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A study on the seismic behaviour of 52-storey steel frame building Ding, Yuming 1999

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A STUDY ON THE SEISMIC BEHAVIOUR OF A 52-STOREY STEEL FRAME BUILDING by Y U M I N G DING B.Sc , Shenzhen University, 1993 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R 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 1999 ©Yuming Ding, 1999 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 C'L// I &nCJ, ntPMt\^ The University of British Columbia Vancouver, Canada Date /few'/ 2&,l<ftf DE-6 (2/88) ABSTRACT Comparison of recorded structural earthquake response and theoretical dynamic analysis is vital to the study of earthquake engineering nowadays. This comparison could be either in the fre-quency domain by utilizing system identification algorithms or in the time domain by comparing response time histories. Not only had seldom previous studies done this comparison, but also less attention was paid to the three-dimensional nonlinear dynamic behaviour despite the fact that it has been generally admitted that three-dimensional response is important in inelastic behavior of buildings. The purpose of this research is to study the seismic behaviour of a well-instrumented 52-storey steel frame building in Los Angeles, California. This building has been subjected to ground mo-tions from several earthquakes among which the 1991 Sierra Madre earthquake and the 1994 Northridge earthquake were selected in this study. During both earthquakes the building re-sponses appeared to remain in the linear range. Detailed frequency domain analyses of the recorded motions from these two earthquakes were conducted to determine the dynamic characteristics of the structure. Three-dimensional nonlin-ear dynamic computer analyses were then employed to evaluate the response of the structure in-duced by other earthquake excitations. Dynamic characteristics and seismic responses from the above two approaches hence could be compared to yield valuable information. The results of this study showed that by performing a linear three-dimensional analysis, the ac-tual response of a building during past earthquakes could be reproduced with confidence. By fur-ther performing a nonlinear three-dimensional analysis, the state and sequence of damage could also be predicted. ii Traditional Nonlinear Static Procedure (NSP, pushover analysis) has the deficiency of excluding the higher mode participation which becomes obvious for highrise buildings. Improvements to the NSP were explored. Because of the torsional response for highrise buildings, two-dimension-al analysis is not feasible to reliably predict their nonlinear response during earthquakes. iii TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES vii LIST OF FIGURES viii ACNOWLEDGEMENTS x Chapter 1 Introduction 1 1.1 General Remarks 1 1.2 Objectives and Scope 3 1.3 Organization 5 Chapter 2 Background of Nonlinear Analysis 7 2.1 Why Nonlinear Analysis 7 2.2 Literature Review 9 Chapter 3 Description of the Figueroa at Wilshire Tower 14 3.1 Introduction 14 3.2 Structural System 16 3.3 Building Materials 21 3.4 Instrumentation 21 Chapter 4 Analyses of Recorded Data 24 4.1 Recorded Earthquake Responses 24 4.2 Time Domain Analysis of Recorded Motions 25 4.3 Frequency Domain Analysis 30 4.3.1 Theory 30 iv 4.3.2 Dynamic Behavior from Seismic Response 34 4.4 Conditioning of Raw Records 41 4.5 Conclusions 44 Chapter 5 Description of the Nonlinear Analysis Program 46 5.1 Introduction 46 5.2 Elements 48 5.3 C A N N Y Hysteresis Models 50 5.4 More information about the program 51 Chapter 6 Three dimensional linear analysis of the Figueroa at Wilshire Tower 54 6.1 Introduction and theory 54 6.2 Development of a linear model 55 6.3 Calibration of the linear model and period sensitivity 58 6.4 Comparison of predicted and recorded time histories 61 6.4.1 Comparison of predicted and recorded time histories during the Sierra Ma-dre earthquake 62 6.4.2 Comparison of predicted and recorded time histories during the Northridge earthquake 64 6.5 Comparison of code design values 66 6.6 Summary 70 Chapter 7 Three-dimensional Nonlinear Analysis of the Figueroa at Wilshire Tower. 71 7.1 Introduction 71 7.2 Description of the Nonlinear model 72 7.3 Time History Analysis 75 7.3.1 Response of the building during the given earthquake excitations 80 7.4 Pushover Analysis 84 7.5 Cyclic Analysis 90 7.6 Conclusions 91 Chapter 8 Summary and recommendations 94 8.1 Summary 94 8.2 Recorded behavior of the FWT and prediction 95 8.3 Recommendations for further research 97 REFERENCES 99 Appendix A Elevations and Details of the Building 102 Appendix B Photos of the Building 108 Appendix C C A N N Y input data file 112 vi LIST OF T A B L E S Table 3.1 Comparative study of perimeter ductile tube versus spine structure 17 Table 3.2 Properties of construction materials 21 Table 4.1 Peak accelerations recorded during two earthquakes 28 Table 4.2 Periods of the first three modes in each direction 39 Table 4.3 Damping ratios of the first three modes in each direction 39 Table 4.4 Scaling constants of residue matrix 40 Table 4.5 Periods and dampings from the conditioned records 43 Table 5.1 Uniaxial hysteresis model options and number of parameters 52 Table 6.1 Estimated weight and mass polar momnt of inertia 57 Table 6.2 Live loads for calibration of the models 58 Table 6.3 Periods of the first three modes in each direction 59 Table 6.4 Damping ratios used for computer analyses 60 Table 7.1 Peak ground accelerations (PGA) of ground motions 76 Table 7.2 Comparison of results 89 vii LIST OF F I G U R E S Figure 3.1 The Figueroa at Wilshire Tower in Los Angeles 15 Figure 3.2 Configuration of braced frames showing different failure mechanisms 18 Figure 3.3 Typical floor plan and details of the columns of the FWT 20 Figure 3.4 Instrumentation plan of the FWT 23 Figure 4.1 Location of the FWT and epicenters of several earthquakes in the vicinity of the building site 24 Figure 4.2 Recorded accelerations during the Sierra Madre earthquake 26 Figure 4.3 Recorded accelerations during the Northridge earthquake 27 Figure 4.4 Response spectra of recorded ground floor motions during the Sierra Madre earth-quake 29 Figure 4.5 Response spectra of recorded ground floor motions during the Northridge earth-quake 29 Figure 4.6 FRFs from the Northridge earthquake 34 Figure 4.7 Comparison of torsional records' power spectra 35 Figure 4.8 Mode shapes of the building from the Sierra Madre earthquake 37 Figure 4.9 Mode shapes of the building from the Northridge earthquake 38 Figure 4.10 Comparison of the coherence functions between recorded motions for " A " level during the Northridge earthquake 42 Figure 4.11 Conditioned records from the Northridge earthquake 43 Figure 5.1 3D multispring column model 49 Figure 6.1 C A N N Y model of the FWT 55 Figure 6.2 Mode shapes for the FWT in each direction 59 Figure 6.3 Comparison of the responses using conventional damping ratios 61 viii Figure 6.4 Comparison of the responses after damping calibration 62 Figure 6.5 Comparison of recorded and computed roof absolute acceleration during the Sierra Madre earthquake 64 Figure 6.6 Comparison of recorded and computed roof absolute acceleration during the Northridge earthquake 65 Figure 6.7 Comparison of recorded and computed roof relative displacement during the Northridge earthquake 65 Figure 6.8 Comparison of recorded and computed 14th story absolute acceleration during the Northridge earthquake 67 Figure 6.9 Comparison of recorded and computed 14th story relative displacement during the Northridge earthquake 68 Figure 7.1 C A N N Y hysteresis model #14 73 Figure 7.2 Time history and response spectrum of the S V A C - A record (Niigata earthquake).. 77 Figure 7.3 Time history and response spectrum of the Joshua Tree Fire station (Landers earth-quake) 78 Figure 7.4 Time history and response spectrum of the Sylmar County Hospital station (Northridge earthquake) 79 Figure 7.5 Maximum inelastic response of the FWT, displacement 80 Figure 7.6 Maximum inelastic response of the FWT, interstory drift 81 Figure 7.7 Maximum inelastic response of the FWT, story shear 82 Figure 7.8 Maximum inelastic response of the FWT, overturning moment 83 Figure 7.9 Roof displacement time history responses, Sylmar County Hospital record . . . . 84 Figure 7.10 Roof acceleration time history responses, Sylmar County Hospital record . . . . 85 Figure 7.11 Base shear versus roof displacement, E-W direction 86 Figure 7.12 Yielding propagation steps in the south perimeter frame in E-W direction . . . . 88 Figure 7.13 Base shear versus roof displacement, cyclic test 92 IX A C K N O W L E D G E M E N T S I am deeply indebted to my supervisor, Dr. Carlos E. Ventura, for his guidance throughout this project, and all the time and effort he put in advising me. Special thanks are due to Dr. Kang-Ning L i of C A N N Y Consultants Pte. Ltd., Singapore for pro-viding his program, C A N N Y - E and valuable assistance on implementing its various features. The financial support for this study was provided by a research grant awarded to Dr. C.E. Ven-tura from the Natural Science and Engineering Research Council of Canada. x Chapter 1 INTRODUCTION 1.1 General Remarks Today, with the help of computers, buildings which have little or no historic precedent are being planned and designed. New structural systems are conceived and applied to extremely tall build-ings in a practical demonstration of the engineer's confidence in the knowledge of structures, theory of analysis and efficiency of computer algorithms. It is therefore important to update the theory with the latest information obtained from recent research and practise. There are three ma-jor factors to consider in the design of all structures: strength, rigidity, and stability. In the design of tall buildings, all three these requirements play an important role. The strength requirement is the dominant factor in the design of low-rise structures. However, as height increases, the rigid-ity and stability requirements become more important, and they are often the dominant factors in the design. These requirements are forcing designers to change from the traditional linear stat-ic techniques to more advanced inelastic dynamic techniques. In certain seismically active regions, severe earthquakes happen frequently, causing the loss of hundreds to thousands of lives. Also, property damage accounts for hundreds of millions of dol-lars. As a consequence of the 1994 Northridge earthquake in Los Angeles, significant research has been carried out in order to: 1) learn the seismic responses from analyzing a large amount of recorded data; and 2) re-evaluate the performance of buildings by detailed dynamic analysis. To achieve the first goal of learning the seismic responses from analyzing the large amount of recorded data, instrumentation is a vital and prerequisite step. Due to the lack of instrumentation 1 in buildings in the first half of the 20th century, the original concept of spectrum design that dom-inates today's design codes is largely based on the recorded data from the 1940 E l Centra earth-quake. It is obvious that the knowledge from one earthquake event could be limited and misleading. Luckily the large amount of recorded strong motion data in recent years provide ex-cellent opportunities to improve engineers' knowledge towards seismic response. One of the objectives from analyzing these large amount of recorded strong motion data must be the better design of future structures. Thus the second goal of re-evaluating the performance of the buildings by detailed dynamic analysis comes to mind. Since the comparison of the recorded seismic response and the responses from computer dynamic analysis could bring improved un-derstanding of seismic response of structures, it is of great interest to carry out a study including both of the purposes mentioned above. This study serves to reach this goal. One of the most important factors which has been avoided in previous research is the three-di-mensional nature of both the excitation and response of the structure during earthquakes. In fact, the three-dimensional behavior of structures during earthquakes has not been studied as exten-sively as that of planar or two-dimensional behavior (Rahmatian, 1997). The unavailability of computer programs capable of three-dimensional inelastic analyses was cited as the excuse of this simplification.(SAC95-04) Experimental studies, although limited, have demonstrated that there is a significant interaction between the responses of a non-symmetric structure along its two principal axes. It has been observed that even structures that appear symmetric often expe-rienced non-uniform connection damage that could make the structures more sensitive to three-dimensional response. With the appearance of sophisticated computer programs and powerful personal computers, a near future tendency is to utilize three-dimensional inelastic analyses for seismic design of structures. 1.2 Objectives and Scope The goal of this thesis is to study and understand the seismic behavior of a steel high-rise build-ing. In order to accomplish this, the following objectives are identified: 1) conduct a system iden-tification of the recorded earthquake motions, 2) investigate the building behavior during past earthquakes through construction of a lumped plasticity computer model and calibrate the model using recorded motions, 3) evaluate the inelastic structural response of the building during se-vere earthquakes shaking, and 4) evaluate the inelastic static response of the building for three lateral load patterns. The building in this study is modeled based on specifications and informations included in the design drawings and personal contacts between the designer of the building Dr. Banavalkar, P.E. of C B M Engineers Inc. of Houston, Texas and Dr. C.E. Ventura, P.Eng., Assoc. Prof, of the de-partment of Civil Engineering in the University of British Columbia, Vancouver. The emphasis of the study conducted here is on the linear and non-linear global response and behavior of the structure, rather than on any of its elements. Foundation and the interaction between the founda-tion and the soil are not modeled in the computer analysis. The main emphasis of the inelastic response is on the evaluation of the effects of near field impulse-type ground motions. In order to achieve the objectives of the study, the following tasks were performed: 3 1. The instrumentation scheme of the Figueroa at Wilshire Tower (FWT) is described, and the strong-motion response records obtained from the building during the 1991 Sierra Madre and the 1994 Northridge earthquakes are described and analyzed. 2. The modal frequencies are determined for the three directions: two translational and one rota-tional, by using system-identification techniques. Also, results are compared with those obtained from the computerized structural analysis conducted by the structural designers. 3. The damping ratios and contributions of each mode to the response are derived, and a number of dynamic characteristics and interesting features are further investigated for future dynamic analysis. (More attention has been paid to the frequency and damping variation between two earthquakes and the modal contributions.) 4. A structural model is set up using computer program C A N N Y (version E) and the linear dy-namic characteristics such as frequencies and mode shapes are verified. The time history re-sponses of the structural model are calculated using the recorded base motions as inputs to the model and are compared with recorded earthquake responses at different floors. Since the build-ing responded linearly during the two earthquakes considered, the comparison of computer and recorded earthquake motions is used to verify the credibility of the computer program results. 5. Analysis is then conducted to understand the three-dimensional inelastic behavior of the build-ing during severe shaking. 4 1.3 Organization A short discussion about linear versus nonlinear analysis, together with a literature review on nonlinear modeling of steel frame buildings are covered in Chapter 2. In Chapter 3 a complete description of the building studied (FWT) is given. Some of the topics discussed here are: the structural system and the reason it was chosen, construction materials, seismic instrumentation plan. Detailed Strong Motion Data analysis results, including time domain analysis and frequency do-main analysis, are presented in Chapter 4. Derived modal frequencies, mode shapes, modal dampings are shown in tables or figures. A general review of the nonlinear analytical program (CANNY-E) used in this study and some of its unique features are presented in Chapter 5. Chapter 6 is concerned with a three-dimensional linear-elastic modeling of the structure. Three-dimensional analyses of the structure are performed for two input ground motions: one from the Sierra Madre earthquake and the other from the Northridge earthquake. The purpose of these analyses is to calibrate and verify the computer model with the real structure. The nonlinear three-dimensional analysis of the building is discussed in Chapter 7. The devel-opment of the model, the underlying assumptions, and finally the response of the structure dur-ing several severe earthquake excitations are presented here. In addition to the time history 5 analysis, pushover and cyclic analysis of the building are also conducted. Finally, a summary of the results of this study followed by recommendations for further studies are given in Chapter 8. 6 Chapter 2 BACKGROUND OF NONLINEAR ANALYSIS 2.1 Why Nonlinear Analysis In the90s, many countries have experienced a number of major earthquake disasters such as the 1994 Northridge earthquake and the 1995 Kobe earthquake. These earthquakes have exposed many important problems that adversely impacted the urban areas around the world in terms of the well-being and safety of individual citizens, the operation of essential social services and life-line systems, and the productivity and economic vitality of the business, financial and industrial sectors. Every earthquake provides many valuable lessons to the earthquake engineering profession. The event in Northridge was no exception to this. What surprised engineering professionals was the damage to modern steel buildings. Until this earthquake, steel buildings were thought to be among the most seismically resilient forms of construction possible. In Los Angeles area, how-ever, welded steel frame buildings suffered many brittle fractures in their connections during the Northridge earthquake. And many of this type of structures collapsed during the motions expe-rienced in Kobe. It is therefore necessary to study the behavior of buildings during these earthquakes in order to improve the knowledge of seismic design. In past design practise, results from linear elastic anal-ysis performed using computer programs were dominantly used to determine member strength demands. The extensive damage and collapse of structures in these two earthquakes proved the deficiency of this approach to predict the building response. The implementation of a perfor-7 mance-based design procedure necessitates reliable analytical predictions of important seismic demands (strength, deformation, and energy dissipation demands). Research needs to be per : formed to develop reliable analytical techniques and tools that permit performance evaluation of three-dimensional systems and its components at all levels of performance, ranging from elastic behavior to degradation leading to collapse. Therefore it is of critical importance to accurately model three-dimensional inelastic building responses. The focus of these type of research need to cover the following important issues: • 3-D modeling procedures • Reliable modeling of strength and stiffness irregularities in plan and elevation • Modeling of deteriorating systems Nonlinear analysis, attempting to represent more realistically the behavior of a building struc-ture, has its own shortcomings. Traditionally it is labour intensive and expensive to implement. Furthermore, nonlinear analysis requires a qualitative interpretation of many inelastic parame-ters in order to capture the response of the structure, while linear analysis could provide reason-able member strength demands by assigning appropriate values to the member stiffness. The nonlinear models expected to capture the true behavior of structural members are difficult to construct. A complete realistic relation between the complex mechanics of material nonlin-earities and the three-dimensional structural response of a system have not been well established through analytical work and experimental testing. Therefore, decisions must be made on which aspects of nonlinearities are significant, and how to capture them in the analytical model. 