building during a time history analysis, Rα=0 = 2.0, α = 0.0. As higher modes became more significant, as they do for taller buildings, the correlation between axial load and moment at the base of each coupled wall became less distinct (Figure 4.29). Higher modes caused more complex dynamics along the height of the building. While the coupling beams in the 5 storey building yielded in relative unison to the base of the shear walls, the coupling beams in the taller buildings yielded progressively throughout the time-history analyses. Therefore, there was less correlation between the flexural behaviour at the base of the shear walls to the coupling forces being developed along the height of the building. Similar asymmetrical behaviour was observed when considering base moment- curvature response at the base of the shear walls, shown in Figure 4.29 and Figure 4.30, for the 5 and 30 storey coupled shear wall buildings, respectively.   101   Figure 4.29: Base moment as a function of curvature in the tension wall of a 5 storey coupled wall building during a time history analysis, Rα=0 = 2.0, α = 0.0.   102   Figure 4.30: Base moment as a function of curvature in the tension wall of a 30 storey coupled wall building during a time history analysis, Rα=0 = 2.0, α = 0.0.  4.10.9 Coupling Beam Response The hysteretic flexural response of both the mid-height and roof level coupling beams in the 5 storey building is given in Figure 4.31 and Figure 4.32, respectively. All five coupling beams in this building reached their design yield strengths of 3827kNm before yielding to varying extents, with the most severe yielding occurred in the mid-height coupling beams.    103   Figure 4.31: Moment as a function of curvature for a third storey coupling beam in a 5 storey coupled shear wall building during a time-history analysis, Rα=0 = 2.0, α = 0.0.   104   Figure 4.32: Moment as a function of curvature for a fifth storey coupling beam in a 5 storey coupled shear wall building during a time-history analysis, Rα=0 = 2.0, α = 0.0. The coupling beam properties had significant impacts on the cyclic performance of coupled wall systems. As is apparent from Figure 4.31 and Figure 4.32, the inelastic curvatures developed were very large. It was assumed that the coupling beams would be properly detailed and thus maintain their strength through the imposed displacement ductility demands - strength degradation in the coupling beams was not considered.  From previous studies of coupling beams, it has consistently been shown that diagonal reinforcement significantly improves the cyclic performance of coupling beams (ATC, 2010). Drifts of 8% were achieved in coupling beams with diagonal reinforcement before the onset of strength degradation. Coupling beams with a clear span to depth ratio of less than four can typically accommodate a diagonal reinforcement scheme (ATC, 2010). The coupling beams used in had clear spans of 2.44 meters (8 feet) and a depth of 1.67 meters (5.5 feet). Therefore, they would be capable of housing diagonal reinforcement and the large ductility capacities considered were reasonable (ATC, 2010).   105  4.10.10 Ductility Capacity The curvature capacities of the different core wall sections were determined from static pushover analyses and an ultimate concrete strain of 0.0035. These limiting capacities are given in Table 4.13 for the coupled shear wall sections. The curvature capacity of the coupled shear walls was highly dependent upon Rα=0. It was also dependant, to a much lesser extent, upon α. For simplicity, this dependency upon α is ignored and a single approximate curvature capacity is presented for each Rα=0 and building height. Table 4.13: Base curvature capacities (rad/km) of the compression wall in the coupled shear wall systems. Number of Storeys, N Relative strength factor, Rα=0 1.0 2.0 4.0 6.0 5 2 3 4 5 10 1.5 2 2.5 3 20 1.3 2 1 1.1 30 1.0 1.5 0.8 1.0 40 0.8 0.8 0.8 0.8 50 0.5 0.6 0.7 0.7  Global drift limits are established to prevent instability and to ensure post-earthquake access to buildings (Tremblay, Léger, & Tu, 2001). Acceptable inter-storey drift limits have been established at 2.5% for normal occupancy buildings and 1% for post-disaster buildings by the NBCC (2010). The tall buildings initiative established global acceptance criteria comprised of a maximum transient inter-storey drift of 3% and a maximum residual inter-storey drift of 1% (PEER, 2010). This residual drift limit is meant to limit post-earthquake deformations which result in a reduced serviceability of buildings.  In addition to shear walls, it was important to consider the curvature capacities which could be expected in the coupling beams. Because the coupling beams are integral to the SFRS in coupled shear wall buildings, they are required to maintain their strength through cyclic yielding. In the ATC (2010) report on tall buildings, it was found that coupling beams could be expected to reach rotations of approximately 0.08 radians before significant strength degradation. From experimental testing, Kwak and Kwan (2003) found that deep reinforced concrete coupling beams were typically capable of achieving rotations of 0.07 radians before significant strength degradation. The coupling beams used in this study were relatively   106  deep, LCB/hCB =1.45, therefore it is more appropriate to expect rotational capacities of approximately 0.07 radians.  Rotational capacity is easily converted to curvature capacity, thus allowing comparison with the coupling beam hystereses presented in the following chapter in terms of curvature. First, an approximate plastic hinge length at the end of the coupling beams was determined with Equation 22, developed to approximate reinforced concrete beam behaviour by Bohl and Adebar (2011). The equation estimates the length of the plastic hinge from the shear span, z, the bar diameter, db, and the yield strength of the steel, fy. Given the dimensions of the coupling beams used in the coupled wall models in this study, and an assumed bar diameter of 20 millimeters, the approximate length of the plastic hinge was determined to be 0.26 meters at each end of the coupling beam. This accounts for approximately 10% of the total length of the coupling beams.   ݈௣ = 	0.12ݖ + 0.014݀௕ ௬݂ (22)  The approximate curvature capacity of the coupling beams was calculated from this plastic hinge length, lp, and the more conservative rotational capacity estimate of 0.07 radians, as shown in Equation 23. This produces a curvature capacity of 260 radians per kilometer in each coupling beam.   ߶஼ = 	 ߠ஼݈௣  (23)     107  Chapter 5 - Results and Discussion  It is important to recall the goal of this study, which was to investigate the significance of seismic ratcheting in buildings with different SFRSs, α, and Rα=0. In order to quantify the ratcheting, three building metrics were considered: maximum roof drifts, maximum base curvatures and residual roof drifts. Maximum roof drifts and maximum base curvatures were used to measure the inelastic yielding which occurred in different buildings. These were useful metrics for quantifying the degree of damage sustained by the building as well as the likelihood of instability and collapse. The residual roof drifts were also useful because indicated the serviceability state of the building - elevators can only operate within a narrow drift tolerance and may not be operational in an otherwise stable building due. Furthermore, buildings with large residual roof drifts would present a hazard to rescue workers as there would concerns regarding instability and collapse from possible aftershocks.  These three performance metrics - which will be generically denoted as ∆ - were considered between buildings. In the SDOF study, two amplification factors, γ and β, were considered. It was realized however that only the latter amplification factor, β, was needed to define the influence of PLDs. Therefore, the inelastic amplification factor, γ, is no longer considered. In addition to the relative amplification factor, β, the additional demand indicator, δ, was considered. These two parameters were each advantageous for comparing different types of response quantities.  Both parameters compared performance metrics in a building with PLDs to an equivalent building with PLDs. The relative amplification factor, β, shown in Equation 24, was effective to relate maximum base curvatures and maximum roof drifts.   ߚ = ߂(ܴఈୀ଴,ߙ) ߂(ܴఈୀ଴,ߙ = 0) (24)  The second parameter, the additional demand, δ, measured the additional demands induced by the PLDs by subtracting the deformation demands present in a corresponding building without PLDs. It was effective at demonstrating the influence of PLDs on residual roof drifts and is presented in Equation 25.   ߜ = 	߂(ܴఈୀ଴,ߙ) − ߂(ܴఈୀ଴,ߙ = 0) (25)   108  These two parameters were used to compare equivalent buildings with and without PLDs subjected to the same ground motion scaled to induce the same maximum elastic overturning base moments as the Vancouver UHS. The typical response of the building to the ten ground motions was then determined by taking the mean value from the set of ten ground motions.  5.1 CANTILEVERED SHEAR WALL BUILDINGS  Mild to moderate amplifications in maximum roof displacement and maximum base curvature were observed across the entire analysis domain of cantilevered shear wall buildings. No collapse cases were identified for the cantilevered shear walls. Rα=0 = 6.0 was only obtainable in the 5 storey buildings (due to overstrength from wall axial loads for taller buildings) and Rα=0 =4.0 was only achieved for 5 and 10 storey buildings. 5.1.1 Hysteretic Behaviour In order to investigate the cantilevered shear wall response to PLDs, hysteretic moment-curvature responses are considered for several characteristic buildings.  Figure 5.1 through Figure 5.4 show the hysteretic response of the 5 storey cantilevered shear wall to the same ground motion (ground motion 2) for different Rα=0 values. In each figure, the wall response is presented both with vertical columns and with inclined columns corresponding to α = 0.4. In each figure, Rα=0 of the building increases from 1.0 to 2.0, 4.0, and finally 6.0. Because the strength of the building decreases from one figure to the next, so too does the inclination of the columns which corresponds to α = 0.4. Rα=0 = 1.0, 2.0, 4.0, and 6.0, correspond to column inclinations of 15.1°, 7.6°, 3.8°, and 2.5°, respectively. In Figure 5.1 neither the building without PLDs nor with PLDs yields (Rα=0 = 1.0). For the unique case of Rα=0 the PLD actually reduced the curvature demands at the base of the shear wall because the additional reinforcing steel added to account for the PLDs stiffened the response of the shear wall. The base curvature demands are within the capacity of the elastic 5 storey cantilevered shear wall section, which was found to be 1.0 rad/km    109   Figure 5.1: Base moment versus curvature for a 5 storey cantilevered shear wall building with and without a PLD corresponding to α = 0.4, Rα=0 = 1.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. In Figure 5.2 through Figure 5.4, the inelastic base curvatures are approximately twice as large in the building with PLDs corresponding to α = 0.4 compared to the regular vertical columns building. Each of these increased maximum base curvature demands correspond approximately to β = 2.0, therefore for this 5 storey building configuration Rα=0 did not affect β significantly.  However, when considering residual base curvatures, it is clear that increasing Rα=0 in buildings with inclined columns resulted in degraded performance. Regardless of the Rα=0, the regular 5 storey building returned to roughly zero residual inelastic curvatures at the end of ground motion 2. However, when considering buildings with inclined columns, increasing Rα=0 produced progressively larger residual base curvatures as indicated in Figure 5.2 through Figure 5.4.  When Rα=0 > 1.0, the 5 storey cantilevered building with inclined columns experienced many load cycles which took it beyond its curvatures capacity of 1.0 rad/km. These cycles which drove large inelastic base curvatures could result in crushing of the concrete would likely cause damage to the shear wall.   110    Figure 5.2: Base moment versus curvature for a 5 storey cantilevered shear wall building with and without a PLD corresponding to α = 0.4, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.   Figure 5.3: Base moment versus curvature for a 5 storey cantilevered shear wall building with and without a PLD corresponding to α = 0.4, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.    111   Figure 5.4: Base moment versus curvature for a 5 storey cantilevered shear wall building with and without a PLD corresponding to α = 0.4, Rα=0 = 6.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. In the 10 storey cantilevered shear wall buildings, Rα=0 ≥ 4.0 were not achievable due to inherent overstrength resulting from axial load and minimum steel requirements in the shear wall. The 10 storey cantilevered shear wall buildings with this maximum achievable Rα=0 = 4.0 were considered because it produced the largest - and therefore most critical – inelastic base curvatures. In Figure 5.5, the moment- curvature response of this building, subjected to three different α corresponding to vertical columns (α = 0.0), columns at an inclination of 2.2° (α = 0.4), and columns at an inclination of 4.5° (α = 0.8). Note the increase in strength achieved through additional reinforcing steel added for the PLDs.  The inelastic curvature demands increase as the inclination of the columns increase; these demands are in excess of the sectional curvature capacity of approximately 0.8 rad/km. Although all three hysteretic responses produce inelastic curvatures which exceed the curvature capacity of the section, the cases with PLDs contain more loading cycles which exceed the curvature capacity of the section (Figure 5.5). The repeated application of large curvature demands would result in spaling and crushing of the concrete at the base of the shear wall. Residual curvatures demonstrate even greater sensitivity to PLDs – increasing from approximately 0 rad/km, to 0.1 rad/km, 1.2 rad/km, and then 2.5 rad/km for α = 0.0, 0.2, 0.4, and 0.6, respectively.   112   Figure 5.5: Base moment versus curvature for a 10 storey cantilevered shear wall building with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. Figure 5.5-demonstartes the effect of an important assumption made in this study. Additional forward yield strength is provided through reinforcing steel to resist the PLDs resulting in the four different yield plateaus indicated in Figure 5.5. It was important to investigate how this assumption influenced the results of this study; three otherwise consistent buildings where additional strength was not provided are shown in Figure 5.6. Larger inelastic curvatures are produced when the cantilevered shear wall is not strengthened to account for the additional PLDs. In the case with α = 0.6, collapse occurs after the building ratchets dramatically in one direction as a result of the PLD. Such a result clearly indicates the importance of accounting for the additional PLDs by adding strength to the SFRS.   113   Figure 5.6: Base moment versus curvature for a 10 storey cantilevered shear wall building with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The forward yield strength of this cantilevered wall not increased to account for the presence of the PLD. The initial and final resting states are identified with hollow and solid circles, respectively. Finally, let us move on to taller buildings (20, 30, 40, and 50 storeys) for which Rα=0 never exceeded 2.0 due to overstrength from the relatively large axial loads in the shear walls. Figure 5.7 and Figure 5.8 show the hysteretic responses of 30 and 50 storey cantilevered shear wall buildings, respectively. For each building, Rα=0 = 2.0 (maximum possible), and four different α are considered. For the 30 storey cantilevered shear wall with Rα=0 = 2.0, α of 0.0, 0.2, 0.4, and 0.6 correspond to column inclinations of 0°, 1.8°, 3.6° and 5.3°, respectively. To achieve these same three α, the columns in the 50 storey building were inclined at 0°,1.4°, 2.7° and 4.1°; respectively. In each case, the forward yield strength was increased to account for additional PLDs – this increase is clearly visible from the yield plateaus shown in Figure 5.7 and Figure 5.8. It is clear that relative to the short 5 and 10 storey buildings, the application of the PLD has only a moderate effect on the inelastic base curvature demands. While the curvature demands did increase moderately as a result of PLDs, there were relatively few cycles which exceeded   114  the curvature capacities of the 30 and 50 storey walls which were 0.5 rad/km and 0.3 rad/km, respectively.  Figure 5.7: Base moment versus curvature for a 30 storey cantilevered shear wall building with varying PLDs, Rα=0 = 2.0, and subjected to ground motion 23. The initial and final resting states are identified with hollow and solid circles, respectively.   115   Figure 5.8: Base moment versus curvature for a 50 storey cantilevered shear wall building with varying PLDs, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. The short 5 and 10 storey buildings experienced more loading cycles which exceeded their respective curvature capacities. The improved behaviour exhibited by of the tall (20, 30, 40, and 50 storey) buildings appears to a result of the increased axial loads in the shear walls of these tall buildings. Large axial loads provided two mitigating effects which appear to prevent seismic ratcheting. The first effect of large axial loads was that the tall buildings had smaller achievable Rα=0 due overstrength. This reduced the inelastic curvatures observed – both with and without PLDs. Secondly; the increased contribution of axial load to wall strength in taller buildings changed the hysteretic behaviour of the wall. The moment-curvature behaviour shifted from Clough-shaped, as it was for shorter buildings, to flag-shaped. The pinching behaviour associated with flag-shaped hystereses inhibited large inelastic curvatures from developing.    116  While the maximum inelastic base curvatures did not significantly increase with the PLDs, there was a moderate effect on the residual curvatures in the buildings. For both buildings shown in Figure 5.7 and Figure 5.8, relatively large residual base curvatures resulted from the largest PLDs, corresponding to α = 0.8, yet the residual base curvatures at α = 0.8 were significantly more moderate. This indicates that there is likely a PLD threshold for each building above which the building will have difficulties to returning to its initial state. This supposed threshold, and the behavioural trends identified above will be further examined in in the context of β and δ.  5.1.2 Maximum Base Curvature The maximum base curvature induced at the base of the shear wall is a good indicator of the ductility demands. β determined for maximum base curvatures are presented in this section - Figure 5.9 through Figure 5.11 give β as a function of both building period and α. Each figure compares buildings with consistent Rα=0 and consequently each figure, taken independently, provides an indication of the correlation of β with the building period and α. When considered together, Figure 5.9 through Figure 5.11 provide an indication of the minimal effect of Rα=0 on the maximum base curvature demands induced in each shear wall.   117   Figure 5.9: β for inelastic base curvatures in cantilevered shear wall buildings, Rα=0 = 2.0. The amplifications seen were moderate, with β approaching 2.0 for the 5, 40 and 50 storey buildings at Rα=0 = 2.0. Similar magnitudes of amplifications were seen for the 5 and 10 storey buildings at higher Rα=0 as shown in Figure 5.10and Figure 5.11. As discussed, Rα=0 greater than 2.0 were not achievable for tall cantilevered shear wall buildings due to inherent overstrength.   118   Figure 5.10: β for inelastic base curvatures in cantilevered shear wall buildings, Rα=0 = 4.0.   119   Figure 5.11: β for inelastic base curvatures in cantilevered shear wall buildings, Rα=0 = 6.0. Figure 5.12 shows the amplifications in base curvatures seen in each of the ten individual ground motions as a result of a moderate α = 0.6, as well as the mean response. As anticipated, there was variability between ground motions. In general, this variability was moderate - the β produced by individual ground motions were always less than 3.0.   120   Figure 5.12: β for inelastic base curvatures in cantilevered shear wall buildings subjected to each of the ten ground motions with the mean, µ, and mean plus one standard deviation, µ + σ, plotted, Rα=0 = 2.0, α = 0.6.  5.1.3 Maximum Roof Drift In addition to maximum base curvatures, the maximum roof drifts of each building were also considered. The β for roof drifts are presented in the same format as were the amplifications in base curvatures; Figure 5.13 through Figure 5.15 show the amplifications seen for Rα=0 = 2.0, 4.0, and 6.0, respectively. As was the case for the maximum base curvature demands, β produced when roof drift demands were considered were quite moderate – in fact, β produced from both metrics correlated very well throughout the entire domain of the both types of shear wall buildings (cantilevered and coupled). Mild to moderate α (0.05 – 0.4) were found to produce β ≤ 1.5 for all achievable Rα=0 and all periods. Even when α was increased to produce relatively large PLDs (0.6 and 0.8), the resulting β were still below 2.0 for all but one analysis case; amplifications of ~2.05 were observed for the 5 storey building with Rα=0 = 4.0.   