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Seismic safety assessment of base isolated buildings using lead-rubber bearings Zhang, Hongzhou 2018

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SEISMIC SAFETY ASSESSMENT OF BASE ISOLATED BUILDINGS USING LEAD-RUBBER BEARINGS by  Hongzhou Zhang  B.Eng., Jilin University, 2014  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)  April 2018  © Hongzhou Zhang, 2018 ii  Abstract  Base isolation using lead-rubber bearing (LRB) has been well-developed and widely-implemented in high seismic zones worldwide. During strong earthquake shaking, LRB is designed to move horizontally and meanwhile carry large axial load. One of the main design challenges is to prevent the LRB from buckling. Although detailed component behavior of LRB under combined axial and shear loads has been well investigated, the seismic performance of base isolated building with LRB has not been systematically examined. In this study, the seismic performances of two prototype buildings, each with different LRB geometric properties, structural periods, and axial loads, were systematically examined. To properly account for the buckling response of the LRB under combined axial and shear loads, robust finite element models of the prototype buildings were developed using the state-of-the-art LRB buckling model implemented in OpenSees. Nonlinear time history analyses were conducted using ground motions selected and scaled based on the 2015 National Building Code of Canada. As shown by the result, when the LRB is designed without accounting the axial and shear interaction, this leads to high probability of failure of the LRB, which can be difficult and expansive to fix. In some situations, this might lead to the collapse of the base isolated building. To mitigate the failed probability of the LRB during strong earthquake shaking, a simple amplification factor of 2.5 is proposed to amplify the design axial load calculated from the combined gravity and earthquake loads when the coupled axial and shear interaction of LRB is not explicitly modeled.   iii  Lay Summary  Lead-rubber bearing (LRB) has been implemented worldwide as an effective seismic isolation device to protect the buildings from earthquake shaking. This thesis aims to study the failure behavior of buildings isolated using LRBs based on advanced numerical modeling techniques. Two prototype buildings in Vancouver, BC, Canada were used as the case study. The buildings were designed based on different combinations of parameters, and analyzed numerically using earthquake records selected based on the 2015 National Building Code of Canada. An amplification factor was proposed for practical design in order to guarantee the low probability of LRB failure under severe earthquakes. iv  Preface  This thesis is the original, independent work of Hongzhou Zhang. The author was responsible for the literature review, model development, computational analysis, data processing, and results presentation. The thesis was drafted by the author and revised based on the comments of Professor Tony Yang, at the University of British Columbia. v  Table of Contents  Abstract .......................................................................................................................................... ii Lay Summary ............................................................................................................................... iii Preface ........................................................................................................................................... iv Table of Contents ...........................................................................................................................v List of Tables .............................................................................................................................. viii List of Figures .................................................................................................................................x List of Symbols ............................................................................................................................ xii List of Abbreviations ...................................................................................................................xv Acknowledgements .................................................................................................................... xvi Dedication .................................................................................................................................. xvii Chapter 1: Introduction ................................................................................................................1 1.1 Seismic Isolation ............................................................................................................. 1 1.2 Stability of LRB .............................................................................................................. 5 1.3 Design Provisions ........................................................................................................... 7 1.4 Motivation and Objectives .............................................................................................. 8 1.5 Scope of the Work .......................................................................................................... 9 1.6 The Organization of this Thesis .................................................................................... 10 Chapter 2: Mechanical Behavior of LRBs.................................................................................12 2.1 Interactions Between Horizontal and Vertical Directions ............................................ 12 2.2 Buckling without Shear Deformation ........................................................................... 14 2.3 Buckling with Shear Deformation ................................................................................ 16 vi  2.4 Advanced Numerical Model ......................................................................................... 17 Chapter 3: Design Procedure for LRB Isolated Buildings ......................................................20 3.1 Capacity Spectrum Method........................................................................................... 20 3.2 Bilinear Model Design .................................................................................................. 21 3.3 LRB Design .................................................................................................................. 25 Chapter 4: Prototype Buildings ..................................................................................................28 4.1 Bilinear Model Parameters ........................................................................................... 29 4.2 Superstructure Design ................................................................................................... 32 4.3 LRB Properties.............................................................................................................. 35 4.4 Numerical Modeling Approach .................................................................................... 37 Chapter 5: Seismic Hazard Analysis ..........................................................................................40 5.1 Seismicity in Southwestern British Columbia .............................................................. 40 5.2 Probabilistic Seismic Hazard Analysis ......................................................................... 44 5.3 Seismic Hazard Model for NBCC 2015 ....................................................................... 45 5.4 Ground Motion Selection .............................................................................................. 46 Chapter 6: Results of Nonlinear Time History Analysis ..........................................................57 6.1 Structural Response without Accounting for the Buckling Failure of LRB ................. 57 6.2 Structural Response Accounting for the Buckling Failure of LRB .............................. 58 6.3 Response of Safe Model ............................................................................................... 61 Chapter 7: Summaries and Conclusions....................................................................................64 7.1 Conclusions ................................................................................................................... 64 7.2 Future Study .................................................................................................................. 65 Bibliography .................................................................................................................................66 vii  Appendices ....................................................................................................................................71 Appendix A LRB Geometric Properties ................................................................................... 71  viii  List of Tables  Table 3.1 Damping reduction factor [17] ..................................................................................... 23 Table 4.1 Site specific spectrum for Vancouver City Hall, 2/50 [27] .......................................... 29 Table 4.2 Bilinear model parameters ............................................................................................ 31 Table 4.3 Calculation of link shear (Building A) ......................................................................... 34 Table 4.4 Calculation of link shear (Building B) .......................................................................... 34 Table 4.5 Structural member sections ........................................................................................... 35 Table 4.6 LRB properties .............................................................................................................. 36 Table 4.7 LRB Design axial force ................................................................................................ 36 Table 4.8 Building periods ............................................................................................................ 39 Table 5.1 The site specific spectrum for Vancouver City Hall, 2/50 [27] .................................... 47 Table 5.2 Dominant earthquake sources for Vancouver City Hall ............................................... 50 Table 5.3 Suite of selected crustal ground motions ...................................................................... 52 Table 5.4 Suite of selected subcrustal ground motions................................................................. 53 Table 5.5 Suite of selected subduction ground motions ............................................................... 54 Table 6.1 Ratio of axial load demand from CSM and time history analyses ............................... 57 Table A.1 LRB geometric properties (Building A – 3Tf – tr = 3mm) ........................................... 71 Table A.2 LRB geometric properties (Building A – 3Tf – tr = 9mm) ........................................... 72 Table A.3 LRB geometric properties (Building A – 3Tf – tr = 15mm) ......................................... 73 Table A.4 LRB geometric properties (Building A – 4Tf – tr = 3mm) ........................................... 73 Table A.5 LRB geometric properties (Building A – 4Tf – tr = 9mm) ........................................... 74 Table A.6 LRB geometric properties (Building A – 4Tf – tr = 15mm) ......................................... 74 ix  Table A.7 LRB geometric properties (Building A – 5Tf – tr = 3mm) ........................................... 75 Table A.8 LRB geometric properties (Building A – 5Tf – tr = 9mm) ........................................... 