8 Despite these long time difficulties faced by both practicing engineers and academic researchers, recent enhancements in computational techniques and the development of more sophisticated models allow structural analysis to have integrated tools, based on components' behavior, which are capable of determining the behavior of three-dimensional structures in a realistic manner. 2.2 Literature Review Literature on three-dimensional nonlinear modeling of tall steel frame structures is limited. One of the most comprehensive books on the structural analysis of tall buildings (Taranath, 1988) in-cluded only the geometric nonlinearities of tall buildings (such as P-delta effect) but not their material nonlinearities. SAC 95-04 report on steel moment frame buildings are the most detailed studies after the Northridge earthquake. However, all of the inelastic analyses in these reports were limited to two-dimensions. The unavailability of computer programs capable of three-di-mensional analyses was quoted as the reason. Henceforth the literature review on two-dimensional inelastic analyses at the member level are discussed briefly below. The determination of the inelastic member properties is an essential step in the evaluation of earthquake response. Early studies of the response of multistory buildings were based on shear-beam idealizations (Penzien, 1960). The entire structure was modeled as a series of single lateral degree of freedom systems, one at each floor, having the global hysteretic characteristics representative of the cyclic inelastic response of the corresponding floor. A par-allel component element was introduced by Clough et al. (1965) and allowed for a bilinear mo-ment-rotation relation. The element consists of two components: a fully elastic component to 9 represent strain hardening and an elasto-perfectly plastic component to represent yielding. The element stiffness matrix is the summation of the stiffness matrix of both components. Kanaan and Powell (1973) used a modification of Clough's parallel component model for their beam column element. The reduction of section plastic moment capacity due to an axial force was introduced by using an interaction formula. The one component series model was formally introduced by Giberson (1969). This element consists of a linear elastic element with one equiv-alent nonlinear rotational spring attached to each end. The inelastic deformations of the member are lumped into the end springs. Chen and Powell (1982) developed a lumped plasticity beam-column element, where it is as-sumed that inelastic behavior is concentrated in plastic hinges at the element end and the remain-ing part of the element remains elastic. In this model it is difficult to establish the general parameters required for the force-deformation relationships and the four dimensional yield sur-faces, which are governed by the cross-sectional dimensions and the hysteretic force-deforma-tion characteristic of the member material. Hsu (1974) and Takayanagi and Schnobrich (1979) proposed a multiple spring model for analyzing wall members. The member is divided into sev-eral subelements along its axis, each represented by a nonlinear spring. A few analytical models on inelastic element behavior have been developed in recent years. One study provided comprehensive information was by Kim (1995). In his study Kim divided ana-lytical models developed in the past two decades into three categories in accordance with an in-creasing level of refinement and complexity. These models are: 10 1. Global models. In these models the nonlinear response of a structure is concentrated at selected degrees of freedom. Each degree of freedom has the hysteretic characteristics of the inter story response. Such models are used in the preliminary design phase. Member forces and local inelastic deformation demands are not available. 2. Discrete element models. In these models the structure is modeled as an assembly of interconnected elements that describe the hysteretic behavior of structural members. Corre-spondingly, two types of element formulation are possible: lumped plasticity models and distrib-uted plasticity models. 3. Microscopic finite element models. In these models members and joints are discretized into a large number of finite elements. Ideally, analytical models should be based on an accurate representation of material behavior taking into account the controlling states of stress or strain and identifying the main factors which influence the hysteretic behavior of each critical region in any structural component dur-ing the earthquake response. At the same time these models should be computationally efficient. In practice, it is very difficult to find an analytical model satisfying both an accurate representa-tion of material behavior and computational efficiency. Although lumped plasticity models sac-rifice accuracy to some degree, these models have the capability to reasonably compromise between simplicity and accuracy in nonlinear seismic response studies. Kim's study concen-trates on this second class of models. He developed lumped plasticity models to better represent the experimental data. Then he used distributed plasticity models (fibre models) to check the re-sults against lumped plasticity models. His study included lumped plasticity models for bare steel beams and columns, composite beams, and column panel zones. 11 Since Kim's study was on steel moment frames, models for bracing element were not included. However, Jain and Goel (1978) denoted a multi-linear hysteretic model EL9 for computer pro-gram DRAIN-2D based on empirical data by Jain (1978). In Jain's study small square tube and angle specimens, with or without gusset plate connections, were subjected to large cyclic static and dynamic axial displacements. A new hysteresis model was proposed for steel tubular mem-bers which includes reduction in compressive strength and increase in member length with num-ber of cycles. It was concluded that in concentric braced frames end moments and axial forces are equally im-portant for bracing members having slenderness ratio of 60, whereas, elastic buckling dominates for slenderness ratio of more than 120. In eccentric frames, end moments dominate over axial forces for bracing members with slenderness ratio of 60 or less. End moments and axial forces are equally important for those with slenderness ratio between 60 to 120 and elastic buckling dominates for slenderness ratio more than 120. Effective slenderness ratio seems to be the most influential parameter in determining the hysteresis behavior of bracing members. In FEMA-274, a typical force versus axial deformation response of a steel brace is shown and the residual compression force for the brace is about 20% of the buckling load. It also states that this percentage is about the same for many brace configurations. Therefore, it further recom-mends that for a more simplified Nonlinear Static Procedure analysis, the axial force-deforma-tion behavior of a brace in compression could be modeled as an elasto-plastic element with the yield force equal to the residual force. However, an elastic analysis will also need to be done to 12 determine the maximum axial force delivered to the column, the beam, and the beam-column connections. Since structures designed according to current building codes are expected to deform well into the inelastic range during severe earthquake ground motions, constructing a sophisticated model for structural analysis has become increasingly important. Due to the complex interactions be-tween the various components of real structures, their dynamic characteristics up to failure can not be identified solely from dynamic tests of scaled models. Moreover, the cost of such tests is often substantial, particularly for large scaled specimens. Historically these difficulties have been overcome by static tests on components and on down-scaled subassemblages of structures under cyclic load reversals. Results from these tests are used in the development and calibration of hysteretic models that permit the extrapolation of the limited test data to other cases and to the dynamic response of complete structures. In this study computer program C A N N Y - E (Li, 1996) is chosen because of its ability of perform-ing both three-dimensional time history elastic and inelastic analysis. Lumped plasticity models were constructed using this program. Detail introduction of the program is given in Chapter 5. 13 Chapter 3 DESCRIPTION OF THE FIGUEROA AT WILSHIRE TOWER 3.1 Introduction An innovative steel structural system that emerged in the 1980s is the so-called spine or outrigger structure, which is considered to be an especially appropriate structural system for tall buildings in high seismic areas. The spine consists of vertical or inclined elements, and shear membranes in the form of braced frames, walls or vierendeel girders. The Figueroa at Wilshire Tower (FWT) is a good example of this type of structure in which the spine is the lateral load resisting system that provides stiffness and stability (Banavalkar, 1991). The FWT is a 52-story steel frame building in Los Angeles, California. The building was designed in 1988, constructed in 1988-90 and instrumented to measure seismic motions after construction. Ac-celeration sensors recorded the building's motions during many earthquakes among which the 1991 Sierra Madre earthquake and the 1994 Northridge earthquake were chosen for this study. These records provided detailed and important information about the building behavior during both earth-quakes. The FWT consists of a 52-story steel frame office tower and five levels of enlarged basement as un-derground parking. The floor plans of the tower are not perfectly square. On each floor, the tip of every corner was clipped and the middle third of each side was notched. Above the 36th story, at clusters of about five stories, the corners of the floors were clipped further to provide a stepback view to the exterior of the building. A n overview of the building is shown in Fig. 3.1 and typical floor plan is shown in Fig. 3.3. 14 Figure 3.1 The Figueroa at Wilshire Tower in Los Angeles 15 3.2 Structural System A comparative study of perimeter ductile tube (popular) versus spine structure (innovative) is shown in Table 3.1. The structural designer of the FWT considered three different structural schemes (two of which are perimeter tubular structure and the other is spine structure) for the building and the spine structure was finally chosen because its many advantages. The structural system of the FWT has three main components: a braced core, twelve columns (8 on the perimeter and 4 in the core), and eight 914-mm (36-inch) deep outrigger beams at each floor connecting the inner and outer columns. The core, about 17.37 m (57 feet) by 21.34 m (70 feet), is concentrically braced between the " A " level (the level just below the ground level) and the 50th story (between 1st story and 2nd story is an exception due to architectural reason). Mo-ment resisting connections are used at the intersection of beams and columns. The outrigger beams, about 12.19 m (40 feet) long, link the four core columns to the eight perimeter columns to form a ductile moment resisting frame. The outrigger beams are laterally braced to prevent lateral torsional buckling and are effectively connected to the floor diaphragm by shear studs to transmit the horizontal shear force to the frame. The interior core is concentrically braced. The sizes of mechanical ducts and door openings into the core dictate the configuration and sizes of braces (Fig. 3.2). In order to achieve redundancy in carrying gravity loads, the most important design criteria of braces is that in the event of buck-ling of one of the compression diagonals, the horizontal members supporting the floor loads do not form a mechanism resulting in catastrophic failure of the floor. For bracing types II and III, 16 Table 3.1 : COMPARATIVE STUDY OF PERIMETER DUCTILE TUBE VERSUS SPINE STRUCTURE (Banavalkar, 1992) CRITERIA PERIMETER TUBE SPINE STRUCTURE I. Structural Performance a) Wind load. b) Seismic event. c) Redundancy & ductility. d) Torsional stiffness. e) Stiffness & strength. a) 100 year storm. b) Maximum probable (lower level event). c) Very ductile & redundant. d) Torsionally stiff. e) Very good. a) 100 year storm. b) Maximum credible (upper level event). c) Ductile & redundant. d) Depends upon composi-tion of the spine. e) Very good. II. Planning a) Flexibility in core & lease space. b) Perimeter planning. c) Articulation of the tower. d) Entry way. e) Underground parking. a) Complete flexibility. b) Too many columns & restrictive in planning. c) Restrictive. d) Transfer girders & soft story. e) Very inefficient. Penetra-tion of too many columns. a) Disciplined layout of core & mechanical system. b) Complete freedom for open spaces along perime-ter. c) Very flexible. d) Complete freedom & con-tinuity. e) Efficient. III. Cost a) Quantity of steel. b) Cost/ton. c) Erection time. d) Fireproofing & enclosure for columns. e) Underground parking. a) Higher steel tonnage. b) May be lower. c) Longer because more erection pieces. d) Larger number. e) Cost prohibitive. Possibly additional basement required. a) Less steel tonnage. b) Competitive. c) Shorter time to erect. Few pieces to erect. d) Minimum quantity. e) Cost effective. IV. Innovations a) Composite concrete columns. b) Up & down construction. a) Not practical. b) Impractical. a) Cost effective. b) Very plausible. 17 different failure mechanisms shown in Figure 3.2 are considered. The bracing members are sized in such a way that the gravity load carrying capacity of the system is not impaired. For bracing type I, in which case the bracing is in tension under gravity load, no failure mechanism needs to be checked. B R A C I N G T Y P E I B R A C I N G T Y P E HI CORE BRACING FIGUEROA AT WILSHIRE [ M i l l i l I I l INACTIVE B R A C I N G T Y P E H r i i i i i i I I 1 1 1 1 INACTIVE r l -r i y ' \ A INACTIVE 5) Figure 3.2 Configuration of braced frames showing different failure mechanisms The outrigger beams have to perform three functions: 1) they have to support the design floor loads; 2) the outrigger beams along with core and perimeter columns have to act as a ductile moment resisting frame to carry a minimum of 25% design code level forces without 18 the presence of interior core bracing; 3) the stiffness of the beam should be such so as to create effective linkage between the interior core and the perimeter columns to provide effective overturning resistance to the seismic loads (Banavalkar, 1991). In high-rise buildings, it is always cost effective to minimize floor-to-floor height. The depth of outrigger beams is therefore dictated by the restriction on the ceiling cavity. To achieve these goals, the beams were offset into the floor and were notched at midspan to allow for the passage of the mechanical ducts. These features eventually have effects on the inelastic modelling of the structure. Heavily loaded perimeter and core columns provide overturning resistance to the entire structure generated by lateral loads. The perimeter column capacity was checked against full plastification of outrigger beams. Perimeter columns are standard I-sections, while core columns are built-up sections with crucifix shape. The floors, acting as diaphragms, are connected to the frames by shear studs to transfer lateral shear forces to the frame. The long span floor framing is structured in such a way that the main columns participating in the lateral load resisting system are heavily loaded by gravity load. Composite galvanized floor deck is 51 mm deep, except at ground floor and level " A " where it is 76 mm deep. In this way, all lateral forces were expected to be resisted by the braced core and the four moment resisting frames. Typical floor plan are in square shape with 47.45 m sides. Typical story heights are 3.96 m. A typical story plan and column details of the building are shown in Figs. 3.3. Selected elevations and details of the building are presented in Appendix A. 19 Perimeter Perimeter column column Outrig beam Core columi — | a . g r 1 Outrigger Ibeam Bracing lCore column B Perimetei ' • uutrigger beam Bracin Outrigger beam 1 Outrigger 1 beam 1 Bracing 1 Outrigger column s Perimetei Core column Outriggei \ ^ ^ ^ beam 1 Bracing TCore column l Outrigger lbeam / column s Peri n colu r 13.84 m -.' , -eter Pen in coll 17.37m neter mn 13.84 m Typical Floor Plan 1016mm 89-102mm thick-1016(914)mm 457mm f 1 32-76mm web thickness 2 38-102mm flange thickness Built-up Column Figure 3.3 Typical floor plan and details of the columns of the FWT 20 Basement of the building is enlarged and has irregular floor plan. Columns in basement are steel section encased in concrete. A 610-mm-thick (2-ft.) concrete wall supporting the parking ramps is not part of the lateral force resisting system for the tower, but it does provide lateral stiffness. The building foundation is concrete spread footings (2.74 to 3.35 m thick), supporting the steel columns with 127mm thick concrete slab on grade. The type of soil is very stiff shale or sand-stone and has allowable bearing pressure of 718.2 kpa (15000 lbs/sqft). 