text:
This action might not be possible to undo. Are you sure you want to continue?
Prentice-Hall International Series in Civil Engineering and Engineering Mechanics
William J. Hall, Editor
Electrowatt Engineering Services Ltd.
PRENTICE-HALL, INC., Englewood Cliffs, N.J. 07632
Library of Congress Cataloguing in Publication Data
WOLF, JOHN P. (date) Dynamic soil-structure interaction.
(Prentice-Hall international series in civil engineering and engineering mechanics) Bibliography: p. Includes index. 1. Soil dynamics. 2. Structural dynamics. 3. Foundations. 4. Earthquake engineering. I. Title. II. Series: Prentice-Hall civil engineering and engineering mechanics series. TA710.W59 1985 624.1'5136 83-25477 ISBN 0-13-221565-9
Editorial/production supervision and interior design: Theresa A. Soler Manufacturing buyer: Anthony Caruso
1985 by Prentice-Hall, Inc., Englewood Cliffs, New Jersey 07632
All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher.
Printed in the United States of America
10 9 8 7 6 5 4 3 2
Prentice-Hall International, Inc., London Prentice-Hall of Australia Pty. Limited, Sydney Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro Prentice-Hall Canada, Inc., Toronto Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo Prentice-Hall of Southeast Asia Pte. Ltd., Singapore Whitehall Books Limited, Wellington, New Zealand
1.1 1.2 1.3 1.4 1.5 1.6 1.7
Objective of Soil-Structure Interaction Analysis Types of Prescribed Loadings, in Particular Seismic Excitation 2 Effects of Soil-Structure Interaction 4 Assumed Linearity 7 Types of Problems Encompassed 8 Direct and Substructure Methods 9 Organization of Text 11 Summary 11
2.1 2.2 2.3 2.4 2.5 2.6
Equations of Motion in the Time Domain 13 Transformation to Modal Amplitudes 14 Equations of Motion for Harmonic Excitation 14 Correspondence Principle 15 Discrete Fourier Transform 16 Method of Complex Response 17
Formulation in Total Displacements
3.1.1 3.1.2 3.1.3 Flexible Base, 18 Rigid Base, 23 Special Foundations, 26
Kinematic and Inertial Interactions
3.2.1 3.2.2 Flexible Base, 28 Rigid Base, 30
Applied Loads and Their Characterization
3.3.1 Rotating Machinery, 33 3.3.2 Impact, 33 3.3.3 Earthquake, 34
Introductory Example
3.4.1 Statement of Problem, 38 3.4.2 Equations of Motion of Coupled System, 42 3.4.3 Equivalent One-Degree-of-Freedom System, 43 3.4.4 Dimensionless Parameters, 45 3.4.5 Parametric Study, 46
Summary Problems
50 52
General Considerations
4.1.1 4.1.2
Frequency Content and Spatial Variation of Applied Loads, 70 Decoupling of Subsystem, 71
Spatial Variation of Seismic Loads
4.2.1 4.2.2
Free-Field Motion, 72 Kinematic Motion of Surface Structure with Rigid Base, 74 4.2.3 Kinematic Motion of Embedded Structure with Rigid Base, 76 4.2.4 Kinematic Motion of Structure with Flexible Base, 76 4.2.5 Axisymmetric Structure, 76 4.2.6 Example, 79
Direct Discretization
4.3.1 Finite-Element Model, 83 4.3.2 Models for Impact Load, 84 4.3.3 Hyperbolic Cooling Tower for Seismic Load, 88 4.3.4 Nuclear Structures for Seismic Load, 90 4.3.5 Frames with Shear Panels for Seismic Load, 93 4.3.6 Models Based on Generalized Displacements, 94
Reduction of Number of Dynamic Degrees of Freedom 95
4.4.1 Transformed Equations of Motion, 95 4.4.2 Mass Lumping Followed by Static Condensation, 96 4.4.3 Substructure-Mode Synthesis, 98
Summary Problems
101 102
One-Dimensional Wave Equation
5.1.1 Significance of Wave Propagation, 114 5.1.2 Statement of Problem, 114 5.1.3 Equation of Motion, 116 5.1.4 Types of Waves, 117 5.1.5 Dynamic-Stiffness Matrix of Finite Rod, 118 5.1.6 Dynamic-Stiffness Coefficient of Infinite Rod, 120 5.1.7 Rate of Energy Transmission, 123 5.1.8 Material Damping, 123 5.1.9 Convergence of Dynamic-Stiffness Coefficient of Finite Rod to That of Infinite Rod, 128 5.1.10 Free-Field Response, 131 5.1.11 Dynamic-Stiffness Coefficient of Site, 132
Three-Dimensional Wave Equation in Cartesian Coordinates 133
5.2.1 Equation of Motion in Volumetric Strain and in Rotation Strains, 133 5.2.2 P-Wave,136 5.2.3 S-Wave,137 5.2.4 Material Damping, 139 5.2.5 Total Motion, 139
Dynamic-Stiffness Matrix for Out-of-Plane Motion
Types of Waves, 140 Transfer- and Dynamic-Stiffness Matrices of Layer and of Half-Space, 141 5.3.3 Special Cases, 143 5.3.4 Loaded Layer, 144 5.3.5 Rate of Energy Transmission, 146 5.3.1 5.3.2
Dynamic-Stiffness Matrix for In-Plane Motion
5.4.1 5.4.2 Types of Waves, 146 Transfer- and Dynamic-Stiffness Matrices of Layer and of Half-Space, 148 5.4.3 Special Cases, 151 5.4.4 Loaded Layer, 153
5.4.5 Rate of Energy Transmission, 156
Three-Dimensional Wave Equation in Cylindrical Coordinates 156
5.5.1 Equation of Motion in Volumetric Strain and in Rotation Strains, 157 5.5.2 Solution Using Fourier Series Circumferentially and Bessel Function Radially, 159 5.5.3 Dynamic-Stiffness Matrix, 164
Summary Problems
164 166
Definition of Task
6.1.1 Three Aspects When Determining Seismic Environment, 179 6.1.2 Location of Control Point, 180
Amplification, Dispersion, and Attenuation
6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 Dynamic-Stiffness Matrix of Site, 182 Site Amplification for Body Waves, 183 Surface Waves, 185 Amplifications, 186 Apparent Velocity and Decay Factors, 187
6.3.1 6.3.2 6.3.3 6.3.4 6.3.5
Incident SH-Waves, 189 Incident P-Waves, 189 Incident SV-Waves, 191 Rayleigh Waves, 192 Displacements and Stresses versus Depth, 193
Single Layer on Half-Space
6.4.1 6.4.2 6.4.3 6.4.4 6.4.5
SH-Waves, 195 Love Waves, 198 Physical Interpretation of Variables,200 P- and SV-Waves, 202 Rayleigh Waves, 203
Parametric Study of Out-of-Plane Motion
6.5.1 Scope of Investigation, 204 6.5.2 Vertically Incident SH-Waves, 205 6.5.3 Inclined SH-Waves, 207
Amplification Within, 207 Amplification Outcropping, 207
6.5.4 Love Waves, 210
Dispersion and Attenuation, 210 Displacements and Stresses versus Depth, 214 Scattered Motion, 218
251 In-Plane Displacements and Stresses versus Depth. 275 7. 282 7.2.1 Description of Site and of Control Motion.and SV-Waves.6. 273 7. 257 6.6 6.5 6. 241 6.8. 292 7.and SV-Waves. MODELING OF SOIL 273 7.7.and SY-Waves. 281 7.1 Weighted Residual Formulation.2 Green's Influence Function for Two-Dimensional Case.8. 219 Vertically Incident SV.7 Soft Site 242 6. 238 Scattered Motion.5 Nondimensionalized Spring and Damping Coefficients. 275 7.6. 221 Incident SY-Waves.5 Scattered Motion.22.214.171.124. 244 Vertically Incident and Inclined P.8 Rock Site 259 6. 224 Combinations of Incident P.1 6. 261 6. 232 Displacements and Stresses versus Depth. 228 6.5 Sommerfeld's Radiation Condition.2.7.6 Use of Analytical Solution.1 6. 259 6. 253 Scattered Motion.7 Description of Site and of Control Motion.2 6. 263 6. 232 Dispersion and Attenuation.3 6.6 Parametric Study of In-Plane Motion 6.1.7. 249 Rayleigh Waves. 285 7.4 6.2.2. 265 Summary 266 Problems 271 7.4 Assumed Wave Patterns.and P-Waves.Contents ix 6.4 Rayleigh Waves.1.2 Elementary Boundaries.3 Green's Influence Function for Axisymmetric Case.1 General Considerations 273 7.2. 259 6.2 Love Waves.1 Dynamic-Stiffness Matrices of Soil.3 Local Boundaries.7.2 6.7.4 Element Size and Numerical Integration.1.8.3 Rayleigh Waves.6.4 Consistent Boundaries.3 219 Scope of Investigation. 242 Vertically Incident and Inclined SH-Waves.8. 279 7.7. 287 7.2 Dynamic-Stiffness Coefficients of Surface Foundation 282 7.1. 279 7. 221 Incident P-Waves.1. 245 Love Waves. 293 . 220 Inclined P.
