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Numerical and experimental investigation of seismic torsional response of single storey ductile structures Dusicka, Peter 2000

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N U M E R I C A L A N D EXPERIMENTAL INVESTIGATION OF SEISMIC TORSIONAL RESPONSE OF SINGLE STOREY DUCTILE STRUCTURES by PETER DUSICKA B.A.Sc. The University of British Columbia, 1997 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES Department of Civil Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A April, 2000 ©Peter Dusicka, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date 4prt£ DE-6 (2/88) ABSTRACT In a structure's response to earthquake motion, the torsional response in combination with lat-eral motion can induce detrimental damage to its lateral load resisting system. Structural de-signers of buildings in seismically active areas generally take advantage of the inelastic behaviour of the structural elements to resist lateral seismic motions, yet the current codes world wide use elastic structural properties as basis for determining the torsional contribution to the response. It is clear that there is a need to better understand the inelastic response of tor-sionally susceptible structural systems. In response to this need, a collaborative research project between the University of British Columbia and the University of Auckland was initi-ated in 1998. A number of objectives guided the research, which concentrated on the response of single storey models. The study compared a recently proposed mechanism-based design (MBD) approach, intended for the design of ductile buildings, to the current design building code provisions in Canada and New Zealand. A n idealised numerical model was designed for a range of structural character-istics. The design requirements and the response of the models obtained from time history non-linear dynamic analyses were compared. Several issues were identified in the M B D approach. An investigative numerical study into the significance of strength characteristics, such as strength distribution and the level of design ductility was carried out. Stiffness-to-strength re-lationships were developed to contrast the response of elastically identical structures, which have the same total base shear capacity, but also have different distribution of strength. The re-ABSTRACT sponse of models of varying levels of design ductility were also included in the analysis. The influence of various system parameters on the inelastic torsional response was observed. An experimental model was developed and evaluated on the shake table to test the torsional response of ductile structural systems. A number of stiffness and strength eccentric model ar-rangements were tested into the inelastic range using the University of British Columbia shake table facility. Both uni-directional as well as bi-directional ground motions were used in the experiments. Finally, the response of the shake table model was compared to the response of the idealised model, similar to that used in the numerical analyses. TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES x LIST OF FIGURES xi ACKNOWLEDGEMENTS xiv CHAPTER 1 - INTRODUCTION 1.1 Torsion in Structures 1 1.2 Historical Background 2 1.3 Objectives 6 1.4 Scope of Work 7 CHAPTER 2 - DESIGN METHODS AND RELATED RESEARCH 2.1 Introduction 9 2.2 Code Based Design 9 2.2.1 Equivalent Static Method 10 2.2.1.1 National Building Code of Canada 10 2.2.1.2 New Zealand Standard Loadings Code 12 2.2.2 Response Spectrum Method 12 2.2.2.1 National Building Code of Canada 13 2.2.2.2 New Zealand Standard Loadings Code 14 iv Table of Contents 2.2.3 Discussion of Code Base Design Approach to Torsion 14 2.3 Mechanism-Based Design Method 15 2.3.1 Element and System Properties 16 2.3.2 Torsionally Unrestrained Systems 18 2.3.3 Torsionally Restrained Systems 20 2.3.4 Discussion of theMBD Method 22 2.4 Review of Related Research 23 2.4.1 Reference System 23 2.4.2 Accidental Eccentricity 24 2.4.3 Torsional to Lateral Stiffness 25 2.4.4 Lateral Resisting Elements 26 2.4.5 Earthquake Directionality 27 CHAPTER 3 - NUMERICAL COMPARISON OF DESIGN METHODS 3.1 Introduction 28 3.2 Structural Model 28 3.2.1 Mass Distribution 30 3.2.2 Stiffness Distribution 31 3.2.3 Earthquake Ground Motion 32 3.3 Torsionally Balanced Structural Systems 33 3.3.1 Design Ductility 34 3.3.2 Base Shear and Element Strengths 35 3.3.3 Artificial Strength Eccentricity 37 3.4 Torsionally Unbalanced Structural Systems 38 3.4.1 Element Strengths in CBD Models 39 3.4.2 Element Strengths in M B D Models 41 3.4.3 Comparison of Strength Distribution 42 v Table of Contents 3.5 Seismic Response 44 3.5.1 Numerical Analysis 44 3.5.2 Response of Torsionally Balanced Systems 46 3.5.2.1 Description of the Torsional Response 46 3.5.3 Comparison o f M B D and CBD Response 48 3.5.4 Effects of Rotational Mass Moment of Inertia 50 3.5.5 Response of Torsionally Unbalanced Systems 51 3.5.5.1 Description of Response 51 3.5.5.2 Comparison o f M B D and C B D Response 54 3.5.5.3 Effects of Rotational Mass Moment of Inertia 58 3.6 Summary 60 CHAPTER 4 - INFLUENCE OF STRENGTH CHARACTERISTICS 4.1 Introduction 62 4.2 Stiffness-to-Strength Relationships 63 4.3 Structural Model 67 4.3.1 Element Stiffness 68 4.3.2 Element Strength 69 4.4 Seismic Response 70 4.4.1 Earthquake Ground Motion 70 4.4.2 Numerical Analysis 72 4.4.3 Comparison of Systems of Different Strength Distributions 73 4.4.4 Relative Location of Centre of Mass 77 4.4.5 Stiffness and Strength Eccentricity Contributions 78 4.4.6 Influence of Design Ductility 79 4.4.7 Influence of Transverse Elements 79 4.5 Summary 82 vi Table of Contents CHAPTER 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL 5.1 Introduction 84 5.2 Test Equipment 84 5.2.1 Shake Table 85 5.2.2 Reaction Arm 86 5.3 Structural Model Layout 88 5.3.1 Design Philosophy 88 5.3.2 Lateral Load Resisting System 88 5.3.3 Modelling Stiffness and Strength Eccentricity 90 5.3.4 Modelling the Floor Diaphragm 91 5.4 Instrumentation Layout 92 5.4.1 Displacement Transducers 93 5.4.2 Accelerometers 93 5.4.3 Data Acquisition 94 5.5 Test Protocol 94 5.5.1 Cyclic Quasi-static Tests 94 5.5.2 Free Vibration Tests 96 5.5.3 Earthquake Simulation 96 CHAPTER 6 - RESULTS OF EXPERIMENTAL TESTS 6.1 Introduction 98 6.2 Model Configurations 98 6.2.1 Component Test Setups 99 6.2.2 Dynamic Test Setups 100 6.3 Results of Quasi-static Tests 102 vii Table of Contents 6.3.1 Hysteresis Behaviour 102 6.3.2 Stiffness and Strength Distribution 105 6.4 Results of System Identification Tests 106 6.4.1 Natural Frequencies of Vibration 107 6.4.2 Damping 108 6.5 Seismic Response 110 6.5.1 Shake Table Earthquake Motion 110 6.5.2 Maximum Demands on Lateral Resisting Components I l l 6.5.3 Inelastic Deformation and Permanent Drift 114 6.6 Summary 116 CHAPTER 7 - COMPARISON OF EXPERIMENTAL AND NUMERICAL RESPONSE 7.1 Introduction 117 7.2 Numerical Model 117 7.2.1 Lateral Load Resisting Elements 117 7.2.2 Mass Distribution 118 7.2.3 Analysis and Calibration of the Model 119 7.3 Comparison of Response 120 7.4 Summary 122 CHAPTER 8 - CONCLUSIONS AND RECOMMENDATIONS 8.1 Conclusions 123 8.2 Recommendations for Further Study 125 REFERENCES 127 viii Table of Contents APPENDIX 1 - STRENGTH DISTRIBUTION F R O M APPLICATION OF DESIGN METHODS 131 APPENDIX 2 - RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS 135 APPENDIX 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS 147 APPENDIX 4 - EXPERIMENTAL M O D E L CONSTRUCTION DRAWINGS 178 APPENDIX 5 - RECORDED D A T A FROM SHAKE T A B L E EXPERIMENTS 182 APPENDIX 6 - PHOTOGRAPHS 208 ix LIST OF TABLES Table 3.1: Variations of the Rotational Mass Moment of Inertia 31 Table 3.2: Comparison of Design Ductility for C B D and M B D 35 Table 3.3: Non-linear Spectral Acceleration (g) of Elasto-plastic Models using E L C M O D Record 36 Table 6.1: Stiffness and Strength Characteristics Obtained from Cyclic Tests 105 Table 6.2: Stiffness and Strength Distribution for Dynamic Setup Configurations . . . 106 Table 6.3: Natural Frequencies of Vibration from System Identification Tests 107 Table 6.4: Damping Ratios Obtained from Displacement Decay 110 Table 7.1: Natural Frequencies of Vibration of Numerical Models 119 x LIST OF FIGURES Figure 1.1: J.C. Penny Department Store Before the Earthquake 3 Figure 1.2: J.C. Penny Department Store After the Earthquake 4 Figure 2.1: Idealised Elasto-plastic Element and System Behaviour 17 Figure 2.2: Torsionally Unrestrained System 18 Figure 2.3: Torsionally Restrained System 20 Figure 3.1: Plan View of the Structural Model 29 Figure 3.2: Acceleration Time History of E L C M O D 32 Figure 3.3: Comparison of Elastic Response Spectra of E L C M O D and NZS4203:92 . . 33 Figure 3.4: Comparison of Element Design Strengths for Torsionally Balanced Models 37 Figure 3.5: Design Spectra for Torsionally Unbalanced Models according toNZS4203:92 42 Figure 3.6: Comparison of Average Element Strengths for M B D and CBD-NZ Models 43 Figure 3.7: Strength Distribution of Models Designed Using CBD-NZ and C B D - C N . . 44 Figure 3.8: Time History for Torsionally Balanced Model Subjected to E L C M O D . . . . 47 Figure 3.9: Ductility Demands for Torsionally Balanced Models, T n = 0.8 sec 49 Figure 3.10: Time History for Torsionally Unbalanced Model I Subjected to E L C M O D 52 Figure 3.11: Time History for Torsionally Unbalanced Model II Subjected to E L C M O D 53 Figure 3.12: Ductility Demands for Torsionally Unbalanced Models, T n = 0.8 sec . . . . 55 Figure 3.13: Diaphragm Rotations for Torsionally Unbalanced Models, T n = 0.8 sec . . 56 xi List of Figures Figure 3.14: Ductility 4 Spectrum Comparison of E L C M O D and NZS4203:92 58 Figure 3.15: Average Strength Eccentricity in Models Designed to Code Based Methods 60 Figure 4.1: Element Dimensions and Locations of CR and C V 65 Figure 4.2: Examples of Stiffness-to-Strength Relationships for Two Locations of C M 66 Figure 4.3: Model Layout in Plan View 67 Figure 4.4: Time History of San Fernando Earthquake Recorded at 234 Figueroa Street 71 Figure 4.5: Response Spectra of San Fernando Earthquake at 234 Figueroa Street . . . . 72 Figure 4.6: Normalised Ductility Demands on Elements 1 and 2 for T n = 0.8 sec and x C M = 0 75 Figure 4.7: Normalised Ductility Demands on Elements 1 and 2 for T n = 0.8 sec and x C M / D = 0.2 76 Figure 4.8: Normalised Demands on Transverse Elements 3 and 4 for T n = 0.8 sec a n d x C M / D = 0.2 81 Figure 5.1: Shake Table Layout 86 Figure 5.2: Photograph of Load Cell Attached Between Reaction Arm and Frame . . . . 87 Figure 5.3: Clamp Used in Moment Resisting Connection 89 Figure 5.4: Clamp Arrangement Used to Reduce the Effective Length of the Element . 91 Figure 5.5: Model and Instrumentation Schematic 92 Figure 5.6: Deformation History Used in Cyclic Tests 95 Figure 5.7: Hammer Used for Impact Test 97 Figure 6.1: Schematic Plan View Representation of Component Setup for Cyclic Tests 99 Figure 6.2: Schematic Plan View Representation of Model Setup for Dynamic Tests . 101 Figure 6.3: Hysteresis Diagrams for the More Flexible and Weaker Component CS1 . 103 List of Figures Figure 6.4: Hysteresis Diagrams for the Stiffer and Stronger Component CS2 104 Figure 6.5: Free Vibration Displacement Time History of Setup DS4 in the EW direction 109 Figure 6.6: Time History of Shake Table Earthquake Input Motions I l l Figure 6.7: Maximum Relative Displacement of Each Lateral Resisting Component. . 112 Figure 6.8: Maximum Ductility Demands of Each Lateral Resisting Component 113 Figure 6.9: Photograph of Moment Hinge Zone Identified by Brittle White Wash . . . . 114 Figure 6.10: Permanent Drift Values of Each Lateral Resisting Component 115 Figure 7.1: Numerical Representation of the Experimental Model 118 Figure 7.2: Ductility Demands Comparison of the Numerical and Shake Table Models 121 xiii A CKNOWLEDGEMENTS The postgraduate study of the author would not be possible without the assistance from the U B C Professional Partnership Program sponsored by BC Hydro and the Natural Sciences and Engineering Research Council of Canada. Their support is gratefully acknowledged. The topic of this study was inspired by the work of Dr. T. Paulay from The University of Can-terbury along with the agreement of cooperation from Dr. C. E. Ventura and Dr. B. J. Davidson, from the the University of British Columbia and the University of Auckland respectively. The help and understanding of the departments of civil engineering and the international student's offices in both academic institutions assisted in making the opportunity of study abroad a suc-cessful reality. The role of the supervisors extended beyond the specific realm of the research topic. Their guidance, assistance as well as academic and social discussions were enjoyed and are greatly appreciated. Additional thanks go to my university friends Hyder J., Peter L. , Jason I. and Sherrill R. who provided for an entertaining environment in and outside of campus. Special thanks and deep gratitude are extended to my parents, who have always supported my endeavours. Without their courage and perseverance in life changing circumstances as well as their continuous loving support, many of my academic and life opportunities would not be pos-sible. xiv Chapter 1 INTRODUCTION 1.1 TORSION IN STRUCTURES A specific topic in the study of the dynamic behaviour of structures subjected to earthquake induced ground motion is torsional response. The effect of torsional response on the lateral load resisting system can be observed by the rotation of the floor diaphragm in addition to lateral displacement in the plan of the structure. The source of the torsional behaviour can be traced to several causes. When subject to lateral ground motion, the level of the structure's torsional response depends mainly upon its mass, stiffness and strength distribution. Any irregularities or discontinuities of those properties can lead to a response that is not in simple translation. Furthermore these characteristic properties often vary from the design intent to the as-built reality. The differences can be caused by the variation in the material properties, inaccuracy in the member dimensions, quality of construction or other changes that can occur throughout the design, construction and life of the structure. Even when assuming that every structural property can be predicted, torsional response can still occur. A l l design codes encourage structural engineers to design buildings that are symmetric, both in plan and elevation. However, more often the variability in the architectural and func-t 1 Chapter 1 - INTRODUCTION tional purpose of the structure dictate that unsymmetric layout becomes unavoidable in the de-sign. Modern codes incorporate a number of methods by which additional strength is provided in the design of the lateral resisting elements to account for the effects of torsional behaviour on the structure's response. The additional strength required in all of these provisions is formulated from analyses of elastic structures. Professor Thomas Paulay, one of the early pioneers of the capacity design procedure (Park & Paulay, 1975), has recently published a number of papers describing a new approach to designing structures for torsional response (Paulay, 1997 a-b & 1998 a-e). This mechanism based design approach considers the inelasticity of a structure de-signed for ductile response at the ultimate limit state. Inspired by this newly proposed method-ology, which is radically different from current code approaches, this research set forth to investigate the torsional behaviour of ductile structural systems. 1.2 HISTORICAL BACKGROUND The earthquake engineering science is filled with unknowns and uncertainties. The develop-ments in earthquake structural engineering are mainly driven by learning from past failures. Structural engineering failures are rarely attributed to a single cause. Most often a combination of several factors contribute to the damage of a structure even when subjected to the large de-mands imposed by a severe ground shaking. Reconnaissance reports and examination of the impact of past events on the infrastructure have identified several damaging cases that were at-tributed mainly to the torsional response of the structure. 2 Chapter 1 - INTRODUCTION One of the first recorded observations of torsional response was made after the Alaska earth-quake at Anchorage in 1964 (Steinbrugge, 1965). One of the worst damaged buildings was the J.C.Penny department store shown before the event in Figure 1.1 and directly after in Figure 1.2. Although the building was square in plan, during this earthquake the lateral resist-ing system was not nearly as symmetric. It was determined that the failure mechanism causing the extensive damage was due to the torsional response of the building, which was the result of the unsymmetric stiffness distribution throughout the structure. The damage after the earth-quake was so severe that the structure had to be demolished to its foundations. Figure 1.1: J.C. Penny Department Store Before the Earthquake* * Godden Collection, Earthquake Engineering Research Center, University of California, Berkeley 3 Chapter 1 - INTRODUCTION Figure 1.2: J .C. Penny Department Store After the Earthquake * Godden Collection, Earthquake Engineering Research Center, University of California, Berkeley Many lessons relating to geotechnical and structural earthquake engineering were learned from the aftermath of the Mexico City earthquake of 1985. One of these was the observation that a large proportion of the structures that failed were characterised by unsymmetric distribution of strength and stiffness. The damage statistics indicated that up to 15% of the failures exhibited a pronounced stiffness asymmetry (Esteva, 1987). The majority of these were reinforced con-crete frame buildings. Plain brick masonry was used to fill the full height of those building sides that were facing the adjoining properties. These walls were designed as in-fill or as shear walls. The lateral system therefore consisted of two or three adjoining rigid in-filled walls while the remaining were conventional moment resisting frames. The observed damage pattern 4 Chapter 1 - INTRODUCTION suggested that large deflections as a consequence of torsional response had a significant role in the dynamic behaviour and the resulting damage and collapse. The effects of unsymmetric structural layout were also observed in the India-Nepal earthquake in 1988 (Thiruvengadam & Watson, 1992). The Surgical Ward-Medical facility at Derbhanga, which was designed and built structurally symmetric, suffered minimal damage. In contrast, the Industrial Department Building was severely damaged due to a heavy torsional response despite a symmetric plan shape. This response was caused by the stiffening effects of a rigid staircase that was located at one end of the building. More recently, similar observations were made after the Kobe earthquake in 1995 (Seible et. al., 1995; EQE, 1995). A five storey office building near the downtown core failed in the second story. The L-shaped structural walls in the rear of the building caused an unsymmetric distribution of stiffness with respect to the distribution of mass at that level. The result was a large torsional response resulting in excessive demands on the second floor, which ultimately collapsed. Other instances of failures or extensive damage were common in the ground floor level of numerous other structures in the area. A number of these were narrow multi-story buildings with open store fronts, which are common in this densely populated city. The irreg-ular distribution of shear walls and concrete frames resulted in a dynamic response that caused the structure to rotate as well as sway under the earthquake ground motion. From all of these observations, it is evident that the torsional response of buildings continues to challenge the structural engineering community. 5 Chapter 1 - INTRODUCTION 1.3 OBJECTIVES Research investigating the torsional response has been active since these undesirable effects became evident in Anchorage in 1964. Initially, these have concentrated on the elastic response which continues to be the basis of the current code torsional provisions. Current trends in the design of structures in zones of high seismicity are utilising the ductile behaviour of structural elements, i.e. the inelastic response. The primary goal of this research is to better understand the torsional inelastic response of ductile structures. To achieve this goal, a collaborative study between the University of British Columbia in Canada and the University of Auckland in New Zealand was initiated. The lessons learned could be extended for the study of more complicated structural systems and may ultimately lead to appropriate design code implementation. The objectives of the research were • to investigate the applicability of a newly proposed design strategy using numerical comparison to the current provisions of building codes in Canada and New Zea-land; • to investigate the influence of strength characteristics, such as stiffness-to-strength relationships and varying levels of inelasticity, on the demands of the lateral load resisting elements; • to develop a cost effective experimental model to simulate the inelastic response of ductile structural systems for uni-directional as well as bi-directional motion; and • to compare the results obtained from the experimental model to a simplified com-puter model, similar to the one used for numerical analyses. 6 Chapter 1 - INTRODUCTION The research pooled the international expertise from the two participating universities. The work completed at the University of Auckland concentrated on the numerical modelling and analysis. The work at the University of British Columbia consisted mainly of the development and testing of the experimental model by taking advantage of the shake table facility. 1.4 SCOPE OF W O R K Improving the current building code provisions to account for inelastic torsional response is the ultimate long range goal. In pursuit of this goal, the first stage of the study, which is summa-rised in this thesis, concentrates on the seismic response of single storey structural systems. The earthquake response of a single storey model is often indicative of the response of multi-storey buildings with similar dynamic characteristics. This encompasses a wide range of structures, including those multi-storey buildings with a so called "weak storey". The weak storey build-ings usually have a relatively flexible lateral resisting system in the first storey when compared with the rigid mass contained in the upper floors. In the design of buildings it is commonly assumed that the floor diaphragm is rigid. This as-sumption was also adopted in this study with the realisation that floor flexibility can also influ-ence the torsional response and ultimate demands on the lateral load resisting elements. The variability in recorded time history records from different locations, even from the same source earthquake, introduces additional complexity to the investigation of earthquake engi-neering problems. As with any numerical or experimental time history analysis, the general ap-plicability of the results increases with the number of time history records used. Due to time 7 Chapter 1 - INTRODUCTION and resource constraints, a limited number of earthquake time history records were included in this study. An additional source of torsion can also come from the ground. In earthquake engineering, it is recognised that the ground motion is not entirely in translation, but may also contain a twist-ing component. From studying instrumented buildings in a number of California earthquakes, some buildings were found to have significant increases in response due to base rotation (De La Llera and Chopra 1994 b). However, the study of the effects of this torsional ground input is beyond the scope of the research presented here. 8 Chapter 2 DESIGN METHODS AND RELATED RESEARCH 2.1 INTRODUCTION Recognising the need for an effective approach to seismic design of ductile structures, the new mechanism based design approach, which incorporates torsional effects along with ductile re-sponse, was recently proposed by Paulay (1997 a-b, 1998 a-e). For contrast, this chapter re-views the provisions of current building codes and the proposed mechanism based design strategy. The concepts of the traditional design methods adopted by modern codes and the re-cently developed methods based on mechanism based design methodology of ductility and dis-placement considerations are presented. These design methods represent a simplified approach to address an inherently complex problem. To appreciate the complex nature of torsional re-sponse combined with inelastic behaviour, an overview of the more recent research efforts into the investigation of seismic response of torsionally susceptible structures is also reviewed. 2.2 CODE BASED DESIGN Before describing the newly proposed design approach, it is important to review the current code provisions for design, focusing the attention on the torsional provisions. Design codes worldwide use a combination of factors to determine the lateral forces for seismic design. Dif-ferent methods of determining these forces are available to the engineer ranging from a simple 9 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH equivalent static approach to more complex dynamic analysis. The type of method used gener-ally depends on the importance of the structure. Torsional code provisions are similar in most codes concerned with lateral loads imposed by earthquakes. Considering the geographical lo-cation of the universities participating in this research, the relevant design code provisions are reviewed for Canada (Associate Committee on the National Building Code, 1995) and New Zealand (Standard Association of New Zealand, 1992). The advantages and shortcomings of these approaches are discussed. 2.2.1 Equivalent Static Method Despite the dynamic nature of earthquakes, a simple quasi-static design method can be used in design when following the building codes from around the world. The method is based upon assumptions of regularity of the building and is therefore inherently limited to building struc-tures without large discontinuities or irregularities. 2.2.1.1 National Building Code of Canada In the National Building Code of Canada, NBCC:95, the base shear is also calculated based primarily on the structure's natural period of vibration and weight, W. The design base shear is defined as V y^JL.jj w h e r e Ve = vS-I-F-W (2.1) where the elastic base shear, Ve, is multiplied by the structural performance factor, U, and di-vided by a force modification factor, R. The force modification factor endeavours to account for the ductility and the redundancy of the structural system. Ve is calculated from the elastic 10 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH design response spectra using the seismic response factor, S. The factors v, / and F represent the zonal velocity ratio, importance factor and the foundation factors, respectively. The member design strengths are obtained by applying a set of lateral forces, whose sum equals the design base shear, at each level of the building. In addition to the floor lateral force Fx, a torsional moment, Tx, is also applied. This torque arises from considering two types of eccen-tricities; the stiffness eccentricity and the accidental eccentricity. The stiffness eccentricity, ex, is the distance measured perpendicular to the direction of seismic loading between the centre of mass and the centre of rigidity. To account for the complexity of torsion, this distance is increased or decreased by 50%, whichever produces the worst effect in the element under consideration. The magnitude of the accidental eccentricity is equal to 0.1 • Dnx, where D^, is the plan width dimension of the building at level x. The worst case load-ing can therefore be expressed by the four possible combinations of the load cases to determine the additional torque. Tx = Fx-[(\±0.5)ex±(0ADnx)] (2.2) The equivalent static approach in its simplicity can only be applied to structures without sig-nificant irregularities. NBCC:95 does not provide a clear definition of an irregularity. Other recommendations such as those contained in the National Earthquake Hazards Reduction Pro-gram, NEHRP (Federal Emergency Management Agency, 1991), can be followed. For irregu-lar structural systems, where the equivalent, static approach is not recommended, both codes advise the use of a dynamic analysis. 