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Three-dimensional nonlinear analysis of dynamic soil-pile-structure interaction for bridge systems under… Rahmani, Amin 2014

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.Three-dimensional nonlinear analysis of dynamicsoil-pile-structure interaction for bridge systems underearthquake shakingsbyAmin RahmaniB.Sc., Civil Engineering, Sharif University of Technology, Iran, 2007M.Sc., Geotechnical Engineering, Sharif University of Technology, Iran, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Civil Engineering)The University of British Columbia(Vancouver)November 2014c© Amin Rahmani, 2014AbstractBridge designers have adopted simple approximate methods to take into accountsoil-structure-interaction (SSI) in dynamic analysis of bridge systems. The mostpopular one is the substructuring method in which the response of the foundationsoil and its interaction with the pile foundation and the abutment system are rep-resented by a set of one-dimensional springs and dashpots. While this methodhas been widely used in practice, it has never been validated by comparing theresults with those obtained from full-scale analyses. This thesis aims to evaluatethe substructuring method and to quantify the level of associated errors for the usein bridge engineering. To this end, the baseline data required for the evaluationprocess is provided by full-scale nonlinear dynamic analysis of the bridge systemssubjected to earthquake shaking using continuum modeling method. This involvesdetailed modeling of the foundation soil, pile foundations, abutment system, andthe whole bridge structure. Three representative bridge systems with two, three,and nine spans are simulated. In all models, nonlinear hysteretic response of thefoundation soil and the bridge piers are accounted for in the analyses using ad-vanced constitutive models. The numerical model of the bridge is validated bysimulating the seismic response of the Meloland Road Overpass for which exten-sive measured data exist over past earthquake events. Subsequently each one ofthe three bridge systems is also simulated using the substructuring method. Com-paring the obtained results with the baseline data indicates that the substructuremodel may not be sufficiently reliable in predicting the bridge response. In par-ticular the method is shown to misrepresent the spectral responses of the bridge,pier deflections, shear forces and bending moments induced at the pier base, andlongitudinal and transverse forces induced to the abutments. The substructuringiimethod is shown to suffer from several fundamental drawbacks that cannot be sim-ply resolved. Using the recent advances in constitutive modeling of geotechnicaland structural materials, and in computational tools and high-performance parallelcomputing, this thesis shows that large-scale continuum models can gradually be-come a powerful and significantly more reliable alternative for proper modeling ofseismic SSI in bridge engineering.iiiPrefaceIn 2010, the Natural Sciences and Engineering Research Council of Canada (NSERC)funded a strategic project at the University of British Columbia (UBC) to study theSoil-Structure-Interaction in Performance Based Design of Bridges. The projectbelonged in the strategic target area of Safety and Security and the proposed re-search topics dealt with Risk and Vulnerability and Resiliency of Systems. Pro-fessors Ventura, Taiebat, and Finn, from the department of Civil Engineering atUBC were leading the project. As a part of this project, the present study aims toevaluate the numerical modeling approach widely used in practice for seismic soil-structure-interaction (SSI) analysis of bridge systems. The outputs of this thesisaid the advancement of the state of practice in this area.I, Amin Rahmani, am the principle contributor to all eight chapters of thisthesis. I was responsible for all major areas of concept formation, data collectionand analysis, and writing the chapters. Some parts of the findings of this thesishave been published in a journal and three conferences so far.• Rahmani A., Taiebat, M., and Finn, W.D.L. (2014). “Nonlinear dynamicanalysis of Meloland Road Overpass using three-dimensional continuummodeling approach”, Soil Dynamics and Earthquake Engineering, vol. 57,pp. 121-132.This paper includes a version of chapter 4 and sections 5.2 and 5.3 of chapter5. I conducted all the numerical analyses and wrote the first draft of themanuscript. Professors Taiebat and Finn provided technical assistance inrevising the manuscript.• Bebamzadeh A., Rahmani A., Taiebat M., Ventura C.E., and Finn W.D.L.iv(2014). “Performance-based seismic design of bridges using high perfor-mance computing tools”, Proceedings of the 10th National Conference inEarthquake Engineering, Earthquake Engineering Research Institute, An-chorage, AK, Paper ID: 1727, 10 pages.This paper is a version of section 5.2 of chapter 5 and appendix A. I con-ducted all the numerical analyses and wrote some parts of the manuscript.Dr. Bebamzadeh wrote major parts of the manuscript. Revisions were madeby Professor Taiebat.• Rahmani A., Taiebat, M., Finn, W.D.L., and Ventura, C.E. (2012). “Deter-mination of dynamic p-y curves for pile foundations under seismic loading”,Proceedings of the Fifteenth World Conference on Earthquake Engineering.Lisbon, Portugal, Paper ID: 5228, 8 pages.• Rahmani A., Taiebat, M., Finn, W.D.L., and Ventura, C.E. (2012) “Evalua-tion of p-y curves used in practice for seismic analysis of soil-pile interac-tion”, Proceedings of GeoCongress, State of the Art and Practice in Geotech-nical Engineering (GSP 225). Hryciw, RD, Athanasopoulos-Zekkos, A,Yesiller, N, eds., American Society of Civil Engineers, pp. 1780-1788.The above two papers are versions of chapter 6, and I conducted all thenumerical analyses and wrote major parts of the manuscript. Revisions weremade by Professors Taiebat and Finn.Additional papers are under preparation to publish the remainder of the thesisfindings.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives and original features . . . . . . . . . . . . . . . . . . 41.3 Structure of presentation . . . . . . . . . . . . . . . . . . . . . . 51.4 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Existing approaches for dynamic SSI analysis of bridges . . . . . 102.2.1 Continuum model (Direct method) . . . . . . . . . . . . . 102.2.2 Substructure model (Multistep method) . . . . . . . . . . 122.3 Modeling soil-pile interaction . . . . . . . . . . . . . . . . . . . . 162.3.1 API guidelines for soil-pile interaction . . . . . . . . . . . 162.3.2 Macro-element models for soil-pile interaction . . . . . . 202.3.3 Spring model used in practice . . . . . . . . . . . . . . . 22vi2.4 Modeling soil-pile cap interaction . . . . . . . . . . . . . . . . . 222.5 Modeling embankment-abutment interaction . . . . . . . . . . . . 232.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.7 Tables and figures . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Substructure model for analysis of bridge systems . . . . . . . . . . 393.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 First step: free-field response analysis . . . . . . . . . . . . . . . 403.3 Second step: effective linearization of backbone curves . . . . . . 413.4 Third step: calculating pile cap dynamic stiffness . . . . . . . . . 423.5 Fourth step: calculating pile cap kinematic motions . . . . . . . . 443.6 Fifth step: dynamic analysis of the bridge model . . . . . . . . . . 443.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.8 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 Elements of continuum modeling in OpenSees . . . . . . . . . . . . 484.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Numerical simulation platform . . . . . . . . . . . . . . . . . . . 494.3 Constitutive modeling of materials . . . . . . . . . . . . . . . . . 494.3.1 Sandy soils . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.2 Clayey soils . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.3 Concrete material . . . . . . . . . . . . . . . . . . . . . . 524.3.4 Reinforcing steel material . . . . . . . . . . . . . . . . . 534.4 Modeling soil-pile interaction . . . . . . . . . . . . . . . . . . . . 534.5 Analysis procedure . . . . . . . . . . . . . . . . . . . . . . . . . 544.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.7 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 Continuum modeling of bridge systems: validation and baseline data 605.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 Analysis of the two-span Meloland Road Overpass . . . . . . . . 625.2.1 Description of the bridge . . . . . . . . . . . . . . . . . . 625.2.2 Model development and simulation details . . . . . . . . 635.2.3 Dimensions and mesh refinement of the soil domain . . . 65vii5.2.4 Input earthquake shakings . . . . . . . . . . . . . . . . . 665.3 Validation of the continuum modeling method . . . . . . . . . . . 685.3.1 Predominant period of the bridge . . . . . . . . . . . . . 685.3.2 Spectral responses of the bridge structure . . . . . . . . . 685.3.3 Other seismic responses of the bridge . . . . . . . . . . . 695.4 Additional baseline data . . . . . . . . . . . . . . . . . . . . . . 715.4.1 Analysis of a prototype three-span bridge system . . . . . 725.4.1.1 Description of the bridge . . . . . . . . . . . . 725.4.1.2 Model development and simulation details . . . 725.4.1.3 Input earthquake shakings . . . . . . . . . . . . 735.4.1.4 Simulation results . . . . . . . . . . . . . . . . 735.4.2 Analysis of a nine-span bridge system . . . . . . . . . . . 745.4.2.1 Description of the bridge . . . . . . . . . . . . 745.4.2.2 Model development and simulation details . . . 765.4.2.3 Input earthquake shakings . . . . . . . . . . . . 775.4.2.4 Simulation results . . . . . . . . . . . . . . . . 785.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.6 Tables and figures . . . . . . . . . . . . . . . . . . . . . . . . . . 816 Evaluation of the API p-y springs: static and dynamic analysis . . . 1196.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.2 Static analysis of single piles . . . . . . . . . . . . . . . . . . . . 1206.2.1 Spring method for static problems . . . . . . . . . . . . . 1206.2.2 Evaluation of the spring method . . . . . . . . . . . . . . 1216.2.3 Further comments on the application of the spring method 1216.3 Dynamic analysis of single piles . . . . . . . . . . . . . . . . . . 1236.3.1 Spring method for seismic problems . . . . . . . . . . . . 1236.3.2 Continuum modeling method . . . . . . . . . . . . . . . 1246.3.3 Evaluation of the spring method . . . . . . . . . . . . . . 1246.3.4 Further comments on the application of the spring method 1256.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.5 Tables and figures . . . . . . . . . . . . . . . . . . . . . . . . . . 129viii7 Simulation of the bridge systems using the substructuring method:analysis and evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 1397.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.2 Analysis of the two-, three-, and nine-span bridges . . . . . . . . 1407.2.1 Free-field motions . . . . . . . . . . . . . . . . . . . . . 1407.2.2 Dynamic stiffnesses of the pile group and the abutment . . 1417.2.3 Kinematic input motions . . . . . . . . . . . . . . . . . . 1427.2.4 Developing the global model of the bridges . . . . . . . . 1437.3 Evaluating the substructure model of the bridges . . . . . . . . . . 1447.3.1 Seismic responses of the two-span bridge . . . . . . . . . 1447.3.2 Seismic responses of the three-span bridge . . . . . . . . 1457.3.3 Seismic responses of the nine-span bridge . . . . . . . . . 1487.4 Discussion on the evaluation of the substructuring method . . . . 1507.4.1 Representing soil response . . . . . . . . . . . . . . . . . 1517.4.2 Input ground motions . . . . . . . . . . . . . . . . . . . . 1537.4.3 Mass of soil domain . . . . . . . . . . . . . . . . . . . . 1547.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1557.6 Tables and figures . . . . . . . . . . . . . . . . . . . . . . . . . . 1578 Summary and future research . . . . . . . . . . . . . . . . . . . . . 1918.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . 194Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Appendix A Importance of parallel computing . . . . . . . . . . . . . 209A.1 Analysis execution time . . . . . . . . . . . . . . . . . . . . . . . 210A.2 Analysis costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211Appendix B Description of the static tests performed on single piles . . 214Appendix C Description of the seismic centrifuge tests performed onsingle piles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224ixAppendix D Results obtained from the continuum models of the three-and nine-span bridges . . . . . . . . . . . . . . . . . . . . . . . . . . 226D.1 Results for the three-span bridge . . . . . . . . . . . . . . . . . . 226D.2 Results for the nine-span bridge . . . . . . . . . . . . . . . . . . 251Appendix E Results obtained from the substructure models of the three-and nine-span bridges . . . . . . . . . . . . . . . . . . . . . . . . . . 270E.1 Dynamic stiffnesses at the bridge supports . . . . . . . . . . . . . 270E.2 Results for the three-span bridge . . . . . . . . . . . . . . . . . . 275E.3 Results of the nine-span bridge . . . . . . . . . . . . . . . . . . . 292xList of TablesTable 2.1 P-multipliers suggested in AASHTO (2012) . . . . . . . . . . 27Table 2.2 Recommended API t-z curves for piles embedded in clays andsands (API, 2007). . . . . . . . . . . . . . . . . . . . . . . . . 27Table 2.3 Recommended API Q-z curves for piles embedded in clays andsands (API, 2007). . . . . . . . . . . . . . . . . . . . . . . . . 28Table 5.1 Input parameters for the soil constitutive models (Kwon and El-nashai, 2008). . . . . . . . . . . . . . . . . . . . . . . . . . . 81Table 5.2 Input parameters for the concrete material used in fiber beam-column element. . . . . . . . . . . . . . . . . . . . . . . . . . 81Table 5.3 Input parameters for the deconvolution analysis in ProShake(2003). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Table 5.4 Summary of information for the selected ground motions (ap-plied to the three-span bridge). . . . . . . . . . . . . . . . . . 82Table 5.5 Undrained shear strength (Su) values in the soil profiles 1 and 2. 82Table 5.6 Input parameters for the soil constitutive model used in the nine-span bridge model. . . . . . . . . . . . . . . . . . . . . . . . . 83Table 5.7 Predominant period of the nine-span bridge system in the lon-gitudinal and transverse directions. . . . . . . . . . . . . . . . 83Table 5.8 Summary of information for the selected ground motions thatare linearly scaled to the UHS of Vancouver (applied to thenine-span bridge). . . . . . . . . . . . . . . . . . . . . . . . . 83xiTable 6.1 Brief description of the 27 experimental static tests performedon laterally loaded single piles (further details of the tests arepresented in Appendix B. . . . . . . . . . . . . . . . . . . . . 129Table 6.2 Input parameters for the PIMY constitutive model used for sim-ulating the soil in test no. 12. . . . . . . . . . . . . . . . . . . 130Table 6.3 Brief description of two centrifuge tests performed on singlepiles subjected to earthquake shakings (further details of thetests are presented in Appendix C. . . . . . . . . . . . . . . . 130Table 6.4 Input parameters for the soil constitutive models used for simu-lating the soil behavior in the dynamic centrifuge tests. . . . . 130Table 7.1 Spring and dashpot coefficients that represent the dynamic stiff-nesses of the pile group and the abutment system of MRO dur-ing the 1979 Imperial Valley and 2010 El Mayor-Cucapah earth-quakes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Table E.1 Constants of the springs that represent the 6×6 stiffness matrix(K) of the pile group and the abutment system for the three-spanbridge system under the ten earthquake events. . . . . . . . . . 271Table E.2 Constants of the dashpots that represent the 6×6 damping ma-trix (C) of the pile group and the abutment system for the three-span bridge system under the ten earthquake events. . . . . . . 272Table E.3 Constants of the springs that represent the 6×6 stiffness matrix(K) of the pile group and the abutment system for the nine-spanbridge system under the five earthquake events. . . . . . . . . . 273Table E.4 Constants of the dashpots that represent the 6×6 damping ma-trix (C) of the pile group and the abutment system for the nine-span bridge system under the five earthquake events. . . . . . . 274xiiList of FiguresFigure 1.1 (a) The Cypress Viaduct, Interstate 880 in Oakland, Califor-nia after the 1989 Loma Prieta earthquake (photo credit: U.S.Geological Survey), and (b) the Santa Monica Freeway, LosAngeles, after the 1994 Northridge earthquake (photo credit:Associated Press/Lois Bernstein). . . . . . . . . . . . . . . . 8Figure 2.1 Application of 3D continuum modeling method (direct method)in the literature for simulating (a) Pointer Street Overpass, (b)a prototype concrete bridge, (c) Meloland Road Overpass, and(d) Humboldt Bay Middle Channel Bridge . . . . . . . . . . . 29Figure 2.2 Three-step solution to account for soil-structure-interaction indynamic analyses (Kausel et al., 1978). . . . . . . . . . . . . 30Figure 2.3 Application of substructuring method in the literature for sim-ulating SSI for bridge systems. . . . . . . . . . . . . . . . . . 30Figure 2.4 The graphs presented in API (2007) for determining; (a) initialsubgrade reaction, k, and (b) coefficients C1, C2, and C3. . . . 31Figure 2.5 Characteristic shapes of the API (2007) p-y curves for softclays under static loading (after Matlock, 1970). . . . . . . . . 32Figure 2.6 Characteristic shapes of the (a) t-z, and (b) Q-z curves as pre-sented in API (2007). . . . . . . . . . . . . . . . . . . . . . . 33Figure 2.7 Effects of soil nonlinearity and inertial interaction on lateralstiffness of foundation system (Finn, 2005). . . . . . . . . . . 34xiiiFigure 2.8 Eight sets of springs and dashpots which are used in numer-ical modeling of a boundary value problem to represent thedynamic 6×6 stiffness matrix (K). . . . . . . . . . . . . . . . 34Figure 2.9 Comparing ultimate lateral soil resistance recommended in APIwith the values measured in experimental tests (Zhang et al.,2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.10 Schematic of the element proposed by Boulanger et al. (1999);presenting the components of the element and the correspond-ing force-deflection responses. . . . . . . . . . . . . . . . . 36Figure 2.11 Schematic of the element proposed by Taciroglu et al. (2006);presenting the components of the element and the correspond-ing force-deflection responses. . . . . . . . . . . . . . . . . 37Figure 2.12 Schematic of the element proposed by Allotey and El Naggar(2008); presenting the standard and direct reload curves, thegeneral unload curve and typical two-way cyclic response ofthe element. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 2.13 Schematic of the spring model used in practice for dynamicSSI analysis of piles. . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.1 Schematic of five consecutive steps of substructuring method;(a) first step: determination of the depth-varying time historiesof displacements, (b) second step: effective linearization ofthe backbone curves, (c) third step: determination of the 6×6stiffness matrix (K), (d) fourth step: determination of the kine-matic input motion at the pile cap, and (e) fifth step: dynamicanalysis of the bridge model (Shamsabadi, 2013). . . . . . . 47Figure 4.1 Schematic of the pressure-dependent multi-yield surface (PDMY)model in stress space and its typical response in stress-strainspace (Yang et al., 2003). . . . . . . . . . . . . . . . . . . . . 57Figure 4.2 Schematic of the pressure-independent multi-yield surface (PIMY)model in stress space and its typical response in stress-strainspace (Gu et al., 2011). . . . . . . . . . . . . . . . . . . . . . 57xivFigure 4.3 Schematic of stress-strain response of the Kent-Scott-Park modelfor concrete material. . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.4 Schematic of stress-strain response of the bilinear inelastic modelfor steel material. . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.5 Description of methodology used for soil-pile interaction mod-eling, (a) plan view , and (b) finite element mesh. . . . . . . . 59Figure 5.1 Meloland Road Overpass(MRO), (a) overall photo, (b) the pier,(c) the abutment, and (d) configuration of the accelerometers(CESMD, 2013). . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 5.2 Schematic representation of soil layers at the MRO site (di-mensions are not to scale). . . . . . . . . . . . . . . . . . . . 85Figure 5.3 The developed finite element continuum model of the MROincluding the structural and geotechnical components of thebridge (visualized by GiD, 2013). . . . . . . . . . . . . . . . 85Figure 5.4 Modeling of the pier using fiber-section beam column elements(a) fiber discretization of the pier cross-section, and (b) cyclicmoment-curvature response. . . . . . . . . . . . . . . . . . . 86Figure 5.5 (a) Time history of acceleration at top of the left abutment, and(b) time history of shear forces induced at the pier base in thelongitudinal and transverse directions; sensitivity of the bridgestructure response to the model dimensions. . . . . . . . . . . 87Figure 5.6 Time histories of acceleration on top of the two soil columnscompared to those at points next to the lateral boundaries ofthe continuum model (a) on the ground surface, and (b) on topof the embankment. . . . . . . . . . . . . . . . . . . . . . . . 88Figure 5.7 (a) Time history of acceleration at top of the left abutment, and(b) time history of shear forces induced at the pier base in thelongitudinal and transverse directions; sensitivity of the bridgestructure response to the mesh refinement. . . . . . . . . . . . 89xvFigure 5.8 Time histories of the recorded and computed accelerations atthe ground surface in the longitudinal and transverse directionsduring the; (a) 1979 Imperial Valley earthquake, and (b) 2010El Mayor-Cucapah earthquakes. . . . . . . . . . . . . . . . . 90Figure 5.9 Spectral values of the recorded and computed accelerations atthe ground surface in the longitudinal and transverse directionsduring the; (a) 1979 Imperial Valley earthquake, and (b) 2010El Mayor-Cucapah earthquakes. . . . . . . . . . . . . . . . . 91Figure 5.10 Acceleration response spectrum of motions at different loca-tions of the bridge for the damping ratio of 5% during the;(a) 1979 Imperial Valley earthquake, and (b) 2010 El Mayor-Cucapah earthquake. . . . . . . . . . . . . . . . . . . . . . . 92Figure 5.11 Displacement response spectrum of motions at different lo-cations of the bridge for the damping ratio of 5% during the(a) 1979 Imperial Valley earthquake, and (b) 2010 El Mayor-Cucapah earthquake. . . . . . . . . . . . . . . . . . . . . . . 93Figure 5.12 Time histories of pier top displacements relative to the pile capin longitudinal and transverse directions during the, (a) 1979Imperial Valley earthquake, and (b) 2010 El Mayor-Cucapahearthquake. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Figure 5.13 Time histories of pile cap rocking in longitudinal and trans-verse directions during the, (a) 1979 Imperial Valley earth-quake, and (b) 2010 El Mayor-Cucapah earthquake. . . . . . . 95Figure 5.14 Moment-curvature response and time histories of bending mo-ment at the pier base during the, (a) 1979 Imperial Valley earth-quake, and (b) 2010 El Mayor-Cucapah earthquake. . . . . . . 96Figure 5.15 Force-deflection response at left and right abutments and atthe pier base in the longitudinal direction during the, (a) 1979Imperial Valley earthquake, and (b) 2010 El Mayor-Cucapahearthquake. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97xviFigure 5.16 Force-deflection response at left and right abutments and atthe pier base in the transverse direction during the, (a) 1979Imperial Valley earthquake, and (b) 2010 El Mayor-Cucapahearthquake. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Figure 5.17 Bending moment distribution, Myy (about the transverse axisof the bridge), at the instance of maximum pile displacementsduring the, (a) 1979 Imperial Valley earthquake, and (b) 2010El Mayor-Cucapah earthquake. . . . . . . . . . . . . . . . . . 99Figure 5.18 Bending moment distribution, Mxx (about the longitudinal axisof the bridge), at the instance of maximum pile displacementsduring the, (a) 1979 Imperial Valley earthquake, and (b) 2010El Mayor-Cucapah earthquake. . . . . . . . . . . . . . . . . . 100Figure 5.19 (a) Schematic of the bridge superstructure and the underlyingsoil layers (dimensions are not to scale), and (b) 3D finite ele-ment continuum model of the three-span bridge system (visu-alized by GiD, 2013). . . . . . . . . . . . . . . . . . . . . . . 101Figure 5.20 Bending moment distribution, Mxx, at the instance of maxi-mum pile displacements during the 1979 Imperial Valley earth-quake for the, (a) two-span bridge (MRO), and (b) three-spanbridge (pile foundations of pier 1); comparing the bending mo-ment distribution along the pile foundations of the two bridge. 102Figure 5.21 Acceleration response spectrum of the motions (fault-parallelcomponent), for 5% damping, (a) events No. 1 to 5, and (b)events No. 6 to 10. . . . . . . . . . . . . . . . . . . . . . . . 103Figure 5.22 Acceleration response spectrum of the motions (fault-normalcomponent), for 5% damping, (a) events No. 1 to 5, and (b)events No. 6 to 10. . . . . . . . . . . . . . . . . . . . . . . . 104Figure 5.23 Maximum seismic responses of the bridge during the the tenearthquake events including, (a) maximum pier drifts, (b) max-imum shear forces induced at the pier base (pier 1), (c) maxi-mum bending moments induced at the pier base (pier 1), and(d) forces induced to the abutment system in longitudinal andtransverse directions. . . . . . . . . . . . . . . . . . . . . . . 105xviiFigure 5.24 Schematics of the nine span bridge supported on the, (a) soilprofile 1, and (b) soil profile 2 (dimensions are not to scale). . 106Figure 5.25 Shear wave velocity profile for bridge sites 1 and 2. . . . . . . 107Figure 5.26 3D finite element continuum model of the nine-span bridgesystem including the structural and geotechnical componentsof the bridge (visualized by GiD, 2013). . . . . . . . . . . . . 108Figure 5.27 (a) 3D fiber cross-section of the eight piers, (b) moment-curvatureresponse of the section about axis x (longitudinal direction),and (c) moment-curvature response of the section about axis y(transverse direction). . . . . . . . . . . . . . . . . . . . . . . 109Figure 5.28 (a) 3D fiber cross-section of the pile foundations, and (b) moment-curvature response of the section in the longitudinal and trans-verse directions. . . . . . . . . . . . . . . . . . . . . . . . . . 110Figure 5.29 Acceleration response spectrum of the motions linearly scaledto the uniform hazard spectrum (UHS) of Vancouver in theperiod range of 0.1 to 1.0 s; (a) fault-parallel components, and(b) fault-normal components . . . . . . . . . . . . . . . . . . 111Figure 5.30 Acceleration response spectrum of the selected ground mo-tions and computed motions at the ground surface of the con-tinuum model (fault-parallel components); evaluating the de-convolution process by comparing the responses for seven groundmotions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure 5.31 Acceleration response spectrum of the selected ground mo-tions and computed motions at the ground surface of the con-tinuum model (fault-normal components); evaluating the de-convolution process by comparing the responses for seven groundmotions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Figure 5.32 Time histories of relative displacements of the pier top with re-spect the pier base for piers 1, 4, 6, and 8 during the 1978 Tabasearthquake when the bridge is supported on, (a) soil profile 1(soft soil), and (b) soil profile 2 (stiff soil). . . . . . . . . . . . 114xviiiFigure 5.33 Time histories of shear forces induced at the base of piers 1, 4,6, and 8 during the 1978 Tabas earthquake when the bridge issupported on, (a) soil profile 1 (soft soil), and (b) soil profile 2(stiff soil). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure 5.34 Time histories of bending moments induced at the base of piers1, 4, 6, and 8 during the 1978 Tabas earthquake when thebridge is supported on, (a) soil profile 1 (soft soil), and (b)soil profile 2 (stiff soil). . . . . . . . . . . . . . . . . . . . . . 116Figure 5.35 Comparing bending moment distribution, Myy, in the longitu-dinal direction at the instance of maximum pile displacementsduring the 1978 Tabas earthquake for the, (a) pier 4 in soil pro-file 1, (b) pier 8 in soil profile 1, (c) pier 4 in soil profile 2, (d)pier 8 in soil profile 2. . . . . . . . . . . . . . . . . . . . . . 117Figure 5.36 Comparing bending moment distribution, Mxx, in the trans-verse direction at the instance of maximum pile displacementsduring the 1978 Tabas earthquake for the, (a) pier 4 in soil pro-file 1, (b) pier 8 in soil profile 1, (c) pier 4 in soil profile 2, (d)pier 8 in soil profile 2. . . . . . . . . . . . . . . . . . . . . . 118Figure 6.1 Schematic of the spring method used in practice for static anal-ysis of laterally loaded single piles. . . . . . . . . . . . . . . 131Figure 6.2 Level of error in prediction of maximum displacement of pilehead and maximum bending moment along the pile shaft forthe tests listed in Table 6.1; the positive error indicates overes-timation of the measured response, and the negative error indi-cates underestimation of the measured response. (note: bend-ing moments were not reported in tests 8, 10, 13–16, and 20–26).131Figure 6.3 Finite element mesh of test No. 12 (visualized by GiD, 2013). 132Figure 6.4 (a) Pile head load vs pile head deflections, and (b) pile headload vs maximum bending moments in Test 12; comparing themeasured values with those computed by the continuum andspring methods. . . . . . . . . . . . . . . . . . . . . . . . . 132xixFigure 6.5 Schematic of the spring method used for dynamic analysis of asingle pile, (a) first step: site response analysis, and (b) secondstep: dynamic analysis of the pile. . . . . . . . . . . . . . . . 133Figure 6.6 Time histories of input acceleration in the centrifuge test of (a)Gohl (1991), and (b) Wilson (1998). . . . . . . . . . . . . . . 134Figure 6.7 Finite element meshes of a single pile system in the centrifugetest of (a) Gohl (1991), and (b) Wilson (1998). The finite ele-ment meshes are halved along the direction of shaking (x axis)due to the symmetry along y axis. . . . . . . . . . . . . . . . 135Figure 6.8 Acceleration response spectra (for damping ratio of 5%) at theground surface in (a) the centrifuge test of Gohl (1991), and (b)the centrifuge test of Wilson (1998); comparing the measuredresponses with those computed in the continuum model. . . . 136Figure 6.9 Acceleration response spectra (for damping ratio of 5%) at thesuperstructure in (a) the centrifuge test of Gohl (1991), and (b)the centrifuge test of Wilson (1998); comparing the measuredand the computed responses in the continuum and the springmodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Figure 6.10 Bending moment profile at the instance of pile maximum dis-placement in (a) centrifuge test of Gohl (1991), and (b) cen-trifuge test of Wilson (1998); comparing the measured and thecomputed responses in the continuum and the spring models. . 137Figure 6.11 Acceleration response spectra (for damping ratio of 5%) at thesuperstructure in (a) the centrifuge test of Gohl (1991), and (b)the centrifuge test of Wilson (1998); comparing the measuredand the computed responses in the spring models A and B. . . 137Figure 6.12 Bending moment profile at the instance of pile maximum dis-placement in (a) the centrifuge test of Gohl (1991), and (b) thecentrifuge test of Wilson (1998); comparing the measured andthe computed responses in the spring models A and B. . . . . 138xxFigure 6.13 Natural vibration periods of the first three modes of the soil-pile system in (a) the centrifuge test of Gohl (1991), and (b) thecentrifuge test of Wilson (1998); comparing the natural periodsin the continuum model and the spring model. . . . . . . . . . 138Figure 7.1 Numerical model of the soil columns representing the free-field conditions in the foundation soil and the embankmentused for the seismic site response analysis. . . . . . . . . . . 158Figure 7.2 Examples for the effective linearization of the backbone curvesto obtain the secant stiffnesses; (a) linearization of the API p-y curve in the transverse direction at the depth of 2.0 m, (b)linearization of the API t-z curve at the depth of 2.0 m, (c)linearization of the API Q-z curve at the pile tip, and (d) lin-earization of the backbone curve at top of the abutment systembased on the guidelines of Caltrans (2013). . . . . . . . . . . 159Figure 7.3 Substructure models of the two-, three-, and nine-span bridgesystems including the bridge superstructure supported on lin-ear spring and dashpot models. All three models are created inOpenSees finite element program (McKenna and Fenves, 2001). 160Figure 7.4 Schematic of the channel locations at the MRO site (channels2, 3, 7, and 9 record the transverse responses, and channel 4records the longitudinal responses). . . . . . . . . . . . . . . 161Figure 7.5 Acceleration response spectra (for 5% damping) of the MROstructure for the (a) 1979 Imperial Valley earthquake, and (b)2010 El Mayor-Cucapah earthquake; comparing the measuredresponses with those estimated by the continuum and the sub-structure models. . . . . . . . . . . . . . . . . . . . . . . . . 162Figure 7.6 Acceleration response spectra (for 5% damping) of the mo-tions at the abutment top (point A in Fig. 7.4), and at thefree-field of the approach embankment (point B in Fig. 7.4),in the transverse direction computed in the continuum modelduring the (a) 1979 Imperial Valley earthquake, and (b) 2010El Mayor-Cucapah earthquake. . . . . . . . . . . . . . . . . . 163xxiFigure 7.7 Time histories of bending moment for (a) the 1979 ImperialValley earthquake, and (b) the 2010 El Mayor-Cucapah earth-quake; comparing the responses in the continuum and the sub-structure models. . . . . . . . . . . . . . . . . . . . . . . . . 164Figure 7.8 Time histories of pier top relative displacements with respectto the pier base for (a) the 1979 Imperial Valley earthquake,and (b) the 2010 El Mayor-Cucapah earthquake; comparingthe responses in the continuum and the substructure models. . 165Figure 7.9 Dynamic force-deflection responses at left and right abutmentsand at the pile cap in the transverse direction for (a) the 1979Imperial Valley earthquake, and (b) the 2010 El Mayor-Cucapahearthquake; comparing the responses in the continuum and thesubstructure models. . . . . . . . . . . . . . . . . . . . . . . 166Figure 7.10 Acceleration response spectrum (for 5% damping) of the mo-tions computed at the center of the deck in the longitudinaldirection for the ten earthquake event; comparing the resultsobtained from the substructure model with those obtained fromthe continuum model. . . . . . . . . . . . . . . . . . . . . . . 167Figure 7.11 Acceleration response spectrum (for 5% damping) of the mo-tions computed at the center of the deck in the transverse di-rection for the ten earthquake event; comparing the results ob-tained from the substructure model with those obtained fromthe continuum model. . . . . . . . . . . . . . . . . . . . . . . 168Figure 7.12 (a) Maximum pier drifts, (b) maximum base shear forces, (c)maximum bending moments for pier 1, and (d) maximum forcesinduced to the abutment system; comparing the results of thesubstructure model with those of the continuum model for theten earthquake events. (a), (b), and (d) show bridge responsein the longitudinal direction, and (c) shows the response aboutthe transverse axis of the bridge. . . . . . . . . . . . . . . . . 169xxiiFigure 7.13 (a) Maximum pier drifts, (b) maximum base shear forces, (c)maximum bending moments for pier 1, and (d) maximum forcesinduced to the abutment system; comparing the results of thesubstructure model with those of the continuum model for theten earthquake events. (a), (b), and (d) show bridge responsein the transverse direction, and (c) shows the response aboutthe longitudinal axis of the bridge. . . . . . . . . . . . . . . . 170Figure 7.14 Exceedance probability of the relative difference in estima-tion of, (a) maximum pier drifts, (b) maximum base shearforces, (c) maximum bending moments for pier 1, and (d) max-imum forces induced to the abutment system in longitudinaland transverse directions for the ten earthquake events. Thecurves represent the best fit to the data. . . . . . . . . . . . . 171Figure 7.15 Acceleration response spectrum (for 5% damping) of the mo-tions computed at the center of the deck in soil profile 1 in thelongitudinal direction for all five earthquake events; compar-ing the results of the continuum model, the substructure modeland the fixed model. . . . . . . . . . . . . . . . . . . . . . . 172Figure 7.16 Acceleration response spectrum (for 5% damping) of the mo-tions computed at the center of the deck in soil profile 1 in thetransverse direction for all five earthquake events; comparingthe results of the continuum model, the substructure model andthe fixed model. . . . . . . . . . . . . . . . . . . . . . . . . . 173Figure 7.17 Acceleration response spectrum (for 5% damping) of the mo-tions computed at the center of the deck in soil profile 2 in thelongitudinal direction for all five earthquake events; compar-ing the results of the continuum model, the substructure modeland the fixed model. . . . . . . . . . . . . . . . . . . . . . . 174Figure 7.18 Acceleration response spectrum (for 5% damping) of the mo-tions computed at the center of the deck in soil profile 2 in thetransverse direction for all five earthquake events; comparingthe results of the continuum model, the substructure model andthe fixed model. . . . . . . . . . . . . . . . . . . . . . . . . . 175xxiiiFigure 7.19 Exceedance probability of the pier 4 drift for the nine-spanbridge in (a) soil profile 1 in the longitudinal direction, (b) soilprofile 1 in the transverse direction, (c) soil profile 2 in thelongitudinal direction, and (d) soil profile 2 in the transversedirection; comparing the results of the continuum model, thesubstructure model and the fixed base model. The curves rep-resent the log-normal distribution best fitted to the data. . . . . 176Figure 7.20 Exceedance probability of maximum shear forces induced atthe base of pier 4 of the nine-span bridge in (a) soil profile 1in the longitudinal direction, (b) soil profile 1 in the transversedirection, (c) soil profile 2 in the longitudinal direction, and (d)soil profile 2 in the transverse direction; comparing the resultsof the continuum model, the substructure model and the fixedbase model. The curves represent the log-normal distributionbest fitted to the data. . . . . . . . . . . . . . . . . . . . . . . 177Figure 7.21 Exceedance probability of maximum bending moments inducedat the base of pier 4 of the nine-span bridge in (a) soil profile 1about the transverse direction, (b) soil profile 1 about the lon-gitudinal direction, (c) soil profile 2 about the transverse di-rection, and (d) soil profile 2 about the longitudinal direction;comparing the results of the continuum model, the substruc-ture model and the fixed base model. The curves represent thelog-normal distribution best fitted to the data. . . . . . . . . . 178Figure 7.22 Exceedance probability of maximum forces induced to the abut-ment 1 of the nine-span bridge in (a) soil profile 1 in the longi-tudinal direction, (b) soil profile 1 in the transverse direction,(c) soil profile 2 in the longitudinal direction, and (d) soil pro-file 2 in the transverse direction; comparing the results of thecontinuum model, the substructure model and the fixed basemodel. The curves represent the log-normal distribution bestfitted to the datas. . . . . . . . . . . . . . . . . . . . . . . . . 179xxivFigure 7.23 Time histories of relative displacements of the top of pier 4with respect to the pier base for the nine-span bridge systemsupported on (a) soil profile 1 (soft soil), and (b) soil profile 2(stiff soil) during the 1989 Loma Prieta earthquake; comparingthe responses obtained from the continuum, the substructure,and the fixed base models. . . . . . . . . . . . . . . . . . . . 180Figure 7.24 Time histories of base shear force of pier 4 for the nine-spanbridge system supported on (a) soil profile 1 (soft soil), and(b) soil profile 2 (stiff soil) during the 1989 Loma Prieta earth-quake; comparing the responses obtained from the continuum,the substructure, and the fixed base models. . . . . . . . . . . 181Figure 7.25 Time histories of bending moment induced at the base of pier4 for the nine-span bridge system supported on (a) soil pro-file 1 (soft soil), and (b) soil profile 2 (stiff soil) during the1989 Loma Prieta earthquake; comparing the responses ob-tained from the continuum, the substructure, and the fixed basemodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Figure 7.26 (a) Force-deflection response in the transverse direction, and(b) moment-rotation response along the axis x at the pile capof the MRO; comparing the response in the continuum model,the substructure model, and the reported values in the literature(Douglas et al., 1991; Zhang and Markis, 2002a). . . . . . . 183Figure 7.27 (a) Force-deflection responses in the longitudinal direction, and(b) force-deflection responses in the transverse direction at theabutment of the MRO; comparing the response in the contin-uum model, the substructure model, and the reported valuesin the literature (Douglas et al., 1991; Wilson and Tan, 1990a;Zhang and Markis, 2002a). . . . . . . . . . . . . . . . . . . 184Figure 7.28 Schematic of the substructure model of the MRO in whichstiffness of the springs at the pile cap and at the abutmentsare determined from the pushover analysis of the bridge con-tinuum model. Unloading-reloading paths are simulated usingthe Masing rule. . . . . . . . . . . . . . . . . . . . . . . . . . 185xxvFigure 7.29 Acceleration response spectrum (for 5% damping) of the mo-tions at five different locations of the MRO during the 2010El Mayor-Cucapah earthquake; investigating the level of im-provement of substructure model when stiffness of the springsare obtained from the bridge continuum model. . . . . . . . . 186Figure 7.30 Time histories of pier top displacements with respect to thepile cap of the MRO during the 2010 El Mayor-Cucapah earth-quake; investigating the level of improvement of substructuremodel when stiffness of the springs are obtained from the bridgecontinuum model. . . . . . . . . . . . . . . . . . . . . . . . . 187Figure 7.31 Time histories of bending moment induced at the pier baseof the MRO during the 2010 El Mayor-Cucapah earthquake;investigating the level of improvement of substructure modelwhen stiffness of the springs are obtained from the bridge con-tinuum model. . . . . . . . . . . . . . . . . . . . . . . . . . 188Figure 7.32 Comparing the acceleration response spectrum (for 5% damp-ing) of the motions computed in the continuum model at (a) thetop of the abutment wall, and (b) the top of the pile cap withthe acceleration response spectrum of the motions computed inthe free-field, i.e., the input motions for the substructure model. 189Figure 7.33 Time histories of bending moment induced at the pier base ofthe MRO during the 2010 El Mayor-Cucapah earthquake; in-vestigating the effect of the soil mass on the overall responseof the bridge structure (the assigned soil mass at the two endsof the bridge deck in Case I: 0.0, Case II: 353.3 ton, and CaseIII: 706.6 ton). . . . . . . . . . . . . . . . . . . . . . . . . . 190Figure A.1 Schematic of the parallel computing process. . . . . . . . . . 212Figure A.2 Analysis execution time of the MRO continuum model underthe 1979 Imperial Valley earthquake using different number ofprocessors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 213xxviFigure A.3 Cloud cost and analysis execution time of the MRO continuummodel for a suite of ten ground motions using different numberof processors. . . . . . . . . . . . . . . . . . . . . . . . . . . 213Figure B.1 Schematic of the 27 tests listed in Table. 6.1. . . . . . . . . . 215Figure C.1 Schematic of the centrifuge tests which are simulated usingspring and continuum modeling approaches (dimensions arenot to scale). . . . . . . . . . . . . . . . . . . . . . . . . . . 225Figure D.1 Time histories of fault-parallel and fault-normal componentsof the input ground motions for the three-span bridge. TheFault parallel and fault normal components are applied in lon-gitudinal and transverse directions, respectively. . . . . . . . . 227Figure D.2 Time histories of relative displacements of the pier top withrespect to the pier base (for pier 1) of the three-span bridgein the longitudinal and transverse directions for the three-spanbridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231Figure D.3 Time histories of shear forces induced at the pier base (pier1) of the three-span bridge in the longitudinal and transversedirections using the continuum modeling method. . . . . . . . 235Figure D.4 Time histories of bending moments induced at the pier base(pier 1) of the three-span bridge in the longitudinal and trans-verse directions using the continuum modeling method. . . . . 239Figure D.5 Time histories of forces induced at the abutment (abutment 1)of the three-span bridge in the longitudinal and transverse di-rections using the continuum modeling method. . . . . . . . . 243Figure D.6 Comparing acceleration response spectra (for 5% damping) ofthe motion computed at the middle of the three-span bridgedeck with ARS of the input motion in the longitudinal direction. 