321 7.1 Significance of Boundary-Element Method. 301 Two-Dimensional Versus Three-Dimensional Modeling.1. 334 7.1 7.5 Pile Foundation. 369 Equation of Motion in Frequency Domain. 373 Base Response Motion Relative to Free Field.1 Direct Analysis of Total Structure-Soil System 369 8.4. 329 7.4. 313 Analytical Solution of Illustrative Example.2 Example to Illustrate Basic Concepts of Boundary-Element Method.1 Scope of Investigation.184.108.40.206 Green's Influence Functions. 321 System Free Field.2.2 . 318 Reference Soil System.6.5.5. 312 7. 375 8. 325 7. 318 Generalization to Three Dimensions for System Ground. 319 System Excavated Part.7 Dynamic-Stiffness Coefficients of Adjacent Foundations 336 Summary 340 Problems 343 369 8. 333 7.2 Green's Influence Function.6 Embedded Rectangular Foundation 327 7. 313 Weighted-Residual Technique.2 Substructure Analysis with Flexible Base 373 8.1 8.4. 371 Basic Equation of Motion in Total Displacements.4 Properties of Dynamic-Stiffness Coefficients of Excavated Part and System Free Field. ALTERNATIVE FORMULATION OF EQUATION OF MOTION 8.3 Dynamic-Stiffness Coefficients of Soil Domains Calculated by Boundary-Element Method.3 Dynamic-Stiffness Coefficients.6. 316 188.8.131.52. 328 7.2 Equation of Motion in Time Domain. 327 7.2 7. 307 Scattered Motion. 322 7. 329 7.1.4 Two-Dimensional Rigid Basemat (Strip Foundation) 293 Three-Dimensional Rigid Basemat (Disk Foundation) 301 7.1 8.3 Complete Set of Results.6.6. 310 7.3 7.5 Dynamic-Stiffness Coefficients of Embedded Foundation 312 7.5 Parametric Study System Ground.x Contents 7.6 Parametric Study System Free Field.
3.1 Evaluation of Interaction Effects 405 9. 387 Quasi-static Transmission of Base Response Motion.3.5 Quasi-static Transmission of Free-Field Input Motion.4. 380 379 8.2 Inclined SH-Waves. 393 8.1. 423 .4 Quasi-static Transmission of Base Response Motion.3 Transformation of Stiffness Matrix.2 8. 384 8. 389 8.4. 411 9. ENGINEERING APPLICATIONS 405 9. 384 8. 415 9.3 Quasi-static Transmission of Scattered Motion. 395 Summary 396 Problems 398 9. 391 8.4 8.3 8.2 Equation of Motion in Time Domain Using Convolution Integrals.5 Bridge Structure. 386 8.1.Contents xi 8.3 Depth of Layer.4.3 Inclined P.1 Dimensionless Parameters.1.and SV-Waves.3.1 8.4 Rod with Exponentially Increasing Area. 414 9.3.1 Basic Equation of Motion in Total Displacements.5 Disk on Half-Space and on Layer.2. 407 9. 405 9. 409 9.1 Types of Nonlinearities.5.2 Effects of Horizontally Propagating Waves 412 9.1.4 Approximate Formulation in Time Domain 8. 392 8.1 Investigated Structures.5. 388 Transformation to Modal Amplitudes of Fixed-Base Structure. 408 9. 388 Transformation to Modal Amplitudes of Total System. Transformation to Modal Amplitudes of Fixed-Base Structure.2.2 Equivalent One-Degree-of-Freedom System.2. 406 9.6 Second Mode.5. 390 8.2.4 387 Basic Equation of Motion in Total Displacements. 419 9.1.5 Analysis of Nonlinear Structure with Linear Soil (Far Field) 390 8.4. 386 8.3.4 Rayleigh Waves. 376 Quasi-static Transmission of Base Response Motion.5.2 Base Response Motion Relative to Scattered Motion.3 Substructure Analysis with Rigid Base 383 8.3 8.5 Transformation to Modal Amplitudes of Fixed-Base Structure.126.96.36.199. 383 8.4 Mass of Base.2.
4 Concluding Remarks 9.xii Contents 9.3. 428 9.3. 442 Recorded Field Performance.3 Examples from Actual Practice 9.3.3 Horizontally Propagating Waves on Hyperbolic Cooling Tower.4 Horizontally Propagating Waves on Nuclear Island with Aseismic Bearings. 436 9.3 441 Needfor Adequate Consideration.3.4.1 426 Through-Soil Coupling of Reactor Building and of Reactor-Auxiliary and Fuel-Handling Building.2 9. 441 Modeling Aspects.2 Pile-Soil-Pile Interaction.4. 433 9.4. 426 9.1 9. 443 Summary 447 Problems 450 REFERENCES AND ACKNOWLEDGMENTS INDEX 456 459 .
The effect of soil-structure interaction is recognized to be important and cannot. The structure will interact with the surrounding soil. the presence of the structure in the soil will change the dynamic system from the fixed-base condition. makes use of such familiar concepts as the directstiffness method of structural analysis and also applies such recent developxiii . Even the seismic-design provisions applicable to everyday building structures permit a significant reduction of the equivalent static lateral load compared to that applicable for the fixed-base structure. For the design of critical facilities. In the case of a soft site. The calculation can thus be restricted to the structure excited by this specified motion. This has led to a situation where the analysis of soil-structure interaction has become a highly controversial matter. A uniform approach to analyze both the free-field response as well as the actual interaction analysis is presented in this text.. especially nuclear-power plants. the free-field motion at the site in the absence of the structure is strongly affected. First. leading to a further change of the seismic motion at the base. This text on soil-structure interaction deals with both the free-field response and the actual interaction analysis. two important modifications arise for the same incident seismic waves from the source.PREFACE In the seismic analysis of a structure founded on rock. arising from unbalanced mass in rotating machinery). in general.g. Second. the motion experienced by the base is essentially identical to that occurring in the same point before the structure is built. This rigorous procedure. very complex analyses are required which are based on recent research results. based on wave propagation. be neglected. some of which have not been fully evaluated. Soil-structure interaction is also important for other loading cases (e.