11 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH 2.2.1.2 New Zealand Standard Loadings Code In the New Zealand Loadings Code, NZS4203:92, the design base shear, V, is calculated pri-marily from the first natural period of vibration, Th the ductility, //, and the weight of the struc-ture, Wt. Other factors are incorporated leading to the calculation of the design base shear as V = c • Wt where c = ch(Tv u.) • Sp- R • Z • Lu (2.3) The factor ch(TJ,JU) is obtained from a design spectrum which has incorporated the ductility value and depends on the soil properties at the site. The factors Sp, R, Z and Lu account for the structural performance, risk factor, zone and the limit state factors respectively. To obtain member design strengths, a set of lateral loads, whose sum is equal to the base shear, is applied at the centre of mass on each level of the building. In addition to this lateral floor load Fx, a torque equalled to ±0.1 • B • Fx is applied to account for the accidental torsional effects. The distance B is the plan dimension of the building at the floor height considered as measured perpendicular to the direction of loading. Note that in a system where there is no stiffness ec-centricity, the torsional provisions accounting for accidental eccentricity of NBCC:95 are iden-tical to the ones used inNZS4203:92. The added torque results in extra strength to account for a possible torsional component of the ground motion, inelastic actions, variations of stiffness with time and other uncertainties in the estimate of dead and live load locations. 2.2.2 Response Spectrum Method In instances when dynamic analysis for determining the seismic design forces is necessary, the codes recommend the elastic response spectrum analysis. This method of finding the natural 12 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH periods, the associated modes of vibration and the resulting forces is common the world over. The obtained elastic values are then scaled to account for the system ductility. With the advent and continuing improvement of computers, many computer software packages have been de-veloped to assist the designers with the detailed calculations. The elastic spectrum analysis takes into account the inherent stiffness eccentricities of the system. However, the procedure for incorporating the accidental torsional effects does vary in the respective codes. 2.2.2.1 National Building Code of Canada In Canada, a statically applied torque is used in addition to the dynamic analysis to account for the accidental eccentricity. First, the response spectrum analysis is done with the mass assumed to be located at the expected centre of mass. The obtained member forces are scaled using the ratio of the elastic base shear in the dynamic analysis and the base shear from the equivalent static method. This scaling of the response spectrum values accounts for the reduction of elastic forces due to ductility and ensures that the elastic base shear from the dynamic analysis is not less than the one recommended in the equivalent static method. Second, a static torque Tx is applied at each level as follows Tx = Fx.{±0.\.Dnx) (2.4) where Fx is the design inertia force obtained from the dynamic analysis and Dm is plan width dimension at that level. Similar to the equivalent static approach, this torque provides addition-al element strengths in order to account for the accidental eccentricity. The design member 13 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH strengths are therefore the addition of the worst condition from the combination of the dynamic analysis and the static torque. 2.2.2.2 New Zealand Standard Loadings Code The design guidelines and the current practice in New Zealand approach the problem of incor-porating the accidental eccentricity by varying the location of the centre of mass in the dynamic analysis. The distance of the mass displacement equals 0.1 • B, as measured from the antici-pated nominal location of the centre of mass. In the dynamic analysis, the designer often cannot intuitively predict which of the locations of mass result in the most adverse effect on a specific member. Hence, multiple analyses are re-quired to account for the different possible locations of the centre of mass. The maximum de-mand requirement for each member is obtained from all of these analyses. These element forces are scaled using the design spectrum scaling factors Sml and Sm2. These factors account for the ductility and also ensure that the total base shear from the response spectrum does not become less than the base shear calculated using the equivalent static method. 2.2.3 Discussion of Code Base Design Approach to Torsion Conveniences as well as limitations can be observed in using the current code approach to pro-vide sufficient strength against the torsional response of structures under earthquake loading. The consequences of applying the code provisions, in either the equivalent static or the re-sponse spectrum methods, are additional strength requirements that are applied to the lateral load resisting elements. The resulting structure is therefore designed to a total strength capac-Chapter 2 - DESIGN METHODS AND RELATED RESEARCH ity, which is larger than that due to the system translation alone. However, having the additional strength does not provide for a clear mechanism of failure. Overall, the torsional provisions in the equivalent static method are simple to implement. How-ever, a limitation arises from the static application of a force to approximate the dynamic de-mands on the structure. Dynamic properties that are characteristic in the torsional behaviour are not taken into account. For example, two structures having identical mass, stiffness and strength eccentricities but different torsional mass moment of inertia would not be expected to respond similarly. The most relevant limitation of the code is that, regardless of the design code and regardless of the type of analysis, the code provisions are always based on the elastic properties of the struc-ture. This contradicts the design intent. In areas of high seismic risk, modern structures are de-signed to behave in a ductile manner because it would not be economical to implement elastically behaving systems. 2.3 MECHANISM-BASED DESIGN METHOD This section reviews the recently proposed mechanism-based seismic design strategy for the torsional seismic response of ductile buildings as proposed by Paulay (1997 a-b, 1998 a-e). This alternate method was formulated in an effort to address some of the shortcomings of cur-rent code specifications. The method focuses on the displacements of the resisting elements at the ultimate limit state based on an applied static force. A rotation of the diaphragm is consid-ered as part of the response. Similar to the concepts of capacity design in structural systems 15 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH subject to lateral displacements, the essence of the design strategy lies in identifying a viable plastic failure mechanism. A limit in the system ductility is determined based on the arrangement of the resisting elements and their respective yield capacity. The total base shear is calculated using this reduced system ductility capacity by following the procedures for the equivalent static method currently used in the codes. The strength of the individual elements is assigned to resist the design base shear without additional concerns of stiffness eccentricity. The aim of the design method is to estab-lish the location of the centre of strength, at least approximately, so as to result in similar dis-placement ductility demands in all of the translatory elements. In this methodology, a structural system is classified as either torsionally unrestrained or tor-sionally restrained. Both structural systems are outlined in the following sections, however analyses of only the torsionally unrestrained systems are described in this report. Upon identi-fying the relevant system properties, the basis of the reduced displacement capacity for each of the two systems is presented. 2.3.1 Element and System Properties Before describing the mechanism based design approach, several system properties are re-viewed. A lateral resisting system usually consists of elements with various magnitudes of yield displacements in any one direction of resistance. Given a number of elasto-plastic ele-ments of nominal strength Vni and stiffness kt, a system yield displacement in translation, i.e. without any rotation, can be expressed by summing over all of the elements in each principal direction using the symbol £ , as 16 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH n i (2.5) An example of a two element system and the resulting system yield displacement in translation is shown in Figure 2.1. Once element 1 yields, the actual system strength deflection curve fol-lows the dashed line until element 2 yields. With no post-yield stiffness of the elements, the system response also does not have any post yield stiffness. The system behaviour is idealised into a bi-linear response shown by the solid line by extending the elastic stiffness to the inter-polated system yield displacement A . In this generalisation, it is assumed that all lateral resisting elements yield. Also, much like the individual elements, the system displacement is idealised using elasto-plastic behaviour. The location of the centre of strength, C V , becomes an important property of the structure. Consid-ering Xj to be the distance of an element to C M , then the distance measured from C V to C M , referred to as the strength eccentricity evx for the system, can be calculated as idealised system 7 element 1 element 2 A, A2 Lateral Displacement Figure 2.1: Idealised Elasto-plastic Element and System Behaviour 17 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH ev x = y (2-6) ±-> * ni 2.3.2 Torsionally Unrestrained Systems The torsionally unrestrained structural systems are described as those incapable of resisting ad-ditional torque once all of the elements, which are parallel to the line of earthquake motion, be-come inelastic. The limiting structural ductility capacity can be calculated by assuming that one element does not yield while at least one other element reaches its full ductility capacity. An example of the torsionally unrestrained system, where the distances from the centre of mass to elements 1 and 2 are identified by the fraction of the total distance D using a and p respec-tively, is shown in Figure 2.2(a). Each element's strength is assigned according to equilibrium (a) Plan (b) Displacement Profile Figure 2.2: Torsionally Unrestrained System 18 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH criteria, i.e. Vx = (3 • X Vni and V2 = a • ^ Vni, and the elements have no post yield stiff-ness. The system yield displacement at C M can be calculated from Equation 2.5 as L \ y = $-Ay]+a-Ay2 (2.7) where AyJ and Ay2 are the yield displacements of element 1 and 2 respectively. The limiting system ductility can be obtained from the geometry profile at the ultimate limit state assuming rigid diaphragm action as shown in Figure 2.2(b). It is further assumed that one of the elements yields just before the other. For simple systems this may result in an element having an actual strength in excess of the nominal design strength, a real and probable occurrence. In the case of the assumed elasto-plastic element behaviour and the location of the non-yielding element, the resulting system ductility from the displacement profile becomes one of "•A = (P - Ay\ + a • V-Mmox ' Ayl)/Ay w h e r e element 2 yields first (2.8) V-A = (P • V-AXmax ' A y \ + a ' A y i ) / A y w h e r e element 1 yields first (2.9) The system ductility restriction can be calculated for systems with several lateral load resisting elements using similar type of analysis. Further considerations of post yield stiffness can be uti-lised to increase the expected system ductility capacity. In such cases, both of the elements could become inelastic at the ultimate limit state. 19 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH 2.3.3 Torsionally Restrained Systems The torsionally restrained systems are those in which a torque applied to the diaphragm can be resisted even after the elements that are parallel to the line of earthquake yielding become plas-tic. A n example is shown in Figure 2.3(a). The torsional rotation is controlled by the transverse elements. As in the case of torsionally unrestrained systems, the limiting system ductility is cal-culated by laterally pushing the system to a point where at least one of the elements reaches its full displacement ductility capacity. I +c\ J© CM ® ® (a) Plan V1T CM © (b) Displacement Profile Figure 2.3: Torsionally Restrained System To achieve this, a lateral force equal to the design base shear, Vbase, is applied at the centre of mass. It is assumed that all of the lateral resisting elements become plastic at the ultimate limit state. At this stage, the yielded elements have no contribution to the torsional stiffness or tor-sional strength. Hence, a torque, Q - evx • X Vni > results due to the strength eccentricity in 20 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH the system. The torque is resisted only by the transverse elements, which are assumed to behave elastically. The resulting angle of rotation of the rigid diaphragm can therefore be calculated as 2X« e,» = ^ - v 2 (2-io) where kTi represents the elastic stiffness of the transverse elements. Under uniform translation of the rigid diaphragm, the system yield displacement, A , can be calculated as outlined in Section 2.3.1, "Element and System Properties". For this example, considering the displace-ments profile shown in Figure 2.3(b), where element 1 reaches its displacement ductility ca-pacity first, the limiting system ductility at C M is determined to be = T - = (»A\max- Ay\+X\ ^ T u V \ C 2- 1 1) y Note that the origin of the co-ordinate system is assumed at C M and that an anticlockwise ro-tation is taken as positive. By considering the displacement profile at the ultimate limit state, this procedure can be used for other torsionally restrained systems. Since the torsional behav-iour is controlled by the elastically behaving transverse elements, the effects of post yield stiff-ness of the yielding elements are negligible. It should be noted that torsionally restrained mechanisms could degrade into torsionally unrestrained when the torsional resistance of the transverse elements is exhausted due to the rotational demand or due to the influence of earth-quake motion in the transverse direction. 21 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH 2.3.4 Discussion of the MBD Method Similar to the equivalent static method, the M B D approach is computationally simple to imple-ment. The use of the fundamental concepts makes this method viable as a design procedure in the realm of potential code provisions. The M B D method offers an approach to the design of lateral load resisting elements by incor-porating the torsional concepts with the inelastic properties of the structure as well as consid-ering a failure mechanism. This approach is consistent with the concepts of capacity design, a methodology that originated in New Zealand and is now popular world wide. Using these con-cepts, the designer chooses the way in which the structure fails and details the components ac-cordingly. The M B D method effectively extends these concepts to torsionally susceptible structural systems. The concepts of the M B D approach can be applied to a wide range of structural systems, in-cluding those termed as irregular by current code standards. However, it should be noted that the method relies only on the application of a static force to represent the lateral inertia forces. As a result, the torsional dynamic properties are not incorporated in the methodology, which has similar limitations to that of the equivalent static method. Through personal correspond-ence with the author, Dr. Paulay has acknowledged the omission of the rotational mass moment of inertia from the M B D methodology and its possible consequences on ultimate ductility de-mands (Paulay 1999). Encouraged by his response and the initial numerical investigations, the investigation into the effects of rotational mass moment of inertia was also incorporated into this study. 22 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH 2.4 REVIEW OF RELATED RESEARCH A number of researchers world wide have decided to tackle the challenging subject of torsional structural response. The main focus of these numerical studies was the verification of code pro-visions, which are based on elastic analysis, for ductile structures. No reference to experimental testing was found during the literature review. Single storey structural models with rigid dia-phragm were used in the majority of these studies. As reflected in the code provisions, reviewed in "Code Based Design" on page 9, an increase in stiffness eccentricity results in an increase in the expected torsional influence on the re-sponse. In the research on torsional behaviour, the stiffness eccentricity has most often been normalised to the base dimension between the outer most lateral load resisting elements. This normalisation allows the results to be applicable for structures with similar eccentricity to base dimension ratios. A l l other research has focused on stiffness eccentricity as the governing source of torsional behaviour, but the M B D methodology, reviewed in Section 2.3, "Mecha-nism-Based Design Method", is solely concerned with strength eccentricity. Even with these relatively simple structural systems, many factors other than the magnitude of the stiffness ec-centricity influence the overall torsional seismic behaviour. 2.4.1 Reference System To quantify the torsional response, researchers found it convenient to compare the observed de-mand to a reference system. A symmetric model, one where the centre of mass and centre of rigidity are located at the geometric centre, has traditionally been used. Due to the symmetric balance, the model will always respond in pure translation. In effect, the system behaves as a single degree of freedom system in each direction of lateral resistance. A deviation in demand 23 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH on any of the lateral load resisting elements is used as a measure of the torsional contribution to the overall response. A similar reference system is an analytically created model that is torsionally balanced (Wong & Tso, 1994; Tso & Zhu, 1993). This system only differs from the symmetric model when three or more lateral load resisting elements are used in the direction of ground motion. A tor-sionally balanced system has an identical stiffness distribution to the eccentric model, but the centre of mass is moved to coincide with the centre of rigidity. Furthermore, the strength of the lateral elements is assumed directly proportional to their stiffness and hence the centre of strength also coincides with the centre of mass. Again, this system also responds only in trans-lation. 2.4.2 Accidental Eccentricity There are numerous sources of accidental eccentricity ranging from uncertainty in stiffness and material properties to foundation rotational motion input. The structural deformation increase due to stiffness uncertainty for example may be up to 10 % for reinforced concrete and 5 % for steel buildings, but are usually less (De La Llera and Chopra 1994 a). The majority of the studies focused on verifying existing code procedures for a variety of struc-tural systems. The structural systems were designed based on code provisions, which often in-cluded the additional strength from incorporating accidental eccentricity. This inclusion introduced a difference in opinion on the appropriate element strength for the reference model. Some researchers (Correnza et. al., 1992) believe that to be consistent, accidental eccentricity allowance should also be incorporated into the reference model. Others point out that since the 24 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH model is used as a device to quantify the torsional effects, the model itself need not be designed according to codes (Wong & Tso, 1994). Regardless of the reference model used, a number of different design procedures can be used to not only design the structure, but also incorporate the accidental eccentricity. For example, the provisions for accounting for accidental torsion in the response spectrum method vary be-tween codes reviewed earlier, i.e. between New Zealand and Canadian codes. Both incorporate the accidental eccentricity by effectively increasing the strengths of the lateral load resisting elements and hence limiting the ultimate ductility demands. However, Wong and Tso (1994) have concluded that although both approaches are viable methods for incorporating accidental eccentricity, the static torque method used in NBCC:95 is preferred over the one in which the location of mass is varied as implemented in NZS4203:92. The conclusions were that the static torque method provides a more consistent level of strength margin in the lateral load resisting elements and is computationally easier to implement because only one dynamic analysis is re-quired to obtain the element strengths. 2.4.3 Torsional to Lateral Stiffness Many studies have shown that an important relationship influencing the torsional response ex-ists between the natural lateral frequency (nL and the natural rotational frequency coe of the structure. These findings are not reflected in any of the code provisions. A parameter widely used to characterise this relationship for a system of mass m, radius of gyration r, system lateral stiffness KL and system torsional stiffness KQ, is the uncoupled torsional to lateral frequency ratio Q., calculated as 25 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH where m • r K< e 2 CD L (2.12) This ratio can be used to categorise structural systems from torsionally flexible for low values, i.e. Q < 1, to torsionally stiffer systems with higher values of Q . A low value of Q implies the torsional stiffness of the structure is small compared to the lateral stiffness. This can be caused for example by the lateral load resisting planes located close to the geometric centre. Structural systems of low values of torsional to lateral frequency ratio are not recommended and should be avoided in design (Humar and Kumar 1998). The torsional to lateral frequency ratio was also found to influence the contribution of the ac-cidental eccentricity due to stiffness uncertainty (De La Llera and Chopra 1994 a). The conclu-sions were based mostly on the response of elastic systems and included new method of including accidental torsion in design. 2.4.4 Lateral Resisting Elements Mono-symmetric structural systems consist of structural eccentricity in only one principal di-rection. The complex nature of investigating torsional response restricted most research efforts to these mono-symmetric systems. Some studies did not incorporate any transverse elements, i.e. elements oriented in the symmetric direction of the structural system (Tso & Zhu, 1992; Chandler et. al., 1992). In these cases, only unidirectional ground motion was incorporated in the analyses. Elements in both principal directions of lateral resistance contribute to the defini-tion of elastic torsional stiffness. Hence, the inelastic response of systems is influenced signif-icantly by the contribution of the transverse elements (Goel & Chopra, 1990). 26 Chapter 2 - DESIGN METHODS AND RELATED RESEARCH Three elements for each of the two principal directions of lateral resistance were often adopted to resist the earthquake induced motion. With this setup the value of Q was varied by adjusting the stiffness of individual elements. However, the number of elements has only a limited influ-ence on the response of a systems with similar stiffness and strength characteristics (Goel & Chopra, 1990) and other parameters, such as radius of gyration, can be used to vary Q . 2.4.5 E a r t h q u a k e D i r e c t i o n a l i t y Initially, the research has considered unidirectional ground motion in assessing the response of eccentric structural systems. More recently, the inclusion of both components in the time his-tory evaluation of the response has been adopted. A fully uni-directional approach neglecting transverse elements and loading, has been found to give reasonable and conservative estimates of critical deformations, but may have underestimated the ductility demand in some cases. Ac-curate evaluation of inelastic dynamic torsional effects in eccentric structures require both of the horizontal earthquake components to be considered (Chandler et. al., 1997). 27 Chapter 3 NUMERICAL COMPARISON OF DESIGN METHODS 3.1 INTRODUCTION Numerical analyses were used to evaluate the proposed mechanism-based design methodology for torsionally susceptible ductile structures. The M B D procedure was targeted for the adoption into current code provisions and therefore an approach was taken to compare numerical models designed based on M B D to those designed to current code provisions in Canada and New Zea-land. The nominal design requirements as well as the response to an earthquake record were considered. 3.2 S T R U C T U R A L M O D E L A single storey structural model was used for the numerical analyses. The seismic response of a single storey model is indicative of more complex structural systems, such as a multi-storey building and is often the first step in an attempt to identify the major issues before embarking on the analysis of more complex problems. One of the limitations of this study is the compar-ison of only torsionally unrestrained structural systems as defined Section 2.3, "Mechanism-Based Design Method". The arrangement of the structural components of the numerical model is shown in Figure 3.1. 28 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS y i CM CR x I 1 CV e ctD D x CO Figure 3.1: Plan View of the Structural Model In this model, a rigid rectangular slab of dimensions Bx and By is used to connect the massless lateral load resisting elements 1, 2 and 3. The out-of-plane stiffness and strength of each ele-ment is assumed to be negligible. Hence, elements 1 and 2 provide lateral stiffness and strength in the y-direction and element 3 in the x-direction only. The dimension D is the plan distance measured between the extreme locations of the lateral load resisting elements and the distance a • D, as measured from element 1, defines the location of the lateral load resisting elements with respect to the location of the centre of mass CM. A uni-directional earthquake ground motion was considered in the direction parallel to the y-axis only and hence the transverse element 3 was assumed to remain elastic and relatively stiff when compared to elements 1 and 2. The transverse element 3 was always placed coincidental with the location of CM in the y-direction. Hence, any stiffness eccentricity er or any strength eccentricity ev existed in the x-direction only. 29 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS 3.2.1 Mass Distribution The total mass, M, was assumed to be concentrated at the location of CM. The rotational mass moment of inertia IQ, computed with respect to CM, was included in the analysis. For a thin rigid rectangular plate of uniformly distributed mass, the rotational mass moment of inertia can be expressed as (Bx2 + Bv2) Floors with the same mass but with different plan shapes or mass distributions can have differ-ent values of rotational mass moment of inertia. As a consequence, it is important to investigate the sensitivity of different designs to variations in the torsional inertia. To this end, three dif-ferent values of I0 were considered. The nominal value corresponded to the situation where Bx = By = D. An increase as well as a decrease in the magnitude were considered in the study. The decrease was representative of a numerically small I0 such as to simulate the near absence of torsional inertia. Given a constant mass, the relative value can be geometrically compared to Bx- By = D/2. The increase in the value of I0 was chosen to be a significant increase from the nominal and can be geometrically visualised as a model where BY = 2-B., = 2-D. For a constant mass and constant distance between the lateral load re-sisting elements, the relative values of the rotational mass moment of inertia are summarised in Table 3.1. 30 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS Table 3.1: Variations of the Rotational Mass Moment of Inertia Model Variations Rotational Mass Moment of Inertia Diaphragm Dimensions Decreased I0 / -!