DESIGN OF GRAVITY–LOAD FRAMES FOR SEISMIC DEFORMATION DEMANDS IN HIGHRISE CONCRETE BUILDINGS WITH SHEAR WALLS IN CANADA by  Guillermo Sainz Albanez  B. Sc., Instituto Tecnológico y de Estudios Superiores de Occidente, 2019 B. Sc., Instituto Tecnológico y de Estudios Superiores de Occidente, 2012  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  January 2024  © Guillermo Sainz Albanez, 2024 ii  The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled: Design of Gravity–Load Frames for Seismic Deformation Demands in Highrise Concrete Buildings with Shear Walls in Canada  submitted by Guillermo Sainz Albanez in partial fulfilment of the requirements for the degree of Master of Applied Science in Civil Engineering  Examining Committee: Perry Erwin Adebar, Professor, Dept. of Civil Engineering, UBC Supervisor  Carlos Estuardo Ventura, Professor, Dept. of Civil Engineering, UBC Supervisory Committee Member   iii  Abstract A building can be divided into two main components: the seismic-force resisting system (SFRS), responsible for withstanding seismic forces, and the remaining structure, often referred to as the gravity-load resisting frame (GLRF). Although the GLRF is not designed to resist seismic forces directly, it must be designed to accommodate the seismic deformation demands of the entire structure. The Canadian building code, specifically CSA A23.3 Clause 21.11, outlines requirements for designing the GLRF. These requirements are relatively new and have not seen widespread adoption in engineering practice. Design engineers have faced challenges in efficiently performing the required calculations. This thesis explores various methods to apply the specified interstorey drift envelope from CSA A23.3 to shear wall buildings' GLRF. It offers practical recommendations for using the computer program ETABS to impose the necessary lateral displacements. The thesis includes a comprehensive building design case study that demonstrates the proposed procedure and examines the design criteria for the GLRF in typical shear wall buildings, exploring various GLRF configurations. Additionally, the 2020 edition of the National Building Code of Canada (NBCC) introduces new requirements for designing the GLRF of buildings exceeding 30 metres in height in seismic category SC4 (regions of high seismicity, such as the lower mainland of BC). A case study is conducted to assess the implications of these new requirements, which have been adopted in BC in March 2024.      iv  Lay Summary In Canada, when designing concrete highrise buildings, structural engineers incorporate and detail structural walls to provide the necessary lateral stiffness and strength for the entire building. Meanwhile, gravity columns and gravity walls located around the core are responsible for supporting vertical gravity loads. However, during seismic events, the entire building sways as a whole. Past earthquakes have demonstrated that the failure of these gravity load resisting columns and walls can lead to the collapse of the entire structure. To address this issue, the Canadian code presents guidelines and requirements so structural engineers may provide satisfactory designs for these columns and walls to prevent these kinds of failures. The contents of this thesis aim to serve as complement to the Canadian code and will assist engineers in determining the demands affecting gravity columns, ensuring adequate capacities and resistances that withstand the forces generated by earthquakes of varying intensities, while they continue resisting their tributary gravity loads.    v  Preface Guillermo Sainz was responsible for conducting all the analyses presented in this document, as well as generating all the analytical models required in ETABS. The design of the gravity columns also fell within his responsibilities. Additionally, Guillermo was responsible for the post-processing of data and interpretation of results.  Prof. Perry Adebar carried out the design of the seismic-force resisting system (SFRS), as presented in the 4th edition of the Cement Association of Canada Concrete Design Handbook. He also dimensioned the thickness of the slabs considered in this study. The equations used to generate the 2D displacement profiles to impose the necessary demands induced in the gravity load resisting frame were also developed by him. Prof. Adebar was deeply involved in the supervision of this project.   vi  Table of Contents  Abstract ......................................................................................................................................... iii Lay Summary ............................................................................................................................... iv Preface .............................................................................................................................................v Table of Contents ......................................................................................................................... vi List of Tables ............................................................................................................................... xii List of Figures ............................................................................................................................. xiv List of Symbols ...........................................................................................................................xxv List of Abbreviations ............................................................................................................... xxix Acknowledgements ....................................................................................................................xxx Dedication ................................................................................................................................. xxxi Chapter 1: Introduction ................................................................................................................1 1.1 Background ................................................................................................................. 1 1.2 Goals and Objectives .................................................................................................. 4 1.3 Methodology ............................................................................................................... 5 1.4 Thesis outline .............................................................................................................. 6 Chapter 2: Archetype Building and Structural Design ..............................................................9 2.1 Building Description ................................................................................................... 9 2.2 Wind Analysis ........................................................................................................... 16 2.3 Seismic Analysis ....................................................................................................... 28 vii  2.4 Design of the SFRS ................................................................................................... 35 Chapter 3: Methodology for Applying CSA A23.3 Clause 21.11 ............................................41 3.1 Introduction ............................................................................................................... 41 3.2 Alternative Procedures Investigated ......................................................................... 41 3.2.1 Series of Pushover Analyses ......................................................................... 41 3.2.2 Imposing Displacements with Infinitely Rigid Slabs ................................... 42 3.2.3 Lateral Loads to Achieve Target Drifts ........................................................ 44 3.2.4 Concentrated Moments to Achieve Target Drifts ......................................... 44 3.3 CSA A23.3 Clause Requirements ............................................................................. 46 3.3.1 Simplified Analysis of Buildings .................................................................. 47 3.3.2 Summary of Requirements ........................................................................... 48 3.3.3 Description of GLRF .................................................................................... 53 3.4 Applying Displacement Profiles in ETABS ............................................................. 54 3.4.1 2D Displacement Profiles ............................................................................. 56 3.4.2 Imposing 2D Displacements in 3D Models .................................................. 60 3.5 Modelling of Slab-Column Frames in ETABS ......................................................... 64 3.5.1 Columns Modelling ...................................................................................... 64 3.5.2 Slabs Modelling (Above Grade) ................................................................... 64 3.5.3 Plastic Hinges for Effective Beam Width ..................................................... 68 3.5.4 Slabs Modelling (Below Grade) and Boundary Conditions ......................... 72 viii  3.6 Effects of the Seismic Induced Deformations on Columns of GLRF ...................... 73 3.6.1 Axial Forces on Columns.............................................................................. 73 3.6.2 Bending Moments on Columns .................................................................... 74 3.6.3 Shear Forces on Columns ............................................................................. 74 Chapter 4: Application of CSA A23.3 Clause 21.11 to Case Studies ......................................76 4.1 Introduction ............................................................................................................... 76 4.2 Displacements at the top of GLRF............................................................................ 76 4.2.1 Displacements in Coupled – wall direction .................................................. 77 4.2.2 Displacements in Cantilever – wall direction ............................................... 78 4.3 Displacement profiles for GLRF .............................................................................. 79 4.3.1 Displacement profiles for Coupled – wall direction ..................................... 79 4.3.2 Displacement profiles for Cantilever – wall direction .................................. 82 4.4 Effective widths and nominal flexural capacities for the slabs................................. 84 4.5 Flexural analysis and design of Columns – Uniform GLRF .................................... 86 4.5.1 Corner columns for Coupled – wall direction............................................... 86 4.5.2 Interior columns Coupled – wall direction (linear analysis) ......................... 88 4.5.3 Interior columns Coupled – wall direction (nonlinear analysis) ................... 89 4.5.4 Corner columns for Cantilever – wall direction ........................................... 93 4.5.5 Interior columns Cantilever – wall direction (linear analysis)...................... 95 4.5.6 Interior columns Cantilever – wall direction (nonlinear analysis) ................ 96 ix  4.6 Shear analysis and design of Columns – Uniform GLRF......................................... 99 4.6.1 Corner column for Coupled – wall direction .............................................. 100 4.6.2 Interior column Coupled – wall direction (nonlinear analysis) .................. 101 4.6.3 Corner column for Cantilever – wall direction ........................................... 103 4.6.4 Interior column Cantilever – wall direction (nonlinear analysis) ............... 104 4.7 Summary of demands of Columns - Uniform GLRF ............................................. 106 4.8 Ductile redesign of Columns – Uniform GLRF ..................................................... 107 4.9 Flexural analysis and design of Columns – GLRF with Transfer slab ................... 110 4.9.1 Interior column Coupled – wall direction (nonlinear analysis) .................. 111 4.9.2 Interior column Cantilever – wall direction (nonlinear analysis) ............... 115 4.10 Shear analysis and design of Columns – GLRF with Transfer slab ....................... 119 4.10.1 Interior column Coupled – wall direction (nonlinear analysis) .................. 119 4.10.2 Interior column Cantilever – wall direction (nonlinear analysis) ............... 121 4.11 Summary of demands of Columns – GLRF with Transfer slab ............................. 123 4.12 Ductile redesign of Columns – GLRF with Transfer slab ...................................... 124 Chapter 5: Applying NBCC 4.1.8.23 to Case Study Building ................................................128 5.1 Introduction ............................................................................................................. 128 5.2 Ground Motions with 2,475-Year and 475-Year Return Period............................. 128 5.3 475-Year Analytical Model .................................................................................... 130 5.3.1 Initial assumptions for effective stiffnesses ................................................ 132 x  5.3.2 Refined effective stiffness for structural walls ........................................... 132 5.3.3 Refined effective stiffness for gravity columns .......................................... 137 5.4 2,475-Year and 475-Year Responses ...................................................................... 138 5.4.1 Effective Stiffness ....................................................................................... 139 5.4.2 Natural periods and mode shapes ............................................................... 139 5.4.3 Displacements ............................................................................................. 139 5.5 Influence of Uncracked Rigid Core ........................................................................ 143 5.5.1 Uncracked structural walls in the Coupled Direction. ................................ 143 5.5.2 Uncracked structural walls in the Cantilever direction. .............................. 145 5.5.3 Lateral demands on Coupled – wall direction ............................................ 150 5.5.4 Lateral demands on Cantilever – wall direction ......................................... 151 5.6 Observations on low demands in Gravity Columns ............................................... 152 Chapter 6: Final Discussion and Conclusions .........................................................................154 6.1 Background ............................................................................................................. 154 6.2 Design of Prototype Building ................................................................................. 155 6.3 Implementation of Simplified Analysis Procedure in CSA A23.3 Clause 21.11 ... 156 6.4 Observations on Different Configurations of GLRF of the Archetype Building ... 158 6.4.1 Uniform GLRF............................................................................................ 158 6.4.2 GLRF with Thick Transfer Slab ................................................................. 160 6.5 Pilot Study on Additional Performance Requirements of NBCC Article 4.1.8.23 . 162 xi  6.6 Limitations and Recommendations for Future Work ............................................. 164 References ...................................................................................................................................167 Appendices ..................................................................................................................................168 Appendix A Gravity Design ............................................................................................... 168 A.1 Gravity Loads.............................................................................................. 168 A.2 Slabs ............................................................................................................ 170 A.3 Columns ...................................................................................................... 171 Appendix B Seismic Analysis ............................................................................................ 176 B.1 Site Properties ............................................................................................. 176 B.2 Importance Factor and Seismic Category ................................................... 178 B.3 Structural Configuration and Irregularities ................................................. 179 B.4 Methods of Analysis and Direction of Loading .......................................... 179 B.5 SFRS Force Modification Factors .............................................................. 180 B.6 Torsional Sensitivity ................................................................................... 180 B.7 Dynamic Procedure ..................................................................................... 184 Appendix C Wind Analysis ................................................................................................ 194 C.1 Dynamic Procedure ..................................................................................... 194 C.2 Gust Factors ................................................................................................ 198 C.3 Wind Loading ............................................................................................. 201  xii  List of Tables  Table 2.1 Coupling beam depths. ................................................................................................. 14 Table 2.2 Distribution of concrete compression strengths for the SFRS. ..................................... 14 Table 2.3 Natural periods and mode shapes for wind analysis. .................................................... 17 Table 2.4 Natural periods and mode shapes for seismic analysis. ................................................ 30 Table 2.5 Summary of Seismic Demands according to NBCC 2020. .......................................... 31 Table 3.1 Expressions for recommended displacement profiles resulting in interstorey drift profiles defined in CSA A23.3 Clause 21.11. ............................................................................................ 57 Table 4.1 Nominal bending moment capacities in slabs for Coupled direction analysis. ............ 84 Table 4.2 Nominal bending moment capacities in slabs for Cantilever direction analysis. ......... 85 Table 4.3 Summary of demand-to-capacity ratios for columns part of Uniform GLRF. ........... 106 Table 4.4 Summary of demand-to-capacity ratios for columns part of GLRF with Transfer slab...................................................................................................................................................... 123 Table 5.1 Effective stiffnesses for different models. .................................................................. 139 Table 5.2 Natural periods and mode shapes for different models. ............................................. 139 Table 5.3 Spectral displacements for different models. .............................................................. 140 Table 5.4 Maximum displacements resulting from dynamic linear analysis. ............................. 140 Table 5.5 Comparative on natural periods and mode shapes on different 475-year models. ..... 148 Table 5.6 Summary of factors contributing to low demands on gravity columns. ..................... 153 Table A.1 Superimposed Dead Loads for Gravity Design. ........................................................ 168 Table A.2 Live Loads for Gravity Design. ................................................................................. 169 Table B.1 Equivalent lateral loads – Coupled-wall Direction. ................................................... 182 xiii  Table B.2 Equivalent lateral loads – Cantilever-wall Direction. ................................................ 183 Table B.3 Natural periods and mode shapes for seismic analysis based on SFRS alone. .......... 185 Table B.4 Natural periods and mode shapes for seismic analysis based on SFRS and GLRF. .. 186 Table B.5 Summary of Seismic Demands according to NBCC 2020. ....................................... 187 Table C.1 Natural periods and mode shapes for wind analysis. ................................................. 195 Table C.2 Wind loading along the coupled-wall direction. ........................................................ 202 Table C.3 Wind loading along the cantilever-wall direction. ..................................................... 203  xiv  List of Figures  Figure 2.1 Plan view of levels below grade. ................................................................................. 10 Figure 2.2 Plan view of levels above grade. ................................................................................. 11 Figure 2.3 Elevation of the prototype building on the left-side, and a 3D view of the analytical model in ETABS on the right-side................................................................................................ 12 Figure 2.4 Plan view of the SFRS. ................................................................................................ 13 Figure 2.5 Design of gravity columns for Uniform Gravity-Load Frame. ................................... 15 Figure 2.6 Wind loads resultants acting on a building; values of Cp from NBCC 2020. ............. 19 Figure 2.7 Calculation for the gust effect factor for the windward face in the coupled-wall direction. ....................................................................................................................................... 20 Figure 2.8 Calculation for the gust effect factor for the windward face in the cantilever-wall direction. ....................................................................................................................................... 21 Figure 2.9 Dynamic pressure profiles: windward pressure (blue) and leeward pressure (red); on the left side for the coupled-wall direction, and on the right side for the cantilever-wall direction........................................................................................................................................................ 22 Figure 2.10 Lateral loads resultant from dynamic wind analysis: coupled-wall direction (blue) and cantilever-wall direction (red). ...................................................................................................... 23 Figure 2.11 Shear demands due to wind loading in coupling beams. ........................................... 25 Figure 2.12 Demands acting on walls due to wind loading; Coupled-wall direction. .................. 26 Figure 2.13 Demands acting on walls due to wind loading; Cantilever-wall direction. ............... 27 Figure 2.14 Uniform Hazard Spectrum for Vancouver City Hall corresponding to an event with 2% probability of exceedance in 50 years. ................................................................................... 28 xv  Figure 2.15 Isometric and elevation views of analytical model in ETABS for Seismic Analysis........................................................................................................................................................ 29 Figure 2.16 Shear demands due to seismic loads in coupling beams: no accidental torsion (light blue); with accidental torsion (dark blue). .................................................................................... 32 Figure 2.17 Demands acting on walls due to seismic loading; Coupled-wall direction. .............. 33 Figure 2.18 Demands acting on walls due to seismic loading; Cantilever-wall direction. ........... 34 Figure 2.19 Summarize of shear demands in coupling beams: with accidental torsion due to 1.0 E (dark blue); without accidental torsion due to 1.0 E (light blue); and due to 1.4 W (red dashed-line). .............................................................................................................................................. 36 Figure 2.20 Summarize of demands acting on walls in Coupled-wall direction: due to 1.0 E (blue line); due to 1.4 W (red dashed-line). ........................................................................................... 37 Figure 2.21 Summarize of demands acting on walls in Cantilever-wall direction: due to 1.0 E (blue line); due to 1.4 W (red dashed-line). ........................................................................................... 38 Figure 2.22 Layout of vertical reinforcement for wall piers at grade level. ................................. 40 Figure 3.1 First attempt of imposing displacements in ETABS; On the left-side the deformed shape of the analytical model; on the right-side incorrect bending moments at the columns. ............... 43 Figure 3.2 Trial for modified interstorey drift envelope for cantilever walls, and the corresponding curvature profile and concentrated bending moments required. ................................................... 45 Figure 3.3 Envelope of minimum interstorey drift ratios over building height for Coupled and Cantilever walls from CSA A23.3 ................................................................................................ 49 Figure 3.4 Maximum calculated induced bending moment for column members from CSA A23.3....................................................................................................................................................... 50 xvi  Figure 3.5 Requirements for tied columns meeting dimensional limitations according to Clause 21.11.3.3.3..................................................................................................................................... 51 Figure 3.6 Requirements for moderately ductile columns according to Clause 21.11.3.3.3. ....... 52 Figure 3.7 Requirements for ductile columns according to Clause 21.11.3.3.3. .......................... 53 Figure 3.8 Displacement profile that would result in the envelope of interstorey drift ratios for cantilever walls. ............................................................................................................................ 55 Figure 3.9 Recommended two-part displacement profiles giving interstorey drift profiles defined in CSA A23.3 Clause 21.11 for Coupled – walls. ........................................................................ 58 Figure 3.10 Recommended two-part displacement profiles giving interstorey drift profiles defined in CSA A23.3 Clause 21.11 for Cantilever – walls. ..................................................................... 59 Figure 3.11 Example of three models imposing displacements on a frame on the Coupled – wall direction. ....................................................................................................................................... 61 Figure 3.12 Example of the assignation of a displacement profile assigned along the full height of the building for the Cantilever – wall direction. ........................................................................... 62 Figure 3.13 Isometric views of three different frames with displacement imposed along the full height............................................................................................................................................. 63 Figure 3.14 Modifications allowed to model EBW. Extracted from Appendix C of 2020 LATBSDC Guidelines. ................................................................................................................. 65 Figure 3.15 Limitations on Effective Beam Width; extract from Appendix C of 2020 LATBSDC Guidelines. .................................................................................................................................... 