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Analytical study to investigate the seismic performance of single story tilt-up structures Olund, Omri 2008

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i   ANALYICAL STUDY TO INVESTIGATE THE SEISMIC PERFORMANCE OF SINGLE STORY TILT-UP STRUCTURES   by OMRI OLUND P. Eng, B.Sc. Civil Engineering, University of British Columbia, 2001    A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Civil Engineering)     THE UNIVERSITY OF BRITISH COLUMBIA April 2009   © Omri Olund, 2009   ii  ABSTRACT This report describes an analytical study to investigate the seismic performance of single-story tilt-up structures with steel deck roof diaphragms. A review of current practice in North America for the seismic design of tilt-up structures reveals two points of interest; the flexibility of the roof diaphragm is not considered in calculation of the fundamental building period for the design, and that a force-based approach is used for seismic design that does not incorporate the principles of capacity-design currently used in other building systems, such as moment frames, braced frames, and shear walls. To explore the application of capacity design for tilt-up structures, three possible failure mechanisms are investigated and compared: rocking of wall panels, sliding of wall panels, and frame action for buildings with wall panels incorporating large openings.  Based on results of an industry survey, two building archetypes are created to represent the most common building types found in seismically active areas; one including solid panels and the second incorporating panels with large openings.  Consideration of the sliding mechanism suggests it would be difficult to incorporate into common applications due to building geometry irregularities, difficulties in estimating sliding resistance, and permanent deformation resulting from the mechanism. Analytical results from this study are compared with findings from previous research; the most interesting comparison showed the period from the analytical model to be within 10% of the estimated building period from ASCE 41 when the weight of the out-of-plane walls are considered in the estimate.  The rocking mechanism and frame mechanism are studied further by carrying out a preliminary assessment of seismic performance factors (R-values) utilizing concepts from the ATC-63 Methodology.  Various analyses, including non-linear time history analyses for a suite of earthquakes, are carried out on 3D models of the building archetypes. Based on analysis results, the adequacies of some building components are evaluated, including the strength of the roof   iii  deck connectors and the strength of wall panel to roof connections both in-plane and out-of- plane.  Further research is required to provide a recommendation for R-values, however, preliminary recommendations are provided and limitations of the study are discussed. iv  TABLE OF CONTENTS Abstract ......................................................................................................................... ii Table of Contents ........................................................................................................ iv List of Tables ............................................................................................................... ix List of Figures ............................................................................................................... x Acknowledgements ................................................................................................... xiv 1 INTRODUCTION ..................................................................................................... 1 1.1 Overview ............................................................................................................................................................. 1 1.2 Current Practice in the Design of Tilt-up Structures ..................................................................................... 4 1.2.1 Design of Tilt-up Panels for Vertical and Out-of-Plane Loading............................................................... 4 1.2.2 Design of Tilt-up Panels for In-Plane Loading .......................................................................................... 7 1.2.3 Design of Connections ............................................................................................................................. 11 1.2.4 Connecting Panels for Vertical, Out-of-Plane and In-Plane Loads .......................................................... 14 1.2.5 Design of Roof System ............................................................................................................................ 22 1.2.6 U.S. Perspective ....................................................................................................................................... 25 1.2.7 Discussion of Current Design Methods.................................................................................................... 26 1.3 Previous Research ............................................................................................................................................ 27 1.3.1 Roof Diaphragm ....................................................................................................................................... 27 1.3.2 Wall Panels with Openings ...................................................................................................................... 31 1.3.3 Building System ....................................................................................................................................... 33 Table of Contents   v  1.4 Research Aims.................................................................................................................................................. 35 1.4.1 Evaluate Previous Research on Building System ..................................................................................... 35 1.4.2 Investigate Alternatives for Capacity Design ........................................................................................... 35 1.4.3 Quantify Building Performance for Selected Mechanisms ...................................................................... 36 1.4.4 Thesis Organization ................................................................................................................................. 36 2 ASSESSMENT METHODOLOGY ........................................................................ 37 2.1 General ............................................................................................................................................................. 37 2.2 Seismic Performance Factors ......................................................................................................................... 39 2.3 Seismic Hazard ................................................................................................................................................ 40 2.3.1 Ground Motion Record Sets ..................................................................................................................... 40 2.3.2 Ground Motion Record Scaling ............................................................................................................... 42 2.4 Archetypical Systems....................................................................................................................................... 44 2.4.1 Archetypical System 1: Solid Wall Panels ............................................................................................... 48 2.4.2 Archetypical System 2: Wall Panels with Openings ................................................................................ 50 2.5 Non-linear Analysis Methods .......................................................................................................................... 51 2.5.1 Software ................................................................................................................................................... 51 2.5.2 Simulated and Non-Simulated Deterioration / Collapse Mechanisms ..................................................... 52 2.5.3 Non-linear Model Calibration .................................................................................................................. 53 2.5.4 Incremental Dynamic Analysis ................................................................................................................ 54 2.6 Collapse Fragility and Uncertainties .............................................................................................................. 56 2.7 Median Collapse Adjustment for Spectral Shape ......................................................................................... 57 2.8 Evaluation and Acceptance Criteria .............................................................................................................. 58 3 INVESTIGATION OF MECHANISM ALTERNATIVES ......................................... 60 Table of Contents   vi  3.1 Analysis Model Configuration ........................................................................................................................ 60 3.1.1 Conventional Building ............................................................................................................................. 60 3.1.2 Model 1: Sliding Mechanism ................................................................................................................... 62 3.1.3 Model 2: Rocking Mechanism ................................................................................................................. 65 3.1.4 Model 3: Frame Mechanism .................................................................................................................... 65 3.2 Model Verification ........................................................................................................................................... 67 3.2.1 Model 1: Sliding Mechanism ................................................................................................................... 67 3.2.2 Model 2: Rocking Mechanism ................................................................................................................. 71 3.2.3 Model 3: Frame Mechanism .................................................................................................................... 75 3.3 Time History Analysis Results ........................................................................................................................ 80 3.3.1 Model 1: Sliding Mechanism ................................................................................................................... 81 3.3.2 Model 2: Rocking Mechanism ................................................................................................................. 82 3.3.3 Model 3: Frame Mechanism .................................................................................................................... 84 3.4 IDA Results ...................................................................................................................................................... 87 3.4.1 Model 1: Sliding Mechanism ................................................................................................................... 87 3.4.2 Model 2: Rocking Mechanism ................................................................................................................. 88 3.4.3 Model 3: Frame Mechanism .................................................................................................................... 89 3.5 Comparison of Rocking and Sliding Mechanisms ........................................................................................ 90 3.6 Possible Connection Details for Rocking Mechanism................................................................................... 91 3.7 Incorporating a Rocking Mechanism for Panels with Openings ................................................................. 97 3.8 Evaluation of Previous Research .................................................................................................................. 100 3.8.1 Ductility Demands of Walls vs. Roof .................................................................................................... 100 3.8.2 Ductility Demands on Legs of Frame Panels ......................................................................................... 101 3.8.3 Seismic Demands on Roof Diaphragm Due to Out-of-Plane Response of Wall Panels ........................ 102 4 QUANTIFICATION OF SEISMIC PERFORMANCE FACTORS ......................... 105 Table of Contents   vii  4.1 Model 4: Rocking Mechanism ...................................................................................................................... 105 4.1.1 Simulated and Non-Simulated Collapse ................................................................................................. 108 4.1.2 IDA Results, Collapse Statistics and Uncertainty .................................................................................. 109 4.1.3 Acceptance Criteria and Evaluation of R ............................................................................................... 113 4.2 Model 5: Frame Mechanism ......................................................................................................................... 115 4.2.1 Simulated and Non-Simulated Collapse ................................................................................................. 116 4.2.2 IDA Results, Collapse Statistics and Uncertainty .................................................................................. 116 4.2.3 Acceptance Criteria and Evaluation of R ............................................................................................... 119 4.3 Model 6: Frame Mechanism – Eccentric Building ..................................................................................... 121 4.3.1 Simulated and Non-Simulated Collapse ................................................................................................. 122 4.3.2 IDA Results, Collapse Statistics and Uncertainty .................................................................................. 122 4.3.3 Acceptance Criteria and Evaluation of R ............................................................................................... 126 4.3.4 Comparison of IDA Results from Rocking, Frame and Eccentric Models ............................................ 127 5 CONCLUSIONS AND RECOMMENDATIONS ................................................... 130 5.1 Summary of Observations ............................................................................................................................ 130 5.2 Recommendations and Future Research ..................................................................................................... 134 6 REFERENCES .................................................................................................... 137 APPENDIX A. ANALYSIS OF CONVENTIONAL BUILDING .................................... 142 APPENDIX B. DESIGN NOTES AND MODEL PROPERTIES FOR ARCHETYPICAL SYSTEM 1: SOLID WALL PANELS .......................................................................... 165 APPENDIX C. DESIGN NOTES AND MODEL PROPERTIES FOR ARCHETYPICAL SYSTEM 2: PANELS WITH OPENINGS ................................................................... 197 Table of Contents   viii  APPENDIX D. SAMPLE CALCULATION FOR COLLAPSE STATISTICS FOR ARCHETYPICAL SYSTEM 1: SOLID WALL PANELS ............................................. 216  ix  LIST OF TABLES Table 1.1   Deck Test Specimens – Fastening Configurations (Essa, Tremblay and Rogers 2003)  ...................................................................................................................................................... 29 Table 1.2   Results from Monotonic Testing (Essa, Tremblay and Rogers, 2003) ...................... 29 Table 1.3   Results from Cyclic Testing (Essa, Tremblay and Rogers 2003)............................... 30 Table 2.1  Summary of Ground Motion Records (ATC-63, 2008) .............................................. 41 Table 2.2  Industry Survey Results – Typical Single Story Tilt-up Building Attributes.............. 45 Table 2.3  Spectral Shape Factor for Different R Factors ............................................................ 58  List of Figures   x  LIST OF FIGURES Figure 1.1  Concrete Tilt-up Wall Panels Ready for Concrete Placement ..................................... 2 Figure 1.2  2005 NBCC Design Spectrum ..................................................................................... 9 Figure 1.3  Standard Tilt-up Connectors ...................................................................................... 13 Figure 1.4  Joist Pocket Connection - EM1 (Weiler Smith Bowers, 2008) ................................. 16 Figure 1.5  Tie Strut Connection for Out-of-Plane Deck Forces (Weiler Smith Bowers, 2008) . 16 Figure 1.6  Slab to Panel Connection (Weiler Smith Bowers, 2008) ........................................... 17 Figure 1.7  Panel on Dropped Footing (Weiler Smith Bowers, 2008) ......................................... 18 Figure 1.8  Deck Connection for In-Plane Forces (Weiler Smith Bowers, 2008) ........................ 19 Figure 1.9  Design Forces for Panel Sliding / Overturning .......................................................... 19 Figure 1.10  Design Forces for Roof Diaphragm ......................................................................... 23 Figure 1.11  Schematic of Test Setup ........................................................................................... 28 Figure 1.12  Monotonic and Quasistatic Cyclic Loading Protocols ............................................. 28 Figure 1.13  Panel Geometry ........................................................................................................ 31 Figure 1.14  Test Specimen Reinforcement ................................................................................. 32 Figure 2.1  Seismic Performance Factors - Canadian Practice ..................................................... 39 Figure 2.2  Seismic Performance Factors - US Practice ............................................................... 40 Figure 2.2  IDA Results for Different Scaling Procedures ........................................................... 43 Figure 2.3  Typical Roof Design for All Building Archetypes .................................................... 47 Figure 2.4  Roof Diaphragm Zones .............................................................................................. 48 Figure 2.5  Solid Wall Panels – Concrete Outline, Reinforcement, and Connections ................. 49 List of Figures   xi  Figure 2.6  Wall Panels with Openings – Concrete Outline, Reinforcement, and Connections .. 51 Figure 2.7  IDA Results for One Earthquake Record ................................................................... 54 Figure 2.8  IDA Results for Five Earthquake Records ................................................................. 55 Figure 2.9  Fragility Curve Based on IDA Results for 22 Earthquake Records ........................... 56 Figure 3.1  Two-Panel Model for Pushover Analyses .................................................................. 61 Figure 3.2  Model Used to Investigate Sliding System ................................................................ 63 Figure 3.3  Model Used to Investigate Frame Mechanism ........................................................... 66 Figure 3.4  First Mode (Period = 0.58 seconds) ........................................................................... 68 Figure 3.5  Sliding Mechanism - Pushover Analysis Along Short Axis of Building ................... 69 Figure 3.6  Sliding Mechanism – Displaced Shape for Pushover Along Short Axis of Building 70 Figure 3.7  Sliding Mechanism - Pushover Analysis Along Long Axis of Building ................... 71 Figure 3.8  Rocking Mechanism - Pushover Analysis Along Short Axis of Building ................. 72 Figure 3.9  Rocking Mechanism – Displaced Shape for Pushover Along Short Axis of Building  ...................................................................................................................................................... 73 Figure 3.10  Rocking Mechanism - Pushover Analysis Along Long Axis of Building ............... 74 Figure 3.11  Frame Mechanism – Force-Displacement Plot for Beam-Column Subassembly: Comparison of Analytical and Experimental Results (Dew et al., 2001) ..................................... 76 Figure 3.12  Frame Mechanism – Panel Leg with Geometry and Reinforcement from Archetypical System 2: (a)Force-Displacement Plot  (b)Perform Model of Leg ......................... 77 Figure 3.13  Frame Mechanism – Two Legs Connected vs. One Leg: (a)Moment-Curvature Plot; (b)Force-Drift Plot ........................................................................................................................ 79 Figure 3.14  Frame Mechanism – Pushover Analysis Along Short Building Axis ...................... 80 Figure 3.15  Sliding Model – Wall and Roof Drifts for Northridge Earthquake;  (a) Sa(T1)=0.1g; (b) Sa(T1)=1.0g ............................................................................................................................ 81 Figure 3.16  Sliding Model – Breakdown of Energy Dissipation ................................................ 82 List of Figures   xii  Figure 3.17  Rocking Model – Wall and Roof Drifts for Northridge Earthquake, Sa(T1)=1.0g . 83 Figure 3.18  Rocking Model – Breakdown of Energy Dissipation .............................................. 84 Figure 3.19  Frame Model – Wall and Roof Drifts for Northridge Earthquake, Sa(T1)=3.0g .... 85 Figure 3.20  Frame Model – Breakdown of Energy Dissipation .................................................. 86 Figure 3.21  Sliding Model – IDA Drift Results for 8 Ground Motion Records: (a)End Walls; (b)Centre of Roof .......................................................................................................................... 87 Figure 3.22  Rocking Model – IDA Drift Results for 8 Ground Motion Records: (a)End Walls; (b)Centre of Roof .......................................................................................................................... 89 Figure 3.23  Frame Model – IDA Drift Results for 8 Ground Motion Records: (a)End Walls; (b)Centre of Roof .......................................................................................................................... 90 Figure 3.24  Sliding and Rocking Models: IDA Drift Results for 8 Ground Motion Records: (a)End Walls;  (b)Centre of Roof ................................................................................................. 91 Figure 3.25  Rocking Mechanism – Consideration at Panel to Panel Interface - Elevation View from Inside of Building ................................................................................................................ 92 Figure 3.26  Rocking Mechanism – Consideration at Panel to Panel Interface - Detail at Panel Interface ........................................................................................................................................ 93 Figure 3.27  Rocking Mechanism – Possible Connection Details: Elevation of Building ........... 95 Figure 3.28  Rocking Mechanism – Possible Base Connection Details: Tie-down and Shear Pin  ...................................................................................................................................................... 96 Figure 3.27  Rocking Mechanism for Panels with Openings – Possible Connection Details: (a)Elevation of Building; (b)Details ............................................................................................. 99 Figure 3.28  Median Drift Centre of Roof and End Walls for 8 Ground Motion Records: (a)Sliding Model; (b)Rocking Model ......................................................................................... 100 Figure 3.29  Inelastic Wall Displacement at Sa(T1) = 3.0g:  Analysis Results vs. Predicted .... 102 List of Figures   xiii  Figure 3.32  Seismic Demands on Roof Diaphragm due to Out-of-Plane Response of Wall Panels for Sa(T1) = 0.5g:  Analysis Results vs. Common North American Practice vs. ASCE41- 06 Approximation ....................................................................................................................... 103 Figure 4.1  Rocking Model – 2 Adjacent Panels Connected at End Walls ................................ 106 Figure 4.2  Rocking Model - Pushover Analysis Along Short Axis of Building (2 panels connected) ................................................................................................................................... 107 Figure 4.3  Rocking Model – IDA Results for 22 Ground Motion Record Pairs: (a)Drift at End Wall;  (b)Drift at Centre of Roof; (c)In-Plane Deck Forces at End Walls; (d)Out-of-Plane Wall to Roof Connection Forces ............................................................................................................. 110 Figure 4.4  Rocking Model – Collapse Statistics: (a)Roof Drift; (b)In-Plane Deck Forces; (c)Out- of-Plane Deck Forces .................................................................................................................. 113 Figure 4.5  Frame Model – IDA Results for 22 Ground Motion Record Pairs: (a)Drift at End Wall;  (b)Drift at Centre of Roof; (c)In-Plane Deck Forces at End Walls; (d)Out-of-Plane Wall to Roof Connection Forces ............................................................................................................. 117 Figure 4.6  Frame Model – Collapse Statistics: (a)Roof Drift; (b)In-Plane Deck Forces; (c)Out- of-Plane Deck Forces .................................................................................................................. 119 Figure 4.7  Eccentric Model – IDA Results for 22 Ground Motion Record Pairs: (a)Drift at End Wall;  (b)Drift at Centre of Roof ................................................................................................ 123 Figure 4.8  Eccentric Model – IDA Results for 22 Ground Motion Record Pairs: (a)In-Plane Deck Forces at End Wall with Frame Panels; (b) In-Plane Deck Forces at End Wall with Solid Panels; (c) Out-of-Plane Deck Forces ........................................................................................ 124 Figure 4.9  Eccentric  Model – Collapse Statistics: (a)Roof Drift; (b)In-Plane Deck Forces; (c)Out-of-Plane Deck Forces ...................................................................................................... 126 Figure 4.10  Comparison of IDA Results for the Rocking Model, Frame Model, and Eccentric Model: (a)Drift at End Walls with Openings; (b)Drift at Centre of Roof (c)In-Plane Deck Forces; (d) Out-of-Plane Deck Forces ..................................................................................................... 128    xiv  ACKNOWLEDGEMENTS  My sincerest thanks go to my research supervisor, Dr. Ken Elwood, for his guidance and support throughout this research endeavour.  Thanks also to Dr. Perry Adebar, for his insight and for completing second review duties. I would also like to acknowledge the generous support of the Portland Cement Association and the Cement Association of Canada which provided funding to the author.  Support and input from Kevin Lemieux of Weiler Smith Bowers is also greatly appreciated.  1  1 INTRODUCTION 1.1 Overview Tilt-up construction is widely used in the United States and Canada to construct warehouses, office buildings, schools and other types of buildings.  Use of the tilt-up method of construction began in the 1940’s as a way of constructing concrete wall buildings with considerably less formwork than is necessary for casting the walls in place, thus offering cost advantages over other types of construction. Other advantages include the durability of the concrete walls in comparison with other types of walls and the efficiency of the system in that the concrete walls act as cladding, vertical load carrying elements, and as components of the lateral load resisting system (LLRS).  The most common application of tilt-up construction is for single story commercial and industrial structures. Until recently, the design of tilt-up structures was not specifically considered in Canadian building and material codes.  Canadian codes first addressed the design of tilt-up structures in the Design of Concrete Structures Standard issued in 1994, CAN/CSA A23.3-94 (CSA 1994). Currently, tilt-up structures in Canada are designed in accordance with the requirements of the 2005 National Building Code of Canada (NBCC) and the Design of Concrete Structures Standard CAN/CSA A23.3-04.  The Concrete Design Handbook, Third Edition (CAC 2006) offers some guidance and examples for the design of tilt-up structures.  In the latter publication, within Section 13, titled “Tilt-up Concrete Wall Panels”, the following is stated: “Further development of this design standard is required in areas including: • Service Load Deflections • Analysis and Design for Seismic Requirements ” Considering their extensive use in seismically active areas, there has been limited research into the seismic performance of tilt-up building systems, and most of the research that has been Chapter 1 Introduction   2  conducted on this topic has focussed on tilt-up buildings with timber deck diaphragms (Hamburger and McCormick, 1994). Tilt-up structures are essentially constructed in three steps.  The floor slab and foundation are constructed initially, with connectors embedded as required for future fit-up with wall panels. The reinforced concrete wall panels are then cast in a horizontal a position on the building floor slab, with connectors embedded as required for future fit-up with the floor slab and roof diaphragm. After the concrete has gained sufficient strength, the panels are tilted (lifted) by crane and set on the foundations; typically between setting pins cast into the foundations. The panels are then held in place with temporary bracing until the roof system is constructed and connected to the wall panels.  The roof system is typically constructed of either wood or steel. For steel roof systems, sheet metal decking is used in conjunction with open web steel joists and structural steel girders to transfer vertical loads to walls and columns.   Figure 1.1  Concrete Tilt-up Wall Panels Ready for Concrete Placement  In an earthquake, the metal roof decking is designed and constructed to act as a diaphragm to transfer inertial loads from the roof mass and part of the mass of the walls moving out-of-plane into the end walls oriented parallel to the direction of motion.  Connections between the walls and the roof decking are designed to transfer the inertial forces estimated for the design earthquake.  Connections between wall panels moving in-plane are designed to resist inertial forces transferred from the roof deck, as well as inertial forces generated within the in-plane Chapter 1 Introduction   3  walls themselves, such that there is a sufficient margin of safety against sliding or overturning. For a building designed in accordance with current practices, it is difficult to determine in what manner failure of the system will occur for lateral loads.  The roof diaphragm and the wall connections are designed using similar force modification factors (R-values), leading to uncertainty as to whether the roof deck or the wall connections would yield first.  Also, some of the wall connections, specifically the connections between the walls and the floor slab and footing, are subject to both lateral loads and uplift when a lateral load is applied to the building. Although the wall to slab connections have been tested for lateral loads, there have been no tests for uplift.  In order to accurately predict the behaviour of these connections, it is necessary to understand the strength and stiffness interactions that exist for combined uplift and lateral loads. Experimental testing was carried out on these connections as part of this study in order to investigate these interactions.  A detailed description of the testing program and results is provided in Frank Devine’s Master’s Thesis (Devine, 2008) The portion of the research discussed herein consists of an analytical study to investigate the seismic performance of single-story tilt-up structures with steel deck roof diaphragms.  Some of the possible failure mechanisms for these structures are investigated and compared, including rocking of wall panels, sliding of wall panels, and frame action for wall panels with openings. The rocking mechanism and frame mechanism are studied further by incorporating them into the design of a typical single story tilt-up structure designed and assessing the performance of the structure by utilizing concepts from the ATC-63 Methodology (ATC, 2008), The sections below include a brief review of current practices in the design of tilt-up structures, a discussion of previous related research, research aims, and guidelines used to form the basis for the research methodology.  Also included are a detailed description of the research methodology, the specific building systems studied and associated analytical models, results from analyses conducted, as well as some conclusions and recommendations based on the study. Chapter 1 Introduction   4  1.2 Current Practice in the Design of Tilt-up Structures 1.2.1 Design of Tilt-up Panels for Vertical and Out-of-Plane Loading In CAN/CSA A23.3-04, tilt-up panels are described in Clause 23.1.3 as “slender vertical flexural slabs that resist lateral wind or seismic loads and are subject to very low axial stresses.” This clause also states, “Because of their high slenderness ratios, they shall be designed for second- order P-∆ effects to ensure structural stability and satisfactory performance under specified loads.”  Tilt-up concrete wall panels are designed as load bearing elements that span vertically from the floor slab to the roof.  Panels are designed to resist vertical loads imposed by the roof system. Typically, for the purposes of wall panel design, loads from roof joists are considered as uniformly distributed line loads applied at an eccentricity to the centreline of the panel.  A minimum eccentricity of half the panel thickness is prescribed by the code to account for accidental bearing irregularities. Tilt-up panels are also designed to accommodate out-of-plane seismic and wind loads.  Out-of- plane seismic forces for the design of individual wall panels are determined using the 2005 NBCC Clause 4.1.8.17 for Elements of Structures.  The equivalent out-of-plane seismic force for design of panel reinforcement is calculated as follows (CAC, 2006): ppeaap WSISFV ***)2.0(**3.0=  (1.1) Where:  Sa(0.2) =  5% Damped spectral response acceleration at a period of 0.2 seconds.  Value based on published climatic data within 2005 NBCC. Fa =  Acceleration based site coefficient that is dependent on soil conditions and Sa(0.2). p xrp p R AAC S ** =  Where 0.7 ≤ Sp ≤ 4.0    Ax = Height Factor = n x h h *21+ Chapter 1 Introduction   5  hx, ,hn = Height above the base level.  For out-of-plane forces, hx is taken as the center of mass of the panel. Ie =  Importance factor for earthquakes, equals 1.0 for normal importance buildings.    Wp =  Weight of panel Category 1 of 2005 NBCC Table 4.1.8.17 applies for the design of tilt-up panels for out-of-plane bending:  Cp =  Component risk factor.  Usually taken as 1.0. Ar =  Dynamic amplification factor.  For short period buildings with flexible walls, this is equal to  1.0.  If the natural frequency of the component is close to the fundamental period o the building, this factor could increase to as much as 2.5 Rp =  Response factor associated with the ductility of the component. Normally 2.5 for design of reinforced tilt-up wall panels out-of- plane. For a typical building in Vancouver with foundation Site Class D: Sa(0.2) = 0.94 Fa =  1.1 hx =  0.5* hn Ax = n n h h*5.0 *21+  = 2.0 Sp = 5.2 0.2*0.1*0.1  = 0.8 Ie = 1.0 Vp = 0.3*1.1*0.94*1.0*0.8*Wp =  0.25*Wp (0.62*Wp  for  Ar = 2.5) Chapter 1 Introduction   6  Although the value of Ar is normally taken as 1.0, it is important to note that if the flexibility of the deck diaphragm is considered, the fundamental period of the building may approach the fundamental period of the wall panels, resulting in considerably greater amplification of seismic demands in the wall panels.  Often, out-of-plane seismic loads will exceed wind pressure. Out-of-plane wind loads for the design of wall panels are determined based on the 2005 NBCC Clause 4.1.7 for Wind Loads.  Appendix A includes an example of the design of a typical tilt-up building, including the design of panels for out-of-plane wind and seismic loads. Moments from vertical load eccentricity are added to bending moments induced by out-of-plane transverse loads (wind or seismic), and the combined moments are modified to account for P-∆ effects.  