{"http:\/\/dx.doi.org\/10.14288\/1.0392505":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Applied Science, Faculty of","type":"literal","lang":"en"},{"value":"Civil Engineering, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Slotboom, Christian","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2020-07-22T18:42:55Z","type":"literal","lang":"en"},{"value":"2020","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Applied Science - MASc","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"Self-centring Cross-Laminated Timber (CLT) walls are a low damage seismic force resisting system, which can be used to construct tall wood buildings. This study examines two approaches to model self-centring CLT walls, one that uses lumped plasticity elements, and another that uses fibre-based elements. Finite element models of self-centring CLT walls are developed using the Python interpreter of Opensees, OpenSeesPy, and tested under monotonic and reverse cyclic loading conditions. Outputs from the analysis are compared with data from two existing experimental programs. Both models accurately predict the force displacement relationship of the wall in monotonic loading. For reverse cyclic loading, the lumped plasticity model could not capture cyclic deterioration due to crushing of CLT. Both models slightly overpredict the post-tension force. Sensitivity analyses were run on the fibre model, which show the wall studied is not sensitive to the shear stiffness of CLT. OpenSeesPy models are also created of a two-story structure, which is tested dynamically under a suite of ground motions. The structure is based on a building tested as part of the NHERI TallWood initiative. During testing the foundation of the building was found to be inadvertently flexible. To determine the appropriate model parameters for this foundation, calibrations were performed by running a sequence of OpenSeesPy analyses with an optimization algorithm. Outputs from the lumped plasticity and fibre models were compared to experimental results, which showed that both could capture the global behaviour of the system with reasonable accuracy. Both models overpredict peak post-tension forces. The suite of analyses is then run again on the building to predict the performance with a rigid foundation. Cyclic deterioration is more significant for the building with a rigid foundation, and as a result the fibre mode is more accurate.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/75246?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"NUMERICAL ANALYSIS OF SELF-CENTRING CROSS-LAMINATED TIMBER WALLS by Christian Slotboom BA.Sc., The University of British Columbia, 2015 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2020 \u00a9 Christian Slotboom, 2020 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, a thesis entitled: Numerical Analysis of Self-centring Cross-laminated Timber Walls submitted by Christian Slotboom in partial fulfillment of the requirements for the degree of Master of Applied Science in Civil Engineering Examining Committee: Terje Haukaas, Professor, Civil Engineering, UBC Supervisor Carlos Molina Hutt, Assistant Professor, Civil Engineering, UBC Supervisory Committee Member Marjan Popovski, Lead Scientist, FPInnovations and Adjunct Professor, Wood Science, UBC Supervisory Committee Member iii Abstract Self-centring Cross-Laminated Timber (CLT) walls are a low damage seismic force resisting system, which can be used to construct tall wood buildings. This study examines two approaches to model self-centring CLT walls, one that uses lumped plasticity elements, and another that uses fibre-based elements. Finite element models of self-centring CLT walls are developed using the Python interpreter of Opensees, OpenSeesPy, and tested under monotonic and reverse cyclic loading conditions. Outputs from the analysis are compared with data from two existing experimental programs. Both models accurately predict the force displacement relationship of the wall in monotonic loading. For reverse cyclic loading, the lumped plasticity model could not capture cyclic deterioration due to crushing of CLT. Both models slightly overpredict the post-tension force. Sensitivity analyses were run on the fibre model, which show the wall studied is not sensitive to the shear stiffness of CLT. OpenSeesPy models are also created of a two-story structure, which is tested dynamically under a suite of ground motions. The structure is based on a building tested as part of the NHERI TallWood initiative. During testing the foundation of the building was found to be inadvertently flexible. To determine the appropriate model parameters for this foundation, calibrations were performed by running a sequence of OpenSeesPy analyses with an optimization algorithm. Outputs from the lumped plasticity and fibre models were compared to experimental results, which showed that both could capture the global behaviour of the system with reasonable accuracy. Both models overpredict peak post-tension forces. The suite of analyses is then run again on the building to predict the performance with a rigid foundation. Cyclic deterioration is more significant for the building with a rigid foundation, and as a result the fibre mode is more accurate. iv Lay Summary Self-centring Cross-Laminated Timber (CLT) walls are a new structural system that is intended to have low damage during earthquakes. To design buildings that use self-centring CLT walls, it is important to have computer models that can accurately predict how these wall elements respond to loads. In this research, two ways of modelling self-centring CLT walls are examined. A variety of simulations are run on each model, and including static tests, where forces are applied to the structure, and dynamic tests, where earthquakes are simulated. Different wall systems are examined with each modelling method, including single walls, and a two-story building. To assess their accuracy, the predictions made by the computer models are compared to physical experiments. Understanding the accuracy of the two models investigated is the main value of this work and will allow engineers to make better design decisions in the future. v Preface The research presented in this thesis is completed by me and supervised by Dr. Terje Haukaas, Professor at the University of British Columbia in Vancouver. Co-supervision is provided by Dr. Carlos Molina Hutt, Professor at the University of British Columbia in Vancouver. Co-supervision is also provided by Dr. Marjan Popovski, Lead Scientist at FPInnovations in Vancouver and Adjunct Professor at the University of British Columbia in Vancouver. I am responsible for developing and conducting the research program, with continued advice provided by my supervisory team. Recommendations for numerical modelling is provided by Dr. Haukaas, Dr. Molina Hutt, and Dr. Popovski. Recommendations regarding computer programing is provided by Dr. Haukaas. Support for technical details relating to Cross-Laminated Timber is provided by Dr. Popovski. For the numerical analyses described in chapters 4 and 5, I created the nonlinear models in OpenSeesPy, and completed the analyses. I also created the Python programs necessary to pre-and post-process the data resulting from the analyse. I am responsible for writing the text of the thesis, with editing provided by Dr. Haukaas, and Dr. Molina Hutt, and Dr. Popovski. The Journal paper is being written as an iterative process, with feedback provided by my supervisory team. vi Table of Contents Abstract ................................................................................................................................... iii Lay Summary ......................................................................................................................... iv Preface ...................................................................................................................................... v Table of Contents ................................................................................................................... vi List of Tables ........................................................................................................................ viii List of Figures .......................................................................................................................... x Acknowledgements ............................................................................................................ xviii Dedication ............................................................................................................................. xix Chapter 1: Introduction ........................................................................................................ 1 1.1 Research Motivations................................................................................................ 3 1.2 Scope and Objectives ................................................................................................ 4 1.3 Thesis Outline ........................................................................................................... 5 Chapter 2: Research Background ........................................................................................ 6 2.1 CLT as a Lateral Force Resisting System ................................................................. 6 2.2 Self-centring Walls System Concept ...................................................................... 13 2.3 Self-centring Walls as a Resilient Lateral Force Resisting System ........................ 19 2.4 In-plane Properties of CLT ..................................................................................... 25 Chapter 3: Analytic and Numerical Models of Self-centring Wall systems ................... 33 3.1 Section Analysis using the Monolithic Beam Analogy .......................................... 33 3.2 Lumped Plasticity Models ...................................................................................... 37 3.3 Multi-Spring Models ............................................................................................... 40 3.4 Fibre-based Models ................................................................................................. 42 3.5 Energy Dissipation Devices .................................................................................... 45 Chapter 4: Component Level Modelling of Wall Systems ............................................... 50 4.1 Experimental Data from FPInnovations and Ganey ............................................... 50 vii 4.2 Wall Archetypes and Scope of Study ..................................................................... 55 4.3 Prediction and Measurement of Stiffness for Self-Centring CLT Walls ................ 58 4.4 Prediction of Shear Modulus and Post-Tension Tendon properties ....................... 68 4.5 Analysis of walls using the Monolithic Beam Analogy ......................................... 70 4.6 Overview of Nonlinear Numerical Models............................................................. 79 4.7 Model Initialization ................................................................................................. 87 4.8 Model Parameter Calibration .................................................................................. 89 4.9 Numerical Modelling Results ................................................................................. 93 4.10 Parametric Studies ................................................................................................ 106 4.11 Discussion of Wall Models ................................................................................... 112 Chapter 5: Seismic Performance of Wall Systems in Buildings .................................... 116 5.1 NHERI Study Overview ....................................................................................... 116 5.2 Overview of Building Model ................................................................................ 124 5.3 Calibration of Base Beam Properties .................................................................... 133 5.4 Pre-Analysis Checks and Data Processing ........................................................... 137 5.5 Results for Models with Beam .............................................................................. 140 5.6 Results for Models with a Rigid Foundation ........................................................ 147 5.7 Discussion of Results ............................................................................................ 155 Chapter 6: Conclusion ....................................................................................................... 159 6.1 Research Findings ................................................................................................. 159 6.2 Suggestions for Future Work ................................................................................ 161 Bibliography ........................................................................................................................ 163 Appendices ........................................................................................................................... 171 Appendix A : Additional Comparison Figures for the Fibre Model of a Two-story Building with a Flexible Foundation ............................................................................................... 171 Appendix B : Additional Comparison Figures for the Lumped Plasticity Model of a Two-story Building with a Flexible Foundation. ...................................................................... 186 Appendix C : Additional Comparison Figures for the Model of a Two-story Building run With a Rigid Foundation and Single Earthquakes ............................................................ 201 viii List of Tables Table 4-1: CLT geometric properties. .................................................................................... 52 Table 4-2: Axial damper properties. ....................................................................................... 52 Table 4-3: U-shaped flexural plate material and geometric properties. .................................. 52 Table 4-4: Scope of experimental data used in study. ............................................................ 57 Table 4-5: Measured initial tangent slope. .............................................................................. 63 Table 4-6: Modulus of elasticity as measured by different methods. ..................................... 66 Table 4-7: Summary of used material properties. ................................................................... 68 Table 4-8: Stiffness reduction factor on post-tension tendon assembly stiffness. .................. 69 Table 4-9: Measured shear modulus for CLT. ........................................................................ 69 Table 4-10: Element assignments for Archetype I fibre model .............................................. 80 Table 4-11: Element assignments for Archetype I lumped plasticity model. ......................... 81 Table 4-12: Element assignments for Archetype III fibre model ........................................... 83 Table 4-13: Element assignments for Archetype II lumped plasticity model. ....................... 83 Table 4-14: Element assignments for Archetype III fibre model ........................................... 85 Table 4-15: Element assignments for Archetype III lumped plasticity model ....................... 86 Table 4-16: U-shaped flexural plate damper parameters. ....................................................... 87 Table 4-17: Time taken to complete numerical analysis. ..................................................... 106 Table 4-18: Summary of used material properties. ............................................................... 107 Table 5-1: CLT geometric properties. .................................................................................. 119 Table 5-2: U-shaped flexural plate material and geometric properties. ................................ 119 Table 5-3: Post-tension tendon properties. ........................................................................... 119 Table 5-4: Limit states for 2 story building. (Wichman 2018). ............................................ 121 Table 5-5: Groundmotions considered.................................................................................. 122 ix Table 5-6: Element assignments for the 2 story fibre model. ............................................... 126 Table 5-7: Calibrated Material Properties............................................................................. 130 Table 5-8: Effective beam stiffness. ..................................................................................... 136 Table 5-9: Key limit states for the fibre model (FM) and lumped plasticity model (LPM). 148 x List of Figures Figure 2-1: The significant loading directions for timber (Kretschmann 2010). ...................... 7 Figure 2-2: Diagram of CLT, modified from Gagnon and Pirvu (2011). ................................. 8 Figure 2-3: Experimental setup and results from the initial phases of the SOFIE project (Lauriola and Sandhaas 2006). ............................................................................................... 10 Figure 2-4: Overview of (a), the model used for CLT, (b) the results of the model compared with wall experimental results (Pei et al. 2013). ..................................................................... 12 Figure 2-5: Summary of (a) the building tested, and (b) the results of dynamic testing on the building (Pei et al. 2013). ........................................................................................................ 13 Figure 2-6: Self-centring wall system. .................................................................................... 14 Figure 2-7: Components of deformation for a rocking wall system. ...................................... 14 Figure 2-8: Key limit states for the pushover of a self-centring wall system ......................... 15 Figure 2-9: Shear force (V), post-tension force (P), and base contact (C) for a the elastic range of a self-centring wall. ............................................................................................................ 16 Figure 2-10: Position of the neutral axis as rocking develops. ............................................... 18 Figure 2-11: Hysteretic behaviour of self-centring walls. ...................................................... 19 Figure 2-12: 6 Story PRESSS structure shake table test (Nakaki et al. 1999)........................ 20 Figure 2-13: Experimental results for press-lam walls with and without u-shaped flexural plate dampers (Iqbal et al. 2015). ................................................................................................... 22 Figure 2-14: Summary of buildings numerically modeled in Sarti's FEMA P695 study (Sarti et al. 2016) .................................................................................................................................. 23 Figure 2-15: (a) Architectural rendering of Frameworks building, (b) structural renderingand (c) lateral load Resisting System for Frameworks Building (Zimmerman and Mcdonnell 2017)................................................................................................................................................. 25 Figure 2-16: In-plane loading for CLT. .................................................................................. 26 xi Figure 2-17: CLT normal forces with (a) no slip and (b) slip, image modified from Danielsson et al. (2017). ............................................................................................................................ 27 Figure 2-18: Components of CLT Shear Deformation (Danielsson et al. 2017). .................. 27 Figure 2-19: Testing of CLT shear properties for the Frameworks building (Zimmerman and Mcdonnell 2017). .................................................................................................................... 29 Figure 2-20: Crushing of CLT at wall toe. ............................................................................. 30 Figure 2-21: Measurement made in Chen et al. (2018). ......................................................... 31 Figure 3-1: Monolithic beam analogy..................................................................................... 34 Figure 3-2: Predictions of monolithic beam analogy vs. results. ............................................ 36 Figure 3-3: Summary of lumped plasticity model and experimental results for post-tensioned moment frame (Palermo et al. 2005). ..................................................................................... 38 Figure 3-4: Numerical model vs. experimental data for lumped plasticity models of LVL walls. (Sarti 2015). ............................................................................................................................ 39 Figure 3-5: Diagrams of (a), a typical multi-spring element and (b), a typical wall model using multi-spring elements (Iqbal et al. 2015). ............................................................................... 41 Figure 3-6: Summary of experimental wall setup, fibre wall model, and analysis results for base shear vs. drift and post-tension force vs. drift (Perez et al. 2008). ................................. 44 Figure 3-7: A single U-shaped flexural plate, and a paired wall system using U-shaped flexural plates. ...................................................................................................................................... 46 Figure 3-8: Hysteresis for U-shaped flexural plate damper (Baird et al. 2014). .................... 48 Figure 3-9: Axial fuse damper connection and mount (Sarti 2015). ...................................... 49 Figure 4-1: Diagram of experimental setup for a single wall. ............................................... 51 Figure 4-2: Test results for (a) a single wall without a damper and (b) a single wall with axial dampers from walls studied by FPInnovations. ...................................................................... 53 Figure 4-3: Test results for walls from Ganey\u2019s study for (a) base shear vs actuator drift and (b) post-tension force vs actuator drift (Ganey 2015). ............................................................ 55 xii Figure 4-4: Wall archetypes (a) single wall, (b) wall with axial dampers, (c) paired wall with U-shaped flexural plate dampers. ........................................................................................... 56 Figure 4-5: CLT as a Composite Section. ............................................................................... 59 Figure 4-6: (a) Moment and rotation relationship over height, (b) measured moment curvature relationship (Ganey 2015). ...................................................................................................... 60 Figure 4-7: Measuring initial stiffness for CLT panel. ........................................................... 61 Figure 4-8: Measured Force - Deformation Relationship using compression tests (Ganey 2015). ...................................................................................................................................... 65 Figure 4-9: Components of CLT assembly deformation. ....................................................... 68 Figure 4-10: Section analysis algorithm. ................................................................................ 70 Figure 4-11: Imposed rotations for (a) a trial neutral axis location greater than the post-tension tendon location, and (b) less than the post-tension tendon location. ...................................... 71 Figure 4-12: Shear vs. drift using the monolithic beam analogy. ........................................... 77 Figure 4-13: PT force vs. drift using the monolithic beam analogy. ...................................... 78 Figure 4-14: Base contact vs. drift using the monolithic beam analogy. ................................ 79 Figure 4-15: Wall Archetype I OpenSees node diagram for the Fibre Model (FM) and lumped plasticity model (LPM). .......................................................................................................... 80 Figure 4-16: Wall Archetype II OpenSees node diagram for the fibre model (FM) and lumped plasticity model (LPM). .......................................................................................................... 83 Figure 4-17: Calibrated axial damper material. ...................................................................... 84 Figure 4-18: Wall Archetype III node diagram for the fibre model (FM) and lumped plasticity model (LPM). .......................................................................................................................... 85 Figure 4-19: Calibrated U-shaped flexural plate damper material. ........................................ 87 Figure 4-20: Summary of Algorithm to Initialize PT Force. .................................................. 88 Figure 4-21: Summary of Algorithm to Calibrate Material Property. .................................... 90 Figure 4-22: Overview of Objective Function Algorithm. ..................................................... 90 xiii Figure 4-23: Summary of algorithm to calibrate material property. ....................................... 92 Figure 4-24: Objective function for Optimization Analysis. .................................................. 92 Figure 4-25: Shear Force vs. Drift for Fibre Model Tests ...................................................... 95 Figure 4-26: Shear force vs. drift for FPInnovations fibre model tests. ................................. 96 Figure 4-27: Post-tension force vs. drift for Ganey\u2019s tests. .................................................... 97 Figure 4-28: Post-tension force vs. drift for FPInnovations tests, fibre model....................... 99 Figure 4-29: Drift vs. base contact for lumped plasticity models of Ganey\u2019s tests. ............. 100 Figure 4-30: Shear force vs. drift for lumped plasticity model tests. ................................... 101 Figure 4-31: Shear force vs. drift for lumped plasticity tests of Ganey\u2019s walls. .................. 102 Figure 4-32: Post-tension force vs. drift for FPInnovations lumped plasticity tests. ........... 103 Figure 4-33: Post-tension force vs. drift for Ganey\u2019s lumped plasticity tests. ..................... 104 Figure 4-34: Drift vs. base contact for Ganey lumped plasticity models. ............................ 105 Figure 4-35: Parametric study of number of fibres on: (a) shear, (b) PT force, (c) base contact................................................................................................................................................ 108 Figure 4-36: Parametric study of CLT flexural stiffness on: (a) shear, (b) PT force, (c) base contact. .................................................................................................................................. 109 Figure 4-37:Parametric study of CLT compression stiffness on, (a) shear, (b) post-tension force, (c) base contact. .......................................................................................................... 110 Figure 4-38: Effect of CLT flexural and compression stiffness on (a) shear force, (b) post-tension force, (c) base contact. .............................................................................................. 110 Figure 4-39: Effect of CLT yield stress, (a.) shear force, (b.) post-tension Force, (c.) base contact. .................................................................................................................................. 111 Figure 4-40: Effect of CLT shear stiffness on, (a) Shear force, (b) post-tension force, (c) base contact. .................................................................................................................................. 112 Figure 5-1: Overview of two-story building (Griesenauer 2018). ........................................ 117 Figure 5-2: Slotted connection for column base (Griesenauer 2018). .................................. 118 xiv Figure 5-3: OpenSees model of 2 story building, and expanded groups of nodes. .............. 125 Figure 5-4: Overview of wall base elements. ....................................................................... 127 Figure 5-5: Moment-rotation backbone for lumped plasticity model. .................................. 128 Figure 5-6 Equivalent bar elements. ..................................................................................... 130 Figure 5-7: Main regions of performance for a rocking wall system. .................................. 132 Figure 5-8: Summary of algorithm to calibrate material property. ....................................... 135 Figure 5-9: Idealized flange model. ...................................................................................... 136 Figure 5-10: Objective function for Optimization Analysis. ................................................ 137 Figure 5-11: Pushover of two story building, for the fibre model (FM) and lumped plasticity model (LPM). ........................................................................................................................ 138 Figure 5-12: Filtered vs. unfiltered shear force. ................................................................... 140 Figure 5-13: GM2 fibre model results. ................................................................................. 141 Figure 5-14: GM9 fibre model results. ................................................................................. 142 Figure 5-15: GM12 fibre model results. ............................................................................... 142 Figure 5-16: GM2 lumped plasticity model.......................................................................... 143 Figure 5-17: GM9 lumped plasticity model.......................................................................... 144 Figure 5-18: GM12 lumped plasticity numerical analysis. ................................................... 144 Figure 5-19: Peak design parameters from the fibre and lumped plasticity model compared to experiment............................................................................................................................. 146 Figure 5-20: Comparison of the base shear and inter-story drift for the lumped plasticity model (LPM) and fibre model (FM) for a two story building with a rigid foundation. .................. 150 Figure 5-21: GM2 comparison of fibre model (FM) and lumped plasticity model (LPM) for non-sequential runs. .............................................................................................................. 151 Figure 5-22: GM9comparison of fibre model (FM) and lumped plasticity model (LPM) for non-sequential runs. .............................................................................................................. 152 xv Figure 5-23: GM12 comparison of fibre model (FM) and lumped plasticity model (LPM) for non-sequential runs. .............................................................................................................. 152 Figure 5-24: Peak design parameters from the fibre and lumped plasticity (LP) model compared to experiment. ....................................................................................................... 153 Figure 5-25: Rayleigh Damping forces for numerical models with a rigid foundation. ....... 154 Figure A-1: System Hysteresis vs. experimental data for fibre model, GM1-8. .................. 172 Figure A-2: System Hysteresis vs. experimental data for fibre model, GM9-12. ................ 173 Figure A-3: System shear vs. experimental data for fibre model, GM1-8. .......................... 174 Figure A-4: System shear vs. experimental data for fibre model, GM9-12. ........................ 175 Figure A-5: Roof drift vs. experimental data for fibre model, GM1-8. ................................ 176 Figure A-6: Roof drift vs. experimental data for fibre model, GM9-12. .............................. 177 Figure A-7: Floor drift vs. experimental data for fibre model, GM1-8. ............................... 178 Figure A-8: Floor drift vs. experimental data for fibre model, GM9-12. ............................. 179 Figure A-9: East wall outer Post-tension force vs. experimental data for fibre model, GM1-8................................................................................................................................................ 180 Figure A-10: East wall inner post-tension force vs. experimental data for fibre model, GM9-12........................................................................................................................................... 181 Figure A-11: East wall outer post-tension force vs. experimental data for fibre model, GM1-8................................................................................................................................................ 