121   Figure 5.13: β for roof drifts in cantilevered shear wall buildings, Rα=0 = 2.0.    122   Figure 5.14: β for roof drifts in cantilevered shear wall buildings, Rα=0 = 4.0.   123   Figure 5.15: β for roof drifts in cantilevered shear wall buildings, Rα=0 = 6.0. As was the case when base curvatures were considered, the variability in the response amplifications observed was quite modest, as indicated in Figure 5.16. For the parametric combination shown, the amplifications never exceeded 2.0 for a single ground motion.   124   Figure 5.16: β for roof drifts in cantilevered shear buildings subjected to each of the ten ground motions with the mean, µ, and mean plus one standard deviation, µ + σ, plotted, Rα=0 = 2.0, α = 0.6.  5.1.4 Residual Roof Drifts Significant residual drifts were observed in the cantilevered shear wall buildings when PLDs were present. As discussed earlier, residual roof drifts have implications on the serviceability of damaged buildings after earthquakes. Buildings which are visibly leaning pose a risk to surrounding areas and severely hamper rescue efforts to the surrounding areas (PEER, 2010). It was important to quantify the effect that PLDs have on this metric.  The nature of residual roof drifts is quite different from maximum base curvatures and maximum roof drifts; this metric centered tended to achieve very small values. Therefore, the relative amplification factor, β, was not effective at quantifying additional displacement demands; relative these small values, any significant residual roof drift was, by comparison, extremely large, which produced extremely large   125  β. Therefore, residual roof drifts required a different approach than that taken for maximum base curvatures and maximum roof drifts. Rather than comparing maximum quantities, it was now necessary to compare quantities which could be extremely small; therefore, the additional demand, δ, was used to compare residual roof drifts.  When Rα=0 ≥ 2.0 were considered, moderate values of δ were observed. In general, δ was found to increase as the period (height) of the building decreased. As the building height increased, the buildings tended to exhibit flag-shaped hystereses; this behaviour was effective at mitigating large residual roof drifts. In short buildings with smaller periods, the fat hysteretic loops exhibited were ineffective at restoring the buildings to their initial resting state.  Short buildings were often able to return to their initial state when no PLDs were present, however for most cases, the addition of PLDs often prevented the same building from recovering the inelastic displacements imposed by each ground motion. As a result, the 5, 10, 20, and 30 storey buildings all show large increases when this metric was considered, as shown in Figure 5.17. This is especially true for higher relative strength factors of 4.0 and 6.0, shown in Figure 5.18 and Figure 5.19, respectively.  Maximum acceptable residual inter-storey drifts limits have been established by PEER (2010) at 1%; therefore, the additional global residual roof drifts demonstrated in the 5, 10, and 20 storey buildings are significantly large. These large values are particularly significant as they are reflective of increases in the global – rather than inter-storey - residual roof drifts; it is likely that residual inter-storey drifts in exceedance of this acceptable tolerance of 1% are present in individual storeys (PEER, 2010).   126   Figure 5.17: δ, the additional residual roof drift, expressed as a percentage of building height, produced in cantilevered shear wall buildings, Rα=0 = 2.0.   127   Figure 5.18: δ, the additional residual roof drift, expressed as a percentage of building height, produced in cantilevered shear wall buildings, Rα=0 = 4.0.   128   Figure 5.19: δ, the additional residual roof drift, expressed as a percentage of building height, produced in cantilevered shear wall buildings, Rα=0 = 6.0. As was the case for maximum base curvatures and maximum roof drifts, there was variability in the residual roof drifts between ground motions. This variability was significantly smaller than the variability in the maximum base curvatures and the maximum roof drifts, as indicated in Figure 5.20, which shows δ as a function of building height for each ground motion, with α = 0.6 and Rα=0 = 2.0. Few ground motions drastically exceed the mean result at any point within the analysis domain. Furthermore, the scatter of the δ produced from individual ground motions was quite small about the mean result. One explanation for this is that residual roof drifts are highly dependent upon the building, and less dependent upon the ground motion. This is logical - it is often the properties of the shear wall which dictate a building’s ability to recover to zero drift (ATC, 2010; PEER, 2010).   129   Figure 5.20: δ, the additional residual roof drift, expressed as a percentage of building height, produced in cantilevered shear wall buildings subjected to each of the ten ground motions with the mean, µ, and mean plus one standard deviation, µ + σ, plotted, Rα=0 = 2.0, α = 0.6.  5.1.5 Discussion Moderate degradation in building performance was found to result from the presence of PLDs in cantilevered shear wall buildings. This degradation was demonstrated by increased maximum base curvatures, maximum roof drifts, and residual roof drifts for almost all cantilevered shear wall buildings considered.  As shown in Figure 5.22, there was variability in β for different ground motions, and for different building periods/heights. It is important to note that this variability increased as the building performance became more severely affected; the variability of β at small PLDs was moderate relative to the scatter at large α. The variability of the response amplifications is effectively shown in Figure 5.21 and Figure 5.22.   130  Each figure indicates the β produced from every single cantilevered shear wall building analyzed. In Figure 5.21, the data is organized by Rα=0 and α, with all considered building periods shown. In Figure 5.22, the data is organized by the period and α, with all considered Rα=0 plotted.  Consider the influence of PLDs on the maximum roof drifts and roof drift demands. The definition of α – as a ratio of building yield strength – meant that the magnitude of the PLDs was inversely proportional to Rα=0; i.e. same α corresponded to half the PLDs in a building with Rα=0 = 4.0 than in a building with Rα=0 = 2.0. A consequence of this approach was that β demonstrated minimal dependency upon Rα=0 and as can be seen in Figure 5.21, β at Rα=0 = 2.0 consistently matched or exceeded those observed at higher Rα=0. In Figure 5.21, 1050 data points are plotted; one for each analysis case resulting from different combinations of the considered α, Rα=0, building heights, and ground motions.  Figure 5.21: β for maximum roof drifts from each time-history analysis conducted on the cantilevered shear wall buildings. Each data point represents β from a single building and ground motion (630 data points). The data is organized by Rα=0 and α; no differentiation is made between buildings with different periods. The maximum base curvature demands and maximum roof drift demands in cantilevered shear wall buildings were correlated with α; these demands were greatest in buildings with large PLDs. β displayed   131  mild dependence on building height; in general, the buildings with the smallest and largest fundamental periods were affected the most severely by PLDs, as is shown in Figure 5.22. The 20 and 30 storey buildings consistently showed the smallest amplifications in maximum base curvatures and roof drifts, however, degraded performance was observed in all cantilevered shear wall buildings regardless of height.  The variation of β with building period was caused, in large part, by changes in the hysteretic behaviour as a result of different axial loads. Shorter buildings (with smaller axial loads), demonstrated Clough hysteresis which was prone to ratcheting. As the height of the buildings increased, so too did axial loads, resulting in flag-shaped hysteresis. This hysteretic behaviour inhibited ratcheting and was responsible for the mild β observed in the 20 and 30 storey buildings. What then was the driving mechanism behind the amplifications seen for the 40 and 50 storey buildings? The answer to this is again apparent from investigation of the hysteresis of the different buildings. In the 10 storey building shown in Figure 5.5, additional reinforcing steel had a very distinct effect on the forward yield strength of the wall due to the well-defined yield plateau. This was not true of the 40 and 50 storey buildings, which had less well defined transitions to yielding, as can be seen in Figure 5.8. The high-rise buildings had to displace to relatively large base curvatures before they achieved the additional moment resistance provided by additional reinforcing steel. Therefore, the presence of PLDs promoted increased base curvatures before the longitudinal steel yielded; resulting in the moderate β observed in the 40 and 50 storey walls.  To a large extent, the variation in β with building height was a result of the specific core sections considered. For a given building height, there are a wide range of shear walls that will satisfy seismic design requirements; various thicknesses, lengths, strengths, and stiffness could be used. If smaller cores were selected, the effect of axial load would have been less significant and Clough behaviour would be more prevalent in high-rise buildings than it was in this study. If larger core sections were selected, the flag-shaped hysteresis observed in the 40 and 50 storey buildings may have also governed the 5, 10, 20, and 30 storey buildings. Although there are infinite possible shear walls which could be used in each building, this study considered a single shear wall at each height. The behaviour at each building height was dependent upon this single assumed shear wall section. This variation in the considered shear wall geometry at different heights likely influenced the β produced. Therefore, given this influence and the narrow range of shear wall sections considered, an appropriate and conservative approach would be to develop design limits for PLDs independent of building height. The variations observed in this study are, to a large extent, a function of the specific shear wall section considered at each building height.   132   Figure 5.22: β for maximum roof drifts from each time-history analysis conducted on the cantilevered shear wall buildings. Each data point represents β from a single building and ground motion (630 data points). The data is organized by the period and α; no differentiation is made between Rα=0 = 2.0, 4.0, and 6.0. The residual roof drifts exhibited in the cantilevered shear wall buildings demonstrated different correlations than did the maximum base curvatures and roof drifts. The final resting state of each cantilevered shear wall buildings was highly dependent on the axial load in the wall, as is clear from Figure 5.23.   133   Figure 5.23: Hysteretic behaviour and residual demands in four 5 storey cantilevered wall buildings with varying axial loads corresponding to wall self-weight (Wwall) alone, wall self-weight plus 10% of floor- weight (Wfloor), wall self-weight plus 20% of floor-weight, and wall self-weight plus 30% of floor-weight, Rα=0 = 4.0, α = 0.4. The initial and final resting states are identified with hollow and solid circles, respectively. Residual roof drifts demonstrated a clear dependence on the axial load in the shear walls; a lower bound axial load was considered in this study – the walls only supported their own self-weight. Buildings with greater axial load in the cantilevered shear walls would be less sensitive to PLDs and could be expected to produce smaller residual roof drifts, therefore the increases in the residual roof drifts demonstrated in this study are conservative.  The degree of axial load, and thus pinching behaviour, was correlated to the height of the walls, therefore, the increases in the residual roof drifts were also well correlated to the height of the building. Tall buildings exhibited very small increases in the residual roof drifts because the relatively large inherent axial loads promoted pinching behaviour which effectively returned the building to its initial state. The   134  shorter 5, 10, and 20 storey walls were less prone to recovering inelastic displacements, and as a result, the addition of inclined columns caused relatively large increases in the residual roof drifts. This is shown in Figure 5.24, in which the smaller periods (shorter buildings) consistently produce the largest increases in the residual roof drifts at each considered α.  Figure 5.24: δ, the additional residual roof drift, expressed as a percentage of building height, produced from each time-history analysis conducted on the cantilevered shear wall buildings. Each data point represents δ from a single building and ground motion (1050 data points). The data is organized by the period and α; no differentiation is made between Rα=0 = 2.0, 4.0, and 6.0.  5.2 COUPLED SHEAR WALL BUILDINGS  The behaviour of coupled shear wall buildings was more complex than that of the cantilevered shear wall buildings. Coupling beams significantly affected the global response and changing axial loads in the coupled shear walls drastically affected their stiffness and strength. The coupled wall buildings exhibited larger values of β and δ than did the cantilevered wall buildings.   135  5.2.1 Hysteretic Behaviour It was found that the moment-curvature response at the base of the compression shear wall provided the best indication of the overall building response. The hysteretic response of 5 storey buildings with and without PLDs corresponding to α = 0.4 and various values of Rα=0 are shown in Figure 5.25 through Figure 5.28. In Figure 5.25 (fully elastic response), the additional reinforcing steel - to account for PLDs - stiffened the elastic response resulting in smaller base curvatures. The compression shear walls exhibit asymmetrical stiffness due to the compressive and tensile axial loads associated with positive and negative curvatures, respectively. When the wall experienced positive curvature, axial compression was applied from coupling effects, when negative curvatures were imposed, so too was an axial tensile load.  Figure 5.25: Base moment versus curvature in the compression wall of a 5 storey coupled shear wall building with and without PLDs corresponding to α = 0.4, Rα=0 = 1.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. When the strength of the 5 storey building was reduced to achieve Rα=0 = 2.0, an unexpected result was observed, as shown in Figure 5.26. The compression wall in the building with PLDs did not yield; however, the wall in the regular building with α = 0.0 did yield. This result is a direct consequence of the approach taken to model the coupled shear wall buildings. As with the cantilevered wall buildings, longitudinal reinforcement steel was added to the coupled walls to accommodate the dead load demands resulting from the PLD. However, in the coupled shear wall buildings, only ~30% of the overall overturning resistance was derived from flexural strength at the base of the walls, the remaining ~70% came the coupling forces (coupling ratio of 0.7). When modelling the coupled wall buildings, it was assumed that engineers would add reinforcing steel to the base of the shear walls rather than asymmetrically detail the coupling beams to account for PLDs. As a result, when α increases, the flexural strength of the coupled shear walls increased; however, the strength of the coupling beams was kept   136  constant. Therefore, as α increased, the flexural strength at the base of the shear walls became relatively large compared to the coupling beams. The compression wall did not yield for the case with a PLD corresponding to α = 0.4, shown in Figure 5.26, however the coupling beams yielded more significantly than their counterparts in the building with no PLD (α = 0.0). Again, there was significant stiffness asymmetry in the walls due to variable axial loads.  Figure 5.26: Base moment versus curvature in the compression wall of 5 storey coupled shear wall buildings with and without PLDs corresponding to α = 0.4, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. When Rα=0 = 4.0, the yielding at base of compression wall became more significant both with and without PLDs, as can be seen in Figure 5.27. For the 5 storey buildings, it is at this Rα=0 that PLDs corresponding to α = 0.4 first begin to have significant impacts on residual curvatures (and thus residual roof drifts). The building with vertical columns effectively recovers to its initial state after the ground motion has finished; however, the building with PLDs has a large residual curvature, almost as large as the maximum inelastic curvatures imposed during the ground motion. Although the residual base curvatures differed as a result of the PLDs, it is interesting to note that the maximum base curvatures induced in the compression wall did not. Both the regular building (α = 0.0) and building with PLDs (α = 0.4) experienced approximately equivalent maximum base curvatures. One final observation that can be made from Figure 5.27 and Figure 5.28 is that the flexural strength of the walls in the reverse direction (negative curvatures) is very small. As stated earlier, coupled shear walls experience axial tension when put into negative curvature; consequently shear walls in buildings with large Rα=0 - and thus little reinforcing steel - had very small flexural strengths.   137   Figure 5.27: Base moment versus curvature in the compression wall of 5 storey coupled shear wall buildings with and without PLDs corresponding to α = 0.4, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. Finally, two 5 storey coupled shear wall buildings with Rα=0 = 6.0 are considered. To achieve this large Rα=0 reinforcing steel was removed from the base of the coupled walls. The hysteretic moment curvature response at the base of the compression wall in this relatively weak building is given in Figure 5.28. When PLDs were applied, the building experienced the largest residual base curvatures relative to the other 5 storey coupled wall buildings considered. These large residual base curvatures were a result of extensive seismic ratcheting which created base curvatures in excess of 9 rad/km. The curvature capacities (Section 4.10.10) of the shear walls considered in Figure 5.27 were approximately 4 rad/km and 5 rad/km for the shear walls considered in Figure 5.28. For both buildings with PLDs shown in Figure 5.27 and Figure 5.28, the curvature capacities of the sections were exceeded by many of the load cycles - this would result in severe cyclic strength degradation.   138   Figure 5.28: Base moment versus curvature in the compression wall of 5 storey coupled shear wall buildings with and without PLDs corresponding to α = 0.4, Rα=0 = 6.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. In addition to examining the influence of Rα=0, it is useful to examine the effect of varying PLDs in buildings with constant Rα=0. Figure 5.29 shows the hysteretic response of a 30 storey coupled wall building with Rα=0 = 4.0 and α = 0.0, 0.2 and finally 0.4. As shown, the maximum base curvatures induced in the compression shear wall increased with α; at a rate greater than the corresponding PLDs. Similar to the maximum base curvatures, the residual base curvatures also increased significantly with the PLDs.  For both considered α, there were significant base curvatures induced by the PLDs (in the forward direction corresponding to the PLDs). These base curvatures developed despite the tendency of this particular ground motion to promote base curvatures in the opposite direction, demonstrated by the relatively large negative curvatures which develop in the regular buildings (α = 0.0).  One may notice that there is no increase in the flexural strength of the compression wall as a result increasing PLDs. This is because there was inherent overstrength in the shear wall for Rα=0 = 4.0 - similar to the overstrength in the cantilevered shear wall buildings which prevented large Rα=0 from being achieved in the high-rise buildings. However, unlike the cantilevered wall systems, the majority of the strength of the coupled shear wall buildings was derived from the coupling beams and not the flexural strengths of the shear walls. Therefore, although the coupled shear walls were stronger than necessary for high Rα=0, the coupled shear wall buildings were still able to achieve high Rα=0 through reduced coupling forces (reduced coupling beam yield strength). A consequence of this was that the tall coupled shear wall   139  buildings occasionally had coupling ratios below 0.7; in these buildings, the flexural strength at the base of the shear walls had a greater relative contribution to the overall overturning moment resistance.  