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Probabilistic analysis software for structural seismic response Sjoberg, Brian David 2003

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Probabilistic Analysis Software for Structural Seismic Response by Brian David Sjoberg B.Sc, The University of Calgary, 1988 M.Sc., The University of Calgary, 1999  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF  D O C T O R O F PHILOSOPHY  in T H E 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  T H E U N I V E R S I T Y O F BRITISH C O L U M B I A October 2003 © Brian David Sjoberg, 2003  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  'tfSt*r  The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  11  ABSTRACT  The evaluation of seismic reliability of building structures is a complex and computationally expensive process since it requires, at the most fundamental level, the evaluation of the probabilistic dynamic response of a given structure to the stochastic dynamic action of an earthquake. Because of the difficulty of determining the response of a structure in a statistical sense, past estimates of the seismic reliability of existing structures, and typical structural systems, have been largely qualitative in nature. With the movement of many national building codes towards more performance-based design measures, a need was identified for a more quantitative method of evaluating structural reliability under seismic loading.  To meet this need, a new software application called PSResponse  was developed  that gives engineers and researchers the ability to rigorously evaluate the probable effect of a wide range of ground motion characteristics and structural model parameters, each with their own random nature, on the dynamic response of a structure. The mathematical modeling methods forming the foundation of the software architecture were selected following  a comprehensive  review  of random vibration methods and numerical  procedures that assessed their suitability for analyzing the probabilistic seismic response of civil engineering structures. That review determined that the frequency-domain based random vibration methods are too restrictive in their inherent assumptions to confidently apply their results to real structures experiencing realistic earthquakes.  Instead, a  numerical time-history approach incorporating the Monte Carlo method provides a robust, accurate and straightforward means of evaluating the probabilistic response of a structure without regard to the degree of non-linearity in the restoring force, complexity of the structural system, nature of the variability in structural properties or nature of the random excitation process.  As part of the software development process, a new algorithm for parameter identification  of the well-known  BWBN,  or Bouc-Wen, hysteresis model  was  developed, which included a modification to the function controlling pinching behaviour  Ill  to simplify the parameter identification process. The number of pinching parameters was reduced from six to three, which has the added benefit that the role of each of the three new parameters is more easily understood than the relationship between the six parameters of the original pinching function.  Following development of the beta version of PSResponse, two case studies were completed that demonstrated the capabilities of the software as a research and analysis tool.  These case studies provided for the first time a probabilistic analysis of the  importance of the hysteresis assumption in inelastic analysis, the accuracy of the wellknown equal displacement observation in structural dynamics and the relative effect of random structural properties on elastic dynamic response.  Results showed that the  hysteretic behaviour of a structure needs to be accurately modeled, particularly in shorter natural period structures, to provide an accurate probabilistic description of response and hence a good estimate of seismic structural reliability. Also, the equal displacement principle is valid in the sense that elastic peak displacement provides a generally conservative first approximation of inelastic peak displacement, which in turn results in a generally conservative prediction of reliability. Finally, case study results showed that the characteristics and randomness of ground motion records has a much larger influence than structural randomness on the probabilistic dynamic response of a structure. Therefore, once a suitable seed record has been selected, the peak response probability distributions for a given structural model could be applied to a real structure with reasonable confidence since the assumed level of uncertainty in the structural parameters needs to be only approximately correct.  However, for strength related limit state  evaluations related to peak response, structural variability still has an important effect.  IV  ACKNOWLEDGEMENTS  I gratefully acknowledge the support of my supervisor, Dr. Helmut Prion, throughout the course of this project. His encouragement, financial support and generosity in providing resources to aid in software development were much appreciated.  I would also like to thank the members of my advisory committee; Dr. Ricardo Foschi, Dr. Frank Lam, Dr. Reza Vaziri and Dr. Carlos Ventura who acted as both my committee members and teachers throughout my years at U B C . generosity with their time were always appreciated.  Their teaching efforts and  In particular, the comments and  suggestions of Dr. Foschi and Dr. Ventura during the final development phase of PSResponse were encouraging and insightful.  Finally, I would like to thank my fiancee, Karen Evans, both for her support and understanding throughout my graduate studies, as well as her invaluable assistance in the development of PSResponse.  To Barry & Elaine Sjoberg  VI  TABLE OF CONTENTS  Page ABSTRACT  ii  ACKNOWLEDGEMENTS  iv  DEDICATION  v  TABLE OF CONTENTS  vi  LIST O F T A B L E S  ix  LIST O F F I G U R E S  x  C H A P T E R 1: I N T R O D U C T I O N  1  1.1 1.2 1.3 1.4  GENERAL OBJECTIVE SCOPE ORGANIZATION  1 4 5 5  C H A P T E R 2: L I T E R A T U R E R E V I E W  7  2.1 INTRODUCTION 2.2 M A T H E M A T I C A L M O D E L I N G OF S T R U C T U R A L B E H A V I O U R 2.2.1 Analytical Methods 2.2.1.1 Markov Methods Based on the Fokker-PlanckKolmogorov Equation 2.2.1.1.1 General Random Vibration Theory and Markov Process Theory 2.2.1.1.2 • Solution Methods for the Fokker-Planck Kolmogorov Equation 2.2.1.1.2.1 Galerkin Based Methods 2.2.1.1.2.2 Finite Element Method 2.2.1.1.2.3 Closure Techniques 2.2.1.1.2.4 Stochastic Averaging 2.2.1.1.2.5 Numerical Diffusion 2.2.1.1.3 First-Passage Problem Solution Methods 2.2.1.1.3.1 Assumption of Independent Events 2.2.1.1.3.2 Markov Process Theory 2.2.1.2 Perturbation Method 2.2.1.3 Equivalent Linearization Method 2.2.1.4 Functional Series Representation  7 8 9 9 :...9 17 18 18 19 21 24 27 28 33 34 36 40  vii  2.2.1.5 Decomposition Method 41 Numerical Methods 44 2.2.2.1 Direct Monte Carlo Simulation 45 2.2.2.2 Selective Monte Carlo Simulation 46 2.2.2.2.1 Double and Clump 47 2.2.2.2.2 Russian Roulette and Splitting 48. 2.2.2.2.3 Latin Hypercube Sampling 49 2.2.2.3 Response Surface Method 50 2.3 E A R T H Q U A K E GROUND MOTION M O D E L S 53 2.3.1 Filtered White-Noise and Filtered Poisson Process Models 54 2.3.2 Spectral Representation Method 57 2.3.3 Stochastic Wave Theory 60 2.3.4 A R M A Models 60 2.3.5 Wavelet Models 62 2.4 HYSTERESIS M O D E L S 65 2.4.1 General Hysteresis Models 65 2.5 SEISMIC S T R U C T U R A L RESPONSE A N D RELIABILITY STUDIES....70 2.2.2  CHAPTER 3: NUMERICAL ALGORITHM DEVELOPMENT 3.1 INTRODUCTION 3.2 SELECTION OF ANALYSIS M E T H O D 3.3 N U M E R I C A L M O D E L S A N D SOLUTION M E T H O D S 3.3.1 Numerical Time-Stepping Method 3.3.2 Time-Stepping Overshoot Problem 3.3.3 Structural Model 3.3.4 Hysteresis Model 3.3.4.1 Modification to the B W B N Model 3.3.4.2 Numerical Solution of Hysteresis Model 3.3.4.3 Consideration of Degrading Natural Frequency 3.3.4.4 Parameter Identification 3.3.4.5 Parameter Adjustment for M D O F Structures 3.3.5 Fourier Analysis and Power Spectrum Estimation 3.3.5.1 FFT Algorithm 3.3.5.2 Power Spectrum Algorithm 3.3.6 Acceleration Record Filtering 3.3.7 Random Number Generation  CHAPTER 4: SOFTWARE FRAMEWORK AND USER-INTERFACE 4.1 4.2 4.3 4.4  75 75 75 77 77 79 80 82 82 85 88 90 96 96 97 98 104 105  109  INTRODUCTION COMPUTATIONAL FRAMEWORK USER-INTERFACE STRUCTURE SOFTWARE FEATURES  109 109 112 114  4.4.1 4.4.2  116 117  General Input Parameters Dialog Box Multiple Earthquake Analysis Parameters Dialog Box  4.4.3 4.4.4 4.4.5 4.4.6 4.4.7 4.4.8 4.4.9 4.4.10 4.4.11 4.4.12 4.4.13 4.4.14 4.4.15 4.4.16 4.4.17 4.4.18 4.4.19 4.4.20  Single-Degree-of-Freedom Properties Dialog Box Multiple-Degree-of-Freedom Shear Properties Dialog Box Multiple-Degree-of-Freedom Frame Properties Dialog Box Probability Distribution Example Dialog Box Hysteresis Parameter Identification Dialog Boxes Modal Damping Parameters Dialog Box M D O F Damping Parameters Dialog Box Oscillation Motion Parameters Dialog Box Single Ground Motion Parameters Dialog Box Ground Motion Generation Parameters Dialog Box (Filtered) Ground Motion Generation Parameters Dialog Box (Real) Elastic Responses Dialog Box SDOF Inelastic Responses Dialog Box M D O F Inelastic Responses Dialog Box Plot Modal Results Dialog Box Plot SDOF Results Dialog Box Plot M D O F Results Dialog Box Print Results Dialog Box  118 119 120 121 122 125 127 128 129 131 133 135 136 137 138 141 143 146  CHAPTER 5: SOFTWARE VERIFICATION AND CASE STUDIES  151  5.1 INTRODUCTION 5.2 S O F T W A R E VERIFICATION 5.2.1 Elastic Analysis 5.2.1.1 Single-Degree-of-Freedom Systems 5.2.1.2 Multiple-Degree-of-Freedom System 5.2.2 Inelastic Analysis 5.2.2.1 Single-Degree-of-Freedom Systems 5.2.2.2 Multiple-Degree-of-Freedom System 5.3 C A S E STUDIES 5.3.1 Case Study #1 5.3.2 Case Study #2  151 151 154 154 156 159 159 164 170 171 184  CHAPTER 6: SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 6.1 S U M M A R Y 6.2 CONCLUSIONS 6.3 R E C O M M E N D A T I O N S FOR F U R T H E R R E S E A R C H  REFERENCES  210 210 218 222  223  IX  LIST O F T A B L E S  Page Table 3.1  Power Spectrum Frequency Resolution  101  Table 5.1  Comparison of Peak Deformation Responses of SDOF Systems  156  Table Table Table Table Table Table Table Table Table Table Table Table Table Table  Comparison of Elastic Peak Responses of Five-Storey Shear Frame 158 Comparison of Inelastic Peak Responses of SDOF Systems 163 Comparison of Inelastic Peak Responses of Five-Storey Shear Frame 165 Peak Response Statistics of a Five-Storey Elastic Shear Frame 172 Elastic Peak Response Distribution Comparison Statistics 177 Statistics for the Percentage Difference between R H A & RSA Results .... 179 Hysteresis Model Descriptions 186 SDOF Peak Deformation Response Statistics 187 Gumbel Distribution Parameters for Peak Deformation 199 Displacement (Drift) Limit Exceedence Probabilities 200 Displacement (Drift) Limit Reliability Indices 201 Percentage Difference Statistics Inelastic vs. Elastic Peak Deformation...204 Gumbel Distribution Parameters for Percentage Difference Results 205 Zero Percent Difference Exceedence Probabilities 206  5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15  X  LIST O F F I G U R E S  Page Fig. Fig. Fig. Fig.  2.1 2.2 2.3 2.4  System Response and White Noise Approximation First-Passage Time and Domain Barriers in the Phase Plane Filtered White-Noise Power Spectrum Example Amplitude Modulating Function  14 28 55 56  Fig. 3.1 Fig. 3.2 Fig. 3.3  B W B N Model Pinching Function Comparison Exact Solution of Non-Pinching, Non-Degrading B W B N Model Square Window, Bartlett Window and Fourier Transforms  86 87 102  Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.  Computational Framework - Initialization , Computational Framework - Multiple Record Analysis (Filtered) Computational Framework - Multiple Record Analysis (Real) Computational Framework - Run Multiple Records Computational Framework - Solvers Computational Framework - Generate Random Properties User- Interface Framework General Input Parameters Dialog Box Multiple Earthquake Analysis Parameters Dialog Box Single-Degree-of-Freedom Properties Dialog Box Multiple-Degree-of-Freedom Shear Properties Dialog Box Multiple-Degree-of-Freedom Frame Properties Dialog Box Probability Distribution Example Dialog Box Hysteresis Parameter Identification Dialog Box (Quasi-Static Data) Hysteresis Parameter Identification Dialog Box (Acceleration Data) Modal Damping Parameters Dialog Box Modal Properties Pop-Up Dialog Box M D O F Damping Parameters Dialog Box Oscillation Motion Parameters Dialog Box Single Ground Motion Parameters Dialog Box Ground Motion Statistics Pop-Up Dialog Box Ground Motion Generation Parameters Dialog Box (Filtered) Ground Motion Generation Parameters Dialog Boxes (Real) Elastic Responses Dialog Box Storey Selection Example Dialog Box SDOF Inelastic Responses Dialog Box M D O F Inelastic Responses Dialog Box Plot Modal Results Dialog Box Input Motion Time-History Pop-Up Dialog Box Input Spectrum Pop-Up Dialog Box Response Spectra Pop-Up Dialog Box Modal Response Time-History Example Pop-Up Dialog Box  110 Ill Ill 112 113 114 115 116 117 118 119 120 121 122 124 125 126 127 128 129 130 131 133 135 136 136 137 138 139 140 140 141  4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32  XI  Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.  4.33 4.34 4.35 4.36 4.37 4.38 4.39 4.40 4.41 4.42 4.43 4.44 4.45  Plot SDOF Results Dialog Box Hysteresis Parameter Identification Dialog Box (Quasi-Static Data) Hysteresis Parameter Identification Dialog Box (Acceleration Data) Plot M D O F Results Dialog Box M D O F Response Time-History Example Pop-Up Dialog Box M D O F Response Time-History Example Pop-Up Dialog Box M D O F Response Time-History Example Pop-Up Dialog Box Print Results Dialog Box Analysis Summary Report Random Natural Frequency Array Printed Output Example Response Time-History Printed Output (Elastic Analysis) Response Spectra Data Printed Output Example Multiple Record Analysis Printed Output (Elastic Analysis)  Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.  5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9  SDOF Elastic Deformation Response - PSResponse vs. SAP2000 155 Five-Storey Elastic Shear Frame 156 Five-Storey Shear Frame Elastic Response - PSResponse vs. SAP2000.... 157 SDOF Inelastic Deformation Response - PSResponse vs. SAP2000 160 SDOF Inelastic Shear Response - PSResponse vs. SAP2000 161 SDOF Hysteresis Loops - PSResponse vs. SAP2000 162 SDOF Dissipated Energy - PSResponse 164 Five-Storey Inelastic Shear Frame 165 Five-Storey Shear Frame Inelastic Deformation Response PSResponse vs. SAP2000 166 Five-Storey Shear Frame Inelastic Shear Response PSResponse vs. SAP2000 167 Five-Storey Shear Frame Hysteresis Loops - PSResponse vs. SAP2000 ... 168 Five-Storey Shear Frame Dissipated Energy - PSResponse 169 Base Shear Response Histograms 173 Base Moment Response Histograms 174 Storey 5 Displacement Response Histograms 175 Peak Response Distributions 176 Error Distribution for C Q C & SRSS vs. R H A Peak Responses 183 E l Centro Response Spectra 184 Cyclic Test of Parallam® Column and Best-Fit Hysteresis Model 185 SDOF Hysteretic Response with T = 8.0 sec, Damping = 2% 187 Displacement Response Histograms, T = 0.1 sec, Damping = 2% 189 Displacement Response Histograms, T = 0.2 sec, Damping = 2% 190 Displacement Response Histograms, T = 0.4 sec, Damping = 2% 191 Displacement Response Histograms, T = 0.5 sec, Damping = 2% 192 Displacement Response Histograms, T = 0.6 sec, Damping = 2% 193 Displacement Response Histograms, T = 1.0 sec, Damping = 2% 194 Displacement Response Histograms, T = 3.0 sec, Damping = 2% 195 Displacement Response Histograms, T = 8.0 sec, Damping = 2% 196 Hysteresis Model Comparison of Peak Deformation Distributions 198 Percentage Difference Histograms Inelastic vs. Elastic 207  Fig. 5.10 Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.  5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.30  141 142 143 143 144 145 145 146 148 149 149 150 150  Xll  Fig. 5.31  Percentage Difference Distributions Inelastic vs. Elastic  208  Introduction  Chapter 1  1  CHAPTER 1  INTRODUCTION  GENERAL  1.1  The field of earthquake engineering has existed since the Middle Ages when many European, Chinese and Japanese structures were constructed with post and beam systems employing complex joinery and/or diagonal bracing for lateral load resistance. These construction methods,  as well as lightweight  timber materials, structural  redundancy and even passive control devices, allowed structures to sustain large deformations under earthquake loading and effectively dissipate the input seismic energy. Using these techniques, some churches and temples that were constructed as much as 1000 years ago still stand today (Popovski 2000).  The  successful  seismic performance of these ancient structures is more a  testament to the craftsmanship ofthe original builders and generations of experience than to a fundamental understanding of seismic loading on soils and structures. The field o f earthquake engineering, as we know it today, really began developing in the latter half o f the 20 century as seen from the first inclusion of seismic loading provisions in the main th  text ofthe National Building Code of Canada (NBCC) in 1953. These provisions, which specify both the seismic design forces and the design and detailing requirements for lateral load resisting structural systems, have been updated approximately every five years since 1965 resulting in the current 1995 version of the N B C C .  In the current version of the N B C C the seismic design provisions provide estimates of peak ground acceleration and peak ground velocity for various regions o f the country resulting from an earthquake that has a 10 percent probability of exceedence in 50 years.  These ground motion parameters are then used in conjunction with simple  Chapter 1  Introduction  2  formulae to determine a distribution of static lateral forces for which the structure should be designed. The resulting forces are considered equivalent to the dynamic forces of an earthquake and any structure designed to resist these forces should be able to resist minor levels of earthquake ground motion without damage, resist moderate levels of earthquake ground motion without structural damage but possibly with some non-structural damage, and resist major levels of earthquake ground motion without collapse but with some structural as well as non-structural damage. These qualitative levels of performance are contingent on the design allowing for stress reversals, providing adequate member ductility and providing connections with adequate strength and resilience.  Included in  the various formulae used to determine the design lateral forces are the use of several factors to account for; inelastic behaviour (R), relative importance of the structure (I), and site soil effects (F).  The prescriptive procedures and various factors outlined in the 1995 N B C C are based on years of past proven experience and are easy to apply. As it concerns a matter of life safety, this approach has been justified on the grounds that any major change in building practice can lead to unexpected risks.  It was this view that led to the  introduction of a calibration factor, the so-called 'over-strength' factor (U = 0.6), in the 1990 N B C C to maintain the same seismic design forces when a new formulation for base shear was adopted. The drawback of a prescriptive procedure is that it oversimplifies a very complex problem and it does not allow the reliability of a design to be quantified.  In  1992,  the  Canadian National Committee  on Earthquake Engineering  (CANCEE), which has the responsibility of preparing and recommending the seismic loading provisions of the N B C C , recognized that major changes would be necessary for the 2000 N B C C . A resolution was approved which stated (Heidebrecht et al. 1995):  "that C A N C E E place a very high priority on a major redevelopment of the seismic loading provisions of the 2000 edition of N B C C , with particular emphasis on:  Chapter 1  Introduction  3  1. Developing a format suitable for utilizing seismic hazard expressed as uniform hazard spectra, 2. Evaluating the appropriateness of the level of protection (expressed as the minimum lateral seismic force) by comparison with that used in various U.S. codes (UBC, S E A O C and NEHRP), including comparisons of seismic hazard determined at points along the Canada-U.S. border.  and that C A N C E E maintain close linkages with various U.S. code development projects in order to benefit from their experience".  To implement this resolution, C A N C E E established the N B C C 2000 Task Force to generate a redevelopment plan and coordinate code development work. In 1993, the task force published a list of six major issues that needed to be addressed:  1. Seismic loading format suitable for utilizing spectral ordinates determined from seismic hazard analysis, 2. Evaluation of the current level of protection, 3. Role of different design (or performance) levels in the code, 4. Development of direct site spectra to recognize different site soil conditions, 5. Development of design requirements for low to moderate seismic hazard zones, 6. Explicit or implicit recognition of over-strength in seismic design.  In recognizing these issues, C A N C E E is clearly moving the N B C C towards a performance-based design code, perhaps with the intent of establishing  specific  probabilistic performance targets for a code designed structure in a seismic event. To enable the evaluation of structural performance, the current design objectives of: no damage, moderate structural damage and prevention of collapse in the event of minor, moderate and major levels of earthquake ground motion, respectively, need to be expressed as limits on quantitative terms such as: a damage index, overall lateral deflection or inter-storey drift.  In addition, the evaluation of performance, and  Chapter 1  Introduction  4  specifically the level of protection, implies the use of reliability-based methods to express the risk of non-performance (i.e. exceedence of limits) in probabilistic terms.  1.2  OBJECTIVES  Evaluating the reliability or risk of non-performance of a structural system under seismic loading requires, at the most fundamental level, a method to evaluate the probabilistic dynamic response of a given structure to the stochastic dynamic action of an earthquake. Once the response of the structure is known in a statistical sense, the task of determining structural reliability is a matter of using the statistical description of structural response to determine the probability of exceeding the chosen limits on the quantitative terms being used to assess structural performance.  Evaluation of the probabilistic response of a structure to stochastic dynamic loading is a complex and computationally expensive process.  For this reason past  estimates of the seismic reliability of existing structures, and typical structural systems, have been largely qualitative in nature. With the movement towards more performancebased design codes a need exists for a more rigorous and convenient method of evaluating structural reliability under seismic loading.  This need is the basis for the  present research project, which has the following objectives:  1. Evaluate the various analytical and numerical methods that have been developed to predict the response of linear and non-linear systems under stochastic dynamic actions. 2. Identify a method suitable for probabilistic analysis of the seismic response of civil engineering structures. 3. Develop software to enable application of the chosen probabilistic analysis method for use by engineers and researchers in evaluating structural reliability under seismic loading.  Chapter 1  1.3  Introduction  5  SCOPE  Since the objectives of this study relate to the evaluation of structural reliability under seismic loading, it is given that the study and related software development focus on  probabilistic dynamic response  acceleration events.  of  structures  to  earthquake-induced ground  Dynamic loading in the form of wind-induced vibration, blast  loading or impact loading is not considered, although the mathematical descriptions of the structural response to those types of loading are very similar. In addition, the type of structures that are considered in this study are those that may be represented by typical lumped mass models consisting of shear walls or frames as the lateral load resisting elements.  Irregular structures consisting of complex geometry with significant  distributed mass are outside the scope of this investigation.  1.4  ORGANIZATION  This thesis contains six chapters that are divided as follows; Chapter 2 contains a literature review of the research related to different aspects of modeling the response of linear and non-linear systems to stochastic dynamic loading. Mathematical modelling techniques are subdivided into the frequency domain based analytical methods and time domain based numerical methods and each is reviewed for its merits and limitations. Following the description of the numerical methods, which require sample functions of the input stochastic process (i.e. earthquake) and a hysteresis model to describe the inelastic restoring force in the dynamic equation of motion, earthquake ground motion models and hysteresis models are reviewed. Chapter 2 concludes with a summary of past seismic structural response and reliability studies. Chapter 3 summarizes the reasons for choosing the numerical approach in modeling the probabilistic response of structural systems  and  details  the  development  Computational issues associated  of  the  required numerical algorithms.  with time-history analysis, hysteresis  modeling,  earthquake generation and filtering, random number generation and solution of the structural dynamics eigenvalue problem are discussed.  Chapter 4 outlines the basic  framework ofthe software application that was developed, tentatively titled  PSResponse,  Introduction  Chapter 1  6  along with a description of the structure of the user interface overlaying the computational framework.  Following the description of the overall architecture of the  software, each of the key features available to the user and the appearance of the user interface are illustrated with screen captures taken from each type of dynamic analysis. The screen captures are accompanied in most cases by an explanation of the required and optional user inputs. Chapter 5 summarizes the verification process that was carried out to establish the accuracy of the software and presents two case studies that were done to demonstrate the capabilities of the program as a research and analysis tool. Finally, Chapter 6 summarizes the development of PSResponse, summarizes the conclusions that were drawn from the case studies of Chapter 5 and discusses possible future developments for the software.  Literature Review  Chapter 2  CHAPTER  7  2  LITERATURE REVIEW  2.1  INTRODUCTION  The application of probabilistic methods in the field of structural dynamics has followed from the original engineering application of these methods in the aerospace industry for the study of random phenomena such as; engine noise related acoustic fatigue failure, fluctuating airframe stresses associated with wind gusts, and landing gear stresses (Lin 1967). Probabilistic methods and their underlying theory originated in the initial work of physicists on the theory of Brownian motion, which was developed in the early years of the 20 century. Since that time, researchers in the fields of physics, th  engineering and mathematics have expanded the theory and application of probabilistic methods into a vast field of its own that crosses back and forth over traditional discipline boundaries. Section 2.2 of this chapter summarizes the key research related to the mathematical modelling of probabilistic structural behaviour that has been carried out over the past 50 years. A complete listing of all relevant research was not attempted due to the sheer volume of work that has been done on the subject. Rather, attention was focused on highlighting the work that most directly formed the background for further research in each area up to the present time. Sections 2.3 and 2.4 summarize the models that have been developed by researchers for the generation of artificial ground motion time-histories and representation of a non-linear, hysteretic restoring force in the dynamic equation of motion. These models are not included in Section 2.2 since they are not probabilistic  Chapter 2  Literature Review  8  methods per se, instead they are a necessary component of the numerical methods detailed in Section 2.2.  The final section of this chapter briefly reviews past seismic structural response and reliability studies that have utilized one or several of the mathematical modeling techniques summarized in Sections 2.2 - 2.4.  2.2  M A T H E M A T I C A L MODELLING OF STRUCTURAL BEHAVIOUR  The unpredictable nature of earthquakes in their arrival time, magnitude, duration and frequency content means that the dynamic loading on a structure cannot be described by a deterministic function of time. Seismic loading can only be defined in a statistical sense, a so-called random excitation, and the resulting structural response to this loading is also a random function of time.  The analysis of structural response to random  vibration is termed stochastic dynamic analysis, or more commonly, random vibration  analysis.  In the event that seismic structural response stays within the linear-elastic range, as in the case of small earthquakes, the structural restoring force is easy to model and the response statistics or probability densities can be obtained by well-developed linear random vibration methods in the frequency domain. However, to study the performance of structures under severe ground motion, where inelastic response behaviour occurs as a result of plastic deformation of structural elements and connections, linear random vibration methods no longer apply and the changing nature of the structural restoring force needs to be taken into consideration.  It is well known that when a structure  becomes inelastic the restoring force becomes highly non-linear and hysteretic whereby it depends on the prior history of motion of the system and whether the deformation is increasing or decreasing. In addition, the restoring force may deteriorate in strength or stiffness, or both, as the random vibration progresses.  With this reality in mind, the  modeling of the restoring force and the subsequent response analysis of an inelastic  Literature Review  Chapter 2  9  structure to random vibration is a difficult problem for which very few exact solutions, in the probabilistic sense, exist.  The various analytical and numerical methods that have been developed to predict the response of non-linear systems under stochastic dynamic loading are reviewed in the following sections.  The merits and limitations of each method are included for the  purpose of identifying which one may best suit the modeling of civil engineering structures.  2.2.1  Analytical Methods  Analytical procedures are all termed random vibration methods. As in the case of linear elastic systems, the analysis is carried out in the frequency domain, which, at least in principle, yields the complete response statistics to a random loading.  2.2.1.1  Markov Methods Based on the Fokker-PIanck-Kolmogorov Equation  2.2.1.1.1  General Random Vibration Theory and Markov Process Theory  A Markov stochastic process is termed a one-step-memory random process since it has the property that its present state is only dependent on its immediate past state (Lin 1967). For a discrete random process X(t), this property is expressed in the following relationship between the conditional probability functions:  p(x„, t„\x„-i,  tn-ll  X2, t2i Xj,  ti) = p(x , t„\x .j, n  n  t„-i)t„>t„-l>  t2>ti  [1]  The conditional probability function appearing on the right-hand side of [1] is called the transition probability function ofthe Markov process X(t).  A discrete Markov process,  usually called a Markov chain, is completely defined by its first probability function p(x/, t/) and the transition probability function.  If the initial state of a Markov process is  known, p(Xo = xo) - 1, which is a common situation in many practical applications, the process is then completely characterized by its transition probability.  Chapter 2  Literature Review  10  In the case of structural dynamics, where random processes are continuously valued, the transition probability function is called the transition probability density and analogous to the discrete case, a continuous Markov process, which is sometimes called a diffusion process, is specified in any one of three equivalent ways:  1. by the first probability density p(xi, tj) and the transition probability density p(x2, t2\Xl,  tj)  2. by the second probability densityp(x/, tp, X2, t ) = p(x/, t/) p(x , t \x\, t\) 2  2  2  3. by the transition probability density if the initial value of the random process is known at least with probability 1.  The transition probability density of a Markov process is governed by an integral equation that is given as follows. For an arbitrary continuous scalar random process,  p(xi, t \xi, ti) = fp(x , t ; x, t\xi, ti) dx 2  2  2  = fp(x, t\xi, ti) p(x , t \x, t; x,, ti) dx 2  2  [2]  However, if X(t) is Markovian and the transition probability density is denoted by q , Eq. x  [2] reduces to:  q (x , t |xi, ti)= 1 q (x, t|x tj) q (x , t |x, t) dx x  2  2  x  h  x  2  2  [3]  Equation [3] is known as the Chapman-Kolmogorov-Smoluchowski equation for a onedimensional random process and it is readily extended to a vector random process. Solutions to this integral equation, whether scalar or vector-valued, are frequently obtained by solving an equivalent partial differential equation called the Fokker-PlanckKolmogorov (FPK) equation that describes the evolution of the transition probability density function. For a one-dimensional Markov process this equation is given by:  Literature Review  Chapter 2  - ^ + - ( ^ J - ^ ( ^ J dt dx ' 2\dx 1  x  K  x  2K  x !  + - ^ 3\dx i  r  11  ( Q j - - - ^ ( / J ^ )  3K  x !  4! dx"  V  +  =0  [4]  x  where the coefficients A, B, C, D etc. are called the derivative moments, which give the rates of various moments of the increment in X(t) conditional on X(t) = x. The derivative moments can be written as:  A = A(x,t) = lim — E[x{t + At)- X{t)\ X{t) = x] A<-0 A?  B = B(x,t)= lim — E{x(t + At)-X(t)]  \X(t) = x  2  Ar->0  [5]  C = C{x,t) = lim — E\x(t + At)- X{t)J \ X(t) = x) A(->0  D = D(x,t) =  lun-^Elx(t  + At)-X(t)]  4  \ X(t) = x)  The F P K equation for the transition probability density of a Markov vector random process is identical except that each scalar transition probability density q is x  replaced by the vector transition probability density qf j with the number of dimensions x  equal to the number of state variables in the underlying stochastic process.  The F P K  equation is also known as the Kolmogorov forward equation where the adjective 'forward' refers to the fact that the time derivative in the equation is a derivative with respect to the later time.  There is a corresponding Kolmogorov backward equation,  which is the adjoint of the forward equation, where the time derivative is with respect to the earlier time.  It has been shown (Lin 1967) that the response vector of a non-linear system, whether single degree of freedom or multi-degree of freedom, under the excitation of a shot-noise or a filtered shot-noise is a Markov vector process. A shot-noise S(t) is a random process in which the mean and auto-covariance functions are given by:  Chapter 2  Literature Review  12  (t) = E[s{t)]=0  Ms  >^fe>0=^fe) [S(*X I S(t )] = £ M ,  *SS fe ,t )=*2 2  2  fe, f ) = R fe, r )  K  2  SS  ss  2  )-  M  s  fe )\S(t ) - fl fe )]} 2  [6]  s  - fe )p fe) s  * ( f r ) = £[s(r)s(f + r)] K  p  2  where K2 is the second cumulant function, which is equivalent to the second central moment function, Rss is the auto-correlation function which in the case of a zero mean process such as a shot-noise is equivalent to the auto-covariance function, I(t) is the intensity function of the shot-noise and 6(x) is the Dirac delta function. In the case of a weakly stationary shot-noise, which by definition has a mean function that is independent of time and a correlation function that is dependent only on the time difference r = ti - t2, the intensity function is time invariant and therefore constant which results in the auto-correlation function being an impulse at time t - 1. The spectral density of the weakly stationary shot-noise, which by the well-known WienerKhintchine theorem is the Fourier transform of the auto-correlation function  RSS(T),  is  then given by:  [7]  Therefore, the spectral density is constant. A weakly stationary random process with a constant spectral density is called a white-noise.  The physical interpretation of a  constant spectral density is that the energy content in the random process is uniformly distributed over the entire frequency range. The mean-square value of a random process, which is a measure of the average energy content, is given by:  Chapter 2  Literature Review  E\x (t)\=x 2  13  = l i m - \x dt = R  2  2  0 also R-xx ( ) T  =  2faxx{a>)cos(o)r)dco 0  [8]  CO  R (0) = 2\<!> (a))da> xx  xx  0  and  a  =x -(xj  2  2  Therefore, the mean-square value of a white-noise, which is the area under the spectral density curve Oxx(o), is equivalent to the variance of the zero mean process and is unbounded. Since a physically realizable random process cannot have an infinite average energy the white-noise process is a mathematical idealization. Similarly, non-stationary shot-noise and filtered shot-noise, which is shot-noise passed through an appropriate linear filter, are also mathematical idealizations and, strictly speaking, physically impossible.  However, response analysis in which the excitation is modeled as non-  stationary shot-noise,  filtered shot-noise or white-noise,  leading to a Markovian  response vector {Z, Z}, which represents the random displacement Z(t) and random velocity Z(t) of each degree of freedom, can give meaningful results.  To determine whether or not the mean-square output of a system computed by approximating an actual excitation spectral density by a white-noise spectral density is a good approximation to the actual mean-square output, a single-degree-of-freedom linear structure will be considered. Classical linear random vibration theory states:  S  Y  (a>) = H{CO)H'{CO)-S  x  (CO) = \H(af  •S  X  {a)  [9]  where Sy(a>) is the output spectral density, Sx(co) is the input spectral density and \H(a>)\  2  is the system transfer function or transmittancy function or frequency response function. In the case of an SDOF system, the system transfer function is given by:  Literature Review  Chapter 2  l"M =m 2  2  ( 2  (ffl  0  14  y , ' -co ) +{2£co (o) 2  [ 1 0 ]  v  0  For a white-noise input, where the spectral density is a constant = So, the mean-square output is then given by:  E[Y }= ]sAco]dco=\\H(cofs {co)dc ^R {0) 2  x  A  Yr  =-  ^  T  [11]  Therefore, the mean-square output of a damped system (£, > 0) is finite even when the mean-square input is infinite as is the case for an ideal white-noise. Also, for a lightly damped system, which is applicable to most practical civil engineering structures, the transfer function is sharply peaked at the undamped natural frequency, ©o, of the structure and the system acts like a narrow-band filter. Therefore, the major contribution to the integral in Eq. [11] is obtained in the vicinity of the natural frequency of the structure and the value of the input spectral density, Sx(co), outside that vicinity is unimportant. Assuming the spectral density of the actual excitation is slowly varying in the vicinity of the structure's natural frequency, it may be reasonably approximated by a white-noise spectral density (see Fig. 2.1).  System Response Actual Excitation White Noise  Frequency  Figure 2.1: System Response and White Noise Approximation  Chapter 2  Literature Review  15  For the assumption of a non-stationary shot noise as an approximation to the actual excitation it is not possible to state in mathematical terms the conditions under which the approximation is acceptable, as was done in the case of the white-noise idealization. The case of a non-stationary shot-noise, however, is relevant particularly in seismic response analysis since earthquake ground motions are non-stationary random processes.  The non-stationary nature of earthquakes is clearly evident in a typical  acceleration-time record, which shows a general trend of first increasing and then decreasing in intensity with time. Also, the correlation in the ground motion for two time instants, say, one second apart is clearly not constant throughout the record. A stationary process by definition has an auto-correlation function which is constant for a given time separation x. If the non-stationary shot-noise excitation is modeled as a sequence of random impulses with independent arrival times (i.e. Poisson distributed with a non stationary arrival rate) and independent amplitudes, then the characteristics of the structure determine whether it will respond in the same manner to the random impulses as under a real earthquake. The effect would be approximately the same if the average time spacing between independent impulses substituting for the actual earthquake record were 'short' as sensed by the structure. The response of a single-degree-of-freedom structure to an impulse has the form of damped free vibration, so a useful measure of the 'shortness' of the average time spacing between independent impulses is to compare that spacing with the damped natural period of vibration ofthe structure. The assumption that a real earthquake can be modeled as non-stationary shot noise will be valid if the damped natural period is, say, an order of magnitude longer than the average time spacing between impulses.  When the damped natural period of a structure is not much longer than the time separation for which the earthquake record is essentially uncorrelated, a model more general than non-stationary shot-noise is required to model the earthquake ground motion. In this case, however, the response of the structure to the excitation is no longer a Markov vector process.  With this understanding, the superposition of random pulses  with an assumed pulse shape and independent arrival times and amplitudes can be used in this situation and the general theory of random pulses can then be applied. It should be  Chapter 2  Literature Review  16  noted that shot-noise is simply a special case of the superposition of random pulses in which the pulse shape is given by the impulse function 5(t - x). More detail regarding the modeling of earthquake ground motions using random pulses is given in Section 2.3. Using random pulse theory, the variance function of a random pulse train with a sinewave pulse shape is shown to increase with time to a maximum and then decrease with time to zero. This shows that the average energy in the simulated excitation increases and then decreases with time in the same manner as a typical earthquake record. From this evidence it is concluded that random pulse trains can be used to model earthquakes in the event that non-stationary shot-noise is not a suitable model due to the short damped natural period of a given structure. In this case, however, the response of the structure is no longer Markovian.  Given that an actual excitation can reasonably be modeled as a shot-noise, either stationary or non-stationary, the response of a non-linear system is a Markov vector process, as was noted previously. In the case of a single degree of freedom non-linear system, the governing differential equation is given by:  [12]  where g is a non-linear function of the displacement Z and velocity Z. Let Z = Yj and Z = Y , the differential equation of motion is then equivalent to two first-order equations 2  given by:  Y Y ]=  2  [13]  Y =-g(Y„Y )+S{t) 2  2  where the shot-noise S(t) has replaced F(t). If S(t) is Gaussian, then the FPK equation governing the transition probability density q{Y}(Y, t|Yo, to) is given by:  =0  [14]  Literature Review  Chapter 2  17  where I(t) is the intensity function of the shot-noise. A Gaussian or normal process is a random process whose probability distribution is completely defined by its mean function, px(t), and its covariance function, K x(t). Equivalently, a Gaussian random X  process may be defined as one in which the cumulant functions higher than second order are equal to zero. With this property, a weakly stationary Gaussian process is also strongly stationary since all moments higher than second order are zero. The assumption of a Gaussian random process is frequently used in random vibration analysis since many real phenomena can be satisfactorily modeled using that assumption.  This is a  consequence of the Central Limit Theorem, which states that when a random process is the sum of a large number of independent random processes, it approaches a normal process, regardless of the distribution of the individual constituents, as the number of independent constituents increases without limit. In the case of a shot-noise, the process tends to a Gaussian distribution as the arrival rate of the impulses tends to infinity. Unfortunately, even with the assumption of a Gaussian process, which is further limited to the assumption of a white-noise process, no exact solution has been found for the FPK equation, Eq. [14].  2.2.1.1.2  Solution Methods for the Fokker-Planck-Kolmogorov Equation  A number of solution techniques have been developed to approximate the transition probability density function in the FPK equation.  The assumption of a  stationary response, in which the first term in the equation is neglected, was initially used (Caughey 1963) to develop an analytical solution for the joint probability density between the stationary displacement and velocity of a single-degree-of-freedom system. However, this method relied on a non-hysteretic assumption for the non-linear stiffness function and the stationary assumption was too restrictive in scope for short-term transient processes. For this reason, a number of more general numerical methods were developed to solve the FPK equation, some of which are briefly summarized below.  Chapter 2  2.2.1.1.2.1  Literature Review  18  Galerkin Based Methods  The Galerkin method is one of the weighted residual methods that are used for approximating the solution to a differential equation governing the behaviour of a continuous system (Bathe 1996).  The Galerkin method uses trial functions that are  identical with the weighting functions and the integral of the product of the weighting functions and the residual is set to zero to determine the coefficients for each trial function.  Stratonovitch (1964) and Atkinson (1973) used eigenfunctions of the FPK  equation of a linear system as trial functions while Bhandari and Sherrer (1968) used Hermite polynomials as trial functions to represent the transition probability density function under a white-noise excitation. In each case the solution assumed a stationary response for a weakly non-linear one or two-degree-of-freedom system, for which the associated F P K equation is of two and four dimensions respectively.  Wen (1975)  extended these results using Hermite polynomial trial functions to include the transient response using a filtered Gaussian shot-noise to take the non-stationarity and spectral content ofthe excitation into consideration. Later Wen (1976) extended his results to hysteretic systems using a smooth differential equation model for hysteresis first proposed by Bouc (1963) that was compatible with the Markov-Galerkin formulation of the FPK equation (see Sec. 2.4).  