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Inelastic response of SDOF systems subjected to subduction and crustal ground motions Mattman, Dominic Willy 2006

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INELASTIC RESPONSE OF SDOF SYSTEMS SUBJECTED TO SUBDTJCTION AND CRUSTAL GROUND MOTIONS by DOMINIC W I L L Y M A T T M A N B.A.Sc, University of British Columbia, 2004 A THESIS SUBMITTED IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F APPLIED SCIENCE in T H E F A C U L T Y O F G R A D U A T E STUDIES (Civil Engineering) T H E UNIVERSITY O F BRITISH C O L U M B I A August 2006 © Dominic Wil ly Mattman, 2006 ABSTRACT Structural engineers are increasing turning to nonlinear static procedures to gain better insight into the performance of the structures they design. K e y to these procedures is estimating the displacement demand of the design level earthquake. The Coefficient Method, from F E M A 356 ( A S C E , 2000) and recently updated in F E M A 440 ( A T C , 2005), estimates this target displacement. This method was developed through statistical studies of S D O F systems utilizing crustal ground motions. Therefore, to expand the applicability of the Coefficient Method to areas subjected to subduction earthquakes such as the Pacific Northwest of North America, this study examines the behaviour of S D O F systems with 6 different hysteretic models, 50 periods of vibration, 9 relative strength values and 88 ground motions (40 from crustal sources and 48 from subduction sources). Three of the hysteretic models included in the study exhibit strength degradation with one model incorporating cyclic strength degradation while the other two included in-cycle strength degradation. Furthermore, the frequency of instability of the in-cycle strength degrading systems was examined. The greatest differences between the subduction and crustal ground motions were observed in: the inelastic displacement ratios of the E P P systems (up to 3 times higher and convergence to the "equal displacement rule" at higher periods), the inelastic displacement of the cyclic strength degrading systems when compared to E P P systems and the maximum permissible relative strength value for the in-cycle strength degrading model. With the subduction records, a moderate increase in the frequency of instability of the in-cycle strength degrading systems was also noted. Negligible differences were seen in the inelastic displacements of the cyclic and in-cycle strength degrading systems provided the later remained stable. A simple S D O F model was used to compare the response of a system retrofitted only for strength with the response of the original weak, but ductile, system. The results showed an increased displacement in the retrofitted system unless there was a significant increase n in strength, the negative slope of the response was not too steep and the system had short period of vibration. i i i TABLE OF CONTENTS Abstract »' Table of Contents iv List of Tables vi List of Figures vii Aknowledgements x » 1 Introduction 1 1.1 Background 1 1.2 Objectives 4 1.3 Overview 4 2 Literature Review 6 2.1 Nonlinear Static Procedures 6 2.2 Inelastic Displacement Ratios of Bilinear and Stiffness Degrading Systems... 12 2.3 Strength Degrading Systems 15 3 Ground Motions 19 3.1 Crustal Ground Motions 20 3.2 Subduction Ground Motions 25 3.3 Peak Ground Acceleration and Displacement 30 4 Hysteretic Models & Analysis Parameters 32 4.1 Elastic Perfectly-Plastic 32 4.2 Bouc-Wen 32 4.3 Stiffness Degrading 33 4.4 Stiffness and Cycl ic Strength Degrading (CSD) 34 4.5 Stiffness and In-Cycle Strength Degrading (ISD) 35 4.6 Stiffness and In-Cycle Strength Degrading with Residual Strength (ISDR).. . . 36 4.7 Common Analysis Parameters 38 5 SDOF Analyses with Crustal and Subduction Ground Motions 39 5.1 Elastic Perfectly Plastic 39 5.1.1 Linear Regression 43 5.1.2 Variability of EPP Results 47 5.2 Bouc-Wen 49 iv 5.3 Stiffness Degrading 52 5.4 Target Displacements 55 6 SDOF Analyses with Strength Degrading Systems 65 6.1 Stiffness and Cycl ic Strength Degrading Systems 65 6.2 Stiffness and In-Cycle Strength Degrading Systems 68 6.3 Comparison o f Cycl ic and In-Cycle Strength Degrading Systems 71 6.4 Instability of Stiffness and In-Cycle Strength Degrading Systems 74 6.4.1 Maximum Force Reduction Factors 76 6.5 Stiffness and In-Cycle Strength Degrading with Residual Strength Systems .. 82 7 Conclusions 93 7.1 Impact of Subduction Ground Motions 96 7.2 Future Research 97 References 99 APPENDIX A. Ground Motions 103 A . 1 Ground Motion Processing 103 A . 2 Crustal Ground Motions - Time Histories 104 A.3 Crustal Ground Motions - Pseudo-Acceleration Response Spectrum 114 A . 4 Subduction Ground Motions - Time Histories 121 A.5 Subduction Ground Motions - Pseudo-Acceleration Response Spectrum 133 APPENDIX B. Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 141 APPENDIX C. Stiffness and In-Cycle Strength Degrading with Residual Strength and Equivalent EPP Displacement Ratios 171 v LIST OF TABLES Table 2-1: Values of C2 coefficient from F E M A 273 ( A T C / B S S C , 1997) 8 Table 2-2: Coefficents for Equation 2-13 (Ruiz-Garcia and Miranda, 2003) 13 Table 3-1: Crustal ground motions site class C ( A T C , 2005) 21 Table 3-2: Crustal ground motions site class D ( A T C , 2005) 22 Table 3-3: Summary of Tokachi-Oki ground motion properties with epicentral distance between 71 and 152km 27 Table 4-1: Analysis parameters 38 Table 5-1: Coefficients of [5 in C; sub coefficient for site class C 44 Table 5-2: Coefficients of f3 in C/ sub coefficient for site class D 44 v i LIST OF FIGURES Figure 2-1: Force-displacement envelope of an elastic and inelastic system 7 Figure 2-2: Values of C/ from Eq. 2-5 for various relative strengths 10 Figure 2-3: Values of C2 from Eq. 2-7 for various relative strengths 10 Figure 2-4: Idealization of base shear-displacement response from nonlinear static analysis 11 Figure 2-5: Force-displacement relationship of in-cycle strength degrading systems (Miranda and Akkar, 2003) 17 Figure 2-6: Lateral strength required to prevent instability (Miranda and Akkar, 2003). 17 Figure 3-1: Cascadia Subduction Zone and tectonic setting of southwestern B C (Onur et. al, 2006) 19 Figure 3-2: Acceleration spectra of crustal ground motions recorded on site class C soils. 23 Figure 3-3: Acceleration spectra of crustal ground motions recorded on site class D soils. 23 Figure 3-4: Displacement spectra of crustal ground motions recorded on site class C soils. 24 Figure 3-5: Displacement spectra of crustal ground motions recorded on site class D soils. 24 Figure 3-6: Acceleration spectra of subduction ground motions recorded on site class C soils 28 Figure 3-7: Acceleration spectra of subduction ground motions recorded on site class D soils : 28 Figure 3-8: Displacement spectra of subduction ground motions recorded on site class C soils 29 Figure 3-9: Displacement spectra of subduction ground motions recorded on site class D soils 29 Figure 3-10: Peak ground acceleration (PGA) vs. distance for subduction and crustal ground motions 30 Figure 3-11: Peak ground displacement (PGD) vs. distance for subduction and crustal ground motions 31 v i i Figure 4-1: EPP hysteretic response 32 Figure 4-2: Bouc-Wen hysteretic response for (a) n = 5.0 and (b) n = 0.5 33 Figure 4-3: Stiffness degrading hysteretic response. 34 Figure 4-4: Diagram of cyclic strength degrading hyseteric rules (not drawn to scale for clarity) 34 Figure 4-5: Stiffness and cyclic strength degrading hysteretic response (CSD) with strength degradation curve 35 Figure 4-6: Stiffness and in-cycle strength degrading hysteretic response (ISD) 36 Figure 4-7: Stiffness and in-cycle strength degrading hysteretic response with residual strength (ISDR) ; 37 Figure 4-8: Stiffness and in-cycle strength degrading hysteretic response with residual strength (ISDR) model parameters 37 Figure 5-1: Mean inelastic displacement ratios for EPP hysteretic model for crustal ground motions 40 Figure 5-2: Mean inelastic displacement ratios for E P P hysteretic model for subduction ground motions 41 Figure 5-3: Ratio of the mean inelastic displacement ratios for subduction ground motions to the mean inelastic displacement ratios for crustal ground motions 42 Figure 5-4: Ratio of inelastic displacement ratio for subduction ground motions from Figure 5-2 to C, in F E M A 440 43 Figure 5-5: C/ s u o coefficient 45 Figure 5-6: Comparison of C/ coefficient from F E M A 440 and Cj Sub coefficient 46 Figure 5-7: Mean ratio of the inelastic displacement of E P P systems subjected to subduction ground motions to the C/ sub coefficient 47 Figure 5-8: Coefficient of variation of inelastic displacement ratios for E P P hysteretic model for crustal ground motions 49 Figure 5-9: Coefficient of variation of inelastic displacement ratios for E P P hysteretic model for subduction ground motions 49 Figure 5-10: Mean inelastic displacement ratios for the Bouc-Wen hysteretic model for crustal ground motions (n = 5.0) 50 V l l l Figure 5-11: Mean inelastic displacement ratios for the Bouc-Wen hysteretic model for subduction ground motions (n = 5.0) 50 Figure 5-12: Mean inelastic displacement to EPP ratios for the Bouc-Wen hysteretic model for subduction ground motions in = 5.0) 51 Figure 5-13: Mean inelastic displacement ratios for the Bouc-Wen hysteretic model for subduction ground motions (n = 0.5) 52 Figure 5-14: Mean inelastic displacement to E P P ratios for Bouc-Wen hysteretic model for subduction ground motions in = 0.5) 52 Figure 5-15: Mean inelastic displacement to EPP ratios for stiffness degrading hysteretic model for crustal ground motions 53 Figure 5-16: Mean inelastic displacement to E P P ratios for stiffness degrading hysteretic model for subduction ground motions 54 Figure 5-17: Ratio of increased displacement from stiffness degradation from subduction ground motions to C2 in F E M A 440 54 Figure 5-18: Mean ratio of the target displacement from F E M A 440 for site class C to the inelastic displacement of EPP systems subjected to subduction ground motions 57 Figure 5-19: Mean ratio of the target displacement from F E M A 440 for site class D to the inelastic displacement of E P P systems subjected to subduction ground motions 58 Figure 5-20: Mean ratio of the inelastic displacement of E P P systems subjected to subduction ground motions to the target displacement from F E M A 440 for site class C 61 Figure 5-21: Mean ratio of the inelastic displacement of E P P systems subjected to subduction ground motions to the target displacement from F E M A 440 for site class D 62 Figure 5-22: Mean ratio of the inelastic displacement of EPP systems subjected to subduction ground motions to the target displacement using the C / Sub coefficient for site class C 63 Figure 5-23: Mean ratio of the inelastic displacement of E P P systems subjected to subduction ground motions to the target displacement using the C/ Sub coefficient for site class D 64 ix Figure 6-1: Mean ratio of inelastic displacements of C S D (ACSd) and E P P ( A e p p ) models for crustal ground motions 66 Figure 6-2: Mean ratio of inelastic displacements of C S D (ACSd) and EPP ( A e p p ) models for subduction ground motions 66 Figure 6-3: Ratio of C S D to E P P displacements for crustal ground motions divided by the C2 coefficient from F E M A 440 68 Figure 6-4: Ratio of C S D to E P P displacements for the subduction ground motions divided by the C2 coefficient from F E M A 440 68 Figure 6-5: Backbone curve for ISD model with instability limit 69 Figure 6-6: Mean inelastic displacement to E P P ratios for stiffness and in-cycle strength degrading (ISD) hysteretic model for crustal ground motions 70 Figure 6-7: Mean inelastic displacement to E P P ratios for stiffness and in-cycle strength degrading (ISD) hysteretic model for subduction ground motions 70 Figure 6-8: Ratio of the inelastic displacements from the ISD model to the C S D model for crustal ground motions 72 Figure 6-9: Ratio of the inelastic displacements from the ISD model to the C S D model for subduction ground motions 72 Figure 6-10: Hysteretic response of C S D and ISD models (T = 0.45s, R = 4.0, Aisd/Acsd = 1.23, Los Angeles, Hollywood Storage Building, Northridge 1994) 73 Figure 6-11: Hysteretic response of C S D arid ISD models (T = 1.0s, i? = 4.0, Aisc/Acsd = 1.00, Gilroy #2, Keystone Rd. , Morgan H i l l 1984) 73 Figure 6-12: Frequency of instability for ISD model and crustal ground motions 74 Figure 6-13: Frequency of instability for ISD model and subduction ground motions.... 74 Figure 6-14: Frequency of theoretically stable ISD systems exceeding a ductility of 10 for crustal ground motions 76 Figure 6-15: Cumulative frequency of instability to determine Rmax 77 Figure 6-16: Maximum force reduction factor for crustal ground motions site class C. . . 78 Figure 6-17: Maximum force reduction factor for subduction ground motions site class D . ; 79 Figure 6-18: Design maximum force reduction factor for crustal ground motions 81 Figure 6-19: Design maximum force reduction factor for subduction ground motions... 81 x Figure 6-20: Backbone curve of stiffness and in-cycle strength degrading with residual strength (ISDR) hysteretic model and equivalent E P P model 82 Figure 6-21: Mean inelastic displacement ratios for ISDR model, a = -0.10 and / = 0.1, 0.8 83 Figure 6-22: Mean inelastic displacement ratios for ISDR model, R = 4.0 and a= -0.1, -0.8 84 Figure 6-23: Tri-linear backbones of ISDR model 84 Figure 6-24: Mean ratio of ISDR model to equivalent E P P system for crustal ground motions recorded on site class D , RR = 5 86 Figure 6-25: Mean ratio of ISDR model to equivalent E P P system for subduction ground motions recorded on site class D , RR = 5 87 Figure 6-26: Mean ratio of ISDR model to equivalent E P P system for crustal ground motions recorded on site class D plotted against y 89 Figure 6-27: Mean ratio of ISDR model to equivalent E P P system for subduction ground motions recorded on site class D plotted against y 89 Figure 6-28: Mean ratio of ISDR model to equivalent E P P system for crustal ground motions recorded on site class D , y=0.5 91 Figure 6-29: Mean ratio of ISDR model to equivalent E P P system for subduction ground motions recorded on site class D , y=0.5 92 x i AKNOWLEDGEMENTS Special thanks must be given to my supervisor Dr. Ken Elwood, whose support and guidance have made this research possible. I would also like to thank Dr. Perry Adebar for his insightful feedback. I am grateful to the Natural Sciences and Engineering Research Council of Canada ( N S E R C ) for funding this research and their continued support of graduate students. Tuna Onur, Garry Rogers and John Cassidy from the Pacific Geoscience Centre were invaluable with their tremendous knowledge and expertise in selecting ground motions for this study. This is one of the most crucial elements that made this study possible. I would also like to thank Kevin Riederer, Chris Meis l , Soheil Yavari , Hamid Karimian, Houman Ghalibafian, Martin Turek, Arnaud Charlet, T i m Matthews, Aaron Korchinski and Vignesh Ramadhas for their friendship and wisdom throughout my time at U B C . M y family and friends have had a great influence on me, and have provided countless examples of success through hard work. Y o u have all been a great inspiration to me. Lastly, I am eternally grateful to my wife Oana for the endless support and encouragement with which you have provided me. x i i 1 INTRODUCTION 1.1 Background The behavior of structures under the applied forces and displacements induced by earthquake ground motions is highly complex and generally a nonlinear response. Many factors that may not greatly affect the response of a building are ignored to enable structural engineers to provide a solution which approximately approaches the response of the structure. The goal of the research presented is to improve the knowledge base available to engineers from which they can begin to make their assumptions and predictions of the ultimate response of the structure. The hierarchy of analysis techniques for modeling the response of structures contains four different approaches with increasing levels of sophistication. The simplest approach used in practice is a linear-static analysis. When performing this type of analysis, the engineer w i l l utilize the linear properties of the structural elements and apply a static load of the predicted earthquake forces which are based on a design level of spectral acceleration at the period of the structure. The 1995 National Building Code of Canada ( N B C C ) (Canadian Commission on Building and Fire Codes, 1995) included this type of analysis with modifications to the design forces to account for the nonlinear response of the lateral force resisting system. While this method is the simplest and is generally considered to be conservative for "regular" structures, it is not the best tool available for predicting the response of structures which deviate from a "regular" structure. In the 2005 N B C C (Canadian Commission on Building and Fire Codes et al, 2005), unless the structure can be shown to free of irregularities specifically outlined in the code, the minimum level of analysis for the seismic performance of structures has been changed to a linear-dynamic analysis. In a linear-dynamic analysis, the engineer again assumes a linear response of the structure, but now performs a dynamic analysis using earthquake ground motions or more commonly using a design response spectrum. In this analysis, multiple mode shapes of the structure can be determined and combined with the spectral acceleration at the period 1 Introduction Chapter 1 of the individual mode shapes to provide an improved prediction of the forces and displacements that w i l l be imposed. The next improvement is to consider the nonlinear behavior of the structure when the seismic forces are applied. Nonlinear-static procedures include the nonlinear behavior of structural elements when subjected to a static load meant to represent the earthquake. This load distribution can be the same as that assumed in a linear-static analysis or the results from a linear-dynamic analysis can be utilized. The static load is then continually increased by increasing the loads by a common multiplier and the displacements of the structure are recorded until the structure reaches the target displacement level. The behavior of the structure is then evaluated at the target displacement. The target displacement is a function of the linear and nonlinear properties of the structure as well as the desired level of performance (e.g. Immediate Occupancy, Life Safety and Collapse Prevention which are defined in the publication by the Federal Emergency Management Agency ( F E M A ) in the United States entitled FEMA 356: A Prestandard and Commentary for the Seismic Rehabilitation of Buildings ( A S C E , 2000)). The fourth level of analysis is the nonlinear-dynamic analysis or sometimes referred to as a nonlinear time-history analysis. In this analysis, the model of the structure has incorporated the nonlinear behavior of the structural elements and is subjected to recorded or synthetic ground motions. The response of the structure is then calculated throughout the duration of the earthquake. A couple of the fundamental issues in this analysis are the number of ground motions to use and the selection of these records. While records are preferably selected within an appropriate magnitude and distance to represent the hazard for the structure, the engineer must ensure that the response spectra from the ground motions are a reasonable fit for the seismic hazard spectrum. However, even when this approach is followed, the variability between ground motions w i l l cause different responses with some being more severe than others and therefore, the structural design is strongly influenced by the ground motions selected. 2 Chapter I Introduction A s engineers strive to better understand the performance of the structures they design, the use of performance-based approaches has dramatically increased. This has resulted in an increase in the use of nonlinear-static procedures increasing dependence on the methods for estimating the target displacement. However, there is considerable uncertainty in determining the target displacement and this w i l l be the primary focus of this study. Currently, the most common methods of calculating the target displacement are the Coefficient Method in F E M A 356 ( A S C E , 2000), and the Capacity Spectrum Method in ATC-40: The Seismic Evaluation and Retrofit of Concrete Buildings, Volume 1 and 2 ( A T C , 1996). Recent efforts by researchers have lead to improvements of these methods to predict the target displacement and the modified procedures have been published as FEMA 440: Improvement of Nonlinear Static Seismic Analysis Procedures ( A T C , 2005). Significant work was conducted in the calculation of the ratio of inelastic displacement to the elastic displacement for different hysteretic systems with constant strength for the improvement of the Coefficient Method. While this research has provided valuable insight and refinement for this method, the source of the ground motions were typically recorded from crustal earthquakes in California. In an effort to extend the applicability of the Coefficient Method in F E M A 440 to areas where the seismic hazard is not solely dominated by crustal earthquakes, this research study undertaken has utilized subduction ground motions recorded during the 2003 Tokachi-Oki earthquake off the coast of Japan to asses the effectiveness of the Coefficient Method in predicting the nonlinear displacements of a structure subjected to such an event. This is particularly of concern along the west coast of North America where major cities such as Vancouver, Seattle, Victoria and Olympia are within 150km of the Cascadia Subduction Zone. The current study also investigates the effects of strength degradation on the inelastic displacements of S D O F systems. Two types of strength degradation are included: cyclic strength degradation which occurs between cycles and in-cycle strength degradation which occurs within one cycle of loading. 3 Chapter 1 1.2 Objectives Introduction • To assess the effectiveness of the Coefficient Method in predicting the nonlinear displacements associated with subduction events and suggest modifications to the method i f necessary. • To provide input to practicing engineers on the applicability and necessity of considering the nonlinear displacements associated with subduction events. • To compare the nonlinear response of structures which exhibit in-cycle strength degradation as opposed to cyclic strength degradation. • To compare the nonlinear displacements of systems with retrofits that improve the strength of the system but do not improve the ductility. 1.3 Overview A summary of the development of the Coefficient Method, recent research related to inelastic displacement ratios of S D O F systems and research involving strength degrading systems and their stability is presented in the second chapter. The third chapter details the seismicity of the Western portion of North America, with specific reference to the Cascadia Subduction Zone. The source and characteristics of the ground motions used in this study are also detailed and related to the seismic zones mentioned. The six different hysteretic models used in the analysis are detailed with examples of their behaviour in the fourth chapter. Chapters five and six present the results of 1.8 mil l ion single-degree-of-freedom (SDOF) analyses utilizing the hysteretic models from chapter four and the ground motions in 4 Chapter I Introduction chapter three. Chapter five presents the inelastic displacement ratios for the Elastic Perfectly-Plastic (EPP) and Bouc-Wen (Babar and Noori , 1985) models. The inelastic displacements for the stiffness degrading model are compared to the inelastic displacements of the E P P system. Comparisons between the results for the subduction and crustal records are made in addition to the comparisons between the subduction results and the Coefficient Method in F E M A 440. Chapter six examines the results of the strength degrading hysteretic models and compares the two types of strength degradation. Discussion of the instability of the in-cycle strength degrading systems is also included in this chapter. The main conclusions and recommendations of this research are presented along with recommendations for areas of further research in chapter seven. 5 Chapter 2 2 LITERATURE REVIEW Literature Review The implementation of performance-based approaches to evaluating structures has become more prevalent in structural engineering practice. One of the methods to assess the nonlinear performance is through the use of nonlinear-static analysis. The benefits of performing this type of analysis is that a more realistic behaviour of the structure is assumed when compared with either linear-static or linear-dynamic analyses and a reduced level of complexity when compared with a nonlinear-dynamic analysis. Nonlinear-dynamic analysis is also gaining in use among the engineering community but w i l l not become common practice until significant issues pertaining to ground motion selection and statistical justification for the results from these analyses can be established. Since the use of nonlinear-static procedures is growing rapidly with recent improvement to the two most common procedures for estimating the target displacement, it would seem prudent to examine these procedures for their suitability to areas where the seismic hazard includes subduction and subcrustal earthquakes. Therefore, this study is focused on improving the displacement predictions from the Coefficient Method included within F E M A 440. 2.1 Nonlinear Static Procedures The Coefficient Method was original introduced to the engineering community in the publication o f FEMA 273: NEHRP Guidelines for the Seismic Rehabilitation of Buildings ( A T C / B S S C , 1997). Since the publication of F E M A 273, the use of this nonlinear static procedure has gained wide acceptance amid the engineering community. The first step of the procedure is to estimate the base shear force versus the lateral displacement of a control node, commonly referred to as the "pushover curve". This node is typically taken at the top of the structure and by gradually multiplying the lateral load distribution by a common factor the pushover curve can be established. The performance of the structure is then assessed at the so-called "target displacement", which is calculated through Eq . 6 Chapter 2 Literature Review 2-1 in the Coefficient Method. The target displacement, St, is determined by multiplying the spectral displacement, defined by Sa, the spectral acceleration at the effective period, Te, and the effective period itself, by four coefficients, Co, Cy, C2 and C3, which attempt to account for the nonlinear response of the structure under dynamic loading. The first coefficient, Co, is the coefficient to relate the spectral displacement from the equivalent S D O F system to the roof displacement of a multiple degree of freedom procedure is that the response must be dominated by the first mode of the building otherwise a linear dynamic analysis must also be performed. The second coefficient, Cy ; aims at capturing the effects of a nonlinear dynamic procedure and the displacements associated with them when compared to those of a linear elastic response. This increase in displacement is illustrated in Figure 2-1. The values for Cy are determined through Eq . 2-2 and are a function of the ratio of the elastic force demand to the strength of the system defined in Eq . 2-3, R, and the period at which the acceleration spectrum changes from constant acceleration to constant velocity, To. St — C0C]C2C:3Sa i s (2-1) system, however, it should be noted that a requirement for using this nonlinear static Force Elastic Displacement — • Figure 2-1: Force-displacement envelope of an elastic and inelastic system. 