67 Figure 3.16 Example of correct assignation of rigid diaphragm for a storey level above grade. . 68 Figure 3.17 Negative bending moment diagrams due to different load combinations. ................ 70 Figure 3.18 Example of plastic hinge definition to be assigned to an EBW frame in ETABS. ... 71 xvii  Figure 3.19 Analytical model in ETABS with plastic hinges defined and assigned to EBW frames only. .............................................................................................................................................. 72 Figure 3.20 Shear force demand adjusted for column where Mf exceeds Mp. ............................. 75 Figure 4.1 Displacements at the top of GLRF in the Coupled direction. ..................................... 77 Figure 4.2 Displacements at the top of GLRF in the Cantilever direction. .................................. 78 Figure 4.3 Displacement profiles assigned for frame located along axes 4 and 5 in the Coupled direction. ....................................................................................................................................... 80 Figure 4.4 Elevations of analytical models for frame in Coupled direction. ................................ 81 Figure 4.5 Displacement profiles assigned for frame D in the Cantilever direction. ................... 82 Figure 4.6 Elevations of analytical models for frame in Cantilever direction. ............................. 83 Figure 4.7 Comparative of probable bending moment capacities of thick slab and columns. ..... 85 Figure 4.8 SFD and BMD for Columns L4 and D4 for Coupled direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). ............................ 86 Figure 4.9 PM interaction diagram for Columns L4 and D4 for Coupled direction. ................... 87 Figure 4.10 SFD and BMD for Columns L6 and D6 at the top; AFD for Column L6 at the bottom left, AFD for Column D6 at the bottom right for Coupled direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). ....................................... 88 Figure 4.11 Demands for Column L6; SFD and BMD at the top, AFD at the bottom for Coupled direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). .................................................................................................................................. 89 Figure 4.12 PM interaction diagram for Column L6 for Coupled direction. ................................ 90 xviii  Figure 4.13 Demands for Column D6; SFD and BMD at the top, AFD at the bottom for Coupled direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). .................................................................................................................................. 91 Figure 4.14 PM interaction diagram for Column D6 for Coupled direction. ............................... 92 Figure 4.15 SFD and BMD for Columns D12 and D4 for Cantilever direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). ................... 93 Figure 4.16 PM interaction diagram for Columns D12 and D4 for Cantilever direction. ............ 94 Figure 4.17 SFD and BMD for Columns E11 and E5 at the top; AFD for Column E11 at the bottom left, AFD for Column E5 at the bottom right for Cantilever direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). ....................................... 95 Figure 4.18 Demands for Column E11; SFD and BMD at the top, AFD at the bottom for Cantilever direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). .................................................................................................................................. 96 Figure 4.19 PM interaction diagram for Column E11 for Cantilever direction............................ 97 Figure 4.20 Demands for Column E5; SFD and BMD at the top, AFD at the bottom for Cantilever direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). .................................................................................................................................. 98 Figure 4.21 PM interaction diagram for Column E5 for Cantilever direction.............................. 99 Figure 4.22 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Columns L4 and D4 for Coupled direction. ......................................................... 100 Figure 4.23 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Column L6 for Coupled direction. ....................................................................... 101 xix  Figure 4.24 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Column D6 for Coupled direction. ....................................................................... 102 Figure 4.25 PM (probable capacity) interaction diagram on the left-side; shear demand and shear resistance for Columns D12 and D4 for Cantilever direction. .................................................... 103 Figure 4.26 PM (probable capacity) interaction diagram on the left-side; shear demand and shear resistance for Column E11 for Cantilever direction. .................................................................. 104 Figure 4.27 PM (probable capacity) interaction diagram on the left-side; shear demand and shear resistance for Column E5 for Cantilever direction. .................................................................... 105 Figure 4.28 Cross-section of redesigned Column L6 for Coupled direction. ............................. 107 Figure 4.29 PM interaction diagram for redesigned Column L6 for Coupled direction. ........... 108 Figure 4.30 PM (probable capacity) interaction diagram on the left-side; shear demand and shear resistance for redesigned Column L6 for Coupled direction. ..................................................... 109 Figure 4.31 Demands for Column L6 with Transfer slab; SFD and BMD at the top, AFD at the bottom for Coupled direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). ............................................................................................. 111 Figure 4.32 PM interaction diagram for Column L6 with Transfer slab for Coupled direction. 112 Figure 4.33 Demands for Column D6 with Transfer slab; SFD and BMD at the top, AFD at the bottom for Coupled direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). ............................................................................................. 113 Figure 4.34 PM interaction diagram for Column D6 with Transfer slab for Coupled direction. 114 Figure 4.35 Demands for Column E11 with Transfer slab; SFD and BMD at the top, AFD at the bottom for Cantilever direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). ............................................................................................. 115 xx  Figure 4.36 PM interaction diagram for Column E11 with Transfer slab for Cantilever direction...................................................................................................................................................... 116 Figure 4.37 Demands for Column E5 with Transfer slab; SFD and BMD at the top, AFD at the bottom for Cantilever direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). ............................................................................................. 117 Figure 4.38 PM interaction diagram for Column E5 with Transfer slab for Cantilever direction...................................................................................................................................................... 118 Figure 4.39 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Column L6 for Coupled direction. ....................................................................... 119 Figure 4.40 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Column D6 for Coupled direction. ....................................................................... 120 Figure 4.41 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Column E11 for Cantilever direction. .................................................................. 121 Figure 4.42 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Column E5 for Cantilever direction. .................................................................... 122 Figure 4.43 Cross-section of redesigned Column E11 with Transfer slab for Cantilever direction...................................................................................................................................................... 124 Figure 4.44 Elevation of redesigned Column E11 with Transfer slab for Cantilever direction. 125 Figure 4.45 PM interaction diagram for redesigned Column E11 with Transfer slab for Cantilever direction. ..................................................................................................................................... 126 Figure 4.46 PM (probable capacity) interaction diagram on the left-side. Shear demands and shear resistance for redesigned Column E11 with Transfer slab for Cantilever direction. .................. 127 Figure 5.1 Spectral accelerations for seismic events with different return periods. ................... 129 xxi  Figure 5.2 Spectral displacements for seismic events with different return periods. ................. 130 Figure 5.3 3D view and elevations along the coupled and cantilevered walls of analytical model for 475 year-return-period........................................................................................................... 131 Figure 5.4 Response-2000 models for each wall pier in the positive Coupled – wall direction; Wall -1 at the top, Wall -2 at the middle and Wall -3 at the bottom. .................................................. 133 Figure 5.5 Bending moment – curvature responses in the Coupled – wall Direction. ............... 134 Figure 5.6 Response-2000 models for each wall pier in the Cantilever – wall direction; Walls 1 and 2 at the top and Wall -3 at the bottom. ................................................................................. 136 Figure 5.7 Bending moment – curvature response for column D4 at grade level; Coupled-wall Direction. .................................................................................................................................... 137 Figure 5.8 Bending moment – curvature response for column E11 at grade level; Cantilever-wall Direction. .................................................................................................................................... 138 Figure 5.9 Comparative on interstorey drift ratios in the Coupled-wall Direction. .................... 141 Figure 5.10 Comparative on interstorey drift ratios in the Cantilever-wall Direction................ 142 Figure 5.11 Bending moment – curvature response for Wall -1 subjected to an axial compression of 2,586 kN in the Coupled-wall Direction. ............................................................................... 143 Figure 5.12 Bending moment – curvature response for Wall -2 subjected to an axial compression of 19,673 kN in the Coupled-wall Direction. ............................................................................. 144 Figure 5.13 Bending moment – curvature response for Wall -3 subjected to an axial compression of 41,594 kN in the Coupled-wall Direction. ............................................................................. 144 Figure 5.14 Bending moment – curvature response for Wall -1 subjected to an axial compression of 21,142 kN in the Cantilever-wall Direction. .......................................................................... 145 xxii  Figure 5.15 Bending moment – curvature response for Wall -2 subjected to an axial compression of 19,772 kN in the Cantilever-wall Direction. .......................................................................... 146 Figure 5.16 Bending moment – curvature response for Wall -3 subjected to an axial compression of 22,939 kN in the Cantilever-wall Direction. .......................................................................... 146 Figure 5.17 3D views of analytical models in ETABS for 475-year analysis; on the left-side the SFRS + GLRF model; on the right-side the SFRS Only model. ................................................ 147 Figure 5.18 Normalized interstorey drifts ratio due to RSA with equal base shears; on the left-side Coupled-wall Direction, on the right-side Cantilever-wall Direction. ....................................... 149 Figure 5.19 Normalized demands due to RSA with equal base shears.; Coupled-wall Direction...................................................................................................................................................... 150 Figure 5.20 Normalized demands due to RSA with equal base shears; Cantilever-wall Direction...................................................................................................................................................... 151 Figure A.1 Plan view of roof level showing location of Chillers and Mechanical Room. ......... 169 Figure A.2 Critical spans and overhangs at the slabs; Level L12 to Level L30. ........................ 170 Figure A.3 Tributary area of columns above grade level. .......................................................... 171 Figure A.4 Axial force demands and resistance of columns along the entire height of the building: combination 1 (darkest blue line); combination 2 (gray-blue line); combination 3 (light blue line); and maximum axial resistance (red dashed line). ....................................................................... 174 Figure A.5 Summary of columns sections per level of the building. ......................................... 175 Figure B.1 Seismic Hazard Tool showing site designation based on shear wave velocity and location of site. ............................................................................................................................ 177 Figure B.2 Design spectral accelerations provided by the Seismic Hazard Tool. ...................... 177 xxiii  Figure B.3 Uniform Hazard Spectrum for Vancouver City Hall corresponding to an event with 2% probability of exceedance in 50 years. ........................................................................................ 178 Figure B.4 Extract of Table 4.1.8.9 from NBCC 2020. .............................................................. 180 Figure B.5 Reduced spectral acceleration for the Coupled-wall direction (Rd =4.0 and Ro =1.7)..................................................................................................................................................... 188 Figure B.6 Reduced spectral acceleration for the Cantilever-wall direction (Rd =3.5 and Ro =1.6)..................................................................................................................................................... 188 Figure B.7 Shear demands due to wind loading in coupling beams: no accidental torsion (light blue); with accidental torsion (dark blue). .................................................................................. 189 Figure B.8 Demands acting on walls due to seismic loading; coupled-wall direction. .............. 190 Figure B.9 Demands acting on walls due to seismic loading; cantilever-wall direction. ........... 191 Figure B.10 Spectral acceleration for both directions to check for lateral deflections. .............. 192 Figure B.11 Maximum storey drift ratios for seismic analysis: ratios corresponding to the Coupled direction (blue); ratios corresponding to the Cantilever direction (red). .................................... 193 Figure C.1 Full and partial wind loading from NBCC 2020. ..................................................... 195 Figure C.2 Values of Cp for main structural system from NBCC 2020. .................................... 197 Figure C.3 Calculation for the gust effect factor for the windward face in the coupled–wall direction. ..................................................................................................................................... 200 Figure C.4 Calculation for the gust effect factor for the windward face in the cantilever–wall direction. ..................................................................................................................................... 201 Figure C.5 Dynamic pressure profiles: pressure acting on windward face (blue), pressure acting on leeward face (red); on the left side for the Coupled-wall direction, and on the right side for the Cantilever-wall direction. ........................................................................................................... 204 xxiv  Figure C.6 Lateral loads resultant from dynamic analysis: Loads acting on Coupled direction (blue), loads acting on Cantilever direction (red). ...................................................................... 205 Figure C.7 Elevation views of the deformed shape of the model due to wind loading in each direction of analysis. ................................................................................................................... 206 Figure C.8 Shear demands due to wind loading in coupling beams. .......................................... 207 Figure C.9 Demands acting on walls due to wind loading; Coupled-wall direction. ................. 208 Figure C.10 Demands acting on walls due to wind loading; Cantilever-wall direction. ............ 209  xxv  List of Symbols Ag = gross area of the cross-section of the member. Ast = area of reinforcement. Ave = effective area of the cross-section for shear. B = maximum value of Bx. In the context of wind analysis, it refers to background turbulence factor. Bx = ratio at level x used to determine torsional sensitivity. b = effective width of slab intervening in frame action between vertical supports. Ce  = exposure factor Cg  = gust factor Cp  = external pressure coefficient Ct  = topographic factor c1 = dimension of the column or wall parallel to the direction of the analysis supporting effective beam width. D = dead load. Dnx = plan dimension of the building at level x perpendicular to the direction of seismic loading being considered. dv  = effective shear depth. E = earthquake load. Ec = modulus of elasticity of the concrete. Es = modulus of elasticity of the non-prestressed reinforcement. Ft = portion of V to be concentrated at the top of the structure. Fx = lateral force applied to level x. xxvi  f’c = nominal concrete compressive strength.  fy = specified yield strength of non-prestressed reinforcement.  h = overall thickness of the member. hn = height of the structure above grade level in metres. hi, hx = height in metres above the base to level i or x. In the context of wind analysis, it refers to storey height. Ig = gross moment of inertia of the cross-section of the member. Ie = effective moment of inertia of the cross-section of the member. IE = earthquake importance factor of the structure. IW = wind importance factor of the structure. L = live load. l1 = center to center span of the effective beam width parallel to the direction of the analysis. Me = maximum elastic bending moment. Mf = factored bending moment demand. Mr = factored bending moment resistance. Mn = nominal bending moment resistance. Mp = probable bending moment resistance. Mv = factor to account for higher modes effects on base shear. M2 = bending moment acting around local axis 2. M3 = bending moment acting around local axis 3. Pf = factored axial load demand. Pr, max = maximum axial load resistance calculated for columns. Pro = factored axial load resistance at zero eccentricity. xxvii  p  = specified external pressure, considered positive when acting towards the structure, and negative when acting away from the surface. q  = reference velocity pressure. Rd = ductility-related force modification factor. Ro = overstrength-related force modification factor. S = snow load. S(T) = design spectral acceleration expressed as ratio to gravitational acceleration. Ta = fundamental lateral period of vibration of the structure in sec. W = seismic weight. Wi, Wx = portion of W that corresponds to level i or x. w = effective width. wi = storey width. V = shear force acting at the base of the structure. Vf = factored shear load demand. Vr = factored shear load resistance. Vs30 = average shear wave velocity in m/sec. V2 = shear force acting parallel to local axis 2. V3 = shear force acting parallel to local axis 3. αw = section property reduction factor used for wall effective stiffness properties. α1 = ratio of average stress in rectangular compression block to the specified concrete strength. δmax  = maximum displacements at the extreme points of the structure for each level. δave  = average of the displacements reported at the extreme points of the structure. xxviii  ϕc = material factor for concrete equal to 0.65. ϕs = material factor for reinforcement steel equal to 0.85. ΔfRdRo = NBCC 2020 design displacement at the top of GLRF, obtained from linear dynamic analysis.    xxix  List of Abbreviations 2D: two dimensions. 3D: three dimensions. AFD: axial force diagram. BMD: bending moment diagram. CB: coupling beams. CSA: Canadian Standards Association. EBW: effective beam width. GLRF: gravity-load resisting frame. LATBSDC: Los Angeles Tall Buildings Structural Design Council. NBCC: National Building Code of Canada. RSA: response spectrum analysis. SFD: shear force diagram. SFRS: seismic-force resisting system. ULS: ultimate limit state.    xxx  Acknowledgements I would like to express my profound gratitude to my research supervisor, Dr. Perry Adebar. Collaborating with him has been, and will continue to be, a pivotal moment in my career and professional development in Canada. His unwavering guidance and patience have been instrumental throughout this entire process. He exemplifies perpetual learning and dedicated societal contribution. To my former employers and mentors, Guillermo Coronado, Daniel Gonzales, and especially Carlos Pastor, I extend my heartfelt appreciation. Their encouragement not only led me to pursue my master's degree but also fostered the knowledge and skills I cultivated under their guidance. I reserve special appreciation for my friend Sergio Godinez, who shares my passion for concrete tall buildings. His availability for technical discussions was invaluable, aiding me in navigating various stages of this work. I would also like to extend my gratitude to my friends both in Mexico and Canada. Their friendship provided the fuel and motivation needed to complete this pivotal chapter in my life. Most importantly, I dedicate special recognition to my father, Alejandro; my mother, Gabriela; my brother, Pablo; my sister, Adriana; and my partner, Paula. Their boundless love, care, and patience made this adventure possible. It is through their unwavering support that I learned to believe in myself and pursue my goals with unwavering passion and integrity.     xxxi  Dedication          To my family and my beautiful Canadian wildflower.    1  Chapter 1: Introduction 1.1 Background Like many regions that border the Pacific Ocean, the west coast of Canada is situated within the Pacific Ring of Fire, a geographical area characterized by heightened volcanic and seismic activity along the edges of the Pacific Ocean. Specifically, the southwestern coast of the province of British Columbia is exposed to various types of seismic sources capable of triggering subduction, intraslab, and crustal earthquakes. Meanwhile, in the urban areas of British Columbia, in response to the high demand for housing, a favored solution has been the construction of tall concrete high-rise buildings.   In Canada, the preferred layout of high-rise buildings includes a central core made up of reinforced concrete structural walls. Usually, the core will enclose the elevator and stairwell shafts. The core is typically continuous from the top of the building to the foundation. The building’s floor slabs are supported by the core and by columns or load-bearing walls positioned around the perimeter of the floor plan, but unlike the core, these members may not be continuous along the entire height of the building. To allow for discontinuity, a thick slab or transfer slab may be introduced at the storey where the location of the columns or the bearing walls changes.  The National Building Code of Canada (NBCC) (National Research Council of Canada, n.d.) requires that the seismic force resisting system (SFRS) shall be designed to resist 100% of the earthquake loads and their effects due to the design-level ground motion having a 2% probability of exceedance in 50 years. In a typical high-rise core wall building in western Canada, the SFRS consists of ductile coupled walls in one direction of the core and ductile (cantilever) shear walls in 2  the perpendicular direction of the core. The core of the building is the SRFS and must be designed to resist 100% of the seismic demands.   NBCC also requires that all structural framing elements not considered to be part of the SFRS must be investigated and shown to behave elastically or to have sufficient non-linear capacity to support their gravity loads while undergoing earthquake-induced deformations due to the design level ground motions having a 2% probability of exceedance in 50 years. The floor slabs and the perimeter columns and bearing walls that support the floor slabs are the “structural framing elements not considered to be part of the SFRS” in a core wall building and are more commonly referred to as the gravity-load resisting frame (GLRF). A new requirement in the 2020 edition of NBCC (Article 4.1.8.23.(4)) is that for Normal Importance Category buildings in Seismic Category SC4 with a height above grade of more than 30 m, the structural framing elements not considered to be part of the SFRS shall be designed to ‘behave elastically’ for the ground motions having 10% probability of exceedance in 50 years. Highrise core wall buildings in the Vancouver area must meet this additional requirement once the new edition of the BC Building Code (based on the 2020 NBCC) come into effect in March 2024.  For concrete buildings, the detailed procedures used to meet the NBCC requirement that the GLRF be investigated and shown to behave elastically or to have sufficient non-linear capacity to support their gravity loads while undergoing earthquake-induced deformations due to the design level ground motion, is given in Clause 21.11 of Canadian Standard CSA A23.3-19, Design of Concrete Structures (CSA (Canadian Standards Association), 2019). Clause 21.11 includes general analysis 3  requirements to determine the forces and deformations induced in structural members not considered to be part of the SFRS due to seismic demands on the SFRS as follows: (a) the complete structure is displaced laterally to the design displacements incorporating the effects of torsion, including accidental torsion, and accounting for foundation movements, (b) the inelastic displacement profile of the SFRS must be accounted for  (c) upper-bound effective stiffnesses are used to determine the induced forces, and (d) the additional interstorey drifts resulting from shear strains in the plastic hinge regions of the SFRS must be accounted for.  The requirement in (b) that the inelastic displacement profile of the SFRS be accounted for because yielding of the SFRS causes concentration of deformations at plastic hinge locations, makes the general analysis procedure too difficult to implement for most structural engineering design offices in Canada. CSA A23.3 Clause 21.11 also provides a simplified analysis procedure as follows: (a) The shear force and bending moments induced in members of a gravity-load resisting frame are determined at each level by subjecting the two-dimensional gravity-load resisting frame (2D GLRF) to a prescribed interstorey drift ratio for that level that depends on the type of SFRS (shear wall or coupled wall) and the design lateral deflection at the top of each 2D GLRF. All 2D GLRF must be investigated in each direction of loading. (b) The additional vertical load that is induced in vertical-load resisting members due to lateral deformation of the structure is determined by summing the shear forces from all horizontal members supported by the vertical-load resisting member for each level and summing the contribution for all levels above the level of interest. 