The stiffness of the both the 30 storey wall shown in Figure 5.29 and the 50 storey wall shown in Figure 5.30 wall was less asymmetrical than that of the shorter 5 storey walls; this was because the axial dead load was large relative to the coupling forces in the tall shear walls. For short 5 storey buildings, the live axial load was so significant it was able to pull the shear wall into tension when curvatures became negative. This was not the case for the tall walls, where relatively large shear wall sections combined with increased height produced dead axial loads much greater than the coupling forces induced by the coupling beams.  Figure 5.29: Base moment versus curvature in the compression wall of 30 storey coupled shear wall buildings with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.   140  The hysteretic response of the compression wall in a 50 storey coupled shear wall building was very similar to that of the compression wall in the 30 storey building, as is shown in Figure 5.30. The only significant difference in the observed behaviour was that the 50 storey building had more significant pinching behaviour which inhibited the large residual base curvatures which were present in the 30 storey buildings. Again, it is significant that PLDs induced such large inelastic base curvatures because the ground motion alone would tend to promote greater demands in the opposite direction as illustrated by regular buildings (α = 0.0) case in Figure 5.30.  As was the case for short coupled shear wall buildings, the addition of inclined columns promoted inelastic curvatures in excess of the curvature capacities of the shear wall sections, which were given in Section 4.10.10. For the 30 storey building in Figure 5.29, the curvature capacity of the section shown was approximately 0.8 rad/km. The curvature demands did not exceed this capacity when the gravity system did not contain an irregularity (α = 0.0), however, when PLDs were applied, the inelastic curvature demands greatly exceeded this capacity. Demands of 1.2 rad/km and 3.4 rad/km were imposed for the ‘α= 0.2’ and the ‘α = 0.4’ cases, respectively. The same behaviour was true of the 50 storey building shown in Figure 5.30, although the demands did not exceed the capacity as significantly, the ‘α = 0.4’ case still produced demands in excess of the 0.7 rad/km capacity of the shear wall section.   141   Figure 5.30: Base moment versus curvature in the compression wall of 50 storey coupled shear wall buildings with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. Finally, it was important to examine the hysteretic behaviour of the coupling beams and investigate how their properties dictate the global response of the coupled shear wall buildings. As discussed, the strength of the coupling beams was not adjusted to account for the PLDs. The Clough hysteretic model used was prone to seismic ratcheting; Figure 5.31 shows distinct ratcheting behaviour, and a very clear tendency for this ratcheting to ‘run away’. Note that the residual base curvatures did not simply increase linearly with the PLD, but rather at a greater, almost exponential, rate. Relative to the base of the shear walls, the coupling beams experienced large curvatures were large; however, these demands were generally well within the curvatures capacities of the coupling beams considered which were approximately 260 rad/km (Section 4.10.10).   142   Figure 5.31: Moment versus curvature in coupling beams on the fifth storeys of 10 storey coupled shear wall buildings with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.  5.2.2 Maximum Base Curvature Base curvatures were considered in the compression shear wall rather than the tension wall because the compression wall provided a greater portion of the overall overturning moment resistance than did the tension shear wall; therefore, the performance of the compression wall was more important. Furthermore, the compression shear walls behaviour was more consistent than the tension walls, which were dominated by tensile coupling forces. PLDs would tend to ratchet the building laterally towards the compression shear wall; therefore, maximum base curvatures in the wall occur when it is in compression. Conversely, for the same lateral building ratcheting, the tension shear wall is in axial tension. As we saw from the hysteretic plots in Section 5.2.1, the curvatures at the base of the tension shear walls were poorly   143  correlated to overall building behaviour. Therefore, the maximum base curvature at the base of the compression wall was a more appropriate metric to consider.  β are given in Figure 5.32 through Figure 5.34 for Rα=0 = 2.0 to Rα=0 = 6.0, respectively. In Figure 5.32 and Figure 5.33, β is given for Rα=0 = 2.0 and 4.0, respectively. Figure 5.32 gives the best illustration of the effect of PLDs on coupled shear wall buildings of different heights. β shows clear dependence on both α and the height of the coupled shear wall buildings. The short 5 storey buildings were barely influenced by PLDs, whereas in the tall 50 storey building, large β were produced. The base curvature demands were clearly amplified by α and tended to ‘run away’ and lead to collapse when α was large. α ≥ 0.4 produces multiple collapses and extremely large β for the maximum base curvature demands.  Figure 5.32: β for inelastic base curvatures in the compression wall of coupled shear wall buildings, Rα=0 = 2.0. The mid-rise 5, 10, and 20 storey buildings shown in Figure 5.33 demonstrated larger β when buildings with Rα=0 = 4.0 were considered. The increase of Rα=0 from 2.0 to 4.0 induced greater inelastic displacement - leading to larger β than were seen in Figure 5.32. However, this trend did not continue for   144  building heights greater than 20 storeys; β actually decreased for the 30, 40, and 50 storey buildings. The reason for this is simple, although, the coupled shear wall buildings were designed to achieve Rα=0 = 4.0, the flexural strength in the shear walls was disproportionally large due overstrength (from large axial loads). Therefore, although the global response of tall buildings was degraded by large α, the base curvatures in the walls did not reflect this degradation due to their relatively high flexural strength. This was similar to the overstrength phenomenon observed in some cantilevered shear wall buildings; however, unlike cantilevered shear wall buildings, it did not prevent coupled shear wall buildings from achieving high Rα=0.  Figure 5.33: β for inelastic base curvatures in the compression wall of coupled shear wall buildings, Rα=0 = 4.0. When Rα=0 = 6.0 was considered (Figure 5.34), the relative flexural strength at the base of shear walls in the 20, 30, 40, and 50 storey buildings even greater – coupling ratios greater than 0.7. Therefore, in these buildings, β for maximum base curvatures were less than would be expected for corresponding buildings with coupling ratios of 0.7. Due to the large flexural strength of base of these shear walls relative to the coupling beams; inelasticity was concentrated in the coupling beams rather than the shear walls. Despite   145  this influence, there were still large β in the maximum base curvatures across the entire range of building heights considered (excluding the 5 storey building which performed quite well).  Figure 5.34: β for inelastic base curvatures in the compression wall of coupled shear wall buildings, Rα=0 = 6.0. Figure 5.35 shows β from each individual ground motion considered for buildings with Rα=0 = 2.0 and α = 0.2. This shear wall was not governed by axial load; the relative flexural strengths of the wall are consistent with a coupling ratio of 0.7. The variability between ground motions is moderate, with some scatter about the mean and no significant outlying results from any one ground motion.   146   Figure 5.35: β for inelastic base curvatures in the compression wall of coupled shear wall buildings subjected to each of the ten ground motions with the mean, µ, and mean plus one standard deviation, µ + σ, plotted, Rα=0 = 2.0, α = 0.2.  5.2.3 Maximum Roof Drift In this section, the β for the maximum roof drifts are considered in the same manner in which the maximum base curvatures were investigated in Section 5.2.2. As was the case for cantilevered shear wall buildings, β were similar between the two different performance metrics (maximum base curvatures and maximum roof drifts). β for Rα=0 = 2.0, 4.0, and 6.0 are given in Figure 5.36, Figure 5.37, and Figure 5.38, respectively. In general, β increased with α, and to a lesser extent with building height. For all Rα=0  > 1.0 (i.e. inelastic) considered, there was no significant correlation between β and Rα=0. From the results considered, it is clear that α of 0.4 represents an approximate threshold, above which β become exponentially larger.   147   Figure 5.36: β for roof drifts in coupled shear wall buildings, Rα=0 = 2.0.    148   Figure 5.37: β for roof drifts in coupled shear wall buildings, Rα=0 = 4.0.   149   Figure 5.38: β for roof drifts in coupled shear wall buildings, Rα=0 = 6.0. The variability in the response was dependent upon the magnitude of Rα=0 and α. When the mean β were mild, as they were buildings with for Rα=0 = 2.0 and α = 0.2, the variability in the amplifications between ground motions was small, as shown in Figure 5.39. However, if the mean β increased, so too did the variability between individual ground motions; some ground motions resulted in collapse while others did not. Some ground motions produced β much greater than the mean, while others showed only mild amplifications.   150   Figure 5.39: β for roof drifts in coupled shear wall buildings subjected to each of the ten ground motions with the mean, µ, and mean plus one standard deviation, µ + σ, plotted, Rα=0 = 2.0, α = 0.2.  5.2.4 Residual Roof Drift The third response metric considered; the residual roof drift of the coupled wall buildings are presented in this section. When considering residual roof drifts, it was necessary to use δ rather than β, which compared the absolute increase in the residual roof drifts, expressed as a percentage of the building height. When interpreting these results, it is important to recall that an δ of 2% could be of both of an increase from 0.5% to 2.5% residual roof drift or an increase from 5% to 7% (although such large residual roof drifts were never observed for regular buildings without PLDs). δ for coupled shear wall buildings with Rα=0 = 2.0, 4.0, and 6.0 are given in Figure 5.40, Figure 5.41, and Figure 5.42, respectively.   151   Figure 5.40: δ, the additional residual roof drift, expressed as a percentage of building height, produced in coupled shear wall buildings, Rα=0 = 2.0.    152   Figure 5.41: δ, the additional residual roof drift, expressed as a percentage of building height, produced in coupled shear wall buildings, Rα=0 = 4.0.   153   Figure 5.42: δ, the additional residual roof drift, expressed as a percentage of building height, produced in coupled shear wall buildings, Rα=0 = 6.0. The residual roof drifts were increased by PLDs in all coupled shear wall buildings which experienced inelastic behaviour (i.e. Rα=0 > 1.0). This is different from the cantilevered shear walls; for which only the short 5, 10, and 20 storey buildings experienced additional residual roof drift demands as a consequence of PLDs. The reason for this difference is that the coupled shear wall buildings were not governed by the flag-shaped moment-curvature hysteresis at the base of the shear walls, but rather by the Clough hysteresis in the coupling beams. This Clough behaviour was prone to ratcheting and resulted in more widespread residual roof drifts than were seen for the cantilevered wall buildings, even at relatively small Rα=0 = 2.0. Given that acceptable residual inter-storey drift limits have been set at 1%, (Section 4.10.10) the additional residual roof drift demands calculated for many cases in Figure 5.40 through Figure 5.42 are very significant. The coupled shear wall buildings with Rα=0 > 2.0 and α > 0.2 would not be in serviceable condition after the imposed ground motions.    154  As was the case for the previous two response quantities; maximum base curvatures and maximum roof drifts, there was a PLD threshold above which the residual roof drifts tended to run away. At α ≤ 0.4, the residual roof drifts were moderate and conducive to post-earthquake serviceability; however at α ≥ 0.4 and above, the residual roof drifts became exceedingly large and serviceability degraded severely.  The variability in the additional demands (Figure 5.43) was consistent with that observed for β of the maximum base curvatures and the maximum roof drifts. This variability was small when the mean δ were moderate, however, as δ became more severe, so too did the differences between the residual roof drifts caused by individual ground motions.  Figure 5.43: δ, the additional residual roof drift, expressed as a percentage of building height, produced in coupled shear wall buildings subjected to each of the ten ground motions with the mean, µ, and mean plus one standard deviation, µ + σ, plotted, Rα=0 = 2.0, α = 0.2.    155  5.2.5 Discussion At each building height considered, the performance of the coupled shear wall buildings was markedly worse than that of the corresponding cantilevered shear wall models. All three response quantities considered showed significant increases in imposed demand as a consequence of PLDs, these increased demands were consistently larger in the coupled shear wall building than in the cantilevered wall models.  In can be concluded that the coupled wall buildings were more sensitive to inclined columns than the cantilevered wall buildings were. This sensitivity was primarily the result of the coupling beams in the coupled wall model. These coupling beams were modelled with a Clough hysteretic model, an appropriate approach given their negligible and unpredictable axial loads (ATC, 2010). This Clough hysteretic behaviour of the coupling beams dominated the global building response, despite the flag-shaped behaviour exhibited by the flexural response at the base of the coupled shear walls. It has been observed that it is this Clough hysteretic behaviour that is most sensitive to seismic ratcheting and was the driver behind the observed inelastic response amplifications.  A second compounding influence which made ratcheting behaviour in the coupled walls more prominent was that high Rα=0 could be achieved for all considered building heights. Rα=0 was limited in tall cantilevered shear wall buildings due to overstrength from axial load; however, this was not true for coupled shear wall buildings. When the hysteretic behaviour of these taller walls was considered, there were cases in the 10, 20, 30, 40, and 50 storey buildings which demonstrated significant ratcheting when α was 0.2 and larger. In general, the performance of the high-rise coupled shear wall buildings was most severely degraded by PLDs; however, the importance of building height on the observed demands should not be over-estimated because this relationship was inconsistent. Variations between different building heights were influenced by the specific shear wall sections considered. β for the maximum roof drifts in each coupled shear wall buildings considered are shown in Figure 5.44. All building heights were significantly influenced by PLDs; variation between buildings with different heights was insignificant relative to α. Although the 5 storey coupled shear wall building with a period of 0.5 seconds consistently performed well regardless of the PLDs, all of the other building periods considered showed large additional displacement demands.   156   Figure 5.44: β for the maximum roof drifts produced in the compression wall from each time-history analysis conducted on the coupled shear wall buildings. Each data point represents β from a single building and ground motion (1260 data points). The data is organized by the period and α; no differentiation is made between Rα=0 = 2.0, 4.0, and 6.0. As was the case for cantilevered shear wall buildings, there was significant variability in the response from each ground motion, and Rα=0 considered. Buildings which exhibited the most degraded performance also demonstrated the most variability between ground motions. This variability increased dramatically as α increased beyond 0.2 (Figure 5.45). Beyond this threshold of 0.2, identical buildings would perform relatively well to some ground motions; however, when subjected to a different ground motion, instability and collapse would occur. β across the suite of considered ground motions and Rα=0 were relatively consistent for constant α, when this α was 0.2 or smaller. Once α exceeded this threshold, large variation existed between analysis cases and collapses occurred.   157   Figure 5.45: β for the maximum roof drifts produced in the compression wall from each time-history analysis conducted on the coupled shear wall buildings. Each data point represents β from a single building and ground motion (1260 data points). The data is organized by Rα=0 and α, with the relationship to the period ignored. The residual roof drifts in the cantilevered wall models showed a clear inverse correlation to the building height; this trend was not demonstrated by coupled shear wall buildings, as can be seen in Figure 5.46. This can be explained by the dominance of the Clough behaviour produced by the coupling beams in the coupled wall systems. This Clough behaviour dominated the global response and inhibited recovery to the initial resting state. As can be seen in Figure 5.46, the increases in residual roof drifts were correlated with α. The additional residual roof drifts frequently caused by PLDs often exceeded the recommended maximum residual inter-story drifts of 1% when α > 0.2.   158   Figure 5.46: δ, the additional residual roof drift, expressed as a percentage of building height, produced from each time-history analysis conducted on the coupled shear wall buildings. Each data point represents δ from a single building and ground motion (1260 data points). The data is organized by the period and α; no differentiation is made between Rα=0 = 2.0, 4.0, and 6.0.    159  Chapter 6 - Case Studies  For the main body of the study, analyses were conducted with ten horizontal ground motion records from crustal earthquakes on buildings with inclined columns over the full height and coupled wall buildings with a coupling ratio of 0.7; however, it was of interest to explore additional design and loading conditions. With this aim, five additional case studies were completed. Each with a relatively narrow scope, focusing on a single building configuration: the 30 storey coupled shear wall building with Rα=0 = 4.0 and PLDs corresponding to α = 0.2. This archetype building was selected for use in the case studies because it produced β < 1.60 in the study described above; however, when α exceeded 0.2, severe inelastic displacement demands and collapses were observed in this building. Therefore, this building was supposed to have relatively large sensitivity to any additional demands and served as an ideal archetype for comparison with the five case studies. A sixth case study considered steel braced frame buildings.  6.1 GRAVITY SYSTEM IRREGULARITY  In the study by Best, Elwood, and Anderson (2011), three different gravity system irregularities were identified; these same three irregularities are considered in this case study. These irregularities were inclined columns along the entire buildings height, inclined columns over the ground floor lobby, and eccentric floor spans; an irregularity which applies greater weight to one side of the shear wall than the other.. For each loading mechanism, the irregularity was scaled such that the same base overturning moment was applied to the building. However, each of the three gravity system irregularities, shown in Figure 6.1, imposes a different bending moment distribution along the height of the shear wall. These distributions, shown in Figure 6.2, result from the loading mechanism imposed on the shear wall by each gravity system irregularity,    160   Figure 6.1: Three gravity system irregularities considered; fully inclined (left), inclined lobby (center), and eccentric floor spans (right) (Best, Elwood, & Anderson, 2011).   Figure 6.2: Bending moment distribution from each gravity system irregularity - fully inclined (left), inclined lobby (center), and eccentric floor spans - along the height of the shear wall in a 5 storey building (right) (Best, Elwood, & Anderson, 2011). Although the bending moment distributions vary significantly over the height of the building, it is primarily the variation in the plastic hinge region that influences the inelastic response. This is because, although coupling beams may yield all any heights, inelastic behaviour only occurred in this plastic hinge at the base of the shear wall. The distribution of moment demands in this region influence the inelastic demands imposed; however, the sensitivity of the displacement demands to this distribution was poorly   161  understood. From Figure 6.2, it can be seen that the moment demands at the top of the plastic hinge differ, depending on type of irregularity in the gravity system. Eccentric floor spans produce a uniform bending along the height of the plastic hinge while fully inclined columns produce a slightly smaller bending moment at the top of the hinge and inclined columns over the lobby alone impose the smallest moment demands at the top of the plastic hinge.  