Roesset. Veletsos. The author has been influenced over the years by the research published by many authorities. but which lead to new insights into the various aspects. It is offered for advanced undergraduate and graduate students in civil engineering. Somaini. Sincere apologies are offered to anybody who might feel offended by being left out. D. The results of vast parametric studies are included.xiv Preface ments as the boundary-element method. von Arx. Then problems are formulated which not only allow the student to corroborate full understanding of the analytical techniques. J. As a prerequisite some knowledge of structural dynamics is essential which can be acquired in a one-term course. At the end of each chapter a summary is included. and B. Bucher. As a rule. Examples from actual practice demonstrate that these methods are being applied. P. M. The problems thus form an important part of the text. Weber for their dedicated efforts. in many cases quite important on their own merit. simple approximate methods are developed which nevertheless capture the essential features of soil-structure interaction. for its financial support. P. Obernhuber. J. The reader who has no intention of solving the problems in detail will find it advantageous to glance through them. a detailed solution procedure is provided and the results. Lysmer. the author would sincerely like to thank Messrs. Wolf . K. and A. The important contributions of the author's colleagues and students are gratefully acknowledged. John P. Luco. E. to name just a few (in alphabetical order): Professors E. are presented in the form of figures. Credit references have thus been restricted to those necessary for copyright reasons. which can be used directly by the analyst. Skrikerud. Besides the rigorous procedure. the author is indebted to Electrowatt Engineering Services Ltd. Kausel. G. This allows the analyst to perform preliminary calculations with simplified models to determine the key parameters before starting with complicated computations. G. Many aspects of this field are so well established that it is difficult to assign credit for them. Finally. J. In particular. Darbre. The course on which this text is based has been taught for the past few years at the Swiss Federal Institute of Technology in Zurich. S.
It must also be considered that in many important cases (e. The dimensions of the structure are finite.. and highly developed methods for solving them are readily available. where the response is expected to have died out from a practical point of view. This leads to a finite domain for the soil which can be modeled similarly to the structure. In general. The soil is a semi-infinite medium.1 OBJECTIVE OF SOIL-STRUCTURE INTERACTION ANALYSIS Structural dynamics deals with methods to determine the stresses and displacements ofa structure subjected to dynamic loads. This need to model the unbounded foundation medium properly distinguishes soil dynamics from structural dynamics. For static loading. an unbounded domain. can be introduced. However. The fundamental objective of the analysis of soil-structure interaction is 1 . The fictitious boundary would reflect waves originating from the vibrating structure back into the discretized soil region instead of letting them pass through and propagate toward infinity. The total discretized system. can then be analyzed straightforwardly. however. earthquake excitation) the loading is applied to the soil region around the structure. The corresponding dynamic equations of motion of the discretized structure are then formulated. the structure will interact with the surrounding soil. for dynamic loading. this means that the former has to be modeled anyway. a fictitious boundary at a sufficient distance from the structure. It is thus not permissible to analyze only the structure. consisting of the structure and the soil.g. this procedure cannot be used. It is thus rather straightforward to determine a dynamic model with a finite number of degrees of freedom.INTRODUCTION 1 1.
.1----1 f. A specified time-varying load acts on a structure embedded in layered soil. The procedure normally followed can be characterized as follows: In the control . Earthquakes are caused by a sudden energy release in a volume of rock lying on a fault. Today's state of the art of earthquake engineering allows only the influence of the local site conditions on the seismic input motion to be taken into account.J7- ~ T--~~-+--------4)-- ~------+---------~. The dynamic response of the structure and. and definitely the most complicated to analyze. The discretized model of the structure and of the soil domain is shown schematically.g. blast loadings. not the case). Even if all details of how the source mechanism works and the data of the travel path of the seismic waves to the site were available (which is. the crash of an aircraft onto a nuclear power plant (which can govern the design. of course. of the soil is to be calculated.~ 1/ Figure 1-1 Fundamental objective of analysis of soil-structure interaction. 1. to a lesser extent. illustrated in Fig. The semiinfinite soil domain. Probably the most important loading. 1-1. it would still be impossible to model all aspects because of the size of the dimensions compared to those of the structure..2 Introduction Chap. In any event. taking into account the radiation of energy of the waves propagating into the soil region not included in the model. represented by a layered half-space. and so on. impact loads [e. 1 ~~~~==~~r-- t-----t'r. although the probability of occurrence is small)]. represents an energy sink. IN PARTICULAR SEISMIC EXCITATION Many types of time-varying loads acting directly on the structure can arise: periodic loads originating from rotating machinery in buildings. is earthquake excitation. This source is normally located a large distance away and at a significant depth from the site. the many uncertainties involved make it meaningless to analyze the complete earthquake-excitation problem.2 TYPES OF PRESCRIBED LOADINGS. which acts primarily on the soil.
In this evaluation the type of the structure will play an important role. the local geology. the motion in the horizontal direction is uniform.g. Starting from this control motion in one point. All this should allow the source mechanism.g. based on past earthquakes of the region. which will. i.e. 1. also be influenced by the opinion of the engineering seismologist as described above.g. The seismic motion acting on the structure is thus not known before the soil-structure interaction analysis is performed. To be able to do this. the seismic hazard of the region is assessed. For a potentially hazardous structure such as a nuclear power plant this probability will be selected to be very small. of course.Sec. Quite arbitrary assumptions with far-reaching consequences have to be made. 1-1). These assumptions. such as the duration of the motion and the frequency content. the peak ground acceleration). the transmission path. It is outside the scope of this work to discuss these very important aspects of earthquake engineering in any depth. without the excavation and without the structure). The largest uncertainties arise in these preliminary phases.. a hospital or a fire station). Excavating the soil and building the structure cause changes in the dynamic system. However. Besides influencing the free-field response. thus modifying the seismic motion of the free field. the earthquake motion throughout the free field (characterized by its spatial and temporal variation) is calculated. the motion in the control point at the surface is assumed to arise from vertically propagating waves. To analyze soil-structure interaction.. will affect the characteristics of the free-field response as far as the amplitudes and the frequency content away from the control point are concerned. for seismic excitation. The structure will interact with the soil along the embedment. or at least at a similar site. These activities leading to the definition of a design motion in a selected control point precede the actual soil-structure interaction analysis.. this can again be achieved only by making quite arbitrary and stringent assumptions regarding the wave pattern in the control point. analyzing soil-structure . are selected-mostly empirically-by the engineering seismologist. In summary. for example. in Particular Seismic Excitation 3 point located at the surface of the so-called free field (site prior to construction. all solutions are compatible with the local soil conditions. the variation of the freefield motion with depth can be calculated (Fig. It will be chosen to be somewhat larger but will still have a dominant effect for a structure which has to remain fully operational during an earthquake (e. The structural engineer or the licensing authority specifies the acceptable probability that the earthquake used for design will be exceeded during the life of the structure. and the soil conditions at the site to be taken into account very approximately. the acceleration as a function of time) is specified (Fig. 1-1). the local soil conditions will also affect the actual interaction of the soil-structure system. that which is assumed to characterize the motion (e.2 Types of Prescribed Loadings. The other parameters. the earthquake motion (e.. it is sufficient to think of the prescribed motion as being derived from an observed record at this very site. In this case. If. As will become apparent when discussing the site response. This then allows the determination of the most important parameter.
it can be assumed that in the rock. (e) inertial interaction. the actual interaction. " ". The particle motion is shown in Fig. Only qualitative statements are. 1. when the structure is inserted into the seismic environment of the free field. a horizontal motion that propagates vertically is selected. the incident seismic waves arriving from the source of the earthquake are the same for both structures. I L!:==. and second. 1 interaction consists of two distinct parts: first.. (c) free field. however. As the distance between the two structures is small. of course. (b) outcropping rock. 1-2 as solid arrows having lengths proportional to the earthquake excitation. determining the free-field response of the site. b c E d . possible at this stage. (a) Sites. the dynamic response of a structure founded on rock is compared to that of the same construction. (d) kinematic interaction. The two identical structures with a rigid base (consisting of the basemat and the side walls) are shown in Fig.3 EFFECTS OF SOIL-STRUCTURE INTERACTION To illustrate the salient features of soil-structure interaction. The control point is chosen at the free surface of the rock (point A). I .4 Introduction Chap. The loadings applied directly to the structure are. but embedded in soil.::::!J (c 777777c I. In the following. calculating the modification of the seismic motion. The soil layer rests on top of the rock. contained as special cases in the general formulation.. 0 e \_ . For the sake of simplicity. 1-2a. i i Figure 1-2 Seismic response of structure founded on rock and on soil.--. . mostly seismic excitation will be examined.
