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Seismic performance of post-and-beam wood buildings Li, Minghao 2009

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SEISMIC PERFORMANCE OF POST-AND-BEAM WOOD BUILDINGS by Minghao Li B. A. Sc., Tongji University, China, 2000 M. A. Sc., Tongji University, China, 2003 A THESIS SUBMITTED TN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Forestry) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) January 2009 © Minghao Li, 2009 ABSTRACT This thesis presents a study to evaluate the seismic performance of post-and-beam (P&B) wood assemblies and buildings of Japanese style using computer modeling, experimental studies and probabilistic-based approaches. A numerical model called “PB3D” is proposed to predict the lateral response of the P&B buildings under static or dynamic loads. Special techniques are used to reduce the problem size and improve computational efficiency with reasonable prediction accuracy. This model simplifies a P&B building into a combination of 2D assemblies (e.g. shear walls, floor/roof diaphragms) while capturing the global structural responses of interest (e.g., inter-story drift and floor/roof acceleration). A mechanics-based wood shear wall model is implemented to represent the hysteretic properties of symmetric/nonsymmetric P&B walls. Roof/floor diaphragms are modeled as structural frames with calibrated equivalent diagonal braces in order to consider the influence of the diaphragm in-plane stiffness on the building performance. Experimental studies have been conducted to study the behavior of 2D assemblies and buildings. The engineering characteristics of single-brace P&B walls have been evaluated by monotonic and reversed cyclic tests. The contribution of additional gypsum wallboards to the wall lateral resistance has also been studied. An in-plane pushover test has been conducted to study the in-plane stiffness of a floor diaphragm. Two one-story P&B buildings have been tested under biaxial static loads and one-directional seismic 11 loads, respectively. The established test database as well as a test database of a two-story P&B building provided by a research institute in Japan has been used to verify the “PB3D” model. Using the response surface method with importance sampling and considering the uncertainties involved in seismic ground motions, structural mass, and response surface fitting errors, seismic reliability analyses have been conducted to estimate the seismic reliabilities of a series of shear walls, a one-story building and a two-story building. System effect on the shear wall reliability has also been studied. The framework presented in this thesis provides a useful tool to assess the seismic performance of the P&B wood buildings and to aid the performance-based seismic design of these structural systems. 111 TABLE OF CONTENTS ABSTRACT.ii TABLE OF CONTENTS iv LIST OF TABLES vii LIST OF FIGURES ix ACKNOWLEDGEMENT xiv CHAPTER 1 INTRODUCTION 1 1.1 Problem description 1 1.2 Objectives and organization 4 CHAPTER 2 BACKGROUND AND LITERATURE REVIEW 7 2.1 Introduction 7 2.2 Computer modeling 7 2.2.1 Shear wall models 8 2.2.1.1 Nonlinear shear springs 8 2.2.1.2 Finite element wall models 11 2.2.2 Building models 12 2.3 Experimental studies 16 2.3.1 Shear wall and diaphragm tests 17 2.3.2 Building tests 20 2.4 Current code guidelines for seismic design 22 2.5 Reliability-based seismic analysis 29 2.5.1 Seismic fragility analysis 29 2.5.2 Fully coupled seismic reliability analysis 31 2.6 Summary 33 CHAPTER 3 FORMULATION OF POST-AND-BEAM BUILDING MODEL - “PB3D” 35 3.1 Introduction 35 3.2 Formulation of structural assemblies 37 3.2.1 “Pseudo-nail” shear wall model 37 3.2.2 Equivalent diagonal-braced roof/floor diaphragms 44 3.3 Coordinate transformation of elements in 3D space 50 3.4 Formulation of static “PB3D” model 52 3.4.1 Principle of virtual displacement 53 3.4.2 Formulation of internal & external virtual work 54 3.4.3 Formulation and solution of a building system 56 iv 3.5 Formulation of dynamic “PB3D” model .60 3.5.1 Formulation of mass matrix and damping matrix 61 3.5.2 Time-stepping method - Newmark’s integration 63 3.6 Summary 65 CHAPTER 4 EXPERIMENTAL STUDIES ON SHEAR WALLS AND FLOOR DIAPHRAGM 66 4.1 Introduction 66 4.2 Diagonal-braced shear wall test 67 4.2.1 Specimens and materials 67 4.2.2 Loading setup and instrumentation 69 4.2.3 Test observations and failure modes 72 4.2.4 Test results and discussions 78 4.3 Floor diaphragm test 86 4.3.1 Specimens and materials 86 4.3.2 Loading setup and instrumentation 88 4.3.3 Test program 92 4.3.3.1 Phase 1 92 4.3.3.2Phase2 93 4.3.3.3Phase3 94 4.3.3.4Phase4 94 4.3.4 Test results and discussions 95 4.4 Summary 99 CHAPTER 5 EXPERIMENTAL STUDIES ON ONE-STORY BUILDINGS 101 5.1 Introduction 101 5.2 Test specimens and materials 101 5.3 Static test 105 5.3.1 Loading setup and instrumentation 105 5.3.2 Test observations and results 108 5.4 Dynamic test 113 5.4.1 Instrumentation and measurement 113 5.4.2 Test program 116 5.4.3 Test results and discussions 118 5.4.3.1 Hammer impact tests 118 5.4.3.2 Square-wave impulse tests 119 5.4.3.3 Shake table tests 121 5.5 Summary 129 CHAPTER 6 “PB3D” MODEL VERFICATION 132 6.1 Introduction 132 6.2 Model verification by one-storey buildings 132 V 6.2.1 Static model verification 136 6.2.2 Dynamic model verification 138 6.3 Model verification by a two-storey building 141 6.3.1 Building description 141 6.3.2 Dynamic model verification 146 6.4 Summary 158 CHAPTER 7 SEISMIC RELIABLITY ANALYSIS OF POST-AND-BEAM SHEAR WALLS AND BUILDINGS 160 7.1 Introduction 160 7.2 Earthquake hazards 161 7.3 Response surface method with importance sampling 164 7.4 Approaches to seismic reliability estimation 171 7.5 Seismic reliability analyses of P&B shear walls 172 7.5.1 Shear wall descriptions 173 7.5.2 Seismic reliability of shear walls 180 7.6 Seismic reliability analyses of P&B buildings 196 7.6.1 Seismic reliability of a one-story building 196 7.6.2 Seismic reliability of a two-story building 202 7.6.3 System effect on seismic reliability of shear walls in buildings 209 7.6.3.1 A single-brace wall in the one-story building 210 7.6.3.2 A double-brace wall in the two-story building 211 7.7 Summary 214 CHAPTER 8 CONCLUSIONS AND FUTURE RESEARCH 216 8.1 Conclusions 216 8.2 Future research 220 BIBLIOGRAPHY 223 vi LIST OF TABLES Table 2.1 Multiplier of required length of effective shear walls in BSL (cm/rn2) 27 Table 4.1 Material list of single-brace wall with GWB sheathing 68 Table 4.2 Single-brace wall test programs 71 Table 4.3 Progressive failure observations of bare-framed walls 75 Table 4.4 Progressive failure observations of GWB-sheathed walls 75 Table 4.5 Summary of test results of bare-framed walls 84 Table 4.6 Summary of test results of GWB-sheathed walls 84 Table 4.7 Material list of the floor diaphragm 88 Table 4.8 Test phases with associated floor configurations 92 Table 4.9 Summary of the floor diaphragm test results 97 Table 5.1 List of wood members in the one-story building 105 Table 5.2 Test sequence 116 Table 5.3 Summary of hammer impact test results 119 Table 5.4 Summary of square-wave impulse test results 120 Table 6.1 “Pseudo-nail” model parameters of the single-brace wall 133 Table 6.2 Effective shear wall length and eccentricity ratio of the two-story building.. 143 Table 6.3 Types of shear walls and quasi-shear walls used in the two-story building.... 143 Table 6.4 “Pseudo-nail” model parameters of the walls in two-story building 147 Table 7.1 Test result summary of cyclic tests of 1 .82m walls 176 Table 7.2. “Pseudo-nail” parameters of single-brace 0.91m walls 177 Table 7.3 “Pseudo-nail” model parameters of l.82m walls 177 Table 7.4 Historical earthquake records used for reliability analysis 180 Table 7.5 Mean and standard deviation of peak drift response of 1 .82m long TG-PL wall 185 Table 7.6 RS coefficients of peak wall drift (mean) 189 Table 7.7 RS coefficients of peak wall drift (standard deviation) 189 vii Table 7.8 RS fitting errors of mean and standard deviation of peak drifts 189 Table 7.9 Seismic failure probability of the P&B shear walls (event) 194 Table 7.10 Seismic failure probability of the P&B shear walls (annual, 0.2/year) 194 Table 7.11 Mean and standard deviation of peak inter-story drift of the one-story building 197 Table 7.12 RS coefficients of peak inter-story drift of the one-story building 201 Table 7.13 RS fitting errors of peak inter-story drift of the one-story building 201 Table 7.14 Seismic failure probability of one-story building (event) 202 Table 7.15 Seismic failure probability of one-story building (annual, 0.2/year) 202 Table 7.16 Stmctural mass levels of the two-story building 203 Table 7.17 Mean and standard deviation of peak inter-story drift of the two-story building 203 Table 7.18 RS coefficients of peak inter-story drift of the two-story building 207 Table 7.19 RS fitting errors of peak inter-story drift of the two-story building 207 Table 7.20 Seismic failure probability of the two-story building (event) 208 Table 7.21 Seismic failure probability of the two-story building (annual, 0.2/year) 209 Table 7.22 Seismic failure probability of one-story building and shear wall (event) 211 Table 7.23 Summary of the wall response and the wall carried equivalent mass 212 Table 7.24 Seismic failure probability of two-story building and shear wall (event) 214 viii LIST OF FIGURES Figure 2.1 Example of a nonlinear shear spring for wood shear walls 9 Figure 3.1 An example of a P&B wood building (with diagonal-braced walls) 36 Figure 3.2 Simulation of a symmetric wood shear wall via a “pseudo-nail” model 38 Figure 3.3 Compression-only embedment properties and the hysteretic rules 39 Figure 3.4 Calibration procedures for a symmetric wood shear wall 41 Figure 3.5 Typical shear walls in P&B buildings 42 Figure 3.6 Simulation of a nonsymmetric wood shear wall via a “pseudo-nail” model .. 43 Figure 3.7 Calibration input curve for a nonsymmetric wood shear wall 43 Figure 3.8 An example of a P&B floor diaphragm 45 Figure 3.9 Coordinate system of a 3D beam element 47 Figure 3.10 Schematic of the beam element under torsion 47 Figure 3.11 Global coordinate system and element local coordinate system 51 Figure 3.12 Schematic of a “PB3D” model 53 Figure 3.13 Modified Newton-Raphson iterations 58 Figure 4.1 Configuration of a single-brace wall 69 Figure 4.2 Elevation of loading set up and transducer locations 70 Figure 4.3 Wall positions and loading types 72 Figure 4.4 Reversed cyclic test protocol 72 Figure 4.5 Out-of-plane buckling of the diagonal brace 74 Figure 4.6a Failure mode A: nail withdrawal from BP plate 76 Figure 4.6b Failure mode B: complete pull out of nails from BP plate 76 Figure 4.6c Failure mode C: nails broken due to low cycle fatigue failure 76 Figure 4.6d Failure mode D: Nails punching through GWB 77 Figure 4.6e Failure mode E: GWB detached from wood frame 77 Figure 4.6f Failure mode F: GWB nails broken due to low cycle fatigue failure 77 ix Figure 4.6g Failure mode G: wood splitting in sill/top plate (perp. to grain tens.) 78 Figure 4.6h Failure mode H: wood crushing in sill/top plate (perp. to grain comp.) 78 Figure 4.7a Load-drift curve of Wall A-i 79 Figure 4.7b Load-drift curve of Wall A-2 79 Figure 4.7c Load-drift curve of Wall A-3 80 Figure 4.7d Load-drift curve of Wall A-4 80 Figure 4.7e Load-drift curve of Wall B-i 80 Figure 4.7f Load-drift curve of Wall B-2 81 Figure 4.7g Load-drift curve of Wall B-3 81 Figure 4.7h Load-drift curve of Wall B-4 81 Figure 4.8a Method of defining yield strength and stiffness 82 Figure 4.8b Method of defining ultimate strength P 83 Figure 4.9 Energy dissipation in the reversed cyclic tests 85 Figure 4.10 Schematic of the floor diaphragm 88 Figure 4.11 Schematic of loading and transducer definitions along y direction 89 Figure 4.12 Schematic of loading and transducer definitions along x direction 90 Figure 4.13a Test phase 1: in-plane pushing on bare-framed floor along y direction 93 Figure 4.13b Test phase 2: in-plane pushing on partially sheathed floor along y direction 93 Figure 4.1 3c Test phase 3: in-plane pushing on fully sheathed floor along y direction ... 94 Figure 4.13d Test phase 4: in-plane pushing on fully sheathed floor along x direction ... 95 Figure 4.14a Test phase 1 — load vs. deformation curves 96 Figure 4.14b Test phase 2— load vs. deformation curves 96 Figure 4.14c Test phase 3 — load vs. deformation curves 96 Figure 4.14d Test phase 4 — load vs. deformation curves 97 Figure 5.1 a Exterior elevation of long sides (east and west) 103 Figure 5.1 b Exterior elevation of short side (south) 103 x Figure 5.1 c Exterior elevations of short side (north) 104 Figure 5.2 Shear wall layout of the one-story building 104 Figure 5.3 The one-story P&B building in the static test 106 Figure 5.4 Actuator loads along two horizontal directions of the building 107 Figure 5.5 Locations and definitions of sensors in the static test 108 Figure 5.6a Failure modes at diagonal brace connection 109 Figure 5 .6b Nail withdrawal at diagonal brace connection on top corner of the wall 110 Figure 5.7a Deformation curves of diagonal brace connections of east and west walls.. 111 Figure 5.7b Deformation curves of diagonal brace connections of north and south walls 111 Figure 5.8 Wall load-drift curves under the biaxial pushover loads 113 Figure 5.9 Twisting of the roof diaphragm (lOx deformation) 113 Figure 5.10 The one-story P&B building in the shake table test 114 Figure 5.11 Locations and definitions of sensors in the shake table test 115 Figure 5.12 Hammer impact locations on the top plates of east and south walls 117 Figure 5.13 An example of acceleration response in hammer impact test 119 Figure 5.14 An example of acceleration response in the square-wave impulse test 120 Figure 5.15 Fourier amplitude spectrum of the acceleration response 121 Figure 5.16 Table acceleration input with scaled PGA of 0.069g 122 Figure 5.17 Acceleration response of roof (north end and south end) 122 Figure 5.1 8a Table acceleration input in KOBE 1 OONS- 1 test 123 Figure 5.18b Table velocity input in KOBE100NS-1 test 123 Figure 5.1 8c Table displacement input in KOBE 1 OONS- 1 test 124 Figure 5.19 Fourier amplitude spectrum of input accelerogram in KOBE 1 OONS- 1 test 124 Figure 5.20 Acceleration response spectrum of table input (5% damping ratio) 125 Figure 5.21 Slide responses of diagonal brace connections in KOBE100NS-1 test 126 Figure 5.22 Acceleration responses of roof diaphragm in KOBE100NS-1 test 126 xi Figure 5.23 Drift responses of north wall and south wall in KOBE100NS-1 test 127 Figure 5.24 Structural damage in KOBE100NS-2 test 128 Figure 5.25 Slide responses of diagonal brace connections in KOBE100NS-2 test 128 Figure 5.26 Roof acceleration responses in KOBE 1 OONS-2 test 129 Figure 5.27 Drift responses of north and south walls in KOBE100NS-2 test 129 Figure 6.1 “PB3D” model of the tested single-story building 133 Figure 6.2 “Pseudo-nail” model predicted ioops vs. test results 134 Figure 6.3 “Pseudo-nail” model predicted energy dissipation vs. test results 134 Figure 6.4 Calibration of equivalent braces of the roof using ANSYS software 136 Figure 6.5 “PB3D” model predicted wall response against test results 138 Figure 6.6 Model predicted story drift responses vs. test results 139 Figure 6.7 Model predicted roof acceleration responses vs. test results 140 Figure 6.8 Fourier amplitude spectra of roof acceleration response (one-story building) 140 Figure 6.9 Shake table test on a two-story P&B building 141 Figure 6.10 Shear wall layout of the two-story building 142 Figure 6.11 Table input accelerogram in shake table test of the two-story building 145 Figure 6.12 “PB3D” model of the two-story P&B building 146 Figure 6.13 Model predicted base shear force responses vs test results 149 Figure 6.14 Model predicted floor acceleration responses vs test results 150 Figure 6.15 Model predicted roof acceleration responses vs test results 151 Figure 6.16 Definitions of sensors to measure inter-story drifts (DG1DG6, DG9DG14) 152 Figure 6.17 Inter-story drift response of the first story (DG1DG6) 154 Figure 6.18 Inter-story drift response of the second story (DG9DGl4) 156 Figure 6.19 Fourier amplitude spectra of floor acceleration responses (two-story building) 157 xli Figure 6.20 Fourier amplitude spectra of roof acceleration responses (two-story building) 157 Figure 7.1 Configuration of the 1 .82m long wall frame 175 Figure 7.2 1.82 m long double-brace, OSB-sheathed and plywood-sheathed walls 175 Figure 7.3 Cyclic test results vs “pseudo-nail” model predictions (0.91m walls) 177 Figure 7.4 Cyclic test results vs “pseudo-nail” model predictions (1.82 m walls) 179 Figure 7.5 Earthquake records used in the seismic reliability analysis 183 Figure 7.6 IDA curves of 1 .82m long TG-PL wall 186 Figure 7.7 Confidence curves of 1 .82m long TG-PL wall 188 Figure 7.8 Response surface fitted data vs model simulation data (0.91 m walls) 190 Figure 7.9 Response surface fitted data vs model simulation data (1.82 m walls) 193 Figure 7.10 IDA curves under five mass levels of one-story building 199 Figure 7.11 Confident curves of peak inter-story drift of the one-story building 200 Figure 7.12 Response surface fitted data vs model simulation data (one-story building) 201 Figure 7.13 IDA curves under five mass levels of two-story building 205 Figure 7.14 Drift confident curves of the two-story building under five mass levels .... 207 Figure 7.15 Response surface fitted data vs model simulation data (two-story building) 208 Figure 7.16 Location of the 1.82 m long double-brace wall in two-story building 212 xli’ ACKNOWLEDGEMENT I would like to express my gratitude to my research supervisor, Dr. Frank Lam for his academic guidance and financial support. Without his help, I would not have been able to accomplish my Ph.D. degree. His advice, encouragement, and patience through the past five years have helped me to develop research skills in timber engineering. A special thank goes to Dr. Ricardo 0. Foschi for his kind help and advice. He was always available for discussions and for sharing his knowledge. Also, I would like to thank Dr. J. David Barrett for his general guidance in timber engineering and Dr. Minjuan He for leading me to this research field. A special thank also goes to Dr. Shiro Nakajima from Building Research Institute of Japan and Mr. Minoru Okabe from Center for Better Living of Japan for providing the test database and technical information about the Japanese building systems. Finally I would also like to thank my friends and colleagues in the Timber Engineering and Applied Mechanics group, Larry Tong, George Lee, Fluijun Yan, Dr. Dominggus Yawalata, Dr. James Gu, Xiaobin Song, Dr. J. P. Hong, Hiba Anastas and others. xiv To my wife, Zehui Liu my son, Ryan Li and my parents, Boqing Li and Meihua Shen xv CHAPTER 1 INTRODUCTION 1.1 Problem description Influenced by highly developed temple architecture introduced from China in the seventh century, Japanese dwelling has been based on traditional post-and-beam (P&B) timber structural systems for a long history due to the extraordinary merits of wood (e.g., high strength-to-weight ratio, good constructability, reliable perfonnance, wood resource availability and renewability, comfort, and architectural aesthetics). Nowadays, employing modem construction and manufacturing technologies, P&B structures are still the most common structural types for low-rise residential and commercial buildings in Japan. Similar to the lightframe systems in North America, the Japanese P&B buildings are also box-type structures consisting of two-dimensional horizontal and vertical assemblies such as walls, floors, ceilings and roofs. They usually employ timber members with relatively larger cross sections for wall posts and roof/floor beams compared with the lightframe systems. The uniqueness of the P&B buildings is the presence of traditional mortise-and-tenon joinery reinforced by specially designed metal hardware. Nail connections are also widely used to connect framing members and sheathing panels thus enhancing the overall structural integrity and ductility. As to lateral resistance, conventional P&B buildings adopted plastered mud walls with very low lateral capacity, but modem P&B buildings commonly adopt diagonal-braced walls or structural-panel- sheathed walls with much enhanced capacity by metal fasteners. Although wood structures have a good reputation of resisting earthquake loads or wind loads, their structural integrity during extreme natural disasters is not necessarily guaranteed. Over a long history, the structural safety of the P&B buildings against severe earthquakes is always a great concern for the Japanese people. According to the plate- tectonic theory, Japan is located along a triple plate conjunction of the Pacific, Philippine, and Eurasian plates. At this site, the Pacific plate is sliding under the Eurasian plate forming a deep-sea trench. This subduction activity gives rise to the frequent occurrences of hazardous shallow earthquakes. Despite the awareness of this situation, under unanticipated earthquakes with high intensity, the Japanese people still suffered huge property and life losses due to the inadequate seismic capacity of structures. For example, in the 1995 Hyogo-ken Nanbu (Kobe) earthquake, a large number of traditional P&B buildings in the densely-populated downtown and suburban Kobe were partially destroyed or even collapsed. The reason for the poor structural performance lies in some design related issues and the deterioration of building components. Under the severe ground shaking, large inertial forces were imposed onto the buildings carrying heavy gravity loads. Structural insufficiencies, such as weak first stories, large openings in walls, irregular layout of wall systems, and inadequate provision of inter-story and foundation anchorage, greatly weakened the lateral capacities of the buildings (Prion and Filiatrault, 1996; Arima, 1998). Poor quality maintenance of the buildings was another main culprit where decay in the structural components greatly reduced the lateral resistance. Therefore, 2 the earthquake experience in Japan raised substantial concerns about the safety of the P&B buildings in future earthquakes. Significant efforts are needed to develop a knowledge base of their seismic performance. In the last two decades, research on wood buildings has covered many topics with various levels of complexity. Special attention has been directed to studying the seismic performance of shear walls which are believed to play the most important role to provide the lateral resistance of wood buildings. Experimental studies have been conducted on wood members, assemblies and building systems. Meanwhile, a variety of computer models have also been developed to simulate the structural response of wood systems under different loading scenarios. However, for complicated wood buildings, it is still a challenging job to develop a robust and versatile computer model with good computational efficiency and prediction accuracy. In earthquake-prone regions, codes and standards are required to guide structural designs. Most of current seismic design codes aim to achieve a balance between economy and safety by prescribing the minimum requirements with the intention to prevent structural collapse and life losses in major earthquakes. The force-based seismic design approach is widely used in the building codes such as Building Standard Law of Japan (2000), International Building Code (2003), and National Building Code of Canada (2005). The basic idea is to calculate the equivalent base shear by multiplying dead load, spectral response acceleration, and a series of modification factors. Then the total lateral force is distributed among the different building stories. These lateral forces are resisted 3 by roof, floor diaphragms and shear walls of individual stories and then transferred to foundation by anchorage. Although the force-based approach is convenient for practical design purposes, it does not holistically reflect the real behavior of a building under seismic loads. Multiple performance expectations corresponding to different seismic hazard levels are not generally addressed. For example, the 1994 Northridge earthquake in California caused few casualties and collapses of wood buildings. However, the insured property loss was reported as high as US$ 10 billion due to severe structural and nonstructural damages. Therefore, it is of great interest to reassess the seismic safety levels of wood buildings and introduce the performance-based design philosophy to satisfy the multiple performance expectations. 1.2 Objectives and organization Most of earthquakes are unpredictable and their intensities cannot be controlled. But the devastating action of earthquakes on structures can be mitigated. Fortunately, knowledge in seismological science and building engineering enables us to understand what is required of an engineering structure to achieve good seismic performance. Recent earthquake disasters in Japan have raised the level of safety concerns among occupants, builders, designers and product suppliers. More efforts are needed to prepare for and mitigate earthquake risks with a long term strategy. Canada is one of the important timber product suppliers to Japan and occupies a significant market share in the building construction market. It is of special interest for Canadian forest product industry and researchers to get involved in the collaborative research on the performance of 4 Canadian timber products used in the Japanese building systems. In recent years, scientific liaisons have been established between the University of British Columbia and the Building Research Institute in Japan. The objective of the collaboration is to achieve a better understanding of the seismic performance of the Japanese P&B buildings and to provide a knowledge-based tool to help design and construct safer buildings. As part of the collaborative research, this study focuses on computer modeling and laboratory testing of the P&B shear walls, diaphragms, and buildings to assess the seismic performance of the P&B buildings systematically. A three-dimensional finite element model - “PB3D” has been developed to predict the lateral response of the P&B buildings subject to static or dynamic loads. To improve computational efficiency, this model incorporates simplified floor/roof diaphragms and shear walls while maintaining reasonable prediction accuracy. The roof/floor diaphragms are modeled as framing systems with calibrated equivalent diagonal braces to consider the in-plane stiffness, thus taking into account the influence of the diaphragm flexibility on the load-sharing among the walls and diaphragms. A versatile mechanics-based “pseudo-nail” wall model is used to represent the hysteretic characteristics of wood shear walls. Experimental studies have been carried out to study the behavior of P&B shear walls, a floor diaphragm and full scale buildings. The established test database is used to verify the “PB3D” model. In the last part, reliability-based seismic analyses considering the uncertainties arising from earthquake ground motions, structural carried mass and response surface fitting errors were performed to assess the seismic reliability of a series of shear walls and two 5 buildings with respect to multiple performance expectations. In brief, this thesis consists of the following chapters: 1) background and literature survey; 2) formulation of the P&B building model — “PB3D”; 3) experimental studies on shear walls and a floor diaphragm; 4) experimental studies on one-story buildings; 5) “PB3D” model verification; and 6) seismic reliability analyses on shear walls and buildings; and 7) conclusions and future research. 6 CHAPTER 2 BACKGROUND AND LITERATURE REVIEW 2.1 Introduction Seismic design should reflect a proper understanding of structural characteristics and the nature of earthquake loads. However, it is very formidable to evaluate the seismic performance of wood structures due to the inherent complexity of the structural systems and earthquake ground motions. Such complexity involves the inherent variation of wood properties, structural geometries and configurations, construction quality and the state of maintenance. And, earthquake ground motions are highly unpredictable due to a series of intervening influential factors, such as earthquake source mechanism, source-to-site transmission path properties, epicentral distance, and site conditions. Knowledge in structural dynamics and earthquake engineering needs to be well understood and integrated in order to build rational wood structures with good seismic performance. In recent years, especially since the 1994 Northridge earthquake and the 1995 Kobe earthquake, researchers have conducted a great deal of research to study the seismic performance of wood structures via experimental studies and computer modeling. Meanwhile, codes and regulations have been updated with newly obtained knowledge and design philosophy. This chapter will present a review of the related research. 2.2 Computer modeling A variety of computer models, simplified or detailed, have been developed to study the structural behavior of wood systems on a component level or a system level. When a 7 wood building is subject to seismic loads, horizontal inertia forces need to be transferred safely and effectively by structural components to foundation via a continuous load transfer path. Within this load path, shear walls function as the most important links to provide the lateral resistance. Therefore, a proper representation of shear wall characteristics is essential to accurately capture the seismic response of wood buildings. 2.2.1 Shear wall models In wood buildings, a shear wall can be generally defined as an in-plane or plate-type structural unit designed to transmit force in its own plane. Two types of shear walls (structural-panel-sheathed walls and diagonal-braced walls) are commonly used in modern wood buildings. A lot of efforts have been made to study the engineering characteristics of shear walls such as strength, stiffness, ductility, hysteresis, and energy dissipation. A series of wood shear wall models have been proposed to simulate the wall behavior. Summary of some previous research on wood shear wall modeling can also be found in the theses of Dolan (1989), Durham (1998), He (2002) and Gu (2006). 2.2.1.1 Nonlinear shear springs Nonlinear shear springs can be used to represent the global load-drift behavior of specific wood shear walls (Ewing et al., 1980; Kivell et al., 1981; Stewart, 1987; Sakamoto and Ohashi, 1988; Kamiya, 1988; Ceccotti and Vignoli, 1990; Kawai, 1998; and, Folz and Filiatrault, 2004a). They are essentially single degree-of-freedom (DOF) hysteresis models with good computational efficiency since a shear wall is simplified to a single DOF spring element. The load-deformation relationship of a nonlinear shear spring, 8 represented by some mathematical formulas, is purely a phenomenon-based description of the hysteresis of a shear wall. The mathematical formulas, such as piecewise linear equations or exponential equations, can be explicitly defined by fitting load-drift curves from a shear wall test or a detailed shear wall modeling result. Meanwhile, the loading and unloading paths of the shear spring under hysteretic loads need to be defined explicitly. Folz and Filiatrault (2001) proposed a single DOF lightframe shear wall model with a load-drift backbone curve represented by Eq. (2.1) which was originally proposed by Foschi (1977). Six unknown parameters F0,K0,i , , 6 need to be fitted (Fig. 2.1). Under cyclic loading, an additional set of parameters (zero-displacement intercept F1, degraded stiffness K, and stiffness parameters r3 and r4) need to be fitted. sgnô .(Fo+r1Kost).[l—exp(—K0I/ )] sgn 6 F + r2K0 6— sgn(5). 5,, I (2.1) 0 8IHSFI Fo Figure 2.1 Example of a nonlinear shear spring for wood shear walls 9 Another interesting nonlinear shear spring model is the Bouc-Wen-Baber-Noori model (BWBN) (Baber and Wen 1981, Baber and Noon, 1986). This model can simulate the hysteresis of a shear wall by using a set of differential equations instead of the explicit piecewise or exponential functions. A total of 13 parameters need to be fitted. Strength and stifihess degradation as well as the pinching effect can be represented by this model. For specific wood shear walls, these nonlinear spring models can be fitted in a good agreement with test results. However, their applications have some limitations due to the lack of versatility. For example, if the loading protocol changes, the fitted model may lose its validity and the model parameters need to be recalibrated. It is well recognized that in reversed cyclic tests, the shear wall response highly depends on test protocols (He et al., 1998, Gatto and Uang, 2003). When a shear wall is subject to a long sequence of demanding loading cycles, the major failure mode tends to be the low cycle fatigue failure of nail connections. However, this failure mode is rarely observed in real earthquakes. He et al. (1998) thus proposed a new cyclic protocol with a substantially shortened loading sequence to closely represent real earthquake conditions. It was shown that the new cyclic protocol could effectively reflect shear wall performance under seismic loads since the failure modes were more consistent with the real seismic behavior. Therefore, for the empirically fitted shear spring models, it is advisable to select an appropriate protocol which can represent a wide range of the real earthquake situations. The hysteresis from a wood shear wall test and the hysteresis from a nail connection test show similarities in strengthlstiffness degradation and pinching effect, due to the interaction between the metal fasteners and the surrounding wood medium, as well as the formation of gaps (Dolan and Madsen, 1992a, 1992b). It is well known that the hysteretic behavior of a wood shear wall is dominated by the behavior of connections which involves the nonlinear characteristics of wood under compression and elasto-plastic metal fasteners. Therefore, Gu and Lam (2004) proposed a wood shear wall model, called “pseudo-nail” model, to represent the lateral behavior of a wood shear wall via a single nail acting upon a nonlinear compression-only embedment medium. The hysteretic algorithm that permits the calculation of a single nail behavior was adapted to represent the wall behavior. Details about this shear wall analog model will be introduced in Chapter 3 of this thesis. 