In the study by Best, Elwood, and Anderson (2011), it was found that the greater the overall distribution of bending moments applied along the plastic hinge, the greater the amplifications. Eccentric floor spans imposed the largest displacement demands while inclined columns over the lobby imposed the smallest. Fully inclined columns produced demands slightly smaller than the eccentric floor span irregularity; the additional demands associated with eccentric floor spans decreased with building height and were negligible for high-rise buildings. As building height increases, the variation in the bending moment profile imposed by fully inclined columns over the height of the plastic hinge becomes very small; a distribution similar to that imposed by the eccentric floor span irregularity. If current construction trends continue, fully inclined columns are the irregularity most likely to impose large PLDs; therefore it was the main consideration in the study by Best, Elwood, and Anderson (2011), as well as this study.  In order to validate this approach it was necessary to confirm that each irregularity imposes similar inelastic demands. In the first study, three different irregularities in the gravity system were considered: fully inclined columns, inclined columns over the ground floor lobby, and eccentric floor spans. The hysteretic response of the 30 storey coupled shear wall building with each irregularity is shown in Figure 6.3, and Figure 6.4. In each figure, otherwise identical buildings are configured with a different irregularity and subjected to one of the ten ground motions considered for the archetype building case.   162   Figure 6.3: Base moment versus curvature in the compression wall of three 30 storey coupled shear wall buildings with PLDs corresponding to α = 0.2, Rα=0 = 4.0, and subjected to ground motion 35. Each building has one of the three irregularities: fully inclined columns, inclined columns over the lobby, and eccentric floor spans. The initial and final resting states are identified with hollow and solid circles, respectively.  Figure 6.4: Base moment versus curvature in the compression wall of three 30 storey coupled shear wall buildings with PLDs corresponding to α = 0.2, Rα=0 = 4.0, and subjected to ground motion 164. Each building has one of the three irregularities: fully inclined columns, inclined columns over the lobby, and eccentric floor spans. The initial and final resting states are identified with hollow and solid circles, respectively.   163  As one can see from the hysteretic behaviour at the base of the compression wall, the performance resulting from each of three types of irregularities is very similar. On average, the inclined lobby irregularity produced the greatest base curvature demands; however, both the eccentric floor span and fully inclined column irregularities produced the greatest demands for isolated ground motions.  The greatest base curvature demands coincided with irregularities which applied high concentrated moment demands at the base of the plastic hinge; inclined columns over the lobby and fully inclined columns produced the greatest and second greatest base curvature demands, respectively. Although, the eccentric floor span irregularity had the greatest overall applied moment distribution throughout the plastic hinge region, irregularities which applied concentrated moment demands at the base of plastic hinge (Figure 6.2) resulted in the largest maximum base curvature demands (Table 6.1). Table 6.1: Mean and maximum β for the maximum base curvature demands in a 30 storey coupled shear wall building with each of the three types of structural irregularities. α = 0.2, Rα=0 = 4.0, subjected to ten ground motions. Gravity System Irregularity βmean, Maximum Base Curvatures βmax, Maximum Base Curvatures Fully Inclined (Archetype) 1.72 2.85 Inclined Lobby 2.01 2.81 Eccentric Floor Spans 1.56 2.92  When maximum roof drifts were considered, the performance trend of the three types of irregularities was consistent with that uncovered by Best, Elwood, and Anderson (2011); that is, inclined columns over the lobby produced the mildest amplifications with fully inclined columns causing significantly larger demand amplifications. The eccentric floor span irregularity produced the greatest demands. In the study by Best, Elwood, and Anderson (2011), the difference in performance between the fully inclined column and eccentric floor span irregularities was found to be negligible. The results from this case study indicate that the difference in demands is larger than originally thought, although still mild. The largest mean β in maximum roof drift demands resulted from the eccentric floor span irregularity, as shown in Table 6.2. This change can be explained by returning to the reason why the 30 storey coupled shear wall building was selected for this case study. It had previously performed just within the threshold of acceptable behaviour, beyond which demand amplifications tended to run away. For this reason, it was particularly sensitive to the additional demands imposed on the structure from the eccentric floor spans rather than the   164  fully inclined columns. The eccentric floor span irregularity resulted in mean β 14% greater than those from the archetype with fully inclined columns (Table 6.2). This is significant and warrants additional consideration, both in terms of additional studies and in determining acceptable performance limits. Table 6.2: Mean and maximum β for the maximum roof drift demands in a 30 storey coupled shear wall building with each of the three types of structural irregularities. α = 0.2, Rα=0 = 4.0, subjected to ten ground motions. Gravity System Irregularity βmean, Maximum Roof Drifts βmax, Maximum Roof Drifts Fully Inclined (Archetype) 1.60 1.98 Inclined Lobby 1.10 1.21 Eccentric Floor Spans 1.83 2.27  Additional base curvatures and roof drifts were imposed on the structure by the two additional irregularities considered: inclined columns over the lobby and eccentric floor spans, respectively. These additional demands are significant and warrant further consideration. It is probable that buildings with smaller PLDs would be less sensitive to additional demands from this eccentric floor span irregularity.  6.2 STRENGTHENED COUPLING BEAMS  Two main assumptions were made regarding the coupling beams in the coupled shear wall models. First, it was assumed that coupling beams would generally be designed with symmetrical strength, despite the asymmetrical demands imposed by a PLD. Second, it was thought that engineers would typically account for the additional demands imposed by a structural irregularity by adding strength at the base of the shear walls rather than in the coupling beams. Therefore, in the coupled shear wall buildings, the strength of the coupling beams was not influenced by PLDs, as can be seen in Equation 26. The second assumption resulted in disproportionate increase in yielding of the coupling beams relative to the base of the shear walls when the buildings were subjected to PLD demands. Concentration of yielding in the coupling beams - which were found to catalyze ratcheting - may have produced greater displacement demands than would otherwise be induced in buildings with coupling beams strengthened to account for PLDs. The archetype case, which was considered for the results presented in section 5.2, had a coupling ratio of 0.7 and will be referred to as the ‘CR = 0.7’ case.    165   ܯ௬ = ܥܴ ∗ ܯா ∗ ܮ஼஻2 ∗ ܰ ∗ ܴ ∗ ܮ  (26)  In order to investigate the sensitivity of the observed displacement amplifications to the strength of the coupling beams; a second case study was conducted. In this case study, symmetrical strength was maintained in the coupling beams, however, the coupling beams were given additional strength. The strength of the coupling beams was increased and the flexural strength at the base of the shear walls was decreased - an approach which allowed the shear walls to maintain constant overturning moment resistance, while achieving coupling ratios of 0.77 and 0.84. These higher coupling ratios correspond to coupling beam strengths of 10% and 20% greater than those used in the archetype building. It is important to note that while the coupling beams were made stronger, the additional strength was accounted for by reducing the flexural yield strength at the base of the shear walls accordingly; Rα=0 = 4.0 was still achieved by the coupled shear wall buildings.  The hysteretic behaviour at the base of the compression wall is shown in Figure 6.5 and Figure 6.6 for the three coupling ratios considered. The hysteresis of the archetype building (coupling ratio = 0.70) is compared to the hysteresis of both buildings with coupling ratios of 0.77 and 0.84. As the coupling ratio increased, there were subtle changes in the hysteretic behaviour at the base of the shear walls. These subtle changes were the result of the interaction of two counteracting behaviours; decreased flexural strength at the base of the shear walls and decreased ratcheting. As a consequence, there were no significant or consistent effects on the maximum base curvature demands as a result of increasing the coupling ratio, as indicated by Figure 6.5 and Figure 6.6.   166   Figure 6.5: Base moment versus curvature in the compression wall of three 30 storey coupled shear wall buildings with coupling ratios of 0.70, 0.77, and 0.84, α = 0.2, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.  Figure 6.6: Base moment versus curvature in the compression wall of three 30 storey coupled shear wall buildings with coupling ratios of 0.70, 0.77, and 0.84, α = 0.2, Rα=0 = 4.0, and subjected to ground motion 124. The initial and final resting states are identified with hollow and solid circles, respectively.   167  When β for the base curvatures from each coupling ratio were compared, it was found that the amplifications were very consistent between the different coupling ratios cases (Table 6.3). This can be explained as the result of two conflicting behaviours; the reduced ratcheting caused by increased coupling beam strength, and a shift of inelasticity behaviour to the base of the shear walls. In other words, although less yielding occurred, a greater portion of this yielding occurred in the flexural response of the shear walls. Table 6.3: Mean and maximum β for maximum base curvatures in the 30 storey coupled shear wall buildings with three considered coupling ratios for α = 0.2, Rα=0 = 4.0, subjected to ten ground motions. Coupling Ratio βmean, Maximum Base Curvatures βmax, Maximum Base Curvatures 0.70 (Archetype) 1.72 2.85 0.77 1.50 2.27 0.84 1.57 2.44  While strengthening the coupling had negligible effects on the curvature demands at the base of the shear walls, it was found to significantly decrease the maximum roof drift demands. The higher the coupling ratios considered in this case study were less prone to ratcheting. Of the three coupling ratios considered, the archetype building with a coupling ratio of 0.70 was the most prone to ratcheting. This is shown in Figure 6.7 and Figure 6.8, which shows the roof drifts in the three considered 30 storey coupled shear wall buildings.   168   Figure 6.7: Roof drift versus time in the 30 storey coupled shear wall buildings with three considered coupling ratios, α = 0.2, Rα=0 = 4.0, and subjected to ground motion 38.  Figure 6.8: Roof drift versus time in the 30 storey coupled shear wall buildings with three considered coupling ratios, α = 0.2, Rα=0 = 4.0, and subjected to ground motion 59.   169  When the β in the roof drift demands from each of the three considered coupling ratios was examined, it was found that the largest coupling ratio produced the smallest β. The archetype building with a relatively low coupling ratio of 0.7 was most prone to developing large roof drifts demands. β for the roof drifts of the 30 storey building with each of the three cases are given in Table 6.4. Table 6.4: Mean and maximum β for maximum roof drifts in the 30 storey coupled shear wall buildings with three considered coupling ratios for α = 0.2, Rα=0 = 4.0, subjected to ten ground motions. Coupling Ratio βmean, Maximum Roof Drifts βmax, Maximum Roof Drifts 0.70  (Archetype) 1.60 1.98 0.77 1.50 1.92 0.84 1.42 1.83  Strengthening the coupling beams shifted inelastic behaviour from the coupling beams to the bases of the shear walls. By decreasing the amount of yielding in the coupling beams, performance of the coupled shear wall buildings was improved. Coupling beams exhibit Clough hysteretic behaviour - prone to seismic ratcheting - while the base of the shear walls demonstrate flag-shaped hysteresis with pinching. This pinching inhibited seismic ratcheting, therefore strengthening the coupling beams reduced the maximum roof drift demands.  In conclusion, by shifting inelastic behaviour away from the coupling beams in a coupled wall system, the seismic ratcheting phenomenon was mitigated. Furthermore, it can be concluded that the ratcheting phenomenon is extremely sensitivity to the extent of yielding in the coupling beams, which are the main catalyst for ratcheting in the coupled wall systems. Despite this finding, it must be recognized that if coupling beam strength is selected to remain elastic for a design earthquake, a slightly larger than design ground motion would cause yielding of coupling beams which might lead to hysteretic behaviour in the beams which is prone to ratcheting.  6.3 SEISMIC ZONE  Thus far, the analyses on the shear wall buildings in this study have been done with ground motions scaled to impose overturning moment demands corresponding to the Vancouver UHS. Canada is a large country, with hugely different seismic hazards. The Vancouver UHS was considered in this study because   170  Vancouver has both a large number of buildings in the mid- to high-rise height range, and a relatively large seismic hazard. Other cities in Canada – with different seismic hazards – are also of concern as they are also be governed by NBCC code provisions. To determine in which seismic zones NBCC clauses should affect three additional cities: Victoria, Calgary, and Montreal - each subject to greatly different seismic hazards - are considered in this third case study.  Victoria, for example, has a UHS (Figure 6.9) with spectral accelerations exceeding those in Vancouver. Victoria currently has a relatively small downtown – the tallest buildings are approximately 20 storeys. However, these modest heights are within the considered scope of this study and the large spectral accelerations posed by the UHS make Victoria an ideal candidate of consideration in this final case study.  In addition to Victoria, the seismic hazard in Calgary and Montreal were also considered (Figure 6.9). Calgary has a small seismic hazard, with PGA approximately an order of magnitude less than in Vancouver. However, Calgary’s downtown core is developed and has a large number of mid- and high- rise buildings, new buildings in this rapidly growing city will be affected by any NBCC provisions adopted. Relative to Calgary and other cities in eastern Canada, Montreal has a relatively large seismic hazard and is a major urban center. It was prudent to consider the severity of seismic ratcheting in Calgary and Montreal.  The archetype 30 storey coupled shear wall building was again considered for analysis in Victoria and Vancouver; however, Rα=0 = 4.0 was not achievable due to the low seismic demands in Calgary and Montreal. Therefore, the archetype 30 storey building is first investigated in Vancouver and Victoria and a second 10 storey coupled wall building (Rα=0 = 2.0, α = 0.2) is later considered in all four cities.    171   Figure 6.9: Acceleration response spectra for ten ground motions scaled to match the maximum elastic base overturning moment imposed by the UHS in Vancouver, Victoria, Calgary, and Montreal on the archetype 30 storey cantilevered shear wall building (the 30 storey archetype building was not analyzed in Calgary and Montreal however the spectra are included for completeness and consistency). Each linearly scaled spectrum is indicated with a dashed line with the mean of the ten records in bold and the UHS indicated with a solid line. Victoria has greater seismic demands than Vancouver; therefore, the 30 storey building in Victoria had greater yield strengths - maintaining consistent Rα=0. This additional strength was achieved through additional longitudinal steel at the base of the shear walls. When the response of the 30 storey coupled shear wall building was compared between Victoria and Vancouver, the cases in Victoria produced moderately greater displacement demands than in Vancouver. When regular buildings without PLDs were considered, as shown in Figure 6.10, the response in each city was quite similar; although mildly greater moments and corresponding curvatures occurred at the base of the building located in Victoria. When PLDs corresponding to α = 0.2 were considered, the performance of the building in Victoria was more severely degraded than was the building in Vancouver. Maximum base curvatures increased to   172  approximately 1.8 rad/km in the Victoria case compared with approximately 1.4 rad/km in the Vancouver case.  Figure 6.10: Base moment versus curvature in the compression wall of two 30 storey coupled shear wall buildings subjected to ground motion 2 linearly scaled to impose maximum elastic base overturning moments corresponding to the UHS in Vancouver and Victoria, α = 0.0, Rα=0 = 4.0. The initial and final resting states are identified with hollow and solid circles, respectively.  Figure 6.11: Base moment versus curvature in the compression wall of two 30 storey coupled shear wall buildings subjected to ground motion 2 linearly scaled to impose maximum elastic base overturning moments corresponding to the UHS in Vancouver and Victoria, α = 0.2, Rα=0 = 4.0. The initial and final resting states are identified with hollow and solid circles, respectively. For this building, it was found that the greater seismic demands in Victoria resulted in greater inelastic displacement demand amplifications than were observed in Vancouver. Additional inelastic displacement   173  demands were mild – the maximum and mean β produced from the ten records each increased by approximately 1%, as shown in Table 6.5. Table 6.5: Mean and maximum β for maximum roof drifts in the 30 storey coupled shear wall building when located Vancouver and Victoria, α = 0.2, Rα=0 = 4.0, subjected to ten ground motions. Seismic Zone βmean, Maximum Roof Drifts βmax, Maximum Roof Drifts Vancouver (Archetype) 1.60 1.98 Victoria 1.62 2.00  Due to inherent overstrength, the 30 storey building – considered above –was unable to achieve Rα=0 = 4.0 in Calgary and Montreal due to the significantly decreased maximum elastic base overturning moment demands. Inherent strength was provided by minimum steel and axial load at the base of the shear walls which prevented the building from achieving high Rα=0 in these cities (this was the same overstrength issue which limited the scope of the cantilevered shear wall study). In order to compare the influence of PLDs on ratcheting in these cities (Calgary and Montreal), it was necessary to consider a different building than that used for other case studies, and considered for Victoria above. The 10 storey coupled shear wall building with Rα=0 = 2.0 and α = 0.2 was considered instead. The hysteretic response of the 10 storey coupled shear wall building to the same ground motion scaled for each of the four cities is shown in Figure 6.12 and Figure 6.13, with and without PLDs, respectively.    174   Figure 6.12: Base moment versus curvature in the compression wall of four 10 storey coupled shear wall buildings subjected to ground motion 2 linearly scaled to impose maximum elastic base overturning moments corresponding to the UHS in Vancouver and Victoria, α = 0.0, Rα=0 = 2.0. The initial and final resting states are identified with hollow and solid circles, respectively.   175   Figure 6.13: Base moment versus curvature in the compression wall of four 10 storey coupled shear wall buildings subjected to ground motion 2 linearly scaled to impose maximum elastic base overturning moments corresponding to the UHS in Vancouver and Victoria, α = 0.2, Rα=0 = 2.0. The initial and final resting states are identified with hollow and solid circles, respectively. As shown in Figure 6.12 and Figure 6.13, buildings located in the four cities are subjected to hugely different seismic demands. The magnitudes of the demands vary greatly; however the effect of PLDs appears similar in the four considered cases. In each 10 storey building, the presence of PLDs resulted in dramatic increases in residual base curvature demands and moderate increases in maximum base curvature demands. The displacement demand amplifications were similar between the four different cities; however, regions with the lowest seismic demands generally exhibited greater β. Montreal exhibited the largest mean β when maximum roof drifts were considered and Victoria demonstrated the smallest β, as shown in Table 6.6.   