247Figure D.7 Comparing acceleration response spectra (for 5% damping) ofthe motion computed at the middle of the three-span bridgedeck with ARS of the input motion in the transverse direction. 249xxviiFigure D.8 Time histories of fault-parallel and fault-normal componentsof the input ground motions for the nine-span bridge (decon-volved motions at the depth of 30.0 m). . . . . . . . . . . . . 252Figure D.9 Time histories of relative displacement of the pier top with re-spect to the pier base of piers 1, 4, 6, and 8 for the nine-spanbridge supported on soil profile 1. . . . . . . . . . . . . . . . 254Figure D.10 Time histories of relative displacement of the pier top with re-spect to the pier base of piers 1, 4, 6, and 8 for the nine-spanbridge supported on soil profile 2. . . . . . . . . . . . . . . . 256Figure D.11 Time histories of shear force induced at the bases of piers 1, 4,6, and 8 of the nine-span bridge supported on soil profile 1. . . 258Figure D.12 Time histories of shear force induced at the bases of piers 1, 4,6, and 8 of the nine-span bridge supported on soil profile 2. . . 260Figure D.13 Time histories of bending moment induced at the bases of piers1, 4, 6, and 8 of the nine-span bridge supported on soil profile 1. 262Figure D.14 Time histories of bending moment induced at the bases of piers1, 4, 6, and 8 of the nine-span bridge supported on soil profile 2. 264Figure D.15 Time histories of forces induced at the abutment (abutment 1)of the nine-span bridge supported on soil profile 1. . . . . . . 266Figure D.16 Time histories of forces induced at the abutment (abutment 1)of the nine-span bridge supported on soil profile 2. . . . . . . 268Figure E.1 Time histories of relative displacements of the pier top withrespect to the pier base (for pier 1) of the three-span bridge inthe longitudinal and transverse directions; comparing the re-sults obtained from the substructure model with those obtainedfrom the continuum model. . . . . . . . . . . . . . . . . . . . 276Figure E.2 Time histories of shear forces induced at the pier base (pier 1)of the three-span bridge in the longitudinal and transverse di-rections; comparing the results obtained from the substructuremodel with those obtained from the continuum model. . . . . 280xxviiiFigure E.3 Time histories of bending moments induced at the pier base(pier 1) of the three-span bridge in the longitudinal and trans-verse directions; comparing the results obtained from the sub-structure model with those obtained from the continuum model. 284Figure E.4 Time histories of forces induced at the abutment (abutment 1)of the three-span bridge in the longitudinal and transverse di-rections; comparing the results obtained from the substructuremodel with those obtained from the continuum model. . . . . 288Figure E.5 Time histories of relative displacement of the pier top with re-spect to the base of pier 4 of the nine-span bridge supportedon soil profile 1; comparing the results obtained from the sub-structure and the fixed base model with those obtained fromthe continuum model. . . . . . . . . . . . . . . . . . . . . . . 293Figure E.6 Time histories of relative displacement of the pier top with re-spect to the base of pier 4 of the nine-span bridge supportedon soil profile 2; comparing the results obtained from the sub-structure and the fixed base model with those obtained fromthe continuum model. . . . . . . . . . . . . . . . . . . . . . . 295Figure E.7 Time histories of shear force induced at the base of pier 4 ofthe nine-span bridge supported on soil profile 1; comparing theresults obtained from the substructure and the fixed base modelwith those obtained from the continuum model. . . . . . . . . 297Figure E.8 Time histories of shear force induced at the base of pier 4 ofthe nine-span bridge supported on soil profile 2; comparing theresults obtained from the substructure and the fixed base modelwith those obtained from the continuum model. . . . . . . . . 299Figure E.9 Time histories of bending moment induced at the base of pier 4of the nine-span bridge supported on soil profile 1; comparingthe results obtained from the substructure and the fixed basemodel with those obtained from the continuum model. . . . . 301xxixFigure E.10 Time histories of bending moment induced at the base of pier 4of the nine-span bridge supported on soil profile 2; comparingthe results obtained from the substructure and the fixed basemodel with those obtained from the continuum model. . . . . 303Figure E.11 Time histories of forces induced at the abutment 1 of the nine-span bridge supported on soil profile 1 in the longitudinal andtransverse directions; comparing the results obtained from thesubstructure and the fixed base model with those obtained fromthe continuum model. . . . . . . . . . . . . . . . . . . . . . . 305Figure E.12 Time histories of forces induced at the abutment 1 of the nine-span bridge supported on soil profile 2 in the longitudinal andtransverse directions; comparing the results obtained from thesubstructure and the fixed base model with those obtained fromthe continuum model. . . . . . . . . . . . . . . . . . . . . . . 307xxxChapter 1IntroductionScience can amuse and fascinate us all, but it is engineering thatchanges the world. — Isaac Asimov (1988)1.1 OverviewIn past earthquakes, bridge structures have suffered severe damages mainly becauseof inadequate seismic designs. Fig. 1.1a shows the collapse of the Cypress Viaductof Interstate 880 in Oakland, California after the 1989 Loma Prieta earthquake.The main cause of the collapse was the failure of the bridge piers due to lackof reinforcing rebars in the pier cross section. Fig. 1.1b shows the failure of thepiers and the deck of the Santa Monica Freeway in Los Angeles after the 1994Northridge earthquake. These bridges could have survived those earthquakes ifgeotechnical and structural demands (i.e., deflections and induced forces) wereappropriately estimated. This indeed requires detailed numerical modeling of bothgeotechnical and structural components for which the seismic interaction and thenonlinear hysteretic response are taken into account in analyses.During an earthquake event, a structure and the underlaying soil considerablyinfluence the seismic response of each other. A structure supported on soft groundand the same structure supported on rock respond differently under an identicalearthquake motion. The structure on the former soil may undergo relatively smaller1forces but larger displacements because the foundation of the structure can trans-late and rotate. This phenomenon is called soil-structure-interaction (SSI) whichconsists of two components: kinematic and inertial interactions. At the onset ofearthquake excitation, kinematic interaction occurs because of the differential lat-eral and vertical displacements of the ground which induce lateral and verticalforces and bending moments to the foundation of the structure. Inertial interactionoccurs because of the inertia forces of the structure which are transferred to thefoundation and ultimately to the soil.Dynamic SSI was first studied by Sezawa and Kanai (1935) who modeled athin cylindrical rod (representative of a structure) with a hemispherical tip at thebase embedded in a homogenous elastic half-space. Since then, various studieswere performed in order to achieve better understanding of the mechanisms in-volved in SSI. In the past four decades, significant progress has been made in thisfield mainly for the needs of the offshore and nuclear power industries. In somecases SSI reduces structural demands, and in other cases SSI increases the struc-tural demands according to the literature. Its effects are most significant for stiffstructures supported on relatively soft soils and smallest for soft structures foundedon relatively stiff soils (Kramer, 1996). Inclusion of SSI in numerical analyseshas been found to be essential for realistic simulation of many civil engineeringproblems not only in the research community but also in the engineering practice.Two approaches are frequently used to simulate soil-structure systems: springand continuum modeling methods. In the spring method, the stiffness and damp-ing of the foundation soil is accounted for in analyses by a set of linear/nonlinearuncoupled springs and dashpots. This results in substantial savings of time forcomputation and enables the engineers to perform parametric studies of the prob-lem at hand. Although simple, the approach suffers several disadvantages. Finn(2005) showed that the idealization of the soil domain with uncoupled springs anddashpots results in the inability of the approach to appropriately simulate the kine-matic and inertial interactions. Furthermore, characterization of the spring modelsis always associated with considerable level of uncertainty. For pile foundations,which are of the main interest in this thesis, a common practical approach is to fol-low the guidelines of American Petroleum Institute (API, 2007). Although theseguidelines are based on the results of static or slow cyclic loading tests, they have2been widely used in practice for seismic problems. This is in spite of the findings ofseveral researchers such as Murchison and O’Neill (1984), Gazioglu and O’Neill(1984), Finn (2005), and most recently Choi et al. (2013) who argued against thevalidity of these curves and reported significant levels of error in estimation ofstatic or seismic responses of pile foundations when API curves are adopted inthe analyses. Nevertheless, practitioners still prefer to use these springs not only insmall-scale SSI problems such as a single pile but also in large-scale problems suchas bridge system because the approach is very simple, requires less computationalefforts, and other efficient alternatives are lacking.In the continuum modeling method, structural components and the underlyingsoil domain are modeled in a fully coupled manner without relying on any ancillarymodels such as springs. Therefore, compared to the spring method it potentiallyprovides more powerful means for obtaining realistic estimates of kinematic andinertial interactions. Successful implementation of this technique mainly relies onproviding appropriate constitutive models that are capable of simulating the actualnonlinear hysteretic behavior of the foundation soil and the structural elements.Nonlinear dynamic analysis of continuum models can be a challenging and tediousprocess, which requires significant computational capabilities. One can expeditethe large-scale simulations by taking advantage of high performance computingresources, such as multicore computers, computer clusters, and cloud computingservices.With the advances in the continuum modeling method, the validity of the springmethod was argued in the research communities. Finn (2005) used both methodsfor dynamic analysis of soil-pile interaction, and he concluded that the API springsare very unreliable in predicting the seismic response of piles. The study showedthat continuum model of the soil-pile system better estimates the kinematic and in-ertial interactions. Despite the limitations and uncertainties of the spring method,it has been widely used in practice. After four decades of intensive studies therestill exists a large gap in SSI simulation tools used between SSI specialists andpracticing civil engineers (Tyapin, 2007). From reviewing the literature about dy-namic SSI analysis of bridges, the existing gaps in knowledge can be categorizedas follows:3• Bridge systems have been rarely modeled using 3D continuum modelingmethod because of the complexity of the analysis and the computational de-mands.• The accuracy of the spring method has been assessed for static and dynamicanalyses of small-scale problems such as pile foundations, while its accuracyhas never been adequately investigated for the analysis of large-scale SSIproblems such as bridges.1.2 Objectives and original featuresThe spring method has been shown to be computationally efficient for analysis ofbridge systems (e.g., Tongaonkar and Jangid (2003); Zhang and Markis (2002b);however, it has never been validated by comparing the results with field measure-ments or those of full-scale analyses. The only serious attempt to evaluate themethod was made by Zhang and Markis (2002b) who studied the seismic responsesof highway overcrossings in California using the spring method. They showed thatthe spring method adequately captures the recorded responses at the middle ofthe bridge. The evaluation was limited to only one point at the bridge structure,and the validity of the computed responses were not assessed at other locations ofthe bridge. Therefore, a research study is needed to comprehensively evaluate thespring method and quantify the levels of error in order to inform bridge engineersof the validity of their simulation results.The first objective of this research is to develop full-scale three-dimensional(3D) continuum models of three different multi-span bridges in order to gener-ate the required benchmark data for evaluation of the state of practice for dynamicanalysis of bridges. The other purpose of this investigation is to illustrate the poten-tial for further practical applications of the large-scale continuum models in bridgeengineering with the aid of recent advances in material constitutive modeling andparallel computing environments. More specific components and original featuresof this part of the thesis are as follows:• Advanced constitutive models developed in previous studies are used to sim-ulate nonlinear hysteretic behavior of the foundation soil and the bridgepiers.4• Finite element formulation with an implicit time integration scheme is adoptedfor nonlinear time history analysis.• A high-performance parallel computing tool is used to speed up the execu-tion time of analysis.• The 3D continuum model is validated by simulating the seismic responseof a well-instrumented bridge during past earthquake events recorded at thebridge site.The second objective is to simulate the same bridge systems using spring method,namely substructuring method, and test the adequacy of the method. The state ofpractice is followed for obtaining the constants of springs and dashpots to modelSSI at piles and abutments. Original features of this module are as follows:• A step-by-step procedure for determination of the kinematic input motionsand the pile group and the abutment dynamic stiffnesses is presented.• Guidelines of API (2007), AASHTO (2012), and Caltrans (2013) are used tocharacterize the springs at piles and abutments.• The substructuring method is used for dynamic analysis of the same multi-span bridge systems. The results are compared with the results of the cor-responding continuum models, and the level of errors in predicting seismicresponse of the bridges is quantified.1.3 Structure of presentationThe organization of the thesis for fulfilling the research objectives is as follows:Chapter 2 provides a literature review of the most relevant research works onnumerical methodologies used for simulation of SSI.Chapter 3 explains the common practical approach, namely the substructuringmethod, using springs for simulating seismic performance of bridge systems. Astep-by-step procedure of developing the bridge substructure model is presented.Chapter 4 presents the elements of the continuum modeling method used in thisthesis. The finite element formulation and the finite element program are firstly dis-cussed, and thereafter, the constitutive models used for simulating the stress-strain5response of soil, concrete, and steel are described in detail. The methodology ofsimulating soil-pile interaction in a continuum model is explained, and the analysisprocedures are presentedChapter 5 describes the procedure of developing and analyzing the 3D contin-uum models of two-, three-, and nine-span bridge. The constitutive models used tosimulate the stress-strain response of soils, concrete and steel are discussed. Thecontinuum models are used to compute the baseline data which is required for theevaluation of the substructuring method. Methodologies used for the bridge sim-ulation are validated by simulating the seismic response of Meloland Road Over-pass, located in California, US, when subjected to the 1979 Imperial Valley and2010 El Mayor-Cucapah earthquakes. Amongst the recorded earthquake eventsat the bridge site, these two events caused the strongest shaking. The three-spanbridge model estimates seismic responses of a prototype bridge supported on thesoil profile in the MRO site. The bridge is subjected to a suite of actual groundmotions with different PGAs and predominant periods. The ground motions areselected in a way that induce different levels of inertial and kinematic interactions.The nine-span bridge model estimates seismic responses of a prototype large-scalebridge supported on two soil profiles representing soft and stiff grounds in Vancou-ver, Canada. The bridge is subjected to ground motions which are linearly matchedto the uniform hazard spectrum (UHS) of Vancouver in the period range of 0.1 to1 s. The selected UHS has a probability of exceedance of 2% in 50 years.Chapter 6 evaluates the API p-y springs used for analysis of laterally-loadedsingle pile foundations. As part of the substructuring method, the API curves playa key role in determination of pile group stiffnesses. To this end, twenty-seven full-scale static field tests and centrifuge tests performed on single piles are simulatedusing API springs. In addition, two dynamic centrifuge tests performed on singlepiles subjected to earthquake shakings are simulated. The API springs, when usedfor simple static and dynamic SSI problems, are preliminary evaluated by compar-ing the results of the analyses against the recorded measurements.Chapter 7 first presents the procedure of developing and analyzing the sub-structure models of the two-, three-, and nine-span bridge systems. All steps ofthe substructuring method for these bridges are explained in detail. The chapterfollows by investigating how the results from the substructuring method compare6with those from the more exact continuum model analyses conducted in Chapter 5.The capabilities of the substructuring method in predicting the spectral response ofthe bridge, the pier deflections, the shear forces and the bending moments inducedat the pier base, and the forces induced to the abutments are evaluated against theresponses from the continuum models. The levels of differences are quantified, anddetailed comments are made on the application of the substructuring method.Finally, Chapter 8 presents a summary of the thesis, the conclusions drawnfrom this research, and directions for some future works.71.4 Figures(a)(b)Figure 1.1: (a) The Cypress Viaduct, Interstate 880 in Oakland, Californiaafter the 1989 Loma Prieta earthquake (photo credit: U.S. GeologicalSurvey), and (b) the Santa Monica Freeway, Los Angeles, after the 1994Northridge earthquake (photo credit: Associated Press/Lois Bernstein).8Chapter 2Literature reviewIf I have seen farther it is by standing on the shoulders of Giants.— Sir Isaac Newton (1855)2.1 IntroductionDue to the common perception that to ignore soil-structure-interaction (SSI) is con-servative, the SSI provisions in codes are still voluntary and are often neglected inpractice. In the past decades, several studies investigated the effects of SSI on theoverall seismic response of the structures. Some reported the beneficial effectsand some reported the detrimental effects of SSI. Kappos et al. (2002) showedthat inclusion of SSI in the numerical analysis of irregular R/C bridges bridges re-duces the induced seismic forces. On the contrary, Tongaonkar and Jangid (2003)reported that the abutment deformations of a three-span bridge system may be un-derestimated if SSI is not included in the numerical analysis. Jeremic et al. (2004)found that depending on the characteristics of the input earthquake motion, thefoundation soil and the structure, the inclusion of SSI in the analysis can have ei-ther beneficial or detrimental effects on the seismic response of structure. Finn(2005) studied the SSI effects on a three-span AASHTO model bridge and showedthat kinematic and inertial interactions are not simulated appropriately if the bridgestructure is fixed at supports without including SSI. He found that fixed base mod-els are adequate only when the ratio of the superstructure stiffness to the founda-9tion stiffness is small so that the effect of inertial interaction on system frequencyis negligible compared to the effect of kinematic interaction.Recognizing the importance of SSI effects, design guidelines such as FEMA273 (1997), AASHTO (2012), and Caltrans (2013) suggest that the flexibility ofthe foundation should be included in the numerical model. However, these de-sign guidelines do not prescribe how to account for SSI in the numerical analyses.Kramer (1996) divided the methods for the analysis of SSI into two categories: di-rect method and multistep method. These methods are discussed in the followingsection.2.2 Existing approaches for dynamic SSI analysis ofbridges2.2.1 Continuum model (Direct method)In this method, the general equation of motion of a soil-structure system (Eq. 2.1) issolved in a fully-coupled manner, without resorting to any independent calculationsof ground and structure responses. The method is referred to as three-dimensional(3D) continuum modeling approach hereafter in this thesis.MU¨+CY˙+KY= 0 (2.1)Where M, C, and K are the whole system mass, damping and stiffness matrices, re-spectively, U is the absolute displacement vector, and Y is the relative displacementvector with respect to the base of the system.The finite element method potentially provides one of the most powerful meansfor solving the above equation using standard step-by-step numerical integrationprocedures. To obtain realistic estimates of inertial and kinematic interactions be-tween structures and the foundation soils, advanced constitutive models can beadopted in solving the equation. Many researchers such as Roscoe and Burland(1968), Prevost (1985), Pastor and Zienkiewicz (1986), Jefferies (1993), Newsonand Davies (1996), Beaty and Byrne (1998), Yang et al. (2003), Dafalias and Man-zari (2004), Dafalias et al. (2006), Taiebat and Dafalias (2008), Gu et al. (2011),10and Seidalinov and Taiebat (2014) did valuable research on constitutive model-ing of soils and proposed models by which soil nonlinear hysteretic stress-strainrelationship is properly approximated. Use of these constitutive models, whenproperly calibrated against site conditions, results in appropriate estimates of thestiffness matrix (K) and the deflection vector (Y).The 3D continuum modeling method is rarely used in bridge engineering, be-cause the nonlinear analysis has known to be tedious, time consuming, and in somecases impractical. There are very few studies in the literature studying the seismicresponse of bridges using the continuum modeling approach. The work of Mc-Callen and Romstad (1994) is one of the first studies in this field. They developeda large three-dimensional finite element model for the Painter Street Overpass us-ing equivalent linear method to represent the nonlinear hysteric behavior of thefoundation soil. Their 3D continuum model is shown in Fig. 2.1a. Jeremic et al.(2009) studied the influence of non-uniform soil conditions on a prototype concretebridge. In their study, the whole bridge system was not simulated owing to compu-tational limitations; the bridge deck was modeled with linear elastic beam-columnelements, and the bridge abutments were not simulated under the assumption thatthe bridge deck was disconnected from the abutments (see in Fig. 2.1b). Kwonand Elnashai (2008) modeled the Meloland Road Overpass (MRO) for which thegeotechnical components, including the embankments, abutments, and pile groups,were modeled in one platform, and the structural components, including the bridgedeck and the pier, were modeled in another platform (see in Fig. 2.1c). Nonlineardynamic analysis was conducted in a sequential manner by applying the outputsfrom one platform as the inputs to the other one. Elgamal et al. (2008) developeda continuum model of the Humboldt Bay Middle Channel Bridge using nonlinearmodels of soil and structural materials and studied the effects of permanent grounddeformation on seismic response of the bridge (see in Fig. 2.1d). Later, Lu et al.(2011) used the same continuum model of the bridge and showed that computa-tional challenges of the nonlinear dynamic analysis can be greatly reduced usinghigh-performance computing techniques. Using a parallel computing environment,they reduced analysis execution time from 24 to 9 hours.112.2.2 Substructure model (Multistep method)This method, also called multistep method, relies on the principles of superpositionassuming a linear elastic system and isolates the two components of SSI: kinematicinteraction and inertial interaction. Based on this, the Eq. 2.1 is equivalent to thefollowing two matrix equations (Kausel et al., 1978),M1U¨1 +CY˙1 +KY1 = 0 (2.2)MY¨2 +CY˙2 +KY2 =−M2U¨1 (2.3)where U1 = Y1 +Ug, U = U1 +Y2, Y = Y1 +Y2, and M = M1 +M2, M1 is themass of the system excluding the mass of the foundation-structure, and M2 is themass of the system excluding the mass of the soil, Ug is the input ground motionapplied to the base of model. In Eq. 2.2, mass of the foundation-structure systemis excluded but its stiffness is included in matrix K. The solution of this equationis the kinematic motion of the foundation-structure system (Y1) because no iner-tial interaction occurs between the massless system and the foundation soil. Theresulting response referred to as the kinematic interaction. The resulting Y1 is thenused in Eq. 2.3 to calculate Y2 which is the deformations due to inertial interac-tions. This equation does not include the mass of the foundation-structure system,and applies fictitious inertial forces, i.e., M2U¨1, to the structure. The resultingresponse is referred to as the inertial interaction.According to Kausel et al. (1978), if the combination of foundation and struc-ture is assumed to be rigid, the massless foundation-structure system moves as arigid body in the kinematic interaction analysis (Eq. 2.2). Therefore, Eq. 2.2 in factrepresents a massless rigid foundation subjected to the same input seismic load-ing as the global system. Provided that the assumptions of linear system and rigidfoundation are pertinent, Kausel et al. (1978) proposed the following three-stepsolution for practical purposes:• Determination of the motion of the massless foundation which includes trans-lational and rotational components. This step is called kinematic interactionanalysis.• Determination of the foundation dynamic stiffnesses, i.e., frequency-dependent12impedances.• Dynamic analysis of the structure supported on the impedances and sub-jected to the motions determined in the first step. This step is called inertialinteraction analysis.The first two steps require continuum modeling of the problem which maymake the three-step solution unappealing for practical applications. FollowingKausel et al. (1978), the first step can be reasonably simplified by adopting one-dimensional wave-propagation theory, without considering the presence of the rigidfoundation, and the second step can be simplified by using proposed equations forcalculation of dynamic stiffnesses based on analytical solutions. This three-stepsolution is referred to as substructuring method in the literature. The method is themost popular approach in both research communities and engineering practice toaddress SSI because of its simplicity and computational efficiency.This section mainly focuses on description of the fundamentals of the multi-step method applicable for simulation of SSI for any type of structure. Detailsof the kinematic and inertial interaction analyses (Eqs. 2.2 and 2.3) for numericalmodeling of a bridge system will be explained later in Chapter 3.Zhang and Markis (2002b) presented a systematic procedure on the basis ofsubstructuring method for the seismic response analysis of highway overcross-ings located in California. The schematic configuration of their model is shownin Fig. 2.3a. Tongaonkar and Jangid (2003) studied the effects of SSI on the peakresponses of a three-span continuous deck bridge seismically isolated by the elas-tomeric bearings (see their model in Fig. 2.3b). A 3D analysis of an AASHTOmodel bridge (Fig. 2.3c) was conducted by Finn (2005) using the substructuringmethod and demonstrated the strong dependence of the response on the coupledinertial interaction of the superstructure. As shown in Fig. 2.3b,c a stick model isused to simulate the bridge superstructure. The stick model is a collection of beamelements with equivalent cross-section properties representing the bridge deck, thegirders, and the piers. Shamsabadi et al. (2007) studied the seismic response of atypical two-span overpasses using the substructuring method. The bottom of thecolumn was modeled as a pin connection without considering SSI, but the SSImodel at both abutments in the longitudinal direction consists of a nonlinear spring13in series with a gap element (see in Fig. 2.3d).The idea of substituting the continuum soil domain with springs and dashpotswas first suggested by Lysmer (1965) assuming a vertical spring and a verticaldashpot for a single degree of freedom (SDOF) system. Although the model issimple, the constants of springs and dashpots are very difficult to determine. Usingthe continuum modeling method (direct method), Finn (2005) investigated the lat-eral stiffness of a 4×4 pile foundation with a single degree of freedom on the topof the pile group representing the bridge structure. Fig. 2.7 illustrates how signifi-cantly the lateral stiffness of a foundation system may vary due to both nonlinearityof soil and inertial interaction between the superstructure and the foundation soil.Nevertheless, for practical purposes, linear response analyses are the most commonanalysis types in design of bridges (Lam et al., 2007). In these analyses a linearstiffness representation of the foundation is incorporated in the numerical model.Traditionally, the pile foundation and the supporting soil media are analyzed inthe frequency domain where the piles and the soil are assumed to be elastic. Thedynamic stiffness of the pile foundation is established as an output of the analy-sis. The dynamic stiffness (K) is composed of real component, K, and imaginarycomponent, ωC given by,K= K+ iωC (2.4)where K and C can be interpreted as spring and dashpot constants.The dynamicstiffness is a function of angular frequency of the input motion (ω).The dashpot coefficient reflects two types of damping in the system: radiationdamping and material damping. The former is the dissipated energy due to thepropagation of motion away from the foundation, and the latter is the dissipatedenergy due to the yielding of soil material and significant hysteretic behavior. Whatfollows is a brief description of the methodologies suggested in the literature aswell as codes/guidelines to calculate spring and dashpot coefficients of foundationand abutment systems.The dynamic stiffness of a 3D pile foundation is composed of 6 diagonal ele-ments representing lateral, vertical, rocking, and torsional impedances. There arealso four more off-diagonal elements which represent the coupling effects betweenthe lateral displacements and rocking of the foundation. Eq. 2.5 presents the ele-14ments of the 6×6 dynamic stiffness matrix,K=K11 0 0 0 K15 00 K22 0 −K24 0 00 0 K33 0 0 00 −K42 0 K44 0 0K51 0 0 0 K55 00 0 0 0 0 K66+iωC11 0 0 0 C15 00 C22 0 −C24 0 00 0 C33 0 0 00 −C42 0 C44 0 0C51 0 0 0 C55 00 0 0 0 0 C66(2.5)where K11 and K22 are the lateral stiffnesses along the x and y axes, respectively,and K33 is the vertical stiffness along the z axis. K44 and K55 are the rockingstiffnesses about the y and x axes, respectively, and K66 is the torsional stiffnessabout the z axis. K15 = K51 and K24 = K42 are the cross coupling stiffnesses, andω is the angular frequency of the input motion. Similar definitions apply to thecorresponding damping coefficients presented in Eq. 2.5. For numerical modelingof a boundary value problem, the stiffness and damping matrices can be representedby eight one-dimensional elements each composed of a linear spring in parallelwith a dashpot. Fig. 2.8 illustrates the outline of the springs and dashpots for alldegrees of freedom.According to Gazetas (1991), dynamic stiffness of a foundation system is afunction of the shape of the foundation, material properties of the underlying soil,the amount of embedment (i.e., surface/embedded mat foundation, piled founda-tion), and the frequency of the input dynamic load. Using combined analytical-numerical methods, Kausel et al. (1978), Gazetas (1991) and Mylonakis et al.(2006) presented practical series of tables and charts for estimation of translational,rotational, and cross-coupling spring and dashpot coefficients for mat foundations15and single piles embedded in elastic soil deposits. The presented equations becamethe basis for the procedure proposed by Makris et al. (1994) to obtain dynamicstiffness of pile-supported foundation system of bridges. The proposed values areapplicable for simulating simple SSI problems where the foundation soil remainselastic during static or seismic loading. However, none of these methods providesany simple procedure to establish stiffness matrices that vary with the deflection ofthe foundation. To represent the nonlinear inelastic response of the foundation soiland its interaction with the superstructure, Zafir (2002) and Lam et al. (2007) usedsecant stiffness values at a representative displacement level expected during theearthquake. Details of this methodology will be discussed later in Chapter 3.The beam on a nonlinear Winkler foundation (BNWF) approach can be con-sidered as an improvement on the linear spring models. In this approach, the foun-dation is simulated as a beam supported on a series of discrete nonlinear springs(representing the underlying soil). The approach is mainly used in practice to ac-count for nonlinear response of soil-structure systems. Although simple and prac-tical, characterization of nonlinear springs is difficult and challenging. AmericanPetroleum Institute (API, 2007) and California Department of Transportation (Cal-trans, 2013) provide some ready-to-use nonlinear backbone curves for the analysesof soil-pile interaction and embankment-abutment interaction, respectively. In thefollowing sections, these guidelines together with other existing approaches forsimulating SSI at pile foundations and abutments are discussed.2.3 Modeling soil-pile interaction2.3.1 API guidelines for soil-pile interactionFor laterally loaded piles in sands, API (2007) recommends the following equation,namely p-y curve, at any specific depth H,p = Apu tanh(kHyApu) (2.6)where p is the lateral resistance per unit length of pile at lateral pile deflection y,k is the modulus of subgrade reaction that can be determined from Fig. 2.4a as a16function of internal friction, φ , and pu is the ultimate soil lateral resistance, A isa constant depending on whether the loading is static or slow cyclic. Two typesof soil resistance are assumed to exist when pile pushes into the soil: (a) wedgefailure: near the ground surface, a passive wedge resists the lateral movement ofthe pile; (b) flow failure: at some distance below the ground, soil tends to flowhorizontally around the pile rather than move upward in a wedge due to excessiveoverburden pressure. Based on these two failure mechanisms, API (2007) rec-ommends the smaller of the values given by the following two equations as theultimate soil lateral resistance (pu),{pus = (C1H +C2D)γ ′H, wedge failurepud =C3Dγ ′H, flow failure(2.7)where γ ′ is effective soil weight, D average pile diameter from surface to depth,C1, C2, and C3 are coefficients that can be determined from Fig. 2.4b as a functionof φ .To determine p-y curves for laterally loaded piles in soft clays API (2007)recommends using the following relationp =0.5pu(yyc)1/3,yyc< 8pu,yyc≥ 8(2.8)where yc = 2.5εcD, εc is the strain which occurs at one-half the maximum stresson laboratory unconsolidated undrained compression tests of undisturbed soil sam-ples, and D is the pile diameter. The initial stiffness that is implied by the p-y curve,Eq. 2.8, is infinite. Fig. 2.5 presents the schematic shape of the p-y for soft clays.According to API (2007), pu is the smaller of the values given by the followingequations,{pus = 3cD+ γ ′XD+ JcX , X < XR wedge failurepud = 9cD, X ≥ XR flow failure(2.9)where c is undrained shear strength for undisturbed clays, X is the depth belowground surface, J is a dimensionless empirical constant with values ranging from170.25 to 0.5, and XR is the depth at which the type of failure mechanism changesfrom wedge to flow. Given that pus = pud , XR can be calculated as follows,XR =6Dγ ′DcJ(2.10)The recommendations of API (2007) are all based on the measurements of fieldtests performed on piles under static and slow cyclic loading. It is to be noted thatthe piles were not tested under fast cyclic or seismic loadings. API (2007) does notprovide any equation for p-y curves of piles embedded in stiff clays, but it refers tothe procedure proposed by Reese et al. (1975). Details about their procedure andthe proposed methodologies for determining p-y curves in rocks, multi-layered soildeposits, and slopes can be found in Reese et al. (2004).API (2007) provides guidelines for determining t-z and Q-z curves, where tis the axial pile shear stress, z is the pile axial displacement, and Q is the tip endbearing. These guidelines are based on the study of Meyer et al. (1975). Therecommended nonlinear t-z and Q-z backbone curves for sands and clays are pre-sented in Table 2.2 and Table 2.3, respectively. The characteristic shapes of the t-zand Q-z are presented in Fig. 2.6.Other design guidelines such as FEMA 451 (2006), AASHTO (2012) andCanadian Foundation Engineering Manual (2006) recommends API backbone curvesfor characterizing the nonlinear springs. Although these backbone curves are widelybeing used in practice, several researchers have reported the unreliability of themfor different piles and soil conditions. Murchison and O’Neill (1984) and Gaziogluand O’Neill (1984) can be considered as the first researchers studying the range ofapplicability of API curves. Murchison and O’Neill (1984) studied 24 full-scaletests on piles in cohesionless soils; 14 static tests and 10 slow cyclic tests on singlepiles. They concluded that the API curves are not adequate for the analysis of staticor slow cyclic loading tests. Gazioglu and O’Neill (1984) conducted similar studieson 30 full-scale tests in clayey soils; 21 static and 9 slow cyclic tests. They reportedthat deflections and bending moments are poorly estimated when API curves areused in their numerical analyses. In addition, Zhang et al. (2005) showed that APIrecommendations for calculating ultimate resistance of cohesionless soils underes-18timate the actual ultimate lateral resistance at shallower depth but overestimate theactual ultimate lateral resistance at deeper depths. Fig. 2.9 shows the distributionof the measured maximum earth pressure pmax over the API ultimate soil lateralresistance with depth. It is clearly shown that the measured values are poorly pre-dicted. McGann et al. (2011) also reported that the initial stiffness and ultimatesoil resistance are both significantly overestimated if the recommendations of APIare employed for static analysis of a single pile embedded in cohesionless soils.Kim and Jeong (2011) concluded that for piles embedded in clayey soil, using APIcurves results in significant overestimation of pile lateral displacements and bend-ing moment profile. Based on the results of some dynamic centrifuge tests, Choiet al. (2013) reported that the API curves are significantly different from the ex-perimental ones; the ultimate soil resistance is underestimated, while the subgradereaction modulus is overestimated at the small deflections of piles.Another challenge in determination of the foundation dynamic stiffness is toobtain reasonable estimates of pile-soil-pile interaction. When piles act in a group,soil-pile interaction reduces the lateral resistance of the individual piles so that thegroup will generally exhibit less lateral capacity compared to the sum of the lat-eral capacities of the individual piles. The state of practice is to reduce the soillateral resistance p by a constant factor called p-multiplier. API (2007) does notprovide any guidelines to account for pile group effects. Table 2.1 presents thep-multipliers recommended in American Association of State Highway and Trans-portation Officials (AASHTO, 2012). In the table, row 1 refers to the leading rowin the direction of loading, and the next rows refer to the trailing rows. The recom-mendations are only for pile groups with S/D of 3 and 5, where S is the distancebetween center of piles and D is pile diameter. There is no consideration for thenumber of piles, fixity of the pile heads, and the properties of the foundation soil.In practice, for cyclic and seismic problems, the average of the p-multipliers areusually used for considering pile group effects because the direction of loadingchanges repeatedly. This average value is called group reduction factor. The valid-ity of these factors has been argued in the study of Fayyazi et al. (2014). The studyshowed that the guidelines of AASHTO overestimate the group reduction factors,hence the lateral resistance, in larger pile groups and larger spacings, especially forfixed pile head conditions.19Although unreliability of API curves for static problems is well-established,they are also being utilized for modeling soil-structure systems subjected to earth-quake loads. Matlock and Foo (1978) extended the BNWF concept to seismicproblems. They performed dynamic analyses in two consecutive steps: (i) site re-sponse analysis where depth-varying ground motion time histories are calculatedalong the pile foundations, and (ii) nonlinear dynamic analysis of pile where thedepth varying ground motion time histories are applied to the supports of nonlinearsprings. Later, Novak and Sheta (1980) concluded that for dynamic/seismic prob-lems the soil around the pile shaft should be separated into two different zones:the near field (plastic zone) where strong nonlinear soil-pile interaction occurs andthe far field where the soil behavior is primarily linear elastic. Based on theseconcepts, Badoni and Makris (1996), El Naggar and Novak (1996), Wang et al.(1998), Boulanger et al. (1999), Gerolymos and Gazetas (2005), Taciroglu et al.(2006) and Allotey and El Naggar (2008) each proposed macro-element modelswhich were shown to adequately simulate soil-pile interaction. In the following,the macro-element models proposed by Boulanger et al. (1999), Taciroglu et al.(2006), and Allotey and El Naggar (2008) are briefly described.2.3.2 Macro-element models for soil-pile interactionTo obtain realistic estimates of dynamic soil-pile interaction, a macro-elementmodel should be an assembly of four basic elements: a drag element to accountfor friction between the pile and the soil, a gap element to account for contact be-tween the pile and the soil, a dashpot to account for radiation damping, and anelastoplastic p-y, t-z, Q-z element to account for hysteretic response of soil.Boulanger et al. (1999):The macro-element model is composed of an elastic, plastic, and gap elements inseries. The elastic element consists of an elastic spring and a damper in parallelto simulate radiation damping in the system. The plastic element is a nonlinearspring which is rigid (inactive) in the range of−Cr pu < p <Cr pu, where Cr is ratioof p/pu when plastic yielding first occurs in virgin loading, Cr was recommendedto be 0.35 for soft clays and 0.20 for sands. The rigid range of p, which is 2Cr pu,20remains constant and translates with plastic yielding (kinematic hardening). Thegap element consists of a nonlinear closure spring in parallel with a nonlinear dragspring. The closure spring allows for a smooth transition in the load-displacementbehavior as the gap opens or closes. The parameters of the elastic and plastic el-ements are determined using the guidelines of API (2007). Fig. 2.10 shows theschematic of the proposed model and typical cyclic response of its elements.Taciroglu et al. (2006):This model consists of three elements placed in parallel:• Leading-face element is an assembly of a gap subelement and an elastoplas-tic p-y subelement in series. The leading-face element simulates the inter-action between the pile and the soil when the pile pushes into the soil. Theelement is rigid (inactive) under tensile forces. The initial stiffness and theultimate soil lateral resistance are obtained from the guidelines of API.• Rear-face element is identical to the leading-face element. The only dif-ference is the direction of the resisting forces. This element simulates thesoil-pile interaction at the opposite side.• Drag element for which the governing equation is a one-dimensional classi-cal, rate-independent elastic perfectly plastic models. The element simulatesthe frictional forces along the soil-pile interface.These elements and their typical cyclic responses are illustrated in Fig. 2.11.In summary, the macro-element model incorporates hysteretic response of the soil,drag forces, and gap at the soil-structure interface but it does not take into accountradiation damping in the system.Allotey and El Naggar (2008):The model is capable of accounting for different types of unloading and reloadingcurves, cyclic soil stiffness and strength degradation due to increased soil nonlin-earity, sliding, gap, and radiation damping. It is dominantly a compression model,therefore, two elements are needed at each depth for simulating the soil-pile inter-action. Each element is composed of a backbone curve, generalized unload curves21(GUC), standard reload curves (SRC), and direct reload curves (DRC) as shown inFig. 