182 Figure A-12: East wall outer post-tension force vs. experimental data for fibre model, GM9-12........................................................................................................................................... 183 Figure A-13: U-shaped flexural plate deformation vs. experimental data for fibre model, GM1-8............................................................................................................................................. 184 Figure A-14: U-shaped flexural plate deformation vs. experimental data for fibre model, GM9-12........................................................................................................................................... 185 Figure B-1: System hysteresis vs. experimental data for fibre model, GM1-8. ................... 187 xvi Figure B-2: System hysteresis vs. experimental data for lumped plasticity model, GM9-12................................................................................................................................................ 188 Figure B-3: Shear vs. experimental data for lumped plasticity model, GM1-8. ................... 189 Figure B-4: Shear vs. experimental data for lumped plasticity model, GM9-12. ................. 190 Figure B-5: Roof drift vs. experimental data for lumped plasticity model, GM1-8 ............. 191 Figure B-6: Roof drift vs. experimental data for lumped plasticity model, GM9-12. .......... 192 Figure B-7: Floor drift vs. experimental data for lumped plasticity model, GM1-8. ........... 193 Figure B-8: Floor drift vs. experimental data for lumped plasticity model, GM9-12. ......... 194 Figure B-9: East wall outer post-tension force vs. experimental data for lumped plasticity model, GM1-8. ...................................................................................................................... 195 Figure B-10: East wall outer post-tension force vs. experimental data for lumped plasticity model, GM9-12. .................................................................................................................... 196 Figure B-11: East wall inner post-tension force vs. experimental data for lumped plasticity model, GM1-8. ...................................................................................................................... 197 Figure B-12: East wall inner post-tension force vs. experimental data for lumped plasticity model, GM1-8. ...................................................................................................................... 198 Figure B-13: U-shaped flexural plate displacement vs. experimental data for lumped plasticity model, GM1-8. ...................................................................................................................... 199 Figure B-14: U-shaped flexural plate displacement vs. experimental data for lumped plasticity model, GM9-12. .................................................................................................................... 200 Figure C-1: System hysteresis for fibre and lumped plasticity model GM1-8. .................... 202 Figure C-2: System hysteresis for fibre and lumped plasticity model GM9-13. .................. 203 Figure C-3: Shear force for fibre and lumped plasticity model GM1-8. .............................. 204 Figure C-4: Shear force for fibre and lumped plasticity model GM9-13. ............................ 205 Figure C-5: Roof drift for fibre and lumped plasticity model GM1-8. ................................. 206 Figure C-6: Roof drift for fibre and lumped plasticity model GM9-12. ............................... 207 xvii Figure C-7: Floor drift for fibre and lumped plasticity model GM1-8. ................................ 208 Figure C-8:: Floor drift for fibre and lumped plasticity model GM9-13. ............................. 209 Figure C-9: East wall inner post-tension force for fibre and lumped plasticity model GM1-8................................................................................................................................................ 210 Figure C-10: East wall outer post-tension force for fibre and lumped plasticity model GM9-13........................................................................................................................................... 211 Figure C-11: East wall outer post-tension force for fibre and lumped plasticity model GM1-8................................................................................................................................................ 212 Figure C-12: East wall outer post-tension force for fibre and lumped plasticity model GM9-13........................................................................................................................................... 213 Figure C-13: U-shaped flexural plate displacement for fibre and lumped plasticity model GM1-8............................................................................................................................................. 214 Figure C-14: U-shaped flexural plate displacement for fibre and lumped plasticity model GM9-13........................................................................................................................................... 215 xviii Acknowledgements I would like to first thank my supervisor, Professor Terje Haukaas, for his guidance and financial support. Without him believing in me, this research would not have been possible. Thanks also go to my co-supervisors, Dr. Carlos Molina-Hutt, for offering his time to my Mitacs project, and Dr. Marjan Popovski, for his financial support and technical expertise. I\u2019d also like to thank my family, for their support and interest in my education throughout the years. Special thanks go to my parents for nurturing my compulsion to ask why. I want to extend thanks Adam Gerber for introducing me to the subject of my thesis and providing me with feedback throughout for my research. I would also like to extend thanks to people in industry who offered their time to make recommendations about my research, including Armin Bebam Zadeh, Lucas Epp, and Robert Jackson. This research was also enabled by the community of graduate students at UBC, who tolerated my questions and helped keep me sane. Special thanks to Mike Fairhurst and Fabricio Bagatini Cachuco for providing a significant amount of technical support and help with learning the beast that is OpenSees. xix Dedication For the prosecution knowledge seeker. Oh no you found me 1 Body Chapter 1: Introduction In modern years, timber structures are being pushed to ever higher heights. For most of the 20th century, wood products were primarily used in light frame construction with few exceptions. However, new products developed in the past twenty years have made it possible for timber to capture a larger segment of the building market (Brandner et al. 2016). Driven by a global desire to reduce carbon emissions, timber is increasingly used in tall buildings and high-performance seismic applications. This trend is exemplified by the design and construction of buildings such as Brock Commons in Vancouver Canada, or the Mj\u00f8st\u00e5rnet in Brumunddal Norway. These tall timber buildings are the first of their kind and the cumulation of decades of research into tall timber. In tandem to advances in the timber sector, structural engineers have become increasingly concerned with quantifying the performance of buildings during seismic events. Advances in numerical modelling have made it practical to predict a building\u2019s behaviour with increased accuracy. This has enabled engineers to use performance-based design to think beyond collapse prevention and estimate the life-cycle cost of a structural system. Examining buildings from a life-cycle perspective has also led to higher scrutiny of contemporary building systems. Many modern building systems rely on the yielding of structural components to dissipate seismic energy. This yield mechanism often occurs in components that are part of the main structural system, such as the rebar in reinforced concrete, or hold-down connectors in timber walls. Damage caused by yielding of these members can be difficult and costly to repair, making them less appealing from a lifecycle cost perspective. This was highlighted by the poor performance of buildings during the 2011 Christchurch earthquake (Pampanin et al. 2011). While many structures withstood the earthquake without 2 collapsing, there was widespread damage to the surviving structures. This led to a significant portion of the existing building stock being demolished. To decrease the lifecycle cost of buildings, new innovative solutions are needed. As response to these two pressures, self-centring Cross-Laminated Timber (CLT) wall systems have emerged as a timber structural system that can be used in tall buildings. When coupled with external energy dissipation devices, self-centring CLT walls have the potential to be a low damage seismic force resisting system with high ductility. As the name suggests, self-centring walls can centre themselves after seismic forces are applied, reducing damage and repair costs. The external energy dampers add ductility to the system and can be easily replaced post-earthquake. One challenge that exists with self-centring wall systems is that they are difficult to analyze. When force is applied to the system, the response of the system is highly nonlinear. In the past this has required specialized models to assess forces in the system. These specialized modelling approaches often use methods which are not well known in the structural engineering community. For self-centring CLT walls to be widely adopted, there needs to be an improved understanding of how to model self-centring systems with widely adopted structural models. This document presents a study of contemporary structural analysis models for self-centring CLT walls, using two widely available nonlinear models. 3 1.1 Research Motivations While self-centring wall systems using precast concrete or Laminated Veneer Lumber (LVL) have been researched significantly, systems using CLT are still relatively new. There are still many unknowns about how models tested in other materials will perform when applied to CLT systems. While some studies have modeled CLT walls under static loads (Akbas et al. 2017; Ganey 2015), at the time of writing there are no studies that numerically model the dynamic performance of self-centring CLT walls. Many contemporary studies of self-centring walls have used multi-spring numerical models to analyze these systems. However, these models can be difficult to implement, and many practicing engineers are unfamiliar with them. In CLT there is considerably less experience with models using fibre and lumped plasticity element formulations. These models are attractive because they are easy to use and can be easily implemented in many commercial software packages. Comparing the results of these models to experimental data will give a better understanding of how accurate these models are, and where they can be successfully employed. A recent development in earthquake structural engineering is the Python interpreter of the finite element software package Open System for Earthquake Engineering Simulation (OpenSees) (Mckenna 2011). By extending OpenSees to Python with OpenSeesPy (Zhu et al. 2018), it is possible to run finite element simulations and process data in a more streamlined manner. This has the potential to reduce development time and enable sophisticated optimization and decision-making. In part, this thesis is also a proof of concept that demonstrates some of the capabilities that are possible using OpenSeesPy. Almost all of the data handling and analyses presented in this thesis are completed leveraging Python and OpenSeesPy. 4 1.2 Scope and Objectives The objective of this research is to develop a better understanding of how lumped plasticity and fibre-based nonlinear models can be used to analyze self-centring CLT walls. To achieve this goal, the following sub-objectives are completed: \u2022 Analyze data from experimental studies of self-centring CLT walls to see if any trends exist in the in-plane stiffness of CLT. \u2022 Improve understanding of how to model the static behaviour of self-centring CLT systems by: o Analyzing wall systems from two contemporary research studies, using both lumped plasticity and fibre-based numerical models. o Comparing experimental results with numerical results from both wall models. o Performing a sensitivity analysis on key parameters for self-centring CLT walls. \u2022 Improve understanding of how contemporary of fibre-based and lumped plasticity models capture the dynamic behaviour of self-centring CLT systems by: o Analyzing a two-story building tested under dynamic loads using both lumped plasticity and fibre-based numerical models. o Comparing results of the model to experimental data. o Inferring the performance of the building in conditions different than the experiment. \u2022 Leverage the new Python interpreter of OpenSees to reduce development time and automate analyses. 5 1.3 Thesis Outline To address the scope of work, research is presented in the following structure: Chapter 2 describes existing research on self-centring CLT systems. A history is given of CLT as a structural material, and self-centring walls as a resilient lateral force resisting system. Chapter 3 describes the relevant analysis models for self-centring CLT walls. Existing research on the in-plane properties CLT is presented. Existing research is also presented on numerical models for self-centring CLT systems, as well as common seismic dampers. Chapter 4 presents research into the analytic models of self-centring CLT walls at a component level. Experimental data from two existing research programs on self-centring CLT walls is reviewed, and the CLT stiffness in each study is analyzed. Walls are then modeled using fibre and lumped plasticity models. Results from this analysis are compared with experimental results. Sensitivity studies for key analysis parameters are then presented. Chapter 5 presents research into the dynamic analysis of a two-story building using self-centring walls. Experimental data from an existing research program is compared with fibre and lumped plasticity numerical models of the building. The building is analyzed under two states: one where it has a flexible foundation, and one where it has a rigid foundation. A calibration analysis is completed to determine the properties of the flexible foundation. Data for dynamic models with the flexible foundation are compared to experimental data, while the model without a beam is used to predict the performance of the building. Chapter 6 summarizes the key findings of the research program and makes recommendations for future work. 6 Chapter 2: Research Background In this Chapter, an overview of past research into mass timber technolog, and self-centring wall systems is provided. The history of modern mass timber is discussed, focusing on developments in mass timber and CLT. Prior research into lateral force resisting systems using CLT is also presented, with a focus on major experimental programs. An overview of the lateral force-deformation relationship for self-centring walls systems is presented, followed by a review of major experimental programs studying self-centring walls. Finally, contemporary research on the in-plane properties of CLT is discussed. 2.1 CLT as a Lateral Force Resisting System CLT has recently emerged as the material of choice for many high-performance timber structural systems. This surge in popularity for CLT was enabled by research that chiefly took place over the past twenty years. To understand why CLT has become a popular building material, it is important to first understand the mechanical properties and manufacturing process for wood. Wood has been used as saw lumber for most of the history of wood as a structural material. These structural elements were directly cut from tree wood with little modification. Wood is an anisotropic and orthotropic material; the strength and stiffness of wood vary significantly depending on the direction and type of loading applied to it. Figure 2-1 depicts three significant directions of loading for a saw lumber member: parallel to the wood fibres, transverse to the fibre, and radial to the fibres (Kretschmann 2010). When loaded parallel to the grain of the fibres, wood has strength and stiffness to weight rations that are high when compared to other structural materials. However, the out of plane strength properties of wood in the tangential 7 and radial directions are significantly weaker. This restricts how saw timber members can be loaded efficiently. Figure 2-1: The significant loading directions for timber (Kretschmann 2010). There are several other factors that have traditionally limited wood as a structural material. Wood has mechanical properties that are highly variable when compared to other building materials (Kretschmann 2010). Microscopic natural variations that occur during the life of a tree can have a significant influence on the strength and stiffness of timber members. In addition, macroscopic defects such as knots can create stress concentrations that limit the overall strength of a member. Wood is sensitive to moisture, and the moisture ratio of a timber member will influence the strength of timber and cause it to expand or contract (Kretschmann 2010). Perhaps most importantly, the size of structure members that can be made from solid wood is limited by the size of the trees they are cut from. In the past, this bounded the type of structural member which were made from wood. Because saw lumber is relatively small in size, wood was typically not used in larger buildings. In the early 2000s, advances in manufacturing technology made it possible to economically create larger timber structural elements. These so-called mass timber elements are composed of a wood product such as wood fibres, veneers, or boards, bound together into a larger 8 structural element, typically with an adhesive. Combining timber elements in this way helped overcome many of the limiting factors on timber-based construction. Larger timber members allowed the strength of wood to be better utilized, as knots or other defects could easily be removed or placed in non-critical areas. The resulting structural members were also more dimensionally stable, and less prone to swelling and shrinking with moisture. Larger timber members take longer to combust, and therefore slow the rate of fire growth and retain their strength for longer during a fire. Combined, these factors allow for mass timber to be used in a much greater range of applications than saw lumber. Originally developed in Austria and Germany, CLT was one of the more popular products to emerge during the early 2000s. CLT is a plate-like mass timber product, manufactured by gluing together orthogonally alternating layers of timber into a solid panel. Figure 2-2 depicts a typical CLT cross section with five layers. Each layer of timber is glued together along the top and bottom surface of each board, for example the boards between layer \u201cd1\u201d and \u201cd2\u201d would be glued. Typically, the edges between boards in the same layer are not glued together, for example the boards in the layer \u201cd2\u201d, would not be glued. Panels of CLT generally have three, five, or seven layers, which range in thickness from approximately 10mm to 40mm. Figure 2-2: Diagram of CLT, modified from Gagnon and Pirvu (2011). 9 A typical CLT panel will have width from one to four metres, and lengths up to 18 metres. By stacking layers in alternating directions, panels can have a \u201ctwo-way action\u201d, similar to reinforced concrete (Mohammad et al. 2012). Panels have a \u201cstrong\u201d axis that runs in the direction of the top and bottom layer of CLT and a \u201cweak\u201d axis that runs perpendicular to those layers. In Figure 2-2, B-B is a section cut of the strong axis. By varying the size and number of strong and weak axis layers, the properties of CLT in each direction can be controlled. This allows CLT to be a versatile material that can be tailored for many different applications. Because it has relatively high in plane stiffness, CLT showed a lot of promise as a lateral force resisting member. In the early 2000s, several major research programs studied laterally loaded CLT, and helped to bring prominence to CLT on a worldwide stage. Initial tests highlighted the importance of connectors for CLT structures. One prominent initial study of laterally loaded CLT were tests performed at the University of Ljubljana by Dujic et al. (2006). Three possible boundary conditions were considered for the CLT walls: Rocking response, where one edge of the panel base was free to rotate at the base; combined rocking and shear, where the panel was fixed at both base edges; and shear response, where the base was fixed and rotation of the cantilever at the top end was also fixed. The study found that the ultimate strength of walls was strongly influenced by the boundary condition considered. This finding is a key aspect of CLT behaviour and implies that connector performance will significantly impact wall strength. Knowledge about the performance of CLT under lateral loads was further expanded by the \u201cFiemme house constructive system\u201d, or the SOFIE project, undertaken by the Trees and Timber Institute of the National Research Council of Italy. This project studied the seismic performance of multistory buildings made from CLT and culminated in the seismic testing of 10 a seven story CLT building. As part of the SOFIE initiative, quasi-static cyclic and pseudo dynamic tests were completed on CLT wall components and building specimens. These tests showed that the connection detailing had a significant impact on the strength and ductility of both the wall and building the system (Lauriola and Sandhaas 2006). Figure 2-3 shows a typical test setup for the CLT wall specimens. Also depicted in Figure 2-3 is a typical force-deformation experiment hysteresis, where the shear force applied is plotted as a function of the actuator drift during testing. The wall systems were found to have relatively high stiffness, and the ability to dissipate energy through ductility. CLT walls were observed to have a \u201cpinching\u201d type behaviour when loaded cyclically. Pseudo dynamic tests on a one-story building model were also completed as part of the study. These building level tests confirmed that the wall specimens in buildings performed similarly to the wall component tests. Figure 2-3: Experimental setup and results from the initial phases of the SOFIE project (Lauriola and Sandhaas 2006). Once component level data were gathered, 1D shake-table tests were completed at the National Research Institute for Earth Science and Disaster Resilience (NIED) in Japan (Ceccotti and Follesa 2006). Three buildings that were three stories tall were tested. These tests reaffirmed the ability of CLT to resist lateral loads and provided valuable data about the progression of damage in CLT structures. Finally, a seven story building was seismically tested at the NIED 11 facility in Japan (Ceccotti and Carmen 2013). The building performed similar to expectations, with minimal residual drift noted after the suite of earthquakes used. While most research prior to the early 2000s was initiated in European countries, by the 2010s interest in CLT had reached a more global market. In North America there were several influential research programs that further advanced CLT research. With these programs came a shift in focus, as research began to explore the application of CLT in high seismic regions for mid-rise and high-rise buildings. Beyond verifying performance, researchers were now investigating methods for design and analysis that were specific to CLT. In Canada, a significant amount of research on CLT products was completed by the institute FPInnovations, Canada\u2019s forest product research institute. In 2011 FPInnovations released their CLT handbook, which provided comprehensive information about CLT, including state of the art information about lateral loads (Gagnon and Pirvu 2011). Popovski et al. (2010) completed a comprehensive series of quasi-static tests on laterally loaded CLT wall panels at FP Innovation\u2019s laboratory in Vancouver Canada. This program tested a suite of 32 walls under monotonic and cyclic loading conditions, with the objective of testing a broad range of timber connectors. The study resulted in a better understanding of the hysteretic behaviour of CLT walls using common connector types, as well as the strength and stiffness properties of those connectors. Follow up studies included pseudo dynamic tests on a full scale two-story CLT house (Popovski et al. 2014), and efforts to standardize analysis and design models for CLT planar walls and wall assemblies (Gavric and Popovski 2014). Outside of FPInnovations, research programs continued in Europe and the USA. Gavric et al. (2015) examined the lateral load behaviour of single and coupled CLT wall panels at the IVALSA Tree and Timber Institute. In this study behaviour of single and coupled walls was 12 characterized based on the stiffness of their connectors. Based on these findings, analytic models were proposed for the backbone of various wall configurations. During testing it was observed that CLT panels deformed little in-plane, compared to deformation from rigid body motion. Work also progressed on numerical models for CLT systems. Pei et al. (2013) proposed a hysteretic model for the lateral force-drift relationship for CLT walls. This model was validated against test data from FPInnovations wall tests and found to be accurate. The model was then used to complete a performance-based design for a six story CLT apartment building located in Los Angeles. This building was tested dynamically, and the resulting interstory drifts at various shaking intensities were used to propose a R factor for the structure. The proposed R-factor for the building was 4.5. Figure 2-5a shows the pinching shaped hysteretic model for the structure, and several of the important parameters for the model. The predictions of this model compared to experimental results are shown in Figure 2-5b, and it can be seen that the model can reasonably match the data. Figure 2-4: Overview of (a), the model used for CLT, (b) the results of the model compared with wall experimental results (Pei et al. 2013). (a) (b) 13 Figure 2-5a shows the geometry of the building tested in the numerical study. Figure 2-5b shows a cumulative density function for the maximum drift observed in the experimental suite of motions, at each intensity level. Figure 2-5: Summary of (a) the building tested, and (b) the results of dynamic testing on the building (Pei et al. 2013). 2.2 Self-centring Walls System Concept Self-centring wall systems with seismic dampers is a low-damage lateral force resisting system. The system is appealing mainly because of two properties: self-centring walls can re-centre after a seismic event, and dampers in the system can be easily repaired. Figure 2-6 shows a rocking self-centring CLT wall and its key features. There are two critical structural components for self-centring walls: the shear wall itself and a post-tension tendon. The shear wall resists lateral load, while the post-tension tendon connects the top of the wall to the foundation and is used to apply an initial\/restoring force to the system. The base of the shear wall will have connection detailing so that rotation is permitted but sliding along the base is not. The post-tension tendon are given an initial stress, which applies a compression force to the wall system. Because the wall can rotate at the base connection, the initial force applied by post-tension tendon is critical for the systems initial stiffness. When lateral force is initially (a) (b) 14 applied to the system, moment is carried by a redistribution of the compression forces at the base of the wall. More initial force will mean that the wall will maintain this initial stiffness for longer. The post-tensioned tendon is also essential for the system to re-centre. While the post-tension tendon remains elastic, it applies a restoring force that will allow the wall to return a centred position. Because rotation is permitted at the walls base, each wall will have three predominant components of deformation: flexural, shear, and rigid body movement. These components are shown in Figure 2-7. Figure 2-6: Self-centring wall system. Figure 2-7: Components of deformation for a rocking wall system. Shear wall Post-tension tendon Flexure Shear Rocking Total 15 Figure 2-8 depicts the shear force deformation relationship for a typical self-centring rocking wall system without dampers. There are four main regions of the force deformation curve. In the first region the entire base of the wall is under compression and behaves linearly. As lateral force is applied to the system the base of the wall will eventually begin to uplift at one of the wall edges, creating a gap between the wall and the foundation. The point where uplift initiates and the base begins to rigidly rotate is termed the decompression point. After the decompression point, there is a transition into a second linear plateau as the wall begins to uplift in region two. If energy dampers are included in the system, they typically will yield in this region. Region three is characterized by a softening of the second linear slope as the wall begins to crush at the rocking toe. In region four, the post-tension tendon has begun to yield, and the systems stiffness drops dramatically. The ultimate failure of the system will occur either when the toe of the wall crushes, or when the post-tension tendon ruptures. For most of the systems studied, yielding of the post-tension tendon is the ultimate limit state. Figure 2-8: Key limit states for the pushover of a self-centring wall system Figure 2-9 overviews a cycle where lateral force is applied to a self-centring wall system, then released. In this figure, the shear force applied to the system is denoted \u201cV\u201d, the force in the Region 1 Region 2 Region 3 Region 4 Drift (%) Base Shear 16 post-tensioned tendon, \u201cP\u201d, the amount of base in contact \u201cC\u201d and displacement at the top of the wall \u201cu\u201d. Deformations in Figure 2-9 have been exaggerated for clarity. Figure 2-9: Shear force (V), post-tension force (P), and base contact (C) for a the elastic range of a self-centring wall. V C P u u u V C P u u u V C P u u u V C P u u u V C P u u u V C P u u u Rest Elastic response Post-tension elongation begins Decompression Load is removed Second elastic slope 17 In the initial state of rest, the post-tension tendon is in tension and the wall is in compression. When the initial lateral load is applied to the wall system, the lateral force-deformation response of the wall is elastic. This is because the applied bending moment is carried by a redistribution of compression force at the base of the wall. As more lateral force is applied, this elastic response continues will continue until compression stress at the wall toe opposite to the load direction becomes zero. The decompression moment of the wall, \ud835\udc40\ud835\udc51\ud835\udc52\ud835\udc50, can be calculated based on simple statics: \ud835\udc40\ud835\udc51\ud835\udc52\ud835\udc50 =\ud835\udf0e\ud835\udc43\ud835\udc47\ud835\udc3c\ud835\udc66=\ud835\udc470\ud835\udc3c\ud835\udc34\ud835\udc66 (2.1) where: \ud835\udf0e\ud835\udc43\ud835\udc47 is the initial post tension stress at the base of the wall; I is the moment of inertia for the wall; y is the distance to the edge of the wall from the wall centroid ; \ud835\udc470 is the initial post tension force applied to the wall; A is the cross sectional area of the wall. After the decompression load is reached, a gap between the wall and foundation will begin to open at the wall toe. As the gap continues to open, the wall system will rigidly rotate at the wall base. This rigid rotation increases the lateral deformation and causes the shear force displacement curve to soften. Analysis of forces in the wall base is difficult after uplift begins. Plain sections no longer remain plane and there is no strain compatibility between the section and post-tension tendon. Analysis techniques for walls past decompression will later be discussed in Chapter 3. As more load is applied, the wall will continue to rotate at the base connection, and less of the wall is in contact with the foundation. Eventually the neutral axis will reach a post-tension tendon and any additional rotation will cause that tendon to elongate. Figure 2-10 shows a self-centring wall prior to the neutral axis reaching the post-tensioned cable, and after it reaches the 18 post-tension cable. Uplift reaching the post-tension tenon will stop the softening of the lateral force-deformation curve, creating a second linear slope. As more lateral force is applied to the system, the tension force in that post-tension tendon will begin to increase. This will also increase the compression force at the base of the CLT wall. If load is removed when the wall is in this zone, the force in the post-tension cable will pull the system back to a centred position, with no damage occurring to the wall. Figure 2-10: Position of the neutral axis as rocking develops. If seismic dampers are included, they will generally be designed such that they yield after decompression of the system. As the system undergoes deformation cycles, the dampers will yield and begin to dissipate seismic energy from the system. Dampers will generally be sized so that the initial post-tension force is great enough to overcome the yield force. Figure 2-11 depicts the resulting \u201cflag shaped\u201d hysteretic behaviour as a self-centring wall goes through a cycle of load. The flagged shaped hysteresis is a combination of the self-centring elastic behaviour, Figure 2-11a, and the hysteretic damper behaviour, Figure 2-11b. Because the seismic dampers are external to the structural element, they can readily be repaired. 