2.2.1.2 Finite element wall models To extend the versatility of wood shear wall models, researchers also proposed some numerical models of varying complexities. van de Lindt (2004) enumerated a series of numerical woodframe models introduced from 1982 to 2001. These models paid special attention to the proper implementation of sheathing-to-framing connections since these connections normally govern the behavior of woodframe walls. Foschi (1977) developed a FE wood diaphragm model in which plate elements were used to model sheathing panels, beam elements were used to model frame members, and nonlinear spring elements with calibrated exponential force-displacement relationship were used to model nail connections. Foschi (2000a, 2000b) improved the nail connection model by developing a mechanics-based model to simulate the response of 11 metal fasteners interacting with wood medium. This model was then implemented into the PBWALL program (Foschi, 2005) to calculate the static and dynamic response of the Japanese P&B shear walls. Itani and Cheung (1984) developed a FE model to study wood diaphragms. Nail connections along sheathing panels were modeled as smeared nonlinear springs. Dolan (1989) developed a detailed FE model to simulate panel-sheathed shear walls. Beam and plate elements were used to model framing members and sheathing panels. Bilinear springs were used to model the connections between framing members. Bilinear contact elements were used to model the gaps between adjacent sheathing panels. 3D nonlinear springs were used to model panel-frame nail connections. Kasal and Leichti. (1992) developed a static FE model for lightframe walls using the commercial software package ANSYS. Wall Studs and sheathing panels were modeled as coupled shell elements. Nonlinear characteristics of nail connections were lumped at common nodes of sheathing panels and wall studs and were represented by 3D nonlinear springs. The properties of the springs were calibrated by the test results. Richard et al. (2003) also developed a FE lightframe shear wall model in which nail connections were modeled as nonlinear springs with the backbone force-displacement curve as shown in Figure 2.1. The loading and unloading rules were modified into exponential curves instead of multi-linear curves. 2.2.2 Building models To understand the structural performance of a wood building on a system level, 12 researchers developed a variety of building models using different modeling techniques. For nonlinear dynamic problems of wood buildings, simulation accuracy and computational efficiency are of primary importance for a viable and reliable wood building model. “Pancake” models are commonly used to model the seismic response of buildings through a degenerate 2D planar analysis. A “pancake” model simplifies a building into a planar pancake system with floor and roof diaphragms superimposed on top of each other. Nonlinear shear springs, representing shear walls, are then implemented between floor and roof diaphragms. If rigid roof/floor diaphragms are assumed, only two translational degrees-of-freedom (DOFs) and one rotational DOF are taken into account for each diaphragm level. By this means, the system DOFs have been greatly reduced and computational efficiency can be signifciantly improved. It should be noted that “pancake” models neglect many structural details with the intention to capture the global response of buildings (e.g., inter-story drifts, inter-story shear forces, roof/floor accelerations). The vertical loading effect is usually not considered. This is acceptable when only lateral response of buildings are calculated under seismic loads. Folz and Filliatrault (2004a) developed a “pancake” model with the assumption of rigid diaphragms to simulate the seismic response of a two-story lightframe building tested under the CUREE-Caltech woodframe project. Oyamada et a!. (2004) also developed a “pancake” model with the assumption of rigid diaphragms to study the effect of torsional oscillation in the Japanese wood buildings. In these two “pancake” models, 13 nonlinear single DOF shear springs fitted with different mathematical curves were used to represent the shear walls. For a “pancake” model with rigid diaphragms, the structural response is dominated by the underlying shear wall system. In contrast, considering the in-plane stiffness of roof/floor diaphragm, a “pancake” model can be refined with floor/roof diaphragms modeled in a more detailed manner. For example, Filiatrault et a!. (2003) used plane- stress quadrilateral elements with calibrated in-plane stiffness to simulate lightframe floor/roof diaphragms. Additional beam elements with small bending stiffness are used on four floor edges to connect the floor elements and the single DOF shear wall springs. Detailed numerical models have also be developed to simulate the response of wood buildings by modeling individual structural components (e.g., framing members, sheathing panels, metal fasteners, and contacts between sheathing panels). To improve computational efficiency and feasibility, submodelling techniques have been used to reduce the system complexicty. Tarabia and Itani (1997) developed a FE-based 3D model to analyze a one-story lightframe building under seismic loads. The building was idealized as a combination of generic diaphragm components. Connections between the diaphragm components were lumped into equivalent intercomponent springs along two lateral and one withdrawal directions. The diaphragm component was in fact a super-element containing five basic elements: framing elements, sheathing elements, sheathing interface elements, framing connection elements and frame-sheathing connection elements. The springs for the 14 intercomponent connections and the frame-sheathing connections consisted of exponential backbone curves and multi-linear loading/unloading curves. He et al. (2001) and Lam et a!. (2002) developed a detailed FE model - LightFrame3D to predict the static and dynamic response of general lightframe buildings. In this model, a lightframe building is also composed of generic super-elements which consider three basic elements: framing element, panel element and panel-frame connection element. The LightFrame3D model is capable of analyzing lightframe buildings with the following features: single side sheathing or double side sheathing, openings in shear walls, nonlinear nail connections and rigid insulation, in-plane concentrated loads and out-of-plane distributed loads applied simultaneously. A uniqueness of this model is the implementation of the mechanics-based representation of the load-deformation characteristics of panel-frame nail connections developed by Foschi (2000a, 2000b), which is believed to improve the accuracy and robustness of the model. Mosalam et al. (2002) developed a FE model to simulate the seismic response of an asymmetric three-story lightframe building tested in the CUREE-Caltech woodframe project using the software package SAP 2000. Calibrated shell and frame elements were used to model the shear walls and the floor/roof diaphragms. Collins et al. (2005) developed a FE model to study a one-story lightframe building under cyclic loads by using the software package ANSYS. This model used beam elements and shell elements to model linear-elastic framing members and sheathing panels, respectively. The hysteresis of shear walls was modeled by two equivalent 15 diagonal nonlinear single DOF springs (Dolan, 1989, Kasal and Xu, 1997). This model provided a good estimation of the higher order response parameters such as energy dissipation. Tsuda and Miyizawa (2004) developed a 3D FE model using the software package RUAUMOKO to predict the seismic response of a two-story Japanese P&B wood building. This model simplified panel-sheathed shear walls into equivalent diagonal springs with nonlinear characteristics calibrated from shear wall pushover tests. The “Wayne Stewart” degrading hysteretic model was used to represent the diagonal springs. In case of window/door opening, a special nonlinear spring with different tension and compression properties was used to model the connections between the opening lintel and sill with the adjacent wall posts. Although individual wall posts were modeled, it was still difficult to predict properly the axial force response in the wall posts. 2.3 Experimental studies Experimental studies can provide straightforward and valuable information about structural behavior despite of relatively high testing costs. In the U. S., the 1994 Northridge earthquake brought forth a necessity to conduct a series of research projects, such as the CUREE-Caltech woodframe project, to advance the engineering of lightframe structures and to improve the efficiency of construction technology for targeted seismic performance levels (Filiatrault et al., 2000). The CUREE-Caltech woodframe project carried out a series of laboratory tests on shear wall systems, a one-story building, a two story building, and a three-story building. In Japan, since the 1995 Kobe earthquake, a 16 number of research projects have been carried out to study the seismic performance of wood structural systems by establishing the state-of-the-art test facilities and conducting full-scale tests. In Canada, the University of British Columbia also conducted a series of structural tests to study the seismic performance of structural assemblies or buildings in the lightframe systems and the P&B systems. 2.3.1 Shear wall and diaphragm tests van de Lindt (2004) summarized a total of 31 experimental programs from 1983 to 2001, which evaluated the engineering characteristics of lightframe shear walls with different lengths, sheathing types, connectors, and loading protocols. Most of these walls were constructed by 38 x 89 mm lumbers, plywood/OSB panels, and nail connections. A series of 2.44 m x 2.44 m walls were tested by Dolan (1989), Tissell (1993), and Durham et al. (2001). Other walls with larger sizes were tested by Karacabeyli and Ceccotti (1996), Lam et a!. (1997). As part of the CUREE-Caltech woodframe project, Pardoen et al. (2003) tested 52 one-story high walls and four two-story high walls to establish a database of the engineering parameters of common wall systems and to quantifr seismic performance levels of wall surface finishes. A total of 59 lightframe walls with different aspect ratios were tested by APA — The Engineered Wood Association. The test database was used to verify the current methodology to estimate shear wall and diaphragm deformations (Skaggs and Martin, 2004). In Japan, a large number of P&B shear walls have been tested. Sugiyama et a!. (1988a) conducted racking tests on 11 single-brace and double-brace walls of various 17 lengths. The effect of calcium silicate siding boards on shear wall capacity was studied. It was also concluded that the load-carrying capacity of a building is larger than that of a summation of individual walls due to the system effect. Hayashi (1988) also conducted racking tests on conventional P&B walls sheathed with plywood panels. In particular, the wall opening ratio was studied and it was found that the wall stiffness and strength decreased when the opening ratio increased. Kawai (1998) performed monotonic, cyclic and pseudo-dynamic tests on 12 types of panel-sheathed walls sheathed with plywood panels and gypsum wallboards (or siding boards) and six diagonal-braced walls, in order to evaluate the performance of the walls with different lateral resistance mechanisms. Yasumura (2000) conducted static and dynamic tests on 11 plywood-sheathed walls to investigate the racking strength, stiffness as well as the influence of opening, blocking and hold-down devices. It was found that blocking members play an important role to transfer shear forces between the horizontally installed plywood panels. And test results also showed a considerable decrease of strength in the walls with no hold-down devices installed. Yamaguchi et al. (2000) conducted static and dynamic tests on 12 woodframe walls built with Canadian S.P.F. lumber and plywood panels. The test results showed that the force-deformation curves of the walls were affected by the loading rate and the test protocols. It was found that the monotonic, reversed cyclic and pseudo-dynamic tests with slow loading rate showed smaller wall strength than the rapid loading tests, which implied the time-dependent characteristics of nailed shear walls. Center for Better Living of Japan (2001) conducted monotonic and cyclic tests to study the strength, stiffness, 18 hysteretic behavior of a total of 24 double-brace, plywood-sheathed and OSB-sheathed walls constructed with three different wood species. It was found that walls built with Canadian Hem-fir (Tsuga heterophylla) had relatively higher strength and stiffness than the walls built with other two species. Okabe et al. (2002) tested 11 GWB-sheathed walls to evaluate the wall characteristics. The effect of different types of GWBs and metal fasteners on the wall performance was investigated. Komatsu et a!. (2004) conducted a series of cyclic tests to study the lateral performance of a type of P&B walls sheathed with prefabricated small-size mud shear panels, which are easy to use in construction. Shake table tests were carried out by Yamada et a!. (2004) to compare the seismic performance of four types of P&B shear walls: frame-only; plaster-sheathed, double- brace, and plywood-sheathed. Yasumura et a!. (2006) conducted pseudo-dynamic tests on a series of one- and two-story high plywood-panel-sheathed P&B walls. The test results were used to verify a building model. At the University of British Columbia, static and cyclic tests have been conducted to study the lateral performance of three types of 1.82 x 2.73 m Japanese P&B walls: double-brace, four-braced, and OSB-sheathed (Stefansescu, 2000). These walls adopted the traditional Japanese mortise-and-tenon joinery. The test results demonstrated the dependence of the wall behavior on the test protocols. Also it was found that the OSB sheathed walls dissipated much more energy than the braced walls using the same cyclic loading protocols. Very limited experimental work on the in-plane behavior of roof/floor diaphragms 19 has been reported in literature. Phillips et al. (1993) studied how a roof diaphragm affected the lateral load distribution among the shear walls in a one-story lightframe building. Kamiya et al. (1998) tested three P&B floor diaphragms to study the shear force concentration around floor openings and the axial forces in framing members. Simplified equations to determine the shear forces were also introduced. Filiatrault et al. (2002) studied the in-plane stiffness of a floor diaphragm in a two-storey lightframe building. The influence of nail schedules, panel-edge blocking, adjacent walls above and below the floor was investigated. 2.3.2 Building tests In the last decade, most of full-scale wood building tests were performed on shake tables to simulate the seismic response under earthquake loads. In the CUREE-Caltech woodframe project, shake table tests were conducted to study three full-scale lightframe buildings: a simplified box-type one-story building, a two- story single family house, a three-story apartment building (Filiatrault et al. 2000). In Japan, a limited number of wood buildings have been tested in recent years. Aoki et al. (2000) studied the effect of asymmetric arrangement of shear walls in conventional P&B structures. Thirty half-scaled buildings, with the same amount of braced walls but different plan layouts and diaphragm flexibility, were tested statically and dynamically. It was found that the symmetric arrangement of walls as well as the high in-plane stiffness of horizontal diaphragms was conducive to limit the shear wall deformations and reduce the torsional effects of the buildings. 20 Takahiro et al. (2000) conducted shake table tests on eight one-story 3.64 m x 455 m buildings with rigid diaphragms to study the torsional effect with different layouts of shear walls. It was found that high eccentricity ratio caused significantly larger deformations of the walls. Another three buildings with soft diaphragms were also tested to study the effect of diaphragm flexibility on the building performance. Yasumura et al. (2004) reported a pseudo dynamic test on a plywood-sheathed two- story building and a series of plywood-sheathed r walls. The objective was to study the behavior of a shear wall functioning in a building, instead of as an isolated structural unit. Koshihara et al. (2004) and Miyake et al. (2004) reported a series of shake table tests on four full-scale two-story P&B buildings subject to different combinations of the 1995 Kobe (JR Takatori) ground motion components (E-W only; N-S only; combined E-W and N-S; and combined E-W, N-S, and U-D). It was found that the combined ground motions along two horizontal directions imposed larger deformations in the buildings than one- directional ground motion input. It was also found that the initial stiffness of the buildings, the diagonal brace connections, and the bending strength of wall posts also affected whether the building collapsed or not when it was undergoing very large ground motions. Miyazawa (2004) also reported a series of shake table tests on six full-scale two storey buildings tested from 1995 to 2003. Four of them adopted the traditional braced walls or mortared walls while the other two adopted sheathed woodframe walls. The test results indicated that the ordinary wood dwelling houses are able to resist the strong 21 ground motions like the Kobe JMA record and it is able to construct stronger buildings to resist the earthquakes with even higher intensity than the Kobe earthquake. Shimizu et al. (2004) conducted shake table tests of three 5.88 m x5.46 m two-storey P&B buildings with different types of shear walls (e.g., braced walls, plastered mud walls and plywood-sheathed walls). The diagonal brace connection failure and the brace buckling failure were observed in the building with braced walls. The test observations also indicated that diagonal braces might lose the load-carrying capacity at drift ratio of 1/30. Also in the Hyogo Earthquake Engineering Research Center of Japan, some full- scale wood P&B buildings have been tested on the world largest shake table “E-Defense” since 2005. The main objective was to study the seismic performance of the existing conventional P&B buildings built with old building codes and the new buildings built with updated seismic design code. 2.4 Current code guidelines for seismic design The force-based seismic design codes for wood buildings are intended to provide adequate strength under the seismic hazard level associated with the life safety performance expectation. These codes use elastic spectral accelerations to determine the required elastic lateral resistance. The required lateral resistance is then calculated by multiplying reduction factors considering the system ductility and material overstrength, thus allowing inelastic deformations under large earthquake excitations. For example, the International Building Code (2003) stipulates a simplified method 22 to calculate the equivalent seismic force at level x of a lightframe building not exceeding three stories v 1.2SDs w (2.2) where w is the portion of the effective seismic weight of the structure at level x; SDS is the design elastic acceleration at short period; and, R is the response modification factor. In the Building Standard Law of Japan (BSL) (2000), the seismic design force Q of the it/i storey of a n-story building is calculated by Q, =Z•R1 A1 .C0 w, (2.3) where w, is the total weight of ith and the above stories; Z is the seismic zoning coefficient; R, is the design spectral coefficient; A. is the lateral shear distribution factor; and, C0 is the standard shear coefficient (0.2 or higher). Filiatrault and Folz (2002) enumerated some drawbacks of the force-based seismic design for wood buildings: 1) Empirical equations to evaluate fundamental period may not be appropriate for wood buildings; 2) Force reduction factors assigned for wood structure are in fact difficult to justify; 3) No consensus has been reached on the appropriate definition of yield and ultimate displacement for lateral load-resisting systems; 4) The implication that maximum displacement that the corresponding elastic structure would undergo should be equal to the displacement of the actual inelastic structure when the reduction factor R is introduced is not appropriate for wood structures; 23 and, 5) Deformation limit states are not taken into account. Some of these drawbacks are closely related with the truth that wood buildings do not have well-defined yielding points and they demonstrate the nonlinearity over the whole loading history. So, it might be inappropriate to use the concept of “yielding” for wood buildings. Structural deformations might be a better choice for the design criteria since they are closely related to the structural damages caused by seismic loads. The force-based design aims to protect the life safety and cannot guarantee damage controls for wood buildings under multiple seismic hazard levels. For example, in the 1994 Northridge earthquake, a vast majority of wood buildings performed very well from a structural standpoint and there were relatively few structural failures which caused life losses. But most of the financial losses were caused by the non-structural failures. The newly evolving paradigm of performance-based designs provides building designers and owners with flexibility to reduce property and life losses through better defined performance objectives and the corresponding limit states. In the U. S., the seismic design of lightframe wood construction is still on the move to the performance-based engineering. The performance-based design allows the owners to identify an acceptable level of risk for a particular hazard level, and it encourages the designers to develop innovative solutions which meet the expectations of the owners. However, to eventually realize the performance-based seismic design of wood buildings, the wood engineering community must address two important issues: 1) performance expectations; and 2) numerical model improvement (van de Lindt, 2008). 24 The performance expectations must be agreed upon and the correlations between damage descriptors and they should be rigorously developed through testing or experience. Attempt has been made to define such correlations. For example, the U. S. Federal Emergency Management Agency (FEMA) Prestandard and Commentary for Rehabilitation of Buildings provided a framework for the performance-based seismic retrofit of existing structures including woodfrarne buildings (FEMA, 2000). The FEMA Prestandard proposed the connections between performance objectives and seismic hazard levels. It uses the inter-story drift as the damage descriptor for woodframe buildings under seismic loads. For example, 3% inter-storey drift can be used for the indicator of structural collapse; 2% for the indicator of life safety issues; 1% for the indicator of immediate occupancy. Most of current models for wood components, connections, shear walls or roof/floor assemblies are reasonably accurate. As they are integrated into a complex building system, the prediction accuracy may decrease. The finite element based detailed 3D models are usually very time-consuming to perform nonlinear seismic analysis even with the current computational speed. Thus, it is necessary to develop building models with good computational efficiency and reasonable accuracy. The National Building Code of Canada (2005) is the first objective-based building code which comprises objectives, functional statements and acceptable and alternative solutions. Though there is no broadly agreed methodology for the performance-based design, the commonly discussed elements of objectives and functional statements are 25 quite compatible with this objective-based design code. While sharing many characteristics with the performance-based codes, the objective-based code has some differences. For example, the mixture of performance and prescriptive provisions were retained as acceptable solutions. For seismic design, no quantitative performance criteria are currently included. Future developments of the performance-based design in defining the acceptable criteria can also be incorporated in the objective-based design code. The Building Standard Law of Japan (BSL) (2000) also stipulates the type and the level of the required performance for a designed building. It prescribes two seismic performance expectations: 1) to satisfy the life safety requirement with respect to major earthquakes with a 475-year return period; and 2) to satisfy the damage limitation requirement with respect to moderate earthquakes with a return period of 30-50 years (service life). To evaluate the seismic performance of a building, this code employs the response spectrum analysis and the equivalent linearization technique which simplifies a building into an equivalent single DOF system (Kawai et al., 2006). To design a wood building, the BSL prescribes the minimum amount of shear walls. The required amount of shear walls for each story is simply calculated by (2.4) where L is the total effective length of shear walls (in cm), Ai is the floor area (in m2), F is the multiplier of required amount of shear wall length depending on the number of the building stories and the roof type, as given in Table 2.1. For example, in a two-story building with a heavy roof and a floor area of 50 m2, the first story must have at least 26 1650 cm long effective shear walls. Table 2.1 Multiplier of required length of effective shear walls in BSL (cm/rn2) Single- Two-story building Three-story building building 1st story 2nd story 1st story 2nd story 3rd story Light roof 11 29 15 46 34 18 Heavy roof 15 33 21 50 39 24 The effective length of a shear wall is calculated by multiplying the actual wall length and its shear wall multiplier, which is determined by the shear wall load-carrying capacity. The building code specifies different shear wall multipliers for shear walls with different lateral resistance mechanisms. For example, a diagonal-braced wall with one 45 mm x 90 mm brace has a shear wall multiplier of 2.0; a wall with one side sheathed by 7mm thick plywood panels has a shear wall multiplier of 2.5; and a wall with both sides sheathed by 7mm thick plywood panels has a shear wall multiplier of 5.0. A unit shear wall multiplier corresponds to the shear wall capacity of 1.98 kN/m. Designers and building owners can also choose to build a wood building with better seismic performance than the code minimum by increasing the effective shear wall length. In Japan, a so called “Quality Assurance Law” (QAL) can also be used to design a wood building using a completely different structural calculation method than the BSL. In the QAL, a wood building designed according to the BSL can be ranked as a Rank 1 building To design a wood building with higher ranking, such as Rank 2 and Rank 3, more detailed structural calculations than the BSL are required and the interactions between structural components and the level of the structural integrity need to be taken into 27 account. In the BSL, the eccentricity ratios along two horizontal directions of a building (x and y) are calculated by e. e Rex = —- and Rey = X (2.5) rex where and ex are the projected lengths of the distance between the rigidity center S(SX, S) and the gravity center G(Gx, G) on x axis and y axis, respectively, simply calculated by ex =S —G and ey =S,, —G (2.6) And the coordinates of S and G are calculated by (A..x) (A.Y) G = and G = (2.7) in which, A1 is the area of each divided square or rectangular floor segment with coordinates (X, )“) at its geometric center. — (i, . x1) d — (ia, 2 8X — an — Lxi where L1 and L1 are the effective lengths of individual shear walls along x direction and y direction, with coordinate X, or I”1, respectively The equivalent torsion radii rex and rey in Eq. (2.5) are defined by rex = and r = (2.9) where L1 and L, are effective length of individual shear walls along x and y direction, respectively. Kr is the torsional stiffness and is defined by 28 Kr ={LXI .fr, _s )2}+{L .(x, _s)2} (2.10) The building code prescribes that the eccentricity ratio for a wood building should be less than 0.30 since a very unbalanced shear wall layout will lead to significant structural twisting which will greatly increase structural deformations under lateral loads. The code recommends that it is advisable to have the eccentricity ratio of less than 0.15. 2.5 Reliability-based seismic analysis A proper understanding of the uncertainties involved in ground motion characteristics, structural properties, site soil conditions, and construction practice is required to estimate the safety reliability of existing and newly-built structures against seismic hazards. The knowledge of structural dynamics and reliability methods makes it possible to evaluate structural seismic reliabilities with respect to different performance expectations considering many uncertainty sources. In the last decade, however, limited work has been reported on the seismic reliability study of wood shear walls due to the difficulties in characterizing those uncertainties, defining the performance criteria, and developing computer models. Usually, the displacement-based approach is employed to estimate the seismic reliability of wood shear walls since shear wall deformation is closely related to the structural damage levels. Alternatively, the strength-based approach was also used by van de Lindt et al. (2005). 2.5.1 Seismic fragility analysis Particularly, when the influence of one source of uncertainties (e.g., ground motions) is much larger than the other sources of uncertainties, a so-called fragility analysis can be 29 used to uncouple a seismic reliability analysis by separating the structural response from the hazard. In a fully coupled seismic reliability analysis, the general formulation of the failure probability can be described as: Pf = P(LSIM = a)fJM(a)cla (2.11) where P(LSIM = a) is the conditional limit state probability, IM is the seismic intensity measure which can be the spectral acceleration, peak ground acceleration (PGA), and other parameters to characterize the hazard levels; fIM (a) is the probability density function of earthquakes with the intensity measure a. The discrete form of Eq. (2.11) is Pf =P(LSIM=a)PJM(a) (2.12) where JM (a) is the probability of the seismic hazard with the intensity measure a. The conditional limit state probability P(LSIM a) is defined as the “fragility”. One may find from Eq. (2.11) and Eq. (2.12) that the fragility analysis is an essential component of a fully coupled seismic reliability analysis. In case that a fully coupled analysis may not be viable, e.g., when a fully probabilistic description of earthquake hazards is not available, a structural fragility analysis can be used to evaluate the seismic reliability given the specified hazard intensities. This method is still very instructive for regulation makers and code builders. Plus, it is less complicated than a fully coupled reliability analysis since it separates the response analysis from the hazard analysis. Rosowsky and Ellingwood (2002) proposed a fragility analysis methodology to evaluate the probable response of lightframe construction subjected to different natural 30 hazard levels. Rosowsky (2002) reported a study of developing a risk-based (or fragility) methodology to guide the seismic design of 2.44 m x 2.44 m wood frame walls, under the CUREE-Caltech project. Sensitivity studies were also performed to investigate the relative contributions of different uncertainties such as nail connection properties, damping, nail spacing, and the layout of sheathing panels. van de Lindt and Walz (2003) conducted a seismic fragility analysis on 1.22 m x 2.44 m lightframe walls considering the different suites of ground motions on different sites. The effect of uncertainties in the shear wall hysteresis on the resistance side was also investigated. Gu et al. (2006) compared seismic reliabilities of Japanese wood shear walls constructed with three different wood species under three gravity load levels. Confidence curves were also obtained for the comparison purpose. 2.5.2 Fully coupled seismic reliability analysis A fully coupled seismic reliability analysis should consider as many uncertainties as possible. It is supposed to provide the most reliable and comprehensive safety assessment of a structure under seismic loads. Generally, the structural failure probability is best estimated by the Monte Carlo Simulation (MCS) with variance reduction techniques. Usually, the MCS requires a large number of performance function evaluations and is not practicable for the nonlinear dynamic problems of complicated systems. This is a motivation for response representation through response surfaces. With them, MCS, first order reliability method or second order reliability method (FORM/SORM), Importance sampling (IS), local interpolation, etc., can be efficiently used to estimate failure 31 probabilities. The application of the response surface method will be discussed in detail in Chapter 7 of the thesis. Researchers have used fully coupled reliability approaches to estimate the seismic reliability of wood structures. Ceccotti and Foschi (1998) coupled dynamic analyses with FORM to calibrate a seismic design factor of wood shear walls in the National Building Code of Canada. PGA and some of the wall hysteretic parameters were selected as random variables with assigned coefficient of variation (COV) based on the engineering judgment. Foliente (2000) used MCS to estimate the seismic reliability of a series of shear walls. A modified BWBN hysteretic shear wall model was used. The study was conducted for seismic hazards in Tokyo, Japan. Also using MCS and the BWBN model, Paevere and Foliente (2000) found that the hysteretic pinching and stiffness degradation have a significant influence on the peak response of wood shear walls as well as the reliability estimations. Zhang (2003) used the designed experiment and neural network to study the seismic reliability of lightframe walls with respect to multiple performance expectations. The randomness in ground motions, carried mass and nailing spacing were considered. Sjogerg et al. (2004) developed a probability-based software package to assess the reliability of wood structures under seismic loads. MCS was used to get the probabilistic distribution of the dynamic response of single DOF or simple multiple DOF structures subject to a large quantity of randomly generated seismic records. Foschi (2005) also developed a framework to estimate the seismic reliability analysis of the Japanese P&B shear walls using the neural network method. Rosowsky et al. (2005) used MCS to 32 evaluate the strength and reliability of lightframe walls subject to combined axial and transverse loads. Sensitivity studies were also performed to investigate the influence of system parameters on the wall reliability. Wang and Foliente (2006) reported a study on the seismic reliability of low-rise nonsymmetric woodframe buildings. They used the inter-storey drift as the performance criteria to formulate the performance function and a simple power-law formula was developed to represent the drift demand through the spectral displacement of earthquake records. It was found that the uncertainties due to ground motions and structural modeling are the major sources of variations in the estimated structural demand. 