3.3 Building Materials From the general notes on design plans, all structural steel framing including columns use A S T M A-572 (grade 50). For properties of structural steel and other materials please refer to Table 3.2. Table 3.2: Properties of construction materials Member Nominal Strength (MPa) Expected Strength (MPa) Modulus of Elasticity E(MPa) A S T M A-572 (GRADE 50) fy=345 380 200,000 Large tower footings and water tank fc '=34.5 N A 28,960 Slab on grade fc =20.7 N A 22,753 A l l other concrete fc =27.6 N A 25,510 3.4 Instrumentation The Figueroa at Wilshire Tower was instrumented by the California Strong Motion Instrumen-tation Program(CSMIP) in 1990. Twenty sensors in total were installed on different levels of the building to measure the global building response to earthquakes, including the translational, tor-21 sional, and vertical motions. For the locations of the sensors, please see Fig. 3.4. The purpose of this instrumentation is to measure the global building response to earthquakes, specially the vertical motion of the building at the base, torsional, and translational motions at the base and upper floors. The mode shapes provided by the structural engineers are used for op-timizing the sensor locations in order to determine the antinodes of: a) the first three modes of lateral motion in each direction, and b) the first two modes of torsional motion. It is expected that the contributions of the higher modes to the total response of this building are significant. To measure torsional motion of the floor, pairs of sensors are placed in the N-S direction rather than the E-W direction because the eccentricity is larger in the E-W direction. The floor locations of the sensors are based on the mode shapes provided by the structural designer according to the computer analysis. It is worth noting that an initial sensor layout proposal included two vertical sensors to measure the rocking motion at the " A " level (CSMIP, 1990), but these sensors were dropped because: 1) the four columns at the core are continuous from the base to the roof; 2) the soil is very stiff and the tower is very flexible, the rocking contribution to the total motion is expected to be very small; and 3) the rocking motion would be too small to be measured by accelerometers. No ad-ditional sensor to measure torsion is placed at the roof level because the lever arm between sen-sors is small; the distance from the center to the east side of the core is less than 10 m. 22 23 Chapter 4 ANALYSES OF RECORDED DATA 4.1 Recorded Earthquake Responses The two earthquakes considered in this study are the 1991 Sierra Madre earthquake and the 1994 Northridge earthquake. The building experienced its first strong ground motion during the Sierra Madre earthquake, which had its epicenter about 25 km north-east of the building. No damage was reported after this event. The epicenter of the 1994 Northridge earthquake was located about 30 km north-west of the building. Again, no damage was reported during this earthquake. Northridge( 1994) San Fernando( 1971) Sierra Madre( 1991) F W T Whittier(1987) Long Beach(1933) Figure 4.1 Location of the FWT and epicenters of several earthquakes in the vicinity of the building site (SCEC_DC, 1999) 24 The locations of the FWT and epicenters of several earthquakes including those happened before the construction of the FWT are shown in Fig. 4.1. The epicenters of the Sierra Madre earthquake and the Northridge earthquake have similar distance to the location of the FWT. However, their directions have 90 degree difference. The acceleration response records obtained during the Sierra Madre earthquake and the Northridge earthquake from the instrumentation scheme are shown in Fig. 4.2 and Fig. 4.3. Re-corded peak acceleration values are included at the right side of figures above each record. 4.2 Time Domain Analysis of Recorded Motions The acceleration records are shown in Fig. 4.2 and Fig. 4.3. The peak acceleration derived from the processed data are summarized in Table 4.1. In this table S M represents the Sierra Madre earthquake and N R represents the Northridge earthquake. At the basement, the recorded peak horizontal accelerations are 0.08g in E-W direction and 0.09g in N-S direction during the Sierra Madre earthquake. The peak horizontal acceleration for the superstructure is 0.23g recorded by the N-S accelerometer at the roof (53rd floor). Note that the peak horizontal acceleration in the N-S direction at the 14th floor is 0.14g which equals to that at the 49th floor. These acceleration values are higher than at any other recording stories, specially the 35th floor. This is a clear evidence of participation of higher modes in the building response. In the Northridge earthquake, the recorded peak horizontal acceleration at the basement are 25 1 Level E: Center of Tower - Up 2 " " - F 3 4 E. End of Tower - N 5 Level A: Center of Tower - B . _ -^1 H -^/WV^ E. End of Tower - N 8 14th Floor: Center - E .9 " - N 10 22nd Floor: " - E 11 " " - N 12 " E. Wall - N 13 ~ 35th Floor: Center - £ 14 ._ 15 E. Wall - N 16 49th Floor: Center - E 17 " _ N 18 " Max. Accel.(g) 0.04 E. Wall - N 19 Roof: renter - ft 20 " " _ N T r i g g e r Time: 14:43:59.5 GMT 0.08 Q.09. 0.07 0.10 0 12 JLJJL 0.12 0.14 Structure Reference O r i e n t a t i o n N-355" 1 2 3 * 10 15 Sec. Figure 4.2 Recorded accelerations during the Sierra Madre earthquake (Huang, et al., 1991) 27 Table 4.1: Peak accelerations (g) recorded during two earthquakes Story S M S M S M N R N R N R N-S ' E-W Torsional N-S E-W Torsional Roof 0.28 0.20 N A 0.40 0.23 N A 49th 0.15 0.12 0.04 0.13 0.13 0.06 35th 0.13 0.10 0.03 0.21 0.11 0.03 22nd 0.15 0.13 0.05 0.19 0.12 0.06 14th 0.17 0.14 N A 0.19 0.12 N A " A " Level 0.14 0.12 0.02 0.17 0.13 0.04 " E " Level 0.11 0.09 0.05 0.14 0.11 0.08 0.1 lg in E-W direction and 0.14g in N-S direction. The peak horizontal acceleration for the su-perstructure is 0.4lg recorded by the N-S accelerometer at the roof (53rd floor). Linear-elastic response spectra of horizontal ground motions recorded at the ground floor of the building (5% damping) are provided in Fig. 4.4 and Fig. 4.5. These response spectra will be help-ful in understanding the frequency range of energy input for each ground motion. The response spectra of the Sierra Madre earthquake show the peak of energy input between 0.2 sec to 0.6 sec. The response spectra of the Northridge earthquake show the peak of energy input between 0.5 sec to 5 sec. 28 Si erra. Madre , G , 200/.SEC •Pscittto V c l o c i ' i y . inni/s 0 - 00 '•• 0.01 \ 1 "•-'V -••vj ~ - -- Y , ' x;xrY~£>j \ - - - \ ^ r s h r x l " A P -i \ ^ • Y . k Y Y - p Y X, 1 S X Y i s v s . : ; \ , Y X 1 Y 1 1 t- - x. s < Y s Y V \ \ \ V-Y \ Y x Y 7 T y i ^ \ ^ N ^ x\ X \ \ Y l: V Y • ^ \ : X • V \ x. Ik fsX • • / \ S :<\ X \ _ i • =Yr-v- s i IK \ , X - Y-iH ' Y \ s \ \ - | H ^ Y 1 \ \ \ : N- Y / N x x ^ X \ "S \ \ \ \ Y > Y \ x -x Y \ Y N "r '*** V ? T — ^ - \ - V Y, —^^  Y ^. N ^ Y , L! JYil~ \ \ .....1..:: \ i : 1 x ^ K *•> X < Y ^ • . Y ! M l | l •M K ; i x Y k Y i ' , Y Y Y A N \ ; \ V 1 1 L Y Y x s , \ Y x S V . Y V Y \ \_ i I S i e r r a Madre. G. 2007SEC I 0 .00 P o r i i H l . .S.-.ODnd*. 5.00 .1 Damiiiii} 5 .00 !i l)am|)ine Figure 4.4 Response spectra of recorded ground floor motions during the Sierra Madre earthquake Northridge Earthquake, GAL, 0.01 SEC P t c u O o V e l o c i t y , em/s Northr idge Earthquake, GAL, 0.0ISEC P x c u d o V e l o c i t y , c m / * 1 0 . 0 0 P c r i t i d . S e c o n d ) o d . S e c o n d s 5 . 0 0 S D a m p i n g 5 . 0 0 * D a m p i n g Figure 4.5 Response spectra of recorded ground floor motions during the Northridge earthquake 29 4.3 Frequency Domain Analysis Detailed frequency domain analyses of recorded responses were performed to determine the dy-namics characteristics of the building. The probable natural frequencies, damping ratios and mode shapes of the FWT were estimated based on a Fourier Transform analysis of recorded mo-tions. The results are presented in section 4.3.2. 4.3.1 Theory In this study, the natural periods, damping ratios and mode shapes were identified using comput-er program ME'scope (The ME'scope, 1998). The strong motion records were used to calculate Frequency Response Functions (FRF's) and coherence functions between recorded accelera-tions. The accelerations frequency response function is the Fourier transform of the unit-impulse response function which is usually used to describe the dynamic properties of physical systems. From the frequency response functions the natural frequencies and the mode shapes are derived. The physical interpretation of the frequency response function is straightforward. For a harmonic excitation at a particular frequency, it is intuitively obvious that the ratio of output motion to in-put force would be greater at the resonant frequency than at other frequencies. Based on the same reasoning, when looking at the FRFs of the building response, the peaks may indicate possible modes of the structure. The frequency response functions are calculated by taking the record at the base as the input into the structure and all other measurements as the outputs. In ME'scope, a MIMO(Multi-Input-Multi-Output) model is constructed as a frequency domain 30 model, where Linear Spectra (Fourier Transforms) of multiple inputs are multiplied by elements of a FRF matrix to yield the Linear Spectra of multiple outputs. It can be written as: (Y(co)} = [H(co)]{X(co)} (4.1) {Y(co)} is the vector of linear spectra of outputs; [H(co)] is the FRF matrix; {X(co)} is the vector of linear spectra of inputs and; co is an independent frequency variable (in rad/sec). Each FRF in the MIM O model defines the dynamic properties between a single input DOF and a single output DOF. FRF measurements are usually made under controlled conditions, where the test structure is artificially excited using one or more shakers. A FRF measurement is typi-cally formed as the ratio of output motion over input force, where the measured output motion can be acceleration, velocity, or displacement. Strictly speaking, when an acceleration record is used as the input rather than a force time histo-ry, the result is not a frequency response function but rather a dimensionless ratio of input to out-put motion. If the mass of the building is known, the dimensionless FRF could be divided by the mass to yield the standard FRF, since they both are output motion divided by input base shear. To obtain FRFs, estimates of the Cross and Input Auto Spectrum matrices are calculated first. Time domain windows and overlap signal processing can be used during the calculation of these matrix estimates. Next, the FRF matrix is formed by multiplying the Cross Spectrum matrix by the inverse of the input Auto Spectrum matrix. This is done by solving the equation below: [H(co)] = [{Y(co)} x {X(co)} T] x [{X(co)} x { X ( c o ) } V (4.2) 31 where the superscript "T" denotes "transpose of the vector". Modal parameters are identified by curve-fitting the FRFs. Curve-fitting is a process of matching a parametric model of a FRF to the measured data. The unknown parameters of the parametric model are the modal frequencies, modal damping and modal residues (mode shape components) such as shown in equation 4.3. N [H(ffl)] i L j - c o - p ( k ) j . c o - p * ( k ) J (4.3) where: H(co) = FRF matrix (n by n) R(k) = Residue matrix for the kth mode (n by n) p(k) = pole location for the kth mode = - cr(k) + jco(k) rj(k) = modal damping for the kth mode (rad/sec, or Hz) co(k) = modal frequency for the kth mode (rad/sec, or Hz) n= number of DOFs of the FRF model *- denotes the complex conjugate j - denotes the imaginary axis in the complex plane (= J - i ) N = number of modes considered in the analysis The outcome of curve-fitting is a set of modal parameters for each mode that is identified in the frequency range of the measurements. Modal parameter estimation is done in the following steps: 1) Determine the number of modes in a frequency band by visual inspection of the FRFs, 2) Estimate modal frequency and damping for the modes in a frequency band, 32 3) Estimate modal residues for the modes with frequency and damping estimates in a band, 4) Derive mode shapes from the modal parameters. When analyzing the response of building to earthquakes, we always have to deal with the noise problem. The recorded signals are not as "clean" as those generated by the computer simulation or even from a lab experiment. Hence the observation of frequencies could be difficult. Past re-search experience provides us with a number of useful tools. We know from existing theory that when there is a natural frequency, the data will exhibit the following features: 1. the real component of the FRF will be zero; 2. the imaginary component of the FRF will peak; 3. the magnitude of the FRF will peak; 4. the phase of the FRF will be -90 degrees; 5. the coherence will usually drop; 6 the vibration shape will exhibit properties of the mode shape. 4.3.2 Dynamic Behavior from Seismic Response selected amplitudes of FRFs derived from the Northridge records are shown in Fig. 4.6. These FRFs are between the acceleration records in the E-W direction of the building. The amplitude of the FRFs does not have unit since the FRFs are derived by (output acceleration)/(input accel-eration). The FRFs from the real earthquake records are different from those derived from lab testing or theoretical input. However, at least three peaks are clearly observed from these FRFs therefore implies three possible natural frequencies for the structure. A MIMO system identifi-cation procedure has been implemented and the following observations are pertinent to the find-33 300 30.0 Log 3.00 300m 30.0m 22F/Level A 2.60 0.00 500r 0.00 500m 300 1.00 1.50 Hertz 2.50 3.00 1.00 1.50 Hertz 2.00 2.50 3.00 49F/Level A 1.50 Hertz 2,S"0 3.00 Figure 4.6 FRFs from the Northridge earthquake (Note: m stands for milli) ings of this study. 1. The acceleration records recorded during both the Sierra Madre earthquake and the Northridge earthquake have been given in Fig. 4.2 and 4.3. The torsional acceleration record is then obtained by subtracting the centre location signal from the edge location signal on the same level (e.g. channel 4 - channel 3). The comparisons of the torsional acceleration records and their power spectra between the " A " level and the " E " level (see Fig 4.7) show that the magnitude of the " E " level torsional record is much larger than the magnitude of the " A " lev-el torsional record. This observation goes against the usual dynamic assumption of a rigid body basement for ordinary building design (Schueller, 1976). However for high-rise build-34 30 E 2 20 o a) Q. ?' I 10 0 dotted: level " E " solid: level " A " ,1 s •i; V Mr 1 Hz 3 Hz 5 Hz 7 Hz 9 Hz 11 Hz 13 Hz 15 Hz Frequency Figure 4.7 Comparison of torsional records' power spectra ings the rigid body assumption may not hold because of the complexity of the basement. For the FWT this discrepancy could be caused by: 1) the isolated column foundation which made the slab at the " E " level be more flexible than that at the " A " level therefore the signals are much more prominent at the " E " level and 2) the long concrete wall (shown in Fig 3.4) sup-porting the parking ramp which may significantly reduce the torsional response. Furthermore, the power spectra show high energy level at the " E " level between 6Hz to 7Hz and between 8.9Hz to 9.2Hz. These high frequencies do not contribute significantly to high-rise building analyses, which require low frequencies (long periods). Hence the modal anal-ysis will be based on the records from the " A " level as the input record. This in effect isolates the superstructure as the system to be identified and excludes the effect of any soil-structure interaction. The translational acceleration records do not show such behavior as of the tor-35 sional records. 2. The building periods derived from the ME'scope program are shown in Table 4.2. For refer-ence, the periods used for design of the building (Banavalkar, 1992) are also included in the table. The last column in the table shows the percentage change of derived periods between the Sierra Madre earthquake to the Northridge earthquake. The maximum change is less than 13 percent. While the natural periods in the E-W direction between two earthquakes remain almost constant (the -1.7% change for N-S(l) is negligible), the natural periods in the N-S direction during the Northridge earthquake are longer than those during the Sierra Madre earthquake. There are several possible reasons for the observed tendency such as an increase of building mass and/or a decrease of building stiffness. Previous studies (Safak and Celebi, 1991) also showed that larger excitation tends to lengthen the periods of the structure. In this case, the Northridge earthquake has a relatively larger amplitude of excitation than the Sierra Madre earthquake. The first two mode shapes in each direction of the building from the Sierra Madre earthquake and the Northridge earthquake are shown in Fig. 4.8 and Fig. 4.9. 3. The natural periods of this high-rise building in both translational directions are around 6 sec-onds. Therefore a low frequency of 0.171Hz(5.85sec) must be detectable in the FRFs. In or-der to use the FRFs to identify the fundamental natural frequency, the frequency step for the 36 E-W elevation N-S elevation Torsion (top view) Figure 4.8 Mode shapes of the building from the Sierra Madre earthquake computed FRFs must be less than 0.015Hz. However signal processing theory states that the frequency step should be the inverse of the total time of the time domain signal. To reach a frequency step of 0.0125Hz, a minimum span of 80 seconds must be considered for the time-history records. 4. The trend of damping ratios in both earthquakes is that the lowest frequency has the highest damping ratio, which can be explained by that the first mode of each direction bears the ma-jor part of energy dissipation. Damping ratios are shown in Table 4.3. 37 E-W elevation N-S elevation Torsion (top view) Figure 4.9 Mode shapes of the building from the Northridge earthquake Signal processing theory using Fourier Transform depends on the assumption of linear sys-tem response, which has traditional viscous damping. However the actual damping mecha-nism is far more complicated and the viscous damping has a serious deficiency in that the energy loss per cycle at a fixed displacement amplitude is dependent upon the response fre-quency. Experiments on structural metals indicate that the energy dissipated internally in cy-clic straining of the material is essentially independent of the cyclic frequency (Chopra, 1995). 38 Table 4.2: Periods(sec) of the first three modes in each direction Direction Design value Sierra Madre Northridge Percent change E-W(l) 6.73 5.47 6.06 +10.8 N-S(l) 6.59 5.85 5.75 -1.7 Torsional(l) 6.36 4.55 4.82 +5.9 E-W(2) 2.08 1.71 1.86 +8.8 N-S(2) 2.11 1.74 1.74 0 Torsional(2) N . A . 1.62 1.82 +12.4 E-W(3) 1.13 0.92 0.96 +4.3 N-S(3) 1.18 0.91 0.91 0 Torsional(3) N . A . 0.94 1.04 +10.6 Table 4.3: Damping ratios of the first three modes in each direction Direction Sierra Madre Northridge E-W(l) 6.7% 1.6% N-S(l) 4.4% 5.5% E-W(2) 0.01% 2.0% N-S(2) 3.2% 0.5% E-W(3) 0.14% 1.2% N-S(3) 1.9% 1.2% When the records of Northridge earthquake are used to compute the FRFs, discrepancies be-tween the imaginary part and the magnitude of the FRFs are observed. This implies the ex-istence of complex damping. Therefore the damping ratios derived here can only be treated as approximation of damping effects evaluated as equivalent viscous damping. The negative damping ratios derived from torsional FRFs are not within the normal range (viscous damp-ing is always positive in order to dissipate energy) therefore have been eliminated. 5. The FRFs can also be expressed in terms of modal pole locations and mode shapes: modes H(co) = £ k= 1 •A(k) r{u(k)Hu(k ) }T 1 , A * ( k ) r { u * ( k ) H u n k ) i T r J t t -P (k ) jco-p*(k) (4.4) A(k) in the equation is the scaling constant for mode k and {u(k)} is the "kth" modal vector. A(k) can be used to resemble the modal participation factor times the dynamic amplification factor. It accounts for both the structural system characteristics and the excitation informa-tion. For each earthquake the frequency components are unique therefore the dynamic am-plification factor is unique. Scaling constants derived by this MIMO model are shown in Table 4.4. Table 4.4: Scaling Constants of Residue Matrix Mode Sierra Madre Northridge E-W N-S TORSION E-W N-S TORSION First 2.9 6.8 0.018 0.5 6.2 1.1 Second 6.5 8.3 0.016 5.0 2.8 2.4 Third 0.5 6.2 0.025 3.0 2.7 0.13 The magnitudes are compared in each direction. During both earthquakes the second modes in the E-W direction have larger magnitude than that of the other modes, which implies big-ger contribution to the building response. 