The disadvantage of the Galerkin based approach is the low rate of convergence for highly non-linear systems and the complexity of the integrals involved in cases where the non-linearities do not take the form of polynomials. To improve the convergence rate for this type of solution of the FPK equation, Soize (1988) proposed a method that allowed for the steady-state solution of systems of higher order, say, 10 to 20.  2.2.1.1.2.2  Finite Element Method  The Finite Element Method can be regarded as an extension of the classical weighted residual methods. Using a Finite Element solution method for the stationary FPK equation, Langley (1985) and Langtangen (1991) integrated the weighted residual  Chapter 2  Literature Review  19  statement for the problem to produce the weak form of the FPK equations and then chose weighting (or shape) functions defined for finite regions (or elements) of the problem domain. In this way, a set of linear equations was constructed in which the unknowns are the values of the joint probability density function at a number of points (or nodes) in the domain.  These equations are then solved by standard matrix methods to yield the  stationary transition probability density function of the response process.  Spencer and Bergman (1985) developed a Finite Element algorithm to solve the FPK equation for smooth hysteretic stationary systems and then later applied the algorithm to the transient FPK equation and obtained a solution for the evolution of the transition probability density function for two classical non-linear second order oscillators, the Duffing Oscillator and the Van der Pol Oscillator, subjected to an additive white-noise excitation (Bergman and Spencer 1991). An external excitation, which in general is represented by g(Z, Z) -F(t), is characterized as additive if the accompanying coefficient g(Z, Z) is just a constant, whereas a multiplicative excitation is displacement and/or velocity dependent resulting in a non-constant coefficient g(Z, Z).  The drawback of the Finite Element solution method is that in practice it is limited to single-degree-of-freedom systems due to the high computational effort required for systems of higher than two dimensions.  Also, as with all FPK equation  based solution methods, the excitation is assumed to be a stationary Gaussian process (i.e. white-noise).  2.2.1.1.2.3  Closure Techniques  The problem of closure arises frequently when analyzing the response of a nonlinear system to random excitation.  The differential equations that describe certain  moments of the non-stationary response contain higher moments and when additional equations for the higher moments are derived they contain even higher moments.  A  closure technique refers to a procedure by which the infinite hierarchy of differential equations governing the statistical moments of a random vibration response process is  Chapter 2  Literature Review  20  truncated at some order. The higher order terms in the remaining lower order moments are then expressed in terms of the lower order moments.  In the case of a stationary  response, the differential equations reduce to a set of algebraic equations.  The derivation of the infinite hierarchy of differential equations for moments or cumulants of the response process utilizes the F P K equation in terms of the Ito type equation.  Ito (1951) showed that a Markov diffusion process X(t) is governed by a  stochastic differential equation given by:  dX(t)= ju(x,t)dt + cr(x,t)dW(t)  [15]  where p and a are the drift and diffusion coefficients of the process and W(t) is a unit Wiener process describing Brownian motion, which is a Gaussian random unit (a = 1) process with stationary independent increments and is therefore a Markov process. Equation [15] describes the motion of a mechanical system under random (white-noise) excitation and leads directly to the FPK equation through the use of the time derivative of the moment generating function of the response process X(t) (Solnes 1997).  The  associated infinite hierarchy of differential equations governing the statistical cumulants ofX(t), are given by (Wu 1987):  = 2E[(x - K )ju(x, t)] + E[CT  2  x  (X, t)]  [16] =  3 [ju(x,t)[{x E  K  2  )  2  - K \+ 2  3E[(x -  AT, > 7  2  (x,t)]  dK (t) _ 4  dt  As stated previously, in the case of a stationary response, dic ldt = 0, and Eqs. [16] n  reduces to a set of algebraic equations.  Chapter 2  Literature Review  21  The simplest closure scheme for solving Eq. [16] is Gaussian Closure in which the expressions of the order > 2 are expressed in terms of the first two cumulants and then the first two equations are solved.  Iyengar (1978) used Gaussian Closure to study the  response of a hysteretic system with a smooth restoring force. To improve this closure scheme, non-Gaussian properties have been considered either in terms of approximating the unknown probability density with a truncated Gram-Charlier or Edgeworth series (Crandall 1980), which are general purpose probability distributions, or by including addition cumulants of higher order than 2 (Wu and Lin 1984). Suzuki and Minai (1987) utilized a non-Gaussian closure  technique  to thoroughly analyze  the  response  characteristics of inelastic systems including displacement, velocity, maximum response, cumulative plastic deformation and low cycle fatigue damage factor.  Their procedure  used a series expansion of the joint probability density function of the response state vector in terms of a product of normal gamma density and orthogonal polynomials. The state vector included specified quantities concerned with the white-noise shaping filter for seismic excitation, the hysteretic structure and also with measures of structural damage.  Although closure techniques apply to transient non-stationary response processes and also to systems with multiple degrees of freedom, the major shortcoming of this method is the significant increase in computational effort required for a modest increase in the number of degrees of freedom of the system as well as the restriction of the excitation to a white-noise process.  2.2.1.1.2.4  Stochastic Averaging  The principle of Stochastic Averaging is to simplify the equations describing slowly fluctuating response quantities by time-averaging the rapidly fluctuating response quantities. This principle is a non-trivial extension of the Krylov-Bogoliubov averaging method for deterministic excitations since it involves accounting for the averaged effect of a random excitation multiplied by a correlated response (Zhu 1988).  Stochastic  Averaging Methods in random vibration analysis can be viewed as a combination of this  Chapter 2  Literature Review  Stochastic Averaging principle and the FPK equation method.  22  By averaging certain  response quantities, the FPK equation is simplified or even reduced in dimension, which reduces the difficulties in solving it.  There are three methods of Stochastic Averaging, namely, the Standard Stochastic Averaging Method, the averaging method of coefficients in the FPK equation, and the Generalized Stochastic Averaging Method. The standard method was developed first by Stratonovitch (1964) and it applies to narrow-band responses, which are represented as sinusoidal oscillations with slowly varying amplitude and phase. Approximate equations for the slowly varying quantities are obtained by time-averaging the rapid fluctuations. In the second method, developed by Khasminskii (1963), the drift and diffusion coefficients in the FPK equation are averaged with respect to time. In the third method, also developed by Stratonovitch (1964) and alternatively known as the Stochastic Averaging Method of the energy envelope, the response variables are divided into rapidly varying quantities and slowly varying quantities and approximate equations for the latter are obtained by averaging the rapid fluctuations of the former. For a single-degree-offreedom system the rapidly varying quantity is the displacement and the slowly varying quantity is the energy envelope.  The application of the Stochastic Averaging Methods, which were first developed to analyze non-linear phenomena in radio engineering, to mechanical and structural systems began in the late 1970's for predicting the response, deciding the stability and estimating the reliability of non-linear systems subject to random external and parametric excitation. A parametrically excited system is one in which the effective stiffness and/or damping parameters are forced to vary with time. The most important property of such systems is that for periodic parametric excitation there are ranges of excitation amplitude and frequency for which the response remains bounded (stable) and ranges for which the response grows without limit (unstable). This type of excitation, however, is not relevant to the analysis of civil engineering structures and will not be considered further. In the case of externally excited structures, the central idea of the Stochastic Averaging Methods is that if the typical structural response time is much longer than the excitation  Chapter 2  Literature Review  23  correlation time, the excitation in effect acts as independent pulses and the response of the system can be described in terms of a scalar quantity, usually the energy content of the system, which is approximately a one-dimensional Markov process. Therefore, the dimension of the problem is greatly reduced and the solution of the F P K equations becomes much simpler. For a single-degree-of-freedom narrow-band (lightly damped) structural system subjected to a wide-band excitation, the Standard Stochastic Averaging Method may be used to transform the state vector {Z, Z} to a pair of slowly varying processes expressed in terms of the amplitude a(t) and phase angle (p(t) as follows:  Z{t) = a(t)cos(cot+ <p(a,t)) Z(t) = -coa(t)sm(cot + (p(a,t))  By time-averaging over the period of oscillation the phase angle cp can be eliminated i.e. uncoupled from the FPK equation, which results in a(t) being a one dimensional Markov process. Therefore, as stated previously, the dimension of the problem is reduced and the solution of the FPK equation becomes much simpler.  Iwan and Lutes (1968) applied Stochastic Averaging Methods to non-linear systems with bilinear hysteresis and showed that it gives inaccurate results for systems with large non-linearities.  However, it was later shown that the Krylov-Bogoliubov  technique, which forms the basis of the Stochastic Averaging Method, may seriously overestimate the energy dissipation capacity of elasto-plastic or nearly elasto-plastic systems (Wen 1980). This results in a large underestimation of the root-mean-square (RMS) response of the system in a certain response range. Later, Roberts (1978) came to the conclusion that Stochastic Averaging was applicable to an oscillator with a bilinear restoring force-displacement characteristic.  Extending that work, Roberts and Spanos  (1986) applied Stochastic Averaging Methods to smooth hysteretic systems and obtained good results for narrow-band systems.  Cai and Lin (1988) used a similar approach,  which was applicable to either bilinear or smooth hysteretic systems, without the restriction that the response be a narrow-band process.  In this approach, which was  termed Equivalent Non-linearization, the original system that cannot be solved exactly is  Chapter 2  Literature Review  24  replaced by a substitute non-linear system for which an exact solution is known. To find the best non-linear approximation, the mean-square error is minimized. The application of this procedure is quite restricted since exact analytical solutions exist only for a very limited class of problems, most of which are not applicable to practical engineering cases.  Further developments in applying averaging techniques to randomly excited systems have been given by several authors using both Stochastic Averaging and QuasiConservative  Averaging  Methods.  Quasi-conservative  averaging  was  originally  developed by Landa and Stratonovitch (1962) and Khasminskii (1964) in which the equation of motion of a system is replaced two first-order equations for energy and displacement.  The original method, which was only applicable under Gaussian white-  noise, was extended by Roberts (1982) and others to include non-white  additive  excitation and then further extended by Cai (1995) using Roberts' scheme to include multiplicative excitations.  Lin and Cai (2000) recently addressed the problem of  multiple-degree-of-freedom  systems with both high and low damping modes, and/or  strongly non-linear stiffness under non-white stochastic additive and multiplicative excitation.  2.2.1.1.2.5  Numerical Diffusion Techniques  Numerical Diffusion Techniques, also known as Cell Mapping Methods, were developed from the theory of point-to-point mapping of dynamic systems attributed to Poincare in the 19 century. For a dynamic system governed by: th  z(t) = F(t,z{t))  [18]  where z is a real-valued ^-dimensional vector and F is a real-valued vector function that is explicitly periodic in t, the governing equation may be integrated over one period to relate the state of the system at the end of one period to the state at the end of the next period. Viewed in this manner, the governing equation for the system takes on the form:  Chapter 2  Literature Review  25  z(n + l)=G(z(n))  [19]  in which a point z(n) in the state space, or phase plane {Z, Z ) , is mapped by G after one period into a point z(n+l).  Such a point-to-point mapping dynamic system is called a  point map or a Poincare map in the mathematical literature.  A Poincare map is a  mathematical idealization of the dynamic system since, considering physical limitations on measurement accuracy and the inherent round-off error in numerical evaluation, there is a limit beyond which two values of a state variable cannot be differentiated and therefore must be treated as the same. For this reason the state variables must be treated as having discrete values which leads to the idea of considering the state space not as a continuum of points but rather as a collection of very small intervals or cells. The theory of point-to-point mapping then becomes one of cell-to-cell mapping, which can be used to study the global behaviour of real non-linear dynamic systems governed by ordinary differential equations.  Cell Mapping Methods, in the context of analyzing dynamic  systems through discretization, may be viewed as discretizing the dependent state variables, whereas classical stepwise time integration is a procedure to discretize the independent time variable and Finite Element analysis is a procedure to discretize the independent spatial variables.  Hsu (1980) and Hsu and Guttalu (1980) developed an algorithm for analyzing the behaviour of non-linear dynamic systems using cell-to-cell mapping, which was termed simple cell mapping, since each cell could only be mapped to one other cell, called an image cell, during each iteration of the algorithm. This work was later extended to a generalized cell mapping (GCM) algorithm (Hsu 1981, Hsu et al. 1982) which allowed for the mapping of a cell to multiple image cells, each image cell possessing a fraction of the total probability of occurrence. The probabilities Py of mapping cell i to cell j in one mapping step are contained in a transition probability matrix P, which completely controls the evolution process of the dynamic system.  Using the theory of discrete  Markov chains and knowing the initial state ofthe dynamic system, the probability ofthe system being in a given future state (cell) after n mapping steps is completely determined.  Chapter 2  Literature Review  26  This can be shown by using the discrete form of the Chapman-KolmogorovSmoluchowski equation, Eq. [3], which is given by:  PU"=ZZ H-KJ  0<m<n  P  ?  [20]  k  where the «-step transition probability P"^ is defined as the probability of being in cell k after n steps starting from cell j. Given in the form of function mapping, Eq. [20] is rewritten as:  P (n)=f P (n-l),^n i  j  ijPj  = l,2...  [21]  7=1  where Pi(n) represents the probability of the system being in therthcell at time nx and Py the probability of the system being in therthcell at time x when the system is initially in the y'th cell with probability one. Unfortunately, for non-linear stochastic systems, the one-step transition probability matrix Py involving all i's and fs is rarely available and simulation methods are normally required to determine the conditional probability density function of the response process.  Hsu and Chiu (1986) used Monte Carlo  simulation to construct a histogram estimator of Py by generating a large number of sample trajectories of time duration T out of each cell in the phase plane. However, when a large number of cells are used and a large number of sample trajectories are simulated out of each cell, the Monte Carlo method becomes quite computationally intensive. To avoid the time-consuming simulation of Py, Sun and Hsu (1990) proposed a Gaussian approximation for the conditional probability density function of moving from y'th cell to the rth cell when the time x is sufficiently small. This Gaussian approximation was allowed to vary in shape with the initial starting point (cell j) to properly capture the global non-linear system behaviour. The Generalized Cell Mapping Method is widely applicable to many types of systems that are either weakly or strongly non-linear and either lightly or heavily  Chapter 2  Literature Review  27  damped. It can provide both transient and steady-state solutions of system response and may even be applied to stochastic systems which don't necessarily admit the FPK equation. The restriction on the GCM method is the assumption of a discrete Markov chain of mapping steps, which is based on the system excitation being a white-noise process. Also, the applicability of GCM to degrading hysteretic systems is not known. 2.2.1.1.3  First-Passage Problem Solution Methods  For structural reliability calculations the probability distribution of the time to first-passage (exit) of a safe domain for the response process is of considerable interest. This type of problem is commonly referred to as a. first-passage problem. There are three different types of safe domains that are commonly used to characterize the firstpassage problem (see Fig. 2.2). In the first case, the safe domain is characterized by a single barrier or threshold level z = b, called a type-B barrier, which is typically described quantitatively in terms of the RMS response level. The second case is similar to the first except that the safe domain is characterized by the double barrier z = ±b, called a type-D barrier. In the third case, a passage level.for the envelope process A(t) rather than the process Z(t) itself is considered. An envelope-passage level is called a type-E barrier.  For a damped dynamic system subject to random excitation an exact solution to the first-passage problem remains to be found for any of the barrier types and approximate solution methods must be used. There are two analytical solution strategies that are used to obtain approximate statistics for the time to first-passage.  The first  strategy relies on the special nature of independent random events with exponential distributions while the second strategy employs Markov theory and does not require the assumption of independent events.  Chapter 2  Literature Review  28  Figure 2.2: First-Passage Time and Domain Barriers in the Phase Plane  2.2.1.1.3.1  A s s u m p t i o n of Independent Events  The assumption of independent events leads to various expressions for the first crossing density, p\{T), which is the density function describing the probability that the response process surpasses a given threshold for the first time (since t = 0) during the interval T < t < T + dT.  The form of each expression, assuming a reasonably high  threshold, is given by (Crandall 1970):  {T) = ae-  aT  Pi  [22]  where a is called the limiting decay rate of the first crossing density, which depends on which type of event is assumed to occur independently.  The possible choices are;  independent threshold crossings, independent peaks, independent envelope crossings, and independent envelope peaks.  Chapter 2  Literature Review  29  The simplest approximation to the first crossing density is obtained by assuming independent threshold crossings of type-B or type-D barriers.  This is equivalent to  assuming that the arrival of failures (i.e. threshold crossings) is rare enough that they can be considered independent events. In this case, the number of failures n within the time interval [0, t] is a Poisson process, which is described by:  PM=M^-  [ 2 3 ]  where v is the mean threshold crossing rate. The mean crossing rate for a type-D barrier is simply twice the mean crossing rate for a type-B barrier. The probability of no failures {n = 0) in the time interval [0,t] is given by:  P[0,t]=e- '  [24]  v  The probability of failure in the time interval [0,t] is then given by:  P [0,t] = l-P[0,t] = l-e-"  [25]  F  This result may be interpreted as the probability that the first-passage time is equal to or less than t, which is the cumulative distribution function (CDF) of T. The first crossing density then follows from differentiation of Eq. [25]:  P ,(r) = u f  w  [26]  Equation [26] can be used to compute the statistical properties of the first-passage time T. In particular, the mean and variance of the time until first crossing are given by:  Chapter 2  Literature Review  co  1  o  v  E[T]= J(p,(0*  30  [27]  *>(Th\(t-E[Tf (t)dt Pl  o  The threshold crossing rate v is determined from the well known Rice's formula, which for a type-B barrier, is given by:  [28]  where z = b is the threshold level and fzz is the joint density function of z and z. When information about threshold crossings only from below (v ) is required, termed the +  upcrossing rate, the lower limit of integration in Eq. [28] is changed to zero and v = '/ +  2  v. Again, for a type-D barrier the threshold crossing rate and upcrossing rate is simply twice that for a type-B barrier.  In the special case of a stationary Gaussian process with zero mean, the upcrossing rate of threshold z = b is given by:  v +=  1  z-exp  u  2n o-  z  y  -  [29]  When the threshold b is zero, the problem is known as the zero-crossing problem, which counts the number of loading cycles.  For a stationary normal zero mean process, the  expected of rate of zero crossings from below is given by:  Literature Review  Chapter 2  31  The assumption of independence of threshold crossings is not well suited to narrow-band processes such as the response of lightly damped dynamic systems. Once a sample function of a narrow-band random process crosses a given threshold level b or \b\, the probability is high that the following excursion will produce another crossing. ^Therefore, for every threshold crossing of an envelope A(t), there may be several threshold crossings of the narrow-band process Z(t), which is enclosed by the envelope, and consequently, independence of the narrow-band crossings is lost. This phenomenon is referred to as clumping. The importance of taking clumping into consideration is measured by the average clump size for which expressions have been derived by Lyon (1961) and Racicot (1969). With the clumping phenomenon present, an improved estimate for the first crossing density of a narrow-band random process can be obtained by considering the envelope A(t) and assuming that the envelope crossings of the type-E barrier are independent (Lin 1967). When there are many excursions in each clump, the time of an envelope threshold crossing, which must precede the first crossing in each clump, is nearly the same as the time ofthe first crossing ofthe clump. For this reason there is little difference between the results for type-D and type-E barriers. Using the expected rate of threshold crossings of the envelope process, expressions analogous to Eq. [26] and Eqs. [27] are obtained for first crossing density and first-passage time T of the envelope process.  For a stationary Gaussian random process, the expected type-E barrier  upcrossing rate ofthe envelope process is given by:  +  where  bo  (  exp|^-  b  [31]  J  Chapter 2  Literature Review  cr =cr 2  = JO  2 z  Z Z  32  (co)dco  [32]  OZZ(OJ) is the spectral density and co is a representative mid-band frequency of the m  narrow-band random response process (i.e. natural frequency). Apartfromthe time to first-passage, the probability distribution of the peaks of the random process is also of interest. A peak value in a sample function z(t) of a continuously valued random process Z(t), which is also continuous with respect to time, occurs when z(t) = 0. Therefore, analysis of the peak distribution is a zero-crossing problem of the first derivative z(t).  For a stationary zero mean Gaussian process, the  probability density of the peaks is given by:  / x  Vl-a  Pp\ ) = — ^ a  O  z  2  e  yflTT  x  ba  j  P v  [  [2(7 (l - a) 2  Z  2<r,  l + erf  exp  2(7z  )  Va  [33] For a narrow-band random process, a = 1 and the probability density for the peak magnitude reduces to:  Pp( ) = — F b  e x  P  v  2 o  V  [34] j  Equation [34] represents the Rayleigh distribution, which is a special case of the Extreme Value Type III distribution known as the Weibull distribution. This same result for a narrow-band process is obtained by considering that the proportion of cycles for which Z  Chapter 2  Literature Review  33  > b is simply v /v , where v is the upcrossing rate given by Eq. [29] and v 0  at which cycles occur given by Eq. [30].  0  is the rate  The ratio is equivalent to 1 - F (b) and P  therefore by differentiation p?(b) in Eq. [34] is obtained.  2.2.1.1.3.2  Markov Process Theory  Under the assumption of a Markov response process a number of researchers have proposed approximate solutions to the first-passage problem.  Among these, for a  discrete random process, the random walk model (Toland and Yang 1971) is well known. For the continuous case, first-passage problem solutions are based on the FPK equation governing the evolution of the transition probability density function in the phase plane. For first-passage problems, the transition probability density function, which describes the instantaneous joint distribution of z and z over the phase plane, is called the probability mass.  This term is derived from visualizing the joint distribution as a  distribution of mass over the phase plane.  During the evolution of a random process  beginning at t = 0, the probability mass spreads out from the initial starting distribution and its centre advances along a clockwise spiral trajectory in the phase plane. The rate at which probability mass crosses an absorbing boundary (type-B, type-D or type-E) and is lost defines the first-passage probability density.  There are two approaches for determining the rate of loss of probability mass, one is based on the Numerical Diffusion Technique (Sec. 2.2.1.1.2.5) and the other uses the Finite Element Method (Sec. 2.2.1.1.2.2). The application of Numerical Diffusion to the first-passage problem was pioneered by Crandall et al. (1966) and later extended by Sun and Hsu (1988) using their Generalized Cell Mapping Algorithm. The Finite Element solution of the first-passage problem was initially developed using a Petrov-Galerkin method to solve the backward Kolmogorov equation, which is the formal adjoint of the forward Kolmogorov equation or FPK equation (Bergman and Spencer 1983, Spencer 1986). A solution for the first-passage problem was later developed directly from the solution of the FPK equation using a Bubnov-Galerkin method (Spencer and Bergman 1991).  Chapter 2  Literature Review  34  Results from the Numerical Diffusion and Finite Element studies of the firstpassage problem indicate that the general form of the first-crossing density given by Eq. [22], which is based on the assumption of independent events, is valid for both linear and non-linear systems.  2.2.1.2  Perturbation Method  This method may be applied to any continuous or discrete multiple-degree-offreedom system having small non-linearities such that the governing equations of the system may be expressed in a solvable linear form. Specifically, the solution to a nonlinear set of equations is expanded in terms of a small scaling parameter e that characterizes the magnitude of the non-linear terms involved. Perturbation theory has been used for deterministic vibration analysis for many years and was generalized to the case of stochastic excitation, in particular non-linear systems, by Crandall (1963).  In the case of a single-degree-of-freedom system having small non-linearities r\, the equation of motion may be written as:  z + ]d + a> z + sr/(z,z) = F(t)  [35]  0  where y, ©o, and s are constants and s «  1. If the perturbations are of the order s or  smaller then the response may be written as a power series in s as follows:  z(t)= z {t)+ez {t) + e z (t)+ • • • 2  0  x  2  [36]  Taking derivatives of the above series expansion, the resulting expressions for z, z and z may be substituted into the equation of motion of the non-linear system, Eq. [35]. Equating terms of the same order in e results in a set of linear equations of motion for zo, zi, z  2  One could write as many equations as desired for higher orders of 8. Clearly,  Chapter 2  Literature Review  35  the accuracy of the power series expansion of the response depends on the number of terms retained in the series.  However, the mathematical problems and uncertain  convergence criteria associated with finding solutions including higher than first-order terms are usually too complicated to be of practical use.  Therefore, typically, the  Perturbation Method is limited to first-order perturbation in which only terms of the order s are included.  The  solution for the displacement responses z (t), zj(t), z (t) 0  2  , which are  summed to yield the complete solution, are obtained with the use of the convolution integral, which gives the response of a linear dynamic system to a series of impulses. Infinitesimally short impulses are used to represent the arbitrarily varying force F(t) since the system response to a unit impulse is known and superposition may be used on the linearized system to determine the response to a series of impulses. Taking an example from Branstetter et al. (1988), if r\(z, z) - zz in Eq. [35], then the resulting linear equations for terms of order zero and one in s are:  z + yz .+ co z = F(t) 0  a  0  0  [37]  z\ +yz + co z = -z z x  The  0  x  0  0  solution for the first of Eqs.  [37] is given by the convolution integral as follows:  [38] — CO  where h() is the impulse response function of the linear system. For a single-degree-offreedom system, the convolution integral, of course, specializes to Duhamel's integral. Knowing zo(t) from Eq. [38], the second of Eqs. [37] is solved using the following convolution:  00  z,(r)= \h(t-  r)(-z (r)i (r))/r 0  0  [39]  Chapter 2  Literature Review  36  In this way, as many terms as desired in the series expansion for z(t) may be evaluated, however, as stated previously, typically only first-order order terms in s are considered.  If z(t) is stochastic, then the mean function and the auto-correlation function for the response of the example system are determined from the expected value of each convolution as follows (Lin 1967):  E[z(t)] = E[Z (t)] + ^[zAt)] 0  co  co  - r]dr - e $E[Z (r)z (r)]ft(f -r]dr  E[z(t)] = \E[F(t)\i(t  0  Q  [40] E[z(t + r)z(t)] = E[Z (t + T)Z (f)] + EE\ Z (t + T)zAt)]+£E[Z (t)Z {t 0  Rzzfe)= * z z 0  0  0  0  }  + T)]  fe)+ A z*fe)+* z , z fe). R  0  0  Although the Perturbation Method is applicable to multiple-degree-of-freedom systems and is not confined to a white-noise assumption for the excitation, it is limited to weakly non-linear systems and as such is not applicable to ductile, hysteretic systems.  2.2.1.3  Equivalent Linearization M e t h o d  The method of Equivalent Linearization was originated in the 1930's for the treatment of non-linear systems under deterministic excitations. It was first extended to the case of random excitation independently by Botoon (1954) and Caughey (1960) and later generalized for multiple-degree-of-freedom systems by Iwan (1973).  In this  method, the stochastic equation governing a non-linear system is replaced by an 'equivalent' linearized version which introduces a random error between the true nonlinear and linearized systems. This error is minimized, usually in a mean-square sense, by setting to zero the partial derivatives of the expected value of the squared error with respect to the coefficients appearing in the linearized equation. These partial derivatives define a set of equations, which are then solved for the required coefficients.  Chapter 2  37  Literature Review  In the case of a single-degree-of-freedom stochastic system, the governing nonlinear differential equation is given by:  z + h(z,z)=G(t)  [41]  Assuming the function h(z, z) containing non-linear terms related to the system damping and stiffness may be approximately written as the sum of two linear components, one pertaining to damping and the other to stiffness, the governing differential equation is linearized to:  z + az + J3z = G(t)  [42]  The error introduced by the linearization is then the difference between the non-linear h function and the two linear components:  e = h{z,z)-az-pz  [43]  This error is a random process and must be minimized for the best prediction of system response. The usual means of minimizing the error is to minimize the mean-square error, which is accomplished by requiring that:  dp  L  J  Substituting the error equation, Eq. [43], into the partial derivatives of Eqs. [44], the coefficients a and P which minimize s are given by (Lin 1967):  Chapter 2  Literature Review  38  E[z ]ff [Z • h(z,z)]- E[ZZ]E[Z • h(z,z)] 2  a =  E[z ]E[z ]-(E[zzf 2  2  [45] E[Z ]E[Z • h(z,z)]-E[ZZ]E[Z • h(z,z) 2  P=  E[z ]E[z ]-(E[zzf 2  2  Note that Eqs. [45] are not explicit expressions for a and P since the expectations appearing on the right-hand sides depend on a and p.  Evaluation of a and p is  simplified if the excitation G(t) is assumed to be stationary and Gaussian with a zero expectation and the non-linearities in stiffness and damping are separable, i.e.:  h{z,z) = fXz) +  [46]  f (z) 2  However, these conditions are not essential for determining a and (3. In general, the linearized system coefficients are functions of the unknown, usually non-Gaussian, response statistics and other statistics involving the restoring force. For this reason an iterative solution procedure is generally required to determine a and p.  The first application of Equivalent Linearization techniques to hysteretic systems subjected to random excitation was given by Caughey (1960) who modelled a bilinear system using the Krylov-Bogoliubov (K-B) assumption of slowly varying parameters. However, similar to the results of Iwan and Lutes (1968), the K - B assumption may lead to serious underestimates in the RMS response of the system (see Sec. 2.2.1.1.2.4). To eliminate reliance on the K - B assumption, Wen (1980) and Baber and Wen (1981) incorporated Bouc's (1967) smooth hysteresis model, which models a hysteretic, deteriorating system with non-linear differential equations, to obtain a solution for the system linearization coefficients.  Their solution for the response statistics of both  stationary and non-stationary hysteretic degrading systems utilized a convenient procedure suggested by Atalik and Utku (1976) to estimate the linearization coefficients in multiple-degree-of-freedom  systems.  The modelling and solution procedure  developed by Wen was later extended to systems under non-zero mean excitation (Baber  Chapter 2  Literature Review  39  1984) and also to systems exhibiting pinching of the hysteresis loops (Baber and Noori 1985) .  With these improvements, the Equivalent Linearization Method utilizing the  smooth hysteresis model has been successfully applied to response and damage prediction of a variety of structural systems under seismic excitation.  The accuracy of the Equivalent Linearization Method, as measured against exact FPK solutions of certain systems and Monte Carlo simulation, is generally very good with an error of less than 20 percent considered representative (Branstetter et al. 1988). Furthermore, unlike most approximate analytical methods, the accuracy of Equivalent Linearization is relatively independent of the severity of the non-linearity, be it of geometric or material source. The error in mean-square response remains quite small even for large non-linearities (Roberts 1981). Caution is necessary, however, when the excitation spectral content is such that the power spectral density function vanishes rapidly as the frequency goes to zero.  In this case, which is typical of earthquake  excitation, the method tends to underestimate the displacement response.  The error  depends largely on the characteristics of the system restoring force and the excitation in the low frequency range. It is negligible when the power spectral density function is non-zero at zero frequency but for an earthquake excitation model with a power spectral density which goes to zero at zero frequency, the R M S response could be underestimated by 2 0 - 3 0 % (Wen 1989).  In addition to the caution necessary when using Equivalent Linearization for seismic response analysis, it should be noted that the assumption of a Gaussian input results in an assumed Gaussian response of the linearized non-linear system.  This  assumption is not correct for a non-linear system, which is known to have a nonGaussian response to a Gaussian input. The result of an assumed Gaussian response is that it may significantly misrepresent the frequency of high response levels to extreme loads, which contribute most to first-passage and fatigue failures. Instead of assuming a Gaussian distribution for the response variables, the accuracy of the method may be improved by including higher order non-Gaussian effects. Full probability distributions have been estimated from non-linear response moments with Gram-Charlier and  Chapter 2  Literature Review  40  Edgeworth series (Crandall 1980), however, these series can behave erratically yielding negative probability densities and crossing rates for significantly non-linear systems. To alleviate this problem, Winterstein (1988) proposed a Hermite moment model in which response moments (skewness, kurtosis etc.) are used to form non-Gaussian response contributions made orthogonal through a Hermite series to minimize mean-square error. The Hermite moment model predicts full probability distributions of the response and its extremes as well as crossing rates and fatigue damage rate.  If used with observed  moments from a response time history, the model corrects for non-linearity without the need to fully specify or analyze a precise non-linear model. In analytical studies, the Hermite moment model can be combined with various moment estimation techniques such as Moment Closure (see Sec. 2.2.1.1.2.3).  2.2.1.4  Functional Series Representation  The Functional Series Representation Method is similar to the Perturbation Method in that it is limited to systems in which the non-linearities are small. Given that the response of a linear system may be expressed in the form of a convolution as given in Eq. [38], it has been shown (Wiener 1958) that the convolution may be generalized to a Volterra series expression when the system is non-linear:  Z  W = £]••• J*.fe.'2 • «=1  )• F(* ~ 'i )• - (tF  K Yh-dt  [47]  n  _oo -co  The function h may be regarded as therathdegree impulse response function. Bedrosian n  and Rice (1971) later obtained therathdegree frequency response function H in terms of n  system parameters using a system with harmonic input of the form:  [48]  Chapter 2  Literature Review  41  Using this input and substituting into the Volterra series expression and then into the equation of motion, the «th degree frequency response function H is obtained. Once H n  n  is determined, h„ in Eq. [47] is evaluated using the Fourier transform pair relationship given by the following:  [49] //>„...,co ) n  = J - jA.fr,,...,/.)-^-"—-'-^, . . . A . -OO  —CO  Once h„ is evaluated the response of the Volterra series expression, and therefore the non-linear system response, is completely determined. This method may also be used to evaluate the response statistics of weakly non-linear systems.  2.2.1.5  Decomposition Method  This method is an operator-based technique originally proposed by Adomian (1983) and later extended by Benaroya (1984) to analyze non-linear problems in structural dynamics. This method, which has received limited attention in the literature, does not require any of the assumptions concerning the characteristics of the forcing function and/or system non-linearity which are used in the Fokker-Planck, Equivalent Linearization, Perturbation and Functional Series Methods. It is particularly suited to systems with random properties, for example, systems in which stiffness is a random function of time.  As summarized by Branstetter et al. (1988), consider the following  linear system where the stiffness k(t) is a function of time only, and is therefore not dependent on the load:  mz(t)+ cz(t) + k{t)z{t)= F{t)  [50]  where m and c are constants and k(t) is a random process. Define the linear operator L as:  Chapter 2  Literature Review  d d L = m— - + c— + k(t) dt dt  42  2  [51]  2  Then the equation of motion in operator form is:  L[z(t)] = F(t)  [52]  The system stiffness may be written as the sum of deterministic and random parts where the deterministic part is the mean value of k(t), given by E[k(t)], and the random part, denoted by K(t), is the random fluctuation about the mean:  k(t)=E[k(t)]+K(t)  [53]  Similarly, the operator L may be separated into a deterministic part D and random part R resulting in L = D + R, where:  D = m^ + c— + E[k{t)] dt dt R = K(t) T  2  1  W  J  [54]  Substituting L=D + R into the operator form of the equation of motion results in:  z = D- F-D~'Rz ]  [55]  Now, let z be written as the sum of a series:  z(t)=± (t) Zi  1=0  Substituting Eq. [56] into Eq. [55], the equation of motion becomes:  [56]  Chapter 2  Literature Review  43  z^D' F-D~ R(z +z,+..) x  [57]  x  0  The series expression for z(t), Eq. [56], may be regrouped as follows:  z {t) = D' F x  0  z,(t) = -D-'Rz  0  =  z (t) = -D~ Rz,  -D' RD~ F x  x  [58]  =D' RD' RD- F  x  X  2  X  X  Therefore, Eq. [56] may be re-written as:  z(t)=±{-iy(D- R)D- F x  [59]  x  1=0  In practice, of course, this series expression for the solution of the non-linear system will be truncated to a finite number of terms, which introduces some error.  Finally, the  deterministic operator D has to be inverted to allow computation of the response statistics.  If the inverse D'  x  exists and has a corresponding Green's function git,x),  sometimes called the weighting function, then Eq. [59] may be written in integral form as:  z(t) = \g(t,T)F{r)dT-  0  \g{t,T)K(T)z(r)dT  + A z (t)+ x  x  A z (t) 2  2  [60]  0  where z\(t) and zj(t) solve the homogeneous equation L[z] = 0 and A\ and Ai are determined by initial conditions. Statistics of the response may be computed using Eq. [60], which was derived without using any assumptions on the nature of the forcing function.  Chapter 2  2.2.2  Literature Review  44  Numerical Methods  In contrast to the analytical random vibration methods described in Section 2.2.1, which apply in the frequency domain, numerical methods are time domain based. The time-history approach to finding the response statistics of a dynamic system is generally more accurate and robust than any of the random vibration methods described previously. The inherent assumptions in each of the analytical procedures, for instance the whitenoise assumption of the Markov based methods or the assumption of weak non-linearity in the Perturbation and Decomposition Methods, are not required to obtain response statistics using time domain based numerical methods. In the context of civil engineering structures, dynamic analysis of a structure subjected to a simulated load time-history, using a numerical integration scheme, will yield a response time-history. An ensemble of these can be used to estimate the response statistics of the system at any time in the response process without regard to the nature of the restoring force (i.e. degree of nonlinearity), complexity of the structural system (i.e. number of degrees of freedom) or nature of the random excitation process.  This generality is the reason that numerical  methods are frequently used to verify results obtained using other analytical random vibration methods.  The greatest drawback to the numerical approach is the computational cost. One time-history calculation corresponds to a single sample of response, whereas the more efficient random vibration methods yield, at least in principle, the complete statistical response of the system in their solution.  Also, for structural reliability problems,  computation of the response statistics is only the first step in a two-step process. The second step, which makes use of the response statistics, is the approximation of the probability of failure using any of a variety or combination of methods: distribution fitting of response statistics, a response surface approach, F O R M / S O R M , Direct Monte Carlo simulation, and Selective Monte Carlo simulation using variance reduction techniques.  The computational cost of generating time-history response statistics is  dependent on the reliability analysis method used to solve for the probability of failure. With this in mind, the origin of the numerical approach to solving complex stochastic  Chapter 2  Literature Review  45  non-linear dynamic systems is summarized below followed b y a brief description o f the techniques that were subsequently developed to reduce computation time.  2.2.2.1  Direct M o n t e C a r l o Simulation  The principle o f Monte Carlo simulation seems to have originated with Buffon's Needle Experiment i n the 18 century. th  Buffon's original experiment was to drop a  needle o f length L at random on a grid o f parallel lines o f spacing D where D > L from which the value o f n could be inferred by observing the number o f intersections between needle and lines. The approximation o f n is given by:  [6,] RD  1  i  where N is the number o f needle drops and R is the number o f intersections.  Buffon's Needle Experiment and a few other applications o f random sampling pre-date the naming and systematic development o f the Monte Carlo Method, which began i n about 1944. The name is taken from Monte Carlo, Monaco where the Roulette wheel, which is a simple random number generator, is synonymous with the city. The development o f Monte Carlo simulation as a research tool stems from work on the atomic bomb during the Second World W a r by mathematicians  John von Neumann and  Stanislaw U l a m . This work involved a direct simulation o f the probabilistic problems concerned with random neutron diffusion i n fissile material. Since that time Monte Carlo simulation has been applied to a wide variety o f mathematical problems involving stochastic and dynamic systems i n fields ranging from economics to the natural sciences to engineering.  Although Monte Carlo methods quite often offer the only available solution to a complex stochastic system, Direct Monte Carlo simulation is not well adapted to solving dynamic reliability problems. The reason for this is that the probability o f failure being  Chapter 2  Literature Review  46  estimated governs the number of sample responses or realizations that are required. Assuming a binomial distribution for the quantity being estimated, the relative error in a Monte Carlo estimate is given by (Melchers 1999):  [62]  where k is the standard normal variable associated with a given confidence interval, n is the number of realizations and p is the expected probability of occurrence. Note that Eq. [62] does not depend on the dimension of the state space (i.e. the number of degrees of freedom in the structural system). Since, typically, dynamic systems under service must be very reliable, failures or malfunctions are rare events and therefore, from Eq. [62], the number of realizations required for an accurate estimate of a small probability of failure is very large. Assuming that the observed value of the probability of failure of a system is required to be within 5% of the true value with 95% confidence, then for a system with an expected probability of failure of 10" , 1.54 x 10 realizations would be required. 