« xo\ 4 B* = By = l Nominal I0 hi Bx=By=D Increased IQ hs = 2-5 • Io2 B = 2 • B = 2 • D A y 3.2.2 Stiffness Distribution An idealised bi-linear elasto-plastic element behaviour, similar to that shown in Figure 2.1, was assumed for all lateral load resisting elements. The post yield stiffness was assigned to be nu-merically zero. The total elastic lateral stiffness, K, in the direction parallel to the earthquake input was computed based on the predetermined natural period of vibration, T, in the direction of earthquake motion as follows: (3.2) This total stiffness was distributed among the lateral load resisting elements to create the de-sired stiffness eccentricity er such that the centre of rigidity CR was consistently located be-tween element 1 and CM. For structural systems where the centre of mass coincides with the centre of rigidity, the stiff-ness eccentricity is by definition equal to zero. The modes of vibration for these systems are therefore in pure translation or pure rotation. They are referred to as torsionally balanced (Tso et. al. 1992, 1993) because while elastic, they respond to a uni-directional ground motion 31 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS in simple translation. The structural systems where a stiffness eccentricity does exist are termed as torsionally unbalanced and result in modes of vibration that are a combination of translation and rotation. For the torsionally unbalanced models considered for the analysis, the creation of stiffness ec-centricity influenced the natural periods of vibration from the nominal values assumed in Equation 3.2. Similarly, the mode shapes in these cases changed from simple translation in one mode and pure rotation in the other, to a combination of translation and rotation for both of the modes of vibration. For this reason, Tn is referred to as the nominal period of torsionally bal-anced models, while Tx and T2 are the actual first and second periods of vibration obtained using modal analysis for the torsionally unbalanced models. 3.2.3 Earthquake Ground Motion In this part of the study, only the response to unidirectional earthquake ground motion was con-sidered. The ground acceleration time history was applied in the y-direction, parallel to the ori-entation of elements 1 and 2 of the model shown in Figure 3.1. The synthetically generated acceleration earthquake E L C M O D record, shown in Figure 3.2, was used for the analysis. 0.4 0 4 8 12 16 Time (sec) Figure 3.2: Acceleration Time History of E L C M O D 32 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS The record originated from the California 1940 E l Centro NS component where the data has been modified in frequency and magnitude such that the 5% damped elastic acceleration re-sponse spectrum approximated the NZS4203:92 intermediate soil elastic seismic hazard accel-eration spectrum as shown in Figure 3.3. 0.0 ^ 1 • • -0 1 2 3 4 Period, T (sec) Figure 3.3: Comparison of Elastic Response Spectra of E L C M O D and NZS4203:92 The E L C M O D spectrum follows especially closely for the period range T> 0.45 sec. However, it should be noted that, although the approximation was made for the elastic spectrum, the val-ues associated with the acceleration ductility response spectra of E L C M O D may have a greater variability when compared to the associated NZS4203:92 ductility response spectra. 3.3 TORSIONALLY BALANCED STRUCTURAL SYSTEMS As the definition of torsionally balanced structures suggests, no stiffness eccentricity exists. The system is considered to be structurally symmetric and therefore the equivalent static meth-33 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS od can be adopted for the design. In these conditions, the design calculations used to account for the accidental eccentricity are common to both NBCC:95 as well as NZS4203:92. For com-parison, only two different approaches of determining base shear and assigning element strength were considered; the code based design, CBD, and the mechanism based design, M B D , methods. Several aspects of the seismic response of the torsionally balanced systems were investigated. The following variables were incorporated in the torsionally balanced structural models: • location of the lateral elements with respect to C M as defined by a = 0.3, 0.5, 0.7; • nominal natural periods of vibration, Tn = 0.2, 0.4, 0.8, 1.4, 2.0 sec; • rotational mass moment of inertia Iol, Io2 and Io3. 3.3.1 Design Ductility The structural ductility value p, in the New Zealand loadings standard can be as high as 10, but is also restrained by provisions in the respective material standards. For example, a reinforced concrete ductile moment frame is allowed a maximum ductility factor of 6.0. On the other hand, the maximum value of the force modification factor in the Canadian building code is 4.0. Again the value depends on the material and also on the redundancy of the structural system. For this study, the maximum element design ductility of p. = 4.0 was assumed. In the case of the mechanism based design, the ductility was appropriately reduced for the mod-els as described in Section 2.3, "Mechanism-Based Design Method". For comparison, the sys-tem ductility of the code design can also be thought of as effectively reduced. The effective 34 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS reduction arises from the imposed increase in element strengths, which result from the appli-cation of the torsional provisions. After the design of a torsionally balanced model, shown in Figure 3.1, based on code based design, the required total strength becomes 1.2 times the value of the design base shear. Hence, models where the equal displacement concept can be applied, i.e. for T > 0.7 seconds, the effective design ductility becomes p./1.2. The comparison of the effective design ductility values for the respective methods are shown in Table 3.2. Table 3.2: Comparison of Design Ductility for CBD and M B D Location of lateral elements w.r.t. C M Design Ductility CBD* M B D a = 0.3 3.3 1.9 a = 0.5 3.3 2.5 a = 0.7 3.3 1.9 * effective ductility only for CBD models where equal displacement concept is applicable 3.3.2 Base Shear and Element Strengths In order to focus on the torsional effects alone, each torsionally balanced model was designed to respond to the time history earthquake acceleration record with a lateral ductility demand equal to the design ductility i f it only translated. Any deviation from this reference value can therefore be attributed to the torsional response of the structure. The total design base shear of a structure of mass M was calculated using the formula Vbm = sa<LTlV).M (3.3) 35 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS where sa{T, p.) is the non-linear spectral acceleration of E L C M O D . E L C M O D record approx-imates the elastic design spectrum, but may not necessarily approximate the inelastic spectra. Using the non-linear spectral values of E L C M O D itself allowed for models in each nominal period range to reach similar levels of inelasticity. The sa(T, \x) values used in the design are summarised in Table 3.3. The individual strengths were assigned to the lateral load resisting elements as outlined in Section 2.2, "Code Based Design", and Section 2.3, "Mechanism-Based Design Method", for the two respective methods. Table 3.3: Non-linear Spectral Acceleration (g) of Elasto-plastic Models using E L C M O D Record Nominal Natural Period of Vibration 0.2 sec 0.4 sec 0.8 sec 1.4 sec 2.0 sec M B D u. = 1.9 (a = 0.3,0.7) 0.623 0.579 0.333 0.172 0.126 vx = 2.5 (a = 0.5) 0.478 0.455 0.249 0.145 0.096 CBD yi = 1.9 for all a 0.274 0.236 0.160 0.110 0.058 A comparison of member strengths as determined by the two design methods for each value of a is shown in Figure 3.4. In all of the cases considered, the element strength requirements are higher when M B D was used. The difference in strength was significantly increased for models where C M is unsymmetrical with respect to the lateral load resisting elements, i.e. a * 0.5 . This is mainly due to the large reduction in the design ductility imposed by the M B D method for these cases. For example, the element strength for the model designed using M B D where Tn = 0.4 sec. and a = 0.3 is 2.1 times larger for element 1 and 1.8 times larger for element 2 than the equivalent model designed using CBD. 36 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS 0.2 0.4 0.8 1.4 2.0 Period, sec (a) a = 0.3 0.2 0.4 0.8 1.4 2.0 Period, sec (b) a = 0.5 0.2 0.4 0.8 1.4 2.0 Period, sec (c) a = 0.7 • Element 1, MBD m Element 1, CBD - Element 2, MBD • Element 2, CBD Figure 3.4: Comparison of Element Design Strengths for Torsionally Balanced Models 3.3.3 Artificial Strength Eccentricity For torsionally balanced structures there are situations where, at least numerically, no strength eccentricity exists resulting in a response that is in pure translation. For example, the optimal solution for the mechanism based design method assumes that the centre of strength coincides with the centre of mass. In real structures this is unlikely to happen as variability in steel place-ment and dead and live load distributions for example will create a structure that is different from the design model. In order to investigate the torsional response of the code based as well as the mechanism based designs, small artificial strength eccentricities were introduced to these nominally balanced structures. Several ways of introducing a strength eccentricity into a model exist. The one cho-sen was to vary the values of individual element strengths so as to result in the desired magni-tude of the strength eccentricity. The location of the centre of mass remained constant. 37 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS Increasing the strength of one element was proportionally compensated by decreasing the strength of the other. This approach permitted a better comparison between models of constant total base shear for the varying values of ev. The values of the artificial strength eccentricities used were, ev/D = 0, 0.05 and 0.10. For consistency, the artificial strength eccentricity was introduced by increasing the strength of element 1 and decreasing the strength of element 2. This resulted in C V always shifting in the direction of element 1. From Figure 3.4, the element strengths of structural models where a = 0.3 and 0.7 are identical in magnitude, but reversed between the two elements. This was expected from the symmetry in structural geometry as these models are mirror images about the y-axis in the nominal design. The two cases were kept separate because introducing the ar-tificial strength eccentricity created models that were no longer mirror images with respect to the location of CV. 3.4 TORSIONALLY UNBALANCED STRUCTURAL SYSTEMS Torsionally unbalanced structural systems are those that have a centre of stiffness that does not coincide with the centre of mass, that is er * 0. Models representing these structural systems were designed to have a predetermined magnitude of stiffness eccentricity. To achieve this, the model stiffness was assigned based on a nominal period of vibration and the elements' stiffness were adjusted as described in Section 3.2.2, "Stiffness Distribution". The following variables were incorporated in the torsionally unbalanced models: • location of the lateral elements with respect to C M as defined by a = 0.3, 0.5, 0.7; 38 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS • nominal natural periods of vibration, Tn = 0.4, 0.8, 1.4 sec; • stiffness eccentricity, e/D = 0, 0.1, 0.2 and 0.3; • rotational mass moment of inertia Io}, Io2 and Io3. The structural models were designed according to the code based design provisions of Canada and New Zealand as well as the M B D method. 3.4.1 Element Strengths in C B D Models The building codes restrict the application of the equivalent static method to structures without large irregularities. Hence for consistency, the elastic response spectrum dynamic analysis was used to determine the relevant strengths for all code base design models. The NZS4203:92 in-termediate soil elastic design spectrum, i.e. p. = 1.0 shown in Figure 3.5, was used in the dy-namic analyses. The peak modal responses were combined using the complete quadratic combination, CQC, rule to obtain the maximum probable values. Although the method of calculating the elastic response spectrum dynamic analysis is univer-sal, the approach of incorporating the accidental torsional effects varies between the NZS4203:92 and NBCC:95. The New Zealand standard shifts the location of the centre of mass while the Canadian building code applies a static torque to account for the accidental eccentric-ity as reviewed in Section 2.2.2, "Response Spectrum Method". For comparison, both of these design approaches were considered separately. They are referred to as CBD-NZ where the cen-tre of mass was shifted and C B D - C N where the static torque was applied to account for the ac-cidental torsional effects. 39 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS In both codes, the element forces obtained from the dynamic analysis are scaled such that the base shear from the response spectrum dynamic analysis is not less than the base shear obtained from the equivalent static method as outlined in "Equivalent Static Method" on page 10. As a result, an additional variation should be noted between CBD-NZ and CBD-CN when account-ing for ductility. CBD-NZ uses the ductility p = 4.0 response spectrum in Figure 3.5 to calcu-late the equivalent static base shear, while CBD-CN linearly scales the elastic base shear by dividing by R, the force reduction factor. This approach is consistent with the provisions of NZS4203:92 andNBCC:95 for the two respective methods. The difference becomes irrelevant for natural periods T> 0.7 sec, due to the equal displacement principle that is incorporated for calculation of the ductility response spectra in NZS4203:92 in this period range. Since the modes of vibration of torsionally unbalanced structural systems are a combination of translation and rotation, the period that has the dominant translational component was used to calculate the equivalent static base shear. For example, some models with larger value of rota-tional mass moment of inertia exhibited first period of vibration whose mode was dominated by rotation. Hence, in these cases the second period was used to obtain the equivalent static base shear. Any strength eccentricity in the models designed using the code based methods were a result of applying the respective design procedures. No additional strength eccentricity was intro-duced in the torsionally unbalanced cases. 40 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS 3.4.2 Element Strengths in MBD Models Unlike the current codes, the mechanism based design approach does not impose any regularity restrictions in its application. It does not incorporate the effects of torsional moment of inertia nor does it recognise any torsional response contribution from the stiffness eccentricity. In-stead, the location of C V becomes important. Hence, the calculations for determining the base shear and assigning the element strengths are similar to those for torsionally balanced cases. The reduced design ductility values listed in Table 3.2 for M B D were also adopted for the tor-sionally unbalanced cases. For M B D , the base shear was calculated using Equation 3.3. In order to be consistent with the intended code methodology, the non-linear spectral accelerations sa(T, u,) were derived from the elastic response spectrum of NZS4203-.92 denoted by Ch(T, 1). These values were calcu-lated using the equations obtained from the commentary part of NZS4203:92 and are shown in Figure 3.5 for the relevant design ductilities. C,(0.45, 1) sa(T,\i) = - for 7<0.45 sec (3.4) r ( u - l ) - 0 . 4 5 -0.7 + 1 C (T 1) sa(T,\i) = — for 0.45 <T< 0.7 sec (3.5) -(p-l)-r - y ' 0.7 + 1 C (T 1) sa(T, a.) - h ' for T> 0.7 sec (3.6) Similarly to calculating the equivalent base shear, which was used to scale the response spec-trum results, the base shear obtained for the M B D method was based on the dominant transla-tional period of vibration of the model and the associated sa(T, u) value. The individual 41 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS strengths were assigned to the lateral load resisting elements as outlined in Section 2.3.2, "Tor-sionally Unrestrained Systems". 0 1 2 3 4 Period, T (sec) Figure 3.5: Design Spectra for Torsionally Unbalanced Models according to NZS4203:92 The M B D procedure for torsionally unrestrained structural systems implied not permitting strength eccentricity and therefore ev/D = 0 was adopted for all models designed using M B D . Any rotational response was therefore due to the stiffness eccentricity only. No artificial strength eccentricity was introduced into any torsionally unbalanced models. 3.4.3 Comparison of Strength Distribution The individual element strengths are summarised for each model in Appendix 1. The element strengths of the M B D models were generally greater than the code based design cases. Figure 3.6 compares the element strengths of M B D to CBD-NZ. These design element strengths were averaged over the period range and a values. The comparison is separated into the different values of the rotational mass moment of inertia and increasing stiffness eccentric-42 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS ity. For the low values of the rotational mass moment of inertia, both of the element strengths for the M B D models are consistently greater. The difference increases for element 2 and de-creases for element 1 for the high value of the rotational mass moment of inertia, Io3. Nonethe-less, on average the total base shear capacity of M B D models was 146% of that required by CBD-NZ. O - ^ C N r n O ^ C N C ! O ^ - C N c n o o o o o o o o o o o o Stiffness Eccentricity, er/D • lo, • • Io2 • • I03 • Figure 3.6: Comparison of Average Element Strengths for M B D and C B D - N Z Models The characteristics of the strength distribution can be observed from the value of the strength eccentricity ev. For the CBD methods, the distribution of strengths that resulted from the re-sponse spectrum analyses depended not only on the natural period of vibration and stiffness ec-centricity, but also on the mass moment of inertia. As a result, the strength distribution between the two elements varied for each arrangement. Figure 3.7 illustrates the strength distribution for models designed using response spectrum methods of CBD-NZ and C B D - C N . In general, for both codes an increase in the rotational mass moment of inertia resulted in an increase in the strength eccentricity in the design of torsionally unbalanced models. The strength eccentricity is measured from the location of C M as illustrated in Figure 3.1. A posi-43 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS • b2 • b3 s 55 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -0.05 f - I •0.10 I o--Q • CE2 0.3 -0.15 Q > <u >. o •c c tt) 8 LU •a £ 55 .? "5 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -0.05' -0.10 -0.15 & b l • b2 Ob3 0.1 ose ^ B Q > 0.30 0.25 0.20 0.15 0.10 • b3 I | ° — £—|—e— 0.00 -0.051 -0.10 -0.15 CPl 0.3 Design Stiffness Eccentricity, e/D Design Stiffness Eccentricity, e/D Design Stiffness Eccentricity, e/D raj a =0.3 #;a = 0.5 f c ; a = 0.7 Figure 3.7: Strength Distribution of Models Designed Using C B D - N Z and C B D - C N tive strength eccentricity corresponds to the centre of strength to be located between element 1 and C M . Hence, the response spectrum design methods emphasised an increase in strength for the stiffer element 1. Conversely, M B D does not depend on the value of the rotational mass moment of inertia nor the stiffness eccentricity. 3.5 SEISMIC RESPONSE The code and mechanism based designs of the structural models were evaluated using non-lin-ear numerical analysis. The observed response and ultimate demands that resulted from the ap-plied ground acceleration are summarised and the observations outlined. 3.5.1 Numerical Analysis The computational speed and capacity of computers have evolved to the point where a structure can be numerically modelled and analysed in a timely and cost effective manner. A non-linear 44 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS time history dynamic response analysis was used to determine the ultimate ductility demands on the lateral load resisting elements for models, which were designed based on code and mechanism based design methods. This computationally intensive procedure was achieved with the aid of commercially available computer software. The numerical analysis was conducted using two types of structural analysis programs: • DRAIN-2DX v4.1L (University of California Berkeley, 1993) is a computer pro-gram, which is mainly used for research purposes. It is capable of two-dimensional non-linear dynamic analysis. • SAP2000 v7.1 (Computers and Structures Inc., 1998) is a commercially developed computer program, which is commonly used in structural engineering practice. Among its many features, is the capability to perform three-dimensional non-linear dynamic analysis. DRAIN-2DX was used for the investigation of torsionally balanced structural systems. In these analyses, considerations were made for stiffness and mass proportional Rayleigh damping, which was calibrated for 5.0% of critical damping in the first two modes of vibration. SAP2000, which became available for this research after the study was already initiated, was adopted for the evaluation of all remaining cases. The preference was given to SAP2000 main-ly because of its data output capabilities and wide spread use in the structural engineering de-sign practice. Modal damping was used to apply 5.0% damping on all fundamental modes of vibration. 45 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS 3.5.2 Response of Torsionally Balanced Systems This section describes the results of the analyses for torsionally balanced models, i.e. models with no stiffness eccentricity. 3.5.2.1 Description of the Torsional Response An appreciation of the response of the torsionally balanced models subjected to the prescribed ground acceleration can be gained from the computed time history values. In artificial situa-tions, such as the case with numerical modelling, the torsionally balanced models responded in only lateral motion while in the elastic range. Hence, any observed rotation was the result of the artificial strength eccentricity of the model when one or both of the elements yielded. Consider the response shown in Figure 3.8 of an elastically symmetric structure where a = 0.5, which is designed using M B D . In this example, the periods are T1 = 0.80 sec in translational and T2 = 0.65 sec in pure torsional modes of vibration. The location of C M coincided with the location of CR, exactly between elements 1 and 2. The only eccentricity was the artificially cre-ated strength eccentricity ev/D = 0.1, resulting in C V located between C M and element 1. For illustration, the first five seconds of the time history response have been separated to show the displacements of elements 1 and 2 as well as the rotation of the diaphragm and force in each of the elements. A yield histogram is used to show when each of the elements become plastic. While in the elastic range at the start of the response, the model translated without rotation. At time t = 1.91 sec, element 2 reached its yield strength and the diaphragm started to rotate. In the later part of the response when both elements were again in the elastic range, the model vi-brated harmonically in rotation as well as translation. The motion decayed in magnitude due to 46 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS damping. At the end of the response, a residual rotation of the diaphragm resulted from the dif-ferential permanent deformation of the lateral elements. Similar behaviour was observed for all other CBD and M B D strength eccentric torsionally balanced models. (a) 0 < t < 25 sec (b) 0 < t < 5 sec Figure 3.8: Time History for Torsionally Balanced Model Subjected to E L C M O D * * model designed using MBD, Tn = 0.8 sec, a = 0.5, Ig = Io2, ev/D = 0.1 Post yield stiffness was not incorporated into the model. Yet, contrary to the initial assumption o f M B D , the force in element 1 continued to increase after element 2 has yielded. In this case, element 1 also reached its yield strength on the next load reversal. A n explanation for this dis-47 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS crepancy in the mechanism design philosophy and the observed behaviour is discussed further in Section 3.5.4, "Effects of Rotational Mass Moment of Inertia". 3.5.3 Comparison of MBD and CBD Response The code based and mechanism based design approaches for torsionally balanced models were directly compared based on the ultimate ductility demand. The absolute maximum ductility de-mands with respect to the artificial strength eccentricity on both of the lateral load resisting el-ements are shown in Appendix 2. Each set summarises the results for one of the nominal natural periods of vibration. For illustration, a typical set of results is shown in Figure 3.9 for models with Tn= 0.8 sec. The graphs are divided based on a and the assigned value of the ro-tational mass moment of inertia. For the torsionally balanced models, which have no strength eccentricity, the response was in pure translation. In essence, these models behaved as a single degree of freedom systems re-sulting in identical demand on each of the two elements. This was the case for nominally de-signed M B D models. In the case of CBD where a * 0.5, an inherent strength eccentricity resulted in the nominal design as a consequence of the accidental eccentricity provisions. Where strength eccentricity existed, the diaphragm started to rotate when one or both of the el-ements yielded. Because CV was artificially moved towards element 1, the ductility demand on element 2 increased while the demand on element 1 decreased. The variance in the ductility demand between the two elements became larger with the increasing strength eccentricity. Models designed to both the CBD and M B D methods displayed similar trends. In general, the change in the ductility demand with the strength eccentricity, i.e. the slope of the line, was sim-48 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS 49 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS ilar between the two design approaches. This indicates that both M B D and C B D are similarly sensitive to any particular change in strength eccentricity. The observed ductility demands on the elements that were designed according to M B D were consistently lower than those elements that were designed according to CBD. The difference in ductility demand can be mainly attributed to the larger design base shear and consequently the larger element strength allocation for these M B D models. 3.5.4 Effects of Rotational Mass Moment of Inertia Neither the equivalent static methods of the code based design nor the mechanism based design approach incorporate the effects of rotational mass moment of inertia. However, from the nu-merical results in Appendix 2, it is evident that this characteristic property does influence the inelastic system response. Considering each value of a , the difference in ductility demand between the two elements de-creased with an increase in the rotational mass moment of inertia in both designs. For some cas-es of low mass moment of inertia, i.e. 7 o l , the ductility demand on element 2 exceeded ductility demand p. = 8.0, a value two times larger than the design intent. Such an increase in ductility was not observed in cases of higher mass moment of inertia for any values of artificial strength eccentricity. One of the assumptions of the mechanism based design approach for the torsionally unre-strained systems that have no post-yield stiffness was that once element 2 yielded, element 1 would not reach its yield strength. However, the results indicate that element 1 did indeed yield in the majority of the cases considered. A general trend can be observed regardless of the nat-50 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS ural period considered, such that an increase in the mass moment of inertia resulted in an in-crease in the ductility demand of element 1 and a decrease in demand on element 2. This decreased the difference between the demands on element 1 and element 2 in any given model resulting in a less rotational response. This observation suggests that in torsionally balanced systems, an increase in the rotational mass moment of inertia provides an increase in the tor-sional restraint against rotation for the inelastic response. The result is a more even distribution of ductility demands among the lateral elements. 