40 4.4 Conditioning of Raw Records During the system identification process the coherence functions between any two of the input record signals are observed to be close to 1. (A coherence function of unity between pairs of in-put records implies complete linear dependence.) Hence the records from the two translational directions are highly correlated. This correlation might affect the results of the modal analysis. To reduce the correlation, computer program DADiSP (DADiSP 4.1, 1997) was used to imple-ment the conditioning algorithm developed by Bendat and Piersol (1993) to obtain conditioned records from the original ones. Details of this conditioning algorithm are presented in Chapter 10 of their book "Engineering Applications of Correlation and Spectral Analysis To obtain conditioned records, the equivalent of least-squares prediction operations are used to remove the linear effects of one or more input records from the data. In physical terms, system identification using conditioned records is equivalent to determining the dependence between one of the inputs and the output when the other input is turned off, assuming all correlated effects between the two inputs originated from the other input are turned off. The coherence functions are reduced significantly. Comparison of the coherence between the original and the conditioned input records is shown in Fig. 4.10. The dotted line is for the coher-ence function of the original records, while the solid line is the coherence function of the condi-tioned records. It is clear that the conditioning process reduces the dependence of two input records significantly. Frequency domain system identification theory require that the input records are not correlated. 41 dotted: before condition solid: after condition 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7Hz 8Hz 9Hz 10Hz Frequency Figure 4.10 Comparison of the coherence functions between recorded motions for " A " level during the Northridge earthquake The conditioning process performed above will help to justify the use of the frequency domain system identification theory. However, since the magnitude of the input record signals are also reduced after the conditioning, these records are difficult to be utilized on the analyses of modal dampings and mode shapes. The conditioned records are shown in Fig. 4.11. After removing the correlations from the second and the third acceleration records, it is clear that the general mag-nitude is reduced to less than half of the original records. This may cause difficulty when deriv-ing modal damping and mode shapes. Conditioned records are also analyzed for the modal frequencies and damping ratios. Results are shown in Table 4.5. There are small variations between the modal frequencies derived from the 42 0.00 10.0 20.0 30.0 40.0 Seconds 50.0 60.0 70.0 80.0 100 40.0--20.0 -S0.0 -14& Torsional o.oo loci Sao 3b~7o 40~7J srlo ioTo 7bTJ soTJ" Seconds Figure 4.11 Conditioned acceleration (cm/sec*sec) records from the Northridge earthquake Table 4.5: Periods(sec) and dampings(%) from the conditioned records Direction Periods Damping Ratios Sierra Madre Northridge Sierra Madre Northridge E-W(l) 5.62 5.94 0.3 -4.5 N-S(l) 5.88 5.79 4.5 -3.6 Torsional(l) 4.55 4.40 -3.4 12.5 E-W(2) . 1.72 1.81 0.3 -1.7 N-S(2) 1.69 - 1.87 2.1 -0.5 Torsional(2) 1.58 1.74 -0.5 -0.8 E-W(3) 0.92 1.00 3.0 0.8 N-S(3) 0.90 0.96 3.5 -4.7 Torsional(3) 0.97 0.97 -0.1 1.3 43 original records and those from the conditioned records. These variations are acceptable in this study. However most of the damping ratios from the conditioned records are negative. Therefore improvements on the method of conditioning shall be investigated in the future. 4.5 Conclusions The study of the dynamic behavior of the Figueroa at Wilshire Tower during the two earthquakes yields valuable information. Using signal processing techniques, the periods, damping ratios and mode shapes of the building were found. Values are close to the analytical results obtained by the structural designer. The tendency of frequency variation between two earthquakes has been observed. This variation will be considered in later linear time history analysis of the building. The damping ratio has long been an argument to the theoretical analysis of real structures. From this study, it is shown that damping ratio can be identified from the real earthquake records. The modal participation factor and the dynamic amplification factor are very important in decid-ing the dynamic response of the structure for certain earthquake excitations. The magnitude of the Residue function is shown to represent the modal participation factor times the dynamic am-plification factor in this study. During both earthquakes the second modes in the E-W direction have larger magnitude than that of the other modes, which implies a bigger contribution of the second mode to the building response. Since the dynamic amplification factor heavily depends on the excitation, the lack of certain frequency components in the earthquake excitation may cause the related mode shape to disappear in the building response. In this study no lack of fre-44 quency components is observed. Conditioning of original records was performed to remove correlation between input excitations. Coherence functions between any two input excitations are significantly reduced. These condi-tioned records are the signals that are more appropriate to be used to compute FRFs. However the magnitudes of input excitations were also reduced, causing problems in deriving mode shapes and modal damping ratios. Improvement of this conditioning technique needs to be in-vestigated. The estimated parameters will be used to assess the reliability of parameters in computer models for the building studied. Details will be presented in Chapter 6. 45 Chapter 5 DESCRIPTION OF THE NONLINEAR ANALYSIS PROGRAM 5.1 Introduction C A N N Y - E (Li, 1996) is a general purpose computer program for 3-dimensional nonlinear static and dynamic analyses of reinforced concrete and steel building structures. The program is based on a lumped plasticity model and has the capacity for analyzing large structures. Some general features of the program are summarized in the following paragraphs. The structure is idealized as a number of nodes, or joints, connected by a number of deformable elements. The program has both rigid and flexible floor modeling capabilities. Geometric non-linearities, i.e. large deformation, are not included. Although the analysis is limited to small de-formations, P-delta effects can be considered. In the analysis every member in the structure is treated as a massless straight linear element. The mass of structure is lumped at structural joints or concentrated at the center of gravity of every floor level i f a rigid floor slab is assumed. Initial conditions can be defined before the start of static and dynamic analyses. The nonlinear dynamic analysis is conducted step by step in a specified small time interval. The time interval is selected considering the characteristics of the input motion and taking into con-sideration the following: (1) the nonlinear relations between forces and resultant displacements; (2) the compatibility relation of displacements at each structural joint; (3) the equilibrium of forces at each structural joint; and (4) differential relation among time response functions. The current version (CANNY-E , 1996) of the program used for this study does not iterate during time 46 steps to account for the stiffness changes, but the unbalanced force at the end of each time step is calculated and added to the next time step. The equation of motion includes all degrees of freedom with or without mass of the structure. The program does not implement any static condensation technique. The skyline matrix tech-nique is used in the program to assemble and memorize the structural matrices. Matrix decom-position due to the change of stiffnesses is carried out by the Square-root-free Choleski method. An internal optimum renumbering system was developed and utilized in the program to mini-mize the total size of the skyline matrix. Such approach reduces the required size of computer core memory and speeds up the computation. The incremental equation of motion is expressed as: [M]{AX} + [C]{AX} + [K]{AX} = {AFe} - {Fu} + {AFt} (5.1) where, [M] is the diagonal mass matrix of the system, [C] = a k [K] + a m [ M ] + a 0 [ K 0 ] is a Rayleigh-type viscous damping matrix, {AX} is the vector of displacement increments of the structure relative to the base, {AX} is the increment of the velocity vector, {AX} is the increment of acceleration vector, [K] is the instantaneous stiffness matrix, [ K 0 ] is the initial stiffness matrix, {AF e } is the increment of earthquake inertia force vector, {F u } is the unbalanced force vector at the beginning of the present time interval, {AF t } is the incremental time-varying external load vector. 47 Two numerical procedures, the Newmark's Beta-Method and the Wilson's Theta-Method are employed and available in the program to solve the equations of motion. Linear variation of ac-celeration is assumed in the numeric integration of the equations of motion during a specified time interval. Newmark's Beta-Method was used in this study because of its stability on conver-gence. 5.2 Elements The elements used in the program to represent the structural members are beams, columns, shear panels, trusses, cables and constraint spring elements. A beam element is limited to uniaxial bending with optional shear and axial deformation. The inelastic flexural deformation of the beam element is assumed to be concentrated at its ends, and represented by the rotation of two nonlinear bending springs. The shear and axial deformations of beam are approximated by independent shear and axial springs placed at midspan. The beam element does not include the interactions among the bending, shear and axial forces. The beam axial deformation option can not be included if it is in a rigid floor slab. Columns are idealized by one of the three models described here: a one-component model for uniaxial bending, a biaxial bending model, or a multi-spring model. Interaction between the axial load and bending moment can be taken into account by only the multi-spring model. Shear and torsional deformation of the column element can be included in the analysis. 48 One of the important features of this program is its multispring column model (Fig. 5.1). It 3D ELEMENT MODEL 3D COLUMN MODEL A, B Multiple axial spring Figure 5.1 3D multispring column model (after L i , 1996) 49 developed based on a modification of the model originally proposed by Lai et al.(1984) to sim-ulate the flexural behavior of reinforced concrete columns under varying axial load and bi-direc-tional lateral load reversal. The column was idealized to be a linear element with its length equal to the column clear height and two multi-spring elements with zero length at the base and the top. The spring properties were determined based on the material properties and section ge-ometries. The number of steel and concrete springs were selected according to the section shape and location of reinforcing bars. A link element connects two nodes and subjects them to tension and compression with no bend-ing. The link element can be defined in a three-dimensional space or a two-dimensional plane. The cable element has a start node and a terminal node. Between the start node and the terminal node, middle nodes can be specified to change its direction in space. The cable has stiffness in tension only. The constraint spring element is a simple one-component element. It is used to change any one of the displacement components at a node. It can be used to analyze a substructure or partial frames without changing the properties of structural members. 5.3 C A N N Y Hysteresis Models The program includes a number of hysteresis models expressing nonlinear force-displacement relationships. Some are used for one-component models to simulate the inelastic behavior of uniaxial bending, shear and axial deformation, and some used for multi-spring model to repre-sent the behavior of biaxial-bending and axial force interaction. 50 The uniaxial hysteresis models included in the program are summarized in Table 5.1. A detailed description of each hysteresis model is presented in the C A N N Y - E users' manual. 5.4 More Information about the Program Program C A N N Y - E is comprised of three programs running in DOS interface: " P R E C A N N Y " , " C A N N Y " , and " P S C A N N Y " . P R E C A N N Y is a pre-processing program for reading the input data file and preparing it for the main program. It reads the free format text data file and performs memory allocation, automatic renumbering, and initialization of element and structural matrices. It generates a binary data file, called " C A N N Y data file". C A N N Y is the main program where all the numerical computations are carried out. It stores the calculated results into a binary file, called "Binary result file". P S C A N N Y is a post-processing program which reads the calculated results from the binary re-sult file and transforms them into text-format file. A n additional graphical interface and animation module for the program called V C A N N Y is used to check the contents of the C A N N Y data file and show peak values of forces for each el-ement. The program uses the computer core memory and part of the hard disk as virtual memory if re-quired to store the stiffness matrices and conduct the analysis. It does not have any limitation 51 Table 5.1: Uniaxial hysteresis model options and number of parameters Hysteresis Number (HN) Hysteresis Model Bilinear Trilinear Tension/compression only Stiffness degradation Strength deterioration ex, • i-H m Softening Number of Parameters 1 Linear elastic model X X 0 X X X X 0 IT Tension only elastic model X X 0 X X X X 0 1C Compression only elastic mode X X 0 X X X X 0 2 degrading bilinear model 0 X X 0 X X X 1 3 Modified Clough model 0 X X 0 X X X 1 4 Degrading bi/tri-linear model 0 0 X 0 X X X 0 5 Cross-peak bilinear model 0 X X 0 X X X 2 6 Cross-peak bi/tri-linear model 0 0 X 0 X X X 2 7 Cross-peak pinching model 0 0 X 0 X 0 X 3 8 Origin-oriented model 0 0 0 0 X X X 0 9 Peak-oriented model 0 0 X 0 X X X 0 10 Bilinear slip model 0 X X X X 0 X 2 11 Bi/tri-linear slip model 0 0 X X X 0 X 2 12 Bilinear elastic model 0 X 0 X X X X 0 13 Bi/tri-linear elastic model 0 0 0 X X X X 0 14 C A N N Y simple model 0 0 0 0 0 X X 2 15 C A N N Y sophisticated model 0 0 0 0 0 0 0 7 16 Maeda bi/tri-linear model 0 0 X 0 X X X 2 18 Axial stiffness model 1 0 1 19 Axial stiffness model 2 1 0, functional x, not functional —, not applicable 52 regarding to the size of the structure, provided that sufficient core memory in hard drive space is available. The program is very efficient regarding to the computational time for an analysis. A time history analysis of the building studied subjected to 180 seconds (18,000 time steps) of shaking takes about 3 hours on a Pentium 266 PC. The C A N N Y program has been applied in a number of structural analyses of actural buildings and research studies. It has been used in the three-dimensional nonlinear dynamic seismic anal-yses of a seven story reinforced concrete building (Rahmatian 1998) in Los Angeles. A computer model was set up by Khoshnevissan (Khoshnevissan 1998) using C A N N Y for the analysis of a bent at Oak Street bridge in Vancouver. Sewell (Sewell 1999) used a C A N N Y model to study the behavior of reinforced concrete shear wall. C A N N Y program has been used to analyze single and multi-story steel plate shear walls under simulated seismic loading by Rezai (Rezai 1999). Currently a low-rise steel frame buildings in Burbank, California is being investigated by Ba-khtavar (1999) using C A N N Y . Several new commercial buildings in Japan have been analyzed and designed using C A N N Y (Li, 1999). 53 Chapter 6 THREE-DIMENSIONAL LINEAR ANALYSIS OF THE FIGUEROA AT WILSHIRE TOWER 6.1 Introduction and Theory The primary objective of performing a three-dimensional linear analysis is to understand the be-havior of the FWT during the Sierra Madre and Northridge earthquakes, where the building is believed to have responded as a linear elastic system. The results from this analysis also help to determine the extent of contribution of the different structural components as well as the non-structural components to the overall response of the building. The linear model and the related analytical studies are used to form a basis for the development of the nonlinear model. The mode superposition combined with the response spectrum method is one of the most popular techniques for the elastic analysis of structures. However, it is not directly applicable to inelastic analysis because the principle of superposition is no longer applicable. Direct integration time history analysis overcomes this drawback. This method consists of a step-by-step direct integra-tion in which the time domain is discretized into a number of small time increments 5t, and for each time interval the equations of motion are solved with the displacements and velocities of the previous step serving as initial functions. Direct integration time history analysis is not normally employed as an analysis tool in practical design of high-rise buildings because it requires large computational effort. However, recent en-hancements in computational techniques and the development of more sophisticated models al-54 low popular application of time history analyses, based on components' behavior, which are capable of determining the behavior of three-dimensional structures in a more realistic manner. 