75 Table A.9 LRB geometric properties (Building A – 5Tf – tr = 15mm) ......................................... 75 Table A.9 LRB geometric properties (Building B – 1.5Tf – tr = 3mm) ........................................ 76 Table A.11 LRB geometric properties (Building B – 1.5Tf – tr = 9mm) ...................................... 76 Table A.12 LRB geometric properties (Building B – 1.5Tf – tr = 15mm) .................................... 77 Table A.13 LRB geometric properties (Building B – 2Tf – tr = 3mm) ......................................... 77 Table A.14 LRB geometric properties (Building B – 2Tf – tr = 9mm) ......................................... 78 Table A.15 LRB geometric properties (Building B – 2Tf – tr = 15mm) ....................................... 78 Table A.16 LRB geometric properties (Building B – 2.5Tf – tr = 3mm) ...................................... 78 Table A.17 LRB geometric properties (Building B – 2.5Tf – tr = 9mm) ...................................... 79 Table A.18 LRB geometric properties (Building B – 2.5Tf – tr = 15mm) .................................... 79  x  List of Figures  Figure 1.1 The effect of period shift and damping of seismic isolation ......................................... 2 Figure 1.2 Low-damping rubber bearing and lead-rubber bearing ................................................. 3 Figure 1.3 The different hysteresis behaviors of elastomeric bearings (a) Low-damping rubber bearing [3]; (b) High-damping rubber bearing [4]; (c) Lead-rubber bearing [5] ............................ 3 Figure 1.4 Friction pendulum system [1] ........................................................................................ 4 Figure 1.5 Triple pendulum bearing [6] .......................................................................................... 4 Figure 1.6 The dynamic stability test hysteresis loop of an elastomeric bearing [9] ...................... 6 Figure 2.1 A simple mechanical model for elastomeric bearings [14] ......................................... 13 Figure 2.2 Overlapping area method ............................................................................................ 17 Figure 2.3 Physical model of LeadRubberX ................................................................................. 18 Figure 2.4 Example inputs for LeadRubberX ............................................................................... 19 Figure 3.1 Capacity spectrum method [25]................................................................................... 21 Figure 3.2 Bilinear model vs Sa*Mass - Sd plot ............................................................................ 22 Figure 3.3 The iterative design procedure for bilinear models ..................................................... 24 Figure 3.4 Damping ratio optimization ......................................................................................... 25 Figure 3.5 Plan dimensions for the calculation of maximum total displacement [4] ................... 27 Figure 4.1 Isometric view of Buildings A and B .......................................................................... 28 Figure 4.2 Plan layout of Buildings A and B ................................................................................ 29 Figure 4.3 Optimization of damping ratio .................................................................................... 30 Figure 4.4 Bilinear models ............................................................................................................ 31 Figure 4.5 Eccentrically braced frame (EBF) [28] ....................................................................... 32 xi  Figure 4.6 Free-body diagram showing the link shear and frame shear of EBF .......................... 33 Figure 4.7 OpenSees Navigator models........................................................................................ 38 Figure 4.8 Effect of axial load (P) on shear behavior using LeadRubberX .................................. 38 Figure 5.1 Simplified Seismic Hazard Map for Small (1-2 Story) Structures [38] ...................... 42 Figure 5.2 The Cascadian subduction zone [39] ........................................................................... 43 Figure 5.3 Cascadian earthquake source [40] ............................................................................... 43 Figure 5.4 Generating the design spectrum from the PSHA results [43] ..................................... 45 Figure 5.5 The site specific spectrum for Vancouver City Hall, 2/50 .......................................... 47 Figure 5.6 Seismic hazard deaggregation at Vancouver City Hall [27] ....................................... 48 Figure 5.7 Period range of ground motion scaling for isolated buildings [19] ............................. 49 Figure 5.8 Site-specific spectrum and scenario-specific period ranges (TRS) for ground motion selection (a) Building A, and (b) Building B ................................................................................ 50 Figure 5.9 Ground motion selection for Building A (a) crustal; (b) subcrustal; (c) subduction... 55 Figure 5.10 Ground motion selection for Building B (a) crustal; (b) subcrustal; (c) subduction . 56 Figure 6.1 Illustration of LRB failure recognition (a) time series of absolute shear displacement; (b) time series of axial load and buckling load; (c) shear hysteresis; (d) axial hysteresis ............ 59 Figure 6.2 Amplification factor vs probability of failure for each period .................................... 60 Figure 6.3 Failure distribution of crustal, subcrustal, and subduction earthquakes ...................... 61 Figure 6.4 Inter-story drift ratio of Building A (a) X direction; (b) Y direction .......................... 62 Figure 6.5 Inter-story drift ratio of Building B (a) X direction; (b) Y direction ........................... 63 Figure 6.6 Link shear (a) Building A; (b) Building B .................................................................. 63  xii  List of Symbols  KV: vertical stiffness Ab: bonded rubber area Ec: compression modulus Tr: total rubber thickness uh: shear deformation r: radius of gyration of the bonded rubber area KV0: axial compressive stiffness at zero shear deformation  KH: horizontal stiffness KH0: horizontal stiffness at zero axial load Gr: rubber shear modulus P: axial load Pcr: critical buckling load Pcr0: critical buckling load of LRB under zero horizontal displacement AS: shear area including the effect of the rigidity of steel shims IS: moment of inertia including the effect of the rigidity of steel shims I: moment of inertia of the area of bonded rubber h: height of LRB including rubber layers and steel shims  Er: rotational modulus S: shape factor of single rubber layer F: diameter modification factor K: bulk modulus of rubber xiii  S: shape factor of single rubber layer F: diameter modification factor K: bulk modulus of rubber D2: diameter of LRB D1: diameter of lead core tr: single rubber layer thickness Ar: reduced area Q: characteristic strength K1: the initial stiffness K2: post-elastic stiffness Dmax: maximum displacement Sa: spectral acceleration Sd: spectral displacement ξeff: equivalent damping ratio B: damping reduction factor WD: energy dissipated per hysteresis cycle Keff: effective stiffness Fyl: yield strength of lead DTmax: maximum total displacement e: actual eccentricity plus 5% accidental eccentricity Tp: post-elastic period of isolated building Tf: fixed-base period W: building weight xiv  Fx: distributed base shear at level x Vs: base shear for the superstructure Vu: link shear D: dead load L: live load S: snow load E: earthquake load TR: period range for ground motion selection TRS: scenario specific period range  xv  List of Abbreviations  AASHTO: American Association of State Highway and Transportation Officials ASCE:  American Society of Civil Engineers  CSA: Canadian Standards Association CSM: capacity spectrum method EBF: eccentrically braced frame LRB: lead-rubber bearing MCE: maximum considered earthquake NBCC: National Building Code of Canada PEER: Pacific Earthquake Engineering Research Center PSHA: probabilistic seismic hazard analysis  SDOF: single degree of freedom SFRS: seismic force resisting system xvi  Acknowledgements  First, I would like to express my sincere gratitude to my supervisor, Professor Tony Yang. His patience, motivation, and immense knowledge have guided me in exploring the world of innovative engineering technologies. Without his selfless support and insightful advice, this research would not have been possible. I also would like to thank Prof. Wei Guo for reviewing my thesis and providing valuable advice.  I would like to offer my thanks to my research colleagues, Yuanjie Li, Dorian Tung, and Lisa Tobber, for their valuable guidance and advice which helped me with the technologies I had needed to investigate for my research. My gratitude also extends to my other friends in the department including Dongbin Zhang, Tianyi Li, Yuxin Pan, and Tianxiang Li for their help with my studies.  My deepest gratitude goes to my parents for their unconditional support and unwavering belief in me. Above all, I would like to thank my wife, Mi Zhou, for her endless love and support.  xvii  Dedication  This thesis is dedicated to my parents and brother.1  Chapter 1: Introduction  1.1 Seismic Isolation Seismic isolation is a well-developed and widely-implemented technique used to mitigate structural and non-structural damage under severe earthquake shaking. This technique decouples the superstructure from the substructure by interposing components with low horizontal stiffness under the superstructure. The soft layer (i.e., the isolation system) results in a lengthened fundamental period as compared with the period of the same building with fixed base, significantly reducing the seismic force transmitted to the superstructure. Some isolation systems also provide additional damping to the structure, thus further reducing the seismic demand. The effect of seismic isolation can be simply illustrated in Figure 1.1, where the spectral acceleration (Sa) decreases with the increase of the period (T). Compared to a conventional fixed-base building, the superstructure of an isolated building can remain elastic during earthquakes due to its reduced demand. The response of an isolated low- or mid-rise building under horizontal ground motion is dominated by the first dynamic mode, where its lateral deformation is concentrated in the isolated system, causing the superstructure to move like a rigid body. Another benefit of seismic isolation is that the reduced acceleration of the superstructure will protect the building’s non-structural components. Isolated buildings’ superior seismic performance allows them to achieve full operation during severe earthquakes.     2   Figure 1.1 The effect of period shift and damping of seismic isolation  Seismic isolation has been studied intensively and implemented in numerous bridges, buildings, and infrastructures over the last three decades. A variety of seismic isolation hardware types have been developed, and these can be classified into two main categories: elastomeric bearings and sliding bearings [1]. Elastomeric bearings are made from laminated rubber layers and steel shims. The configurations of elastomeric bearings allow them to move horizontally with low stiffness while possessing a substantial vertical load carrying ability sufficient to support the superstructure. The rubber used in this type of bearing can be either low- or high-damping. Invented in New Zealand in 1976 [2], the lead-rubber bearing (LRB) have been extensively implemented throughout the world. LRBs differ from low-damping rubber bearings only by the lead core inserted in their central holes. This lead core provides stable energy dissipation following its yielding during earthquakes. Figure 1.2 reveals the configuration of low-damping rubber bearing and lead-rubber bearing. The hysteresis behavior of diverse elastomeric bearings is shown in 3  Figure 1.3. Compared to low- and high-damping rubber bearings, lead-rubber bearings have a more stable bilinear hysteresis behavior able to provide predictable damping.      