3.5.5 Response of Torsionally Unbalanced Systems This section describes the results of the analyses for torsionally unbalanced systems, i.e. those that do have a stiffness eccentricity. 3.5.5.1 Description of Response In order to illustrate the response of the torsionally unbalanced systems, consider the time his-tory response, shown in Figure 3.10. In this example, the natural periods of vibration are Tj = 0.84 sec. for the mode of vibration that is dominated by translation and T2 = 0.63 sec. for the one dominated by torsion. The mass distribution is symmetric with respect to the two lateral load resisting elements, i.e. a = 0.5. The element strengths were designed using M B D , result-ing in identical strengths for the two lateral load resisting elements. Hence, the source of rota-tion in the response of the model was the stiffness eccentricity e/D = 0.3. The centre of rigidity was located between the centre of mass and element 1. For illustration, the first five seconds of the time history response have been separated to show clearly the displacements of elements 1 and 2 as well as the resulting rotation of the diaphragm. A yield histogram is used to show when each of the elements become plastic. 51 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS element 2 element 1 E 0.15 § 0.10 5 0.05 8 o.oo element 1 element 2 0.15 element 1 >^ Q -]Q element 2, $ 0.05 1 8" o.oo £ -0.05 (j -0.15 Bem.1 > I IJ 1 1, J Time (sec) 2° I IN 1 1 i*l 1 • 20 25 Bem.1 .9 >- IL Bem.2 (a) 0 < t < 25 sec IFTWI (b) 0 < t < 5 sec Figure 3.10: Time History for Torsionally Unbalanced Model I* Subjected to E L C M O D * model designed using MBD, Tn = 0.8 sec, a = 0.5 , /„ = Io2, e/D = 0.1 As a result of the stiffness eccentricity, the model started to respond in both rotation as well as translation prior to any yielding of the lateral load resisting elements. If the ground motion at this point stopped, the floor diaphragm would return to its initial equilibrium position. The model continued to respond to the ground acceleration, such that element 1 first reached its yield strength. Element 2 remained elastic until the demand on element 1 decreased and the load direction reversed. This sequence was not common among the various structural systems. For example, a model that differed only in the decreased value of the rotational mass moment 52 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS of inertia responded such that due to the combination of translation and rotation, element 2 was the first to yield. The response of this case is shown in Figure 3.11. element 1 0.15 element 2 . 0.10 i 0.05 0.00 0.05 0 ° - 1 0 ] element 1 0.15 i Time (sec) Bem.1 0 Bem.2 10 Time (sec) 15 20 25 (a) 0 < t < 25 sec Bem.1 2 >-Bem.2 0" IT (b) 0 < t < 5 sec Figure 3.11: Time History for Torsionally Unbalanced Model II Subjected to E L C M O D * model designed using MBD, Tn = 0.8 sec, a = 0.5, / ( ) = Io], e/D = 0.1 In order to investigate how yielding of the lateral elements effects the rotation of the dia-phragm, the elastic rotational response was also included in the plots. In the response shown in Figure 3.10, the yielding of the elements appears to decrease the rotation as compared to the 53 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS elastic response. On the other hand in Figure 3.11, the yielding of the elements appears to en-hance the rotational response. Torsionally unbalanced models, which were designed to CBD, most often had both stiffness as well as strength eccentricities. The response was therefore more difficult to interpret because both eccentricities contributed to the torsional behaviour. Nonetheless, the overall response still followed a similar sequence of initial elastic rotation and further inelastic contribution. In most of the cases, the equilibrium position of the diaphragm after the event indicated a residual drift in translation as well as rotation due to the different yielding of the elements. 3.5.5.2 Comparison of MBD and CBD Response The code based and mechanism based design approaches for accounting for torsion were di-rectly compared based on the ultimate ductility demands. The absolute maximum ductility de-mands on both lateral load resisting elements as well as the absolute maximum rotation of the diaphragm for the torsionally unbalanced models are summarised in Appendix 2. Each set con-tains the results for one of the nominal natural periods of vibration. For instance, a typical set of values is shown in Figure 3.12 for the element ductility demand and Figure 3.13 for the max-imum rotation of the diaphragm in models with nominal natural period Tn =0.8 sec. The graphs are divided according to the value of a , and the value of the rotational mass moment of inertia. The results include all three approaches to design of the torsionally unbalanced struc-tural systems; CBD-NZ, C B D - C N and M B D . The mechanism based design approach tends to produce lower element ductility demands when compared to the code based design methods. This trend is especially visible for lower and 54 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS 00 o i l 'pUBLU9Q Xjl[flOnQ ri 'pUBUJ3Q Aji[ipnQ ri 'puBiusQ /fyi[tpnQ rl 'pueuisQ A)||i;onrj ri 'puBiuSQ A)|| |jonQ ri 'pireuiSQ XjjinonQ ri 'puBLU3Q Aji|nonQ ri 'pueuraQ Xji|i]onQ m u s 55 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS (pej) uoi]e;oy luSejqdBjQ (pBj) UOUBJOy Ul3BJL|dB|Q (pei) uonejoji uiSEiijdEiQ (pei) uoiimo'y uiSBjiiduiQ (psj) uoijEioy uiSEjqdEiQ (pej) U O J I E I O ^ ui8EJi|dBig (puj) U O I I E J O ^ LuSeji|dB|, (pej) uoijeioy uiSejqdeiQ (pej) uone;oy uiSejqdeiQ - i—l s-3 56 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS intermediate values of rotational mass moment of inertia, i.e. I o l and I o 2 . This conservatism is the result of the large strength increase in the members that were designed using the M B D ap-proach. In comparison to the models designed using the C B D methods, the demands on any one element in M B D appear to be less variable with respect to the stiffness eccentricity than in CBD. This characteristic is associated with the consistency of strength distribution in the M B D models, i.e., all have zero strength eccentricity. In general, the maximum response of the models designed by the two different approaches of incorporating accidental eccentricity, CBD-NZ and CBD-CN, indicate similar trends. In some cases, models designed using the code based methods exceed the design ductility \x = 4.0. However, this result does not necessarily indicate inadequacy in the current code provisions. One of the factors contributing to this discrepancy can be traced to the design earthquake used in the analysis. The E L C M O D record approximates the elastic design spectrum, but does not attempt to approximate any of the ductility design spectra. The input acceleration time history can be observed to impose greater demands than is expected in the response spectrum design. In the case of the M B D , this can be shown by considering models which responded in pure translation, i.e. those where er/D = 0. The ductility demand in those cases exceeds the design ductility listed in Table 3.2. A direct comparison of the ductility p = 4.0 spectrum of E L C M O D and the design spectrum is shown in Figure 3.14. The demand variability from the artificial earthquake record, especially at the lower period, can be observed. When comparing the maximum ductility demands of the torsionally balanced and the torsion-ally unbalanced models included in Appendix 2, the demands on the elements increase at a much greater rate for the relative value of strength eccentricity as opposed to the rate of in-57 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS ^ 0.5 0.0 * 1 • • -0.0 0.5 1.0 1.5 2.0 Period, T (sec) Figure 3.14: Ductility 4 Spectrum Comparison of E L C M O D and NZS4203:92 crease for values of stiffness eccentricities. From this observation it is evident that strength dis-tribution for the torsionally unrestrained structural systems has a greater influence on the ultimate ductility demands than the stiffness distribution. This observation supports one of the main concepts of the proposed mechanism based design philosophy. The values of the maximum absolute rotation of the diaphragm of the torsionally unbalanced models are also shown in Appendix 2. The maximum rotation in models designed using M B D does not appear to be significantly different from those designed using the C B D methods. No conclusive trend was observed in the absolute maximum diaphragm rotations for the different design methods. 3.5.5.3 Effects of Rotational Mass Moment of Inertia The response of the torsionally unbalanced systems is more complex than the torsionally bal-anced. In order to focus on the effects of the rotational mass moment of inertia for the unbal-58 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS anced cases, consider the response of the M B D models. These models have a consistent strength distribution, moreover they have no strength eccentricity. The observed response in-dicated large differences in ductility demand between elements 1 and 2 for the larger value of rotational mass moment of inertia Io3. When both elements are designed to the same ductility, the element at the flexible edge appears to be under-utilised. For these torsionally unbalanced cases, the mass has started to rotate before the elements yielded. In general, the response of the torsionally unbalanced systems suggests that the mass moment of inertia becomes a driving rather than a resisting effect. The opposite effect was observed in the torsionally balanced sys-tems, which had an artificially induced strength eccentricity. In those cases, when the model became inelastic and tended to rotate, the initially non-rotating mass provided an inertial re-straint in torsion. For the code based design methods, the distribution of strengths that resulted from the response spectrum analyses depended not only on the stiffness eccentricity, but also on the value of the rotational mass moment of inertia. As described in Section 3.4.3, "Comparison of Strength Dis-tribution", the characteristics of the strength distribution among the elements can be observed from the value of the strength eccentricity. The values of the strength eccentricity summarised in Figure 3.15 were averaged over the period range and a values for the models designed using the code based approach. It is clear that the code based design methods emphasised an increase in strength for the stiffer element. As a result, compared with models designed to M B D , the response indicated a more even distribution of ductility demands on the lateral load resisting elements. These observations illustrate that the rotational mass moment of inertia needs to be included in any design procedure that intends to account for torsion. 59 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS 1 n CBD-NZ • CBD-CN m m. m\ m m m 1 o MJ f l O — <N rn O — CN c i -o o o o o o-l d d d o d o Stiffness Eccentricity, e ID • Io1 • • Io2 • • Io3 • Figure 3.15: Average Strength Eccentricity in Models Designed to Code Based Methods 3.6 SUMMARY The mechanism based approach of incorporating torsion shows potential for possible imple-mentation in design codes due to the simple application and the consideration of non-linear be-haviour. However, several issues were exposed when models that were designed using the mechanism based methodology were compared to those that were designed based on the cur-rent code provisions of incorporating torsion. • In M B D , a large increase in design strength in the lateral elements was observed as compared to the values suggested by the current code provisions. Such difference would lead to a proportional increase in the structural costs. • One of the initial assumptions in M B D for systems that have elements with no post yield stiffness is that when all but one of the elements become plastic, no restraint in the system exists. The numerical analyses demonstrated that in the majority of the cases considered both of the lateral elements yielded in the torsionally unre-60 Chapter 3 - NUMERICAL COMPARISON OF DESIGN METHODS strained models. This indicates that the initial M B D assumption is overly conserva-tive for the types of structural systems considered. The response of torsionally balanced models indicated that - an increase in strength eccentricity resulted in torsional response and caused an increase in the difference of ductility demands between the two ele-ments; and - an increase in the value of rotational mass moment of inertia resulted in a decrease in the system rotation and allowed for a more equal distribution of ductility demands among the elements. Thus, in the unique case of a tor-sionally balanced system, the moment of inertia acted as a torsional restraint when some or all of the lateral elements yielded. The response of torsionally unbalanced models indicated that - the ductility demands for models based on M B D tended to be conservative for low and intermediate values of rotational mass moment of inertia; and - an increase in the rotational mass moment of inertia for the M B D designed models resulted in an increase in the difference in ductility demand between the lateral resisting elements. 61 Chapter 4 INFLUENCE OF STRENGTH CHARACTERISTICS 4.1 INTRODUCTION In Chapter 3, several issues were identified concerning the mechanism-based design method-ology for the range of structural models considered. As a result, this method was not further investigated in the realm of design code provisions. The comparative study did point out that using different design methods leads to a different strength distribution in structural systems, which in turn influences the ultimate demands of inelastically responding lateral load resisting elements. The next, step in investigating the torsional response of ductile structures, which is discussed in this chapter, focuses on the influence of the strength characteristics on the re-sponse. Independent of any code provisions, the strength of a lateral element is varied based on its stiffness for a simple realistic example. The approach of previous numerical studies of inelastic torsion, similar to the one adopted in Chapter 3, have focused solely on verifying the adequacy of current code provisions through numerical modelling (Chandler et. al. 1997; Humar & Kumar 1998; Tso & Zhu 1992; Wong & Tso 1994). In most of these studies, the strength of the lateral resisting elements was based on elastic design calculations, such as the minimum requirements of equivalent static or response spectrum methods. Thus, the strength of the elements became a function of a design procedure 62 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS and was based on the elastic properties of the structure. However, the relationship between an element's stiffness and its strength is dependant upon the type of lateral resisting system, mem-ber dimensions and material properties. In a design procedure, this inherent relationship be-tween stiffness and strength, is divorced from one another. A number of simple relationships between an element stiffness and strength are outlined. The seismic response of a single storey model of a range of system parameters to encompass a wide variety of structural systems was analysed using time history dynamic analysis. Previously, re-search has focused on highly ductile structural systems, i.e. where the target ductility demand ranged from p. = 5 to 6. In engineering practice, structures are also designed for lower ductil-ity values. To investigate the influence of strength distribution in these cases, models of lower design ductility were also incorporated into this study. 4.2 STIFFNESS-TO-STRENGTH RELATIONSHIPS The strength of any structural element is related to its stiffness mostly through common varia-bles such as its physical dimensions, even for reinforced concrete walls (Paulay 1998 a) and frames (Priestley, 1998). Buildings are likely to have diverse stiffness-to-strength relationships of their lateral load resisting elements due to the different dimensions and material property possibilities. This is further complicated by a possible mixed systems of different lateral load resisting element combinations, such as combining frame and shear wall arrangements. For the purpose of this investigation, commonly used definitions of stiffness and strength were used. Consider an elasto-plastic rectangular element " i " of elastic modulus E, yield stress fy, section width bh depth hi and element length Z,-. The lateral bending stiffness and strength of each el-63 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS ement can be defined proportional to these properties for fix-fix or fix-pin boundary conditions as k; cc (4.1) 2 V;oc (4.2) Assuming consistent material properties, in this formulation the stiffness and strength are re-lated based on the geometric dimensions only. In a structural system, two elements in any one principal direction are sufficient to define the centre of rigidity CR and the centre of strength C V as illustrated in Figure 4.1. By varying the dimensions alone, different locations of CV can exist for the same location of CR. For example, with the proportionalities of element strength from Equation 4.2 and stiffness from Equation 4.1, CV coincides with CR for any ratio bx/b2, assuming hx/h2 = L2/Lx = 1.0. This relationship is referred to as stiffness-to-strength distribution A . On the other hand, C V will not coincide with CR for any ratio L2/Ll where hl/h2 = bx/b2 = 1.0 except for L2/Lx = 1.0. This relationship is referred to as strength distributions. Also, C V will not coincide with CR for any ratio of hx/h2 where b\/b2 = L2/Lx = 1.0 except for hx/h2 =1.0 and is referred to as strength distribution C. Other variations are possible even for the simple relationships described in this example. Each location of CR has a unique location of C V depending on the stiffness-to-strength rela-tionship used. Given the location of C M , the stiffness eccentricity er and strength eccentricity ev can be calculated. Figure 4.2 illustrates the resulting relationship between er and ev based on 64 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS Figure 4.1: Element Dimensions and Locations of C R and C V the locations of CR and C V for stiffness-to-strength distributions A , B and C for two specific locations of C M . One where the centre of mass is located exactly between the two elements, i.e. where xCM = 0, and the other where the mass is closer to element 2, i.e. where xCM/D = 0.2. In these examples, the ratio of only one of the dimensions was varied from 0 to 1.0, while the other ratios remained unity. A positive value of stiffness eccentricity implies that CR is located in the positive x-direction from C M , i.e. between C M and element 2. Simi-larly, a positive value of strength eccentricity implies that C V is located in the positive x-direc-tion from C M . For strength distribution A , the location of CR coincided with C V in all cases. For strength distribution B and xCM = 0, C V was significantly closer to C M than CR. For distribution B, the stiffness of an element increases at a significantly greater rate than its strength with a de-65 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS Stiffness Eccentricity, er/D Stiffness Eccentricity, er/D Figure 4.2: Examples of Stiffness-to-Strength Relationships for Two Locations of C M creasing effective length L. An element that is stiffer is also stronger. In the case of xCM = 0, element 2 is the stiffer and the stronger than element 1 for all positive values of er. In the case of xCM/D = 0.2, element 2 is always stiffer and stronger than element 1 for all values of e/D > -0 .2 . A number of relationships between er and ev of note exist for xCM/D = 0.2 and strength dis-tribution B. A n elastically balanced system, i.e. where er = 0, can have an eccentricity in the inelastic range, namely a significant strength eccentricity. On the other hand, a system with no strength eccentricity, i.e. where ev = 0, can have significant stiffness eccentricity. Eccentric-ity values between these two latter cases result in stiffness and strength eccentricities of oppo-site sign. Hence, although CR is located between C M and element 2, C V is between C M and element 1. Negative stiffness eccentricities are associated with larger values of negative strength eccentricity, i.e. < |e v |. 66 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS 4.3 STRUCTURAL M O D E L The structural model used for the analysis consisted of a rigid rectangular diaphragm of mass M connected to massless lateral load resisting elements, similar to that used in Chapter 3. The coordinate axis as well as other labelling notations have been modified to accommodate the ad-ditional complexity introduced by including the transverse elements and considering different locations of centre of mass with respect to the lateral load resisting elements. The arrangement of structural components used for the numerical model is shown in Figure 4.3. I C M ® © D/2 s^tlCR . CVi " ""CM D Q Q x Figure 4.3: Model Layout in Plan View Two lateral load resisting elements were used in each of the two principal directions. The ele-ments in the model were symmetrically placed about the reference coordinate axis at a distance D and X • D, where X represents the ratio of the spacing between the elements in the two prin-cipal directions. Three element arrangements were considered, A. = 0, 0.5 and 1.0. The out-of-plane stiffness and strength of each element was assumed to be negligible. Hence, elements 1 67 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS and 2 provided lateral stiffness and strength in the y-direction and elements 3 and 4 in the x-direction only. The total mass M of the model was assumed to be concentrated at C M . In general, dead and live load distributions often result in various locations of C M . Hence, the position of C M was defined with respect to the coordinate axis by the distance xCM. Due to the complexity and the number of parameters already considered, the mass was placed equidistant between the ele-ments 3 and 4 resulting in a mono-symmetric model. Two positions of C M measured with re-spect to the lateral load resisting elements were investigated in this study, xCM/D = 0 and 0.2. 4.3.1 Element Stiffness In this study, the nominal translational period of vibration was assumed identical in the two principal directions of resistance. A wide range of nominal lateral periods was considered, T = 0.2, 0.4, 0.8, 1.4 and 2.0 seconds. The model elastic lateral stiffness K was then computed using the predetermined value of Tn and M. This total stiffness was redistributed between ele-ments 1 and 2 to achieve the desired stiffness eccentricity er in the y-direction while being evenly distributed between elements 3 and 4 in the x-direction. A range of stiffness eccentric-ities -0.2 < er/D < 0.2 and their corresponding strength eccentricities were considered. The ratio of the nominal translational period Tn to the nominal rotational period TQ calculated as (4.3) 68 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS is an important parameter governing the torsional response (Humar & Kumar, 1998), where KE is the elastic torsional stiffness of the lateral elements and r is the radius of gyration about C M . A constant nominal value of Q = 1.0 was chosen. An increase in X results in an increase in KQ . In the cases considered, an appropriate adjustment in the value of the radius of gyration was made to maintain the nominal value of Q . 4.3.2 Element Strength The resisting elements were idealised to have a bi-linear force deformation relationship. The post yield stiffness was set to 3% of the elastic. This value reflected the one used in previous numerical studies (Wong and Tso, 1994) and falls within the range for common reinforced con-crete elements, i.e. less than 6% (Paulay, 1998 a). To attain one of the objectives of investigating inelastically responding structural systems of varying degrees of inelasticity, the design ductility \xd was varied as \xd = 2 , 4 and 6. The model base shear capacity Vb was determined for each principal direction from the non-linear spectral acceleration Sa(Tn, \id), which was based on the design ductility \xd and nominal pe-riod of vibration Tn. Vb = Sa{Tn,vxd)-M (4.4) The base shear was then numerically distributed to the lateral load resisting elements based on the element stiffness to reflect the two different stiffness-to-strength relationships. Two of the three stiffness-to-strength relationships identified as A and B in Figure 4.2 were considered in this investigation. Stiffness-to-strength relationship C was not incorporated in this study be-69 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS cause the relationships A and B represent the extreme cases for the different locations of C V for each location of CR. This approach resulted in a comparison of structural models that were elastically identical but had a significantly different distribution of strength, i.e. different loca-tions of CV. Since the objective of this study was not verifying conformance to design code procedures, no additional or accidental eccentricity was incorporated in establishing the design strengths of the lateral load resisting elements. 4.4 SEISMIC RESPONSE The structural models were evaluated using non-linear time history numerical analysis. A bi-directional ground acceleration input was used for the non-linear time history dynamic analy-sis. The earthquake ground motions used and the resulting response are outlined in the follow-ing section. 4.4.1 Earthquake Ground Motion The selected ground motions were obtained during the 1971 San Fernando earthquake as re-corded at 234 Figueroa Street in Los Angeles, California. The two horizontal components of this record, N37E and S53E shown in Figure 4.4, were applied in the analyses for the y-direc-tion and x-direction ground acceleration input respectively. This set of acceleration time histories was chosen for a number of reasons. The corresponding elastic response spectrum of each record is similar to the Newmark and Hall design spectrum (Newmark & Hall, 1982), which is often used as basis in modern building codes. The peak 70 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS o < -0.2 " 0 5 10 15 20 25 30 35 40 Time (sec) o < -0.2 " 0 5 10 15 20 25 30 35 40 Time (sec) Figure 4.4: Time History of San Fernando Earthquake Recorded at 234 Figueroa Street ground accelerations recorded at approximately 41 km from the epicentre on stiff soil is nearly identical in magnitude for the two components; (amax)miE = ®-2®g and (amax)S53E = 0.19g. This characteristic is desirable in the investigation of transverse ele-ment's contribution in the torsional resistance. Furthermore, this earthquake record has been used as one of a number of records in previous studies on torsional response of asymmetric structures (Humar & Kumar, 1998; Tso & Zhu, 1994; Wong & Tso, 1992). Considering that only one earthquake record was used and in an effort to reduce the uncertainty in the correlation between any single design spectrum and the earthquake ground input, the in-elastic response spectra of the earthquake itself were used for the design for the two principal directions of earthquake resistance. The inelastic and elastic spectra are shown in Figure 4.5. 71 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS These design ductility spectra were obtained from the response of an inelastic single degree of freedom SDF system with 5% damping and 3% strain hardening. 