Investigation of soil-flexible foundation-structure interaction for incident plane SH waves

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Content INVESTIGATION OF SOIL-FLEXIBLE FOUNDATION-STRUCTURE INTERACTION FOR INCIDENT PLANE SH WAVES by Vlado Gicev A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillm ent of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CIVIL ENGINEERING) May 2005 Copyright 2005 Vlado Gicev Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3180303 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3180303 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I would like to express my sincerest thanks to Professor M ihailo Trifunac for his masterful guidance and help throughout the research, without which this thesis would not exist. He has been not only a great scientific advisor but also a friend whose company made my life in the U.S. easier. Equally, I would like to thank Professor M arija Todorovska, who brought me to USC, and financially supported my research from her grants, and whose advice was very useful in my studies. I am grateful to Professors V.W. Lee and W. Proskurowski for their superb teaching of the classes related to my research and for their enthusiasm in giving advice whenever I asked them. I would like to thank Professor F.J. Sanchez Sesma from Universidad National Autonoma de Mexico for advice regarding numerical treatment of artificial boundaries. The teaching experience I had as a teaching assistant in the Civil Engineering Department at USC was very valuable for my career, and I want to express my gratitude to the Department of Civil Engineering for giving me this opportunity. Finally, I want to thank my beloved daughter Sofija and my wife Maja for their endless patience. I dedicate this dissertation to them. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ACKNOWLEDGEMENTS................................................................................ ii LIST OF T ABLE S........................................................................................... v LIST OF FIGURES......................................................................................... vi ABSTRACT................................................................................................ xiii CHAPTER I INTRODUCTION 1.1 Numerical m e th o d s ......................................................... 1 1.2 Soil-structure In te ra ctio n ................................................. 5 1.3 O rganization................................................................... 7 CHAPTER II COMPUTATIONAL MODEL 2.1 Numerical schemes and grid parameters: In tro d u ctio n ............................................ 9 2.2 Numerical s c h e m e ......................................................... 14 CHAPTER III ARTIFICIAL BOUNDARY 3.1 Artificial boundaries: A R eview ........................................ 22 3.1.1 Elementary b o un d a rie s.......................................... 23 3.1.2 Consistent (global) boun d a rie s................................24 3.1.3 Imperfect (local) boun d a rie s................................... 34 3.1.3.1 Paraxial boun d a rie s.................................... 35 3.1.3.2 Viscous b o un d a rie s....................................41 3.1.3.3 M ulti-directional boun d a rie s........................ 42 3.1.3.4 Expansion boun d a rie s.................................45 3.1.3.5 Extrapolation b o un d a rie s.............................45 3.1.4 C o n clu sio n .......................................................... 49 3.2 Rotated viscous artificial b o un d a ry................................... 50 3.3 Numerical te s ts ............................................................. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER IV SOIL-STRUCTURE INTERACTION WITH A FLEXIBLE FOUDATION: STEADY-STATE ANALYSIS 4.1 In tro d u ctio n ................................................................... 73 4.2 Numerical e xa m p le ........................................................ 74 4.2.1 Input and grid param eters....................................... 74 4.2.2 R e su lts................................................................ 78 4.3 C o n clu sio ns.................................................................. 98 CHAPTER V SOIL-STRUCTURE INTERACTION WITH A FLEXIBLE FOUNDATION: TRANSIENT ANALYSIS 5.0 Introduction. 1-D M o d e l................................................104 5.1 Input and grid parameters for the 2-D m odel..................... 112 5.2 Energy distribution in the s y s te m .................................... 117 5.3 C o n clu sio n .................................................................. 139 CHAPTER VI INPUT PARAMETERS FOR STRUCTURAL DESIGN 6.0 Introduction and M o d e l................................................ 144 6.1 R e su lts ....................................................................... 148 6.2 C o n clu sio n .................................................................. 169 CHAPTER VII NONLINEAR ANALYSIS 7.0 M o d e l......................................................................... 170 7.1 Distribution of the energy and the permanent s tra in s ....................................................174 7.2 Average displacem ents at the contacts and distribution of the permanent s tra in s ............................... 186 CHAPTER VIII SU M M AR Y .......................................................................... 207 BIBLIOGRAPHY.........................................................................................210 APPENDIX I FINITE DIFFERENCE FORMULAE FOR CHARACTERISTIC POINTS...................................................... 216 APPENDIX II INPUT ENERGY FOR THE STRUCTURE...................................... 227 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES TABLE 1, MAXIMA OF FOUNDATION DISPLACEMENT AND RELATIVE DISPLACEMENT............................................................ 98 TABLE 2. ERROR IN ENERGY CALCULATIONS FOR TWO DIFFERENT BUILDINGS....................................................... 119 TABLE 3. DEPENDENCE OF THE ERROR FROM THE GRID PARAMETERS.............................................................. 135 TABLE 4. ENERGY DISTRIBUTION OF THE FIELD REACHING THE FOUNDATION...................................................... 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES Fig. 2.1 SOIL-FOUNDATION-STRUCTURE S YSTEM ...................................... 10 Fig. 2.2 THE MODEL WITH THE ARTIFICIAL BOUNDARY............................... 11 Fig. 2.3 APPROXIMATION OF THE FOUNDATION......................................... 17 Fig. 2.4 CHARACTERISTIC GRID POINTS.................................................... 18 Fig. 2.5 TYPICAL COMPUTATIONAL C E LL................................................... 20 Fig. 3.1 DISPERSION RELATIONS FOR THE PARAXIAL BOUNDARY................. 37 Fig. 3.2 WAVE PROPAGATING TOWARD ARTIFICIAL BOUNDARY y = C ........................................................................ 38 Fig. 3.3 EXTRAPOLATION BOUNDARY........................................................ 47 Fig. 3.4 DECOMPOSITION OF THE PROBLEM.............................................. 53 Fig. 3.5 THE LEFT BOTTOM CORNER OF THE M O D E L.................................. 57 Fig. 3.6 TEST MODEL - BUILDING LOADED ON THE T O P ..............................61 Fig. 3.7 TEST EXAMPLE: HOLLYWOOD STORAGE BUILDING .......................... 70 Fig. 3.8 TEST EXAMPLE: HOLIDAY INN HO TEL............................................ 71 Fig. 4.1 MODEL: HOLLYWOOD STORAGE BUILD IN G..................................... 75 Fig. 4.1a TIME HISTORY OF THE DISPLACEMENT AT POINT 0 FOR SOME FREQUENCIES pf = 5 0 0 m / s , y = 3 0 ° ...................... 79 Fig. 4.2 RESPONSE AT THE BUILDING - FOUNDATION CONTACT NORMALIZED BY FREE SURFACE RESPONSE ^ = 3 0 0 mis, y = 3 0 ° ............................................................. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4.3 RESPONSE AT THE BUILDING - FOUNDATION CONTACT NORMALIZED BY FREE SURFACE RESPONSE 1 3f = 5 0 0m is , y - 3 0 ° ............................................................. 81 Fig. 4.4 RESPONSE AT THE BUILDING - FOUNDATION CONTACT NORMALIZED BY FREE SURFACE RESPONSE p f = 3 0 0 mis, y = 6 0 ° ............................................................. 82 Fig. 4.5 RESPONSE AT THE BUILDING - FOUNDATION CONTACT NORMALIZED BY FREE SURFACE RESPONSE /3f = 5 0 0 mis, y = 60°............................................................. 83 Fig. 4.6 RELATIVE RESPONSE................................................................... 85 Fig. 4.7a THE EFFECT OF THE BUILDING AS ADDED MASS TO THE HALF SPACE..................................................................... 87 Fig. 4.7b TIME HISTORIES OF DISPLACEMENTS, VELOCITIES, AND ACCELERATIONS AT THE TOP AND BOTTOM OF THE BUILDING............................................................................. 88 Fig. 4.8a DISPLACEMENT FOR SOME CHARACTERISTIC FREQUENCIES fif =300m/s, y = 30°.................................... 90 Fig. 4.8b DISPLACEMENT FOR SOME CHARACTERISTIC FREQUENCIES p f =500m/s, y = 30°.................................... 91 Fig. 4.8c DISPLACEMENT FOR SOME CHARACTERISTIC FREQUENCIES fif = 300m/ $ , y = 6 0 ° .................................... 92 Fig. 4.8d DISPLACEMENT FOR SOME CHARACTERISTIC FREQUENCIES pf = 5 Q Q w /5 , y = 6 0 ° .................................... 93 Fig. 4.9 TIME HISTORIES OF THE RESPONSE TO THE HALF-SINE PULSE AT 0 (SOLID LINE) AND AT O’ (DASHED L IN E )..................... 102 Fig. 5.0a 1-DTEST: M O D E L ...._ _ D A D ...................................................... 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 Fig. 5.0b AN ARTIFICIAL BOUNDARY FOR 1-D WAVE PROPAGATION............. 106 Fig. 5.0c PROPAGATION OF TRAPEZOIDAL PULSE THROUGH SHEAR B E A M ............................................................ 109 Fig. 5.0d RESPONSE IN THE MIDDLE OF THE 1-D M O D E L........................... 110 Fig. 5.1 MODEL WITH COMPONENTS OF THE MOTION IN THE S O IL ..................................................................113 Fig. 5.2 FILTERED HALF-SINE PU LSE....................................................... 115 Fig, 5.3a BALANCE OF ENERGY FOR WHOLE MODEL OF HOLLYWOOD STORAGE BUILDING......................... .................... 123 Fig. 5.3b BALANCE OF ENERGY FOR WHOLE MODEL OF HOLIDAY INN HO TEL.................................................................. 124 Fig. 5.4 TIME HISTORIES OF DISPLACEMENTS AT THE TOP AND BOTTOM OF THE BUILDING ................................................. 125 Fig. 5.4a FOURIER TRANSFORM OF HALF-SINE PULSE FOR DIFFERENT DURATIONS OF THE P U LS E .................................128 Fig. 5.5a THREE COMPONENTS OF THE ENERGY IN THE MODEL AS FUNCTIONS OF TIME: HOLLYWOOD STORAGE B UILD IN G .............. 131 Fig. 5.5b THREE COMPONENTS OF THE ENERGY IN THE MODEL AS FUNCTIONS OF TIME: HOLIDAY INN HO TEL................................. 132 Fig. 5.5c DISTRIBUTION OF ENERGY REACHING THE FOUNDATION: HOLLYWOOD STORAGE BUILDING............................................... 133 Fig. 5.5d DISTRIBUTION OF ENERGY REACHING THE FOUNDATION: HOLIDAY INN HO TEL.................................................................. 134 Fig. 5.6a DISPLACEMENT OF THE SOIL ISLAND AT THE END OF THE ANALYSIS: HOLLYWOOD STORAGE BU LDIN G .........................141 Fig. 5.6b DISPLACEMENT OF THE SOIL ISLAND AT THE END OF THE ANALYSIS: HOLIDAY INN HO TEL........................................... 142 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.0 THE MODEL FOR TRANSIENT ANALYSIS.........................................145 Fig. 6.1a AMPLITUDES OF THE NORMALIZED AVERAGE DISPLACEMENT AT THE CONTACTS y = 0 ...................................150 Fig. 6.1b AMPLITUDES OF THE NORMALIZED AVERAGE DISPLACEMENT AT THE CONTACTS y = 3 0 ° .............................. 151 Fig. 6.1c AMPLITUDES OF THE NORMALIZED AVERAGE DISPLACEMENT AT THE CONTACTS / = 6 0 ° ............. 152 Fig. 6.1 d AMPLITUDES OF THE NORMALIZED AVERAGE DISPLACEMENT AT THE CONTACTS y = 85° .............................. 153 Fig. 6.2a AMPLITUDES OF THE NORMALIZED AVERAGE VELOCITIES AT THE CONTACTS y = 0 .........................................158 Fig. 6.2b AMPLITUDES OF THE NORMALIZED AVERAGE VELOCITIES AT THE CONTACTS y = 3 0 ° ......................................159 Fig. 6.2c AMPLITUDES OF THE NORMALIZED AVERAGE VELOCITIES AT THE CONTACTS y = 60° ..................................... 160 Fig. 6.2d AMPLITUDES OF THE NORMALIZED AVERAGE VELOCITIES AT THE CONTACTS y = 85° ...................................... 161 Fig. 6.3a NORMALIZED AMPLITUDES AT LEFT AND RIGHT ENDS OF THE BUILDING - FOUNDATION CONTACT y = 0 ........................162 Fig. 6.3b NORMALIZED AMPLITUDES AT LEFT AND RIGHT ENDS OF THE BUILDING - FOUNDATION CONTACT y = 3 0 ° .................... 163 Fig. 6.3c NORMALIZED AMPLITUDES AT LEFT AND RIGHT ENDS OF THE BUILDING- FOUNDATION CONTACT y = 6 0 ° ..................... 164 Fig. 6.3d NORMALIZED AMPLITUDES AT LEFT AND RIGHT ENDS OF THE BUILDING - FOUNDATION CONTACT y = 8 5 ° .................... 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6,4 MODEL: DISK SITTING ON HALF S PAC E....................................... 167 Fig. 6,5 TIME HISTORIES OF THE MOTION AT LEFT AND RIGHT ENDS OF THE BUILDING - FOUNDATION CONTACT J3f = 500m/s ......... 168 Fig. 7.1 MODEL WITH NONLINEAR S O IL ................................................... 171 Fig. 7.1a CONSTITUTIVE LAW o - s FOR NONLINEAR S O IL........................ 172 Fig. 7.2a ENERGY DISTRIBUTION IN MODEL WITH NONLINEAR SOIL: HOLLYWOOD STORAGE BUILDING y = 3 0 ° ................................... 176 Fig. 7.2b ENERGY DISTRIBUTION IN MODEL WITH NONLINEAR SOIL: HOLLYWOOD STORAGE BUILDING y = 6 0 ° ................................... 177 Fig. 7.3a ENERGY DISTRIBUTION IN MODEL WITH NONLINEAR SOIL: HOLIDAY INN HOTEL ^ = 3 0 ° ..................................................... 178 Fig. 7.3b ENERGY DISTRIBUTION IN MODEL WITH NONLINEAR SOIL: HOLIDAY INN HOTEL r = 6 0 ° ..................................................... 179 Fig. 7.4a PERMANENT STRAIN DISTRIBUTION IN THE SOIL ISLAND: HOLLYWOOD STORAGE BUILDING................................................ 182 Fig, 7.4b PERMANENT STRAIN DISTRIBUTION IN THE SOIL ISLAND: HOLIDAY INN HO TEL................................................................... 183 Fig. 7.5a TIME HISTORIES OF DISPLACEMENTS AT THE TOP AND THE BOTTOM OF THE BUILDING FOR NONLINEAR S O IL ................... 184 Fig. 7.5b DEVELOPMENT OF THE PERMANENT STRAIN IN TIME AT P O IN T S .................................................................... 185 Fig. 7.6a NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS OF THE MODEL WITH NONLINEAR SOIL y = 0 ............................... 187 Fig. 7.6b NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS OF THE MODEL WITH NONLINEAR SOIL y = 3 0 ° ............................ 188 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.6c NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS OF THE MODEL WITH NONLINEAR SOIL y = 6 0 ° ............................ 189 Fig. 7.6d NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS OF THE MODEL WITH NONLINEAR SOIL y = 8 5 ° ............................ 190 Fig. 7.7b NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS: CASE OF LARGE NONLINEARITY IN THE SOIL y = 3 0 ° ..............192 Fig. 7.7c NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS: CASE OF LARGE NONLINEARITY IN THE SOIL y = 6 0 ° .............. 193 Fig. 7.7d NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS: CASE OF LARGE NONLINEARITY IN THE SOIL y = 8 5 ° .............. 194 Fig. 7.8a DISTRIBUTION OF THE PERMANENT STRAIN JUST AFTER THE WAVE HAS PASSED THE FOUNDATION y = 0 .......................... 196 Fig. 7.8b1 DISTRIBUTION OF THE PERMANENT STRAIN JUST AFTER THE WAVE HAS PASSED THE FOUNDATION y = 3 0 ° , 77 = 0 . 6 .... 197 Fig. 7.8b2 DISTRIBUTION OF THE PERMANENT STRAIN JUST AFTER THE WAVE HAS PASSED THE FOUNDATION 7 = 3 0 ° , 77 = 1 . 8 .....198 Fig. 7.8c1 DISTRIBUTION OF THE PERMANENT STRAIN JUST AFTER THE WAVE HAS PASSED THE FOUNDATION y = 6 0 ° , 77 = 0 . 6 .....199 Fig. 7.8c2 DISTRIBUTION OF THE PERMANENT STRAIN JUST AFTER THE WAVE HAS PASSED THE FOUNDATION y = 6 0 ° , 77 = 1 . 8 .....200 Fig. 7.8d DISTRIBUTION OF THE PERMANENT STRAIN JUST AFTER THE WAVE HAS PASSED THE FOUNDATION y = 8 5 ° ..................... 201 Fig. 7.9 DISPLACEMENT IN THE SOIL AFTER THE PULSE HAS PASSED THE FOUNDATION......................................... 203 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.10 DISPLACEMENT IN THE SOIL AFTER THE PULSE HAS PASSED THE FOUDATION FOR DIFFERENT INCIDENT ANGLES AND 77 = 1 .................................................................... 206 Fig. II INPUT ENERGY REACHING THE FOUNDATION................................ 228 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT A numerical scheme is used for sim ulation of SH wave propagation through three different media: soil, a sem i-circular flexible foundation, and a structure. The response is studied for two types of input: monochromatic steady-state wave and half-sine pulse. The steady-state solution is presented for the example of the Hollywood Storage Building for two angles of incidence and for two different foundation stiffness. The displacem ent and strain am plitudes for the flexible foundation are generally larger than the am plitudes for the rigid foundation, except in the frequency range close to the natural frequencies of the building on a fixed base. The transient response to a half-sine wave is analyzed with emphasis on three aspects of the problem: 1. Energy distribution 2. Response at the contacts 3. Response when the soil is nonlinear. It is shown that the distribution of the maximum energy in the building and the scattered energy from the foundation are invariant with the duration of the pulse. Also, the input energy reaching the foundation is independent of the angle of incidence. The results of the energy distribution are illustrated for the Hollywood Storage building and the Holiday Inn hotel in Van Nuys, both in the Los Angeles metropolitan area. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The response at the contacts (soil-foundation and foundation-building) is studied for the Holiday Inn hotel for four angles of incidence and three foundation stiffness. The results are presented as normalized average displacem ents and as functions of dim ensionless frequencies. The constitutive law of nonlinear soil is assumed to be ideally elasto-plastic. Three levels of nonlinearity are considered, and the energy distribution and the distribution of the permanent strain are shown graphically for the above-mentioned two buildings. Through analysis of the response at the contacts, it is shown that the response in the presence of small and intermediate nonlinearity generally does not differ appreciably from the linear response. There are significant differences between the response experiencing large nonlinearity and the linear response. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I INTRODUCTION 1.1 Numerical methods In the era of fast computers, obtaining solutions to many previously unsolvable problems became a reality, especially for problems involving partial differential equations (PDE), in which the analytical solutions exist only for the sim plest conditions. By utilizing numerical methods, one can solve a problem from initial time to some desired tim e at all spatial points. The m ost popular numerical methods for solving PDEs are the finite element method (FEM) and the finite difference method (FD). Usually, FEM uses im p licit schemes in which the unknown quantities at all spatial points are obtained sim ultaneously for each time step by solving a system of linear algebraic equations. M ost finite difference schemes are explicit, wherein the solution is obtained from the solution of the previous tim e step and the equations are uncoupled. Solving a full linear system of Nth order requires 0(N 2) operations, while for the uncoupled system the order of com plexity is 0(N). Thus, the explicit schemes are preferable in the numerical analyses, especially for large-scale problems (where N is big). The systems occurring in the im p licit schemes are usually banded and sym m etric, and so the order of complexity is much smaller than 0(N Z ) but still bigger than the one for explicit schemes. Also, the im plicit schemes are unconditionally stable, which is not the case with the explicit schemes. Further, the finite elements as a numerical tool are more suitable than finite 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. differences for m odeling complicated and irregular geometries. On the other hand, for large- scale problems, as inseismological practice, for example, the explicit schemes are preferable because they are cheaper and easier to im plem ent in numerical algorithms. Our goal in this work is to utilize a numerical scheme for simulation of wave propagation through bounded and unbounded media and to study the phenomena accompanying this wave passage. The idea of the numerical methods for PDEs is to replace the derivatives with small but finite differences at discrete space and time increments. Using an iterative procedure, the solution advances in tim e and, for the wave equation, in space as well. This iteration can go on forever, and it is up to the particular application when the algorithm should be ended. From many numerical schemes for hyperbolic PDE described in the literature (e.g., Sm ith,1985; Sod, 1985; Katsaounis & Levy,1999; Levy et al., 2000), the Lax-Wendroff (Lax & W endroff,1964) scheme is chosen. This is an explicit scheme, with second-order accuracy, both in time and in space o(At2,Ax2). The problems we wish to study, can be classified into three groups: • Heterogeneities and discontinuities of the medium • M odeling of the free surfaces • Artificial boundaries. According to Moczo (1989) and Zahradnik et al. (1993), the computational finite difference schemes that are used in applications of wave propagation, can be divided as follows: 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • Homogeneous schemes in which the boundary conditions of continuity of the displacem ents and the stresses are applied explicitly at the contact points, • Heterogeneous schemes in which all of the points in the inner computational domain, including the contact points, are computed with the same formula. In the earlier works, the wave-propagation problems were formulated with the second-order wave equation in terms of the displacement. Alterman and Karal (1968) used the homogeneous formulation to solve elastic wave propagation in layered media. At the contact points, using the continuity of the stresses and displacements, they consider additional rows of fictitious points, which they use for com puting the displacem ents at the contact. After obtaining the displacem ents at the contact, the motion in the next m edium can be computed using the displacem ent at the boundary points from the previous medium. Boore (1972) proposed the heterogeneous scheme. At an interface point m, he approximated the derivative by d ( d u \ Mrn+l/2 (^ m + 1 dx (1.1) A x When this derivative is used in the second-order wave equation directly, the displacem ent at the boundary point m is obtained without explicitly considering the stress-continuity f f boundary condition /u— = n — , where x is the normal of the contact at the dx dx i V Ji considered point and the subscripts represent the two media. Boore treated the free surface as a dw special interface with pi2 = 0 and the boundary condition at the free surface, — = 0 , where dn Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. n is normal of the free surface. This approximation of the free surface, known as vacuum formalism, was used by Zahradnik and Urban (1984) in studying variation of the ground motion due to the presence of a mountain. Other schemes, as in Vidale and Clayton (1986), Levander (1988), and Hayashi et al. (2001), use special formulae for com puting the needed functions at the points of the free surface. Kummer et al. (1987) approximated the mixed derivatives that appear in the equations of P (pressure) and SV (shear, in plane of propagation) wave motion by expanding the first derivative in a specific direction in terms of a Taylor series. For example Sf{x,y)^ d f [ ^ y }) ( ( A x 2 a3 / ( s ,.„ y ,) dy dy dxdy 2 dx23y Zahradnik et al. (1993) tested the above scheme and three other schemes for their behavior at discontinuities. Moczo (1989) used a heterogeneous scheme with variable grid spacing in the vertical direction and the Reynolds artificial boundary (Reynolds, 1978) for solving a sedimentary basin. Virieux (1984) introduced the first-order finite difference scheme for SH (shear, normal to the plane of propagation) waves by replacing the displacem ent field with the particle velocity and the shear stress field. With this new scheme, he solved a quarter plane, a sedimentary basin, and a salt dome problem for im pulsive and plane wave excitation. Levander (1988) proposed 0(k2,h4), a staggered grid. Lin (1996) proposed Zwas’s scheme (Eilon et al.,1972), used in gas dynamics, for solving crack problems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. At present, authors are m ostly concerned with developing m ultigrid and 3-D schemes, as in Hayashi et al. (2001), Ohminato and Chouet (1997), Wang and Schuster (2001), Graves (1996) that provide high-order accuracy in space 0(k2,h4 ). Accurate higher-order schemes, especially in space, can be tailored by using more neighboring points in the stencil, as shown by Dablain (1986). 1.2 Soil-Structure Interaction The particular problem studied in this work is the soil-structure interaction with flexible foundation. By its nature, this is a 3-D problem because both the superstructure and the foundation are 3-D media. For sim plicity, in this work only a two-dim ensional representation of the problem w ill be studied by taking one dim ension (the length) of the structure and the foundation as being infinite. For this 2-D model, we w ill study only the anti-plane response caused by the propagation of SH waves. Wong and Trifunac (1975) studied the w all-soil-w all interaction with the presence of two or more shear walls, and Abdel-Ghaffar and Trifunac (1977) studied the soil-bridge interaction both with a sem i-cylindrical rigid foundation and for an input plane SH wave. Other studies were conducted to analyze the dependence of the interaction on the shape of the rigid foundation. Wong and Trifunac (1974) solved the interaction of the shear wall erected on the elliptical rigid foundation for shallow and deep embedment. Westermo and Wong (1977) studied three different boundary models for soil-structure interaction of an embedded sem i- 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. circular rigid foundation and showed the differences in their dynamic behavior. They concluded that without a transmitting boundary all of the models develop resonant behavior and that the introduced damping in the soil cannot adequately model the radiation damping. Luco and Wong (1977) studied a rectangular foundation welded to an elastic half space and excited by a horizontally propagating Rayleigh wave. V.W. Lee (1979) solved a 3-D interaction problem consisting of a single mass supported by an embedded hemispherical, rigid foundation for incident plane P, SV, and SH waves, in spherical coordinates. W hile considerable research has been carried out on the phenomena of interaction with a rigid foundation, only several recent publications deal with a flexible foundation. Todorovska et al. (2001) solved an interaction of a dike on a flexible embedded foundation, and Hayir et al. (2001) described the same dike but in the absence of a foundation. Todorovska (2001) gave the estimate that for the ratio of the stiffness of the foundation and the soil — > 2 0 , and for M s pf = ps the model with an absolutely rigid foundation is approximately valid for many analyses. Aviles et al. (2002) analyzed in-plane motion of a 4 degrees of freedom model with 3 DOF (horizontal, vertical, and rocking) at the flexible foundation and 1 DOF (horizontal) in the superstructure. They described the dependence of the system properties (the effective period and damping) with the change of the geometry of the model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.3 Organization This work is organized in two parts. The first part presents the theory, while the second part illustrates applications and results. The theoretical part consists of Chapters II and III. 1. The second chapter in the thesis describes the model and the parameters of the numerical scheme. First, a short review of the grid parameters is given, after which the derivation of the numerical scheme is presented. Appendix I in which the finite difference equations are presented for characteristic points in the model, belongs to this chapter. 