Applied Science, Faculty of
Civil Engineering, Department of
DSpace
UBCV
Carvajal Uribe, Juan Carlos
2011-06-17T22:42:02Z
2011
Doctor of Philosophy - PhD
University of British Columbia
This research thesis is product of a joint study between the Ministry of Transportation and Infrastructure (BCMoT) and the University of British Columbia (UBC) to evaluate the effect of Embankment-Abutment-Structure Interaction (EASI) in the estimation of seismic demands of Integral Abutment Bridges (IABs).
IABs consist of a continuous concrete deck integrated with abutments supported on flexible foundations. These structures have become very popular due to the elimination of costly and maintenance prone expansion joints and bearings. Analytical studies and strong-motion earthquake data have shown that the seismic response of the approach embankments in the far field affects the response of IABs. However, current seismic analysis procedures neglect the far-field embankment response because of the complexity in modeling this type of dynamic interaction. Therefore, a simple and accurate model that allows bridge designers to include EASI in the calculation of the seismic demands of IABs is needed.
This thesis develops a simple dynamic model, called 3M-EASI, for calculating the seismic response of IABs taking into account EASI. The proposed model consists of two far-field embankment components connected to the bridge structure component by spring-dashpot elements that represent the near-field components. The main contribution of this thesis is the development of the far-field embankment component using equivalent-linear analysis. The 3M-EASI model was verified with time-history analyses of 2D continuum soil finite element models of full-height IABs using the computer program ABAQUS.
The analyses indicated that the far-field embankment response affects the response of IABs if the following conditions act simultaneously: (a) the near-field stiffness is greater than 0.4 times the bridge stiffness, and (b) the period of the far-field embankment components is longer than 0.7 times the period of the bridge-near-field system. The 3M-EASI model is shown to be rational, accurate, computationally efficient, and easy to implement in bridge design.
https://circle.library.ubc.ca/rest/handle/2429/35577?expand=metadata
SEISMIC EMBANKMENT-ABUTMENT-STRUCTURE INTERACTION OF INTEGRAL ABUTMENT BRIDGES by JUAN CARLOS CARVAJAL URIBE M.A.Sc., Autonomous National University of Mexico − UNAM, 2000 B.Sc., Industrial University of Santander − UIS, Colombia, 1996 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) June, 2011 Juan Carlos Carvajal Uribe, 2011 ii ABSTRACT This research thesis is product of a joint study between the Ministry of Transportation and Infrastructure (BCMoT) and the University of British Columbia (UBC) to evaluate the effect of Embankment-Abutment-Structure Interaction (EASI) in the estimation of seismic demands of Integral Abutment Bridges (IABs). IABs consist of a continuous concrete deck integrated with abutments supported on flexible foundations. These structures have become very popular due to the elimination of costly and maintenance prone expansion joints and bearings. Analytical studies and strong-motion earthquake data have shown that the seismic response of the approach embankments in the far field affects the response of IABs. However, current seismic analysis procedures neglect the far-field embankment response because of the complexity in modeling this type of dynamic interaction. Therefore, a simple and accurate model that allows bridge designers to include EASI in the calculation of the seismic demands of IABs is needed. This thesis develops a simple dynamic model, called 3M-EASI, for calculating the seismic response of IABs taking into account EASI. The proposed model consists of two far-field embankment components connected to the bridge structure component by spring-dashpot elements that represent the near-field components. The main contribution of this thesis is the development of the far-field embankment component using equivalent-linear analysis. The 3M-EASI model was verified with time-history analyses of 2D continuum soil finite element models of full-height IABs using the computer program ABAQUS. The analyses indicated that the far-field embankment response affects the response of IABs if the following conditions act simultaneously: (a) the near-field stiffness is greater than 0.4 times the bridge stiffness, and (b) the period of the far-field embankment components is longer than 0.7 times the period of the bridge-near-field system. The 3M-EASI model is shown to be rational, accurate, computationally efficient, and easy to implement in bridge design. iii TABLE OF CONTENTS ABSTRACT…………………………………………………………………………………… ii TABLE OF CONTENTS……………………………………………………………………... iii LIST OF TABLES……………………………………………………………………………. vii LIST OF FIGURES…………………………………………………………………………...viii ACKNOWLEDGMENTS……………………………………………………………………. xii DEDICATION……………………………………………………………………………….. xiv 1 INTRODUCTION…………………………………………………………………………. 1 1.1 Standard Bridges………………………………………………………………………...1 1.2 Integral Abutment Bridges (IABs)…………………………………………………....... 2 1.3 Temperature-Induced Displacement in IABs………………………………………....... 5 1.4 Modeling the Seismic Response of IABs………………………………………………. 6 1.4.1 Terminology for Dynamic Analysis of IABs…………………………………….. 7 1.4.2 Time-History Analysis with Continuum Soil Models……………………………... 8 1.4.3 Time-History Analysis with Frame-Spring-Dashpot Models……………………... 9 1.4.4 Pseudo-Static Analysis with Frame Models…………………………………….... 10 1.4.5 Pseudo-Static Analysis with Frame-Spring Models……………………………… 12 1.4.6 IABs with Approach Embankment Models………………………………………. 13 1.5 Objective and Organization of Thesis………………………………………………… 13 2 LITERATURE REVIEW…………………………………………………………………16 2.1 Wilson and Tan (1990a,b)…………………………………………………………….. 18 2.1.1 Far-Field Embankment Model…………………………………………………… 19 2.1.2 Near-Field Embankment Model………………………………………………….. 21 2.1.3 Discussion………………………………………………………………………… 22 2.2 Zhang and Makris (2002a,b)………………………………………………………….. 23 2.2.1 Far-Field Embankment Model…………………………………………………… 23 2.2.2 Validation of the Far-Field Embankment Model………………………………… 26 iv 2.2.3 Near-Field Stiffness Model………………………………………………………. 27 2.2.4 Embankment-Abutment-Structure Interaction (EASI)…………………………… 28 2.2.5 Discussion………………………………………………………………………… 29 2.3 Inel and Aschheim (2004)……………………………………………………………...30 2.3.1 Capacity Curves of Approach Embankments…………………………………….. 30 2.3.2 Discussion………………………………………………………………………… 32 2.4 Kotsoglou and Pantazopoulou (2007, 2009)………………………………………….. 32 2.4.1 Embankment-Bridge Model……………………………………………………… 33 2.4.2 IABs with Flexible Decks………………………………………………………… 34 2.4.3 Discussion………………………………………………………………………… 35 2.5 Summary………………………………………………………………………………. 36 3 THE 1ME MODEL………………………………………………………………………. 38 3.1 Differential Equation of Motion………………………………………………………. 38 3.2 Discretization of the Differential Equation of Motion………………………………... 41 3.2.1 Dimensionless Shape Functions Ψn(z)…………………………………………….. 43 3.2.2 Modal Contribution Analysis…………………………………………………….. 44 3.2.2.1 H = 10 m and Vs = 150 m/s………………………………………………….. 50 3.2.2.2 H = 10 m and Vs = 60 m/s…………………………………………………… 52 3.2.2.3 Discussion……………………………………………………………………. 53 3.3 The 1ME Model for Linear Analysis…………………………………………………. 55 3.3.1 The 1ME Model with Constant Vs……………………………………………….. 55 3.3.1.1 Example 1: Calculation of relative displacement time histories……………. 58 3.3.1.2 Example 2: Calculation of peak response quantity profiles………………… 59 3.3.2 The 1ME Model with Linear Variation of Vs……………………………………. 61 3.3.3 The 1ME Model with Degradation of Gin………………………………………... 63 3.4 The 1ME Model for Non-Uniform Embankments……………………………………. 66 3.5 The 1ME Model for Equivalent Linear Analysis……………………………………... 71 3.5.1 Effective Shear Strain of the Approach Embankment……………………………. 72 3.5.2 Procedure for Calculating the Equivalent Linear Response……………………… 73 3.5.2.1 Example 3: Calculation of the Equivalent Linear Response………………... 75 v 3.6 Summary………………………………………………………………………………. 76 4 VERIFICATION OF THE 1ME MODEL…………………………………………….... 78 4.1 Physical Properties and Input Ground Motion………………………………………... 79 4.2 Low Soft Approach Embankment…………………………………………………….. 82 4.3 Low Stiff Approach Embankment…………………………………………………….. 87 4.4 High Soft Approach Embankment……………………………………………………. 91 4.5 High Stiff Approach Embankment……………………………………………………. 94 4.6 Summary………………………………………………………………………………. 96 5 THE 3M-EASI MODEL…………………………………………………………………. 98 5.1 Integral Abutment Types……………………………………………………………… 98 5.2 Dynamic Response of IABs for Very Low Amplitude Motion……………………….100 5.3 Dynamic Response of IABs for Earthquake Motion……………………………….... 101 5.4 The Bridge Structure Component……………………………………………………. 103 5.5 The Near-Field Embankment Component…………………………………………… 104 5.6 The Far-Field Embankment Component……………………………………………... 108 5.6.1 Damping vs Location of the Embankment Response…………………………… 108 5.6.2 Approach Embankment Length…………………………………………………. 112 5.7 Verification of the 3M-EASI Model………………………………………………... 114 5.7.1 Continuum Soil Finite Element Model………………………………………….. 115 5.7.2 The Bridge Structure Component……………………………………………….. 116 5.7.3 The Far-Field Embankment Component………………………………………… 117 5.7.4 The Near-Field Embankment Component………………………………………. 118 5.7.5 Results: Fundamental Period……………………………………………………. 120 5.7.6 Results: Displacement Time Histories………………………………………….. 126 5.7.7 Results: Peak Response Quantities……………………………………………… 128 5.7.1.1 Peak Relative Displacement………………………………………………… 128 5.7.1.2 Peak Abutment Force……………………………………………………….. 132 5.7.1.3 Peak Foundation Force……………………………………………………… 133 5.8 Parametric Analysis with the 3M-EASI Model……………………………………… 135 vi 5.9 Summary……………………………………………………………………………... 139 6 CONCLUSIONS AND FUTURE WORK……………………………………………... 142 6.1 Future Work………………………………………………………………………….. 144 REFERENCES……………………………………………………………………………... 146 Annex A: Procedure for Calculating the Equivalent Linear Properties of the 1ME (Far-Field) model…………………………………………………….…... 150 Annex B: Procedure for Calculating the Parameters of the 3M-EASI model………...…... 152 vii LIST OF TABLES 3.1 Relative modal contribution analysis of a shear beam with H = 10 m and Vs = 150 m/s………... 50 3.2 Relative modal contribution analysis of a shear beam with H = 10 m and Vs = 60 m/s…………. 52 3.3 Physical properties of two embankments with Vs = 150 and 60 m/s…………………………….. 58 3.4 Peak response quantities for two embankments with Vs = 150 and 60 m/s……………………… 59 3.5 Physical properties of and infinitely long approach embankment………………………………... 75 3.6 Calculation of the equivalent linear properties of the approach embankment……………………. 75 4.1 Physical properties of four uniform approach embankments for ProShake analysis…………….. 80 4.2 Equivalent linear analysis of the low soft embankment using the 1ME model…………………... 82 4.3 Equivalent linear analysis of the low stiff embankment using the 1ME model………………….. 87 4.4 Equivalent linear analysis of the high soft embankment using the 1ME model…………………. 91 4.5 Equivalent linear analysis of the high stiff embankment using the 1ME model…………………. 94 4.6 Summary of the profile errors obtained with the 1ME model……………………………………. 97 5.1 Abutment type coefficient (AT) for calculating the near-field stiffness KAB…………………… 107 5.2 Physical properties of the uniform approach embankments…………………………………….. 115 5.3 Parameters of the bridge structure component…………………………………………………... 116 5.4 Parameters of the far-field embankment components per unit length…………………………... 117 5.5 Parameters of the near-field embankment component…………………………………………... 119 5.6 Fundamental period T1 of IABs obtained with four different models……………………………122 viii LIST OF FIGURES 1.1 Deterioration of deck joints and expansion bearings in standard bridges…………………………. 2 1.2 Cross-section of a single-span IAB supported on piled foundations………………………………. 3 1.3 Abutment-superstructure connection for two types of IABs………………………………………. 3 1.4 Steel H-Pile foundation for supporting an integral abutment……………………………………… 4 1.5 Temperature-induced displacement patterns in IABs……………………………………………... 5 1.6 Near and far field of an approach embankment……………………………………………………. 7 1.7 2D continuum FE model of an IAB for time history analysis……………………………………... 8 1.8 Frame-spring-dashpot model of an IAB for time history analysis………………………………… 9 1.9 Frame model of an IAB for pseudo-static analysis……………………………………………….. 11 1.10 Pseudo static analysis in an IAB frame model to calculate seismic demands…………………... 11 1.11 Frame-spring model of an IAB for pseudo-static analysis……………………………………… 12 1.12 Pseudo static analysis in an IAB frame-spring model to calculate seismic demands…………… 12 1.13 The 3M-EASI model for calculating seismic demands of IABs………………………………... 15 2.1 Bridge collapse due to out-of-phase movement between the approach embankment and the bridge structure during the 2010 Chile Earthquake…………………………………………... 16 2.2 Collapse of the near-field of an approach embankment during the 2010 Chile Earthquake……... 17 2.3 Collapse of the far-field of an approach embankment during the 2010 Chile Earthquake……….. 17 2.4 Two-dimensional representations of approach embankments in the far-field……………………. 19 2.5 Strong-motion instrumentation of MRO………………………………………………………….. 20 2.6 Identification of the fundamental transverse frequency of the MRO approach embankments…... 21 2.7 Cross-section of an infinitely long embankment…………………………………………………. 24 2.8 Normalized shear modulus and damping coefficient as a function of shear strain………………. 25 2.9 Kinematic response functions of MRO approach embankments…………………………………. 26 2.10 Transverse crest response of MRO embankment computed with the truncated shear-beam…… 27 2.11 Longitudinal crest response of MRO embankment computed with the truncated shear-beam…. 27 2.12 Frame-spring-dashpot model of the MRO………………………………………………………. 28 2.13 Generic embankment cross section……………………………………………………………… 30 2.14 Normalized embankment capacity curves for common embankment configurations…………... 31 2.15 Embankment-Bridge model for calculating the seismic response of IABs……………………... 33 ix 2.16 Substructures for analysis of IABs with flexible decks…………………………………………. 35 3.1 Uniform soil beam in the far field of an approach embankment…………………………………. 38 3.2 Internal forces in a differential section of a shear beam caused by a ground motion…………….. 39 3.3 Visualization of the location of the degree of freedom un(z=H,t)………………………………........ 42 3.4 Harmonic mode shape functions ψn(z) for solving the equation of motion……………………….. 45 3.5 Equivalent SDOF system of the vibration mode ψn(z) of a shear beam…………………………... 45 3.6 Equivalent SDOF systems for calculating the seismic response…………………………………. 46 3.7 5% damped displacement response spectrum of the Loma Prieta ground motion……………….. 47 3.8 Pseudo acceleration response spectrum of the Loma Prieta ground motion……………………… 48 3.9 Pseudo static analysis for calculating internal forces……………………………………………... 48 3.10 Location of modal periods T1, T2 and T3 in the 5% damped response spectrum of the Loma Prieta ground motion for two shear beams with different shear wave velocity Vs………. 53 3.11 Approach embankment with constant shear wave velocity……………………………………... 56 3.12 Relative displacement time histories at the top of two embankments with Vs = 150 and 60 m/s using the 1ME model……………………………………………………………………. 59 3.13 Peak response quantity profiles for two embankments with Vs = 150 and 60 m/s using the 1ME model…………………………………………………………………………………... 60 3.14 Approach embankment with shear wave velocity ratio rVs = 1.5………………………………. 62 3.15 Equivalent initial shear modulus Gin of an embankment with linear variation of Vs………….... 62 3.16 Equivalent initial shear modulus Gin of an embankment with rVs = 1.5………………………... 63 3.17 Shear modulus degradation profile of Gin for DG = 0.6………………………………………… 64 3.18 Constant degraded shear modulus Gdeg of an embankment with cosine degradation of Gin…….. 65 3.19 Constant degraded shear modulus Gdeg of an embankment with DG = 0.6……………………... 66 3.20 Two-dimensional representations of an infinitely long approach embankment………………… 67 3.21 3D FE model of a non-uniform approach embankment………………………………………… 67 3.22 Fundamental mode shapes of a non-uniform approach embankment…………………………… 68 3.23 Fundamental period ratio of a non-uniform shear beam………………………………………… 69 3.24 Density reduction factor for a non-uniform shear beam………………………………………… 70 3.25 Equivalent linear properties Gsec and ξ for representing the hysteretic behavior of soils during cyclic shear loading……………………………………………………………………… 71 3.26 Equivalent linear properties Gsec and ξ for a clay soil with PI = 30……………………………... 72 3.27 Displacement time histories u(t) obtained with the 1ME model for EF = 1……………………... 76 x 4.1 Shear wave velocity Vs profiles of four uniform approach embankments……………………….. 79 4.2 Discretization of an approach embankment for ProShake analysis………………………………. 80 4.3 Vucetic-Dobry (1991) modulus reduction and damping curves………………………………….. 81 4.4 Initial and matched ground motions………………………………………………………………. 81 4.5 Initial and matched response spectra……………………………………………………………... 82 4.6 Output time histories at the top of the low soft embankment…………………………………….. 83 4.7 Peak response quantity profiles in the low soft embankment…………………………………….. 84 4.8 Degradation profile of the initial shear modulus in the low soft embankment…………………… 86 4.9 Equivalent linear properties of the low soft embankment………………………………………... 86 4.10 Output time histories at the top of the low stiff embankment…………………………………… 88 4.11 Peak response quantity profiles in the low stiff embankment…………………………………... 88 4.12 Degradation profile of the initial shear modulus in the low stiff embankment…………………. 89 4.13 Equivalent linear properties of the low stiff embankment………………………………………. 90 4.14 Output time histories at the top of the high soft embankment…………………………………... 91 4.15 Peak response quantity profiles in the high soft embankment…………………………………... 92 4.16 Equivalent linear properties of the high soft embankment……………………………………… 93 4.17 Output time histories at the top of the high stiff embankment………………………………….. 94 4.18 Peak response quantity profiles of the high stiff embankment………………………………….. 95 4.19 Equivalent linear property profiles in the high stiff embankment………………………………. 96 5.1 Integral abutment bridge with stub abutments……………………………………………………. 98 5.2 Integral abutment bridge with full-height abutments…………………………………………….. 99 5.3 Integral abutment types…………………………………………………………………………… 99 5.4 Cross section of a full-height IAB in the longitudinal direction of the deck……………………. 101 5.5 Components of the 3M-EASI model for calculating the seismic response of IABs…………….. 102 5.6 Input ground motion and output displacement motions in the 3M-EASI model…………………102 5.7 Representation of the bridge structure using a single-degree-of-freedom system………………. 103 5.8 Pushover of the frame model for calculating the bridge stiffness KB…………………………….104 5.9 Pushovers on integral abutments for calculating the near-field stiffness………………………... 105 5.10 Near-field stiffness KAB of full-height abutments……………………………………………… 106 5.11 Near-field stiffness KAB of stub abutments…………………………………………………….. 107 5.12 1ME models for calculating the response at the top z=H of the embankment…………………. 108 5.13 Equivalent 1ME models for calculating the far-field response at z = H……………………….. 109 5.14 Equivalent damping ξz for calculation of far-field response at z ≠ H………………………….. 110 xi 5.15 Equivalent damping ξz for calculation of the far-field response at z ≤ H……………………… 111 5.16 Fundamental period TE of the 1ME model in the 3M-EASI model……………………………. 114 5.17 2D plane strain continuum soil FE model of a single-span full-height IAB…………………… 115 5.18 Pushover on the FE model for calculating KAB……………………………………………….. 118 5.19 Near-field stiffness obtained from pushover analyses on continuum soil FE models…………. 119 5.20 The 1M model………………………………………………………………………………….. 120 5.21 The 1M-ASI model…………………………………………………………………………….. 121 5.22 The 3M-EASI model…………………………………………………………………………… 121 5.23 Fundamental period T1 of IABs with embankments A (Gdeg = 22.1 MPa)…………………….. 122 5.24 Fundamental period T1 of IABs with embankments B (Gdeg = 8.9 MPa)……………………... 123 5.25 Fundamental mode shape of the ABAQUS model when T1 = TE……………………………… 123 5.26 Fundamental mode shape of the ABAQUS model when T1 > TE……………………………… 123 5.27 Equivalent area of the added mass of the near field……………………………………………. 125 5.28 Distance N from the abutment vs bridge-embankment period ratio…………………………… 125 5.29 Relative displacement time history of the 1M-ASI model for LB = 15 m……………………… 126 5.30 Relative displacement time history of the 3M-EASI model for LB = 15 m……………………. 126 5.31 Relative displacement time history of the 1M-ASI model for LB = 40 m……………………… 127 5.32 Relative displacement time history of the 3M-EASI model for LB = 40 m……………………. 128 5.33 Peak relative displacement……………………………………………………………………... 129 5.34 Relative displacement time history of the ABAQUS model for LB = 10 m…………………… 129 5.35 Location of the effective far-field embankment response……………………………………… 131 5.36 Peak abutment force……………………………………………………………………………. 132 5.37 Relative displacement time history of the ABAQUS model for LB = 30 m…………………… 133 5.38 Peak foundation force………………………………………………………………………….. 134 5.39 Peak relative displacement ratio DEASI/DASI for IABs with 0.1 ≤ Kr ≤ 0.8 and ξz/ξ = 0.5……... 136 5.40 Peak relative displacement ratio DEASI/DASI for IABs with 1 ≤ Kr ≤ 8 and ξz/ξ = 0.5…………. 138 5.41 Peak relative displacement ratio vs EASI index……………………………………………….. 139 xii ACKNOWLEDGMENTS I would like to thank my supervisor Dr. Carlos Ventura for the research guidance, financial and logistic support, and for the different research projects I had the opportunity to participate in the Earthquake Engineering Research Facility (EERF) at UBC. These projects consolidated my research studies comprehensively. The process of making sense of ambient noise was also significant learning from Dr. Ventura. It introduced me to the world of structural dynamics which is the signature of this thesis. His patience, vision, and friendship were fundamental for concluding this PhD thesis. I will be always in debt with my co-supervisor Dr. Liam Finn for his deep interest, attention, dedication, passion, and practical guidance on shaping the final product of this research thesis for implementation in engineering practice. Without his financial and logistic support, patience, time, insatiable appetite for technical discussion, and advice, this thesis would not have had the expected results. I am honored of having the opportunity of working with him. I would also like to thank Dr. Anoosh Shamsabadi from the California Department of Transportation (Caltrans) for the practical advice and efforts he put on this study. His visits to UBC in Vancouver for discussing the advance of this research played an important role for the implementation of the proposed model in bridge design practice. My appreciation extends to Dr. Donald Anderson from UBC for his specific and valuable comments for improving the implementation of the 3M-EASI model. His experience in bridge analysis and design and approval to the proposed model were instrumental for gaining confidence on the research results using the 3M-EASI model. The financial support for this project was provided by the Ministry of Transportation of British Columbia (BCMoT) under Professional Partnership Program. I gratefully thank Mrs. Sharlie Huffman of BCMoT for her support and motivation for this research study. xiii My gratitude will be always to the Institute of Engineering of the National University of Mexico – UNAM. The academic preparation, research training, and professional experience I received in Mexico during 8 years have great influence in my research study, which is reflected in this thesis. Dr. Miguel Romo, Dr. Victor Taboada, and Dr. Eulalio Juarez Badillo were fundamental for continuing my career as a researcher in geotechnical engineering. Likewise, my appreciation extends to my colleagues Carlos Gutierrez, Jose Marquina, Ismael Vega, Gabriel Palmerin, and Adonay Sandoval for sharing their knowledge and experience in foundation design. The pressure over the course of this research work would have been unbearable without the passionate debates I had with my good friends and colleagues Jose Centeno, Hugon Juarez, Bishnu Pandey, and Ruth Calimbo. After all these years, fortunately, we never came up with any practical and easy-to-implement solutions for the world’s problems. Finally, this life project is the result of Hilda and Amanda, who shaped my education, reinforced my courage, and opened my mind to believe in what matters. I am indebted to my mother Hilda for the financial and emotional support since we both started this dream in 1997. xiv To Hilda and Amanda 1 Chapter 1 INTRODUCTION Integral Abutment Bridges (IABs) are a cost-effective solution for highway agencies due to the elimination of deck joints and bearings. Analysis of the recorded response of IABs indicates that the seismic response of the approach embankments in the far field plays a significant role in the response of the bridge structure. However, current bridge design procedures neglect the effects of this dynamic interaction because of the complexity in coupling the available far-field embankment models with the bridge structure. This thesis develops a simple dynamic mass-spring-dashpot system for IABs, called the 3M- EASI model, which takes into account the inertial and kinematic interaction of the near and the far field of the approach embankments with the bridge structure. This is called Embankment-Abutment-Structure Interaction (EASI). The main contribution of this research study is the development and verification of a simple dynamic far-field embankment component, called the 1ME model, using equivalent linear analysis. The main advantage of the proposed 3M-EASI model is its easy implementation in bridge design procedures. 1.1 Standard Bridges One of the most important aspects of bridge design is the reduction of deck joints and expansion bearings. Joints deteriorate due to impact from repeated live loads as well as from temperature-induced cyclic loading which causes expansion and contraction of the bridge deck 2 (Figure 1.1a). Bearings get filled with dirt and rocks and ultimately fail to function or split and rupture due to unanticipated movements (Figure 1.1b). Joints and bearings are expensive to buy, install, maintain, repair, and replace. The most frequently encountered maintenance problem in bridges involves corrosion due to leaking expansion joints and seals that permit salt-laden run-off water from roadway surface to attack the bearings and supporting structures. a) deck joint b) expansion bearing Figure 1.1 Deterioration of deck joints and expansion bearings in standard bridges Because of the underlying problems, highways agencies have been eliminating joints and bearings where possible and are finding out that jointless bridges perform well without the continual maintenance issues inherent in joints. The longitudinal movement induced in jointless bridges by temperature changes, creep and shrinkage is then accommodated by special provisions at the abutments (Mistry, 2000). 1.2 Integral Abutment Bridges (IABs) Integral abutment bridges, or jointless bridges, are single or multi-span structures with a continuous concrete deck and approach slabs, integral with abutments supported on flexible foundations (Figure 1.2). Expansion joints and bearings at the ends of the deck are replaced with isolation joints at the ends of the approach slabs. In these types of bridges, the road surfaces are continuous from one approach embankment to the other and the abutments are cast integral with the deck as shown in Figure 1.3. 