6  9  Clearly, without the use of supercomputers and parallel processing (Johnson et al. 1997) it is impractical to generate this many samples for realistic problems where each response computation may require dynamic analysis of a non-linear structural system.  2.2.2.2  Selective Monte Carlo Simulation  The slow convergence of the Direct Monte Carlo estimate, where the error decreases in proportion to ri \ has led directly to the so-called 'variance reduction' v  techniques used in Selective Monte Carlo simulation to reduce the computational cost of estimating low probability events. A number of techniques such as: Antithetic Variates (Ayyab and Haldar 1984), Conditional Expectation (Ayyub and Chia 1991) and Importance Sampling (Kahn 1956) were developed for static reliability problems but are not well suited to dynamic systems. This may be illustrated by considering the case of Importance Sampling in which an importance sampling probability density function h(x) is located on the limit state surface at the point of maximum likelihood (or the design  Chapter 2  Literature Review  47  point) o f the joint probability density function^*) o f the random variables in standard normal space.  Parenthetically, the design point is usually not known prior to the  reliability analysis and must be located by application o f numerical maximization techniques, a search algorithm or more recently by application o f a neural network. Once positioned, the sampling density is used to increase the number o f random samples that fall into the failure domain in the standard normal space thereby reducing the variance of the Monte Carlo estimate o f the failure probability for a given number o f trials. Note that to offset the expected distortion o f the original joint density function by the sampling distribution, the weight o f each sample must be modified by \lh(x). The difficulty in applying this technique to a dynamic system is two-fold. Firstly, unlike the static case, the joint density function o f the random variables is not known, instead there is only a finite sample o f realizations.  Second, the application o f Importance Sampling to a  dynamic system would require a description ofthe sampling density function with respect to time, which greatly increases the problem complexity.  To handle dynamic reliability problems, two variance reduction techniques have been developed that model the flow o f probability into the low probability region o f the phase plane.  2.2.2.2.1  Double and C l u m p  Since the flow o f probability into the low probability region is o f paramount interest in reliability analysis, Pradlwarter et al. (1994) developed a procedure they named 'Double and C l u m p ' ( D & C ) which provides a means to increase the sample size o f important realizations falling in the failure domain. The basic idea is that in the event that a sample realization is identified as important at a given time step, based on specific energy and weight criteria, the state vector i n the phase plane is doubled. Doubling o f a realization at a certain time step x simply means that an identical copy o f the state vector is made and its weight is halved. The effect o f doubling can only be observed at a time t > x since starting at t = x, the loading increments are assumed to be independent in both doubled state vectors.  Hence, two different random paths in the state space w i l l be  Chapter 2  Literature Review  48  observed due to the differences in loading for t > T. In order to preserve the total number of realizations, those trajectories that do not meet the energy and weight criteria for doubling are clumped together and the weight of each realization is combined. The varying magnitude of the weights of all realizations, which change discontinuously in time, represents a generalization of Direct Monte Carlo simulation in which all weights are equal (-  \lri) and constant in time.  In Importance Sampling, weight can be  interpreted as the ratio between the joint density and the sampling density (f[x)/h(x)), which changes with position in the standard normal space but not as a function of time.  To ensure accuracy of the Double and Clump procedure it is important that the statistics of the samples are not significantly affected.  Doubling has no effect on the  sample statistics but clumping does since it is impossible to maintain the same amount of information with fewer samples.  Clumping to the mean value of two state vectors  ensures identical first moments before and after clumping, however, higher moments are generally slightly underestimated after clumping.  The distortion of higher moments  depends mainly on the distance in the phase plane between the realizations to be clumped. Therefore, in order to minimize distortion, only realizations close to each other are permitted to  clump.  This proximity requirement, however,  increases  the  computational cost of D & C since all realizations, which are simulated simultaneously, must be searched repeatedly during the time stepping procedure for a sufficiently close partner to clump with.  2.2.2.2.2  Russian Roulette and Splitting  To alleviate the problem of searching all realizations for a clumping partner in D & C , Pradlwarter and Schueller (1997) proposed a similar Selective Monte Carlo method called 'Russian Roulette and Splitting' (RR&S) which was an adaptation for stochastic dynamic systems of a method first developed and applied for estimating low probability events in neutron shielding problems. In RR&S, instead of searching for two close realizations suitable for clumping, an 'unimportant' realization is simply killed off and the weights of all other realizations are normalized with respect to the probability of  Chapter 2  Literature Review  49  survival of each realization before the killing of the unimportant sample.  To ensure  preferential survival of important realizations, which are those that enter an important region in the phase plane, a higher survival probability is associated with them. The name Russian Roulette is derived from this probabilistic method of determining which realizations are killed, while Splitting was used to describe the doubling of important particles in the original neutron shielding problems.  In a general complex stochastic dynamic system it may be difficult to determine a measure to distinguish between important and unimportant regions. Pradlwarter and Schueller  (1999) proposed  'Distance  For this reason,  Controlled' Monte Carlo  simulation, which combines RR&S with an evolutionary technique to determine the realization selection criterion. This evolutionary technique is somewhat different from the use of Genetic Algorithms, which was proposed by Johnson et al. (1996).  The  proposed method is shown to predict extremely low probability events as well as the ability to analyze complex dynamic systems for which other methods do not appear to be suitable.  2.2.2.2.3  Latin Hypercube Sampling  Latin Hypercube Sampling is a technique that provides a constrained sampling scheme, instead of the purely random sampling of Direct Monte Carlo simulation, to reduce the variance of a Monte Carlo estimator for a given number of random samples (Imam and Conover 1980, Ayyub and Lai 1989).  For this reason, Latin Hypercube  Sampling is called a selective sampling scheme.  In traditional random sampling, random numbers between 0 and 1 are generated and these are then used to generate random variables according to the prescribed distribution function for each variable. Typically, the inverse transformation method is used with the cumulative distribution function (CDF) to map the random numbers to the generated random variables. Other procedures for generating random variates include the Composition Method and the Acceptance-Rejection Method attributed to von Neumann  Chapter 2  Literature Review  50  (Rubinstein 1981). In Latin Hypercube Sampling, the density function or CDF of each variable is divided into n non-overlapping intervals of equal probability, where n is the number of random values that have been chosen for Monte Carlo simulation.  A  representative value from each interval is then chosen for each variable and randomly matched without replacement to a value for each of the other random variables. In this way, the entire range of each random variable is represented in the set of variables to be used in a Monte Carlo simulation. For example, if 10 Monte Carlo simulations were used to estimate the mean value of a performance function that contained 5 random variables, then the density function of each random variable would be divided into 10 intervals and 10 sets of 5 values would be generated. Each set is obtained by randomly matching one interval value from each of the 5 variables together.  A  proposed variant of Latin Hypercube Sampling called Updated Latin  Hypercube Sampling, which further reduces the variance in Monte Carlo estimates of commonly used statistical parameters such as mean, standard deviation, coefficient of variation and CDF, was developed by Florian (1992). This was followed by the work of Huntington and Lyrintzis (1998) who proposed two techniques to even further improve the performance of Latin Hypercube Sampling, at the cost of significantly longer computation time.  2.2.2.3  Response Surface Method  Response surface methodology (RSM) is a collection of statistical analysis methods that examines the relationship between experimental response and variations in the values of the input variables.  Developed by research scientists performing  experiments in biology and agriculture, it is intended to create and analyze statistical models of processes that are difficult to study directly for reasons of complexity, or because the underlying mechanism controlling the process is not well understood or the response data is expensive to produce (Myers 1976).  In the context of structural  reliability analysis, the functional relationship between the structural response and the input variables is approximated by a response surface model, which is then used in a  Chapter 2  Literature Review  51  conventional reliability analysis method such as F O R M / S O R M  or Monte Carlo  simulation.  Using R S M to study the influence of variables on an outcome or process consists of two phases, response surface design followed by the analysis phase. Response surface design is the process of deciding on an experimental strategy that determines the number of variables and what combination of variable levels should be used in an experiment to generate an outcome.  Each outcome, which is then used to fit the surface, involves a  certain cost or computation time, therefore, the response surface design should be as efficient as possible at fitting the surface in the area of interest. Since the actual form and degree of the surface are not known ahead of time, only discrete experimental outcomes are known, there is little guidance in the selection of the approximating surface, however, a second-order polynomial is typically used.  The general form of a second order  approximating function is given by:  1=1  1=1  1=1 7=1  where E(y) is the expected value of the response, k is the number of independent variables, p,- and Py are the regression coefficients and x, is the ith variable. Box and Wilson (1951) introduced an efficient class of designs for fitting second-order surfaces called central composite designs, which consists of a 2  k  factorial design, with each  variable at the two normalized levels of-1 and +1, augmented by 2k axial points and n  2  center points for a total of 2 +2k + n k  2  N points. This design results in a significantly  reduced number of experimental points from a 3* design, with each variable at three levels, which would normally be required to fit a second-order surface.  Once the N design points have been chosen, the regression coefficients p, and p  y  are determined using the method of least squares, which minimizes the total error between the predicted response from Eq. [63] and the actual experimental outcome. The total error resulting from the use of the fitted response surface is comprised of two  Chapter 2  Literature Review  elements, pure error and lack-of-fit  error.  52  Pure error comes from the intrinsic  randomness of the system and cannot be eliminated, while lack-of-fit error is the result of the inability of the response surface model to represent the true response with a simple polynomial expression.  For the response surface model to be considered an accurate  representation of the true response, it is required that the lack-of-fit error can be neglected.  The means of measuring lack-of-fit error, so that a decision criterion for  accepting a response surface model may be established, is provided by the analysis of variance technique (ANOVA).  Using A N O V A ,  Faravelli  (1989) suggested an  expression involving the pure error and lack-of-fit error for evaluating the goodness-offit of a response surface.  Bohm and Bruckner-Foit (1992) proposed the use of two  alternative criteria based on the lack-of-fit error to validate a response surface, stating that the measure suggested by Faravelli had no theoretical justification. The increased computational effort required by the Bohm and Bruckner-Foit criteria led Yao and Wen (1996) to propose an empirical measure based on Faravelli's expression that reduced computation time.  The analysis phase of R S M involves the use of techniques such as: canonical  analysis, method of steepest ascent and method of ridge analysis to analyze the experimental information in the second order response model and draw conclusions regarding the influence of variables and the predicted outcome as the input variables change.  Chapter 2  2.3  Literature Review  53  E A R T H Q U A K E GROUND MOTION MODELS  The use of any of the numerical methods of Section 2.2.2 requires sample functions of the stochastic process, field or wave that is to be used as input to the system being analyzed.  The generated  sample  functions. must accurately describe  the  probabilistic characteristics of the corresponding stochastic process, field or wave which may be either stationary or non-stationary, homogeneous or non-homogeneous,  one  dimensional or multi-dimensional, uni-variate or multi-variate, and Gaussian or nonGaussian.  For purposes of definition, a stochastic process is an infinite population or  ensemble whose samples are functions  of time only, together with information  concerning relative probabilities of sample values.  A stochastic field is similar to a  stochastic process except that the samples are functions of space rather than time, whereas a stochastic wave model incorporates probabilistic information regarding both time and space to generate sample functions. In the case where there is uniformity in the random process, field or wave, it is described by the designations stationary in time and homogeneous in space. A stationary process is one whose probability distributions across the ensemble are invariant with respect to translations in the origin of time. Similarly, a random field is homogeneous with respect to a particular spatial coordinate if its probability distributions are invariant with respect to translations of the origin along the axis of that coordinate.  A stationary or homogeneous random process or field can be described in terms of its spectral density, which is related to its auto-correlation function through the wellknown Wiener-Khintchine theorem (see Sec. 2.2.1.1.1).  Spectral decomposition of a  random process is extremely useful because the auto-correlation and spectral density functions provide average amplitude and frequency information about sample processes and, in the case of a linear time-invariant system, they provide the corresponding statistics of the stochastic dynamic response.  Non-stationary random processes and  fields, however, are more difficult to model since the concept of spectral density does not apply due to the fact that the auto-correlation function is no longer a function of the time-shift (T = t\ - t{) only, but depends on two independent time arguments t\ and t . 2  Chapter 2  Literature Review  54  Several techniques have been developed to approximate a non-stationary process including: (1) the use of a generalized spectral density, defined by a double Fourier transform (Lin 1967), (2) the use of an evolutionary power spectrum (Priestley 1965, 1967), which may be used to describe relatively slow changes in frequency content of a non-stationary process, (3) modeling the non-stationary random process as a nonstationary shot-noise (Lin 1967), or (4) modeling the non-stationary random process as an 'equivalent' stationary random process modulated by a deterministic amplitude variation (Bolotin 1960).  In the fields of earthquake engineering and seismology, a large number of stochastic models for generating artificial ground acceleration records have been proposed. These models may be roughly divided into five categories, which are listed in order of earliest to most recent: (1) filtered white-noise and filtered Poisson process models, (2) spectral representation method, (3) stochastic wave theory, (4) autoregressive and moving average (ARMA) models, and (5) wavelet models.  Selected  references from a review of the first three catergories by Shinozuka and Deodatis (1988) and a review of A R M A models by Kozin (1988) are given in the following summary of each category.  2.3.1  Filtered White-Noise and Filtered Poisson Process Models  The first ground motion models (Housner 1947, Bycroft 1960) were stationary white-noise processes, which have a constant Fourier amplitude spectrum (or power spectral density). Later, it was recognized from analyses of strong-motion records that the energy content of ground motion is not uniformly distributed over all frequencies, but is concentrated in certain frequency regions. To incorporate this, Tajimi (1960), using the work of Kanai (1957), proposed the filtered white-noise model, which accounted for local site properties and a dominant frequency in the ground motion. The Kanai-Tajimi filter model, which has been used extensively in the past to describe strong ground motion, was later modified by Clough and Penzien (1975) to remove the inconsistency of unbounded ground velocity and displacement at zero frequency. This modified version  Chapter 2  Literature Review  55  of the non-white Kanai-Tajimi model is given by the following equation for the power spectral density of the ground acceleration (see Fig. 2.3):  CO ^  f  1 + 4^  G  sM=s -  K^H  0  J  [64]  2\  2\  CO  1K^Gj  where the first bracket represents the low-pass Kanai-Tajimi filter and the second bracket represents the high-pass Clough-Penzien filter. So is the constant white-noise spectrum scaled to the energy of the ground motion,  is the predominant frequency of  the ground motion, which is indicative of the geological character of the local subsoil, COG is the equivalent damping based on the hardness of the subsoil,  and co^ are empirical  parameters that are determined by matching actual ground motion recordings from the site to ensure the correct frequency content of the artificial earthquake. More recently, seismologists have developed a wide variety of theoretical Fourier amplitude spectrum models that are based on the physical parameters of the earthquake source and medium such as: magnitude, distance, fault dimension, attenuation parameters, velocity, wave propagation velocity etc.  (Solnes 1997).  shear-wave  These models are highly  specialized and, as such, outside the scope of this study and therefore, will not be reviewed further.  04 0  1  20  1  40  1  60  1  80 Frequency (rad/s)  Figure 2.3: Filtered White-Noise Power Spectrum  1  100  Literature Review  Chapter 2  56  Following the work of Kanai and Tajimi, a variety of time-modulating functions were introduced to produce non-stationary ground motion models that reflected the time varying intensity (or amplitude non-stationarity) typical of real earthquake ground motion accelerograms.  Since an earthquake motion is essentially an evolutionary  process, the filtered output sample functions have to be amplitude modulated to resemble the time evolution of the real motion, that is, show a build-up phase, a strong motion phase and an attenuating tail (see Fig. 2.4).  The proposed time-modulating models  include: time-modulated harmonics (Bogdanoff et al. 1961), filtered modulated whitenoise (Bolotin 1960, Housner and Jennings 1964, Amin and Ang 1968, Iyengar and Iyengar 1969, Ruiz and Penzien 1971) and the fdtered modulated Poisson process (Cornell 1960, Shinozuka and Sato 1967, Lin 1963, 1965).  In each model there are  typically a number of constants that completely define the time-modulating envelope function, including the total duration of the earthquake. These constants depend on the magnitude of the earthquake, the distance from the causative fault and the focal depth.  0 -K 0  ,  ,  ,  1  5  10  15  20  Time (sec) Figure 2.4: Example Amplitude Modulating Function (Amin and Ang 1968)  From a physical interpretation standpoint, the filtered Poisson process model is closer to representing an actual earthquake than the filtered white-noise model since it consists of the sum of a series of independent impulses arriving at Poisson distributed times.  The two models can be made identical up to the second moment, however, by  imposing an impulse arrival rate v that is a certain function of the white-noise spectral  Chapter 2  Literature Review  57  amplitude. Also, it can be shown that the white-noise models are Gaussian due to the Central Limit Theorem and the filtered Poisson process models, although in general nonGaussian, are asymptotically Gaussian as the impulse arrival v - » oo (Lin 1967). This non-Gaussian property can be used in applications where earthquake records indicate a significant deviation from Gaussian behavior.  2.3.2  Spectral Representation M e t h o d  The Spectral Representation Method is perhaps the most widely used approach to generate sample functions of a stochastic process, field or wave. Although the concept of the method has existed for some time (Rice 1944, 1945), it was Shinozuka and Jan (1972) and Shinozuka (1972) who first applied it for simulation purposes including multidimensional, multi-variate and non-stationary cases. representation  simulates  a  1D-1V (one-dimensional  In its simplest form, spectral and uni-variate)  stationary  stochastic process, which corresponds to seismic ground motion with a single horizontal component, using the following series:  [65]  where Sxx((£>) is the known one-sided power spectral density of the stochastic process, Aco is the frequency interval used to discretize the power spectrum and  are independent  random phase angles uniformly distributed over the range 0 - » 2TT. A n upper cut-off frequency co = A/Aoo is implied in Eq. [65] beyond which Sxxi®) may be assumed to be M  zero for mathematical or physical reasons.  Using this series, the auto-correlation  function and expected value of the simulated process converge to those of Sxx(u>), and the process becomes asymptotically Gaussian by virtue of the Central Limit Theorem, as the number of terms N increases. In addition, the simulated process is ergodic, at least to the second moment, regardless of the size of N. This makes the method directly applicable to time-domain analysis in which the ensemble average can be evaluated in terms of the temporal average. It should be noted that the sample functions generated using Eq. [65]  Chapter 2  Literature Review  58  will be periodic with the sampling period T = 27T/ACO. This periodicity may be eliminated by randomizing the frequencies co, either by adding a small random frequency 8co, (Shinozuka and Jan 1972) or by considering the frequencies co, to be random variables with a probability density modeled after the spectral density of the process (Solnes 1997). The latter technique automatically concentrates the random frequencies around the peaks in the spectral density of the stochastic process.  The  extension of the 1D-1V stationary case to a non-stationary stochastic  process, to more realistically simulate seismic ground motion, may be accomplished by using the evolutionary power spectrum developed by Priestley (1965, 1967). In this case, the spectral representation series becomes:  f(t) = 42 -f,pA  2  (f,©,.) ** (coj )Aco 5  • cos(<V + ^ )  [66]  7=1  where A(t,(Oj) is the evolutionary modulating function and all other terms are the same as the stationary case given by Eq. [65].  Similar to the stationary case, the simulated  process converges to the target evolutionary power spectrum and is asymptotically Gaussian as N —> co. The process, however, is no longer ergodic since by definition only a stationary process can be ergodic.  Also, it should be noted that the Fast Fourier  Transform (FFT) technique developed by Cooley and Tukey (1965), which is used to determine the spectral amplitudes Sxxity) ° f the target process (Yang 1972), no longer applies when using an evolutionary power spectrum. However, for the special case when A(t,(Oj) ~ A(t), i.e. the modulating function is independent of frequency, the nonstationary stochastic process becomes a uniformly modulated non-stationary stochastic process and the FFT technique again applies. This situation is equivalent to modeling the non-stationary random process  as a stationary random process  modulated by a  deterministic amplitude variation function as discussed previously. A more direct use of the FFT technique to generate sample functions of Gaussian random processes was proposed by Wittig and Sinha (1975), which substantially reduces computation time in comparison with using Eqs. [65] or [66].  In this method, discrete frequency functions  Chapter 2  Literature Review  59  that correspond to the Fourier transform of the target process are generated.  Sample  functions are then obtained by taking the inverse Fourier transform of the discrete frequency functions using the FFT technique.  The more complex cases of simulating an n D - l V non-homogeneous stochastic field and an nD-mV homogeneous stochastic field were also developed by Shinozuka and Jan (1972). A multi-variate stochastic field may be used, for example, to simulate different seismic ground motions at various locations in a large-scale structure (Kareem et al. 1997). The relationship between the stochastic processes at each location, which may be separately modulated, is defined by a coherence function,  which is the  frequency-domain equivalent of the time-domain based correlation function.  Various  researchers have extended the simulation of multi-variate stochastic fields to include among other things:  non-Gaussian properties (Yamazaki and Shinozuka 1988),  simulation of non-stationary vector processes using an FFT-based approach (Li and Kareem 1991), spatially incoherent ground motions (Ramadan and Novak 1993), and simulation of ground motion time-histories compatible with prescribed response spectra (Hao et al. 1989, Abrahamson 1993, Deodatis 1996, Zhang and Shinozuka 1996).  Even more complex  non-stationary stochastic  process  models have been  developed to simultaneously represent the amplitude and frequency non-stationarity of seismic ground motion. Frequency non-stationarity is due to the different arrival times of the P (primary or push) waves, S (secondary or shear) waves and surface (Rayleigh and Love) waves that propagate at different velocities through the earth's crust. Several studies have shown that non-stationarity in frequency content can have a significant effect on the response of both linear and non-linear structures (Yeh and Wen 1990, Papadimitriou 1990, Conte 1992). To account for this effect, a number of filtered whitenoise process based models, filtered Poisson process based models, and spectral representation based models have been proposed (Kubo and Penzien 1979, Safak and Boore 1986, Lin and Yong 1987, Grigoriu et al. 1988, Fan and Ahmadi  1990,  Papadimitriou 1990, Yeh and Wen 1990, Conte and Peng 1997, Nakayama and Fujiwara 1997).  Literature Review  Chapter 2  2.3.3  60  Stochastic W a v e T h e o r y  Stochastic wave theory, which was developed by Deodatis and Shinozuka (1989), is an extension of the Spectral Representation Method. This technique attempts to more realistically simulate seismic ground motion by describing it as a stochastic wave arising from a propagating seismic wave. It is intended for seismic response analysis of largescale structures extending over a wide spatial area such as water and gas transmission systems and large-span bridges (Deodatis et al. 1990, Zhang et al. 1991).  2.3.4  A R M A Models  Auto-regressive moving average models are one of a family of stationary time series models that includes: auto-regressive (AR), moving average (MA), and mixed auto-regressive and moving average models (ARMA), as well as their extension to a particular class of non-stationary random processes, the auto-regressive integrated moving average model (ARTMA).  A time series model is one in which a sequence of  values are generated representing possible observations of a random process at discrete values of time. The model parameters are then estimated on the basis of a comparison of estimated statistics of the generated sequence and the statistics of the actual observations of the random process, which are treated as sample functions drawn out of an ensemble of infinite possibilities. This procedure is very similar to the filtered white-noise model in that the time series is generated by passing a discrete white-noise, which provides the required sequence of values from a random process, through a linear filter. The general form of a linear time series is given by (Nigam and Narayanan 1994):  x = fj, + a +y/ a _ +y/ a _ +... t  where p and  t  x  t  x  2  t  [67]  2  are fixed parameters. The series (...a,-\, a a,+\, ...) is the white-noise h  sequence of identically distributed and independent random shocks with zero mean and constant variance a . 2  a  In this form, the series x, is represented as the weighted sum of the  Chapter 2  Literature Review  61  current and past disturbances. Equation [67] may be rearranged, however, to express the time series in terms of the current disturbance and all previous observations of the process x : t  X, = 7t X _ x  t  x  +  7T t-2 +••• + « , + S  [68]  X  2  where the weights 7t, are functions of the v|/,- weights and 5 is a constant which is a function of p and  The general linear process of Eq. [68] has an infinite number of  terms but, for practical purposes, only a finite number of weighting terms are given a non-zero value. The auto-regressive (AR) model is given by Eq. [68] with TT, = 0 for i > p.  The moving average (MA) model is given by Eq. [67] with v|/, = 0 for / > q. The  mixed auto-regressive moving average (ARMA) model of order p, q, denoted by A R M A (p,q) is the sum of the A R and M A models given by:  x, = n x „ x  t  x  +... + 7T x,_ +a,+Sp  p  i//,a,_, - . . . - y/ a,_ q  [69]  q  where the negative sign in Eq. [69] is introduced by convention. The inclusion of both auto-regressive and moving average terms typically results in a model that has fewer terms than would be necessary for a model of pure A R or pure M A form. The variables in Eq. [69] are all scalar variables, which corresponds to a so-called single input-single output linear model.  A R M A models may also be extended to multiple input-multiple  output linear systems, in which case the observations x and the random terms a, become t  vectors and the coefficients 7t, and VJ/,- become matrices.  The basic problem of modeling an observed time series x by the A R M A model is t  the estimation of the coefficients (ft,, \|/,) and determining the best model order (p, q) to fit the observed data. The estimation of parameters is usually based upon least squares or maximum likelihood methods (see Box and Jenkins 1970), while determination of the best choice for model order is commonly based on two criteria developed by Akaike (1979); the 'final prediction error' criterion (FPE) and the 'Akaike information criterion'  Chapter 2  Literature Review  62  (AIC). The application of A R M A models to simulate the observed time series of seismic ground accelerations, which are treated as single input-single output systems, was first developed by Liu (1970). Since that time a considerable volume of literature has been devoted to earthquake modeling using A R M A models, see for example Polhemus and Cakmak (1981), Chang et al. (1982), Safak (1989), and a review article by Kozin (1988).  The main advantage of A R M A models over the use of filtered white-noise process, filtered Poisson process or spectral representation models is the reduced computation time and computer memory requirements. The digital generation of sample functions of a random process is accomplished by recursively obtaining its sample values at discrete times once the model coefficients (TC,, VJ/,) are estimated. The model requires only the generation of a sequence of independent Gaussian random variates a . For the t  simulation of structural response to the sample ground motion, only the storage of the appropriate model coefficients in the computer memory, the generation of the whitenoise sequence and the recursive computation of the time series is required. Another advantage of the method is that the time series can be generated in real time, which can be used in random vibration experiments.  2.3.5  Wavelet Models  The most recent development with applications to simulation of non-stationary multi-variate processes is the use of wavelet functions, which were initially developed for analyzing seismic data in oil exploration studies. For engineering purposes, Newland (1994a, 1994b) applied Daubechies' wavelet (Daubechies 1992) for analyzing vibration signals and developed Discrete Wavelet Transform (DWT) and Fast Wavelet Transform (FWT) computational algorithms, which have emerged as powerful tools to analyze the temporal variations in frequency content of non-stationary processes.  For purposes of modeling seismic ground acceleration, the wavelet representation of a zero mean process^) with non-stationary characteristics is summarized as follows (Basu and Gupta 1997, 1998):  Chapter 2  Literature Review  CO  1  63  CO  \\—Wj{a,b)w , (t)dadb  f{t) = ^  a b  if/  -CO-CO  (t-b}  1  a\  \  [70]  a )  2  <2n _i  where the parameter b has the physical significance o f localizing the wavelet basis function \\i(t) at t = b and the parameter a captures the local frequency content. For numerical evaluation o f the integrals i n E q . [70], discretization parameters a and Ab are used, resulting i n cij = a  and bj = (j ~  j  The step changes at a = Uj and b = bj are  defined as:  {b  -  M  Ab. = ^  b  , ) + (  —^ 2  lJ  7  Aa = ;  2  b  i -  b  i  J  = A6 [71]  2  ^  (Ty  The discretized version o f the time-history o f the ground acceleration is then given by:  '  J  J  A  [72]  K=  f  o  — cr a)  Chapter 2  Literature Review  64  The choice o f the wavelet basis function \y(t) that should be used to model f[t) in the above equations primarily depends on how suitable a given basis function is for the dynamic system.  The referenced papers (Basu and Gupta 1997, 1998) used a slightly  modified form o f the Littlewood-Paley ( L - P ) basis function given by:  / \ nn=  i  1  sino7zr-sin;# ;  '  , [73]  r  The application o f wavelet-based functions can be extended to a wavelet-based random vibration theory to predict the stochastic dynamic response statistics o f a linear time-invariant system.  This is exactly analogous to the well-developed linear random  vibration methods i n the frequency domain.  Chapter 2  2.4  Literature Review  65  HYSTERESIS MODELS  Response analysis of any dynamic system, whether linear and deterministic or non-linear and stochastic, requires a model that governs the relationship between input and response.  For dynamic mechanical and structural systems undergoing inelastic  deformation, this relationship is given by the equation of motion, which includes a term expressing the hysteretic restoring force as a function of system displacement.  This  hysteretic restoring force depends not only on the instantaneous displacement of the system, but also on the time history of the response and it may deteriorate in strength, stiffness or both as dynamic oscillation progresses. time-dependent  Models describing this hysteretic,  behaviour of yielding dynamic systems are classified  as either  phenomenological or mechanics-based. Phenomenological models describe the nature of the load-deformation relationship based on observations but do not necessarily explain the behaviour, while mechanics-based models are based on the properties of the individual elements that comprise the system.  Mechanics-based models provide better  insight into how material properties affect system response but they are inconvenient to use in inelastic dynamic analysis of complete structural systems made up of multiple members (Foliente et al. 1998).  2.4.1  General Hysteresis Models  Early  hysteresis  models  to  describe  inelastic  structural behaviour were  phenomenological and typically bilinear, including the well-known elastoplastic model and the Ramberg-Osgood model.  The Ramberg-Osgood (1943) model describes the  force-displacement curve envelope curve (or backbone curve or skeleton curve) by a three-parameter polynomial, and allows smooth transition from the elastic to the plastic region and some freedom in the shape of the hysteresis. However, as in the case of the bilinear model, it was difficult to include system deterioration in the model. Later models proposed by Clough and Johnston (1966), Takeda et al. (1976) and others, extended the basic bilinear system to include deterioration by using a set of empirical rules. The use of piece-wise linear models of the force-displacement relationship, governed by a set of  Chapter 2  Literature Review  66  empirical stiffness rules related to displacement, is well-suited for time-history analysis of structural response by means of step-by-step numerical integration. This has resulted in a wide variety of models being developed for various materials and types of loading (see, for example, the 3-D non-linear dynamic structural analysis program CANNY (Li 1996), which has twenty uni-axial hysteresis models).  99  However, for analytical  treatment of system response to random vibration, the use of empirical rules is difficult to put in a mathematically tractable form and, therefore, non-linear differential equations are used to model the force-displacement relationship.  Differential equation models are still phenomenological in nature but they are given in a mathematically explicit form so that an analytical solution of the system response may be obtained. The first smooth hysteretic restoring force model was proposed by Bouc (1967) and later generalized by Wen (1976), however, this model did not include the pinching and degradation behaviour exhibited by many hysteretic systems that sustain damage under large deformation. For this reason, extensions to the model were proposed by Baber and Wen (1981) who incorporated stiffness and/or strength degradation as a function of hysteretic energy dissipation, and Baber and Noori (1986) who incorporated pinching behaviour using a 'slip-lock' element. This model, which has since become known as the Bouc-Wen-Baber-Noori (BWBN) hysteresis model, introduces a state variable z and separates the restoring force into non-hysteretic and hysteretic components. For a single-degree-of-freedom (SDOF) non-linear system, the equation of motion in standard form is then given by:  u + 2^co u + ao) u + (l-a)a) z = f(t) 2  0  0  2  0  [74]  where £, is the system damping ratio, coo is the natural frequency, a is the ratio of postyielding to pre-yielding stiffness, j\t) is the mass normalized forcing function, typically assumed to be zero mean, and z is the 'hysteretic force', which is described by the nonlinear differential equation:  67  Literature Review  Chapter 2  yu z  z = h(z)-  [75]  V  The parameters A, P y and « are the hysteresis shape parameters (if n = oo, the bilinear elastoplastic case is obtained), v and r| are the strength and stiffness degradation parameters, and h(z) is the pinching function introduced by Baber and Noori (1986). Note that for systems in which the loading fluctuates in such a manner that the • displacement is cyclic but does not change sign, there is a tendency of the BWBN model to introduce some artificial drift into the system response. This problem can be corrected by adding two more terms to the governing equation (Casciati 1987) but it is not particularly important for systems undergoing random excitation (Wen 1989). The SDOF system response model given by Eqs. [74] and [75] states that the rate of increase of the restoring force depends on the state of the system (in terms of u and z) as well as whether it is in a loading or unloading stage, due to the absolute value signs. Also, for a given time-history of displacement, the restoring force is completely specified by the differential equation i.e. there are no empirical rules. However, the model also forces pinching to occur at zero load, which doesn't necessarily reflect the true behaviour of all structural systems, some of which pinch at a residual force level. To generalize the BWBN model, Foliente (1995) added an additional constant parameter q that sets the pinching level as a fraction of the ultimate value of z. This model is referred to as the modified BWBN model and is given by the following:  h(z) = l - £ , exp -  (zsgn(u)-qz )  [76]  u  where sgn() is the signum function, and z is the ultimate value of z given by: u  1 V(P+Y).  [77]  Chapter 2  Literature Review  68  and the parameters that control the severity and rate of pinching, respectively, are:  £ = 6 [l-exp(-/wr)] 0  [78]  Strength and stiffness degradation are modeled, respectively, by:  v = 1+Ss u  [79] rj = l + S e n  Finally, strength and stiffness degradation, as well as pinching, are controlled by the hysteretic energy dissipation given by the following:  '/  e = (l - a)a> jziidt 2  0  [80]  The modified B W B N model, which has been extended to bi-axial hysteretic systems (Park et al. 1986), is very flexible and can produce a wide variety of hysteresis shapes to model the behaviour of hysteretic degrading systems. In addition, standard damping ratios for linear systems can be used for response analysis since the B W B N model simply adds a hysteretic element to a mechanical model that contains linear viscous damping and a linear spring. Once the parameters of the hysteresis model are identified, dissipated energy can be obtained from the hysteresis trace of the response. The identification or estimation of suitable model parameters for a particular set of system materials and configurations is known as a system identification problem. A simple technique to determine the parameters from test or field data, which is based on least square minimization, was developed for the uni-axial and bi-axial B W B N model by Wen and Ang (1987) and Sues et al. (1988). Other common methods, which were cited  Chapter 2  Literature Review  69  by Foliente et al. (1998), include: sequential regression analysis (Masri and Caughey 1979, Masri et al. 1982, Masri et al. 1987), spectral method (Roberts et al. 1995), Newton's method, Gauss' method, and the extended Kalman filtering technique.  The  general theoretical aspects o f hysteretic system identification have been reviewed by M i n a i and Suzuki (1987).  Recently, Dobson et al. (1998) presented a Boolean model for mechanical hysteretic systems that uses a set o f Boolean statements to provide a piece-wise n o n linear representation o f the loading and unloading curves over an arbitrary number o f sub-intervals. B y significantly increasing the number o f sub-intervals, the modeling o f hysteresis can be accomplished by using a database o f arrays, which eliminates the need for system identification when the hysteresis loop is constructed from experimentally sampled static or dynamic data.  Chapter 2  2.5  70  Literature Review  SEISMIC S T R U C T U R A L R E S P O N S E A N D R E L I A B I L I T Y STUDIES  The assessment of structural response under dynamic loading has received considerable attention in the literature over the past 30 years.  With the advent of  increasingly powerful personal computers, researchers are now able to conveniently analyze complex analytical random vibration models or quickly perform time-history analysis, which has led to a wealth of literature relating to structural response under dynamic loading.  Considering only those studies that relate to non-linear structural  response under seismic loading, a very brief review of some past work that has utilized one or several of the analytical or numerical techniques summarized in Sections 2.2 - 2.4 is given below. This review does not list those studies that include a structural response analysis to illustrate the use of a proposed analytical or numerical technique. Those studies have already been cited in the relevant section describing the technique.  A large majority of the literature deals with seismic response analysis of steel and concrete structures, some of which were considered deterministic and others that were treated as having variable mechanical properties. These studies are listed in order of the type of analysis that was used beginning with the Monte Carlo method, which seems to be the most popular non-linear stochastic dynamic analysis technique. L i u (1969) used fifty artificial earthquake records based on the 1940 E l Centro earthquake and elastoplastic and bilinear degrading hysteresis loops to collect SDOF earthquake response statistics; Meskouris and Kratzig (1987) used three records and a Takeda-type hysteretic law to investigate a five-storey concrete frame; O'Connor and Ellingwood (1987) used 20 California earthquake records in a Latin Hypercube Sampling scheme and an elastoplastic hysteresis loop to determine SDOF steel frame reliability indices based on limit states defined by displacement ductility and a low-cycle fatigue damage index; Conte et al. (1991) used two sets of 100 A R M A model records and several piece-wise linear hysteresis models to evaluate SDOF response statistics and perform a sensitivity study on structural parameters; Seya et al. (1993) used eighteen artificial records in a Latin Hypercube Sampling scheme and a bilinear hysteresis loop to generate fragility curves for five displacement ductility-based limit states for a hypothetical five-storey steel  Chapter 2  Literature Review  71  building; Bolotin (1993) used twenty artificial records to investigate the effect of random peak ground accelerations and spectral characteristics as well as random structural parameters on the response of a sixteen-story building; Lee (1996) used six California records and a bilinear hysteresis loop to investigate the significance of peak ground acceleration to peak ground velocity ratio on the displacement ductility demand of a tenstorey frame structure; Collins et al. (1996) used a uniform hazard spectra approach based on 1292 artificial records generated for the Los Angeles area to evaluate structural performance of SDOF frames using displacement ductility-based limit states. This work included a proposed reliability-based seismic design procedure using displacementbased performance criteria and an equivalent system methodology for application to M D O F structures; Han and Wen (1997) used 88 records recorded around the world and a proposed 'equivalent non-linear system' approach to evaluate the performance of seven multi-storey structures; Marek et al. (1997) performed a sensitivity analysis on structural random variables using 100 artificial records in a Latin Hypercube Sampling scheme; Bagchi (1999) used four records generated by the Geological Survey of Canada from uniform hazard spectra to evaluate the performance of concrete frame structures in Victoria and Montreal.  