1. To gain insight into how the soil affects the dynamic response of the structure. For the structure on the rock. 1-2b would hardly differ from the control motion of the rock in point A. but not always (depending on its frequency content).. This rigidbody motion will result in accelerations (leading to inertial loads) which will vary over the height of the structure." Figure 1-2 also illustrates the main effects of taking soil-structure interaction into consideration. As the rock is very stiff. The rigid base will experience some average horizontal displacement and a rocking component. 1-2c). For the structure founded on soft soil.the motion of the site in the absence of the structure and of any excavation. in contrast to the applied accelerations in the case of a structure founded on rock.~First. resulting in motions in points D and E which differ from that in C. thus resulting in horizontal displacements that increase toward the free surface of the site. Third. 1-2c). a significant rocking component will arise for an embedded structure. an overturning moment and a transverse shear acting at the base will develop. 1-2d). During the earthquake. 1-2c. the motion in this fictitious rock outcrop shown in Fig. This wave will propagate vertically through the soil layer. Second. This geometric averaging of the seismic input motion will be the result of the so-called kinematic-interaction part of the analysis. as discussed later in great detail. the inertial loads applied to the structure will lead to an overturning moment and a transverse shear acting at point 0 (Fig. excavating and inserting the rigid base into the site will modify the motion (Fig. 1-2a)will be different from the control motion in the control point A because of the coupling of the structure-soil system. the translational component will in many cases be larger than the control motion and. the motion of the base of the structure in point 0 (Fig.Sec. Because of the amplification of the site (free-field response). If there were no soil on top of the rock in point C of Fig. the motion is amplified. These will cause deformations in the soil and thus. the seismic-input motion acting on the structuresoil system will change (Fig. 1-2d).g. no rocking motion arises at the base. what is called the free-field response. once again. in point B) will be the same. in addition. The input acceleration resulting in the applied horizontal inertial loads will be constant over the height of the structure. First. the motion throughout the rock (e. For a given control motion. is modified (Fig. 1-2e). Points D and E are nodes in the free field which will subsequently lie on the structure-soil interface (the base) when the structure has been built.3 Effectsof Soil-Structure Interaction 5 From a practical point of view. this (horizontal) motion can be applied directly to the base of the structure. This part of the analysis will henceforth be referred to as "inertial interaction. the seismic response of the structure depends only on the properties of the structure. Each frequency component of the motion is . these two stress resultants will not lead to any (additional) deformation at the base. The presence of the soil layer will reduce the motion in point C (Fig. The resulting horizontal displacement of the base is thus equal to the control motion. modify the motion at the base. In general. it is convenient to distinguish between the three following effects.
it will be negligible for a flexible structure founded on firm soil. For this approximate interaction analysis. The shape of the vibrational mode will also be changed. The displacement at the top of the structure relative to its base may be larger because of the foundation rocking if one takes soil-structure interaction effects into account. resulting in a significantly smaller seismic input "felt" by the structure. There are also some exceptional cases where the simplified interaction effects will govern the design. only the material damping of the soil will act. in general. as. 1-2e). be significantly below that applicable for the fixed-base structure. 1-2c) and the geometric averaging due to the embed- . This could influence the spacings between adjacent buildings and in very extreme cases also increase the second-order effects. 1-2e). In certain cases. In this case. an acceleration time history which is quite different from the control motion. for example. the control motion is directly used as input motion in the final dynamic system (Fig. impossible to determine a priori whether the interaction effects will increase the seismic response. when designing structures. it is possible that no waves propagate away from the structure. 1 affected differently. especially at the top of a tall structure. the presence of the soil in the final dynamic model (Fig. If. it is very important to determine for a specific site whether the loss of energy by radiation of waves can actually take place. the radiation of energy of the propagating waves away from the structure wiIfresult in an increase of the damping of the final dynamic system (Fig. that reduction in seismic forces which results from considering the approximate soil-structure interaction analysis be used. the fundamental frequency will be moved below the range of high seismic excitation. In any soil-structure analysis. On the other hand. Second. Third. for example. It is intuitively plausible that the soil-structure interaction increases the more flexible the soil is and the stiffer the structure is. leading to a strongly reduced response. For a soil site approaching an elastic half-space. however. and thus to a conservative design. It is obvious from the many opposing effects that it is. 1-2d) is neglected when the interaction analysis is performed. the response will in many circumstances be smaller than the fixedbase response obtained from an analysis that neglects all interaction effects. the total overturning moment and the total transverse shear. This amplification of the seismic motion is held responsible for the fact that structures founded on a deep soft-soil sitehave been damaged more severely in actual earthquakes than have neighboring structures founded on rock. resulting in. in general. decreasing the fundamental frequency to a value which will. The fixed-base analysis leads to larger values of the global response. and no beneficial effect on the seismic response is to be expected. Economic considerations normally dictate that.6 Introduction Chap. The introduced rocking of the base will affect the response. It is important to stress that this approximate method of calculating the interaction. Fig. this increase will be significant. For a soil site consisting of a shallow layer. the first effect discussed above (the change of the seismic-input motion. 1-2e)will make the system more flexible. a method that neglects the free-field site analysis (Fig. The implication of this reduction will depend on the frequency content of the seismic-input motion.
The parameters appearing in these relationships can be selected for a particular site. nuclear power plants) which cannot be designed on the basis of the global response alone. which will eventually allow permanent settlements to be calculated. The important aspects of soil-structure interaction can. Varying the important parameters in the analysis should lead to response predictions which will consequently result in a safe design.. are developed. This can be achieved by iteration.4 Assumed Linearity 7 ment effect (Fig. superposition is used. But already when determining the free-field site response by starting from the motion in the control point. however. This means that a linear system is actually calculated. 1. such as the properties of the soil or the conditions of the embedment. be modeled approximately in an appropriate model. possible to select the material properties so that they are compatible with the strains reached during the excitation. possible to model the radiation characteristics of the unbounded foundation domain with the same accuracy as the finite dynamic model of the structure and of the soil represented in the model. soil-structure interaction is always analyzed. It also makes it necessary to estimate additional key parameters. It is true that taking the flexibility of the underlying soil into account when calculating the seismic response of a structure does complicate the analysis considerably. This simplification of performing only a linear analysis should also be seen in connection with the unsatisfactory way of specifying the seismic motion in the control point. which . however. Within the framework of linear analyses. adjusting the material properties at the end of each iteration. can be justified for an important structure. calculating a series of linear systems. considering all effects. It should also be remembered that linear soil-structure interaction has to be fully understood before nonlinear methods. The fact that only linear or such quasi-linear analyses are possible shows how primitive today's state of the art of soil-structure interaction for seismic excitation really is. The results will be physically logical. but will require engineering judgment to evaluate them.4 ASSUMED LINEARITY Soils behave strongly nonlinearly when excited by the quite high earthquake levels of interest to structural engineers. The reason is actually conceptional. The total solution equals the sum of the responses of the free field and of the interaction part when the structure is inserted into the seismic environment of the free field. For special structures (e. 1. however.g. The cost of performing such an analysis. although higher than for a linear calculation. As explained above. it is. Improved procedures should be used even if uncertainties do exist. not yet possible. It is. An exact deterministic solution is indeed impossible. although quite promising nonlinear constitutive models for soil do exist. of course. soil-structure interaction for seismic excitation implicitly assumes the law of superposition to be valid. 1-2d). which are difficult to determine. True nonlinear analyses are.Sec. is inconsistent.
Buried structures (caissons. The structures can be embedded. 1. be inclined and can vary from flexible to rigid. with depth but remain constant within the individual layers. for example. More than one structure can be analyzed at the same time.5 TYPES OF PROBLEMS ENCOMPASSED Although it is not possible to model all details of geometry and material of a practical problem. both consisting of isotropic viscoelastic material with hysteretic damping (Fig. It is to be expected that the resulting design will be sufficiently conservative. The soil consists of horizontal layers resting on a half-space.8 Introduction Chap. pipe systems) can be analyzed. The shape of the soil-structure interface is quite general: the basemat and the walls can. thus taking the through-soil coupling effects of nearby structures into account. 1 forms the input to the soil-structure interaction analysis. The three-dimensional nature of the problem is properly represented. The properties vary Figure 1·3 Soil-structure interaction system. computationally efficient procedures do exist for calculating all aspects of soil-structure interaction when the soil can be modeled as a horizontally layered half space (Fig. 1-3). the engineer will vary parametrically the soil properties and study the sensitivity of the response of the dynamic system for extreme cases. tunnels. In performing the analysis. 1-3). the following essential features can be captured. If the soil in the vicinity of the . The linear structures can exhibit hysteretic damping. The number and the location of the battered and vertical piles of different lengths and different dimensions of the cross section can be freely chosen. Pile foundations are permissible. This configuration is called a layered half-space. As will be shown later.