19 Figure 2-11: Hysteretic behaviour of self-centring walls. Continued loading will follow the second linear curve until a failure limit state is reached. The failure will limit state of the wall will either be a compression failure in the toe of the wall or yielding of the post-tension tendon. Which failure occurs first depends on the type system and detailing. For walls made of timber, some timber yielding is generally permitted before the wall ultimately fails in compression. 2.3 Self-centring Walls as a Resilient Lateral Force Resisting System Research into self-centring walls as a lateral force resisting system was first pioneered by the joint US-Japan Precast Seismic Structural System (PRESSS) research program (Priestley 1991). This program examined the performance of post-tensioned, precast concrete moment frame and shear wall systems. The objective of the PRESSS program was to develop a seismically efficient, low damage precast concrete structural system as well as recommendations for analysis and design of that system. This program was broken into three distinct phases. During the first phase, various system concepts were evaluated to determine which ones were most suitable for testing in the program (Nakaki and Englekirk 1991). The second phase consisted of experimental studies performed on components of the precast system, such as wall systems. Notable outcomes of phase II included the development of (b) Seismic damper (a) Self-centring wall (c) Combined hysteresis 20 analysis and design methods for post-tensioned precast concrete walls (Kurama 1997; Kurama et al. 1999) and testing of various seismic dampers (Schultz and Magana 1996). The third and final phase of the program concluded with the design and testing of a five story post-tensioned structure, depicted in Figure 2-12 (Nakaki et al. 1999). Pseudo-dynamic tests were used to simulate earthquake activities at four different hazard levels: 33%, 50%, 100%, and 150% of the design level earthquake (Priestley et al. 1999). These hazard levels resulted in drifts up to 2.5%, and it was found that the system sustained low damage for the earthquakes simulations considered. Figure 2-12: 6 Story PRESSS structure shake table test (Nakaki et al. 1999). After the PRESSS program concluded, there was continued interest in self-centring systems using other materials, particularly in timber. In New Zealand, extensive research efforts were 21 made to create self-centring systems using the Laminated Veneer Lumber (LVL) at the University of Canterbury. These wall and moment frame lateral force resisting systems were referred to as Press-lam. LVL was considered as a potential alternative to precast concrete because it has a relatively high stiffness and can be premanufactured. Initial component level tests on moment frames showed that Press-lam moment frames with axial dampers had a stable flag shaped hysteresis similar to post-tensioned concrete systems (Palermo et al. 2005). Other studies used reverse cyclic loading tests to examine the performance of single (Newcombe 2008) and paired wall systems constructed using LVL (Iqbal et al. 2007) as cited in Sarti (2015). These tests confirmed that LVL was a viable material for self-centring walls, and paired wall tests showed U-shaped flexural plate dampers could be used to control the energy dissipation of the system. Later experimental studies at the University of Canterbury focused on analysis and connection design of different system configurations. Investigations by Newcombe helped adapt analysis and design techniques from precast concrete to LVL systems (Newcombe 2008, 2011). In addition, Newcombe performed unidirectional and bidirectional pseudo dynamic testing on a 2\/3 scale Press-lam building models (Newcombe 2011). This building test demonstrated that the performance of post-tensioned wall and moment frame connections in buildings was comparable to component level tests. The biaxial building tests also showed that the structure behaved similarly under bidirectional and unidirectional loading. Armstrong investigated the connection analysis and design of post-tensioned moment frames using various axial dampers (Armstrong 2015). The performance of paired walls with U-shaped flexural plate dampers was investigated by Iqbal et al. (2015). These experiments verified that LVL systems using these dampers could dissipate energy in a stable way, while 22 retaining a reentering behaviour. Figure 2-13 overviews the so called \u201cflag shaped\u201d hysteretic behaviour of an LVL paired wall system, where the applied lateral force is plotted against displacement at the top of a wall. Results are shown for a pair wall system with and without dampers. Figure 2-13: Experimental results for press-lam walls with and without u-shaped flexural plate dampers (Iqbal et al. 2015). Sarti experimentally investigated different damper and connection details for Pres-Lam wall and frame systems (Sarti 2015). As part of this research program, Sarti also compared the analytic models in contemporary research to experimental data and proposed modifications to better fit that data. This and other studies formed a base of knowledge which Sarti used to complete a FEMA P-695 (ATC 2009) study on a suite of Pres-Lam buildings (Sarti et al. 2016). It was proposed that the Pres-lam system had a response modification factor, system overstrength factor, and deflection amplification factor of 7, 3.5, and 7.5 respectively. Figure 2-14 overviews the suite of buildings studied in Sarti\u2019s FEMA P-695 procedure. 23 Figure 2-14: Summary of buildings numerically modeled in Sarti's FEMA P695 study (Sarti et al. 2016) Following the success of research into LVL self-centring systems in New Zealand, there was interest in developing self-centring wall systems using CLT in North America. CLT buildings using self-centring walls could capitalize on the in plane strength of CLT, as well as existing manufacturing infrastructure and experience with the material. A major research initiative for self-centring CLT was the NHERI TallWood Project funded by the U.S. National Science Foundation (Pei et al. 2017). Currently ongoing, the objective of this program is to develop the knowledge and experience needed to design self-centring CLT structures in the range of 8-20 stories. As part of a planning phase of the project, Ganey conducted a series reverse cyclic wall tests on CLT rocking wall specimens at the University of Washington (Ganey 2015). Seven specimens were loaded up to 10% drift, revealing the important limit states of the system. Different design parameters such as post-tension force was varied to see their impact on building performance. 24 In a later study, Akbas used data from these wall model tests to develop nonlinear models of the wall systems (Akbas et al. 2017). Uniaxial shake-table tests of a two story self-centring CLT structure were completed (Pei et al. 2019; Wichman 2018). It was observed that the building experienced very low damage during all earthquakes considered. Future components of the project may include the design and shake table test of a 10 story self-centring CLT building (Pei et al. 2017). Outside of the NHERI TallWood framework, there are several other notable programs studying CLT systems. Ongoing research into self-centring CLT panels is being undertaken at FPInnovations. One significant study took place at FPInnovations in Vancouver, where Chen et al. (2018, 2020) tested a series of self-centring CLT walls. These walls were tested using reverse cyclic loading with both axial buckling restrained brace and U-shaped flexural plate dampers. Research into self-centring CLT also took place at the consulting company KPFF as part of a ongoing effort to design the Frameworks building (Zimmerman and Mcdonnell 2017, 2018). This twelve-story building was designed for Portland Oregon, a high seismic active zone. Figure 2-15 overviews the structural system, showing the CLT wall coupled with u-shaped flexural plate dampers to boundary column elements in the gravity system. The Frameworks building was designed with a performance-based design approach and made use of site-specific testing of CLT walls and connections that are not publicly available. Completion of this design demonstrated that self-centring CLT walls could be used as a viable solution for buildings in this height range. 25 Figure 2-15: (a) Architectural rendering of Frameworks building, (b) structural renderingand (c) lateral load Resisting System for Frameworks Building (Zimmerman and Mcdonnell 2017) 2.4 In-plane Properties of CLT While CLT has been extensively researched as a lateral load resisting system, there is still ongoing research into how individual panels of CLT deform when loaded in plane. This gap in knowledge likely exists because for many smaller structures CLT behaves as a rigid body when loaded laterally. For shorter walls, rigid body motion of the wall due to the deformation of the connections tends to be much larger than the deformation in the CLT panel. However, as CLT is used in larger structures and connection detailing improves, the in-plane strength of CLT is increasingly becoming important. For CLT, two loading conditions are particularly significant: the bending and shear stiffness, and the local axial stiffness of CLT loaded in compression. Figure 2-16 shows these loading conditions in a), laterally loaded CLT section, and b), the compression loading condition at the toe of a CLT wall. (a) (b) (c) 26 Figure 2-16: In-plane loading for CLT. It is often the case that the boards in each layer of CLT are not glued together along their edges. For these panels without edge gluing, the in-plane normal forces from bending must be transferred across a transverse layer of CLT. Figure 2-17 denotes this phenomenon on the bending forces across a CLT cross section. To ensure section compatibility, the normal forces must be transferred between layers. Figure 2-17a shows the a cross section with perfect bonding between layers, while Figure 2-17b depicts a cross section with only partial bonding. Because the bond between layers is imperfect, there will be some interface slip between layers and the section will have a reduced bending and shear stiffness. While studying the strength and deflection of CLT beams, Flaig proposed analytic models for the shear stress and stiffness of CLT members loaded in plane (Flaig 2013). Figure 2-18 depicts the two extra components of shear deformation proposed in this model, with \u0263\ud835\udc66\ud835\udc65 accounting for shear formation from slippage, and \u0263\ud835\udc61\ud835\udc5c\ud835\udc5f accounting for shear deformation from torsion of the crossing layer. Equations for modified shear modulus were also proposed by Flaig: a) In-plane bending and shear b) In-plane Compression 27 Figure 2-17: CLT normal forces with (a) no slip and (b) slip, image modified from Danielsson et al. (2017). Figure 2-18: Components of CLT Shear Deformation (Danielsson et al. 2017). \u0263\ud835\udc61\ud835\udc5c\ud835\udc5f =12\ud835\udc49\ud835\udc4f3\ud835\udc3e\u22191\ud835\udc5a3\u22191\ud835\udc5b\ud835\udc50\ud835\udc4e (2.2) \u0263\ud835\udc66\ud835\udc65 =2\ud835\udc49\ud835\udc4f3\ud835\udc3e\u2219 (1\ud835\udc5a\u22121\ud835\udc5a3) \u22191\ud835\udc5b\ud835\udc50\ud835\udc4e (2.3) \ud835\udc3a\ud835\udc52\ud835\udc53\ud835\udc53,\ud835\udc36\ud835\udc34 =6\ud835\udc495\ud835\udc34\ud835\udc54\ud835\udc5f\ud835\udc5c\ud835\udc60\ud835\udc60(\u0263\ud835\udc61\ud835\udc5c\ud835\udc5f + \u0263\ud835\udc66\ud835\udc65)=\ud835\udc3e\ud835\udc4f25\u2219\ud835\udc5b\ud835\udc36\ud835\udc34\ud835\udc61\ud835\udc54\ud835\udc5f\ud835\udc5c\ud835\udc60\ud835\udc60\u2219\ud835\udc5a2(\ud835\udc5a2 + 1) (2.4) \ud835\udc3a\ud835\udc52\ud835\udc53\ud835\udc53,\ud835\udc36\ud835\udc3f\ud835\udc47 = (1\ud835\udc3a\ud835\udc59\ud835\udc4e\ud835\udc5a+1\ud835\udc3a\ud835\udc52\ud835\udc53\ud835\udc53,\ud835\udc36\ud835\udc34 )\u22121 (2.5) where: \ud835\udc49 is the cross-section shear force; \ud835\udc3e is the slip modulus (N\/mm3); \ud835\udc5a is the number of boards in a CLT layer; \ud835\udc5b\ud835\udc50\ud835\udc4e is the number of glue lines in a CLT layer; \ud835\udc61\ud835\udc54\ud835\udc5f\ud835\udc5c\ud835\udc60\ud835\udc60 is the total (a) (b) 28 thickness of the CLT section; \ud835\udc61\ud835\udc61\ud835\udc5c\ud835\udc5f is the total thickness of the CLT section; \ud835\udc4f is the width of every CLT layer; and \ud835\udc34\ud835\udc54\ud835\udc5f\ud835\udc5c\ud835\udc60\ud835\udc60 is the thickness of the CLT section. These equations are highly dependent on the geometry of the panels within the CLT member, and a slip modulus which is equal to the deformation per unit of stress. Later studies helped verify Flaig\u2019s analytic equations for shear stress (Danielsson et al. 2017), (Jele\u010d et al. 2019). Another model for the effective shear modulus of CLT was proposed by Moosbrugger et al. (2006). Analytical equations were proposed that modeled shear deformation CLT in terms of two components of deformation: a shear and torsional mechanism. In addition, an empirical equation was to fit the results from a series of finite element studies. In a later study, Bogensperger et al. (2010) adopted the empirical equation for predicting the in-plane stiffness of CLT. Brandner et al. (2015) compared predictions made by the empirical equation against test data for small scale CLT specimens. For these small specimens, it was found that the equation predicted shear stiffness within a maximum error of approximately 20%. \ud835\udc3a090 =\ud835\udc3a0,\ud835\udc5a\ud835\udc52\ud835\udc4e\ud835\udc5b1 + 6\ud835\udefc\ud835\udc47 (\ud835\udc61\ud835\udc59,\ud835\udc5a\ud835\udc52\ud835\udc4e\ud835\udc5b\ud835\udc64\ud835\udc59)2 (2.6) \ud835\udefc\ud835\udc47 = \ud835\udc5d (\ud835\udc61\ud835\udc59,\ud835\udc5a\ud835\udc52\ud835\udc4e\ud835\udc5b\ud835\udc64\ud835\udc59)\ud835\udc5e (2.7) where: \ud835\udc3a090 is the gross section modulus across the section, \ud835\udc3a0,\ud835\udc5a\ud835\udc52\ud835\udc4e\ud835\udc5b is the shear mean modulus of each layer; \ud835\udc64\ud835\udc59 is the long edge length of CLT lamella; \ud835\udc61\ud835\udc59,\ud835\udc5a\ud835\udc52\ud835\udc4e\ud835\udc5b is the mean thickness of the CLT layers; \ud835\udc5d and \ud835\udc5e are empirically defined coefficients that depend on the number of CLT layers. 29 As part the consulting company KPFF\u2019s private research program on CLT walls for the Frameworks building, experimental testing of CLT panels was completed at Oregon State University (Zimmerman and Mcdonnell 2017). Measurements of shear stiffness were calculated and compared against the predictions from Flaig\u2019s equations, as well as Brandner\u2019s adoption of Moosbrugger\u2019s empirical equation. Figure 2-19 depicts the results of the CLT wall testing at different stress levels, against the predictions. It was found that both equations predicted dramatically different stiffnesses for the CLT panels, and that the neither equation was particularly accurate when compared to the test data. Figure 2-19: Testing of CLT shear properties for the Frameworks building (Zimmerman and Mcdonnell 2017). Based on the following research, there is still significant uncertainty about the stiffness of CLT when loaded in plane. For the shear stiffness models examined, it is unclear if they generalize well to walls. In all studies examined, the effect of interlayer slip on bending stiffness was not considered or assumed to be negligible. At the time of writing, no research is known that explicitly considers the effect of slip on bending stiffness. 30 An additional area of ongoing research is the in-plane compression strength and stiffness of CLT at the toe of CLT walls. As a CLT wall is loaded laterally, a compression force will develop at the toe of that wall. If this stress locally exceeds CLT\u2019s compression strength, the wall will begin to crush, as depicted in Figure 2-20. For traditional CLT wall systems using hold-down type connectors, the contribution of compression on wall performance is often negligible. Hold-down connections typically yield before compression failure occurs, and hold-down connections deformations are often far greater than compression deformation. However, in new applications of CLT such as self-centring wall systems, there is a large axial force applied to the wall panel. For these wall systems, the in-plane compression of CLT will have a significant effect on the global system, and the toe compression is being studied with increased scrutiny. Figure 2-20: Crushing of CLT at wall toe. The in-plane compression properties of CLT are often determined using a compression test. Tests are performed by applying an axial force to a small-scale specimen of CLT, and monitoring the resulting force deformation relationship. For CLT, the force deformation curve is similar in shape to other timber products: the curve is initially elastic until a yield force is reached. After the force reaches the yield point value, the member begins to crush and deformations dramatically increase. While the yield force and shape of the force-deformation 31 curve is known in general, it is less clear what length deformations should be measured over. Depending on what length deformations are measured over, dramatically different estimates of the elastic modulus will be made. This type of axial test does not necessarily reflect the loading condition As part of a study on self-centring CLT wall systems, Chen et al. (2018) completed a series of compression tests on CLT specimens at FP Innovation\u2019s Vancouver Campus. Figure 2-21 depicts an image of the setup used to measure compression properties. For some classes of specimen two trials were considered: one where the specimen was loaded directly in the centre of the specimen, and another where load was only applied to the edge of the specimen. For each specimen, measurements were made across the entire specimen, and within 25.4mm of the actuator head. Global estimates of the modulus of elasticity used deformation over the entire specimen, while local estimates only used deformation measured close to the actuator head. It was found that the local modulus of elasticity was dramatically different from the global modulus of elasticity. There was also a significant different in elastic modulus measured for test specimens that were loaded along their edge. Figure 2-21: Measurement made in Chen et al. (2018). 32 Several other notable studies have tested the in-plane compression properties of CLT, with an emphasis on CLT walls. In a student research project at Lehigh University, Duskin et al. (2017) tested a series of five CLT specimens loaded under in-plane compression. The CLT specimens were loaded in their centre and stress strain data was reported over three different zones. As part of his 2015 thesis, Ganey also performed in-plane compression tests on a series of CLT specimens (Ganey 2015). 33 Chapter 3: Analytic and Numerical Models of Self-centring Wall systems When loaded laterally self-centring walls have an initial linear slope, followed by a softening as the system decompresses and rotates rigidly. To be able to accurately model self-centring wall systems, it is crucial to have accurate models of the rocking interface at the walls foundation. The following Section provides background information on the analytic and numerical modelling techniques used to predict the behaviour of self-centring walls. Four popular analysis methods are considered: section analysis using the monolithic beam analogy; numerical analysis using multi-spring models; numerical analysis using fibre-based models; numerical analysis using lumped plasticity models. An overview of contemporary models for seismic U-shaped flexural plate and axial fuse dampers are also included. 3.1 Section Analysis using the Monolithic Beam Analogy Section analysis of the base connector is challenging because there is no strain compatibility between the wall section and the post-tension tendon. In lieu of analytic equations, a trial and error section analysis procedure was proposed by Pampanin et al. (2001). For a single member with no dampers, a rough outline of the section analysis procedure is as follows: 1. Choose a trial rotation (\ud835\udf03\ud835\udc5b). 2. Choose a trial neutral axis depth for the connection (\ud835\udc50\ud835\udc5a). 3. Estimate strain in the post-tension tendon(s) and at the compression block based on the prior trial neutral axis and rotation. 4. Calculate the stress in each member using stress-strain relationships 5. Calculate section forces by summing the stresses over relevant structural members. 6. Iterate the neutral axis until force equilibrium has been reached 34 7. Evaluate the connections external moment. By completing this process for multiple imposed base rotations, a monotonic moment rotation relationship for the section can be calculated. An in-depth section analysis using the monolithic beam analogy is presented in Section 4.5. While this procedure works schematically, a method of calculating the compression strain across a wall section is still required. To estimate strain in the compression region, Pampanin et al. (2001) proposed use of a compatibility condition called the monolithic beam analogy. In the monolithic beam analogy, it is assumed that the stress block in the self-centring member is the same as the stress block in the plastic hinge region of a monolithic member. By equating the rigid body rotation with the known rotation of a plastic hinge region, it is possible to estimate the strain peak strain in the compression block. Once the peak strain is known, a linear strain distribution is assumed between the edge of the wall and the neutral axis. This process is outlined in Figure 3-1 and the following equations: Figure 3-1: Monolithic beam analogy. Plastic hinge \ud835\udee5\ud835\udc52 \ud835\udee5\ud835\udc5d \ud835\udee5\ud835\udc52 \ud835\udee5\ud835\udc5f Monolithic wall Self-centring wall 35 \ud835\udee5\ud835\udc52 + \ud835\udee5\ud835\udc5d = \ud835\udee5\ud835\udc52 + \ud835\udee5\ud835\udc5f (3.1) \ud835\udee5\ud835\udc5d = (\ud835\udc3f \u2212\ud835\udc3f\ud835\udc5d2) \ud835\udf03\ud835\udc5d = (\ud835\udc3f \u2212\ud835\udc3f\ud835\udc5d2) \ud835\udc3f\ud835\udc5d(\ud835\udf19\ud835\udc62 \u2212 \ud835\udf19\ud835\udc52) (3.2) \ud835\udf19\ud835\udc62 =\ud835\udf00\ud835\udc50\ud835\udc50 (3.3) \ud835\udee5\ud835\udc5f = \ud835\udc3f\ud835\udf03\ud835\udc5f (3.4) where: \ud835\udee5\ud835\udc52 is the elastic wall deformation; \ud835\udee5\ud835\udc5d is the wall deformation from plastic hinge rotation; \ud835\udee5\ud835\udc5f is the wall deformation from rigid body motion; \ud835\udc3f is the wall length; \ud835\udc3f\ud835\udc5d is the plastic hinge height; \ud835\udf03\ud835\udc5d is the plastic hinge rotation; \ud835\udf03\ud835\udc5f is the rigid body rotation; \ud835\udf19\ud835\udc62 is the total curvature at plastic hinge; \ud835\udf19\ud835\udc52 is the elastic curvature at elastic hinge; \ud835\udf00\ud835\udc50 is the peak strain in the wall section; \ud835\udc50 is the neutral axis depth. Initial trials of the monolithic beam analogy showed that it was able to accurately capture the force deformation of self-centring precast concrete moment frames (Pampanin et al. 2001) and shear walls (Palermo et al. 2005). The Monolithic beam analogy was also successfully applied to other materials. Newcombe (2008) adapted the monolithic beam analogy to accurately predict the performance of LVL wall systems, and Sarti (2015) had success using this method for LVL wall systems. However, it was noted that a reduced section modulus was needed to obtain accurate results (Newcombe 2011). For wall to foundation connections, it was proposed that the stiffness at the rocking interface was 55% of the parallel to grain stiffness. Ganey (2015) showed that the analysis procedure could accurately predict the behaviour of CLT walls. Ganey also compared the post-tension force and neutral axis location predicted by the Monolithic beam analogy to experimental results. It was found that the post-tension force 36 was slightly over-predicted, and neutral axis location was under-predicted. Figure 3-2 shows sample results from Ganey\u2019s studies. Figure 3-2: Predictions of monolithic beam analogy vs. results. While the monolithic beam analogy has been proven accurate at predicting shear and post-tension forces in self-centring wall connections, there are a few notable challenges with the model. In particular, it is not always clear what height to use for the plastic hinge region of the monolithic beam. The plastic hinge height used is often based on empirical data, and as such, the model may be more difficult to use in applications where there is less data. For self-centring walls, cyclic deterioration will occur if crushing of the wall system occurs at the rocking interface. The monolithic beam analogy only produces a monotonic backbone curve and does not make predictions about the cyclic backbone deterioration. As such, it is not able to predict progressive damage of the wall system as it passes through multiple cycles of 37 loading. For this progressive damage, numerical methods are needed to understand the behaviour of self-centring walls under dynamic or reverse cyclic loading at higher drifts. Some questions have also been raised about the accuracy of the Monolithic beam analogy in timber members. Newcombe (2015) presented an alternative approach using a winkler spring analogy. This approach assumes that displacement at the rocking interface varies linearly at the base of the rocking wall. A series of effective springs are then defined at the base of the wall and used to calculate the force across the section. Kovacs (2016) later used this method to successfully model the LVL walls used in Sarti\u2019s experimental program. A challenge with using this model is that the length of the winkler springs must be empirically calibrated from data. 3.2 Lumped Plasticity Models The simplest and most convenient numerical model for a self-centring wall base connection is the lumped plasticity model. In general, lumped plasticity models predict nonlinearity using a predefined force-deformation relationship close to a region of member nonlinearity. While these models do not consider detailed information about stresses and strains, they are often able to predict global relationship between force and deformation accurately. Defining the force-deformation relationship in a lumped plasticity element requires prior knowledge of the backbone curve. Generally, information about the backbone curve will either come from analytic equations or empirical test data. Degradation rules can also be implemented to change the backbone throughout the analysis. For a self-centring wall, the base connection can be modeled with a rotational spring close to the rocking interface. The moment-rotation relationship of this spring will correspond to the rigid body rotation of the base against the moment applied. This relationship is generally 38 modeled with a multilinear elastic material, which allows the system to re-centre after a load is applied. The force-deformation relationship for the rotation spring can be derived by using the monolithic beam analogy (Pampanin et al. 2001). Lumped plasticity models of wall systems typically do not explicitly model the post-tension force within the system. However, the post-tension force can be recovered using the original rotation-force relationship for the post-tension bars. Several notable studies have used lumped plasticity models to accurately predict the response of self-centring wall systems. Palermo et al. (2005) used lumped plasticity models to model self-centring precast concrete walls. These walls used a multilinear elastic rotation spring, coupled with a hysteresis spring to model dampers in the system. Figure 3-3 shows the output numerical model against the experimental data. It can be observed that the model accurately matched the monotonic backbone and could reasonably capture the hysteresis. However, it is not able to match changes in initial slope as the specimen completed multiple cycles. Figure 3-3: Summary of lumped plasticity model and experimental results for post-tensioned moment frame (Palermo et al. 2005). In an experimental program at the University of Canterbury, Sarti (2015) used lumped plasticity models to calculate the response of LVL walls. Figure 3-4 shows numerical data 39 plotted against experimental data from one of the tests. The models were able to capture lateral force-drift response in the range of conditions considered with reasonable accuracy. Degradation was less significant in these tests. Another notable study using lumped plasticity models to analyze LVL systems was completed by Di Cesare et al. (2019). In this study, experimental data from seismic tests of a 2\/3rd scale, self-centring three story moment fame were used to validate the performance of numerical models. It was found that the lumped plasticity models were able to estimate the dynamic response of the structure in the range of conditions considered. Figure 3-4: Numerical model vs. experimental data for lumped plasticity models of LVL walls. (Sarti 2015). A challenge with lumped plasticity models is that there is no rotation of the section prior to the decompression moment for the wall. Numerically, this leads to a base connector that will very quickly go from being almost perfectly rigid to relatively flexible. This sudden change in stiffness has the potential to lead to convergence problems, or unrealistically large damping forces. There is little guidance about what initial stiffness should be used for the rotational spring. If the base interface is too stiff, the numerical challenges listed above may affect performance, while if it is too flexible the numerical model will be inaccurate. When studying 40 LVL walls, Sarti used an initial rotation equal to 1\/10 of the rotation where the post-tension cable began to elongate. This was reported to yield results which were both stable and reasonably accurate. If a multi-linear elastic material is used to model the base connector deformation, it will not be possible to capture progressive damage at the rocking interface. This potentially restricts the application of lumped plasticity models to scenarios where a significant amount of deterioration is not expected. How important progressive damage is to the dynamic performance of a wall system will depend on the type of a material and loading considered. It is unclear if capturing cyclic deterioration is important for typical applications of CLT. At the time of writing, there are no material models available to predict the cyclic deterioration of self-centring wall elements using lumped plasticity systems. There are also no known studies using lumped plasticity models to simulate CLT wall components or buildings loaded dynamically. 