2. The third chapter starts with a review of the artificial boundaries. In the second section of this chapter, the artificial boundary algorithm is derived, and in the third section it is illustrated using two numerical examples. The applications part consists of four chapters, which deal with particular applications of the method. 3. The fourth chapter deals with steady-state aspects of the soil-structure interaction, with a flexible foundation. In the first section, some aspects of the soil-structure interaction for a rigid circular foundation are reviewed. In the second section, the input and the grid parameters for this application are introduced, and after that the results and the observations are presented. 4. The fifth chapter deals with the distribution of energy in the system. First, it provides some insight about the interaction system for pulse-like input with a sim ple 1-D model. In the second section, the input and the grid parameters are explained. Then, in the third section, the 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. distribution of the energy for two different buildings is studied, and the lim itations of the accuracy of the model are discussed. In the last section, conclusions about the energy distribution are presented. 5. In the sixth chapter, input parameters for energy-based engineering design of earthquake-resistant structures are given in the form of averaged amplitudes of the response (displacem ent and velocity) at the contacts soil-foundation and building-foundation, and as functions of the dimensionless frequency of the ground motion. Also, the displacements at two bottom corners of the building are given as functions of dim ensionless frequency. These results are presented for four different incident angles and for three different foundation stiffnesses. 6. Chapter VII illustrates some aspects of the response when the soil is nonlinear. In the first section, assumptions on the constitutive law a = cr(s), with internal definition of three kinds of nonlinearity, are presented. The next section deals with the balance and distribution of energy in the nonlinear system, and the last section shows the average displacem ents at the contacts, as functions of dim ensionless frequency, and compares the results with those of the linear case. Finally, general conclusions and a plan for future work are presented. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER II COMPUTATIONAL MODEL 2.1 Numerical schemes and grid parameters: Introduction The problem under consideration is the soil-foundation-structure interaction of a 2-D rectangular structure with a sem icircular foundation embedded in linear or nonlinear soil. The geom etric and material properties are given in Fig. 2.1. The physical domain of the problem is infinite in the soil, and for computational purposes, with an imposed artificial boundary (efgh in Fig. 2.2), the problem is defined in the finite domain 0 = n 5 U Q / U n i , which consists of three sub-domains. The foundation is flexible, with finite density pf and shear wave velocity p f . Moreover, without loss of generality, the foundation density is taken to be equal to the soil density pf = ps for sim plification of the numerical scheme. When a continuous problem of wave propagation is approximated with an explicit discrete scheme, the grid spacings and the tim e step must be chosen in such a way as to properly represent the waveform. First, the grid must satisfy the stability criterion, which requires that the eigenvalues of the system matrix are not greater than 1. For a second-order scheme in two dim ensions for the SH case, it is known (M itchell, 1969) that the stability condition is given by the Courant number : / = A n a x A / (2 .1) 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 2.1 SOIL-FOUNDATION-STRUCTURE SYSTEM Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig.2.2 THE MODEL WITH THE ARTIFICIAL BOUNDARY f t . Pb Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For a homogeneous grid, A x = Ay , the stability condition becomes 8 At 1 r ' max ^ A < (2 .2) Ax V 2 ’ where = m&x(pf J s,pb). An approximation of the continuous-wave propagation problem by a discrete grid leads to errors in the solution. The most important one is the grid dispersion, because it causes the velocity of the wave propagation to be a function of the grid spacing. Alford et al. (1974), Dablain (1986), and Fah (1992) studied the effect of different parameters on the grid dispersion. A measure of accuracy is the ratio between the numerical and the physical velocity of propagation r - , which ideally should be 1. The parameters that influence this error are: • The density of the grid m = A / A x (m - number of points per wavelength), • The Courant number % • The angle of the wave incidence 6. It was shown (Alford et al.,1974; Dablain,1986; Fah,1992) that the error increases when m decreases, % decreases, and 9 is close to 0 or nil. To increase the accuracy of the numerical schemes, usually researchers use higher- order approximations of the space derivatives. In this way, for sm aller m the same accuracy is achieved as for bigger m with a lower-order approximation. For example, for the L"1 order derivative in x: 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f x + ( 2 / - 1 ) — - / x-(2l-Y )-~ (2.3) A x where (L = 2 ,4 ,6 ,...) and the coefficients d2 A 1 can be obtained from the Taylor series. For second-order approximation, Moczo (1989) recommended m = 12, Alford et al. (1974) have shown that with m = 11 points for a half-power wavelength for second-order approximation and with m = 5.5 points for a half-power wavelength for fourth-order approximation, the results for diffraction around a corner are very sim ilar. Fah (1992) concluded that for m > 10 the error due to the grid dispersion is less than 1%. Levander (1988) was the first to use a fourth-order scheme. Dablain (1986) compared o ( a / 2, A x 2), o ( a / \ A x 4), and a very high o(At4, A x 10) scheme and showed that for the same achieved accuracy the ratios between these densities are m2 2 : m2A : mAl0 = 8 : 4 : 3 . sampling and showed how to maximize the spatial frequency so that the error is sm aller than the initially adopted adm issible error. Also the relations between the order of the spatial operator and the required grid points per wavelength (m) for five adm issible errors in the group velocity Eg r = 0.0003; 0.001; 0.003; 0.01; 0.03 were plotted. Holberg showed that for the standard operator (two samples, second-order approximation), and adm issible error Eg r = 0 .0 1 , the required number of points, per wavelength, that w ill keep the error within this bound is m = 16. For Egr = 0.03 and for the same operator the required m = 9. This shows that the c ^c o Holberg (1987) analyzed the error in the group velocity — involved with the spatial dk 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. recommendations from the previous authors involve error in the group velocity smaller than 3%. For higher accuracy, m grows rapidly, so for Egr = 0 .0 0 0 3 , m » 100 points/wavelength. Also, it was noted that the errors at the interfaces are controlled by the direction of the transmitted and the reflected wave. 2.2 Numerical scheme For our problem, the system of three partial differential equations (for u , v , and w) describing the dynamic equilibrium of an elastic body is reduced to the third equation only Introducing the new variables v = — , ex z = — , and ey z = — , and dividing (2.4) with p , dt dx dy the order (of 2.4) is reduced to the system of three first-order PDE: (because u - v = — = 0 ). Neglecting the body forces in the z direction (Fz = 0), this dz equation is: dt2 dx dy y (2.4) dw (2.5) w h e re : 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The first equation in (2.5) represents the dynamic equilibrium of forces in the z direction with neglected body force Fz > while the second and the third equations give the relations between the strains and the velocity. The abbreviations er = sr7, cr = r r„ , * J C J C Z 1 x XZ ’ sy = sy2, and a y = ry z are used later in the text. Instead of using velocity-stress formulation as in m ost previous studies, we use a velocity-strain-stress formulation, because in the nonlinear analysis it is more convenient to u p d a te r = t{s) than s = e{r). The equation (2.5) can be seen as a conservation law, by which the time rate of change of the quantity U on a differential area dA-dx-dy is equal to the sum of the differences of the fluxes F (in the x direction) and G (in the y direction) on the boundaries of that area. Moreover, the equation (2.5) is the m ost general mathematical representation of our physical problem, in which we dw only assume that the strains are small so that we have geom etric linearity — = tmeB * el3 dxi (i = 1,2). With this formulation, we can study the nonlinear response of the structure due to d'v material nonlinearities — - = jui3 * const (i = 1,2). dei3 A review of the artificial boundaries w ill show that the exact artificial boundaries are usually defined with a circular shape for 2-D problems or a spherical shape for 3-D problems. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Naturally, the coordinate systems that best describe these shapes are polar for 2-D problems or spherical for 3-D ones. Because of the rectangular shape of the structures, it is obvious that the polar coordinate formulation for the considered problem is not suitable, and thus the Cartesian formulation is used. For defining the grid spacing, in our problem we have an additional requirement for modeling the sem icircular foundation. The soil and the foundation subdomains, Qs and Qf , are discretized with square grids, while for the structure, Q b , we w ill use the rectangular grid. Because of the Cartesian formulation, the sem icircle is approximated with a symm etric hexagon with an axis of symmetry at x = 0. This requires an even number of grid intervals in the x direction. For different grid densities mf - L ! A x , this approximation is shown in Fig. 2.3, where L = 2a is the width of the structure. The vertical grid spacing in the structure is obtained R from the formula Ayb = — Ay , which prevents the dispersion relation in the vertical direction A at the contact foundation-structure. The horizontal spacing A x is constant along the whole grid. The time step A t , as was mentioned above and in Lin (1996), should be as big as possible to provide that x in (2.1) can be as close as possible to 1. In the numerical examples, At is computed from % = 0 .9 5 . The Taylor series expansion of the field U(t) at the point (x,y) gives; U(t + At) = U(t) + ( ^ \ A f + - ( ^ - = | A f 2 + o ( a / 3) . (2.7) \d t Jt ° ^ v J 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 0 A X /L 3 4 A X /L 3 8 A X /L Fig. 2.4 CHARACTERISTIC GRID POINTS Lb= 2 a y“ oA2 ' I D I L I ^ T j At r / f F , Bi / c / \ y 1 4 X1 / b 1 / 3 / / 2 G1 c Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Using the relation (2.5), the tim e derivatives of U are substituted with the space derivatives of the fluxes, and (2.7) becomes: U(t + At) = U(t) + r dF DG^ ] ------ dx dy A t + A t2 d 2 [dx w h e re : f A dF^+ dG dx dy ^ dy f B dF dG 1 ------ dx dy + 0(At2} (2.8) d F 0 5 a , pdex 0 and dG 0 0 do-y pdsy ~dU~ 1 0 0 dU 0 0 0 0 0 0 1 0 0 (2.9) For the problem at hand, from numerical tests it seems that the Zwas’ numerical scheme proposed by Lin (1996) becomes unstable when the upgoing field from the soil and the foundation and the downgoing one from the structure meet at the point H (Fig. 2.4). The problem is solved num erically for steady-state and transient half-sine excitation using the finite difference Lax-Wendroff method (Lax & Wendroff, 1964) for approximating the system of three first-order partial differential equations (2.5). The typical cell (i,j) for central- difference approximation is shown in Fig. 2.5. The points denoted by numbers are not grid points; the quantities at the points 1 to 4 are obtained as mean values of the quantities in the two neighboring grid points (denoted by circles), while the quantities for points 5 to 8 are obtained as mean values of quantities in the neighboring four grid points. The finite-difference equations for characteristic points in the model (Fig. 2.4), are given in Appendix I. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 2.5 TYPICAL COMPUTATIONAL CELL o- ( i - U ) o - O ' -o- ( i+ 1 , j + l ) 8 4 7 1 (i.j) 3 5 6 2 o - O - ( i+ 1 J) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. At the points of the physical boundaries, B x = { M ( x ,j/) |y = 0 ,|x |> a and B 2 = {a^(x, j ) | y = H b, |x| < a , the boundary condition a y = 0 is prescribed. To customize these points to the numerical scheme given with the equations 1 .1 A-I.3A in Appendix I and to calculate for the velocities and the strains (stresses) in the x direction, we introduce a row of fictitious points at the distance Ays above and another one at the distance Ayb above B2. To satisfy the boundary condition o y - 0 for the boundary point (i,j), we use the vacuum form alism to update the quantities of the fictitious points so that for the fictitious point ( i,j+ 1 ) ^ T ;'+ 1 ™ ’T _ / 1 1 ^ y i - l , j + l ^ y i , j + 1 ~~ ^ y i , j - l ' ^ y i + \ , j+ \ ~ M i , j + ll 2 ® in every time instant. In a sim ilar way, we proceed along the physical boundaries G, = {P(x, _y) | x = -a,y > 0 and G2 = {Q(x,y) | x = a, y > 0 }, where o x = 0 is prescribed. For example, for G1 we introduce a column of fictitious points x = - a - A x , y > 0 with prescribed values as fo llo w s : = V;+1J ’ = ~ ( J x i+ l,j~ l ’ ~~ ~ < Jx i+ \,j ’ & x i- l, j+ l ~ ~ ( J x i+ l,j+ l ’ /V l/2 ,/ “ ® ' The values of the velocities and the strains (stresses) in the y direction can be computed from the equations 1 .1 A - I.3A. For G2, the x indices take opposite signs relative to the previous numerical boundary conditions. The artificial boundaries are a constitutive part of the model, but because of their com plexity and the considerable research done in this field the next chapter is dedicated to the subject of artificial boundaries only. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER III ARTIFICIAL BOUNDARY 3.1 Artificial boundaries: A Review For dynamic analysis of a problem defined in an infinite domain in terms of discrete methods, there is need for a boundary called an “absorbing,” “artificial," or “ nonreflecting" transparent boundary. For practical reasons, the computational domain has to be finite, and the role of these artificial boundaries is to replace the effect of the truncated domain. It is obvious that this task can be accom plished only if we can solve for the unknown quantities at the absorbing boundaries. Kausel and Tassoulas (1981) classified the boundaries that occur in wave propagation problems into three g ro u p s : 1. elementary (nontransmitting) boundaries, 2. consistent (global) boundaries, and 3. im perfect (local) boundaries In essence, there is no sharp separation between the global and local boundaries because many of the local boundaries are obtained just by truncation of the infinite series obtained at some global boundaries. Nevertheless, the classification here is done with the ideas the authors use in developing the boundary. 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1.1 Elementary boundaries At the points on an elementary boundary, either the displacem ents or the stresses are prescribed. In the first case, we have a Dirichlet, and in the second a Neumann boundary condition. These two conditions occur at the existing physical boundaries, where either the boundary is stress-free so that the prescribed stresses (spatial derivatives of the displacement) are zero or the boundary is fixed so that the prescribed displacem ents are zero for all time. At the artificial boundaries, the Dirichlet and Neumann nonzero conditions can be applied only for the sim plest problems in which an analytical solution exists at the boundary. This is a case for a pure half-space problem in which the solution can be obtained from the ray theory in any point of the domain, or for 1-D wave propagation problem (Fujino & Hakuno, 1978) in which at the artificial-boundary point the solution can be uniquely defined from the solution of the neighboring point, with shifting in time. These two examples are trivial, but they can be used as test examples for accuracy of the actual numerical schemes that describe an artificial boundary. If, at the boundary, zero displacem ent (fixed boundary) or zero stress (free boundary) are prescribed, the boundary behaves as a perfect reflector: that is, the energy reflected back into the inner domain is equal to the incident energy on the boundary. The fixed boundary reflects the incident field out of phase, and the free boundary reflects it in phase. Smith (1974) used the above properties of the fixed and free boundary and constructed an absorbing boundary by solving at the boundary twice, first im plying the fixed, then the 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. stress-free boundary condition, and then taking the average as a solution. As pointed out by Kausel (1988) this boundary is a perfect absorber if only one boundary interface exists in the model. In the case of more than one boundary in the model, this boundary fails because the waves are reflected more than once. 3.1.2 Consistent (global) boundaries During the past decade, due to the need for highly accurate solutions to the problems involving infinite domains, the use of consistent boundaries became attractive. These boundaries are perfect absorbers, but they cannot be readily applied in ‘marching in time' procedures because of their nonlocality, both in time and space. This nonlocality comes from the terms that appear in the boundary equations in the transformed (Laplace or Fourier) space. The inverse transform of these terms back into the physical space does not yield regular, local differential operators, but rather some pseudo differential operators (Tsynkov, 1998). The result is that the solution at a boundary point depends upon the time history of the solutions in all of the boundary points. It should be pointed out that this is not so for 1-D problems, in which the boundary condition can be obtained readily in the physical space. Furthermore, for Courant number = l there is even no truncation error from the numerical scheme (Dablain,1986), as can Ax be tested for the case of linear and nonlinear shear beam sitting on a 1-D linear half space. 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tsynkov (1998) provided a detailed review of the existing global and local artificial boundaries and showed how to use the difference potential method for solving problems in aerodynamics. One of the first global boundaries is that developed by Engquist and Majda (1979). The key point of this boundary is to eliminate the incom ing (reflected) field on the boundary. The solution of the linear wave equation : for the plane wave can be written as u = u(oot + kxx + kyy ) , and the dispersion relation of the co2 solution is —j = k2+ky . The constant-phase surfaces are planes given with the equation cot + kxx + kyy = C . These planes travel in the space in direction {-kx -ky). Now, if the computational domain is x > 0 and the artificial boundary is at x = 0, then solving the dispersion relation for kx we have For the given setup above and for the positive radical, the incom ing (reflected) waves from the boundary travel in the positive x direction and should have kx < o . This corresponds to the minus sign in the expression for kx and the opposite for the outgoing waves. The Fourier (3.2) ( 2 \ d 2U transform of (3.1) in ta nd y gives the relation: - — ~ k 2 v u = — - , , dx 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where u{a>, x, ky) = j j u(t, x, y)e l{m t+ k ,y > dtdy t,y or from (3.2) d 2u , 2. ^ — T + kxii = 0, dx with solution (3.3) u Cx e - ^ X + C2e '^x = C 1 m(1) +C2 u(2). (3.4) As mentioned above, the mode with negative exponent is incom ing (reflected from the boundary) and should be eliminated from the solution at the boundary. This means that the solution at the boundary should be parallel (equal up to the constant) with the outgoing mode e ^ x (the mode traveling in direction -x ). This linear dependence between the solution and the outgoing mode at the artificial boundary at x = 0, for the second-order ordinary differential equation (ODE) (3.3) is given with the zero-valued Wronskian of second order as follows: W = u u{2 ) du dum dx dx = 0 . x=0 Replacing m(2) = e^ x and using (3.2), the boundary condition at x = 0 in the transformed space becomes (3.5) The boundary condition (3.5) can be obtained also by factorization of the operator (3.3): dx2 - + k l d Y d u = | — + ik dx dx - ikr \u 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note that the square root in (3.5) appears as an irrational term for which the inverse Fourier transform back to the physical space does not have a derivative of u. In utilizing this boundary for numerical computation, the square root is rationalized by using a series of Pade' approximants (Clayton & Engquist,1977). A considerable amount of research was done on constructing artificial boundaries using so-called Dirichlet to Neumann (DtN) mapping on the artificial boundary (Givoli & Keller, 1990;. Grote & Keller, 1996; Givoli, 2001). The idea of this approach is to express the normal derivative of the solution in terms of the solution itself at the boundary. Givoli and Keller (1990) used DtN for solving problems in 2-D elastodynamics using circular artificial boundary r = R. Starting with the Helmholtz decom position of the displacem ent vector field and using the Sommerfeld radiation condition for the irrotational and rotational potentials l i m r 1/2(<t> r -ik LQ >) = 0 l i m r 1/2( T r - ik LW) = 0 . (3.6a) r-~ »cQ J r - » oo The solution for the displacem ent in polar coordinates is found as a series of Hankel functions of first kind. Next, the radial and transverse components of the displacem ent at the boundary r = R are expanded in a single Fourier series along a circular coordinate. Matching the sim ilar terms of these two sets of series, the coefficients of the series involving Hankel functions are found, and with that the displacem ent field at the boundary r = R. Using this solution and the relations between the tractions and the displacements, the traction in polar coordinates at r = R is obtained. The final step is to go back to Cartesian coordinates with the well-known orthogonal transformation. In this way, the computational domain becomes annulus with the artificial 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. boundary as the outer boundary and the inner boundary with arbitrary shape, where the displacem ents and/or tractions are prescribed. This boundary was im plem ented in FE (finite element) formulation (Givoli and Keiler.1990). The effect of the boundary condition on the standard FE scheme is the adding of additional matrix Kb , the entries of which are an infinite series along the circular coordinates in the standard stiffness matrix. Kallivokas and Lee (2004) developed an artificial boundary with elliptic shape. They analyzed the scattering problem in an infinite fluid domain from a scatterer with arbitrary shape with a prescribed Neumann condition at its boundary. The com putational domain was enclosed with an e lliptic artificial boundary. The main concern was to satisfy the Sommerfeld radiation condition (3.6a), which in terms of the pressurep (the shear potential for fluid is zero) in physical space is given by Ximrl,1(p r + P-) = 0 . (3.6b) /--><» ‘ Q First, the governing wave equation of the problem is Laplace-transformed in time in the inner domain £ 1 Next, the authors consider the auxiliary problem in the outer domain Q + : s2P(x,s;t)=c2AP(x,s-,t), x e O + ; P(x,s;t) = p(x,t), x e T a , (3.7) where F a is the characteristic equation of the artificial boundary (ellipse) and P(x,s;t) is an auxiliary field. It was shown by using the Duhamel’s principle that the Laplace-transform of the solution in the outer domain is the Laplace transform of the solution of the auxiliary problem (3.7) 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y?(x, 5 ) = je stP (x , 5; t)d t, x e f i 4 (3.8) and also on the boundary pv(*,s) = \e~s tPv(iL,s;t)dt, x e Fa (3.8a) For the auxiliary field, the asymptotic expansion from geometrical optics is chosen as follows: , x e f T (3.9) P(x,$;f) = e c*(X)jP k= 0 _(s + r(x))a (3.9) will satisfy the second equation in (3.7) if and only if for x e Ta j ( x ) = 0 ^ (0)( x ,r ) = /? ( x ^ ) A(k)(x,f) = 0, * > 1. Taking the normal derivative of (3.9) and using (3.8a) we get 0 0 A (*,*) = — XV A ^ S )+ Y u ^ k)^ s) ~ ■ x e r ~ k = 0 (3.10) _(s + r(*))a The nonnegative parameter y(x) acts as damping and is used to provide stability for the higher-order absorbing boundary conditions. By introducing (3.9) into (3.7) and matching the terms with the same powers of (s + y), the equations for the unknown functions j ( x ) and A(k)(x, t) are obtained. Using the Fermi-type coordinate system, the normal derivatives of these functions can be expressed in terms of the curvature and the arc coordinate. The m-th order of the boundary (m = 0,1,2,3,...) is set by keeping m terms in the equation (3.10) and 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. truncating terms k > m + 1. The authors present the first four orders of the boundary (m = 0,1,2,3). Lee and Kallivokas (2004), using a second-order boundary of the previously mentioned boundary and image theory, show how to solve a scattering problem in a half space with an absorbing boundary of elliptical shape. The above two boundaries are classified as being of the global type because they are derived through truncation of the infinite series (3.10). All global methods involve high-order derivatives in time and in tangential direction on the artificial boundary. For implementation of these boundaries in finite element formulation, one needs to develop special finite elements that have high-order regularity in tangential direction at the artificial boundary. An alternative approach, which reduces the order of the boundary condition, so called arbitrary high-order condition (AHOC), was proposed by several authors. The trade-off in this approach involves new unknowns on the boundary. In this approach, the originally developed high-order boundary is replaced by an equivalent low-order one. Givoli (2001) used AHOC for steady state (Helmholtz equation) and tim e-dependent (wave equation) problems. The Helmholtz equation is e lliptic PDE in which only the spatial derivatives exist. He considered a k-th order artificial boundary in the form - f ^ = £ r « - (3 1 1 ) or If the artificial boundary involves only the tangential derivatives of higher order, then (3.11) has the form 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The order of (3.12) is reduced from k to 1 by involving k new unknowns: Vn = M (3.13) If only even tangential derivatives occur in the original artificial high-order boundary, the order of the boundary can be reduced to 2. The matrix form of the m odified boundary, for both odd and even tangential derivatives appearing on the artificial boundary is and for artificial boundaries involving only even derivatives For some local boundaries where a high order of radial derivatives occurs, as in the boundary of Bayliss and Turkel (1980), the m odified reduced-order boundary has the form The matrices Y and Z are symmetric, U = (u ^ v2 vk )T is the unknown vector and e1 = (1 0 0 . ...0 )T is a k-order vector. The approach for constructing AHOC for a tim e-dependent case (wave equation) is sim ilar to the steady-state case, with the difference that (3.12) involves a double series because the time derivatives should be reduced also. The number of the additional unknowns becomes — e, = YU + ZU". dr 1 (3.15) du dr e, = Y U + Z — U dr (3.16) 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (k*m ), where k is the highest derivative in the tangential direction and m is the highest derivative in time. Because of their non-rectangular shape, most of the global boundaries cannot be utilized in standard finite difference formulations. Grate and Keller (1996) proposed an artificial boundary and a finite difference algorithm for treating a spherical artificial boundary with a spherical grid. Their artificial boundary has the form (3.17) where % n m = z nm {t) is the solution of the ordinary differential e q u a tio n : — — -z (r)= A z (t) + u (t) z (o) = 0. i, nmx J « nmV / nm V / nm V / at (3.18) The n nxm matrices A „ = [A ^ ] are defined as fo llo w s : - n(n + 1) /(2 a)J i f i = 1 A 'n =<(n + /)(« +1 - z)/(2z) i f i = j +1 (3.19) 0 otherwise and Yn m (O, (p) = [(2« + 1)(« - j/w|)/ 4?r(« + |w |)}/2e'm (0 P jm i(cos(9) (3.20) is the nm-th spherical harmonic normalized over a unit sphere and 0 ° J (3.21) 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is a vector function having as the only nonzero entry the inner product 2 z r j r iYnm Mr=a)= \\Y nm {0,(p)iia,e,(p,t)§m6ded(p. (3.22) 0 0 The authors used the leap-frog FD (finite difference) scheme with the artificial boundary r = a and the algorithm as follows: 0. Initialize Uk at t = 0 and t = At. Set z ° n m = 0 and z\m = 0 1. Compute Uk+1 at all inner points using the FD scheme 2. Using the adopted FD scheme and FD approximation of the artificial boundary (3.17), compute U k a + A r and U*+1. 