The other methods of analysis have not been used nearly as frequently as the Monte Carlo technique. Suzuki and Araki (1997) used the Response Surface Method to evaluate the reliability of a two-storey frame structure with random structural properties using the 1940 E l Centro and 1995 Kobe earthquake records and a bilinear hysteresis loop; Casciati and Faravelli (1985) also used the Response Surface Method to generate seismic fragility curves for a four-storey frame structure with random structural properties.  Several other authors have used the analytical Equivalent Linearization  Method to evaluate structural performance: Sues et al. (1985) analyzed a four-storey steel frame building and a seven-storey concrete building using the B W B N hysteresis model and random structural properties; Wen and Eliopoulos (1994) analyzed a fivestorey frame to determine the response statistics and then used the statistics to calculate probabilities of exceeding certain inter-storey drift limits given a certain seismic hazard; Foliente et al. (1996) evaluated the accuracy of Equivalent Linearization incorporating  Chapter 2  Literature Review  72  the modified B W B N model against 200 Monte Carlo response samples and then analyzed the sensitivity of the response to changes in structural properties and hysteresis shape parameters. Finally, a number of authors have used stochastic finite element solutions to determine the reliability of structures under dynamic loads (Langley 1985, Mahadevan and Mehta 1991, Faravelli 1992, Bucher and Brenner 1992, Brenner and Bucher 1996).  Research on the seismic performance of timber structural systems has not received the same attention in the literature as that for steel and concrete structures. This may be due to its less common usage in large-scale structures and perhaps because of the difficulty in characterizing timber structural behaviour, which besides being non-linear may also be influenced by the rate and duration of loading. O f the research that has been done, some of the recent work is summarized below.  Chui and Ni (1995) investigated the effect of the moment-rotation characteristics of circular bolted timber connections on the seismic performance of a timber frame. Using a DRAIN-2D moment-rotation hysteresis model, the frame response to the 1940 El Centro and 1971 Orion Blvd. earthquakes for a range of hysteresis model parameters was determined.  The study showed that very stiff connections likely result in brittle  failure of the timber members, while normally stiff connections would result in a connection failure at large displacement.  The shape of the hysteresis loop was also  shown to influence response levels, with highly pinched loops resulting in larger displacement responses.  Latendresse et al. (1995) used a mechanics-based  finite-  element approach to model multiple dowel connections using the analogy of a beam on a deformable foundation. Experimental hysteresis loops under cyclic loading were shown to match well with the analytical model. Yasumura (1996) performed a series of cyclic lateral load tests on wood framed shear walls, glulam braced frames, glulam moment resisting frames and glulam arched frames to determine a 'behaviour' factor q, which is a function of the displacement ductility. It appears that the behaviour factor is identical to the well-known force reduction factor R in the 1995 N B C C , since it is the same function of ductility as that proposed by Newmark and Hall (1973) i.e. q = R = V(2p-1). Using the behaviour factor, a timber structural system of each type was designed and tested using  Chapter 2  Literature Review  73  the 1940 E l Centra, 1952 Taft and 1995 Kobe earthquakes to evaluate its seismic performance.  Results showed that the maximum displacement response was usually  within the collapse limit of each structure. Frenette et al. (1996) tested a two-storey Parallam® moment resisting frame using six consecutive earthquake records, which were taken from the  1992  Landers earthquake and scaled to different peak ground  accelerations. The purpose was to assess the elastic characteristics of the frame and then observe its performance during a major earthquake, an aftershock and subsequent major earthquakes.  Experimental results compared well with analytical predictions using  DRAIN-2DX and DPSA, a material properties based non-linear finite element program. Foliente et al. (1998) summarized the system identification process for estimating the parameters in the modified B W B N hysteresis model and then used experimental data from shear wall pseudo-dynamic tests in Japan to estimate the parameters. Using the 1989 Loma Prieta and 1995 Kobe earthquakes, a comparison was made between the time history response using a non-degrading, non-pinching hysteresis model and the timehistory response using the modified B W B N model. It showed that displacement response might be severely underestimated if a non-degrading, non-pinching hysteresis model is used. Ceccotti and Karacabeyli (1998) evaluated the required 'Action Reduction Factor' for moment resisting timber frames, which is analogous to the force reduction factor R in the 1995 N B C C . Using eight earthquake records mainly from Italy, it was determined that an A R F of 2 is appropriate for one-storey buildings with timber moment resisting frames. Hockey et al. (1999) summarized several techniques for the reinforcement of timber connections to increase ductility for earthquake resistance.  These techniques  included: fibreglass reinforcement, glued plywood reinforcement, application of a truss connector plate, and the insertion of glued-in or threaded rods transverse to the wood grain. Popovski et al. (1999) evaluated the dynamic response of braced timber frames with various brace connections. Using DRAIN-2DX, with a hysteresis model based on experimental shake table tests, a time-history analysis was done using five earthquake records from around the world.  From these analyses, the influence of different  connection details on the seismic response of braced timber frames was determined. In addition, a conservative force reduction factor R of 1.5 was suggested for bolted connections, while for glulam riveted connections an R value of 2.0 was suggested.  Chapter 2  Literature Review  74  Finally, Karacabeyli and Ceccotti (1999) evaluated the dynamic performance of a fourstorey wood shear wall structure using 28 earthquake records, six of which were real records and the remaining 22 were records that had been modified to fit the Vancouver area design spectrum. Results showed that the current 1995 N B C C force modification factor R = 3 for plywood nailed shear walls is appropriate.  Chapter 3  Numerical Algorithm Development  75  CHAPTER 3  NUMERICAL ALGORITHM DEVELOPMENT  3.1  INTRODUCTION  The development of a software application to enable more rigorous structural reliability analysis under seismic loading began by selecting a suitable probabilistic analysis method from among the available methods outlined in Chapter 2. Once the decision was made to use the numerical analysis approach, the task of outlining the basic structure of the software and deciding what functionality should be included was the next step in the process of creating and integrating the required algorithms. Chapter 3 begins with a brief summary of the reasons for adopting the numerical analysis approach followed by several sections that detail key numerical components, existing model modifications and solution methods that form the foundation of the chosen structure and functionality of PSResponse. In each section, relevant computational issues such as; running time, error control, convergence, proper numerical estimation and random number generation are discussed.  It should be noted that the details and discussion in each of the following sections, and the sections themselves, are not intended to be a complete description of the algorithmic structure of PSResponse (see Chapter 4).  The intent of this chapter is to  provide the important technical details behind the algorithms that form the software.  3.2  SELECTION OF ANALYSIS METHOD  The decision regarding which probabilistic analysis method should be used to determine the statistical dynamic response of structures undergoing earthquake induced excitation was based on the key requirements of robustness and flexibility. The chosen  Chapter 3  Numerical Algorithm Development  76  analysis method had to allow for significant complexity in the structural system for analysis of multi-storey structures, while also allowing for a large degree of nonlinearity in system restoring forces to accommodate the hysteretic nature of inelastic structural seismic response.  In addition, the stochastic input excitation, in the form of  earthquake induced ground acceleration, should reflect the true nature of the random excitation  process,  rather than a mathematically  tractable  idealization.  These  requirements tended to eliminate all the frequency domain based analytical methods, with the exception of the Equivalent Linearization Method, due to their various serious limitations regarding the nature of the restoring force, structural complexity or type of random excitation process.  The Markov based methods each required that the input excitation be a stationary Gaussian process (i.e. white noise), which is a poor representation of a real earthquake spectrum, and most of the methods (Galerkin Method, Finite Element Method, Closure Technique) have the disadvantage  of slow convergence  or large computational  requirements for highly non-linear or multiple-degree-of-freedom systems. In the case of the Numerical Diffusion Method, the applicability of Generalized Cell Mapping (GCM) to degrading hysteretic systems, which characterizes most structural systems, is not known.  Other analytical methods of determining the statistical response of structures to stochastic excitation, such as the Perturbation Method and the Functional Series Representation Method, are not confined to a white-noise assumption for the excitation and are even applicable to multiple-degree-of-freedom systems. However, each of these methods are limited to weakly non-linear systems and as such are not applicable to ductile, hysteretic structural systems.  The Equivalent Linearization Method has been successfully applied to response and damage prediction of a variety of highly non-linear structural systems under seismic excitation. However, response results may tend to be underestimated using this method when the excitation spectral content is such that the power spectral density function  Chapter 3  Numerical Algorithm Development  77  vanishes rapidly as the frequency goes to zero, which is typical of earthquake excitation. In addition, the assumption of a Gaussian input excitation results in an assumed Gaussian response of the linearized non-linear system. This assumption is not correct for a nonlinear system, which is known to have a non-Gaussian response to a Gaussian input. The result of an assumed Gaussian response is that it may significantly misrepresent the frequency of high response levels to extreme loads, which contribute most to firstpassage and fatigue failures.  With these limitations in mind, the decision to forego the frequency domain based methods in favour of a time-domain based approach was made.  The time-history  approach to finding the response statistics of a dynamic system is generally more accurate and robust than any of the analytical random vibration methods since there are no limitations or assumptions required to obtain response statistics using time domain based numerical methods.  This generality is the reason that numerical methods are  frequently used to verify results obtained using other analytical random vibration methods.  3.3  NUMERICAL MODELS AND SOLUTION METHODS  Having chosen to use a time-history approach for probabilistic dynamic response analysis, the development of algorithms to solve the general differential equation of motion for random earthquake loading required numerical models and solution methods in four major areas: earthquake generation, hysteresis modeling, structural modeling and an overall numerical time-stepping method. The noteworthy technical details in each major area are summarized in the following.  3.3.1  Numerical Time-Stepping Method  Since analytical solution of the equation of motion is not possible for arbitrarily varying excitations and non-linear systems, a numerical time-stepping method is required to integrate the differential equation, or system of equations, governing the  Chapter 3  Numerical Algorithm Development  response of the system.  78  There are a number of types of time-stepping procedures  applicable to both linear and non-linear systems that are categorized by the assumption made in interpolating the system response and their inherent stability. The stability of a numerical procedure refers to its ability to return a bounded solution for a given timestep length. Procedures that "blow up" in the presence of numerical round-off and return meaningless results if the time-step is longer than some stability limit are conditionally stable, while those that lead to bounded solutions regardless of the time-step length are  unconditionally stable.  For response analysis of single-degree-of-freedom  (SDOF) systems, whether  linear or non-linear, the stability criterion does not usually control the choice of timestepping procedure since the time-step required for numerical accuracy is considerably smaller than the stability limits of the conditionally stable procedures. response analysis of multiple-degree-of-freedom  However, for  (MDOF) systems, unconditionally  stable procedures are generally necessary to avoid the excessive computational demands of conditionally stable procedures, which require an extremely short time-step to remain within the stability limit of higher modes of response.  Since M D O F analysis capability  was a definite requirement of the software application, an unconditionally stable timestepping procedure was needed.  Selection of which procedure to use for algorithm  development was based on a review of time-stepping procedures (Chopra 1995) that compared the solution results for a linear system using two procedures; Newmark's Average Acceleration Method and Wilson's Method. That review showed that Wilson's Method introduces numerical damping into the dynamic system, whereby the system displacement amplitude decays with time even in the absence of system damping. No amplitude decay is introduced by Newmark's Average Acceleration Method. Also, the tendency for numerical methods to elongate or shorten the natural period of response is reduced in the Newmark Method as compared to Wilson's Method, both of which induce a period elongation.  In addition, a second Newmark procedure called the Linear  Acceleration Method, which is conditionally stable, induces an even smaller period elongation than the Average Acceleration Method and is, therefore, more accurate than its unconditionally stable counterpart.  For this reason, using the Linear Acceleration  Chapter 3  Numerical Algorithm Development  79  Method is preferred over the other methods in situations where a conditionally stable procedure is allowable.  With these properties in mind, both Newmark methods were selected for response analysis. This was done because there is no difference in the time-stepping algorithm between the two methods except for the value of two constants, y and B. Therefore, the superior accuracy of the Linear Acceleration Method could be utilized and provision made to switch to the Average Acceleration Method when required for stability reasons.  3.3.2  Time—Stepping Overshoot Problem  Integration of system response in a time-stepping procedure can give rise to significant error when the transitions in the force-deformation relationship associated with velocity sign changes are not followed closely.  When the calculated response  velocity V j + i at time z+1 changes sign with respect to velocity V j this indicates that at some point during the time-step the response displacement either started increasing or decreasing. If the point at which the velocity went to zero during the time-step is not identified in the numerical procedure then the step-by-step path of the force-deformation relationship will "overshoot" the true path and the calculated displacement at the end of time-step z'+l will be either too large or too small. These departures from the exact path will occur at each reversal of velocity, leading to errors in the numerical results.  To minimize the overshoot problem, the Newmark time-stepping algorithm uses a variable time-step when velocity sign changes are detected. The sign of the velocity is tested after each time-step and in the event of a sign change the integration is reset back to the beginning of the time-step.  The time-step is then repeatedly bisected until the  absolute value of the response velocity at the end of the reduced time-step is less than a preset fraction of the peak response velocity. The peak response velocity is assumed to be 10% ofthe peak ground velocity, which is conservative except for extremely short and extremely long natural period structures, based on a typical response spectrum. Assuming a peak response velocity based on peak ground velocity, rather than actually  Chapter 3  Numerical Algorithm Development  80  calculating it, is done to eliminate the need to determine the peak response velocity from an initial guess of an appropriate minimum velocity and then iteratively adjusting the minimum velocity to be a fraction of the updated value of peak response velocity.  3.3.3  Structural Model  The type of structures that the software application is intended to model are those that may be represented as typical lumped mass idealizations consisting of shear walls or frames as the lateral load resisting elements.  These idealized structures are further  simplified to lateral-degree-of-freedom only models prior to response analysis to facilitate rapid calculation of the response time-history.  Also, calculation of the  displacement time-history is based solely on integration of the governing differential equation of motion, consequently, the second-order effect on lateral displacement produced by the vertical load acting on the structure in its displaced configuration is not considered in the displacement calculation.  The mathematical representation of any type of structure is contained in its mass and stiffness matrices.  For both types of structures considered in the software  application, categorized as shear and frame, the mass matrix is diagonal owing to the assumption of lumped floor masses.  Shear structures are considered to have rigid  horizontal beams, slabs or diaphragms, which eliminates all rotational degrees of freedom and renders the stiffness matrix a lateral stiffness matrix automatically. Frame structures retain the rotational degrees of freedom at the intersection of horizontal and vertical elements since the horizontal elements are not considered rigid. The stiffness matrix is then reduced to a lateral stiffness matrix by eliminating the rotational degrees-offreedom through static condensation since it is assumed that no mass is associated with the rotational degrees-of-freedom.  The static condensation process employs an L U  decomposition and back-substitution procedure for the required matrix inversions.  Chapter 3  Numerical Algorithm Development  81  For modal analysis and information purposes, the natural frequencies of the structure are determined by solving the structural dynamics eigenvalue problem given by the following:  k<z) = cy m^  [1]  2  The eigensystem in Eq. [1] is solved by first transforming it into the standard eigenvalue problem by pre-multiplying by m" . This gives: 1  A<(> = 44>  [2]  1  2  where A = m" k and X = co . In general, A is not symmetric although m and k are both symmetric matrices.  The eigensystem is then solved using a Jacobi procedure that  requires that A be a symmetric matrix, which is achieved by Ay = Ay  where A = m ~ k m /2  [3]  /z  and y = m"< ' t> and X = co . 2/  2  The mode shapes ( § ) associated with the natural frequencies (co = VA. ) are n  n  determined by pre-multiplying the eigenvector matrix (y ) by m \ v  n  n  The resulting mode  shapes are normalized such that the generalized modal masses M  n  are 1.  Re-  normalization of the mode shapes to 1 is straightforward. The distribution of damping in the structural model is determined by whether the system is linear or non-linear. Classical damping is assumed for linear systems to allow for modal analysis.  For non-linear systems, damping may be specified as Rayleigh  damping or as a multiple of a single baseline damping value for each storey.  Chapter 3  3.3.4  Numerical Algorithm Development  82  Hysteresis M o d e l  Modeling of the hysteretic restoring force in the equation of motion is done using the Bouc-Wen-Baber-Noori (BWBN) hysteresis model (Sec.  2.4.1), which was  modified to reduce the number of parameters that need to be identified (see Sec. 3.3.4.4). The B W B N model is used because it is able to produce a wide variety of hysteresis shapes, including the pinching and degradation behaviour exhibited by many hysteretic systems, without the use of piece-wise linear equations governed by numerous empirical rules relating stiffness to displacement.  3.3.4.1  M o d i f i c a t i o n to the B W B N model  The B W B N hysteresis model introduces a state variable z into the equation of motion and separates the restoring force into non-hysteretic and hysteretic components. For an SDOF system, the equation of motion in standard form is then given by:  ii + 2<^co u + aco u + (l - a)co z = f(t) 2  0  [4]  2  0  where £, is the system damping ratio, coo is the natural frequency, a is the ratio of postyielding to pre-yielding stiffness,/^) is the mass normalized forcing function and z is the hysteretic displacement, which is described by the non-linear differential equation:  [5]  where the parameters A, p y and n are the hysteresis shape parameters, v and r\ are the strength and stiffness degradation parameters, and h(z) is the pinching function given by the following:  Chapter 3  Numerical Algorithm Development  83  (zsgnfc) -qz )  2  h(z)= l-<f, exp  u  [6]  ti  where sgn() is the signum function, q is a constant parameter that sets the pinching level as a fraction of the ultimate value of z, and z is the ultimate value of z. The parameters u  that control the severity and rate of pinching, respectively, are:  £ = £„[l-exp(-/>ff)] [7]  #2 =(^0+VX^ + ^ l ) Finally, strength and stiffness degradation are modeled, respectively, by:  v = 1+£ £ u  [8] /7 = 1 +  5e n  where s is the dissipated hysteretic energy.  In total there are thirteen separate parameters in the BWBN model that must be identified, although two of the parameters, A and n, are typically set to 1. The identification or estimation of suitable model parameters for a given structural configuration and material type is known as a system identification problem, which rapidly increases in difficulty as the number of parameters increases. To simplify the system identification problem the existing pinching function, which utilizes six parameters, was modified to have only three parameters. The remaining non-unity parameters; a, (5, y, 8 and S are more fundamental and were not altered. The revised V  n  pinching function is given by:  h(u,z) = f  [(/? + /)• z] +u-[\ + sgnfc)- sgn(ti)] - [l - / ] • [lO - [fi + y)-u} 2  f  2  [9]  Chapter 3  Numerical Algorithm Development  where / = exp(-^ • s) and sgn( ) is the signum function as before.  84  In this modified  pinching function the role of each of the three new parameters; <p, ju and v, is more easily understood than the relationship between the six parameters of the original pinching function.  The parameter (p controls the overall rate of increase in pinching as damage  cycles progress, the parameter ju controls the rate of stiffness recovery throughout the loading phases of each cycle and the parameter v controls the rate of stiffness recovery during the increasing displacement portion of each loading phase.  In modifying the pinching function the assumption was made that pinching occurs at zero restoring force, which is equivalent to the assumption that q = 0 in the original pinching function. Therefore, in effect, the pinching function has been reduced from a five parameter model to a three parameter model in its modified form. The assumption that pinching occurs at or very near zero restoring force was based on the observation that significant hysteretic pinching in overall cyclic structural behaviour is largely the result of structural damage associated with localized failures such as cracking and connection degradation due to material crushing.  This structural damage decreases the initial  stiffness at the very beginning of each loading cycle, until increasing displacement closes the cracks and connection gaps at which point stiffness begins to increase. Individual fasteners and reinforcement within the structure may continue to exhibit typical yielding behaviour where stiffness at the beginning of each loading cycle is equivalent to the elastic stiffness until yield is reached; however, the overall behaviour of a structure includes the cumulative effect of damage throughout the entire structure, which degrades initial stiffness at the outset of each loading cycle.  Figure 3.1 shows a comparison of the original and modified pinching functions in the B W B N model for three types of cyclic behaviour corresponding to a structure subjected to an increasing amplitude sinusoidal displacement. The first two cases assume that pinching occurs at zero restoring force, with subsequent stiffness recovery or no stiffness recovery (Fig. 3.1, top and middle), while the third case illustrates pinching at a non-zero restoring force (Fig. 3.1, bottom).  The hysteretic data was generated by  Chapter 3  Numerical Algorithm Development  85  selecting the original parameters of the B W B N model such that the structure undergoing the sinusoidal displacement exhibited significant yielding and stiffness degradation behaviour and then the modified pinching function parameters were fitted to that hysteresis loop.  From Figure 3.1 it can be seen that the modified pinching function  provides a good fit to the original data when pinching occurs at zero restoring force, however, as expected, it increasingly underestimates the hysteretic force at the beginning of a loading cycle as the cycles progress when pinching occurs at a non-zero restoring force.  Therefore, response displacement may tend to be overestimated using the  modified pinching function in applications that exhibit pinching behaviour at a significant force.  3.3.4.2  Numerical Solution of Hysteresis Model  Determining the hysteretic displacement z in the differential equation of motion at each time-step in the Newmark procedure requires solution of Eq.[5], which is a firstorder, non-linear ordinary differential equation for which no exact solution exists. Therefore, a numerical solution method was required, which had to be incorporated into the Newton-Raphson iteration scheme in the Newmark Method. Initially, a fourth-order Runge-Kutta algorithm (Burden and Faires 1985) was developed and tested against the exact solution for hysteretic displacement z as a function of displacement u that exists for the non-degrading, non-pinching B W B N hysteresis model. Without the pinching and stiffness degradation terms in the model, the equation becomes a linear ordinary differential equation that has a piece-wise continuous exact solution defined by four equations that depend on the sign of velocity and displacement. Testing indicated that the numerical solution provided a very good approximation of the exact solution. Figure 3.2 shows a typical non-pinching, non-degrading hysteresis loop, with the associated exact equations, for one cycle of a sinusoidal input displacement.  Numerical Algorithm Development  Chapter 3  Pinching Without Stiffness Recovery  D i s p l a c e m e n t (m)  Pinching With Stiffness Recovery 0,04  o. m  5  o  •ffi >-  I  0 04 D i s p l a c e m e n t (m)  Pinching at Non-Zero Force  as a. (A b -Q2 u  I,  -0.04D i s p l a c e m e n t (m) Original P i n c h i n g Function  M o d i f i e d P i n c h i n g Function  Figure 3.1: BWBN Model Pinching Function Comparison  86  Numerical Algorithm Development  Chapter 3  87  z = [A/((3+Y)][1-exp(((3+ )(-u))] Y  z = [A/(p+ )][exp((P+Y)(u-u2))-1] Y  Figure 3.2: Exact Solution of Non-Pinching, Non-Degrading BWBN Model  The Runge-Kutta scheme is not usually the most computationally efficient method for solving non-linear ordinary differential equations but it was the starting point for developing a solution method because it succeeds in virtually all applications. The RK scheme is a one-step method where the approximation for z \ involves information i+  from the previous step z, only, solution methods that use information from several previous steps are termed multi-step predictor-corrector  methods. Predictor-corrector  methods are more computationally efficient than Runge-Kutta for many smooth systems but they are more difficult to start-up because of the need for more past information. A predictor-corrector  solution  scheme called the Adams Fourth-Order  Predictor-  Corrector Method (Burden and Faires 1985) was tested against the Runge-Kutta scheme and found to give the same accuracy (-0.01% - 0.02% difference) for the final value of z for the same number of sub-steps / within each time-step of a cyclic motion record. The  Chapter 3  Numerical Algorithm Development  88  Adams Method has the added advantage that it runs approximately 25% faster.  In this  method, the fourth-order Runge-Kutta scheme is used to obtain starting values for the four-step Adams-Bashforth predictor method, which is then corrected by one iteration of the three-step Adams-Moulton method.  The number of predictor-corrector steps  following start-up depends on the level of accuracy required of the numerical solution. Implementation of the Adams Fourth-Order Method within the Newton-Raphson iterative process of the Newmark Method was initially completed using 100 predictorcorrector steps within each Newton-Raphson iteration.  This number of steps was  selected as a good balance between accuracy and run-time following a comparison of numerical hysteretic displacement results with the exact solution for a non-pinching, non-degrading system undergoing cyclic oscillation.  Displacement error varied from  0.04% using 1000 predictor-corrector steps to 1.43% using five steps. The error for 100 steps was 0.32%.  Following testing of the completed Newmark algorithm using real earthquake time-histories as the input excitation, it was determined that response analysis time could be improved significantly if the number of steps in the Adams Method solution algorithm could be reduced, while maintaining accuracy.  Therefore, a new Adams Method  algorithm was developed that uses adaptive step sizes for solving the O D E describing the hysteresis loop.  Step size is adjusted according to the error estimate determined by  comparing the predicted and corrected values of the dependent variable (i.e. z) after each cycle of the predictor-corrector process.  Step size is iteratively reduced by a factor of  one-half until the error is within the prescribed tolerance and iteratively increased by a factor of two when the error is less than 1% ofthe prescribed tolerance.  3.3.4.3  Consideration of D e g r a d i n g N a t u r a l Frequency  Degradation in stiffness as the structural system response reaches the inelastic range has the effect of increasing the natural period of the structure, which affects the way the structure responds to subsequent excitation.  The feasibility of modeling the  Chapter 3  Numerical Algorithm Development  89  degradation of natural frequency during non-linear response was assessed by calculating the dynamic response of systems in which the original, undamaged natural frequency was multiplied by the square-root of the stiffness ratio at the beginning of each time-step. The stiffness ratio was defined as the ratio of current hysteretic stiffness to original hysteretic stiffness. Initial attempts to update the natural frequency after each iteration of the Newton-Raphson procedure resulted in convergence problems and instability in the response calculation.  Assessment of the degrading natural frequency model showed that a minimum allowable stiffness ratio was required to prevent convergence problems associated with large displacements during time-steps where hysteretic stiffness approached zero.  This  situation, which may arise at response velocity sign changes or in the pinched region of the hysteresis loop, causes the stiffness ratio to approach zero and, consequently, natural frequency goes to zero. In this event, post-yield stiffness, given by the third term in Eq. [4], also approaches zero and extremely large deflections are incurred resulting in model convergence problems and instability.  Limiting the reduction in natural frequency by adding a minimum allowable stiffness ratio to the dynamic analysis algorithm tended to eliminate the convergence problem but added an unwanted additional parameter to the hysteresis model. In fact, from a system identification perspective, updating the natural frequency throughout a non-linear dynamic analysis effectively adds two parameters to the hysteresis model; the minimum stiffness ratio and an implicit parameter that is similar to the stiffness degradation parameter (S ). n  Consequently, the system identification process for the  degrading natural frequency model will identify values of; a, /?, y, 8 , 8 , q>, JU and v that V  n  differ from those that would be identified without consideration of natural frequency degradation. However, the end result is the same, a phenomenological hysteresis model that provides a close match between model response and an input real response record or input pseudo-static test data.  This implies that there is no need to include natural  frequency degradation in the dynamic analysis algorithm since inclusion only alters the  Chapter 3  Numerical Algorithm Development  90  hysteresis model parameters to match the given input at the cost of an additional hysteresis parameter and possible convergence problems.  It should be noted that accounting for the inelastic response phenomena of natural period elongation within the phenomenological hysteresis model may increase the risk of model error when hysteresis parameters are estimated from pseudo-static test data. In a pseudo-static test there is little or no dynamic effect present in the response, therefore, the true impact of a degrading natural frequency is not reflected in the input data used for hysteresis parameter identification.  This may in turn cause the pseudo-static model  parameters to underestimate the true dynamic displacement response which now includes the actual consequence of a degrading natural frequency.  This risk of underestimating  the response displacement is not really the fault of accounting for period elongation within the hysteresis model, nor the fault of the system identification procedure itself, which only seeks to duplicate the input data, rather it is an unavoidable consequence of using pseudo-static data to calibrate a dynamic model.  3.3.4.4  P a r a m e t e r Identification  As stated in Sec. 3.3.4.1, the determination of appropriate values for the modified hysteresis model parameters; a, ft, y 8 , 8 , cp, ju and v contained in Eqs.[4, 5, 8, 9] is V  n  known as a system identification problem, or more accurately, as a parameter estimation problem.  System identification is the general name given to a wide field that seeks to  infer a mathematical or algorithmic model of a dynamic system based on observed data from the system.  Parameter estimation is the second stage of the two-stage system  identification process that determines the numerical values of all the parameterized elements of the mathematical model identified in the first stage.  In the present  application, the model from the first stage of the system identification process is the newly modified version of the B W B N hysteresis model, which may be called a gray-box within the overall system framework governed by the equation of motion.  The term  gray-box is used to describe a model in which the adjustable parameters have a physical interpretation such as the stiffness degradation parameter in the B W B N model. A model  Chapter 3  Numerical Algorithm Development  91  in which the parameters are simply vehicles for adjusting the fit to the available data and do not reflect physical considerations in the system is referred to as a black-box.  There are two common approaches to minimizing the error between a predictive model and the observed data on which it is based (Ljung 1999). termed the prediction-error  The first approach,  approach, is to form a scalar-valued norm or criterion  function that measures the size of the prediction error and then choose the parameters that minimize the norm or function. This approach contains several well-known procedures such as the Least Squares Method and the Maximum Likelihood Method.  The other  approach, termed the correlation approach, is to require that the prediction error be uncorrelated with a given data sequence. In other words, the model parameters must be chosen such that the prediction error at every step in the sequence of observed data is independent of the previous steps. This approach is the basis for various instrumentvariable methods, which are most suited to auto-regression type (ARX) system models that describe the relationship between input and output using a linear difference equation containing previous values of the output variable. Using this type of model is equivalent to treating the entire system as a black-box, which by definition casts the parameter identification problem completely in the observation space, in terms of the observed input and output, without any reference to the underlying mechanics of the system.  Given that the parameters to be identified apply only to the gray-box hysteresis model and not an overall black-box system, the prediction-error approach was adopted for developing an algorithm to identify hysteresis parameters from input data.  For  reasons of simplicity, the quadratic norm of the Least Squares Method was chosen over the likelihood function of the Maximum Likelihood Method to measure the size of the prediction error associated with a given trial parameter vector 6.  Therefore, the  generalized error norm is given by:  [10]  Chapter 3  Numerical Algorithm Development  92  where l(.) is the quadratic norm, 1(e) = 'A e , Z is the observed dataset and the prediction 2  N  error sequence e is given by:  [11]  For predictor models that are a linear function of the parameters to be identified, the prediction error sequence becomes:  [12]  s{t,9)=y{t)-<p {t)-e T  and the resulting least-squares norm, given by substitution into Eq. [10], becomes a quadratic function in 6 that can be minimized analytically to determine the optimal parameter set. In general, however, Eq. [10] cannot be minimized analytically, as is the case for the gray-box hysteresis model, therefore, the minimum has to be found by iterative numerical techniques.  Numerical minimization methods can be divided into three groups; (i) methods using values of the function VN only, (ii) methods using values of the function V and its N  gradient, and (iii) methods using values of the function VN, its gradient and its Hessian, which is the second derivative matrix or curvature matrix. For the special case of the Least Squares Method, the gradient of the criterion function is given by:  [13]  where y/(t,6) is the dxp  gradient matrix ofy(t\6) with respect to 0. This leads to a family  of search routines given by:  [14]  Chapter 3  Numerical Algorithm Development  where Rj  l)  93  is a d x d matrix that modifies the search direction and jUf/ is the step size. If l)  RN® is defined as the identity matrix, Eq. [14] becomes the steepest-descent method, which corresponds to the second group of methods listed previously. This method has the drawback that it is fairly inefficient close to the minimum of Eq. [10], which is the reason the Hessian is used in the third group of methods, to improve efficiency.  The  Hessian for the Least Squares Method is given by:  where y/'(t,6) is the d x d Hessian of e(t,6). The third group of methods utilizes the Hessian as the search direction matrix R, which then makes Eq. [14] a Newton method. It is quite costly, however, to compute all the terms of \|/' so the second term in Eq. [15] is typically ignored, which is equivalent to replacing the Newton-Raphson method with the modified Newton-Raphson method.  This is permissible since a good estimate of the  Hessian is only required in the vicinity of the minimum for a Newton method and the term multiplying the second derivative in Eq. [15] is the random error e(t,d), which can have either sign and should in general be uncorrelated with the model. Therefore, the second derivative terms tend to cancel out when summed from t — 1...N. The actual implementation of a search scheme using the Hessian is commonly done using a method called the Levenberg-Marquardt  procedure that smoothly varies between the two  extremes of Eq. [14], the steepest descent method and the inverse-Hessian method. In this procedure, the step-size is set equal to unity {ji  (i> N  = 1) and the search direction  matrix, R, is modified by adding a parameter to the Hessian as follows:  *i?  W = II A'W  )• V '  )+ M  [16]  where A, is a positive scalar that is used to control the convergence in the iterative scheme. In a search region far from the minimum value of the norm, A, is set to a large value, which effectively reduces the step-size and turns the search direction towards the  Chapter 3  Numerical Algorithm Development  gradient as in the steepest-descent method.  94  As the minimum is approached, X is  iteratively reduced to zero, which smoothly switches the procedure to the inverseHessian method.  While the notion of using a parameter estimation method that identifies the optimal model parameter vector 6 through an automated search for the minimum of the least-squares norm is appealing, there is a practical difficulty in implementing such a scheme.  To use the formulas given previously, the gradient of the prediction, y/(t,6),  must be calculated at each step of the input data sequence.  There are a number of  methods for doing so, the choice of which depends on the model structure, but for each method the computational effort required to compute the gradient can be significant. The only alternative to expending that computational effort is to implement a Group 1 method, which uses values of the function F^only, in conjunction with a specific search pattern. Since the gray-box hysteresis model contains eight parameters to be estimated, resulting in an eight-dimensional gradient for a Group 2 or Group 3 method, the Group 1 method, combined with a directed search, was used to develop an algorithm for estimating the hysteresis model parameters.  It is an inherent feature of iterative search routines that only convergence to a local solution is guaranteed, in this case the local minimum of V (0, Z ). N  N  To find the  global minimum, which gives the optimal parameter vector 6, there is no other way than to start the iterative minimization routine at different feasible initial values and compare the results.  For a physically parameterized model, such as the gray-box hysteresis  model, feasible initial values and an associated search pattern may be determined from physical insight into the model structure. Consideration of the physical meaning of each of the modified B W B N hysteresis parameters showed that the dominant parameters are the post-yield stiffness parameter, a, and the two parameters that control the nondegrading shape of the hysteresis loop, /? and y. From that starting point, a three-stage algorithm was developed that begins by evaluating combinations of those three parameters that satisfy both Eq. [ 4 ] , the differential equation of motion, and Eq. [5], the differential equation of hysteretic displacement, for a given input data record. The input  Numerical Algorithm Development  Chapter 3  95  data is taken from either a pseudo-static cyclic displacement test or an acceleration response time-history of the structure for which the hysteresis parameters are to be identified.  The actual number of parameter combinations that satisfy both differential  equations depends on the yield displacement of the structure but the upper limit on the possible number of combinations evaluated for least-squares error is set at 3,960,000. This covers values of a that range from 0.01 to 0.99 and values of B and y that correspond to a yield displacement range of 0.007 - 2.000 m.  Once the least-squares error combination of the three dominant parameters is roughly determined, the second stage of the algorithm evaluates trial values of the pinching parameters, cp, pi and v, while refining the precision of a, B and y. As many as 69,300 parameter combinations are evaluated for least-squares error, depending on the initial values of a, B and y identified in the first stage.  The decision to identify the  pinching parameters before the stiffness degradation parameter, 8 , which overlap in their n  physical role in the hysteresis model, was made following comparisons of model fits with each parameter type identified first. Identifying pinching parameters before the stiffness degradation parameter is perhaps intuitive since the stiffness reduction associated with pinching is considerably larger, but localized to a portion of each loading phase, than the stiffness reduction associated with the stiffness degradation parameter, which applies throughout the hysteresis cycle.  Therefore, attempting to fit the stiffness degradation  parameter before the pinching parameters results in identifying a range of possible values with similar and relatively large least-squares errors that adequately model the stiffness reduction in a certain portion of the loading phase but do a poor job in the remainder of the loading phase.  The third stage of the parameter estimation algorithm finalizes the hysteresis model parameter vector by evaluating the least-squares error for 289 trial values of the strength and stiffness degradation parameters, 8 and 8 , in combination with the other V  parameters identified in the second stage.  n  Chapter 3  3.3.4.5  Numerical Algorithm Development  96  Parameter Adjustment for M D O F Structures  For non-linear dynamic analysis of a multiple-degree-of-freedom structure, the hysteresis model parameters, which are identified from structural test data corresponding to a certain stiffness and yield strength, must be adjusted for each different storey stiffness and yield strength. This adjustment is straightforward for the parameters related to yield strength, y and /?, which are simply modified by the ratios of storey stiffness to test structure stiffness and storey yield strength to test structure yield strength to maintain the proper storey yield displacement. The post-yield stiffness parameter, a, is assumed to remain constant. The remaining hysteresis parameters control pinching behaviour and strength and stiffness degradation as a function of the dissipated energy, which is calculated as the area of the hysteresis loop. The assumption that is made in adjusting these parameters is that a similar structural system with a different stiffness and/or yield strength that experiences a similar level of energy dissipation, normalized with respect to the calculated yield energy, will exhibit a similar degree of pinching and strength and stiffness degradation as the response history progresses.  