3.3 Multi-Spring Models As is seen in Section 3.2, lumped plasticity models struggle with capturing cyclic deterioration of wall elements due to crushing at the wall base. To more accurately capture the behaviour of the rocking interface, the multi-spring model were developed by Spieth et al. (2004). A typical multi-spring element is overviewed in Figure 3-5, along with a wall model using multi-spring elements. The multi-spring element captures gap-opening at the rocking interface by using a series of contact springs at the base of the wall. These springs have no tensile stiffness, and compression stiffness that is dependant on the material being modeled. Generally the stiffness of a spring will depend on the effective area it represents, the materials modulus of elasticity, and the effective length of the spring. The effective length of each spring will need to be 41 calibrated based on experimental data. A rigid beam element is then used to connect each spring, and enforce the assumption that plane sections remain plane. Figure 3-5: Diagrams of (a), a typical multi-spring element and (b), a typical wall model using multi-spring elements (Iqbal et al. 2015). After the initial development of this model, multi-spring models were quickly adopted in many research studies. It was found that the system could accurately model the behaviour of concrete systems (Spieth et al. 2004). Adoptions in LVL also found success, and many studies in New Zealand used this analysis method. Both Newcombe (2011) and Sarti (2015) used this method to accurately predict the behaviour of self-centring moment frames and wall components. Iqbal successfully employed multi-spring models to predict the performance of it coupled LVL walls with dampers (Iqbal et al. 2015). When predicting the seismic performance factors for post-tensioned wall systems, Sarti et al. (2016) used multi-spring models. In CLT, Ganey (2015) used the multi-spring numerical model to successfully predict the performance of single and paired CLT walls using dampers from his experimental study. Multi-spring models have several major advantages over the models discussed prior. Using multi-spring models, it was possible to capture progressive damage to the wall at the rocking interface by using degrading materials in the spring elements. It was also possible to capture (a) (b) 42 the force-moment interaction at the base of the wall system. As an added benefit, in multi-spring models the force in the post-tension cable will also be directly simulated, as opposed to having the performance measured implicitly. While the multi-spring model is accurate, it is relatively difficult to implement, and requires that the user creates and records many elements in their model. As with other models, the height of the spring needs to be calibrated based on experiment. Often it is set equal to the high over which compression damage occurs in experiment. The multi-spring model can be unwieldy to use, as it requires that many elements are created and monitored throughout the experiment. 3.4 Fibre-based Models Fibre-based numerical models are a commonly used method of estimating the response across the cross section for a structural element. In a fibre model, the cross-section properties are defined using a number of discrete points across the section. Each of these fibres will have a pair of local coordinates, area, and material associated with it. Fibre sections generally enforce the assumption that plane sections remain plane, and the strain at each point of the fibre section are equal to the sum a normal strain, and rotation strain. In two dimensions, the strain at the nth fibre is defined as follows: \ud835\udf00\ud835\udc5b = \ud835\udf000 + \ud835\udc66\ud835\udc5b\ud835\udf190 (3.5) where: \ud835\udf00\ud835\udc5b is strain at fibre \u201cn\u201d; \ud835\udf000 is the normal strain across the section wall deformation; \ud835\udf190 is the curvature across the section; \ud835\udc66\ud835\udc5b is the distance from the neutral axis. The stress in each fibre is determined using the material stress strain relation for that fibre. The net forces and stiffness across the section are calculated by summing the contribution of each 43 fibre. Fibre sections are often used in beam column elements. For fibre sections in beam column elements, the normal strain and curvature across the section will depend on the section location within the element, and the elements curvature distribution. In self-centring elements, the rocking interface can be modeled with a fibre-based beam column element. The sections in this beam column will have material properties associated with the material it is modelling, and no tension strength. The lack of tensile strength allows the wall to simulate uplift at the connection interface as the gap at the base of the wall opens. Similar to other models, the height of the fibre element will need to be determined based on experiments. Generally, the height of the beam column element is chosen based on estimates of height of compression damage in the toe of the wall. Fibre models were first used to model self-centring systems in reinforced concrete. Kurama (1997) developed fibre-based models for precast concrete sections. It was noted that the fibre section would capture global behaviour with gap opening at the connection interface, but not local effects. This is because of the assumption that plane sections remain plane, which is not true for a concrete section. In a later study Perez et al. (2008) used the fibre models developed by Kurama to model a self-centring precast concrete wall. It was found that the fibre numerical model could accurately simulate the systems shear displacement relationship, as well as the post-tension force in the structure. This finding was validated for both monotonic and cyclic loading. Figure 3-6 overviews the experimental set up, fibre model, and fit of data for the study by Perez et al. (2008). The height of the beam-column element in this study was set equal to height of the confined concrete failure zone. 44 In CLT, Akbas et al. (2017) used fibre elements to simulate the suite of walls tested in Ganey\u2019s (2015) thesis. A comparison of numerical and experimental values showed that fibre models were able to accurately model the shear, drift, and post-tension force in the walls studied. Figure 3-6: Summary of experimental wall setup, fibre wall model, and analysis results for base shear vs. drift and post-tension force vs. drift (Perez et al. 2008). Conceptually, both multi-spring models and fibre models function very similarly. Both calculate the response of the rocking interface using on the contributions of a finite number of 45 areas, and both enforce the assumption plane sections remain plane. However, there often are a few subtle differences between the two models, depending on the platform used to run numerical simulations. Internally, now finite element software, such as OpenSees, treats fibre sections differently than other elements. For example, the in the OpenSees \u201cforceBeamColumn\u201d element, fibre sections undergo a sub-element integration before the global iteration begins (Scott et al. 2008). It is important for practitioners to understand how the software they are using handles fibre elements, and any implications that may have on their models. Because of the assumption plane sections remain plane, fibre sections will not be able to predict the local stress and strain the compression toe with great accuracy. However, global behaviour can be predicted with accuracy. Compared to multi-spring elements, fibre sections are generally easier to implement. Many existing software packages have the option of using fibre-based sections. There is also a considerable amount of industry experience using fibre elements, making them more appealing to general practitioners. For self-centring wall systems and buildings made out of CLT, there is currently no experience using fibre-based models on dynamic systems. 3.5 Energy Dissipation Devices Self-centring walls are often paired with energy dissipation devices to increase the ductility of the system. There are two types of dampers commonly used with self-centring wall systems: and U-shaped flexural plate and bucking restrained brace axial dampers. U-shaped flexural plates are a hysteretic seismic damper that was first proposed by Kelly et al (1972). These dampers are steel plates that have been formed into a \u201cU\u201d shape and are typically used to connect two wall systems as depicted in Figure 3-7. As lateral force is applied to the 46 paired wall system, the relative displacement between each wall will cause the dampers to yield, absorbing seismic energy. Figure 3-7: A single U-shaped flexural plate, and a paired wall system using U-shaped flexural plates. Expanding on the research of Kelly et al., Baird et al. (2014) proposed several equations to predict the plastic moment (\ud835\udc40\ud835\udc5d), plastic force (\ud835\udc39\ud835\udc5d), yield force (\ud835\udc39\ud835\udc66), yield deformation (\u2206\ud835\udc66), and damper stiffness (k) based on analysis using energy methods. Because of their predictable hysteresis, it is possible to predict the hysteresis of U-shaped flexural plates using a Ramberg-Osgood steel model. Baird also used finite element models were also used to predict the Ramberg-Osgood R factor (R), which controls the transition between the initial and second slope. \ud835\udc40\ud835\udc5d = \ud835\udf0e\ud835\udc66\ud835\udc4d =\ud835\udf0e\ud835\udc66\ud835\udc4f\ud835\udc62\ud835\udc61\ud835\udc6224 (3.6) \ud835\udc39\ud835\udc5d =2\ud835\udc40\ud835\udc5d\ud835\udc37\ud835\udc62=\ud835\udf0e\ud835\udc66\ud835\udc4f\ud835\udc62\ud835\udc61\ud835\udc6222\ud835\udc37\ud835\udc62 (3.7) \ud835\udc39\ud835\udc66 =23\ud835\udc39\ud835\udc5d (3.8) 47 \u2206\ud835\udc66=27\ud835\udf0b\ud835\udc39\ud835\udc6616\ud835\udc38\ud835\udc4f\ud835\udc62(\ud835\udc37\ud835\udc62\ud835\udc61\ud835\udc62)3 (3.9) \ud835\udc580 =\ud835\udc39\ud835\udc66\u2206\ud835\udc66=16\ud835\udc38\ud835\udc4f\ud835\udc6227\ud835\udf0b(\ud835\udc61\ud835\udc62\ud835\udc37\ud835\udc62)3 (3.10) \ud835\udc45 = 7.1 ln\ud835\udc61\ud835\udc62\ud835\udc37\ud835\udc62+ 29.5 (3.11) These formulas are defined in terms of the U-shaped flexural plate: yield stress (\ud835\udf0e\ud835\udc66); elastic modulus (\ud835\udc38); plastic section modulus (Z); thickness (\ud835\udc61\ud835\udc62); width (\ud835\udc4f\ud835\udc62); and curve diameter (\ud835\udc37\ud835\udc62). Figure 3-8 overviews an experimental test of a U-shaped flexural plate (Baird et al. 2014). It can be observed that the damper has a stable hysteresis that contains a relatively large area, making it attractive for energy dissipation. U-shaped flexural plate dampers are appealing because they can be manufactured relatively easily. It is also possible to achieve a number of different damping levels by varying the geometric properties of the dampers. U-shaped flexural plate dampers were successfully used with reinforced concrete paired wall systems in the PRESSS program (Priestley et al. 1999). Following this implementation, many research programs that adopted these dampers for paired walls in LVL self-centring walls (Iqbal et al. 2007, 2016; Newcombe et al. 2011; Sarti 2015). Many notable experimental of studies also exist of CLT walls using U-shaped flexural plates (Chen et al. 2020; Ganey 2015; Wichman 2018). 48 Figure 3-8: Hysteresis for U-shaped flexural plate damper (Baird et al. 2014). Axial dampers using buckling restrained braces are also another damper system commonly used with self-centring walls at their base. These wall fuses are composed of a mild steel bar, which is encased in such a way that the fuse will not be able to buckle when compression load is applied to them. Standard ways of preventing buckling are by surrounding the fuse with an epoxy or metal container. Because the dampers are restrained against buckling, they will have a stable hysteresis as they undergo tension compression cycles. Axial fuse energy dissipators are typically placed at the base of wall elements and dissipate energy by yielding as the self-centring wall rocks at the base interface. Figure 3-9 depicts a typical axial fuse damper and wall mount for these dampers. 49 Figure 3-9: Axial fuse damper connection and mount (Sarti 2015). The contribution of the stiffness and yield strength of most dampers can be calculated based on simple statics. However, for some fuse types it is necessary to consider the contribution of anti-buckling system. The hysteresis of fuses can typically be captured using a standard Guiffr\u00e8-Menegotto-Pinto steel model (1973). Axial type fuses have seen various implementations in self-centring systems made from reinforced concrete, (Amaris Mesa 2010), LVL (Marriott 2009; Newcombe 2011; Palermo et al. 2006; Sarti 2015), and CLT (Chen et al. 2018; Kramer et al. 2015). 50 Chapter 4: Component Level Modelling of Wall Systems The goal of this chapter is to apply contemporary modelling techniques to capture the behaviour of self-centring CLT walls, with and without seismic dampers. First, experimental data is obtained from two experimental programs that study self-centring CLT walls. An overview of each experimental program is given and information about material properties is noted from both studies. Using two studies instead of one allows for the identification of consistent trends through different studies, and will give a more complete understanding of variability and uncertainty. Next, analytical and numerical models are created for the walls studied in the experimental program. Emphasis is placed on fibre-based and lumped plasticity models, which are available commercially available finite element software programs. Calibrations of the numerical models are conducted to identify appropriate values of selected material parameters. Subsequently, key design parameters, such as the base shear and post-tension force, are compared between the calibrated models and the experimental data. 4.1 Experimental Data from FPInnovations and Ganey Experimental data for wall panels is drawn from two sources: a 2018 study by the research institute FP Innovation in Canada (Chen et al. 2018) and a 2015 study by Ryan Ganey at the University of Washington (Ganey 2015). In the study by FPInnovations, seventeen single and coupled self-centring CLT walls were tested with and without seismic dampers. In the experiments, the post-tension force of each wall was varied together with the size and location of the damper. Three different post-tension force levels were considered: 44.5kN, 89kN, and 133.5kN. In addition, two types of dampers were used: buckling restrained axial fuses and U-shaped flexural plates. The wall panels were tested using monotonic and cyclically loading protocols. 51 Figure 4-1: Diagram of experimental setup for a single wall. The height, length, width, and layer information for the CLT panels considered is outlined in Table 4-1. The transverse and parallel layers of CLT had different sizes, and boards within a layer did not have edge-gluing. The CLT used in FPInnovations experiment was reported to comply with ANSI\/APA PRG 320 Standard E1 grade, while in Ganey\u2019s study, the CLT was constructed at the university of Washington from Douglas-fir Larch No.2 boards. Figure 4-1 depicts the experimental setup for a single CLT wall. At the top of the wall, an actuator was used to apply force to the specimen. Wall panels had a hole in the centre of the wall, through which ran a 20mm diameter post-tension cable. These post-tension cables had a modulus of elasticity of 205GPa, a yield strength of 900MPa, and an ultimate strength of 1,100MPa. At the base of the wall, shear keys prevented sliding and out of plane movement of the wall on the foundation. The geometric and material properties of the axial and U-shaped flexural plate dampers used are outlined in Table 4-2 and Table 4-3, respectively. Several axial dampers were tested under cyclic loading, while the U-shaped flexural plate dampers were not tested. 52 Table 4-1: CLT geometric properties. Experiment Designation Number of Laminations Lamentation Thickness (mm) Wall Actuator Height (mm) Width (mm) Length (mm) Height (mm) FPInnovations (F) 3 (Strong) 2 (Weak) 33 (Strong) 18 (Weak) 143 1000 3000 2900 University of Washington, Ganey (G) 3 (Strong) 2 (Weak) 33 (Strong) 33 (Weak) 165 1220 4420 4120 Table 4-2: Axial damper properties. Variable FPInnovations Yield Stress (MPa) 300 Young\u2019s Modulus (GPa) 200 Diameter (mm) 12.7 Length (mm) 550 Table 4-3: U-shaped flexural plate material and geometric properties. Variable Ganey FPInnovations Yield Stress (MPa) 410 200 Young\u2019s Modulus (GPa) 200 200 Width (mm) 102 140 Thickness (mm) 9.5 6.35 Diameter (mm) 104 100 Yield Force (kN) 17.9 5.8 Plastic Force (kN) 18.9 11.3 Stiffness (kN\/mm) 3.1 1.33 Using the actuator to apply loads, specimens were studied under monotonic and cyclic loading protocols. For tests using cyclic loads, the load protocol B from the ASTM standard E2126 (ASTM 2015) was used. During testing, the reported variables were base shear, drift at the actuator, and post-tension tendon force. Other properties of the wall such as base uplift and U-shaped flexural plate deformation were recorded but not made publicly available. The post-tension force was measured through a load cell at the top of each panel, while horizontal load 53 and displacements were measured by a load cell at the actuator, and actuator displacement. Loading was stopped if the post-tension tendon force reached 85% of the yield force, or if any of the dampers failed. The maximum lateral drift considered varied between 2% for 3%. The walls studied performed as expected, with no specimens failing prematurely. Figure 4-2a shows a typical stress strain curve for single wall specimens, while Figure 4-2b shows a specimen with axial dampers. Monotonic tests generally had a stiffer backbone than walls tested cyclically. However, a similar trend was observed in the overall shape of the response. During cyclic tests of single walls, there was not much plastic deformation observed. This suggests that the CLT was not significantly crushing in the experiment. For walls with dampers a pinching hysteresis is observed, suggesting significant yielding of the dampers. A slight asymmetry was noted in both the shear and post-tension force response, positive cycles reached higher drifts and lower forces than negative cycles. This effect was mostly insignificant. For the cyclic test show in Figure 4-2a, the load difference between positive and negative cycles was approximately 5%. The asymmetric observed was likely due to the flexing of the column connected to the actuator. Figure 4-2: Test results for (a) a single wall without a damper and (b) a single wall with axial dampers from walls studied by FPInnovations. (a) (b) 54 In Ganey\u2019s study, eight experiments were performed on single and coupled self-centring CLT walls. The walls tested were built from two individual CLT panels, joined using steel splice plates and self-tapping screws. The following wall configurations were studied: three single walls with unique post-tension forces; two tests that were repeats of failed single wall tests; a single wall test using a CLT foundation; a single wall test with a structural composite lumber core; and a test of a coupled wall system with U-shaped flexural plate dampers. The two repeated tests occurred due to failure of the CLT bearing plate at the top of the wall. The three unique post-tension levels considered were: 103kN, 267kN, and 334kN. As with the FPInnovations study, Table 4-1 outlines the material and geometric properties of the wall. Ganey\u2019s test used a setup like the one depicted in Figure 4-1. An actuator applied force near the top of the specimen, and shear keys prevented sliding and out of plane movement. The post-tension tendon diameter varied between 32mm and 36mm, and had a nominal modulus of elasticity, yield strength, and ultimate strength of 220GPa, 900MPa, and 1,100MPa respectively. The geometric and material properties of the U-shaped flexural plate dampers are outlined in Table 4-3. Material tests were completed on the CLT material, post-tension tendon steel, as well as U-shaped flexural steel plate. In Ganey\u2019s study, wall panels were tested using cyclically applied loads. The load protocol for cyclic tests was made using the guiding document ACI ITG-5.1-07 (ACI 2008). Experiments were subjected to the full loading protocol, with a maximum drift ranging between 10% and 12%. Due to the large drifts applied, readings from the actuator had to be corrected to account for vertical displacement of the wall. Figure 4-3 depicts a typical force deformation curve for the shear and PT force of a single wall specimens. Yielding of the post-tension cable occurred at approximately 4% drift. The test specimens showed significant plastic deformation before 55 the post-tension yield point. This suggests that crushing of the CLT had a significant effect on the overall response. Crushing starts to become more pronounced between the 2% and 3% drift cycle. There was a large asymmetry observed between cycles in the positive direction and cycles in the negative direction, in Figure 4-3 the positive and negative shear forces differ by approximately 25%. This asymmetry suggests significant deformation occurred in the loading column, or foundation. Figure 4-3: Test results for walls from Ganey\u2019s study for (a) base shear vs actuator drift and (b) post-tension force vs actuator drift (Ganey 2015). 4.2 Wall Archetypes and Scope of Study From the studies outlined in Section 4.1, several experiments were selected to compare their results against analytic and numerical models. The walls chosen were experiments that performed as expected and had few additional complicating factors such as using multiple damper types or a different foundation material. For FPInnovations study, experimental data was only made publicly available for a limited number of wall specimens, and the chosen experiments were drawn from that pool. The chosen experiments were split into three archetypes: (I) single walls without dampers as shown in Figure 4-4a, (II) single walls with (a) (b) 56 axial dampers as in Figure 4-4b, and (III) paired walls with U-shaped flexural plate dampers as in Figure 4-4c. For Archetypes II and III, different damper configurations were considered, beyond what is depicted in Figure 4-1. A total of 11 experimental walls were modeled, with six walls being from Archetype I, three from Archetype II, and two from Archetype III. Figure 4-4: Wall archetypes (a) single wall, (b) wall with axial dampers, (c) paired wall with U-shaped flexural plate dampers. Table 4-4 summarizes key design parameters that were varied through each test. The first column of the table is the experimental designation of the wall, this is a unique identifier for each wall that is used to reference it throughout this study. For convenience, each wall is also classified into one of the archetypes outlined in Figure 4-4; walls that share an archetype are expected to have similar behaviour. The post-tension force and area denotes the initial force in the post-tension tendon, as well as the area of that post-tension tendon. Together these variables control the walls initial decompression point, and how much total drift can be applied to the system before the post-tension tendon yields. A high initial stress will lead to a stiffer structure that yields at lower maximum drifts. Combined, the post-tension force and area will give the (a) (b) (c) I II III 57 initial stress in the tendon. In Table 4-3, this stress is reported as a percentage of the yield stress to show how close each specimen is to yielding. For the paired CLT walls, each wall was pretensioned to the same PT force. However, in Ganey\u2019s study one of the walls was overloaded. Table 4-4: Scope of experimental data used in study. Experiment Designation Archetype PT Force (kN) PT Area (mm2) PT Stress (% yield) Max Drift (%) Dampers (#) Loading Type F1 I 45.5 316 0.16 2.5 - Static F2 I 89 316 0.32 2.5 - Static F3 I 133.5 316 0.47 2.5 - Static F4 II 89 316 0.32 2.5 4 Static F5 II 89 316 0.32 2.5 4 Cyclic F6 II 44.5 316 0.16 2.5 4 Cyclic F7 III 89 (x2) 316 (x2) 0.32 2.5 4 Cyclic G1 I 267 806 0.32 4 - Cyclic G2 I 100 1020 .1 4 - Cyclic G3 I 343 806 0.4 4 - Cyclic G4 III 343, 394 806 (x2) 0.4, .46 4 2 Cyclic The maximum drift considered for the simulations is 4% for Ganey\u2019s walls. In FPInnovations studies, the maximum drifts considered were the maximum experimental value considered, which fall below the 4% threshold. This limit is imposed for two reasons: in most experiments the post-tension tendon yields near 4% drift; and many guidelines for performance-based design do not permit interstory drifts beyond approximately 4%. For example, the \u201cAlternative Procedure For Seismic Analysis And Design Of Tall Buildings Located In The Los Angeles Region\u201d (LATBSDC 2018) permits a mean peak drift of 3.5% for a suite of ground motions, and maximum drift of 4.5% for individual ground motions in that suite of motion. Because yielding of the post-tension tendon will lead to unacceptable drift, accurate simulation of the wall beyond the post-tension yield point has bene considered unnecessary for most common applications. 58 The number of dampers in each structure is also reported, for Archetype II this denotes the number of axial dampers, and for Archetype III it denotes the number of U-shaped flexural plate dampers. The quantity of dampers affects the overall hysteretic properties of the system. Walls with more damping have a larger area filled by their hysteresis and as such more energy dissipated. Finally, the loading type of each experiment is noted, where static denotes walls where load is applied monotonically, and cyclic denotes walls where load is applied using a reverse cyclic loading protocol. 4.3 Prediction and Measurement of Stiffness for Self-Centring CLT Walls Once the experimental program had been chosen, data from each study is analyzed to determine the material properties of the CLT wall system. Of particular interest is the elastic modulus of the CLT section. Because a full analysis of the CLT cross section is quite involved, the behaviour of a section is often summarized using a single section modulus, \ud835\udc38\ud835\udc50. In the following section, four methods are followed to determine the elastic modulus of each CLT section from design guides and experimental data. The following methods are considered for measuring the CLT section modulus: Simplified code predictions; calculations using experimental measurements and Timoshenko beam theory; experimental measurements on moment curvature relationship; and experimental measurements on compression. It is possible to determine elastic properties of a layered cross section using basic composite theory with the thickness and the elastic modulus of each layer. Figure 4-5 denotes a typical CLT cross section, where the strong axis layers are running in plane, and the weak axis layers are running perpendicular to the strong axis layers. 59 Figure 4-5: CLT as a Composite Section. For the section in Figure 4-5, composite cross section modulus \ud835\udc38\ud835\udc50 would be defined as follows: \ud835\udc38\ud835\udc50 =\ud835\udc38||\ud835\udc51||\ud835\udc5b|| + \ud835\udc38\u22a5\ud835\udc51\u22a5\ud835\udc5b\u22a5\ud835\udc51||\ud835\udc5b|| + \ud835\udc51\u22a5\ud835\udc5b\u22a5 (4.1) where: \ud835\udc51|| is the depth of each strong axis layer; \ud835\udc5b|| is the number of strong axis layers; \ud835\udc38|| is the elastic modulus of each strong axis layer; \ud835\udc51\u22a5 is the depth of each weak axis layer; \ud835\udc5b\u22a5 is the number of weak axis layers; \ud835\udc38\u22a5 is the elastic modulus of each weak axis layer. For a CLT panel the layer thickness of parallel and transverse are known, and the elastic modulus of each layer can be estimated based on information about the saw lumber used. The ANSI\/APA technical standard PRG320 (APA 2017) provides guidance about the material properties for CLT panels. In the FPInnovations study, the CLT used had a grade of E1, resulting in a longitudinal layer stiffness of 11.7GPa. In Ganey\u2019s study, the CLT was constructed from Douglas-fir Larch No.2 and better boards, leading to a longitudinal stiffness estimate of 11GPa. The section modulus can also be predicted experimentally, using measurements of moment\u2013curvature relationship and Euler-Bernoulli beam theory. The average curvature across a section is calculated by measuring the rotation at two points on the beam. The difference of the rotation measurements is then divided by the distance between the measurement points. The average moment can be determined by assuming a linear distribution of internal moment and using experimental measurements on applied force. Figure 4-6a denotes a possible way of measuring the composite modulus using the applied force, and two rotation meters. Multiple \ud835\udc38|| \ud835\udc38\u22a5 60 measurements of moment and curvature can then be taken through the experiment to predict the average result. Figure 4-6b shows a linear regression of measured moment curvature points, which was used by Ganey (2015) to predict the CLT elastic modulus in his thesis. Figure 4-6: (a) Moment and rotation relationship over height, (b) measured moment curvature relationship (Ganey 2015). The relationship between flexural moment and curvature at a cross section can then be calculated using the following relationship: \ud835\udc40\ud835\udc4e\ud835\udc63\ud835\udc54 = \ud835\udc38\ud835\udc3c \u2219 \ud835\udf19\ud835\udc4e\ud835\udc63\ud835\udc54 (4.2) \ud835\udc40\ud835\udc4e\ud835\udc63\ud835\udc54 =\ud835\udc402 + \ud835\udc4012 (4.3) \ud835\udf19\ud835\udc4e\ud835\udc63\ud835\udc54 =\ud835\udf032 \u2212 \ud835\udf031\u210e2 \u2212 \u210e1 (4.4) Where: \ud835\udc40\ud835\udc4e\ud835\udc63\ud835\udc54 is the average moment between the points of measurement, \ud835\udf03\ud835\udc56 is the measured rotation at measuring point \u201ci\u201d; \ud835\udc40\ud835\udc56 is the predicted bending moment at measuring point \u201ci\u201d, \u210e is the height between measuring points rotation; \ud835\udc38\ud835\udc3c is the flexure modulus of rigidity The elastic section modulus \ud835\udc38\ud835\udc66 is finally calculated by dividing the flexural rigidity, (\ud835\udc38\ud835\udc3c)\ud835\udc5b\ud835\udc52\ud835\udc61 by the gross modulus of inertia, \ud835\udc3c\ud835\udc54\ud835\udc5f\ud835\udc5c\ud835\udc60\ud835\udc60. Moment Rotation \ud835\udf031 \ud835\udf032 \ud835\udc401 \ud835\udc402 (a) (b) 61 \ud835\udc38\ud835\udc50 =(\ud835\udc38\ud835\udc3c)\ud835\udc5b\ud835\udc52\ud835\udc61\ud835\udc3c\ud835\udc54\ud835\udc5f\ud835\udc5c\ud835\udc60\ud835\udc60 (4.5) In Ganey\u2019s experimental program, moment curvature measurements were taken during the experiment to predict a final section modulus. The elastic modulus reported for each test was unavailable in FPInnovations study. Because the initial response of a self-centring CLT wall will theoretically be linear, it is possible to estimate an elastic modulus from experimental measurements of the initial tangent stiffness of test specimens. Figure 4-7 shows the initial slope \u201ck\u201d of the self-centring wall, which are measured based on displacement at the actuator, and the experimental the shear force. When measuring this initial slope, what point to use can be subjective. In this study, k is calculated by tracing the initial slope of the shear force displacement graph with a tangent line, then measuring the slope of that tangent line. In Ganey\u2019s dataset, the tangent slopes used were the ones reported from his study. For experimental data from FPInnovations, a different technique was used to calculate the tangent the slope, and the values used in this report were measured independently. Figure 4-7: Measuring initial stiffness for CLT panel. k\ud835\udc43 \ud835\udc62 Displacement at Actuator Shear Force 62 Once experimental measurements of stiffness were made, the wall is then modeled as a Timoshenko beam. For Timoshenko beams, the displacement at the end of cantilever is the sum of the shear and flexural components of deformation as follows: \ud835\udc62 = \ud835\udc62\ud835\udc60 + \ud835\udc62\ud835\udc53 =\ud835\udc43\ud835\udc3f\ud835\udc34\ud835\udc60\ud835\udc3a\ud835\udc50+\ud835\udc43\ud835\udc3f33\ud835\udc38\ud835\udc50\ud835\udc3c\ud835\udc54 (4.6) \ud835\udc58 =\ud835\udc43\ud835\udc62=1\ud835\udc3f\ud835\udc34\ud835\udc60\ud835\udc3a\ud835\udc50+\ud835\udc3f33\ud835\udc38\ud835\udc50\ud835\udc3c\ud835\udc54 (4.7) Where: \ud835\udc43 is the applied force; \ud835\udc3f is the cantilever length for the wall; \ud835\udc38\ud835\udc50\ud835\udc3c is the flexure modulus of rigidity; \ud835\udc3a\ud835\udc50 is the effective shear modulus for the section; \ud835\udc34\ud835\udc60 is the shear area for the cross section; \ud835\udc62\ud835\udc60 is the flexural displacement; \ud835\udc62\ud835\udc53 is the shear displacement. Equation (4.7) can then be solved for the flexural modulus of rigidity, assuming shear the modulus known. If there are reliable measurements for the shear modulus of CLT, \ud835\udc3a\ud835\udc50, those can be directly substituted into equation (4.7). The flexural modulus of rigidity will then be calculated as follows: \ud835\udc38\ud835\udc3c = (3\ud835\udc3f2(1\ud835\udc58\ud835\udc3f\u22121\ud835\udc34\ud835\udc60\ud835\udc3a\ud835\udc50))\u22121 (4.8) However, if shear data are not available, it can be estimated in terms of the composite section modulus using simplified \u201crule-of-thumb\u201d equations (4.9). A typical estimate for Gc is between 1\/12th and 1\/20th of the section modulus. The shear modulus of rigidity is then be written in terms of the flexure modulus of rigidity (4.11), and the new equation can be substituted into and solved for the section elastic modulus (4.12). 