2.12. Similar to the previous models, the guidelines of API (2007) are adoptedfor deriving the backbone curve. DRC simulates soil reactions to the pile movingin gap zones, and GUC and SRC simulate the unloading and reloading curves de-rived from the backbone curve and the extended Masing rule (Pyke, 1979) beyondthe gap zones.Radiation damping is modeled using a nonlinear dashpot placed in parallel withthe nonlinear spring described above. The damping coefficient at each time step iscalculated as a function of the current stiffness of the spring.2.3.3 Spring model used in practiceIn practice, the drag and gap components are often ignored for simplicity. Inaddition, radiation damping is usually neglected for problems in which the soil-structure system undergoes strong shaking. Chako (1995) showed that radiationdamping is negligible compared to material damping when soil behavior is highlyhysteretic with large plastic deformations. Allotey and El Naggar (2008) also re-ported that radiation damping does not influence the seismic response of a sin-gle pile considerably. The schematic of the nonlinear spring model is shown inFig. 2.13. To simulate soil hysteretic behavior, API curves are used as the back-bones for the nonlinear springs to simulate the first loading path, and subsequentloading paths are simulated using the Masing rule. Based on the Masing rule, theunloading and reloading curves is the same as the backbone curve, with the originshifted to the loading reversal point, but is enlarged by a factor of 2.0.2.4 Modeling soil-pile cap interactionThe lateral stiffness of an embedded pile cap can be derived following the pro-cedure used in the study of Zafir (2002). In this procedure, the initial stiffness iscalculated using the procedure presented by Gazetas (1991), and ultimate soil resis-tance is calculated assuming a passive wedge type failure in front of the pile cap.Eq. 3.1 presents the nonlinear force-deflection relationship at the pile cap (Zafir,2002),F =∆1/Kmax +R f∆/Fp(2.11)22where F is the load at deflection ∆, Kmax is the initial stiffness of the pile cap, Fpis the ultimate passive soil resistance, R f is the ratio between the actual and thetheoretical ultimate force and is given by R f = 1−Fp/(Kmax∆max), where ∆maxis the deflection at the ultimate passive soil resistance. Based on the behavior ofretaining walls, ∆max varies from 0.002H to 0.04H (H is the thickness of the pilecap).2.5 Modeling embankment-abutment interactionCaltrans (2013) recommends both nonlinear and bilinear backbone curves for inte-gral abutments in the longitudinal direction, however, it recommends just bilinearbackbone curves for seat-type abutment. These curves were derived from the forcedeflection results of large-scale abutment testings at the University of California,Davis (Maroney, 1995) and the University of California, Los Angeles (Stewartet al., 2007). The guidelines recommended by Caltrans (2013) to determine force-deflection responses for integral and seat-type abutments are presented in the fol-lowing.Integral abutment:According to Caltrans (2013), the nonlinear force-deflection response of integralabutments can be obtained as follows,F(y) =8y1+3yH1.5, for cohesionless embankments8y1+1.3yH, for cohesive embankments(2.12)where F(y) is the induced load in kip per ft of wall width, y is the wall deflectionin inch, and H is the hight of the wall in inch. In the transverse direction of thebridge, Caltrans (2013) recommends a conservative stiffness value of 40.0 kip/inch(7000.0 kN/m) per pile, ignoring the effects of wing-walls.No guideline has been provided for determining the coefficient of dashpots toaccount for energy dissipation at abutments. Since the bilinear backbone curves arenot used in this thesis for integral-type abutments, only nonlinear force-deflectionbackbone curves are presented.23Seat-type abutment:Based on a bilinear idealization of the force-deflection response, the backbonecurve is governed by the initial stiffness, Kabut in kip/in, for the abutment and thepassive pressure force resisting the movement of the abutment, Pu in kip. Thesetwo parameters can be calculated by the following equations,Kabut = KiH5.5w (2.13)Pu = 5.0hbw5.5Ae (2.14)where Ki is 50.0 kip/inch/ft, w is the width of the back-wall in ft, H is the height ofthe back-wall in ft, hbw is the height of the deck seating on the back-wall in ft, andAe is the effective abutment wall area given by Ae = hbww. Caltrans (2013) does notprovide any recommendation for determination of the force-deflection response inthe transverse direction.The number of studies investigating the SSI response of abutments are not asmany as those investigating the SSI for shallow and pile foundations. The modelsproposed for simulating embankment-abutment interaction consist of an equivalentlinear spring in parallel with a linear viscous dashpot. They were not as advancedas those proposed to simulate soil-pile interaction.The study of Wilson and Tan (1990b) is one of the first studies in this field.They proposed two equivalent linear springs which represented the transverse andvertical stiffnesses of an integral abutment. They derived the constants of thesprings based on linear plane-strain analysis of a typical trapezoidal-shaped em-bankment cross-section in the frequency domain. Eq. 2.15 gives their expressionsto calculate the transverse stiffness (Kt) and the vertical stiffness (Kv) of an abut-ment system,Kt =2SLwingln(1+2SH/B)GdegKv =4(1+ν)SLwingln(1+2SH/B)(2.15)where S is the slope of the embankment(1H : SV ), Lwing is the wing length of the24abutment, H is the height of the embankment, B is the width of the embankmentat the top, ν is the poisson’s ratio of the embankment material, and Gdeg is thedegraded shear modulus calculated as follows,Gdeg = (4HFw1.18(H/B)0.08)2ρ (2.16)where Fw is the fundamental transverse frequency of the trapezoidal-shaped em-bankment cross-section and is given by Fw = 1.18(H/B)0.08√Gmax/ρ/4H, andGmax and ρ are the low-strain shear modulus and the mass density of the embank-ment material, respectively.Zhang and Markis (2002a) conducted forced-vibration dynamic analyses (infrequency domain) of 3D finite element model of a bridge embankment and pro-posed a step-by-step procedure to obtain the dynamic stiffness of abutments (i.e.,spring and dashpot coefficients) in the transverse direction. They concluded thatthe values obtained for the transverse dynamic stiffnesses can also be used withconfidence to represent the longitudinal dynamic stiffnesses.2.6 SummaryBoth beneficial and detrimental effects of SSI on structural demands have beenreported in the literature. SSI effects are insignificant when the ratio of superstruc-tural stiffness to foundation stiffness is small. Existing guidelines suggest includingSSI in numerical analyses, without providing any prescription on how to model it.Direct method or 3D continuum modeling method is rarely used for nonlineardynamic SSI analysis of bridge systems either in the research community or inpractice. This is mainly because the analysis requires significant computationaleffort which is very expensive and time consuming. However, it has been shownthat the efficiency of the method can be improved with the aid of recent advancesin computational tools and parallel computing environments.Multistep method or substructuring method consists of three major steps: de-termining the kinematic input motions by conducting site response analysis; de-termining the dynamic stiffness of the foundation either by conducting pushoveranalysis on the soil-foundation system or by using available analytical equations;25and finally performing dynamic analysis of the superstructure supported on springsand dashpots.Because of its simplicity, the substructuring method is more appealing to theengineering practice compared to the direct method. However, appropriate charac-terization of the springs and dashpots is very difficult and uncertain. The relatedrecommendations in design guidelines are based on the results of static or slowcyclic loading tests in which no inertial interaction occurs between structure andthe foundation soil. It is however well-established that inertial interaction consid-erably changes the foundation stiffness. The substructuring method has never beenproperly validated for dynamic analysis of bridges by comparing the results withfield measurements or those of full-scale analyses.No matter how significant the level of error is, the springs and dashpots arewidely used in practice to simulate SSI for bridges. This potential problem callsfor further investigation in this area to quantify the level of errors and to provideinsight to the engineering community about the associated errors and the alternativesolutions.262.7 Tables and figuresTable 2.1: P-multipliers suggested in AASHTO (2012)Pile spacing in the Row 1∗ Row 2 Row 3 and higherdirection of loading3D 0.80 0.40 0.305D 1.00 0.85 0.70∗ Row 1 is the leading row in a pile group in the direction of loading.Table 2.2: Recommended API t-z curves for piles embedded in clays andsands (API, 2007).Soil type z/D t/t∗maxClay 0.0016 0.300.0031 0.500.0057 0.750.0080 0.900.0100 1.00∞ 0.70 to 0.90Soil type z (inch) t/tmaxSand 0.000 0.000.100 1.00∞ 1.00∗ tmax is the skin friction capacity (API, 2007).27Table 2.3: Recommended API Q-z curves for piles embedded in clays andsands (API, 2007).Soil type z/D Q/Q∗pClay and sand 0.002 0.250.013 0.500.042 0.750.073 0.900.100 1.00∗ Qp is the total end bearing (API, 2007).28(a) McCallen and Romstad (1994) (b) Jeremic et al. (2009)Note: Dimension of bridge is exaggerated.xyzMassStructural modelGeotechnical model(c) Kwon and Elnashai (2008) (d) Elgamal et al. (2008)Figure 2.1: Application of 3D continuum modeling method (direct method)in the literature for simulating (a) Pointer Street Overpass, (b) a proto-type concrete bridge, (c) Meloland Road Overpass, and (d) HumboldtBay Middle Channel Bridge .29Figure 2.2: Three-step solution to account for soil-structure-interaction in dy-namic analyses (Kausel et al., 1978).free-field motionsrecorded crest motionsViscoelastic embankmentsand elastic support at the center bent(a) Zhang and Markis (2002b) (b) Tongaonkar and Jangid (2003)Figure 26. Stick model of the bridge with the foundation springs and dashpots.(c) Finn (2005) (d) Shamsabadi et al. (2007)Figure 2.3: Application of substructuring method in the literature for simu-lating SSI for bridge systems.30= ultimate resistance (force/unit length), lbs/in. = Coefficients determined from Figure 6.8.6-1 from surface to depth, ) relationships formore definitive, by(6.8.7-1)for cyclic or static loading condi-------------- y 3.0 0.8 H---–" #Figure 6.8.7-1—Relative Density, %F028 29 30 36 40 4520Sand abovethe watertable40 60 80 100300250200150100500k (lb/in3)F´, Angle of Internal FrictionRelative Density, %VeryLoose LooseMediumDense DenseVeryDenseSand belowthe watertable(a)------------- y 3.0 0.8 H---–" #Figure 6.8.6-1—Coefficients as Function of  ´54321020 25 30 35 401009080706050403020100Values of Coefficients C1 and C 2Values of Coefficients C3C2C1C3Angle of Internal Friction, F´, degF(b)Figure 2.4: The graphs presented in API (2007) for determining; (a) initialsubgrade reaction, k, and (b) coefficients C1, C2, and C3.31Figure 2.5: Characteristic shapes of the API (2007) p-y curves for soft claysunder static loading (after Matlock, 1970).321.00 0.01 0.02 0.03 0.04 0.050.80.6Z/D0 0.01 0.02 0.03 0.04 0.05Z, inches0.4t/tmax0.20tRES = 0.9 tmaxtmax = ftRES = 0.7 tmaxRange of tRESfor claysClay: Sand:ClaySandZ/D t/tmax Z, inch t/tmax0.000.00160.00310.00570.00800.01000.02000.000.300.500.750.901.000.70 to 0.900.70 to 0.900.000.100.001.001.00(a)z/D t/tmax0.0020.0130.0420.0730.1000.250.500.750.901.00Q/Qp = 1.0z/D zu = 0.10 x Pile Diameter (D)Figure 6.7.3-1—Pile Tip-load—Displacement (Q-z) curve(b)Figure 2.6: Characteristic shapes of the (a) t-z, and (b) Q-z curves as pre-sented in API (2007).33Figure 2.7: Effects of soil nonlinearity and inertial interaction on lateral stiff-ness of foundation system (Finn, 2005).1 (axis x)3 (axis z)K11C11K33C33K15C15Pier base2 (axis y)3 (axis z)1 (axis x)2 (axis y)K11C11K22C22Pier baseK55 C55K22C22K33C33C24Pier baseK44 C44K24K66 C665 4 6 Figure 2.8: Eight sets of springs and dashpots which are used in numericalmodeling of a boundary value problem to represent the dynamic 6×6stiffness matrix (K).34Figure 2.9: Comparing ultimate lateral soil resistance recommended in APIwith the values measured in experimental tests (Zhang et al., 2005).35FIG. 16. Characteristics of Nonlinear p-y Element: (a) Com-A dynamic beam on a nonlinear Winkler foundation (or dy-) analysis method was evaluated against a set ofcentrifuge model tests involving pile-supported structures in aprofile of soft clay (6 m thick) overlying dense sand. Ninedifferent earthquake shaking events were performed on twomodel configurations, including five scaled Kobe motions (at the base) and four scaled Santa Cruz motionsat the base). A baseline set of analysis pa-rameters was selected using procedures commonly used inpractice. The analyses consisted of performing 1D equivalent-linear site response analyses to calculate the dynamic responseanalyses tocalculate the dynamic response of the structural models. Im-analyses include the useof series hysteretic/viscous damping to represent radiationdamping (Wang et al. 1998) and the inclusion of gapping ef-fects (after Matlock et al. 1978). Calculated and recorded re-sponses of the two single-pile-supported systems and soil pro-Reasonable agreement was obtained between the dynamicanalyses and the centrifuge model data over the wide rangeof shaking intensities and earthquake motions covered in thisstudy. Peak superstructure motions (accelerations and displace-ments) were underestimated on average by about 15–20%when using the baseline set of analysis parameters. Similardifferences were obtained for peak pile bending momentsalong the pile length. When the recorded free-field motionsFigure 2.10: Schematic of the element proposed by Boulanger et al. (1999);presenting the components of the element and the corresponding force-deflection responses.36Drag element Leading-face elementRear-face element Combined elementFigure 2.11: Schematic of the element proposed by Taciroglu et al. (2006);presenting the components of the element and the corresponding force-deflection responses.37Figure 2.12: Schematic of the element proposed by Allotey and El Naggar(2008); presenting the standard and direct reload curves, the generalunload curve and typical two-way cyclic response of the element.Free-field ground motionAPI backbone curveMasing rulePile elementdFFigure 2.13: Schematic of the spring model used in practice for dynamic SSIanalysis of piles.38Chapter 3Substructure model for analysisof bridge systemsEngineering is the art or science of making practical.— Samuel C. Florman (1976)3.1 IntroductionIn the previous chapter it was shown that the analysis of a linear elastic soil-structure system can be conducted in two separate steps; the kinematic interac-tion and the inertial interaction. This method of analysis is called substructuringmethod. Based on the concepts of the substructuring method, Kausel et al. (1978)proposed a three-step solution for numerical analysis of a general SSI problemwith linear elastic behavior. The solution did not provide any specific procedureto determine the dynamic stiffness matrix and the kinematic input motion for apile-supported bridge system under an earthquake shaking. There are small num-ber of studies that provide detailed explanation of the substructuring method foranalysis of a pile-supported bridge system. In the study of Zhang and Markis(2002b), linear elastic system of pile foundation and abutments were analyzed us-ing frequency-domain methods in order to determine the dynamic stiffness matrixat the pile cap and at the abutments. Furthermore, in most of the related studies,39the motions recorded or computed in the free-field are used as the kinematic in-put motions, and the kinematic interaction between the pile foundations and thesupporting soil are neglected.As part of the present study the latest state of engineering practice in dynamicanalysis of bridge system was reviewed in consultation with the expert bridge engi-neers from the California Department of Transportation. As a result a design man-ual was prepared which describes highlights of the overall analysis approach forsimulating seismic SSI in bridge systems (Shamsabadi, 2013). This design man-ual does not explain details of the simulation procedure such as the methodologyof calculating free-field motions and calculating the damping matrices represent-ing the energy dissipation at the bridge supports. This thesis presents the analysisapproach, i.e., the substructuring method, with five consecutive steps in a moreelaborate and organized way. A brief description of each step for the analysis of atypical pile-supported bridge system is presented in the following sections.3.2 First step: free-field response analysisSite response analysis is conducted in the free field both for the foundation soil andthe embankment in order to determine depth-varying time histories of displacementin the absence of the bridge structure (Fig. 3.1a). The state of practice for siteresponse analysis is to use variants of the computer program, Shake (1972), inwhich an equivalent linear approach is utilized to obtain reasonable estimates ofground nonlinear, inelastic response. The method of analysis used in Shake (1972)cannot allow for nonlinear stress-strain behavior because its representation of theinput motion by a Fourier series and use of transfer functions for solution of thewave equation rely on the principle of superposition, which is only valid for linearsystems. In the equivalent linear approach, linear analyses are performed with soilproperties that are iteratively adjusted to be consistent with an effective level ofshear strain induced in the soil.Since this thesis focuses on the validity of spring and dashpot models in pre-dicting dynamic SSI, it is essential to minimize the level of approximation in thefree-field site response analysis. To better approximate the nonlinear, inelastic re-sponse of soil, the stress-strain behavior of each element of soil should be tracked40directly in the time domain. Therefore, nonlinear time history analysis is per-formed for a continuum model of the soil profile for which nonlinear hystereticstress-strain behavior of soils is taken into account using advanced constitutivemodels. The constitutive models used for modeling soil behavior will be discussedlater in Chapter 4. The continuum model of the soil profile is composed of a stackof 3D brick elements representing a column of soil. Nodes at the same elevationare tied to one another in all three translational degrees of freedom. The modelsimulates one-dimensional (1D) wave propagation in the soil profile. Self weightanalysis is first performed to obtain the initial state of stress, and then the inputground motion is applied to the nodes at the base of the soil column in the form ofdisplacements.The outputs of this step are the depth-varying time histories of displacementin the free-field of foundation soil and the embankment and the correspondingmaximum displacements (∆n).3.3 Second step: effective linearization of backbonecurvesAPI (2007) p-y, t-z, and Q-z nonlinear backbone curves for the pile foundations, thenonlinear backbone curve representing the lateral stiffness of the pile cap, and Cal-trans (2013) load-deflection curves for the abutment systems are determined fol-lowing the procedure presented in Chapter 2. The guidelines of AASHTO (2012),as presented in Table 2.1, should be used to determine the group reduction factorin order to account for group effects in the pile groups. Since the response of thesoil-structure system is assumed to be linear elastic in the substructuring method,secant stiffnesses for the interaction of the piles, the pile cap, and the abutmentshave to be calculated to approximate the nonlinear response.Fig. 3.1b illustrates the procedure of calculating the lateral and vertical secantstiffnesses for pile foundations. Following Shamsabadi (2013), lateral secant stiff-nesses of piles (kH,n), the pile cap, and the abutment systems can be calculated atthe maximum lateral displacement (∆n) of the ground in the free-field. To deter-mine the vertical secant stiffness (kV,n), it is necessary to first calculate the settle-ment of the pile group under the tributary weight of the deck, the pier, and the pile41cap. The computer program GROUP v.8 (2012a) is adopted to analyze the pilegroup that is supported on API nonlinear p-y, t-z, and Q-z springs. The tributaryweight of the deck, the pier, and the pile cap are vertically applied to the model.The computed settlement (S) is used to calculate the secant stiffnesses from the t-zand Q-z curves.3.4 Third step: calculating pile cap dynamic stiffnessAs presented in Eq. 2.5, the dynamic stiffness (K) of a 3D pile foundation is com-posed of six diagonal elements representing the lateral, vertical, rocking, and tor-sional dynamic stiffnesses, and four off-diagonal elements representing the cou-pling effects between the lateral displacements and rocking of the foundation. Onemay use existing practical tables and charts to obtain the related spring and dashpotconstants but those are applicable for simulating simple SSI problems where thefoundation soil remains elastic during static or dynamic loadings. The nonlinearbehavior of the soil-structure system may be more properly accounted for in theanalysis by using the secant stiffness values. In the studies of Lam et al. (1998)and Lam et al. (2007) the secant stiffness values were used to better approximatethe hysteretic loading, unloading and reloading behavior of soil-pile group system,and based on this they ultimately provided guidelines for numerical modeling offoundations under earthquake shakings. As mentioned earlier, in the present thesisthe secant stiffness is calculated at the corresponding maximum displacement ofthe soil layers in the free field excluding the effects of the superstructure seismicresponses.The 6×6 stiffness matrix of a pile group (K) is the sum of the stiffness matricesat each individual pile head. For a pile group composed of m piles, the components42of the stiffness matrix are calculated as follows:K11 =m∑n=1K11n , K22 =m∑n=1K22n , K33 =m∑n=1K33nK44 =m∑n=1K44n +m∑n=1K33n× y2nK55 =m∑n=1K55n +m∑n=1K33n× x2nK66 =m∑n=1K66n +m∑n=1(K11n× y2n +K22n× x2n)K42 = K24 =m∑n=1K24n , K51 = K15 =m∑n=1K15n(3.1)where Ki jn (i, j=1...6) represents horizontal (K11n and K22n), vertical (K33n), rocking(K44n and K55n) and cross-coupling stiffnesses (K42n = K42n and K15n = K51n) ofthe nth pile, xn and yn are the coordination of the nth pile on the x and y axes,respectively. The process of calculating the 6×6 stiffness matrix is done in thecomputer program, GROUP v.8 (2012a), by simulating the pile group supportedby the equivalent linear springs (Fig. 3.1c). Lateral and vertical forces and bendingmoments are concurrently applied to the pile cap in all directions. Since the systemis linear elastic, the magnitude of the forces is not important in calculating the 6×6stiffness matrix. This modeling process is similar to that used in the design manual(Shamsabadi, 2013).Shamsabadi (2013) as well as the existing design guidelines, i.e., API (2007),AASHTO (2012), and Caltrans (2013), do not provide any procedure to determinethe dashpot constants. In the present study, the dashpot constant is determinedfollowing the approach used in the study of Gazetas (1991) where the coefficientof dashpots is calculated by the following equation,Ci j = 2βKi j/ω (3.2)where β is damping ratio, Ki j is an element of the 6×6 stiffness matrix of the pilegroup (i, j = 1...6), and ω is the angular frequency of the input dynamic load.Several researchers such as Werner (1994), Zhang and Markis (2002b), and Lee43et al. (2011) made valuable efforts to quantify the level of energy dissipation for asoil-structure system under a seismic loading. They ended up with different rangesof damping ratios implying that the determination of the dashpot coefficients is oneof the major challenges in the substructure model. Traditionally, the damping ratioof bridge systems is assumed to be 5% without considering the energy dissipationat the bridge boundaries (Lee et al., 2011). Using system identification techniques,Werner (1994) reported damping ratios ranging from 19 to 26% for a bridge sys-tem. Zhang and Markis (2002b) concluded the damping ratio of about 50% in thelongitudinal direction and 10 to 20% in the transverse direction for typical highwayovercrossings. Lee et al. (2011) reported a damping ratio in the order of 25% inboth longitudinal and transverse directions. In the present study, the damping ratiois approximated to be 25% in both longitudinal and transverse directions accordingto the study of Lee et al. (2011).3.5 Fourth step: calculating pile cap kinematic motionsIn this step, kinematic input motion is determined on top of the pile cap. Ac-cording to Fan et al. (1991), the motion on the pile cap would be the same as thefree-field motion if the system is subjected to a low-frequency earthquake shaking.Following this, Makris et al. (1994) used the free-field motions as the kinematicinput motion in their study. According to Shamsabadi (2013), possible effects ofkinematic interactions on the pile cap motion can be considered by simulating amassless pile group in absence of the bridge structure. For practical purposes,the response of the foundation soil is modeled by the equivalent linear springs anddashpots that were obtained in the second step. The depth-varying time histories ofdisplacements, which were obtained in the first step, are then applied to the groundnodes of the springs (Fig. 3.1d). The output of the analysis is the time history ofdisplacements of the pile cap which is called the kinematic input motion.3.6 Fifth step: dynamic analysis of the bridge modelIn this step, a time history analysis is performed to obtain the seismic responseof the bridge structure supported on the equivalent linear springs and dashpots(Ki j and Ci j) which were obtained in the second and third steps. In this study, to44minimize the sources of error and to stay focused on the dynamic SSI responses atthe supports, the bridge deck is fully simulated without relying on any simplifiedmodels such as the stick model. Fig. 3.1e present the schematic of the model. Themass of the pile cap is assigned to the base of the pier as a lumped mass as follows,M =M11 0 0 0 0 00 M22 0 0 0 00 0 M33 0 0 00 0 0 I44 0 00 0 0 0 I55 00 0 0 0 0 I66(3.3)where M11 = M22 = M33 are the mass of the pile cap, and I44, I55, and I66 are themass moment inertia of the pile cap about the x, y, and z axes, respectively. Themass moment inertia of a square pile cap with a width of B and thickness of h isgiven by,I44 = I55 =M(B2 +h2)12I66 =MB26(3.4)In addition, the mass of abutment system is assigned to the two ends of thebridge deck in three translational directions. The mass moment inertia of the abut-ment system is not assigned in the rotational directions on the base of the assump-tion that the abutment system does not move in those directions. The masses of thefoundation soil and the embankment is not included in the model, because thereis no information about the volume of soil affected by the response of the bridgestructure under a given earthquake shaking.The kinematic input motions in the form of displacements are applied to theground nodes of the springs and dashpots located at the two ends of the bridgedeck and the pier base. The input motions to be applied to the two ends of thebridge deck and the pier base were already obtained in the first and fourth steps,respectively.453.7 SummaryThe substructure method for dynamic analysis of bridges is explained in detail.The method, which is similar to the state of practice in Caltrans, is presented infive consecutive steps.In the first step, the time histories of ground displacement are calculated in thefree-field of the foundation soil and the embankment, and the profiles of maximumground displacements are determined. In the second step, the secant stiffnessesalong the piles, the pile caps and the abutments are determined. For a given back-bone curve, the lateral and vertical secant stiffnesses are defined as the stiffness atthe maximum displacement of soil in the free-field and the settlement of the pilegroup, respectively. In the third step, the secant stiffnesses are then used to create a3D numerical model of the pile group. The pile group is concurrently subjected tolateral and vertical forces and bending moments. The magnitudes of the forces andbending moments are not important because the model response is linear elastic.The output of the analysis is the 6×6 stiffness matrix (K) and the 6×6 dampingmatrix (C). The damping matrix accounts for both material and radiation damp-ing. In the fourth step, kinematic input motion at the pile cap is computed for amassless pile group, excluding the stiffness and mass of the bridge superstructure.In the fifth and final step, the global model of the bridge structure supported on theequivalent linear springs and dashpots is subjected to the kinematic input motionsin the form of displacements.In the present study, errors caused by site response analysis and bridge su-perstructure modeling (stick model) are minimized to narrow the sources of errordown to those caused by the methodologies used for simulating the dynamic SSI.463.8 Figures(c) Third stepFixed Lateral laod Moment Massless pile cap and pilesOutput: kinematic input motion(d) Fourth stepVertical loadInput motion(a) First step Ground surfaceLayer 1Layer nLayer 2Layer 3...yp∆1 kH,1Depth-varying time histories of displacement in the free-field ∆1 : Maximum disp.∆2 ∆n yp∆2 kH,2yp∆n kH,nEquivalent linear springs(e) Fifth stepDepth-varying disp.time histories (from the first step) kH,1kH,2kH,nLateral secant stiffness* ** This spring is used when the pile cap is embedded.kV,1kV,2kV,n.(b) Second step Weight of structureAPI p-y curveVertical secant stiffnessAPI t-z curveAPI Q-z curveztSkV,1zt kV,2zQ kV,nSSS: pile group settlementx yz 6x6 dynamic stiffness matrix (     ) represented as shown in Fig. 2.8. Kinematic input motion (fourth step) Kinematic input motion (first step)Dynamic stiffnesses along the axes x and y(c) Third step  (d) Fourth step ** This spring is used when the pile cap is embedded.Bridge deckOutput: 6x6 stiffness matrix (K)Output: depth varying time histories of disp. Output: secant stiffness values for the piles and the abutmentOutput: seismic response of the bridgedF k∆ Longitudinal & transverse secant stiffnessesCaltrans curve∆ : Maximum disp. at top of the embankmentFigure 3.1: Schematic of five consecutive steps of substructuring method; (a)first step: determination of the depth-varying time histories of displace-ments, (b) second step: effective linearization of the backbone curves,(c) third step: determination of the 6×6 stiffness matrix (K), (d) fourthstep: determination of the kinematic input motion at the pile cap, and (e)fifth step: dynamic analysis of the bridge model (Shamsabadi, 2013).47Chapter 4Elements of continuum modelingin OpenSeesMake things as simple as possible, but not simpler.— Albert Einstein (1879–1955)4.1 IntroductionThis chapter introduces the methodologies used to better approximate the kine-matic and inertial interactions between the bridge structure and the foundation soil.The numerical simulation platform is first introduced. It is of interest to utilize themost efficient modeling approaches as well as computational tools. Amongst ex-isting numerical methods, finite element method is the most popular tool used byengineers to analyze problems in continuum mechanics. In this method, the spatialdomain of the problem in hand is subdivided into a number of subdomains, calledfinite elements. The size of the elements can be equal or different. Therefore, ageometrically complex domain of the problem can be represented as a collectionof finite elements. The present work employs the finite element method to ana-lyze the static and seismic responses of bridge systems. The advanced constitutivemodels used to simulate the nonlinear hysteretic response of the foundation soiland the bridge piers are presented. The linear elastic models used to simulate the48rest of the bridge structural components are not presented due to their simplicity.The methodology of simulating soil-pile interaction in the continuum model is ex-plained in detail. Finally the procedure of dynamic time history analysis of thesoil-structure system is presented.4.2 Numerical simulation platformThe finite element program, Open System for Earthquake Engineering Simula-tion (OpenSees) developed by McKenna and Fenves (2001), is adopted to formu-late the finite element model. OpenSees employs object-oriented methodologies tomaximize modularity and extensibility for implementing constitutive models, solu-tion methods, and data processing and communication procedures (McKenna andFenves, 2001). The program consists of a wide range of inter-related libraries ofmaterial constitutive models, elements, solution algorithms, integrators, and equa-tion solvers. These libraries are independent allowing great flexibility in combin-ing the libraries to solve different types of engineering problems. This also allowsearthquake engineering researchers to build upon each other’s contributions anddevelopments. In OpenSees, the numerical model is introduced as a domain ob-ject, which is composed of all analysis components such as node, material models,elements, boundary conditions, loading patterns, and equation solvers. This do-main is created by the user for each problem through an interface script using Tclprogramming language.4.3 Constitutive modeling of materialsA constitutive model is a mathematical description to link the strain and stress in-crements in a given material. Each constitutive model simplifies the actual materialresponse to a certain level. The level of this simplification could have significanteffects on realistic prediction of the overall numerical model response in a bound-ary value problem. While it is important to seek simplicity as much as possible, inorder to have a representative simulation in analysis of boundary value problemsit is vital to have adequate complexity in constitutive modeling of the materialsinvolved1.1Make things as simple as possible, but not simpler. – Albert Einstein49In the recent decades, a plethora of constitutive models with different levelsof sophistication are developed to model the nonlinear response of soils. Thosemodels are generally in the categories of classical plasticity, hypo-plasticty, nestedsurface plasticity, and bounding surface plasticity models. Some of these mod-els also incorporate other useful frameworks such as critical state soil mechanics.A number of these models have been used for simulation of cyclic response ofsoils with some success. The nested surface and bounding surface plasticity mod-els appear to be the most successful ones in modeling of the cyclic response ofsoils under multidimensional loading. Among the bounding surface models forsoils (Dafalias, 1986) one can list the SANISAND (Manzari and Dafalias, 1997,Dafalias and Manzari, 2004, Dafalias et al., 2004, Taiebat and Dafalias, 2008)and SANICLAY (Dafalias et al., 2006, Taiebat, Dafalias and Peek, 2010, Taiebat,Dafalias and Kaynia, 2010, Seidalinov and Taiebat, 2014) family of models. Someexamples for the recent uses of SANSIAND and SANICLAY models in simulationof boundary value problems are Taiebat et al. (2007), Jeremic´ et al. (2008), Taiebat(2008), Cheng and Jeremic´ (2009), Taiebat, Jeremic´, Dafalias, Kaynia and Cheng(2010), Shahir and Pak (2010), Taiebat et al. (2011), and Rahmani and Pak (2012).As for the nested surface models for soils Prevost (1985), one can list the PDMYYang et al. (2003) and PIMY Gu et al. (2011) models that are robustly implementedin OpenSees finite element program (see more in http://cyclic.ucsd.edu/opensees).This class of models has been extensively used in various geotechnical earthquakeengineering problems. Some examples are Zeghal and Elgamal (1994), Yang andElgamal (2001), Elgamal et al. (2002), Elgamal, Yang, Lai, Kutter and Wilson(2005), Elgamal, Lu and Yang (2005), Ilankatharan and Kutter (2008), Elgamalet al. (2008), Kwon and Elnashai (2008), Armstrong et al. (2012), Chang et al.(2013), and Kolay et al. (2013). While the stress-strain response of natural mate-rials such as soils are a lot more complex compared to those in man-made mate-rial, there are several available models for characterizing the constitutive responseof concrete and steel, for which the literature review is beyond the scope of thepresent study. In the following section, the models used for simulating stress-strainresponse of all materials including soils, concrete and steel are presented.504.3.1 Sandy soilsHysteretic nonlinear behavior of the sandy layers are simulated using an elasto-plastic constitutive model proposed by Yang et al. (2003). The model includes aDrucker-Prager yield surface with a non-associative flow rule and a robust devia-toric kinematic hardening rule. This constitutive model is called pressure depen-dent multi-yield model (PDMY) in OpenSees.In the PDMY model, the stress-strain relation is linear and isotropic inside theyielding surface, and nonlinearity results from plasticity (Yang et al., 2003). Theplasticity is formulated based on the multisurface-plasticity framework of Prevost(1978). The yield surface is conical in principle stress space and is given as,f =32[s− (p′+ p′0)α] : [s− (p′+ p′0)α]−M2(p′+ p′0)2 = 0 (4.1)where s = σ ′− p′δ is the deviatoric stress tensor (σ ′ is effective Cauchy stresstensor, δ is second-order identity tensor), p′ is the mean effective stress, p′0 a smallpositive constant used to limit the size of the yield surface at p′ = 0, α is a second-order deviatoric tensor that defines the yield surface center in deviatoric space, Mdefines the yield surface size, and the symbol : denotes a doubly contracted tensorproduct.The hardening zone is defined by a number of similar yield surfaces which aregenerated by a shear stress-strain hyperbolic backbone curve given as,τ = Gγ1+γγr(p′rp′)d(4.2)where τ and γ are the octahedral shear stress and strain, respectively; G is theshear modulus; γr = τmax/Gr, in which τmax is the maximum shear strength whenγ approaches ∞ and Gr is low-strain shear modulus at p′r. The schematics of theyield surfaces in the stress space and its typical response in the stress-strain spaceare shown in Fig. 4.1. Details about the hardening rule and the flow rule can befound in the study of Yang et al. (2003).514.3.2 Clayey soilsThe clayey layers are simulated using a pressure-independent version of the PDMYmodel with von Mises type yield surface, associative flow rule, and a deviatorickinematic hardening rule (Gu et al., 2011). The model is called pressure indepen-dent multi-yield model (PIMY) in OpenSees framework. The schematics of theyield surfaces in the stress space and its typical response in the stress-strain spaceare shown in Fig. 4.2.The multi-yield-surface plasticity model employs the concept of a field of plas-tic moduli to achieve a better representation of the material plastic behavior undercyclic loading conditions (Gu et al., 2011). Each yield surface of this multi-yield-surface J2 plasticity model is defined in the deviatoric stress space as,f =√32(τ−α) : (τ−α)−K = 0 (4.3)where K is the size of the yield surface which defines the region of constant plasticshear moduli. The yield surfaces are generated by a shear stress-strain hyperbolicbackbone curve given as follows,τ = Gγ1+γγr(4.4)where,γr =γmaxτmaxGγmax− τmax. (4.5)An associative flow rule is used to compute the plastic strain increments. De-tails about the formulation of the model including the hardening rule and the flowrule cab be found in Gu et al. (2011).4.3.3 Concrete materialThe uniaxial Kent-Scott-Park model (Mander et al., 1988) is used for constitu-tive modeling of concrete material. In this model, tensile strength is zero andunloading-reloading stiffness degrades with increasing strain. The model is calledConcrete01 in OpenSees framework. Fig. 4.3 shows the stress-strain response of52the model. The stress-strain response is described by concrete compressive strengthat 28 days (σmax), concrete crushing compressive strength (σu), strain at maximumcompressive strength (εY ), and strain at crushing strength (εu), and initial slope ofthe response (E) is given by 2σmax/εY .4.3.4 Reinforcing steel materialIn this thesis, a simple constitutive model is used for modeling the nonlinear in-elastic behavior of steel. The stress-strain response of reinforcement steel is simu-lated using a uniaxial bilinear inelastic model with kinematic hardening, equivalentto the one-dimensional J2 plasticity model with linear kinematic and no isotropichardening. The model is called Steel01 in OpenSees framework. The schematicof the model is presented in Fig. 4.4b. The bilinear stress-strain response is de-scribed by steel Young’s modulus (E), yield strength (σY ), strain hardening ratio(b) which is the ratio between post-yield tangent and initial elastic tangent. Theunloading-reloading paths are governed by using the concept of the Masing rule.4.4 Modeling soil-pile interactionSoil domain is modeled using solid eight-node brick elements that each containseight gauss points with three degrees of freedom. The pile is simulated using dis-placement based beam-column elements with six degrees of freedom. In order tocreate a model of a complete soil-pile system, pile and soil finite elements have tobe coupled. To this end, solid elements in the region physically occupied by thepiles are first removed, and the pile beam-column elements are then placed in mid-dle of this opening. At each elevation the pile nodes are horizontally connected tothe soil nodes using eight rigid beam-column elements as shown in Fig. 4.5. Thecomputed forces in these elements are used to obtain p-y curves in which p is thesummation of forces in the direction of loading, and y is the pile deflection. Theconnectivity between the rigid element node and the soil node (both nodes havethe same coordination in the finite element mesh) is provided only for translationaldegrees of freedom, while rotational degrees of freedom for the rigid element areleft unconnected. Same thing applies to the connection between the rigid elementnodes and the pile nodes. Irrespective of the shape of the surrounding medium ad-53jacent to the piles, the rigid beams are used for connections between that mediumand the piles to enforce compatibility between soil and pile deflections. Using thismethodology to model soil-pile interaction, the physical volume of the pile ele-ments is simulated in the finite element mesh and all deformation modes of a pile(axial, bending, shearing) are transferred to the surrounding soil (Jeremic et al.,2009).Under loading, the interface between two dissimilar media, i.e., pile and soil,can experience different modes such as relative slip or debonding between the twomedia. However, slippage and debonding between pile and soil are assumed tobe negligible for simplicity of the study. For a more realistic analysis of SSI undermore severe demands, instead of using rigid connections at the interface of soil andpile as used here, one would need to simulate the possible slippage and gappingmechanisms by using advanced interface elements4.5 Analysis procedureThe analysis of the soil-structure system is carried out in three stages: In the firststage the finite element model excludes any structural elements, and the weight ofthe soil is applied to all solid elements to obtain the initial states of stress. In thesecond stage, the soil elements at the region occupied by the piles are removed todefine the physical volume of the piles; then the structural components of the bridgeare placed at once, and self weight analysis is performed to bring the full systemto equilibrium. In the third and final stage, the fixities at the base of the finiteelement mesh are removed in both longitudinal and transverse directions, and thecorresponding time histories of displacements (i.e., the input earthquake shaking)are applied to the base in these two horizontal directions using the multi-supportexcitation pattern in OpenSees.Static nonlinear analysis is used for the first and second stages, and transientdynamic nonlinear analysis with time steps of 0.01 s is used for the third stage ofthe analysis. The parallel-computing environment in OpenSees, namely the “Sin-gle Parallel Interpreter” application or OpenSeesSP (McKenna and Fenves, 2008),is employed to speed up the execution time of analyses. The system solver MUl-tifrontal Massively Parallel sparse direct Solver (MUMPS) is adopted to solve the54large sparse system of equations in the analysis. A maximum of 50 iterations isset for each analysis step to achieve the prescribed tolerance, i.e., to reach a normdisplacement increment less than 0.001. The penalty approach is used enforce con-straint equations in the analysis. Krylov-Newton algorithm is adopted to determinethe sequence of steps taken to solve the nonlinear equation. For dynamic analyses,Newmark time-stepping method with the parameters γ=0.70 and β=0.36 are usedto integrate the equations of motion. The energy conserving form of the Newmarkmethod (i.e., γ = 0.5 and β = 0.25) may develop high frequency noises. In this the-sis, a parametric study showed that such high frequency noises can be effectivelyeliminated by using the Newmark parameters of γ=0.70 and β=0.36.4.6 SummaryThis chapter described the methodology used in OpenSees program to simulatesoil-pile system using continuum modeling method. Detailed information was pro-vided for OpenSees computer program, the constitutive models, the methodologyused for modeling soil-pile interaction and the analysis procedure.Advanced constitutive models are used to simulate nonlinear hysteric responseof the soil material under static and seismic loading. Sandy and clayey materialsare simulated using pressure dependent multi-yield model (PDMY) and pressure-independent multi-yield model (PIMY), respectively. Both models are formulatedbased on the multisurface-plasticity framework. The former model includes aDrucker-Prager yield surface with a non-associative flow rule and a robust devi-atoric kinematic hardening rule, and the latter model includes a von Mises typeyield surface, associative flow rule, and a deviatoric kinematic hardening rule.The uniaxial Kent-Scott-Park mode, namely Concrete01 in OpenSees, is usedfor constitutive modeling of concrete, and the uniaxial bilinear inelastic model,namely Steel01 in OpenSees, is adopted for constitutive modeling of steel.Soil solid elements are connected to pile beam-column elements by eight rigidbeam elements to simulate the physical volume of the pile in the continuum model.Slippage and gap between the pile and the surrounding soil are neglected in thisstudy.The study employs OpenSeesSP, the parallel computing environment of OpenSees,55in order to speed up the execution time of analyses.564.7 FiguresFigure 4.1: Schematic of the pressure-dependent multi-yield surface(PDMY) model in stress space and its typical response in stress-strainspace (Yang et al., 2003).Figure 4.2: Schematic of the pressure-independent multi-yield surface(PIMY) model in stress space and its typical response in stress-strainspace (Gu et al., 2011).57stressstrainεY εult σultσmaxECompressionTensionFigure 4.3: Schematic of stress-strain response of the Kent-Scott-Park modelfor concrete material.stressstrainσYECompressionTensionσYbxEbxEFigure 4.4: Schematic of stress-strain response of the bilinear inelastic modelfor steel material.58Equal DOFs in threetranslational directionsRigid beam columnconnectionSoil elementsPile node(pinned to the rigid beamelement)(a)Surrounding soil elementsRigid beam-column elementsPile elementEqual translational DOFs(b)Figure 4.5: Description of methodology used for soil-pile interaction model-ing, (a) plan view , and (b) finite element mesh.59Chapter 5Continuum modeling of bridgesystems: validation and baselinedataThe Golden Gate Bridge is a giant moving math problem.