Using this assumption, the  energy-related hysteresis parameters for each storey are adjusted such that the product of the parameter and the constant terms multiplying the normalized dissipated energy is constant (see Eqs. [8] and [9]).  3.3.5  Fourier Analysis and Power Spectrum Estimation  Generation of artificial earthquakes for Monte Carlo analysis of structural dynamic response requires a frequency spectrum source that contains the spectral characteristics of the type of earthquake being studied for its impact on structural response. This frequency spectrum source may be a white-noise Gaussian process that is filtered to give the required spectral shape or, alternatively, it may be determined from an actual earthquake record. When using an actual record the power spectral density, which describes the contribution of the individual harmonic components in a random signal, must be determined from a Fourier transform analysis of the discretely sampled ground acceleration data. This analysis is typically performed by the well-known fast Fourier  Numerical Algorithm Development  Chapter 3  97  transform or FFT algorithm. The salient details of the FFT algorithm and an associated spectral analysis algorithm that were adapted for use in the software application being developed in this project are summarized in the following.  3.3.5.1  F F T Algorithm The FFT algorithm, first developed by Cooley and Tukey in 1965, is based on a  recursive application of the Danielson-Lanczos Lemma, which was developed in 1942. The Danielson-Lanczos Lemma divides the discrete Fourier transform of a function consisting of N discretely sampled points into the sum of two discrete Fourier transforms, each of length TV/2. Each individual transform is made up of the even-numbered and odd-numbered points in the original N, respectively.  Through recursive application of  the Danielson-Lanczos Lemma, the original Fourier transform is repeatedly subdivided in half until transforms of unit length are obtained, assuming the original N is an integer power of two.  If the length of the original dataset is not a power of two, it must be  padded with zeroes up to the next power of two. With the input data subdivided down to transforms of unit length, the Fourier transform of the single data point is simply the identity operation, in other words, the transform of the point is the point itself.  The  computational efficiency of the FFT algorithm over the slow Fourier transform comes from a data indexing system that is based on recording the pattern of even and odd data points during recursive application of the Danielson-Lanczos Lemma as a binary number. Using a bit reversal procedure, the original data is rearranged into bit-reversed order that allows for a highly efficient method of recombining the one-point transforms into two-point transforms, the two-point transforms into four-point transforms and so on until the final transform is obtained.  The efficiency of the FFT algorithm reduces the computational effort required to estimate a Fourier transform of length N from an 0(N ) 2  process using the slow Fourier  transform algorithm to an 0(N log2 A ) process. This difference is best understood when 7  illustrated by the example of a function with TV = 10 discrete points. The computational 6  effort involved in using the slow algorithm is 50,000 times greater than that using the  Chapter 3  Numerical Algorithm Development  98  FFT, which in terms of computation time translates to one second versus 14 hours. For an earthquake acceleration record with 7V = 1500 discrete points, the computational effort in estimating the Fourier transform is decreased by a factor of 142 when using the FFT algorithm.  It is worth noting that other classes of FFT algorithms exist, such as those that subdivide the initial dataset of length N to some small power of two, rather than to the trivial transform of unit length. These are called base-4 FFT's or base-8 FFT's. There are also FFT algorithms for datasets of length /V that are not a power of two. They work by using relations analogous to the Danielson-Lanczos Lemma that subdivide the initial problem into successively smaller problems, not by factors of two, but by whatever small prime factors happen to divide N. The larger the largest prime factor of N is, the slower this method becomes until it reverts to the slow Fourier transform when N is prime and no subdivision is possible.  One example of this method is the Winograd Fourier  transform class of algorithms that, for some values of N, may be up to twice as fast as the simpler F F T algorithms based on dataset lengths of an integer power of two.  This  advantage in speed, however, is offset by considerably more complicated data indexing and the fact that the operation cannot be done in-place, where.the original dataset is replaced by its Fourier transform.  3.3.5.2  Power Spectrum Algorithm  Following computation of the Fourier transform of a function or dataset, the estimation of the associated power spectral density (PSD) depends on the type of normalization applied to one of several possible descriptions of the functions total power. For a function c(t) sampled at Appoints to produce values CQ, C\...CN-\,  total power may  be described by:  [17]  Chapter 3  Numerical Algorithm Development  IjkOfd-ltW 0  1  7  V  99  [18]  2  7=0  ])c(0| ^« AXK-|  2  2  0  [19]  7=0  These descriptions of total power are termed the sum squared amplitude, mean squared amplitude and time-integral squared amplitude, respectively.  If the estimation of the  power spectral density of c(t) is done using a periodogram estimator of the power spectrum, then the estimate is defined at NI2 + 1 frequencies as:  / (o)= p(/o)=T| |c r ,  J  T  0  N /?  1  |2 +  \„ C  |2  ( A ) = ^ r R l I "-*I  J  (*=1,2...M2-1)  [20]  ^(/ ) = W , 2 ) = ^ r | c r w  e  w/2  where Ck is the discrete Fourier transform of the function c(t) given by:  »Z\  2mjk/  C =Yu i  C e  (* = 0, 1..JV-1)  k  [21]  7=0  a n d i s defined only for the zero and positive frequencies as follows:  f^^  =  /VA  ^Tr  2  /V  (7t = 0,l.../V/2)  [22]  The normalization associated with the periodogram estimate of the power spectral density is such that the sum of the NI2 + 1 values of P is equal to the mean squared amplitude of the function c(t). This can be seen by comparing Eq. [18] with Parseval's theorem, which is given by:  Chapter 3  Numerical Algorithm Development  A M  1  100  A M  [23] l  k=0  y  n=0  As seen from Eqs. [20], the PSD is defined over the frequency range from zero to the Nyquist critical frequency, f  c  = 1/(2A) where A is the sampling time interval.  Integration of the PSD beyond the Nyquist frequency is unnecessary since, according to the sampling theorem, any power outside of the Nyquist interval will be aliased into the interval by the act of discrete sampling. In other words, the frequency components of a function that is not bandwidth limited to less than the Nyquist frequency are aliased, or falsely translated, into the interval (-f , f ) automatically by discrete sampling. This is a c  c  result of the fact that estimation of the Fourier spectrum from the sampled data, which is taken from a continuous analogue signal, results in an estimated spectrum that is periodic with period 1/A and symmetric about the zero frequency position.  Any frequencies  higher than f in the original signal distort the calculated spectrum by aliasing towards c  higher frequencies, which distorts the Fourier coefficients for frequencies below the Nyquist frequency. However, if a continuous function h(t) that is sampled at an interval A happens to be bandwidth limited to frequencies smaller in magnitude than f , then the c  function h(t) is completely determined by its samples, h„. In fact, h(t) is given explicitly by the formula:  [24]  The spectrum distortion associated with aliasing can only be dealt with prior to sampling, once the signal has been discretely sampled there is little that can be done to remove aliased power. Therefore, to overcome aliasing, provision must be made ahead of time to sample the signal at a rate that gives at least two points per cycle of the highest frequency present, which is either known ahead of time from the natural bandwidth limit of the signal or else enforced by analog filtering of the continuous signal with a lowpass linear filter prior to sampling. Following data collection it is a simple matter to determine  Chapter 3  Numerical Algorithm Development  101  if the signal has been properly sampled, if the Fourier transform does not approach zero as the frequency approaches f from below then it is likely that frequency components c  outside the Nyquist range have been translated into the critical range. In this event, the Fourier transform cannot be assumed to be zero beyond the Nyquist frequency.  From the perspective of earthquake spectrum estimation and dynamic structural response analysis, the issue of aliasing is not particularly worrisome since a lightly damped structure, which is applicable to most practical civil engineering structures, acts like a narrow-band filter resulting in a sharply peaked response at the undamped natural frequencies of the structure. This means that the major contribution of the power spectral density to the dynamic response is obtained in the vicinity of the natural frequencies of the structure and the value of the power spectral density outside that vicinity does not significantly affect the response (see Eq. 2.11 and Fig. 2.1).  Therefore, under the  assumption that the earthquake ground acceleration has been sampled to capture frequencies up to perhaps 25 Hz, the spectrum distortion associated with aliasing will have little if any effect on even the higher modes of response, which will typically have natural frequencies much less than 25 Hz.  Of more concern in calculating the periodogram estimator of a power spectrum are the issues of accuracy in the face of frequency leakage and minimizing the variance of the estimator.  The first issue of frequency leakage between the discrete frequency  components in the periodogram is a consequence  of the values of P(f ) k  in the  periodogram estimate not being exactly equal to the continuous P(f) at frequency k since fk is supposed to be representative of a whole frequency bin extending from halfway from the preceding discrete frequency to halfway to the next one. This problem is addressed by data windowing in which the original data is multiplied by a window function Wj that changes smoothly from zero to a maximum and then back to zero as j ranges from 0 to N. A smooth window function merely improves on the square window function inherent in any finite sample length N that has been obtained, in effect, by multiplying an infinite record by zero except during the total sampling time, T = NA, when it is multiplied by 1. Thus the square window is in essence an ideal lowpass filter with a stopband or cutoff  Chapter 3  Numerical Algorithm Development  102  frequency equal to \IT. By the convolution theorem, the Fourier transform of the product of the data and the window function is equal to the convolution of the data's Fourier transform with the window's Fourier transform. Without data windowing, the Fourier transform of the inherent square window, which has large side lobes beyond the cutoff frequency due to the abrupt changes in window amplitude between zero and 1, results in a convolved Fourier transform with substantial components beyond the cutoff frequency. This in turn leads to leakage in the calculated power spectrum. Using a smooth window function counteracts frequency leakage by reducing the side lobes of the convolved Fourier transform (see Fig. 3.3).  The window function that has been incorporated into  the power spectrum algorithm of the software application is known as the Bartlett window, which has a triangular form given by the following:  Figure 3.3: Square Window, Bartlett Window and Fourier Transforms  Chapter 3  Numerical Algorithm Development  103  The second issue in calculating the periodogram estimator of a power spectrum, which is minimizing the variance of the estimate, is achieved by a frequency averaging technique that segments the data into K segments of 2M data points that are separately FFT'd to produce K individual periodogram estimates. These K periodograms are then averaged to obtain a PSD estimate at M + 1 frequencies between zero and the Nyquist frequency (f ). c  This averaging process reduces the variance of the PSD estimate by a  factor that depends on how the data in each segment is overlapped with adjacent segments. The best method of overlapping depends on whether variance reduction is being done to obtain the smallest spectral variance from a fixed amount of computation or whether variance reduction is being done to obtain the smallest spectral variance per data point. In the first case it is best to segment the data without any overlapping, which requires 2KM data points, and results in variance reduction by a factor K. In the second case, which minimizes variance from a fixed number of available sampled data points, the segments should be overlapped by one half of their length resulting in variance reduction by a factor of approximately 9KI\ 1 for (K + \)M points.  Since the spectral analysis algorithm is intended to calculate the power spectrum of earthquake ground acceleration data that has already been recorded, the second method of variance reduction using overlapping segments of 2M data points was the one that was incorporated into the calculation of the periodogram estimator.  The value of M, which  determines the number of frequency values between zero and f , c  is set within the  algorithm based on the time-step of the input ground acceleration record. Table 3.1 lists the power spectrum frequency resolution for different input time-steps.  Table 3.1: Power Spectrum Frequency Resolution I lmc Step ibjc)  >0.04  512  > 0.02  1024  > 0.01  2048  <0.01  4096  Chapter 3  Numerical Algorithm Development  104  The number of data segments, K, is determined from the frequency resolution, M, and the number of data points in the acceleration record, which is doubled in length prior to determining K. The acceleration record is then padded with zeroes to provide a total of (K+ \)Mpoints for spectrum estimation.  3.3.6  Acceleration Record Filtering An  algorithm to frequency-filter the input earthquake record, as well as any  generated earthquake records, was developed to allow for the removal of a chosen number of high frequency and low frequency components in the record(s). This filtering option was included in the software application to ensure that the ground acceleration record being used for structural dynamic response analysis is truly representative of a real earthquake.  The ground displacement in a real earthquake, for example, typically  oscillates around a zero mean and comes to rest with a small or zero final displacement. For  certain records that contain low frequency components, however, the calculated  ground displacement may oscillate around a dominant low frequency or exhibit significant drift over the duration of the earthquake, resulting in a final displacement that is seriously in error.  The  filter that is used for removing unwanted frequencies from the acceleration  record is a cosine-type window that transitions between 0 and 1 as follows:  y =-  y~  f^flaw.i 9  +  V  Jlow_\)' r  J low-.  window(f ) =  flow 1  r  2  J low  flow 2 — f  2  y = r> ^ v+ ( fJ-high f _ 1 /j ) r- - — r ^ Jhighjl ~ Jhighl h  1 + sin(y)  i  g  h  fl  <  low  2  1  n y =  f  <  f^fl high_2  OV  fhigh 1  <  — fhu high_\  f  <  fh  high _ 2  [26]  Chapter 3  where  Numerical Algorithm Development  fiowj, fiowj., fhighj  and fhighj. are the window transition frequencies.  105  Using this  window, the structure of the filtering algorithm is based on a visual inspection of the calculated ground displacement over the time-history of the acceleration record. The unfiltered ground displacement time-history is displayed onscreen in the software user interface and the user then adjusts the low frequency and high frequency window transitions until the resulting ground displacement time-history, which overlays the unfiltered time-history, is deemed acceptable.  To aid in determining the effect of  removing high and low frequency acceleration components; the minimum, maximum, average and final ground velocities and displacements are summarized in the visual display of the displacement time-history. This allows the user to, for example, adjust the cosine window transitions to filter the acceleration record such that the final ground velocity is close to the physically required value of zero. The calculation of the ground velocity and ground displacement time-histories from which the minimum, maximum, average and final values are determined is done in the frequency domain to reduce the accumulation of error inherent in the double integration procedure required in the time domain.  3.3.7  Random Number Generation  A reliable source of random numbers is essential for any sort of stochastic modeling or Monte Carlo analysis to ensure that a random quantity is as close to truly random as possible within the confines of a deterministic computer.  An informal  definition of randomness in the context of computer-generated sequences is that the deterministic algorithm that produces a random sequence should be different from, and in all measurable respects statistically uncorrelated with, the computer program that uses its output.  The computational issues related to generating reliable random sequences are  summarized in the following along with a description of the algorithms that were incorporated into the software application for use in randomizing structural properties and generating artificial earthquake time-histories.  Chapter 3  Numerical Algorithm Development  106  The simplest method of generating random numbers for use in a computer program is to use the routine that has likely been provided in the language the program is written in. While this is a convenient means of generating random numbers it can lead to serious violations of the assumption of uncorrelated random sequences when a large number of random numbers is required. System-supplied random number generators are typically linear congruential generators, which generate a sequence of integers I\, h, h, each between 0 and m - 1, by the recurrence relation:  = alj + c (mod m)  [27]  where m is the modulus and a and c are positive integers called the multiplier and the increment, respectively.  The recurrence given by Eq. [27] will eventually repeat itself  with a period that has a maximum length m for properly chosen values of m, a and c. Improperly chosen values of the modulus, multiplier and increment will result in a considerably shorter period length.  This then leads to two shortcomings of using a  system-supplied random number generator, firstly, some computer manufacturers have made exceedingly poor choices for m, a and c (Press et al. 1999), thereby seriously shortening the repeat period.  Second, the value of m, which is the largest possible  random number, is often not very large on many computers. The American National Standards Institute (ANSI) standard for the C language, which was used to develop the software application, requires only that the system-supplied routine rand() return an integer that is at least 32,767'.  This number is far too small for the random number  intensive process of generating earthquake time-histories for Monte Carlo analysis. For example, generating a single Poisson process type earthquake similar to the 116 second Chile Llollelo event, sampled at 200 Hz, requires approximately 70,000 random numbers, which means that the entire random number table would be used more than twice for each artificial record that was generated. Clearly, the records are then highly correlated and not close to being truly random as required in a Monte Carlo analysis that may use thousands of earthquake records.  Chapter 3  Numerical Algorithm Development  107  To eliminate the potential problems associated with system-supplied random number generators and machine specific choices for the recurrence equation parameters, a number of portable random number generators that can be implemented in various programming languages on various machines have been developed. The generator that was implemented in the software application is based on a multiplicative congruential algorithm given by:  I  J+l  = aij  (mod m)  [28]  Using this simple recurrence relation, Park and Miller (1988) developed a Minimal Standard generator that is based on the choices:  a = 7 =16807 5  m=2  31  -1 = 2147483647  This Minimal Standard generator, first proposed in 1969, has a period of 2  [29]  31  - 2 ~ 2.1 x  10 and has passed all theoretical statistical tests since its inception. The portability of 9  this generator to essentially any programming language on essentially any machine is due to its ability to work with numbers generated by Eqs. [28, 29] that exceed the maximum value for a 32-bit integer, which is the limit for a high-level language. Using a 64-bit product register in Assembly language would allow the equations to be used directly but the implementation would not be portable between machines.  The implementation of the Park and Miller Minimal Standard random number generator in the software application returns a uniform random deviate between 0.0 and 1.0, exclusive of the endpoint values.  A small improvement on the basic Minimal  Standard algorithm, called the Bays-Durham shuffle, is included in the generator to remove subtle low-order serial correlations present in the basic generator. This shuffling algorithm shuffles the output such that the y'fh value in the sequence, Ij, is output not on the y'fh call, but rather on a randomized later call, which occurs at j + 32 on average. This generator has been shown to pass all statistical tests up to the point where the number of calls starts to become on the order of 5% ofthe period m. Therefore, for applications that  Chapter 3  Numerical Algorithm Development  108  require less than approximately 100,000,000 random numbers in a single calculation this generator has no known flaw. Clearly, this is well within the maximum demand imposed by the sort of Monte Carlo analysis mentioned previously. As an aside, there are other well-accepted random number generators with much longer periods that could have been incorporated into the software application, however, their relative execution times are significantly longer.  These generators, which have periods of ~ 2.3 x 10  18  and beyond,  provide random number sequences that are, for all practical purposes, impossible to repeat on existing computers.  These types of random number generators are used in  cryptographic systems.  Since the randomization of structural properties requires random deviates that follow distributions other than the uniform distribution, provision was made to generate random numbers that also follow the normal, lognormal, Gumbel, Frechet and Weibull distributions. Random sequences with each of these distributions are generated from a sequence of random numbers uniformly distributed between 0.0 and 1.0 using the wellknown inverse transformation method. This method uses the inverse of the cumulative distribution function (CDF) of a variable to map the uniform random numbers to values that follow the required distribution.  Chapter 4  Software Framework and User Interface  109  CHAPTER 4  S O F T W A R E F R A M E W O R K AND U S E R - I N T E R F A C E  4.1  INTRODUCTION  Following development of the key solution algorithms and numerical components that form the foundation of PSResponse, which was outlined in Chapter 3, the overall architecture of the software that links all the sub-components together with the user interface was constructed.  Chapter 4 summarizes that architecture beginning with a  flow-chart description of the core computational framework in Section 4.2.  In Section  4.3, the structure of the user-interface that overlays the computational framework is briefly described followed by a more detailed look at the user-interface in Section 4.4. In this more detailed look, the key software features available to the user are illustrated using screen captures taken from various types of dynamic analysis.  4.2  COMPUTATIONAL FRAMEWORK  The calculating engine of PSResponse consists of approximately 79 algorithms linked together in an object-oriented framework that can be loosely divided into six groupings for illustrative purposes.  These six groupings are shown in Figure 4.1 to  Figure 4.6. With the exception of the User Input box in Figure 4.1 and the Store Results box in Figure 4.5, each box in each figure corresponds to a separate algorithm that is linked or called by other algorithms in the manner shown. For purposes of simplicity and clarity, some algorithms are shown in more than one figure to avoid too many cross-links between figures.  It should be clear that the computational framework illustrated in Figure 4.1 to Figure 4.6  shows all the algorithms and associated  links to provide the entire  Software Framework and User Interface  Chapter 4  110  functionality of PSResponse. The actual logical path that a particular PSResponse analysis would follow depends, of course, on the options selected and type of dynamic analysis performed.  User Input  Set Structural Properties  Set Method  Fill Stiffness Matrix  Calculate Natural Frequenices  w  Condense Stiffness Matrix  Calculate Elastic Properties  LU Decomp  * \ LU Backsub  Spectrum Type  Identify Hysteresis Parameters  I Run Filtered Spectrum Records x Fig. 4 2 -  Jacobi Get Random Properties Parameters  Fill Hyst Data Array  Calculate Modal Components Run Real Spectrum Records Fig 4.3  t Solvers (Syst Ident) . Fig. 4 5 -  Calculate Spectra  Run Single Record  Fill Earthquake Record  Solvers Fig 4 5 Filter Ground Motion  Spectrum  Figure 4.1: Computational Framework - Initialization  Real F F T  ->|  FFT  |  Chapter 4  Software Framework and User Interface  Get Record Parameters  111  r—P\ Set Amplitude Modulation Set Filtered Spectrum  Run Filtered Random Frequency Records Run Filtered Spectruin Records  Run Filtered Random Phase Records Run Recoids Fig 4 4  Run Filtered Poisson Records Run Filtered IFFT Records T T Generate Uniform Random Numbers  Adjust Record  i  T Calculate Ground Motion  w  Real FFT  -•  Figure 4.2: Computational Framework - Multiple Record Analysis (Filtered Spectrum)  Get Record Parameters  Fill Real Data Array Run Real Spectrum Records  FFT  Calculate Ground Motion  i i  Real FFT  Run Real IFFT Records  Spectrum  Run Real Random Phase Records  Set Amplitude Modulation  Run Real Random Frequency Records Run Real Poisson Records  \  Run Records Fig 4 4  Figure 4.3: Computational Framework - Multiple Record Analysis (Real Spectrum)  FFT  Chapter 4  Software Framework and User Interface  Run Records  112  Select Modal Responses  Create Real Random Phase Record  Select MDOF Responses  Create Filtered Random Phase Record  Create Filtered IFFT Record  Select S D O F Responses  Create Real Random Frequency Record  Generate Gaussian Random Numbers  Generate Random Structural J Properties \ Fig 4 6  J Create Real IFFT ! Record  Create Filtered Random Frequency Record Adjust Record  Create Real Poisson Record  Solvers Fig. 4 5  Spectrum  FFT  Create Filtered Poisson Record  Filter Multiple Ground Motions  Generate Uniform Random Numbers  Real FFT  Figure 4.4: Computational Framework - Run Multiple Records  4.3  USER-INTERFACE STRUCTURE  The user-interface that overlays the computational framework is based on a wizard manager architecture that guides the user through a series of Windows-driven input and output dialog boxes. The wizard manager algorithm, which acts as the link  Chapter 4  Software Framework and User Interface  113  between the dialog boxes and the computational framework, determines the dialog box sequence, passes information between dialog boxes, passes input data to computational algorithms and stores both input data and output arrays.  The basic structure of the  algorithm, in terms of the general order in which user input is collected, is outlined by the simplified flow-chart in Figure 4.7. Although not shown, there are a total of 36 separate paths through the flow-chart in which the flow-chart boxes typically represent a single dialog box in the user-interface.  In some instances, however, several dialog boxes are  associated with a single flow-chart box.  Solver Systjdent  Calculate Ground Motion  Real FFT  FFT  Newmark_Syst_ldent  •W Solver Elastic  Solvers  Newmark Elastic  Calculate Modal Responses  Check Stability  Solver SDOF  H  -H  Solver_MDOF  fe  w  -W Adams Variable  RK4  Calculate CQC Correlations  Store Results  Newmark SDOF  Calculate SDOF Responses  Newmark_MDOF  w  Adams Variable MDOF  j _T_  LU Decomp  Calculate MDOF Responses •  LU Backsub  Figure 4 . 5 : Computational Framework - Solvers  Fill Matrix  RK4  MDOF  Chapter 4  Software Framework and User Interface  114  Generate Gaussian Random Numbers Generate Random Value Generate Uniform Random Numbers Multiply Distribution Parameters Generate Random Structural Properties  LU Decomp Fill Stiffness Matrix  Condense Stiffness Matrix LU Backsub . Jacobi  Calculate Natural Frequencies Eigensort Calculate Modal Components Calculate C Q C Correlations  Figure 4.6: Computational Framework - Generate Random Properties  4.4  SOFTWARE  FEATURES  The user-interface, outlined schematically in Figure 4.7, is best illustrated using screen captures of the dialog boxes and onscreen plots available to the user as well as sample printouts of dynamic analysis results.  These screen captures along with  explanatory notes, which are shown in the following sections without further comment, will serve to present the key features of the dialog boxes and the analysis options available.  Chapter 4  Software Framework and User Interface  115  Structural Input D a m p i n g Input SDOF I : G e n e r a l Input  Multiple  Shear  Hysteretic  Parameters  Analysis  Structure  Input  Modal Damping  Non-Classical Damping Frame Structure  Choose Distribution Choose Distributions  B a s e M o t i o n Input Select Responses Oscillation Motion Elastic  Output R e s u l t s  Responses Single Earthquake  Plot SDOF N  Inelastic  .y' R e s p o n s e s Real Spectrum  Print MDOF ~j  Inelastic  *;  Responses  White-Noise  Legend  Spectrum  — •  R e q u i r e d Input Choose  •  Optional Path  L_J"  Figure 4.7: User- Interface Framework  Storeys  Chapter 4  4.4.1  Software Framework and User Interface  116  General Input Parameters Dialog Box  •Analysis Types <" Oscillation Response <• Single Earthquake Response  Numerical Solution Methods  -  C Newmark's Average Acceleration Method <• Newmark's Linear Acceleration Method  C Multiple Earthquake Response Structure Type*— — <• Single-Degree-of-Freedom C Multiple-Degree-of-Freedom (Shear Structure) r  Numerical dynamic analysts is subject to prescribed error. DefauH tolerances may be reduced for greater accuracy or increased for shorter computation time. TOLERANCE:  1.00  (0.01-100)  Muttiple-Degree-ol-Ffeedom IFrame Structure!  Dynamic Analysis Types (* Elastic Response Inelastic Response  Unit Systems  f? SI C Imperial  Cancel  Next>  Figure 4.8: General Input Parameters Dialog Box  Option Oscillation Response Single Earthquake Response Multiple Earthquake Response S ingle-Degree-of-Freedom Multiple-Degree-of-Freedom (Shear) Multiple-Degree-of-Freedom (Frame) Elastic Response Inelastic Response Newmark Average Acceleration Method Newmark Linear Acceleration Method Tolerance  Notes Input motion is specified as a superposition of sine waves for structural and connection test protocols. Detailed response analysis of a single earthquake including response spectra option and onscreen response time4iistory plots. Summary level response results for multiple earthquakes. 1-D structure withfixedbase. 2-D structure with rigid floor diaphragms. 2-D structure with flexible floor beams. Elastic response is assumed regardless of displacement. Inelastic response is modeled using input hysteretic data. Unconditionally stable numerical solution method. This method is automatically selected for inelastic MDOF analysis. Default numerical solution method. Tolerance relative to encoded default values. Thisfieldis used to multiply the default values.  Chapter 4  4.4.2  Software Framework and User Interface  117  Multiple Earthquake Analysis Parameters Dialog Box  •Earthquake Generation Methods (• Inverse FFT <"* Fftered Poisson Process Spectral Representation:  1  .  luitil ia°iiiJii,i 1 1 I".  •Ii  II  i  M  <" Random Phases C Random Phases and Frequencies  r Earthquake Spectrum Sources \  f  Fiered Gaussian White-Noise  i  I  <• Earthquake Recotd  r Structural Model Properties <* Deterministic C  Random  Cancel  Next >  Figure 4.9: Multiple Earthquake Analysis Parameters Dialog Box  Option Inverse FFT  Filtered Poisson Process  Random Phases  Random Phases and Frequencies  Filtered Gaussian White-Noise Earthquake Record  Notes Earthquakes are generated using an inverse Fast Fourier Transform process where the components of the input earthquake frequency spectrum are randomized with uniform random numbers. Generated earthquakes are modeled as the sum of a series of independent impulses arriving at Poisson distributed times. The independent pulses have random frequencies with a probability density based on the spectral density of the input earthquake record. Phase angles of the independent pulses are uniformly randomly distributedfrom0 — » 2n. Earthquakes are generated using the Spectral Representation method with harmonic frequency phases uniformly randomly distributed from 0 —* 2n. All generated earthquakes have an autocorrelation function that matches the input earthquake record. Same as above except the harmonicfrequenciesare also random variables with a probability density based on the spectral density of the input earthquake record. This tends to concentrate random frequencies around the peak of the input power spectrum. The input spectral density for generating earthquakes is constructed by shaping a constant white-noise spectrum with a low-pass Kanai-Tajimi filter and high-pass Clough-Penzien filter. The input spectral density for generating earthquakes is taken from  Chapter 4  Software Framework and User Interface  the calculated power spectrum of an input earthquake record. A l l structural properties and damping values are constant. Structural properties and damping values are random variables following a selected probability distribution. A random structure is generated for each earthquake response analysis.  Deterministic Random  4.4.3  118  Single—Degree—of-Freedom Properties Dialog Box  Natural Frequency™-• (" Specify Directly ** Specify Mass and Stiffness  Mass  J0.00  Stiffness  JO tSO  (kN/m)  Calculate Frequency  Frequency  jO.OO  (Hz)  Structural Damping Value  000  IX critical) Cancel  Next >  Figure 4.10: Single-Degree-of-Freedom Properties Dialog Box  Option Specify Directly Specify Mass and Stiffness  Notes The natural frequency of the structure is input directly. This option will disable the Mass and Stiffness fields and Calculate Frequency button. The natural frequency of the structure is determined from input values for mass and stiffness. The Calculate Frequency button updates the natural frequency.  Chapter 4  4.4.4  Software Framework and User Interface  119  M u l t i p l e - D e g r e e - o f - F r e e d o m Shear Structure Properties D i a l o g B o x  • Shear Structure Geometry-  (2-20)  Number of Storeys  •BaseBne Values  -  Storey Mass  0.00  (kg)  Storey Lateral Stiffness  0.00  (kN/m)  Storey Height  0.00  N  t- Baseline Multiples Baseline values ate multiplied by the following specified factors to give the mass, storey stiffness and storey height distribution of the shear structure. If mote than one shear wal is present on a given storey, the multiplication factor should account for the total lateral stiffness. Yield multiples are the relative yield strength ratios. Baseline yield strength is determined later from input data. Storey  Mass  Stiffness  Height  Yield  1  1.00 1.00 1.00 1.00  1.00 1.00 1.00 1.00  1.00 1.00 1.00 1.00  1.00 1.00 1.00 1.00  2 3 4  Cancel  Next >  Figure 4.11: Multiple-Degree-of-Freedom Shear Structure Properties Dialog Box  Field Baseline Values  Baseline Multiples  Notes Baseline values for storey mass, storey lateral stiffness and storey height are used to allow a regular structure to be quickly specified as a pattern of baseline multiples. Baseline values are multiplied by the specified baseline multiples that default to 1.00. Yield multiples are the relative yield strengths of each storey in which the baseline yield strength is determined later from input hysteretic data. The Yield column is disabled for elastic analysis.  Software Framework and User Interface  Chapter 4  4.4.5  120  Multiple-Degree-of-Freedom Frame Structure Properties Dialog Boxes  'tilt®? Qa^l^i^fomS&tiS) Frame Geometry fit  n 2oj  Numbe< ol Storeys Y>  Hatha of Beys  (1-5) EICT  W  Baseline Values  ~™ ~  IM 2  The following baseline values wi be rnuOipted by usei-specified (actorstogive the mass, stiffness, storey height and bay length distribution of the frame structure. Storey Mass  000  Beam Stiffness  000  (El - kN'mZ)  Column Stiffness  (0.00  (El • kfTm2)  Stcwey Height  fooo™  M  000  M  Bay Length  i  M  1  LkH  KB  Cancel  Next>  -  Storey Mass  Beam Stiffness  Column SSfness  Storey Height  Bay Length  Bays  (kg)  (El - kN'm2)  (El - kN~m2)  (m|  (m)  2  5000.  100.  100.  3.  5.  i ; Storeys  [ H i  (kg)  Baseline Frame Properties  Elcs  2  • Baseline Multiples The baseline frame properties will be multiplied by the following factorstogive the mass, stiffness, storey height and bay length distribution of the frame structure. Yield multiples are the relative yield strength ratios. Baseline yield strength is d? If more than one frame is present, the mutWcation factor for each beam and column stiffness should account for the total stjfness of each storey and bay.  1.00  ji.00  Mass2|l.00 Haght2|l.00 Yield 2|1.00  |l.00  |l.00  jl.00 h.oo  |1.00  Mass1|l.00 Height 1  (1.00  Yield 1 h 00  Length 1  (Too™  Length 2  I'.oo  fuJ0~ Cancel  Ne«t>  Figure 4.12: Multiple-Degree-of-Freedom Frame Structure Properties Dialog Boxes  Chapter 4  Software Framework and User Interface  Field Baseline Values  Baseline Multiples  4.4.6  121  Notes Baseline values for storey mass, beam stiffness, column stiffness, storey height and bay length are used to allow a regular structure to be quickly specified as a pattern of baseline multiples. Baseline values are multiplied by the specified baseline multiples that default to 1.00. To simplify the specification of baseline multiples, the inputfieldsare arranged within a schematic of the frame structure. Yield multiples are the relative yield strengths of each storey in which the baseline yield strength is determined later from input hysteretic data. The Yieldfieldsare disabled for elastic analysis.  Probability Distribution Example Dialog Box  Probability Distributions *•  Normal  *~ Lognormal Uniform f* Gumbel (Extreme Type I) C  Ftechet (Extreme Type II)  <" Weibull (Extreme Type III) <~ None -Distribution ParametersEnter parameters relative to deterministic value of N A T U R A L F R E Q U E N C Y . Standard deviation  0.00  Parameter 2 Parameter 3 Cancel  Next >  Figure 4.13: Probability Distribution Example Dialog Box  Option  Notes A Probability Distribution dialog box will appear for each structural parameter that applies to the current analysis. For example, it will appear Probability Distributions twice for a SDOF system that is specified by natural frequency, once for the natural frequency and once for the specified damping. The parameter fields specify the shape characteristics of the chosen probability distribution. Values should be entered with respect to the baseline structural properties entered previously, in the chosen unit system. Distribution Parameters In the case of modal damping, parameter values are entered with respect to the lowest selected mode of response. The mean value of the distribution is taken as the baseline value.  Software Framework and User Interface  Chapter 4  4.4.7  122  Hysteresis P a r a m e t e r Identification Dialog Boxes  110/"^^  •4*1  Inelastic Response Data Types C Quasi-Static Force vs. Displacement C Acceleration Response * Use Default Hysteresis Parameters Post Yield Stiffness p§15  [X)  Yield Displacement 13.00  (in)  Cancel  Next>  Force vs Displacement T est Types <• S tand-Alone T ension/Compression Connection T est C' Full-Scale Frame/Shear Structure Test Hysteresis data must be stored in a text file containing the displacement data in one column and the corresponding force data in a second column. The columns must be space or tab seperated. Filename Data Sampling Frequency Representative Storey Number \ Data Units System: kN vs. m  C kips vs. in  Connection Application Brace Fuse Connection Brace End Connection Brace Angle  (o 00 •  (deg. from horia.)  Cancel  Next >  Figure 4.14: Hysteresis Parameter Identification Dialog Box (Quasi-Static Data)  Option/Field Quasi-Static Force vs. Displacement Acceleration Response  Notes Hysteresis parameter identification is based on pseudo-static cyclic displacement test data of an SDOF structural system or connection. Hysteresis parameter identification is based on acceleration response data taken from a shake-table test of an SDOF structure.  Chapter 4  Software Framework and User Interface  Use Default Hysteresis Parameters  Stand-Alone Tension/Compression Test  Full-Scale Frame/Shear Structure Test  Filename  Data Sampling Frequency  Representative Storey Number  Data Units System Brace Fuse Connection  Brace End Connection Brace Angle  123  Post-yield stiffness is specified as a percentage of initial stiffness. Yield displacement may be specified as any value greater than zero. A yield displacement equivalent to the peak elastic displacement response will result in a slightly different dynamic response, as compared to the elastic response, due to the continuous nonlinear nature of the smooth hysteresis model. Hysteresis loops are assumed to remain undamaged with no pinching and no progressive strength or stiffness degradation. For MDOF structures, the specified yield displacement applies to thefirststorey. The dialog box with this option will appear if the QuasiStatic... option is selectedfromthe previous dialog box. It is assumed that the specified cyclic displacement data is taken from a connection test in which the brace-type connection is subjected to cyclic tension and compression forces. Choosing this option will enable the Connection Application options at the bottom of the dialog box. It is assumed that the specified cyclic displacement test data corresponds to the lateral displacement of the top of a full-scale frame or shear wall. Choosing this option will disable the Connection Application options at the bottom of the dialog box. Test data must be stored as a textfilewith displacement data in one column and the corresponding force data in a second column. Columns must be space or tab seperated. Maximum column width is 12 characters and data may be specified in exponential format. Number of pseudo-static force vs. displacement data points recorded per second. This field is enabled when an MDOF structure has been specified for dynamic analyses. The representative storey number indicates which frame or shear wall stiffness and yield strength multiple to associate with the identified hysteresis parameters. The hysteresis parameters for the other storeys are adjusted according to the storey stiffnesses and yield strength multiples specified for the other storeys. Units system of the input hysteretic.data. Brace with a single yielding element or connection, typically at mid-brace. All inelastic deformation is concentrated at this one point. Lateral displacement of the top of the frame and the corresponding lateral force are determined by adjusting the input data according to the input brace angle. Same as above except the brace has yielding end connections. Therefore, lateral displacement is calculated as twice the adjusted input displacement to account for two yielding connections. Angle of brace in degreesfromhorizontal.  Chapter 4  Software Framework and User Interface  124  I B see Acrelerahon Rer crd I he base acceleration record must be a text hie containing the time-history of base acceleration in a single data column. Record Name: —  j  :  Time-Step  "  y|  V  \/\/  \/  S  Browse... J  (sec)  Acceleration Units: ~~ (taction o l g  <" mm/s/s  <~ in/s/s  <•  f"° cm/s/s  r  (  m/s/s  • Response Acceleration Record-  it/s/s -  -  -  The response acceleration record must be a text file with the time-history data in a single column. Data must be taken from an SDOF structure with the same properties specified previous^ or representative of one storey in an MDOF structure. RecordName: Browse... )  \N\\\N\\\N\  (Hz)  Data Sampling Frequency j Test Damping |  [X critical)  Test Mass (~~  (kg)  Representative Storey Number j Acceleration Units: r  fraction of g  & m/s/s  <~ mm/s/s  C  in/s/s  (~ cm/s/s  c  ft/s/s  <Bxi.  Cancel  Next>  1  Figure 4.15: Hysteresis Parameter Identification Dialog Box (Acceleration Data)  Option/Field Base Acceleration Record  Time-Step Acceleration Units  Response Acceleration Record  Data Sampling Frequency  Notes The dialog box with this option will appear if the Acceleration Response option is selectedfromthe previous dialog box. The base acceleration record must be stored as a textfilecontaining the timehistory data in a single column. Maximum column width is 1 2 characters and data may be specified in exponential format. Time span between base acceleration time-history data points. The inverse of the time-step must be an integer multiple of the response acceleration data sampling frequency. Unit system of input base acceleration record. The response acceleration record must be stored as a text file containing the time-history data in a single column. Data must be takenfroman SDOF structure with the properties specified earlier or representative of one storey in an MDOF structure. Maximum column width is 12 characters and data may be specified in exponential format. Number of response acceleration time-history data points recorded per second. The data samplingfrequencymust be an integer multiple  Chapter 4  Software Framework and User Interface  Test Damping  Test Mass  Representative Storey Number  Acceleration Units  4.4.8  125  of the inverse of the base acceleration time-step. Estimated damping of the SDOF test structure. This damping level does not have to match the modal or storey damping level(s) specified for the structural system being modeled with the identified hysteresis parameters. Thisfieldis enabled when an MDOF structure has been specified for dynamic analyses. The test mass is defined as the gravity load placed on the SDOF test structure for shake-table testing. It is used to determine the natural frequency of the test structure for response analysis in the hysteresis parameter identification process. Thisfieldis enabled when an MDOF structure has been specified for dynamic analyses. The representative storey number indicates which frame or shear wall stiffness and yield strength multiple to associate with the identified hysteresis parameters. The hysteresis parameters for the other storeys are adjusted according to the storey stiffnesses and yield strength multiples specified for the other storeys. Unit system of input response acceleration record.  