3 deck and girders foundation: one row of piles (see Fig. 1.4) approach embankment integral abutment (see Fig. 1.3) approach slab isolation joint Figure 1.2 Cross-section of a single-span IAB supported on piled foundations I girder Box girder Abutment Abutment Deck Deck Wingwall a) cast-in-place prestressed concrete I girders b) prestressed concrete Box girder Figure 1.3 Abutment-superstructure connection for two types of IABs The effect of forces parallel to the bridge longitudinal direction is minimized by designing the abutments and their foundation flexible and less resistant to longitudinal movements of the bridge. Only a single row of steel (H or small diameter pipe) piles is generally used to provide vertical support to abutments and minor resistance to longitudinal forces (Figure 1.4). Short and medium span bridges have also been designed with full-height integral abutments supported on narrows spread footings capable of providing small rotations. Fully rigid frames with fixed or near fixed foundations have been used where the cyclic thermal strains of the bridge deck are very small or negligible. The feasibility of an IAB is generally conditioned to the following requirements: Overall length of the superstructure less than 150 m. 4 Types of superstructures: steel I girders, steel box girders, cast-in-place prestressed concrete I girders, and prestressed concrete box girders. Concrete deck and skew less than 35°. Angle subtended by a 30 m arc along the structure less than 5°. Abutment height and wingwall length less than 6 and 7 m, respectively. Sub-soil conditions not susceptible to liquefaction, slip failure, sloughing or boiling. Figure 1.4 Steel H-Pile foundation for supporting an integral abutment The limitation placed on the total length of the bridge structure is mainly a function of local soil properties, seasonal temperature variations, resistance of abutment foundations to longitudinal movements, and the type of superstructure being considered. IABs with overall length longer than 150 m have been constructed in the United States. To date, Tennessee holds the record for the longest jointless bridge: a pre-cast concrete bridge on Tennessee State Route 50 over Happy Hollow Creek. It is a nine-span AASHTO 72 inches bulb-tee girder bridge with composite deck. Its overall length measures 360 m. The state of Colorado has built the longest integral bridge with cast-in-place concrete beams, with a span of 290 m, and the longest steel girder integral bridge, with a span of 318 m (Hassiotis et al, 2006). The absence of problems associated with deck joints and bearings offer a significant maintenance cost savings over the life of these types of bridges. However, there are other 5 additional advantages in the use of IABs: Elimination of uplift due to dead loads, which results in greater end-span ratios. Increase in the reserve of capacity to resist potentially damaging overloads, by redistributing loads along the continuous and full-depth diaphragm at bridge ends. Elimination of unseating of the girders at the supports during a seismic event, making IABs the preferred structures in more active seismic regions. Simplicity in the construction sequence. Although many IABs have been built in North America in the last four decades, no uniform national standards exist for static and seismic design of this type of structure in the USA or Canada. Design practices and assumptions concerning limits of thermal movement, soil pressure, pile design, and seismic demands vary considerably among the highway agencies. These decisions are based largely on past experiences (Kunin, 2000). 1.3 Temperature-Induced Displacement in IABs Repeating temperature changes of the bridge create longitudinal movements in the deck (contraction and expansion) and rotation of the abutment walls about their base (Figure 1.5). The magnitude of the wall movement is governed by the bridge length, deck type, foundation type, and changes in bridge temperature. contraction expansion a) winter position b) summer position Figure 1.5 Temperature-induced displacement patterns in IABs The complex soil-structure interaction mechanism involving relative movement between the abutments and adjacent backfill soil due to thermal variations may cause a serious problem in IABs. There are two important consequences of this movement: 6 Seasonal and daily cycles of expansion and contraction of the bridge deck can lead to an increase in earth pressure behind the abutment. This can result in the horizontal resultant earth force on each abutment being significantly greater than for which an abutment would typically be designed and represents a potentially serious long-term source of integral bridge problems. The second important consequence is the soil deformation adjacent to each abutment. It has been postulated that settlement troughs occur as a result of the soil slumping downward and toward the back of each abutment. In many cases this is addressed by incorporation of an approach slab into the bridge design, whereby the slab is intended to span the void created underneath it. However, there is also evidence to suggest that such a slab is unnecessary and that regular maintenance of the bridge surface can be sufficient to largely overcome this problem. A vast amount of research studies and technical reports are available in the literature concerning the modeling, analysis, design, construction, laboratory and field tests, monitoring, and performance evaluation of integral abutments and their foundations due to temperature- induced displacements. The most relevant studies about this topic are found in England et al, 2000; Faraji et al, 2001; Dicleli et al, 2005; Fennema et al, 2005; Hassiotis et al, 2006; and Clayton et al, 2006. The results of the studies have focused on proposing: Earth pressure distribution models for design of abutment walls. Procedures to calculate vertical and horizontal displacements of the backfill soil. Non-linear spring elements for modeling of abutment-backfill interaction. 1.4 Modeling the Seismic Response of IABs The exclusion of joints in IABs causes the bridge structure to be in constant interaction with the approach embankments. Therefore, the seismic response of the structure and the embankments is coupled, which makes the modeling stage and the calculation of seismic 7 demands a challenging task for bridge designers. The following is a short summary of the terminology and modeling techniques used in practice for calculating seismic demands in integral abutment bridges. 1.4.1 Terminology for Seismic Analysis of IABs From a dynamic point of view, the main components that control the seismic response of an IAB are the bridge structure, the near field, and the far field of the approach embankments. The first type of interaction between the bridge structure and the approach embankments occurs in the near field, also called Abutment-Backfill Interaction (ABI). The near field is defined as the section of the approach embankment where the deformation of the soil is affected by the displacement of the abutment. This section, shown in Figure 1.6, extents up to a distance equal to 3 times the height h of abutment, approximately. The near field adds stiffness to the abutment because of the backfill soil stiffness and it also connects the seismic response of the far field of the approach embankment with the bridge structure. Approach Embankment Far Field firm ground Near Field Bridge Structure h 3h Figure 1.6 Near and far field of an approach embankment Beyond the near field is the far field, which is the section of the approach embankment that is not affected by the abutment displacement or by the seismic response of the bridge structure. The interaction of the far field with the near field and the bridge structure is called Embankment-Abutment-Structure Interaction (EASI). This second type of interaction is related to the effect that the seismic response of the approach embankments in the far field has on the seismic response of the bridge structure. The main effect of EASI is the modification of the ground motion that is applied to the near field. 8 1.4.2 Time-History Analysis with Continuum Soil Models Figure 1.7 illustrates a 2D continuum Finite Element (FE) model of an IAB for calculating the seismic response of the bridge structure in the longitudinal direction of the deck. The model takes into account the coupling between the bridge structure and the near and far field of the approach embankments. The bridge structure and its foundation are usually modeled with beam-column elements to create a frame-type configuration. The approach embankments, on the other hand, are modeled with 2D continuum plane strain soil elements (Yang et al, 2004; Zhang et al, 2008). The boundary conditions of the FE model are assumed as rollers and the ground motion is applied at the base of the soil mesh. embankment mesh embankment meshbridge structure Figure 1.7 2D continuum FE model of an IAB for time history analysis The continuum FE model shown in Figure 1.7 is probably one of the closest representations to the physical model of an IAB. Some of the requirements for calculating seismic demands in the bridge structure using this type of model are: Specialized software capable of modeling structural and continuum soil elements, non- linear time history analysis for soil materials, surface contact interactions, etc. High time demand for the creation and solution of the FE model and for the post- processing, interpretation, organization, and revision of the output data. Specialized bridge designers with good knowledge in structural dynamics, soil mechanics, and also with experience in FE modeling. Because of the high demand of computational and human resources for analysis of IABs with continuum soil elements, engineering practice rarely uses this type of model. 9 1.4.3 Time-History Analysis with Frame-Spring-Dashpot Models The model shown in Figure 1.8 represents the more approximate procedure for calculating seismic demands in IABs in current practice without the need of continuum soil elements. The procedure consists of replacing the continuum soil elements with spring-dashpot components that are attached to a frame model of the bridge structure and its foundation. Then a dynamic analysis is carried out by applying a set of input motions to the spring-dashpot components. One of the challenges in this type of analysis is determining the force-displacement characteristics of the springs and the viscous coefficient of the dashpots. in pu t m ot io ns soil deck and girders piles abutment Figure 1.8 Frame-spring-dashpot model of an IAB for time history analysis The output data obtained from this type of analysis is in the form of time-history records of a response quantity at different locations in the model. For example, total displacement at the center of the deck, shear force at the top of the foundation, axial force of the spring-dashpot components, etc. The bridge designer then has to decide what level of response to use in design; i.e. mean, median or peak response values. The spring-dashpot components of the model in Figure 1.8 represent the kinematic response of the backfill and foundation soil. On the other hand, the input motions applied to the spring- dashpot components represent the seismic response of the approach embankments in the far field. A key step in this type of analysis is the determination of the input motions. The input motions are obtained by modeling separately the approach embankments with specialized software for soil response analysis and by calculating the seismic response in the 10 form of acceleration records at different locations along the height of the embankments for a given ground motion. The final result is a set of time-history records (the input motions) that are applied to the ends of the spring-dashpot components. This type of procedure is rarely used in practice because the amount of input motions to handle in the frame-spring-dashpot model may be large, which makes the analysis a time- consuming task. Another reason is that some of the commercial software used in bridge design does not accept a set of input motions that contains different acceleration records. Therefore, the most common procedure used in practice neglects the seismic response of the approach embankments in the far field and applies the ground motion to the bridge foundation and to the spring-dashpot components. One technical reason to justify this procedure is the assumption that the approach embankments do not modify significantly the ground motion during a seismic event. This assumption, however, has already been proved not to be valid based on analytical studies and on strong-motion earthquake data obtained from instrumented approach embankments (Wilson and Tan, 1990a; Zhang and Makris, 2002b). 1.4.4 Pseudo-Static Analysis with Frame Models Pseudo-static analyses are the most common technique used by bridge designers due to the low demand of computer time. This type of analysis requires the response spectrum of a ground motion, or a design response spectrum (CAN/CSA-S6-06), and the fundamental period (T1) of the bridge. The calculation of T1 depends on the modeling considerations for taking into account the soil-structure interaction. The simplest approach to calculate seismic demands in IABs (i.e. relative displacement of the deck, abutment force, and foundation force) consists of creating a frame model with beam- column elements for the bridge structure and the foundation as shown in Figure 1.9. 11 deck and girders abutment piles LE seismic earth pressure Figure 1.9 Frame model of an IAB for pseudo-static analysis The structural design of piles is carried out using the equivalent cantilever method (Abendroth and Greiman, 1989; Greigman et al, 1987) as a column with a fixed base at some distance LE below the base of the abutment. The equivalent pile length LE is calculated with Equation 1.1, where EP and IP are the pile modulus of elasticity and moment of inertia with respect to the plane of bending, respectively, and kh is coefficient of sub-grade reaction of the soil. 4 h PP E k I E 4 L = 1.1 The pseudo-acceleration demand A1 of the bridge is obtained from the response spectrum with the fundamental period T1 of the frame model (Figure 1.10a). The seismic demands in the bridge structure are calculated with a static analysis by applying a load F along the deck equal to M times A1, where M is the mass of the deck and girders (Figure 1.10b). T1 A1 F = M A1 a) response spectrum b) load in the frame model Figure 1.10 Pseudo static analysis in an IAB frame model to calculate seismic demands The seismically induced lateral earth pressure of the backfill soil on the abutments (Figure 1.9) is usually assessed with the Mononobe-Okabe (M-O) method (Mononobe and Matsuo, 1929). It has been found that at very low levels of acceleration the induced pressures are in general 12 agreement with those predicted by the M-O method. However, as the accelerations increase to those expected in regions of moderate seismicity, the induced pressures are larger than those predicted by the M-O method. The deviation is attributed to the abutment flexibility, backfill soil properties, and the seismic response of the approach embankment (Green et al, 2003). 1.4.5 Pseudo-Static Analysis with Frame-Spring Models A more refined approach to calculate seismic demands in IABs with pseudo-static analysis consists of replacing the backfill and foundation soil with springs as shown in Figure 1.11. deck and girders abutment piles soil Figure 1.11 Frame-spring model of an IAB for pseudo-static analysis Figure 1.12 illustrates the procedure to calculate the seismic demands in the bridge structure, which is similar to the one shown in Figure 1.10. F = M A1 T1 A1 a) response spectrum b) load in the frame-spring model Figure 1.12 Pseudo static analysis in an IAB frame-spring model to calculate seismic demands It should be noted that T1 of the frame-spring model in Figure 1.12a is shorter than T1 of the frame model in Figure 1.10a because of the addition of stiffness to the bridge structure given 13 by the soil-spring components. This would cause the seismic demands obtained with the two frame-based models (Figure 1.10 and 1.12) to be different. The model shown in Figure 1.11 does not take into account the seismic response of the approach embankments in the far field for calculating the seismic demands of the bridge structure. Despite the fact that neglecting the far-field embankment response does not represent the seismic response of IABs or other bridges as well, the model shown in Figure 1.11 is probably the most common procedure used in current bridge design practice. 1.4.6 IABs with Approach Embankment Models Three relevant studies conducted by Wilson and Tan (1990a, 1990b), Zhang and Makris (2002a, 2002b), and Kotsoglou and Pantazopoulou (2009) have addressed the calculation of the seismic response of IABs considering the near- and the far-field embankment response. These research works are discussed in Chapter 2. The studies concluded that the participation of the far field in the calculation of the seismic response of the bridge structures is important. The above studies are a significant contribution to the modeling and understanding of the seismic response of IABs. However, the proposed procedures for coupling the far-field embankment models with the bridge structure require a high demand of computational and human resources. This has resulted in the continuing use of bridge models that neglect the seismic response of the approach embankments in the far field. 1.5 Objective and Organization of Thesis The University of British Columbia (UBC) and the Ministry of Transportation and Infrastructure (BCMoT) initiated in 2007 a joint project to study the effects of the near and the far field of the approach embankments on the seismic response of IABs. 14 The main interest of UBC and BCMoT, and therefore the objective of the research thesis, is to provide an accurate and simple dynamic model to bridge engineers for calculating seismic demands in IABs taking into account EASI. The organization of the thesis for developing a simple dynamic model for IABs is as follows: Chapter 2 provides a literature review of the most relevant research work on modeling the near- and far-field response of approach embankments and their effects on the seismic response of IABs. Chapter 3 describes the analytical development of a single mass-spring-dashpot system, called the 1ME model, for calculating the seismic response of approach embankments in the far field using equivalent-linear analysis. Chapter 4 presents the verification of the 1ME model for four types of approach embankments using the computer program ProShake. Chapter 5 describes the proposed three-mass system, called the 3M-EASI model, for calculating the seismic response of IABs in the longitudinal direction of the deck using equivalent-linear analysis. The proposed model takes into account the interaction of the near and far field of the approach embankments with the bridge structure. The 3M-EASI model, shown in Figure 1.13, is verified with time-history analyses of continuum soil finite element models using the computer program ABAQUS. The chapter also includes the estimation of seismic demands in the bridge structure neglecting the far-field embankment response for comparison with the estimations obtained with the 3M-EASI model. Chapter 6 summarizes the conclusions of this research thesis and suggests future work on the seismic response of IABs using the 3M-EASI model. 15 ground motion Far Field Bridge StructureNear Field Near Field Far Field Figure 1.13 The 3M-EASI model for calculating the seismic response of IABs Annex A includes the step-by-step procedure for calculating the equivalent linear properties of the 1ME (far field) model. Annex B includes the step-by-step procedure for calculating the parameters of the 3M-EASI model. 16 Chapter 2 LITERATURE REVIEW A crucial factor affecting the seismic performance of a bridge is the seismic response of the approach embankments. Figure 2.1 shows the unseating of the end girders of a three-span overpass bridge during the 2010 Chile Earthquake. The collapse was caused by a poor estimation of the peak relative displacement between the bridge structure and the approach embankments, which exceeded the seating length of the abutment. Figure 2.1 Bridge collapse due to out-of-phase movement between the approach embankment and the bridge structure during the 2010 Chile Earthquake This type of failure is mainly caused by the difference between the fundamental periods of vibration of the bridge structure and the approach embankment, which causes an out-of-phase movement not usually taken into account in bridge design due to the complexity of calculating the seismic response of the approach embankments. 17 The functionality of the bridge is also compromised if the seismic performance of the approach embankments is poor. This is the case of the bridge shown in Figures 2.2 and 2.3 which was closed after the 2010 Chile Earthquake due to the collapse of the near- and the far- field of one of the approach embankments. This case history illustrates the importance of calculating the seismic response of the approach embankments during the geotechnical design. Figure 2.2 Collapse of the near-field of an approach embankment during the 2010 Chile Earthquake Figure 2.3 Collapse of the far-field of an approach embankment during the 2010 Chile Earthquake 18 The 1964 Alaska Earthquake showed that even well-compacted engineered backfills can be vulnerable to excessive settlements. This was the case of Copper River Bridge 2, which was closed due to a 1 m settlement of the backfill (near field) caused by the out-of-phase motion between the abutment and the approach embankment, alternately compacting and sloughing off material (Yashinsky and Karshenas, 2003). Recognizing that the soil response must be taken into account in bridge design, engineering practice includes in the modeling stage the effects of Soil-Structure Interaction (SSI) by enhancing the structural models with springs-dashpot components that represent the soil behavior, as discussed in Chapter 1. SSI has been studied extensively in the last 30 years and a considerable number of spring-dashpot based models are available in the literature (Wolf, 1985; Cakmak, 1987; Kolar and Nemec, 1989; Bull, 1994). These types of models have focused mainly on modeling soil-foundation-structure interaction and abutment-backfill interaction for seat-type abutments (Shamsabadi, 2007; Zhang et al, 2008). Contrary to the work done on SSI in standard bridges, the models proposed in the literature for calculating the seismic response of Integral Abutment Bridges (IABs) are very few. This may be associated to the complexity of coupling the dynamic response of the approach embankment models with the bridge structure. This chapter reviews the most relevant research studies on modeling the seismic response of the near- and the far-field of the approach embankments and the procedures for coupling the proposed models with the bridge structure. 2.1 Wilson and Tan (1990a,b) Wilson and Tan performed the first study that presented a simple analytical model to estimate the near-field stiffness for integral abutments. The two-part analytical model was developed to assist in seismic response analysis of short and medium span highway bridges. 19 The first part of the model proposes a procedure for determination of the degraded shear modulus Gdeg of the approach embankment based on the identification of its fundamental transverse frequency of vibration Fw in the far-field. The second part of the model provides an expression for calculating the near-field stiffness using the identified Gdeg. The proposed model is validated with strong-motion earthquake records obtained from the Meloland Road Overpass (MRO) in California for the 1979 Imperial Valley earthquake. 2.1.1 Far-Field Embankment Model Wilson and Tan (1990a) formulated a simple model for calculating the fundamental frequency of vibration of an approach embankment in the transverse direction of the bridge. The basis of the proposed model is an analogy between the fundamental frequency of a truncated wedge (representing a cross section of the approach embankment, as in Figure 2.4a) and the fundamental frequency of a uniform shear beam with the same height H, shear modulus G, density ρ, and Poisson’s ratio ν, as shown in Figure 2.4b. The uniform shear beam is used as the basis of comparison because of the simplicity of its frequency equation. B H HG, ρ, ν G, ρ, ν1 S u(t) a) truncated wedge b) uniform shear beam Figure 2.4 Two-dimensional representations of approach embankments in the far-field In order to develop the simplified relationship between the two models, the frequency of the wedge model was computed using a finite element model and this frequency was compared to that of the uniform shear beam. The finite element computations were carried out for a number of truncated wedges of different heights H, top widths B, and side slopes S. The study led to the relationship given in Equation 2.1, where Fw is the fundamental transverse frequency of the truncated wedge and Fsb is the frequency of the uniform shear beam. The equation was calibrated for 1 ≤ S ≤ 3 and 0.2 ≤ H/B ≤ 1. 20 ( ) sb08.0w F H/B .181 F = where 4H /G Fsb ρ= 2.1 The authors assumed that the seismic response of the approach embankment in the far-field for a given ground motion üg(t) can be calculated with a single-degree-of-freedom (SDOF) linear system with frequency Fw and damping ratio ξ, as indicated with Equation 2.2. g(t)(t) 2 w(t)w(t) u u)F (2 u )F ξ(4 u &&&&& =π+π+ 2.2 Fw and ξ are strain-dependent properties that depend on the effective shear strain γef of the approach embankment. However, no expression or procedure was proposed to calculate γef. Instead, Fw and ξ are estimated with a system identification technique using the ground motion üg(t) and the recorded embankment response ür(t). The proposed technique systematically varies Fw and ξ in Equation 2.2 until the analytical response ü(t) matches the recorded response ür(t). Wilson and Tan (1990b) applied their system identification technique to the instrumented approach embankments of Meloland Road Overpass (MRO) using strong-motion earthquake data of the 1979 Imperial Valley earthquake (Figure 2.5). Figure 2.5 Strong-motion instrumentation of MRO (after Wilson and Tan, 1990b) Figure 2.6 shows a comparison of the identified Fw of the MRO approach embankments using different time windows. Analysis of the initial 0-4s time window indicates that the approach 21 embankments had an initial fundamental frequency of 2.52 Hz. This frequency dropped substantially to 1.39 Hz during the interval of most intense shaking (4-8s). In the third interval (8-12s), the frequency recovered to 2.08 Hz, which is closer to the initial value. The reduction of Fw during the segment of most intense shaking is consistent with observations that the degraded shear modulus Gdeg of soils decreases as the effective shear strain γef increases. 2.52 1.39 2.08 1.51 0 0.5 1 1.5 2 2.5 3 0 - 4 4 - 8 8 - 12 0 - 12 Time Window (s) F w ( H z) Figure 2.6 Identification of the fundamental transverse frequency of the MRO approach embankments The frequency of 1.51 Hz obtained with the 0-12s time window is very close to the frequency identified from the 4-8s time window (1.39 Hz). The reason for this is that the far-field embankment response tends to be governed by the larger amplitude motions; therefore, the 4-8s segment exerts a dominate influence on the 0-12s optimization calculations. 2.1.2 Near-Field Stiffness Model Equation 2.3 gives the proposed expressions to calculate the total transverse Kt and total vertical Kv stiffness of the near field. Kt and Kv are also referred in the literature as the abutment stiffnesses or the stiffness of the Abutment-Backfill Interaction (ABI). deg wing t G H/B) 2S(1ln L 2S K += deg wing v G H/B) 2S(1ln L )S4(1 K + ν+= 2.3 22 Lwing is the wing length of the abutment and Gdeg is the degraded shear modulus calculated with Equation 2.4, in which Fw is obtained with the proposed identification technique. ( ) H/B .181 F H4 G 2 08.0 w deg ρ = 2.4 Kt and Kv are incorporated as spring components in the FE frame model of the bridge structure for modeling the abutment-backfill interaction. The calculation of seismic demands is obtained following the methodology explained in Sections 1.4.3 or 1.4.5. The frame-spring model of the MRO showed that in the fundamental transverse mode, the translations of the ends of the bridge deck contribute significantly to the total maximum displacement at the midspan. In contrast to this, relatively small translational motions were observed at the ends of the bridge deck in the vertical modes. These results suggest that the stiffness of the near-field has a significant greater influence on the shape of the first transverse mode than it has on influencing the shapes of the vertical modes. 