0.2 0.3 0.4 0.5 0.7 1 2 3 4 Period, T (sec) 0.2 0.3 0.4 0.5 0.7 1 2 3 4 Period, T (sec) Figure 4.5: Response Spectra of San Fernando Earthquake at 234 Figueroa Street* * Log scale was used for period values 4.4.2 Numerical Analysis Time history non-linear dynamic analysis was used to calculate the response to the bi-direc-tional earthquake ground acceleration using SAP2000 (Computers and Structures Inc., 1998) 72 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS non-linear structural analysis software. The maximum displacement of each lateral resisting el-ement was obtained. For each element " i " the ductility demand p ; was calculated by compar-ing the maximum displacement ( & m a x ) t to the yield displacement (Ay). as (Amax)i To quantify the torsional response, the element displacement as well as ductility demands were normalised as a percentage of the response of a SDF system, which had the same total base shear capacity Vb and total stiffness K for that direction of ground motion. This was achieved by comparing the element demand to the SDF displacement or ductility demand \aSDF as appropriate. In effect, the response of a SDF system represents the response of a one storey structure where C M coincides with CR as well as C V and therefore responds only in transla-tion. The difference between the observed demands in the model and the SDF demands repre-sent an increase or decrease as compared to the response in simple translation and was quantified as A, - A n n r normalised displacement demand = — (4.6) ASDF normalised ductility demand = (4.7) V^SDF 4.4.3 Comparison of Systems of Different Strength Distributions The response of the models, which were characterised by the two stiffness-to-strength distribu-tions A and B, were compared based on the maximum ductility demands on the lateral load re-73 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS sisting elements. The results for all of the elements' displacement and ductility demands are summarised in Appendix 3. For illustration, the ductility demands on elements 1 and 2 for mod-els where Tn = 0.8 sec and xCM = 0 is shown in Figure 4.6: and for Tn = 0.8 sec and xCM/D = 0.2 in Figure 4.7. Although only one earthquake record was used in the analyses, a number of distinctive behaviours and relevant trends were observed. Any vertical line can be used to compare models of identical elastic properties and total base shear capacity. The only difference is the distribution of strength among the elements, which depends on the stiffness-to-strength distribution used. For the relatively small eccentricities considered, a significant difference in response resulted between the two stiffness-to-strength distributions. In the case of xCM = 0, the weaker and more flexible element in stiffness-to-strength distri-bution A was observed to have a larger ductility demand than the stiffer and stronger element. The flexible element was element 1 for the positive values of er and element 2 for the negative values. This trend was particularly evident for higher values of design ductility, i.e. models of larger inelastic response. For stiffness-to-strength distribution B, the two elements were more closely approximating the demands of a SDF system. In the case of xCM/D = 0.2, the models with stiffness-to-strength distribution A indicated similar trends as in xCM = 0 cases. An increase in stiffness eccentricity resulted in an increase in demand. However, stiffness-to-strength distribution B was significantly different. For all values of eccentricity considered, the stiffer and stronger element 2 had almost always a much larger demand than element 1. Furthermore, the ductility demand on the element did not change as dramatically with an increase in er and remained relatively constant in many of the models. 74 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS Q •c c o •c \ 1 \ 1 II 3. \ 1 d 1 II << 1 s 1 puBiusQ XjijponQ puBuiarj Xjiijpnrj Has^/(jas^) e o Q o •C c <o o rt OJ £ II 1! •u A II << PUBUI3Q XjijipnQ PUBUI3Q X j I J i p n Q jas^/(das^) puBiuaa Xjqipna >• •c c ID O Q [3 3 fa S1U9UI3[3 3SJ3ASUBJX U33MPQ 30UB1SIQ UI 3SB3J0UI 75 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS das^/(jas^) puBuiarj Xjjinona o •c e u o dasfydastf-tf) pUBUPQ XjIIUOnQ Q pUBUPQ XjIJipnQ iastf/(dastf-tf) pUBlU3Q /CniiptiQ Q u '3 •n c o pUBUPQ XjI[ipriQ c c3 pUBUPQ / C j I l i p t i Q pUEUPQ /fyiipnQ puBiuarj Xjinpnrj c o o C/3 I 75 jas^/(aas^) pueupQ XjinpnQ bl S}U9UI9ig 9SJ9ASUEJX U93Mpg 30UB}Sia Ul 3SB3J0UT 76 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS This lack of sensitivity with respect to stiffness eccentricity indicates that an optimal value of stiffness and strength eccentricity may exist, whereby the ductility demand on the elements is less affected by stiffness eccentricity in the structural system. 4.4.4 Relative Location of Centre of Mass The recorded demands for the same values of stiffness eccentricity varied for the different lo-cations of xCM. Furthermore, the demands varied for the same absolute values of stiffness ec-centricity in some cases. For the case of xCM/D = 0, the demands on all of the elements were mirror images about the y-axis due to symmetry in geometry as well as stiffness and strength distributions. A n increase in the absolute value of a stiffness eccentricity resulted in an increase in ductility demand on one or both of the elements. For the case of xCM/D = 0.2, symmetry in response about the y-axis was not observed. Considering stiffness-to-strength distribution B for xCM/D = 0.2, symmetry in stiffness ec-centricity about the y-axis was not followed by symmetry in strength eccentricity. For example, the value e/D = 0.1 corresponded to e/D = -0.09 whereas e/D - -0.1 corresponded to e/D = -0.17. Based on the observations made earlier, where the value of strength eccen-tricity was found to contribute to the inelastic response, different demands were expected for positive and negative values of the same absolute value of stiffness eccentricity. However, stiffness-to-strength distribution A , where stiffness and strength eccentricities are of equal val-ue, also resulted in different responses for the same absolute value of stiffness and strength ec-centricity. In general, the demand increased at a greater rate for positive increase in stiffness eccentricity for positive values of xCM. The demands on the lateral elements for inelastic tor-77 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS sional response were influenced by the location of C M as defined with respect to the lateral load resisting elements. 4.4.5 Stiffness and Strength Eccentricity Contributions While elastic, systems with no stiffness eccentricity respond in pure translation. In the inelastic response however, rotation of the diaphragm does occur in structural systems where the ele-ments yield at different displacements as discussed in Section 3.5.2.1, "Description of the Tor-sional Response". The resulting combination of translation and rotation in the response induced additional demands on some or all of the elements as observed for models with xCM/D = 0.2 and er = 0. In all of these elastically balanced cases, the stiffer and stronger element, i.e. ele-ment 2, was found to have an increase in demand while the more flexible and weaker element a decrease in demand. Similarly, the models with stiffness-to-strength relationship B where xCM/D = 0.2 and er/D = 0.2 had a very small strength eccentricity ev/D = 0.02. Due to the insignificant strength eccentricity, no additional rotations were introduced while both elements have yield-ed. Nonetheless, this structural system had a rotational component in the response from the start of the ground motion and due to rotational momentum, the motion continued into the inelastic range. The maximum demands occurred in the inelastic part of the response. From these obser-vations it is clear that the strength eccentricity alone is not the sole contributor to the torsional response in ductile structures. 78 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS 4.4.6 Influence of Design Ductility A significant difference in demand between stiffness-to-strength distribution A and B can be observed for values of lower design ductility, i.e. \xd < 4 , for elastically identical systems. This illustrates that a significant contribution due to the inelastic response was also observed for sys-tems of high strength and therefore limited inelastic behaviour. One can expect the difference in the response from one strength distribution to another to di-minish as the structural system approaches purely elastic behaviour, i.e. \xd —» 1.0 . This tran-sition is expected to occur at much smaller design ductility values than considered here. Similarly, structural systems with design ductility much greater than 6 are expected to have low contribution to the torsional response from stiffness eccentricity as the response would be mostly inelastic. However, the higher values of design ductilities used in this study, which re-flect some of the maximum values used in current building codes, were also found to have a significant contribution due to the elastic stiffness eccentricity. For the range of systems con-sidered, both stiffness as well as strength eccentricity contribute to the inelastic torsional re-sponse. For the inelastic torsional response of ductile structural systems, the strength distribution seems more important than magnitude of base shear capacity assuming that the structure can accommodate the resulting displacements. 4.4.7 Influence of Transverse Elements The bi-directional ground motion imposes inelastic deformations on elements in both of the principal directions. An increase in distance between the transverse elements 3 and 4, i.e. an increase in X, was found to generally decrease the demands on elements 1 and 2. However, the 79 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS transverse elements provided a minimal level of restraint in the response mostly because they resisted the ground motion in the transverse direction at a similar level of inelastic demand. The source of the torsional response originated from the different stiffness and strength values of elements 1 and 2. Due to the resulting rotations of the rigid diaphragm the demand on ele-ments 3 and 4 were also affected. The results are presented in Appendix 3 for all of the cases considered and an example is shown in Figure 4.8. For the transverse elements, normalised dis-placement and normalised ductility demands are one and the same because the stiffness and the strength of elements 3 and 4 are always the same. The magnitude of increase in demand on el-ements 3 and 4 is significantly smaller than the increase in demand of elements 1 and 2 for the same model. In an opposite trend from the demand on elements 1 and 2, an increase in X resulted in an over-all increase in ductility demand on the transverse elements. Due to the geometric layout of the transverse elements, a rotation of the rigid diaphragm imposes larger displacements on ele-ments that are farthest from the centre of rotation. Hence, the transverse elements in models with large value of X are more affected than those where X = 0. 80 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS o II -4 aasV/(jasV-V) 3 jasV/(HasV-V) piTBUISfJ dsiQ 5^  jasV/^asV-V) PUBUI9Q "dSIQ o" il 2" 8 jasV/(HasV-V) pUBlUSQ dSIQ >. •C c o o jasV/(jasV-V) pUBUISQ dSIQ 'o •c c o o [3 HasV/(jasV-V) pUBUlSQ dsiQ Q u dasV/(jasV-V) PUBU13Q dSIQ HasV/(jasV-V) PUBUI3Q dSIQ Q •c c u o dasV/(aasV-V) PUBUI3Q dSIQ fi 00 s-3 SJU3UI3I3 3SJ3ASUBJX U39/Yq3g 30UBJSIQ UI 3SB3J0UT. 87 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS 4.5 SUMMARY Two variations of element stiffness-to-strength relationships were analysed in a single storey model using time history dynamic analysis for one set of bi-directional earthquake ground mo-tions. In the investigation of these ductile models, which were elastically identical but had dif-ferent strength distributions, a number of trends in the demands were observed. • In some cases, the strength eccentricity of the model caused the ductility demand to remain relatively constant despite a changing stiffness eccentricity. An optimal value of stiffness and strength eccentricity may therefore exist, whereby the ductil-ity demand on the lateral load resisting elements is less affected by the stiffness eccentricity of the structural system. • The relative location of the lateral elements with respect to C M was found to influ-ence the demands on the elements. The demand increased at a greater rate for posi-tive increase in stiffness eccentricity and positive values of xCM. • Both strength eccentricity as well as stiffness eccentricity contribute to the overall torsional response. • Strength eccentricity was also found to influence the response in models of limited design ductility. The strength distribution in these structural systems seems more important than magnitude of base shear capacity. 82 Chapter 4 - INFLUENCE OF STRENGTH CHARACTERISTICS A n increase in the distance between the transverse elements was found to only mar-ginally decrease the demands on elements 1 and 2, but it did increase the demands on the transverse elements 3 and 4. 83 Chapter 5 EXPERIMENTAL SETUP AND TEST PROTOCOL 5.1 INTRODUCTION Torsional seismic response, elastic as well as inelastic, has been studied using numerical inves-tigations since the adverse effects of torsion were first noticed in the Anchorage Earthquake in 1964. To date, little work has been done experimentally and no reference was found pointing to any experimental study of inelastic torsional response. The objective of the research conducted at The University of British Columbia was to take ad-vantage of the shake table facilities in order to develop and test an experimental structural mod-el. An inexpensive modular model was designed for the first of a kind experimental testing in inelastic torsional response from earthquake induced motion. This chapter reviews the equip-ment used for the experimental work and for collection of data from the experiment. The model design process and construction is described and the testing procedures outlined. 5.2 TEST EQUIPMENT Dynamic testing of the model was conducted using The Earthquake Engineering Laboratory at The University of British Columbia. This facility was upgraded in 1996 from a single axis hor-izontal motion to a multi-axis motion simulator. The facility incorporates a digital computer control and data acquisition system to perform a wide range of tests and analyses. 84 Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL 5.2.1 Shake Table The laboratory's main component is the shake table, illustrated in Figure 5.1. The table which is 3 m x 3 m (10 ft x 10 ft) in plan and 0.4 m (1 ft 4 in) deep, sits in a pit such that the top of the table is at an elevation of the laboratory floor. The table was fabricated from aluminum in a cellular structure to maximise the strength to weight ratio. The total weight of the table is 21 kN (47 kip) and it has a payload capacity of 159 kN (35.7 kip). A grid of holes 38 mm (1.5 in) in diameter provide anchor points for attaching the test specimens. The shake table along with its components were designed to have a fundamental vibrational frequency of ap-proximately 40 Hz, as not to interfere with the operation of the table, which usually operates below 20 Hz frequency. Three hydraulic actuators, one acting in the East-West (EW) direction and two acting in the North-South (NS) direction, drive the table. Oil is supplied by 265 1/min (70 Gal/min) pump at a pressure of 20.5 MPa (3000 psi). The stalling force capacity of the actuators is 165 kN (37 kip) in the y-direction and 90 kN (20 kip) for each of the actuators in the x-direction. The force reactions are resisted by the reinforced concrete wall foundations of the pit. Four hinged links support the table vertically to minimise lateral frictional resistance. This arrangement al-lows for replicating earthquake ground motion in three degrees of freedom with a maximum displacement of ±76 mm (3 in) and maximum velocity of 100 cm/sec (40 in/sec) in all direc-tions. The shake table is controlled by a signal processing subsystem, driven by a Multi-shaker Con-trol Software. The software performs a closed-loop control of the excitation to produce motions with high degree of accuracy to the requested input for models of varying mass and stiffness. 85 Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL Reaction Column Pit Outline NS Actuators 90 kN Each Anchor Holes PLAN VIEW Reaction Column Rigid sr Arm Rigid Links Hinged at 1 Top and Bottom SECTION A-A Figure 5.1: Shake Table Layout 5.2.2 Reaction A r m A rigid arm, which was pin-connected to a reaction column located independent of the shake table, was employed for cyclic testing. The arm, illustrated in the section view of Figure 5.1, was hinged at the rigid column and supported vertically by a brace which carried the self weight. The arm was secured to the top of the test frame using a pin bracket. This arrangement 86 Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL resulted in a tension-compression only system for the arm and did not introduce any additional vertical forces. The shake table was moved in the EW direction to the desired displacement, creating a quasi-static relative displacement between the top frame and the shake table. The load resisted by the reaction arm during testing was measured using a 44.5 kN (10 kip) Sensotec Precision Pancake Thin Model 41 load cell. The load cell was mounted directly between the frame and the reac-tion arm as shown in Figure 5.2. Figure 5.2: Photograph of Load Cell Attached Between Reaction Arm and Frame 87 Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL 5.3 STRUCTURAL M O D E L LAYOUT To experimentally simulate inelastic response, effort was placed on the selection of material and on the design of the model. The details of the design are contained in the construction draw-ings in Appendix 4. 5.3.1 Design Philosophy The governing philosophy in the design of the experimental model was to employ a relatively inexpensive model which is easily constructed and easily repaired after inelastic deformations are achieved. A n additional factor in deciding on the model design was the convenience to re-arrange the lateral load resisting elements and the location of the additional mass. 5.3.2 Lateral Load Resisting System From lessons learned in past earthquakes and current design methodology of capacity design, "weak beam and strong column" approach is preferred for the design of ductile structural sys-tems. However, "strong beam weak column" is practical and acceptable design practice for sin-gle storey structures. Considering the bi-directional earthquake ground motion, along with satisfying the design philosophy of rapid and cost-effective replacement of damaged parts, the inelastic yielding of the vertical elements were chosen for this model. Desirable material properties of the lateral load resisting system were sought such that large inelastic displacements could be achieved with force deformation characteristics that closely approximate bi-linear response. Mild steel that deforms in bending moment became an ideal choice. Rectangular flat vertical bars were chosen to resist the lateral motion. The bars cross-es Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL section was 6.3 mm (0.25 inches) by 38.1 mm (1.5 inches) and they were cut to 1215 mm (4 ft) lengths. Clamps, which can accommodate flat bars of various size, were developed to resist the bending moment due to displacement of the top frame relative to the shake table. Each clamp consisted of a plate, 220x 152x 12 welded to a hollow rectangular structural steel section HSS16 x 51 x 8 . Holes were drilled and tapped to accommodate 5/8" bolts (imperial size was chosen due to cost considerations) used to secure the flat bars. This moment resisting connec-tion, illustrated in Figure 5.3, was used for both the top as well as the bottom of the frame to connect the mild steel vertical bars. The assembly resisted moments in the strong and weak di-rections of the rectangular bar as illustrated by M s t r o n g and M w e a k respectively. Figure 5.3: Clamp Used in Moment Resisting Connection Using bolts to secure the flat bar in all directions in the clamp provided for a reliable connection and at the same time a quick replacement procedure. Not using welding or introducing other irregularities in the yielding elements allowed for the development of moment hinge zones without cracking or other undesirable deformations. To protect the bolts and isolate the yield-89 Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL ing hinge zones to a predetermined location, steel rectangular spacers were used between the bar and the bolts in the clamp. The clamp was bolted to the floor diaphragm at the top as well as the matching base at the shake table elevation using two bolts per clamp. The effective length of the rectangular bar, measured between the top and bottom clamps, was 910 mm. The moment resisted by the bending of the flat bar was transferred via the clamp into the frame di-aphragm. 5.3.3 Modelling Stiffness and Strength Eccentricity Two ways of introducing stiffness and strength eccentricity exist in the experimental model de-scribed above. The stiffness and strength characteristics of the lateral load resisting system can be changed. Or, the location of the mass can be moved to create the eccentricity. Both of these approaches were utilised in the experimental model. Changing the characteristic properties of the lateral resisting element allows for the added flex-ibility of manipulating stiffness and strength by different amounts. As described in Section 4.2, "Stiffness-to-Strength Relationships", decreasing the effective length for rectangular moment resisting element drastically increases the stiffness of the element, while only marginally in-creasing the strength. Decreasing the effective length was achieved by an additional clamp which was bolted to the bottom frame 180 mm higher, as illustrated in Figure 5.4. This resulted in 20% decrease to 730 mm in effective length for the stiffer and stronger configuration of the rectangular bar. Two Z55 x 55 x 6.4 angles bolted the two clamps together and a steel link provided the restraint in the strong moment direction. 90 Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL M, M, additional clamp 'weak 'strong 180 mm L55x55x6.4 steel link Figure 5.4: Clamp Arrangement Used to Reduce the Effective Length of the Element Calculating the location of CR and C V , the mass could be positioned such that only strength or stiffness or both eccentricities exist in the model as appropriate. 5.3.4 Modelling the Floor Diaphragm The model consisted of two matching welded frames, one representing the floor diaphragm and the other providing the connection to the shake table. For illustration, see Figure 5.5. To form each frame, steel channels 050x12 were welded to form a stiff diaphragm 1300 mm square. The governing design consideration for choosing the size of the channels was the necessity of sufficient depth to provide for an effective moment transfer between the clamp and the frame while limiting the inelastic response to the mild steel bars. The frame also needed to support the weight of the additional mass necessary to provide inertia to yield the lateral resisting ele-ments. This additional mass consisted of steel plates 1500 x 600 mm 2 of 1 / 4 " (6.3 mm) and 1 / 2 " (12.7 mm) depths that were securely clamped to the top frame. 91 Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL Holes were drilled in the web of the channels, such that lateral elements could be installed using the clamp system at numerous configurations. The bottom matching welded frame was secured to the shake table by 8 - 5 / 8 " bolts to provide a rigid base for the model. 5.4 INSTRUMENTATION LAYOUT The main objective was to gather the relative displacement of the single storey model at the locations of the lateral load resisting elements. Also , system identification tests required for ac-celeration to be measured. Strategically placed sensors were used to obtain displacements and acceleration measurements. The instrumentation schematic is shown in Figure 5.5. Figure 5.5: Model and Instrumentation Schematic 92 Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL The shake table accelerations and displacements were recorded from the instrumented actua-tors that drive the table. In addition, displacement and accelerations of the base were measured for assurance that movement of the base did not occur with respect to the table. Furthermore, the sensors at the base could be used to detect any rotational momentum transfer between the model and the shake table. This was not expected to occur due to the small rotational moment of inertia of the model relative to the table. In the future, this instrumentation layout can also be implemented to introduce and monitor torsional ground motion input of the shake table. A l -though this is possible using the shake table facility at U B C , this complexity was not introduced in this study. 5.4.1 Displacement Transducers Celesco PT-101 displacement transducers with 254 mm (10 in) and 2540 mm (100 in) range were used to measure the horizontal motion in the EW and NS directions respectively. The dis-placement transducers were mounted independent of the shake table and connected to the frame using a thin steel cable. The displacement response of the diaphragm was monitored by two displacement transducers mounted on each of two sides of the model. They are labelled on the schematic as "e", "f ' and "g", "h" for east-west and north-south directions respectively. As-suming that the diaphragm is rigid, the displacement of any point could be calculated from the recorded data. 5.4.2 Accelerometers Accelerometers were used to measure the acceleration of both the model diaphragm as well as the model base. Four tri-axial ICSensors O E M Model 3021 accelerometers, labelled "O", "P", "Q" and " M " on the schematic, were used to monitor accelerations in the EW, NS and vertical 93 Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL directions. They are rated for acceleration range of up to ±5 g. Two uni-directional ICSensors O E M Model 3110 accelerometers, labelled " N " and " L " for EW and NS direction respectively, were also attached to the base. These sensors have a rated range of ±2 g with 1 % accuracy. 5.4.3 Data Acquisition A Labview-based computerised data acquisition system was used to record all of the sensor da-ta. A n electrical signal generated by the sensors, displacement transducers or accelerometers, was transmitted over cable into an amplifier. The signal was then amplified and filtered using a signal conditioner to improve the quality of the recorded electrical signal. A l l data recorded in the dynamic tests used a low pass filter set to 30 Hz, incorporating Butterworth order 2 co-efficients. The data was stored on the data acquisition computer at 200 samples per second in ASCII format. 5.5 TEST PROTOCOL Prior to conducting the earthquake simulation, characteristic dynamic properties of the speci-mens were obtained. Due to the replacing of yielded elements, a level of consistency from one setup to another was desired. A number of component and system identification tests were car-ried out before each dynamic test. 5.5.1 Cyclic Quasi-static Tests Uncertainty in material properties and interaction between the components of the frame re-quired preliminary component testing to obtain information on load vs. displacement charac-teristics of the lateral load resisting system. Rather than performing specific material tests, such 94 Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL as cupon tension tests, from which stiffness and strength of the system can be inferred, cyclic quasi-static tests were directly performed on selected components of the system. This approach allowed for testing not only the moment yielding flat bars independently, but also the interac-tion of the clamp and the frame. Standard testing procedures were followed based on the guidelines in ATC-24 (Applied Tech-nology Council, 1992). Three cycles of push and pull input were used for each deformation control parameter 8, as shown in Figure 5.6 where ni signifies the number of cycles N in duc-tility value i. Six cycles, which were less than the yield displacement Sy, were used before in-elastic displacements were recorded. The pattern was repeated with increase displacement up until the maximum possible displacement of the table of ±76 mm (3 in) was reached. N Figure 5.6: Deformation History Used in Cyclic Tests •displacement limited by shake table capacity 95 Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL 5.5.2 Free Vibration Tests Impact testing is a standard method of identifying the dynamic properties of test specimens by analysing the free vibration after the model was struck. The top frame was struck by a hammer, which is instrumented to record the load. Based on the response of the system, as measured by the instrumentation, frequency response functions can be calculated. Natural frequencies were identified using this method by striking the model at the centre of mass. In addition to the impact hammer tests, free vibration tests were also conducted using the shake table. A nearly instantaneous displacement of 4 mm was introduced by the shake table. In ef-fect, this test simulates pulling the structure to a set displacement and releasing it. The differ-ential displacement between the mass and base of the model resulted in free vibration of the model which was much larger in displacement magnitude than the vibrations caused by the im-pact hammer. In addition to obtaining natural frequencies, damping was also extracted from these tests using the amplitude decay method. 