— John van der Zee (1936)5.1 IntroductionThe need for realistic numerical simulations in geotechnical earthquake engineer-ing applications necessitates developing three-dimensional (3D) continuum mod-els using a series of high fidelity geotechnical/structural models. 3D continuummodels have been rarely used in practice since nonlinear dynamic analysis oflarge-scale models requires major computational efforts that can be tedious, timeconsuming, and in some cases impractical. However, recent advances in high-performance computing software and hardware with the aid of parallel computingenvironments permit the analysis of large-scale problems such as bridge systems.In the recent decade, 3D continuum models have been widely used for simulat-ing small-scale soil-structure systems such as retaining walls and pile foundations(e.g., Cheng and Jeremic´, 2009; Shamsabadi et al., 2010; McGann et al., 2011).60Also the extensive database available from experimental tests (e.g., Wilson et al.,2000; Abdoun et al., 2003; Brandenberg et al., 2005; Dashti et al., 2010) has fa-cilitated the validation and application of these computational models. However,large-scale soil-structure systems such as bridges on pile foundations have beenrarely modeled using continuum modeling method.In this chapter, the continuum modeling method used for developing and ana-lyzing 3D continuum models of bridge systems is discussed. The approach is vali-dated by comparing the resulting seismic responses from a continuum model withthose recorded for the two-span Meloland Road Overpass (MRO) located in Cali-fornia, US, during the past earthquake events. In order to expand the numericallygenerated baseline data for evaluation of the substructuring method, in addition tothe generated data as described above for the MRO, the validated continuum modelis also extended to study two more prototype bridge systems:• The first one is a three-span bridge system with the same soil profile as theMRO bridge site. A suite of ground motions with a wide range of peakground accelerations and predominant periods are applied to the bridge sys-tem to induce different levels of kinematic and inertial interactions betweenthe bridge structure and the supporting soils.• The second one is a prototype nine-span bridge system located at two sitesrepresenting soft and stiff grounds in Vancouver, Canada. The bridge systemis subjected to a suite of ground motions linearly matched to the uniformhazard spectrum (UHS) of Vancouver with a exceedance probability of 2% in50 years in the period range of 0.1 to 1.0 s. The main goal here is to providereference data by which the applicability of the substructuring method isevaluated for a typical nine-span bridge system that is located in Vancouver,Canada.The main goals of this chapter can be listed as follows; (i) illustrate the potentialfor further practical applications of the large-scale continuum models with the aidof recent advances in parallel computing environments, and (ii) provide baselinedata for the next part of the thesis in which the validity of the current state ofpractice for dynamic analysis of bridge systems is assessed.615.2 Analysis of the two-span Meloland Road Overpass5.2.1 Description of the bridgeThe MRO is a two-span integral abutment bridge built in 1971 near El Centro,California, US, as part of Highway 8. Below is a summary of the structural charac-teristics and instrumentation of this bridge. The bridge deck has a length of 64.0 m,width of 10.0 m and depth of 1.733 m (Fig. 5.1a). The deck section is box girdercomposed of four vertical webs with a thickness of 0.2 m. The pier at the centerof the deck is 5.0 m in height above the ground surface with a diameter of 1.52 m(Fig. 5.1b). The pier is reinforced by a total of 18 longitudinal rebars with a diam-eter of 0.057 m. The pier foundation is composed of a 4.6 m by 4.6 m pile cap witha thickness of 2.0 m supported by 25 vertical timber piles (a 5×5 pile group) withlengths of 15.0 m and diameters of 0.32 m at the top and 0.20 m at the bottom. Asshown in Fig. 5.1c, the abutment is of integral type with no deck joints and bear-ings. The height of the back walls is about 3.0 m with a thickness of 0.46 m, andeach side of the abutment has two 6.0 m long wing walls with a thickness of 0.3 m.The abutments are supported by seven vertical timber piles (a 7×1 pile group) withlengths of 18.0 m and diameters of 0.32 m at the top and 0.20 m at the bottom. Theside slope of the embankment is 1V:2H, and the slope in front of the back walls is1V:1.5H. The bridge is instrumented with 29 accelerometers on the structure and3 accelerometers at a free-field site (CESMD, 2013). Fig. 5.1d depicts the locationof the instruments.The site geology is classified as deep soft alluvium with Vs30 of 192 m/s (CESMD,2013). The shear wave velocity is approximately 140 m/s near the ground surfaceand 730 m/s at a depth of 150 m (Anderson, 2003). Fig. 5.2 shows the soil profileat the bridge site. The soil layering follows the works of Maragakis et al. (1994)and Kwon and Elnashai (2008). The embankment soil material is composed ofone layer of medium clay for which the cohesion is 20.0 kPa and the density is1.6 ton/m3. The underlying soil is composed of five layers of clays and silty sands.The clayey layers are located at 0–2.7, 6.0–10.7, and more than 15.0 m below theground surface with cohesion values of 35.9, 76.6, and 86.2 kPa, and densities of1.5, 1.8, and 1.8 ton/m3, respectively. The in-between layers are silty sands with62friction angle of 33◦, and density of 1.9 ton/m3. The elastic properties of the soillayers, i.e. the shear and bulk moduli, are selected from the work of Kwon andElnashai (2008) which in turn were adopted from the standard values suggested byYang et al. (2008) based on the corresponding cohesion and friction angle valuesfor the clay and sand layers.5.2.2 Model development and simulation detailsThe continuum model of the bridge is developed using OpenSees finite elementanalysis framework (McKenna and Fenves, 2001). The finite element mesh is vi-sualized using the computer program GiD (2013) and is shown in Fig. 5.3. Allcomponents of the bridge including the 5×5 and the 7×1 timber pile groups, thepier, the deck, the back walls, the wing walls, and the soil domain are modeled.Solid eight-node brick elements with eight gauss points are used to model the soildomain and the pile cap. Each node of the solid elements has three translationaldegrees of freedom. Four-node shell elements with three translational and three ro-tational degrees of freedom at each node (MITC4, Bath (1984)) are used to modelthe back walls, wing walls, and bridge deck. The thickness of the shell elements isconsidered to be 0.46 m and 0.3 m for the back walls and wing walls, respectively;for the bridge deck, an equivalent thickness of 1.2 m is used in order to have sim-ilar moment of inertia to the original section. 3D fiber section displacement basedbeam-column elements with six degrees of freedom, are used to model the bridgepier and the piles. The simulated cross-sections of the pier and the piles are circularwith diameters of 1.52 and 0.32 m, respectively. The tapered configuration of thepiles is neglected for simplicity.The piles are rigidly connected to the pile cap. Same thing applies to the con-nection of the pier to the pile cap and the deck. The abutments of MRO are ofintegral type, and therefore the abutment walls and the deck are also rigidly con-nected in the numerical model. To connect the shell elements of the abutment wallsto the solid elements of the embankment, the corresponding nodes with the samecoordinates are constrained to have the same displacements using the equalDOFconstraint available in OpenSees. Soil-pile interaction is modeled following theprocedure presented in Chapter 4.63The nodes at the base of the mesh are fixed in all directions, and the nodeswith equal elevations at the four lateral boundaries are constrained to have equaldisplacements.Advanced nonlinear hysteretic models are used for constitutive modeling of thefoundation soil. Pressure dependent multi-yield (PDMY) and pressure independentmulti-yield (PIMY) constitutive models are used to simulate sandy and clayey lay-ers, respectively. The details about the models were presented in Chapter 4.Kwon and Elnashai (2008), who also simulated the MRO, reported that resultsof field tests (e.g., suspension logging) and laboratory tests (e.g., resonant column,cyclic torsional shear, and dual specimen direct simple shear) performed as partof the ROSRINE project (Anderson, 2003), were not enough for characterizing thesoil models. Very different values of shear wave velocities were obtained from eachtest. Therefore, the unknown values are extracted from the typical values given byYang et al. (2008) who provided a quick reference for selecting parameter valuesof the constitutive models for soft, medium, and stiff clays and for loose, medium,medium-dense, and dense sands. Table 5.1 presents the input parameters for theconstitutive modeling of sands and clays, which are consistent with those selectedin Kwon and Elnashai (2008) for simulation of the same bridge site.Elastic behavior is assigned to the timber piles, the concrete pile cap, the deck,and the abutment walls because in seismic design of bridge systems these com-ponents are capacity-protected so that damage is not allowed. Following Kwonand Elnashai (2008), Young’s moduli of 12.4, 24.8, and 24.8 GPa are used for thetimber piles, the pile cap and the abutment walls, respectively. The Young’s mod-ulus of cracked concrete section, E=20.0 GPa (Dendrou et al., 1985), is assumedfor the pier and the deck. The density of the concrete material is assumed to be2.45 ton/m3. In the design process, the bridge piers are usually allowed to yieldsince any possible damage in extreme events can be easily detected and repaired.This necessitates the use of an appropriate nonlinear elastic-plastic model for thepier materials. The uniaxial Kent-Scott-Park model (Mander et al., 1988) is usedfor constitutive modeling of the concrete material. Due to the lack of data forcharacterizing the pier material, the input parameters of confined and unconfinedconcrete are extracted from the typical values and presented in Table 5.2. For thelongitudinal rebars, the bilinear elastic model, presented in Chapter 4, is used with64Young’s modulus of 200.0 GPa and yield strength of 455.0 MPa. Fig. 5.4 shows thefiber discretization of the pier cross-section and the corresponding cyclic moment-curvature response under an axial load of 9.6 MN (tributary weight of the bridgedeck). From the moment-curvature analysis, the plastic bending moment of thebridge pier is 15.0 MN.m.The procedure of the analysis was discussed in detail in Chapter 4. Dynamicanalysis of the described continuum model would be too slow if only one CPUwere used. To this end, the parallel-computing environment in OpenSees, namelyOpenSeesSP, is employed. A total of ten 2.4 GHz CPUs with 12 GB of RAM onWindows Server 2012 64–bit Operating System are concurrently allocated for eachanalysis. Using this technique, the analysis runtime decreased by a factor of 5. Itis noted that no matter how efficient the parallel algorithm is, there is an upperlimit on the usefulness of adding more CPUs. This is referred to as Amdahl’s law(Amdahl, 1967). For the continuum model of the MRO, ten CPUs appeared to bethe upper limit. Further details about the importance of parallel computing and itsfeatures are presented in Appendix A.5.2.3 Dimensions and mesh refinement of the soil domainThe lateral boundaries of the model should be placed at a location where the 3Deffects due to presence of the bridge are negligible, and it should be ensured thatfree-field conditions at the lateral boundaries of the finite element mesh are appro-priately captured. To this end, the following two steps are taken:In the first step, to minimize the effects of the lateral boundaries on the seismicresponse of the bridge, appropriate dimensions of the continuum model are deter-mined as follows. Trial dynamic analyses are performed for the continuum modelswith the soil domain dimensions of 79.0 m×30.0 m×20.0 m, 99.0 m×50.0 m×20.0 m,and 119.0 m×70.0 m×20.0 m (in directions x, y, and z shown in Fig. 5.3). The first3 s of the 1986 San Salvador earthquake with PGA of 0.85g, i.e. the major partof the motion including the PGA, is applied to the base of the model in both lon-gitudinal and transverse directions. This strong earthquake is selected in order tomobilize the highly nonlinear hysteretic behavior of the employed materials. Thetime histories of acceleration at top of the left abutment and the time histories of65shear force at the pier base are presented in Fig. 5.5, showing almost no differ-ence between the results of the second and third models. Accordingly, the soildomain dimensions of 99.0 m×50.0 m×20.0 m are considered as sufficient for theanalyses.In the second step, to make sure that free-field conditions at the lateral bound-aries are properly captured, the soil response at these boundaries is investigated.Two columns of soil, one with a height of 20.0 m representing the foundation soil,and the other one with a height of 28.0 m representing the embankment and foun-dation soil are subjected at the base to the same ground motion as above. Thenodes at the same elevations are tied to one another in all three translational de-grees of freedom. The resulting acceleration time histories on top of these twosoil columns are compared to those at points next to the lateral boundaries of thesoil-bridge continuum model on the ground surface and on top of the embankment.Presented results in Fig. 5.6 confirm that the model properly captures the free-fieldconditions.Furthermore, it is important to determine the appropriate level of mesh refine-ment in the soil domain. To do this, trial dynamic analyses are performed on mod-els with three levels of mesh refinements consisting of 23,688, 31,844, and 38,245solid elements. The soil domain dimensions are the same for all three meshes (i.e.,99.0 m×50.0 m×20.0 m), and the models are subjected to the same earthquakeas above. The resulting time histories of acceleration at top of the left abutmentand time histories of shear force at the pier base are presented in Fig. 5.7, showingalmost no difference between the results of the second and third levels of meshrefinement. In summary, the continuum model includes a total of 41,177 nodes,3996 beam-column elements, 1931 shell elements, and 31,844 solid elements rep-resenting a soil domain of 99.0 m long (in direction x), 50.0 m wide (in directiony), and 20.0 m deep (in direction z).5.2.4 Input earthquake shakingsThe seismic responses of the MRO during the 1979 Imperial Valley earthquakewith PGA of 0.32g and the 2010 El Mayor-Cucapah earthquake with PGA of 0.21gare of interest in this study since these two events caused the strongest shaking66amongst those recorded at the bridge site. During both events free-field motionswere recorded at the ground surface. These motions should be deconvolved to adepth of 20.0 m before being applied at the base of the continuum model. Theinput motion deconvolution is quite difficult and affected by many factors. In thepresent study, deconvolution analysis was performed using the equivalent linearprogram ProShake (2003). To this end, a 20.0 m deep soil column is modeled inProShake with comparable properties to those in the free-field of the continuummodel, i.e., with same profiles of density and shear wave velocity, and with repre-sentative modulus reduction and damping curves as will be discussed shortly. Therecorded earthquake motions in the free-field are applied as outcrop motions to thesurface of soil column in the ProShake model. The resulting in-slab motions at thebase of the ProShake model are applied in form of displacement time histories tothe base of the continuum model in the third stage of analysis (Mejia and Dawson,2006).The above deconvolution process for each earthquake would result in a com-puted motion at the ground surface in the free-field of the continuum model (i.e.,at the lateral boundaries) that differs from the corresponding recorded one in termsof their PGA and spectral values. This is because the nonlinear hysteretic behav-ior of soil is modeled quite differently in the ProShake model and the continuummodel: in the former model it is approximated by equivalent linear shear modulusand damping as functions of shear strain, while in the latter model it is simulated bythe advanced elastoplastic constitutive models with additional numerical damping,particularly important for small strains, that is provided by Newmark time-steppingmethod (see in Chapter 4).In order to have a deconvolution process which results in comparable computedmotions at the ground surface of the continuum model to those recorded at the free-field, iterative process is used for estimation of the modulus reduction and dampingcurves of the soil layers in the ProShake model. This is achieved by changing thechoice of these curves from the available curves in the library of ProShake forthe two medium sand layers, the one medium clay layer, and the two stiff claylayers in this problem (see Fig. 5.2). Table 5.3 presents the input parameters forthese sublayer in the ProShake model for the first and last iterations. Note that inProShake the soil profile is modeled with 20 sublayers, each one with a thickness67of 1.0 m, underlain by an infinite sublayer with the same properties as those of the20th sublayer.Time histories of the recorded and computed (from the last iteration) acceler-ations at the ground surface using the above method for the 1979 Imperial Valleyand 2010 El Mayor-Cucapah events are presented in Figs. 5.8(a,b). For better com-parison, the corresponding spectral values are also presented in Figs. 5.9(a,b). Thecomparisons suggest that these deconvolved ground motions qualify as appropriateinput motions for the analyses. From the deconvolution process, the PGAs of theinput motions at the depth of 20.0 m are obtained to be 0.30g and 0.16g, for thetwo earthquakes, respectively.5.3 Validation of the continuum modeling method5.3.1 Predominant period of the bridgeTo obtain the predominant period of the bridge system, the continuum model isexcited at its base by harmonic motions with periods of 0.1, 0.2, 0.3, 0.35, 0.4, 0.5,and 1.0 s. The period which causes the largest displacements and shear forces isconsidered to be the predominant period of the system. This period appears to be0.35 s ( f =2.86 Hz) in both longitudinal and transverse directions and is close to theidentified periods reported in the previous studies. For instance, using system iden-tification techniques for this bridge, Wilson and Tan (1990a) reported the periodof 0.40 s ( f =2.50 Hz) and Kwon and Elnashai (2008) reported the periods rangingfrom 0.31 to 0.34 s ( f =2.94 to 3.22 Hz) in the transverse direction.5.3.2 Spectral responses of the bridge structureFig. 5.10 depicts the acceleration response spectra (ARS) of the measured andcomputed motions for the damping ratio of 5% in channels 2, 3, 4, 7, and 9 forthe two earthquake events (see Fig. 5.1d for the location and the direction of thesechannels). During the Imperial Valley earthquake, the recorded spectral accelera-tions at the pile cap (channels 2 and 4) and at the bridge deck (channels 3, 7, and 9)are predicted adequately considering the inherent uncertainties in the motions, ma-terial properties, and the approximations embedded in the deconvolution process.68During the El Mayor-Cucapah earthquake, the spectral responses of the recordedmotions in channels 2 and 4 are predicted adequately, but in channels 3, 7, and9 the maximum spectral response is predicted at period of about 0.56 s which is0.12 s larger than the corresponding recorded one. In addition to the uncertain-ties in determination of the input parameters, this level of difference can be partlyattributed to the ignorance of gap mechanism is the simulation of soil-pile andembankment-abutment interactions. It should be noted that if the recorded groundsurface motions were used as the input motions as was done by Kwon and Elnashai(2008), the prediction of the spectral response would be much worse. The use ofdeconvolved motions as the input shakings considerably improves the results of thecontinuum model, especially during the El Mayor-Cucapah earthquake.Fig. 5.11 shows the displacement response spectra at the aforementioned chan-nels for the two earthquake events. During both events, there is a quite good agree-ment between the displacement response spectra of the recorded and computedmotions. Generally, the level of error in prediction of the displacement responsespectra is lower compared with the acceleration response spectra due to frequencyfiltering during the double integration of the acceleration values. It can be con-cluded that the 3D continuum model is generally capable of simulating the longi-tudinal and transversal seismic response of the MRO despite the uncertainties inthe input motions and the material properties.5.3.3 Other seismic responses of the bridgeThe only available measurements for the MRO during the 1979 Imperial Valleyand 2010 El Mayor-Cucapah earthquake events are the accelerations at differ-ent locations of the bridge. The validated continuum model is used to computeother seismic responses of the bridge, such as the time histories of pier deflec-tions, shear forces and bending moments induced at the base of the pier, and theforce-deflection responses at the abutments.Fig. 5.12 shows the time histories of longitudinal and transverse relative dis-placements of the pier top with respect to the pile cap during the Imperial Valleyand El Mayor-Cucapah earthquakes. Due to the relatively flexible transverse con-figuration of the bridge, the deflections of the bridge pier are larger in the trans-69verse direction, especially during the El Mayor-Cucapah earthquake at which thetransverse peak displacement (i.e., 0.022 m) is approximately twice as large as thelongitudinal one (i.e., 0.012 m). As shown in this figure, during both events thelateral relative displacements of the pier are small, and very low levels of perma-nent deformations are observed at the end of the seismic shaking. This implies theprimarily elastic behavior of the pier during the seismic shakings.Fig. 5.13 depicts the time histories of the pile cap rocking in both longitudi-nal and transverse directions. Very low levels of pile cap rocking are noted duringboth earthquake events. The peak rocking is about 0.006 rad (0.34◦) and 0.003 rad(0.17◦) during the Imperial Valley and El Mayor-Cucapah earthquakes, respec-tively. This low level of pile cap rocking is mainly attributable to the high rotationalstiffness of the embedded pile cap and the underlying 5×5 pile group.The time histories of induced bending moments at the base of the pier and themoment-curvature response are shown in Fig. 5.14. The higher levels of deflectionsand rocking in the transverse direction compared with those in the longitudinal di-rection cause the larger bending moments as observed in this figure. The maximumbending moment is about 8.0 and 5.0 MN.m during Imperial Valley and El Mayor-Cucapah earthquakes, respectively. These maximum moments are much less thanthe plastic bending moment of the pier cross-section (i.e., 15.0 MN.m). Accord-ing to the moment-curvature response, the highest level of inelastic behavior isobserved in the transverse direction during the Imperial Valley earthquake and thelowest level is observed in the longitudinal direction during the El Mayor-Cucapahearthquake.Fig. 5.15 shows the longitudinal force-deflection response at the top of theleft and right abutments and at the base of the pier. Due to the higher intensityof the seismic shaking, the forces during the Imperial Valley earthquake are ap-proximately twice as large as the corresponding one during the El Mayor-Cucapahearthquake. Low levels of hysteretic nonlinear behavior are observed at the pierbase where the average slope of the force-deflection loops is about 350.0 MN/mwhile at the abutments the slope of the loops varies in a wide range implying signif-icant levels of nonlinear hysteretic behavior. Since all components of the abutmentare elastic except the embankment, the observed nonlinear hysteretic behavior isattributable to yielding of the embankment material.70The force-deflection response in the transverse direction is presented in Fig. 5.16.Due to the symmetric configuration of the pier and the underlying foundation in thelongitudinal and the transverse directions the slope of the force-deflection loops isthe same (i.e., about 350.0 MN/m), while at the abutments the slope of the loops inthe transverse direction is smaller than the longitudinal one because of higher lev-els of nonlinear hysteretic behavior. The most prominent feature of the transverseforce-deflection response at the abutments is the larger reduction of the backfillstiffness as a function of displacement after unloading or reloading occurs. Thelarge level of stiffness reduction results in large plastic deformations and large hys-teretic loops in the transverse direction, as shown in the figure.Figs. 5.17 and 5.18 show the bending moment distributions along the pileslocated along the diagonal of the pile cap at the instance of the maximum dis-placement of the pile cap about the transverse and longitudinal axes of the bridge,respectively. During both events, due to the large translational and rotational stiff-nesses of the embedded pile cap, the underlying piles experience very low levelsof bending moments. The significant contribution of embedded pile caps to trans-lational and rotational stiffnesses has been reported in the previous studies of Zafirand Vanderpool (1998) and Mokwa and Duncan (2003) where pile cap stiffnesswas reported to be more than 50% of the total pile group stiffness. The effectsof the pile cap noted above are responsible for the small bending moments in thepile group under the center pier where the maximum moment is only 20.0 kN.m.Therefore, similar to the bridge pier all piles remained elastic during both earth-quake events.5.4 Additional baseline dataFor the MRO model, small forces were found at the pile foundations due to the verystiff embedded pile cap. This implies negligible SSI at the pile foundations underthe pier in that model (see in Fig. 5.17), and therefore limiting the evaluation of thespring and dashpot models almost only to those representing the lateral stiffness ofthe pile cap and the abutment system. In order to more appropriately engage theAPI springs of the pile group under the pier in the dynamic analysis of the globalbridge model, prototype three- and nine-span bridges are also modeled with some71changes compared to the MRO model. This would also expand the numericallygenerated baseline data for evaluation of the substructuring method. Details ofthese prototype bridge models are described in the following subsections.5.4.1 Analysis of a prototype three-span bridge system5.4.1.1 Description of the bridgeThe bridge system including the structural and geotechnical components are iden-tical to that of the MRO model, except that this bridge has three spans with a spanlength of 45.0 m, and the pile cap is located 0.10 m above the ground surface.Fig. 5.19a presents the schematic of the three-span bridge. The bridge deck has alength of 135.0 m, width of 10.0 m, and depth of 1.73 m. The abutment systemsand the embankments including the geometry and mechanical properties are thesame as those of the MRO model.5.4.1.2 Model development and simulation detailsThe same procedure as described in Section 5.2 is used to develop the continuummodel including the elements, constitutive models, and analysis procedure. Thefinite element mesh of the three-span bridge is shown in Fig. 5.19b. The cross-section of the piers are the same as that in MRO. The tributary weight of the bridgedeck on each pier is 13.95 MN, and from the moment-curvature analysis the plasticbending moment of the bridge pier is calculated to be 15.72 MN.m.Following the procedure discussed in Section 5.2, sensitivity analyses are car-ried out to determine the appropriate dimensions and refinement of the finite el-ement mesh so that free-field conditions at the lateral boundaries of the meshare captured. The continuum model includes a total of 6671 beam-column ele-ments, 4388 shell elements, and 40,800 solid elements representing a soil domainof 182.0 m long (in direction x), 66.0 m wide (in direction y), and 20.0 m deep (indirection z). A total of ten 2.4 GHz CPUs with 12 GB of RAM are concurrentlyallocated for each analysis. Details about the parallel processing can be found inAppendix A.Fig. 5.20 compares the range of bending moments of the pile foundations un-72derlying the MRO pier with that of the pile foundations underlying pier 1 of thethree-span bridge. The input earthquake shaking for both models is the 1979 Im-perial Valley earthquake. It is shown that the maximum bending moment inducedat the piles of the three-span bridge is about 8 times larger than that induced at thepiles of MRO.Predominant period of the bridge system is determined by applying a suit ofharmonic motions with periods of 0.2, 0.3, 0.35, 0.4, 0.45, 0.5, and 1.0 s. Theperiod which causes the largest displacements and shear forces is considered to bethe predominant period of the system. This period is 0.4 sec in the longitudinaldirection and 0.45 sec in the transverse direction.5.4.1.3 Input earthquake shakingsThe bridge system is subjected to ten actual earthquake events with the peak groundaccelerations (PGA) varying in the range of 0.22 to 0.82g and the predominantperiods varying in the range of 0.1 to 1.0 s. Table 5.4 lists the selected earthquakeevents, and Figs. 5.21 and 5.22 compare the spectral response of the fault-paralleland fault-normal components of the motions for 5% damping. The motions areselected in such a way that the bridge undergoes different levels of inertial andkinematic interactions. Fault parallel and fault normal components are applied inlongitudinal and transverse directions, respectively.5.4.1.4 Simulation resultsThe generated baseline data includes the pier drifts, the shear forces, and the bend-ing moments induced at the pier base, the forces induced to the abutments and thespectral responses of the bridge superstructure for the selected earthquake events.The force induced to the abutment is the total force that is transferred from the deckto the abutment system including the back-wall, the wing-walls, and the embank-ment. In this section, the maximum of pier drifts, the shear forces, the bendingmoments, and the forces induced to the abutment in each event are presented anddiscussed. The time histories of the responses for all ten earthquake events arepresented in Appendix D.Fig. 5.23a shows the maximum pier drifts in both longitudinal and transverse73directions for each event. The pier drift varies in the range of 0.35 to 1.5%, andgenerally the drift is larger in the transverse direction than that in the longitudinaldirection because of the lower stiffness of the bridge structure in the transversedirection. Fig. 5.23b presents the maximum shear forces induced at the pier baseshowing that the maximum base shear forces varies in the range of 2.5 to 5.5 MN.Fig. 5.23c illustrates the variation of the maximum bending moments induced atthe pier base during the ten events. The maximum bending moment is equal tothe plastic bending moment, i.e., Mp = 15.76 MN.m, in events 6, 7, 8, 9, and 10.Fig. 5.23d shows the maximum longitudinal and transverse force applied to bridgedeck during the ten events. The longitudinal forces are considerably larger than thetransverse forces in all events. This may be partly due to the larger stiffness of theabutment in the longitudinal direction compared to that in the transverse direction.5.4.2 Analysis of a nine-span bridge system5.4.2.1 Description of the bridgeThe bridge system has nine spans with the length of 37.0 m and the width of10.0 m. The bridge deck is composed of a 0.17 m thick slab, four longitudinalI girders along the deck, and eight transverse I girders located at top of the piers.The deck is supported on 8 piers and two abutments at the two ends. The piersare located 37.0 m apart in the longitudinal direction. There are two expansionjoints above the third and sixth piers, and therefore the bridge deck is subdividedinto three continuous parts connected at the joints. The cross-section area (A), themoment of inertia about axis y (Iy) and the moment of inertia about axis z (Iz) forthe longitudinal girders are 0.8 m2, 0.05 m4, and 0.01 m4, respectively. For thetransverse girders, A, Iy, and Iz are assumed to be 0.1 m2, 0.07 m4, and 0.0002 m4,respectively. There are two expansion joints above the third and sixth piers. Theheight of all eight piers is 10.0 m, and the cross-section of each pier is assumedto be rectangular with a length of 2.5 m (in the transverse direction) and a widthof 1.0 m (in the longitudinal direction). Each pier is reinforced by a total of 44longitudinal rebars, and the cross-section area of each rebar is 1000.0 mm2. Thefoundation underneath each pier is composed of a 6.0 m by 6.0 m pile cap with a74thickness of 1.2 m supported by 25 vertical square piles (a 5×5 pile group). Eachpile is 0.40m wide and 15.0 m long and is reinforced by 12 longitudinal rebars eachwith cross-section area of 500.0 mm2. The abutment system is classified as seat-type abutments with a geometry identical to that of the MRO. The embankment,supporting the back-wall and the wing-walls, is 10.0 m high, and the slope of theembankment on the sides and in front of the abutment back-wall is assumed to be1.0V:1.0H. The walls are supported by 7 vertical square piles (a 7×1 pile group).The piles material and cross-section are the same as those of the piles under thepiers.The nine-span bridge system is assumed to be supported on two idealized soilprofiles with depths of 30.0 m. The embankment material is the same in the twosoil profiles. The embankment is composed of 10.0 m high clay material withunit weight of 20.0 kN/m3, shear wave velocity of 123.5 m/s, and undrained shearstrength of 30.0 kPa. Fig. 5.24 presents schematic of the bridge supported on thetwo soil profiles. The first soil profile consists of a 11.0 m thick soft clay layerunderlain by a 19.0 m thick stiff clay layer. The second soil profile consists ofonly one layer of stiff clay. The properties of the two soil profiles are identicalat the depths larger than 11.0 m below the ground surface. The properties of thesoft and stiff soil layers are selected in a way that follows the soil classificationsof the Geological Survey of Canada (GSC). According to GSC, the shear wavevelocity varies in the range of less than 180.0 m/s for soft soil layers (site class E)and in the range of 360.0 to 760.0 m/s for stiff soil layers (site class C). Fig. 5.25shows the shear wave velocity profiles for sites 1 and 2. The shear wave velocityprofile is directly calculated from parabolic distribution of shear modulus which isgiven as, G = G0(σ ′v/patm)0.5 , where σ ′v is the vertical effective stress, and patmis the standard atmospheric pressure , i.e. 101.0 kPa, G0 is selected in a way thatthe shear wave velocity is less than 180.0 m/s for the soft soil and in the rangeof 360.0 to 760.0 m/s for the stiff soil. Therefore, G0 is assumed to be 40,000.0and 400,000.0 kPa for the soft and stiff soil layers, respectively. Undrained shearstrength (Su) of the two soil profiles are presented in Table 5.5. The undrainedshear strengths in the first 11.0 m below the ground surface in soil profiles 1 and2 are in the range of 20.0 to 30.0 kPa and 72.0 to 129.0 kPa, respectively. For thedepths below 11.0 m, the undrained shear strengths for the two soil profile are the75same and varies in the range of 186.0 to 395.0 kPa. According to the GSC, soilprofile 2 is classified as site class C in which 360 < Vs < 760 m/s and Su > 100.0.5.4.2.2 Model development and simulation detailsThe bridge deck slab (with the length of 333.0 m and the thickness of 0.17 m) ismodeled using 3D linear elastic shell elements. The longitudinal and transverseI girders are all modeled using 3D linear elastic beam-column elements as shownin Fig. 5.26. In the present study, the expansion joints above the third and sixthpiers are simulated as follows; the deck, including the slab and the longitudinal Igirders, is subdivided into three continuous parts which are connected by perfecthinges dictating equal displacements with no constraint on the rotations.Nonlinear fiber sections are adopted for simulating all piers and pile founda-tions. The same material parameters presented in Table 5.2 are used for constitu-tive modeling of the concrete and the steel. Fig. 5.27 shows the fiber discretizationof the pier cross-section and the corresponding cyclic moment-curvature responseunder the axial load of 7.0 MN (tributary weight of the bridge deck). From themoment-curvature analysis, the plastic bending moment of the pier sections is11.4 MN.m in the longitudinal direction (Mp,y=11.4 MN.m) and is 25.6 MN.min the transverse direction (Mp,x=25.6 MN.m). The same material parameters forconcrete and steel are assigned to the material of the pile foundations. Fiber de-scretization of the pile cross-section and its cyclic moment-curvature response un-der the axial load of 0.30 MN are shown in Fig. 5.28. The plastic bending momentof the piles is 0.42 MN.m (Mp,y=Mp,x=0.42 MN.m).The abutment back-wall and the wing-walls are modeled similar to those forthe two and three span bridges. The abutment system is classified as seat-typeabutments, and therefore, the bridge deck is connected to the abutment back-wallthrough perfect hinges using equalDOF constraints for the three translational de-grees of freedom. In the present model, the existing gap between the deck and theback-wall (typical seat-type abutment) is not modeled for simplicity.The finite element mesh of the soil domain is 387.0 m long (along the x axis),110.0 m wide (along the y axis), and 30.0 m deep (along the z axis). The soildomain is modeled using 113,352 eight-node brick elements in which each node76has three translational degrees of freedom. Details of the finite element mesh isshown in Fig. 5.26. Definition of the pressure dependent soil parameters is unnec-essary in this stage of the study. Rather, focus is maintained on reproduction of thesoil hysteretic elastoplastic shear response. To this end, the pressure independentmulti-yield (PIMY) constitutive model, presented in Chapter 4, is used in analy-ses. The PIMY model provides a convenient way of capturing the key nonlineardynamic hysteretic characteristics of soil response that would adversely affect thebridge structure. Table 5.6 presents the input parameters used for the constitutivemodeling of the soil profile 1, the soil profile 2, and the embankment.The procedure of the analysis was already discussed in detail in Chapter 4. Tospeed up the execution time of the dynamic analysis, a total of eighteen 2.4 GHzCPUs with 20 GB of RAM are concurrently used. Further details about the parallelprocessing can be found in Appendix A.The procedure used to determine the predominant period of the bridge systemis the same as that used for the MRO and the three-span bridge. Table 5.7 presentsthe predominant periods of the nine-span bridge supported on soil profile 1 and 2.5.4.2.3 Input earthquake shakingsThe nine-span bridge model is subjected to five ground motions linearly matchedto the uniform hazard spectrum (UHS) of Vancouver with the exceedance proba-bility of 2% in 50 years for site class C in the period range of 0.1 to 1.0 s. Thisperiod range includes the predominant periods of the bridge system supported onthe two different soil profiles. Table 5.8 presents a brief description of the selectedground motions, and Fig. 5.29 compares the spectral response of the scaled mo-tions, both fault-parallel and fault-normal components, with the UHS (2% in 50years) of Vancouver .The selected motions have to be deconvolved to the depth of 30.0 m to deriveincident earthquake motions along the base of the continuum model. The proce-dure adopted for the deconvolution analysis can be found in Section 5.2.4. In thisprocedure, a 30.0 m deep soil column is modeled in ProShake (2003). The de-convolution should be conducted in a soil profile that is classified as site class C,as defined in Geological Survey of Canada (GSC), because the selected ground77motions have been linearly scaled to the UHS which is derived for a site classC. The soil profile 2 represents this site class, and therefore the properties of thesoil profile 2 is used in the ProShake model. The deconvolution analysis is per-formed iteratively to obtain comparable computed motions at the ground surfaceof the continuum model to the selected ground motions. This is achieved by chang-ing the choice of modulus reduction and damping curves of the soil layers in theProShake model. The modulus reduction curve proposed by Sun et al. (1988) forclays with PI = 40− 80 at the depth of 0.0 to 7.0 m and PI > 80 at the depth of7.0 to 30.0 m and the damping curves (upper bound) proposed by Sun et al. (1988)for the whole soil profile are used in the last iteration for all five earthquake events.Figs. 5.30 and 5.31 compare the acceleration response spectra of the motions com-puted at the ground surface of the continuum model with those of the selectedground motions for the fault-parallel and fault-normal components, respectively.The comparisons suggest that these deconvolved ground motions qualify as appro-priate input motions for the analyses. Fault parallel and fault normal componentsare applied in longitudinal and transverse directions, respectively.5.4.2.4 Simulation resultsThe time histories of the input motions, shear forces and bending moments inducedat the base of piers, the forces induced to the abutment system, and the accelerationresponse spectra of the motions computed at the middle of the bridge deck arepresented in Appendix D. In this section, the seismic response of the bridge inevent No. 1 (1978 Tabas earthquake) is only discussed. Fig. 5.32 shows the timehistories of relative displacements of the pier top with respect to the pier base forpiers 1, 4, 6, and 8 of the bridge supported in soil profile 1 and soil profile 2. Ineach of the soil profiles, the relative deflections of all four piers are similar withpeak value of about 8.0 cm implying that the bridge deck moves rigidly during theearthquake shaking. Since the soil profile 2 is much stiffer than the soil profile 1,the time history of pier deflections in soil profile 2 is generally smaller than that insoil profile 1.Fig. 5.33 shows the time histories of shear forces induced at the base of piers1, 4, 6, and 8 for the two soil profiles. In each of the soil profiles, the shear forces78induced at the base of all four piers are the same. The bridge supported on soilprofile 2 appears to undergo smaller base shear forces compared to those of thebridge supported on soil profile 1.The time histories of bending moments induced at the base of piers 1, 4, 6, and8 for the two soil profiles are presented in Fig. 5.34. In soil profile 1, the maximumbending moment in longitudinal and transverse direction is about 37 and 50 % ofthe plastic bending moment of the pier cross section, respectively. In soil profile 2,these values are 36 and 43 %, respectively. The pier cross section remains elasticduring the earthquake event. This is also observed for the rest of the selectedearthquake events.Fig. 5.35 compares the bending moment distribution along the piles underly-ing the piers 4 and 8 in the longitudinal direction, Myy, at the instance of maximumpile displacements. The maximum bending moment is about 80.0 kN.m in soilprofile 1. This value is only 20% of the plastic bending moment of the pile cross-section. In soil profile 2, the maximum bending moment induced to all piles isabout 20.0 kN which is less than 10% of the plastic bending moment . Since soilprofile 2 is much stiffer than soil profile 1, piles embedded in soil profile 2 undergosmaller bending moments. Fig. 5.36 shows the bending moment distribution in thetransverse direction, Mxx. In soil profile 1, the maximum bending moment reaches45% of the plastic bending moment. In soil profile 2, the maximum bending mo-ment reaches 15% of the plastic bending moment. All piles remain elastic duringthe earthquake shaking. This is also observed for the rest of earthquake events.5.5 SummaryThree different bridge systems; two-, three-, and nine-span bridges were simulatedusing 3D continuum modeling method in which nonlinear hysteretic responses offoundation soils and the critical structural components of the bridge, i.e., the piers,are taken into account. The main goals were; (i) demonstrating the potential ofthe large-scale continuum modeling for practical applications, and (ii) providingbaseline data for the next step of the research which focuses on evaluating theexisting simplified approaches used in practice for dynamic analysis of bridges.The methodologies used in the 3D continuum models including finite element79formulation, boundary conditions, system solvers and constitutive models used forsimulating the soil and structural material behaviour are validated by simulatingseismic responses of Meloland Road Overpass (MRO) during the 1979 ImperialValley and 2010 El Mayor-Cucapah earthquakes. The continuum model satisfac-torily captured the recorded responses of the bridge.The baseline data required for evaluation of the substructuring method used forthe design of bridges was provided as follows:• The two-span bridge located at the MRO bridge site was shaken by twoearthquake motions recorded at the bridge site.