2.1.3 Discussion Wilson and Tan introduced three general assumptions in their research work: a) The near-field stiffness is proportional to Gdeg. b) Gdeg is obtained from the far-field embankment response. c) The far-field embankment response can be calculated with a SDOF linear system. The major contribution of these assumptions is the simplicity for modeling the near- and the far-field response of approach embankments using linear analysis, which is convenient for easy implementation in bridge design procedures. However, the authors did not propose a procedure for calculating the far-field embankment response of non-instrumented approach embankments. This is a problem for bridge design because Gdeg is determined from the far- field embankment response. 23 Another key factor that was not solved by Wilson and Tan is the dynamic coupling between the far-field embankment model and the bridge structure. Neither the far-field embankment model nor its seismic response (i.e total acceleration) is taken into account in the FE model of the bridge structure. Thus, the motion applied at the near-field spring components is the ground motion. As mentioned in Section 1.4.3, this procedure does not represent the seismic response on IABs. Based on Wilson and Tan’s work it is recommended that their model be enhanced with the following developments in further investigations: A simple procedure to calculate the far-field embankment response using a SDOF linear system. A procedure to couple the SDOF far-field embankment model with the near-field model. 2.2 Zhang and Makris (2002a,b) Zhang and Makris also developed a near-field stiffness model for integral abutments using the degraded shear modulus Gdeg. However, their major contribution was the determination of Gdeg using a proposed procedure for calculating the far-field embankment response for any ground motion. The procedure was validated with strong-motion earthquake data of the MRO in California for the 1979 Imperial Valley earthquake. 2.2.1 Far-Field Embankment Model Zhang and Makris (2002a) formulated a multi-modal linear model for calculating the far-field response in the transverse direction of the approach embankment. The model is based on the development of a kinematic response function, which is the solution of the differential equation of motion of a truncated shear beam (Figure 2.7) in the frequency domain. The solution takes into account the modes of vibration of a homogeneous truncated shear beam. 24 Figure 2.7 Cross-section of an infinitely long embankment (after Zhang and Makris, 2002a) Equation 2.5 gives the kinematic response function I(ω) of the truncated shear beam, defined as the ratio of the amplitude of the crest motion cu to the amplitude of the base motion gou . go oo2oo1 go c )( u )kz(Yc )zk(Jc u u 1 I +=+=ω 2.5 where [ ] H))(k(zY )(kz)/Y(kzJ H))(k(zJ u c ooo1o1oo go 1 +−+= 2.6 and 1 o1 o1 2 c)kz(Y )kz(J c = ( )η+ρ ω= i1/G k deg ( )cb c o BB HB z −= 2.7 where ω is the circular frequency, Gdeg is the degraded shear modulus, η is the hysteretic damping coefficient, ρ is the density, Bc and Bb are the crest and bottom width, respectively, of the cross section of the approach embankment, H is the height, and Jo, J1, Yo and Y1 are the zero and first order Bessel functions of the first and second kind, respectively. The displacement-time history at the crest of the embankment (t)cu is calculated with the kinematic response function I(ω) and the Fourier transform of the ground motion ug(ω). 25 ∫ ∞ ∞− ω ωω δωπ= e u I 2 1 u ti)(g)((t) c 2.8 The proposed procedure to calculate the far-field embankment response is as follows: step 1: evaluate I(ω) assuming initial values of Gdeg and η step 2: compute uc(t) step 3: calculate the effective shear strain as )H/u( 0.65 cmaxef =γ step 4: obtain Gdeg and η from the averaged curves of Figure 2.8 using γef step 5: repeat steps 2-4 until γef converges Figure 2.8 Normalized shear modulus and damping coefficient as a function of shear strain (after Zhang and Makris, 2002a) 26 2.2.2 Validation of the Far-Field Embankment Model Figure 2.9 plots with a continuous bold line (shear beam) the amplitude of the kinematic response function I(ω) of the MRO north approach embankment computed with Equation 2.5 and converged values of γef = 0.52 %, Gdeg = 2 MPa, and η = 0.52 for the 1979 Imperial Valley Earthquake. F1 F2 Figure 2.9 Kinematic response functions of MRO approach embankments (after Zhang and Makris, 2002a) Zhang and Makris’ model (called shear beam in Figure 2.9) is a 1D approximation of a 3D approach embankment. The solution of the shear beam approximation was compared with the I(ω) obtained with a 2D and a 3D FE model. The approach slope along the longitudinal direction (4.4%) of the embankment is also included in a 3D model (tapered). The peak values of I(ω) at F1 = 1.4 Hz and F2 = 3.4 Hz in Figure 2.9 represent the first two natural frequencies of the truncated shear beam. The results indicate that the shear-beam approximation captures most of the transverse response of the approach embankment in comparison to the FE models. Figure 2.10 plots the transverse crest response (total acceleration and relative displacement time histories) of the MRO north embankment when computed with Equation 2.8. The figure 27 indicates that an equivalent-linear analysis with the 1D truncated shear beam model yields results on the crest response that are in satisfactory agreement with the recorded motions. a) total acceleration b) relative displacement Figure 2.10 Transverse crest response of MRO embankment computed with the truncated shear-beam (after Zhang and Makris, 2002a) The shear-beam approximation also provides an acceptable prediction of the longitudinal response using the kinematic response function I(ω) obtained in the transverse direction of the embankment as shown in Figure 2.11. Figure 2.11 Longitudinal crest response of MRO embankment computed with the truncated shear-beam (after Zhang and Makris, 2002a) 2.2.3 Near-Field Stiffness Model Equation 2.9 provides the near-field transfer function for integral abutments in the transverse Kt, longitudinal Kl, and vertical Kv direction of the bridge (Kt = Kl = Kv = K). The real and 28 imaginary part represents the stiffness and the viscous coefficient, respectively, of a spring- dashpot component. ( ) ( ) ( ) ( ) 3coooooooo o1ooooo1 deg SHB0.7 )kz(Y )Hz(kJ H)k(zY )(kzJ )kz(Y H)k(zJ H)k(zY )(kzJk )i(1G K +−+ +−+η+= 2.9 Gdeg and η are the converged values obtained from the far-field embankment response. For practical purposes the authors recommended to obtain a frequency-independent value of K. This is done by plotting the real and imaginary part of Equation 2.9. Then the spring and dashpot’s values are selected from the plots at the low-frequency interval. This frequency- independent spring value is close to the one obtained with the Wilson and Tan’s model, given in Equation 2.3. 2.2.4 Embankment-Abutment-Structure Interaction (EASI) Zhang and Makris (2002b) proposed applying the far-field embankment response at the ends of the near-field spring-dashpot components to take into account EASI in the calculation of the seismic response of the bridge structure (Figure 2.12). Figure 2.12 Frame-spring-dashpot model of the MRO (after Zhang and Makris, 2002b) The proposed procedure was examined by comparing the computed time response with records from the MRO. It was concluded that the earthquake response of IABs can be realistically estimated if the amplified crest motions are taken into account in the FE model. 29 2.2.5 Discussion Zhang and Makris’ model is a rational development for the proper seismic analysis of IABs using frame-spring-dashpot models. The most important concepts introduced in their research work are: The far-field embankment response can be calculated with an equivalent-linear analysis. The motion applied at the end of the near-field spring-dashpot components is the far- field embankment response. However, the complexity in the implementation of the proposed model is not suitable for current bridge design procedures. The main reasons for this are: Commercial software for structural analysis does not have the proposed model implemented. The proposed EASI model does not couple the far-field embankment model with the structure. Therefore, it is not possible to identify the mode shapes and the natural frequencies of the whole dynamic system. Identification of modal shapes and natural frequencies of the whole dynamic system (bridge structure and approach embankments) is useful in bridge design procedures for calculating seismic demands using spectrum-based analysis. It is recommended that Zhang and Makris’ procedure be improved with the following developments in further investigations: A simpler seismic model for calculating the far-field embankment response. A simple dynamic model that couples the far-field embankment model with the bridge structure. A parametric study to understand the effects of EASI on IABs. 30 2.3 Inel and Aschheim (2004) Inel and Aschheim proposed a procedure to estimate the displacement demand of short IABs taking into account the force-displacement characteristics of the approach embankments. The procedure is based on the calculation of the fundamental transverse period of the bridge- embankment system and the capacity curves of the embankments. The study indicates that the displacement demand of the bridges depend to a large extend on the embankment flexibility. 2.3.1 Capacity Curves of Approach Embankments Figure 2.13 shows the discretization of the cross section of an approach embankment using lumped masses and shear springs. Figure 2.13 Generic embankment cross section (after Inel and Aschheim, 2004) The capacity curve of the embankment is obtained with a pushover analysis. In the simulation, the lateral forces are distributed along the embankment height in proportion to the lumped masses and the assumed parabolic deflected shape of the embankment during strong shaking. A key factor for calculating the total force (or base shear) with the pushover analysis is the effective length of the embankment L’ (distance perpendicular to the cross section). L’ is obtained from a calibration procedure using a FE model of the bridge structure and strong- motion earthquake records. The authors applied this procedure to the Meloland Road and Painter Street Overcrossings, in California, for several earthquakes of variable intensity. It was found that L’ varied from 2 to 16 m and it is ground motion intensity dependent. 31 Figure 2.14 plots the normalized capacity curves of approach embankments for different values of Plasticity Index (PI), height H’, and crest width w’. In the curves, the embankment resistance per effective embankment length L’ is normalized by the embankment width at the mid-height wavg, and the low strain shear modulus Gmax. ∆/H’ is the ratio of the crest displacement and the embankment height. Figure 2.14 Normalized embankment capacity curves for common embankment configurations (after Inel and Aschheim, 2004) The authors assumed that the bridge-embankment system can be approximated by an equivalent SDOF system in which the mass is given by the bridge superstructure and the approach embankments (proportional to L’) and the stiffness and strength by the embankment capacity curve. These parameters are used for calculating the fundamental period and the displacement demand of the bridge structure using time-history or response spectrum based analyses. 32 2.3.2 Discussion The concept of embankment capacity curves introduced by Inel and Aschheim is an interesting tool for modeling the force-displacement characteristics of IABs. The most important conclusion of their research work is: The displacement demands experienced by the columns of short IABs are primarily a function of the flexibility of the approach embankments. However, the proposed model combines the seismic response of the bridge structure, the near field, and the far field of the approach embankments in one SDOF system, which does not represent the rational understanding of how the different components interact with each other. This is the main reason why the effective length of the embankment, obtained from the calibration of the model, has a significant variation (2 m ≤ L’ ≤ 16 m) and it is ground motion intensity dependent. Some of the limitations of Inel and Aschheim’s model are: It is valid only for short IABs. The development of new capacity curves for other types of approach embankments requires a calibration procedure using strong-motion earthquake records. 2.4 Kotsoglou and Pantazopoulou (2007, 2009) Kotsoglou and Pantazopoulou developed a SDOF embankment-bridge model that takes into account EASI. The model is based on the calculation of the fundamental transverse mode shape of the embankment for the boundary conditions imposed by the bridge structure. The seismic response of the bridge structure is estimated with a substructuring procedure. 33 2.4.1 Embankment-Bridge Model Figure 2.15a shows the proposed 3D approach embankment model for calculating the seismic response of the bridge in the transverse direction. The lumped mass and elastic spring attached at the end of the embankment represent the inertial and kinematic interaction of the bridge structure with the embankment. Figure 2.15b shows the fundamental transverse mode Φ(z,y) of vibration of the embankment-bridge system described mathematically with Equation 2.10. x, y and z coordinates are aligned with the transverse, longitudinal and vertical directions, respectively, of the embankment. a) embankment-bridge model b) fundamental mode shape Figure 2.15 Embankment-Bridge model for calculating the seismic response of IABs (after Kotsoglou and Pantazopoulou, 2007) λ−λ− λ−−λ− π=Φ )sin(y ) sin(L ) cos(L)cos(y z 2H cos y)(z, 2.10 L and λ are obtained from solving a set of differential equations for the given boundary conditions imposed by the values of the lumped mass and the spring. The authors did not include in their research paper (2007) a closed-form solution of L and λ. Equation 2.11 represents the equation of motion of the proposed SDOF embankment-bridge model in the transverse direction, where the subscripts “emb” and “bridge” refer to bridge structure and embankment, respectively. ux is the relative transverse displacement of the lumped mass, which assumes that the bridge deck is rigid. üg is the ground motion. 34 gxxembx1 u I u K u C u M &&&&& −=++ 2.11 where bridgeemb1 M 0.5 M M += bridgeemb K 0.5 K K += 2.12 ξ= K M2 C embembemb bridgeemb M 0.5 I I += 2.13 ∫ ∫ δδΦρ= L 0 H 0 )y,z( 2 emb y z B M ∫ ∫ δδΦρ= L 0 H 0 )y,z(emb y z B I 2.14 δδδ ΦδΦ+δδδ ΦδΦ−= ∫ ∫∫ ∫ L 0 H 0 2 )y,z( 2 )y,z( L 0 H 0 2 )y,z( 2 )y,z(degemb y zy y z z G B K 2.15 In Equations 2.12 to 2.15, Kbridge is the equivalent linear stiffness given by the non-linear behavior of the bents and the abutments of the bridge structure, B is the average width of the cross section of the embankment, and Gdeg and ξ are the equivalent linear properties of the approach embankment. 2.4.2 IABs with Flexible Decks If the bridge deck is considered flexible in the transverse direction, then Mbridge and Kbridge in Equation 2.12 have to be multiplied by a dimensionless factor (α < 1) that gives the effective mass and stiffness that participate in the interaction with the embankment. α depends on the deflected shape of the deck and it is calculated from the seismic response of the bridge structure with a FE model. The inertial and kinematic contribution of the approach embankments to the bridge response is represented by lumped masses (Memb), springs (Kemb), and dashpots (Cemb) attached to both ends of the deck. 35 The global embankment-bridge system is therefore divided into three substructures: a) two SDOF embankment-bridge models and b) a FE model of the bridge structure as shown in Figure 2.16. Figure 2.16 Substructures for analysis of IABs with flexible decks (after Kotsoglou and Pantazopoulou, 2009) The seismic response of each substructure is calculated iteratively to update the values of α, L and λ. The procedure is considered convergent when the calculated displacement time- histories at the contact nodes coincide. 2.4.3 Discussion Kotsoglou and Pantazopoulou’s model is rational and accurate. The idea of considering one SDOF system that takes into account EASI for IABs is appealing for engineering practice. However, the model is not easy to implement in practice due to the iterative procedure of solving a set of differential equations to calculate the parameters λ and L that determine the fundamental transverse mode shape. The proposed model includes the near field and the far field in one single model. However, the shear strain γ of the soil is different for each field. The model does not specify how and where 36 the effective shear strain γef is calculated in the approach embankment to obtain the equivalent linear properties Gdeg and ξ. It is concluded that including the bridge structure, the near field, and the far field of the approach embankment in a SDOF system complicates the calculation of the seismic response of IABs and makes its implementation difficult in bridge design procedures. 2.5 Summary Strong-motion earthquake data and analytical studies indicate that the far-field embankment response has a significant effect on the displacement demands of IABs and it should not be ignored in design. Therefore, the general consensus is that Embankment-Abutment-Structure Interaction (EASI) should be included in the modeling stage of the bridge. Several models have been proposed for calculating the seismic response of IABs taking into account EASI. The implementation of these models, however, is complicated for current bridge design procedures. This is the main reason why the estimation of seismic demands in IABs usually neglects the far-field embankment response. It is concluded then, that a simple model for calculating and coupling the seismic response of the approach embankments in the far field with the bridge structure is needed in current practice. A new model, called 3M-EASI, that takes into account the inertial and kinematic interaction of the near and the far field of the approach embankments with the bridge structure is developed in this research thesis. The basic assumptions of the 3M-EASI model are based on Wilson and Tan’s work: The far-field embankment response can be calculated with a SDOF system using the shear beam model and an equivalent-linear analysis. 37 The near-field can be represented by a spring component using the degraded shear modulus Gdeg obtained from the far-field embankment response. The main advantage of the 3M-EASI model is the easy implementation in bridge design procedures. 38 Chapter 3 THE 1ME MODEL This chapter describes the development of a SDOF system, called the 1ME model, for calculating the seismic response of approach embankments in the far field. The 1ME model is developed by discretizing the differential equation of motion of a uniform shear beam using the Galerkin method. The first mode of vibration of the beam is assumed to be the approximate solution of the displacement field in the embankment. The shape effects of non- uniform embankments are accounted by modifying the period of vibration of an equivalent uniform embankment with a dimensionless factor. The nonlinear response of the 1ME model is approximated by an equivalent-linear analysis. 3.1 Differential Equation of Motion Figure 3.1 shows the idealization of a uniform soil beam located in the far field of an approach embankment of height H and supported on firm ground. Far Field firm ground H near field Figure 3.1 Uniform soil beam in the far field of an approach embankment 39 Figure 3.2 illustrates the displaced position of the uniform soil beam at a given time t due to the displacement ground motion ug(t) applied at the base, which is assumed to be rigid. The beam is called uniform because its cross section area A and density ρ are assumed to be constant. The shear beam model is chosen for representing the dynamic behavior of the approach embankment because of its accuracy and simplicity (Chapter 2). H z ut ug(t) u(z,t) ut(H,t) A V(z,t) V(z+δz,t) F(z,t)δz rigid base Figure 3.2 Internal forces in a differential section of a shear beam caused by a ground motion The origin of the coordinate system (z,ut), where z is the vertical coordinate and ut is the total horizontal displacement, is located at the base of the beam. The thick dashed line represents the rigid body displacement ug(t), or ground displacement, of the beam. The bold line represents the total displacement ut(z,t) = ug(t) + u(z,t) given by the ground displacement ug(t) and the relative displacement u(z,t) caused by the deformation of the beam. ut(z,t) and u(z,t) depend on the location z in the beam and on the time t. The total displacement ut(z,t) creates body F(z,t) and shear V(z,t) forces along the beam as shown in Figure 3.2. The equilibrium condition of these forces in a differential section of height δz of the beam is expressed with Equation 3.1. 40 0 V V F )t,zz()t,z()t,z( =−+ δ+ 3.1 F(z,t) is the inertial force created by the total acceleration üt(z,t) = üg(t) + ü(z,t) of the differential section as indicated in Equation 3.2, where ü(z,t) = δ2u(z,t)/δt2 is the relative acceleration and üg(t) is the ground acceleration. ( ) z u u A F t)(z,g(t))t,z( δ+ρ= &&&& 3.2 V(z,t) and V(z+δz,t), given in Equation 3.3, are the shear forces at the ends of the differential section. τ(z,t) is the shear stress calculated with Equation 3.4, which assumes the beam material to have shearing characteristics of a Kelvin-Voigt solid. A V )t,z()t,z( τ= ( )A V )t,z()t,z()t,zz( ττ δ+=δ+ 3.3 t)(z,)z(t)(z,)z()t,z( G γγτ η+= & where z u )t,z()t,z( δ δ=γ t z u )t,z( 2 )t,z( δδ δ=γ& 3.4 G(z) is the depth variable shear modulus, η(z) is the depth variable viscous coefficient, γ(z,t) is the shear strain, and )t,z(γ& is shear strain velocity. The subscript (z,t) indicate that the response quantity is depth and time dependent. Introducing Equations 3.2 to 3.4 in Equation 3.1 gives the Equation 3.5, which is the differential equation of motion of a uniform shear beam. 0 u A t z u A z u A G z t u A )t(g t)(z, 2 )z( t)(z, )z(2 t)(z, 2 =ρ+ δδ δη+δ δ δ δ−δ δρ && 3.5 41 3.2 Discretization of the Differential Equation of Motion The Galerkin method is a widely used technique in computational mechanics to find an approximate solution of a differential equation by transferring it into a discrete counterpart. The method satisfies the differential equation over a finite domain in an integral or average sense rather than at every single point (Cook et al, 2001). Equation 3.6 expresses the Galerkin statement where DE is the differential equation of motion of the uniform shear beam, u(z,t) is a proposed solution of the relative displacement field, and 0 and H are the limits of the finite domain, which in this case is the height of the embankment. ( ) 0 z u DEH 0 t)(z, =δ∫ 3.6 For simplicity in finding an approximate solution of DE, the viscous component will be neglected η(z) = 0 for now but included later on as a function of the viscous damping ratio ξ of a SDOF system. Equation 3.7 represents the Galerkin statement of the undamped equation of motion of the uniform shear beam where the symbols ′ and ü refer to first variation ′ = δ/δz and acceleration ü = δ2u / δt2, respectively. ( ) 0 z u u A ''u AG u A H 0 t)(z,(t) g)t,z()z(t)(z, =δ ρ+−ρ∫ &&&& 3.7 The proposed solution u(z,t) can be expressed as a sum, or superposition, of various partial solutions un(z,t) as indicated in Equation 3.8. ∑∞ = = 1n t)n(z,t)(z, u u where t)n(H,n(z)t)n(z, u u ψ= and 1 (H)n =ψ 3.8 un(z,t) is proposed as the product of a function ψn(z) that depends only on the coordinate z and a function un(H,t) that depends only on the time t. ψn(z) is a dimensionless shape function that 42 satisfies the boundary conditions of the physical model and its value is equal to 1 at z = H. un(H,t), called the degree of freedom, is calculated at z = H as shown in Figure 3.3. The calculation of the relative displacement un(z,t) at z ≠ H is done by using un(H,t) and ψn(z). H z ut ug(t) un(z,t) un(H,t) ψn(z) un(H,t) degree of freedom Figure 3.3 Visualization of the location of the degree of freedom un(H,t) Introducing Equation 3.8 in 3.7 gives Equation 3.9, which is the superposition of the equations of motion of the undamped SDOF systems un(H,t). ( )g(t)nt)n(H,nt)n(H,n 1n u I u K u M &&&& −=+∑∞ = 3.9 where ∫ δψρ= H 0 2 )z(nn z A M ( )∫ δψψ−= H 0 )z(n )z( )z(nn z ''G A K ∫ δψρ= H 0 )z(nn z A I 3.10 Mn is called the generalized mass, Kn is the generalized stiffness, and In is the generalized load factor. The solution of the differential equation of motion, considering energy dissipation, can be easily obtained from Equation 3.9 by expressing the generalized viscous coefficient Cn in terms of the viscous damping ratio ξn as indicated in Equations 3.11 and 3.12. 43 ( )g(t)nt)n(H,nt)n(H,nt)n(H,n 1n u I u K u C u M &&&&& −=++∑∞ = 3.11 where nnnn K M 2 C ξ= 3 .12 Equation 3.13 is an equivalent representation of Equation 3.11 but expressed in terms of the period of vibration Tn and the excitation factor EFn that scales the ground motion üg(t). ( ) −= π+ ξπ+∑∞ = g(t)nt)n(H, 2 n t)n(H, n n t)n(H, 1n u FE u T 2 u T 4 u &&&&& 3.13 where n n n K M 2 T π= n n n M I EF = 3.14 The solution of Equation 3.13 for each degree of freedom un(H,t) depends only on the parameters Tn, ξn, EFn, and the ground motion üg(t). This type of time-dependent differential equation is solved with time-steeping methods (i.e. Newmark’s method) already coded in commercial computer programs, many of which are available in the internet for free. The solution obtained from these computer programs are time histories of different response quantities (i.e acceleration, displacement, etc) for each degree of freedom un(H,t). The complete solution u(z,t) is then obtained by superposition of the time histories of all the degrees of freedom un(z,t) considered in the approximate solution as indicated in Equation 3.8. 3.2.1 Dimensionless Shape Functions Ψn(z) The calculation of Mn, Cn, Kn, In, Tn, and EFn depends of the shape functions Ψn(z). These functions must satisfy the boundary conditions of the physical model expressed in Equations 3.15 and 3.16. un(0,t) = 0 → zero relative displacement at the base 3.15 τn(H,t) = 0 → zero shear stress at the top (free surface) 3.16 44 Taking into account the considerations of Equations 3.17 to 3.19 and the boundary conditions of Equations 3.15 and 3.16, it is concluded ψn(z) must satisfy the conditions of Equation 3.20. un(z,t) = ψn(z) un(H,t) and un(H,t) ≠ 0 → non trivial solution 3.17 τn(z,t) = G(z) γn(z,t) and G(z) ≠ 0 → non trivial solution 3.18 γn(z,t) = ψ′n(z) un(H,t) where ψ′n(z) = δψn(z) / δz 3.19 0 ' and 0 )H(n)0(n =ψ=ψ 3.20 Equation 3.21 includes a set of harmonic shape functions ψn(z) that satisfy the conditions of Equation 3.20 for odd values of the integer n. .. 1,3,5nfor H z 2 nosc H2 n H z 2 nins )z(n)z(n ' = π π=⇒ π= ψψ 3.21 Figure 3.4 plots the shape functions ψn(z) and ψ′n(z) of Equation 3.21 for n = 1, 3 and 5. ψn(z) are known as the mode shapes of vibration and they determine the relative displacement profile in the beam (Figure 3.4a). ψ′n(z), on the other hand, determine the shear strain and shear stress profiles in the beam (Figure 3.4b). 3.2.2 Modal Contribution Analysis The solution of the differential equation of motion is based on the superposition of the partial solutions un(z,t), which depend on ψn(z) and on the mechanical and geometrical properties of the beam. ψn(z) are the mode shapes of vibration and each one contributes to the total response of the beam. The modal contribution analysis will be done for a homogenous uniform shear beam with the properties indicated in Equation 3.22, where B and L are the width and length of the cross section of the beam, respectively, and G is the constant shear modulus with depth. G G L B A (z) == 3.22 45 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ψn(z) D im en si on le ss H ei gh t = z / H mode 1: n=1 mode 2: n=3 mode 3: n=5 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ψ'n(z) / (nπ/2H) D im en si on le ss H ei gh t = z / H mode 1: n=1 mode 2: n=3 mode 3: n=5 a) shape function for relative displacement b) shape function for shear strain and stress Figure 3.4 Harmonic mode shape functions ψn(z) for solving the equation of motion Introducing Equations 3.21 and 3.22 in Equations 3.10 and 3.12 gives the discrete properties of the equivalent SDOF systems un(H,t) as indicated with Equation 3.23 and visualized in Figure 3.5. 