5.5.3 Earthquake Simulation The final test of the model was subjected to an earthquake simulation. The shake table is capa-ble of reproducing ground motion in two principal directions, east-west (EW) and north-south (NS). Each setup was subjected to an earthquake record in three individual test directions as follows: • EW component only; • NS component only; and • EW and NS components simultaneously. 96 Chapter 5 - EXPERIMENTAL SETUP AND TEST PROTOCOL F i g u r e 5.7: H a m m e r used f o r I m p a c t Tes t Each of the seismic simulation tests were expected to yield some or all of the lateral force re-sisting elements. The damaged elements were replaced as needed for the following series of tests. 97 Chapter 6 RESULTS OF EXPERIMENTAL TESTS 6.1 INTRODUCTION From the observations made in the numerical analyses, a number of experimental model con-figurations were chosen for the shake table tests. Component quasi-static tests were carried out to find the elastic as well as the inelastic properties of the lateral load resisting elements. Dy-namic system identification tests were used to obtain the elastic dynamic properties of the mod-el before a uni-directional as well as bi-directional earthquake ground motions were applied to excite the model into the inelastic range. The configuration of the test setups is described and the results of the tests presented. 6.2 M O D E L CONFIGURATIONS In experimental testing, time constraints for construction of the model and inhibitive costs of materials restrict the amount of tests that can be conducted. This is further highlighted with the damage associated with inelastic testing, whereby parts of the model must be replaced after each test. Careful consideration of these factors lead to the selection of test setups for the quasi-static and dynamic tests. 98 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS 6.2.1 Component Test Setups Quasi-static cyclic tests were used to obtain the elastic stiffness, yield strength and post-yield stiffness of the lateral load resisting elements. Two types of lateral elements were used to create stiffness and strength eccentricities. One is stiffer and stronger when compared to the other, which is more flexible and weaker. The difference in stiffness and strength between these two elements was achieved by decreasing the effective length of the bars as described in Section 5.3.3, "Modelling Stiffness and Strength Eccentricity". To test the characteristics of each of these lateral resisting element components separately, two different test configurations were used as illustrated in Figure 6.1. Component setup CS1 was used to test the resistance of the flexible and weaker elements and CS2 was used for the stiffer and stronger elements. JL D = 1050 mm J . D = 1050 mm Setup: CS1 Setup: CS2 | Stiffer and Stronger 0 More Flexible and Element Weaker Element Figure 6.1: Schematic Plan View Representation of Component Setup for Cyclic Tests 99 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS The rectangular bar elements were symmetrically mounted to resist the force generated by the reaction arm, which is oriented in the EW direction, in only translation. Each rectangular bar can resist a lateral force in-plane or out-of-plane, corresponding to the larger and smaller values of the second moment area, respectively. In each component setup, two elements were resisting the lateral force in-plane and two elements were resisting the force out-of-plane. The distance between the elements resisting in plane was 1050 mm. 6.2.2 Dynamic Test Setups Mono-symmetric models similar to those considered for the numerical analyses were used. Eight lateral load resisting elements were included to form the lateral load resisting system. Two of these elements were the stiffer and stronger elements acting in-plane in the EW direc-tion. Four specific setup configurations of the lateral load resisting elements and locations of C M were chosen for the experimental dynamic testing as illustrated in Figure 6.2. An additional mass of 700 kg in the form of rectangular 1500 x 600 mm 2 steel plates was se-cured at the geometric centre of the top frame for DS1 and DS3, while it was placed such that the overall location of C M of the model was at approximately the same location as CR for DS2 and DS4. The location of CR was determined from the cyclic test results discussed in Section 6.3.2, "Stiffness and Strength Distribution". Hence, DS2 and DS4 had no stiffness ec-centricity. Longitudinal lateral load resisting elements are referred to those elements that resist the lateral forces in-plane in the EW direction and transverse lateral load resisting elements refer to those that resist in the NS direction. Dynamic setups DS1 and DS2 have the transverse elements lo-100 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS JL I JL y+ 131 mm T IT T D = 1050 mm Setup: DS1 J] ^ • D = i050 mm Setup: DS2 y* LJ i i i mn n r r D = 1050 mm D = 1050 mm Setup: DS3 Setup: DS4 | Stiffer and Stronger [] More Flexible and 9 Centre of Mass Element Weaker Element Figure 6.2: Schematic Plan View Representation of Model Setup for Dynamic Tests cated coincidental with the centre of mass in the NS direction, while DS3 and DS4 have the transverse elements spaced at the same distance as the longitudinal elements, represented by dimension D = 1050 mm. Each of the four setups was subjected a uni-directional earthquake motion in each of the mod-el's principal directions and also to a bi-directional earthquake motion. These three dynamic 101 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS tests are identified based on the direction of earthquake shaking in the test; i.e. EW, NS and EW&NS. 6.3 RESULTS OF QUASI-STATIC TESTS The quasi-static cyclic tests were used to obtain the static characteristic properties of the lateral load resisting elements. Of particular interest was the hysteresis behaviour and the resulting elastic stiffness, post-yield stiffness and the yield strength. Each dynamic setup consisted of a combination of the component setups. The individual elements may be rearranged, but act ef-fectively as one component of the same stiffness and strength characteristics. For example, the elements in DS1 resisting lateral loads in the EW direction consist of one CS1 component on the south side and one CS2 component on the north side. The results of the quasi-static tests are therefore presented for the component setup as a whole, and was not separated into the char-acteristic properties of one bar element. 6.3.1 Hysteresis Behaviour Each of the component setup configurations were tested two times for consistency. The steel was obtained in one batch order from the supplier in an effort to minimise variance in the ma-terial. The results of the cyclic tests are shown in Figure 6.3 for CS1 and Figure 6.4 for CS2. A good agreement between the two tests of the same component arrangement was found. The mild steel bars were able to undergo large lateral displacements without fracturing. As can be expected for a mild steel rectangular bar in bending, the hysteresis loops were well defined and 102 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS o -100 o p. -100 Displacement (mm) -6-Displacement (mm) gure 6.3: Hysteresis Diagrams for the More Flexible and Weaker Component CSl" * maximum displacements were governed by shake table displacement limits of ±76 mm 103 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS o [il o PH Displacement (mm) -6-Displacement (mm) Figure 6.4: Hysteresis Diagrams for the Stiffer and Stronger Component CS2 * maximum displacements were governed by shake table displacement limits o f ± 7 6 m m 104 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS consistent in each test. The shape of the force deformation curve can be confidently approxi-mated by a bi-linear response curve. 6.3.2 Stiffness and Strength Distribution The stiffness and strength of each of the component configurations was obtained from the re-sults of the cyclic tests. These properties, as summarised in Table 6.1, were calculated by ap-proximating the force deformation behaviour with a bi-linear response. Table 6.1: Stiffness and Strength Characteristics Obtained from Cyclic Tests Component Setup Elastic Stiffness (kN/mm) Post-yield Stiffness (kN/mm) Yield Strength (kN) C S l a 0.131 0.015 3.31 CSlb 0.131 0.015 3.30 CS2a 0.217 0.020 4.13 CS2b 0.220 0.021 4.17 two tests, identified as a and b, were made for the same component setup The stiffness and strength comparison between the two tests of the same component setup were within 1 % in elastic stiffness as well as yield strength values. By reducing the effective length of the mild steel bars from CS1 to CS2, the stiffness increased approximately by 67 % while the strength increased by only 26 % when compared to CS 1. Note that the out-of-plane yield displacement of the transverse elements was never reached in the cyclic tests. The inelastic behaviour was confined to the bars resisting the lateral force in-plane. Hence, the post-yield stiffness of the overall component behaviour consists of the ine-lastic post-yield behaviour of the longitudinal bars resisting in-plane and the elastic response 105 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS of the transverse bars resisting out-of-plane. The post-yield stiffness was estimated at 11 % and 9 % of the elastic stiffness for CS1 and CS2 respectively. The stiffness and strength distribution in the model can be calculated for each dynamic setup based on the values of stiffness and strength of the two tested components. The resulting stiff-ness and strength eccentricity for the dynamic setups is summarised in Table 6.2, using the la-belling convention of Figure 4.3. For elastically torsionally balanced setups DS2 and DS4, the location of C M , which coincides with CR for these cases, was determined to be 0.125 • D = 131 mm from the geometric centre. Table 6.2: Stiffness and Strength Distribution for Dynamic Setup Configurations Component Setup Location of C M from axis origin xCM/D Stiffness Eccentricity er/D Strength * Eccentricity ev/D DS1 0 0.125 0.057 DS2 0.125 0 -0.068 DS3 0 0.125 0.057 DS4 0.125 0 -0.068 * negative values of strength eccentricity indicate location of CV between CM and element 1 using notation in Figure 4.3 6.4 RESULTS OF SYSTEM IDENTIFICATION TESTS System identification tests were necessary to obtain the elastic dynamic properties of the mod-els before subjecting it to the earthquake ground motion. 106 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS 6.4.1 Natural Frequencies of Vibration The natural vibrational frequencies of each dynamic model were obtained by calculating the power spectrum function of the acceleration records using Mathcad's Signal Processing Exten-sion Pack (Mathsoft, 1999). Three impact hammer tests and one initial displacement free vi-bration test were performed for each model. The natural frequencies were identified and are summarised in Table 6.3. Table 6.3: Natural Frequencies of Vibration from System Identification Tests Dynamic Test Setup Intended Direction of Earthquake Shaking Natural Frequencies of Vibration (Hz) Impact Hammer Test (average over 3 tests) Free Vibration Initial Displacement Test EW Direction NS Direction EW Direction NS Direction DS1 EW 3.0, 4.0* 2.9 3.0,4.0 3.0 NS 3.0, 4.1* 2.9 no data 2.9 EW&NS 3.0, 4.1 2.9 3.0, 4.0 2.7 DS2 EW 3.4 3.0 3.4 2.9 NS 3.6 2.9 4.0 3.1 EW&NS 3.6 3.0 3.3 no data DS3 EW 3.4, 4.8 2.7 3.3,4.6 3.0 NS 3.3,4.7 3.0 2.9,4.2 2.9 EW&NS 3.4, 4.8 2.8 3.0, 4.2 2.3 DS4 EW 4.0 3.1 3.4 2.9 NS 3.6 3.1 3.2 2.4 EW&NS 3.5 3.2 3.4 2.6 * not averaged, data from only 1 impact hammer test were available 107 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS Two frequencies were identified for the EW direction in DS1 and DS3. Due to the elastic ec-centricity in these models, the two frequencies corresponded to mode shapes with significant translational and rotational components. The lower of the two frequencies corresponded to a mode that always had a larger translational component, while the higher frequency correspond-ed to a mode that had a higher torsional component. The experimental model was rebuilt for every test. The low variation in the identified natural frequencies in each set of dynamic tests indicates a good consistency and capability of the setup to be replicated for the three separate inelastic tests. Also, satisfactory correlation between the natural frequencies obtained using impact hammer tests and those obtained using the free vi-bration tests were also found. 6.4.2 Damping Damping was obtained by measuring the displacement decay in the free vibration of the model when a nearly instantaneous initial displacement was introduced. To estimate the overall sys-tem damping, only the elastically torsionally balanced models DS2 and DS4 were used. While elastic, as was the case in the free vibration tests, the response of these models was mainly in translation and was well suited for measuring displacement decay. The damping ratio C, was estimated using the well known logarithmic decrement formulation (Chopra, 1995) as (6.1) 108 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS where j is the number of cycles past cycle i of the free vibration and A is the maximum dis-placement in a particular cycle. An example of the motion decay found in the models is shown in Figure 6.5. 5 10 15 20 Time (sec) Figure 6 .5: Free Vibration Displacement Time History of Setup D S 4 in the E W direction The damping ratios calculated for the two principal directions of the model are summarised in Table 6.4. These values are much lower than the commonly used values in the design of build-ings, which range between 3% and 5% but can be as low as 2 %. The low damping of the model can be attributed to the material used and the simplicity of the model. In general, steel buildings have lower damping ratios than concrete buildings where microcracks assist in energy dissipa-tion. Also, the experimental model represented a building's structural system alone and had no energy dissipation contribution from non-structural components. On average, the damping ratio is 0.45 % and 0.33 % for the EW and NS directions respectively. The main difference in damping between the two principal directions is that the total stiffness of the model in the EW direction is higher than in the NS direction. 109 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS Table 6.4: Damping Ratios Obtained from Displacement Decay Model Test Setup Intended Direction of Earthquake Shaking Damping Ratio C, (%) Direction of Excitation EW Direction of Excitation NS DS2 EW 0.41 0.35 NS 0.49 0.27 EW&NS 0.40 0.35 DS4 EW 0.43 0.33 NS 0.48 0.37 EW&NS 0.47 0.30 6.5 SEISMIC RESPONSE Each setup was subjected to three earthquake ground motions as per test protocol. From the ob-tained time history displacement records, the maximum and permanent displacements were ob-tained. The recorded data from the instrumentation are included in Appendix 5. 6.5.1 Shake Table Earthquake Motion The horizontal components N37E and S53E of the earthquake time history recorded at 234 Figueroa Street in Los Angeles during the 1971 San Fernando earthquake described in Section 4.4.1, "Earthquake Ground Motion", were used as inputs for the shake table motion. The N37E component was always used for the EW motion of the shake table and the S53E component was always used for the NS motion. To excite the model to the inelastic range, the amplitude of the motions were scaled 3.5 times higher. The scaled time history records exceeded the shake table displacement limitation of ±76 mm (±3 in). Digital filtering was used to reduce the contribution of the low frequency 110 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS components of the waveform and decrease large displacement without affecting the frequency content in the range of interest. A cut-off frequency of 1 / 3 Hz was chosen as it was well below the natural frequencies of the models. A high pass filter incorporating Butterworth order 8 co-efficients was applied for the N37E component and Butterworth order 4 coefficients filter for S53E component. The resulting time history records are shown in Figure 6.6. The maximum acceleration of the two components was similar, at approximately 0.6 g. o O o < 0.5 0.0 -0.5 EW Shake Table Motion 10 15 20 25 Time (sec) 30 35 40 •3B G O s-i Jin "3 o o < 0.5 0.0 -0.5 10 15 20 25 Time (sec) 30 35 40 Figure 6.6: Time History of Shake Table Earthquake Input Motions 6.5.2 Maximum Demands on Lateral Resisting Components Maximum displacements were extracted from the top frame transducers and relative values were calculated to determine the deformation of the lateral resisting components. The lateral displacement demands are summarised in Figure 6.7. The results are presented for the three 111 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS shake table tests that were conducted for each dynamic setup. It should be noted that in DS1 and DS2, the transverse elements were located between transducers "g" and "h". The maxi-mum recorded displacements for these two transducers do not generally occur at the same time. Assuming a rigid diaphragm, a linear interpolation for the duration of the entire recording was necessary to obtain the maximum displacements on the transverse elements. For DS3 and DS4, the transducer displacements directly indicated the displacements of all the lateral load resist-ing components. o u ca E W N S E W & N S D i r e c t i o n o f E a r t h q u a k e S h a k i n g JDST E W N S E W & N S D i r e c t i o n o f E a r t h q u a k e S h a k i n g o E W N S E W & N S D i r e c t i o n o f Ea r thquake S h a k i n g fDST E W N S E W & N S D i rec t i on o f Ea r thquake S h a k i n g I—| L o n g i t u d i n a l N o r t h I I C o m p o n e n t L o n g i t u d i n a l S o u t h C o m p o n e n t T r a n s v e r s e E a s t L_! C o m p o n e n t * r T r a n s v e r s e W e s t C o m p o n e n t * Figure 6.7: Maximum Relative Displacement of Each Lateral Resisting Component* * due to the l ayou t o f the t ransverse e lements fo r D S 1 a n d D S 2 , o n l y 1 v a l u e o f d e m a n d w a s c a l c u l a t e d a n d s h o w n as the T ransve rse Eas t c o m p o n e n t . 112 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS From the elastic stiffness and yield strength values in Table 6.1, where an idealised bi-linear response was assumed, the ductility demands on each of the lateral load resisting components can be calculated. The maximum calculated ductility demands are shown in Figure 6.8. The ex-perimental model was found to be very effective in modelling significant inelastic response. Ductility demands of more than p > 3 were reached in all of the setups. T 3 C OJ Q o 3 Q EW NS EW&NS Direction of Earthquake Shaking [D3T EW NS EW&NS Direction of Earthquake Shaking 5.0 •a c 4.0 i 3.0 S 2.0 9 a 1.0 5.0 -6 c 4.0 1 3.0 >, ictili 2.0 —' Q 1.0 DS2 EW NS EW&NS Direction of Earthquake Shaking T DS4 • U I* ' Hi ' Hi" II EW NS EW&NS Direction of Earthquake Shaking • Longitudinal North Component Longitudinal South i - - , Transverse East Component L_! Component* Transverse West Component * Figure 6.8: Maximum Ductility Demands of Each Lateral Resisting Component* * due to the layout of the transverse elements for DS1 and DS2, only 1 value of demand was calculated and shown as the transverse East component. Torsional response was observed for all of the setups, including DS2 and DS4 where no stiff-ness eccentricity existed. The longitudinal component on the north side of the model was the stiffer and the stronger component. The displacement and especially ductility demands were 113 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS significantly higher for this component in the strength only eccentric cases of DS2 and DS4. The longitudinal south component had larger demands for the stiffness eccentricity dominated cases o f D S l andDS3. 6.5.3 Inelastic Deformation and Permanent Drift Each steel bar was coated with whitewash to visually identify inelastic deformations in the steel. The brittle white wash cracked and flaked away for large strains in the steel and was an effective identifier of plastic hinge zones. The hinge zones formed at the top and bottom of the bar, near the clamps. The deformation occurred via moment in bending, not shear deforma-tions, as can be seen from the horizontal strips in the white wash in Figure 6.9. Figure 6.9: Photograph of Moment Hinge Zone Identified by Brittle White Wash 114 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS The inelastic deformations in the steel bars resulted in permanent displacements of the dia-phragm at the end of each test. Steel bars that yielded were replaced for the next test. The per-manent drift values are summarised for each of the components in Figure 6.10. In general, larger drifts were recorded for the strength only eccentric models of DS2 and DS4. 40 J , 30 e I 20 <L> O C3 a. 10 5 o 40 J, 30 c E 20 o o DST EW NS EW&NS Direction of Earthquake Shaking j DS3 1 J 1 nil EW NS EW&NS Direction of Earthquake Shaking 40 30 : g 20 -: -plac plac 10 : 5 DST EW NS EW&NS Direction of Earthquake Shaking EW NS EW&NS Direction of Earthquake Shaking • Longitudinal North • Longitudinal South i - - , Transverse East IJ Ii Transverse West Component Component Component * I II Component * Figure 6.10: Permanent Drift Values of Each Lateral Resisting Component* * due to the layout of the transverse elements for DS1 and DS2, only 1 value of demand was calculated and shown as the transverse East component. 115 Chapter 6 - RESULTS OF EXPERIMENTAL TESTS 6.6 SUMMARY A n effective experimental model was developed for modelling inelastic response of torsionally susceptible structural systems. The model has a very low damping ratio, but does possess the following attributes: • the model can be replicated to have similar elastic dynamic properties as shown in the system identification tests; • the hysteresis behaviour of the lateral load resisting system closely approximates a bi-linear force deformation behaviour; • variations in the distribution of stiffness and strength were effectively achieved; and • the lateral load resisting system was capable of reaching large ductility values for uni-directional as well as bi-directional motion. Torsional response was observed for all models, including those that had no stiffness eccentric-ity. These strength eccentric models had significantly higher demands on the stiffer and strong-er element. The results of these experiments can be used as a bench mark for additional tests involving different ground motion records or more detailed experimental models. 116 Chapter 7 COMPARISON OF EXPERIMENTAL AND NUMERICAL RESPONSE 7.1 INTRODUCTION Complementary to testing the experimental model, an investigative study was carried out to compare the results obtained from the shake table testing to the idealised computer model used for the numerical analyses in Chapter 3 and 4. The numerical model was assigned properties obtained from the experimental tests and was subjected to earthquake ground motion using non-linear time history analysis. The details of the numerical models representing the experi-mental test setups are outlined and demands on the lateral load resisting components compared. 7.2 NUMERICAL M O D E L The experimental models described in Section 6.2.2, "Dynamic Test Setups", were modelled using the simplified two dimensional numerical model described in Section 4.3, "Structural Model". A n illustration is shown in Figure 7.1. 7.2.1 Lateral Load Resisting Elements Four massless lateral load resisting elements were used for the numerical model to resist the lateral loads for the two principal directions. Elements 1 and 2 in Figure 7.1 correspond to the longitudinal lateral load resisting components on the south and north side of the shake table 117 Chapter 7 - COMPARISON OF EXPERIMENTAL AND NUMERICAL RESPONSE Experimental Model Numerical Model Figure 7.1: Numerical Representation of the Experimental Model model respectively. They were spaced at a distance D = 1050 mm. Elements 3 and 4 corre-spond to the transverse elements on the east and west side respectively. The value of X = 0 was used for the numerical representation of DS1 and DS2 while A- = 1.0 was used for DS3 and DS4. The resisting elements were idealised to have a bi-linear force deformation behaviour. For stiff-ness and strength characteristics assigned to the numerical model, the average values for each of the component setups listed in Table 6.1 were used. Elements 1, 3 and 4 were assigned av-erage characteristics of CS1 while the stiffer and stronger element 2 was assigned characteris-tics ofCS2. 7.2.2 Mass Distribution A rigid diaphragm was assumed to connect the massless lateral load resisting elements. From the physical dimensions of the top frame diaphragm and the steel plates used for additional 118 Chapter 7 - COMPARISON OF EXPERIMENTAL AND NUMERICAL RESPONSE mass, the total mass was calculated at approximately 820 kg and the rotational mass moment of inertia about the centre of mass at 200 kg-m2. The rotational mass moment of inertia did not significantly change when the centre of mass was shifted for DS2 and DS4. 7.2.3 Analysis and Calibration of the Model Initial elastic analyses of the numerical model resulted in calculated frequencies that were low-er than those recorded in Table 6.3. It was found that decreasing the estimated value of mass and the associated rotational mass moment of inertia by 10 % resulted in a very close correla-tion between the calculated frequencies and the frequencies recorded experimentally. The cal-culated natural frequencies of the numerical model are summarised in Table 7.1. Table 7.1: Natural Frequencies of Vibration of Numerical Models Dynamic Test Setup Natural Frequencies of Vibration (Hz) EW * Direction NS Direction Pure Rotational DS1 3.1,4.0 3.0 -DS2 3.5 3.0 3.6 DS3 3.3,4.9 3.0 -DS4 3.5 3.0 4.8 includes frequencies that are a combination of translation and rotation Non-linear time history analysis was carried out using the earthquake ground motion recorded by the shake table accelerometer. A damping ratio C, = 0.4 % was used for all of the analyses. 119 Chapter 7 - COMPARISON OF EXPERIMENTAL AND NUMERICAL RESPONSE 7.3 COMPARISON OF RESPONSE Maximum displacements were recorded and ductility demands calculated as shown in Figure 7.2. The ductility demands on each lateral resisting component are shown for the dy-namic setups and directions of ground motion used in the experimental testing. For compari-son, the demands shown in Figure 6.8 are reproduced and re-arranged for side-to-side comparison with the numerically obtained values. The ductility demands were lower for the numerical model in all of the cases. In general, with the exception of DS3 in bi-directional motion EW&NS, the demands on the individual lateral load resisting components were consistent in shape between the numerical and the shake table demands on each of the lateral load resisting elements. The observed trends in response indi-cated the largest demand on the weaker longitudinal component on the south side of the model in DS1 and DS3, which have both stiffness as well as strength eccentricity. The maximum de-mand is reversed for the strength only eccentric models of DS2 and DS4. This holds for unidi-rectional, as well as bidirectional ground motion in both the numerical as well as the experimental cases. Numerically, a computer model can have perfectly aligned lateral elements of exactly the same characteristics subjected to a consistently applied level of motion. Experimentally variations are inevitable as small differences in mass distribution, section properties or calibration of the recorded instruments do occur. This can be observed, for example, from the small differences in demand for the NS direction in the shake table results as opposed to the precision observed in the numerical results. 120 Chapter 7 - COMPARISON OF EXPERIMENTAL AND NUMERICAL RESPONSE 1 4 a 3 i 2 3 Q 1 DS1 - Numerical Model i I 5i I 4 & 3 1 2 a Q 1 I 4 a 3 >. 