3. Compute z ^ 1 (the authors used trapezoidal rule of Runge-Kutta 2-step method) and go to 1. Using the above FD algorithm, the authors solved three problems, a) Time harm onic source in full space b) Scattering of low frequency plane wave vertically im pinging upon a spherical obstacle c) Radiation from a circular piston on a sphere. Then they compared results with the local Bayliss-Turkel (1980) boundary (BT). In all of the problems, the obstacle has the radius r0 = 0.5 with artificial boundary at r = 1. For all of the problems, the proposed algorithm gives accurate results. The BT gives accurate result for the second problem and fails for the first and the third ones. This is expected because the scattered field from a sphere for small wave numbers is alm ost spherical and BT is derived to annihilate such waves. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Hagstrom et al. (2003) developed a high-order boundary based on the m ulti-pole expansion of the outgoing field for solving the convective wave equation. The starting point is the Bayliss-Turkel boundary. Gachter and Grote (2003) derived and implemented in FE form ulation an exact spherical boundary for 3-D elastic waves. Premrov and Spacapan (2004) presented an interesting idea for improving local asym ptotic low-order DtN mapping, for problem s involving higher modes. Tsynkov (1998) described the construction of exact boundary conditions using the difference potential method on the 3-D problem of transonic fluid flow near a wing. The artificial boundary in this method can have arbitrary shape. 3.1.3 Imperfect (local) boundaries The main advantage of the local artificial boundaries is that they are local in space and time - that is, the solution at a boundary point depends only upon the solutions in several neighboring points in several time steps backwards. The other advantage for the tim e step procedures is that they are defined in the time domain and are not frequency dependent. Even though they are not perfect absorbers, because of the above properties the local boundaries are widely used in practical applications. The local absorbing boundaries can be subdivided in several classes depending upon the idea used in their development. We distinguish: 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • Paraxial boundaries • Viscous boundaries • M ultidirectional boundaries • Boundaries based on expansion of the outgoing field • Extrapolation boundaries. 3.1.3.1 Paraxial boundaries One of the most widely used artificial boundaries is the paraxial boundary developed by Clayton and Engquist (1977) which is based on the global boundary of Engquist and Majda. The two-dim ensional scalar wave equation is For wave propagating in the positive direction of an axis, the minus sign should be used in the corresponding term. Here, A is the amplitude of the scalar and is a constant. Plugging (3.24) into (3.23) and performing the differentiation, we obtain the dispersion relation (3.23) where the scalar field has form P(x, z, t) = Ae (3.24) (3.25) 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If we want to “ cut" the field in the y direction, say by imposing artificial boundary y = const, we should express the wave number ky PK C O ±, 1- P lk 0 3 (3.26) By using the Pade rational approximation of the square root (Clayton & Engquist, 1977), we can obtain the first three paraxial boundary conditions in the frequency domain. For a positive sign in (3.24), those are 2^ A1 : A2 A3 PK C O co PK C O 1 + 0 PK C O U P K A ® . 3 (PK + 0 PK C O 4 A , and (3.27) (3.28) 4 1 co 1 1 r P K^2 v 0 j O r (5k co (3.29) A graphical representation of the dispersion relations A1, A2, and A3, together with the \P j C O dispersion relation of the wave equation k) +k\ = A 2 = — , is given in Fig. 3.1. From the figure and from the relations (3.27), (3.28), and (3.29), it is obvious that the error is sm aller as kxbecomes sm aller or as the angle 0 between the ray and the normal of the boundary becomes smaller (because kx = & s in 6). In the ultimate case in which 0 = 0, the wave is perfectly absorbed in all three approximations. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 3.1 DISPERSION RELATIONS FOR THE PARAXIAL BOUNDARIES 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 3 .2 WAVE PROPAGATING TOWARD ARTIFICIAL BOUNDARY y = C Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The equations (3.27), (3.28), and (3.29) are not applicable in finite difference schemes, but they can be transformed to the tim e domain. For example, if we want to use the A2 approximation for the boundary y = const, and if the propagation is in a positive x and y direction (Fig.3.2), the scalar field can be represented as p = (3 30) We want to match (3.28) with a proper relation of the partial derivatives of P. The desired derivatives are Ptt=-co2P Pyt- k ycoP Pxx= - k 2 xP. The equation (3.28) in time domain becomes = (3.31) P 2 P 1 tt ^ 1 tt In a sim ilar way, we can represent the A1 and A3 boundaries, so that the three boundaries in the time domain are A1 : P + - - P t =0 (3.32) y P A 2 : P ^ + L . P ' - L P " ^ (3.33) P 2 A 3 : PyK- P - . p ^ + l p ^ i l . p a = 0 , (3.34) as given by Clayton and Engquist (1977). The reflection coefficients for paraxial boundaries are r = (3.35) C1 + cos 0 ) 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where j is the order of the boundary. It is obvious that for 9 - > n l 2 , r — > l , which corresponds to total reflection, but, as was pointed out by the authors, in practical problems these components of the field w ill be absorbed by the next, perpendicular boundary before interaction with the inner field. Nevertheless, one should expect some amount of reflection. Using factorization of the differential operators in the 2-D wave equation, Reynolds (1978) has derived a sim ilar boundary, which for absorbing boundary x = const is given as 1 d2u d2u p d2u 0 + — r- + — ------ — — 0 , (3.36) /? dxdt dx 1 + p dy where p = p • At/Ax . The reflection coefficient is cos9 - cos2 0 ---------- • sin2 6 R = --------------------- ^ -------- - . (3.37) cos# + cos2 9 + ...sin2 6 1 + p It can be seen that for p = 1 the reflection coefficient R is the same as r in (3.35) for a paraxial boundary of order 2. For different parameters p and different angles 0, Reynolds tabulated and graphed the reflection coefficient R given in (3.37). From the table, for 0 = tc/6, the least reflection gives p = 1. For 0 = tc/4 the equation (3.36) is a perfect absorber for p = 1 / V 2 , and for 0 = t c /3 it is a perfect absorber for p - 1 /2 . 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1.3.2 Viscous boundaries This type of local artificial boundaries, based on Lysmer and Kuhlemeyer (1969), is often used in practical computations because of its sim plicity. For a SH wave, the principal ti strain (in the direction of propagation) is given by s = — , and the strain in the direction P perpendicular to the boundary (Fig.3.2) is 1J s = cos 0. (3.38) y p M ultiplying (3.38) by G = p ■ ft2, we obtain the viscous boundary in the form t + pficos0- — = O . (3.39) dt This expresses the dynamic equilibrium at the boundary y = C. The wave numbers in (3.30) can be expressed in terms of the direction of propagation of the wave (the angle 0 in Fig. 3.2), as follows: k = £ sin0 = — sin 0 kv - — cos# . P y p The equation (3.30) in a more general form can be written as n . / xsin0 + ycos0'\ P = f ( t - k xx - k yy) = f t ------------ - --------- V P ) and the scalar field P must satisfy the wave equation P (3.40) 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If we take sym bolically the square root of the differential operators, we can w rite : - P xx ' (3.42) n ir i fit xsin # Recalling that x = - f — t = --------- , sin 0 f i we can write - Ptt dx2 f i 1 tt d2t sin2 0 n Performing the required differentiation in (3.42), we obtain for a wave propagating in positive y (positive sign in 3 .4 2 ): Taking the scalar field P in (3.43) as the displacem ent field u and m ultiplying (3.43) by G = pfi2 we obtain the viscous boundary (3.39). The viscous boundary is a perfect absorber for the waves with normal incidence, in which case the viscous boundary is identical with the paraxial boundary of order 1 (3.32). 3.1.3.3 M ulti-directional boundaries An approach to m ulti-directional boundaries can start from the concept of viscous boundaries. If a SH wave propagates in direction (see Fig.3.2), the relation between the particle velocity and the strain is given by " a | c o s # d V Q Kdy f i d t) (3.43) 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. du - du ^ = ^ 3-44) at dy1 which is obtained from the 1-D wave equation written in the y, direction. Because we want the derivative with respect to y, the RHS (right-hand side) of (3.44) should be modified. Using the relation y = - ^ — , the derivative on the RHS of (3.44) becomes cos 6 du__ d u _ _dy__ _ J _ _ 'du_ Re p I a c i ng this in (3.44), we again obtain the equation (3.43), dyx dy dyx cos 0 dy where the scalar field is u. From this comes the idea of the m ulti-directional boundary. Since in the general case the incidence of the waves onto the boundary is from different angles, Higdon (1986; 1991) approximated this continuous incidence with a finite set of m angles of incidence. The absorbing boundary then can be written in form: ( \d d looser S — H B —- v j / dt dy \u = 0 , (3.45) where a3 are presumed angles of incidence. Thus, the components of the wave that impinge upon the boundary with angles of incidence equal to a ] w ill be perfectly absorbed. Theoretically, as m - » qo , this boundary is a perfect absorber for any scalar wave field. Also, it is seen that in a special case, a . = 0 V 1 < j < m , this boundary becomes identical with the A1 paraxial boundary (3. 32). A question arises as to how many angles, m, should be included in (3.45) and what their values should be for a successful absorption. From numerical tests, Higdon (1991) concluded that m = 2 is a practical choice but that the values of a } are problem dependent. 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For example, if the waves near normal incidence are expected, all aj in (3.45) should be taken equal to 0. But from numerical tests, Higdon concluded that the amount of reflection is not very sensitive to the values of these angles. Apparently, the range of these values must be |a ; | < ^ , which follows from the geometry (Fig. 3.2) and the assumed direction of propagation. The reflection coefficient for this boundary is *.=n 7=1 c o s a ; - COS 0 cosay +cos0 which is a big improvement compared with the viscous and A1 boundaries, because all of the m ultipliers in the product are sm aller than 1 and the product is sm aller than any m ultiplier. The advantages of this boundary are • It is sim ple (it is composed of derivatives of the first order and can easily be implemented). • There is no need for corner conditions as in paraxial boundaries. • The difference approximation uses values at the existing grid points compared with the extrapolation boundary where “ m onitoring" (non-existing grid points) should be introduced. The main drawback is the problem dependence of the number m and the values of a , . 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1.3.4 Expansion boundaries Bayliss and Turkel (1980) derived artificial boundary conditions based on the expansion of the outgoing pressure for 2-D flow as follows: / , (t ~r,0,<j>) ;=i ( d d T + {d t dr J we obtain M ultiplying (3.46) by r m and applying operator Lm = Lm (rm p ) = o ( r m ~ 1). To have the operator operating only on p, the authors involved new operators B.P = I \ i + — /= 1 V r J (3.46) (3.47) with leading term of the error being o(r 2 m l). 3.1.3.5 Extrapolation boundaries Again, the idea of 1-D wave propagation is used for this type of boundaries. The total wave field can be represented as a sum of components propagating in different directions £ (Liao & Wong, 1984): U(x,y,z,t) = Y ;UM i -$)■ (3-48) 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The displacem ent of a point P in time t + At, due to the wave component propagating in direction can be predicted as (Fig. 3.3): - c(t + A t) ) = u ((g p - cAt)-ct) = u(fgp - 2cAt)- c(t - A f) ) = ... . (3.49) Therefore, the displacem ent of P in time t + A t is expressed as the displacem ent in points E on a distance kcAt (k = 1 ,2 ,...) from P backward along the direction of propagation, in times t, t - A t , t - 2 A t ...... It can be seen that (3.49) cannot be applied in practical computations because there are infinitely many directions of propagation and the components of the field cannot be separated into the points E, even if we suppose that a grid can be chosen with these points. We can pick the points on a normal erected from P and introduce m onitoring points. These points play the role of points that lie on the same wave front as the points E, at distances kc'A t (k = 1 ,2 ,...) from the point P in direction of the normal. However, because the direction of propagation is not known, these points cannot be located on the same wavefront as the points E. What can be done is to locate these monitoring points at equal distances d so that the error from their exact location is ke0 ( k = 1 ,2 ,...). Then, using the backward differences, the Newton- Gregory extrapolation formula can be used to obtain the displacem ent in P in time t + A t as u = u0 + An 0 + A2 u0 + .. + ANu0 + o { sqH ), (3.50) where A are backward differences in space (in the direction of the norm al-x) and time Au0 = w0 - u_x A2 u0 = u0 - 2u_x + w_2 = Au0 - Aw_j .... Am u0 = Am ~ 1 u0 - An ^ lu_l . The displacements are then evaluated as 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Rg. 3.3 EXfRAPOLAllONBOUKDW ! Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. u_k = u\xp ~(k + l)d, yp,t-kA t]. (3.51) Because in general the monitoring points A k are not the grid points, performing the extrapolation (3.50) requires that the values of the displacem ents (3.51) should be interpolated at all of the monitoring points from the neighboring grid points. This is the main drawback of this boundary, A sim ilar idea for an extrapolation boundary was used by Liao et al. (1978) in solving a circular rigid disc welded to the surface of a homogeneous isotropic halfspace. Many other authors have proposed local artificial boundaries that can be viewed as m odifications of the above five. Kausel (1988) presented a review of the local artificial boundaries and concluded that all of them can be expressed with sim ilar equations at least in the continuous domain, as we saw above for paraxial, viscous, and m ulti-directional boundaries. He suggested that for finite difference applications the paraxial type boundaries seem to be the most convenient and that for finite element applications extrapolation boundaries are most suitable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1.4. Conclusion In practical computation of wave propagation problems in an infinite or sem i-infinite domain there is need to use artificial boundaries. They are a set of points that bound the computational domain but physically are just points of the full space. These artifacts can be classified into two main groups: • Exact (perfect) boundaries that are perfect absorbers but are global both in time and space and for practical implementation should be truncated. These are more suitable for FE form ulation because of their non-rectangular shape. The FE formulations are more flexible for m odeling nonrectangular geometries than the FD schemes. • Local boundaries that are m ostly derived for rectangular shapes. Their time and space derivatives are of low order and so are easily im plem ented in numerical algorithms. The first group uses m ostly DtN maps to replace the Sommerfeld radiation condition at the artificial boundary. Because in their original formulation they involve high-order derivatives in time and in the tangential direction, for practical computation the exact infinite series are truncated, which means that some error is introduced. Some boundaries deal with reducing the order of the derivatives by involving additional unknowns. M ost of these boundaries are derived for full space. Finally, the behavior of the boundary cannot be predicted in a case in which the incoming plane wave and the outgoing cylindrical wave meet at the boundary. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The local boundaries are not exact, and the error (reflection from the boundary) depends upon the distance from the scatterer. The further away the boundary is imposed, the sm aller the error w ill be. Nevertheless, local boundaries are popular in FD applications, which involve rectangular domains and com plicated outgoing fields with wide frequency ranges. 3.2 Rotated viscous artificial boundary The artificial boundary 8 in the model is a viscous type of boundary rotated to absorb the wave field com ing under angle 0 in the point m e 8 (Fig. 2.2). The idea for this boundary comes from the consideration of spherical (3-D) or circular (2-D) propagation of the scattered wave field. It is also related to the case of the Higdon m ulti-directional boundary with m = 1 in (3.45) and with different a x at different boundary points. Based on many numerical tests with different boundaries, this one appears to give the best results. For example, the second-order Clayton-Engquist boundary and the two-directional Higdon boundary lead to instability at the corner points for some com binations of angles of incidence and durations of the pulse in our transient analysis. This is a local boundary, and some small residual reflected field is expected, but it is stable for all incident angles and frequencies considered in this work. The residual field is larger for higher frequencies when the diffraction is more pronounced and when the ray theory involves larger errors. Nevertheless, this artificially reflected field is negligible compared with the solution, and the model gives satisfactory results that can be seen in the next section and in Chapter V. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To im plem ent this artificial boundary, the soil island R = Qs^jQ f { where Q s is the region occupied by soil and O , is the region occupied by the foundation) is divided into two regions: Rl -\h4{x,y)\xa < x < x d,yb< y < y a) with boundaries IT j = abu h e c d , and, R2 = R \R 1 with inner boundary T2 =lkvjkjvj ji and an outer boundary that is the artificial boundary 8 = ef 'ufg'ugh (Fig. 2.2). At the top, these two regions are bounded by the half space. We want the finite difference solution in region R ., to be the solution for the total field and in region R2 to be the solution for the scattered field only. The two rows (2,3) next to the bottom artificial boundary f g , the two columns (2,3) next to the left artificial boundary e f , and the two columns next to the right artificial boundary gh are inserted in the model (Fig. 3.4) for practical purposes. As will be shown later, the analytical (half-space) solution is evaluated on the boundaries r 1 and r 2. The role of the inserted rows and columns is to decrease the error arising from the difference between the solution in the discrete domain (FD solution for the total field) and the solution in the continuous domain (half space, analytical solution). To solve in R1 ( we first should solve on T^ for the total field. This can be accomplished if we know the total field in all neighboring points of the boundary TT including the points on r 2. For that purpose, in the finite difference (scattered) solution on F 2 e R2 the analytical (half space) solution should be added. The procedure for solving in R2 is sim ilar. First, we subtract the analytical (half-space) solution from the finite difference (total) solution on ^ e R}to obtain the scattered field on r h and then, using finite differences, the solution for the scattered 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. field is obtained on r 2. In this way, the problem under consideration is replaced by two auxiliary problems (Fig.3.4). Assuming that the upgoing displacem ent field wu (t) at a point f and the angle of incidence y are known, we proceed for each time step (refer to Fig.3.4) as follows 1. Update the analytical solution for the half space at the points on the lines: • ab - the fourth column of the model • be - the fourth row of the model • cd - the third to the last column in two consecutive time steps t - A t = (k-1)A t and t = kA t k = 1 ,2,3 ,.... ,T/At, where T is the time at the end of the analysis and A t is the time step. For example, on the line ab (the fourth column) the solution for the displacements is ( 0 = w»(t- tQ ) H ( t- t0) + wu (t - tx)H(t - t x), where wu is the prescribed displacem ent of the upgoing field at the point f(0,0) (Fig.2.2), and < i= g , - C / - l ) A y + g , + (4 -l)A x are a(riva, ,jmes c c c c 8 at the point (4,j) of the incident and reflected waves, respectively; cx = and im y 8 c = are the phase velocities in x and y, respectively; Ax-Ay are spatial steps for the y cos y soil; H s is the height of the soil island in the model; and Fl() is the Heaviside function. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 3.4 DECOMPOSITION O F THE P R O B LE M CO. C l cCl cCl dZ C l cc. cc £ ax cr C C l ax CL o C D 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We now perform the above for the fifth column, the fifth row, and the second-to-last colum n at time t = kAt. The velocities and the stresses for the half-space solution at the boundary (lines ab, be, and cd) are: w k — w k ~l w k, . — w k . w k — w k / \ At ' ^ Ac yh } Ay ; 1 1 where pis is the shear modulus of the soil, x y - i = — and j = —L are discrete spatial coordinates in the x and y directions, respectively, A r Ay k = — is the discrete time coordinate, At v.j is the velocity at the point in time t = k ■ A t , and are the shear stresses xx z and xy 2 at the point ( x ,.,^ .) in tim e t = k- At. The above notation for the velocities and the stresses with the discrete spatial coordinates as subscripts and the discrete time coordinate as a superscript w ill be used in the following text. 2. From the finite difference solution on the lines ab, be, and cd (obtained by solving region R,), subtract the above half-space solution. This difference gives us the scattered field on ^ (the boundary condition for region R2). 3. Solve the second and the third rows and columns, as well as the next-to-last and the second-to-last columns using FD scheme. This solution is the scattered field in R2. 4. Solve for the points at the absorbing boundary B -e f'u fg ^ jg h . The absorbing boundary is a local, viscous type of boundary, consisting of rotated dashpots as shown on 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig.3.4 for the point m. For example, the velocities at ef (the wave travels in the direction of negative x) are computed from the scalar equation dv dv a dv dx dv — - P ~ r - P ~— r ~ P ' dt ds dx ds co s < f> • dx where s is the spatial coordinate in the direction of propagation of the wave, H , - (m -1 )dy (3.2.1) 6 = tan -i V 2 is the polar coordinate, (3.2.2) and ( f > = 6 is the incident angle of the outgoing field (the angle between the normal to the boundary at m and the polar ray Om) (Fig.3.4). The outgoing components of the stress u x and cr are computed when the velocity in the equation (3.2.1) is replaced with the desired stress. With sim ilar equations, the velocities and the stresses are computed at all of the points of the absorbing boundary s. The finite difference approximations of (3.2.1) with a coordinate system as in Fig.2.2 are as follows. On the left boundary: D [ — D ! COS,<j) S k. = 0 •.j < j) = 9 On the bottom boundary: D [ - ~ ^ — D COS0 n S u= 0 , = On the right boundary: D [ + -@ — D x c o s ^ S -, = 0, (j>~n - 9 (3.2.3a) (3.2.3b) (3.2.3c) S. ■ in the equations (3.2.3) stands for v, < j x and ay 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The three points at the left {(o,o);(o,A yJ,(A x,o) and the three points at the right bottom corners {(Lm ,o), (Lm , Ays), (Lm - Ax,o)} are treated as in Clayton and Engquist (1977). For example, for the left bottom corner (Fig. 3.5 where h stands for A x ) it is assumed that the outgoing scalar field im pinges upon the boundary under angle 6 = — in the points P, 4 Q, and R and has the form S = S0ei{m t+ k'x+ k> y\ (3.2.4) Considering simultaneous 1-D wave propagations in the x and y directions, we proceed as follows: ®L = cx^ = yl2JS — (3.2.5) dt dx dx ®L = C — = > 1 2 /? — , (3.2.6) dt dy dy where cx and cy are the phase velocities of the outgoing wave im pinging upon the corner under angle ^ = ~ - Summing equations (3.2.5) and (3.2.6) we obtain ^ a s - f a s + a f l (3 2 7) (3 dt dx dy) It is seen from Fig.3.5 that the spatial derivatives cannot be approximated with central difference and that to get the explicit scheme the temporal derivative must be approximated by backward 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 3.5 THE LEFT BOTTOM CORNER OF THE MODEL y? ?W(0,2h) R(0,h) T(h,h) Q(h,0) V (2 h ,0 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. difference. For the coordinate system in Fig.3.5 (in Clayton & Engquist.1977, y is directed downward), the finite difference approximation of (3.2.7) is V 2 f i \ D *-D * -D l Sk A = 0 , (3.2.8) where k is the discrete temporal coordinate, ft is the shear wave velocity in the soil, and the subscript A stands for the point P, Q, or R. First, the points Q and R are solved. For example, for the point R, keeping in mind that Ays = Ax and referring to Fig.3.5, from (3.2.8) we have Q ik _ Sr = T i 1 . At + 2 — ft Ax (3.2.9) The scalar at the point Q is computed in a sim ilar way. Having Sk R and Sk , the scalar in P is computed as « • k | p = 7 T T a 7 ■ + 2 (3 Ax Ax (3.2.10) S in the above equations stands for v, < j x and a y of the scattered field, and all of these dynamic quantities must be resolved with the described procedure at the six corner points to obtain a stable solution. It is important to note that the three corner points give the necessary transition between the boundary conditions on the vertical and horizontal boundaries (and vice versa), which makes the boundary stable. Also it should be pointed out that the quantities S in the six corner points are always computed in one time step of retarded time. They are computed 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in the beginning of the algorithm , but because these points belong to the boundary 8 , the procedure is presented here. 5. Update the analytical solution for the half space in the points on the lin e s : • kl - the third column of the model • kj - the third row of the model • ji - the second to last column in two consecutive time steps t - A t = (k-1) A t and t = kAt k = 1 ,2 ,3 , T /A t For example on the line kl (the third column) the solution for the displacem ents i s : M 'S . ; ( 0 = ( / - t0 ) H ( t ~ t 0) + Wu (t - tx )H (t - t x ) , ( j - l) A v (3 - l) A x , H - ( j - l) A y + H (3 - l) A x . . where t0 = ^ ^ L— and tx - —- — - — — + ----- 1 — are arrival c c c c y x y x B times at the point (3,j) for the incident and reflected waves, respectively, with cx = , sin y c = — , and H() is the Heaviside function. y C O S f We now perform the above for the fourth column, the fourth row, and the next-to-last column at time t = kAt. The velocities and stresses for the half-space solution at the lines ab, be, and cd are wk —wk~l Wk ■ —Wk- Wk■ , —wk . . k W i , j W 1,] _ .. '+ 1 . J „ k _ ,, r r i , J +1 , y hJ v _ a f i ' ................ , 0 - f is.................... !’; dt ' X ,,J 5 dx y 'J ' 4 dy 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6. Add the finite difference solution for the points on the lines kl, kj, and ji (obtained from solving in region R2) to the above solution for the half space. This sum is the total field on r 2 (the boundary condition for region FR). 7. Solve for the points {M(x,y)\M gR^. 8. If k < = T/dt return to step 1. 9. End. 3.3 Numerical tests To test the model, and in particular the artificial boundary, the structure (Fig.3.6) is loaded at the top (jx'| <a,y' = H b) with excitation in the form of a half-sine displacem ent pulse with duration td = O.ks and amplitude A = 0 .5 m : u0 - Asm — . (3.3.1) td This pulse is Fourier transformed into the frequency space, and only the components with Fourier am plitudes larger than 3% of the Fourier am plitude of the zero-th component (k = 0) are kept. For the last kept component (the largest wave number), the corresponding frequency is found from com s x =coc = f3b- . This is the cut-off frequency with which the pulse (3.3.1) is low-pass filtered using an Ormsby filter (Trifunac, 1971). All of the top points of the structure, 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 3.6 TEST MODEL - BUILDING LOADED ON THE TOP LO y Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. at / = H b are loaded with the same excitation. As a measurement of the error, we take the difference between the input and the sum of radiated energy and the energy in the building, divided by the input energy _\E„r -(E n l +Eb\ E. inp (3.3.2) where the input energy is computed using the formula for continuous space (Aki & Richards,1980): Tr]+ t„ E inP = P b f i b A \v 2dt, (3.3.3) 0 where pb is the density of the building, f5b is the shear wave velocity of the building, A = 2a • 1 is the area normal to the direction of the wave propagation, v is the velocity at the points of the section A-A, Td is duration of the filtered pulse, and ta = is the travel time Pb of the wave from the top of the building to section A-A. To avoid the singularity at the line of the application of the load, the input energy is computed at section A-A (Fig. 3.6). The duration of the pulse is chosen to satisfy the relation: td +(2-nf +\)-Et< 2(<H b E5Ay- b ) , (3.3.4) where td is the duration of the unfiltered pulse, the left-hand side is the total duration of the filtered pulse Td , and the right-hand side of is the travel time at section A-A of the pulse to and from the building-foundation contact. After the pulse is com pletely applied at the top section 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / - H bt this section behaves like a free boundary. The equation (3.3.3) in discrete space is E m p = p bfibAxAt (Trl+ta) / A t f ,.2 , ..2 k =1 — ; — + 2 > a ^ i=L+l y (3.3.5) where k is the discrete time coordinate, and i is the discrete spatial coordinate in the x direction. The sets of points x = ( L - l ) A x , y > H s and x = (M - l) A x ,y > H s are the left and right free boundaries of the building (Fig.3.6). The radiated energy is computed as T / d t 1 (3.3.6) where (/, j) e 8 , T is the time at the end of the analysis, ^ ; is the angle of incidence of the outgoing ray at the point (i,j) described in step 4 in section 3.2 and vl hk is the velocity at the point (x ,,yj) in tim e t-k-At. In the first test example, the properties of the Hollywood Storage building described in Duke et al. (1970) are used. According to their model, the properties of the building, the soil, and the foundation (Fig. 3.6) for east-west (longitudinal) response are • Radius of the foundation: a = 7 .8 m • Height of the building: H b = 4 5 .6 m • Fundamental frequency: / 0 =\.%5Hz M M Ratios of the masses: — - = 1.4 — - = 1.0, M . M . 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where M b and M 0 are the masses of the building and the foundation per unit length, and M s is the mass of the soil occupying the volume of the foundation, also per unit length. With the fundamental frequency and the height of the building known, the shear wave velocity of the building is fib = 4 • H b ■ f 0 = 3 3 7 .4 4m / s . From the depth profile of the shear wave velocities, close to the surface (in the first 100 - 150 feet of soil) the shear wave velocity varies from 500 to 1,200 ft/s. In our test example, we adopted f3s = 2 5 0 m/s, which is approximately the mean value for the above measured velocity. The density of the soil is taken as ps = 2000 kg/m3. The distribution of the energy is shown in Fig. 3.7. Ein p is the cumulative input energy, Era d is the cumulative radiated energy measured at the points of the artificial boundary 8 = ef yj fgyjgh (Fig.3.6), and Eb\s the instantaneous energy in the building computed as where sfj = s2 ^ + s2 ^ is square of the resultant strain at the point u V y (3.3.7) i — for x = L ,M ^ ■ = 2 Ax otherwise H H h t — ^ + — * - + i A y s A y 6 otherwise and pb = pbPl is the shear stiffness of the building. 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The energy was measured for two different model sizes (Fig.3.6): 1. 7/ = — = 4a s 2 2. 77 = — = 5a . 2 At the end of the analysis (T = 4s) the values of the energy and the computed error (3.3.2) are: For model size 1: Em p = 51024137, Era d = 50924467, Eb = 45447 e = 0.16% For model size 2: Em p =51024137, Em d = 50781637, Eb = 47317 ^ = 0.38%. For our second test example, the properties of the Holiday Inn hotel described in Blume and Assoc. (1973) are used. The dimensions and the properties of the building, the soil, and the foundation (Fig. 3.6) in the transversal direction of the building are: • Radius of the foundation: a = 9.55 m • Height of the building: H b = 20.0 m • Shear wave velocity of the building: /?h =lOO mis • Shear wave velocity of the soil: / ^ = 250mis • Shear wave velocity of the foundation: ^ = 3 0 0 mis • Density of the building: pb = 270 kg/m3 • Density of the soil and the foundation: ps- p f = 2000kg/m3. Again the energy was measured for two different model sizes 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. H = ^ = 4a 2 2. H = ^ = 5a. 2 The distribution of the energy versus tim e is shown in Fig. 3.8. At the end of the analysis (T = 14.59s), the values of the energy and the computed error (3.3.2) were: For model size 1: Ein p = 1244629J, Era d = 1224142J, Eb = 8208.7 £ = 0.99% For model size 2: Em p = 1244629J, Era d = 1221035J, Eb = 8459,7 e = 1.22% . With these test examples, it can be seen that the model gives satisfactory results when the artificial boundary is located far enough from the foundation. With this setting, we assume that the artificial boundary “ sees" the foundation as a point source generating cylindrical outgoing waves, which then allows us to make an approximation of the incident angle of the waves, < f > , relative to the normal on the boundary 8 . As can be seen from figures 3.7 and 3.8, the building radiates the energy partially when the pulse reaches the building-foundation contact. Then, one part of the pulse is transmitted into the foundation and one part is reflected back in the building. In the figures, this interval of time corresponds first to the sharp decrease of the instantaneous energy in the building, and second, after travel time t, from the building-foundation contact to 8, to the sharp increase of the radiated energy. After the pulse has passed the contact (its reflected part is com pletely in the building), there is a state of “ constant energy” in the building and, after time t , , a constant 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cumulative energy, which has passed through 8. This state is represented by flat parts of the curves Eb = E h(t) and Era d = E rad( / ) . From Fig. 3.7 for the Hollywood Storage building (HSB), the decrease of energy in the building can be represented with the ratio of the am plitudes (measured directly from the plot) of the flat parts as follows: £ l = « = 2 . 1 ; ^ = - L = 2 . 1 4 ; ^ = i ± = 2 . 0 3 ; S t = M = 2 . 0 9 .......... E „ 3 E is 1.4 E u 0.69 E u 0.33 or generally Ebi * 2 . 1 . (3.3.8a) EU + l Recalling that the energy in some instant of tim e is proportional to the square of the velocity, when we take the square root of (3.3.8a) we have - ^ - * 1 . 4 5 . (3.3.9a) v , - +i We can construct an equivalent single-degree-of-freedom (SDOF) oscillator with a natural frequency equal to the fundamental natural frequency of the building with the damping ratio — , (3.3.10) 2 n U V *h where 8 = In — = I n - 1 - is the logarithm ic decrement, ui is the r positive (negative) peak V,+l displacement, and uM is ( i+ 1 ) s l positive (negative) peak displacem ent. The damping ratio of the equivalent SDOF oscillator for HSB computed from (3.3.10) is £h s b = 0 .0 5 9 . (3.3.11a) 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In a sim ilar way, from Fig.3.8, for the Holiday Inn hotel (HIH) we obtain ~ r ~~w 1-17 (3.3.8b) Eb i+ l Vi *1 .0 8 (3.3.9b) v < + i Ch s b - 0 .0 1 2 . (3.3.11b) The fundamental natural frequencies of the above buildings can be obtained from the solution of 7 C 0 the wave equation for a building on a fixed base as a > = — — , where p b is the shear wave 2 H b velocity in the building and H b is the height of the building. We then have ®h s b ~ ~ T ( C ' ~ 0-68 a H IH = 2 .57i (C ■ (o)m H = 0 .0 9 4 . For example, if both SDOF oscillators have initial displacem ent uQ = l we can compute the time 1 In ci in which the am plitude w ill decrease to 1/a (a > 1). From e~(< a t = - = > / = — . a £a > For example, the time in which the amplitude will decrease to 1/4 of the initial displacem ent (a = 4) for the Hollywood Storage building is In < 4 tH S B = -------= 2 .0 4 5 (3.3.12a) 0.68 and for the Holiday Inn hotel it is t„m = - ^ = 14.75 5 . (3.3.12b) Hm 0.094 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For these examples, where p f p f is close to p s (3s, the damping ratio can be evaluated by the use of the reflection and transmission coefficients at the contacts. From the boundary conditions of continuity of the stress and displacem ent at the contact between building and foundation, the reflection coefficient from building to building is (Fig.3.6) 1 _ P f 0 f * * = - ? ? - ■ P -1 1 3 ) 1 + - P b P b and the transmission coefficient from building to foundation is (3 1 1 4 ) 1 + — — - Pb Pb The energy remaining in the building after the wave has passed the contact is E™ = k 2 refE °ld . (3.3.15) From (3.3.15), -p o ld r p _ ^bi F n e w F k ^ b ^ b i+ 1 re f (3.3.16) For HSB kre f = -0 .6 5 1 , for HIH kre f = - 0 . 9 1 4 , and from (3.3.16), = 2 .3 5 8 (3.3.17a) \^bi+l J H S B = 1 .1 9 7 . (3.3.17b) F V b i+ l J h i h 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 3.7 TEST EXAMPLE: HOLLYWOOD STORAGE BUILDING 5 0 Q Q 4 0 0 0 3 0 0 0 2000 1000 3 . 0 4 . 0 2.0 1.0 0.0 U_1 5 0 0 0 4 0 0 0 H = Lm / 2 = 5a 20QQ 1000 { . — , 0 4 . 0 3 . 0 2.0 1.0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 3.8 TEST EXAMPLE: HOLIDAY INN HOTEL L U 1 2 5 0 1000 7 5 0 5 0 0 2 5 0 0 8.Q 1D .0 12.0 1 4 . 0 2.0 4 . 0 6.0 0.0 L U 1 2 5 0 1000 7 5 0 5 0 0 2 5 0 0 12.0 1 4 . 0 6.0 10.0 2.0 0.0 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparing (3.3.8a,b) and (3.3.17a,b), we can see that these ratios of energies are close. The equations (3.3.17a,b) involve errors because they do not account for the reflection of the wave from foundation to foundation at the foundation-soil contact and later transmission of that reflected wave from the foundation to the building. The equation (3.3.16) w ill be accurate if the foundation properties are the same as the soil properties. The curve End=E rad{t) represents the cumulative energy passing through the artificial boundary 8. The increases along the curve are equal to the dissipated energy from the building during each passing of the wave through the building-foundation contact. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER IV SOIL-STRUCTURE INTERACTION WITH A FLEXIBLE FOUNDATION: STEADY-STATE ANALYSIS 4.1 Introduction Trifunac (1972) presented an analytical solution of the interaction of an infinitely long shear wall supported by a rigid sem i-circular foundation embedded in linear homogeneous half space and excited by plane SH waves with arbitrary incidence. The am plitude of the foundation motion A was found in terms of Bessel and Hankel functions, and several important conclusions were drawn: 1. The amplitude of the foundation motion does not depend upon the angle of incidence. 2. When the excitation frequency is equal to any natural frequency of a structure with a fixed base, A becomes zero. 3. The relative displacem ent of the structure |i?Mz| = |H'r | = |-H '0, - w 0| (Fig.4.1) with zero structural damping at resonant frequencies is finite, unlike the same structure with neglected interaction, where the relative displacem ent goes to infinity with the time. This difference is caused by radiation of wave energy into the half space. As shown in Trifunac (1972), the relative displacem ent of the structure (wall) is: 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If c o s (k„H b) = c o s fe ” + ^ = 0 (n = 0 ,1 ,2 ,...), then £ „ = — is the wave 2 Pb number of the n-th natural frequency of the structure. From (4.1), with neglected interaction, A = 1 and \Ruz\ - » oo. But if the interaction is considered, then because A = 0 at kn, the first term in (4.1) becomes undefined (0/0), and the finite lim it at 0 exists. The steady-state solutions for the foundation motion and the relative motion for different com binations of ratios (explained in Section 3.3) and e = ^ sHh are given. M h M s M s p b a and M 0 are the masses of the building and the foundation per unit length, and M s is the mass of the soil occupying the volume of the foundation, also per unit length. / 3 S and (3b are the shear wave velocities of propagation in the soil and in the building, respectively, H b is the height of the building, and a is the radius of the sem icircular foundation, s is a dim ensionless parameter. 4.2 Numerical example 4.2.1 Input and grid parameters Using the model described in Chapters II and III, the steady-state response for the Hollywood Storage building (Duke et al.,1970) is obtained. The model with the geometry and the material properties of the constitutive parts is shown in Fig.4.1. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4.1 MODEL HOLLYWOOD STORAGE BUILDING CO y -*..> X Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lm/2 = 5a Recalling the algorithm presented in Chapter III, the half-space analytical solution must be computed for every time step on the curves T ^ a b ^ jb c u c d and r 2 j i (Fig.2.2, Fig.3.4). The excitation is a monochromatic sinusoidal plane wave of the form: u(xi , y j , t) = /4 [s in (0 ( / - tx ))H(t - tx) + sin (fl(t - 1 2 ))H (t - 1 2)] (4 2 ) where: A - 0 .5m is am plitude of the wave, x sin y y cos^k tx = - * - + - i — — is travel time of the wave from the origin (point f in p p Fig.2.2) to the considered point, x s in r ( H + y A c o s y l = — - + is travel time of the wave from the origin to the free P P surface and from the free surface to the considered point, y is the angle of incidence, and H() is the Heaviside function. The natural frequencies of the fixed-base building for the elastic modes in the y direction are: / „ = * 2 ” ~ ( Hz ) n = 1 , 2 , 3 , . . . . ( 4 . 3 ) 4 / 7 b The first three natural frequencies in the y direction are / 01 = 1.85 H z , / 0 2 = 5.55 H z , and / 0 3 = 9.25 H z . The modes in the x direction represent the torsional response of the structure. From the solution of the linear wave equation and the boundary condition in the x direction, the 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. J flJ T characteristic numbers in the x direction are kx m = - — (m = 0 ,1 ,2 ,...) with corresponding h angular natural frequencies com = kx m f3b. The lowest elastic mode m = 1 (m = 0 corresponds to rigid mode) has natural frequency ® 10 = 2 1 .6 3 • # . The first subscript in the natural frequencies denotes the number of the mode (including the rigid-body mode 0) in the x direction, and the second one, the number of the mode in the y direction. The analysis is performed for the frequency range of the input motion 0.5Hz < O < 6 .0 Hz, the am plitude A = 0.5 m, and for incident angles y = 30° and y = 6 0 ° . The m inim um wavelength is = AQ = 6 H z = & 4 2 m > L b =2a = \5,6m, \ 2k and so A x is chosen from the criterion for proper m odeling of the foundation (Fig.2.3): (mP = 1 2 = — -— Ax = — 1 fL a ^ m a X 12 The vertical spacing for the finite difference grid in the building is obtained as 6 Ayb - — - A x . The effective horizontal and vertical spacing for the cells B (Fig.2.4) is P s /\y Ax b = Ay B = — j=. The time step is then obtained from (2.1) as V 2 At % (4.4) 2v(i,y) J 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and for the given properties the point B’ (Fig. 2.4) has the biggest denominator and the sm allest £ A t. The artificial boundary is located at H = — = 5a . s 2 With the above parameters our model is com pletely defined. 4.2.2 Results We generate motion in the system by using u(xi,yJ,t) as given by the equation (4.2). A point is in rest until the wave arrives, and then experiences a sudden change in velocity from 0 to AD. The situation is sim ilar at all of the points in the model. The transition of the regime from rest to harmonic steady state is present in the first phase of the analysis (Fig.4 .1a) until the transients go out of the system. For our numerical example, the steady-state regime for any frequency is established after 7 - 8 s from the beginning of the analysis, when the am plitudes of the motion become constant. In Fig. 4.1a, the tim e histories of the displacem ent at point 0 are shown for several input frequencies for the stiffer foundation f}f = 5 0 0mis and for the angle of incidence y = 3 0 °. In Figures 4.2 to 4.5, the dynam ic am plification factor versus input frequency is shown for three points on the foundation-structure contact (Fig. 4.1) for angles of incidence y = 30° and y - 6 0 °, and for foundation stiffnesses fif =300m/s and fif = 5QQm/s. For comparison, am plitude A plots for the rigid foundation are also shown. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4.1a TIME HISTORY OF THE DISPLACEMENT AT POINT 0 FOR SOME FREQUENCIES p ( = 500 rg/s, y = 30 1.5 = 1.Hz E o.o - 1.5 0.0 4.0 8.0 12.0 1.5 n = 1-85HZ E o 5 0.0 1.5 4.0 0.0 8.0 12.0 1.5 0.0 - 1.5 q = 1.675Hz 1.5 0.0 - 1.5 0.0 4.0 8.0 12.0 Q = 3.5Hz 4.0 ^ 8.0 12.0 o = 5.525Hz Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Fig. 4 .2 RESPONSE AT THE BUILDING - FOUNDATION CONTACT NORMALIZED BY FREE SURFACE RESPONSE |A| 2.0 £ = 1.4 H = 45.6 m p b= 337.44 m/s a = 7.8 m p s= 250 m/s = 300 m/s rigid foundation 0.5 0.0 6.0 3.0 4.0 5.0 0.0 1.0 2.0 f (Hz) 00 O Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Fig. 4 .3 RESPONSE AT THE BUILDING - FOUNDATION CONTACT NORMALIZED BY FREE SURFACE RESPONSE 2.0 U !!b = 1.4 H = 45.6 m 337'44 m/s = 250 m/s a - 7.8 m 5 0 500 m/s rigid foundation 0.5 0.0 0.0 1.0 2.0 3.0 5.0 4.0 6.0 f (Hz) o o Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Fig. 4 .4 RESPONSE AT THE BUILDING - FOUNDATION CONTACT NORMALIZED BY FREE SURFACE RESPONSE H = 45.6 m p b- 337-44 m/ s a = 7.8 m r s= 250 m/s 300 m /s D O bO Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Fig. 4 .5 RESPONSE AT THE BUILDING - FOUNDATION CONTACT NORMALIZED BY FREE SURFACE RESPONSE 2.0 ^ = 1.4 H = 45.6 m Pb= 337-44 m/s a = 7.8 m p s= 250 m/s .5 I A 1 1.0 = 500 m/s rigid foundation 0.5 o.o 0.0 1.0 3.0 2.0 4.0 5.0 6.0 f (Hz) 00 u> For all cases, at small frequencies, the am plitudes of all three points (A, 0, B) are the same and approach 1 as f approaches 0. This can be explained by the fact that for an input wave much longer than the foundation dim ension and of such small frequency that the inertial forces become negligible the whole system just rides on the wave and the effect of the interaction is negligible. Relative displacem ents are negligible in this frequency interval (Fig.4.6). As the input wavelength becomes smaller, and as the input frequency approaches the natural frequency of the building, the am plitudes at the three different contact points begin to differ. This difference is partially caused by the horizontal wave passage through the foundation (differential motion as discussed in Trifunac & Todorovska, 1997), which causes a torsional response by the foundation, and partially by the soil-structure interaction. The sm aller the incident angle, the larger is the relative contribution of the soil-structure interaction to the response, and the larger the incident angle, the larger is the relative contribution of the wave passage. In extreme cases, when the incident angle y = 0 there is no effect of the differential motion, and when the incident angle y = ^ the effect of wave passage is most prominent. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4 .6 RELATIVE RESPONSE 10.0 7.5 5.0 2.5 0.0 3.0 4.0 1.0 2.0 5 . 0 6.0 0.0 f (Hz) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From the plots in Fig. 4.2 to 4.5 it can be seen that for the same angle of incidence, and for the softer foundation, there is bigger difference between the motions of the left and right ends of the building. Also, for the same foundation stiffness, the difference between the motions at A and B is larger for the larger angle of incidence. The first observation results from the fact that as the foundation is stiffer it resists the incident wave deformations more, and the solution is closer to the solution for the rigid foundation. The second observation comes from the fact that the phase wavelength in the horizontal direction = cxT = - ^ —T is sm aller for the sin;r larger incidence angle y and for smaller wavelengths the differential motions are more pronounced. In this frequency range, where the frequencies of the input motion are sm aller than the sm allest natural frequency of the building, the response of the model can be seen as a response of the half space, with the building as an added mass. If the building and foundation did not exist, the soil could be considered as an equivalent single-degree-of-freedom oscillator (SDOF) with mass ms and stiffness ks, which under the steady-state excitation oscillates with an amplitude equal to one. The forces resisting the motion of undamped SDOF are the inertial (d'Alam bert force), F a , and the elastic, F sE, forces, which at any time have the same direction. The total force resisting the motion for this “ oscillator" can be written as F = F, + F . s 1 e Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4.7a THE EFFECT OF THE BUILDING AS ADDED MASS TO THE HALF SPACE smax r/7 7 c) r r r r a) r /7 7 b) m. = 0 r m 7 7 7 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4.7b TIME HISTORIES OF DISPLACEMENTS, VELOCITIES, AND ACCELERATIONS AT THE TOP AND BOTTOM OF THE BUILDING 7 . 5 5 .Q bottom 2 . 5 E 0.0 - 2 . 5 ■ 5 .0 - 7 . 5 2.0 2.2 7 5 . 0 5 0 . 0 bottom 2 5 . 0 E - 2 5 . 0 - 5 0 . 0 - 7 5 . 0 2.0 2.2 7 5 0 . 0 5 0 0 . 0 2 5 0 . 0 0.0 2 5 0 . 0 bottom ■ 5 0 0 .0 ■ 7 5 0 .0 2.2 2.0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If we consider the system with a building placed on the soil, the equivalent system becomes a two-degree-of-freedom oscillator (TDOF), as shown in Fig. 4.7a, with the building represented by the top oscillator with mass mh and stiffness kb , and corresponding forces Fb = Fb I +FbB, where Fb I is the inertial force and Fb E is the elastic force in the building, Because the frequency of the input motion is lower than the fundamental frequency of the building, the deformation in the building is small and so is the elastic force Fb E with respect to the inertial force Fb I and the elastic force of the half space FsE. In Fig.4 .7a, five early stages of the system excited by harmonic motion for t > 0 are shown, when the steady-state regime is still not established. The Fig.4.7a is supplemented by Fig.4.7b, where the time histories of the displacements, velocities, and accelerations in the points 0 (at the building-foundation contact) and O'(at the top of the building) (see Fig.4.1) are shown between the third and fourth positive peaks when the steady-state regime is still not established. If the soil is rigid, there is no interaction, and the system frequency is equal to the frequency of the building. Because of the finite stiffness of the soil, the system frequency is always sm aller than the fundamental frequency of the building. In our example, the system natural frequency is « 1 .6 7 5 H z . The am plitudes for all cases and for all three points are the largest for this frequency. In Fig. 4.6, this frequency range is characterized with rapid growth of the relative displacement. The displacem ents for y = 3 0 ° and for the peak frequency Q iy s & 1.675 Hz are shown in the second plots in Fig. 4.8a,b for p f -30Qmls and p f = 5 0 0 mis 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4.8a DISPLACEMENT F O R SOME CHARACTERISTIC Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4.8b DISPLACEMENT F O R SOME CHARACTERISTIC Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4.8c DISPLACEMENT F O R SOME CHARACTERISTIC Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 d DISPLACEMENT F O R SOME CHARACTERISTIC Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. respectively. The displacem ents for the same frequency and y = 60° are shown in the second plots in Fig. 4.8c,d for p f = 300 mis and p f = 5 0 0 mis respectively. After reaching the maximum values near 1.68 Hz, all three curves decrease. At the fundamental frequency of a building on a fixed base, the am plitude at the m iddle point 0 approaches zero for a rigid foundation, while the ends have small, nonzero values larger and more separated for a softer foundation and for angle y = 60°. The relative response of the building (Fig. 4.6, Table 1) is the largest for frequencies between the frequency at which the maximum foundation motion occurs and the fundamental frequency of the building on its fixed base, for all cases. Immediately after the natural frequency of the building, and approximately until the curve for the rigid foundation reaches its second maximum, it seems that for all cases the flexibility of the foundation acts like a damper and that the am plitude of the rigid foundation is larger than the am plitude at the m iddle point of flexible foundations. Also, in this frequency range only the amplitude of the left end is larger than the am plitudes in the center and at the right end. This is more pronounced for a flexible foundation and for the larger y. For all other frequency ranges, the am plitude at the point 0 of the flexible foundation is larger than the corresponding am plitude of the rigid foundation. The frequency range between the two natural frequencies of the building (2.5 Hz < / < 5 Hz) for p f = 500 m i s , is characterized by fairly flat displacements for the points 0 and B, while the curve for point A follows more closely the curve for the rigid 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. foundation. This is especially pronounced for y = 60°, which suggests that the center of torsion is closer to point A. For the softer foundation, the am plitudes of displacem ents of points 0 and B increase, while for point A they are approximately flat (i.e., they do not depend upon frequency). In this frequency range, the wavelength range is 100m>A>50m. The quarter of the wavelength kx as an indicator of the differential motion at the surface is in range O C m 1 0 ^ m i > — — > — ^— . The length of the building Lb =2a = \5.6m is either sm aller than or sin y 4 sin y 12 5 12 5 close to this range for y < s in '1 — « 5 3 °, or it is in the range for y > sin "1 — — « 5 3 °. 