2.6 Summary This chapter introduced background information and literature survey on wood shear walls, diaphragms and buildings. The complicated nature of wood systems (e. g., the variability of wood properties, the extensive use of metal fasteners, the system load- sharing effect, and construction methods) and the high uncertainties in loading make it very challenging to predict their structural response accurately and efficiently. Using detailed FE modeling techniques, it is still very difficult to predict accurately the response of individual structural components. However, the structural global response such as roof/floor accelerations, inter-story drifts, and energy dissipations can be captured relatively easily and accurately. For nonlinear dynamic analyses, an accurate representation of the hysteresis of shear walls is of primary importance since the lateral behavior of a wood building is governed by shear walls. Experimental studies on wood 33 shear walls, horizontal diaphragms and buildings were also reviewed. A large portion of the test work focused on studying the strength, stiffness and hysteretic behavior of shear walls. A series of shake table tests on full-scale buildings have been conducted to assess the seismic performance on a system level. Recent advancement of design codes related to the seismic design of wood systems have also been introduced. Performance-based design philosophy is the current trend. Considering the uncertainties in the demand and capacity of wood structures, a reliability- based method is the most appropriate way to evaluate the safety levels of wood buildings under seismic loads. Reliability-based studies on wood shear walls and buildings using different reliability methods have also been reviewed. It is evident that a reliability-based assessment of seismic performance of wood buildings requires a robust computer model with reasonable accuracy and good computational efficiency. One of the objectives of this thesis work is to deliver such a model to fulfill the task of assessing seismic performance of the Japanese P&B wood systems. 34 CHAPTER 3 FORMULATION OF POST-AND-BEAM BUILDING MODEL - “PB3D” 3.1 Introduction The Japanese post-and-beam (P&B) wood buildings are essentially three dimensional frames covered by sheathing materials. Usually, the P&B buildings employ timber members with relatively larger cross sections than the lightframe buildings. For example, 105 mm xl 05 mm members are commonly used as wall posts and sill plates in the P&B systems. And for top plates and roof/floor beams, timber members with even larger cross sections are used. Figure 3.1 shows an example of a two-story P&B building which adopts diagonal-braced walls as the shear walls to provide lateral resistance. The uniqueness about the Japanese P&B buildings is the extensive use of traditional mortise and-tenon joinery to connect framing members. Specially designed metal hardware, such as metal straps or plates, is widely used to reinforce some important connections. Nail connections are also extensively used to fasten framing members and sheathing panels. These features make it very challenging to model the structural response, especially the nonlinear dynamic response, using general structural analysis software. 35 Figure 3.1 An example of a P&B wood building (with diagonal-braced walls) Due to the complicated nature of P&B buildings, the finite element (FE) method was used for structural modeling. A FE model idealizes an actual engineering structure (or a physical model) into a mathematical model which can be effectively formulated and solved by a standard procedure. In general, a more detailed or refined FE model tends to give a more accurate simulation. However, for a complicated structure with a large number of DOFs, a detailed nonlinear FE model is very computationally intensive because it needs to run iterations for each load step; and, a nonlinear time history analysis usually consists of hundreds or thousands of load steps. Therefore, it is necessary to seek a good balance between model complexity, computational efficiency, and simulation accuracy. 36 In this thesis work, a FE model — “PB3D” has been developed to model the structural response of the P&B buildings. The main objective is to predict the global seismic response in terms of inter-storey drifts, inter-story shear forces, roof/floor acceleration with good computational efficiency and reasonable accuracy. Special techniques have been employed to reduce the problem size and improve the computational efficiency while capturing those key characteristics of the seismic response. 3.2 Formulation of structural assemblies The P&B buildings consist of two major subsystems: vertical wall assemblies and horizontal roof/floor assemblies. Accordingly, in the “PB3D” model, a P&B building degenerates into a combination of the horizontal and vertical assemblies. Shear walls are simply represented by a recently developed mechanics-based single DOF hysteresis model which considers the lateral load-drift characteristics only; and for the roof/floor diaphragms, 3D beam elements are used to model framing members and bar elements are used to model the equivalent diagonal braces, accounting for the in-plane stiffness of diaphragms. The contribution of partition walls to the lateral resistance of a building is neglected because they have less structural significance compared with the shear walls. For the same reason, the out-of-plane resistance of shear walls is also neglected. 3.2.1 “Pseudo-nail” shear wall model As mentioned in the literature review, a wood shear wall model, called “pseudo-nail” model has been proposed by Gu and Lam (2004), to represent the lateral behavior of a wood shear wall via a single nail acting upon a nonlinear compression-only embedment 37 medium (Fig. 3.2). The hysteretic algorithm to represent a single nail behavior (Foschi, 2000a, 2000b) was adapted to represent the wall behavior. P(W) Figure 3.2 Simulation of a symmetric wood shear wall via a “pseudo-nail” model In the single nail algorithm (Foschi, 2000a), a nail is modeled by a series of elasto plastic steel beam elements with the specified yielding strength of 250 MPa and the modulus of elasticity of 200 GPa. The surrounding embedment medium is modeled as nonlinear compression-only springs with an exponential force-deformation relationship p(w) (Fig. 3.3 (a)). Five parameters (Qo, Qi, Q2, K, and Dm) are used to define (w), as shown in Eq. (3.1) to Eq. (3.3). JP(W)=(Q0+ QiWX1_e0) W (3 1)1F(w)= T’max2 fW > Dmax where ‘3max = (0 + Q1 Xi — e’°Q0) (3.2) — logO.8 33 [(Q2 - i•0)D ]2 (•) F 4- 4- 4- - 38 P(w) P(W) (a) (b) Figure 3.3 Compression-only embedment properties and the hysteretic rules A set of loading and unloading rules for the embedment medium under cyclic loads are defined as follows. The previously experienced maximum deformation is point P, denoted by (F , W) on the backbone curve (Fig. 3.3 (b)). An associated point D0 is defined as D0 W0 — ] / K, representing the gap size due to the unrecoverable nonlinear deformation of embedment medium. Given a new displacement W, three situations need to be examined: 1) If W D0, no contact occurs between the nail and the embedment medium, and the reaction force (w) = 0; 2) If D0 <W W0, contact starts, but the embedment medium deforms in the linear elastic range with the stiffness K, and the reaction force p(w) = K(W — D0); and 3) If W > W0 , the new displacement exceeds the previous experienced maximum displacement 147, and the reaction force (w) is evaluated on the backbone curve. The maximum displacement W0 needs to be updated to the current W, and the gap size needs to be updated by D0 = W — P / K. P0 0 D Q2Drn w 0 Do Wo W 39 It should be noted that the nail will deform to both sides under hysteretic loads. Thus, W and D0 experienced on either side need to be updated as long as the new displacement exceeds W0 on that side. To model the behavior of a shear wall, the required parameters in the algorithm, such as the embedment medium properties (Qo, Qj, Q2, K, and Dm) and the nail geometry (length L and diameter D), must be relevant to the hysteresis of the wall. Once calibrated for this purpose, the algorithm is said to represent the wall as a “pseudo-nail” shear wall model. The calibration procedure for a shear wall is described as follows. First, a shear wall load-drift curve, which can be obtained from a monotonic test, is required as the input curve. This curve consists of an up ramp and a softening part, up to approximately 80% of the maximum load or lower, followed by a down ramp to zero displacement. A backbone curve from a shear wall reversed cyclic test, spliced with an unloading curve can also be used as the input curve (Fig. 3.4a). Then, using searching algorithms such as random search, hill-climbing, and genetic algorithm (Gu and Lam, 2004; Gu, 2006), the model parameters can be identified by minimizing the squared errors between model predicted load-drift curve and the input curve (Fig. 3.4b). To check the model accuracy, comparison can be made between the model predicted hysteresis and the test results under the same load history (Fig. 3.4c). 40 0 -J Drift (c) Figure 3.4 Calibration procedures for a symmetric wood shear wall In this study, the random search method was used to identify the shear wall model parameters. This method needs to first define the initial upper bound and the lower bound Drift (a) 41 of the parameters. Then all trial solutions within the upper bound and the lower bound are generated randomly. After locating the best solution with smallest squared error, a second random search starts within refined lower bound and upper bound until the best solution is obtained. Diagonal-braced walls and structural-panel-sheathed walls are commonly used in the P&B buildings. As shown in Fig. 3.5, the first one is a single-brace wall which behaves differently depending on whether the diagonal brace is under tension or compression. Nonsymmetric response is expected in its hysteresis ioops. The second one is a double- brace wall. And the last one is a structural-panel-sheathed wall. The latter two types are structurally symmetrical walls. HiNH H’7 11111 ‘øø Figure 3.5 Typical shear walls in P&B buildings The original “pseudo-nail” model can only address symmetric walls by calibrating one set of embedment medium parameters (Qo, Qi, Q2, K, and Dm) and the nail length L and diameter D. To model the hysteresis of nonsymmetric single-brace walls, the original single nail algorithm was then modified in this thesis work. An additional set of embedment medium parameters (Qo, Qi, Q2, K, and D) was introduced to account for 42 P2(W) Figure 3.7 Calibration input curve for a nonsymmetric wood shear wall It should be noted that the current “pseudo-nail” model cannot fully address the observed strength degradation of wood shear walls experiencing large deformations after the maximum load during a set of load cycles. The strength degradation becomes more 43 the different responses when the wall deforms to one side and the other side (Fig. 3.6). Therefore, one full cycle of load-drift curve of a nonsymmetric wall is needed for the model calibration (Fig. 3.7). The two sets of embedment medium parameters and the nail geometry can be calibrated using the same search algorithms. F FH.. JI(i -÷ WA P1(W) —* :4-M4’ —b -* -: —I—i Figure 3.6 Simulation of a nonsymmetric wood shear wall via a “pseudo-nail” model -—-—Test curve —Model calibration input Unloading \ 3 Max load Backbone 80% Max. lobd or lower — Unloading 80% Mm. load Backbone or lower Mm. load Drift apparent when a wood shear wall undergoes large deformations and connections experience significant nonlinear deformations such as nail withdrawal and nail pull- through. The presented shear wall model is an analog of a single nail connection which only takes into account the lateral interactions between the nail and the surrounding wood embedment. This nail withdrawal mode however has not been implemented in the current model. If the “nail” would be allowed to withdraw, the strength degradation can be addressed to predict more accurately the response of wood shear walls with large deformations. This is a limitation in the current model. Compared with the empirically fitting shear wall models, such as the single DOF model presented by Folz and Filiatrault (2001) and the BNBW model, the “pseudo-nail” model has the following advantages: 1) this model is mechanics-based and considers the interactions between wood medium and metal fasteners which usually govern the wood shear wall behavior; 2) the model is independent of loading protocols and is adapted to any loading history; and 3) this model only needs an up-ramp and a down-ramp of load- drift curve as the input for calibration and can predict the shear wall response under cyclic loading or dynamic loading. In Chapter 6 and Chapter 7, this model will be verified by comparing its predictions with the test results from reversed cyclic tests and dynamic tests. 3.2.2 Equivalent diagonal-braced rooflfloor diaphragms Roof/floor diaphragms in the P&B systems are wood frames sheathed by structural panels with a large number of frame-panel nail connections (Fig. 3.8). A special feature of 44 the P&B diaphragms is the installation of wood or metal short braces at the diaphragm corners to strengthen the corner connections and improve the in-plane integrity of the diaphragm. It is well recognized that the in-plane stiffness of horizontal diaphragms will affect the distribution of lateral forces among the underlying walls in a building. If the diaphragm is very rigid, lateral forces will be distributed based on the relative stiffness of walls; if the diaphragm is very soft, lateral forces will be distributed based on the tributary area of individual walls. Therefore, the in-plane stiffness of roof/floor diaphragm will affect the design of shear wall systems and affect the seismic performance of the entire building. _____________ -Corner __ ____ ______________________ brace 1 , J, Figure 3.8 An example of a P&B floor diaphragm In the “PB3D” model, the contribution of floor joists, roof rafters, sheathing panels and panel-frame connections to the in-plane stiffness of roof/floor diaphragms is considered by the equivalent diagonal braces (truss bar elements) with calibrated cross sections. The floor/roof beams with relatively large cross sections are modeled by beam / Sheathing panel — Floor beam a.. Joist [Girth 45 elements. By this means, the number of DOFs in roof/floor diaphragms is significantly reduced. Three-dimensional Euler-Bernoulli beam elements are used to model floor beams, roof purlins, and roof support posts in the diaphragms. The beam element considers normal stresses on the cross section due to axial force and bending moments as well as shear stresses due to beam torsion. The following assumptions are made in formulating the beam element: 1) Sections which are initially plane and normal to the centroidal axis remain so during deformation; 2) Small deformations are assumed, and material is isotropic and follows the Hooke’s law; and 3) The P-A effect is considered due to the bending deflection amplification caused by the axial compressive force. As shown in Fig. 3.9, the beam element has two nodes I and J, and each node has 6 degrees-of-freedom ( u, v, v’, w, w’, q5): u: axial displacement along x direction, the axis of the beam; v: lateral displacement along y direction; v’: rotation around z axis; w: lateral displacement along z direction; w’: rotation around y axis; and b: torsion around the x axis. 46 w V Figure 3.9 Coordinate system of a 3D beam element The normal strain consists of three components: 1) compression or tension due to axial loads; 2) deflection amplification due to compressive loads (P-A effect); and 3) bending along two transverse directions. It is defined by ãu 1(ãw”2 1(8v2 ã2w a2v S I —z————y——— (3.4)X ôx 2 ox,) 2 . Ox,) at2 at2 The shear strain due to longitudinal torsion is defined by = (3.5) where p is the distance from the centroid of cross section; ç is the twist angle which varies along the beam length (Fig. 3.10). This is an approximate torsion formulation for the rectangular cross section because the warping torsion effect is neglected. Figure 3.10 Schematic of the beam element under torsion The material property is assumed to be linear elastic and the stress-strain relationship is: cr=Es (3.6) z dx 47 r—Gy (3.7) The nodal DOF vector of the beam element is: aT = (3.8) The non-zero items in the shape function vectors for translations and rotations are given from Eq. (3.9a) to Eq. (3.12b), in which is the variable in natural coordinate system and Ax is the length of a beam element. 1) Shape function vector of u(x) — N0 N01 =O.5—O.5 (3.9a) N07 0.5 + O.5 (3.9b) 2) Shape function vector of v(x) — L0 L02 =O.5—O.75+O.25 (3.lOa) L03 =O.25(1_4r_gr2 +3) (3.lOb) L08 =O.5+O.75—O.25 (3.lOc) L09 =O.25(_1_+2 +3)- (3.lOd) 3) Shape function vector of w(x) — M0 M04 =O.5—O.75+O.25c (3.lla) M05 =O.25(1__2+3)- (3.llb) M10 =O.5+O.75—O.25 (3.llc) M11 =O.25(_1_+2 +3) (3.lld) 4) Shape function vector of qS(x) — K0 48 K06 =O.5—O.5 (3.12a) K12 =0.5+0.54 (3.12b) Thus, the displacements formulated in the shape function vectors and the nodal DOF vector are: u(x)= Na (3.13a) w(x)= Ma (3.13b) v(x) = La (3.13c) (x)= Ka (3.13d) And u , v , v , w , w , can be easily obtained by calculating the first or second order derivatives of the basic displacement vectors. Du ôN’ 2ãN’ —= a=— a=Na ôx aX AX (3.13e) ôv 2ãL’ = —a = ———a = La ax 8x AX (3.130 — ____ 2 2 a a=La (3.13g)T 2 — ax2 T aw aM 2 6M0 a=Ma (3.13h)—= a=— ax aX Axô — ___ _ 2 N2 a=I—I a=M’a (3.13i) ox2 LAx) a2 (3.13j) Ox Ox Axa Now Eq. (3.4) and Eq. (3.5) can be rewritten as: 49 a =(N —yL _zM+.2aT( iM’ +L1L)a (3.14) r —pKa (3.15) The 3D bar element can be easily formulated since it considers the axial deformation only. The stiffness matrix can be obtained by modifying the stiffness matrix of the beam element to the components related to the axial deformation. 3.3 Coordinate transformation of elements in 3D space The 3D beam elements and bar elements can be arbitrarily oriented in the structural model. Coordinate transformation needs to be carried out in 3D space when the stiffness matrix, load vectors, displacement vectors, and boundary conditions are transformed from the element local coordinate system to the structural global coordinate system, or vice versa. Thus, in the “PB3D” model, the transformation matrices for the elements are constructed. The beam element and the bar element has two nodes I and J with global coordinates (x1,y, , z,) and , , Each node has 6 degrees-of-freedom ( u, v, v’, w, w’, qS). The element stiffness matrix Ke is first evaluated in the element local coordinate system. Multiplied by the transformation matrix T, it is assembled into the global stiffness matrix. The stiffness matrix transformation is described as Kg =TTKeT (3.16) The displacement transformation from the element local coordinate system to the global coordinate system is 50 ag =TTae (3.17) And the transformation of the nodal force vector ‘e to the global vector ‘Pg is ‘Pg =TT’P (3.18) In Eq. (3.16) to Eq. (3.1 8), the coordinate transformation matrix T is T=rT3X3 9 1 (3.19)[ 0 T33J C C (3.19a) _cxz_ cy czz_ where is the direction cosines between a global coordinate axis and a element local coordinate axis. Following the right hand rule, the local coordinate system is defined in Fig. 3.11, in which axis is directed from node I to node J, j axis is perpendicular to both axis and z axis, and axis is perpendicular to both axis and j7 axis. z Figure 3.11 Global coordinate system and element local coordinate system Let i,j,k denote the unit vectors along x, y, z axes in the global coordinate system and ii, V, W denote the unit vectors along , , axes in the element local coordinate system. The relationship between ii, ,W and 1, j, k is shown as follows: ii = C,i + C,j + Ck (3.20a) Cx:? x x 51 v=kxii (3.20b) W=iix (3.20c) where direction cosines C 2 — 2 2 = 1 (3.21 a)(1—)+(y—y1)+(—1) j I 2 2 =m (3.21b) (_1) z. —z. cx = 2 2 =n (3.2lc) +(_,) Then, Eq. (3.20a), Eq. (3.20b) and Eq. (3.20c) can be rewritten as ii 7 =T j (3.22) W k m n where T3 = —- -L 0 (3.23) --- D D D in which D = g12 + m2 (3.23a) 3.4 Formulation of static “PB3D” model By integrating the structural elements for shear walls and roof/floor diaphragms, the “PB3D” model can be formulated following standard FE procedures. This model is essentially a “pancake” model with roof and floor diaphragms superimposed on top of each other (Fig. 3.12). In between, the single DOF “pseudo-nail” shear walls are installed to transfer the lateral loads from the top diaphragm to the bottom diaphragm and finally to the foundation. The bottom nodes of the first-story shear walls are fixed, representing the rigid connections with the foundation. The model does not take into account the 52 structural vertical response since vertical ground motions usually have a much smaller influence on the seismic performance of wood buildings than horizontal ground motions. Past earthquake experiences also showed that vertical ground motions rarely caused the failure of wood buildings. And most of wood buildings are well designed to cany vertical loads. Beam element Brace element “Pseudo-nail” wall Figure 3.12 Schematic of a “PB3D” model 3.4.1 Principle of virtual displacement The fundamental of the displacement-based finite element solution is the principle of virtual displacement, which can be defined as follows: if a structural element or system is in equilibrium, for any virtual small displacements a compatible with the boundary conditions, the total internal virtual work ÔWT is equal to the total external virtual work SWE. This principle can be expresses as: oWE =0 (3.24) 53 The total internal virtual work is = (3.25) in which & is the corresponding virtual strain; is the stress in equilibrium with the applied loads. The total external virtual work is $TfdV+ (3.26) V _ body surftice node in which 5à is a continuous displacement field over the body and is compatible with the system constraints; f8 are body forces; 8a5 are the displacement field c?a evaluated at the surface S; f3 are surface forces; and Sa, the virtual displacement at the node i where a nodal load R is applied. 3.4.2 Formulation of internal & external virtual work The internal virtual work of the 3D Euler-Bernoulli beam element with torsion is (8Wbean? ) = jlcr(s)& + (3.27) where & = (N1 — yL2 — zM2)+6aT(MIMr + LIL) = -yL _zM 4aT(MiM +LiL)1] r(y)= GpKa Eq. (3.27) can be rewritten in term of the shape functions in the element coordinate system: 54 (bearn)i = T[L J(EANNT + EIZL2L+ EIM2M +G1K + T[JEA{(M1M+L1L +i[N1aT +(M1M + L1 aTMiM + LiL)}da (3.28) where L is beam length, A is cross section area, I. and 1. are moments of inertia of the cross section about two neutral axes y and z; J is polar moment of inertia of the cross section, and a is the element nodal displacement vector. The first integral of Eq. (3.28) accounts for the internal virtual work due to axial loading, bending along two major axes on the cross section, and torsion. This term is linear elastic and is independent of the beam deformations. The second integral is displacement-dependant and accounts for the nonlinear P-A effect due to axial loading. For the bar element, the internal virtual work is (bar)j T[LSEANNTd]a (3.29) The internal virtual work in shear walls represented by the “pseudo-nail” models is calculated as follows: (5wj1 =F(a)QT5a (3.30) where F(a) is the shear wall restoring force given the system displacement vector a. It should be noted that the wall restoring force is dependent on the loading history. In the “PB3D” model, the “pseudo-nail” algorithm has been implemented to obtain the shear wall restoring force. QT is the coordinate transformation vector depending on the shear 55 wall orientation. For shear walls along x direction and shear walls along y direction, respectively, the model has Q = (i,o,o,o,o,o,—i,o,o,o,o,o) and Q = (o,i,o,o,o,o,o,—i,o,o,o,o) (3.31) The external virtual work is simply presented by (SWj =ÔàTR (3.32) where R is the external force vector. 3.4.3 Formulation and solution of a building system The principle of virtual displacement for a P&B building can be formulated as (8W.i;,eam)j +(T’Vbar)i +(sw)1—6àTR=O (3.33) Rewriting Eq. (3.33) as (3.34) where ‘I’ is the system out-of-balance force vector, presented by ‘P_Pbearn bar +‘I’ —R (3.35) in which ‘P beam = [ S(EANINIT + EIL2L+ EJM2M + GC1K +[1EA{(MiM +LIL)IN +-[NiaT +(M1M +LiL1aTJMiM (3 .35a) Wbar =[1yEANiNdja (3.35b) =F(a)QT (3.35c) Since &i is arbitrary, to achieve the system equilibrium, the out-of-balance force vector V 56 must vanish, i.e. ‘V = 0 (3.36) In the “PB3D” model, given the system displacement vector, Eq. (3.35a), Eq. (3.35b) and Eq. (3.35c) can be evaluated to obtain the nodal force vectors of the elements. Then, the system out-of-balance force vector can be assembled following the standard FE procedure. During the assembling, special attention has been paid to the coordinate transformation as described in Eq. (3.17) and Eq. (3.18). Given the system displacement vector, the system tangent stiffness matrix [v’i’] can also be constructed by assembling the individual element stiffness matrices. For the beam element: [V’vb]= 8’V =I-J(EAN1NIT +EL,L2L+EIM2M+GJK1K’) +L-JEA{(M1M+L1LI0N+L[N1a+(M1M+L1Li0a](M+L1Lj}d (3.37a) For the bar element: [V’Vbar = = J(EANINI )i (3 .37b) For the single DOF shear wall element: a’VSS = Ô1QT (3.37c) 6a 3a Again, the coordinate transformations also need to be performed, as described in Eq. (3.16) and Eq. (17). To solve for the displacement vector a, the modified Newton-Raphson iteration is in 57 the “PB3D” model (Fig. 3.13). Given the original system displacement vectora0, the system out-of-balance force vector ‘V and the system tangent stiffness matrix[V’V0]can be evaluated. Then, a linear solution is performed and an updated displacement vector a1 can be obtained by a1=a0+[V’V](—W) (3.38) If the convergence criteria are not satisfied, an updated out-of-balance force vector ‘I’ is evaluated with the updated displacement vector while the system stifihess matrix remains unchanged. Then, another linear solution is performed. This iterative procedure continues until the problem converges r-EEEE:::: a0 a12 Figure 3.13 Modified Newton-Raphson iterations For each load step, the integrals in the out-of-balance force vector and tangent stiffness matrix need to be integrated numerically. In the “PB3D” model, Gaussian quadrature is used to fulfill the numerical integration of the polynomial expressions. The integrals in Eq. (3.35) and Eq. (3.37) are expressed in polynomials f(cgr) . One dimensional integral in the natural coordinate system by the Gaussian quadrature can be described as: 58 = (3.39) where , is the location of integration point; n is the order of integration or the number of the Gauss points; W is the weighting factor; f(,) is the polynomial evaluated at Gauss point . In the Gaussian quadrature, n unequally spaced sampling points are required to integrate exactly a polynomial of order at most (2n — i). In the “PB3D” model, the external loads can be applied in the force control mode or the displacement control mode. In the program coding, the imposed nodal displacement is applied as the regular displacement boundary conditions by modifying the corresponding rows in the out-of-balance vector ‘1’ and the stiffness matrix V’I’. However, when the applied force exceeds the structural capacity and the buckling occurs, the stiffness matrix might be ill-conditioned. Therefore, in the “PB3D” model, to obtain the post-peak structural response, one needs to use the displacement control mode. Within each load step, the specified convergence criteria (or tolerances) are checked. The convergence criteria will influence the computational time and accuracy. If the convergence criteria are set too large, the solution might be inaccurate because the errors would be accumulated over the load steps. However, if the convergence criteria are set too strict, the solution might never converge or too much computational effort is needed. The”PB3D” model uses three absolute convergence criteria: (3.40a) L\a1N (3.40b) 59 N”1 N x Na, 8 (3 .40c) where ‘V, is the out-of-balance force vector after the ith iteration within one load step; Aa1 is the incremental displacement vector; and ‘V x Aa1 is the incremental energy. Usually, 8F’ and 5E are small numbers (e.g., 6F =1 N; 5d =0.1 mm; and 5E =0.1 N.mm). The “PB3D” model also uses three relative convergence criteria: NN (3.40d)II”'0N IIAaN (3.40e)Nao N” x (3.40f)N”0 N x Nzao II where ‘V and Aa0 are the initial out-of-balance force vector and the incremental displacement vector of the first iteration in the load step. The relative criteria s, s and 4 can be set a small number such as 0.001. The “PB3D” model requires either of the absolute criteria and the relative criteria to be satisfied. For some problems, it might be difficult to satisfy the three criteria simultaneously. The model relaxes the convergence criteria and requires satisfying any two of the three convergence criteria for a converged solution. 3.5 Formulation of dynamic “PB3D” model The equation of motion of a nonlinear multi-DOF system can be formulated as: Mii+Cá+f(a,ã)=P(t)=—Müg (3.41) 60 where M and C are the mass matrix and the damping matrix; a, a, a the displacement velocity and acceleration vector of the finite element assemblage. Üg is the ground acceleration vector; f(a, a) is the resistant force vector which is a function of deformation history and velocity history. The positive velocity refers to the increasing deformation and the negative velocity refers to the decreasing deformation. The resisting force vector in a nonlinear system is not single-valued and based on the time history of the deformation. 3.5.1 Formulation of mass matrix and damping matrix In the “PB3D” model, the structural carried mass is lumped onto structural nodes based on the tributary area. Thus, the system mass matrix is essentially a diagonal matrix. Since only horizontal ground motions are considered, two components along the horizontal x and y directions are nonzero for the nodes with lumped mass. The mass matrix now becomes m1 m1 Yi M= = (3.42) m, m1 Structural damping is defined as the process by which the vibration of a structure steadily diminishes in amplitude. It reflects the mechanism of energy dissipation in a vibrating system. In structural damping, kinetic energy and strain energy are dissipated by various mechanisms. Usually, when a structure is subject to large seismic loads, most of 61 energy is dissipated by the structural nonlinearity. However, other energy dissipation mechanisms should also be considered in structural modeling because decaying motion is observed even though a structure vibrates in an apparent linear elastic range. Thus, the concept of viscous damping is introduced in the structural dynamic analysis. The damping mechanism in wood buildings is very complicated and not completely understood. Viscous damping can be used to represent approximately the energy dissipation when a wood structure deforms in a linear elastic range. Under severe ground motions, most of input energy is dissipated by the nonlinear hysteresis in the structural units such as shear walls. The contribution of viscous damping may diminish with the increase of structural deformations. Therefore, the selection of the viscous damping coefficient may not as critical as the proper representation of the structural nonlinearity in the dynamic analyses of wood buildings subject to large seismic loads. The classical Rayleigh damping matrix including a mass-proportional damping term and a stiffness-proportional damping term is used in the “PB3D” model. It is defined by C=aM+J3K (3.43) where M is the mass matrix and K is the stiffness matrix; a and fi are the constants to be determined from the damping ratios corresponding to two unequal frequencies of oi, and from different vibration modes (Chopra, 2000). If both modes have the same damping ratio , which is reasonable based on experimental data, then 2aa 2/ ‘ ;,8= (3.44) (O+O) (O,+(O 62 In the “PB3D” model, a and ,B are estimated by the first and second vibration modes of the building. The damping ratio can be estimated based on test database or expert opinions. For example, damping ratio between 5% and 7% was recommended for wood structures with nailed or bolted connections (Newmark and Hall, 1982). 