Figure 1.2 Low-damping rubber bearing and lead-rubber bearing    (a) (b)  (c) Figure 1.3 The different hysteresis behaviors of elastomeric bearings (a) Low-damping rubber bearing [3]; (b) High-damping rubber bearing [4]; (c) Lead-rubber bearing [5] 4  The sliding bearing is a frictional isolation device utilizing the friction between the slider and sliding surface as demonstrated in Figure 1.4. The geometry of the sliding surface in the friction pendulum system can provide a restoring force and stable bilinear hysteresis behavior during earthquakes. An improved sliding bearing is called triple pendulum bearing as shown in Figure 1.5. The triple pendulum system has multi-stage behavior where different stiffness and damping ratio can be achieved at different displacement levels. This type of bearing can be designed to satisfy different performance targets based on small, moderate, and severe earthquakes.   Figure 1.4 Friction pendulum system [1]   Figure 1.5 Triple pendulum bearing [6]  5  The effectiveness of seismic isolation has been examined by the performance of isolated buildings during earthquakes. A well-known example is the base-isolated computer center of the Ministry of Post and Telecommunications in Japan. The isolation system of the building was implemented using a combination of low-damping rubber bearings, lead-rubber bearings, and steel dampers. During the 1995 Hanshin-Awaji Earthquake, the peak acceleration experienced by the superstructure of the computer center were reduced to about 1/4 and 1/3, respectively, in the two horizontal directions of that on the foundation. The excellent performance of the isolated buildings showed the effectiveness of seismic isolation. After 1995, the number of isolated buildings in Japan increased dramatically [7]. As a further example, the USC Hospital in Los Angeles performed well during the strong earthquake ground shaking of the 1994 Northridge Earthquake. The drift ratios of the superstructure were less than 10% of the limit, and the building was undamaged [8]. Additionally, the Japan Society of Seismic Isolation collected the response data of isolated buildings in Japan during and after the 2011 Tohoku Earthquake. The records show that all isolated buildings remained undamaged and that their occupancy continued after the earthquake.      1.2 Stability of LRB During earthquakes, elastomeric bearings experience large degrees of horizontal deformation to reduce the response of the superstructure as caused by their low stiffness levels. However, these large displacements are always accompanied by high compressive axial loads resulting from the superstructures’ gravity loads and the seismic overturning forces, which cause buckling failure in the elastomeric bearing. Figure 1.6 illustrates buckling failure of an elastomeric bearing tested by 6  Sanchez et al. [9]. Due to P-Δ effect, the buckling load capacity of the elastomeric bearing decreases with the increase in shear deformation. When the buckling occurs, the bearing will lose its shear and axial resistance. Isolation systems are normally placed at the base or lower levels of buildings and the axial loads transmitted to them are significant. In such situations, the buckling of the isolators during earthquakes will cause severe damage to buildings and immense life losses. Therefore, the design of elastomeric bearings to prevent buckling under severe earthquakes is crucial.   Figure 1.6 The dynamic stability test hysteresis loop of an elastomeric bearing [9]  As a type of elastomeric bearing, the LRB is facing the same challenge of buckling failure under large shear deformation. The theory to calculate the buckling load of LRBs can be dated back to the works done by Haringx [10] in the 1960s. In this study, the influence of the axial force on the horizontal stiffness and the buckling load of a slender elastic column under combined flexural and shear deformation was derived analytically. Gent [11] applied Haringx’s theory in predicting the critical buckling load of elastomeric bearings and verified his theory with experimental tests. The 7  results revealed that Haringx’s theory can be used to model the buckling behavior of elastomeric bearings under combined axial and shear loads well. Koh and Kelly [12, 13] further improved the Haringx’s theory and developed a detailed viscoelastic model for elastomeric bearings for dynamic application. They have conducted detailed numerical simulation and verified against experimental dynamic tests. The result shows the detailed viscoelastic model can accurately model the dynamic behavior of elastomeric bearings. However, the detailed viscoelastic model was too complicated for typical structural engineering applications. For this reason, Koh and Kelly [14] developed a simplified two-spring model to simulate the stability effects of elastomeric bearings. The two-spring model provided close approximation to the detailed model and it was validated using experimental tests [14, 15]. Based on the theory developed, the two-spring model was not able to simulate the load capacity of the LRBs under large horizontal deformation. To overcome this inefficiency, Buckle and Liu [3, 16] developed the overlapping area method to calculate the axial capacity of LRBs under large horizontal deformation. According experimental tests, this method provided a conservative prediction of the buckling load [3, 16].  1.3 Design Provisions Many of the building and bridge codes worldwide have considered the stability of isolators during the seismic design. The American building code ASCE-7 [17] and bridge code AASHTO [18] have different clauses for the vertical load stability of isolators (clause 17.2.4.6 and 12.3, respectively). ASCE-7 indicates that the isolator should be designed to resist combined axial load (from dead, live and earthquake) when the shear deformation reached the maximum total displacement. The AASHTO code presents more stringent requirements, whereby the isolators are 8  required to satisfy the following two criteria: (1) That the axial capacity of the isolator at zero shear deformation shall have a compression capacity larger than three times the sum of the unfactored dead and live loads; and (2) That the isolator shall be designed to resist a combined dead, live, and earthquake loads, at the amplified maximum total displacement. The Canadian building code [19] and bridge code [20] have similar requirement, where the isolator shall has sufficient load carrying capacity at the target shear deformation. The Chinese code [21], on the other hand, considers the stability issue by limiting the compressive pressure and lateral deformation on each isolator.  1.4 Motivation and Objectives One of the design challenges of LRB is the ability to carry axial load under large shear deformation, where the LRB could buckle. LRBs are usually installed at the base of a building, hence the buckling failure of LRBs could result in difficult structural repair and hefty financial losses. Therefore, it is crucial to mitigate the buckling failure of LRBs under the maximum considered earthquake (MCE) shaking.   Typically speaking, the force-deformation response in shear direction of the LRB is modeled using a simple bilinear model without considering the axial and shear coupling effects. According to the previous research, the axial stiffness of the LRB decreases with the increase of shear deformation. Apart from that, the axial force will result in the reduction of shear stiffness [14]. Therefore, the simple bilinear model will overestimate the behavior of the LRB and ignore the potential buckling failure. For overcoming this deficiency, Kumar developed a state-of-the-art 3D LRB model [22]. This model adopted the algorithm of the two-spring model and overlapping area method to 9  simulate the axial and shear coupling as well as buckling effect of LRBs. Although detailed component behavior of LRB under combined axial and shear loads has been well investigated, the seismic performance of base isolated building with LRB has not been systematically examined.  The first objective of this research is to systematically study the seismic performance of building isolated by LRBs with different LRB geometric properties, structural periods, and axial loads. The study is based on the nonlinear time history analyses on robust finite element models developed using the advanced LRB buckling model in OpenSees. The second objective is to propose a simple amplification factor to amplify the design axial load of LRB for practical design when the coupled axial and shear interaction of LRB is not explicitly modeled. The amplification factor aims at mitigating the probability of LRB buckling failure for isolated buildings under severe earthquake shaking.    1.5 Scope of the Work The scope of work for this study is as follows: 1) Review the theories on the mathematical approaches to the LRB buckling effect. 2) Present a design procedure for seismic isolation systems with bilinear behavior according to the capacity spectrum method. 3) Develop a design method to calculate the LRBs’ geometry properties in accordance with the determined bilinear model and target axial capacity when the LRBs are subjected to the maximum total displacement. 10  4) Implement the design procedure in designing two prototype buildings isolated by LRBs. Each building will be designed according to different groups of parameters including its target period, the single rubber layer thickness, and the axial capacity. 5) Develop detailed finite element models of the prototype buildings in OpenSees [23] utilizing the advanced 3D mechanical model for LRBs, “LeadRubberX” [22], which takes LRB buckling behavior into consideration.  6) Conduct nonlinear time history analyses on the developed models using ground motions selected and scaled according to the 2015 National Building Code of Canada (NBCC). Identify failure events resulted by the buckling of LRB. 7) Summarize the results and provide recommendations.  1.6 The Organization of this Thesis Chapter 2 of this thesis reviewed the theories on the mechanical properties of LRBs, including the coupled vertical-horizontal response and the buckling effect. The numerical model used in this study was also introduced. The approach to design the bilinear model for the LRB system according to the target period and damping ratio is explained in Chapter 3. In Chapter 4, the design procedure to determine the geometric properties of the LRBs based on the bilinear model and axial capacity, is specified. Chapter 5 introduces the seismic hazard in Vancouver, BC, Canada, and provides ground motion selection for the analysis based on hazard analysis. In Chapter 5, two prototype buildings for location at Vancouver City Hall are designed using LRBs at their base according to different target periods, single rubber layer thicknesses, and axial capacity. The time 11  history analysis results are summarized and presented in Chapter 6. Finally, conclusions are drawn in Chapter 7.    12  Chapter 2: Mechanical Behavior of LRBs  This chapter will introduce the well-established theory on the mechanical behavior of lead-rubber bearings. The coupling of horizontal and vertical responses and the calculation of the buckling load under large shear deformation levels have already been discussed. However, the nonlinear behavior of LRBs under tension is not the focus of this study, and has not been included in this chapter. A robust numerical model that includes all theories on the mechanical behavior of LRBs is also introduced. This model is applied in this study to model LRBs and account for buckling behavior.  