6 DIRECT AND SUBSTRUCTURE METHODS Conceptionally. For the embedded structure shown in Fig. The force-displacement relationship of the degrees of free- = FREE FIELD INTERACTION Figure 1-4 Seismic soil-structure interaction with substructure method. It will be shown that these motions are needed only in those nodes at which the structure is subsequently inserted (shown as circles). the number of dynamic degrees of freedom in the soil region is high. the substructure method proceeds as follows. This finite soil domain can be regarded as part of the structure. As the law of superposition has to be implicitly assumed to be valid in a soil-structure interaction analysis anyway. 1. whereby those located along the embedment on the soil-structure interface are indicated as circles. 1. the analysis can still be performed. resulting in a large computer-storage requirement and in significant running time. The free-field response of the site (without any embedment) is calculated first.6 Direct and Substructure Methods 9 structure-soil interface is irregular. The nodes of the dynamic model of the structure are shown. (This direct procedure would even allow certain nonlinear material laws of the soil to be considered. the unbounded soil is analyzed as a dynamic subsystem. The various aspects of the substructure method are explained in great depth in other chapters of this text. probably the easiest way to analyze soil-structure interaction for seismic excitation is to model a significant part of the soil around the embedded structure and to apply the free-field motion at the fictitious boundary discussed in connection with Fig. The interaction part which then follows has two steps.Sec.) However. it is computationally more efficient to use the substructure method. In the first. No equations are developed here. only a rough idea is needed to understand the advantages. . In this introduction. 1-1. 1-4.
It is possible to analyze the structure with much more detail than the soil system. in order to determine the detailed response. The formulation can. In the direct method. 1 dom of these same nodes. The transfer functions allow the first few natural frequencies and the corresponding approximate damping ratios to be determined. The important parameters are more easily identified and their influence on the response more readily understood. In simple cases. Material damping can also easily be introduced for harmonic excitation. in which the responses of the structure and the soil are determined simultaneously. Once the transfer functions have been determined. using the so-called complex-response method. is beyond the scope of this text. For instance. an axisymmetric solution can be used for the soil in connection with a full three-dimensional analysis of the nonaxisymmetric structure to determine the torsional response. (Quasi-)linear behavior of the structure-soil system is postulated. In the second step. The intermediate results can be interpreted and "smoothed" to take uncertainties into account. Using the substructure method allows the complicated soil-structure system to be broken down into more manageable parts which can be more easily checked. the dynamic stiffness taking the radiation of energy into account will be frequency dependent and complex. can also be calculated by completely different methods. The accuracy in the intermediate and higher-frequency ranges can be checked easily. This means that the properties of the coarser model have to be determined by a systematic reduction procedure from those of the finer one. which works in the time domain using convolution operators. whereby one applies the resulting motion of the base of the structure. The advantages of working in the frequency domain are important. however. results applicable to one of the subsystems can be used without any computation. the structure can be modeled only very coarsely. be expanded straightforwardly to apply also to the case in which the nonlinearity is restricted to the structure. It is then often necessary to model the structure more precisely in a second stage. a spring-dash pot system.10 Introduction Chap. it is also possible to calculate efficiently the final response for another seismic environment. the dynamic stiffness of the soil does not have to be recalculated. is determined. The unbounded soil medium can best be described for harmonic excitation. If the free-field motion is changed. the structure supported on this spring-dash pot system is analyzed for a loading case which depends on the free-field motion. These so-called dynamic-stiffness coefficients of the soil can physically be interpreted as a generalized spring. In some cases. The same applies if the structure is modified during the design process. The models of the structures used in the two stages. however. which will be in contact with the structure. the soil and the structure. parametric studies can be restricted to one part. As will be shown later. . The two subsystems. A thorough treatment of this procedure. This means that linear soilstructure interaction analysis is best handled in the frequency domain. have to be consistent.
is also included. the vertically incident and inclined body waves as well as the surface waves. Vast parametric studies. 1 Summary OF TEXT 11 1. Examples from actual practice follow. is discussed in Chapter 6. which will make up the seismic motion. A structure excited by a dynamic load (e. An approximate method. Chapter 7 deals with the modeling of the other substructure. SUMMARY 1. including those arising from horizontally propagating waves.7 ORGANIZATION The fundamentals of a discrete dynamic system are summarized in Chapter 2. the boundary-element method is used to calculate the dynamic-stiffness coefficients of surface and embedded foundations. which has to be determined before the actual interaction analysis can be performed. the seismic motion at the base of the structure is affected in .g. The various types of waves. the (limited) recorded field performance is compared to the results of the analysis. the important effects of soil-structure interaction are evaluated approximately. The various possibilities of modeling the structure are discussed in Chapter 4. and thus linearity as well. which are found to be in good agreement. the soil is an unbounded domain whose radiation condition has to be taken into account in the dynamic model. 3. the soil. Using a simple introductory example. are evaluated. calculating that modification of the seismic motion which results from the actual interaction when the structure is inserted into the seismic environment of the free field. Analyzing soil-structure interaction for seismic excitation consists of two distinct parts: First. In contrast to the structure. Chapter 9 contains engineering applications. Finally. determining the free-field response of the site and second. This leads to the dynamic-stiffness matrix of the layered half-space assembled from those of the individual soil layers and of the rock. which also include the analysis of actual sites. a seismic motion) interacts with the surrounding soil. This then allows the basic equation of motion for the analysis of soil-structure interaction by the substructure method to be derived in Chapter 3. In Chapter 5 the fundamentals of wave propagation in Cartesian and cylindrical coordinates are established. The seismic free-field response of the site. which works directly in the time domain.. 2. are addressed. After reviewing the various possibilities. When comparing a structure embedded in soil with the same structure founded on rock. Alternative formulations of the equation of motion are derived in Chapter 8. The interaction effects.Chap. are evaluated to be able to demonstrate the importance of the free-field response of a site. The law of superposition is assumed.
(Quasi-)linear three-dimensional analysis of soil-structure interaction is possible for one or more structures with flexible or rigid bases of arbitrary shape embedded in a horizontally layered half-space with material damping. Besides affecting the seismic input. . the unbounded soil is first examined and can be represented by a generalized spring. 5. In the substructure method. usually amplified. the presence of the soil also makes the dynamic system more flexible and the radiation of energy of the propagating waves away from the structure increases the damping of the total dynamic system.12 Introduction Chap. 4. on which the structure is then supported for the interaction analysis. resulting in an averaging of the translation and in an additional rotational component (kinematic interaction). excavation of the soil and insertion of the base of the (massless) structure into it will alter the seismic input. the free-field response is modified. Second. Third. the inertial loads determined from this seismic input will further change the seismic motion along the base (inertial interaction). 1 three ways: First.
The matrices [M].1 EQUATIONS OF MOTION IN THE TIME DOMAIN The elastic continuum is divided into elements interconnected only at a finite number of nodes. which is a function of time. acting in the direction of the displacements in all nodes. the established textbooks in structural dynamics and in the theory of vibrations should be studied. and the static-stiffness matrix. If the loading case consists partly of prescribed motions as a function of time at support nodes. including all derivations and a discussion of the assumptions and limitations. In this case the elements in {R} corresponding to these displacements represent the unknown 13 . the equations are also formulated for these degrees of freedom. This leads to [M]{P} + [C]{r} + [K]{r} = {R} (2. For a thorough treatment of this subject.1) The vector {r}. the equations of motion are established by assembling the element matrices. The vector {R} denotes the prescribed loads. [C]. 2. which are a function of time. which are constant for a linear system. Using the finite-element method to perform the spatial discretization. Only those relations that are actually used in this text are specified. contains the displacements of all unconstrained degrees of freedom of all nodes. The main objective is to define the nomenclature that is used in this text. and [K] represent the mass matrix. A derivative with respect to time is denoted as a dot.FUNDAMENTALS OF DISCRETE DYNAMIC SYSTEM 2 The fundamental equations of a discrete dynamic system are summarized in this chapter. the damping matrix.