A strength limit state is exceeded when the maximum factored bending moment (including primary combined moment due to applied loadings, and secondary moment due to P- ∆ effects) exceeds the factored moment resistance of the concrete section.  This limit state is usually evaluated at the centre of the panel, where imposed moments are the greatest. CAN/CSA A23.3-04 has adopted the “moment magnifier” method of analysis for evaluation of the P-∆ effects.  The limitation to this method is that it is only applicable when axial compression loads are less than 0.1*Ag*fc’, where Where: Ag =  Gross Cross-sectional area of the concrete panel   fc’ =  Specified Unconfined Compressive Strength of Concrete  Utilizing the moment magnifier method, the factored moment is calculated as follows: botfwfft f f PP eP hw M δ**)( 2 * 8 * 2         ∆+++=       (1.2) Where: Mf =  Factored moment including P-∆ effects   wf =  Specified unconfined compressive strength of concrete h =  Wall height   Ptf =  Factored axial load at top of panel Chapter 1 Introduction   7  e =  Axial load eccentricity at top of panel Pwf =  Factored panel weight above mid height   ∆o =  Initial deflection at panel mid height δb =  Moment magnification factor  The moment magnification factor, fb f b K P − = 1 1δ Where: Pf =  Factored axial load at mid height = Ptf + Pwf   Kbf =  Panel bending stiffness = 2 *5 **48 h IE crc    Ec =  Concrete elastic modulus Icr =  Cracked section moment of inertia Appendix B includes an example of the design of a typical tilt-up building, with P-∆ effects considered utilizing the method described above. There is also a limitation on the panel height to thickness ratio, depending on the reinforcement configuration used in the cross-section.  For a single mat of reinforcement, the maximum height to thickness ratio is 50; for a double mat, the maximum is 65. In addition to the above, CAN/CSA A23.3 also states that transverse deflections due to non- seismic service loads must be less than h/100. The NBCC 2005 requires that interstory drift due to seismic loading must be smaller than 2.5% drift.  This design requirement is not considered in the Concrete Design Handbook. 1.2.2 Design of Tilt-up Panels for In-Plane Loading The main sources of in-plane loads for tilt-up wall panels are wind and seismic loads.  As for out-of-plane loads, wind loads are calculated in accordance with the 2005 NBCC Clause 4.1.7 on Wind Loads. Chapter 1 Introduction   8  In Canada, seismic demands for tilt-up structures are determined using the 2005 NBCC Clause 4.1.8 on Earthquake Load and Effects.  Code provisions require that the design base shear for a building, V, be calculated as follows:  od ve RR WMSIV * *** =  (1.3)  Where:  V =  Design Base Shear (kN)  Ie =  Importance Factor  S =  Spectral Acceleration (g)  Mv = Higher Mode Shear Factor  W = Weight of the building (kN)  Rd = Force Reduction Factor based on ductility  Ro = Force Reduction Factor based on over-strength  The spectral acceleration is the design spectrum value at the building natural period.  In the 2005 NBCC, the design spectrum is determined as follows:  S(T) = Fa*Sa(0.2) for T≤0.2 s   = Fv*Sa(0.5) or Fa*Sa(0.2), whichever is smaller for T = 0.5 s.   = Fv*Sa(1.0) for T = 1.0 s.   = Fv*Sa(2.0) for T = 2.0 s.   = Fv*Sa(2.0) / 2 for T ≥ 4.0 s. Fa and Fv are factors prescribed by the code depending on the building foundation conditions and the period.  Seismic acceleration values (Sa) are provided at four structural periods, and are based on an earthquake with an annual probability of excedence of 1 in 2475.  Once the design Chapter 1 Introduction   9  spectrum is constructed, spectral accelerations for structure periods between the values provided can be interpolated from the design spectrum.  Figure 1.2 below illustrates the design spectrum for Vancouver for a site with a very dense soil or soft rock foundation (Site Class C, Fa=1, Fv=1):   Figure 1.2  2005 NBCC Design Spectrum Most tilt-up structures satisfy the code requirements for “Regular Structures” and as such, seismic demands are evaluated using the “Equivalent Static Force Procedure”. In addition, for regular structures with an Rd value greater than or equal to 1.5, the base shear can be reduced as follows:  od e RR WSIV * *)2.0(* 3 2 ∗=  (1.4)  The 2/3 cutoff is illustrated in Figure 1.1 above. For various types of buildings, the 2005 NBCC code prescribes formulae to calculate the period to be used in determining the spectral acceleration. Of the building types considered in the code, Chapter 1 Introduction   10  a tilt-up building most closely resembles a concrete shear wall building.  The formula provided to calculate the fundamental period for a concrete shear wall building is as follows: 4 3 *05.0 hT =  (1.5) Where:  h = height of the shear wall in meters For a typical wall panel with a height of 9 m, Eqn. 1.5 would give a period of vibration equal to 0.26 seconds. It is interesting to note that for a typical tilt-up structure, the period calculated in accordance with the 2005 NBCC code can be substantially shorter than the 1st mode period results from an eigenvalue analysis of a typical single story tilt-up building with a steel deck diaphragm.  This discrepancy will be discussed further in Section 3.2. The higher mode shear factor, Mv, is equal to unity for buildings with a fundamental period smaller than 1.0 seconds.  This factor is included to account for the effect of higher modes on the response of the structure, and does not apply for relatively short buildings with a small period. The weight of the building, W, used in determining the base shear typically consists of the roof weight, including 25% of the design snow load, the full dead weight of the in-plane walls, and half the weight of the out-of-plane walls.  Only half the weight of the out-of-plane walls is considered since the behaviour of the walls in the out-of-plane direction is assumed to be similar to simply supported beams subjected to a uniformly distributed load, i.e. half the inertial force is assumed to be transferred to the base of the panel, and half is assumed to be transferred to the diaphragm. For the design of tilt-up structures, the force reduction factor based on ductility, Rd, varies between 1.0 and 1.5 depending on the component being designed.  An Rd value of 1.5 is used for tilt-up structures for calculation of the base shear.  The force reduction factor based on over- strength, Ro, is1.3, consistent with the NBCC 2005 requirements for conventional construction.  Chapter 1 Introduction   11  1.2.3 Design of Connections In tilt-up construction, vertical and lateral loads are transferred to the wall panels and between panels by various types of connections.  For tilt-up structures within the scope of this work, connections are used to transfer gravity loads from beams and joists to the panels, lateral forces between the roof deck diaphragm and the panels, shear forces between panels, shear forces between the panels and the floor slab or foundation. The design of connections for tilt-up structures is carried out in accordance with conventional design practices for embedded connections.  Guidelines for the design of tilt-up connections are provided within the Concrete Design Handbook (CAC 2006) for Canadian practice, and within “Connections for Tilt-up Wall Construction” (PCA 1987) for design in the United States.  The descriptions provided below of typical practice in the design and construction of tilt-up connections are based on information from the guideline documents referenced above. Connections are designed to resist forces greater than the imposed loads.  Typical connections used in the tilt-up industry are cast-in-place concrete in-fill anchors, drilled-in anchors, and welded embedded metal connectors. Cast-in-place concrete in-fill sections are constructed by leaving a blockout between the panels to be connected and extending the panel rebar beyond the face into the blockout. To connect the panels, concrete is placed in the blockout.  Cast-in-place concrete in-fill sections provide more continuous (and greater) load transfer than discrete drilled or embedded connections. However, these types of connections are seldom used since they are more expensive than other types of connections due to the additional forming and concrete work required. Drilled-in expansion or adhesive anchors are used in the tilt-up industry mainly for supporting light loads or for repairs.  They are not as ductile as cast-in-place anchors, and are thus not as suitable for seismic applications.  In addition, expansion anchors are problematic in thin panel sections, especially when large edge distance is required. The type of connectors most often used in the tilt-up industry are welded embedded metal connectors, due to their cost advantage relative to other connectors.  The strength and ductility of embedded connectors vary, depending on the configuration of the connector, the type of embedment anchor used, and the extent of embedment. Chapter 1 Introduction   12  In recent years, efforts have been made to standardize the types of welded embedded connectors most often used in the tilt-up industry in Canada.  In 1998, testing (Lemieux et al., 1998) was conducted on five standard connection types most commonly used in Canadian practice.  The purpose of the testing was to verify strength values used in design, and to evaluate the performance of the connectors in order to establish appropriate force modification factors to be used in seismic design.  The test specimens were subjected to both monotonic and cyclic loading protocols.  The five standard connection types tested were labelled as EM1, EM2, EM3, EM4, and EM5.   The connector details and their assigned design strengths are shown in the Figure 1.3 below, taken from Section 13 of the Concrete Design Handbook (CAC, 2006):  Chapter 1 Introduction   13    Figure 1.3  Standard Tilt-up Connectors The EM1 joist seat embedded connector shown above is used to transfer loads from open web steel joists to the wall panels.  The joist seat is supported by and field welded to an EM1 connector located within a blockout maintained in the concrete wall panel. The connector Chapter 1 Introduction   14  consists of an embedded angle with two 15M reinforcing bars welded to it and embedded in the concrete to provide anchorage. The EM2, EM3, and EM4 embedded connectors consist of steel plates with welded steel studs embedded into the concrete.  These connectors are used in various ways.  The EM2 connector has two shear studs and is most often used to connect the roof diaphragm perimeter chord angle to the wall panels, as well as to connect the wall panels to the floor slab or footings. The EM3 connector has four shear studs and is commonly used to connect small beams or channels to the wall panels.  The EM3 connector has 8 shear studs and is typically used to connect larger beam or brace connection to wall panels.  For seismic design utilizing the above studded embedded connectors with the design capacities provided in the figure above, the CAN/CSA A23.3-04 code recommends a force modification factor for ductility, Rd = 1.0, and a force modification factor for overstrength, Ro = 1.3. The EM5 edge connector consists of an embedded steel angle with a continuous 20M reinforcing bar welded to it and embedded into the concrete for anchorage.  It is most commonly used to transfer shear loads between panels and to transfer shear loads between panels and the slab. Based on testing (Lemieux et al. 1998), the EM5 connector exhibits a more ductile response than other types of embedded connectors.  It is important to note that the 1998 testing program was carried out for loading applied in shear only, and did not consider uplift loads or interaction between shear and uplift.  Tests including these considerations are currently being conducted at the University of British Columbia (Devine 2008). For seismic design utilizing the EM5 connector with the design capacity provided in the figure above, the CAN/CSA A23.3-04 code recommends a force modification factor for ductility, Rd = 1.5, and a force modification factor for overstrength, Ro = 1.3. 1.2.4 Connecting Panels for Vertical, Out-of-Plane and In-Plane Loads In the design of tilt-up structures, individual panels must be connected to the roof diaphragm, to each other, and to the foundation in order to provide an adequate load path for the design loads. Chapter 1 Introduction   15  For vertical loading transfer from the roof system to the wall panels, the open web steel joists are supported on EM1 connectors at formed pockets in the wall panel.  The joist seats are welded to the EM1 angle to secure the joists. Out-of-plane loads used for the design connections are similar to those described in Section 1.2.1, except for assumptions relating to seismic loads.  The main difference in the code requirements occurs in calculation of the force factor, Sp.  For the same example building as was used in Section 1.2.1 the following calculation illustrates how out-of-plane seismic forces are calculated differently for the design of connections: Category 21 of 2005 NBCC Table 4.1.8.17 applies to flexible components with non-ductile material or connections, which most closely describes tilt-up panels with rigid connections.  This is because the reinforced concrete section of the wall panels in out-of-plane bending are considered to be much more ductile than the connections.  The Ar, Rp and Sp values used for Equation (1.1) are modified as follows: Ar =  2.5 (Dynamic amplification factor) Rp =  1.0 (Ductility factor) Sp = 0.1 0.2*5.2*0.1  = 5.0, but limited to 4.0. (Note the increase from 0.8 in the previous calculation of Sp for the design of wall reinforcement.) In this case, the tributary weight of the panel is used in Equation (1.1) to determine the out-of- plane connection design force. In common practice, the tributary weight of the panel is assumed to be half the panel weight. This results in an increase in the calculated seismic force Vp from 0.25*Wp to 0.62*Wp.  This means that for tilt-up panels designed in accordance with the 2005 NBCC, the out-of-plane seismic forces used to design the connections are approximately five times the forces used to design the panel reinforcing. To connect the wall panels to the roof system for out-of-plane loading, two types of connections are employed in combination.  The out-of-plane resistance of the EM1 connection to the open web steel joists is considered.  Refer to Figure 1.4 below for an illustration of this connection. Chapter 1 Introduction   16   Figure 1.4  Joist Pocket Connection - EM1 (Weiler Smith Bowers, 2008) For panels that do not have open web steel joists framing in, tie struts are provided, connected to the panel with an EM1 connection and connected to the roof with deck fasteners.  Refer to Figure 1.5 below for an illustration of this connection.  Figure 1.5  Tie Strut Connection for Out-of-Plane Deck Forces (Weiler Smith Bowers, 2008) Chapter 1 Introduction   17  To transfer out-of-plane loading from individual panels at the base to the floor slab, two methods are used.  In some cases, EM2 or EM3 connections embedded in the wall panel are field welded to EM5 connectors embedded in the floor slab.  Refer to Figure 1.6 below for an illustration of this connection.   Figure 1.6  Slab to Panel Connection (Weiler Smith Bowers, 2008) In other cases, a section of the floor slab is cast around dowels extending from the wall and slab after the wall has been erected. In addition to the above, the footings are typically cast with rebar extending out to act as locating pins to facilitate erection.  These bars extending from the footing may provide some modest additional out-of-plane resistance for the base of the panel.  Friction between the panels and the footings also provides some additional resistance to out-of-plane loading at the base of the panels.  Refer to Figure 1.7 below for an illustration of this connection. Chapter 1 Introduction   18   Figure 1.7  Panel on Dropped Footing (Weiler Smith Bowers, 2008) For in-plane load transfer between the wall panels and the roof system, the roof diaphragm perimeter angle is periodically welded to embedded EM2 or EM3 connectors.  Sufficient connection is provided to accommodate the maximum shear load from the roof diaphragm. Refer to Figure 1.8 below for an illustration of this connection.  Chapter 1 Introduction   19   Figure 1.8  Deck Connection for In-Plane Forces (Weiler Smith Bowers, 2008) Connections between panels and between panels and the footings or floor slab are typically designed to resist panel sliding and overturning.  The following methodology is presented in the Concrete Design Handbook (CAC 2006) examples to design panel connections for in-plane loads:  Figure 1.9  Design Forces for Panel Sliding / Overturning Width, b H ei gh t, h EM3 to EM5 Connection EM5 to EM5 Connection Vroof Vin-plane walls Wroof WPanel VP/P VP/S Vertical Reaction VP/P WPanel VHold-down VP/P WPanel Wroof Wroof VP/P Chapter 1 Introduction   20  Figure 1.9 above illustrates the forces considered in designing tilt-up panel connections to resist panel sliding and overturning due to in-plane seismic loads.  The figure shows only the forces considered in the design and does not present a complete free body diagram (FBD), which would include vertical forces on the panel to slab connectors.  The forces shown are for an end wall of a building, comprised of three identical panels.  For design purposes, the panels and connectors are assumed rigid. For overturning considerations, moments are taken about the bottom corner of each panel individually.  It is important to note that the panel to slab connections are not considered in the calculation of overturning resistance. For example, the number of connections required to resist overturning moments for the three panels would be calculated as follows: Overturning moment, RoRd hVhVMo planewallsinroof * 1 * 2 **       += −  (1.6) Resisting moment, bVpanelsbWWpanelsbVM holddownroofpppr *3*2 *)(2**/ +++=  (1.7) Where   Vroof =  Seismic shear force transmitted from the roof diaphragm (Typically consists of the weight of the roof including 25% the design snow load as well as half the weight of the out-of-plane walls multiplied by the seismic base shear coefficient)   Vin-plane walls =  Seismic shear force from the in-plane walls (Typically consists of the weight of the in-plane walls multiplied by the seismic base shear coefficient)   Rd =  1.5   Ro =  1.3 (Seismic force modification factors established based on connector testing) Vp/p =   Number of panel to panel connectors * connector resistance Chapter 1 Introduction   21  (Typically EM5 embedded connectors are used for panel to panel connections.  Connectors are embedded in adjacent panels and field welded.) Wp =  Weight of panel Vhold-down = Maximum hold-down weight at corner (Approximately half the weight of the adjacent connected out-of- plane panel is assumed to assist in resisting the overturning moment) In the calculation of overturning moment in Equation 1.6 above, the seismic force for the in- plane walls is applied at the centre of gravity. It is important to note that in reality, the acceleration response of a building in a seismic event is more accurately modeled with a triangular distribution, which would result in the seismic force being applied at two thirds the height of the in-plane walls. Based on Equation 1.7 above, sufficient numbers of panel-to-panel connectors are provided such that the overturning demands due to the applied loads can be resisted.  The following equations illustrate how the panels are designed for sliding considerations: [ ] RoRd VVH planewallsinroofapplied * 1 * − +=  (1.8) µ*)(/ WroofWpVH spresisting ++=  (1.9) Where  Rd =  1.0   Ro =  1.3 (Seismic force modification factors established based on connector testing) Vp/s =   Number of panel to slab connectors * connector resistance (Typically EM5 connectors are embedded in the slab and EM3 connectors are embedded in the panel, and the two connectors are field welded after panel erection.) Chapter 1 Introduction   22  µ =  0.5  (Coefficient of friction between the panels and footing) Sufficient numbers of panel to slab connectors are provided such that the applied horizontal forces can be accommodated. Using the above design methodology, it is clear that seismic loads less than or equal to the design loads can be accommodated.  However, it is not certain how the system will behave for seismic loads greater than the design loads. In reality, the connections at the base will undergo both shear and uplift loads.  Also the magnitude of the loads applied on the connectors will depend on the relative stiffness’ of the panel to panel connectors and the panel to slab connectors in uplift and shear.  It is important to understand the behaviour of a structural system post-yield in order to ensure that the system can behave in a ductile manner and prevent brittle collapse for ground motions more severe than the design earthquake. 1.2.5 Design of Roof System The roof system for tilt-up structures is typically constructed of either wood or steel.  Wood systems consist of plywood decking supported on wood joists, wood beams and steel columns. Although wood roof systems have been used extensively in the past, steel roof systems are currently used for most new tilt-up construction.  For this reason, this study will focus on steel roof systems.  Steel roof systems consist of steel decking supported on open web steel joists, steel girders and steel columns.  To carry the vertical loads, the steel decking is designed to span between the open web steel joists, which are supported by either a steel girder or one of the concrete wall panels.  The girders are most often supported on steel columns, though occasionally they are framed into wall panels due to building layout considerations. To transfer lateral loads due to earthquakes and wind from the out-of-plane walls and roof into the in-plane walls, the steel roof decking is designed to act as a diaphragm. The steel decking is connected to underlying joist members either with pins, screws or by puddle welds.  Side laps of adjacent decking panels are typically fastened by screws, but may also be fastened by button punching or welding.  A steel angle is placed around the perimeter of the deck and fastened to the steel decking by screws, welds, or pins.  Figure 1.10 below illustrates the typical free body diagram used to determine the design forces on the deck and perimeter chord. Chapter 1 Introduction   23     Figure 1.10  Design Forces for Roof Diaphragm The decking itself is used to transfer shear loads, while the perimeter angle is used to resist axial loads.  The perimeter angle is also used to transfer shear loads from the decking into the in-plane wall panels.  The perimeter angle is welded to embedded connectors in the perimeter wall panels (usually EM2 or EM3 connectors), typically at a spacing of about 4 ft.  From the above figure, the design shear load for the deck would be as follows: 2 * max LqV =  (1.10) To design for seismic loads on the roof diaphragm, standard practice in North America is to assume a tributary mass of half the out-of-plane walls and the mass of the roof is participating in the first mode response.  The design axial load for the perimeter angle would be as follows Shear Force Diagram: x Mmax Vmax Bending Moment Diagram: B L q Chapter 1 Introduction   24  B Lq B MCdesign *8 * 2 max ==  (1.11) In Canada, steel deck diaphragms are designed in accordance with a document titled “Design of Steel Deck Diaphragms – 3rd Edition” (Canadian Sheet Steel Building Institute, 2006).  This document endorses  two methods used to determine the shear capacity of deck sections, one based on the Tri-Services Method (S.B. Barnes and Associates, 1973), and one based on the Steel Deck Institute (SDI) Method (Steel Deck Institute, 2004).  The Tri-Services Method was developed by S. Barnes and Associates and is based on a series of full-scale tests of steel deck panels from which empirical equations were developed for strength and stiffness.  This method has limited applicability and is subject to the following restrictions: • Deck connections to the supporting structure must be welded with 12mm (0.5in) minimum effective diameter. • Side-lap connections between deck sheets must be button punched or seam welded. • Sheet thickness must be at least 0.76mm (0.030 in or 22ga). The maximum thickness is 1.52mm (0.060 in or 16ga). • Each deck unit must be attached to the framing member by at least two welds. • Side lap attachments have a maximum spacing of 0.9m (3ft). • The original tests were based only on horizontal assemblies. The SDI method was developed by Dr. L.D. Luttrell based on analytical work and tests conducted at West Virginia University.  In this method, the ultimate capacity of the diaphragm is limited by any one of four failure modes: • Fastener failure along the outer panel edge. • Fastener failure around interior panel. • Failure of the corner fasteners. • Plate-like shear buckling. Chapter 1 Introduction   25  Some of the variables selected during design of the deck include the thickness of the deck, the profile of the deck, the type of fasteners used and the fastening pattern. The Tri-Services method has often been used in Canada due to its conformity to standard construction practices and due to its adoption by the Canadian Sheet Steel Building Institute. However recent testing (Tremblay et. al, 2003) has indicated that welded and button punched deck connections do not perform favourably when compared to screwed and pinned connections under applied cyclic loading.  This has led to a shift in Canadian practice to the more frequent use of screwed and pinned connections in seismically active areas, requiring that the design be carried out using the SDI method.  For the purposes of this study, the design of steel deck diaphragms has been based on using pinned and screwed deck fasteners, and has been carried out in accordance with the SDI method.  Hilti Profis DF Dia software (Hilti Corporation, 2006) was used to carry out the design of the steel deck diaphragm.  Hilti has conducted extensive testing on various fasteners and has recently proposed a modification factor to apply to SDI calculated strength and stiffness based on test results (Hilti Corporation, 2008). For seismic loads, force modification factors are applied depending on the type of fasteners used. If pins are used to fasten the deck to the underlying members, and screws are used to fasten the deck sheet side laps, a ductility factor, Rd = 1.5 and an over-strength factor, Ro = 1.3 are typically used. 1.2.6 U.S. Perspective The design and construction of tilt-up buildings in the U.S. is done in much the same way as it is in Canada.  The main reference manual used in design is “The Tilt-Up Construction and Engineering Manual – 6th Edition” (Tilt-up Concrete Association 2005).  Currently, the governing building code used in design of tilt-up structures is the International Building Code (IBC 2006), which references the concrete design code ACI-318 (2005) issued by the American Concrete Institute (ACI) for concrete design. There are no major differences in design provisions between U.S. and Canadian practices for non-seismic loading.  In addition, most construction details are very similar.  However, the consideration of seismic loads in the design of tilt-up structures is carried out slightly differently. Chapter 1 Introduction   26  The following is a brief summary to highlight the differences in the approach used to calculate seismic demands in accordance with IBC 2006, in comparison to the NBCC 2005: The fundamental period is calculated as follows: 4 3 *02.0 hT =  (1.12) Where:  h = height of the wall panels in feet Note that Equation (1.12) above, is identical to Equation (1.5) used in Canada when compared with the same units.  Similar to Canadian practice, calculation of the building period does not account for the flexibility of the roof diaphragm. The IBC 2006 uses a deterministic approach to obtain the Maximum Credible Earthquake (MCE) for a given location.  The Design Earthquake (DE) is defined as two-thirds of the MCE. It turns out that for most locations in the US, the MCE is governed by the 2% in 50 year earthquake which is the same earthquake return period used for the NBCC 2005.  Also, since tilt- up buildings generally fall within the short period cut-off defined in the NBCC 2005, the earthquake demands prescribed by the NBCC 2005 are multiplied by 2/3 to obtain the design base shear.  As such, the earthquake demands based on US and Canadian codes are equivalent. Force modification factors (R values) are treated slightly differently in US codes than in the NBCC 2005.  In US codes, there is no distinction between factors accounting for overstrength and factors accounting for ductility (Ro and Rd in the NBCC 2005).  A single R value is prescribed, depending on the lateral load resisting system.  Within the IBC 2006, tilt-up structures fall in the category of load bearing ordinary precast shear walls, for which an R value of 3.0 is prescribed.  1.2.7 Discussion of Current Design Methods The methods described above for seismic design of tilt-up structures constitute a force-based approach.  The structural system is designed to accommodate seismic forces calculated in Chapter 1 Introduction   27  accordance with the building code.  One major drawback of the approach described above is that there is little consideration for capacity design principles commonly incorporated into other structural systems currently in use.  In essence, there is no clear, stable failure mechanism for the system, since the standard connectors used do not have sufficient ductility in the direction in which they are loaded to allow a stable mechanism to form. Another problem with the current design approach is that the code calculated fundamental period of the building currently used to establish seismic demands does not account for the flexibility of the steel roof deck, and thus does not accurately predict the fundamental period for a typical single story tilt-up structure. 1.3 Previous Research 1.3.1 Roof Diaphragm A reasonable assessment of the strength and stiffness of the roof deck diaphragm is important in this study in order to ensure that: • The diaphragm for the archetypical building is appropriately designed • The stiffness of the diaphragm is reasonably estimated and incorporated into the analysis model • The strength of the deck is appropriately estimated when compared with demands from the analysis. There has been considerable research on the behaviour of corrugated cold-formed steel deck diaphragms and many tests have been conducted, though most have been performed under monotonically increasing load (Nilson, 1960; Easley and McFarland, 1969; Luttrell and Ellifritt, 1970; Easly, 1977; Steel Deck Institute (SDI), 1981; Klingler, 1996; Lemay and Beaulieu, 1986). Seismic loading is inherently cyclical, and therefore the results of these tests may not be applicable for this study. More recently, several tests have been conducted incorporating cyclical loading.  In one study (Essa, Tremblay and Rogers, 2003), 18 large scale tests were carried out on diaphragm assemblies made with 22ga (0.76mm) and 20ga (0.91mm) thick metal deck sheets using various types of fasteners in various configurations.  The tests were performed using a Chapter 1 Introduction   28  cantilever type configuration for the test setup, with the steel deck diaphragm in a horizontal plane.  Both cyclic and monotonic testing was conducted.  Figure 1.11 below (Essa, Tremblay and Rogers 2003) illustrates the test setup used.  Figure 1.11  Schematic of Test Setup Nine different configurations of fasteners and deck thicknesses were included in the program. Two specimens were constructed for each configuration; one was tested with monotonic loading and one with cyclic loading. The loading protocols used are illustrated in Figure 1.12 below (Essa, Tremblay and Rogers 2003).  Figure 1.12  Monotonic and Quasistatic Cyclic Loading Protocols Chapter 1 Introduction   29  In the above figure, D1 and D2 are determined from the monotonic testing to be used in the cyclic testing.  D1 is the displacement assuming the specimen remains elastic based on the secant stiffness up to the peak load. D2 is the actual displacement at the peak load. The fastening configurations and corresponding deck thicknesses tested are shown in the table below [Essa, Tremblay and Rogers 2003].  Of particular interest are Test No.’s 4, 7, 17 and 18, which incorporate B-Deck nestable deck profile with nailed (Hilti) deck to frame fasteners and screwed side lap fasteners, since this configuration was adopted for this study.  These have been outlined in Table 1.1 below. Table 1.1   Deck Test Specimens – Fastening Configurations (Essa, Tremblay and Rogers 2003)  The table below provides the results for the monotonic testing [Essa, Tremblay and Rogers 2003].    The results of interest are outlined. Table 1.2   Results from Monotonic Testing (Essa, Tremblay and Rogers, 2003)  Chapter 1 Introduction   30  In the above table, the SDI* values of strength and stiffness were based on prior monotonic testing [Rogers and Tremblay, 2003].   It is believed there is an error in the table in the title of the third column from the left.  It seems the intent of the authors was to compare strength results from testing to SDI calculated strengths, not SDI* calculated strengths.  The results for the cyclic testing are provided in the table below.  The results of interest are outlined in red. Table 1.3   Results from Cyclic Testing (Essa, Tremblay and Rogers 2003)  From the tables above, it can be observed that the stiffness of the 22ga (0.76mm) deck is approximately equal to (0.92)*( 0.838)*(SDI calculated stiffness) or 0.77*SDI stiffness, and that the stiffness of the 20ga (0.91mm) deck is approximately equal to (1.128)*( 0.797)*(SDI calculated stiffness) or 0.90*SDI stiffness.  For the purposes of this study, a ratio of 0.8*SDI calculated stiffness was selected for modeling the deck diaphragm. Additional tests have been conducted recently by Hilti on proprietary powder actuated fasteners and screw connectors  and results are described in a draft report (Hilti Corporation 2008). Within the report, Hilti proposes a modification factor to be applied to the shear strength of the deck as calculated using the SDI method.  Also, Hilti provides different strength values to be used with the SDI equations that are dependent on the thickness of the base metal to which the deck is attached.  For the deck configuration used in this study, the modifications proposed in the Hilti report result in a strength increase of 25%.  For the purposes of this study, the deck was designed using the conventional SDI approach, but the higher strength proposed by Hilti was used to compare with the deck demands from the non linear analyses.  Refer to Section 2.4 for more details of the roof deck configuration used in this study. Chapter 1 Introduction   31  1.3.2 Wall Panels with Openings Part of this study involves investigating the behaviour of tilt-up buildings incorporating wall panels with large openings.  The behaviour of tilt-up wall panels with large openings was investigated in a recent study [Dew, Sexsmith and Weiler, 2001], in which 6 different specimens with 3 different hinge zone tie spacings were investigated.  The geometry of the wall panel represented by the test specimens is shown in Figure 1.13 below [Dew, Sexsmith and Weiler, 2001].  Figure 1.13  Panel Geometry The actual test specimens used were one quarter of the panel shown in the figure above.  The extents of the representative specimen were established at locations where inflection points were thought to occur based on prior analysis.  The layout of reinforcing steel used in the various specimens is as shown in Figure 1.14 below [Dew, Sexsmith and Weiler, 2001]. Chapter 1 Introduction   32   Figure 1.14  Test Specimen Reinforcement As shown in Figure 1.14 above, 3 different hinge zone tie spacings were used for the leg and the beam: 100mm, 200mm and 300mm.  The test panel shown above has many similarities to the panels used in this study, including the layout of reinforcing steel, the aspect ratio of the leg, and the overall dimensions. Within the section on discussion of test results, it is indicated that the hinge region was found to be approximately equal to the depth of the member.  The paper also indicated that for all cases, the reinforcement in the horizontal beam did not appear to yield, nor did the shear reinforcement in the leg. Chapter 1 Introduction   33  It is observed in the paper that the specimens with a tie spacing of either 100mm or 200mm maintained the design flexural strength of the specimen even on the third cycle of their load sequence, while the specimens with a tie spacing of 300mm did not.  The design flexural strength was defined using stress block methods assuming an ultimate concrete stain of 0.0035. Displacement ductility was determined for the various tests.  