2.1 Interactions Between Horizontal and Vertical Directions The horizontal and vertical responses of LRBs are coupled. This effect can be considered from two aspects: the influence of shear stiffness on axial loads and the effects of axial stiffness on shear displacements. The two-spring model developed by Koh and Kelly [14] was a simplified model using two linear springs to account for the shear and flexural behavior of elastomeric bearings as shown in Figure 2.1. The two-spring model was found to provide a close approximation to the previously developed detailed model and was validated using experimental tests [14, 15]. 13   Figure 2.1 A simple mechanical model for elastomeric bearings [14]  The vertical stiffness of the LRB can be derived from the two-spring model as follows:  𝐾𝑉  =  𝐴𝑏𝐸𝑐𝑇𝑟[1 +3𝜋2(𝑢ℎ𝑟)2]−1= 𝐾𝑉0 [1 +3𝜋2(𝑢ℎ𝑟)2]−1 2.1 where  Ab = bonded rubber area Ec = compression modulus of the bearing Tr = total rubber thickness uh = shear deformation r = radius of gyration of the bonded rubber area KV0 = axial compressive stiffness at zero shear deformation   The horizontal stiffness can be expressed as a function of the axial load and critical buckling load: 14   𝐾𝐻  =  𝐺𝑟𝐴𝑏𝑇𝑟[1 − (𝑃𝑃𝑐𝑟)2] = 𝐾𝐻0 [1 − (𝑃𝑃𝑐𝑟)2] 2.2 where  Gr = rubber shear modulus P = axial load Pcr = critical buckling load KH0 = horizontal stiffness at the zero axial load  2.2 Buckling without Shear Deformation The critical buckling load of an LRB without horizontal displacement can be derived from the two-spring model as [4]:  𝑃𝑐𝑟0 = √𝑃𝑆𝑃𝐸 2.3 where PE and PS are given by:  𝑃𝐸 =𝜋2𝐸𝑟𝐼𝑆ℎ2 2.4  𝑃𝑆 = 𝐺𝑟𝐴𝑆 2.5 where AS and IS are the shear area and moment of inertia including the effect of the rigidity of the steel shims, and are calculated as:  𝐴𝑆 = 𝐴𝑏ℎ𝑇𝑟 2.6  𝐼𝑆 = 𝐼ℎ𝑇𝑟 2.7 where  I = moment of inertia of the area of bonded rubber 15  h = height of the LRB including its rubber layers and steel shims Er = rotational modulus of the bearing Er is given as Ec/3 for circular and spare bearings, and where Ec is provided as:  𝐸𝑐 = (16𝐺𝑟𝑆2𝐹+43𝐾)−1 2.8 where  S = shape factor of a single rubber layer F = diameter modification factor K = bulk modulus of rubber  The geometric properties mentioned above are derived as follows for a circular LRB:  𝑆 =𝐷2 − 𝐷14𝑡𝑟 2.9  𝐹 =𝑑2 + 1(𝑑 − 1)2−1 + 𝑑(1 − 𝑑)ln (𝑑) 2.10  𝑑 =𝐷2𝐷1 2.11  𝐼 =𝜋(𝐷24 − 𝐷14)64 2.12  𝐴𝑏 =𝜋(𝐷22 −𝐷12)4 2.13  𝑟 = √𝐼𝐴𝑏 2.14 where  D2 = diameter of the LRB 16  D1 = diameter of the lead core tr = single rubber layer thickness  2.3 Buckling with Shear Deformation When the LRB is subjected to shear displacement uh, the critical buckling load is reduced. The reduced buckling load is accounted for by the overlapping area method developed by Buckle and Liu [16]. This method calculates the reduced buckling load based on the overlapping area between the upper- and lowermost layers of the bearing under shear deformation, and provides a conservative prediction of the buckling load [3, 16]. The reduced buckling load of an LRB under shear deformation is given as:  𝑃𝑐𝑟 = 𝑃𝑐𝑟0𝐴𝑟𝐴𝑏 2.15   where Ar is the reduced area as shown in Figure 2.2 and is calculated as:  𝐴𝑟 =𝐷224(𝜑 − sin𝜑) 2.16  𝜑 = 2 cos−1 (𝑢ℎ𝐷2) 2.17 17   Figure 2.2 Overlapping area method  However, LRBs will not lose their stability when the overlapping area is equal to zero, as suggested by experimental tests [3, 9, 15]. A piecewise function of LRBs’ reduced critical buckling load was proposed by Warn et al. [15], and is given as:  𝑃𝑐𝑟 ={  𝑃𝑐𝑟0𝐴𝑟𝐴𝑏     (𝐴𝑟𝐴𝑏≥ 0.2)0.2𝑃𝑐𝑟0      (𝐴𝑟𝐴𝑏< 0.2) 2.18  2.4 Advanced Numerical Model The mathematical approach to considering LRBs’ mechanical properties has been implemented in OpenSees by Kumar [22-24]. The algorithm was programmed in the element known as LeadRubberX in OpenSees. This three-dimensional element possesses two nodes and 12 degrees of freedom. The physical model of this element is shown in Figure 2.3. This robust numerical model has been validated by experimental tests [24]. 18   Figure 2.3 Physical model of LeadRubberX   This numerical model includes various LRB mechanical behavior patterns, such as coupled shear and axial directions, varying buckling load, and nonlinear in tension. The inputs of the model require only a knowledge of the LRBs’ material and geometric properties. Figure 2.4 shows the input needed to define a LeadRubberX element. The coupling of the two horizontal directions is accounted for by a coupled bidirectional model, while the coupling of vertical and horizontal directions is accounted indirectly using the coupling equations in Section 2.1. The buckling effects summarized in Sections 2.2 and 2.3 are explicitly considered in this model. The recorder for this element can register the instantaneous values of the critical buckling load, which are displacement-dependent. In the numerical analysis for this study, the LRBs’ recorded buckling loads were checked at every time series step and then compared with the real-time axial loads in order to recognize failure events. The analysis was stopped with any exceeding of the buckling load, thus conserving a large amount of time in the analysis process. 19   Figure 2.4 Example inputs for LeadRubberX   20  Chapter 3: Design Procedure for LRB Isolated Buildings  This chapter begins by introducing the capacity spectrum method (CSM) for designing the LRBs’ bilinear model parameters using the target spectrum. The procedure uses the assumed target damping ratio of the isolation system and isolated fundamental period as inputs, and calculates the key parameters of the bilinear model utilizing an iterative process. The second part of this chapter presents a procedure used to determine the geometry of an LRB required to achieve the designed bilinear model and the target axial capacity under the maximum total displacement.   3.1 Capacity Spectrum Method The capacity spectrum method (CSM) is a graphical approach to compare the structural capacity with the seismic demand on the structure as illustrated in Figure 3.1 [25]. In this method, the capacity of a structure can be generated by a force-displacement relationship. Then the structure can be idealized to an equivalent Single-Degree-Of-Freedom (SDOF) system by converting the base shear and the roof displacement of the structure to spectral acceleration and spectral displacement, respectively. The seismic demand is defined by a damped spectral displacement – spectral acceleration plot. An intersection point can be found when the capacity of the structure and the seismic demand are plotted together, which provides an inelastic strength and deformation demand for the structure. The CSM provides a simple way to design an inelastic structure and a visual assessment of how the structure will perform under earthquake shaking.    21   Figure 3.1 Capacity spectrum method [25]  3.2 Bilinear Model Design Without considering the axial and shear coupling behavior of the LRB, the seismic response of the base isolated building with LRB can be designed with bilinear model using CSM. With the location of the prototype building selected, the seismic demand of the prototype site can be represented using the demand curve as shown in Figure 3.2. The vertical axis is represented using spectral acceleration (Sa) multiplied by the mass of the structure, while the horizontal axis is represented using the spectral displacement (Sd). The response of a low- or mid-rise isolated building is dominant by the fundamental structural period because of the low stiffness of isolation system. Therefore, in the capacity spectrum method, an isolated building can be idealized as a single-degree-of-freedom structure with the mass of the superstructure and the stiffness of the isolation system. Because the vertical axis represents the base shear and the horizontal axis represents the isolator deformation, the bilinear response of the LRB (known as the capacity curve) and the demand curve can be plotted together as shown in Figure 3.2.  22   Figure 3.2 Bilinear model vs Sa*Mass - Sd plot  A bilinear model can be fully characterized by four key parameters: (1) the characteristic strength (Q); (2) the initial stiffness (K1); (3) the post-elastic stiffness (K2); and (4) the maximum displacement (Dmax) [26] which is shown in Figure 3.2. The procedure of bilinear model design starts with the assumed target fundamental period (Tp) of the isolated building computed by the post-elastic stiffness of the isolation system (according to NBCC [19]) and the target equivalent viscous damping ratio (ξeff) of the isolation system.   Based on ξeff, the demand curve is reduced from the 5% damped demand curve using:  𝑆𝑎(𝜉𝑒𝑓𝑓)  =  𝑆𝑎(5%)𝐵 3.1 23   𝑆𝑑(𝜉𝑒𝑓𝑓)  =  𝑆𝑑(5%)𝐵 3.2 where B is the damping reduction factor listed in  Table 3.1.  Table 3.1 Damping reduction factor [17] Equivalent viscous damping ratio, ξeff B ≤ 2% 0.8 5% 1.0 10% 1.2 20% 1.5 30% 1.7 40% 1.9 ≥ 50% 2.0  Based on Tp, K2 can be determined by Tp and the weight of the superstructure (W):  𝐾2 = (2𝜋𝑇𝑝)2𝑊𝑔 3.3 As an approximate rule of thumb, K1 is taken as 10K2. With a trial of Q value, Dmax can be determined by finding the intersection point of the capacity curve (bilinear line) and the reduced demand curve. Equation 3.4 - 3.7 show the derivation of Dy, Keff, and WD (energy dissipated per cycle) as functions of the key parameters.   𝐷𝑦  =  𝑄𝐾1 − 𝐾2 3.4  𝐾𝑒𝑓𝑓  =  𝐾2 +𝑄𝐷𝑚𝑎𝑥 3.5  𝑊𝐷  =  2𝜋𝜉𝑒𝑓𝑓 𝐾𝑒𝑓𝑓 𝐷𝑚𝑎𝑥2 3.6 24   𝑄 =  𝑊𝐷4(𝐷𝑚𝑎𝑥 − 𝐷𝑦) 3.7 However, Q can be derived using Equation 3.7 which is coupled with the previous equations. Therefore, an iterative procedure (illustrated in Figure 3.3) is necessary to calculate the bilinear design parameters. In this procedure, the target post-elastic period and damping ratio are considered as inputs. Then, the demand curve can be determined by reducing the 5% damped demand curve based on the target damping ratio. Using the initialized Q, the other bilinear model parameters can be calculated. However, another Q is also computed according to the derived parameters. If the Q is not closed enough to the value in the previous step, the procedure will be recalculated based on the new value of Q. The iterative procedure will provide the final solution if the results between two successive steps are sufficiently close in value.   Figure 3.3 The iterative design procedure for bilinear models  25  In this study, the target damping ratio is selected such that the base shear transmitted to the superstructure is minimized. Figure 3.4 shows an illustration of damping optimization, where the three bilinear models share the same isolation period but different damping ratio. In this case, system B has the lowest base shear; hence, the corresponding damping ratio is selected as the most optimal one.  Figure 3.4 Damping ratio optimization  3.3 LRB Design The parameters that define an LRB are: 1) the material properties of rubber, such as shear modulus of rubber (Gr) and bulk modulus of rubber (K); 2) shear yield strength of lead (Fyl); 3) the geometric properties of the LRB, including the lead diameter (D1), the total diameter (D2), the steel shim thickness (ts), the single rubber layer thickness (tr), and the number of rubber layers (nr). Once the bilinear model of an LRB has been defined, different combinations of geometric properties can 26  result in the same bilinear behavior, but with different axial capacity. In this study, Gr, K, ts, and Fyl are assumed to be constants and the influence of the tr and nr on the buckling capacity of the LRB is analyzed.  