7 Chapter 2 Literature Review l + (R-l)f *— i f T <T 0 else C, =1.0 R e-where R-———* — VyIW c0 (2-2) (2-3) However, the magnitude of Cj in the nonlinear static procedure does not need to be greater than the values of this coefficient associated with the linear static procedure (i.e. for Te less than 0.1s, Ci =1.5 and for T >To, Ci is taken as unity with linear interpolation between these limits). The effect of the stiffness degradation, strength degradation and pinching are included through the inclusion of d- It was felt that the reduced size of the hysteresis loops would increase the inelastic displacements especially in structures which undergo significant damage. Therefore, the systems which exhibit this hysteretic behaviour are listed as Framing Type 1 and other systems are Framing Type 2 and the Q coefficient for Framing Type 1 is a function of the Performance Level ( A T C / B S S C , 1997). The values for this coefficient are shown in Table 2-1 for both types of framing. Table 2-1: Values of C2 coefficient from F E M A 273 (ATC/BSSC, 1997) T< 0.1 second T>T0 Performance Level Framing Type 1 Framing Type 2 Framing Type 1 Framing Type 2 Immediate Occupancy 1.0 1.0 1.0 1.0 Life Safety 1.3 1.0 1.1 1.0 Collapse Prevention 1.5 1.0 1.2 1.0 P-A effects are included as the nonlinear displacements can increase significantly when the system exhibits a negative post-yield slope. Correspondingly, the Q coefficient is 1.0 i f the system has a nonnegative post-yield slope but is otherwise determined by Eq . 2-4 where a is the ratio of post-yield stiffness to the elastic stiffness of the effective system. 8 Chapter 2 Literature Review C 3 =1.0 + (2-4) With the increasing usage of F E M A 273, there was a desire to transform the guidelines into a codified standard. A s a step towards this end goal, the guidelines, F E M A 273, and the commentary, F E M A 274 ( A T C / B S S C , 1997), were combined into a prestandard with the intent of adopting this document as a standard. The resulting document is the widely used F E M A 356 ( A S C E , 2000). In F E M A 356, the Coefficient Method has only minor changes from the version found in F E M A 273. The most significant change was the addition of more detailed values for the Co coefficient by adding additional factors for buildings where the inter story drift decreases with height. The other change was seen in Eq . 2-3, where XICo was replaced by a coefficient denoted as Cm which is defined as an effective mass factor that ranged from 0.8 to 1.0 which resulted in slightly higher R values than in F E M A 356. The Coefficient Method has become a widely used technique for estimating the target displacement that w i l l be induced by an earthquake. With increasing feedback from engineers using the procedure and to improve the areas where judgment was required to develop the method, there was a need to conduct research to improve the Coefficient Method to develop more realistic predictions. Thus the foundation for what has become F E M A 440 ( A T C , 2005) was established. The result of the research was improved formulae for calculating the coefficients Cj and C2 along with the replacement of the C3 coefficient with a strength limitation based on the post-yield stiffness of the structure. The improved formulae for Ci and C2 are presented in Eqs. 2-5, 2-6 and 2-7, where a is a constant which calibrates Eq . 2-5 for different site classes, and plots of these equations are shown in Figure 2-2 and Figure 2-3. C, =1 + (2-5) 9 Chapter 2 Literature Review where R = m-S„ (2-6) C2 =1 + 800 R-l (2-7) o XT o Figure 2-2: Values of C; from E q . 2-5 for various relative strengths. 0 2 3 Figure 2-3: Values of C2 from E q . 2-7 for various relative strengths. The increase of inelastic displacements which result from a negative post-yield slope are very difficult to quantify through the use a coefficient as a small decrease in strength can result in a significant increase in displacement. Therefore, to account for loss in strength that occurs in systems that experience negative post-yield stiffness and to account for the difference between the amount of the strength degradation due to cyclic degradation and the amount due to in-cycle degradation such as P -A effects, a limitation of the amount of 10 Chapter 2 Literature Review force reduction allowed, Rmax, has been included in F E M A 440. The equation for the strength limitation is: (2-8) where ae - ap_A + X(a2 -ap_&) (2-9) t = l + 0.\5ln(T) (2-10) In Eq. 2-8, the limiting value of strength reduction is calculated using the yield displacement, Ay, the yield force, Vy, the displacement associated with maximum base shear, Ad, the effective post-elastic strength, ae, which is defined in Eq . 2-9 and the exponent t defined in Eq . 2-10. The effective post-elastic strength is defined by the negative stiffness associated with P-A effects, CCP-A, the negative stiffness of the response, a2 and a factor to account for the impulsive nature of near-fault ground motions, X. The negative stiffness is calculated by connecting the line from the maximum base shear, Vd, through the response at 60% of the yield base shear. Near-fault ground motions cause more in-cycle strength degradation that can lead to dynamic instability and accordingly, X is taken as 0.8 for near-fault ground motions and 0.2 otherwise. The variables used in Eq. 2-8 to 2-10 are displayed in Figure 2-4. Base shear 0.6 V. Actual force-displ curve \ Ad Displacement Figure 2-4: Idealization of base shear-displacement response from nonlinear static analysis (ATC, 2005). 11 Chapter 2 Literature Review Since P-A effects have been accounted for through the limitation on the force reduction factor, the C$ coefficient from Eq. 2-1 has been removed and the revised equation for calculating the target displacement in F E M A 440 is shown in Eq . 2-11. 2.2 Inelastic Displacement Ratios of Bilinear and Stiffness Degrading Systems The new formulae incorporated with in F E M A 440 are the result of various studies on the inelastic displacement ratios of S D O F systems. These studies were undertaken to better understand the effects of different hysteretic behaviors on the ratio of the inelastic displacement of the structure to the elastic displacement. This work has become the foundation from which the improvements to the Coefficient Method were created. One of the most influential studies in the creation of F E M A 440 was work done by Ruiz-Garcia and Miranda. Ruiz-Garcia and Miranda (2003) conducted a study to compare the inelastic displacement of S D O F systems with constant relative strength to the elastic displacement of these systems. A total of 64,800 inelastic displacement ratios were calculated from the results of S D O F nonlinear dynamic analyses on Elastic Perfectly-Plastic (EPP) systems with 50 periods, 7 constant relative strength levels subjected to 216 ground motion records. These ground motions were from 12 different earthquake events and recorded on three different site classes of soil (B, C and D). It should be noted that the 12 earthquakes which provided the ground motions were all shallow, crustal earthquakes that occurred in California. The results of the analyses indicated that the ratios of the inelastic displacement to the elastic displacement, CR, were on average greater than 1.0 for periods less than 1.0s and were approximately equal to 1.0 for periods greater than 1.0s. While this holds for the mean values, there was significant dispersion seen within the results especially at shorter periods where the inelastic displacement ratios were the (2-11) 12 Chapter 2 Literature Review highest. The study found that the period at which the transition from where the inelastic displacements are approximately equal to the elastic displacements appeared to be a function of the relative strength of the system as wells as the site class of the soil. The period at which this transition occurs increases with decreased strength (i.e. higher R value) and softer soils. After performing regression analyses on the results obtained, Ruiz-Garcia and Miranda proposed the simplified expression seen in Eq . 2-12 to estimate the inelastic displacement ratio given the period of the structure, T, the relative strength, R, and the coefficients and variables defined in Table 2-2 . 1 1 a(TITs)b c (R-l) (2-12) Site Class a b c Ts(s) B 42 1.60 45 0.75 C 48 1.80 50 0.85 D 57 1.85 60 1.05 It was also found during this study that the inelastic displacement ratios of weaker systems were affected by differences in magnitude. When results were grouped into three magnitude ranges, the stronger systems, tended to show little difference between the results for the different magnitudes, whereas for the weaker systems, a significant difference was seen as the average inelastic displacement ratios for the smaller magnitude events were smaller than those associated with the larger magnitude events. In addition, the results were grouped into three different distance ranges, but no significant differences were seen between the different distance ranges. These results expand on their previous work (Ruiz-Garcia and Miranda, 2002a and 2002b). This research supplemented the work done by Miranda (2000) on the inelastic displacement ratios of S D O F systems with E P P hysteretic behavior using a constant ductility. The trends seen in this work are similar to those seen in Ruiz-Garcia and 13 Chapter 2 Literature Review Miranda (2003) but are marginally smaller due to a systematic underestimation of inelastic demands when using constant ductility as opposed constant relative strength. Miranda (2000) utilized 264 ground motions recorded in California with magnitudes ranging from M s = 5.8 to M s = 7.8. Again, minor differences were seen when the results were grouped by magnitude, specifically at shorter periods, and little effect was seen when the results were grouped by distance. Work by Krawinkler et al. (2003) supports these conclusions as they calculated acceleration response spectra normalized by the spectral acceleration at a given period. These spectra showed some difference between the groupings of magnitude and distance but the correlations were not significant or consistent across the range of periods. There were also differences in the response of bilinear systems with a positive post-yield slope of 3% of the initial stiffness but again the differences were not major or consistent. Other researchers have reached similar conclusions with positive post-yield systems (Chopra and Chintanapakdee, 2004) Gupta and Krawinkler (1998) studied the effect of stiffness degradation on the inelastic displacements of S D O F systems through the use of a pinched hysteretic model with stiffness degradation. They found a small amplification of the inelastic displacement of a stiffness degrading system relative to an E P P system for medium and long periods. This amplification increased significantly as the period decreased. Similar results were also seen in a separate study by Gupta and Kunnath (1998) but it was noted that these displacements were also sensitive to the amount of stiffness degradation and that more severe degradation leads to a further increase in the inelastic displacement. Ruiz-Garcia and Miranda (2005) also investigated the effect of stiffness degradation on the inelastic displacement of S D O F systems. The three stiffness degrading models used in their study consisted of a modified Clough model, Takeda model and origin-oriented model. The primary difference between these models is the rate of unloading stiffness degradation which varies from none in the modified Clough model to severe in the origin-oriented model. A l l of these models showed increased inelastic displacement relative to the E P P model for periods less than 1.0s, but the displacements from the modified Clough model and Takeda model, which exhibits moderate unloading stiffness 14 Chapter 2 Literature Review degradation, were similar. The greatest difference was seen in the origin-oriented model which incorporated severe stiffness degradation and showed an increased inelastic displacement relative to the E P P model even at 3.0s which was the extent of the analyses. 2.3 Strength Degrading Systems The effect of strength degradation on the response of S D O F systems has been studied by several researchers. This research has included hysteretic models which incorporate strength degradation that occurs between cycles of loading (i.e. cyclic strength degradation) as well as degradation which occurs within a loading cycle (i.e. in-cycle strength degradation). The effect of cyclic strength degradation was included in the study by Gupta and Kunnath (1998). Their results also showed that as the level of degradation increased from a moderate level of strength degradation to a severe level, the inelastic displacements could increase by up to 50% in the short period range. Ruiz-Garcia and Miranda (2005) studied two different stiffness and strength degrading models which exhibited cyclic strength degradation. When the inelastic displacements from these models are compared with an EPP system, the displacement of the cyclic strength degrading models exceed the EPP displacements for periods less than 0.7s and increase with larger relative strength values. These results support the findings of Gupta and Kunnath (1998) and Pekoz and Pincheira (2004) by showing an increase in the level of inelastic displacement as the level of strength degradation became more severe. Song and Pincheira (2000) also studied the effect of in-cycle and cyclic strength degradation on S D O F systems and this research was supplemented by Pekoz and Pincheira (2004, 2005). The hysteretic model used in these studies included a positive post-yield slope followed by a negative post-yield slope that degraded to a residual strength of 10% of the yield force before becoming plastic. The strength degradation occurs within a cycle of loading but the primary degradation occurs between cycles as the negative slopes used in these studies never exceeded 5% of the elastic stiffness. The hysteretic model also included the effects of pinching and unloading strength degradation. Song and Pincheira (2000) only utilized 12 ground motions with 8 recorded 15 Chapter 2 Literature Review from crustal earthquakes and 4 from subduction earthquakes although this distinction is not discussed. The number of ground motions used was increased in the subsequent research to 60 in Pekoz and Pincheira (2004) and 102 in Pekoz and Pincheira (2005) although the majority of these latter two sets are ground motions from crustal sources. The results of these analyses show that the inelastic displacements increase relative to non-degrading system as the amount of strength degradation (a combination of cyclic and in-cycle degradation) increases. These studies showed that the parameters which affected this the most were the negative post-yield slope and the unloading stiffness of the model. However, only three values of this negative slope were considered with the steepest being 5% of the elastic stiffness. Song and Pincheira investigated a number of S D O F systems that degraded down to the residual strength or exceeded the displacement required to achieve this residual force level. I f a system met either of those criteria, it was considered collapsed. A t short periods, the majority of the ground motions caused the system to collapse but a limiting period above which no collapse was seen was determined for each of the systems. Systems with severe degradation collapsed for the majority of the ground motions for periods as large as 1.0s. Pekoz and Pincheira (2005) also examined the effect of the residual strength level and showed that the displacements of the S D O F systems decreased as the residual strength level increased. Four levels of residual strength were examined, 10, 25, 50 and 75 percent of the yield strength, and the inelastic displacements decreased by up to 55% from a residual strength of 10% to 75% of the yield strength. The issue of instability for in-cycle strength degradation has been investigated by researchers such as Bernal (1992) and Miranda and Akkar (2003). While Bernal examined instability on the characteristics of the system and the properties of the ground motions (peak ground displacement, peak ground velocity, peak duration), Miranda and Akkar investigated the lateral strength required in order to have the system remain stable based on the spectral ordinates of the ground motions. Stability was defined as an inelastic displacement smaller than the theoretically instability point, Asi, as defined in the force-displacement relationship illustrated in Figure 2-5. This force deformation 16 Chapter 2 Literature Review relationship is defined by the yield force, Fy, yield displacement, Ay, initial stiffness, K, and the negative slope ratio, a. The lateral strength required to prevent instability was recorded for 72 crustal ground motions and mean relationships of this lateral strength were established for 10 values of negative post-yield slope. It was observed that the lateral strength required to maintain stability increased as the post-yield slope steepened and for slopes as steep as or steeper than the initial stiffness, the S D O F systems had to remain elastic to prevent instability. The sensitivity of the results to the period of vibration was most prominent in the mildest slopes. Figure 2-6 shows the mean lateral strength required to prevent instability, Rc, as a function of period. F Figure 2-5: Force-displacement relationship of in-cycle strength degrading systems (Miranda and Akkar, 2003). 18 16 14 12 10 8 6 4 2 0 0.0 0.5 1.0 1.5 2.0 2,5 3.0 PERIOD |8] Figure 2-6: Lateral strength required to prevent instability (Miranda and Akkar, 2003). Gupta and Krawinkler (1998) also investigated the effect on a negative post-yield slope. They determined that a negative post-yield slope increased the displacements with decreasing lateral strength and period of vibration. A s seen by Miranda and Akkar, these inelastic displacements also increase as the negative slope becomes steeper. However, - - - • a = 0.(O a«=0.W - - - • a » 0.0» a = 0.15 - - - • a - 0.3O o= 1.00 17 Chapter 2 ; Literature Review these increased displacements from the negative post-yield slope are decreased by the presence of a pinched hysteretic behaviour. These results for two levels of strength decay support the conclusions reached by Miranda and Akkar . 18 Chapter 3 3 GROUND MOTIONS Ground Motions The Cascadia Subduction Zone extends off the west coast of North America from northern California to southern British Columbia where the oceanic Pacific and Juan de Fuca Plates are being subducted by the continental North American Plate (Figure 3-1). This tectonic setting creates three distinct earthquake sources: shallow crustal earthquakes within the overriding continental plate, deep earthquakes referred to as subcrustal events which occur in the subducting oceanic plate and megathrust events which occur at the interface between the continental and oceanic plates. A s the oceanic plate is being subducted, a locked portion of the fault is typically formed and when this locked portion releases, large magnitude megathrust or subduction earthquakes occur. These events typically have a magnitude large than 8.0 and have a longer duration of strong motion than associated with crustal events (Onur, Rogers, Cassidy & Bird , 2006). The mean return period of this subduction event is on the order of 600 years (Adams, 1990) with the last occurrence Jan. 17, 2006 (Satake et al., 1996). Given the current point in the earthquake cycle, there is an 11% probability of a subduction event occurring within the next 50 years (Onur and Seemann, 2004). Such large magnitude events may result in higher levels of seismic demand as well as an increased duration and number of loading cycles compared to the smaller magnitude events typically associated with crustal sources. Therefore, to compare the effect of ground motions recorded from a subduction source, two sets of ground motions, one from crustal sources and another from subduction sources were included in this study. Subcrustal earthquakes are not considered in this study since they w i l l typically not determine the magnitude of the design spectrum subduction regions. Figure 3-1: Cascadia Subduction Zone and tectonic setting of southwestern B C (Onur et. al, 2006). 19 Chapter 3 3.1 Crustal Ground Motions Ground Motions The crustal ground motion suite consists of 20 ground motions recorded on National Earthquake Hazard Reduction Program (NEHRP) site class C and 20 ground motions from site class D . These records are from 8 earthquakes recorded in California with magnitudes ranging from 6.0 to 7.5. The peak ground accelerations range from 66 cm/s (0.07 g) to 557 cm/s (0.57 g) while the peak ground displacements are from 0.7 cm to 38.7cm. Table 3-1 and Table 3-2 show the detailed properties of each of the 20 ground motions from each site class. These ground motions are the same records used in the calibration of the displacement modification procedure included within F E M A 440 ( A T C , 2005) for the two site classes being studied. These ground motions were included in this study to enable statistical comparisons of the results from the subduction ground motions with the results published in F E M A 440. Furthermore, these ground motions provided a basis to investigate the effects of strength degradation. The results w i l l then be compared with the values obtained using the subduction ground motion suite to determine the whether a particular result is common to crustal and subduction records or whether the result is exacerbated by the subduction events. The pseudo-spectral accelerations and spectral displacements of the crustal ground motions have been plotted in Figure 3-2 through Figure 3-5. The mean of the 20 ground motions is plotted along with the spectrum for each of the records and the 2%/50 yr exceedance design spectra for Vancouver, Victoria and Seattle. The Vancouver and Victoria spectra are based on the N B C C 2005 building code while for Seattle the spectra is constructed using the N E H R P procedure in F E M A 356 and the spectral accelerations at periods of vibration of 0.2s and 1.0s from the U S G S (USGS, 2002). The mean spectral acceleration of the crustal ground motions is lower than the design spectra for Vancouver, Victoria and Seattle for both site class C and D , but that is to be expected since the design earthquake is a rare and extreme event that is more representative of the stronger ground motions which reach or exceed these design spectra. This mean displacement spectra are lower than the design spectra for periods of vibration greater than 1.0s but are reasonably close to the design displacements for periods shorter than 1.0s. 20 Chapter 3 Ground Motions Table 3-1: Crustal ground motions site class C ( A T C , 2005) Date Earthquake Name Magnitude (Ms) Station Name Station Code Comp. (deg) Distance (km) PGA (cm/s2) PGV (cm/s) PGD (cm) NEHRP Site Class* 10/15/79 Imperial Valley 6.8 El Centra, Parachute Test Facility 5051 315 14.2 200.2 20.0 7.8 C 02/09/71 San Fernando 6.5 Pasadena, CIT Athenaeum 80053 90 31.7 107.9 14.7 6.6 C 02/09/71 San Fernando 6.5 Pearblossom Pump 269 21 38.9 133.4 4.8 1.4 C 06/28/92 Landers 7.5 Yermo, Fire Station 12149 0 23.2 240.3 57.5 37.5 C 10/17/89 Loma Prieta 7.1 APEEL 7, Pulgas 58378 0 47.7 153.0 18.9 6.9 C 10/17/89 Loma Prieta 7.1 Gilroy #6, San Ysidro Microwave site . 57383 90 19.4 166.9 14.9 2.9 C 10/17/89 Loma Prieta 7.1 Saratoga, Aloha Ave. 58065 0 13 494.5 50.3 14.9 C 10/17/89 Loma Prieta 7.1 Gilroy, Gavilon college Phys Sch Bldg 47006 67 11.6 349.1 21.0 5.5 C 10/17/89 Loma Prieta 7.1 Santa Cruz, UCSC 58135 360 17.9 433.1 20.6 6.7 C 10/17/89 Loma Prieta 7.1 San Francisco, Dimond Heights 58130 90 77 110.8 11.6 3.8 C 10/17/89 Loma Prieta 7.1 Freemont Mission San Jose 57064 0 43 121.6 12.1 4.8 C 10/17/89 Loma Prieta 7.1 Monterey, City Hall 47377 0 44.8 71.4 3.7 1.1 C 10/17/89 Loma Prieta 7.1 Yerba Buena Island 58163 90 80.6 66.5 8.5 2.8 C 10/17/89 Loma Prieta 7.1 Anderson Dam (downstream) 1652 270 21.4 239.4 20.4 6.8 c 04/24/84 Morgan Hill 6.1 Gilroy Gavilon college Phys Scl Bldg 47006 67 16.2 95.0 2.7 0.6 c 04/24/84 Morgan Hill 6.1 Gilroy #6, San Ysidro Microwave Site 57383 90 11.8 280.4 33.4 5.1 c 07/08/86 Palm Springs 6 Fun Valley 5069 45 15.8 126.5 7.9 1.0 c 01/17/94 Northridge 6.8 Littlerock, Brainard Canyon 23595 90 46.9 70.6 6.7 1.3 c 01/17/94 North ridge 6.8 Castaic Old Ridge Route 24278 360 22.6 557.2 43.1 8.0 c 01/17/94 Northridge 6.8 Lake Hughes #1, Fire station #78 24271 0 36.3 84.9 10.3 3.3 c Chapter 3 Ground Motions Table 3-2: Crustal ground motions site class D ( A T C , 2005) Date Earthquake Name Magnitude station Name Station Comp. Distance PGA PGV PGD Site (MSJ Code (deg) (km) (cm/s2) (cm/s) (cm) Class 06/28/92 Landers 7.5 Yermo, Fire Station 22074 270 24.9 240.0 57.8 38.7 D 06/28/92 Landers 7.5 Palm Springs, Airport 12025 90 37.5 87.2 13.5 5.5 D 06/28/92 Landers 7.5 Pomona, 4th and Locust 23525 0 117 65.5 13.9 6.4 D 01/17/94 Northridge 6.8 Los Angeles, Hollywood Storage Bldg. 24303 360 25.5 381.4 29.7 6.4 D 01/17/94 Northridge 6.8 Santa Monica City Hall 24538 90 27.6 866.2 47.9 13.6 D 01/17/94 Northridge 6.8 Los Angeles, N. Westmoreland 90021 0 29 388.8 21.3 2.1 D 10/17/89 Loma Prieta 7.1 Gilroy 2, Hwy 101 Bolsa Road Motel 47380 0 12.7 360.0 34.9 6.6 D 10/17/89 Loma Prieta 7.1 Gilroy 3, Sewage Treatment Plant 47381 0 14.4 544.5 28.7 7.2 D 10/17/89 Loma Prieta 7.1 Hayward, John Muir School 58393 0 57.4 166.5 14.0 4.0 D 10/17/89 Loma Prieta 7.1 Agnews, Agnews State Hospital 57066 0 28.2 157.6 21.8 11.3 D 10/01/87 Whittier Narrows 6.1 Los Angeles, 116 , h St School 14403 270 22.5 388.5 22.4 1.6 D 10/01/87 Whittier Narrows 6.1 Downey, County Maintenance Bldg 14368 180 18.3 193.2 25.1 3.7 D 10/15/79 Imperial Valley 6.8 El Centra #13, Strobel Residence 5059 230 21.9 136.2 10.3 6.0 D 10/15/79 Imperial Valley 6.8 Calexico, Fire Station 5053 225 10.6 269.6 21.1 6.5 D 04/24/84 Morgan Hill 6.1 Gilroy #4, 2905 Anderson Rd 57382 360 12.8 341.4 18.0 2.9 D 04/24/84 Morgan Hill 6.1 Gilroy #7, Mantnilli Ranch.Jamison Rd 57425 0 14 183.0 7.3 0.7 D 04/24/84 Morgan Hill 6.1 Gilroy #2, Keystone Rd. 47380 90 15.1 208.0 8.7 1.9 D 04/24/84 Morgan Hill 6.1 Gilroy #3 Sewage Treatment Plant 47381 90 14.6 196.2 13.3 3.1 D 02/09/71 San Fernando 6.5 Los Angeles, Hollywood Storage Bldg. 135 90 21.2 207.0 28.6 18.6 D 02/09/71 San Fernando 6.5 Vernon, Cmd Terminal 288 277 40.9 104.6 17.5 20.6 D Chapter 3 Ground Motions 23 24 Chapter 3 3.2 Subduction Ground Motions Ground Motions The approach taken by the Geological Survey of Canada in addressing the seismic hazard associated with a subduction event from the Cascadia Subduction Zone has been to use a deterministic scenario due to the fact that the magnitude range for such events is relatively constrained. The probability in the hazard assessment is determined from the uncertainty in the time of occurrence or return period of the earthquake, not the ground motion (Adams, 2003). Adopting this approach, it was desired to identify a previous subduction event similar to that expected for the Cascadia Subduction Zone, and use ground motions from such an event to investigate the displacement demands on S D O F systems (Onur, Rodgers et al., 2006). The ground motions selected for this study were recorded during the 2003 Tokachi-Oki Earthquake which occurred near the island of Hokkaido in Northern Japan and had a Moment Magnitude of 8.0. This earthquake is very close in magnitude to what is expected from the Cascadia Subduction Zone There were 48 records available within 150km of the epicenter from the Kyoshin Network (K-Net) and the Kiban-Kyoshin Network (Kik-Net). These records were recorded on N E H R P site class C, D and E sites, although only the results from site classes C and D have been included in this study. (It should be noted that shear wave velocities were not available for all sites to a depth of 30m, and therefore, to calculate the average shear wave velocity for this range, the last known value was used to extrapolate the results to 30m. This is a conservative assumption as typically the shear wave velocity increases with depth and therefore the actual average shear wave velocity would be somewhat larger than this value.) Table 3-3 summarizes the properties of each ground motion with an epicentral distance between 71 and 152 km. In the Pacific Northwest, major urban areas including Victoria, Vancouver, Seattle, and Portland are all within or just outside this distance range from the Cascadia Subduction Zone. Furthermore, significant damage has been observed within or beyond this distance range for previous subduction earthquakes (2004 Sumatra, 1985 Mexico, 1964 Alaska, and 1960 Chile). While epicentral distance is not the ideal parameter for determining the distance with subduction sources, it provides a reasonable distance parameter which is easily obtained. 