63 \ud835\udc3a\ud835\udc50 \u2245\ud835\udc38\ud835\udc5016 (4.9) \ud835\udc3c =\ud835\udc4f\u210e312=\ud835\udc34\ud835\udc5b\ud835\udc52\ud835\udc61\u210e212 (4.10) \ud835\udc34\ud835\udc60\ud835\udc3a =56\ud835\udc34\ud835\udc5b\ud835\udc52\ud835\udc61 \u2219 \ud835\udc3a\ud835\udc50 =58\u22191\u210e2 \ud835\udc38\ud835\udc3c (4.11) \ud835\udc38\ud835\udc3c = (\ud835\udc58\ud835\udc3f (8\u210e25+\ud835\udc3f23)) (4.12) For the studies examined, the initial tangent slopes measured are as overviewed in Table 4-5. The elastic modulus for the section is calculated using equation (4.8), using an estimate of 350MPa for the shear modulus. Table 4-5: Measured initial tangent slope. Study Tangent Modulus (kN\/mm) F1\/F2\/F3 2.5 G1 2.2 G2 1.7 G3 2.6 If compression tests data is available, the section modulus for CLT can be measure by using the elastic portion of a force deformation curve. Because the specimen dimensions are known, the section modulus is calculated using the applied force and change in length as follows: \ud835\udc38\ud835\udc50 =\ud835\udf0e\ud835\udf16=\ud835\udc3f\ud835\udc60\ud835\udc43\u2206\ud835\udc3f\ud835\udc60\ud835\udc34 (4.13) where: \ud835\udc3f\ud835\udc60 is the length of the specimen; \u2206\ud835\udc3f\ud835\udc60 is the change in length of the specimen; \ud835\udc34 is the cross-section area of the specimen. 64 Figure 4-8 denotes the stress strain relationship for material tests conducted by Ganey, using a Tinius-Olsen testing machine. Tests were done on CLT specimens with layering similar to the layering used in Ganey\u2019s CLT wall tests. These CLT specimens were cut into a column, and load was applied to the entire cross section. The output force and displacement were measured across the whole specimen and reported by the testing machine. The stress strain curve is characterized by a low initial stiffness, followed by an elastic zone and ending in a yield plateau. The low initial stiffness occurs due to localized imperfections crushing and is excluded from calculations. Measurements were for the 5ply CLT specimen used to determine the cross-section For FPInnovations, compression tests were run on specimens that had different layup dimensions from the CLT walls tested. Several testes were completed, but only data from the test specimen CLT-A is used in this study. These CLT specimens tested were cut into bands 267mm long, 500mm tall, and 89mm wide. Load was applied eccentrically, along a 100mm band at the edge of the specimen. This eccentric loading condition is similar to how a CLT wall would be loaded as it rocks. Both local and global measurements of displacement were reported for the CLT. Global measurements were made over the full height specimen, and local measurements were within 25.4 mm height of the applied load. The tested specimens had a different cross section than the CLT walls. Therefore, to convert the section modulus measured for the specimen into a modulus for the CLT walls, the stiffness of longitudinal layers is calculated using equation (4.1). The longitudinal modulus is then used with the CLT wall dimensions and equation (4.1) to find the equivalent stiffness for that CLT wall. 65 Figure 4-8: Measured Force - Deformation Relationship using compression tests (Ganey 2015). The modulus of elasticity calculated for each of the four methods is summarized in Table 4-6, broken down by each study. Important to note is that the different predictors on section modulus experimental data do not converge on a single value, highlighting the difficulty in choosing appropriate model parameters. For the experimental study by FPInnovations, the reported results are an average for walls F1-F3. For Ganey\u2019s study there is more variance, and the single wall tests are reported separately. For both studies, the code prediction and compression test data were the same for each experiment in a study. There is a high degree of variation between the CLT stiffness calculated by each method. In both studies the elastic modulus predicted by code estimates is significantly higher than those measured experimentally. While some difference between prediction and measurement is expected for timber materials, the degree of variation observed here is unlikely to be caused by natural material uncertainty. In addition, the experimentally measured elastic moduli were clustered much more closely, suggesting the actual modulus of elasticity is lower than predicted by code equations. 66 Table 4-6: Modulus of elasticity as measured by different methods. Method FPInnovations F1-F3 (GPa) G1 (GPa) G2 (GPa) G3 (GPa) Code Prediction 8.6 6.75 6.75 6.75 Moment-Curvature Measurements - 3.4 3.2 5.0 Tangent Slope Measurements 2.4 2.8 2.3 3.5 Compression Test Measurement (Global) 5.6 3.05 3.05 3.05 Compression Test Measurement (Local) 1.6 - - - This difference between code predictions and experimental results difference was noted in Ganey\u2019s study, but the root cause of it was ultimately considered unknown. Ganey suggested a manufacturing error may have caused the decrease in CLT stiffness, however both studies show a similar trend. Because this behaviour is noted in both studies, there could be an underling phenomenon whose effects are not fully understood. For CLT, each method of measuring the section elastic modulus has caveats associated with them, and these caveats could explain the contradictory measurements. Both measurements of moment-curvature and tangent slope predict the section modulus using the bending stiffness of the section, while compression tests measure the axial stiffness. A potential challenge with estimating the section modulus using flexural measurements is that modulus of elasticity is indirectly measured through the flexural rigidity of the section. First the section rigidity, EI, is measured, then the elastic modulus is estimated using the gross modulus of inertia. This could be problematic, because if edge gluing is not present between boards in the same layer, the section stiffness could be lower than the gross section stiffness (Danielsson et al. 2017). This occurs due to slip between boards within a layer, which in turn reduces the effective modulus 67 of inertia to be less than the gross modulus of inertia. In this case, using the gross modulus of inertia would overestimate the effective modulus of inertia, and underestimate of the elastic modulus for the section. In compression test measurements, the elastic modulus is calculated using force and deformation relationships, both of which can be measured with a great deal of certainty. Interlayer slip is unlikely to influence compression tests, however, knowing what length to use while measuring the system stiffness is challenging. As can be seen by the local vs. global data presented by FPInnovations, the local and global compression modulus of elasticity for a specimen can vary greatly. Currently, it is unclear how much these local effects influence the performance of CLT walls. Using the full modulus of elasticity may overestimate the actual stiffness for the system. A consequence of the challenges noted above, is that it is not necessarily the case that flexural rigidity and axial rigidity for CLT will have the same properties. Throughout this study, these values have not been considered the same. There is some historical precedent for using a different flexural and axial elastic modulus in post-tensioned systems. When studying LVL systems, both Newcombe and Sarti used a \u201cconnection modulus\u201d, which locally reduced the elastic modulus of the LVL section at the local connection. Table 4-7 below depicts the final material properties used in the numerical and analytic models. Due to the large range in possible values, the material properties chosen were ultimately based on what fit the data best. For the FPInnovations study, these values were later validated using the calibration process described in Section 4.8. Sensitivity studies in section 4.10 also had an influence on the final properties chosen. Based on the compression test data from Ganey, the yield force of CLT is taken to be 25MPa. This value is used for both studies. 68 Table 4-7: Summary of used material properties. Study Designation E Compression (GPa) E Flexure (GPa) F 2.6 2.6 G 3.0 3.6 4.4 Prediction of Shear Modulus and Post-Tension Tendon properties To accurately model walls, it is also necessary to know the shear stiffness of the CLT panel, and the post-tension tendon stiffness. Because the post-tension cable is steel, it has material properties are known quite reliably. However, the post-tension cable consists of several components in parallel, and these additional components may affect the assembly stiffness. Figure 4-9 shows the notable components of the system: the connection to the foundation, the tendon itself, the post-tension saddle, and local effects at the top of the CLT panel. Considering all these factors, the total stiffness of the assembly is less than just the cable alone. The total deformation of the system can be described as follows: Figure 4-9: Components of CLT assembly deformation. \ud835\udc62\ud835\udc5b\ud835\udc52\ud835\udc61 = \ud835\udc62\ud835\udc53\ud835\udc51 + \ud835\udc62\ud835\udc5d\ud835\udc61 + \ud835\udc62\ud835\udc4f\ud835\udc5f + \ud835\udc62\ud835\udc60\ud835\udc51 (4.14) where the deformation components are defined as follows: \ud835\udc62\ud835\udc53\ud835\udc51 is the deformation in the foundation connection; \ud835\udc62\ud835\udc5d\ud835\udc61 is the deformation in the post-tension cable; \ud835\udc62\ud835\udc60\ud835\udc51 is the deformation \ud835\udc62\ud835\udc53\ud835\udc51 \ud835\udc62\ud835\udc5d\ud835\udc61 \ud835\udc62\ud835\udc60\ud835\udc51 \ud835\udc62\ud835\udc4f\ud835\udc5f 69 in the post-tension \u201csaddle\u201d; \ud835\udc62\ud835\udc4f\ud835\udc5f is the deformation due to localized bearing at the post-tension saddle. In this study, a net reduction factor on the elastic modulus for the tendon is used in lieu of more accurate information. Table 4-8 overviews the stiffness reduction factors used on the post-tension tendon. These factors were chosen based on what value best matched the results of the numerical models tested in Section 4.6. Table 4-8: Stiffness reduction factor on post-tension tendon assembly stiffness. Study Designation \ud835\udf19\ud835\udc5d\ud835\udc61 \ud835\udc38\ud835\udc60 (GPa) F 1.0 205 G 0.85 220 The shear modulus of CLT must also be defined in order to model the system accurately. In Ganey\u2019s study, the shear modulus was measured experimentally using linearly potentiometers. Results of these measurement are depicted in Table 4-9. Based on these measurements, 350MPa is chosen as a conservative lower bound estimate of shear stiffness. This value is very similar to the shear modulus determined used by Zimmerman and Mcdonnnell (2017). A value of 350MPa is also used as a rough estimate for FPInnovations study, because no data were available. In Section 4.10, it is shown through parametric studies that the walls analyzed in this experiment are not sensitive to the CLT shear modulus assumptions. Table 4-9: Measured shear modulus for CLT. Method G1 (MPa) G2 (MPa) G3 (MPa) Shear Modulus 315 365 900 70 4.5 Analysis of walls using the Monolithic Beam Analogy Before the numerical models were created, a section analysis is run on Archetype I walls using the monolithic beam analogy. This analysis is used to get a general sense of wall performance and understand if the material properties were chosen appropriately. Figure 4-10 depicts the algorithm used to do a section analysis with the Monolithic Beam analogy. A function in the programing language Python is written to run the analysis, where the components within the dashed line of figure 4-10 are contained within that function. Figure 4-10: Section analysis algorithm. The first step in the analysis procedure is to input model properties that will remain constant throughout the analysis. These static properties include the geometry of the wall, as well as the Choose Trial N.A. Calculate Strains with Member Compatibility Calculate Section Forces Check Equilibrium Impose Rotation Evaluate Moment Capacity Input Static Properties Output Backbone Curve 71 material properties for the CLT and post-tension tendon. These static properties were used as defined in Sections 4.2, 4.2, and 4.4. Separate material properties were considered for CLT in compression and flexure. The analysis is then initiated by imposing a rotation on the base connection, then choosing a trial neutral axis location for that rotation. Figure 4-11 shows two possible neutral axis neutral axis lengths for a similar imposed rotation. If the neutral axis is smaller than the location of the post-tension tendon, the bar will elongate and the compression force on the cross section will increase. The final neutral axis length chosen is the one that satisfies equilibrium on the section. Through the process, rotation is incremented monotonically in fixed steps and the trial neutral axis rotation is chosen based on a Newton algorithm. The algorithm is ended once a final rotation has been reached. Figure 4-11: Imposed rotations for (a) a trial neutral axis location greater than the post-tension tendon location, and (b) less than the post-tension tendon location. At step n of the sequence, the imposed rotation is as follows: \ud835\udf03\ud835\udc56\ud835\udc5a\ud835\udc5d,\ud835\udc5b = \ud835\udc5b \u2219 \ud835\udc51\ud835\udc45 (4.15) where: \ud835\udc51\ud835\udc45 is a change in rotation. \ud835\udc50\ud835\udc61\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc59 \ud835\udf03\ud835\udc56\ud835\udc5a\ud835\udc5d,\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc59 \ud835\udf03\ud835\udc56\ud835\udc5a\ud835\udc5d,\ud835\udc5b(\ud835\udc4e) (\ud835\udc4f) 72 For the CLT wall section, a linear distribution of strain is assumed across the portion of wall in contact with the foundation. The maximum strain point is calculated according to the monolithic beam analogy, using equation (4.17). The plastic hinge height is considered to be 0.3m, approximately twice the width of the CLT panel, based on guidance from Akbas (2017). At each step the post-tension tendon is checked for deflection according to equation (4.18). If the neutral axis is smaller than the post-tension tendon location, there is no change in post-tension force. \u025b\ud835\udc50,\ud835\udc5a = (\ud835\udf03\ud835\udc56\ud835\udc5a\ud835\udc5d\ud835\udc3f\ud835\udc50\ud835\udc4e\ud835\udc5b\ud835\udc61(\ud835\udc3f\ud835\udc50\ud835\udc4e\ud835\udc5b\ud835\udc61 \u2212\ud835\udc3f\ud835\udc5d2 ) \ud835\udc3f\ud835\udc5d+ \ud835\udf11\ud835\udc66) \ud835\udc50\ud835\udc5b,\ud835\udc5a (4.16) \u025b\ud835\udc50\ud835\udc59\ud835\udc61 = \u025b\ud835\udc50,\ud835\udc5a\ud835\udc66 (4.17) \u2206\ud835\udc3f\ud835\udc5d\ud835\udc61,\ud835\udc5a = {0 \ud835\udc53\ud835\udc5c\ud835\udc5f \ud835\udc50\ud835\udc5b,\ud835\udc5a \u2264 \ud835\udc66\ud835\udc5d\ud835\udc61(\ud835\udc66\ud835\udc5d\ud835\udc61 \u2212 \ud835\udc50\ud835\udc61\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc59)\ud835\udf03\ud835\udc56\ud835\udc5a\ud835\udc5d,\ud835\udc5b \ud835\udc53\ud835\udc5c\ud835\udc5f \ud835\udc66\ud835\udc5d\ud835\udc61 < \ud835\udc50\ud835\udc5b,\ud835\udc5a } (4.18) \u025b\ud835\udc5d\ud835\udc61,\ud835\udc5a =\u2206\ud835\udc3f\ud835\udc5d\ud835\udc61,\ud835\udc5a\ud835\udc3f\ud835\udc50\ud835\udc4e\ud835\udc5b\ud835\udc61+ \u025b\ud835\udc5d\ud835\udc61,0 (4.19) where: \ud835\udc50\ud835\udc5b,\ud835\udc5a is the trial neutral axis \u2018m\u2019 at rotation step \u2018n\u2019; \ud835\udf11\ud835\udc66 is the decompression curvature of the cross section; \u025b\ud835\udc50\ud835\udc59\ud835\udc61 is the strain at location \u2018y\u2019 of the cross section; \u025b\ud835\udc50,\ud835\udc5a is the maximum strain considered across the cross section for trial neutral axis \u2018m\u2019; \u025b\ud835\udc5d\ud835\udc61,0 is the initial strain applied to the post-tension cable; \u025b\ud835\udc5d\ud835\udc61,\ud835\udc5a is the strain in the post-tension tendon for trial neutral axis \u2018m\u2019; \ud835\udc3f\ud835\udc5d is the plastic hinge height for the wall at a point y on the cross section; \u2206\ud835\udc3f\ud835\udc5d\ud835\udc61,\ud835\udc5a is the change in length of the post-tension cable for neutral axis \u2018m\u2019; \ud835\udc3f\ud835\udc50\ud835\udc4e\ud835\udc5b\ud835\udc61 is the height of the wall; and other variables as previously defined. 73 From the input material strains, the force in each member can be calculated using the stress strain relationships described in equations (4.20) and (4.21). Both the post-tension tendon and CLT are checked for yielding and will have their forces plateau if yielding does occur. Before the yield strain is reached in the extreme edge of the CLT section, forces across the section were calculated exactly using a triangular stress distribution (4.22). After the extreme edge of the CLT yields, a numerical integration scheme is used to approximate forces across the CLT section (4.23). The post-tension force is calculated based on the stress multiplied by the area (4.24). \ud835\udf0e\ud835\udc50\ud835\udc59\ud835\udc61,\ud835\udc5a = {\u025b\ud835\udc50\ud835\udc59\ud835\udc61,\ud835\udc5a\ud835\udc38\ud835\udc50\ud835\udc59\ud835\udc61 \ud835\udc53\ud835\udc5c\ud835\udc5f \u025b\ud835\udc50\ud835\udc59\ud835\udc61 \u2264 \u025b\ud835\udc66,\ud835\udc50\ud835\udc59\ud835\udc61\ud835\udf0e\ud835\udc66,\ud835\udc50\ud835\udc59\ud835\udc61 \ud835\udc53\ud835\udc5c\ud835\udc5f \u025b\ud835\udc66,\ud835\udc50\ud835\udc59\ud835\udc61 \u2264 \u025b } (4.20) \ud835\udf0e\ud835\udc5d\ud835\udc61,\ud835\udc5a = {\ud835\udf19\ud835\udc5d\ud835\udc61\ud835\udc38\ud835\udc5d\ud835\udc61 \u025b\ud835\udc5d\ud835\udc61,\ud835\udc5a \ud835\udc53\ud835\udc5c\ud835\udc5f \u025b\ud835\udc5d\ud835\udc61 \u2264 \u025b\ud835\udc66,\ud835\udc5d\ud835\udc61\ud835\udf0e\ud835\udc66,\ud835\udc5d\ud835\udc61 + \ud835\udefc\ud835\udc5d\ud835\udc61\ud835\udf19\ud835\udc5d\ud835\udc61\ud835\udc38\ud835\udc5d\ud835\udc61(\u025b\ud835\udc5d\ud835\udc61,\ud835\udc5a \u2212 \u025b\ud835\udc66,\ud835\udc5d\ud835\udc61) \ud835\udc53\ud835\udc5c\ud835\udc5f \u025b\ud835\udc66,\ud835\udc5d\ud835\udc61 \u2264 \u025b\ud835\udc5d\ud835\udc61 } (4.21) \ud835\udc39\ud835\udc50,\ud835\udc5a = \u2212\ud835\udc38\ud835\udc50\ud835\udc59\ud835\udc61\u025b\ud835\udc50\ud835\udc59\ud835\udc61,\ud835\udc5a\ud835\udc50\ud835\udc5b,\ud835\udc5a2 (4.22) \ud835\udc39\ud835\udc50,\ud835\udc5a = \u222b \ud835\udf0e\ud835\udc50\ud835\udc51\ud835\udc34\ud835\udc50\ud835\udc61\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc590\u2245 \u2211 \ud835\udf0e\ud835\udc56,\ud835\udc50\ud835\udc59\ud835\udc61\u2206\ud835\udc34\ud835\udc41\ud835\udc56=0 (4.23) \ud835\udc39\ud835\udc5d\ud835\udc61,\ud835\udc5a = \ud835\udc34\ud835\udc5d\ud835\udc61\ud835\udf0e\ud835\udc5d\ud835\udc61,\ud835\udc5a (4.24) where: \ud835\udf0e\ud835\udc50\ud835\udc59\ud835\udc61,\ud835\udc5a is the CLT stress as a function of strain at step \u2018m\u2019; \ud835\udf0e\ud835\udc56,\ud835\udc50\ud835\udc59\ud835\udc61 is the stress in the CLT section at location \u2018i\u2019; \u025b\ud835\udc66,\ud835\udc50\ud835\udc59\ud835\udc61 is the yield strain of CLT; \ud835\udf0e\ud835\udc66,\ud835\udc50\ud835\udc59\ud835\udc61 is the yield stress of CLT; \ud835\udf19\ud835\udc5d\ud835\udc61 is the stiffness modifier factor for CLT; as defined in Table 4-8, \ud835\udc38\ud835\udc5d\ud835\udc61 is the reported elastic modulus for the PT cable; \ud835\udefc\ud835\udc5d\ud835\udc61 is the post yield stiffness ratio for the PT cable; \u025b\ud835\udc66,\ud835\udc5d\ud835\udc61 is the yield strain of the post-tension cable; \ud835\udf0e\ud835\udc66,\ud835\udc5d\ud835\udc61 is the yield stress of the post-tension cable; \ud835\udc3f\ud835\udc5d is the 74 plastic hinge height for the wall at a point y on the cross section; \u2206\ud835\udc3f\ud835\udc5d\ud835\udc61,\ud835\udc5a is the change in length of the post-tension cable for neutral axis \u2018m\u2019; \ud835\udc3f\ud835\udc50\ud835\udc4e\ud835\udc5b\ud835\udc61 is the height of the wall. Once the forces in the post-tension tendon and CLT section are calculated, equilibrium of tension and compression forces on the section are calculated to define a residual function. The next trial neutral axis location is chosen using a newton algorithm to minimize the residual value. The residual is differentiated with respect to the neutral axis location, and the quotient of the current residual and the derivative of the residual is then used to approximate the change in neutral axis. This process is repeated until the residual is sufficiently small, and the final value are taken as the neutral axis location at this imposed rotation. \ud835\udc45\ud835\udc5b,\ud835\udc5a = \ud835\udc39\ud835\udc50,\ud835\udc5a + \ud835\udc39\ud835\udc5d\ud835\udc61,\ud835\udc5a (4.25) \ud835\udc45\ud835\udc5b,\ud835\udc5a = \u2212\ud835\udc38\ud835\udc50\ud835\udc59\ud835\udc612(\ud835\udf03\ud835\udc56\ud835\udc5a\ud835\udc5d,\ud835\udc5b\ud835\udc3f\ud835\udc50\ud835\udc4e\ud835\udc5b\ud835\udc61(\ud835\udc3f\ud835\udc50\ud835\udc4e\ud835\udc5b\ud835\udc61 \u2212\ud835\udc3f\ud835\udc5d2 ) \ud835\udc3f\ud835\udc5d+ \ud835\udf11\ud835\udc66) \ud835\udc50\ud835\udc5b,\ud835\udc5a2 +\ud835\udc34\ud835\udc50\ud835\udc59\ud835\udc61\ud835\udf19\ud835\udc5d\ud835\udc61\ud835\udc38\ud835\udc5d\ud835\udc61\ud835\udf03\ud835\udc56\ud835\udc5a\ud835\udc5d,\ud835\udc5b(\ud835\udc66\ud835\udc5d\ud835\udc61 \u2212 \ud835\udc50\ud835\udc5b,\ud835\udc5a)\ud835\udc3f\ud835\udc50\ud835\udc4e\ud835\udc5b\ud835\udc61 (4.26) (\ud835\udc51\ud835\udc45\ud835\udc51\ud835\udc50)\ud835\udc5b,\ud835\udc5a= \u2212\ud835\udc38\ud835\udc50\ud835\udc59\ud835\udc61 (\ud835\udf03\ud835\udc56\ud835\udc5a\ud835\udc5d,\ud835\udc5b\ud835\udc3f\ud835\udc50\ud835\udc4e\ud835\udc5b\ud835\udc61(\ud835\udc3f\ud835\udc50\ud835\udc4e\ud835\udc5b\ud835\udc61 \u2212\ud835\udc3f\ud835\udc5d2 ) \ud835\udc3f\ud835\udc5d+ \ud835\udf11\ud835\udc66) \ud835\udc50\ud835\udc5b,\ud835\udc5a \u2212\ud835\udc34\ud835\udc50\ud835\udc59\ud835\udc61\ud835\udf19\ud835\udc5d\ud835\udc61\ud835\udc38\ud835\udc5d\ud835\udc61\ud835\udf03\ud835\udc56\ud835\udc5a\ud835\udc5d,\ud835\udc5b\ud835\udc3f\ud835\udc50\ud835\udc4e\ud835\udc5b\ud835\udc61 (4.27) \ud835\udc45\ud835\udc5b,\ud835\udc5a(\ud835\udc51\ud835\udc45\ud835\udc51\ud835\udc50 )\ud835\udc5b,\ud835\udc5a\u2245 \u2206\ud835\udc50\ud835\udc5b,\ud835\udc5a (4.28) \ud835\udc50\ud835\udc5b,\ud835\udc5a+1 = \ud835\udc50\ud835\udc5b,\ud835\udc5a + \u2206\ud835\udc50\ud835\udc5b,\ud835\udc5a (4.29) where: \ud835\udc45\ud835\udc5b,\ud835\udc5a is the residual at rotation step \u2018n\u2019 and neutral axis step \u2018m\u2019; (\ud835\udc51\ud835\udc45\ud835\udc51\ud835\udc50)\ud835\udc5b,\ud835\udc5a is the derivative of the residual with respect to the neutral axis location, at rotation step \u2018n\u2019 and 75 neutral axis step \u2018m\u2019; \u2206\ud835\udc50\ud835\udc5b,\ud835\udc5a is the Change in neutral axis between step \u2018m\u2019 and \u2018m+1\u2019 for rotation \u2018n\u2019; and all other variables are as previously defined. Once equilibrium has been reached, the section forces and rotations are then recorded. In particular, the resisting moment of the connection is defined as the difference between the compression block centroid, and the post-tension tendon as a moment arm. The deflection at some height will be a summation of the rotation, flexure, and shear components of deflection, as described in equation (4.32). \ud835\udc40\ud835\udc5b = (\ud835\udc66\ud835\udc5d\ud835\udc61 \u2212\ud835\udc50\ud835\udc5b3) \ud835\udc47\ud835\udc5b (4.30) \ud835\udc49\ud835\udc5b =\ud835\udc40\ud835\udc5b\u210e (4.31) \ud835\udc62\ud835\udc5b = \ud835\udc62\ud835\udc5f\ud835\udc5c\ud835\udc61 + \ud835\udc62\ud835\udc53 + \ud835\udc62\ud835\udc60 = \u210e\ud835\udf03\ud835\udc56\ud835\udc5a\ud835\udc5d,\ud835\udc5b + \ud835\udc49\ud835\udc5b (\ud835\udc43\u210e33\ud835\udc38\ud835\udc50\ud835\udc3c\ud835\udc54+\ud835\udc3f\u210e\ud835\udc34\ud835\udc60\ud835\udc3a\ud835\udc50) (4.32) where: \u210e is the height of deflection measurement. The results of base shear vs. system drift are shown in Figure 4-12. Because Ganey\u2019s experiment is cyclic, and the monolithic beam analogy is monotonic, it is most appropriate to compare the analysis results to the backbone curve of the experimental data. FPInnovations\u2019 data can be directly compared to the analysis because those walls were also tested monotonically. The analytical model agrees reasonably well with experiment, with results being better for FPInnovations\u2019 study than Ganey\u2019s study. The static pushover experiments F1, F2 and F3 had analytic results that were very close to measurements made during the experiment across all drifts. In Ganey\u2019s experiments, the analytic model departed from the backbone curve at higher drifts. For G1 and G2, the section analysis overestimated shear force, while in G3 it underestimated the force. 76 The post-tension force and neutral axis location are reported for single walls. The predicted post-tension force is depicted in Figure 4-13. For the walls in FPInnovations\u2019 study, the post-tension force matched experimental results, but were slightly stiffer. In Ganey\u2019s experimental study, the post-tension force matched the experiment more closely than shear, but is also slightly too stiff for all walls. The neutral axis plots in Figure 4-14 were significantly different than the experimental values. Data are only available for walls from Ganey\u2019s study. The studied walls decompressed at a lower drift than the experiment, and less of the wall is in contact with the base. At higher drifts, the match between experiment and analysis is improved. A similar trend is observed for all three experiments. 77 Figure 4-12: Shear vs. drift using the monolithic beam analogy. (F1) (F2) (F3) (G1) (G2) (G3) 78 Figure 4-13: PT force vs. drift using the monolithic beam analogy. (F1) (F2) (F3) (G1) (G2) (G3) 79 Figure 4-14: Base contact vs. drift using the monolithic beam analogy. 4.6 Overview of Nonlinear Numerical Models Next, numerical models of Archetypes I, II and II, are created using the finite element analysis program OpenSees (Mckenna 2011), through the Python interface OpenSeesPy. These models were developed using prior numerical models as a guidance (Akbas et al. 2017). For each wall archetype, two types of numerical models were defined: a fibre model and a lumped plasticity model. These models are distinguished from each other by how they account for uplift of the CLT wall base. In the fibre model, uplift is captured using a fibre section with no tensile strength. The fibre model explicitly simulates force in the post-tension tendon element and can capture CLT compression damage. Measuring the tendon force and CLT damage theoretically allows the fibre model to track damage throughout the experiment. (G1.) (G3.) (G2.) 80 The lumped plasticity model implicitly accounts for uplift of the base connection, using a calibrated moment-rotation spring. The monolithic beam analogy is used to create the moment-rotation backbone curve for these models. Like the monolithic beam analogy, the lumped plasticity model is not able to capture progressive damage in the structure. In this model, the post-tension tendon is not modeled explicitly, and post-tension forces are calculated from a derived post-tension rotation relationship. Figure 4-15 below denotes the node diagrams for the numerical models of wall Archetype I, while Table 4-10 and Table 4-11 show element assignments. Comparing the two models, it is expected that fibre model will be more accurate. However, knowing how well the lumped plasticity model performs is important, because these models are very accessible and can be less computationally intensive. Figure 4-15: Wall Archetype I OpenSees node diagram for the Fibre Model (FM) and lumped plasticity model (LPM). Table 4-10: Element assignments for Archetype I fibre model Element Node Assignment Fibre ForceBeamColumn N1-N2 Timoshenko Beam N2-N3, N3-N4 Co-Rotational Truss N4-N5 Wall Configuration FM Diagram LPM Diagram N1 N2 N3 N4 N5 N1\/N2N3 N4N5 81 Table 4-11: Element assignments for Archetype I lumped plasticity model. Element Node Assignment Rocking Connection Zero-length Element N1-N2 Timoshenko Beam N2-N3, N3-N4, N4-N5 In the fibre model, the first node is placed at the base of the wall, and restrained in the x, y and rotation degree of freedom. The second node is defined at the height of the CLT damage zone. The height of the damage zone is taken as 0.3m, approximately twice the thickness of the wall, based on recommendations from Akbas et al. (2017). Connecting node one and two is a \u201cforceBeamColumn\u201d element. This element is defined with a fibre section that had dimensions equal to that of the CLT wall modeled, with a total of 50 fibres used across the cross section. Each fibre is assigned an \u201cElastic Perfectly-Plastic Gap\u201d material to capture the both uplift in tension and yield damage of the structure in compression. For each study, the elastic modulus of CLT is defined as described in Section 4.2. The compressive yield force used for the CLT is 25MPa. A total of three integration points are distributed evenly along the element length, and a P-Delta transformation is used. Above the CLT damage zone, nodes three and four are placed at the actuator load height, and top of the wall. Connecting these nodes is an elastic Timoshenko beam element, with cross section variables equal to the CLT section dimensions. The flexural and shear properties for the beam are based on section 4.2 and 4.4, respectively. The node five is then placed .25m below node one, and a corotational truss element is used to connect the elements. The extra length added by placing node five lower than node one accounts for the additional length of the post-tension tendon. During the analysis, the corotational truss element is used to apply the post-tension force to the structure, by using a steel02 material with an initial stress. The initial 82 stress of the material must be chosen such that, when released, the structure will settle at a force near to the target post-tension force. The process of choosing an initial stress for each wall is described in Section 4.8. In the lumped plasticity model, node one is similarly defined at the base of the wall, while node two is placed directly on top of it. Between these two coincident nodes, a calibrated zero length element is used to simulate the moment rotation relationship of the base connection in the CLT wall. For each wall and post-tension level, the backbone curve function defined in Section 4.5 is run, and the output moment rotation relationship is calculated. Points on the backbone curve are then used to create multi-linear elastic material. Nodes three, four and five are defined at the damage height, the actuator height, and the height of the wall, respectively. Timoshenko beam elements connect these nodes, with properties defined similarly to the fibre model. Numerical models for Archetype II are defined as in Figure 4-16 as well as Table 4-12 and Table 4-13. For Archetype II, most of the elements are as assigned in Archetype I. However, node three is now defined at the top of the damper height. From this node, rigid beam elements are defined perpendicular to the wall length to simulate the spacing between the axial dampers. The spacing between each damper is 0.16m for F4, 0.16m for F5 and 0.25m for F6. At the end of these rigid links, two nodes are defined, one connected to the rigid link, and another that is fixed. Between these two nodes, two calibrated zero length elements are defined, representing the two axial dampers on each side of the wall. 83 Figure 4-16: Wall Archetype II OpenSees node diagram for the fibre model (FM) and lumped plasticity model (LPM). Table 4-12: Element assignments for Archetype III fibre model Element Node Assignment Fibre ForceBeamColumn N1-N2 Timoshenko Beam Element N2-N3, N3-N4, N4-N5 Co-Rotational Truss N5-N6 Rigid Element N8-N3, N3-N10 Axial Damper Zero-length Element N7-N8 (x2), N10-N9 (x2) Table 4-13: Element assignments for Archetype II lumped plasticity model. Element Node Assignment Rocking Connection Zero-length Element N1-N2 Timoshenko Beam N2-N3, N3-N4, N4-N5, N5-N6 Rigid Element N8-N4, N4-N10 Axial Damper Zero-length Element N7-N8 (x2), N10-N9 (x2) Wall Configuration FM Node Diagram LPM Node Diagram N1 N2 N4 N5 N6 N3 N8 N7 N10 N9 N3N5N6N1\/N2 N4 N8 N7 N10 N9 84 Each zero-length element for the dampers is defined with a steel02 material. The material properties of each damper is then calibrated against tests of bucking restrained braces conducted at FPInnovations (Chen et al. 2018). The material property used is depicted in Figure 4-17, overlaid against test results. Based on the experimental testing done, a yield force of 44kN is used, as well as a stiffness of 44kN\/mm. The transition parameters were then modified to best match the hysteresis observed in experimental testing. In the experiment, a ratcheting effect is observed, where the displacement changes by approximately 1mm as the load passes through zero. This ratcheting is considered unique to the experiment and is not included by the material model. The final material model matched experiment reasonably closely. Figure 4-17: Calibrated axial damper material. Finally, numerical models for Archetype III are defined as in Figure 4-18 as well as Table 4-14 and Table 4-15. In Archetype III, there are two wall components modeled with the same node and element setup, and each wall connected by U-shaped flexural plate dampers. Individual walls have key elements as assigned in previous archetypes. However, there are additional 85 nodes defined at the damper heights connected with Timoshenko beams elements. In both studies, the height of the dampers is at one third and two thirds of the wall height. In the centre of the two walls, two nodes are defined on top of each other at the height of each damper. Rigid elements connect nodes between the two walls to nodes on the wall system. These are used to account for the offset between the CLT wall and the damper. Finally, calibrated zero-length elements are used to model the U-shaped flexural plate. Figure 4-18: Wall Archetype III node diagram for the fibre model (FM) and lumped plasticity model (LPM). Table 4-14: Element assignments for Archetype III fibre model Element Node Assignment Rocking Connection Zero-length Element N1-N2, N8-N9 Timoshenko Beam N2-N3, N3-N4, N4-N5, N5-N6, N9-N10, N10-N11, N11-N12, N12-N13 Co-Rotational Truss N6-N7, N13-N14 Rigid Element N3-N15, N15-N10, N4-N17, N18-N11 U-shaped flexural plate Zero-length Element N15-N16 (x2), N17-N18 (x2) Wall Configuration FM Node Diagram LPM Node Diagram N1 N2 N5 N6 N7 N3 N4 N8 N9 N12 N13 N14 N10 N11 N15, N16 N17, N18 N1\/N2 N3 N6 N7 N4 N5 N8\/N9 N10 N13 N14 N11 N12 N15, N16 N17, N18 86 Table 4-15: Element assignments for Archetype III lumped plasticity model Element Node Assignment Rocking Connection Zero-length Element N1-N2, N8-N9 Timoshenko Beam N2-N3, N3-N4, N4-N5, N5-N6, N6-N7, N9-N10, N10-N11, N11-N12, N12-N13, N13-N14 Rigid Element N4-N15, N15-N11, N5-N17, N18-N12 U-shaped flexural plate Zero-length Element N15-N16 (x2), N17-N18 (x2) The zero length elements used a steel02 material to match the behaviour of U-shaped flexural plate dampers predicted by experiment. Test data for the U-shaped flexural plate dampers was available in Ganey\u2019s experimental program, but not for FPInnovations. Based on the U-shaped flexural plate tests, material properties were chosen for the steel02 material, such that it matched experimental results. Figure 4-19 shows the calibrated material properties overlaid with the actual material properties. To extrapolate Ganey\u2019s U-shaped flexural plate test data to FPInnovations walls, predictions about the design parameters for the dampers were made based on equations (3.6) and (3.10). For Ganey\u2019s experimental data, the ratio between the predicted and calibrated material properties is calculated. These ratios were then used to scale the results to predictions made for FPInnovations\u2019 U-shaped flexural plate dampers. For each experimental program, the U-shaped flexural plate dampers had dimensions and material properties summarized in Table 4-3. The ultimate strength of the U-shaped flexural plate is used as the yield strength for the steel02 material. It is found that the stiffness is slightly less than predicted by equations, and the plastic force is slightly more than predicted by equations. 