3.5.2 Time-stepping method - Newmark’s integration Newmark’s integration is used to solve the equation of motion due to its unconditional stability during solution. Given a short time interval At, the incremental form of Eq. (3.41) is MAü1 + CAn1 + (Af), AP, (3.45) where (Af), can be approximated by (Af), (K,)Au, (3.46) Rewrite Eq. (3.45) as (k.)TAuI =AP (3.47) where (K.) =K.) +—‘—C+ 1 M (3.48) ‘ T ‘ T fiAt ,8(At)2 and AP. AP. + + Lcn + rIM + AtI-— — 1’Clü. (3.49)I 8At fi )‘ L2 k2fl ) ]‘ Herein, j3 and y define the variation of acceleration. The “PB3D” model uses y = 1/2 and ,6 = 1 / 4, which corresponds to the constant average acceleration within At. To solve Eq. (3.47), the modified Newton-Raphson iterations are also used. The procedure of using Newmark’s method to perform a nonlinear time history 63 analysis can be summarized as follows: 1) Initialize the displacement, velocity and acceleration vectors of the system, select time interval At and calculate the mass matrix and the damping matrix; 2) For each time step i, calculate the incremental load vector AF, determine the tangent stifihess K1, calculate K1, solve for Au, using original or modified Newton-Raphson iteration, then calculate Az, and Au,, finally obtain u,1, 1i÷ and ii11; and 3) Repeat for the next step, just replace i by i + 1, till the end of required loading history. The average acceleration method is the most robust method for a step-by-step dynamic analysis of complicated systems like the P&B buildings. The only problem with this method is that the short-period (or high-frequency) vibration modes, which are smaller than the time step, might oscillate indefinitely after they are excited. These high- mode oscillations (also called spurious oscillations) can be reduced by adding the stiffness proportional damping matrix, which is included in the classical Rayleigh damping matrix introduced in the previous section. The modal damping ratio for a vibration mode with a frequency of co is =‘/3a,, (3.50) It is large for high frequencies and small for low frequencies. Thus, it will damp out the short periods to prevent them from spurious oscillations during the step-by-step time integration. 64 3.6 Summary This chapter presented a finite element model - “PB3D” to model the lateral response of the P&B wood buildings under static loads and dynamic loads. Formulations of three types of structural elements in this model (beam element, bar element and shear wall element) have been introduced. The shear wall element was represented by the mechanics-based “pseudo-nail” model, which can be calibrated by a shear wall test or a detailed FE shear wall model. Although the “pseudo-nail” model considers the lateral behavior of a shear wall only, the input load-drift curve for model calibration has implicitly embraced the influence with or without hold-down devices. The beam element and the bar element were used to construct the equivalent roof/floor diaphragms, taking into account the in-plane stiffness. The procedures of coordinate transformation in 3D space were also introduced. Based on the principle of virtual displacement, the static and dynamic equations of the model were formulated. In a static analysis, the “PB3D” model can simulate the structural lateral response using the force control model or the displacement control mode. In a dynamic analysis, the lumped mass matrix and the Rayleigh damping matrix were used. Newmark’s time-stepping integration was used to solve for the equation of motion. This model can simulate the seismic response of the P&B buildings under hi-axial horizontal ground motions. 65 CHAPTER 4 EXPERIMENTAL STUDIES ON SHEAR WALLS AND FLOOR DIAPHRAGM 4.1 Introduction Shear walls are important structural assemblies in wood buildings to carry vertical and lateral loads simultaneously. Under seismic loads, they are also the major structural units to provide lateral resistance and assure the safety of buildings. In the Japanese P&B buildings, diagonal braced (sujikai) walls are commonly used as shear walls. A typical diagonal braced wall consists of vertical, horizontal and diagonal wood framing members which are connected by mortise and tenon joinery and reinforced by metal fasteners. Continual research is required to achieve a better understanding of the lateral behavior of the diagonal braced walls since the previous earthquakes such as the Kobe earthquake caused disastrous consequences in some P&B buildings due to the inadequate lateral capacity of shear wall systems. The in-plane stifffiess of roof/floor diaphragms is another influential factor which affects the lateral performance of wood buildings. If a diaphragm is very rigid, lateral wind/earthquake loads will be distributed among the underlying walls based on their relative stiffness. If a diaphragm is very flexible, lateral loads will be distributed based on the tributary area of individual walls. In reality, wood roof/floor diaphragms are neither completely rigid nor completely flexible. It is beneficial to know the level of in-plane stiffness of roof/floor diaphragm and how it affects the seismic performance of the entire 66 building. The P&B floor diaphragms consist of beams with relatively larger cross sections and floor joists with smaller cross sections. Structural panels or planks are sheathed on the frame top and nail connections are used to connect panels and the frame. On the floor corners, knee braces are also installed to enhance the floor integrity. Special metal fasteners are used to reinforce some important framing connections (e.g., corner connections). Roof diaphragms in the P&B buildings have similar structural configurations as floor diaphragms except that they usually have short posts to support roof purlins (beams) to form an inclined roof surface. In this study, in-plane stiffness tests have not been conducted on roof diaphragms. A total of eight single-brace walls and a floor diaphragm built with Canadian Hem-fir (Tsuga heterophylla) were tested in the Timber Engineering and Applied Mechanics Laboratory at the University of British Columbia. The engineering characteristics of the walls such as strength, stiffness, failure modes, and energy dissipation were studied. The contribution of gypsum wailboards (GWB) to the shear wall lateral capacity was also studied. The in-plane stiffness, load path and load-sharing mechanism of the floor diaphragm were investigated. The contributions of different floor components to the in- plane stiffness were evaluated progressively. 4.2 Diagonal-braced shear wall test 4.2.1 Specimens and materials The eight single-brace walls were divided into two groups (Group A and Group B). Four walls in Group A were bare-framed, and the other four walls in Group B were 67 additionally sheathed with a 15.9 mm thick GWB on one side. All wood members were kiln-dried Canadian Hem-Fir, grade stamped as Canada Tsuga E120 (CFLA, 2005), with a specified mean modulus of elasticity of 12 GPa. The measured moisture content ranged from 12% to 15%. JAS (Japanese Agricultural Standard) certified metal fasteners were used except for the 48 mm long GWB nails, which were purchased from a local Canadian supplier. Table 4.1 gives the material list of the wall additionally sheathed with a GWB. Table 4.1 Material list of single-brace wall with GWB sheathing LengthMember Dimension (mm) Quantities(mm) Sill/top plate 105 x105 1510 2 Wood Post 105 x 105 2730 2 members Stud 30 x105 2730 1 Brace 45 x 90 2745 1 BP Plate Z-mark No.41-i 2 CP.T Plate Z-mark No.38-I 2 V.P Plate Z-mark No.36-i 2 HD-B20 Z-mark No. 82-1 4 ZN65 nail 3.5 65 40Metal N75 nail 13.8 75 2fasteners ZN9O nail 14.I 90 16 GWB nail c12.6 48 85 ScrewLSl2 t12 110 16 BoltMl2 t12 65 2 BoltMl6 16 165 4 Gypsum wallboard 910 x 2730 1 Figure 4.1 shows the configuration of the wall, which consists of 105 mm x 105 mm posts (Hashira) and sill/top plates (Dodai/Dohsashi), a 45 mm x 90 mm diagonal brace (Sujikai), and a 30 mm x105 mm wall stud (Mabashira). Four hold-down devices (HD B20 Z-mark) were installed to provide overturning restraint. The GWB nails were spaced at 100 mm on center along the panel perimeter and 200 mm in the field. 68 105x 105 Figure 4.1 Configuration of a single-brace wall 4.2.2 Loading setup and instrumentation Figure 4.2 shows the elevation of the test set-up and the location of transducers. A horizontal load was applied on the top of the wall via a MTS 243 .25T hydraulic actuator with a stroke of +1-254 mm, a capacity of 161.9 kN in compression and 98.8 kN in tension. A C-channel steel spreader beam with cross section of 76 mm x 127 mm was mounted on the top of the wall. The loading head of the actuator was connected with one end of the spreader beam using a swivel joint to release any moment effect. No vertical dead load was applied on the top of the wall. The base of the actuator was mounted onto a strong steel H-shaped reaction frame. The top plate of the wall was secured with the spreader beam via two hold-down devices on both ends. Four caster rollers were also 69 installed to allow the free movement of the spreader beam in the direction parallel to the applied load while preventing the out-of-plane movement of the wall. The bottom plate of the wall was also fastened with another C channel steel beam with the cross section of 76 nun x 127 mm by two hold-down devices. The holes in the top/bottom plate which accommodated for the hold-down bolts were drilled 1.0 mm larger than the bolt diameter. A total of seven LVDT (Linear Variable Differential Transformer) transducers and one string pot were installed to measure the wall deformations at different locations, as shown in Fig. 4.2. CR1 measured the horizontal displacement of the sill plate. The string pot CH2 measured the wall drift at the top plate level. CR3 through CR6 measured the vertical deformations of the connections between wall post and top/sill plate where hold- down devices were installed. CH7 and CR8 measured the deformations of diagonal connections. The load and displacement readings were sampled at a rate of 2 Rz. Spreader beam Figure 4.2 Elevation of loading set up and transducer locations I #2$ #4Actuator #6 Reaction column 70 Since the single-brace wall behaves differently under pushing or pulling, in each wall group, monotonic tests were carried out on two walls, one subjecting the brace to tension and the other subjecting the brace to compression. A list of the wall specimens with the corresponding loading protocols are given in Table 4.2. Figure 4.3 also shows the schematics of the loading types and wall positions of the individual walls. In the monotonic test, the wall was pushed over until the deformation angle reached 1/15 rad (1/15 of the wall height) and the loading rate was 0.13 mmlsec. The reversed cyclic tests used the CBL (Center for Better Living in Japan) protocol, which consists of 12 groups, with three identical cycles in each group (CBL, 2001). The drift amplitude of each group was ±1/600, ±1/450, ±1/300, ±1/200, ±1/150, ±1/100, ±1/75, ±1/50, ±1/30, ±1/24, ±1/20, and ±1/15 rad (Fig. 4.4). The CBL protocol is a modified PEO (Performance Evaluation Organizations) protocol to test a wood shear wall with hold-downs, based on the BSL of Japan. The original PEO protocol consists of eight CBL cycle groups from 1/600 rad to 1/50 rad plus a final pushover (BCJ, 2000). The cyclic loading rate was 1 mm/sec. Table 4.2 Single-brace wall test programs Wall No. Loading type Sheathing A-i MonotonicTa N/A A-2 Cyclic N/A A-3 MonotonicCb N/A A-4 Cyclic N/A B-I MonotonjcTa GWB B-2 Cyclic GWB B-3 Monotonic-d’ GWB B-4 Cyclic GWB Note: aDiagonal brace was loaded under tension; and bDiagonal brace was loaded under compression. 71 Figure 4.3 Wall positions and loading types 200 150 100 E Z 50 w E 0 w U . -50 -100 -150 -200 Figure 4.4 Reversed cyclic test protocol 4.2.3 Test observations and failure modes For the bare-framed walls in Group A, the diagonal brace connection zones experienced a lot of damage, such as nail withdrawals from BP metal plates, and wood splitting and crushing in the sill/top plates. These failure modes were also observed in the GWB-sheathed walls in Group B. It was found that in the monotonic test of the bare framed wall A-3, the diagonal brace experienced the out-of-plane buckling failure under Cycle No. 72 compression (Fig. 4.5); this failure mode was also observed in the braced walls of a two storey P&B building in a racking test (Sugiyama et al., 1988b). However, this failure mode was not observed in the cyclic tests of walls A-2 and A-4. This might be explained by the fact that under the demanding test protocol with a long sequence of cycles and equal amplitude groups, the gradually increased cyclic loads caused progressively accumulated nonlinear deformations in the diagonal bracing connections. These actions loosened the connections and provided some space for the diagonal brace movements under tension and compression. In the GWB-sheathed walls, GWB nail withdrawal started when the wall drift reached 1/50 rad. At a drift of 1/20 rad, most of the GWB nails were pulled out from the wood members. All the tested walls showed very low strength and stiffness at a drift of 1/15 rad. Under such large deformations, it is believed that these walls lost most of the load-carrying capacity, and the structural collapse was imminent. This was especially true if a dead load was applied on the top of the wall, the wall would experience a large P-A effect at such as a high drift level. Another interesting observation was that, in the cyclic tests, some nails experienced the low cycle fatigue failure, which was not observed in the monotonic tests and cannot be observed generally in actual seismic situations. This also agrees with the observations from the lightframe wall tests (Lam et al., 1997). 73 Pushover load Figure 4.5 Out-of-plane buckling of the diagonal brace Tables 4.3 and 4.4 give the observed progressive failure modes with increasing wall drift demands. The symbol “0” in the tables indicates the occurrence of the failure mode. Failure modes from A to I are described as follows: A: nail withdrawal from BP plate at diagonal bracing connection (Fig. 4.6a); B: complete pull-out of the nails from BP plate at diagonal brace connection (Fig. 4.6b); C: breakage of nails at diagonal bracing connection due to low cycle fatigue (Fig. 4.6c); D: punching-through of GWB nails (Fig. 4.6d); E: detachment of GWB from wall frame (Fig. 4.6e); F: breakage of GWB nails due to low cycle fatigue (Fig. 4.60; G: wood splitting in sill/top plate (perpendicular to grain tension failure) (Fig. 4.6g); H: wood crushing in sill/top plate (perpendicular-to-grain compression failure) (Fig. 4.6h); 74 I: out-of-plane buckling of diagonal braces (Fig. 4.5). Table 4.3 Progressive failure observations of bare-framed walls Wall Drift Failure mode No. amount A B C D E F G H 1/30 0 0 A-l 1/24 1/20 0 1/30 0 0 A-2 1/24 0 1/20 0 1/30 0 A-3 1/24 0 1/20 0 1/30 0 0 A-4 1/24 0 1/20 0 Table 4.4 Progressive failure observations of GWB-sheathed walls Wall Drift Failure mode No. amount A B C D E F G H 1/50 0 B1 1/30 01/24 0 1/20 0 0 1/50 0 B2 1/30 01/24 0 1/20 0 0 1/50 0 B3 1/301/24 0 1/20 0 1/50 0 B4 1/30 01/24 0 1/20 0 0 75 Figure 4.6a Failure mode A: nail withdrawal from BP plate Figure 4.6b Failure mode B: complete pull out of nails from BP plate Figure 4.6c Failure mode C: nails broken due to low cycle fatigue failure 76 Figure 4.6d Failure mode D: Nails punching through GWB Figure 4.6e Failure mode E: GWB detached from wood frame Figure 4.6f Failure mode F: GWB nails broken due to low cycle fatigue failure 77 Figure 4.6g Failure mode G: wood splitting in sill/top plate (perp. to grain tens.) Figure 4.6h Failure mode H: wood crushing in sill/top plate (perp. to grain comp.) 4.2.4 Test results and discussions The hysteresis from a wood shear wall test typically has some special features such as the pinching effect, strength and stiffness degradation. Usually, the load-drift curve of a wood shear wall shows significant nonlinearity over the entire loading history, making it difficult (or sometimes even conceptually misleading) to define the yield strength for wood shear walls. However, for convenience, the yield strength of wood shear walls is still defined following some specific rules for design purpose. Under seismic loads, energy dissipation is another important criterion to evaluate the seismic performance of a 78 z0-2 -J z : : bo-i - 50 iO0 i0 2( : - - -- 181 Figure 4.7b Load-drift curve of Wall A-2 wood shear wall. Usually, if the hysteresis embraces a large area, a high capacity of energy dissipation is expected. Figures 4.7a to 4.7h show the load-drift curves of the eight walls tested in this study. - 14-- - = V: -1bo - 0 100 150 2 —6--I- -io4 18 I I Drift (mm) 0 Figure 4.7a Load-drift curve of Wall A-i .10 z 0 -2 -J Drift (mm) 79 40 z 0 -J Drift (mm) Figure 4.7c Load-drift curve of Wall A-3 z 0-j- -j Drift (mm) Figure 4.7d Load-drift curve of Wall A-4 40 - - A - - - - 2 JO 1150 1100 -o fob ----t 14- 18- I __ Drift (mm) Figure 4.7e Load-drift curve of Wall B-i -2 iN zE*5E -iso :1O: 5 iôo io —14-— \ I 14-— \ I to-- \I 0 to- 14—-n 40 - 0 Icu 02 —I 80 z . o2 -j Drift (mm) Figure 4.7f Load-drift curve of Wall B-2 .40 ft I 14-- I4I I I \ I LN 6 - - - - - 2-- Figure 4.7g Load-drift curve of Wall B-3 4- E 50 Drift (mm) Figure 4.7h Load-drift curve of Wall B-4 -J -240 -150 -100 -50 0 50 100 150 2 -- z:z:zzz::zzz: 0 Wall drift (mm) 40 z o -2 -J 81 The method used to define the yield strength, stiffness, the ultimate strength and the ductility ratio followed the Japanese wood shear wall test and evaluation method (BCJ, 2000). A monotonic pushover curve or a backbone curve from a cyclic test can be used to determine the shear wall characteristics. As shown in Fig. 4.8a, the procedure to define the yield strength P, and stiffness K is given as follows: 1) first, locate four points (peak load Pmax, 0.9Pmax, 0.4Pmax, and 0.lPmax) on the load-drift curve; 2) draw a straight line through the point 0.9Pm and the point 0.4Pmax and calculate its slope KT; 3) draw a tangent line to the load-drift curve which has the same slope KT; and, 4) draw a straight line through the point 0.4Pm and the point 0.1 Pm, which intersects with the tangent line. The y coordinate of this intersection point is the yield strength P,, of the shear wall. On the load-drift curve, the associated x coordinate is the yield drift D. The stiffness of the shear wall K is simply defined as P/D. 0 -J 4- -4 Figure 4.8a Method of defining yield strength and stiffness Figure 4.8b shows the method to define the ultimate strength P. First, the shear wall stiffness K needs to be defined following the above procedure. Then, the point 0.8Pmax On 82 Pmax I I I [ I I D Drift the softening (post-peak load) portion of the load-drift curve needs to be located. By equating the dissipated energy from zero displacement to the post-peak 0.8Pm with the area enclosed by the wall stiffness line, horizontal P line, and the vertical O.8PmaX line (shadowed area in Fig. 4.8b), one can calculate the ultimate strength P. Pmax Drift D Figure 4.8b Method of defining ultimate strength P The ductility factor p is defined as (4.1) PU/K where D is the drift corresponding to the ultimate strength, P, is the yield strength, and K is the stiffness. Another factor called “structural property factor” D is defined as D = 1 (4.2) /2p-1 The allowable shear wall resistance is defined by aIlow =MIN{P, 0.2P/D, 2Pm/3 1!/12o} (4.3) where P,, is the peak load, P11120 is the load at the drift ratio of 1/120 rad. Tables 4.5 and 4.6 give the test summaries of the engineering characteristics of the walls in Group A and 83 0 -J Group B, respectively. Table 4.5 Summary of test results of bare-framed walls A-2 A-4Wall Number A-i + - A-3 + - Avg. Pmax(kN)a 6.71 7.29 11.41 9.98 11.13 6.64 8.86 2/3Pmax(kN) 4.47 4.86 7.61 6.65 7.42 4.43 5.91 P(kN)b 3.85 4.08 6.10 5.79 6.23 3.47 4.92 Dy(mm)C 35.57 33.29 28.50 32.54 33.40 27.07 31.73 K(kN/mm)d 0.108 0.123 0.214 0.178 0.187 0.128 0.156 Pu (kN) n/a 6.40 10.05 n/a 10.08 6.00 8.13 Du (mm) n/a 129.4 120.9 n/a 125.3 106.7 120.6 p n/a 2.49 2.58 n/a 2.33 2.28 2.42 O.2P1frJ; n/a 2.55 4.10 n/a 3.87 2.26 3.20 p11120(kN) 2.66 2.99 4.93 4.15 4.54 3.07 3.72 Pallow (kN) 2.66 2.55 4.10 4.15 3.87 2.26 3.20 Table 4.6 Summary of test results of GWB-sheathed walls B-2 B-4Wall Number B-i B-3 Avg. + - + Pmax(kN)a 12.93 12.15 15.74 16.16 15.16 11.13 13.88 2/3Pmax(kN) 8.62 8.10 10.49 10.77 10.11 7.42 9.25 P, (kN)b 6.46 7.66 9.08 9.80 7.88 6.04 7.82 Dy(mm)C 35.57 33.29 28.50 32.54 33.40 27.07 31.73 K (kN/mm)d 0.223 0.227 0.249 0.228 0.321 0.287 0.256 Pu (kN)e n/a 9.50 12.97 n/a 13.03 9.42 11.23 Du (mm) n/a 104.9 125.5 n/a 101.4 83.9 103.9 p n/a 2.51 2.41 n/a 2.50 2.56 2.50 o.2p.fEi n/a 3.81 5.07 n/a 5.21 3.82 4.48 Pu120 (kN) 5.38 5.90 5.96 5.57 7.63 6.21 6.11 PalIow (1(N) 5.38 3.81 5.07 5.57 5.21 3.82 4.48 Note: apeak load; byjeld strength; °Drift corresponding to yield strength; dStiffness; eUltimate strength; Drift corresponding to ultimate strength; and at drift ratio of 1/120 In the monotonic tests, the actuator load did not drop to 80% of the peak load when the prescribed maximum drift was reached. Therefore, based on the Japanese method, the information about the ultimate strength F0, and ductility factors 1u could not be retrieved. 84 Only the peak load, yield strength, and stiffness were evaluated. The average allowable resistance Pajiow, which is used to guide the shear wall design in Japan, and the average stiffness K of the bare-framed walls in Group A was 3.2 kN and 0.156 kN!mm, respectively. The average Paiiow of the GWB-sheathed walls in group B was 4.48 kN, 40% higher than that of the bare-framed walls; and, the average stiffness K of the GWB sheathed walls was 0.256 kN/mm, 64% higher than that of the bare-framed walls. Figure 4.9 shows the energy dissipation of individual walls in the cyclic tests. On average, the total dissipated energy of the GWB-sheathed walls was 44% higher than that of the bare- framed walls. The test results indicated that the additional GWB contributed significantly to the strength, stiffness and energy dissipation of the single-brace walls. 6 9 12 15 18 21 24 Cycle No. Figure 4.9 Energy dissipation in the reversed cyclic tests Based on the test results of the walls, the following conclusions can be drawn: 1) Gypsum wallboards significantly improved the lateral capacity of the single-brace walls in terms of the peak load, strength, stiffness, and energy dissipation; 2) Under the lateral loads, the maj or failure modes were nail withdrawal in the diagonal 85 1000 E 800 600 : 400 2’ 200 0 C Iii 0 • WaIIA-2 • WaIIA-4 -..•--. WaIIB-2 0 3 27 30 33 36 connections, wood perpendicular to grain failure in sill/top plates, and out-of-plane buckling of the brace under compression. Besides, in the walls with an additional GWB, GWB nail withdrawal and GWB detachment were also observed; 3) In the cyclic tests, low-cycle fatigue failure of nail connections was observed in the diagonal connections and GWB connections while this failure mode was not observed in the monotonic tests; and 4) Hold-down devices performed very well in the tests and provided a good anchorage to prevent the wall uplifting. 4.3 Floor diaphragm test 4.3.1 Specimens and materials Floor diaphragms in the P&B buildings have hierarchical framing members including girders, joists, and sometimes secondary beams if the floor size is large. A P&B wood building can have variable configurations of floor diaphragms with/without openings and plan offsets. In this study, a 3.64 m x 5.46 m floor segment, selected from a typical two-storey P&B building, was tested in the Timber Engineering and Applied Mechanics Laboratory at the University of British Columbia (Fig. 4.10). This floor diaphragm was constructed following Japanese building practice. All timber members were kiln-dried Canadian Hem-fir, grade stamped as Canada Tsuga E120. The measured moisture content ranged from 12% to 15%. JAS certified metal connectors were used. In a real building, floor edges are supported by the underlying walls. Thus, in this floor diaphragm, the floor side beams adopted relatively smaller members with cross section of 86 105 mm x 105 mm. For the floor central beams with a span of 3.64 m, larger members with cross section of 105 mm x 270 mm were used. Floor joists were small members with cross section of 45 x 90 mm and were supported by the central beams and the side beams at both ends. Floor joists had a span of 1.82 m and were spaced at 303 mm. Short braces with cross section of 90 mm x 90 mm were installed at four floor corners to improve the diaphragm integrity. This floor diaphragm had three sections including two end sections and one central section. Each section was 1.82 m wide and 3.64 m long. On the top of the floor frame, structural plywood panels with thickness of 16 mm were installed with common CN5O nails (50.8 mm in length, 2.9 mm in diameter) spaced at 150 mm on panel edge and 300 mm on field. On each side of the floor, three short steel rods with diameter of 25 mm or 32 mm were used to tie down the side beams with the strong concrete floor in the laboratory. Timber members with mortise and tenon joinery were processed with the computer numerical controlled (CNC) machine Hundegger K2. The material list of floor members and metal fasteners is given in Table 4.7. 87 n I - — CD — C D k) - - 1 \Q C C C — x x x x x k) i — 0 C C - C C ‘ J • 4 C L I t’ J a x — 00 C rj’j o CD (I ) — . - + . C) C CD CD o o c2 - _ j 3 I 3I g - y lcD I0 cà lo C) H — — . 11 Q o — . 1a o i (0 1 CD - o m - c: < N CD — . C, ) C = . qQ_ — . - CD z CD C — . - (C Cl) (,) CD CD CC ) — E ?‘ CD c 1= C) E 0) a CD — CD Cl) a C CiD C- t CD — . - * C) - ‘ C) H — - — . CD CD 00 CD 0) (IQ CC ) C, ) ‘ - z “ • ci c C, ) C C) CD CD CD - C - rJD C) C, ) c CD - t CD o — CC ) C, ) CD o — - C Q- C — . — . CC ) CD - - U I 00 ) C 0- - C - - © C C C C :: ;1 iiii ii E E E E E E E / - - - — CD o lC D CJ ’I ‘ < I- , — 1C /) O l CD X I — ID - C I CD 36 40 acceptable. The in-plane stiffness of the floor diaphragm along two perpendicular directions was tested via a PARKER 08.00 actuator with a stroke of ±125 mm and a capacity of 665 kN. The actuator was mounted onto a strong steel H column and pushed the center of a spreader beam (Fig. 4.11 and Fig. 4.12). A steel hollow-section beam was used as the spreader beam to distribute the actuator load equally to two loading points along the floor edge. Underneath the spreader beam, ball bearing plates were placed to reduce the friction between the spreader beam and the supporting guides. In this test, the loading rate in the force control mode was 3 kN/min. The loading rate in the displacement control mode was about 2 mmlmin. 4 6 7 Figure 4.11 Schematic of loading and transducer definitions along y direction 89 #3 4= #1 4= Figure 4.12 Schematic of loading and transducer definitions along x direction Five LVDT transducers and two string pots were installed to measure the floor deformations at various locations (Fig. 4.11 and Fig. 4.12). The load and displacement readings were sampled at a rate of 4 Hz. Sensor channels were set up as follows: chi and ch3 measured the horizontal displacements of both supporting ends; ch2 measured the horizontal mid-span displacement; ch4, ch5, ch6 and ch7 measured the shear deformations of two floor end sections where shear deformations would predominate according to the load scenario and floor geometry. The calculation of diaphragm deflection and shear deformation from the direct displacement measurement is described as follows. Suppose the load is applied along y direction as shown in Fig. 4.11, let A, denote the measurement of the ith transducer, the floor mid-span deflection can be obtained by deducting the average of end support displacements (chl and ch3) from the measured mid-span displacement (ch2), which is #7 #6 #5 90 A —A2 (4.4) 2 Thus, the global diaphragm in-plane stiffness along y direction is: F K =_L (4.5) where F is the applied load along y direction. The shear strain of the floor end section where ch4 and ch5 are installed is: A45Vu +u (4.6) bd where A45 is the average of the measurements of ch4 and ch5, b and d are the length and width of the floor end section, respectively. Thus, the shear deformation is: A=yx1 (4.7) where L is the floor span. And the shear stiffness of the end section is: F/2 K = (4.8) Thus, the shear rigidity GA of the floor is defined as F /2 GA = (4.9) ‘I Rearranging Eq. (4.9) with Eq. (4.7) and Eq. (4.8), one has GA=K3•- (4.10) Similarly, the shear deformation, shear stiffness and shear rigidity of the other end section where ch6 and ch7 are installed can also be calculated by Eq. (4.6) to Eq. (10). 91 4.3.3 Test program Four test phases were carried out in the floor diaphragm test along x and y loading directions with progressively added sheathing panels (Table 4.8). Each test phase was carefully monitored to avoid the occurrence of significant structural damage. Otherwise, the next test phase could present misleading information due to the accumulated damage and stiffness degradation. Table 4.8 Test phases with associated floor configurations Test Load Floor config.phase Mode Direction 1 Disp. control (20mm) y Frame only 2 Force control (45kN) y Partially sheathed 3 Force control (45kN) y Fully sheathed 4 Force control (45kN) x Fully sheathed 4.3.3.1 Phase 1 Only the timber frame of the floor diaphragm was constructed. No sheathing panel was installed. The load was applied along y direction which was the “weak” direction of the floor (Fig. 4.13a). Considering the relatively low in-plane stiffness of the bare-framed floor, the displacement control mode was used for loading. Based on the preliminary calculation, the maximum actuator stroke was set to be about 20 mm, about 1/273 of the floor length. The floor was expected to deform within a linear elastic range without structural damage. 92 _—____‘.{ Figure 4.13a Test phase 1: in-plane pushing on bare-framed floor along y direction 4.3.3.2 Phase 2 Structural plywood panels were then attached onto the end sections of the floor (Fig. 4.1 3b). This test phase was to evaluate the contribution of the partially sheathed plywood panels to the in-plane stiffness. The force control mode was used for loading. The maximum actuator load was 45 kN based on the preliminary estimation. Meanwhile, the floor was carefully monitored during the loading to assure that no significant damage would occur. The set-up of instrumentation was the same as test phase 1. Figure 4.13b Test phase 2: in-plane pushing on partially sheathed floor along y direction 93 4.3.3.3 Phase 3 The floor diaphragm was fully sheathed by plywood panels (Fig. 4.1 3c). The test phase was to evaluate the in-plane stiffness of the complete floor diaphragm along y direction. Loading and instrumentation was the same as test phase 2. Again, the force control mode was used for loading and the maximum actuator load was about 45kN. Figure 4.13c Test phase 3: in-plane pushing on fully sheathed floor along y direction 4.3.3.4 Phase 4 The in-plane stiffness of the fully sheathed floor was evaluated along x direction which was the strong direction of the floor (Fig. 4.1 3d). End supports with the steel rods were relocated to the long sides of the floor. Loading and instrumentation were rearranged to accommodate for the new loading scheme. The force control mode was used for the loading and the maximum actuator load was also 45kN. 94 Figure 4.13d Test phase 4: in-plane pushing on fully sheathed floor along x direction 4.3.4 Test results and discussions The test results of four test phases in terms of load-deflection curves are shown from Fig. 4.14a to Fig. 4.14d. Each figure has three load-deformation curves. One represents the relationship between the actuator load and the floor mid-span deflection. The other two represent the relationship between the shear force and the shear deflection of two end sections of the floor. Due to the loading and structural symmetry, each support end of the floor was assumed to carry half of the applied load. Therefore, two end sections of the floor would carry the same amount of in-plane shear force. The summary of the test results in each test phase is given in Table 4.9. 95 50 ____________________ 40 30 20 10 0 0 5 10 15 20 25: Deformation (mm) Figure 4.14a Test phase 1 — load vs. deformation curves 50 —Mid-span 40 —Endsectioni I —Endsection2 30 20 - - H 10 ----F ------- H---- 0 5 10 15 20 25 Deformation (mm) Figure 4.14b Test phase 2 — load vs. deformation curves 50 —Mid-span 40 —Endsectioni —End section2 30 F CU F I20 10 0 0 5 10 15 20 25 Deformation (mm) Figure 4.14c Test phase 3 — load vs. deformation curves 96 —Mid-span —Endsectioni - —Eridsection2 4- -- 0 50 —Mid-span 40 - —End sectioni - - —End section2 30 L. = 20 10 - 0 0 5 10 15 20 25. Deformation (mm) Figure 4.14d Test phase 4—load vs. deformation curves Table 4.9 Summary of the floor diaphragm test results Mid- Section 1 Section 2GlobalTest Actuator span Stiffness Shear Shear Shear Shearphase load (kN) Def. stiffness rigidity stiffness rigidity(kN/mm)(mm) (kN/mzn) (kN) (kN/mm) (kN) 1 11.24 21.2 0.53 0.318 577 0.305 555 2 45.19 4.7 9.6 8.162 14855 6.722 12234 3 45.16 3.67 12.3 9.050 16471 7.846 14280 4 45.09 0.6 75.0 49.889 60515 57.233 69424 It was found that the floor deformed almost in a linear elastic range in test phase 1 and no structural damage was observed. In test phases 2 and 3, the load-deformation curves show some nonlinear deformations in the floor under the load of 45 kN. Nevertheless, no significant structural damage was observed. Special attention should be given to the test results in test phase 4. Very high in-plane stiffness was found along the x direction, which was the strong direction of the floor. Considering the floor aspect ratio, the concept of “deep” beam to estimate the diaphragm stiffness may not be appropriate along the x direction. It was found that the loading effect in test phase 4 was highly concentrated to the floor area close to the loading points. At the peak load of 45 kN, the 97 actuator stroke was 7.5 mm. Thus, the floor side beam directly loaded by the spreader beam should experience approximately the deflection of 7.5 mm. However, the measured deflection of the side beam on the opposite side of the floor was only 0.6 mm. Therefore, the measured floor in-plane stiffness along the x direction should be used with caution because the floor in the test phase 4 did not behave as a “deep” beam, but a much complicated two-dimensional planar structure with high load concentration. These test observations were also consistent with the finite element modeling results of lightframe floor diaphragms with different aspect ratios (Wang et al., 1999). The FE modeling results showed that in a floor diaphragm with a high aspect ratio, the tensile chord at the far end experienced even no loads. In this test, due to the large depth of the floor (x direction loading), a secondary beam formed close to the loading edge and a strip of panel worked as the tension chord. Thus, the “tensile” chord at the far end experienced very small deflection. Some conclusions can be drawn from the test results of the floor diaphragm: 1) The floor diaphragm showed very high in-plane stiffness with sheathed plywood panels along two perpendicular directions; 2) The floor diaphragm seemed to have a high in-plane load carry capacity because no significant structural damage has been observed in the floor diaphragm subject to a load as high as 45 kN; 3) Plywood panels and nail connections play a very important role to increase the in plane stiffness of the floor diaphragm. Along the weak y direction, the in-plane stiffness 98 of the fully sheathed floor was 22 times higher than that of the bare-framed floor; and the shear stiffness and rigidity of two end sections of the floor was 25 times higher than that of the bare-framed floor; and 4) The floor aspect ratio also plays an important role to distribute the internal forces within the diaphragm and can influence the interpretation of the in-plane stiffness of the diaphragm. 