25 Chapter 3 Ground Motions However, when the slip distribution from the Tokachi-Oki Earthquake was overlaid on the Cascadia Subduction Zone, the epicenter of the Tokachi-Oki Earthquake is slightly farther than the expected point of energy release for the Cascadia Subduction Zone (Onur, 2006). Therefore, the distance range of ground motions used could be extended in future research. The method used to process these ground motions is described in A P P E N D I X A . The ground motions range in peak ground acceleration from 66 cm/s 2 (-0.07 g) to 969 cm/s (~1.0 g), and have a range of peak ground displacements of 5.7 cm to 54.7 cm. Since, in this study, the concern is in calculating ratios of displacement demands (e.g. E P P vs. Elastic) for strengths normalized to the intensity of the ground motion, the variability in seismic demands apparent in Table 3-3 for sites o f similar distance and site class is not of significant concern. Figure 3-6 and Figure 3-7 show the pseudo-acceleration spectra for the subduction ground motions and Figure 3-8 and Figure 3-9 plot the displacement spectra. The mean spectral accelerations of the site class C subduction ground motions exceed the design spectra for Vancouver and Victoria for periods of vibration larger than 1.0s. For the site class D records, the mean spectral acceleration is greater than or equal to the Vancouver spectrum for periods larger than 0.2s and 1.0s for the Victoria spectrum. The mean ground motions are smaller than the Seattle spectrum for all periods as the seismic hazard in Seattle is dominated by the potential for a large magnitude event on the Seattle Fault which runs beneath the city. This crustal earthquake at a short distance provides a higher level of demand than a subduction event at larger distance. While the spectral accelerations from the subduction ground motions may be smaller than the design spectra for Vancouver and Victoria for periods less than 1.0s, the mean displacement spectra for the subduction ground motions in the period range in negligibly different from the design spectra. For periods greater than 1.0s, the mean displacement spectrum is greater than that of the design spectra for Vancouver and Victoria up to a period of approximately 4.5s for the site class C records. On site class D , the displacement demand is very close to the design spectra for Vancouver and Victoria until a period of 4.0s. 26 Chapter 3 Ground Motions The mean spectral accelerations and displacements of the subduction ground motions are significantly higher than the mean values of the crustal ground motions sets. This increased level o f demand, combined with the additional number of loading cycles, w i l l affect the inelastic displacement demands of the S D O F systems considered in this study. This increased level o f demand is a combination of the larger magnitude of the subduction event as well as the increased long period frequency content of the subduction ground motions ( C R E W , 2005). Table 3-3: Summary of Tokachi-Oki ground motion properties with epicentral distance between 71 and 152km. Station Name Station Distance Component (EW) Component (NS) NEHRP Code (km) PGA PGV PGD PGA PGV PGD Site Class* (cm/s2) (cm/s) (cm) (cra/s1) (cm/s) (cm) ERIMOMISAKI, K-Net HKD112 71 113.5 11.1 6.1 141.7 12.1 5.9 D MEGURO, K-Net HKD113 74 205.1 16.1 5.8 156.2 13.5 7.7 C ERIMO, K-Net HKD1I1 81 66.0 12.0 8.6 66.9 13.7 8.7 D HIROO, K-Net HKD100 84 969.0 49.3 14.1 809.5 40.7 8.5 D SAMANI, K-Net HKD110 102 174.9 28.3 13.1 215.8 39.1 20.2 D TAIKI, K-Net HKD098 103 345.5 91.4 31.5 365.3 75.3 38.7 C SAMANI, Kik-Net HDKH07 104 197.0 39.7 20.3 169.5 27.2 15.1 C TAIKI, Kik-Net TKCH08 109 499.0 45.5 15.4 415.6 22.6 6.1 D URAKAWA, K-Net HKD109 117 238.5 33.0 12.8 184.5 35.6 15.3 D URAHORO, K-Net HKD091 119 374.4 61.3 20.3 391.8 54.0 18.9 E CHOKUBETSU, K-Net HKD086 120 800.5 102.8 54.7 732.7 65.2 24.8 D TOYOKORO, Kik-Net TKCH07 123 404.4 76.4 21.9 365.8 93.4 33.9 D NAKASATSUNAI, K-Net HKD096 128 199.0 33.9 16.0 176.9 24.2 10.5 C SHIRANUKA, K-Net HKD085 131 277.6 47.1 22.8 257.0 39.2 17.8 E SHIRANUKA-S, Kik-Net KSRH09 134 387.2 69.0 33.9 368:5 43.1 18.5 D KUSHIRO, K-Net HKD077 136 410.4 44.1 11.3 314.3 35.6 11.8 D MITSUISHI, K-Net HKD108 136 165.8 15.9 7.8 161.2 20.7 11.4 D IKEDA, K-Net HKD092 138 608.9 53.1 15.5 436.3 49.4 26.0 D OBIHIRO, K-Net HKD095 146 190.6 36.1 19.4 148.4 37.1 27.2 C A K A N , K-Net HKD084 148 353.8 40.5 19.4 352.3 35.6 20.4 C NOYA, K-Net HKD107 148 75.5 16.0 9.6 103.3 24.2 7.4 D AKAN-S, Kik-Net KSRH02 148 405.2 40.8 20.3 373.6 40.1 18.7 D MEMURO, Kik-Net TKCH06 149 144.3 39.2 20.6 163.5 31.1 17.9 D TSURUI-S, Kik-Net KSRH07 152 493.8 40.8 19.9 338.6 36.4 17.9 D *Average shear wave velocity was extrapolated for some sites. 27 28 Chapter 3 Ground Motions NBCC Spectrum - Vancouver NBCC Spectrum - Victoria NEHRP Spectrum - Seattle Mean of C Ground Motions Figure 3-8: Displacement spectra of subduction ground motions recorded on site class C soils. NBCC Spectrum - Vancouver 1 - i NBCC Spectrum - Victoria - - NEHRP Spectrum - Seattle • Mean of D Ground Motions 0.5 1.5 2.5 T(s) 3.5 4.5 Figure 3-9: Displacement spectra of subduction ground motions recorded on site class D soils. 29 Chapter 3 3.3 Peak Ground Acceleration and Displacement Ground Motions Peak ground acceleration (PGA) and displacement (PGD) have often been use to quantify the amount of seismic demand from a given earthquake. The P G A and P G D of each of the crustal and subduction ground motions have been plotted in Figure 3-10 and Figure 3-11. The P G A of the crustal ground motions are very similar in magnitude to those recorded from the Tokachi-Oki Earthquake albeit that the P G A values are at much larger distance from the source with the subduction records. With the crustal ground motions, the magnitude of the P G A decreases as the distance from the source increases. This is also observed in the subduction records, but the rate of decay is much slower as the magnitude of the subduction ground motions at 150 km from the epicenter is still significant. The P G D values from the crustal and subduction ground motions are also similar in magnitude. A s was observed with the P G A values, the P G D values of the subduction records differ from the crustal values in that the magnitude of these P G D s occur at much shorter distances than the subduction records. 1200.0. 1000.0 800.0 J . 600.0 < O Q_ 400.0 200.0 0.0 B I T " • Crustal Site Class C • Crustal Site Class D a Subduction Site Class C • Subduction Site Class D A t I 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 Distance (km) Figure 3-10: Peak ground acceleration (PGA) vs. distance for subduction and crustal ground motions. 30 Chapter 3 Ground Motions • Crustal Site Class C a Crustal Site Class D A Subduction Site Class C • Subduction Site ClassD ^ > • • • • • n i 0 • • • M • • • • A • • • • • • i • • • • .0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 Distance (km) Figure 3-11: Peak ground displacement (PGD) vs. distance for subduction and crustal ground motions. 31 Chapter 4 Hysteretic Models & Analysis Parameters 4 HYSTERETIC MODELS & ANALYSIS PARAMETERS This chapter details the S D O F analyses conducted for this study. Each of the six hysteretic models used is described along with the parameters used to define it. The range of parameters used in the S D O F model is also presented. 4.1 Elastic Perfectly-Plastic The Elastic Perfectly-Plastic (EPP) hysteretic model is the well-known model which exhibits an elastic response to loading until the yield force is reached when the response changes to a plastic behaviour. This model is used to quantify the amount of inelastic deformation in a structure that does not experience stiffness or strength degradation. This model forms the basis for the Cy coefficient in the displacement modification procedure in F E M A 440. The behaviour of this model is examined, but more importantly it is the basis to which other hysteretic models are compared. Figure 4-1 illustrates the force-deformation response of the E P P model. Figure 4-1: E P P hysteretic response. 4.2 Bouc-Wen The Bouc-Wen model used in this study is described by Baber and Noori (1985) and implemented within OpenSees (OpenSees, 2005). This model is similar in shape to the E P P model but differs by having a gradual transition from the elastic to plastic portion of the response. This rounded transition is a better approximation to the actual response of a 32 Chapter 4 Hysteretic Models & Analysis Parameters structure than the more idealistic response assumed in the E P P model; however, the model is significantly more complex and not frequently used in engineering practice. One of the main parameters to be considered when using this model is the degree of rounding in the model defined by the parameter n. A s n increases, the response becomes less rounded and approaches the E P P system and as n decreases the elastic portion of the curve decreases and the transition occurs over a larger displacement. The Bouc-Wen model was used with n = 5.0 and n = 0.5 and the hysteretic behaviour of these systems is illustrated in Figure 4-2. Figure 4-2: Bouc-Wen hysteretic response for (a) n= 5.0 and (b) n=0.5. 4.3 Stiffness Degrading The third hysteretic model used is a stiffness degrading model similar to the modified Clough model (Clough, 1966; Mahin and L i n , 1983). This peak oriented model allowed the loading stiffness of the system to degrade as the model was loaded to larger inelastic displacements. The unloading stiffness of the model can be degraded in OpenSees but was not degraded and remained equal to the elastic stiffness in this study. Stiffness degrading models are typically used to model the behaviour of ductile reinforced concrete structures. This is an important model for construction in Vancouver and Victoria as the majority of high- and mid-rise structures in these areas are constructed of this material. The hysteretic response of the stiffness degrading model is illustrated in Figure 4-3. 33 Chapter 4 Hysteretic Models & Analysis Parameters 4.4 Stiffness and Cyclic Strength Degrading (CSD) The first of the strength degrading models used in this study is the stiffness and cyclic strength degrading (CSD) model. This model was created by the author to degrade the strength of the S D O F system between cycles of loading along a predefined degradation curve. The model is a peak oriented model so that when the model is subject to reloading it aims for the largest inelastic displacement and the maximum strength level. However, when the reloading curve intersects the degradation curve, the response becomes plastic and continues to displace without an increase in strength. This strength level is then the new maximum strength level that is used for defining the reloading stiffness. The degradation curve, which occurs after the yield displacement has been exceeded, is a negative slope that is 5% of the elastic stiffness followed by a residual strength that is 5% of the yield strength. This residual strength was included to improve the numerical stability of the analyses. A diagram of the model parameters is included in Figure 4-4 and a sample force-deformation response of this model has been plotted in Figure 4-5. This model is referred to as the C S D model in Chapter 6. Figure 4-4: Diagram of cyclic strength degrading hyseteric rules (not drawn to scale for clarity). 34 Chapter 4 Hysteretic Models & Analysis Parameters F Figure 4-5: Stiffness and cyclic strength degrading hysteretic response (CSD) with strength degradation curve. 4.5 Stiffness and In-Cycle Strength Degrading (ISD) In-cycle strength degradation occurs in structural elements such as shear critical and flexure-shear reinforced concrete columns (Elwood and Moehle, 2003), concentrically braced steel frames (Uriz and Mahin, 2004) and F R P reinforced masonry walls (ElGawady, Lestuzzi and Badoux, 2005). This type of degradation is different from the previous model in that the degradation of strength occurs within a single loading cycle as opposed to between cycles. The importance of the difference between these types of degradation was highlighted in F E M A 440. In-cycle strength degradation may lead to larger inelastic displacements due to the loss of strength during loading. This loss of strength requires the structure to accelerate to maintain equilibrium which then causes further displacements. Therefore, the dynamic stability of these systems is a concern which is not present in cyclic strength degrading systems. The backbone of this model was defined as an elastic response followed by a negative slope which was 5% of the elastic stiffness similar to the response in Figure 2-5. This rate of degradation is the same as that incorporated within the cyclic strength degrading (CSD) model. For consistency with the cyclic strength degrading model and for numerical stability of the response, a residual strength of 5% of the yield strength was included in the model. This hysteretic response is presented in Figure 4-6. This model is referred to as the ISD model in chapter 6. 35 Chapter 4 Hysteretic Models & Analysis Parameters Figure 4-6: Stiffness and in-cycle strength degrading hysteretic response (ISD). 4.6 Stiffness and In-Cycle Strength Degrading with Residual Strength (ISDR) The last hysteretic model used also includes the effect of in-cycle strength degradation but differs from the stiffness and in-cycle strength degrading model by including a residual level of strength. This residual strength in the system can come from many sources such as: the strength of the unretrofitted masonry wall (ElGawady et al, 2005), the gravity load resisting system, or a redundant lateral force resisting system. The initial portion of the hysteretic backbone is the same for the two previous models but diverge once the yield strength is reached. The ISD model had a post-yield slope of -0.05 times the initial stiffness whereas the post-yield slope can be varied in this model. Ten values of the ratio of post-yield stiffness to the initial stiffness of the system, a, were used to evaluate the importance of this parameter on the inelastic displacement: -0.05, -0.10, -0.20, -0.40, -0.60, -0.80, -1.00, -1.25, -1.50, -1.75 and -2.00. Since significant levels of negative post-yield stiffness were used, only four relative strength values were used: R = 1.5, 2.0, 3.0 and 4.0. The other parameter which was varied in the ISDR model was the ratio of residual strength to the yield strength of the system, y. Nine values of y were considered, ranging from 0.1 to 0.9. A plot of the hysteretic behaviour has been included in Figure 4-7. 36 Chapter 4 Hysteretic Models & Analysis Parameters Figure 4-7: Stiffness and in-cycle strength degrading hysteretic response with residual strength (ISDR). B y varying the residual strength and the post-yield slope, structures that consist of multiple lateral load resisting elements that may provide some residual strength can be modeled. Another application of this model is for comparing the results of a weak ductile system that has been retrofitted to improve the strength but exhibits in-cycle strength degradation until it reaches the strength of the original system. To compare with an equivalent E P P system, certain variables must be defined. The relative strength value of the ISDR system, R, is the value previously defined in Eq . 2-6 while the relative strength value for the equivalent EPP , RR, is defined in Eq . 4-1 and Figure 4-8 with m being the mass of the system, SA, the spectral acceleration at the period of vibration, FR is the residual strength of the system, FY is the yield strength of the ISDR system and y is the residual strength ratio. R^mSJL = R l w h e r e r = FIL FA r (4-1) F/Fv 1.0 Elastic Response aK In-Cycle Strength Degrading System (R = mSJFy) Equivalen&EPP I System (Rk - mSjfyFy) AEPP ^In-cycle Figure 4-8: Stiffness and in-cycle strength degrading hysteretic response with residual strength (ISDR) model parameters. 37 Chapter 4 Hysteretic Models & Analysis Parameters 4.7 Common Analysis Parameters A l l o f the nonlinear time history analyses were conducted using the OpenSees program (OpenSees, 2005). The S D O F model consisted of a single truss element with lumped mass and stiffness subjected to the crustal and subduction ground motions. For each hysteretic model, analyses were conducted for constant strength systems where the relative strength is defined by Eq . 2-6, which defines the yield strength of the system based on the spectral acceleration of the ground motion. The stiffness of the system was defined so that specific periods of vibration are achieved. Nine relative strength values were used for all models excluding the ISDR model: 1.0, 1.5, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0 and 8.0. The ISDR model had only four relative strength values: 1.5, 2.0, 3.0 and 4.0. Fifty periods of vibration were used ranging from 0.05s to 2.0s at intervals of 0.05s and from 2.1s to 3.0s at intervals of 0.1s. These periods were used for each of the hysteretic models with the exception of the elastic perfectly-plastic model which included an additional 20 analyses conducted between 3.1s and 5.0s also at an interval of 0.1s. A l l of the systems included 5% viscous damping. The following sections describe the six hysteretic models used. Table 4-1 summarizes the parameters used in the study. Table 4-1: Analysis parameters Source Number Hysteretic Period Values R Values y Values a Values Type of GM's Models No. Range No. Range No. Range No. Range EPP 70 0.05s to 5.0s Bouc-Wen 1.0... 8.0 1 uctio i 48 Stiffness Degrading 0.05s to 3.0s 9 - -Subd CSD 50 Subd ISD 1 -0.05 ISDR 4 1.5... 4.0 9 0.1 ... 0.9 10 -0.05 ... -2 EPP 70 0.05s to 5.0s Bouc-Wen 1.0... 8.0 +s 40 Stiffness Degrading 0.05s to 3.0s 9 - -I— o CSD 50 ISD 1 -0.05 ISDR 4 1.5... 4.0 9 0.1 ... 0.9 10 -0.05 ... -2 38 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions 5 SDOF ANALYSES WITH CRUSTAL AND SUBDUCTION GROUND MOTIONS The Single-Degree-of-Freedom (SDOF) analyses were conducted using the crustal and subduction ground motions described in Chapter 3. The parameters defining the analyses and the hysteretic models used to define the system are defined in Chapter 4. The results of these analyses are discussed below. 5.1 Elastic Perfectly Plastic The first model used for analysis was the elastic-perfectly-plastic (EPP) hysteretic model. The inelastic displacement o f the EPP system was recorded and compared to the elastic spectral displacement similar to the research conducted by Ruiz-Garcia and Miranda (2003) and others (Krawinkler et al, 2003, Chopra and Chintanapakdee, 2004, Miranda, 2000). The mean of the ratio of the inelastic displacement to the elastic spectral displacement is shown in Figure 5-1 for the 20 crustal ground motions for each site class C and D , and Figure 5-2 for the 12 site class C and 32 site class D subduction ground motions. The inelastic displacement of the S D O F systems are significantly higher than the elastic displacement in the short period range for both site class C and D crustal and subduction ground motions. The inelastic displacement ratios exceed a factor of three for short periods. These high levels of inelastic displacement ratios are of concern, but are mitigated by two different factors. First, structures with very short periods have very small elastic spectral displacements and thus the absolute inelastic displacements experienced by the structure may still be within tolerable limits. Second, many structures which have very short periods have a response that is greatly affected by the interaction between the surrounding soil and the structure. Since F E M A 440 requires the effects of soil-structure interaction be included in the analysis, the inelastic displacements calculated from the coefficient method w i l l be reduced. 39 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions A s the period of the S D O F systems increase, the inelastic displacements decrease and approach a value of 1.0 which indicates that the inelastic displacement is the same as the elastic displacement. Above this period the so-called "equal displacement rule" provides a good approximation of the inelastic displacement. The C/ coefficient from F E M A 440, which accounts for the inelastic behaviour of the structure, has a value of 1.0 for periods of vibration greater than 1 .Os. Considering the mean inelastic displacement ratios for the crustal ground motions, the transition to "equal displacement" region occurs at a period of approximately 1.2s for site class C and 1.4s for site class D . However, the mean inelastic displacement ratios for the subduction ground motions can be considerably higher than those observed for the same system subjected to the crustal ground motions. Since the inelastic displacement ratios are higher for the subduction records, it is logical that the equal displacement rule would become applicable at a higher period of vibration. The mean inelastic displacement ratios for the subduction records approach the elastic spectral displacements at periods greater than 3.0s for relative strength values, R, greater than 2.0. In fact, it can be seen that the equal displacement approximation would underestimate the mean response by approximately 25% for a ductile system with R equal to 4 and a period of 2.3s. Figure 5-2 shows that the transition to the use of the equal displacement rule happens at short periods for R = 1.5 and R =2.0, but happens between approximately 3.0s and 3.5s for the other relative strength values used in this study. 40 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions 2.5 T(s) Figure 5-2: Mean inelastic displacement ratios for E P P hysteretic model for subduction ground motions. The study of inelastic displacement ratios conducted by Ruiz-Garcia and Miranda (2003) found a positive correlation between the magnitude of the earthquake ground motion and the inelastic displacement ratio existed for short periods of vibration. While the inelastic displacement ratios of the subduction records are generally higher than those from the crustal records for all periods of vibration, the larger inelastic displacement ratios at the shorter periods may be the result of the larger magnitude of the Tokachi-Oki Earthquake compared with that of the crustal earthquakes. The larger inelastic displacement ratios in the analyses with higher periods of vibration may be due to either the richer long period content found in subduction records ( C R E W , 2005) or an extension of the effect of the higher magnitude event. Since most large magnitude (M w >8) earthquakes result from subduction sources, it is unclear, but not necessarily significant, whether the increase in the inelastic displacement ratio is due to a magnitude dependence or a source dependence. 41 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions The mean inelastic displacement ratios of the subduction ground motions has been divided by the mean inelastic displacement of the crustal ground motions and shown in Figure 5-3. The mean ratio of the subduction records is up to 3 times higher than the crustal records for periods less than 1 .Os. For periods greater than 1 .Os, the mean inelastic displacement ratio is still significantly higher than the crustal inelastic displacement ratio although the effect is more pronounced in the site class C records than in the site class D records. The effect of the subduction ground motions is more prominent as the strength of the system decreases (i.e. R increases). The inelastic displacement ratios for the R = 8 system are much larger when subjected to the subduction ground motions whereas for the R = 1.5 systems, the effect of the subduction ground motions is negligible for period of vibration greater than approximately 0.25s. ^ ? 2 < ~ i 1 < 0 < R = 1.5 R = 2.0 -— R = 3.0 R = 4.0 — R = 6.0 — R = 8.0 Site Class C 1 T(s) o o - Q 3 W 3 2.5 2 1.5 1 0) < 0.5 R = 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 R = 8.0 1 T(s) Figure 5-3: Ratio of the mean inelastic displacement ratios for subduction ground motions to the mean inelastic displacement ratios for crustal ground motions. The effect o f the subduction ground motions, whether through the enriched long period content or simply magnitude effects, increases the inelastic displacement ratios when compared to those seen through the use of crustal ground motions. This point is further evidenced by the results presented in Figure 5-4 which compares the inelastic displacement ratios of the subduction records with the corresponding values of the Ci coefficient found in F E M A 440. The C ; coefficient is intended to capture the increased displacements associated with inelastic behaviour but does not include the effects of strength or stiffness degradation and was derived based on the analyses conducted by several researchers using crustal ground motions (e.g. Ruiz-Garcia and Miranda 2002, 42 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions 2003, Aydinoglu and Kacmaz, 2002, Ramirez et al, 2002, Chopra and Chintanapakdee, 2004). For periods of vibration less than 0.5s, the C/ coefficient does a poor job of capturing the inelastic displacement ratios of the EPP systems using the subduction ground motion set. This underestimation is somewhat mitigated by the effects of soil-structure interaction which must be considered in following F E M A 440. While the Cj coefficient does a better job of predicting the response for periods larger than 0.5s, there is still significant underestimation, particularly for the site class C ground motions. For a ductile system with R equal to 4, C/ underestimates the inelastic displacement by factors of 2.5 and 3 at a period of 0.2s, and by factors of 1.4 and 1.5 at a period of 1.0s for site classes C and D, respectively. Since C/ is equal to unity for periods greater than 1.0s (the start of the "equal displacement" region for crustal ground motions), the ratio plotted in Figure 5-4 for T > 1.0s represents the inelastic displacement ratio for these systems. The underestimations of the mean inelastic displacement ratios by the F E M A 440 C/ factor will result in an unconservative estimate of the inelastic demand for regions affected by subduction ground motions. 5.1.1 Linear Regression As it was observed in Figure 5-1 and Figure 5-2, the inelastic displacement ratios of the EPP systems subjected to the subduction ground motions were greater than those of the 43 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions same systems subjected to the crustal ground motions. Figure 5-4 illustrates that the inelastic displacement ratios of the E P P systems were not well captured by the Ci coefficient in F E M A 440 and therefore to achieve better predictions of the target displacement, an improvement of the Cy coefficient has been calculated. B y examining the results presented in Figure 5-4, it was felt that a linear approximation could be made to the data to determine a correction factor to the current Cy coefficient in F E M A 440 for regions affected by subduction ground motions. Therefore, this linear approximation was constructed from a period of 0.5s to the point where the values of (Aepp/Ae)/Ci were consistently less than 1.05 (i.e. Aepp/Ae was never more than 5% greater than Cy). A least squares linear regression was performed to determine the slope and the intercept of the correction factor. The resulting equation for the improved Ci coefficient is shown in Eq . 