• The three-span bridge supported on the MRO bridge site was shaken by asuite of ten ground motions with the peak ground acceleration (PGA) varyingin the range of 0.22 to 0.82g and the predominant period varying in the rangeof 0.1 to 1.0 s.• The nine-span bridge was supported on two different soil profile: one con-sists of a very soft layer underlain by a stiff layer and the other consists ofone layer of stiff soil layer that represents the site class C in Vancouver. Thetwo bridge models were shaken by five ground motions which are linearlymatched to UHS (2% in 50 years) of Vancouver for site class C in the periodrange of 0.1 to 1.0 s.Parallel computing techniques were adopted to speed up the execution time ofanalyses. Using this technique, the analysis runtime decreased by a factor of about5 for analyses of the continuum models under the present study.805.6 Tables and figuresTable 5.1: Input parameters for the soil constitutive models (Kwon and El-nashai, 2008).Parametera Clay model Sand modelLayer 1b Layer 3 Layer 5 Embankment Layers 2 and 4Gr (MPa) 60.0 150.0 150.0 19.0 75.0Br (MPa) 300.0 750.0 750.0 90.0 200.0cu (kPa) 35.9 76.6 86.2 20.0 0.0φ (◦) 0.0 0.0 0.0 0.0 33.0γmax 0.1 0.1 0.1 0.1 0.1φPT (◦) - - - - 27.0n - - - - 0.5d1 - - - - 0.4d2 - - - - 2.0c - - - - 0.07a Gr , low-strain shear modulus; Br , low-strain bulk modulus; cu, undrained shear strength; γmax, octahedral shearstrain at which the maximum shear strength is reached; n, a constant defining variation of shear modulus as offunction of mean effective confinement; φ , soil friction angle; φPT , phase transformation angle; d1 and d2,dilation parameters; c, contraction parameter.b See Fig. 5.2 for the soil layers.Table 5.2: Input parameters for the concrete material used in fiber beam-column element.Parameter Confined concrete Unconfined concretef ′c, compressive strength, (kPa) 34474.0 27600.0εc, strain at compressive strength 0.004 0.002f ′cu, crushing strength, (kPa) 21000.0 0.0εcu, strain at crushing strength 0.014 0.00881Table 5.3: Input parameters for the deconvolution analysis in ProShake(2003).Layers∗ ρ (ton/m3) Vs (m/s) G/Gmax curve Damping curveFirst iteration1 1.5 195.0 Clay – PI=10–20 (Sun et al., 1988) Clay – average (Sun et al., 1988)2 and 4 1.9 198.0 Sand (Seed and Idriss, 1970) – average Sand (Seed and Idriss, 1970) – average3 1.8 286.0 Clay – PI=40–80 (Sun et al., 1988) Clay – average (Sun et al., 1988)5 1.8 286.0 Clay – PI>80 (Sun et al., 1988) Clay – average (Sun et al., 1988)Last iteration1 1.5 195.0 Clay – PI=20–40 (Sun et al., 1988) Clay – lower bound (Sun et al., 1988)2 and 4 1.9 198.0 Sand (Seed and Idriss, 1970) – average Sand (Seed and Idriss, 1970) – lower bound3 and 5 1.8 286.0 Clay – PI=40–80 (Sun et al., 1988) Clay – lower bound (Sun et al., 1988)∗See Fig. 5.2 for the soil profile.Table 5.4: Summary of information for the selected ground motions (appliedto the three-span bridge).Event No. PEER-NGA No. Earthquake Name Recording station PGA∗ (g)1 1148 Kocaeli (1999) Arcelik 0.222 1116 Kobe (1995) Shin-Osaka 0.243 1787 Hector Mine (1999) Hector 0.344 721 Superstition Hills-02 (1987) El Centro Imp. Co. Cent 0.365 1244 Chi-Chi (1999) CHY101 0.446 960 Northridge-01 (1994) Canyon Country-W Lost Cany 0.487 1111 Kobe (1995) Nishi-Akashi 0.518 829 Cape Mendocino (1992) Rio Dell Overpass-FF 0.559 767 Loma Prieta (1989) Gilroy Array #3 0.5610 1602 Duzce (1989) Bolu 0.82∗ PGA is the peak acceleration of the geometric mean of the fault normal and fault parallel components of a motion.Table 5.5: Undrained shear strength (Su) values in the soil profiles 1 and 2.Undrained shear strength, Su (kPa)Depth (m) soil profile 1 soil profile 20-5 20.0 72.55-6 20.0 79.06-7 20.0 93.07-11 30.0 129.011-15 186.0 186.015-20 251.0 251.020-25 323.0 323.025-30 395.0 395.082Table 5.6: Input parameters for the soil constitutive model used in the nine-span bridge model.Parameter Soil profile 1 Soil profile 2 EmbankmentGr (MPa) G∗ G∗ 30.0Br (MPa) B∗∗ B∗∗ 140.0cu (kPa) Su∗∗∗ Su∗∗∗ 30.0φ (◦) 0.0 0.0 0.0γmax 0.1 0.1 0.1∗ G is calculated using the equation G = G0(σ ′v/patm)0.5, where G0 is assumed to be 40,000.0 kPa for the softsoil layer and 400,000.0 kPa for the stiff soil layer.∗∗ B = 2G(1+ν)/3(1−2ν), where ν is Poisson’s ratio and assumed to be 0.4 for clays.∗∗∗ Su is obtained from Table 5.5.Table 5.7: Predominant period of the nine-span bridge system in the longitu-dinal and transverse directions.Bridge site Predominant periodLongitudinal direction Transverse directionSoil profile 1 0.40 0.60Soil profile 2 0.30 0.40Table 5.8: Summary of information for the selected ground motions thatare linearly scaled to the UHS of Vancouver (applied to the nine-spanbridge).Event No. PEER-NGA No. Earthquake Name Recording station Scaling factor1 139 Tabas, Iran (1978) Dayhook 1.0362 291 Irpinia, Italy-01 (1980) Rionero In Vulture 3.1873 802 Loma Prieta (1989) Saratoga-Aloha Ave 0.9964 975 Northridge-01 (1994) Glendora-N Oakbank 5.6115 3471 Chi-Chi, Taiwan-06 (1999) TCU075 4.68183(a) (b) (c)(d)Figure 5.1: Meloland Road Overpass(MRO), (a) overall photo, (b) the pier,(c) the abutment, and (d) configuration of the accelerometers (CESMD,2013).842.0 m4.0 mLayer 1: Medium ClayLayer 2: Medium SandLayer 3: Stiff ClayLayer 4: Medium SandLayer 5: Stiff ClayEmbankment: Soft Clay Embankment: Soft Clay4.0 m5.0 m5.0 m8.0 mρ =1.5 (ton/m3)ρ =1.9 (ton/m3)ρ =1.9 (ton/m3)ρ =1.8 (ton/m3)ρ =1.8 (ton/m3)ρ =1.6 (ton/m3) ρ =1.6 (ton/m3)1V:1.5H1V:1.5HFigure 5.2: Schematic representation of soil layers at the MRO site (dimen-sions are not to scale).xyz99.0 m50.0 m20.0 m2 spans @ 32.0 mLeft abutmentRight abutment5x5 pile group 7x1 pile group7x1 pile group15.0 m18.0 mFigure 5.3: The developed finite element continuum model of the MRO in-cluding the structural and geotechnical components of the bridge (visu-alized by GiD, 2013).85(a) 0.02  0.01 0 0.01 0.02 20 1001020Curvature (1/m)Moment (MN.m)(b)(b)Figure 5.4: Modeling of the pier using fiber-section beam column elements(a) fiber discretization of the pier cross-section, and (b) cyclic moment-curvature response.86Acceleration (g)0 0.5 1 1.5 2 2.5 3−0.500.5Time (sec)Transverse dir.−0.500.5  Longitudinal dir.79x30mx20m 99x50mx20m 119x70mx20m(a)Base Shear Force (MN)0 0.5 1 1.5 2 2.5 3−404Time (sec)Transverse dir.−404  Longitudinal dir.79x30mx20m 99x50mx20m 119x70mx20m(b)Figure 5.5: (a) Time history of acceleration at top of the left abutment, and (b)time history of shear forces induced at the pier base in the longitudinaland transverse directions; sensitivity of the bridge structure response tothe model dimensions.87Acceleration (g))0 0.5 1 1.5 2 2.5 3−0.600.6Time (sec)Transverse dir.−0.600.6  Longitudinal dir.Soil ColumnContinuum model(a)Acceleration (g))0 0.5 1 1.5 2 2.5 3−0.600.6Time (sec)Transverse dir.−0.600.6  Longitudinal dir.Soil ColumnContinuum model(b)Figure 5.6: Time histories of acceleration on top of the two soil columnscompared to those at points next to the lateral boundaries of the con-tinuum model (a) on the ground surface, and (b) on top of the embank-ment.88Acceleration (g)0 0.5 1 1.5 2 2.5 3−0.500.5Time (sec)Transverse dir.−0.500.5  Longitudinal dir.15988 elements 26004 elements 31540 elements(a)Base Shear Force (MN)0 0.5 1 1.5 2 2.5 3−404Time (sec)Transverse dir.−404  Longitudinal dir.15988 elements 26004 elements 31540 elements(b)Figure 5.7: (a) Time history of acceleration at top of the left abutment, and (b)time history of shear forces induced at the pier base in the longitudinaland transverse directions; sensitivity of the bridge structure response tothe mesh refinement.89Acceleration (g)0 5 10 15 20 25 30−0.300.3Time (sec)Transverse dir.−0.300.3  Longitudinal dir. RecordedContinuum model(a)Acceleration (g)0 5 10 15 20 25 30−0.300.3Time (sec)Transverse dir.−0.300.3  Longitudinal dir. RecordedContinuum model(b)Figure 5.8: Time histories of the recorded and computed accelerations at theground surface in the longitudinal and transverse directions during the;(a) 1979 Imperial Valley earthquake, and (b) 2010 El Mayor-Cucapahearthquakes.90Acceleration response spectrum (g)0 1 2 3 400.51Period (sec)Transverse dir.00.51Longitudinal dir. RecordedContinuum model(a)Acceleration response spectrum (g)0 1 2 3 400.51Period (sec)Transverse dir.00.51Longitudinal dir. RecordedContinuum model(b)Figure 5.9: Spectral values of the recorded and computed accelerations at theground surface in the longitudinal and transverse directions during the;(a) 1979 Imperial Valley earthquake, and (b) 2010 El Mayor-Cucapahearthquakes.91Acceleration response spectrum (g)0 1 2 3 40123Period (sec)Channel090123 Channel070123 Channel040123 Channel030123 Channel02  RecordedContinuum model(a)Acceleration response spectrum (g)0 1 2 3 40123Period (sec)Channel090123 Channel070123 Channel040123 Channel030123 Channel02  RecordedContinuum model(b)Figure 5.10: Acceleration response spectrum of motions at different locationsof the bridge for the damping ratio of 5% during the; (a) 1979 ImperialValley earthquake, and (b) 2010 El Mayor-Cucapah earthquake.92Displacement response spectrum (m)0 1 2 3 40123Period (sec)Channel090123 Channel070123 Channel040123 Channel030123 Channel02  RecordedContinuum model(a)Displacement response spectrum (m)0 1 2 3 40123Period (sec)Channel090123 Channel070123 Channel040123 Channel030123 Channel02  RecordedContinuum model(b)Figure 5.11: Displacement response spectrum of motions at different loca-tions of the bridge for the damping ratio of 5% during the (a) 1979Imperial Valley earthquake, and (b) 2010 El Mayor-Cucapah earth-quake.93Rel. displacement (m)0 5 10 15 20 25 30−0.04−0.0200.020.04Time (sec)Transverse dir.−0.04−0.0200.020.04 Longitudinal dir.(a)Rel. displacement (m)0 5 10 15 20 25 30−0.04−0.0200.020.04Time (sec)Transverse dir.−0.04−0.0200.020.04 Longitudinal dir.(b)Figure 5.12: Time histories of pier top displacements relative to the pile capin longitudinal and transverse directions during the, (a) 1979 ImperialValley earthquake, and (b) 2010 El Mayor-Cucapah earthquake.94Pile cap rocking (0.001 x rad)0 5 10 15 20 25 30−606Time (sec)Transverse dir.−606 Longitudinal dir.(a)Pile cap rocking (0.001 x rad)0 5 10 15 20 25 30−606Time (sec)Transverse dir.−606 Longitudinal dir.(b)Figure 5.13: Time histories of pile cap rocking in longitudinal and transversedirections during the, (a) 1979 Imperial Valley earthquake, and (b)2010 El Mayor-Cucapah earthquake.95Bending moment (MN.m)0 5 10 15 20 25 30Time (sec)Transverse dir.−2 0 2−808Curvature (0.001x1/m)Longitudinal dir.−808(a)Bending moment (MN.m)0 5 10 15 20 25 30−808Time (sec)Transverse dir.−2 0 2−808Curvature (0.001x1/m)−808 Longitudinal dir.−808(b)Figure 5.14: Moment-curvature response and time histories of bending mo-ment at the pier base during the, (a) 1979 Imperial Valley earthquake,and (b) 2010 El Mayor-Cucapah earthquake.96Deflection (m) −0.04 −0.02 0 0.02 0.04Right abutment−0.04 −0.02 0 0.02 0.04Pier base−0.04 −0.02 0 0.02 0.04−6−3036Force (MN)Left abutment(a)Deflection (m) −0.04 −0.02 0 0.02 0.04Right abutment−0.04 −0.02 0 0.02 0.04Pier base−0.04 −0.02 0 0.02 0.04−6−3036Force (MN)Left abutment(b)Figure 5.15: Force-deflection response at left and right abutments and at thepier base in the longitudinal direction during the, (a) 1979 ImperialValley earthquake, and (b) 2010 El Mayor-Cucapah earthquake.97Deflection (m) −0.04 −0.02 0 0.02 0.04Right abutment−0.04 −0.02 0 0.02 0.04Pier base−0.04 −0.02 0 0.02 0.04−6−3036Force (MN)Left abutment(a)Deflection (m) −0.04 −0.02 0 0.02 0.04Right abutment−0.04 −0.02 0 0.02 0.04Pier base−0.04 −0.02 0 0.02 0.04−6−3036Force (MN)Left abutment(b)Figure 5.16: Force-deflection response at left and right abutments and at thepier base in the transverse direction during the, (a) 1979 Imperial Val-ley earthquake, and (b) 2010 El Mayor-Cucapah earthquake.98−50 −40 −30 −20 −10 0 10 20 30−18−16−14−12−10−8−6−4−20Bending moment, Myy (kN.m)Depth (m)  Pile APile BPile CPile DPile EGround surface Pile headPile tip(a)−50 −40 −30 −20 −10 0 10 20 30−18−16−14−12−10−8−6−4−20Bending moment, Myy(kN.m)Depth (m)  Pile APile BPile CPile DPile E(b)Figure 5.17: Bending moment distribution, Myy (about the transverse axis ofthe bridge), at the instance of maximum pile displacements during the,(a) 1979 Imperial Valley earthquake, and (b) 2010 El Mayor-Cucapahearthquake.99−50 −40 −30 −20 −10 0 10 20 30−18−16−14−12−10−8−6−4−20Bending moment, Mxx (kN.m)Depth (m)  Pile APile BPile CPile DPile EGround surfacePile tipPile head(a)−50 −40 −30 −20 −10 0 10 20 30−18−16−14−12−10−8−6−4−20Bending moment, Mxx(kN.m)Depth (m)  Pile APile BPile CPile DPile E(b)Figure 5.18: Bending moment distribution, Mxx (about the longitudinal axisof the bridge), at the instance of maximum pile displacements dur-ing the, (a) 1979 Imperial Valley earthquake, and (b) 2010 El Mayor-Cucapah earthquake.1002.0 m4.0 mLayer 1: Medium ClayLayer 2: Medium SandLayer 3: Stiff ClayLayer 4: Medium SandLayer 5: Stiff ClayEmbankment: Soft Clay Embankment: Soft Clay4.0 m5.0 m5.0 m8.0 mρ =1.5 (ton/m3)ρ =1.9 (ton/m3)ρ =1.9 (ton/m3)ρ =1.8 (ton/m3)ρ =1.8 (ton/m3)ρ =1.6 (ton/m3) ρ =1.6 (ton/m3)1V:1.5H1V:1.5HXZ3 spans @ 45 m = 135 m0.1 m above the ground surface(a)xyz182.0 m66.0 m20.0 m5x5 pile group 25x5 pile group 17x1 pile group 17x1 pile group 2Abutment 2Abutment 1Pier 1Pier 23 spans @ 45.0 m18.0 m 15.0 m(b)Figure 5.19: (a) Schematic of the bridge superstructure and the underlyingsoil layers (dimensions are not to scale), and (b) 3D finite elementcontinuum model of the three-span bridge system (visualized by GiD,2013).101−200 −150 −100 −50 0 50 100−18−16−14−12−10−8−6−4−20Bending moment, Mxx (kN.m)Depth (m)  Pile APile BPile CPile DPile EPile headPile tipGround surfaceTwo−span bridge (MRO)(a)−200 −150 −100 −50 0 50 100−18−16−14−12−10−8−6−4−20Bending moment (kN.m)Depth (m)  Pile APile BPile CPile DPile EPile tipGround surfaceThree−span bridge(b)Figure 5.20: Bending moment distribution, Mxx, at the instance of maximumpile displacements during the 1979 Imperial Valley earthquake for the,(a) two-span bridge (MRO), and (b) three-span bridge (pile founda-tions of pier 1); comparing the bending moment distribution along thepile foundations of the two bridge.1020 1 2 3 400.511.522.53Period (sec)Acceleration response spectrum (g)  NGA 1148NGA 1116NGA 1787NGA 721NGA 1244(a)0 1 2 3 400.511.522.53Period (sec)Acceleration response spectrum (g)  NGA 960NGA 1111NGA 829NGA 767NGA 1602(b)Figure 5.21: Acceleration response spectrum of the motions (fault-parallelcomponent), for 5% damping, (a) events No. 1 to 5, and (b) events No.6 to 10.1030 1 2 3 400.511.522.53Period (sec)Acceleration response spectrum (g)  NGA 1148NGA 1116NGA 1787NGA 721NGA 1244(a)0 1 2 3 400.511.522.53Period (sec)Acceleration response spectrum (g)  NGA 960NGA 1111NGA 829NGA 767NGA 1602(b)Figure 5.22: Acceleration response spectrum of the motions (fault-normalcomponent), for 5% damping, (a) events No. 1 to 5, and (b) eventsNo. 6 to 10.1041 2 3 4 5 6 7 8 9 1000.511.52Event No.Maximum pier drift (%)  Longitudinal dir.Transverse dir.(a)1 2 3 4 5 6 7 8 9 100246810Event No.Maximum shear force (MN)  Longitudinal dir.Transverse dir.(b)1 2 3 4 5 6 7 8 9 100510152025Event No.Maximum bending moment (MN.m)  Plastic moment, Mp=15.76 (MN.m)Myy, about the transverse axisMxx, about the longitudinal axis(c)1 2 3 4 5 6 7 8 9 1005101520Event No.Maximum force (MN)  Longitudinal dir.Transverse dir.(d)Figure 5.23: Maximum seismic responses of the bridge during the the tenearthquake events including, (a) maximum pier drifts, (b) maximumshear forces induced at the pier base (pier 1), (c) maximum bendingmoments induced at the pier base (pier 1), and (d) forces induced tothe abutment system in longitudinal and transverse directions.10511.0 m Soft clay layerStiff clay layerSoft Clay19.0 m10.0 mρ =1.8 (ton/m3)ρ =2.0 (ton/m3) 1V:1H 1V:1HSoft Clayρ =2.0 (ton/m3)Longitudinal I girders9 spans @ 37 m = 333 mDeck slabXZρ =1.5 (ton/m3)4.0 mExpansion joint Expansion joint15.0 m0.1 m above the ground surface(a) Soil profile 1Stiff clay layerSoft Clay30.0 m10.0 mρ =1.8 (ton/m3)ρ =2.0 (ton/m3) 1V:1H 1V:1HSoft Clayρ =2.0 (ton/m3)Longitudinal I girders9 spans @ 37 m = 333 mDeck slabXZExpansion joint Expansion joint15.0 m0.1 m above the ground surface(b) Soil profile 2Figure 5.24: Schematics of the nine span bridge supported on the, (a) soilprofile 1, and (b) soil profile 2 (dimensions are not to scale).1060 100 200 300 400 500 600 700 800−30−25−20−15−10−50Shear wave velocity (m/s)Depth (m)  Soil profile 1Soil profile 2Figure 5.25: Shear wave velocity profile for bridge sites 1 and 2.107xyz387.0 m110.0 m30.0 m9 spans @ 37.0 m5x5 pile group7x1 pile groupLongitudinal I girdersTransverse I girdersLongitudinal I girdersPier 1Pier 2Pier 3Pier 4Pier 5Pier 6Pier 7Pier 8Abutment 2Abutment 1Expansion jointExpansion joint15.0 m22.0 mFigure 5.26: 3D finite element continuum model of the nine-span bridge sys-tem including the structural and geotechnical components of the bridge(visualized by GiD, 2013).108(a)−0.03 −0.015 0 0.015 0.03−40−2002040Curvature (1/m)Moment (MN.m)(b)−0.03 −0.015 0 0.015 0.03−40−2002040Curvature (1/m)Moment (MN.m)(c)Figure 5.27: (a) 3D fiber cross-section of the eight piers, (b) moment-curvature response of the section about axis x (longitudinal direction),and (c) moment-curvature response of the section about axis y (trans-verse direction).109(a)−0.03 −0.015 0 0.015 0.03−1−0.500.51Curvature (1/m)Moment (MN.m)(b)Figure 5.28: (a) 3D fiber cross-section of the pile foundations, and (b)moment-curvature response of the section in the longitudinal and trans-verse directions.1100 1 2 3 400.511.52Period (sec)Acceleration response spectrum (g)  NGA 139NGA 291NGA 802NGA 975NGA 3471UHS−Vancouver(a)0 1 2 3 400.511.52Period (sec)Acceleration response spectrum (g)  NGA 139NGA 291NGA 802NGA 975NGA 3471UHS−Vancouver(b)Figure 5.29: Acceleration response spectrum of the motions linearly scaled tothe uniform hazard spectrum (UHS) of Vancouver in the period rangeof 0.1 to 1.0 s; (a) fault-parallel components, and (b) fault-normal com-ponents .1110 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  NGA 139 Selected motionComputed motion(a)0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  NGA 291 Selected motionComputed motion(b)0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  NGA 802 Selected motionComputed motion(c)0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  NGA 975 Selected motionComputed motion(d)0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  NGA 3471 Selected motionComputed motion(e)Figure 5.30: Acceleration response spectrum of the selected ground motionsand computed motions at the ground surface of the continuum model(fault-parallel components); evaluating the deconvolution process bycomparing the responses for seven ground motions.1120 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  NGA 139 Selected motionComputed motion(a)0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  NGA 291 Selected motionComputed motion(b)0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  NGA 802 Selected motionComputed motion(c)0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  NGA 975 Selected motionComputed motion(d)0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  NGA 3471 Selected motionComputed motion(e)Figure 5.31: Acceleration response spectrum of the selected ground motionsand computed motions at the ground surface of the continuum model(fault-normal components); evaluating the deconvolution process bycomparing the responses for seven ground motions.113Rel. displacement (m)0 5 10 15 20 25 30−0.08−0.0400.040.08Time (sec)Transverse dir.  Pier 1Pier 4Pier 6Pier 8−0.08−0.0400.040.08 Longitudinal dir.  Pier 1Pier 4Pier 6Pier 8(a)Rel. displacement (m)0 5 10 15 20 25 30−0.08−0.0400.040.08Time (sec)Transverse dir.  Pier 1Pier 4Pier 6Pier 8−0.08−0.0400.040.08 Longitudinal dir.  Pier 1Pier 4Pier 6Pier 8(b)Figure 5.32: Time histories of relative displacements of the pier top with re-spect the pier base for piers 1, 4, 6, and 8 during the 1978 Tabas earth-quake when the bridge is supported on, (a) soil profile 1 (soft soil), and(b) soil profile 2 (stiff soil).114Shear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.  Pier 1Pier 4Pier 6Pier 8−2−1012 Longitudinal dir.  Pier 1Pier 4Pier 6Pier 8(a)Shear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.  Pier 1Pier 4Pier 6Pier 8−2−1012 Longitudinal dir.  Pier 1Pier 4Pier 6Pier 8(b)Figure 5.33: Time histories of shear forces induced at the base of piers 1, 4, 6,and 8 during the 1978 Tabas earthquake when the bridge is supportedon, (a) soil profile 1 (soft soil), and (b) soil profile 2 (stiff soil).115Bending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.  Mp=25.6 MN.mPier 1Pier 4Pier 6Pier 8−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Pier 1Pier 4Pier 6Pier 8(a)Bending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.  Mp=25.6 MN.mPier 1Pier 4Pier 6Pier 8−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Pier 1Pier 4Pier 6Pier 8(b)Figure 5.34: Time histories of bending moments induced at the base of piers1, 4, 6, and 8 during the 1978 Tabas earthquake when the bridge issupported on, (a) soil profile 1 (soft soil), and (b) soil profile 2 (stiffsoil).1160 100 200 300 400 500−18−16−14−12−10−8−6−4−20Bending moment, Myy (kN.m)Depth (m)  Soil profile 1; pier 4Mp=420 kN.mPile APile BPile CPile DPile E(a)0 100 200 300 400 500−18−16−14−12−10−8−6−4−20Bending moment, Myy (kN.m)Depth (m)  Soil profile 1; pier 8Mp=420 kN.mPile APile BPile CPile DPile E(b)0 100 200 300 400 500−18−16−14−12−10−8−6−4−20Bending moment, Myy (kN.m)Depth (m)  Soil profile 1; pier 4Mp=420 kN.mPile APile BPile CPile DPile E(c)0 100 200 300 400 500−18−16−14−12−10−8−6−4−20Bending moment, Myy (kN.m)Depth (m)  Soil profile 1; pier 8Mp=420 kN.mPile APile BPile CPile DPile E(d)Figure 5.35: Comparing bending moment distribution, Myy, in the longitudi-nal direction at the instance of maximum pile displacements during the1978 Tabas earthquake for the, (a) pier 4 in soil profile 1, (b) pier 8 insoil profile 1, (c) pier 4 in soil profile 2, (d) pier 8 in soil profile 2.1170 100 200 300 400 500−18−16−14−12−10−8−6−4−20Bending moment, Mxx (kN.m)Depth (m)  Soil profile 1; pier 4Mp=420 kN.mPile APile BPile CPile DPile E(a)0 100 200 300 400 500−18−16−14−12−10−8−6−4−20Bending moment, Mxx (kN.m)Depth (m)  Soil profile 1; pier 8Mp=420 kN.mPile APile BPile CPile DPile E(b)0 100 200 300 400 500−18−16−14−12−10−8−6−4−20Bending moment, Mxx (kN.m)Depth (m)  Soil profile 1; pier 4Mp=420 kN.mPile APile BPile CPile DPile E(c)0 100 200 300 400 500−18−16−14−12−10−8−6−4−20Bending moment, Mxx (kN.m)Depth (m)  Soil profile 1; pier 8Mp=420 kN.mPile APile BPile CPile DPile E(d)Figure 5.36: Comparing bending moment distribution, Mxx, in the transversedirection at the instance of maximum pile displacements during the1978 Tabas earthquake for the, (a) pier 4 in soil profile 1, (b) pier 8 insoil profile 1, (c) pier 4 in soil profile 2, (d) pier 8 in soil profile 2.118Chapter 6Evaluation of the API p-ysprings: static and dynamicanalysisNo amount of experimentation can ever prove me right; a singleexperiment can prove me wrong.— Albert Einstein (1879–1955)6.1 IntroductionThe accuracy of the 6×6 stiffness matrix, representing the flexibility of the soil-pile foundation system, mainly relies on the appropriateness of the API (2007)backbone curves. Therefore, it is essential to first evaluate those backbone curves.In the following, the API guidelines are used for both static and dynamic analysisof single piles. In addition, 3D continuum modeling method is used for simulatingthe same problems. The results are compared to assess the validity of the p-ymodels.1196.2 Static analysis of single piles6.2.1 Spring method for static problemsThe practical approach for simulating the static response of a laterally-loaded sin-gle pile is formulated based on the concept of beam on a nonlinear Winkler foun-dation (BNWF) in which the single pile is simulated by beam elements supportedon a set of discrete nonlinear springs. The schematic of the method is presentedin Fig. 6.1. The static equilibrium equation for an elastic beam supported by non-linear springs under a lateral load is solved to obtain the static response of the pilefoundation. This equation is given by,d2dx2(EpIpd2ydx2)− khy−W = 0 (6.1)where EpIp is the flexural stiffness of the pile, y is the lateral pile deflection, x isthe direction along the pile, W is any external distributed lateral load along thelength of the pile, and kh is the secant stiffness which is defined as the ratio ofthe soil resisting force to the lateral pile deflection, y. This is obtained using thenonlinear back-bone curves recommended in API (2007). The API curves for pilesembedded in sand, soft clay, and stiff clay were already presented in Chapter 2.The computer program, LPILE v.6 (ENSOFT Inc., 2012b), is commonly used inthe engineering practice to solve the static equilibrium equation. Following thestate of practice the same computer program is used in this thesis to simulate thestatic response of a single pile.Since 1984 several researchers have argued the validity of API p-y curves (e.g.,Murchison and O’Neill, 1984, Gazioglu and O’Neill, 1984, Zhang et al., 2005, Mc-Gann et al., 2011, and most recently Choi et al., 2013). To confirm the conclusionsof the previous studies, 27 static experimental tests performed on free-head singlepiles embedded in sandy and clayey soils are simulated using the spring method.Table 6.1 presents a brief description of the tests. Further details about the soilproperties, the cross-section and the length of the piles are presented in AppendixB. In all tests, the pile undergoes only lateral loading that is applied at the pilehead. The first 10 tests were performed in dry/saturated sands and the rest were120performed in soft and dry/saturated stiff clays.6.2.2 Evaluation of the spring methodFig. 6.2 shows the level of error in estimation of the responses of the pile, i.e.,the pile head displacement and the maximum bending moment along the pile, foreach test. The error is defined as the subtraction of the computed response fromthe measured response divided by the measured response. The positive error in-dicates overestimation of the measured response, and the negative error indicatesunderestimation of the measured response. In sands, the pile response is generallyunderestimated for piles with a diameter greater than 60 cm, while the response isoverestimated for piles with a diameter less than 60 cm. In clays, no specific trendis noted. The error varies in the range of 5 to 150% in estimation of the pile headdisplacement and varies in the range of 5 to 90% in estimation of the maximumbending moment. In all cases except tests 3 and 4, the error in estimation of themaximum bending moment is smaller than the one in estimation of the maximumpile head displacement.6.2.3 Further comments on the application of the spring methodThe drawbacks of the spring method for a static problem can be listed as follows;(i) the backbone p-y curves are case-specific, and they may not appropriately rep-resent the nonlinear inelastic response of the soil surrounding the pile, and (ii) thediscrete springs are one-dimensional, and act independently when pile is loaded.The springs may not appropriately simulate the response of the three-dimensionalsoil domain.To better investigate the drawbacks of the method, continuum modeling methodis used to simulate one of the field tests. To this end, test 12 is selected. This testwas performed on a single pile with a diameter of 319 mm and a wall thicknessof 12.7 mm, and a length of 12.8 m. The pile was embedded in a single layer ofsoft clay with an undrained shear strength of 14.4 kN/m2, and a submerged unitweight of 5.5 kN/m3. Soil elements at the distance greater than 8D (D: pile diame-ter) from the pile are coarser, while the elements at the distance less than 8D fromthe pile are finer. Fig. 6.3 shows the finite element mesh which is halved along the121direction of loading due to the symmetry of the model along the axis y. The pilestiffness, cross-sectional area of the pile and the applied static loads are halved.The methodology used to simulate soil-pile interaction was already discussed indetail in Chapter 4. The finite element mesh contains 6525 elements to represent asoil domain with a length and width of 10.0 m and depth of 16.0 m which is largeenough to minimize the effects of boundaries on the response of the pile. Boundaryconditions are set in the following way,• The nodes located at the base of the mesh are fixed in all directions.• The nodes at equal depths on the lateral boundaries are constrained to haveequal displacements.• The nodes on the plane which halves the model constrained along the axis y.• All other internal nodes are free to move in any direction.The nonlinear inelastic stress-strain response of the soil is simulated using thePIMY constitutive model. The input parameters for the model are presented inTable. 6.2.The results from the pushover analysis of the continuum model are used todetermine the p-y curves. The determination of p-y curves from the results of apushover analysis was already presented in Section 4.4. These curves are usedin the spring method, and the results are compared with the measured values andthe results of the continuum model. Fig. 6.4 presents the pile head load versusthe pile head deflections and the maximum bending moments along the pile. Thecontinuum model satisfactorily captures the measured pile head deflections andmaximum bending moments. The spring model using API p-y curves estimates thepile head deflections and the maximum bending moments with errors of 92% and15%, respectively. It is noted that if the p-y curves are derived from the continuummodel and used as the backbone curves in the spring model, the pile response ispredicted satisfactorily with an error less than 2%. This clearly implies that thespring method can be an efficient tool for static analysis of pile foundations onlyif appropriate backbone curves are defined for the springs. The spring method willbe evaluated for dynamic soil-pile interaction problems in the following section.1226.3 Dynamic analysis of single piles6.3.1 Spring method for seismic problemsThe state of practice to model soil-pile interaction for a single pile subjected toseismic loadings consists of two consecutive steps as shown in Fig. 6.5. In thefirst step, site response analysis is carried out to determine the depth-varying timehistories of absolute displacement, and in the second step dynamic analysis of thepile is carried out by applying the time histories of absolute displacements to theground nodes of the springs. The dynamic equilibrium equation of an elastic beamsupported by non-linear springs is given by,EpIp∂ 4y∂x4 +m(∂ 2y∂ t2 +∂ 2yg∂ t2)+ c(∂y∂ t −∂y f f∂ t)+ kh (y− y f f ) = 0 (6.2)where EpIp is the flexural stiffness of the pile, m is the pile mass per unit length,y is the relative displacement of the pile with respect to the base excitation (yg),y f f is the relative free-field displacement with respect to the base excitation, c isthe damping coefficient, and kh is the secant stiffness which is defined as the ratioof the soil resisting force to the relative displacement of the pile (y). To determinekh at each loading step, API p-y curves are used as back-bone curves, and Masingrule is used to model the subsequent hysteretic curves.The validity of the spring method in predicting seismic response of a single pileis assessed by simulating two centrifuge tests conducted by Gohl (1991) on a singlepile in sands and by Wilson (1998) on a single pile in clays. Table 6.3 presents abrief information about the tests. Detailed description of the tests are presented inAppendix C. Amongst the performed series of tests, test No. 12 in the centrifugetest of Gohl (1991) and Csp5 event B in the centrifuge test of Wilson (1998) are ofthe interest of this study. Fig. 6.6 shows the time histories of acceleration appliedto the base of the soil container in the two centrifuge tests. The peak accelerationin the former test is 0.15g and in the latter is 0.14g.The spring model of the centrifuge tests is developed in OpenSees (McKennaand Fenves, 2001) to solve the dynamic equilibrium equation, i.e., Eq. 6.2. The123depth-varying time histories of absolute displacement are computed using the pro-cedure presented in Section 3.2. To this end, the study remains focused on theevaluation of the soil-pile interaction modeling.6.3.2 Continuum modeling methodThe two centrifuge tests are also simulated using the 3D continuum modelingmethod. The finite element meshes are shown in Fig. 6.7. The lumped mass on thepile head represents the superstructure. Due to the symmetry, the model is halved atthe line of symmetry along the center-line of the pile, and stiffness, cross-sectionalarea of the pile and the mass of the superstructure are halved. The mesh for the firstcentrifuge test consists of 2704 solid elements representing a soil domain with thelength of 31.49 m, the width of 10.49 m, and the depth of 12.0 m (Fig. 6.7a). Themesh for the second centrifuge test consists of 5400 solid elements representing asoil domain with the length of 54.2 m, the width of 20.2 m, and the depth of 17.5 m(Fig. 6.7b). The finite element meshes are large enough to minimize the effects ofboundaries on the response of the pile. The whole soil deposit in the container issimulated in both continuum models. Boundary conditions are set in the same wayas presented in Section 6.2.3.The input parameters for the single sandy layer in the centrifuge test of Gohl(1991) and clay and sandy layers in the centrifuge test of Wilson (1998) are pre-sented in Table 6.4. The input parameters for the sand in the first centrifuge testare obtained from the study of Yang et al. (2003) where the PDMY model was cal-ibrated for Nevada sands with Dr=40%. In the second centrifuge test, low-strainshear modulus of clays is derived from available Torvane measurements (i.e., cu,undrained shear strength) using the equation G = 200cu (Bowles, 1996). The pa-rameters of the underlying sandy layer are extracted from the values reported byPopescu and Prevost (1993) for Nevada sands with Dr=80%.6.3.3 Evaluation of the spring methodFig. 6.8 compares the spectral response of the motions measured and computed atthe free-field ground surface in both centrifuge tests. The free-field in the contin-uum model is considered as the distance greater than 8D. There is a good agree-124ment between the measured and computed spectral responses in all periods.Fig. 6.9 shows the acceleration response spectra of the measured and computedmotions of the superstructure in the two centrifuge tests. Fig. 6.10 presents themeasured and computed bending moment profiles at the instance of maximum pilehead displacement in the two centrifuge tests. It is shown that the continuum modelsatisfactorily captures the pile responses (i.e., the spectral response and the bendingmoment distribution); while the spring model poorly predicts the pile responses. Inthe first centrifuge test, the spectral response of the superstructure is considerablyoverestimated for the periods less than 1.0 s, and the maximum bending moment isoverestimated with an error of 72%. In the second test, the spectral response of thesuperstructure is significantly underestimated in the period range of 1.0 to 1.6 s,and the maximum bending moment is underestimated with an error of 47%.6.3.4 Further comments on the application of the spring methodSome of the major drawbacks of the spring method, when used for a seismic prob-lem, can be listed as follows: (i) the nonlinear hysteretic response of the soil maynot be adequately simulated during the site response analysis, and therefore, thedepth-varying time histories of the displacements may not represent the displace-ments of the free-field, (ii) the discrete springs are one-dimensional, and act inde-pendently during the excitation. The springs may not appropriately simulate theresponse of the three-dimensional soil domain and the radiation damping, (iii) thebackbone p-y curves are case-specific, and they may not appropriately representthe nonlinear hysteretic response of the soil and the resulting material damping,(iv) the mass of the soil domain that is affected by the soil-pile interaction is notaccounted for in the analysis. Boulanger et al. (1999) and Allotey and El Naggar(2008) reported that among the existing drawbacks of the method, the input free-field motions are the major one. In this study, since the error in calculations ofthe input free-field motions has been minimized (see in Fig. 6.8) the errors can beattributed to the second and third drawbacks.Fig. 6.11 compares the measured spectral response at the superstructure withthe computed ones for which two different spring models are used; (i) model A, aspring model that employs the API (2007) guidelines, and (ii) model B, a spring125model that employs the p-y backbone curves derived from the pushover analysisof the continuum model. The use of appropriate p-y backbone curves does notresult in any improvement in prediction of the measured spectral response. Thiscan be partly due to the inappropriate simulation of damping in both models. Al-though the backbone curves are adequately determined in model B, the subsequentunloading-reloading paths that represents the hysteric response of the soil and thematerial damping are poorly simulated. Fig. 6.11 shows that the spectral responseobtained from model B is larger than that obtained from model A in both centrifugetests. This implies that the material damping in model B is less that the materialdamping in model A. However, significant improvement is noted in prediction ofthe maximum bending moment distributions along the pile. As shown in Fig. 6.12,by using the model B the error in prediction of the maximum bending moment de-creases from 72 to 16 %, and from 47 to 30% for the former and latter centrifugetests, respectively.As mentioned earlier, one of the drawbacks of the spring method is the missingmass of the soil surrounding the pile. To investigate the effects of the missing masson the overall response of the soil-pile system, an eigenvalue analysis is performedto determine the modal periods of four different models: (i) the model A, (ii) themodel B, (iii) model C, the continuum model which satisfactorily captures themeasured responses in the centrifuge tests, and (iv) model D, the same continuummodel with massless soil domain. Fig. 6.13 compares the natural vibration periodsof the models A, B, C, and D in their first three modes. There is quite a significantdifference between the second and third natural periods of models A, B, and Dwith model C. This difference is very small when comparing the natural periodsof the the spring models (models A and B) with those of the massless continuummodel (model D) . Therefore, the observed difference is due to the missing massof the soil domain in the spring model. The differences in the first mode is smallercompared to those in the second and third modes implying that mass of the soildomain does not affect the first natural period of the soil-pile system. In addition,the use of appropriate p-y backbone curves in the spring model may slightly reducethe difference in the first mode (see in Fig. 6.13b).1266.4 SummaryIn this chapter, twenty-seven full-scale tests on single piles were simulated usingthe spring method in which API backbone curves were used to characterize thenonlinear springs. The error was in the range of 5 to 150% in estimation of pilehead displacements and was in the range of 5 to 90% in estimation of maximumbending moments. In sands, the pile head displacements and maximum bend-ing moment were generally underestimated for piles with a diameter greater than60 cm, while they were overestimated for piles with a diameter less than 60 cm.In clays, no specific trend was noted. The significant levels of error imply thatthe API backbone curves are case-specific and cannot be generalized for differentconditions of piles and soils.For simulation of single piles subjected to static loading, if appropriate p-ybackbone curves were used to characterize the nonlinear springs, the spring methodmight satisfactorily predict static response of a single pile. Determination of appro-priate backbone curves for different problems can be impractical in the engineer-ing practice. In this study, appropriate p-y backbone curves were directly obtainedfrom the results of a pushover analysis of the continuum model. For seismic prob-lems, the spring method is composed of two separate steps; (i) site response analy-sis, and (ii) dynamic analysis of the pile supported on a series of nonlinear springs.In order to evaluate the spring method, two dynamic centrifuge tests performedon single piles in clays and sands were simulated following API guidelines. Thespring model poorly predicted the seismic responses of the piles (i.e., spectral re-sponse of the superstructure and bending moment distribution). The spring modeloverestimated the maximum bending moment by 72% for the pile in sand and un-derestimated it by 47% for the pile in clays.For the two dynamic centrifuge tests studied here, use of appropriate p-y curvesdid not improve the prediction of the seismic response of the pile implying that thespring method suffers some other drawbacks too. The discrete springs are one-dimensional, and act independently during the excitation. The springs may notappropriately simulate: (i) the unloading-reloading paths and the resulting materialdamping, (ii) the continuous three-dimensional configuration of the soil domain,(iii) the mass of that portion of the soil domain which is affected by the kinematic127and inertial interactions between the bridge system and the soil.1286.5 Tables and figuresTable 6.1: Brief description of the 27 experimental static tests performed on laterally loaded single piles(further details of the tests are presented in Appendix B.Test no. Test type∗ Pile type Pile diameter (cm) Soil type Reference1 field steel pipe 41.0 saturated sand Mansur et al. (1964)2 field steel pipe 5.0 saturated sand Parker et al. (1970)3 field steel H-shape (16WF26) - dry sand Mason and Bishop (1953)4 field steel H-shape (14H17) - dry and saturated sand Murchison (1983)5 field steel pipe 61.0 saturated sand Cox et al. (1974)6 field steel pipe 27.0 dry sand Brown et al. (1988)7 static centrifuge steel pipe 122.0 dry sand Georgiadis et al. (1991)8 static centrifuge steel pipe 43.0 dry sand McVay et al. (1995)9 static centrifuge steel pipe 72.0 dry sand Mezazigh and Levacher (1998)10 field steel pipe 51.0 dry and saturated sand Murchison (1983)11 field steel pipe 32.0 soft clay Matlock (1970)12 field steel pipe 32.0 soft clay Meyer (1979)13 field steel pipe 11.5 soft clay Gill (1968)14 field steel pipe 22.0 soft clay Gill (1968)15 field steel pipe 32.5 soft clay Gill (1968)16 field steel pipe 41.0 soft clay Gill (1968)17 field steel pipe 102.0 soft clay Kim and Jeong (2011)18 field drilled shaft 240.0 soft clay Kim and Jeong (2011)19 field steel pipe 64.0 saturated stiff clay Reese et al. (1975)20 field steel pipe 11.5 saturated stiff clay Gill and Demars (1970)21 field steel pipe 22.0 saturated stiff clay Gill and Demars (1970)22 field steel pipe 32.5 saturated stiff clay Gill and Demars (1970)23 field steel pipe 41.0 saturated stiff clay Gill and Demars (1970)24 field drilled shaft 90.0 saturated stiff clay Halloway (1978)25 field steel pipe 41.0 dry stiff clay Price and Wradle (1981)26 field steel pipe 85.0 dry stiff clay Jun et al. (2008)27 field steel pipe 76.0 dry stiff clay Reese and Welch (1975)∗ In all tests, the lateral load is monotonically applied at the pile head of a free-head single pile.129Table 6.2: Input parameters for the PIMY constitutive model used for simu-lating the soil in test no. 12.Parameter∗ valueGr (MPa) 1.08Br (MPa) 2.34cu (kPa) 14.4φ (◦) 0.0γmax 0.1∗ Gr , low-strain shear modulus; Br , low-strain bulk modulus; cu, undrained shear strength; γmax, octahedral shearstrain at which the maximum shear strength is reached; and φ , soil friction angle.Table 6.3: Brief description of two centrifuge tests performed on single piles subjected toearthquake shakings (further details of the tests are presented in Appendix C.Test no. Test type Pile type Pile diameter (cm) Soil type Reference1 dynamic centrifuge steel pipe 57.0 dry sand Gohl (1991)2 dynamic centrifuge steel pipe 67.0 saturated clay and sand Wilson (1998)Table 6.4: Input parameters for the soil constitutive models used for simulat-ing the soil behavior in the dynamic centrifuge tests.Parameter∗ Gohl (1991) Wilson (1998)Sand Clay (0–1.5m) Clay (1.5–3.0m) Clay (3–4.5m) Clay (4.5–6.0m) SandGr (MPa) 33.0 0.5 1.3 1.8 2.4 41.5Br (MPa) 22.0 2.33 6.1 8.4 11.2 90.0cu (kPa) 0.0 2.5 6.5 9.0 12.0 0.0φ (◦) 35.0 0.0 0.0 0.0 0.0 39.5γmax 0.1 0.1 0.1 0.1 0.1 0.1φPT (◦) 26.5 - - - - 27.0n 0.5 - - - - 0.5d1 0.5 - - - - 0.6d2 100.0 - - - - 3.0c 0.18 - - - - 0.05∗ Gr , low-strain shear modulus; Br , low-strain bulk modulus; cu, undrained shear strength; γmax, octahedral shearstrain at which the maximum shear strength is reached; n, a constant defining variation of shear modulus as offunction of mean effective confinement; φ , soil friction angle; φPT , phase transformation angle; d1 and d2,dilation parameters; c, contraction parameter.130ypAPI p-y curveFixedLateral Loadpukk: initial subgrade reaction moduluspu: ultimate lateral soil resistanceFigure 6.1: Schematic of the spring method used in practice for static analysisof laterally loaded single piles.