2 BLH Mn ρ= H2 BLG 2 n K 2 n π= nn 2 GLB n C ξρπ= π ρ= n BLH2 In 3.23 ρ , G , ξn un(H,t) H L Kn üg(t) In üg(t) Mn un(H,t) Cn a) shear beam vibrating with the mode ψn(z) b) equivalent SDOF system Figure 3.5 Equivalent SDOF system of the vibration mode ψn(z) of a shear beam 46 The product In üg(t) in Figure 3.5b is an input force time history, which is usually not accepted in that form in most commercial software for seismic analysis. Instead, the programs take the acceleration ground motion üg(t) and calculate the input force as the product Mn üg(t). One way to solve this problem is by expressing the equation of motion as a function of the parameters Tn, ξn and EFn (Equation 3.13), which are calculated with Equation 3.24. G n H4 Tn ρ= π= n 4 EFn 3.24 Figure 3.6a shows the SDOF system with the parameters Mn, Kn, Cn, In and Figure 3.6b shows the equivalent SDOF with the parameters Tn, ξn, and EFn. In and üg(t) in Figure 3.6a have units of mass (kg) and acceleration (m/s2), respectively, so the product In üg(t) has units of force (N). On the other hand, in Figure 3.6b the parameter EFn is a dimensionless scalar so the product EFn üg(t) is a constant scaled expression of the ground motion with units of acceleration (m/s2). The model in Figure 3.6a solves the differential equation of motion in units of force while the model in Figure 3.6b solves it in units of acceleration. Kn In üg(t) Mn un(H,t) Cn EFn üg(t) Tn un(H,t) ξn üg(t) Tn (un(H,t))us EFn ξn = = a) mass-spring-dashpot system b) Tn-ξn system with scaled üg(t) c) system with unscaled üg(t) Figure 3.6 Equivalent SDOF systems for calculating the seismic response The value of EFn varies for each vibration mode, which creates many scaled ground motions EFn üg(t) to analyze. This is mathematically correct but not practical from an engineering point of view. A convenient way to obtain the same response of the SDOF system of Figure 3.6b is by first calculating the response with the unscaled ground motion üg(t) and then multiplying the unscaled response quantities (un(H,t))us by the scaling factor EFn as shown in Figure 3.6c. This 47 equivalence is based on the principle that the response of a linear elastic system is proportional to the magnitude of the excitation. One of the advantages of the SDOF system of Figure 3.6c is that allows studying the contribution of each vibration mode to the total response u(z,t) by using the response spectrum of the ground motion üg(t). The response spectrum of a given ground motion üg(t) is a plot of the peak value of an output quantity (i.e. displacement, acceleration, etc) as a function of the natural period T and damping ratio ξ of a SDOF system (Chopra, 1999). Figure 3.7b shows, for instance, the 5% damped relative displacement response spectrum D of the Loma Prieta ground motion üg(t) (Figure 3.7a) for SDOF systems with periods that vary from T = 0 to 2 s. Figure 3.7b indicates, for example, that the peak relative displacement of two SDOF systems with T = 0.4 and 1 s is D = 1.63 and 7.04 cm, respectively. -0.16 -0.12 -0.08 -0.04 0 0.04 0.08 0.12 0.16 0 5 10 15 20 25 30 35 40 Time (s) A cc el er at io n (g ) ξ = 5% 0 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T (s) D (c m ) a) Loma Prieta ground motion üg(t) b) relative displacement response spectrum Figure 3.7 5% damped displacement response spectrum of the Loma Prieta ground motion A response quantity that is widely used in seismic design is the pseudo acceleration PSA, defined by Equation 3.25. PSA is usually expressed in units of gravity (1 g = 9.81 m/s2). D T 2 PSA 2 π= 3.25 48 Figure 3.8 plots the 5% damped pseudo acceleration response spectrum of the Loma Prieta ground motion. The figure indicates, for instance, that the pseudo acceleration of a SDOF system with T = 0.4 s is PSA = 0.41 g. On the other hand, if the period of the SDOF system is T = 1 s then PSA = 0.28 g. ξ = 5% 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T (s) PS A (g ) Figure 3.8 Pseudo acceleration response spectrum of the Loma Prieta ground motion The usefulness of the pseudo acceleration PSA is based on the fact that the peak internal seismic forces, Finternal = K D, of the vibrating system with period T can be obtained by applying a static force, Fstatic = M PSA, to the system as shown in Figure 3.9. The static force Fstatic will cause the displacement D in the system, which is the peak relative displacement obtained from the dynamic response of the SDOF subjected to the ground motion üg(t). K MFstatic = M PSA D Finternal = K D Figure 3.9 Pseudo static analysis for calculating internal forces 49 Based on the concept of pseudo acceleration PSA and the principle of modal superposition, the peak relative displacement upeak(z) of the shear beam is obtained with Equation 3.26, where the modal peak relative displacement is obtain as Dn = (Tn/2π)2 PSAn. ∑∑ ∞ = ∞ = πρ π=≈ ψ 1,3,5n 3 n 2 3 1,3,5n nn)z(npeak(z) n PSA H z 2 nins G H16 D EF u 3.26 Other response quantities of the beam can also be obtained with PSAn as indicated with Equations 3.27 and 3.28. The symbol approximation ≈ is used in these equations due to the fact that the peak response for each vibration mode may not occur at the same time t. ∑∑ ∞ = ∞ = πρ π=≈γ ψ 1,3,5n 2 n 2 1,3,5n nn)z(npeak(z) n PSA H z 2 ncos G H8 D EF ' 3.27 ∑τ ∞ = π π ρ=γ≈ 1,3,5n 2 n 2peak(z)peak(z) n PSA H z 2 ncosH8 G 3.28 The maximum values of Equations 3.26 to 3.28 occur at the top of the beam z = H for the relative displacement, and at the base z = 0 for the shear strain and shear stress as indicated with Equations 3.29 and 3.30. From theory of structural dynamics (Chopra, 1999) it can be demonstrated that the peak total acceleration üt is approximately equal to PSA if ξ ≤ 20 %. Based on this approximation, Equation 3.31 gives the peak üt at the top of the beam. ∑∞ = πρ π≈ 1,3,5n 3 n 2 3peak(H) n PSA 2 nins G H16 u 3.29 ∑∞ = ρ π≈γ 1,3,5n 2 n 2peak(0) n PSA G H8 ∑τ ∞ =π ρ≈ 1,3,5n 2 n 2peak(0) n PSAH8 3.30 ∑∞ = π π≈ 1,3,5n nt peak(H) n PSA 2 nins4 u&& 3.31 50 The above equations indicate that the contribution of each vibration mode depends on PSAn, the odd integer n, and the sign function sin(nπ/2) = ±1. The value of PSAn depends on the modal period Tn, the modal damping ξn, and the ground motion üg(t). Two cases based on the shear wave velocity Vs of the approach embankment are considered for analyzing the contribution of each mode to the total response. The assumed height H and Vs of the beam correspond to average properties of approach embankments found in practice. 3.2.2.1 H = 10 m and Vs = 150 m/s The first case of the modal contribution analysis considers a shear beam with H = 10 m, shear wave velocity Vs = 150 m/s, and üg(t) = Loma Prieta ground motion (Figure 3.7a). In this case, the pseudo acceleration demands PSAn are taken from the 5% damped response spectrum (Figure 3.8) for each modal period Tn, calculated using Equation 3.32. nVs H4 Tn = where ρ= G Vs 3.32 Table 3.1 presents the relative modal contribution analysis for the first five vibration modes with respect to the contribution of mode 1 using Equations 3.29 to 3.31. The peak shear stress was omitted from the analysis because its relative contribution is the same that the one of the shear strain. Table 3.1 Relative modal contribution analysis of a shear beam with H = 10 m and Vs = 150 m/s Mode n Tn (s) PSAn (g) un / u1 γn / γ1 ütn / üt1 1 1 0.27 0.34 1.00 1.00 1.00 2 3 0.09 0.22 -0.02 0.07 -0.22 3 5 0.05 0.16 0.00 0.02 0.10 4 7 0.04 0.16 0.00 0.01 -0.07 5 9 0.03 0.16 0.00 0.01 0.05 Total = 0.98 1.11 0.87 Mode 1 - Total = 0.02 -0.11 0.13 Relative Modal Contribution 51 Table 3.1 indicates that the relative contribution of each vibration mode to the total response depends on the response quantity being considered. For instance, if the response quantity of interest is the relative displacement, the contribution of modes 2 to 5 is negligible (< |±0.02|) in comparison to the contribution of mode 1 (= 1.0). This is due to the fact that the modal displacement is proportional to PSAn by strongly reduced by the factor ±1/n3 (+ for n = 1, 5, 9; – for n = 3, 7), as indicated in Equation 3.29, so the response is controlled by the fundamental mode (n = 1). The alternance of the ± sign in the summation of the modal displacement makes the total contribution of the five modes to have a value of 0.98, which is just 0.02 lower than the contribution of the mode 1. This means that if the relative displacement of the beam were calculated with only the fundamental mode, then the response would be overestimated by 2%. A similar tendency is observed with the shear strain γ, in which the mode 2 has a slight contribution of 0.07 in comparison to the contribution of mode 1. The modal contribution of this response quantity is always positive and proportional to PSAn but still reduced significantly by the factor 1/n2 for modes 2 and higher (n ≥ 3 → 1/n2 ≤ 0.11), as indicated in Equation 3.30. Therefore, the fundamental mode is still the one that controls the response of the beam. Table 3.1 indicates that the total contribution of the five modes for the shear strain is 1.11, which is just 0.11 higher than the contribution of the mode 1. This means that if the shear strain (or shear stress) were calculated with only the fundamental mode, then the response would be underestimated by 11%. A different situation occurs with the total acceleration üt, in which the relative contribution of mode 2 (|–0.22|) and mode 3 (|0.10|) is not negligible. As indicated in Equation 3.31, the modal contribution of the total acceleration is proportional to PSAn and to the factor ±1/n, which does not reduced significantly the contribution of mode 2 (n = 3 → 1/n = 0.33) in comparison to the one of mode 1 (n = 1 → 1/n = 1.0). However, it is worth noting that the alternance of the ± sign in the summation of the modal contributions makes the total contribution of the five modes to have a value of 0.87, which is just 0.13 lower than the contribution of the fundamental mode. This means that if the total acceleration were calculated with only the mode 1, then the response would be overestimated by 13%. 52 The modal contribution analysis of Table 3.1 indicates that the response of the beam is strongly controlled by the fundamental mode. The error in the calculation of the peak relative displacement, peak shear strain, and peak total acceleration of the beam using only the mode 1 is 2%, -11%, and 13%, respectively, where the + and − sign means over and underestimation, respectively. 3.2.2.2 H = 10 m and Vs = 60 m/s Table 3.2 presents the relative modal analysis of the same beam analyzed in Table 3.1 but with a lower shear wave velocity Vs = 60 m/s. This low value of Vs represents the extreme case of a very soft approach embankment. Table 3.2 Relative modal contribution analysis of a shear beam with H = 10 m and Vs = 60 m/s Mode n Tn (s) PSAn (g) un / u1 γn / γ1 ütn / üt1 1 1 0.67 0.31 1.00 1.00 1.00 2 3 0.22 0.43 -0.05 0.15 -0.46 3 5 0.13 0.17 0.00 0.02 0.11 4 7 0.10 0.20 0.00 0.01 -0.09 5 9 0.07 0.17 0.00 0.01 0.06 Total = 0.95 1.19 0.62 Mode 1 - Total = 0.05 -0.19 0.38 Relative Modal Contribution The main effect of reducing Vs in this case is that the modal periods Tn are now longer than the ones calculated in Table 3.1. Therefore, the pseudo acceleration demands PSAn and the relative modal contributions for each modal period Tn change as shown in Table 3.2. The modal contribution analysis for this case indicates again that the seismic response of the beam is strongly controlled by the fundamental mode, especially for the relative displacement and the shear strain. The error in the calculation of the peak relative displacement, peak shear strain, and peak total acceleration of the beam using only the mode 1 is 5%, -19%, and 38%, respectively. The considerable overestimation of the total acceleration for this case is analyzed in the next section. 53 3.2.2.3 Discussion The modal contribution analyses in Tables 3.1 and 3.2 indicate that the peak relative displacement of the beam can be calculated with a very good level of accuracy (error < 5%) by using only the fundamental mode. The analyses also showed that the calculation of shear strain using only the first mode is acceptable (error < 19%) from an engineering point of view due to the fact that the uncertainty in the determination of the soil properties is usually high. The calculation of the total acceleration using only the fundamental mode overestimates the response of the beam in Table 3.1 with an error of 13%, which is considered acceptable. On the other hand, the overestimation using only the mode 1 in Table 3.2 is significant (error = 38%). The level of overestimation is related to the location of the first two modal periods T1 and T2 of the beam in the pseudo acceleration response spectrum of the ground motion. Figures 3.10a and b show the location, in the PSA response spectrum, of the first three modal periods of the beams (T1, T2 and T3) analyzed in Tables 3.1 and Table 3.2, respectively. 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 T (s) PS A (g ) 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 T (s) PS A (g ) T1T2 T1T2T3 T3 a) H = 10 m and Vs = 150 m/s b) H = 10 m and Vs = 60 m/s Figure 3.10 Location of modal periods T1, T2 and T3 in the 5% damped response spectrum of the Loma Prieta ground motion for two shear beams with different shear wave velocity Vs 54 In the first case, the PSA of the modal periods T2 and T3 is smaller than the one of fundamental mode T1. Therefore, the modal contribution of the 2nd and 3rd modes is small in comparison to the contribution of mode 1. This is the reason why the overestimation of the total acceleration using only the fundamental mode is just 13% in Table 3.1. This tendency of small contribution by the higher modes generally occurs when T1 is located in the zone of highest amplification of the spectrum as shown in Figure 3.10a. A different situation occurs in Figure 3.10b. In this case the PSA of T2 is bigger than the pseudo acceleration of the modal periods T1 and T3. Thus, the contribution of 2nd mode is now important in comparison to the contribution of the fundamental mode. This is why the overestimation in this case is bigger than the one analyzed in Figure 3.10a. This tendency of important contribution by the 2nd mode to the total acceleration of the beam generally occurs when T1 is located in a zone of medium or low amplification of the spectrum and T2 is in the zone of highest amplification as shown in Figure 3.10b. The contribution of the modal periods T3 and higher (n ≥ 5) is usually not important because the factor 1/n (n ≥ 5 → 1/n ≤ 0.2) reduces significantly any amplification given by PSAn. It is concluded from the modal contribution analyses that calculating the seismic response of a homogeneous uniform shear beam using only the fundamental mode: It is accurate for the relative displacement. It is acceptable for the shear strain and shear stress. It is acceptable for the total acceleration if T1 is in the zone of highest amplification of the PSA response spectrum. It may overestimate significantly the total acceleration if T2 and T1 are in the zone of highest and lowest amplification of the PSA response spectrum, respectively. 55 3.3 The 1ME Model for Linear Analysis It was shown in Chapter 2 that the seismic response of an approach embankment in the far field can be calculated using the shear beam model. In addition, in section 3.2.2 it was proved that the seismic response of the shear beam is strongly controlled by the fundamental mode and that the error in the estimation of the peak response quantities using only this mode was acceptable from an earthquake engineering point of view. The above opens the opportunity for calculating the far-field embankment response using only a SDOF system based on the fundamental mode. Equation 3.33 gives the shape functions Ψ(z) and Ψ′(z) that are used for calculating the parameters of the SDOF system that represents the far field of the approach embankment. These functions are taken from Equation 3.21 for n = 1 (mode 1). π=ψ H z 2 ins )z( ⇒ ππ=ψ H z 2 osc H2 )z( ' 3.33 The SDOF system will be called “the 1ME model” in this thesis. The main advantage of the 1ME model is that no modal superposition is required for calculating the far-field embankment response. The 1ME model will be developed for three general cases. The first two cases take into account different initial distributions of the shear wave velocity Vs(z) with depth in the embankment. The third one assumes a degradation profile of the shear modulus G(z) with depth resulting from the nonlinear response of the soil of the embankment. 3.3.1 The 1ME Model with Constant Vs The first case to analyze assumes that the initial shear wave velocity in the embankment is constant with depth, Vs(z) = Vs, as shown in Figure 3.11a. Figure 3.11b plots the variation of 56 the initial shear modulus, obtained as G = ρVs2, which is also constant, G(z) = G, along the dimensionless height z/H. This is the case already analyzed in section 3.2.2 for a homogeneous uniform shear beam and it is appropriate for low engineered embankments. 0 0.2 0.4 0.6 0.8 1 0.8 0.9 1 1.1 1.2 Vs(z) / Vs D im en sio nl es s H ei gh t = z / H 0 0.2 0.4 0.6 0.8 1 0.8 0.9 1 1.1 1.2 G(z) / G D im en sio nl es s H ei gh t = z / H a) shear wave velocity profile b) shear modulus profile Figure 3.11 Approach embankment with constant shear wave velocity Equation 3.34 represents the equation of motion of the 1ME model in which ME is the effective mass of the embankment, CE is the viscous coefficient, ξ is the damping ratio, KE is the embankment stiffness, and EF is the scaling factor of the ground motion üg(t). This equation calculates the far-field response at the top (z = H) of the approach embankment. ( ) g(t)Et)(H,Et)(H,Et)(H,E u FE M u K u C u M &&&&& −=++ 3.34 where ρ= 2 BLH ME ξρπ= G2 LB CE GH8 BL K 2 E π= π= 4 EF 3.35 Note that the calculation of the parameters of this equation requires the dimensions B and L of the uniform approach embankment. Equation 3.36 is the equivalent representation of Equation 3.34 in terms of the fundamental period of the embankment TE, the damping ratio ξ, and the scaling factor EF. 57 ( ) g(t)t)(H, 2 E t)(H, E t)(H, u FE uT 2 u T 4 u &&&&& −= π+ ξπ+ 3.36 where Vs 4H G H4 TE =ρ= 3.37 Note that the calculation of the parameters of Equation 3.36 does not require B and L. This indicates that the far-field embankment response, in terms of acceleration ü, velocity u& , and displacement u, is independent of B and L but strongly dependent on H, ρ, G, and ξ. Equation 3.36 calculates the seismic response at the top of the embankment (z = H) as a relative displacement time history u(H,t). The peak value of u(H,t) and the fundamental mode shape Ψ(z) can be used to calculate the peak value of other response quantities at any depth (z ≤ H) in the embankment. Equations 3.38 to 3.41 calculate the peak relative displacement, peak shear strain, peak shear stress, and peak total acceleration profiles based on either the peak relative displacement D or on the pseudo acceleration PSA. These equations already include the scaling factor EF = 4/π. π π= H z 2 ins D 4 u )z(peak or π π= H z 2 ins PSA T u 3 2 E )z(peak 3.38 π=γ H z 2 cos H D 2 )z(peak or π π=γ H z 2 cos H 2 PSA T 2 2 E )z(peak 3.39 π=τ H z 2 cos H DG 2 )z(peak or π π ρ=τ H z 2 cos PSA H8 2)z(peak 3.40 π −π+= H z 2 ins PGAPSA 4 PGA u )z(peak&& 3.41 58 The advantage of Equations 3.38 to 3.41 is that D or PSA can be easily obtained from the response spectra of the ground motion üg(t) or from design response spectra included in building and highway codes. 3.3.1.1 Example 1: Calculation of relative displacement time histories Table 3.3 presents the physical properties of two embankments with the same height H and density ρ but with different initial shear wave velocity Vs. The shear modulus is calculated as G = ρVs2. The period of vibration TE is calculated with Equation 3.37 and the damping ratio is assumed to be ξ = 5%. The embankments are subjected to the Loma Prieta ground motion üg(t), plotted in Figure 3.7a. The scaling factor for üg(t) is EF = 4/π ≈ 1.27. Table 3.3 Physical properties of two embankments with Vs = 150 and 60 m/s Vs H ρ G TE ξ m/s m tonne/m3 MPa s % 150 10 2 45 0.27 5 60 " " 7.2 0.67 " Figure 3.12 plots the relative displacement time histories at the top of the embankments u(z=10m,t) using the 1ME model. The time histories were obtained by solving Equation 3.36 with the computer program SeismoSignal, which can be downloaded for free from the website http://www.seismosoft.com/en/SeismoSignal.aspx. The figure shows that the soft embankment (Vs = 60 m/s) responds with bigger displacements than the ones of the stiff embankment (Vs = 150 m/s). For instance, the peak relative displacement of the soft embankment is upeak(z=10m) = 4.36 cm while the one of the stiff embankment is upeak(z=10m) = 0.76 cm. The difference in the seismic response of the two embankments is due to the difference in their periods of vibration TE. In this example the period of the soft embankment (TE = 0.67 s) is 2.5 times longer than the period of the stiff embankment (TE = 0.27 s). 59 -4 -3 -2 -1 0 1 2 3 4 5 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time (s) u ( z= 10 m ) (c m ) Vs = 150 m/s Vs = 60 m/s Figure 3.12 Relative displacement time histories at the top of two embankments with Vs = 150 and 60 m/s using the 1ME model 3.3.1.2 Example 2: Calculation of peak response quantity profiles Table 3.4 presents the calculation of the peak response quantities of the two embankments analyzed in Table 3.3 using the 1ME model. PSA and D are obtained from the response spectra plotted in Figures 3.8 and 3.7b, respectively, for TE = 0.27 and 0.67 s. The peak response quantities are calculated with Equations 3.38 to 3.41 at z = 10 m for the total acceleration üt(z=10m) and relative displacement u(z=10m), and at z = 0 m for the shear strain γ(z=0m) and shear stress τ(z=0m). Table 3.4 Peak response quantities of two embankments with Vs = 150 and 60 m/s Vs TE ξ PGA PSA D üt(z=10m) u(z=10m) γ(z=0m) τ(z=0m) m/s s % g g cm g cm % kPa 150 0.27 5 0.16 0.34 0.60 0.43 0.76 0.12 54.1 60 0.67 5 0.16 0.31 3.42 0.39 4.36 0.68 49.3 Peak Response QuantityResponse Spectra Figure 3.13a plots the peak relative displacement profiles upeak(z) calculated with Equation 3.38. As mentioned earlier, the soft embankment (Vs = 60 m/s) responds with bigger displacements than the ones of the stiff embankment (Vs = 150 m/s). A similar behavior is 60 observed in Figure 3.13b with the peak shear strain profiles γpeak(z) calculated with Equation 3.39. It is concluded from this example that upeak(z) and γpeak(z) are strongly dependent on D. 0 2 4 6 8 10 0 1 2 3 4 5 upeak (cm) z (m ) Vs = 150 m/s Vs = 60 m/s 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8γpeak (%) z (m ) Vs = 150 m/s Vs = 60 m/s a) peak relative displacement b) peak shear strain 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 ütpeak (g) z (m ) Vs = 150 m/s Vs = 60 m/s 0 2 4 6 8 10 0 10 20 30 40 50 60τpeak (kPA) z (m ) Vs = 150 m/s Vs = 60 m/s c) peak total acceleration d) peak shear stress Figure 3.13 Peak response quantity profiles of two embankments with Vs = 150 and 60 m/s using the 1ME model Figure 3.13c plots the peak total acceleration profiles calculated with Equation 3.41. The figure shows that the profiles are similar for both embankments even though their natural periods TE are very different. This is due to the fact that the pseudo acceleration PSA is similar for both embankments (PSA = 0.34 and 0.31 g) as shown in Figure 3.10 with TE = T1. 61 A similar situation occurs with the peak shear stress profiles calculated with Equation 3.40 and plotted in Figure 3.13d. In this case, the difference in the peak shear stress at the base of the embankments (z = 0 m) is just 4.8 kPa. It is concluded from this example that ütpeak(z) and τpeak(z) are strongly dependent on PSA. 3.3.2 The 1ME Model with Linear Variation of Vs The second case to analyze assumes that the initial shear wave velocity increases linearly with depth from the top of the embankment as indicated with Equation 3.42. Vstop is the shear wave velocity at z = H, Vsbase is the shear wave velocity at z = 0, and rVs ≥ 1 is the ratio between Vsbase and Vstop. The linear variation of Vs(z) gives a parabolic distribution of G(z) with depth as indicated with Equation 3.43 where Gtop = ρVstop2 is the shear modulus at z = H. This case is appropriate for high approach embankments. Equations 3.42 and 3.43 are plotted in Figure 3.14 for rVs = 1.5. ( ) + −−= 1 1 H z rVs1 Vs sV top (z) where top base Vs Vs rVs = 3.42 ( ) 2 top )z( 1 1H z rVs1 G G + −−= 3.43 The stiffness KE of the 1ME model for this case is obtained with Equation 3.45 by introducing Equation 3.43 in 3.44 (see Kn in Equation 3.10). ( )∫ δψψ−= H 0 )z( )z( )z(E z ''G LB K where π=ψ H z 2 ins )z( 3.44 in 2 E G H8 BL K π= where ++= 2topin rVs 28 15 rVs 3 1 23 3 G G 3.45 Equation 3.45 shows that the stiffness KE of an approach embankment with a parabolic variation of G(z) is equivalent to KE of an embankment with constant G (Equation 3.35) but 62 with the difference that G = Gin, which is called the equivalent initial shear modulus of the embankment. This equivalence is very useful because it allows using Equations 3.34 to 3.41 to obtain the far-field embankment response. 0 0.2 0.4 0.6 0.8 1 1 1.1 1.2 1.3 1.4 1.5 Vs(z) / Vstop D im en sio nl es s H ei gh t = z / H rVs = 1.5 0 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 G(z) / Gtop D im en sio nl es s H ei gh t = z / H rVs = 1.5 a) shear wave velocity profile b) shear modulus profile Figure 3.14 Approach embankment with shear wave velocity ratio rVs = 1.5 Figure 3.15a plots Equation 3.45 for approach embankments with 1 ≤ rVs ≤ 2. The figure shows, for example, that the equivalent initial shear modulus of an embankment with rVs = 1.5 is Gin ≈ 1.84Gtop. For the case rVs = 1 ⇒ Gin = Gtop. 1 1.4 1.8 2.2 2.6 3 1 1.2 1.4 1.6 1.8 2 rVs G in / G to p 0 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 rVs ( z / H ) G in a) equivalent initial shear modulus b) depth of equivalent initial shear modulus Figure 3.15 Equivalent initial shear modulus Gin of an embankment with linear variation of Vs 63 Replacing Gin (Equation 3.45) for G(z) in Equation 3.43 gives the depth at which G(z) = Gin as indicated with Equation 3.46. This equation is plotted in Figure 3.15b for embankments with 1 ≤ rVs ≤ 2. The figure shows that the variation of (z/H)Gin with rVs is very small, so for practical purposes (z/H)Gin ≈ 0.3. Equation 3.46 has a singularity for the case rVs = 1 due to the fact that G(z) = Gtop = Gin for all z/H. ( ) 0.3 /GG 1 rVs1 1 1 H z topin Gin ≈− −−= 3.46 Figure 3.16a plots the parabolic variation of G(z) of an embankment with rVs = 1.5. The gray dot shows the location z/H = 0.30 at which G(z) = Gin = 1.84Gtop. Figure 3.16b shows the equivalent embankment with a constant shear modulus G(z) = 1.84Gtop that can be used to calculate the far-field response of the embankment of Figure 3.16a. 0 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 G(z) / Gtop z / H rVs = 1.5 Gin = 1.84Gtop 0 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 G(z) / Gtop z / H rVs = 1.5 a) embankment with parabolic G(z) b) equivalent embankment with constant Gin Figure 3.16 Equivalent initial shear modulus Gin of an embankment with rVs = 1.5 3.3.3 The 1ME Model with Degradation of Gin The third case considered assumes a cosine variation of the shear modulus G(z) with depth in the embankment as indicated with Equation 3.47. 64 π−= H z 2 cosD 1 G G Gin )z( where in base G G G 1 D −= 3.47 This case represents an embankment with an initial constant shear modulus Gin which is degraded with depth due to the nonlinear response of the soil during a seismic event. The proposed cosine function assumes that the degradation of Gin is proportional to the mobilized shear strain γ(z) in the embankment (see Equation 3.39). DG is the degradation factor of Gin at the base of the embankment (z = 0) and it varies from 0 (no degradation → Gbase = Gin) to 1 (total degradation → Gbase = 0). Figure 3.17 plots Equation 3.47 for DG = 0.6. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 G(z) / Gin z / H DG = 0.6 Figure 3.17 Shear modulus degradation profile of Gin for DG = 0.6 Figure 3.17 shows that the degradation of Gin at the top of the embankment (z/H = 1) is zero G(z=H) = Gin. This is due to the fact that the mobilized shear strain at the top of the embankment (free surface) is also zero γ(z=H) = 0. On the other hand, the value of G(z) at the base of the embankment is G(z=0) = 0.4Gin, which means that Gin has degraded 60%. The stiffness KE of the 1ME model for this case is obtained with Equation 3.48 by introducing Equation 3.47 in 3.44. 