4.4 Summary This chapter presented the experimental studies on a total of eight single-brace shear walls and a floor diaphragm in the P&B wood buildings. The lateral performance of the single-brace walls was evaluated by monotonic pushover and reversed cyclic tests. The in-plane stiffness of a floor diaphragm was evaluated by in-plane pushover tests. All test specimens were built following Japanese building practice, such as the processing of mortise and tenon joinery and the application of JAS certified metal hardware. All the wood members were Canadian Hem-fir, grade stamped as Canada Tsuga E120. In the shear wall tests, four walls were only bare-framed and the other four walls were additionally sheathed with a GWB. The test results indicated a significant contribution of the GWB to the lateral performance of the single-brace walls in terms of the strength, stiffness, energy dissipation under the same loading protocols. These findings implied that a certain amount of GWBs in a P&B building might provide additional safety margins under seismic loads or wind loads. A seismic design which excludes the contribution of GWBs might be a conservative design. 99 In the floor diaphragm tests, with progressively added plywood panels on the frame, the contributions of the sheathing panels and nail connections to the floor in-plane stiffness were assessed quantitatively in terms of the global in-plane stiffness, shear stiffness and rigidity. The test results indicated the great importance of the sheathing panels and panel-frame nail connections to the floor in-plane stiffness. The test results also indicated the relatively high in-plane stiffness of the floor diaphragm along two perpendicular directions. It should be noted that the floor diaphragms in the real P&B buildings might have many different configurations. The aspect ratio, openings, plan offsets will also significantly affect the load distribution among the floor components. More experimental studies are required to achieve a better understanding of the behavior of the P&B floor diaphragms. The established test database will be further used to calibrate the “pseudo-nail” shear wall models and the equivalent diagonal-braced floor diaphragms which are implemented in the “PB3D” model. 100 CHAPTER 5 EXPERIMENTAL STUDIES ON ONE-STORY BUILDINGS 5.1 Introduction Building behavior cannot be estimated by simply adding up individual structural components or assemblies without taking into account their interactions and functions as a system. To evaluate the building performance on a system level, a straightforward approach is to perform full-scale tests. In this study, two full-scale one-story P&B buildings were tested at the University of British Columbia. The first building was tested under biaxial lateral pushover loads. The second one was tested on a uniaxial shake table to simulate the seismic response under uniaxial earthquake excitation. In both tests, a digital data acquisition system was used to record the structural response at different locations, such as wall drifts, connection deformations, and roof accelerations. The failure modes, load sharing mechanism, strength and stiffness characteristics of the buildings were investigated. The test results will also be used to verify the proposed “PB3D” model. 5.2 Test specimens and materials Two identical one-story P&B buildings with a plan size of 5.46 m x 3.64 m were constructed following Japanese building practice. Figures 5.1 a, 5.1 b and 5.1 c show the building elevations. Figure 5.2 shows the layout of the shear walls. The geometry of the building in these figures is shown in millimetres. The storey height was 2.73 m. Each side of the building consisted of two 0.91 m long unsheathed single-brace walls with one at 101 each end. Two longer side walls had the same wall configuration while one shorter side wall had a larger door opening and the other short side had a smaller window opening. A inclined roof with a slope of 21.8° was assembled to provide the horizontal diaphragm action and carry vertical dead load. Eaves, 0.6 m in width, were constructed on both sides of the roof. Overhangs, 0.6 m in width, were built into both ends of the roof. The roof was made with heavy timber members. Plywood panels with thickness of 12.5 mm were used as the roof sheathing. Framing members with mortise and tenon joinery were processed by the CNC machine Hundegger K2. JAS certified nails and metal hardware were used to tie the major members together. To prevent the structural overturning under large lateral loads, a total of 12 hold-down devices were installed on the wall post-sill plate connections and four hold-downs were installed on the corner post-top plate connections. All framing members were kiln-dried Canadian Hem-fir, grade stamped as Canada Tsuga E120 except for the wall top plates with cross section of 105 mm x 210 mm. The moisture content of the wood members ranged between 12% and 15%. A material list of the wood members for one building is given in Table 5.1. 102 300 300 C = = = = 910 910 3640 Figure 5.1 b Exterior elevation of short side (south) Figure 5.1 a Exterior elevation of long sides (east and west) 103 Figure 5.1 c Exterior elevations of short side (north) Note: .. 45x90 mm single-brace walls Figure 5.2 Shear wall layout of the one-story building 104 Table 5.1 List of wood members in the one-story building Dimension Length/ThicknessMember Quantities(mm x mm) (mm) Roof plywood panel 2400x1200 12.5 6 Roof rafter 60x45 3620 26 105x105 360 6 Roof short post 105x 105 740 6 105x105 1100 3 Roof beam 105x105 4840 5 210x105 5420 3 Top plate 210x105 4840 2 210x105 1820 2 Wall post 105xl05 2560 14 Diagonal brace 90x45 2600 8 Brace post l05x30 2520 8 Window/door lintel 105x45 1780 11 105x45 880 15Wall shortpost 105x45 500 18 105x105 5420 2Sill plate 105x105 3760 2 According to the BSL, the expected gravity loads from roof tiles and ceiling fixtures are about 100 kg/rn2. Therefore, based on the total projection area (6.66 m x 4.84 m) of the roof, six boxes of steel punching with a total mass of 3200 kg were securely fastened onto the top of the roof to represent the distributed dead load. Hold-down devices were secured with the strong rectangular steel tube beams connected with the strong floor in the lab or the concrete beams on the shake table. 5.3 Static test 5.3.1 Loading setup and instrumentation The first building was tested statically in the Timber Engineering and Applied Mechanics Laboratory at the University of British Columbia (Fig. 5.3). Biaxial pushover loads were applied via two MTS hydraulic actuators. One was MTS 243.30T with a stroke of ±250 mm and a capacity of 254 kN in compression and a capacity of 162 kN in 105 tension and the other one was MIS 243.35T with a stroke of ±250 mm and a capacity of 365 kN in compression and a capacity of 240 kN in tension. Each actuator was connected with a steel spreader beam via a swivel joint in order to apply a uniformly distributed load on the top plate of the wall (Fig. 5.4). A concrete strong wall and an H-shaped strong steel column were available in the lab to provide the reaction supports for the actuators. The displacement control mode was used for loading and the loading rate was about 0.5 mmlsec. Figure 5.3 The one-story P&B building in the static test 106 Figure 5.4 Actuator loads along two horizontal directions of the building A total of 13 LVDT displacement transducers and two string pots were installed to measure the deformation of the building, such as wall drifts, diagonal connection deformation, and corner post uplifting (Fig. 5.5). Transducers Chi to Ch4 measured the wall drifts of four side walls. Ch5 to Ch12 measured the deformations of diagonal connections at the lower corners. String pots Ch13 and Ch14 were used to measure the 107 distortion of the roof diaphragm. The load and displacement readings were sampled at a rate of 2 Hz. N #4 #15 #1/7 Figure 5.5 Locations and definitions of sensors in the static test 5.3.2 Test observations and results The building was pushed along two horizontal directions simultaneously until the actuator displacement reached 1/24 of the wall height and then retracted to the original zero position. Although some damage of structural components was observed, the building retained its structural integrity. Most of the damage occurred in the diagonal brace connections. Similar to the failure modes observed in the single-brace shear wall tests introduced in Chapter 4, the major failure modes of this building were the nail withdrawal from the BP plates and wood tension perpendicular to grain failure in the 108 still/top plates due to the high tensile force in the diagonal braces (Fig. 5.6a). One interesting observation was the nail withdrawal in one of the compression brace to top plate connections in the north-west corner of the west wall (Fig. 5.6b). However, in the shear wall tests, nail withdrawal is not a typical failure mode for the diagonal brace connection in compression since the close contact and wood crushing between the end of diagonal brace and the sill/top plate or wall post does not allow the further development of nail withdrawal from the BP plate. This phenomenon might be explained by the twisting action of the roof diaphragm which caused the out-of-plane deformation of the BP plate and the nail withdrawal as well. Figure 5.6a Failure modes at diagonal brace connection / :hdrawal / .i perpendicL 109 Nail withdrawal Figure 5.6b Nail withdrawal at diagonal brace connection on top corner of the wall In this building, the single-brace walls provided most of the lateral resistance and the critical diagonal brace connections carried very high concentrated loads. Significant nonlinear deformation was observed in these connections. Figures 5.7a and 5.7b show the relationship between the deformation of the diagonal brace connections and the corresponding actuator load. As expected, the diagonal brace connections under tension underwent much larger deformation than the diagonal connections under compressions. For example, transducer Ch5 recorded 19 mm elongation while the deformation of the connections under compressions were less than 6mm. Test data recorded from transducer Ch8 was corrupted during the test and was therefore discarded. 110 z0 -J 1 2 z . CD 0 -J a- 2 CD C.) —Ch7 -30--- —Ch8 - —Ch9 2- - ChlO -E -4 4 8 1 1 Deformation (mm) Figure 5.7a Deformation curves of diagonal brace connections of east and west walls Deformation (mm) Figure 5.7b Deformation curves of diagonal brace connections of north and south walls The peak deformations recorded by string pots Ch13 and Ch14 between the opposite corners of the roof were 1.67 mm and 1.08 mm, respectively. Therefore, the average shear strain was 0.00045 based on the roof geometry 3.64 m x 5.46 m and the shear deformation was only 2.48 mm, which means that the distortion of the roof diaphragm under the biaxial loads was very small. The roof diaphragm performed more like a rigid diaphragm due to the contributions of the roof sheathing panels and four short corner braces intended to enhance the roof integrity. Figure 5.8 shows the load-drift curves of 111 four side walls. The y coordinate denotes the actuator load carried mainly by two side walls parallel to the loading direction. The load-drift curves showed that the building behaved in a fairly ductile manner. Peak loads had not been reached; i.e. some load capacity margin was still available. Based on the observed damage, it seems however that reaching the peak load was imminent. The load-drift curves indicated that this building had the nonsymmetrical horizontal deformation under the symmetrical loading. Structural twisting occurred during testing. Figure 5.9 shows the in-plane twisting of the roof diaphragm when the actuator loads reached their peaks, 31.5 kN from east to west and 27.6 kN from north to south. One observation was that the north wall and the south wall were stiffer than the east wall and the west wall under the same amount of actuator loads although they were shorter than the east and the west wall. This can be explained by the “effective” length of the P&B shear walls which governs the lateral resistance of the total wall systems. In this building, the effective shear wall length of each side wall is actually the total length of two single-brace walls multiplied by a shear wall multiplier. Thus, four side walls had the same effective shear wall length. However, the nonsymmetrical structural response was mainly caused by the different capacities of individual walls due to the inherent variations of material properties and construction quality. 112 35 30 —West W&I i:: E _ 5 20 40 60 80 100 120 Wall drift (mm) Figure 5.8 Wall load-drift curves under the biaxial pushover loads A w Figure 5.9 Twisting of the roof diaphragm (lOx deformation) 5.4 Dynamic test 5.4.1 Instrumentation and measurement The second building was constructed and tested dynamically in the Earthquake Engineering Laboratory at the University of British Columbia (Fig. 5.10). This laboratory is equipped with a servo-controlled uniaxial shake table, 6 m x 6.5 m in a plan size. It consists of a rectangular steel frame covered by a series of reinforced concrete beams to 3.64 m 5.46 m 113 form a platform. Strong bolts are embedded in the concrete beams to secure the test specimen. An actuator with payload capability of 296 kN (66.7 kips) and a maximum stroke of ±305 mm (±12 in) is mounted under the table to drive the system. The loading system is the MTS 458 servo-controlled system. Real-time analysis and control software DASYLab V7.00.00 are used as the digital data acquisition system. In this test, a series of strong steel tubes were also used to accommodate the connections between the hold- down devices of the building and the shake table. Figure 5.10 The one-story P&B building in the shake table test A total of six LVDT displacement transducers, five string pots, two triaxial accelerometers and one uniaxial accelerometer were installed to record the deformations and accelerations of the structure at different locations as well as the table movement under the dynamic loads. A total of 19 channels were set up to record the building response including a separate channel for hammer impact tests (Fig. 5.11). In Fig.5.11, “P” 114 stands for LVDT transducers; “SP” stands for string pot; and “AiD” and “A3D” stand for uniaxial and triaxial accelerometers, followed by a channel serial number. The sensors were set up as follows: string pots cHi and cH2 measured the absolute displacement of the walls along the shaking direction; string pots cH3 and cH4 measured the distortion of the roof diaphragm; transducers cR5 to cH8 measured the deformation of the diagonal brace connections at the lower corners of the single-brace walls along the shaking direction; transducers cR9 and cH 10 measured the wall post uplifting; accelerometers cH ii to cH 16 measured the accelerations on the top of the walls (the roof ceiling level) along the shaking direction; string pot cR17 measured the shake table displacement and accelerometer cR1 8 measured the table acceleration. All the readings were sampled at a rate of 500 Hz to capture the structural response with enough accuracy. Figure 5.11 Locations and definitions of sensors in the shake table test S A3D-1 #14,#1 5,#1 6 Ground motion 115 5.4.2 Test program Since only one building was built for the shake table test, it was desirable to retrieve as much information as possible from this building. Four test runs were scheduled (Table 5.2). The hammer impact tests and the square-wave impulse test are non-destructive tests to estimate the natural period of damped free vibration of the building as well as the damping ratio at different test stages. The data collected from the shake table test allows the study of the response of the building under seismic loads. Table 5.2 Test sequence First run Second run Third run Fourth run 1-1 Hammer impact 2-1 Hammer impact 3-1 Hammer impact 4-1 Hammer impact 1-2 Square-wave 2-2 Square-wave 3-2 Square-wave 4-2 Square-wave impulse impulse impulse impulse 2-3 Shake table 3-3 Shake table1-3 Shake table (KOBEI OONS-1) (KOBE100NS-2) 1) The hammer impact tests were conducted at different locations of the building. Six locations on the top plates of the south wall and the east wall were selected to perform the impact test (Fig. 5.12). Each impact location was struck three times in the horizontal direction by an impact hammer. The time interval between each strike was about five seconds. The decaying acceleration response was recorded by the accelerometers mounted on the top plates of the walls. These digital data were filtered to remove the instrument and background noise. The damping ratio can be evaluated by: (5.1) 2y a1÷ Where a, is the acceleration amplitude in the ith cycle, a11 is the acceleration amplitude 116 in the (i+j)th cycle. souUi end Center North end West end Center East end \ __ East wall South wall Figure 5.12 Hammer impact locations on the top plates of east and south walls 2) In the square-wave impulse test, the damped free vibration of the building was triggered by a square-wave impulse. Each square-wave test consisted of a couple of impulses and one decayed response of the building was obtained from each impulse. Using Eq. (5.1), the damping ratio could be easily calculated. And the natural period of damped free vibration was determined from the vibration curves by counting the time period for a single cycle of vibration. 3) The shake table tests used the modified 1995 Kobe JMA N-S record. In the first shake table test, the peak ground acceleration (PGA) was scaled down to a low level so that no structural damage would be expected and the structure was expected to deform almost in a linear elastic range. This test stage was also used to check if all the test facilities and the data acquisition system functioned well. The second shake table test (KOBE100NS-1) used the full-scale Kobe JMA N-S record with PGA of 0.82 g. with the intention to assess the seismic performance of the building subject to severe ground ( I c II - ‘ ‘V/ 117 shaking. Under such intensive ground motions, a typical wood building would have some significant nonlinear deformations, especially in some critical connections with concentrated loads. Structural damage might be accumulated and the stiffness degradation would be expected in the building. If the building would not collapse in the KOBE100NS-l test, the KOBE100NS-2 test would continue to assess the residual seismic capacity of the building again with the full-scale Kobe JMA N-S record. If the building would experience a lot of significant damage in the KOBE 1 OONS- 1 test, it would be retrofitted as much as possible, for example, replacing the damaged components and fixing the critical connections. 5.4.3 Test results and discussions 5.4.3.1 Hammer impact tests Hammer impact tests were carried out at each test run. The natural period and the damping ratio were estimated by the decaying acceleration curves from the accelerometers mounted on the top of the walls (Fig. 5.13). Table 5.3 gives the test results on six locations in terms of period T and damping ratio . It was found that the impact hammer did not have enough power to trigger the vibration of the entire building because it was hand driven. Therefore, the dynamic characteristics obtained in these tests were unfortunately not representing those of the building but the impacted structural components such as the wall top plates. It was obvious that the recorded frequency was much higher than that of a normal wood building. For example, the striking on the center of the top plate of the south wall gave a period of only 0.0042 sec while a normal timber 118 building has a fundamental period between 0.1 to 0.5 second. The recorded average damping ratio was about 4.9 %, showing the energy dissipation property of those timber members under damped free vibrations. 2 1.5 I ----1i--f 0.5 0 J I Lo.$\j\Jo.06 0M9 Q.12 0. -1 —--——- -1.5 Time (sec) Figure 5.13 An example of acceleration response in hammer impact test Table 5.3 Summary of hammer impact test results Impact 1st Run 2nd Run 3rd Run 4th Run location T(s) (%) T(s) (%) T (s) (%) T(s) (%) S. end 0.0104 5.6 0.0103 5.1 0.0104 4.5 0.0104 5.5East wall Center 0.0117 4.0 0.0118 4.3 0.0117 4.5 0.0120 5.5 N.end 0.0121 5.0 0.0105 5.7 0.0120 5.0 0.0101 4.2 h E. end 0.0105 4.4 0.0106 4.0 0.0108 4.8 0.0109 3.7OUt Center 0.0043 4.7 0.0042 5.7 0.0043 5.1 0.0044 5.1 wa W.end 0.0103 4.4 0.0132 4.8 0.0105 5.2 0.0103 6.1 Average 0.0099 4.7 0.0101 4.9 0.01 4.8 0.0097 5.0 5.4.3.2 Square-wave impulse tests In each test run, square-wave impulse tests were also conducted to evaluate the natural period of damped vibration of the building as well as the damping ratio. Again, the decaying acceleration response recorded by the accelerometers A3D- 1 and A3D-2 were analyzed in the time domain and the frequency domain through the fast Fourier transform (FFT). Raw digital data were filtered by a low-pass filter to remove the high 119 frequency noise. Eq. (5.1) was also used to determine the damping ratio. For each test run, one square-wave test consisted of three impulses, each having one decaying curve. Table 5.4 shows the average values of the fundamental period and damping ratios of the building in four test runs. Figure 5.14 gives an example of the decaying acceleration response measured by A3D-1 mounted on the top of the west wall along the shaking direction. And Fig. 5.15 shows the corresponding Fourier amplitude spectrum. Usually the frequencies corresponding to the peak amplitudes can be regarded as the natural frequencies. In this case, the fundamental frequency was approximately 3.7 Hz. Table 5.4 Summary of square-wave impulse test results Freq. domain Time domain Damping ratioTest run (Hz) T (see) (%) 1 3.68 0.285 4.7 2 3.68 0.274 4.4 3 3.04 0.313 3.9 4 2.74 0.327 3.9 0.02 0.015 0.01 a) 0.005 0 0 __ a, •! -0.005 C., -0.01 -0.015 -0.02 Time (sec) 1 I I I I I - - ::rzz Figure 5.14 An example of acceleration response in the square-wave impulse test 120 25 (0 20 C., 0 . 11: 15 Figure 5.15 Fourier amplitude spectrum of the acceleration response The square-wave impulse test results from the first run and the second run showed that the natural period and the damping ratio of the building did not change much after the first shake table test because it was a non-destructive test and was not supposed to change the dynamic characteristics of the building. However, during the KOBE100NS-1 test with PGA of 0.82 g, some nonlinear deformation occurred and slight stiffness degradation was observed, as indicated by the square-wave impulse test results. The recorded natural period increased from 0.28 sec to 0.31 sec after the KOBE100NS-1 test. Again, after the KOBE1 OONS-2 test, the natural period increased from 0.31 sec to 0.33 sec. However, the damping ratio did not change. In this study, the square-wave impulse test was a more reliable method to evaluate the natural period and the damping ratio of the building because the vibration of the entire building was triggered, thus making it possible to measure the system dynamic parameters. 5.4.3.3 Shake table tests In the first shake table test, the input table acceleration had a small PGA of 0.069 g 121 0 5 10 Frequency (Hz) 20 0.1 0.08- 0.06 0.04 0.02 -0.06 - -0.08 - -0.1 0.1 0.08 0.06 : 0.04 0.02 a, • -0.02 -0.04 -0.06 -0.08 -0.1 (Fig. 5.16). Under such low ground motions, no damage would be expected in this building. As shown in Fig. 5.17, the measured peak roof acceleration response by the accelerometers A3D-1 and A3D-2 was only 0.079 g, 14% higher than the peak table input acceleration. The peak wall drift was only 2.7 mm, 0.1% of the storey height. The structural deformation was very small and no damage was observed. Time (sec) Figure 5.16 Table acceleration input with scaled PGA of 0.069g North end - -IH end - - ----i Li --i- ll I L1 i - F. • 15 20- • I I I ‘EEEETEEHEH Time (sec) Figure 5.17 Acceleration response of roof (north end and south end) In the KOBE100NS-l test, the input table acceleration had a high PGA of 0.82 g. Figures 5.1 8a to 5.1 8c show the table input of the acceleration, velocity and displacement. 122 The peak table velocity was 68 cm/sec and the peak table displacement was close to 20 cm. Figure 5.1 8a Table acceleration input in KOBE 1 OONS- 1 test zir 1==EJ Figure 5.18b Table velocity input in KOBE100NS-1 test 80 60 0 40 E , 20 o 0 0 -20 -40 I -60 -80 Time (sec) 123 Time (sec) Figure 5.18c Table displacement input in KOBE100NS-1 test Figure 5.19 shows the Fourier amplitude spectrum of this accelerogram which shows that the input energy was concentrated between 1 Hz and 7 Hz and reaches peak at the frequency of 6 Hz. The acceleration response spectrum with 5% damping ratio also shows that a linear system with natural period of 0.2 second would have the peak response (Fig. 5.20). Since the building had the natural period of 0.28 second, which is near the main frequency content of the input, one would expect high structural response from the excitation. 0 E U -C 4- Cu I ‘a) I -o U- Figure 5.19 Fourier amplitude spectrum of input accelerogram in KOBE100NS-1 test 124 20 15 - E . 10 - . -5- .0 Cu -15 - -20 700 600 500 400 300 200 100 0 F I F F I I I F I I I I I -4 -I -I- I F I I I I I I I I I F I I I I I F I F I I I I I F I I I I I F I I I I I I I I I I — — -4 4- U F [ F 0 2 4 6 8 10 12 Frequency (Hz) 14 16 18 20 Period (sec) Figure 5.20 Acceleration response spectrum of table input (5% damping ratio) This building demonstrated very good seismic performance in the KOBE 1 OONS- 1 test. No significant structural damage was observed. The measured peak deformation of the critical diagonal brace connections was about 4.3 mm under tension and 1.4 mm under compression (Fig. 5.21). As recalled in the biaxial pushover test of the first building, the recorded peak deformation of diagonal brace connections was around 19 mm under tension and 5 mm under compression, much higher than the deformation recorded in this test. The measured peak roof acceleration was about 0.873 g and the magnification factor was 1.065 with respect to PGA of 0.82 g (Fig. 5.22). The maximum wall drift parallel to the ground shaking was about 37 mm, 1.4% of the storey height (Fig. 5.23). It was found that the south wall and the north wall which were parallel to the shaking direction had almost the same acceleration and drift response. The maximum difference of the drift response from these two walls at the same moment was only 1.2 mm. The maximum in-plane twist angle of the roof diaphragm was only 0.0002 degree which could be negligible. This implied that the difference between the stiffness of these 125 two walls was very small under the applied seismic loads. Despite the different opening size in the south wall and the north wall, the same shear wall effective length and the similar lateral stiffness of these single-brace walls led to the symmetric structural response in this test. 5 3 E -1 -3 -5 Figure 5.21 Slide responses of diagonal brace connections in KOBE100NS-l test EflEEH E Figure 5.22 Acceleration responses of roof diaphragm in KOBE100NS-l test 0.8 —Northend 0.6 —Southend EJfr ____ C.) -0.6 -0.8 Time (see) 126 50 40 30 _____________ 20 E 10 -- 20 -20 -30 -- I -40 -50——-- —— Time (sec) Figure 5.23 Drift responses of north wall and south wall in KOBE100NS-1 test Since the KOBE 1 OONS- 1 test did not cause significant structural damage in the building. The KOBE 1 OONS-2 test was carried out with the same seismic input as used in the KOBE 1 OONS- 1 test with the intention to evaluate the remaining seismic capacity of the building after it had experienced one severe earthquake attack. No structural retrofitting was done. In this test, the building experienced larger nonlinear deformation. Significant nail withdrawal from the BP plates was observed due to the tension of the diagonal brace (Fig. 5.24). The measurements from cR5 to cH8 showed that the maximum deformation of the diagonal brace connections was 5.7 mm under tension (Fig. 5.25), 33% greater than that in the KOBE100NS-1 test. And the maximum deformation under compression was 2.0 mm, 43% greater than that in the KOBE100NS-1 test. The recorded peak acceleration on the roof level was about I .28g and the magnification factor was 1.56 with respect to PGA of 0.82 g (Fig 5.26). Peak wall drift along the shaking direction increased to 46.6 mm, 1.8% of the story height (Fig. 5.27). The peak wall drift was 26% higher than that in the 127 • -NohwaII -SouthwaII -- I_ i- - - - - - - (1- • fri KOBE100NS-1 test. These observations indicated the stiffness degradation of the building. The square-wave impulse test followed by this test also showed the elongation of the natural period due to the structural “softening”. Nevertheless, this building showed very good seismic performance under the second severe simulated ground shaking. The peak wall drift ratio was smaller than 2%, a commonly recommended performance limit for wood buildings with respect to the life safety performance expectation. Figure 5.24 Structural damage in KOBE100NS-2 test Time (sec) Figure 5.25 Slide responses of diagonal brace connections in KOBE100NS-2 test 28 1.5 1.2 0.9 0.6 0.3 0 0 <-0.6 -0.9 -1.2 -1.5 Figure 5.27 Drift responses of north and south walls in KOBE100NS-2 test 5.5 Summary This chapter presented the experimental studies on two identical full-scale one-storey P&B wood buildings which were constructed with Canadian Hem-fir, grade stamped as Canada Tsuga E120, following Japanese building practice. The objective was to study the lateral performance of the P&B buildings on a system level. The first building was tested under monotonic pushover loads along two perpendicular horizontal directions. Structural response such as wall drifts, connection -j —Northend ——South end H Time (sec) Figure 5.26 Roof acceleration responses in KOBE100NS-2 test 50 40 30 20 .gio = -10 -20 - -30- -40- -50 - SouthwaII —Northwall ‘z’zzzzz:z: ----0 2 — — — — — T I I I I I Time (sec) 129 deformation as well as failure modes was carefully measured and investigated. Test observations and results showed that the single-brace walls play the most important role to provide the lateral resistance of the building. Most of nonlinear deformation and damage occurred within the diagonal brace connection zones as observed in the single- brace shear wall tests. Ideally, the tested building was structurally symmetric because the four side walls had the same effective shear wall length. However, the test results showed the twisting of the roof diaphragm due to the stiffness difference between individual walls. This was caused by the variations among the wall mechanical properties and the construction quality. The second building was tested dynamically on a shake table. The input accelerogram was the modified 1995 Kobe JMA N-S record with PGA of 0.82 g. This building was severely shaken twice with the accelerogram. It was found that the building performed very well in the first strong shaking (KOBE100NS-1 test) and no severe structural damages were observed. The recorded peak wall drift was about 37 mm, 1.4% of the story height. No repair work was done after the first strong shaking. In the second strong shaking (KOBE100NS-2 test), the building had larger deformations. Significant nail withdrawals were observed in the diagonal brace connections in the walls. The measured diagonal brace connection deformations were about 30% more than the measurements in the first strong shaking. The peak wall drift increased to 46.6 mm (1.8% of the story height) and was 26% higher than that in the KOBE100NS-1 test. The square wave test results also showed the elongation of the natural period due to the structural 130 “softening” in the KOBE100NS-2 test. Nevertheless, the peak wall drift was still lower than the commonly used inter-story drift criterion of 2% for major earthquake attacks. Therefore, it is believed that this building has demonstrated good seismic performaxice and good capability to absorb large earthquake energy. 131 CHAPTER 6 “PB3D” MODEL VERFICATION 6.1 Introduction In this chapter, the “PB3D” model predictions were compared with a test database in order to check the validity and robustness of the model. The test database includes the test results of the static test and the shake table test of two one-storey P&B buildings presented in Chapter 5 and a shake table test of a two-story P&B building tested in Japan and provided by the Building Research Institute of Japan. The building response of interest, such as inter-story drifts, roof/floor acceleration, and base shear, was chosen for the comparisons and verifications. 6.2 Model verification by one-storey buildings A “PB3D” model was built to simulate the structural response of the one-story buildings tested under the static pushover loads and the seismic loads. Figure 6.1 shows the schematics of the model. In the dynamic analysis, the structural mass, which includes the gravity load of 100 kg/rn2 and the self-weight of the roof assembly, was lumped onto the roof nodes based on their tributary areas. A total of eight “pseudo-nail” walls were used to model eight single-brace walls. The model parameters of the single-brace wall were calibrated based on the test database of the single-brace walls presented in Chapter 4. Table 6.1 gives the calibrated twelve “pseudo-nail” model parameters including two sets of embedment properties to consider the nonsymmetrical response of the wall under pushing or pulling. Figure 6.2 and Figure 6.3 show that the model prediction of load-drift 132 hysteresis and energy dissipation agreed reasonably well with the test results. However, it can be noted that the “pseudo-nail” model did not account for the observed strength degradation upon the repeated drift demands in the same cycle group. The discussion about this limitation of the “pseudo-nail” model has been presented in section 3.2.1 of Chapter 3. Figure 6.1 “PB3D” model of the tested single-story building Table 6.1 “Pseudo-nail” model parameters of the single-brace wall Single-brace wall Parameters Brace under Brace under tension compression Qo(kN/mm) 0.5872 1.7355 Qi (kN/mm2) 0.06 0.0021 Q2 1.399 1.07 K (kM/mm2) 0.0983 0.0787 Dmax(mm) 58.236 87.236 L(mm) 101.25 D(mm) 4.13 Zt4Y / 133 Wall A Model is—- - .1 .