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27−80−4004080120160Test NumberError (%)  DeflectionBending moment× 100Computed − Measured Error (%) = MeasuredFigure 6.2: Level of error in prediction of maximum displacement of pilehead and maximum bending moment along the pile shaft for the testslisted in Table 6.1; the positive error indicates overestimation of themeasured response, and the negative error indicates underestimation ofthe measured response. (note: bending moments were not reported intests 8, 10, 13–16, and 20–26).13110.0 m16.0 m5.0 mSurrounding soil elementsFive rigid beam-column elementsPile elementx yzLoading dir.Figure 6.3: Finite element mesh of test No. 12 (visualized by GiD, 2013).0 0.05 0.1 0.15 0.2020406080100Pile head deflection (m)Lateral load at pile head (kN)  MeasuredContinuum modelSpring model (API, 2007)Spring model (derived from the continuum model)(a)0 50 100 150 200 250 300020406080100Maximum bending moment (kN.m)Lateral load at pile head (kN)  MeasuredContinuum modelSpring model (API, 2007)Spring model (derived from the continuum model)(b)Figure 6.4: (a) Pile head load vs pile head deflections, and (b) pile head loadvs maximum bending moments in Test 12; comparing the measuredvalues with those computed by the continuum and spring methods.132Input motion(a) Ground surfaceLayer 1Layer NLayer 2Layer 3....Depth-varying time histories of displacement in the free-field .(a)(b) Time history of absolute displacement (computed in the first step)dFAPI backbone curveMasing ruleGround node(b)Figure 6.5: Schematic of the spring method used for dynamic analysis of asingle pile, (a) first step: site response analysis, and (b) second step:dynamic analysis of the pile.1330 5 10 15 20 25 30−0.2−0.100.10.2Time (sec)Acceleration (g)(a)0 5 10 15 20 25 30−0.2−0.100.10.2Time (sec)Acceleration (g)(b)Figure 6.6: Time histories of input acceleration in the centrifuge test of (a)Gohl (1991), and (b) Wilson (1998).13431.5 m12.0 m10.5 mx yz Single layer of sand(a)54.2 m17.0 m20.2 m(Clay layer)(Sand layer)6.0 m11.0 mx yz(b)Figure 6.7: Finite element meshes of a single pile system in the centrifugetest of (a) Gohl (1991), and (b) Wilson (1998). The finite elementmeshes are halved along the direction of shaking (x axis) due to thesymmetry along y axis.1350 0.5 1 1.5 2 2.5 3 3.5 400.511.5Period (sec)Acceleration Response Spectrum (g)  MeasuredContinuum model(a)0 0.5 1 1.5 2 2.5 3 3.5 400.511.5Period (sec)Acceleration Response Spectrum (g)  MeasuredContinuum model(b)Figure 6.8: Acceleration response spectra (for damping ratio of 5%) at theground surface in (a) the centrifuge test of Gohl (1991), and (b) thecentrifuge test of Wilson (1998); comparing the measured responseswith those computed in the continuum model.0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration Response Spectrum (g)  MeasuredContinuum modelSpring model(a)0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration Response Spectrum (g)  MeasuredContinuum modelSpring model(b)Figure 6.9: Acceleration response spectra (for damping ratio of 5%) at thesuperstructure in (a) the centrifuge test of Gohl (1991), and (b) the cen-trifuge test of Wilson (1998); comparing the measured and the com-puted responses in the continuum and the spring models.136−200 0 200 400 600 800−12−9−6−30Bending moment (kN.m)Depth (m)  MeasuredContinuum modelSpring model(a)−200 0 200 400 600 800−12−9−6−30Bending moment (kN.m)Depth (m)  MeasuredContinuum modelSpring model(b)Figure 6.10: Bending moment profile at the instance of pile maximum dis-placement in (a) centrifuge test of Gohl (1991), and (b) centrifugetest of Wilson (1998); comparing the measured and the computed re-sponses in the continuum and the spring models.0 0.5 1 1.5 2 2.5 3 3.5 400.511.522.53Period (sec)Acceleration Response Spectrum (g)  MeasuredSpring model A (API, 2007)Spring model B (derived from the continuum model)(a)0 0.5 1 1.5 2 2.5 3 3.5 400.511.522.53Period (sec)Acceleration Response Spectrum (g)  MeasuredSpring model A (API, 2007)Spring model B (derived from the continuum model)(b)Figure 6.11: Acceleration response spectra (for damping ratio of 5%) at thesuperstructure in (a) the centrifuge test of Gohl (1991), and (b) the cen-trifuge test of Wilson (1998); comparing the measured and the com-puted responses in the spring models A and B.137−200 0 200 400 600 800−12−9−6−30Bending moment (kN.m)Depth (m)  MeasuredSpring model A (API, 2007)Spring model B (derived from the continuum model)(a)−200 0 200 400 600 800−12−9−6−30Bending moment (kN.m)Depth (m)  MeasuredSpring model A (API, 2007)Spring model B (derived from the continuum model)(b)Figure 6.12: Bending moment profile at the instance of pile maximum dis-placement in (a) the centrifuge test of Gohl (1991), and (b) the cen-trifuge test of Wilson (1998); comparing the measured and the com-puted responses in the spring models A and B.0 1 2 3 4 500.20.40.60.811.21.4Mode numberNatural period (sec)  Spring model A (API, 2007)Spring model B (derived from the continuum model)Continuum model CContinuum model D (massless soil domain)(a)0 1 2 3 4 500.20.40.60.811.21.4Mode numberNatural period (sec)  Spring model A (API, 2007)Spring model B (derived from the continuum model)Continuum model CContinuum model D (massless soil domain)(b)Figure 6.13: Natural vibration periods of the first three modes of the soil-pilesystem in (a) the centrifuge test of Gohl (1991), and (b) the centrifugetest of Wilson (1998); comparing the natural periods in the continuummodel and the spring model.138Chapter 7Simulation of the bridge systemsusing the substructuring method:analysis and evaluationEssentially, all models are wrong, but some are useful.— George E.P. Box (1987)7.1 IntroductionThe most common approach used in practice for simulation of the seismic responseof bridge systems is the substructuring method which is formulated on the basis ofsuperposition principles. Assuming linear elastic SSI responses, the stiffness andinput motions are calculated without accounting for the effects of the seismic re-sponses of the bridge superstructure. Then, the bridge superstructure is supportedon a sets of equivalent linear springs and dashpots, and the system is subjected toearthquake shaking at the supports. This procedure was explained in five consecu-tive steps in Chapter 3.In this chapter, the five steps of the substructuring method are followed to sim-ulate the seismic responses of the two-, three-, and nine-span bridge systems whichwere presented in Chapter 5. For each bridge system, the free-field motions, the139kinematic input motion at the pile cap, dynamic stiffness of the pile groups under-neath the piers, and ultimately the 3D substructure model are determined follow-ing the five-step procedure. The seismic responses that are of interest here includethe pier drifts, shear forces and bending moments induced at the pier base, load-deflection response at the bridge supports, and the spectral responses of the bridgestructure. The results are compared to the corresponding responses simulated bythe continuum model, and some comments about the validity of the substructuringmethod are provided.7.2 Analysis of the two-, three-, and nine-span bridges7.2.1 Free-field motionsAs part of the substructuring method, a site response analysis is required to obtainthe depth-varying time histories of the ground displacement in the free field (see inFig. 3.1a). For each one of the bridge systems that are simulated in this study, twosite response analyses are needed; one for the embankment without accounting forthe SSI effects of abutment system and the other for the foundation soil withoutaccounting for the SSI effects of the pile foundations and the bridge structure ontop.The finite element model of the free-field soil profile consists of a column of3D brick elements for which advanced constitutive models are defined to obtainrealistic estimates of soil nonlinear hysteric behavior. The constitutive models werealready discussed in Chapter 4. Boundary conditions are set in the following way:• The nodes located at the base of the mesh are fixed in all three directionsduring the self-weight analysis stage. The fixity of base nodes are then re-moved in the horizontal directions for applying the seismic shakings at thebase.• The nodes at equal depths are constrained to have equal displacements in allthree directions.Fig. 7.1 shows the finite element mesh of two soil columns simulating the free-field of the foundation soil and the embankment. The input motion is applied to the140base nodes in the form of displacement using the multi-support excitation pattern(the same method used for shaking the continuum model). The outputs of theanalyses are the depth-varying time histories of the ground displacement and theprofile of the maximum ground displacement.7.2.2 Dynamic stiffnesses of the pile group and the abutmentThe force-displacement backbone curves at various depths of the pile foundationsand the at the abutments are determined based on the recommendations of API(2007) and Caltrans (2013). The lateral force-displacement backbone curve atthe pile cap is determined following the procedure presented by Zafir (2002), i.e.,Eq. 3.1. The equivalent secant stiffnesses are then calculated at the correspondingmaximum ground displacements. As an example, the procedure is illustrated inFig. 7.2 with reference to the MRO site during the 1979 Imperial Valley earthquake.Fig. 7.2a shows the API p-y curve at the depth of 2.0 m for the pile foundationsunderneath the bridge pier. The secant stiffness is calculated at the displacement of0.023 m which is the maximum ground displacement in the free-field of the foun-dation soil in the transverse direction of the bridge during the 1979 Imperial Valleyearthquake. The secant stiffness is calculated to be 75.53/0.023 = 3,284.0 kN/m.Accounting for the group reduction factor, the secant stiffness to be used in calcu-lating the 6×6 stiffness matrix decreases to 3284.0×0.42 = 1,379.0 kN.m, where0.42 is the average of the p-multipliers recommended in AASHTO for differentrows of a pile group with spacing of 3D. Fig. 7.2b presents the API t-z curve atthe depth of 2.0 m for the pile foundations underneath the bridge pier. To linearizethe API t-z, the settlement of the pile group needs to be calculated. To this end,the pile group, which is supported on nonlinear API p-y, t-z and Q-z springs, ismodeled in the computer program GROUP v.8 (2012a). The tributary weight ofthe deck, the pier, and the pile cap (i.e., 9.73 MN) are vertically applied to themodel. The output of this analysis is the settlement of the pile cap. The secantstiffness is calculated at the displacement of 0.004 m which is the calculated set-tlement of the pile group under the vertical load of 9.73 MN. The secant stiffnessis 10.725/0.004 = 2,680.0 kN/m. Fig. 7.2c shows the API Q-z curve at the piletip (depth of 17.0 m). The secant stiffness is calculated at the displacement of1410.004 m which is the settlement of the pile group. The secant stiffness at the piletip is 30.2/0.004= 7,750.0 kN/m. Fig. 7.2d presents the nonlinear backbone curverecommended in Caltrans for simulating longitudinal response of the integral abut-ment system. The secant stiffness is calculated at the displacement of 0.002 mwhich is the maximum displacement of the embankment top in the free-field withrespect to that at the depth of 3.0 m, i.e., the depth of the button of the abutmentwall. The secant stiffness is calculated to be 821.0/0.002 = 410,500.0 kN/m.The secant stiffnesses are then used to create a 3D numerical model of the pilegroup in the computer program, GROUP. Lateral and vertical forces and bendingmoments are then applied to the pile cap all together (see in Fig. 3.1c). Since thesystem is linear elastic, the magnitude of the forces is not important in calculatingthe 6 × 6 stiffness matrix. Torsional stiffness of the system is beyond the scope ofthis research. Therefore, torsional stiffness of the piles are not taken into accountin the analysis. The calculated 6 × 6 stiffness matrix is composed of five diagonalelements representing the stiffnesses in the horizontal, vertical, and two rotationaldirections; and four off-diagonal elements representing the coupling effects be-tween horizontal displacements and rocking of the pile cap. The elements of the6×6 damping matrix are calculated using Eq. 3.2. Following Lee et al. (2011), thedamping ratio is assumed to be 25% in both longitudinal and transverse directions.Table 7.1 presents the secant stiffnesses and the corresponding dashpot coefficientsrepresenting the equivalent dynamic stiffnesses of the MRO during the 1979 Im-perial Valley and the 2010 El Mayor-Cucapah earthquakes. The same procedure isapplied to determine the spring and dashpot constants for the three-span and nine-span bridge systems under the selected ground motions. Tables E.1, E.2, E.3, andE.4 in Appendix E present the constants of the springs and dashpots that are usedfor the simulation of the three and nine-span bridge systems.7.2.3 Kinematic input motionsTo obtain the kinematic motion on top of the pile cap, a massless finite elementmodel of the pile group supported by a series of equivalent linear springs and dash-pots is created in OpenSees framework. In this model, the constants of the lateraland vertical springs at different depths are the secant stiffness values which are142determined by linearizing the API p-y, t-z and Q-z curves. The constant of thedashpots are then calculated using the Eq. 3.2. The depth-varying time histories ofdisplacement in the free-field are then applied to the ground nodes of the springsin the massless pile group (see in Fig. 3.1d). A dynamic time-history analysis isperformed to obtain the kinematic motion at top of the pile cap.The kinematic motion at the abutment is obtained from the site response analy-sis of the embankment without accounting for the SSI effects. The time histories ofdisplacement at the top of the soil column are used as the kinematic input motionat top of the abutment system.7.2.4 Developing the global model of the bridgesThe elements and materials used to model the bridge deck and the piers are iden-tical to those used in the continuum models. In practice however, it is common tosimplify the bridge deck by simulating equivalent linear beam-column elements,called a stick model of the bridge. This simplification is not used in the presentwork to maintain the focus of the study on the evaluation of the SSI models usedat the bridge supports. The bridge superstructure is supported by the equivalentlinear springs and dashpots at the base of the piers and top of the abutments. Thebridge deck is fixed in the vertical direction at the abutments based on the assump-tion that the settlement of the abutment system including the walls and the piles isnegligible.Fig. 7.3 shows the substructure models of the two-, three-, and nine-span bridgesystems developed in the OpenSees finite element framework. The translationaland rotational springs and dashpots are modeled using the zerolength element,and the cross-coupling ones are simulated using the zerolengthCoupled elementas available in OpenSees. A uniaxial linear elastic constitutive model in parallelwith a uniaxial linear viscous constitutive model is assigned to the zerolength andzerolengthCoupled elements to represent the dynamic stiffness.The input motions are applied to the free ends of the springs in the form ofdisplacements using the multi-support excitation pattern in OpenSees. With theaid of the multi-support excitation pattern, it is possible to apply different timehistories of displacements at the pile caps and at the abutments.1437.3 Evaluating the substructure model of the bridges7.3.1 Seismic responses of the two-span bridgeThe time histories of acceleration were recorded at various locations of the two-span bridge, i.e., MRO, during the 1979 Imperial Valley and 2010 El Mayor-Cucapah earthquakes. The location of five of the channels is shown in Fig. 7.4.Acceleration response spectrum (for the damping of 5%) of the recorded and com-puted motions at these locations are compared in Fig. 7.5. The results obtainedfrom the continuum model of the bridge is also presented in the figure. Duringthe Imperial Valley earthquake, the spectral response at all channels is satisfacto-rily predicted by the substructure model. However, during the El Mayor-Cucapahearthquake the spectral response is poorly predicted. For example, the spectral re-sponses at channels 3, 7 and 9 are significantly overestimated for the periods vary-ing in the range of 0.5 to 1.0 s. The level of error during the Imperial Valley earth-quake is negligible compared to that during the El Mayor-Cucapah earthquake. Forclarity, the spectral response of the abutment in the transverse direction (point Ashown in Fig. 7.4) is compared to the spectral response of the embankment top(point B) where the SSI effects on the response of the soil is negligible. The resultsare shown in Fig. 7.6. During the former earthquake the spectral responses aresimilar while during the latter earthquake the spectral response of the abutment topis different from that of the free-field. This may imply that the kinematic and in-ertial interactions between the bridge structure and the embankment are significantduring the El Mayor-Cucapah earthquake, however they are insignificant duringthe Imperial Valley earthquake. It can be concluded that the springs and dashpotswhich are supposed to simulate the kinematic and inertial interactions do not workproperly.Fig. 7.7 compares the computed time histories of bending moment at the pierbase using the continuum and the substructure models during the Imperial Valleyand El Mayor-Cucapah earthquakes. Using the substructure model, the maximumbending moment is overestimated during the two events by a factor of 2.0 and3.0, respectively. The error during the El Mayor-Cucapah earthquake is largercompared to that during the Imperial Valley earthquake. The substructure model144predicts yielding of the pier cross section at the times t=4.0 s and t=12.0 s duringthe Imperial Valley and El Mayor-Cucapah earthquakes, respectively. However,the continuum model predicts an elastic response of the pier in both earthquakes.Fig. 7.8 shows the time histories of pier top relative displacements with respect tothe pier base. The pier displacements are also significantly overestimated over timeduring both events. The plastic deformations are relatively smaller during the ElMayor-Cucapah earthquake because of the lower intensity of the event.Fig. 7.9 presents dynamic force-deflection responses at the abutments and atthe pile cap in the transverse direction during both events. Due to the similarityof the results, the longitudinal responses are not presented here. The linear springsand dashpots are shown to be incapable of predicting the hysteretic force-deflectioncurves which results in inaccurate simulation of the actual stiffness and damping ofthe system. The dynamic stiffness is highly overestimated at the pile cap while it isunderestimated at the abutments. This was expected since the spring and dashpotconstants are derived based on some semi-empirical equations which may be validjust for a specific problem. In addition, these values are derived under static orslow cyclic loading conditions, and inertial interaction between the superstructureand the foundation soil is not accounted for. The significant effects of inertialinteraction on lateral stiffness of foundations were already shown in Fig. 2.7.7.3.2 Seismic responses of the three-span bridgeThe substructuring method is evaluated by comparing the results against those ob-tained from the continuum model of the three-span bridge system under ten differ-ent earthquake events. Spectral responses of the motions computed at the centre ofthe deck in the longitudinal and transverse directions during the ten events are pre-sented in Figs. 7.10 and 7.11, respectively. The spectral response in the superstruc-ture model is generally overestimated, especially in the transverse direction. Theobserved discrepancies may be partly due to the use of equivalent linear springs anddashpots assumed in the substructure model where nonlinear hysteretic behavior ofthe foundation soil and the resulting material damping are poorly simulated. Oneway to reduce the level of overestimation is to increase the dashpots constants inthe substructure model. For real cases, determining the accurate value of damping145seems to be very difficult and impractical.The time histories of the pier displacements, the shear forces, and the bendingmoments induced at the pier base, the forces induced to the abutments resultingfrom the analysis of the substructure model are presented in Appendix E. Figs. 7.12and 7.13 compare the maximum of the the pier drifts, the shear forces and bend-ing moments induced at the pier base, and the forces induced to the abutmentspredicted by the substructure model with those predicted by the continuum modelin the longitudinal and transverse directions, respectively. The force induced tothe abutment is the total force transferred from the deck to the abutment systemincluding the back-wall, the wing-walls, and the embankment. The substructuremodel generally overestimates the pier drifts. The level of overestimation is moresignificant in the transverse direction especially in the events No. 2, 5, 6, and 10where the maximum transverse pier drift is overestimated by a factor of about 3(see in Fig. 7.13a). The shear forces at the base of the pier is satisfactorily pre-dicted in the longitudinal direction by the substructure model in all events exceptin the event No. 10 where the longitudinal shear force is about 30% underestimated(see in Fig. 7.12b). In the transverse direction, the base shear forces predicted inthe continuum model are poorly predicted. Similar observations are noted whencomparing the maximum bending moments. The substructure model wrongly pre-dicts the yielding of the entire pier section. This is more severe in the event No. 2(see in Fig. 7.13c). The forces induced at the abutment in the longitudinal directionare highly overestimated in the the longitudinal direction. For example, in eventNo. 8 the maximum force is overestimated by a factor of 3 (see in Fig. 7.12d). Thelevel of overestimation is relatively less in the transverse direction.For more illustration, the difference in estimation of the peak pier drifts, shearforces and bending moments induced at the base of pier, and forces applied to theabutment system during each event are presented and discussed. The differencebetween the peak values is defined as the subtraction of the values obtained fromthe substructure model from the values obtained from the continuum model dividedby the value obtained from the continuum model. Multiplying the absolute valueof this ratio by 100 gives the relative percentage difference which is called relativedifference, R, hereafter. For each set of results, exceedance probability of a givenrelative difference is calculated. The exceedance probability is the probability that146a certain value is going to be exceeded. The relative differences are sorted fromleast to greatest, and the corresponding exceedance probability is defined as 1−(i−1)/N, where i is 1,2,3...10 and N is the total number of events (i.e., N = 10).Due to the similarity of the results for piers 1 and 2 as well as abutments 1and 2, only responses of pier 1 and abutment 1 (shown in Fig. 5.19) are presented.Fig. 7.14a shows the exceedance probability of the pier drift in both longitudinaland transverse directions during the ten events. In all events, the relative differ-ence (R) in predicting the maximum pier drift varies in the range of 0 to 225%.Exceedance probability of the relative difference of 50% is about 40% in the longi-tudinal direction and 90% in the transverse direction. This significant level of dif-ference between the results is partly due to the yielding of the pier cross section andthe resulting plastic deformations. Fig. 7.14b,c show the exceedance probability ofrelative difference in estimating the maximum shear forces and maximum bendingmoments induced at the base of the pier 1 in both longitudinal and transverse direc-tions. The relative difference in estimation of the maximum shear forces varies inthe range of about 0 to 30% and 2 to 100% in the longitudinal and transverse direc-tions, respectively. The relative difference in estimation of the maximum bendingmoments varies in the range of about 0 to 25% and 0 to 75% about the transverseand longitudinal axes of the bridge, respectively. As shown in the figure there is ahigh probability that the shear forces and bending moments induced at the pier baseare poorly estimated in the substructure model. Fig. 7.14d shows the exceedanceprobability of relative difference in estimation of maximum forces induced to theabutment during the ten events. If substructure model is used, exceedance proba-bility of the relative difference of 50% is about 60% in the longitudinal directionand 18% in the transverse direction. The observed discrepancies can be attributedto the incorrect estimation of the abutment longitudinal and transverse dynamicstiffnesses. The SSI responses at bridge abutment are highly nonlinear hysteretic,and the springs and dashpots appear not to be suitable to represent the seismic re-sponse of the abutment-embankment of the bridge. The other point that is noted inFigs. 7.14a–d is that the relative differences are mainly positive. This implies thatthe substructure model generally overestimates the bridge response.1477.3.3 Seismic responses of the nine-span bridgeThe capabilities of the substructure model in predicting the time histories of dis-placement and forces are assessed by comparing the results with those obtainedfrom the continuum model and the fixed base model in which the bridge super-structure is fixed at the supports without considering any SSI effects in the anal-ysis. Figs. 7.15 and 7.16 show the spectral responses of the motions recorded atthe middle of the bridge deck in soil profile 1 for the five earthquake events in thelongitudinal and transverse directions, respectively. Both substructure and fixedbase models poorly predict the spectral responses. According to Fig. 7.15, thesubstructure model considerably underestimates the spectral response in the longi-tudinal direction for the periods greater than 0.4 s, however it predicts the spectralresponse for the periods less than 0.4 s. Similar to the substructure model, thefixed base model considerably underestimates the spectral response for the periodsgreater than 0.4 s, but it overestimates the spectral response for the periods lessthan 0.4 s. According to Fig. 7.16, the spectral response in the transverse directionis considerably overestimated for the periods less than 0.6 sec and underestimatedfor the periods greater than 0.6 sec using both models. It is to be noted that thesubstructure model satisfactorily predicts the spectral responses during Events no.1 and 3 for the periods less than 0.6 sec. Figs. 7.17 and 7.18 present the spectralresponses of the motions recorded at the middle of the bridge deck in soil profile2 for the five earthquake events in the longitudinal and transverse directions, re-spectively. It is shown that even if the foundation soil is stiff, both the substructuremodel and the fixed base model poorly predict the spectral response, but in generalthe substructure model predicts the spectral responses better than the fixed basemodel. In soil profile 2 (with stiffer soil), the substructure model overestimates thespectral response in the longitudinal direction in the period range of 0.3 to 0.7 s(see in Fig. 7.17). The model generally overestimates the spectral response in theperiod range of 0.4 to 0.6 s (see in Fig. 7.18).Fig. 7.19 compares the exceedance probability of maximum drifts of pier 4 forthe five earthquake events. Results of the continuum model are compared withthose of the substructure and fixed base models in the longitudinal and transversedirections. In soil profile 1, the maximum pier drifts of all events are satisfactorily148predicted in the longitudinal direction while poorly predicted in the transverse di-rection. The same is observed for the bridge in soil profile 2. Due to the cantileverconfiguration of the bridge in the transverse direction, effects of SSI on the bridgeresponse are larger, and therefore, the level of difference between the results ofsubstructure model and the continuum model is larger.The exceedance probability of maximum shear forces induced at the base ofpier 4 in soil profiles 1 and 2 is shown in Fig. 7.20. In both soil profile, the sub-structure and fixed base models accurately predict the maximum shear forces ofthe continuum model in the longitudinal direction. However, the maximum shearforces are overestimated in the transverse direction. The level of overestimation ismore significant when the fixed base model is used for the bridge analysis.Fig. 7.21 compares the computed maximum bending moments induced at thebase of pier 4 in soil profiles 1 and 2. The maximum bending moments predicted bythe substructure and fixed base models in the longitudinal direction are very closeto those predicted by the continuum model. The maximum bending moments areconsiderably overestimated in the transverse direction. The median of the maxi-mum bending moments is overestimated by a factor of 1.5 and 2 in the transversedirection in soil profile 1 and soil profile 2, respectively. The substructure modelpredicts the response of the continuum model better compared to the fixed basemodel.Peak longitudinal and transverse forces induced to abutment 1 are presented inFig. 7.22. The forces in the transverse direction are significantly overestimated bythe substructure and fixed base models in the both soil profiles. Using the substruc-ture model, there is exceedance probability of 50% that the maximum transverseforces are overestimated by a factor of 4 in both soil profiles. Both models satis-factorily capture the maximum forces in the longitudinal direction in soil profile 1,while they overestimate the maximum forces in soil profile 2.The time histories of the pier displacements, the shear forces, and the bendingmoments induced at the pier base, the forces induced to the abutments resultingfrom the analysis of the substructure model and the fixed base model during allearthquake shakings are presented in Appendix E. In this section, the seismic re-sponse of the bridge in event no. 3 (1989 Loma Prieta earthquake) is discussed.Fig. 7.23 shows the time histories of pier top relative displacements with respect to149the base for pier 4. The figure includes three sets of results that are obtained fromthe continuum, substructure, and the fixed base models. The time histories of pierdeflection are poorly predicted using the substructure and the fixed base models ofthe bridge in soil profile 1. The pier deflections are relatively better predicted insoil profile 2. The substructure model and the fixed base model predict similar timehistories of deflections. This implies that the dynamic stiffnesses calculated at thebridge piers and the abutment using the API, AASHTO, and Caltrans guidelinesare very large. The flexibility of the foundation soil, the pile groups, the embank-ment, and the abutment systems is not properly represented using the springs anddashpots.The time histories of shear forces induced at the base of pier 4 obtained fromthe continuum, substructure, and fixed base models of the bridge under the 1989Loma Prieta earthquake are compared in Fig. 7.24. For the bridge supported onsoil profile 1, the time histories of base shear forces in both directions are poorlypredicted. For the bridge supported on soil profile 2, the shear forces in the longitu-dinal direction are satisfactorily predicted, but those in the transverse direction arepoorly predicted. This is partly because of the larger SSI effect in the transversedirection. Fig. 7.25 shows the time histories of bending moment induced at thebase of pier 4 during the 1989 Loma Prieta earthquake. The time histories of thebending moment in soil profile 1 are poorly predicted using both the substructureand the fixed base models. Yielding of the pier cross-section is predicted in thetransverse direction by the substructure and fixed base model while the maximumbending moment, computed in the continuum model, is about 60% of the plasticbending moment. Similar to the time histories of pier deflection and base shearforce, the time histories of bending moment are relatively better predicted in soilprofile 2.7.4 Discussion on the evaluation of the substructuringmethodIn the previous sections, results from the substructure models of three differentbridge systems were compared to those from the continuum models, and the dif-ferences between the results were quantified. The major causes for the observed150differences in the results stem from many simplifying assumptions in the formu-lation of the substructuring method. In particular the formulation of the substruc-turing method appears to lack representing the: (i) nonlinear hysteretic responseof soil, (ii) proper input ground motions at bridge supports, and (iii) mass of thesoil domain supporting the bridge structure. These three modeling aspects are ex-plained in the following sections.7.4.1 Representing soil responseSeismic performance of a structure highly depends on its natural vibration peri-ods which in turn depend on the stiffness of the structure at its supports. In thesubstructuring method, the actual nonlinear response of soil at the bridge supportsis simplified by linear elastic response with equivalent secant stiffness. It is nec-essary to investigate how the secant stiffness values compare with the nonlinearforce-deflection in the corresponding direction.A pushover analysis is performed on the continuum model of the two-spanbridge (i.e., MRO model) to determine the actual force-deflection response at thepile cap and the abutments. The pushover analysis is performed on the continuummodel by pushing the bridge deck by 10.0 cm simultaneously in both longitudi-nal and transverse directions. The force-deflection responses at the pile cap andthe abutment-deck connection are of the interest. Figs. 7.26a,b show the nonlinearbackbone curve at the pile cap in the transverse direction and the rotational direc-tion (about the axis x) obtained from the pushover analysis. The correspondingsecant stiffness value used in the substructure model and those suggested by Dou-glas et al. (1991) and Zhang and Markis (2002a) for the MRO are also presentedin the figure. Their suggested values are based on frequency-domain analysis ofthe 5 × 5 pile group of the MRO. The maximum deflection and rocking of the pilecap under the 1979 Imperial Valley and 2010 El Mayor-Cuapah earthquake arealso presented in the figure. As shown in Fig. 7.26a, the secant stiffness values areall smaller than the actual stiffness of the pile group, and they do not adequatelyrepresent the stiffness of the pile group for the deflections less than the maximumdeflection, 0.04 m. In the rotational direction (Fig. 7.26b), the value used in thesubstructure model is very close to the actual initial stiffness, however, the pro-151posed values are considerably smaller, especially that proposed by Douglas et al.(1991).Figs. 7.27a,b show the nonlinear backbone curves at the abutment in the longi-tudinal and transverse directions obtained from the pushover analysis. The nonlin-ear backbone curves are compared to the secant stiffness, used in the substructuremodel as well as the stiffness values for MRO reported in the literature (Douglaset al., 1991; Wilson and Tan, 1990a; Zhang and Markis, 2002a). The reportedstiffness values in these studies are all based on the frequency-domain analysis ofthe embankment-abutment system of the MRO. The maximum deflections of theabutment system in the longitudinal direction (i.e., 0.015 m) and the transverse di-rection (i.e., 0.02 m) under the 1979 Imperial Valley and 2010 El Mayor-Cuapahearthquake are also presented in the figure. The secant stiffness used in the sub-structure model is significantly larger in the longitudinal direction and significantlysmaller in the transverse direction compared to the actual stiffness of the abutmentsystem. Using the approach of Wilson and Tan (1990a) and Zhang and Markis(2002a), the stiffness of the abutment appeared to be considerably underestimated.In summary the significant differences among the stiffness values shown inFigs. 7.26 and 7.27 imply the unreliability of the secant stiffness values in predict-ing the actual nonlinear response of the pile foundations and the abutment.Another limitation of the presented substructuring method is that it uses a con-stant dynamic stiffness matrix to represent the flexibility of the foundation sys-tem. In reality this stiffness varies at different levels of deformation. The variationof the stiffness matrix by deformation of the ground-bridge system may be takeninto account in the analysis by using the force-deflection curves obtained from thepushover analysis of the continuum model; the damping in the system may bemodeled by using the Masing rule.This aspect is also studied for the MRO model. The schematic of the model isshown in Fig. 7.28. The seismic response of the bridge under the 2010 El Mayor-Cucapah earthquake is investigated here. Fig. 7.29 shows the acceleration responsespectra (for the damping of 5%) of the recorded and computed motions at fivedifferent locations of the MRO. It compares the recorded response with the onesfrom the continuum model, the substructure model with constant dynamic stiff-ness (springs linearized from design guidelines), and the substructure model with152variable dynamic stiffness (nonlinear springs from continuum model). The figureshows that the substructure models may not capture the bridge response even ifappropriate stiffness values are assigned at the bridge supports. Using appropri-ate stiffness values at the bridge supports, the level of error in prediction of themeasured responses at channel 3 relatively decreases. However, the modified sub-structure model is still incapable of capturing the bridge response in channels 7 and9.Figs. 7.30 and 7.31 show the time histories of pier top displacement with re-spect to the pier base and the time histories of bending moment induced at the pierbase, respectively. In general, the level of differences decreases when appropriatesprings are used at the supports. For example, the maximum bending moment inthe transverse direction is overestimated by a factor of 2.2 which was 3.0 if springswere characterized based on the guidelines of API (2007), AASHTO (2012), andCaltrans (2013). However, the difference is not still acceptable from practical pointof view. This can be partly because the unloading-reloading paths are simulatedusing an empirical hardening rule without accounting for the inertial interactions.This way of modeling may not adequately represent the nonlinear hystertetic re-sponse of soil and the resulting material damping. In addition, the nonlinear springsare one-dimensional, and act independently during the excitation. Therefore, thesprings may not appropriately represent the continuous three-dimensional configu-ration of soil domain and the coupling effects of the soil response in one directionon one another.7.4.2 Input ground motionsThe other source of error in the substructuring method is the input ground motionsthat are applied to the bridge model at its supports. The nonlinear hysteretic re-sponse of soil may not be adequately simulated in the site response analysis. Inaddition, the effects of inertial interaction are not accounted for in determinationof the input motions for the substructure model. Fig. 7.32 compares the spec-tral response of the motions computed on the pile cap and top of the abutment inthe continuum model of the bridge with those used as input motions for the sub-structure model. As shown in Fig. 7.32a, spectral response of the motion in the153free-field, i.e., input kinematic motion, is considerably different from that at theabutment for the periods less that 0.7 s. Same thing is noted at the pile cap in theperiod range of 0.2 to 0.4 s.7.4.3 Mass of soil domainThe mass of soil domain that is affected by the soil-pile and embankment-abutmentinteractions is not accounted for in the analysis of the substructure model. Asexplained in Section 6.3.4, this results in a modal response significantly differentfrom that of a real bridge system.In order to investigate the effects of this mass of soil on the overall responseof the bridge, different masses of soil domain are assigned to the pier base and thetwo ends of the bridge deck in the MRO model. It appeared that for this particularbridge the mass of soil around the pile group under the pier does not have a signif-icant effect on the overall response, therefore, only the soil mass of the approachembankment is discussed here. Three different levels of soil mass are assigned tothe bridge substructure model. The height of the embankment affected by SSI inall three cases is assumed to be 3.0 m (same as the height of the abutment backwall), and the lengths are assumed to be 0 in Case I, 10t in Case II, and 20t in CaseIII, where t is the thickness of the abutment backwall. Assuming these volumesof soil, the added masses are calculated to be 0 , 353, and 706 ton, respectively.Fig. 7.33 compares the resulting time histories of bending moment induced at thepier base. It appears that the additional soil mass significantly changes the time his-tory of bending moment about the longitudinal axis, while no considerable changesis noted about the transverse axis. The bending moments about the longitudinalaxis increases by a factor of about 2 between 9 to 11 s if the mass of embankmentis included in the numerical model. This implies that the mass of the supportingsoil may significantly influence the seismic response of the bridge structure, andtherefore it is necessary to include the soil mass in the numerical model of a bridgesystem.1547.5 SummaryIn this chapter, the two-, three-, and nine-span bridge systems were simulated usingthe substructuring method. Five consecutive steps of the method were followed forthe analysis of the bridges during every single earthquake event. Depth-varyingtime histories of displacement in the free-field were determined by a nonlineartime history analysis in OpenSees finite element framework. The numerical modelconsisted of a stack of 3D solid elements in which the stress-strain response of thesoil material was governed by advanced constitutive models such as PDMY andPIMY models as available in OpenSees. The backbone curves recommended inAPI (2007) for piles and Caltrans (2013) for abutments were first linearized at themaximum ground displacements to determine the secant stiffness values. The 6×6stiffness matrix representing the stiffness of the pile group was then determinedby a pushover analysis of the pile group in the compute program, GROUP v.8.The kinematic input motion to be applied to the pier base was determined throughdynamic analysis of the massless pile group subjected to the depth-varying timehistories of displacements at the free end of the springs.Substructure model of the two-span bridge (the MRO) poorly predicted the de-flections and induced forces to the bridge structure during the 1979 Imperial Valleyand 2010 El Mayor-Cucapah earthquakes (shown in Figs. 7.5 to 7.9). Comparingthe spectral responses of the computed motions with the measured motions at thebridge structure, the substructure model did not adequately capture the seismic re-sponse of the bridge especially during the El Mayor-Cucapah earthquake wherethe effects of SSI were significant. The substructure model predicted large pierdeflections and yielding of the pier cross section during both events while the pierwas predicted to remain elastic using the continuum model. Maximum bendingmoments at the pier base was overestimated during the two events by factors of 2.0and 3.0, respectively (Fig. 7.7).Analyses of the three-span bridge showed that if substructuring method is usedfor the analyses there is a high probability