5-1 and plotted in Figure 5-5. Table 5-1 and Table 5-2 show the coefficients for site class C and D for various values of relative strength R. Clsub = CJ = 1 + 1+ R-l aT\ R-l aT2 V C(0.5J)' R for 0.2s <T< 0.5s b- — for 0.5s <T< 1.0s ^ = ^ = 1.0 b v RJ >1.0 for T> 1.0s (5-1) Table 5-1: Coefficients of /7in Cisub coefficient for site class C R Coefficient <2 3 4 5 6 7 8 b 1 1.4 1.6 1.8 2.0 2.1 2.4 c 0 0.42 0.65 1.1 1.6 2.1 3.1 Table 5-2: Coefficients of Bin Cjsub coefficient for site class D R Coefficient <2 3 4 5 6 7 8 b 1 1.3 1.5 1.6 1.7 1.8 1.9 c 0 0.36 0.61 0.87 1.3 1.8 2.2 44 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions " 4 O " 4 o 3 0 Figure 5-5: C; s u b coefficient. The C; sub coefficient incorporates a correction factor, /?, which increases the C/ coefficient in F E M A 440. /? accounts for the increased inelastic displacement ratios of the E P P systems that were subjected to the subduction ground motions. Since the lowest period considered in the regression to determine /? was 0.5s, the value of this factor remains constant for periods smaller than this period. The improved coefficient can be compared with the values from F E M A 440 which were presented in Figure 2-2 to observe the increase in the predicted inelastic displacement. The values determined using F E M A 440 and the C\ Sub coefficient are presented in Figure 5-6 for a relative strength value of 4.0 on both site class C and D soils. The Cj SUb coefficient is much larger than the Cj coefficient from F E M A 440 at short periods. The difference between the two equations decreases as the period of vibration of the system increases until they meet at a period greater than 3.0s. Aside from the increased value of the coefficient, the increased period at which the inelastic displacements become equal to the elastic displacements is significantly larger than for the original coefficient. In F E M A 440, the C/ coefficient is taken as 1.0 for all periods greater than 1.0s while the Cisub reaches 1.0 between T= 1.0s and T = 3.8s. 45 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions d~ 2 C FEMA 440 1 sub R = 4 Site Class o 0 C 1 FEMA 440 1 sub R = 4, Site Class D 1 2 3 0 1 2 T (s) T (s) Figure 5-6: Comparison of C, coefficient from FEMA 440 and Cj sub coefficient. The improved Ci coefficient from Eq. 5-5 was then compared to the inelastic displacement ratios of the subduction ground motions which resulted in the plots in Figure 5-7. When compared with Figure 5-6, it is clear that the relatively simple correction factor results in a considerably better estimation of the inelastic displacement ratio for the subduction ground motions. Only a minor underestimation occurs for a few relative strength values for periods greater than 0.75s for site class C and 0.5s for site class D . The mean inelastic displacement ratios then increase for periods less than these periods with significant underestimations for R equal 3.0 or greater. Furthermore, the dramatic improvement is evident by the sudden decrease in the ratio of the inelastic displacement ratios to the Cy SUb coefficient from before 0.5s, where the regression was stopped, to periods greater than 0.5s where the ratio is very close to one or less than one for all TandR. 46 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions 0.5 0 Site Class C Mean of 12 Ground Motions Figure 5-7: Mean ratio of the inelastic displacement of E P P systems subjected to subduction ground motions to the C, s u b coefficient. 5.1.2 Variability of EPP Results The mean results of these analyses show that there is a significant effect of the subduction ground motions on the inelastic displacement ratio for the EPP system. This information should also be complimented by discussion of the dispersion of the results. Figure 5-8 shows the coefficient of variation for the inelastic displacement ratios using the crustal ground motions, for which the mean was presented in Figure 5-1. For both site class C and D , the coefficient of variation is quite small for the stronger systems with lower R values while the variation generally increases as the relative strength of the EPP system decreases. The variability of the results is not uncommon when dealing with earthquakes and thus it must be recognized that the actual inelastic displacements seen by the structure could be on the order of 50% higher than those calculated using an 47 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions approximate procedure intended to provide a mean estimate of the response. The coefficient of variation for the site class C records is slightly smaller than that for the site class D records for the short periods (T < 0.5s), but the coefficient o f variation is similar between the two sets of records for larger periods. Figure 5-9 presents the coefficient of variation of the inelastic displacement ratios from the subduction ground motions. The coefficient of variation for the site class D records is roughly the same as that seen for the crustal ground motions from the same site class. The coefficient of variation for the site class C results is slightly smaller than that for site class D and also smaller than the coefficient of variation for the crustal ground motions on the same site class at short periods. This is a result of the smaller number of ground motions, 12 for the site class C subduction records compared to the 32 site class D subduction records and 20 crustal ground motions for both site classes. A s the number of ground motions increases the coefficient of variation w i l l approach the variability of the process being modeled (Ang and Tang, 1975). Due to limited data recorded from subduction events, the number of records used is smaller than in previous studies (Ruiz-Garcia and Miranda, 2003), but is consistent with the number of records used for each site class in the development of F E M A 440 ( A T C , 2005). When that coefficient of variation of the inelastic displacement ratios are compared with those published for studies which have used large suites of ground motions (Ruiz-Garcia and Miranda, 2003, Ruiz-Garcia and Miranda, 2002), the variability noticed in Figure 5-8 and Figure 5-9 is consistent with that observed when much larger suites of ground motions are considered. These results show that the longer duration ground motions from the higher magnitude subduction event do not result in higher levels of variability than the crustal ground motions despite the larger inelastic displacements and the increase in the number of yielding events. 48 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions 1.5 1.25 1 I 0.75 to 0.5 0.25 0 0 Site Class C R = 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 R = 8.0 1 2 T ( s ) 1.5 1.25 | 0.75 to 0.5 0.25 0 1 2 T ( s ) Figure 5-8: Coefficient of variation of inelastic displacement ratios for E P P hysteretic model for crustal ground motions. 1.5 1.25 „ 1 1 0.75 0.5 0.25 0 0 Site Class C T(s) 1.5 1.25 ^ 1 CD < | 0.75 1 0 0.5 0.25 0 0 T(s) Figure 5-9: Coefficient of variation of inelastic displacement ratios for E P P hysteretic model for subduction ground motions. 5.2 Bouc-Wen The second hysteretic model used in the analyses is the Bouc-Wen hysteretic model described in Chapter 4. The mean inelastic displacement ratios of the Bouc-Wen systems with n = 5.0 using the crustal ground motions are plotted in Figure 5-10 and using the subduction ground motions in Figure 5-11. Since the hysteretic behaviour of this model is very similar to the E P P model, it is not surprising to observe nearly identical results from to those in Figure 5-1 and Figure 5-2. To identify how similar these results are, the inelastic displacement of the Bouc-Wen system was divided by the inelastic displacement of the EPP systems and the mean results of this ratio are shown in Figure 5-12 for the 49 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions subduction ground motions. The rounding of the hysteretic behaviour is more indicative of the behaviour o f a real structure than the highly idealized EPP behaviour but it becomes abundantly clear that the effect o f the small rounding negligibly affects the inelastic displacement of the system. The results from the crustal ground motions have not been presented but indicate the same trends as in Figure 5-12. 3 2.5 2 1 -s 1.5 I 1 0.5 0 Mean of 20 Ground Motions Site Class C 1 3 2.5 2 <D 3 ~5 1.5 ] 1 0.5 0 Mean of 20 Ground Motions Site Class D 1 T(s) T(s) Figure 5-10: Mean inelastic displacement ratios for the Bouc-Wen hysteretic model for crustal ground motions (n = 5.0). 3 2.5 2 CD a ^ 1.5 a 1 0.5 0 Mean of 12 Ground Motions Site Class C 0 1 3 2.5 2 -4 1.5 x> 1 0.5 0 Site Class D 1 T (s) T (s) Figure 5-11: Mean inelastic displacement ratios for the Bouc-Wen hysteretic model for subduction ground motions (n = 5.0). 50 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions a. CD 3 2.5 2 1.5 1 0.5 R= 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 R = 8.0 CL CD Mean of 12 Ground Motions Site Class C 3 2.5 2 1.5 1 0.5 R = 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 R = 8.0 Mean of 32 Ground Motions Site Class D 0 1 2 3 0 1 2 3 T(s) T(s) Figure 5-12: Mean inelastic displacement to E P P ratios for the Bouc-Wen hysteretic model for subduction ground motions (// = 5.0). To investigate the effect of the amount of rounding or the n parameter of the Bouc-Wen model, additional analyses were conducted using n - 0.5 which produces an extremely rounded hysteresis as shown in Figure 4-2. When the inelastic displacement of this Bouc-Wen system is plotted in Figure 5-13, there is an increase in the inelastic displacements for very short periods, but in general there is very little change. This can be visually confirmed by comparing Figure 5-12 to Figure 5-14 which compares the inelastic displacements of the n = 0.5 Bouc-Wen systems to the inelastic displacements of the EPP systems. There is increased variation with n = 0.5 and significantly higher displacements for very short periods than the inelastic displacements of the E P P systems. With the extremely rounded hysteresis, the rounding begins at a low level of force. Therefore, since the yield displacement of the short period systems is very small, i f the system enters this rounded portion of the curve, the displacement w i l l be much higher than i f the structure remains elastic. Aside from the very short periods, in general the displacements of the Bouc-Wen system are not significantly different than the inelastic displacements of the EPP systems. Therefore, by using to values of n to bound the parameters of the Bouc-Wen model, it can be seen that the influence of the rounded hysteretic behaviour, which is more realistic of the actual behaviour of a structure, provides inelastic displacements which are negligibly different from the inelastic displacements of an equivalent idealized E P P system even for large degrees of rounding. 51 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions 3 2.5 2 9 ! 3 ~5 1.5 -a 3 1 0.5 0 0 Mean of 12 Ground Motions Site Class C 1 T(s) 3 2.5 2 o 3 -* 1.5 3 1 0.5 0 Mean of 32 Ground Motions Site Class D 1 T(s) Figure 5-13: Mean inelastic displacement ratios for the Bouc-Wen hysteretic model for subduction ground motions (« = 0.5). R = 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 R = 8.0 Mean of 12 Ground Motions Site Class C 2.5 CL C L <D ^ 1.5 1 0.5 R = 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 R = 8.0 Mean of 32 Ground Motions Site Class D : 0 0 1 2 3 0 1 2 3 T (s) T (s) Figure 5-14: Mean inelastic displacement to E P P ratios for Bouc-Wen hysteretic model for subduction ground motions (« = 0.5). 5.3 Stiffness Degrading The third model considered contains stiffness degradation within the hysteretic model to model the effect of damage on the structure through repeated loading. The model utilized is a peak-oriented stiffness degrading model similar to the Modified Clough (Mahin and Lin, 1983, Clough, 1966) model that is well established in literature. The C2 coefficient found within F E M A 440 is intended to incorporate the increased inelastic displacements that result from stiffness degradation and strength degradation that happens between cycles. Therefore, when examining the inelastic displacements of the stiffness degrading model, they should be compared with the inelastic displacements of the equivalent EPP system. Thus, the mean ratio of the inelastic displacement of the stiffness degrading 52 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions system to the inelastic displacement of the EPP system is presented in Figure 5-15 for the crustal ground motions and Figure 5-16 for the subduction ground motions. The effect of the stiffness degradation is significant for periods of vibration which are less than 1 .Os. This maximum value of this ratio for site class C and D is at T = 0.05s, the shortest period of analysis, and for the strongest system with R = 1.5. However, i f the shortest periods are removed, the increase in the inelastic displacement from the EPP to the stiffness degrading model are fairly small and for periods greater than 1.0s and 0.55s for site class C and D respectively for the subduction ground motions and 0.5s and 0.8s respectively for the crustal records, the inelastic displacement from the EPP system is greater than or equal to that of the stiffness degrading system. When these results from the crustal records are compared with the subduction records, the results from the subduction ground motions seem to show slightly increased values of this ratio. This is easily identified by the maximum value seen which was 2.5 for R = 1.5 whereas the maximum value for the crustal ground motions is only 2.0 for R = 1.5. In addition, where the inelastic displacement of the EPP system was greater than or equal to the inelastic displacement of the stiffness degrading system with the subduction ground motions, the difference between these values is even less with the crustal ground motions. 3 2.5 2 Q . a ^ 1.5 1 0.5 0 3 2.5 2 a. Q. a> 3 1.5 •D Mean of 20 Ground Motion; Site Class C 1 0.5 0 R = 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 — R = 8.0 Mean of 20 Ground Motions Site Class D 0 1 2 3 0 1 2 3 T (s) T (s) Figure 5-15: Mean inelastic displacement to EPP ratios for stiffness degrading hysteretic model for crustal ground motions. 53 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions CL C L 3 2.5 2 1.5 1 0.5 0 R= 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 R = 8.0 Site Class C Mean of 12 Ground Motions 3 2.5 2 a. Q. o ^ 1.5 1 0.5 0 R = 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 R = 8.0 Site Class D. Mean of 32 Ground Motions 0 1 2 3 0 1 2 3 T ( s ) T (s) Figure 5-16: Mean inelastic displacement to E P P ratios for stiffness degrading hysteretic model for subduction ground motions. To determine the degree to which the increased displacements are captured by the d coefficient in F E M A 440, the mean ratio of the inelastic displacement from the stiffness degrading systems divided by the inelastic displacement of the EPP model was divided by the value of C2 for each of the periods used in the analysis. The C2 coefficient from F E M A 440 provides a conservative estimate of this ratio with the exception of the very short period range wifri stronger systems as the C2 coefficient is small for lower R values and thus doesn't capture the increased displacement observed within the analyses. The comparison of the results presented in Figure 5-16 to the d coefficient in F E M A 440 is shown in Figure 5-17. o 3 2.5 2 1.5 1 0.5 0 Site Class C Mean of 12 Ground Motions 3 2.5 O" 2 31.5 1 0.5 0 Site Class D ; Mean of 32 Ground Motions 0 1 2 3 0 1 2 3 T (s) T (s) Figure 5-17: Ratio of increased displacement from stiffness degradation from subduction ground motions to C2 in F E M A 440. 54 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions 5.4 Target Displacements To calculate the target displacement of a structure using F E M A 440, not only is the value of Ci required, but also the values of the other coefficients and the elastic spectral displacement expected. The inelastic displacements of the E P P system were not captured by the Ci coefficient while the effect of stiffness degradation was captured by the C2 coefficient. Therefore, the inelastic displacements of the E P P systems were compared with the target displacements from F E M A 440 to evaluate whether the higher values of the Ci coefficient for subduction ground motions discussed in 5.1 w i l l have an effect on the design displacements for three cities affected by the Cascadia Subduction Zone. The elastic spectral accelerations for Vancouver and Victoria were calculated from the 2005 N B C C which uses a 2%/50yr Uniform Hazard Spectrum. The accelerations for Seattle were determined from the spectrum defined by the 2002 U S G S maps for 2%/50yr exceedance (USGS, 2002) for the spectral acceleration at 0.2s and 1.0s. The Cy coefficient from F E M A 440 was used to determine the target displacement for each location. These spectral displacements are plotted in Figure 3-8 and Figure 3-9 and the Cy coefficient used is plotted in Figure 2-2. When the mean of the ratio of the target displacement from F E M A 440 to the inelastic displacement of the E P P systems subjected to the individual subduction ground motions is plotted in Figure 5-18 and Figure 5-19 for Vancouver, the target displacement calculated by F E M A 440 is greater than the inelastic displacement of the E P P system for R less than or equal to 4.0 for site class C and for all R for site class D with the exception of very short periods. For Victoria, the target displacement is greater than the inelastic displacement for all values of R, both site class C and D , and all periods with the exception of very short periods. Similarly, the target displacement for Seattle is greater than the inelastic displacement for almost all periods but differs from the results observed for Victoria in that the target displacement is significantly higher and greater than 2.0 times the inelastic displacement for the majority of periods and R values. 55 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions These results are counterintuitive as the inelastic displacement ratios for the subduction ground motions are larger than the Cj coefficient as shown in Figure 5-4 and the average spectral displacement of the subduction ground motions is larger than or equal to the N B C C spectra for Vancouver and Victoria, then the product of these two should result in the inelastic displacement of the E P P systems from the subduction ground motions being greater than the target displacement calculated from F E M A 440. Figure 5-18 and Figure 5-19 suggest that the target displacements are conservative, however, these results, which are a function of how this ratio is calculated, are incorrect as presented. While the above results indicate that F E M A 440 is predicting target displacements that are on average larger than the displacements from the inelastic S D O F analyses with the E P P model, this method of analyzing the data is somewhat flawed. For a given period, T, and relative strength value, R, the target displacement calculated by Eqs. 2-5 though 2-8 would predict a single value for 8t and what is shown in Figure 5-18 and Figure 5-19 is the mean or expected value of the constant St divided by the random variable Aineiastic. A t first this may seem like an appropriate way to evaluate this ratio and ultimately no different from taking the mean of the ratio of the random variable AineiasiiC divided by the constant St; however, as w i l l be discussed below it can actually produce drastically different results. If we let the ratio of A i n e i a s t i c divided by St be represented by the random variable X of which outcomes of the random variable are JC„ then the mean of Xcm be represented as: E[X] = jux=~Y,xi (5-2) n ;=i Where E[X] is the expectation of the random variable X, / / x is the mean of the random variable, n is the number of outcomes and x, is the ith outcome of the random variable. Therefore, it follows that i f we take the inverse of this random variable (i.e. the ratio of St to Aineiastic as shown in Figure 5-18 and Figure 5-19) it can then be represented as: 56 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions J I i i i i i L 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 T(s) A A R= 1.5 R = 2.0 „ R = 3.0 ** R = 4.0 R = 6.0 R = ft 0 Seattle - Site Class C Mean of 12 Ground Motions i i i 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 T(s) Figure 5-18: Mean ratio of the target displacement from F E M A 440 for site class C to the inelastic displacement of E P P systems subjected to subduction ground motions. 57 58 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions This form of the mean of a random variable as displayed in Eq . 5-3 is equivalent to 1/H where H is called the Harmonic Mean while the more common formulation of the mean in Eq . 5-2 is known as the Arithmetic Mean. Clearly these two formulations w i l l not result in the same value and it can be shown that the Arithmetic Mean is always greater than the Harmonic Mean. To better understand the difference between the means, the Harmonic mean can also be presented as a weighted average as in Eq . 5-4 with w, being the weighting factor which is equal to i/x,-. Thus, when the value of the weighting factor is substituted into Eq . 5-4 the result is Eq . 5-5 which simplifies to the expression in Eq . 5-3. n n v 1=1 1 (5-5) Therefore, i f x,- is less than 1.0 (i.e. the target displacement predicted by F E M A 440 is larger than Aineiastic), then the weighting factor has a value greater than 1.0. If x, is larger than 1.0 (i.e. the target displacement from F E M A 440 under predicts the inelastic displacement), then the weighting factor is less than 1.0. Thus, when the approximate method or calculated value is conservative, the weighting factor is larger than the weighting factor for the cases when the approximate method is unconservative. This biases the results towards the conservative side and may lead to a false conclusion that the method is acceptable when in fact it is unconservative. Thus the ratio of the random variable to the constant value should be used to provide unbiased results of the mean value. A similar phenomenon was identified by Miranda (2001) in the approximation of the mean inelastic displacement ratios for constant ductility. If the mean ratio of the inelastic displacement divided by the target displacement is plotted as in Figure 5-20 and Figure 5-21, it can be seen that very different results are 59 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions observed than taking the inverse of the ratios plotted in Figure 5-18 and Figure 5-19. For site class C records, the Vancouver N B C C acceleration spectrum and for periods up to between 4.0s and 4.5s, the inelastic displacements of the E P P systems now are greater than the target displacement calculated by F E M A 440 with the exception of periods between 0.15s and 0.75s for R = 1.5 and R = 2.0. For R = 4.0, the mean ratio of the inelastic displacement to the target displacement is approximately 1.5 for periods ranging from 0.2s to 3.0s. For site class D , the target displacement provides a better prediction for all periods but still underestimates by up to 1.5 times for periods greater than 0.2s. However, the target displacement provides a conservative estimate for periods larger than 2.5s which is significantly shorter than for site class C. For site class C in Victoria, the target displacement provides a better estimate than for Vancouver due to the higher displacement demand for Victoria. However, for R = 4.0, the inelastic displacement ranges between 1.0 and 1.5 times the target displacement for periods ranging from 0.2s to 4.3s. A s with Vancouver, the target displacement begins to provide a conservative estimate at higher periods. This happens between periods of 4.0s and 4.5s for site class C and 2.5s and 3.0s for site class D . For Seattle, the result of taking the mean of the inelastic displacement of the E P P system divided by the target displacement may provide slightly different numerical results, but the general conclusion has not changed. For values of relative strength less than 8.0, the target displacement still provides a conservative estimate of the inelastic displacement seen by the EPP systems subjected to the subduction ground motions for both site class C and D . This is primarily due to the fact that the seismic hazard in Seattle is dominated by the Seattle Fault as discussed in Chapter 3. 60 61 3 2.5 2 1 1.5 _ro CD C <" 1 0.5 WO Seattle - Site Class D Mean of 32 Ground Motions R= 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 R = 8.0 0.5 1 1.5 2.5 T(s) 3.5 4.5 Figure 5-21: Mean ratio of the inelastic displacement of E P P systems subjected to subduction ground motions to the target displacement from F E M A 440 for site class D . When the C; sub coefficient was used to calculate the target displacement for systems in Vancouver and Victoria, the results are an improvement over the results in Figure 5-20 and Figure 5-21. The results for site class C in Vancouver and Victoria are shown in 62 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions Figure 5-22 while the values for site class D are plotted in Figure 5-23. For site class C, it can be seen that the target displacement, S, sub, is still underpredicting the inelastic displacement of the EPP systems subjected to the subduction ground motions, however it is still a dramatic improvement over the target displacements used in Figure 5-20 and Figure 5-21. The result of the discrepancy between the target displacement and the inelastic displacement is due primarily to the fact that the subduction ground motions have a mean spectral displacement which is greater than the spectral displacement for Vancouver and Victoria from the 2005 NBCC for periods greater than approx. 0.8s (Figure 3-8). This is not the case for site class D. As shown in Figure 3-9, the average displacement spectrum of the subduction ground motions is very close to that of the design spectra for Vancouver and Victoria. Therefore, with the Ci SUb coefficient, the target displacement for site class D is generally very close to the inelastic displacements of the SDOF systems for periods greater than 0.5s. Figure 5-22: Mean ratio of the inelastic displacement of E P P systems subjected to subduction ground motions to the target displacement using the Cj s u b coefficient for site class C . 63 Chapter 5 SDOF Analyses with Crustal and Subduction Ground Motions 3 -2.5 J 2 \ 1.5 g D 5 1 0.5 0 Vancouver - Site Class D Mean of 32 Ground Motions Figure 5-23: Mean ratio of the inelastic displacement of E P P systems subjected to subduction ground motions to the target displacement using the C, s u b coefficient for site class D. 64 Chapter 6 SDOF Analyses with Strength Degrading Systems 6 SDOF ANALYSES WITH STRENGTH DEGRADING SYSTEMS In the analysis of a structure, strength degradation within the force-displacement relationship can have a significant impact on the inelastic displacement demand. Poorly detailed reinforced concrete structures ( A T C , 2005), concentrically braced frames (Uriz and Mahin, 2004), F R P reinforced masonry walls (ElGawady, Lestuzzi and Badoux, 2005) are just a few examples of structural systems which are susceptible to strength degradation under cyclic loading. Therefore, to better understand the effects of strength degradation, three hysteretic models have been analyzed using the crustal and subduction ground motions suites. Each of the hysteretic models discussed in this chapter include stiffness degradation in addition to the different forms of strength degradation. First, the effect of strength degradation which occurs between cycles of loading (cyclic degradation) is examined. The second and third models incorporate strength degradation which occurs within one cycle of loading (in-cycle degradation), with the third model incorporating a specified level of residual strength after degradation. 