15.6 15.6 The foundation experiences substantial differential motion and strong torsional excitation, while the effect of interaction is sm aller because the natural periods of the building and of the system are outside this range (Fig.4.6). This explains why the right end of the building (point B) has large am plitudes and why the difference in the displacem ents for the right and left ends is the largest in this interval. Such trends are more pronounced for larger y when the wavelength of incident motion kx is smaller. The effect of the differential motion,' if the foundation has the same properties as the X soil, is the largest when -^- = Lb - 15.6 m or Xx =cs x -T = - ^ - - — = 62Am (4.5) sin y f (cx is the phase velocity in the soil, and T is the period of the input motion), which 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. corresponds to the frequency of the input motion / = , r f — (4.6) 6 2 .4 - sin y For incident angle y = 6 0 °, / = 4.63 Hz and for y = 3 0 ° the frequency / is larger and is not in the considered frequency range. Because of the presence of the foundation, which is stiffer than the soil, there is scattering of the incoming wave from the soil-foundation contact (more scattering for higher input frequencies). Also, because of the larger wave velocity in the foundation, the phase wavelength is longer than that computed in equation (4.5). Finally, the effect of the building as added mass increases the motion at the building-foundation contact. While for small input frequencies (sm aller than the first natural frequency of the building) the effect of the building as an added mass is dom inant compared with the effects of scattering from the foundation and the differential motion, in this frequency range (2 .5 Afe < f < 5 Hz), the effects of the scattering and of the differential motion are substantial in the response of the building-foundation contact due to higher input frequency and sm aller phase wavelength Xx relative to the length of the building Lb. For stiffer foundations (Figures 4.3 and 4.5), the effect of the scattering from the soil-foundation contact, which decreases the am plitudes of the motion at the building- foundation contact, is larger than the same effect for the softer foundation. Also, because of the larger velocity of propagation in the foundation, the effect of the differential motion at the building-foundation contact is sm aller for a stiffer foundation. The outcome of these two facts is the absence of peak am plitude in Figures 4.3 and 4.5 in this frequency range. 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the softer foundation, the effect of the scattering from the soil-foundation contact is sm aller due to sm aller difference in impedances between the soil and the foundation. Also, the effect of the differential motion at the building-foundation contact is stronger due to smaller phase wavelength Xx (smaller p f ). The sum of the effects of the building as an added mass and the differential motion on the building-foundation contact is stronger than the effect of the scattering at the end of the considered frequency range. The outcome is the appearance of peak am plitudes of the displacem ent at points 0 and B (larger than 1) for input frequencies close to the end of the considered frequency range (Figures 4.2 and 4.4). These peak am plitudes are larger for larger angles of incidence because of the stronger effect of the differential motion (Eq. 4.5 and 4.6). The above trends can be seen in the plots of the motion of the model in Figures 4.8a,b,c,d for Q = 3 .5 Hz (approximately where all curves meet) and at 0 = 5 Hz (where the effect of the torsion is most pronounced in this frequency range). Reaching the second natural frequency, the displacem ent at the m iddle point of the building-foundation interface approaches zero, while the displacements of the end points are not zero (the right end has bigger amplitudes), which is the consequence of torsional excitation. Finally, Table 1 provides the extreme values of the foundation motion, the relative response, and their frequencies for the m iddle point 0 on the contact between the building and the foundation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE 1. MAXIMA OF FOUNDATION DISPLACEMENT AND RELATIVE DISPLACEMENT Pf (m /s) r (•) ^ A m a x ) (Hz) ^m ax ) V rmax ) (Hz) w rmax (m) 300 30 1.675 1.554 1.75 10.89 500 30 1.675 1.338 1.75 10.84 300 60 1.675 1.562 1.75 10.90 500 60 1.675 1.346 1.75 10.86 4.3 Conclusions From the above analysis, it can be seen that most of the general trends for the rigid foundation also hold for the flexible foundation, except the one that the foundation motion does not depend upon the incident angle. The reason for this difference is the dependence of the response on the wave passage through the foundation and the generated differential motions in the foundation-building contact. For a flexible foundation, the am plitudes A are larger except in the narrow frequency range imm ediately after the natural frequencies of the building on a fixed base. This increase is mostly due to the fact that the rigid foundation scatters more of the incident energy and so the radiation damping caused by scattering is larger. This means that the equivalent single-degree- 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of-freedom oscillator (SDOF) has a larger damping ratio £ for the system with a stiffer foundation. To find the damping ratio of the SDOF, the system is loaded with a transient half-sine pulse as an input displacem ent function. The duration of the pulse is chosen so that the frequency of the input motion is close to the system frequency D,iy s & 1.675 H z, where the differences between the am plitudes for different combinations of foundation stiffness p f and incident angles y are the largest. The am plitude of the pulse is A = 0.5 m, and its dim ensionless frequency is 2a am " = T = (4 J ) where a is the radius of the fo u n d a tio n ,^. is the shear wave velocity in the soil, and a > is the frequency of the ground motion. For c o = 1.6Hz& , from (4.7) 77 = 0 .1 . The procedure for numerical computation is explained in the next Chapter V. The tim e histories of the motion for points 0 and 0 ’ (see Fig. 4.1) are illustrated in Fig. 4.9 with solid and dashed lines, respectively, for the com binations of incident angles y and foundation stiffnesses p f corresponding to those presented in Figures 4.2 to 4.5. The first peaks in the building that depend upon the scattering of the incident wave from the foundation, for different cases of foundation stiffness and angles of incidence, are as follows: 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. p f = 3 Q 0 m ls , y = 30° . uo,m s x = 1.8528 to 2. j3f = 300 m / 5 , f = 60° M0,m ax = 1.8408 m 3. = 500 m /5 , ^ = 30° «o lm a x = 1.8398 m 4. = 500 m is, y = 60° «o lm g x = 1.8340 m . The scattering from the foundation is larger for a stiffer foundation (sm aller amplitudes in the building), but the variation is small for this example. Also, the scattering is slightly stronger for U' a larger incident angle. The ratios — ^ (the superscript denotes the ordered number of the K positive amplitudes i= 1 ,2 ,3 ,...) for the first case (in which the analysis lasted for 8 s) are: u\ 1.8528 ul 1.3851 1.338 and further (for i= 2,3 ,... 10) R = — ~ ^ ~ r — 1.377; 1.395; 1.411; 1.421; 1.415; 1.407; 1.405; 1.413; 1.408 . ul ; It can be seen that for the first three periods the ratio changes due to reflection of the pulse from the building-foundation contact back into the building. After the fourth period, the ratio of the consecutive am plitudes becomes fairly constant. In Fig. 4.9, we illustrate eight positive peaks, so we can find seven amplitude ratios. The last three ratios (fifth, sixth, and seventh) and their average values for all of the cases above are: 1. fif =300m/s, y ~ 30° R5 =1.421, R6 =1.415, R7 =1.407 R{l) = 1.414 2. - 300/w/ s , y - 60° R5 -1.412, R6 =1.423, R7 =1.414 R{2 ) = 1.416 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3. (3f = 500 m !s , y = 30° R5 =1.428, R6 =1.439, R7 =1.428 W(3 ) =1.432 4. p f =500m /s, r = 60° W5 = 1.429 , i?6 = 1.439, R7 = 1.423 i?(4) =1.430. $ The corresponding damping ratios £ = — (<5 = In i? is a logarithm ic decrement) of the 2n equivalent single-degree-of-freedom oscillator are £ = 0.055; = 0.055 ; C3 = 0.057 ; = 0.057. We can conclude that the stiffer foundation radiates the energy faster and that the radiation does not depend upon the incident angle. In Chapter III, from the diagram of the instantaneous energy in the building, we found that the damping ratio is roughly £B S B = 0.0 5 9 . The ratio obtained here is more precise because the flat parts of the curve representing the instantaneous energy in the building in terms of time are not ideally flat. As the foundation becomes more flexible, more of the incident energy is transmitted from the soil to the foundation, which causes larger am plitudes of displacem ent of the foundation. Also, the larger amplitudes in a flexible foundation are the consequence of the presence of the elastic forces generated in the flexible foundation and in the contact points by the wave passage, both of which do not contribute in a rigid foundation. At low frequencies, the elastic forces are negligible and the differences in the foundation motion for different angles of incidence and the same foundation stiffness are negligible. As the frequencies become higher, after the first natural fixed-base frequency, the dependence of the foundation motion on the incident angle is more pronounced. 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4.9 TIME HISTORIES O F THE RESPONSE T O THE HALF-SINE z z . o uu nz CO c O h- c o Q _i o CO. o I — <c !_ i_ l CO _I Z D CL. 00 ° C O O O 00 o o o o o (UU) n go ca 00 o o o o o 00 E & S ca 00 a o o o 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The dependence of the foundation motion on the foundation stiffness is obvious at ail frequencies. As the foundation becomes stiffer, the amplitudes of the center (0) and end contact points (A and B) become closer and converge to the amplitudes of the rigid foundation. As given in Table 1, the maximum amplitudes occur at the same frequency for any stiffness and for any incident angle. The peak of the relative response of the building occurs at the system frequency O.(wrm sx)<con 0 and is relatively independent of the incident angle and foundation stiffness. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER V SOIL-STRUCTURE INTERACTION WITH A FLEXIBLE FOUNDATION: TRANSIENT ANALYSIS 5.0 Introduction. 1-D Model In the beginning of this chapter we give a brief review of the solution of the linear and nonlinear shear beam sitting on linear soil. This is the sim plest model of soil-structure interaction, but it is useful to get some further insight into this phenomenon. The 1-D model is shown in Figure 5.0a, together with the assumed displacem ent pulse in the soil. The densities of the soil and of the beam are the same: pb = ps = 2700 kg/m 3. The velocity of propagation of the shear waves in the soil is f3s = 3 0 0 m is and in the building pb = 1 0 0 m is . As pointed out in Dablain (1986), a numerical scheme on the order of accuracy <9(Ar2,A x2)has an exact solution for = and with the ratio of the spatial A x z\x 3 interval — - = — we can meet this requirement. In the model in Fig. 5.0a, A x 6 = 0.1 m and ^ Ps Axs =0.3 m. The prescribed incoming displacem ent in the soil is in the form of a trapezoidal pulse with the properties given in Fig. 5.0a. The absorbing boundary is the one explained in Fujino and Hakuno (1978) (Fig.5.Ob). In Fig. 5.0b, the horizontal axis is time and the vertical axis is space. The column consisting of points 1, 2, 3 represents a time step. 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 5.0a 1-D TEST : MODEL AND LOAD X A a C\J CD. a e / i ca cn C\J < e ' en C N J x cn CD o X < CO trf / 3 td /3 , t / 3 t(s) x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X/AX Fig. 5.0b AN ARTIFICIAL BOUNDARY FOR 1-D WAVE PROPAGATION (3,k - 3) (3,k- 2) (3,k - 1) (3,k) (2,k - 1 ) (1 ,k - 3) t/A t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Point 1 is a dum m y point where the assumed displacem ent is applied. We assume that this displacem ent travels upward in each time step. Point 2 is the boundary point of the model where the quantities of the motion are updated in each time step, as w ill be shown, and point 3 is the first spatial point where the motion is computed by finite differences. The motion at each point results from two com ponent motion - from the wave going upward, and the wave going downward. To update the quantity of motion at boundary point 2 in time step k, we proceed as follows. The total motion at 2 is u(2,k) = T u(2, k)+ 1 u(2,k) , (5.0a) where the arrows denote the direction of the wave propagation ( T upward and I downward). The motion at point 1 is assumed from the upgoing wave u(l, t / A /) = T u(l, t / Ar) = m0( / / At), then T u(2,k) - w (l,k - \) = u0(k - I). (5.0b) The com ponent of motion from the wave traveling downward is I w(2, k) =4< w (3 ,& - 1 ) . (5.0c) From u(3, k - 1) = T u(3, k - l ) + i u(3, k - 1), it follows that 1 u(3, k - 1) = m(3,k - 1 ) - T u(3,k- 1) . (5,0d) The motion at point 3 in time step ( k - 1) from the wave traveling upward is the motion at 2 from the wave traveling upward in the previous tim e step (k - 2). From equation (5.0b), the motion at 2 from the wave traveling upward in time step (k - 2) is the assumed motion in time step (k - 3), so if T u(3 ,k - 1) = f u(2,k -2 ) = u0(k-3) equation (5.0c) becomes 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -I u(l,k) =4- u(3,k- 1) = u(3,k - 1 ) - u 0(k — 3 ). (5.0e) Inserting (5.0b) and (5.0e) into (5.0a), the motion at point 2 is u(2,k) = u0(k - \) + u ( 3 ,k -l) -n 0(k-3) \/k . (5.Of) Equation (5.Of) is the boundary condition at the artificial boundary point 2, where u stands for velocity, strain or stress in our formulation. For the linear case, at the contact one part of the incom ing wave is transmitted into the other medium and one part is reflected back in the same m edium . The corresponding coefficients are obtained from the boundary conditions of continuity of the displacements and stresses at the contact. For a transmitted wave from medium B to m edium A, and for a reflected wave from medium B back to m edium B, the above coefficients are: = — f j - < 5 ° i > P a l + P a P a 1+A p bPb and, sim ilarly, P a P a P b P P a P a *^=— (50.2) 1 + PbPb For the opposite direction of propagation, the numerator and the denom inator in the fractions exchange places. 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 5 .0 c PROPAGATION OF TRAPEZOIDAL PULSE THROUGH SHEAR BEAM 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^^+^08+/:+:/./+./^.:^70^:/^/:/:.$3A 555555555555M545555555555555555555555555555 Fig. 5.0d RESPONSE IN THE MIDDLE OF THE 1-D MODEL LINEAR BEAM NONLINEAR BEAM Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Once the wave enters the beam, there are m ultiple reflections from the top with no loss of energy in the linear case and m ultiple reflections and transmissions at the contact with soil with loss of energy (amplitude) computed from (5.0.1), where medium B is the beam and medium A is the soil. For the values given above, the coefficient of the reflected wave in the beam is kre jS ^ B = - 0 . 5 , and the coefficient of the transmitted wave into the soil (the loss in the beam) is ktrB ^ A = 0 .5 . This means that the wave in the beam after reflecting from the contact w ill have an amplitude equal to half of the wave am plitude before reflection. The wave that entered the beam (transmitted from the soil into the beam with coefficient of transmission 3 K a ~>b - ~ ) have amplitude A = (-\)nK A ^ u ^ - T n. (5.0.3) In Fig.5.0c the displacem ent in the beam is plotted for time intervals St = 0 .0 0 7 5 . The bottom line (x = 0) is the contact between the soil and the beam, while the top of the beam is at x = 29.7 m. The propagation of the pulse is shown in the top plot for a linear beam, and in the bottom plot for a nonlinear beam. It can be seen that the first nonlinear zone occurs close to the top, where the reflected wave from the top meets the incoming wave from bottom. The solution at point M (x = l5m) is given in Fig.5.0d. For the linear case, the displacem ent at M in time occurs in couples of equal responses. The first pulse corresponds with the wave propagating upward, and the second is the same wave reflected from the top and propagating downward. The next two pulses at M are with opposite am plitude and are twice as 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. small. This is because of the reflection from the soil-beam contact. Using equation (5.0.3), the response can be found at any time and at any point on the beam. The 2-D models are more complicated, but in a general sense the 2-D soil-structure interaction (SSi) system works in a sim ilar way. 5.1 Input and grid parameters for the 2-D model The input is a plane SH wave with prescribed motion at Tv as given in Fig. 5.1 (for clarity, only the inner region with the artificial boundary 8 of the model in Fig.2.2 is shown): u(xi,y J ; t) = A v (x ,.,^ .)e r „ where A-O.Sm sm sm \Jd VO sin y yj cos^ (5.1) • + P is the am plitude of the half-sine pulse, is the duration of the half-sine pulse, is the duration of the filtered pulse, is the arrival time of the wave going upward, x s iny (H + v )cosr r, = 1 -- T- + — ^ — i is the arrival time of the wave going downward, P P 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 5.1 MODEL W ITH COMPONENTS OF THE MOTION IN THE SOIL zr. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y is the angle of incidence, and H() is the Heaviside function. Spatially, in the direction of propagation this plane SH wave has the form: w(s) = A sin - -- - - , (5.2) o where s is the spatial coordinate in the direction of propagation, and is the velocity of propagation in the soil. The length L w = p std 0 is discretized into 2 ” intervals, As = ^ , and the spatial pulse (Eq. 5.2) is then Fourier transformed. To find the optimal sam pling interval, we (3 (X) should m inim ize the error in the numerical group velocity — . dk The propagation of the pulse consists of propagation of infinitely many modes. If the modes propagate with different velocities, dispersion of the solution w ill occur. The sampling interval can be found by trial and error to satisfy the numerical group velocity requirement 1 — = 2"M . = — = /? , but m inim izing the error in the group velocity can increase the error Ak 1 At 2 ” As in the phase velocity. Using the recommendation in Fig.5 of Holberg (1987), we proceed as follows. The half-sine pulse is sampled by small enough sam pling interval by taking n = 10, for example. Then the half-sine pulse is Fourier transformed. The biggest Fourier amplitude is at zero spatial frequency kx (Fig.5.4a), and as the frequency /c; = j = 1,2,3,...,2n'1 2 As 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 5.2 FILTERED HALF-SINE PULSE E 0 . 5 0 0 . 2 5 0.0 0 . 3 0 . 4 0.2 0.0 0.1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. increases, the Fourier am plitudes decrease. For em pirically adopted percentage p, we can find the highest mode y m ax for which the Fourier amplitude satisfies The corresponding wave number for this mode is kJ m s x = 2nKjm m , and the corresponding Section 2.1, twelve points for this shortest wavelength are chosen in the grid. The highest angular frequency corresponding to the highest mode is co^ = J3,kjmsx. We use this frequency as the cut-off frequency in low-pass filtering of the pulse. The time dependence of the motion, is filtered using a low-pass Ormsby filter (Trifunac, 1971) with a cut-off frequency of <ymax. From numerical testing, it was realized that for very high cut-off frequencies (co > 2 0 0 rad!s) the solution deteriorates. The maximum frequency allowed in the system w ill be c o = 2 0 0 rad/s, even though for short pulses higher frequencies 0 m ax are obtained from the procedure explained above (Fig.5.4a). In Fig. 5.2, the filtered pulse with td 0 = 0 .0 6 2 4 s and a cut-off frequency co = 200 rad Is is shown. Again, point f(0,0) is taken as a point with prescribed displacem ent in the form of a filtered half-sine pulse, and the input at all points on is computed using (5.1). (5.3) wavelength is A Following the recommendations for the grid parameters given in /m ax (5.4) 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2 Energy distribution in the system In Chapter 3 we tested our model for a building when the input displacem ent was provided at the top. Now, the model is loaded from below, with input motion described with equation (5.1). In Appendix II we show that for the model studied here the energy that reaches the foundation does not depend upon the angle of incidence, so the results presented here are only for incident angle y = 30 °. From the law of conservation of energy, the energy entering the system (solid arrows in Fig.5.1) should be equal to the energy going out of it. Using the energy formula for plane waves (Aki & Richards, 1980) the input energy is computed as E,n p = p j s cos^vE(t)\t , (5.5) Uj,t where ^ i s the angle between the normal erected at a point on IT , and the radial direction centered at 0 (see Fig. 5.1). The input energy enters the model along a. The left boundary (x = 0), due to the incident wave going upward and the reflected wave from the half space going downward, and b. The bottom boundary (y = 0), due to the incident wave going upward. The outgoing energy from the model is: 1. Ev Due to the reflected plane-wave field reflected from the half space, going downward toward the bottom and the right boundaries, and due to the incident plane-wave field going upward toward the right boundary (x = L J , for all points on these two boundaries except the points in the shadow of the foundation (the points that belong 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to the interval a'b' in Fig.5.1). The com ponent of the energy ^ is shown with dotted arrows in Fig.5.1 and is computed from equation (5.5) with velocities from the half space solution. 2. E2: Due to scattering from the foundation and the building, computed from (5.5) with velocities obtained as a difference between the total and the half-space solution. 3. E3: Due to the energy radiated from the building through the foundation in the form of a cylindrical wave field (dashed circle with dashed arrows in Fig. 5.1) computed from (5.5) with velocities obtained as a difference between the total and the half-space solution. The numerical tests are performed for models of the Hollywood Storage building (HSB) (Duke et al., 1970) and the Holiday Inn hotel (HIH) (Blume & Assoc.,1973), with geometry and material properties given in Section 3.3. For different dim ensionless frequencies 77 = — = 2a = 0 .05; 0.5; 1; 2 the error is computed from: The results of the error calculations are illustrated in Table 2 together with the dim ensionless depth of the soil island, H s l a , as a parameter describing the size of the model (Fig.5.1). The length of the model is again Lm = 2 -Hs. In Eout, besides the measured cumulative energy flowing through the curve ^ , the residual energy in the building is included, 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which is of the order of several hundredths to several tenths of a percent of the cumulative outgoing energy at the end of the analysis. This energy is computed as an instantaneous quantity (3.3.7) rather than as cumulative one. In the calculations, as the frequency r\ increases, the percentage p in (5.3) that determines the number of significant modes (for which the Fourier am plitudes are larger than p i 100 of the largest Fourier am plitude) increases as well. This is done for practical reasons because at higher r\, following the discretization procedure in 5.1, the grid spacing decreases tremendously due to the increase in the frequency, so that for the frequencies in the table p = 0.5; 3; 3 ;3 percent, respectively. TABLE 2. ERROR IN ENERGY CALCULATIONS FOR TWO DIFFERENT BUILDINGS Hollywood Storage Building Holyday Inn Hotel T 1 H /a E,np(KJ) U K J ) e (% ) T| H /a E|np(K J) U K J ) s (% ) 0.05 5.0 95750.7 95637.5 0.12 0.05 25.0 506697.4 502050.9 0.92 0.50 5.0 908950.8 909775.3 -0.09 0.50 6.0 1103821.0 1088928.0 1.35 1.00 5.0 1618250.0 1645651.1 -1.69 1.00 5.0 1758780.0 1766575.0 -0.44 2.00 4.7 1124830.0 1155606.0 -2.74 2.00 4.7 1623363.0 1729975.0 -6.57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As can be seen from Table 2, for the longest pulse under consideration, the HIH model requires a large soil island to achieve satisfactory accuracy, while the results for HSB are accurate with a relatively small soil island in the model H s = 5a . Computing the energy with the soil island H s = 5a , the above error in the HIH model is about 11%. It is obvious that for this building, with a relatively small soil island, the model is not accurate enough. The error comes from the assumption that the outgoing field has a cylindrical form propagating in a radial direction from point 0 in Fig.5.1 as a center and that it reaches IT at point M ^ ) with angle f T ^ Z JL < f> = ta n '1 ■ x. H. J This assumption obviously does not hold for long pulses for HIH when the building is soft and the ratio of the masses ^ - « l . The spatial gradients of the m otion quantities are small, and practically the model moves as a rigid body. In this case, the shear stress at the building-foundation contact in the vertical direction is small, and the variation of the shear stress in the horizontal direction is small, too. The building attracts a small amount of energy, and these conditions are close to the free surface condition. This causes the scattered field to be closer to that of the half-space solution - that is, a plane wave with an angle of propagation equal to the incident angle y. The cylindrical pattern is not formed, and the error comes from the difference between the angles y and 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A remedy for this case is to put the artificial boundary that is the main source of error as far away from the building as possible. In that way, the outgoing field decreases substantially, so that the error from the artificial boundary is not so pronounced. On the other side, for the stiff building at the building-foundation contact the boundary conditions differ. The building attracts more energy and acts as a sink for a field reaching it. The scattered field is m ostly due to the two corner points of building-foundation-soil contact and the other contact points along the building-foundation contact, which all transm it the incom ing field in the building. This sudden change in the rigidity causes the two corner points to become sources of additional field that changes the pattern of the reflected field even, for long wavelengths, from plane to cylindrical waveforms. For shorter pulses, which “ feel” better the contrast between the soil and the foundation, the resulting outgoing field is always cylindrical. The assumption for com puting the angles 4 > and the appropriate flow areas at the points on , are satisfied and the errors in the Table 2 are negligible even for relatively small soil islands. The shorter pulses excite the building with a higher frequency, and the response of the building results from superposition of its higher modes. The results of the energy distribution in the whole model are shown on Figure 5.3a for HSB and in Figure 5.3b for HIH. The dashed line represents the input energy, the solid line indicates outgoing energy measured at r , , and the dotted line represents the instantaneous energy in the building. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Trifunac (1972) showed that “ light” and “ soft” buildings on heavy soil (small — - ) M s respond as buildings on fixed bases. The radiation damping is small, and the energy is trapped in the building. The situation for a stiff building is opposite. The building releases the energy faster through the contact with the foundation, and after a few periods the am plitude at the top of the building decreases rapidly. The above observations are further illustrated in Fig.5.4, where the time histories of displacem ents at points A (x '= - a, y ’ = 0) and C (x’ = - a, y ’ = Hb ) are shown for different dim ensionless frequencies. The motion at contact point A is shown with a solid line, and the motion at the top of the building C is indicated with a dotted line. It is obvious that the stiff HSB has greater radiation damping, and after approximately 5 s the motions in the foundation and the building practically cease. However, for the HIH building the situation is different. To compare the input dim ensionless frequency 77 = with the 4 1 ; fundamental frequency of the building coN, the latter is written as / L „ _ 4 h _ = 4 L P* N 2H b a 2H b V b a ’ a where % = a--* is the dim ensionless fundamental frequency of the building. For the 2 H b P , buildings under consideration, we have rjH S B = 0 .1 1 5 4 4 for the Hollywood Storage building and rjH IH = 0 .0 9 5 5 for the Holiday Inn hotel. 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Fig. 5.3a BALANCE OF ENERGY FOR W HOLE MODEL HOLLYWOOD STORAGE BUILDING 100 t) = 0.05 U-J 5 0 0 0.0 3 . 