6.2 Development of a linear model Computer program C A N N Y is used for the three-dimensional linear analysis of the FWT. The program is chosen because of its ability to perform both linear and non-linear time-history anal-yses. The building is modeled as a combination of braced frame and moment frame consisting of 14 main column lines. The wireframe model of the building is shown in Fig. 6.1. Z X 1-53F B896 C618 L472 BD V I E W l i Figure 6.1 C A N N Y model of the FWT The in-plane stiffness of the floor system was considered to be high (the thickness of the com-posite slabs at each floor is 50mm), and therefore, the floor diaphragms were assumed to be rigid. Under this constraint, each diaphragm can be shown to consist of three in-plane degrees of free-55 dom, one rotational and two translational. The building was assumed to be fixed at the ground level. This assumption is based on the infor-mation obtained from the analysis of the recorded motions in Chapter 4. A l l the beam and col-umn centre lines were considered to coincide with column line coordinates. The masses of the beam and column members associated with each floor were calculated from the specified geo-metric dimensions of elements and combined with the mass of slab to determine the total story mass at corresponding floor. The estimated final mass and mass moment of inertia were lumped at the centre of mass at each floor. The estimated weight of the structure at each story level is shown in Table 6.1. The total mass of the building was estimated to be about 50927 tonnes. This information was provided by the structural designer (Banavalkar 1991). Beams, columns and bracing members were modeled in C A N N Y by beam element, column el-ement and link element. In beam and column elements, properties of flexural rotation in both ends, shear deformation and axial deformation were numbered. These numbered properties were defined later by different hysteresis models. Properties of built-up section were calculated ac-cording to steel design handbook (handbook, 1985). For elastic analysis, all the hysteresis mod-els chosen are simple linear models. For link element, only property of axial deformation need to be defined. Contribution of panel zone of beam column connection was accounted by the rigid zone length at the element start and end node. To evaluate the contribution of gravity load frames, an additional model was constructed taking into account the perimeter gravity load columns. Comparison of periods (periods derived by the 56 Table 6.1: Estimated weight and mass polar moment of inertia Floor Weight (kN) Rotational Inertia (kN • m 2 ) 2nd 16808 8362769 3rd 11938 4774000 4th 10894 4218800 5th-6th 10471 3994320 7th-9th 10204 3801671 10th-13th 10007 3733875 14th-17th 9808 3665661 18th-19th 9710 3631765 20th 10646 3690051 21st-30th 9599 3570164 31st 10354 3568513 32nd-33rd 9657 3517663 34th-35th 9486 3447384 36th 10369 3749583 37th-42nd 9078 3104672 43rd 8505 2693336 44th 8433 2657367 45th 8589 2670596 46th 8348 2657780 47th-48th 7703 2204690 49th 8316 2504407 50th 9550 2907479 51st 10845 2200801 52nd 8050 1391108 53rd/Roof 3055 234813 57 two models are within 5 percent difference), mode shapes and responses showed that effect of accounting gravity load frames is negligible on changing the dynamic characteristics of the com-puter model. Therefore the computer model shown in Fig. 6.1 does not include gravity load frames. 6.3 Calibration of the linear model and period sensitivity. The live load contributions to the dynamic mass used to calibrate the models to study the re-sponse of the building during the Sierra Madre and the Northridge earthquakes are shown in Ta-ble 6.2. The corresponding computed fundamental periods are also included for reference. Table 6.2: Live loads for calibration of the models Model Description Fundamental Periods(sec) E-W/ N-S/ Torsional Sierra Madre Self Weight + 50% Live Load 5.74/5.38/4.63 Northridge Self Weight + 80% Live Load 6.00/5.61/4.75 The periods obtained from the C A N N Y model are compared with those from the analysis of the recorded earthquake response in Table 6.3. The computer model yields periods that are in good agreement (within 10%) with the observed building periods from the Northridge earthquake. Views of mode shapes for the first six modes of the Northridge model are presented in Fig. 6.2. The first mode shapes in both X and Y direction show the shape of shear type structure. The second mode shapes in both X and Y direction are typical and have an inflection point at about 2/3 height of the structure. The second torsional mode shape shows a large translational curve below the mid-height of the structure. 58 XI YI Tl X2 Y2 T2 Figure 6.2 Modes shapes for the FWT in three directions Table 6.3: Periods(sec) of the first three modes in each direction Direction C A N N Y model (SM) Sierra Madre C A N N Y model (NR) Northridge E-W(l) 5.74 5.85 6.00 6.06 N-S(l) 5.38 5.47 5.61 5.75 Torsional(l) 4.63 4.55 4.75 4.82 E-W(2) 1.65 1.71 1.85 1.86 N-S(2) 1.67 1.74 1.77 1.74 Torsional(2) 1.60 1.62 1.71. 1.82 E-W(3) 0.93 0.92 0.97 0.96 N-S(3) 0.91 0.91 0.97 0.91 Torsional(3) 0.92 0.94 1.04 1.04 Damping ratios (Rayleigh damping) are critical to the analytical response of the building. For low-rise building analysis, 5 percent critical damping is usually applied to the first mode and the 59 second mode. If the same 5 percent damping ratios for the first mode and the second mode are applied to this high-rise building, the analytical response from the computer model is significant-ly different from the recorded response. Comparison of the responses using the Northridge records in the E-W direction on the 49th story is presented in Fig. 6.3. Dotted line is the recorded acceleration time history, while solid line is the computed acceleration time history. Because of the high damping ratios for the first mode and the second mode, response caused by the higher modes was significantly reduced and the amplitude of the response was significantly lower than the recorded response. Maximum response of the computer model was only about 1/5 of the maximum recorded response. After careful comparison of the analytical responses due to different damping ratios, 5 percent critical damping for mode 1 and 2 percent critical damping for mode 9 were used for the com-puter model. Comparison of the analytical response and recorded response is shown in Fig.6.4. The recorded motion used is the same as in the previous comparison. The analytical response can capture high frequencies of the recorded response very well. The maximum amplitude of the an-alytical response is very close to the recorded maximum amplitude. Damping ratios for mode 1 to mode 12 are thus derived and summarized in Table 6.4. Table 6.4: Damping ratios used for computer analyses Mode 1 2 3 4 5 6 7 8 9 10 11 12 Period (sec) 6.00 5.61 4.75 1.85 1.77 1.71 1.04 0.98 0.97 0.74 0.68 0.66 Damping Ratio 0.05 0.047 0.041 0.021 0.021 0.021 0.020 0.020 0.020 0.022 0.023 0.023 60 dotted: recorded solid: computed -1.5 ' ' ' ' ' ' 5 Figure 6.3 Comparison of the responses using conventional damping ratios 6.4 Comparison of predicted and recorded time histories. The three-dimensional responses of the building during the Sierra Madre earthquake and the Northridge earthquake were computed analytically. The purpose of such an analysis was to cor-relate the recorded responses with the analytical results, and to assess the effects of various mem-bers on the overall three-dimensional behavior of the structure for small to medium levels of excitation. The response of the building was computed for the 60 seconds of the Sierra Madre record and the 180 seconds of the Northridge record. During both analyses two translational and one for tor-sional ground excitations were used as inputs. 61 dotted: recorded solid: computed 1.5 r ; ; -1.5 1 ' 1 ' ' ' 1 5 Figure 6.4 Comparison of the responses after damping calibration 6.4.1 Comparison of predicted and recorded time histories during the Sierra Madre earth-quake Figure. 6.5 shows the recorded and computed absolute acceleration and relative displacement at the 49th floor during the Sierra Madre earthquake. Acceleration were chosen at the 49th floor instead of the roof because the top three floors consist only the core of the building. A good match between the two responses implies that the model is able to predict the frequency content of the actual response of the structure. 62 oes,oes/ui oes,oes/ui oes.oes/pej 63 6.4.2 Comparison of predicted and recorded time histories during the Northridge earth-quake Figure. 6.6 and Figure. 6.7 show the recorded and predicted absolute accelerations and relative displacements at the 49th floor during the Northridge earthquake. The same as that for the Sierra Madre earthquake, acceleration were chosen at the 49th floor. The correlation between predicted and recorded accelerations are better than that for displace-ments. This difference may be due to the way in which displacements are inferred from recorded accelerations. The integration routine may not produce the same results as those from the direct time history analyses. Figure. 6.8 and Figure. 6.9 show the first 35 seconds (for comparison clarity) of recorded and predicted absolute accelerations and relative displacements at the 14th floor during the Northridge earthquake. This second comparison of motions also provides good match between recorded time histories and computed time histories. A good match between the two responses implies that the model is able to reproduce the frequen-cy content of the actual response of the structure. And since the time history responses of this computer model fit well with the responses from two major earthquakes that the building expe-rienced, the computer model can be treated as a reliable representation of the structure system of the building during typical earthquake excitations. This observation sets the base of next step of the analysis. 64 oes,oes/ui MS.MS/UJ oes,oes/pej 65 E-W -0.15 I 1 ! 1 I I I I I I 0 20 40 60 80 100 120 140 160 180 Time, sec N-S E 8 ss CL w b 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 •XI— \lvtj\N AAA) m v v ' —V—V— v - — V — M — V -I 0 20 40 60 80 100 120 140 160 180 Time, sec Figure 6.7 Comparison of recorded and computed roof relative displacement during the Northridge earthquake 6.5 Comparison of code design values In order to achieve a better understanding of the code design values in comparison with the dy-namic analysis values, two code design procedures are followed to derive the elastic base shear for the building. The building code used in this building's structural design is the City of Los Angeles Building Code, 1985 edition. The structural design also complied with the 1987 SEAOC recommended force. These codes conform to Uniform Building Code(UBC) 82 section 91.2312. Earthquake Regulations. These design regulations were used to evaluate the elastic static seismic force. 66 Tima_sec „ 1 -2 0 Wft^MH 15 25 35 Tim«_»ec Figure 6.8 Comparison of recorded and computed 14th floor absolute acceleration during the Northridge earthquake According to UBC82, minimum earthquake forces for structures should be calculated as: V - Z I K C S W (6.1) Z is the numerical coefficient dependent upon the zone. I is the occupancy importance factor. K is the numerical coefficient for structural type. C is the numerical coefficient derived from the natural period. S is the numerical coefficient for site-structure resonance. W is the total dead load of the structure. 67 -0.10 15 25 35 Ting i k Figure 6.9 Comparison of recorded and computed 14th floor relative displacement during the Northridge earthquake For Los Angels located in Zone No. 4, numerical coefficient Z = 1. Occupancy Importance Fac-tor I for this building was chosen as 1. For a dual bracing system consisting of a ductile moment-resisting space frame and braced frames, the horizontal force factor is K = 0.80. Since the natural period of the building is about 5.65 seconds, the numerical coefficient C is C = — - — = 0.0281. 15 • JT 68 From the general notes on the site plans, the type of soil under this building is very stiff shale or sandstone with bearing value of 15000 lbs/sq.ft. In such case numerical coefficient for site-struc-ture resonance S was chosen as 1. The total weight of the building was estimated to be 50927 tons. Therefore the total seismic force for the building is: V = 1 x 1 x 0.80 x 0.0281 x 1 x 50927 x 9.8 = 11198kN (6-2) NBCC1990 as a reference was also used to evaluate the seismic lateral force in order to compare design guidelines. In NBCC1990, the equivalent lateral seismic force representing elastic re-sponse shall be calculated as: V e = v S - 1 - . F - W (6.3) Since the zonal velocity ratio v for Los Angels is not available, v = 0.3 (for Victoria) is used. The seismic response factor S is S = — = ^ = 0.63 . JT V5T65 The seismic importance factor I was chosen as 1. The foundation factor F was also chosen as 1. Therefore the elastic lateral seismic force V c = 0.3 x 0.63 x 1 x 1 x 50927 x 9.8 - 94328 kN. For ductile braced frame force modification factor R should be chosen as 3.0. Hence the mini-V mum lateral seismic force is V = — • U = l^2J± . n.6 - 18866 kN. R 3.0 The two design codes are from different countries and different period of time. Design base shear from N B C C is shown to be higher than U B C (in about 40% reduction). However, it should be 69 noticed that the N B C C used an empirical load reduction factor (R) of 3 to account for the duc-tility of the steel braced frame. The base shear derived is not the maximum base shear that the building can resist during earthquakes. 6.6 Summary In this chapter it has been shown that a linear model and linear analysis have reasonably captured the recorded seismic response of the FWT building during past small to medium level earth-quakes. The building was modeled as a combination of braced frame and moment frame consist-ing of 14 main column lines. After live load and damping ratio calibration of the model, the periods derived from the computer model were shown to be in good agreement (within 10%) with the observed building periods from the Northridge earthquake. The response of the building was then computed for the 60 seconds of the Sierra Madre record and the 180 seconds of the Northridge record. During both analyses two translational and one for torsional ground excitations were used as inputs. A good match between the computed responses and the recorded responses implied that the model was able to reproduce the frequency content of the actual response of the structure. Therefore, the computer model can be treated as a reliable representation of the structure system of the building during typical earthquake excitations In doing this analytical study confidence is gained toward the computer model based on recorded seismic responses. Hence the linear model and the related analytical studies will be used to form a basis for the development of the nonlinear model. 70 Chapter 7 THREE-DIMENSIONAL NONLINEAR ANALYSIS OF THE FIGUEROA AT WILSHIRE TOWER 7.1 Introduction This chapter describes the nonlinear analyses of the Figueroa at Wilshire Tower (FWT). Planar pushover and cyclic analysis of the three-dimensional model of the building were performed first. Then, three-dimensional nonlinear time history analyses of the building were conducted to simulate the building response during three earthquake excitations. Finally, comparison between the prediction from pushover analysis and responses from time history analyses was used to modify the traditional method of pushover analysis in order to obtain better prediction from pushover analysis for the ground motions considered. The purpose of performing nonlinear analyses is to determine the response and behavior of the building during severe shaking. Therefore it is extremely important to choose the variables de-fining the inelastic properties of independent elements. In-depth research on previous studies and national/local guidelines was carried out to verify the credibility of these variables. On the other hand, there is no intention of simulating any pre-defined behavior of the building by artificially changing the properties and variables of the structure model. The model is an as-built model and all the responses and conclusions discussed later are not manipulated by any form of number-picking. 71 7.2 Description of the Nonlinear model. A detailed finite element model is developed based on structural drawings of the building. It has the same basic features as the linear model. The building is modeled as a combination of braced frames and moment frames consisting of 14 main column lines. The model of the building has been shown in Fig. 6.1. Floors are modeled as rigid diaphragms with three degrees of freedom for each floor. The building is assumed to be fixed at the base. The estimated mass of each floor is lumped at the center of mass of the corresponding diaphragm (mass of columns is included in floor masses), and all members are assumed to be massless elements. Additional nonlinear model features are discussed below. Hysteresis behavior of each member is simulated by a hysteresis model expressing the nonlinear force-displacement relationship. A l l flexural members are modeled by a simple bilinear model, shown in Fig. 7.1 ( C A N N Y hysteresis model #14). A more sophisticated model is available in C A N N Y (model #15) but its use signif-icantly increases the computational effort. Also the study of general responses of tall buildings does not require the most sophisticated model as discussed in Chapter 2. Yielding strengths are calculated based on the expected steel yielding strength. From the general notes on the building drawings, all steel framing including columns are A S T M A-572 (grade 50). Nominal yielding strength for A-572 (grade 50) is 345MPa (50ksi). Expected steel yielding strength is assumed based on average values such as those cited in the Interim Guidelines (FEMA 1995) in accor-dance with the SAC reports: Fy = 380MPa (55ksi). The post-yielding stiffness factor is chosen as 0.02 in reference to the SAC reports. Beyond the 72 (a) Outside loop (b) Bi-linear rule for smaller loop Figure 7.1 C A N N Y hysteresis model #14 initial yield strength, strain hardening can increase member and connection strengths, although this increase will depend upon the member's resistance to local or lateral/torsional buckling or other modes of failure. The increase in strength due to strain hardening is generally limited to a small value that only causes a modest increase in member strength. 73 Strain hardening is sometimes specified to help maintain numerical convergence of nonlinear analyses, but in such cases it is important to keep the value to a minimum to avoid masking true (physical) instabilities. The strain hardening factor for the compression side of bracing elements was chosen as 0.001. The unloading stiffness degradation factor y was chosen as 0.3. The unloading stiffness was de-termined as following: where, d m a x > d y , d ' m a x < d ' y . The strength deterioration factor Xu was chosen as 0.75. The re-loading yielding strength was evaluated considering the attained ductility as following: where, u is the ductility. When A,u = 0.75 was chosen, there would be 50% strength remained at ductility u = 3. The axial force-deformation behavior of bracing members in compression are modeled as an elasto-plastic element with the yield force equal to the residual force (FEMA 274). However, an elastic analysis is necessary to determine the maximum axial force delivered to the column, the beam, and the beam-column connections. (7.1) (7.2) 74 The shear stiffness of beams and the axial stiffness of columns are assumed to be linear elastic and are modeled by linear elastic hysteresis model ( C A N N Y model #1). There is no panel zone element in computer program C A N N Y . Though it is feasible to use other software to analyze panel zone after using C A N N Y to compute global response of the building structure, this was not carried out during this study. First, the difficulty in defining panel zone behavior makes it hard to construct a realistic panel zone element. Dr. Krawinkler of Stanford university provided a report in SAC 95-04 with similar conclusion. Please refer to Literature Re-view for further information. Secondly, this research is focused on global response of the build-ing structure therefore the detail modeling of panel zone elements is not necessary (contribution of rigid end caused by the panel zone is accounted in the computer model by defining the length of the rigid end in both beam and column elements). Convergence was not a problem for this lumped plasticity model during time history analysis. The small time step (typically 0.01 or 0.005 second) of input acceleration records may be the reason. However, during pushover and cyclic analyses the program crashed frequently when a significant number of elements started to yield and the stiffness matrix changed significantly. Therefore, during certain critical pushover stages, a smaller time step of 0.001 second was used. 7.3 Time History Analysis Since this building remained in the elastic range during the past earthquakes, there is no real non-linear response records available. In order to observe the nonlinear response of the building in possible stronger earthquakes, three recorded earthquake motions from elsewhere were used to 75 compute the building responses. First, a set of records from the S M A C - A station during the 1964 Niigata (Japan) earthquake was used as input excitation. These records show a major energy input around the first mode period (4.5 to 6 seconds). The second set of recorded motions used as input excitation was the Joshua Tree Fire station records during the 1992 Landers Earthquake in California. These records have a duration of at least 40 seconds of strong shaking. This set was selected because it may be of practical interest to observe the responses of high-rise buildings during this type of long duration shaking. Last, the Sylmar County Hospital station record obtained during the 1994 Northridge earthquake was used as input excitation. This is a short duration, strong impulse, near-source ground motion that may be very damaging to tall buildings. Table 7.1 shows the peak ground accelerations of the three excitation records used in the time Table 7.1: Peak ground accelerations (PGA) of ground motions Motion Niigata(x) Niigata(y) Landers(x) Landers(y) Sylmar(x) Sylmar(y) PGA cm/sec* sec 129 171 268 278 578 319 PGA g 0.13 0.18 0.27 0.28 0.59 0.33 PGV (cm/sec) 46 61 27 28 78 42 PGD (cm) 33 44 8 8 16 9 Note: PGA is Peak Ground Acceleration, PGV is Peak Ground Velocity, and PGD is Peak Ground Displacement. history analyses. Figure 7.2, 7.3, 7.4 show the time history plots of the ground motions used and their response spectra in comparison with the U B C 94 (as reference) design spectrum. 