4  (c) For buildings with cantilever walls or coupled walls, over the height of the plastic hinge region of the SFRS, the minimum curvature demand on all gravity-load columns and walls shall not be taken less than the curvature demand on the SFRS walls.   Further discussion of the simplified analysis procedure and other pertinent clauses in CSA A23.3 is presented in Chapter 3.  1.2 Goals and Objectives The simplified procedure described in CSA A23.3 Clause 21.11 is simpler for design offices to implement than the general analysis procedure, but still has many challenges. The high-level objective of this thesis is to develop guidance for design engineers on the implementation of the simplified procedure in Clause 21.11 to typical highrise concrete shear wall buildings in Canada. The simplified procedure defines an interstorey drift envelope but does not specify the displacement profiles that are to be used. One of the objectives of this thesis is to investigate how different displacement profiles will influence the demands on the GLRF. Consulting engineers typically use commercial structural engineering software ETABS (CSI Computers and Structures Inc., 2019) to conduct their seismic analysis of highrise buildings. Another important objective of this thesis is to investigate the best way to impose a displacement profile on buildings using ETABS. Once a suitable procedure is developed to implement the simplified analysis procedure in CSA A23.3 Clause 21.11, the next objective is to investigate common arrangements of the GLRF to determine if the Clause 21.11 requirements may require significant design changes. The various 5  GLRF configurations investigated and assessed in this study included: i) a uniform GLRF configuration with slender slabs, and ii) a GLRF with a transfer slab at the quarter height. The final objective of this thesis is to conduct a pilot study related to the new requirements in the 2020 edition of NBCC Article 4.1.8.23 for Normal Importance Category buildings in Seismic Category SC4 with a height above grade exceeding 30 metres – structural framing elements not considered part of the SFRS must be designed to 'behave elastically' when subjected to ground motions with a 10% probability of exceedance in 50 years. The first step was to determine appropriate values for effective stiffness, not only for the SFRS but also for the GLRF. CSA A23.3-19 does not prescribe these values for this level of demand. Once the values for effective stiffness are refined, and additional seismic analysis is conducted, the objective is to assess whether the gravity-load columns will remain elastic or require further design changes. 1.3 Methodology Developing an archetype building was a crucial step in this study. This archetype building is meant to represent a typical high-rise concrete building in Canada with a typical layout for the core of buildings, playing the role of the SFRS, and the slabs and gravity-load columns being part of the GLRF. Analytical models based on this prototype building were generated in ETABS, with the goal to perform dynamic wind and seismic analysis to estimate the demands affecting the core along its height, as well as the maximum displacement at roof level due to seismic loading.  Prior to Clause 21.11 analysis, additional models were generated to explore all the tools and options that ETABS offers to develop the simplest way to accurately conduct the simplified analysis of buildings outlined in Clause 21.11 of CSA A23.3. The definitive procedure that was 6  selected, which is discussed in detail in Chapter 3, involves applying 2D displacement profiles using axially rigid elements pulling each storey level, until reaching the displacements that result in the target interstorey drifts. Once these interstorey drifts were imposed on the building, for both coupled and cantilever wall directions, the demands on the gravity-resisting columns were investigated.   Finally, in the context of the additional performance requirements, introduced in the new edition of NBCC, a pilot study was conducted using the same archetype building model with a uniform GLRF (slender slabs and columns). This study required additional dynamic seismic analysis based on spectral accelerations of different magnitude. It also involved a series of iterations to refine the effective stiffnesses of the SFRS and GLRF. The consequences of the demands of this magnitude were observed in different framing members not part of the SFRS to determine whether the structure remains elastic or not.  1.4 Thesis outline This thesis is organized into six chapters, presenting the following content:  Chapter 2: Archetype Building and Structural Design introduces the building set as case study, illustrating its geometry, considered gravity loads, material properties, as well as sections used to define all the members involved in both, the Gravity Load Resisting Frame (GLRF) and the Seismic Force Resisting System (SFRS). A summary of the wind and seismic analyses and lateral loading are presented to justify the design of the SFRS.  7  Chapter 3: Methodology for Applying CSA A23.3 Clause 21.11 presents a comprehensive summary of conducting lateral analysis on the GLRF in accordance with Clause 21.11. The summary begins with an overview of all the requirements that must be addressed in this analysis, as well as the tools already outlined in CSA A23.3 standard. Next, it provides a detailed description of the 2D displacement profiles proposed to impose the required displacements as specified in the Simplified Analysis, also part of Clause 21.11. Following that, the summary concludes with guidelines and recommendations for modeling slab-column connections to limit the forces transferred from the slabs to the building's columns. Finally, it offers guidance on how to accurately assess whether a particular column possesses an adequate level of resistance or ductility to accommodate the displacement demands resulting from ground motions.  Chapter 4: Applying CSA A23.3 Clause 21.11 to Case Study aims to demonstrate the practical application of the methodology outlined in Chapter 3 to the archetype development discussed in Chapter 2 of this thesis. Building upon the foundation laid out in Chapter 3, readers will enhance their understanding of conducting the Simplified analysis of buildings, as presented in Clause 21.11.2.2, by the conclusion of this chapter. Additionally, this chapter also presents the results of analyses conducted on various GLFR configurations for the same building (one with slender and flexible columns and slabs, and a second one where a thick transfer slab is included). This chapter concludes by highlighting the most critical locations for each direction of the SFRS analysis and provides examples of redesigned columns.  Chapter 5: Applying NBCC 4.1.8.23 to Case Study In this chapter, recommendations are provided for analyzing the same archetype building presented in Chapter 2. The goal is to 8  determine whether the SFRS and GLRF will have the capacity to remain elastic after defining seismic loads corresponding to ground motions with a 10% probability of exceedance in 50 years.  Chapter 6: Summary and Conclusions summarizes the contributions of this work, presents its conclusions and limitations, and suggests recommendations for further improvements on this work.   9  Chapter 2: Archetype Building and Structural Design This chapter presents all the relevant information of the prototype building's layout and the proposed members for both the Seismic Force Resisting System (SFRS) and the Gravity-Load Resisting Frame (GLRF). Additionally, it presents a summary of the analyses and design processes conducted for the GLRF and SFRS.  2.1 Building Description The archetype building evaluated in this study is a 30-storey residential structure with 5 levels for parking below grade level. This structure is representative of the most common typology of highrise concrete buildings in the urban areas of British Columbia in Canada. Figure 2.1 shows a plan view of the levels below grade (parking levels), where the overall dimensions of the plan at these elevations are 45.70 m x 45.70 m. Whereas Figure 2.2 illustrates the layout of the levels located above grade level (offices/residential), and the overall dimensions of the slab for those levels are 25.90 m x 25.90 m. The elevation along the coupled-wall direction, shown in Figure 2.3, will provide a clearer idea of the level's distribution along the height of the building.   10   Figure 2.1 Plan view of levels below grade. 11   Figure 2.2 Plan view of levels above grade. 12   Figure 2.3 Elevation of the prototype building on the left-side, and a 3D view of the analytical model in ETABS on the right-side.  13  Taking a closer look at the core of the building, which is designed to resist not only to withstand gravity forces but also lateral forces due to wind gusts and ground motions, Figure 2.4 shows the lengths and thicknesses of each wall pier, as well as the overall dimensions of the core. The SFRS in the coupled-wall direction (x-direction) will be designed as Ductile Coupled Walls. For the cantilever-wall direction (y-direction), the structural walls will be designed as Ductile Shear Walls, according to the National Building Code of Canada 2020 (NBCC 2020).  Figure 2.4 Plan view of the SFRS. 14  The coupling beams CB1-2 and CB2-3 shown in the previous figure will have the same thicknesses as the flanges of the core. Table 2.1 presents more detailed information about how the depths of the coupling beams vary along the height of the project. Table 2.1 Coupling beam depths.  For this archetype all the reinforcement is considered to be grade 400 MPa. Additionally, for the SFRS, it was proposed to vary the concrete compression strengths along the height of the building: from the foundation to level L10 𝑓𝑐’ = 45 MPa; from level L11 to level L20 𝑓𝑐’  = 35 MPa; from level L21 to the Penthouse 𝑓𝑐’  = 30 MPa. Table 2.2 presents a summary of the variation just described. Table 2.2 Distribution of concrete compression strengths for the SFRS.  For the realization of this project, the gravity loads to be part of the dead load were the following: • Self-weight of the structure assuming 24 kN/m3.  • Cladding weight of 0.72 kPa of wall area. For a clear height of 2.59 m, this would be 1.9 kN/m along the perimeter of each level above grade. CB1-2 CB2-3Penthouse 190 4,880 2640* 2240*L2 - L30 190 2,590 595 695L1 250 4,210 2,375 2,275P1 250 4,270 2,335 2,435P2 - P5 250 2,750 815 915Coupling beam depths [mm]* Opening height increased to satisfy maximum coupling beam depth.LevelThickness of slab above [mm]Clear storey height [mm]Level L21 to Penthouse 30Level L11 to Level L20 35Level L1 to Level L9 45Parking P5 to Parking P1 45Range of Levels    [MPa]  ’15  • Superimposed dead load (including partitions) of 1.25 kPa, for all floors above the first level, 1.75 kPa for the first level, and 0.5 kPa for the parking levels below grade. • Mechanical room on the roof with a weight of 194 kN. • Chillers on the roof with a weight of 749 kN.  As for the live load, a distributed load (acting uniformly) of 1.9 kPa was assumed for all levels above the first floor, 4.8 kPa for the first floor, and 2.4 kPa for the parking levels below grade. Finally, 1.7 kPa was considered for the snow load on the roof. Relevant information regarding the design of the gravity columns may be found in subsection A.3 of Appendix A. This includes details about the tributary areas for each column, as shown in Figure A.3,  as well as the load take-down, leading to the axial load demands and maximum resistances represented graphically in Figure A.4.  Figure 2.5 summarizes the different sections that will be used as columns for this archetype building. For further details, please refer to Appendix A.  Figure 2.5 Design of gravity columns for Uniform Gravity-Load Frame.  16  For lateral analyses (wind and seismic), the dead loads to be considered acting on the structure when it is also being subjected to lateral loads, and hence, being part of the seismic weight, are as follows: • Self-weight of the structure assuming 24 kN/m3.  • Cladding weight of 0.72 kPa of wall area. For a clear height of 2.59 m, this would be 1.9 kN/m along the perimeter of each level above grade. • Superimposed dead load (including partitions) of 0.75 kPa. • Mechanical room on the roof with a weight of 194 kN. • Chillers on the roof with a weight of 499 kN. The total seismic weight of the building was calculated to be 154,545 kN.   2.2 Wind Analysis For wind analysis, a new analytical model in ETABS (CSI Computers and Structures Inc., 2019) was generated using the same mass as the one used for the seismic analysis presented in the previous subsection. However, different effective stiffnesses were defined according to Table N9.2.1.2 in the Commentary to Clause 9.2.1.2 from CSA A23.3 for the Ultimate Limit State (ULS). For the coupling beams with diagonal reinforcement, Ave = 1.2 · (0.45 · Ag), and Ie = 0.35 · Ig were considered. It is worth mentioning that, while the Table from the Commentary specifies an effective stiffness of 0.4 for the shear stiffness of the beams, a decision was made to maintain a value of 0.45 (from Table 21.1 in CSA A23.3). This decision was based on the fact that, it is not logical to consider a higher stiffness for seismic demands (where the member will yield, and the stiffness will decrease) than for wind analysis. 17  For walls, Ave = 0.75 · Ag, and Ie = 0.75 · Ig were used. To be conservative with the design of the core, the contribution to lateral stiffness and lateral strength coming from the columns was neglected for the wind analysis. The periods resulting from a modal analysis considering the stiffnesses and assumptions mentioned above are shown in the following Table 2.3. Table 2.3 Natural periods and mode shapes for wind analysis.  According to NBCC 2020 in 4.1.7.2-(3), this archetype should be classified as a very dynamically sensitive building. The correct way to analyze this structure for wind demands would be to perform a wind tunnel procedure. However, for research purposes, it was decided to carry out a dynamic wind analysis (considering full loading only), following section 4.1.7.8 of the code. In essence, the dynamic procedure consists of calculating a set of static loads that will be applied to the structure. However, these loads are based on dynamic pressure profiles. To illustrate how the pressure profiles are computed, please refer to the following Equation. 𝑝 = 𝐼𝑤𝑞𝐶𝑒𝐶𝑡𝐶𝑔𝐶𝑝  ( 1 )  Where, p = specified external pressure, considered positive when acting towards the structure, and negative when acting away from the surface. Iw = importance factor for wind load (considered equal to 1.0, Table 4.1.7.3 in the NBCC). 1 5.23 69 0 02 3.98 0 68 03 2.28 0 0 79Mode Period [sec]Mass Participation [%]x - dir. (coupled)y - dir. (coupled)z - dir. (torsion)18  q = reference velocity pressure (for Vancouver City Hall equal to 0.45 kPa, Table C-2 in the NBCC). Ce = exposure factor (considered as 1.33 for the windward and 0.94 for leeward) Ct = topographic factor (considered equal to 1.0) Cg = gust factor Cp = external pressure coefficient The external pressure coefficient is calculated for all the faces (windward, leeward, lateral) exposed to the wind gusts; it is based on the height/width ratio of the building. The calculation of the gust factor is the main difference between the static and the dynamic procedures, examples of its computation are shown in Figure 2.7 and Figure 2.8. For more detailed information regarding its calculation please refer to Appendix B. As stated in the NBCC, the net wind load will be the algebraic sum of all the pressures or suctions acting on each face. For a typical building, we would have the following scenario. 19   Figure 2.6 Wind loads resultants acting on a building; values of Cp from NBCC 2020.  As seen in the previous Figure 2.6, all faces of the building will be subjected to suctions due to vortexes created by the wind current. The windward face will be the only one being pushed by the wind. Following the example shown, for the main direction of analysis, parallel to the wind direction, the effects taking place on the windward face (pressure pushing) and the leeward face (suction pulling) will add to each other. Based on all these factors that were introduced and calculated, the pressure profiles were generated for the windward and leeward faces in each direction of analysis. These profiles are shown in 20  Figure 2.9. The blue line represents the pressure calculated for the windward face, whereas the red line represents the pressure applied to the leeward face.  Figure 2.7 Calculation for the gust effect factor for the windward face in the coupled-wall direction.  21     Figure 2.8 Calculation for the gust effect factor for the windward face in the cantilever-wall direction.   22   Figure 2.9 Dynamic pressure profiles: windward pressure (blue) and leeward pressure (red); on the left side for the coupled-wall direction, and on the right side for the cantilever-wall direction.01020304050607080901000.00 0.50 1.00 1.50Elevation [m]Pressure [kPa]01020304050607080901000.00 0.50 1.00 1.50Elevation [m]Pressure [kPa]23  After adding the pressures shown above and multiplying them by the corresponding tributary areas, the lateral loads can be obtained. The following Figure 2.10 shows the lateral loads that were applied to the analytical model in each direction.   Figure 2.10 Lateral loads resultant from dynamic wind analysis: coupled-wall direction (blue) and cantilever-wall direction (red).  01020304050607080901000 50 100 150Elevation [m]Lateral Loads [kN]24  Introducing the demands that the wind loads will cause to the SFRS, Figure 2.11 illustrates the demands in each coupling beam along the height of the building. To show how the demands are being distributed along the height of the core, Figure 2.12 and Figure 2.13 show the shear and bending moment demands for each direction of analysis.25    Figure 2.11 Shear demands due to wind loading in coupling beams. 01020304050607080901000 200 400 600 800 1,000 1,200Elevation [m]Coupling Beam (CB1-2) Shear Forces, V2 [kN]01020304050607080901000 200 400 600 800 1,000 1,200Elevation [m]Coupling Beam (CB2-3) Shear Forces, V2 [kN]26    Figure 2.12 Demands acting on walls due to wind loading; Coupled-wall direction. -200204060801000 2,000 4,000 6,000 8,000 10,000Elevation [m]Shear Force, V3 [kN]-200204060801000 50,000 100,000 150,000 200,000 250,000Elevation [m]Bending Moment, M2 [kN·m]27    Figure 2.13 Demands acting on walls due to wind loading; Cantilever-wall direction. -200204060801000 2,000 4,000 6,000 8,000 10,000 12,000Elevation [m]Shear Force, V2 [kN]-200204060801000 50,000 100,000 150,000 200,000 250,000Elevation [m]Bending Moment, M3 [kN·m]28  2.3 Seismic Analysis The seismic analysis was performed in accordance with the National Building Code of Canada 2020 (NBCC 2020). The location was assumed to be Vancouver City – Hall (Latitude: 49.261 Longitude: -123.114). The site was designated based on the average shear wave velocity, Vs30 = 500 m/s. The spectral accelerations shown below correspond to an event with a 2% probability of exceedance in 50 years.  Figure 2.14 Uniform Hazard Spectrum for Vancouver City Hall corresponding to an event with 2% probability of exceedance in 50 years.  As described above, the weight considered to compute the inertial forces induced by ground motions was 154,545 kN. Reduced section properties were used to account for concrete cracking in accordance with Clause 21.2.5.2. For the coupling beams in ductile coupled walls, designed according to CSA A23.3:2019 Clause 21.5.8.2 (coupling beams with diagonal reinforcement),    1.010.6990.400.2480.06830.03040.000.200.400.600.801.000.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Spectral acceleration [1/g]Natural period [sec]29  Ave = 1.2 · (0.45 · Ag), and Ie = 0.25 · Ig. The factor of 1.2 compensates for the fact that ETABS uses a shear area of Ave = (5/6) · Ag. The reduction factor αw was initially assumed to be equal to Ro in Equation 21.1 of Clause 21.2.5.2. This resulted in Ave = 0.5 · Ag and Ie = 0.5 · Ig for the walls. Geometric nonlinearity was included by defining P-delta effects (iterative based on loads). Figure 2.15 shows a 3D view and elevations of the SFRS.   Figure 2.15 Isometric and elevation views of analytical model in ETABS for Seismic Analysis.   30  Table 2.4 Natural periods and mode shapes for seismic analysis.  Noting that the SFRS has a height of  90.15 m above grade level, and that, according to NBCC 2020, this structure meets all the requirements to be classified as regular, Table 2.5 presents a complete summary of all seismic force used to ensure that the SFRS will have adequate levels of lateral stiffness and resistance. Since the structure is not torsional sensitive, an accidental eccentricity was defined as +/- 5% of the plan dimension of the building for the Response Spectrum Analysis (RSA).             1 6.44 69 0 02 4.82 0 67 03 2.56 0 0 79Mode Period [sec] x - dir. (coupled)y - dir. (coupled)z - dir. (torsion)Mass Participation [%]31  Table 2.5 Summary of Seismic Demands according to NBCC 2020. NBCC Reference Parameter  Coupled Wall Cantilever Wall    (a) Seismic Weight, W = 154,545 kN 154,545 kN    (b) Fundamental period from ETABS, T = 6.44 sec 4.82 sec  4.1.8.12.(5) & 4.1.8.12.(6) (c) Design elastic base shear, Ved =    Elastic base shear, Ve = 13,800 kN 18,391 kN   4.1.8.5.(1); 4.1.8.9 (d) IE = 1.0; RdRo =  4.0 x 1.7 = 6.8 3.5 x 1.6 = 5.6  4.1.8.12.(7) (e) Design base shear, Vd = VedIE/RdRo = 2,031 kN 3,285 kN  4.1.8.11.(3).(c) & 4.1.8.11.(3).(d).(iii) (f) Empirical period Ta = 2 x 0.05hn0.75             = 2 x 1.463 sec   2.93 sec 2.93 sec   4.1.8.4.(9) (g) S(2.93 sec) = 0.146 0.146  4.1.8.11.(6) (h) Mv =  1.02 1.42  4.1.8.11.(2) (i) Min. lateral earthquake force,                 V(Ta = 2.93 sec) = 3,375 kN 5,706 kN   4.1.8.12.(8) (i) Scaled design base shear                    (regular structure) Vd = 0.8V =  2,700 kN 4,565 kN     (k) Scaling factor for design forces (j)/(e) = 1.33 1.39  4.1.8.11.(2) (l) Min. lateral earthquake force                   (for deflection), V(Ta = 4.0 sec) = 2,125 kN 2,580 kN   4.1.8.12.(11) (m) Scaled design base shear                            Vd = 0.8V (for deflection) =  1,700 kN 2,064 kN     (n) Scaling factor for deflection forces (m)/(e) = 1.00 1.00  4.1.8.13.(2) (o) Total multiplier for deflections                   (n) x RdRo/IE =  6.80 5.60    Once the RSA was performed, the demands on each member part of the SFRS were captured to then proceed with the design process. The dark blue line in Figure 2.16 shows the shear demands resulting from RSA with an accidental torsion defined, whereas the light blue line represents the demands that do not include this accidental torsion. Figure 2.17 and Figure 2.18 present the demands affecting the wall piers for each direction of analysis along their entire height.  32   Figure 2.16 Shear demands due to seismic loads in coupling beams: no accidental torsion (light blue); with accidental torsion (dark blue). 01020304050607080901000 200 400 600 800 1,000 1,200Elevation [m]Coupling Beam (CB1-2) Shear Forces, V2 [kN]01020304050607080901000 200 400 600 800 1,000 1,200Elevation [m]Coupling Beam (CB2-3) Shear Forces, V2 [kN]33    Figure 2.17 Demands acting on walls due to seismic loading; Coupled-wall direction. -200204060801000 1,000 2,000 3,000 4,000Elevation [m]Shear Force, V3 [kN]-200204060801000 20,000 40,000 60,000 80,000 100,000Elevation [m]Bending Moment, M2 [kN-m]34    Figure 2.18 Demands acting on walls due to seismic loading; Cantilever-wall direction. -200204060801000 2,000 4,000 6,000 8,000 10,000Elevation [m]Shear Force, V2 [kN]-200204060801000 50,000 100,000 150,000Elevation [m]Bending Moment, M3 [kN-m]35  2.4 Design of the SFRS The same SFRS will be responsible for providing sufficient levels of lateral strength and stiffness to face any kind of lateral load. Figure 2.19 to Figure 2.21 present a summary of the lateral demands affecting each component of the SFRS. The red line represents the demands caused by wind loads, amplified by the corresponding load factor set by NBCC 2020 (1.4 W). Similarly, the blue lines indicate the lateral demands resulting from seismic loads (1.0 E).   36   Figure 2.19 Summarize of shear demands in coupling beams: with accidental torsion due to 1.0 E (dark blue); without accidental torsion due to 1.0 E (light blue); and due to 1.4 W (red dashed-line). 01020304050607080901000 200 400 600 800 1,000 1,200Elevation [m]Coupling Beam (CB1-2) Shear Forces, V2 [kN]01020304050607080901000 200 400 600 800 1,000 1,200Elevation [m]Coupling Beam (CB2-3) Shear Forces, V2 [kN]37    Figure 2.20 Summarize of demands acting on walls in Coupled-wall direction: due to 1.0 E (blue line); due to 1.4 W (red dashed-line). -200204060801000 2,000 4,000 6,000 8,000 10,000Elevation [m]Shear Force, V3 [kN]-200204060801000 50,000 100,000 150,000 200,000 250,000 300,000Elevation [m]Bending Moment, M2 [kN-m]38    Figure 2.21 Summarize of demands acting on walls in Cantilever-wall direction: due to 1.0 E (blue line); due to 1.4 W (red dashed-line).-200204060801000 2,000 4,000 6,000 8,000 10,000 12,000Elevation [m]Shear Force, V2 [kN]-200204060801000 50,000 100,000 150,000 200,000 250,000Elevation [m]Bending Moment, M3 [kN-m]39  As shown in Figure 2.19,  Figure 2.20, and Figure 2.21, the design of the rigid core at grade level is governed by wind loads. In the most recent edition of the code (NBCC 2020), these demands remain unchanged from those considered during the design of this archetype, which was previously presented as a design example in Chapter 11 of the 4th edition of the Concrete Design Handbook. Therefore, the same design of the core for the plastic hinge will be adopted in the present study and used as a reference to conduct the analyses outlined in Chapter 5 of this thesis. The detailed layout of the vertical reinforcement for the structural walls is provided in Figure 2.22.                40   Figure 2.22 Layout of vertical reinforcement for wall piers at grade level.   41  Chapter 3: Methodology for Applying CSA A23.3 Clause 21.11 3.1 Introduction This chapter presents all the steps necessary to correctly perform the simplified analysis of buildings, as required by CSA Standard A23.3 Design of Concrete Structures Clause 21.11. These steps are part of a new methodology developed as a result of this work.  3.2 Alternative Procedures Investigated Before presenting the recommended methodology, information is presented about the alternative solutions that were investigated as part of this study.  3.2.1 Series of Pushover Analyses The first option arose from a thorough comprehension of the interstorey drift ratios specified in Clause 21.11.2.2, which represent the maximum drift that each level of the GLRF may experience in the event of ground accelerations of a certain intensity. Consequently, the optimal approach would involve leveraging the tools available in ETABS to define a nonlinear static load case (pushover) for each level of the GLRF that requires examination. This necessity stems from a limitation within the software, which permits the monitoring of only one master node's displacement during each nonlinear static analysis.  