87 Table 4-16: U-shaped flexural plate damper parameters. Variable Ganey FPInnovations Prediction Calibration Prediction Calibration Plastic Force (kN) 18.9 21.1 8.29 9.3 Stiffness (kN\/mm) 3.0 2.6 1.36 1.13 Ratio of Yield Slope to Initial Slope (%) - 1.5 - 1.5 Transition Parameter cR1 - .88 - .88 Figure 4-19: Calibrated U-shaped flexural plate damper material. 4.7 Model Initialization Before the models could be run, several initializations are completed. For the fibre model, the first step in each analysis is to determine the initial stress needed to achieve the target post-tension force. To automatically initialize the post-tension stress, a program is created in Python that would run multiple OpenSeesPy analyses in a sequence. A sketch of the algorithm is depicted in Figure 4-20 below, where the blue box represents the initialization function in 88 Python. The program first inputs a trial stress and creates an OpenSeesPy wall model with that stress. This wall model is then analyzed to determine the force in the post-tension cable as the model settles. The if the trial force is not within a certain tolerance of the target force, an optimization algorithm is used to change the trial stress, and the process is repeated. Once the trial force is within a certain tolerance of the target force, it is returned for use in the main analysis. Figure 4-20: Summary of Algorithm to Initialize PT Force. For the lumped plasticity model, the backbone curve first had to be initialized. The curve from the monolithic beam analogy section analysis is calculated. Because the moment-rotation relationship from the monolithic beam analogy does not include elastic rotation, the initial portion of the curve has no slope. To initialize the curve, the first moment point is modified to be zero instead of the original starting value. The positive curve is then mirrored, and negative moment-rotation points are added to the curve so that cyclic loads can be applied. Run Analysis Check \u0e2b\ud835\udc39\ud835\udc5d\ud835\udc61,\ud835\udc56 \u2212 \ud835\udc39\ud835\udc5d\ud835\udc61\u0e2b < \ud835\udc61\ud835\udc5c\ud835\udc59 Input Trial Stress Return Final Stress Build Model Change Trial Stress 89 The OpenSees models were analyzed under similar loading conditions to the test models. Force is applied at the actuator height for all three archetypes. For Archetype III, a rigid link is used to ensure that both actuator nodes had the same displacement. For static loads, force is applied using a load control algorithm until the target displacement is reached. In the case of cyclic loads, displacements were applied to the structure using the load protocol from each experiment. A Newton algorithm is used to solve the system of equations. If the analysis failed at a certain step, smaller load steps were applied to the structure 4.8 Model Parameter Calibration As is shown in Section 4.2, there is a large amount of uncertainty about what material properties to use for CLT. To determine what material properties best fit the data, an optimization algorithm is completed on the static pushover models F1, F2, and F3. The purpose of this analysis is to minimize the difference between outputs from the numerical models, and experimental data. The fibre model is used for this calibration because it is considered more accurate than the lumped plasticity model. A sketch of the algorithm used to complete this analysis is overviewed in Figure 4-21, where the blue box represents the calibration function. First a trial parameter is chosen for the elastic modulus of CLT. Because the optimization analysis is done with FPInnovations\u2019 test data, a single parameter is used for both the compression and flexural modulus of elasticity. The experiments F1, F2, and F3 are then analyzed with the chosen parameter. Results from the shear-displacement curve are then compared to experimental data with a sampling function. The sampling function and it is inputs and outputs are overviewed in Figure 4-22. 90 Figure 4-21: Summary of Algorithm to Calibrate Material Property. Figure 4-22: Overview of Objective Function Algorithm. \ud835\udc45 =\u2211 (\ud835\udc662,\ud835\udc56 \u2212 \ud835\udc661,\ud835\udc56)2\ud835\udc41\ud835\udc56=0\ud835\udc41 (4.33) Run Analysis Get Objective Function Value Trial Parameter Return Parameter with Lowest Objective Function Build Model Change Trial Parameter Sample Shifted Data Output Objective Function Value Shift Curves into Common Domain Input Analysis and Experiment Curves 91 where: \ud835\udc45 is the residual for a single curve; \ud835\udc661,\ud835\udc56 is the y value of sample point \u201ci\u201d on the first curve; \ud835\udc662,\ud835\udc56 is the y value of sample point \u201ci\u201d on the second curve; \ud835\udc66\ud835\udc41 is the number of sample points. First the backbone curves for the experiment and analysis are read. These curves are then shifted into a common data frame between the maximum and minimum value of the experiment. A fixed amount of points are taken between the two maximum points at regular spacing. For all the values in the domain of the sample space, linear interpolation is used with the analysis curves to determine the range of the sample space. A residual function is then defined using the square of the difference between the sample experiment curve and the sample analysis curve (4.33). This residual is then divided by the number of samples, to prevent any one wall from being favored in the optimization analysis. The total residual is the sum of the residuals for all three walls. Once a residual has been determined for an elastic modulus, the process of running all three analyses and sampling the result is repeated with a different elastic modulus. To choose the next elastic modulus, a golden section algorithm is used. Bounding values for the analysis were 2GPa, and 8GPa. Use of the golden section algorithm assumes that the function only has one local minimum within the sample space. For the range of variables considered this assumption will likely hold valid. The returned objective function for the optimization analysis is shown in Figure 4-24. The results of the analysis predict that an elastic modulus of 2.6GPa will return the optimal results. The objective function is smooth in the sampled region. 92 Figure 4-23: Summary of algorithm to calibrate material property. Figure 4-24: Objective function for Optimization Analysis. Run Analysis Get Objective Function Value Trial Parameter Return Parameter with Lowest Objective Function Build Model Change Trial Parameter 93 4.9 Numerical Modelling Results For each of the twelve wall models used in this study, two numerical analyses were run in OpenSees. Key parameters from the numerical studies are then recorded and compared with outputs from experiment. The measured parameters of the study include: actuator drift, base shear, post-tension force, and percentage of base contact. For both the experimental and numerical measurements, interstory drift is calculated by dividing horizontal displacements at the actuator by the height of the actuator. For the experimental program, shear force is calculated using load cells at the actuator location. In the numerical program, shear force is recorded using the horizontal reaction force at the base of the shear wall. For the fibre model, the horizontal reaction force for the post-tension cable is also recorded, and the net reaction force is the sum of the wall and post-tension base shear. Force in the post-tension cable is calculated by monitoring the force in the cable element itself. For experimental walls, the force is monitored with a load cell at the top of the wall, for numerical walls the force in the element is recorded. In the lumped plasticity numerical model, a function is created using the post-tension force \u2013 rotation relationship. This function is then used with numerically calculated rotation values to evaluate the post-tension force. Experimental readings of the percentage of base in contact with the foundation is available for Ganey\u2019s experimental program. These readings were made based on a series of potentiometers at the base of the wall model. Linear interpolation between the potentiometer readings is used to determine the location of uplift. In the lumped plasticity model, the base contact is calculated using the rotation of the base element, and the predefined relationships between rotation and base contact. In the fibre model, the percentage of base in contact with the foundation is 94 calculated by outputting the normal and rotation components of section strain. Using these two values, the neutral axis location can be calculated by finding the point of zero strain: \ud835\udf00 = \ud835\udf000 + \ud835\udc66\ud835\udf19 (4.34) where: \ud835\udc66 is the distance from the centre of the cross section; \ud835\udf00 is the strain at a cross section location y; \ud835\udf000 is the normal component of section strain; \ud835\udf19 is the rotational component of section strain. Figure 4-25 shows the results of the fibre model for Ganey\u2019s experimental program. For these experiments, CLT crushing played a more significant role in the wall panel behaviour. In the cyclic single wall tests, the fibre model is able to match both the backbone curve, and the deterioration path of the CLT walls. For all cases studied, the experiment had more asymmetry than the analysis models. Generally, one side of the numerical curve matched the experiments better than the other. For G1 and G2, negative cycles were stiffer than experiment, and for G3 and G4, positive cycles were less stiff than the experiment. In all three models, the initial elastic range occurred up to approximately 0.5% drift. Figure 4-26 depicts the shear force for the fibre model compared against experimental data from the FPInnovations test program. For the static pushover tests, the fibre model is slightly stiffer than the experiment, but able to capture the overall shear behaviour quite closely. For models from Archetype II, the static backbone followed a similar trend to the experimental static backbone but is stiffer between the ranges of 0.5% and 2% drift. The cyclic backbone for tests F5 and F6 is slightly less stiff than the experiment, but followed the trend closely. The unloading stiffness slope departed from experimental values, leading to hysteretic loops that were less full than the experiment. For Archetype III, test F6 matched the experimental trend 95 and values well. The unloading slope is slightly different, showing a more pronounced flag shaped behaviour. The numerical model is notably more elastic at low drifts than the experiment is, suggesting the experiment experienced permanent deformation. For the model G4 of Archetype III, the experiment matched the general trend well. However, the experimental curve is much more asymmetric than the numerical curve. For the final drift cycles, upon reloading the numerical increased in stiffness in a convex curve, while the experimental curve is close to linear. In experiment G4, it can be observed that both the experiment and numerical analysis transition from a flag shaped hysteresis to more of a pinching shaped hysteresis. Figure 4-25: Shear Force vs. Drift for Fibre Model Tests (G1) (G2) (G3) (G4) 96 Figure 4-26: Shear force vs. drift for FPInnovations fibre model tests. (F7) (F1) (F2) (F3) (F4) (F5) (F6) 97 Next, the post-tension tendon force is compared with experimental results, as depicted in Figure 4-28. For every archetype in FPInnovations\u2019 experimental data set, the post-tension force is greater in the numerical models than it is in the experiments. In the static pushovers, this difference is most pronounced at higher drift, however the results overall matched the experiment closely. For F5, the fibre model post-tension tendon force followed a \u201cbow shaped\u201d hysteresis, similar to the experiment. The numerical model in F5 initially had stiffness comparable to the experiment, then after approximately 1% drift, departed along a steeper slope. In Archetype III, the fibre model matched the experimental trend quite closely. Notably, the fibre model is able to capture differences in the post-tension slope between the loading and unloading direction. Figure 4-27: Post-tension force vs. drift for Ganey\u2019s tests. (G1) (G2) (G3) (G4) 98 The fibre model post-tension force in Ganey\u2019s experiments is showcased in Figure 4-27. For both Archetypes I and III, the post-tension force shown matches the experimental trend well. Similar to the FPInnovations study, the post-tension force predicted by the numerical analysis is greater than the experimental force. The increase in stiffness is most pronounced at higher drifts. In models G1 and G2, no yielding occurs in the PT bar. In G3, yielding occurs at approximately 4% drift, similar to experiment. In model G4, yielding occurs at approximately 3.5% drift, which is earlier than what is observed in the experiment. For experimental tests, there is an abrupt drop in the post-tension force as it transitions from loading to unloading. This drop in post-tension force is more pronounced in upper cycles and is not seen in the numerical model. For experiment G4, data is not available for the south wall of the experiment due to a failed load cell. Finally, Figure 4-29 depicts the amount of base in contact with the foundation for G1, G2 and G3 in the fibre model. Results show that the initial uplift of the wall occurred much earlier than what is measured experimentally for models. In the numerical models, uplift begins at very low drifts, while in the experiment the wall remains in contact with the foundation up to a drift of approximately 0.5%. The numerical curves have an initial decompression slope that is similar to experimental values, but experimental curve softened at a higher rate. Beyond approximately 2% drift, the numerical results aligned more closely with experiments. 99 Figure 4-28: Post-tension force vs. drift for FPInnovations tests, fibre model. (F1) (F2) (F3) (F5) (F7) North (F7) South 100 Figure 4-29: Drift vs. base contact for lumped plasticity models of Ganey\u2019s tests. Next, the results of the numerical lumped plasticity model in base shear is depicted in Figure 4-30. The lumped plasticity numerical model matched well with experimental results for the static pushovers F1, F2, and F3. For the models of Archetype II, F4, F5, and F6, the lumped plasticity model could not capture the experimental trend, and there is a significant deviation for both static and cyclic tests. The deviation between the experiment and the model was less significant in F4, where the initial post-tension force was lower. For Archetype III, the lumped plasticity model could match the experiments shear force trend, suggesting that crushing was not significant for these walls. (G1) (G2) (G3) 101 Figure 4-30: Shear force vs. drift for lumped plasticity model tests. (F7) (F1) (F2) (F3) (F4) (F5) (F6) 102 Figure 4-31 presents the results of the lumped plasticity model against the experiments studied in Ganey\u2019s experiments. Up to cycles of approximately 2% drift, the lumped plasticity model captured the general trend quite well. However, beyond the CLT yield point, the lumped plasticity model begins to depart from the experiment during unloading and reloading. For experiment G4 from Archetype III this trend is repeated. However, lower drifts were less accurate and hysteretic loop enclosed significantly less area. Figure 4-31: Shear force vs. drift for lumped plasticity tests of Ganey\u2019s walls. Figure 4-32 shows the post-tension tendon force for the lumped plasticity model, compared to FPInnovations\u2019 experimental data. As with the fibre model, the lumped plasticity model is generally stiffer than predicted in experiment. For Archetype III, the lumped plasticity walls, the post-tension force is recovered using the rotation to post-tension force relationship force (G1) (G2) (G3) (G4) 103 for single walls. This assumption will lead to some directional inaccuracies. The vertical force transferred by the U-shaped flexural plates will increase the post-tension force for drifts in one direction, and decrease it for drifts in the opposite direction. However, despite this, the post-tension force predicted for Archetype III is quite close to experimental values. Figure 4-32: Post-tension force vs. drift for FPInnovations lumped plasticity tests. (F1) (F2) (F3) (F7) North (F7) South 104 Figure 4-33 shows the post-tension tendon force for the lumped plasticity model, compared to Ganey\u2019s experimental data. In all walls considered, the post-tension force is considerably more stiff than experiment at drifts beyond approximately 1%. The lumped plasticity model also could not capture the cyclic deterioration of the post-tension tendon, as expected. Figure 4-33: Post-tension force vs. drift for Ganey\u2019s lumped plasticity tests. Figure 4-34 summarizes the amount of base in contact with the foundation, compared with the experimental results for Ganey\u2019s study. The amount of base in contact with the foundation is recovered from the original rotation neutral axis relationship for individual CT walls. As with the fibre numerical model, the walls begin to decompress sooner that the experimental values and begin to converge at higher drifts. (G1) (G2) (G3) (G4) 105 Figure 4-34: Drift vs. base contact for Ganey lumped plasticity models. Finally, the time taken to run each OpenSees analysis is compared in select studies. These results are shown to give a general indication of the speed of each model. There are many factors that go into the speed of a program, so the following section should only be taken as a very rough guideline to highlight trends in analysis speed. Table 4-17 depicts the difference between run times for the lumped plasticity model and the fibre-based model. To simulate realistic scenarios a maximum drift of 2.5% is considered, as beyond this point the lumped plasticity model is inaccurate. The pre-analysis time is not considered. For each analysis, three runs were completed and the results were averaged. For test G4, there is a significant increase in time for the fibre model. This is because a much smaller displacement increment test is (G1) (G2) (G3) 106 needed for the model to converge. Results show that the fibre-based model takes approximately four times longer to run than the lumped plasticity model. Table 4-17: Time taken to complete numerical analysis. Study Analysis time (s) Fibre Model Lumped Plasticity Model G1 0.50 0.12 G2 0.44 0.11 G3 0.40 0.12 G4 35.20 5.50 F7 30.20 18.82 4.10 Parametric Studies Due to the uncertainty surrounding material properties, a series of parametric studies are carried out on key variables for the fibre-based numerical model. For each study, a variable of interest is chosen, then a sequence of OpenSeesPy analyses are run using different values of that variable. Study G1 is taken to be the baseline model, with base material properties outlined in Table 4-18. Each OpenSees analysis completed is a static pushover. For each sequence of pushovers, the shear force, post-tension force, and amount of base in contact is compared to each other and experiment values. The maximum and minimum values presented in Table 4-18 were chosen based on the values of CLT observed in Section 4.3. The flexural and compression elastic modulus values range from one gigapascal, which is close to the lower bound observed in compression tests, to nine gigapascal, which is close to the value predicted by code. The shear modulus varied from 50MPa, to approximately 650MPa. These values were based on the range of shear stiffness 107 values observed in Zimmerman and Mcdonnell (Zimmerman and Mcdonnell 2017), as depicted in Figure 2-19. In this study, it is observed that experimentally measured values of shear stiffness ranged from 350MPa to 950MPa, and predictions from equations ranged from 300MPa, to 500MPa. The lower values of 50MPa, and 200MPa used in this parametric, are considered unlikely to occur in practice. For the yield stress of CLT, the range of values considered were calculated by taking the and adding \u00b140% to that range. Table 4-18: Summary of used material properties. Parameter Base Value Minimum Value Maximum Value Flexural Elastic Modulus (GPa) 3.5 1.0 9.0 Compression Elastic Modulus (GPa) 3.0 1.0 9.0 Shear Modulus (MPa) 350 50 650 Compression Yield Strength Modulus (MPa) 25 15 35 Number of fibres 100 10 100 The first parametric study completed is on the number of fibers in the fiber section. Figure 4-35 shows the results of this parametric study. For the wall section tested, it can be observed that runs using between 20 and 100 fibers had very similar results. 108 Figure 4-35: Parametric study of number of fibres on: (a) shear, (b) PT force, (c) base contact. Figure 4-36 depicts a parametric study where the flexure stiffness of the Timoshenko beam members is varied. A range of one to nine gigapascals is considered in the analysis, with intervals of two gigapascals. For all three of the response measures, the difference between each study is greater for low values of flexural stiffness. Between one and three gigapascals, there is a large jump in the response. However, between five and nine gigapascals there is very little difference between the response curves. This behavior suggests that as the flexure behaviour of the wall becomes stiffer, rotation of the base connection begins to govern the response of the overall system. (a) (b) (c) 109 Figure 4-36: Parametric study of CLT flexural stiffness on: (a) shear, (b) PT force, (c) base contact. Figure 4-37 depicts a parameter study where the compression stiffness of CLT panels is varied. A similar range of one to nine gigapascals are used for the experiment. There is a large jump in the systems stiffness between one and three gigapascals, and little change in stiffness between five and nine gigapascals. Regarding the percentage of the base in contact with the foundation, the run with one gigapascal matched experimental results most closely. Figure 4-38 shows a parametric study where both the flexural modulus of elasticity and compression modulus of elasticity are varied together. For the shear force and post-tension force, the run using three gigapascals matched results most closely. (a) (b) (c) 110 Figure 4-37:Parametric study of CLT compression stiffness on, (a) shear, (b) post-tension force, (c) base contact. Figure 4-38: Effect of CLT flexural and compression stiffness on (a) shear force, (b) post-tension force, (c) base contact. (a) (b) (c) (a) (b) (c) 111 A parametric study is also completed on the yield stress of CLT in Figure 4-39. Analysis runs of took place between a lower bound of 15MPa and upper bound of 35MPa, with intervals of 5MPa used for each trial. As expected, the initial portion of each curve is unchanged between runs. Beyond 1% drift the curves were affected, and changes between CLT strengths resulted in a relatively uniform change between each curve. Figure 4-39: Effect of CLT yield stress, (a.) shear force, (b.) post-tension Force, (c.) base contact. Finally, Figure 4-40 depicts a parametric study completed on the model, using the shear modulus of CLT as the design variable. The shear modulus is varied between 50MPa, and 650MPa in increments of 150MPa. Based on the analyses completed, the models showed a low (a) (b) (c) 112 sensitivity to shear modus. Runs in the range of 200MPa to 650MPa had nearly identical results, while there is a large jump between 50MPa and 200MPa. Figure 4-40: Effect of CLT shear stiffness on, (a) Shear force, (b) post-tension force, (c) base contact. 4.11 Discussion of Wall Models Based on the calibration analysis and parametric studies completed, the material properties chosen in Section 4.2 were appropriate for the numerical models. Results from both the section analysis and numerical studies suggest that the CLT wall stiffness is significantly less than predicted by the grade of CLT. Ultimately, the material properties chosen were not based on any single test, but on what fit the data best. While the wall models could match the experiments, it would have been difficult to predict what material parameters to use prior to testing . Based on the studies examined, more information is needed to accurately predict the (a) (b) (c) 113 compression and shear modulus of elasticity for CLT. However, while by no means definitive, some rough guidelines are proposed based on the trends observed. A connection modulus, or ratio of connection stiffness to global stiffness, can be found by dividing the CLT stiffness predicted by PRG 320 by the parameters employed in the model. For Ganey\u2019s study the connection modulus is 0.45, for FPInnovations the connection modulus is 0.3. Looking at measured predictors, for FPInnovations and Ganey\u2019s study, the elastic modulus calculated using the tangent slope data lead to reasonably satisfactory results. FPInnovations tangent measurement is very close to the used material property, while Ganey\u2019s tangent measurements were slightly less accurate. However, using the tangent stiffness to predict the elastic modulus is not practical, because it requires that walls are tested. Looking at other predictors, the compression modulus of elasticity for CLT has potential to lead to good results. In Ganey\u2019s experiments, the compression modulus of elasticity is equal to the used modulus of elasticity, and accurately predicted wall performance. In contrast, the compression modulus of elasticity from FPInnovations study is significantly different than the modulus of elasticity in the calibrated numerical models. The calibrated modulus of elasticity, 2.6Ga, is between the compression global value of 5.6GPa and local value of 1.6GPa. A possible reason for the difference in accuracy between each experiment is that Ganey and FP Innovation\u2019s compression tests were completed very differently. In FPInnovations compression tests, load is applied along the specimen edges, while load is applied concentrically in Ganey\u2019s. Because the calibrated material property is between the local and global modulus of elasticity, it is possible that over some predefined length the modulus of elasticity would yield more accurate results. 114 The fibre-based numerical model is able to accurately capture the shear and post-tension force for all archetypes with reasonable accuracy. The shear force is similar to the experimental shear force for all models, while the post-tension force is generally stiffer than experiment. The largest deviations from the experiment occurred in Ganey\u2019s study at high drifts, where the PT cable prematurely yielded due to extra stiffness in the model. It is possible that the difference in post-tension stiffness between experiment and numerical analysis is caused by localized deformations at the top of the CLT panel. Despite the differences in post-tension cable force, the fibre model is able to accurately capture wall behaviour. Comparing the shear and post-tension force for the fibre and lumped plasticity numerical model, the fibre model is more accurate. This result is expected, as the lumped plasticity model used a multi-linear elastic property which does not degrade. While the lumped plasticity curve quickly diverged from experiment past the yielding of CLT, in the linear elastic range of CLT it had similar accuracy with the fibre model. This can be seen in F1, F2, and F3, and F7, where the lumped plasticity model accurately matched the experiments where cyclic loads had been applied. It is also highlighted in the tests G1, G2 and G3, where the lumped plasticity model is quite close to experiment at drifts less than 2%. For all experiments in Archetype I and III, the lumped plasticity model is able to capture the experiment backbone curve with similar accuracy to the fibre model. Based on these observations, there are two scenarios where the lumped plasticity model can be used accurately for Archetypes I and III: static pushovers, and analyses where yielding of the CLT is not significant. The Lumped plasticity model was notably inaccurate walls in Archetype II. This discrepancy could be caused by the lumped plasticity model inaccurately predicting rotation close to the base of the wall. 115 For both model types, the amount of base in contact with the foundation is significantly different than for experiment. Both the initial decompression point, and the base-contact drift curve were underestimated by the numerical analyses. There are several possibilities that may account for the observed difference between experiment and analysis. Because data are only reported from one source, Ganey\u2019s study, it is possible that this effect is unique to that experiment. The strain distribution across the cross section may also be incorrect. Currently it is assumed that strain is linearly distributed across the base, however the real behaviour at the base of the wall may be more complex. It is also possible that localized compression effects are influencing the CLT stiffness, making it less than expected and increasing the amount of base in contact with the foundation. Regions of the wall may not be in proper contact with the foundation, reducing the effective stiffness of the section. Finally, the elastic perfectly plastic material property may not be able to accurately capture the initial portion of the stress strain curve. For component level walls, there is not a significant increase in analysis speed between the fibre and lumped plasticity numerical model. The fibre model did run approximate four times slower than the lumped plasticity model. However, this performance increase will likely be less significant for bigger models, where inversion of the global stiffness matrix will take more time. 116 Chapter 5: Seismic Performance of Wall Systems in Buildings In the following chapter, the lateral system for a two-story structure with a self-centring CLT walls is analyzed using numerical models from Chapter 3. Experimental data are taken from the NHERI TallWood shake table tests (Pei et al. 2019), where a building was tested under a suite of ground motions. First an overview of the building and experimental program is given. During testing, it was found that the steel beam supporting each rocking wall deformed significantly. This extra flexibility influenced the dynamics of the structure and is modeled to accurately capture the buildings dynamic behaviour. The numerical models used on the building are also described. Both lumped plasticity and fibre models were used to analyze the lateral system of the structure. A calibration analysis is run to determine what properties to use for the flexible beam foundation. The calibrated base material is then applied to the numerical models of the system. Results from these analyses are then compared against experimental data for key analysis parameters, including base shear, post-tension force, and interstory drift. To predict what the performance of the building would have been on a rigid foundation, a second analysis is then completed without the flexible foundation beam. The suite is run twice, once sequentially, and once non-sequentially. A comparison is also given between the lumped plasticity model, and the fibre model for individual earthquakes in the test suite. Finally, a discussion is then given on the results of the numerical analysis and the performance of each model. 