176  Table 6.6: Mean and maximum β for maximum roof drifts in the 10 storey coupled shear wall building when located in each of four considered Canadian cities, α = 0.2, Rα=0 = 2.0, subjected to ten ground motions. Seismic Zone βmean, Maximum Roof Drifts βmax, Maximum Roof Drifts Vancouver  1.26 1.68 Victoria 1.23 1.68 Calgary 1.31 1.63 Montreal 1.42 1.73  Although the four cases considered all exhibited similar demand amplifications, it is clear from Figure 6.12 and Figure 6.13 that these amplifications correspond to hugely different demands; buildings in Vancouver and Victoria experience relatively large base curvatures while curvatures in Montreal are more moderate and those in Calgary are very mild. The different base curvature demands shown by each considered city indicate that design limits may not be appropriate for all seismic regions of Canada. To validate this, the roof drift demands imposed in the four considered cities are presented in Table 6.7. Table 6.7: Mean roof drifts in the 10 storey coupled shear wall building when located in each of four considered Canadian cities, α = 0.0 and 0.2, Rα=0 = 2.0, subjected to ten ground motions. Seismic Zone Mean Roof Drift (%), α = 0.0 Mean Roof Drift (%), α = 0.2 Vancouver  0.36 0.46 Victoria 0.41 0.51 Calgary 0.04 0.05 Montreal 0.15 0.22  Buildings in all four cities experienced relatively moderate roof drifts – the largest mean roof drifts occurred in Victoria and were only 0.51% with PLDs present. The seismic demands in Vancouver and Montreal produced similar mean rood drifts of 0.46% and 0.22%, respectively. The mean roof drift demands of 0.05% induced in the building in Calgary were significantly smaller than the other three cities considered. In Calgary, the mean roof drift increased from 0.04% without PLDs to 0.05% when PLDs corresponding to α = 0.2 were applied. Such a small increase in roof drift demands (~0.01%) would have negligible effects on the stability of the building. Therefore, it would be logical to exempt regions with low seismic hazards – such as Calgary – from design limits which set an upper bound to allowable PLDs.   177  6.4 SUBDUCTION GROUND MOTIONS  The ten ground motions considered in the main study were from crustal earthquakes. However, a significant portion of the seismic hazard in the lower mainland of British Columbia is derived from the potential for a large subduction event. Crustal earthquakes produce ground motions with relatively short durations and few cycles when compared with larger subduction earthquakes. Subduction events typically occur over hundreds of kilometers along plate boundaries, and as a result, they produce ground motions that can last several minutes and complete many loading cycles. The number of loading cycles may be an important factor contributing to seismic ratcheting in buildings with PLDs. Therefore, a fourth case study was conducted with two ground motions from subduction events. The first, shown in Figure 6.14, was recorded from the 2010 Chile Earthquake which had a magnitude of 8.8. This ground motion was recorded at a distance of 209km from the epicenter and exhibits peak ground accelerations in the horizontal direction, PGAH, of PGAH = 6.84m/s2 during its 180 second duration.  Figure 6.14: Subduction ground motion, Chile 2010, magnitude 8.8, Stn: Angol, ANGO, UCS (CESMD, 2012; Seismosignal, 2012).   178  The second subduction ground motion used was from the 2011 Great Tohoku Earthquake which occurred of the eastern coast of northern Honshu. This subduction event had a magnitude of 9.0 and the ground motion was recorded at a distance of 290km from the epicenter. The PGAH in the ground motion is 11.9m/s2 and the duration is approximately 282 seconds.  Figure 6.15: Subduction ground motion, Tohoku 2011, magnitude 9.0, Stn: Imaichi, TCG009, KNET. (CESMD, 2012; Seismosignal, 2012). Each ground motion was linearly scaled to induce the same maximum elastic base overturning moments as the Vancouver UHS. This scaling approach was adopted to allow direct comparison of response quantities with the results from the crustal ground motions discussed above. For most period ranges this was considered conservative since subduction event demands are not expected to exceed the UHS in Vancouver except potentially at long periods (NBCC 2010).  The base moment-curvature response of the building when subjected to the February 2010 Chile Earthquake is given in Figure 6.16. The presence of the PLD produced larger maximum curvatures; however, the building maintained stability and excessive seismic ratcheting did not occur. The maximum   179  roof drifts were large, producing β = 2.03 – just slightly larger than the maximum β induced in the same building from the most severe of the ten crustal ground motions, β = 1.98.  Figure 6.16: Base moment versus curvature in the compression wall of a 30 storey coupled shear wall building with and without PLDs corresponding to α = 0.2 subjected to the 2010 Chile subduction ground motion, Rα=0 = 4.0. The initial and final resting states are identified with hollow and solid circles, respectively. The response to ground motion from the March 2011 Tohoku Earthquake is given in Figure 6.17 with and without PLDs corresponding to α = 0.2. As with the ground motion from the Chile Earthquake, the presence of seismic ratcheting was not observed. The addition of the PLD to the building did amplify the maximum roof drifts induced; producing β = 1.62. Maximum base curvatures were less than 0.5 rad/km. The maximum curvatures were within the ductility capacity of the section which was determined to be 0.8 rad/km.    180   Figure 6.17: Base moment versus curvature in the compression wall of a 30 storey coupled shear wall building with and without PLDs corresponding to α = 0.2 subjected to the 2011 Tohoku subduction ground motion, Rα=0 = 4.0. The initial and final resting states are identified with hollow and solid circles, respectively. For both ground motions considered, the performance of the building was moderately degraded by PLDs. At α = 0.2, the buildings performed in a manner consistent with the ten crustal ground motions and did not significantly exceed the approximate curvature capacity at the base of the shear walls of 0.8 rad/km (Section 4.10.10). Although, relative to crustal records, subduction records have many cycles and long duration, this did not appear to produce additional ratcheting. The 2011 Tohoku ground motion produced β = 1.62 and the 2010 Chile ground motion produced β = 2.03; although slightly larger, both values were congruous with the range of β produced by the ten crustal ground motions. This case study demonstrated that subduction records with numerous cycles can result in approximately 30% greater amplifications in displacement demands compared to crustal ground motions.  6.5 VERTICAL GROUND MOTIONS  In the studies presented thus far, vertical motion was neglected and the buildings were subjected to only horizontal ground motion; however, it has been hypothesized that vertical ground motion could have a compounding interaction with PLDs which would drastically increase structural demands. A final study investigated the behaviour of buildings with PLDs to two ground motions with vertical components. The horizontal components were linearly scaled to induce the same maximum elastic base overturning moments as the Vancouver UHS and the same scaling factors were applied to the vertical components.   181   To characterize this source of risk, two new ground motions were used for this case study. Seismic design codes typically characterize the peak ground accelerations in the vertical direction, PGAV, as 2/3 of the peak ground acceleration in the horizontal plane, PGAH. However, depending on the depth and the orientation of the ground motion, this ratio can vary significantly. Two ground motions were selected which represented the potential variability. Ground motion 549, which was obtained from a crustal event in California had PGAV/PGAH = 0.56; approximately 2/3. Ground motion 1, obtained from a recording station near the February 2011 Christchurch earthquake, had PGAV/PGAH = 1.44. The magnitude 6.3 event which occurred in Christchurch is notorious for its large vertical accelerations; therefore this record represents an extreme upper bound for PGAV/PGAH. Together, these two ground motions are a basic represented the potential variability in the vertical components of ground motions. Ground motion 549 and 1 are given in Figure 6.18 and Figure 6.19, respectively, which both the horizontal and vertical components of each ground motion indicated.  Figure 6.18: Ground motion 549, Stn: USGS Bishop – LADWP South (PEER, 2012).   182   Figure 6.19: Ground motion 1, Stn: REHS Christchurch Resthaven (CESMD, 2012). For each of the two ground motions, the hysteretic response at the base of the compression wall in the 30 storey coupled shear wall building was recorded. This hysteretic response was considered both with and without PLDs, and with and without vertical ground motion included in the analysis. In this way, the compounding influence of vertical motion and PLDs could be distinguished from the displacement demands imposed by the additional vertical motion and PLDs acting independently. Ground motion 1 - which has a more significant vertical component than ground motion 549 - exhibited a greater change in the observed response when vertical ground motions were considered; however, this effect was small compared to the demands from the horizontal component alone (Figure 6.20). The addition of the vertical component did not translate to increased base curvature demands. Figure 6.21 shows response of the same building to ground motion 1, both with and without PLDs considered. Once again, the hystereses differed only mildly as a result of vertical excitation - ground motion 1, which had a larger vertical component, was more significantly affected than ground motion 549. However, the effect was mild and imposed additional base curvature demands of less than 5%. For both ground motions considered, the addition of   183  the vertical ground had similar effects on the buildings with PLDs as it did on buildings without PLDs; evidence against the supposed compounding effect that had been feared. For both considered ground motions, the inclusion of vertical components had negligible effects on the maximum base curvatures demands at the base of the shear walls in the coupled buildings.  Figure 6.20: Base moment versus curvature in the compression wall of a 30 storey coupled shear wall building with and without PLDs corresponding to α = 0.2 and subjected to ground motion 549 with and without the vertical component, Rα=0 = 4.0. The initial and final resting states are identified with hollow and solid circles, respectively.    184   Figure 6.21: Base moment versus curvature in the compression wall of a 30 storey coupled shear wall building with and without PLDs corresponding to α = 0.2 and subjected to ground motion 1 with and without the vertical component, Rα=0 = 4.0. The initial and final resting states are identified with hollow and solid circles, respectively. The building was subjected to each ground motion with, and without, the vertical component included. For both ground motions, the analysis with the vertical component produced β for maximum roof drifts that was less than 5% greater than the analysis without the vertical component, as shown in Table 6.8. Table 6.8: β for the maximum roof drift demands imposed in each of the four considered cases for each ground motion with and without vertical ground motion considered, α = 0.2, Rα=0 = 4.0. Vertical Ground Motion Ground Motion 549, PGAV/PGAH = 1.44 Ground Motion 1, PGAV/PGAH = 0.56 Excluded 1.56 2.09 Included 1.55 2.12    185  In addition to comparing the maximum base moment curvatures at the base of the walls and the maximum roof drift demands imposed, it was also prudent to consider the demands imposed upon the base of the columns. This was to insure that the combination of PLDs and vertical motion don’t have the potential to produce severe increases in the vertical demands. The gravity system (i.e. floor slab and columns) were defined to have a fully elastic response, therefore it was only possible to compare the maximum force demands at the base of the columns under different structural configurations. The maximum compressive and tensile demands on the base of the columns are shown in Table 6.9 for the columns on the compression side of the building and in Table 6.10 for the columns on the tension side of the building. All of the presented data are from analyses which included vertical ground motion. Table 6.9: Maximum axial demands at the base of the exterior columns on the compression face of the 30 storey coupled shear wall building with Rα=0 = 4.0. Applied Demand Ratio, α Maximum Compression, Ground Motion 549, (kN) Maximum Tension, Ground Motion 549, (kN) Maximum Compression, Ground Motion 1, (kN) Maximum Tension, Ground Motion 1, (kN) 0.0 -226 000 -18 500 -312 000 64 200 0.2 -206 000 -44 000 -324 000 81 900  The PLDs interacted with the vertical component of ground motion producing differences in the force demands in the gravity columns. The effect of this interaction was moderate; for ground motion 549, PLDs actually decreased the maximum compressive demands. For ground motion 1, which had relatively large vertical ground motion, PLDs produced additional demands; however, these additional demands were moderate; axial compressive demands increased by 4% and tensile demands increased by 28%.  The effects of PLDs on the axial demands in the columns on the tension side of the coupled wall building were consistent with those observed on the compression column. The effects on force demands were very mild; some maximum axial demands decreased, while others increased. The largest deviation occurred for the axial compressive demands resulting from ground motion 549, where the maximum force increased by 14% when PLDs corresponding to α = 0.2.    186  Table 6.10: Maximum axial demands at the base of the exterior columns on the tension face of the 30 storey coupled shear wall building with Rα=0 = 4.0. Applied Demand Ratio, α Maximum Compression, Ground Motion 549, (kN) Maximum Tension, Ground Motion 549, (kN) Maximum Compression, Ground Motion 1, (kN) Maximum Tension, Ground Motion 1, (kN) 0.0 -226 000 -18 500 -312 000 64 200 0.2 -257 000 -15 600 -319 000 62 300  6.6 STEEL BRACED FRAME BUILDINGS  Steel braced frame buildings - common in eastern Canada – were considered in a final case study. Three two-dimensional building models; 8, 12, and 16 storeys tall, originally developed by Izvernari et al (2007), were used to represent mid-rise buildings, which are typical of steel braced frame construction. The three buildings heights considered; 30.8 meters, 46.0 meters, and 61.2 meters, are shown in Figure 6.22, Figure 6.23, and Figure 6.24, respectively (Izvernari, 2007; Izvernari, Lacerte, & Tremblay, 2007).  In the height range used, both moderately ductile (Type MD) and limited ductility (Type LD) systems were considered (Izvernari, 2007; Izvernari, Lacerte, & Tremblay, 2007. The strengths and PLDs on each of the three buildings were varied through the same domain as was used for the concrete shear wall buildings. For all analyses, the inclined column irregularity was considered along the full height of the building. The inclination required to achieve α = 0.4 at Rα=0 = 2.0 in each of the three building heights is indicated in Figure 6.22, Figure 6.23, and Figure 6.24 (right). The inclination corresponding to various considered α at different Rα=0 for each of the three considered building heights are given in Appendix A.3.    187   Figure 6.22: Structural configuration of 8 storey steel braced frame model with vertical columns (left) and with inclined columns, α = 0.4, Rα=0 = 2.0 (right) (Izvernari, Lacerte, & Tremblay, 2007).  Figure 6.23: Structural configuration of 12 storey steel braced frame model with vertical columns (left) and with inclined columns, α = 0.4, Rα=0 = 2.0 (right) (Izvernari, Lacerte, & Tremblay, 2007).  Figure 6.24: Structural configuration of 16 storey steel braced frame model with vertical columns (left) and with inclined columns, α = 0.4, Rα=0 = 2.0 (right) (Izvernari, Lacerte, & Tremblay, 2007).   188  6.6.1 SFRS Strength and Demand Relationships A different approach was used to achieve target Rα=0 than that used for the shear wall buildings. As opposed to adjusting the strength of the SFRSs to achieve target Rα=0, as was done for the concrete shear wall buildings, the ground motions applied to each building were scaled instead. The strength of the steel braced frame in each building was left unaltered for all analysis cases. This was a convenient divergence from the earlier approach due to the complexity of the steel braced frame models which were used. A second major departure from the r approach taken for the shear wall buildings was that the weights of the buildings were not scaled to achieve desired periods.  Base shear forces were considered in the steel braced frame buildings rather than base overturning moments, which were used in the flexural dominated shear wall buildings. Rα=0 and α, were used to manipulate the relationship between the shear strength of the SFRS and the elastic shear demands in the steel braced frame system. Because of the different approach taken for the steel braced frame models, there are differences from the shear wall buildings which must be recognized.  The shear wall buildings considered constant seismic demands and variable wall strengths to achieve different Rα=0. This approach was undesirable for the steel braced frames because of their complexity. Instead, a simpler approach of maintaining a constant the base shear yield strength, Vy, and manipulating the maximum elastic base shear demands, VE max, was used.  As a consequence of this new approach, it was not possible to reduce the reverse yield strength of the SFRS to account for the strengthening effect of the PLD – the approach which was used for the shear wall buildings. To maintain as much consistency as possible with the shear wall buildings, the relationship between the forward yield strength, My+ (Vy), and the elastic seismic demands, ME max (VE max), was kept constant to that considered in the shear wall buildings. The elastic base shear demands, VE max, were adjusted to achieve each target Rα=0, and α. This new formulation is presented in Equation 28 from its previous form shown in Equation 27.   ௬ܸ = ாܸ	௠௔௫(1 + ߙ)ܴఈୀ଴  (27)   ாܸ	௠௔௫ = ௬ܸܴఈୀ଴(1 + ߙ) (28)    189  As with the base shear yield strength, the approach taken to calculate PLDs, VPEL, at the base of the building was kept consistent the shear wall buildings. The same relationship used for the shear wall buildings, simply rearranged in terms of base shear, is given in Equation 29. By substituting in Equation 28, a simple relationship was obtained to calculate the required dead base shear demands for each applied load ratio, α. This relationship is given in Equation 30. One may note that although the PLDs no longer have a dependence Rα=0, the maximum elastic base shear demands are now dependant on Rα=0 - thus preserving the relationship between VE max and VPEL.  The relationships between the demands and capacities in the steel braced frame models are shown in Figure 6.25. Note that the relationships are consistent with those used for the shear wall buildings (Figure 4.5) with one important difference, the yield strength of the SFRS in the reverse direction. Not only was it no longer possible to reduce this reverse yield strength to account for the strengthening effect of the PLD, but the reverse yield strength was in fact strengthened by a magnitude equivalent to the PLD, even though the demands in the reverse direction were actually reduced.   ௉ܸா௅ = ߙ ாܸ	௠௔௫ܴఈୀ଴  (29)   ௉ܸா௅ = ߙ ௬ܸ(1 + ߙ) (30)    190   Figure 6.25: Schematic of the general parametric relationship between the demands and capacities of the steel braced frame systems. The relative magnitudes shown correspond to a building with Rα=0 = 2.0 and α = 0.5.  6.6.2 Building Design For a comprehensive description of the design procedure taken for each of the three braced frame models adopted, please see the original work published by Izvernari, Lacerte, & Tremblay (2007). The two- dimensional models were representations of three dimensional buildings supported laterally by two identical braced frames at opposite sides of the floor plan. In each of the models, the building mass was lumped at the three nodes on the concentrically braced frame at each floor level. The weight of the building was distributed between the concentrically braced frames and the seven gravity columns according to their respective tributary areas (Figure 6.26). The building mass and weight were derived   191  from the load sources indicated in Figure 6.26. Details regarding the weight distribution in each steel braced frame building are provided in Table 4.14.   