1 2 EW NS EW&NS Direction of Earthquake Shaking DS2 - Numerical Model Lk EW NS EW&NS Direction of Earthquake Shaking DS3 - Numerical Model 1 EW NS EW&NS Direction of Earthquake Shaking DS4 - Numerical Model EW NS EW&NS Direction of Earthquake Shaking DS1 - Shake Table Model I -o c a 3 « 2 EW NS EW&NS Direction of Earthquake Shaking IDS2 - Shake Table Model I I EW NS EW&NS Direction of Earthquake Shaking I 4 & 3 1 2 3 Q DS3 - Shake Table Model I. EW NS EW&NS Direction of Earthquake Shaking -o a DS4 - Shake Table Model EW NS EW&NS Direction of Earthquake Shaking • Longitudinal North • Longitudinal South i - - , Transverse East .j lU Transverse West Component Component L_! Component 1 11 Component Figure 7.2: Ductility Demands Comparison of the Numerical and Shake Table Models 121 Chapter 7 - COMPARISON OF EXPERIMENTAL AND NUMERICAL RESPONSE 7.4 SUMMARY The shake table tests were modelled using the idealised computer model previously used for the numerical studies. Using the physical dimensions, stiffness and strength properties of the shake table model, the numerical model was calibrated and analysed using time history inelas-tic dynamic analysis. Lower magnitude of ductility demands were recorded for the numerical model. A consistent correlation between the observed trends and shape of the ultimate demand values on the individual elements was observed. These consistencies between the numerical and experimental results induce additional confidence in both the computer modelling as well as the shake table testing technique developed to investigate the inelastic response of torsion-ally susceptible structural systems. 122 Chapter 8 CONCLUSIONS AND RECOMMENDATIONS 8.1 CONCLUSIONS Outlining specific changes in the current building codes remains a long range goal of the re-search in inelastic torsional seismic response. Nonetheless, increased level of understanding of ductile torsional response was gained as all of the objectives of this part of the research were achieved. In addition to the detailed summary conclusions at the end of each relevant chapter, the following are the generalised observations. A number of issues were identified with the mechanism-based design (MBD) approach to tor-sion for the range of structural systems considered. The method imposes a significant increase in strength in some systems when compared to the values suggested in current codes. The ro-tational mass moment of inertia was found to significantly affect the dynamic behaviour. From the response of torsionally unbalanced models, an increase in the rotational inertia increased the difference in demand between the lateral resisting elements. In the unique case of torsion-ally balanced models, an increase in the rotational mass moment of inertia was observed to de-crease the torsional response. One of the initial M B D assumptions of the lack of restraint in some systems was found to be very conservative when compared to the observed response. The influence of strength characteristics on the torsional seismic response was studied using realistic examples of different stiffness to strength distributions. From the observations in the 123 Chapter 8 - CONCLUSIONS AND RECOMMENDATIONS response, it is evident that elastically identical systems of different strength distribution have very different response to earthquake ground motion. Both stiffness as well as strength eccen-tricity contribute to the overall inelastic torsional response in ductile structural systems. Even models with no stiffness eccentricity were found to have a significant torsional response due to the strength eccentricity alone. Models of design ductility as low as p. = 2 were also affected by the different distributions of strength. The relative location of the lateral elements with re-spect to the centre of mass influenced the demands on the lateral elements. A n experimental model capable of ductility demands in the range considered in the numerical investigations was developed and tested for uni-directional as well as bi-directional ground motion. The lateral resisting system has hysteresis load-deformation relationship which ap-proximates a bi-linear curve. The model has very low damping ratio, but can be easily adapted to simulate inelastic response of structural systems of wide range of stiffness, strength and mass distributions. The ductility demands on the lateral elements of the shake table model were com-pared to the demands on an idealised computer model, similar to that used in numerical analy-ses. The consistent level of correlation in the results injected confidence in the numerical as well as the experimental models used to study inelastic response of torsionally susceptible duc-tile structures. 124 Chapter 8 - CONCLUSIONS AND RECOMMENDATIONS 8.2 RECOMMENDATIONS FOR FURTHER STUDY The study of inelastic torsional response of structures subjected to earthquake ground motion is challenging due to the many factors that need to be considered. Through better understanding of the torsional response obtained during this study, a number of additional areas of future re-search were identified. So far, only mono-symmetric models, which have stiffness and strength eccentricity in only one principal direction, were considered. An additional investigation is needed to consider the more common case of structural systems that have asymmetry in both of the principal direc-tions. In these cases, the effects could be significantly influenced by bi-directional ground mo-tion. Ultimately a transition into investigating the issues identified in this study in the response of multi-storey systems is necessary. Only a limited number of earthquake ground motions were used. The time history dependency of the findings contained in this thesis can be alleviated by conducting similar analysis tech-niques using a range of time history records. Of particular interest would be the extent of the influence of strength properties using ground motions of different characteristics, such as sin-gle pulse or near field events. A rigid floor diaphragm was assumed to connect the lateral load resisting elements in the anal-yses. No reference was found that investigated the effects of diaphragm flexibility in the con-text of torsional response. This convenient modelling idealisation is commonly made, yet different degrees of diaphragm flexibility could potentially influence the torsional response of both elastic as well as ductile structural systems. 125 Chapter 8 - CONCLUSIONS AND RECOMMENDATIONS Strength characteristics were found to influence the torsional response. A n effort was placed in modelling the structural elements using bi-linear force deformation response in both the nu-merical as well as the experimental parts of the study. It is recommended that a sensitivity study is carried out to investigate the influence of the hysteresis model used as well as the post-yield stiffness ratio of the lateral load resisting elements. This research has broken new ground by developing and testing inelastic torsional response us-ing an experimental model. Additional ability to simulate different structural system configu-rations was incorporated into the model, which can be adopted to investigate many of the issues discussed above. The effectiveness of the model can be improved by modelling structural sys-tems of higher damping ratios. Through duplication of some of the existing components, mod-elling multi-storey structures is also possible. 126 REFERENCES Applied Technology Council, 1992. ATC-24: Guidlines for Cyclic Testing of Components of Steel Structures. Redwood City, CA. Associate Committee on the National Building Code, 1995. National Building Code of Cana-da, NBCC. National Research Council of Canada. Ottawa, Ontario. Chopra, Ani l K. , 1995. Dynamics of Structures, Theory and Applications to Earthquake Engi-neering, Prentice Hall, Inc. Chandler, A . M . , Correnza J. C. and Hutchinson G. L. , 1997. Inelastic Response of Code-de-signed Ecentric Structures Subject to Bi-directional Loading, Structural Engineering and Me-chanics, Vol.5, No. 1,51-58. Correnza J. C , Hutchinson G. L . and Chandler A . M . , 1992. A Review of Reference Models for Assessing Inelastic Seismic Torsional Effects in Buildings, Soil dynamics and Earthquake En-gineering, Vol . 11, 465-484. De La Llera J. C. and Chopra A . K. , 1995. Estimation of Accidental Torsion Effects for Seismic Design of Buildings, Journal of Structural Engineering. ASCE, Vol . 121, 102-114. De La Llera J. C. and Chopra A . K. , 1994 a. Accidental Torsion in Buildings due to Stiffness Uncertainty, Earthquake Engineering and Structural Dynamics. Vol . 23, 117-136. De La Llera J. C. and Chopra A. K. , 1994 b. Accidental Torsion in Buildings due to Base Ro-tation Excitation, Earthquake Engineering and Structural Dynamics. Vol . 23, 1003-1021. Drain 2DX- User Guide, Report No. U C B / S E M M 93/17 & 18, Department of Civil Engineer-ing, University of California Berkeley, December 1993. Dusicka, P., Davidson, B. J. and Ventura, C. E., 2000. Investigation into the Significance of Strength Characteristics in Inelastic Torsional Seismic Response, Proceedings of the Twelveth World Conference on Earthquake Engineering. Auckland, New Zealand. EQE International, 1995. The January 17, 1995 Kobe Earthquake, An EQE Summary Report, http://www.eqe.com/. 127 REFERENCES Esteva, L. , 1987. Earthquake Engineering Research and Practice in Mexico After the 1985 Earthquakes, Bulletin of the New Zealand National Society for Earthquake Engineering. Vol . 20, No. 3, 159-200. Federal Emergency Management Agency, 1991. NEHRP Recommended Provisions for the De-velopment of Seismic Regulations for New Buildings, Washington, DC: F E M A . Goel, R. K. and Chopra, A . K. , 1991. Inelastic Seismic Response of One-Storey, Asymmetric-Plan Systems: Effects of System Parameters and Yielding, Earthquake Engineering and Struc-tural Dynamics. Vol . 20, 201-222. Goel, R. K. and Chopra, A . K. , 1990. Inelastic Seismic Response of One-Storey, Asymmetric-Plan Systems: Effects of Stiffness and Strength Distribution, Earthquake Engineering and Structural Dynamics. Vol . 19, 949-970. Humar J. L . and Kumar P., 1998. Torsional Motion of Buildings Druing Earthquakes. I. Elastic Response. II. Inelastic response, Canadian Journal of Civil Engineering, Vol . 25, 898-934. Paulay, T., 1998 a. A Mechanism-Based Design Strategy for the Torsional Seismic Response of Ductile Buildings, European Earthquake Engineering. No.2, 12-28. Paulay, T., 1998 b. A Simple Seismic Design Strategy Based on Displacement and Ductility Compatibility, Asia-Pacific Workshop on Seismic Design & Retrofit of Structures. Chinese Taipei, 207-221. Paulay, T., 1998 c. Mechanism in Ductile Building Systems as Affected by Torsion, New Zea-land National Society for Earthquake Engineering Conference. Wairakei, Taupo, March 1998, 111-118. Paulay, T., 1998 d. Twist in NZS4203:1992, Journal of the Structural Engineering Society New Zealand. Vol . 11, No. 2, September 1998. Paulay, T., 1998 e. Torsional Mechanism in Ductile Building Systems. Earthquake Engineering and Structural Dynamics, Vol . 27, 1101-1121. Paulay, T., 1997 a. Are Existing Seismic Torsion Provisions Achieving the Design Aims? Earth-quake Spectra, Vol . 13, No. 2. 128 REFERENCES Paulay, T., 1997 b. A Behaviour-based Design Approach to Earthquake-induced Torsion in Ductile Buildings. Seismic Design Methodologies for the Next Generation of Codes, Fajfar & Krawinkler. Balkema, Rotterdam. Paulay, T. 1999. Personal Correspondence dated June and July 1999. Priestley M.J .N, 1998. Brief COmments on Elastic Flexibility of Reinforced Concrete Frames and Significance to Seismic Design, Bulletin of the New Zealand National Society for Earth-quake Engineering. Vol.31, No.4, 246-259. Park, R. and Paulay, T., 1975. Ductile Reinforced Concrete Frames - Some Comments on the Special Provisions for Seismic Design ofACI318-71 and on Capacity Design, Bulletin of the New Zealand National Society for Earthquake Engineering. Vol.8, N o . l , 70-90. Newmark, N . M . and Hall, W. J. (1982), Earthquake Spectra and Design, Earthquake Engineer-ing Research Institute, EERI. Mathcad 2000, 1999. Calculation Software for Technical Professionals, Mathsoft Inc. SAP2000 - Analysis Reference Version 7.0, 1998. Integrated Finite Element Analysis and De-sign of Structures, Computers and Structures Inc. Safak E. and Celebi M . , 1990. New Techniques in Record Analyses: Torsional Vibrations, Pro-ceedings of Fourth U.S. National Conference on Earthquake Engineering. Palm Springs, Cali-fornia. Seible F., Priestley M.J.N, and MacRae G., 1995. The Kobe Earthquake of January 17, 1995, Report No. SSRP- 95/03. Standard Association of New Zealand, 1992. Code of Practice for General Structural Design and Design Loadings for Buildings, NZS 4203. Wellington, New Zealand. Steinbrugge, K . V . , 1965. Structural Engineering Aspects of the Alaskan Earthquake of March 27, 1964, Proceedings of the Third World Conference on Earthquake Engineering. Auckland and Wellington, New Zealand, S50-S75. Thiruvengadam, V . and Watson, J .C , 1992. Post Earthquake Damage Studies on the Perform-ance of Buildings During Bihar (India)-Nepal Earthquake on 21st August 1988, Proceedings / Tenth World Conference on Earthquake Engineering. Madrid, Spain, 67-72. 129 REFERENCES Tso, W. K. and Wong, C. M . , 1995. Seismic Displacement of Torsionally Unbalanced Build-ings, Earthquake Engineering and Structural Dynamics, Vol . 24, 1371-1387. Tso, W. K . and Wong, C. M . , 1993. An Evaluation of the New Zealand Code Torsional Provi-sions, Bulletin of the New Zealand National Society for Earthquake Engineering. Vol.26, No.2, 194-207. Tso, W. K. and Zhu, T. J., 1992. Design of Torsionally Unbalanced Structural Systems Based on Code Provisions I: Ductility Demand, Earthquake Engineering and Structural Dynamics, Vol . 21,609-627. Wong, C. M . and Tso, W. K. , 1994. Inelastic Seismic Response of Torsionally Unbalanced Sys-tems Designed Using Elastic Dynamic Analysis, Earthquake Engineering and Structural Dy-namics. Vol . 23, 777-798. 130 Appendix 1 STRENGTH DISTRIBUTION FROM APPLICATION OF DESIGN METHODS Included are calculated element strengths and the resulting strength distribution from applying different design methods. The results are summarised for the three periods of vibration consid-ered, T = 0.4, 0.8 and 1.4 sec. The numerical model used is illustrated below. y C M © hcR — 1 ! 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in CM o o o m r- co CN CJ) CJ) O) h -o o o o ti ti ti ti m C M co CO O CO CO O T- O O ti ti ti ti C O T- 00 o i - CM i - T-o o o o 00 00 C D •<-C D C D in C D o o o o d ti ti ti s \r co in co CN o o o o m T- co m o o o CO CD 00 CN 00 CD CD CD O O O O ti O ti ti C D CD O T-00 O T C D O i - T- T-ti ti ti ti O 00 00 •<- O C D O T- T- O 1-d d d d S O ^ - CM in co i - f O O T- 1-d ti ti d fit g> ± 3 O •*-' o co o LU o o o o o o o o o o d d o o o o o o o o o o o o ti o ti ti o o o o o o o o o o o o ti ti ti ti o o o o o o o o o o o o o o o o o o o o o ti d d d o o o o o o o o o o o o d d d d o o o o o o o o o o o o o o o o o o o o o ti ti d d o o o o o o o o o o o o d d d d U) c £ CD c E in CN 00 o C O C D C N o o o ti d ti d CO c g) CO cu O LU T— in C N o CN T— 00 C N C N C N T— ti ti d ti 00 CM 00 T- •<- CO T- 1- O o d d 00 CM 00 i - - i - CO v T- O o d d T- r~- oo CM CM T- o co CM CN C N i -d d d d m co co CN co co oo oo o o o o ti ti ti ti in CD o oo 00 CD CN o o o ti ti ti d T— o CD o CN o CM CN CN T— T— ti ti o o CD in cj) •<- o co co in o) T- o oo o o o i - CO O) T-CN T- CD 00 CN CM i - i -d ti d ti in 1- m oo CO O) CO s o o o o d d d d in CD oo CD C D co o o o 1- co CM CN CM CM CN CN o d d O CN CD O CN CN CN CO O O O O O CM CD O CM CN CN CO O O O O T- m C D CM CM CO f CM CM CM CM d d ti ti m CD o in CO CO o o O O T- T-ti ti ti ti to •o o Q) CO E: -o 5 > < s n oo in co C N co co •<*• o d d o CM CO o co o co 00 T-N CO CM iO CN CN CN i -C0 CO CO CO ti ti ti ti O O CD CO O CN 00 CO 00 00 00 o ti ti ti ^ S I O O O ) lO N O N CO CO CO CN ti d d ti O O r- CD o C N r-- in oo oo co co ti ti ti ti C O o CO m C D C D ti d d d o in t o T- 00 00 CO CM O C O C D CN o in CD 00 00 CD ti ti d CO CO 00 CO T- CO CD T-N co in i n ti d d d O CO O h -O TJ- CN CN 00 00 CD O o t i t i c o m co o co in O S S o d d r- in v-CN T- CN T- CO oo o o o o O CO M O M N N N d d ti ti co CN oo in n oo T- s O O CN -3" O O CN CO O 00 CO CO 00 N S CD ti ti ti ti s C M co w CN 00 CN CO T- O i - CM in ••= m <J « i= Q £ c -t. t 0) D J= o co o LU O i - (M o d d o T- CM co d d ti ti o i - CN CO ti ti d d o T- CM o d d O i - CM co d d d d O T- C N CO d ti ti d O v- CM o d d o T- CM C O ti ti ti d O i - CM CO ti ti ti ti e o = » 9 0 = » Z 0 = » C'0 = » 90 = 1 0 Z'0 = » C0 = » 90 = » Z'0 = » 133 Appendix 1 - STRENGTH DISTRIBUTION FROM APPLICATION OF DESIGN METHODS fit 2 « ® CO O LU CO 00 CO CO LO CO o o o c> ci d O LO CO o O T- CM CN o o o o o ' d c i d CO O ) N O ) CO 1- T- T-o o o o d o d o CO CO CO T f LO O O o o o o d 9 ffi ° O C O O o o d d c o T f o r-c o —^ CO o O O O i -d d d d £3 i o o o O o T-P d d O CN CO CO O T f 00 T-O i - T- CM d d d d CO T f T f CO CO O J CD 00 O T- CN CN d d d d tn c £ CD c E CD 0) 3) ^ c O ) tn CD Q o o c o CO CO CO o o o c o o o CO LO o o o o d d LO LO CM c o LO LO LO T f o o o o d d d d t f ) CM N ( N LO LO T f T f O O O O d d d d CO CO 00 T f h ~ CO CD o o o o d d d d r-~ T f i - o c o CO c o c o o o o o d o d o CO o LO LO CO T f CM LO o o o o d d d d LO O CD LO T f o o o o d d CO CO LO LO r-- LO o o o o d d d d T f LO LO o o o CO CO CM T f h~ r~- CO o o o o d d d d CD CO T- T f CO CO T f T f o o o o d d d d r-~ c o c o c o CM CN o o o o d d c o o c o N O ) O O O i -d d d i o c o c o CM LO T f T f T f o o o o d d d d L O T f m s oi o O O O T-d d d d c o o CN oo h - CD CO CO o o o o d d d d N O ) O ( O c o L O oo o o o o d d d d fit g>i= Q CO 8 LU O T f C M T f T - O O O O o d d O •<- T f LO O CM CN CO O O O O d d d d O CD T f CN T f LO LO LO O O O O d d d d cp T- co J ; o o i P o o 9 d d O CO CM h -O CM i - O O O O i -d d d d O J O J T T C M I CO T f O O O O i -d d d d fc; C M O J 8 ° ? o d d o i n ( M K O CN O ) CO O T— CM d d d d r T f LO LO CO LO LO O J O T— CM CN d d d d 2 to c £ CD c E CD 0) C O c gj CO CD Q T f LO c o CO CM LO o o o o d d d d CM i n CO LO 1^  CO LO O o o o d d d d T- N- O LO T f T f O O O LO LO T f O O O CM S CO N S CD CO LO O O O O d d d d S S L O r-CO CO CO CO O O O O d d d d o CO T— T f CO CN LO o o o o d d d d CD T f O J CO oo LO o o o o d d d d CO O J CD LO T f CO O O O o d d o o o o d d ( O O I D S CO S CD LO O O O O d d d d O 1- T f oo T f T f CO c o o o o o d d d d L O o r~-CO CN 1-o o o O J o LO CD 00 O J o o o CO i - O J h -LO T f CO CO o o o o d d d d CO 00 CD CO LO CO 00 o O O O T-d d d d O J T— t^- CN CD CD LO CD O O O O d d d d I O T- X- T-CO L O N O ) o o o o d d d d ±= o g» J = Q £ § "2" 55 8 LU o o o o o o o o o o d d o o o o o o o o o o o o d d d d o o o o o o o o o o o o d d d d o o o o o o o o o o d d o o o o o o o o o o o o d d d d o o o o o o o o o o o o d d d d o o o o o o o o o o d d o o o o o o o o o o o o d d d d o o o o o o o o o o o o d d d d 2 co c i I CD CD CO C g> CO CD Q 00 LO LO T f LO LO T f o o o o d d d d LO 00 LO T f CO CN o T— o o d d d X— LO I P*- CO LO o o o o d d i - m N 1^  CO LO o o o o d d LO T- 1- CM CO CO CM i -o o o o OO CO CN 00 LO LO LO T f o o o o d d d d o o o LO LO LO CO o o o LO CO CO i - 00 T- T- O o d d T f O J T- o N CO CO LO O O O O d d d d T f O J 1- o S CO CD LO o o o o d d d d LO CD CD LO CO CN v- O o o o o 00 T f O LO 10 LO LO T f O O O O d d d d 00 O ) CN LO LO CD o o o LO O ) T f CO CO T f T f CO T - LO t~~ r-~ oo oo o o o o d d d d T f CD i - LO N - 00 00 o o o o d d d d L O o o o CO T f L O CD o o o o 00 O T f O J LO CO CO CD o o o o d d d d ° £ 8 CO r- io .2 O aj IS - £ 3 > T f 00 CM CN o o T f CO CD h - LO d d d d o CD LO o o LO 00 o T f T f T f T f CO T f O J CD CO T f T - | CO LO o T f LO 00 T f O J LO 00 CN CO CM 00 CO LO LO T f d d d d O T f LO CM O CO CM N T f T f LO CO 00 O CO T f LO oo CM CM i -c o T f T f T f O T f 00 i - T- O O J CO N f CO T f O O J o CN T— O J O J O T f CO O S CM T f CO CO O T f LO LO O CD O J CN T f CO CN CN O T f CN LO O CD OO O J T f CO CN i -O O O O O LO T f CD CM O LO T f T f O N M O T f T f CO O o co o r~-O S i - O ) T f T f CO I*-CN CM O N O CO O ) CO 1-00 CO CN T-O O J CO 00 00 00 T- LO CM CO T f O h - O J f - CD O J 00 O J T-^ ^ ^ C M CO f CD 4= Q £ c < t CD CD J5 CJ LU O i - CM o d d o T- CN CO d d d d O T- CM CO d d d d O T- CM o d d O i - CM CO d d d d O i - CM CO d d d d O T- CM o d d O 1- CM CO d d d d O 1- CM CO d d d d £ 0 = » 90 = » Z0 = » £'0 = » 90 = » Z'0 = » £0 = *> 90 = » Z0 = » 134 Appendix 2 RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS Maximum demands on the lateral load resisting elements 1 and 2 for systems designed to dif-ferent design methods are summarised. The numerical model layout is illustrated below. a D D B x Numerical Model used for Design Method Comparison The results are organised as follows: • Ductility demands for torsionally balanced models for periods T = 0.2, 0.4, 0.8, 1.4 and 2.0 sec. • Ductility demands for torsionally unbalanced models for periods T = 0.4, 0.8 and 1.4 sec. • Diaphgragm rotations for torsionally unbalanced models for periods T = 0.4, 0.8 and 1.4 sec. 135 Appendix 2 - RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS a a rl 'pueuisrj X;i|ijonQ rl 'pueiuSQ X;iijpnQ rl 'puEiuSQ ^ )i|nong rl 'pueujSQ X)i|iionQ u u 6 Q" a CQ CQ rl 'pueiuSQ X;i|i)onQ rl 'pueiuSQ Xji[iionQ U © - a o <u a _« 13 s o a) S-o H s-« s • a o SB c « s-Q - a c s Q a o o •3 & a a Y36 , c Appendix 2 - RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS rl 'pireuiSQ ityiponQ rl 'pUElUSQ Xll a a 3. a rl 'pireupQ X)i|ipnQ a rl 'puEiusQ Xjiiponrj 3. rl 'puEUi3aX;i[ipnQ rl 'pueuraQ Xji|ijonQ tu a a s a pq m rl 'pusuiSQ X;i[ipriQ rl 'puBiuaQ X;i|ipnQ rl 'puBiuaQ X)i[ipnQ U © 0) -a o -S "« a o •»* &e s. o H o JS a .2f Q a s-« s o !Z> T3 fl es S Q fl o u CS a a o U <s r4 x 13 fl ii C a 737 Appendix 2 - RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS HI r*i <N — © * © « - i - 3 ' r * i ( N —> O >G "~i "> ri — O rl 'puEuraQ X)![ipnQ rl 'pireuiSQ Xji|i}onQ rl 'puEiusQ /tylfpriQ 138 Appendix 2 - RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS a rl 'puBiuaQ Xji[i)DnQ rl 'puBiu3(jX)!|nona rl 'puBiusrj /tyupnQ rl 'puBiusQ Anuona rl 'puBiusQ X;i|i)onQ rl 'puEurag Xii|iionrj 1 1 1 s X i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I i X. 1 1 / : ] n" 1 i V i i > \\ 1 \ i ] i J i i V 1 — 1 — i 1—LI —W -1 3. a UJ JZ a e 3. CQ S g g E S a> o W W a cf oq m rl 'puEuiaa X;i|!iona rl 'puEiuSQ X;i|ipnQ rl 'puBiusQ Xji|ijonQ -a o T3 eu u C J5 *« a o in S-O H s-.o o c .SP at Q c u s-o -a S o Q fl © a & o U x 'O fl o a a < 139 Appendix 2 - RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS ri 'pusuraQ XjnnonQ ri 'pueuiSQ Xji|i;onQ rl 'pireuiSQ X;i|i)Dna T-8 u W rl 'puEuiSQ X;i|iionQ rl 'puEiuaQ Xj;[ijon(T VI m CN — rl 'pireuraQ jtyijionrj a a rl 'puEurarj Xii|ijonQ 1 T m fN — O rl 'puEiusfj XjinpnQ v-i r-i (N — rl 'pireuiag Xj[|ipnQ a w a § a ca S o u CQ CQ 2 S u © r i -a o -o e> fl JS "3 M fl _© o H i-.o -a o c wo « fl cu s-W5 T3 fl E Q fl o 'S OS a E o U un r i ,x -3 e cu a < 140 Appendix 2 - RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS CU <0 V ! CI <N — ' O *0 t/1 d CN — © V D W i ^ - f n r N — O *o m t o CN — o v o v - i ^ r m f N — O t o CN — O rl 'puBLUSQ Xji|ipnQ rl 'pireuiSQ Xii|iionrj rl 'pireuisQ iCiiijiong 747 Appendix 2 - RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS CU c/> 00 © rl 'pireiuag XjipionQ rl 'pireuisQ X)i[iiong rl 'puEiuag iC.}i|iionrj ri 'pireiuarj XjfiipnQ ri 'puBiu3Q /ty|iiong rl 'pueiusQ ^ )i|ipnQ rl 'pireiusQ X)i[i}onrj ri 'pireiusrj Xji|[)onQ ri 'puEiusQ X)i[iionrj r i T3 S CU a < 142 Appendix 2 - RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS T—Mi \- J—tr 'Si 1-5 II rl 'pireurarj X)i[i)Dna rl 'puemsrj X;i]ipnQ rl 'pireuraQ X;i|i;onQ VI f ) CN ~-rl 'pireurarj /()i|ipnQ rl 'pireurarj XjjinDnQ rl 'puBurarj Xii|ijonrj 1 iP j> \ i ' i V i . / N I ; ' - / l \ i • —UL c 8 u rl 'pueurag Xii|ipng Tf m CN — rl 'pueuratj Xinjiona 00 r i s a a 743 Appendix 2 - RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS (pEj) UOIJEJOJ UlSEJUdBIQ (pej) UOI)EJCV>J luSEjqdEiQ (pEj) UOI)BJO>J UlSEJl|dElQ (pEj) UOI)BJO>J mSsjudBiQ 3. o ,c ( p B i ) UOI}E;0>I O l S E J l l d B I Q o 5 S S S w N N Q Q o u e 1) B u s • Q Q CQ PQ U U B S <D a) w 3 a . a ffl CQ (pBJ) UOI)E)0>J luSEJUdEirj (PEJ) UOI)B}0>J luSBll[dB!Q (psj) uouEwa luSEjqdEiQ ON r i c a a < 144 Appendix 2 - RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS (pEj) UOITOO^ I luSBjqdEiQ (pel) uoiTOcry ui§E.i i |dEiQ (pBj) UOi;BJO'JJ UlSEJlldBIQ (pBj) UOHB;O>I ui§BJi|dB;a (pBj) UOI)B}ffH UI§BJl[dBlQ (pEj) UOIJBJO"5J uiSEjqdEifj (pBj) UOI;E}(TH iu3BJi|dBiQ (pBj) UOI;B;O>I uiSEji|dBiQ H -3 c « a. < 145 Appendix 2 - RESPONSE COMPARISON OF DIFFERENT DESIGN METHODS 146 Appendix 3 RESPONSE OF SYSTEMS WITH VARYING S TIFF NESS- TP'S TRENG TH DISTRIBUTIONS Maximum demands on the lateral load resisting elements for systems of xCM/D = 0, 0.2 and T= 0.2, 0.4, 0.8, 1.4 and 2.0 sec. are summarised. The displacement as well as ductility de-mands are normalised to the response of a single degree of freedom system in the direction of the element's resistance. The summary for the numerical model illustrated below is shown in the following order: • Displacement Demands of Elements 1 and 2 • Ductility Demands of Elements 1 and 2 • Demands of Elements 3 and 4. I 0 , C M © © D/2 sHcR -- > x I I C V e X CM D (N s Q X Numerical Model used for Comparison of Different Stiffness-to-Strength Distributions 147 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS dasV/(HasV-V) HasV/(HasV-V) •c c CD o 3 I co I •a o s aasV/(jasV-V) aasV/CjasV-V) •n c o Q •c c u 'So sV/(jasV-V) sV/(jasV-V) \ o i o Q •C c CD o {*> CO Q •c c CD O co ' 3 t3 -t-» c e <u -s CO" 09 3 3 X ) crt crt 5 Q j= x: ti to c c <u u J 3 J 3 0 0 C O 6 ta3 3 3 r2 £ a 5 •B • bh bo| c -u u t3 J 3 C O C O A 6 I I cn cn crt crt 1) CD ,e ,e ta ta i i i 1 i i aasV/(jasV-V) sV/(iasV-V) jasV"/(dasV-V) 148 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS aasV/^asV-V) t3 •c II •a 3. d II << £ •c c o V O II IE T 3 3 . II << HasV/(jasV-V) Q u 3 c to o jasV/(jasV-V) jasV/(jasV-V) jasV/(jasV-V) il II •a 3 . o" II —^v •c c o CN II •a 3. O -3 2? o m CD HasV/(dasV-V) jasV/^asV-V) HasV/(jasV-V) 149 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS Q CD >^ O ia sV/(HasV-V) jasV/(aasV-V) sV/(dasV-V) o -I r-cS> O O ia sV/(jasV-V) (3 jasV/(jasV-V) •c c CD O 3 jasV/(jasV-V) 3. o" I <^ cj> 8 •4 o> o 8 \ J CN 2? CD CN I •o 3. d I << -4 O IT) o o •c c CD O t i l CD CD •c CD (3 sV/(jasV-V) HasV/(jasV-V) jasV/(jasV-V) 750 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS jasV/(jasV-V) dasV/(jasV-V) jasV/(jasV-V) 151 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS •c c CD O jasV/(aasV-V) sV/(dasV-V) da sV/(dasV-V) dasV/(dasV-V) £ si o V O •i H 1 tf5 dasV/(dasV-V) Q ai~ 'o •c c CD O dasV/(dasV-V) [3 dasV/(jasV-V) dasV/(jasV-V) dasV/(dasV-V) o 752 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS •c c CJ C J o •C Q C J C J •c (3 jasV/(jasV-V) jasV/(iasV-V) aasV/(dasV-V) o •c c CD C J •c e C J C J •c c CO o iasV/(jasV-V) jasV/(jasV-V) aasV/(jasV-V) Q ej >J cj •c c cj C J Q C J '5 •c c C J C J i3 o •c C C J C J jasV/(jasV-V) jasV/(jasV-V) jasV/(jasV-V) 753 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS jasV/(HasV-V) HasV/OiasV-V) a s£ o •C c ID o fi o fi •c c u o fi \ l \ \ w / / n o // A jasV/(dasV-V) jasV/(jasV-V) Vr CN II •a d II •c c CD o fi •c c CD O fi ' o •c c CD O fi jasV/(dasV-V) dasV/(jasV-V) Q CD fi a CD >i '5 •c c CD CD fi C O I <55 Q CD '5 •c c CD CD fi rt tN G C CD E CD u tq | CQ 03 c c .2 .2 3 3 £ .-2 c/l c/l 5 a X ! . "Sb bo e e CD CD to to A A tU <D rt CN | C C I CD CD | £ CD PJ PJ < < a o 3 3 •c -c a a "So to c c CD CD is to CO CO A A ta ta I I jasV/(dasV-V) jasV/(jasV-V) jasV/(jasV-V) 154 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS II O < Ms u 8 - C r dasV/(dasV-V) jasV/(HasV-V) Q CJ •c c CJ o Q CJ CO HasV/^asV-V) jasV/^asV-V) II IS. • i d II •a o •C e cj CJ •c c CJ CJ Q CJ c CJ CJ jasV/(aasV-V) dasV/(dasV-V) Q CJ '5 •c c CJ o Q CJ c2 -*-» C cj E CJ B O 3 3 5 3 JS J3 "So to c c CJ CJ 43 43 oo c/3 6 6 — f S C £3 CJ CJ £ E OJ _CD 3 3 Q Q to to c c CJ CJ 43 43 00 CZ) A 6 CJ iasV/(jasV-V) jasV/(jasV-V) dasV/(dasV-V) 755 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS jasV/^asV-V) jasV/(jasV-V) jasV"/(dasV-V) 756 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS nasV/(aasV-V) jasV/(jasV-V) Q CJ ' 3 •c c CJ o Q ni" >^  ' 3 •c c cj o VO II II •o 3. d II s 8 dasV/(jasV-V) jasV/(jasV-V) c CJ CJ Q CJ ' 3 •c c CJ CJ (3 vo II -a II 8 8 f5>—r-HasV/(dasV-V) HasV/(dasV-V) Q CJ >^  ' 3 •c c cj CJ Q CJ ' 3 •C c CJ CJ C CJ E CJ UJ UJ c o 3 3 X> X> Q ( x: X J I oo ob c c CJ CJ 43 43 co &o 6 A c c CJ CJ E E CJ CJ e O 3 3 - O X ) 5 3 ' 3 •c + J c CJ CJ ul Q CJ >, ' 3 •c e CJ CJ i i Q CJ ?