5.1 NHERI Study Overview Experimental data for the building studied is based on a structure that was tested as part of the NHERI TallWood Project (Pei et al. 2017, 2019). For this sub-task of the ongoing project, a 117 two-story building was constructed and tested at UC Sand-Diego\u2019s NHERI shake-table facility. Figure 5-1 shows the building as tested, and a model of the structural elements in that building. A summary of the major components of the building is listed in the following chapter. A full description of the building and the design process is described in Wichman (2018). Figure 5-1: Overview of two-story building (Griesenauer 2018). The height of the first and second floor of the building was 3.66m and 3.05m respectively, and the foundation beams were approximately 0.95m tall. At each level, a 6.1m by 17.7m diaphragm was supported by a series of glulam beams and columns. The first-floor diaphragm Column, discontinuous Foundation beam Column, continuous Self-Centring wall pair 118 was a CLT panel, while second floor diaphragm was a composite between a CLT panel and a concrete topping. The outer columns were discontinuous and connected to the bottom of the beams, while the inner columns were continuous. The gravity system members had special connections to minimize damage as the CLT walls rocked, as well as limit their contribution to the strength and stiffness of the lateral system. For column bases, slotted connections were used for bolt holes to permit base rocking, as shown in Figure 5-2. At the beam and column interface, a connection was used that permitted rotation between the beams and column. The seismic mass of the building was reported to be 41.6 metric tons for the first floor, and 42.8 metric tons for the roof. Figure 5-2: Slotted connection for column base (Griesenauer 2018). Lateral force in the building was resisted by two pairs of self-centring walls, for a total of four shear walls. The seismic system was decoupled from the gravity system and did not resist any gravity loads. To facilitate this decoupling, the floor diaphragms were connected to CLT walls with a special slotted joint designed to only transfer shear force. This connection also allowed for uplift of wall elements to not be transferred into the diaphragms. At each level, the floor diaphragms braced the rocking wall against out of plane movement. Table 5-1 overviews the 119 geometric and material properties of each CLT shear wall. An elastic modulus of 8.5GPa was reported for the CLT (Wichman 2018), based on test data that is publicly available at the time of publishing this document. For each of the two wall pairs, five U-shaped flexural plate dampers were placed at heights from the wall base of 2.3m, 3.3m, 4.4m, 5.5m and 6.5m. Table 5-2 overviews the U-shaped flexural plate material and geometric properties. Table 5-1: CLT geometric properties. Number of Laminations Lamentation Thickness (mm) Wall CLT Grade Width (mm) Length (mm) Height (mm) 3 (Strong) 2 (Weak) 35 (Strong) 35 (Weak) 175 1520 7320 E2 Table 5-2: U-shaped flexural plate material and geometric properties. Variable Value Yield Stress (MPa) 350 Young\u2019s Modulus (GPa) 200 Width (mm) 115 Thickness (mm) 9.5 Diameter (mm) 92 Yield Force (kN) 13.1 Plastic Force (kN) 19.7 Stiffness (kN\/mm) 4.8 Table 5-3: Post-tension tendon properties. Yield Stress (MPa) Ultimate Stress (MPa) PT Force (kN) PT Area (mm2) Initial Stress (% yield) 625 825 53.5 (x4) 316 (x4) 0.30 Each wall was pre-tensioned to the foundation using four 19mm diameter post-tension cables. On either side of each panel, two cables were offset by approximately 130mm from the panel 120 centreline. Table 5-3 summarizes the initial post-tension force and material properties for each post-tension cable. The elastic modulus of the cable was not reported, so it is taken to be 215GPa. This was based on the modulus of elasticity for post-tensioned cables in Ganey\u2019s tests (Ganey 2015). At the top of each CLT panel, a post-tension saddle was used to connect the PT cables to the wall. Each cable was pre-tensioned to an initial force of 53.4kN, which corresponded to 0.39 of the cables yield force. At the foundation of the building, the gravity system was supported by steel beams that were cantilevered off the base of the shake table. Similarly, the CLT walls were supported by a hollow steel section that was bolted to the top of the shake table. The each CLT wall rested on top of the flange of the hollow beam section, where shear tabs prevent sliding or out of plane motion. A connection for the post-tension cables was welded to the base beam on either side of each wall. The lateral system of the building was designed to achieve a specific performance at four different performance targets: (1), wind loading; (2), immediate occupancy; (3), limited repair; (4), Collapse prevention. Whether the building performed as expected was determined by the status of key structural components in the lateral system. The structural members monitored include the U-shaped flexural plate dampers, CLT walls, and post-tension tendons. Interstory drift was also considered as a limit state for the structure. A description of each performance level, and their key limit states is provided in Table 5-4. These objectives were chosen based on the LA Tall Buildings Alternate Design Procedure (LATBSDC 2018) and recommendations made by Ganey (2015). Because the building was relatively small, additional modifications to these performance objectives were made. These included: allowing \u201cminor\u201d yielding of U- 121 shaped flexural plate dampers during the service level hazard and permitting \u201cminor\u201d crushing to occur at the base of the CLT wall after 1% drift. Table 5-4: Limit states for 2 story building. (Wichman 2018). Performance Level Description Seismic Hazard Level Limit States 1 Wind Load - No decompression of CLT walls. 2 Immediate Occupancy 50% in 30 years (SLE) Minor yielding of the U-shaped flexure plates, no yielding of post-tension tendon, no crushing of the CLT. 3 Limited Repair 10% in 50 years (DBE) Yielding of the U-shaped flexural plate permitted, no yielding of the post-tension tendon, minor crushing of CLT. Inter-story drift less than 2% 4 Collapse Prevention Maximum Considered (MCE) Crushing of the CLT panel permitted, minimal yielding of PT cable permitted. Inter-story drift less than 4%. After design and construction was completed, fourteen unidirectional ground motions tests were sequentially run on the building to verify its performance at each of the seismic hazard levels. United States Geological Survey hazard data corresponding to a location in Oakland California with class B soil were used for the site-specific seismic hazard for the structure. The fourteen ground motions tested were created by scaling records of four unique ground motions events: 1979 Imperial Valley, 1989 Loma Prieta, 1987 Supension hill, 1994 Northridge. Once the records were scaled to match the target site specific spectra, an additional level of scaling was used to account for the range of possible outputs from the shake table. A full description of the seismic hazard assessment can be found in Sarah Wichman\u2019s 2018 thesis (Wichman 2018). The sequence of seismic tests as they were run on the building is overviewed in Table 5-5. Table 5-5 also overviews the properties of each test, including the base event name, 122 seismic hazard level, and effective spectra acceleration at a period of 0.9s. This period corresponds to the measured fundamental period for the building. Table 5-5: Groundmotions considered. Ground-motion Event Name Hazard Intensity Effective PGA (g) GM1 Loma Prieta SLE 0.16 GM2 Loma Prieta SLE 0.18 GM3 Northridge SLE 0.19 GM4 Superstition Hills SLE 0.13 GM5 Northridge DBE 0.53 GM6 Northridge* DBE 0.52 GM7 Imperial Valley SLE 0.13 GM8 Northridge* DBE 0.53 GM9 Loma Prieta DBE 0.52 GM10 Superstition Hills DBE 0.44 GM11 Loma Prieta MCE 0.62 GM12 Northridge MCE 0.73 GM13 Superstition Hills MCE 0.63 GM14 Northridge 1.2 x MCE 0.85 *GM run twice back-to-back After each earthquake test was run, a white noise follow up test was run to estimate the natural frequency of the building. At the beginning of each test day, the building was also inspected to assess damage to the structure, or the effect of changes. Inspections occurred more frequently after larger earthquakes. During some of the inspections, modifications were made to the structure. Most notably, attempts were made to stiffen the foundation of the rocking wall after GM5 was run. On either side of the beam, pieces of scrap metal were welded to the base beam close to the outer edge of each wall. The post-tension cables were also re-tensioned if their force was different from their initial force by \u00b11.3 kN. Re-tensioning was only required after earthquakes with higher intensities and was performed after GM8 and GM13. 123 After testing, it was observed that the foundation beam for the CLT wall had deformed downward significantly as the wall rocked. This deformation was most pronounced at the edges of the CLT wall, where rocking forces were greatest. The downwards deformation persisted after the stiffeners were added to the structure and became more pronounced as earthquake intensity increased. After the experiment was completed, permanent deformation of the wall was noted, indicating the beam foundation beam had yielded. Little torsion was observed in the experiments, indicating that the centre of mass and rigidity for the structure were likely quite close. This result was expected as the structure was very symmetric. It also was observed that structure had deviated from the original design performance objectives. Notably, there was much less damage observed in the CLT wall foundation and post-tension tendon than expected. For all earthquakes considered, no significant crushing of CLT was observed during testing, even at 1.2 the MCE level. Similarly, yielding of the PT bars did occur, but only during the final 1.2 MCE level test. The recorded drifts of the structure were also higher than expected for DBE tests, with the tests averaging 2.24% (Wichman 2018). The lack of damage in the structure suggests that deformation of the base beam had influenced the dynamics of the structure and protected main structural members from damaging forces. The outputs of the structure were recorded using the following instrumentation. Load cells were used to record the post-tension force in each CLT wall cable. Shear force in the CLT walls was measured using two methods: accelerometers measured the acceleration at different locations of each floor, and strain measurements made at the wall diaphragm connector. Notably, both methods of measurement lead to different base shear results. Potentiometers were used to measure the downward displacement of the CLT foundation beam throughout the 124 experiment. Absolute displacement at each floor was measured using readings from potentiometers attached to fixed points outside of the shake-table. 5.2 Overview of Building Model To evaluate how the wall models described in Section 3.5 perform when loaded dynamically, the two story building described in Section 5.1 is modeled in OpenSees. Like the wall component models, both fibre and lumped plasticity numerical models of the building were used. For each numerical model, simplifications were made to reduce the building model\u2019s complexity and improve computational speed. In the numerical models, only the lateral system and the major components of that system were included. The gravity system of the building is not modeled. This simplification is possible because the gravity system is connected to the foundation using slotted bolt holes that prevent transfer of shear force. While some lateral force transfer to the gravity system is inevitable, the amount of shear passed would be difficult to quantify without component level testing of the column connections. The effect of modelling this extra stiffness would likely have a negligible effect on the overall dynamic behaviour of the building. To include the overturning effects of out of plane gravity loads, a leaning column is added to the model. More simplifications were possible because the building was observed not sensitive to torsion during testing. Throughout experimentation, the rotation of each diaphragm was negligible, and approximately half of the seismic force was transferred to each wall pair. Because of the buildings uniform response to loads, only two dimensions and one wall pair is considered in the model. It is assumed that seismic load would transfer evenly to each wall, so only half of the total seismic mass for the building is modeled. To account for the additional moment caused by the drift of the gravity system, a leaning column is included in the model. 125 Figure 5-3: OpenSees model of 2 story building, and expanded groups of nodes. Wall Configuration Node Diagram N101 N104 N105 N107 N106 N108 N109 N110 N111 G1 G2 G3 G4 G1 East Wall West Wall G3 G4 G2 N103 N114 N102 N113 N301 N302 N112 N115 N116 N402 N403 N404 N401 N406 126 Figure 5.2 depicts the OpenSees model of each wall system. Nodes are given an identification for each wall, using the hundreds digit as a label. The node groups are defined as follows: group \u201c100\u201d belongs to the West wall; \u201c200\u201d corresponds to the East wall; \u201c300\u201d corresponds to U-shaped flexural plate nodes; and \u201c400\u201d corresponds to leaning column nodes. Nodes in the East wall are placed at the same height as the West wall and have the same element assignments. Nodes in each wall pier have the same horizontal location, and there is a horizontal gap between each wall pier equal to the wall length plus a 25mm gap. Table 5-6 overviews the elements assigned between each node. Nodes in the second wall are not listed as they mirror those in the first wall. Figure 5-4 overviews the base elements of the structure in the case of the fibre model. Table 5-6: Element assignments for the 2 story fibre model. Element Base Node Assignment Foundation Element N102-N103 Rocking Connection Zero-length Element N103-N104 Timoshenko Beam N104-N105, N105-N106, N106-N107, N108-N109, N109-N110, N110-N111, N111-N112 Co-Rotational Truss N113-N115, N114-N116 Rigid Beam Element N101-N102, N112-N115, N112-N116, N102-N113, N102-N114, N401-N402, N404-N405 Flexible Rotation ZLE N402-N403, N403-N404, N405-N406 Rigid Truss Element N207-N403, N211-N406, U-shaped flexural plate Zero-length Element N301-N302, N303-N304, N305-N306, N307-N308, N309-N310 127 Figure 5-4: Overview of wall base elements. The first node of the model is fixed at grade level. A rigid element is used to account for the height of the base beam, which is supported on the shake table. Connected to the top of the rigid beam element is a zero-length section element, which models the flexural response of the base beam. A fibre section with length and width equal to the base of the CLT wall is assigned to the zero-length element. Each fibre in the section is assigned an elastic material, whose modulus of elasticity would be calibrated to best fit the experimental data. An in-depth explanation of the calibration process for this material can be found in Section 5.3. Using a zero-length section to model the base beam can give good average results for the base deformation, however it has several limitations. Notably, there is no one-to-one correspondence between the forces at the bottom of the wall, and forces applied to the steel beam surface. This loss of fidelity means that models with the flexible beam will not be able to capture the compression force at the bottom of the wall with full accuracy. However, because the average properties across the section are similar to the real beam, accuracy of global results such as base shear and drift should be preserved. Because the foundation beam uses an elastic material, the model is not able to capture the yielding of the base beam. To account for the Rigid Foundation Element Flexible Foundation Element Rocking Element 128 yielding of the base beam, two different elastic moduli are considered for the flexible zero length element, one for pre-yield and one for post-yield. Next the rocking element is connected to the top of the flexible foundation element. The type of rocking element used depends on if the model is a fibre or lumped plasticity model. An elastic modulus and yield strength of 8.5GPa and 25MPa is used for the base contact element in both models. For the fibre-based numerical model, the rocking element is an OpenSees \u201cforceBeamColumn\u201d element. This element is defined as described in Section 4.6, with a height of 0.3m. For the lumped plasticity model, the rocking element is a zero-length element with a custom material assigned to the moment direction of that element. The monolithic beam analogy is used to create a moment-rotation backbone curve for the wall system. The Python program that is defined Section 4.6 is used to automate the analysis. The geometry and materials properties presented in this Section were used in the section analysis. Figure 5-5 overviews a linearized moment-rotation relationship derived for a single wall. The initial decompression point occurs at 0.0001rad and a moment of 107kNm, the first instance of CLT crushing occurs at 0.006rad, and yielding of the first bar occurs at 0.024 rad. Also included is a plot of the rigid rotation to post-tension force relationship. Figure 5-5: Moment-rotation backbone for lumped plasticity model. 129 After the base elements are created, nodes are then defined at the height of each U-shaped flexural plate, as well as the location of each story. Connecting these nodes is a set of Timoshenko beam elements, with geometry defined by the CLT section. While it is observed in Section 3.5 that the flexural and compressive elastic modulus for CLT may differ, a value of 8.5GPa is used for both elastic modulus in lieu of more detailed data. No measurements were available on the CLT shear properties, so a shear modulus of 350MPa is also used for the Timoshenko beam element. As noted in Section 4.10, the behaviour of the CLT section is not sensitive to shear, so this value is deemed sufficient as a reasonable lower bound. At each floor level, the nodes of the east and west walls have their horizontal degree of freedom linked so they have the same displacement. The seismic mass of each floor is applied to the node at floor height, and evenly distributed between each wall pairs. At the node between the rigid foundation element and flexible foundation element (Nodes 101 and 201), two rigid elements are defined to account for the offset of each post-tension cable. Similarly, two rigid elements are placed at to the top of the CLT wall. Between the nodes at the end of these rigid elements are corotational truss elements that model the post-tension tendons. These were assigned a steel02 material and given an initial stress which would create the pretension force. As noted in Section 4.4, the post-tension tendon will likely be less stiff than the predicted nominal stiffness. To account for this reduction in stiffness, a stiffness modifier of 0.85 is applied to the elastic modulus of the post-tension bar. This modifier corresponds with the stiffness modifier used to model walls in Ganey\u2019s study. To simplify calculations, the two bars on either side of the wall were combined into a single bar, reducing the total number of bar elements from four to two for each wall. Figure 5-6 depicts this simplification, looking at a cut of the cross section. The new equivalent bars have a diameter 130 of 26.8mm, and initial post-tension force of 107kN. All other properties of this equivalent bar remained unchanged. Figure 5-6 Equivalent bar elements. After each wall is created, two nodes were defined at the centre of each wall, at the height of the U-shaped flexural plate nodes. Rigid beam elements were used to connect the centre nodes to the wall nodes. A zero-length element is defined between these two nodes, with material properties calibrated to match the expected U-shaped flexural plates. Equations (3.6) to (3.10) were used to predict the damper properties, then the modifiers from Ganey\u2019s study is used to determine the final material properties. Table 5-7 summarized the material properties calculated using nominal values, and the final adjusted material properties. Table 5-7: Calibrated Material Properties. Variable NHERI Calculated Value Used Value Damper Stiffness (MPa) 4.8 4.1 Damper Plastic Force (kN) 19.7 22.0 Finally, the leaning column is created. A node is defined at a height equal to the top of the foundation element, and fixed in the translation degrees of freedom. Nodes are then defined at the height of each floor. These floor nodes are connected using rigid beam elements that have zero length elements at either end. The zero length elements are rigid in the x and y translational degrees of freedom, but very flexible in the rotation degree of freedom. These elements allow Four bars Two equivalent bars 131 the leaning column to act like a pseudo truss, without the stiffness matrix becoming singular. At each floor node, a point load equal to the weight of that floor is then assigned. Finally, rigid truss elements connect the leaning column to the CLT wall element, at the height of the floor. The structure is analyzed using Rayleigh damping to account for unmodeled energy dissipation. A damping ratio of 2% is used, based on estimates by Wichman (2018). This damping value is measured using a free vibration test and small displacements; therefore, the effective value of hysteretic damping is not subtracted from it. The coefficients on the mass and stiffness matrices were estimated using the following equations (Chopra 2012): \ud835\udc4e0 = \ud835\udf012\ud835\udf14\ud835\udc56\ud835\udf14\ud835\udc57(\ud835\udf14\ud835\udc56 + \ud835\udf14\ud835\udc57) (5.1) \ud835\udc4e1 = \ud835\udf012(\ud835\udf14\ud835\udc56 + \ud835\udf14\ud835\udc57) (5.2) where: \ud835\udc4e0 is the damping coefficient on the mass matrix; \ud835\udc4e1 is the damping coefficient on the stiffness matrix; \ud835\udf01 is the critical damping ratio; \ud835\udf14\ud835\udc56 is the first frequency of interest; \ud835\udf14\ud835\udc57 is the second frequency of interest. Because the structure is only two stories tall, the dynamic response of the building is expected to mostly occur its first mode. To prevent overestimation of damping when the structure goes nonlinear, the two targeted frequencies were the first mode of the structure and an effective period based on a secant to 2% drift. These tangent slops are shown in figure Figure 5-7. The frequency and secant slope values are based on eigenvalues and pushovers completed in Section 5.4. The secant period is calculated based on the ratio of stiffness between the initial stiffness and secant stiffness as follows: 132 Figure 5-7: Main regions of performance for a rocking wall system. \ud835\udf142 = \ud835\udf141\u221a\ud835\udc3e2\ud835\udc3e1 (5.3) where: \ud835\udc3e1 is the stiffness of initial slope; \ud835\udc3e2 is the stiffness at a 2% secant slope; \ud835\udf141 is the initial fundamental frequency for building; \ud835\udf142 is the effective period at 2% drift. For the models including a base beam element the frequencies of interest were .85Hz and 1.3Hz respectively. For models without the beam element, the frequencies of interest were 1Hz and 2.6Hz respectively. The calculated Rayleigh damping coefficients were then applied to the mass matrix, and tangent stiffness matrix. During initial runs of the nonlinear model, unrealistically large spikes in Rayleigh damping were observed for the rocking element. To reduce the effect of spurious damping, the Rayleigh damping factors for both the lumped plasticity and fibre rocking element were set to zero. However, unrealistically large damping forces were still measured in the model without the base. A more in depth discussion of the Rayleigh forces in the structure is given in Section 5.6 and 5.7. K2 K1 Drift (%) Base Shear 1 2 3 4 5 133 Before the dynamic analysis of the building is can be run, the model is first initialized. For the fibre model, forces in the post-tension cable are initialized using a sequence of analyses as described in Section 4.8. After the iteration is complete, the correct material properties are used to build the structure. Ground motions are then run sequentially on the building until the entire suite of motions is complete. Each ground motion is applied as a uniform excitation with acceleration, velocity, and displacement components. The ground motion data used is the acceleration, velocity and displacement are the output values reported by the shake table, as opposed to the raw input signal. For each ground motion in the sequence, the initial state of the model is equal to the final state of the model at the end of the previous ground motion. A Hilber-Hughes-Taylor time integrator is used, with a factor of 0.8 and time step of 2.5ms. If the analysis failed at a certain step time, the size of the time step is reduced, and the that step is attempted again. This process is repeated several times until the time step is 1\/1000ths of the original time step size, at which point the analysis would terminate. 5.3 Calibration of Base Beam Properties As discussed in Section 5.2, the top surface of the foundation beam is noted to deform significantly during testing. This deformation influenced the dynamics of the building and must be accounted for to accurately model the system. Detailed information about the full geometry and material properties of the beam are not known, making accurate predictions difficult. Further complicating matters, the beam did not have a uniform stiffness, as stiffeners were added in some locations across the cross section. In lieu of additional finite element modelling, an approach is adopted that could leverages the existing Python model infrastructure. To account for this flexible foundation, a zero-length fibre section element is attached to the wall base. The materials properties of this element were considered unknown, and would be 134 determined using an optimization analysis, run on the fibre model of the building. This optimization analysis follows a similar procedure to the calibration described in 4.8 for the elastic modulus of CLT. Figure 5-8 overviews the algorithm used to calculate the elastic modulus which returned optimal results. First, a trial value for the foundation elastic modulus is chosen. For that trial value, twelve ground motions were run on the building. The shear force of the model is then calculated and compared to the experimentally measured shear force. At a number of time steps, the experiment and shear data are shifted into a common data frame. For each of those points, the difference between the numerical and experimental curve is calculated. A residual is then defined, which is equal to the sum of the squares for each sample point, divided by the number of samples. A final residual for the trial value of elastic modulus is then calculated by summing the residuals for each ground motion of interest. To account for inelastic deformation of the foundation that occurred during earthquakes GM5, two elastic moduli were defined: one for pre-yielding behaviour, and one for post-yielding behaviour. For pre-yielding behaviour, sampling is only done on the first four earthquakes. For post-yield behaviour, sampling is done on GM5 to GM11. Ground motion 12 is found to be an outliner, and not included in the sample function for improved accuracy. After each trial is completed, an optimization rule is used to choose a new elastic modulus. The optimal solution is the trial modulus of elasticity which had the lowest residual value. 135 Figure 5-8: Summary of algorithm to calibrate material property. It is important to make sure the output values made physical sense in addition to matching the experimental data. To give reasonable bounds on this calibration analysis, a simplified analysis is performed on top surface of the flange. In this analysis, the top surface of the steel member is modeled as a beam, and load is distributed across the centre of the beam. While the top flange will have some out of plane bending, this bending is not considered to be significant because of the high ratio between total wall length, 3.04m, and the width of the flange, .43m. Two restraint conditions are considered for the beam model: one where the beam is simply supported, and another where the supports of the beam are fixed against rotation. The actual support condition will have a partial rotational stiffness, so it is reasonable to believe any equivalent stiffness will fall between these two values. Figure 5-9 shows the beam with the two boundary conditions considered. Run seismic analysis Check if all ground motions are run Return optimum value Build model Change trial elastic modulus Update objective function value Check if objective function is at minimum Trial elastic modulus 136 Figure 5-9: Idealized flange model. The foundation beam had an approximate flange thickness of 16mm, length between supports of 430mm, and load is applied is applied across a 175mm strip in the middle of the top surface. The elastic modulus is assumed to be 200GPa. The deflection at the edge of the CLT section vs. the total force applied is then used to estimate a stiffness per unit width of foundation. Table 5-8 summarized the bounding stiffnesses calculated using these effective beam properties. Table 5-8: Effective beam stiffness. Method Pin boundary condition Fixed boundary condition Value (GPa) 50 1650 Figure 5-10 summarizes the objective function for both sets of optimization analysis. The analysis returned an optimal beam stiffness of 530MPa, and 380MPa for pre and post yield, respectively. To compare each analysis suite side by side, the objective function is normalized by dividing each value by the peak value of the function. It can be observed that the optimal values of the objective function are relatively close to the peak values. This result was unexpected, likely occurred due to noise in the shear data, which will cause errors that are relatively constant through each run. The objective function appears to be smooth in the range of values considered, with a clearly defined optimal value. Flange cross section Idealized systems 137 Figure 5-10: Objective function for Optimization Analysis. 