Figure 6.26: Steel braced frame building overview: a) plan view of 8 and 12 storey buildings, b) braced frame elevation for the 8 storey building, © Izvernari, Lacerte, & Tremblay (2007), by permission. In the design of each of the three buildings, site class C soil conditions were assumed and the brace member sizes were selected such that the member forces produced a target base shear. This base shear was equal to that which would be calculated if the simplified design approach prescribed by the NBCC was adopted. The mass and weight distributions of the three buildings considered are provided in Table 4.14 (Izvernari, 2007; Izvernari, Lacerte, & Tremblay, 2007). Table 4.14: Building properties, MD: Moderately Ductile, LD: Limited Ductility, (Izvernari, Lacerte, & Tremblay, 2007). Storeys, N Height, H (m) Type Rd Building Weight, W (MN) Total Floor Weight (MN) Floor Weight on Inclined Columns (MN) T1 (s) T2 (s) V/W (%) 8 30.8 MD 3.0 118 15 8 1.53 0.54 6.32 12 46.0 LD 2.0 180 15 8 2.15 0.74 6.65 16 61.2 LD 1.58 242 15 8 2.53 0.83 7.74    192  6.6.3 Bracing Members The brace elements were modelled with nonlinear beam-column elements which were comprised of 345MPa steel. Steel properties were represented by the uniaxial Giuffré-Menegotto-Pinto (Steel02) material model which accounted for Bauschinger effects (Izvernari, 2007; Izvernari, Lacerte, & Tremblay, 2007; OpenSees, 2012). Each brace was discretized into eight elements along the length which allowed an out of plane camber to amplitude of L/250 mid-length – allowing bracing members to buckle out plane. Both HSS- and W-sections were selected for the bracing members, depending on the desired yielding/buckling force. Gusset plates were modelled with rigid elements at the end of each brace (Izvernari, 2007; Izvernari, Lacerte, & Tremblay, 2007). Cyclic strength degradation was not considered in the hysteretic models of the SFRS an approach which yields un-conservative estimates inelastic performance.  6.6.4 Beams and Columns The beams and columns were modelled with elastic beam-column elements representative of W-sections. Each element was assigned a plastic hinge length equal to the depth of the member at each end. Where also connected to a diagonal bracing member, the connections were assumed to be fully fixed. Where a connection existed between a column and a beam element, the connection was assigned flexural strength 10% that of the weakest connecting member (Izvernari, 2007; Izvernari, Lacerte, & Tremblay, 2007).  Gravity columns with similar properties were grouped together for simplicity, resulting in seven gravity columns, each representative of a different type of column. The types of columns in each building are: interior (one type), exterior (three types), corner (one type), and columns connected to the bracing bent acting in the perpendicular direction (two types) (Izvernari, 2007; Izvernari, Lacerte, & Tremblay, 2007).The gravity columns, consisting of W-section members, were represented with and elastic beam- column elements. These columns were pinned at the base and assigned splices with 10% of the strength of the weakest connecting member. The gravity columns were laterally constrained to the brace bent at each floor level with a rigid diaphragm. 6.6.5 Ductility Capacity Lateral inter-storey drifts calculated are to be a maximum of 1% for post-disaster buildings, 2% for schools, and 2.5% for all other buildings (NBCC, 2010). The gravity system components must then be designed to accommodate the imposed lateral deflections without losing their load bearing capacity.   193  These maximum limits ensure that excessive P-∆ effects do not develop and that lateral instability does not lead to collapse.  Residual inter-storey and roof drifts are also detrimental to the serviceability of steel braced frame buildings after earthquakes. Visibly leaning structures pose a hazard to rescue workers and can slow down the recovery of a community after a significant earthquake. Furthermore, excessive inter-storey drifts can prevent elevators from remaining operational and prevent access to otherwise healthy buildings. PEER (2010) recommends a maximum residual inter-storey drift of 1% in order to maintain building operation.  6.6.6 Extent of Seismic Ratcheting In this case study, the influence of Rα=0 on the seismic performance of the steel braced frame buildings is first explored by examining the hysteretic response of the 8 storey building with and without PLDs to the same ground motion at the four considered Rα=0. The hysteresis presented are all for a single ground motion, crustal record 2; however, other crustal ground motions were considered, with similar behavioural trends observed. Due to the simplified approach taken for the steel braced frame models – in which the seismic demands were scaled rather than the building strength to achieved different Rα=0 - the author has refrained from comprehensively defining the extent of seismic ratcheting observed in terms of Rα=0, T, and α, as was done for shear wall buildings. Rather, the hysteretic behaviour of the three considered steel braced frame buildings is examined under different loading conditions to develop a understanding of the sensitivity of this SFRS to ratcheting.  Building cases with Rα=0 = 1.0 corresponded to elastic conditions; therefore, there was no significant yielding. Two such buildings are considered in Figure 6.27 both with, and without, PLDs corresponding to α = 0.4. For both considered α, the steel brace frame buildings remained largely in the elastic range for the duration of the analyses, with only mild yielding and insignificant residual drifts. Due to the definition of VE max and VPEL, the PLDs associated with α = 0.4 required smaller seismic demands in order to maintain consistent total base shear demands in the forward direction for varying α. The yield strength of the SFRS the steel braced frame buildings was not adjusted to account for PLDs; therefore, both the stiffness and strength are constant regardless of α and Rα=0.   194   Figure 6.27: Base shear versus first storey drift in the 8 storey concentrically braced frame building with and without PLDs corresponding to α = 0.4, Rα=0 = 1.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. When buildings with Rα=0 > 1.0 – inducing inelastic behaviour - were considered, it was found that the 8 storey steel braced frame models exhibited fat hysteretic loops with a distinct transition from elastic to plastic deformation. This behaviour was similar to the elastic perfectly plastic hysteretic model considered in the SDOF study conducted. This was significant because this hysteretic behaviour is similar to the coupled shear wall buildings, and one that is susceptible to significant ratcheting producing large residual deformations.  In Figure 6.30, it can be seen that even before the introduction of PLDs to the SFRS, the 8 storey steel braced frame building is prone to developing residual deformation demands. Pinching behaviour - which was shown to restore the high-rise cantilevered shear wall buildings to their original states – was not observed. Residual drift demands are relatively large compared to the maximum drifts imposed. When PLDs were applied, the lack of a restoring influence lead to significant ratcheting and large residual deformation demands shown in Figure 6.28 when α = 0.4.   195   Figure 6.28: Base shear versus first storey drift in the 8 storey concentrically braced frame building with and without PLDs corresponding to α = 0.4, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. When greater still Rα=0 were considered, such as those shown in Figure 6.29 and Figure 6.30, it is clear that increased seismic demands caused a greater number of loading cycles to achieve yield in the SFRS. The consequence of this was that when PLDs were applied, the structure ratcheted laterally during each load cycle and developed corresponding large residual drift demands.  Figure 6.29: Base shear versus first storey drift in the 8 storey concentrically braced frame building with and without PLDs corresponding to α = 0.4, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.   196   Figure 6.30: Base shear versus first storey drift in the 8 storey concentrically braced frame building with and without PLDs corresponding to α = 0.4, Rα=0 = 6.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. Below, the effect of PLDs on each of the three steel braced frame building heights are examined by considering four α (0.0, 0.2, 0.4, and 0.6) and three Rα=0 (2.0, 4.0, and 6.0) at each building height. The 8 storey steel braced frame building is shown in Figure 6.31, Figure 6.32, and Figure 6.33. At low Rα=0 (2.0) PLDs produced only minor ratcheting for all considered α. None of the considered α produced additional first storey drift demands, however significant residual first storey drifts were produced. When buildings with Rα=0 = 4.0 and Rα=0 = 6.0 were considered, significant ratcheting was observed at all α, with clear increases in both maximum and residual first storey drift demands for incrementally larger PLDs. Similar to the shear wall buildings considered, the dependence of residual demands on α is much greater than on Rα=0.   197   Figure 6.31: Base shear versus first storey drift in the 8 storey concentrically braced frame building with varying PLDs, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.   198   Figure 6.32: Base shear versus first storey drift in the 8 storey concentrically braced frame building with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.   199   Figure 6.33: Base shear versus first storey drift in the 8 storey concentrically braced frame building with varying PLDs, Rα=0 = 6.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. When 12 and 16 storey buildings were considered, similar hysteretic behaviour to the 8 storey buildings was observed, that is, fat hysteretic loops with distinct transitions between elastic and plastic behaviour. Again, three Rα=0 (2.0, 4.0, and 6.0) were considered at four different α (0.0, 0.2, 0.4, 0.6). Significant ratcheting was observed for all cases with α ≥ 0.4 and Rα=0 ≥ 4.0 as can be seen in Figure 6.34 through Figure 6.39. For these 12 and 16 storey buildings, Rα=0 has an influence on the maximum first storey drifts imposed – larger demand amplifications were produced in buildings with Rα=0 greater than 2.0.   200   Figure 6.34: Base shear versus first storey drift in the 12 storey concentrically braced frame building with varying PLDs, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.   201   Figure 6.35: Base shear versus first storey drift in the 12 storey concentrically braced frame building with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.   202   Figure 6.36: Base shear versus first storey drift in the 12 storey concentrically braced frame building with varying PLDs, Rα=0 = 6.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.    203   Figure 6.37: Base shear versus first storey drift in the 16 storey concentrically braced frame building with varying PLDs, Rα=0 = 2.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.   204   Figure 6.38: Base shear versus first storey drift in the 16 storey concentrically braced frame building with varying PLDs, Rα=0 = 4.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively.   205   Figure 6.39: Base shear versus first storey drift in the 16 storey concentrically braced frame building with varying PLDs, Rα=0 = 6.0, and subjected to ground motion 2. The initial and final resting states are identified with hollow and solid circles, respectively. Significant ratcheting was produced in the steel braced frame buildings regardless of the building height and the extent of seismic ratcheting correlated strongly with α. When Rα=0 =2.0, the effect on the maximum inelastic demands was minimal; a result which was influenced by the simplified approach taken for the steel braced frame models. Regardless, significant increases in the residual first storey drift demands were observed for all considered Rα=0. Cases with Rα=0 =4.0 and Rα=0 =6.0 consistently produced significant amplifications in the maximum first storey drift demands for all considered building heights.  All three considered building heights exhibited similar hysteretic behaviour – fat hystereses with no pinching and a low propensity to self center after each ground motion. This hysteretic behaviour is similar to that exhibited by the coupled shear wall buildings – a building type that was found to be severely affected by PLDs.   206  Chapter 7 - Recommendations  This study has demonstrated that seismic ratcheting can occur under design level ground motions for both cantilevered and coupled shear wall buildings with PLDs. In addition, a case study considering steel braced frame buildings identified behaviour which is prone to ratcheting and demonstrated increased deformation demands as a result of PLDs.  Seismic ratcheting due to PLDs is not currently considered within the scope of the NBCC. It is suggested that PLDs be recognized as a new structural irregularity within the NBCC and treated in a manner similar to how torsional sensitivity and vertical stiffness irregularities are currently addressed (NBCC Table 4.1.8.6). Furthermore, the scope of the current study should be expanded upon in future studies to consider increased seismic demands, additional SFRS, and the effectiveness and dependability of mitigating measures used by engineers to counteract PLDs.  7.1 CURRENT STUDY  When considering maximum base curvatures and maximum roof drifts, β showed inconsistent dependence upon the height (period) of the buildings. In general, high-rise coupled shear wall buildings - with large fundamental periods - ratcheted more significantly than the mid-rise coupled shear wall buildings, which had relatively small fundamental periods. However, this was not true of the cantilevered shear wall buildings considered, which showed smaller deformation demand amplifications at greater building heights (periods). Such dependence on building height was inconsistent, and to a certain extent, dependent upon the specific building models considered. Regardless, relative to α, the correlation of β with building height was insignificant.  The dependence of β on Rα=0 was minimal due to the definition of α. A convenient consequence of this was that deformation demand amplifications were well correlated with α, regardless of Rα=0. β at low α (< 0.2) were small and exhibited moderate variability; however, at increased α (> 0.2), β became relatively large and variable. As was the case with building height, the dependence of β on Rα=0 was insignificant relative to the strong dependence on α for the shear wall buildings considered.    207  The lack of significant dependence of the observed ratcheting behaviour on Rα=0 indicate that the allowable α should be independent of the design ductility (Rd). Furthermore, the inconsistent relationship with building period for the considered SFRSs suggests that the α limit should not be related to the building period (height).  This study has, however, demonstrated that ratcheting behaviour is highly dependent on the SFRS, or more precisely, the hysteretic behaviour exhibited by the SFRS. Coupled shear wall buildings and steel braced frame buildings – each with reasonably fat hysteresis – are more prone to ratcheting than cantilevered shear wall buildings, which exhibit flag-shaped hysteresis. Therefore, it is proposed that the allowable α be lower for SFRS with fat hysteresis compared with flag-shaped or nonlinear elastic hysteresis.  In order to define a limit above which standard elastic design procedures can no longer reliably estimate deformation demands, only a maximum allowable value of α need be defined for each SFRS. Above this limit, nonlinear analysis is required to accurately capture the performance of the building.  To assist in selecting appropriate limits for α, Figure 7.1 and Figure 7.2 present β for maximum roof drifts produced by all considered α in the cantilevered shear wall and coupled shear wall buildings, respectively. In each of these figures, all considered Rα=0 and building periods are shown for each α. In this way, the variation of the ratcheting phenomenon across all considered buildings with a given α can be clearly seen. For both SFRSs, it is evident that increasingly large α correspond to both larger deformation demand amplifications and greater variability throughout the range of considered building cases.   208   Figure 7.1: β in cantilevered shear wall buildings at each α for all considered Rα=0, periods, and ground motions.  Figure 7.2: β in coupled shear wall buildings at each α for all considered Rα=0, periods, and ground motions.   209  The two figures which are shown indicate the range of behaviour observed in this study at each considered α; however, such detail is not needed to establish appropriate limits for each SFRS. Figure 7.3 provides the mean β value (solid line) for the worst performing building (i.e. considering all periods and Rα=0) and the maximum β (dashed line) for all ground motions and buildings considered, for the cantilevered and coupled shear wall buildings. It is clear from this figure that in order to insure equivalent deformation amplifications in the two SFRS, a smaller limit on the allowable α must be established for coupled shear walls buildings than for cantilevered shear wall buildings.  Figure 7.3: Deformation amplifications in cantilevered and coupled shear wall buildings corresponding to the mean β (solid line) for the worst performing building (considering all periods and Rα=0) and the maximum β (dashed line) for all ground motions and buildings considered. Recognizing that any limit on β will be arbitrary, it is suggested here to limit the mean β and the maximum β to approximately less than 1.5 and 2.5, respectively. Beyond these values, variability in the deformation demands increases dramatically and amplifications are difficult to estimate with accuracy. Consulting Figure 7.3, these limits on β suggest maximum allowable α values of 0.1 and 0.4 for coupled shear wall and cantilevered shear wall buildings, respectively. Steel braced frame buildings demonstrated a propensity to ratchet, similar to coupled shear wall buildings, therefore, the more conservative limit of 0.1 may be appropriate for this SFRS.  The case study conducted in section 6.3, identified that although the deformation amplifications are relatively consistent for buildings in different seismic regions, the low seismic demands in some regions   210  (e.g. Calgary) produce such mild deformations that limits on α are not warranted. Therefore, such limits need not apply to all seismic regions within Canada; i.e. they may exclude those with small seismic hazards. It is recommended to limit the allowable α only in regions with IEFaSa(0.2) ≥ 0.50; including cities such as Vancouver, Victoria, Ottawa, Montreal, and Quebec. It is suggested that regions with IEFaSa(0.2) < 0.50 be exempt from limits on α. Therefore buildings with PLDs in cities such as Calgary, Edmonton, Winnipeg, Toronto, and Halifax shall not require nonlinear dynamic analyses if they exceed the α limits above.  Below these limits, displacement demand amplifications are moderate and have mild variation. Nonetheless, PLDs below these α limits still produce increased inelastic deformation demands which warrant consideration. The inelastic amplification factor, γ, indicates the ratio of the maximum inelastic deformation demands in buildings with PLDs, ∆(R α=0,α) to the maximum deformation demands in equivalent elastic systems without PLDs, ∆(R α=0=1,α=0). These maximum elastic deformation demands are already calculated by engineers using standard elastic design procedures. Therefore, such an amplification factor is a useful method to estimate the increased inelastic deformation demands which result from PLDs during seismic loading.  The inelastic amplification factor is provided in Equation 31. Plotted as a function of α, this factor is given for the maximum roof drifts in cantilevered shear wall buildings in Figure 7.4 and Figure 7.5, and coupled shear wall buildings in Figure 7.6 and Figure 7.7. The first of each pair of figures presents γ with the dependence on period ignored, while the latter in each pair considers γ with the dependence on Rα=0 ignored.   ߛ = ߂(ܴఈୀ଴,ߙ) ߂(ܴఈୀ଴ = 1,ߙ = 0) (31)     211   Figure 7.4: γ corresponding to the mean γ for the worst performing cantilevered shear wall building (considering all periods) as a function of α for different Rα=0.  Figure 7.5: γ corresponding to the mean γ for the worst performing cantilevered shear wall building (considering all Rα=0) as a function of α for different periods.   212   Figure 7.6: γ corresponding to the mean γ for the worst performing coupled shear wall building (considering all periods) as a function of α for different Rα=0.  Figure 7.7: γ corresponding to the mean γ for the worst performing coupled shear wall building (considering all Rα=0) as a function of α for different periods.    213  When considering the inelastic amplification factor, γ, presented above, it is clear that this factor is affected in a manner similar to β by increased α values – greater α correspond to larger γ, for both the cantilevered and coupled shear wall buildings. Furthermore, γ is moderate for both cantilevered and coupled buildings with low α; however, γ increase dramatically for coupled shear wall buildings when α increases passed ~0.2.  For both cantilevered and coupled shear wall buildings, γ shows clear dependence upon Rα=0. At almost every α considered, γ corresponding to Rα=0 = 6.0 exceed γ corresponding to Rα=0 = 4.0, which in turn exceed γ corresponding to Rα=0 = 2.0.  