= The diagonal matrix [C] contains as elements the modal damping ratios As will be demonstrated later (Section 8.4) the equa(2.7) (2.6a) (2. 2.} and the corresponding natural frequencies OJ. 2.OJ. + OJ.Y} + [Q]{y} = [<I>Y{R} The vector [<I>Y{R} is the generalized modal load vector. Assembling all suitably scaled {epj} in [<1>] and all OJ~ as the elements on the diagonal matrix [Q].5 still represents a coupled system.3a) (2.1) are formulated as (coordinates). 2 reaction forces. If it is assumed that the orthogonality condition also applies to the damping matrix.8) the response (after the initial transient) {r} = {u} exp (iOJt) . 2. In general.5 is rewritten as {Y} or formulated + 2[C][Q]1/2{Y} + [Q]{y} = [<I>Y{R} {ep. Classical modes thus do not exist.Y{R} (2.14 Fundamentals of Discrete Dynamic System Chap.2) The vector {epJ and OJ~ represent the jth eigenvector and eigenvalue. (2. 2. Eq. + 2C. The radiation of energy due to the propagation of waves away from the region of interest belongs to this type of damping (see Section 5.1).: y.y. {P} exp (iOJt) will be of a similar form: (2.3 EQUATIONS OF MOTION FOR HARMONIC EXCITATION 'i' For a harmonically varying load with the excitation {R} = frequency OJ.2 TRANSFORMATION TO MODAL AMPLITUDES The free-vibration mode shapes {ep.3b) where [I] represents the unit matrix. Eq.5) {Y} + [<I>Y[C][<I>]{. 2.1. Viscous damping is assumed in Eq. respectively.4). Introducing the transformation {r} = [<I>]{y} where {y} denotes the vector of the modal amplitudes tions of motion (Eq. of the undamped system follow from the solution of the eigenvalue problem (2. The number of selected mode shapes can be selected smaller than the number of degrees of freedom. 2.y.6b) for each modal amplitude y. the orthogonality conditions are as follows [<I>Y[M][<I>] = [I] [<I>Y[K][<I>] = [Q] (2. the orthogonality condition for damping is (approximately) satisfied for the structure but not for the complete structure-soil system.
15) . where. As the total system will always have some damping. r for the displacement) from the corresponding amplitude in the frequency domain (u).co2)z. [S] is not singular for any co. by using the correspondence principle. this distinction is not enforced if no confusion can arise. . Ifno viscous damping is present.2.g. However. [S] (Eq. 2. {tP. + 2ico'jco.14) [K*] = [K](1 2'i) + applies. With {u} Eq.11) (2.12) + ico[cI>)T[e][cI>] condition is also valid for [C]. 2.9 is transformed ([n] or if the orthogonality to = [cI>]{z} . The transformation to modal amplitudes can also be performed. 2. if in a parametric study the displacement amplitudes u as a function of frequency are discussed in depth for many cases. to be able to simplify the nomenclature in a specific example.10) can be expressed by (2. The equations of motion (Eq.1). Different damping ratios for the two types of body waves are discussed in Section 5.. For instance. = (co. 2. and a maximum value is also shown for a transient excitation. 2.4 CORRESPONDENCE PRINCIPLE The material damping occurring in the soil and the structure involves a frictionalloss of energy. respectively.4 Correspondence Principle 15 The vectors {P} and {u} represent the (complex) amplitudes of the loads and of the displacements. 2.13) In principle. This so-called (linear) hysteretic damping is independent of frequency.co2[I]){z} = [cI>)T{P} (2. It can be introduced into the solution. The latter states that the damped solution is obtained from the elastic one by replacing the elastic constants by the corresponding complex ones.10) The amplitudes {u} follow from Eq.V{P} (2. when working in the frequency domain. is the damping ratio.1) are formulated as (2.Sec.9) [S]{u} = {P} where the dynamic-stiffness [S] matrix [S] is specified as = [K] + ico[e] - co2[M] (2. The viscous-damping matrix [e] representing the radiation of energy in the soil will also be a (much more complicated) function of the elastic constants (see Section 5. This means that for the staticstiffness matrix (2. Umax (and not rmax) will be used to designate this quantity. different letters are used in this text to distinguish a variable in the time domain (e.9 by direct solution.
) -2ni ~ .17b) is specified as exp 2ni~ Sea = -2 n 0-1 /~O L: P(wJ (I") n j = 0.. for example. The Fourier transform pair formulated for one element of the load vector is specified as pew) R(t) = r~~ 1 R(t) exp (-iwt) dt (2.. With t) = j I1t = j'I'[n and co. . in the definition of the amplitude per unit to of the load component at frequency ca i= P(w)/(2n)]. 2. that is. 2.17a) (2.ISa) and the inverse transform R(t) (Eq. 2 After transformation to modal amplitudes. [wj(l this case is specified as = + 2'i) - W2]Z) {tP)f{P} (2. Eq.n - I (2.. where for the efficiency of the calculation n is selected as a power of 2. Taking full advantage of this .. This so-called quiet zone will allow the free vibration components of the response of the system to be damped out when the new period of the load starts.. This integral is an extension of the Fourier series to the case of a load having continuous frequency components. Some derivations use this factor in the other equation.16 Fundamentals of Discrete Dynamic System Chap. this numerical-analysis procedure can also be used to calculate periodic loads as. n - I (2. = l dw = '2n/T. In other formulas for the Fourier transform pair. The term dw P(w/)/(2n) represents the Fourier coefficient for the frequency WI. . The period T is divided into n equal increments of I1t each. where the frequency increment (and the lowest frequency) dw = 2n/T. . the signs of the arguments of the exponential functions are reversed. the load has to be assumed to be periodic with a finite period T. 1.1Sb) As the nonperiodic load is actually replaced by a suitably chosen periodic one. The frequency span is also divided into the same number of increments dw.17b) = 2n _~ pew) exp (iwt) dco f+~ The normalization factor 1/(2n) is introduced into the equation for R(t).16) 2. a lot of duplication will occur when evaluating the exponential functions. In the numerical-analysis procedure.17a formulated as a finite sum of discrete terms equals P(w/) = dt )~ R(t) exp 0-1 (. arising from machine vibrations. . When evaluating the discrete Fourier transforms. I.= 0.5 DISCRETE FOURIER TRANSFORM The Fourier integral of the excitation represents a nonperiodic load which can be regarded as a periodic one of infinite duration. To minimize errors the load period is extended by including an interval of zero load.
2. Using the inverse transform of Eq. 2.)} = -2 .)}.22) the displacements in the time domain at discrete points {r(t. The frequency increment to solve Eq.21) The soil's contribution to [K] and [C] will in general depend on COl (Sections 5. 2. 2. the equations of motion in the frequency domain are equal to (2. which is called the Fast Fourier Transform. The material damping of the structure can be included by using [K*] instead of [K] (Eq.20) with (2. 2.1 and 9. the amplitudes of the loads in the frequency domain {P(col)} calculated at all discrete frequencies COl are determined (Eq.Sec.20 is selected as quite large. The equations of motion in the frequency domain are never solved for all points of the Fourier transform.6 METHOD OF COMPLEX RESPONSE By applying the discrete Fourier transform to the loads discretized at discrete points {R(t.1Sa). whereby the analytically available solutions for systems having one or two degrees of freedom (anchored at the calculated frequencies) can be used. In analogy to Eqs.-1 I ~O 11: L: u(co l) exp 211:i- ([j) n (2. Intermediate values follow by interpolation.10. 2.20 for {u(coI)} results in the amplitudes of the displacements in the frequency domain. a very efficient scheme results.1Sb.14). . but formulated for the displacements ilco {r(t. Solving Eq. using a binary representation and applying the results of one step immediately in the next.9 and 2. The highest frequency COrnax contained in the load equals COrnax = "2 ilco = Llt n 11: (2. 2.19) 2.1).)} follow. possibly smaller in the neighborhood of the first few peaks of the response occurring at the natural frequencies of the system.6 Method of Complex Response 17 fact.