For a given tr and nr, the lead diameter (D1), the total rubber thickness (Tr), the bonded rubber area (Ab), and total diameter (D2) can be calculated based on the bilinear model parameters using the following equations:  𝐷1 = √4𝑄𝜋𝐹𝑦𝑙 3.8  𝑇𝑟  =  𝑛𝑟𝑡𝑟 3.9  𝐴𝑏  =  𝑇𝑟𝐾2𝐺𝑟 3.10  𝐷2  =  √4𝐴𝑏𝜋+ 𝐷12 3.11  Once the geometry of an LRB has been defined, the buckling load of the LRB with shear deformation can be calculated according to the procedure shown in Chapter 2. It should be noted that the capacity of LRB should be calculated as the critical buckling load under DTmax, which is the maximum total displacement of the LRB considering the torsional response of the building as given by [4]:  𝐷𝑇𝑚𝑎𝑥 = 𝐷𝑚𝑎𝑥 (1 + 𝑦12𝑒𝑏2 + 𝑑2) 3.12 where  e = actual eccentricity plus 5% accidental eccentricity 27  y = distance to a corner perpendicular to the direction of the seismic loading b, d = dimensions of the building plan  The equation is generated based on the assumption that the seismic load KeffD is acting on the mass center which has a distance from the center of stiffness of e as shown in Figure 3.5. For a b × d rectangular plan, assuming that the isolators are evenly distributed at the base of the building, the torsional stiffness of the isolation system is given as Keff (b2 + d2) / 12. The corresponding rotation caused by the seismic load is calculated as:  𝜃 =  𝐾𝑒𝑓𝑓𝐷𝑒𝐾𝑒𝑓𝑓 (𝑏2 + 𝑑212 )=12𝐷𝑒𝑏2 + 𝑑2 3.13 Thus, the additional displacement resulting from the torsional effect is given by:  12𝐷𝑒𝑏2 + 𝑑2𝑦 3.14   Figure 3.5 Plan dimensions for the calculation of maximum total displacement [4] 28  Chapter 4: Prototype Buildings  Two office buildings for location at Vancouver City Hall, BC, Canada were selected as the prototype buildings for this study. Figure 4.1 shows the dimensions of the prototype buildings. Building A is a 5-story building with a floor area of 45 m by 63 m and a total building height of 18.85 m. Building B is a 15-story building with a floor area of 27 m by 27 m and a total building height of 55.35 m. The site class was assumed to be in Class C according to NBCC [19]. LRBs were used at the base level as the isolation devices. As shown in Figure 4.2, the LRBs were grouped into three types, based on the locations (Type A = center, Type B = side, and Type C = corner). The superstructure was designed using perimeter steel eccentrically braced frames (EBFs) shown in Figure 4.2.   Figure 4.1 Isometric view of Buildings A and B 29   Figure 4.2 Plan layout of Buildings A and B  4.1 Bilinear Model Parameters The values of spectral acceleration of the site-specific spectrum with a hazard level of a 2% probability of exceedance in 50 years (2/50), and a damping ratio of 5% are shown in Table 4.1. The Sa values were generated by Natural Resources Canada (NRC) in a site-specific seismicity hazard report [27] for Vancouver City Hall, with site Class C. According to NBCC [19], the design spectrum was constructed using the Sa and the corresponding Sd on T = 0.2, 0.5, 1, 2, 5, and 10 sec.   Table 4.1 Site specific spectrum for Vancouver City Hall, 2/50 [27] T [sec] PGA 0.05 0.1 0.2 0.3 0.5 1 2 5 10 Sa [g] 0.366 0.446 0.678 0.844 0.851 0.753 0.424 0.257 0.081 0.029  30  Each prototype building was designed based on three different target periods (Tp). For building A, 3, 4, and 5 times the fixed based period were selected as the target isolated periods; for building B, 1.5, 2, and 2.5 times the fixed base period were chosen. The NBCC requires that the isolation period should be larger than three times the fixed base period. However, this requirement resulted in unpractically large LRBs for Building B. The LRBs designed based on the selected target periods mentioned above had reasonable sizes.   As explained in Chapter 3, the target damping ratio should also be defined prior to the design of the bilinear model. The target damping ratio can be optimized by minimizing the base shear for the superstructure. Figure 4.3 illustrates the relationship between the damping ratio and the normalized based shear for each period. In this case, for a target period, a single target damping ratio was selected at the bottom of each curve. The selected damping ratio is listed in Table 4.2.  Figure 4.3 Optimization of damping ratio 31  Based on the target period and damping ratio, the bilinear model parameters can be calculated using the capacity spectrum method and the iterative process in Chapter 3. The design results are shown in Figure 4.4 and Table 4.2.  Figure 4.4 Bilinear models  Table 4.2 Bilinear model parameters Building Tf [sec] W [kN] Tp ξeff Vs/W Q/W K2/W [m-1] Dmax [m] DTmax [m] A 0.63 66564 3Tf 23% 0.215 0.083 1.127 0.117 0.140 4Tf 21% 0.165 0.058 0.634 0.171 0.205 5Tf 21% 0.134 0.047 0.406 0.217 0.260 B 1.63 49231 1.5Tf 21% 0.171 0.060 0.673 0.167 0.192 2Tf 21% 0.128 0.045 0.379 0.222 0.255 2.5Tf 21% 0.099 0.035 0.242 0.268 0.308  32  4.2 Superstructure Design Eccentrically braced frames were selected as the seismic force resisting system (SFRS) for the superstructure of the prototype buildings. An EBF application in a real project is presented in Figure 4.5. The EBF has high elastic stiffness and high ductility at large story drift. The plastic mechanism of the EBF is controlled by a small segment on the beams called the link [28].   Figure 4.5 Eccentrically braced frame (EBF) [28]  According to NBCC [19], the superstructure of an isolated building should be designed to remain elastic during earthquakes. The eccentrically braced frames were designed using the base shear of 3Tf and 1.5Tf for buildings A and B, respectively, since they provided the maximum base shear among the three target periods for each building. The base shear was distributed vertically along the height of the superstructure as expressed by:  𝐹𝑥 = 𝑉𝑠ℎ𝑠𝑊𝑥∑ ℎ𝑖𝑊𝑖𝑁𝑖=1 4.1 33  where Wx and Wi are the weights at levels x and i, hx and hi are the respective heights of the structure above the isolation level.   Regarding 10% of the eccentricity of the lateral seismic force, the base shear force transmitted to the EBFs on one side is equal to 0.6Fx. The link shear can be obtained from the free body diagram of half of a segment of the frame as shown in Figure 4.6.   Figure 4.6 Free-body diagram showing the link shear and frame shear of EBF  The link shear can thus be calculated as:  𝑉𝑢 = 𝑉𝑖ℎ𝑖𝐿 4.2 where Vi and hi are the frame shear and story height, respectively; and L is the bay width.   Table 4.3 and Table 4.4 present the results of the link shear calculation of the EBFs.   34  Table 4.3 Calculation of link shear (Building A) Level Wx (kN) hx (m) Fx (kN) Fx per EBF (kN) Vx (kN) Vu (kN) 5th 7299 3.65 3861.098418 1158.329525 1158.33 469.76697 4th 11795 3.65 4057.104292 1217.131288 2375.461 963.38133 3rd 11853 3.65 3098.024923 929.407477 3304.868 1340.3077 2nd 11853 3.65 2118.995402 635.6986206 3940.567 1598.1188 1st 11882 4.25 1142.754965 342.8264894 4283.393 2022.7136 Base 11882 - - - - -  Table 4.4 Calculation of link shear (Building B) Level Wx (kN) hx (m) Fx (kN) Fx per EBF (kN) Vx (kN) Vu (kN) 15th 1923.0183 55.35 599.182495 404.4481841 404.4482 164.02621 14th 3082.1283 51.7 897.0142231 605.4846006 1009.933 409.58385 13th 3082.1283 48.05 833.685366 562.737622 1572.67 637.80522 12th 3082.1283 44.4 770.3565088 519.9906435 2092.661 848.69031 11th 3082.1283 40.75 707.0276517 477.2436649 2569.905 1042.2391 10th 3139.9326 37.1 655.7711573 442.6455312 3012.55 1221.7565 9th 3139.9326 33.45 591.2545879 399.0968468 3411.647 1383.6124 8th 3139.9326 29.8 526.7380185 355.5481625 3767.195 1527.807 7th 3139.9326 26.15 462.2214491 311.9994782 4079.195 1654.3401 6th 3139.9326 22.5 397.7048797 268.4507938 4347.646 1763.2118 5th 3203.4602 18.85 339.929419 229.4523578 4577.098 1856.2675 4th 3203.4602 15.2 274.1075421 185.0225909 4762.12 1931.3044 3rd 3203.4602 11.55 208.2856652 140.592824 4902.713 1988.3226 2nd 3203.4602 7.9 142.4637883 96.16305713 4998.876 2027.3221 1st 3232.9072 4.25 77.34642386 52.2088361 5051.085 2385.2347 Base 3232.9072 - - - - -  35  The links were designed according to the clauses in CSA S16-14 [29]. Once the link sections were obtained, the beams, columns, and braces were capacity designed based on the link resistance.  Table 4.5 shows the structural member sections of the superstructure for buildings A and B.   Table 4.5 Structural member sections Building Level EBF Gravity Column Link Beam Column Brace A 4 - 5 W410x67 W610x113 W250x58 W310x107 W250x67 2 - 3 W530x165 W690x192 W310x143 W360x162 W250x101 1 W610x241 W760x257 W310x226 W360x196 W250x101 B 11 - 15 W610x113 W690x192 W360x262 W310x202 W250x101 6 - 10 W690x240 W690x289 W360x677 W310x313 W310x202 1 - 5 W760x314 W840x359 W360x1068 W360x347 W310x313  4.3 LRB Properties Based on the bilinear model parameters calculated in section 4.1, the characteristic strength and post-elastic stiffness of the isolation systems were distributed to each isolator based on the tributary area. In this case, the bilinear model of each single LRB was obtained. Table 4.6 lists the LRB properties used in this study. The rubber shear modulus was applied based on different LRB locations.     36  Table 4.6 LRB properties Property Notation [unit] Value Rubber shear modulus Gr [MPa] 1.1 (Type A), 0.6 (Type B), 0.3 (Type C) Rubber bulk modulus K [MPa] 2000 Lead yield strength Fyl [MPa] 10 Single shim thickness ts [mm] 2 Single rubber layer thickness tr [mm] 3, 9, 15  The design axial force (PD) of the LRB was calculated based on a load combination in NBCC [19]:   𝑃𝐷 = 1.0𝐷 + 1.0𝐸 + 0.5𝐿 + 0.25𝑆 4.3 where D, L, and S are dead, live, and snow loads, respectively; E was considered as the axial load on LRB caused by overturning from the static equivalent earthquake load. Table 4.7 lists the design axial force for the LRBs in the prototype buildings.   Table 4.7 LRB Design axial force  Building A Building B Tp 3Tf 4Tf 5Tf 1.5Tf 2Tf 2.5Tf Type A 2787 2672 2598 9285 8708 8304 Type B 1691 1565 1485 7538 6527 5820 Type C 957 869 812 5217 4351 3745 (unit: kN)  As presented in the previous chapter, different combinations of LRB geometric parameters result in the same bilinear curve. In this study, the single rubber thickness (tr) was selected as 3, 9, and 37  15 mm to represent the typical range of rubber thickness used in the industry. With each of the selected single rubber thicknesses (example tr = 3 mm), different number of nr would result in different axial capacity. In this study, the number of nr was increased (hence the buckling capacity of the LRB would increase accordingly) to identify the axial capacity (Pcr) needed to resist the factored design load shown in Table 4.7, when the shear deformation reached the maximum total displacement. In this study, nr was selected to achieve different levels of capacity, which can be generalized as an amplification factor defined as:  𝐴𝑚𝑝𝑙𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 =  𝑃𝑐𝑟𝑃𝐷 4.4   4.4 Numerical Modeling Approach The prototype buildings were modeled in 3D in OpenSees Navigator [30] as shown in Figure 4.7. The LRBs were modeled using the advanced numerical element called “LeadRubberX” [22, 24] which has 2 nodes and 12 degrees of freedom. This element requires only the geometric and material properties of an LRB. Figure 4.8 shows the hysteresis of a LeadRubberX element under different axial loads. The result shows that the behavior of the LRB changes significantly with the presence of different axial loads. The superstructures were designed to remain elastic according to NBCC  [19]. Hence these elements were modeled using elastic elements. The peak forces in these elements were checked to ensure that the forces do not exceed the elastic limit. The floor above the isolator was modeled using a rigid elastic slab [31]. Rigid diaphragm constraints were used to model the floors on the superstructures.  