At first consideration, it would appear that building period influenced the γ produced in the cantilevered shear wall study. Upon reflection however, this apparent relationship is explained by the different achievable Rα=0 at each cantilevered building height (period) - recall that the flexural strength of these cantilevered systems were governed by axial load at some heights and that greater Rα=0 values were achievable in the 5 and 10 storey buildings than at greater heights (periods). Therefore, the apparent relationship with building period shown in Figure 7.5 is actually a result of the relationship with Rα=0.  For coupled shear wall buildings, the effect of building period (height) on γ (Figure 7.7) is inconsistent and the result of two counteracting influences. First, negating the influence of PLDs (α = 0), γ is typically larger for shorter buildings with smaller periods. However, increasing α promotes an opposing trend - that is: taller coupled shear wall buildings with greater periods exhibit larger β. Together, these two conflicting behaviours result in the lack of a clear relationship between building period and γ shown for coupled shear walls.  Based on these observations, it is appropriate to apply a simple prescriptive inelastic amplification factor γ, which can be used to estimate the amplified inelastic demands which result from PLDs within the allowable α range proposed. The factor, presented in Equation 32 and Figure 7.8, amplifies elastic deformation demands to provide approximations of inelastic deformation demands for cases within the α limits proposed for both cantilevered and coupled shear wall buildings.   ߛ = 1.5 (32)    214   Figure 7.8: Prescriptive γ proposed for design as well as γ corresponding to the mean γ for the worst performing cantilevered and coupled shear wall building (considering all Rα=0 and periods). Using these limits of α, and proposed inelastic amplification factor, γ, the red underlined additions shown in Figure 7.9 are proposed for Table 4.1.8.6 (Structural Irregularities) given in in the NBCC. Additions to the associated Section 4.1.8.10 (Additional System Restrictions) are given in Figure 7.10.    215  Table 4.1.8.6. Structural Irregularities(6) Forming Part of Sentence 4.1.8.6.(1)  Type Irregularity Type and Definition Notes 1 Vertical Stiffness Irregularity Vertical stiffness irregularity shall be considered to exist when the lateral stiffness of the SFRS in a storey is less than 70% of the stiffness of any adjacent storey, or less than 80% of the average stiffness of the three storeys above or below. (1) (3) (7) 2 Weight (mass) Irregularity Weight irregularity shall be considered to exist where the weight, Wi, of any storey is more than 150 percent of the weight of an adjacent storey. A roof that is lighter than the floor below need not be considered. (1) 3 Vertical Geometric Irregularity Vertical geometric irregularity shall be considered to exist where the horizontal dimension of the SFRS in any storey is more than 130 percent of that in an adjacent storey. (1) (2) (3) (7) 4 In-plane Discontinuity in vertical lateral force-resisting element An in-plane offset of a lateral load-resisting element of the SFRS or a reduction in lateral stiffness of the resisting element in the storey below. (1) (2) (3) (7) 5 Out-of-Plane Offsets Discontinuities in a lateral force path, such as out-of-plane offsets of the vertical elements of the SFRS. (1) (2) (3) (7) 6 Discontinuity in Capacity - Weak Storey A weak storey is one in which the storey shear strength is less than that in the storey above. The storey shear strength is the total strength of all seismic-resisting elements of the SFRS sharing the storey shear for the direction under consideration. (3) 7 Torsional Sensitivity- to be considered when diaphragms are not flexible. Torsional sensitivity shall be considered to exist when the ratio B calculated according to Sentence 4.1.8.11(9) exceeds 1.7. (1) (3) (4) (7) 8 Non-orthogonal Systems A “Non-orthogonal System” irregularity shall be considered to exist when the SFRS is not oriented along a set of orthogonal axes. (5) (7) 9 Permanent Lateral Demand on SFRS Excessive permanent lateral demand on the SFRS shall be considered to exist where the ratio α calculated according to Sentence 4.1.8.10(4) exceeds 0.1. It is permitted to increase the limit on α to 0.4 for cantilevered shear walls. (3) (7) Notes: To Table 4.1.8.6.A - Requirements for Irregular Structures (1) See Article 4.1.8.7. (2) See Article 4.1.8.15. (3) See Article 4.1.8.10. (4) See Sentences 4.1.8.11.(9), (10), and 4.1.8.12.(4) (5) See Article 4.1.8.8 (6) One storey penthouses with a weight less than 10% of the level below need not be considered in the application of this table. (7) See Appendix A   Figure 7.9: Proposed additions to Table 4.1.86 in the NBCC to account for the new structural irregularity of PLFs on the SFRS.    216  Shall  4.1.8.10. Additional System Restrictions 1) Except as required in Clause (2)(b), structures with a Discontinuity in Capacity - Weak Storey as described in Table 4.1.8.6. - Type 6, are not permitted unless IEFaSa(0.2) is less than 0.2 and the forces used for design of the SFRS are multiplied by  Rd Ro. 2) Post-disaster buildings shall: a) Not have any irregularities conforming to Table 4.1.8.6. - Type 1, 3, 4, 5, 7, and 9 for cases where IEFaSa(0.2) is equal to or greater than 0.35, b) Not have an irregularity conforming to Table 4.1.8.6. - Type 6 - Discontinuity in Capacity - Weak Storey, and c) Have an SFRS with a Rd of 2.0 or greater. 3) For buildings having fundamental lateral periods, Ta, of 1.0 sec or greater and where IEFvSa(1.0) is greater than 0.25, walls forming part of the SFRS shall be continuous from their top to the foundation and shall not have irregularities of Type 4 or 5 as defined in Table 4.1.8.6. 4) Buildings with Type 9 irregularity as defined in Table 4.1.8.6. shall be permitted where IEFaSa(0.2) is less than 0.5, and shall be exempt from the following considerations. The ratio  for Type 9 irregularity as defined in Table 4.1.8.6 shall be determined according to the following equation for each orthogonal direction, determined independently:  = PLD / Qy where:  PLD = permanent lateral demand on the SFRS at the base of the yielding system (Appendix A)  Qy = expected yield strength of SFRS (Appendix A) The expected yield strength of the SFRS may be taken as equal to Ro multiplied by the minimum lateral earthquake force as determined in section 4.1.8.11 or 4.1.8.12, as appropriate. Buildings with excessive permanent lateral demands as defined in Table 4.1.8.6 - Type 9 irregularity are not permitted unless determined to be acceptable based on nonlinear dynamic analysis studies (Appendix A). For buildings with permanent lateral demands not exceeding the limits on  in Table 4.1.8.6. – Type 9 irregularity, design displacements as determined in 4.1.8.13 shall be amplified by 1.5.  Figure 7.10: Proposed additions to Section 4.1.8.10 in the NBCC to account for the new structural irregularity of PLFs on the SFRS. If a building design results in a PLD exceeding the allowable α values, standard elastic design procedures are expected to significantly underestimate the displacement demands. In such cases the observed demands amplifications were both large and variable; therefore, nonlinear dynamic analyses, with careful attention to the hysteretic properties of the yielding components, are required to determine if the design is prone to a ratcheting behavior. Due to the critical importance of hysteretic behaviour on seismic ratcheting, validation of the inelastic models used for yielding components with experimental tests is highly encouraged. Acceptance criteria such as those presented in FEMA (2009) may be used to evaluate the performance of the building using nonlinear dynamic analysis.  In order to counteract PLDs created by inclined columns, eccentric floor spans, or other gravity system irregularities, engineers may attempt to create restoring lateral demands. This technique is appropriate if lateral demands are resolved within the gravity system at each floor level – such as inclining interior   217  columns in a direction opposite to inclined exterior columns. For such cases, the SFRS is not subjected to PLDs and the ratcheting phenomenon investigated herein is not anticipated. In some buildings, it may be possible to resolve the PLDs within the SFRS – by employing techniques such as post-tensioning within a shear wall. Considering the sensitivity of the results in this study to small changes in the hysteretic behaviour of the SFRS, and the highly nonlinear response of the SFRS during strong ground shaking, any approach which resolves PLDs within the SFRS should be validated using nonlinear dynamic analyses. Such design techniques to address PLDs were outside the scope of this study.  Additional SFRSs, such as moment frames were not considered in this study and steel braced frames were only briefly considered in a case study. The performance of these SFRSs with PLDs needs to be thoroughly investigated and addressed within the NBCC. In particular, SFRSs with fat hysteresis – shown to be extremely prone to ratcheting - such as buckling restrained braces need to be considered. In the absence of such studies and considering the generally fat hysteresis expected for such systems, it would be prudent to impose a conservative allowable value for α, such as that proposed above for coupled shear walls.  7.2 FUTURE STUDY  This study has considered cantilevered and coupled shear wall buildings in significant detail while also touching on steel braced frame buildings. The results of this study emphasize the significance of PLDs on the seismic response of structures and demonstrate that seismic ratcheting is sensitive to building properties and loading conditions. Different properties with suspected influence on behaviour were briefly considered in case studies. Some of these case studies, such as that which considered vertical ground motions, indicate that no further study is warranted. Others, however, identify areas which demand additional attention, such the behaviour of other SFRS, and the effect of subduction ground motions. The following areas, the effect of which requires future consideration, are discussed in sections 7.2.1 through 7.2.8:  Axial Load in Shear Wall  Shear Dominated Walls  Shear Wall Section Geometry  Additional SFRSs  Coupling Beam Strength  Unique Mitigating Measures  Refined Material Models  Seismic Hazard   218  7.2.1 Axial Load in Shear Wall The hysteretic behaviour of each SFRS considered (coupled shear walls, cantilevered shear walls, and steel braced frames) governed the performance of each building type. This study identified the significance of the different hysteretic behaviour of various SFRSs; however, it didn’t consider the range of hysteretic behaviour within each SFRS. Shear wall buildings, in particular, exhibit widely different hysteretic behaviour which is dependent on the axial load in the shear walls. To simplify the study, a single conservative lower bound axial load was considered – resulting in hysteretic behaviour more prone to ratcheting and producing relatively large demand amplifications. However, if a larger axial load was considered, it is likely that the performance of the shear wall buildings would be improved and seismic ratcheting would be inhibited. Although the range of possible hysteretic behaviour was identified and the worst case axial load assumed, the sensitivity of seismic ratcheting to axial load in the shear walls was not extensively explored or quantified. Future studies should further explore the influence of axial load on seismic ratcheting and refine the proposed α limit for different axial loads.  7.2.2 Shear Wall Section Geometry Similar to axial load, only a single concrete shear wall section was analyzed at each building height. The hysteretic behaviour of concrete shear walls is highly dependent upon the shear wall section considered. Wall properties, such as wall length, wall thickness, concrete strength, and steel area result in different hysteretic behaviour. In this study, it was found that seismic ratcheting was very sensitive to the hysteretic behaviour of the SFRS – therefore the performance of shear wall buildings with PLDs is sensitive to core wall geometry – a consideration ignored in this study. In addition to different core wall geometry, additional types of shear walls should be considered – such as planar and T-shaped walls. T-shaped shear walls, in particular, could be especially sensitive to PLDs given that the hysteretic response of these walls is already inherently asymmetric.  7.2.3 Coupling Beam Strength In this study, asymmetrical yield strength was assigned to the shear wall buildings to account for the additional demands from PLDs. In both the cantilevered shear wall and coupled shear wall buildings this additional strength was added at the base of the shear walls through increased reinforcing steel. In coupled shear wall buildings, however, the majority of the overturning moment resistance is provided by coupling beams up the height of the building. In this study, these coupling beams were not assigned asymmetrical strength to account for PLDs – a conservative assumption resulting in a decrease in the   219  effective coupling ratio and increased yielding of the coupling beams. Coupling beams exhibited fat hysteresis that is particularly susceptible to seismic ratcheting and was found to be the driving factor behind the relatively poor performance of coupled shear wall buildings. It is probable that if the strength of the coupling beams was adjusted to account for PLDs, the building behaviour would be improved– as identified in the case study which considered different coupling ratios. Providing asymmetrical yield strengths in the coupling beams would likely result in even greater improvements to performance than were shown in this case study which simply increased the coupling ratio. In the case study, increasing the strength in the forward direction shifted inelasticity away from the coupling beams, and to the base of the shear walls – an effect which mitigated ratcheting. However, by maintaining a lower strength in the reverse direction, it is likely that coupling beams would be more prone to yielding in reverse and as a result, better able to recover their inelastic deformations and return to their initial state. The effect of asymmetrical coupling beam strength, as well as different coupling ratios should be investigated in future studies.  7.2.4 Refined Material Models This study used fibre sections to model axial-flexural interaction along the height of the plastic hinge (section 4.9.3 and 4.10.6). Using this fibre section model, each shear wall was discretized into three material models: unconfined concrete, confined concrete, and longitudinal reinforcing steel. Both concrete models exhibited strength degradation from deformation demands; however neither model considered cyclic strength degradation – an effect which could have significant impacts on the performance of buildings with lower strengths. Steel fibres were modeled with a Clough hysteretic model (bilinear backbone) which negated both strain hardening and bar fracture. Future consideration of mid- and high-rise shear wall buildings should utilize material models which incorporate these effects.  7.2.5 Shear Dominated Walls This study considered two building types in detail: cantilevered shear wall buildings and coupled shear wall buildings. The shear wall models considered were dominated by flexural demands and for simplicity shear deformations were ignored. Low-rise shear wall buildings – not considered in this study – are governed by shear demands which should be included in future work. It is not known how shear- dominated walls and frames will interact with PLDs - shear strength and ductility in concrete structures are relatively poorly understood concepts. Therefore, such shear dominated structures should be within the scope of future work, with special attention given to shear-flexural interaction.   220   7.2.6 Additional SFRSs This study focused extensively on the behaviour of cantilevered and coupled shear wall buildings. Although steel braced frame buildings were also considered, this analysis was cursory and limited to a single case study with significant modeling simplifications. Based on the behaviour observed in this case study – fat hysteresis and drastically increased residual deformation demands as a result of PLDs - this SFRS demands comprehensive attention in future studies.  In addition to steel braced frames, other SFRSs which are governed by shear deformations in the lower stories, such as steel plate shear walls, steel moment frames, and systems which incorporate buckling restrained braces should also be investigated – all of which are typically governed by lateral drift demands.  In eastern Canada – an area where these SFRS are commonly used - seismic demands are relatively small; therefore, relatively small PLDs can easily produce large α. As a consequence there is potential for buildings to develop α which are larger than the range considered in this study (0 – 0.8) and this should be reflected in the scope of additional research.  7.2.7 Unique Mitigating Measures There are different techniques which engineers may use to counteract PLDs imposed by an irregularity in the gravity system. Such techniques include, but are not limited to, measures such as post-tensioning, asymmetric weight application to concrete shear walls, or opposite inclination of interior columns in order to mitigate demands imposed by exterior inclined columns.  Such mitigating measures were outside of the scope of this study; therefore, a conservative approach is proposed to account for such strategies. It is suggested that measures which resolve PLDs within the gravity system at each floor level (e.g. opposite inclination of interior columns) may be used to allow buildings to satisfy α limits. However, if PLDs are resolved within the SFRS of a building (e.g. through post-tensioning or applying asymmetrical weight to the shear wall) then it is recommended that a nonlinear dynamic analysis be conducted. This conservative approach is proposed for two reasons. First, the restoring force provided by post-tensioning systems could be decreased by damage during inelastic seismic response. Furthermore, the force distributions from a mitigating technique may differ from the distribution of PLDs along the height of the building. In this study, only the magnitude of the PLD at the base of the building was considered in proposed applied demand ratio, α, limits. The effect of large scale   221  variation in the demands along the entire height of the building were outside the scope and may have unanticipated effects on the nonlinear dynamic response.  7.2.8 Seismic Hazard In this study, only the seismic hazard from site class C soil conditions and the Vancouver UHS spectrum were considered. Future studies should consider the sensitivity of seismic ratcheting to a broader range of ground motions scaled to different seismic demands.  Ten crustal ground motions were used for the majority of the analyses conducted. In the future, this work should be expanded to consider additional ground motions. In particular, ground motions from eastern Canada – which tend to have a greater energy in the low frequency range – should be used. Records should be selected which when un-scaled, produce seismic demands comparable to the each considered UHS. More research should be conducted regarding subduction ground motions. Although the case study conducted indicated that the additional risk is moderate, the scope of this case study was limited to a single building configuration and two subduction ground motions.  The main body of this study considered buildings in Vancouver, and the associated seismic demands. A brief case study was conducted which demonstrated that the deformation demand amplifications in Victoria, Calgary, and Montreal were similar to those in Vancouver. Focusing on estimating limits for the NBCC, this study investigated the performance of buildings subjected to 2%/50 year ground motions. Considering the sensitivity of systems with PLDs to collapse, future studies should use Incremental Dynamic Analysis based on a consistent probabilistic framework for all SFRS with and without PLDs (Vamvatsikos & Cornell, 2002). Limits on PLDs should be established based on the probability of collapse determined according to the methodology outline by Federal Emergency Management Agency document P695 (FEMA, 2009).    222  Chapter 8 - Summary and Conclusions  Architectural irregularities in the gravity system of buildings impose PLDs on the SFRS which raised concerns within the SCED. The effect of PLDs resulting from inclined columns on the inelastic seismic performance of cantilevered and coupled shear wall buildings was considered at six heights from 5 to 50 storeys. Using nonlinear dynamic time history analyses, each building was analyzed across a domain of ten crustal ground motions, six PLDs, and three relative strengths. This study demonstrated that PLDs acting on SFRSs can lead to seismic ratcheting behaviour. This seismic ratcheting results in increased inelastic deformation demands and in some cases, collapse.  The extent of seismic ratcheting is highly dependent on the type of SFRS, or more accurately, the hysteretic properties of the SFRS. SFRSs with flag-shaped hysteresis – such as cantilevered shear walls – are moderately sensitive to PLDs. Such SFRSs benefit from the restoring effect of axial load in the shear wall which inhibits seismic ratcheting. As a result, residual deformation demands in cantilevered shear wall buildings are small and such systems are resilient against seismic ratcheting.  