To differentiate between the various subsystems. 2.1 Flexible Base To derive the basic equations of motion. by g (for ground). The dynamic system consists of two substructures. superscripts are used when necessary. and e is used to designate the excavated soil. In the following it is appropriate to work also with other subsystems for the soil (Fig. the actual structure and the soil with excavation. is denoted by f. the remaining nodes of the structure by s.21 is deleted. 3-1). The soil without excavation. the so-called free field. The word "total" (superscript t) expresses that the motion is referred to an origin that does not 18 . it is sufficient to examine a single structure embedded in soil for earthquake excitation (Fig. In the substructure method no nodes are introduced in the interior of the soil. which is a function of the discrete value of the frequency (0. The amplitudes of the total displacements are denoted by {ut}. Subscripts are used to denote the nodes of the discretized system.BASIC EQUATION OF MOTION J 3. 3-2).20 and 2. the soil with excavation.1 FORMULATION IN TOTAL DISPLACEMENTS 3.1. The subscript used to indicate a discrete value of (0 in Eqs. The nodes located on the structure-soil interface are denoted by b (for base). The structure's base consisting of the basemat and of the adjacent walls is assumed at first to be flexible. The structure is discretized schematically as shown. The structure is indicated by s (when used with a property matrix). the other substructure. The dynamic equations of the motion are formulated in the frequency domain.
is assumed to be constant throughout the structure. 2. respectively. which is a bounded system. The vector symbols are deleted in Fig.15) [S] = [K](l + 2ei) - w2[M] (3.Sec. The order of this vector equals the number of dynamic degrees offreedom of the total discretized system. Some formulas. move. The hysteretic-damping ratio which is independent of frequency. . is calculated as (Eq. The formulation can straightforwardly be expanded to the case of nonuniform damping. 3. The vector rut} can be decomposed into the subvectors {u!} and {un. however. The dynamic-stiffness matrix [S] of the structure. e.1) where [K] and [M] are the static-stiffness and mass matrices.1 Formulation in Total Displacements 19 ut s utb Figure 3-1 Structure-soil system. 5 f e Figure 3-2 Reference subsystems. cannot be written so concisely. [S] can also be decomposed into the submatrices [Sss]. 3-1 for the sake of conciseness.
For the reference system of the free field. The vector {uK} denotes the displacement amplitudes of the soil with excavation for the earthquake excitation.2 leads to (3. as for this . [S{bl and {un are the dynamic-stiffness matrix and the vector of displacement amplitudes. The dynamic-stiffness matrix of the soil [Sfbl (Fig. 3 [SsbJ. Setting {P. The equations of motion of the structure are formulated as [Sss][Ssbl] [ [Sbs][SbbJ {{u~}} = {{Ps}} {un {Pb} (3. 3. and [SbbJ. f o o 000 0 FREE FIELD SCATTERED Figure 3-3 Dynamic-stiffness matrix and earthquake excitation referred to different reference systems of soil.3) Both substructures contribute to the dynamic equilibrium equations of the nodes b lying on the structure-soil interface.1.20 Basic Equation of Motion Chao. To avoid using unnecessary symbols. 3. [Sfbl could be determined by eliminating all degrees of freedom not lying on the structure-soil interface of a mesh of the soil extending to infinity. the superscript s (for structure) is used only when confusion would otherwise arise. follows analogously from Eq. Conceptionally. The dynamic-stiffness matrix [Sbbl of the excavated soil. using the properties of the soil. vanish. arising from the soil with excavation. 3-3) is not as easy to determine as that of the structure.} = {OJin Eq. respectively. a bounded domain. For the displacement amplitudes {un. respectively.2) where {Ps} and {Pb} denote the amplitudes of the loads and of the interaction forces with the soil. For earthquake excitation. as the soil is an unbounded domain. the nodes not in contact with the soil (the subscript s stands for the structure) are not loaded. The contribution of the soil is discussed first. the interaction forces acting in the nodes b.
1 Formulation in Total Displacements 21 loading state.4 can be formulated for this special case.Sec.6 in Eq. 3. the equations system are of motion of the total structure-soil (3. the forces [Sfb]{Ug} are not zero.3 and 3.7 leads to [S{b]{Un = [Sfb]{Ug} (3. Substituting Eq. This so-called "scattered" motion is not easy to determine. 3.2. They are equal to [Sffb]({U1} . 3-3). The free-field system results when the excavated part of the soil is added to the soil with excavation (Fig.5) In this formulation.4. (3. The interaction forces of the soil will thus depend on the motion relative to {un. 3. [SZb] = [Sbb]' and {un = {un.8 in Eq.9) In this formulation in total displacements. that is.4) Combining Eqs. 3-2 and 3-3). The influence of an exterior boundary with an applied earthquake motion also has to be taken into account when calculating the forces in nodes b. the line with the nodes b (where the motion is equal to fum is a free surface. Including the contribution of the structure ({Pb} in Eq.fum. Introducing Eq. Although for the substructure of the soil with excavation. With [SbS] = [0]. 3.6) By stipulating that the "structure" consist of the excavated part of the soil only. the equations of motion for the nodes in contact with the soil (subscript b) are formulated as (3.3. the line that will subsequently form the structure-soil interface is a free surface (Figs. the motion in the nodes (which will subsequently lie on the structure-soil interface) of the ground with the excavation. the free-field motion. This also holds for the assembly process of the dynamic-stiffness matrices (3. the earthquake excitation is characterized by {un. Eq. 3. This is explained further in Section 8.7) ([Sbb] [Sfb]){Un = [Sfb]{Ug} + results.5 results in the discretized equations of motion (3. where the basic equations of motion are rederived starting from those of the direct formulation of soilstructure interaction. 3. the load vector is expressed as the product of the dynamic-stiffness matrix of the free field [S{b] (discretized in the . as discussed above. 3.8) This equality of forces is quite a remarkable result in its own right.2). It is desirable to replace {un by {un. which does not depend on the excavation (with the exception of the location of the nodes in which it is to be calculated).
For instance. it is helpful to think of the generalized spring as having a length approaching zero.1). a regular system/is formed on which the load vector depends.11) This represents the equations of motion of a discretized system (consisting of the structure and in the embedded region of the difference of the structure and of the soil) supported on a generalized spring described by [Stb]' The excitation consists of a prescribed support motion {un acting at the end of the generalized spring that is not connected to the structure. Eq.5.9 is reformulated as (3.3. [Stb] should be easier than [Sfb] to calculate (see Section 7. The load acts only on the nodes b at the base of the structure.2. By adding the excavated part of the soil (system e) to the irregular systemg.3). the nodes s not connected with the soil are unloaded. This physical interpretation is remarkable. 3.10. is equal to (3. which is also valid for applied loads and not only prescribed motions.6. Only outgoing waves that propagate toward infinity arise (see Section 7. 3 nodes at which the structure is subsequently inserted) and of the free-field motion {un in the same nodes.10) The difference of the property matrices of the structure and of the soil in the embedded region [SZb] .4 for further applications of this concept. The amount of interaction of the embedded part described by the lefthand side of Eq. where the matrix and vector symbols as well as the subscripts have been omitted. The generalized spring represents the stiffness and damping of the layer built in at its base at the top of the bedrock. For a better understanding of the foregoing. the soil is not excavated (Fig. 3-4. 3-3). The applied support motion is the free-field response at the free surface of the layer and not at its base. 3. 3. It should be emphasized that in the system/for which [Stb] is calculated. 3. [Stb] and {un are discussed in depth in Chapters 7 and 6.9 depends on [SZb] + [SKb]' which.22 Basic Equation of Motion Chap. The interaction part of soil-structure analysis is thus formulated as a so-called source problem. . This is shown schematically in Fig. It is important to note that the support motion to be applied at the end of the generalized spring is the free-field response {un at the structure-soil interface and not at some (fictitious) boundary at a depth where the underlying medium can be regarded as very stiff. See Problems 3. after making use of Eq. with the source being located on the structure-soil interface. As the boundary of the free-field region is regular. The derivation of the basic equations of motion is based on substructuring with replacement. and 3. The vector {un has only to be calculated in those nodes b which subsequently will lie on the structure-soil interface. As expected for seismic excitation. Using Eq. consider a structure supported on the surface of a layer of soil resting on rigid bedrock. 3.[S~b] is thus of importance. respectively. Procedures to determine [Sfb].