Modal Damping Parameters Dialog Box  CalculatedNatuia) Frequencies The following are the deterministic natural frequencies of the structure (Hz): Mode  Freq  1  0.50  2  1.46  3  2.30 Modal Properties —xs&sxr-  - Modal Damping---The distribution of structural damping is assumed to allow for modal analysis (classical damping). Any combination of modes from 1 to 5 may be selected. Modal Damping Ratios:  P  Model  Damping JO00  i YZ//777777777777777777777777/?7'n  R? Mode 2  Damping pOO0™~~  {%)  W  Damping foOO  (*)  Mode 3  Cancel  Figure 4.16: Modal Damping Parameters Dialog Box  J  Next >  Software Framework and User Interface  Chapter 4  Field Calculated Natural Frequencies  Modal Properties Button  Modal Damping Ratios  Notes Calculated naturalfrequenciesfor translational modes only. Thesefrequencies,which are given in Hz, are determined from the baseline structural properties and baseline multiples that were specified previously. Generates a pop-up dialog box that displays mode shape data, modal expansion of floor masses, effective modal masses and effective modal heights. See Figure 4.17. Modes to be included in the elastic response analysis are selected from a scroll box along with a corresponding damping level for each selected mode. If structural properties have been specified as random, a Probability Distribution dialog box will follow this dialog box. The parameters for the selected damping probability distribution should be entered relative to the lowest selected mode.  Mode shapes normalized lo unity: (Modes in columns ordered horn top storeytobottom storey) 1.0000 -0.9190 0.7635 -0.5462 0.2846 0.9190 -0.2846 -0.5462 1.0000 -0.7635 0.7635 0.5462 -0.9190 -0.2846 1.0000 0,5462 1 0000 0.2846 -0.7635 -0.9190 0.2846 0.7635 1.0000 0.9190 0.5462 Mode shapes normalized to give unit generalized modal masses: (Modes in columns ordered from top storey to bottom stoiey) 0.0028 0.0026 -0.0021 -0.0015 0,0008 0.0026 0.0008 0.0015 0.0028 -0.0021 0.0021 -0.0015 0.0026 -0.0008 0.0028 0.0015 -0.0028 -0.0008 -0.0021 -0.0026 0.0008 -0.0021 -0.0028 0.0026 0.0015 Modal expansion of floor masses (kips/g): (Modes in columns ordered from top stoiey to bottom stoiey) 125.17 115.03 95.57 68.37 35.63  -36.21 -11.22 21.52 33.41 30.09  15.86 -11.34 -19.09 5.91 20.77  126  -6.32 1.50 11.57 -4.03 -3.29 5.28 -8.83 -4.86 1 0.63 2.89  Effective modal masses (kips/g): 439.76 43.5912.11 3.750.78 Effective modal heights (ft): 42.16-14.44 9.16-7.13 6.25  OK  Figure 4.17: Modal Properties Pop-Up Dialog Box  Software Framework and User Interface  Chapter 4  4.4.9  127  M D O F Damping Parameters Dialog Box  Calculated Natural Frequencies The following are the deterministic natural frequencies of the stiucture (H2): Mode  Freq  1  0.50  2  1.46  Damping Parameters Rayleigh Damping Mode  X Critical  Mode  % Critical  0.00  F  0.00  Update Ratios Mode  % Critical  1  0.00  2  0.00  H \w/,y//////.  C Damping Distribution  Cancel  Next >  Figure 4.18: MDOF Damping Parameters Dialog Box  Field/Option Calculated Natural Frequencies  Rayleigh Damping  Damping Distribution  Notes Calculated natural frequencies for translational modes only. These frequencies are determined from the baseline structural properties and baseline multiples that were specified previously. Inelastic dynamic analysis precludes modal analysis, which results in the need to specify the damping matrix. Rayleigh damping is used to construct a classical damping matrixfrommodal damping ratios. Two modal damping ratios must be specified, which then determines the damping ratios for the remaining modes. The Update Ratios button calculates the damping ratios for each non-specified mode. If structural properties have been specified as random, a Probability Distribution dialog box will follow this dialog box. The parameters for the selected damping probability distribution should be entered relative to the lowest specified mode. A non-classical damping matrix is constructed from the specified damping for each storey. If structural properties have been specified as random, the parameters for the selected damping probability distribution should be entered relative to the specified baseline value.  Software Framework and User Interface  Chapter 4  128  4.4.10 Oscillation Motion Parameters Dialog Box  -Oscillation Motion Plot-  -Peak Amplitude Types<"* Acceleration (g) & Displacement (m)  4H44  - Oscillation Parameters—-  —  The input oscillation motion is a periodic function formed by the superposition of individual sine waves Number of Sine Waves  y  Input M otion Duration  [ll  22  Isec)  66 Time  4 4  88  11.0  Update Rot  Sine Wave Description Each sine wave is described by a peak amplitude (m), frequency (He), rise time to peak amplitude and subsidence time to zero amplitude. Wave  Ampl  Freq  Rise  Sub  1  1  1  3  3 < 8acK  Cancel  Next>  Figure 4.19: Oscillation Motion Parameters Dialog Box  Field/Option Acceleration Displacement Number of Sine Waves Input Motion Duration Ampl Freq Rise Sub Update Plot  Notes Base motion is specified as a time-history of acceleration given in terms of g. Base motion is specified as a time-history of displacement. The input oscillation motion is formed by the superposition of up to nine sine waves. Duration must be a multiple of 0.01 sec. Maximum duration is 99 seconds. Amplitude of sine wave. This is specified in the chosen unit system (m, in. or g) Frequency of sine wave in Hz. Time when peak amplitude is reached. Default is zero, indicating no rise time. Duration of interval during which amplitude decreases from maximum to zero at the end of the motion. Default is zero, indicating no subsidence time. Updates and displays the calculated superposition of sine waves.  Chapter 4  Software Framework and User Interface  129  4.4.11 Single Ground Motion Parameters Dialog Box  • Ground Motion Record The input ground motion may be selected from a list of available records or specified by a file containing ground motion acceleration data. Earthquake Record: Browse...  (sec)  Time-Step JO.OO Acceleration Units: f  fraction of g  <~ mm/s/s  C in/s/s  cm/s/s  <r ft/s/s  ft m/s/s  Motion Properties •Record Options P" Fiter Record (Cosine Window) Low Frequency Window Transition (0.00  to [0,01  (Hz)  High Frequency Window Transition 1100.00  to II 00.01  (Hz)  W Calculate Response Spectra Damping Levels J3  Level  X Critical  1  0.00  2  0.00  3  0.00  Cancel  Next>  Figure 4.20: Single Ground Motion Parameters Dialog Box  Field/Option Earthquake Record" Time-Step Acceleration Units Motion Properties Button Filter Record  Notes A single ground motion record is used for response analysis and subsequent onscreen plotting of results. The earthquake record must be stored as a text file containing the time-history data in a single column. Maximum column width is 12 characters and data may be specified in exponential format. Time span between earthquake time-history data points. Unit system of earthquake acceleration record. Generates a pop-up dialog box that displays summary statistics and a time-history plot of the unfiltered ground motion as well as the filtered ground motion if the Filter Record option is chosen. See Figure 4.21. This dialog box may be opened and closed repeatedly to display the results of adjusting filter window transitions. This option enables a cosine-type window filter for removal of high  Chapter 4  Software Framework and User Interface  and low frequency content in the earthquake record to ensure that the ground acceleration record being used for structural dynamic response analysis is truly representative of a real earthquake. Specifies thefrequencyrange in which the cosine-window varies from 0 to 1. Frequencies below the lower limit are removed and frequencies above the upper limit are unaltered. Specifies thefrequencyrange in which the cosine-window varies from 1 to 0. Frequencies above the upper limit are removed and frequencies below the lower limit are unaltered. This option enables the calculation of the response spectra for the specified earthquake record at up to nine damping levels. The spectra are formedfromthe maximum elastic response at 351 natural periods between 0.01 sec and 50 sec. The periods are evenly distributed in three full decades and a truncated fourth decade defined as follows: 0.01-0.1 sec, 0.11-1.0 sec, 1.1-10 sec, 10-50 sec. The spectra are displayed onscreen following response analysis. See Figure 4.31.  Low Frequency Window Transition High Frequency Window Transition  Calculate Response Spectra  Unfiltered Ground Motion Statistics Minimum ground velocity: -7.243 in/s Maximum ground velocity: 3.203 in/s Average ground velocity: -1.136 in/s Final ground velocity: 0.147 in/s Minimum ground displacement: -22.563 in Maximum ground displacement: 1.473 in Average ground displacement: -9.254 in Final ground displacement: -22.031 in Unliltered. Filtered-  KERNN21E G r o u n d D i s p l a c e m e n t 226e+001 r  ,1'.50e*001 7.52e+000 O.OOe+000  5  130  10 15 20 25 30 35 40 45 50 55 . „ .Time (*), „  cm Figure 4.21: Ground Motion Statistics Pop-Up Dialog Box  Chapter 4  Software Framework and User Interface  131  4.4.12 G r o u n d M o t i o n Generation Parameters Dialog Box - Filtered Spectrum  Basic Ground Motion Parameters  Record Length  Generated Record Sequence  (sec)  Number of Records  Max Acceleration [0.3  Number of Saved Records  1000  Random Seed j 12345  EF~  Non-Repeatable  - Frequency Options Use Upper Cut-off Frequency P  Fier Records (Cosine Window)  Maximum Frequency Low Frequency WindowTransWon High Frequency Window Transition  Amplitude Modulation Parameters  rggj"""  to jo.01  (Hz)  JirjorjO  to hoocn  (Hz)  White-Noise Filter Parameters  Modulation isbased on a second-order increasing function and exponential decreasing function  The power spectrum for white-noise records is based on fiering the white-noise with a highpass and ground motion (lowpass) Tier  Time to Peak Acceleration  (sec)  Lowpass (Ground Motion) Fitter:  Peak Acceleration Period  (sec)  Set the exponential decay rate with an acceleration value following the peak acceleration period FractionofMaxAcceleration j0.1 Occurrence Time  J20  Update Plot  (0:001-1) (sec)  j  12.0 18.0 Time (s)  Fundamental Frequency (firm ground • 2.5 Hz) Damping Ratio (firm ground «= 80S)  2.50  (Hz)  [60,00 {% critical)  Highpass Filter: Fundamental Frequency (firm ground « 0,65 Hz)  ...  Damping Ratio (firm ground « 50%)  (Hz) [X critical)  Update Plot  30.0  10  15  Freq (Hz) Cancel  Next >  Figure 4.22: Ground Motion Generation Parameters Dialog Box - Filtered Spectrum  Field/Option Record Length Maximum Acceleration Number of Records Number of Saved Records Generated Record Sequence  Notes Duration of earthquake. Specifies the peak amplitude in the acceleration time-history. Number of generated earthquakes. Specifies the number of random earthquake time-histories to be saved in a text file. Maximum number is 50. This option specifies the random nature of the generated earthquake records. The three types of record sequences are as follows: Identical : the same earthquake time-history is used for each  Chapter 4  Software Framework and User Interface  Use Upper Cut-off Frequency  Filter Records  Low Frequency Window Transition High Frequency Window Transition  Time to Peak Acceleration  Peak Acceleration Period  Fraction of Max Acceleration  Occurrence Time White-Noise Filter Parameters  132  dynamic analysis. This permits evaluation of the probabilistic response of structures in which randomness is limited to structural properties only. Repeatable : for a given random seed, the same sequence of earthquake time-histories will be generated each time a multiple record analysis is done, assuming the same Basic Ground Motion, Amplitude Modulation and White—Noise Filter parameters are specified. If structural properties have been specified to be random, the sequence of random structures is not repeated. This permits comparison of the relative effect on probabilistic structural response of earthquake record randomness vs. various degrees of structural property randomness. Non-Rep eatable : the sequence of generated earthquake timehistories and the possible sequence of random structural properties are not repeatable. Specifying this truly random type of multiple record analysis allows results to be combined with analyses performed at different times and/or on different computers. This option truncates the input frequency spectrum at the specified frequency to reduce computation time in generating earthquake records using the Spectral Representation method or Filtered Poisson method. This option is disabled when earthquakes are generated using the inverse Fast Fourier Transform process. This option enables a cosine-type window filter for removal of high and low frequency content in the earthquake record to ensure that the ground acceleration record being used for structural dynamic response analysis is truly representative of a real earthquake. Specifies the frequency range in which the cosine-window varies from 0 to 1. Frequencies below the lower limit are removed and frequencies above the upper limit are unaltered. Specifies thefrequencyrange in which the cosine-window varies from 1 to 0. Frequencies above the upper limit are removed and frequencies below the lower limit are unaltered. Temporal amplitude modulation of the generated ground acceleration record is based on a second-order increasing function followed by a period of constant maximum acceleration and then an exponentially decreasing function. The parameters specifying the temporal amplitude modulation are treated as deterministic and therefore are not randomized for each record generated. The constant maximum acceleration is normalized to unity by the specified peak acceleration. The time when peak acceleration amplitude is reached determines the second-order increasing function. Specifies the duration of the constant maximum acceleration period in the generated earthquake record. The exponential decay rate in acceleration amplitude is determined by a single point following the peak acceleration period. The specified fraction of maximum acceleration sets the amplitude (ordinate) of the single point relative to the normalized maximum value of unity. Sets the time (abscicca) of the single point determining the exponential decay rate. The fundamental frequencies and damping ratios of the highpass and lowpassfilterscan be adjusted to shape the input frequency spectrum to the desiredfrequencycontent.  Chapter 4  Software Framework and User Interface  4.4.13 Ground Motion Generation Parameters Dialog Boxes - Real Spectrum  3 • Ground Motion Spectrum Source Artificial records wl be generated using the spectrum of the input ground motion This motion may be selected from a list of available lecords or specified by a text file contatnrng (he time-history of ground acceleration in a single data column. Earthquake Record:  ~3  ]C: \Quakes\elcentio.txt Time-Step  JoT6  Browse...  (sec)  Acceleration Units: I* fraction of g  C mm/s/s  inA/s  i~ m/s/s  f  ft/s/s  cm/Vs  Generation Parameters  -  Number of Artificial Records Number of Saved Records  pOD |b"  ~~~ (50 Max.)  (~ Identical Record Sequence *  Repeatable Record Sequence  Seed 11234567  <~ Non-Repeatabte Record Sequence R Filer Records (Cosine Window) Low Fieq Window Transition 0.00 High Freq Window Transition h 00.00  to  (Hz)  0.01  > il 00.01  ,c  (Hz)  Next>  Cancel  Frequency Truncation Option The maximum possible frequency in the generated records is 25 Hz. (7 Use an upper cut-off frequency of  |12  (Hz)  •Duration Truncation Option  - —  The maximum acceleration in the record is 3.12867 m/s/s at t - 2.04 sec  W  Truncate ground motion at  jTiT"  Cancel  (sec)  Next>  Figure 4.23: Ground Motion Generation Parameters Dialog Boxes - Real Spectrum  133  Chapter 4  Software Framework and User Interface  134  Notes Field/Option The input spectral density for generating earthquakes is taken from the calculated power spectrum of the specified earthquake record. A deterministic envelope function describing the temporal amplitude modulation of the record is also calculated Earthquake Record and used for each generated record. The earthquake record must be stored as a textfilecontaining the time-history data in a single column. Maximum column width is 12 characters and data may be specified in exponential format. Time-Step Time span between earthquake time-history data points. Acceleration Units Unit system of earthquake acceleration record. Number of Artificial Records Number of generated earthquakes. Specifies the number of random earthquake time-histories to be Number of Saved Records saved in a text file. Maximum number is 50. The input earthquake time-history is used for each dynamic analysis. This permits evaluation of the probabilistic response Identical Record Sequence of structures in which randomness is limited to structural properties only. For a given random seed value, the same sequence of earthquake time-histories will be generated each time a multiple record analysis is done. If structural properties have been specified to be random, the sequence of random structures is not Repeatable Record Sequence repeated. This permits comparison of the relative effect on probabilistic structural response of earthquake record randomness vs. various degrees of structural property randomness. The sequence of generated earthquake time-histories and the possible sequence of random structural properties are not Non-Repeatable Record Sequence repeatable. Specifying this truly random type of multiple record analysis allows results to be combined with analyses performed at different times and/or on different computers. This option enables a cosine-type windowfilterfor removal of high and lowfrequencycontent in the earthquake record to ensure that the ground acceleration record being used for Filter Records structural dynamic response analysis is truly representative of a real earthquake. Specifies thefrequencyrange in which the cosine-window High Frequency Window Transition varies from 1 to 0. Frequencies above the upper limit are removed and frequencies below the lower limit are unaltered. Specifies the frequency range in which the cosine-window Low Frequency Window Transition varies from 0 to 1. Frequencies below the lower limit are removed and frequencies above the upper limit are unaltered. This option truncates the calculated power spectrum of the earthquake at the specified frequency to reduce computation time in generating earthquake records using the Spectral Frequency Truncation Option Representation method or Filtered Poisson method. This option is disabled when earthquakes are generated using the inverse Fast Fourier Transform process. This option truncates all generated acceleration time-histories at the specified time to reduce dynamic response computation Duration Truncation Option time. The occurrence time of maximum acceleration in the record is provided to aid in setting a truncation time.  Software Framework and User Interface  Chapter 4  135  4.4.14 Elastic Responses Dialog Box  Response Quantities Storey Displacement F  Storey Relative Acceleration  (~ Base Shear V  Base Moment  r  Storey Drift  P Stoiey Relative Velocity f~ Storey Total Acceleration T~ Stotey Shear T~ Storey Moment  Response Quantity Summary Types—  Selected Times  ff Peak Value (RHA 8, RSA)  0.00  V* Time to Threshold Value  0.00 0.00  P Response at Selected Time(s)  0.00 Cancel  Newt>  Figure 4.24: Elastic Responses Dialog Box  Field/Option Response Quantities  Peak Value  Time to Threshold Value  Response at Selected Time(s)  Notes Any combination of response quantities may be selected. Each selection may have a different combination of specified storeys. Once all elastic response quantities have been selected, a Storey Selection Dialog Box will appear for each response quantity. See Figure 4.25. This option is enabled for multiple record analysis only. The peak value of each selected elastic response quantity for each selected storey for each earthquake time-history analysis will be included in an output file. See Figure 4.45. Peak values are determinedfromthe response history analysis (RHA) as well as a response spectrum analysis (RSA), which combines maximum modal values using the absolute sum (ABSSUM), square-root-of-sum—of-squares (SRSS) and complete quadratic combination (CQC) modal combination rules. This option is enabled for multiple record analysis only. Selecting this option allows a threshold value for each selected response quantity to be specified for each selected storey in the Storey Selection Dialog Box. See Figure 4.25. The time of the first crossing of the specified threshold is recorded for each earthquake response analysis. If the threshold is not reached, the threshold time is recorded as 0.0. The recorded times are included in an output file. See Figure 4.45. This option is enabled for multiple record analysis only. Selecting this option turns on a scroll box where a maximum of 20 response times may be entered. The value of each selected response quantity for each selected storey at the specified times will be recorded. The recorded values are included in an output file. See Figure 4.45.  Chapter 4  Software Framework and User Interface  W Storey 1  Threshold Value J1.0  r  Storey 2  Threshold Value j  r  Storey3  Threshold Vai»  f  136  (in) " fin)  ~ ~ ~ ~  T~ Storey 4 W Storey 5  Threshold Value [2,0  Cancel  (in)  Next>  Figure 4.25: Storey Selection Example Dialog Box  4.4.15 SDOF Inelastic Responses Dialog Box  -1  •vr9jfVlirj'5jr>:Hj/!:!ni;-Ji l  -Response Quantities fi- Displacement  r* Damping Force (per kg)  f~ Relative Velocity •  V  J  Relative Acceleration  f" Hysteretic Restoring Force (per kg)  Hysteretic Displacement  F~ Total Restoring Force (per kg)  -  V  f~ Hysteretic Energy (per kg)  V  Linear Restoiing Force (per kg)  'Hyaerelic Dwgtam  f Response Quantity Summary Types I j  Selected Times  Peak Value  0.00  fv Time to Threshold Value 0.00  P? Response at Selected Time(s)  0.00  <  Back  Cancel  Next>  Figure 4.26: SDOF Inelastic Responses Dialog Box  Field/Option Response Quantities  Notes Any combination of response quantities may be selected. The force and energy response quantities are expressed per unit mass since a single-degree-offreedom structure may be specified by natural frequency only, in which case mass is not known. The response quantities are defined as follows: Displacement: lateral displacement of lumped mass with respect to the ground.  Chapter 4  Software Framework and User Interface  Peak Value  Time to Threshold Value Response at Selected Time(s)  137  Hysteretic Displacement: displacement z in Equation [3.4]. Relative Velocity : velocity of lumped mass with respect to the ground. Relative Acceleration : acceleration of lumped mass with respect to the ground. Damping Force : product of damping and relative velocity. Linear Restoring Force : product of initial stiffness per unit mass, displacement and post-yield stiffness. See Equation [3.4]. Hysteretic Restoring Force : product of initial stiffness per unit mass, hysteretic displacement and (1 - post-yield stiffness). See Equation [3.4]. Total Restoring Force : sum of linear restoring force and hysteretic restoring force. Hysteretic Energy : Summed product of hysteretic restoring force and displacement. Hysteretic Diagram : this option enables an onscreen plot of the response hysteresis loop for a single earthquake analysis. This option is disabled for multiple record analysis. See Section 4.4.14. Since the dynamic structural response to each earthquake time-history is assumed to be inelastic, the peak value of each selected elastic response quantity for each selected storey is determined from the response history analysis (RHA) only. Response spectrum analysis does not apply. See Section 4.4.14. Although an SDOF structure is by definition a single storey structure, when this option is selected a Storey Selection Dialog Box will appear for each selected response quantity to permit threshold values to be specified. See Figure 4.25. See Section 4.4.14.  4.4.16 M D O F Inelastic Responses Dialog Box  r Response Quantises r~ Displacement  I  r  Diift  F~ Damping Force  r  Relative Velocity  f~ Linear Restoring Force  -  Inertia Force  f~ Relative Acceleration  P  f" Hysteretic Displacement  P  Total Restoring Force  r  r  Total Moment  Total Shear  Hysterelic Resloring Force  I"" Hystetetie Energy Response Quantity Summary Types  Selected Times  0.00 0.00 0.00 0.00  P Peak Value P Time to Threshold Value P Response at Selected Timels)  Cancel  &4  Next)  Figure 4.27: MDOF Inelastic Responses Dialog Box  Chapter 4  Software Framework and User Interface  Field/Option Response Quantities Peak Value Time to Threshold Value Response at Selected Time(s)  Notes See Section 4.4.15 for definitions of response quantities. For multiple-degree-of-freedom structures, with known storey masses, the force and energy response quantities are expressed as total force and total energy. See Section 4.4.15. See Section 4.4.14. See Section 4.4.14.  4.4.17 Plot Modal Results Dialog Box  IjipriH-i'  —  Input Motion Trme-History r  Full Record Window 252 FuD Record Window  Input Motion Spectrum <• Full Spectium Window <*" 252 FuJ Spectrum Window Response Spectra Full Spectrum Window <"* One Decade Window Response Time Histories Full Record Window C 2 5 * Full Record Window Storey Displacement i i  Ptot  |  Rot  |  Plot  Rot  Cancel  138  Next >  Figure 4.28: Plot Modal Results Dialog Box  Chapter 4  Button/Option  Input Motion Time-History  Input Motion Spectrum  Response Spectra  Response TimeHistories  Software Framework and User Interface  139  Notes The Plot Modal Results dialog box is displayed only if the Oscillation Response option or Single Earthquake Response option was chosen along with the Elastic Analysis option from the initial General Input Parameters dialog box. This button activates a pop-up dialog box that displays the time-history of the input ground acceleration, calculated velocity and calculated displacement. The Full Record Window option displays the time-history in a scale that fits the entire record across the width of the pop-up dialog box. For a more detailed display, select the expanded scale given by the 25% Full Record Window option. Once the pop-up dialog box has been displayed, it may be closed and re-opened repeatedly with either time-scale. See Figure 4.29. Displays the power spectrum of the input earthquake motion. The power scale is normalized to the maximum power value in the record. This button is disabled if the Oscillation Response option was selected in the General Input Parameters dialog box. See Figure 4.30. Displays the response spectra for displacement, pseudo-velocity, pseudoacceleration, ratio of pseudo-velocity to relative velocity and ratio of pseudoacceleration to relative acceleration for each damping level specified previously. This button is disabled if the Calculate Response Spectra option was not checked in the Single Ground Motion Parameters dialog box. See Figure 4.31. Displays a separate pop-up dialog box for each storey that was selected for a given response quantity. The pop-up dialog box displays the individual modal responses as well as the total response. See Figure 4.32.  Ground Acceleration (al 3.19e001 1 59e-001 0 OOe+000 •1 59e-GTJ1 -3l9e-001  10  15  20  25  30  35  25  30  35  Time (s)  Ground Velocity Iin/s| 1.42e+001  7lie+0rjp OiOOe+000  711e+000  1 42e+001  15 Time (s)  20  Ground .Displacement (inl 8 32e-000i 4lEetOOO! 0 00e+000 •4.16e+000 •8 32e+000  Figure 4.29: Input Motion Time-History Pop-Up Dialog Box  Chapter 4  Software Framework and User Interface  140  Power Spectrum 10r  j ,  os-  .  -  IliiJiillJllliiSiS IIIISMlf lilljIIJIftl^SllflllllJSJaS^ !  _ 06-  !  05-  - ''  .  i  04  25  50  75  ,  1 00 1 25 Frequency {Hz|  1 50  175  20.0  22.5  25 0 '  Figure 4.30: Input Spectrum Pop-Up Dialog Box  Deformation finl 0 0 2 — 2 51etO01 202 — 1 88e+0Q1 5,0 2» 1 26e*001 6 28e+000 -  :  > 0 03  W0  0 30 '10' Peiiod Is)  J  V.  30  10.0,  300  1000  .C  mo  30.0  1000  100 . . . 30 0  1000  Pseudo-Velocity fin/sl  0 0 2 — 1.07e*O02 2.0 2 — 8 016*001 , 502 5 34e*001 2 S7ei-001 003  0 10  0 30 1 0 Period (t)  Pseudo-Acceleration fql.  f 'QOZ _ ~3.78e*C00 202 — 2.83e+000 502 1.89e*O00 944e-001 .  „ , „ 0 03  0 1 0...  030  . 1  0,„J.,3.0  Figure 4.31: Response Spectra Pop-Up Dialog Box  .v  Chapter 4  Software Framework and User Interface  141  Mode 1  15 . Time ( » ) ,  20  25  30  35  20  25  30  35  30  35  Mode 2 I 6 ?8e*000 1  3 39e»000  I aooe»ooo -3 39e*000 •6 78e»000 10  15 , Time (s)  Total Response  25  .  Figure 4.32: Modal Response Time-History Example Pop-Up Dialog Box  4.4.18 Plot S D O F Results Dialog Box  General  -  ~  -  - -  Input Motion T*n«-H*sto»j< Fu8 R e c o r d W i n d o w <""' 25X Fut! R e c o r d W i n d o w Input Motion Spectrum FuS Spectrum W i n d o w *~  2 5 X Full Spectnim W i n d o w  Hytt«  * Patwarter identification  Plot  •• R e s p o n s e Time Histories <* FiM R e c o r d Window  2 S S Fua R e c o r d W r * d o w  Displacement  Plot  Hytteretic Displacement  Hysteretic Eneigry Eper kfl)  Plot  Hytteietic Diagiam  • ;>. :  I  Cenc^el  Plot  |  Next >  Figure 4.33: Plot SDOF Results Dialog Box  Chapter 4  Software Framework and User Interface  Button/Option Input Motion Time-History Input Motion Spectrum  Hysteresis Parameter . Identification  Response Time-Histories  142  Notes The Plot SDOF Results dialog box is displayed only if the Oscillation Response option or Single Earthquake Response option was chosen along with the Inelastic Analysis option from the initial General Input Parameters dialog box. See Section 4.4.17. See Section 4.4.17. Displays a comparison of the input hysteresis data with the model hysteretic response for the same displacement or acceleration timehistory. If the Quasi-Static Force vs. Displacement option was specified for input hysteresis data, only the input and model hysteresis loops are displayed. If the Acceleration Response option was specified for input hysteresis data, the input and model acceleration response time-histories are displayed along with the associated model hysteresis loop. The Hysteresis Parameter Identification button is disabled if the Use Default Hysteresis Parameters option was selected in the initial Hysteresis Parameter Identification dialog box. See Figure 4.34 and Figure 4.35. Displays the response time-history for the selected response quantity. See Section 4.4.19.  Force per unit mass fkN/kql vs. Displacement [m| Input— Model—  1 20e-003  9 01e-004  6 01e-004  3.00e-004  -3.00e-004  -6.01 e-004:  -3 01e-004  •1 20e-003 •0.16'  -0 1 2  -0 08  -0 04 00 Displacement  0 04  0 08  0 12  0.16  Figure 4.34: Hysteresis Parameter Identification Dialog Box (Quasi-Static Data)  Chapter 4  Software Framework and User Interface  Response Acceleration Time History fmVs/sl, Input— 4 74-tOOO, Model—  50  100  15 0 Tims (*)  20 0  250  300  350  Force per unit mass fkN/kql vs. Displacement fml  Sit 9 e-003 •008  002  -0  00  Displacement  Figure 4.35: Hysteresis Parameter Identification Dialog Box (Acceleration Data) 4.4.19 Plot M D O F Results D i a l o g B o x  •GeneraiInput, Motion Tana-History ti- FJRecordWindDw <" 25XFul Record Window Input Motion Spectrum FuJ Spectrum Window *~ 25£ FuS Spectrin Widow  Response Tm-jts HtjJories ** Fufi Record Window <~ 25* Full Record Window Displacement  Ptot  j  Shew FVxc© Moment Hystorefc Energy HysteceXic Diagram  j  Cancel  Figure 4.36: Plot MDOF Results Dialog Box  143  Software Framework and User Interface  Chapter 4  Button/Option  144  Notes The Plot MDOF Results dialog box is displayed only if the Oscillation Response option or Single Earthquake Response  Input Motion Time-History Input Motion Spectrum Hysteresis Parameter Identification Response Time-Histories  option was chosen along with the Inelastic Analysis option from the initial General Input Parameters dialog box. See Section 4.4.17. See Section 4.4.17. See Section 4.4.18. Displays the response time-history of each selected storey for a given response quantity on a single pop-up dialog box. See Figure 4.37, Figure 4.38 and Figure 4.39.  . t i n y (III.I Storey 1 3 69e-001 ' 1 85e-001 0 QOe+000 1 s-i-urji 3G9e001  15 Time (s)  20  Storey 2 (  7 26e-001 3.63e-001 0 OOe+000 3 63e001 7 2Ee  001  15 Time ( i |  20  25  30  351  Storey 3 I1.06e+000 ! 5.30e-001 ! O.OOe+000 •5 30e001 •1.06e*000:  fl/fl  \AnAf\M A IU art JV A * M a j l M WW f\ 10  15  20  .  25  30  35  Figure 4 . 3 7 : MDOF Response Time-History Example Pop-Up Dialog Box (Displacement)  Chapter 4  Software Framework and User Interface  145  ...n.M Sloiev 1 > 4 78e+002 3 58e+002 2 33e*002: 1 19e+002 10  •"15 Time (s)  20  25  35  Storev 2  15  20  Time (SE]  Storev 3  Figure 4.38: MDOF Response Time-History Example Pop-Up Dialog Box (Energy)  Storev 3  02  05  08  1  0  7  2.5  34  Storev 4  •07  -03 0.0 Displacement Storey.S „„  08  ' 1  Figure 4.39: MDOF Response Time-History Example Pop-Up Dialog Box (Hysteresis)  Chapter 4  Software Framework and User Interface  \ 46  4.4.20 Print Results Dialog Box  '•" General  -  -  -  M Anafeisis Summary f~ Time-History of Input Motion I"* Frequency Spectrum of Input Motion F " Hytrteesfe Parameter idmtiiisaiicn  Oamtii'avt  )~ Random Natural Frequency Array  r  ;•- Multiple Earthquake Response—  -  f ? Multiple Record Analysis Results f>7 Generated Records  r  <  - - M •'.  Figure 4.40: Print Results Dialog Box  Option Analysis Summary Time-History of Input Motion Frequency Spectrum of Input Motion  Notes Summarizes all input data. See Figure 4.41. Base motion acceleration, velocity and displacement timehistory data. Calculated power spectrum data from input earthquake motion. This option is disabled if Oscillation Response was selected in the General Input Parameters dialog box.  Hysteresis Parameter Identification Comparison  Hysteresis model comparison data used in the onscreen plots of Section 4.4.18 and Section 4.4.19. This option is disabled if the Elastic Analysis option was selected in the General Input Parameters dialog box or the Use Default  Hysteresis Parameters option was selected in the initial Hysteresis Parameter Identification dialog box.  Calculated modal frequencies for each random structure generated in a multiple record analysis. This option is Random Natural Frequency Array  Time-History of Response(s)  disabled if the Oscillation Response option or Single Earthquake Response option was selected in the General  Input Parameters dialog box or structural properties were specified as deterministic in the Multiple Earthquake Analysis Parameters dialog box. See Figure 4.42. Time-history data used in the onscreen plots of Section  Chapter 4  Software Framework and User Interface  147  4.4.17, Section 4.4.18 and Section 4.4.19. If the Elastic Analysis option was selected in the General Input  Parameters dialog box, the data is saved in a separate text file for each selected storey of each selected response quantity. See Figure 4.43. If the Inelastic Analysis option was selected, the data for each selected storey of a given response quantity is saved in one text file. This TimeHistory of Response(s) option is disabled if the Multiple Earthquake Response option was selected in the General Input Parameters dialog box.  Response spectra data used in the onscreen plots of Section 4.4.17. This option is disabled if the Multiple Response Spectra Data  Multiple Record Analysis Results  Earthquake Response option or the Inelastic Analysis option was selected in the General Input Parameters dialog box or the Calculate Response Spectra option was not checked in the Single Ground Motion Parameters  dialog box. See Figure 4.44. Peak values, threshold times and response values at selected times for each selected response quantity for each earthquake time-history analysis. This option is disabled if the Oscillation Response option or Single Earthquake Response option was selected in the General Input  Parameters dialog box. See Figure 4.45. Time-history acceleration data and frequency spectrum data for the specified number of saved artificial earthquake records. This option is disabled if the Oscillation Generated Records  Response option or Single Earthquake Response option was selected in the General Input Parameters dialog box  or the number of saved records was set at zero in the Ground Motion Generation Parameters dialog box.  Temporal amplitude modulation data corresponding to the onscreen plot  in the  Ground Motion  Generation  Parameters (Filtered Spectrum) dialog box. This option is Input Amplitude Modulation Data  disabled if the Oscillation Response option or Single Earthquake Response option was selected in the General Input Parameters dialog box or the Earthquake Record  input spectrum option was selected in the Multiple Earthquake Record Analysis Parameters dialog box.  Input frequency spectrum data corresponding to the onscreen plot  in the  Ground Motion  Generation  Parameters (Filtered Spectrum) dialog box. This option is Input Filtered White-Noise Spectrum Data  disabled if the Oscillation Response option or Single Earthquake Response option was selected in the General Input Parameters dialog box or the Earthquake Record  input spectrum option was selected in the Multiple Earthquake Record Analysis Parameters dialog box.  Chapter 4  File  Software Framework and User Interface  Ecft  p c»  View  H  Insert  a  RESPONSE General  -  Format  Help  rit  %j  is,  Stochastic  Input  148  Dynamic  A n a l y s i s Summary R e p o r t  ^  Parameters  ', ,| : .| ;  Structure type : Hultiple-Degree-of-Freedom system Dynamic a n a l y s i s type : E l a s t i c N u m e r i c a l s o l u t i o n method : L i n e a r a c c e l e r a t i o n Solution tolerance multiplier : 1.00 Units system chosen : Imperial Structural  Properties  MDOF s y s t e m m o d e l l e d a s : Number o f s t o r e y s : S Structural  [ h j =,>  parameter(s)  structure  distribution  (kips/g)  Deterministic  floor  Baseline Baseline  storey storey  lateral stiffness (kips/in) height (ft) : 12,000  Baseline Baseline Baseline  storey storey storey  mass m u l t i p l e s : 1.000 1.000 1.000 1.000 1 . 0 0 0 s t i f f n e s s m u l t i p l e s : 1.000 1 . 0 0 0 1 . 0 0 0 1 . 0 0 01 . 0 0 0 height m u l t i p l e s : 1.000 1.000 1.000 1.000 1.000  natural  Hode s h a p e s n o r m a l i z e d (Hodes  i n columns  :  :  Baseline  Deterministic  mass  Shear  100.000  frequencies to  ordered  unity from  (Hz)  :  :  0.SOO  31.540  1.459 2.300  : top storey  to bottom  storey)  1.0000  -0.9190  0.763S  -0.5462  0.2846  0.9190  -0.2846  -0.S462  1.0000  -0.7635  0.7635 0.S462 0.2846  0.5462 1.0000 0.7635  -0.9190 0.2846 1.0000  -0.2846 -0.7635 0.9190  1.0000 -0.9190 0.5462  Hode  shapes normalized  (Modes  i n columns  2.9S5  to  give  unit  ordered  from  top storey  g e n e r a l i z e d modal masses  For Help^pressFl_  Figure 4.41: Analysis Summary Report  to bottom  storey)  3.370  Chapter 4  File  Software Framework and User Interface  Edit  View  Insert  149  Format  Natural frequency unit • H z Record  .  Hode  1  Hode  2  Hode  3  Hode  4  Hode  5  1  0.18661  1 34421  2  14333  2  75378  3  35925  2  0.S002S  1 48418  2  38466  3 01480  3  37318  3  0.49529  1 46078 •  2  2 4797  2  80518  3  38426  4 5  0 . 5 3 507  1 51690  2  43447  3  04032  3  39290  0.46997  1 41316  2  19707  2  87995  3  39887  6  0.50500  1 53320  2  37374  2  97304  3  3 6897  7  0.50289  1 51469  2  45742  2  99141  3  47517  6  0.51223  1 46880  2  42199  2  98649  3  54023  9  0.49491  1 38339  2  15615  2  85615  3  18944  10  0.52100  1 43408  2  30652  2  92850  3  48821  11  0.S0496  1 45617  2  26181  2  95716  3  37419  12  0.47051  1 39245  2  26113  3 09155  3  51494  For Help, press Ft.  Figure 4.42: Random Natural Frequency Array Printed Output  H 9 File  Edit  Output  View  Insert  Format  response  units  Help  :  in  T i m e (s) 0 . 000e+000  Hode 1 0 . 000000e+000  2 . OOOe -002 4 . OOOe - 0 0 2  - 3 . 020394e - 0 0 4 - 1 . 377908e -003  6.OOOe - 0 0 2 8 . OOOe -002  - 3 . 133410e -003 - S . 327846e -003  1. OOOe - 0 0 1  - 8 . 293011e -003  1. 200e -001  - 1 . 264193e -002 - 1 . 862458e -002  3 . 2 7 1 0 9 7 e - 003 4 . 7 0 4 4 3 9 e - 003 6. 3 2 1 7 3 9 e - 003 7 . 8 4 2 2 2 7 s - 003  1. 400e - 0 0 1 1. 600e - 0 0 1 1. 800e - 0 0 1 2 . OOOe - 0 0 1  - 2 . S 7 7 5 8 9 e -002 - 3 . 327581e -002  Hode 2  Hode 3  Total 0 .OOOOOOe+OOO  0 . 000000e+000  0 .000000e+000  8 . 6 2 3 2 7 1 e - 005 3 . 8 9 B 6 9 8 e - 004 8 . 72 6 5 7 0 e - 004  - 3 . 7 1 1 1 7 8 e - 005 - 1 . 6 S 6 5 1 2 e - 004 - 3 . 6 2 2 3 2 6 e - 004  1. 4 4 9 8 7 7 e - 003  - 5 . 8 0 8 8 6 4 e - 004  -2 . S 2 9 1 8 5 e - 0 0 4 -1 .153 689e - 0 0 3 -2 .622986e -003 -4 .458856e -003  2 . 198727e- 003  - 8 . 4 5 S 1 4 7 e - 004  - 6 . 9 3 9 7 9 8 e -003  -1 . 2 1 0 8 2 S e - 003  -1 . 0 5 8 1 6 6 e -002  - 1 . 6 7 8 4 0 2 e - 003 -2 . 1 5 4 6 S 6 e - 003 -2 . 5 0 4 5 0 0 6 - 003  -1 . 5 S 9 8 5 5 e -002 - 2 . 1 6 0 8 8 1 6 -002 - 2 . 7 9 3 8 0 8 e -002  - 4 . 074166e -002 - 4 . 8660S7e -002  9 . 1 1 7 1 8 9 e - 003  - 2 . 6 6 0 0 9 9 e - 003  - 3 . 4 2 8 4 5 7 a -002  2 . 200e - 0 0 1  1. 0 2 5 6 3 l e - 002  - 2 . 6 6 6 4 4 3 e - 003  2 . 400e - 0 0 1  - S . 794847e -002  1. 149S60e- 002  - 4 . 1 0 7 0 7 0 e -002 - 4 . 9 0 7 5 1 5 6 -002  2 . 600e - 0 0 1  - 6 . 896677e -002  1. 2 9 0 8 3 4 e - 002  - 2 .622 2 7 8 e - 003 - 2 . 5 8 6 2 7 5 6 - 003  2 . OOOe - 0 0 1  - 8 . 098102e -002  1. 4 2 5 0 5 0 e - 002  - 2 . 5 0 6 6 9 9 c - 003  3 . OOOe - 0 0 1  - 9 . 317560e -002  1. 52 6 1 4 0 e - 002  3 . 200e - 0 0 1  - 1 . 0540976 - 0 0 1  1. S 8 8 7 4 2 e - 002  - 2 . 2 9 7 5 4 1 e - 003 - 1 . 9 6 8 0 2 6 e - 003  3 . 400e - 0 0 1  1. 6 0 7 8 8 1 e - 002  3 . 600e - 0 0 1 3 . 800e - 0 0 1  - 1 . 174991e - 0 0 1 - 1 . 293071e - 0 0 1 - 1 . 412986e - 0 0 1  4 . OOOe - 0 0 1 4 . 200e - 0 0 1  - 1 . 542 344e - 0 0 1 - 1 . 688670e - 0 0 1  4 . 400e - 0 0 1  - 1 . 847443e - 0 0 1  4 . 600e - 0 0 1 4 flfino - n m  - 1 . 990253e - 0 0 1 -•> -nm  - S . 8 6 4 4 7 0 e -002 - 6 . 9 2 3 7 2 l e -002 - 8 . 0 2 1 1 7 4 e -002 - 9 . 149034e -002 -1 . 0 2 9 5 1 6 e - 0 0 1 -1 .144664e -001  002  -1 . S 3 1 3 9 4 e - 003 -1 . 0 1 4 8 3 5 e - 003  002 002  - 5 . 2 4 8 5 1 5 e - 004 -1 . 9 9 6 7 2 1 c - 004  002  -1 . 5 4 5 1 3 4 e - 004  -1 .395804e -001 -1 . 5 4 4 9 4 S e - 0 0 1  1. 4 2 9 7 2 5 e - 002 1 . 3 3 7 9 0 4 B - 002 1 11 >*QT « ; » - nn?  - 3 . 1 0 9 1 0 3 6 - 004 - 3 . 4 4 6 1 9 2 e - 004 - 1 11?Q1 «;<•- nn<t  -1 .707S80e -001 -1 .8599096 -001 -1 -nm  1. S S S S S S e 1. 5 3 7 9 6 7 6 1. 4 8 5 3 7 5 e 1. 4 5 2 7 0 9 e -  -1 . 2 6 4 4 3 8 e - 0 0 1  For Help, press Fi  Figure 4.43: Example Response Time-History Printed Output (Elastic Analysis)  Chapter 4  File  Software Framework and User Interface  Edit  View  Insert  Format ft  Spectral  -  Analysis  Damping L e v e l  Help  - """"  " ta  "  •  150  ICG  %j  1 of 3  = O.OOe+OOO %  Period  Hax  Disp  Spec-Vel  Pseudo-Vel  Spec-Accel  (in/s)  (in/s)  (in/s/s)  (in/s/s)  782e-004 779e-004 065e-004  4.942e-002 6.525e-002  2 .376e-001 2 . 730e-001  1 . 493e+002 l .S59e+002  1 .493e+002 1 .SS9e+002  8.298e-002  3 . 17Se-001  1 .663e+002  1 .663e+002  1.30e-002 1.40e-002  400e-004 8 469e-004  1.047e-001 1.241e-001  3. S77e-001 3.801e-001  1 .729e+002 1 .706e+002  1 .729e+002 1 .7O6e+002  1.SOe-002 1.60e-002  1 072e-003 1 243e-003 1 507e-003  1.595e-001 1.884e-001  4. 4916-001 4. 882e-001  1 .e81e+002 1 .917e+002  1 .8816+002 1 .917e+002  2.328e-001  5. 569e-001  2 .0S8e+002  2 .058e+002  1 664e-003  2.604e-001  1 973e-003 2 2 50e-003 2 532e-003  3.148e-001  2 .027e+002 2 .lS8e+002  2 .027e+002 2 .158e+002  2.OOe-002 2.10e-002  3.601e-001 4.086e-001  5. 808e-001 6. S 2 S e - 0 0 1 7. 0676-001 7. 574e-001  2 .220e+002 2 . 2 66e+002  2 .220e+002 2 . 2 66e+002  2.20e-002  2 852e-003  4.612e-001  2 . 32 6e+002  2 . 32 6e+002  3 169e-003  e. 1 4 5 e - 0 0 1  2.30e-002 2.40e-002  5.141e-001  8. 658e-001  2 . 36Se+002  2 . 3 65e+002  3 SOSe-003  2.SOe-002  3 848e-003  S.706e-001 6.2S8e-001  9. 18Se-001 9. 671e-001  2 .40Se+002 2 .431e+002  2 .40Se+002 2 . 43 l e + 0 0 2  (in)  (s) 3 4 S 7  l.OOe-002 1.10e-002 1.20e-OOZ  1.70e-00Z 1.80e-002 1.90e-002  For Help, press F l „ „ „ . . .  ,  —,—  ,• , _ „ , « .  Pseudo-Accel  v.i  ..,_>,.,,..„.  Figure 4.44: Response Spectra Data Printed Output  File  •  Eat  View  Insert  Format Help  y a ! »  @,  [Elastic Analysis Results  A!  Range 1 - Storey 1 Displacement (in) : Threshold Value • I.000 Range 2 » Storey S Displacement (in) : Threshold Value - 2.000 Record  Range I  -2 2 1 1  3  -a  4 S 6 7 8 9 10 11 12 13 14 IS 16 17 18 19  2 -2 2 1 3 2 -2 2 2 2 4 -2 3 -1 -2 2  RHA SS73e+000 7644e+000 0699e+000 8424e+000 1777e+000 0890e+000 2441e+0OD 128Se+OO0 0963e+000 S907e+000 0493e+000 OlS3e+000 8108e+000 2074e+000 4643e+000 7700e+000 4437e+000 5912e+O00 6152e+000  SRSS COC 2 6457e+OQ0 2 6S42e+000 1 74S9e+000 1 7S12e+000 2 1217e+000 2 1184e+000 2 6921e+000 2 701Se+000 2 0123e+000 1 9997e+000 1 9971e+000 1 9937e+000 1 2907E+000 1 293Se+000 2 89S4e+000 2 8948e+000 2 3380e+000 2 3270e+O00 2 2386e+O00 2 2373e+000 2 0848e+OO0 2 O964e+O00 2 0563e+000 2 0499C+000 2 6300e+000 2 6246e+O0O 4 0034e+000 3 9983e+000 2 2941e+000 2 2874e+000 4 0S18e+000 4 0461e+000 1 4718e+000 1 4649C+000 2 7029e+000 2 7018e+000 2.0861e+000 2 0813e+000  3  2 3  3 3 3 2 3 3 3 3 2 3 S 3 4 2 3 3  ABSSUH 6297e+000 4S67e+O00 02776+000 S807e+000 1099e+000 0134e+000 0il9e+000 8420e+000 389le+000 2810e+000 0224e+000 9084e+000 S69Se+O00 2717e+000 48S5e+000 9900e+000 4077e+000 989Se+O0O 032Se+000  Threshold Time 2. 1200e+000 2. 2000e+000 2. 3000e+000 2.0800e+000 S. 9400e+000 2. 3000e+000 2. 6000e+000 2. OSOOe+000 2.8200e+000 2. 6600e+000 2.4000e+000 2. lBOOe+000 2. 7000e+000 1. 8800e+000 4. S000e+000 1. 9400e+000 3. lBOOe+000 2 .1600e+000 1. S600e+000  T - 1.000 s -2 3041e -001 - 3 1204e -001 -4 971Se -002 -1 6406e -001 6 0393e -002 2 4784e -002 -9 2714e -002 -2 4409e -001 3 9301e -002 -2 0272e -001 1 0304e -001 -B SS3Be -002 -9 321Se -002 7 3997e -002 3 9S02e -001 -2 3177e -001 -9 8886e -002 1 553 6e-001 3 7601e -001  r -  5.000  -9 7960e-001 -8 .13446-002 -1 <311e-001 -1 758Se+000 -2 4762e-001 8 22Sle-001 1 .6194e-001 3 4169e-001 3 1070e-001 2 .9164e-001 6 .4S46e-001 8 .0695e-001 2 07S6e+000 2 .1496e+000 9 .lS72e-002 -6 .3898e-001 1 .1904e+000 8 .0433C-001 -1 .06S4e+000  |Fw,H«lp, .pressf )„_..  Figure 4.45: Example Multiple Record Analysis Data Printed Output (Elastic Analysis)  Chapter 5  CHAPTER  Software Verification and Case Studies  151  5  S O F T W A R E VERIFICATION AND C A S E STUDIES  5.1  INTRODUCTION  The final stage in any software development process is, of course, verification of the efficacy of the software application. In the case of numerical software, this requires that the calculated output match, to within some subjective tolerance, the corresponding output from an accepted benchmark, which may be a theoretical result, experimental data or results from a similar software model.  Once a suitable benchmark has been  established and the software has been verified to adequately match that benchmark, it may be released for use as a beta version with the expectation that user testing will identify useful upgrades and problems not identified in initial testing.  The issues related to selecting a suitable benchmark for evaluating structural dynamic analysis results are discussed at the beginning of Section 5.2 followed by several subsections that then use the chosen benchmarks to evaluate various elastic and inelastic PSResponse analysis results. Section 5.3 then summarizes two case studies that were completed as the first application of the beta release of PSResponse. The first case study analyzes the effect of random properties on the dynamic response of a five-storey elastic structure while the second case study analyzes the well-known equal displacement observation in structural dynamics as well as the effect of hysteresis model properties on displacement response.  5.2  SOFTWARE VERIFICATION  Verification of numerical software requires that an accepted benchmark be established against which calculated results may be compared. In the case of structural  Chapter 5  Software Verification and Case Studies  152  dynamic analysis, particularly non-linear time-history analysis, establishing a benchmark must typically rely on experimental data or reasonable agreement between independent software applications, rather than on theoretical results. This is because a complete theoretical structural dynamic response over the time-history of an excitation is not available per se in circumstances involving a non-linear force-deformation relationship and/or response velocity sign reversals.  