Displacement ductility is defined as follows:         ∆ ∆ = yield ultDuctility  (1.13) Where:  ∆ult = Ultimate displacement    ∆yield =Yield displacement In this paper, the yield displacement was assumed to occur when a bending moment equal to the design flexural strength was applied to the specimen, while the ultimate displacement was assumed to occur when the specimen could no longer maintain the design flexural strength. The calculated ductility of the specimens was used to estimate an appropriate force reduction factor for the specimens.  Since the fundamental period of the tilt-up panels was expected to be in the order of 0.2 seconds, which is significantly less than the period at the peak spectral acceleration in the design spectrum, it was assumed that the equal energy principle could be applied to determine an appropriate force reduction factor for the specimens.  The equal energy principle states the force reduction factor is calculated as follows: ( ))1*2 −= DuctilityR  (1.14) A force reduction value (Rd*Ro) of 2.0 is commonly used for the design of tilt-up frame panel legs.  Based on the test results, this paper indicated that a force reduction value of 2.0 is reasonable for the specimens tested if a maximum tie spacing of 200mm is used.  As such, these values were incorporated in this study. 1.3.3 Building System There has been some previous work carried out to investigate the behaviour of the overall building system for tilt-up structures subjected to earthquakes.  One recent study (Adebar, Guan Chapter 1 Introduction   34  and Elwood, 2004) investigated the amplification of inelastic drifts in concrete tilt-up frames due to steel deck roof diaphragms designed to remain elastic during the design earthquake.  The above study proposed a simplified approach to estimate the inelastic drifts in concrete tilt-up panels with openings for walls subjected to in-plane seismic loading, which is described in the following equation: )1(*)( we wy dwi V V TS −=∆  (1.15) Where:  ∆wi = Inelastic Wall Displacement Sd(T) = Spectral displacement at the centre of the roof at the fundamental period of the building Vwy = Wall strength Vwe = Maximum elastic demand for the wall This approach was validated with results from nonlinear analyses of simplified models of various configurations for tilt-up buildings incorporating panels with openings. Another study (Hawkins, Wood and Fonseca, 1994) investigated the possibility of using a detailed analytical model to reproduce the measured response of an instrumented tilt-up warehouse located in Hollister, California, subjected to the 1989 Loma Prieta Earthquake.  The study reported good correspondence between results from the analysis and the measured response, and was helpful in developing the analytical model used for the current work. The American Society of Civil Engineers (ASCE) Standard 41-06, titled “Seismic Rehabilitation Standard” provides guidelines for estimating both building period and the out-of-plane seismic load distribution on the roof diaphragm for single story structures with flexible diaphragms.  It appears that the guidelines provided in the above standard are not incorporated either in American or Canadian practice for the design of tilt-up structures.  The ASCE 41-06 approximations for building period and out-of-plane loading on the roof diaphragm will be compared with the analysis results from this study within Section 3.2. Chapter 1 Introduction   35  1.4 Research Aims 1.4.1 Evaluate Previous Research on Building System Within this study, a simplified approach previously proposed (Adebar, Guan and Elwood, 2004) to estimate the inelastic drifts in concrete tilt-up panels with openings is compared with results from non-linear analyses using a detailed finite element model.  Other conclusions from the above study, including the effect of yielding in-plane walls on the roof displacement are also investigated with a detailed model. In addition, results from the analyses are compared with the following items: • Determination of building period based on NBCC 2005 and ASCE 41-06, and • Seismic demands on the roof diaphragm due to out-of-plane response of wall panels based on ASCE 41-06 and common North American practice. 1.4.2 Investigate Alternatives for Capacity Design The concept of capacity design is considered in the seismic design of most modern structures. Capacity design essentially requires that there be a clearly identifiable, ductile mechanism in the lateral load resisting system of a building, thus providing occupants with ample warning prior to failure. Current methods for the design of tilt-up structures do not incorporate the principles of capacity design.  Panels are connected together to resist code-prescribed seismic forces. Connections are designed with R values selected based on results from testing of individual connectors.  The roof diaphragm is designed with an R value selected based on results from testing of steel deck diaphragms.  It is difficult to assess how tilt-up buildings designed to current practice would behave in an earthquake, since the interactions between the various components are so complex.  One objective of this study is to investigate alternatives for failure mechanisms that could be incorporated into the seismic design of tilt-up structures to meet the intent of capacity design.  Sliding, rocking and frame mechanisms are considered for tilt-up panels. Given that this study focuses on the behaviour of the wall panels, a failure mechanism resulting from yielding of the roof deck diaphragm is not considered. Chapter 1 Introduction   36  1.4.3 Quantify Building Performance for Selected Mechanisms R-values for buildings have traditionally been selected based on results from component testing, historical performance of building systems in past earthquakes, and judgement. It is more difficult to select R-values for new building systems without historical data on performance in major earthquakes.  One methodology to assess R-values for new building systems has been recently proposed by the Applied Technology Council (ATC) in the U.S. and is described in a document titled “Quantification of Building Seismic Performance Factors - ATC-63 Project Report – 90% Draft” (ATC, 2008).  The intent of the ATC-63 Methodology is to provide a rational approach to quantify building system performance and select R-values.  In this study, aspects of the ATC-63 Methodology will be used as a guideline to assist in selecting R-values for building designs incorporating the various failure mechanisms considered. 1.4.4 Thesis Organization To address the objectives outlined above, Chapter 2 will describe the assessment methodology used to quantify building performance. Chapter 3 will describe an investigation of the various mechanism alternatives considered for tilt-up structures, including results from analyses, an evaluation of how the analysis data compares with previous research, and proposed connection details. Chapter 4 will describe the results from analyses conducted to quantify building performance.  Chapter 5 will provide conclusions and recommendations with regards to this study.   37  2 ASSESSMENT METHODOLOGY 2.1 General The assessment methodology used in this study to quantify tilt-up building performance for selected mechanisms is based on concepts from the ATC-63 Methodology (ATC 2008), which was developed to quantify building system performance and response parameters for use in seismic design.  The assessment methodology used in this study does not follow all of the steps of the ATC-63 Methodology. The stated objective of the ATC-63 Methodology is to provide a “rational basis for establishing global seismic performance factors (SPF’s), including the response modification coefficient (R Factor), of new seismic force-resisting systems proposed for construction and inclusion in model building codes and resource documents. It also provides a more rational basis for re-evaluation of the SPF’s of existing seismic force-resisting systems.” R factors are used in both Canadian and U.S. building codes to estimate demands for seismic load resisting systems that are designed using linear methods but are expected to respond beyond the linear range for the design earthquake.  In the 2005 National Building Code of Canada, R factors are separated into an overstrength term, Ro and a ductility term, Rd, and the two terms are multiplied together to determine the overall reduction to elastic demands.  R factors have a tremendous effect on the design requirements for a given building.  For example, for the design of a tilt-up building to be located in Vancouver B.C., typical values for Ro and Rd would be 1.3 and 1.5 respectively for the majority of the components of the seismic load resisting system. This means a reduction in the forces determined from linear elastic analysis by a factor of (1.3)*(1.5) = 2.0. Traditionally, R factors have been derived based on judgement, observations of the performance of existing structures in previous earthquakes, and observations from component testing. However, many recently defined seismic force resisting systems have not been exposed to significant earthquakes, and their abilities to meet the design requirements are uncertain.  In Chapter 2 Assessment Methodology   38  addition, assessment of R factors by the traditional approach likely leads to considerable variability in seismic performance between the various seismic force resisting systems. The intent of the ATC-63 Methodology is to ensure that different seismic force resisting systems have a similarly low probability of collapse for the design earthquake. The ATC-63 Methodology is based on applicable design criteria and requirements of the ASCE 7-05 “Minimum Design Loads for Buildings and Other Structures” provisions.   However for the purposes of this study, the ATC-63 Methodology is adapted to structures designed in accordance with the 2005 NBCC provisions. The ATC-63 Methodology involves the following process: • Representative structures are identified to capture the various types of applications for the seismic force resisting system in practice.  Refer to Section 2.4 for the representative structures used in this study. • The representative structures are designed in a manner consistent with how they would be designed in practice, with trial R values.  Refer to Section 2.4 for more detail on the design of the buildings investigated in this study. • Analytical models of the representative structures are then developed, incorporating all non-linear component behaviour required to simulate collapse. Refer to Section 2.5 for general aspects of the non linear modeling. • Non-linear Incremental Dynamic Analysis (IDA), (Vamvatsikos and Cornell, 2002) is carried out for each model for a suite of earthquake records. Refer to Section 2.5 for a description of the IDA procedure. • The earthquake intensity required to cause collapse is determined for each earthquake record, and collapse statistics are generated   (Refer to Section 2.6). • If the probability of collapse at the design earthquake intensity is sufficiently low, the R factor used to design the structure is deemed appropriate.  If not, the R-factor is modified, the strength of the seismic force resisting system is adjusted accordingly and the process is repeated until an appropriate R factor is determined. Chapter 2 Assessment Methodology   39  The sections below describe the ATC-63 Methodology in more detail.  2.2 Seismic Performance Factors Within the ATC-63 Methodology, Seismic Performance Factors include the response modification coefficient related to ductility (R factor in the ASCE7-05 and Rd in the 2005 NBCC) and the system over-strength factor (Ωo in the ASCE7-05 and Ro in the 2005 NBCC). Hereafter within this report these two factors will be referred to as Rd and Ro. ATC-63 can also be used to evaluate the displacement amplification factor, Cd.  However the overall conclusion from the report is that, Cd = R. Figure 2.1 and Figure 2.2 below illustrates conceptually how SPF’s are considered within the ATC-63 Methodology, and how they are incorporated in both Canadian and U.S. practice.                Figure 2.1  Seismic Performance Factors - Canadian Practice  Sp ec tr a l A cc el er a tio n  (g)  Spectral Displacement Cs Smax SMT Ro SDMT Design Earthquake RD Legend: RD      =  Force reduction factor based on ductility Ro      = Force reduction factor based on over-strength SDMT  =  Drift due to design earthquake SDCT  =  Drift due to collapse level earthquake ŜCT     =  Spectral acc. at collapse level earthquake SMT    = Spectral acceleration at design level earthquake CS       =  Spectral acceleration used for design (=SMT/RdRo) CMR  = Collapse Margin Ratio  ( = ŜCT /SMT) CMR ŜCT Collapse Level Earthquake SDCT SDMT RdRO Chapter 2 Assessment Methodology   40   Figure 2.2  Seismic Performance Factors - US Practice Within Figure 2.1 and Figure 2.2 above, CMR refers to the collapse margin ratio, which is a ratio of the median spectral acceleration observed at collapse of the structure from the analysis divided by the design spectral acceleration.  A minimum CMR value is required in order to ensure a sufficiently low probability of collapse at the design earthquake. This will be discussed further in Section 2.6 on Collapse Fragility. 2.3 Seismic Hazard 2.3.1 Ground Motion Record Sets The earthquake records considered in the ATC-63 Methodology consist of twenty-two ground motion record pairs from sites located more 10km from fault rupture.  Within the ATC-63 document, they are referred to as “Far-Field” records.  All of the records used in the methodology are from large magnitude events in the PEER NGA database, varying from a  Chapter 2 Assessment Methodology   41  Magnitude 6.5 to 7.6 on the Richter scale.  The ground motion records selected represent various site conditions and source mechanisms.  A maximum of two record pairs are used from each earthquake, so as to avoid event bias.  A sufficient number of records are considered to allow statistical evaluation of record-to-record variability and collapse fragility.  Table below provides a summary of the ground motion records used within this study. Table 2.1  Summary of Ground Motion Records (ATC-63, 2008) M Year Name Name Owner Component 1 Component 2 PGAmax (g) PGVmax (cm/s) 1 6.7 1994 Northridge Beverly Hills - Mulhol USC NORTHR/MUL009 NORTHR/MUL279 0.52 63 2 6.7 1994 Northridge Canyon County - WLC USC NORTHR/LOS000 NORTHR/LOS270 0.48 45 3 7.1 1999 Duzce, Turkey Bolu ERD DUZCE/BOL000 DUZCE/BOL090 0.82 62 4 7.1 1999 Hector Mine Hector SCSN HECTOR/HEC000 HECTOR/HEC090 0.34 42 5 6.5 1979 Imperial Valley Delta UNAMUCSD IMPVALL/H-DLT262 IMPVALL/H-DLT362 0.35 33 6 6.5 1979 Imperial Valley El Centro Array #11 USGS IMPVALL/H-E11140 IMPVALL/H-E11230 0.38 42 7 6.9 1995 Kobe, Japan Nishi-Akashi CUE KOBE/NIS000 KOBE/NIS090 0.51 37 8 6.9 1995 Kobe, Japan Shin-Osaka CUE KOBE/SHI000 KOBE/SHI090 0.24 38 9 7.5 1999 Kocaeli, Turkey Duzce ERD KOCAELI/DZC180 KOCAELI/DZC270 0.36 59 10 7.5 1999 Kocaeli, Turkey Arcelik KOERI KOCAELI/ARC000 KOCAELI/ARC090 0.22 40 11 7.3 1992 Landers Yermo Fire Station CDMG LANDERS/YER270 LANDERS/YER360 0.24 52 12 7.3 1992 Landers Coolwater SCE LANDERS/CLW-LN LANDERS/CLW-TR 0.42 42 13 6.9 1989 Loma Prieta Capitola CDMG LOMAP/CAP000 LOMAP/CAP090 0.53 35 14 6.9 1989 Loma Prieta Gilroy Array #3 CDMG LOMAP/G03000 LOMAP/G03090 0.56 45 15 7.4 1990 Manjil, Iran Abbar BHRC MANJIL/ABBAR-L MANJIL/ABBAR-T 0.51 54 16 6.5 1987 Superstition Hills El Centro Imp. Co. CDMG SUPERST/B-ICC000 SUPERST/B-ICC090 0.36 46 17 6.5 1987 Superstition Hills Poe Road (temp) USGS SUPERST/B-POE270 SUPERST/B-POE360 0.45 36 18 7.0 1992 Cape Mendocino Rio Dell Overpass CDMG CAPEMEND/RIO270 CAPEMEND/RIO360 0.55 44 19 7.6 1999 Chi-Chi, Taiwan CHY101 CWB CHICHI/CHY101-E CHICHI/CHY101-N 0.44 115 20 7.6 1999 Chi-Chi, Taiwan TCU045 CWB CHICHI/TCU045-E CHICHI/TCU045-N 0.51 39 21 6.6 1971 San Fernando LA - Hollywood Stor CDMG SFERN/PEL090 SFERN/PEL180 0.21 19 22 6.5 1976 Fiuli, Italy Tolmezzo - FRIULI/A-TMZ000 FRIULI/A-TMZ270 0.35 31 ID No. Earthquake Recording Station NGA Record Information (File Names - Horizontal Records) Recorded Motions       Chapter 2 Assessment Methodology   42   2.3.2 Ground Motion Record Scaling To carry out incremental dynamic analysis, further described in Section 2.5.4, earthquake records are scaled up until the collapse capacity of the structure is determined. In the ATC-63 Methodology the earthquake record scaling is done in two steps.  In the first step, the records are normalized by their peak ground velocities in an effort to remove variability between records due to inherent differences in event magnitude, distance to source, source type and site conditions, without removing the record-to-record variability required for IDA.  In the second step, the records are collectively scaled so as to determine the collapse capacity for the structure and IDA plots are generated. The scaling method recommended by ATC-63, specifically the first step in the scaling process, is limiting in terms of statistical analysis of the data.  All of the IDA graphs are plotted in terms of spectral acceleration at the first mode of the structure vs. drift.  When records are normalized in terms of their peak ground velocities, the “normalized” records result in very different spectral accelerations at a given period.  This means that when IDA results are plotted for a given scaling factor, data points from all of the records will have very different spectral acceleration and drift values.  As such it is impossible to determine a median IDA curve from the data if the records are scaled in this manner.  For the purposes of the ATC-63 Methodology, a median IDA curve is not required since failure of the structure is typically modelled explicitly, and the methodology only requires the spectral acceleration at median collapse to be determined.  Within this study however, it is necessary to obtain the median IDA curve, since collapse of the structure is not explicitly modelled but must be determined by comparing results from analyses with capacity limits. For this reason, the scaling method recommended by ATC-63 was not used. In order to enable extraction of a median IDA curve from the data, it was elected to normalize the records by the spectral acceleration at the first period of the structure for the purposes of this study.  This means that for each of the 22 record pairs applied to the model, each of the records applied in the direction of the first mode of the structure would result in the same spectral acceleration. This method of scaling provides some consistency in the plotted data and allows extraction of a meaningful median IDA curve.  During the course of the study it was determined Chapter 2 Assessment Methodology   43  that regardless of how the records are normalized, the IDA curves generated are essentially the same since the entire record set is collectively scaled from a relatively small intensity earthquake through to earthquake intensity sufficient to cause collapse.  In other words, normalizing of the records has no effect on variability between records.  Figure 2.3 below illustrates the results from the two methods of scaling. Sa(T1) vs. Roof Drift (Records Normalized by Sa[T1]) 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 0 0.01 0.02 0.03 Drift Sa (T1 )  ( g) MUL009 LOS000 BOL000 HEC000 H-DLT262 H-E11140 NIS000 SHI000 Sa(T1) vs. Roof Drift (Records Normalized by Peak Ground Velocity) 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 0 0.01 0.02 0.03 Drift Sa (T1 )  ( g) MUL009 LOS000 BOL000 HEC000 H-DLT262 H-E11140 NIS000 SHI000  Figure 2.3  IDA Results for Different Scaling Procedures The figure above presents IDA results from two different analysis runs.  The two models analyzed have some differences, hence the slightly different IDA curves for individual records. However, the purpose of presenting the two plots above is to illustrate the differences in results due to the scaling method used.  The chart on the left presents IDA results for records normalized to the first mode spectral acceleration.  Note that all of the data points for each collective scaling factor are at the same first mode spectral acceleration, thus allowing statistical analysis at each collective scaling. The chart on the right presents IDA results for records normalized to the Peak Ground Velocity. Note that at each collective scaling factor there is no consistency either in the first mode spectral acceleration or in drift, and hence it is much more Chapter 2 Assessment Methodology   44  difficult to carry out statistical analysis at each collective scaling factor and obtain a meaningful median IDA response curve. 2.4 Archetypical Systems Seismic force resistance systems are often used in more than one type of building and can be used in a broad range of applications.  In order to evaluate the collapse performance of a seismic force resisting system, the various applications of that system must be characterized into a set of typical structures that can then be studied.   The ATC-63 Methodology refers to these typical structures as Archetypical systems.  For the purposes of this study, a survey was conducted amongst tilt-up designers in both the U.S. and Canada to assess the most common characteristics of single story tilt-up structures with steel deck diaphragms.  The results of the survey are summarized in the table below.  Chapter 2 Assessment Methodology   45  Table 2.2  Industry Survey Results – Typical Single Story Tilt-up Building Attributes  Typical Roof Diaphragm Attributes: 1 Percentage of steel, wood, and hybrid deck systems:      Steel: 75%      Wood: 0%      Hybrid: 25%  (US)      Steel: 99%      Wood: 0%      Hybrid: 0%  (Canada) 2 Percentage of connections used for deck:      Puddle Welds: 80%      Mech fasteners: 20% (US)      Puddle Welds: 25%      Mech fasteners: 75% (Canada) 3 Deck profile: 1.5" wide rib type B deck painted most common 4 Deck thicknesses used: 16ga, 18ga, 20ga, 22ga 5 Range of deck thicknesses: 2 6 Percentage of out-of-plane wall mass apportioned to the roof diaphragm. 50% Typical Building Attributes: 1 Height of Tilt-up Walls (pin to pin) 30 ft 2 Ratio of Height to Thickness of Tilt-up walls 50 3 Structural thickness (excludes reveal) 7.25 in 4 Weight of Panels 60000 lb 5 Plan dimensions of buildings 100x200 ft x ft 6 Span between LFRS supporting diaphragm Maximum of 2 x width in high seismic zones 7 Joist Span 50 ft 8 Joist Spacing 6.25 ft 9 Joist depth 42 in 10 Girder Span 37.5 ft 11 Girder depth 30 in 12 Roof Dead Weight including joists, diaphragm, insulation, ballast (psf) 20 lb/ft 2 Embedded Connectors 1 Number of Connectors between panels As required 2 Number of Panel to Floor Slab Connections (per panel) 4 feet o.c. 3 Number of Panel to Footing Connections (per panel) 0 4 Number of Floor Slab to Footing Connections (per panel) As required 5 Spacing of Diaphragm chord angle to wall panel connections Joist spacing or 4 ft  Chapter 2 Assessment Methodology   46   The survey results reported above indicate only the most common value for each building attribute.  The actual survey results contain a range of the most likely characteristics for each attribute.   Based on the survey results above, two archetypical building systems were established in order to incorporate the most common building characteristics that would have the greatest impact on the seismic force resisting system.  It is important to note that not all building sizes have been considered in this study, as would be required to fully conform to the ATC-63 Methodology. Also, in the ATC-63 Methodology, the R factor is evaluated based on the combined results of all the archetypes, while within this study, R factors are evaluated for each archetype individually.  This is done because the two archetypes considered have very different mechanisms. One building archetype incorporates solid wall panels, while the other incorporates panels with openings.  Each of the archetypical buildings were designed using conventional procedures based on the 2005 NBCC for a location in Vancouver, BC with foundation Site Class D.  The following characteristics are common for both building archetypes: • The building plan dimensions are 30.48m (100ft) x 60.96m (200ft). •  The roof deck system is 38mm (1.5”) deep, Type B steel decking.  18ga and 20ga decking is used. • Deck fasteners consist of No. 12 screws at deck sheet side laps and end laps and Hilti pins to connect deck sheets to underlying steel members. Hilti Profis software (Hilti 2008) was used to design the deck fasteners in accordance with Steel Deck Institute requirements and Canadian safety factors. • The roof deck perimeter angle was designed in accordance with SDI requirements.  The perimeter angle is welded to embedded connectors in wall panels at 4ft centre to centre spacing. • Joists are 1050mm (3’-6”) deep with L/L chord members.  A catalogue produced by Omega was used to design joists and determine chord properties. Chapter 2 Assessment Methodology   47  • Steel beams and columns are used for the gravity load system.  Columns are also used adjacent to wall panels where beams frame in, such that beams do not transfer gravity loads to the wall panels. • Wall panels are 7.62m (25ft) and 9.144m (30ft) high.  The weight of each panel is limited to 27,300kg (60,000lb).  This is the typical limitation used in the Vancouver area to limit the size of crane required, and is in the middle of the range of panel weights indicated in the survey of companies in the US. The figure below provides a sketch of the typical roof design for all of the building archetypes.  Figure 2.4  Typical Roof Design for All Building Archetypes Chapter 2 Assessment Methodology   48  The design of the steel deck diaphragm was separated into zones based on required resistance. Zones 1 and 2 were used for the design of the roof and are called up in the above sketch.  The diaphragm zones are illustrated in Figure 2.5 below.  Figure 2.5  Roof Diaphragm Zones 2.4.1 Archetypical System 1: Solid Wall Panels Archetypical System 1 is based on a building with solid tilt-up wall panels that are 184mm (7.25”) thick.  This is a popular thickness since it is the actual width of a 2”x8” piece of lumber, allowing for easy forming of the panels, and also because it provides for a reasonable height to width ratio of approximately 50 for a 9.144m (30ft) high panel.  The building layout for the archetype is much simpler than a “real” building layout, and does not include features such as re- entrant corners, loading bays, office floors, etc.  This was done in order to investigate the overall seismic response of the building system.  The design notes for this archetype building are included in Appendix A.  Figure 2.6 below illustrates details for concrete outline, reinforcement and connections. Chapter 2 Assessment Methodology   49   Figure 2.6  Solid Wall Panels – Concrete Outline, Reinforcement, and Connections The sections shown in the above figure are referenced from Figure 2.4.  It can be observed in the above figure that panel to panel connections are only provided on the short axis of the building, not the long axis.  This is because panels on the long axis of the building are subjected to less base shear than panels on the short axis, due to less tributary roof mass and out-of-plane wall Chapter 2 Assessment Methodology   50  mass acting per unit length.  Calculations to determine the number of connectors are included in the design notes for this archetype building (Appendix B). 2.4.2 Archetypical System 2: Wall Panels with Openings Archetypical System 2 is based on a building with tilt-up wall panels with openings 5.62m (18’- 5”) wide x 3.625m (11’-11”) high.  The panels are 240mm (9.5”) thick as required to accommodate the out-of-plane loading.  The dimensions of the panels openings and legs were selected such that the aspect ratio of the panel legs was the same as that used for recent experimental testing of panel legs (Dew et. al., 2001).  This was done to ensure that the experimental results could be used to calibrate the numerical model. The openings in the panels are incorporated since they are often required for tilt-up buildings in practice to allow for loading docks, or for the front side of commercial or retail buildings.  The design notes for this archetype building are included in Appendix B. Figure 2.7 below illustrates details for concrete outline, reinforcement and connections (the sections shown are referenced from Figure 2.4.) Chapter 2 Assessment Methodology   51   2.5 Non-linear Analysis Methods 2.5.1 Software Perform 3D, Version 4.0.3 (Computers and Structures, 2007) was used to carry out the analyses. This is commercially available, Windows-based structural analysis software that is specifically  Figure 2.7  Wall Panels with Openings – Concrete Outline, Reinforcement, and Connections Chapter 2 Assessment Methodology   52  designed for non-linear time history analyses and performance assessment for 3D structures. The software contains an element library sufficiently comprehensive to adequately model most conventional structures and components and provides a graphical interface for pre and post processing.  In addition, it allows the user to set up multiple earthquake analysis runs in succession and provides the option to apply three different earthquake records on three different axes simultaneously.  The software uses an event-to-event solution strategy to carry out analyses. The software does have some limitations with regards to modelling interactions between element responses in two directions or between two force resultants.  It is able to model moment/compression interactions and shear/shear interactions commonly found in practice. However, it does not allow a generalized interaction relationship to be entered.  This limits the user in that results from testing of connections or other components cannot be directly incorporated if the interaction of responses in different directions does not match the conventional interaction relationships used in Perform.  This limitation is discussed further in Section 3.1.1 within the context of this study. 2.5.2 Simulated and Non-Simulated Deterioration / Collapse Mechanisms Within the ATC-63 Methodology, “simulated collapse mechanisms” are those responses that are incorporated within the time history analyses and lead to the occurrence of a collapse mechanism forming within the analysis. “Non-simulated collapse mechanisms” are those responses that are not incorporated within the analyses and must be tracked separately or indirectly by interpreting the results of the analysis. Within this study, simulated collapse mechanisms include failure of legs for frame panels with openings, and failure of panel to panel connectors.  Other behaviour modes, such as sliding and rocking of the wall panels, are directly modeled and directly affect the response of the overall model, though they do not cause instability in the structural system and thus cannot be considered a collapse mechanism.  Non-simulated collapse mechanisms include the failure of the roof diaphragm, failure of the connections between the roof diaphragm and the concrete wall panels, and collapse of the steel columns used to support gravity loads.  To assess whether structural collapse can be considered to have occurred based on the response of these Chapter 2 Assessment Methodology   53  components, the analysis results are reviewed to determine at what point in the analyses (if any) the failure threshold for these components was surpassed. At the outset of this study, it was established that the non-linear response of the roof deck diaphragm and associated connections would not be incorporated in the analysis.  This is because the focus of this study is on the non-linear response of the wall panels.  In the context of the seismic force resisting system, the wall panels and the roof diaphragm act as two systems in series.  This means that if either one of the elements yield, it will likely reduce the demands on the other element in series (i.e. if the roof diaphragm yields, the walls likely will not yield, thus making it difficult to study the behaviour of the walls).  This behaviour is intuitively expected for a series system and was also observed in a previous study (Adebar et al., 2004).  It was decided that within this study, the roof diaphragm would be modeled as a linear system, and the response would be tracked and the demands reported. However, the response would not be considered to initiate collapse of the structure, even if the failure threshold of the roof (designed in accordance with conventional practice) was exceeded. 2.5.3 Non-linear Model Calibration The models used to carry out the analysis are verified in several ways.  Where component response is based on test results, models are constructed of the individual components and a non- linear static analysis is carried out for the component model to ensure that the response hysteresis from the model adequately matches the test results. For the overall model a static gravity analysis is initially carried out and the weight of the building as determined by hand calculation is compared to the dead load results from the analysis.  A modal analysis is carried out and the fundamental period determined from analysis is compared to the fundamental period determined using the ASCE 41-06 (ASCE, 2006) equation for a one-story building with a single span flexible diaphragm.  A non-linear static pushover analysis is carried out and the response is compared with the response determined based on hand calculation.  Section 3.2 describes the model verification in more detail. Chapter 2 Assessment Methodology   54  2.5.4 Incremental Dynamic Analysis The collapse capacity of the structure is assessed using Incremental Dynamic Analysis (IDA), (Vamvatsikos and Cornell, 2002).  In this method of analysis, multiple time history analyses are carried out for a given earthquake record or a set of earthquake records.  In each analysis, the intensity of the applied ground motion is increased until collapse is detected in the structure.  In the ATC-63 Methodology, results from IDA are described in terms of the first mode spectral acceleration of the structure for a given earthquake record at a given scaling and the corresponding roof drift of the structure determined from analysis.  Figure 2.8 below provides an illustration of the concept of IDA for one earthquake ground motion.  Figure 2.8  IDA Results for One Earthquake Record The points on the plot at which simulated and non-simulated collapse occur are labelled.  For the above plot, the collapse capacity is reached when non-simulated collapse occurs (illustrated with a star), which is common for the tilt-up buildings investigated in this study. The figure below provides an illustration of the concept of IDA for five earthquake ground motions. Sacollapse  Roof Drift Sa (T 1) N o n - Si m u la te d Co lla ps e Si m u la te d Co lla ps e Sp ec tr a l A cc el er a tio n  Chapter 2 Assessment Methodology   55   Figure 2.9  IDA Results for Five Earthquake Records Within Figure 2.9 above, results are plotted from IDA for five different earthquake records. The spectral acceleration at the collapse level intensity is plotted with a star for each earthquake record. The most significant point in the figure above is the median collapse capacity of the structure, labelled as ŜCT.  This is used essentially to anchor the fragility curve for the structure and to obtain the Collapse Margin Ratio (CMR). The ATC-63 Methodology requires that the earthquake record pairs be applied to the model simultaneously in two orthogonal directions, and once in one direction and then rotated 90 degrees. This essentially requires 44 IDA runs to be carried out.  For this study, it was decided that earthquake record pairs would be applied simultaneously to the model in orthogonal directions, but that the directions would not be rotated; i.e. only 22 IDA runs would be carried out. ŜCT SMT  Roof Drift Sa (T 1) Sp ec tr a l A cc el er a tio n  Design Earthquake (AEF = 1/2475) Chapter 2 Assessment Methodology   56  2.6 Collapse Fragility and Uncertainties Within the ATC-63 Methodology, the capacities obtained from the IDA for the twenty two earthquake records are fitted to a log-normal cumulative distribution.  This plot is referred to as a “fragility curve”.  An illustration is shown in the figure below.  Figure 2.10  Fragility Curve Based on IDA Results for 22 Earthquake Records The points on the plot represent the collapse capacity determined for each earthquake record. The solid line in the figure above is based on the collapse data for twenty two  earthquake records fitted to a lognormal cumulative distribution, which is characterized by two parameters: the median collapse capacity, ŜCT and the standard deviation of the natural logarithm, β. By definition, the median collapse capacity corresponds to a 50% probability of collapse, and therefore governs the location of the lognormal cumulative distribution plot. The variability term controls the slope of the curve; the greater the variability, the more shallow the slope (i.e. higher probability of collapse at lower spectral acceleration values). Chapter 2 Assessment Methodology   57  Within the ATC-63 Methodology, the standard deviation of the natural logarithm, β, is identified by a subscript that refers to the source of the variability.  