^ ' 3 •c c CJ cj (3 60 00 c c CJ CJ 4= 43 co cz> A A crt c/l CJ CJ C C I I I 1 I I jasV/(jasV-V) jasV/(jasV-V) jasV/(jasV-V) 157 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS 158 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS II II T3 A <o I << o —' •c c o fi I •a =L © © Q •c c u o fi o •c e u o fi j a s ^ / ( j a s ^ ) jastf/(aastf-tf) das^/( jas^) •c +^  c o fi Q •c C O fi CO CO si j a s ^ / ( j a s ^ ) j a s ^ / ( j a s ^ ) •c c u o fi •n c u o fi •c fi iaslt/iiasli-ti) j a s ^ / ( j a s ^ ) 759 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS ias^/( ias^) •c c o fi c cj cj fi II -o A 1 \ 1 \ 1 vs m" 1 d 1 II 1 s 1 Y o d — e r a Q CJ fi c cj CJ fi c <u E _u 3 w | ca i c o Q Q tb tb c c CJ u to to cz) oo CJ jas^/(jas^) _cu c o 3 X) X) a a •c c cj cj fi C cj O fi C/J CO tb tb c c 0 0 0 0 o 6 I I I 1 I I das^XJas^) das^ XJastf-tf) 760 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS Q cj >l 'o •c c o CJ ias^/(ias^) dastf/(jastf-tf) jas^/(das^) o I \ c O C ' / © 6 •c c cj o i i Q CJ •c C CJ CJ i i Q CJ •c C cj CJ i i Has»/(jas"-») CJ CJ •c C cj CJ i i CJ i i jas^/(das^) jas^/(jas^) 161 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS 162 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS c cj CJ fi •c c <D O fi fi das'1*/(iasli-ti) das^/(das r f -^) fi fi •c c CJ CJ fi j a s ^ / ( j a s ^ ) fi fi fi iastf/(Hastf-tf) ^ / ( d a s ^ ) 763 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS das^/(Has r f -^) di jas^/(jas^) Q •c di 2? o m w; 11 11 ©> O di -4 8 jas^/(aas^) •c c o di Q >^  '3 •c di di iasd/iadslM) aasltfaasli-li) jas^/ ( jas^) 764 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS dastf/(dastf-tf) Q CJ fi CJ •c fi Q cj — •c c fi •c c 1) cj fi jas /^(das^-"f) fi fi as « 3 3 Q Q 0 0 too c c <U CJ Ja 4= CZ) CZ) A U CJ ta ta c CJ E P J P J < < c o 3 3 X I X l Q Q •c c CJ CJ fi fi •c c CJ CJ fi on cn 60 SO c c cj cj J= J 3 CZ> CZ) A CD CJ ta ta jas r f/(das^) das^/(aas^) 765 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS Q di di di dastf/(iastf-rf) j a s ^ / ( j a s ^ ) j a s ^ / ( j a s ^ ) i3 di di Hastf/(jastf-tf) jas^/(jasrf-^) j a s ^ / ( j a s ^ ) di Q di Q o •c c u o di j a s ^ / ( a a s ^ ) dastf/(iastf-tf) j a s ^ / ( i a s ^ ) 766 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS fi fi Q 1) O fi d a s ^ / ( j a s ^ ) j a s ^ / ( j a s ^ ) jastf/(dastf-tf) c u o fi fi fi jas^/(das^) d a s ^ / ( H a s ^ ) j a s ^ / ( j a s ^ ) Q fi Q fi Q u >i 'o •c c 1) o fi dastf/(dastf-rf) das**/(aaslM) 167 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS 3. o" I << o o i l . t d jasV/(jasV-V) •c e ID O i i I II o s jasV/(jasV-V) Q CD 'o •c c tD CD i i jasV/(jasV-V) •c e CD O i i to to CD I 55 3. o" II ^S1 O "4 CN - C r jasV/(iasV-V) Q CD c tD O i i da SV/(JQSV-V) Q CD >1 'o •c c CD O i i jasV/(jasV-V) •c c CD o i i •c c <D o i i CN I —^ r Ni ,\ -a 3. ! 1 d I o •c c ca CD i i a CD 5^  '3 •c c CD o i i jasV/(jasV-V) jasV/(jasV-V) jasV/(jasV-V) 768 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS A o" II o o t d CN -c>-dasV/(HasV-V) ' o •c c o fi dasV/(dasV-V) dasV/(dasV-V") Q a? fi W W I so" I e .2 3 3 XI X> a a bo bo c c 4= U CO 00 II O II P i i i t d HasV/(jasV-V) A o" II '3" iasV/(dasV-V) o •c fi o •c c <u o fi iasV/(jasV"-V) P i i i jasV/^asV-V) 'o •C c <D o fi o •c e o u fi aasV/(dasV-V) •c c o fi Q u >1 o •C c <u o fi ti ta w 5 3 X I a a bb t b e c OO 00 I I o o ta3 f T I I 169 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS VO I II T3 3-o" II c< —^^  O v = <vi i n jasV/(jasV-V) cj •c c CJ o i i Q cj >v 'o •c c CJ i i a CJ •n c CJ CJ i i CO CO a> sV/(jasV-V) jasV/(jasV-V) CN T d 3 •v^ OV *—2-ja sV/(jasV-V) i i aasV/(jasV-V) CJ •c C CJ o i i - M l • v t t \ / / II •a 3. \ \ \i I I V T d II << t < 1 f .^3 aasV/(jasV-V) CJ i i CO CO <D CN II •a 3. cT II << 'a •4 civ ja •c c cj CJ i i •c c cj o i i •c c cj CJ i i I do jasV/(jasV-V) jasV/(jasV-V) 170 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS II II •a o" II O 4 -3 -sV/OasV-V) —i—t-i r-•4 t d jasV/(jasV-V) 8 jasV/(dasV-V) T - T I r-Q •c c CD o fi o •C fi •c c CD O fi dasV/(jasV-V) sV/(jasV-V) ja sV/(dasV-V) •c c CJ cj fi •c c <D o fi CJ fi ja sV/(dasV-V) dasV/(aasV-V) jasV/(dasV-V) •c c cj o fi fi ro T* c c CJ CJ E E oa pa" Q Q ob ab c c CJ CJ 43 43 G O C O c c CJ CJ E E CJ CD S 5 Q Q bO CJ) c c CD CD 43 43 G O G O ti ta i i f 1 i i u CU a CU a 171 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS VO II II - D 3. <o II ,—^ o •4 Q co '3 •c c u o co cn tD vo II d II £ •c c to o il I 55 Q to i i CO CO tD dasV/(jasV-V) £asV/(jasV-V) jasV/(dasV-V) T o -4 t d C N - © -HasV/(dasV-V) dasV/(jasV-V) jasV/(jasV-V) Q to >v •c c CO o i i i i i i CO CO tD I 55 jasV/^asV-V) jasV/(HasV-V) jasV/(jasV-V) 772 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS -r-n r-r<5 o •C fi crt crt CJ Q cj >. o •c c cj cj fi CO on Q fi aasV/(aasV-V) jasV/(jasV-V) dasV/(jasV-V) 5Q dasV/(iasV-V) •c fi •c e cj o fi I do fi aasV/^asV-V) jasV/(iasV-V) cJ> P i l l f Q CN d •c c CJ cj fi —«—r CN II XI in" d II << fi on on •c c cj CJ fi jasV/(dasV-V) iasV/(dasV-V) jasV/(dasV-V) 173 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS 3. c f II << /—V O o> t Q jasV/(iasV-V) 3. c f II 1 ci> f p aasV/(HasV-V) |(N II 3. c f II << 1? P i l l CN - © -jasV/(jasV-V) Q cj di Q CJ >. '5 •c c o di dasV/(aasV-V) r ' T • i — i — JQSV/(JQSV-V) Q CJ CJ •c c CJ CJ di di CO CO CJ Q CJ '3 •c c CJ o di jasV/(jasV-V) aasV/(jasV-V) di •c c CJ c j di >v '3 •c c CJ o di c c CJ CJ E E ' CJ JD 3- w « 05" c e .2 .2 '-4-* 3 3 J O J O 'C 'C C/i C/l 5 5 -5 c c CJ CJ b b 00 00 A A * J -»-• e c CJ CJ E E U CJ U 5 Q Q J5 -C ab ab c c CJ CJ b b oo oo A A I I * 1 I I u CU SB iasV/(jasV-V) jasV/^asV-V) 774 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS II o" II CJ c CJ CJ fi Q CJ 'o •c C u o fi iasV/(HasV-V) jasV/^asV-V) jasV/(jasV-V) © " il << X ) CN T d •TV P i | i — i -t P jasV/(jasV-V) o •c c cj CJ fi II •a o jasV/(jasV-V) •c c CJ cj fi jasV/(iasV-V) fi II © " II 1? o i n CN t P jasV/(aasV-V) •c c CJ cj fi rns-i t N II -o A m " d ,\ w ; M : k / / p 3 ' h P i l l jasV/(jasV-V) O CN <r l - d 1 • i CN ' •c c cj o fi jasV/(jasV-V) •c c CJ fi 775 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS 3. o" II O P P i l l r-f d jasV/(jasV-V) o II c<5 t p HasV/OasV-V) l<N II o~ II /—V o in t P 1) di di HasV/(dasV-V) jasV/(jasV-V) di Q CO di o •C di HasV/(jasV-V) jasV/(jasV-V) Q >v di Q CO >v di o •c di c c CO CO E E _co _CD W W aa eS 3 3 x> X> Q Q 6 0 6 0 c c CO CO 43 43 0 0 0 0 A A I I t/1 c/1 VI C/l CO CO IS is do do f l •<* c c CO CO E E CO CO s s < -< c c o o 3 3 JO X) a Q js x: to 60 c c CO CO 43 43 oo oo A A c c IS IS oo oo I I f f I I u ii VI jasV/(iasV-V) jasV/(jasV-V) HasV/(dasV-V) 776 Appendix 3 - RESPONSE OF SYSTEMS WITH VARYING STIFFNESS-TO-STRENGTH DISTRIBUTIONS © II 8 P i i i t d CJ fi fi jasV/(jasV-V) jasV/(aasV-V) jasV/(jasV-V) © " II P i l l f d fi e g o •c fi fi jasV/(jasV-V) jasV/(dasV-V) jasV/(jasV-V) o" II 'c? P i l l jasV/(jasV-V) fi jasV/(jasV-V) fi jasV/(jasV-V) fi 177 Appendix 4 EXPERIMENTAL MODEL CONSTRUCTION DRAWINGS Construction drawings developed for cost estimate, manufacturing and machining of all parts of the experimental model are included. The model consists of 2 identical frames, 16 clamps, steel bars and additional mass. The steel bars, measuring 0.25 in wide, 1.5 in deep and 4 feet long (imperial size was used due to cost and availability), and mass did not need manufacturing and are not included in these construction drawings. A three dimensional sketch is included be-low for illustration. Three Dimensional Sketch of Experimental Model Assembly 178 Appendix 4 - EXPERIMENTAL MODEL CONSTRUCTION DRAWINGS FRAME - PLAN NOTES: 1. FRAMES TO BE FULLY WELDED. 2. TOLERANCE TO BE ± 1 m m . THE UNIVERSITY OF BRITISH COLUMBIA EARTHQUAKE ENGINEERING RESEARCH LABORATORY INELASTIC TORSION EXPERIMENT PETER DUSICKA SCALE: NTS UNITS: mm (UNO) 1/3 179 Appendix 4 - EXPERIMENTAL MODEL CONSTRUCTION DRAWINGS 1300 (SPACING 150 TYP.) SECTION 'A' MODIFY TO SUIT DETAIL 1 NOTES- THE UNIVERSITY OF BRITISH COLUMBIA EARTHQUAKE ENGINEERING RESEARCH LABORATORY 1. FRAMES TO BE FULLY WELDED. 2. TOLERANCE TO BE ±1mm. INELASTIC TORSION EXPERIMENT PETER DUSICKA SCALE: NTS 9 / , UNITS: mm (UNO) / ^ 180 Appendix 4 - EXPERIMENTAL MODEL CONSTRUCTION DRAWINGS I—=*— 1_50 220 SECTION 'B' MOUNTING CLAMP (24) REQUIRED NOTES- THE UNIVERSITY OF BRITISH COLUMBIA EARTHQUAKE ENGINEERING RESEARCH LABORATORY 1. FRAMES TO BE FULLY WELDED. 2. TOLERANCE TO BE ± 1 m m . INELASTIC TORSION EXPERIMENT PETER DUSICKA SCALE: NTS -< , , UNITS: mm (UNO) ^ / ^ 181 Appendix 5 RECORDED DATA FROM SHAKE TABLE EXPERIMENTS Recorded data from the shake table actuators and the instrumented model subjected to shake table motion is summarised. Due to the number of channels, two pages are used to present re-corded data for each test. Labels used for each sensor are illustrated in the sketch of the instru-mentation layout below. Shake Table and Instrumentation Layout 182 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS The data is presented for the following tests: • Setup: DS1, uni-directional EW • Setup: DS 1, uni-directional NS • Setup: DS1, bi-directional EW&NS • Setup: DS2, uni-directional EW • Setup: DS2, uni-directional NS • Setup: DS2, bi-directional EW&NS • Setup: DS3, uni-directional EW • Setup: DS3, uni-directional NS • Setup: DS3, bi-directional EW&NS • Setup: DS4, uni-directional EW • Setup: DS4, uni-directional NS • Setup: DS4, bi-directional EW&NS 183 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 15 30 Time (sec) 45 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 60 75 Displacement (mm) - "B" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "C" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "D" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - " E " EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "F" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "G" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "H" NS 15 30 Time (sec) 45 60 75 LLUJL Acceleration (g) - "L" EW' 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" EW 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" NS 15 30 Time (sec) 45 60 75 Appendix 5.1: Recorded Instrumentation for DS1, Uni-directional E W 184 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" vertical 15 30 Time (sec) 45 60 75 Acceleration (g) - "N" NS 15 30 Time (sec) 45 60 75 Wm* f i l l Acceleration (g) - "O" EW 15 30 Time (sec) 45 60 75 Acceleration (g) - "O" NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "O" vertical M M H M M M H M f \ 30 Time (sec) 45 15 30 Time (sec) 45 60 75 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 |l4l^ «ll|il||lp<»Ml'l»'l|l||li|i<|)»|llll Acceleration (g) - "P" NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "P" vertical >»HIH>H'»«| Willi 30 Time (sec) 45 15 30 Time (sec) 45 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 60 75 Acceleration (g) - "Q" NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "Q" vertical 15 30 Time (sec) 45 60 75 Appendix 5.2: Recorded Instrumentation for DS1, Uni-directional E W (cont.) 185 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 100 50 0 -50 -100 100 50 0 -50 -100 Displacement (mm) - "A" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "B" EW mtmmiwMiwmimmttmlmmmmmmt< >:% • M 15 30 Time (sec) 45 60 75 Displacement (mm)'- "C" NS 15 30 Time (sec) 45 60 75 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 Displacement (mm) - "D" NS Srft*«'W»«*»y •vVw**^  *i ***** 15 30 Time (sec) 45 60 75 Displacement (mm) - "E "EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "F" EW 15 30 Time (sec) 45 60 75 <'iiii'iifWya>iiriii>iii minM Displacement (mm) - "G" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "H" NS 15 30 Time (sec) 45 60 75 tWmwi*vt*mimi ***** Acceleration (g) - "L" EW 15 30 Time (sec) 45 60 75 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" EW MMMMMHMMMMNPMMMMM4 15 30 Time (sec) 45 60 75 kli 111 j 1 J Ml _ . 1 Acceleration (g) - "M" NS 15 30 Time (sec) 45 60 75 Appendix 5.3: Recorded Instrumentation for DS1, Uni-directional NS 186 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" vertical 15 30 Time (sec) 45 60 75 Acceleration (g) - "N" NS 15 30 Time (sec) 45 60 75 1 m<> »miii'iiM Acceleration (g) - "O" EW 15 30 Time (sec) 45 60 75 15 30 Time (sec) 45 Acceleration (g) - "O" NS 60 75 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "O" vertical 15 30 Time (sec) 45 60 75 omitiinwii Acceleration (g) - "P" EW 30 Time (sec) 45 D 15 30 Time (sec) 45 60 75 Acceleration (g) -"P" vertical D 15 30 Time (sec) 45 60 75 Acceleration (g) -"Q" EW 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 30 Time (sec) 45 15 30 Time (sec) 45 60 75 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "Q" vertical 15 60 75 30 Time (sec) 45 Appendix 5.4: Recorded Instrumentation for DS1, Uni-directional NS (cont.) 187 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 100 50 0 -50 -100 100 50 0 -50 -100 Displacement (mm) - "A" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - " B " EW 1. „m f •• 15 30 Time (sec) 45 60 75 Displacement (mm) - "C" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "O" NS 15 30 Time (sec) 45 60 75 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 1.0 0.5 0.0 -0.5 -1.0 splacement (mm) - "E" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "F" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "G" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "H" NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "L" EW 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" EW mmm+mmmmmmm*mmimm4 15 30 Time (sec) 45 60 75 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" NS »'»wiiwni><iNww»iiiwiMi*ii mmmi\ 15 60 30 Time (sec) 45 Appendix 5.5: Recorded Instrumentation for DS1, Bi-directional E W & N S 75 188 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 15 30 Time (sec) 45 15 15 15 15 30 Time (sec) 45 30 Time (sec) 45 30 Time (sec) 45 30 Time (sec) 45 30 Time (sec) 45 Acceleration (g) - "M" vertical 60 75 Acceleration (g) - "IM" NS 60 75 Acceleration (g) - "O" vertical 60 75 Acceleration (g) - "P" vertical 30 Time (sec) 45 60 75 Acceleration (g) - "Q" vertical 60 75 30 Time (sec) 45 Appendix 5.6: Recorded Instrumentation for DS1, Bi-directional EW&NS (cont.) 189 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 Displacement (mm) - "A" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - " B " EW 15 30 Time (sec) 45 60 75 Displacement (mm) - " C " N S 15 30 Time (sec) 45 60 75 Displacement (mm) - "D" NS t\* iW 4 WM—IHH 15 30 Time (sec) 45 60 75 Displacement (mm) - "E" EW 100 50 0 -50 -100 100 50 0 -50 -100 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 30 Time (sec) 45 15 30 Time (sec) 45 60 75 Displacement (mm) - "G" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "H" NS 15 30 Time (sec) 45 60 75 Sensor Malfunction Acceleration (g) - "L" EW 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" EW 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" NS 15 30 Time (sec) 45 60 75 Appendix 5.7: Recorded Instrumentation for DS2, Uni-directional E W 190 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" vertical 15 30 Time (sec) 45 60 75 Acceleration (g) - "N" NS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 30 Time (sec) 45 15 30 Time (sec) 45 60 75 |MNI>#<W«<(<iwioniiiii<M<iiiW>l>ii>i<>i » Acceleration (g) - "O" NS D 15 30 Time (sec) 45 60 75 Acceleration (g) -"O" vertical " — " • ' — ~ — ' - —• • — -1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 30 Time (sec) 45 15 30 Time (sec) 45 60 75 Acceleration-(g) - "P" NS D 15 30 Time (sec) 45 60 75 Acceleration (g) -"P" vertical 30 Time (sec) 45 15 30 Time (sec) 45 60 75 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 ii Min i> Acceleration (g) - "Q" NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "Q" vertical 15 60 75 30 Time (sec) 45 Appendix 5.8: Recorded Instrumentation for DS2, Uni-directional E W (cont.) 191 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 Displacement (mm) - "A" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "B" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "C" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "D" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - " E " E W 15 30 Time (sec) 45 60 75 Displacement (mm)' - "F" EW 15 30 Time (sec) 45 60 75 Displacement (mm)' - "G" NS ' • A w i n j v ^ 15 30 Time (sec) 45 60 75 Displacement (mm) - "H" NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "L" EW 15 30 Time (sec) 45 60 75 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" EW 15 30 Time (sec) 45 60 75 . . . i J . M l l Acceleration (g) - "M" NS 15 60 30 Time (sec) 45 Appendix 5.9: Recorded Instrumentation for DS2, Uni-directional NS 75 192 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" vertical MHIMHHBHaMllMH^ 15 30 Time (sec) 45 60 75 Acceleration (g) - "N" NS 15 30 Time (sec) 45 60 75 Ac celeration (g) - "0 " EW 15 30 Time (sec) 45 60 75 15 30 Time (sec) 45 Acceleration (g) - "O" NS 60 75 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "O" vertical 15 30 Time (sec) 45 60 75 Acceleration.(g) - "P" EW 30 Time (sec) 45 D 15 30 Time (sec) 45 60 75 Acceleration (g) - "P" vertical D 15 30 Time (sec) 45 60 75 Acceleration (g) - "Q" EW 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 15 30 Time (sec) 45 60 75 15 30 Time (sec) 45 Acceleration (g) - "Q" NS motmrnmu m mm mm 60 75 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "Q" vertical 15 60 75 30 Time (sec) 45 A p p e n d i x 5.10: R e c o r d e d I n s t r u m e n t a t i o n f o r D S 2 , U n i - d i r e c t i o n a l N S (cont.) 193 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 Displacement (mm) - "A" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "B "EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "C" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "O" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - " E " E W 15 30 Time (sec) 45 60 75 Displacement (mm) - "F" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "G" NS 30 Time (sec) 45 15 30 Time (sec) 45 60 75 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" EW n»lNillMWI»HiltlHli>IKINW>H>lH»ll 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" NS 15 30 Time (sec) 45 60 75 Appendix 5.11: Recorded Instrumentation for DS2, Bi-directional EW&NS 194 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" vertical 15 30 Time (sec) 45 60 75 Acceleration (g) - "N" NS I'lHWHMW H>W»WI|l||l|)l|IWimill»ll»l 15 15 15 15 15 15 30 Time (sec) 45 30 Time (sec) 45 60 75 Acceleration (g) - "O" vertical 30 Time (sec) 45 30 Time (sec) 45 60 75 Acceleration (g) - "P" NS mtmmtmm 30 Time (sec) 45 60 75 Acceleration (g) - "P" vertical 30 Time (sec) 45 30 Time (sec) 45 60 75 Acceleration (g) - "Q" NS »l»IP|l»l)»>»»<l»IWl^ l#IIM»»4W*iW 30 Time (sec) 45 60 75 Acceleration (g) - "Q" vertical 60 75 30 Time (sec) 45 Appendix 5.12: Recorded Instrumentation for DS2, Bi-directional E W & N S (cont.) 195 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Displacement (mm) - "A" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "B" EW 15 30 Time (sec) 4 5 60 75 Displacement (mm) - "C" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "D" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "E" EW 15 30 Time (sec) 45 60 75 Displacement (mm)-"F" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "G" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "H"NS 15 30 Time (sec) 45 60 75 • Acceleration (g) - "L" EW 15 30 Time (sec) 45 60 75 Acceleration (g) - "M' EW 15 30 Time (sec) 45 60 75 Acceleration (g) - "M' NS 15 30 Time (sec) 45 60 75 Appendix 5.13: Recorded Instrumentation for DS3, Uni-directional E W 196 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" vertical 0 15 30 Time (sec) 45 60 75 Acceleration (g) - "N"NS 0 15 30 Time (sec) 45 60 75 Acceleration (g) - "O" EW 0 15 30 Time (sec) 45 60 75 Acceleration (g) - "O" NS 0 15 30 Time (sec) 45 60 75 Acceleration (g) - "O" vertical 0 15 30 Time (sec) 45 60 75 Acceleration (g) - "P" EW 0 15 30 Time (sec) 45 60 75 Acceleration (g) - "P" NS 0 15 30 Time (sec) 45 60 75 Acceleration (g) - "P" vertical 0 15 30 Time (sec) 45 60 75 Acceleration (g) - "Q" EW 0 15 30 Time (sec) 45 60 75 Acceleration (g) - "Q" NS 0 15 30 Time (sec) 45 60 75 Acceleration (g) - "Q" vertical 0 15 30 Time (sec) 45 60 75 Appendix 5.14: Recorded Instrumentation for DS3, Uni-directional E W (cont.) 197 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Displacement (mm) - "A" EW 15 30 Time (sec) 4 5 60 75 Displacement (mm) - " B " EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "C" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "D" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "E" EW *********** mmmmmmkmm»*m^immmma»mHmm§mmmn* w » n i m i i » i w n i < 15 30 Time (sec) 45 60 75 Displacement (mm)" - "F" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "G" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "H"NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "L" EW 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" EW <mmmmnmmmmn 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" NS 15 30 Time (sec) 45 60 75 Appendix 5.15: Recorded Instrumentation for DS3, Uni-directional NS 198 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) -"M" vertical 0 15 30 Time (sec) 45 60 75 Acceleration (g) -"N" NS 0 15 30 Time (sec) 45 60 75 Acceleration (g) -"O" EW 0 15 30 Time (sec) 45 60 75 Acceleration (g) -"O" NS 0 15 30 Time (sec) 45 60 75 Acceleration (g) -"O" vertical 0 15 30 Time (sec) 45 60 75 - Acceleration (g) -"P" EW 0 15 30 Time (sec) 45 60 75 Acceleration (g) -"P" NS 0 15 30 Time (sec) 45 60 75 Acceleration (g) -"P" vertical 0 15 30 Time (sec) 45 60 75 Acceleration (g) -"Q" EW 0 15 30 Time (sec) 45 60 75 Acceleration (g) -"Q" NS 0 15 30 Time (sec) 45 60 75 Acceleration (g) -"Q" vertical 0 15 30 Time (sec) 45 60 75 Appendix 5.16: Recorded Instrumentation for DS3, Uni-directional NS (cont.) 199 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 Displacement (mm) - "A" EW 15 30 Time (sec) 45 60 75 Displacement (mm)'-"B" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "C" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "D" NS 15 30 Time (sec) 45 60 75 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Displacement (mm) - " E " E W 15 30 Time (sec) 4 5 60 75 Displacement (mm) - "F" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "G" NS 15 30 Time (sec) 4 5 60 75 Displacement (mm) - "H" NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "L" EW 3 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" EW IMM|M 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" NS 15 60 30 Time (sec) 4 5 Appendix 5.17: Recorded Instrumentation for DS3, Bi-directional E W & N S 75 200 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" vertical 15 30 Time (sec) 45 60 75 Acceleration (g) - "N" NS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 15 15 m»mM»H*#H»W mm* 15 30 Time (sec) 45 30 Time (sec) 45 60 75 Acceleration (g) - "O" vertical 30 Time (sec) 45 30 Time (sec) 4 5 60 75 Acceleration (g) - "P" vertical 30 Time (sec) 45 30 Time (sec) 4 5 60 75 Acceleration (g) - "Q" vertical 15 60 75 30 Time (sec) 4 5 Appendix 5.18: Recorded Instrumentation for DS3, Bi-directional E W & N S (cont.) 201 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 Displacement (mm) - "A" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "B" EW ! 1—' 1 15 30 Time (sec) 45 60 75 Displacement (mm) - " C " N S 15 30 Time (sec) 45 60 75 Displacement (mm) - "D" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - " E " E W 100 50 0 -50 -100 100 50 0 -50 -100 30 Time (sec) 45 15 30 Time (sec) 45 60 75 Displacement (mm) - "G" NS i l l M M » M I 15 30 Time (sec) 45 60 75 Displacement (mm) - "H" NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "L" EW 15 30 Time (sec) 45 60 75 •Jfm Acceleration (g) - "M" EW 15 30 Time (sec) 45 60 75 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" NS 15 60 30 Time (sec) 45 Appendix 5.19: Recorded Instrumentation for DS4, Uni-directional E W 75 202 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" vertical 15 30 Time (sec) 45 60 75 Acceleration (g) - "N" NS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 15 30 Time (sec) 45 30 Time (sec) 45 60 75 Acceleration (g) - "O" NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "O" vertical 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 15 30 Time (sec) 45 30 Time (sec) 45 60 75 Acceleration (g) - "P" NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "P" vertical 15 30 Time (sec) 45 30 Time (sec) 45 60 75 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "Q" NS mini **imimi*<mmmm*t.— 15 30 Time (sec) 45 60 75 Acceleration (g) - "Q" vertical 15 30 Time (sec) 45 60 75 Appendix 5.20: Recorded Instrumentation for DS4, Uni-directional E W (cont.) 203 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 100 50 0 -50 -100 100 50 0 -50 -100 Displacement (mm) - "A" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "B" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "C" NS 15 30 Time (sec) 45 60 75 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 100 50 0 -50 -100 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Displacement (mm) - "D" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "E" EW 15 30 Time (sec) 45 60 75 Displacement (mm) - " F " EW 15 30 Time (sec) 45 60 75 Displacement (mm) - "G" NS ' - ^ f O T 15 30 Time (sec) 45 60 75 Displacement (mm) - " H " NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "L" EW mum*mmm*mmm* 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" EW 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" NS 15 60 30 Time (sec) 45 Appendix 5.21: Recorded Instrumentation for DS4, Uni-directional NS 75 204 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "M" vertical 15 30 Time (sec) 45 60 75 Acceleration (g) - "N" NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "O" EW 15 30 Time (sec) 45 30 Time (sec) 45 60 75 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "O" vertical 15 30 Time (sec) 45 60 75 Acceleration (g) - "P" EW 15 30 Time (sec) 45 60 75 Acceleration (g) - "P" NS <P)lliNIHII»lll»l*»i«IW^ 15 30 Time (sec) 45 60 75 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 Acceleration (g) - "P" vertical 15 30 Time (sec) 45 60 75 Acceleration (g) - "Q" EW ||<|II|I»>I>IIH|I»H 30 Time (sec) 45 D 15 30 Time (sec) 45 60 75 Acceleration (g) - "Q" vertical 3 15 30 Time (sec) 45 60 75 1.0 0.5 0.0 -0.5 -1.0 Appendix 5.22: Recorded Instrumentation for DS4, Uni-directional NS (cont.) 205 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 100 50 0 -50 -100 100 50 0 -50 -100 Displacement (mm) - "A" EW 15 30 Time (sec) 45 60 75 Dis splacement (mm) - "1 3" EW 30 Time (sec) 45 15 30 Time (sec) 45 100 50 0 -50 -100 100 50 0 -50 -100 60 75 . i Displacement (mm) - "D" NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "E" EW 100 50 0 -50 -100 100 50 0 -50 -100 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 30 Time (sec) 45 15 30 Time (sec) 45 60 75 Displacement (mm) - " G " NS 15 30 Time (sec) 45 60 75 Displacement (mm) - "H" NS 15 30 Time (sec) 45 60 75 Acceleration (g) - "L" EW tmumwm •mmmmm mm nmm :— 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" EW 15 30 Time (sec) 45 60 75 Acceleration (g) - "M" NS 15 60 30 Time (sec) 45 Appendix 5.23: Recorded Instrumentation for DS4, Bi-directional E W & N S 75 206 Appendix 5 - RECORDED DATA FROM SHAKE TABLE EXPERIMENTS 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 15 30 Time (sec) 45 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 1.0 0.5 0.0 -0.5 -1.0 30 Time (sec) 45 15 30 Time (sec) 45 15 30 Time (sec) 45 30 Time (sec) 45 15 30 Time (sec) 45 15 30 Time (sec) 45 15 30 Time (sec) 45 15 30 Time (sec) 45 Acceleration (g) - "M" vertical 60 75 Acceleration (g) - "N" NS 60 75 Acceleration (g) - "O" NS 60 75 Acceleration (g) - "O" vertical 60 75 Acceleration (g) - "P" NS 60 75 Acceleration (g) - "P" vertical 60 75 Acceleration (g) - "Q" EW 60 75 Acceleration (g) - "Q" NS ^iiw»i)i^ii)ww»iiw>^ii>i»>i)iiilii»<Nii) vn,w<»* :— 15 30 Time (sec) 45 60 75 Acceleration (g) - "Q" vertical 15 60 75 30 Time (sec) 45 Appendix 5.24: Recorded Instrumentation for DS4, Bi-directional E W & N S (cont.) 207 Appendix 6 209 Appendix 6 - PHOTOGRAPHS Appendix 6 - PHOTOGRAPHS Appendix 6 - PHOTOGRAPHS l l l l I l l l III I l l s DS1 EW DS1 NS DS1 EW&NS Appendix 6.7: Deformed Bars After DS1 Test for EW, NS and EW&NS Directions Appendix 6.8: Deformed Bars After DS2 Test for EW, NS and EW&NS Directions 212 Appendix 6 - PHOTOGRAPHS Appendix 6.9: Deformed Bars After DS3 Test for EW, NS and E W & N S Directions Appendix 6.10: Deformed Bars After DS4 Test for EW, NS and E W & N S Directions 213 

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