Ultimately, the demands used to assess the column's capacity would be derived from the envelopes of all the diagrams resulting from each pushover case. While this option would provide a deeper understanding of the core principles underlying the Simplified analysis of buildings and yield highly accurate demand estimations, it was promptly discarded. The reason for this decision lies 42  in the sheer volume of work required to analyze even a single frame part of the GLRF frame. The workload associated with this option would be prohibitively high.   3.2.2 Imposing Displacements with Infinitely Rigid Slabs The next option explored involved utilizing the Ground Displacement command, which is a component of the ETABS toolbox, to define and assign demands to node-objects. However, as illustrated in Figure 3.1, while the deformed shape of the SFRS appears correct, it is evident that there is no deformation in the horizontal members of the analytical model. This lack of deformation results in no induced bending moment demands on the building's slabs and inaccurate induced bending moments on the columns.  The utilization of the Ground Displacement command is a part of the final methodology presented in the subsequent subsections. It is crucial to note that in this approach, the nodes where the imposed displacements have been defined must all be disconnected from the diaphragm at that storey level of the model. Figure 3.16 provides an illustration of this issue, which will be further discussed in the following subsections of this chapter.   43   Figure 3.1 First attempt of imposing displacements in ETABS; On the left-side the deformed shape of the analytical model; on the right-side incorrect bending moments at the columns.    44  3.2.3 Lateral Loads to Achieve Target Drifts Given the understanding that this particular software appeared to handle forces more effectively as demands rather than displacements (or, at least, this was our belief at the time), we devised a new strategy involving an iterative process. This approach relied on updating the lateral loads applied to the diaphragm at each storey level of the analytical model until the resulting drift matched the values specified in the interstorey envelopes outlined in Clause 21.11.2.2.  The foundation of this approach was based on the concept of storey stiffness, defined as the storey shear at a specific level divided by the drift (relative displacement) of that same level. However, implementing this strategy proved to be challenging for several reasons: i) The definition of storey stiffness is most applicable to structures where their response is primarily governed by shear, such as frame structures, rather than structural walls. ii) The number of iterations required varied significantly depending on the direction of analysis (coupled-wall or cantilever-wall) and the stiffness of the gravity system. iii) Achieving convergence was not always possible due to the substantial lateral loads needed at the base of the structure to attain the high shear deformations depicted in the interstorey envelopes specified in Clause 21.11.2.2.  3.2.4 Concentrated Moments to Achieve Target Drifts Drawing lessons from the challenges encountered in the previously discussed process, a similar strategy was tested, this time with two significant adjustments: i) Instead of applying lateral loads to different diaphragms, concentrated moments were applied to each wall segment of the SFRS, at different elevations. 45  ii) The interstorey drift ratio envelopes were modified by excluding the high shear deformation at the base of the structure. If this strategy proved successful, an additional analysis would be conducted to assess the demands resulting from the shear deformation at the base. Figure 3.2 provides a visual summary of the concepts just described that would have applied to the cantilever-wall direction.  Figure 3.2 Trial for modified interstorey drift envelope for cantilever walls, and the corresponding curvature profile and concentrated bending moments required.  While this strategy did reduce the number of iterations, it did not take long before its drawbacks became apparent: i) Applying concentrated moments proved to be somewhat cumbersome, as they had to be individually assigned to each wall (as opposed to a single lateral load assigned to the diaphragm). Moreover, distributing bending moments became complex as it depended on the flexural stiffness 46  of each wall. It's worth noting that what is depicted in Figure 3.2 assumes constant flexural rigidity, condition that is not face in the majority of real-life projects. ii) Additionally, different configurations of the GLRF were considered, and it became evident that the stiffer the GLRF, the more it resisted deformation imposed by the SFRS. One potential solution to this issue would involve artificially modifying the GLRF's stiffness and then amplifying the demands on the gravity system to assess whether these members meet the requirements specified in Clause 21.11. However, this approach only added further complexity to the process.  After exploring all of these options and conducting the necessary assessments, a definitive methodology is presented in the subsequent pages of this chapter. This methodology is the recommended approach for implementing the framework outlined in CSA A23.3.  3.3 CSA A23.3 Clause Requirements With the purpose of ensuring that all members of the GLRF, which until this point have been designed to resist gravity forces only, will also have an adequate level of strength and/or capacity of ductility to withstand the lateral displacement demands induced by ground motions. The general requirements presented in Clause 21.11.2 of the main body of CSA A23.3 are essential to conduct an adequate analysis, serving as a tool to study the demands on the GLRF. These requirements can be summarized as follows: • The model must be laterally displaced so that the rooftop of the GLRF reaches the design displacements (ΔfRdRo). These displacements result from the seismic analysis conducted for the SFRS, and they must consider the effects of torsion and foundation movements. • Yielding of the SFRS, leading to an inelastic displacement profile. 47  • Upper-bound estimates for effective stiffnesses of all members forming the GLRF, ensuring safe estimations of forces in these members. • Shear deformations of the SFRS at the location of the plastic hinge.  To fulfill all the requirements listed above, it may be necessary, to perform a more comprehensive analysis. For this reason, CSA A23.3 offers a Simplified Analysis, which proves to be a valuable tool, addressing three out of the four requirements effectively.  3.3.1 Simplified Analysis of Buildings When the response of the building is clearly governed by structural walls as part of the SFRS, whether they are coupled walls or cantilever walls, CSA A23.3 provides envelopes resulting from numerous nonlinear dynamic analyses conducted on shear wall buildings. These envelopes also account for shear deformations of the SFRS that occur at the plastic hinge region once the SFRS has yielded (Adebar et al., 2010, 2014).  After performing an analysis for each level of the GLRF, where each frame's level is subjected to the maximum displacement it may experience, the effects on each member forming the GLRF can be studied. As complementary guidelines to the code mentions (EGBC, 2022) , the SFRS imposes demands on the GLRF due two main reasons: • High axial stiffness of the floor system: The presence of reinforced concrete slabs forces the GLRF to adopt the same deflected shape as the SFRS. This becomes particularly critical in the plastic hinge region, as the curvature demands of the structural walls result in significant bending moment demands on the gravity columns. 48  • Frame action: The arrangement of the core as a vertical member on one end, a gravity column on the other, and a horizontal slab connecting these two supports forms a frame. During lateral swaying of the building, forces are transferred from the slab to the columns. This aspect becomes particularly significant when the spans between the core and the columns are relatively short.  3.3.2 Summary of Requirements  To carry out a Simplified Analysis correctly, certain key ingredients are required. CSA A23.3, in Clause 21.11, provides the following ingredients: • Design displacements ΔfRdRo: These displacements are the product of the seismic analysis of the SFRS, considering appropriate levels of effective stiffness to ensure safe estimations of displacements. Total displacements should be considered, accounting for foundation movements. • Envelopes of minimum interstorey drift ratio as a function of the global ratio (see Figure 3.3). Utilizing the design displacements mentioned above, the appropriate magnitude of lateral displacements can be calculated for each level of interest within a particular GLRF.  49   Figure 3.3 Envelope of minimum interstorey drift ratios over building height for Coupled and Cantilever walls from CSA A23.3  Since this analysis is meant to be done with linear models, columns must always be kept as linear elastic members. However, this is not the case for the slabs, as mentioned below. CSA provides another set of important clauses to work with linear columns, which may exhibit unrealistically huge forces. These clauses are: • Clause 21.11.3.1, which states that to limit the shear demand on a column, the shear force shall be adjusted at every location where the bending demand Mf exceeds the probable bending capacity of the column Mp (ϕc = 1.0; ϕs = 1.25). • Clause 21.11.3.2, which states that all demands that the slab will transfer to the columns shall be limited by their nominal flexural capacity Mn (ϕc = 1.0; ϕs = 1.0). 50  • Clause 21.11.3.3.3 introduces a table of maximum induced moments to assess if the columns (or walls) possess an adequate level of ductility based on the level of axial load present in the member and its level of detailing. To expand upon the last item mentioned above, Clause 21.11.3.3.3 provides tools for assessing the adequacy of ductility in columns. Figure 3.4 below summarizes the limits set by CSA A23.3 for induced bending moments in columns based on their level of detailing.  Figure 3.4 Maximum calculated induced bending moment for column members from CSA A23.3  To illustrate graphically the different requirements for different levels of detailing in columns (that correspond to different levels of ductility), Figure 3.5 below shows the requirements for a column classified as a tied column, which would allow for an amplification of the flexural resistance of the cross-section between 1.5 to 2.0, depending on the level of axial compression present in the column. 51   Figure 3.5 Requirements for tied columns meeting dimensional limitations according to Clause 21.11.3.3.3.  Aiming for a higher level of ductility and, consequently, more detailed reinforcement, Figure 3.6  illustrates the requirements for a column designed as a moderately ductile member. The left side of the figure shows the vertical spacing requirements for ties in the column's elevation, while the right side depicts the horizontal spacing requirements for ties in a cross-section. This level of detailing allows for an amplification factor of the flexural resistance between 2.0 and 3.0. 52   Figure 3.6 Requirements for moderately ductile columns according to Clause 21.11.3.3.3.  Finally, Figure 3.7 depicts the requirements for a column designed as a high-ductile or ductile member. It's important to note that the same limitations shown in the cross-section of Figure 3.4 apply to this type of column, with the exception that the minimum dimension of the cross-section must be at least 300 mm. Additionally, there are other requirements not shown in the figure, such as the necessity to maintain a vertical reinforcement quantity between 1% and 6% of the cross-sectional area. The trade-off of having such a high density of reinforcement is that it results in an amplification factor of the flexural resistance ranging from 3.0 to 5.0. 53   Figure 3.7 Requirements for ductile columns according to Clause 21.11.3.3.3.  3.3.3 Description of GLRF According to the Canadian approach for seismic analysis and design of buildings, the structure must include a SFRS capable of resisting 100% of the inertial loads induced by ground motions. NBCC provides a list of different options for SFRS in Table 4.1.8.9. To complement the structural solution, GLRF is also necessary to bear the greatest portion of gravity loads and transfer them to the foundation. To illustrate this, consider a typical highrise concrete building. The SFRS often consists of a rigid core used for elevators and stairways. In one 54  horizontal direction of analysis, the coupled-wall system provides the necessary lateral stiffness and strength, while the cantilever walls fulfill the same function in the perpendicular direction. Additionally, the combination of flat slabs with gravity columns and/or bearing walls integrates the GLRF.  3.4 Applying Displacement Profiles in ETABS As mentioned at the beginning of this chapter, the envelopes provided by CSA A23.3 indicate the maximum displacements that each level may experience when the building sways laterally and the SFRS has yielded. However, it is important to note that these maximum displacements do not necessarily occur simultaneously. To illustrate this idea, Figure 3.8 depicts the displacement profile resulting from the envelope of minimum interstorey drift ratio given by CSA A23.3 for cantilever walls. The concept mentioned above has been discussed previously.   55   Figure 3.8 Displacement profile that would result in the envelope of interstorey drift ratios for cantilever walls.  Based on the previous paragraph and Figure 3.8 above, if we perform an analysis where all the storey levels are subjected to the deformations from the envelopes shown in Clause 21.11 (Figure 3.3), we would be forcing the structure to exhibit an impossible deformed shape. In the case of a linear analysis (slab modeled as elastic members), this would result in excessively high demands captured from the gravity columns, particularly in terms of axial demands, which would be overly conservative.   56  To address this problem, the methodology presented below breaks down the process into two stages: 1. Separating the displacement profiles into two groups with the goal of estimating the axial loads on the columns. Subsequently, a third displacement profile is added, continuous along the entire height, to capture shear forces and bending moments without unreal discontinuities. 2. Incorporating nonlinearity into the models by defining plastic hinges in the slabs. These plastic hinges serve to limit the demands transferred to the columns to the nominal flexural capacity of the slabs (Clause 21.11.3.2). The combination of these two steps leads to a better estimation of demands on the columns forming part of the GLRF. It is worth mentioning that in some cases, regardless of whether displacements are being imposed along the bottom or top portion of the building, it may be observed that all the slabs are reaching their nominal capacity. In such scenarios, the need to separate the displacement profiles disappears. However, this study lacks the evidence to define the conditions under which this scenario is likely to occur.  3.4.1 2D Displacement Profiles The recommended method for dividing the displacement profiles for both directions, coupled and cantilever walls, has been previously presented (Adebar & Sainz Albanez, 2023). The expressions are shown in Table 3.1. 57  Table 3.1 Expressions for recommended displacement profiles resulting in interstorey drift profiles defined in CSA A23.3 Clause 21.11.   The graphical representations of these displacement profiles for the coupled-wall direction are given in Figure 3.9. 58   Figure 3.9 Recommended two-part displacement profiles giving interstorey drift profiles defined in CSA A23.3 Clause 21.11 for Coupled – walls.  Similarly, Figure 3.10 provides the graphical representation of the displacement profiles for the cantilever-wall direction.       59   Figure 3.10 Recommended two-part displacement profiles giving interstorey drift profiles defined in CSA A23.3 Clause 21.11 for Cantilever – walls.          60  3.4.2 Imposing 2D Displacements in 3D Models In this subsection, a series of general steps will be described to impose a set of 2D displacement profiles onto full 3D models generated in the commercial software ETABS. The starting point of this methodology is capturing the total displacements at the top of the GLRF (global drifts) for both directions of analysis. By using these displacements as inputs and employing the set of equations shown in Table 3.1, it is possible to calculate three different displacement profiles for each frame of the GLRF.  Please note that since three displacement profiles will be assigned to each frame: i) displacements imposed along the bottom half, ii) displacements imposed along the top half, and iii) displacements imposed along the full height, it will be necessary to work with three different models per frame. To illustrate this, Figure 3.11 shows an example for a frame oriented parallel to the coupled-wall direction of analysis.   61   Figure 3.11 Example of three models imposing displacements on a frame on the Coupled – wall direction.  To impose the displacements, it is necessary to define axially rigid members. These members are fictitious and will have one node connected to the column of the frame of interest, while the other node shall be connected to a pinned support. The displacements will then be assigned as "demands" to all the nodes with pinned supports that connect with the fictitious members. ETABS provides the Ground Displacement command as the tool to perform this task. Figure 3.12 illustrates an example of a displacement profile assigned along the full height of the shear wall building. 62   Figure 3.12 Example of the assignation of a displacement profile assigned along the full height of the building for the Cantilever – wall direction.  Since 2D displacement profiles are being assigned to 3D models, special attention needs to be directed toward preventing unreal torsional moments from affecting each storey level. This can be easily achieved by examining the plan symmetry of the project at hand (see Figure 3.13). 63   Figure 3.13 Isometric views of three different frames with displacement imposed along the full height.  In Figure 3.13 above, it is evident that for the building on the left-hand side, the displacements can be directly imposed on the central frame since it is fairly aligned with the center of rigidity of each level. However, on the right-hand side, to prevent torsional moments, the displacements need to be imposed on two different frames within the same model (these frames are not aligned with the centre of rigidity).  All the displacements imposed are stored in the software as "demands," linked to a specific linear load case (load pattern). To utilize these displacements in defining a nonlinear static case (pushover 64  case), the same load case must be specified as a reference. This can be achieved by selecting Load Pattern as the Load Type, and to ensure that all storey levels reach the level of displacements imposed, Full Load needs to be specified as the Load Application method. These settings can be configured in the Load Case Data window.  3.5 Modelling of Slab-Column Frames in ETABS 3.5.1 Columns Modelling For this procedure, columns need to be modeled as linear frame elements, with one frame per storey level, connecting them with slabs or beams from a particular level, as well as columns above or below. It is essential to consider appropriate levels of effective stiffness. In Clause 21.11.3.3.3 of CSA A23.3, it is specified that columns should be modeled with an effective stiffness Ec·Ie = 1.0·Ec·Ig. However, some may find this approach to be too conservative. Based on a comprehensive review of different references, including ASCE 41, CSA A23.3, and ACI 318, it is proposed to use an effective stiffness Ec·Ie = 0.7·Ec·Ig for all columns, as long as they do not experience a low level of axial compression (or net axial tension forces).  3.5.2 Slabs Modelling (Above Grade) Important attention needs to be given to how to model the slabs and their level of cracking, as this will dictate the magnitude of demands transferred into the columns, particularly the axial forces. Engineers may find guidance in the literature on how to better model slabs when the structure is subjected to lateral loading. The methodology presented here refers to the Equivalent Beam Width (EBW) procedure (Hwang & Moehle, 2000), which is also recommended in Appendix C of the LATBSDC Guidelines (Los Angeles Tall Buildings Structural Design Council, 2020). 65  The goal of the EBW method is to accurately represent the stripe of the slab that will be framing into the supports (walls or columns) with an equivalent frame member defined with the same thickness as the real slab but an effective width.  According to the LATBSDC Guidelines, it is highly important to explicitly model the slab to capture the coupling effect (outrigger effect) between the rigid core and the gravity columns. This effect will have a significant impact in cases where: • The span between the core and the columns is less than 6.10 m (20 ft), or • The span between columns is less than 3.05 m (10 ft).  To make the implementation of this modeling more feasible, minor modifications to the analytical model are allowed.  Figure 3.14 Modifications allowed to model EBW. Extracted from Appendix C of 2020 LATBSDC Guidelines.  66  Figure 3.14 above shows, on the left-hand side, the real floor plan of the building, while on the right-hand side, it can be observed that some columns were aligned with the core, and others were merged into an equivalent column.  To calculate the effective width of the frame members used to represent the slab stripes, the expression relies on the dimensions of the supports and the spans. For interior frames, the expression is as follows:  𝑏 =  2 · 𝑐1 +𝑙13  ( 2 )  Where, c1 = the dimension of the support parallel to the direction of analysis. l1 = the span length existing between supports (center to center) parallel to the direction of analysis. If the frame of interest is an exterior frame, the previous equation needs to be divided by a factor of 2. It is important to note that for each frame, two different widths will need to be modeled for each half of the span, ensuring that the effective width does not exceed the tributary width of the column or wall. For more clarity, please refer to Figure 3.15. 67   Figure 3.15 Limitations on Effective Beam Width; extract from Appendix C of 2020 LATBSDC Guidelines.  When performing this study (Hwang & Moehle, 2000), realized that by modeling the slab using the Equivalent Beam Width (EBW) procedure, the model would provide a good estimate for the uncracked stiffness of the element. However, when effects of cracking need to be considered, including a factor of 1/3 would lead to a more accurate assumption Ec·Ie = 0.33·Ec·Ig of for these frame members.  Once all the slabs have been modeled as equivalent EBW members, rigid diaphragms should be assigned to all the nodes of each storey level in the analytical model. The designer must ensure that the node with the pinned support (where the imposed displacements were assigned) is not connected to the diaphragm. For more clarity, please refer to Figure 3.16.  68   Figure 3.16 Example of correct assignation of rigid diaphragm for a storey level above grade.  3.5.3 Plastic Hinges for Effective Beam Width As mentioned at the beginning of this chapter, Clause 21.11.3.2 requires that a limit be set to the forces being transferred from the slabs to the columns so that they do not exceed the nominal capacity of the slabs. This can be accomplished by defining and assigning plastic hinges to the EBW frame members described previously.  When the refined design of the slab is not yet completed, a method to establish the values needed to be defined in the software as nominal capacities for negative bending moments is as follows: According to the Canadian approach, for gravity loads design, the slab should have enough factored resistance to meet the demands (Mr ≥ Mf) caused by the load combination: 1.25D + 1.50L (NBCC 2020, Table 4.1.3.2.-A).  69  For a lightly reinforced concrete section, the ratio between nominal to factored capacity is equal to 1 / 0.85 = 1.18. However, remaining on the conservative side, this factor can be rounded up to 1.25. This means that a method to estimate the nominal capacity of the slab could be as follows. Given that, 𝑀𝑟 = 𝑀𝑓  ( 3 )  And, 𝑀𝑛 = 1.25 · 𝑀𝑟  ( 4 )  Which means that the load combination leading to the nominal flexural capacity would be,  1.25 · (1.25𝐷 +  1.50𝐿) = 1.56𝐷 + 1.88𝐿  ( 5 )  However, when a ground motion occurs, the structure is already sustaining gravity loads. To maintain a conservative approach in this procedure, it is assumed that when the inertial forces induced by ground motions occur, the structure is only supporting dead load (1.0D). In this case, the flexural capacity defined for negative bending moments is the difference between the load combination resulting in the nominal capacity and the bending moment resulting from the gravity loads present on the structure at the same time when the seismic loads occur.   Assuming the uniform loads specified in subsection 2.1 of Chapter 2 act on the EBW frame members representing different slabs, bending moment diagrams (BMD) similar to those in Figure 3.17 can be obtained. To effectively capped the transfer of forces from the slabs to the gravity columns, the bending moments associated with the combinations of 1.56D + 1.88L and 1.0D need 70  to be subtracted (resulting in 111.8 kN·m – 47.3 kN·m = 64.5 kN·m). This value defines the flexural capacity in ETABS, as illustrated in Figure 3.18.        Figure 3.17 Negative bending moment diagrams due to different load combinations.   1.25D + 1.50L 1.56D + 1.88L 1.0D 1.0E 71   Figure 3.18 Example of plastic hinge definition to be assigned to an EBW frame in ETABS.  Note that there is a small variation in the reported bending nominal capacity in Figure 3.17 corresponding to the load case 1.0E, and the one highlighted in the figure above. The reason for this variation is that when plastic hinges are defined, a small secondary slope is considered to facilitate convergence in the analysis.  To estimate nominal capacities for positive bending moments, it is reasonable to consider the minimum amount of reinforcement required to provide structural integrity. 72   Figure 3.19 Analytical model in ETABS with plastic hinges defined and assigned to EBW frames only.  3.5.4 Slabs Modelling (Below Grade) and Boundary Conditions Since the total displacements needed to start the set of analyses for Clause 21.11 already include the effects of foundation movements, it is important to remain consistent with the model presented for the Simplified analysis of buildings. Ensuring that no translational displacement occurs below 73  the plastic hinge region (grade level) is crucial. To achieve this, the model in ETABS should include the following: 1. Lateral supports on the retaining walls. 2. Slabs modeled as membrane elements with rigid diaphragms assigned. The combination of these two items will prevent any lateral displacement at any node below grade level while allowing the columns located at these levels to rotate as much as they need to.  3.6 Effects of the Seismic Induced Deformations on Columns of GLRF 3.6.1 Axial Forces on Columns As shown in Table 4.1.2.3.