The variability in the solid curve is a result of variability in response between different earthquakes, and is termed “Record to Record” variability, and is expressed as βRTR. The dotted line shown in Figure 2.10 incorporates variability from modeling uncertainties (βMDL), uncertainties related to the degree to which the design and characterisation of the structure arch-type represent actual structures in the field, (referred to as uncertainties related to design requirements, (βDR), and uncertainties related to test data (βTD).   The ATC-63 Methodology assumes a fixed value for the record-to-record uncertainty, βRTR=0.4. However, within this study, the βRTR is determined directly from the results of the analysis.  For the remaining sources of variability, the uncertainty is quantified by assigning values based on a quality rating as follows:   β=0.2 for “superior”, β=0.3 for “good”, β=0.45 for “fair”, and β=0.65 for “poor”. The various sources of uncertainty are combined as follows: 2222 TDDRMDLRTRTOT βββββ +++=  (2.1) The dotted line represents the collapse fragility curve for the structure incorporating all sources of variability.  From the collapse fragility curve, the probability of structural collapse at the design earthquake level can be determined. 2.7 Median Collapse Adjustment for Spectral Shape The spectral shape of rare ground motions is peaked near the period of interest, causing these ground motions to be less damaging than other records when scaled to the same intensity (Baker and Cornell, 2006).  The ground motion records used in the ATC-63 Methodology are from earthquakes of magnitude 6.5 to 7.6 on the Richter scale and do not incorporate this spectral shape, and are thus likely to result in under-prediction of the collapse capacity of a structure (i.e. conservative prediction of the probability of collapse).  To correct for this effect, the ATC-63 Methodology provides an adjustment factor, termed a “spectral shape factor” for the median collapse spectral acceleration determined from IDA.  The spectral shape factor depends on the seismicity of the region of interest, and the anticipated degree of softening of the initial building period during response.  Within the 70% draft of ATC-63, the building deformation capacity was Chapter 2 Assessment Methodology   58  assumed to be related to the R factor for the structural system, which is incorporated in the method to determine the spectral shape factor.  For high seismicity regions (areas of interest within this study), the spectral shape factor from the 70% draft of ATC-63 was as follows: Table 2.3  Spectral Shape Factor for Different R Factors  R Factor 2 2.5 3 4 Spectral Shape Factor 1.00 1.05 1.1 1.2  As can be observed from the above table, spectral shape has little influence on structural collapse capacity for the range of R-factors likely to be relevant within this study (i.e. 1 ≤ R ≤ 3). The manner in which the Spectral Shape Factors are determined has changed for the 90% Draft of the ATC-63 Methodology; ductility is used instead of the R factor.  However this cannot be used for the rocking or sliding system, since there is no bound on ductility.  The spectral shape factors from the 70% Draft have been adopted for this study. 2.8 Evaluation and Acceptance Criteria The collapse capacity of a structure is evaluated in accordance with the methodology outlined above.  Once a collapse fragility curve is developed for a structure, the probability of structural collapse at the design earthquake level can be determined.  If this probability is above the desired value, the R-factor used to design the structure arch-type is reduced, the structure is re-designed, and the analysis process is repeated.  Once the probability of collapse is above the desired value, the R-factor used can be assumed to be adequate. For the purposes of this study, a maximum probability of collapse of 0.10 is adopted for the design earthquake with an annual excedence frequency of 1 in 2475.  It is important to note that this criteria is more stringent that recommended by the ATC-63 Methodology.  Within the ATC- 63 Methodology, the acceptable maximum probability of collapse is 0.2 within a performance Chapter 2 Assessment Methodology   59  group and 0.1 on average for the performance group.  Since there are no performance groups within this study, the more stringent requirement has been adopted.  60  3 INVESTIGATION OF MECHANISM ALTERNATIVES 3.1 Analysis Model Configuration Based on testing of standard EM5 wall-to-slab connectors (Devine, 2008), it is likely that strong ground shaking would result in failure of the wall-to-slab connections with limited energy dissipation at the connection.  Failure of the connection may subsequently lead to either sliding or rocking of the wall panels on the foundation.  This section describes efforts to explore the nonlinear response of tilt-up buildings after failure of the slab-to-wall connection, and to investigate the preferred mode of response for the design of new tilt-up buildings. Three types of energy dissipating mechanisms were investigated for tilt-up structures:  sliding of wall panels, rocking of wall panels, and frame mechanisms.  In addition, an attempt was made to investigate the response of a building designed according to current practice.  In order to investigate alternatives for an energy dissipating mechanism for tilt-up buildings, the two archetypical buildings discussed in Section 2.4 were considered.  The design of the wall panels and connections was modified to ensure the building response displayed the desired mechanism. 3.1.1 Conventional Building To investigate the response of a building designed in accordance with current practice, Archetypical System 1 was considered.  Available information on existing connections was gathered (refer to Section 1.2.3) and the system was initially studied by conducting pushover analyses on two adjacent wall panels connected to each other and to the floor slab of the building.   Figure 3.1 below illustrates the two-panel model. Chapter 3 Investigation of Mechanism Alternatives   61     Figure 3.1  Two-Panel Model for Pushover Analyses Combined backbone curves were constructed for EM3 to EM5 connections and EM5 to EM5 connection based on available test data for individual connectors (see Section 1.2.3). While gathering available information on connector test data, it was determined that there had not been any tests conducted on the EM3 to EM5 connections at the base of the panels for uplift due to rocking.  It is expected that there would be an interaction for both strength and stiffness of this connection between uplift and shear. Due to the lack of this information, a meaningful analysis of a building designed according to current practice was not possible.  An attempt was made to conduct a pushover analysis of the above model, based on an assumed behaviour for the EM3 to EM5 connector in uplift and an assumed interaction relationship between uplift and shear. Unfortunately, considerable difficulty was encountered in attempting to accurately model the bi- directional response expected for the EM3 to EM5 connection with different load-deformation Chapter 3 Investigation of Mechanism Alternatives   62  relationships in each direction and interaction between the two directions in Perform (Computers and Structures, 2006).  A discussion of the types of elements that were used and the problems encountered is included in Appendix A. After the data is available from tests on the uplift performance of the EM3 to EM5 connection (Devine 2008), future analytical research should focus on the development of an interaction model and the assessment of current tilt-up construction with slab-to-wall connections. 3.1.2 Model 1: Sliding Mechanism To investigate the response of a building designed to have a failure mechanism characterized by in-plane sliding of the wall panels, Archetypical System 1 was considered but the connection layout was modified.  It was assumed that all panels were connected to adjacent panels and with no connections at the base.  A 3D model was constructed in Perform to carry out the analyses. The model is illustrated in Figure 3.2 below. Chapter 3 Investigation of Mechanism Alternatives   63   Figure 3.2  Model Used to Investigate Sliding System As can be observed in the figure above, the roof deck diaphragm is modeled using an equivalent truss system.  The expected stiffness of the roof deck diaphragm was determined using the Hilti Profis software (Hilti, 2006), which follows the SDI procedures, and by considering results from testing of steel deck panels in pure shear (Tremblay et. al, 2003).  The properties of the equivalent braces were determined to match the expected stiffness of the deck diaphragm.   The procedure used to determine the equivalent brace properties is described in Appendix B.  The properties of the joists were incorporated into the model.  Beam elements were used with the Chapter 3 Investigation of Mechanism Alternatives   64  moment of inertia of the full joist, and axial properties of the top chord of the joist only (the bottom chord is assumed not to participate in the deck diaphragm behaviour).  The properties of the deck in the direction perpendicular to the joist span direction were incorporated into the model as tension/compression members with axial properties from an equivalent deck area.  The perimeter angle was also incorporated into the model as a beam element. Girders and columns were incorporated as beam elements.  The steel columns were assumed to be fixed at the base since they normally are supported on a concrete pedestal below the slab level. The roof system was assumed to be elastic for the purposes of this study.  This assumption was made to ensure that non-linear behaviour occurred in the walls, which are the focus of this study, and not in the roof diaphragm. When two non-linear components (with limited strain hardening) are acting in series, as is the case in a typical tilt-up building, if one component yields, the other will remain elastic.  Nonlinear behaviour of the steel deck diaphragm is outside the scope of this study. Vertical loads from the roof were treated as concentrated loads on the joists at each joist node. As shown in the above figure, joists span between the girders and the wall panels.  Girders span between columns.  Columns were provided adjacent to end wall panels to ensure that vertical loads were not transferred from the girders to the end wall panels.  This is consistent with conventional design practice. The wall panels were modeled as fibre elements with elastic-perfectly-plastic (EPP) material properties for the concrete and reinforcing steel fibres, with an Elastic Modulus of 30000MPa for concrete and 200000MPa for steel.  At the base of the wall panels, contact elements were provided (Friction Pendulum elements used in Perform) to model the friction contact between the wall panels and the footings.  The properties for these elements include a large stiffness in compression, a very small stiffness in tension, and a large stiffness in shear.  The shear resistance is proportional to the applied vertical compression load.  The constant of proportionality is the friction factor, which was selected to ensure the lateral force required to initiate wall sliding was the same as the lateral force required to initiate wall rocking in the rocking model (see Section 3.1.3).  This was done to allow direct comparison between the results of the two mechanisms.  A friction factor of 0.42 was used for comparison purposes. Chapter 3 Investigation of Mechanism Alternatives   65  The wall panels were fixed in out-of-plane translation along the base.  This assumption was considered valid since in practice, wall panels are erected using “locating pins” (refer to Figure 1.7) consisting of short stubs of large diameter reinforcing steel extending up from the footing used as a guide during installation of the panels.  Adjacent panels were connected using short beam elements to model the EM5-EM5 connections, incorporating the backbone curve (refer to Appendix B) developed for those connections. A Rayleigh damping value of 3% was used for the model at vibration periods of 0.2 and 1.5 times the first mode period.  The values for the periods were adopted based on a recommendation by the ATC-63 Methodology.  The damping value was adopted based on a recommendation in Chopra (2000). 3.1.3 Model 2: Rocking Mechanism To investigate the response of a building designed to have a failure mechanism characterized by in-plane rocking of the wall panels, Archetypical System 1 was considered but the connection layout was modified.  It was assumed that all panels were not connected to adjacent panels and thus able to rock independently.  The panels were also assumed to be connected at the base with connections that would stop sliding from occurring but would allow rocking of the panels to occur (refer to Section 3.6). A 3D model was constructed in Perform to carry out the analyses. The model used was very similar to the one used to study the sliding mechanism. The main differences are that the corner nodes of the panels are fixed in horizontal translation (in-plane and out-of-plane displacement) and there are no EM5-EM5 connections between panels for the rocking model.  This allows the individual panels to rock on their corner nodes. 3.1.4 Model 3: Frame Mechanism To investigate the response of a building designed to have a failure mechanism characterized by in-plane bending of the legs in wall panels with openings, Archetypical System 2 was considered. A 3D model was constructed in Perform to carry out the analyses.  As shown in the layout of Archetype 2 (Figure 2.7), panels with openings are incorporated along the short axis of the building.  To model the panels with openings, fibre elements are used with EPP material properties for concrete and steel fibres, as well as for shear behaviour.  For walls in the long axis Chapter 3 Investigation of Mechanism Alternatives   66  of the building, rocking panels were assumed.  The rocking panels were modeled as described previously.  Figure 3.3 below illustrates the analytical model used.  Figure 3.3  Model Used to Investigate Frame Mechanism The frame mechanism was considered separately from the sliding or rocking mechanism, since it only applies to buildings with large openings in wall panels.  As such, the yield strengths of the concrete and reinforcing steel in the panel legs were not modified to result in a base shear equal to the sliding and rocking models.  The material properties were based on realistic values. One important limitation in the modeling of the frame mechanism was the lack of available information on the behaviour of the panel to slab connections (refer to EM5 connection shown in Chapter 3 Investigation of Mechanism Alternatives   67  Figure 1.6) at the base.  Although some information is available on the behaviour of the connections in the horizontal directions, both in-plane and out-of-plane (Lemieux et. al. 1998), there is no information available on their behaviour in uplift.  An experimental program is currently underway at UBC (Devine 2008) to investigate the behaviour of EM5 panel to slab connections in uplift and to investigate interaction relationships between shear and uplift.  For the modeling carried out in this study, a pinned base was assumed at each of the panel legs. 3.2 Model Verification The analytical models used to carry out the investigations were verified in several ways.  Where possible, results from simple analyses were compared with hand calculations.  This was done for gravity loads, as well as for pushover analysis.  The fundamental mode determined from modal analysis was compared with a simple estimate using the ASCE 41-06 equation (ASCE, 2007) in Section 3.2.1.  In addition, the displaced shape of the models was observed for various load cases in an effort to detect erroneous behaviour. 3.2.1 Model 1: Sliding Mechanism Several aspects of the model were checked.  The calculated weight of the building (9243 kN) was compared to the dead load reaction from the analysis (9210 kN).  Results from a modal analysis indicated the first mode period of the model was 0.58 seconds, with 56% of the total building mass participating in this mode. The first mode occurs in the direction of the short axis of the building.  The next important mode for the model was the third mode, with a period of 0.33 seconds, with 38% of the mass participating.  The third mode occurs in the direction of the longer axis of the building.  Figure 3.4 below illustrates the first mode shape as determined using Perform. Chapter 3 Investigation of Mechanism Alternatives   68   Figure 3.4  First Mode (Period = 0.58 seconds) The first mode period of the structure was also estimated using the equation recommended by ASCE 41-06 for buildings with flexible diaphragms:  5.0)78.01.0(1 dwT ∆+∆=  (3.1) Where ∆w and ∆d are in-plane wall and diaphragm displacements in inches, due to a distributed lateral load in the direction under consideration, equal to the weight of the diaphragm.  Using the above equation, the first mode period was calculated to be 0.41 seconds, which is considerably shorter than the first mode period determined from the modal analysis.  However, when the ASCE 41-06 estimate was modified by considering half the weight of the out-of-plane walls, the resulting period was 0.63 seconds, which is approximately 9% longer than the period determined from modal analysis.  The inclusion of half the weight of the out-of-plane walls was thought to be reasonable, since it is included in the seismic demands in standard practice.  Also, from observing the shape of the first mode (Figure 3.4) it is obvious that some of the inertial forces from the mass of the out-of-plane walls must be transferred into the diaphragm, and hence, will participate in the mode. Chapter 3 Investigation of Mechanism Alternatives   69  It is interesting to note both the first mode period determined from analysis (0.58 seconds) and using the ASCE 41-06 formula (0.63 seconds) are considerably longer than the first mode period determined using the NBCC 2005 formula (0.26 seconds) or the IBC 2003 formula (0.11 seconds). Pushover analyses were conducted for the sliding model in the direction of each primary axis of the building.  The shape of the load distribution used for the pushover analyses was based on the fundamental mode in the direction of each primary axis.  For each pushover analysis, the total base shear resisted by the in-plane wall panels was plotted against the drift at the middle of the roof.  The base shear resisted by the in-plane walls is expressed as a percentage of the assumed tributary building weight for the direction of interest including the weight of the roof, half the weight of the out-of-plane walls, and the full weight of the in-plane walls – identical to the tributary weight assumed in practice for calculation of the design base shear.  This was done to allow comparison of the pushover plot with the design base shear.  To check the results from Perform, the ultimate base shear for the building was estimated using hand calculations.  Figure 3.5 below illustrates the results of the pushover analysis in the direction of the short axis of the building.  The roof drift is expressed in terms of the displacement at the middle of the roof divided by the height of the building.  Figure 3.5  Sliding Mechanism - Pushover Analysis Along Short Axis of Building Chapter 3 Investigation of Mechanism Alternatives   70  As can be observed, the results from Perform match closely with the ultimate base shear estimated using hand calculations. Figure 3.6 below illustrates the displaced shape of the model during the pushover analysis in direction of the short axis of the building.  Figure 3.6  Sliding Mechanism – Displaced Shape for Pushover Along Short Axis of Building As can be observed in Figure 3.6 above, the response of the model in the direction of the long axis and short axis of the building has been decoupled.  There are no connections provided between adjacent corner panels.  This is necessary to ensure a sliding mechanism can develop. Figure 3.7 below illustrates the results of the pushover analysis in the direction of the long axis of the building. Chapter 3 Investigation of Mechanism Alternatives   71   Figure 3.7  Sliding Mechanism - Pushover Analysis Along Long Axis of Building As can be observed from Figure 3.7 above, the results for pushover analysis in the long axis of the building are similar to results for analysis in the short axis of the building.   The ultimate base shear matches closely with the estimate using hand calculations.  In addition to the above, the response of the model for time history analysis was verified to ensure that the appropriate types of energy dissipation were being exhibited.  This will be discussed further in Section 3.3. 3.2.2 Model 2: Rocking Mechanism The model used to investigate the rocking mechanism was also verified.  The results from both elastic analysis with dead loads and modal analysis were identical to results from the sliding model.  This was expected, considering that no aspects of the model were modified that would cause a change in elastic response. Pushover analyses were conducted for the rocking model in the direction of each primary axis of the building.  The shape of the load distribution used for the pushover analyses was based on the Chapter 3 Investigation of Mechanism Alternatives   72  fundamental mode in the direction of each primary axis.  For each pushover analysis, the total base shear resisted by the in-plane wall panels was plotted against the drift at the middle of the roof.  As for the sliding model, the base shear was expressed as a percentage of the assumed tributary weight, incorporating the full weight of the roof, half the weight of the out-of-plane walls, and the full weight of the in-plane walls (as is commonly done in practice). To check the results from Perform, the ultimate base shear for the building was estimated using hand calculations.  Figure 3.8 below illustrates the results of the pushover analysis in the direction of the short axis of the building.  Figure 3.8  Rocking Mechanism - Pushover Analysis Along Short Axis of Building As can be observed from Figure 3.8 above, the results from Perform match closely with the ultimate base shear estimated using hand calculations.  The lateral load capacity of the in-plane walls at yield corresponds to a base shear of approximately 0.15g, compared to the NBCC 2005 design elastic base shear of 0.68g for building period calculated using the code prescribed formula and for Site Class D. In effect, this design is based on an assumed RdRo value of approximately 4.5 for earthquake loading in the direction of the short axis of the building, where Chapter 3 Investigation of Mechanism Alternatives   73  RdRo is defined as the design elastic base shear divided by the yield strength of the building (i.e. RdRo =Sa(T1)Wtributary/Vy). Since the calculated period of the building is relatively short (0.26 seconds), the NBCC 2005 limits the base shear to two-thirds of the peak design spectral acceleration. It is interesting to note that if the first mode period from the model is used (approximately 0.6 seconds) instead of the NBCC 2005 calculated period, the design base shear reduces to 0.64g, resulting in a slightly smaller assumed RdRo value of 4.3.  Figure 3.9 below illustrates the displaced shape of the model during the pushover analysis in direction of the short axis of the building.  Figure 3.9  Rocking Mechanism – Displaced Shape for Pushover Along Short Axis of Building Similarly to the sliding model, the response in the direction of the long axis and short axis of the building has been decoupled.  There are no connections provided between adjacent corner panels.  This is necessary to ensure a rocking mechanism can develop. Chapter 3 Investigation of Mechanism Alternatives   74  The post-yield stiffness exhibited in the pushover results illustrated in Figure 3.8 is due to bending of the roof diaphragm perimeter angle at locations between adjacent in-plane wall panels.  This behaviour is realistic considering that in conventional design practice, the roof perimeter angle is typically welded to embedded connectors 2 ft from the edge of the panel. Rotation of an individual panel would cause a vertical displacement at the edge of the panel relative to the adjacent panel.  This in turn would force the roof perimeter angle to bend. The bending demands on the perimeter angle presents one of the difficulties in implementing a rocking mechanism as will be discussed further in Section 3.6.  Figure 3.10 below illustrates the results of the pushover analysis in the direction of the long axis of the building.  Figure 3.10  Rocking Mechanism - Pushover Analysis Along Long Axis of Building The results for pushover analysis in the long axis of the building are similar to results for analysis in the short axis of the building and the same observations apply, except for the yield capacity.  The lateral load capacity of the in-plane walls at yield corresponds to a base shear of approximately 0.28g, compared to an elastic design base shear of 0.68g. In effect, this design is based on an assumed RdRo value of approximately 2.4 for earthquake loading in the direction of the long axis of the building. Chapter 3 Investigation of Mechanism Alternatives   75  3.2.3 Model 3: Frame Mechanism Various aspects of the model used to investigate the frame mechanism were verified.  The calculated weight of the building (9053 kN) was compared to the dead load reaction from the analysis (9010 kN).  Results from a modal analysis indicated the first mode period of the model was 0.61 seconds, with 55% of the total building mass participating in the mode. The first mode occurs in the direction of the short axis of the building.  The next important mode for the model was the third mode, with a period of 0.34 seconds, with 42% of the mass participating.  The third mode occurs in the direction of the longer axis of the building.  The two modal periods and associated mass participation given above are very similar to the modal periods determined for Models 1 and 2.  This is expected for the third mode of the building, since for all the models, solid panels are used on the long axis of the building.  For the first mode, this suggests that even for a building incorporating panels with large openings, the elastic response is predominantly governed by the roof deck diaphragm. For the frame model, results from static reversed-cyclic analysis of a beam-column subassembly (refer to Figure 1.14 for details of the geometry and reinforcing steel) were compared with experimental test results from Dew et al. (2001), as illustrated in Figure 3.11 below.  Chapter 3 Investigation of Mechanism Alternatives   76   Figure 3.11  Frame Mechanism – Force-Displacement Plot for Beam-Column Subassembly: Comparison of Analytical and Experimental Results (Dew et al., 2001) The analytical model matches the experimental data reasonably well for strength and stiffness. The displacement capacity displayed by the Perform model is less than that shown by the experimental results.    The discrepancy here is likely due to a difference in the ultimate strain for the steel reinforcement.  An ultimate strain of 0.05 was selected for the Perform 3D model, based on the limit recommended by ASCE 41 (ASCE, 2007) for reinforcing steel bars undergoing cyclical loading.  Similar to many other limits in ASCE 41, 0.05 is likely conservative, and hence it is not surprising that the test was able to exceed the displacement capacity implied by the model. For Archetypical System 2, a larger leg was required to meet the design requirements than was used in Dew et al. (2001).  The leg dimensions used in this current study are proportionally similar to the leg used in the experimental study, with width: thickness ratios of 4.2 for both and length: width ratios of 3.6 for both.  In Figure 3.12 below, the results from Perform 3D are compared with results from hand calculations using a tri-linear moment curvature model Chapter 3 Investigation of Mechanism Alternatives   77  (Ibrahim, A.M.M and Adebar, P., 2004) for the leg of a frame panel used in Archetypical System 2 (refer to Figure 2.7 and Figure 3.3).  Figure 3.12  Frame Mechanism – Panel Leg with Geometry and Reinforcement from Archetypical System 2: (a)Force-Displacement Plot  (b)Perform Model of Leg A reasonably good match can be observed between the two methods for initial stiffness.  The strength is slightly higher for the Perform model.  This is because Perform calculates the moments, stresses, strains and other response quantities at the centre of each element, resulting in a reduction in the cantilevered height for the leg by half the height of the element closest to the fixed end.  For a cantilevered height of 3.625m and an element height of 0.463m, the strength increase is 3.625/(3.625-(0.463)(0.5)) = 1.07, or 7%. A larger displacement capacity can be observed in the Perform results than in the results from the hand calculations.  The ultimate displacement obtained from hand calculations using the Tri- linear moment-curvature model was determined based on the following assumptions: (a) (b) Chapter 3 Investigation of Mechanism Alternatives   78  • Distance to section neutral axis determined based on compressive stress block assuming four of the five rows of bars yielding in tension. • Ultimate curvature obtained by considering the ultimate concrete strain of 0.0035 divided by the distance to the neutral axis. • Inelastic displacement determined assuming plastic hinge length equal to the section depth. • Elastic displacement determined using cracked moment of inertia. The material models used in Perform include strength degradation and can accommodate strains well beyond the ultimate strains with reduced strengths. Also, the Perform model accounts for cyclical degradation by decreasing the stiffness of the reinforcing steel with each cycle.  Both of these reasons account for the greater displacements observed in the Perform model. In the archetypical building and in common practice, legs of adjacent panels are connected together with panel to panel connectors, providing shear transfer between the legs and effectively doubling the width of the leg section, thereby changing its characteristics dramatically.  Figure 3.13 below illustrates both a moment-curvature plot and a force-displacement plot to compare the behaviour of two legs from adjacent frame panels connected together with the behaviour of one panel leg, based on results from hand calculations. Chapter 3 Investigation of Mechanism Alternatives   79   Figure 3.13  Frame Mechanism – Two Legs Connected vs. One Leg: (a)Moment-Curvature Plot; (b)Force-Drift Plot There is an increase of approximately 200% in the strength per leg, and a corresponding decrease in the displacement capacity when the panel legs are connected together.  This due to two reasons: • When two legs are connected together, the depth of the section is double that of a single leg, and as a result the rebar at the extreme tension fibre must undergo twice as much strain in order to facilitate the same curvature. • The panel-to-slab connections increase the base fixity dramatically, such that the legs are effectively fixed at both ends. This significantly changes the response of the overall structure from the original design intent, i.e. the Rd and Ro values used in design are effectively cut in half.  Figure 3.14 below shows the pushover curve for the entire model in the direction of the short axis of the building. (a) (b) Chapter 3 Investigation of Mechanism Alternatives   80   Figure 3.14  Frame Mechanism – Pushover Analysis Along Short Building Axis Due to the connectivity between adjacent panels, the resistance of the building to lateral loading increases considerably. The design base shear for the building for in-plane flexural strength of the panel legs (based on Rd=1.5, Ro=1.3) was 0.35g.  As can be seen in Figure 3.14 above, the actual base shear resisted by the building is on the order of 0.87g.  This is effectively a 250% increase in strength. The Pushover analyses were conducted for the frame model only in the direction of the short axis of the building, since the long axis has solid panels similar to the rocking model. 3.3 Time History Analysis Results To observe the distinct characteristics of each mechanism, the results from a time history analysis for one earthquake applied in the direction of the short axis of the building was considered for each model.  The record considered was from the Northridge earthquake and was recorded at the Beverly Hills - Mulhol recording station.  It constituted the first record in the set Chapter 3 Investigation of Mechanism Alternatives   81  recommended for use in the ATC-63 Methodology (see Table 2.1) and was obtained from the PEER-NGA database (http://peer.berkeley.edu/nga/).  Two scaling factors were considered in order to capture the response of the models in both the linear and nonlinear ranges.  The record was scaled in amplitude only such that the 3% damped spectral acceleration at the first mode period was equal to 0.1g (i.e. Sa(T1) = 0.1g) to investigate the linear response, and 1g to initiate nonlinear response.  The scaling factors required to achieve these intensities were 0.0625 and 0.625 respectively.  The breakdown of energy dissipation mechanisms was investigated for each model. 3.3.1 Model 1: Sliding Mechanism The response of the sliding model for the record described above can be illustrated by considering the drift at the centre of the roof (displacement at mid-span divided by the height of the roof) and the drift at the top of the end wall panels (displacement at top of end walls divided by the height of the end walls). These two drifts are shown plotted in Figure 3.15 below for Sa(T1) = 0.1g and 1.0g.  Figure 3.15  Sliding Model – Wall and Roof Drifts for Northridge Earthquake;  (a) Sa(T1)=0.1g;   (b) Sa(T1)=1.0g It can be observed in Figure 3.15 (a) above that the roof essentially governs the response, and the walls remain elastic.  The maximum roof drift is approximately 0.14% (13mm). There is no (a) (b) Chapter 3 Investigation of Mechanism Alternatives   82  residual displacement.  In Figure 3.15 (b) above, the walls exhibit some nonlinear behaviour. Approximately 7 seconds into the earthquake, the end walls begin to slide on the foundation. The maximum roof drift is approximately 0.8% (73mm). A residual drift of approximately 0.2% (18mm) is observed.  The roof diaphragm oscillates about the end walls.  A breakdown of the dissipated energy is illustrated in Figure 3.16 below.  Figure 3.16  Sliding Model – Breakdown of Energy Dissipation As expected the majority of the energy dissipation consists of inelastic energy dissipated by the sliding action of the walls (70%). The remaining energy dissipated is due mainly to damping through movement of the building mass and deformation of the roof diaphragm (Alpha-M and Beta-K energy respectively). 3.3.2 Model 2: Rocking Mechanism The response of the rocking model for the Northridge MUL009 record scaled to Sa(T1) = 0.1g is essentially the same as for the sliding model, with the roof governing the response and the walls remaining within the linear range of response.  As expected for linear response, the maximum roof drift matches the observed maximum roof drift for the sliding model. Chapter 3 Investigation of Mechanism Alternatives   83  The drifts at roof centre and top of the end wall panels are shown plotted in Figure 3.17 below for Sa(T1) = 1.0g.  Figure 3.17  Rocking Model – Wall and Roof Drifts for Northridge Earthquake, Sa(T1)=1.0g It can be observed in the figure above that the walls exhibit considerable nonlinear behaviour. Similar to the sliding model, nonlinear response commences approximately 7 seconds into the earthquake as the end walls begin to rock on the foundation.  Once the earthquake intensity increases at about 8 seconds, the walls and roof maintain essentially the same displacement for the majority of the analysis run.  The maximum roof drift is approximately 3.2% (290mm). There is no residual drift; the wall panels always rock back to their original position. It is important to note that the maximum roof drift is approximately four times the maximum roof drift observed for the sliding model.  This is due to the fact that the rocking mechanism does not dissipate as much energy as the sliding mechanism.  A breakdown of the dissipated energy is illustrated in Figure 3.18 below.  Chapter 3 Investigation of Mechanism Alternatives   84     Figure 3.18  Rocking Model – Breakdown of Energy Dissipation Zero inelastic energy is present since the rocking mechanism is nonlinear-elastic and does not result in the dissipation of hysteretic energy. Approximately 80% of the total energy dissipated consists of Alpha-M viscous energy dissipated by movement of the building mass.  