76 77 78 m 3 CD •a 3 cr J3 cd <u <u 00 T3 M o I—H a, cn O O U S3 .S CD r < > •f s 1 ) ) < < 3 -+-» o CD cn CD cn S3 O a, cn CD VH C as O +J cn IS CD s •<* CD f-i 3 bo 79 7.3.1 Responses of the building during the selected earthquake excitations Results of the inelastic, time-history analyses are shown in Fig. 7.5, 7.6, 7.7 and 7.8 in terms of lateral displacement, interstory drift, story shear and overturning moment. In these plots x-direc-tion corresponds to the East-West direction of the building (see Fig.3.4) while y-direction corre-sponds to the North-South direction of the building. .0 -0.6 -0.2 0.2 0.6 1.0 Displacement_x Figure 7.5 Maximum inelastic responses of the FWT, displacement The plot of maximum lateral displacements shows the significant effect of the Niigata motion on the response of this building. Maximum displacement at roof level in the N-S direction reach 1.8 m. The maximum displacement plot also shows clearly the combination of at least two mode shapes in the responses of this building for the Landers and Northridge motions. 80 Figure 7.6 Maximum inelastic responses of the FWT, interstory drift The input motions selected produce different envelopes of interstory drift in each direction of the building as shown in Fig. 7.6. The largest drifts occur for the Niigata motion in the North-South direction (y-direction). The maximum value for the interstory drift is 1.5% and it occurs at the 23rd story. Since interstory drift ratio is a major factor on damage, damages of the building will be expected the most severe on the 23rd story in the case of the Niigata type of ground excita-tions. Except for the y-direction (N-S) of the Niigata response, the FWT generally behaves well. The.consistent shapes of the interstory drift eliminate any weak story mechanism for the build-ing. It can be observed that in contrast with the variation of the story displacements, with uniform 81 -50000 -30000 -10000 10000 30000 50000 Shear x -50000 -30000 -10000 10000 30000 50000 Shear_y Figure 7.7 Maximum inelastic response of the FWT, story shear shape the maximum story shear increases down from the roof story to the 49th story (similar to the expected distribution for low-rise buildings). For the next 20 stories below there is no signif-icant change of maximum story shear. This contradicts the code-defined invert-triangular lateral load distribution along the height of the building. It is also a clear sign of higher modes effects. The maximum story shears in the lower part of the building show the usual increase towards the ground floor. However, one still could notice that in some cases the tip of the curve is growing outward instead of growing inward. This can only be explained by a lateral load pattern which is nearly rectangular or trapezoidal. The overturning moment plots for the Landers and the Northridge motions have peculiar onion-like shapes. The floors between the 20 to 25 story experienced the largest overturning moment 82 Niigata Landers Northridge -3.0E+006 -1.0E+006 1.0E+006 3.0E+006 Moment x ::!=:;::z::z::=:;=:/i/: ' 1 -r:X:ljtz. I'J ij 'EEEEEEEEfyEEE i / |i \\":r EEEE. Niigata Landers Northridge EEWEEEE] M :=fr::! • \ \ \!\ ; m -~iY\\ 1 \ i / <!• : ; 7 [ii ' L : 11 : Fi::F : i | :FfEE / i ;i Wr • ;l ; / ; ; '.1 ' / ' l :i/i::::::::r::::::::..,:;::i,:;:^ '': 1/ i . i — i— i i l / )i M : :......., :V..|.,... : F A\F 52 50 48 46 44 42 40 38 36 34 32 £.30 Q 28 In 26 24 22 20 18 16 14 12 10 8 6 4 2 -6e+006 -3e+006 1.4e-009 3e+006 6e+006 Moment_y Figure 7.8 Maximum inelastic response of the FWT, overturning moment during these two earthquakes. The overturning moment for the Niigata motion in the x (East-West) direction show similar behavior, but in the y (North-South) direction the moment distri-bution is as generally expected. Roof displacement time history responses from the Sylmar County Hospital excitation are plot-ted in Fig. 7.9. The pulse effect of the Sylmar ground motion is apparent in the displacement time history show, where the maximum amplitude reached 0.66m on one cycle. A permanent dis-placement of more than 0.09m is also indicated. The displacement time history for the other two excitations does not appear to be any permanent displacement at the end of the time history even though many beam and bracing members have experienced large yielding. Comparing the time 83 E-W E S 0.5 <u | 0 ra g- -0-5 b -1 10 N-S 20 30 40 Time, sec 50 60 Figure 7.9 Roof displacement time history responses, Sylmar County Hospital history plot of input ground excitations, it infers to the strong impulse, short duration of the Syl-mar record as the major reason for residue displacement. Roof acceleration time history respons-es are plotted in Fig. 7.10. 7.4 Pushover Analysis In order to estimate the lateral resistance of the building at ultimate load, static nonlinear (push-over) analyses were conducted. The results of pushover analysis are sensitive to the choice of 84 E-W 20 30 40 50 Time, sec Figure 7.10 Roof acceleration time history responses, Sylmar Country Hospital load patterns, particularly for this case of high-rise building. A predetermined lateral force pat-tern from building code was applied incrementally in a step-wise manner in each direction inde-pendently. This lateral force pattern from building code assumes the lateral force distribution as an inverted-triangular pattern which is in accordance with the first mode shape of multi-degree-of-freedom building structures. Later on, other lateral force patterns used to perform pushover analyses will be presented to achieve better responses when comparing to the time history anal-yses. The building was analyzed based on applying the selected lateral force pattern until a specified 85 target roof displacement was reached. A target roof displacement of 4.5m (3 percent of the height of building) and an increment of 0.1m to 0.001m (depend on the condition of convergence) was selected for this study. The base shear versus roof displacement for pushover analysis of the building in the east-west direction is shown in Fig. 7.11. Significant yielding in the structure occurs at a roof deflection of •o 20000kN 1.0m 2.0m 3.0m roof_displacement 40000kN 1 20000kN 10000kN Figure 7.11 Base shear versus roof displacement, E-W direction approximately 1.5m. First yielding members are the outrigger beams between the 23rd to the 26th floors when the roof displacement reached 0.883m. The first group of yielding bracing members are those between the 22rd to the 26th floors when the roof displacement reached 1.13 m. Maximum base shear is 4.71 x 10 kN, while maximum overturning moment is 86 6.72 x 1 0 6 k N - m . Figure 7.12(b) shows the yielding propagation in the south perimeter frame in the east-west di-rection. Outrigger beams start yielding at around the 22nd floor and propagate upward. Bracing members start yielding at around 4th story and propagate upward. The north perimeter frame has similar steps. The same target displacement and increment were selected for the analysis in the north-south di-rection. The yielding propagation steps in the south perimeter frame in the north-south direction are similar with those in the east-west direction which shows that this building has similar strength in both lateral direction. After completing the analysis for the code-defined lateral load pattern, a second lateral pattern defined in FEMA-273 was adopted. This pattern, often termed the uniform pattern, shall be based on lateral forces that are proportional to the total mass at each floor level. FEMA-273 is the first guideline to introduce the uniform pattern as one of the necessary load patterns for push-over analysis. The same target displacement as that used for code-defined lateral pattern was selected for the uniform load pattern. Figure 7.12c shows the yielding propagation steps in the south perimeter frame in the north-south direction. Bracing members start yielding at around 15th story. The yielding sequence of the building when subjected to a near-field type time history motion, such as the Sylmar record, is presented in Figure 7.12a. It is clear that when comparing results from 87 88 triangular and uniform lateral load pattern analyses, a closer yielding sequence to that obtained from the Sylmar motion analysis is obtained by the uniform load pattern. A third lateral pattern was investigated in order to make the response from the pushover analysis better match the response from the time history analysis. This proposed load pattern has a trap-ezoidal shape. Long side of the trapezoid doubles the short side. The yielding propagation (brac-ing members start yielding at around 18th story) in Fig. 7.12d clearly show that the best comparison of yielding steps is between this load pattern and the time history response subjected to a near-field type input motion. Table 7.2 is a summary of the maximum base shear and overturning moments for the FWT based Table 7.2: Comparison of Results Event Base Shear (kN) Overturning Moment (value x 10 6 kN • m) V V / V d M M / M d Design 32716 1.0 2.92 1.0 Time History Analysis Northridge 14182 0.4 0.53 0.2 Time History Analysis Niigata 50961 1.6 6.43 2.2 Time History Analysis Joshua 39623 1.2 2.08 0.7 Time History Analysis Sylmar 68121 2.1 2.31 0.8 Pushover Analysis Triangular 14325 0.4 2.09 0.7 Pushover Analysis Uniform 73283 2.2 8.02 2.7 Pushover Analysis Trapezoidal 56043 1.7 6.80 2.3 Note: Vd = design base shear = 32716 kN, Md = design overturning moment = 2.92 x 10 kN*m. 89 on design values, four time history analyses and three pushover analyses. In the table it is clear that the Northridge (elastic response based on the FWT's recorded motion during the Northridge earthquake) event pushed the structure well within the design range. Base shear during this earth-quake is only 40 percent of the design value. Different types of earthquakes have produced great-ly different base shear and overturning moment demands, ranging from 70 percent to 220 percent of the design value. While the Sylmar excitation caused the largest base shear for the structure, the Niigata excitation demanded the highest overturning moment resistance. For pushover analysis, a trapezoidal lateral load pattern is a compromise between the traditional invert-triangular pattern and the F E M A defined uniform pattern. Uniform pattern provides the upper bound but may be too conservative. Its base shear and overturning moment demands are more than four times of those by invert-triangular pattern. When comparing with the time history results the proposed trapezoidal lateral load pattern remains in close range of both base shear and overturning moment demands for the types of ground motions considered. 7.5 Cyclic Analysis Cyclic analysis, a static nonlinear analysis under load reversal, was also conducted. The model was subjected to several load reversals, so that it would capture the dynamic characteristics of the member elements as well as the whole structure. A predetermined lateral load pattern was applied incrementally in a step-wise manner. In this analysis, a lateral load pattern of invert-tri-angle was assumed in each direction. 90 The analysis was performed based on displacement control. The first target displacement was 1 m at roof. Then the structure model was unloaded and pushed back to -1 m roof displacement. In this range the structure is within linear response. No hysteresis loop energy dissipation was observed during the load reversal. The second target displacement of 2 m pushed the structure into yielding. Displacement increases rapidly while base shear generally remained at the 45,000 kN level. After unloading the structure to its initial position, a -2 m roof displacement was used for the cyclic analysis. It is obvious that the stiffness of the structure has changed due to the el-ement unloading stiffness degradation. A third target displacement of 4 m roof displacement pushed the structure well into inelastic response. The building's base shear capacity has been reached. The total strength of the structure was seen deteriorated. Large area of hysteresis loop showed the advantage of steel structure in dissipating energy. The base shear versus roof displacement for the cyclic response of the building in the east-west direction is shown inFig. 7.13. The maximum base shear capacity for this building is shown to be about 47,000 kN at roof displacement of around 2 m. Cyclic analysis in the north-south direc-tion shows similar base shear / roof displacement curve. This observation is consistent with the uni-directional pushover analyses. 7.6 Conclusions Based on the design specifications, a three-dimensional nonlinear model of the building has been developed and analyzed using different analytical techniques. Time history analyses using three distinguish types of severe shakings recorded in past earthquakes were conducted to study the 91 Figure 7.13 Base shear versus roof displacement, cyclic test inelastic behavior of the building. Afterwards, the pushover analysis using traditional inverted-triangular lateral load pattern shows the possible sequence of damage greatly different from that predicted by the three-dimensional nonlinear time history analysis under the Sylmar excitation. An improved lateral load pattern is therefore derived and yields better fit with the dynamic re-sponses for the type of ground motion considered. By applying recent enhancements in computational techniques and the development of more so-phisticated models this chapter provides valuable information about the dynamic behavior of high-rise steel buildings. Envelop plots of lateral displacement, interstory drift, story shear and overturning moments present interesting features that have not been shown in previous research. 92 And the importance of choosing ground excitations for dynamic analyses is clearly demonstrated by the responses of the FWT under the three distinctive types of excitations considered in this study. 93 Chapter 8 SUMMARY AND RECOMMENDATIONS 8.1 Summary The goal of this thesis was to study and understand the seismic behavior of a steel high-rise building. The primary objectives of the study have been achieved. These include: 1) system iden-tification of the recorded earthquake motions, 2) investigation of the building behavior during past earthquakes through construction of a lumped plasticity computer model and calibration of the model using recorded motions, 3) evaluation of the inelastic structural response during se-vere shaking, and 4) evaluation of the inelastic static response of the building for three lateral load patterns. The subject of this study is an instrumented 52-story steel braced and moment frame building in Los Angeles which has experienced several earthquakes after construction was finished in 1990. Among the recorded building seismic responses those from the 1991 Sierra Madre earthquake and the 1994 Northridge earthquake are investigated. The dynamic characteristics of the structure were determined by utilizing system identification algorithm on the recorded responses during both earthquakes. Afterwards three-dimensional lin-ear elastic time history analyses of the building are conducted using information obtained from the system identification analyses. The linear computer model was calibrated to better represent the dynamic behavior of the building during past earthquakes. 94 Nonlinear computer model was constructed to predict structural responses during future severe earthquakes. First, three distinctive types of earthquake excitations were used in the nonlinear time history analyses for the prediction of building behavior. Next, pushover and cyclic analyses were considered. Three lateral load patterns were discussed to improve the yielding sequence ob-tained from the pushover analyses. 8.2 Recorded Behavior of the F W T and Prediction The following conclusions are drawn regarding the behavior of the building during the earth-quakes. These are based on the discussion of the building responses and on the analytical studies of thie structure. • Linear analysis is able to reproduce the response of the building when the dynamic character-istics of the structure are well defined and the computer model calibrated. The correlation be-tween the computed and recorded responses of the building during the 1991 Sierra Madre earthquake and the 1994 Northridge earthquake is a clear evidence for the observation. The good match implies that the calibrated linear model is a good representation of the dynamic behavior of the building during past earthquakes. • Pushover analyses and time history analyses performed on the nonlinear lumped plasticity model disclose interesting features of the structure during future possible earthquakes. Three distinctive types of earthquake excitations were used. The first set of records from the 1964 Niigata (Japan) earthquake have a major energy input around the first mode period of the 95 studied building (4.5 to 6 seconds). The second set of records are the Joshua Tree Fire station records during the 1992 Landers Earthquake which have a long duration of at least 40 seconds of strong shaking. The last set of records are from the Sylmar County Hospital station during the 1994 Northridge earthquake. This is a short duration, strong impulse, near-source ground motion that may be very damaging to tall buildings. • Envelope of the maximum displacement shows the significant effect of the Niigata motion on the response of this building. The Niigata earthquake has the largest magnitude among the three records considered. Because of the major energy input around the first mode period in this record the maximum displacement plot has the conventional inverted-triangular shape which is the typical first mode shape of buildings. The maximum displacement plot also shows clearly the combination of at least two mode shapes in the building responses for the other two excitations. • The input motions selected produced different envelopes of interstory drift in each direction of the building as shown in Chapter 7. The largest drifts occur for the Niigata motion in the mid-height of the building. Therefore severe damage may be expected to happen in these area if Niigata type of ground motions are experienced. • Maximum story shear increases down from the roof to the 49th story. For the next 20 stories below there is no significant change of maximum story shear resulting from the Sylmar and Joshua Tree motions. This again contradicts the code-defined inverted-triangular lateral load distribution along the height of the building. It is also a clear sign of higher modes effects. 