-.-,-,i, o 3 6 9 12 15 18 21 24 27 30 33 36 Cycle No. Wall A-4 Test 16--- 12 8 4z -200 -8 -12 12 - -- ;?---.-_ 8- ////7..4//j //‘ 4-z 0 50 100 150 200 .S-200 50 100 150 200 Figure 6.2 “Pseudo-nail” model predicted loops vs. test results -12 - 800 H7O0- 600 500 0 4- 0. 300- 2’ 200- a) w 100. 0 Figure 6.3 “Pseudo-nail” model predicted energy dissipation vs. test results The size of the equivalent diagonal braces in the roof diaphragm can be calibrated via an experimental study or a detailed finite element model. In this study, the commercial software package ANSYS was used for its calibration. One quarter of the roof diaphragm was modeled due to the structural symmetry along both horizontal directions. The roof beams and rafters were modeled by the 3D Timoshenko beam elements with the assumption of linear isotropic material property. The MOE of the wood members was 12 GPa. The plywood roof sheathing panels, 12.5 mm in thickness, were 134 modeled with shell elements with the assumption of linear orthotropic material property. The MOE along two in-plane perpendicular directions were 10.2 GPa and 6.8 GPa and the in-plane shear modulus was 600 MPa. The panel-frame nail connections along two lateral perpendicular directions were modeled by linear springs with stiffness of 130 N/mm. Nail withdrawal along the vertical direction of the panels was not considered. In the equivalent diagonal braced roof diaphragm, rafters, plywood panels and nail connections were replaced by two cross diagonal braces. The brace geometry was adjusted until the same in-plane stiffness was achieved. In this study, the diagonal braces were calibrated to be 30 mm x 30 mm Hem-fir lumber and were pin connected with the roof beam nodes (Fig. 6.4). N0AL OULtJTION STEP=1 sus =1 TI91 UY (AVO) R5ys0 .075221 003092 SNX =. 074101 AN —,0O3Ot2 01400 005494 RO2F tJIA10RAG9) MUa.V53S .032552 049423 .055595 135 NODAL SOLUTION Figure 6.4 Calibration of equivalent braces of the roof using ANSYS software 6.2.1 Static model verification In this analysis, the displacement control mode was used for loading, which was consistent with the static test. Specified displacement was enforced onto the top of four side walls along the two horizontal directions simultaneously until it reached the peak drift of 105 mm, close to the peak drift recorded in the static test. Then, the structure was retracted to the initial zero position. The whole loading history was a half cycle of the reversed cyclic loops. The reaction forces along two directions were calculated based on the system equilibrium and were compared with the actuator loads recorded in the test. In this model, due to the loading and structural symmetry, the structural response was expected to be symmetrical and no torsional effect of the roof diaphragm was observed. However, the test results indicated that the lateral forces were not evenly distributed 5ThP= 1 SUS =1 TI1=1 Uy AVU R5YNO DNX - .075919 SOUl ORX .O74i1 AN — 113504 009150 NC EUr,oL5NT OO.0CE ANALYSIS 035934 04940 04117 136 among the walls due to the stiffness difference between the walls, thus leading to the twisting of the roof diaphragm. Figure 6.5 shows the comparison between the model predictions and the test results in terms of the load-drift relationship along east-west direction. The model prediction indicated higher stiffness and higher peak load than the test results. The peak load applied on both south wall and north wall was about 36 kN, 14% higher than the test result. It might be explained by the inherent variation of the materials and the construction details between the walls in the shear wall tests and the walls in the building test. Besides, it should be noted that in the shear wall tests, for each single-brace wall, four hold-down devices were installed at four corners, while only three hold-down devices were installed for each single-brace wall in the building, as shown in Fig. 5.3. Plus, at the corners of the building, two single-brace walls shared the corner post. Under the biaxial loading, one corner post was subject to much higher uplifting force than the others. The above facts might account for the reduced lateral strength and stiffness of the walls in the building. Another important reason was that in the shear wall tests, no vertical dead load was applied on the top of the wall while a gravity load of 100 kg/m2 was applied on the roof of the building and each single-brace wall carried 400 kg dead load approximately. This would also introduce the significant P-A effect in the walls and reduce the lateral stiffness and strength of the building. 137 Wall drift(mm) Figure 6.5 “PB3D” model predicted wall response against test results 6.22 Dynamic model verification A dynamic analysis was also performed to simulate the seismic response of the building in the shake table test. Based on the vibration test results of the building, the viscous damping ratio of 4.7% was used to construct the damping matrix of the model. The total structural mass of 3200 kg was lumped on the roof nodes based on their tributary areas. The input accelerogram was the recorded table accelerogram with PGA of 0.82 g, as shown in Fig. 5.18a. Due to the structural symmetry and the loading symmetry, the predicted structural dynamic response was also symmetrical. No torsional effect was predicted, as observed in the shake table test. The building oscillated like a single DOF system with the mass on the top under the uniaxial ground motion. The model predictions agreed well with the test results in terms of the response of inter-story drift and roof acceleration. The model predicted peak inter-story drift was about 41.5 mm, 1.5% of the story height. The peak drift of the walls recorded in the test was about 37 mm, 1.3% of the story height. The 138 50 40 30 20 E 10 -20 -30 -40 -50 50 40 30 20 10 0 E E -20 -30 -40• -50 model predicted peak inter-story drift was 12% higher than the test result (Fig. 6.6). The model predicted peak roof acceleration was about 0.928 g, 6.3% higher than the recorded peak roof acceleration of 0.873 g in the shake table test (Fig. 6.7). The acceleration responses at the north end of the roof predicted by the “PB3D” model and recorded in the test were processed by the FFT to obtain the Fourier amplitude spectra and to compare the frequency components. Figure 6.8 also shows a good agreement between the model predictions and the test results in the frequency domain. - Test (north wall) 1H Model - - 1 ---I- • -20 2 r + 5 Time (sec) Test (south waiiT Model 20-- iEt:’zzzz:zzz 5 Time (sec) Figure 6.6 Model predicted story drift responses vs. test results 139 0.8- 0.6- 0.4 0) C.) <0 •1- 0 0 -0.4 - -0.6 - -0.8 - 700 600 2 £ 500 400 300 e 200 I 2 100 0 0.8 0.6 0.4 0) C.) <0 -0.4 -0.6 -0.8 ,-;. Test(northend) Model _.__;__J. . - 1 EJlZJli:z:E Time (sec) Test(south end) - L - - - E1E1zzz::zz::::z f_q7•ff( Time (sec) Figure 6.7 Model predicted roof acceleration responses vs. test results Test (north end) -— I I I Model I I I I I I I I I I I I r I I I I I I [ I I I I I I IjFtk I I I I ,cj I r I I I I I r I I - I • I I I I I I • I I I I I 0 2 4 6 8 10 12 14 16 18 201 Frequency (Hz) Figure 6.8 Fourier amplitude spectra of roof acceleration response (one-story building) 140 6.3 Model verification by a two-storey building A shake table test on a two-story P&B building was conducted in Japan and the test results were provided by the Building Research Institute of Japan for this study (Fig. 6.9). This test database was also used to verify the “PB3D” model by comparing the model predictions with the test results in terms of the time-history response of base shear, inter- story drifts and roof/floor accelerations at different locations of the building. Figure 6.9 Shake table test on a two-story P&B building 6.3.1 Building description This two-story building was designed to comply with the minimum requirements in the BSL. It had a plan size of 7.28 m x 7.28 m and a total construction area of 106 m2. The main timber frame was built with the European whitewood glulam. The height of the first story was 2.95 m and the height of the second story was 2.83 m. The exterior shear walls were sheathed by 7.5 mm thick plywood panels, stamped as JAS grade 2 panels. The interior walls were diagonal-braced walls with 45 mm x 90 mm diagonal braces. 141 GWBs with thickness of 12.5 mm were also used for some exterior and interior walls. Figure 6.10 shows the wall layouts of the first story and the second story. I I I I I I I I I I I I I lP. ‘i -JI III III I / IILl ILl j— I III L-———•———-J I I- ..i I I I L I I 4- I - + 4 I!’ y I- t I 7280 I First story Second story Note: — shear wall (with 7.5mm thick plywood panels): shear wall (with one 45x90 mm diagonal brace); quasi-shear wall (with 12.5mm thick GWB). Figure 6.10 Shear wall layout of the two-story building Table 6.2 gives the effective shear wall lengths and the eccentricity ratios of each story along two horizontal directions. In the BSL, the calculation of the effective shear wall lengths for a wood building considers only the contributions of the strictly defined “shear walls” which usually refer to the diagonal-braced walls or the walls fully sheathed with structural panels. However, the Quality Assurance Law (QAL) considers the contributions from the so called “quasi-shear walls” which usually refer to the structurally less important walls, such as walls sheathed with GWB, particleboard, mud walls, and walls partially sheathed with structural panels satisfying certain requirements. 142 The quasi-shear walls usually have lower capacities and smaller shear wall multipliers. But their contributions to the structural lateral resistance cannot be neglected in order to make a reasonably accurate prediction of the seismic response of the buildings. This building can be ranked as a building with Rank 1 seismic performance because the effective shear wall lengths of the first story just met the requirement of a Rank 1 building. The eccentricity ratio of the first story was 0.20, smaller than the code recommended limit of 0.30. Table 6.2 Effective shear wall length and eccentricity ratio of the two-story building Required Existing effective . . . EccentricityType Storey & direction effective shear shear wall length ratio wall length (m) (m) Building 2nd storey X 1 1 13 16.84 (151%) 0.14 Standard y 16.84 (151%) 0.14 Law St x 19.11 (109%) 0.20(BSL) 1 storey 17.49 18.66 (107%) 0.15 Quality 2 storey X 14.51 22.84 (157%) 0.15 Assurance y 22.84(157%) 0.15 Law St x 25.16 (101%) 0.19 ‘R k i 1 storey 24.87an ‘ y 25.84 (104%) 0.15 Note: numbers in the parentheses are the ratio of the existing amount and the required amount. Table 6.3 gives a list of shear walls (plywood-sheathed / diagonal-braced) and quasi- shear walls (GWB-sheathed) with different lengths used in this building. Table 6.3 Types of shear walls and quasi-shear walls used in the two-story building No. Wall type Length (m) Shear wall multiplier 1 Plywood-sheathed only 0.91 2.5 2 GWB-sheathed only 0.91 0.5 3 Plywood + GWB sheathed 0.91 3.0 4 Single 45x90mm diagonal brace 0.91 2.0 5 Double 45x90mm diagonal braces 1.82 2.0 143 In the floor diaphragm, 105 mm >< 210 mm and 105 mm x 270 mm timber members were used as the floor beams and 45 x90 mm members were used as the floor joists. Most of the floor beams were spaced at 1820 mm with a span of 3640 mm and the floor joists were spaced at 303 mm. On the top of the floor frame, structural plywood panels with thickness of 16 mm were installed with common CN5O nails (50.8 mm in length, 2.9 mm in diameter) spaced at 150 mm on the panel edge and 300 mm on the field. An inclined roof system with a slope of 26° was built in this building. The roof frame consisted of 105 mm x 270 mm tie beams, 105 mm x 210 mm purlins, 105 mm x 105 mm support posts, and 45 mm x 60 mm rafters. The tie beams and purlins were spaced at 1820 mm. The rafters were spaced at 300 mm. Plywood panels with thickness of 12.5mm were used as the roof sheathing. The roof system was a tile-covered heavy roof with self weight of 74.3 kN. Taking into account the half weight of the walls in the second story, the total roof mass was about 82.9 kN, approximately 160 kg/rn2 with respect to the total roof area. The floor system was loaded with additional dead loads of 55.4 kN. Plus the self weight of the floor and half of weight of the walls in the first story and the second story, the total floor mass was about 93.1 kN, approximately 180 kg/rn2with respect to the total floor area. The input ground motions were the modified 1995 Kobe JMA records with PGA of 0.72 g along x direction, 0.57 g along y direction, and 0.29 g along the vertical direction (Fig. 6.11). 144 0.8 0.6 - 0.4 - — 0.2 - 0 00 0 -0.4 - -0.6 - -0.8 0.8 0.6 0.4 C) oO 0 -0.4 -0.6 -0.8 7E L± X direction (horiztonal) J Time (sec) Y direction (horizontal) __________________ 0.8 - — I ---—--—--—--------- I 0.6 0.4 0.2 - : L i1.tAAN.ftAa,. - -: -0.4 -0.6 -0.8 Time (sec) Z direction (vertical) Time (sec) Figure 6.11 Table input accelerogram in shake table test of the two-story building 145 6.32 Dynamic model verification A “PB3D” model was built to simulate the seismic response of the two-story building (Fig. 6.12). Since the configuration of the floor diaphragm of this building was similar to the tested floor diaphragm introduced in Chapter 4, in the model, the diagonal braces of the floor diaphragm were assumed to be 37 mm x37 mm lumber. Also in the model, the diagonal braces of the roof diaphragm were assumed to be 30 mm x30 mm lumber, which were the same as used in the one-story building model presented in the preceding part of this chapter. Figure 6.12 “PB3D” model of the two-story P&B building In this study, due to the lack of test data on the plywood-sheathed 0.91 m walls built with European whitewood glulam, the “pseudo-nail” model parameters of these walls were calibrated based on the assumption that the load-carrying capacity of the 0.91 m 146 wall is 50% of that of the 1.82 m wall because in the P&B buildings, a panel-sheathed 1.82 m wall is structurally symmetric and consist of two identical 0.91 m wall segments. The test data on the 1.82 m plywood-sheathed walls, 1.82 m GWB-sheathed walls, and 1.82 m double-brace walls, for calibration purpose, were provided by Japanese colleagues (CBL, 2001; Okabe, 2002). For a wall sheathed with both plywood and GWB, it was assumed that the wall capacity was the sum-up of a plywood-sheathed wall and a GWB sheathed wall. To calibrate the nonsymmetric single-brace walls built with European whitewood glulam, the test results of the single-brace walls presented in Chapter 4 were used. The load-drift curve for the whitewood single-brace wall was obtained by adjusting the curves of the Hem-fir single-brace wall based on the ratio of E values between whitewood and Hem-fir. Only a small number of single-brace walls were used in the building, which would not introduce much error for the “PB3D” model prediction. Table 6.4 gives the “pseudo-nail” model parameters of the five types of walls in this building. Table 6.4 “Pseudo-nail” model parameters of the walls in two-story building 0.91 m 0.91m single-brace0.91m 0.91m 1.82m Parameter plywd GWB plywd+G brace brace double- sheathed sheathed under under brace sheathed tens. comp. Qo 1.111 0.733 1.571 0.5562 1.6387 5.906(kN/mm) Qi 2 0.005 0.0001 0.009 0.049 0.0019 0.0021(kN/mm) Q2 1.282 1.522 1.222 1.399 1.07 1.192 K 2 0.141 0.103 0.162 0.0854 0.0725 0.179(kN/mm) Dinax (mm) 63.48 38.03 70.24 56.33 84.19 58.04 L(mm) 140.34 80.10 150.41 103.67 129.55 D (mm) 5.38 3.69 6.52 4.09 6.22 147 In the seismic analysis, the roof mass of 160 kg/rn2 and the floor mass of 180 kg/rn2 were applied and lumped onto the roof/floor nodes based on their tributary areas. Damping ratio of 5.0% was used. The modified 1995 Kobe JMA record (x and y components) were used as the input excitations, as shown in Fig. 6.11. Since the “PB3D” model camiot consider vertical ground motions, the ground motion along z direction was neglected in this study. In the shake table test, this building was subject to very intensive ground motions and experienced very large deformation at around 5 seconds after the shaking started. The recorded maximum inter-story drift of the first story exceeded 150 mm. The building had significant twisting and the collapse was imminent. It was also found that some sensors could not record the large deformation which exceeded their measuring limits. Therefore, only the responses in the first ten seconds were used for comparison purpose. Figure 6.13 shows the comparisons between the model predicted base shear forces and the test results along the x and y directions. The predicted peak base shear along the x direction was about 96.3 kN, 8.7% lower than the recorded 105.5 kN in the test. The predicted peak base shear force along the y direction was about 111.7 kN, 16% lower than the recorded 132.9 kN in the test. 148 Base shear - X direction 100• — 50 z 0 0 0 LI -50 -100 -150 150 Test Model :f_ - - Time (sec) DI Figure 6.13 Model predicted base shear force responses vs test results Figures 6.14 and 6.15 show the comparisons between the predicted acceleration responses along x and y directions at the mass centers of the floor and roof diaphragms and the test results. The predicted peak acceleration of the floor along x direction was 0.83 g, 7.8% higher than the recorded 0.77 g in the test; the predicted peak floor acceleration along y direction was 0.76 g, 15% higher than the recorded 0.66 g in the test; the predicted peak roof acceleration along x direction was 0.82 g, 5.1% higher than the recorded 0.78 g in the test; and, the predicted peak roof acceleration along y direction was 0.98 g, 17% higher than the recorded 0.84 g in the test. 149 0.8 0.6 0.4 — 0.2 ., 0 C., <-0.2 -0.4 -0.6 -0.8 Time (sec) Time (sec) Figure 6.14 Model predicted floor acceleration responses vs test results —1 Roof acceleration - X direction H -Test -- Model ____ - Zr:: E E E E E E E E Time (sec) Floor acceleration - X direction 0.8 0.6 0.4 0.2 0 C.) -0.4 -0.6 -0.8 H _____ Test _ ___ 7 . / 2 4- --6 8 1 :——I • - - - Floor acceleration - Y direction Test H Model --f :• ; ri,.(. . ? 4T, i- i• fr _ .. -. S ‘‘ -- - - ‘- -- - - - -6- 8 . 0.8 0.6 0.4 .0.2 C.) <-0.2 -0.4 -0.6 -0.8 150 1 Roof acceleration-Y direction 0.8- Test 0.6 Model 0.4 _ - -1 Time (sec) Figure 6.15 Model predicted roof acceleration responses vs test results The inter-story drift response at 12 different locations was also compared against the test results. Figure 6.16 shows the locations of the sensors to measure the inter-story drifts of the first story and the second story. Figure 6.17 shows the comparisons between the predicted inter-story drift responses of the first story against the test results. Reasonable agreements can be seen within the first five seconds before the building experienced the substantially large deformation of more than 150 mm, around 5% of the story height, well above the commonly recommended collapse prevention criterion of 3%. Both the model predictions and the test results showed very significant twisting of the building due to the irregular layout of the wall system and the severe ground shaking. The model predicted peak inter-story drift of the first story was about 200 mm, also indicating the near collapse state of the building. Figure 6.18 shows the comparisons between the predicted inter-story drift responses of the second story against the test results. The model predicted peak inter-story drift of the second story was 51.7 mm, 6% lower than the recorded peak inter-story drift of 54.8 mm in the test. 151 P DG6 —- DC5 _________ --s---- ___ - Dd3 .First story Figure 6.16 Definitions of sensors to measure inter-story drifts (DG1DG6, DG9—DG14) 200 150 100 50 E I -50 -100 -150 -200 200 150 100 50 1 -100 -150 -200 DG2 - X direction Time (sec) I I ---.- t ,DCl 7 — —e-———-- -m-—---o---— I I ii c — - I r;— ;-—-‘-- IN DC9 \ I ,“ .-—i ili I_ -I L I I I I —-I ÷ I I I I rizzii DG2 DCI 4 P —I DG1 0 r DC4 I I — DG1 2 DG13 m DG11 Second story DGI- X direction Test ModeI 1’ - - -r\- - I i / / \ \:/ •. -——- ‘ - \________________ —, k.— -- -/‘ . . -2 4 i i8 1 0 Time (sec) Model S... j/-- - 152 DG3 - X direction 200 ___________ 150 Test - Model 100 ----- •——-__ 50 2 0 --------— -50. -100 -150 -200 __ _____________________ Time (sec) DG4- Y direction 200 150 100 E 50 go 50 -100 -150 -200 Test • Model — - - k- - - - - i’•i I I Time (sec) DG5 - Y direction 200 150 Test 100 ModeI — - --- /___E I _ — I I • -. • • -50 -100 -150 -200 -—-----— I I Time (sec) 153 200 150 100 50 E ______ _________________ L.. -50 —100 -150 -200 Time (sec) Figure 6.17 Inter-story drift response of the first story (DO 1 DG6) DG6 - Y direction Test -- Model z’z:zzz DG9- X direction 154 DGII- X direction 60 _______ Test 1 40 Model .20 —-- :_____ E I\/ ) \E ,‘\ /.‘0 —- - —— — 2 :6.. 8 1 -20 — -------, -40 - - -60 — —__________________ 155 Time (sec) C.) I LL DGI4- Y direction 60 40 20 E E -20 -40 -60 Time (sec) Figure 6.18 Inter-story drift response of the second story (DG9DGl 4) The acceleration responses at the mass centers of the floor and roof diaphragms predicted by the “PB3D” model and recorded in the test were processed by the FFT to obtain the Fourier amplitude spectra and to compare the frequency components. Figures 6.19 and 6.20 show a reasonably good agreement between the model predictions and the test results in the frequency domain. Floor acceleration - X direction Model - i 0 24 6810121416 1820 Frequency (Hz) 600 500 400 300 200 100 0 156 Floor acceleration - Y direction 600— - I Test 2 500 Model -- 400 - 300 -f-- 2 200 -4- - -- - - 100 0 i820 Frequency (Hz) Figure 6.19 Fourier amplitude spectra of floor acceleration responses (two-story building) Roof acceleration - X direction 6001 ___ Test E 500 Model1 400 300 2 200 __ I100 - 1o L1 j ‘p’ 0 d ____________ ____ — _____ 0 2 4 6 8 10 12 14 16 18 20 Frequency (Hz) Roof acceleration - Y direction 600 - Test 500 Model1I I I _ ______ 400 a 300 2 I 200 --—--—--‘ __ L L__J :; ‘•I 100 LI.. I — I I 0 0 2 4 6 8 10 12 14 16 18 201 Frequency (Hz) Figure 6.20 Fourier amplitude spectra of roof acceleration responses (two-story building) 157 6.4 Summary This chapter presented the comparison between the “PB3D” model predictions of two P&B buildings (e.g., base shear force, roof/floor acceleration, inter-story drifts) and the test results. The first building was a one-story building subject to biaxial static loads and uniaxial ground motions. The second building was a two-story building subject to tn- axial ground motions. The “pseudo-nail” model parameters of the single-brace walls in the one-story building were calibrated based on the test results of the O.91m single-brace walls. The equivalent diagonal braces for the roof diaphragm were calibrated by a detailed finite element model developed in the commercial software package ANSYS. By integrating the wall models and the equivalent roof diaphragm, the “PB3D” model was able to simulate the static and dynamic response of the one-story building. Comparisons between the model simulation results and the test results showed 12% error in the peak inter-story drift response and 6.3 % error in the roof acceleration response. The “pseudo-nail” model parameters of the walls in the two-story building were calibrated based on a shear wall test database established at the University of British Columbia as well as a test database provided by Japanese colleagues. The “PB3D” model of the two-story building considered two horizontal ground motions and neglected the vertical ground motion although the building was subject to triaxial seismic inputs in the shake table test. The model predictions showed around 5% 16% errors compared with the test results in terms of the peak base shear responses and the peak roof/floor 158 acceleration responses. The time-history responses of the inter-story drifts also agreed well with the test results within the first five seconds. However, the inter-story drift data recorded by some transducers (e.g., DG 2, DG3 and D04) were corrupted after five seconds because the building experienced very large deformation and was near collapse, which also makes it difficult for comparison after five seconds. The CUREE-Caltech Woodframe Project International Benchmark Study reported five numerical models to predict the behavior of a two-story woodframe building subject to uniaxial ground motions (Folz and Filiatrault, 2004b). Compared with the prediction error ranges of those models, it is believed that the ‘PB3D” model predictions of these two buildings showed a good agreement with the test results. 159 CHAPTER 7 SEISMIC RELIABLITY ANALYSIS OF POST- AND-BEAM SHEAR WALLS AND BUILDINGS 7.1 Introduction When it comes to estimating the seismic safety of buildings in a quantitative manner, the probabilistic or reliability-based approach represents not just a better option but a necessity because of the high uncertainties inherent in earthquakes and structural systems. These uncertainties involve the characteristic of earthquake ground motions (e.g., peak ground accelerations, frequency contents, duration of shaking), structural mass (e.g., live loads), material properties, construction types and maintenance quality. In the last decades, seismic reliability studies have mainly focused on steel structures and reinforced concrete structures. Limited work has been reported on the seismic reliability study of wood structures due to the difficulties in characterizing those uncertainties, defining the performance criteria, and developing computer models. In this chapter, attempt was made to estimate the seismic reliability of a series of P&B shear walls and a one-story P&B building and a two-story P&B building by running computer simulations using the “pseudo-nail” wall model and the “PB3D” building model. The response surface method (RSM) with importance sampling (IS) was used to approximate the structural seismic response with various combining uncertainties and to formulate the explicit limit state functions, thus estimating the probability of failure with respect to different performance expectations. 160 7.2 Earthquake hazards The major seismic risk to buildings is the structural vibrations caused by ground shaking, which can result in nonstructural and structural damage or even the worst scenario, structural collapse. In earthquake engineering, the structural response to the time variation of ground motion is of the great interest. Detailed structural response may include the time-histories of deformation, internal forces, and stresses. Since they are closely related to the extent of damage a structure may experience in an earthquake, they must be controlled by rational designs. For structural design in seismic-prone regions, it is a common practice to define the seismic hazard levels using probabilistic concepts. In North America, three seismic hazard levels are usually defined based on 50%, 10% and 2% exceedance probabilities within 50 years, the service life of common structures in many design codes. Their corresponding return periods are 75 years, 475 years and 2475 years, respectively. Accordingly, three performance levels are also stipulated: immediate occupancy, life safety and collapse prevention. Most building codes are mainly concerned with the life safety performance level. However, the trend towards the performance-based engineering in recent years aims to consider multi-performance levels with supporting test methods, analytical tools and rational understanding of structural limit states. In Japan, the seismic design code also implements a performance-based engineering framework. As introduced in the literature review, two performance objectives are defined: life safety and damage limitation (Kawai et al., 2006). To satisfi the life safety objective, one should design a 161 building such that neither the entire building nor individual story collapses. The damage limitation objective can be satisfied by controlling the damage level of the building where some permanent deformation is acceptable to dissipate the input earthquake energy without causing life safety problems. Two sets of earthquakes (maximum earthquakes and once-in-a-lifetime earthquakes) are considered, each having different exceedance probabilities. The maximum earthquakes with respect to the life safety performance level have a return period of about 475 years. These earthquakes are usually severe earthquakes. The once-in-a-lifetime earthquakes with respect to the damage limitation performance level have a return period of 30 50 years. These are usually moderate earthquakes. The term “seismic hazard” stands for the probability that the seismic intensity measure at a site exceeds a specified value during a period of time. The simplest probabilistic model to model the occurrence of earthquakes during a period of time is the Poisson process. Although the Poisson process provides an elementary model of the occurrence of earthquakes, it has been found to be consistent with historical earthquakes that are of engineering interest in structural applications (Algermissen 1983). In the U. S., the Poisson process is also used to map the seismic hazards and model the rate of earthquake occurrence (USGS, 2008). The Poisson process has the probability mass function (Pinto et a!. 2004). Pr{k}= e (7.1) where k is the number of earthquakes; t is the period of time (e.g. in years); and 2 is the 162 annual arrival rate of earthquakes. If no earthquakes occur over the time period t, Eq. (7.1) can be rewritten as: Pr(0)=e (7.2) Thus, the probability of occurrence of one earthquake or more is P.(k1)=l.0—e’ (7.3) Now consider the annual probability, i.e. t = 1 year. If the probability of occurrence of an earthquake with a specified intensity is E’ then the annual arrival rate of such an earthquake is2E This is called the compound Poisson procedure. Therefore, the annual exceeding probability becomes a =1.0—e (7.4) Given the statistics of seismicity at one site, the corresponding seismic intensity measures corresponding to different hazard levels can be determined using the Poisson process. Assumed that the annual arrival rate of earthquakes is 2 = 0.2 / year (one earthquake every five years) and the intensity measure such as PGA follow a lognormal distribution with mean a of 0.25 g and coefficient of variation (COV) Va of 0.55, two seismic hazard levels are considered: (1) Once-in-a-life earthquakes with a return period of 50 years. These are moderate earthquakes and are consistent with the exceedance probability of 63.5 % within a 50 year window. The annual exceedance probability of occurrence of an earthquake with the PGA greater than a1 is 163 Fa(aG >aGdl)=1.O—exp(—O.2FE(aG >aGd1))= Thus, PE(aG > a3 )o.ioi and the corresponding standard normal variate is RN = 1.271. Therefore aGdl = a 2 exp(RN Jin(i + v ))= 0.42gql + v (2) Major earthquakes with return period of 475 years. These are severe earthquakes and are consistent with the exceedance probability of 10% within a 50 year window. The annual exceedance probability of occurrence of an earthquake with the PGA greater than aGd2 is Pa(aG >aGd2)=1.O—p(—0.2PE(aG >aGd2)) Thus, PE(a > aGd2)0.0105 and the corresponding standard normal variate isRN = 2.307. Therefore, a,2 = 2 exp(RN ijin(i + v ))= 0.72g.ji + Va The Poisson process described here can be also used to relate the structural failure probabilities on the event base to the structural failure probabilities on the annual base or over a certain period of time, e.g., a service life of 50 years. 7.3 Response surface method with importance sampling The probability estimations of the structural performance of shear walls or buildings can be carried out using the Monte Carlo Simulation (MCS) with variance reduction techniques. For a well-designed structure with low failure probability F1, the crude MCS requires a large number of simulations to achieve reasonable estimations. For nonlinear dynamic problems of complicated systems, this approach might not be suitable since it is very computational intensive. In order to improve computational efficiency, researchers 164 have developed other approaches such as the RSM, local interpolation and neural network to estimate the seismic reliability of nonlinear systems. The RSM was originally proposed by Box and Wilson (1954) in order to find the operating conditions of a chemical process in which some of the responses were optimized. Later on, it was further applied to solve structural problems aiming to approximate the relationship between the structural input (loads and other structural parameters) and the structural output (response). This method replaces the actual structural response with explicit functions of the random variables of interest. In the case of studying the seismic performance of a building, the random variables can include ground motions, gravity load, and structural properties. By performing dynamic analyses on the sampling points of the random variables, this method can facilitate the seismic reliability analysis of complicated nonlinear structures by reducing the required intensive computational effort. The structural response R such as displacements, internal forces, inter-story drifts or damage indices can be represented as a function of a series of intervening random variables X1, X2,X3 .. R(X)=R(X,23...,X) (7.5) For a nonlinear dynamic problem, the implicit function R can be replaced by an explicit function using the RSM: F(X)=F(X1,23...,X) (7.6) Polynomial functions are commonly used for the explicit function: 165 F(X)=F0+aT(X—X)+(X—b(X (7.7) where F0 is the structural response evaluated at the random variable vector X0, a and b are unknown coefficient vectors: I a=a1,a23...,a) b1 (7.9) At least 2n+1 sampling points including X0 are needed to determine the coefficient vectors a and b. Wong (1984) proposed a complete quadratic polynomial response surface using the information obtained around the mean of random variables. The selected sampling points were located between p and p ± ka. p and a are vectors of mean and standard deviation of the random variable vector X, and k is an arbitrary factor (e.g. k = 1,2,3...). By this means, the characteristics of the fitted response surface are located around the mean of the random variables. To improve the prediction accuracy, Bucher and Bourgund (1990) proposed an adaptive interpolation scheme to reformulate the response surface. Following the above scheme to obtain the original response surface (x), the reliability index fi and the associated design point Xd corresponding to (x) can be determined. An update on the new anchorage point for the updated response surface is then obtained as Xa=P+(Xd_P) (7.10) g(p)- g(Xd) This process helps to locate the anchorage point closer to the true failure surface g(x). 