65 deg 2 E G H8 BL K π= where π−= Gindeg D3 8 1 G G 3.48 Equation 3.48 shows that the stiffness KE of an approach embankment with a cosine variation of G(z) is equivalent to KE of an embankment with constant G = Gdeg. Therefore, Equations 3.34 to 3.41 can be used to obtain the far-field embankment response. Gdeg is called the equivalent degraded shear modulus of the embankment. Figure 3.18a plots Equation 3.48 for 0 ≤ DG ≤ 1. The figure shows, for example, that the constant degraded shear modulus of an embankment with DG = 0.6 is Gdeg ≈ 0.49Gin. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 DG G de g / G in 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 DG ( z /H ) G de g a) constant degraded shear modulus b) depth of constant degraded Gdeg Figure 3.18 Constant degraded shear modulus Gdeg of an embankment with cosine degradation of Gin Figure 3.18a shows an inconsistency for DG = 1 (Gdeg ≈ 0.15Gin) due to the fact that a total degradation of the soil at the base of the embankment would seismically isolate it (Gdeg = 0). This is not captured with Equation 3.48 because Gdeg represents an average value of degradation. It is therefore recommended to use the equation for DG ≤ 0.8 and to interpolate Gdeg linearly between DG = 0.8 (Gdeg ≈ 0.32Gin) and DG = 1 (Gdeg = 0) as shown with the dashed line in the figure. 66 Replacing Gdeg (Equation 3.48) for G(z) in Equation 3.47 gives the depth at which G(z) = Gdeg as indicated with Equation 3.49. This equation is plotted in Figure 3.18b which shows that (z/H)Gdeg ≈ 0.35 for 0 ≤ DG ≤ 1. .350 H z degG ≈ 3.49 Figure 3.19a plots the cosine degradation of Gin of an embankment with DG = 0.6. The gray dot shows the location z/H = 0.35 at which G(z) = Gdeg = 0.49Gin. Figure 3.19b shows the equivalent embankment with a constant degraded shear modulus G(z) = 0.49Gin that can be used to calculate the seismic response of the embankment of Figure 3.19a. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 G(z) / Gin z / H DG = 0.6 Gdeg = 0.49Gin 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 G(z) / Gin z / H DG = 0.6 a) embankment with cosine G(z) b) equivalent embankment with constant Gdeg Figure 3.19 Constant degraded shear modulus Gdeg of an embankment with DG = 0.6 3.4 The 1ME Model for Non-Uniform Embankments Figure 3.20a shows an infinitely long approach embankment which cross section is non- uniform with depth. The seismic response of this type of embankment is calculated with Zhang and Makris’ model. As discussed in section 2.2, the model is not easy to implement in bridge design procedures. 67 B H TE 1 S Gdeg , ρ TE Gdeg , ρred a) non-uniform b) equivalent uniform Figure 3.20 Two-dimensional representations of an infinitely long approach embankment A simple way to solve the problem is by obtaining an equivalent uniform embankment that has the same height H and fundamental period TE of the non-uniform embankment as shown in Figure 3.20b. This solution was proposed by Wilson and Tan and it requires a calibration process using FE models as discussed in Section 2.1. Their solution, however, was developed only for the transverse direction of the embankment and for few cases of the ratio B/H. The equal-period solution for non-uniform embankments can be easily implemented in the 1ME model. The equivalent uniform embankment is obtained with the assumption of equal H, Gdeg, and TE but different density ρred as shown in Figure 3.20. ρred is determined from a calibration process using 3D FE models as the one shown in 3.21. transverselo ngit udin al B L H Figure 3.21 3D FE model of a non-uniform approach embankment 68 The FE models are used to obtain the fundamental mode shape and the corresponding period of vibration of non-uniform embankments in the transverse and the longitudinal direction as shown in Figures 3.22a and b, respectively. a) transverse direction b) longitudinal direction (cross section) Figure 3.22 Fundamental mode shapes of a non-uniform approach embankment The displacement field of the FE models is restrained in both the vertical and transverse direction when obtaining the period in the longitudinal direction. In the same way, the period in the transverse direction is obtained by restraining the displacement field of the models in both the vertical and longitudinal direction. This restraining conditions ensures that the FE models behave as infinitely long shear beams (L → ∞). The geometrical properties of the models are varied to create a data base of fundamental periods of approach embankments with different cross sections. In this case, the height H and length L of the FE models is kept constant, L = H = 10 m, while the side slope S and the crest width B is varied, S = 1, 1.5, 2, and 0.1 ≤ B/H ≤ 16. The physical properties of the models are ρ = 2 tonne/m3, Gdeg = 20 MPa, and ν = 0.3. Figure 3.23 plots the data base of fundamental periods for different configurations of B/H and S (no labeled in the figure) obtained with the computer program ABAQUS (Simulia Inc.). The periods of the FE models (Tnu) in each direction are normalized with respect to the period of a uniform shear beam (Tu) calculated with Equation 3.37 (G = Gdeg). The subscripts “nu” and “u” stand for non-uniform and uniform, respectively. 69 0.7 0.75 0.8 0.85 0.9 0.95 1 0 2 4 6 8 10 12 14 16 B / H T n u / T u Longitudinal Transverse Fitted curve Figure 3.23 Fundamental period ratio of a non-uniform approach embankment Figure 3.23 shows that the shape effects reduce the fundamental period of vibration of non- uniform embankments in comparison to the one of uniform embankments → Tnu/Tu ≤ 1. For instance, the period of a triangular embankment (B/H = 0) is Tnu = 0.72 Tu. On the other hand, the period of a non-uniform embankment with B/H = 2 is Tnu = 0.92 Tu. The shape effects are practically negligible for B/H ≥ 4 → 0.96 ≤ Tnu/Tu ≤ 1. The figure also shows that for a given value of B/H, the difference in the period ratio Tnu/Tu between the longitudinal and transverse direction is very small (< 5%). This is consistent with Zhang and Makris’ work that showed that the seismic response of the embankment in the longitudinal and transverse direction is practically the same. Another observation in Figure 3.23 is that Tnu/Tu is practically insensitive to the side slope S, which is also consistent with Wilson and Tan’s model. Equation 3.50 represents the fitted curved of the data points of Tnu/Tu for both directions. (B/H)96.0 1 (B/H) 98.0 72.0 T T u nu + += 3.50 The period of the 1ME model for non-uniform embankments TE is obtained by multiplying the period of a uniform embankment with the period ratio Tnu/Tu as indicated in the first part of 70 Equation 3.51. This is equivalent to change the density ρ of the embankment for a reduced density ρred that gives the same TE as indicated in the second part of the equation. ρ= u nu deg E T T G 4H T → deg red E G H4 T ρ= 3.51 ρred is reduced by multiplying ρ with the dimensionless density reduction factor Rρ given in Equation 3.52 and plotted in Figure 3.24. ρρ=ρ R red where 2 (B/H)96.01 (B/H)98.072.0 R + +=ρ 3.52 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 B / H R ρ Figure 3.24 Density reduction factor for a non-uniform approach embankment The figure shows that the density reduction factor varies from 0.52 ≤ Rρ ≤ 1, with the lower limit for the case of a triangular embankment B/H = 0. It is concluded from Figure 3.24 that from a practical point of view a non-uniform approach embankment can be considered as a uniform embankment if B/H > 4 since the reduction of the density is less than 10% (Rρ > 0.9). 71 3.5 The 1ME Model for Equivalent Linear Analysis The equivalent linear model is an approximation of the actual nonlinear behavior of the soil. The model assumes that the hysteretic behavior the soil exhibits during cyclic loading can be represented by the equivalent linear properties Gsec (secant shear modulus) and ξ (viscous damping ratio). Both properties depend on the cyclic shear strain amplitude γc. Figure 3.25 visualizes the equivalent linear properties Gsec and ξ of a soil material subjected to a given cyclic shear strain amplitude γc. Aloop is the area of the hysteresis loop and it represents the energy dissipated in one cycle. γ τ γc Gsec c c sec G γ τ= Aloop τc 2 csec loop G A 2 1 γπ=ξ Figure 3.25 Equivalent linear properties Gsec and ξ for representing the hysteretic behavior of soils during cyclic shear loading Considerable attention has been given to the determination of Gsec and ξ for different soil types using laboratory tests (Seed and Idriss, 1970; Iwasaki et al, 1978; Ishibashi and Zhang, 1993). These equivalent properties are presented in the form of modulus reduction and damping ratio curves. Figure 3.26 shows, for example, the dimensionless modulus reduction Gsec/Gmax and ξ curves of a clay soil with plasticity index PI = 30 (Vucetic and Dobry, 1991). Gmax is the maximum value of Gsec at a very low shear strain amplitude γc < 1x10-4 %. Figure 3.26a shows that the equivalent linear modulus degrades, Gsec/Gmax < 1, as the shear strain increases, γc > 1x10-3 %. The magnitude of the degradation of Gsec represents the level 72 of nonlinearity in the soil. Figure 3.26b, on the other hand, shows that the equivalent damping ratio ξ increases as γc increases. This is a consequence of the loss of stiffness of the soil which increases the energy dissipation. 0 0.2 0.4 0.6 0.8 1 0.0001 0.001 0.01 0.1 1 10γc (%) G se c / G m ax 0 5 10 15 20 25 0.0001 0.001 0.01 0.1 1 10γc (%) ξ ( % ) a) shear modulus reduction curve b) damping ratio curve Figure 3.26 Equivalent linear properties Gsec and ξ for a clay soil with PI = 30 3.5.1 Effective Shear Strain of the Approach Embankment The characterization of Gsec and ξ is based on simple harmonic loading with shear strain amplitude γc. The time history of shear strain for a typical earthquake motion, however, is highly irregular with a peak value γpeak that may only be approached by few spikes in the record. Therefore, for a shear strain amplitude γc = γpeak the harmonic loading represents a more severe condition than the transient record, although their peak values are identical. As a result, it is common to characterize the strain level of the transient record with the effective shear strain γef which has been empirically found to be γef ≈ 0.65 γpeak (Idriss and Sun, 1992). Equivalent linear analysis is based on computing the effective shear strain γef of the soil to determine Gsec and ξ from the shear modulus reduction and damping curves. As shown in Figure 3.13b, the peak shear strain γpeak(z) varies with depth in the embankment. Therefore, it is necessary to determine the depth z at which γef will be calculated for representing the strain level of the embankment. 73 Section 3.3.3 showed that the equivalent constant degraded modulus Gdeg of an approach embankment subjected to a cosine degradation of G(z) is obtained at z = 0.35H. This is then the proposed depth in Equation 3.53 for calculating the effective shear strain of the embankment. )H35.0z(peakef 0.65 =γγ = 3.53 Introducing Equation 3.39 in 3.53 gives γef of the approach embankment as indicated with Equation 3.54. This equation already includes the scaling factor EF = 4/π. (%) H D 111 ef =γ or (%) H PSA T 2.8 2 E ef =γ 3.54 3.5.2 Procedure for Calculating the Equivalent Linear Properties Since the computed γef of the embankment depends on the values of Gsec/Gmax and ξ, an iterative procedure is required to ensure that the equivalent linear properties used in the analysis are compatible with the computed strain levels. The proposed iterative procedure for calculating the equivalent linear properties of an infinitely long approach embankment (L → ∞) with the 1ME model operates as follows: step 1: specify H, B, ρ, Gtop, rVs, Gsec/Gmax and ξ curves, and üg(t) step 2: calculate ρred with Equation 3.52 ρρ=ρ R red where 2 (B/H)96.01 (B/H)98.072.0 R + +=ρ 3.52(bis) step 3: calculate Gin with Equation 3.45 ++= 2topin rVs 28 15 rVs 3 1 23 3 G G 3.45(bis) 74 step 4: assume γef(i) = 1x10-4 % and obtain Gsec/Gmax and ξ for γc = γef(i) step 5: calculate Gdeg = Gin (Gsec/Gmax) step 6: calculate TE with Equation 3.37 for ρ = ρred and G = Gdeg deg red E G H4 T ρ= 3.37(bis) step 7: obtain D or PSA by solving Equation 3.36 for TE, ξ, and EF = 1 g(t)(t) 2 E (t) E (t) u uT 2 u T 4 u &&&&& −= π+ ξπ+ 3.36(bis) step 8: obtain γef(i+1) with Equation 3.54 (%) H D 111 ef =γ or (%) H PSA T 2.8 2 E ef =γ 3.54(bis) step 9: calculate the tolerance in the convergence with Equation 3.55 (%) 1 100 (i) ef 1)(i ef tol −γ γ= + 3.55 step 10: if tol ≤ 5 % → end if tol > 5 % → obtain Gsec/Gmax and ξ for γc = γef(i+1) → repeat steps 5 to 10 Equation 3.55 calculates the difference between the computed effective shear strain values in two successive iterations and expresses it in percentage. 75 3.5.2.1 Example 3: Calculation of the Equivalent Linear Response Tables 3.5 and 3.6 presents the physical properties and the application of the procedure proposed in section 3.5.2, respectively, for calculating the equivalent linear properties of an infinitely long approach embankment (L → ∞) subjected to the Loma Prieta ground motion. Table 3.5 Physical properties of and infinitely long approach embankment Ground Motion H B ρ Vstop Gtop rVs Gsec/Gmax ξ üg(t) m m tonne/m3 m/s MPa Loma Prieta 10 13 2 100 20 1.5 Fig. 3.7a Physical Properties Property Curves Fig. 3.26 Table 3.6 Calculation of the equivalent linear properties of the approach embankment Iteration ρred Gin Gsec/Gmax Gdeg TE ξ D PSA γef tol i tonne/m3 MPa MPa s % cm g % % 0 1.8 36.8 - - - - - - 1.E-04 - 1 " " 1.0 36.8 0.281 1.03 1.282 0.65 0.142 1.E+05 2 " " 0.486 17.88 0.404 9.54 1.466 0.36 0.163 14.4 3 " " 0.461 16.97 0.414 9.97 1.528 0.36 0.170 4.2 Table 3.6 indicates that the equivalent linear properties of the embankment at the beginning of the procedure (iteration 1) are Gdeg = 36.8 MPa and ξ = 1.03 %. These properties are obtained by assuming that γef = 1x10-4 %, which has to be verified by calculating D from the seismic response of the 1ME model for TE = 0.281 s and ξ = 1.03 % (Figure 3.27). The seismic response of the 1ME model indicates that effective shear strain of the embankment, γef = 0.142 %, is much bigger than the one initially assumed, which means that there is not consistency between the current equivalent linear properties and the computed γef. This is reflected by the high value of tol = 1x105 > 5 %. Thus, Gsec/Gmax and ξ have to be updated for γc = 0.142 %. The updated properties are Gdeg = 17.88 MPa and ξ = 9.54 %. 76 -1.5 -1 -0.5 0 0.5 1 1.5 2 5 6 7 8 9 10 11 12 13 14 15 Time (s) u ( t) (c m ) Iteration 1 Iteration 3 Di=1 Di=3 Figure 3.27 Displacement time histories u(t) obtained with the 1ME model for EF = 1 Iteration 2 shows that consistency between the updated equivalent linear properties and the computed γef has improved significantly but convergence has not been reached yet (tol = 14.4 % > 5%). Therefore, the properties are updated again for γc = 0.163 %. Iteration 3 indicates that convergence has been reached (tol = 4.2 < 5 %) with the equivalent linear properties Gdeg = 16.97 MPa and ξ = 9.97 %. 3.6 Summary This chapter developed a SDOF system, called the 1ME model, for calculating the seismic response of a homogeneous uniform approach embankment in the far field. The initial parabolic distribution of the shear modulus G(z) and the assumed cosine degradation profile of G(z) during the seismic response of a non-homogenous embankment are replaced by a constant value Gdeg with depth in an equivalent homogeneous embankment. Similarly, the shape effects of non-uniform embankments are taken into account by obtaining an equivalent uniform embankment which density is reduced so that it matches the fundamental period. Expressions for obtaining the equivalent homogeneous uniform embankment were developed in this chapter. 77 The nonlinear response of the soil is approximated by an equivalent linear analysis. The equivalent linear properties (Gdeg and ξ) of the approach embankment are obtained with a proposed step-by-step procedure that is simple to execute and converges in a few iterations. The 1ME model is accurate for calculating the relative displacement and acceptable for the shear strain, shear stress, and total acceleration. However, the model may overestimate significantly the total acceleration if the second and first periods of vibration are in the zone of highest and lowest amplification of the PSA response spectrum, respectively. The chapter also developed simple expressions for calculating total acceleration, relative displacement, shear strain, and shear stress profiles in the embankment. These types of profiles are useful for seismic design of the approach embankments. Expressions for calculating the discrete properties of the 1ME model (ME, CE, and KE) based on the equivalent-linear properties (Gdeg and ξ) were also developed. These discrete properties are necessary for coupling the 1ME model with the near-field and the bridge structure components in the proposed 3M-EASI model, discussed in Chapter 5. 78 Chapter 4 VERIFICATION OF THE 1ME MODEL The differential equation of motion of a uniform shear beam has been used extensively since the early 70’s for ground response analyses. The equation is solved in the frequency domain for the case of a linear elastic soil deposit subjected to a harmonic input motion. The solution is based on the calculation of transfer functions that depend on the shear modulus G and the damping ratio ξ of the soil layers. Mapping from the time to the frequency domain, and vice versa, of the input motion and output motions is carried out using the Fourier transform algorithms (Kramer, 1996). The actual nonlinear response of the soil deposit can be approximated by the frequency domain solution if strain-dependent equivalent linear properties Gγ and ξγ are used in the analysis. Since the computed shear strain γ in the soil layers depends on the assumed values of Gγ and ξγ, an iterative procedure is required until consistency between the updated equivalent linear properties and γ is satisfied. This procedure, already discussed in Chapter 3, is called equivalent linear analysis and it has been implemented in the well known computer programs SHAKE (Schnabel et al., 1972) and ProShake (EduPro Civil Systems, 2010). The nonlinear response of the 1ME model is also approximated by equivalent linear analysis; therefore, the model will be verified with ProShake analyses for four uniform approach embankments. Time histories and peak response quantity profiles are used as the output data for comparison purposes. 79 4.1 Physical Properties and Input Ground Motion Figure 4.1 plots the four cases to analyze in the verification of the 1ME model. The first two cases are low homogeneous embankments with height H = 7 m, shear wave velocity Vstop = 100 and 200 m/s, and shear wave velocity ratio rVs = 1. These embankments are named low soft and low stiff in Figure 4.1a. The last two cases are high non-homogeneous embankments with H = 14 m, Vstop = 100 and 200 m/s, and rVs = 1.5. These embankments are named high soft and high stiff in Figure 4.1b. The assumed physical properties represent limit cases of H and Vs of approach embankments found in practice. 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 Shear Wave Velocity Vs (m/s) D im en sio nl es s H ei gh t = z /H 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 Shear Wave Velocity Vs (m/s) D im en sio nl es s H ei gh t = z /H High Soft rVs = 1.5 High Stiff rVs = 1.5 Low Soft rVs = 1 Low Stiff rVs = 1 a) low embankments H = 7 m b) high embankments H = 14 m Figure 4.1 Shear wave velocity Vs profiles of four uniform approach embankments Figure 4.2 shows the assumed discretization of 10 layers for each embankment to calculate the seismic response with ProShake. The input motion is applied at the base, which is assumed to be rigid, and considered as an outcrop motion. The time history used for comparison purposes is the total acceleration at the top of the embankments. Peak values of the relative displacement, shear strain, shear stress, and total acceleration were computed in each layer to compare them with the peak response quantity profiles obtained with the 1ME model. Table 4.1 presents the physical properties of each layer of the four embankments to be analyzed with ProShake. The shear wave velocities of the high embankments were calculated from Figure 4.1b at the midpoint of each layer. The Plasticity Index PI of the soft and stiff embankments is PI = 50 and 30, respectively. 80 Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6 Layer 7 Layer 8 Layer 9 Layer 10 H z output ∆H w av e pr op ag at io n input Rigid Base Figure 4.2 Discretization of an approach embankment for ProShake analysis Table 4.1 Physical properties of four uniform approach embankments for ProShake analysis Layer γ ∆H Vs PI Vs PI ∆H Vs PI Vs PI kN/m3 m m/s m/s m m/s m/s 10 20 0.7 100 50 200 30 1.4 103 50 205 30 9 " " 100 " 200 " " 108 " 215 " 8 " " 100 " 200 " " 113 " 225 " 7 " " 100 " 200 " " 118 " 235 " 6 " " 100 " 200 " " 123 " 245 " 5 " " 100 " 200 " " 128 " 255 " 4 " " 100 " 200 " " 133 " 265 " 3 " " 100 " 200 " " 138 " 275 " 2 " " 100 " 200 " " 143 " 285 " 1 " " 100 " 200 " " 148 " 295 " H = 7 H = 14 γ : unit weight, ∆H: thickness, Vs: shear wave velocity, PI: plasticity index Low Embankments High Embankments Soft Stiff Soft Stiff Figure 4.3 plots the modulus reduction Gsec/Gmax and damping ξ curves that represent the characterization of the equivalent linear properties of the clay-sandy soil of the embankments. 81 0 0.2 0.4 0.6 0.8 1 0.0001 0.001 0.01 0.1 1 10γc (%) G se c / G m ax PI = 30 PI = 50 0 5 10 15 20 25 0.0001 0.001 0.01 0.1 1 10γc (%) ξ ( % ) PI = 30 PI = 50 a) modulus reduction curves b) damping curves Figure 4.3 Vucetic-Dobry (1991) modulus reduction and damping curves The verification of the 1ME model is done with a ground motion that matches the 5% damped Uniform Hazard Spectrum, UHS, for Vancouver, British Columbia, Site C (Adams and Halchuk, 2003). This motion is obtained with a scaling technique known as spectral matching, which requires an initial acceleration time history and a target spectrum. In this case, the initial time history is the Loma Prieta ground motion plotted in Figure 3.7a. Figures 4.4a and b plot the initial and the matched acceleration time histories, respectively, obtained with the computer program RSPMatch (Abrahamson, 1998). The figures show that the peak ground acceleration of the initial time history (PGA ≈ 0.16 g) has been scaled up almost 3 times in the matched time history (PGA ≈ 0.45g). -0.48 -0.36 -0.24 -0.12 0 0.12 0.24 0.36 0.48 0 5 10 15 20 25 30 35 40 Time (s) A cc el er at io n (g ) -0.48 -0.36 -0.24 -0.12 0 0.12 0.24 0.36 0.48 0 5 10 15 20 25 30 35 40 Time (s) A cc el er at io n (g ) a) initial time history b) matched time history Figure 4.4 Initial and matched ground motions 82 Figures 4.5a and b plot the target, the initial, and the matched pseudo acceleration and relative displacement response spectra, respectively. The figures show that the response spectra of the initial time history have been scaled up in the period interval 0 to 2 s to match the UHS. ξ = 5% 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T (s) PS A (g ) Target: UHS Initial Matched ξ = 5% 0 2 4 6 8 10 12 14 16 18 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T (s) D (c m ) Target: UHS Initial Matched a) pseudo acceleration response spectra b) relative displacement response spectra Figure 4.5 Initial and matched response spectra 4.2 Low Soft Approach Embankment Table 4.2 presents the equivalent linear analysis of the low soft embankment using the 1ME model and the procedure proposed in Section 3.5.2. The density reduction factor is Rρ = 1 due to the fact that the embankments are uniform. The calculation of the relative displacement time history and its peak displacement D was done with the computer program SeismoSignal. Table 4.2 Equivalent linear analysis of the low soft embankment using the 1ME model Iteration Gin Gsec/Gmax Gdeg TE ξ D PSA γef tol i MPa MPa s % cm g % % 0 20.4 - - - - - - 1.E-04 - 1 " 1.0 20.4 0.280 0.96 3.416 1.75 0.542 5.E+05 2 " 0.366 7.46 0.463 11.09 2.515 0.47 0.399 26.4 3 " 0.430 8.77 0.427 9.84 2.389 0.53 0.379 5.0 4 " 0.441 8.99 0.422 9.64 2.439 0.55 0.387 2.1 83 Table 4.2 shows that convergence is reached with 4 iterations. The equivalent linear properties of the embankment are Gdeg = 8.99 MPa and ξ = 9.64%, which represents a degradation of 56% of the initial shear modulus Gin. The period of vibration of the embankment increased from TE = 0.28 s in iteration 1 to TE = 0.422 s in iteration 4, which represents an increase of 51% in the initial TE. Figure 4.6 plots the total acceleration time histories of the equivalent linear response at the top of the low soft embankment obtained with ProShake and the 1ME model. The figure shows that the 1ME model reproduces with a good level of accuracy the seismic response of the embankment obtained with ProShake. The similarity of both time histories is not only in the period of vibration but also in the peak values of the acceleration. In this case, the overestimation of the peak total acceleration at t ≈ 9.02 s with the 1ME model is only 6%. -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 5 6 7 8 9 10 11 12 Time (s) To ta l A cc el er at io n (g ) ProShake 1ME model Figure 4.6 Output time histories at the top of the low soft embankment The error in the estimation of the profile (per) of a given response quantity (RQ) in the embankment with the 1ME model is calculated with Equation 4.1. ( ) ( ) ( )∑ ∑ = = − = 10 1 ayerl oShakePrlayer 10 1 ayerl oShakePrlayerME1layer RQ RQ RQ per 4.1 84 Figure 4.7a plots the peak total acceleration profiles obtained with ProShake and the 1ME model. The irregular shape of the ütpeak profile obtained with ProShake is the result of the contribution of higher modes of vibration to the response of the embankment. The smooth shape of the profile obtained with the 1ME model, on the other hand, is due to the fact that only the fundamental mode of vibration is considered in the response. This is the reason why the 1ME model overestimates ütpeak in the embankment. The overestimation, however, is acceptable from an engineering point of view since the profile error is per = 13%. 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ütpeak (g) z (m ) ProShake 1ME model 0 1 2 3 4 5 6 7 0 0.5 1 1.5 2 2.5 3 3.5 upeak (cm) z (m ) ProShake 1ME model a) total acceleration b) relative displacement 0 1 2 3 4 5 6 7 0 10 20 30 40 50 60 70τpeak (kPa) z (m ) ProShake 1ME model 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 γpeak (%) z (m ) ProShake 1ME model c) shear strain d) shear stress Figure 4.7 Peak response quantity profiles in the low soft embankment Figure 4.7b shows that the peak relative displacement profile obtained with the 1ME model is very similar to the one obtained with ProShake. This is a consequence of the fact that the 85 relative displacement in the embankment is strongly controlled by the fundamental mode. The error in the estimation of upeak with the 1ME model is per = 12%. Figure 4.7c shows that the 1ME model overestimates the peak shear strain γpeak for z > 2.2 m and underestimates it for z < 2.2 m with a profile error of per = 17%. Figure 4.7d plots the peak shear stress τpeak profile obtained with the 1ME model, which is very similar to the one obtained with ProShake. The error in the estimation of τpeak is per = 6%. The analysis of the peak response quantity profiles in Figure 4.7 indicates that the calculation of the seismic response of the low soft embankment (TE = 0.42 s) with the 1ME model has a good level of accuracy from an engineering point of view, especially for the relative displacement and the shear stress. Equation 4.2 calculates the degradation factor DG of the embankment after the equivalent linear analysis has converged. According to Table 4.2, Gdeg/Gin = 0.441; therefore the degradation factor of the low soft embankment is DG = 0.65. −= in deg G G G 1 1.17 D 4.2 Figure 4.8 plots the equivalent linear shear modulus profile of the low soft embankment obtained with the assumed cosine degradation of Gin for the 1ME model (Equation 3.47 with Gin = 20.4 MPa and DG = 0.65) and with ProShake. The value of G at the top of the embankment is Gtop = Gin = 20.4 MPa. The figure shows that the assumed cosine degradation of Gin is a good approximation to the actual degradation profile obtained with ProShake for a multilayer system. Figure 4.9a plots the constant degraded shear modulus, Gdeg = 8.99 MPa, of the 1ME model. It is shown in the figure that Gdeg is approximately the value of G(z) obtained with ProShake at z ≈ 2.5 m, which is z = 0.35H. This location agrees with the one found with Equation 3.49, which assumes that the degradation of Gin with depth has a cosine distribution. 