38   Figure 4.7 OpenSees Navigator models   Figure 4.8 Effect of axial load (P) on shear behavior using LeadRubberX  39  Mass was assigned as lumped masses on the master nodes at each floor, where the masses in two horizontal directions and a torsional mass at each floor were included. The master nodes at each floor were shifted at 5% of the building plan dimensions to take accidental eccentricity into account. Gravity loads were applied to columns using the load combination shown in Equation 4.3. 2.5% stiffness-proportional damping was assigned to the first isolation mode (Tp) shown in Table 4.8.  Table 4.8 Building periods Building Tf [sec] Model Tp [sec] A 0.63 Building A - 3.0Tf 2.1 Building A - 4.0Tf 2.8 Building A - 5.0Tf 3.4 B 1.63 Building B - 1.5Tf 2.9 Building B – 2.0Tf 3.7 Building B - 2.5Tf 4.5       40  Chapter 5: Seismic Hazard Analysis  This chapter summarizes the source and mechanism of the earthquakes contributing to the seismic hazard risk in Southwestern British Columbia; the procedure for ground motion selection for the time history analyses of the prototype buildings will be demonstrated. The ground motion selection procedure was based on the NBCC 2015 Structural Commentary using the target spectrum, and the deaggregation results provided by Natural Resource Canada (NRC).  5.1 Seismicity in Southwestern British Columbia The prototype building is located in Southwestern British Columbia, a part of the Ring of Fire area where a tremendous number of earthquakes occur each year. Vancouver is in Canada’s region of highest seismic hazard, as seen in Figure 5.1. This region is near the Cascadia subduction zone, stretching from Southern British Columbia to Northern California, where several tectonic plates are in collision, as shown in Figure 5.2. The Juan de Fuca Plate, the Explorer Plate, and the Gorda Plate are subducting beneath the North American Plate at a rate of 2-4 cm/yr [32], and this process produces three types of earthquake: crustal, subcrustal, and subduction. Figure 5.3 illustrates the sources of Cascadian earthquakes.  Crustal Earthquakes occur within the North American Plate and are caused by a collision between blocks of the continental crust [33]. This type of earthquake generally occur at depths of less than 35 km [34]. Although crustal earthquakes are commonly seen in this region, very few of them are 41  large enough to cause damage. Two large crustal earthquakes have taken place during British Columbia’s history, M7.3 in 1946 and M7.0 in 1918; both occurred on Vancouver Island [34].   Subcrustal earthquakes take place within the oceanic plates as they descend beneath the North American Plate, as shown in Figure 5.3. This type of earthquake generally has a depth of below 30 km and a magnitude less than M7.5 [35]. Two main sources of subcrustal events are located along the west coast of Vancouver Island and beneath Puget Sound [36]. The most severe subcrustal earthquakes in this region are concentrated in the Puget Sound area. Examples include the 2001 M6.8 in Nisqually, the 1965 M6.5 in Seattle, and the 1949 M6.8 in Olympia [35].       The last type of earthquake, the subduction earthquake, occurs when the interface between the oceanic and North American plate rupture due to accumulated energy in the “locked” zone [36]. This type of earthquake is the rarest and most damaging. They can be as large as M9 and lead to severe aftershocks and destructive tsunamis [35]. Geological evidence shows that an M9 subduction earthquake occurred in 1700 at the southwestern coast of Vancouver Island; this was the most recent subduction earthquake to take place in the Cascadian subduction zone [37].    42    Figure 5.1 Simplified Seismic Hazard Map for Small (1-2 Story) Structures [38] Vancouver 43   Figure 5.2 The Cascadian subduction zone [39]  Figure 5.3 Cascadian earthquake source [40]  44  5.2 Probabilistic Seismic Hazard Analysis Probabilistic seismic hazard analysis (PSHA) is a procedure used to quantify the uncertainties of ground shaking in order to assess the future seismic risk of a site. The system entails the following steps: 1) Identifying all seismic sources that could cause damage to the site; 2) Identifying the distribution of the rate of the earthquakes with magnitudes exceeding every considered level; 3) Identifying the distribution of the distance from the earthquakes to the site; 4) Predicting the distribution of the ground motion intensity as a function of magnitude, distance, and other factors.; 5) Combining all the information above and calculating the rate of exceedance for different ground motion intensities [41].  One of the main results generated from PSHA is the uniform hazard spectrum for the site. PSHA provides the relationship between the rate of exceedance and the spectral acceleration for each period, following which the data for certain rate levels can be plotted versus the corresponding period. The process is illustrated in Figure 5.4.   Hazard deaggregation is another crucial result of PSHA. Deaggregation can be calculated by dividing the total hazard range into different magnitudes and distances for earthquakes under consideration for the site. These results help to provide a better understanding of the predominant seismic source for buildings with different periods, leading to a better selection of ground motions for time history analyses [42]. Particularly for the Vancouver area where the seismic source is complicated, as specified in the previous subsection, it is crucial to select ground motions from diverse sources for their dominated period range based the deaggregation results. 45   Figure 5.4 Generating the design spectrum from the PSHA results [43]  5.3 Seismic Hazard Model for NBCC 2015 In NBCC 2015, the seismic design values are provided in terms of 5% damped spectral acceleration at a hazard level of 2% probability of exceedance in 50 years (2% in 50 years). The design values are developed based on Canada’s 5th Generation seismic hazard model [44].   In western Canada, several seismic sources have been updated for the 5th Generation model, including three fault sources (the Juan de Fuca, Explorer and Winona segments) in the Cascadia subduction zone, an updated treatment of the Queen Charlotte Island faults, and five added faults 46  in the Yukon and Alaska region [45]. In addition to the seismic source, ground motion prediction equations (GMPEs) are also a key component of the new seismic hazard model. GMPEs predict the spectral acceleration of ground motions as functions of magnitude and distance. The database for the ground motion records has grown rapidly, leading to an improvement in the GMPEs. In order to provide a simple, efficient, and flexible solution for considering the epistemic uncertainty in the GMPEs, a three-equation approach is suggested for the new seismic hazard model. This approach uses three representative GMPEs (i.e., central, lower, and upper) rather than a weighted combination of equations [46].    5.4 Ground Motion Selection The ground motion selection for this study was based on the procedure summarized in the Structural Commentary of NBCC 2015. The Commentary has specified two methods for determining the target response spectrum for ground motion selection. Method A uses a single target spectrum, while two or more site-specific scenario target response spectra are required for Method B. Method A was chosen for this study. The values for the spectral acceleration of the site specific spectrum with a hazard level of 2% probability of exceedance in 50 years (2/50) and a damping ratio of 5% for the site are shown in Table 5.1 and Figure 5.5. The Sa values were generated by NRC as Site Class C in a site specific seismicity hazard report [27] for Vancouver City Hall.     47  Table 5.1 The site specific spectrum for Vancouver City Hall, 2/50 [27] T [sec] 0 0.05 0.1 0.2 0.3 0.5 1 2 5 10 Sa [g] 0.366 0.446 0.678 0.844 0.851 0.753 0.424 0.257 0.081 0.029    Figure 5.5 The site specific spectrum for Vancouver City Hall, 2/50  The site specific seismic hazard deaggregation of Vancouver City Hall was requested from NRC [27]. Figure 5.6 shows the seismic hazard deaggregation of 2/50 for the site at periods of 0.5, 1, 2, and 5 seconds. Based on the site specific seismic hazard analysis, the site has three dominant seismic hazard sources, namely, the crustal, subcrustal, and subduction hazard source, and the characteristic distance and magnitude ranges for earthquakes at each source are listed in Table 5.2. It can be seen from the deaggregation results that crustal and subcrustal earthquakes have a 48  dominant presence at low period ranges of up to 2 seconds, while subduction earthquakes make significant contribution to the hazard at long periods of greater than 1second.   Figure 5.6 Seismic hazard deaggregation at Vancouver City Hall [27]  According to NBCC 2015, as specified in Figure 5.7, the ground motions should be scaled within a period range (TR) of between 0.2T1 and 1.5T1 for isolated buildings where T1 is the fundamental period of the isolated building computed using the post-elastic stiffness of the isolation system. As 49  specified in Chapter 4, the post-elastic periods (Tp) of the prototype buildings with different isolation systems ranges from 2.1 to 3.4sec for Building A and 2.9 to 4.5sec for Building B. For each building, in order to select ground motions for all the design periods, the total period range should cover all the individual TR of each Tp. Therefore, the period range for Building A is defined as 0.2*2.1 - 1.5*3.4sec (0.42 – 5.1sec), and 0.2*2.9 - 1.5*4.5sec (0.58 – 6.75sec) for Building B. Scenario specific period ranges (TRS) were defined for all the seismic sources that contribute to the hazard. The TRS may overlap one another, and should cover the period range TR. Table 5.2 and Figure 5.8 show the specified TRS for each seismic source.   Figure 5.7 Period range of ground motion scaling for isolated buildings [19]     50  Table 5.2 Dominant earthquake sources for Vancouver City Hall Source Distance Magnitude TRS    Building A Building B Crustal ≤ 35km ≤ M7.5 0.42 - 1.2sec 0.58 - 1.2sec Subcrustal 35 – 140km ≤ M7.5 0.42 - 1.5sec 0.58 - 1.5sec Subduction 120 – 300km ≥ M8.0 1 - 5.1sec 1 - 6.75sec   (a)  (b) Figure 5.8 Site-specific spectrum and scenario-specific period ranges (TRS) for ground motion selection (a) Building A, and (b) Building B 51  For each seismic source, 11 sets of ground motion records, including two horizontal components and a vertical component, were selected. Crustal records were selected from the PEER NGA-West2 database [47], while subcrustal and subduction records were selected from the S2GM database [48]. For each pair of horizontal ground motions, a geometric mean of the 5% damped response spectrum was generated and amplitude scaled to match the target spectrum within the corresponding TRS. The scaled spectra of the selected ground motions are plotted in Figure 5.9 and Figure 5.10. The information on the ground motions is summarized from Table 5.3 to Table 5.5.     