166 Then, a new response surface using Xa as the new center point can be obtained by running additional 2n+l samples. In doing so, the vector X has been transferred to the standard normal space. In addition to the evaluation at X, this updated RSM requires 4n+3 sampling points in total. Now, consider the application of RSM in the seismic reliability analysis of the P&B structures in this study. Peak wall drift or peak inter-story drift, which is a rational indicator for structural damage under seismic loads, is commonly used to formulate the performance function: G=5—A(M,aQ,r,y) (7.11) where is the structural capacity, which in this case is the tolerable wall/inter-story drift limit with respect to certain performance criteria; A is the peak wall/inter-story drift demand, also a function of a series of random variables which are, in this study, the structural carried mass M; earthquake PGA aG, earthquake frequency contents r, and the response surface fitting error y. The frequency content r of ground motions is difficult to quantify and be linked to the performance functions. It was assumed that the characteristics of earthquakes except for PGA can be fully represented by a suite of earthquake records. First, a response database needs to be established by running the seismic simulations repetitively for the selected combinations of random variables over a defined domain. For each combination, a time-history analysis was performed for each earthquake record. A 167 set of mean As,,7 and standard deviation 0licm of the peak structural response (wall drift or inter-story drift) can be obtained over the suite of earthquake records. Therefore, for all the combinations of random variables, a number of sets of Asm and crm can be established. Then, the quadratic response surfaces, represented by Eq (7.12a) and Eq. (7.12b), can be used to fit the mean and the standard deviation of the peak structural response over the specified domain of the random variables, respectively. A boundary condition has been applied herein since the wall drift response will vanish if the carried mass Mor PGA a is equal to zero (Möller and Foschi, 2003). tX,5 = aG(alM +a2M2)+ a(a3M+a4M2) (7.12a) L\rs = aG (b1M +b2M2)+ a(b3M +b4M2) (7.12b) where aG is PGA; Mis the carried mass; a1 to a4 and b1 to b4 are coefficients which can be evaluated by the squared error F between the response fitting and simulations: N,G 2 F —aGJ(alM+a2Mj2)_a(a3j+a4Mj2)} (7.13) 7=1 j=1 where A,, is the mean of the peak drifts from the simulations over the suite of earthquake records with mass M, and PGA aGJ; NM and NaG are the total numbers of sampling points for mass and PGA, respectively. The squared error is minimized by 168 N1 NQ = —2 {A — aGJ(alMI +a2M,2 )_a(a3M1+a4M,2)}aGJMI = 0 a1 /=1 1 N1 N,G = —2 {z\ — aJ (a, +a2M,2)_a(a3M1+a4M12)}aGJM = 0 aa2 ,= (7.14)ÔF N1 = —2 {z — aGJ(alM, +a2M1)— a(a3M1+a4M12)}ajM = 0 a3 i=1 j=1 8F N1 NG = —2 — aGI (a1M +a2M, )— a(a3M1+a4M12)}ajM =0 a4 1=1 j=1 Solving Eq. (7.14), a1 to a4 can be obtained for the response surface of Ars. Similarly, coefficients b1 to b4 can be solved for the response surface ofo. Now taking into account the response surface fitting errors, the mean and standard deviation of the peak response are adjusted to =[a(a1M+22) a(a341_yj (7.15a) o =[aG(blM+bM2)+a(b2)J1_yU) (7.15b) where and y are the random variables representing the response surface fitting errors, which are assumed to follow the normal distribution. The fitting errors of the ith combination of the random variables can be calculated by = lXrsIXs,n (7.16a) Ars yl rs srn (7.16b) rs For all the combinations of the random variables, the mean and the standard deviation of the fitting errors can be obtained. It is assumed that the peak response follows a lognormal distribution. Therefore, Eq. (7.11) can be rewritten as 169 G = aH — 2 exp(RN Jin(i + (7.17) + where aU is the specified drift capacity (a is drift ratio limit and H is the wall height or the story height); A is the mean of peak drift demand; v is the COy, o / A; and RN is the normal variate RN(0,1) . A and o are given in Eq. (7.15a) and Eq. (7.15b), respectively. Now, Eq. (7.17) involves five random variables: PGA a, carried mass M, two response surface fitting errors and y; and the normal variate RN (0,1). Once this explicit response surface is obtained, the “design point” with the associated reliability index /9 can be calculated by FORM. When the accuracy of reliability estimation is influenced by the high nonlinearity of the performance function in some cases, the IS can be further applied by centering the sampling distribution near the failure domain, thus improving the accuracy of estimation. This method samples in the most important region which is around the design point Xd. Rewrite the integration function of failure probability in the sampling method, one has p1 =JI(x)[f(x)/h(x)]h(x)dx (7.18) where index 1(x) is defined such that 1(x) 0 if structure is safe and 1(x) = 1 if structure fails; f(x) is the joint density function of random variables X, centered at the means; h(x) is the joint density function of random variables X, centered at the design point Xd. And this function can be a normal joint density function. Thus, the failure probability can be estimated by the average of the function I(x)[f(x)/ h(x)] over the new sampling domain around the design point: 170 p1=__I(x)[f(x)/h(x)] (7.19) 7.4 Approaches to seismic reliability estimation Seismic reliability of a structure can first be estimated at a constant seismic hazard level (e.g., at a constant a0). For a specified seismic hazard level, this is a conditional reliability problem, also known as “fragility”. Then, the fragility can be integrated over the entire range of possible hazard levels to obtain the total reliability or failure probability of the structure because the hazard levels are also random. This is a two-step process to estimate structural reliability, as shown in Eq. (2.11) or Eq. (2.12) in Section 2.5.1. The fragility analysis provides a powerful tool to perform a risk analysis on a given structure with respect to certain hazard levels. For example, to estimate the seismic reliability of a wood shear wall at one site, deterministic simulations need to be performed over an ensemble of site-specific earthquake records which have been scaled to a certain hazard level. By fitting the seismic responses using a lognormal distribution or other distributions, its fragility curve can be generated. Then, in order to obtain the total structural failure probability, the fragility analysis with respect to different hazard levels also needs to be conducted. The fragility method also has some disadvantages. First, it typically considers the uncertainty involved in seismic hazards only. However, in some cases, other uncertainties also need to be considered. For example, due to the possible irregular distribution of structural mass, individual wood shear walls in a building might carry different amount of 171 mass which will significantly change their fundamental periods as well as the seismic demands. Second, when the fragility analysis is used as a tool to guide the performance- based design, they might not be as efficient as a fully coupled reliability method. For example, in order to establish the optimal nail spacing or the maximum structural mass for a wood shear wall to meet target reliabilities, the fragility analysis needs to be run repetitively for the wall with different nail spacing or structural mass. For a highly nonlinear problem, this search might be very computational intensive. Considering the same problem in a fully coupled analysis, the nail spacing or the structural mass can be treated in advance as variables in the performance function. And the optimal values can be obtained relatively efficiently using an inverse reliability approach (Foschi et al., 2002). This thesis uses the response surface method which is a fully-coupled method to estimate the seismic reliability of the P&B walls and buildings. To generate the response database for the response surface fitting, simulations at the sampling points of the random variables need to be run over the selected suite of earthquake records. Although the performance function only considers the influence of ground motions and structural mass, this approach can be easily extended to consider the influence of other structural input variables of interest and is conducive to the future research on the performance-based engineering of the P&B wood systems. 7.5 Seismic reliability analyses of P&B shear walls Seismic reliability of two types of 0.91 m long shear walls and eight types of 1.82 m 172 long shear walls commonly used in the P&B buildings were studied. “Pseudo-nail” wall models, calibrated for each type of the walls, were used to simulate the seismic response of individual walls with different combinations of random variables. The peak wall drift response was approximated by the explicit response surfaces Eq. (7.1 5a) and Eq. (7.1 5b). Using the RSM with IS, the seismic reliabilities of the shear walls with respect to four levels of performance expectation were estimated. 7.5.1 Shear wall descriptions The two types of 0.91m long walls were single-brace walls built with Canada Tsuga (TG, Tsuga heterophylla). One was the bare-framed single-brace wall and the other one had the same configuration but additionally sheathed with a GWB on one side. The information about these two types of walls including the configurations and the test results has been introduced in Chapter 4. The eight types of 1.82 m long walls had different shear resistance mechanisms and different wood species for the wall frames. Reserved cyclic tests of these walls were conducted by the Center for Better Living of Japan (CBL, 2001). Figure 7.1 shows the frame configuration of the walls. On top of the frame, diagonal double-brace walls, oriented strand board (OSB)-sheathed walls and plywood-sheathed walls were constructed (Fig. 7.2). In these walls, all the top plates were built with Douglas fir (Pseudotsuga menziesii) timber members. For the double-brace walls and the plywood- sheathed walls, framing members except for the top plate were built with three different wood species: Japanese Sugi (SG, Cryptomeria japonica), Canada Tsuga (TG, Tsuga 173 heterophylla), and European Whitewood (WW, Picea abies) glulam. For the OSB sheathed walls, only two species (SG and TG) were used to build the wall frames. Therefore, a total of eight types of walls were studied considering the combination of wood species and construction types. And for each type of the walls, three replicates were tested. In the walls, the SG members were not JAS graded; the TG members were JAS graded as E120; and the WW glulam members were JAS graded E85-F300. For the double-brace wall, two 45 mm x 90 mm members were installed as the diagonal braces connected with the adjacent posts and sill/top plates via standard BP2 metal plates and ZS50 nails (zinc-coated spiral nails with length of 50 mm and diameter of 5.2 mm). For the plywood-sheathed wall, JAS Grade 2 larch plywood panels with thickness of 9.5 mm were used. For the OSB-sheathed wall, JAS Grade 4 OSB panels with thickness of 9 mm were used. All the structural-panel-sheathed walls adopted common N50 nails spaced at 150 nm-i to connect the panels and the timber frames. It should also be noted that the OSB-sheathed wall had two 0.91 m x 2.73 m panels, but the plywood-sheathed wall had four panels installed onto four segments of the wall frame with additional horizontal blocking members because the regular size of plywood panels in Japan is limited to 0.91 m x 1.82 m. Therefore, approximately 14% more nail connections were used in the plywood-sheathed wall than the OSB-sheathed wall. 174 1 1 1 1 1 N 30x105 0 r) Post 105x 105 S—HD2O /‘i—down Sill plate r—j — — — 11/105x105 Figure 7.1 Configuration of the 1.82m long wall frame •1 _liHrl ift Figure 7.2 1.82 m long double-brace, OSB-sheathed and plywood-sheathed walls The reversed cyclic tests of the 1.82 m long walls also used the CBL protocol, as shown in Fig. 4.4. The cyclic loading rate was 1 mm/sec. The backbone curve of the hysteresis was used to evaluate the shear wall yield strength, stiffness, ultimate strength, and ductility ratio according to the Japanese wood shear wall test and evaluation method (BCJ, 2000). Table 7.1 gives the summary of the test results based on the average of three wall replicates for each type of the walls. It was found that the structural-panel-sheathed walls compared favorably with the double-brace walls in terms of strength, stiffness and 175 1820 --J 1 455 455 - 455 455 1 \ Top plate 105x1 80 N Brace 450 OSB panel 9mm I-J Plywood panel 9.5mm ductility. The plywood-sheathed walls also had higher strength than the OSB-sheathed walls because the 14% more nail connections provided the extra contribution. Although other arrangements of plywood panels are possible, in general, they are not used in the Japanese P&B construction. Table 7.1 Test result summary of cyclic tests of 1 .82m walls SG- SG- SG- TG- TG- TG- WW- WW Brace OSB Plywd Brace OSB Plywd Brace Plywd P(kN) 9.71 7.61 8.84 12.77 9.00 11.52 11.07 8.52 D(mm) 27.26 12.94 20.40 29.70 15.02 18.92 28.00 12.33 K (kN/mm) 0.357 0.603 0.435 0.430 0.617 0.609 0.397 0.706 P11(kN) 15.00 11.95 13.99 19.28 14.20 19.93 17.62 13.60 D11(mm) 90.9 120.5 166.3 97.5 116.1 122.1 93.6 149.7 P 2.17 6.10 5.16 2.16 5.18 3.73 2.09 7.78 0.55 0.30 0.33 0.55 0.34 0.39 0.56 0.26 0.2]J2,u—1(kN) 5.46 7.93 8.55 7.04 8.48 10.12 6.28 10.33 2/3Pinax (lcN) 10.78 8.82 10.44 14.03 10.96 15.37 13.38 10.22 Pinax(kN) 16.17 13.24 15.65 21.04 16.44 23.06 20.07 15.33 11120(kN) 8.15 9.76 9.24 9.97 10.50 12.60 9.16 10.75 Paiiow 5.46 7.61 8.55 7.04 8.48 10.12 6.28 8.52 Note: Paiiow = MIN{P, 0.2F J2p —1 , 2/3Pmax, P11120} Tables 7.2 and 7.3 give the calibrated “pseudo-nail” wall model parameters for the 0.91m walls and the 1.82m walls, respectively. Figure 7.3 and Figure 7.4 show the comparisons between the model hysteretic loops and the cyclic test results of these walls. The test results presented here are the average load-drift curves of the wall replicates for each type of the walls. Overall, good agreement can be observed between the model predictions and the test results. 176 Table 7.2. “Pseudo-nail” parameters of single-brace 0.91m walls Model Bare-framed wall Wall with gypsum boardsBrace under Brace under Brace under Brace underParameter . tension compression tension compression Qo(kN/mm) 0.5872 1.7355 16.4355 5.9872 Qi (kN/mm2) 0.06 0.002 1 -0.14 0.001 Q2 1.399 1.07 1.03 1.108 K (kNImm2) 0.0983 0.0787 0.1246 0.1283 Drnax (mm) 58.24 87.23 99.33 83.13 L(mm) 101.25 121.25 D (mm) 4.13 5.08 TG-i BRACE ;—Testa 12- DO 50100150 2 -12 - —————-46 —_________ 2 •0 -2 Drift (mm) t 0. TG-1 BRACE (with drywall) —Testa.g F TG-1 BRACE 2 U —----—-——— -—-——--——1 — —Modefl 12-j )O 50 100 150 2 -8 - -12 Drift (mm) L_TMode2a/t 2 479I k 7/1/1/’ 6 - -12 - TG-1 BRACE (with drywall) z Drift (mm) Figure 7.3 Cyclic test results vs “pseudo-nail” model predictions (0.91m walls) Table 7.3 “Pseudo-nail” model parameters of 1 .82m walls SG- TG- WW- SG- TG- WW- SG- TGParameter Brace Brace Brace Plywd Plywd Plywd OSB OSB Q0(kN/mm) 5.966 7.806 5.906 0.957 1.703 1.905 1.515 1.585 Qi (kN/mm2) 0.0026 0.0021 0.0021 0.016 0.022 0.0001 0.011 0.007 Q2 1.146 1.185 1.192 1.765 1.124 1.519 1.601 1.229 K(kNIrnm2) 0.221 0.188 0.179 0.078 0.171 0.175 0.138 0.143 D15 (mm) 56.33 54.54 58.04 47.26 73.26 51.51 34.74 58.57 L(mm) 101.59 131.03 129.55 270.68 201.25 220.62 240.70 242.73 D(mm) 5.08 6.25 6.22 9.26 9.31 8.21 8.11 8.61 50 100 150 2 Drift (mm) 177 • -25-- Drift (mm) 100 50 SG-BRACE 25—- .—Testag 20- 1001502 Drift (mm) SG-BRACE 2 •0 0 -200 20 : -/-j 50 100 150 2 -15- ‘0 -20 SG-OSB -25 —TeStag 20 - 2 0 -150 100 150 2 -15 -20 - 0 SG-OSB —Model 20. .200 -150 150 200 Drift (mm)Drift (mm) SG-PLYWOOD Ii ,_.Io_ — - —. Jo 50 100 150 -20 SG-PLYWOOD : 25--- —--Model 20j -____ ---y_il,i’ -200 00 Drift (mm) 150 2 Dlift(mm) TG-BRACE 25- —Model 20 TG-BRACE I !200 50 100150 20 1200 Drift (mm) 178 100 150 2 -25--- -----— Drift (mm) TG.OSB .. TG-OSB •— 25 —•-----—-•-...——..----- 25- —Tesla 20 —Model 20 - 2 0 150 150 200 5100 150 200 .--— — - -....——— Drift (mm) Drift (mm) . TG-PLYW000 TG-PLYWOOD i r —Model 2 50 100 150 200 200 / 100 150 2 0 Dr(mm) * Drift (mm) WJg-BRACE •1 20 Figure 7.4 Cyclic test results vs “pseudo-nail” model predictions (1.82 m walls) 179 WW-BRACE 25- _______ [2 r Model; 100 150 2 z Drift (mm) 260 100 150 2010 Drift (mm) WW-PLYWOOD —Tesla 20 - .0 -J 260 100 WW-PLYWOOD . 25 • —Model 20 __-- l / 1 j L-z 200 -2Ø0 71 7’ió’o - 50 100 Drift )mm) Drift (mm) 2010 7.5.2 Seismic reliability of shear walls The uncertainties considered in this study involve the randomness of ground motions, structural mass and the response surface fitting errors. The mechanical properties of the walls were assumed to be deterministic. The variability in the fitted “pseudo-nail” model parameters have not been considered since a limited number of the walls were tested and the test results could not provide enough information to account for the model variability. Table 7.4 gives the suite of Japanese historical earthquake records used in this study. The first six records were provided by the Building Research Institute of Japan. The other records were selected from the internet-based Kyoshin Network strong ground motion database (K-NET, 2004). Table 7.4 Historical earthquake records used for reliability analysis No Event Year Component PGA (g) Station 1 Tokyo 1956 NS 0.0755 Tokyo 101 2 Northern Miyagi-oki 1962 EW 0.0485 Sendai 501 3 Head land of Echizen 1963 EW 0.0255 Osaka 205 4 Tokachi-oki 1968 EW 0.1866 Hachinohe 5 Miyaki-oki 1978 NS 0.2634 Tohoku 6 Hyogo-kenNanbu 1995 NS 0.8365 JMAKobe 7 Hyogo-ken Nanbu 1995 NS 0.2432 Shin Osaka 8 Hyogo-ken Nanbu 1995 EW 0.6155 Takatori 9 Southern Akita Pref. 1996 NS 0.473 6 Naruko 10 Kagoshima Pref. 1997 EW 0.5034 Miyanojyo 11 Izuohshima 2000 EW 0.2522 Nijima 12 Akinada Geiyo 2001 NS 0.4083 Hojyo Olrecord -Tokyo 1956 0.1 0.05 1.) 0 U -0.05 -0.1 15 Time (s) 180 0.1 O2record N. Miyagi-oki 1962 0.05 - -0.05 -0.1 Time (s) O3record Headland of Echizen 1963 0.1 0.05 - -0.05 -0.1 Time (s) O4record - Tokachi-oki 1968 0.5 0.4H t!f -0.4 - -0.5 Time (s) O5record - Miyaki-oki 1978 0.5 0.4 JI Time (s) O6record - Hyogo-ken Nanbu (JMA Kobe) 1995 1.0 U U Time (s) 181 O7record - Hyogo-ken Nanbu (Shin Osaka) 1995 0.5 — 0.4 - 0.3 0.2 - 0.1 - u 0.0 U <-0.1 -0.2 -0.3 -0.4 -0.5 1.0 0.8 - 0.6 - 0.4 - 0.2 - 0.0 U <-0.2 U -0.4 - -0.6 - -0.8 - -1.0 - 1.0 0.8 0.6 0.4 0.2 o 0.0 0 <-0.2 -0.4 -0.6 -0.8 -1.0 1.0 0.8 0.6 0.4 0.2 - c., 0.0 0 -0.4 -0.6 -0.8 -1.0 0.5 0.4 0.3 - 0.2 0.1 c., 0.0 U < -0.1 0 -0.2 -0.4 -0.5 &t 30 Time (s) O8record - Hyogo-ken Nanbu (Takatori) 1995 15 20 25 30 Time (s) O9record - S. Akita Pref. 1996 Time (s) lOrecord - Kagoshima Pref. 1997 1:5 Time (s) lirecord - lzuohshima 2000 1 12 Time (s) 15 182 1.0 l2record - Akinada Geiyo 2001 -------------- 20 25 30 Time (s) Figure 7.5 Earthquake records used in the seismic reliability analysis In P&B wood buildings, depending on the layouts of wall systems and the number of stories, individual shear walls will carry different amount of mass. For example, in the tested one-story building at the University of British Columbia, the shear walls carried approximately a mass of 880 kg/rn. And in the two-story building tested in Japan, the shear walls in the second story carried approximately a mass of 850 kg/rn and the walls in the first story carried approximately a mass of 1410 kg/rn. In this study, to cover the possible range of the input variables, five structural mass levels (550 kg/rn, 825 kg/rn, 1100 kg/m, 1650 kg/rn and 2200 kg/rn) and ten PGA levels (0.05 g, 0.1 g, 0.15 g, 0.2 g, 0.3 g, 0.4 g, 0.5 g, 0.6 g, 0.7 g, and 0.8 g) were selected as the sarnpling points of the input variables. Since the response surface is represented by the quadratic functions (Eq. 7.1 2a and Eq. 7.1 2b), the number of the selected sampling points is more than sufficient to establish the coefficients of the quadratic functions. Alternatively, a statistical based design of computer experiment approach (Koehler and Owen, 1996; Sacks et al. 1989) can be used to select these representative variables and construct the response database. In this study, a total of 50 combinations of mass and PGA were calculated. Similar to the incremental dynamic analysis (IDA) (Varnvatsikos and Cornell, 2001), for each mass 183 0.8 - 0.6 0.4 -. 0.2 0.0 - -0.2 0 -0.4 -0.6 -0.8 -1.0 level, the shear wall is subject to twelve ground motion records, each scaled to multiple levels of seismic intensity. Therefore, the relationship between the structural response and the intensity level can be thoroughly investigated. For each combination of random variables, time-history analyses were performed for the suite of earthquake records. In this study, for the two types of O.91m single-brace walls, the time-history analysis was performed twice because these walls had different response depending on whether the diagonal brace was under tension or compression. The average response of two analyses was used to estimate their seismic reliabilities. For each type of the walls, with each combination of random variables, a set of mean and standard deviation of the peak wall drift responses over the suite of earthquake records were obtained. The 1.82 m plywood-sheathed Canada Tsuga (TG-PL) wall is presented here as an example. Table 7.5 gives the mean and standard deviation of the peak wall drift responses with respect to different combinations of PGA and structural mass. Figure 7.6 shows the IDA curves of the wall subject to the twelve earthquake records under five mass levels. Figure 7.7 shows the confidence curves of the wall under each mass level. The TG-PL wall had much higher seismic demand with increase of the structural mass. And the variability of the wall drift responses among the earthquake records also increased significantly with the increase of the carried mass and the PGA levels. 184 Table 7.5 Mean and standard deviation of peak drift response of 1 .82m long TG-PL wall PGA 550kg/rn 825kg/rn 1100kg/rn 1625kg/rn 2200kg/rn (\ CT (1 (1 /1 CT (7g, (mm) (rnrn) (rnm) (mm) (mm) (mm) (mm) (mm) (rnrn) (rnm) 0.05 0.43 0.23 0.64 0.24 0.89 0.26 1.18 0.25 1.64 0.47 0.10 0.76 0.30 1.14 0.32 1.68 0.43 2.47 0.64 3.84 1.26 0.15 1.05 0.36 1.65 0.46 2.49 0.61 4.90 1.63 6.74 2.59 0.20 1.37 0.46 2.36 0.72 3.69 0.84 7.81 2.72 11.29 4.94 0.30 2.20 0.75 4.47 1.23 8.16 2.77 15.42 6.57 25.87 15.25 0.40 3.59 1.03 7.66 2.27 14.22 5.35 28.58 15.83 46.23 32.04 0.50 5.55 1.48 12.74 4.54 22.12 9.29 46.39 30.56 77.51 52.03 0.60 7.94 2.23 19.12 7.39 32.46 15.90 69.09 51.26 111.57 78.05 0.70 11.59 3.66 26.64 11.15 45.71 35.46 113.05 79.93 124.72 90.43 0.80 14.75 4.29 36.09 16.87 64.00 53.72 129.05 89.74 137.01 101.44 Mass level I (55OkgIm) - IDA curves 0 20 40 60 80 100 120 140 160 180 200 Interstory drift (mm) Mass level 2 (825kg1m) - IDA curves 0.8r 0.7 zzzzz:zzzizz 0.4 - --------—- -—------- ----4 --------—- - ---4-----0 I I I I I I I °- 0.3 z:zzziz 0 20 40 60 80 100 120 140 160 180 20q Interstory drift (mm) 185 Mass level 3 (llOOkgIm) - IDA curves o 20 40 60 80 100 120 140 160 180 200 Interstory drift (mm) Mass level 4 (1625kg1m) - IDA curves ° 0 20 40 60 80 100 120 140 160 180 200 Interstory drift (mm) Mass level 5 (2200kgIm) - IDA curves 0 20 40 60 80 100 120 140 160 180 20 Interstory drift (mm) Figure 7.6 IDA curves of 1 .82m long TG-PL wall 186 Mass level I (550kg/rn) - Confidence curves 0.6 — - —16% 9] -- EEEEEEEEEzz 0 20 40 60 80 100 120 140 160 180 200 Interstory drift (mm) Mass level 2 (825kg/rn) - Confidence curves 0.8 _____ 0.7 ,z 50%L 0.6 -/- / - —0.5 < 0.4 : zzzzzzzzzziz4zz 0.1 0.0 0 20 40 60 80 100 120 140 160 180 20d Interstory drift (rnrn) Mass level 3 (1100kg/rn) - Confidence curves 0.8 I 0.7 + 0.6 — - -16% 0.5 ---- — - -84%K 0 20 40 60 80 100 120 140 160 180 20 Interstory drift (mrn) 187 Mass level 4 (1625kg1m) - Confidence curves o 20 40 60 80 100 120 140 160 180 200 Interstory drift (mm) Mass level 5 (2200kgim) - Confidence curves 0 20 40 60 80 100 120 140 160 180 20 Interstory drift (mm) Figure 7.7 Confidence curves of 1 .82m long TG-PL wall The coefficients a to a4 and b1 to b4 for the fitted quadratic response surfaces of all the shear walls are given in Tables 7.6 and 7.7, respectively. The corresponding response surface fitting errors and y are given in Table 7.8. In these tables, abbreviations “BR”, “PL” and “OS” stand for “diagonal-braced”, “plywood-sheathed” and “OSB sheathed”, respectively. Figures 7.8 and 7.9 show the comparison between the fitted response surfaces and the model simulations of the O.91m walls and the 1.82 m walls. The “x” coordinates represent the seismic simulation results and the “y” coordinates represent the response surface fitted results. If all the data points are located on the 45° 188 straight lines, it represents a perfect response surface fitting without introducing any fitting errors. Table 7.6 RS coefficients of peak wall drift (mean) MeanWall type a1 a2 a3 a4 0.91 TG-BR -0.2226e-2 0.4805e-4 0.2730 -0.1 138e-3 m TG-BR(GWB) -0.1791e-1 0.2983e-4 0.1139 -0.1639e-4 SG-BR -0.4167e-1 0.6099e-4 0.2721 -0.1174e-3 TG-BR -0.3478e-1 0.4194e-4 0.1642 -0.4871e-4 WW-BR -0.4819e-l O.5540e-4 0.2155 -0.8197e-4 1.82 SG-PL -0.5536e-l 0.5527e-4 0.2330 -0.8743e-4 m TG-PL -0.3731e-l 0.2718e-4 0.1116 -0.1112e-4 WW-PL -0.6774e-1 0.5523e-4 0.2267 -0.7789e-4 SG-OS -0.6886e- 1 0.611 7e-4 0.2603 -0.973 8e-4 TG-OS -0.6576e-1 0.5469e-4 0.2233 -0.7659e-4 Table 7.7 RS coefficients of peak wall drift (standard deviation) Standard deviationWall type 01 03 V4 0.91 TG-BR -0.2420e-1 0.4824e-4 0.1741 -0.9670e-4 m TB-BR(GWB) -0.1997e-1 0.1716e-4 0.5624e-1 -0.6125e-5 SG-BR -0.8106e-1 0.7608e-4 0.2617 -0.1286e-3 TG-BR -0.6981e-1 0.5326e-4 0.1546 -0.5455e-4 WW-BR -0.7952e- I 0.6369e-4 0.1982 -0.8342e-4 1.82 SG-PL -0.9073e-1 0.6809e-4 0.2248 -0.9413e-4 m TG-PL -0.4467e-1 0.2363e-4 0.9090e-1 -0.6583e-5 WW-PL -0.8421e-1 0.5961e-4 0.2095 -0.7987e-4 SG-OS -0.8926e-1 0.6854e-4 0.2369 -0.1018e-4 TG-OS -0.8010e-1 0.5628e-4 0.1964 -0.7359e-4 Table 7.8 RS fitting errors of mean and standard deviation of peak drifts Peak drift mean Peak drift stdevWall Type mean stdev mean stdev 0.91 TG-BR 0.1115 0.1850 0.2033 0.3152 m TG-BR(GWB) 0.0056 0.2705 0.1764 0.3181 SG-BR 0.1208 0.2320 0.2088 0.3730 TG-BR 0.0510 0.2611 0.1429 0.4548 WW-BR 0.0713 0.2858 0.2042 0.4096 1.82 SG-PL 0.0497 0.3380 0.3145 0.5966 m TG-PL 0.0565 0.2820 0.2215 0.3166 WW-PL 0.1216 0.3031 0.2656 0.4154 SG-OS 0.0920 0.3623 0.2597 0.4209 TG-OS 0.1143 0.2990 0.2490 0.4326 189 /4/ 7-.. . . / • •;/.. 200- 200- 180 - 160- 160- 140- 140-. _120- • E120- 100- 80 80- °eo 60 40 40 20 : 20 0 0 0 20 40 60 80 100 120 140 160 180 200 -20 Input (mm) TG-1 -BR-GWB (stdev) 0 20 40 60 80 100 120 140 160 180 200 Input (mm) Figure 7.8 Response surface fitted data vs model simulation data (0.91 m walls) E 120 E 100 80 60 40 SG-BRACE (mean) 0 20 40 60 80 100 120 I40 160 180 200 Input (mm) SG-BRACE (stdev) 200 180- 160 7 140 // 8 120 j 100. &80 7’ • 60- // • 40 / •••._ 201V 0 20 40 60 80 100 120 140 160 180 200 Input (mm) TG-1-BR (mean)200 - 180 - 160 140- O 120 - 8 100 - 80 60- 40 .,/ 20/ TG-1-BR (stdev) 200 - 180 - 160 140 - 8 120 -- 8 100 0 80- 60- • 40- • 0 20 40 60 80 100 120 140 160 180 200 Input (mm) 20 - 0 TG-1 -BR-GWB (mean) 0 20 40 60 80 100 120 140 160 180 200 Input (mm) 180- 200 180 160 140 20 190 SG-OSB (mean) SG-OSB (stdev) 200- 200- 180- 180 160- 160 140- 140- 120- E120 6 6100- . 80- 0 80- ‘ 0 20 40 60 80 100 120 140 160 180 200 0 -20 Input (mm) SG-PLYWOOD (mean) SG-PLYW000 (stdev) 200- 200 180j 180 160- 160 1401 140 120 E120 E 6 _100- •100 . 80 80 60- 60 40 40 20 20 0 0 -20 0 20 40 60 80 100 120 140 160 180 200 __________________________________________________________ Input (mm) TG-BRACE (mean) 200 140 ;120- /4’ 6 6100- /_‘ . [°60 40 20 -20 20 40 60 80 100 120 140 160 180 200 Input (mm) 4 4 . 20 40 60 80 100 120 140 160 180 200 Input (mm) 20 40 60 80 100 120 140 160 180 200 Input (mm) TG-BRACE (stdev) 200 180 160 1 l40 / 120_j . 100- 80 0 60- • • 40- . 201.Z -20 20 40 60 80 100 120 140 160 180 200 Input (mm) 191 TG-OSB (mean) TG-OSB (stdev) 200- 200] 180- - 180- // 160- 7 160 // 140H 140- 120 120 • // ioo 7’ • _100J 7. • V. 7 • D 80— / . D • ‘• 0 j // •.°6o1 ,“ 60 • ••A •/ • 40 • •‘ 40 i 2oj,4V’ ___________________________ -20 Ô 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Input (mm) I Input (mm) TG-PLYWOOD (mean) 200 - ./140 /// E120- / 1: 60 - 40- .y 20 0- 0 20 40 60 80 100 120 140 160 180 200 Input (mm) 200 180 160 140 120 E gioo S- 80 60 40 20 -20 20 40 60 80 100 120 140 160 180 20 Input (mm) TG-PLYWOOD (stdev) 200 / El20 7’ E - I .l00 88O 7 .. 60 0 20 40 60 80 100 120 140 160 180 200 Input (mm) WIN-BRACE (mean) • • WIN-BRACE (stdev) 200 180 1601 140120J . 100 4 80-1 7” ..4/ • 4oJ •7’7” 20 I or 0 20 40 60 80 100 120 140 160 180 200 Input (mm) 192 WN-PLYWOOD (mean) 200 - WN-PLYWOOD (stdev) 200 - 180 160 140 E 120- E120 E ioo- 80 80- 60- 60 40 40 • 20 0 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Input (mm) Input (mm) Figure 7.9 Response surface fitted data vs model simulation data (1.82 m walls) Case studies on the seismic reliabilities of these walls were conducted. It was assumed that the structural mass applied on the top of the walls followed a lognormal distribution with mean of 1100 kg/rn and COV of 0.1, approximately 1000 kg on a 0.91 m long wall and 2000 kg on a 1.82 m long wall. The PGA statistics were assumed to be a lognormal distribution with mean of 0.25 g and COV of 0.55. This, coupled with an annual arrival rate of earthquakes of 0.2/year, results in PGA of 0.77 g with a return period of 475 years and PGA of 0.43 g with a return period of 50 years. Such an assumption on PGA values corresponds to very high seismicity. The performance criteria in terms of peak wall drift ratio or peak inter-story drift ratio were defined as follows: 0.5% for serviceability; 1% for immediate occupancy; 2% for life safety; and 3% for collapse prevention. The failure probabilities with associated reliability indices for individual shear walls with respect to different performance expectations were estimated using the reliability analysis software RELAN (Foschi, etc., 2007). Tables 7.9 and 7.10 give the summaries of 193 . 180 160 140 - the reliability estimations on the event base and on the annual base, respectively. Pie is the event failure probability with associated reliability index respectively. Using Eq. (7.4), the annual failure probability fa with associated reliability index /a can be obtained. Table 7.9 Seismic failure probability of the P&B shear walls (event) Serviceability Immediate Occu. Life Safety Collapse Prey. Wall type (0.5%) (1%) (2%) (3%) Pie /Je Pie lie Pie lie Pie lie TG BR 0.5696 -0.175 0.2744 0.600 0.0861 1.365 0.0356 1.8030.91 - (0.5708) (-0.178) (0.2768) (0.592) (0.0930) (1.322) (0.0382) (1.772) m TG-BR 0.2937 0.542 0.1013 1.274 0.0239 1.979 0.0086 2.380 (GWB) (0.2977) (0.531) (0.0999) (1.282) (0.0235) (1.985) (0.0081) (2.430) BR 0.4048 0.241 0.1721 0.946 0.0527 1.619 0.0229 1.998SG- (0.3999) (0.254) (0.1698) (0.955) (0.0530) (1.617) (0.0230) (1.995) TG BR 0.2799 0.583 0.1024 1.268 0.0278 1.914 0.0114 2.275(0.2685) (0.617) (0.0977) (1.295) (0.0266) (1.933) (0.0110) (2.291) WWBR 0.3154 0.481 0.1227 1.162 0.0350 1.812 0.0147 2.179(0.3023) (0.518) (0.1184) (1.183) (0.0330) (1.838) (0.0146) (2.182) SG PL 0.2977 0.531 0.1195 1.178 0.0356 1.805 0.0153 2.1611.82 - (0.2765) (0.593) (0.1137) (1.207) (0.0329) (1.839) (0.0142) (2.190) n PL 0.1257 1.147 0.0438 1.708 0.0117 2.268 0.0047 2.595TG- (0.1148) (1.201) (0.0394) (1.757) (0.0108) (2.298) (0.0045) (2.613) WWPL 0.1977 0.850 0.0787 1.414 0.0241 1.976 0.0106 2.303(0.1878) (0.886) (0.0735) (1.450) (0.0218) (2.018) (0.0104) (2.310) SG os 0.2391 0.709 0.1107 1.223 0.0414 1.734 0.0212 2.029(0.2310) (0.736) (0.1061) (1.248) (0.0386) (1.767) (0.0195) (2.065) TG os 0.2045 0.826 0.0805 1.402 0.0242 1.973 0.0106 2.304(0.1928) (0.868) (0.0744) (1.444) (0.0219) (2.015) (0.0103) (2.316) Note: results in parentheses were obtained from IS. Table 7.10 Seismic failure probability of the P&B shear walls (annual, 0.2/year) Serviceability Immediate Occu. Life Safety Collapse Prey. Wall type (0.5%) (1%) (2%) (3%) P0 lie Pie /i Pi Pia Pa TG BR 0.1077 1.239 0.0534 1.613 0.0171 2.118 0.0071 2.4520.91 - (0.1079) (1.238) (0.0539) (1.608) (0.0184) (2.088) (0.0076) (2.428) m TG-BR 0.0570 1.580 0.0201 2.052 0.0048 2.590 0.0017 2.929 (GWB) (0.0578) (1.574) (0.0198) (2.058) (0.0047) (2.597) (0.0016) (2.948) 194 SG BR 0.0778 1.420 0.0338 1.828 0.0105 2.308 0.0046 2.605 - (0.0769) (1.426) (0.0334) (1.833) (0.0105) (2.308) (0.0046) (2.605) TG BR 0.0544 1.604 0.0203 2.048 0.0055 2.543 0.0023 2.834 - (0.0523) (1.623) (0.0194) (2.060) (0.0053) (2.556) (0.0022) (2.848) WWBR 0.0611 1.546 0.0242 1.974 0.0070 2.457 0.0029 2.759 - (0.0587) (1.566) (0.0234) (1.988) (0.0066) (2.478) (0.0029) (2.759) SG PL 0.0578 1.574 0.0236 1.985 0.0071 2.452 0.0031 2.7371.82 - (0.0538) (1.609) (0.0225) (2.005) (0.0066) (2.478) (0.0028) (2.770) m TG PL 0.0248 1.963 0.0087 2.378 0.0023 2.834 0.0009 3.121 - (0.0227) (2.000) (0.0078) (2.418) (0.0022) (2.848) (0.0009) (3.121) WWPL 0.0388 1.765 0.0156 2.155 0.0048 2.590 0.0021 2.863 - (0.0369) (1.788) (0.0146) (2.181) (0.0044) (2.620) (0.0021) (2.863) SG os 0.0467 1.678 0.0219 2.016 0.0082 2.400 0.0042 2.636 - (0.0451) (1.694) (0.0210) (2.034) (0.0077) (2.423) (0.0039) (2.661) TG os 0.0401 1.750 0.0160 2.144 0.0048 2.590 0.0021 2.863 - (0.0378) (1.777) (0.0148) (2.175) (0.0044) (2.620) (0.0021) (2.863) Note: results in parentheses were obtained from IS. Based on the seismic reliability analyses of the walls, some conclusions can be drawn: 1) The walls built with Japanese SO seemed to have the lowest seismic reliability. The walls built with Canadian TO seemed to have the highest seismic reliability; 2) The plywood-sheathed or OSB-sheathed walls have higher seismic reliabilities than the double-brace walls; 3) Comparisons between the 0.91 m single-brace walls with or without GWB indicated the significant contribution of GWB to the wall seismic reliability. The failure probability of the GWB-sheathed single-brace wall drops to 53%, 38%, 28%, and 24% of that of the bare-framed single-brace wall with respect to the four performance expectations, respectively; and 4) For each type of the walls, the trend of the reliability indices from the serviceability performance criterion to the collapse prevention criterion shows a significant monotonic 195 increase, indicating that the shear walls have a much higher failure probability with respect to the performance expectations such as serviceability or immediate occupancy. 