About 15% of the total energy dissipated consists of Beta-K viscous energy dissipated by deformation of the roof and wall panels.  One form of energy dissipation that is not incorporated in Perform is the energy that is dissipated when the panel hits the foundation during rocking.  This would represent an additional form of energy dissipation in a real rocking structure. 3.3.3 Model 3: Frame Mechanism The response of the frame model for the Northridge MUL009 record scaled to Sa(T1) = 0.1g is essentially the same as for the sliding and rocking models, with the roof governing the response and the walls remaining elastic.  As expected for elastic response, the maximum roof drift Chapter 3 Investigation of Mechanism Alternatives   85  matches the observed maximum roof drift for the sliding and rocking models.  The frame model does not exhibit any yielding of the in-plane walls for a spectral acceleration Sa(T1) = 1.0g, due to the greatly increased strength of the panels legs resulting from connection of adjacent panels. The response of the frame model for the Northridge MUL009 record scaled to Sa(T1) = 3.0g is illustrated in Figure 3.19 below.  Figure 3.19  Frame Model – Wall and Roof Drifts for Northridge Earthquake, Sa(T1)=3.0g It can be observed in Figure 3.19 above that the walls exhibit considerable inelastic behaviour. Observing Figure 3.14, it is evident that the yield drift for the in-plane walls is approximately 0.3% (27mm). Within Figure 3.19 it can be observed that at approximately 3 seconds into the earthquake, the walls begin to oscillate at a drift of approximately 0.3% (27mm).  At approximately 7 seconds, the wall drift increases to approximately 0.4% (37mm), and eventually at approximately 12 seconds into the earthquake, the walls fail in-plane due to straining of the rebar past the ultimate allowable strain. The response shown above is characteristic of the frame model response when the earthquake intensity is scaled up enough to cause yielding of the walls. Typically, if the walls yield during an analysis run, the run is terminated due to the model Chapter 3 Investigation of Mechanism Alternatives   86  exceeding the maximum allowable strain specified for the reinforcing steel.  For the analysis results plotted in Figure 3.19 above; prior to termination of the analysis run at about 12.5 seconds, the maximum wall drift is approximately 1.3% (120mm) and the maximum roof drift is approximately 3.4% (310mm).   It is important to note that the maximum wall drift is considerably smaller than the drift shown in Figure 3.12 for the reversed-cyclic loading of the beam-column subassembly, but is consistent with the maximum wall drift shown in Figure 3.14. A breakdown of the dissipated energy is illustrated in Figure 3.20 below.  Figure 3.20  Frame Model – Breakdown of Energy Dissipation Almost 50% of the total energy dissipated consists of Alpha-M viscous energy dissipated by movement of the building mass.  Only 15% of the total energy dissipated consists of inelastic energy dissipated by the in-plane walls, and about 15% consists of Beta-K viscous energy dissipated by deformation of the roof and wall panels.  It is interesting to note that the energy dissipated by inelastic deformation of the in-plane walls actually makes up a small percentage of the total energy dissipated.   This is likely due in part to the manner in which the panels are Chapter 3 Investigation of Mechanism Alternatives   87  connected, resulting in less ductility in the response of the panel legs than would normally be expected. 3.4 IDA Results The first eight earthquake records from the list recommended by the ATC-63 Methodology (Table 2.1) were selected for IDA to allow comparison between the different mechanisms.  The records were applied in the direction of the short axis of the building only.  The records were scaled as discussed in Section 2.3.2.  To illustrate the IDA results, first mode spectral acceleration was plotted against centre roof drift and top of end wall drift. 3.4.1 Model 1: Sliding Mechanism The IDA results for drift at the top of the end wall and at the centre of the roof for the sliding model are shown in Figure 3.21 below.  Figure 3.21  Sliding Model – IDA Drift Results for 8 Ground Motion Records: (a)End Walls; (b)Centre of Roof In Figure 3.21 above, it can be observed that the model remains elastic until the ground motions are scaled up to a first mode spectral acceleration of approximately 0.25g, at which point the wall panels begin to slide.  This is consistent with the pushover curve for this model, in which the total base shear required to cause the panels to slide was 0.25g.  The median IDA curve (a) (b) Chapter 3 Investigation of Mechanism Alternatives   88  appears to have a flattening trend, but does not completely flatten out for the range of drifts shown above. This is because the pushover curve for the sliding model is essentially elastic perfectly plastic (EPP) and does not exhibit any strength degradation (Vamvatsikos and Cornell, 2002).  The nonlinear response of many EPP systems can be approximated with the equal displacement principle, which states that for a given earthquake demand, the total displacement response would be the same for a yielding EPP system as for an equivalent elastic system with stiffness equal to the initial stiffness of the EPP system.  If the sliding model behaved in accordance with the equal displacement principle, the IDA plot would be linear.  The observed median IDA plot indicates that the sliding model does not behave strictly in accordance with the equal displacement principle but also does not exhibit the flattening behaviour found in systems that have a significant negative post-yield slope.  The continual positive slope of the IDA curves suggests that collapse of a tilt-up building cannot be defined by a flattening of the IDA curve, and must be identified by other indicators of collapse that are not modelled directly, such as drift limits for the gravity system. 3.4.2  Model 2: Rocking Mechanism The IDA results for drift at the top of the end wall and at the centre of the roof for the rocking model are shown in Figure 3.22 below.   Chapter 3 Investigation of Mechanism Alternatives   89   Figure 3.22  Rocking Model – IDA Drift Results for 8 Ground Motion Records: (a)End Walls;  (b)Centre of Roof In Figure 3.22 above, it can be observed that the model remains within the linear range of response until the ground motions are scaled up to a first mode spectral acceleration of approximately 0.25g, at which point the walls begin to rock.  This is consistent with the results for the sliding model and with the pushover curve. (Recall that the sliding and rocking models were calibrated such that the yielding strength of both systems was identical.)  The response of the roof does not appear to change significantly as the walls begin to rock. 3.4.3 Model 3: Frame Mechanism The IDA results for drift at the top of the end wall and at the centre of the roof are shown in Figure 3.23 below for the frame model. It can be observed that the model remains elastic until the ground motions are scaled up to a first mode spectral acceleration of approximately 2.0g, at which point some of the ground motions result in yielding of the reinforcing steel in the panel legs.  This is consistent with the results for the pushover curve, though there is more variability in results from the various ground motions than was observed for the sliding and rocking models. (a) (b) Chapter 3 Investigation of Mechanism Alternatives   90   Figure 3.23  Frame Model – IDA Drift Results for 8 Ground Motion Records: (a)End Walls; (b)Centre of Roof It is important to note that the ultimate drift values plotted above for each ground motion do not necessarily represent the complete response of the model for that ground motion, since in many cases the analysis reached a point at which the strain limit of 0.1 was exceeded in the reinforcing steel,  causing the analysis to stop.  The strain limit for the reinforcing steel was established since the accuracy of the analysis results was questionable beyond this point. It does not represent the strain capacity of the steel.  Strength loss in the reinforcing steel occurs at a strain of 0.05, at which point the strength steeply degrades to 0.1% of the yield strength.  The analysis continues with very low rebar strength until the strain in the reinforcing steel reaches 0.1, at which point the analysis is halted. 3.5 Comparison of Rocking and Sliding Mechanisms Figure 3.24 below provides a comparison between the median response of the rocking and sliding models.  It can be observed that the rocking model undergoes significantly greater displacements in both the walls and the roof once the walls yield. (a) (b) Chapter 3 Investigation of Mechanism Alternatives   91   Figure 3.24  Sliding and Rocking Models: IDA Drift Results for 8 Ground Motion Records: (a)End Walls;  (b)Centre of Roof In comparison of the two mechanisms, it is important to consider that the building archetype used to form a basis for the analytical models is a very simple system. In reality, buildings are not perfectly rectangular and can have re-entrant corners, or some of the walls may not be parallel to each other.  When these complications in building geometry are considered, the practicability of the sliding mechanism becomes questionable.  Another problem is that the actual friction factor between the wall panels and the foundation may be difficult to predict.  In addition, due to the residual displacement inherent in the sliding mechanism, earthquake damage to a building designed to fail in this manner would be difficult to repair.  The rocking mechanism is more practicable for solid panels since it inherently does not involve a residual displacement; rocking panels always return to their original position.  Based on the observations above, the rocking mechanism and frame mechanism are investigated further using concepts from the ATC- 63 Methodology (refer to Section 2). 3.6 Possible Connection Details for Rocking Mechanism One of the problems inherent in the rocking mechanism is that adjacent panels that are rocking will have a differential vertical displacement at their interface.  This movement would likely (a) (b) Chapter 3 Investigation of Mechanism Alternatives   92  cause the roof perimeter angle to yield in bending and may cause local damage to the roof in the vicinity of the interface between the rocking panels. One possible method to mitigate this problem is to end the roof perimeter angle at the wall either side of the interface between panels and introduce an additional angle to connect the nearest adjacent wall connections (EM2) with a single bolt (effectively a pin) at each connection.  This would allow free rotation of the angle when the panels rock.  The sketches in Figure 3.25 and Figure 3.26 below illustrate this concept.  Figure 3.25  Rocking Mechanism – Consideration at Panel to Panel Interface - Elevation View from Inside of Building  Chapter 3 Investigation of Mechanism Alternatives   93    Figure 3.26  Rocking Mechanism – Consideration at Panel to Panel Interface - Detail at Panel Interface Another difficulty with the rocking mechanism is that the type of panel to slab connections currently used in common practice (EM5 connections – refer to Figure 1.6) would have significant interaction between uplift and shear.  In order for the rocking mechanism to be feasible in practice, the behaviour of the panel to slab connections in uplift and shear would have to be de-coupled.  One possible solution to this problem is to provide two sets of connections: • One set of connections would provide lateral resistance only, in both the in-plane and out-of-plane directions, without providing any resistance in uplift.  A shear pin concept could be used to implement this type of connection between the panel and the footing.  A Chapter 3 Investigation of Mechanism Alternatives   94  smooth round bar could be embedded in the footing and extended a short distance into a sleeve in the panel.  If the shear pin connections were near the middle of the panel, they would undergo less uplift movement. • Another set of connections would provide uplift resistance only, without providing any lateral resistance.  A tie-down concept could be used to implement this type of connection between the panel and footing.  A threaded bar could be embedded in the footing and extend up to an embedded connection in the panel where the threaded bar would be bolted onto a welded bolt-box assembly.   Alternatively to tie-down anchors, adjacent panels can be connected together in order to provide uplift resistance.  Although more practicable, increasing uplift resistance by connecting adjacent panels causes greater differential vertical displacements at the interface between  rocking panels for a given horizontal displacement, thus causing more damage to the roof than the tie-down alternative. Figure 3.27 and Figure 3.28 below illustrate possible details for the shear pin and tie-down connections proposed above.  Chapter 3 Investigation of Mechanism Alternatives   95    Figure 3.27  Rocking Mechanism – Possible Connection Details: Elevation of Building  Chapter 3 Investigation of Mechanism Alternatives   96   Figure 3.28  Rocking Mechanism – Possible Base Connection Details: Tie-down and Shear Pin Chapter 3 Investigation of Mechanism Alternatives   97   3.7 Incorporating a Rocking Mechanism for Panels with Openings Considering the greatly increased in-plane strength resulting from connection to adjacent panels, there is some uncertainty as to whether a frame mechanism could develop for buildings incorporating panels with large openings similar to Archetype 2.  There is limited information on the panel to slab connections, which may fail prior to a frame mechanism forming, and would certainly affect the response. Considering this uncertainty, it may be worthwhile to consider incorporating a rocking mechanism for panels with openings.  Results described in Chapter 4 are utilized, in which a preliminary RdRo value for rocking is assessed to be 2.1. It is important to note that the assessment of RdRo for the rocking mechanism was done based on a model with 2 adjacent panels connected together and rocking as pairs.  This type of arrangement is not practicable for panels with large openings, since the outside leg of each “rocking pair” of panels would be required to accommodate the lateral forces transferred from both panels and would have to be strengthened considerably.  Alternatively, the rocking mechanism can be incorporated for individual panels using the tie-down and shear pin concept as illustrated in Figure 3.27 and Figure 3.28 above.  The RdRo value of 2.1 assessed in Section 4 is still considered valid, since the tie-down and shear pin concept would provide additional energy dissipation and would likely increase the assessed RdRo value.  Within this section the rocking mechanism is incorporated for individual panels with openings.  Some preliminary sizes of tie-down anchors and shear pins are provided, based on the details illustrated in Figure 3.28 above and design forces used to establish Archetype 2. For Archetype 2, the design in-plane elastic base shear for each leg of each panel was determined to be 161kN/ leg (refer to Appendix C) based on Rd =1.5 and Ro = 1.3.  The corresponding base shear for RdRo =2.1 would be as follows: (161kN/leg)(1.5)(1.3)/(2.1) = 150 kN/leg.  The design of the shear pin is as follows: 1. Material Properties: Round bar conforming to CAN/CSA G40.21, 300W steel,  Yield strength, Fy=300MPa, Ultimate Strength, Fu=450MPa 2. Consider as pin according to CAN/CSA S16-01:  Vr = 0.66*øs*Fy*As, where øs=0.9 Chapter 3 Investigation of Mechanism Alternatives   98  3. The required steel area to provide sufficient shear resistance is 840 mm2. 4. A 35mm (1.375”) diameter bar would suffice, with As= 960 mm2. 5. The design shear strength of the bar, Vr = 171kN 6. The nominal shear strength of the bar, Vu = 253 kN (assuming Fu=450MPa and øs=0.9) 7. Concrete embedment would have to be designed accordingly. The design of the tie-downs is as follows: 1. Material Properties: Round bar conforming to CAN/CSA G30.18, 400W hot-rolled threaded steel bar,  Yield strength, fy=400MPa, assumed ultimate strain, ɛu=0.1 2. Panel Geometry:  height = 9.144m, width = 7.62m, weight = 284kN 3. Taking moments about the corner of a panel, the overturning moment is approximately equal to, Mo = 2*(150kN/leg)*(9.144m height) = 2743 kNm. 4. Tie-down force required at each leg is approximately, T=[(Mo – (284kN)*0.5*7.62m] / 7.62m, so T = 218 kN at each corner of the panel. 5. Consider nominal yield resistance (since preferably this would be the “fuse”),  Tr=fy*As, so the required steel area to provide sufficient force is 545 mm2. 6. 2- 20mm (0.75”) diameter bars would suffice at each corner of the panel, with combined steel area, As= 600 mm2. 7. A free stressing length of 1200mm would provide uplift displacement capacity of approximately Duplift = 1200mm*ɛu = 120mm. 8. Hook development length would be required in the footing. Assuming 30MPa concrete strength, side cover greater than 60mm and end cover greater than 50mm, the hook development length would be approximately 250mm. 9. EM4 embedded connection with 150mm long studs is required to connect the tie-downs to the panel.  The design shear resistance is 265kN (refer to Figure 1.3), with an expected ultimate resistance of approximately 440kN. Chapter 3 Investigation of Mechanism Alternatives   99  10. Appropriate steel detailing would be required to support the tie-down nut and bearing plate at the EM4 embedded connection. The details described above are illustrated in Figure 3.29 below.   Figure 3.29  Rocking Mechanism for Panels with Openings – Possible Connection Details: (a)Elevation of Building; (b)Details (a) (b) Chapter 3 Investigation of Mechanism Alternatives   100   3.8 Evaluation of Previous Research 3.8.1 Ductility Demands of Walls vs. Roof One interesting phenomenon that occurs in nonlinear series systems is that when one component yields, as the applied loading is scaled up, the displacement in the yielded component comprises a progressively larger proportion of the total displacement in the system.  In tilt-up structures, the walls and the roof behave as a series system. Intuitively, if the applied earthquake loading is progressively scaled up, one would expect that whichever component yields first, (i.e. the walls or the roof), the displacement of that component will comprise a progressively larger proportion of the total displacement. This phenomenon was observed in previous analytical research conducted on tilt-up structures [Adebar et al., 2004] and was also observed in this study.  Figure 3.30 below illustrates the median results from IDA using eight different ground motion records for the sliding and rocking models.  Displacements at the centre of the roof and at the top of the end walls are plotted for each model.  Figure 3.30  Median Drift Centre of Roof and End Walls for 8 Ground Motion Records: (a)Sliding Model; (b)Rocking Model (a) (b) Chapter 3 Investigation of Mechanism Alternatives   101  As can be observed, the walls yield at approximately Sa(T1) = 0.25g.  As the earthquake demand is scaled up past this yield point, the displacement of the walls comprises a progressively larger portion of the total displacement.  For example, at Sa(T1) = 0.5, results for the sliding model indicate the walls displace to a drift of 0.003, while the roof displaces to a drift of 0.007, i.e. the walls take up about 43% of the total displacement.  However, at Sa(T1) = 1.5, the walls displace to a drift of 0.012, while the roof displaces to a drift of 0.016, i.e. the walls take up about 75% of the total displacement.  Similar results can be observed for both the sliding mechanism and the rocking mechanism (plotted above), as well as for the frame mechanism (not shown). 3.8.2 Ductility Demands on Legs of Frame Panels One simple method that has been proposed [Adebar et al., 2004] to estimate the inelastic displacement demands for legs of frame panels is described in the following equation:  )1(*)1( we wy dwi V V TS −=∆  (3.2) where  ∆wi  = Inelastic wall displacement    Sd(T) = Spectral displacement at centre of roof    Vwy  = Wall strength    Vwe = Maximum elastic demand on walls The data obtained from the nonlinear dynamic analysis of the frame model for 15 ground motions was plotted against the above expression to determine how well Equation 3.2 can predict the displacement demands on the walls (refer to Figure 3.31 below). Chapter 3 Investigation of Mechanism Alternatives   102   Figure 3.31  Inelastic Wall Displacement at Sa(T1) = 3.0g:  Analysis Results vs. Predicted There seems to be reasonable agreement between the analysis data and the estimate proposed [Adebar et al., 2004] for a spectral acceleration of 3.0g.  Comparisons were made at the relatively high intensity spectral acceleration since there was insufficient yielding of the in-plane walls to make a meaningful comparison at lower earthquake intensities. 3.8.3 Seismic Demands on Roof Diaphragm Due to Out-of-Plane Response of Wall Panels As part of this study, the seismic demands on the roof diaphragm due to out-of-plane response of the walls panels determined from analysis and compared with demands calculated according to common practice in North America (as described in Section 1.2.5), and with demands calculated according to ASCE 41-06 recommendations.  ASCE 41-06 recommends the following equation for determining the demands on a flexible deck diaphragm:                −= 2 215.1 dd d d L x L Ff  (3.3) Chapter 3 Investigation of Mechanism Alternatives   103  Where: fd = Inertial load per foot   Fd = Total inertial load on a flexible diaphragm   x = Distance from the center line of flexible diaphragm   Ld = Distance between lateral support points for diaphragm In plotting Equation 3.3 above, the total inertial load, Fd, was assumed to be the mass of the roof diaphragm and half the mass of the out-of-plane walls, multiplied by Sa(T1) = 0.5g. Comparisons were made at a first mode spectral acceleration of 0.5g and are plotted in Figure 3.32 below.  Figure 3.32  Seismic Demands on Roof Diaphragm due to Out-of-Plane Response of Wall Panels for Sa(T1) = 0.5g:  Analysis Results vs. Common North American Practice vs. ASCE41-06 Approximation Chapter 3 Investigation of Mechanism Alternatives   104  The plot includes results from analysis of the frame model for all 22 earthquake records recommended by the ATC-63 Methodology (refer to Table 2.1), scaled to a first mode spectral acceleration of 0.5g.  The results were processed by taking an average of the maximum out-of- plane connection forces at the top of each roof panel for panels on the long axis of the building. In all of the models, the walls along the long axis of the building are connected to the roof at four nodes, as shown in Figure 3.3.  At interface locations between panels, the corner node of only one of the panels is connected to the roof, in order to allow the panels to rock independently of one another.  Due to this eccentricity in the roof-to-panel connections, the connection forces are not evenly distributed within each panel.  Thus it was considered more appropriate to take an average of the connection forces obtained at the four roof-to-panel connections. It can be observed within Figure 3.32 that the demands from the median analysis results were typically higher than demands calculated using common North American practice by up to 50 percent for panels near the middle of the building.  It is also apparent that the results were higher than the ASCE 41-06 parabolic shear load approximation near the ends of the diaphragm. The most accurate representation of the median analysis results would be obtained by considering the envelope of forces from the constant shear distribution and the parabolic shear distribution (i.e. use the constant shear distribution to determine out-of-plane forces near the ends of the diaphragm and parabolic shear distribution to determine out-of-plane forces near the middle of the diaphragm).  It is noted that there is considerable scatter evident in the results, particularly toward the high demands.  105  4 QUANTIFICATION OF SEISMIC PERFORMANCE FACTORS As discussed in Section 1.2.7, the current methods used for seismic design of tilt-up structures do not incorporate capacity design principles commonly used in the design of other structural systems.  No clear, stable mechanisms are typically identified for tilt-up systems during design, and there is considerable uncertainty as to what types of mechanisms would form in many existing tilt-up structures. In Section 3, several possible mechanisms were identified and evaluated.  Rocking and sliding mechanisms were incorporated into Archetypical System 1; a frame mechanism was considered for Archetypical System 2.  In this section, the rocking mechanism and the frame mechanism are considered further.  In order to incorporate a rocking mechanism into design, an assessment must be made of an appropriate R-factor.  For the frame mechanism, the current R-factors used for design are based on component testing, without consideration of the overall response (within this section, R is assumed equivalent to RdRo from the NBCC 2005.)  In order to make a preliminary assessment of an appropriate R-factor for the rocking mechanism, and to confirm the current R-factors for the frame mechanism, concepts from the ATC-63 Methodology were used. 4.1 Model 4: Rocking Mechanism To investigate the response of a building designed to have a failure mechanism characterized by in-plane rocking of the wall panels, Archetypical System 1 was considered but the connection layout was modified.  Two adjacent panels were connected together at the end walls (in-plane walls along the short axis of the building), such that two sets of two adjacent panels could rock independently at end walls.  Each set of two panels was assumed to have connections at the base that would stop sliding from occurring but would allow rocking (refer to shear pin connection in Figure 3.28.)  In addition, connections at corners were removed such that panels could rock independently in different directions.  Figure 4.1 below illustrates the assumed connection layout.  The sections shown in the figure are referenced from Figure 2.4. Chapter 4 Quantification of Seismic Performance Factors   106   Figure 4.1  Rocking Model – 2 Adjacent Panels Connected at End Walls The model is similar to the one used to make comparisons between the different mechanisms (refer to Section 1 for details).  The only difference being the additional connections between the end wall panels.  The change in the model was verified to ensure reasonable behaviour by conducting a pushover analysis in the direction of the short axis of the building and comparing the results to hand calculations.  Figure 4.2 below illustrates the results of this analysis. Chapter 4 Quantification of Seismic Performance Factors   107    Figure 4.2  Rocking Model - Pushover Analysis Along Short Axis of Building (2 panels connected) The results indicate good agreement between the Perform and hand calculations.  The yield strength of this model is 0.3W, where W is the seismic weight of the building.  The NBCC 2005 elastic design base shear based on the first mode period calculated according to the code formula and Site Class D in Vancouver is 0.68W.  The NBCC 2005 design base shear used incorporates the two-thirds cut-off rule since it was also incorporated in design of the building archetypes.  If the ASCE 41 formula (Equation 3.1) is used to determine the fundamental period, the elastic design base shear for a Site Class D building in Vancouver would be 0.64W.  This effectively means the design of the in-plane walls is based on an RdRo value of 2.3.  It is important to note that R-values are structure specific and not location specific.  The assessment in this study uses the Vancouver design base shear to evaluate whether an RdRo value of 2.3 is adequate for a rocking model.  Vancouver was selected as the location since it has a relatively high design base Chapter 4 Quantification of Seismic Performance Factors   108  shear.  If it is established that this RdRo value is adequate for the Vancouver design base shear, it should be reasonable to use this RdRo value for other design base shear values. 4.1.1 Simulated and Non-Simulated Collapse The rocking of the walls is the only mechanism that is simulated in the modeling.  Non- simulated collapse mechanisms include yielding of the roof diaphragm, rupture of the out-of- plane wall to roof connectors, and excessive deformation of the gravity columns.  Non-simulated collapse mechanisms were evaluated by tracking the forces and deformations associated with the mechanisms and comparing with calculated capacities. The roof diaphragm was designed based on SDI provisions with recommended safety factors. The nominal strength of the roof diaphragm was evaluated based on removing SDI safety factors and considering results from recent testing [Hilti, 2008].  The nominal strength for the decking arrangement used in Zone 2 (refer to Figure 2.5) was determined to be 70 kN/m, compared to a factored design strength of 31 kN/m.  Refer to Appendix B for detailed calculations. The nominal capacity of the out-of-plane wall to roof connections was determined by considering NBCC 2005 design forces and removing safety factors from design capacities of connectors.  It was assumed that designers would typically provide sufficient connections and drag struts into the deck to transfer the NBCC 2005 design out-of-plane forces. The factored resistance of out-of-plane wall to roof connections was determined based on testing (Lemieux et. al, 1998).  The average nominal strength determined from the test results was multiplied by 0.6 to obtain the factored design strengths shown in the Concrete Design Handbook (CAC 2006) and illustrated in Figure 1.3.   In removing this safety factor, the nominal strength of the out-of-plane connections was estimated to be 42 kN/m, compared to a design capacity of 25 kN/m. Allowable lateral deflections for the gravity columns was determined by considering the interaction between the unfactored design axial load and the second order moment that occurs in the gravity columns when the roof is displaced.  The maximum allowable roof drift before the gravity columns could no longer adequately support the unfactored design axial load was estimated to be 0.03.  A maximum allowable drift of 0.03 was considered appropriate, since the NBCC 2005 requires that the interstory drift for a structure must be limited to 0.025.  It is Chapter 4 Quantification of Seismic Performance Factors   109  recognized that the assessment of the gravity columns was based only on force capacities and that deformations at collapse may be much larger. Within this study, the primary acceptance criteria used to determine the suitability of a selected R value was the criteria for a maximum roof drift of 0.03.  The other criteria for non-simulated collapse, including in-plane yielding of the roof deck diaphragm and failure of the out-of-plane panel to roof connectors were monitored in order to determine if higher design strengths would be required in order to ensure the roof drift criteria could be met. 4.1.2 IDA Results, Collapse Statistics and Uncertainty Incremental dynamic analysis was carried out for the 22 ground motion record pairs recommended by the ATC-63 Methodology (Table 2.1).  Each record pair was applied simultaneously to the model, in orthogonal directions.  The IDA results are shown plotted in Figure 4.3 below. Chapter 4 Quantification of Seismic Performance Factors   110   Figure 4.3  Rocking Model – IDA Results for 22 Ground Motion Record Pairs: (a)Drift at End Wall;  (b)Drift at Centre of Roof; (c)In-Plane Deck Forces at End Walls; (d)Out-of- Plane Wall to Roof Connection Forces It is interesting to note that as the end walls yield the slope of the IDA curve for the in-plane deck forces in Figure 4.3c increases, indicating the forces imparted to the deck is limited by the rocking of the walls.  This confirms previous research (Adebar et al., 2004) that indicated that in (a) (b) (d) (c) Chapter 4 Quantification of Seismic Performance Factors   111  a series system, when one component yields, it shields the other components from greater forces. The collapse data for the model is established for each non-simulated collapse mechanism by considering the first mode spectral acceleration at which the IDA curve for each individual earthquake record exceeds the collapse threshold.  The median value from the collapse data is then used to define a lognormal cumulative distribution to describe the collapse statistics for each non-simulated collapse mechanism.  Refer to Appendix D for sample calculations on how the collapse statistics are determined. As discussed in Section 2.6, the ATC-63 Methodology requires consideration of variability due to modeling uncertainties, uncertainty in how representative the archetype design is to actual structures (termed “design requirements”), and uncertainty relating to the amount of test data. Values of variability are assigned to each of these uncertainties and are used to establish an aggregate variability for each non-simulated collapse mechanism.  This aggregate variability is then incorporated into the lognormal distribution used to fit the collapse data, in order to adjust the curve.  An increase in the degree of uncertainty judged to be in the analysis causes a corresponding increase in the amount of adjustment that is applied to the lognormal distribution, resulting in a more conservative distribution.   For the rocking model, the following values were adopted: • Record to record uncertainties, βRTR= 0.37 for the non-simulated collapse due to drift, 0.36 for in-plane deck forces, and 0.47 for out-of-plane deck forces. (This value was calculated directly from the collapse data for each non-simulated collapse mode, and is equal to the standard deviation of the natural log of the collapse spectral acceleration results) • Modeling uncertainties, βMDL=0.45 was selected.  The model was judged to be a “fair” representation of the actual structure.  The reason for the relatively high uncertainty assigned to modeling is that the model does not incorporate the non-linear response of the roof deck diaphragm.  Also, for the frame and eccentric models, there is no consideration of the behaviour of the panel to slab connections. • Design requirements, βDR=0.3 was selected.  As a result of the survey conducted (refer to Table 2.2) with various tilt-up contractors and consultants in the US and Canada, there is Chapter 4 Quantification of Seismic Performance Factors   112  good confidence that the building archetype reasonably represents typical construction for single story tilt-up structures found in practice. • Test Data, βTD=0.3 was selected.  The stiffness and capacity of the roof diaphragm, as well as the capacity of the panel to deck connections are based on extensive testing, and thus there is good confidence in the numbers used for the stiffness of the deck and for the nominal strengths in order to define non-simulated collapse of the structure. Based on the values selected above for various sources of uncertainty, the aggregate variability was determined using Equation 2.1 to be 0.72 for collapse due to drift, 0.71 for collapse due to in-plane deck forces, and 0.77 for collapse due to out-of-plane deck forces.  These values were then incorporated to obtain the adjusted lognormal cumulative distribution for each non- simulated collapse mode, which was then used to evaluate the R factor based on the selected acceptance criteria.  In Figure 4.4 below, the collapse data is plotted for each non-simulated collapse mode (plotted as individual points), along with the lognormal distribution (plotted as a solid line), the adjusted lognormal distribution (plotted as a dash-dot line), as well as the NBCC 2005 base shear for a Vancouver building on Site Class D (plotted as a dashed line). Chapter 4 Quantification of Seismic Performance Factors   113   Figure 4.4  Rocking Model – Collapse Statistics: (a)Roof Drift; (b)In-Plane Deck Forces; (c)Out-of-Plane Deck Forces As can be observed in Figure 4.4 above, for all three non-simulated collapse modes considered, the lognormal distribution fits the collapse data reasonably well.  The collapse data for the out- of-plane deck force collapse mode exhibits some scatter at the tail of the distribution. 4.1.3 Acceptance Criteria and Evaluation of R As discussed previously in Section 2.8, for the purposes of this study, a maximum probability of collapse of 0.1 is adopted for the design earthquake with an annual excedence frequency of 1 in (a) (c) (b) Chapter 4 Quantification of Seismic Performance Factors   114  2475.    For the roof drift failure mode, using the adjusted cumulative distribution shown as the dash-dot line in Figure 4.4 (a), it was determined that there was a 10% probability of exceeding a roof drift of 3% at Sa(T1)=0.62g.  