52  Table 5.3 Suite of selected crustal ground motions Source No. Scale Factor Event M Rjb (km) Rrup  (km) Vs30 (m/s) Horizontal 1 Filename Building A Building B Crustal 1 1.86 1.84 Northridge, CA 6.69 12.38 13.35 402.16 RSN1083_NORTHR_GLE170 Crustal 2 1.97 2.00 Imperial Valley, CA 6.53 15.19 15.19 471.53 RSN164_IMPVALL.H_H-CPE147 Crustal 3 3.41 3.32 Irpinia, Italy 6.20 22.68 22.69 574.88 RSN302_ITALY_B-VLT000 Crustal 4 1.40 1.45 Corinth, Greece 6.60 10.27 10.27 361.40 RSN313_CORINTH_COR--L Crustal 5 3.23 3.31 Chuetsu, Japan 6.80 23.63 29.25 640.14 RSN4869_CHUETSU_65042NS Crustal 6 2.28 2.21 Chuetsu, Japan 6.80 20.60 25.33 375.22 RSN5270_CHUETSU_NIG024NS Crustal 7 2.22 2.12 Chalfant Valley, CA 6.19 21.55 21.92 370.94 RSN548_CHALFANT.A_A-BEN270 Crustal 8 1.64 1.79 San Fernando,CA 6.61 19.33 22.63 450.28 RSN57_SFERN_ORR021 Crustal 9 3.92 3.72 Joshua Tree, CA 6.10 21.73 22.30 396.41 RSN6875_JOSHUA_5071045 Crustal 10 1.35 1.31 Loma Prieta, CA 6.93 19.97 20.34 561.43 RSN755_LOMAP_CYC195 Crustal 11 1.49 1.54 Landers, CA 7.28 17.36 17.36 396.41 RSN881_LANDERS_MVH045  53  Table 5.4 Suite of selected subcrustal ground motions Source No. Scale Factor Event M Epicentral Distance (km) Hypocentral Distance (km) Vs30 (m/s) Horizontal 1 Filename Building A Building B Subcrustal 12 5.99 5.78 Geiyo, Japan 6.40 79.34 94.32 379.97 Geiyo_EHM0010103241528-EW Subcrustal 13 1.30 1.25 Geiyo, Japan 6.40 39.72 59.18 266.97 Geiyo_EHM0160103241528-EW Subcrustal 14 6.50 6.50 Michoacan, Mexico 7.30 103.57 108.70 403.41 Mich_UNIO9701_111_S90E Subcrustal 15 2.46 2.58 Miyagi, Japan 7.20 124.03 110.31 205.38 Miyagi_Oki_MYG0060508161146-EW Subcrustal 16 4.48 4.21 Miyagi, Japan 7.20 244.35 247.93 548.06 Miyagi_Oki_TCG0060508161146-EW Subcrustal 17 2.06 2.08 Nisqually, WA 6.80 53.40 70.60 327.66 Nisqually_1416a_a-125 Subcrustal 18 2.10 2.12 Nisqually, WA 6.80 45.30 66.50 347.17 Nisqually_1421a_a-200 Subcrustal 19 3.32 3.33 Nisqually, WA 6.80 28.40 60.59 312.00 Nisqually_1437a_a-270 Subcrustal 20 1.93 2.09 Olympia, WA 6.90 28.49 74.70 485.51 Olympia_OLY0A-356 Subcrustal 21 3.19 3.45 Shonshu, Japan 6.40 41.02 65.45 501.42 SHonshu_EHM0050103241528-EW Subcrustal 22 1.42 1.53 Shonshu, Japan 6.40 33.77 61.17 560.58 SHonshu_EHM0080103241528-EW  54  Table 5.5 Suite of selected subduction ground motions Source No. Scale Factor Event M Epicentral Distance (km) Closest Distance (km) Vs30 (m/s) Horizontal 1 Filename Building A Building B Subduction 23 4.01 3.89 Hokkaido, Japan 8.00 221.98 225.92 542.18 Hokkaido_HKD0390309260450-EW Subduction 24 2.63 2.76 Hokkaido, Japan 8.00 224.90 164.08 648.89 Hokkaido_HKD0540309260450-EW Subduction 25 3.10 3.20 Hokkaido, Japan 8.00 182.26 123.35 512.21 Hokkaido_HKD0930309260450-EW Subduction 26 2.23 2.30 Hokkaido, Japan 8.00 183.55 188.29 384.05 Hokkaido_HKD1040309260450-EW Subduction 27 4.53 4.34 Hokkaido, Japan 8.00 80.95 91.20 460.82 Hokkaido_HKD1110309260450-EW Subduction 28 3.62 3.53 Hokkaido, Japan 8.00 264.34 267.66 455.12 Hokkaido_HKD1180309260450-EW Subduction 29 3.15 3.09 Hokkaido, Japan 8.00 221.56 160.86 602.51 Hokkaido_HKD1270309260450-EW Subduction 30 2.93 2.86 Tohoku, Japan 9.00 409.95 212.96 378.84 Tohoku_CHB0221103111446-EW Subduction 31 3.61 3.43 Tohoku, Japan 9.00 383.05 186.25 409.63 Tohoku_GNM0081103111446-EW Subduction 32 3.21 3.19 Tohoku, Japan 9.00 427.81 230.71 618.97 Tohoku_KNG0051103111446-EW Subduction 33 3.00 2.93 Tohoku, Japan 9.00 400.65 203.72 411.33 Tohoku_TKY0061103111446-EW  55   (a)  (b)  (c) Figure 5.9 Ground motion selection for Building A (a) crustal; (b) subcrustal; (c) subduction 56   (a)  (b)  (c) Figure 5.10 Ground motion selection for Building B (a) crustal; (b) subcrustal; (c) subduction 57  Chapter 6: Results of Nonlinear Time History Analysis  6.1 Structural Response without Accounting for the Buckling Failure of LRB The nonlinear time history analyses were conducted based on the selected ground motions. The axial demands of LRBs from the analysis, where the buckling effect was not considered, were gathered. The results were calculated in the form of mean plus standard deviation of the peak response for all ground motions. Table 6.1 shows the average demand (mean plus standard deviation) for all isolators obtained from the nonlinear time history analyses over the design axial force presented in Table 4.7. The results show that the demand from dynamic analysis is close to the design axial load calculated using the CSM. This indicates that the static procedure as presented in Session 2.1 can be used to efficiently estimate the axial demand as compared with the nonlinear time history analysis when bilinear model is used. However, when the buckling behavior of LRB is explicitly modeled, as indicated in the next sub-session, the LRB designed using bilinear model (without accounting the buckling behavior) has high probability of failure. This could result to difficult structural repair and hefty repair costs.  Table 6.1 Ratio of axial load demand from CSM and time history analyses Building A Building B Tp = 3Tf Tp = 4Tf Tp = 5Tf Tp = 1.5Tf Tp = 2Tf Tp = 2.5Tf 1.04 0.99 0.97 1.09 0.97 0.93   58  6.2 Structural Response Accounting for the Buckling Failure of LRB During the nonlinear time history analyses, the instant critical buckling load capacity of the LRBs were calculated and compared with the real time axial demands in the LRBs. If the compressive force of any LRB exceeded the critical buckling load, LRB failure is recognized. Figure 6.1 illustrates the time series when the capacity (buckling load) and the demand (axial force) crossed, and the failed LRB recorded in the shear and axial hysteresis. The result clearly shows that the buckling load of the LRB decreased with the increase of lateral displacement.   Failure of the isolated building is defined when the first isolator failed. The probability of failure is defined as the ratio of the number of failure events over the 33 ground motions included in this study. Error! Reference source not found. show the probability of failure of the base isolated buildings, when the axial capacity is calculated using the amplification factor times the design loads presented in Table 4.7 (which can be calculated using CSM shown in Chapter 3 or directly from time history analysis, when the LRB is modeled using bilinear model). The result shows if the capacity of the isolator, at the maximum total displacement, is calculated using bilinear model, the base isolated building could have high probability of collapse (35% to 55% probability of collapse for Building A, and 30% to 35% probability of collapse for Building B). Besides, as the isolation period increases, the probability of collapse generally reduces. In this study, 10% was assumed as the acceptable collapse probability for practical design. In this case, the result shows an amplification factor of 2.5 is needed to ensure that both prototype buildings have acceptable probability of collapse. One interesting finding is that to achieve the same performance objective, under the same amplification factor, the LRBs designed with thinner single rubber layer thickness had higher probability of collapse. 59   (a)  (b)   (c) (d) Figure 6.1 Illustration of LRB failure recognition (a) time series of absolute shear displacement; (b) time series of axial load and buckling load; (c) shear hysteresis; (d) axial hysteresis 60        Figure 6.2 Amplification factor vs probability of failure for each period 61  Figure 6.3 shows the percentage of the failure from each of the seismic hazard source. The result shows that, for the short building (Building A), all the three hazard sources make significant impact for the probability of collapse. As the isolation period increases, the influence of the crustal and subcrustal earthquake decreases. As for the tall buildings (Building B), only the subduction source earthquake makes a significant impact.  Building A – 3Tf Building A – 4Tf Building A – 5Tf    Building B – 1.5Tf Building B – 2Tf Building B – 2.5Tf    Figure 6.3 Failure distribution of crustal, subcrustal, and subduction earthquakes  6.3 Response of Safe Model As the increasing of the amplification factor applied in the design of LRB, the probability of failure reduces accordingly. When there is no failure recognized in the model, it refers to a “safe model”. Crus.9%Subc.28%Subd.63%Crus.3%Subc.7%Subd.90%Crus.0%Subc.2%Subd.98%Crus.0%Subc.1%Subd.99%Crus.0%Subc.0%Subd.100%Crus.0%Subc.0%Subd.100%62  The dynamic response of the superstructures is presented to demonstrate the effectiveness of the design procedure.   The inter-story drift ratio of the superstructure for Building A and B is plotted in Figure 6.4 and Figure 6.5, respectively. The EBF link shear is shown in Figure 6.6. For the models with the same period, the results of models with different single rubber layer thickness are extremely close to each other, hence only the results for tr = 3mm are presented. Both the inter-story drift and the link shear were calculated using the mean plus standard deviation of the peak responses. The building with longer period has smaller response because the base shear decreases with the increasing of isolation period. The EBF links are found to remain elastic for both Building A and Building B, thus satisfy the requirement of NBCC [19].       (a) (b) Figure 6.4 Inter-story drift ratio of Building A (a) X direction; (b) Y direction 63      (a) (b) Figure 6.5 Inter-story drift ratio of Building B (a) X direction; (b) Y direction    (a) (b) Figure 6.6 Link shear (a) Building A; (b) Building B  64  Chapter 7: Summaries and Conclusions  7.1 Conclusions LRB, as a commonly-used seismic isolation system, has been implemented worldwide. As shown by previous research, LRB is vulnerable to buckling when subjected to combined large axial and shear loads. Although detailed buckling behavior of the LRB has been well investigated, the seismic performance of the base isolated building with LRB has not been systematically studied. In this study, the probability of failure for two prototype buildings, each with different LRB geometric properties and axial loads, was systematically examined. State-of-the-art LRB buckling model implemented in OpenSees was used to check the probability of failure of the prototype buildings under a range of earthquake shaking intensities. Based on the detailed nonlinear time history analyses, the following conclusions are drawn: • As shown in the nonlinear time history analysis results, when the LRB is modeled as bilinear model, the axial demand from dynamic analysis is close to the estimated demand calculated using equivalent static force procedure presented in Session 2. • If the axial demand of the LRB is obtained directly from the equivalent static design load or time history analysis using bilinear model, the base isolated building could have high probability of failure.  • As the isolation period increases, the probability of failure decreases accordingly.  • A simple amplification factor of 2.5 is proposed to increase the required axial capacity of the LRB (when the demand is calculated from the bilinear model) so as to ensure the probability of failure of the isolated building is within 10%. 65  • Subduction earthquakes dominate the failure of isolated buildings according to the analysis. For sites that are influenced by subduction hazard such as Vancouver, it is crucial to include subduction ground motions in the analyses.  7.2 Future Study The recommendations for future study are listed as follows: • The same procedure in this research can be repeated in case study of buildings with different height, and buildings with irregular shape, which will expand the spectrum and provide more comprehensive recommendations on LRB design. • The failure criteria used in this study is simplified and conservative, a more realistic and detailed failure criterion can be proposed in the future study. • The future study may also utilize the robust LRB model in OpenSees to investigate the effect of other LRB properties such as nonlinear tension behavior and lead degradation.     66  Bibliography    [1] G. P. Warn and K. L. Ryan, "A review of seismic isolation for buildings: historical development and research needs," Buildings, vol. 2, pp. 300-325, 2012. [2] W. Robinson and A. 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Rogers, "Fifth generation seismic hazard model input files as proposed to produce values for the 2015 National Building Code of Canada," Geol. Surv. Canada Open File, vol. 7576, p. 15, 2014. [45] T. I. Allen, J. Adams, and S. Halchuk, "The seismic hazard model for Canada: Past, present and future," presented at the Proceedings of the Tenth Pacific Conference on Earthquake Engineering, 2015. [46] G. M. Atkinson and J. Adams, "Ground motion prediction equations for application to the 2015 Canadian national seismic hazard maps," Canadian Journal of Civil Engineering, vol. 40, pp. 988-998, 2013. [47] PEER. NGA-West2 -- Shallow Crustal Earthquakes in Active Tectonic Regimes. 