5.4 Pre-Analysis Checks and Data Processing Before the dynamic analysis is run, a series of checks were completed to ensure that the models behaved as expected. Two analyses were done on each of the four models: an eigenvalue analysis to check the fundamental frequency of the building, and a pushover to check the monotonic backbone curve. In the eigenvalue analysis it is found that the first mode of the structure occurred at 1.3 Hz for the model with a beam. Both the fibre and lumped plasticity model had similar frequencies. This value is quite close to the experimental frequency, which is measured to be in a range from approximately 1.1Hz - 1.2Hz for the majority of testing. The model without the beam had a first mode fundamental frequency of 2.6 Hz. The large difference in frequency for the model with a rigid foundation emphasizes that the foundation 138 beam had a large impact on the system dynamics. The lumped plasticity and fibre model had similar results. Pushovers were also completed on both models, using forces distributed as according to the first mode response for the building. For the model with the foundation beam, the higher initial stiffness for the beam is used in the pushover. Figure 5-11 shows the results of the pushover analysis, plotting base shear against total roof drift for the fibre and lumped plasticity models. Also included in this plot is experimental data for the peak shear and peak drift of each earthquake. In the model with the beam, the both fibre and lumped plasticity numerical models were quite close to experimental results, with larger earthquakes being more flexible. This result matches expectations, as the structure should become more flexible than the monotonic backbone after repeated load cycles. For the model without the foundation beam, the backbone curve is stiffer than experimental results, as expected. In both models, the fibre and lumped plasticity fibre numerical model had similar backbone curves. Figure 5-11: Pushover of two story building, for the fibre model (FM) and lumped plasticity model (LPM). Before dynamic measurements from numerical models could be compared, the experimental data had to be processed. Four significant measurements the from experiment were used: acceleration and displacement at each floor, force in the post-tension tendons, and 139 displacement of U-shaped flexural plate dampers. For models with the beam, peak downward displacement of foundation is also compared with experimental values. For each wall pair, data from the post-tension cables were averaged. Because the numerical model combined post-tension cables on the North and South side of each wall, experimental data for cables on either side of the wall were summed. The absolute displacement value of each floor is averaged from all the potentiometers at the floor in question. These values were then converted to interstory drifts by subtracting the displacement of the floor below and dividing by the height of each floor. Displacement in each U-shaped flexural plate damper were taken by averaging the displacement in each wall pair. During the test, two independent methods were used to calculate base shear: one using acceleration data, and one using measurements of strain in the diaphragm. The results from both methods were contradictory, but the acceleration values used in this study because it is considered more accurate. The shear force is calculated by multiplying the acceleration of each floor by the seismic mass of that floor. Because there were multiple accelerometers at each story, measurements from all accelerometers were averaged into a single reading. The original ground motions acceleration data exhibited high frequency spikes in acceleration. These spikes were likely caused by the impact of the wall as it rocked. To make the output shear more realistic, a Savitzky\u2013Golay filter is used to smooth data and reduce high frequency noise. Figure 5-12 shows a sample of the filtered vs unfiltered data for GM5. 140 Figure 5-12: Filtered vs. unfiltered shear force. 5.5 Results for Models with Beam The suite of ground motions is run on the two story OpenSees models. For models with the beam, only the first twelve ground motions were considered because the post-tension cables yielded during GM12 in the numerical model. After the suite of motions is run on the two-story structure, the fibre and lumped plasticity numerical models were compared with the processed data from the experimental results. To showcase the suitability of each model, time history plots are presented from three earthquakes, one for each intensity category. GM2 is used for the SLE hazard level, GM9 for the DBE hazard level, and GM12 for the MCE hazard level. The time range presented is cropped to the most intense shaking during the earthquake. A summary of the numerical plots for all models is presented in Appendix A and Appendix B, and an in-depth discussion of the results is given in Section 5.7. Each figure compares the system hysteresis of force vs. top floor interstory drift, followed by shear force, top floor interstory drift, and post-tension force plotted against time. Figure 5-13 shows the results of 141 ground motion two for the fibre-based numerical model. It can be observed that the numerical results follow the experimental trend quite closely for the design parameters presented. Notably, the local maximums\/minimums for shear and drift were approximately 10% less than the experimental peak values. Because the first mode governs the response of the building, the interstory drift for each floor is similar to the building\u2019s net drift. Figure 5-13: GM2 fibre model results. Figure 5-14 and Figure 5-15 overview the results of GM9 and GM12 respectively for the fibre-based numerical model. Like earthquake in the SLE hazard level, the DBE and MCE hazard level earthquakes follow the experimental trend of data quite well. For some of the ground motions examined there are notable departures from the experimental trend in shear and drift near to the end of the earthquake. In terms of post-tension force, the numerical model reached much larger forces than the experimental data, but followed the general trend closely. 142 Figure 5-14: GM9 fibre model results. Figure 5-15: GM12 fibre model results. 143 Similarly, Figure 5-16, Figure 5-17, and Figure 5-18 depict the results of the analytical model compared against experimental results for the lumped plasticity model for ground motion two, nine and twelve respectively. The observed trend of shear and drift vs. time is similar to experimental values, but the peaks of each cycle were generally underpredicted. The lumped plasticity model is not able to capture the general trend for the post-tension force, and many of the smaller peaks were smoothed over. Figure 5-16: GM2 lumped plasticity model. 144 Figure 5-17: GM9 lumped plasticity model. Figure 5-18: GM12 lumped plasticity numerical analysis. 145 The peak results from certain demand parameters were also compared between the numerical models and experimental results. Understanding how peak values compare to experiment is important because peak values alone are often used as predictors of performance. The following design parameters were examined: shear force, post-tension force, interstory drift on the first and second floor, and U-shaped flexural plate deformation. For the post-tension force, the peak reading used is the largest value from all post-tension cables. Measurements for the U-shaped flexural plate dampers are taken from the third damper. Also included is the peak downward deformation of the rocking wall to help evaluate the accuracy of the foundation model used. Figure 5-19 compares the peak values measured in the numerical models to the peak values measured in the experiment. Based on the close trend between the numerical and experimental data, and the similarity of peak values, the shear force and drift of the model could be predicted with reasonable accuracy. However, in nearly all cases the peak shear predicted by the analysis is less than the shear in experiment. This is especially noticeable for design level earthquakes. For both the drift of the structure and displacement of the U-shaped flexural plate dampers, the fibre and lumped plasticity models match experimental results quite closely. This indicates that both models were able to predict the displacement profile of the structure. Peak drift values for the first and second floor were quite similar, which reinforces that the first model of the structure is governing the structures dynamic behaviour. 146 Figure 5-19: Peak design parameters from the fibre and lumped plasticity model compared to experiment. Regarding the post-tension force, there is a high variance between numerical and experimental values, with being both underpredicted for some earthquakes and overpredicted for others. This finding indicates that the numerical models were poor predictors of the post-tension force in the building. While the numerical models overpredicted the peak downwards deformation of the foundation element, the peak results were generally similar to experimental measurements. This finding 147 helps to validated that simplified foundation model used could reasonably model the downward displacement of the real foundation. Comparing the overall performance of the lumped plasticity and fibre-based numerical models, both lead to similar results. The similarity of both models throughout the entire sequence of earthquakes is possible because the foundation prevented damage. 5.6 Results for Models with a Rigid Foundation To predict the performance of the building with a rigid foundation, the suite of ground motions is run again using a model with no flexible foundation element. Like the model with the beam, each ground motion is applied directly after each other in a sequence. The performance of the building is then checked against the original performance objectives outlined in Table 5-4. The U-shaped flexural plate damper performance objective is considered to have been exceeded if \u201cminor yielding\u201d is found in at least three out of five dampers. Minor yielding is defined to be deformations greater than 20% of the yield deformation, or in this case a displacement of 3.4mm. In the fibre-based numerical model, crushing of the CLT is determined by evaluating strains in the CLT section. If strain in a fibre exceeded the yield strain of 2.6 mm\/m, it is considered to have crushed. Yielding of PT members is calculated by checking if the post-tension element had a force greater than the yield force (350kN). For the lumped plasticity model, the performance objectives for CLT crushing and post-tension bar yielding were based on rotations of the element. At a rotation of 0.006 rad or more the CLT is considered to have yielded. At a rotation of 0.024 rad or more, the post-tension cable is considered to have yielded. A summary of the performance of the building during each earthquake is overviewed in Table 5-9. In this table, the target limit states for each intensity level are displayed, and a \u201cx\u201d denotes that the building exceeded the requirements of this limit 148 state. Limit states are grouped by the seismic intensity level, and a passing earthquake will not experience any limit states from that group. For example, the SLE GM3 and MCE GM11 met their respective performance targets for both the fiber model and lumped plasticity model, while the SLE GM2 and MCE GM12 did not. Table 5-9: Key limit states for the fibre model (FM) and lumped plasticity model (LPM). SLE DBE MCE U-shaped flexural plate yielding CLT Crushing 2% Drift Post-tension Yielding 4% Drift Fibre Model Target Limit States GM1 x - - - - GM2 x - - - - GM3 - - - - - GM4 - - - - - GM5 x x x - - GM6 x x x - - GM7 x x - - - GM8 x x x - - GM9 x x - - - GM10 x x - - - GM11 x x - - - GM12 x x x x - Lumped plasticity Model Target Limit States GM1 x - - - - GM2 x - - - - GM3 - - - - - GM4 - - - - - GM5 x x x - - GM6 x x x - - GM7 - - - - - GM8 x x x - - GM9 x x - - - GM10 x x - - - GM11 x x - - - GM12 x x x x - Based on the results of Table 5-9, the building without the flexible foundation performed according to the original performance objectives, with a few notable exceptions. For the SLE hazard level, some U-shaped flexural plate yielding did occur during in GM1 and GM2. 149 However, the average of these deformations is 3mm for the fibre model, and 3.2mm which is still relatively close to the yield deformation. The building did not fulfil the drift requirements for the DBE hazard level earthquakes, and several earthquakes at SLE hazard level. It should be noted that there is bias in results, as all the earthquakes which exceeded the 2% drift requirement were Northridge ground motions. The average drift is 2.0 and 1.9 respectively for the fibre mode and lumped plasticity model in the design level earthquakes. The first yielding of CLT occurred during GM5, and all subsequent design level earthquakes had some CLT yielding. For the design level earthquakes, the average peak strain is 130mm\/m, much larger than the yield strain of 2.6mm\/m. These large strain values indicate upon yielding, the outer most fibres experienced significant demands during the analysis. However, it\u2019s not clear if the output strains from the fibre model correlate well with experimental data. The first occurrence of post-tension yielding also coincided with the first MCE earthquake. Both the lumped plasticity model and fibre-based model had almost identical results for each performance objective. The results of the lumped plasticity model and fibre numerical model are compared to each other to see if they converge. It is observed that the fibre and lumped plasticity models initially have a similar dynamic performance but begin to diverge from each other at later earthquakes. Figure 5-20 highlights this phenomenon by comparing the shear and drift of for both models during several earthquakes. It can be observed that for GM5 the fibre-based, and lumped plasticity numerical models aligned quite closely. However, for the similar GM6, the shear and drift results diverged significantly. For GM10 the shear and drift response completely diverge for most of the earthquake. This result highlights how crushing of the CLT plays a much more important role for the suite of earthquakes with a rigid foundation. 150 Figure 5-20: Comparison of the base shear and inter-story drift for the lumped plasticity model (LPM) and fibre model (FM) for a two story building with a rigid foundation. 151 Because the suite of ground motions is run sequentially, damage due to crushing will build up between each earthquake run. This make the comparison between the two lumped plasticity and fibre models somewhat unrealistic for later earthquakes, as these earthquakes will have gone through many more cycles than expected. To understand the differences between of both models for single earthquakes, the suite is run again on each model but instead clearing the model after each run. The entire suite of 14 ground motions is run on each model. For the 1.2xMCE hazard level earthquake GM14, data is not presented as the numerical model was non-convergent. Figure 5-21 compares the response of the lumped plasticity model to the fibre model for earthquakes for GM2, Figure 5-22 for GM9 and Figure 5-23 for GM12. Images overviewing the entire suite of ground motions is are presented in Appendix C. Figure 5-21: GM2 comparison of fibre model (FM) and lumped plasticity model (LPM) for non-sequential runs. 152 Figure 5-22: GM9comparison of fibre model (FM) and lumped plasticity model (LPM) for non-sequential runs. Figure 5-23: GM12 comparison of fibre model (FM) and lumped plasticity model (LPM) for non-sequential runs. 153 For the suite of motions on a rigid foundation, Figure 5-24 compares the peak values calculated by the fibre model, to peak values calculated in the lumped plasticity model. Both models predicted similar peak values. However, it can be observed that the lumped plasticity model generally underestimated most important quantities for the DBE and MCE hazard levels. Figure 5-24: Peak design parameters from the fibre and lumped plasticity (LP) model compared to experiment. During the analysis of the building with a rigid foundation, large rayleigh forces were observed in the design level and maximum considered earthquakes. These large spikes in Rayleigh 154 damping coincide with by large spikes in stiffnesses that occur when CLT wall quickly passes through the centred position for the wall. Similar rayleigh forces were not noted in the model with the beam, likely because the beam smoothed the walls transition between the first and second slope. Figure 5-25 depicts the rayleigh forces observed in the lumped plasticity and fibre-based numerical models for GM3 and GM6. The overturning moment over time is plotted against the sum of all the Rayleigh moment forces in the structure. Given the size of the damping forces, it is likely that they influenced the dynamics of the system. A more in-depth discussion of these forces is given in Section 5.7 Figure 5-25: Rayleigh Damping forces for numerical models with a rigid foundation. GM3 Fibre GM3 Lumped Plasticity GM6 Fibre GM6 Lumped Plasticity 155 5.7 Discussion of Results Based on the analysis of the building model with a beam, the calibrated beam properties proposed in Section 5.3 could reasonably model the building with a flexible foundation. However, by picking an elastic modulus for the foundation that is based on the best \u201cfit\u201d for the experimental data, there is risk of overfitting the beam element and missing other important modelling concerns. This risk has been partially mitigated by appropriately bounding values for the elastic modulus of the beam. Comparing measurements of the downward deformation of the beam between the numerical analysis and experiment confirmed that the numerical model is within reasonable bounds of the experimental values. Using these properties, both the fibre and lumped plasticity numerical model could accurately capture the global drift and shear of the system. Both models slightly underpredicted drifts and shear forces in the model. This underprediction of peak values could be explained if the Rayleigh damping used is too large, or there is too much energy dissipated by the U-shaped Flexural plate dampers. When predicting shear and drift, generally the fibre model is more accurate. This is expected as the lumped plasticity model cannot capture cyclic deterioration. Based on the observed results of this study, post-tension forces generated by fibre and lumped plasticity model should be viewed with some scrutiny. The lumped plasticity model struggles at capturing the trend in post-tension force for the structure. Generally, the fibre numerical model is more accurate, and could capture the general trends in the post-tension force. However, both numerical models were unable to accurately capture the peak post-tension force in the structure. Even with a reduced stiffness of 85%, the numerical model generally overpredicted the peak force in the post-tensioned tendons. Based on the wall tests presented in Section 4.9, it is 156 expected that the post-tension force could be overestimated by approximately 20%. However, the observed differences were much larger. One possible reason for this difference is that flexing of the foundation beam lead to lower post-tension force. Given the significant difference between analytic and experimental values, as well as the known flexibility of the beam element, it seems likely that the foundation had some influence. Another possibility is that a component in the post-tension cable element which has not been accounted for is adding flexibility to the system. This uncertainty surrounding the post-tension tendon stiffness could be improved with experimental testing. Currently there is a lack of available data about the stiffness of the post-tension assembly. One possible way of testing the stiffness of the post-tension wall assembly would be to measure the force and deformation at the top of the as the CLT wall system it is initially tensioned. When the flexible foundation element is removed, it is found that the building performed according to the original performance objectives, with a few exceptions. Notably, the peak strain in the CLT is much larger than the yield strain. This highlights a potential challenge with using the occurrence of yielding as a limit state. The performance of the fibre model for the suite of earthquakes run sequentially with a rigid foundation indicated that walls which experience some yielding damage still retain the majority of their strength. Using peak strains to the of CLT alone may be overly punitive, as there is no indication of the amount of wall that has crushed. It is also unclear if the either model can accurately predict the strain distribution at the bottom of the CLT wall. For the building examined, achieving no crushing of CLT during design level earthquakes would likely be challenging. 157 For the model with a rigid foundation, the fibre and lumped plasticity model significantly diverged as the suite of ground motions progressed. However, when each earthquake is run as a single event, the performance is similar between the fibre and lumped plasticity numerical models. Peak results for both models were reasonably similar across all measured demand parameters. However, the lumped plasticity model generally underestimated values. Based on the results of the fibre model, cyclic deterioration due to crushing at the wall base can have a significant effect on the performance of CLT walls. Because of this, the fibre model is clearly more accurate than the lumped plasticity model for the building model considered. The performance of the lumped plasticity model could be improved if there is a suitable rule to capture the cyclic deterioration due to crushing of the CLT. Currently, the fibre-based numerical model uses an elastic perfectly plastic material, however this material property is much simpler than the real behaviour. Because crushing is important to the system dynamics, the fibre model could be improved by using a material property that is tuned more closely to the actual behaviour of timber. The large damping forces that were observed in the rigid foundation models were observed in both the lumped plasticity and fibre-based model. These spikes in damping likely occur because of large changes in stiffness that occur in elements as they transition between the first and second slope for the load-deflection curve of the wall. Removing Rayleigh damping in key elements only partially reduced the spikes of damping in the dynamic models. Other measures that could have be taken include smoothing the transition between the first and second slope. Algorithms could be employed that test the model for problematic conditions, such as very sudden stiffness changes, then removes damping as appropriate. It is possible that using damping models, such as modal damping, could improve model performance. When Rayleigh 158 damping must be used, great care should be taken to ensure to make sure the damping forces haven\u2019t significantly affected the performance of the structure. 159 Chapter 6: Conclusion In this study, two sets of nonlinear models were used to analyze self-centring wall systems: one using lumped plasticity elements, and another using fibre-based elements. The simulations completed include pushover and reverse cyclic tests of individual wall systems, and dynamic tests of a two-story building using self-centring CLT walls as a lateral force resisting system. Results of the wall system were compared to data from existing experimental studies of self-centring CLT wall systems. Predictions were also made about how both models would perform if the building had a rigid foundation. Optimization analyses were used to calibrate properties of the nonlinear models to better match experimental data. For each portion of the analyses described above, key findings are discussed in the following Section. Recommendations are also presented for future work. 6.1 Research Findings For the component level wall tests, the key findings are as follows: \u2022 There is uncertainty with regards to the appropriate choice of elastic modulus and shear modulus for the CLT panel in self-centring CLT walls. Information from compression tests are likely the best possible indicator of the material performance. \u2022 Results from pushover tests on self-centring wall systems show that both the fibre and lumped plasticity model could capture the shear-displacement behaviour of the wall systems. \u2022 Results from the reverse cyclic tests show that the fibre model could capture the behaviour of wall models. Reverse cyclic tests on the lumped plasticity model show 160 that it is inaccurate beyond the yield point of CLT, because it cannot capture cyclic deterioration. \u2022 In both models, the amount of base in contact with the foundation is generally underestimated, and post-tension tendon force overestimated. \u2022 Sensitivity analyses performed on pushover analyses of the fibre-based model showed that shear stiffness did not significantly affect global results. Overpredicting elastic modulus for the flexural section of CLT generally did not affect the systems pushover results. The system is more sensitive to the elastic modulus and yield strength at the base of the wall. For the nonlinear dynamic analysis of the two-story building with a flexible foundation, the key findings were as follows: \u2022 The flexible foundation significantly influenced the dynamics of the two-story test building, even before the building yielded. \u2022 Both the fibre and lumped plasticity model could reasonably capture the shear and drift of the structure. \u2022 The lumped plasticity model struggled to capture the post-tension force. Peak values of the post-tension force were overestimated by both models. For the nonlinear dynamic analysis of the two-story building with a rigid foundation, the key findings were as follows: \u2022 Both models predict that the building with a rigid foundation would perform similar to the intended performance objectives. However, both models predicted some crushing of CLT at design level earthquakes. 161 \u2022 When single earthquakes were run on the model, both the fibre and lumped plasticity model had similar peak results. \u2022 Cyclic deterioration due to crushing had a significant impact on the global behaviour of the structure. \u2022 Large Rayleigh damping forces were observed in the models with a rigid foundation. Caution should be used when employing Rayleigh damping with CLT rocking structures. \u2022 OpenSeesPy is successfully paired with Python programs to automate portions of the analysis. 6.2 Suggestions for Future Work Based on the knowledge gained during this study, the following recommendations for future studies are proposed. For experimental studies on self-centring CLT walls: \u2022 There is a need for a model that can accurately predict the material properties of mass timber panels used in self-centring walls. \u2022 The stiffness of the entire post-tension assembly should be investigated. Current models of the self-centring walls system overpredict the systems post-tension force. It is currently unclear if the deformation of elements within the post-tension assembly affect the overall stiffness of the system. For nonlinear modelling of self-centring CLT walls: \u2022 A deterioration model could be implemented for self-centring lumped plasticity elements. While very appealing from a simplicity standpoint, lumped plasticity models are currently limited by their inability to capture cyclic deterioration. 162 \u2022 The performance of wall models using two dimensional elements could be investigated. These models are theoretically able to capture the local stress and strain at the base of the wall with more accuracy. Results could then be compared with one dimensional models, to see if there are any differences in predictions. \u2022 The effect of using different damping models on dynamic analyses could be investigated. In the models examined, Rayleigh damping resulted in very large spikes of forces. It is unclear if these large damping forces are specific to the Rayleigh damping model used. It is also unclear how significantly the effected model results. To improve the model of the two-story building, it is recommended that: \u2022 A full three-dimensional analysis is performed on the building, including modelling of the structural components in the gravity system of the building. 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(2018). \u201cFramework \u2013 Innovation in Re-Centering Mass Timber Wall Buildings.\u201d Eleventh U.S. National Conference on Earthquake Engineering, Los Angeles, California. 171 Appendices Appendix A: Additional Comparison Figures for the Fibre Model of a Two-story Building with a Flexible Foundation In the following section, results for the fibre-based numerical model is compared to experimental data for the suite of twelve ground-motions considered. Earthquakes are run sequentially, where the end condition of one earthquake is the starting condition for the next. Experimental data for GM5 is unavailable, due to failure of measuring devices during experiment. 172 Figure A-1: System Hysteresis vs. experimental data for fibre model, GM1-8. 173 Figure A-2: System Hysteresis vs. experimental data for fibre model, GM9-12. 174 Figure A-3: System shear vs. experimental data for fibre model, GM1-8. 175 Figure A-4: System shear vs. experimental data for fibre model, GM9-12. 176 Figure A-5: Roof drift vs. experimental data for fibre model, GM1-8. 177 Figure A-6: Roof drift vs. experimental data for fibre model, GM9-12. 178 Figure A-7: Floor drift vs. experimental data for fibre model, GM1-8. 179 Figure A-8: Floor drift vs. experimental data for fibre model, GM9-12. 180 Figure A-9: East wall outer Post-tension force vs. experimental data for fibre model, GM1-8. 181 Figure A-10: East wall inner post-tension force vs. experimental data for fibre model, GM9-12. 182 Figure A-11: East wall outer post-tension force vs. experimental data for fibre model, GM1-8. 183 Figure A-12: East wall outer post-tension force vs. experimental data for fibre model, GM9-12. 184 Figure A-13: U-shaped flexural plate deformation vs. experimental data for fibre model, GM1-8. 185 Figure A-14: U-shaped flexural plate deformation vs. experimental data for fibre model, GM9-12. 186 Appendix B: Additional Comparison Figures for the Lumped Plasticity Model of a Two-story Building with a Flexible Foundation. In the following section, results for the lumped plasticity numerical model is compared to experimental data for the suite of twelve ground-motions considered. Earthquakes are run sequentially, where the end condition of one earthquake is the starting condition for the next. Experimental data for GM5 is unavailable, due to failure of measuring devices during experiment. 187 Figure B-1: System hysteresis vs. experimental data for fibre model, GM1-8. 188 Figure B-2: System hysteresis vs. experimental data for lumped plasticity model, GM9-12. 189 Figure B-3: Shear vs. experimental data for lumped plasticity model, GM1-8. 190 Figure B-4: Shear vs. experimental data for lumped plasticity model, GM9-12. 191 Figure B-5: Roof drift vs. experimental data for lumped plasticity model, GM1-8 192 Figure B-6: Roof drift vs. experimental data for lumped plasticity model, GM9-12. 193 Figure B-7: Floor drift vs. experimental data for lumped plasticity model, GM1-8. 194 Figure B-8: Floor drift vs. experimental data for lumped plasticity model, GM9-12. 195 Figure B-9: East wall outer post-tension force vs. experimental data for lumped plasticity model, GM1-8. 196 Figure B-10: East wall outer post-tension force vs. experimental data for lumped plasticity model, GM9-12. 197 Figure B-11: East wall inner post-tension force vs. experimental data for lumped plasticity model, GM1-8. 198 Figure B-12: East wall inner post-tension force vs. experimental data for lumped plasticity model, GM1-8. 199 Figure B-13: U-shaped flexural plate displacement vs. experimental data for lumped plasticity model, GM1-8. 200 Figure B-14: U-shaped flexural plate displacement vs. experimental data for lumped plasticity model, GM9-12. 201 Appendix C: Additional Comparison Figures for the Model of a Two-story Building run With a Rigid Foundation and Single Earthquakes In the following section, a comparison is given between the lumped plasticity model and fibre model for the two-story structure with a rigid foundation, where earthquakes are run non-sequentially. 202 Figure C-1: System hysteresis for fibre and lumped plasticity model GM1-8. 203 Figure C-2: System hysteresis for fibre and lumped plasticity model GM9-13. 204 Figure C-3: Shear force for fibre and lumped plasticity model GM1-8. 205 Figure C-4: Shear force for fibre and lumped plasticity model GM9-13. 206 Figure C-5: Roof drift for fibre and lumped plasticity model GM1-8. 207 Figure C-6: Roof drift for fibre and lumped plasticity model GM9-12. 208 Figure C-7: Floor drift for fibre and lumped plasticity model GM1-8. 209 Figure C-8:: Floor drift for fibre and lumped plasticity model GM9-13. 210 Figure C-9: East wall inner post-tension force for fibre and lumped plasticity model GM1-8. 211 Figure C-10: East wall outer post-tension force for fibre and lumped plasticity model GM9-13. 212 Figure C-11: East wall outer post-tension force for fibre and lumped plasticity model GM1-8. 213 Figure C-12: East wall outer post-tension force for fibre and lumped plasticity model GM9-13. 214 Figure C-13: U-shaped flexural plate displacement for fibre and lumped plasticity model GM1-8. 215 Figure C-14: U-shaped flexural plate displacement for fibre and lumped plasticity model GM9-13. 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