-A NBCC 2020, to examine the total effects of axial loads on columns, the prescribed load combination is 1.0D + 1.0E + 0.5L + 0.25S. For the portion of the demand resulting from earthquake loads (imposed displacement profiles), the axial load considered is the maximum between the axial load from the model where the displacements were imposed along the bottom half of the building and the axial load from the model where the displacements were imposed along the top half of the building. This is particularly crucial if, the horizontal members (beams or slabs) are kept as linear members.  An exception could be made for cases where all the slabs are yielding at their connections with the column, independently of where the displacements are being imposed. In such cases, the axial load may be taken from the model where the displacements were imposed along the full height.  It is important to keep in mind that the higher the level of axial compression on a column, the lower its curvature capacity. Columns that will exhibit the greatest axial force demands will be the 74  ones closer to the core of the building, as this causes the slab connecting the column with the core to have a greater effective stiffness.  3.6.2 Bending Moments on Columns Considering that the bending moments transferred from the slabs to the columns are negligible, the bending moment demands that are important to observe are the ones resulting from the imposed displacements along the full height only (1.0E). This prevents capturing any discontinuity in the demands read from the column due to how the displacements are imposed.  As mentioned previously, the procedure that CSA A23.3 offers to examine if a column possesses an adequate level of ductility is Clause 21.11.3.3.3. Based on demand-capacity ratios (Mf/Mr), the level of axial load in the column, and the level of detailing in the cross-section of the column, we can assess if the column would accommodate the lateral displacement demands while bearing its corresponding gravity load.  The critical locations for the columns that need to be examined are the ones at grade level (plastic hinge region for the core) and the columns connecting (above and below) with horizontal members with high flexural stiffness, such as a transfer girder or a thick slab.  3.6.3 Shear Forces on Columns Similar to the bending moment demands, the shear forces affecting the gravity columns (or walls) need to be taken from the model with imposed displacements along the full height of the building only (1.0E). However, according to Clause 21.11.3.1, before examining the effects of these 75  demands on the columns, a previous step of filtering is required. First, the locations where the bending moment demands on the column exceed the probable flexural capacity (Mp) of the column must be identified. Since the column cannot admit a moment greater than its maximum capacity, the shear force needs to be adjusted accordingly. Figure 3.20 illustrates the application of this Clause.  Figure 3.20 Shear force demand adjusted for column where Mf exceeds Mp.  Once the shear forces have been adjusted for every location where the Mp is exceeded, the resulting shear force diagram may be taken as true and used to examine the effects on the vertical members of the GLRF.  Due to the presence of high axial compression, reinforced concrete sections benefit from increased shear force capacity. Therefore, the critical columns that require special attention are the ones located on the "tension side of the building," meaning the columns with a minor level of axial compression (or even net axial tension forces). These columns will require a higher ratio of transverse reinforcement. 76  Chapter 4: Application of CSA A23.3 Clause 21.11 to Case Studies 4.1 Introduction This chapter presents the application of the methodology outlined in Chapter 3 to the archetype development discussed in Chapter 2 of this thesis.  The focus of this examination was on the frames within the GLRF that exhibited the most significant lateral displacements at the roof level. Building upon the foundation laid out in Chapter 3, the following content of this chapter will provide guidance to engineers and consultants on conducting the simplified analysis of buildings, as presented in Clause 21.11.2.2.  In this chapter, the results of analyses conducted on two distinct GLFR scenarios are presented. The first scenario features a uniform structure, with all gravity columns and slabs being relatively slender members. The second scenario involves a 1.2 m-thick transfer slab located at a quarter of the building's height (Level L7). The interaction diagrams displayed in the subsections below encompass demands observed from Level L1 to L11, corresponding to the sections where the most substantial demands were encountered.  4.2 Displacements at the top of GLRF The displacements depicted in Figure 4.1 and Figure 4.2, observed at the upper portion of the GLRF, are outcomes of the Response Spectrum Analysis (RSA) carried out during the seismic analysis and design of the SFRS. This analysis constitutes a part of the content discussed in Chapter 2. These displacements take into consideration accidental torsion effects. However, no analyses 77  were conducted to refine the effective stiffness of the SFRS, nor to evaluate the structural drift below the plastic hinge.  For enhanced clarity, readers are encouraged to consult Figure 2.2, which serves as a reference for each column sampled in this study. 4.2.1 Displacements in Coupled – wall direction  Figure 4.1 Displacements at the top of GLRF in the Coupled direction.  78  The highlighted columns represent those that displayed the most significant global drifts in the context of this analysis direction. As displacement magnitude directly influences the demands imposed on columns, these specific columns were selected to pinpoint critical locations along their height. The subsequent sections of this chapter present these demands.  4.2.2 Displacements in Cantilever – wall direction  Figure 4.2 Displacements at the top of GLRF in the Cantilever direction. 79  In a manner analogous to the approach taken for the coupled-wall direction, the highlighted columns above correspond to those exhibiting the greatest global drifts within the cantilever-wall analysis direction.  4.3 Displacement profiles for GLRF As a component of this new approach for implementing Clause 21.11.2.2, three displacement profiles were generated to provide a conservative estimation of the axial forces within the gravity columns.  4.3.1 Displacement profiles for Coupled – wall direction To illustrate the 2D displacement profiles relevant to the coupled-wall analysis direction, refer to Figure 4.3, which displays the displacement profiles corresponding to the frame situated along axes 4 and 5. Additionally, Figure 4.4 presents the deformed shape along with the formation of plastic hinges, depicting the outcome of each model and its respective displacement profile. 80   Figure 4.3 Displacement profiles assigned for frame located along axes 4 and 5 in the Coupled direction. 81   Figure 4.4 Elevations of analytical models for frame in Coupled direction.82  4.3.2 Displacement profiles for Cantilever – wall direction Figure 4.5 portrays the 2D displacement profiles that were employed for the frame situated along axis E in the cantilever-wall analysis direction. Complementing this, Figure 4.6 visually presents the deformed shapes and plastic hinge formations specific to each model and displacement profile.  Figure 4.5 Displacement profiles assigned for frame D in the Cantilever direction.   83   Figure 4.6 Elevations of analytical models for frame in Cantilever direction.84  4.4 Effective widths and nominal flexural capacities for the slabs As outlined in the preceding chapter, the approach employed to model the slabs involved replacing them with frame elements possessing equal thickness as the slabs yet featuring an effective width. An integral aspect of this novel methodology involves constraining the bending moments of the slabs at the column faces. The dimensions of the effective cross-section of the frame member representing the slab, along with the maximum flexural capacities of the slabs at their connection with the gravity columns, are summarized in Table 4.1 and Table 4.2. These capacities align with the provisions specified in Clause21.11.3.2 Table 4.1 Nominal bending moment capacities in slabs for Coupled direction analysis.  Width [mm] Thickness [mm]3,772.50 190.00 31.27 39.374,302.50 190.00 92.09 57.535,350.00 190.00 107.63 67.723,764.20 190.00 31.27 40.113,772.50 190.00 31.27 77.514,302.50 190.00 92.09 112.405,350.00 190.00 107.63 132.503,764.20 190.00 31.27 78.904,272.50 190.00 31.31 75.074,302.50 190.00 92.09 113.975,350.00 190.00 107.63 133.864,264.20 190.00 31.31 75.944,272.50 250.00 43.55 83.244,302.50 250.00 170.51 121.805,350.00 250.00 192.35 140.044,264.20 250.00 43.55 86.24Coupled DirectionLevelsDimensions Positive BM [kN-m]Negative BM [kN-m]L12 to L29L2 to L11L1L3085  Table 4.2 Nominal bending moment capacities in slabs for Cantilever direction analysis.  In the scenario involving the transfer slab, no hinges were incorporated into the analytical models. This decision was made based on the observation that the substantial thickness of the transfer slab would prompt the connected columns to experience yielding before the slab itself. Thus, introducing plastic hinges within the slab would likely yield unconservative lateral load estimations for the columns. Refer to Figure 4.7 for an illustration depicting the probable flexural capacities of all members connected at the joint where the transfer slab is located.  Figure 4.7 Comparative of probable bending moment capacities of thick slab and columns.  Width [mm] Thickness [mm]3,620.00 190.00 31.25 34.635,237.50 190.00 107.57 55.883,620.00 190.00 31.25 68.025,237.50 190.00 107.57 109.694,120.00 190.00 31.30 64.575,237.50 190.00 107.57 111.194,120.00 250.00 43.54 73.565,237.50 250.00 192.24 115.19Cantilever DirectionLevelsDimensions Positive BM [kN-m]Negative BM [kN-m]L12 to L29L2 to L11L1L3086  4.5 Flexural analysis and design of Columns – Uniform GLRF This section presents the outcomes of the analysis and design process for the gravity columns highlighted in Figure 4.1 and Figure 4.2, corresponding to the coupled and cantilever directions, respectively. It's important to note that all axial force diagrams (AFD), shear force diagrams (SFD), and bending moment diagrams (BMD) presented in this section exclusively depict the effects of the applied displacements (1.0 E).   4.5.1 Corner columns for Coupled – wall direction The following demands were exhibit by the Columns L4 and D4 for the coupled – wall direction.  Figure 4.8 SFD and BMD for Columns L4 and D4 for Coupled direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line).  87  As the columns positioned at the corners do not directly frame with the rigid core, there exists no member bridging these two elements. Consequently, following the completion of these analyses, the axial force present in these columns was determined to be zero. This is the underlying reason why an axial force diagram (AFD) is absent from Figure 4.8.   Figure 4.9 PM interaction diagram for Columns L4 and D4 for Coupled direction.  To evaluate the adequacy of the columns' flexural and shear capacities, the axial load was combined according to Article 4.1.3.2 of NBCC 2020. As evident in Figure 4.8, the most significant demands occur at grade level. According to Clause 21.11.3.3.3, the columns at this elevation (refer to Figure 2.5) could potentially be assigned an amplification factor for their flexural capacity, ranging between 1.0 and 1.5 based on the axial load level. However, Figure 4.9 clearly illustrates that these columns lack the necessary level of ductility. 88  4.5.2 Interior columns Coupled – wall direction (linear analysis)                Figure 4.10 SFD and BMD for Columns L6 and D6 at the top; AFD for Column L6 at the bottom left, AFD for Column D6 at the bottom right for Coupled direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). 89  4.5.3 Interior columns Coupled – wall direction (nonlinear analysis)            Figure 4.11 Demands for Column L6; SFD and BMD at the top, AFD at the bottom for Coupled direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). 90  Above, in Figure 4.10 the column demands derived from a linear analysis are illustrated. In contrast, Figure 4.11 showcases the column demands arising from a nonlinear analysis. It is evident that the shear force and bending moment at ground level exhibit similarities in both figures. However, a notable difference emerges in the axial force diagram (AFD). In the context of this particular archetype, the axial force variation's uniformity throughout the column's height can be attributed to a specific observation: regardless of the location of the displacement profiles within the coupled-wall direction, all the slabs connecting with the columns have reached their yielding point. This fact is visually demonstrated in Figure 4.4.  Figure 4.12 PM interaction diagram for Column L6 for Coupled direction.  Figure 4.12 provides a visual representation of the reduction in the amplified factored flexural resistance for Column L6, attributed to the increased level of axial compression. 91                   Figure 4.13 Demands for Column D6; SFD and BMD at the top, AFD at the bottom for Coupled direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). 92   Figure 4.14 PM interaction diagram for Column D6 for Coupled direction.  Lastly, as shown in Figure 4.13 and Figure 4.14 above, the figures depict the demands and response of Column D6. 93  4.5.4 Corner columns for Cantilever – wall direction  Figure 4.15 SFD and BMD for Columns D12 and D4 for Cantilever direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line).   Continuing in a similar manner as for Columns L4 and D4, Figure 4.15 demonstrates the shear force and bending moment demands for Columns D12 and D4, specifically when the displacements are being applied for the cantilever direction of analysis. Notably, the demands for these columns are oriented parallel to the minor local axis of their cross-sections. 94   Figure 4.16 PM interaction diagram for Columns D12 and D4 for Cantilever direction.  Figure 4.16 provides evidence that these columns possess the necessary flexibility to accommodate the induced demands arising from the lateral displacements in this particular direction of analysis.  95  4.5.5 Interior columns Cantilever – wall direction (linear analysis)                Figure 4.17 SFD and BMD for Columns E11 and E5 at the top; AFD for Column E11 at the bottom left, AFD for Column E5 at the bottom right for Cantilever direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). 96  4.5.6 Interior columns Cantilever – wall direction (nonlinear analysis)             Figure 4.18 Demands for Column E11; SFD and BMD at the top, AFD at the bottom for Cantilever direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). 97  The same observations previously mentioned for Columns L6 and D6 in the coupled direction also apply to Columns E11 and E5. The key distinction is that the axial force diagram (AFD) in Figure 4.18 highlights a more pronounced variation in axial load when the displacements are applied along the top of the building. This distinction becomes evident in Figure 4.6, which substantiates this point by revealing that not all slabs connecting with columns reach their yielding point.  Figure 4.19 PM interaction diagram for Column E11 for Cantilever direction.  Upon observing Figure 4.19, it becomes evident that the present column design exhibits a satisfactory level of flexural resistance. 98                Figure 4.20 Demands for Column E5; SFD and BMD at the top, AFD at the bottom for Cantilever direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line).  99   Figure 4.21 PM interaction diagram for Column E5 for Cantilever direction.  Similar to Column E11, Figure 4.21 confirms that column E5 meets the necessary requirements to withstand the lateral displacement demand sin the cantilever direction.  4.6 Shear analysis and design of Columns – Uniform GLRF In accordance with Clause 21.11.3.1, all columns selected for this study underwent an examination to determine their capacity to withstand the shear demands previously illustrated in the figures above.    100  For the sake of clarity, the shear demands and resistances depicted in the subsequent figures have been plotted along their clear height of the column. However, as per Clause 11.3.6.4, the demands that need to be considered to ensure adequate shear resistance were captured at a distance dv from the face of the supports. In the case of these columns, this distance was 720 mm.  4.6.1 Corner column for Coupled – wall direction  Figure 4.22 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Columns L4 and D4 for Coupled direction.  Regarding Columns L4 and D4 located at the corners, the bending demands at grade level exceeded the probable flexural capacity (Mp) of the columns. Consequently, it became necessary to restrain the shear demand (Vf,) to a level that aligns with the shear force associated when the column reaches its maximum flexural capacity Mp for that level of axial load at one of its ends. 101  4.6.2 Interior column Coupled – wall direction (nonlinear analysis)  Figure 4.23 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Column L6 for Coupled direction.  102   Figure 4.24 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Column D6 for Coupled direction.  It is intriguing to observe that while the columns situated at the lower levels of the building benefit from enhanced protection due to a tighter spacing of ties (150 mm), as depicted in Figure 4.22, Figure 4.23, and Figure 4.24, the columns within the coupled direction still fell short of satisfying the necessary shear resistance.   103  4.6.3 Corner column for Cantilever – wall direction  Figure 4.25 PM (probable capacity) interaction diagram on the left-side; shear demand and shear resistance for Columns D12 and D4 for Cantilever direction.  In the cantilever-wall direction of analysis, Figure 4.25 demonstrates that Columns D12 and D4 located at the corners exhibit adequate shear resistance.  104  4.6.4 Interior column Cantilever – wall direction (nonlinear analysis)  Figure 4.26 PM (probable capacity) interaction diagram on the left-side; shear demand and shear resistance for Column E11 for Cantilever direction.   105   Figure 4.27 PM (probable capacity) interaction diagram on the left-side; shear demand and shear resistance for Column E5 for Cantilever direction.  Regarding Columns E11 and E5, which are part of interior frames, Figure 4.26 and Figure 4.27 provide confirmation that the original design, along with ties having a 150 mm separation, enables these columns to withstand the demands arising from lateral displacements in this analysis direction.       106  4.7 Summary of demands of Columns - Uniform GLRF Table 4.3 below offers a concise overview of the demand-to-capacity ratios for the column locations highlighted at the beginning of this chapter, representing those with the most significant demands. Although the columns in the cantilever direction exhibited demand-to-capacity ratios below 1.0, their inclusion in the summary serves to facilitate a comparison between the two analysis directions.  Table 4.3 Summary of demand-to-capacity ratios for columns part of Uniform GLRF.      Direction Column Level Mf / Factor · Mr Vf / Vr Ps / f'c ·AgCoupled L4 P1 0.89 1.52 0.21Coupled L4 L1 1.18 1.12 0.26Coupled D4 P1 0.89 1.52 0.21Coupled D4 L1 1.18 1.12 0.26Coupled L6 P1 0.87 1.04 0.27Coupled L6 L1 1.39 0.95 0.34Coupled D6 P1 0.87 1.45 0.21Coupled D6 L1 1.17 1.12 0.26Cantilever D12 P1 0.14 0.05 0.21Cantilever D12 L1 0.19 0.04 0.26Cantilever D4 P1 0.14 0.05 0.21Cantilever D4 L1 0.19 0.04 0.26Cantilever E11 P1 0.43 0.44 0.26Cantilever E11 L1 0.62 0.35 0.32Cantilever E5 P1 0.40 0.44 0.20Cantilever E5 L1 0.52 0.34 0.25Uniform GLRF107  4.8 Ductile redesign of Columns – Uniform GLRF Given that Column L6 exhibited the most notable capacity exceedance (1.39), it was selected as an example to illustrate the process of enhancing column ductility. It's important to note that no subsequent analysis involved the alteration of cross-sectional dimensions, nor did it encompass adjustments to the seismic increment. Similarly, no distinct demands were considered for this column due to the heightened flexural stiffness resulting from the modification.  Figure 4.28 Cross-section of redesigned Column L6 for Coupled direction.  To enhance the ductility of this column, the necessary step is to increase the width of the column. This adjustment will ensure that the cross-section aligns with the dimensional specifications stipulated in Clause 21.4.2.2.   108   Figure 4.29 PM interaction diagram for redesigned Column L6 for Coupled direction.  With the increase in width of the cross-section, there is a corresponding increase in the factored capacity Mr. Nonetheless, the primary advantage of meeting the dimensional requirements lies in the notable rise of the amplification factor, evident in Figure 4.29 as compared to Figure 4.12.  109    Figure 4.30 PM (probable capacity) interaction diagram on the left-side; shear demand and shear resistance for redesigned Column L6 for Coupled direction.  By reducing the tie spacing from 150 mm to 120 mm, Figure 4.30 serves to justify that the column now possesses sufficient shear resistance.        110  4.9 Flexural analysis and design of Columns – GLRF with Transfer slab A thick transfer slab was integrated into the analytical model within ETABS. The transfer was modeled at level L7, which stands at an elevation of 21.14 m above grade level. While the seismic weight was kept unchanged for this scenario for simplicity, the increased weight was accounted for the gravity loads acting on the columns. The forthcoming sections of this chapter exclusively present diagrams related to interior Columns L6 and D6 for the coupled-wall direction, and Columns E11 and E5 for the cantilever-wall direction. This selection is made since the columns positioned at the corners, lacking a frame connecting them with the core, yield the same outcomes as the results detailed above.     111  4.9.1 Interior column Coupled – wall direction (nonlinear analysis)                 Figure 4.31 Demands for Column L6 with Transfer slab; SFD and BMD at the top, AFD at the bottom for Coupled direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). 112   Figure 4.32 PM interaction diagram for Column L6 with Transfer slab for Coupled direction.  The demands that surpass the column’s ductility, ranked from the most to least severe, are as follows: i) the cross-section immediately above the thick slab, ii) the cross-section directly below the transfer slab, and iii) the cross-section located at grade level. It’s important to note that the amplification factor of 1.36 was applied to the demand with an axial compression below 8,400 kN (0.4·f’c Ag). For the other two demands, an amplification factor of 1.0 was employed in accordance with Clause 21.11.3.3.3. Notably, the inclusion of the thick slab led to a notable increase in axial load within the column, as evident in Figure 4.31.  113                   Figure 4.33 Demands for Column D6 with Transfer slab; SFD and BMD at the top, AFD at the bottom for Coupled direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). 114    Figure 4.34 PM interaction diagram for Column D6 with Transfer slab for Coupled direction.  In contrast to Column L6, an amplification factor of 1.50 was applied uniformly to all the demands shown in Figure 4.34. This decision was based on the consideration that the axial compression level for all these demands remained below 4,200 kN (0.2·𝑓𝑐′ Ag).    115  4.9.2 Interior column Cantilever – wall direction (nonlinear analysis)                Figure 4.35 Demands for Column E11 with Transfer slab; SFD and BMD at the top, AFD at the bottom for Cantilever direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). 116   Figure 4.36 PM interaction diagram for Column E11 with Transfer slab for Cantilever direction.  The same comment previously mentioned for Column L6 in the coupled direction also holds true for Column E11 in terms of the applied amplification factor.  117                   Figure 4.37 Demands for Column E5 with Transfer slab; SFD and BMD at the top, AFD at the bottom for Cantilever direction: displ. at bottom (dark blue); displ. at top (light blue); and displ. along full height (red dashed-line). 118   Figure 4.38 PM interaction diagram for Column E5 with Transfer slab for Cantilever direction.  Similarly, the comment made for Column D6 in the coupled direction is applicable to Column E5 in terms of the amplification factor.       119  4.10 Shear analysis and design of Columns – GLRF with Transfer slab The diagrams displayed below, illustrating the demands and capacities, pertain to the columns situated directly above the thick slab (level L8). It's crucial to remember that at this elevation, the column cross-sections are fortified against shear demands through the use of ties with a specified separation of 250 mm.  4.10.1 Interior column Coupled – wall direction (nonlinear analysis)  Figure 4.39 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Column L6 for Coupled direction.   120   Figure 4.40 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Column D6 for Coupled direction.   121  4.10.2 Interior column Cantilever – wall direction (nonlinear analysis)  Figure 4.41 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Column E11 for Cantilever direction.  122   Figure 4.42 PM (probable capacity) interaction diagram on the left-side; shear demands and shear resistance for Column E5 for Cantilever direction.  Across all the scenarios depicted above, where the bending moment demands from linear columns (Mf) significantly exceed the probable bending moment capacity (Mp), the adjustment in the shear force demand (Vf) becomes more pronounced.        123  4.11 Summary of demands of Columns – GLRF with Transfer slab Similar to the approach taken for the Uniform GLRF, Table 4.4 offers a summarized overview of the demand-to-capacity ratios for the columns examined following the inclusion of the transfer slab. Notably, the increase in axial load for the columns positioned beneath the thick slab has led to some demands that were originally within the yielding surface of the interaction diagrams now being displaced outside this region. For instance, consider the case of Column E11 at level L1 as demonstrated in Table 4.3. Initially, it displayed a flexural demand-to-capacity ratio of 0.62. However, due to the increased axial compression resulting from the thick slab, this ratio spiked to 1.25. Table 4.4 Summary of demand-to-capacity ratios for columns part of GLRF with Transfer slab.   Direction Column Level Mf / Factor · Mr Vf / Vr Ps / f'c ·AgCoupled L6 P1 1.60 0.96 0.44Coupled L6 L1 3.30 0.72 0.57Coupled L6 L7 2.64 3.00 0.50Coupled L6 L8 2.74 2.58 0.26Coupled D6 P1 1.03 1.90 0.09Coupled D6 L1 1.15 1.16 0.11Coupled D6 L7 1.63 6.48 0.05Coupled D6 L8 2.55 2.74 0.20Cantilever E11 P1 0.69 0.44 0.42Cantilever E11 L1 1.25 0.35 0.54Cantilever E11 L7 1.87 2.61 0.47Cantilever E11 L8 2.02 2.44 0.25Cantilever E5 P1 0.56 0.58 0.09Cantilever E5 L1 0.55 0.48 0.10Cantilever E5 L7 1.18 5.62 0.05Cantilever E5 L8 2.02 2.54 0.19GLRF with Transfer Slab at Level L7124  4.12 Ductile redesign of Columns – GLRF with Transfer slab Figure 4.43 illustrates the implications of enhancing the capacity of Column E11 to attain the necessary ductility for accommodating demands arising from lateral displacements. In the presented redesign, the column was treated as a moderately ductile member. Consequently, not only did the cross-sectional width need to be increased, but the level of detailing followed stricter requirements. This change resulted in a significant increase in the quantity of transverse reinforcement.  