0 4 . 0 5 . 0 1.0 2.0 5 0 0 t ( s ) 0.0 Ein -out T _LJ - 4 — - 1.0 2.0 r\ = 0.5 3 . 0 4 . 0 J t (s) 5 . 0 1 5 0 0 1000 U J 5 0 0 -out 0.0 1.0 r i = 1 2.0 3 . 0 4 . 0 1000 — D L J_ I 5 0 0 jt (S) o 0.0 1 .0 2.0 i— i to U J iq. 5.3b BALANCE O F ENERGY F O R WHOLE MODEL OF o x o C O C N J o U_1 LU o o o o _ I o X ( m ) 3 o C O o t o o L U o o o ( m ) 3 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Fig. 5.4 TIME HISTORIES OF DISPLACEMENTS AT THE TOP . AND BOTTOM OF THE BUILDING 2.0 Hollywood Storage bldg. n = 0.05 1.0 0.0 -1.0 -2.0 0.0 2.0 4.0 2.0 Holiday Inn n = 0 .05 1.0 '0.0 -1.0 t(s ) -2.0 12.0 15.0 18.0 0.0 3.0 6.0 9.0 2.0 1.0 -1.0 -2.0 4.0 2.0 0.0 2.0 Hollywood Storage bldg. n = 0.5 1.0 E ' o . o n - 1.0 - 2.0 2.0 4.0 0.0 i— > to In Fig. 5.4, it is obvious that for 77 = 0.05 < r}mK < r/H S B both buildings respond with their fundamental modes only, and this presents the idea that the response of a building can be replaced with a single-degree-of-freedom damped oscillator. For 77 = 0.05 at HIH, after t2 = 17.5 s some small motion at A can still be seen, and at C the am plitude is still about 29% of the initial amplitude. The first am plitude of the displacem ent at the top occurs at T = 2.1 sec, so in time t = 1,-% = 15.4 s the amplitude decreases to 0.29 of the initial am plitude. This result agrees fairly well with the result obtained in Eq. 3.3.12b, where we found that the required time for am plitude decrease from 1 m to 0.25 m is t = 14.75 sec. In the plots for 77 = 0.5 it can be seen that the buildings respond with higher modes. In the plot for HIH, it is interesting to notice that after the input ground motion has passed the foundation the motion at point A is practically zero until the reflected pulse from the top of the building comes again to point A, with one part of it being reflected back to the building and one part being radiated, at which time the motion at A ceases again. From Table 2 it can also be seen that as the pulse becomes shorter the error increases which is m ostly the consequence of the Saint-Venant principle at the points of application of the load - curve TV While the half space solution for the velocity jum ps from 0 to — A (Eq. 5.1), the velocity of the field computed with finite differences changes gradually. When these two fields meet at 1% the outgoing velocity computed as a difference between the total field and the analytical (half space) field jum ps, as well. This jum p is greater for larger velocities (sm aller td and bigger iq), and that is the main reason the outgoing energy at larger r\ is always greater than the input one. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The other reason for the error is that while alm ost the whole temporal Fourier transform of the pulse for lower r\ is contained in frequencies smaller than c o = 200rad/s, higher r\ values have a substantial amount of the response for a > 2 0 0 rad Is (Fig. 5.4a), and the big part of this frequency response is filtered out. This can cause errors in the transmitted and reflected field at the contacts (Holberg, 1987). One more reason for the error is the diffraction around the corners of the foundation. The diffraction violates the assumption in the half-space analytical solution (5.1) that the waves propagate in directions of straight rays normal to their fronts so that the outgoing diffracted waves impinge upon the boundary with angles different than expected. To gain better insight about the energy distribution, only the rays (energy) reaching the foundation should be considered. This energy is extracted from the total energy in the model by subtracting the half-space outgoing energy on T j \ [« ',& '] from the total input energy. In Fig.5.1, this corresponds to subtracting the “ dotted-arrows” energy from the “ solid-arrows" energy. In this way, the scattered + radiated energy (the second and the third parts of the outgoing energy) should balance the input energy on the interval [a1 , 5 ']. Theoretically the difference between (1) the difference of the input (solid arrows) energy (Eq. 5.5) and the first part of the outgoing energy (dotted arrows); (see page 110), Ej - E in p - E1 , and (2) the sum of the second and the third parts of the outgoing energy (page 111), E n = E 2 + E 3, gives us the instantaneous energy in the soil: E(t) = E1 - E n . (5.7) 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 5.4a FOURIER TRANSFORM O F HALF-SINE PULSE 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Figures 5.5a and 5.5b the energies Eb and En are given together with the instantaneous energy in the building Eb for r t = 0.05, 0.5, 1, and 2 for HSB and HIH, respectively. Also, the duration of the pulse td 0 and the error computed from (5.6) at the end of the analysis e(%) are given when Eb is virtually zero. To provide a more detailed view (i.e., to avoid the peak of the energy Ej), only the energy E: after the pulse has left the model is shown as a constant in time. These plots are given in Fig. 5.5c and Fig. 5.5d for HSB and HIH, respectively. We should mention here that the results for high q depend upon the choice of grid parameters, especially on the parameter p in Eq. (5.3). For the shortest duration of the pulse, 7 7 = 2 , and for the highest input frequency allowed in the model (from numerical tests ®max = 2 0 0 rad Is), from Fig. 5.4a we can see that a significant part of the frequency response is filtered. Because we allow temporal frequencies up to 200 rad/s in the model, we should choose higher p in sampling the grid (Eq. 5.3) to prevent the dispersion relation of the wave propagation in the soil. In Table 3, the error in energy calculation is presented for different choices of the cut-off frequency and the parameter p. Taking sm aller p in (5.3) and cut-off frequency coc = 200 rad/s, we obtain a large velocity at r , , and severe error in the balance of energy occurs in the computation, m ostly due to the previously mentioned Saint-Venant principle. A remedy is to decrease the input velocity by specifying a lower cut- off frequency in the filter. A question arises as to how large the cut-off frequency should be in the low-pass 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. filtering of the pulse. One good indicator is given in the next chapter (Fig.6 .1b for our example for HIH), where it w ill be seen that after some input r |g the response does not change with increasing r|, indicating that the grid behaves as a low-pass filter, with the highest frequency being In our example one can estimate j]g » 1 .7 , and from (5.8) co g = 139.8 rad I s . The plots in Fig. 5.5 are obtained from models with the most accurate com bination of grid parameters (p,coc) for each building. For sm aller 77, the grid parameters (p,a>c) are kept the same as in Table 2. The best insights about the energy distribution can be obtained for the cases in which 2 H the pulse is short enough to be com pletely contained in the building - e.g. when td 0 < — - , Pb In our examples, this is a case for r\ = 0 .5 ,1 , and 2 for both buildings. The energy distribution is shown for these cases in Table 4. From the plots, it can be seen that the HSB attracts more energy (the dotted lines) but radiates it faster for any input frequency. This behavior is m ostly due to the coefficient of transmission between the building and the foundation, given by (5.0.1). The HSB has kb . closer to 1 than HIH, which means that a larger amount of flow occurs in both directions at the building-foundation contact and also sm aller reflection. For HIH, the situation is the opposite. 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Fig. 5.5a THREE COMPONENTS OF THE ENERGY IN THE MODEL AS FUNCTIONS OF TIME: HOLLYWOOD STORAGE BUILDING 4 0 6 0 0 t d 0 = 0 .6 2 4 s s = 0 .25 % 4 0 0 20 200 t ( S ) o 4 . 0 7 . 0 0.0 1.0 2.0 3 . 0 5 . 0 6.0 0.0 8 0 0 6 0 0 4 0 0 200 - 0.0 E , E„ Eh T] = I td 0 = 0 .0 3 1 2 s e = - 1 .7 6 % 3 0 0 200 100 -4-.... 1.0 J ( S ) 1.0 E , E„ Eh 2.0 E , Eh Eh !f\ rj = 0.5 ■ d O = 0 .0 6 2 4 s - 1.8 8 % M s) 3 . 0 4 . 0 5 . 0 T| = 2 td 0 = 0 .0 1 5 6 s s = - 0 .55 % ,t(s) 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 5.0 Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Fig. 5.5b THREE COMPONENTS OF THE ENERGY IN THE MODEL AS FUNCTIONS OF TIME: HOLIDAY INN HOTEL 4 0 0 200 0.0 1000. ^ 8 0 0 I i t 6 0 0 4 0 0 200 0 5 . 0 r) = 0.05 d O 0.764 s E„ Eh s = - 0 .2 3 % 8 0 0 6 0 0 4 0 0 200 ; t ( S ) 10.0 1 5 . 0 20.0 0.0 X] = 1 0 = s — - 3.51 % td 0 = 0.0382 s 5 0 0 4 0 0 3 0 0 200 100 t(S) 0.0 1.0 2.0 3.0 4.0 5.0 0.0 0.5 td 0 = 0.0764 s s = 5.45 % L 1.0 2.0 _______ L ... _ . . i. 3 . 0 4 . 0 E , R = - E „ E b £ = - ,t(s) 5 . 0 6 . 0 7 . 0 0.0191 s 0.58 % (s) 1.0 2.0 3.0 4.0 5.0 UJ ro Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Fig. 5 .5 c DISTRIBUTION OF ENERGY REACHING THE FOUNDATION: _ HOLLYWOOD STORAGE BUILDING — D L U 1 5 r| = 0 .05 td0 = 0 .6 2 4 s s = 0 .2 5 % 1 0 5 t ( s ) o 4 . 0 7 . 0 1.0 2.0 3 . 0 5 . 0 6.0 0.0 t| = 0.5 t d 0 = 0 .0 6 2 4 s s - - 1 .8 8 % LU 100 5 0 ; ( s ) 0 1.0 2.0 3 . 0 4 . 0 5 . 0 0.0 7 5 5 0 0 .0 1 5 6 s L U 0 .5 5 % 2 5 t ( s ) t ( s ) o 5.0 1.0 2.0 3.0 4.0 0.0 200. L n = 0 .0 3 1 2 s 100 = - 1 . 7 6 % 2.0 3.0 4.0 1.0 0.0 L O L k > o I— <c Q :zi id o HI i— CD Z I HI O <c LU c c > - CD C C L U i LU LU q _ J — O Q in o ^ eg c n r Q (— i y i o q m “O L O L O C D CP ( m ) 3 C O C O L O r^- o C D * C D CO Cv J CD L U C P 00 C O L O C D CD L U o ( m ) 3 C O C X I oo CO C D C D L O CO o ( m ) 3 ( m ) 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE 3. DEPENDENCE OF THE ERROR FROM THE GRID PARAMETERS B u ild in g T 1 P (% ) co c (rad / 5) E ,(M J ) Eh (M J ) e (% ) HSB 1 4 200 245.41 272.46 - 11.02 HSB 1 4 160 205.43 209.04 -1.76 HSB 2 5 200 181.18 230.17 -27.04 HSB 2 5 160 130.91 149.77 -14.4 HSB 2 8 140 77.84 78.27 -0.55 HIH 2 4 200 258.74 370.84 -43.33 HIH 2 4 160 189.32 224.76 -18.72 HIH 2 8 135 113.17 105.48 6.80 HIH 2 8 145 128.89 129.64 -0.58 TABLE 4. ENERGY DISTRIBUTION OF THE FIELD REACHING THE FOUNDATION B uilding 1 1 E , (MJ) Ebmax ^3 (MJ) £fcmax (% ) E j — (% ) E , HSB 0.5 146.5 61.8 42.2 57.8 HSB 1.0 205.4 86.9 42.3 57.7 HSB 2.0 77.8 32.0 41.2 58.8 HIH 0.5 147.0 20.7 14.1 85.9 HIH 1.0 276.4 37.1 13.1 86.9 HIH 2.0 128.9 17.5 13.6 86.4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a f u f 6h As the ratio of the mechanical impedances — = —— - is closer to 1, and as the angle of incidence is closer to zero, the building attracts and radiates more energy. That the percentage of the energy distribution does not depend upon the input frequency can be seen from Table 4, where for any duration of the pulse HSB attracts (radiates) about 42% of the energy reaching the foundation and the rest (about 58%) is imm ediately scattered from the soil-foundation and building foundation contacts. The HIH building “ sees" only about 13.5% of the input energy reaching the foundation, and the rest (about 86.5 %) is scattered from the contacts. The scattered energy, E2, never enters the building and for design of earthquake resistant structures, it is im portant to learn how to increase the percentage of this part of the outgoing energy (last column of Table 4). The scattered energy depends prim arily upon the coefficients of reflection and transmission at the soil-foundation and building-foundation contacts (Eqs. 5.0.1 and 5.0.2 for 1-D problem), which depend upon the properties of the media meeting at the contacts. Assuming a 1-D propagation and particle velocity of the incident wave equal to 1, the energy reaching the building in unit time per unit area is e , p j a = p j M ; '■ K ~ hf . (5.9) where is the coefficient of transmission from soil to foundation, and 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. K 1 + PbPb P fPf is the coefficient of transmission from foundation to building. Using these coefficients in (5.9) and assuming that p f - p s, we have 1 + Ps P, s J 1 + PbPb \ 2 4pbPb PfPf 1 + PbPb PfPf (5.10) Substituting — — = r , (5.10) becomes PbPb = ( c / )2 - ,2 4pbPb'r ( i+ ry (5.11) From (5.11) we can analyze three lim iting cases for (5.11): 1. p b pb » p fP r , lim e . = 0 . Because of the empty space between the elements of the r->0 structure, the structure is usually “ softer” and “ lighter” than the foundation, so this case is only of theoretical interest. It corresponds to a structure floating on fluid. 2. p b pb p f p f , lim eb = • pf Pf . This is more realistic, but a rare case. 3. p b pb « Pf/3ft lim e , = (k‘ l f f ■ 4■ p b(3b. This is the m ost realistic case, ^ J r-> c o As can be seen from the above lim iting cases, the energy is com pletely scattered from the building-foundation contact for case 1, and the energy reaching the building is zero. This case cannot exist in the m odeling of real structures because the density of the structures is smaller than the density of the foundation, and, as a consequence, the impedance of the 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. structure is sm aller than the impedance of the foundation. When the impedances of the structure and the foundation are equal (case 2), the building-foundation contact does not exist, and the wave motion is not altered between the foundation and the structure. The energy passing through unit area in unit time in the structure is equal to that in the foundation, which means that 100% of the energy entering the foundation is transmitted into the building. Finally, which applies to m ost buildings, there is loss of energy due to scattering at the building- foundation contact, and the percentage of the energy in the foundation entering the building is sm aller than that in case 2. The influence of the soil-foundation contact on eb is expressed by 1 + . P .J Pf where p = is the ratio of the impedances of the foundation and the soil for the same densities for these two media. We can distinguish three lim iting cases for D in (5.12): a. » Bf ,p - » 0 : lim D = 4 r s r J r p^o b. Bf - > fi,,p - » 1: l im £> = 1 J ' p - + 1 c. p s « fif ,p -> oo : lim D = 0 . p —>co With combination of the cases 2, 3, a, b, and c, the system soil-foundation-structure can be designed to m inim ize the energy eb entering the structure (5.11). From the above lim iting 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cases, the best com bination to m inim ize eb (i.e., to maximize the energy scattered from the contacts) is combination 3c which means a “ light" and “soft” structure on “ heavy” and "stiff” foundation embedded in “ soft" soil. In the real world, usually the properties of the structure and the soil are known in advance. There are two approaches to achieve the com bination 3c: • Design a stiff foundation, so that the lim iting cases 3 and c are fullfiled. • Design the weaker contacts between the building and the soil. For example, if the soil is stiff, a layer of soft material around the foundation can be inserted before the filling of the foundation, so that case c is simulated; if the building is stiff, soft material acting as a base isolator can be inserted above the foundation. The scattered energy E2 also depends upon the angle of incidence y , as w ill be shown in Chapter VII. As y becomes larger, the sm aller is the amount of energy entering the building and the larger is the scattered energy from the soil-foundation and building-foundation contacts. Finally, in Figures 5.6a and 5.6b the solution in the soil island for FISB and HIH, respectively, at the end of the analysis is shown normalized to 2A/50, or in scale 1:0.02, to visualize the radiation. 5.3 Conclusion From the above analysis, it can be concluded that for the same amount of energy reaching the foundation (the input energy), when the material properties of the foundation are 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. close to the material properties of the soil, a building that has coefficient ktr closer to 1 will attract more energy and radiate it out faster. The error in the solution arises m ostly because of the error in the scattered field, E2, due to the consequence of the Saint-Venant principle. It states that if some distribution of forces acting on some region of the body is replaced by a different distribution of statically equivalent forces acting on the same region, then the effects of the two different distributions on the parts of the body sufficiently far away from the region of application of the forces are essentially the same. At the region of application of the forces, different distributions cause different effects. The error depends mainly upon the duration of the pulse and is more pronounced for shorter pulses (larger velocities). The overlapping of the scattered field and the half-space field at V, is inevitable for any size of the soil island, and it overestimates the outgoing energy, especially the energy E2 for ground motion frequency O = — > 100 rad I s . ho To keep the balance of energy, we need to filter out the higher frequencies. To use higher frequencies (velocities) of the input, we need either faster com puters or higher-order accurate finite difference schemes, especially with respect to space coordinates. For example, for t j = 2 and p = 3%, the algorithm generates a grid with 3 2 1*37 0 spatial points. The ordinary PC with a Pentium IV processor needs roughly about 40 hours (the speed depends upon the speed of the hard-disk controller and the available space on the hard disk) for simulation of Te n d = 2.5 s. To take more modes into the analysis, for example for p = 2%, we 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Fig. 5.6a DISPLACEM ENT OF THE SOIL ISLAND AT THE END OF THE ANALYSIS: HOLLYW OOD STORAGE BUILDING Fig. 5.6b DISPLACEMENT O F T H E S O IL ISLAND A T T H E E N D OF 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. need spatial grid of 481 * 555 points which is 1 ,52 = 2.25 more points than the grid for p = 3%. Also, to preserve the Courant condition (Eq. 2.1), A t should be made 1.5 times smaller. Knowing that the order of com plexity is O(N), where N is the total number of spatial points, the simulation for p = 2% should last about 3.375 tim es longer than for p = 3%. From tests, it was noticed that runs for bigger N values, lasted longer than expected. Probably this is because of the larger number of l-O operations and the fact that the controller needs more time to find a record in larger files. As shown in Appendix II (II.9), the input energy brought to the foundation depends only upon the soil properties, the am plitude of the pulse, and t|, and it is the same for two different buildings sitting on the same soil and having the same r\. Indeed, in our examples this is the case when r| is so small that all of the modes involved in the grid have temporal frequency ®max = A A nax - 200rad I s . For example, for 7 7 = 0 .0 5 , HIH has Ez = 1 6 .4 0 0 M J and HSB has.E'j. = 16.385 M J . For 77 = 0.5 , when p = 3%, the highest mode for HSB has dual , c o ^ ) = (o .8 0 5 5m~l , 2 0 1 .3 4 rad/s). As was shown in Table 4, although ® m ax is slightly larger than the cut-off frequency coc ~ 2 0 0 rad Is again the input energies are essentially equal. It is obvious that for 77 > 0.5 the input energies are not equal because of the filtering of the original pulse and the loss of the higher modes. The shorter pulse (HSB) at the same t] loses more significant modes, and that is the reason E fS B < E fm at higher r|. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER VI INPUT PARAMETERS FOR STRUCTURAL DESIGN 6.0 Introduction and Model In engineering practice, the response spectrum method is a popular tool for structural design because it is sim ple and does not depend upon the details in the structure (Trifunac, 2003). The stiffness and the mass matrix can be diagonalized using eigenvectors of the system so that an uncoupled system of second-order ordinary differential equations can be obtained. When the excitation is the ground motion, the right-hand side of this system includes acceleration of ground motion. It is obvious that these input parameters should be well defined to get an accurate prediction of the response of the system. The codes usually prescribe the spectral acceleration as a function of the period and of the soil properties at the site. Solving for the system response, first the vector of maximum modal displacem ents is obtained, from which, using the square root of the sum of squares technique (SRSS), the vector of m axim um displacem ent shears and moments is obtained. A different approach to the design problem is to evaluate m axim um local drift in the structure in terms of the peak velocity of the velocity pulse entering the structure (Trifunac et a l, 2001). The building starts to respond to the ground motion with its fundamental mode 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig, 6 .0 THE MODEL FOR TRANSIENT ANALYSIS y □c Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 H after tim e ^ (H b is the height of the building, and p b is shear wave velocity in the Pb building) has passed from the entrance of the pulse into the building. For early transient response in time t(o < ? < / , ) when the pulse is still in the building, designing with the response spectrum method is inappropriate because the representation of the displacem ent field in the building requires representation in terms of many modes, and thus for studying the response in this time interval the wave propagation approach is natural and also the most efficient and direct. As further discussed in Trifunac e ta l. (2001) the velocity am plitude of the pulse when it enters the building depends upon the soil, the foundation, and the building properties, and because of local dissipation of the energy due to soil-structure interaction (for the SH case this involves scattering from the soil-foundation and building-foundation contacts) the amplitude of the velocity pulse entering the building is reduced and is always sm aller than the amplitude in the soil. Our goal in this chapter is to find the am plitudes with which the pulse is entering the system building-foundation. For that purpose, the peak average displacem ents and the peak average velocities at the building-foundation and soil-foundation contacts are studied for a range of dim ensionless frequencies 0.05 < rj< 2 when the incident wave excitation is a half sine pulse. All of these quantities are normalized with respect to the peaks in half space so that the ratios give the effect of the soil-foundation-structure interaction and enable us to evaluate 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the peak velocities entering the structure. The curves are constructed with 40 points with an increm ent A ij = 0 .0 5 . The model (Fig.6.0) has the same soil properties as the model in Chapter 5. The example building has the properties of the Holiday Inn hotel, and the foundation has density equal to the soil density. The analysis is performed for three different velocities of propagation of the SH waves in the foundation: f3f = 250, 300, and 500 m/s. To avoid interference of the incom ing pulse in the structure and the reflected pulse from the top of the structure, the height of the structure Hb is computed from the condition ' ^ j L > td + — . The size of the soil island Pb Pf is taken as H s = 4a, The pulse is Fourier transformed in space, and only the frequencies that have response am plitudes larger than p(%) of the frequency response of the zero-th mode (k = 0) are used. The percent p varies depending upon the angle of incidence, and it is taken as p = 5, 6, 6, 7 for y = 0°, 30°, 60°, and 90°, respectively. For proper m odeling of the foundation, the m inim um number of spatial intervals per length of the foundation is mf = 1 8 / 2a, and the maximum number of intervals per length of the foundation is mf = 4 0 /2a (Fig. 2.3). For the highest spatial frequency km sx , the corresponding highest temporal frequency of the pulse is ®max = Psk m ax • This is the cut-off frequency for the Ormsby low-pass filter, so the higher frequencies are filtered out from the analysis. The analysis ends when the pulse has com pletely passed point B in Fig. 6.0. 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.1 Results In figures 6.1a,b,c,d the normalized am plitudes of the average displacem ents at the soil-foundation (dashed lines) and building-foundation (solid lines) contacts are shown for three different foundation stiffness. The averaging is done in every tim e step k as: where: N is the number of points on the soil-foundation contact, M is the number of points on the building-foundation contact, f k = — is the current tim e step, At wf is the displacem ent of the i-th point on the soil-foundation contact at tim e step k, and w * is the displacem ent of the j-th point on the building-foundation contact at time step k. The amplitudes of the quantities (6.1) and (6.2) fo ra certain m are (6.1) (6.2) maxw Vic I (6.3) maxw vi t I (6.4) 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The am plitude at the free surface of the half-space solution is K = 2 - m a x K (6.5) where u \ is the prescribed displacem ent in time step k at the point x = 0, y = 0, and the normalized am plitudes are For small r|, the am plitudes do not depend upon the angle of incidence y or upon the stiffness of the foundation and all curves approach 1 as r\ approaches 0. The pulse occupies all contact points, and in addition the displacem ent at all of the contact points is approximately agreement with the energy distribution for long pulses when there is virtually no scattering from the foundation and all of the energy reaching the foundation reaches the building-foundation contact. In Fig. 6.1a, when the incidence is vertical, as the pulse becomes shorter a s _ f decreases, which is a consequence of the fact that the pulse does not occupy all contact points simultaneously. When the displacem ent at the bottom points of the contact reach their peak amplitudes, the pulse has not yet reached the points close to the building-foundation contact, and later, when the other points reach their peak am plitudes, the pulse has passed the former points and their displacem ents are small. The coefficient a s _ f does not depend much (6.6) (6.7) equal because of the small spatial gradient which is due to small v. This is in 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.1a AMPLITUDES OF THE NORMALIZED AVERAGE DISPLACEMENT AT THE CONTACTS 1.0 y — 0 0 . 9 0.8 300 0 . 7 0.6 < t 0 . 5 500 0 . 4 0 . 3 500 250 0.2 contact building - foundation contact soil - foundation 0.1 0.0 0.6 o.a 1.0 1.2 1 . 4 1.6 1.8 2.0 0.0 0.2 1 1 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.1 b AMPLITUDES OF THE NORMALIZED AVERAGE DISPLACEMENT AT THE CONTACTS 0 . 9 0 . 7 0.6 0 . 5 500 0 . 4 0 . 3 .... 2 5 0 '-JafcT: 300 0.2 contact building - foundation contact soil - foundation 0.1 0.0 0 . 4 0.6 0.8 1.0 1.2 1 . 4 1.6 1.8 2.0 0.0 0.2 T j Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.1c AMPLITUDES OF THE NORMALIZED AVERAGE DISPLACEMENT AT THE CONTACTS 1.0 0 . 9 - 0 . 7 0.6 s <C <£ I I °-5 a 0 . 4 0 . 3 - 0.2 0 .1 0.0 p f = 250 m/s p f = 300 nYs p { = 500 rrVs contact building - foundation contact soil - foundation 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1.8 2.0 1 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.1 d AM PLITUDES OF THE NORMALIZED AVERAGE DISPLACEM ENT AT THE CONTACTS y = 85° 0.9 0.8 0.7 <C 0.5 0.4 0.3 pf = 250 m/s p ( = 300 m/s pf = 500 m/s contact building - foundation contact soil - foundation 0.2 0.0 2.0 0.8 0.4 0.6 0.2 0.0 'n Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. upon the foundation stiffness because there is a trade-off. Because of the larger velocity of propagation, as the foundation become stiffer, more points on the soil-foundation contact reach the zone of peak displacem ents simultaneously. This is not so for the building-foundation contact because for vertical incidence, the pulse reaches the points simultaneously. For vertical incidence, the curves a b _f (ri) depend upon the stiffness of the foundation (parameter J3f ). When the stiffness of the foundation is equal to the soil stiffness, there is no scattering from the foundation, and in the previous chapter it was shown that for a soft building the reflected field from the building-foundation contact is alm ost equal to the incident field. The system behavior is close to the half-space behavior, and a ^ f is close to one. For short pulses, there is some decrease of a ^ d u e to scattering from the building-foundation contact, especially at the corner points, but the am plitudes are still about 90% of the half-space amplitudes. As the foundation become stiffer, there is a decrease of a b _f due to scattering from the, foundation, and a b _ f is the sm allest for the stiffest foundation. It can be seen that for very short pulses a b _f becomes virtually constant and does not decrease with increasing iq. As we concluded in the previous chapter, the foundation behaves like a low-pass filter. As the pulse reaches the cut-off frequency of this “filter", every further increase in the frequency (i.e., decrease in the pulse duration) is irrelevant, and beyond some iq (in our example 77 = 1 .7 ) the responses a b _ f and a s _ f are practically constant. 