7.6 Seismic reliability analyses of P&B buildings The seismic reliability analyses on a one-story building (tested at the University of British Columbia) and a two-story P&B building (data provided by the Building Research Institute of Japan) were carried out. The structural forms of these two buildings have been introduced in Chapter 5 and Chapter 6, respectively. The “PB3D” models were used to simulate the seismic response of the buildings with different combinations of random variables. The uncertainties considered herein also involved the randomness of ground motions, carried mass and the fitting errors of the RSM, same as the shear wall reliability analyses. The twelve earthquake records, given in Table 7.4, were also used to represent the characteristics of earthquake ground motions. Using the RSM with IS, the seismic reliabilities of these two buildings with respect to four performance expectations were estimated. 7.6.1 Seismic reliability of a one-story building To estimate the seismic reliability of the one-story building, five structural mass levels (120kg/rn, 140kg/rn2,160kg/rn2,180kg/rn2,and 200kg/rn) and eight PGA levels (0.lg, 0.2g, 0.3g, 0.4g, 0.5g, 0.6g, 0.7g, and 0.8g) were selected as the sampling points of the random variables. Thus, a total of 480 nonlinear time-history analyses were conducted for this building. In the seismic simulation, uniaxial ground motion was applied along the x direction of the building. For each combination of PGA and structural mass, over the 196 twelve earthquake records, a set of mean A and standard deviation of the peak inter-story drift response can be obtained. Table 7.11 gives 40 sets of mean z\ and standard deviation crcm corresponding to 40 combinations of PGA and structural mass. Table 7.11 Mean and standard deviation of peak inter-story drift of the one-story building PGA Mass level 1 Mass level 2 Mass level 3 Mass level 4 Mass level 5 EU 6 6 4U 6 1U Ug, (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) 0.10 1.65 0.29 2.03 0.36 2.69 0.47 3.28 0.57 3.73 0.65 0.20 5.32 0.97 6.71 1.25 7.88 1.52 9.02 1.98 10.37 2.59 0.30 10.30 1.75 12.37 2.22 14.48 3.05 16.97 4.17 19.49 5.38 0.40 15.78 2.73 18.95 3.78 22.66 5.38 26.63 7.31 30.94 9.48 0.50 22.00 4.01 26.66 5.68 32.13 8.28 37.92 11.28 43.60 14.68 0.60 28.70 5.65 35.29 8.51 42.52 12.20 50.03 16.66 57.25 21.44 0.70 36.41 7.26 45.15 11.47 54.49 16.59 63.81 22.73 74.48 32.05 0.80 44.67 9.67 55.39 15.25 67.28 22.70 81.32 35.34 94.86 47.63 1 I I I I I F I = 0 50 100 150 200 Under each structural mass level, the IDA curves for the twelve earthquake records were plotted to present the relationship between PGA and the peak inter-story drift, as shown in Fig. 7.10. Each line represents the peak inter-story drift response with respect to different PGA levels over a single record. Figure 7.11 shows the confidence curves under different mass levels which were calculated from the statistics over the twelve earthquake records. Mass level I - IDA curves 0.8 0.7 0.6 0.5 <0.4 0 o 0.3 0.2 0.1 - 0.0 L Max Interstory Drift (mm) 197 Mass level 2 - IDA curves 0 50 100 150 200 Max Interstory Drift (mm) Mass level 3 - IDA curves 0.8 0.7 - - 0.6 0.5 < 0.4 I F a. 0.3 : I- 0.2 0.1 0.0 F 100 150 20( Max Interstory Drift (mm) 0 50 Mass level 4 - IDA curves 0.8 0.7 0.6 0.5 <0.4C, a.. 0.3 0.2 0.1 0.0 0 50 100 150 20( Max Interstory Drift (mm) 198 Mass level 5 - IDA curves 0.0 A 0 50 100 150 200 Max Interstory Drift (mm) Figure 7.10 IDA curves under five mass levels of one-story building Mass level 1-IDA curves __7_ 0 50 100 150 20 Max Interstory Drift (mm) Mass level 2 - IDA curves 0.8 ——16%:0.6 / — -84%: 0.5 ----/ n 7 //y —. 50% 0.3 -/-‘ 0.2 0.1 0.0. 0 50 100 150 200 Max Interstory Drift (mm) 199 Mass level 3 - IDA curves o 50 100 150 200 Max Interstory Drift (mm) Mass level 4 - IDA curves — 0 50 100 150 200 Max Interstory Drift (mm) Mass level 5 - IDA curves — 0 50 100 150 20( Max Interstory Drift (mm) Figure 7.11 Confident curves of peak inter-story drift of the one-story building The coefficients a to a4, b1 to b4 for the fitted quadratic response surfaces of the mean and standard deviation of the peak inter-story drift responses are given in Table 200 7.12. And the corresponding response surface fitting errors and y are given in Table 7.13. Figure 7.12 shows the comparisons between the fitted response surface and the model simulations, which indicates very good response surface fitting. Table 7.12 RS coefficients of peak inter-story drift of the one-story building Mean Stdev a1 a2 a3 a4 b3 b4 0.1965 -0.164e-3 0.1122 0.211e-2 0.1386 -0.973e-3 -0.4193 0.419e-2 Table 7.13 RS fitting errors of peak inter-story drift of the one-story building Mean of peak inter-story Stdev of peak inter-story drift drift mean stdev mean stdev 0.009462 0.03565 -0.01964 0.084091 One-story building (mean) One-story building (stdev) 100 100 80 80 E60 &60 40 040 20 20 0 0 0 20 40 60 80 100 0 20 40 60 80 100 lnput(mm) lnput(mm) Figure 7.12 Response surface fitted data vs model simulation data (one-story building) The same assumptions about the seismic hazard as used in the shear wall reliability analysis were taken. The PGA of the earthquakes follows a lognormal distribution with mean of 0.25 g and COV of 0.55. The structural mass of the building followed the lognonnal distribution with COV of 0.1. The seismic reliability of the building with respect to three structural mass levels was 201 estimated. Table 7.14 gives the reliability estimations with respect to four performance expectations under each structural mass level. fe and are the event failure probability and the associated reliability index. Given the earthquake annual arrival rate of 0.2/year, the annual failure probability P1 and the reliability index /a of the building are also given in Table 7.15. Table 7.14 Seismic failure probability of one-story building (event) Serviceability Immediate Occu. Life Safety Collapse Prey. Mass(kg/m2) (0.5%) (1%) (2%) (3%) Pie lie Pie lie Pie lie Pie lie 140 0.2264 0.751 0.0537 1.610 0.0083 2.395 0.0024 2.826(0.2175) (0.781) (0.0492) (1.652) (0.0073) (2.441) (0.0021) (2.858) 160 0.2959 0.536 0.0818 1.393 0.0152 2.165 0.0049 2.585(0.2846) (0.569) (0.0736) (1.450) (0.0136) (2.207) (0.0044) (2.620) 180 0.3647 0.346 0.1141 1.205 0.0245 1.969 0.0086 2.382(0.3533) (0.376) (0.1024) (1.268) (0.0225) (2.005) (0.0081) (2.405) Table 7.15 Seismic failure probability of one-story building (annual, 0.2/year) Serviceability Immediate Occu. Life Safety Collapse Prey. Mass(kg/m2) (0.5%) (1%) (2%) (3%) Pia Pa Pfa Pa Pa Pia lie 140 0.0443 1.703 0.0107 2.301 0.0017 2.929 0.0005 3.291(0.0426) (1.721) (0.0098) (2.334) (0.0015) (2.968) (0.0004) (3.352) 160 0.0575 1.576 0.0162 2.139 0.0030 2.748 0.0010 3.090(0.0553) (1.596) (0.0146) (2.181) (0.0027) (2.782) (0.0009) (3.121) 80 0.0703 1.474 0.0226 2.003 0.0049 2.583 0.0017 2.9291 (0.0682) (1.489) (0.0203) (2.048) (0.0045) (2.612) (0.0016) (2.948) Note: numbers in parentheses are the results calculated by IS. 7.6.2 Seismic reliability of a two-story building Using the RSM with IS, the seismic reliability of the two-story building was also estimated. The uncertainties also involved the randomness of ground motions, structural mass and the RSM fitting errors. Twelve earthquake records in Table 7.4 were also used to conduct the seismic simulations. Five structural mass levels (Table 7.16) and eight 202 PGA levels (0.lg, 0.2g, 0.3g, 0.4g, 0.5g, 0.6g, 0.7g, and 0.8g) were selected as the sampling points of the random variables. Table 7.16 Structural mass levels of the two-story building Mass level Floor (kg/rn2) Roof (kg/rn2) 1 140 120 2 160 140 3 180 160 180 200 4 200 _____ 5 ___________ 220 ____________________ In this building, the first story had almost the same amount of effective shear wall length along the x and y directions but a higher eccentricity ratio along the x direction (see Table 6.2). Therefore, uniaxial earthquake load was applied along the x direction of the building. For each combination of PGA and structural mass, over the twelve earthquake records, a set of mean Asm and standard deviation °m of the peak inter-story drift response was obtained. Table 7.17 gives 40 sets of mean Asm and standard deviation u corresponding to 40 combinations of PGA and structural mass. Table 7.17 Mean and standard deviation of peak inter-story drift of the two-story building PGA Mass level 1 Mass level 2 Mass level 3 Mass level 4 Mass level 5 (\ p ci p ci p ci p ci p cig, (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) 0.10 2.57 0.47 2.99 0.59 3.30 0.75 3.65 0.87 4.17 0.99 0.20 6.79 1.20 7.91 1.70 9.40 2.48 10.98 3.32 12.58 4.21 0.30 12.96 2.91 15.68 4.48 18.57 6.13 21.65 8.26 24.75 10.59 0.40 20.70 6.20 25.27 9.01 30.06 12.50 34.78 16.13 39.00 19.65 0.50 30.46 10.77 37.25 15.74 43.99 20.97 49.92 26.24 56.57 34.35 0.60 42.03 17.57 51.34 25.24 60.61 33.69 69.55 44.73 76.55 51.95 0.70 55.92 27.19 69.97 40.91 83.42 56.31 90.48 63.06 99.12 73.07 0.80 76.29 45.18 92.78 61.86 107.50 79.87 117.13 88.43 125.56 96.90 Under each structural mass level, the IDA curves for the twelve earthquake records were plotted to present the relationship between the PGA and the peak inter-story drift 203 100 150 200 Max Interstory Drift (mm) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 100 150 Max Interstory Drift (mm) 200 250 300 Mass level 3 - IDA curves L L L L 50 100 150 200 250 Max Interstory Drift (mm) response (Fig. 7.13). Figure 7.14 shows the confidence curves with respect to each mass level which were calculated from the statistics over the twelve earthquake records. Mass level I - IDA curves 0 50 250 300 0 Mass level 2 - IDA curves 0 z 0.8 0.7 0.6 0.5 <0.4 0 0- 0.3 0.2 0.1 0.0 0 300 204 Mass level 4 - IDA curves 0.8 0.7 0.3 0.2 0.1 0.0 0 50 100 150 200 250 300, Max Interstory Drift (mm) Mass level 5 - IDA curves 100 150 200 250 300 Max Interstory Drift (mm) Figure 7.13 IDA curves under five mass levels of two-story building 0.8 __________ 0.7 0.6 0.5 <0.4C, 0- 0.3 0.2 0.1 0.0 0 50 100 150 200 250 300, Max Interstory Drift (mm) 0 50 Mass level I - Confidence curve -J .---/-- - 205 0.8 0.7 0.6 0.5 < 0.4C, a.. 0.3 0.2 0.1 100 150 200 Max Interstory Drift (mm) 50 100 150 Max Interstory Drift (mm) 100 150 200 Max Interstory Drift (mm) Mass level 2 - Confidence curve 0 50 250 300 0.0 0.8 0.7 0.6 0.5 <0.4 0.3 0.2 0.1 0.0 Mass level 3 - Confidence curve -4 0 200 250 300 Mass level 4 - Confidence curve 300 206 Mass level 5 - Confidence curve 0 50 100 150 200 250 300 Max lnterstory Drift (mm) Figure 7.14 Drift confident curves of the two-story building under five mass levels The coefficients a1 to a4, b1 to b4 for the fitted quadratic response surfaces of the peak inter-story drift of the building are given in Table 7.18. And the corresponding response surface fitting errors and y are given in Table 7.19. Figure 7.15 shows the comparisons between the fitted response surface and the model simulations. Table 7.18 RS coefficients of peak inter-story drift of the two-story building Mean StdeV a1 a2 a3 a, b2 b4 -0.1511 0.0015 1.1030 -0.0017 -0.4236 0.0015 d53 0.0004 Table 7.19 RS fitting errors of peak inter-story drift of the two-story building Mean of peak inter-story Stdev of peak inter-story drift drift mean stdev mean s$eV -00133 00598 00099 00998 207 100 Two-story building (mean) Two-story building (stdev) 100 80 80 I E6° g6o 40 040 20 20 0 0 0 20 40 60 80 100 0 20 40 60 80 100 lnput(mm) lnput(mm) Figure 7.15 Response surface fitted data vs model simulation data (two-story building) The same assumptions about the earthquake hazard were taken. The PGA follows a lognormal distribution with mean of 0.25 g and COV of 0.55. The structural mass of the building followed a lognormal distribution with COV of 0.1. The seismic reliabilities of the two-story building were estimated under three levels of structural mass. Table 7.20 gives the event failure probability fe with associated reliability index /e with respect to four performance expectations. Given the earthquake arrival rate of 0.2/year, the annual failure probabilities ‘fa with associated reliability indices ,6 are given in Table 7.21. Table 7.20 Seismic failure probability of the two-story building (event) Mass Serviceability Immediate Occu. Life Safety Collapse Prey. (kg/rn2) (0.5%) (1%) (2%) (3%) Ft. Rf. Pie lie Pie lie Pie /3e Pie lie 60 140 0.2923 0.547 0.1024 1.268 0.0278 1.915 0.0116 2.2701 (0.2811) (0.580) (0.0892) (1.345) (0.0248) (1.963) (0.0104) (2.313) 180 160 0.3665 0.341 0.1363 1.097 0.0387 1.766 0.0166 2.129(0.3417) (0.408) (0.1224) (1.163) (0.0347) (1.816) (0.0152) (2.166) 200 180 0.4445 0.139 0.1756 0.932 0.0522 1.624 0.0230 1.996(0.4179) (0.207) (0.1626) (0.984) (0.0477) (1.668) (0.0210) (2.033) . 208 Table 7.21 Seismic failure probability of the two-story building (annual, 0.2/year) Mass1k / 2 Serviceability Immediate Occu. Life Safety Collapse Prey.g n (0.5%) (1%) (2%) (3%) Fl. Rf. Pfa Ba Pla Ba Pia Ba Pia Ba 160 0.0568 1.582 0.0203 2.048 0.0055 2.543 0.0023 2.834140 (0.0547) (1.601) (0.0177) (2.104) (0.0049) (2.583) (0.0021) (2.863) 180 1 0.0707 1.471 0.0269 1.928 0.0077 2.423 0.0033 2.71660 (0.0661) (1.505) (0.0242) (1.974) (0.0069) (2.462) (0.0030) (2.748) 200 80 0.0851 1.372 0.0345 1.818 0.0104 2.312 0.0046 2.6051 (0.0802) (1.404) (0.0320) (1.852) (0.0095) (2.346) (0.0042) (2.636) Note: the numbers in parenthesis are the results calculated by IS. Based on the seismic reliability analyses of the one-story building and the two-story building, some conclusions can be drawn: 1) The seismic reliability of the one-story building seemed to have higher seismic reliability than the two-story building since the one-story building was a symmetrical structure and had relatively more shear walls with respect to its construction area; 2) The increased structural mass level significantly increased the failure probability of the buildings. For example, on the event base, an increase of 20 kg/rn2 structural mass would lower the reliability index by 0.15—M.20 with respect to the life safety performance expectation; and 3) In these two buildings, the trend of the reliability indices from the serviceability performance criterion to the collapse prevention criterion showed a significant monotonic increase, also indicating that the buildings have a much higher failure probability with respect to the performance expectations such as serviceability or immediate occupancy. 7.6.3 System effect on seismic reliability of shear walls in buildings The seismic performance of shear walls in building systems compare favorably with 209 the shear walls functioning separately as a single piece due to the system effect such as load-sharing among shear walls. However, no prior work has been reported in the literature on evaluating the contribution from the system effect in the P&B buildings. In this study, based on the seismic simulation database of the shear walls and the buildings, attempt was made to study the influence of the system effect on the seismic reliability of two types of P&B shear walls. 7.6.3.1 A single-brace wall in the one-story building The one-story building has four identical 0.91 m long single-brace walls along x or y direction. Due to the loading symmetry and the structural symmetry, each wall carried a quarter of the total mass based on their tributary areas. If three structural mass levels (140 kg/m2, 160 kg/m2, 180 kg/rn2) are considered, then, accordingly, each shear wall carries 700 kg, 800kg and 900 kg, which correspond to 769 kg/rn, 879 kg/rn, and 989 kg/rn with respect to the wall length of 0.91 m. The seismic reliability analysis of the single-brace wall and the one-story building used the same assumptions about the seismic hazard. PGA follows a lognormal distribution with mean of 0.25 g and COV of 0.55. The structural mass follows a lognormal distribution with COV of 0.1. Under the three structural mass levels, the seismic reliabilities of the building and the single-brace wall with respect to the four performance expectations have been estimated. Table 7.22 gives the comparisons between the building reliabilities and the wall reliabilities evaluated by FORM on an event base. It was found that the single-brace wall had much lower seismic reliabilities 210 than the building under the three load levels with respect to four performance expectations. Table 7.22 Seismic failure probability of one-story building and shear wall (event) Serviceability Immediate Occu. Life Safety Collapse Prey. Mass level (0.5%) (1%) (2%) (3%) Pie lie Pfe lie Pie lie Pie lie Building 0.2264 0.751 0.0537 1.610 0.0083 2.395 0.0024 2.826(140 kg/rn) Wall 0.3830 0298 0.1515 1.030 0.0412 1.737 0.0162 2.140(769kg/rn) Building 0.2959 0.536 0.0818 1.393 0.0152 2.165 0.0049 2.585(160kg/rn) Wall 04493 0127 01901 0878 00542 1.605 00217 2.019(879kg/rn) Building 0.3647 0.346 0.1141 1.205 0.0245 1.969 0.0086 2.382(180kg/rn) Wall 05117 -0.029 0.2311 0.735 0.0691 1.482 0.0282 1.908(989kg/rn) 7.6.3.2 A double-brace wall in the two-story building In the two-story building, the seismic reliability of a 1.82 m long double-brace wall in the first story was estimated in order to compare its performance in the building and in a separate manner. The location of the wall is shown in Fig. 7.16. Its seismic response was retrieved from the simulation database. The summary of the wall drift response with respect to eight specified PGA levels and three structural mass levels of the building is given in Table 7.23. Given the PGA, mean and standard deviation of the wall drift response in the building, using Eq. (7.1 2a) and Eq. (7.1 2b), the equivalent carried mass by the wall can be estimated. The mean and standard deviation of the estimated mass was then used to estimate the seismic reliability of the wall functioning in the building. 211 -- II EEEiI - _-_i \1.2m t----21 d ‘ubIe-braceI - I1 — -—- ---—-D-- 7280 First story Figure 7.16 Location of the 1.82 m long double-brace wall in two-story building Table 7.23 Summary of the wall response and the wall carried equivalent mass Building Avg. Stdev. mass PGA Mean Stdev Eqv. massa Eqv. rnassb eqv. eqv. (kg/m2) (g) (mm) (mm) (kg/rn) (kg/rn) mass mass Fl. Rf. (kg/rn) (kg/rn) 0.1 2.62 0.55 1079 1163 0.2 7.30 1.73 1035 1028 0.3 14.82 4.48 1027 933 0.4 24.24 8.94 1000 862160 140 974 890.5 35.94 15.61 976 827 0.6 49.83 25.07 954 825 0.7 68.38 40.98 970 914 0.8 90.31 60.62 989 997 0.1 2.90 0.71 1116 1186 0.2 8.73 2.50 1125 1092 0.3 17.67 6.12 1141 1031 0.4 28.89 12.43 1134 1017180 160 1115 650.5 42.53 20.79 1123 1000 0.6 59.03 33.66 1117 1046 0.7 81.72 56.34 1163 1225 0.8 101.47 72.59 1125 1204 0.1 3.21 0.85 1154 1206 0.2 10.28 3.30 1215 1153 0.3 20.66 8.25 1252 1142 0.4 33.51 16.01 1260 1153200 180 1236 690.5 48.40 26.08 1248 1158 0.6 68.76 45.67 1286 1325 0.7 88.43 62.32 1261 1342 0.8 113.89 85.88 1280 1338 212 aThe equivalent mass was calculated based on the response surface of drift mean (Eq. 7.12a); bThe equivalent mass was calculated based on the response surface of drift standard deviation (Eq.7.l2b). Alternatively, the wall carried mass can be estimated by distributing the total mass among the shear walls based on the effective shear wall length of individual walls. In other words, the structural mass assigned to each shear wall is proportional to its contribution to the overall lateral resistance of the building. Each shear wall is treated as a separate unit to provide the lateral resistance. This study considered only the x component of ground motions. Thus, the mass distribution was based on the effective shear wall length of the walls along the x direction. The total effective shear wall length along the x direction of the walls in the first story was 19.11 m according to the BSL of Japan. The effective shear wall length of the double-brace wall was 3.64 m. If the total structural mass is M, then, the mass on the double-brace wall is about 0.1 9M. Accordingly, this wall will carry about 70% more mass than the equivalent mass estimated by the wall response in the building. The seismic reliability analyses of the double-brace wall carrying the equivalent mass and the distributed mass used the same assumptions about the seismic hazard. PGA follows a lognormal distribution with mean of 0.25 g and COV of 0.55. The distributed mass follows a lognormal distribution with COV of 0.1. The equivalent mass follows a lognormal distribution with the mean and standard deviation given in Table 7.23. Table 7.24 gives the comparisons of the event-based seismic reliability of the double brace wall carrying the equivalent mass and the distributed mass with respect to four 213 performance expectations. The failure probabilities with associated reliability indices were evaluated by FORM. Again, three mass levels of the building were considered. The wall carrying the equivalent mass had much higher seismic reliability, which indicates the great benefits of the system effect to the shear wall functioning in the building system. Table 7.24 Seismic failure probability of two-story building and shear wall (event) Building .Serviceability Immediate Life Safety Collapse Prey. (kg/rn2) (kI) (0.5%) Occu. (1%) (2%) (3%) Fl. Rf. Pie lie Pie lie Pie lie Pie 160 140 Eqv. 974 0.2469 0.684 0.0931 1.321 0.0264 1.936 0.0111 2.286Distr. 1660 0.6004 -0.255 0.3213 0.464 0.1136 1.208 0.0508 1.637 180 160 Eqv. 1115 0.3198 0.468 0.1270 1.141 0.03697 1.787 0.0157 2.153 Distr. 1881 0.6811 -0.471 0.4120 0.222 0.1648 0.975 0.0776 1.421 200 180 Eqv. 1236 0.3866 0.288 0.1620 0.986 0.0485 1.660 0.0207 2.040Distr. 2102 0.7417 -0.649 0.4963 0.009 0.2254 0.754 0.1135 1.208 77 Summary This chapter presented the seismic reliability analyses of a series of P&B shear walls (two types of 0.91 m long single-brace walls, and eight types of 1.82 m long double-brace and structural-panel-sheathed walls), a one-story P&B building and a two-story P&B building with respect to four performance expectations (serviceability, immediate occupancy, life safety and collapse prevention) using the RSM with IS. The peak wall drift and the peak inter-story drift were chosen as the performance criteria since they are rational indicators to assess the seismic damage in wood structures. The randomness of earthquake ground motions, structural carried mass and response surface fitting errors was considered in the formulation of the performance functions. The “pseudo-nail” model and the “PB3D” model were used to perform the seismic simulations of the shear walls 214 and the buildings. The seismic reliability analyses of the shear walls indicate that the diagonal-brace (single-brace and double-brace) walls have lower reliabilities than the structural-panel- sheathed (plywood-sheathed and OSB-sheathed) walls. The 0.91 m long single-brace wall has lower reliabilities than the 1.82 m long double-brace wall. And the plywood-sheathed walls have higher reliability than the OSB-sheathed wall because more nail connections were used in the plywood-sheathed walls. Within a class of walls with the same lateral resistance mechanism, the Japanese sugi walls have the lowest reliability and the Canadian tsuga walls have the highest reliability. The seismic reliability analyses of the P&B buildings indicate that the seismic reliability of a shear wall functioning in a building system is significantly higher than the shear wall functioning as a separate unit due to the system redundancy and system effect. Therefore, the estimated seismic reliabilities of the shear walls should be at a lower bound considering its actual performance in the building system. The seismic reliabilities of the shear walls and the buildings show a tendency that from the serviceability performance expectation to the collapse prevention performance expectation, the structures had increased reliabilities. It implies that for the P&B walls and the buildings, it is more likely to exceed the serviceability performance limit state compared to the collapse prevention limit state under seismic attacks during their service life. 215 CHAPTER 8 CONCLUSIONS AND FUTURE RESEARCH 81 Conclusions This thesis presents a study to assess the seismic performance of the Japanese post- and-beam (P&B) wood buildings. Experimental studies and computer modeling were used to investigate the behavior of the structural assemblies (e.g., shear walls and floor/roof diaphragms) as well as the buildings. The response surface method with importance sampling was used to estimate the seismic reliabilities of a series of P&B shear walls and two P&B buildings to study their seismic performance in a probabilistic- based manner. A finite element model “PB3D” was proposed to simulate the lateral response of the P&B buildings under static or dynamic loads. This model is essentially a “pancake” model with roof/floor diaphragms superimposed on each other. Shear walls, represented by nonlinear shear springs, were implemented to connect the roof/floor diaphragms and transfer the lateral loads to the foundation. The “PB3D” model is intended to capture the characteristics of the global response of the P&B systems under seismic loads such as base shear, inter-story drifts and roof/floor accelerations, etc. These deformation and acceleration demands can be used to evaluate the damage caused by earthquake ground motions and set up the relevant performance functions for seismic safety estimations. To improve computational efficiency, a series of modeling techniques were used to simplify the complexity of the buildings while maintaining reasonable simulation 216 accuracy. For the P&B shear walls (e.g., diagonal-braced walls and structural-panel- sheathed walls), a versatile mechanics-based wood shear wall model called “pseudo-nail” model was used to represent the load-drift hysteresis of the walls. The original “pseudo- nail” model has been modified to model structurally nonsymmetric walls, such as the single-brace walls, by introducing an additional set of model parameters. Another important feature of the “PB3D” model is to consider the effect of the in-plane stiffness of roof/floor diaphragms on the lateral performance of the buildings. Calibrated equivalent diagonal braces were used to represent the in-plane contributions of the sheathing panels, floor joints, roof rafters, and nail connections. By doing so, the system DOFs have been greatly reduced. The “PB3D” model consists of three types of elements: nonlinear shear spring elements represented by the “pseudo-nail” models, 3D beam elements for roof/floor beams and bar elements for the equivalent diagonal braces. In the dynamic analysis, structural mass was lumped onto the floor/roof nodes based on their tributary areas. Rayleigh damping matrix was used to consider the energy dissipation of the structures deforming approximately within the linear elastic range. The damping ratio of the “PB3D” model was determined by test results or by judgment when a test database is not available. At the University of British Columbia, experimental studies have been conducted to study the performance of two types of P&B walls, a floor diaphragm, and two one-storey buildings. Timber members in these test specimens were Canadian Hem-fir, grade stamped as Canada Tsuga E120. In the shear wall tests, a total of eight 0.91 m long 217 single-brace walls were tested under monotonic and reversed cyclic loads. A Japanese cyclic loading protocol was used. The contribution of gypsum wall boards to the shear wall lateral capacity was also studied by the comparative tests of the walls in two groups. Significant enhancement of strength and stiffness was observed in the walls additionally sheathed with a gypsum wall board. In the floor diaphragm test, the in-plane stiffness of a P&B floor diaphragm was evaluated progressively by adding structural members at different test stages. Under the specified pushover loads, no significant structural failure was observed and the floor diaphragm deformed within the linear elastic range approximately. Test results show that the floor diaphragm had high in-plane stiffness. Two one-story P&B buildings were constructed following Japanese building practice. The first one was tested under biaxial monotonic pushover loads applied at the roof level. The second one was tested on a shake table subjected to the modified 1995 Kobe JMAN S earthquake record. Good seismic performance was observed in the building subject to the severe ground shaking with PGA of 0.82 g. The test database of the one-story buildings was used to verify the “PB3D” model. The shear wall test results were used to calibrate the “pseudo-nail” model parameters incorporated in the building models. The floor diaphragm test results were used to calibrate the equivalent diagonal braces for floor diaphragms. For roof diaphragms, detailed FE models were built to calibrate the equivalent braces using the commercial software package ANSYS. A “PB3D” model was then built to simulate the static and dynamic response of the one-storey building under static loads and seismic loads. 218 Meanwhile, a test database of a two-story building provided by the Building Research Institute of Japan was also used to verify the “PB3D” model. The model predictions of these two buildings agreed reasonably well with the test results. Considering the randomness involved in earthquake ground motions, structural carried mass, and the response surface fitting errors, seismic reliability analyses were conducted on ten types of P&B shear walls with different lengths, wood species and lateral resistance mechanisms, a one-story P&B building and a two-storey P&B building. A suite of Japanese historical earthquake records were used to perform the seismic simulations with different combinations of the random variables. In formulation of the performance functions, peak wall drift response was used as the performance criteria for shear walls and peak inter-story drift response was used as performance criteria for buildings since they are rational indicators to assess the damage levels of timber structures. Case studies were carried out to estimate the seismic reliabilities of the shear walls and the buildings with respect to four performance criteria: serviceability, immediate occupancy, life safety, and collapse prevention. The system effect on the shear wall reliability was also studied by comparing the performance of the selected shear walls functioning in the building or in a separate manner. The “PB3D” model and the framework of the seismic reliability assessment presented in this study provide a useful tool to aid the performance-based seismic design of the P&B structures and to achieve a balance between the seismic safety and the construction cost effectiveness. 219 8.2 Future research The “PB3D” model does not consider the effect of vertical ground motions on the structural response. To improve its completeness, the model can be further developed to simulate the structural vertical response. Therefore, knowledge about the structural characteristics along vertical direction is needed. For example, the behavior of hold-down devices, the effect of gravity loads and other mechanisms to prevent structural overturning needs to be further studied. Currently, the “pseudo-nail” wall model cannot fully address the strength degradation of wood shear walls under reversed cyclic loads. The wall model can be further developed to represent this characteristic by adding the mechanisms causing this phenomenon such as nail withdrawal. This thesis presented the seismic reliability assessment on specific P&B walls and buildings. In order to design a P&B structure to meet certain performance expectations, a library of “pseudo-nail” models needs to be established in future to represent a wide range of P&B walls constructed with different wood species, nails, nail spacing, etc. As to the building system, aspect ratio of roof/floor diaphragms, openings or offsets in building plan or elevations, irregular layout of wall systems, contributions of partition walls, need to be further investigated via experimental studies and computer modeling. Since shear walls play the most important role for the lateral resistance of the buildings, the enhancement of wall lateral strength and stiffness as well as the energy dissipation under hysteretic loads will greatly improve the lateral capacity of the buildings. Any further studies on the design details of wall configurations, reinforced connections such as 220 diagonal brace connections are greatly encouraged for this purpose. It is believed that shear walls should have better seismic performance in the building system than in a separate manner due to the system redundancy and load-sharing mechanism. This study assessed the seismic reliabilities of only two types of shear walls functioning in two specified buildings. It is strongly suggested that more research work needs to be done to assess the influence of the system effect on shear walls in a probabilistic manner. Earthquake ground motion is an important concern for the structural seismic safety and is considered as the major source of uncertainties during structural service life. In this study, it was assumed that the selected ensemble of historical records could represent the overall characteristics of the earthquakes statistically. Apparently, more studies are needed to select more representative earthquake records for seismic simulations and seismic reliability analysis. It will be very interesting to study the influence of frequency contents on the dynamic response in a quantitative manner. A rational and feasible seismic reliability method is important for the accuracy of reliability estimations of engineering structures. Practically, Monte Carlo Simulation is not feasible to estimate the seismic reliability of complicated wood systems such as the P&B buildings. The RSM with IS was used in this study and it turned out to be an efficient method. 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