96 The maximum story shears in the lower part of the building show the usual increase towards the ground floor. However, it can still be noticed that in several cases the tip of the curve is growing outward instead of growing inward. This can only be explained by a lateral load pat-tern which is nearly rectangular or trapezoidal. • The overturning moment plots for the Sylmar and Joshua Tree motions have peculiar onion-like shapes. The floors between the 20th to 25th story experienced the largest overturning mo-ment for these two excitations. Since the overturning moment resistance of the building is de-signed to be taken by the outrigger beams, it may suggest the outrigger beams between these stories will reach their yield strength first. • A proposed trapezoidal lateral load pattern provides the best comparison of yielding propa-gation steps for the building between the pushover analyses and the time history analyses un-der the Sylmar type of excitation considered. 8.3 Recommendations for Further Research • In the future, the development of more sophisticated models for steel members may be em-ployed into the 3-D nonlinear computer analysis, accounting for better representative of the steel element properties. The capacity and behavior of each element, especially in the inelas-tic range, are determined by its expected material strength. An analysis based on the actual material strengths obtained from experimental tests of samples taken from the building, rather than the specified values, will probably result in a better correlation between predicted and 97 future possible earthquakes responses. A more detailed computer model including the expanded basement foundation and the inter-action between the foundation and the soil would provide additional insight into the perfor-mance of the building. Due to the limitation of the computer program C A N N Y , damping used for the dynamic anal-ysis was restricted to Rayleigh-type damping. When features of the program include other types of damping, it is recommended that other types of damping be implemented into the computer model to simulate more realistically the energy dissipation in the structure. v Effect of gravitational frames was investigated for linear dynamic analysis of the structure. Including the gravitational frames for the nonlinear dynamic time history analysis may pro-vide more useful information to assess the contribution of the gravitational frames. Ductility demand and collapse mechanism of the structure were not covered in this study. It would be interesting to carry out further research on these two aspects. Conditioning of input records are important to apply system identification algorithm. How-ever magnitude of the conditioned records were reduced, caused difficulty in the system iden-tification process. Therefore improvements on the method of conditioning shall be investigated in the future. 98 R E F E R E N C E S Bakhtavar, M , 1999, "Seismic Behavior of Low-rise Steel Frame Structures", Thesis in progress, U B C . Banavalkar, P.V., 1991, "Spine Structures Provide Stability in Seismic Areas", Modern Steel Construction, January, pp. 13-19. Banavalkar, P.V., 1992, Personal communication with Dr. C.E. Ventura, June. Bendat, J.S. and Piersol, A . G . , 1993, "Engineering Applications of Correlation and Spectral Analysis", John Wiley & Sons, Inc., Toronto, 458 p. Celebi, M . , and Safak, E., 1991, "Seismic Response of Transamerica Building. I: Data and Pre-liminary Analysis." J. Struct. Engrg., A S C E , 117(8), pp. 2389-2404. Chen, P.F. and Powell, G.H., 1982, "Generalized Plastic Hinge Concepts for 3D Beam-Column Elements", EERC Reports No. 82/20, University of California, Berkeley, Nov. 1982. Chopra, A . K . , 1995, "Dynamics of Structures: Theory and Applications to Earthquake Engineer-ing", Prentice Hall, Upper Saddle River, N.J., 729 p. Clough, R.W., Benuska, K . L . and Wilson, E.L., 1965, "Inelastic Earthquake Response of Tall Buildings", Proceedings of the 3rd WCEE, New Zealand, Vol . II, pp. 65-89. Clough, R.W. and Penzien, J., 1993, "Dynamics of Structures", McGraw-Hill, London, 738 p. CSMIP, 1990, "Instrumentation of the Figueroa at Wilshire Building (#24602)", CSMIP internal building instrumentation data report, Sep. 1990, 4 p. "DADiSP 4.1", 1997, User's Guide, DSP Development Corporation, Cambridge, M A . Federal Emergency Management Agency, 1997, "The NEHRP Guidelines for Seismic Rehabil-itation of Buildings", F E M A 273, Washington, D.C. Giberson, M.F. , 1969, "Two Nonlinear Beams with Definition of Ductility", Journal Structural Division, A S C E , Vol . 95, No. ST7, July 1969, pp. 137-157. Handbook of Steel Construction, 1985, Canadian Institute of Steel Construction, Willowdale, Ontario. Hsu, L.W., 1974, "Behavior of Multi-story Reinforced Concrete Walls during Earthquakes", Ph.D. Thesis, Dept. of Civil Eng., University of Illinois, Urban, 1974. Huang, M . , Shakal, A. , Darragah, R., Ventura, C , Cao, T., Sherburne, R., Fung, P., Wampole, J., DeLisle, M . , and Petersen, C , 1991, "CSMIP Strong-Motion Records from the Sierra Madre, 99 California Earthquake of 28 June 1991," Calif. Dept. of Conservation, Division of Mines and Geology, Office of Strong Motion Studies, Report OSMS 91-03, Sacramento, C A , 108 pages. Jain, A.K.,1978, "Hysteresis Behavior of Bracing Members and Seismic Response of Braced Frames with Different Proportions", Ph.D. Thesis, University of Michigan, Ann Arbor, M L , 332 P-Jain, A . K . and Goel, S.C., 1978, "Hysteresis Models for Steel Members Subjected to Cyclic Buckling or Cyclic End Moments and Buckling - Users Guide for DRAIN-2D", Report No. U M E E 78R6, Department of Civil Engineering, University of Michigan, Ann Arbor, MI. 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Journal of Structural Engineering, A S C E , 117(8), pp. 2405-2425. 100 Schueller, W., 1976, "High-rise Building Structures", Rainbow-Bridge Book Co., Ltd. Sewell, C , 1999, "Inelastic Response of Torsionally Unbalanced Buildings Designed Using Elastic Dynamic Analysis", Thesis in progress, UBC. Shakal, A. , Huang, M . , Darragah, R., Cao, T., Sherburne, R., Malhotra, P., Cramer, C , Sydnor, R., Grazier, R., Maldonado, G., and Petersen, C , Wampole, J., 1994, "CSMIP Strong-Motion Records from the Northridge, California Earthquake of January 17,1994," Calif. Dept. of Con-servation, Division of Mines and Geology, Office of Strong Motion Studies, Report OSMS 94-07, Sacramento, C A , 308 p. Southern California Earthquake Center Data Center ( S C E C D C ) , 1999, "Historical Earthquakes in the Los Angeles Basin Area", source: http://www.scecdc.scec.org/labasin.html, material dat-ed Jan. 1999, retrieved Feb. 1999. Takayanagi, T. and Schnobrich, W.C., 1979, "Nonlinear Analysis of Coupled Wall Systems", Earthquake Engineering and Structural Dynamics, Vol . 7, 1979, pp. 1-22. Taranath, B.S., 1988, "Structural Analysis and Design of Tall Buildings", McGraw-Hill Book Company, New York, N Y . , 739 p. "The ME'Scope", 1998, Operating Manual Version 4.0, Vibrant Technology, Jamestown, CA. , 293 p. Uniform Building Code (UBC), 1982, International Conf. of Building Officials, Whittier, CA. , 780 p. Uniform Building Code (UBC), 1994, International Conf. of Building Officials, Whittier, CA. 101 A p p e n d i x A Elevations and Details of the Building 102 Perimeter! column Perimeter | column Perimeter column -» Perimeter column Outriggi beam Core column) ' Outrigger beam Bracing Outrigger -heam Core column Outriggei beam 13.84 m Bracing Bracing Perimeter column 17.37m Outrigger beam Core column Outrigger beam Bracing Outrigger beam . TCore column lOutrigger lbeam Penrieter coll mn 13.84m Perimetei) "column s ?! Perimeter column Typical Floor Plan 9RAV1TV LOAO DESIGN C R I T E R I A A B S ( M i - M g ) < M p n A B S ( M 2 + M g 2 ) < M p n | M g 3 < M p n METAL' DECK V u = V p +/- V g l o r V g 2 or V g 3 M p n = plastic moment capacity of notched section M g or V g = factored moment or shear W'Q HEADEO STUDS Y1 X CONTINUOUS ANGLE .OUTRIGGER BEAM II GRAVITY P R E V E N T O N OF T H R E E HINGE FORMATION S E C T I O N T H R U O U T R I G G E R B E A M F I G U E R O A AT WILSHIRE Figure a.l. FWT, typical floor plan and details 103 13.84m u 1 7 J 7 m 1.1.84m 5.42m 5.42m 5.72m 7.53m Figure a.2. FWT, Frame A Elevation 104 Figure a.3. FWT, Frame B Elevation 105 13.84m 17.37 m 13.84m Figure a.4. FWT, Frame C Elevation 106 107 A p p e n d i x B Photos of the Building 108 100 A p p e n d i x C CANNY input data file 1998.6, dynamic analysis of Figueroa at Wilshire a steel frame highrise in L.A. // analysis assumptions and output options title = DYNAMIC ANALYSIS OF FIGUEROA AT WILSHIRE force unit = kN length unit = m time unit = sec gravity acceleration is 9.805 including floor rotation of rigid floor level including P-Delta effect output of overall response at floor levels /* output of period time history /*static analysis // control data of static response response limit 10.0 master DOFs for analysis control : 49F XY-translation binary format output of response results at every 0-step /•destination base shear factor 0.20 by increment 0.01 /•destination base shear factor 0.21 by increment 0.001 destination displacement 1.5 by increment 0.01 destination displacement 2.5 by increment 0.001 destination displacement 4.5 by increment 0.005 /•destination base shear factor 0.20 by increment 0.01 /•destination base shear factor 0.21 by increment 0.001 /•destination displacement 1.0 by increment 0.1 /•destination displacement -1.0 by increment 0.1 /•destination displacement 1.0 by increment 0.1 /•destination displacement 2.0 by increment 0.01 /•destination displacement -1.5 by increment 0.01 /•destination displacement -2.0 by increment 0.005 /•destination displacement 4.0 by increment 0.001 /•destination displacement -4.0 by increment 0.001 /•destination displacement 18.0 by increment 0.2 /•destination displacement -18.0 by increment 0.2 /•destination displacement 0 by increment 0.001 // control data of dynamic response integration 2-step for one acceleration data start time 0.0, end time 180.0 check peak displacement 0.05 113 response limit 5.0 master DOFs for analysis control: 49F XY-translation binary format output of response results at every 0-step Newmark method using Beta-value 0.25 /•damping constant 0.053 to mass matrix [M] /*damping constant 0.00168 to stiffness matrix [K] damping coefficient 0.1012 proportional to mass matrix [M] damping coefficient 0.00321 proportional to stiffness matrix [K] scale factor 0.01 to X-EQ file = e:\ding\fw-c\nry2.txt scale factor 0.01 to Y-EQ file = e:\ding\f\v-c\nrx2.txt scale factor 0.01 to R-EQ file = e:\ding\fw-c\nr-r22.txt // node locations XI and X7, Y l and Y3, IF to 53F X8 and X9, Y l and Y3, IF to 50F XI and X7, Y4 and Y5, IF to 50F Y1.X4, IF to 17F Yl,X4,32Fto43F Y1,X2, 19Fto32F Y1,X3,44F to 49F Y1,X5, 18Fto31F Y1.X5, 50F to 53F Y1,X3,50F to 53F Y1,X6,45F to 49F Y2, XI and X7, 2F to 48F Y2,X1 andX7, 50F to 53F Y3, X2, 3F to 4F Y3,X2, 31Fto 44F Y3, X3, 4F to 29F Y3, X3, 44F to 49F Y3,X4, lFto3F Y3, X5, 30F to 43F Y3, X5, 50F to 53F Y3,X3, 50F to 53F Y3, X6, 3F to 49F /*node X8 and X9, Y4 and Y5, IF to 50F // node degrees of freedom node Y l , X4, 2F to 17F, eliminate Y-Z rot node Y l , X4, 32F to 43F, eliminate Y-Z rot node Y l , X2, 19F to 32F, eliminate Y-Z rot node Y l , X3, 44F to 49F, eliminate Y-Z rot node Y l , X5, 18F to 3IF, eliminate Y-Z rot node Y l , X5, 50F, eliminate Y-Z rot node Y l , X6,45F to 49F, eliminate Y-Z rot node Y2, XI and X7, 2F to 48F, eliminate X-Z rot node Y2, XI and X7, 50F, eliminate X-Z rot node Y3, X2, 3F to 4F, eliminate Y-Z rot node Y3, X2, 3 IF to 44F, eliminate Y-Z rot node Y3, X3,4F to 29F, eliminate Y-Z rot node Y3, X3,44F to 49F, eliminate Y-Z rot node Y3, X4,2F to 3F, eliminate Y-Z rot node Y3, X5, 30F to 43F, eliminate Y-Z rot node Y3, X5, 50F, eliminate Y-Z rot 114 node Y3, X6, 3F to 49F , eliminate Y-Z rot /*node XI to X9, Y l to Y5, IF, eliminate all component //--53F(r: 52F(r 51F(r: 50F(r 49F(r 48F(r 47F(r: 46F(r 45F(r 44F(r 43F(r 42F(r 41F(r 40F(r 39F(r 38F(r 37F(r 36F(ri 35F(n 34F(n 33F(ri 32F(n 31F(r: 30F(r: 29F(r: 28F(r: 27F(r: 26F(r; 25F(r; 24F(r: 23F(r: 22F(r 21F(r 20F(r 19F(r 18F(r 17F(r 16F(r 15F(r 14F(r 13F(r 12F(r: HF(r: 10F(r: floor level gid floor, above 52F above 51F gid floor, gid floor, gid floor gid floor, gid floor gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, gid floor, above 5 OF above 49F above 48F above 47F above 46F above 45F' above 44F above 43 F' above 42F' above 41F above 40F above 39F above 38F above 37F above 36F above 35F above 34F above 33F above 32F above 3IF above 3 OF above 29F above 28F above 27F above 26F above 25F above 24F above 23 F above 22F above 21F above 20F above 19F above 18F above 17F' above 16F above 15F above 14F above 13F above 12F above 1 IF above 10F : Z=218.0 G(8.7,10.5) :Z=213.0 G(8.6,10.4) :Z=207.0 G(9.0,10.0), :Z=202.0 G(8.7,10.7) :Z=194.0G(8.7,10.7) : Z=190.1 G(8.7,10.7), : Z=186.1 G(8.7,10.7) :Z=182.2 G(8.7,10.7) Z=178.2 G(8.6,10.84), W=9439 Rj=2670596, Z=174.2 G(8.7,10.7) Z=170.2 G(8.7,10.7) Z=166.3 G(8.7,10.7) Z=162.3 G(8.7,10.7) W=3360 Rj=234813, W=8855 Rj= 1391108, W=11930 Rj=2200801, W= 10555 Rj=2907479, W=9147 Rj=2504407, W=8473 Rj=2204690, W=8473 Rj=2204690, W=9178 Rj=2657780, W=9273 Rj=2657367, W=9355 Rj=2693336, W=9988 Rj=3104672, W=9988Rj=3104672, W=9988 Rj=3104672, W=9988Rj=3104672, W=9988 Rj=3104672, W=9988 Rj=3104672, W=11399Rj=3749583, W=10436 Rj=3447384, W=10436 Rj=3447384, W=10626Rj=3517663, W= 10626 Rj=3 517663, W=11398 Rj=3568513, W=10559Rj=3570164, W=10559Rj=3570164, W=10559Rj=3570164, W=10559Rj=3570164, W=10559 Rj=3570164, :Z=158.3 G(8.7,10.7), :Z=154.3 G(8.7,10.7), :Z=150.4 G(8.7,10.7), :Z=146.5G(8.7,10.7), :Z=142.5G(8.7,10.7), :Z=138.5 G(8.7,10.7), : Z=134.5 G(8.7,10.7), : Z=130.6G(8.7,10.7), :Z=126.7 G(8.7,10.7), : Z=122.7 G(8.4,10.2), : Z=118.7 G(8.7,10.7), :Z=114.7 G(8.7,10.7), :Z=110.8G(8.7,10.7), :Z=106.9 G(8.7,10.7), : Z=102.9 G(8.7,10.7), : Z=98.9 G(8.7,10.7), W=10559 Rj=3570164, : Z=95.0 G(8.7,10.7), W=10559 Rj=3570164, : Z=91.0 G(8.7,10.7), W= 10559 Rj=3570164, : Z=87.0 G(8.7,10.7), W=10559 Rj=3570164, : Z=83.0 G(8.7,10.7), W=10559 Rj=3570164, : Z=79.0 G(9.0,l 1.2), W=11706 Rj=3690051, : Z=75.0 G(8.7,10.7), W=10680 Rj=3631765, : Z=71.1 G(8.7,10.7), W=10680 Rj=3631765, : Z=67.2 G(8.7,10.7), W=10780 Rj=3665661, : Z=63.2 G(8.7,10.7), W=10780 Rj=3665661, : Z=59.2 G(8.7,10.7), W=10780 Rj=3665661, : Z=55.2 G(8.7,10.7), W=10780 Rj=3665661, : Z=51.3 G(8.7,10.7), W=l 1006.5 Rj=3733875, : Z=47.4 G(8.7,10.7), W=l 1006.5 Rj=3733875, Z=43.4 G(8.7,10.7), W=l 1006.5 Rj=3733875, above 9F): Z=39.4 G(8.7,10.7), W=l 1006.5 Rj=3733875, 9F(rigid floor, above 8F): Z=35.4 G(8.7,10.7), W=l 1224 Rj=3801671, 8F(rigid floor, above 7F): Z=31.4 G(8.7,10.7), W=l 1224 Rj=3801671, 7F(rigid floor, above 6F): Z=27.4 G(8.7,10.7), W=l 1224 Rj=3801671, 6F(rigid floor, above 5F): Z=23.4 G(9.0,10.4), W=l 1511 Rj=3994320, 5F(rigid floor, above 4F): Z=19.5 G(9.0,10.4), W=l 1511 Rj=3994320, 4F(rigid floor, above 3F): Z=15.0 G(9.1,10.3), W=11974 Rj=4218800, 115 3F(rigid floor, above 2F): Z=10.4 G(9.2,13.0), W=13138 Rj=4774000, 2F(rigid floor, above IF): Z-6.10 G(7.5,l 1.7), W=18488 Rj=8362769, lF(footing floor, fixed support): Z=0 // frame locations X1:0,X2: 4.10, X3: 6.1, X4: 8.69, X5: 11.3, X6: 13.30, X7: 17.37, X8: 31.21, X9: -13.84 Y l : 0, Y2: 10.82, Y3: 21.23, Y4: 33.35, Y5: -11.71 // element data : beam frame XI and X7. frame XI and X7; frame XI and X7, frame XI and X7. frame XI andX7: frame XI and X7 frame X7 , Y frame X7 , Y frame X7 , Y frame X7 , Y frame X7 , Y frame XI and X7; frame XI and X7 frame XI and X7 frame frame frame frame frame frame frame frame XI XI XI XI XI XI XI XI frame Y3 ,x frame Y3 ,x frame Y3 ,x frame Y3 ,x frame Y3 ,x frame Y3 ,x frame Y3 ,x frame Y3 ,x frame Y3 ,x frame Y3 ,x frame Y3 ,x frame Y3 ,x frame Y3 ,x frame Y l andY3 frame Y l and Y3 frame Y l and Y3 frame Y l ,x frame Y l ,x frame Y l ,x frame Y l ,x Y l to Y3, 2F LU06 RU06 SU806 .46 .46 Y l to Y3, 3F to 6F LU08 RU08 SU808 .46 .46 Y l to Y3, 7F to 8F LU09 RU09 SU809 .46 .46 Y l to Y3, 9F to 12F LU10 RU10 SU810 .46 .46 Y l to Y3, 13Fto 16F LU11 RU11 SU811 .46 .46 Y l to Y3, 17F to 19F LU12 RU12 SU812 .46 .46 to Y3, 20F LU07 RU07 SU807 .46 .46 to Y3, 21F LU11 RU11 SU811 .46 .46 to Y3, 22F to 25F LU12 RU12 SU812 .46 .46 to Y3, 26F to 35F LU13 RU13 SU813 .46 .46 to Y3, 36Fto47F LU14RU14 SU814 .46 .46 Y1 to Y3, 48F to 49F LU 15 RU 15 SU815 .46 .46 Yl toY3, 50F LU88 RU88 SU888 .46.46 Y l to Y3, 5IF to 53F LU02 RU02 SU802 .46 .46 to Y3, 20F to 22F LU12 RU12 SU812 .46 .46 to Y3, 23F to 30F LU13 RU13 SU813 .46 .46 toY3,31F LU09 RU09 SU809 .46.46 toY3, 32F LU12 RU12 SU812 .46.46 to Y3, 33F to 35F LU13 RU13 SU813 .46 .46 to Y3, 36F to 44F LU14 RU14 SU814 .46 .46 toY3,45F LU09 RU09 SU809 .46.46 to Y3,46F to 47F LU14 RU14 SU814 .46 .46 to X7, 2F LU87 RU87 SU887 .46 .46 to X7, 3F LU15 RU15 SU815 .46 .46 to X7, 4F LU13 RU13 SU813 .46 .46 to X7, 5F to 6F LU15 RU15 SU815 .46 .46 toX7, 7FtolOF LU16RU16SU816 .46.46 to X7, 1 IF to 14F LU17 RU17 SU817 .46 .46 toX7, 15Fto 18F LU18RU18SU818.46.46 to X7, 19F LU19 RU19 SU819 .46 .46 to X7, 20F LU 14 RU 14 SU814 .46 .46 to X7, 21F to 24F LU19 RU19 SU819 .46 .46 to X7, 25F to 30F LU20 RU20 SU820 .46 .46 to X7, 3IF to 43F LU16 RU16 SU816 .46 .46 to X7, 44F to 49F LU20 RU20 SU820 .46 .46 XI toX7, 50F LU88 RU88 SU888 .46.46 XI to X7, 51F to 52F LU89 RU89 SU889 .46 .46 X1 to X7, 53F LU02 RU02 SU802 .46 .46 toX7,2F LU86 RU86 SU886 .46.46 toX7,3Fto8F LU12RU12SU812 .46.46 toX7, 9Ftol lF LU13RU13SU813.46.46 toX7, 12Fto 18F LU14RU14SU814.46.46 116 frame Y l , XI to X7, 19F to 24F LU19 RU19 SU819 .46 .46 frame Y l , XI to X7, 25F to 30F LU20 RU20 SU820 .46 .46 frame Y l ,XltoX7,31F LU17 RU17 SU817 .46.46 frame Y l , XI to X7, 32F to 35F LU16 RU16 SU816 .46 .46 frame Y l , XI to X7, 36F to 41F LU17 RU17 SU817 .46 .46 frame Y l , XI to X7, 42F to 44F LU18 RU18 SU818 .46 .46 frame Y l , XI to X7, 45F to 49F LU20 RU20 SU820 .46 .46 frame X7 , Y5-Y1, 2F LU92 RU92 SU892 .46.46 frame XI and X7, Y5-Y1, 3F to 4F LU03 RU03 SU803 .46 .46 frame XI and X7, Y5-Y1, 5F to 49F LU05 RU05 SU805 .46 .46 frame XI and X7, Y5-Y1, 50F LU94 RU94 SU894 .46 .46 frame XI , Y5-Y1, 2F LU92 RU92 SU892 .46 .46 frame X7 , Y3 to Y4, 2F LU01 RU01 SU801 .46 .46 frame XI and X7, Y3 to Y4, 3F to 4F LU03 RU03 SU803 .46 .46 frame XI and X7, Y3 to Y4, 5F to 49F LU05 RU05 SU805 .46 .46 frame X1 and X7, Y3 to Y4, 50F LU94 RU94 SU894 .46 .46 frame XI , Y3 to Y4, 2F LU01 RU01 SU801 .46 .46 frame Y l and Y3, X9-X1, 2F LU02 RU02 SU802 .46 .46 frame Y l and Y3, X9-X1, 3F to 4F LU03 RU03 SU803 .46 .46 frame Y l and Y3, X9-X1, 5F to 36F LU04 RU04 SU804 .46 .46 frame Y l and Y3, X9-X1, 37F to 49F LU05 RU05 SU805 .46 .46 frame Y l and Y3, X9-X1, 