6.1 Stiffness and Cyclic Strength Degrading Systems The first hysteretic model used in the S D O F analyses which incorporated strength degradation is the stiffness and cyclic strength degrading (CSD) model. A s described in Chapter 4, this model differs from other cyclic strength degrading models in that the strength degradation has been defined so that a direct comparison between the cyclic and in-cycle strength degrading models can be made. The degradation curve has a slope of 5% of the elastic stiffness as illustrated in Figure 4-4. The inelastic displacement of this model (A c sd) was normalized by the inelastic displacement of an E P P system ( A e p p ) subjected to the same ground motion. The mean of this ratio has been plotted in Figure 6-1 for the crustal ground motions and in Figure 6-2 for the subduction ground motions. S D O F systems with the C S D model show larger 65 Chapter 6 SDOF Analyses with Strength Degrading Systems displacements than the EPP systems. It is observed in results from the crustal ground motions that for periods less than 1 .Os, this ratio increases rapidly for the weaker systems (i.e. higher R values). The increased inelastic displacements are much larger for the site class D records than the site class C records for systems with R greater than 2.0. A s the period increases, the displacement of the C S D systems for both site classes approaches the displacement of the E P P systems, with convergence occurring at a different periods for each relative strength value. For the crustal ground motions, regardless of site class or relative strength value, the displacement of the cyclic strength degrading system is approximately equal the displacement of the EPP system for periods larger than 1.0s. 66 Chapter 6 SDOF Analyses with Strength Degrading Systems The inelastic displacements from the C S D model are significantly higher when the systems are analyzed using the subduction ground motions. Despite the larger displacements there is more consistency between the site classes as the site class C results being only slightly larger than the site class D results. For example, the ratio of the inelastic displacement of the C S D system to the E P P system reaches 3.0 for R = 3.0 at a period of approximately 0.5s for both site class C and D with the subduction ground motions but this ratio is nearly 1.0 for site class C and 1.5 for site class D with the crustal ground motions. With the higher inelastic displacements, the ratio of the inelastic displacements from the subduction ground motions approaches unity for all R values between 2.0s and 2.3s for both site class C and D ground motions (in contrast to convergence to unity at 1.0s for the crustal ground motions). While the effect of the site class on these results is difficult to determine, it is clear that the effect of the subduction ground motions must be considered when assessing the displacement demands for a cyclic strength degrading system. These results are for one value of the negative slope of the degradation curve (5% of the elastic stiffness) and thus to expand the applicability of these results, further analyses should be conducted. The C2 coefficient in F E M A 440 is intended to capture the increased displacements from the C S D model. Figure 6-3 and Figure 6-4 compare the ratios presented in Figure 6-1 and Figure 6-2 to the C2 coefficient. The C2 coefficient does a very good job of capturing this effect for the site class C crustal ground motions but not for the site class D records. This coefficient accounts for an increase in the displacement of the C S D system, but the increase is not sufficient for the site class D crustal records. When the C2 coefficient is applied to the subduction ground motions, the coefficient underestimates the increased displacement from the C S D model as the coefficient is 1.0 for periods larger than 0.7s and the value for periods less than 0.7s in too small to capture this increase. A s was seen in Figure 6-2, ACSd/Aepp did not approach unity until periods much larger than 0.7s and the application of this coefficient should be extended to larger periods for the subduction ground motions in addition to increasing its value. It appears from these figures that, the degradation incorporated within the C S D model is more severe than that used to define the C2 coefficient as this would account for the underestimation of the ratios seen. To 67 Chapter 6 SDOF Analyses with Strength Degrading Systems address this, the amount of degradation should be quantified so that larger coefficients can be used for more severe degradation. 3 2.5 CNI y 2 1 0.5 0 0 Site Class C . • Mean of 20 Ground Motions 1 T(s) o 3 2.5 2 T3 m < 1 0.5 0 0 Site Class D .'• Mean of 20 Ground Motions 1 T(s) Figure 6-3: Ratio of CSD to E P P displacements for crustal ground motions divided by the C2 coefficient from F E M A 440. •a in O 3 2.5 2 1.5 1 0.5 0 Site Class C Mean of 12 Ground Motions 1 T(s) o T3 O 3E <J. 3 2.5 2 1.5 1 0.5 0 0 R = 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 — R = 8.0 Site Class P . ; Mean of 32 Ground Motions 1 T(s) Figure 6-4: Ratio of CSD to E P P displacements for the subduction ground motions divided by the C2 coefficient from F E M A 440. 6.2 Stiffness and In-Cycle Strength Degrading Systems The second strength degrading model used in this study was a stiffness and in-cycle strength degrading (ISD) model. This model was included to enable a comparison between strength degradation which occurs between cycles and degradation that happens within any given cycle of loading. This distinction is important and as it is believed that in-cycle strength degradation can lead to dynamic instability while cyclic strength 68 Chapter 6 SDOF Analyses with Strength Degrading Systems degradation w i l l not ( A T C , 2005). A s detailed in Chapter 4, the backbone of the hysteretic response incorporates a negative post-yield slope equal to 5% of the initial stiffness. To capture dynamic instability within the results of the S D O F analyses with the ISD model, the point at which a system becomes unstable must be defined. The displacement level at which the negative slope would degrade to zero strength i f no residual strength was included was considered the instability limit for the ISD model. This instability limit is presented in Figure 2-5 but is plotted again in Figure 6-5 for clarity. A s with the C S D model, the inelastic displacements of the ISD model (AjSd) are compared to the inelastic displacements of the EPP model ( A e p p ) in Figure 6-6 for the crustal ground motions and Figure 6-7 for the subduction ground motions. These results have excluded those systems which have become unstable as previously defined (systems experiencing instability w i l l be discussed in more detail below). For the crustal ground motions, the inelastic displacements of the ISD systems are reasonably close to the displacements of the E P P systems with the same period of vibration. This observation seems to hold when the ISD systems were analyzed using the subduction ground motions. •^instability Figure 6-5: Backbone curve for ISD model with instability limit. 69 Chapter 6 SDOF Analyses with Strength Degrading Systems 3 2.5 2 1.5 1 0.5 R = 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 R = 8.0 3 2.5 a 2 CL ^ 1-5 1 Mean of 20 Ground Motions Site Class C 0.5 R = 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 R = 8.0 Mean of 20 Ground Motions Site Class D 0 1 2 3 0 1 2 3 T (s) T (s) Figure 6-6: Mean inelastic displacement to E P P ratios for stiffness and in-cycle strength degrading (ISD) hysteretic model for crustal ground motions. CL CL <D 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Motions Site Class C 3 2.5 2 1.5 1 0.5 0 Mean of 32 Ground Motions Site Class D 0 1 2 3 0 1 2 3 T (s) T (s) Figure 6-7: Mean inelastic displacement to E P P ratios for stiffness and in-cycle strength degrading (ISD) hysteretic model for subduction ground motions. While there is a slight increase in these ratios due to the subduction ground motions, the more interesting result from Figure 6-7 is the termination of the mean results as the period decreases. This is highlighted by the results for R = 8.0 where the plot of the mean results terminates at T = 0.3s. The reason for the termination of these results, as well as the results for other values of relative strength, is that all of the ground motions used caused the S D O F systems to exceed the previously defined instability limit. This termination of results is not seen in the results from the crustal ground motions which would indicate a greater frequency of instability with the subduction ground motions. The frequency of instability in these analyses is discussed further in section 6.4 o f this report. 70 Chapter 6 SDOF Analyses with Strength Degrading Systems The inelastic displacements of the ISD system are in general slightly larger than the inelastic displacements of the EPP system with the largest displacements for short periods. However, given that the structure remains stable, this increase is minimal. 6.3 Comparison of Cyclic and In-Cycle Strength Degrading Systems The C S D model was defined so that the rate of strength degradation would make for a direct comparison with the result of the stiffness and ISD model. Since the rates of degradation follow the same pattern as seen in Figure 4-5 and Figure 4-6, the comparison then focuses on the nature of the strength degradation and its impact on the inelastic displacements. The comparison w i l l focus on results where instability was not observed for the ISD model. The inelastic displacements of the ISD systems were divided by the inelastic displacements of the C S D systems and the mean of this ratio is plotted in Figure 6-8 and Figure 6-9 for the crustal and subduction ground motions respectively. The ratio of the displacements for the two types of strength degradation can be considered as 1.0 for practical purposes. Both of these figures show that the inelastic displacements of the ISD system are only slightly larger than those of the corresponding C S D system. This result indicates that ISD systems do not result in drastically higher inelastic displacements than those associated with cyclic strength degradation. However, it must be mentioned, that the premise for this argument is that the ISD system does not experience instability which would result in the significant damage to the structure. 71 Chapter 6 SDOF Analyses with Strength Degrading Systems X3 w O <j 3 2.5 2 1.5 1 0.5 0 R : R-R-R : R • R : 1.5 2.0 3.0 4.0 6.0 8.0 Mean of 20 Ground Motions Site Class C 3 2.5 2 8 ^ 1.5 1 0.5 0 R • R •• R . R : R: R : 1.5 2.0 3.0 4.0 6.0 8.0 Mean of 20 Ground Motions Site Class D 0 1 2 3 0 1 2 3 T (s) T (s) Figure 6-8: Ratio of the inelastic displacements from the ISD model to the CSD model for crustal ground motions. 3 2.5 2 w u x> 1.5 <r 1 0.5 0 R • R R • R • R • R •• 1.5 2.0 3.0 4.0 6.0 8.0 Mean of 12 Ground Motions Site Class C 3 2.5 2 "3 1.5 1 0.5 0 R = 1.5 R = 2.0 R = 3.0 R = 4.0 R = 6.0 R = 8.0 Mean Of 32 Ground Motions Site Class D: 0 1 2 3 0 1 2 3 T (s) T (s) Figure 6-9: Ratio of the inelastic displacements from the ISD model to the CSD model for subduction ground motions. To illustrate the differences in the hysteretic response of the two types of degradation, the force-displacement relationships of the two models have been plotted in Figure 6-10 and Figure 6-11 for two SDOF systems. The first figure shows how the in-cycle strength degradation causes an increased inelastic displacement during a large loading cycle from the crustal ground motion considered. When the cyclic degrading system is subjected to the same ground motion, the inelastic displacement is smaller as the system becomes plastic during the given loading cycle and thus the ratio of the inelastic displacements, Asd/Axd, is 1.23 for this particular system and ground motion. When the level of inelastic displacement is not significantly larger than the yield displacement, the effect of the type of strength degradation is minimal as seen in Figure 6-11. In this second example, the 72 Chapter 6 SDOF Analyses with Strength Degrading Systems inelastic displacement of the in-cycle strength degradation is the same as that recorded for the same system with cyclic strength degradation. While these results present only one level of strength degradation, the trends presented are very interesting. The inelastic displacements of the ISD system are only slightly larger than the displacements from an equivalent C S D system provided the ISD system remains stable. Thus, the distinction between the types of strength degradation is not important in terms of the inelastic displacements, but is important in establishing the stability of the system. Further analyses need to be conducted to compliment these results for additional levels of strength degradation. cu o o Ll-CU o -0.02 0 0.02 0.04 0.06 0.08 Displacement (m) -0.02 0 0.02 0.04 0.06 0.08 Displacement (m) Figure 6-10: Hysteretic response of CSD and ISD models (T = 0.45s, R = 4.0, Ais/Acsd = 1.23, Los Angeles, Hollywood Storage Building, Northridge 1994). Backbone Cyclic Model -0.02 0 0.02 Displacement (m) -0.02 0 0.02 Displacement (m) Figure 6-11: Hysteretic response of CSD and ISD models {T=1.0s,R = 4.0, Ais/Acsd = 1.00, Gilroy #2, Keystone Rd., Morgan Hill 1984). 73 Chapter 6 SDOF Analyses with Strength Degrading Systems 6.4 Instability of Stiffness and In-Cycle Strength Degrading Systems One of the primary concerns with systems that exhibit negative post-yield behaviour is dynamic stability. A given system which exhibits this behaviour may have satisfactory performance in a nonlinear static analysis but be subject to instability under dynamic loading. Thus it is not sufficient to present the mean ratios of the inelastic displacements of the ISD systems to the EPP systems without discussing the number of analyses which were subjected to dynamic instability. The inelastic displacement at which instability was determined was presented in Figure 6-5. The percentage of ground motions that resulted in instability for the ISD system was determined and presented in Figure 6-12 and Figure 6-13 for the crustal and subduction ground motions respectively. 0 1 2 3 0 1 2 T(s) T(s) Figure 6-12: Frequency of instability for ISD model and crustal ground motions. T (s) T (s) Figure 6-13: Frequency of instability for ISD model and subduction ground motions. 74 Chapter 6 SDOF Analyses with Strength Degrading Systems The first observation from these figures is that as the period decreases, the number of ground motions which cause instability increases and thus the instability frequency increases until it reaches 100% at T = 0.05s. For stronger systems (i.e. R = 1.5, 2.0), virtually none of the systems become unstable until the period of vibration becomes less than 0.5s, while for weaker systems, some ground motions cause instability even at periods of 3.0s. The crustal ground motions recorded on site class C soils have fewer ground motions that cause instability at a given period than the site class D crustal ground motions. The crustal records from both site classes have lower frequencies than the subduction ground motions. The greater frequency of instability with the subduction ground motions is present across the periods used in the analysis and is evidenced by the lack of instability from any strength level for periods larger than 2.5s using the crustal ground motions whereas instability is seen even at 3.0s for the site class D subduction records. The frequencies of instability are higher for the site class C subduction records than for the site class D subduction records, whereas with the crustal ground motions the site class D records result in higher frequencies of instability than the site class C records. This result may be affected by the smaller number of site class C subduction ground motions and thus more subduction records are required before conclusions can be made about the effect of site class. While differences are present between the crustal and subduction data sets, they are not significant enough to indicate an effect of the subduction ground motions. The rates of instability are very important in determining the inelastic displacements of the ISD model, but even i f the structure remains theoretically stable, significant damage could still occur. The number of ground motions which caused the ISD model to exceed a ductility of 10 but not exceed the theoretical instability limit was recorded and the percentage of the ground motions for which this occurred is plotted in Figure 6-14 for the crustal ground motions. Similar results were seen for the subduction ground motions but were not included. The maximum percentage of ground motions which exceeded a ductility of 10 but remained "stable" is 30% for an R o f 6.0 and 40% for an R of 8.0 while the percentage of ground motions is low for all other strength levels. Therefore, for weak systems (R = 6.0, 8.0) not only must instability be considered but also the ductility 75 Chapter 6 SDOF Analyses with Strength Degrading Systems demand on the ISD system. Systems which exceed a ductility of 10 have suffered an extensive amount of damage and while they may be "stable", the applicability o f nonlinear static procedures is questionable at this level. 0 1 2 3 0 1 2 3 T (s) T (s) Figure 6-14: Frequency of theoretically stable ISD systems exceeding a ductility of 10 for crustal ground motions. 6.4.1 Maximum Force Reduction Factors The high frequencies of instability for both the crustal and subduction ground motions must be considered when analyzing a structure and a maximum level of relative strength should be defined in order to reduce the probability of the structure becoming instable under dynamic loading. In the previous section, the instability of the ISD system was examined and highlighted the high frequencies of instability for short period structures. Figure 6-12 showed that when a system with a period of 1.0s is subjected to a crustal record, the probability of the system becoming unstable would be zero for R < 6.0 and 20% for R = 8.0 on site class C. On site class D , the same system would have a probability of zero for R < 3.0 and 5%, 15% and 35% for R =4.0, 6.0 and 8.0 respectively. Therefore, to limit the probability of instability of a system with in-cycle strength degradation, a maximum R value can be defined (Miranda and Akkar, 2003; A T C , 2005). 76 Chapter 6 SDOF Analyses with Strength Degrading Systems To complement the results obtained with the ISD model, additional analyses were conducted using the stiffness and in-cycle strength degrading with residual strength (ISDR) model presented in section 4.6. The results from these analyses, with a residual strength of 0AFy, were used to quantify the maximum values of the force reduction factor (relative strength value, R) to reduce the probability of instability. The same instability limit that was applied in Section 6.2 was used again to determine instability of the ISDR model. B y counting the number of analyses which had inelastic displacements exceeding the instability limit, a cumulative probability of instability could be determined for each period of vibration and post-yield slope. The maximum R value was taken to be the value of R which causes less than 50% of the analyses to reach instability. Since there are only four discrete values of R at which the analyses were conducted, the relative strength value that was associated with a failure probability of less than 50% was selected. Figure 6-15 shows the frequency of instability for the subduction site class D ground motions for one period and two different post-yield slopes as the relative strength of the system changes. The maximum relative strength value is 3.0 for the system with a = -0.20 and 2.0 for a = -0.40. For some values of negative post-yield slope, an instability cumulative frequency of 50% was never achieved for the relative strength values considered, and thus further analyses should be conducted to refine these maximum R values. a = -0.2, T= 1, Site Class D Mean of 32 cy 1 c cu •eq 0.8 Li-CD > 0.6 to um 0.4 O — 0.2 TO "55 0 a = -0.4, T= 1; Site Class D Mean of 32 bid 1 2 3 R R Figure 6-15: Cumulative frequency of instability to determine 77 Chapter 6 SDOF Analyses with Strength Degrading Systems B y calculating the maximum relative strength values for each combination of T and a, the relationship between the instability of the system and the negative post-yield slope can be better understood. Figure 6-16 and Figure 6-17 show the values Rmax o f the systems across the 50 periods and 10 negative post-yield slopes for the crustal and subduction ground motions. A s it was observed before in Figure 6-12 and Figure 6-13, the inelastic displacement of the ISD systems is greater than the instability limit for all R values at short periods and thus as the period decreases Rmax goes to 1.0 for all a values. If Rmax is 1.0, the system w i l l become unstable unless it remains elastic. For the mildest post-yield slopes, Rmax increases rapidly to 4 which is the largest R value considered in the analyses and therefore could be significantly higher than this value as seen in Figure 6-12Figure 6-12 and Figure 6-13. A s a changes to a steeper slope, the maximum relative strength also decreases across all periods. For periods larger than 0.5s, it can be seen that Rmax decreases from 4 with a = -0.05, -0.10 to 3 at a = -0.20 to 1 as a deceases to -2.0 for the subduction ground motions. These reductions in the Rmax value occur at shorter period values for the crustal ground motions. For example, for the crustal ground motions with a = -0.20, the system achieves an Rmax o f 4 at a periods just above 1.0s, while when subjected to the subduction ground motions, it only achieves this level at a period of 2.3s. For steeper negative post-yield slopes (e.g. a = -0.80 to -2.00), the maximum relative strength values vary between an Rmax o f 1.5 and 1.0 indicating that i f the system has a degrading slope steeper than -0.80 times the elastic stiffness, then i f the system does not remain elastic it w i l l l ikely become unstable. 4 3 X ra E m 2 1 o a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 rt/w Site Class C, 20 Ground Motions 1 2 T(s) ra E m 2 a = -0.80 — a = -1.00 — a = -1.25 a = -1.50 — a = -2.00 ird\ 1A1 Site Class C, 20 Ground Motions 1 2 T(s) Figure 6-16: Maximum force reduction factor for crustal ground motions site class C . 78 Chapter 6 SDOF Analyses with Strength Degrading Systems a = -0.80 a = -1.00 a = -1.25 a = -1.50 Hummm Q ~ —2 Q0 ff " . Site Class D, 32 Ground Motions 0 1 2 3 " 0 1 2 3 T (s) T (s) Figure 6-17: Maximum force reduction factor for subduction ground motions site class D. The maximum relative strength values determined and plotted in Figure 6-16 and Figure 6-17, provide useful information, but make practical application of the data challenging. It is difficult to say that using an R value of 3 is appropriate with a = -0.20 at T = 0.7s when the data indicates that Rmax is 2 either side of this point for the subduction ground motions. Therefore, to provide a conservative approximation to this data and create a more idealized relationship, the first time Rmax decreased when moving from higher periods to lower periods, the lower value was selected as the new maximum R. Rmax was not allowed to increase for smaller periods. The result of this idealization for the crustal ground motions is shown in Figure 6-18, while the corresponding plot for the subduction ground motions is in Figure 6-19. Due to the finite number of analyses performed, the results indicate a sudden increase in Rmax for a slight change in period while analyses using non-integer values would result in smooth relationships between Rmax and T. Therefore, judgment should be used when selecting Rmax from Figure 6-18 and Figure 6-19, particularly considering that these Rmax values are for a probability of failure of less than 50%. When the results from the analyses utilizing the site class C subduction ground motions are compared with the results from the site class D records presented in Figure 6-19, the step functions have been shifted to the right (higher periods) for site class C, especially for the steepest negative slopes. This decrease in the Rmax value at a given period is a result of the higher inelastic displacements seen in the site class C ground motions. The 4 3 X CO a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 4 3 X CO Site Class D, 32 Ground Motions 79 Chapter 6 SDOF Analyses with Strength Degrading Systems Rmax values from the crustal ground motions provide more consistent results between the site classes. The higher displacements in the subduction site class C records have the greatest effect on the mildest degrading slopes. This can be observed in the higher rates of instability for these records in Figure 6-13. The effect of the increased displacements in the site class C ground motions is much smaller for the intermediate post-yield slopes but has virtually no effect i f a is -0.8 or steeper. When the results of the crustal ground motions are compared with the same results from the subduction ground motions, the steps in the plots indicating a larger Rmax in general occur at higher periods than with the subduction records. Again, this is l ikely the result of the higher level of demand found with the subduction ground motion suite. These shifts may not seem significant, but i f a structure with a period of 1.0s and designed with R = 3.0 is checked against the results from the crustal ground motions, the plots would say that the structure had a less than 50% chance of becoming unstable for both site classes. However, i f that structure was then checked against the results of the subduction ground motions, the Rmax value has decreased to 2.0 and thus the structure would have a greater than 50% chance of becoming unstable. Thus, the subduction ground motions, whether through increased number of cycles or simply the increased magnitude of the ground motions, increases the occurrences of instability in the ISD model. 80 Chapter 6 SDOF Analyses with Strength Degrading Systems E or 2 E or 2 a = -0.05 a - -0.10 01 = -0.20 a = -0.40 — a = -0.60 Site Class C, 20 Ground Motions 1 2 T(s) a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 Site Class D, 20 Ground Motions to E or 2 E or 2 a = -0.80 a = -1.00 a = -1.25 a = -1.50 — a = -2.00 1 Site Class C, 20 Ground Motions 1 2 a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 X Site Class D, 20 Ground Motions 0 1 2 3 0 1 2 3 T (s) T (s) Figure 6-18: Design maximum force reduction factor for crustal ground motions. E or 2 to E or 2 a = -0.05 a = -0.10 a - -0.20 a = -0.40 a = -0.60 r Site Class C, 12 Ground Motions 1 2 T(s) 0 t a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 E or 2 TO E or 2 a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 J Site Class C, 12 Ground Motions 1 2 T(s) Site Class D, 32 Ground Motions 0 0 = -0.80 (X = -1.00 a = -1.25 a = -1.50 a = -2.00 Site Class D, 32 Ground Motions 1 2 3 0 1 2 T (s) T (s) Figure 6-19: Design maximum force reduction factor for subduction ground motions. 81 Chapter 6 SDOF Analyses with Strength Degrading Systems 6.5 Stiffness and In-Cycle Strength Degrading with Residual Strength Systems The third strength degrading model used in this analytical study into the effects of strength degradation combines the in-cycle strength degradation seen in the ISD model with a residual level of strength. A s described in Chapter 4, this hysteretic model was utilized with the residual strength and the post-yield stiffness varied. The parameters of this model were defined in Figure 4-8 but are presented here again in Figure 6-20 for clarity. The stiffness and in-cycle strength degrading model with residual strength (ISDR) is a representation of the main structural system for a building which originally had a relative strength value of RR with a ductile response similar to that of an EPP system which was retrofitted to increase the strength of the system so that the relative strength is R but then exhibits a negative post-yield slope. The ISDR model was included to determine the effect on the inelastic displacement the system experiences i f this type of strength only retrofit is completed. F/Fv R = ?RA> 1.0 «;• Elastic Response aK In-Cycle Strength Degrading System (R = mSJFy) i K Equivalent'-EPP I System (R$ = mSJyFy) AEPP Ai„.cycie Figure 6-20: Backbone curve of stiffness and in-cycle strength degrading with residual strength (ISDR) hysteretic model and equivalent E P P model. The inelastic displacements of the systems were compared to the elastic spectral displacements of the ground motions. The mean values of this ratio are plotted in Figure 6-21 for two values o f the ratio of peak strength to residual strength, y, and one value o f negative post-yield slope, a, while the remaining plots for the other combinations of y and a are presented in A P P E N D I X B . When the inelastic displacement ratios shown in 82 Chapter 6 SDOF Analyses with Strength Degrading Systems Figure 6-21, Figure 6-22 and A P P E N D I X B are compared with the ratios seen for the E P P systems in Figure 5-2, the values for the ISDR system with a high residual strength, y = 0.8, are found to be very similar to those for the E P P systems whereas it is clear that as the residual strength decreases to y = 0.1, the inelastic displacement ratios increase significantly. The inelastic displacement for a system with R = 4.0 and T = 2.0s, has increased from about 1.25 for y = 0.8 to 2.0 for y = 0.1. To illustrate this further, the inelastic displacement ratios for 3 systems with varying levels of residual strength (y), but with the same relative strength (R), are plotted in Figure 6-22. The influence of the residual strength and the post-yield stiffness can clearly be observed as the system with the post-yield stiffness ratio of -0.10 shows a gradual increase in the inelastic displacement ratio as y decreases from 0.8 to 0.2. However, when the ratio of the elastic stiffness to the post-yield stiffness is -0.80, the increase in the inelastic displacement ratio is much more dramatic even in the decrease from y o f 0.8 to 0.4. The increase in the inelastic displacement is even more significant as y decreases to 0.2 as the inelastic displacement ratio is larger than 3.0 for periods less than 1.7s. 3 2.5 a- 2 W A I -2 1.5 -o co 3 1 0.5 0 Mean of 32 Ground Motions Site Class D Y = 0.8, a = -0.1 : 2.5 W A r-2 1.5 "D CO V) 3 1 0.5 0 Mean of 32 Ground Site Class D y= 0.1, a = -0.1 0 1 2 3 0 1 2 3 T (s) T (s) Figure 6-21: Mean inelastic displacement ratios for ISDR model, a= -0.10 and y= 0.1, 0.8. 