86 0 1 2 3 4 5 6 7 0 3 6 9 12 15 18 21 G (MPa) z (m ) ProShake Cosine Degradation DG = 0.65 Gin = 20.4 MPa Figure 4.8 Degradation profile of the initial shear modulus in the low soft embankment Figure 4.9b plots the constant damping ratio, ξ = 9.64%, of the 1ME model. The figure shows that ξ is approximately the value of ξ(z) obtained with ProShake at z ≈ 2.5 m ≈ 0.35H. 0 1 2 3 4 5 6 7 0 3 6 9 12 15 18 21 G (MPa) z (m ) ProShake 1ME model 0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 ξ (%) z (m ) ProShake 1ME model a) shear modulus b) damping ratio Figure 4.9 Equivalent linear properties of the low soft embankment It is concluded from this analysis that the seismic response of an embankment with equivalent linear properties G(z) and ξ(z) that vary with depth can be calculated with a good level of accuracy by obtaining an equivalent homogeneous embankment with properties Gdeg and ξ that are constant with depth (1ME model). 87 4.3 Low Stiff Approach Embankment Table 4.3 presents the equivalent linear analysis of the low stiff embankment using the 1ME model. The table shows that convergence is reached again with 4 iterations. The equivalent linear properties of the embankment are Gdeg = 41.05 MPa and ξ = 9.22%, which represents a degradation of 50% of Gin. The period of the embankment increased from TE = 0.14 s in iteration 1 to TE = 0.197 s in iteration 4, which means an increase of 41% in the initial TE. Table 4.3 Equivalent linear analysis of the low stiff embankment using the 1ME model Iteration Gin Gsec/Gmax Gdeg TE ξ D PSA γef tol i MPa MPa s % cm g % % 0 81.5 - - - - - - 1.E-04 - 1 " 1.0 81.5 0.140 1.03 0.526 1.08 0.083 8.E+04 2 " 0.579 47.21 0.184 8.06 0.740 0.88 0.117 40.7 3 " 0.516 42.06 0.195 9.01 0.804 0.85 0.128 8.7 4 " 0.504 41.05 0.197 9.22 0.814 0.84 0.129 1.2 Figure 4.10 plots the total acceleration time histories of the equivalent linear response at the top of the low stiff embankment obtained with ProShake and the 1ME model. The figure shows that the accuracy of the 1ME model in reproducing the time history obtained with ProShake is very good. In this case, the 1ME overestimates in 13% the peak total acceleration at t ≈ 7.6 s. Figure 4.11a shows that the ütpeak profile obtained with the 1ME model closely reproduces the profile obtained with ProShake. As mentioned before, the 1ME model overestimates the total acceleration in the embankment because the contribution of higher modes is neglected in the response. The profile error of ütpeak with the 1ME model is per = 11%. Figure 4.11b shows that the 1ME model slightly underestimates the peak relative displacement profile of the embankment with an error of per = 7% with respect to the ProShake profile. 88 -0.8 -0.4 0 0.4 0.8 1.2 5 6 7 8 9 10 11 12 Time (s) To ta l A cc el er at io n (g ) ProShake 1ME model Figure 4.10 Output time histories at the top of the low stiff embankment 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 1.2 ütpeak (g) z (m ) ProShake 1ME model 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 1.2 upeak (cm) z (m ) ProShake 1ME model a) total acceleration b) relative displacement 0 1 2 3 4 5 6 7 0 20 40 60 80 100τpeak (kPa) z (m ) ProShake 1ME model 0 1 2 3 4 5 6 7 0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 γpeak (%) z (m ) ProShake 1ME model c) shear strain d) shear stress Figure 4.11 Peak response quantity profiles in the low stiff embankment 89 Figure 4.11c plots the peak shear strain profiles. In this case, the 1ME model overestimates γpeak for z > 1.8 m and underestimates it for z < 1.8 m with an error of per = 14%. Figure 4.11d shows that the peak shear stress profile τpeak obtained with the 1ME model is practically the same that the one obtained with ProShake. The average error in the estimation of this response quantity with the 1ME model is per = 5%. The analysis of this case shows again that the accuracy of the 1ME model in calculating the seismic response of an approach embankment is good (per ≤ 14%), especially for the peak relative displacement and the peak shear stress. Figure 4.12 plots the equivalent linear shear modulus profile obtained with the cosine distribution and ProShake. In this case DG = 0.58. The figure shows again that the degradation profile of Gin obtained with the cosine distribution for the 1ME model is a good approximation to the degradation profile obtained with ProShake. 0 1 2 3 4 5 6 7 0 15 30 45 60 75 90 G (MPa) z (m ) ProShake Cosine Degradation DG = 0.58 Gin = 81.5 MPa Figure 4.12 Degradation profile of the initial shear modulus in the low stiff embankment Figures 4.13a and b plot the constant degraded shear modulus, Gdeg = 41.05 MPa, and constant damping ratio, ξ = 9.22%, respectively, of the 1ME model. The figures show again that equivalent linear properties obtained with the 1ME model are approximately the values of G(z) 90 and ξ(z) obtained with ProShake at z = 0.35H, which verifies the assumed cosine distribution for representing the degradation of Gin in the embankment with the 1ME model. 0 1 2 3 4 5 6 7 0 15 30 45 60 75 90 G (MPa) z (m ) ProShake 1ME model 0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 ξ (%) z (m ) ProShake 1ME model a) shear modulus b) damping ratio Figure 4.13 Equivalent linear properties of the low stiff embankment The calculation of the seismic response using the 1ME model for the low stiff embankment had a better level of accuracy than the one for the low soft embankment. This is due to the fact that the fundamental period of the stiff embankment, TE = 0.2 s, is located in the zone of highest amplification of the response spectrum and the period of the second mode, T2 = TE/3 = 0.07 s, is in the zone of low amplification (see Figure 4.5a). Therefore, the contribution of the second mode is not significant and the response of the embankment is practically controlled by the first mode, which is the basic assumption of the 1ME model. On the contrary, the fundamental period of the soft embankment, TE = 0.42 s, is located in the zone of moderate amplification of the response spectrum and the period of the second mode, T2 = TE/3 = 0.14 s, is in the zone of high amplification. Therefore, the contribution of the second mode is not negligible, especially for calculating the total acceleration, and the response of the embankment is partially influenced by the second mode, which is not captured with the 1ME model. This is the reason why the accuracy of the 1ME model is better for the stiff embankment than for the soft embankment. A similar case was already discussed in Section 3.2.2.3 (see Figure 3.10). 91 4.4 High Soft Approach Embankment Table 4.4 presents the calculation of the equivalent linear properties of the high soft embankment using the 1ME model. The analysis indicates that the converged properties are Gdeg = 17.16 MPa and ξ = 9.29% and that the period of the embankment is TE = 0.61 s. Gin = 37.4 MPa is the equivalent constant shear modulus that represents the initial parabolic distribution of G(z). Table 4.4 Equivalent linear analysis of the high soft embankment using the 1ME model Iteration Gin Gsec/Gmax Gdeg TE ξ D PSA γef tol i MPa MPa s % cm g % % 0 37.4 - - - - - - 1.E-04 - 1 " 1.0 37.4 0.413 0.96 4.006 0.94 0.318 3.E+05 2 " 0.473 17.71 0.601 9.01 4.352 0.49 0.345 8.6 3 " 0.459 17.16 0.610 9.29 4.402 0.48 0.349 1.1 Figure 4.14 plots the total acceleration time histories obtained with ProShake and the 1ME model. It is shown in the figure that the 1ME model calculates the seismic response of the embankment with a good level of accuracy. The error in the estimation of the peak total acceleration at t ≈ 9.1 s is just −5%. -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 4 5 6 7 8 9 10 11 12 13 Time (s) To ta l A cc el er at io n (g ) ProShake 1ME model Figure 4.14 Output time histories at the top of the high soft embankment 92 Figure 4.15a plots the peak total acceleration profiles obtained with ProShake and the 1ME model. Once again, the 1ME model overestimates ütpeak in most of the height of the embankment with an error of per = 9%. Figure 4.15b shows that the calculation of the peak relative displacement with the 1ME model has a high level of accuracy in comparison to the ProShake profile (per = 4%). Figure 4.15c plots the peak shear strain profiles. In this case, the 1ME model underestimates γpeak for z > 5 m and overestimates it for z < 5 m with a profile error of per = 12%. 0 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ütpeak (g) z (m ) ProShake 1ME model 0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 upeak (cm) z (m ) ProShake 1ME model a) total acceleration b) relative displacement 0 2 4 6 8 10 12 14 0 20 40 60 80 100 120 τpeak (kPa) z (m ) ProShake 1ME model 0 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 γpeak (%) z (m ) ProShake 1ME model c) shear strain d) shear stress Figure 4.15 Peak response quantity profiles in the high soft embankment 93 Figure 4.15d shows how the 1ME model slightly overestimates the peak shear stress τpeak profile obtained with ProShake. The error in the estimation of τpeak is per = 7%. Figures 4.16a and b plot the constant degraded shear modulus, Gdeg = 17.16 MPa, and constant damping ratio, ξ = 9.29%, respectively, of the 1ME model for comparison to the equivalent linear properties obtained from ProShake. The figures show that the properties of the 1ME model represent average values of the ProShake properties at the lower part of the profiles. 0 2 4 6 8 10 12 14 0 4 8 12 16 20 24 G (MPa) z (m ) ProShake 1ME model 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 ξ (%) z (m ) ProShake 1ME model a) shear modulus b) damping ratio Figure 4.16 Equivalent linear properties of the high soft embankment The variation of G(z) obtained with ProShake in Figure 4.16a can be visualized as a combination of the initial parabolic distribution of G(z) (rVs = 1.5) and a degradation profile of G(z) similar to a cosine distribution due to the seismic response of the embankment. This is the reason why the G(z) profile for this type of embankment does not have the same shape that the ones analyzed for low embankments (Figures 4.9a and 4.13a) in which the initial distribution of G(z) is constant with depth (rVs = 1). It is concluded from this analysis that the seismic response of non-homogeneous embankments with a complex distribution of G(z) with depth can be acceptably calculated (per ≤ 12%) with an equivalent homogenous embankment using the 1ME model. 94 4.5 High Stiff Approach Embankment Table 4.5 presents the calculation of the equivalent linear properties, Gdeg = 78.78 MPa and ξ = 8.83%, of the high stiff embankment using the 1ME model. The degradation of the equivalent initial shear modulus Gin is 47% and the equivalent period is TE = 0.285 s. Table 4.5 Equivalent linear analysis of the high stiff embankment using the 1ME model Iteration Gin Gsec/Gmax Gdeg TE ξ D PSA γef tol i MPa MPa s % cm g % % 0 149.7 - - - - - - 1.E-04 - 1 " 1.0 149.7 0.207 1.03 1.097 1.03 0.087 9.E+04 2 " 0.570 85.35 0.274 8.19 1.374 0.74 0.109 25.2 3 " 0.526 78.78 0.285 8.83 1.326 0.66 0.105 3.5 Figure 4.17 plots the total acceleration time histories at the top of the high stiff embankment obtained with ProShake and the 1ME model. It is shown in the figure that the 1ME model calculates the seismic response of the embankment with a high level of accuracy. The error in the estimation of the peak total acceleration at t ≈ 8.7 s is just −1%. -1 -0.5 0 0.5 1 4 5 6 7 8 9 10 11 12 13 Time (s) To ta l A cc el er at io n (g ) ProShake 1ME model Figure 4.17 Output time histories at the top of the high stiff embankment 95 Figure 4.18a plots the peak total acceleration profiles obtained with ProShake and the 1ME model. The figure shows that the 1ME model overestimates ütpeak for 4 m < z < 12 m. The error in the calculation of ütpeak is per = 6%. Figure 4.18b shows that the calculation of the peak relative displacement in the embankment with the 1ME model is accurate and with a slight underestimation for z > 7 m. The error in the estimation of upeak is per = 5%. Figure 4.18c shows that the 1ME model underestimates the peak shear strain γpeak for z > 5 m and slightly overestimates it for z < 5 m. The profile error is per = 10%. Figure 4.18d plots the peak shear stress profiles. The 1ME model estimates τpeak in the embankment with a high level of accuracy (per = 3%). 0 2 4 6 8 10 12 14 0 0.2 0.4 0.6 0.8 1 ütpeak (g) z (m ) ProShake 1ME model 0 2 4 6 8 10 12 14 0 0.4 0.8 1.2 1.6 2 upeak (cm) z (m ) ProShake 1ME model a) total acceleration b) relative displacement 0 2 4 6 8 10 12 14 0 20 40 60 80 100 120 140 160 τpeak (kPa) z (m ) ProShake 1ME model 0 2 4 6 8 10 12 14 0 0.04 0.08 0.12 0.16 0.2 γpeak (%) z (m ) ProShake 1ME model c) shear strain d) shear stress Figure 4.18 Peak response quantity profiles of the high stiff embankment 96 Figure 4.19a plots the shear modulus profile G(z) obtained ProShake and the constant degraded shear modulus, Gdeg = 78.78 MPa, obtained with the 1ME model. Gdeg is an average value of G(z) at the lower part of the embankment. Figure 4.19b also shows that the constant damping ratio, ξ = 8.83%, of the 1ME model represents an average value of ξ(z) obtained with ProShake at the lower part of the embankment. 0 2 4 6 8 10 12 14 0 20 40 60 80 100 120 G (MPa) z (m ) ProShake 1ME model 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 ξ (%) z (m ) ProShake 1ME model a) shear modulus b) damping ratio Figure 4.19 Equivalent linear property profiles in the high stiff embankment The calculation of the seismic response of the high stiff embankment with the 1ME model showed a better level of accuracy than the one of the high soft embankment. The reason for this is that the period of the stiff embankment (TE = 0.285 s) is located in a zone of higher amplification than the period of the soft embankment (TE = 0.61 s). 4.6 Summary This chapter verified the calculation of the far-field embankment response with the 1ME model using ProShake analyses. The comparison of the acceleration time histories at the top of the embankments indicated that the calculation of the equivalent dynamic properties (TE and ξ) of the embankments with the 1ME model is very accurate in comparison to the ProShake results. 97 Table 4.6 summarizes the equivalent dynamic properties (TE and ξ) and the profile errors of the approach embankments analyzed in the verification of the 1ME model. Table 4.6 Summary of the profile errors obtained with the 1ME model Embankment type initial equivalent initial equivalent ütpeak upeak γpeak τpeak Low Soft 0.28 0.42 0.96 9.6 13 12 17 6 Low Stiff 0.14 0.20 1.03 9.2 11 7 14 5 High Soft 0.41 0.61 0.96 9.3 9 4 12 7 High Stiff 0.21 0.29 1.03 8.8 6 5 10 3 Mean Value = 9.8 7.0 13.3 5.3 üt: total acceleration, u: relative displacement, γ: shear strain, τ: shear stress Profile Error (%)TE (s) ξ (%) The verification indicated that the 1ME model has a good level of accuracy in comparison to ProShake, especially for the shear stress (per ≈ 5%) and the relative displacement (per ≈ 7%) profiles. These response quantities are fundamental for the seismic design of the approach embankments in the far field. The 1ME model also showed that the total acceleration is overestimated because higher modes of vibration of the embankment are neglected in the proposed model. However, the error in the estimation (per ≈ 10%) is acceptable from an engineering point of view. A similar situation occurs with the shear strain (per ≈ 13%). 98 Chapter 5 THE 3M-EASI MODEL This chapter describes the proposed 3M-EASI model for calculating the seismic response of IABs. The model takes into account the inertial and kinematic interaction of the far and near field of the approach embankments with the bridge structure. The 3M-EASI model is verified for single-span full-height IABs in the longitudinal direction of the deck using the computer program ABAQUS. 5.1 Integral Abutment Types Integral Abutment Bridges (IABs) in British Columbia are practically new and most of them are located along Highway 19 in Vancouver Island. The majority of the bridges are single- span structures with a wide variety of design solutions for the superstructures (Carvajal et al, 2008). The structural solutions for the abutments, however, are basically of two types: stub abutments (Figure 5.1) and full-height abutments (Figure 5.2). Figure 5.1 Integral abutment bridge with stub abutments 99 Figure 5.2 Integral abutment bridge with full-height abutments Stub abutments are short walls (height ≈ 3 m, including deck end diaphragm) supported on a single row of flexible piles embedded in the approach embankment (Figure 5.1). Due to the low bending capacity of the piles, a hinge (or pin connection) is formed at the base of the abutment wall (Figure 5.3a) during the thermal expansion of the deck or the seismic response of the bridge structure. These types of abutments do not contribute to the stiffness of the bridge structure due to the high flexibility and low bending capacity of the piles. piles abutment spread footing abutment pin connection pin connection a) stub abutment b) full-height abutment Figure 5.3 Integral abutment types Full-height abutments are reinforced concrete walls (5 ≤ height ≤ 11 m, 1 ≤ thickness ≤ 2.5 m) supported on spread footings (Figure 5.2). These types of abutments are detailed at the base so that the wall-footing joint behaves mechanically as a pin connection (Figure 5.3b) to accommodate the thermal expansion of the deck. Contrary to stub abutments, full-height abutments may add a significant amount of stiffness to the bridge structure due to the wall thickness and the location of the pin connection. 100 From a dynamic point of view, the integral abutment type plays a significant role in single- span or multi-span IABs with bents of low stiffness. For example, the stiffness of the bridge structure of a single-span IAB with stub abutments (Figure 5.1) is practically negligible due to the high flexibility of the piles. Therefore, the seismic response of the structure is strongly dependent on the abutment-backfill interaction and the far-field embankment response due to the fact that the bridge structure does not have enough stiffness to resists the displacement imposed by the approach embankments. This is also valid for multi-span bridges with flexible bents, which is the case of the Meloland Road Overpass studied in Chapter 2. A different situation occurs for single-span IABs with full-height abutments (Figure 5.2). In this case, the stiffness of the bridge structure is given by the abutment walls and is not negligible. Thus, the bridge has enough stiffness to resist the displacement imposed by the far- field embankment response and its response depends on the combined effects of the seismic response of the bridge structure and the approach embankments. 5.2 Dynamic Response of IABs for Very Low Amplitude Motion Carvajal et al. (2008) conducted a series of Ambient Vibration (AV) tests on IABs and their approach embankments to determine the dynamic properties of these structures. Periods of vibration, mode shapes, and damping ratios were identified for each IAB in the vertical and transverse direction of the deck. However, no dynamic properties could be identified in the longitudinal direction using AV data. Analysis of the recorded data in the far field of the embankments indicated that the response of stiff soils for low amplitude motion is controlled by surface Rayleigh waves (Carvajal and Ventura, 2009), so no information about the fundamental periods could be obtained for the far field. The conclusion of the AV testing campaign was that the effects of the far-field embankment response on the bridge structure can be neglected for very low amplitude motion. 101 5.3 Dynamic Response of IABs for Earthquake Motion Figure 5.4 shows the components that affect the seismic response of an IAB: The Bridge Structure The Near Field of the approach embankments The Far Field of the approach embankments The bridge structure shown in Figure 5.4 is a single-span IAB with full-height abutments pin connected to the firm ground and with homogeneous approach embankments. This is the predominant structure type found in British Columbia and it will be used as the reference for analyzing the seismic response of IABs. Embankment Bridge Structure Far FieldNear Field H firm ground Embankmentabutment Figure 5.4 Cross section of a full-height IAB in the longitudinal direction of the deck The pin support conditions of the abutments in Figure 5.4 represent the connection between the abutment and the spread footings firmly anchored to the ground to avoid rocking or sliding of the foundation. Therefore, Soil-Foundation-Structure Interaction (SFSI) is not considered in the verification of the 3M-EASI model and it is out of the scope of this thesis. Figure 5.5 shows the proposed mass-spring-dashpot system for calculating the seismic response of IABs. The system is called the 3M-EASI model, in which 3M stands for three masses and EASI stands for Embankment-Abutment-Structure Interaction. The proposed model takes into account the inertial and kinematic interaction of the near and the far field of the approach embankments with the bridge structure. 102 ME MEMB KAB CAB Far Field Bridge StructureNear Field KAB CAB KE CE KB CB KE CE Far FieldNear Field Figure 5.5 Components of the 3M-EASI model for calculating the seismic response of IABs The 3M-EASI model has the same components that the IAB shown in Figure 5.4 has. The far field of each approach embankment is calculated with the 1ME model. The near field is modeled with a spring-dashpot component and it is the link between the far field and the bridge structure. Finally, the bridge structure is modeled with a SDOF system based on the fact that the mass of the bridge is concentrated in the superstructure and the axial deformation of it is negligible. The determination of the discrete properties M, K and C for each component is described in the next sections. The seismic response of the 3M-EASI model is obtained by applying the input ground motion üg(t) at the supports as shown in Figure 5.6. The output motion is obtained in the form of time histories; for example, one acceleration-time history üt(t) for each mass and one force-time history for each spring-dashpot component and support. ME MEMB üg(t) üg(t) üg(t) üt(t) üt(t)ütB(t) Figure 5.6 Input ground motion and output acceleration motions in the 3M-EASI model 103 The 3M-EASI model is developed for calculating the seismic response of IABs in the longitudinal direction of the bridge. This is due to the fact that the flexibility of the bridge structure is usually higher in this direction, thus the biggest displacement demands also occurs in the longitudinal direction. 5.4 The Bridge Structure Component Figure 5.7a shows the frame model of the bridge structure of a single-span IAB with full- height abutments. The frame is modeled with beam-column elements and the abutments are pin connected to the firm ground. Figure 5.7b shows the equivalent representation of the frame model using a SDOF system, which is computationally more efficient than the frame model. uB(t) represents the relative displacement of the deck with respect to the supports. MB, KB and CB are the mass of the superstructure, the bridge structure stiffness, and the viscous coefficient, respectively. MB KB CB uB(t)uB(t) superstructure abutment a) frame model of the bridge structure b) equivalent SDOF system Figure 5.7 Representation of the bridge structure using a single-degree-of-freedom system The stiffness of the bridge structure KB is obtained with a pushover analysis of the frame model as sown in Figure 5.8, where F is the force applied in deck and uB is the deformation of the structure. Equation 5.1 gives the parameters of the bridge structure component, where KB is obtained from the pushover analysis and ξB is the damping ratio of the bridge structure. 104 F uB Figure 5.8 Pushover of the frame model for calculating the bridge stiffness KB uresupestruct MB → B B u F K = BBBB KM2 C ξ= 5.1 The determination of the parameters MB, KB, CB with Equation 5.1 is a standard procedure in dynamic analysis of bridge structures (Chopra, 1997; Priestley and Calvi, 1996). 5.5 The Near-Field Embankment Component Figure 5.9 shows the idealization of three limit conditions of integral abutments when interacting with the backfill soil (near field). F is the force imposed by the superstructure on the abutment and uB is the displacement at the top of the abutment. The first condition is a full-height abutment in which the abutment wall is flexible and the superstructure is rigid (Figure 5.9a). This situation occurs in slender abutment walls of short and wide bridges with wing walls that are not integral with the abutment wall. Thus, the superstructure-abutment joint translates in the longitudinal direction but its rotation is practicably negligible. This is represented in Figure 5.9a with a fixed support at the top of the abutment. The second condition is a full-height abutment in which the abutment wall is rigid and the superstructure is flexible (Figure 5.9b). This situation occurs in thick abutment walls of long 105 and narrow bridges with wing walls that are integral with the abutment wall. Therefore, the abutment rotates as a rigid body with respect to its pinned support. The final condition is a stub abutment (Figure 5.9c). In this case, the low resistance of the flexible piles in the longitudinal direction is negligible in comparison to the abutment-backfill passive resistance. Thus, the abutment just translates longitudinally as a rigid body since the deck would prevent rotation of the abutment. uB F uB F uB F H h Backfill soil reaction piles a) flexible full-height b) rigid full-height c) stub Figure 5.9 Pushovers on integral abutments for calculating the near-field stiffness The near-field stiffness (KAB) is determined with pushover analyses for the conditions shown in Figure 5.9. The abutment wall and the backfill soil are modeled with elastic plane strain FE models (abutment width Babut = 1 m) for different abutment heights (H and h) and shear modulus (Gdeg) of the backfill soil. Gdeg is obtained from the far-field embankment response using the 1ME model (Section 3.5.2). KAB is calculated with Equation 5.2 using pushover simulations. Kabut is the abutment stiffness neglecting the interaction with the backfill. For example, Kabut is zero for the conditions shown in Figures 5.9b and c due to the fact that the abutment has no resistance to displacement if the backfill is removed. For the case of Figure 5.9a → Kabut ≠ 0. abut B AB K u F K −= 5.2 106 Figure 5.10 plots the KAB data obtained from the pushover simulations using the computer program ABAQUS and the Equation 5.2 for the case of a flexible and a rigid full-height abutment in plane strain condition. The abutment thickness is 1 m, the Young modulus of the concrete is E = 28.5 GPa, and the Poisson ratio is ν = 0.2. The embankment height H was varied from 5 to 11 m and the shear modulus Gdeg from 5 to 63 MPa (50 ≤ Vs ≤ 175 m/s, ν = 0.3, ρ = 2 tonne/m3). Flexible (AT = 1.26) 0 20 40 60 80 0 10 20 30 40 50 60 70 Gdeg (MPa) K A B (M N /m ) FE data Fitted curve Rigid (AT = 0.94) 0 20 40 60 80 0 10 20 30 40 50 60 70 Gdeg (MPa) K A B ( M N /m ) FE data Fitted curve a) flexible b) rigid Figure 5.10 Near-field stiffness KAB of full-height abutments for a 1 m width The FE data in Figure 5.10 indicate that KAB is linearly proportional to Gdeg and practically insensitive to H (not labeled in the figure). Therefore, from a practical point of view, KAB can be approximated with a straight line of slope AT = 1.26 and 0.94 for the flexible and rigid full- height abutments, respectively. Figure 5.11 plots the KAB data obtained from the pushover simulations for the case of a stub abutment with height h = 2 and 3 m. The embankment height H was varied from 9 to 13 m. The FE data indicate again that KAB is linearly proportional to Gdeg and slightly sensitive to H (not labeled in the figure). Therefore, from a practical point of view, KAB can be approximated with a straight line of slope AT = 0.96 and 1.16 for h = 2 and 3 m, respectively. Equation 5.3 gives the approximate expression for calculating the near-field stiffness using the abutment type coefficients (AT) obtained from the pushover simulations. AT is summarized in Table 5.1 and Babut is the abutment width. 