Figure 4.43 Cross-section of redesigned Column E11 with Transfer slab for Cantilever direction.  As depicted in the Figure above, longitudinal reinforcement was added to support the additional ties required. The increase is the column’s width is also evident.     125   Figure 4.44 Elevation of redesigned Column E11 with Transfer slab for Cantilever direction.  Figure 4.44 serves to provide a clearer depiction of the requisites for enhancing a member’s ductility. Notably, this Figure underscores the increase in horizontal reinforcement. Not only were the rebar diameters modified from 10M to 15M, but confinement reinforcement was also required in accordance with Clause 21.2.8.2. It is important to highlight that due to the column’s short length outside of the confinement regions, the decision was made to maintain the same tie spacing as required within the confinement regions. 126   Figure 4.45 PM interaction diagram for redesigned Column E11 with Transfer slab for Cantilever direction.  As a result of striving for a more ductile design, the substantial increase in the amplified factored resistance is quite evident. Notably, the amplification factor of 2.93 shown above, corresponds to the demand captured above the transfer slab at level L8 (Pf ≈ 5,000 kN). For the remaining demand that surpasses the factored resistance with an axial compression exceeding 9,600 kN (0.4·f’c Ag), an amplification factor of 2.0 would be applied.    127   Figure 4.46 PM (probable capacity) interaction diagram on the left-side. Shear demands and shear resistance for redesigned Column E11 with Transfer slab for Cantilever direction.  At this elevation, despite possessing a higher probable flexural capacity Mp, the shear demand required to be adjusted to correspond to Mp acting at the lower end of the column. The revised arrangement of ties, as illustrated in in Figure 4.43 and Figure 4.44, provides sufficient shear capacity to effectively withstand the demands at this location.    128  Chapter 5: Applying NBCC 4.1.8.23 to Case Study Building 5.1 Introduction In its 2020 edition, NBCC introduced Article 4.1.8.23. This particular article adds further performance requirements that are applicable to post-disaster buildings, high importance buildings, and a specific subset of normal importance buildings.  Specifically, for Normal Importance Category buildings located in Seismic Category SC4, which is the case for many buildings located in the urban areas of British Columbia, with a height above grade of more than 30 metres, the structural framing elements not considered to be part of the SFRS must be designed to ‘behave elastically’ when subjected to ground motions with a 10% probability of exceedance in 50 years. This chapter presents every step of the process that was carried out to apply the additional performance requirements to the prototype building with a uniform GLRF.  5.2 Ground Motions with 2,475-Year and 475-Year Return Period  Based on the characteristics of the archetype building outlined in Chapter 2, Article 4.1.8.23-4 in NBCC 2020 mandates an additional seismic analysis for this building. This analysis involves utilizing spectral accelerations with a 10% probability of exceedance over a 50-year period, equivalent to a return period of 475 years.  For context, Figure 4.1 provides a visual representation of these reference spectral accelerations, which are to be employed in the supplementary analysis. This figure also offers a comparison with the previously discussed spectral accelerations from Chapter 2. The latter accelerations were 129  employed in analyzing the structure's response to an event with a 2% probability of exceedance over a 50-year period, corresponding to a return period of 2,475 years. Moving forward, Figure 5.2 illustrates a comparison of spectral displacements. These displacements correspond to events with varying return periods and thus differing intensities.  Figure 5.1 Spectral accelerations for seismic events with different return periods.  Both spectra presented above were generated employing the Log-Log interpolation method outlined in the 'Notes to Part 4' section of the NBCC 2020. The highlighted spectral accelerations depicted in the figures align with the characteristic periods stipulated by the code. 1.010.6990.400.2480.0683 0.03040.5080.3420.1850.0970.0234 0.009140.00.20.40.60.81.00.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Spectral acceleration [1/g]Natural Period, T [sec]2,475 years 475 years 130   Figure 5.2 Spectral displacements for seismic events with different return periods.  5.3 475-Year Analytical Model Unlike the seismic analysis conducted to design the SFRS (with a 2,475-year return period), this supplementary analysis requires the incorporation of the GLRF in the model to be used for this pilot study. To achieve this, a new model was developed. This new model closely resembles the ones utilized for all the analyses presented in Chapter 4 of this thesis. However, certain distinctions have been considered, such as the inclusion of an unrestrained structure below grade level and the absence of strut members employed to enforce displacements. For a clearer understanding of this model, Figure 5.3 provides visual representations extracted from the ETABS model. 94399247424755521469614522701002003004005006007008000.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Spectral displacement [mm]Natural Period, T [sec]2,475 years 475 years 131   Figure 5.3 3D view and elevations along the coupled and cantilevered walls of analytical model for 475 year-return-period. 132  5.3.1 Initial assumptions for effective stiffnesses As CSA A23.3 had not yet determined the recommended effective stiffnesses for the 10% in 50 year analysis, the following effective stiffnesses were assumed in the current study: • Structural walls → 0.90·Ig and 0.90 Ag  • Coupling beams → 0.35·Ig and 0.40 Ag • Gravity columns → 0.70·Ig and 1.00 Ag  •  Slabs → Elastic Beam Width model, as in chapter 4 with 0.33·Ig These values were defined into the model as starting points, initiating the iterative process aimed at enhancing and refining the effective stiffnesses of walls and columns mainly.  5.3.2 Refined effective stiffness for structural walls Given that the structural walls within the SFRS exhibit considerably greater lateral stiffness in comparison to the gravity columns, they hold a dominant influence over the building's lateral response. As a consequence, prioritizing the enhancement of their flexural stiffness levels took precedence before addressing the columns. To guide the process, Equation 21.2 from CSA A23.3 was selected as the preferred method to establish the updated stiffness values. For convenience, the equation is provided below: 𝛼𝑤 = 1.0 − 0.35 (𝑀𝑒𝑀𝑛− 1.0) , 𝑏𝑢𝑡 0.5 ≤ 𝛼𝑤 ≤ 1.0  ( 6 )  Where,  𝑀𝑒 = 𝑀𝑓 · 𝑅𝑑𝑅𝑜  ( 7 )  133  Following the calculation of axial loads affecting each wall pier in both coupled and cantilever directions of analysis and building upon the design of the core at the plastic hinge location outlined in Chapter 2, Response-2000 (Bentz & Collins, 2000) was employed as the computational tool to determine the nominal flexural capacities of each wall.                           Figure 5.4 Response-2000 models for each wall pier in the positive Coupled – wall direction; Wall -1 at the top, Wall -2 at the middle and Wall -3 at the bottom. 134  Considering that the axial compressions exhibited in the walls were 3,760 kN for Wall – 1; 19,691 kN for Wall – 2; and 40,402 kN for Wall – 3, the following nonlinear bending moment – curvature responses were obtained.  Figure 5.5 Bending moment – curvature responses in the Coupled – wall Direction.  Recalling that these walls are interconnected by rigid diaphragms, it's imperative that they possess compatible curvatures. To achieve this, the nominal flexural capacities were combined, corresponding to a curvature of 8 rad/km. Additionally, the supplementary moment generated by the axial forces inherent in the walls, multiplied by the lever arm to the core's centroid, was factored in. The culmination of these considerations yields a comprehensive total nominal flexural capacity in this direction, denoted as Mn = 182,365 kN·m.  135   In the course of linear dynamic analysis, where forces were reduced by an overstrength factor (RdRo = 1.30), the core encountered a demand for bending moment, denoted as Mf = 140,116 kN·m. This translated to a maximum elastic moment, denoted as Me = 182,151 kN·m. Plugging these values into Equation 21.2 in CSA A23.3-19 (Equation 6 shown above) yielded an αw value of 1.0.  A similar methodology was pursued for the cantilever walls. Figure 5.6 showcases the Response-2000 models that were devised to capture moment-curvature relationships for each individual wall. In these models, axial compressions were considered as follows: 21,142 kN for Wall-1, 19,772 kN for Wall-2, and 22,939 kN for Wall-3.  Upon calculation, the cumulative nominal flexural capacity emerged as Mn = 559,743 kN·m. In contrast, the demands extracted from the ETABS model tallied Mf = 213,758 kN·m, equating to a corresponding Me = 277,885 kN·m. Substituting these values into Equation 21.2 resulted in an αw of 1.18, thus confirming the validity within the limit of 1.0.  Based on these results, an initial inference can be drawn: the structural walls exhibit a moderate level of cracking. However, this inference was further substantiated through individual assessments for each wall. These specific confirmations will be discussed in a later section of this chapter. It is worth noting that after conducting the analysis while assuming the structural walls remained uncracked in both directions, the model was rerun to extract updated demands. This enabled the verification of αw values. 136   Figure 5.6 Response-2000 models for each wall pier in the Cantilever – wall direction; Walls 1 and 2 at the top and Wall -3 at the bottom. 137  5.3.3 Refined effective stiffness for gravity columns In the case of the gravity columns, a more iterative approach was essential due to the utilization of a nonlinear bending-moment curvature response. Similar to the procedure for the walls, a series of iterations ultimately concluded in the consideration of the columns as uncracked (EcIe = 1.0). Taking reference from the previously presented plan view above grade level, as featured in Chapter 2 of this thesis, an examination of the coupled direction of analysis reveals that column D4 exhibits the greatest demand, registering Mf = 310 kN·m at grade level.   Figure 5.7 Bending moment – curvature response for column D4 at grade level; Coupled-wall Direction.  In the context of the cantilever direction of analysis, it was observed that column E11 exhibit the greatest flexural demand of Mf = 188 kN·m, at grade level. 138   Figure 5.8 Bending moment – curvature response for column E11 at grade level; Cantilever-wall Direction.  As evident from Figure 5.7 and Figure 5.8, the curvatures corresponding to the bending moments that align with the demands extracted from the ETABS model vividly illustrate that the columns remain well within the elastic range of their response.  5.4 2,475-Year and 475-Year Responses To dive deeper into the research and gain a comprehensive understanding of underlying factors contributing to the notably low demands derived from the 475-year analysis, a succinct comparative exercise was undertaken.  139  5.4.1 Effective Stiffness Table 5.1 provides a concise overview of the effective stiffness employed for the seismic analysis showcased in Chapter 2, as well as the stiffness values computed in the preceding discussion. Table 5.1 Effective stiffnesses for different models.   5.4.2 Natural periods and mode shapes Table 5.2 illustrates a comprehensive comparison of the natural periods and corresponding mode shapes across each model. Table 5.2 Natural periods and mode shapes for different models.   5.4.3 Displacements Table 5.3 provides a consolidated presentation of the spectral displacements corresponding to the periods outlined above, encompassing each of the analytical models. MemberWalls 0.5 Ig 0.5 Ag 1.0 Ig 1.0 AgCoupling Beams 0.25 Ig 0.45 Ag 0.25 Ig 0.45 AgSlabs (EBW) - - 0.33 Ig 1.0 AgColumns - - 1.0 Ig 1.0 Ag2,475 years 475 years1 6.44 69 0 0 4.11 71 0 02 4.82 0 67 0 3.20 0 70 03 2.56 0 0 79 2.05 0 0 80Period [sec]Mass Participation [%]Modex - dir. (coupled)y - dir. (cantileverz - dir. (torsion)2 % probability of exceedance in 50 years 10 % probability of exceedance in 50 yearsPeriod [sec]Mass Participation [%]x - dir. (coupled)y - dir. (cantileverz - dir. (torsion)140  Table 5.3 Spectral displacements for different models.   From the analytical models, the displacements were extracted at the top of the GLRF, specifically at level L30. Notably, the displacements arising from the 475-year seismic analysis are significantly reduced—by 75% for the coupled-wall direction and 70% for the cantilever direction—compared to the displacements stemming from the 2,475-year seismic analysis. Table 5.4 Maximum displacements resulting from dynamic linear analysis.   Upon observing Table 5.1, a discernible trend emerges: the stiffness of the analytical model for the 475-year analysis surpasses its counterpart more than twofold. Conversely, in the context of the 2,475-year analysis, the effective stiffness allocated to the walls amounts to merely half of their initial stiffness. In this scenario, the contribution originating from the GLRF also plays a role. As a result, the displacements experienced by the walls in both analysis directions are notably diminished in comparison to those obtained from the seismic analysis based on an event featuring a 2% probability of exceedance within a 50-year span. Model Mode 1 Mode 2475 years 133 1192,475 years 524 415Ratio 0.25 0.29Spectral Displacements [mm]Model Coupled Direction Cantilever Direction475 years 197 1802,475 years 782 629Ratio 0.25 0.29Displacement at Roof Level [mm]141  Given that the walls remain structurally intact, there exists no requirement to amplify the interstorey drifts in accordance with Clause21.11 of CSA A23.3. With the aim of presenting a lucid comparison of the foregoing discussions, Figure 5.9 and Figure 5.10 visually depict the interstorey drift ratios for each analysis direction.  Figure 5.9 Comparative on interstorey drift ratios in the Coupled-wall Direction.  142   Figure 5.10 Comparative on interstorey drift ratios in the Cantilever-wall Direction.         143  5.5 Influence of Uncracked Rigid Core Prior to embarking on the endeavor to quantitatively assess the extent to which the structural walls contribute to maintaining the relatively low demands on the gravity columns, a comprehensive validation process was conducted. This involved crafting a nonlinear bending moment-curvature response for each individual wall pier across both analysis directions.  5.5.1 Uncracked structural walls in the Coupled Direction. Revisiting the models created in Response-2000 to represent the coupled walls (refer to Figure 5.4) and updating the axial forces exhibited in the walls due to the change in stiffness, the subsequent moment-curvature responses were derived.  Figure 5.11 Bending moment – curvature response for Wall -1 subjected to an axial compression of 2,586 kN in the Coupled-wall Direction.  144   Figure 5.12 Bending moment – curvature response for Wall -2 subjected to an axial compression of 19,673 kN in the Coupled-wall Direction.  Figure 5.13 Bending moment – curvature response for Wall -3 subjected to an axial compression of 41,594 kN in the Coupled-wall Direction. 145   5.5.2 Uncracked structural walls in the Cantilever direction. Following a similar methodology to that employed for the coupled walls, the models depicted in Figure 5.6 were utilized to extract the moment – curvature responses using Response-2000.   Figure 5.14 Bending moment – curvature response for Wall -1 subjected to an axial compression of 21,142 kN in the Cantilever-wall Direction.  146   Figure 5.15 Bending moment – curvature response for Wall -2 subjected to an axial compression of 19,772 kN in the Cantilever-wall Direction.  Figure 5.16 Bending moment – curvature response for Wall -3 subjected to an axial compression of 22,939 kN in the Cantilever-wall Direction. 147  An intriguing observation emerges from the analysis: across both analysis directions, Wall – 2 was the sole wall pier that exhibited a comparatively higher curvature. Nonetheless, as mentioned earlier in this chapter, the interconnection of all walls through the slab—possessing substantial axial stiffness—enables Wall – 1 and Wall – 3 to effectively offset the influence of Wall – 2. With the confirmation that the rigid core remains within the elastic range, endeavors were undertaken to construct a novel model. This model isolated the SFRS in an uncracked state, while retaining the precise seismic weight.  Figure 5.17 3D views of analytical models in ETABS for 475-year analysis; on the left-side the SFRS + GLRF model; on the right-side the SFRS Only model. 148  Table 5.5 provides an overview of the variations in the natural periods upon disregarding the contribution originating from the GLRF. Notably, the variations in the two principal directions of analysis did not surpass the threshold of 10%. Table 5.5 Comparative on natural periods and mode shapes on different 475-year models.  Following the execution of a modal analysis, the subsequent step involved conducting Response Spectrum Analysis to ascertain the magnitude of the disparities in the reported demands within the core. To facilitate a clearer comparison, the base shears in the SFRS Only model were adjusted through scaling to align with the base shears in the SFRS + GLRF model.      1 4.11 71 0 0 4.52 70 0 02 3.20 0 70 0 3.47 0 69 03 2.05 0 0 80 2.08 0 0 80ModeSFRS OnlyPeriod [sec]Mass Participation [%]x - dir. (coupled)y - dir. (cantileverz - dir. (torsion)SFRS + GLRFPeriod [sec]Mass Participation [%]x - dir. (coupled)y - dir. (cantileverz - dir. (torsion)149   Figure 5.18 Normalized interstorey drifts ratio due to RSA with equal base shears; on the left-side Coupled-wall Direction, on the right-side Cantilever-wall Direction.  The normalized interstorey drift ratios depicted above serve as evidence that the GLRF exerts minimal influence over the control of lateral displacements within the structure. The most notable discrepancy accounts for approximately a mere 10% variation.    150  5.5.3 Lateral demands on Coupled – wall direction Mirroring the approach undertaken for the interstorey drift ratios, a comparative analysis was conducted for the demands extracted from the rigid core within the context of the coupled-wall direction of analysis. Figure 5.19 presents the shear force and overturning moment diagrams, mapped across the height of the core above grade level.  Figure 5.19 Normalized demands due to RSA with equal base shears.; Coupled-wall Direction.      151  5.5.4 Lateral demands on Cantilever – wall direction Next, we introduce the shear force and overturning diagrams that correspond to the cantilever – wall direction of analysis.  Figure 5.20 Normalized demands due to RSA with equal base shears; Cantilever-wall Direction.  Upon a thorough examination of Figure 5.19 and Figure 5.20, a compelling deduction emerges. The substantial lateral stiffness and strength of the uncracked walls, play a significant and pivotal role in governing the overall lateral response of the building.   152  5.6 Observations on low demands in Gravity Columns After conducting the sets of analyses detailed in the preceding sections of this chapter, the exploration has led to the identification of five fundamental factors that collectively contribute to the notably subdued demands affecting the gravity columns part of the uniform GLRF of this archetype building. Among these factors, two are a direct outcome of the stipulations set forth by NBCC 2020 for this supplementary analysis: • Reduced spectral accelerations and consequently diminished spectral displacement resulting from a seismic event with a 475-year return period, in contrast to the 2,475-year return period. • Article 4.1.8.23 of NBCC 2020, in this context, presents no requisites for scaling adjustments within this seismic analysis. •  The remaining three factors, closely linked to the structural characteristics of this building, include: • Low global drift ratios. • Substantial absorption of lateral demands by the structural walls. • Absence of the need to amplify interstorey drifts in accordance with Clause 21.11 in CSA A23.3.  The first two factors are a direct outcome of the system's high lateral stiffness, with the primary contribution emanating from the structural walls. The third factor is an inherent consequence of the uncracked condition of the structural walls. Given the absence of damage or plastic deformation within these elements, the requirement for interstorey drift amplification to account for nonlinear deformation concentrations is obviated. 153  In essence, by considering the structural walls as uncracked, a potent defensive mechanism is established for the gravity columns, effectively channeling the bulk of the demands towards the walls. Additionally, the rigid structural walls serve as a safeguard, shielding the GLRF from significant lateral displacements. A summarized overview of the discussions is encapsulated in Table 5.6. Table 5.6 Summary of factors contributing to low demands on gravity columns.     Contributions SourceSmall spectral displacements / accelerations NBCC 2020No scaling of forces requirement for NBCC 4.1.8.23 NBCC 2020Small lateral displacements at roof level High lateral stiffness from uncracked walls and columnsWalls resisting most of the demands Walls remaining uncracked No amplification of interstorey drift (Clause 21.11) No inelastic flexural and shear demands Main Contributions to Low Demands on Gravity Columns154  Chapter 6: Final Discussion and Conclusions 6.1 Background  NBCC mandates that the SFRS must be designed to resist 100% of earthquake demands from ground motions having a 2% probability of exceedance in 50 years. In high-rise core wall buildings, the SFRS typically consists of ductile coupled walls in one direction and ductile (cantilever) shear walls in the other direction. NBCC also requires that all members that are part of the GLRF (not part of the SFRS), must either behave elastically or have sufficient non-linear capacity to support the applied gravity loads while undergoing earthquake-induced deformations. Clause 21.11 in CSA A23.3 provides further guidance on how to analyze and design the GLRF members under the seismic deformation demands. The 2020 edition of NBCC requires that all buildings in Seismic Category SC4 with a height above 30 metres must have structural framing elements not part of the SFRS (GLRF) that behave elastically for ground motions with a 10% probability of exceedance in 50 years.  The objectives of this thesis can be summarized very simply as (i) investigate different procedures for implementing the analysis requirements of CSA A23.3 Clause 21.11 on a typical core wall building using ETABS and developing recommendations on how to implement the recommended procedure; (ii) study a number of different arrangements of GLRF to determine whether the Clause 21.11 requirements are likely to require design changes for a typical GLRF, and (iii) conduct a pilot study on the new NBCC requirements for the GLRF to remain elastic for ground motions having a 10% probability of exceedance in 50 yrs. A summary of the work that was done for each of these high-level objectives and the conclusions that were arrived at are summarized in the 155  sections below. In order to undertake the current study, a prototype building design had to be completed and this is described in the first section below.  6.2 Design of Prototype Building The archetype building used in the studies conducted for this thesis is based on the example provided in Section 11.5 of Chapter 11, of the 4th edition of the Cement Association of Canada Concrete Design Handbook. The core layout (SFRS) and overall geometry of the building were kept the same. Special attention in this study was devoted to the design of the GLRF. After establishing the column layout for this archetype building, the columns were designed to resist the demands resulting from their tributary area and load takedown. The design of the columns followed the relevant sections outlined in CSA A23.3-19.  To assess the demands on the core, dynamic wind and seismic analyses were performed according to the updated 2020 edition of NBCC. It was observed that, in the context of the wind analysis, the demands remained the same as the ones that were specified by NBCC 2015. Specifically, the reference velocity pressure (q) remained at 0.45 for Vancouver City Hall.  In terms of seismic analysis, the 2020 edition of NBCC allows engineers to generate a more accurate Uniform Hazard Spectrum (UHS) using the specific value of Vs30 for the site rather than the site class and using log-log interpolation for the spectrum. As a result of these two changes in the definition of the design spectrum, the seismic demands for the prototype building considered in this study decreased by almost 30% compared to the design according to the 2015 NBCC.  156  The design of the core at grade was primarily governed by wind loads (refer to Figures 2.19 to 2.21), and the detailed design of the walls and grade (Figure 2.22) was kept the same as in the example in the handbook. This consistency is particularly significant for the work conducted in the pilot study discussed in Chapter 5, where the effective stiffnesses of the SFRS were refined based on the nominal flexural capacities (Mn) of the walls.  For all the work involving Clause 21.11, a crucial aspect was obtaining the estimation of the maximum displacements at the top of the GLRF (as shown in Figure 4.1 and Figure 4.2). This was achieved by conducting linear dynamic (modal response spectrum) analysis using the updated hazard values corresponding to NBCC 2020.  6.3 Implementation of Simplified Analysis Procedure in CSA A23.3 Clause 21.11 Canadian Standard CSA A23.3 Clause 21.11 specifies the requirements for the design of reinforced concrete GLRF subjected to earthquake-induced deformations. The simplified analysis procedure prescribes an envelope of interstorey drifts that depends on the lateral displacements at the top of the GLRF.  One of the main contributions of this thesis was developing a simplified procedure to impose the interstorey drift envelope onto a building using the commercial software ETABS, which is widely preferred by consulting engineering firms. As described in Chapter 3 of this thesis, a number of different approaches were examined. Some of these approaches were: i) conducting a static non-linear analysis (pushover) per storey level, ii) the application of equivalent lateral loads that would require iterations to achieve the target interstorey drift, iii) the application of concentrated 157  moments and forces on the walls part of the SFRS, while reducing the stiffness of the GLRF, among others. These options were ultimately discarded, primarily due to the significant time investment required to obtain satisfactory results.  The key to the definitive procedure used for imposing displacement demands on the various models of the prototype building involved the use of axially rigid elements that were disconnected from the diaphragms at each storey level. These elements effectively displaced each storey level to the desired displacement, achieving the target interstorey drift as outlined in the simplified analysis of buildings in Clause 21.11.2.2. To achieve this, each of the axially rigid elements required the definition of a pinned support at one of its ends where a displacement was introduced as a demand using the Ground Displacement command.  