In these situations, the theoretical response  assumes perfect conformity to the true force-deformation curve, which requires perfect convergence in the iterative residual force process of each time-step, and perfect identification of response velocity sign reversal points. Clearly, a certain accepted level of numerical error will always be present at every time-step and will tend to accumulate over the duration of a time-history analysis due to the recursive nature of time-stepping methods. Therefore, since the theoretical dynamic response of a structure cannot be separated from the inherent calculation error, software verification must be based on comparison with experimental data or similar software models. Of the two remaining verification options, the next best choice would typically be verification against experimental data since comparing two software models is subject to compounding of each individual model error. This may result in a situation where the difference between two models is deemed acceptable while the real error in the subject software, as compared to a nominally theoretical result, is the unacceptable sum of the individual model errors. Verification against experimental data avoids the problem of error compounding and is the ultimate test of any mathematical model. However, for this software, comparison of a calculated acceleration response time-history against shaketable data would largely be an exercise in evaluating the efficiency of the built-in system identification algorithm. This is because for a given experimental acceleration response time-history, the software will fit the parameters of the smooth hysteresis model such that the calculated acceleration response time-history matches the input as closely as possible (Sec. 3.3.4.4).  While a close match validates the fundamental algorithmic  structure and shows that a certain combination of hysteresis parameters was able to match the input acceleration record, it does not necessarily verify that for a different set of structural hysteresis parameters, which may be entered directly to model a desired yield  Chapter 5  Software Verification and Case Studies  153  displacement, the calculated dynamic response is correct. O f course, one may identify the hysteresis parameters from experimental data collected using a selected base motion history and then compare the calculated and experimental response of a nominally identical structure using the same hysteresis parameters and a different base motion history. This, however, will inevitably result in differences that can be attributed to one or several of the following causes: (i) the nominally identical structure was not identical and, therefore, was modeled with a mismatched set of hysteresis parameters; (ii) the structure was identical but the identified hysteresis parameters are not valid for a different base motion history; (iii) the software model is flawed. The first possible error source is likely not significant and may be quantified by comparing the experimental response of several nominally identical structures using the same base motion history as was used for hysteresis parameter identification. This allows the effect of small structural differences on dynamic response to be evaluated independently of a software flaw or the effect of a different base motion history. The second and third possible error sources are much more likely to be responsible for any observed differences between an experimental response and the calculated response, unfortunately, it is impossible to separate the two. Even if a new set of hysteresis parameters is identified from the experimental response using the different base motion history, it is unclear whether the difference in the hysteresis parameter sets is strictly due to an inherent need to adjust the parameters for each different base motion history or there is a flaw in the software. Therefore, the only way to verify the accuracy of PSResponse is to directly enter a set of basic hysteresis parameters corresponding to a specific yield displacement and compare results with a similar software application and other published results. While this approach is subject to the aforementioned error compounding problem, as well as difficulties in setting up identical dynamic analyses using different software models, the intent is simply to perform a so-called "sanity check" to verify that PSResponse provides dynamic analysis results that are reasonably similar to an accepted industry standard software package.  Following the decision to use the results of a similar software package as the verification benchmark, an accepted industry standard software package needed to be chosen from a list of possible candidates including; SAP, ETABS, DRAIN, CANNY, and  Chapter 5  Software Verification and Case Studies  RAUMOKO  154  among others. Based on consideration of the industry acceptance of each  candidate as well as preliminary attempts to use each software package,  SAP2000  Nonlinear 8.1.2 (Computers & Structures Inc. 2003) was chosen as the software benchmark.  It is the most recent version of a widely-used engineering analysis and  design package, first developed in 1975, which is capable of performing highly sophisticated non-linear time-history analysis.  Finally, as a further verification check, results from each software application were checked against results published by Chopra (1995).  A l l results, which were  obtained from both elastic and inelastic dynamic analysis of single-degree-of-freedom (SDOF) and multiple-degree-of-freedom (MDOF) structures, are summarized in the following sections.  5.2.1  Elastic Analysis  Verification of elastic dynamic analysis results is a relatively straightforward process, as compared to inelastic dynamic analysis results, since elastic analysis does not involve an iterative residual force process and does not depend on the assumed hysteretic behaviour of the structure. Without those sources of discrepancy, elastic analysis results should be almost identical amongst all software applications.  5.2.1.1  Single-Degree-of-Freedom Elastic Systems  Beginning with SDOF systems, Figure 5.1 and Table 5.1 show the  PSResponse  and SAP2000 calculated deformation response of several natural frequency and damping combinations to the north-south component of the 1940 Imperial Valley, California earthquake recorded at E l Centro, hereafter referred to as the E l Centra ground motion. Included in Table 5.1 are Chopra's peak deformation values taken from results given as Figure 6.4.1.  There is very good agreement between all results, with SAP2000  Chopra being virtually identical while PSResponse  and  gave slightly different peak  deformations. The difference is likely due to the way in which the overshoot problem  Chapter 5  Software Verification and Case Studies  155  (Sec. 3.3.2) is dealt with in each set of results. It appears that the SAP2000 and Chopra results do not reflect a subdivision of time-steps to determine the point of zero velocity when a velocity sign reversal occurs.  T=2sec damp='  8 6  A A /WW 1 W / V / WJ i -2, vV / v t M W / W\t\ v v wV ; 5  4  2  A  i\  A  o  l  \ !  I / S  \ /  \ r  n  \  /  \ / 'A/  -6  -8  Figure 5.1: Elastic Deformation Response of SDOF Systems - PSResponse vs. SAP2000  Chapter 5  Software Verification and Case Studies  When this feature of  156  PSResponse was disabled, all three sets of results were virtually  identical.  Table 5.1: Comparison of Peak Deformation Responses of SDOF Systems (in.)  5.2.1.2  Swtiin  PSResponse  .s\i'2inm  Chopra  T = 0.5 sec, C, = 2%  -2.70  -2.67  -2.67  T = 1 sec, C = 2%  -5.72  -5.97  -5.97  T = 2 sec, £ = 2%  -7.06  -7.47  -7.47  T = 2 sec, <; = 0%  9.29  9.92  9.91  T = 2 sec, C, = 5%  5.43  5.37  5.37  Multiple-Degree-of-Freedom Elastic System  To verify elastic analysis results for M D O F systems, the five-storey shear frame shown in Figure 5.2 was analyzed to allow comparison with detailed results given by Chopra as Example 13.2.6 and Example 13.8.2. Figure 5.3 and Table 5.2 illustrate the close agreement between each set of software results and Chopra's published results for several selected response quantities related to the El Centra ground motion.  C, = 5% for all natural modes m = 100 kips/g  m  h = 12 ft. k = 31.54 kips/in.  m  h, k  m  h, k  m  h, k  rigid beams  h, k  Figure 5.2: Five-Storey Elastic Shear Frame  Chapter 5  Software Verification and Case Studies  157  Base Shear 80 60 40 tn .0-  20 0 -• -20 0 -40  -  -60 -80 B a s e Moment 3000 2000 1000  2  Illll / M. 1 \ -1000  -2000  V  0  \\l A Aiff  y Ll V  A AA A A \J  \J20 • \ /  \J  25^  -3000  Storey 5 Shear  Figure 5.3: Elastic Responses of Five-Storey Shear Frame - PSResponse vs. SAP2000  A ^30  Chapter 5  Software Verification and Case Studies  158  Table 5.2: Comparison of Elastic Peak Responses of Five-Storey Shear Frame Mode MIHI.II  Result  ( (iinhinaliiiii  Source  Mode 1  Mode 2  Mode 3  Mode 4  Mode 5  \M^I  SRSS  RHA  Base Slieiir (kips)  Base Moment  Storey 5  Store> 5  (kip*fi)  Shear (kips)  Displacement (in.)  PSResponse  61.021  2572.637  17.368  6.797  Chopra  60.469  2549.400  17.211  6.731  PSResponse  24.174  -349.153  -20.084  -0.922  Chopia  24.533  -354.330  -20.382  -0.936  PSResponse  -9.864  -90.370  -12.919  -0.239  Chopra  -9.867  -90.402  -12.923  -0.239  PSResponse  -2.908  20.739  4.892  0 055  Chopia  -2.943  20.986  -4.951  0.055  PSResponse  -0.553  -3.460  -1.U02  -0.009  Chopra  -0.595  -3.718  -1.141  -0.010  PSResponse  98.601  3037 9  56 404  S 026  Chopra  98 407  3018.8  56 608  "7.9^1  PSResponse  66.470  2598.7  29.985  6.866  Chopra  66.066  2575.6  30.074  6.800  PSResponse  66.396  2596.8  30.701  6.S61  Chopra  66 5<P  2572.2.  29.338  6."93  PSResponse  74.030  2587.5  34.970  6.836  Chopra  73.278  2593.2  35.217  6.847  SAP2000  74.376  -2517.6  33.716  -6.806  M  The preceding elastic dynamic analysis results given in Table 5.1, Figure 5.1, Table 5.2 and Figure 5.3 indicate that there is very good agreement between each independent set of results for both single-degree-of-freedom  elastic systems and a  multiple-degree-of-freedom elastic system. This is taken as evidence that PSResponse results meet an acceptable standard of accuracy for elastic dynamic analysis of structures.  Chapter 5  5.2.2  Software Verification and Case Studies  159  Inelastic Analysis  Unlike elastic dynamic analysis, inelastic dynamic analysis depends on the assumed hysteretic behaviour of the structure, assumed post-yield behaviour and convergence of an iterative residual force process at each time-step. comparison of results with SAP2000  To allow a  and results published by Chopra (1995),  PSResponse hysteretic behaviour for each time-history analysis was constrained to remain undamaged with no pinching and no progressive strength or stiffness degradation. Yield displacement or drift, which must be specified in PSResponse when the default non-degrading, non-pinching behaviour is selected, was determined for each analysis from the non-linear dynamic analysis examples given by Chopra, which serve as PSResponse verification cases.  In SAP2000, yielding behaviour is based on a user-  specified yield stress for each structural element such as a beam or column. These yield stresses were adjusted in each analysis to limit all shear forces to the elastoplastic limits imposed in the Chopra examples.  5.2.2.1  S i n g l e - D e g r e e - o f - F r e e d o m Inelastic Systems  Beginning again with SDOF systems, Figure 5.4, Figure 5.5, Figure 5.6 and Table 5.3 show the PSResponse and SAP2000 calculated responses of systems with identical elastic properties but different normalized yield strengths to the El Centro ground motion, where the normalized yield strength is equivalent to the ratio of yield displacement to peak elastic displacement. Included in Table 5.3 are Chopra's peak values taken from results given in Section 7.4.1. Note that SAP2000 results are not given for the case in which normalized yield strength = 0.125 because the software indicated that the system had collapsed under the E l Centro ground motion. To illustrate the relative hysteretic behaviour of each system, Figure 5.7 shows the calculated energy dissipation for each normalized yield strength, which is the cumulative area of the associated hysteresis loop. Results are presented for PSResponse only since the corresponding results from SAP2000 were not available.  Figure 5.4: Inelastic Deformation Response of SDOF Systems - PSResponse vs. SAP2000  Chapter 5  Software Verification and Case Studies  Figure 5.5: Inelastic Shear Response of SDOF Systems - PSResponse vs. SAP2000  161  Chapter 5  Software Verification and Case Studies  Figure 5.6: Hysteresis Loops of SDOF Systems - PSResponse vs. SAP2000  162  Chapter 5  Software Verification and Case Studies  163  Table 5.3: Comparison of Inelastic Peak Responses of SDOF Systems (T = 0.5 sec, Damp = 5%) Norm Yield = 1  Norm Yield = 0.5  Norm Yield = 0.25  Norm Yield = 0.125  Displ  Slu-iir  Displ  Shear  Displ  Shear  Displ  Shiiir  (in.)  (kips)  (in.)  (kips)  (in.)  (kips)  (in.)  (kips)  PSResponse  -2.25  -13.45  -1.48  5.84  -1.54  -3.37  -2.09  -1 ~4  Chopra  -2.25  -13.45  1.62  6.73  1.75  3.36  2.07  1.68  SAP2000  -2.24  -13.38  1.63  6.64  -1.90  3.32  -  -  As expected, the preceding inelastic dynamic analysis results in Table 5.3, Figure 5.4, Figure 5.5 and Figure 5.6 show greater differences between software applications than the elastic analysis results presented previously (Sec. 5.2.1.1) due to the increased number of assumptions and sources of numerical error inherent in inelastic dynamic analysis.  The primary difference between PSResponse and SAP2000,  other than the  overshoot issue identified previously, is in the "sharpness" of the yield point, which is clearly illustrated in Figure 5.6. The SAP2000  shear forces exhibit classic elastoplastic  behaviour, which was also used by Chopra, while the PSResponse shear forces follow smooth curves and do not have well-defined yield points. This is because the default hysteresis parameters in PSResponse were selected to reflect the yield behaviour of most structures, which tends to follow a smooth, continuous curve without abrupt changes in stiffness.  For structures that do have a well-defined yield point, such as certain SDOF  steel structures, the sharpness of the yield point could be increased in PSResponse with a minor change to the default hysteresis parameters.  In spite of the difference in yield behaviour between models, there is still reasonably good agreement between PSResponse, SAP2000 and the published results of Chopra.  Since the intent was simply to perform a "sanity check" to verify that  PSResponse provides inelastic dynamic analysis results that are similar to a standard software package, the reasonably good agreement between calculated responses is again taken as evidence that PSResponse results meet an acceptable standard of accuracy for single-degree-of-freedom inelastic structures.  Chapter 5  Software Verification and Case Studies  164  Figure 5.7: Dissipated Energy of SDOF Systems - PSResponse  5.2.2.2  Multiple-Degree—of-Freedom Inelastic System  The final stage in the PSResponse software verification process was the simulation of a multiple-degree-of-freedom structure undergoing inelastic deformation. To verify inelastic dynamic analysis results for M D O F systems, a five-storey shear frame, shown in Figure 5.8, was again analyzed to allow comparison with results given by Chopra in Section 19.1.2 and Section 19.1.3. Table 5.4, Figure 5.9, Figure 5.10 and Figure 5.11 show the PSResponse and  SAP2000  calculated storey deformation and shear  responses of the frame structure to the El Centro ground motion.  To again illustrate  relative hysteretic behaviour, Figure 5.12 shows the energy dissipation at each storey  Chapter 5  Software Verification and Case Studies  165  level for PSResponse only, since the corresponding results from SAP2000 were again not available. £ = 5% Rayleigh damping in first two natural modes m = 100 kips/g  h= 12 ft.  L  m  k = 87.08 kips/in. yield = 26.05 kips  m  k = 146.2 kips/in. yield = 43.60 kips  m  k = 190.6 kips/in. yield = 57.15 kips  m  k = 220.2 kips/in. yield = 66.80 kips k = 234.9 kips/in. yield = 72.55 kips  rigid beams  ssssssssss  sss  Figure 5.8: Five-Storey Inelastic Shear Frame  Table 5.4: Comparison of Inelastic Peak Responses of Five-Storey Shear Frame Storey 1  3  5  OlMlltll\  PSResponse  SAP2000  Chopra  Displacement (in.)  -0.-8  -1.05  -1.20  Drift (in.)  -0.78  -1.05  -1.20  Shear (kips)  69.56  72.02  72.55  Displacement (in.)  -1 38  -2.28  -2.05  Drift (in.)  -0.71  -1.23  Shear (kips)  60.92  65 79  66.80  Displacement (in.)  -2 12  -2.60  -2.40  Drift (in.)  -0.80  1.05  Shear (kips)  52.60  57.69  57.15  Displacement (m)  -2 63  -2 50  -2.40  I)iill i in <  -0.59  11 1  Shear (kips)  40 28  44 46  43.60  Displacement (in.)  -3 08  -3.48  -3 :u  Drift (in.)  0.68  -1.02  -0.81  Shear (kips)  25.30  25.67  26.05  -  Figure 5.9: Inelastic Deformation Response of Five-Storey Shear Frame - PSResponse vs. SAP2000  Figure 5.10: Inelastic Shear Response of Five-Storey Shear Frame - PSResponse vs. SAP2000  Figure 5.11: Hysteresis Loops of Five-Storey Shear Frame - PSResponse vs. SAP2000  Figure 5.12: Dissipated Energy of Five-Storey Shear Frame - PSResponse  Chapter 5  Software Verification and Case Studies  170  Similar to SDOF systems, the preceding inelastic dynamic analysis results for a five-storey shear structure subjected to the El Centro ground motion show greater differences between software applications than the elastic analysis results of a similar structure (Sec. 5.2.1.2). This is again due to the additional sources of numerical error and increased number of assumptions inherent in inelastic dynamic analysis as compared to elastic analysis.  Differing assumptions regarding yield behaviour and the overshoot  problem, which were identified previously (Sec. 5.2.2.1), as well as systemic differences in setting up a dynamic analysis between SAP2000 analyses will never be exactly comparable.  and PSResponse mean that the  Nevertheless, each independent set of  dynamic analysis results for a multiple-degree-of-freedom inelastic structure, and the published results of Chopra, are in reasonably close agreement.  This standard of  accuracy is considered acceptable for the intended purpose of PSResponse, which is to provide fast, summary-level dynamic analysis results for determining the probabilistic response of linear and non-linear systems under stochastic dynamic loading. To that end, it is worth noting that PSResponse was two orders of magnitude faster than SAP2000 in calculating the inelastic dynamic response of the five-storey shear structure subjected to the El Centro ground motion. Calculation time for PSResponse was less than one second on a 1.5 GHz Pentium 4 computer with 256 M B R D R A M while SAP2000,  with its much  more detailed finite-element basis, took approximately 83 seconds.  5.3  C A S E STUDIES  The first application ofthe beta release oi PSResponse was the completion of two case studies that were selected from the myriad possibilities in structural dynamics largely to demonstrate the capabilities of the program as a research and analysis tool. These case studies analyzed, from a probabilistic point of view, three general questions pertinent to structural dynamics and earthquake engineering; the relative effect of random structural properties on the dynamic response of a structure, the appropriateness of the well-known equal displacement observation in structural dynamics and the effect of hysteresis model properties on displacement response.  Results from the case studies  provide answers to those questions for the chosen structural models, input variable ranges  Chapter 5  Software Verification and Case Studies  171  and output response quantities without attempting to be completely general in nature. Also, comparison of certain results with some of the seismic structural response and reliability studies cited in Section 2.5 was left as future work.  5.3.1  Case Study #1  To assess the relative effect of structural property variability on the random response of a structure subjected to random earthquake ground motions, the five-storey elastic structure used for model verification in Section 5.2.1.2, and shown in Figure 5.2, was subjected to a sequence of 1000 generated records, using the E l Centro ground motion as the seed, at five different levels of variability in the storey mass, storey stiffness, storey height and modal damping. To ensure that the frequency content of the generated records did not result in unrealistic ground displacements, the peak elastic displacement of a long period SDOF structure (T = 12.0 seconds) was determined for each record in the sequence using; no filtering, filtering with a low frequency window transition of 0.05-0.075 Hz and filtering with a low frequency window transition of 0.10.15 Hz. In each case there was no significant difference in the statistics describing the distribution of the peak displacements. The variability levels of the structural properties were characterized by Normal distributions with a coefficient of variation (COV) of 0.0, 0.1, 0.2, 0.3 and 0.4 in each of the structural parameters with the exception of storey height where the COV's were set at 0.0, 0.007, 0.014, 0.021 and 0.028, corresponding to standard deviations of 1 to 4 inches in the 12 foot storey height. The dynamic responses that were chosen for probabilistic analysis were limited to; peak base shear, peak base moment and peak fifth storey displacement.  Table 5.5 lists the basic statistics for each set of peak results that were generated using the same earthquake sequence as well as statistics for the peak results of the deterministic structure (COV = 0.0) subjected to a second sequence of 1000 earthquakes. The second earthquake sequence was used to verify that the probability distribution of each response quantity was not affected by a particular random sequence of earthquakes. For reference purposes, the peak elastic responses of the five-storey structure to the El  Chapter 5  Software Verification and Case Studies  172  Centro ground motion are bracketed in Table 5.5 below each response quantity in column one.  The actual frequency distributions corresponding to each set of calculated statistics in Table 5.5 are given as histogram plots in Figure 5.13 - Figure 5.15.  These  plots are overlaid with the associated Gumbel distribution, given as a solid line, and Normal distribution, given as a dashed line, to illustrate the fit of each type of distribution to the data. Note that the discontinuity in some of the Normal distributions is the result of lumping together all responses below the point ofthe discontinuity.  Table 5.5: Peak Response Statistics of a Five-^Storey Elastic Shear Frame Response Quantity Base  Statistic  Scq. #1 0.0  cov -  Seq. #2 COV -0.0  Seq. #1 COV = 0.1  Seq. #1 COV-0.2  Seq. #1 COV = 0.3  Scq. #1 COV = 0.4  Min.  27.40  28.16  31.96  27.99  19.79  13.67  Shear  Max.  155.35  185.68  173.67  193.67  169.14  290.82  (kips)  Avg.  73.33  74.37  73.41  73.92  73.92  76.30  (74.03)  Stdev.  20.58  22.31  21.25  22.36  24.07  29.35  Base  Mm.  994 03  1153 60  988 00  945 33  832 28  493 37  Moment  Max  6X29 30  7128 50  6705 30  7882.40  6643.50  12248 00  (kip'ft)  Avg.  2843.27  2878 58  2851 44  2819 01  277" 07  2^18 96  (258"'.53)  Stdev.  877.92  915.03  902 10  933 67  973.03  1 m.05  Storey 5  Min.  2.63  3.05  2.66  2.75  2.69  2.59  Displ.  Max.  18.04  18.83  19.21  21.11  28.70  37.46  (in.)  Avg.  7.51  7.61  7.61  7.77  8.32  8.83  (6.84)  Stdev.  2.32  2.42  2.42  2.57  3.25  3.89  From Figure 5.13, Figure 5.14 and Figure 5.15 it is apparent that the Gumbel distribution is a better description of the peak response data in each case than the Normal distribution, as expected, since the Gumbel distribution is an Extreme Value-Type I distribution. Therefore, using the Gumbel distribution as a basis for comparison, Figure 5.16 summarizes the individual distributions for each set of results to illustrate the effect of increasing variability in structural parameters on the probability distribution of the chosen elastic response quantities.  Chapter 5  Software Verification and Case Studies  Figure 5.13: Base Shear Response Histograms  173  Chapter 5  Software Verification and Case Studies  Deterministic Properties  174  Deterministic Properties Record Sequence 2  . Record Sequence 1  • 120 •  8 8 § 8 83 3 3 8 8 3 kip*ft  s> a s  1  S S 8 « S w Wp*ft  Property COV= 1 0 % Record Sequence 1  Property COV - 2 0 % Record Sequence 1  100 £  1,3 •  ao60  § 8 '8 a  ift w fl ^  H si ff  kipft  Property COV = 30% Record Sequence 1  Kipft  Figure 5.14:  Base Moment Response Histograms  f  n  o o o g g o g o g o g g o o o o  WO  Chapter 5  Software Verification and Case Studies  Deterministic Properties Record Sequence 1  175  De'ermr, stic Properties Record Secucnce 2  'Yir? 120 100  100  It  c 31  80 '  iiiiii!  GO  60  Ilk-  o  m ro i f o  n  P«lP|||||§i|fi  Property COV = 10% Record Sequence 1  Property COV = 20% Record Sequence 1  KG • 120  MiO • '.100 •  \  80-  60-  at>V  J  1111  Property COV =30% Record Sequence 1  Property C O V = 40% Record Seqijence\1  "- ... -  ^  -  m -  a  «« .- <" •» ™ i . M P. M  8 0  -  »M  m  Figure 5.15: Storey 5 Displacement Response Histograms  in  i'»  M ryi  mm  Chapter 5  Software Verification and Case Studies  176  Base Shear  0  25  50  75  100  125  150  4000  5000  6000  kips Base Moment 140  0  1000  2000  3000  kip*ft Storey 5 Displacement  ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ COV = 0%, Sequence 1 COV = 20%  COV = 0%, Sequence 2 COV = 30%  Figure 5.16: Peak Response Distributions  COV = 10% COV = 40%  Chapter 5  Software Verification and Case Studies  177  To assess whether the individual distributions are statistically equivalent or significantly different, several statistical tests were performed to compare each set of peak results with those of the deterministic structure (COV = 0.0).  These tests evaluate  whether two distributions possess the same mean {Student's t-test), and the same variance (F-test) and, more generally, whether two distributions are equivalent based on the maximum difference between the cumulative distribution functions  (Kolmogorov-  Smirnov test). Table 5.6 lists the statistics comparing the response distribution for each level of structural variability with that of the deterministic structure, including the statistics comparing the response distributions of the deterministic structure using the two different generated record sequences.  Table 5.6: Elastic Peak Response Distribution Comparison Statistics Response Quantity Base Shear (kips)  Slalistir F t QKS  Base  1  Moment  t  (kip*ft)  QKS  Storey 5  F  Displ.  t  (in.)  QKS  Seq.#2  Seq. #1  Seq. #1  Seq. #1  ( <>\ -0.1  CON' = 0.2  ( <)\ -0.3  1.1751  1.0659  1.1803  1.3681  2.0333  -1.0834  -0.0890  -0.6113  -0.5891  -2.5671  0.6785  0.6406  0.8840  0.0809  0.0003  1.1311  1.2284  1.6657  1 5939  2 6h29  (ON  0.0  1.0863  '" •'• 1.0559  Seq. #1 ( ON  0.4  -0.8805  -0.2051  0  0  7161  0.S840  0.3344  0.0089  0.0000  1.0863  1.0917  1.2236  1.9620  2.8190  -0.8804  -0.8978  -2.3359  -6.4176  -9.0789  0.7161  0.7888  0.0525  0.0000  0.0000  5988  Comparing the statistics of Table 5.6 with the critical values of F (0.8496, 1.1771) and t (-2.57823, 2.5783) at the 1% level of significance, it appears that a coefficient of variation of 10% in the storey mass, storey stiffness, storey height and modal damping does not significantly affect the elastic response distribution of peak base shear, peak base moment or peak fifth storey displacement for a random sequence of El Centro type earthquakes. At the C O V = 20% level of variability, the peak fifth storey displacement distribution was marginally affected while the peak base shear and peak base moment distributions remain unaffected.  At the C O V = 30% level of variability, the peak fifth  storey displacement distribution and the peak base shear distribution were both  Software Verification and Case Studies  Chapter 5  178  significantly affected while the peak base moment distribution was only marginally affected, however, this may be a result of the smaller variability used for storey height. At the C O V = 40% level of variability, all response distributions were significantly affected.  These observations are supported by the Kolmogorov-Smirnoff test statistic  QKS for each set of peak responses, which may be interpreted as the probability that the underlying distribution of the peak responses is the same as that of the deterministic structure. In each case where the response distribution appears to be affected at a certain level of parameter variability, there is a corresponding sharp decrease in the Q  KS  statistic  indicating that it is highly unlikely that the underlying distributions are equivalent. Lastly from Table 5.6, the distribution comparison statistics for the deterministic structure using the two different sequences of random earthquakes indicate that there is no significant difference in the distributions of any of the three types of peak responses. This suggests that the assumption that the probability distribution of each response quantity is not affected by a particular random sequence of earthquakes is valid.  Keeping in mind that only one particular type of earthquake and one basic pattern of storey mass, storey stiffness, storey height and modal damping were modeled, the foregoing results seem to indicate that the randomness of the generated ground motions accounts for the majority of the observed range in a given peak response while structural randomness has a relatively minor effect.  This then suggests, or perhaps confirms the  prevailing opinion, that careful attention needs to be paid to the characteristics of the ground motion records used when analyzing the dynamic response behaviour of a structure.  Once a suitable seed record or suite of seed records has been selected,  however, the peak response probability distributions for a given structural model could be applied to a real structure with reasonable confidence since the assumed level of uncertainty in the structural parameters needs to be only approximately correct.  Finally, to assess the accuracy of determining peak responses for a multipledegree-of-freedom system from a response spectrum analysis (RSA) procedure, the peak time-history (RHA) responses in each set of results were compared with modal combinations  using three common modal combination rules; the absolute sum  Chapter 5  Software Verification and Case Studies  179  (ABSSUM) rule, the square-root-of-sum-of-squares (SRSS) rule and the complete quadratic combination (CQC) rule.  Table 5.7 lists the basic statistics for each  distribution of percentage differences between the R H A peak response and the corresponding R S A peak responses. For reference purposes, the percentage differences between the R H A and RSA results for the E l Centro ground motion are bracketed in Table 5.7 below each modal combination type in column two.  Table 5.7: Statistics for the Percentage Difference between RHA & RSA Results Response  Base Shear  Modal  Statistic  Combo.  (% Diff)  Scq. #1  Seq. #2  Seq. #1  Seq. #1  Seq. #1  Seq. #1  ( o\  COV  COV  COV  COV  ( o\  0.(1  0.0  0.1  0.2  0.3  11.4  Min  -28.32  -27.42 '  -29.63  -27.90  -31.15  -30.17  SRSS  Max  31.11  34.20  30.64  38.27  30.34  49.16  (-10.21%)  Avg  -1.54  -1.49  -1.33  -2.31  -2.66  -3.40  Stdev  9.78  9.79  9.70  9.46  9.71  9.81  Min  -28.90  -27.46  -30.21  -27.99  -33.65  -31.73  CQC  Max  30.97  33.94  29.77  37.51  29.62  48.74  (-10.31%)  Avg  -1.54  -1.49  -1.33  -2.29  -2.59  -3.34  Stdev  9.79  9.79  9.69  9.48  9.73  9.85  Min  11.39  9.16  5.09  4.47  1.42  1.19  ABSSUM  Max  111.54  106.78  125.70  111.22  129.86  126.06  (33.19%)  Avg  43.03  42.69  43.26  41.08  40.48  37.11  Stdev  16.25  17.20  16.81  17.24  19.57  19.73  Min  -17.95  -17.21  -20.05  -15.25  -25.96  -25.84  Max  23.10  16.62  20.80  16.19  19.43  20.12  Avg  -1.03  -0.94  -1.13  -0.78  -1.45  -1.53  Stdev  5.19  5.10  5.01  4.67  5.04  5.43  -17.04  -19.83  -15.23  -23.76  -26.36  Min \ . -17.72 Base  CQC  Max  23.37  16.77  20.85  16.10  19.27  19.90  Moment  (0.36%) ijiliiillli^^^^Bil  Avg  -1.03  -0.94  -1.13  -0.78  -1.45  -1.53  Stdev  5.19  5.11  5.02  4.68  5.02  5.46  "Min •  2.24  1.92  1.27  1.61  0.32  1.27  Max  54.09  48.28  52.37  54.81  71.34  69.73  Avg  15.52  15.37  15.14  15.06  14.97  16.14  Stdev  8.05  7.79  7.93'  8.02  9.05  11.16  \ ABSSUM  Chapter 5  Response  Software Verification and Case Studies  Modal Combo.  Statistic ("'„ ttifO  180  Seq.#l  Seq. #2  Seq. #1  Seq. #1  Seq. #1  Seq. #1  COV  COV  COV  COV  COV  < o\  0.1  0.2  0.3  0.4  0.0  Min  -17.95  -17.21  -18.06  -20.86  -22.49  -18.88  SRSS  Max  23.11  16.62  21.04  16.22  16.66  18.48  (0.44%)  Avg  -1.03  -0.94  -1.11  -0.92  -1.12  -0.91  Stdev  5.19  5.10  5.01  4.81  4.72  4.66  Min  -17.72  -17.04  -17.87  -21.06  -22.01  -18.68  Storey 5  CQC  Max  23.37  16.77  21.11  16.09  16.39  18.16  Displ.  (0.37%)  Avg  -1.03  -0.94  -1.11  -0.92  -1.12  -0.91  Stdev  5.19  5.11  5.01  4.83  4.71  4.65  Min  2.24  1.92  1.20  2.24  0.84  0.87  ABSSUM  Max  54.09  48.28  55.12  48.39  53.13  54.43  (17.41%)  Avg  15.52  15.37  15.22  14.78  13.60  12.91  Stdev  8.05  7.79  7.84  7.62  7.87  8.46  Comparing any distribution statistic in Table 5 . 7 across a row shows that the statistic remains approximately constant at each level of structural variability. This was not unexpected since the modal combination rules are not direct functions of any type of uncertainty in the structural parameters, rather they are functions of individual peak modal responses, and in the case of the C Q C rule, a correlation coefficient that depends only on modal frequencies and damping.  Therefore, given a random structure, the  associated natural frequencies and random damping determine both the peak R H A response and R S A results for a given earthquake motion without regard to the structural variability that generated the random structure.  Although not directly influenced by  structural variability, the accuracy of each set of RSA results is certainly affected by the variation in each random structure but in a manner that is difficult to predict.  Each  unique set of natural frequencies and modal damping results in a different relationship between the magnitudes of the peak modal responses and, therefore, a different level of influence of the higher modes of response, which in turn has a direct effect on the accuracy of the R S A results.  For a certain random structural realization in which the  fundamental mode dominates the total response, the R S A results will tend to be quite accurate whereas another structural realization with more significant higher modes of  Chapter 5  Software Verification and Case Studies  181  response will tend to show larger errors in the RSA results. The difficulty in predicting the effect of a certain level of structural randomness on R S A accuracy is due to the jagged nature of the response spectrum of a single earthquake.  Individual modal  responses may be negligibly or significantly affected by large or small random variations in modal frequencies and damping, particularly at certain periods, which results in an unpredictable effect on the modal combination and hence an unpredictable effect on RSA accuracy.  Furthermore, the relationship between the accuracy of the C Q C results as  compared to the SRSS and A B S S U M results is influenced by the degree of separation between the natural frequencies for a given structure. Structures with well-separated natural frequencies will exhibit little or no difference between C Q C and SRSS peak responses with the SRSS peak response becoming relatively less accurate for more closely spaced natural frequencies.  The observed distribution statistics in Table 5.7 for the A B S S U M results, which are an upper-bound on the peak response, show an average overestimation of the peak base shear of approximately 42%, and an average overestimation of the peak base moment and peak fifth storey displacement of approximately 15%.  Clearly, the  A B S S U M modal combination rule gives a very conservative estimate of peak response. The SRSS and C Q C results for each response type across the range of structural variability were virtually identical, which indicates that the natural frequencies of the five-storey shear frame were well-separated at each level of variability.  The average  error for the peak base shear was in the range of -1.3% to -3.4%, the average error for the peak base moment was in the range of -0.8% to -1.5%, and the average error for the peak fifth storey displacement was in the range of-0.9% to -1.1%. The error is largest for the peak base shear because the higher mode responses are likely more significant relative to the first mode as compared to the peak base moment and peak fifth storey displacement (see Table 5.2). Although the SRSS and C Q C peak responses consistently underestimated the R H A peak responses in this limited study, this is not a general trend for response spectrum analysis. The variance in each set of errors was observed, as noted previously, to be approximately constant across the range of structural variability in Table 5.7, therefore, it is well represented for each response quantity by the results for the  Chapter 5  Software Verification and Case Studies  182  deterministic structure. Figure 5.17 shows the histogram plots for the SRSS and CQC errors in peak base shear, peak base moment and peak fifth storey displacement. Included with the histogram plots are the associated Normal distributions to illustrate the close agreement with the observed error distribution.  Using the Normal distributions in Figure 5.17 to represent the SRSS and CQC error distribution, the variance or standard deviation for each response quantity is a measure of the level of confidence that may be applied to an estimate of the accuracy of each modal combination rule. From the basic properties of a Normal distribution, 68.3% of the area is contained within one standard deviation of the mean, which indicates that approximately % of observed results should be no more or no less than one standard deviation different from the average value. Using this property, approximate rules-ofthumb can be established for the accuracy of the C Q C and SRSS modal combination rules for peak base shear, peak base moment and peak fifth storey displacement. Taking the mean errors in Table 5.7 to be approximately zero, the observed standard deviations indicate that A of the time the C Q C and SRSS results for peak base shear will be within 2  10% of the R H A result, while peak base moment and peak fifth storey displacement will be within 5% of the R H A result.  The preceding approximate rules-of-thumb and observations regarding the relative effect of structural variability on peak response distribution were, of course, derived from a very limited study using only one particular type of earthquake and one basic pattern of storey mass, storey stiffness, storey height and modal damping. However, they serve as an example of the type of analysis that may be done using PSResponse  to provide fast, summary-level dynamic analysis results for determining the  probabilistic response of a structure under stochastic dynamic loading.  Chapter 5  Software Verification and Case Studies  183  Base Shear 100  100  80  80  o <D 60  in  13  cr  <D LL  40 20  CM  40 20  U K . ,  U1G m  60  t-  CM  r»  to  oi  m  r-  T-  i  '  LO T-  O) *-  CO CM  0 CO £  % Error Base Moment  ^  a  g. LL  180  180  150  150  120  120 90  90  60  60 30  30  •IMfi  0 CM  i ^ t— - c o o > m T - c o r ^ T - i o c n c o  CM  co  T—  % Error Storey 5 Displacement 180  180  150  150  £F 120  120  <D  g.  90  il  60 30 0 W T CM  - f CM  -  E T-  O  O  i -  J  l  1  il O  T 1  -  C1  90 60 30 l i t * . . O  I  v  -  T-  lO -  0 O) -  CO CM  CO  g  % Error Figure 5.17: Error Distribution for CQC & SRSS vs. RHA Peak Responses  Chapter 5  5.3.2  Software Verification and Case Studies  184  Case Study #2  To assess the appropriateness of the well-known equal displacement observation in structural dynamics and the effect of hysteresis model properties on displacement response, a group of single-degree-of-freedom  structures with eight different natural  periods and 2% damping were subjected to a sequence of 1000 generated records, again using the E l Centro ground motion as the seed, using six different hysteresis models. The natural periods of the structures were selected from the deformation and acceleration spectra for the E l Centro earthquake to cover a reasonably wide range of periods that might be encountered in an actual structure. The periods that were selected were; 0.1, 0.2, 0.4, 0.5, 0.6, 1.0, 3.0 and 8.0 seconds, which are shown on the response spectra in Figure 5.18.  The hysteresis models were derived from experimental data taken from a  cyclic lateral displacement-controlled test of a Parallam® column, which was connected to a rigid base with special hollow steel tubes acting as dowels (Ruxton 2003). The cyclic test data was analyzed using the hysteresis parameter identification feature of PSResponse,  which identifies the best hysteresis parameter set based on a least-squares  error algorithm. The displacement and force time-histories of the cyclic test as well as the experimental hysteresis loop and corresponding best-fit model are shown in Figure 5.19.  Figure 5.18: El Centro Response Spectra  Chapter 5  Software Verification and Case Studies  185  Note from Figure 5.19 that the best-fit model is a compromise between two regions of pinching and stiffness recovery in one direction of loading for the large amplitude displacement cycles.  The large displacement cycles begin as usual with a  Chapter 5  Software Verification and Case Studies  186  pinched region at the beginning of the loading cycle followed by stiffness recovery and a stiffness plateau but this is followed by another region of increasing stiffness in the same half-cycle of loading. This unusual behaviour does not fit well with the behaviour of the modified B W B N hysteresis model that was incorporated into PSResponse, therefore, the hysteresis parameter identification algorithm identified the best-fit compromise between the two regions of stiffness recovery.  The hysteresis models that were derived from the best-fit model of Figure 5.19 are listed in Table 5.8. These models were chosen to represent a broad range of possible inelastic behaviour exhibited by structures with different yield strengths and rates of structural deterioration. The different characteristics of each model were achieved by simply altering the appropriate best-fit hysteresis parameters that had been identified by PSResponse. To illustrate the behaviour of each hysteresis model, Figure 5.20 shows the first five seconds of deformation response to the E l Centro ground motion for a singledegree-of-freedom structure with T = 8.0 seconds and 2% damping for each of the six models.  Table 5.8: Hysteresis Model Descriptions Model  Yield Displacement (in) 0.060  A  Best-fit hysteresis model from test data.  B  Best-fit hysteresis model with degradation parameters doubled.  0.060  C  Best-fit hysteresis model with degradation parameters quadrupled.  0.060  D  No degradation - 100% yield strength of Model A.  0.060  E  No degradation - 50% yield strength of Model A.  0.030  F  No degradation - 25% yield strength of Model A.  0.015  Having established a number of hysteresis models to simulate a range of inelastic behaviour, the deformation response of a single-degree-of-freedom  structure was  determined at the eight natural periods noted previously using the same sequence of 1000 generated earthquake records for each period and hysteresis model combination. Table  Chapter 5  Software Verification and Case Studies  187  5.9 lists the basic statistics for each set of peak deformation results as well as the statistics for the peak deformation of an elastic structure at each period.  El Centra Response (0-5 sec)  IBMHMmN  15  10  z  ' 'it  -10  -15  -0 1  -0 05  0 05  0 1  0 15  02  0 25  Blililiii^^H I  HystA  HystB  HystC  •HystD  HystE  Hyst F i  Figure 5.20: Hysteretic Response of SDOF Structure with T = 8.0 sec, Damping = 2%  Table 5.9: SDOF Peak Deformation Response Statistics (m)  lllB  Statistic  Elastic  Hyst A  Hyst B  Ihstt  Hyst D  Hvst E  Hyst F  Min.  0.00083  0.00082  0.00082  0.00082  0.00082  0.00084  0.00082  0.1  Max.  0.00236  0.00225  0.00225  0.00227  0.00222  0.00223  0.00220  sec.  Avg.  0.00141  0.00140  0.00140  0.00140  0.00140  0.00140  0.00140  Stdev.  0.00024  0.00023  0.00023  0.00023  0.00023  0.00023  0.00023  Min. ,"  0.00457  0.00461  0.00461  0.00462  0.00460  0.00160  0 00127  0.2  Max.  0.02312  0.01526  0.01557  0.01434  001656  001447  0.01469  sec.  Avg.  0.01012  0.00883  000877  000864  0.00894  0.00841  000-96  Sldev.  0.00252  0 00176  0.00174  0.00169  000181  0 00158  0 00142  Chapter 5  Software Verification and Case Studies  188  1 las tic  Hyst A  ll\sl 1$  Hyst C  Hyst D  Hyst E  Hyst V  Min.  0.01784  0.01439  0.01438  0.01400  0.01468  0.01324  0.01206  0.4  Max.  0.08166  0.05231  0.05515  0.05547  0.05180  0.04943  0.06213  sec.  Avg.  0.03709  0.02910  0.02910  0.02932  0.02919  0.02842  0.03114  Stdev.  0.00928  0.00539  0.00544  0.00556  0.00540  0.00529  0.00722  Mm  0 02416  0 02116  0 02100  0 02033  0.02116  0.01919  0.01882  Max  0 11331  0 06816  0 068?2  0 07154  0.06709  0.07387  0.09819  \\»  0 05788  0 04141  0 041 *8  004151  0.04149  0.04033  0.04548  Stde\  0 01540  0 00772  0 00783  0 O0S24  0.00758  0.00814  0.01251  Min.  0.03165  0.02344  0.02300  0.02343  0.02462  0.02462  0.02226  0.6  Max.  0.15964  0.09256  0.09318  0.10498  0.09295  0.10506  0.13715  sec.  Avg.  0.07457  0.05124  0.05121  0.05153  0.05128  0.05090  0.05841  Stdev.  0.02054  0.01039  0.01063  0.01136  0.01023  0.01238  0.01824  Mm  003881  0 03608  0.0  0.03541  0 03559  003524  0 03503  1.0  Max  0.25019  015670  0.16267  0 17988  0.15322  0.21269  0 340~4  sec  A\g.  0 11296  0 07759  0 0^748  0.07848  0 07799  0 08316  0.10252  Stdev.  0.03412  0.01984  0 02001  0.02139  0 01994  0.02628  0.03952  Min.  0.06733  0.05492  0.05501  0.05435  0.05706  0.06479  0.07339  3.0  Max.  1.11230  0.81400  0.82257  0.82161  0.82328  1.05070  1.28130  sec.  Avg.  0.34202  0.21224  0.21426  0.22266  0.21450  0.25422  0.30196  Stdev.  n 153^0  0.09276  0.09787  0.10225  0.09410  0.12333  0.15232  Mm.  0 09298  0.07416  0.07930  0.07801  0.07947  0.08089  0.07906  8.0  Max  1.30990  2.53050  2.53110  3.04320  2.51870  3.24960  3.7210O  sec.  A vn  0 42480  0.43666  0.45300  0.50538  0.43297  0.56794  0.71517  ;-\ Stdev.' ' ' 0.2018"  0.28548  0.30441  0.34712  0.28002  0.39477  0.50414  T  05 set.  •iiiiiiiB  Statistic  • -  MA\  The actual frequency distributions corresponding to each set of calculated statistics in Table 5.9 are given as histogram plots in Figure 5.21 - Figure 5.28. These plots are once again overlaid with the associated Gumbel distribution, given as a solid line, and Normal distribution, given as a dashed line, to illustrate the fit of each type of distribution to the data.  Again note that the discontinuity in some of the Normal  distributions is the result of lumping together all responses below the point of the discontinuity.  Chapter 5  Software Verification and Case Studies  Hysteresis Model A  189  Hysteresis Model B T 120  I, so ff c  <!> 'Ui'.'  A ill  A'-  I  iiiiiiiiiniiiin^u Hysteresis Model C  N  _  .Hysteresis Model'D  , -'.-'80  ff  [. c  mm  g. -!= J)  IlSw 20 !  1J4*. i-  x10'm  r- *~  r-  x10  v  4  m  Figure 5.21: Displacement Response Histograms, T = 0.1 sec, Damping = 2%  N  N  («)  IN N  N  IN  Chapter 5  Software Verification and Case Studies  Hysteresis Model A  190  Hysteresis Model B ..140  -  ,,140  120  ' 100  K  I-  ftrt  I1.IJ  1  <  ill Ift  O  N  Willie  x10" m  Hysteresis Model C  Hysteresis Model D  A  §  80  if  60  .»u -'• .ST \  5  ti  jllllllllllllk  _  "20*  •<> Hysteresis Model E  - xicr m  Hysteresis Model F  x10-m  Figure 5.22: Displacement Response Histograms, T = 0.2 sec, Damping = 2%  lIIPIPISI  Chapter 5  Software Verification and Case Studies  Figure 5.23: Displacement Response Histograms, T = 0.4 sec, Damping = 2%  191  Chapter 5  Software Verification and Case Studies  Figure 5.24: Displacement Response Histograms, T = 0.5 sec, Damping = 2%  192  Chapter 5  Software Verification and Case Studies  Figure 5.25: Displacement Response Histograms, T = 0.6 sec, Damping = 2%  193  Chapter 5  Software Verification and Case Studies  Hysteresis Model A '  Hysteresis Model B  200  ffiipit  194  200  A  180 160  160  140  140  tr ' »  .140  .120  c  -' 120 •  • f i q p  100  so  • *>• 60  k... M" -.tO ^** 5  CD <s> <si a)  xio'mjr*  „"* .