that the bridge seismic response is poorlyestimated (shown in Figs. 7.10 to 7.14). There is a probability of 50% that therelative difference in estimation of maximum pier drifts, shear forces at the pierbase, and forces induced to the abutments exceeds 90, 50, and 60%, respectively.155The substructure model incorrectly predicted yielding of the pier with large plasticdeflections. The spectral responses of the deck were considerably overestimatedwhich implied incorrect simulation of damping in the system.A larger bridge system with nine spans is modeled in two different sites. Fromthe analysis of the bridge system it was noted that the substructuring method is un-reliable in predicting the actual displacements, forces induced to the bridge super-structure, and spectral responses of the bridge supported on a soft ground (shownin Figs. 7.15 to 7.25). The level of unreliability of the substructuring method wasalso shown to be significant for the bridge supported on a stiff ground where SSIeffects on the response of the bridge are relatively less. In general, the fixed basemodel of the bridge appeared to be less reliable than the substructure model evenwhen SSI was negligible.Although the substructure model is computationally cost-effective, determina-tion of spring and dashpot constants is very difficult. This study showed that char-acterization of the springs and dashpots based on recommended empirical equa-tions can be associated with significant levels of error. The use of these simplemodels may cause inadequate simulation of the foundation flexibility and energydissipation. It was shown that even if the constants of the springs were derivedfrom the continuum model, the substructure model of the bridge poorly predictedthe seismic response of the bridge because the following components of the modelwere not adequately simulated: (i) the unloading-reloading paths and the resultingmaterial damping, (ii) the continuous three-dimensional configuration of the soildomain and the coupling effects of the soil response in various directions, and (iii)the mass of that portion of the soil domain which is affected by the kinematic andinertial interactions between the bridge system and the soil.1567.6 Tables and figuresTable 7.1: Spring and dashpot coefficients that represent the dynamic stiffnesses of the pile group and the abutmentsystem of MRO during the 1979 Imperial Valley and 2010 El Mayor-Cucapah earthquakes.Stiffness DampingParameter∗ Imperial Valley El Mayor-Cucapah Parameter Imperial Valley El Mayor-CucapahPile group K11 (MN/m) 272.8 480.6 C11 (MN.s/m) 7.6 13.4K22 (MN/m) 260.0 554.9 C22 (MN.s/m) 7.2 15.5K33 (MN/m) 10740.5 10740.5 C33(MN.s/m) 291.2 291.2K44 (MN.m/rad) 19116.0 20559.7 C44 (MN.s/rad) 532.4 572.6K55 (MN.m/rad) 19299.3 20279.2 C55 (MN.s/rad) 537.5 564.8K15 = K51 (MN/rad) 643.8 1156.37 C15 =C51 (MN.s/(rad.m)) 17.9 32.2K24 = K42 (MN/rad) 572.1 1038.86 C24 =C42 (MN.s/(rad.m)) 15.9 28.9Abutment K11 (MN/m) 410.5 404.4 C11 (MN.s/m) 11.4 11.3K22 (MN/m) 48.9 48.9 C22 (MN.s/m) 1.4 1.4∗ K11 and K22 are the lateral stiffnesses along the x and y axes, respectively.K33 is the vertical stiffness along the z axis.K44 and K55 are the rocking stiffnesses about the y and x axes, respectively.K15 = K51 and K24 = K42 are the cross coupling stiffnesses.157x yzFree-field model for the embankmentFree-field model for the foundation soilL 1; Foundation soil profile L 1; Foundation soil profileL 2; Embankment soil profileTied in three translational directionsTied in three translational directionsFor the two and three-span bridge systems:L1=20.0 mL2=8.0 mFor the nine-span bridge system:L1=30.0 mL2=10.0 mFigure 7.1: Numerical model of the soil columns representing the free-fieldconditions in the foundation soil and the embankment used for the seis-mic site response analysis.1580 0.005 0.01 0.015 0.02 0.025 0.03020406080100y (m)P (kN)(0.023, 75.53)API p−y curve atthe depth of 2.0 mMax. ground disp.at the depth of 2.0 m obtainedfrom the site response analysisSecant stiffness(a)0 0.005 0.01 0.015 0.02051015z (m)t (kN)Settlement of the pile groupunder the tributary weight ofthe bridge structure(0.004,10.725)API t−z curve at thedepth of 2.0 mSecant stiffness(b)0 0.01 0.02 0.03 0.04 0.05020406080z (m)Q (kN)Settlement of the pile groupunder the tributary weight ofthe bridge structureSecant stiffnessAPI Q−z curve at the pile tip(0.004,30.2)(c)0 0.05 0.1 0.15 0.20200040006000800010000y (m)F (kN)Caltransbackbone curveSecant stiffness(0.002, 821.0)Max. disp. at top ofthe embankment obtianed fromthe site response analysis(d)Figure 7.2: Examples for the effective linearization of the backbone curves toobtain the secant stiffnesses; (a) linearization of the API p-y curve in thetransverse direction at the depth of 2.0 m, (b) linearization of the APIt-z curve at the depth of 2.0 m, (c) linearization of the API Q-z curveat the pile tip, and (d) linearization of the backbone curve at top of theabutment system based on the guidelines of Caltrans (2013).1592 spans @ 32.0 m3 spans @ 45.0 m9 spans @ 37.0 mx yzNine-span bridgeThree-span bridgeTwo-span bridgeDynamic stiffnesses along the axes x and y 6x6 dynamic stiffness matrix (     ) represented as shown in Fig. 2.8.Figure 7.3: Substructure models of the two-, three-, and nine-span bridge sys-tems including the bridge superstructure supported on linear spring anddashpot models. All three models are created in OpenSees finite ele-ment program (McKenna and Fenves, 2001).1602.0 m4.0 mLayer 1: Medium ClayLayer 2: Medium SandLayer 3: Stiff ClayLayer 4: Medium SandLayer 5: Stiff ClayEmbankment: Soft Clay Embankment: Soft Clay4.0 m5.0 m5.0 m8.0 mρ =1.5 (ton/m3)ρ =1.9 (ton/m3)ρ =1.9 (ton/m3)ρ =1.8 (ton/m3)ρ =1.8 (ton/m3)ρ =1.6 (ton/m3) ρ =1.6 (ton/m3)1V:1.5H1V:1.5HChannel07 Channel09Channel03Channel02 Channel04Point A Point BFigure 7.4: Schematic of the channel locations at the MRO site (channels 2,3, 7, and 9 record the transverse responses, and channel 4 records thelongitudinal responses).161Acceleration response spectrum (g)0 1 2 3 40123Period (sec)Channel090123 Channel070123 Channel040123 Channel030123 Channel02  RecordedContinuum modelSubstructure model(a)Acceleration response spectrum (g)0 1 2 3 40123Period (sec)Channel090123 Channel070123 Channel040123 Channel030123 Channel02  RecordedContinuum modelSubstructure model(b)Figure 7.5: Acceleration response spectra (for 5% damping) of the MROstructure for the (a) 1979 Imperial Valley earthquake, and (b) 2010 ElMayor-Cucapah earthquake; comparing the measured responses withthose estimated by the continuum and the substructure models.1620 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  Abutment top (Point A)Embankment free−field (Point B)(a)0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  Abutment top (Point A)Embankment free−field (Point B)(b)Figure 7.6: Acceleration response spectra (for 5% damping) of the motionsat the abutment top (point A in Fig. 7.4), and at the free-field of theapproach embankment (point B in Fig. 7.4), in the transverse directioncomputed in the continuum model during the (a) 1979 Imperial Valleyearthquake, and (b) 2010 El Mayor-Cucapah earthquake.163Bending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.0 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.0 MN.m  Continuum modelSubstructure model(a)Bending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.0 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.0 MN.m  Continuum modelSubstructure model(b)Figure 7.7: Time histories of bending moment for (a) the 1979 Imperial Val-ley earthquake, and (b) the 2010 El Mayor-Cucapah earthquake; com-paring the responses in the continuum and the substructure models.164Rel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.Continuum modelSubstructure model(a)Rel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.Continuum modelSubstructure model(b)Figure 7.8: Time histories of pier top relative displacements with respect tothe pier base for (a) the 1979 Imperial Valley earthquake, and (b) the2010 El Mayor-Cucapah earthquake; comparing the responses in thecontinuum and the substructure models.165Deflection (m) −0.04 −0.02 0 0.02 0.04Right abutment−0.04 −0.02 0 0.02 0.04Pier base−0.04 −0.02 0 0.02 0.04−6−3036Force (MN)Left abutmentContinuum modelSubstructure model(a)Deflection (m) −0.04 −0.02 0 0.02 0.04Right abutment−0.04 −0.02 0 0.02 0.04Pier base−0.04 −0.02 0 0.02 0.04−6−3036Force (MN)Left abutmentContinuum modelSubstructure model(b)Figure 7.9: Dynamic force-deflection responses at left and right abutmentsand at the pile cap in the transverse direction for (a) the 1979 Impe-rial Valley earthquake, and (b) the 2010 El Mayor-Cucapah earthquake;comparing the responses in the continuum and the substructure models.1660 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longitudinal dir.Event No. 1 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longitudinal dir.Event No. 2 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longitudinal dir.Event No. 3 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longitudinal dir.Event No. 4 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longitudinal dir.Event No. 5 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longitudinal dir.Event No. 6 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longitudinal dir.Event No. 7 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longitudinal dir.Event No. 8 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longitudinal dir.Event No. 9 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longitudinal dir.Event No. 10 Continuum modelSubstructure modelFigure 7.10: Acceleration response spectrum (for 5% damping) of the mo-tions computed at the center of the deck in the longitudinal directionfor the ten earthquake event; comparing the results obtained from thesubstructure model with those obtained from the continuum model.1670 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 1 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 2 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 3 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 4 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 5 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 6 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 7 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 8 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 9 Continuum modelSubstructure model0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 10 Continuum modelSubstructure modelFigure 7.11: Acceleration response spectrum (for 5% damping) of the mo-tions computed at the center of the deck in the transverse directionfor the ten earthquake event; comparing the results obtained from thesubstructure model with those obtained from the continuum model.1681 2 3 4 5 6 7 8 9 1000.511.522.533.54Event No.Maximum pier drift (%)  Longitudinal dir. Continuum modelSubstructure model(a)1 2 3 4 5 6 7 8 9 1002468Event No.Shear force (MN)  Longitudinal dir. Continuum modelSubstructure model(b)1 2 3 4 5 6 7 8 9 100510152025Event No.Maximum bending moment, M yy (MN.m)  Plastic moment, Mp=15.76 (MN.m)About the transverse dir. Continuum modelSubstructure model(c)1 2 3 4 5 6 7 8 9 10048121620Event No.Force at the abutment (MN)  Longitudinal dir. Continuum modelSubstructure model(d)Figure 7.12: (a) Maximum pier drifts, (b) maximum base shear forces, (c)maximum bending moments for pier 1, and (d) maximum forces in-duced to the abutment system; comparing the results of the substruc-ture model with those of the continuum model for the ten earthquakeevents. (a), (b), and (d) show bridge response in the longitudinal di-rection, and (c) shows the response about the transverse axis of thebridge.1691 2 3 4 5 6 7 8 9 1000.511.522.533.54Event No.Maximum pier drift (%)  Transverse dir. Continuum modelSubstructure model(a)1 2 3 4 5 6 7 8 9 1002468Event No.Shear force (MN)  Transverse dir. Continuum modelSubstructure model(b)1 2 3 4 5 6 7 8 9 100510152025Event No.Maximum bending moment, M xx (MN.m)  Plastic moment, Mp=15.76 (MN.m)About the longitudinal dir. Continuum modelSubstructure model(c)1 2 3 4 5 6 7 8 9 10048121620Event No.Force at the abutment (MN)  Transverse dir. Continuum modelSubstructure model(d)Figure 7.13: (a) Maximum pier drifts, (b) maximum base shear forces, (c)maximum bending moments for pier 1, and (d) maximum forces in-duced to the abutment system; comparing the results of the substruc-ture model with those of the continuum model for the ten earthquakeevents. (a), (b), and (d) show bridge response in the transverse di-rection, and (c) shows the response about the longitudinal axis of thebridge.1700 50 100 150 200 250 300 350020406080100R in estimation of pier drift (%)Exceedance probability (%)  Longitudinal dir.TrendlineTransverse dir.TrendlineContinuum − SubstructureContinuumR =(a)0 25 50 75 100 125 150020406080100R in estimation of shear force (%)Exceedance probability (%)  Longitudinal dir.TrendlineTransverse dir.Trendline(b)0 50 100 150 200020406080100R in estimation of bending moment (%)Exceedance probability (%)  Myy, about the transverse axisTrendlineMxx, about the longitudinal axisTrendline(c)0 25 50 75 100 125 150020406080100R in estimation of abutment force (%)Exceedance probability (%)  Longitudinal dir.TrendlineTransverse dir.Trendline(d)Figure 7.14: Exceedance probability of the relative difference in estimationof, (a) maximum pier drifts, (b) maximum base shear forces, (c) maxi-mum bending moments for pier 1, and (d) maximum forces induced tothe abutment system in longitudinal and transverse directions for theten earthquake events. The curves represent the best fit to the data.1710 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 1Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 2Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 3Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 4Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 5Continuum modelSubstructure modelFixed base modelFigure 7.15: Acceleration response spectrum (for 5% damping) of the mo-tions computed at the center of the deck in soil profile 1 in the longitu-dinal direction for all five earthquake events; comparing the results ofthe continuum model, the substructure model and the fixed model.1720 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 1Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 2Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 3Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 4Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 5Continuum modelSubstructure modelFixed base modelFigure 7.16: Acceleration response spectrum (for 5% damping) of the mo-tions computed at the center of the deck in soil profile 1 in the trans-verse direction for all five earthquake events; comparing the results ofthe continuum model, the substructure model and the fixed model.1730 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 1Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 2Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 3Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 4Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 5Continuum modelSubstructure modelFixed base modelFigure 7.17: Acceleration response spectrum (for 5% damping) of the mo-tions computed at the center of the deck in soil profile 2 in the longitu-dinal direction for all five earthquake events; comparing the results ofthe continuum model, the substructure model and the fixed model.1740 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 1Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 2Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 3Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 4Continuum modelSubstructure modelFixed base model0 0.5 1 1.5 2 2.5 3 3.5 402468Period (sec)Acceleration response spectrum (g)  Event No. 5Continuum modelSubstructure modelFixed base modelFigure 7.18: Acceleration response spectrum (for 5% damping) of the mo-tions computed at the center of the deck in soil profile 2 in the trans-verse direction for all five earthquake events; comparing the results ofthe continuum model, the substructure model and the fixed model.1750 0.5 1 1.5 2 2.5020406080100120Drift of pier 4 (%)Exceedance probability (%)  Soil profile 1; longitudinal dir.Continuum modelSubstructure modelFixed base model(a)0 0.5 1 1.5 2 2.5020406080100120Drift of pier 4 (%)Exceedance probability (%)  Soil profile 1; transverse dir.Continuum modelSubstructure modelFixed base model(b)0.05 0.5 1 1.5 2 2.5020406080100120Drift of pier 4 (%)Exceedance probability (%)  Soil profile 2; longitudinal dir.Continuum modelSubstructure modelFixed base model(c)0 0.5 1 1.5 2 2.5020406080100120Drift of pier 4 (%)Exceedance probability (%)  Soil profile 2; transverse dir.Continuum modelSubstructure modelFixed base model(d)Figure 7.19: Exceedance probability of the pier 4 drift for the nine-spanbridge in (a) soil profile 1 in the longitudinal direction, (b) soil pro-file 1 in the transverse direction, (c) soil profile 2 in the longitudinaldirection, and (d) soil profile 2 in the transverse direction; compar-ing the results of the continuum model, the substructure model andthe fixed base model. The curves represent the log-normal distributionbest fitted to the data.1760 2 4 6 8020406080100120Max. shear force; base of pier 4 (MN)Exceedance probability (%)  Soil profile 1; longitudinal dir.Continuum modelSubstructure modelFixed base model(a)0 2 4 6 8020406080100120Soil profile 1; transverse dir.Max. shear force; base of pier 4 (MN)Exceedance probability (%)  Continuum modelSubstructure modelFixed base model(b)0 2 4 6 8020406080100120Max. shear force; base of pier 4 (MN)Exceedance probability (%)  Soil profile 2; longitudinal dir.Continuum modelSubstructure modelFixed base model(c)0 2 4 6 8020406080100120Soil profile 2; transverse dir.Max. shear force; base of pier 4 (MN)Exceedance probability (%)  Continuum modelSubstructure modelFixed base model(d)Figure 7.20: Exceedance probability of maximum shear forces induced at thebase of pier 4 of the nine-span bridge in (a) soil profile 1 in the lon-gitudinal direction, (b) soil profile 1 in the transverse direction, (c)soil profile 2 in the longitudinal direction, and (d) soil profile 2 in thetransverse direction; comparing the results of the continuum model,the substructure model and the fixed base model. The curves representthe log-normal distribution best fitted to the data.1770 10 20 30 40 50 60020406080100120Soil profile 1; Myy, about the transverse axisMax. bending moement; base of pier 4 (MN.m)Exceedance probability (%)  Mp=11.4 MN.mContinuum modelSubstructure modelFixed base model(a)0 10 20 30 40 50 60020406080100120Soil profile 1; Mxx, about the longitudinal axisMax. bending moement; base of pier 4 (MN.m)Exceedance probability (%)  Mp=25.6 MN.mContinuum modelSubstructure modelFixed base model(b)0 10 20 30 40 50 60020406080100120Soil profile 2; Myy, about the transverse axisMax. bending moement; base of pier 4 (MN.m)Exceedance probability (%)  Mp=11.4 MN.mContinuum modelSubstructure modelFixed base model(c)0 10 20 30 40 50 60020406080100120Soil profile 2; Mxx, about the longitudinal axisMax. bending moement; base of pier 4 (MN.m)Exceedance probability (%)  Mp=25.6 MN.mContinuum modelSubstructure modelFixed base model(d)Figure 7.21: Exceedance probability of maximum bending moments inducedat the base of pier 4 of the nine-span bridge in (a) soil profile 1 aboutthe transverse direction, (b) soil profile 1 about the longitudinal direc-tion, (c) soil profile 2 about the transverse direction, and (d) soil profile2 about the longitudinal direction; comparing the results of the contin-uum model, the substructure model and the fixed base model. Thecurves represent the log-normal distribution best fitted to the data.1780 4 8 12 16 20020406080100120Soil profile 1; longitudinal dir.Max. force; abutment 1 (MN)Exceedance probability (%)  Continuum modelSubstructure modelFixed base model(a)0 4 8 12 16 20020406080100120Soil profile 1; transverse dir.Max. force; abutment 1 (MN)Exceedance probability (%)  Continuum modelSubstructure modelFixed base model(b)0 4 8 12 16 20020406080100120Soil profile 2; longitudinal dir.Max. force; abutment 1 (MN)Exceedance probability (%)  Continuum modelSubstructure modelFixed base model(c)0 4 8 12 16 20020406080100120Soil profile 2; transverse dir.Max. force; abutment 1 (MN)Exceedance probability (%)  Continuum modelSubstructure modelFixed base model(d)Figure 7.22: Exceedance probability of maximum forces induced to the abut-ment 1 of the nine-span bridge in (a) soil profile 1 in the longitudinaldirection, (b) soil profile 1 in the transverse direction, (c) soil profile2 in the longitudinal direction, and (d) soil profile 2 in the transversedirection; comparing the results of the continuum model, the substruc-ture model and the fixed base model. The curves represent the log-normal distribution best fitted to the datas.179Rel. displacement (m)0 5 10 15 20 25 30−0.12−0.0600.060.12Time (sec)Transverse dir.−0.12−0.0600.060.12 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(a)Rel. displacement (m)0 5 10 15 20 25 30−0.12−0.0600.060.12Time (sec)Transverse dir.−0.12−0.0600.060.12 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(b)Figure 7.23: Time histories of relative displacements of the top of pier 4 withrespect to the pier base for the nine-span bridge system supported on(a) soil profile 1 (soft soil), and (b) soil profile 2 (stiff soil) during the1989 Loma Prieta earthquake; comparing the responses obtained fromthe continuum, the substructure, and the fixed base models.180Shear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(a)Shear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(b)Figure 7.24: Time histories of base shear force of pier 4 for the nine-spanbridge system supported on (a) soil profile 1 (soft soil), and (b) soilprofile 2 (stiff soil) during the 1989 Loma Prieta earthquake; compar-ing the responses obtained from the continuum, the substructure, andthe fixed base models.181Bending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.Mp=25.6 MN.m−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Continuum modelSubstructure modelFixed base model(a)Bending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.Mp=25.6 MN.m−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Continuum modelSubstructure modelFixed base model(b)Figure 7.25: Time histories of bending moment induced at the base of pier4 for the nine-span bridge system supported on (a) soil profile 1 (softsoil), and (b) soil profile 2 (stiff soil) during the 1989 Loma Prietaearthquake; comparing the responses obtained from the continuum,the substructure, and the fixed base models.1820 0.04 0.08 0.12 0.16 0.2 0.24 0.28051015202530Deflection (m)Force (MN)  Transverse dir.K=260.0 MN/mmax. computed defl. Continuum modelSubstructure modelDouglas et al. (1991)Zhang and Makris (2002)(a)0 1 2 3 4 5 6 7 801020304050Rocking (0.001 x rad)Bending moment (MN.m)  Rocking about axis xK=19,299 MN.m/radK=7611 MN.m/radK=1888 MN.m/rad max. computed rockingContinuum modelSubstructure modelDouglas et al. (1991)Zhang and Makris (2002)(b)Figure 7.26: (a) Force-deflection response in the transverse direction, and(b) moment-rotation response along the axis x at the pile cap of theMRO; comparing the response in the continuum model, the substruc-ture model, and the reported values in the literature (Douglas et al.,1991; Zhang and Markis, 2002a).1830 0.04 0.08 0.12 0.16 0.2 0.24 0.280246810max. computed defl.Deflection (m)Force (MN)  Longitudinal dir.K=430 MN/mK=91.2 MN/mK=20.7 MN/mContinuum modelSubstructure modelDouglas et al. (1991)Zhang and Makris (2002)(a)0 0.04 0.08 0.12 0.16 0.2 0.24 0.280246810Deflection (m)Force (MN)  Transverse dir.K=48.9 MN/mK=34.2 MN/mK=91.2 MN/mK=20.7 MN/mmax. computed defl.Continuum modelSubstructure modelWilson and Tan (1990)Douglas et al. (1991)Zhang and Makris (2002)(b)Figure 7.27: (a) Force-deflection responses in the longitudinal direction, and(b) force-deflection responses in the transverse direction at the abut-ment of the MRO; comparing the response in the continuum model,the substructure model, and the reported values in the literature (Dou-glas et al., 1991; Wilson and Tan, 1990a; Zhang and Markis, 2002a).184Components of dynamic stiffness: a nonlinear spring x yz 6x6 dynamic stiffness matrix (    ) as a function of displacement  Kinematic input motion Kinematic input motionDynamic stiffnesses as a function of displacement along the axes x and ydjFiDerived from the continuum modelMasing ruleKij (i,j=1...6)Figure 7.28: Schematic of the substructure model of the MRO in which stiff-ness of the springs at the pile cap and at the abutments are deter-mined from the pushover analysis of the bridge continuum model.Unloading-reloading paths are simulated using the Masing rule.185Acceleration response spectrum (g)0 1 2 3 40123Period (sec)Channel090123 Channel070123 Channel040123 Channel03  0123 Channel02  RecordedContinuum modelSubstructure model−springs from GuidelinesSubstructure model−springs from continuum modelFigure 7.29: Acceleration response spectrum (for 5% damping) of the mo-tions at five different locations of the MRO during the 2010 El Mayor-Cucapah earthquake; investigating the level of improvement of sub-structure model when stiffness of the springs are obtained from thebridge continuum model.186Rel. displacement (m)0 5 10 15 20 25 30−0.06−0.0300.030.06Time (sec)Transverse dir.−0.06−0.0300.030.06 Longitudinal dir.  Continuum modelSubstructure model−springs from GuidelinesSubstructure model−springs from continuum modelFigure 7.30: Time histories of pier top displacements with respect to the pilecap of the MRO during the 2010 El Mayor-Cucapah earthquake; inves-tigating the level of improvement of substructure model when stiffnessof the springs are obtained from the bridge continuum model.187Bending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.0 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.0 MN.m  Continuum modelSubstructure model−springs from GuidelinesSubstructure model−springs from continuum modelFigure 7.31: Time histories of bending moment induced at the pier base ofthe MRO during the 2010 El Mayor-Cucapah earthquake; investigatingthe level of improvement of substructure model when stiffness of thesprings are obtained from the bridge continuum model.1882.0 m4.0 mLayer 1: Medium ClayLayer 2: Medium SandLayer 3: Stiff ClayLayer 4: Medium SandLayer 5: Stiff ClayEmbankment: Soft Clay Embankment: Soft Clay4.0 m5.0 m5.0 m8.0 m 1V:1.5HPoint CPoint A Point BPoint D0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  Transverse dir.Top of the abutment (Point A)Free−field (Point B), input motionfor the substructure model(a)0 0.5 1 1.5 2 2.5 3 3.5 400.511.52Period (sec)Acceleration response spectrum (g)  Transverse dir.Top of the pile cap (point C)Free−field (point D), input motion for the substructure model(b)Figure 7.32: Comparing the acceleration response spectrum (for 5% damp-ing) of the motions computed in the continuum model at (a) the top ofthe abutment wall, and (b) the top of the pile cap with the accelerationresponse spectrum of the motions computed in the free-field, i.e., theinput motions for the substructure model.1892.0 m4.0 mLayer 1: Medium ClayLayer 2: Medium SandLayer 3: Stiff ClayLayer 4: Medium SandLayer 5: Stiff ClayEmbankment4.0 m5.0 m5.0 m8.0 mEmbankmentL_eff3.0 mL_eff3.0 mCase I:      L_eff = 0Case II:     L_eff = 10tCase III:    L_eff = 20tt: wall thicknessBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.0 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.0 MN.m  Continuum modelSubstructure model−Case ISubstructure model−Case IISubstructure model−Case IIIFigure 7.33: Time histories of bending moment induced at the pier base of theMRO during the 2010 El Mayor-Cucapah earthquake; investigating theeffect of the soil mass on the overall response of the bridge structure(the assigned soil mass at the two ends of the bridge deck in Case I:0.0, Case II: 353.3 ton, and Case III: 706.6 ton).190Chapter 8Summary and future researchScience never solves a problem without creating ten more.— George B. Shaw (1856–1950)8.1 SummaryThe most common method used in practice to simulate seismic performance ofbridge systems is the substructuring method which divides a bridge system intotwo subsystems: the bridge superstructure which typically includes the bridge deckand the piers, and the foundation systems at the bridge supports which include thepile foundations, the abutments and the surrounding soil domain. Soil-structureinteraction (SSI) is taken into account by replacing the foundation system with aconstant dynamic stiffness matrix, which in turn is informed from sets of linearizedsprings and dashpots representing the soil domain. The method is very popularin both engineering practice and research communities, however, the validity ofthe approach has never been examined. This PhD thesis aimed to remedy thisserious omission by testing how results of this widely-used simple practical methodcompare with those from a more exact method, i.e., continuum modeling. The goalis to aid the advancement of the numerical simulation of SSI used in engineeringpractice for the design of bridge systems.The substructuring method was explained and presented in five consecutive191steps based on the current state of practice in Caltrans. The baseline data requiredfor evaluating the substructuring method was provided by analyzing continuummodels of three different bridge systems with two, three, and nine spans each sub-jected to a number of actual earthquake shakings. This involved detailed modelingof the foundation soil, pile foundations, abutment system, and the whole bridgestructure. Elastoplastic constitutive models were used to simulate nonlinear hys-teretic behavior of soil and piers. The continuum model was validated by simulat-ing the seismic responses of the Meloland Road Overpass (MRO) during the 1979Imperial Valley and the 2010 El Mayor-Cucapah earthquakes.The evaluation of the substrucutring method began by evaluating the force-deflection backbone curves (nonlinear springs) recommended in the API guide-lines. This was primarily done because the accuracy of the substructuring methodpartly relies on the accuracy of those backbone curves. To this end, twenty-sevenstatic tests and two dynamic centrifuge tests on single piles were selected to pro-vide the required baseline data. This study showed that using the API guidelines insimulation of static field tests on laterally loaded single pile results in considerablelevels of error which vary in the range of 5 to 150% in estimation of the pile headdisplacements, and in the range of 5 to 90% in estimation of the maximum bendingmoments.The evaluation process continued with simulating two-, three-, and nine-spanbridge systems using the substructuring method, and comparing the computed re-sponses with those obtained from the continuum analyses. It was shown that thedifferences between the results are significant and do not lie in an acceptable range.For example, using the substructuring method for simulation of the MRO the max-imum bending moments at the pier base was overestimated by factors of about2 and 3 during the 1979 Imperial Valley and the 2010 El Mayor-Cucapah earth-quakes, respectively. Similar levels of difference were observed in the analyses ofthe three-span and the nine-span bridge systems.Considering the findings of the present study on deficiencies of proposed springsin API (2007), AASHTO (2012), and Caltrans (2013) guidelines for substitutingthe soil response, which are along the lines of findings from some previous stud-ies, it is not surprising that application of such springs in the seismic SSI analy-sis of bridge systems does not give satisfactory results for different conditions of192the structure and the foundation soil. Moreover, it appears from previous studiesthat there is no logical approach with adequate level of complexity to improve thesubstructuring method. On the other hand, the continuum modeling method wasshown to be significantly more reliable in capturing the seismic response of soil-pile interaction. To sum up, the main contributions of this research are listed asfollows:• The thesis is one of the few documents in which full-scale nonlinear con-tinuum models of small to large-scale bridge systems were developed foranalysis under earthquake shakings (Chapter 5). The methodology was val-idated by simulating the seismic responses of an actual bridge system, i.e.,the Meloland Road Overpass, during the past earthquake events recorded atthe bridge site (Section 5.3).• The thesis elucidated the need for the change of the current state of prac-tice that is used for simulation of seismic SSI for bridge systems. Thiswas claimed based on evaluating the state of practice against three seriesof baseline data: (i) data of twenty-seven static field tests on single piles(Section 6.2), (ii) data of two dynamic centrifuge tests on single piles (Sec-tion 6.3), and (iii) numerical modeling results obtained from continuum anal-yses of two-, three-, and nine-span bridge systems subjected to a wide rangeof actual earthquake shakings (Chapter 7).The presented evaluation of the state of practice for analysis of seismic response ofbridge systems, and the illustrated capabilities of the continuum modeling methodjustify the importance of high fidelity comprehensive analysis of seismic SSI inorder to enhance seismic safety, reduce unwarranted expenses, and increase re-silience of the built environment. Considering ongoing advances in constitutivemodeling of geotechnical and structural materials, computational tools, and paral-lel computing environments, large-scale continuum models can gradually becomea powerful and significantly more reliable alternative for proper modeling of seis-mic SSI in bridge engineering.1938.2 Future researchThere is no prospective approach to improve the spring modeling and in particu-lar the substructuring method for seismic SSI analysis of bridge system. Previousstudies which intended to improve the spring method could not be generalized fordifferent types of SSI systems. This was further elucidated in this thesis. Futureresearch needs to be directed towards improving the continuum modeling methodin terms of simulation methodologies and the required computational efforts. Forinstance, more comprehensive constitutive models can be implemented in the prac-tical continuum modeling numerical codes and used for seismic analysis of bridges,to obtain more realistic estimates of kinematic and inertial interactions between thestructure and the supporting ground. In this study PDMY and PIMY constitutivemodels were used for simulation of sandy and clayey layers, respectively. TheSANISAND and SANICLAY class of constitutive models and their recent evo-lutions at the University of British Columbia, have been shown to be capable ofsimulating several fundamental mechanisms involved in the soil response. Thesemodels are formulated within the framework of critical state soil mechanics, andtherefore ideally there is no need to re-calibrate the parameters of the models fordifferent conditions of a specific soil. The SANISAND class of models are bound-ing surface plasticity models formulated in such a way to generate plastic strainsfor any small changes in the stress-ratio, as one would expect for sands. They offergeneric add-on mechanisms to account for the inherent and evolving anisotropy inthe material. The models are based on an elegant platform for capturing the peakstress-ratio and the contractive/dilative tendency of response in relation to the soilstate parameter. The models also offer a means for extending the range of appli-cation to any type of constant stress-ratio loading, if desired, to better capture theresulting plastic strains. The SANICLAY class of models offer extensive evolu-tion based on the popular Modified Cam-Clay model. They allow for capturing thestress-induced anisotropy in the material. They also offer a destructuration mecha-nism to capture the collapsible response observed in sensitive clays. More recentlyan efficient bounding surface formulation has been incorporated into the model tocapture the plastic strains more realistically under cyclic loadings such as those inthe earthquake events. Both SANISAND and SANICLAY class of models have a194systematic generalization of the formulation to the multi-axial stress-strain space,which is essential in their application for multi-directional loading mechanisms in-duced during earthquakes. These constitutive models have not yet been adopted formodeling large-scale problems such as those in bridge engineering. Application ofthese models in the analysis of bridge systems would result in better simulation ofthe SSI mechanisms, especially in soft grounds and also where the ground is proneto liquefaction.195BibliographyAASHTO (2012), LRFD Bridge Design Specifications, American Association ofState Highway and Transportation Officials, Washington D.C., USA.Abdoun, T., Dobry, R., ORourke, T. and Goh, S. 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(1998), Lateral response of large diameter drilledshafts: 1-15/us 95 load test program, in ‘Proceedings of the 33rd EngineeringGeology and Geotechnical Engineering Symposium’, University of Nevada,Reno, NV, pp. 161–176.Zeghal, M. and Elgamal, A. W. (1994), ‘Analysis of site liquefaction usingearthquake records’, ASCE Journal of Geotechnical Engineering120(6), 996–1017.207Zhang, J. and Markis, N. (2002a), ‘Kinematic response functions and dynamicstiffness of bridge embankments’, Earthquake Engineering and StructuralDynamics 31(11), 1933–1966.Zhang, J. and Markis, N. (2002b), ‘Seismic response analysis of highwayovercrossings including soil-structure interaction’, Earthquake Engineeringand Structural Dynamics 31(11), 1967–1991.Zhang, L., Silva, F. and Grismala, R. (2005), ‘Ultimate lateral resistance to pilesin cohesionless soils’, Journal of Geotechnical and GeoenvironmentalEngineering 131(1), 78–83.208Appendix AImportance of parallel computingIn modern engineering, parallel computing environments has become an essen-tial tool for timely analysis and interpretation of large-scale physical simulations.It allows the aggregation of computer resources to address the computational re-quirements for complex simulations and extensive design variation analyses. Re-cently parallel processing has become available in different structural and geo-chemical engineering software including OpenSees, SAP2000, ABAQUS, AN-SYS, and FLAC.The parallel computing environments use domain decomposition methods wherethe computation is broken into a number of tasks. These tasks include the forma-tion of the system of equations, the solution of the equations and the updating of thenodal response quantities. These tasks may be subdivided into smaller tasks. Forexample, the formulation of the system of equations at the element level may bedivided into the formulation of mass, stiffness and damping matrices, and assem-bly into the system of equations (McKenna, 1997). The schematic of the processthat includes subdividing the problem into several discrete tasks and assigning thetasks to separate CPUs is shown in Fig. A.1.The nonlinear dynamic analysis of the MRO model subjected to an earthquakewith a duration of 20.0 s would take about a week if one single 2.4GHz CPU wereused. This analysis runtime is not acceptable for the purposes of this thesis. Theparallel computing technique and the required resources has to be adopted in thepresent study to reduce the execution time of the analysis. Cloud services provide209a web-based access to a shared pool of configurable processing resources, stor-age resources, as well as applications. Examples of Cloud services are NEEShub(http://nees.org) and Amazon EC2 Cloud Center (http://aws.amazon.com). In thepresent study, two in-house servers each consists of a total of eighty 2.4GHz CPUsprovides the required resources for the parallel computations.A.1 Analysis execution timeOptimally, doubling the number of processing elements should halve the executiontime of an analysis, and doubling it a second time should again halve the execu-tion time. However, most of algorithms used for parallel processing shows a linearspeed up for small numbers of CPUs, which flattens out into a constant value forlarge numbers of CPUs. A program solving a large mathematical or engineeringproblem will typically consist of several nonparallelizable parts and several paral-lelizable parts. Those nonparallelizable parts of the program, no matter how effi-cient the parallel algorithm is, will put an upper limit on the usefulness of addingmore CPUs. This is commonly referred to as Amdahl’s law (Amdahl, 1967) whichstates that if P is the parallelizable parts of a program and (1−P) is the nonparal-lelizable parts, then the maximum speedup (Smax) that can be achieved by using Nprocessors is:Smax =1(1−P)+PN(A.1)In the above equation, as N tends to infinity the maximum speedup is limitedto a constant value; 1/(1−P).Fig. A.2 compares the execution time of nonlinear dynamic analysis of theMRO model under the 1979 Imperial Valley earthquake (duration: 20 s) usingparallel commuting with 1, 4, 8, 10, and 16 processors. The run time is reducedabout 5 times using 8 processors. However, there is no significant reduction in runtime observed when using more than 8 processors. According to Amdahl’s law, thisis due to those sequential portions of the program which can not be parallelized.210A.2 Analysis costsFig. A.3 shows the cost and execution time of a set of ten analyses using differentnumbers of processors. It is assumed that the analyses were conducted in AmazonEC2 Cloud Center. Each EC2 units costs $2.5 per hour. Ten analyses can be exe-cuted simultaneously in parallel using different numbers of EC2 units. Thereforethe execution time is limited by the longest ground motion. By using two Ama-zon EC2 units and assigning four processors for each analyses, the cost is reducedby 40%. No significant cost reduction is observed by using three Amazon EC2and running each ground motions with eight processors. However, by using fiveEC2 and assigning sixteen processors for each ground motions, the total cost isincreased due to the diminishing return of execution time decreases.211CPUCPUCPUCPUProblem Domain decomposition ProcessorsFigure A.1: Schematic of the parallel computing process.2120 2 4 6 8 10 12 14 16 1804080120160Number of processorsAanalysis execution time (h)Figure A.2: Analysis execution time of the MRO continuum model under the1979 Imperial Valley earthquake using different number of processors.050100150200250300350400450500Number of processorsCloud cost ($)  0 12 (1 EC2 unit) 64 (2 EC2 unit) 96 (3 EC2 unit) 160 (5 EC2 unit) 004080120160200Aanalysis execution time (h)Execution timeCloud costCloud c stExecuti n timeFigure A.3: Cloud cost and analysis execution time of the MRO continuummodel for a suite of ten ground motions using different number of pro-cessors.213Appendix BDescription of the static testsperformed on single pilesThe database used for evaluating the API springs for static analysis of piles ispresented in more detail in this Appendix. For each test, the geometry and materialproperties of the single pile and the properties of the surrounding soil layers arepresented in Fig. B.1. The presented values are used for simulation of the pile andthe supporting springs in the computer program LPILE v.6 (ENSOFT Inc., 2012b).The program employs the guidelines of API (2007) to construct the p-y curves.Some important points about these tests are as follows,• All tests, expect Tests no. 7, 8, and 9, were performed on full-scale pilesplaced in the field. Tests no. 7, 8, and 9 were performed on small-scale ofpiles in centrifuge tests where a monotonic load was applied at the pile headwhile the container was spinning.• All tests, except tests no. 3, 4, 18, and 24, were performed on steel pipepiles. The piles were H-shape in tests no. 3 and 4, and drilled shaft pileswere tested in tests No. 18 and 24.2140.11 m16.2 mSand:γsat=19.5 kN/m3φ=42ºPile section:D = 41 cmt = 1.52 cmPile head loadEpIp = 7.0x104 kN.m2(a) Test no. 1 (Mansur et al., 1964)0.20 m2.44 mSand:γsat=17.3 kN/m3φ=40ºPile section:D = 5.1 cmt = 0.16 cmPile head loadEpIp = 15.4 kN.m2(b) Test no. 2 (Parker et al., 1970)12.2 mSand:γdry=15.4 kN/m3φ=35ºPile section:Pile head loadEpIp = 6.6x103 kN.m2b = 0.18 mh = 0.403 mt = 0.0075 m(c) Test no. 3 (Mason and Bishop, 1953)Figure B.1: Schematic of the 27 tests listed in Table. 