107 h = 2 m (AT = 0.96) 0 20 40 60 80 0 10 20 30 40 50 60 70 Gdeg (MPa) K A B ( M N /m ) FE data Fitted curve h = 3 m (AT = 1.16) 0 20 40 60 80 0 10 20 30 40 50 60 70 Gdeg (MPa) K A B ( M N /m ) FE data Fitted curve a) h = 2 m b) h = 3 m Figure 5.11 Near-field stiffness KAB of stub abutments for a 1 m width degabutAB G B TA K = 5.3 Table 5.1 Abutment type coefficient (AT) for calculating the near-field stiffness KAB Flexible Rigid h = 2 m h = 3 m 1.26 0.94 0.96 1.16 1.1 Full-Height Abutment Stub Abutment Median Value One of the advantages of calculating KAB with Equation 5.3 is that no critical embankment length (Lc) is required. This is a different way for calculating KAB in comparison to the near- field embankment models proposed in Chapter 2 for stub abutments where KAB is proportional to Lc but no unified criterion exits for determining its value. For example, for Wilson and Tan: Lc = Lwing and for Zhang and Makris: Lc = 0.7√SBH. Equation 5.4 gives the viscous coefficient CAB of the near-field component. The factor Iabut and the total stiffness of the bridge KIAB depends of the contact conditions between the abutment wall and the backfill soil. ( )BIABIABBabutAB C KM2 I C −ξ= 5.4 where no contact loss → Iabut = 1/2 KIAB = KB + 2KAB 5.4a 108 total contact loss → Iabut = 1 KIAB = KB + KAB 5.4b The total damping ratio of the bridge ξIAB includes the damping of the bridge structure ξB and the added damping from the near field of the approach embankments; therefore, ξIAB > ξB. The calculation of ξIAB is not a simple procedure as Zhang and Makris (2002a) showed in their research work. However, the authors proved with FE simulations that the effect of CAB is not as important as the far-field displacement response and that a good approximation can be obtained if the added damping from the near field is neglected. Therefore, for practical purposes it would be assumed that ξIAB = ξB. 5.6 The Far-Field Embankment Component 5.6.1 Damping vs Location of the Embankment Response Figure 5.12a shows the 1ME model for calculating the far-field response at the top of the embankment (z = H) using the scaling factor EF = 4/π ≈ 1.27 for the ground motion üg(t). However, in the 3M-EASI model the ground motion is not scaled (see Figure 5.6), which means that EF = 1. An approximate way to obtain the far-field embankment response at the top using EF = 1 is by reducing the damping ratio ξz < ξ so that the response is amplified to compensate the reduction of EF from 1.27 to 1 (Figure 5.12b). EF üg(t) u(z=H,t) ξ üg(t) u(z=H,t) TE ξz≈TE a) 1ME model with EF=1.27 b) 1ME model with EF=1 Figure 5.12 1ME models for calculating the response at the top z=H of the embankment 109 Figure 5.13 shows with a grey line the far-field response at the top of an embankment (z = H) using a 1ME model with the following dynamic properties: TE = 0.4 s, ξ = 10%, and EF = 1.27. The peak relative displacement of the response is upeak ≈ -3.1 cm. The same figure shows with a black line the far-field embankment response using a 1ME model with the following dynamic properties: TE = 0.4 s, ξz = 5%, and EF = 1. The peak relative displacement of the response of this model is also upeak ≈ -3.1 cm. -3.2 -1.6 0 1.6 3.2 5 6 7 8 9 10 11 12 Time (s) u ( z= H ,t) ( cm ) EF=1, ξz=5% EF=1.27, ξ=10% upeak TE=0.4s Figure 5.13 Equivalent 1ME models for calculating the far-field response at z = H Figure 5.13 shows that both responses have practically the same damped period of vibration (TED = TE / √1-ξ2) and peak relative displacement upeak. The response of the 1ME model with ξz = 5% shows small variations in the displacement amplitude in comparison to the 1ME model with ξ = 10%. This example shows that it is possible to obtain the far-field response at the top (z = H) of an embankment by using the 1ME model with an unscaled ground motion üg(t), EF = 1, and a reduced damping ratio ξz = 0.5 ξ. The above analysis opens the opportunity for calculating the far-field embankment response at other location z < H using the 1ME model with ξz. This is especially useful in IAB with full- height abutments in which the effective embankment response for the bridge structure is located at z ≤ H as it will be shown later on in the verification of the 3M-EASI model. 110 The first step for calculating the response at z ≤ H is to determine the corresponding excitation factor EF that gives the response at the location z in the embankment. For example, if EF = 1.27 the 1ME calculates the response at z = H. On the other hand, if EF < 1.27 the 1ME model calculates the response at z < H. Equation 5.5 gives EF for a given location z assuming that the response of the embankment is controlled by the first mode of vibration (Figure 5.14a). π= H z 2 ins .271 EF 5.5 The second step is to determine ξz for a 1ME model with parameters TE and EF = 1 that gives the same peak relative displacement upeak of the 1ME model with parameters TE, ξ and EF ≤ 1.27 (Equation 5.5). Figure 5.14b plots ξz/ξ vs EF values for 1ME models with parameters that vary from 0.2 ≤ TE ≤ 0.6 s and 5 ≤ ξ ≤ 15% (not labeled in the figure). The data (grey dots) was obtained following the two steps explained above. The displacement-time histories u(t) and their peak values upeak were calculated with the computer program SeismoSignal for the ground motion used in the verification of the 1ME model (Figure 4.4b). 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 EF z / H 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 0.5 1 1.5 2 2.5 3 ξz / ξ EF Data Fitted curve a) EF vs z/H b) damping vs EF Figure 5.14 Equivalent damping ξz for calculation of the far-field response at z ≠ H 111 Figure 5.14b shows that EF vs ξz/ξ data is TE-ξ dependent (not labeled in the figure) but with a variation that is not significant from an engineering point of view (less than 20%). The data indicates that 1.4 > EF ≥ 1 for 0.5 ≤ ξz/ξ ≤ 1 and 1 ≥ EF > 0.55 for 1 ≤ ξz/ξ ≤ 3. The equivalent damping was kept ξz/ξ ≤ 3 to avoid changes in more than 10% in the damped period of the equivalent response. The fitted curved of the EF vs ξz/ξ data is given in Equation 5.6. −= ξ ξln 38.0 1 FE z 5.6 The expression for obtaining the response at a given location z using the 1ME model with EF = 1 and ξz is given in Equation 5.7 by combining Equations 5.5 and 5.6. −= − ξ ξ ln .30 .790sin .640 H z z1 → π−=ξ ξ H z 2 ins .353 .632exp z 5.7 Equation 5.7 is plotted in Figure 5.15 where it is shown that the equivalent damping ξz of a 1ME model with EF = 1 is ξz ≈ 0.5ξ to obtain the response at the top (z = H) of the embankment. On the other hand, if damping is not reduced, ξz = ξ, the response is calculated at z = 0.58H. Any response at a given location z < 0.58H requires an equivalent damping that is ξz > ξ. For example, for the condition of z = 0.3H → ξz = 3ξ. 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 ξz / ξ z / H Figure 5.15 Equivalent damping ξz for calculation of the far-field response at z ≤ H 112 5.6.2 Approach Embankment Length The calculation of the far-field embankment response using the 1ME model depends on the parameters TE = 4H√ρred/Gdeg and ξ (Equation 3.36), so no dimensions (B,L) of the horizontal cross section area of the embankment are required. This is valid for a 1ME model that is not connected to any other component. However, in the 3M-EASI model the 1ME models are interacting with the near-field and the bridge structure components (Figure 5.5). The kinematic and inertial interaction of the 1ME model with other components modifies its seismic response, which contradicts its basic function as a dynamic element for calculating the far-field embankment response. One way to minimize the modification of the far-field response using the 1ME model is by calculating the response with the parameters ME, CE, and KE shown in Equation 5.8 (see Equation 3.34). In this way, the dimensions B and L are required (Figure 3.21). redE 2 HBL M ρ= zdegredE G2 LB C ξρπ= deg 2 E GH8 BL K π= 5.8 The width B of the approach embankment is easily determined from the geometrical characteristics. For example, B can be equal to the crest width or to 1 m if the analysis is done in plane strain conditions. The length L of the approach embankment, on the other hand, is not easily determined from the geometrical configuration since L → ∞ in the field. Therefore, a criterion is needed for obtaining a finite value of L. Equation 5.9 gives the proposed criteria to find a value of L that does not modify significantly the far-field response of the 1ME model in the 3M-EASI model. BE M 0001 M ≥ and ABE K 0001 K ≥ 5.9 113 The first part of Equation 5.9 is an inertial condition to assure that the embankment mass ME is big enough so that any inertial force induced by the bridge mass MB on ME will not affect the acceleration-time history calculated with the 1ME model. The second part of Equation 5.9 is a kinematic condition to assure that the embankment stiffness KE is big enough in comparison to the near-field stiffness KAB so that any relative displacement induced by the bridge mass MB on ME will not affect the displacement-time history calculated with the 1ME model. According to the criteria of Equation 5.9 and the use of Equation 5.3 for calculating KAB (max. value AT = 1.26), it is concluded that the length of the approach embankment L has to satisfies the conditions given in Equation 5.10, where B, H and ρred are the width, height, and reduced density of the embankment, respectively. B B H 1000 BH M 2000 L abut red B ≥ρ≥ 5.10 Figure 5.16 plots the variation of the period TE of the 1ME model for different ME/MB and KE/KAB ratios when coupled with the near-field and the bridge structure components in the 3M-EASI model. The values of MB and KAB are taken from one of the IAB that will be analyzed in the verification of the 3M-EASI model. The identification of the period TE using the 3M-EASI model can be done with any computer program for structural analysis that has mass and spring elements; for example, SAP, ETABS, CANNY, etc. In this case, the computer program ABAQUS was used. Figure 5.16 shows that the period of the 1ME model converges to TE = 0.27 s as ME → 1000MB and KE → 1000KAB. On the other hand, if ME = 0.1MB (small L) → TE = 0.13 s, which is not the period of the approach embankment in the far-field. As ME and KE increases (by increasing L) the period converges to TE = 0.27 s, which is the period of the 1ME model as it would not be interacting with any other component. 114 0.05 0.10 0.15 0.20 0.25 0.30 0.01 0.1 1 10 100 1000 10000 ME / MB T E ( s) 0.05 0.10 0.15 0.20 0.25 0.30 0.01 0.1 1 10 100 1000 10000 KE / KAB T E (s ) a) mass vs period b) stiffness vs period Figure 5.16 Fundamental period TE of the 1ME model in the 3M-EASI model The analyses shown in Figure 5.16 indicate that the criteria proposed in Equation 5.9 is valid for calculating the far-field embankment response with the 1ME model when coupled in the 3M-EASI model. 5.7 Verification of the 3M-EASI Model The 3M-EASI model will be verified for single-span full-height IABs since this is the predominant integral abutment bridge type found in British Columbia. Another reason is that the research done on seismic response of IABs has been based on bridges with stub abutments using the strong-motion earthquake data recorded on the Meloland Road Overcrossing (Chapter 2); therefore, there is a lack of knowledge on the seismic response of IABs with full- height abutments. 5.7.1 Continuum Soil Finite Element Model Figure 5.17 shows the geometrical characteristics of the FE models used for the verification of the 3M-EASI model. The abutment height, thickness, and width are H = 7 m, 1 m, and Babut = 1 m, respectively. Euler beam elements are used for modeling the abutments with a mesh grid of H/10 = 0.7 m. 115 The superstructure thickness and width are 2 m and 1 m, respectively, and it is modeled with 0.7m-long Euler beam elements. The bridge length LB is varied from 10 to 45 m to analyze eight bridges with different fundamental periods of vibration TB of the bridge structure. 7 m 210 m 10 m ≤ LB ≤ 45 m Figure 5.17 2D plane strain continuum soil FE model of a single-span full-height IAB The abutments and the superstructure are made of concrete, whose mechanical properties are ρ = 2.4 tonne/m3, E = 28.5 GPa, and ν = 0.3. For simplicity in the verification process, it is assumed that the bridge structure remains elastic and that no separation occurs between the abutments and the backfills. The embankment height, length, and width are H = 7 m, L = 30H = 210 m, and B = 1 m, respectively. The soil is modeled with 4-node bilinear plane strain quadrilateral elements and the mesh grid is H/10 = 0.7 m. Both approach embankments have the same physical properties and are uniform. The soil properties are summarized in Table 5.2. Two different set of analyses (A and B) are analyzed for the approach embankments based on the shear wave velocity (Vs = 160 and 100 m/s). The purpose of these two sets of analyses is to study the effect of the variation of fundamental period of the embankments on the response of the bridge structures. Table 5.2 Physical properties of the uniform approach embankments Approach Embankments H ρ Rρ Vstop ν rVs PI Gsec/Gmax ξ m tonne/m3 m/s A 7 2 1 160 0.3 1 30 B 7 2 1 100 0.3 1 50 Physical Properties Property Curve Fig. 4.3 Fig. 4.3 116 The sixteen FE models are solved with the computer program ABAQUS using elastic, plane strain, implicit dynamic analysis and the ground motion of Figure 4.4b. Gravity load is not considered and damping for the bridge structure is modeled with dashpots while the embankments are modeled with Rayleigh damping. The results obtained with the FE simulations will be compared with the ones obtained with the 3M-EASI model is Sections 5.7.5 to 5.7.7. 5.7.2 The Bridge Structure Component Table 5.3 gives the parameters of the bridge structure component obtained with Equation 5.1 for the eight bridges to analyze. The pushover simulations for calculating KB were done with ABAQUS. It is assumed that damping in the bridge structure is ξB = 5 %. Table 5.3 Parameters of the bridge structure component for a 1 m width LB MB KB ξB CB TB m tonne MN/m % MN / m/s s 10 63.5 37.0 5 0.15 0.26 15 87.3 35.8 " 0.18 0.31 20 111.1 34.6 " 0.20 0.36 25 135.0 33.4 " 0.21 0.40 30 159.1 32.3 " 0.23 0.44 35 183.5 31.2 " 0.24 0.48 40 208.3 30.2 " 0.25 0.52 45 233.6 29.2 " 0.26 0.56 The table indicates that the bridge structure stiffness KB decreases as the bridge length LB increases. This is the result of the reduction of the bending stiffness of the superstructure as the bridge length increases. The table also shows that the period of the bridge structure TB increases as LB increases, which is mainly the result of the bridge mass MB being increased as LB increases. 117 5.7.3 The Far-Field Embankment Component Table 5.4 gives the equivalent linear properties (Gdeg and ξ) and the parameters of the far-field embankment component, per unit length, obtained with the procedure of Section 3.5.2 and Equation 5.8 for the two types of approach embankments. Table 5.4 Parameters of the far-field embankment components per unit length Approach Vsdeg Gdeg TE ξ L ME CE KE Embankments m/s MPa s % m tonne MN / m/s MN/m A 104 22.1 0.27 10.7 1 7.1 0.035 3.9 B 66 8.9 0.42 9.6 1 7.1 0.020 1.6 The table shows that the initial Vs of the approach embankments type A has been reduced from 160 to 104 m/s because of the non-linear response of the soil. A similar situation occurs in the embankments type B where Vs reduced from 100 to 66 m/s. The fundamental periods of the embankments type A and B are TE = 0.27 and 0.42 s, respectively. The calculation of the equivalent damping ξz with Equation 5.7 is not done at this stage of the verification since the location z of the effective embankment response is unknown for full- height IABs. Therefore, ξz will be identified later on from the FE results by matching the response of the 3M-EASI model with the peak relative displacement of the deck. If the bridge were an IAB with stub abutments, then ξz = 0.5 ξ since the effective embankment response is located at z = H as it was proved by various research works discussed in Chapter 2. The approach embankment length L is calculated with Equation 5.10, where MB = 233.6 tonne (for LB = 45 m in Table 5.3), H = 7 m, ρred = 2 tonne/m3, and B = Babut = 1 m. The parameters of the far-field embankment models are obtained by multiplying their unit values given in Table 5.4 by 32717. m 7000 m 27173 L ≥≥ 5.10(bis) 118 5.7.4 The Near-Field Embankment Component The parameters of the near-field embankment component are calculated with Equation 5.3 for three idealized limit conditions of the abutment-backfill interaction. For the bridges being analyzed, the conditions vary between a flexible AT = 1.26 (for short bridges) and a rigid AT = 0.94 (for long bridges) full-height abutment. To verify the applicability of the AT values proposed in Table 5.1, KAB will be obtained with pushover simulations on the bridge structure in contact with the approach embankments as shown in Figure 5.18. The shear modulus Gdeg of the soil is given in Table 5.4. uB F EmbankmentEmbankment Figure 5.18 Pushover on the FE model for calculating KAB KAB is obtained with Equation 5.11, where KB is the bridge structure stiffness (Table 5.3). The viscous coefficient CAB is obtained with Equation 5.4 for the condition of no contact loss and ξIAB = ξB = 5%. Table 5.5 shows the parameters of the near-field embankment components for the sixteen cases being analyzed. −= B B AB K u F 2 1 K 5.11 Figure 5.19 plots the KAB values of Table 5.5. For the case of embankments A the stiffness varies from 27.6 to 25.5 MN/m as LB increases. The figure plots with horizontal lines the two limit conditions for a flexible (KAB = 27.8 MN/m) and a rigid (KAB = 20.8 MN/m) full-height abutment with Gdeg = 22.1 MPa. 119 Table 5.5 Parameters of the near-field embankment component for a 1 m width LB ξIAB KAB CAB KAB CAB m % MN/m MN / m/s MN/m MN / m/s 10 5 27.6 0.044 11.2 0.020 15 " 27.3 0.052 11.1 0.024 20 " 26.9 0.059 11.0 0.027 25 " 26.6 0.065 10.9 0.030 30 " 26.3 0.070 10.7 0.033 35 " 26.0 0.076 10.6 0.036 40 " 25.8 0.081 10.5 0.038 45 " 25.5 0.086 10.4 0.040 Embankments A Embankments B For the case of embankments B the stiffness varies from 11.2 to 10.4 MN/m as LB increases. The figure plots with horizontal lines the two limit conditions for a flexible (KAB = 11.2 MN/m) and a rigid (KAB = 8.4 MN/m) full-height abutment with Gdeg = 8.9 MPa. 0 5 10 15 20 25 30 35 5 10 15 20 25 30 35 40 45 50 LB (m) K A B ( M N /m ) Embankments A Embankments B Flexible Rigid Flexible Rigid Figure 5.19 Near-field stiffness obtained from pushover analyses on continuum soil FE models Figure 5.19 shows that in general the bridges behave as IAB with flexible full-height abutments, especially for LB ≤ 30 m. As the bridge length increases, the abutments tend to behave as rigid walls due to the reduction in the bending stiffness of the superstructure. It is concluded that the proposed abutment type coefficients AT in Table 5.1 are a good approximation to the values obtained from pushovers analysis on continuum soil FE models. 120 5.7.5 Results: Fundamental Period The first type of output data to be compared is the fundamental undamped period of vibration T1 of the dynamic systems. Four different models are used in the comparison. The first one is the 1M model (Figure 5.20), which is the bridge structure component analyzed in Section 5.7.2. This model considers only the response of the bridge structure and neglects its interaction with the near and the far field of the approach embankments. This model is used in design of standard bridges and it is considered in the verification process because the seismic response of IABs tends to be strongly controlled by the response of the bridge structure for some embankment conditions. MB Bridge Structure KB CB üg(t) ütB(t) Figure 5.20 The 1M model The second model is the 1M-ASI (Figure 5.21), in which 1M stands for one mass and ASI stands for Abutment-Structure Interaction. This is the model used in bridge design of IABs and it considers the interaction of bridge structure with the near-field embankment components. Since the far-field embankment response is neglected, the input motion applied at the supports of the near-field components is the same ground motion üg(t) applied at the support of the bridge structure component. The third and four models are the 3M-EASI (Figure 5.22) and the continuum soil FE (Figure 5.17), called ABAQUS, respectively, which both consider the interaction of the bridge 121 structure with the near and the far field of the approach embankments. It is assumed that the ABAQUS model is the closest representation to the physical model of an IAB. MB KAB CAB Bridge StructureNear Field KAB CAB KB CB Near Field üg(t) üg(t) üg(t) ütB(t) Figure 5.21 The 1M-ASI model ME MEMB KAB CAB Far Field Bridge StructureNear Field KAB CAB KE CE KB CB KE CE Far FieldNear Field üg(t) üg(t) üg(t) üt(t) üt(t)ütB(t) Figure 5.22 The 3M-EASI model The values of the fundamental undamped periods T1 of the four models are found in Table 5.6 and plotted in Figures 5.23 and 5.24 for embankments A and B, respectively. The difference between the embankments A and B is the equivalent-linear shear modulus Gdeg = 22.1 and 8.9 MPa, which affects their fundamental periods of vibration TE = 0.27 and 0.42 s, respectively. Figure 5.23 shows that T1 of the ABAQUS model with embankments A varies from 0.27 to 0.38 s as LB increases from 10 to 45 m. For the case of LB = 10 m, T1 coincides with the 122 period of the approach embankments in the far field → T1 = TE = 0.27 s. As the bridge length increases T1 becomes longer than TE, which means that the far-field period is the second period of the system → T2 = TE for LB ≥ 15 m. A similar situation occurs with the IABs with embankments B in Figure 5.24 but in this case T1 = TE = 0.42 s for LB ≤ 30 m and T2 = TE for LB > 30 m. It is concluded that the shortest value of T1 in the ABAQUS model is TE. Table 5.6 Fundamental period T1 of IABs obtained with four different models LB 1M 1M-ASI 3M-ASI ABAQUS TE 1M-ASI 3M-ASI ABAQUS TE m s s s s s s s s s 10 0.26 0.16 0.27 0.27 0.27 0.20 0.42 0.42 0.42 15 0.31 0.20 0.27 0.27 " 0.24 0.42 0.42 " 20 0.36 0.22 0.27 0.29 " 0.28 0.42 0.42 " 25 0.40 0.25 0.27 0.30 " 0.31 0.42 0.42 " 30 0.44 0.27 0.28 0.32 " 0.34 0.42 0.43 " 35 0.48 0.30 0.30 0.34 " 0.37 0.42 0.44 " 40 0.52 0.32 0.32 0.36 " 0.40 0.43 0.46 " 45 0.56 0.34 0.34 0.38 " 0.43 0.43 0.48 " Embankments A Embankments B 0.1 0.2 0.3 0.4 0.5 0.6 5 10 15 20 25 30 35 40 45 50 LB (m) T 1 (s ) ABAQUS 1M 1M-ASI 3M-EASI TE Figure 5.23 Fundamental period T1 of IABs with embankments A (Gdeg = 22.1 MPa) Figure 5.25 shows the mode shape of the ABAQUS model for the cases where T1 = TE. It is shown in the figure that the motion of the bridge structure is controlled by the fundamental mode shape of the embankments in the far field, which has a cosine deflected shape as the one assumed for the 1ME model (see Figure 3.13a). 123 0.1 0.2 0.3 0.4 0.5 0.6 5 10 15 20 25 30 35 40 45 50 LB (m) T 1 ( s) ABAQUS 1M 1M-ASI 3M-EASI TE Figure 5.24 Fundamental period T1 of IABs with embankments B (Gdeg = 8.9 MPa) Figure 5.25 Fundamental mode shape of the ABAQUS model when T1 = TE Figure 5.26, on the other hand, shows the mode shape of the ABAQUS model for the bridges where T1 > TE. In this case, the motion of the bridge is controlled by the displacement of the bridge structure and the near field of the approach embankments. Figure 5.26 Fundamental mode shape of the ABAQUS model when T1 > TE The seismic response of the bridge structure is therefore determined by the superposition of the two modes shown in Figures 5.25 and 5.26. The participation of each one depends on the period difference and on the stiffness of the near field, which connects the bridge structure with the far field of the approach embankments. Figures 5.23 and 5.24 show that the period of the 1M model can be longer or shorter than T1 of the ABAQUS model depending of the value of TE. Two cases of interest are LB = 10 m 124 (Fig. 5.23) and LB = 30 m (Fig. 5.24) in which T1 ≈ TE. For these conditions, the bridge structure and the approach embankments vibrate in phase. The period of the 1M-ASI model plotted in Figures 5.23 and 5.24 is always shorter than the one obtained with ABAQUS. This is a consequence of neglecting the far field of the approach embankments. The 1M-ASI model has only one period of vibration. The period of the 3M-EASI model is always shorter than or equal to the one obtained with the ABAQUS model but longer than or equal to the period obtained with the 1M-ASI model → (1M-ASI ≤ 3M-EASI ≤ ABAQUS )T1. The improvement in the estimation of T1, in comparison to the 1M-ASI model, for bridges with T1 = TE is due to the interaction of the bridge structure and the near field with the far-field embankment components. The 3M-EASI model has two periods of vibration T1 and TE. The 3M-EASI model tends to have the same period T1 that the one of the 1M-ASI model when T1 > TE. However, contrary to the 1M-ASI model, the 3M-EASI model has a second period (T2 = TE) which improves the calculation of the seismic response of the bridges in the modal superposition. For some bridges the fundamental period T1 obtained with the 3M-EASI model is shorter than T1 obtained with the ABAQUS model. This is due to the fact that the effective mass of the near field that moves with the superstructure is not added to the mass MB of the bridge structure component. This effective mass is obtained by adding mass to MB until T1 of the 3M- EASI model matches T1 of the ABAQUS model. In this case, it is assumed that the area of the effective mass is triangular with dimensions N and H as shown in Figure 5.27. Figure 5.28 shows the values of N normalized with respect to H, which determines the effective mass of the near field. N is plotted against the ratio between the period of the bridge structure TB (Table 5.6) and the period of the far-field embankment component TE. 125 N H Figure 5.27 Equivalent area of the added mass of the near field 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 TB / TE N / H Embankments A Embankments B Figure 5.28 Distance N from the abutment vs bridge-embankment period ratio Figure 5.28 shows that the added mass of each near-field (MN = ½NHBρ) is period dependent. For example, if the period of the bridge structure is shorter than or equal to the period of the far-field embankment component, TB/TE ≤ 1, then the added mass is zero N = 0 → MN = 0. This is due to the fact that the bridge structure is pushed by the far-field of the approach embankments and it vibrates with the period TE. In this case, the bridge structure tends to behave as a retaining wall by resisting the displacement imposed by the far-field of the approach embankments. On the other hand, if TB/TE > 1, then N varies from 0.8 to 0.55 as TB/TE increases. Now the bridge structure pushes the near-field, which effective mass vibrates in phase with the superstructure. Figure 5.28 shows that N → 0.5H for TB/TE > 2.2. The effective mass of the near-field is not considered in the 3M-EASI model since it is valid only for abutments that are in permanent contact with their backfills. In addition, the analyses 126 shown in the next section indicate that good results are obtained by neglecting this added mass which simplifies the calculation of the parameters of the proposed model. 5.7.6 Results: Displacement Time Histories Figures 5.29 and 5.30 plot the relative displacement time histories of the bridge deck, for the case of a short superstructure with LB = 15 m and embankments A, obtained with the 1M-ASI and 3M-EASI model, respectively. Both models are compared against the ABAQUS model for the time interval of the strongest response. -2 -1 0 1 2 5 6 7 8 9 10 11 12 Time (s) u B (t ) (c m ) ABAQUS 1M-ASI LB = 15 m - Embankments A Figure 5.29 Relative displacement time history of the 1M-ASI model for LB = 15 m -2 -1 0 1 2 5 6 7 8 9 10 11 12 Time (s) u B (t ) (c m ) ABAQUS 3M-EASI LB = 15 m - Embankments A Figure 5.30 Relative displacement time history of the 3M-EASI model for LB = 15 m 127 Figure 5.29 clearly shows that the 1M-ASI model neither captures the period of vibration nor the peak value of the response. This is due to the fact that the seismic response of the bridge is strongly influenced by the far-field embankment response, which is not taken into account in the 1M-ASI model. Figure 5.30, on the other hand, shows that the 3M-EASI model captures both the period of vibration and the peak value of the response with a very good level of accuracy. The improvement in the calculation of the seismic response with the 3M-EASI model is due to the far-field embankment components (1ME model). Figures 5.31 and 5.32 plot the relative displacement time histories of the bridge deck for the case of a long superstructure with LB = 40 m and embankments A obtained with the 1M-ASI and 3M-EASI model, respectively. -3.6 -2.4 -1.2 0 1.2 2.4 3.6 5 6 7 8 9 10 11 12 Time (s) u B (t ) (c m ) ABAQUS 1M-ASI LB = 40 m - Embankments A Figure 5.31 Relative displacement time history of the 1M-ASI model for LB = 40 m The figures show that both models have almost the same period of vibration but the 1M-ASI model still underestimates the peak displacement by 35%, approximately, in comparison to the ABAQUS model. The 3M-EASI model, on the other hand, estimates the peak displacement with a very good level of accuracy. The out-of-phase vibration of the 3M-EASI model from 9 to 11.3 s is due to the fact that the effective mass of the near field is not considered in the model; therefore, T1 is a little shorter 128 (10%) in comparison to the period of the ABAQUS model. However, from a design point of view, the 3M-EASI model is accurate in the sense that the peak displacement is estimated with a very good level of accuracy. -3.6 -2.4 -1.2 0 1.2 2.4 3.6 5 6 7 8 9 10 11 12 Time (s) u B (t ) (c m ) ABAQUS 3M-EASI LB = 40 m - Embankments A Figure 5.32 Relative displacement time history of the 3M-EASI model for LB = 40 m 5.7.7 Results: Peak Response Quantities Three response quantities are obtained for the verification of the models: a) peak relative displacement of the deck (uB), peak abutment force (RAB), and peak foundation force (RF). These response quantities are obtained from the time histories of the bridge structure and the near-field embankment components. 