-O <0T  "40"*',  40  20  20  oi  "a  Nt' ,  b 3 p s>  i x10'm  Hysteresis Model C  •HI  Hysteresis Model D  M  N N  x1(T nt  111111  Hysteresis Model E  H/sterasii Moael F  160 140  120 l i l i  §§J|| 100 G  80  .2 Li.  mi iiiiii  •  80 40 20  o «$^c»j »  <q (vi «q *r o « <i  S 8 S 8  xlirm- .  Figure 5.26: Displacement Response Histograms, T = 1.0 sec, Damping = 2%  ...  J  Chapter 5  Software Verification and Case Studies  Figure 5.27: Displacement Response Histograms, T = 3.0 sec, Damping = 2%  195  Chapter 5  Software Verification and Case Studies  Figure 5.28: Displacement Response Histograms, T = 8.0 sec, Damping = 2%  196  Chapter 5  Software Verification and Case Studies  197  From Figure 5.21 - Figure 5.28 it is apparent that the Gumbel distribution is again a better description of the peak deformation data in each case than the Normal distribution. Therefore, using the Gumbel distribution as a basis for comparison, Figure 5.29 summarizes the individual distributions at each period, including the distribution of peak elastic deformations, to illustrate the effect of the different hysteresis models and their agreement with the elastic distribution across the range of natural periods.  Beginning with the question of the effect of hysteresis model properties on peak displacement response, a number of observations can be made from Figure 5.29 and Table 5.10, which lists the dispersion parameter and mode describing the Gumbel distribution for each set of results. Note that the dispersion parameter is analogous to the variance of a Normal distribution while the mode is the location of the peak in the distribution. Comparing only the hysteresis models with each other, the modes of each distribution are located at essentially the same peak displacement for a given natural period.  There was a tendency for the modes of the lower yield strength models,  Hysteresis E and Hysteresis F, to occur at slightly higher peak displacements, especially at the longer natural periods with their larger peak displacements, but this tendency was relatively small. At natural periods of 0.1 seconds and 0.2 seconds there was negligible difference between the hysteresis models because the peak displacements did not reach yield for any of the models.  With increasing peak displacements at the longer natural  periods, the dispersion of the distributions for the lower yield strength models and Hysteresis C, with degradation parameters quadrupled, was increased with respect to the other models, particularly for Hysteresis F with 25% yield strength.  This tendency was  not particularly significant for Hysteresis C but there were definite differences from Hysteresis A and B, which had smaller degradation parameters. Only at the large peak displacements of T = 3.0 seconds and T = 8.0 seconds did the distribution of Hysteresis B, with degradation parameters doubled, become noticeably different from Hysteresis A. At all natural periods, the distributions of Hysteresis A and D, which represented 100% yield strength with and without degradation parameters, were negligibly different.  Chapter 5  Software Verification and Case Studies  198  T=0 2 s e c  Figure 5.29: Hysteresis Model Comparison of Peak Deformation Distributions  Chapter 5  Software Verification and Case Studies  199  Table 5.10: Gumbel Distribution Parameters for Peak Deformation  HPfli 0.1 sec.  0 2 sec.  0.4 sec.  0 5 sec.  0.6 sec.  1 0 set.  3.0 sec.  8 0 set-  Statistic  Hist \  Ihsl K  H>st (  Hyst D  II>sl F.  Hyst F  Dispersion  5479.1890  5501.9051  5460.1337  5531.1127  5603.9804  5623.0013  Mode  0.0013  0.0013  0.0013  0.0013  0.0013  0.0013  Dispcision  728.5579  737.2292  •'60 2819  708.0143  812.1206  905.2658  Mode  0.0080  0.0080  0.0079  0.0081  0.0077  0.0073  Dispersion  238.0663  235.7321  230.5929  237.6564  242.4594  177.7142  Mode  0.0267  0.0266  0.0268  0.0268  0.0260  0.0279  Dispersion  166.0894  163.7728  155.7097  169.1176  157.5203  102.520.5  Mode  0.0379  0.0379  0.0378  0.0381  0.0367  0.0398  Dispersion  123.4556  120.6492  112.9021  125.3194  103.6185  70.3339  Mode  0.0466  0.0464  0.0464  0.0467  0.0453  0.0502  Dispersion  64 6310  64 lO."  59 9523  64.3189  48 7978  32 4550  Mode  0 008"  0 068 5  0 0689  0 0690  0 0~13  0 0S4  Dispersion  13.8265  13.1043  12.5436  13.6291  10.3990  8.4201  Mode  0.1705  0.1702  0.1766  0.1721  0.1987  0.2334  4 4926  4 2133  3 6948  4.5802  '•'.3.2489  0.3082  0.3160  0J492  0.^069  0.391)3  ' il'Hl  Mode  -7  2.5440 " 0.4883  To enable a quantitative comparison of the effect of each hysteresis model on the distribution of peak deformation response at each natural period, the fitted Gumbel distributions of Figure 5.29 were used to evaluate the reliability index (3 of each hysteresis model associated with structural drift limits of; 0.5%, 1%, 2% and 4%, respectively. This type of analysis is an example of the intended purpose of PSResponse, which is to enable the evaluation of structural reliability under earthquake loading. The displacement limits corresponding to the chosen drift limits were determined from a structural height that was calculated by assuming that the experimental lateral stiffness data of Figure 5.19 was taken from a rigid beam portal frame structure with 0.2 m x 0.2 m timber columns. The resulting frame height of 5.34 m gives displacement  Software Verification and Case Studies  Chapter 5  200  limits of; 0.027 m, 0.053 m, 0.107 m and 0.214 m, respectively, for the chosen drift limits. The performance function for this reliability analysis may then be expressed as:  G = X-x  [1]  where X is the displacement limit and x is the peak displacement random variable. The probability of failure and the associated reliability index may then be simply determined as:  p =P{G<o)=p(x>x)  [2]  f  /? = - « » - ( p )  [3]  ,  /  Evaluation of the exceedence probability in Eq. [2] was done using the cumulative distribution function of the Gumbel distribution given by:  F (x) = exp[- exp(- a(x - b))]  [4]  x  where a is the dispersion of the distribution and b is the mode of the distribution. Table 5.11 lists the calculated probabilities of exceeding each displacement (drift) limit for each hysteresis model at each natural period and Table 5.12 shows the reliability indices associated with the calculated exceedence probabilities.  Table 5.11: Displacement (Drift) Limit Exceedence Probabilities  r  0.1 sec  0.2 sec  1 iinit  Elastic  Hyst A  ll\st Ii  0.027 m  0  0  0  0.053 m  0  0  0.107 m  0  0.214 m  llw(  Hyst D  Hyst fc  Hyst F  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  n.iij- in  0.0001  0  0  0  0  0  0  0 053 m  0  0  o .*  0  0  . 0  0  0.107 m  0  0  0  0  0  0  0  0 214m  0  0  0  ((  0  0  0  '  Chapter 5  T  0.4 sec  0.5 sec  0.6 sec  1.0 sec  3.0 sec  8.0 sec  Software Verification and Case Studies  201  1 imil  1 lasln.  II\st \  Hyst It  ll\st(  Il\sl 1)  II\sl 1  Hyst F  0.027 m  0.8960  0.6035  0.6015  0.6167  0.6111  0.5474  0.6902  0.053 m  0.0604  0.0019  0.0020  0.0024  0.0020  0.0014  0.0115  0.107 m  0  0  0  0  0  0  0  0.214 m  0  0  0  0  0  0  0  0.027 m  0.9994  0 9979  0.9973  0 9954  0 99S5  0 9898  0.9761  0 053 m  0.5697  0J0787  0 0803  0.0895  OWO  0.0735  0.2287  n le m  0 0094  . .0  0  0  •0  0  0.0010  0 214m  "  0  0  0  0  0.027 m  1  1  1  0.9999  1  HBBI  0.053 m  0.8846  0.3634  0.3639  0.3786  0.107 m  0.0714  0.0006  0.0007  0.214 m  0.0001  0  0  0.027 m  1  1  0 053 in  0.9952  0 9361  0 9326  i'.I""" m  0 5046  0.0805  0214m  0 0125  0.027 m  -  0.9987  0.9940  0.3642  0.3634  0.5601  0.0011  0.0005  0.0017  0.0182  0  0  0  0  1  0.9998  0.9985  0 9248  0.9393  0.9134  0.9393  0.0812  0.0966  0 0832  0.1609  0.3846  0 0001  0 0001  0.0002  0.0001  0.0009  0.0150  0.9996  0.9993  0.9985  0.9985  0.9993  0.9974  0.9966  0.053 m  0.9982  0.9938  0.9904  0.9911  0.9937  0.9894  0.9896  0.107 m  0.9822  0.9098  0.8987  0.9089  0.9120  0.9254  0.9449  0.214 m  0.8064  0.4219  0.4307  0.4652  0.4318  0.5739  0.6920  in  0 9991  0.9709  0.9659  0.9627  0.9728  0 9614  0.9606  0 053 m  0.9974  0.9570  0 9516  0.9496  0.9592  0.9498  0.9515  0.107 m  0 9854  0.9153  0.9104  0.9134  0 9T8  0^9187  0.9285  0 214m  0.8826  0.7828  0 7850  0.8075  0.7836  0.8302  0.8659  H\sK  Uyst D  Hyst E  Hvst F  Table 5.12: Displacement (Drift) Limit Reliability Indices T  0.1 sec  0.2 sec  Limit  1 IjstlL  Hssl \  Hsst Ii  0.027 m  8  8  8  8  8  8  8  0.053 m  8  8  8  8  8  8  8  0.107 m  8  8  8  8  8  8  8  0.214 m  8  8  8  8  8  8  8  0.027 m  3 7086  4.7531  4.7946  4 S966  4.6621  5.1166  5.5062  0 053 m  6.2656  7.7  161  7 7706  8  rf#59,16  0.107 m  8  8  8  8  s  8  8  0 214m  8  8  8  8  8  8  S  ;  *f"8  8  Chapter 5  1  0.4 sec  0.5 sec  0.6 sec  1.0 sec  3.0 sec  8.0 sec  Software Verification and Case Studies  202  Limit  1.Instil  H\sl \  ll\st Ii  H\st(  ll\st 1)  H>st K  Hyst h  0.027 m  -1.2590  -0.2625  -0.2573  -0.2969  -0.2821  -0.1192  -0.4964  0.053 m  1.5515  2.8951  2.8777  2.8222  2.8853  2.9784  2.2743  0.107 m  3.9710  5.7328  5.7020  5.6243  5.7238  5.8180  4.8026  0.214 m  6.6620  8  8  8  8  8  7.7571  0.027 m  -3.2187  -2.8584  -2.7823  -2 6016  -2.9094  -2.3180  -1.9785  0.053 m  -0 P56  1.4139  1.4031  1 3439  1.4255  1.4505  0.7433  0.107 m  2.3511  -4.2555  4.2227  4.0973  4.2970  4.1672  3.0835  M.'14 111  4.7051  7.2567  7.2032  7 0082  7 3254  7.0775  5.5136  U.U27 m  -4.1384  -4.1923  -4.0139  -3.6553  -4.3232  -3.0221  -2.5111  0.053 m  -1.1985  0.3493  0.3481  0.3091  0.3474  0.3494  -0.1513  0.107 m  1.4652  3.2512  3.2074  3.0701  3.2790  2.9334  2.0915  0.214 m  3.7377  5.9899  5.9157  5.6982  6.0381  5.4469  4.2668  0.027 m  -4.8320  -4.9423  -4.8473  -4.43"1  -4.9719  -3.5885  -2.9712  0.053 ni  -2.5923  -1.5230  -1.4956  -1.4380  -1 5489  -1.3621  -1.5486  0.107 m  -0.0115  1.4019  1.3974  1.3012  1.3836  0.9906  0.2934  0.214 m  2 2412  3. ^652  3.7489  3.5886  3.7480  3.1064  2.1713  0.027 m  -3.3665  -3.1968  -2.9769  -2.9777  -3.1846  -2.7980  -2.7072  0.053 m  -2.9149  -2.4980  -2.3416  -2.3677  -2.4968  -2.3058  -2.3124  0.107 m  -2.1006  -1.3396  -1.2741  -1.3339  -1.3529  -1.4422  -1.5975  0.214 m  -0.8647  0.1971  0.1745  0.0873  0.1718  -0.1862  -0.5014  0 027 m  -3.1261  -I 8941  -1.8240  -1 7826  -1.9236  -1.7674  -1.7574  0.053 m  -2.7965  -1.7171  -1.6607  -1.6406  -1 7419  -1.6428  -1.6597  0.107 m  -2.1814  -1.3743  ' -1.3432  -1.3621  -1.3905  -1.3966  -1.4646  0.214 m  -1.1883  -0.7815  -0.7890  -0.8688  -0.7844  -0.9549  -1.10"3  1  Examining the reliability indices of Table 5.12 reveals three key trends in the drift exceedence probabilities.  Firstly, for each drift limit at each natural period the elastic  response gives a conservative estimate of drift reliability as evidenced by the smaller p value compared to each of the six hysteresis models.  Secondly, the P values for the  hysteresis models vary significantly at a given natural period and drift limit. Thirdly, the range of variability in the p values for the hysteresis models is dependent on the natural period and chosen drift limit. For example, the reliability index for T = 1.0 seconds and  Software Verification and Case Studies  Chapter 5  203  1% drift (0.053 m) ranged between -1.3621 and -1.5489, while for 2% drift (0.107 m) at the same natural period, the reliability index ranged between 1.4019 and 0.2934.  The latter two key observations indicate that the characteristics of a hysteresis model have a significant effect on the calculated seismic reliability of a structure, with the effect being more or less pronounced depending on the capacity limit that is used to assess seismic reliability. Taken together, these observations show that the hysteretic behaviour of a structure needs to be accurately modeled, particularly in shorter natural period structures, to provide an accurate probabilistic description of response and hence a good estimate of seismic structural reliability.  The natural period dependence of the  relative importance of an accurate hysteresis model can be seen in the basic equation of motion for a structure, which can be written as follows:  ..  .  F{x)  ..  [5]  m  From that form of the equation of motion it is clear that the relative importance of the restoring force F(x) is reduced with increasing mass for a given structural stiffness. Therefore, the shape of the hysteresis loop for longer period structures is relatively less important when determining the probabilistic response of the structure.  Turning to the question of the appropriateness  of the equal displacement  observation in structural dynamics, it is clear from Figure 5.29 that the distributions of elastic peak displacement differ from the inelastic distributions for all natural periods exhibiting a significant response. In general, the modes of the inelastic distributions are located at a lower peak displacement than the modes of the corresponding elastic distributions, which were close to the spectral displacement for the E l Centro ground motion at each period (see Fig. 5.18).  This observed trend suggests that the elastic  response of a structure tends to give a conservative estimate of peak inelastic displacement, which was noted previously in the analysis of drift reliability. To enable evaluation of this observed conservativeness and identify how it changes with natural  Software Verification and Case Studies  Chapter 5  204  period and relative yield strength, the percentage difference between the elastic and inelastic peak displacement for each record in the sequence of 1000 generated ground motions was determined at each natural period using three hysteresis models; Hysteresis D, Hysteresis E and Hysteresis F. These represent non-degrading hysteretic behaviour with 100% yield strength, 50% yield strength and 25% yield strength, respectively. Determining the probabilistic description of the percentage difference between peak elastic and peak inelastic displacement then allows for a reliability analysis of the conservativeness of the equal displacement principle.  Table 5.13 lists the basic statistics for the distribution of each set of percentage difference results and Figure 5.30 shows the histogram plots of the actual frequency distribution of the results for two natural periods. These plots are once again overlaid with the associated Gumbel distribution, given as a solid line, and Normal distribution, given as a dashed line, to illustrate the fit of each type of distribution to the data.  Table 5.13: Percentage Difference Statistics for Inelastic vs. Elastic Peak Deformation Period  0.1 sec.  0.2 sec.  0.4 sec.  0.5 sec.  Statistic  Hyst D  HystE  Hyst I  Min.  -25.64  -25.23  -28.60  Max.  20.45  33.41  34.40  Avg.  -0.46  -0.42  -0.33  Stdev.  4.91  6.34  7.84  Min.  -33.56  -44.89  -55.61  Max.  16.75  22.40  31.25  Avg.  -10.58  -15.28  -19.14  Stdev.  7.40  10.24  13.50  Min.  -52.25  -58.31  -60.70  Max.  31.69  53.06  123.04  Avg.  -19.17  -20.45  -12.08  Stdev.  13.35  17.88  26.25  Min.  -59.69  -67.54  -70.91  Max:  23.38  66.37  114.47  Avg.  -25.62  -27.02  -17.11  Stdev.  14.60  18.71  28.47  Software Verification and Case Studies  Chapter 5  Period  0.6 sec.  1.0 sec.  3.0 sec.  8.0 sec.  205  Statistic  Hyst I)  HystE  Hyst F  Min.  -65.73  -68.49  -68."  Max.  43.85  54.51  140.59  Avg.  -28.39  -28.49  -17.57  Stdev.  15.48  19.82  29.18  Min.  -62.40  V-69.41  -69.26  Max.  44.25  82.71  224.70  Avg.  -28.06  -22.80  -4.36  Stdev. .  17.06  25.06  38.82  Min.  -79.88  -76.08  -80.76  Max.  162.25  269.27  451.71  Avg.  -31.63  -17.62  -0.34  Stdev.  28.61  42.79  58.15  Min.  -67.94  -64.75  -73.19  Max.  211.63  384.39  430.45  Avg.  • 1.55 _  32.84  68.16  Stdev.  37.10  58.93  80.80  From Figure 5.30, the Gumbel distribution once again appears to be a better description of the percentage difference data than the Normal distribution, therefore, using the Gumbel distribution as a basis for comparison, Figure 5.31 summarizes the individual distributions of each set of percentage difference results for each hysteresis model at each natural period. Also, Table 5.14 lists the dispersion parameter and mode describing the Gumbel distribution for each set of results.  Table 5.14: Gumbel Distribution Parameters for Percentage Difference Results Period 0.1 sec.  0.2 sec.  0.4 sec.  Statistic  Hyst 1)  Hyst E  Hyst F  Dispersion  0.26  0.20  0.16  Mode  -2.67  -3.27  -3.86  Dispersion  0.17  0.13  0.09  Mode  -13.91  -19.89  -25.22  Dispersion  0.10  0.07  0.05  Mode  -25.18  -28.49  -23.89  Chapter 5  Software Verification and Case Studies  Period 0.5 sec. 0.6 sec. 1.0 sec. 3.0 sec. 8.0 sec.  206  Statistic  H>stD  Hyst £  HystF  Dispersion  0.09  0.07  0.05  Mode  -32.19  -35.45  -29.92  Dispersion  o.os  0.06  0.04  Mode  -35.36  -37.41  -30.70  Dispersion  0.08  (J.U5  0.03  Mode  -35.74  -34.08  -21.83  0.04  0.03  0.02  Mode  -44.50  -36.88  -26.51  Dispersion  0.03  0.02  0.02  Mode  -15.15  6.32  31.79  [>l-.piT<IUII  Using the fitted Gumbel distributions of Figure 5.31, the probability that the percentage difference between peak inelastic and peak elastic displacement exceeded 0% was evaluated for each hysteresis model (i.e. yield level) at each natural period. This then allows for a reliability estimate of the conservativeness of the equal displacement principle.  Note that a negative percentage difference indicates that the peak elastic  displacement exceeded the peak inelastic displacement, thereby giving a conservative estimate of peak displacement. Table 5.15 lists the calculated exceedence probabilities for each hysteresis model at each natural period.  Table 5.15: Zero Percent Difference Exceedence Probabilities 1-0. L s  1=0.2 s  1=0.4 s  T=0.5 s  1=0.6 s  1=1.0 s  1=3.0 s  1=8.0 s  HystD  0.3924  0.0858  0.0853  0.0575  0.0520  0.0659  0.1272  0.4469  HystE  0.4029  0.0795  0.1215  0.0843  0.0850  0.1603  0.2818  0.6826  HystF  0.4127  0.0871  0.2674  0.2287  0.2285  0.3850  0.4272  0.8092  Chapter 5  Software Verification and Case Studies  Figure 5.30: Percentage Difference Histograms for Inelastic vs. Elastic Peak Deformation  207  Chapter 5  Software Verification and Case Studies  Figure 5.31: Percentage Difference Distributions for Inelastic vs. Elastic Peak Deformation  208  Chapter 5  Software Verification and Case Studies  209  Examining the Gumbel distributions of Figure 5.31 and the associated reliability indices of Table 5.16 it is clear that the peak elastic displacement of a structure generally gave a conservative estimate of peak inelastic displacement (elastic > inelastic).  This  trend was consistent except for the longer natural period structures (T = 3.0 sec, T = 8.0 sec) with reduced yield strength (Hyst E , Hyst F) where very large peak displacements, which exceeded peak ground displacement by a factor of 2 to 3, were commonly observed (see Fig. 5.18 and Fig. 5.29). Also, for a given natural period the probability that peak elastic  displacement  gave a conservative  estimate of peak inelastic  displacement (elastic > inelastic) was reduced as yield strength decreased. This is clearly evident from the increasing dispersion in the percentage difference distributions as yield strength decreased from 100% to 50% to 25% in Hysteresis D, Hysteresis E and Hysteresis F, respectively.  The degree of dispersion consistently increased with  increasing peak displacements as natural period increased.  While it is difficult to make definitive statements regarding the accuracy of the equal displacement principle owing to the wide spread in percentage difference results in Figure 5.31, the preceding observations indicate that the equal displacement principle is generally valid in the sense that elastic peak displacement provides a useful, conservative first approximation of inelastic peak displacement.  Chapter 6  CHAPTER  Summary, Conclusions and Recommendations  210  6  SUMMARY, CONCLUSIONS AND RECOMMENDATIONS  6.1  SUMMARY  The evaluation of seismic reliability of building structures is a complex and computationally expensive process since it requires, at the most fundamental level, the evaluation of the probabilistic dynamic response of a given structure to the stochastic dynamic action of an earthquake. Because of the difficulty of determining the response of a structure in a statistical sense, past estimates of the seismic reliability of existing structures, and typical structural systems, have been largely qualitative in nature. With the movement of many national building codes towards more performance-based design measures, a need was identified for a more quantitative method of evaluating structural reliability under seismic loading.  The overall objective of this study was then to develop a simple, useable software application for probabilistic analysis of the dynamic response of civil engineering structures to random ground motions. Knowing the probabilistic response of a structure, an accurate assessment of a specific reliability measure can be made from the probability of exceeding a chosen threshold. Using this approach, the aim was to provide a tool for engineers and researchers that could be used to evaluate the probable effect of a wide range of ground motion characteristics and structural model parameters, each with their own random nature, on the dynamic response of a structure.  To begin the process of developing the software application, a comprehensive review of random vibration methods and numerical procedures was carried out to identify a suitable method of analyzing the probabilistic seismic response of civil engineering structures.  This review, which included all key research related to the mathematical  Chapter 6  Summary, Conclusions and Recommendations  211  modelling of probabilistic structural behaviour done over the past 50 years, assessed the limitations of each method with regard to the level of complexity in the structural model, degree of non-linearity in system restoring forces and nature of the random excitation process. To provide an accurate, robust and practical means of evaluating structural reliability under seismic loading, the chosen procedure had to allow for highly non-linear response behaviour, realistic stochastic structural models with multiple degrees of freedom, and realistic earthquake motions.  These requirements tended to eliminate all  the frequency-domain based analytical random vibration methods, with the exception of the Equivalent Linearization method, because they are too restrictive in their inherent assumptions to confidently apply their results to real structures experiencing realistic earthquakes.  The Markov-based methods assume a white noise excitation, which is a poor representation of a real earthquake spectrum, and most of them (Galerkin method, Finite Element method, and the Closure Technique) have the disadvantage of slow convergence or large computational requirements for highly non-linear or multiple-degree-offreedom systems. In the case of the Numerical Diffusion method, the applicability of Generalized Cell Mapping (GCM) to degrading hysteretic systems, which characterizes most structures, is not known.  The Perturbation method and the Functional Series Representation method are not confined to a white-noise assumption for the excitation and are even applicable to multiple-degree-of-freedom systems.  However, these methods are limited to weakly  non-linear systems and as such are not applicable to ductile, hysteretic structural systems.  The Equivalent Linearization method has been successfully applied to response and damage prediction of a variety of highly non-linear structural systems under seismic excitation. However, response results may tend to be underestimated using this method when the excitation spectral content is such that the power spectral density function vanishes rapidly as the frequency goes to zero, which is typical of earthquake excitation. In addition, the assumption of a Gaussian input excitation results in an assumed Gaussian  Chapter 6  Summary, Conclusions and Recommendations  212  response of the linearized non-linear system. This assumption is not correct for a nonlinear system, which is known to have a non-Gaussian response to a Gaussian input. The result of an assumed Gaussian response is that it may significantly misrepresent the frequency of high response levels to extreme loads, which contribute most to firstpassage and fatigue failures.  With these limitations in mind, the decision was made to forego the efficiency of the frequency-domain based methods in favour of a robust numerical time-history approach incorporating the Monte Carlo method.  This approach, while  more  computationally demanding than the analytical procedures, allows the probabilistic response of a structure to be evaluated without regard to the degree of non-linearity in the restoring force, complexity of the structural system, nature of the variability in structural properties or nature of the random excitation process.  The decision to adopt a Monte Carlo, time-history approach for determining the probabilistic seismic response of a structure required that several types of component models be incorporated into the overall architecture of the software, which was named PSResponse.  Models for generating and modulating artificial ground motion time-  histories, structural models along with a means of simulating a non-linear, hysteretic restoring force, as well as an overall numerical time-stepping method to solve the differential equation of motion were linked together to form the computational foundation of the software.  In addition, the overall framework of PSResponse required  algorithms for Fourier analysis and power spectrum estimation, a frequency filtering algorithm to ensure that input ground accelerations were truly representative of a real earthquake, a long period random number generator to ensure a reliable source of random numbers essential for Monte Carlo analysis, and algorithms for solving the eigensystem representing the natural frequencies and mode shapes of a multiple-degree-of-freedom structure. In total, the computational engine of PSResponse consists of approximately 79 algorithms linked together in an object-oriented framework.  Chapter 6  Summary, Conclusions and Recommendations  213  The ground motion models that were incorporated into PSResponse are wellknown methods that reproduce the probabilistic characteristics of a specified frequency spectrum while the hysteresis model is a version of the well-known B W B N , or BoucWen, hysteresis model that was specifically modified for this software application. The B W B N model, which uses a smooth differential equation to represent the non-linear component of the restoring force in the equation of motion, was chosen over the use of piece-wise linear equations because it is able to reproduce a wide variety of hysteresis shapes, including pinching and degradation behaviour, without the use of numerous empirical rules governing the relationship of stiffness to displacement. It also allowed for the development of an algorithm within the software that identifies the parameters governing hysteretic behaviour from experimental data provided by the user. Thirteen separate parameters must be identified in the original B W B N model, although two of the parameters are typically set to unity, which is a computationally demanding process since system identification problems rapidly increase in difficulty as the number of parameters increases. To simplify the process, the B W B N model was modified to reduce the number of parameters controlling pinching behaviour from six to three using the assumption that overall structural hysteretic pinching begins at or very near zero restoring.force in each loading cycle. With this modified pinching function, the role of each of the three new parameters is more easily understood than the relationship between the six parameters of the original pinching function.  One parameter controls the overall rate of increase in  pinching as damage cycles progress, a second parameter controls the rate of stiffness recovery throughout the loading phases of each cycle and a third parameter controls the rate of stiffness recovery during the increasing displacement portion of each loading phase. Incorporation of the modified B W B N hysteresis model into the numerical,timestepping procedure required that a separate numerical solution algorithm be linked with the Newton-Raphson iteration scheme in the Newmark Method since the first-order, non-linear ordinary differential equation of the B W B N model has no exact solution.  Following development of the solution algorithms and numerical components that form the computational engine of PSResponse, a Windows user-interface was developed to provide easy access to the software and ensure the integrity of the input data prior to  Chapter 6  analysis.  Summary, Conclusions and Recommendations  214  The user-interface is based on a wizard manager architecture that guides the  user through a sequence of input and output dialog boxes that depends on the type of analysis selected. A wizard manager algorithm, which acts as the link between the dialog boxes and the computational framework, determines which of the 26 dialog boxes are required,  passes information between dialog  boxes,  passes  input  data to  the  computational algorithms and stores both input data and output arrays.  The final phase in the development of PSResponse was verification of the accuracy of a calculated dynamic response against a reliable benchmark, which was chosen to be an accepted  SAP2000  Nonlinear 8.1.2 (Computers & Structures Inc. 2003) because it is  industry standard software  package  sophisticated non-linear time-history analysis.  capable  of performing highly  As a further verification check, elastic  and inelastic analysis results from each software application for both single-degree-offreedom and multiple-degree-of-freedom structures were compared with corresponding results published by Chopra (1995). This comparison showed good agreement between each set of calculated responses for each type of analysis, which was taken as confirmation that PSResponse results meet an acceptable standard of accuracy for the intended purpose of the software.  This purpose is to provide fast, summary-level  dynamic analysis results for determining the probabilistic response of linear and nonlinear systems under stochastic dynamic loading.  To that end, it is worth noting that  PSResponse was two orders of magnitude faster than  SAP2000  in calculating the  inelastic dynamic response ofthe five-storey shear structure subjected to the E l Centro ground motion. Calculation time for PSResponse was less than one second on a 1.5 GHz Pentium 4 computer with 256 M B R D R A M while  SAP2000,  with its much more detailed  finite-element basis, took approximately 83 seconds.  Finally, as a first application of the beta release of PSResponse, two case studies were done to demonstrate the capabilities ofthe program as a research and analysis tool. These case studies analyzed, from a probabilistic point of view, three general questions pertinent to structural dynamics and earthquake engineering; the relative effect of random structural properties on the dynamic response of a structure, the appropriateness of the  Chapter 6  Summary, Conclusions and Recommendations  215  well-known equal displacement observation and the effect of hysteresis model properties on displacement response.  The first case study analyzed the effect of random properties on the dynamic response of a structure by evaluating the probable peak base shear, peak base moment and peak fifth storey displacement of a five-storey elastic structure using a sequence of 1000 generated records and five levels of variability in the storey mass, storey stiffness, storey height and modal damping.  As well, the probable accuracy of the Response  Spectrum Analysis (RSA) procedure for determining the peak responses of a multipledegree-of-freedom system using the absolute sum (ABSSUM), square-root-of-sum-ofsquares (SRSS) and complete quadratic combination (CQC) modal combination rules was evaluated.  Results from the first case study showed that peak elastic response data is well described by the Gumbel distribution, which was then used as the basis for comparison in evaluating the effect of increasing structural parameter variability on the probability distribution of the chosen peak elastic responses. Using a number of statistical tests, the distribution of each peak response was compared at each level of structural variability with the distribution associated with a deterministic structure. From that analysis it was determined that the randomness of the generated ground motions accounts for the majority of the observed range in a given peak response while structural randomness had a relatively minor effect. This then suggests that careful attention needs to be paid to the characteristics of the ground motion records used when analyzing the dynamic response behaviour of a structure. Once a suitable seed record or suite of seed records has been selected, however, the peak response probability distributions for a given structural model could be applied to a real structure with reasonable confidence since a coefficient of variation in the structural parameters of between 0.2 and 0.3 was required before any significant affect on the peak elastic response distributions was observed. Finally, results from the analysis of the accuracy of the R S A procedure showed, as expected, that the A B S S U M modal combination rule gives a very conservative estimate of peak response and that the SRSS and C Q C error distribution is well represented by the Normal  Chapter 6  Summary, Conclusions and Recommendations  distribution.  216  Using the properties of the Normal distribution, approximate rules-of-  thumb were established for the accuracy of the C Q C and SRSS modal combination rules for peak base shear, peak base moment and peak fifth storey displacement. The observed standard deviations in the distributions of the peak responses indicated that; % of the time the C Q C and SRSS results for peak base shear will be within 10% of the R H A result, while peak base moment and peak fifth storey displacement will be within 5% of the R H A result.  The second case study analyzed the equal displacement observation in structural dynamics as well as the effect of hysteresis model properties on displacement response using a group of single-degree-of-freedom structures with eight different natural periods and 2% damping subjected to a sequence of 1000 generated records using six different hysteresis  models. The hysteresis  models  for the structures were derived from  experimental data taken from a cyclic lateral displacement-controlled test of a Parallam® column, which was analyzed using the hysteresis parameter identification feature of PSResponse.  Dynamic analysis results showed that the Gumbel distribution is again a  good description of the random behaviour of peak displacement response and, therefore, it was used as the basis for comparison in evaluating the effect of the different hysteresis models as well as the agreement between peak elastic and peak inelastic displacement.  To  quantify the effect  of different hysteresis  models  on peak inelastic  displacement response, the reliability index [3 of each hysteresis model associated with structural drift limits of; 0.5%, 1%, 2% and 4%, was calculated for each of the eight natural periods.  This type of analysis is an example of the intended purpose of  PSResponse, which is to enable the evaluation of structural reliability under earthquake loading.  The calculated reliability indices showed three key trends in the drift  exceedence probabilities. Firstly, for each drift limit at each natural period the elastic response gives a conservative estimate of drift reliability as evidenced by the smaller (3 value compared to each of the six hysteresis models.  Secondly, the (3 values for the  hysteresis models vary significantly at a given natural period and drift limit. Thirdly, the  Chapter 6  Summary, Conclusions and Recommendations  217  range of variability in the P values for the hysteresis models is dependent on the natural period and chosen drift limit.  The latter two key observations indicate that the characteristics of a hysteresis model have a significant effect on the calculated seismic reliability of a structure, with the effect being more or less pronounced depending on the capacity limit that is used to assess seismic reliability. Taken together, these observations show that the hysteretic behaviour of a structure needs to be accurately modeled, particularly in shorter natural period structures, to provide an accurate probabilistic description of response and hence a good estimate of seismic structural reliability. The natural period dependence of the relative importance of an accurate hysteresis model is a consequence of the decreasing importance of the restoring force F(x) in the equation of motion as structural mass increases for a given structural stiffness.  Therefore, the shape of the hysteresis loop for  longer period structures is relatively less important when determining the probabilistic response of the structure.  Results from the analysis of the equal displacement observation showed that the distribution of elastic peak displacements differed from the inelastic distribution for each hysteresis model for all natural periods exhibiting a significant response. In general, the modes of the inelastic distributions were located at a lower peak displacement than the mode of the corresponding elastic distribution. This observed trend suggests that the elastic response of a structure tends to give a conservative estimate of peak inelastic displacement. To enable evaluation of this observed conservativeness and identify how it changes with natural period and relative yield strength, the distribution of percentage difference between the elastic and inelastic peak displacement was determined for each natural period using three hysteresis models; Hysteresis D, Hysteresis E and Hysteresis F, which represent non-degrading hysteretic behaviour with 100% yield strength, 50% yield strength and 25% yield strength, respectively.  Using Gumbel distributions to describe the distribution of percentage difference results, the probability that the percentage difference between peak inelastic and peak  Chapter 6  Summary, Conclusions and Recommendations  218  elastic displacement exceeded 0% was evaluated for each hysteresis model (i.e. yield level) at each natural period. This then allowed reliability indices to be calculated for the conservativeness of the equal displacement principle. Note that a negative percentage difference indicates that the peak elastic displacement exceeded the peak inelastic displacement, thereby giving a conservative estimate of peak displacement.  The  calculated reliability indices  for the conservativeness  of the equal  displacement principle showed that peak elastic displacement is generally a conservative estimate of peak inelastic displacement (elastic > inelastic) except for longer natural period structures with reduced yield strength.  Also, for a given natural period the  probability that peak elastic displacement will give a conservative estimate of peak inelastic displacement (elastic > inelastic) is reduced as yield strength decreases.  While it is difficult to make definitive statements regarding the accuracy of the equal displacement principle owing to the wide spread in percentage difference results for each hysteresis model at each natural period, the preceding observations indicate that the equal displacement principle is valid in the sense that elastic peak displacement provides a useful, generally conservative first approximation of inelastic peak displacement.  6.2  CONCLUSIONS  Developing a software application for evaluating the probabilistic response of a structural system to the stochastic dynamic action of an earthquake is a challenging process that depends on the successful integration of a number of mathematical modeling techniques that have been developed over the last several decades.  Conclusions  regarding the merits of the modeling methods and program functionality that were integrated into PSResponse as well as the contributions that have been made in the development of this research and analysis tool can be summarized with respect to the objectives that were set out at the beginning of this project. These objectives were:  Chapter 6  1.  Summary, Conclusions and Recommendations  219  Evaluate the various analytical and numerical methods that have been developed to predict the response of linear and non-linear systems under stochastic dynamic actions.  2.  Identify a method suitable for probabilistic analysis of the seismic response of civil engineering structures.  3.  Develop software to enable application of the chosen probabilistic analysis method for use by engineers and researchers in evaluating structural reliability under seismic loading.  The first two objectives were met through a comprehensive review of the random vibration methods and numerical procedures that have been developed since the first application of probabilistic methods in the field of structural dynamics several decades ago. The theory and application of probabilistic methods is a vast field that crosses back and forth over traditional discipline boundaries between engineering, physics, and mathematics and every effort was made to identify and assess all available methods for their suitability in analyzing the probabilistic seismic response of civil engineering structures. That review now serves as a reasonably current state-of-the-art summary of probabilistic methods and the component models required in numerical procedures. From the review it was determined that the frequency-domain based random vibration methods are too restrictive in their inherent assumptions to confidently apply their results to real structures experiencing realistic earthquakes. Instead, a numerical time-history approach incorporating the Monte Carlo method provides a robust, accurate and straightforward means of evaluating the probabilistic response of a structure without regard to the degree of non-linearity in the restoring force, complexity of the structural system, nature of the variability in structural properties or nature of the random excitation process.  The third objective was met by developing a user-friendly, intuitive software tool that successfully integrates all the elements required in stochastic numerical modeling into what is believed to be the first software application of its kind.  Engineers and  researchers now have at their disposal a software application that can rigorously evaluate  Chapter 6  Summary, Conclusions and Recommendations  220  the probable effect of a wide range of ground motion characteristics and structural model parameters, each with their own random nature, on the dynamic response of a structure. This required that an entirely new, standalone application be developed to provide the computational speed necessary for Monte Carlo dynamic analysis of elastic and inelastic structures as well as the convenience of a Windows user-interface for easy access to the software.  To provide researchers the ability to evaluate the probabilistic response behaviour of experimental structural systems and connections, a new algorithm for parameter identification of the well-known BWBN, or Bouc-Wen, hysteresis model was developed for use with experimental data. This algorithm, and the non-linear analysis algorithms in general, also incorporated a modification to the B W B N model that was specifically made for this software application to simplify the parameter identification process.  The  number of parameters controlling pinching behaviour in the B W B N model was reduced from six to three, which has the added benefit that the role of each of the three new parameters is more easily understood than the relationship between the six parameters of the original pinching function.  For engineering design and analysis purposes, the probabilistic seismic response behaviour of a particular structure or structural system may be evaluated simply using the calculated lateral yield displacement of the structure and a specified post-yield relative stiffness.  This allows the approximate probability distribution of seismic response  behaviour to be determined without having experimental data available to fit the B W B N hysteresis model. The accuracy of this approximate distribution, which is based on an estimated yield strength and does not include the effect of degrading hysteretic behaviour, was investigated in a case study that is summarized shortly.  Changes to a  design affecting either the dynamic behaviour of the structure or the random distributions of the structural properties would currently require a new probabilistic analysis of the response behaviour, however, future development of PSResponse could incorporate a neural network or response surface application to reduce the requirement for new probabilistic analyses as design changes are made.  Chapter 6  Summary, Conclusions and Recommendations  221  Following the successful completion of the objectives of this project, a further contribution was made to the fields of structural dynamics and earthquake engineering through the completion of two case studies that also demonstrated the capabilities of PSResponse as a research and analysis tool. These case studies provided for the first time a probabilistic analysis of the importance of the hysteresis assumption in inelastic dynamic analysis, the accuracy of the equal displacement observation and the relative effect of random structural properties on elastic dynamic response.  The general  conclusions that were drawn within the context of the scope of the case studies were as follows:  1.  The hysteretic behaviour of a structure needs to be accurately modeled, particularly in shorter natural period structures, to provide an accurate probabilistic description of response and hence a good estimate of seismic structural reliability.  2.  The equal displacement principle is valid in the sense that elastic peak displacement provides  a  generally  conservative  first  approximation  of  inelastic  peak  displacement, which in turn results in a generally conservative prediction of reliability.  3.  The characteristics and randomness of ground motion records has a much larger influence than structural randomness on the probabilistic dynamic response of a structure.  Therefore, once a suitable seed record has been selected, the peak  response probability distributions for a given structural model could be applied to a real structure with reasonable confidence since the assumed level of uncertainty in the structural parameters needs to be only approximately correct.  However, for  strength related limit state evaluations related to peak response, structural variability still has an important effect.  Chapter 6  6.3  Summary, Conclusions and Recommendations  222  RECOMMENDATIONS FOR FURTHER RESEARCH  There are a virtually unlimited number of studies that could be carried out to evaluate the probable response of a wide range of structural models with various combinations of structural properties possessing different random distributions. These studies could include evaluations of; base isolation systems, optimal mass and damping distributions, first-passage probabilities and the temporal evolution of probabilistic response distributions. These types of analysis are possible with PSResponse but were not included in the case studies done as part of this project.  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