The displacement profiles that were used for this study are outlined in Section 3.3.1. Table 3.1 introduces the equations required to calculate the ratio between the global drift and the interstorey drift, while Figure 3.9 and Figure 3.10 illustrate a graphical representation of these displacement profiles along the height of the building.  As discussed in Chapter 3 as well, when assessing displacement-induced demands affecting the gravity-load columns or load-bearing walls, it is important to ensure that the demands being transferred from the slabs to these vertical members, are being capped according to Clause 21.11.3.2 in CSA A23.3.   158  6.4 Observations on Different Configurations of GLRF of the Archetype Building 6.4.1 Uniform GLRF. For the case in which all the building's slabs were considered slender, with a 250 mm-thick slab at level L1 and 190 mm-thick slabs at all other levels, a comprehensive examination of all the gravity columns was conducted. Detailed studies and assessments (as presented in Chapter 4) were focused on those cases that exhibited the largest displacements at the top of the GLRF. Figure 4.1 and Figure 4.2 highlight the most critical columns for each direction of analysis.  Following a series of linear analyses, the new procedure effectively captured the bending moment and shear force demands for models with a continuous displacement profile assigned along the entire building's height. The models with 2D displacement profiles assigned to either the bottom or top halves of the structure were effective at capturing the acceptable axial load within the columns. However, it became apparent during these experiments that the bending moments and shear forces transferred from the slabs to the columns were unacceptably high. Consequently, a new set of nonlinear analyses was conducted, which required the estimation of nominal flexural capacities of the slabs (as per Clause 21.11.3.2) and their definition in all the models as plastic hinges at each end of the frame members representing the slabs. This adjustment effectively capped the demands being transferred from the slabs to the columns part of the GLRF.  For this archetype building, in the coupled-wall direction of analysis, it was found that, regardless of the displacement profiles used—whether they acted along an isolated half of the building or along the full height—all the slabs were reaching their nominal flexural capacity at their connection with the columns (Figure 4.4). This observation implied that for buildings with a 159  similar layout to the archetype building in the coupled direction, a single set of nonlinear analyses (with displacements imposed along the full height) would be sufficient to estimate the induced demands on the columns.  However, for the cantilever-wall direction of analysis, it was observed that when the displacement profile was assigned along the upper half of the building, not all the slabs were reaching their nominal flexural capacities (Figure 4.6). Therefore, for this direction of analysis in this prototype building, it was meaningful to apply different analyses with displacement profiles assigned to isolated portions of the building to avoid overestimating the axial forces present in the columns, even after conducting nonlinear analyses.  Before completing the assessment of the gravity columns, the shear demand forces (Vf) needed to be filtered according to Clause 21.11.3.1. This was accomplished by identifying all the locations where the bending moment demand (Mf) in the columns exceeded the probable bending moment capacity (Mp) were identified. For those cases, the shear demands acting on the columns were adjusted capping the flexural demands exhibited to their probable flexural capacities.  Upon completing these steps, it was determined that a critical area for this prototype building (and any building with a similar layout) was at the ground level. It was also observed that the original design of the columns within the GLRF provided sufficient resistance for the columns to withstand the demands resulting from lateral displacement in the cantilever direction. In contrast, for the coupled-wall direction, the columns did not possess sufficient ductility to accommodate the 160  demands from lateral displacements occurring in that direction. This can be attributed to two main factors:  1. The structure exhibited greater flexibility in the coupled-wall direction, resulting in higher displacements at the top of the GLRF (global drifts) compared to the cantilever-wall direction.  2. Referring to the envelopes corresponding to the simplified analysis of buildings, at ground level, the ratio between the global drift ratio to the interstorey drift ratio is significantly higher for the coupled-wall direction (1.30) than for the cantilever-wall direction (0.70).  It is also noteworthy that, when assessing the capacities of different columns, regarding flexural capacity, the columns on the 'compression side' of the building exhibited the largest demand-to-capacity ratios. The higher the axial force was in the column, the more it reduced the curvature capacity. Conversely, regarding shear capacity, the highest demand-to-capacity ratios occurred on the 'tension side' of the building, where the levels of axial forces were lower, resulting in a decrease in shear resistance (Vr) calculated according to the General Method as per Clause 11.3.6.4  6.4.2 GLRF with Thick Transfer Slab For the scenario where a 1.20-metre-thick slab was included at level L7 (elevation 21.14 m above grade), a similar procedure was followed as described above. The main difference was that no plastic hinges were defined for the thick transfer slab. This decision was made to avoid leading to an unconservative assessment of the columns, as it would assume that the thick slab would yield before the columns, which is unlikely (Figure 4.7). Since Clause 21.11.3.3.3 encourages analysts 161  and engineers to maintain the column members in their models as linear elements, it was decided to follow the conservative path and consider the frame elements representing the thick slab as linear members in the model.  The columns with the highest demand-to-capacity ratios were once again those that experienced the induced demands due to lateral displacement in the coupled-wall direction. This can be justified by the same two components described above, emphasizing that, in the location where the thick slab was included (around a quarter of the height of the building above grade), the interstorey drift ratio envelopes, part of Clause 21.11.2.2, still present a higher ratio for the relationship between the global drift ratio and the interstorey drift ratio for the coupled-wall direction compared to the cantilever-wall direction at 0.25·hw (1.30 vs. 1.00).  It is noteworthy that cases which were not critical in the scenario without a thick slab became particularly critical, especially in the cantilever-wall direction (see Table 4.3 and Table 4.4). This shift in criticality was attributed to a significant increase in the axial forces exhibited by all the columns located below the thick slab. This increase significantly reduced the curvature capacity in the cross-sections considered for the gravity columns in this building. In a scenario where a thick transfer slab is included as part of the GLRF, both the columns located directly above the thick element and those located directly below should be carefully assessed.  In both scenarios (with and without a thick slab), proposals for column redesigns were presented in alignment with the Canadian Code's vision, which aims to provide a sufficient level of ductility 162  by enhancing the member's detailing to enable it to incur into the inelastic range of deformations with greater stability.  6.5 Pilot Study on Additional Performance Requirements of NBCC Article 4.1.8.23 Additional Performance Requirements included in the new edition of NBCC, especially 4.1.8.23.(4), which states that Normal Importance Category buildings in Seismic Category SC4 with a height greater than 30 metres, such as the archetype building generated for this study (and many high-rise concrete buildings in the urban areas of British Columbia in Canada), NBCC requires that framing members not part of the SFRS shall be designed to behave elastically for a specified earthquake force having a 10% probability of exceedance in 50 years.  This new requirement triggers a seismic analysis that the engineers need to perform, in addition to the seismic analysis conducted to design the SFRS of the building, which is based on ground motions with a 2% probability of exceedance in 50 years, equivalent to a 2,475-year return period. The same archetype building used in the previous study, where the columns within the uniform GLRF were assessed for their resistance and ductility, was considered for this additional performance assessment in a pilot study.  To conduct this additional analysis, a new 3D model was created in ETABS, similar to the ones used for the Clause 21.11 analyses, with the main difference that the structure modeled below grade level was not laterally constrained. In parallel with the linear dynamic seismic analysis, an iterative procedure took place to refine the effective stiffness of the SFRS first and then the GLRF.  163  The iterative process commenced with initial assumptions for effective stiffnesses defined in the ETABS model. After running the Response Spectrum Analysis (RSA), the bending moment demands (Mf) were incorporated into a modified Equation 21.2 from CSA A23.3, along with the nominal flexural capacities (Mn) for each wall in both directions of analysis. The nominal flexural capacities were calculated using the sectional analysis tool Response-2000. After one or two iterations, these equations determined that the walls remained essentially elastic, a conclusion further supported by generating their nonlinear moment-curvature response using Response-2000. This conclusion ruled out the implementation of the simplified analysis of buildings from Clause 21.11.2.2, which incorporates a nonlinear profile displacement due to the concentrations of plastic deformations at the base of the wall. Hence, no amplification of the interstorey drifts was needed.  After determining the effective stiffnesses for the SFRS, an additional iterative process was necessary to define the effective stiffnesses for the gravity columns. Similar to the previous process, the iterations began with initial assumptions, and the stiffness was subsequently refined by using the nonlinear moment-curvature responses (generated by Response-2000) for the cross-sections considered for the columns. After a series of iterations, it was confirmed that the columns remained within the elastic range of their response for this level of demand. To validate this, the interstorey drift ratios resulting from the 475-year analysis were compared with those obtained from the 2,475-year seismic analysis, and the interstorey drift ratios that were amplified according to Clause 21.11 (see Figure 5.9 and Figure 5.10). The result of this comparison revealed that the interstorey drift ratios resulting from these additional analyses were noticeably smaller in comparison to the other two.  164  The fact that the columns presented such low demands after conducting the 475-year seismic analysis was attributed to five factors: i) Reduced spectral accelerations and displacements compared to those resulting from a 2,475-year return period. ii) Article 4.1.8.23 does not require scaling factors for the dynamic base shear. iii) Low global drift ratios. iv) Uncracked walls protecting the gravity columns from lateral forces and lateral displacements. v) The absence of further amplification of interstorey drifts as per Clause 21.11 since the walls also remain elastic.  Due to all these reasons, this additional analysis did not necessitate any modifications to the design of the structure of this archetype building.  6.6 Limitations and Recommendations for Future Work The work presented in this study has developed a methodology that complements the simplified analysis outlined in Clause 21.11.2.2. The effectiveness of this methodology was demonstrated in cases where the GLRF is uniform, and the columns are being connected to the core by slender slabs. In such scenarios, when the structure undergoes lateral swaying, the slabs will tend to yield before the columns, allowing the bending moments to be limited before reaching the columns. This crucial assumption enables the engineer to limit the bending moments in the analytical model by defining plastic hinges in the slabs only (as it was done for this study), while the columns are treated as linear members in the analytical model, which in turn makes it possible to utilize the table of Maximum Calculated Induced Bending Moments as per Clause 21.11.3.3.3. 165   As it was highlighted above, when dealing with a case involving a thick transfer slab, this approach is not viable. In such instances, the columns would yield before the transfer slab, requiring a more conservative strategy. Consequently, the transfer slab also needs to be treated as a linear member, resulting in substantial demands on the columns. Therefore, it is strongly recommended that future research focuses on enhancing Clause 21.11.3.3.3. The proposed enhancement should provide structural engineers with tools and guidance on how to assess the columns ductility capacity, when they need to be included in the model as nonlinear members as well, as opposed to relying solely on bending moments captured from linear members. This adjustment would enable engineers to better estimate the demands on gravity columns or load-bearing walls when a transfer slab or girder is also being included as part of the GLRF.   Furthermore, it is advisable to extend the investigation of applying the simplified analysis of buildings as per Clause 21.11.2.2 to buildings with diverse configurations and assessing its implications for GLRF. While the prototype building used in this study featured a regular geometry with a rigid core at the plan's center and slender perimeter columns, nowadays architectural projects often present greater complexity. These projects frequently demand the inclusion of additional structural walls beyond those forming the rigid core, and these walls may not span the full building height. As a result, it is essential to determine whether Clause 21.11 is the appropriate reference for such scenarios or if supplementary tools or Clauses should be developed.  Regarding Article 4.1.8.23 in NBCC 2020, the investigation conducted in this thesis found that this supplementary analysis did not govern the design in the examined case study. This case 166  involved a scenario where the GLRF was uniform, and no irregularities were present. However, this study recognizes the significance of continuing to explore the conditions under which this additional analysis might gain prominence and influence the design of the structure. This is especially pertinent for structures with specific irregularities, such as the inclusion of transfer slabs or transfer girders.  167  References Adebar, P., Bazargani, P., Mutrie, J., & Mitchell, D. (2010). Safety of gravity-load columns in shear wall buildings designed to Canadian standard CSA A23.3. Canadian Journal of Civil Engineering, 37(11), 1451–1461. https://doi.org/10.1139/L10-075 Adebar, P., Devall, R., & Mutrie, J. G. (2014). Design of Gravity-Load Resisting Frames for Seismic Displacement Demands. Proceedings of the 10th National Conference in Earthquake Engineering. Adebar, P., & Sainz Albanez, G. (2023). Design of Gravity-Load Frames in Shear Wall Buildings for Seismic Deformation Demands: The Canadian Code Approach. In Canadian Conference - Pacific Conference on Earthquake Engineering. Bentz, E. C., & Collins, M. P. (2000). Response-2000 (1.0.5). CSA (Canadian Standards Association). (2019). Design of concrete structures A23.3-19. www.csagroup.org/legal CSI Computers and Structures Inc. (2019). ETABS. EGBC. (2022). Structural Engineering Services for Tall Concrete Building Projects (VERSION 1.0). Hwang, S.-J., & Moehle, J. P. (2000). Models for Laterally Loaded Slab-Column Frames. ACI Structural Journal. Los Angeles Tall Buildings Structural Design Council. (2020). Los Angeles Tall Buildings Structural Design Council (LATBSDC). An alternative procedure for seismic analysis and design of tall buildings. National Research Council of Canada. (n.d.). National Building Code of Canada 2020.   168  Appendices Appendix A  Gravity Design Complementing chapter 2 of this thesis, this appendix describes all the assumptions and procedures followed to carry out the analysis and design for gravity loads for the building selected as the case study. A.1 Gravity Loads As it was mentioned in the main body of this thesis, the loads considered for gravity design differ from the ones considered for lateral design (mass considered for modal analysis and seismic weight). All loads that are considered as dead (permanent) are the following: • Self-weight of the structure. Assuming 24 kN/m3. • Cladding 1.90 kN/m in the perimeter of each storey level above grade. Table A.1 summarizes the magnitudes of superimposed dead loads along the height of the building. Table A.1 Superimposed Dead Loads for Gravity Design. Superimposed Dead Loads Levels Load L1-PH 1.25 kN/m2 P1 1.75 kN/m2 P5-P2 0.50 kN/m2  At roof level (top of level L30) a mechanical room and chillers are also included as part of superimposed dead loads with a distributed weight of 7.2 kPa each. Figure A.1 shows the location of the chillers and the mechanical room, as well as their dimensions. 169   Figure A.1 Plan view of roof level showing location of Chillers and Mechanical Room.  Similar to the superimposed dead loads, the live loads varied along the height of the building. Table A.2 Live Loads for Gravity Design. Live Loads Levels Load L1-L29 1.90 kN/m2 P1 4.80 kN/m2 P5-P2 2.40 kN/m2  Finally, a snow load was considered for the roof top with a magnitude of 1.70 kPa. 170  A.2 Slabs The greatest spans and overhangs that the slabs need to withstand take place from level L12 to level L30. At these levels, the columns have a smaller cross-section.   Figure A.2 Critical spans and overhangs at the slabs; Level L12 to Level L30.  The slabs have a thickness of 250 mm from level P5 to level L1 (all parking levels and the slab located at the top of level at grade), and 190 mm from level L2 to level L30.  171  A.3 Columns To dimension and ensure that the columns would have an adequate level of resistance, an analysis based on the tributary area of each column was carried out. The extra shear generated due to the overhang of the slab was neglected, as were all the bending moments that the slabs would transmit to the columns due to gravity loads. Hence, the columns were designed based on their axial demands only. For practical purposes, it was decided to maintain a single section of columns per level, which is why it was sufficient to perform a load takedown for the column with the greatest tributary area.  Figure A.3 Tributary area of columns above grade level. 172  It is worth noting that NBCC recognizes and distinguishes loads that we can quantify with more precision, and that their effects on the structure will be more permanent (dead loads). Unlike loads that will tend to vary along the lifespan of the structure (live loads). This is part of the reason not only why dead loads and live loads have different load factors, but also the code allows a reduction in the effects due to the live load. NBCC, in article 4.1.5.9 – 4), presents the following Equation to calculate a factor to apply to the live load. 𝑅𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝐹𝑎𝑐𝑡𝑜𝑟 = 0.3 + √9.8 𝐵⁄   ( 8 )  Where B represents the tributary area (in square metres) being supported by a specific member, for a same type of use or occupancy. Please note that every time a change of occupancy happens (live load changes), the tributary area should not be added with tributary area supporting a different live load. According to Table 4.1.3.2.-A in NBCC 2020, there are three cases (or load combinations) that need to be taken in consideration to define the level of demands of every member supporting gravity loads: • 1.40 DEAD. • 1.25 DEAD + 1.50 LIVE + 1.00 SNOW • 1.25 DEAD + 1.50 SNOW + 1.00 LIVE Figure A.4 shows how each one of the cases enlisted above varies along the height of the building (blue lines). For this case study, the load combination governing the design of the column was the second case enlisted above. The column at the bottom of the structure will exhibit an axial demand of 12,392 kN. 173  To calculate the resistance of the columns (red dashed line in Figure A.4), CSA A23.3 offers the required equations in Clause 10.10.4. These equations (applied to this case study) are the following. 𝑃𝑟𝑜 = 𝛼1 · ϕ𝑐 · 𝑓𝑐′(𝐴𝑔 − 𝐴𝑠𝑡) + ϕ𝑠 · 𝑓𝑦 · 𝐴𝑠𝑡  ( 9 )  The expression shown above enables us to calculate the contribution from the concrete and the longitudinal reinforcement towards the maximum axial capacity for compression loads that the cross-section will be able to provide. However, the standard CSA A23.3 recognizes that the columns and bearing walls are members subjected to compression and flexure demands. Hence, the cross-sections need to provide a minimum resistance to withstand bending moment demands. To achieve this, the code sets different limits to the maximum axial capacity of the member. Since the separation of the ties for all the columns in this case study meets the requirements presented in Clause 7.6.5, the maximum capacities that shall be considered for design for gravity loads will be calculated based on the following Equation.    𝑃𝑟,𝑚𝑎𝑥 = (0.2 + 0.002 · ℎ) 𝑃𝑟𝑜 ≤ 0.80 · 𝑃𝑟𝑜  (10)  Where h represents the smaller column dimension. In the Figure A.4 below, it may be confirmed that the columns provide a sufficient level of axial capacity (red dashed line) along the entire height of the building. Figure A.5 provides a summary of all the cross sections used as columns for this project.   174   Figure A.4 Axial force demands and resistance of columns along the entire height of the building: combination 1 (darkest blue line); combination 2 (gray-blue line); combination 3 (light blue line); and maximum axial resistance (red dashed line).  -200204060800 2,000 4,000 6,000 8,000 10,000 12,000 14,000Elevation [m]Axial Force [kN]175   Figure A.5 Summary of columns sections per level of the building.               176  Appendix B  Seismic Analysis Complementing chapter 2 of this thesis, this appendix describes all the assumptions and procedures followed to carry out the seismic analysis for the building selected as the case study. The seismic analysis conducted for this project was performed under the scope of the National Building Code of Canada 2020 (NBCC 2020). Each step of the process will be introduced as a subsection within this appendix.  B.1 Site Properties The 2020 edition of the code allows the specialist to define a site designation based on either the average shear wave velocity (Vs30) or the site class. For this case study, it was decided to use an average shear wave velocity of 500 m/sec as the reference for the site designation. To obtain the design spectral accelerations, the code requires the use of the online tool Seismic Hazard Calculator. To operate this tool, the user needs to provide the shear wave velocity (mentioned previously) and define the location of interest. For this project, the chosen location was Vancouver City Hall, and the following coordinates were provided: • Latitude → 49.261° • Longitude → -123.114° Figure B.1 shows proof of what was stated above. 177   Figure B.1 Seismic Hazard Tool showing site designation based on shear wave velocity and location of site.  With the data defined above, the online tool will provide the following design spectral accelerations, corresponding to 5% damping for this specific site.  Figure B.2 Design spectral accelerations provided by the Seismic Hazard Tool.  178  With these values of spectral accelerations combined with the Log-Log Interpolation method, which is part of the Notes to Part 4 Structural Design (A-4.1.8.4.6), the corresponding Uniform Hazard Spectrum for this site may be generated.  Figure B.3 Uniform Hazard Spectrum for Vancouver City Hall corresponding to an event with 2% probability of exceedance in 50 years. B.2 Importance Factor and Seismic Category Since the importance category for this building was established as normal, Table 4.1.8.5-A in the code specifies that for the seismic analysis under the Ultimate Limit State (ULS), the importance factor (IE) shall be taken as 1.0. Seismic categories shall be defined based on the values of specific spectral accelerations multiplied by the importance factor, IE·S(0.2), and IE·S(1.0). Referring to either Figure B.2 or Figure B.3, the spectral accelerations that would correspond to natural periods of 0.2 and 1.0 seconds would be 1.01 and 0.40, respectively. 1.010.6990.400.2480.06830.03040.000.200.400.600.801.000.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Spectral acceleration [1/g]Natural period [sec]179    As Table 4.1.8.5.-B in the code shows: • IE·S(0.2) = 1.0·(1.01) = 1.01 > 0.75 → Seismic Category SC4. • IE·S(1.0) = 1.0·(0.40) = 0.40 > 0.30 → Seismic Category SC4. The code states that the seismic category shall be taken as the most severe of the categories determined. For our case, both categories are equally severe.  B.3 Structural Configuration and Irregularities Table 4.1.8.6 in the code introduces 10 different types of irregularities that may be present in a building structure. Since the SFRS conceived for this case study is a uniform system (coupled-wall in one direction and cantilever-wall in the perpendicular direction) and its dimensions do not change along the height of the building, combined with a uniform gravity system and an equal distribution of loads on each story level, this building is considered to be regular. However, a specific check was made to confirm that the structure is not torsionally sensitive. This procedure will be presented in one of the following subsections. B.4 Methods of Analysis and Direction of Loading According to section 4.1.8.7, since the Seismic Category defined for this project is SC4, and the structure has been classified as Regular with a maximum height above grade level of 90.15 metres, the Dynamic Analysis Procedure (linear dynamic analysis) will be the method used to carry out the seismic analysis for this building. Regarding the direction of loading, since this building will be provided with a specific SFRS (coupled and cantilever walls) oriented parallel to each main direction in the horizontal plane (X and Y), independent analyses for each direction shall be performed. 180   B.5 SFRS Force Modification Factors Specific lateral resisting systems have been codified, which means that factors related to ductility and overstrength are provided so the engineer may account for the amount of energy that will be dissipated due to inelastic deformations in specific locations of the SFRS. Referring to Table 4.1.8.9 in the code, the factors that are of interest for this case are shown below,    Figure B.4 Extract of Table 4.1.8.9 from NBCC 2020.  