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Figure 6.1b, the same curves are given for incident angle y = 30°. The situation here is more com plicated because the angle of incidence appears as an additional factor. Everything that was said for the soil-foundation (s-f) contact at vertical incidence is relevant here, with slightly larger amplitudes at higher frequencies. This can be explained by diffraction around the foundation, which causes the larger part of the incident field to reach the foundation at higher frequencies. The pulse does not reach all of the points at the building-foundation contact (b-f) sim ultaneously. The average response is due to the scattering from the foundation, and the distribution of the pulse along the contact points (b-f). At sm aller r\, the pulse occupies all of the points at the contact b-f with small spatial variations of the displacem ents due to small v : At small t|, the pulse starts to "feel” first the greater difference in the stiffness between the soil and the foundation. As t | increases and the duration of the pulse decreases, due to the increase in the particle velocity at the contact b-f, sx increases too, so that the spatial variation of the displacem ent is larger. From (6.8), this is the m ost pronounced for the sm allest stiffness of the foundation, where sx is largest, and it is less pronounced for the higher stiffnesses of the foundation, when ex is smaller. At higher rj, again the foundation behaves as a low-pass filter, so that a b _f has alm ost no dependence upon r\. In figures 6.1c and 6.1 d, the dim ensionless amplitudes for angles of incidence y = 6 0 ° and y = 85° are shown, respectively. 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For sm aller rj, we have the same trends as for y = 3 0 ° . The am plitudes at the contact b-f again are the largest for the softer foundation, but the difference is sm aller than in the B f previous case. This is due to larger ex because of sm aller . The change of the s in ^ displacem ent is large along the contact, so that although the pulse occupies all of the contact points, when some of the contact points are in the zone of big displacem ents, the others are not. This is why the am plitudes of ab _ f for these two cases are sm aller than for the case y = 3 0 °. For the case ^ = 8 5 °, it can be seen that there is a further decrease of ah _f in the zone of high frequencies, which means that the foundation still feels the small decrease in td and that the foundation cut-off frequency has shifted to higher frequencies. Everything that has been said about the average am plitudes a holds for the average velocities u in Figures 6.2a,b,c,d, and only the plots are presented. In Figures 6.3a,b,c,d the am plitudes at the left end (x’ = - a, y ’ = 0) and at the right end (x’ = a, y' = 0) of the b -f contact, normalized by the half-space am plitudes are shown as functions of rj. The normalized am plitudes of the left end are shown with a solid line, those of the right end with a dotted line, and the average am plitude for all points of contact b -f with larger dashed lines. For the softest foundation studied here {(3f = (]s = 2 5 0m/s) the am plitude at the left end practically speaking, does not depend upon the incident angle y , and for larger r\ it essentially does not depend on r\ as well. Because there is no foundation, the scattering is due 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to the difference between the soil and the building properties and to the existence of the building. The motion of the right end is different because there is attenuation due to scattering from the contact b -f while the pulse traverses the foundation. This loss is greater for larger incident angles when sx is larger. When the properties of the foundation and the soil differ, the am plitudes at the left and right ends depend upon ^ a n d rj. In figure 6.3a, when the incidence is vertical, the pulse barely recognizes the foundation. The rays traveling close to the left from the left end and close to the right from the right end do not have contact with the foundation and do not experience loss from scattering. The "inner” rays that encounter the foundation undergo loss due to scattering from the foundation. This loss is greater for stiffer foundations. As the incident angle increases, at sm aller r\ the am plitude at the right end becomes larger than the amplitude at the left end. This is especially pronounced for stiffer foundations and larger angles of incidence. To explain this trend, we consider a sim plified model of a stiff disk supported on elastic soil with the wave incidence from the left (Fig. 6.4). The stiffness of the disk is greater than the stiffness of the soil. If the disk had infinite stiffness, points A and B would have the same displacement. In Figure 6.4b (view of the disk from above) after half of the pulse has entered into the disk there is maximum displacem ent at point A. As can be seen from this figure, the elastic forces in the disk resist the imposed motion. In the first quarter-cycle of the m otion of point A, — t, < — , the motion of the disk points is sm aller than the ground motion at point A. The td 4 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.2a AM PLITUDES OF THE NORMALIZED AVERAGE VELOCITIES AT THE CONTACTS 0 . 9 O .S 300 0 . 7 0.6 G O > 0 . 5 0 . 4 500 0 . 3 250 0.2 _500 300" contact b u ild ing-founda tion contact soil - foundation 0.0 1 . 4 2.0 0.6 0.8 0.2 0 . 4 0.0 T| Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6 .2b AM PLITUDES OF THE NORMALIZED AVERAGE VELOCITIES A T THE CONTACTS y = 30° 0 . 9 0.8 0 . 7 0.6 C / O > 0 . 5 0 . 4 500 0 . 3 250 300 0.2 2 5 0 ' 500 contact b u ild ing-founda tion contact soil - foundation 0.0 2.0 0.0 0.2 0 . 4 0.6 0.8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6 .2 c AM PLITUDES OF THE NORMALIZED AVERAGE VELOCITIES AT THE CONTACTS 0 . 9 0 . 7 p, = 250 m/s p f = 300 m/s p f = 500 m/s 0.6 0 . 5 P 0 . 4 0 . 3 0.2 contact b u ild in g -fo u n d a tio n contact soil - foundation 0.0 2.0 0.6 0.8 0.2 0 . 4 0.0 T| 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6 .2d AM PLITUDES OF THE NORMALIZED AVERAGE VELOCITIES AT THE CONTACTS 1.0 85° 0 . 9 0.8 0 . 7 P( = 250 m /s Pf = 300 m/s P , = 500 m/s 0,6 0 . 5 0 . 4 0 . 3 O .Z contact building - foundation contact soil - foundation 0 .1 0.0 1 . 4 1.6 1.8 2.0 1 .0 1.2 0.6 0.8 0.2 0 . 4 0.0 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.3a NORMALIZED AMPLITUDES AT L E F AND RIGHT ENDS OF THE BUILDING - FOUNDATION CONTACT 250 Pf(nVs) 300 0 . 9 250 300 right 0 . 7 left 0.6 c /i C 0 . 5 500 8 0 . 4 0 . 3 0.2 - left end - right end -■ average building - foundation 0.1 0.0 2.0 0 . 4 0.0 0.2 0.6 r i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.3b NORMALIZED AMPLITUDES AT LEFT AND RIGHT ENDS OF THE BUILDING - FOUNDATION CONTACT y = 30° left 250 300 250' .......... right left 0 . 9 300 right 600 left 0 . 7 0.6 right < t 500 0 . 5 300 500' 0 .4 - 0 . 3 0.2 left end right end average building - foundation 0.1 0.0 0 . 4 0.6 0.8 2.0 0.0 0.2 1 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.3c NORMALIZED AMPLITUDES AT LEFT AND RIGHT ENDS OF THE BUILDING - FOUNDATION CONTACT y = 60° left 250 right 300 0 . 9 left 250 right 300 right 0 . 7 500 left 500 o.e <c 0 . 5 0 . 4 0 . 3 0.2 left end ■ ■ right end - average building - foundation 0.0 2.0 1 . 4 0.2 0 . 4 0.6 0.8 0.0 T 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.3d NORMALIZED AMPLITUDES AT LEFT AND RIGHT ENDS OF THE BUILDING - FOUNDATION CONTACT y = 85° 250 left right 300 0 . 9 250' left right 0.8 300 0 . 7 - right 500 500 0.6 left C O <C 0 . 5 II . v 0 . 4 0 . 3 0.2 left end right end — ■ average building - foundation 0.0 2.0 0.6 0.2 0 . 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. elastic forces in the disk dim inish the am plitude at A. On the other side, in Figure 6.4c, as the wave progresses to right, the elastic forces act in the same direction as the soil motion, and this increases displacem ent at A. The elastic forces in this instant of convex deformation of the disk am plify the motion at B. The behavior of our model is sim ilar to the above sim ple example, because in the case of the soft building (e.g., Holiday Inn) it does not change the incom ing half-space field significantly. This model can also explain why the motion at point B (at the right end) in the Figures 4.2 to 4.5 has the largest am plitudes in the steady-state response analysis of the Hollywood Storage building, for intermediate frequencies, in Chapter IV. As the pulse becomes short enough so that the motion at the left end reaches the peak am plitude before the right end starts to move, the am plitude at the left end can become larger because of additional losses of energy due to scattering from the building-foundation contact while the pulse travels from point A to point B. The above example and explanations are further supported by the results shown in Fig. 6.5. In this figure, the time histories of the displacem ents at point A (x1 = - a, y ’ = 0) and point B (x’ = a, y ’ = 0) are shown for y = 6 0 ° and for two values of the dim ensionless frequency, 77 = 0.5 and 77 = 1 .8 . For 77 = 0 .5 , the influence of the elastic forces in the building are so strong for the motion at point A that it cannot reach the zero displacem ent immediately after unloading (passage of the wave). It receives loads from the building near point B; and when the motion at B becomes equal to the m otion at A both points pass the zero line almost simultaneously. For 77 = 1 .8 , the pulse has com pletely passed the left end (point A) before it reaches the right end. The pulses are separated, and the am plitude of the right end is 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.4 MODEL : DISK SITTING ON HALF-SPACE a) _ _ _ _ _ _ _ _ _ _ _ Ai —i B _ _ _ _ _ _ _ _ elastic forces due to disk deformation b) disk pulse rigid disk rigid disk elastic forces due to disk deformation disk pulse Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. oo L O CD O) i n co (lil) M O o Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 slightly sm aller due to the local losses of energy through scattering on the way from point A to point B. 6.2 Conclusion In this chapter, some examples were presented for excitation by strong ground motion pulses to help in the use of such pulses in the design of earthquake-resistant structures. A sim ple, single, half-sine pulse was considered. The strong ground m otion in the free field can be seen as a train of such pulses, and the total foundation response then consists of superposition of the responses of each pulse in the train. With data on how such pulses are m odified in different buildings, the least squares method can be used to construct an empirical formula that w ill describe the pulse am plitudes entering the buildings. The parameters in this formula should include the angle of incidence; the stiffness of the soil, the foundation, and the building; and the geometry of the foundation. Development of such an em pirical equation is beyond the scope of this thesis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER VII NONLINEAR ANALYSIS 7.0 Model In the real world, parts of our model w ill undergo nonlinear deformations and permanent strains during the wave passage. For sim plicity, we here consider nonlinear response in the soil only, while the foundation and the building w ill remain linear. The model is the one shown in Figure 7.1, in which points A and B in the soil, as well as the points 1, 2 ,1 ', 2 ’ and S are allowed to undergo permanent strain when the strain exceeds some prescribed maximum elastic strain sm. We assume that in one direction a point in the soil can yield and in the perpendicular direction it can remain linear. That is, the shear stress in the x direction depends only upon the shear strain in the same direction and is independent of the shear strain in the y direction (and vice versa for the y direction). The motivation for this assumption comes from our sim plified representation of layered soil, which is created by deposition (floods and wind) into more or less horizontal layers. The soil is assumed to be ideally elastoplastic, and the constitutive a - s diagram is shown on Fig. 7.1a. It is assumed that the contacts remain bonded during the analysis and that the contact cells C, D, E, F, G, and H in Figure 7.1 remain linear, as does the zone next to the artificial boundary (the bottom four rows and the left-m ost and right-m ost four columns), as was described in Section 3.2. A question arises how to choose the strain em (Fig.7.1a). 170 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.1 MODEL W ITH NONLINEAR SOIL Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f\Q - 1 A& < 3 civjce' vro's' ,s\°^ = 0 P V « "' o ^ eV te9’ .voAv* d \o ° pVO The velocity in the soil points due to passage of the plane wave (5.4) is n . t U v = w = — A cos— (7.1) and the maximum strain in the direction of propagation of the plane wave is If for a given input plane wave we choose the maximum strain em, given by (7.2), the strains in both directions may remain linear before the wave reaches the free surface or the foundation. We will call this case “ intermediate nonlinearity.’’ If we want to analyze only the nonlinearity due to scattering and radiating from the foundation, we should avoid the occurrence of the nonlinear strains due to the half-space boundary, including both the incom ing and the reflected waves. Then we may choose If the soil is allowed to undergo permanent strains only due to wave passage of incident waves in the full space, then we may choose the maximum strain e„ = max ^ 2nA sin y I t z A cos^^ We call this case “ small nonlinearity. sm < max nAsiny xAcosy . This condition guarantees that in either x or the y direction the soil will undergo permanent strains during the passage of the plane wave. This yielding strain can be written as ^ = C max V, (7.3) 173 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where C is a constant that controls the yielding stress (strain) in the soil. We have the following cases of nonlinearity, depending upon C: 1. C >2: S m all n o n lin e a rity. Permanent strain does not occur until the wave hits the foundation with any angle of incidence. 2. l < C < 2 : In te rm ed ia te n o n lin e a rity. Permanent strain does not occur until the wave is reflected from the free surface or is scattered from the foundation, for any angle of incidence. Permanent strain w ill or w ill not occur after the reflection of the incident wave from the free surface depending, upon the angle of incidence. 3. C < 1 : Large n o n lin e a rity . Permanent strain occurs after reflection from the free surface. Permanent strain may or may not occur before the wave reflects from the free surface, depending upon the angle of incidence. 7.1 Distribution of the energy and the permanent strains In Chapter V the distribution of the energy was illustrated for two different buildings, for SH waves with an angle of incidence y - 3 0 ° , and for four different dim ensionless frequencies. Here we consider the same two buildings with the same dim ensionless frequencies, for two different angles of incidence y = 3 0 ° and y = 6 0 ° , and for intermediate nonlinearity in the soil (C = 1.5). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Figures 7.2a,b, the energy distribution in the system is shown for HSB for angles of incidence of 30° and 60°, respectively, and in 7.3a,b the energy distribution is shown for HIH for the same angles of incidence. In the linear case (Fig.5.3a,b), the energy components that balanced the total input energy entering the model were the total energy exiting the model and some small residual energy in the building at the end of the sim ulation. Here, we have additional com ponent of the energy, which is lost to develop permanent strains at the points of the soil where this nonlinear strain occurs. This energy, called hysteretic energy, is computed from A e t = s - s ^ is the increment of the permanent strain in the x direction at point i, and A s . = S yP f - s‘ y p i is the increment of the permanent strain in the y direction at point i. The hysteretic energy balances the system energy. The numerical computation error can be expressed as E h y * = + a yiAem ) (7.4) 1=1 where: N is the total number of soil points are the stresses at the point i in the x and y directions, respectively (7.5) with same definition for Ejn p and Eo u t as in Chapter V. 175 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. F ig . 7 .2 a E N E R G Y D IS IR IB U T IO N I N M O D E L W I T H N O N L IN E A R SOIL: G O G O 8 II o '' L O S3 d C N J I I I I g r to M ? uu + jJ u f l lJ u # 'i 83 c n i c o to + L i l L l l u f U # 1 ( m ) 3 CO ( m ) 3 ( m ) 3 ( m ) 3 o o o o rd L U _ © LU o o o o o o in o 176 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to o o Q J O CXI |C cb s o o o o - I o C D CD l i f J f u f L U •K - o o o m o o dr- S5 CX I cxj II II ? = r w u f u f = a -CI _Q L U L U L U o i n r * * o o >o o in C M ( m ) 3 (m i) 3 ( m ) 3 o i n C M q d G O u f u f L i# 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C $ 3 c b q LO o uS uS u f ' u f C M II S T S' u J u l a J = - O ( m ) 3 ( m ) 3 oo 40. o .00 u l u J W 178 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. go to a'■ O oo DC <n o o C D eg oc 1 — co Q £d DC <ic Q O I E -Q co r^I O) o I-'- LO LO CD o LU o c \i o o o 1 T 3 O O O m csl ( m ) 3 C O t o r— o ^ 3 - O * 1 -- o o L U L U o o o m o o CD C N J LU L U o o o in ( m ) 3 179 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The energy distribution for HSB is shown in Figures 7.2a and 7.2b for the angles of incidence y = 3 0 ° and y = 6 0 ° , respectively. The thick curve shows the sum E o u t + E h y s that balances the input energy. For y = 3 0 ° and small r\ the hysteretic energy is practically zero. As the incident wave pulse becomes shorter, the hysteretic energy is developed and, for this case the hysteretic energy is between 4 and 6 percent of the input energy. For all rj, the total hysteretic energy is sm aller than the maximal instantaneous energy in the building. For y = 6 0 °, the hysteretic energy is larger than for y = 3 0 ° , about 6 to 9 percent of the input energy and up to 15 percent of the input energy for the sm allest considered r\. In this case, in contrast to the case y = 3 0 °, the hysteretic energy is larger than the maximal instantaneous energy in the building. The maximum Eb is sm aller for y = 6 0 ° than for y = 3 0 ° . As can be seen from the plots, once the hysteretic energy reaches its m axim um it remains flat until the end of the analysis. This is because 100 percent of the permanent strain is developed due to the wave passage in the model, and the scattering from the foundation and the radiation from the building do not contribute to additional permanent strains. The trends are sim ilar for the HIH building (Fig. 7.3a,b) except that here after some time, for case y - 3 0 °, the hysteretic energy starts to grow m onotonically. An explanation of this phenomenon is shown in Figure 7.5a, where the time histories of the displacements in points H(- a,0) (solid line) and T(- a,Hb ) (dotted line) are shown for HSB and HIH, for t\ = 0.5, and for ^ = 3 0 ° and y = 60°. Unlike in Figure 5.4 for the linear case in which point H remains in the initial position after the motion has ceased, for the nonlinear case point H has 180 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. perm anent displacem ent at the end of the analysis even though it is treated as a linear point. As point H develops permanent displacem ent, point T at the top of the building vibrates (dotted curve) around the new equilibrium , which follows the permanent motion of point H (the base). This is more pronounced for ^ = 3 0 ° and for the Holiday Inn hotel, where the radiation dam ping is smaller. The reason for this behavior is that several points close to the soil- foundation contact (around point S in Fig.7.1) undergo permanent strains during the wave passage and are not com pletely unloaded when the radiation from the building takes place. The progress of the permanent strain at the point xs = Xq’ ^ 9 " ^ y s = y g + is illustrated in Fig. 7.5b. In Figures 7.4a and 7.4b, the permanent strains are shown for HSB and HIH. As can be seen from the figures, while the angle of incidence is small, the permanent strain in the x direction can be zero with only the permanent strain in the y direction occurring. The zone of the permanent strain is parallel with the free surface at some depth and has a width that is dependent upon the duration of the pulse. This appearance is m ostly due to the 1-D effect of the wave propagation illustrated in Section 5.0. As the angle of incidence becomes larger, the strain in the y direction eventually vanishes, and the strain in the x direction appears. The largest zone of the permanent strain sx p is along the free surface, where it occurs to some depth. Mostly, this zone is a consequence of the reflection from the half space, where the maximum strain at the free surface is s v = 2 - ^ 2- s i n r . Because the foundation stiffness in this example p s 181 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.4a PERMANENT STRAIN DISTRIBUTION IN THE SOIL ISLAND: HOLLYWOOD STORAGE BUILDING r j = 0.5 y - 30° foundation 30.0 30.0 free surface 15.0 15.0 I J v 0 . 0 1 ---------------------- 7 6 n / \ 0.0 15.0 x(m) n = o.5 y = 6 0 ° o . o 60.0 30.0 45.0 75.0 30.0 45.0 60.0 15.0 0 .0 3 0 .0 3 0 .0 1 5.0 1 5 .0 0 . 0 0 . 0 60.0 .15.0 30.0 45.0 0 .0 . 60.0 75.0 30.0 45.0 15.0 0 .0 r \ = 2 y - 3 0 ° ^ 30,0 15.0 0 .0 4-5.0 1 5 .0 30.0 6 0 .0 D.O 30.0 ,0 0 .0 45.0 60.0 30.0 15.0 0 .0 60.0 ° -° 15' ( r\ = 2 y == 60° 30.0 15.0 0 .0 45.0 15.0 30.0 60.0 0 .0 30.0 .0 0 .0 45.0 60.0 15.D 30.0 0 .0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.4b PERMANENT STRAIN DISTRIBUTION IN THE SOIL ISLAND: HOLIDAY INN HOTEL 45.0 30.0 30.0 15.0 15.0 0.0 45.0 60.0 30.0 75.0 9Q.0 45.0 30.0 60.0 0.0 15.0 30.0 30.0 15.0 15.0 0.0 0.0 15.0 75.0 90.0 0.0 30.0 90.0 4 5 .0 7 5 .0 30.0 6 0 .0 0.0 15.0 3 0 .0 1 5 .0 0.0 1 5 . Q 45.0 75.0 30.0 60.0 0.0 30.0 15.0 0.0 75.0 60.0 45.0 1 5 .0 0.0 30.0 15.D 0 .0 45.0 75.0 15.0 30.0 60.0 0.0 30.0 15.0 0.0 75.0 60.0 30.0 0.0 15.0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Fig. 7.5a TIME HISTORIES OF DISPLACEMENTS AT THE TOP AND THE BOTTOM OF THE BUILDING FOR NONLINEAR SOIL Hollywood Storage Building E o t(s) 3 . 0 4 . 0 2.0 1.0 0.0 Holiday Inn Hotel -1 - Hollywood Storage Building •n = 0.5 0.0 1.0 2.0 3 . 0 4 . 0 t(s) Holiday Inn Hotel 1 E o t(S) t(s ) 2.0 3.0 4.0 5.0 6.0 1 .0 0.0 o o •f*. 10 <C cc co <c D C LU DU LU in T ( w ) ' o CO i I i Q _ S t r CL. <C O LU _ Q L O 05 w CO to (BdM)' £> (BdX)A jD 185 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 0 . 5 is close to the soil stiffness, there is no permanent strain around the foundation. Of course, this does not mean that this w ill be the case when there is a big difference between the soil and foundation properties. 7.2 Average displacem ents at the contacts and distribution of the permanent strains In this section we analyze the model of the Holiday Inn hotel with the geometry and properties given in Section 3.3. In Chapter VI, this analysis was illustrated for the linear case. Here, we consider the case of intermediate nonlinearity with C = 1 and of large nonlinearity with C = 0 .7 . The case of small nonlinearity gives practically the same results for the average displacem ent as does the linear case. It is obvious that for every r\, one would have to change sm in accordance with (7.3) to get the same level of nonlinearity. Together with the curves for the nonlinear case, the results for the linear case (Fig.6.1) are shown for four different angles of incidence: y = 0°,30°,600, and 85°. The normalized average displacements a at the building-foundation contact for nonlinear case are shown with solid dark curves and for the linear case with solid light curves. At the soil-foundation contact, for the nonlinear case these am plitudes are shown with curves with longer dashes, and for linear case they are shown with curves with shorter dashes. In the same figures, the cases for three examples of foundation stiffness are shown: 186 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.6a NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS OF THE MODEL WITH NONLINEAR SOIL y = 0 0 . 9 0.8 300 0 . 7 0.6 <C 0 . 5 500 0 . 4 , 25 0 _ ^ 300 0.2 - contact building - foundation - contact soil - foundation 0.0 0 . 4 2.0 0.2 0.6 0.0 t | Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.6b NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS OF THE MODEL WITH NONLINEAR SOIL y = 30" a £ <C <C II 8 1.0 D .9 O .S 0 . 7 0.6 0 . 5 0 . 4 0 . 3 p ( = 250 m /s p f = 300 rrVs p f = 500 m/s contact building - foundation contact soil - foundation 0.2 0 .1 0.0 1 .0 1 . 4 1.8 0.0 0.2 0 . 4 O .B 0.8 1.2 1.6 2.0 *1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.6c NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS OF THE MODEL WITH NONLINEAR SOIL y = 6 0 ° 1.0 0 . 9 0.8 0.7 0.6 S . < t II °-5 a 0.4 0 . 3 0.2 0.1 0.0 p = 250 m/s p, = 300 nVs p, = 500 m/s contact building - foundation contact soil - foundation I 0,0 0.2 0.4 0.6 0.8 1.0 1.2 1.6 1.8 2.0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.6d NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS OF THE MODEL WITH NONLINEAR SOIL y = 85° 1.0 p f = 250 m /s p f = 300 m /s R, = 500 m /s 0 . 9 0.8 0 . 7 0,6 0 . 5 0 .4 - 0 . 3 0.2 contact building - foundation contact soil - foundation 0.1 0.0 1 . 4 1.6 1.8 2.0 1.0 0 . 4 0.6 0.2 O .Q 190 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • p f = 250m / 5 with the thinnest curve, • p f = 300m / 5 with the thicker curve, • (5f = 500m/ ^ with the thickest curve. For example the normalized average displacem ent at b -f for the nonlinear case with foundation stiffness =300m is is shown with a solid dark curve with intermediate thickness. Again, as explained in section 6 .0 , the interference of the incom ing wave and the reflected wave from the top of the building is avoided by assuming a high enough building, and the analysis is stopped as the wave com pletely passes the right point of the triple soil-foundation-building contact. The distribution of the com ponents of the permanent strain spx,sp y for the case of intermediate nonlinearity C = 1 is shown in the figures 7.8a,b1,b2,c,d at the end of the analyses. The contours of equal strain are drawn with steps of 0.02 in all of the plots in Fig. 7.8. In Figures 7.6a,b,c,d the results for the case C = 1 are shown. When the incidence is vertical (Fig.7.6a), the differences at the contact b -f between the linear and the nonlinear cases are small, and they are largest at sm all r\. In this case, the permanent strains occur only in the y direction, mostly close to the free surface. This results from the interference of the incident and reflected fields from the free surface. The nature of this phenomenon was illustrated for the 1-D case in Section 5.0 (Fig.5.Ob). This nonlinearity appears after the reflection of the wave from the free surface. For sm aller 77, nonlinearity appears close to the contact s -f and is due to scattering especially for a stiffer foundation (Fig.7.8a). For all cases of foundation stiffness, the 191 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.7b NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS CASE: BIG NONLINEARITY IN THE SOIL y - 30° pf = 250 m/s p, = 300 m/s pf = 5 0 0 m/s contact building-foundation contact soil - foundation 0 . 2 0 . 4 0 . 6 0 . 8 192 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig 7 7c NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS: CASE OF LARGE NONLINEARITY IN THE SOIL y - 60° 1.0 0 . 9 ( 3 { = 250 m/s p( = 300 nr/s p, = 500 nYs 0.8 0 . 7 0.6 0 . 5 0 , 3 0.2 contact building - foundation contact soil - foundation 0 .1 0.0 1.2 1 . 4 1.6 1.8 2.0 0 . 4 0.6 0.8 1.0 0.2 0.0 1 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.7d NORMALIZED AVERAGE AMPLITUDES AT THE CONTACTS: CASE OF UR G E NONLINEARITY IN THE SOIL Y = 85" 1.0 0 . 