83 Chapter 6 SDOF Analyses with Strength Degrading Systems 3 2.5 3 "> A r-£ 1.5 "D in W «a 1 0.5 0 Mean of 32 Ground Motions Site Class D R = 4 ,rt = -0.1 0 1 1 0.5 0 Mean of 32 Ground Motions Site Class D R = 4 i O C = -0.8 1 T(s) T(s) Figure 6-22: Mean inelastic displacement ratios for ISDR model, R = 4.0 and a= -0.1, -0.8. The inelastic displacements of the S D O F systems using the (ISDR) hysteretic model were compared with the inelastic displacements of an equivalent E P P system. The equivalent E P P system was defined as an E P P system where the yield strength was equal to the residual strength of the strength degrading model. Therefore, i f R is the relative strength of the ISDR model and /defines the residual strength level then RR, the relative strength level for the EPP system, can be defined as R/y. Figure 6-20 illustrates the relationship of the backbone curves for these two systems and Figure 6-23 shows the comparison of the various backbones of the strength degrading systems and the equivalent EPP system to which they are being compared. The ratio of the inelastic displacement of the ISDR model to the EPP model are shown in Figure 6-24 and Figure 6-25 for the crustal and subduction ground motions from class D , respectively. R = 1 R = 1.5,y = 0.3 R = 2.0, Y = 0.4 R = 3.0, y = 0.6 R = 4.0, Y = 0.8 — R = 5.0,Y = 1.0 1 0.8 ro CO E 0.6 LL 0.4 0.2 0 0 1 2 3 A./A i e Figure 6-23: Tri-linear backbones of ISDR model. 84 Chapter 6 SDOF Analyses with Strength Degrading Systems When the figures are examined, several trends can be identified in the data. The first trend is that the ratio of the inelastic displacement from the ISDR model to the inelastic displacement of an E P P system with an R value that is equal to RR can be higher than 2. This ratio is greater than 1.0 for the majority of combinations of / a n d a. Therefore, i f a strength only retrofit is considered, the inelastic displacement the ISDR system experiences could be twice that seen by the original structure. The second observation from these figures is that as the negative-post yield slope becomes steeper, the inelastic displacement of the ISDR model increases. This can be seen in the ratio plotted in Figure 6-24 and Figure 6-25 as the inelastic displacement of the E P P system is actually greater than that seen by the hysteretic model for short periods and mild amounts of strength degradation. However, for similar periods but with a = -2.0, the displacement of the I S D R model is more than twice the displacement of the equivalent EPP system. This increase in inelastic displacements as the negative post-yield slope becomes steeper is strongest for low values of y, such as the system with y = 0.3, while the ratio of the inelastic displacement o f the ISDR model to the displacement of the equivalent E P P system is essentially invariant to the value of negative post-yield strength degradation for systems with y= 0.8. The reason for this is that with the high residual strength, as soon as the structure has yielded the strength w i l l rapidly degrade down to the residual strength, and thus, only a small variation is seen within the results of theses systems. For the systems with y= 0.8, the ratio of the inelastic displacement to the equivalent E P P system is slightly larger than unity for periods less than 0.6s but is less than unity for the crustal records and approximately equal to unity for the subduction ground motions for periods larger 0.6s. 85 Chapter 6 SDOF Analyses with Strength Degrading Systems 3 2.5 2 1.5 1 0.5 3 2.5 2 1.5 1 0.5 3 2.5 2 1.5 1 0.5 Mean of 20 Ground Motions Site Class D 1 T ( s ) a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 , R = 5 ,Y=0.4 Mean of 20 Ground Motions Site Class D 1 T ( s ) a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 Mean of 20 Ground Motions Site Class D 1 T(s) a = -0.05 a = -0.10 a = -0.20 u = -0.40 a = -0.60 4, R R = 5™T^8 Mean of 20 Ground Motions Site Class D 1 T ( s ) 3 2.5 2 1.5 1 0.5 0 I 3 2.5 2 1.5 1 0.5 3 2.5 2 1.5 1 0.5 0 i 3 2.5 2 1.5 1 0.5 0 0 1.5, R r = 5,Y = Mean Of 20 Ground Motions Site Class D 1 T ( s ) u = -0.80 a = -1.00 a = -1.25 a = -1.50 — a = -2.00 R = 2, R R = 5 ,Y=0.4 : Mean of 20 Ground Motions Site Class D 1 T(s) a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 3, R R = 5 ,Y=0.6 Mean of 20 Ground Motions Site Class D 1 T(s) a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 Mean of 20 Ground Motions Site Class D 1 T(s) Figure 6-24: Mean ratio of ISDR model to equivalent E P P system for crustal ground motions recorded on site class D, RR = 5. 86 Chapter 6 SDOF Analyses with Strength Degrading Systems 3 2.5 2 (Q T3 V? CD 1.5 DC CD O > . 1 < 0.5 0 a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 '1.5,TR R = 5 ,Y=0.3 Mean of 32 Ground Motions Site Class D 1 T(s) a = -0.05 a = -0.10 cx = -0.20 a = -0.40 a = -0.60 R -<r-Mean of 32 Ground Motions Site Class D 1 T(s) a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 Mean of 32 Ground Motions Site Class D 1 T(s) a = -0.05 a = -0.10 a = -0.20 a - -0.40 tx = -0.60 Mean of 32 Ground Motions Site Class D: 1 T(s) 3 2.5 2 1.5 1 0.5 0 3 2.5 2 1.5 1 0.5 0 i 3 2.5 2 1.5 1 0.5 0 I 3 2.5 2 1.5 1 0.5 0 0 Mean of 32 Ground Motions Site Class D 1 T(s) a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 R = 5,y = Mean of 32 Ground Motions Site Class D 1 T(s) a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 R = 3, R R = 5,Y=0.6 Mean of 32 Ground Motions Site Class D 1 T(s) a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 •4, R R = 5,Y=0.8 Mean of 32 Ground Motions Site Class D 1 T(s) Figure 6-25: Mean ratio of ISDR model to equivalent E P P system for subduction ground motions recorded on site class D, RR = 5. 87 Chapter 6 SDOF Analyses with Strength Degrading Systems When comparing the crustal and subduction results of the ratio of the ISDR system to the equivalent EPP system, the ratio from the subduction ground motions tends to be greater than the ratio from the crustal ground motions for milder values of a. This can be seen by comparing the ratios for the a values with shallower slopes than -0.60. However, when the value of a creates a steep slope, the ratio from the crustal ground motions tends to exceed the ratio from the subduction ground motions. This is observed for a = -0.05, -0.10 with R = 1.5, 2.0 in Figure 6-24 and Figure 6-25. These reduced inelastic displacements of the ISDR model in the subduction ground motions tends to decrease as the value o f the residual strength increases. To present this information in another form, the ratio of the inelastic displacement of the ISDR model and an equivalent EPP model is plotted against the amount of residual strength in the system, y. The ISDR systems with four levels of / are compared with an E P P system with a constant value of RR = 5.0 for the crustal and subduction ground motions. Figure 6-26 and Figure 6-27 present this relationship for two periods and the ten values of a used in the analyses. The effect of the strength degradation rate becomes obvious as the steeper rate of strength degradation leads to greater displacements. This can be seen as the ratio of the displacement of the I S D R model to the equivalent EPP model is well below 1.0 for / = 0.3, T = 0.5s when a = -0.05 but increases to over 1.5 when a = -2.0. Furthermore, the larger displacements for steeper slopes is more significant as the residual strength decreases to y = 0.3. The difference in inelastic displacements is negligible when y = 0.8 whereas the spread in the ratio of inelastic displacements increases when y = 0.3. This amplification with steeper slopes is more prominent in shorter period systems. It is observed in Figure 6-26 and Figure 6-27 that increasing the strength of the system becomes beneficial in terms of reducing the inelastic displacement demands on the structure i f the increase in strength is significant (y less than 0.8), the negative post-yield slope remains fairly mild (a greater than -0.20), and the period is less than approximately 1.3s. However, i f these conditions are not met, the inelastic displacement seen by the "improved" structure is at best equal to the inelastic 88 Chapter 6 SDOF Analyses with Strength Degrading Systems displacement of the equivalent EPP system, and at worst experiences an inelastic displacement that is twice the inelastic displacement of the original system. 1 0.8 0.6 0.4 0.2 o1— 0.5 a = -0.05 a = -0.10 a = -0.20 — a = -0.40 *»™ a = -0.60 M e a n of 20 G r o u n d Mo t i ons S i t e ; C l a s s D, R^ = 5, T = 0.5 1.5 2.5 JA 1 0.8 0.6 0.4 0.2 0 ^ 0.5 a = -0.80 — a = -1.00 — „ - -1.25 — a = -1.50 — a = -2.00 M e a n of 20 G r o u n d Mo t i ons Si te C l a s s D. = 5, T = 0.5 Encycle Residual epp 1.5 / A 2.5 Incycle Residual epp 1 0.8 0.6 0.4 0.2 0 ^ 0.5 a = -0.05 a = -0.10 a = -0.20 a = -0.40 — a = -0.60 M e a n of 20 G r o u n d Mo t i ons S i te C l a s s D, R = 5, J = 3 i i D i 1 A, 1.5 2.5 JA 1 0.8 0.6 0.4 0.2 0 ^ 0.5 a = -0.80 — a - -1.00 — a = -1.25 a - -1.50 — a = -2.00 M e a n of 20 G r o u n d Mo t i ons S i te C l a s s D, R = 5, T = 3 , , K , 1.5 2.5 JA Incycle Residual epp Incycle Residual'"epp Figure 6-26: Mean ratio of ISDR model to equivalent E P P system for crustal ground motions recorded on site class D plotted against y. 1 0.8 0.6 0.4 0.2 o1— 0.5 Si te C l a s s D ct = -0.05 a = -0.10 a = -0.20 a = -0.40 «»— a - -0.60 M e a n of 32 G r o u n d Mo t i ons 5,T = 0.5 1.5 2.5 ,/A 1 0.8 0.6 0.4 0.2 o<— 0.5 a = -0.80 a ~ -1.00 a = -1.25 ™™ a = -1.50 — a = -2.00 M e a n of 32 G r o u n d Mo t i ons S i te C l a s s D, R_ = 5 , T = 0.5 i i a i Incycle Residual epp 1.5 2 Incycle Restdua/^epp 2.5 1 0.8 0.6 0.4 0.2 01— 0.5 <x = -0.05 a = -0.10 — a = -0.20 -0.40 —• a = -0.60 M e a n of 32 G r o u n d Mo t i ons S i t e i C l a s s D ; R = 5, T = 3 1 A, 1.5 2.5 ,/A 1 0.8 0.6 0.4 0.2 OL-0.5 a - -0.80 ...— a - -1.00 — a = -1.25 <™~* a = -1.50 —- H = -2.00 M e a n of 32 G r o u n d Mo t i ons S i t e O l a s s D . R = 5, t = 3 1.5 2.5 /A Incycle Residual epp Incycle Residual'"epp Figure 6-27: Mean ratio of ISDR model to equivalent E P P system for subduction ground motions recorded on site class D plotted against y. 89 Chapter 6 SDOF Analyses with Strength Degrading Systems The results from the ISDR model are compared with the equivalent EPP system for a constant amount of residual strength, y = 0.5, in Figure 6-28 and Figure 6-29 where several trends can be observed. The first is that as the strength of the improved system decreases (R increases) the inelastic displacement of the ISDR model increases. With R = 1.5, the inelastic displacement of the ISDR system is less than the inelastic displacement of the equivalent E P P system for shallow rates of strength degradation (a = -0.05 to -0.40) when subjected to the subduction ground motions and slightly larger with the crustal ground motions. However, as the ISDR systems decreases in strength in to R = 4, the ratio of the inelastic displacement increases so that except for a = -0.05, this ratio has is greater than unity for all periods for both the subduction and crustal ground motions. This effect is more pronounced for a between -0.05 and -0.60 as for steeper negative slopes an increase is seen, but it is seen between values of R = 1.5 and R = 2.0. A s the strength decreases further to R = 4.0, the lower strength does not appear to increase the inelastic displacement significantly for a = -0.80 to -2.0. With the amount of residual strength being held constant, the influence o f the post-yield stiffness was approximately constant over the different relative strength values (i.e. each plot in Figure 6-28 and Figure 6-29 show similar results). The ISDR results are less affected by the post-yield slope when the strength of the system is high, R = 1.5, but for the other values of relative strength, the increased displacement as a result of decreasing the post-yield slope appear consistent. Therefore, the inelastic displacements of the ISDR model when compared to an equivalent E P P system are governed by the rate of strength degradation, a , i f it is shallower than -1.00, and the amount of residual strength present in the system. The R value of the system seems to have little effect on the inelastic displacements once it exceeds 2.0. These results are the summary of 8 R and y combinations and further analyses need to be conducted to validate these results. Similar results have been plotted utilizing the subduction and crustal ground motions recorded on site class C soils. These results are plotted in A P P E N D I X C. 90 Chapter 6 SDOF Analyses with Strength Degrading Systems 3,7=0.5; Mean of 20 Ground Motions Site Class D 1 T(s) a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 m^2, R 4,Y=0.5 Mean of 20 Ground Motions Site Class D 1 T(s) a = -0.05 -0.10 a = -0.20 a = -0.40 — , a _ -0.60 Mean of 20 Ground Motions Site Class D 1 T ( s ) O 3 2.5 2 1.5 1 0.5 0 I 3 2.5 2 1.5 a = -0.80 a = -1.00 a = -1.25 -1.50 — a = -2.00 R = 1.5, R =3,7 = 0.5 Mean of 20 Ground Motions Site Class D 1 T(s) 1 0.5 a = -0.80 a = -1.00 a = -1.25 a = -1.50 — a = -2.00 3 2.5 2 1.5 1 0.5 R = 2, R =4,7=0.5 K Mean of 20 Ground Motions Site Class D 1 T(s) a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 R = 3,R =6,7=0.5 Mean of 20 Ground Motions Site Class D 1 T ( s ) a = -0.05 — a = -0.10 a = -0.20 -0.40 , a = -0.60 Mean of 20 Ground Motions Site Class D 0 1 T(s) 2.5 =5= 2 CO 1.5 0.5 R = 4, Mean a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 R = 8 , 7 = of 20 Ground Motions Site Class D 1 T(s) Figure 6-28: Mean ratio of ISDR model to equivalent E P P system for crustal ground motions recorded on site class D, y=0.5. 91 Chapter 6 SDOF Analyses with Strength Degrading Systems a = -0.05 a = -0.10 a = -0.20 a - -0.40 a = -0.60 1.5, R =3,Y=0.5 Mean of 32 Ground Motions Site Class D 1 T(s) a = -0.05 a = -0.10 a = -0.20 a = -0.40 — a = -0.60 RR = 4,Y=0.5 Mean of 32 Ground Motions Site Class D 1 T(s) a = -0.05 a = -0.10 a = -0.20 a = -0.40 — a = -0.60 3, R R = 6 ,Y=0.5 Mean of 32 Ground Motions Site Class D: 1 T(s) a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 •4, R =8,Y=0.5 Mean of 32 Ground Motions Site Class D 1 T(s) a \ or 3 2.5 2 1.5 1 0.5 0 3 2.5 2 1.5 1 0.5 0 3 2.5 2 1.5 1 0.5 0 a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 R = 1.5, R R = 3 ,Y=0.5 Mean of 32 Ground Motions Site Class D 1 T(s) a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 2, R = 4, y= 0.5 Mean of 32 Ground Motions Site Class D 1 T(s) R a = -0.80 a = -1.00 — a = -1.25 a = -1.50 — a = -2.00 0.5 3 .R R = 6,Y: Mean of 32 Ground Motions Site Class D 1 T(s) cx = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 R = 4, R. 0.5 8,Y = Mean of 32 Ground Motions Site Class D 1 T(s) Figure 6-29: Mean ratio of ISDR model to equivalent E P P system for subduction ground motions recorded on site class D, Y=0.5. 92 Chapter 7 7 CONCLUSIONS Conclusions With the increased usage of nonlinear static procedures, estimation of the target displacement has become more critical. The Coefficient Method is one of the most popular procedures for estimating the target displacement, but was developed using only crustal ground motions. The focus of this study was to examine the effects of subduction ground motions on the target displacement. A secondary focus of the study was to investigate the effects of different types of strength degradation on inelastic displacements. The following paragraphs provide a summary of the conclusions reached in this study. The results for the E P P systems used in this study show that the inelastic displacement ratios are significantly increased by the subduction ground motion suite when compared to the crustal ground motions. The inelastic displacement ratios of the E P P systems with the subduction ground motions were as large as 3 times what was seen when the same systems were used with the crustal ground motions. In addition to the increased value of this ratio, the convergence to the "equal displacement rule" occurs at periods of 1.2s and 1.4s for the crustal ground motions on site class C and D , whereas for the subduction ground motions inelastic displacement ratios reach 1.0 for all relative strength values at periods larger than 3.0s. Since the Cy coefficient in the Coefficient Method was developed from statistical studies utilizing crustal ground motions, it is not surprising that this coefficient does not accurately capture the inelastic displacement ratios of the E P P system using the subduction ground motions. To address the underestimation of the inelastic displacement ratios, a linear regression was performed to provide a correction factor /? which is used to increase the C\ coefficient within F E M A 440 for use with subduction ground motions. Significant variability was seen in the inelastic displacement ratios of the E P P models but this variability was on the same order of magnitude as that reported from studies which included many more ground motions. 93 Chapter 7 Conclusions The inelastic displacements of the Bouc-Wen model were compared to the displacements of the E P P model. The rounded hysteretic behaviour, which is considered to be more realistic of actual structures, resulted in elastic displacements which are virtually identical to those seen for the simpler EPP model for both the crustal and subduction ground motions. The stiffness degrading model when subjected to the subduction ground motions had inelastic displacements that were very close to those seen in EPP systems. The minor increase in the displacements is captured by the C2 coefficient from F E M A 440. The stiffness degrading model with the crustal ground motions showed similar results. The target displacement was calculated by the Coefficient Method in F E M A 440 for the spectral hazard in Vancouver, Victoria and Seattle. Since the effect of the stiffness degrading model was minimal, these displacements were compared with the inelastic displacements o f the EPP model and the subduction ground motions. In general, the target displacement under predicted the inelastic displacements of the E P P systems for Vancouver and Victoria when the coefficients from F E M A 440 were used. After applying the correction factor P, the Coefficient Method shows great improvement in predicting the inelastic displacements. The target displacements for Seattle were greater than the inelastic displacements for the subduction records as the seismic hazard in Seattle is dominated by a crustal fault beneath the city. The effect of strength degradation is very important in the analysis of many types of structures. A s the structure is loaded through many cycles, tiie strength begins to degrade. The two types of degradation included in this study are cyclic strength degradation (CSD) which occurs between cycles of loading and in-cycle strength degradation (ISD) which happens during a given loading cycle. The inelastic displacements of the C S D model were much larger than the displacements of the E P P systems for the crustal ground motions. Similar to what was observed for the inelastic displacement ratios for the E P P systems, the ratio of the C S D and E P P inelastic displacements are larger for the 94 Chapter 7 Conclusions subduction ground motions than for the crustal ground motions. Furthermore, the period at which the inelastic displacement of the C S D model becomes approximately equal to the E P P displacements occurs at higher periods for the subduction ground motions. When the results are compared with the C2 coefficient, the increased inelastic displacements of the C S D model relative to the EPP model are poorly captured by this coefficient. This strength degrading model is not the same as was used to create F E M A 440 and results in higher displacements than previous studies. This underestimation of the inelastic displacements of the C S D systems should be addressed in future research. For the ISD model, the inelastic displacements are slightly larger than those for the E P P model at shorter periods but this increase becomes smaller as the period increases. These results do not include the systems which become unstable. These results are consistent between the crustal and subduction ground motions but the subduction ground motions tend to cause a slightly larger number of the ISD systems to become unstable. Unlike other studies, the degradation curve of the C S D model was calibrated to the negative slope o f the ISD model so that a direct comparison between the models could be made. When the ISD displacements are compared with the C S D displacements, they are very close as the ratio of these displacements is approximately 1.0 for all periods. The instability of the ISD model was compared for 10 values of post-yield slope. The maximum relative strength was determined by taking the highest relative strength value which had less than 50% of the ground motions causing failure. The subduction motions resulted in smaller relative strength values than the crustal ground motions for moderate levels of strength degradation while at severe levels of degradation, there was no appreciable difference. The inelastic displacements of the ISDR systems, which are ISD systems with a residual strength, increased as the amount of residual strength decreased or as the negative post-yield slope became steeper. While the subduction ground motions resulted in slightly higher displacement ratios, the difference was minor. The ISDR displacements were compared to an E P P system which had the same amount of strength as the residual in the 95 Chapter 7 Conclusions ISDR model. The ISDR displacements ranged from 0.4 to greater than 2.0 times the displacement of an equivalent E P P system. This comparison intends to show the effect of a strength only retrofit w i l l decrease the inelastic displacements of the "improved" structure (the ISDR system) from those of the original structure (the equivalent E P P system) i f the increase in strength is significant (y less than 0.8), the negative slope is mi ld (a greater than -0.20) and the period of vibration is short (T less than 1.3s). In general, the inelastic displacements of the ISDR model when compared to an equivalent E P P system are governed by the rate of strength degradation, a , i f it is shallower than -1.00, and the amount of residual strength present in the system. Further, the R value of the system seems to have little effect on the inelastic displacements once it exceeds 2.0. 7.1 Impact of Subduction Ground Motions The influence o f the characteristics of the subduction ground motions when compared to the crustal ground motions used could be found in greater and lesser degrees throughout this study. The subduction ground motions had a significant impact on the inelastic displacement ratios of the E P P systems, the inelastic displacement of the C S D systems when compared to the E P P systems and on the maximum relative strength value, Rmax, which can be used with the ISD model for a probability of instability less than 50%. The subduction ground motion set generated a moderate increase in the frequency of instability of the ISD systems when compared to the crustal ground motion results. This was evident as, for moderate levels of strength degradation, Rmax for the subduction ground motions is less than or equal to Rmax calculated utilizing the crustal ground motions. Little impact was seen from the subduction ground motions on the ratio o f the inelastic displacements of the Bouc-Wen and stiffness degrading models when compared to the inelastic displacements of the E P P model. The inelastic displacements o f the ISD model when compared with the E P P model and when compared with the C S D model showed 96 Chapter 7 Conclusions only a minimal change when the subduction ground motions were used instead of the crustal ground motions. A s was discussed earlier, this is contingent on the ISD system remaining stable. The results of the ISDR model show little influence from the type of ground motions used. 7.2 Future Research The research conducted is intended as the staring point for future research into the effects of subduction ground motions on the inelastic behaviour of structures. This research also presented conclusions with strength degrading systems which should be verified and extended with further research. The areas of research identified by this study which required further investigation are detailed below. o The distance definitions for the subduction records used in this study need to be calibrated to the Cascadia Subduction Zone per current research being conducted by the Geological Survey of Canada o The investigation into the effects of subduction ground motions on the non-strength degrading models should be complemented through the use of additional subduction ground motions. o The response of all hysteretic models used in this study should be examined using a variety of subcrustal ground motions to determine the effect of this source type on the inelastic displacements of the systems considered. o Further analyses need to be conducted with additional levels of strength degradation in the C S D and ISD models to extend the comparison between the inelastic displacements seen in these types of strength degrading systems. 97 Chapter 7 Conclusions o To provide a more refined relationship for the relative strength value which w i l l prevent instability, the ISD model must be analysed using intermediate R values between those already used. Higher R values should be considered since, for mi ld values of negative post-yield slope, an instability cumulative frequency of 50% was not reached. o The ISDR model has been compared to an equivalent E P P model for 8 combinations of R and / . Additional E P P analyses should be conducted to validate the conclusions reached with these 8 combinations. 98 References REFERENCES Adams, J., 1990. Paleoseismicity of the Cascadia subduction zone - evidence from turbidites off the Oregon-Washington margin, Tectonics 9, 569-583. Adams, J., and S. Halchuk, 2003. Fourth generation seismic hazard maps of Canada: Values for over 650 Canadian localities intended for the 2005 National Building Code of Canada, Geological Survey of Canada, Open File 4459 Aydinoglu, N . M . and U . Kacmaz, 2002. 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National Building Code of Canada, 2005, National Research Council Canada, Institute for Research in Construction, Ottawa, O N . 99 References Canadian Commission on Building and Fire Codes, 1995. National Building Code of Canada, 1995, National Research Council Canada, Institute for Research in Construction, Ottawa, O N . Clough, R. W . 1966. Effect of Stiffness Degradation on Earthquake Ductility Requirements. University of California, Department of C i v i l Engineering, Berkeley, California. Chopra, A . K . and C. Chintanapakdee, 2004. Inelastic deformation ratios for design and evaluation of structures: single-degree-of-freedom bilinear systems, Journal of Structural Engineering 130 (9), 1309-1319. C R E W , 2005. Cascadia Subduction Zone Earthquakes: A Magnitude 9.0 Scenario, Cascadia Regional Emergency Workgroup, www.crew.org. ElGawady, M . A . , P. Lestuzzi.and M . Badoux. 2004. Experimental Investigation of Retrofitted URM Walls. 10 t h Canadian Masonry Symposium, Banff, A B . Elwood, K . J. and J. P. Moehle. 2003. Shake table tests and analytical studies on the gravity load collapse of reinforced concrete frames. PEER 2003/01, Pacific Earthquake Engineering Research Center, University of California, Berkeley, C A Gupta, A . and H . Krawinkler. 