The 2005 NBCC design base shear for the in-plane walls is 0.68W, corresponding to a 12% probability of exceeding the selected drift limit of 3%.  This does not meet the acceptance criteria adopted for this study, and thus the design of the in-plane walls is not adequate.  The RdRo value of 2.3 which was initially assumed would have to be reduced in order to effectively increase the acceleration corresponding to a 10% probability of exceeding the 3% drift criteria.  If linear interpolation is used to determine the reduced RdRo value, the new value would be approximately equal to (2.3)(0.62g)/(0.68g) = 2.1.  Note that the adjustment for spectral shape is not included in this assessment. It is interesting to note that if the first mode period determined from analysis and based on Equation 3.1 is used instead of the NBCC 2005 code formula, the design base shear reduces to 0.64g, which is very similar to the spectral acceleration corresponding to a 10% probability of exceeding the 3% drift criteria.  If this base shear was used for design, the effective RdRo value would be 2.1. Based on the above, an RdRo value of 2.1 for design of the in-plane walls using a rocking mechanism would meet the acceptance criteria adopted for this study. It should be noted that the means to provide the strength required to achieve the target RdRo value does not need to be by connection of adjacent panels.  A rocking mechanism could also be achieved by other means, including using connections similar to those proposed in Section 3.6. To determine the adequacy of the roof deck for in-plane forces, the median spectral acceleration at which the deck would yield was determined based on data used for Figure 4.3 (c) to be 1.4g. Using the adjusted cumulative distribution for in-plane forces shown in Figure 4.4 (b), the spectral acceleration corresponding to a 10% probability of exceeding the in-plane strength of the roof deck was determined to be 0.55g.  Since the design base shear used for design of the roof deck diaphragm was 0.68g, the design of the roof deck for in-plane forces does not meet the acceptance criteria adopted within this study. In order to meet the acceptance criteria, the strength of the roof for in-plane forces would have to be increased by approximately 25%. Chapter 4 Quantification of Seismic Performance Factors   115  A similar procedure was carried out to determine the adequacy of the out-of-plane panel to roof connections. The median first mode spectral acceleration at which the connections would fail was determined based on data used for Figure 4.3 (d) to be 0.98g.  Using the adjusted lognormal cumulative distribution shown in Figure 4.4 (c), the first mode spectral acceleration corresponding to a 10% probability of exceeding the nominal strength of the out-of-plane panel to roof connections was determined to be 0.39g. The demands on the out-of-plane wall to roof connections calculated according to the NBCC 2005 depend partly on the first mode spectral acceleration of the building and partly on the flexibility of the component being considered (refer to Section 1.2.3).  However as a means of comparison with the IDA plots for the overall structure, the design base shear for the building was used (0.68g). As such, the design of the out-of-plane wall to roof connections does not meet the acceptance criteria adopted within this study. In order to meet the acceptance criteria, the strength of the wall to roof connections for out-of-plane forces would have to be increased by approximately 90%. To summarize, an RdRo value of 2.1 is probably reasonable for the rocking mechanism for this building archetype, though the in-plane diaphragm resistance would have to be increased by 25% and the wall-to-roof connections for out-of-plane loading would have to be increased by approximately 90%. 4.2 Model 5: Frame Mechanism To investigate the response of a building designed to have a failure mechanism characterized by in-plane yielding of the legs of wall panels with large openings, Archetypical System 2 was considered.  The model used to carry out the analyses is identical to the model used to make comparisons between different mechanisms.  Refer to Section 3.4.3 for details. The NBCC 2005 design base shear based on the first mode period calculated according to the code formula and Site Class D is 0.68g.  For design, the legs for the in-plane walls with large openings were designed based on an Rd = 1.5 and Ro = 1.3, resulting in a combined RdRo value of 1.95.  Due to the connections between adjacent panels, the legs of the adjacent end wall panels with large openings behave essentially as a single unit, resulting in much higher in-plane Chapter 4 Quantification of Seismic Performance Factors   116  strengths (refer to Figure 3.14).  Consequently, the yield strength of this model in the direction of the short axis of the building is 0.87g.  This effectively means the design of the in-plane walls is based on an RdRo value less than 1. 4.2.1 Simulated and Non-Simulated Collapse In-plane flexural yielding of legs of wall panels with large openings is the primary mechanism that is simulated in the modeling.  Shear failure is also modelled explicitly.  Non-simulated collapse mechanisms include yielding of the roof diaphragm, rupture of the out-of-plane wall to roof connectors, and excessive deformation of the gravity columns.  Non-simulated collapse mechanisms were evaluated by tracking the forces and deformations associated with the mechanisms and comparing with calculated capacities.  Refer to Section 4.1.1 for a discussion on the non-simulated collapse mechanisms considered and their nominal capacities. 4.2.2 IDA Results, Collapse Statistics and Uncertainty Incremental dynamic analysis was carried out for the 22 ground motion record pairs recommended by the ATC-63 Methodology (Table 2.1).  Each record pair was applied simultaneously to the model, in orthogonal directions.  The IDA results are shown plotted in Figure 4.5 below.     Chapter 4 Quantification of Seismic Performance Factors   117   Figure 4.5  Frame Model – IDA Results for 22 Ground Motion Record Pairs: (a)Drift at End Wall;  (b)Drift at Centre of Roof; (c)In-Plane Deck Forces at End Walls; (d)Out-of- Plane Wall to Roof Connection Forces It can be observed in Figure 4.5 (c) above that the IDA curve for in-plane deck forces remains essentially linear.  There is no increase in slope as observed in Figure 4.3 (c).  This is likely because the in-plane frame walls have a much higher strength than the rocking wall model and (a) (b) (d) (c) Chapter 4 Quantification of Seismic Performance Factors   118  do not exhibit any yielding for the range of spectral accelerations shown in Figure 4.5 (c) above, i.e. there is no “fuse” in the system. The collapse data for the frame model was obtained in the same manner as for the rocking model (refer to Section 4.1.2).  The same values were used for variability, except for variability due to record to record uncertainties, βRTR. This was value was determined (based on the collapse data) to be 0.37 for the non-simulated collapse due to drift, 0.33 for in-plane deck forces, and 0.28 for out-of-plane deck forces, resulting in aggregate variability (βTOT) values of 0.72, 0.70, and 0.68 respectively. In Figure 4.6 below, the collapse data is plotted for each non-simulated collapse mode (plotted as individual points), along with the lognormal distribution (plotted as a solid line), the adjusted lognormal distribution (plotted as a dash-dot line), as well as the NBCC 2005 base shear for a Vancouver building on Site Class D (plotted as a dashed line). Chapter 4 Quantification of Seismic Performance Factors   119   Figure 4.6  Frame Model – Collapse Statistics: (a)Roof Drift; (b)In-Plane Deck Forces; (c)Out-of-Plane Deck Forces  4.2.3 Acceptance Criteria and Evaluation of R Within this study, the primary acceptance criteria for non-simulated collapse used to determine the suitability of a selected RdRo value was the criteria for a maximum roof drift of 0.03.  The (a) (c) (b) Chapter 4 Quantification of Seismic Performance Factors   120  other criteria for non-simulated collapse, including in-plane yielding of the roof deck diaphragm and failure of the out-of-plane panel to roof connectors were monitored in order to determine if higher design strengths would be required in order to ensure the roof drift criteria could be met. Using the adjusted cumulative distribution shown as the dotted line in Figure 4.6 (a) , the spectral acceleration corresponding to a 10% probability of exceeding the 3% drift criteria was determined to be 0.74g.  The 2005 NBCC design base shear for the in-plane walls is 0.68g.  This indicates that the current design used for Archetype 2 meets the acceptance criteria adopted for this study for roof drift. To determine the adequacy of the roof deck for in-plane forces, the median spectral acceleration at which the deck would yield was determined from data used for Figure 4.5 (c) to be 0.94g. This value was used as the median value for a lognormal cumulative distribution similar to the one shown in Figure 4.6.  Using the adjusted cumulative distribution for in-plane forces, the spectral acceleration corresponding to a 10% probability of exceeding the in-plane strength of the roof deck was determined to be 0.38g.  Since the design base shear used for design of the roof deck diaphragm was 0.68g, the design of the roof deck for in-plane forces does not meet the acceptance criteria adopted within this study. In order to meet the acceptance criteria, the strength of the roof for in-plane forces would have to be increased by approximately 80%.  As discussed previously, the increase in the in-plane deck forces in comparison to the rocking model is likely due to the fact that the frame model only yields at a very high earthquake intensity, and thus does not protect the deck from increasing seismic demands in the same manner as the rocking model. A similar procedure was carried out to determine the adequacy of the out-of-plane panel to roof connections. The median first mode spectral acceleration at which the connections would fail was determined from Figure 4.5 (d) to be 1.44g.  Based on the adjusted lognormal cumulative distribution shown in Figure 4.6 (c), the spectral acceleration corresponding to a 10% probability of exceeding the out-of-plane strength of the panel to roof connections was determined to be 0.59g. Using the design base shear for the building as a means of comparison (0.68g), the design of the out-of-plane wall to roof connections does not meet the acceptance criteria adopted within this Chapter 4 Quantification of Seismic Performance Factors   121  study.  The strength of the connections would have to be increased by approximately 15% to meet the acceptance criteria. This indicates that the response of the frame model resulted in lower out-of-plane forces on the panel to roof connections than the rocking model.  Considering that the in-plane deck forces increased considerably, this result may seem counter-intuitive. However, two aspects must be considered: • In the NBCC 2005 Clause 4.1.8.17 regarding seismic demands on elements of structures, non-structural components and equipment, the more flexible a component is, the higher the amplification value, Ar, ascribed to it (refer to Section 1.2.1).  Since the panels are essentially components of the building, it is reasonable that the less flexible frame model results in lower out-of-plane panel to deck forces than the more flexible rocking model. • The panels often do not oscillate in phase in the out-of-plane direction.  In many instances, some panels are flexing in one direction while others are flexing in the opposite direction.  This is due to the difference in the roof stiffness (essentially the out- of-plane lateral support for the wall panel) between the edges of the building and the middle of the building.  Thus it is reasonable that even though the out-of-plane forces may be higher, this may not result in higher in-plane forces overall. In summary, the current RdRo values used with a connection layout commonly used in practice yield results that are within the acceptance criteria adopted within this study for maximum roof drift.  With the current design, the in-plane diaphragm resistance would have to be increased by 80% and the wall-to-roof connections for out-of-plane loading would have to be increased by approximately 15%. 4.3 Model 6: Frame Mechanism – Eccentric Building To investigate the response of a building with large openings on one side and solid panels on the other, Archetypical System 2 was modified by incorporating solid panels at one end wall. The model used to carry out the analyses is similar to Model 3 used to make comparisons between different mechanisms, except that solid panels were incorporated at one end wall that were rigidly connected to each other and at the base.  Refer to Section 3.4.3 for details on Model 3. Chapter 4 Quantification of Seismic Performance Factors   122  The NBCC 2005 design base shear based on the first mode period calculated according to the code formula and Site Class D is 0.68g.  For design, the legs for the in-plane walls with large openings were designed based on an Rd = 1.5 and Ro = 1.3, resulting in a combined RdRo value of 1.95.  Due to the connections between adjacent panels, the legs of the adjacent end wall panels with large openings behave essentially as a single unit, resulting in much higher in-plane strengths (refer to Figure 3.13 and Figure 3.14). 4.3.1 Simulated and Non-Simulated Collapse In-plane flexural yielding of the wall panels with large openings is the primary mechanism that is simulated in the modeling.  Non-simulated collapse mechanisms include yielding of the roof diaphragm, rupture of the out-of-plane wall to roof connectors, and excessive deformation of the gravity columns.  Non-simulated collapse mechanisms were evaluated by tracking the forces and deformations associated with the mechanisms and comparing with calculated capacities.  Refer to Section 4.1.1 for a discussion on the non-simulated collapse mechanisms considered and their nominal capacities. 4.3.2 IDA Results, Collapse Statistics and Uncertainty Incremental dynamic analysis was carried out for the 22 ground motion record pairs recommended by the ATC-63 Methodology (Table 2.1).  Each record pair was applied simultaneously to the model, in orthogonal directions.  The IDA results for roof drift are shown plotted in Figure 4.7 below. Chapter 4 Quantification of Seismic Performance Factors   123   Figure 4.7  Eccentric Model – IDA Results for 22 Ground Motion Record Pairs: (a)Drift at End Wall;  (b)Drift at Centre of Roof The IDA results for deck forces are shown plotted in Figure 4.8 below. (a) (b) Chapter 4 Quantification of Seismic Performance Factors   124   Figure 4.8  Eccentric Model – IDA Results for 22 Ground Motion Record Pairs: (a)In- Plane Deck Forces at End Wall with Frame Panels; (b) In-Plane Deck Forces at End Wall with Solid Panels; (c) Out-of-Plane Deck Forces As can be observed from Figure 4.8 (a) and (b), the wall with solid panels attracts greater in- plane deck forces than the wall with openings. This result is expected since the wall with solid (a) (b) (c) Chapter 4 Quantification of Seismic Performance Factors   125  panels does not yield and is much stiffer.  The in-plane deck forces at the end wall with solid panels will be considered in this assessment. The collapse data for the frame model was obtained in the same manner as for the rocking model (refer to Section 4.1.2).  The same values were used for variability, except for variability due to record to record uncertainties, βRTR. This was value was determined (based on the collapse data) to be 0.36 for the non-simulated collapse due to drift, 0.31 for in-plane deck forces, and 0.29 for out-of-plane deck forces, resulting in aggregate variability (βTOT) values of 0.72, 0.69, and 0.68 respectively. In Figure 4.9 below, the collapse data is plotted for each non-simulated collapse mode (plotted as individual points), along with the lognormal distribution (plotted as a solid line), the adjusted lognormal distribution (plotted as a dash-dot line), as well as the NBCC 2005 base shear for a Vancouver building on Site Class D (plotted as a dashed line). Chapter 4 Quantification of Seismic Performance Factors   126  4.3.3 Acceptance Criteria and Evaluation of R Using the adjusted cumulative distribution shown as the dotted line in Figure 4.9 (a), the spectral acceleration corresponding to a 10% probability of exceeding the 3% drift criteria was determined to be 0.80g.  The 2005 NBCC design base shear for the in-plane walls is 0.68g.  This  Figure 4.9  Eccentric  Model – Collapse Statistics: (a)Roof Drift; (b)In-Plane Deck Forces; (c)Out-of-Plane Deck Forces (a) (c) (b) Chapter 4 Quantification of Seismic Performance Factors   127  indicates that the current design used for Archetype 2 meets the acceptance criteria adopted for this study for roof drift, even if the building is significantly eccentric. To determine the adequacy of the roof deck for in-plane forces, the median spectral acceleration at which the deck would yield near the end wall with solid wall panels was determined from data used for Figure 4.8 (b) to be 0.85g.   Using the adjusted cumulative distribution for in-plane forces (refer to Figure 4.9 (b)), the spectral acceleration corresponding to a 10% probability of exceeding the in-plane strength of the roof deck was determined to be 0.35g.  Since the design base shear used for design of the roof deck diaphragm was 0.68g, the design of the roof deck for in-plane forces does not meet the acceptance criteria adopted within this study. In order to meet the acceptance criteria, the strength of the roof for in-plane forces would have to be increased by approximately 95%. The median first mode spectral acceleration at which the out-of-plane panel to roof connections would fail was determined from data used for Figure 4.8 (c) to be 1.47g.  Based on the adjusted lognormal cumulative distribution shown in Figure 4.9 (c), the spectral acceleration corresponding to a 10% probability of exceeding the in-plane strength of the roof deck was determined to be 0.61g. Using the design base shear for the building as a means of comparison (0.68g), the design of the out-of-plane wall to roof connections does not meet the acceptance criteria adopted within this study.  The strength of the connections would have to be increased by approximately 10% to meet the acceptance criteria. To summarize, the current RdRo values used with a connection layout commonly used in practice lead to results that are within the acceptance criteria adopted within this study for maximum roof drift.  With the current design, the in-plane diaphragm resistance would have to be increased by 95% and the wall-to-roof connections for out-of-plane loading would have to be increased by approximately 10%. 4.3.4 Comparison of IDA Results from Rocking, Frame and Eccentric Models Figure 4.10 below compares median IDA results for the rocking, frame, and eccentric models. Chapter 4 Quantification of Seismic Performance Factors   128   Figure 4.10  Comparison of IDA Results for the Rocking Model, Frame Model, and Eccentric Model: (a)Drift at End Walls with Openings; (b)Drift at Centre of Roof (c)In- Plane Deck Forces; (d) Out-of-Plane Deck Forces There was not much difference in the results for the wall drift between the frame building and the eccentric frame building.  This is likely due to the greatly increased stiffness and strength of (a) (b) (d) (c) Chapter 4 Quantification of Seismic Performance Factors   129  the frame panels resulting from the connection of adjacent frame panels together. The walls did not experience considerable yielding until very high earthquake intensities were applied.  Also, due to the connection layout between adjacent panels, there was limited non-linear response, since ultimate strains in the rebar were attained quickly once yielding occurred. It is interesting to note the differences in the roof responses between the different models.  The roof displacement for the eccentric model is less than that for the frame model.  This is thought to occur because the eccentric building has one end wall that maintains stiffness as the frame panel’s yield, thus reducing displacements at the centre of the roof.  It is evident in Figure 4.10 (c) that the end wall with solid panels attracts more load (higher in-plane deck forces) than the end wall with frame panels.  As expected, the in-plane deck forces for the rocking model are significantly less than for the frame and eccentric models, due to the lower yield base shear exhibited by the rocking model. There are also differences in the out-of-plane wall to roof connection forces for the different models.  The rocking model experiences higher out-of-plane wall to roof connection forces than the other two models, though the in-plane forces decrease.  The rationale for these observations is discussed previously in Section 4.2.3.  This result suggests that an increase in the end wall strength results in lower out-of-plane wall to roof connection forces.   130  5 CONCLUSIONS AND RECOMMENDATIONS The research discussed within this report consists of an analytical study to investigate the seismic performance of single-story tilt-up structures with steel deck roof diaphragms.   Within this section, a summary of observations from this study is provided and recommendations are included. 5.1 Summary of Observations A review of current practice in North America for the seismic design of tilt-up structures revealed the following points of interest: • The calculation of the fundamental building period for determination of the design base shear does not include any consideration of the flexibility of the roof diaphragm, but instead is based solely on the height of the concrete wall panels. • Design for seismic loading is carried out using a strictly force-based approach.  Sufficient panel to roof, panel to panel and panel to slab connections are provided to accommodate the design forces, but there is no explicit consideration of a stable energy-dissipation mechanism that would be expected to occur upon failure of the connections.  This is inconsistent with capacity-design approaches used in current practice for seismic design of other building systems, such as moment frames, braced frames, and shear walls. Some possible energy-dissipation mechanisms for tilt-up structures were investigated and compared, including rocking of wall panels, sliding of wall panels, and frame action for buildings with wall panels incorporating large openings.  In comparing rocking and sliding mechanisms, it was determined that although the sliding mechanism provides significantly more energy dissipation, and hence lower drift demands, it would be very difficult to incorporate sliding into common applications.  Many tilt-up buildings are not perfectly rectangular; re- entrant corners are often required to accommodate a certain building layout, also some walls may not be parallel to one another.  In addition, it would be difficult to consistently estimate the Chapter 5 Conclusions and Recommendations   131  sliding resistance of a panel since it depends on the friction between the panel and footing, which is expected to vary.  Also, sliding results in permanent deformation of the wall panels, which is undesirable.   The rocking mechanism would not result in any permanent deformation of the wall panels, but would likely cause damage to the roof perimeter angle used to transfer shear and axial loads from the roof diaphragm to the tilt-up walls.  Damage is expected due to differential vertical movement at the interface between adjacent panels. One possible way to mitigate this problem would be to stop the roof perimeter angle at the embedded connectors in the wall panels nearest the panel to panel interface and provide a pin ended connector angle between panels. Investigation of the frame mechanism was carried out by considering a tilt-up building incorporating panels with large openings, with sufficient connections provided to resist design forces. It was found that connections between adjacent panels provided sufficient shear transfer between panels to result in legs of adjacent panels effectively acting as a single member.  For panels with openings, this led to flexural strengths double the design values, and in turn, higher in-plane base shear resistance for the overall building. However, the connection of legs also resulted in lower deformation capacity of the legs, leading to a potentially undesirable brittle failure mode. One limitation in the modeling of the frame mechanism was that the panel to slab connections were modeled as a pinned base due to lack of information on the slab to wall connector (Devine 2009). Results from nonlinear analysis using building models prepared for this study were used to evaluate findings from previous research on tilt-up buildings.  The key findings are as follows: • The fundamental period determined from the Perform 3D building model was compared to the formula recommended by ASCE 41-06 (refer to Equation 3.1).  The ASCE 41-06 equation takes into consideration wall and roof displacements resulting from a uniformly distributed lateral load equal to the weight of the roof diaphragm.  It was determined that if the equation was modified by adding half the weight of the out-of-plane walls to the lateral load used to calculate wall and roof displacements, there was a good match between the ASCE 41-06 equation and the first mode period determined from Perfrom 3D (within 10%). Chapter 5 Conclusions and Recommendations   132  • In previous analytical research (Adebar et al., 2004) tilt-up structures were modelled as a two component series system, one component representing the walls while the other represents the roof.  Adebar et al. observed that if one component in the system yields, the displacement of the yielding component comprises a progressively larger portion of the total displacement as the intensity of the applied loading is increased.   This behaviour was also observed in the current study. • In previous analytical research on tilt-up structures with large openings (Adebar et al., 2004), an expression to estimate the inelastic displacement demands for legs of frame panels was proposed (Equation 3.2).  This expression was plotted against data from the current study and was found to fit the data reasonably well.  However, further study should be carried out with alternative arrangements / models to provide further verification for this expression. • For design, seismic loads on the roof diaphragm are typically assumed to have a uniform distribution. ASCE 41-06 recommends a parabolic seismic load distribution for flexible roof diaphragms (Equation 3.3).  The results from this study indicate that using an envelope of the uniform distribution and a parabolic distribution would best match the analysis results. The rocking mechanism and frame mechanism were studied further by incorporating them into the design of a typical single story tilt-up structure and carrying out a preliminary assessment of the performance of the structure utilizing concepts from the ATC-63 Methodology.  Two archetypical systems were established, one which included solid panels at end walls and one that incorporated wall panels with large openings at end walls.  An eccentric model which consisted of a variation of the two systems was also considered, in which one end wall was comprised of panels with large openings, while the other was comprised of solid wall panels.  The following conclusions were obtained from this portion of the study: • For the rocking mechanism, an RdRo value of 2.1 was found to result in a collapse probability of less than 0.1 for the design earthquake. At the target collapse probability, the in-plane demands on the roof diaphragm were found to be approximately 25% higher than the nominal strength, while the out-of-plane demands on the wall to roof Chapter 5 Conclusions and Recommendations   133  connections were found to be approximately 90% higher than the nominal strength of the connections.  This suggests that yielding of the steel deck diaphragm is expected and a nonlinear model of the deck must be used to adequately determine the appropriate R- factor for tilt-up structures. This analysis is beyond the scope of the current study. • For the frame mechanism and for the eccentric model, an RdRo value of 2.0 (based on Canadian practice) for flexural design of the frame panels with a connection layout based on current practice was found to result in a collapse probability of less than 0.1 for the design earthquake. At the target collapse probability, the in-plane demands on the roof diaphragm were found to be approximately 80% higher than the nominal strength for both models.  The out-of-plane demands on the wall to roof connections were found to be approximately 15% higher than the nominal strength of the connections for the frame model, and 10% higher for the eccentric model. Similar to the rocking mechanism, yielding of the steel deck diaphragm is expected and a nonlinear model of the deck must be used to adequately determine the appropriate R-factor for tilt-up structures with openings. • When the results from the three different models were compared, it was evident that the in-plane deck forces tended to increase as the in-plane strength of the end walls increased.  This result was expected, since yielding of the walls would protect the deck from increasing seismic demands. • When the results from the three different models were compared, it was evident that the out-of-plane wall to roof connection forces tended to decrease as the in-plane strength of the end walls increased.  This result may seem counterintuitive considering that the in- plane deck forces tend to increase as the strength of the end walls increases.  Two points to take into account in consideration of this result are: o In the NBCC 2005 Clause 4.1.8.17 regarding seismic demands on elements of structures, non-structural components and equipment, the more flexible a component is, the higher the amplification value, Ar, ascribed to it (refer to Section 1.2.1).  Since the panels are essentially components of the building, it is Chapter 5 Conclusions and Recommendations   134  reasonable that more flexible end walls result in higher out-of-plane panel to deck forces. o The panels often do not oscillate in phase in the out-of-plane direction.  In many instances, some panels are flexing in one direction while others are flexing in the opposite direction.  This is due to the difference in the roof stiffness (essentially the out-of-plane lateral support for the wall panel) between the edges of the building and the middle of the building.  Thus it is reasonable that even though the out-of-plane wall forces may be higher at certain locations, this may not result in higher in-plane forces overall. 5.2 Recommendations and Future Research Based on the observations from the current study discussed above, the following recommendations are made: • The building period used to establish seismic loading for design of tilt-up structures should more accurately reflect the actual period of the building and should incorporate the effects of a flexible roof diaphragm if a metal deck roof diaphragm is used.  ASCE 41-06 has recommended a simple expression (Equation 3.1) to estimate the fundamental period of a single story building with a flexible roof diaphragm.  This equation, with the slight modification to include half the weight of the out-of-plane walls in the tributary weight of the building, matches well with results from analyses from this study, and should be incorporated in design. • Analysis results from this study indicated that the distribution of seismic loading on the roof diaphragm can be more accurately modeled using an envelope of the uniform shear distribution currently used in common North American practice and a parabolic distribution, as recommended in ASCE 41-06. • Consideration of a stable mechanism should be incorporated into the seismic design of tilt-up structures.  One mechanism investigated within this study was rocking of wall panels in-plane.  A preliminary assessment of an appropriate RdRo value indicated that this mechanism could be used with an RdRo value of approximately 2 to achieve an Chapter 5 Conclusions and Recommendations   135  acceptably low probability of collapse for the design earthquake, however further analysis including nonlinearity of the steel deck diaphragm are required prior to the final selection of an RdRo value for rocking.  This mechanism should be investigated further with additional analytical studies and physical testing prior to application in design.  Two aspects that would need to be addressed are: o Design of two sets of panel to slab connections to allow controlled rocking of a wall panel: one set that would provide some uplift resistance without any lateral resistance, and one set that would provide lateral resistance only, without any uplift resistance.  De-coupling of the panel to slab connection behaviour in the horizontal and vertical directions would be necessary for the rocking mechanism to be incorporated. A potential design for these connections has been described in Section 3.6. o Connection details between the wall panel and roof to minimize the damage to the roof diaphragm due to rocking of panels need to be investigated.  A potential design for these connections has been described in Section 3.6. • A frame mechanism was investigated to assess the adequacy of current seismic design practices for tilt-up buildings incorporating panels with large openings.  A preliminary assessment of an appropriate RdRo value indicated that the current RdRo value used in design in Canada (relative to the seismic demand based on the code period) is adequate to achieve an acceptably low probability of collapse.  However, there was a significant limitation in the modeling in that there was insufficient information available on the behaviour of panel to slab connections to accurately model them.  Testing is currently being carried out to investigate their behaviour (Devine, 2008).  Once the experimental program has been completed, further analytical studies should be carried out, incorporating the behaviour of the panel to slab connections. • Analysis results from this study indicated that if either the frame mechanism or rocking mechanism were to be incorporated into the seismic design of a tilt-up structure, the in- plane strength of the roof diaphragm would have to be increased to ensure the mechanism could form prior to yielding of the diaphragm.  Alternatively, yielding of the steel deck Chapter 5 Conclusions and Recommendations   136  diaphragm should be considered in design. For the frame mechanism, the strength of the out-of-plane connections between the wall panels and the roof diaphragm would also have to be increased. • Analysis results from this study indicated that the demands on the out-of-plane panel to roof connections decreased as the strength of the in-plane walls increased.  Also, the demand was higher for panels closer to the centre of the building.   Further studies should be carried out to determine the appropriate design loads for the out-of-plane connections between the wall panels and the roof diaphragm.  137  6 REFERENCES  American Concrete Institute (ACI), 2005, “Building Code Requirements for Structural Concrete”, ACI 318-05, ACI, USA.  Adebar, P., Guan, Z., and Elwood, K., (2004) “Displacement-Based Design of Concrete Tilt-up Frames Accounting for Flexible Diaphragms”, Proceedings of the 13th World Conference on Earthquake Engineering, Vancouver, BC, Canada.  Adebar, P., Gerin, M. (2004) “Accounting for Shear in Seismic Analysis of Concrete Structures,” Proceedings of the 13th World Conference on Earthquake Engineering, Vancouver, BC, Canada.  ASCE/SEI 41-06 (2007) Seismic Rehabilitation of Existing Buildings, American Society of Civil Engineers, Reston, Virginia.  ATC-63 (2008) “Quantification of Building Seismic Performance Factors: 90% draft” Applied Technology Council, Redwood City, California.  Baker, J.W. and Cornell, C.A., (2006), “Spectral Shape, Epsilon and Record Selection”, Earthquake Engineering & Structural Dynamics, 34 (10) 1193-1217.  Bentz, E., (2001), “Response-2000 User Manual, Version 1.1,” Dept. of Civil Engineering, University of Toronto, Toronto, Ontario, Canada. References   138   Canadian Sheet Steel Building Institute (CSSBI), 2006, “CSSBI B13-06, Design of Steel Deck Diaphragms, 3rd Edition”, CSSBI, Canada  Canadian Standard Association (CSA), 2004, “Design of Concrete Structures”, CSA-A23.3-94, CSA, Canada.   Cement Association of Canada (CAC), 2006, “Concrete Design Handbook, 3rd Edition”, CAC, Canada.  Chopra, A.R., 2000, “Dynamics of Structures: Theory and Applications to Earthquake Engineering, Second Edition”, Prentice Hall.  Computers and Structures Inc. (CSI) (2006) “Perform3D Ver. 4.0.3 User Guide”, Berkeley, California.  