9 p ( = 250 m /s p ( = 300 m/s p f = 500 m /s 0.8 0 . 7 0.6 0 . 5 0 . 4 0 . 3 0.2 0.1 contact building - foundation contact soil - foundation 0.0 1.8 1.6 2.0 0.8 1.0 1.2 1 . 4 0.6 0 . 4 0.0 0.2 1 1 194 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. curves for nonlinear response although alm ost equal, are slightly smaller, because of the loss of energy due to nonlinear strains. The difference is largest at sm aller 7 7 when the pulse is longer. At intermediate 77, the difference is the greatest for the stiffest foundation because the nonlinear zone below the foundation is the largest, and it attenuates the incom ing wave (Fig.7.8a). For high 7 7, there is no nonlinear zone around the foundation, and for 77 larger than 1.5 the curves for the linear and nonlinear cases are identical. Close to the left and right boundaries in Fig.7.8 a some zones of very small nonlinearities can be seen. For this case (C = 1), with vertical incidence we are at the lim it between intermediate and large nonlinearities, and any small reflection from the artificial boundary brings the soil points into a nonlinear state even when the wave has not reached the foundation. Nevertheless these permanent strains are small, and they do not affect the overall accuracy in our models. The small nonlinear zone at the bottom of the model is real and, it occurs when the scattered pulse (m ostly the extended part due to filtering) meets the incoming main part of the pulse and the resulting strain slightly exceeds sm. For incident angle y = 3 0 ° both components of the permanent strain exist. In figure 7.8b1, and syp are shown for iq = 0.6, and in Figure 7.8b2 they are shown for i q = 1.8 . As can be seen in the plots for sxp, it is m ostly distributed around the foundation. For the stiffest foundation (the lowest plot), because of the interference of the scattered and the half space fields, there is a substantial am ount of yielding behind the foundation, which decays with softening of the foundation (the second and first plots). As m ight be expected, this strain is larger for shorter pulses because of increased diffraction. Theoretically, at the free surface 195 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig.7.8a D IS T R IB U T IO N O F T H E PE R M A N E N T S T R A IN J U S T AFTER 196 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p = 250 m/s P = 500 rrVs P = 300 m/s 197 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p = 250 m/s o o u > o in o D n 0 0 o o iri _-0 o o a O o a p = 300 m/s p = 500 m/s 198 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. 1 5.0 0.0 0 . 30.0 1 5.0 0.0 0 . 30.0 15.0 0 . 0 - 0. Fig.7.8c1 DISTRIBUTION OF THE PERMANENT STRAIN JUST AFTER THE WAVE HAS PASSED THE FOUNDATION 1 5.0 75.0 30.0 45.0 60.0 y = 6 0 ° *1 = 0.6 60.0 75.0 5.0 30.G 45.0 1 P O c n o C /9 § "CO cn o CD oo TO CO CD CD y p 0 0.0 L - 0.0 30.0 45.0 60.0 15.0 75.0 1 5.0 0 .0 L- 0.0 30.0 45.0 60.0 15.0 75.0 0 15.0 30.0 45.0 60.0 7 5.0 30.0 0.0 L 0.0 45.0 60.0 30.0 75.0 15.0 VO VO Fig.7,8c2 D ISTR IBU TIO N O F T H E PE R M AN EN T S T R A IN J U S T A F T E R THE o C O o d co o 10 c d _ o 00 C O o o O O O o I a = 250 m/s p = 300 m/s pf = 500 m/s o d C D o id _ o o d oo o o o o 2 0 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig.7.8d D ISTR IBU TIO N O F T H E PERM AN EN T S T R A IN J U S T AFTER J 9 in rs o d to o d t o o in • t O O o d m o d n _ o oo o _ o oo o o o 8- Q T ^ pb co Q ( 3 f = 250 m/s ( 3f = 300 m/s P = 500 m/s o 'o to o d to o in co CD II o 'o f O o d o d o in _ o o o _ o qd o o o o 2 0 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. £ =2 v, m a x v sin y m a x / and for C = 1 and y = 3 0 ° we are at the lim it between the c X linear and nonlinear states. Because of the artificial change of the soil properties (the first four columns of points in the grid must be always linear for constructing the boundary) and the im perfect artificial boundary, some small permanent strain in the x direction occurs at the free surface. The biggest strain occurs close to the lowest left corner of the foundation. W hile the strain sx p is generally distributed around the foundation, the strain ^ is m ostly distributed close to the free surface, due to the 1D effect already mentioned for the case of vertical incidence. Although the incidence is not vertical, the vertical components of the motion are large enough to create a big nonlinear zone close to the free surface. As for vertical incidence, this zone is wider but has sm aller intensity for a longer pulse (Fig. 7.8b1) and it is narrower and has greater intensity for a shorter pulse (Fig.7.8b2). The curves for the average displacem ent at the contacts are shown in Figure 7.6b. The displacem ent for the pulse rj = 0.6 is shown in figure 7.9. Because of the nonlinear zones in the soil around the foundation, for nonlinear case the resistance of the soil to the foundation motion is weaker, and for all three cases the contact s-f has greater average displacem ent than for the linear case. On the other hand, while for the stiffest foundation there is no substantial difference at the contact b-f between the linear and nonlinear case, for a softer foundation the displacem ent for the nonlinear case is larger in the entire frequency range. From Fig.7.9 it can be seen that after the yielding of the soil around the foundation the stiffest foundation moves as a rigid body, while in the softer foundations the part of the pulse entering the foundation causes elastic deformations. 2 0 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Fig. 7.9 P = 2 5 0 m /s to o DISPLACEM ENT IN THE SOIL AFTER THE PULSE HAS PASSED THE FOUNDATION In Figures 7.8c1 and 7.8c2, the permanent strain for incident angle y - 6 0 ° is shown, where the component s^ls dominant. Some negligible permanent strain in the y direction can be seen close to the bottom left corner for short pulses and the stiffest foundation (Fig. 7.8c2). In front of the foundation, the nonlinear zone starts from some depth in the soil, depending upon the duration of the pulse, where the permanent strain is the smallest. Advancing to the free surface, the strain magnitude increases, and it is largest at the free surface. For longer pulses (Fig.7.8c1), the zone behind the foundation is nonlinear close to the free surface, with approximately the same thickness and magnitude of strain as in front of the foundation and being the largest close to the right top corners of the foundation. Again this strain concentration can be attributed to the effects of elastic forces, as was shown for the sim ple disk example in the previous chapter. As the duration of the pulse becomes shorter (Fig.7.8c2), for the softer foundations the zone behind the foundation has stronger nonlinearity than the zone in front of the foundation. Also, the softest foundation has the strongest noniinearity behind the foundation for all three considered foundation stiffnesses. This results from the elastic forces at the right end of the contact b-f, which are greatest for the softest foundation and more pronounced for short pulses. What is interesting to note here is that, for the stiffest foundation, behind the foundation the soil remains linear close to the free surface, but a nonlinear zone occurs at some depth. This can be attributed to the diffracted field around the foundation. The stiffer the foundation, the longer is the diffracted field. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The average am plitudes at the contacts a (Fig.7.6c), for nonlinear case are sm aller due to the immediate appearance of permanent strain and the loss of energy of the incoming waves close to the free surface. As for the average amplitudes, what was stated for the case with y = 6 0 ° can be said for y = 85° (Fig.7.6d). The difference between the linear and nonlinear cases is greater for this incidence. In Figure 7.8d it can be seen that the nonlinear zone behind the foundation and close to the free surface appears only for the softest foundation, while for stiffer foundations it is shifted deeper. In Figures 7.7b,c,d the plots of the average displacem ent at the contacts are shown for the case of large nonlinearity C = 0.7 and for ^ = 3 0 ° ,6 0 ° ,8 5 ° together with the corresponding displacem ents for the linear case. Depending upon the angle of incidence, one or both components of the permanent strain appear until the pulse reaches the free surface or the foundation. Everything that was said for intermediate nonlinearity is applicable to this case, except that the differences between the nonlinear and linear cases are greater. Finally, to illustrate attenuation of the pulse due to the energy dissipation from development of permanent strains, Figure 7.10 shows the displacem ent for (3f - 3 0 0 m /s and rt = 1 for different incident angles, at the instant when the pulse has com pletely passed the right corner of the building-foundation-soil contact. The plots are normalized with respect to the half-space am plitude of the pulse. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.10 DISPLACEMENT I N THE SOIL AFTER THE PULSE H A S PASSED THE 206 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER VIII SUMMARY The numerical sim ulation of wave propagation was used to explore the two-dim ensional soil-foundation-structure interaction with a flexible foundation for incident SH-waves. The finite differences numerical method (Lax-Wendroff) was used to solve the wave equation. First, for steady-state, monochromatic incident plane waves, the solution for the foundation motion and the relative displacem ent of the building in terms of the input frequency were studied for the Hollywood Storage building for two angles of incidence and two foundation stiffnesses. The results for the foundation motion were illustrated at three points (left, middle, and right) at the contact of the building and the foundation in order to study the torsional effects. It was shown that relative to the case in which the foundation is absolutely rigid the displacem ent am plitudes for the flexible foundation are generally larger, except for frequencies close to the natural frequencies of the building. The motion at the right end and at the m iddle are generally larger than the motion at the left, as was explained in the Chapter VI. In Chapter V the distribution of the energy for the linear case was considered for two different buildings, the Hollywood Storage building (tall and "rig id ” ) and the Holiday Inn hotel (intermediate height and "soft"). It was shown that the former building attracts more energy and radiates it faster. The plots of energy distribution were illustrated for four different dim ensionless frequencies and for the angle of incidence y = 3 0 ° . It was shown that the input energy reaching the foundation does not depend very much upon the angle of incidence. In Table 4, the 2 0 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. percentage of the maximum energy in the building (radiated) energy and the scattered energy from the foundation were given for both buildings. It was shown that the energy distribution does not depend upon the frequency of the input motion, but rather on the bu ild in g ’s properties. In Chapter VI, average displacem ents and average velocities at the building-foundation and soil-foundation contacts as functions of the dim ensionless frequency were presented. The displacem ents at the left and the right corners of the building-foundation contact as functions of the dim ensionless frequency were also given. These results are important for the designing of earthquake-resistant structures based on the power of incident strong motion pulses. Finally, in Chapter VII, the response of the model with nonlinear soil was studied by considering energy, permanent strain distribution, and the average displacem ents at the contacts. It was shown that normaly the total permanent strain is developed during the wave passage through the soil, although for some angles of incidence in a sm all zone close to the contact large permanent strains in the y direction can develop. The permanent strain in the x direction occurs for larger angles of incidence (closer to the horizontal incidence), and it appears in a zone close to the surface, including the free surface, while the permanent strain in the y direction occurs for sm aller angles of incidence (closer to the vertical incidence), and it appears in a zone at some depth from the free surface. It was shown that the width of the zone and the intensity of the permanent strain are inversely proportional. For longer pulses, the zone is larger with sm aller maximum permanent strains, and for shorter pulses the trend is the opposite. 208 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The numerical sim ulation of the wave propagation is powerful tool for the study of all aspects of soil-structure interaction. Here, we assumed that at the contacts there is no sliding. The next step in this research can be to explore more realistic models where the points on the soil-foundation contact can slide and separate. Further, the nonlinear model can be extended by including nonlinearity in the building and in the foundation. In the future, the soil-structure interaction should be studied in 3D models allowing sliding and gaps at the contacts. Finally, by introducing more “ structures" the response of the structure-soil-structure and of bridge structures can be investigated. Here we studied the response for a sim ple elementary input. This model can be used to study the soil-structure interaction for arbitrary input w = w {t), including real seismograms. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY 1. Abdel-Ghaffar, A.M., & Trifunac M.D. (1977). Anti plane dynam ic soil-bridge interaction for incident plane SH-waves, Proc. 6th World Conf. on Earthquake Ena.. V o l.il. New Delhi, India. 2. Aki, K., & Richards, P. (1980). Quantitative seism ology, theory and methods. (Publication): W.H. Freeman & Co. 3. Alford, R.M., Kelly, K.R., & Boore D.M. (1974). Accuracy of finite-difference modeling of the acoustic wave equation. Geophysics 3 9 . 834 - 842. 4. Alterman, Z. & Caral, F.C. (1968). Propagation of elastic waves in layered media by finite difference methods. Bull. Seism. Soc. of Amer.. 58 (1), 367 - 398. 5. Aviles, J., Suarez, M., & Sanchez-Sesma, F.J. (2002). Effects of wave passage on the relevant dynamic properties of structures with flexible foundation. Earthq. Ena, and Struct. Dynamics, 31, 1 3 9 - 1 5 9 . 6. Bayliss, A., & Turkel, E. (1980). Radiation boundary conditions for wave-like equations, Comm. Pure and Appl. Math. 3 3 . 707 - 725. 7. Blume and Assoc. (1973). 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Geophysics, 43 (6), 1 0 9 9 - 1110. 48. Smith, G.D. (1985). Numerical Solution of Partial Differential Equations. Finite Difference M ethods. Oxford :Clarendon Press. 49. Smith, W.D. (1974). A non-reflecting plane boundary for wave propagation problems. Journal of Computational Physics. 15, 492-503. 50. Sochacki, J., Kubichek, R„ George, J „ Fletcher, W.R. & Sm ithson S. (1987). Absorbing boundary conditions and surface waves. Geophysics 5 2 , 6 0 - 7 1 . 213 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 Sod, G. (1985). Numerical Methods in Fluid Dvnamics.Cambridae. UK Univ. Press. 52. Todorovska, M.I., Hayir, A., & Trifunac, M.D. (2001). Antiplane response of a dike on flexible embedded foundation to incident SH-waves. Soil Dvnam. and Earthq. Ena. 21, 5 9 3 -6 0 1 . 53. Trifunac, M.D. (1971). Zero baseline correction of strong-m otion accelerograms. Bull. Seism. Soc. of America. 61 (5) 1201-1211. 54. Trifunac, M.D. (1972). Interaction of a shear wall with the soil for incident plane SH waves. Bull. Seism. Soc. of America, 62 (1), 63 - 83. 55. Trifunac, M.D. (2003). 70th Anniversary of Biot Spectrum, 23rd Annual Lecture, Indian Society of Earthquake Technology Journal, Paper 431, Vol. 40, N o.1,1 9 -5 0 . 56. Trifunac, M.D., & Todorovska, M .l. (1997). Response spectra and differential motion of columns. Earthquake Eng, and Structural Dvn., 26, (2), 251-268. 57. Trifunac, M.D., Hao, T.Y., & Todorovska, M.l. (2001). On energy flow in earthquaae response. Dept, of Civil Eng., Rep. 01-03, Univ. of Southern California, Los Angeles, California. 58. Tsynkov, S.V. (1998). Numerical solution of problems on unbounded domains. A review. Applied Numerical Mathematics 2 7 , 465 - 532. 59. Vidale, J.E.& Clayton, R.W. (1986). A stable free surface boundary condition for two- dim ensional Elastic Finite Difference Wave Simulation, Geophysics 5 1 . 2247-2249. 60. Virieux, J. (1984). SH-wave propagation in heterogeneous media: Velocity-stress finite difference method, Geophysics 51. 889 - 901. 61. Wang, Y., Xu, J., & Schuster, G.T. (2001). Viscoelastic wave sim ulation in basins by a variable-grid finite-difference m ethod. Bull. Seism. Soc. of Am. 9 1 ,1741 -1 7 4 9 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62. Westermo, B.D. & Wong, H.L. (1977). On the fundamental differences of three basic soil-structure interaction m odels. Proc. 6th World Conf. of Eart. Eng., Vol.II, New Delhi, India. 63. Wong, H.L. & Trifunac, M.D. (1974). Interaction of a shear wall with the soil for incident plane SH waves : Elliptical rigid foundation. Bull. Seism. Soc. of America. 64 (6), 1825 - 1884. 64. Wong, H.L. & Trifunac, M.D. (1975). Two-dim ensional antiplane, building-soil- building interaction for two or more buildings and for incident plane SH waves. Bull. Seism. Soc. of America, 65 (6), 1863 -1 8 8 5 . 65. Zahradnik, J. & Urban, L. (1984). Effect of a sim ple mountain range on underground seism ic motion. Geophys. J.R.astr.Soc. 79, pp.167-183. 66, Zahradnik, J., Moczo, P. & Hron F. (1993). Testing four elastic finite-difference schemes for behavior at discontinuities, Bull. Seism. Soc. of Am erica 83.1 07 -1 2 9. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX I FINITE DIFFERENCE FORMULAE FOR CHARACTERISTIC POINTS Introduction For A, A1 f A2l B and (Fig.2.4) all members of the vector U are computed sim ultaneously. To provide continuity of stresses and strains at the contact, at points C to H the stresses and strains are updated from the displacem ents, after the velocities and the displacem ents in all of the grid points have been computed. The displacem ents in all grid points are computed from the velocities using the rectangular quadrature rule: Points A , At, A2 l W xi+l/2,;'(Vi+ (1.1 A) (p^+W _ 2 c r l ; + a *-i,j)+ (I.2A) 8 A x A yp tJ (< IP” yi+lj+l ~ Gyi-W+l + a yi-l,j-l ) 216 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. * 2 \LO n> 8AxAyPl . ^ + 1 ’;+ 1 Pomfs B , B1 A typical cell of point B (B ^ has 7/8 of its area with material properties of soil (foundation) and 1/8 with material properties of foundation (soil). To avoid this heterogeneity, the cell is modified as shown in Fig.2.4. In this way, a uniform, square cell is obtained with lengths d ^ = db c = d c d = d d a = 2A x'= V2Ax => A x '= ^ = . The stability condition for this V2 cell is / L x A* = ^ l _ /?m a x Ar ^ l Ax' Ax V2 Ax 2 where /?m ax = m a x(fif ,p t ). The condition (1.1 B) is usually critical for obtaining At when J 3 m a x and A x are prescribed. The points labeled with small letters in Fig.2.4 are existing grid points, while the quantities at the m idpoints of the cell sides are computed as mean values from the surrounding grid points. For example, if we want to compute U at point B (x ; , ^ J .) in the soil, we can use 217 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R -- Rt + R. -f R Ra b = i2 ,-_i/2j - 1/2 = ’l :1,J------ — -------------- , and sim ilarly for Rb c and Rad. At the contact m idpoint Rc d = ^ cd ~ i+ \!2 ,j+ \/2 The quantities R are velocities, radial and circular stresses, and radial and circular strains. To obtain the radial and circular components of the stresses at existing points, the orthogonal transformation is used, as follows: cos # sin# - sin # cos # I cr„ (I.2B) UJ where the angle # 1 7 F / 4 for x < 0 I 3tt / 4 for x > 0 The formulae for com puting the quantities in vector U are the same as for the usual Ax uniform cell A, so the formulae (1.1 A)-(I.3A) are used with the substitutions A x '= -= for V 2 A x and isy and of course, a x ,,a y, for a x,a y. In this way, the velocities and the strains in the radial and in circular directions are obtained. The final step is to get back to the strains in Cartesian coordinates: cos# - sin # sin # cos # *y. hj 218 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Points C - horizontal contact soil-foundation: lc+1 k . A t ( k k k A r \ A / V — V H ----------------— (7 + (T — (7 H --------------— 4 A x p . . '+1,;'+0 Xi~1 ’J+0 « + U -0 S -U '-O / y p b l J + l ~ K ,}-1) + • k > l / 2,;+o(vf+U - v 5 ) - p 1 1/2j;+0(v 5 At + 4 /7 ,.; A x The strains are continuous in the x direction and the stresses in y direction are: wk + y ~ wk + ' 0 k+l __ k+\ __ c*^+1 / t ^ — / / J ~ 2Ax **V/+° “ r f * x i , j u xi,j- 0 “ M s^xij k+\ yU j MsMf w , t + 1 u + l ■w ^+1 Ms + P f Ay Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Points D - horizontal structure-foundation contact: vk + 7 = vk.+ > ,j ‘,j At 2AxPij At ( k 4>t (< k k & x i+ \,j+ § -W +0) + A v 4 .(< A&xi+lj'-O ®xi -u -» y Ayb + 4y, + (Ays +Ayb)ph] yiJ+l (a k - a k )+ v yi,j+i yu-i / A f2 2P i]Ax Ayb + Ay + ■ A r2 ■ . hy\ k „nj-oku - A ) - (A - <u )]+ 2Pi j Ax Ayb + Ay + ■ At7 P ll(Ayb+Ays) (1-1D) The strains are continuous in the x direction and the stresses in y direction: w - w P k +1 _ W M . J W i - U ,_<r+l _ _*+l *+ 1 _ u p k +1 x iJ ~ 2 A x ° x ij+ o - M f b x iJ ° x i,i- 0 - M s b xi,j a k+ l = ------------- U ~w k. A PbAys + P f Ayb ^ 2 2 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Points E - vertical soil-foundation contact k+l k . At ( k k k k \ At v. . — v. . h ----- ■ — -—\< 7 - a +a - a H ----------- 1 , 1 'J 4 A yp. yi+oj+i yi+o,j-\ yi-oj+\ / 2 A x/? ) + ^ ' k > 0 ,y +l/2 f e +l - ) - 0.y-l/2 (< ; ~ < - 1 + ■ A / r J A / 2 2 + A / T 1 " ’ k - ° . 2 ' + 1 /2 f e + 1 ~ V * / ) “ P k y i- 0 , J - U 2 f c - V , v - 1 ) ] + 4/? Ay 2pu Ax^ • k + u v k u - v 5 ) - ^ - t / v k y - v f - u ) ] The strains are continuous in the y direction and the stresses in x direction are: jus + p f A x For example, ifx < 0, then w * + 1 - w fc + 1 t + l _ w i J +1 ^ J - l fc+1 _ t+1 fc+1 _ * +l ~ 2Ax ^+°./ " r*f yi,i yi-0,j ~ r s b y i j ■ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Points F - slant soil-foundation contact: As for point B, the cell is modified so that one half of the m odified cell has properties of the soil and another one half has the properties of the foundation. L k - c r k ) + \nk (vk - v k ) - u k (vk - vk )1 + \ ° X 3 4 * '1 2 / ~ A 2 \ry'F \A + \ V 14 v i , j ) Py'F23+ \ v i,j y2 3 / r 2 p u Ax + - A A t2 2 Pi j Ax p u A x (I.1F) The stresses and the strains are obtained directly in the xOy system. For xF < o , W 3 - W u _ W U J- W } _ W 4 ~ W tJ w - w 2 0 * = M s Ax 222 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Points G , G , - comers of the hexagon: For example, for point G, x = - a , as follows: 4AyPiJ K ? = vl + V ^ r ( a y l o,,+i +cry l ojh + a ypp + a w i ~ a ^ J + At 4Ax p t • ( 2 f 7 x L , y + a l p p + ^ x p m - ~ - (JLp) + A t2 4 a .A y 2 V , m' wu M, ~ < J ] + - > * * 1 A / 2 4p u Ay2 A t2 2pu Ax2 where: (I.1 G ) G ypp n V f 1_ 2 ‘ w , — w. . w. , . — w. , . , 1,7+1 i,; + _ 1+1,; >+1 ,7 - * A v , A y , a W , —W ■ , i , j+ \ i,J ~ 1 c r = c r = u yntp ym m 2 A y a xpp=2Mf W< + U + 1 + w )+u - W, ^ wMj -w u 2Ax Ax (I.2G) cr xpm Vxpp+Ps W M J _, - w ^ Ar 223 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. After obtaining velocities and displacem ents at all grid points, the stresses are obtained from the formulae (I.2G). The procedure is sim ilar for the points G' at the bottom ( y = -a ) . Points H - comers on the soil-foundation-structure boundary: There are two points of this kind in the model (x = ±a,y = o). We consider the left point (- a, 0). The boundary conditions for the cell are = Vyi-OJ = 0 <*xu+ 1 = = o (1.1 H) To customize the cell for the numerical scheme, and having the boundary conditions (1.1 H) with linear extrapolation in the fictitious points, we have: Point (i-1 ,j+ 1 ): < j y t - i j - i ' (1.1 Ha) Point (i-1 ,j+ 0 ): a (1.1 Hb) (1.1 He) 224 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. At 2(Ayb+Ays)pu '^Vi+OJ+l + Crj'/-0,;+l +Cr^ / ( < r * + 0 , ; - l + ^ + C T ^pm , + - A / I * \1 . A / 2 ‘° V + CTxpO ~ ° x m l ~ a xmO J J + 2Axp hj 4pu Ayz k.>o,/+ i/2f e + i - vu)-ti+0J-iMj -< ;m )]+ ~ < J 1 + k - v ' , J A/2 4p. .A y A /2 2/7. .Ax2 (I.2H) with: _ Ar _ _ A r (7 — cr > ? > » * M/Mb p f Ayb + p bAys (wfi+j - from the continuity of the stresses in the direction from (1.1 H), l c r xpl A y + A y -(&>,<* * + u-0 + A v 6o-ja .+li/+0) where a . j - 0 = — ( t ■ x _0 from the continuity of the strains in the structure-foundation contact, » + .;+ ^ » + ,, 1 ^xp O Ays + A y - ( A y , ^ +0,;-0 + A > '^ x(+o,;+o ) : crxm0, where °"x!+0J+0 0 from (1.1 H) O' xm l 1 A y + A y ■ (A v,o- b --u-0 +Av6 o -ri_ w +0), where 225 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( ^ x i - i j + o ~ G ’ x i+ ij+ o ■ (1.1 H b ) And from the continuity of stresses in the soil-foundation contact we have: 2 2 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX II INPUT ENERGY FOR THE STRUCTURE From the ray theory, and according to Fig.II, two types of rays reach the foundation and are relevant in the energy distribution: • rays i reach the foundation directly without reflection from the free surface • rays r reach the foundation after reflecting from the free surface The energy brought to the foundation by rays i is dAj is the projection of the differential area of the sem icircular foundation in the direction of propagation of the ray i: dAi -a -d d - cos a . Substituting dA, in (11.1), we have ( 1 1. 1 ) A where: p s is the soil density, J 3 s is the velocity of propagation of the SH wave in the soil v is the particle velocity of the incom ing plane wave, and cos a (II.2) 0 0 where T is the time at the end of the analysis. 2 2 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INPUT ENERG Y REACHING T H E FOUNDATION o Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From geom etry it can be readily seen that a ■ ■ n - y - 0 . and because the cosine is an even function, we can om it the absolute sign. Also 0x= n - y and (11.2) becomes x-y r ”l i 7t~y T E t = aps(5s J cos - - ( / + &/) d6i J v2 dt = apsfis js in ( / + 6i )d61 j v2d t , (11.3) with the solution: i E, = a p sfis-(l + cosr)jv2dt. (11.4) In a sim ilar way, for the energy brought to the foundation by rays r having 8 = + 6 and 02 = y and T is big enough so that the reflected pulse pass the foundation, the integrals in time are equal: i E r = apsp s (l - c o s f)Jv2d t . (11.5) Adding (11.5) to (11.4), the total energy brought to the structure is E t o t = 2 a p j s\ v 2dt. (11.6) It can be seen from (11.6) that the energy reaching the semi circular foundation does not depend upon the incident angle. Also, it is linearly proportional to the diam eter of the foundation, the density, and the shear wave velocity of the soil. 229 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We note finally that for both buildings studied in this thesis, the energy brought to the foundation for the same 7 is the same. This conclusion is straightforward from the definition of 7 7 : 7 7 = — = - ^ — . (11.7) A M o The particle velocity from the definition of the half-sine pulse (5.4) is: v = 7E £C0S^ . (11.8) td O td o Substituting (11.8) in (11.6) and performing the integration, we have n 2 a 2 Eto t = 2 apsPs . M ultiplying the numerator and the denom inator with ^ k e e p in g in 2td0 mind (11.7), the total input energy for the foundation is Etot= n 2A2 p 2 sPsr i . (11.9) It may seem that this equation has the dim ension of force, but we should keep in mind that the third dim ension (length) of the model is taken equal to unity and is therefore omitted in (11.9). 2 3 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Creator Gicev, Vlado (author)

Core Title Investigation of soil-flexible foundation-structure interaction for incident plane SH waves

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School Graduate School

Degree Doctor of Philosophy

Degree Program Civil Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, civil,OAI-PMH Harvest

Language English

Advisor Trifunac, Mihailo (committee chair), Lee, Vincent W. (committee member), Proskurowski, Wlodek (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-331672

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