5.7.7.1 Peak Relative Displacement Figures 5.33a and b show the peak relative displacement of the bridge deck using the four models with embankments A and B, respectively. Figure 5.33a indicates that the 1M model overestimates the displacement for 20 ≤ LB ≤ 45 m with an error that varies from 27 to 37% in comparison to the ABAQUS model. For the cases LB = 10 and 15 m the response of the 1M and the ABAQUS models are practically the same. This is due to the fact that the periods of the bridge structure and the far field of the embankments are very similar T1 ≈ TE = 0.27 s. Therefore, the bridge structure and the far 129 field vibrate in phase with practically no dynamic interaction between them as shown in Figure 5.34. 0 1 2 3 4 5 5 10 15 20 25 30 35 40 45 50 LB (m) u B ( cm ) ABAQUS 1M 1M-ASI 3M-EASI 0 1 2 3 4 5 5 10 15 20 25 30 35 40 45 50 LB (m) u B ( cm ) ABAQUS 1M 1M-ASI 3M-EASI a) embankments A (Gdeg = 22.1 MPa) b) embankments B (Gdeg = 8.9 MPa) Figure 5.33 Peak relative displacement of the bridge deck -1.8 -1.2 -0.6 0 0.6 1.2 1.8 5 6 7 8 9 10 11 12 Time (s) u ( t) ( cm ) Far-Field Bridge Deck LB = 10 m - Embankments A Figure 5.34 Relative displacement time history of the ABAQUS model for LB = 10 m Figure 5.33b shows that the response of the 1M and ABAQUS models is practically the same, even though their fundamental periods T1 are very different (see Figure 5.24). This is an interesting outcome and the explanation for this apparently contradiction is found in the relative value of the near-field stiffness KAB with respect to the bridge structure stiffness KB. The difference between embankments A and B is the equivalent-linear shear modulus Gdeg, which determines the near-field stiffness KAB. For the case of embankments A and B the average value of the abutment stiffness is KAB ≈ 27 and 11 MN/m, respectively (Table 5.5). 130 On the other hand, the average value of the bridge structure stiffness is KB ≈ 33 MN/m (Table 5.3). This means that the ratio of the total near-field stiffness with respect to the bridge structure stiffness is 2KAB/KB ≈ 1.64 and 0.67 for embankments A and B, respectively. The significant difference between the 2KAB/KB values clearly indicates that the influence of the near and the far field of the approach embankments on the seismic response of the bridge structure is less important in the case of IABs with embankments B than in the case of bridges with embankments A. This explains why the response of the bridges structures in Figure 5.33b is practically not affected by the near and the far field of the approach embankments. The results shown in Figures 5.33a and b indicate that the 1M-ASI model underestimates the peak displacement in comparison to the one obtained with the ABAQUS model with an error that varies from 60% (LB = 10 m) to 30% (LB = 45 m) for the case of embankments A. For the case of embankments B, the error varies from 40% (LB = 10 m) to 30% (LB = 45 m). As mentioned before, this underestimation is due to the fact that the 1M-ASI model does not takes into account the far-field embankment response. The calculation of the seismic response is more complex in IABs with full-height abutments than in bridges with stub abutments because of the location of the effective far-field embankment response. For example, stub abutments are short walls supported on flexible piles of negligible stiffness so the abutment-backfill interaction is concentrated close to the top of the embankments. In this case, the location of the effective far-field embankment response is also close to the top. This is the case of the Meloland Road Overpass discussed in Chapter 2 where the models proposed to calculate its seismic response considered the far-field response at the top of the embankments (z = H). On the other hand, full-height abutments are high walls of stiffness that is not negligible. Therefore, the abutment-backfill interaction takes place along the whole height H of the abutment. For this condition, the effective far-field embankment response is located at z ≤ H. However, it is unknown at this stage of the verification what parameters determine this location. 131 In this regard, the location of the effective far-field embankment response was determined by using the 3M-EASI model. This was done by varying the equivalent damping ξz of the far- field embankment components until the peak displacement of the 3M-EASI model matched the peak displacement of the ABAQUS model (Figure 5.33). The identified ξz are plotted in Figure 5.35a with respect to the original damping ξ of the far field calculated in Table 5.4. These ξz/ξ values are plotted against the ratio of the periods of the bridge structure (TB) and the far-field embankment (TE) components. 0 0.5 1 1.5 2 2.5 3 3.5 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 TB / TE ξ z / ξ Embankments A Embankments B Damping rule 0 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 TB / TE z / H Embankments A Embankments B from damping rule a) equivalent damping b) location Figure 5.35 Location of the effective far-field embankment response Figure 5.35a shows that there is a clear tendency in the data, which is represented by the damping rule proposed in Equations 5.12a and b, plotted in the same figure. 1 T Tfor .50 E Bz ≤=ξ ξ 5.12a 3 and 1 T Tfor .501 T T 2.5 z E B E Bz ≤ξ ξ≥+ −=ξ ξ 5.12b Figure 5.35b plots the location of the effective far-field embankment response using Equation 5.7. The figure indicates that the effective location of the embankment response moves downward from the top (z = H) as the ratio TB/TE increases from 1. 132 −= − ξ ξ ln .30 .790sin .640 H z z1 5.7(bis) For the hypothetical case of TB/TE >> 1 the location of the effective far-field embankment response is at the base of the embankment (z = 0), which coincides with the ground motion üg(t). This means that the far-field embankment response can be ignored for IABs with flexible full-height abutments if the period of the bridge structure (TB) is much longer than the period of the approach embankments in the far field (TE). It is concluded from Figure 5.35b that the parameter that determines the location of the effective far-field embankment response for IABs with flexible full-height abutments is the ratio TB/TE. 5.7.7.2 Peak Abutment Force Figures 5.36a and b plot the peak abutment force RAB obtained with the four models. RAB in the ABAQUS model was calculated by integrating the normal stresses of the soil elements in contact with the abutment. For the 1M-ASI and 3M-EASI models RAB was obtained from the forced of the spring of the near-field embankment component. For the 1M model RAB = 0 since this model does not considered any interaction with the approach embankments. 0 0.2 0.4 0.6 0.8 5 10 15 20 25 30 35 40 45 50 LB (m) R A B ( M N ) ABAQUS 1M 1M-ASI 3M-EASI 0 0.2 0.4 0.6 0.8 5 10 15 20 25 30 35 40 45 50 LB (m) R A B ( M N ) ABAQUS 1M 1M-ASI 3M-EASI a) embankments A (Gdeg = 22.1 MPa) b) embankments B (Gdeg = 8.9 MPa) Figure 5.36 Peak abutment force 133 Figure 5.36a and b show that the estimation of RAB with the 3M-EASI model is much better than the one with the 1M-ASI model. The 3M-EASI model not only captures the peak values with a good level of accuracy but also the tendency of RAB obtained the ABAQUS model. For example, in Figure 5.36b the 3M-EASI model is able to follow the reduction of RAB from LB = 10 to 30 m and the subsequent increase of RAB from LB = 30 to 45 m. The 1M-ASI model, on the other hand, misses the tendency of the data. Figure 5.36b shows that RAB → 0 MN when LB → 30 m. This is due to the fact that the period of the bridge structures TB → TE when LB → 30 m (see Figure 5.24). Therefore, as the responses tend to vibrate in phase then the relative displacement between the bridge structure and the far field tends to zero, which causes very small deformation of the near field RAB → 0 MN. This is the case shown in Figure 5.37 for LB = 30 m with embankments B (TE = 0.42 s) and in Figure 5.34 for LB = 10 m with embankments A (TE = 0.27 s). -3.6 -2.4 -1.2 0 1.2 2.4 3.6 5 6 7 8 9 10 11 12 Time (s) u ( t) ( cm ) Far-Field Bridge Deck LB = 30 m - Embankments B Figure 5.37 Relative displacement time history of the ABAQUS model for LB = 30 m 5.7.7.3 Peak Foundation Force Figures 5.38a and b plot the peak foundation force RF obtained with the four models. RF in the ABAQUS model was calculated as the force at the pin support of the abutments in the longitudinal direction of the bridge. For the 1M, 1M-ASI and 3M-EASI models RF was obtained from the force at the support of the bridge structure component. 134 0 0.4 0.8 1.2 1.6 5 10 15 20 25 30 35 40 45 50 LB (m) R F (M N ) ABAQUS 1M 1M-ASI 3M-EASI 0 0.4 0.8 1.2 1.6 5 10 15 20 25 30 35 40 45 50 LB (m) R F (M N ) ABAQUS 1M 1M-ASI 3M-EASI a) embankments A (Gdeg = 22.1 MPa) b) embankments B (Gdeg = 8.9 MPa) Figure 5.38 Peak foundation force Figure 5.38 shows that the estimation of RF with the 3M-EASI model is better than the one with the 1M-ASI model. The level of accuracy of the 3M-EASI model is reasonably good in comparison to the ABAQUS model. Figure 5.38a indicates that RF obtained with the ABAQUS model is not linearly proportional to the peak displacement of the bridge deck plotted in Figure 5.33a. This is due to the complex nature of the dynamic interaction between the flexible abutment and the stiff soil of the near field of the embankments A, which transmits force to the supports due to the pressure distribution along the abutment height. This effect is less important in Figure 5.38b due to the reduction of the shear modulus Gdeg of the embankments. The expression proposed in Section 5.4 for calculating the stiffness KB of the bridge structure component does not considers this complex dynamic abutment-backfill interaction effect. The main reason for neglecting this effect is to keep the calculation of KB simple for bridge engineers using the current analysis procedures. The determination of KB taking into account the abutment-backfill interaction is done by calculating the stiffness with the reaction force at the supports of the abutments for a given displacement of the deck using pushover analysis on the soil continuum FE model shown in Figure 5.17. The analyses (not shown in this thesis) indicated that KB becomes dependent on TB, Gdeg, and TE, which complicates the determination of KB from a practical point of view. 135 5.8 Parametric Analysis with the 3M-EASI Model The 3M-EASI model was verified for single-span IABs with full-height abutments considering two approach embankment types (A and B). The following properties of the components were varied in the analyses: Period of the bridge structure → 0.26 < TB < 0.56 s Bridge structure stiffness → 29 < KB < 37 MN/m Near-field stiffness → KAB ≈ 26.5 (A) and 10.8 (B) MN/m Period of the far-field component → TE = 0.27 (A) and 0.42 (B) s Damping ratio → 0.4 < ξz/ξ ≤ 3.2 A parameter that takes into account the period and stiffness of the bridge structure (TB and KB) and the near-field stiffness (KAB) is the period of the 1M-ASI model (TASI) given in Equation 5.13. KAB is multiplied by two since it is assumed that the no contact loss occurs between the abutments and their backfill soil. B ABB ASI K 2K Kr where Kr 1 T T =+= 5.13 According to Equation 5.13 the period of the bridge structure−near-field system is TASI ≤ TB due to the addition of stiffness KAB ≥ 0 to the bridge structure. Equation 5.14 gives the ratio of the period of the approach embankment in the far field (TE) to the period of the bridge structure−near-field system (TASI). The period ratio TE/TASI takes into account the bridge structure, the near-field, and the far-field embankment components. TE/TASI is a useful index parameter to study the influence of the far-field embankment response on the calculation of the peak relative displacement of the bridge deck. Kr 1 T T T T B E ASI E += 5.14 136 According to Equations 5.13 and 5.14 the verification of the 3M-EASI model considered the following variation in the index parameters. Stiffness ratio (Kr) → Kr ≈ 1.6 (A) and 0.65 (B) Period ratio (TE/TASI) → 1.7 > TE/TASI > 0.8 (A) 2.1 > TE/TASI > 1 (B) The above variation in the index parameters represents just a few cases of all possible combinations of Kr and TE/TASI found in the field. Therefore, a parametric analysis will be carried out with the 3M-EASI model by varying 0.1 ≤ Kr ≤ 8 and 0.2 ≤ TE/TASI ≤ 5. It is worth noticing that TB and KB represent the parameters of a single- or a multi-span bridge structure. Thus, the parametric analysis is not constrained to single-span IABs. Figure 5.39 plots the peak relative displacement obtained with the 3M-EASI model (DEASI) normalized with respect to the one obtained with the 1M-ASI model (DASI). In this figure Kr varies from 0.1 to 0.8. 0.5 1 1.5 2 2.5 3 0.1 1 10 TE / TASI D E A SI / D A SI Kr = 0.1 Kr = 0.2 Kr = 0.4 Kr = 0.6 Kr = 0.8 Figure 5.39 Peak relative displacement ratio DEASI/DASI for IABs with 0.1 ≤ Kr ≤ 0.8 and ξz/ξ = 0.5 137 For simplicity in the analysis, it is considered that the equivalent damping is ξz = 0.5ξ for all cases, which means that the response is calculated at the top of the embankments (z = H). This is the case of IABs with stub abutments. The damping values of the bridge structure and the approach embankments are ξB = 5% and ξ = 10%, respectively. Figure 5.39 shows that DEASI = DASI for TE ≤ 0.2 TASI, which means that the effect of the far- field embankment response on the bridge structure is negligible. This is due to the fact that the embankment displacement is very small in comparison to the displacement of the bridge structure−near-field system. The figure also shows that a resonance condition occurs when the period of the far-field embankment components is very close to the period of the bride structure−near-field system → TE ≈ TASI. The far-field embankment response amplifies the displacement of the bridge deck from 1.1 times DASI for Kr = 0.1 up to 2.8 times DASI for Kr = 0.8. The displacement amplification increases as Kr increases due to the fact that the near-field stiffness KAB, which connects the far field with the bridge structure, also increases. Another displacement amplification occurs when TE > 2 TASI. In this case the displacement of the far-field embankment components is much bigger than the displacement of the bride structure−near-field system. Therefore, the displacement of the bridge structure is strongly controlled by the far-field embankment components, which effect increases as Kr increases. From a practical point of view it is considered that a displacement amplification bigger than 1.25 times DASI is relevant in the estimation of the seismic demands of the bridge structure. Therefore, according to Figure 5.39 the effect of the far-field embankment response is relevant for IABs with Kr ≥ 0.4 and TE > 0.7 TASI, approximately. Figure 5.40 plots the peak relative displacement ratio DEASI/DASI for IABs with 1 ≤ Kr ≤ 8. The general tendency of the displacement amplification of the bridge structure due to the far- field embankment response is very similar to the one previously discussed for Figure 5.39. In this case the effect of the far-field embankment response is relevant for TE > 0.6 TASI. 138 1 1.5 2 2.5 3 3.5 4 0.1 1 10 TE / TASI D E A SI / D A SI Kr = 1 Kr = 1.5 Kr = 2 Kr = 4 Kr = 8 Figure 5.40 Peak relative displacement ratio DEASI/DASI for IABs with 1 ≤ Kr ≤ 8 and ξz/ξ = 0.5 It is concluded from the parametric analysis that the displacement amplification of the bridge structure due to the far-field embankment response can be significantly high for IABs with Kr > 0.4 and TE > 0.7 TASI. Therefore, seismic analyses that neglect the far-field embankment response may underestimate significantly the seismic demands of the bridge structure. Equation 5.15 introduces a parameter called EASI index which takes into account the parameters (Kr and TE/TASI) that affect the displacement amplification of IABs. += ASI E T T Kr 1 Kr Index EASI 5.15 Figure 5.41 plots the peak relative displacement ratio DEASI/DASI of the parametric analysis discussed in Figures 5.39 and 5.40 vs the proposed EASI Index. The figure shows that the displacement amplification (DEASI > 1.25 DASI) occurs for EASI index > 0.2, approximately. 139 0 1 2 3 4 5 6 0.01 0.1 1 10 EASI Index D E A SI / D A SI Figure 5.41 Peak relative displacement ratio vs EASI index It is concluded that the far-field embankment response can be neglected in the estimation of the seismic demands of the bridge structure if EASI index ≤ 0.2. Therefore, time-history analysis using the 1M-ASI model or response spectrum based analysis with TASI can be used for bridge design of IABs. If EASI index > 0.2 then the estimation of seismic demands of the bridge structure requires time history analyses with the 3M-EASI model. The above is summarized as follows. EASI Index ≤ 0.2 → 1M-ASI model → response spectrum EASI Index > 0.2 → 3M-EASI model → time-history analysis 5.9 Summary This chapter described and verified the proposed 3M-EASI model for calculating the seismic response of IABs in the longitudinal direction of the deck. The model takes into account the inertial and kinematic interaction of the near and the far field of the approach embankments 140 with the bridge structure. The analyses showed that the 3M-EASI model is rational, accurate and easy to use. The parametric analysis with the 3M-EASI model concluded that the estimation of the seismic demands of IABs with models that neglect the far-field embankment response may be significantly underestimated if EASI index > 0.2 → Kr > 0.4 and TE/TASI > 0.7. The summary procedure for calculating the parameters of the 3M-EASI model and the seismic response operates as follows: ME MEMB KAB CAB Far Field Bridge StructureNear Field KAB CAB KE CE KB CB KE CE Far FieldNear Field üg(t) üg(t) üg(t) üt(t) üt(t)ütB(t) Figure 5.22(bis) The 3M-EASI model step 1: specify ξB and calculate the parameters of the bridge structure component with Equation 5.1 uresupestruct MB → ushoverp KB → BBBB KM2 C ξ= B B B K M2 T π= 5.1(bis) step 2: specify H, B, ρ, Gtop, rVs, Gsec/Gmax - ξ curves, üg(t) and calculate ρred, Gdeg, ξ, and TE with the procedure of Section 3.5.2 step 3: specify the abutment type and calculates the equivalent damping ξz with Equation 5.12a or b 141 abutments stubor 1 T Tfor .50 E Bz ≤=ξ ξ 5.12a(bis) 3 and 1 T Tfor .501 T T 2.5 z E B E Bz ≤ξ ξ≥+ −=ξ ξ 5.12b(bis) step 4: specify Babut and calculate the length L of the embankment with Equation 5.10 B B H 1000 BH M 2000 L abut red B ≥ρ≥ 5.10(bis) step 5: calculate the parameters of the far-field embankment component with Equation 5.8 redE 2 HBL M ρ= zdegredE G2 LB C ξρπ= deg 2 E GH8 BL K π= 5.8(bis) step 6: specify AT, Iabut and calculate the parameters of the near-field embankment component with Equations 5.3 and 5.4 degabutAB G B TA K = 5.3(bis) Table 5.1(bis) Abutment type coefficient (AT) for calculating the near-field stiffness KAB Median Flexible Rigid h = 2 m h = 3 m Value 1.26 0.94 0.96 1.16 1.10 Full-Height Abutment Stub Abutment ( )BBIABBabutAB C KM2 I C −ξ= 5.4(bis) no contact loss → Iabut = 1/2 KIAB = KB + 2KAB 5.4a(bis) total contact loss → Iabut = 1 KIAB = KB + KAB 5.4b(bis) step 7: apply üg(t) and obtain the peak values of the time-history response of the components of the 3M-EASI model 142 Chapter 6 CONCLUSIONS AND FUTURE WORK Strong-motion earthquake data and analytical studies indicate that the far-field embankment response has a significant effect on the displacement demands of Integral Abutment Bridges (IABs) and it should not be ignored in design. Therefore, the general consensus is that Embankment-Abutment-Structure Interaction (EASI) should be included in the modeling stage of the bridge structure. Several models have been proposed in the literature for calculating the seismic response of IABs taking into account EASI. The implementation of these models, however, is complicated for current bridge design procedures. This is the main reason why the estimation of seismic demands in IABs usually neglects the far-field embankment response. It is concluded then, that a simple model for calculating and coupling the seismic response of the approach embankments in the far field with the bridge structure is needed in current bridge design practice of IABs. This thesis developed a simple mass-spring-dashpot based system, called the 3M-EASI model, for calculating the seismic response of IABs using equivalent linear analysis. The model takes into account the inertial and kinematic interaction of the near and the far field of the approach embankments with the bridge structure. 143 The 3M-EASI model has three basic components: a) the bridge structure, b) the near field, and c) the far field of the approach embankments. The parameters of each component are calculated with a step-by-step procedure proposed in this thesis. One of the main contributions of this research thesis was the development and verification of the far-field embankment component, called the 1ME model, for calculating the seismic response of approach embankments in the far field using equivalent linear analysis. The 1ME model was developed for trapezoidal approach embankments which initial shear modulus distribution with depth is constant or parabolic. The determination of the equivalent linear properties of the 1ME model is done with a proposed step-by-step procedure. Simple expressions for calculating total acceleration, relative displacement, shear strain, and shear stress profiles in the embankment using the 1ME model were also developed in this thesis. These types of profiles are useful for seismic design of the approach embankments. Another important contribution of this thesis was the development of a simple expression for calculating the stiffness of the near-field component for stub and full-height integral abutments using the shear modulus of the 1ME model. The 3M-EASI model was verified for single-span full-height integral abutment bridges using continuum soil finite elements models solved with the computer program ABAQUS. The analyses indicated that the 3M-EASI model is accurate from an engineering point of view. A parametric analysis done with the 3M-EASI model indicated that the far-field embankment response affects the response of IABs if the following conditions act simultaneously: (a) the near-field stiffness is greater than 0.4 times the bridge stiffness, and (b) the period of the far- field embankment components is longer than 0.7 times the period of the bridge-near-field system. 144 The main conclusion of this thesis is that calculating the seismic response of IABs with models that neglect the far-field embankment response may underestimate the peak response quantities for bridge design significantly. The proposed 3M-EASI model is shown to be rational, accurate, computationally efficient, and easy to implement in bridge design procedures with software used in regular design offices. These features make the 3M-EASI model an ideal tool for evaluating uncertainties in the seismic demands of the bridge structure for different ground motions and soil properties. Likewise, the simple step-by-step procedures involved in the 3M-EASI model can be easily incorporated in Highway Code provisions. 6.1 Future Work The main objective of this thesis was to develop a simple and general model for calculating the seismic response of integral abutment bridges taking into account the far-field embankment response. In that sense, the 3M-EASI model achieved the proposed objective. However, the 3M-EASI model was verified only for single-span IABs with flexible full-height abutments, symmetric approach embankment conditions, and in the longitudinal direction of the deck. In addition, only one ground motion was used and no soil-foundation-structure interaction was considered in the verification process. Therefore, it is recommended to verify the 3M-EASI model taking into account the following scenarios: Multi-span IABs with stub and full-height abutments. Soil-foundation-structure interaction in multi-span IABs with piled foundations. 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Earthquake Engineering and Structural Dynamics, 31: 1967-1991. 150 Annex A: Procedure for Calculating the Equivalent Linear Properties of the 1ME (Far-Field) model üg(t) ξ TE üt(t) Figure 5.12a(bis) The 1ME model step 1: specify H, B, ρ, Gtop, rVs, Gsec/Gmax and ξ curves, and üg(t) step 2: calculate ρred with Equation 3.52 ρρ=ρ R red where 2 (B/H)96.01 (B/H)98.072.0 R + +=ρ 3.52(bis) step 3: calculate Gin with Equation 3.45 ++= 2topin rVs 28 15 rVs 3 1 23 3 G G 3.45(bis) 151 step 4: assume γef(i) = 1x10-4 % and obtain Gsec/Gmax and ξ for γc = γef(i) step 5: calculate Gdeg = Gin (Gsec/Gmax) step 6: calculate TE with Equation 3.37 for ρ = ρred and G = Gdeg deg red E G H4 T ρ= 3.37(bis) step 7: obtain D or PSA by solving Equation 3.36 for TE, ξ, and EF = 1 g(t)(t) 2 E (t) E (t) u uT 2 u T 4 u &&&&& −= π+ ξπ+ 3.36(bis) step 8: obtain γef(i+1) with Equation 3.54 (%) H D 111 ef =γ or (%) H PSA T 2.8 2 E ef =γ 3.54(bis) step 9: calculate the tolerance in the convergence with Equation 3.55 (%) 1 100 (i) ef 1)(i ef tol −γ γ= + 3.55(bis) step 10: if tol ≤ 5 % → end if tol > 5 % → obtain Gsec/Gmax and ξ for γc = γef(i+1) → repeat steps 5 to 10 152 Annex B: Procedure for Calculating the Parameters of the 3M-EASI model ME MEMB KAB CAB Far Field Bridge StructureNear Field KAB CAB KE CE KB CB KE CE Far FieldNear Field üg(t) üg(t) üg(t) üt(t) üt(t)ütB(t) Figure 5.22(bis) The 3M-EASI model step 1: specify ξB and calculate the parameters of the bridge structure component with Equation 5.1 uresupestruct MB → ushoverp KB → BBBB KM2 C ξ= B B B K M2 T π= 5.1(bis) step 2: specify H, B, ρ, Gtop, rVs, Gsec/Gmax - ξ curves, üg(t) and calculate ρred, Gdeg, ξ, and TE with the procedure of Annex A 153 step 3: specify the abutment type and calculates the equivalent damping ξz with Equation 5.12a or b abutments stubor 1 T Tfor .50 E Bz ≤=ξ ξ 5.12a(bis) 3 and 1 T Tfor .501 T T 2.5 z E B E Bz ≤ξ ξ≥+ −=ξ ξ 5.12b(bis) step 4: specify Babut and calculate the length L of the embankment with Equation 5.10 B B H 1000 BH M 2000 L abut red B ≥ρ≥ 5.10(bis) step 5: calculate the parameters of the far-field embankment component with Equation 5.8 redE 2 HBL M ρ= zdegredE G2 LB C ξρπ= deg 2 E GH8 BL K π= 5.8(bis) step 6: specify AT, Iabut and calculate the parameters of the near-field embankment component with Equations 5.3 and 5.4 degabutAB G B TA K = 5.3(bis) Table 5.1(bis) Abutment type coefficient (AT) for calculating the near-field stiffness KAB Median Flexible Rigid h = 2 m h = 3 m Value 1.26 0.94 0.96 1.16 1.10 Full-Height Abutment Stub Abutment ( )BBIABBabutAB C KM2 I C −ξ= 5.4(bis) no contact loss → Iabut = 1/2 KIAB = KB + 2KAB 5.4a(bis) total contact loss → Iabut = 1 KIAB = KB + KAB 5.4b(bis) 154 step 7: apply üg(t) and obtain the peak values of the time-history response of the components of the 3M-EASI model
Thesis/Dissertation
2011-11
10.14288/1.0063026
eng
Civil Engineering
Vancouver : University of British Columbia Library
University of British Columbia
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
Graduate
Seismic embankment-abutment-structure interaction of integral abutment bridges
Text
http://hdl.handle.net/2429/35577