In contrast, SFRSs with fat hysteresis – such as coupled shear walls and steel braced frames – are more sensitive to PLDs. The restoring forces in these SFRSs are less dominant than in cantilevered shear walls, and as a result, develop large lateral deformation demands when subjected to PLDs. Such systems tend to produce large residual deformation demands and are prone to instability and collapse.  For the shear wall buildings considered, displacement demand amplifications due to PLDs are not particularly sensitive to the building period or the relative strength of the SFRS. Therefore, design limits to prevent the development of excessive deformation demands need not consider building height or the design ductility, Rd. Deformation demand amplifications can be reliably predicted in each type of SFRS by defining a limit to the applied demand ratio, α, as defined in Equation 33. Beyond this limit, nonlinear dynamic analysis is required to validate performance.   ߙ = ܲܮܦ ܨ௬  (33)  Where PLD is the permanent lateral demand on the SFRS at the base of yielding system and Fy is the yield strength of the SFRS required to resist the earthquake loads with no PLDs. By limiting the applied   223  demand ratio, α, PLDs can be limited to a range in which building performance can be reliably predicted. Coupled shear wall exhibit relatively large deformation demand amplifications and steel braced frame buildings show similar sensitivity to PLDs; therefore, an upper limit of α ≤ 0.1 is proposed for both these SFRSs. Cantilevered shear wall buildings – which demonstrate inherent resistance to seismic ratcheting – may be subjected to larger PLDs corresponding to α ≤ 0.4. This study did not consider other SFRSs such as moment frames and until such research is completed; the more conservative limit of α ≤ 0.1 should probably be used for all SFRSs not considered in this study. There is a gradient of declining performance associated with increasing PLDs; therefore, the exact limits selected for each SFRS are arbitrary and the specific values proposed are subject to consideration by the SCED. Any SFRS with PLDs exceeding the aforementioned limits shall require nonlinear dynamic analyses to validate seismic performance.  As α increases, so do the resulting deformation demand amplifications and the variability of these amplifications between different building configurations and ground motions. The limits placed on α, are, to an extent, discretionary - selected to ensure both moderate and predictable deformation demand amplifications. The proposed limits of α were selected because they limit deformation demand amplifications to a range which is both moderate and had relatively low variability. The worst performing analysis case considered within the allowable α range experienced maximum roof drift demands just 2.2 times greater than the corresponding building without PLDs.  Below the proposed α limits, the variability of the amplifications produced from different ground motions and building periods was relatively consistent. Therefore, the use of a simple inelastic amplification factor, γ = 1.5, is proposed. Such a factor should be used to estimate the inelastic deformation demands in buildings with PLDs below the allowable α limit by amplifying the design deformation demands.  In addition to the main study on cantilevered and coupled shear wall buildings, which considered the influence of building period and relative strength, additional attention was given to the effect of factors such as vertical ground motions and various coupling ratios, by means of case studies. Such case studies were effective at identifying areas which demand greater study. These areas, which should be the focus of future work, include the effect of subduction ground motions – which were shown to promote deformation demand amplifications 30% greater than crustal ground motions. In addition, the sensitivity of deformation demand amplifications to seismic demands should be quantified by incremental dynamic analyses. Additional SFRSs such as moment frames, and shear wall systems dominated by shear behaviour, should be examined and additional attention should be given to steel braced frame models, which were only briefly considered. Finally, the effectiveness and reliability of mitigating measures –   224  such as post-tensioned cables and asymmetric weight application to shear walls – should be investigated before such measures may be allowed to satisfy the limits on α proposed.    225  References  American Society of Civil Engineers, ASCE. (2007). Seismic Rehabilitation of Existing Buildings. ASCE/SEI Standard 41-06.  Applied Technology Council, ATC. (2010). Modeling and acceptance criteria for seismic design and analysis of tall buildings. PEER/ATC 72-1.  Bebamzadeh, A., Ventura, C.E., Pandey, B.H., & Finn, L.W.D. (2012). Design Drift Limits for Performance-based Seismic Assesment and Retrofit Design of School Buildings in British Columbia, Canada. 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Seismic shear demand on wall segments of ductile coupled shear walls. Canadian Journal of Civil Engineering, 27, 3: 506-522.  Elwood, K.J., Matamoros, A.B., Wallace, J.W., Lehman, D.E., Heintz, J.A., Mitchell, A.D., Moore, M.A., Valley, M.T., Lowes, L.N., Comartin, C.D., & Moehle, J.P. (2007). Update to ASCE/SEI 41 concrete provisions. Earthquake Spectra, 23, 3: 493-523.  Federal Emergency Management Agency, FEMA. (2009). Quantification of Building Seismic Performance Factors. National Earthquake Hazards Reduction Program, FEMA P695.  George E. Brown, Jr. Network for Earthquake Engineering Simulation, NEEShub. (2012). National Science Foundation. Retrieved from http://nees.org/home.   226   Izvernari, C. (2007). The Seismic Behaviour of Steel braces with large Sections. PhD thesis, University of Montreal, Montreal, QC.  Izvernari, C., Lacerte, M., & Tremblay, R. (2007). 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Mazzoni, S., McKenna, F., Scott, M.H., & Fenves, G., et al. (2007). OpenSees Command Language Manual. Open System for Earthquake Engineering Simulation (OpenSees). Retrieved from http://opensees.berkeley.edu/wiki/index.php/Command_Manual.  Mitchell, D., Tremblay, R., Karacabeyli, E., Paultre, P., Saatcioglu, M., & Anderson, D. (2003) Seismic force modification factors for the proposed 2005 edition of the National Building Code of Canada. Canadian Journal of Civil Engineering, 30: 308-327.  Moustafa, A., & Takewaki, I. (2010). Deterministic and probabilistic representation of near-field pulse- like ground motion. Soil Dynamics and Earthquake Engineering, 30: 412-422.  National Building Code of Canada, NBCC. (2010). Issued by the Canadian Commission on Building and Fire Codes, National Research Council of Canada.  OpenSees v2.2.0 & v2.3.1. (2012). University of California, Berkeley, California, USA.  Pacific Earthquake Engineering Research Center, PEER. (2010). Tall Buildings initiative: Guidelines for Performance Based Seismic Design of Tall Buildings. Version 1.0, Report No. 2012/05.  Pacific Earthquake Engineering Research Center, PEER. (2012). PEER Strong Motion Database. Retrieved from http://peer.berkeley.edu/peer_ground_motion_database/Strong motion Database.    227  Park, W.S., & Yun, H.D. (2011). Seismic performance of pseudo strain-hardening cementitious composite coupling beams with different reinforcement details. Composites: Part B, 42: 1427- 1445.  Paulay, T., & Priestley, M.J.N. (1992). Seismic design of reinforced concrete and masonry buildings. John Wiley & Sons, Inc., New York, N.Y.  Pina, P.E., Ventura, C.E., Taylor, G., & Finn, L.W.D. (2012). Selection of Ground Motions for the Seismic Risk Assessment of Low-Rise School Buildings in South-Western British Columbia, Canada.  Pique, J. (1976). On the use of simple models in nonlinear dynamic analysis. Massachusetts Institute of Technology, Cambridge, MA, Report R76-43.  Seismosignal v4.1.2. (2012). Seismosoft Ltd, Pavia, Italy.  Tremblay, R., Léger, P., & Tu, J. (2001). Inelastic seismic response of concrete shear walls considering P- delta effects. Canadian Journal of Civil Engineering, 28: 640-655.  Vamvatsikos, D., Cornell, A. (2002). Incremental dynamic analysis. Earthquake Engineering and Structural Dynamics, 31: 491-514.  White, T. (2004). Seismic Demand in High-Rise Concrete Walls. PhD thesis, University of British Columbia, Vancouver, BC.  Yathon, J. (2010). Concrete shear wall sections. Read Jones Christoffersen, Vancouver, BC.  Zhao, Z.Z., & Kwan, A.K.H. (2003). Nonlinear behaviour of deep reinforced concrete coupling beams. Structural Engineering and mechanics, 15, 2: 181-198.    228  Appendices A.1 GROUND MOTIONS  Figure A.1.1: Ground motion 2, Stn: USGS 1095 Taft Lincoln School (Bebamzadeh et al, 2012; Seismosignal, 2012).    229    Figure A.1.2: Ground motion 16, Stn: 9102 Dayhook TR (Bebamzadeh et al, 2012; Seismosignal, 2012).  Figure A.1.3: Ground motion 23, Stn: ENEL 99999 Calitri (Bebamzadeh et al, 2012; Seismosignal, 2012).   230   Figure A.1.4: Ground motion 26, Stn: 6098 Site 2 (Bebamzadeh et al, 2012; Seismosignal, 2012).  Figure A.1.5: Ground motion 35, Stn: CDMG 57007 Corralitos (Bebamzadeh et al, 2012; Seismosignal, 2012).   231   Figure A.1.6: Ground motion 38, Stn: CDMG 57217 Coyote lake Dam (SW Abut) (Bebamzadeh et al, 2012; Seismosignal, 2012).  Figure A.1.7: Ground motion 59, Stn: CDMG 89156 Petrolia (Bebamzadeh et al, 2012; Seismosignal, 2012).   232   Figure A.1.8: Ground motion 124, Stn: USGS 5108 Santa Susana Ground (Bebamzadeh et al, 2012; Seismosignal, 2012).  Figure A.1.9: Ground motion 133, Stn: CUE 99999 Nishi-Akashi (Bebamzadeh et al, 2012; Seismosignal, 2012).   233   Figure A.1.10: Ground motion 164, Stn: Ichinoseki (IWT010) NS (Bebamzadeh et al, 2012; Seismosignal, 2012).   234  A.2 SCALING OF GROUND MOTIONS FOR MDOF ANALYSES  Sample Calculations for 5 Storey Cantilevered Model  Modal Periods (determined from eigenvalue analysis): ௡ܶ = ⎝ ⎜ ⎛ 0.5000.0790.0280.0140.009⎠⎟ ⎞  Mas Participation (Modal Contribution): ܯܴ௡ = ෍ ௝݉߮௝௡ே ௝ୀଵ ෍ ௝݉൫߮௝௡൯ ଶ = ⎝ ⎜ ⎛ 0.7070.2090.0610.0190.004⎠⎟ ⎞ = 1.0ே ௝ୀଵ ൚  Modal Damping Factors (Rayleigh Damping – same as dynamic):  Ϛ௡ = Ϛ ቈ ߱௜ ௝߱൫߱௜ + ௝߱൯߱௡ + ߱௡൫߱௜ + ௝߱൯቉ Where ߱௜ = 12.6	ܪݖ	(݅ = 1) ௝߱ = 223.2	ܪݖ	(݆ = 3) Ϛ = 0.05   235  Ϛ௡ = ⎝ ⎜ ⎛ 0.050.0240.050.0960.148⎠⎟ ⎞ Spectral Accelerations:  Figure A.2.1: Uniform hazard spectrum for Vancouver, British Columbia (NBCC, 2010). ܣ௡ = ܣ( ௡ܶ) ൬ 0.070.02 + Ϛ௡൰଴.ହ = ⎝ ⎜ ⎛ 0.651.190.950.750.62⎠⎟ ⎞ ݃    236  Base Moment Calculation:  ܯ௕௡ = ܯܴ௡ܣ௡෍ ௝݉߮௝௡ℎ௝ே ௝ୀଵ   Modal Combination (SRSS):  ܯ௕ೄೃೄೄ = ට෍(ܯ௕௡)ଶ = 162	307	݇ܰ݉  Ground Motion Scaling Factor Calculation (shown for ground motion 2):  ݈ܵܿܽ݅݊݃	ܨܽܿݐ݋ݎ = 	 ܯ௕ೄೃೄೄ ܯ௕ಶ೗ೌೞ೟೔೎ = 162	307	݇ܰ݉8	725	݇ܰ݉ = 18.6  Where Mb Elastic was determined from linear dynamic analysis    237  Table A.2.1: Ground motion scaling factors for cantilevered shear wall buildings.  Number of Storeys Ground Motion 5 10 20 30 40 50 2 18.57 20.41 19.21 17.83 19.12 23.37 16 8.56 9.48 8.70 11.82 12.98 16.73 23 14.86 8.84 13.99 16.02 24.33 23.15 26 8.05 11.09 14.76 13.17 13.64 19.78 35 4.55 9.68 6.88 10.34 9.71 11.31 38 6.60 9.36 8.23 12.42 6.38 11.94 59 6.21 6.02 11.14 6.92 5.34 5.76 124 16.44 12.89 15.44 33.21 28.36 35.14 133 3.17 11.12 8.45 4.86 8.82 13.80 164 14.98 13.75 9.60 11.76 15.87 25.72     238  Table A.2.2: Ground motion scaling factors for coupled shear wall buildings.  Number of Storeys Ground Motion 5 10 20 30 40 50 2 18.32 21.67 15.39 24.00 20.34 31.13 16 8.78 9.29 9.04 12.57 14.07 19.32 23 14.88 8.61 14.14 14.53 24.64 24.83 26 8.28 11.85 11.89 17.54 19.43 33.67 35 4.44 8.35 9.60 11.45 10.13 20.38 38 6.56 8.93 9.87 9.43 12.06 14.21 59 6.71 5.62 13.70 6.73 6.35 9.82 124 19.98 13.60 15.49 38.62 50.23 28.12 133 3.12 11.06 6.86 9.78 11.38 17.38 164 14.98 14.00 13.38 13.92 16.71 33.65     239  A.3 COLUMNS INCLINATIONS  Sample Calculations for Column Configuration  ݏ௖௢௟ = ߙܯ௬ߩ௖௢௟ ௙ܹ௟௢௢௥ ∑ ݕ௝௡௝ୀଵ  Where, inclination, scol, is defined as:  ݏ௖௢௟ = ݐܽ݊ିଵ ൬ℎ݋ݎ݅ݖ݋݊ݐ݈ܽ	݋݂݂ݏ݁ݐℎ݁݅݃ℎݐ	݋݂	ܿ݋݈ݑ݉݊൰  ߩ௖௢௟ = ܣ௦௟௢௣௘ௗܣ௧௢௧௔௟ = 1.0  ݕ௜ = 4.5݉ + (݊ − 1)(3.5݉)    240  Table A.3.1: Column inclination in the 5 storey cantilevered shear wall models. Applied Demand Ratio, α Relative Strength Factor, Rα=0 1.0 2.0 4.0 6.0 0.0 0° 0° 0° 0° 0.05 1.8° 0.9° 0.4° 0.3° 0.1 3.5° 1.8° 0.9° 0.6° 0.2 7.1° 3.5° 1.8° 1.2° 0.4 14.2° 7.1° 3.5° 2.4° 0.6 21.3° 10.6° 5.3° 3.5° 0.8 28.4° 14.2° 7.1° 4.7°  Table A.3.2: Column inclination in the 10 storey cantilevered shear wall models. Applied Demand Ratio, α Relative Strength Factor, Rα=0 1.0° 2.0 4.0 6.0 0.0 0° 0° 0° 0° 0.05 1.1° 0.5° 0.3° 0.2° 0.1 2.2° 1.1° 0.5° 0.4° 0.2 4.3° 2.2° 1.1° 0.7° 0.4 8.7° 4.3° 2.2° 1.4° 0.6 13.0° 6.5° 3.2° 2.2° 0.8 17.3° 8.7° 4.3° 2.9°     241  Table A.3.3: Column inclination in the 20 storey cantilevered shear wall models. Applied Demand Ratio, α Relative Strength Factor, Rα=0 1.0 2.0 4.0 6.0 0.0 0° 0° 0° 0° 0.05 0.8° 0.4° 0.2° 0.1° 0.1 1.6° 0.8° 0.4° 0.3° 0.2 3.2° 1.6° 0.8° 0.5° 0.4 6.3° 3.2° 1.6° 1.1° 0.6 9.5° 4.8° 2.4° 1.6° 0.8 12.7° 6.3° 3.2° 2.1°  Table A.3.4: Column inclination in the 30 storey cantilevered shear wall models. Applied Demand Ratio, α Relative Strength Factor, Rα=0 1.0 2.0 4.0 6.0 0.0 0° 0° 0° 0° 0.05 0.9° 0.4° 0.2° 0.1° 0.1 1.7° 0.9° 0.4° 0.3° 0.2 3.5° 1.7° 0.9° 0.6° 0.4 7.0° 3.5° 1.7° 1.2° 0.6 10.5° 5.2° 2.6° 1.7° 0.8 14.0° 7.0° 3.5° 2.3°     242  Table A.3.5: Column inclination in the 40 storey cantilevered shear wall models. Applied Demand Ratio, α Relative Strength Factor, Rα=0 1.0 2.0 4.0 6.0 0.0 0° 0° 0° 0° 0.05 0.8° 0.4° 0.2° 0.1° 0.1 1.6° 0.8° 0.4° 0.3° 0.2 3.2° 1.6° 0.8° 0.5° 0.4 6.4° 3.2° 1.6° 1.1° 0.6 9.6° 4.8° 2.4° 1.6° 0.8 12.8° 6.4° 3.2° 2.1°  Table A.3.6: Column inclination in the 50 storey cantilevered shear wall models. Applied Demand Ratio, α Relative Strength Factor, Rα=0 1.0 2.0 4.0 6.0 0.0 0° 0° 0° 0° 0.05 0.7° 0.3° 0.2° 0.1° 0.1 1.4° 0.7° 0.3° 0.2° 0.2 2.7° 1.4° 0.7° 0.5° 0.4 5.4° 2.7° 1.4° 0.9° 0.6 8.1° 4.1° 2.0° 1.4° 0.8 10.8° 5.4° 2.7° 1.8°     243  Table A.3.7: Column inclination in the 5 storey coupled shear wall models. Applied Demand Ratio, α Relative Strength Factor, Rα=0 1.0 2.0 4.0 6.0 0.0 0° 0° 0° 0° 0.05 1.8° 0.9° 0.5° 0.3° 0.1 3.6° 1.8° 0.9° 0.6° 0.2 7.3° 3.6° 1.8° 1.2° 0.4 14.5° 7.3° 3.6° 2.4° 0.6 21.8° 10.9° 5.4° 3.6° 0.8 29.0° 14.5° 7.3° 4.8°  Table A.3.8: Column inclination in the 10 storey coupled shear wall models. Applied Demand Ratio, α Relative Strength Factor, Rα=0 1.0 2.0 4.0 6.0 0.0 0° 0° 0° 0° 0.05 1.0° 0.5° 0.2° 0.2° 0.1 2.0° 1.0° 0.5° 0.3° 0.2 4.0° 2.0° 1.0° 0.7° 0.4 7.9° 4.0° 2.0° 1.3° 0.6 11.9° 5.9° 3.0° 2.0° 0.8 15.8° 7.9° 4.0° 2.6°     244  Table A.3.9: Column inclination in the 20 storey coupled shear wall models. Applied Demand Ratio, α Relative Strength Factor, Rα=0 1.0 2.0 4.0 6.0 0.0 0° 0° 0° 0° 0.05 0.6° 0.3° 0.1° 0.1° 0.1 1.1° 0.6° 0.3° 0.2° 0.2 2.2° 1.1° 0.6° 0.4° 0.4 4.4° 2.2° 1.1° 0.7° 0.6 6.6° 3.3° 1.7° 1.1° 0.8 8.8° 4.4° 2.2° 1.5°  Table A.3.10: Column inclination in the 30 storey coupled shear wall models. Applied Demand Ratio, α Relative Strength Factor, Rα=0 1.0 2.0 4.0 6.0 0.0 0° 0° 0° 0° 0.05 0.5° 0.2° 0.1° 0.1° 0.1 0.9° 0.5° 0.2° 0.2° 0.2 1.9° 0.9° 0.5° 0.3° 0.4 3.7° 1.9° 0.9° 0.6° 0.6 5.6° 2.8° 1.4° 0.9° 0.8 7.5° 3.7° 1.9° 1.2°     245  Table A.3.11: Column inclination in the 40 storey coupled shear wall models. Applied Demand Ratio, α Relative Strength Factor, Rα=0 1.0 2.0 4.0 6.0 0.0 0° 0° 0° 0° 0.05 0.3° 0.2° 0.1° 0.1° 0.1 0.7° 0.3° 0.2° 0.1° 0.2 1.4° 0.7° 0.3° 0.2° 0.4 2.7° 1.4° 0.7° 0.5° 0.6 4.1° 2.0° 1.0° 0.7° 0.8 5.4° 2.7° 1.4° 0.9°  Table A.3.12: Column inclination in the 50 storey coupled shear wall models. Applied Demand Ratio, α Relative Strength Factor, Rα=0 1.0 2.0 4.0 6.0 0.0 0° 0° 0° 0° 0.05 0.3° 0.2° 0.1° 0.1° 0.1 0.7° 0.3° 0.2° 0.1° 0.2 1.3° 0.7° 0.3° 0.2° 0.4 2.7° 1.3° 0.7° 0.4° 0.6 4.0° 2.0° 1.0° 0.7° 0.8 5.3° 2.7° 1.3° 0.9°     246   Table A.3.13: Column inclination in the 8 storey steel braced frame models. Applied Demand Ratio, α Column Inclination 0.0 0° 0.05 0.2° 0.1 0.4° 0.2 0.8° 0.4 1.4° 0.6 1.8° 0.8 2.1°  Table A.3.14: Column inclination in the 12 storey steel braced frame models. Applied Demand Ratio, α Column Inclination 0.0 0° 0.05 0.3° 0.1 0.5° 0.2 0.9° 0.4 1.5° 0.6 2.0° 0.8 2.4°    247  Table A.3.15: Column inclination in the 16 storey steel braced frame models. Applied Demand Ratio, α Column Inclination 0.0 0° 0.05 0.4° 0.1 0.7° 0.2 1.2° 0.4 2.1° 0.6 2.8° 0.8 3.3°     248  A.4 CANTILEVERED SHEAR WALL SECTIONS  Figure A.4.1: Concrete core section for 5 storey cantilevered shear wall model, units in mm (Yathon, 2010).  Figure A.4.2: Concrete core section for 10 storey cantilevered shear wall model, units in mm (Yathon, 2010).    249   Figure A.4.3: Concrete core section for 20 storey cantilevered shear wall model, units in mm (Yathon, 2010).  Figure A.4.4: Concrete core section for 30 storey cantilevered shear wall model, units in mm (Yathon, 2010).   250    Figure A.4.5: Concrete core section for 40 storey cantilevered shear wall model, units in mm (Yathon, 2010).   251   Figure A.4.6: Concrete core section for 50 storey cantilevered shear wall model, units in mm (Yathon, 2010).    252  A.5 CANTILEVERED SHEAR WALL BUILDING DATA  Table A.5.1: Additional cantilevered shear wall data. Number of Storeys, N f’c (MPa) Ec (MPa) Area of Wall (m2) Height of Wall (m) Hinge Length (m) 5 25 22500 7.9 18.5 2.3 10 30 24648 10.0 36 2.75 20 35 26622 20.2 71 3.75 30 40 28460 30.7 106 4.5 40 45 30187 45.0 141 4.5 50 50 31820 62.5 176 4.5  Table A.5.2: Section properties and floor weights for the cantilevered shear wall models. Number of Storeys, N Moment of Inertia of Wall (m4) Effective Stiffness EcIe=0.8EcIg (Nm2) 5 16.0 2.88E+11 10 39.4 7.77E+11 20 126.1 2.69E+12 30 261.4 5.95E+12 40 545.8 1.32E+13 50 1189.5 3.03E+13     253  Table A.5.3: Building weights for the cantilevered shear wall models. Number of Storeys, N Weight of Wall Per Storey (kN) Floor Weight Per Storey (kN) Total Weight Per Typical Storey (kN) Total Building Weight (kN) 5 665 4182 4847 23 999 10 836 3432 4268 42 390 20 1693 2591 4284 85 088 30 2580 1856 4436 132 189 40 3781 1862 5643 224 375 50 5248 3109 8357 416 014  Table A.5.4: Base overturning moments for the cantilevered wall models. Number of Storeys, N Overturning Moment (kNm) 5 162 307 10 270 797 20 549 980 30 957 037 40 1 541 424 50 3 348 235     254  Table A.5.5: Overturning moment yield strength of the cantilevered shear wall buildings. Number of Storeys, N Required Strength for each Rα=0 (kNm) Minimum Strength Limit (kNm) 1.0 2.0 4.0 6.0 5 162 307 81 153 40 577 27 051A 27 000 10 270 797 135 398 67 699 45 133A 53 000 20 549 980 274 990 137 495A 91 663A 200 000 30 957 038 478 519 239 260A 159 506A  400 000 40 1 541 424 770 712 385 356A 256 904A 700 000 50 3 348 234 1 674 117 837 058A 558 039A 1 650 00 ATarget strengths below the minimum achievable strengths for each wall due to the axial load and minimum steel requirements are struck out.    255  A.6 COUPLED SHEAR WALL SECTIONS   Figure A.6.1: Concrete core section for 5 storey coupled shear wall model, units in mm (Yathon, 2010).  Figure A.6.2: Concrete core section for 10 storey coupled shear wall model, units in mm (Yathon, 2010).   256   Figure A.6.3: Concrete core section for 20 storey coupled shear wall model, units in mm (Yathon, 2010).  Figure A.6.4: Concrete core section for 30 storey coupled shear wall model, units in mm (Yathon, 2010).   257   Figure A.6.5: Concrete core section for 40 storey coupled shear wall model, units in mm (Yathon, 2010).  Figure A.6.6: Concrete core section for 50 storey coupled shear wall model, units in mm (Yathon, 2010).   258  A.7 COUPLED SHEAR WALL BUILDING DATA  Table A.7.1: Additional coupled shear wall data. Number of Storeys, N f’c (MPa) Ec (MPa) Area of Wall (m2) Height of Wall (m) Hinge Length (m) 5 25 22500 7.9 18.5 1.15 10 30 24648 10.0 36 1.5 20 35 26622 20.2 71 2.0 30 40 28460 30.7 106 2.25 40 45 30187 45.0 141 2.875 50 50 31820 62.5 176 3.375  Table A.7.2: Calculated section properties and floor weights for the coupled shear wall models. Number of Storeys, N Moment of Inertia of Each Coupled Wall (m4) Effective Stiffness EcIe=0.8EcIg (Nm2) 5 1.3 2.34E+10 10 3.7 7.30E+10 20 11.4 2.43E+11 30 20.8 4.74E+11 40 49.7 1.20E+12 50 90.5 2.30E+12     259  Table A.7.3: Building weights for the coupled shear wall models. Number of Storeys, N Weight of Wall Per Storey (kN) Floor Weight Per Storey (kN) Total Weight Per Storey (kN) Total Building Weight (kN) 5 665 6075 6740 33 461 10 836 8077 8913 88 835 20 1693 8870 10 563 210 662 30 2580 8142 10 722 320 762 40 3781 10 643 14 424 575 583 50 5248 13 445 18 693 932 776  Table A.7.4: Base overturning moments for the coupled wall models. Number of Storeys, N Overturning Moment (kNm) 5 242 300 10 580 021 20 1 302 274 30 2 219801 40 3 666 898 50 7 050 505     260  Table A.7.5: Overturning moment yield strength of the coupled shear wall buildings. Number of Storeys, N Required Strength for each Rα=0 (kNm) 1.0 2.0 4.0 6.0 5 242300 121150 60575 40383 10 580021 290011 145005 96670 20 1302274 651137 325569 217046 30 2219801 1109901 554950 369967 40 3666898 1833449 916725 611150 50 7050505 3525253 1762626 1175084 

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