The Figure shown above, also indicates that for the SFRS selected, there are no limit heights. B.6 Torsional Sensitivity Before fully diving into dynamic analysis using response spectrum analysis, an equivalent static force procedure (static analysis) was carried out with 10% of accidental eccentricity. This was done to determine if the structure was torsionally sensitive or not. The list of all items considered to be part of the seismic weight, as well as the reduced spectra considered for the design of the SFRS in each direction of analysis, will be introduced in the following subsection. For the static analysis, based on rough estimates, the total seismic weight is 181  considered to be 155,088 kN. As will be seen in the next subsection, the empirical period, set by the code, governs both directions of analysis, which, for this case, is Ta = 2.93 sec.  Based on the previous fundamental period, we already know that the base shear of our structure is greater than the minimum value set by the code, which corresponds to a period of 4.0 sec for systems that rely on structural walls. Regarding the upper limits for the base shear, we have the following: • S(0.5) = 0.699 > S(2.93) = 0.1456 • (2/3)·S(0.2) = 0.673 > S(2.93) = 0.1456 Hence, the base shear for both directions of analysis will respect the lower and upper limits established by the code. The lateral loads defined in the analytical model were computed based on the equations presented in section 4.1.8.11, 𝐹𝑥 = (𝑉 − 𝐹𝑡) ·𝑊𝑥 · ℎ𝑥(∑ 𝑊𝑖 · ℎ𝑖𝑛𝑖=1 )  (11)  Where,  Wi, Wx  = portion of seismic weight assigned to level i or x hi, hx  = height above the base (grade level) to level i or x. In metres  Ft = portion of V to be concentrated at the top of the structure. Taken equal to zero where Ta is lower than 0.7 sec. Calculated as 0.07·Ta·V, but does not need to exceed 0.25·V either. V = base shear. The lateral loads calculated for each level in the coupled-wall and cantilever-wall directions, are presented in Table B.1 and Table B.2 respectively. 182  Table B.1 Equivalent lateral loads – Coupled-wall Direction.  Ta 2.93 secV 3,387.07 kNFt 693.66 kNLevel hstorey [m] hx [m] wx [kN] wx · hx [kN·m] Fx [kN] Vx [kN]PH 5.07 90.15 1,553.61 140,057.94 747.98 747.98L30 2.78 85.08 5,796.82 493,193.45 191.27 939.24L29 2.78 82.30 4,992.04 410,844.89 159.33 1,098.57L28 2.78 79.52 4,992.04 396,967.02 153.95 1,252.52L27 2.78 76.74 4,992.04 383,089.15 148.57 1,401.09L26 2.78 73.96 4,992.04 369,211.28 143.18 1,544.27L25 2.78 71.18 4,992.04 355,333.41 137.80 1,682.08L24 2.78 68.40 4,992.04 341,455.54 132.42 1,814.50L23 2.78 65.62 4,992.04 327,577.66 127.04 1,941.54L22 2.78 62.84 4,992.04 313,699.79 121.66 2,063.19L21 2.78 60.06 4,992.04 299,821.92 116.27 2,179.47L20 2.78 57.28 4,992.04 285,944.05 110.89 2,290.36L19 2.78 54.50 4,992.04 272,066.18 105.51 2,395.87L18 2.78 51.72 4,992.04 258,188.31 100.13 2,496.00L17 2.78 48.94 4,992.04 244,310.44 94.75 2,590.75L16 2.78 46.16 4,992.04 230,432.57 89.36 2,680.11L15 2.78 43.38 4,992.04 216,554.70 83.98 2,764.09L14 2.78 40.60 4,992.04 202,676.82 78.60 2,842.69L13 2.78 37.82 4,992.04 188,798.95 73.22 2,915.91L12 2.78 35.04 4,992.04 174,921.08 67.84 2,983.75L11 2.78 32.26 5,058.76 163,195.60 63.29 3,047.04L10 2.78 29.48 5,125.48 151,099.15 58.60 3,105.64L9 2.78 26.70 5,125.48 136,850.32 53.07 3,158.71L8 2.78 23.92 5,125.48 122,601.48 47.55 3,206.26L7 2.78 21.14 5,125.48 108,352.65 42.02 3,248.28L6 2.78 18.36 5,125.48 94,103.81 36.49 3,284.77L5 2.78 15.58 5,125.48 79,854.98 30.97 3,315.74L4 2.78 12.80 5,125.48 65,606.14 25.44 3,341.18L3 2.78 10.02 5,125.48 51,357.31 19.92 3,361.10L2 2.78 7.24 5,125.48 37,108.48 14.39 3,375.49L1 4.46 4.46 6,693.02 29,850.87 11.58 3,387.07155,088.25 6,945,125.93 3,387.07183  Table B.2 Equivalent lateral loads – Cantilever-wall Direction.  Ta 2.93 secV 5,725.76 kNFt 1,172.61 kNLevel hstorey [m] hx [m] wx [kN] wx · hx [kN·m] Fx [kN] Vx [kN]PH 5.07 90.15 1,553.61 140,057.94 1,264.44 1,264.44L30 2.78 85.08 5,796.82 493,193.45 323.33 1,587.77L29 2.78 82.30 4,992.04 410,844.89 269.35 1,857.11L28 2.78 79.52 4,992.04 396,967.02 260.25 2,117.36L27 2.78 76.74 4,992.04 383,089.15 251.15 2,368.51L26 2.78 73.96 4,992.04 369,211.28 242.05 2,610.56L25 2.78 71.18 4,992.04 355,333.41 232.95 2,843.51L24 2.78 68.40 4,992.04 341,455.54 223.85 3,067.36L23 2.78 65.62 4,992.04 327,577.66 214.76 3,282.12L22 2.78 62.84 4,992.04 313,699.79 205.66 3,487.78L21 2.78 60.06 4,992.04 299,821.92 196.56 3,684.34L20 2.78 57.28 4,992.04 285,944.05 187.46 3,871.80L19 2.78 54.50 4,992.04 272,066.18 178.36 4,050.16L18 2.78 51.72 4,992.04 258,188.31 169.27 4,219.43L17 2.78 48.94 4,992.04 244,310.44 160.17 4,379.60L16 2.78 46.16 4,992.04 230,432.57 151.07 4,530.66L15 2.78 43.38 4,992.04 216,554.70 141.97 4,672.64L14 2.78 40.60 4,992.04 202,676.82 132.87 4,805.51L13 2.78 37.82 4,992.04 188,798.95 123.77 4,929.28L12 2.78 35.04 4,992.04 174,921.08 114.68 5,043.96L11 2.78 32.26 5,058.76 163,195.60 106.99 5,150.95L10 2.78 29.48 5,125.48 151,099.15 99.06 5,250.01L9 2.78 26.70 5,125.48 136,850.32 89.72 5,339.72L8 2.78 23.92 5,125.48 122,601.48 80.38 5,420.10L7 2.78 21.14 5,125.48 108,352.65 71.03 5,491.14L6 2.78 18.36 5,125.48 94,103.81 61.69 5,552.83L5 2.78 15.58 5,125.48 79,854.98 52.35 5,605.18L4 2.78 12.80 5,125.48 65,606.14 43.01 5,648.19L3 2.78 10.02 5,125.48 51,357.31 33.67 5,681.86L2 2.78 7.24 5,125.48 37,108.48 24.33 5,706.19L1 4.46 4.46 6,693.02 29,850.87 19.57 5,725.76155,088.25 6,945,125.93 5,725.76184  Once the loads were defined in the analytical model, a factor B needs to be calculated for each level as follows, 𝐵𝑥 =𝛿𝑚𝑎𝑥𝛿𝑎𝑣𝑒  (12)  Where,  B = is the maximum of all Bx values in both orthogonal directions.  δmax = maximum displacements at the extreme points of the structure for each level. δave = average of the displacements reported at the extreme points of the structure. For the coupled-wall direction, B reached a maximum value of 1.07, whereas for the cantilever-wall direction, the maximum value was 1.12. Since both values are less than 1.7, this structure is not classified as torsional sensitive. Hence, the code enables us to perform a linear dynamic analysis, with an accidental eccentricity equal to 0.05·Dnx.  B.7 Dynamic Procedure Same as for the case of wind analysis, to carry on a modal analysis the definition of lateral mass and effective stiffnesses play a key role. To introduce a refined calculation of the seismic weight (lateral mass), the following items were considered: • Self-weight of the structure assuming 24 kN/m3.  • Cladding weight of 0.72 kPa of wall area. For a clear height of 2.59 m, this would be 1.9 kN/m along the perimeter of each level above grade. • Superimposed dead load (including partitions) of 0.75 kPa. • Mechanical room on the roof with a weight of 194 kN. 185  • Chillers on the roof with a weight of 499 kN. • The total lateral weight of the building was calculated to be 154,545 kN. Before starting with the Response Spectrum Analysis (RSA), two models were generated to determine if the GLRF represented an important contribution to the total lateral stiffness of the system. To do this, the natural periods were calculated using two different models. The first model considered just the SFRS, whereas the second one included the slabs and columns as well. According to the National Standard of Canada for Design of Concrete Structures (CSA A23.3:19), reduced section properties were used to account for cracking of concrete in accordance with Cl.21.2.5.2. As the coupling beams in ductile coupled walls are designed according to Cl.21.5.8.2 (coupling beams with diagonal reinforcement), Ave = 1.2·(0.45·Ag), and Ie = 0.25 · Ig. The 1.2 factor compensates for the fact that ETABS uses a shear area of Ave = (5/6)·Ag. The reduction factor αw was initially assumed to be equal to Ro for Equation 21.1 in Cl.21.2.5.2. This gave as result Ave = 0.5· Ag and  Ie = 0.5 · Ig for the walls. For the model where the GLRF was included, the effective stiffness defined for the slabs was Ie = 0.2 · Ig. For the columns Ie = 0.7 · Ig . After declaring P-delta effects (iterative based on loads), the modal analyses were carried out giving the following results. Table B.3 Natural periods and mode shapes for seismic analysis based on SFRS alone. Mode Period [sec] Mass Participation [%] x - dir. (coupled) y - dir. (cantilever) z - dir. (torsion)  1 6.44 69 0 0  2 4.82 0 67 0  3 2.56 0 0 79     186   Table B.4 Natural periods and mode shapes for seismic analysis based on SFRS and GLRF. Mode Period [sec] Mass Participation [%] x - dir. (coupled) y - dir. (cantilever) z - dir. (torsion)  1 5.98 71 0 0  2 4.62 0 70 0  3 2.52 0 0 80   Since the reduction of natural periods were less than 15%, this study considers that the contribution to the GLRF towards the lateral stiffness of the building may be neglected. Hence, the natural periods used for the seismic analysis are the ones reported in Table B.3. When a RSA is performed, several things need to be taken into account: • To provide adequate level of resistance, the natural period of the structure shall be taken as the minimum between the period resulting from the modal analysis, and the upper limit established by the code. For structural walls NBCC 2020 defines a limit for Ta equal to 2 x 0.05hn0.75 • To provide an adequate level of lateral stiffness, the natural period of the structure shall be taken as the minimum between the period resulting from the modal analysis, and the upper limit established by the code. For structural walls NBCC 2020 sets 4.0 sec as a limit for Ta. • For regular structures, the base shear resulting from the RSA, will not be less than 80% of the base shear that would be calculated using a static analysis (this includes the calculation of the higher modes factor, Mv). For structures considered irregular, then the base shear shall not be less than 100% of the base shear product of a static analysis.   187   Table B.5 summarizes everything that was described above. Table B.5 Summary of Seismic Demands according to NBCC 2020. NBCC Reference Parameter  Coupled Wall Cantilever Wall    (a) Seismic Weight, W = 154,545 kN 154,545 kN    (b) Fundamental period from ETABS, T = 6.44 sec 4.82 sec  4.1.8.12.(5) & 4.1.8.12.(6) (c) Design elastic base shear, Ved =    Elastic base shear, Ve = 13,800 kN 18,391 kN   4.1.8.5.(1); 4.1.8.9 (d) IE = 1.0; RdRo =  4.0 x 1.7 = 6.8 3.5 x 1.6 = 5.6  4.1.8.12.(7) (e) Design base shear, Vd = VedIE/RdRo = 2,031 kN 3,285 kN  4.1.8.11.(3).(c) & 4.1.8.11.(3).(d).(iii) (f) Empirical period Ta = 2 x 0.05hn0.75             = 2 x 1.463 sec   2.93 sec 2.93 sec   4.1.8.4.(9) (g) S(2.93 sec) = 0.146 0.146  4.1.8.11.(6) (h) Mv =  1.02 1.42  4.1.8.11.(2) (i) Min. lateral earthquake force,                 V(Ta = 2.93 sec) = 3,375 kN 5,706 kN   4.1.8.12.(8) (i) Scaled design base shear                    (regular structure) Vd = 0.8V =  2,700 kN 4,565 kN     (k) Scaling factor for design forces (j)/(e) = 1.33 1.39  4.1.8.11.(2) (l) Min. lateral earthquake force                   (for deflection), V(Ta = 4.0 sec) = 2,125 kN 2,580 kN   4.1.8.12.(11) (m) Scaled design base shear                            Vd = 0.8V (for deflection) =  1,700 kN 2,064 kN     (n) Scaling factor for deflection forces (m)/(e) = 1.00 1.00  4.1.8.13.(2) (o) Total multiplier for deflections                   (n) x RdRo/IE =  6.80 5.60    Figure B.5 and Figure B.6 offer graphical representation of the reduced spectra defined in ETABS to perform the RSA for each direction. Whereas Figure B.7, Figure B.8, and Figure B.9 present the demands in each component of the SFRS. 188   Figure B.5 Reduced spectral acceleration for the Coupled-wall direction (Rd =4.0 and Ro =1.7)   Figure B.6 Reduced spectral acceleration for the Cantilever-wall direction (Rd =3.5 and Ro =1.6)  0.02140.14560.000.200.400.600.801.000.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Spectral acceleration [1/g]Natural period [sec]0.02600.14560.000.200.400.600.801.000.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Spectral acceleration [1/g]Natural period [sec]189   Figure B.7 Shear demands due to wind loading in coupling beams: no accidental torsion (light blue); with accidental torsion (dark blue). 01020304050607080901000 200 400 600 800 1,000 1,200Elevation [m]Coupling Beam (CB1-2) Shear Forces, V2 [kN]01020304050607080901000 200 400 600 800 1,000 1,200Elevation [m]Coupling Beam (CB2-3) Shear Forces, V2 [kN]190    Figure B.8 Demands acting on walls due to seismic loading; coupled-wall direction. -200204060801000 1,000 2,000 3,000 4,000Elevation [m]Shear Force, V3 [kN]-200204060801000 20,000 40,000 60,000 80,000 100,000Elevation [m]Bending Moment, M2 [kN-m]191    Figure B.9 Demands acting on walls due to seismic loading; cantilever-wall direction. -200204060801000 2,000 4,000 6,000 8,000 10,000Elevation [m]Shear Force, V2 [kN]-200204060801000 50,000 100,000 150,000Elevation [m]Bending Moment, M3 [kN-m]192  As it was mentioned previously, when it comes to providing adequate lateral stiffness to the SFRS (checking for lateral drifts), the loads affecting the model should be the ones coming from the elastic spectrum. Let’s recall that the natural period for both directions is governed by the limit defined in the code, which means that Ta = 4.0 sec. The corresponding spectral acceleration is shown below,  Figure B.10 Spectral acceleration for both directions to check for lateral deflections.  Finally, Figure B.11 shows the maximum storey drifts resulting from the RSA for each direction. The blue line represents the drifts ratio exhibited in the coupled-wall direction (x-direction), whereas the red line represents what is happening in the cantilever-wall direction (y-direction). The limit set by the code in 4.1.8.13-3) of 0.025 is also indicated.  0.09350.000.200.400.600.801.000.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Spectral acceleration [1/g]Natural period [sec]193   Figure B.11 Maximum storey drift ratios for seismic analysis: ratios corresponding to the Coupled direction (blue); ratios corresponding to the Cantilever direction (red).-200204060800.000 0.005 0.010 0.015 0.020 0.025 0.030Elevation [m]Storey Drift Ratio-200204060800.000 0.005 0.010 0.015 0.020 0.025 0.030Elevation [m]Storey Drift Ratio194  Appendix C  Wind Analysis Complementing chapter 2 of this thesis, this appendix describes all the assumptions and procedures followed to carry out the wind analysis for the building selected as the case study. C.1 Dynamic Procedure The definitions for mass and stiffnesses are the backbone of all structural models. As it was mentioned in chapter 2, the items considered as part of the lateral weight for this case study were: • Self-weight of the structure assuming 24 kN/m3.  • Cladding weight of 0.72 kPa of wall area. For a clear height of 2.59 m, this would be 1.9 kN/m along the perimeter of each level above grade. • Superimposed dead load (including partitions) of 0.75 kPa. • Mechanical room on the roof with a weight of 194 kN. • Chillers on the roof with a weight of 499 kN. • The total lateral weight of the building was calculated to be 154,545 kN. Regarding the definition of the effective stiffnesses, we referred to Table N9.2.1.2, in the Commentary to Cl.9.2.1.2 from CSA A23.3, for Ultimate Limit State (ULS).  • For coupling beams with diagonal reinforcement; Ave = 1.2·(0.45·Ag), and Ie = 0.35 · Ig. • For structural walls; Ave = 0.75· Ag and Ie = 0.75 · Ig. • Any contribution coming from the GLRF was neglected for this exercise.      195  The natural periods resultant from a modal analysis are summarized in the Table below. Table C.1 Natural periods and mode shapes for wind analysis. Mode Period [sec] Mass Participation [%] x - dir. (coupled) y - dir. (cantilever) z - dir. (torsion)  1 5.23 69 0 0  2 3.98 0 68 0  3 2.28 0 0 79   Disclaimer: According to NBCC 2020 in 4.1.7.2-(3) this archetype should be classified as a very dynamically sensitive building (its natural frequency is lower than 0.25 Hz), and the correct way to analyze this structure for wind demands would be to perform a wind tunnel procedure. However, for research purposes it was decided to carry out a dynamic wind analysis, assuming full loading only (Figure A.1), following section 4.1.7.8 of the code.  Figure C.1 Full and partial wind loading from NBCC 2020.  196  To illustrate how the calculations were carried out, it is necessary to introduce the following expressions from the code. 𝑤 =∑ℎ𝑖 𝑤𝑖∑ℎ𝑖  (13)  Where,  w = effective width. hi = story height wi = story width (perpendicular to the wind direction at height hi) The pressure profiles to be applied to the model are based on the equation, 𝑝 = 𝐼𝑤𝑞𝐶𝑒𝐶𝑡𝐶𝑔𝐶𝑝  (14)  Where, p = specified external pressure, considered positive when acting towards the structure, and negative when acting away from the surface. Iw = importance factor for wind load (considered equal to 1.0, Table 4.1.7.3 in the NBCC). q = reference velocity pressure (for Vancouver City Hall equal to 0.45 kPa, Table C-2 in the NBCC). Ce = exposure factor Ct = topographic factor (considered equal to 1.0) Cg = gust factor Cp = external pressure coefficient  197  Regarding the exposure factor, the case study was located in rough terrain (exposure B as classified in Commentary I). Hence, the Equation to be used shall be. 𝐶𝑒 = 0.5 (ℎ12.7)0.50, 𝑏𝑢𝑡 0.5 ≤ 𝐶𝑒 ≤ 2.5  (15)  In the equation above, the height h refers to the total height of the building above grade level (90.15 m). The external pressure coefficient needs to be calculated for all the wind-exposed faces of the building, including the windward, leeward, and lateral faces. This calculation is based on the height/width ratio of the building. Figure 2.7 exemplifies what was mentioned earlier.  Figure C.2 Values of Cp for main structural system from NBCC 2020.  198  C.2 Gust Factors The main difference between the Static Procedure and Dynamic Procedure for Wind Analysis resides in the calculation of the gust factor (Cg). Since for the dynamic analysis requires previous calculations for the definition of this factor. 𝐶𝑔 = 1 + 𝑔𝑝𝜎µ  (16)  Where, gp = peak factor, which may be calculated as 𝑔𝑝 = √2ln (𝜈𝑇) +0.577√2ln (𝜈𝑇) , and   𝜎µ= √𝐾𝐶𝑒𝐻(𝐵 +𝑠𝐹𝛽) Where,  ν = average fluctuation rate calculated as 𝑓𝑛𝐷 · √𝑠𝐹𝑠𝐹+𝛽𝐵  T = 3,600 seconds. K = 0.10 (rough terrain) CeH = exposure factor evaluated at reference height, h = H B = background turbulence factor calculated as, 𝐵 =43∫ [11 +𝑥𝐻457] [11 +𝑥𝑤122] [𝑥(1 + 𝑥2)43⁄] 𝑑𝑥914𝐻⁄0  (17)  s = seize reduction factor, which may be calculated as 𝑠 =𝜋3[11+8𝑓𝑛𝐷𝐻3𝑉𝐻] [11+10𝑓𝑛𝐷𝑤𝑉𝐻]  199  F = gust energy ratio calculated as 𝐹 =𝑥02(1+𝑥02)4/3, with 𝑥0 = 1,220 ·𝑓𝑛𝑉𝐻, and β = damping ratio, which may be considered as 0.01 for steel structures or 0.02 for concrete structures.  fnD = natural frequency of vibration of the structure in the along-wind direction [Hz] fn = lowest natural frequency of vibration of the building [Hz] H = height of the building above grade level [m] w = effective width of windward face of the building (defined above) VH = mean wind speed at the top of the structure, 𝑉𝐻 = ?̅?√𝐶𝑒𝐻 [m/sec] Where, 𝑉= reference wind speed at the height of 10 metres, calculated as ?̅? = √2𝐼𝑤𝑞𝜌  Where, Iw = importance factor for wind load. Since this case study is considered to have a normal importance category, for Ultimate Limit State, Iw = 1.0 q = reference velocity pressure [Pa]. As defined above, Vancouver City Hall is being considered. Hence, 0.45 kPa → 450 Pa. ϱ = air density, usually taken as 1.2929 kg/m3 Figure C.3 and Figure C.4 summarize all the calculations listed above for both directions of analysis, coupled-wall and cantilever-wall. 200   Figure C.3 Calculation for the gust effect factor for the windward face in the coupled–wall direction.   201   Figure C.4 Calculation for the gust effect factor for the windward face in the cantilever–wall direction.   C.3 Wind Loading Table C.2 and Table C.3 present the wind loads affecting each storey level of the building in for each direction of analysis. 202  Table C.2 Wind loading along the coupled-wall direction.   q1/50 [kPa] 0.45 CeH, ww 1.33D [m] 25.90 CeH, lw 0.94Iw, ULS 1.00 H/D 2.21Terrain Rough w [m] 24.79Cg 2.73 Cp, ww 0.80Ct 1.00 Cp, lw 0.50PH 5.07 8.23 90.15 2.54 1.33 1.31 0.58 1.89 39.36 39.36 199.56 199.56L30 2.78 25.90 85.08 3.93 1.29 1.27 0.58 1.85 105.94 145.30 403.93 603.49L29 2.78 25.90 82.30 2.78 1.27 1.25 0.58 1.83 132.40 277.69 771.99 1,375.48L28 2.78 25.90 79.52 2.78 1.25 1.23 0.58 1.81 130.88 408.57 1,135.83 2,511.30L27 2.78 25.90 76.74 2.78 1.23 1.21 0.58 1.79 129.33 537.90 1,495.36 4,006.66L26 2.78 25.90 73.96 2.78 1.21 1.19 0.58 1.76 127.75 665.65 1,850.51 5,857.17L25 2.78 25.90 71.18 2.78 1.18 1.16 0.58 1.74 126.15 791.80 2,201.20 8,058.37L24 2.78 25.90 68.40 2.78 1.16 1.14 0.58 1.72 124.51 916.31 2,547.34 10,605.72L23 2.78 25.90 65.62 2.78 1.14 1.12 0.58 1.69 122.84 1,039.15 2,888.84 13,494.56L22 2.78 25.90 62.84 2.78 1.11 1.09 0.58 1.67 121.14 1,160.29 3,225.61 16,720.17L21 2.78 25.90 60.06 2.78 1.09 1.07 0.58 1.65 119.40 1,279.69 3,557.53 20,277.71L20 2.78 25.90 57.28 2.78 1.06 1.04 0.58 1.62 117.62 1,397.30 3,884.51 24,162.21L19 2.78 25.90 54.50 2.78 1.04 1.02 0.58 1.60 115.79 1,513.10 4,206.41 28,368.62L18 2.78 25.90 51.72 2.78 1.01 0.99 0.58 1.57 113.92 1,627.02 4,523.11 32,891.73L17 2.78 25.90 48.94 2.78 0.98 0.96 0.58 1.54 112.00 1,739.02 4,834.48 37,726.21L16 2.78 25.90 46.16 2.78 0.95 0.94 0.58 1.51 110.03 1,849.05 5,140.36 42,866.57L15 2.78 25.90 43.38 2.78 0.92 0.91 0.58 1.49 108.00 1,957.05 5,440.59 48,307.15L14 2.78 25.90 40.60 2.78 0.89 0.88 0.58 1.46 105.90 2,062.95 5,734.99 54,042.14L13 2.78 25.90 37.82 2.78 0.86 0.85 0.58 1.43 103.73 2,166.68 6,023.37 60,065.51L12 2.78 25.90 35.04 2.78 0.83 0.82 0.58 1.39 101.49 2,268.17 6,305.50 66,371.02L11 2.78 25.90 32.26 2.78 0.80 0.78 0.58 1.36 99.15 2,367.32 6,581.15 72,952.17L10 2.78 25.90 29.48 2.78 0.76 0.75 0.58 1.33 96.72 2,464.04 6,850.04 79,802.21L9 2.78 25.90 26.70 2.78 0.72 0.71 0.58 1.29 94.18 2,558.22 7,111.86 86,914.07L8 2.78 25.90 23.92 2.78 0.69 0.67 0.58 1.25 91.50 2,649.73 7,366.24 94,280.31L7 2.78 25.90 21.14 2.78 0.65 0.63 0.58 1.21 88.68 2,738.40 7,612.76 101,893.07L6 2.78 25.90 18.36 2.78 0.60 0.59 0.58 1.17 85.67 2,824.07 7,850.92 109,743.98L5 2.78 25.90 15.58 2.78 0.55 0.54 0.58 1.12 82.44 2,906.51 8,080.10 117,824.08L4 2.78 25.90 12.80 2.78 0.50 0.49 0.58 1.07 78.93 2,985.44 8,299.52 126,123.60L3 2.78 25.90 10.02 2.78 0.50 0.49 0.58 1.07 77.02 3,062.46 8,513.65 134,637.25L2 2.78 25.90 7.24 2.78 0.50 0.49 0.58 1.07 76.96 3,139.42 8,727.58 143,364.83L1 4.46 25.90 4.46 3.62 0.50 0.49 0.58 1.07 100.21 3,239.63 14,448.73 157,813.56P1 0.00 0.00 2.23 0.49 0.58 1.07 50.10 3,289.73 0.00 157,813.56Overturning moment [kN·m]Vi  · hi  [kN·m]Storey  LevelStorey  height [m]Storey  with [m]Storey  elevation [m]Tributary height [m]CeiDynamic pressure, WW [kPa]Dynamic pressure, LW [kPa]Total pressure [kPa]Total force [kN]Storey Shear [kN]203  Table C.3 Wind loading along the cantilever-wall direction.   q1/50 [kPa] 0.45 CeH, ww 1.33D [m] 25.90 CeH, lw 0.94Iw, ULS 1.00 H/D 2.21Terrain Rough w [m] 24.73Cg 2.48 Cp, ww 0.80Ct 1.00 Cp, lw 0.50PH 5.07 7.32 90.15 2.54 1.33 1.19 0.52 1.71 31.80 31.80 161.24 161.24L30 2.78 25.90 85.08 3.93 1.29 1.16 0.52 1.68 92.28 124.08 344.95 506.19L29 2.78 25.90 82.30 2.78 1.27 1.14 0.52 1.66 120.27 244.36 679.31 1,185.51L28 2.78 25.90 79.52 2.78 1.25 1.12 0.52 1.64 118.89 363.25 1,009.83 2,195.34L27 2.78 25.90 76.74 2.78 1.23 1.10 0.52 1.62 117.48 480.73 1,336.44 3,531.78L26 2.78 25.90 73.96 2.78 1.21 1.08 0.52 1.60 116.05 596.79 1,659.07 5,190.85L25 2.78 25.90 71.18 2.78 1.18 1.06 0.52 1.58 114.60 711.38 1,977.64 7,168.49L24 2.78 25.90 68.40 2.78 1.16 1.04 0.52 1.56 113.11 824.49 2,292.09 9,460.58L23 2.78 25.90 65.62 2.78 1.14 1.01 0.52 1.54 111.59 936.09 2,602.32 12,062.89L22 2.78 25.90 62.84 2.78 1.11 0.99 0.52 1.52 110.05 1,046.13 2,908.24 14,971.14L21 2.78 25.90 60.06 2.78 1.09 0.97 0.52 1.50 108.46 1,154.59 3,209.77 18,180.91L20 2.78 25.90 57.28 2.78 1.06 0.95 0.52 1.47 106.85 1,261.44 3,506.80 21,687.71L19 2.78 25.90 54.50 2.78 1.04 0.92 0.52 1.45 105.19 1,366.63 3,799.22 25,486.93L18 2.78 25.90 51.72 2.78 1.01 0.90 0.52 1.43 103.49 1,470.12 4,086.92 29,573.86L17 2.78 25.90 48.94 2.78 0.98 0.88 0.52 1.40 101.75 1,571.86 4,369.78 33,943.63L16 2.78 25.90 46.16 2.78 0.95 0.85 0.52 1.38 99.95 1,671.82 4,647.65 38,591.28L15 2.78 25.90 43.38 2.78 0.92 0.83 0.52 1.35 98.11 1,769.92 4,920.38 43,511.66L14 2.78 25.90 40.60 2.78 0.89 0.80 0.52 1.32 96.20 1,866.12 5,187.83 48,699.49L13 2.78 25.90 37.82 2.78 0.86 0.77 0.52 1.29 94.23 1,960.36 5,449.80 54,149.29L12 2.78 25.90 35.04 2.78 0.83 0.74 0.52 1.27 92.19 2,052.55 5,706.10 59,855.38L11 2.78 25.90 32.26 2.78 0.80 0.71 0.52 1.24 90.07 2,142.63 5,956.50 65,811.88L10 2.78 25.90 29.48 2.78 0.76 0.68 0.52 1.20 87.87 2,230.49 6,200.77 72,012.65L9 2.78 25.90 26.70 2.78 0.72 0.65 0.52 1.17 85.55 2,316.05 6,438.61 78,451.26L8 2.78 25.90 23.92 2.78 0.69 0.61 0.52 1.14 83.12 2,399.17 6,669.69 85,120.95L7 2.78 25.90 21.14 2.78 0.65 0.58 0.52 1.10 80.56 2,479.73 6,893.64 92,014.58L6 2.78 25.90 18.36 2.78 0.60 0.54 0.52 1.06 77.82 2,557.55 7,109.99 99,124.57L5 2.78 25.90 15.58 2.78 0.55 0.49 0.52 1.02 74.89 2,632.44 7,318.18 106,442.75L4 2.78 25.90 12.80 2.78 0.50 0.45 0.52 0.97 71.70 2,704.14 7,517.51 113,960.26L3 2.78 25.90 10.02 2.78 0.50 0.45 0.52 0.97 69.97 2,774.11 7,712.03 121,672.28L2 2.78 25.90 7.24 2.78 0.50 0.45 0.52 0.97 69.91 2,844.02 7,906.37 129,578.66L1 4.46 25.90 4.46 3.62 0.50 0.45 0.52 0.97 91.03 2,935.05 13,090.32 142,668.98P1 0.00 0.00 2.23 0.45 0.52 0.97 45.52 2,980.57 0.00 142,668.98Vi · hi  [kN·m]Storey  LevelStorey  height [m]Storey  with [m]Storey  elevation [m]Tributary height [m]CeiDynamic pressure, WW [kPa]Dynamic pressure, LW [kPa]Total pressure [kPa]Total force [kN]Storey Shear [kN]Overturning moment [kN·m]204  The graphical representation of the pressure profiles along the height of the building is shown below in Figure C.5,  Figure C.5 Dynamic pressure profiles: pressure acting on windward face (blue), pressure acting on leeward face (red); on the left side for the Coupled-wall direction, and on the right side for the Cantilever-wall direction. 01020304050607080901000.00 0.50 1.00 1.50Elevation [m]Pressure [kPa]01020304050607080901000.00 0.50 1.00 1.50Elevation [m]Pressure [kPa]205  As shown in Figure 2.7, the pressure acting on the windward face will push the structure, meanwhile the pressures acting on the leeward face due to the generated vortex will be pulling the structure. The net force results from adding these two pressures since both are deforming the building in the same direction. Figure C.6 gives a graphical representation of the loads defined in the ETABS model for each direction, with the blue line representing the coupled-wall direction and the red line representing the cantilever-wall direction, respectively.  Figure C.6 Lateral loads resultant from dynamic analysis: Loads acting on Coupled direction (blue), loads acting on Cantilever direction (red). 01020304050607080901000 50 100 150Elevation [m]Lateral Loads [kN]206  Figure C.7 below, shows the deformed shape of the analytical model once subjected to the wind loads calculated and shown in Table C.2 and Table C.3 above.  Figure C.7 Elevation views of the deformed shape of the model due to wind loading in each direction of analysis.  207   Figure C.8 Shear demands due to wind loading in coupling beams. 01020304050607080901000 200 400 600 800 1,000 1,200Elevation [m]Coupling Beam (CB1-2) Shear Forces, V2 [kN]01020304050607080901000 200 400 600 800 1,000 1,200Elevation [m]Coupling Beam (CB2-3) Shear Forces, V2 [kN]208    Figure C.9 Demands acting on walls due to wind loading; Coupled-wall direction. -200204060801000 2,000 4,000 6,000 8,000 10,000Elevation [m]Shear Force, V3 [kN]-200204060801000 50,000 100,000 150,000 200,000 250,000Elevation [m]Bending Moment, M2 [kN·m]209    Figure C.10 Demands acting on walls due to wind loading; Cantilever-wall direction. -200204060801000 2,000 4,000 6,000 8,000 10,000 12,000Elevation [m]Shear Force, V2 [kN]-200204060801000 50,000 100,000 150,000 200,000 250,000Elevation [m]Bending Moment, M3 [kN·m]