2013.  [48] A. Bebamzadeh, C. Ventura, and M. Fairhurst. S2GM: Tool for Selection and Scaling of Ground Motions. 2015.       71  Appendices  Appendix A  LRB Geometric Properties Table A.1 LRB geometric properties (Building A – 3Tf – tr = 3mm)  Building A - 3Tf - tr = 3mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.142 0.282 8 1.15 0.100 0.292 11 1.12 0.071 0.295 12 1.14 0.142 0.295 9 1.31 0.100 0.303 12 1.26 0.071 0.306 13 1.28 0.142 0.319 11 1.61 0.100 0.325 14 1.52 0.071 0.328 15 1.56 0.142 0.331 12 1.76 0.100 0.345 16 1.79 0.071 0.348 17 1.84 0.142 0.353 14 2.05 0.100 0.364 18 2.04 0.071 0.367 19 2.12 0.142 0.373 16 2.32 0.100 0.383 20 2.29 0.071 0.376 20 2.26 0.142 0.393 18 2.58 0.100 0.400 22 2.53 0.071 0.394 22 2.53 0.142 0.411 20 2.83 0.100 0.417 24 2.77 0.071 0.411 24 2.80 0.142 0.429 22 3.06 0.100 0.441 27 3.11 0.071 0.427 26 3.07 0.142 0.446 24 3.29 0.100 0.456 29 3.33 0.071 0.443 28 3.33 0.142 0.462 26 3.51 0.100 0.471 31 3.55 0.071 0.458 30 3.58 0.142 0.486 29 3.83 0.100 0.485 33 3.76 0.071 0.473 32 3.83 0.142 0.501 31 4.03 0.100 0.506 36 4.06 0.071 0.487 34 4.08 0.142 0.523 34 4.32 0.100 0.519 38 4.26 0.071 0.501 36 4.32         72  Table A.2 LRB geometric properties (Building A – 3Tf – tr = 9mm) Building A - 3Tf - tr = 9mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.142 0.393 6 1.22 0.100 0.392 7 1.07 0.071 0.411 8 1.16 0.142 0.420 7 1.46 0.100 0.417 8 1.27 0.071 0.435 9 1.36 0.142 0.446 8 1.70 0.100 0.463 10 1.68 0.071 0.458 10 1.56 0.142 0.470 9 1.94 0.100 0.485 11 1.89 0.071 0.480 11 1.76 0.142 0.493 10 2.18 0.100 0.506 12 2.09 0.071 0.521 13 2.17 0.142 0.515 11 2.42 0.100 0.525 13 2.30 0.071 0.540 14 2.37 0.142 0.537 12 2.66 0.100 0.545 14 2.50 0.071 0.559 15 2.58 0.142 0.557 13 2.89 0.100 0.581 16 2.91 0.071 0.577 16 2.78 0.142 0.577 14 3.12 0.100 0.598 17 3.11 0.071 0.611 18 3.19 0.142 0.596 15 3.35 0.100 0.615 18 3.31 0.071 0.628 19 3.39 0.142 0.614 16 3.57 0.100 0.632 19 3.51 0.071 0.644 20 3.59 0.142 0.632 17 3.79 0.100 0.663 21 3.90 0.071 0.659 21 3.80 0.142 0.650 18 4.01 0.100 0.678 22 4.09 0.071 0.690 23 4.20 0.142 0.683 20 4.44 0.100 0.693 23 4.28 0.071 0.704 24 4.40               73  Table A.3 LRB geometric properties (Building A – 3Tf – tr = 15mm) Building A - 3Tf - tr = 15mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.142 0.454 5 1.14 0.100 0.463 6 1.06 0.071 0.494 7 1.18 0.142 0.493 6 1.42 0.100 0.499 7 1.29 0.071 0.527 8 1.40 0.142 0.530 7 1.69 0.100 0.532 8 1.52 0.071 0.559 9 1.62 0.142 0.564 8 1.98 0.100 0.593 10 1.98 0.071 0.588 10 1.84 0.142 0.596 9 2.26 0.100 0.621 11 2.22 0.071 0.617 11 2.06 0.142 0.596 9 2.26 0.100 0.648 12 2.45 0.071 0.644 12 2.29 0.142 0.626 10 2.54 0.100 0.673 13 2.69 0.071 0.670 13 2.52 0.142 0.655 11 2.82 0.100 0.698 14 2.92 0.071 0.719 15 2.97 0.142 0.683 12 3.09 0.100 0.722 15 3.16 0.071 0.742 16 3.20 0.142 0.710 13 3.37 0.100 0.745 16 3.39 0.071 0.765 17 3.43 0.142 0.736 14 3.64 0.100 0.768 17 3.62 0.071 0.787 18 3.66 0.142 0.760 15 3.91 0.100 0.790 18 3.85 0.071 0.808 19 3.89 0.142 0.784 16 4.18 0.100 0.811 19 4.09 0.071 0.829 20 4.11 0.142 0.808 17 4.45 0.100 0.832 20 4.31 0.071 0.849 21 4.34  Table A.4 LRB geometric properties (Building A – 4Tf – tr = 3mm) Building A - 4Tf - tr = 3mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.118 0.401 35 1.03 0.084 0.401 40 1.03 0.059 0.391 39 1.03 0.118 0.436 42 1.28 0.084 0.433 47 1.28 0.059 0.420 45 1.28 0.118 0.468 49 1.52 0.084 0.463 54 1.53 0.059 0.446 51 1.52 0.118 0.503 57 1.78 0.084 0.491 61 1.76 0.059 0.471 57 1.76 0.118 0.535 65 2.02 0.084 0.521 69 2.01 0.059 0.499 64 2.04 0.118 0.565 73 2.25 0.084 0.550 77 2.26 0.059 0.522 70 2.27 0.118 0.601 83 2.52 0.084 0.580 86 2.52 0.059 0.547 77 2.53 0.118 0.632 92 2.75 0.084 0.609 95 2.77 0.059 0.571 84 2.78    74  Table A.5 LRB geometric properties (Building A – 4Tf – tr = 9mm) Building A - 4Tf - tr = 9mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.118 0.503 19 1.04 0.084 0.510 22 1.02 0.059 0.518 23 1.04 0.118 0.550 23 1.32 0.084 0.553 26 1.27 0.059 0.561 27 1.30 0.118 0.583 26 1.53 0.084 0.594 30 1.53 0.059 0.591 30 1.50 0.118 0.625 30 1.81 0.084 0.631 34 1.78 0.059 0.628 34 1.76 0.118 0.654 33 2.01 0.084 0.667 38 2.03 0.059 0.664 38 2.03 0.118 0.692 37 2.27 0.084 0.700 42 2.28 0.059 0.698 42 2.29 0.118 0.727 41 2.53 0.084 0.732 46 2.52 0.059 0.730 46 2.55 0.118 0.761 45 2.77 0.084 0.763 50 2.76 0.059 0.761 50 2.81  Table A.6 LRB geometric properties (Building A – 4Tf – tr = 15mm) Building A - 4Tf - tr = 15mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.118 0.591 16 1.07 0.084 0.594 18 1.00 0.059 0.607 19 1.02 0.118 0.642 19 1.33 0.084 0.655 22 1.30 0.059 0.667 23 1.32 0.118 0.673 21 1.50 0.084 0.698 25 1.53 0.059 0.709 26 1.54 0.118 0.718 24 1.76 0.084 0.738 28 1.75 0.059 0.748 29 1.77 0.118 0.761 27 2.01 0.084 0.788 32 2.06 0.059 0.798 33 2.07 0.118 0.801 30 2.27 0.084 0.824 35 2.28 0.059 0.833 36 2.29 0.118 0.839 33 2.51 0.084 0.858 38 2.50 0.059 0.867 39 2.52 0.118 0.876 36 2.76 0.084 0.901 42 2.80 0.059 0.910 43 2.82        75  Table A.7 LRB geometric properties (Building A – 5Tf – tr = 3mm) Building A - 5Tf - tr = 3mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.107 0.531 101 1.00 0.075 0.516 106 1.01 0.053 0.493 98 1.01 0.107 0.584 123 1.25 0.075 0.561 126 1.26 0.053 0.532 114 1.26 0.107 0.636 147 1.51 0.075 0.605 147 1.50 0.053 0.570 131 1.51 0.107 0.687 172 1.76 0.075 0.650 170 1.76 0.053 0.605 148 1.76 0.107 0.738 199 2.01 0.075 0.692 193 2.00 0.053 0.641 166 2.01 0.107 0.789 228 2.26 0.075 0.735 218 2.25 0.053 0.674 184 2.26  Table A.8 LRB geometric properties (Building A – 5Tf – tr = 9mm) Building A - 5Tf - tr = 9mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.107 0.630 48 1.00 0.075 0.635 54 1.02 0.053 0.633 54 1.02 0.107 0.691 58 1.27 0.075 0.691 64 1.27 0.053 0.683 63 1.27 0.107 0.747 68 1.52 0.075 0.742 74 1.52 0.053 0.730 72 1.51 0.107 0.799 78 1.77 0.075 0.790 84 1.76 0.053 0.774 81 1.75 0.107 0.848 88 2.00 0.075 0.835 94 2.00 0.053 0.820 91 2.02 0.107 0.898 99 2.25 0.075 0.883 105 2.26 0.053 0.860 100 2.26  Table A.9 LRB geometric properties (Building A – 5Tf – tr = 15mm) Building A - 5Tf - tr = 15mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.107 0.721 38 1.01 0.075 0.730 43 1.00 0.053 0.737 44 1.01 0.107 0.792 46 1.28 0.075 0.803 52 1.28 0.053 0.801 52 1.26 0.107 0.849 53 1.51 0.075 0.862 60 1.52 0.053 0.860 60 1.51 0.107 0.910 61 1.77 0.075 0.917 68 1.77 0.053 0.915 68 1.77 0.107 0.967 69 2.03 0.075 0.969 76 2.01 0.053 0.967 76 2.02 0.107 1.021 77 2.28 0.075 1.024 85 2.28 0.053 1.017 84 2.27 76  Table A.10 LRB geometric properties (Building B – 1.5Tf – tr = 3mm) Building B - 1.5Tf - tr = 3mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.204 0.396 9 1.08 0.144 0.457 16 1.03 0.102 0.483 19 1.00 0.204 0.427 11 1.30 0.144 0.505 20 1.30 0.102 0.540 24 1.31 0.204 0.456 13 1.50 0.144 0.550 24 1.54 0.102 0.582 28 1.54 0.204 0.496 16 1.79 0.144 0.591 28 1.78 0.102 0.621 32 1.76 0.204 0.534 19 2.05 0.144 0.638 33 2.05 0.102 0.666 37 2.01 0.204 0.568 22 2.28 0.144 0.674 37 2.25 0.102 0.709 42 2.26 0.204 0.601 25 2.51 0.144 0.724 43 2.54 0.102 0.757 48 2.54 0.204 0.642 29 2.79 0.144 0.764 48 2.76 0.102 0.795 53 2.76 0.204 0.681 33 3.05 0.144 0.809 54 3.01 0.102 0.838 59 3.01 0.204 0.718 37 3.29 0.144 0.851 60 3.25 0.102 0.879 65 3.25  Table A.11 LRB geometric properties (Building B – 1.5Tf – tr = 9mm) Building B - 1.5Tf - tr = 9mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.204 0.521 6 1.03 0.144 0.610 10 1.04 0.102 0.658 12 1.00 0.204 0.590 8 1.42 0.144 0.665 12 1.28 0.102 0.733 15 1.30 0.204 0.622 9 1.60 0.144 0.716 14 1.52 0.102 0.780 17 1.50 0.204 0.652 10 1.79 0.144 0.764 16 1.76 0.102 0.845 20 1.80 0.204 0.709 12 2.15 0.144 0.830 19 2.10 0.102 0.905 23 2.09 0.204 0.735 13 2.32 0.144 0.871 21 2.32 0.102 0.943 25 2.28 0.204 0.786 15 2.66 0.144 0.911 23 2.53 0.102 0.998 28 2.56 0.204 0.810 16 2.82 0.144 0.967 26 2.84 0.102 1.049 31 2.83 0.204 0.856 18 3.14 0.144 1.003 28 3.04 0.102 1.082 33 3.01 0.204 0.878 19 3.29 0.144 1.054 31 3.33 0.102 1.130 36 3.27     77  Table A.12 LRB geometric properties (Building B – 1.5Tf – tr = 15mm) Building B - 1.5Tf - tr = 15mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.204 0.652 6 1.20 0.144 0.740 9 1.10 0.102 0.809 11 1.07 0.204 0.699 7 1.43 0.144 0.816 11 1.39 0.102 0.879 13 1.30 0.204 0.744 8 1.66 0.144 0.851 12 1.53 0.102 0.943 15 1.53 0.204 0.786 9 1.88 0.144 0.917 14 1.81 0.102 1.003 17 1.76 0.204 0.825 10 2.10 0.144 0.979 16 2.09 0.102 1.087 20 2.10 0.204 0.863 11 2.32 0.144 1.037 18 2.37 0.102 1.140 22 2.33 0.204 0.899 12 2.54 0.144 1.065 19 2.50 0.102 1.190 24 2.55 0.204 0.934 13 2.75 0.144 1.119 21 2.77 0.102 1.239 26 2.77 0.204 1.000 15 3.17 0.144 1.170 23 3.03 0.102 1.308 29 3.10 0.204 1.032 16 3.37 0.144 1.219 25 3.28 0.102 1.352 31 3.31  Table A.13 LRB geometric properties (Building B – 2Tf – tr = 3mm) Building B - 2Tf - tr = 3mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.177 0.565 40 1.01 0.125 0.620 56 1.00 0.089 0.635 60 1.00 0.177 0.625 50 1.25 0.125 0.691 70 1.25 0.089 0.704 74 1.26 0.177 0.691 62 1.51 0.125 0.763 86 1.51 0.089 0.771 89 1.51 0.177 0.751 74 1.75 0.125 0.830 102 1.75 0.089 0.837 105 1.76 0.177 0.815 88 2.01 0.125 0.898 120 2.00 0.089 0.901 122 2.01       78  Table A.14 LRB geometric properties (Building B – 2Tf – tr = 9mm) Building B - 2Tf - tr = 9mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.177 0.696 21 1.03 0.125 0.780 30 1.03 0.089 0.825 34 1.01 0.177 0.756 25 1.25 0.125 0.853 36 1.27 0.089 0.905 41 1.25 0.177 0.824 30 1.52 0.125 0.931 43 1.53 0.089 0.988 49 1.53 0.177 0.887 35 1.76 0.125 1.002 50 1.78 0.089 1.056 56 1.76 0.177 0.946 40 2.00 0.125 1.069 57 2.02 0.089 1.129 64 2.01  Table A.15 LRB geometric properties (Building B – 2Tf – tr = 15mm) Building B - 2Tf - tr = 15mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.177 0.802 17 1.03 0.125 0.898 24 1.00 0.089 0.982 29 1.02 0.177 0.887 21 1.31 0.125 1.002 30 1.29 0.089 1.078 35 1.27 0.177 0.946 24 1.52 0.125 1.081 35 1.53 0.089 1.166 41 1.52 0.177 1.019 28 1.78 0.125 1.155 40 1.77 0.089 1.248 47 1.77 0.177 1.087 32 2.04 0.125 1.238 46 2.04 0.089 1.325 53 2.01  Table A.16 LRB geometric properties (Building B – 2.5Tf – tr = 3mm) Building B - 2.5Tf - tr = 3mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.155 0.756 119 1.00 0.110 0.793 146 1.00 0.077 0.786 145 1.01 0.155 0.851 152 1.25 0.110 0.890 185 1.25 0.077 0.875 180 1.25 0.155 0.946 189 1.50 0.110 0.987 228 1.50 0.077 0.962 218 1.50 0.155 1.041 230 1.76 0.110 1.085 276 1.75 0.077 1.050 260 1.75 0.155 1.134 274 2.00 0.110 1.182 328 2.00 0.077 1.139 306 2.01   79  Table A.17 LRB geometric properties (Building B – 2.5Tf – tr = 9mm) Building B - 2.5Tf - tr = 9mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.155 0.869 53 1.01 0.110 0.941 69 1.01 0.077 0.977 75 1.01 0.155 0.967 66 1.26 0.110 1.043 85 1.26 0.077 1.082 92 1.26 0.155 1.056 79 1.50 0.110 1.142 102 1.51 0.077 1.177 109 1.51 0.155 1.150 94 1.76 0.110 1.237 120 1.76 0.077 1.265 126 1.75 0.155 1.242 110 2.01 0.110 1.331 139 2.01 0.077 1.357 145 2.00  Table A.18 LRB geometric properties (Building B – 2.5Tf – tr = 15mm) Building B - 2.5Tf - tr = 15mm Type A Type B Type C D1 D2 nr Amplification factor D1 D2 nr Amplification factor D1 D2 nr Amplification factor [m] [m] [-] [-] [m] [m] [-] [-] [m] [m] [-] [-] 0.155 0.984 41 1.01 0.110 1.083 55 1.02 0.077 1.146 62 1.01 0.155 1.084 50 1.25 0.110 1.194 67 1.27 0.077 1.260 75 1.26 0.155 1.185 60 1.51 0.110 1.296 79 1.50 0.077 1.373 89 1.51 0.155 1.279 70 1.76 0.110 1.398 92 1.75 0.077 1.476 103 1.77 0.155 1.374 81 2.02 0.110 1.499 106 2.01 0.077 1.573 117 2.01  

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