Devine, F. (2008) “Investigation of Concrete Tilt-up Wall to Base Slab Connections,” M.A.Sc. Thesis, Department of Civil Engineering, University of British Columbia, Vancouver, Canada.  Dew, M., Sexsmith, R., Weiler, G. (2001), “Effect of Hinge Zone Tie Spacing on Ductility of Concrete Tilt-up Frame Panels,” ACI Structural Journal, Title n. 98-S78, pp. 823-833.  Easley, J.T., and McFarland, D.E. (1969), “Buckling of Light-Gage Corrugated Metal Shear Diaphragms,” J. Struct. Div., ASCE, 95(7), 1497-1516.  References   139  Easley, J.T., (1977), “The Effect of Cladding Panels in Steel Buildings under Seismic Conditions,” Univ. Degli Studi Napoli Federico II, Italy, 247 pp.  Essa, H.S., Tremblay, R., and Rogers, C.A. (2003), “Behaviour of Roof Deck Diaphragms under Quasi-Static Cyclic Loading,” Journal of Structural Engineering, 129(12), 1658-1666.  Hamburger, R.O., and McCormick, D.L. (1994) “Implications of the January 17, 1994, Northridge Earthquake on Tilt-up and Masonry Buildings with Wood Roofs.” Structural Engineers Association of Northern California (SEAONC), Seminar Papers. San Francisco, CA, pp. 243-255.  Hawkins, N.M., Wood, S.L., Fonseca, F.S., (1994), “Evaluation of Tilt-up Systems”, Fifth U.S. National Conference on Earthquake Engineering, Proceedings, Vol. 3, pp. 687-696.  Hilti Corporation (2006), “Help Manual, Hilti Profis DF Diaphragm Design for Steel Decks Program, Version 1.0”, Document Version 1.0.7, Schaan, Liechtenstein.  Hilti Corporation (2008), “Steel Deck Diaphragms Attached with Hilti X-EDNK22 THQ12, X- EDN19 THQ12 or X-ENP-19 L15 Power-Driven Fasteners and Hilti S-MD 12-14x1 HWH Stitch Sidelaps Connectors – Draft Report”, Schaan, Liechtenstein.  Ibrahim, A.M.M., Adebar, P. (2004), “Effective Flexural Stiffness for Linear Seismic Analysis of Concrete Walls,” Canadian Journal of Civil Engineering, Vol. 31, No. (4), pp. 597-607  International Code Council (ICC), 2006, “International Building Code”, ICC, USA  References   140  Klingler, C.J. (1986), “The Strength and Flexibility of Mechanical Connectors in Steel Shear Diaphragms,” MSc Thesis, West Virginia Univ., Morgantown, W. Va.  Lemay, C., and Beaulieu, D. (1986), “Ètude Expèrimental Sur Des Assemblages De Toitures Mètalliques Sollicitèes en Cisaillement,”  Rep. No. GCI-8604, Universitè Laval, Quèbec (In French)  Lemieux, K., Sexsmith, R., and Weiler, G. (1998) “Behavior of Embedded Steel Connectors in Concrete Tilt-up Panels,” ACI Structural Journal, V. 95, No. 4, July-August, pp. 400-411.  Luttrell, L.D., and Ellifritt, D.S. (1970), “Behaviour of Wide, Narrow, and Intermediate Rib Roof Deck Diaphragms,” Preliminary Rep,-Phase II, West Viginia Univ, Morgantown, W.Va.  National Research Council of Canada (NRCC), 2005, “National Building Code of Canada (NBCC), Part 4”, NBCC 2005, NRCC, Canada.   Nilson, A.H. (1960), “Shear Diaphragms of Light-Gage Steel”, J. Struct. Div. ASCE, 86(11), 111-139.  PEER, 2006, PEER NGA Database, Pacific Earthquake Engineering Research Centre, University of California, Berkeley, California.  Portland Cement Association (PCA), 1987, “Connections for Tilt-up Wall Construction”, PCA, USA.  References   141  S.B. Barnes and Associates (1973), “Seismic Design for Buildings”, TM 5-809-10/NAV FAC P- 355/AFM 88-3, Chap 13, Departments of the Army, the Navy and the Air-Force, USA.  Steel Deck Institute (SDI), 2004, “Diaphragm Design Manual, 3rd Edition”, SDI, Fox River Grove, III, USA.  Tilt-up Concrete Association (TCA), 2005, “The Tilt-up Construction and Engineering Manual, 6th Edition”, Mount Vernon, Iowa, USA.  Tremblay, R., Martin, E., and Yang., W., (2003), “Analysis, Testing and Design of Steel Roof Deck Diaphragms for Ductile Earthquake Resistance”, Journal of Structural Engineering, Vol. 129, No. 12, December, 2003, pp. 1658-1666  Vamvatsikos, D. and Cornell, C.A. (2002) “Incremental Dynamic Analysis”, Earthquake Engineering and Structural Dynamics, 31: 491-514.  Weiler Smith Bowers, Sample Drawings for Tilt-up Building, 2008, Vancouver, B.C., Canada.            142  APPENDIX A. ANALYSIS OF CONVENTIONAL BUILDING Modelling Connections Types: As described in Concrete Design Handbook there are various types of standardized connections used in Tilt-up industry across Canada. This study will focus on panel-to-panel (EM5 to EM5) and panel to slab (EM3 to EM5) connections. Backbone Curves: Backbone curves were produced based on combining test results for individual connections to simulate connectors in series.  Piecewise linear functions were used to approximate test results. The backbone curve for a panel-to-panel connection (EM5 to EM5) is shown in Figure 1 below.            Figure A1: Panel to Panel (EM5-EM5) Shear Panel to Panel (EM5-EM5 shear-shear combined) 0 50 100 150 200 250 300 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Displacement (m) Fo rc e (kN ) Appendix A Analysis of Conventional Building   143    The EM5 to EM5 connection was modelled directly in Perform using a beam element between panels with a shear force-displacement spring incorporated. The backbone curve for a panel-to-slab connection (EM3 to EM5) for forces and displacements in the horizontal direction is shown in Figure 1 below.  The labels in red are used in describing the model of the element.  Figure A2:  Panel to Slab EM3-EM5 Horizontal  There is no test data for EM3-EM5 connections in the vertical (uplift) direction or for interaction between strength and stiffness in the vertical and horizontal directions.  Panel to Slab (EM3-EM5 horizontal combined) 0 50 100 150 200 250 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 Displacement (m) Fo rc e (kN ) Dh2 Phu Dh1 Phe Dh3 Appendix A Analysis of Conventional Building   144   The strength in the vertical direction was approximated as follows: • A failure cone was assumed as shown in Figure3 below, in which the bar bends approximately 50mm from the edge of the angle. • The pullout strength of the resulting concrete wedge is approximately as follows: o Pullout Area Ac = 70mm*100mm – 50mm*20mm = 6000 mm2 o Concrete Strength, fc’ = 30MPa    Figure A3:  Sketch of Assumed Concrete Pullout Failure in the EM5 Connector for Forces in the “Vertical” Direction Dh = Ac*0.3*(fc’)0.5 = 10 kN Appendix A Analysis of Conventional Building   145  • The bending strength of the bars is approximately as follows: o Moment Resistance of Bars,   o Bar Yield Strength, Fy = 400Mpa o Bar Diameter, d = 16mm o Based on bending moment diagram below,  o Equate applied moment and moment resistance, therefore maximum applied force,             Figure A4:  Sketch of Assumed Mechanism in the EM5 Connector for Forces in the “Vertical” Direction  Mr = Fy*Sx = Fy*pi*d3/32 = 0.16 kNm Mf = 0.0125*P  kNm P = (0.16 kNm) / (0.0125m)  = 13 kN P 0.5*P 0.5*P 50mm Mf =  0.5*P*(0.05m)/2  0.0125*P  kNm BMD 15M bar L38x38 Appendix A Analysis of Conventional Building   146  Based on the above assumptions, the strength of the EM5 connection in the vertical direction is in the order of 10% of its strength in the horizontal direction.  The backbone curve for the EM5 connection in the vertical direction can be approximated as follows: • Assume strength in the vertical direction is equal to 10% of the strength in the horizontal direction • Assume initial stiffness in the vertical direction is the same as in the horizontal direction • Assume the vertical displacement at failure 10% of the horizontal displacement at failure The figure below displays the backbone curve for the panel to slab (EM3-EM5) for forces and displacements in the vertical direction.  The labels in red are used to describe the model of the element.          Figure A5: Panel to Slab EM3-EM5 Vertical Backbone Curve Panel to Slab (EM3-EM5 vertical combined) 0 5 10 15 20 25 0.0000 0.0002 0.0004 0.0006 0.0008 Displacement (m) Fo rc e  (kN ) Pvu Dv1 Dv2 Appendix A Analysis of Conventional Building   147  Modelling Horizontal / Vertical Interaction for EM3-EM5 Panel to Slab Connection An attempt was made to model the assumed behaviour of the panel to slab (EM3 to EM5) connection and interaction between the vertical and horizontal response.  Several elements in Perform were tried but results from analyses did not reasonably represent the backbone curves. The section below describes the most promising attempt.  Interaction modeled with the following element: a) A fibre element to satisfy strength compatibility b) A bi-linear moment-rotation spring to ensure horizontal displacements match the backbone curve for horizontal behaviour. c) A bi-linear axial spring to ensure vertical displacements match the backbone curve for vertical behaviour.                Fibre Element E, I, A, Fu Pv Ph L Bi-linear Axial Spring Bi-linear Moment Rotation Spring Appendix A Analysis of Conventional Building   148   Figure A6: Element used to model vertical/horizontal interaction for EM3-EM5 connections  Properties of Fibre Element: • Bending properties used to model horizontal strength • Axial properties used to model vertical strength      Figure A7: Cross-section of fibre element  PERFORM calculates moment of inertia as follows: I = A*y2   (a) To satisfy strength compatibility between axial and moment: For axial (vertical): A PvFu u=       (b) For moment (horizontal): yA LPh yA yLPh I yMFu u * )*5.0(* * **5.0** 2 ===   (c)  Equate equation (b) and (c): uPv LPhy *5.0*=      (d)  Note that (0.5*L) used for the moment arm because the fibre element assumes constant curvature over the element length and utilizes the curvature in the middle of the element to carry out calculations.  y A/2 Appendix A Analysis of Conventional Building   149     Assume: Elastic Modulus,   E = 1.0x 107 kN/m2 Length,    L = 2.0 m Area,     A = 1.0 m2 Calculate: Distance to Neutral Axis, uPv LPhy *5.0*=  = 10 m Moment of Inertia,  2* yAI =   = 1 m4 Ultimate Material Strength, A PvFu u=   = 20 kN/m2 (for fibre element) Strain Compatibility in Fibre Element: • Assume plastic hinge at midpoint of fibre element           Dhtot= Dhe + Dhp Dvtot = Dve + Dvp Pv Ph L Appendix A Analysis of Conventional Building   150    Figure A8: Fibre element displacements  Displacements from Backbone Curves:  Horizontal Elastic, Dhe = Dh1  = 0.0048m   Plastic, Dhp = Dh3-Dh2 = 0.0022m  Vertical Elastic, Dve = Dv1  = 0.00033m   Plastic, Dvp = Dv2 – Dv1 = 0.00037m  Strain in Fibre Element due to Bending:  Plastic Rotation at Hinge, L Dh p p *5.0 =θ    = 0.00223 radians       Figure A9: Strain in Fibre Element Due to Bending   Plastic Strain in fibre, L yp p *θ ε =  = 0.011 Elastic Strain in fibre, E Fu e =ε  = 2.0 x 10 -6  θp εp y Appendix A Analysis of Conventional Building   151  Strain at which strength loss occurs, epu εεε +=  =   0.011          Strain In Fibre Element Due to Axial Loads:  Plastic Strain in fibre, L Dy p =ε  = 0.000183  Elastic Strain in fibre, E Fu e =ε  = 0.000002  Strain at which strength loss occurs, εu = εp + εe = 0.0001853 The plastic strains required to achieve the required horizontal and vertical plastic displacements are not the same.  Use the strain required to match the vertical backbone displacements in the analysis, since it is more conservative (less ductile)  Properties of Bi-linear Axial Spring • Used to model the non-linear characteristics of vertical backbone curve prior to ultimate Initial Stiffness, 1Dv PvKi e=   = 20093 kN/m Yielded Stiffness, 12 DvDv PvPv Ky ey − − =  = 0 kN/m Ratio of Yielded to Initial Stiffness, Ki Ky  = 0   Appendix A Analysis of Conventional Building   152        Since axial spring acting in series with fibre element, must account for axial stiffness of fibre element:              Figure A10:  Axial spring to model stiffness of vertical backbone  Vertical displacement in fibre element at yield, AE LPvDv efe * * =  = 4.0 x 10-6 m Panel to Slab (EM3-EM5 vertical combined) 0 5 10 15 20 25 0.0000 0.0002 0.0004 0.0006 0.0008 Displacement (m) Fo rc e  (kN ) Dvf u Fibre Element linear axial spring Appendix A Analysis of Conventional Building   153  Axial stiffness of fibre element, L AEKfibre *=  = 5.0 x 106  kN/m Stiffness for spring, KfibreKi Ks 11 1 + =   = 60729 kN/m  Check Yield displacement, Ks PvDvDv efe +=1  = 0.00033m   OK   Properties of Bi-linear Moment-Rotation Spring • Used to model the non-linear characteristics of horizontal backbone curve prior to ultimate Initial Stiffness, 1Dv PvKi e=   = 60444 kN/m Yielded Stiffness, 12 DvDv PvPv Ky ey − − =  = 25098 kN/m Ratio of Yielded to Initial Stiffness, Ki Ky  = 0.415  Since bi-linear moment-rotation spring acting in series with fibre element, must account for bending stiffness of fibre element: Appendix A Analysis of Conventional Building   154   Figure A11:  Bi-linear moment-rotation spring to model stiffness of horizontal backbone  Horizontal disp in fibre element at initial yield, Dhfe = θe*(0.5*L) = 2.72 x 10-7 m   Rotation at initial yield, y L e e *εθ =   = 2.72 x 10–7 rad Strain at initial yield, yAE LPhe e ** *5.0* =ε  = 1.36 x 10-6  Horizontal disp in fibre element at ultimate load, )*5.0(* LuDh fu θ=  = 4.0 x 10-7 m Panel to Slab (EM3-EM5 vertical combined) 0 50 100 150 200 250 0.00 0.00 0.00 0.01 0.01 Displacement (m) Fo rc e  (kN ) Dhf Fibre Element Bi-linear Moment- Rotation spring Appendix A Analysis of Conventional Building   155    Rotation at initial yield, y L u u *εθ =   = 4.0x 10–7 rad   Strain at initial yield, yAE LPhu u ** *5.0* =ε  = 2.0 x 10-6 Horizontal stiffness of fibre element, fu u Dh PhKfibre =   = 5.00 x 108 kN/m Initial Stiffness for spring, KfibreKi Ksi 11 1 + =   = 60452 kN/m  Yielded Stiffness for spring, KfibreKy Ksy 11 1 + =   = 25099 kN/m Relationship between rotational and horizontal stiffness: Horizontal Stiffness, 2 * * L M L L M Dh PhKs fu u θ θ ===  Solve for Moment, M = Ks*θ*L2       Figure A12:  Relationship between horizontal and rotational stiffness  Initial rotational stiffness, 2* LKsiMKsi == θθ  = 241807 kNm/rad Yielded rotational stiffness, 2* LKsiMKsy == θθ   = 100397 kNm/rad Check Yield displacement, Ksi PhDhDh efe +=1  = 0.00225 m Dh Ph L θ  Appendix A Analysis of Conventional Building   156  Check Ultimate displacement, Ksi PhPh Ksi PhDhDh euefe )(2 −++=  = 0.0048m Ratio of Yielded to Initial Stiffness, Ki Ky  = 0.415 A pushover analysis of an individual element was carried out in Perform.  The element was pushed individually in the two directions of interest in order to check the hysteretic response.                 Figure A13:  Comparison of results from connection model and vertical backbone curve     EM3-EM5 Vertical Hysteresis -25 -20 -15 -10 -5 0 5 10 15 20 25 -0.0010 -0.0005 0.0000 0.0005 0.0010 Displacement (m) Fo rc e (kN ) Analysis Results Backbone Curve Appendix A Analysis of Conventional Building   157                   Figure A14:  Comparison of results from connection model and horizontal backbone curve  As can be seen from above figures, there is good agreement between the model and the backbone curves.  The envelope of hysteretic response matches closely with the backbone curves. The ductility of the vertical backbone is captured, but the full ductility of the horizontal backbone is not, due to the incompatibility in strains described in the section above.  EM3-EM5 Horizontal Hysteresis -200 -150 -100 -50 0 50 100 150 200 250 -0.010 -0.005 0.000 0.005 0.010 Displacement (m) Fo rc e (kN ) Analysis Results Backbone Curve Appendix A Analysis of Conventional Building   158  Testing of Elements Used to Model EM3-EM5 Panel to Slab Connections with a 2-Panel Model  Model Description In the two-panel model, five EM5-EM5 connectors were used between panels to ensure that there would be little deformation between panels.  Three EM3-EM5 connectors were used per panel, incorporating the assumptions described in the section above.  The stiffness of the panels was increased so that the panels were essentially rigid. The figure below shows a sketch of the two-panel model. The panel to slab connectors are numbered c1 to c6 for tracking the connector response.  Figure A15: Sketch of two-panel model   Loading Protocol: The model was subjected to a point load at a height of 8.5m and pushed to a roof drift of 0.002, then pushed in the opposite direction to a roof drift of –0.002, and then back to zero.  The cycle was then repeated.  The figure below is a screen capture from the PERFORM model at a roof drift of 0.002. c1 c3 c4 c5 c6 c2 Appendix A Analysis of Conventional Building   159          Figure A16: Screen capture of two-panel model at drift of 0.002 from PERFORM  EM3-EM5 Connector Response: The response of connectors c6 and c4 were tracked using three parameters: • Force vs. drift both in the horizontal and vertical directions • Force vs. connector displacement both in the horizontal and vertical directions • Horizontal vs. vertical force  Appendix A Analysis of Conventional Building   160  The figures below illustrate the response.           Figure A17: Connector C6 – Horizontal Force vs. Drift and Horizontal Force vs. Displacement           Figure A18: Connector C6 – Vertical Force vs. Drift and Vertical Force vs. Disp Connector C6: EM3-EM5 Horizontal Force vs. Displacement -20 -10 0 10 20 30 40 50 60 70 -0.015 -0.010 -0.005 0.000 Displacement (m) Fo rc e  (kN ) Connector C6: EM3-EM5 Horizontal Force vs. Drift -20 -10 0 10 20 30 40 50 60 70 -0.0030 -0.0020 -0.0010 0.0000 0.0010 0.0020 0.0030 Drift Fo rc e  (kN ) Connector C6: EM3-EM5 Vertical Force vs. Drift -5 0 5 10 15 20 -0.002 -0.002 -0.001 -0.001 0.000 0.001 0.001 0.002 0.002 Drift Fo rc e  (kN ) Connector C6: EM3-EM5 Vertical Force vs. Displacement -5 0 5 10 15 20 -0.001 0.000 0.001 0.001 0.002 0.002 0.003 Displacement (m) Fo rc e  (kN ) Appendix A Analysis of Conventional Building   161                Figure A19: Connector C6 – Horizontal Force vs. Vertical Force  Discussion: In Figures 15 and 16, it can be observed that when the model is pushed in the first direction (to a drift of 0.002), the EM3 – EM5 element behaves reasonably well.  The response in the vertical and horizontal directions corresponds with the backbone curves until a strength limit is reached due to the interaction between the two. This occurs at a horizontal shear force in the connector of approximately 13 kN (ultimate strength = 200kN) and a vertical force of approximately 19 kN (ultimate strength = 20kN).  This indicates the element obeys the interaction constraints, since 13/200 + 19/20 = 1.0. Once the element reaches this strength limit, both the horizontal and vertical forces drop down to zero. Connector C6: EM3-EM5 Horizontal Force vs. Vertical Force -20 -10 0 10 20 30 40 50 60 70 -5 0 5 10 15 20 Vertical Force (kN) H o riz o n ta l F o rc e  (kN ) Appendix A Analysis of Conventional Building   162  Problems arise when the model is pushed in the opposite direction (to a drift of -0.002).  The element strength should have been reduced to zero as a result of the failure that occurred when the model was pushed in the first direction.  However, it can be seen (on the left side of the curves in Figure 15) that the horizontal force resisted by the element increases, and not with the original stiffness of the element, but with a stiffness that cannot be explained.  Also, the force displacement plot exhibits plateaus and jumps that cannot be readily explained.  The same behaviour can be observed in the vertical direction.            Figure A20: Connector C4 – Horizontal Force vs. Drift and Horizontal Force vs. Displacement      Connector C4: EM3-EM5 Horizontal Force vs. Drift -150 -100 -50 0 50 100 150 -0.0020 -0.0015 -0.0010 -0.0005 0.0000 0.0005 0.0010 0.0015 0.0020 Drift Fo rc e (kN ) Connector C4: EM3-EM5 Horizontal Force vs. Displacement -150 -100 -50 0 50 100 150 -0.02 -0.01 -0.01 0.00 0.01 Displacement (m) Fo rc e (kN ) Appendix A Analysis of Conventional Building   163            Figure A21: Connector C4 – Vertical Force vs. Drift and Vertical Force vs. Displacement            Figure A22: Connector C4 – Horizontal Force vs. Vertical Force Connector C4: EM3-EM5 Horizontal Force vs. Vertical Force -200 -150 -100 -50 0 50 100 150 200 -5 0 5 10 15 20 Vertical Force (kN) H o riz o n ta l F o rc e (kN ) Interactio n Line Connector C4: EM3-EM5 Vertical Force vs. Drift -2 0 2 4 6 8 10 12 14 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003 Drift Fo rc e (kN ) Connector C4: EM3-EM5 Vertical Force vs. Displacement -2 0 2 4 6 8 10 12 14 -0.002 0.000 0.002 0.004 0.006 0.008 0.010 Displacement (m) Fo rc e  (kN ) Appendix A Analysis of Conventional Building   164  Discussion: Connector c4 displays the same type of behaviour as connector c6, except that the horizontal force reaches a greater value and the vertical force in the connector reaches a lesser value at the interaction point.  This is reasonable considering their respective positions. However, connector c4 displays the same unexplained behaviour when the model is pushed to a drift of –0.002. The interaction between the horizontal force and vertical force can be clearly observed in Figure 20, in which the recorded plot closely follows the predicted interaction line for the element.  Due to the problems observed with the element response, this element was not used in the analysis  165  APPENDIX B. DESIGN NOTES AND MODEL PROPERTIES FOR ARCHETYPICAL SYSTEM 1: SOLID WALL PANELS Design of  Solid Panels for Vertical and Out-of-Plane Loading        Assumptions:       1. Building located in Vancouver, B.C., site class D         References:       1. CAN/CSA A23.3       2. "Concrete Design Handbook - Third Edition" by the Cement Association of Canada  Material Properties:  Concrete material factor Φc = 0.65  Reinf Steel material factor Φs = 0.85  Member resistance factor Φm = 0.75  Reinf Steel yield strength fy = 400 MPa   Concrete compressive strength fc' = 30 MPa Modulus of Rupture fr = 0.6*(fc')^0.5   = 3.3 MPa   Concrete tangent modulus Ec = 4500(fc')^0.5   = 24648 MPa   a1 = 0.85-0.0015*fc'   = 0.805   B1 = 0.97-0.0025*fc'   = 0.895  Reinf Steel Elastic modulus Es = 200000 MPa  Concrete unit weight wc = 24 kN/m3   Wall Panel Properties:  Height h = 9144 mm  thickness t = 184 mm  initial deflection ∆o = 25 mm  check span to depth ratio h/t = 50  OK for 1 layer of reinforcing concrete cover c = 32 mm   Loading:  Roof Joist Span L = 15.24 m  Appendix B Design Notes and Model Properties for Archetypical System 1   166  wall self weight qwall = wc*t   = 4.416 kN/m2  Roof Self Wt. qroof = 1.0 kN/m2   Snow Load (NBCC 2005 Cl qsnow = Is*(Ss*(Cb*Cb*Cs*Ca)+Sr)  For tilt-up building in Vancouver: Is = Ss = Cw = Cs = Ca = 1.0, Cb = 0.8   Ss = 1.8 kN/m2   Sr = 0.2  Design Snow Load qsnow = 1.6 kN/m2         Wind Load (NBCC 2005 Cl 4.1.7) qwind = Iw*q*Ce*(Cp*Cg + Cpi*Cgi) 1 in 50 velocity pressure q = 0.48 kN/m2  Importance Factor Iw = 1    Exposure Factor Ce = 1  Exterior Gust Coeff Cp*Cg = 1.3 or -1.5 (positive denotes towards surface, neg denotes away) Interior Pressure Coeff Cpi = 0.3 of -0.45    Interior Gust Coeff Cgi = 2  Inward pressure qwind inward = 1.0*q*1.0*(1.3 + 0.45*2.0)  Outward pressure qwind outward = 1.0*q*1.0*(-1.5 - 0.3*2.0)  Inward pressure governs  Design Wind Load qwind = 1.06 kN/m2         Eccentricity ecc = t/2   = 92 mm         Seismic for Out-of-Plane forces (NBCC 2005 Cl 4.1.8.17) Vp = 0.3*Fa*Sa(0.2)*Ie*Sp*Wp  Soil Modification Factor Fa = 1.1  (For Site Class D) Spectral Acceleration at T=0.2s Sa(0.2) = 0.94 g Importance Factor Ie = 1.0   Sp = Cp*Ar*Ax / Rp  where 0.7<Sp<4.0 Component Risk Factor Cp = 1.0    Dynamic Amplification Factor Ar = 1.0 (Typically used for tilt-up but maybe unconservative) Height Factor Ax = 1+ 2*hx/hn   hx = Height of component above base   = Centre of mass of panel   hn = total height   Ax = 2    Response Factor Rp = 2.5  (for reinforced tilt-up wall panels)  Sp = 0.8  Appendix B Design Notes and Model Properties for Archetypical System 1   167  Seismic for Out-of-Plane forces Vp = 0.248 *Wp         Compare Seismic to Wind (for 1m unit strip) Factored Seismic out-of- plane qseismic = Vp*1.0m*qwall  = 1.10 kN/m  Factored Wind 1.5*qwind = 1.58 kN/m  Wind Governs for Out-of-Plane Loading on Panels   Strength Calculations:  Load Case (4): 1.25D + 1.4W + 0.5S  Factored Load from tributary roof area Ptf = (1.25*qroof+0.5*qsnow)*L/2  = 15.8 kN/m   Factored weight of panel tributary to and above design section Pwf = 1.25*qwall*h/2   = 25.2 kN/m   Axial Load at mid-height Pf = Ptf + Pwf   = 41.0 kN/m   Factored UDL lateral load wf = 1.4*qwind   = 1.5 kN/m2   Check axial load limit Pf / Ag < 0.09*Φc*fc'   0.22 < 1.755 MPa OK         Assume reinforcing on each face (EF):         Reinforcing Steel Provided 15M @ 400 EF  Area of reinf steel (per face) As = 500 mm2/m EF  Reinforcing Steel Ratio p = 2*As/(1000*t)    (both faces)  = 0.0054    Effective depth d = t-c-db/2 mm   = 144 mm   Width of design strip b = 1000 mm   Effective steel area Ase = (Φs*As*fy + Pf*1000) / (Φs*fy)   = 621 mm2/m   Compressive stress block ae = Φs*Ase*fy / (a1*Φc*fc'*1000)   = 13.4 mm   Appendix B Design Notes and Model Properties for Archetypical System 1   168  Resisting Moment Mr = Φs*Ase*fy*(d-ae/2)/10^6   = 29.0 kNm/m         Bending Stiffness (based on As):        Compressive stress block a = Φs*As*fy / (a1*Φc*fc'*1000)   10.8 mm         Distance to neutral axis c = a/B1   = 12.1 mm   Cracked Moment of Inertia Icr = b*c3/3 + Es*As*(d-c)2/Ec   = 7.12E+07 mm4/m    Kbf = 48*Ec*Icr / 5*h2*1000   = 201.4 kNm/m   Moment Magnifier δb = 1/(1-Pf/(Φm*Kbf))   = 1.37   Primary Bending Moment Mb = wf*h2/8 + Ptf*ecc/2 + Pf*∆o   = 17.2 kNm/m   Total Moment Mf = δb*Mb   = 23.6 kNm/m    Mf < Mr  OK   23.6  29.0 kNm/m   Total factored deflection ∆f = Mf / (Φm*Kbf)   = 156 mm          Load Case (3):  1.25D + 1.5S + 0.4W   Factored Load from tributary roof area Ptf = (1.25*qroof+1.5*qsnow)*L/2  = 28.3 kN/m   Factored weight of panel tributary to and above design section Pwf = 1.25*qwall*h/2   = 25.2 kN/m   Axial Load at mid-height Pf = Ptf + Pwf   = 53.5 kN/m  Appendix B Design Notes and Model Properties for Archetypical System 1   169   Factored UDL lateral load wf = 0.4*qwind   = 0.42 kN/m2         Assume reinforcing on each face (EF):         Reinforcing Steel Provided 15M @ 400 EF  Area of reinf steel As = 500 mm2/m EF   Effective steel area Ase = (Φs*As*fy + Pf*1000) / (Φs*fy)   = 657 mm2/m   Compressive stress block ae = Φs*Ase*fy / (a1*Φc*fc'*1000)   = 14.2 mm   Resisting Moment Mr = Φs*Ase*fy*(d-ae/2)/10^6   = 30.6 kNm/m         Bending Stiffness (based on As):        Compressive stress block a = Φs*As*fy / (a1*Φc*fc'*1000)   10.8 mm         Distance to neutral axis c = a/B1   = 12.1 mm   Cracked Moment of Inertia Icr = b*c3/3 + Es*As*(d-c)2/Ec   = 7.12E+07 mm4/m    Kbf = 48*Ec*Icr / 5*h2*1000   = 201.4 kNm/m   Moment Magnifier δb = 1/(1-Pf/(Φm*Kbf))   = 1.55   Primary Bending Moment Mb = wf*h2/8 + Ptf*ecc/2 + Pf*∆o   = 7.05 kNm/m   Total Moment Mf = δb*Mb   = 10.9 kNm/m    Mf < Mr  OK   10.9  30.6 kNm/m         Deflections at Service Loads  Appendix B Design Notes and Model Properties for Archetypical System 1   170         Load Case: 1.0D + 1.0W + 0.5S       (Iw = 0.75, Is = 0.9)   Factored Load from tributary roof area Pts = (1.0*qroof+0.5*Is*qsnow)*L/2  = 13.2 kN/m   Factored weight of panel tributary to and above design section Pws = 1.0*qwall*h/2   = 20.2 kN/m   Axial Load at mid-height Ps = Pts + Pws   = 33.4 kN/m   Factored UDL lateral load ws = 1.0*Iw*qwind   = 0.79 kN/m2   Gross Moment of Inertia Ig = 1000*t3/12   = 519125333.3 mm4/m   N.A. to extreme fibre yt = t/2   92 mm   Cracking Moment Mcr = fr*Ig/yt   = 18.54 kNm/m   Primary Bending Moment Mbs = ws*h2/8 + Pts*ecc/2 + Ps*∆o   = 9.72 kNm/m   Intially assume max service deflection ∆si = h/100  = 91.44 mm         Bending Moment Ms = Mbs + Ps*∆s/1000   = 12.8 kNm/m    Ms < Mcr  Uncracked, therefore Ie = Ig  Stiffness Kbs = 48*Ec*Ie / 5*h2*1000   = 1469 kNm/m   Moment Magnifier δb = 1/(1-Ps/Kbs)   = 1.02   Total Moment Ms = δb*Mbs   = 9.9 kNm/m  Appendix B Design Notes and Model Properties for Archetypical System 1   171    Ms < Mr  OK   9.9  18.5 kNm/m   max service deflection ∆s = Ms/Kbs*1000   = 6.8 mm    ∆s < ∆si  OK         Panel Design Summary         Vertical Reinforcing 15M @ 400 EF  Steel Area As = 1000 mm2/m  Reinforcing Steel Ratio p = As/(1000*t)   = 0.0054         Horizontal Reinforcing 15M @ 400 AF (alternating faces)  Steel Area As = 500 mm2/m  Reinforcing Steel Ratio p = As/(1000*t)   = 0.0027      Design of Solid Panels for In-Plane  Loading Along Long Axis of Building   Assuming building located in Langely, B.C., site class C       Material Properties:   Concrete material factor  Φc = 0.65  Reinf Steel material factor Φs = 0.85  Member resistance factor Φm = 0.75  Reinf Steel yield strength fy = 400 MPa  Concrete compressive strength fc' = 30 MPa  Modulus of Rupture fr = 0.6*(fc')^0.5   = 3.3 MPa  Concrete tangent modulus Ec = 4500(fc')^0.5   = 24648 MPa   a1 = 0.85-0.0015*fc'   = 0.805   B1 = 0.97-0.0025*fc'   = 0.895  Reinf Steel Elastic modulus Es = 200000 MPa  Appendix B Design Notes and Model Properties for Archetypical System 1   172  Concrete unit weight wc = 24 kN/m3  Friction Coeff at Base µ = 0.5      Panel Layout:           Panel Properties:  thickness t = 184 mm  concrete cover cover = 52 mm  Panel Width b = 7620 mm  Height h = 9144 mm  Number of In-plane panels nip = 24  Number of Out of plane panels nop = 8  Total Number of panels ntot = 32  Building dimension (out of plane) Dop = 30.48 m  Building dimension (in plane) Dip = 91.44 m  Roof Area Aroof = Dop*Dip   2787 m2       Loading:  Roof Joist Span L = 15.24 m  Roof Joists in Plane? = 1 (yes = 0, no = 1) wall self weight qwall = wc*t   = 4.416 kN/m2  Roof Self Wt. qroof = 1.0 kN/m2  Snow Load qsnow = Is*(Ss*(Cb*Cb*Cs*Ca)+Sr) For tilt-up building in Langley:      Is = Ss = Cw = Cs = Ca = 1.0, Cb = 0.8   Ss = 2.4 kN/m2   Sr = 0.2   qsnow = 2.1 kN/m2   Roof Weight per panel Wr = qroof*b*L/2   = 58 kN/panel   Appendix B Design Notes and Model Properties for Archetypical System 1   173  Panel Weight Wp = qwall*(b*h)   = 308 kN/panel   Total Building Weight W = qroof*Aroof + ntot*Wp  = 12633 kN   Base Shear Calculation for Rd, Ro = 1.0 (NBCC 2005)   Building Period T = 0.05*h3/4   = 0.26 s        T<2.0 seconds, and R=1.5, Therefire 2/3*S(0.2) applies        Spectral Acceleration S(0.2) = 1.1 g  Soil Modification Factor Fa = 1   Importance Factor I = 1.0   Higher Mode Factor Mv = 1   Seismic Acceleration E = 0.66*I*S*Fa*Mv / (Rd*Ro)  = 0.73 g   Force from Roof Diaphram Vroof = ((qroof+0.25*qsnow)*Aroof + Wp*nop*0.5)*E*0.5  = 1995 kN per one wall line  Panel Shear Vpanel = E*nip*Wp*0.5   2681 kN per one wall line       Total Base Shear Vf = Vroof +Vpanel   = 4675 kN per one wall line       Check Maximum Concrete Shear (Rd, Ro = 1)   Max concrete shear Vrmax = Φc*fc'*t*dv   dv = 0.8*Dip   = 73 m   Vrmax = 262469 kN   Vrmax > Vf OK   Panel Shear Resistance Vc = Φc*B1*0.18*(fc')0.5*t*dv   = 7720 kN        Steel: 15M @ 400 AF  Steel area Av = 200 mm2  spacing s = 400 mm    Vs = Φs*Av*fy*dv*cotθ/s   = 0.486*Av*dv/s    (for θ=35 deg, Φs=0.85, Appendix B Design Notes and Model Properties for Archetypical System 1   174  fy=400MPa)  = 17776 kN        Total Shear Strength Vr = Vc+Vs   = 25496 kN   Vr > Vf OK    Panel Sliding (Base Connections)  Assume welded connections with studded embedments (EM3 in walls to EM5 in slab) Ductility factor Rd = 1 For EM3  Overstrength Ro = 1.3  Required Base Shear for Connector design Vfreqd = Vf/(Rd*Ro) - friction  Vf/(Rd*Ro) - µ*Wp*nip/2   1750 kN per one wall line  Shear resistance VrEM3 = 110 kN / connection in panel (GOVERNS)  VrEM5 = 125 kN / connection in floor slab        No. of connections required per panel:   N = Vfreqd/(VrEM3*nip/2)   = 1.3        PROVIDE 2 BASE CONNECTIONS PER PANEL    Panel Overturning (Wall to Wall Connections)  Assume welded connections with rebar embedments (EM5 to EM5 in walls) Ductility factor Rd = 1.5   Overstrength Ro = 1.3        Overturning Moment Mof = (Vroof*h + Vpanel*h/2) / (Rd*Ro)  = 15639 kNm        Resisting Moments  Panel and Roof Weight Mrweight = (nip/2)*(Wp+Wr)*b/2   = 16723 kNm        End connections Vr = 100 kN    Mrend = Vr*b   = 762 kNm   Total Resisting Moment Mr = Mrweight+Mrend   = 17485 kNm  Appendix B Design Notes and Model Properties for Archetypical System 1   175   Required connection force Vfconn = (Mof - Mr)/(b*(nip/2-1))   = -22 kN/panel   Shear Resistance VrEM5 = 125 kN / connection  No. of connections required  = Vfconn / VrEM5   = -0.2 connections per panel       PROVIDE 0 WALL TO WALL CONNECTIONS PER PANEL        Appendix B Design Notes and Model Properties for Archetypical System 1   176     Design of Solid Panels for In-Plane Loading  on Short Axis of Building        Assumptions:      1. Building located in Vancouver, B.C., site class D        References:  1. CAN/CSA A23.3      2. "Concrete Design Handbook - Third Edition" by the Cement Association of Canada       Material Properties:   Concrete material factor  Φc = 0.65  Reinf Steel material factor Φs = 0.85  Member resistance factor Φm = 0.75  Reinf Steel yield strength fy = 400 MPa  Concrete compressive strength fc' = 30 MPa Modulus of Rupture fr = 0.6*(fc')^0.5   = 3.3 MPa  Concrete tangent modulus Ec = 4500(fc')^0.5   = 24648 MPa   a1 = 0.85-0.0015*fc'   = 0.805   B1 = 0.97-0.0025*fc'   = 0.895  Reinf Steel Elastic modulus Es = 200000 MPa  Concrete unit weight wc = 24 kN/m3  Friction Coeff at Base µ = 0.5      Panel Layout:           Panel Properties:  Appendix B Design Notes and Model Properties for Archetypical System 1   177  thickness t = 184 mm  concrete cover cover = 52 mm  Panel Width b = 7620 mm  Height h = 9144 mm  Number of In-plane panels nip = 8  Number of Out of plane panels nop = 24 Total Number of panels ntot = 32  Building Length (out of plane) L = 60.96 m  Building Width (in plane) w = 30.48 m  Roof Area Aroof = L*w   1858 m2       Loading:  Roof Joist Span Ljoist = 15.24 m  Roof Joists in Plane? = 0 (yes = 0, no = 1) wall self weight qwall = wc*t   = 4.416 kN/m2  Roof Self Wt. qroof = 1.0 kN/m2  Snow Load qsnow = Is*(Ss*(Cb*Cb*Cs*Ca)+Sr) For tilt-up building in Vancouver: Is = Ss = Cw = Cs = Ca = 1.0, Cb = 0.8   Ss = 1.8 kN/m2   Sr = 0.2   qsnow = 1.6 kN/m2   Roof Weight per panel Wr = qroof*b*Ljoist/2   = 0 kN/panel  (No roof weight if joists spanning in plane)  Panel Weight Wp = qwall*(b*h)   = 308 kN/panel   Total Building Weight W = qroof*Aroof + ntot*Wp  = 11704 kN   Base Shear Calculation for Rd, Ro = 1.0 (NBCC 2005)   Building Period T = 0.05*h3/4   = 0.26 s        T<2.0 seconds, and R=1.5, Therefire 2/3*S(0.2) applies   Appendix B Design Notes and Model Properties for Archetypical System 1   178  Spectral Acceleration S(0.2) = 0.94 g  Soil Modification Factor Fa = 1.1   Importance Factor I = 1.0   Higher Mode Factor Mv = 1   Seismic Load E = 0.66*I*S*Fa*Mv / (Rd*Ro)  = 0.68 g   Force from Roof Diaphram Vroof = ((qroof+0.25*qsnow)*Aroof + Wp*nop*0.5)*E*0.5  = 2154 kN per one wall line  Panel Shear Vpanel = E*nip*Wp*0.5   840 kN per one wall line       Total Base Shear Vf = Vroof +Vpanel   = 2994 kN per one wall line       Compare with Forces Due to wind     Wind Load (NBCC 2005 Cl 4.1.7) qwind = Iw*q*Ce*(Cp*Cg + Cpi*Cgi) 1 in 50 velocity pressure q = 0.48 kN/m2  Importance Factor Iw = 1   Exposure Factor Ce = 1  Exterior Gust Coeff Cp*Cg = 1.3 or -1.5 (positive denotes towards surface, neg denotes away) Interior Pressure Coeff Cpi = 0.3 or -0.45   Interior Gust Coeff Cgi = 2  Inward pressure qwind inward = 1.0*q*1.0*(1.3 + 0.45*2.0) Outward pressure qwind outward = 1.0*q*1.0*(-1.5 - 0.3*2.0)  Inward pressure governs  Design Wind Load qwind = 1.06 kN/m2  Base shear due to wind Vw = qwind*h*L/2   = 294 kN per one wall line        Check Maximum Concrete Shear (Rd, Ro = 1) Max concrete shear Vrmax = Φc*fc'*t*dv   dv = 0.8*w   = 24 m   Vrmax = 87490 kN   Vrmax > Vf OK   Panel Shear Resistance Vc = Φc*B1*0.18*(fc')0.5*t*dv   = 2573 kN        Steel: 15M @ 400 AF  Appendix B Design Notes and Model Properties for Archetypical System 1   179  Steel area Av = 200 mm2  spacing s = 400 mm    Vs = Φs*Av*fy*dv*cotθ/s   = 0.486*Av*dv/s    (for θ=35 deg, Φs=0.85, fy=400MPa)  = 5925 kN        Total Shear Strength Vr = Vc+Vs   = 8499 kN   Vr > Vf OK    Panel Sliding (Base Connections)  Assume welded connections with studded embedments (EM3 in walls to EM5 in slab) Ductility factor Rd = 1 For EM3  Overstrength Ro = 1.3  Required Base Shear for Connector design Vfreqd = Vf/(Rd*Ro) - friction  Vf/(Rd*Ro) - µ*Wp*nip/2   1688 kN per one wall line  Shear resistance VrEM3 = 110 kN / connection in panel (GOVERNS)  VrEM5 = 125 kN / connection in floor slab        No. of connections required per panel:   N = Vfreqd/(VrEM3*nip/2)   = 3.84        PROVIDE 4 BASE CONNECTIONS PER PANEL   Panel Overturning (Wall to Wall Connections)  Assume welded connections with rebar embedments (EM5 to EM5 in walls) Ductility factor Rd = 1.5   Overstrength Ro = 1.3        Overturning Moment Mof = (Vroof*h + Vpanel*h/2) / (Rd*Ro)  = 12069 kNm        Resisting Moments  Panel and Roof Weight Mrweight = (nip/2)*(Wp+Wr)*b/2   = 4689 kNm        End connections Vr = 100 kN    Mrend = Vr*b  Appendix B Design Notes and Model Properties for Archetypical System 1   180   = 762 kNm   Total Resisting Moment Mr = Mrweight+Mrend   = 5451 kNm   Required connection force Vfconn = (Mof - Mr)/(b*(nip/2-1))   = 289 kN/panel   Shear Resistance VrEM5 = 125 kN / connection  No. of