1998. Effect of Stiffness Degradation on Deformation Demands for SDOF and MDOF Structures. 6 t h U . S . National Conference on Earthquake Engineering, Seattle, W A . Gupta, B . and S. K . Kunnath, 1998. Effect of Hysteretic Model Parameters on Inelastic Seismic Demands. 6 t h U . S . National Conference on Earthquake Engineering, Seattle, W A . Ibarra, L . F. , and H . Krawinkler. 2005. Global collapse of frame structures under seismic excitations. Report No. 152, John A . Blume Earthquake Engineering Research Center, Department of C i v i l and Environmental Engineering, Stanford University, Stanford, C A . Ibarra, L . F. , R. A . Medina, and H . Krawinkler. 2005. Hysteretic models that incorporate strength and stiffness degradation Earthquake Engineering and Structural Dynamics 34 (12) 1489-1511. Krawinkler, H . , R. Medina and B . A lav i , 2003. Seismic drift and ductility demand and their dependence on ground motions. Engineering Structures 25 (5), 637-653. Lawrence Livermore National Laboratory, 1995. SAC2000. Livermore, C A . http://www.llnl.gov/sac/ 100 References Mahin, S. A . , and J. L i n . 1983. Construction of inelastic response spectra for single-degree-of-freedom systems: Computer program and applications. Report No . U C B / E E R C - 8 3 / 1 7 , Earthquake Engineering Research Center, University of California at Berkeley, Berkeley, California. MathWorks, Inc. 2002. MatLab 7.0. Natick, M A . http://www.mathworks.com Mattman, D . W. and K . J. Elwood. 2006. Inelastic Displacement Ratios of S D O F Systems Subjected to Subduction Earthquake Records. 8 t h U .S . National Conference on Earthquake Engineering. Miranda, E . , 2000. Inelastic Displacement Ratios for Structures on F i rm Sites. Journal of Structural Engineering 126 (10), 1150-1159. Miranda, E . 2001. Estimation of Inelastic Deformation Demands of S D O F Systems, Journal of Structural Engineering 127 (9), 1005-1012. Miranda, E . and S. Akkar. 2003. Dynamic Instability of Simple Structural Systems Journal of Structural Engineering 129 (12), 1722-1726. Onur, T. 2006. Personal Communication. Onur, T., G . Rogers, J. Cassidy and A . Bi rd , 2006. Compilation of a suite of strong motion records for earthquake engineering use in Vancouver, BC, 8th U . S . National Conference on Earthquake Engineering, San Francisco Onur, T. and M . R. Seemann, 2004. Probabilities of Significant Earthquake Shaking in Communities across British Columbia: Implications for Emergency Management, Proceedings 13 t h World Conference on Earthquake Engineering, Vancouver, B . C . , Paper 1065. OpenSees ver. 1.6.2, 2005, Open System for Earthquake Engineering Simulation, (http://opensees.berkeley.edu), Pacific Earthquake Engineering Research Center, University of California, Berkeley, C A . Pekoz, H . A . and J. A . Pincheira. 2004. Seismic Response of Strength and Stiffness Degrading Single Degree of Freedom Systems. 13 t h World Conference on Earthquake Engineering, Vancouver, B . C . , Canada. Paper No . 936. Pekoz, H . A . and J. A . Pincheira. 2005. Effect of Hysteretic Shapes on the Seismic Response of Strength and Stiffness Degrading SDOF Systems. 8 t h U . S . National Conference on Earthquake Engineering, San Francisco, C A . Paper No . 456. 101 References Ramirez, O . M . , M . C . Constantinou, A . S . Whittaker, C.A.Kircher , M . W . Johnson and C .Z . Chrysostomou, 2003. Validation of the 2003 N E H R P Provisions' Equivalent Lateral Force and Modal Analysis Procedures for Buildings with Damping Systems, Earthquake Spectra 19 (4), 981-999. Ruiz-Garcia, J., and E . Miranda. 2002a. Displacement modification factors for evaluation of existing structures. The Twelfth European Conference on Earthquake Engineering, Paper No. 168. Ruiz-Garcia, J., and E . Miranda. 2002b. Estimation of maximum inelastic displacement demands on SDOF systems using approximate methods. 7th U . S . National Conference on Earthquake Engineering, Boston, M A . Ruiz-Garcia, J. and E . Miranda, 2003. Inelastic Displacement Ratios for Evaluation of Existing Structures, Earthquake Engineering & Structural Dynamics 32 (8), 1237-1258. Ruiz-Garcia, J. and E . Miranda. 2005. Performance-Based Assessment of Existing Structures Accounting for Residual Displacements. Report No. 153, John A . Blume Earthquake Engineering Research Center, Department of C i v i l and Environmental Engineering, Stanford University, Stanford, California. Satake, K . , K . Shimazaki, Y . Tsuji, and K . Ueda, 1996. Time and size of a giant earthquake in Cascadia inferred from Japanese tsunami records of January 1700, Nature 379, 246-249. Song, J. K . and J. A . Pincheira. 2000. Spectral Displacement Demands of Stiffness- and Strength-Degrading Systems. Earthquake Spectra 16 (4), 817-851. United States Geological Survey. 2002. Interpolated Probabilistic Ground Motion for the Conterminous 48 States by Latitude Longitude, 2002 Data. Retrieved Mar 19, 2006, from http://eqint.cr.usgs.gov/eq-men/html/lookup-2002-intetp-06.htrnl Ur iz , P. and S. Mahin. 2004. Seismic performance of concentrically braced steel frame buildings. 13 t h World Conference on Earthquake Engineering, Vancouver, B . C . , Canada. 102 Appendix A Ground Motion Processing APPENDIX A. GROUND MOTIONS A.1 Ground Motion Processing The ground motions acquired from the National Research Institute for Earth Science and Disaster Prevention (NIED) through their Kyoshin and Kiban-Kyoshin Seismograph networks (www.k-net.bosai.go.jp and www.kik-net.bosai.go.jp). United States Geological Survey (USGS) Geologic Hazards sites and the Pacific Northwest Seismograph Network (PNSN) locations were uncorrected raw records from the seismographs. Therefore, before using these records in the analysis o f the S D O F systems, the ground motion records needed to be corrected and filtered to eliminate the presence of any linear trends in the acceleration time-history as well as removing high and low frequencies. Each record was taken from its original format and modified to create a record which was a single column o f data in units of g. These changes were necessary for the use of the Seismic Analysis Code (SAC2000) (Lawrence Livermore National Laboratory, 1995) to perform the correcting and filtering of the ground motions. The S AC2000 format of data contains a header with detailed information about the contents of the file and, since the files being input contained no headers, certain properties needed to be entered for the purposes of filtering the data. The timestep of the record needs to be changed to reflect that of the time history as the default timestep is Is. The upper and lower corners required for the bandpass filtering were entered into the program to be used for the records being processed. For the subduction ground motions, these corners were chosen at 0.1 H z and 25Hz. Once each file was read into the program, the mean and any linear trends were removed. The record was then a taper was applied so that the acceleration time history started and ended at zero. The final step in the processing was to apply a Butterworth Filter with four poles and two passes to the time history. The time histories were then written to a text file by SAC2000. A macro was written to apply the above procedures to each file processed. 103 Appendix A Ground Motion Processing The output files from SAC2000 were then loaded in MatLab (MathWorks Inc., 2004) for the purposes of creating pseudo-spectral acceleration, spectral acceleration and spectral displacement plots in addition to integrating the time series in the frequency domain so that the resulting velocity and displacement time histories, which were not required for the analyses, could also be viewed. This was done to determine from a qualitative standpoint whether further processing was required. For the subduction ground motions, the original corners of 0.1 H z and 25 H z were retained for all ground motions. A. 2 Crustal Ground Motions - Time Histories Acceleration 400 200 Figure A - l : Acceleration time-history El Centro, Parachute Test Facility, Imperial Valley 1979. Acceleration 200 Figure A-2: Acceleration time-history Pasadena, CIT Athenaeum, San Fernando 1971. Acceleration 200 Figure A-3: Acceleration time-history Pearblossom Pump, San Fernando 1971. 104 Appendix A Ground Motion Processing Acceleration Figure A-4: Acceleration time-history Yermo, Fire Station, Landers 1992. Acceleration 200 100 ,v> | 0 u -100 -200 Figure A-5: Acceleration time-history APEEL 7, Pulgas, Loma Prieta 1989. Acceleration ' 1: •' 1 1. 1 1: \ I' Peaks: 166 9 : -164.9 :: I" 1 i- i i i i-10 •15: 20 s 25 30 35 40 Figure A-6: Acceleration time-history Gilroy #6, San Ysidro Microwave site, Loma Prieta 1989. Acceleration -500 Figure A-7: Acceleration time-history Saratoga, Aloha Ave. Loma Prieta 1989. 105 Appendix A Ground Motion Processing Acceleration Figure A-8: Acceleration time-history Gilroy Gavilon College Phys Scl Bldg, Loma Prieta 1989. Acceleration 500 -500 !• i : : J Peaks: : 431.8 , j , Uitj,/ , : ,| -433.1 •'i i P 0 10 15 20 s 25 30 35 Figure A-9: Acceleration time-history Santa Cruz, TJCSC, Loma Prieta 1989. Acceleration 100 0 1 ° -100 -200 I; I I' I 10 15 20 s 25 30 35: 40 Figure A-10: Acceleration time-history San Francisco, Dimond Heights, Loma Prieta 1989. Acceleration 200 100 1 0 -100 -200 ! ,! I-' \ l | - :| I I A B M I I :i r ii i-10 15 20 ;25 30 35 40; Figure A - l l : Acceleration time-history Freemont Mission, San Jose, Loma Prieta 1989. 106 Appendix A Ground Motion Processing Acceleration Figure A-12: Acceleration time-history Monterey, City Hall, Loma Prieta 1989. Acceleration Figure A-13: Acceleration time-history Yerba Buena Island, Loma Prieta 1989. Acceleration 400 Figure A-14: Acceleration time-history Anderson Dam (downstream), Loma Prieta 1989. Acceleration Figure A-15: Acceleration time-history Gilroy Gavilon College Phys Scl Bldg, Morgan Hill 1984. 107 Appendix A Ground Motion Processing Acceleration 400 200 1' o -200 -400 Figure A-16: Acceleration time-history Gilroy #6, San Ysidro Microwave Site. Morgan Hill 1984. Acceleration 200 100 -100 -200 Figure A-17: Acceleration time-history Fun Valley, Palm Springs 1986. Acceleration Figure A-18: Acceleration time-history Littlerock, Brainard Canyon, Northridge 1994. Acceleration 1000. 500 --500 Figure A-19: Acceleration time-history Castaic Old Ridge Route, Northridge 1994. 108 Appendix A Ground Motion Processing Acceleration Figure A-20: Acceleration time-history Lake Hughes #1, Fire Station #78, Northridge 1994. Acceleration Figure A-21: Acceleration time-history Yermo, Fire Station, Landers 1992. Acceleration -•10.0' Figure A-22: Acceleration time-history Palm Springs, Airport, Landers 1992. Acceleration Figure A-23: Acceleration time-history Pomona, 4 th and Locust, Landers 1992. 109 Appendix A Ground Motion Processing Acceleration Figure A-24: Acceleration time-history Los Angeles, Hollywood Storage Building, Northridge 1994. Acceleration 1000 500 I 0 o -500 -1000 1 1: .1 | . .1 1. Peaks: 738 7 -866.2 i; IK WU i'«Ju. i:»<**M - tM* -: j 'I**.- _« -• -.-^  _ •• •. • • i i- 1 i 1 i 0 10 15 20 25 30 35 40 Figure A-25: Acceleration time-history Santa Monica City Hall, Northridge 1994. Acceleration '400 •200 , tft | 0. o ,200 -400 'i ' i i i i: ; i 0 10 15 :-s 20 25 30 Figure A-26: Acceleration time-history Los Angeles, N. Westmoreland, Northridge 1994. Acceleration o Figure A-27: Acceleration time-history Gilroy #2, Hwy 101 Bolsa Road Motel, Loma Prieta 1989. 110 Appendix A Ground Motion Processing Acceleration Figure A-28: Acceleration time-history Gilroy #3, Sewage Treatment Plant, Loma Prieta 1989. Acceleration 200 Figure A-29: Acceleration time-history Hayward, John Muir School, Loma Prieta 1989. Acceleration 200 -200' Figure A-30: Acceleration time-history Agnews, Agnews State Hospital, Loma Prieta 1989. 400 Acceleration Figure A-31: Acceleration time-history Los Angeles, 116* St. School, Whittier Narrows 1987. I l l Appendix A Ground Motion Processing Acceleration Figure A-32: Acceleration time-history Downey, County Maintenance Bldg, Whittier Narrows 1987. Acceleration Figure A-33: Acceleration time-history El Centro #13, Strobel Residence, Imperial Valley 1979. Acceleration -400 Figure A-34: Acceleration time-history Calexico, Fire Station, Imperial Valley 1979. Acceleration Figure A-35: Acceleration time-history Gilroy #4, 2905 Anderson Rd, Morgan Hill 1984. 112 Appendix A Ground Motion Processing 200 1 Acceleration Figure A-36: Acceleration time-history Gilroy #7, Mantnilli Ranch, Jamison Rd., Morgan Hill 1984. Acceleration 200 -200 Figure A-37: Acceleration time-history Gilroy #2, Keystone Rd, Morgan Hill 1984. Acceleration Figure A-38: Acceleration time-history Gilroy #3, Sewage Treatment Plant, Morgan Hill 1984. Acceleration 200 -200|--400 Figure A-39: Acceleration time-history Los Angeles, Hollywood Storage Building, San Fernando 1971. 113 Appendix A Ground Motion Processing Acceleration 200 Figure A-40: Acceleration time-history Vernon, Cmd Terminal, San Fernando 1971. A 3 Crustal Ground Motions - Pseudo-Acceleration Response Spectrum Figure A-41: Pseudo-acceleration response spectrum: (a) El Centro, Parachute Test Facility, Imperial Valley 1979, (b) Pasadena, CIT Athenaeum, San Fernando 1971. Figure A-42: Pseudo-acceleration response spectrum: (a) Pearblossom Pump, San Fernando 1971, (b)Acceleration time-history Yermo, Fire Station, Landers 1992. 114 Appendix A Ground Motion Processing P t t a J ( M c ) period {MC) Figure A-43: Pseudo-acceleration response spectrum: (a) APEEL 7, Pulgas, Loma Prieta 1989, (b)Gilroy #6, San Ysidro Microwave site, Loma Prieta 1989. Figure A-44: Pseudo-acceleration response spectrum: (a) Saratoga, Aloha Ave. Loma Prieta 1989, (b) Gilroy Gavilon College Phys Scl Bldg, Loma Prieta 1989. Period (tec) PenKf jMe) Figure A-45: Pseudo-acceleration response spectrum: (a) Santa Cruz, UCSC, Loma Prieta 1989, (b) San Francisco, Dimond Heights, Loma Prieta 1989. 115 Appendix A Ground Motion Processing Figure A-46: Pseudo-acceleration response spectrum: (a) Freemont Mission, San Jose, Loma Prieta 1989, (b) Monterey, City Hall, Loma Prieta 1989. Figure A-47: Pseudo-acceleration response spectrum: (a) Yerba Buena Island, Loma Prieta 1989, (b) Anderson Dam (downstream), Loma Prieta 1989. Figure A-48: Pseudo-acceleration response spectrum: (a) Gilroy Gavilon College Phys Scl Bldg, Morgan Hill 1984, (b) Gilroy #6, San Ysidro Microwave Site. Morgan Hill 1984. 116 Appendix A Ground Motion Processing Figure A-49: Pseudo-acceleration response spectrum: (a) Fun Valley, Palm Springs 1986, (b) Littlerock, Brainard Canyon, Northridge 1994. Figure A-50: Pseudo-acceleration response spectrum: (a) Castaic Old Ridge Route, Northridge 1994, (b) Lake Hughes #1, Fire Station #78, Northridge 1994. Figure A-51: Pseudo-acceleration response spectrum: (a) Yermo, Fire Station, Landers 1992, (b) Palm Springs, Airport, Landers 1992. 117 Appendix A Ground Motion Processing Figure A-52: Pseudo-acceleration response spectrum: (a) Pomona, 4' and Locust, Landers 1992, (b) Los Angeles, Hollywood Storage Building, Northridge 1994. Figure A-53: Pseudo-acceleration response spectrum: (a) Santa Monica City Hall, Northridge 1994, (b) Los Angeles, N. Westmoreland, Northridge 1994. Figure A-54: Pseudo-acceleration response spectrum: (a) Gilroy #2, Hwy 101 Bolsa Road Motel, Loma Prieta 1989, (b) Gilroy #3, Sewage Treatment Plant, Loma Prieta 1989. 118 Appendix A Ground Motion Processing Figure A-56: Pseudo-acceleration response spectrum: (a) Los Angeles, 116' St. School, Whittier Narrows 1987, (b) Downey, County Maintenance Bldg, Whittier Narrows 1987. Period (MC) Perkxt l t tc) Figure A-57: Pseudo-acceleration response spectrum: (a) El Centro #13, Strobel Residence, Imperial Valley 1979, (b) Calexico, Fire Station, Imperial Valley 1979. 119 Appendix A Ground Motion Processing PnwcHtttt Period ( M s ) Figure A-58: Pseudo-acceleration response spectrum: (a) Gilroy #4, 2905 Anderson Rd, Morgan Hill 1984, (b) Gilroy #7, Mantnilli Ranch, Jamison Rd., Morgan Hill 1984. Period « « « Period ( « s ) Figure A-59: Pseudo-acceleration response spectrum (a) Gilroy #2, Keystone Rd, Morgan Hill 1984, (b) Gilroy #3, Sewage Treatment Plant, Morgan Hill 1984. P w M d ( M c ) Period feecf Figure A-60: Pseudo-acceleration response spectrum (a) Los Angeles, Hollywood Storage Building, San Fernando 1971, (b) Vernon, Cmd Terminal, San Fernando 1971. 120 Appendix A Ground Motion Processing A.4 Subduction Ground Motions - Time Histories Acceleration 200 Figure A-61: Acceleration time-history site HKD112, EW component, Tokachi-Oki 2003. Acceleration 200 Figure A-62: Acceleration time-history site HKD112, NS component, Tokachi-Oki 2003. Acceleration 300 Figure A-63: Acceleration time-history site HKD113, EW component, Tokachi-Oki 2003. Acceleration 200 •300 Figure A-64: Acceleration time-history site HKD113, NS component, Tokachi-Oki 2003. 121 Appendix A Ground Motion Processing Acceleration Figure A-65: Acceleration time-history site HKD111, EW component, Tokachi-Oki 2003. Acceleration Figure A-66: Acceleration time-history site HKD111, NS component, Tokachi-Oki 2003. Acceleration 300 Figure A-67: Acceleration time-history site HKD100, EW component, Tokachi-Oki 2003. Acceleration 300 Figure A-68: Acceleration time-history site HKD100, NS component, Tokachi-Oki 2003. 122 Appendix A Ground Motion Processing 200 Acceleration 250 Figure A-69: Acceleration time-history site HKD110, EW component, Tokachi-Oki 2003. Acceleration 200 -200 -400 250 Figure A-70: Acceleration time-history site HKD110, NS component, Tokachi-Oki 2003. Acceleration 300 Figure A-71: Acceleration time-history site HKD098, EW component, Tokachi-Oki 2003. Acceleration "g o 300 Figure A-72: Acceleration time-history site HKD098, NS component, Tokachi-Oki 2003. 123 Appendix A Ground Motion Processing Acceleration 200 •300 Figure A-73: Acceleration time-history site HDKH07, EW component, Tokachi-Oki 2003. Acceleration Figure A-74: Acceleration time-history site HDKH07, NS component, Tokachi-Oki 2003. Acceleration 500 -500-.1 300 Figure A-75: Acceleration time-history site TKCH08, EW component, Tokachi-Oki 2003. Acceleration 500 -500,' 300 Figure A-76: Acceleration time-history site TKCH08, NS component, Tokachi-Oki 2003. 124 Appendix A Ground Motion Processing Acceleration 400 Peaks: 238.5 ;i -172.9 -200 20 40 60 80 :::S 100 120 140^ 160 Figure A-77: Acceleration time-history site HKD109, EW component, Tokachi-Oki 2003. Acceleration 200 Figure A-78: Acceleration time-history site HKD109, NS component, Tokachi-Oki 2003. Acceleration -300 Figure A-79: Acceleration time-history site HKD091, EW component, Tokachi-Oki 2003. Acceleration Figure A-80: Acceleration time-history site HKD091, NS component, Tokachi-Oki 2003. 125 Appendix A Ground Motion Processing Acceleration :250 Figure A-81: Acceleration time-history site HKD086, EW component, Tokachi-Oki 2003. Acceleration 500 0 f ° -sop, -1:000 Will l l i l l i i i i i t r i , - •" •i-Peaks: 436.1 . , -73? 7 , ;j is •i 50 100 150; 200 :250 Figure A-82: Acceleration time-history site HKD086, NS component, Tokachi-Oki 2003. Acceleration 500 I 0 -500 300 Figure A-83: Acceleration time-history site TKCH07, EW component, Tokachi-Oki 2003. Acceleration Figure A-84: Acceleration time-history site TKCH07, NS component, Tokachi-Oki 2003. 126 Appendix A Ground Motion Processing Acceleration ,250 Figure A-85: Acceleration time-history site HKD096, EW component, Tokachi-Oki 2003. Acceleration 200 100 1 0 o .-•too, -200 • •• • • • • vr • •• • • ••••••••••••••[.••• •• •••••••••• • • •  ••• i.- ••••••••• ••••••••••••• i 11 '; Peaks: , • . [:.,..• . • ... J: 169.4 . 1 , - ' l i l i l i i ^ ' - 1 7 6 : 9 " • 0 so; too; 150 200 250 Figure A-86: Acceleration time-history site HKD096, NS component, Tokachi-Oki 2003. Acceleration 300 Figure A-87: Acceleration time-history site HKD085, EW component, Tokachi-Oki 2003. Acceleration 300 Figure A-88: Acceleration time-history site HKD085, NS component, Tokachi-Oki 2003. 127 Appendix A Ground Motion Processing Acceleration -300 Figure A-89: Acceleration time-history site KSRH09, EW component, Tokachi-Oki 2003. Acceleration 400 Figure A-90: Acceleration time-history site KSRH09, NS component, Tokachi-Oki 2003. Acceleration 500 -500 Figure A-91: Acceleration time-history site HKD077, EW component, Tokachi-Oki 2003. Acceleration 400 200 1 ° -200 -400 Figure A-92: Acceleration time-history site HKD077, NS component, Tokachi-Oki 2003. 128 Appendix A Ground Motion Processing Acceleration 250 Figure A-93: Acceleration time-history site HKD108, EW component, Tokachi-Oki 2003. Acceleration 250 Figure A-94: Acceleration time-history site HKD108, NS component, Tokachi-Oki 2003. Acceleration 300 Figure A-95: Acceleration time-history site HKD092, EW component, Tokachi-Oki 2003. Acceleration 500 1 0 -500 Figure A-96: Acceleration time-history site HKD092, NS component, Tokachi-Oki 2003. 129 Appendix A Ground Motion Processing Acceleration 300 Figure A-97: Acceleration time-history site HKD095, EW component, Tokachi-Oki 2003. 200 100 I 0 o -100 ,200 Acceleration ..| . .. , .. . . . .. . .. j . .. Peaks: 148.4 -123.9 i ir .1 i i 0 50 100: 150 s 200 250 300 Figure A-98: Acceleration time-history site HKD095, NS component, Tokachi-Oki 2003. Acceleration 300 Figure A-99: Acceleration time-history site HKD084, EW component, Tokachi-Oki 2003. Acceleration 300 Figure A-100: Acceleration time-history site HKD084, NS component, Tokachi-Oki 2003. 130 Appendix A Ground Motion Processing Acceleration Figure A-101: Acceleration time-history site HKD107, EW component, Tokachi-Oki 2003. Acceleration 200 100> 0 -100 ^ i i i i M I I I Peaks: 103.3 i...v.,^ 81lj35:;....,.,,..;;,.,....; UI..*:<,.AWX • - ^PfMffi V '"•»] " i i !\ i 'i :i i^  ;20 40 60 80 100 120; 140 160 180" 200 ••s Figure A-102: Acceleration time-history site HKD107, NS component, Tokachi-Oki 2003. Acceleration 500 -500 • :l 1. | Peaks: 405.2 -346.1 :-i i:; i' 50 100 150 s 200 250: 300 Figure A-103: Acceleration time-history site KSRH02, EW component, Tokachi-Oki 2003. Acceleration 300 Figure A-104: Acceleration time-history site KSRH02, NS component, Tokachi-Oki 2003. 131 Appendix A Ground Motion Processing Acceleration 200 300 Figure A-105: Acceleration time-history site TKCH06, EW component, Tokachi-Oki 2003. Acceleration 300 Figure A-106: Acceleration time-history site TKCH06, NS component, Tokachi-Oki 2003. Acceleration 500 | 0 -500 300 Figure A-107: Acceleration time-history site KSRH07, EW component, Tokachi-Oki 2003. Acceleration 300 Figure A-108: Acceleration time-history site KSRH07, NS component, Tokachi-Oki 2003. 132 Appendix A Ground Motion Processing A.5 Subduction Ground Motions - Pseudo-Acceleration Response Spectrum — — — • — • — • — • — > — > — i »i 0 i i.. I i —.1 : i .1 j :„ \ 0S 1 1 1 1 1 i 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 . 6 7 8 Figure A-109: Pseudo-acceleration response spectrum site HKD112, Tokachi-Oki 2003: (a) EW component, (b) NS component. Figure A-110: Pseudo-acceleration response spectrum site HKD113, Tokachi-Oki 2003: (a) EW component, (b) NS component. Ported (MC) Period Itec) Figure A - l l l : Pseudo-acceleration response spectrum site HKD111, Tokachi-Oki 2003: (a) EW component, (b) NS component. 133 Appendix A Ground Motion Processing P w x x l f i e c ) P e r i a t f t M C ) Figure A-112: Pseudo-acceleration response spectrum site HKD100, Tokachi-Oki 2003: (a) EW component, (b) NS component. Figure A-113: Pseudo-acceleration response spectrum site HKD110, Tokachi-Oki 2003: (a) EW component, (b) NS component. Figure A-114: Pseudo-acceleration response spectrum site HKD098, Tokachi-Oki 2003: (a) EW component, (b) NS component. 134 Appendix A Ground Motion Processing Period IMC) Period itec) Figure A-115: Pseudo-acceleration response spectrum site HDKH07, Tokachi-Oki 2003: (a) EW component, (b) NS component. »1 ' ' ' ' ' • ' -—> i »•'»! ' ' ' " ' • <" 1 Pi t ied (MC) Period (MC) Figure A-116: Pseudo-acceleration response spectrum site TKCH08, Tokachi-Oki 2003: (a) EW component, (b) NS component. Period <MC> Period (sec) Figure A-117: Pseudo-acceleration response spectrum site HKD109, Tokachi-Oki 2003: (a) EW component, (b) NS component. 135 Appendix A Ground Motion Processing Period (see) Period (wc) Figure A-118: Pseudo-acceleration response spectrum site HKD091, Tokachi-Oki 2003: (a) EW component, (b) NS component. Period(sec) Period (wet Figure A-119: Pseudo-acceleration response spectrum site HKD086, Tokachi-Oki 2003: (a) EW component, (b) NS component. Period tt«c> Period (tee) Figure A-120: Pseudo-acceleration response spectrum site TKCH07, Tokachi-Oki 2003: (a) EW component, (b) NS component. 136 Appendix A Ground Motion Processing Period (eeef period (*te> Figure A-121: Pseudo-acceleration response spectrum site HKD096, Tokachi-Oki 2003: (a) EW component, (b) NS component. Period (eeo Period (tec) Figure A-122: Pseudo-acceleration response spectrum site HKD085, Tokachi-Oki 2003: (a) EW component, (b) NS component. Period (wo Period feec) Figure A-123: Pseudo-acceleration response spectrum site KSRH09, Tokachi-Oki 2003: (a) EW component, (b) NS component. 137 Appendix A Ground Motion Processing Figure A-125: Pseudo-acceleration response spectrum site HKD108, Tokachi-Oki 2003: (a) EW component, (b) NS component. Period G*0 Period fMc) Figure A-126: Pseudo-acceleration response spectrum site HKD092, Tokachi-Oki 2003: (a) EW component, (b) NS component. 138 Appendix A Ground Motion Processing Figure A-127: Pseudo-acceleration response spectrum site HKD095, Tokachi-Oki 2003: (a) EW component, (b) NS component. Figure A-128: Pseudo-acceleration response spectrum site HKD084, Tokachi-Oki 2003: (a) EW component, (b) NS component. "i 5 | $ j 5 J j j o„ —^ -j 5 j j j j -Penot fGeo Period {see} Figure A-129: Pseudo-acceleration response spectrum site HKD107, Tokachi-Oki 2003: (a) EW component, (b) NS component. 139 Appendix A Ground Motion Processing Figure A-130: Pseudo-acceleration response spectrum site KSRH02, Tokachi-Oki 2003: (a) EW component, (b) NS component. Figure A-131: Pseudo-acceleration response spectrum site TKCH06, Tokachi-Oki 2003: (a) EW component, (b) NS component. Figure A-132: Pseudo-acceleration response spectrum site KSRH07, Tokachi-Oki 2003: (a) EW component, (b) NS component. 140 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios APPENDIX B. STIFFNESS AND IN-CYCLE STRENGTH DEGRADING WITH RESIDUAL STRENGTH INELASTIC DISPLACEMENT RATIOS 3 2.5 2 <3 ~o m ff 1.5 TO to ( 0 < 1 0.5 0 Mean of 12 Ground Motions Site Class C Y= 0.1, a =-0.05 1 3 2.5 a) 2 £ 1.5 -a CO CO < 1 0.5 0 V. fl R = 1.5 - R = 2 R = 3 R = 4 /lean of 32 Ground Motions Site Class D y= 0.1, pc = -0.05 T(s) T(s) Figure B - l : Mean inelastic displacement ratios for ISDR model, y = 0.1 and a = -0.05. 3 2.5 CD 2 < y) 0) 1.5 •Q tn y i <3 1 0.5 0 R= 1.5 R = 2 R = 3 R = 4 Mean of 12 Ground Motions Site Class C Y = 0.2, a = -0.05 1 3 2.5 2 £ 1.5 y) y) < 1 0.5 0 Mean of 32 Ground Site Class D Y= 0.2, oc = -0.05 Motions 1 T(s) T(s) Figure B-2: Mean inelastic displacement ratios for ISDR model, y = 0.2 and a = -0.05. 3 2.5 2 1.5 1 0.5 0 R = 1.5 R = 2 R = 3 R = 4 Mean of 12 Ground Motions Site Class C Y= 0.3, oc = -0.05 1 3 2.5 CU 2 < '(O cu 1.5 "O CO CO < 1 0.5 0 R = 1.5 R = 2 R = 3 R = 4 Mean of 32 Ground Site Class D Y = 0.3, a = -0.05 Motions 1 T(s) T(s) Figure B-3: Mean inelastic displacement ratios for ISDR model, y = 0.3 and a = -0.05. 141 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 < m OJ 1.5 "O to tti < 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.4, oc = -0.05 Motions 1 3 2.5 . 2 W A C 2 1.5 "D CO (0 < 1 0.5 0 R = 1.5 R = 2 R = 4 Mean of 32 Ground Motions Site Class D y = 0.4, tx = -0.05 T(s) T(s) Figure B-4: Mean inelastic displacement ratios for ISDR model, y = 0.4 and a = -0.05. 3 2.5 2 1.5 1 0.5 0 R= 1.5 R = 2 R = 3 R = 4 .A \ Mean of 12 Ground Motions Site Class C Y= 0.5, a = -0.05 3 2.5 2 I 1.5 •o CO CO 1 0.5 0 Mean of 32 Ground Site Class D Y= 0.5, a = -0.05 Motions R = 1.5 R = 2 nmm R = 3 R = 4 Mean of 12 Ground Motions Site Class C Y= 0.6, a = -0.05 1 T(s) Mean of 32 Ground Motions Site Class D Y= 0.6, cc = -0.05 1 Figure B-6: Mean inelastic displacement ratios for ISDR model, Y = 0.6 and a = -0.05. 