6.1.21512.2 mSand: Layer 1Pile section:b = 0.36 mh = 0.26 mPile head loadt = 0.011 m0.25 mSand: Layer 3Sand: Layer 2Sand: Layer 4Layer γ* (kN/m3) φº1 17.0 372 18.1 373 18.2 384 19.2 400.31 m0.92 m0.92 mEpIp = 6.05x104 kN.m2* γ=γdry in layer 1γ=γsat in layers 2, 3, and 4(d) Test no. 4 (Murchison, 1983)0.305 m21.0 mSand:γsat=20.2 kN/m3φ=39ºPile section:D = 61 cmt = 0.95 cmPile head loadEpIp = 1.63x105 kN.m2(e) Test no. 5 (Cox et al., 1974)0.305 m5.0 mSand:γsat=15.5 kN/m3φ=38.5ºPile section:D = 27.3 cmt = 0.93 cmPile head loadEpIp = 1.34x104 kN.m2(f) Test no. 6 (Brown et al., 1988)Figure B.1: Continued.2161.25 m9.05 mSand:γdry=16.3 kN/m3φ=36ºPile section:D = 122.4 cmt = 1.725 cmPile head loadEpIp = 2.495x106 kN.m2(g) Test no. 7 (Georgiadis et al., 1991)2.2 m11.1 mSand:γdry=15.2 kN/m3φ=39ºPile section:D = 43 cmt = 1.15 cmPile head loadEpIp = 7.21x103 kN.m2(h) Test no. 8 (McVay et al., 1995)1.6 m12 mSand:γdry=16.1 kN/m3φ=40ºPile section:D = 72 cmt = 6.0 cmPile head loadEpIp = 5.14x105 kN.m2(i) Test no. 9 (Mezazigh and Levacher, 1998)Figure B.1: Continued.21719.2 mSand: Layer 1Pile head loadSand: Layer 3Sand: Layer 2Layer γ* (kN/m3) φº1 17.5 372 17.5 373 17.75 380.92 m2.75 mPile section:D = 51 cmEpIp = 1.21x105 kN.m2t = 1.2 cm* γ=γdry in layers 1 and 2γ=γsat in layer 3(j) Test no. 10 (Murchison, 1983)12.8 mPile head loadDepth (m) γsat (kN/m3) Su (kPa)0 20.0 30.21.14 20.0 32.22.50 20.0 42.3Pile section:D = 32 cmEpIp = 28,730 kN.m2t = 1.27 cm 3.39 20.0 17.53.70 20.0 30.14.30 20.0 23.45.69 20.0 51.87.25 20.0 29.89.47 20.0 32.615.0 20.0 32.6Soft clay(k) Test no. 11 (Matlock, 1970)Figure B.1: Continued.2180.305 m12.8 mSoft clay:γsat=15.5 kN/m3Su=14.4 kPaPile section:D = 32 cmt = 1.27 cmPile head loadEpIp = 28,730 kN.m2(l) Test no. 12 (Meyer, 1979)0.80 mLPile section (Test 13):D = 11.4 cmt = 0.63 cmPile head loadEpIp = 6.23x105 kN.m20.03 mDepth (m) γsat (kN/m3)0 17.85 34.50.6 17.85 57.01.2 17.85 34.51.53 17.85 48.61.7 17.85 53.32.3 17.85 28.43.2 17.85 22.23.73 17.85 23.05.55 17.85 23.0Soft claySu (kPa)Pile section (Test 14):D = 21.9 cmt = 0.702 cmEpIp = 5.26x106 kN.m2L=5.55 mL=6.22 mPile section (Test 15):D = 32.4 cmt = 1.32 cmEpIp = 3.12x107 kN.m2Pile section (Test 16):D = 0.41 cmt = 0.99 cmEpIp = 4.84x107 kN.m2L=5.1 mL=8.14 m(m) Test no. 13, 14, 15, and 16 (Gill, 1968)Figure B.1: Continued.2194.6 mPile head loadLayer γsat (kN/m3) Su (kPa)1 17.7 18.02 17.7 42.0Pile section:D = 90 cmEpIp = 1.03x106 kN.m2Saturated stiff clay0.8 mStiff clay: Layer 1Stiff clay: Layer 21.5 m(n) Test no. 17 (Kim and Jeong, 2011)45 mPile head loadLayer γsat (kN/m3) Su (kPa)1 17.7 18.02 17.7 42.03 17.7 55.0Pile section:D = 240 cmEpIp = 4.234x107 kN.m2Soft clay1.0 mSoft clay: Layer 1Soft clay: Layer 213.5 m5.0 mSoft clay: Layer 3(o) Test no. 18 (Kim and Jeong, 2011)Figure B.1: Continued.2207.0 mPile head loadDepth (m) γsat (kN/m3) Su (kPa)Saturated stiff clay0.31 mTop pile section:D = 64.1 cmEpIp = 493,700 kN.m2t = 2.71 cm8.2 mBottom pile section:D = 61.0 cmt = 0.99 cmEpIp = 168,400 kN.m20 -0.9 18.11.52 19.44.11 20.36.55 20.39.14 20.320.00 20.8257016333333311001100(p) Test no. 19 (Reese et al., 1975)0.80 mLPile section (Test 20):D = 11.4 cmt = 0.66 cmPile head loadEpIp = 6.49x105 kN.m2Depth (m) γsat (kN/m3)0.63 17.85 103.83.04 17.85 66.53.66 17.85 115.5Saturated stiff claySu (kPa)Pile section (Test 21):D = 21.9 cmt = 0.702 cmEpIp = 4.97x106 kN.m2L=3.66 mL=5.18 mPile section (Test 22):D = 32.4 cmt = 1.32 cmEpIp = 2.41x107 kN.m2Pile section (Test 23):D = 0.41 cmt = 0.99 cmEpIp = 4.84x107 kN.m2L=6.71 mL=8.23 m(q) Test no. 20, 21, 22, and 23 (Gill and Demars, 1970)Figure B.1: Continued.2214.6 mPile head loadLayer γsat (kN/m3) Su (kPa)1 19.8 134.02 20.5 101.0Pile section:D = 90 cmEpIp = 1.03x106 kN.m2Saturated stiff clay0.8 mStiff clay: Layer 1Stiff clay: Layer 21.5 m(r) Test no. 24 (Halloway, 1978)16.5 mPile head loadDepth (m) γdry (kN/m3) Su (kPa)0 17 44.14.6 17 85.26.2 17 80.6Pile section:D = 40.6 cmEpIp = 51,378 kN.m2t = 1.0 cm 16.5 17 133.3Dry stiff clay1.0 m(s) Test no. 25 (Price and Wradle, 1981)Figure B.1: Continued.22210.0 mPile head loadLayer γ* (kN/m3) Su (kPa)1 18.8 40.02 18.8 40.03 20.1 105.0Pile section:D = 85.0 cmEpIp = 1.58x106 kN.m24 20.1 105.0Dry/saturated stiff clay0.3 mStiff clay: Layer 1Stiff clay: Layer 2Stiff clay: Layer 3Stiff clay: Layer 44.5 m0.5 m3.0 m* γ=γdry in layer 1γ=γsat in layers 2, 3, and 4(t) Test no. 26 (Jun et al., 2008)12.88 mPile head loadDepth (m) γ* (kN/m3) Su (kPa)Dry/Saturated stiff clayPile section:D = 76.2 cmEpIp = 493,700 kN.m2t = 25.0 cm0 19.40.4 19.41.04 18.85.6 19.16.1 19.112.88 20.7176761051051051635.6 m* γ=γdry at the depth less than 5.6 mγ=γsat at the depth greater than 5.6 m(u) Test no. 27 (Reese and Welch, 1975)Figure B.1: Continued.223Appendix CDescription of the seismiccentrifuge tests performed onsingle pilesThe soil profile in the centrifuge test of Gohl (1991) consisted of a single layer ofdry fine-grained Nevada sand (Dr=40%) with the thickness of 12.0 m. The massdensity and the friction angle of the soil were 15.0 kN/m3 and 35◦, respectively.The single pile was a steel pipe pile with diameter of 57.0 cm and wall thicknessof 1.3 cm at the prototype scale. The pile was extended about 2.0 m above theground surface and carries a superstructure load of 522.0 kN. Fig. C.1a illustratesthe outline of the test. The embedded length of the pile was about 12.0 m. The soilcontainer had a plan area of 31.8×10.56 m and was approximately 15.0 m deepin prototype scale. The soil-pile-superstructure system was spun at a centrifugalacceleration of 60g.The soil profile in the centrifuge test of Wilson (1998) consisted of a layerof saturated soft clay underlain by a layer of saturated dense sand. Based on theTorvane measurements, undrained shear strength of the clay was 2.5, 6.5, 9.0, and12.0 kPa at the depths of 0–1.5, 1.5–3.0, 3–4.5, and 4.5–6.0 m, respectively. Theeffective unit weight of the layer was about 7.75 kN/m3. The effective unit weightand the friction angle of the underlying sand layer (Nevada sand with Dr=80%)were 9.81 kN/m3 and 39.5◦, respectively. The clay layer was 6.1 m thick, and the224sand layer was 11.4 m thick at the prototype scale. The single pile was equivalentto a steel pipe pile with a diameter of 67 cm and a wall thickness of 1.9 cm atthe prototype scale. The outline of the test is presented in Fig. C.1b. The pilewas extended 3.8 m above the ground surface and carries a superstructure load of482.0 kN. The embedded length of pile was about 16.5 m. The container had insidedimensions of 51.6 m long, 20.55 m wide, by 21.0 m deep in prototype scale. Thesoil-pile-superstructure system was spun at a centrifugal acceleration of 30g.12.0 mDry Nevada Sand(Dr=40%)W = 522 kNPile section:D = 0.57 mt = 13 mmRigid box 2.0 mγ=15.1 kN/m3φ=35ºEpIp = 1.72x105 kN.m215.9 m15.9 m(a) Centrifuge test of Gohl (1991)1.5 mSoft clay: layer 2W = 480 kNNevada Sand(Dr =80%) 11.4 m16.5 m3.81 mPile section:D = 0.67 mt = 19 mmLaminar box Soft clay: layer 1Soft clay: layer 4Soft clay: layer 3 1.5 m1.5 m1.5 mLayer1234γsat (kN/m3)18.018.018.018.0Su (kPa)2.56.59.012.0EpIp = 2.82x105 kN.m2 γ=20.0 kN/m3φ=39.5ºSoft clay31.5 m 20.1 m(b) Centrifuge test of Wilson (1998)Figure C.1: Schematic of the centrifuge tests which are simulated usingspring and continuum modeling approaches (dimensions are not toscale).225Appendix DResults obtained from thecontinuum models of the three-and nine-span bridgesD.1 Results for the three-span bridgeThis section presents the time histories of the input ground motions (Fig. D.1), therelative displacements of the pier top with respect to the pier base (Fig. D.2), theshear forces induced at the pier base (Fig. D.3), the bending moments induced atthe pier base (Fig. D.4), and the forces transferred from the deck to the abutmentsystem (Fig. D.5).226Acceleration (g)0 5 10 15 20 25 30−0.8−0.400.40.8Time (sec)Fault−normal−0.8−0.400.40.8 Fault−parallel(a) Event no. 1, 1999 Kocaeli earthquakeAcceleration (g)0 5 10 15 20 25 30−0.8−0.400.40.8Time (sec)Fault−normal−0.8−0.400.40.8 Fault−parallel(b) Event no. 2, 1995 Kobe earthquakeAcceleration (g)0 5 10 15 20 25 30−0.8−0.400.40.8Time (sec)Fault−normal−0.8−0.400.40.8 Fault−parallel(c) Event no. 3, 1999 Hector Mine earthquakeFigure D.1: Time histories of fault-parallel and fault-normal components ofthe input ground motions for the three-span bridge. The Fault paralleland fault normal components are applied in longitudinal and transversedirections, respectively.227Acceleration (g)0 5 10 15 20 25 30−0.8−0.400.40.8Time (sec)Fault−normal−0.8−0.400.40.8 Fault−parallel(d) Event no. 4, 1987 Superstition Hills-02 earthquakeAcceleration (g)0 5 10 15 20 25 30−0.8−0.400.40.8Time (sec)Fault−normal−0.8−0.400.40.8 Fault−parallel(e) Event no. 5, 1999 Chi Chi earthquakeAcceleration (g)0 5 10 15 20 25 30−0.8−0.400.40.8Time (sec)Fault−normal−0.8−0.400.40.8 Fault−parallel(f) Event no. 6, 1994 Northridge-01 earthquakeFigure D.1: Continued.228Acceleration (g)0 5 10 15 20 25 30−0.8−0.400.40.8Time (sec)Fault−normal−0.8−0.400.40.8 Fault−parallel(g) Event no. 7, 1995 Kobe earthquakeAcceleration (g)0 5 10 15 20 25 30−0.8−0.400.40.8Time (sec)Fault−normal−0.8−0.400.40.8 Fault−parallel(h) Event no. 8, 1992 Cape Mendocino earthquakeAcceleration (g)0 5 10 15 20 25 30−0.8−0.400.40.8Time (sec)Fault−normal−0.8−0.400.40.8 Fault−parallel(i) Event no. 9, 1989 Loma Prieta earthquakeFigure D.1: Continued.229Acceleration (g)0 5 10 15 20 25 30−0.8−0.400.40.8Time (sec)Fault−normal−0.8−0.400.40.8 Fault−parallel(j) Event no. 10, 1989 Duzce earthquakeFigure D.1: Continued.230Rel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.(a) Event no. 1, 1999 Kocaeli earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.(b) Event no. 2, 1995 Kobe earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.(c) Event no. 3, 1999 Hector Mine earthquakeFigure D.2: Time histories of relative displacements of the pier top with re-spect to the pier base (for pier 1) of the three-span bridge in the longi-tudinal and transverse directions for the three-span bridge.231Rel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.(d) Event no. 4, 1987 Superstition Hills-02 earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.(e) Event no. 5, 1999 Chi Chi earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.(f) Event no. 6, 1994 Northridge-01 earthquakeFigure D.2: Continued.232Rel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.(g) Event no. 7, 1995 Kobe earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.(h) Event no. 8, 1992 Cape Mendocino earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.(i) Event no. 9, 1989 Loma Prieta earthquakeFigure D.2: Continued.233Rel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.(j) Event no. 10, 1989 Duzce earthquakeFigure D.2: Continued.234Shear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(a) Event no. 1, 1999 Kocaeli earthquakeShear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(b) Event no. 2, 1995 Kobe earthquakeShear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(c) Event no. 3, 1999 Hector Mine earthquakeFigure D.3: Time histories of shear forces induced at the pier base (pier 1) ofthe three-span bridge in the longitudinal and transverse directions usingthe continuum modeling method.235Shear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(d) Event no. 4, 1987 Superstition Hills-02 earthquakeShear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(e) Event no. 5, 1999 Chi Chi earthquakeShear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(f) Event no. 6, 1994 Northridge-01 earthquakeFigure D.3: Continued.236Shear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(g) Event no. 7, 1995 Kobe earthquakeShear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(h) Event no. 8, 1992 Cape Mendocino earthquakeShear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(i) Event no. 9, 1989 Loma Prieta earthquakeFigure D.3: Continued.237Shear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(j) Event no. 10, 1989 Duzce earthquakeFigure D.3: Continued.238Bending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m(a) Event no. 1, 1999 Kocaeli earthquakeBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m(b) Event no. 2, 1995 Kobe earthquakeBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m(c) Event no. 3, 1999 Hector Mine earthquakeFigure D.4: Time histories of bending moments induced at the pier base (pier1) of the three-span bridge in the longitudinal and transverse directionsusing the continuum modeling method.239Bending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m(d) Event no. 4, 1987 Superstition Hills-02 earthquakeBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m(e) Event no. 5, 1999 Chi Chi earthquakeBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m(f) Event no. 6, 1994 Northridge-01 earthquakeFigure D.4: Continued.240Bending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m(g) Event no. 7, 1995 Kobe earthquakeBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m(h) Event no. 8, 1992 Cape Mendocino earthquakeBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m(i) Event no. 9, 1989 Loma Prieta earthquakeFigure D.4: Continued.241Bending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m(j) Event no. 10, 1989 Duzce earthquakeFigure D.4: Continued.242Force at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(a) Event no. 1, 1999 Kocaeli earthquakeForce at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(b) Event no. 2, 1995 Kobe earthquakeForce at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(c) Event no. 3, 1999 Hector Mine earthquakeFigure D.5: Time histories of forces induced at the abutment (abutment 1) ofthe three-span bridge in the longitudinal and transverse directions usingthe continuum modeling method.243Force at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(d) Event no. 4, 1987 Superstition Hills-02 earthquakeForce at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(e) Event no. 5, 1999 Chi Chi earthquakeForce at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(f) Event no. 6, 1994 Northridge-01 earthquakeFigure D.5: Continued.244Force at abutment (MN)−6−3036 Longitudinal dir.0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.(g) Event no. 7, 1995 Kobe earthquakeForce at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(h) Event no. 8, 1992 Cape Mendocino earthquakeForce at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(i) Event no. 9, 1989 Loma Prieta earthquakeFigure D.5: Continued.245Force at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(j) Event no. 10, 1989 Duzce earthquakeFigure D.5: Continued.2460 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longituidnal dir.Event No. 1 Continuum modelInput motion(a) Event no. 1, 1999 Kocaeliearthquake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longituidnal dir.Event No. 2 Continuum modelInput motion(b) Event no. 2, 1995 Kobe earth-quake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longituidnal dir.Event No. 3 Continuum modelInput motion(c) Event no. 3, 1999 HectorMine earthquake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longituidnal dir.Event No. 4 Continuum modelInput motion(d) Event no. 4, 1987 SuperstitionHills-02 earthquake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longituidnal dir.Event No. 5 Continuum modelInput motion(e) Event no. 5, 1999 Chi ChiearthquakeFigure D.6: Comparing acceleration response spectra (for 5% damping) ofthe motion computed at the middle of the three-span bridge deck withARS of the input motion in the longitudinal direction.2470 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longituidnal dir.Event No. 6 Continuum modelInput motion(f) Event no. 6, 1994 Northridge-01 earthquake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longituidnal dir.Event No. 7 Continuum modelInput motion(g) Event no. 7, 1995 Kobe earth-quake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longituidnal dir.Event No. 8 Continuum modelInput motion(h) Event no. 8, 1992 Cape Men-docino earthquake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longituidnal dir.Event No. 9 Continuum modelInput motion(i) Event no. 9, 1989 Loma Prietaearthquake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Longituidnal dir.Event No. 10 Continuum modelInput motion(j) Event no. 10, 1989 DuzceearthquakeFigure D.6: Continued.2480 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 1 Continuum modelInput motion(a) Event no. 1, 1999 Kocaeliearthquake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 2 Continuum modelInput motion(b) Event no. 2, 1995 Kobe earth-quake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 3 Continuum modelInput motion(c) Event no. 3, 1999 HectorMine earthquake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 4 Continuum modelInput motion(d) Event no. 4, 1987 SuperstitionHills-02 earthquake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 5 Continuum modelInput motion(e) Event no. 5, 1999 Chi ChiearthquakeFigure D.7: Comparing acceleration response spectra (for 5% damping) ofthe motion computed at the middle of the three-span bridge deck withARS of the input motion in the transverse direction.2490 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 6 Continuum modelInput motion(f) Event no. 6, 1994 Northridge-01 earthquake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 7 Continuum modelInput motion(g) Event no. 7, 1995 Kobe earth-quake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 8 Continuum modelInput motion(h) Event no. 8, 1992 Cape Men-docino earthquake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 9 Continuum modelInput motion(i) Event no. 9, 1989 Loma Prietaearthquake0 0.5 1 1.5 2 2.5 3 3.5 401234Period (sec)Acceleration response spectrum (g)  Transverse dir.Event No. 10 Continuum modelInput motion(j) Event no. 10, 1989 DuzceearthquakeFigure D.7: Continued.250D.2 Results for the nine-span bridgeThe time histories of the input ground motions (Fig. D.8) that are convolved to thedepth of 30.0 m, the relative displacements of the pier top with respect to the pierbase (Figs. D.9 and D.10 ), the shear forces induced at the pier base (Figs. D.11and D.12), the bending moments induced at the pier base (Figs. D.13 and D.14),the forces transferred from the deck to the abutment system (Figs. D.15 and D.16),and the acceleration response spectra of the motion at the middle of the bridge deck(Figs. ?? and ??) are presented in detail.251Acceleration (g)0 5 10 15 20 25 30−0.4−0.200.20.4Time (sec)Fault−normal−0.4−0.200.20.4 Fault−parallel(a) Event no. 1, 1978 Tabas earthquakeAcceleration (g)0 5 10 15 20 25 30−0.4−0.200.20.4Time (sec)Fault−normal−0.4−0.200.20.4 Fault−parallel(b) Event no. 2, 1980 Irpinia earthquakeAcceleration (g)0 5 10 15 20 25 30−0.4−0.200.20.4Time (sec)Fault−normal−0.4−0.200.20.4 Fault−parallel(c) Event no. 3, 1989 Loma Prieta earthquakeFigure D.8: Time histories of fault-parallel and fault-normal components ofthe input ground motions for the nine-span bridge (deconvolved mo-tions at the depth of 30.0 m).252Acceleration (g)0 5 10 15 20 25 30−0.4−0.200.20.4Time (sec)Fault−normal−0.4−0.200.20.4 Fault−parallel(d) Event no. 4, 1994 Northridge earthquakeAcceleration (g)0 5 10 15 20 25 30−0.4−0.200.20.4Time (sec)Fault−normal−0.4−0.200.20.4 Fault−parallel(e) Event no. 5, 1999 Chi Chi earthquakeFigure D.8: Continued.253Rel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(a) Event no. 1, 1978 Tabas earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(b) Event no. 2, 1980 Irpinia earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(c) Event no. 3, 1989 Loma Prieta earthquakeFigure D.9: Time histories of relative displacement of the pier top with re-spect to the pier base of piers 1, 4, 6, and 8 for the nine-span bridgesupported on soil profile 1.254Rel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(d) Event no. 4, 1994 Northridge earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(e) Event no. 5, 1999 Chi Chi earthquakeFigure D.9: Continued.255Rel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(a) Event no. 1, 1978 Tabas earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(b) Event no. 2, 1980 Irpinia earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(c) Event no. 3, 1989 Loma Prieta earthquakeFigure D.10: Time histories of relative displacement of the pier top with re-spect to the pier base of piers 1, 4, 6, and 8 for the nine-span bridgesupported on soil profile 2.256Rel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(d) Event no. 4, 1994 Northridge earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.1−0.0500.050.1Time (sec)Transverse dir.−0.1−0.0500.050.1 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(e) Event no. 5, 1999 Chi Chi earthquakeFigure D.10: Continued.257Shear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(a) Event no. 1, 1978 Tabas earthquakeShear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(b) Event no. 2, 1980 Irpinia earthquakeShear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(c) Event no. 3, 1989 Loma Prieta earthquakeFigure D.11: Time histories of shear force induced at the bases of piers 1, 4,6, and 8 of the nine-span bridge supported on soil profile 1.258Shear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(d) Event no. 4, 1994 Northridge earthquakeShear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(e) Event no. 5, 1999 Chi Chi earthquakeFigure D.11: Continued.259Shear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(a) Event no. 1, 1978 Tabas earthquakeShear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(b) Event no. 2, 1980 Irpinia earthquakeShear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(c) Event no. 3, 1989 Loma Prieta earthquakeFigure D.12: Time histories of shear force induced at the bases of piers 1, 4,6, and 8 of the nine-span bridge supported on soil profile 2.260Shear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(d) Event no. 4, 1994 Northridge earthquakeShear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Pier 1 Pier 4 Pier 6 Pier 8(e) Event no. 5, 1999 Chi Chi earthquakeFigure D.12: Continued.261Bending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Mxx, about the longitudinal axisMp=25.6 MN.m−24−1201224 Myy, about the transverse axisMp=11.4 MN.m  Pier 1 Pier 4 Pier 6 Pier 8(a) Event no. 1, 1978 Tabas earthquakeBending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Mxx, about the longitudinal axisMp=25.6 MN.m−24−1201224 Myy, about the transverse axisMp=11.4 MN.m  Pier 1 Pier 4 Pier 6 Pier 8(b) Event no. 2, 1980 Irpinia earthquakeBending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Mxx, about the longitudinal axisMp=25.6 MN.m−24−1201224 Myy, about the transverse axisMp=11.4 MN.m  Pier 1 Pier 4 Pier 6 Pier 8(c) Event no. 3, 1989 Loma Prieta earthquakeFigure D.13: Time histories of bending moment induced at the bases of piers1, 4, 6, and 8 of the nine-span bridge supported on soil profile 1.262Bending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Mxx, about the longitudinal axisMp=25.6 MN.m−24−1201224 Myy, about the transverse axisMp=11.4 MN.m  Pier 1 Pier 4 Pier 6 Pier 8(d) Event no. 4, 1994 Northridge earthquakeBending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Mxx, about the longitudinal axisMp=25.6 MN.m−24−1201224 Myy, about the transverse axisMp=11.4 MN.m  Pier 1 Pier 4 Pier 6 Pier 8(e) Event no. 5, 1999 Chi Chi earthquakeFigure D.13: Continued.263Bending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Mxx, about the longitudinal axisMp=25.6 MN.m−24−1201224 Myy, about the transverse axisMp=11.4 MN.m  Pier 1 Pier 4 Pier 6 Pier 8(a) Event no. 1, 1978 Tabas earthquakeBending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Mxx, about the longitudinal axisMp=25.6 MN.m−24−1201224 Myy, about the transverse axisMp=11.4 MN.m  Pier 1 Pier 4 Pier 6 Pier 8(b) Event no. 2, 1980 Irpinia earthquakeBending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Mxx, about the longitudinal axisMp=25.6 MN.m−24−1201224 Myy, about the transverse axisMp=11.4 MN.m  Pier 1 Pier 4 Pier 6 Pier 8(c) Event no. 3, 1989 Loma Prieta earthquakeFigure D.14: Time histories of bending moment induced at the bases of piers1, 4, 6, and 8 of the nine-span bridge supported on soil profile 2.264Bending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Mxx, about the longitudinal axisMp=25.6 MN.m−24−1201224 Myy, about the transverse axisMp=11.4 MN.m  Pier 1 Pier 4 Pier 6 Pier 8(d) Event no. 4, 1994 Northridge earthquakeBending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Mxx, about the longitudinal axisMp=25.6 MN.m−24−1201224 Myy, about the transverse axisMp=11.4 MN.m  Pier 1 Pier 4 Pier 6 Pier 8(e) Event no. 5, 1999 Chi Chi earthquakeFigure D.14: Continued.265Force at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(a) Event no. 1, 1978 Tabas earthquakeForce at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(b) Event no. 2, 1980 Irpinia earthquakeForce at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(c) Event no. 3, 1989 Loma Prieta earthquakeFigure D.15: Time histories of forces induced at the abutment (abutment 1)of the nine-span bridge supported on soil profile 1.266Force at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(d) Event no. 4, 1994 Northridge earthquakeForce at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(e) Event no. 5, 1999 Chi Chi earthquakeFigure D.15: Continued.267Force at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(a) Event no. 1, 1978 Tabas earthquakeForce at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(b) Event no. 2, 1980 Irpinia earthquakeForce at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(c) Event no. 3, 1989 Loma Prieta earthquakeFigure D.16: Time histories of forces induced at the abutment (abutment 1)of the nine-span bridge supported on soil profile 2.268Force at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(d) Event no. 4, 1994 Northridge earthquakeForce at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.(e) Event no. 5, 1999 Chi Chi earthquakeFigure D.16: Continued.269Appendix EResults obtained from thesubstructure models of the three-and nine-span bridgesE.1 Dynamic stiffnesses at the bridge supportsThe constants of the springs and dashpots representing the dynamic stiffnesses atthe pier bases and at the abutments of the three-span bridge, i.e., the pier basesand the abutments, are presented in Tables E.1 and E.2 and those of the nine-spanbridge are presented in Tables E.3 and E.4.270Table E.1: Constants of the springs that represent the 6×6 stiffness matrix (K) of the pile group and the abutmentsystem for the three-span bridge system under the ten earthquake events.Elements of the 6×6 stiffness matrixParameter Event no.1 Event no. 2 Event no. 3 Event no. 4 Event no. 5 Event no.6 Event no. 7 Event no. 8 Event no. 9 Event no. 10Pile group K11 (MN/m) 147.0 92.3 111.1 82.7 94.2 73.7 89.7 73.4 82.0 92.1K22 (MN/m) 122.1 108.8 97.7 92.6 88.4 99.3 778.2 66.5 81.8 84.9K33 (MN/m) 5310.0 5310.0 5310.0 5310.0 5310.0 5310.0 5310.0 5310.0 5310.0 5310.0K44 (MN.m/rad) 8913.0 8901.0 8890.0 8885.0 8881.0 8892.3 8869.1 8854.7 8873.8 8877.1K55 (MN.m/rad) 8933.0 8885.0 8903.0 8874.0 8887.0 8863.5 8882.4 8863.2 8873.6 8885.1K15 = K51 (MN/rad) 179.2 130.7 148.3 121.3 132.5 112.01 128.3 111.7 120.6 130.7K24 = K42 (MN/rad) 158.1 146.3 135.9 131.1 126.9 137.5 116.7 104.6 120.6 123.7Abutment K11 (MN/m) 440.1 414.2 428.0 450.0 430.1 417.7 437.2 399.6 383.4 445.5K22 (MN/m) 49.5 49.5 49.5 49.5 49.5 49.5 49.5 49.5 49.5 49.5271Table E.2: Constants of the dashpots that represent the 6×6 damping matrix (C) of the pile group and the abutmentsystem for the three-span bridge system under the ten earthquake events.Elements of the 6×6 damping matrixParameter Event no.1 Event no. 2 Event no. 3 Event no. 4 Event no. 5 Event no.6 Event no. 7 Event no. 8 Event no. 9 Event no. 10Pile group C11 (MN.s/m) 4.7 2.9 3.5 2.6 3.0 2.3 2.8 2.3 2.6 2.9C22 (MN.s/m) 3.9 3.5 3.1 2.9 2.8 3.2 24.8 2.1 2.6 2.7C33 (MN.s/m) 169.0 169.0 169.0 169.0 169.0 169.0 169.0 169.0 169.0 169.0C44 (MN.s/rad) 283.7 283.3 283.0 282.8 282.7 283.1 281.8 282.5 282.6C55 (MN.s/rad) 284.4 282.8 283.4 282.5 282.9 282.1 282.7 282.1 282.4 282.8C15 =C51 (MN.s/(rad.m)) 5.7 4.2 4.7 3.9 4.2 3.6 4.1 3.6 3.8 4.2C24 =C42 (MN.s/(rad.m)) 5.0 4.7 4.4 4.2 4.1 4.4 3.7 3.3 3.8 3.9Abutment C11 14.0 13.2 13.6 14.3 14.3 13.4 13.9 12.7 12.2 14.2C22 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6272Table E.3: Constants of the springs that represent the 6×6 stiffness matrix (K) of the pile group and the abutmentsystem for the nine-span bridge system under the five earthquake events.Elements of the 6×6 stiffness matrixSoil profile 1 Soil profile 2Parameter Event no.1 Event no. 2 Event no. 3 Event no. 4 Event no. 5 Event no.1 Event no. 2 Event no. 3 Event no. 4 Event no. 5Pile group K11 (MN/m) 40.4 52.3 40.2 42.8 50.5 396.0 423.1 403.4 427.8 446.2K22 (MN/m) 46.1 51.9 39.7 48.1 44.5 411.2 429.6 394.8 385.8 382.6K33 (MN/m) 23,459.0 23,459.0 23,459.0 23,459.0 23,459.0 33,437.9 33,437.9 33,437.9 33,437.9 33,437.9K44 (MN.m/rad) 68,050.5 68,080.4 68,032.8 68055.1 68045.5 97295.9 97311.6 97282.2 97274.5 97272.1K55 (MN.m/rad) 68,036.0 68,066.4 68,033.4 68041.7 68062.0 97283.5 97305.7 97289.8 97309.8 97323.9K15 = K51 (MN/rad) 106.5 124.1 105.1 109.9 121.5 482.9 510.5 481.9 508.9 523.5K24 = K42 (MN/rad) 114.8 126.5 105.8 117.8 112.4 495.3 505.1 489.1 474.4 471.9Abutment K11 (MN/m) 515.7 515.7 515.7 515.7 515.7 515.7 515.7 515.7 515.7 515.7K22 (MN/m) 515.7 515.7 515.7 515.7 515.7 515.7 515.7 515.7 515.7 515.7273Table E.4: Constants of the dashpots that represent the 6×6 damping matrix (C) of the pile group and the abutmentsystem for the nine-span bridge system under the five earthquake events.Elements of the 6×6 damping matrixSoil profile 1 Soil profile 2Parameter Event no.1 Event no. 2 Event no. 3 Event no. 4 Event no. 5 Event no.1 Event no. 2 Event no. 3 Event no. 4 Event no. 5Pile group C11 (MN.s/m) 1.6 2.1 1.6 1.7 2.1 15.8 16.8 16.1 17.1 17.5C22 (MN.s/m) 1.8 2.1 1.6 1.9 1.8 16.4 17.1 15.7 15.4 15.2C33 (MN.s/m) 933.4 933.4 933.4 933.4 933.4 1330.5 1330.5 1330.5 1330.5 1330.5C44 (MN.s/rad) 2707.6 2708.8 2706.9 2707.8 2707.4 3871.3 3871.9 3870.7 3870.4 3870.3C55 (MN.s/rad) 2707.1 2708.3 2706.9 2707.3 2708.1 3870.8 3871.7 3871.0 3871.8 3872.4C15 =C51 (MN.s/(rad.m)) 4.2 4.9 4.2 4.4 4.8 19.2 20.3 19.2 20.2 20.8C24 =C42 (MN.s/(rad.m)) 4.6 5.03 4.2 4.7 4.5 19.7 20.1 19.5 18.9 18.9Abutment C11 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5C22 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5274E.2 Results for the three-span bridgeThis section presents the relative displacements of the pier top with respect to thepier base (Fig. E.1), the shear forces induced at the pier base (Fig. E.2), the bendingmoments induced at the pier base (Fig. E.3), the forces transferred from the deckto the abutment system (Fig. E.4). Each figure includes three sets of results thatare obtained from the continuum model, the substructure model, and the fixed-basemodel of the three-span bridge.275Rel. displacement (m)0 5 10 15 20 25 30−0.2−0.100.10.2Time (sec)Transverse dir.−0.2−0.100.10.2 Longitudinal dir.  Continuum modelSubstructure model(a) Event no. 1, 1999 Kocaeli earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.2−0.100.10.2Time (sec)Transverse dir.−0.2−0.100.10.2 Longitudinal dir.  Continuum modelSubstructure model(b) Event no. 2, 1995 Kobe earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.2−0.100.10.2Time (sec)Transverse dir.−0.2−0.100.10.2 Longitudinal dir.  Continuum modelSubstructure model(c) Event no. 3, 1999 Hector Mine earthquakeFigure E.1: Time histories of relative displacements of the pier top with re-spect to the pier base (for pier 1) of the three-span bridge in the longi-tudinal and transverse directions; comparing the results obtained fromthe substructure model with those obtained from the continuum model.276Rel. displacement (m)0 5 10 15 20 25 30−0.2−0.100.10.2Time (sec)Transverse dir.−0.2−0.100.10.2 Longitudinal dir.  Continuum modelSubstructure model(d) Event no. 4, 1987 Superstition Hills-02 earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.2−0.100.10.2Time (sec)Transverse dir.−0.2−0.100.10.2 Longitudinal dir.(e) Event no. 5, 1999 Chi Chi earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.2−0.100.10.2Time (sec)Transverse dir.−0.2−0.100.10.2 Longitudinal dir.  Continuum modelSubstructure model(f) Event no. 6, 1994 Northridge-01 earthquakeFigure E.1: Continued.277Rel. displacement (m)0 5 10 15 20 25 30−0.2−0.100.10.2Time (sec)Transverse dir.−0.2−0.100.10.2 Longitudinal dir.  Continuum modelSubstructure model(g) Event no. 7, 1995 Kobe earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.2−0.100.10.2Time (sec)Transverse dir.−0.2−0.100.10.2 Longitudinal dir.  Continuum modelSubstructure model(h) Event no. 8, 1992 Cape Mendocino earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.2−0.100.10.2Time (sec)Transverse dir.−0.2−0.100.10.2 Longitudinal dir.  Continuum modelSubstructure model(i) Event no. 9, 1989 Loma Prieta earthquakeFigure E.1: Continued.278Rel. displacement (m)0 5 10 15 20 25 30−0.2−0.100.10.2Time (sec)Transverse dir.−0.2−0.100.10.2 Longitudinal dir.  Continuum modelSubstructure model(j) Event no. 10, 1989 Duzce earthquakeFigure E.1: Continued.279Shear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.  Continuum modelSubstructure model(a) Event no. 1, 1999 Kocaeli earthquakeShear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.  Continuum modelSubstructure model(b) Event no. 2, 1995 Kobe earthquakeShear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.  Continuum modelSubstructure model(c) Event no. 3, 1999 Hector Mine earthquakeFigure E.2: Time histories of shear forces induced at the pier base (pier 1)of the three-span bridge in the longitudinal and transverse directions;comparing the results obtained from the substructure model with thoseobtained from the continuum model.280Shear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.  Continuum modelSubstructure model(d) Event no. 4, 1987 Superstition Hills-02 earthquakeShear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.  Continuum modelSubstructure model(e) Event no. 5, 1999 Chi Chi earthquakeShear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.  Continuum modelSubstructure model(f) Event no. 6, 1994 Northridge-01 earthquakeFigure E.2: Continued.281Shear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.  Continuum modelSubstructure model(g) Event no. 7, 1995 Kobe earthquakeShear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.  Continuum modelSubstructure model(h) Event no. 8, 1992 Cape Mendocino earthquakeShear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.  Continuum modelSubstructure model(i) Event no. 9, 1989 Loma Prieta earthquakeFigure E.2: Continued.282Shear force (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−6−3036 Longitudinal dir.  Continuum modelSubstructure model(j) Event no. 10, 1989 Duzce earthquakeFigure E.2: Continued.283Bending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m  Continuum model Substructure model(a) Event no. 1, 1999 Kocaeli earthquakeBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m  Continuum model Substructure model(b) Event no. 2, 1995 Kobe earthquakeBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m  Continuum model Substructure model(c) Event no. 3, 1999 Hector Mine earthquakeFigure E.3: Time histories of bending moments induced at the pier base (pier1) of the three-span bridge in the longitudinal and transverse directions;comparing the results obtained from the substructure model with thoseobtained from the continuum model.284Bending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m  Continuum model Substructure model(d) Event no. 4, 1987 Superstition Hills-02 earthquakeBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m  Continuum model Substructure model(e) Event no. 5, 1999 Chi Chi earthquakeBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m  Continuum model Substructure model(f) Event no. 6, 1994 Northridge-01 earthquakeFigure E.3: Continued.285Bending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m  Continuum model Substructure model(g) Event no. 7, 1995 Kobe earthquakeBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m  Continuum model Substructure model(h) Event no. 8, 1992 Cape Mendocino earthquakeBending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m  Continuum model Substructure model(i) Event no. 9, 1989 Loma Prieta earthquakeFigure E.3: Continued.286Bending moment (MN.m)0 5 10 15 20 25 30−15−7.507.515Time (sec)Mxx, about the longitudinal axisMp=15.72 MN.m−15−7.507.515 Myy, about the transverse axisMp=15.72 MN.m  Continuum model Substructure model(j) Event no. 10, 1989 Duzce earthquakeFigure E.3: Continued.287Force at abutment (MN)0 5 10 15 20 25 30−6−3036Time (sec)Transverse dir.−8−4048 Longitudinal dir.  Continuum modelSubstructure model(a) Event no. 1, 1999 Kocaeli earthquakeForce at abutment (MN)0 5 10 15 20 25 30−8−4048Time (sec)Transverse dir.−8−4048 Longitudinal dir.  Continuum modelSubstructure model(b) Event no. 2, 1995 Kobe earthquakeForce at abutment (MN)0 5 10 15 20 25 30−8−4048Time (sec)Transverse dir.−8−4048 Longitudinal dir.  Continuum modelSubstructure model(c) Event no. 3, 1999 Hector Mine earthquakeFigure E.4: Time histories of forces induced at the abutment (abutment 1)of the three-span bridge in the longitudinal and transverse directions;comparing the results obtained from the substructure model with thoseobtained from the continuum model.288Force at abutment (MN)0 5 10 15 20 25 30−8−4048Time (sec)Transverse dir.−8−4048 Longitudinal dir.  Continuum modelSubstructure model(d) Event no. 4, 1987 Superstition Hills-02 earthquakeForce at abutment (MN)0 5 10 15 20 25 30−8−4048Time (sec)Transverse dir.−8−4048 Longitudinal dir.  Continuum modelSubstructure model(e) Event no. 5, 1999 Chi Chi earthquakeForce at abutment (MN)0 5 10 15 20 25 30−8−4048Time (sec)Transverse dir.−8−4048 Longitudinal dir.  Continuum modelSubstructure model(f) Event no. 6, 1994 Northridge-01 earthquakeFigure E.4: Continued.289Force at abutment (MN)0 5 10 15 20 25 30−8−4048Time (sec)Transverse dir.−8−4048 Longitudinal dir.  Continuum modelSubstructure model(g) Event no. 7, 1995 Kobe earthquakeForce at abutment (MN)0 5 10 15 20 25 30−8−4048Time (sec)Transverse dir.−8−4048 Longitudinal dir.  Continuum modelSubstructure model(h) Event no. 8, 1992 Cape Mendocino earthquakeForce at abutment (MN)0 5 10 15 20 25 30−8−4048Time (sec)Transverse dir.−8−4048 Longitudinal dir.  Continuum modelSubstructure model(i) Event no. 9, 1989 Loma Prieta earthquakeFigure E.4: Continued.290Force at abutment (MN)0 5 10 15 20 25 30−8−4048Time (sec)Transverse dir.−8−4048 Longitudinal dir.  Continuum modelSubstructure model(j) Event no. 10, 1989 Duzce earthquakeFigure E.4: Continued.291E.3 Results of the nine-span bridgeThis section presents the relative displacements of the pier top with respect to thepier base (Figs. E.5 and E.6), the shear forces induced at the pier base (Figs. E.7and E.8), the bending moments induced at the pier base (Figs. E.9 and E.10), theforces transferred from the deck to the abutment system (Figs. E.11 and E.12).Each figure includes three sets of results that are obtained from the continuummodel, the substructure model, and the fixed-base model of the three-span bridge.292Rel. displacement (m)0 5 10 15 20 25 30−0.12−0.0600.060.12Time (sec)Transverse dir.−0.12−0.0600.060.12Longitudinal dir.  Continuum modelSubstructure modelFixed base model(a) Event no. 1, 1978 Tabas earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.12−0.0600.060.12Time (sec)Transverse dir.−0.12−0.0600.060.12Longitudinal dir.  Continuum modelSubstructure modelFixed base model(b) Event no. 2, 1980 Irpinia earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.12−0.0600.060.12Time (sec)Transverse dir.−0.12−0.0600.060.12Longitudinal dir.  Continuum modelSubstructure modelFixed base model(c) Event no. 3, 1989 Loma Prieta earthquakeFigure E.5: Time histories of relative displacement of the pier top with re-spect to the base of pier 4 of the nine-span bridge supported on soilprofile 1; comparing the results obtained from the substructure and thefixed base model with those obtained from the continuum model.293Rel. displacement (m)0 5 10 15 20 25 30−0.12−0.0600.060.12Time (sec)Transverse dir.−0.12−0.0600.060.12Longitudinal dir.  Continuum modelSubstructure modelFixed base model(d) Event no. 4, 1994 Northridge earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.12−0.0600.060.12Time (sec)Transverse dir.−0.12−0.0600.060.12Longitudinal dir.  Continuum modelSubstructure modelFixed base model(e) Event no. 5, 1999 Chi Chi earthquakeFigure E.5: Continued.294Rel. displacement (m)0 5 10 15 20 25 30−0.12−0.0600.060.12Time (sec)Transverse dir.−0.12−0.0600.060.12Longitudinal dir.  Continuum modelSubstructure modelFixed base model(a) Event no. 1, 1978 Tabas earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.12−0.0600.060.12Time (sec)Transverse dir.−0.12−0.0600.060.12Longitudinal dir.  Continuum modelSubstructure modelFixed base model(b) Event no. 2, 1980 Irpinia earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.12−0.0600.060.12Time (sec)Transverse dir.−0.12−0.0600.060.12Longitudinal dir.  Continuum modelSubstructure modelFixed base model(c) Event no. 3, 1989 Loma Prieta earthquakeFigure E.6: Time histories of relative displacement of the pier top with re-spect to the base of pier 4 of the nine-span bridge supported on soilprofile 2; comparing the results obtained from the substructure and thefixed base model with those obtained from the continuum model.295Rel. displacement (m)0 5 10 15 20 25 30−0.12−0.0600.060.12Time (sec)Transverse dir.−0.12−0.0600.060.12Longitudinal dir.  Continuum modelSubstructure modelFixed base model(d) Event no. 4, 1994 Northridge earthquakeRel. displacement (m)0 5 10 15 20 25 30−0.12−0.0600.060.12Time (sec)Transverse dir.−0.12−0.0600.060.12Longitudinal dir.  Continuum modelSubstructure modelFixed base model(e) Event no. 5, 1999 Chi Chi earthquakeFigure E.6: Continued.296Shear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(a) Event no. 1, 1978 Tabas earthquakeShear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(b) Event no. 2, 1980 Irpinia earthquakeShear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(c) Event no. 3, 1989 Loma Prieta earthquakeFigure E.7: Time histories of shear force induced at the base of pier 4 ofthe nine-span bridge supported on soil profile 1; comparing the resultsobtained from the substructure and the fixed base model with those ob-tained from the continuum model.297Shear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(d) Event no. 4, 1994 Northridge earthquakeShear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(e) Event no. 5, 1999 Chi Chi earthquakeFigure E.7: Continued.298Shear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(a) Event no. 1, 1978 Tabas earthquakeShear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(b) Event no. 2, 1980 Irpinia earthquakeShear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(c) Event no. 3, 1989 Loma Prieta earthquakeFigure E.8: Time histories of shear force induced at the base of pier 4 ofthe nine-span bridge supported on soil profile 2; comparing the resultsobtained from the substructure and the fixed base model with those ob-tained from the continuum model.299Shear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(d) Event no. 4, 1994 Northridge earthquakeShear forces (MN)0 5 10 15 20 25 30−2−1012Time (sec)Transverse dir.−2−1012 Longitudinal dir.  Continuum modelSubstructure modelFixed base model(e) Event no. 5, 1999 Chi Chi earthquakeFigure E.8: Continued.300Bending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.Mp=25.6 MN.m−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Continuum modelSubstructure modelFixed base model(a) Event no. 1, 1978 Tabas earthquakeBending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.Mp=25.6 MN.m−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Continuum modelSubstructure modelFixed base model(b) Event no. 2, 1980 Irpinia earthquakeBending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.Mp=25.6 MN.m−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Continuum modelSubstructure modelFixed base model(c) Event no. 3, 1989 Loma Prieta earthquakeFigure E.9: Time histories of bending moment induced at the base of pier4 of the nine-span bridge supported on soil profile 1; comparing theresults obtained from the substructure and the fixed base model withthose obtained from the continuum model.301Bending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.Mp=25.6 MN.m−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Continuum modelSubstructure modelFixed base model(d) Event no. 4, 1994 Northridge earthquakeBending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.Mp=25.6 MN.m−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Continuum modelSubstructure modelFixed base model(e) Event no. 5, 1999 Chi Chi earthquakeFigure E.9: Continued.302Bending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.Mp=25.6 MN.m−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Continuum modelSubstructure modelFixed base model(a) Event no. 1, 1978 Tabas earthquakeBending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.Mp=25.6 MN.m−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Continuum modelSubstructure modelFixed base model(b) Event no. 2, 1980 Irpinia earthquakeBending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.Mp=25.6 MN.m−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Continuum modelSubstructure modelFixed base model(c) Event no. 3, 1989 Loma Prieta earthquakeFigure E.10: Time histories of bending moment induced at the base of pier4 of the nine-span bridge supported on soil profile 2; comparing theresults obtained from the substructure and the fixed base model withthose obtained from the continuum model.303Bending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.Mp=25.6 MN.m−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Continuum modelSubstructure modelFixed base model(d) Event no. 4, 1994 Northridge earthquakeBending moment (MN.m)0 5 10 15 20 25 30−24−1201224Time (sec)Transverse dir.Mp=25.6 MN.m−24−1201224 Longitudinal dir.Mp=11.4 MN.m  Continuum modelSubstructure modelFixed base model(e) Event no. 5, 1999 Chi Chi earthquakeFigure E.10: Continued.304Force at abutment (MN)0 5 10 15 20 25 30−12−60612Time (sec)Transverse dir.−12−60612Longitudinal dir.  Continuum modelSubstructure modelFixed base model(a) Event no. 1, 1978 Tabas earthquakeForce at abutment (MN)0 5 10 15 20 25 30−12−60612Time (sec)Transverse dir.−12−60612Longitudinal dir.  Continuum modelSubstructure modelFixed base model(b) Event no. 2, 1980 Irpinia earthquakeForce at abutment (MN)0 5 10 15 20 25 30−12−60612Time (sec)Transverse dir.−12−60612Longitudinal dir.  Continuum modelSubstructure modelFixed base model(c) Event no. 3, 1989 Loma Prieta earthquakeFigure E.11: Time histories of forces induced at the abutment 1 of the nine-span bridge supported on soil profile 1 in the longitudinal and trans-verse directions; comparing the results obtained from the substruc-ture and the fixed base model with those obtained from the continuummodel. 305Force at abutment (MN)0 5 10 15 20 25 30−12−60612Time (sec)Transverse dir.−12−60612(d) Event no. 4, 1987 Superstition Hills-02 earthquake(e) Event no. 5, 1999 Chi Chi earthquakeFigure E.11: Continued.306(d) Event no. 4, 1987 Superstition Hills-02 earthquake

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