connections required  = Vfconn / VrEM5   = 2.3 connections per panel       PROVIDE 3 WALL TO WALL CONNECTIONS PER PANEL       Design of Roof Deck Diaphragm and Estimation of Nominal Capacity       Assumptions:      1. Building located in Vancouver, B.C., site class D        References:      1. Deck properties / resistances based on CANAM catalogue   2. Joist properties / resistances based on Omega Joist catalogue   3. Steel Design based on CAN/CSA G40.21, using the CISC Handbook      Building Geometry:      Thickness t = 0.184 m  Height h = 9.144 m  Building length L = 60.96 m  Building width w = 30.48 m  Roof Area Aroof = L*w   1858 m2        Material Properties:  Concrete unit weight wc = 24 kN/m3  Structural Steel Yield Strength Fy = 300 MPa  Structural Steel material factor Φs = 0.9  Structural Steel Elastic Modulus E = 200000 MPa        Loading:  Appendix B Design Notes and Model Properties for Archetypical System 1   181  Roof Self Wt. qroof = 1.0 kN/m2  Snow Load qsnow = Is*(Ss*(Cb*Cb*Cs*Ca)+Sr) For tilt-up building in Vancouver:      Is = Ss = Cw = Cs = Ca = 1.0, Cb = 0.8   Ss = 1.8 kN/m2   Sr = 0.2  Design Snow Load qsnow = 1.6 kN/m2   Specified Roof Load qspecified = qroof+qsnow   = 2.6 kN/m2  Total Factored Roof Load qf = 1.5*qsnow+1.25*qroof   = 3.71 kN/m2        Design Deck for Vertical Loads:      Refer to CANAM Catalogue. Deck profile P-3615 or P-3606 for 18ga, 38mm deep deck Deck span Ld = 1905 mm  Factored Resistance qr = 6.82 kN/m2   qr > qf  OK Allowable specified load (defl) qrs = 3.19 kN/m2   qrs > qspecified  OK                       Check Required Joist Size:      Joist Spacing Sj = 1905 mm  Joist span Ljoist = 15240 mm  Factored Unit Load Qf = qf*Sj   = 7.07 kN/m  Specified Qspecified = qspecified*Sj   = 5.03 kN/m        Required  Joist:  1050mm deep (3'-6") with Chord Combination L/L        Factored Load Capacity (strength) Qr = 12.67 kN/m  Appendix B Design Notes and Model Properties for Archetypical System 1   182   Qr > Qf  OK Specified Load Capacity (deflection) Qrs = 6.22 kN/m   Qrs > Qspecified OK Moment of Inertia Ijoist = 5.70E+08 mm4   = 5.70E-04 m4  Area of Top Chord Achord = 1095 mm2 0.194in top chord  (3/16")        Check Required Beam Size:      Assume simply supported between columns     Yield Stress Fy = 350 MPa  Material Factor phi = 0.9   Assume Beam Span Lbeam = 15240 mm  Tributary width Wbeam = Ljoist  Specified Unit Load Qspecified = Ljoist*qspecified   = 40.23 kN/m  Factored Unit Load Qf = Ljoist*qf   = 57 kN/m  Factored Shear Vf = Qf*Lbeam/2   = 431 kN  Factored Moment Mf = Qf*Lbeam2/8   = 1641 kNm  Unsupported Length Lu = 1900 mm (assume lateral support provided at joists) Estimate Beam Depth h = Lbeam/20   = 762 mm        Use W610x195      Moment Resistance Mr = 1910 kNm   Mr > Mf  OK Moment of Inertia I = 1.68E+09 mm4  Shear Capacity Vr = 1990 kN   Vr > Vf  OK  Deflection at Specified Loads Dservice = 5*Qspecified*Lbeam4 / (384*E*I)  = 84 mm  Allowable Deflection at Specified Loads Dall = Lbeam/180 (For simple span members supporting inelastic roof coverings)  = 85 mm   Dservice < Dall  OK       Check Required Column Size:  Appendix B Design Notes and Model Properties for Archetypical System 1   183  Factored Axial Load Cf = 2*Vf   = 862 kN  Try HSS 254x254x9.5      Unbraced Length kLcolumn = h - joist depth   = 8.09 m  FactoredCompressive Resistance Cr = 1940 kN (CISCHandbook,   pg 4-79)  Cr > Cf  OK       Base Shear  Calculation (NBCC 2005):     For Diaphragm, assume: Rd = 1.5   Ro = 1.3  Building Period T = 0.05*h3/4   = 0.26 s        T<2.0 seconds, and R=1.5, Therefire 2/3*S(0.2)  cutoff applies   Spectral Acceleration S(0.2) = 0.94 g   = 100% of Vancouver Sa(0.2) Soil Modification Factor Fa = 1.1 (Site Class D) Importance Factor I = 1   Higher Mode Factor Mv = 1   Seismic Acceleration E = 0.66*I*S*Fa*Mv / (Rd*Ro)  = 0.35 g   = 0.68 g (for Rd, Ro = 1.0) Unit Load on Roof Diaphragm Along Building Length vroof = ((qroof+0.25*qsnow)*w + wc*t*h)*E  = 29 kN/m        Compare with Forces Due to wind  Wind Load (NBCC 2005 Cl 4.1.7) qwind = Iw*q*Ce*(Cp*Cg + Cpi*Cgi) 1 in 50 velocity pressure q = 0.48 kN/m2  Importance Factor Iw = 1   Exposure Factor Ce = 1  Exterior Gust Coeff Cp*Cg = 1.3 or -1.5 (positive denotes towards surface, neg denotes away) Interior Pressure Coeff Cpi = 0.3 or -0.45  Interior Gust Coeff Cgi = 2  Inward pressure qwind inward = 1.0*q*1.0*(1.3 + 0.45*2.0) Outward pressure qwind outward = 1.0*q*1.0*(-1.5 - 0.3*2.0) Inward pressure governs  Design Wind Load qwind = 1.06 kN/m2  Unit Load on Roof Diaphragm Along Building Length vwind = qwind*h*1.5  = 14 kN/m     Appendix B Design Notes and Model Properties for Archetypical System 1   184  Earthquake Load Governs for Diaphragm Design               Max Shear Load on Diaphragm Vf = vroof*L / (2*w)   = 29.2 kN/m   Max Moment on Diaphragm Mf = vroof*L2 / 8   = 13551 kNm        Fastener Design:      Assumptions:  1. Standard  1.5" deck, intermediate rib  2. Joist spacing @ 6'-3" c/c  3. Support fastening Hilti X-EDN19 or  X-EDNK22 pins at 36/9 fastening pattern  4. Side-lap fastening #12 screw   Distance from End, x (m) Shear Load, Vf (kN/m) Deck Thickness Required Sidelap spacing Shear Resistance Vr (lb/ft) Shear ResistanceVr (kN/m) Shear Stiffness, G' (kN/m) 0 29.2 18ga 4" 2160 31.5 28808 3.81 25.5 18ga 4" 2160 31.5 28808 7.62 21.9 18ga 4" 2160 31.5 28808 11.43 18.2 20ga 6" 1392 20.3 17759 15.24 14.6 20ga 6" 1392 20.3 17759 19.05 10.9 20ga 6" 1392 20.3 17759 22.86 7.3 20ga 6" 1392 20.3 17759 26.67 3.6 20ga 6" 1392 20.3 17759 30.48 0.0 20ga 6" 1392 20.3 17759       Perimeter Angle:  Steel Area Required Asrequired = Mf*1000 / (Φs*Fy*w)   = 1647 mm2  Use L4"x4"x1/2" As = 2430 mm2     Appendix B Design Notes and Model Properties for Archetypical System 1   185                                                       Appendix B Design Notes and Model Properties for Archetypical System 1   186                                                       Appendix B Design Notes and Model Properties for Archetypical System 1   187   CAPACITY BASED ON DRAFT HILTI REPORT (ICC-ES REPORT)    Sne = (2*a1 + 2*a2 +ne)*Qf/L   = 9680 lb/ft    Sni = 2*A*(lambda - 1) +B)*Qf/L  = 26004 lb/ft    Snc = Qf*((N2*B2)/(L2*N2 + B2))0.5  = 4505 lb/ft  Strength Sn = min(Sne, Sni, Snc)     = 4505 lb/ft  Factored Strength (SDI code) 0.6*Sn = 2703 lb/ft  S = c*Sn Nominal Strength   = 4803 lb/ft 70  B = ns*as+1/1296*(4*sum(xp2) +4*sum(xe2))  = 46.440   lambda = 1 - (1.5*Lv)/(240*t0.5) >  = 0.457   = 0.7   as = Qs/Qf   = 0.639   Span Lv = 6.3 ft  Panel Length L = 3*Lv   = 18.9 ft  Side lap connector spacing s = 4 in   ne = ns = 3xlvx12 / s   = 56.7  correlation factor c = 1.066   Deck thickness (18ga) t = 0.0474 in  Moment of Inertia I = 0.284 in4   Qs = 1701 lb   Qf = 2663 lb   a1 = 3   a2 = 3   sum(xe2) = 1656 in2   sum(xp2) = 1656 in2   A = 2   N = 2.333   D = 756  Nominal Buckling Strength Sbuckling = (I*106 / Lv2)   = 7155 lb/ft (Note – much higher than previous 2243 lb/ft) Appendix B Design Notes and Model Properties for Archetypical System 1   188            Notes:    The tables in the Hilti Report could not be included as they are confidential  The latest Hilti table provides different strength values for different base material thicknesses In the original Hilti program, there was no distinction between different values for base material thickness Using the SDI equations modified as per the latest Hilti Report, if a base material thickness smaller than 3/16 inch is used, the strength results are the same as provided by the Hilti software, however, if a base material thicker than 3/16 inch is used, the strengths increase by about 25%   Factored Strength Nominal Strength  (lb/ft) (kN/m) (lb/ft) (kN/m) Base material thickness < 3/16 inch 2127 31 4264 62 Base material thickness > 3/16 inch 2703 39 4802 70                         Appendix B Design Notes and Model Properties for Archetypical System 1   189      Rocking Model Properties (Refer to Figure 3.2):   Material Properties Reinf Steel yield strength fy = 400 MPa  Concrete compressive strength fc' = 30 MPa Concrete tangent modulus Ec = 4500(fc')^0.5   = 24648 MPa  Steel Elastic Modulus E = 200000 MPa  Concrete unit weight wc = 24 kN/m3   Panel Dimensions Height h = 9.144 m 30 ft Thickness t = 0.184 m 7.24 in Width b  = 7.62 m 25 ft Span to depth ratio h/t = 50 Gross Concrete Area per Panel Ag = b*t (Along Horizontal Plane) = 1.402 m2 Overall Bulding Dimensions Length  L = 60.96 m Width w = 30.48 m Joist span Ljoist = 15.24 m   Appendix B Design Notes and Model Properties for Archetypical System 1   190  Dead Loads on Panel: Roof Unit Load qroof = 1.0 kN/m2 Dead Load from tributary roof area Wroof trib = qroof*0.375*Ljoist*b  = 44 kN  Concrete Weight per panel Wp = wc*t*b*h   = 308 kN  Total Building Weight W = qroof*w*L + wc*t*2*(L+w) = 9243 kN Panel Reinforcement: Vertical Reinforcement: 15M @ 400 alternating faces  Steel Steel Area As = 1000 mm2/m  Reinforcing Steel Ratio ρ = As/(1000*t*1000)   = 0.0054        Horizontal Reinforcement: 15M @ 400 alternating faces  Steel Steel Area As = 500 mm2/m  Reinforcing Steel Ratio ρ = As/(1000*t*1000)   = 0.0027  Inelastic General Wall Element Properties Mesh: n x n = 4x4 (Long Axis) 8x8 (End walls)  Vertical Concrete and Reinforcement Layer: No. fibres required for steel = b / (n*bar spacing) = b / (n*0.4m) = 4.8 (use 5) 2.4 (use 3) No. fibres for concrete = 4.8 (use 5) 2.4 (use 3)  Horizontal Concrete and Reinforcement Layer: No. fibres required for steel = h/400/n = 5.7 (use 6) 2.9 (use 3) No. fibres for concrete = 5.7 (use 6) 2.9 (use 3) Inelastic Shear Material:  Shear Strength (Paulay and Priestley - pg 127)  vy = 0.25*(fc')0.5 + ρ*fy Based on Adebar et. al. (2004) Appendix B Design Notes and Model Properties for Archetypical System 1   191  = 2.46 MPa Shear Modulus G ρ*Es Based on Perform 3D User Manual = 1087 MPa   Wall to Footing Element Properties:  Shear resistance proportional to compressive load (no shear resistance when element in tension) Shear friction coefficient u = 0.5 Shear stiffness Ks = 1000000 kN/m Compression Stiffness Kc = Ec*Ac/Hfooting (Assume: -100mm compression zone) = 453514 kN/m                -Hfooting = 1m Use Kc = 500000 kN/m Tension Stiffness Kt = 0.001 kN/m   Roof Deck Perimeter Angle L102x102x12.7 Axial/Bending element with elastic properties Steel Grade 300W Fy = 300 MPa Steel Area As = 2430 mm2 = 0.00243 Appendix B Design Notes and Model Properties for Archetypical System 1   192  Axial Capacity C = As*Fy = 729 kN Moment of Inertia I = 1280000 mm4 = 1.28E-06 m4 Section Modulus S = 32600 mm3 Bending Resistance My = S*Fy = 9.78 kNm     Panel to Panel Contact Element (19mm gap between panels) Compression: Stiffness in Compression Kc = 500000 kN/m gap = 0.019 m Tension: Stiffness in Tension Kt = 0.001 kN/m hook = 10 m            Appendix B Design Notes and Model Properties for Archetypical System 1   193   Panel to Panel Connection Shear Backbone curve based on Devine (2008) Connections are axially rigid   Displacement Force (m) (kN) 0.000 0 0.004 150 0.010 250 0.024 250 0.024 0          Appendix B Design Notes and Model Properties for Archetypical System 1   194    Equivalent Truss for Roof Diaphragm: References: 1. Essa et. al (2003) 2.  Steel Deck Institute  "Diaphragm Design Manual, 3rd Edition" 3. Hilti "Profis" software for diaphragm connection design Assumptions 1. Wide rib, B-deck (nestable) profile, 18ga (1.2mm) and 20ga (0.91mm) 2. Nailed (Hilti) Deck to Frame Connectors at end / interior supports (Hilt X-EDN19 / X-EDNK22-THQ12) 3. Screwed Side Lap Fasteners 4. Results from cyclic tests 4 and 17 from above paper indicate Stiffness from testing is   approximately 0.8*stiffness from SDI calculations 5. Consider secant stiffness (not initial tangent stiffness)   γ   Unit Strength and 0.8*Stiffness based on SDI method Deck Thickness t = 1.2 mm Strength Vr = 31.5 kN/m SDI Stiffness G' = 28808 Stiffness 0.8*G' = 23046 kN/m/rad Strength and Stiffness for Roof Diaphragm Panel length a = 1.905 m Panel width b = 1.905 m Strength Vy = Vr*a = 60 kN Stiffness K = (0.8*G')*a = 43903 kN/rad Shear Angle at Yield γ = Vy/K Appendix B Design Notes and Model Properties for Archetypical System 1   195  = 0.0014 rad Strength and Stiffness for Equivalent Truss:   γ   Brace Angle θ = tan-1(a/b) = 45.0 degrees Brace Length L = a/sin(θ) = 2.694 m Yield Displacement for Brace Dy = γ* b/sin(θ) (Assuming small angle, γ) = 0.004 m Yield Strain for Brace ey = Dy/L = 0.0014 Yield Strength for Brace Ty = Vy/sin(θ) = 85 kN E*A for Brace: Since Dy = Ty*L / (E*A)  E*A = Ty*L  /Dy   = 62089 kN Stiffness for Chords: Assume chord member have properties of equivalent area of deck. Modulus of Elasticity E = 200000 MPa Required  Joist:  1050mm deep (3'-6") with Chord Combination L/L Joist Top Chord Area Ach = 1095 mm2 Joist Moment of Inertia Ijoist = 5.70E+08 mm4 Appendix B Design Notes and Model Properties for Archetypical System 1   196  = 0.0005698 m4 Parallel to Joists (perpendicular to flutes) Area A2 = b*t/2 + Ach = 0.00224 m2 Perpendicular to Joists (parallel to flutes) A3 = a*t/2 = 0.00114 m2 Summary of Deck Equivalent Truss: Deck Thickness Chord Area Zone Gauge Thickness (mm) Fastening Pattern Shear Resistance, Vr (kN/m) Shear Stiffness , G' (kN/m) 0.8*Shear Stiffness, 0.8*G' (kN/m) E*A for Equivalent Brace (kN) Perpend icular to Flutes (m2) Parallel to Flutes (m2) 0 - 22.86m (75ft) from each end 18ga 1.204 36/9 31.5 28808 23046 62089 0.00224 0.00115 22.86m (75ft) to L/2 (150ft) from each end 20ga 0.909 36/9 20.3 17759 14207 38275 0.00196 0.00087     197  APPENDIX C. DESIGN NOTES AND MODEL PROPERTIES FOR ARCHETYPICAL SYSTEM 2: PANELS WITH OPENINGS Design of Panels with Openings for Out-of-Plane Loading       Assumptions:  1. Building located in Vancouver, B.C., site class D        References:  1. CAN/CSA A23.3  2. "Concrete Design Handbook - Third Edition" by the Cement Association of Canada  Material Properties:  Concrete material factor Φc = 0.65  Reinf Steel material factor Φs = 0.85  Member resistance factor Φm = 0.75  Reinf Steel yield strength fy = 400 MPa  Concrete compressive strength fc' = 30 MPa Modulus of Rupture fr = 0.6*(fc')^0.5   = 3.3 MPa  Concrete tangent modulus Ec = 4500(fc')^0.5  = 24648 MPa   a1 = 0.85-0.0015*fc'   = 0.805   B1 = 0.97-0.0025*fc'   = 0.895   Reinf Steel Elastic modulus Es = 200000 MPa Concrete unit weight wc = 24 kN/m3   Panel Properties:  Height h = 9144 mm  thickness t = 240 mm  initial deflection ∆o = 25 mm  concrete cover c = 25 mm  Left leg width b1 = 1000 mm  Opening width b2 = 5620 mm  Right leg width b3 = 1000 mm  Opening height h1 = 3625 mm  Total width b = 7620 mm  Appendix C Design Notes and Model Properties for Archetypical System 2   198                            Loading:  Vertical Load Eccentricity ecc = t/2   = 120 mm  Roof Joist Span L = 15.24 m  Roof Joist Spacing s = 1.905 m  wall self weight qwall = wc*t   = 5.76 kN/m2  Roof Self Wt. qroof = 1.0 kN/m2        Snow Load (NBCC 2005 Cl qsnow = Is*(Ss*(Cb*Cb*Cs*Ca)+Sr) For tilt-up building in Vancouver:     Is = Ss = Cw = Cs = Ca = 1.0, Cb = 0.8   Ss = 1.8 kN/m2   Sr = 0.2  Design Snow Load qsnow = 1.6 kN/m2        Wind Load (NBCC 2005 Cl 4.1.7) qwind = Iw*q*Ce*(Cp*Cg + Cpi*Cgi) 1 in 50 velocity pressure q = 0.48 kN/m2  Importance Factor Iw = 1   Exposure Factor Ce = 1  Exterior Gust Coeff Cp*Cg = 1.3 or -1.5 (positive denotes towards surface, neg denotes away) Appendix C Design Notes and Model Properties for Archetypical System 2   199  Interior Pressure Coeff Cpi = 0.3 of -0.45   Interior Gust Coeff Cgi = 2  Inward pressure qwind inward = 1.0*q*1.0*(1.3 + 0.45*2.0)  Outward pressure qwind outward = 1.0*q*1.0*(-1.5 - 0.3*2.0)  Inward pressure governs  Design Wind Load qwind = 1.06 kN/m2        Eccentricity ecc = t/2   = 4572 mm   Seismic for Out-of-Plane forces (NBCC 2005 Cl 4.1.8.17) Vp = 0.3*Fa*Sa(0.2)*Ie*Sp*Wp  Soil Modification Factor Fa = 1.1  (For Site Class D) Spectral Acceleration at T=0.2s Sa(0.2) = 0.94 g Importance Factor Ie = 1.0   Sp = Cp*Ar*Ax / Rp  where 0.7<Sp<4.0 Component Risk Factor Cp = 1.0   Dynamic Amplification Factor Ar = 1.0 (Typically used for tilt-up but maybe unconservative) Height Factor Ax = 1+ 2*hx/hn   hx = Height of component above base  = Centre of mass of panel   hn = total height   Ax = 2   Response Factor Rp = 2.5  (for reinforced tilt-up wall panels)  Sp = 0.8   Seismic for Out-of- Plane forces Vp = 0.248 *Wp        Compare Seismic to Wind (for 1m unit strip)  Factored Seismic out-of- plane qseismic = Vp*1.0m*qwall  = 1.43 kN/m  Factored Wind 1.5*qwind = 1.58 kN/m  Wind Governs for Out-of-Plane Loading on Panels          Strength Calculations:      Load Case (4): 1.25D + 1.4W + 0.5S         Right Leg   Tributary panel width bt3 = b3+b2/2  Appendix C Design Notes and Model Properties for Archetypical System 2   200   = 3810 mm   Factored Load from tributary roof area Ptf = bt3*(1.25*qroof+0.5*qsnow)*L/2  = 60.1 kN   Factored weight of panel tributary to and above design section Pwf = bt3*1.25*qwall*(h-h1)   = 151.4 kN   Axial Load at design section Pf = Ptf + Pwf  = 211.5 kN   Factored UDL lateral load wf = bt3*1.4*qwind   = 5.63 kN/m    No. 20M bars provided  = 3 bars EF  No. 25M bars provided  = 2 bars EF (as well as one bar in middle of section) Area of reinf steel provided As = 1600 mm2 EF Effective depth d = t-c-db/2 mm   = 202 mm  Width of design strip b = 1000 mm   Effective steel area Ase = (Φs*As*fy + Pf*1000) / (Φs*fy)  = 2222 mm2   Compressive stress block ae = Φs*Ase*fy / (a1*Φc*fc'*b)   = 48.1 mm   Resisting Moment Mr = Φs*Ase*fy*(d-ae/2)/10^6   = 134.4 kNm        Bending Stiffness (based on As):        Compressive stress block a = Φs*As*fy / (a1*Φc*fc'*b)   34.7 mm        Distance to neutral axis c = a/B1   = 38.7 mm        Cracked Moment of Inertia Icr = b*c 3/3 + Es*As*(d-c)2/Ec  Appendix C Design Notes and Model Properties for Archetypical System 2   201   = 3.65E+08 mm4/m    Kbf = 48*Ec*Icr / 5*h2*1000   = 1034.3 kNm/m   Moment Magnifier δb = 1/(1-Pf/(Φm*Kbf))   = 1.37   Max out-of-plane shear due to wind Vwind = wf*h/2  Primary Bending Moment Mb = (wf*h1)*(h-h1)/2 + Ptf*ecc/2 + Pf*∆o  = 65.2 kNm   Total Moment Mf = δb*Mb   = 89.7 kNm    Mf < Mr  OK  89.7  134.4 kNm   Total factored deflection ∆f = Mf / (Φm*Kbf)   = 116 mm   Check for maximum reinforcement  Total steel area As = 4800   As/(b*d) < a1*B1*Φc*fc'*700/ (Φs*fy*(700+fy))  0.0238 < 0.026  OK        Deflections at Service Loads for Leg        Load Case: 1.0D + 1.0W + 0.5S       (Iw = 0.75, Is = 0.9)       Panel width b = 1000 mm  Tributary panel width bt3 = 3810 mm   Factored Load from tributary roof area Pts = bt3*(1.0*qroof+0.5*0.9*qsnow)*L/2  = 50.5 kN   Factored weight of panel tributary to and above design section Pws = bt3*1.0*qwall*h1   = 79.6 kN  Appendix C Design Notes and Model Properties for Archetypical System 2   202   Axial Load at mid-height Ps = Ptf + Pwf   = 130.0 kN   Service UDL lateral load ws = bt3*1.0*lw*qwind   = 3.02 kN/m   Gross Moment of Inertia Ig = b*1000*t3/12   = 1152000000 mm4   N.A. to extreme fibre yt = t/2   120 mm   Cracking Moment Mcr = fr*Ig/yt   = 31.55 kNm   Primary Bending Moment Mbs = ws*h1*(h-h1)/2 + Pts*ecc/2 + Ps*∆o  = 36.46 kNm       Using a triangular concrete stress distribution    n = Es/Ec   = 8.1    kd = (-n*As + ((n*As)^2 + 2*b*n*As*d)^0.5)/b  = 60.6 mm    Icr = b*(kd)^3/3 + n*As*(kd - d)^2  = 333764316.7 mm4        Initially assume ∆s = h/100   = 91.4 mm    Ms = Mbs + Ps*∆s   = 48.4 kNm        Effective Moment of Inertia Ie = Icr + (Ig - Icr)*(Mcr/Ms)^3  = 561074094.0 mm4   Stiffness Kbs = 48*Ec*Ie / 5*h2*1000   = 1588 kNm/m   Moment Magnifier δb = 1/(1-Ps/Kbs)   = 1.09   Total Moment Ms = δb*Mbs  Appendix C Design Notes and Model Properties for Archetypical System 2   203   = 39.7 kNm    Ms < Mr  OK  39.7  48.4 kNm   max service deflection ∆s = Ms/Kbs*1000   = 25.0 mm    ∆s < ∆si  OK   Appendix C Design Notes and Model Properties for Archetypical System 2   204    Design of Panels with Openings for In-Plane Loading         Assumptions:  1. Building located in Vancouver, B.C., site class D         References:       1. CAN/CSA A23.3  2. "Concrete Design Handbook - Third Edition" by the Cement Association of Canada  Material Properties: Concrete material factor Φc = 0.65 Reinf Steel material factor Φs = 0.85 Member resistance factor Φm = 0.75 Reinf Steel yield strength fy = 400 MPa   Concrete compressive strength fc' = 30 MPa Modulus of Rupture fr = 0.6*(fc')^0.5   = 3.3 MPa   Concrete tangent modulus Ec = 4500(fc')^0.5   = 24648 MPa   a1 = 0.85-0.0015*fc'  = 0.805   B1 = 0.97-0.0025*fc'  = 0.895  Reinf Steel Elastic modulus Es = 200000 MPa Concrete unit weight wc = 24 kN/m3 Friction Coeff at Base µ = 0.5 Max concrete strain ec = 0.0035  Panel Layout:  Leg thickness of 240mm required for out-of-plane loading  Minimum 4:1 width/thickness ratio required, so width minimum 960mm.  1000mm selected To maintain same aspect ratio as test panel, leg height increased to 3625mm        Appendix C Design Notes and Model Properties for Archetypical System 2   205        In-Plane Panel Properties:  thickness t = 240 mm  concrete cover cover = 52 mm  Left leg width b1 = 1000 mm   Opening width b2 = 5620 mm   Right leg width b3 = 1000 mm   Panel Width b = 7620 mm   Opening height h1 = 3625 mm   Height h = 9144 mm  Panel C.G. from bottom y = (b*h2/2 - b2*h12/2) / (b*h-b2*h1)   = 5712 mm         Out-of-Plane Panel Properties:  thickness t = 184 mm   Panel Width b = 7620 mm   Height h = 9144 mm     Number of In-plane panels nip = 8  Number of Out of plane panels nop = 16 Total Number of panels ntot = 24  Building Length (out of plane) L = 60.96 m Building Width (in plane) w = 30.48 m   Roof Area Aroof = L*w   1858 m2   Slab properties  Concrete strength fc'slab = 25 MPa  thickness tslab = 200 mm    Loading:  Roof Joist Span Ljoist = 15.24 m  Roof Joists in Plane? = 0 (yes = 0, no = 1) Appendix C Design Notes and Model Properties for Archetypical System 2   206  wall self weight qwall = wc*t   = 5.76 kN/m2  Roof Self Wt. qroof = 1.0 kN/m2 Snow Load qsnow = Is*(Ss*(Cb*Cb*Cs*Ca)+Sr) For tilt-up building in Vancouver:      Is = Ss = Cw = Cs = Ca = 1.0, Cb = 0.8    Ss = 1.8 kN/m2  Sr = 0.2    qsnow = 1.6 kN/m2  Roof Weight per panel Wr = qroof*b*Ljoist/2  = 0 kN/panel  (No roof weight if joists spanning in plane)  Panel with opening Weight Wpo = qwall*(b*h - b2*h1)  = 284 kN/panel       Solid Panel Weight Wps = wc*t*b*h   = 308 kN/panel   Total Building Weight W = qroof*Aroof + nip*Wpo + nop*Wps = 9053 kN  (Calculated)  = 9010 kN (From Perform)  Base Shear Calculation for Rd, Ro = 1.0 (NBCC 2005)  Building Period T = 0.05*h3/4   = 0.26 s         T<2.0 seconds, and R=1.5, Therefore 2/3*S(0.2) applies         Spectral Acceleration S(0.2) = 0.94 g   Soil Modification Factor Fa = 1.1    Importance Factor I = 1.0    Higher Mode Factor Mv = 1    Seismic Load E = 0.66*I*S*Fa*Mv / (Rd*Ro)   = 0.68 g   Force from Roof Diaphram Vroof = ((qroof+0.25*qsnow)*Aroof + Wps*nop*0.5)*E*0.5  = 1734 kN per one wall line  Shear from in-plane panels Vpanels = E*nip*Wpo*0.5 Appendix C Design Notes and Model Properties for Archetypical System 2   207   775 kN per one wall line  Total Elastic Base Shear Vftotal = Vroof +Vpanels  = 2509 kN per one wall line  = 5018 kN for entire building        Compare with Forces Due to wind  Wind Load (NBCC 2005 Cl 4.1.7) qwind = Iw*q*Ce*(Cp*Cg + Cpi*Cgi) 1 in 50 velocity pressure q = 0.48 kN/m2 Importance Factor Iw = 1    Exposure Factor Ce = 1  Exterior Gust Coeff Cp*Cg = 1.3 or -1.5 (positive denotes towards surface, neg denotes away) Interior Pressure Coeff Cpi = 0.3 or -0.45    Interior Gust Coeff Cgi = 2  Inward pressure qwind inward = 1.0*q*1.0*(1.3 + 0.45*2.0) Outward pressure qwind outward = 1.0*q*1.0*(-1.5 - 0.3*2.0) Inward pressure governs  Design Wind Load qwind = 1.06 kN/m2 Base shear due to wind Vw = qwind*h*L/2  = 1177 kN per one wall line   Resistances:    Check Concrete Shear in Legs:  Vf = Vftotal / nip  = 314 kN/leg         leg width b = 1000 mm  effective depth of section d = b - cover -dbties - dbvert/2  925 mm   effective depth for shear dv = 0.9*d   = 833 mm         Check width/thickness  h/t = 4.2 > 4, therefore OK         Max Shear Strength Max Vr = 0.25*Φc*fc'*t*dv   = 974 kN    Max Vr > Vf  OK         Panel Leg shear resistance:      Concrete component Vc = 0.18*Φc*(fc')0.5*t*dv  Appendix C Design Notes and Model Properties for Archetypical System 2   208   = 128 kN   Reinf Steel component   Shear Reinf, 10M ties @ spacing, s =  300 mm c/c spacing Av = 200 mm2    Vs = Φs*Av*fy*dv*cotθ/s   = 269 kN         Shear resistance Vr = Vc + Vs   = 398 kN    Vr > Vf  OK         Base Connections:       Assume welded EM3A connections with studded embedments  Ductility factor Rd = 1    Overstrength Ro = 1.3  Reduced Base Shear Vfreduced = Vftotal/(Rd*Ro)   1930 kN   Shear resistance VrEM3A = 130 kN / connection        Provide 2 connections per panel  Vrconnections = 2*nip*VrEM3A   = 1040 kN         Check concrete bearing where slab is locked into the panels  Bearing Strength Br = 0.85*Φc*fc'*t*tslab   = 663 kN   Friction Vrfriction = 0.5*(Wr + Wpo)*nip / 2  568 kN   Total shear resistance Vr = Vrconnections+Br+Vrfriction  = 2271 kN   Vr > Vfreduced OK  In-plane leg bending:  Ductility factor Rd = 1.5   Overstrength Ro = 1.3  Shear force per leg Vf = Vftotal/(nip*Ro*Rd)   = 161 kN        Moment per leg Mf = Vf*h1  Appendix C Design Notes and Model Properties for Archetypical System 2   209   = 583 kNm        No. of 25M Vertical Reinforcing Bars = 6 bars  No. of 20M Vertical Reinforcing Bars = 6 bars  Total Steel Area Provided Astot = 4800 mm2  Steel Area for Bending As = 2700 mm2   Effective Depth if assume bottom three rows in compression: Number of rows of bars: 5  Bar diameter: 25 mm   Row No. Distance from Bottom Steel Area (mm2) 1 75 1500   2 287 600   3 500 600   4 713 600  5 926 1500    Distance to centroid of steel area from bottom   y =  Sum(Ai*di) / Sum(Ai)   216 mm   Effective Depth:   d =  b-y   d =  784 mm          Compression block a = Φs*fy*As/(Φc*fc'*a1*t)   = 244 mm  Check neutral axis c = a/B1  = 272 mm   check max rebar strain es = ec/c*(b1-c-cover-stirrup)  = 0.009 < 0.05 OK    Moment Resistance Mr = Φs*fy*As*(d-a/2)  = 608 kNm   Mr > Mf  OK   Lateral resistance Vrleg = Mr/h1   = 168 kN         Nominal lateral resistance Vrleg/Φc = 258 kN    Tie spacing in Hinge region s = 8*dbvert   = 200 mm  Appendix C Design Notes and Model Properties for Archetypical System 2   210         Provide 10M ties @ 200 mm c/c spacing in hinge region         Panel Overturning       Ductility factor Rd = 1.5    Overstrength Ro = 1.3         Roof Vroof = Vroof/(Rd*Ro)   = 889 kN         Panel Vpanel = Vpanel/(Rd*Ro)   = 398 kN         Overturning Moment Mof = Vroof*h + Vpanel*y   = 10402 kNm         Resisting Moments  Panel and roof Mrweight = nip/2*(Wpo+Wr)*b/2  = 4328 kNm         End connections Vr = 0 kN   (Hold down weight at corner)  Mrend = Vr*b  = 0 kNm   Total Resisting Moment Mr = Mrweight+Mrend  = 4328 kNm   Required connection force Vfconn = (Mf - Mr)/(b*(nip/2-1))  = 266 kN/panel         Use EM5 connections       Ductility factor Rd = 1.5    Overstrength Ro = 1.3  Shear Resistance VrEM5 = 125 kN / connection  No. of connections required = Vfconn / VrEM5  = 2.1 connections per panel        Provide 3 EM5 Connections per panel (9 total)          Check Beam Header for bending and shear         For overturning analysis at:       Ductility factor Rd = 1.5  Appendix C Design Notes and Model Properties for Archetypical System 2   211  Overstrength Ro = 1.3   Maximum Shear force Vf = Vfconn+Wr/2+Wpo/2  = 408 kN   Horizontal distance from end to C.G. for half panel  x = h*b1 2/2+(h-h1)*(b2/2)*(b/2-b2/4) / (h*b1+(h-h1)*(b2/2))  = 1698 mm   Maximum Moment Mf = Vf*b/2 + (Wr/2)*b/4 + (Wpo/2)*x - Vroofp3*h/(2*nip/2*Rd*Ro) - Vpanel*y/(2*nip/2*Rd*Ro)  = 494 kNm   section width b = h-h1   = 5519 mm   effective depth d = b - cover -dbhor - dbvert/2  = 5444 mm         Steel Area (assume 2 - 20M bars are engaged) As = 600 mm 2         Compression block a = Φs*fy*As/(Φc*fc'*a1*t)   = 54 mm         Moment Resistance Mr = Φs*fy*As*(d-a/2)   = 1105 kNm    Mr > Mf  OK          Check for potential yielding of longitudinal reinforcement         Multiply moment by Rd = 1.5   Ro = 1.3         Maximum Moment Mf = 964 kNm    Mr > Mf  Reinforcing will not yield and anti-buckling ties not required   Use 10M @ 400 EW As = 500 mm2/m  Minimum Reinforcing Asmin = 0.002*Ag = 480 mm2/m  Appendix C Design Notes and Model Properties for Archetypical System 2   212  As > Asmin OK        Check Shear in Header       Multiply moment by Rd = 1.5   Ro = 1.3         Maximum Shear Force Vf = 795 kN         Effective depth for shear dv = 0.9*d   = 4900 mm         Concrete component Vc = 0.18*Φc*(fc')0.5*t*dv   = 754 kN   Reinf Steel component   Shear Reinf, 10M ties @ spacing, s =  400 mm c/c spacing Av = 200 mm2    Vs = Φs*Av*fy*dv*cotθ/s   = 1190 kN         Shear resistance Vr = Vc + Vs   = 1943 kN    Vr > Vf  OK  Appendix C Design Notes and Model Properties for Archetypical System 2   213   Frame and Eccentric Model Properties (Refer to Figure 3.3)  Material Properties: Reinf Steel yield strength fy = 400 MPa Concrete compressive strength fc' = 30 MPa Concrete tangent modulus Ec = 4500(fc')^0.5  = 24648 MPa Steel Elastic Modulus E = 200000 MPa Concrete unit weight wc = 24 kN/m3          Panels with Openings: Panel Dimensions thickness t = 0.24 m  concrete cover cover = 0.052 m  Left leg width b1 = 1 m  Opening width b2 = 5.62 m  Right leg width b3 = 1 m  Panel Width b = 7.62 m  Opening height h1 = 3.625 m  Height h = 9.144 m   Dead Loads on Panel:  Concrete Weight per panel Wpip = wc*t*b*h - b2*h1   = 284 kN             Appendix C Design Notes and Model Properties for Archetypical System 2   214        Leg Vertical Reinforcement:     No. 25M bars 6    No. 20M bars 6  Steel Steel Area As = 4800 mm2 Reinforcing Steel Ratio ρ = As/(b1*t)   = 0.0200       Hinge Zone Horizontal Reinforcement: 10M @ 200 each face  Steel Steel Area As = 1000 mm2/m Reinforcing Steel Ratio ρ = As/(1000*t*1000)  = 0.0042  Inelastic General Wall Element Properties Mesh: n x n = 2x4  Vertical Concrete and Reinforcement Layer: No. fibres required for steel = No. vertical bars /2 = 6.0 No. fibres for concrete = 6.0  Horizontal Concrete and Reinforcement Layer: No. fibres required for steel = h1/200/n = 4.5 (use 6) No. fibres for concrete = 4.5 (use 6) Appendix C Design Notes and Model Properties for Archetypical System 2   215   Beam Vertical Reinforcement: 15M @ 400mm EF  Steel Steel Area As = 1000 mm2/m Reinforcing Steel Ratio ρ = As/(1000*t)  = 0.0042  Horizontal Reinforcement: 15M @ 400mm EF  Steel Steel Area As = 1000 mm2/m Reinforcing Steel Ratio ρ = As/(1000*t)  = 0.0042  Inelastic General Wall Element Properties Mesh: n x n = 8x4  Vertical Concrete and Reinforcement Layer: No. fibres required for steel = b / (n*bar spacing) = b / (n*0.4m) = 2.4 (use 3) No. fibres for concrete = 2.4 (use 3)  Horizontal Concrete and Reinforcement Layer: No. fibres required for steel = (h-h1)/400/n = 3.4 (use 3) No. fibres for concrete = 3.4 (use 3) Connection and roof element properties and concrete shear properties same as rocking model Appendix D Sample Calculation for Collapse Statistics   216  APPENDIX D. SAMPLE CALCULATION FOR COLLAPSE STATISTICS FOR ARCHETYPICAL SYSTEM 1: SOLID WALL PANELS Sample Calculation: Collapse Statistics for Out-of-Plane Roof Forces for Rocking Model Strength of Out-of-Plane panel to roof connectors: NBCC 2005 Seismic Out-of-Plane forces Panel Weight Wp = 308 kN Seismic for Out-of-Plane forces (NBCC 2005 Cl 4.1.8.17) Vp = 0.3*Fa*Sa(0.2)*Ie*Sp*0.5*Wp (Tributary weight of panel assumed to be 0.5*Wp)  Soil Modification Factor Fa = 1.1   (Site Class D) Spectral Acceleration at T=0.2s Sa(0.2) = 0.94 g Importance Factor Ie = 1.0   Sp = Cp*Ar*Ax / Rp  where 0.7<Sp<4.0 Component Risk Factor Cp = 2.0  Dynamic Amplification Factor Ar = 2.5  (Typically used for tilt-up but maybe unconservative) Height Factor Ax = 1+ 2*hx/hn   hx = Height of component above base  = Centre of mass of panel  hn = total height   Ax = 2  Response Factor Rp = 1  (for reinforced tilt-up wall panels)  Sp = 4  NBCC 2005 Seismic Out- of-Plane forces Vp = 0.62 Wp = 191 kN = 25 kN/m 4 - EM1 Connectors Provided at Panels with Joists framing in Appendix D Sample Calculation for Collapse Statistics   217  Design Strength tr = 70 kN/connector for wall thickness > 150mm Total strength Tr = 280 kN / 4 connectors Nominal strength tult = tr/0.6 = 117 kN/connector = 467 kN / 4 connectors Nominal strength per unit length = 61 kN/m At panels without joists, consider nominal strength = NBCC 2005 forces / 0.6 (Testing by Lemieux et. al, 1998, considered the design strength of standard embedded connectors to be 0.6*mean strength) Nominal strength per unit length = 42 kN/m         Consider the following IDA results for Rocking Model Out-of-Plane Panel to Roof Forces: Sa(T1) Earthquake Record Sa(T1) = 0 0.5g 1.0g 2.0g Sa(T1) at(1) Capacity MUL009 0 13 34 68 1.227 L0S000 0 22 43 46 0.972 BOL000 0 18 33 39 2.505 HEC000 0 16 41 66 1.032 H-DLT262 0 17 46 73 0.932 H-E11140 0 41 65 86 0.527 NIS000 0 16 35 67 1.223 SHI000 0 20 62 69 0.763 DZC180 0 16 33 55 1.422 ARC000 0 19 42 62 0.996 YER270 0 12 63 78 0.795 CLW-LN 0 13 30 69 1.301 CAP000 0 49 84 98 0.428 GO3000 0 51 72 101 0.411 ABBAR-L 0 58 88 107 0.364 B-ICC000 0 18 38 35 Not exceeded B-POE270 0 14 32 64 1.308 RIO270 0 21 53 63 0.834 CHY101-E 0 19 59 81 0.792 TCU045-E 0 16 38 46 1.502 PEL090 0 20 53 76 0.832 A-TMZ000 0 14 27 61 1.440 Appendix D Sample Calculation for Collapse Statistics   218  MEDIAN 0 18 43 68 0.986 Note that the Sa(T1) at capacity was determined by linear interpolation between the two IDA data points lesser and greater than than capacity The table below shows the IDA data sorted by increasing Sa(T1) (1)The P[Collapse} based on the raw data is calculated by dividing the number of the record  in sequence from smallest Sa(T1) to largest divided by the total number of records Record No. Eq Record Sa(T1) at which drift criteria exceeded (g) ln(Sa(T1) Raw Data(1) P[Collapse]     0.00   0.00 1 ABBAR-L 0.364 -1.01 0.05 2 GO3000 0.411 -0.89 0.10 3 CAP000 0.428 -0.85 0.14 4 H-E11140 0.527 -0.64 0.19 5 SHI000 0.763 -0.27 0.24 6 CHY101-E 0.792 -0.23 0.29 7 YER270 0.795 -0.23 0.33 8 PEL090 0.832 -0.18 0.38 9 RIO270 0.834 -0.18 0.43 10 H-DLT262 0.932 -0.07 0.48 11 L0S000 0.972 -0.03 0.52 12 ARC000 0.996 0.00 0.57 13 HEC000 1.032 0.03 0.62 14 NIS000 1.223 0.20 0.67 15 MUL009 1.227 0.20 0.71 16 CLW-LN 1.301 0.26 0.76 17 B-POE270 1.308 0.27 0.81 18 DZC180 1.422 0.35 0.86 19 A-TMZ000 1.440 0.36 0.90 20 TCU045-E 1.502 0.41 0.95 21 BOL000 2.505 0.92 1.00 Median 0.972 Mean, µ = LN(Median Sa(T1))  -0.03 Std Dev of LN(Sa(T1))  0.48 Sources of Uncertainty (Based on ATC-63 Methodology): Record to Record Collapse Uncertainty,  BRTR =  0.48 (Same as standard deviation above) Appendix D Sample Calculation for Collapse Statistics   219  Design Requirements-related Collapse Uncertainty,  BDR =  0.3 (0.2, 0.3, 0.45, and 0.65 chosen based on judged rating of superior, good, fair, or poor) Test Data-related Collapse Uncertainty,  BTD =  0.3 (0.2, 0.3, 0.45, and 0.65 chosen based on judged rating of superior, good, fair, or poor) Modeling-related Collapse Uncertainty,  BMDL =  0.45 (0.2, 0.3, 0.45, and 0.65 chosen based on judged rating of superior, good, fair, or poor) Total System Collapse Uncertainty, BTOT =  (BRTR2 + BDR2 + BTD2 + BMDL2)0.5  = 0.78  Data for fitted Lognormal Curves: (1)Adjusted lognormal fitted data modified by incorporating total uncertainty, BTOT (2)Lognormal distribution incorporating mean and standard deviation calculated from raw data Sa(T1) at which drift criteria exceeded (g) Lognormal Fitted Data(1) P[Collapse] Adjusted(2) Lognormal Fitted Data P[Collapse] 0 0.000 0.000 0.1 0.000 0.002 0.2 0.000 0.022 0.3 0.007 0.066 0.4 0.032 0.128 0.5 0.082 0.197 0.6 0.156 0.268 0.7 0.246 0.337 0.8 0.342 0.401 0.9 0.436 0.461 1 0.523 0.514 1.1 0.602 0.563 1.2 0.670 0.606 1.3 0.728 0.645 1.4 0.777 0.680 1.5 0.818 0.710 1.6 0.851 0.738 1.7 0.879 0.763 1.8 0.901 0.785 1.9 0.920 0.804 2 0.934 0.822 2.1 0.946 0.838 Appendix D Sample Calculation for Collapse Statistics   220  2.2 0.956 0.852 2.3 0.964 0.865 2.4 0.971 0.876 2.5 0.976 0.887 2.6 0.980 0.896 2.7 0.984 0.904 2.8 0.987 0.912 2.9 0.989 0.919 3 0.991 0.925 3.1 0.992 0.931 3.2 0.994 0.936 3.3 0.995 0.941 3.4 0.996 0.945 3.5 0.996 0.949 3.6 0.997 0.953 3.7 0.997 0.956 3.8 0.998 0.959 3.9 0.998 0.962 4 0.998 0.965 Sa(T1) for which less than 10% of structures will fail, Sa(T1)10%: 0.36g (for lognormal fitted data with adjustment for uncertainty): This value is obtained by either interpolating from the adjusted lognormal distribution data above or by using the solver function - changing the Sa(T1) value such that the adjusted lognormal cumulative distribution is equal to 0.1,  i.e. LOGNORMDIST(Sa(T1),µ,BTOT) = 0.1 Compare with Design Base Shear: 1 in 2475 Base Shear for Vancouver (Including 1/3 reduction), Sa(T1)des =  0.68g So for the current R value used, the out-of-plane wall to roof connections are overstressed by 0.68/0.36 = 1.89 or 89% Appendix D Sample Calculation for Collapse Statistics   221       Figure D1: Collapse Statistics for Out-of-Plane Deck Forces:

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