50F LU94 RU94 SU894 .46 .46 frame Y l and Y3, X7 to X8, 2F LU02 RU02 SU802 .46 .46 frame Y l and Y3, X7 to X8, 3F to 4F LU03 RU03 SU803 .46 .46 frame Y l and Y3, X7 to X8, 5F to 36F LU04 RU04 SU804 .46 .46 frame Y l and Y3, X7 to X8, 37F to 49F LU05 RU05 SU805 .46 .46 frame Y l and Y3, X7 to X8, 50F LU94 RU94 SU894 .46 .46 /*frame X8 and X9, Y l to Y3, 2F to 50F LU06 RU06 SU806 0 0 /•frame Y4 and Y5, XI to X7, 2F to 50F LU06 RU06 SU806 0 0 /•frame X8 and X9, Y3 to Y4, 2F to 50F LU25 RU25 SU825 0 0 /•frame X8 and X9, Y5-Y1, 2F to 50F LU25 RU25 SU825 0 0 /•frame Y4 and Y5, X7 to X8, 2F to 50F LU25 RU25 SU825 0 0 /•frame Y4 and Y5, X9-X1, 2F to 50F LU25 RU25 SU825 0 0 // element data : link /•X4 to X4 Y l and Y3 IF to 2F AU968 X4-X7 Y l 2F-3F AU907 X4-X7 Y l 3F-6F AU910 X4-X7 Y l 6F-10F AU911 X4-X7 Y l 10F-14F AU912 X4-X7 Y l 14F-16F AU913 X4-X7 Y l 16F-18F AU914 X4-X7 Y l 32F-34F AU917 X4-X7 Y l 34F-39F AU918 X4-X7 Y l 39F-42F AU919 X4-X7 Y l 42F-44F AU920 117 X4-X1 Y l 2F-3F AU907 X4-X1 Y l 3F-6FAU910 X4-X1 Y l 6F-10F AU911 X4-X1 Y l 10F-14F AU912 X4-X1 Y l 14F-16F AU913 X4-X1 Y l 16F-18F AU914 X4-X1 Y l 32F-34F AU917 X4-X1 Y l 34F-39F AU918 X4-X1 Y l 39F-42F AU919 X4-X1 Y l 42F-44F AU920 X1-X2 Y l 18F-21F AU921 X1-X2 Y l 21F-24F AU922 X1-X2 Y l 24F-30F AU923 X1-X2 Y l 30F-31F AU920 X1-X2 Y l 31F-32F AU923 X5-X2 Y l 18F-19F AU917 X5-X2Y1 19F-24FAU918 X5-X2 Y l 24F-28F AU919 X5-X2 Y l 28F-30F AU920 X5-X2 Y l 30F-31F AU918 X5-X2 Y l 31F-32F AU919 X5-X7 Y l 18F-20FAU918 X5-X7 Y l 20F-22F AU919 X5-X7 Y l 22F-29F AU920 X5-X7 Y l 29F-31F AU921 X5-X7 Y l 31F-32F AU923 X3-X1 Y l 44F-49F AU923 X3-X1 Y l 49F-50F AU920 X3-X6 Y l 44F-45F AU922 X3-X6 Y l 45F-49F AU923 X3-X5 Y l 49F-50F AU919 X7-X6 Y l 44F-45F AU931 X7-X6 Y l 45F-47F AU932 X7-X6 Y l 47F-49F AU933 X7-X5 Y l 49F-50F AU916 XI-X2 Y3 2F-3F AU916 X1-X2 Y3 3F-4F AU913 X4-X2 Y3 2F-4F AU916 X4-X6 Y3 2F-4F AU916 X7-X6 Y3 2F-4F AU916 X3-X1 Y3 4F-5F AU912 X3-X1 Y3 5F-8FAU915 X3-X1 Y3 8F-12F AU916 X3-X1 Y3 12F-16F AU917 X3-X1 Y3 16F-21F AU918 X3-X1 Y3 21F-22F AU919 X3-X1 Y3 22F-29F AU920 X3-X1 Y3 29F-30F AU921 X3-X1 Y3 44F-46F AU922 X3-X1 Y3 46F-49F AU923 X3-X1 Y3 49F-50F AU920 X3-X6 Y3 4F-5F AU998 X3-X6 Y3 5F-8F AU940 X3-X6 Y3 8F-12F AU941 X3-X6 Y3 12F-14F AU942 X3-X6 Y3 14F-16F AU917 X3-X6 Y3 16F-24F AU918 X3-X6 Y3 24F-28F AU919 X3-X6 Y3 28F-30F AU920 X3-X6 Y3 44F-46F AU920 X3-X6 Y3 46F-49F AU923 X3-X5 Y3 49F-50F AU919 X7-X6 Y3 4F-6F AU917 X7-X6 Y3 6F-10F AU918 X7-X6 Y3 10F-14F AU919 X7-X6 Y3 14F-16F AU920 X7-X6 Y3 16F-19F AU921 X7-X6Y3 19F-20FAU919 X7-X6 Y3 20F-24F AU922 X7-X6 Y3 24F-30F AU923 X7-X6 Y3 44F-45F AU923 X7-X6 Y3 45F-46F AU931 X7-X6 Y3 46F-47F AU932 X7-X6 Y3 47F-49F AU933 X7-X5 Y3 49F-50F AU916 X1-X2 Y3 30F-41F AU923 XI-X2 Y3 41F-44F AU931 X5-X2 Y3 30F-31F AU920 X5-X2 Y3 31F-34F AU921 X5-X2 Y3 34F-44F AU922 X5-X6 Y3 30F-37F AU933 X5-X6 Y3 37F-44F AU934 X7-X6 Y3 30F-31F AU920 X7-X6 Y3 31F-33F AU921 X7-X6 Y3 33F-35F AU922 X7-X6 Y3 35F-44F AU923 XI Y1-Y2 1F-2F AU996 XI Y3-Y2 1F-2F AU996 XI Y2-Y1 2F-5F AU997 XI Y2-Y1 5F-10FAU998 XI Y2-Y1 10F-13F AU943 XI Y2-Y1 13F-14F AU944 XI Y2-Y1 14F-16F AU945 XI Y2-Y1 16F-19F AU940 XI Y2-Y1 19F-22F AU941 XI Y2-Y1 22F-23F AU946 XI Y2-Y1 23F-31F AU942 XI Y2-Y1 31F-49F AU947 XI Y2-Y3 2F-5F AU997 XI Y2-Y3 5F-10F AU998 XI Y2-Y3 10F-13F AU943 XI Y2-Y3 13F-14F AU944 XI Y2-Y3 14F-16F AU945 XI Y2-Y3 16F-19F AU940 XI Y2-Y3 19F-22F AU941 XI Y2-Y3 22F-23F AU946 XI Y2 -Y3 23F--3 IF AU942 XI Y2 -Y3 31F--49F AU947 XI Y l -Y2 49F--50F AU999 XI Y3 -Y2 49F--50F AU999 X7 Yl-•Y2 1F-2F AU996 X7 Y3 Y2 1F-2F AU996 X7 Y2 •Yl 2F-5F AU997 X7 Y2 •Yl 5F-10F AU998 X7 Y2 •Yl 10F-13F AU943 X7 Y2 -Yl 13F-14F AU944 X7 Y2 •Yl 14F-16F AU945 X7 Y2 -Yl 16F-19F AU940 X7 Y2 •Yl 19F-22F AU941 X7 Y2 -Yl 22F-23F AU946 X7 Y2 •Yl 23F-31F AU942 X7 Y2 -Yl 31F-49F AU947 X7 Y2 •Y3 2F-5F AU997 X7 Y2 -Y3 5F-10F AU998 X7 Y2 -Y3 10F-13F AU943 X7 Y2 -Y3 13F-14F AU944 X7 Y2 -Y3 14F-16F AU945 X7 Y2 -Y3 16F-19F AU940 X7 Y2 -Y3 19F-22F AU941 X7 Y2 -Y3 22F-23F AU946 X7 Y2 -Y3 23F-31F AU942 X7 Y2 -Y3 31F-49F AU947 X7 Y l -Y2 49F-50F AU999 X7 Y3 -Y2 49F-50F AU999 // element data : column /* 4-core-corner columns XI and XI and XI and XI and XI and XI and XI and XI and XI and XI and XI and XI and XI and XI and XI and XI and XI and XI and XI and XI and XI and X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l X7, Y l and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 and Y3 IF to 2F BU90 TU90 SU890 AU990 0 0.42 2F to 4F BU90 TU90 SU890 AU990 0.21 0.42 4F to 6F BU91 TU91 SU891 AU991 0.21 0.42 6F to 8F BU50 TU50 SU850 AU950 0.21 0.42 8Fto 10FBU51 TU51 SU851 AU951 0.21 0.42 10F to 12F BU52 TU52 SU852 AU952 0.21 0.42 12F to 14F BU53 TU53 SU853 AU953 0.21 0.42 14F to 16F BU54 TU54 SU854 AU954 0.21 0.42 16F to 18F BU55 TU55 SU855 AU955 0.21 0.42 18F to 22F BU56 TU56 SU856 AU956 0.21 0.42 22F to 24F BU57 TU57 SU857 AU957 0.21 0.42 24F to 26F BU58 TU58 SU858 AU958 0.21 0.42 26F to 28F BU59 TU59 SU859 AU959 0.21 0.42 28F to 32F BU60 TU60 SU860 AU960 0.21 0.42 32F to 34F BU61 TU61 SU861 AU961 0.21 0.42 34F to 36F BU62 TU62 SU862 AU962 0.21 0.42 36F to 40F BU63 TU63 SU863 AU963 0.21 0.42 40F to 44F BU65 TU65 SU865 AU965 0.21 0.42 44F to 48F BU67 TU67 SU867 AU967 0.21 0.42 48F to 50F BU65 TU65 SU865 AU965 0.21 0.42 50F to 52F BU64 TU64 SU864 AU964 0.21 0.42 120 XI and X7, Y l and Y3, 52F to 53F BU66 TU66 SU866 AU966 0.21 0.42 /* side columns XI and X7, Y4 and Y5, IF to 2F BU93 TU93 SU893 AU993 0 0.42 XI and X7, Y4 and Y5, 2F to 4F BU93 TU93 SU893 AU993 0.21 0.42 XI and X7, Y4 and Y5, 4F to 6F BU70 TU70 SU870 AU970 0.21 0.42 XI and X7, Y4 and Y5, 6F to 8F BU71 TU71 SU871 AU971 0.21 0.42 XI and X7, Y4 and Y5, 8F to 10F BU72 TU72 SU872 AU972 0.21 0.42 XI and X7, Y4 and Y5, 10F to 12F BU73 TU73 SU873 AU973 0.21 0.42 XI and X7, Y4 and Y5, 12F to 14F BU74 TU74 SU874 AU974 0.21 0.42 XI and X7, Y4 and Y5, 14F to 16F BU75 TU75 SU875 AU975 0.21 0.42 XI and X7, Y4 and Y5, 16F to 20F BU76 TU76 SU876 AU976 0.21 0.42 XI and X7, Y4 and Y5, 20F to 22F BU77 TU77 SU877 AU977 0.21 0.42 XI and X7, Y4 and Y5, 22F to 24F BU78 TU78 SU878 AU978 0.21 0.42 XI and X7, Y4 and Y5, 24F to 26F BU79 TU79 SU879 AU979 0.21 0.42 XI and X7, Y4 and Y5, 26F to 28F BU80 TU80 SU880 AU980 0.21 0.42 XI and X7, Y4 and Y5, 28F to 32F BU81 TU81 SU881 AU981 0.21 0.42 XI and X7, Y4 and Y5, 32F to 34F BU82 TU82 SU882 AU982 0.21 0.42 XI and X7, Y4 and Y5, 34F to 36F BU83 TU83 SU883 AU983 0.21 0.42 XI and X7, Y4 and Y5, 36F to 38F BU84 TU84 SU884 AU984 0.21 0.42 XI and X7, Y4 and Y5, 38F to 42F BU85 TU85 SU885 AU985 0.21 0.42 XI and X7, Y4 and Y5, 42Fto44F BU01 TU01 SU801 AU901 0.21 0.42 XI and X7, Y4 and Y5, 44F to 46F BU02 TU02 SU802 AU902 0.21 0.42 XI and X7, Y4 and Y5, 46F to 48F BU24 TU24 SU824 AU924 0.21 0.42 XI and X7, Y4 and Y5, 48F to 50F BU95 TU95 SU895 AU995 0.21 0.42 X8 and X9, Y l and Y3, IF to 2F BU93 TU93 SU893 AU993 0 0.42 X8 and X9, Y l and Y3, 2F to 4F BU93 TU93 SU893 AU993 0.21 0.42 X8 and X9, Y l and Y3, 4F to 6F BU70 TU70 SU870 AU970 0.21 0.42 X8 and X9, Y l and Y3, 6F to 8F BU71 TU71 SU871 AU971 0.21 0.42 X8 and X9, Y l and Y3, 8F to 10F BU72 TU72 SU872 AU972 0.21 0.42 X8 and X9, Y l and Y3, 10F to 12F BU73 TU73 SU873 AU973 0.21 0.42 X8 and X9, Y l and Y3, 12F to 14F BU74 TU74 SU874 AU974 0.21 0.42 X8 and X9, Y l and Y3, 14F to 16F BU75 TU75 SU875 AU975 0.21 0.42 X8 and X9, Y l and Y3, 16F to 20F BU76 TU76 SU876 AU976 0.21 0.42 X8 and X9, Y l and Y3, 20F to 22F BU77 TU77 SU877 AU977 0.21 0.42 X8 and X9, Y l and Y3, 22F to 24F BU78 TU78 SU878 AU978 0.21 0.42 X8 and X9, Y l and Y3, 24F to 26F BU79 TU79 SU879 AU979 0.21 0.42 X8 and X9, Y l and Y3, 26F to 28F BU80 TU80 SU880 AU980 0.21 0.42 X8 and X9, Y l and Y3, 28F to 32F BU81 TU81 SU881 AU981 0.21 0.42 X8 and X9, Y l and Y3, 32F to 34F BU82 TU82 SU882 AU982 0.21 0.42 X8 and X9, Y l and Y3, 34F to 36F BU83 TU83 SU883 AU983 0.21 0.42 X8 and X9, Y l and Y3, 36F to 38F BU84 TU84 SU884 AU984 0.21 0.42 X8 and X9, Y l and Y3, 38F to 42F BU85 TU85 SU885 AU985 0.21 0.42 X8 and X9, Y l and Y3, 42F to 44F BU01 TU01 SU801 AU901 0.21 0.42 X8 and X9, Y l and Y3, 44F to 46F BU02 TU02 SU802 AU902 0.21 0.42 X8 and X9, Y l and Y3, 46F to 48F BU24 TU24 SU824 AU924 0.21 0.42 X8 and X9, Y l and Y3, 48F to 50F BU95 TU95 SU895 AU995 0.21 0.42 /*X4, Y1 and Y3, 1F to 2F BU68 TU68 SU868 AU968 0 0.42 X3 and X5, Y l and Y3, 50F to 52F BU85 TU85 SU885 AU985 0.21 0.42 X3 and X5, Y l and Y3, 52F to 53F BU01 TU01 SU801 AU801 0.21 0.42 XI and X7, Y2, 50F to 52F BU72 TU72 SU872 AU972 0.21 0.42 XI and X7, Y2, 52F to 53F BU74 TU74 SU874 AU974 0.21 0.42 121 /*X8 and X9, Y4 and Y5, IF to 50F BU13 TU13 SU813 AU913 0 0 // stiffness and hysteresis parameters U01 14 2.0e+8 0.0103 C(8200,8205) Y(8235,8235) A(l,l) B(0.02,0.02) P(0.3 0.75) U02 14 2.0e+8 0.00847 C(6900,6900) Y(6907,6907) A(l,l) B(0.02,0.02) P(0.3 0.75) U03 14 2.0e+8 0.00718 C(5900,5900) Y(5920,5920) A(l,l) B(0.02,0.02) P(0.3 0.75) U04 14 2.0e+8 0.00671 C(5500,5501) Y(5541,5541) A(l,l) B(0.02,0.02) P(0.3 0.75) U05 14 2.0e+8 0.00625 C(5100,5109) Y(5199,5199) A(l,l) B(0.02,0.02) P(0.3 0.75) U06 14 2.0e+8 0.00191 C(2300,2300) Y(2303,2303) A(l,l) B(0.02,0.02) P(0.3 0.75) U07 14 2.0e+8 0.00274 C(4400,4400) Y(4402,4402) A(l,l) B(0.02,0.02) P(0.3 0.75) U08 14 2.0e+8 0.00226 C(3700,3702) Y(3772,3772) A(l,l) B(0.02,0.02) P(0.3 0.75) U09 14 2.0e+8 0.00205 C(3400,3400) Y(3480,3480) A(l,l) B(0.02,0.02) P(0.3 0.75) U10 14 2.0e+8 0.0018 C(3100,3103) Y(3143,3143) A(l,l) B(0.02,0.02) P(0.3 0.75) U l l 14 2.0e+8 0.0016 C(2800,2800) Y(2850,2850) A(l,l) B(0.02,0.02) P(0.3 0.75) U12 14 2.0e+8 0.00141 C(2500,2507) Y(2577,2577) A(l,l) B(0.02,0.02) P(0.3 0.75) U13 14 2.0e+8 0.00125 C(2300,2300) Y(2330,2330) A(l,l) B(0.02,0.02) P(0.3 0.75) U14 14 2.0e+8 0.0011 C(2000,2009) Y(2099,2099) A(l,l) B(0.02,0.02) P(0.3 0.75) U15 14 2.0e+8 0.000997 C(1900,1904) Y(1924,1924) A(l,l) B(0.02,0.02) P(0.3 0.75) U16 14 2.0e+8 0.000894 C(1700,1703) Y(1753,1753) A(l,l) B(0.02,0.02) P(0.3 0.75) U17 14 2.0e+8 0.000788 C(1500,1505) Y(1575,1575) A(l,l) B(0.02,0.02) P(0.3 0.75) U18 14 2.0e+8 0.000712 C(1400,1409) Y(1439,1439) A(l,l) B(0.02,0.02) P(0.3 0.75) U19 14 2.0e+8 0.000636 C(1200,1208) Y(1298,1298) A(l,l) B(0.02,0.02) P(0.3 0.75) U20 14 2.0e+8 0.000575 C(l 100,1104) Y(l 184,1184) A(l,l) B(0.02,0.02) P(0.3 0.75) U24 14 2.0e+8 0.00788 C(6400,6402) Y(6452,6452) A( 1,1) B(0.02,0.02) P(0.3 0.75) U25 1 2.0e+8 0.000351 U50 14 2.0e+8 0.0321 C(22800,22806) Y(22846,22846) A(l,l) B(0.02,0.02) P(0.3 0.75) U51 14 2.0e+8 0.0301 C(21400,21400) Y(21404,21404) A(l,l) B(0.02,0.02) P(0.3 0.75) U52 14 2.0e+8 0.0281 C(20000,20000) Y(20010,20010) A( 1,1) B(0.02,0.02) P(0.3 0.75) U53 14 2.0e+8 0.026 C( 18400,18402) Y(18482,18482) A(l,l) B(0.02,0.02) P(0.3 0.75) . U54 14 2.0e+8 0.025 C( 17700,17709) Y(l7799,17799) A(l,l) B(0.02,0.02) P(0.3 0.75) U55 14 2.0e+8 0.0244 C( 17300,17304) Y(17344,17344) A(l,l) B(0.02,0.02) P(0.3 0.75) U56 14 2.0e+8 0.0227 C(16100,16107) Y(16167,16167) A(l,l) B(0.02,0.02) P(0.3 0.75) U57 14 2.0e+8 0.0211 C( 15000,50202) Y(15028,50282) A(l,l) B(0.02,0.02) P(0.3 0.75) U58 14 2.0e+8 0.0199 C(14100,14106) Y(14156,14156) A(l,l) B(0.02,0.02) P(0.3 0.75) U59 14 2.0e+8 0.0188 C( 13300,13300) Y(13397,13397) A(l,l) B(0.02,0.02) P(0.3 0.75) U60 14 2.0e+8 0.0183 C( 13000,13007) Y(13017,13017) A(l,l) B(0.02,0.02) P(0.3 0.75) U61 14 2.0e+8 0.017 C(12100,12100) Y(12107,12107) A(l,l) B(0.02,0.02) P(0.3 0.75) U62 14 2.0e+8 0.0157 C(l 1100,11107) Y(l 1157,11157) A(l,l) B(0.02,0.02) P(0.3 0.75) U63 14 2.0e+8 0.0143 C(10100,10101) Y(10171,10171) A(l,l) B(0.02,0.02) P(0.3 0.75) U64 14 2.0e+8 0.0116 C(09600,09609) Y(09639,09639) A(l,l) B(0.02,0.02) P(0.3 0.75) U65 14 2.0e+8 0.00948 C(07800,07806) Y(07856,07856) A(l,l) B(0.02,0.02) P(0.3 0.75) U66 14 2.0e+8 0.00939 C(07700,07700) Y(07780,07780) A( 1,1) B(0.02,0.02) P(0.3 0.75) U67 14 2.0e+8 0.00887 C(07300,07302) Y(07362,07362) A(l,l) B(0.02,0.02) P(0.3 0.75) U68 14 2.0e+8 0.0602 C(23000,23003) Y(23063,23063) A(l,l) B(0.02,0.02) P(0.3 0.75) U70 14 2.0e+8 0.038 C(31500,31507) Y(31537,31537) A(l,l) B(0.02,0.02) P(0.3 0.75) U71 14 2.0e+8 0.036 C(29800,29807) Y(29867,29867) A( 1,1) B(0.02,0.02) P(0.3 0.75) U72 14 2.0e+8 0.035 C(29000,29000) Y(29070,29070) A(l,l) B(0.02,0.02) P(0.3 0.75) U73 14 2.0e+8 0.031 C(25700,25700) Y(25730,25730) A(l,l) B(0.02,0.02) P(0.3 0.75) U74 14 2.0e+8 0.029 C(24000,24000) Y(24060,24060) A(l,l) B(0.02,0.02) P(0.3 0.75) U75 14 2.0e+8 0.025 C(20709,20709) Y(20759,20759) A(l,l) B(0.02,0.02) P(0.3 0.75) U76 14 2.0e+8 0.0227 C(18803,18803) Y(18823,18823) A(l,l) B(0.02,0.02) P(0.3 0.75) U77 14 2.0e+8 0.0209 C( 17304,17304) Y( 17344,17344) A( 1,1) B(0.02,0.02) P(0.3 0.75) 122 U78 14 2 Oe+8 0 0192 U79 14 2 Oe+8 0 0174 U80 14 2 Oe+8 0 0163 U81 14 2 Oe+8 0 0149 U82 14 2 Oe+8 0 0136 U83 14 2 Oe+8 0 0123 U84 14 2 Oe+8 0 0111 U85 14 2 Oe+8 0 01 U86 14 2 Oe+8 0 021 U87 14 2 Oe+8 0 023 U88 14 2 Oe+8 0 0126 U89 14 2 Oe+8 0 0108 U90 14 2 Oe+8 0 0616 U91 14 2 Oe+8 0 056 U92 14 2 Oe+8 0 0198 U93 14 2 Oe+8 0 037 U94 14 2 Oe+8 0 0094 U95 14 2.0e+8 0 0102 C( 15909,15909) Y( 15939,15939) A(l,l) B(0.02,0.02) P(0.3 0.75) C(14400,14400) Y( 14460,14460) A(l,l) B(0.02,0.02) P(0.3 0.75) C(13509,13509) Y(13549,13549) A(l,l) B(0.02,0.02) P(0.3 0.75) C(12302,12302) Y(12372,12372) A(l,l) B(0.02,0.02) P(0.3 0.75) C(l 1300,11300) Y(l 1309,11309) A(l,l) B(0.02,0.02) P(0.3 0.75) C( 10200,10200) Y( 10209,10209) A(l,l) B(0.02,0.02) P(0.3 0.75) C(09202,09202) Y(09222,09222) A(l,l) B(0.02,0.02) P(0.3 0.75) C(08302,08302) Y(08312,08312) A(l,l) B(0.02,0.02) P(0.3 0.75) C( 14209,14209) Y( 14269,14269) A(l,l) B(0.02,0.02) P(0.3 0.75) C(13003,13003) Y(13093,13093) A(l,l) B(0.02,0.02) P(0.3 0.75) C( 10404,10404) Y( 10474,10474) A(l,l) B(0.02,0.02) P(0.3 0.75) C(08906,08906) Y(08956,08956) A(l,l) B(0.02,0.02) P(0.3 0.75) C(44002,44002) Y(44022,44022) A(l,l) B(0.02,0.02) P(0.3 0.75) C(44002,44002) Y(44022,44022) A(l,l) B(0.02,0.02) P(0.3 0.75) C(14000,14000) Y(14080,14080) A(l,l) B(0.02,0.02) P(0.3 0.75) C(30606,30606) Y(30626,30626) A(l,l) B(0.02,0.02) P(0.3 0.75) C(07808,07808) Y(07818,07818) A(l,l) B(0.02,0.02) P(0.3 0.75) C(08403,08403) Y(08463,08463) A(l,l) B(0.02,0.02) P(0.3 0.75) U801 1 7.7e+7 0.0266 U802 1 7.7e+7 0.0224 U803 1 7.7e+7 0.0193 U804 1 7.7e+7 0.0183 U805 1 7.7e+7 0.0173 U806 1 7.7e+7 0.0107 U807 1 7.7e+7 0.0228 U808 1 7.7e+7 0.0191 U809 1 7.7e+7 0.0174 U810 1 7.7e+7 0.0157 U811 1 7.7e+7 0.0140 U812 1 7.7e+7 0.0125 U813 1 7.7e+7 0.0110 U814 1 7.7e+7 0.00998 U815 1 7.7e+7 0.00904 U816 1 7.7e+7 0.00813 U817 1 7.7e+7 0.00722 U818 1 7.7e+7 0.00638 U819 1 7.7e+7 0.00595 U820 1 7.7e+7 0.00552 U824 1 7.7e+7 0.0213 U825 1 7.7e+7 0.00473 U850 1 7.7e+7 0.0503 U851 1 7.7e+7 0.0472 U852 1 7.7e+7 0.0439 U853 1 7.7e+7 0.0407 U854 1 7.7e+7 0.0391 U855 1 7.7e+7 0.0381 U856 1 7.7e+7 0.0355 U857 1 7.7e+7 0.0351 U858 1 7.7e+7 0.035 U859 1 7.7e+7 0.035 U860 1 7.7e+7 0.035 U861 1 7.7e+7 0.0321 U862 1 7.7e+7 0.0299 123 U863 1 7 7e+7 0.0271 U864 1 7 7e+7 0.0283 U865 1 7 7e+7 0.0232 U866 1 7 7e+7 0.0230 U867 1 7 7e+7 0.0218 U868 1 7 7e+7 0.0629 U870 1 7 7e+7 0.1197 U871 1 7 7e+7 0.1135 U872 1 7 7e+7 0.108 U873 1 7 7e+7 0.0978 U874 1 7 7e+7 0.0908 U875 1 7 7e+7 0.0784 U876 1 7 7e+7 0.0722 U877 1 7 7e+7 0.0664 U878 1 7 7e+7 0.0611 U879 1 7 7e+7 0.0553 U880 1 7 7e+7 0.0516 U881 1 7 7e+7 0.0515 U882 1 7 7e+7 0.0475 U883 1 7 7e+7 0.0429 U884 1 7 7e+7 0.0384 U885 1 7 7e+7 0.0348 U886 1 7 7e+7 0.053 U887 1 7 7e+7 0.043 U888 1 7 7e+7 0.041 U889 1 7 7e+7 0.040 U890 1 7 7e+7 0.19 U891 1 7 7e+7 0.19 U892 1 7 7e+7 0.0542 U893 1 7 7e+7 0.186 U894 1 7 7e+7 0.0413 U895 1 7 7e+7 0.0348 U901 1 2 Oe+8 0.068 U902 1 2 Oe+8 0.057 U907 14 2.0e+8 0.0889 C(30604,18309) Y(30664,18399) A(l,l) B(0.02,0.001) P(0.3 0.75) U910 14 2.0e+8 0.065 C(22301,13404) Y(22391,13434) A( 1,1) B(0.02,0.001) P(0.3 0.75) U911 14 2.0e+8 0.0591 C(20300,12208) Y(20380,12228) A(l,l) B(0.02,0.001) P(0.3 0.75) U912 14 2.0e+8 0.0536 C(18402,l 1009) Y(l8482,11089) A(l,l) B(0.02,0.001) P(0.3 0.75) U913 14 2.0e+8 0.0486 C( 16704,10004) Y( 16774,10064) A(l,l) B(0.02,0.001) P(0.3 0.75) U914 14 2.0e+8 0.0439 C(15103,09006) Y(15143,09086) A(l,l) B(0.02,0.001) P(0.3 0.75) U915 14 2.0e+8 0.0403 C(13800,08304) Y(13890,08334) A(l,l) B(0.02,0.001) P(0.3 0.75) U916 14 2.0e+8 0.0369 C( 12704,07609) Y( 12714,07629) A(l,l) B(0.02,0.001) P(0.3 0.75) U917 14 2.0e+8 0.0331 C(l 1404,06804) Y(l 1424,06854) A(l,l) B(0.02,0.001) P(0.3 0.75) U918 14 2.0e+8 0.0304 C( 10404,06204) Y( 10474,06284) A(l,l) B(0.02,0.001) P(0.3 0.75) U919 14 2.0e+8 0.0275 C(09408,05603) Y(09488,05693) A(l,l) B(0.02,0.001) P(0.3 0.75) U920 14 2.0e+8 0.0251 C(08603,05102) Y(08653,05192) A(l,l) B(0.02,0.001) P(0.3 0.75) U921 14 2.0e+8 0.0227 C(07808,04600) Y(07818,04690) A( 1,1) B(0.02,0.001) P(0.3 0.75) U922 14 2.0e+8 0.0207 C(07105,04201) Y(07135,04281) A(l,l) B(0.02,0.001) P(0.3 0.75) U923 14 2.0e+8 0.0188 C(06400,03804) Y(06490,03894) A(l,l) B(0.02,0.001) P(0.3 0.75) U924 14 2.0e+8 0.0586 C(20208,12106) Y(20228,12136) A(l,l) B(0.02,0.001) P(0.3 0.75) U931 14 2.0e+8 0.0182 C(06202,03708) Y(06262,03758) A(l,l) B(0.02,0.001) P(0.3 0.75) U932 14 2.0e+8 0.0165 C(05603,03406) Y(05693,03416) A(l,l) B(0.02,0.001) P(0.3 0.75) U933 14 2.0e+8 0.0150 C(05101,03007) Y(05161,03097) A(l,l) B(0.02,0.001) P(0.3 0.75) 124 U934 14 2.0e+8 0.0135 C(04608,02800) Y(04668,02801) A(l,l) B(0.02,0.001) P(0.3 0.75) U940 14 2.0e+8 0.0393 C(13509,08109) Y(13549,08129) A(l,l) B(0.02,0.001) P(0.3 0.75) U941 14 2.0e+8 0.0369 C( 12704,07609) Y( 12714,07629) A(l,l) B(0.02,0.001) P(0.3 0.75) U942 14 2.0e+8 0.0321 C(l 1004,06606) Y( 11044,06626) A(l,l) B(0.02,0.001) P(0.3 0.75) U943 14 2.0e+8 0.0503 C( 17304,10400) Y( 17344,10406) A(l,l) B(0.02,0.001) P(0.3 0.75) U944 14 2.0e+8 0.0474 C( 16307,09804) Y(16357,09814) A(l,l) B(0.02,0.001) P(0.3 0.75) U945 14 2.0e+8 0.0417 C(14304,08601) Y(14384,08631) A(l,l) B(0.02,0.001) P(0.3 0.75) U946 14 2.0e+8 0.0340 C(l 1707,07007) Y(l 1727,07037) A(l,l) B(0.02,0.001) P(0.3 0.75) U947 14 2.0e+8 0.0301 C( 10308,06209) Y(10398,06239) A(l,l) B(0.02,0.001) P(0.3 0.75) U950 1 2.0e+8 0.322 U951 1 2.0e+8 0.302 U952 1 2.0e+8 0.281 U953 1 2.0e+8 0.2605 U954 1 2.0e+8 0.25 U955 1 2.0e+8 0.244 U956 1 2.0e+8 0.227 U957 1 2.0e+8 0.217 U958 1 2.0e+8 0.205 U959 1 2.0e+8 0.194 U960 1 2.0e+8 0.188 U961 1 2.0e+8 0.174 U962 1 2.0e+8 0.162 U963 1 2.0e+8 0.147 U964 1 2.0e+8 0.162 U965 1 2.0e+8 0.133 U966 1 2.0e+8 0.132 U967 1 2.0e+8 0.125 U968 1 2.0e+8 0.106 U970 1 2.0e+8 0.29 U971 1 2.0e+8 0.275 U972 1 2.0e+8 0.262 U973 1 2.0e+8 0.237 U974 1 2.0e+8 0.22 U975 1 2.0e+8 0.19 U976 1 2.0e+8 0.175 U977 1 2.0e+8 0.161 U978 1 2.0e+8 0.148 U979 1 2.0e+8 0.134 U980 1 2.0e+8 0.125 U981 1 2.0e+8 0.114 U982 1 2.0e+8 .0.105 U983 1 2.0e+8 0.095 U984 1 2.0e+8 0.085 U985 1 2.0e+8 0.077 U990 1 2.0e+8 0.39 U991 1 2.0e+8 0.35 U993 1 2.0e+8 0.33 U995 1 2.0e+8 0.077 U996 14 2.0e+8 0.070 C(24208,16000) Y(24288,16030) A(l,l) B(0.02,0.001) P(0.3 0.75) U997 14 2.0e+8 0.062 C(21202,12701) Y(21252,12751) A(l,l) B(0.02,0.001) P(0.3 0.75) U998 14 2.0e+8 0.053 C( 18206,10900) Y( 18216,10930) A(l,l) B(0.02,0.001) P(0.3 0.75) U999 14 2.0e+8 0.051 C( 17407,10404) Y(17457,10474) A(l,l) B(0.02,0.001) P(0.3 0.75) 125 // initial load before step by step analysis beam X1&X7 Y l to Y3 2F to 53F, loade 49.4 beam X1&X7 Y3 to Y4 2F to 50F, loade 30.6 beamXl&X7 Y5-Y1 2F to 50F, loade 30.6 beam Y1&Y3 XI to X7 2F to 53F, loade 41.7 beam Y1&Y3 X7 to X8 2F to 50F, loade 34.6 beam Y1&Y3 X9-X1 2F to 50F, loade 34.6 /*beam Y4 and Y5, XI to X7, 2F to 50F, loade 19.9 /*beam Y4 and Y5, X7 to X8, 2F to 50F, loade 17.2 /*beam Y4 and Y5, X9-X1, 2F to 50F, loade 17.2 /•beam X8 and X9, Y l to Y3, 2F to 50F, loade 23.5 /•beam X8 and X9, Y3 to Y4, 2F to 50F, loade 15.3 /•beam X8 and X9, Y5-Y1, 2F to 50F, loade 15.3 // 

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