142 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 CO 2 < c 1.5 •g 01 < 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.7, a = -0.05 Motions 1 3 2.5 o 2 A I-£ 1.5 "D U) CO < 1 0.5 0 Mean of 32 Ground Site Class D y = 0.7, a = -0.05 Motions 1 T(s) T(s) 3 2.5 a 2 < in co 1.5 "D tn < 1 0.5 0 R = 1.5 — R - 2 " ™ " R — 3 R = 4 Mean of 12 Ground Motions Site Class C Y = 0.8, oc = -0.05 Mean of 32 Ground Motions Site Class D Y= 0.8, oc =-0.05 1 T(s) T(s) Figure B-8: Mean inelastic displacement ratios for ISDR model, Y = 0.8 and a = -0.05. 143 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 * 2 W A I -£ 1.5 CO « « 1 0.5 0 \ R B 1.5 R = 2 N"V R = 4 [ \ A re r VI / \ \ A w 3 * ' v L * A s < = : / : : : : : : : V C i Mean of 12 Ground Motions Site Class C 7=0.1, a = -0.1 0 3 2.5 a, 2 < W A r~ £ 1.5 •p to to < 1 0.5 0 Mean of 32 Ground Motions Site Class D y= 0.1, ot = -0.1 1 T(s) T(s) Figure B-10: Mean inelastic displacement ratios for ISDR model, y = 0.1 and a = -0.10. 3 2.5 . 2 £ 1.5 t/> < 1 0.5 0 Mean of 12 Ground Motions Site Class C y= 0.2, a = -0.1 3 2.5 ST 2 W A r-£ 1.5 TJ CO CO < 1 0.5 0 Mean of 32 Ground Site Class D y = 0.2, a = -0.1 Motions 0 1 2 3 0 1 2 3 T(s) T(s) Figure B - l l : Mean inelastic displacement ratios for ISDR model, y = 0.2 and a = -0.10. 3 2.5 2 "i 1.5 "O to to < 1 0.5 0 R = 1.5 R = 2 R = 3 R = 4 Mean of 12 Ground Motions Site Class C y= 0.3, cc = -0.1 3 2.5 2 CO 1.5 CO to < 1 0.5 0 R = 1.5 R = 2 " ™ R = 3 R = 4 w Mean of 32 Ground Motions Site Class D y= 0.3, cc = -0.1 T(s) T(s) Figure B-12: Mean inelastic displacement ratios for ISDR model, y = 0.3 and a = -0.10. 144 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 1.5 1 0.5 0 R = 1.5 R = 2 R = 3 R = 4 Mean of 12 Ground Site Class C y= 0.4, a = -0.1 Motions 0 3 2.5 CD 2 < "in £ 1.5 •a in w <l 1 0.5 0 R = 1.5 R = 2 — R = 3 R = 4 \ ^ Mean of 32 Ground Motions Site Class D y = 0.4, a = -0.1 T(s) T(s) T(s) T(s) Figure B-14: Mean inelastic displacement ratios for ISDR model, y = 0.5 and a = -0.10. 3 2.5 2 <] <h a> 1.5 "O in in <1 1 0.5 0 R= 1.5 R = 2 "™™" R = 3 R = 4 Mean of 12 Ground Motions Site Class C Y= 0.6, a = -0.1 0 1 3 2.5 2 < g 1.5 •o in in < 1 0.5 0 Mean of 32 Ground Motions Site Class D Y= 0-6, a = -0.1 0 1 T (s) T (s) Figure B-15: Mean inelastic displacement ratios for ISDR model, y = 0.6 and a = -0.10. 145 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 146 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 <D 2 < •D a 1.5 •D in < 1 0.5 0 0 \ R= 1.5 R = 2 R = 3 R = 4 Mean of 12 Ground Motions Site Class C y= 0.1, a = -0.2 1 3 2.5 • 2 £ 1.5 "D co V) < 1 0.5 0 Mean of 32 Ground Site Class D y= 0.1, oc = -0.2 Motions 1 T (s) T (s) Figure B-19: Mean inelastic displacement ratios for ISDR model, y = 0.1 and a = -0.20. 147 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 <« 2 "D «9 A f 2 1.5 n v> * 1 0.5 0 0 1 2 3 0 1 2 3 T (s) T (s) Figure B-22: Mean inelastic displacement ratios for ISDR model, y = 0.4 and a = -0.20. in R = 1.5 R = 2 R = 3 R = 4 mm Mean of 12 Ground Motions Site Class C y = 0.4, a = -0.2 Mean of 32 Ground Site Class D y= 0.4, dt = -0.2 Motions 148 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 0 2 -1 in jD 1.5 ~o in 01 -1 1 0.5 0 Mean of 12 Ground Motions Site Class C Y=0.8, a = -0.2 1 3 2.5 . 2 s> 1.5 "D to » < 1 0.5 0 Mean of 32 Ground Site Class D 7=0.8, a = -0.2 Motions 1 3 2.5 2 < in 1.5 T3 in m <J 1 0.5 0 T (s) T (s) Figure B-26: Mean inelastic displacement ratios for ISDR model, y = 0.8 and a = -0.20. 3 2.5 3 . 2 a? 1.5 to to ] 1 0.5 0 R = 1.5 — R = 2 R = 4 Mean of 12 Ground Motions Site Class C y= 0.9, a = -0.2 I R= 1.5 R = 2 R = 3 R = 4 Mean of 32 Ground Motions Site Class D 7= 0.9, a = -0.2 T(s) T(s) Figure B-27: Mean inelastic displacement ratios for ISDR model, 7 = 0.9 and a = -0.20. 149 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 < £ 1.5 (fl w < 1 0.5 0 0 I i . . . . . . \ 1 \ Mean of 12 Ground Motions Site Class C Y= 0.1, a = -0.4 1 3 2.5 a) 2 <] to £ 1.5 tn w < 1 0.5 0 \ - R = 1.5 V. R = 2 — - * f k = 3 R Mean of 32 Ground Motions Site Class D y = 0.1, a = -0.4 1 T (s) T (s) Figure B-28: Mean inelastic displacement ratios for ISDR model, y = 0.1 and a = -0.40. 150 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios Mean of 12 Ground Site Class C y = 0.4, a = -0.4 Motions 3 2.5 •g £ 1.5 to to « 1 0.5 0 0 1 2 3 0 1 2 3 T (s) T (s) Figure B-31: Mean inelastic displacement ratios for ISDR model, y = 0.4 and a = -0.40. 3 2.5 <D 2 < 'in 1.5 •a tn in <J 1 0.5 0 Mean of 32 Ground Motions Site Class D y= 0.4, cc = -0.4 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.5, a = -0.4 1 3 2.5 < 2 W A I -2? 1.5 "O Ui 1 0 < 1 0.5 0 R = 1 5 R = 2 1 R = 3 \ R = 4 A \ o Mean of 32 Ground Motions Site Class D y= 0.5, a = -0.4 0 1 T(s) T(s) Figure B-32: Mean inelastic displacement ratios for ISDR model, y = 0.5 and a = -0.40. 3 2.5 2 1.5 1 0.5 0 R = 1.5 R = 2 \ R = 4 — ' V Mean of 12 Ground Motions Site Class C Y = 0.6, fa = -0.4 0 1 3 2.5 CD 2 < l CO <D 1.5 "O in <i 1 0.5 0 R = 1.5 R = 2 R = 3 — R = 4 Mean of 32 Ground Motions Site Class D Y= 0.6, a = -0.4 0 1 T(s) T(s) Figure B-33: Mean inelastic displacement ratios for ISDR model, y = 0.6 and a = -0.40. 151 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 < w a 1.5 ~o tO to < 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.7, a = -0.4 Motions 1 3 2.5 <• 2 CO ^ _ 2? 1.5 T3 co to < 1 0.5 0 Mean of 32 Ground Site Class D Y= 0.7, ot = -0.4 Motions 1 T(s) T(s) Figure B-34: Mean inelastic displacement ratios for ISDR model, Y = 0.7 and a = -0.40. 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Motions Site Class C Y = 0.8, a = -0.4 1 3 2.5 * 2 £ 1.5 •D LO to ' 1 0.5 0 Mean of 32 Ground Motions Site Class D Y= 0.8, dc = -0.4 1 T (s) T (s) Figure B-35: Mean inelastic displacement ratios for ISDR model, Y = 0.8 and a = -0.40. 3 2.5 3° 2 £ 1.5 •o C73 (0 < 1 0.5 0 3 2.5 2 £ 1.5 •D to <" 1 0.5 0 R= 1.5 — R = 2 """"" R = 3 • * R = 4 Mean of 12 Ground Motions Site Class C Y= 0.9, cc = -0.4 Mean of 32 Ground Site Class D Y= 0.9, cx = -0.4 Motions 1 T(s) T(s) Figure B-36: Mean inelastic displacement ratios for ISDR model, y = 0.9 and a = -0.40. 152 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 CD 2 < co CD 1.5 X) CO IB < 1 0.5 0 Mean of 12 Ground Motions Site Class C y = 0.1, ot = -0.6 1 3 2.5 CD 2 £ 1.5 •p at (0 < 1 0.5 0 Mean of 32 Ground Motions Site Class D Y= 0.1, oc = -0.6 1 T(s) T(s) Figure B-37: Mean inelastic displacement ratios for ISDR model, y = 0.1 and a = -0.60. 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Motions Site Class C Y = 0-2, oc = -0.6 : 3 2.5 2 'co CD 1.5 •u CO CO < 1 0.5 0 R= 1.5 R = 2 V V- R = 3 . R = 4 \ Mean of 32 Ground Site Class D Y= 0.2, a = -0.6 Motions 0 1 2 3 0 1 2 3 T (s) T (s) Figure B-38: Mean inelastic displacement ratios for ISDR model, y = 0.2 and a = -0.60. 3 2.5 2 if> CD 1.5 •o if: CO < 1 0.5 0 Mean of 12 Ground Site Class C y = 0.3, a = -0.6 Motions 1 T(s) 3 2.5 2 < CO CD 1.5 •o CO CO < 1 0.5 0 Mean of 32 Ground Motions Site Class D Y= 0.3, a = -0.6 1 T(s) Figure B-39: Mean inelastic displacement ratios for ISDR model, y = 0.3 and a = -0.60. 153 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 •g w A I -£ 1.5 "O LO W < 1 0.5 0 Mean of 12 Ground Site Class C y = 0.4, a = -0.6 Motions 0 1 3 2.5 CD 2 < CD 1.5 T3 M CO < 1 0.5 0 R = 1.5 R = 2 R = 3 R = 4 Mean of 32 Ground Motions Site Class D Y = 0.4, ex = -0.6 1 T (s) T (s) Figure B-40: Mean inelastic displacement ratios for ISDR model, y = 0.4 and a = -0.60. 3 2.5 CD 2 < CD 1.5 T3 in m < 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.5, a = -0.6 0 1 3 2.5 CD 2 < 'in CD 1.5 "D in in < 1 0.5 0 l ; R= 1.5 R = 2 **m R = 3 • R = 4 Mean of 32 Ground Motions Site Class D y = 0.5, a = -0.6 0 1 T (s) T (s) Figure B-41: Mean inelastic displacement ratios for ISDR model, y = 0.5 and a = -0.60. 3 2.5 2 < in CD 1.5 ~c in <: 1 0.5 0 R = 1.5 R = 2 l V ™»»« R — 3 R = 4 Mean of 12 Ground Motions Site Class C Y= 0.6, oc = -0.6 3 2.5 CD 2 < "w 0J 1.5 ~o <s> </) < 1 0.5 0 1 R = 1.5 R = 2 1 — R = 3 • — • R - 4 Mean of 32 Ground Motions Site Class D y = 0.6, 0c = -0.6 T(s) T(s) Figure B-42: Mean inelastic displacement ratios for ISDR model, y = 0.6 and a = -0.60. 154 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 < to £ 1.5 CO CO < 1 0.5 0 \ A R= 1.5 R = 2 R = 3 R = 4 Mean of 12 Ground Motions Site Class C y= 0.7, bc = -0.6 3 2.5 • 2 < 2 1.5 T3 CO co « 1 0.5 0 \ I R= 1.5 — R = 2 R = 3 R = 4 Mean of 32 Ground Motions Site Class D Y = 0.7, oc = -0.6 T(s) T(s) Figure B-43: Mean inelastic displacement ratios for ISDR model, y = 0.7 and a = -0.60. 3 2.5 2 1.5 1 0.5 0 R = 1.5 — • R = 2 R = 3 R = 4 Mean of 12 Ground Motions Site Class C y= 0.8, oc = -0.6 3 2.5 » 2 "**g W A I -£ 1.5 -o to to < 1 0.5 0 R= 1.5 — R = 2 \ R - 3 • R = 4 Mean of 32 Ground ****** **^*»rr Motions Site Class D y= 0.8, ot = -0.6 0 T(s) T(s) Figure B-44: Mean inelastic displacement ratios for ISDR model, y = 0.8 and a = -0.60. 3 2.5 2 1.5 1 0.5 0 R= 1.5 — R = 2 R = 3 R = 4 Mean of 12 Ground Motions Site Class C y=0.9, oc = -0.6 0 3 2.5 • 2 < W A I -9> 1.5 to to < 1 0.5 0 R = 1.5 — R = 2 R = 3 ' R = 4 Mean of 32 Ground Motions Site Class D y= 0.9, cc = -0.6 1 T(s) T(s) Figure B-45: Mean inelastic displacement ratios for ISDR model, y = 0.9 and a = -0.60. 155 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios Mean ol 12 Ground Motions Site Class C Y= 0.1, a = -0.8 3 2.5 CD 2 < l 'w CD 1.5 T3 CO < 1 0.5 0 RVI.5 R = 2 / . R = 3 ^ \ R = 4 Mean of 32 Ground Motions Site Class D Y= 0.1, a = -0.8 0 1 2 3 0 1 2 3 T (s) T (s) Figure B-46: Mean inelastic displacement ratios for ISDR model, y = 0.1 and a = -0.80. 156 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 2.5 • 2 •g W A I -£ 1.5 ~o < 1 0.5 0 Mean of 12 Ground Site Class C y= 0.4, a = -0.8 Motions 3 2.5 a, 2 w A r-2> 1.5 "D CO CO < 1 0.5 0 R = 1.5 R = 2 \\\ " " " " " R - 3 mm. R = 4 ^ v Mean of 32 Ground Motions Site Class D Y= 0.4, a = -0.8 0 1 2 3 0 1 2 3 T (s) T (s) Figure B-49: Mean inelastic displacement ratios for ISDR model, y = 0.4 and a = -0.80. 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Motions Site Class C Y= 0.5, a = -0.8 3 2.5 • 2 W A I-s 1.5 •a to CO < 1 0.5 0 R = 1.5 - R = 2 . — R = 3 R = 4 \ \ ^ ^ ^ s Mean of 32 Ground Motions Site Class D Y= 0.5, cc = -0.8 0 1 2 3 0 1 2 3 T (s) T (s) Figure B-50: Mean inelastic displacement ratios for ISDR model, y = 0.5 and a = -0.80. 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.6, oc = -0.8 0 1 3 2.5 • 2 g « A f-2 1.5 •o CO to 4 1 0.5 0 i R = 1.5 R = 2 *™™" R = 3 _ R = 4 Mean of 32 Ground Motions Site Class D y= 0.6, cc = -0.8 T(s) T(s) FigureB-51: Mean inelastic displacement ratios for ISDR model, y = 0.6 and a = -0.80. 157 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.7, a = -0.8 Motions 3 2.5 o 2 W A I -2 1.5 CO CO < 1 0.5 0 Mean of 32 Ground Site Class D Y= 0.7, 0c = -0.8 Motions Mean of 12 Ground Site Class C Y= 0.8, oc = -0.8 Motions 1 3 2.5 CD 2 < 'c/j CD 1.5 x: i> < 1 0.5 0 I R= 1.5 R = 2 \ ' R = 3 — - R = 4 —-~ Mean of 32 Ground Motions Site Class D Y= 0.8, a = -0.8 T O ) T(s) Figure B-53: Mean inelastic displacement ratios for ISDR model, y = 0.8 and a = -0.80. 3 2.5 2 <] to is 1.5 ID CO to < 1 0.5 0 Mean of 12 Ground Motions Site Class C y= 0.9, a = -0.8 1 3 2.5 CD 2 < "D 'co CD 1.5 "D CO to < 1 0.5 0 R = 1.5 — R = 2 R = 3 R = 4 Mean of 32 Ground Motions Site Class D Y = 0.9, cc = -0.8 T(s) T(s) Figure B-54: Mean inelastic displacement ratios for ISDR model, y = 0.9 and a = -0.80. 158 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 < £ 1.5 "O CO V) < 1 0.5 0 A V Mean ol 12 Ground Motions Site Class C y= 0.1, a = -1 C = 1 .0 ft 3 2.5 • 2 £ 1.5 M < 1 0.5 0 Mean of 32 Ground Site Class D y = 0.1, a = -1 Motions T(s) T(s) Mean of 12 Ground Motions Site Class C Y= 0.2, a = -1 vy v ^ R = 1.5 \ * . R = 2 A - R = 3 ' v V ' Mean ol Site Cla Y=0.2 , 32 Ground 3S D x = -1 Motions 1 T(s) T(s) Figure B-56: Mean inelastic displacement ratios for ISDR model, y = 0.2 and a = -1.00. 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.3, a = -1 3 2.5 2 < CD 1.5 ~U CO if) < 1 0.5 0 Mean of 32 Ground Site Class D Y= 0.3, oc = -1 Motions 0 1 2 3 0 1 2 T (s) T (s) Figure B-57: Mean inelastic displacement ratios for ISDR model, y = 0.3 and a = -1.00. 159 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 < CO £ 1.5 CO *$ 1 0.5 0 f A V R = 1.5 R = 2 R = 3 R = 4 ' A : W\ A, Mean of 12 Ground Site Class C y = 0.4, a = -1 Motions 3 2.5 * 2 •g A I-£ 1.5 •D CO CO < 1 0.5 0 Mean of 32 Ground Site Class D Y= 0.4, a = -1 0 1 T(s) T(s) Figure B-58: Mean inelastic displacement ratios for ISDR model, y = 0.4 and a = -1.00. 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.5, a = -1 Motions 1 3 2.5 <• 2 is £ 1.5 TJ co co ^ 1 0.5 0 Mean of 32 Ground Motions Site Class D Y=0.5,ct = -1 0 1 T(s) T(s) T(s) Figure B-60: Mean inelastic displacement ratios for ISDR model, y = 0.6 and a = -1.00. 160 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 1.5 1 0.5 0 1 R = 1.5 R = 2 R = 3 R - 4 uA \ ^ > Mean of 12 Ground Motions Site Class C y= 0.7, cc = -1 1 3 2.5 <» 2 •g £ 1.5 "D <o to * 1 0.5 0 R= 1.5 R = 2 R = 3 R = 4 Motions Mean of 32 Ground Site Class D Y= 0.7, a = -1 1 T(s) T(s) Figure B-61: Mean inelastic displacement ratios for ISDR model, Y = 0.7 and a = -1.00. 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Motions Site Class C Y=0.8, oc = -1 1 3 2.5 CD 2 ~o 'c/> CD 1.5 X3 IS) CO < 1 0.5 0 I R= 1.5 <>= R = 2 \ R = 3 R = 4 Mean of 32 Ground Motions Site Class D Y= 0.8, a = -1 T(s) T(s) Figure B-62: Mean inelastic displacement ratios for ISDR model, Y = 0.8 and a = -1.00. 3 2.5 CD 2 < <D 1.5 TD CO (/> < 1 0.5 0 R= 1.5 _ R= 2 R = 3 R = 4 Mean of 12 Ground Motions Site Class C Y= 0.9, oc = -1 3 2.5 CD 2 <] "O 'to 0) 1.5 -o to W < 1 0.5 0 Mean of 32 Ground Motions Site Class D Y= 0.9, a = -1 1 T(s) T(s) Figure B-63: Mean inelastic displacement ratios for ISDR model, y = 0.9 and a = -1.00. 161 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 CO 2 < m £ 1.5 LO to < 1 0.5 0 R ^ 1 . 5 A/T \ /V V \ Mean ol 12 Ground Motions Site Class C Y= 0.1, a = -1.25 3 2.5 M J r-s 1.5 CO w < 1 0.5 0 Mean of 32 Ground Motions Site Class D Y=0.1,oc = -1.25 1 T(s) T(s) Figure B-64: Mean inelastic displacement ratios for ISDR model, y = 0.1 and a = -1.25. 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Site Class C y= 0.2, a =-1.25 Motions 1 3 2.5 o 2 S A C £ 1.5 CO CO < 1 0.5 0 Mean of 32 Ground Motions Site Class D Y= 0.2, cc = -1.25 0 1 3 2.5 2 < •D 'co £ 1.5 CO < 1 0.5 0 T (s) T (s) Figure B-65: Mean inelastic displacement ratios for ISDR model, y = 0.2 and a = -1.25. 3 2.5 * 2 a £ 1.5 -o CO co 3 1 0.5 0 \ R « 14 1 R = 2 \ R = 3 R = 4 Mean of 12 Ground Motions Site Class C Y = 0.3, cx =-1.25 Mean of 32 Ground Motions Site Class D y= 0.3, cc =-1.25 0 1 1 T (s) T (s) Figure B-66: Mean inelastic displacement ratios for ISDR model, y = 0.3 and a = -1.25. 162 Appendix B Stiffness and In-Cycle Strength Degrading with 3 2.5 1) 2 <! *o CO CD 1.5 "O CO CO < 1 0.5 0 n v.. 1 R = 1.5 R = 2 R = 3 R = 4 - A Mean o f 12 Ground Motions Site Class C Y= 0.4, a = -1.25 3 2.5 IB 2 CO E 1.5 •a CO CO < 1 0.5 0 i —f\ R = 1.5 R = 2 R = 3 R = 4 Mean of 32 Ground Motions Site Class D Y= 0.4, a = -1.25 T(s) T(s) Figure B-67: Mean inelastic displacement ratios for ISDR model, y = 0.4 and a = -1.25. 3 2.5 2 < 'co E 1.5 •a CO CO < 1 0.5 0 R= 1.5 R= 2 R = 3 R = 4 Mean of 12 Ground Site Class C y = 0.5, a = -1.25 Motions 1 3 2.5 CD 2 < CO 1.5 • q CO < 1 0.5 0 R = 1.5 R = 2 •'• R = 3 • R = 4 Mean of 32 Ground Motions Site Class D Y= 0.5, <x = -1.25 1 T (s) T (s) Figure B-68: Mean inelastic displacement ratios for ISDR model, y = 0.5 and a = -1.25. 3 2.5 2 < d resic 1.5 CO to < 1 0.5 0 R = 1.5 R = 2 R = 3 R = 4 Mean of 12 Ground Motions Site Class C Y = 0.6, a = -1.25 1 T(s) 3 2.5 CD 2 < d resi( 1.5 CO CO 1 1 0.5 0 Mean of 32 Ground Motions Site Class D y = 0.6, a = -1.25 1 T(s) Figure B-69: Mean inelastic displacement ratios for ISDR model, y = 0.6 and a = -1.25. 163 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 164 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 CD 2 < in CD 1.5 "O in in < 1 0.5 0 V ' Mean of 12 Ground Motions Site Class C y= 0.1, a = -1.5 0 1 3 2.5 CD 2 <] w A r -£ 1.5 ~o IT) m < 1 0.5 0 Mean of 32 Ground Motions Site Class D Y= 0.1, a = -1.5 1 T(s) T(s) 3 2.5 CD 2 in 0) 1.5 TD in in < 1 0.5 0 Figure B-73: Mean inelastic displacement ratios for ISDR model, Y = 0.1 and a = -1.50. 3 2.5 . 2 W A I -s 1.5 CO CO < 1 0.5 0 I % R = 1.5 V4 \ /V \ Mean o: 12 Ground Motions Site Class C Y = 0.2, oc = -1.5 SjBF R = 1.5 ^ » R = 2 V \ R = 3 \ \ / Mean of 32 Ground Motions Site Class D Y = 0.2, oc = -1.5 1 0 1 T(s) T(s) 3 2.5 <D 2 < in 1 1.5 TO CO CO < 1 0.5 0 Figure B-74: Mean inelastic displacement ratios for ISDR model, y = 0.2 and a = -1.50. Mean of 12 Ground Motions Site Class C Y=0.3,a = -1.5 1 3 2.5 CD 2 < '55 g 1.5 •a to to < 1 0.5 0 V \ - R = 1 5 V R = 2 1 ; A — R = 3 \ / \ N W Mean of 32 Ground Motions Site Class D Y= 0.3, a = -1.5 T(s) T(s) Figure B-75: Mean inelastic displacement ratios for ISDR model, y = 0.3 and a = -1.50. 165 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 1.5 1 0.5 0 II R = 1.5 R = 2 R = 3 R = 4 A V - A : ^ Mean Site C y= 0.4 of 12 Ground ass C , a = -1.5 Motions 0 3 2.5 CD 2 < LO s 1.5 Xi CO LO < 1 0.5 0 \ V S r - r \ R= 1.5 R = 2 R = 3 R = 4 ^ ^ ^ ^ Mean of 32 Ground Motions Site Class D Y= 0.4, a = -1.5 T(s) T(s) Figure B-76: Mean inelastic displacement ratios for ISDR model, Y = 0.4 and a = -1.50. 3 2.5 CD 2 < LO 2 1.5 •o LO (0 < 1 0.5 0 Mean of 12 Ground Site Class C Y=0.5,cx = -1.5 3 2.5 CD 2 < 'co CD 1.5 • o CO CO < 1 0.5 0 1 R= 1.5 j l | . .,,„„ R-„2 R = 4 Mean of 32 Ground Motions Site Class D Y= 0.5, cx = -1.5 3 2.5 CD 2 < CO 0J 1.5 Xi CO <" 1 0.5 0 0 1 2 3 0 1 2 T (s) T (s) Figure B-77: Mean inelastic displacement ratios for ISDR model, y = 0.5 and a = -1.50. 3 2.5 • 2 3 ""Ha £ 1.5 LO SO 3 1 0.5 0 Mean of 12 Ground Site Class C y= 0.6, a = -1.5 Mean of 32 Ground Site Class D y= 0.6, a = -1.5 Motions 1 1 T(s) T(s) Figure B-78: Mean inelastic displacement ratios for ISDR model, y = 0.6 and a = -1.50. 166 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Motions Site Class C Y= 0.7, a = -1.5 1 3 2.5 . 2 "> A p £ 1.5 3 1 0.5 0 \ R • 1.5 R = 2 I """"" R = 3 R = 4 \ V V „ - s Mean of 32 Ground Motions Site Class D 7=0.7, a= -1.5 1 T(s) T(s) 3 2.5 CO 2 <] "to CD 1.5 "O (fl 1 0.5 0 Figure B-79: Mean inelastic displacement ratios for ISDR model, y = 0.7 and a = -1.50. 3 2.5 2 •g £ 1.5 ~o CO CO « 1 0.5 0 Mean of 12 Ground Site Class C Y = 0.8, a = -1.5 Motions 11 R = 1.5 • R - 2 R = 3 R = 4 \ IMA,^ ; Mean of 32 Ground Motions Site Class D Y = 0.8, a = -1.5 1 T(s) 1(3) Figure B-80: Mean inelastic displacement ratios for ISDR model, y = 0.8 and a = -1.50. 167 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.1, cx = - 2 Motions 0 1 3 2.5 2 < to £ 1.5 CO (0 < 1 0.5 0 R f 2 R R Mean of 32 Ground Motions Site Class D Y=0.1,cc = - 2 1 T(s) T(s) 32 Figure B-82: Mean inelastic displacement ratios for ISDR model, Y = 0.1 and a = -2.00. 3 2.5 2 1.5 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.2, a = - 2 3 2.5 <D 2 < "co £ 1.5 CO <n <J 1 0.5 0 *\ *A jc— R= 1.5 V Mean of 32 Ground Motions Site Class D Y= 0.2, oc = - 2 0 1 2 3 0 1 2 T (s) T (s) Figure B-83: Mean inelastic displacement ratios for ISDR model, Y = 0.2 and a = 2.00. 3 2.5 a 2 'co £ 1.5 •D CO CO < 1 0.5 0 Mean of 12 Ground Site Class C Y= 0-3, a = - 2 1 3 2.5 <D 2 < CO £ 1.5 -a CO  < 1 0.5 0 Mean of 32 Ground Site Class D Y= 0-3, a = - 2 1 T(s) T(s) Figure B-84: Mean inelastic displacement ratios for ISDR model, y = 0.3 and a = -2.00. 168 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 CD 2 < to CD 1.5 TO CO to < 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.4, a = -2 Motions 1 3 2.5 • 2 W A f £ 1.5 TO CO CO ^ 1 0.5 0 R = 1.5 R = 2 mmm R = 3 R = 4 Mean of 32 Ground Motions Site Class D y = 0.4, cx = -2 T(s) Mean of 12 Ground Motions Site Class C Y= 0.5, a = -2 V R = 1.5 R = 2 R = 3 _ R = 4 Motions Mean of 32 Ground Site Class D Y= 0.5, cc = -2 1 T(s) T(s) Figure B-86: Mean inelastic displacement ratios for ISDR model, y = 0.5 and a = -2.00. 3 2.5 CD 2 <j CO CD 1.5 TO to to < 1 0.5 0 Mean of 12 Ground Site Class C Y= 0.6, a = -2 1 T(s) 3 2.5 CD 2 < 1.5 TO w < 1 0.5 0 l R = 1.5 R = 2 IS — * R = 3 _ R = 4 Mean of 32 Ground Motions Site Class D Y = 0.6, cx = -2 T(s) Figure B-87: Mean inelastic displacement ratios for ISDR model, y = 0.6 and a = -2.00. 169 Appendix B Stiffness and In-Cycle Strength Degrading with Residual Strength Inelastic Displacement Ratios 3 2.5 o 2 s 1.5 to CO « 1 0.5 0 R = 1.5 R = 2 R » 3 R = 4 Mean of 12 Ground Site Class C y= 0.7, tx = -2 Motions 0 1 3 2.5 a" 2 £ 1.5 "D 3 (0 3 1 0.5 0 V R=1.5 R = 2 i l l —-» R = 3 R = 4 l Mean of 32 Ground Motions Site Class D y= 0.7, cc = -2 1 T (s) T (s) Figure B-88: Mean inelastic displacement ratios for ISDR model, y = 0.7 and a = -2.00. 3 2.5 0 2 < £ 1.5 CO CO < 1 0.5 0 R = 1.5 R = 2 •™" R = 3 R = 4 Mean of 12 Ground Motions Site Class C Y= 0.8, oc = -2 1 3 2.5 2 1 « CD 1.5 "O to CO < 1 0.5 0 Mean of 32 Ground Site Class D y= 0.8, a = -2 Motions 1 T (s) T (s) Figure B-89: Mean inelastic displacement ratios for ISDR model, y = 0.8 and a = -2.00. 3 2.5 2 1.5 1 0.5 0 A R= 1.5 R = 2 \ R = 3 • -R = 4 \ Mean of 12 Ground Motions Site Class C y= 0.9, a = -2 0 3 2.5 2 •g £ 1.5 "D CO < 1 0.5 0 1 R= 1.5 R = 2 R M \\ Mean of 32 Ground Motions Site Class D y= 0.9, a = -2 0 T(s) T(s) Figure B-90: Mean inelastic displacement ratios for ISDR model, y = 0.9 and a = -2.00. 170 Appendix C Stiffness and In-Cycle Strength Degrading with Residual Strength and Equivalent EPP Displacement Ratios APPENDIX C. STIFFNESS AND IN-CYCLE STRENGTH DEGRADING WITH RESIDUAL STRENGTH AND EQUIVALENT EPP DISPLACEMENT RATIOS 3 2.5 2 1.5 1 0.5 0 0 a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 1.5, R R = 5 ,Y=0.3 Mean of 20 Ground Motions Site Class C 1 T(s) CO n •g 'in CD DC a o >> o 3 2.5 2 1.5 1 0.5 0 a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 R = 1.5, R R = 5,Y=0.3 Mean of 20 Ground Motions Site Class C 1 T(s) Figure C - l : Mean ratio of ISDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=1.5, RR=5, y=0.3). a. CD 5 •u '5) CD a. _CD u 2.5 1.5 0.5 a = -0.05 a = -0.10 a = -0.20 a = -0.40 — a = -0.60 '=0.4 Mean of 20 Ground Motions Site Class C 1 T ( s ) 2.5 CL CD 5L 2 in CD cr 0J o 1.5 0.5 a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 R = 2, R R = 5,7=0.4 Mean of 20 Ground Motions Site Class C 1 T ( s ) Figure C-2: Mean ratio of ISDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=2, RR=5,7=0.4). 171 Appendix C Stiffness and In-Cycle Strength Degrading with Residual Strength and Equivalent EPP Displacement Ratios 3 2.5 2 1.5 1 0.5 0 a = -0.05 a = -0.10 a = -0.20 a = -0.40 — a = -0.60 CD o R = 3, R R ^ 5 7 y = 0 . 6 Mean of 20 Ground Motions Site Class C 0 1 3 2.5 2 1.5 1 0.5 0 a = -0.80 a = -1.00 a = -1.25 a = -1.50 — _ a = -2.00 R = 3, R r = 5,Y=0.6 \ Mean of 20 Ground Motions Site Class C 0 1 T(s) T(s) Figure C-3: Mean ratio of ISDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=3, RR=5, y=0.6). Et •a 3 2.5 2 S 1.5 1 0.5 0 a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 3 2.5 2 R = 4, R R = 5,Y=0.8 Mean of 20 Ground Motions Site Class C 0 1 S 1.5 0.5 0 a = -0.80 a = -1.00 a = -1.25 a = -1.50 _ a _ -2.00 R = 4, R R = 5,Y=0.8 Mean of 20 Ground Motions Site Class C 0 1 T(s) T(s) Figure C-4: Mean ratio of ISDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=4, RR=5, Y=0.8). Q . CL co a "ro T3 CO tl) CC 3 2.5 2 1.5 1 0.5 0 0 a = -0.05 a = -0.10 a = -0.20 a = -0.40 — a = -0.60 R = 1.5, R R = 3, Y=0.5 Mean of 20 Ground Motions Site Class C 1 T(s) 3 2.5 Si 2 CD Q_ * O >s 1.5 1 0.5 0 a = -0.80 a = -1.00 a = -1.25 a = -1.50 — a = -2.00 R = 1.5, R R = 3, Y=0.5 0 Mean of 20 Ground Motions Site Class C 1 T(s) Figure C-5: Mean ratio of ISDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=1.5, RR=3, Y=0.5). 172 Appendix C Stiffness and In-Cycle Strength Degrading with Residual Strength and Equivalent EPP Displacement Ratios 2.5 2 a 1 1.5 or tD 1 0.5 0 a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 2.5 R = 2 , R R = 4 , Y = 0 . 5 Mean of 20 Ground Motions Site Class C CD CO 1 1.5 0.5 0 a = -0.80 a = -1.00 a = -1.25 a = -1.50 — a = -2.00 R = 2, R R = 4 ,Y=0.5 Mean of 20 Ground Motions Site Class C 0 1 2 3 0 1 2 3 T (s) T (s) Figure C-6: Mean ratio of ISDR model to equivalent E P P model for crustal ground motions recorded on site class C (R=2, RR=4, y=0.5). 3 2.5 ^ 2 CO 1 1.5 ! 1 0.5 0 a = -0.05 a = -0.10 a = -0.20 a = -0.40 — (/. = -0.60 3 2.5 a. cu K 2 03 Mean of 20 Ground Motions Site Class C 0 1 1.5 1 0.5 0 — a = -0.80 a = -1.00 a = -1.25 -1.50 a = -2.00 R = 3, R r = 6 ,Y=0.5 Mean of 20 Ground Motions Site Class C 1 T(s) T(s) Figure C-7: Mean ratio of ISDR model to equivalent E P P model for crustal ground motions recorded on site class C (R=3, R R=6, y=0.5). re 3 "O O) OS ts o >. o 3 2.5 2 1.5 1 0.5 0 a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 Mean of 20 Ground Motions Site Class C 1 2.5 2 1.5 1 0.5 0 — a = -0.80 a = -1.00 a = -1.25 — a = -1.50 — . a = -2.00 R = 4, R R = 8 ,y=0.5 Mean of 20 Ground Motions Site Class C 0 1 T(s) T(s) Figure C-8: Mean ratio of ISDR model to equivalent E P P model for crustal ground motions recorded on site class C (R=4, R R=8, Y=0.5). 173 Appendix C Stiffness and In-Cycle Strength Degrading with Residual Strength and Equivalent EPP Displacement Ratios CD rr a o >1 u 3 2.5 2 1.5 1 0.5 0 a = -0.05 a = -0.10 a = -0.20 a = -0.40 _ a = -0.60 ), Y=0.3 Mean of 12 Ground Motions Site Class C CO CC _CD O > , o c 0 1 3 2.5 2 1.5 1 0.5 0 a = -0.80 a = -1.00 a = -1.25 a = -1.50 — a = -2.00 R = 1.5, R R = 5,y=0.3 Mean of 12 Ground Motions Site Class C 0 1 T(s) T(s) Figure C-9: Mean ratio of ISDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=1.5, RR=5,7=0.3). O . CL < (0 => - g 'w CD a. o 3 2.5 2 1.5 1 0.5 0 a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 Mean of 12 Ground Motions Site Class C 0 1 3 2.5 2 1.5 1 0.5 0 R = 2, R R = 5,y=0.4 Mean of 12 Ground Motions Site Class C 1 T(s) T(s) Figure C-10: Mean ratio of ISDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=2, RR=5,7=0.4). CD o 3 2.5 2 1.5 1 0.5 0 oc = -0.05 a = -0.10 a = -0.20 a = -0.40 — a = -0.60 Mean of 12 Ground Motions Site Class C 3 2.5 2 1.5 1 0.5 cx = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 5, Y = 0.6 Mean of 12 Ground Motions Site Class C 1 1 T(s) T(s) Figure C - l l : Mean ratio of ISDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=3, RR=5, y=0.6). 174 Appendix C Stiffness and In-Cycle Strength Degrading with Residual Strength and Equivalent EPP Displacement Ratios 2.5 ^ 2 "a CD 01 JD O 1.5 0.5 a = -0.05 a = -0.10 a --0.20  = -0.40 a = -0.60 2.5 3 " 0 1.5 R = ' D , y = Mean of 12 Ground Motions Site Class C 0.5 a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 R = 4 , R R = Mean of 12 Ground Motions Site Class C 0 1 0 1 T(s) T(s) Figure C-12: Mean ratio oflSDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=4, RR=5, y=0.8). 2.5 2 $ 1.5 to. 9 1 0.5 a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 = 1.5, R R = 3,Y=0.5 Mean of 12 Ground Motions Site Class C 2.5 ^ 2 03 =1 -a 1.5 0.5 a = -0.80 a = -1.00 a = -1.25 a = -1.50 a = -2.00 R = 1.5, R R = 3,y=0.5 Mean of 12 Ground Motions Site Class C 0 1 1 T(s) T(s) Figure C-13: Mean ratio of ISDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=1.5, RR=3, y=0.5). CD a. o >. o c 3 2.5 2 1.5 1 0.5 0 a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 Mean of 12 Ground Motions Site Class C 1 T(s) 3 2.5 2 1.5 1 0.5 0 0 oc = -0.80 a = -1.00 a = -1.25 a = -1.50 — a - -2.00 R = 2, R R = 4 , Y = 0.5 Mean of 12 Ground Motions Site Class C 1 T(s) Figure C-14: Mean ratio of ISDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=3, RR=6,7=0.5). 175 Appendix C Stiffness and In-Cycle Strength Degrading with Residual Strength and Equivalent EPP Displacement Ratios 2.5 2 1.5 1 0.5 a = -0.05 a = -0.10 a = -0.20 a = -0.40 a = -0.60 Mean of 12 Ground Motions Site Class C 0 1 tt2.5 CU CO Z3 1 1.5 o I 1 " 0.5 0 a = -0.80 a = -1.00 a =-1.25 a = -1.50 a = -2.00 R = 3, R =6, Y=0.5 K Mean of 12 Ground Motions Site Class C 1 T(s) T(s) Figure C-15: Mean ratio of ISDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=4, RR=8, y=0.5). CO TJ CO CD cc o 2.5 2 1.5 1 0.5 0 a = -0.05 a = -0.10 a. — -0.20 a = -0.40 a_ -0.60 R ~ • ' Y ra •a "co CD £ o >1 o Mean of 12 Ground Motions Site Class C 0 1 3 2.5 2 1.5 1 0.5 0 a = -0.80 a = -1.00 a = -1.25 a = -1.50 a _ -2.00 R = 4, R R = 8 ,7 : Mean of 12 Ground Motions Site Class C 1 T(s) T(s) Figure C-16: Mean ratio of ISDR model to equivalent EPP model for crustal ground motions recorded on site class C (R=4, RR=8,7=0.5). 176 

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