It represents a simple.2.1. They are discussed only because valuable physical insight can be gained from the equations.2.13) . It is worth mentioning that {un has no real existence (i.1 Formulation in Total Displacements 23 Figure 3-4 Physical interpretation ments (flexible base). 3.2 Rigid Base The base. At the same time the physical significance of the terms appearing in the equations can be discussed. 3. it does not occur in the real soil-structure system). 3.9 describes the basic equations of motion.12) It is. unnecessary to determine this seismic motion of the soil modified by the excavation {un (Fig.Sec.e. In this case. This compatibility constraint on the structure-soil interface leads to a slight modification of the formulation. formulated for a structure with a flexible base. but general procedure to calculate even the most general case of soil-structure interaction. 3-3). All other formulations which are derived in Sections 3.. 3-1 is shown. The same structure-soil system of Fig.9 can be used. but with a rigid base in Fig. of basic equation of motion in total displace- Equation 3. can be assumed to be rigid for many practical applications (see Section 4. 3-5. however. the total motion at the base {un can be expressed as a function of the total rigid-body motions of a point 0 {u~} as M} = [A]{u~} (3.3). and 8. consisting of the basemat and the adjacent walls. 8. as Eq. which work with relative displacements.8 could be used to calculate the "scattered" earthquake excitation {un: (3.3 are not more powerful. Equation 3.
Obviously. For a rigid base. the free-field motion is unchanged. 3. 3-6). Structure-soil system with For a three-dimensional base. {u~} contains the amplitudes of three displacements and three rotations. the number of degrees of freedom is reduced. The matrix [A] represents the kinematic transformation with geometric quantities only. the motion of the structure-soil interface which represents the boundary between the two subsystems thus depends only on {u~}. 3 ut • Figure 3-5 rigid base. 9 1!::===2:===~ S~o II II II 0 II II II ug Figure 3-6 Reference soil system with excavation and rigid structure-soil interface.9 and premultiplying by the transposed transformation matrix defined by Eq. In the substructure of the soil with excavation (system g). Introducing this transformation of variables. The base is massless (dashed line).14) in Eq.14 leads to [Sss] [ [Sos] [Sso] ] [S~o] [S~o] {{u~}} { {u~} = [AY[Stb]{Ut} {OJ } (3. {{u~}} = [[I] {u~} + [A] ] {{u~}} {u~} (3. the compatibility constraints of the rigid base are enforced (Fig.24 Basic Equation of Motion Chap. 3.15) .1 the [A]-matrix for a two-dimensional case is presented. In Problem 3. Compared to a flexible base.
Substituting this equation in Eq.8.19) This equation can. it is meaningful to calculate the seismic motion of the ground system (soil accounting for the excavation) with the compatibility constraints of the rigid base enforced. The seismic motion of this reference subsystem shown in Fig.16b) (3.1 Formulation in Total Displacements 25 where [S80] [S~o] [S~o] = [S'b][A] = (3.Sec.o].o] ] {{u~}} [ [So.16a) (3. premultiplying Eq. 3-6).5 for a structure with a rigid base.16c) [AY[Sbb][A] = [AY[SEb][A] [S~o]. 3. and [So.16a and b are thus not explicitly used. deleting all matrices associated with the structure. 3.oY are the dynamicstiffness submatrices of the structure with a rigid base. a rotational component is also present in {u~} even for vertically propagating shear waves. [S8'] : [S. The discretized structure specified by [S'] is supported . Equation 3.16c results in [S~o]{un = [AY[S{b]{Ut} or {un = [S~o]-I[AY[S{bl{ut} by [AY. of course.20) as illustrated in on a generalized This equation in total motion can be physically interpreted Fig. As {u{} along the walls of the embedded structure varies with depth. acts in point 0 of the rigid base. and using (3.18) (3. and setting {u~} = {un. 3. [S~o] describes the amplitudes of the three forces and moments acting in point 0 which lead to unit amplitudes of displacements and rotations in the same point of the rigid base connected to the soil. The load vector [AY[S{b]{Ut}. 3-6 is denoted as {un and represents a scattered wave motion. 3. To gain insight into the physical significance of the load vector. Equations 3.] 1 [S~o] [S~o] {u~} + = [S~o]{un {{O} } (3. No need really exists to calculate the motion of any other reference soil system. Substituting Eq.15. which in a three-dimensional case consists of three forces and three moments.18 in Eq.13 : (3. As in the case of a flexible basemat. be derived directly from Eq. 3.15 leads to the equivalent of Eq. 3. the load vector depends on the free-field motion {un in those nodes that will subsequently lie on the structure-soil interface. [S.17) {uff} = [A]{un [I] denotes the unit matrix.15 represents the equations expressed in total motion. 3-7. 3.] = [S. The compatibility constraints of the rigid base for the soil-subsystem g are formulated analogously as in Eq. [S~o] represents the dynamic-stiffness matrix of the soil with excavation for a rigid structure-soil interface (Fig. They are normally directly established when discretizing the structure by selecting models that take account of the geometric constraints of the rigid base. In a general three-dimensional case. 3.
15) not only apply to the single embedded structure shown in Figs. 3. 3-8. The vector and matrix symbols. 3 email@example.com and 3.1. an irregular soil region around the structure exists. The end of the generalized spring not connected to the structure is excited by {un. for a pile found a- .5). In this case. They are sometimes called driving forces. FREE FIELD SCATTERED Figure 3-7 Physical interpretation ments (scattered waves). The nodes b lie on a line separating the irregular soil region and possibly the base of the structure from the regular horizontally layered soil region. but can also be used to analyze the more general structural configurations discussed in Chapter 1 in connection with Fig. this line will coincide with the base of the structure and thus form the structure-soil interface. 3-1 and 3-5. In this case. 3.3 Special Foundations It is assumed that the soil on the exterior of the line with nodes b consists of a layered half-space. as well as the subscripts. 1-3. This interpretation is also valid with minor adjustments for the structure with a flexible base (Eq. 3-7. have been deleted in Fig. the bounded irregular soil region is regarded as part of an expanded structure. The basic equations of motion (Eqs.5 can be interpreted analogously for a structure with a flexible base. 3-20. This allows efficient formulations to be used to calculate [Slb] and {ut}. Equation 3. which is calculated from {ut} and [Stb]. of basic equation of motion in total displace- spring characterized by [S~o]. the basic equation (Eq. {un contains the degrees of freedom of the nodes lying on the structure-soil interfaces of all structures. however. 3. as discussed in Chapter 1. By setting {u~} = {OJin Eq.26 Basic Equation of Motion Chap. Normally. as shown in Fig. 3.9) can still be used. it can be deduced that [S~o]{un represents the amplitudes of the forces exerted on the rigid base in point 0 by the seismic motion when the base is kept fixed. If.
. Analogously.}.Sec. This case is illustrated schematically in Fig. the other degrees of freedom of all structures are assembled in {u. 3. This also follows from the equation of motion at the base (from Eq. system with tion of the nodes along all piles. The matrices [SIb] and [Slb] are full. and {un. The dynamic degrees of freedom are {u~}. The load vector corresponding to the dynamic-equilibrium equations of point 0 equals [AY([S~6]{U~} + [S~b]{Um. the structure is built in at its base with the prescribed motions {un. {u~}. and part of it rigid. Each structure contributes to the dynamic-stiffness matrix of the total structural configuration. no soil-structure interaction occurs. In this case. The equations of motion of the built-in structure with prescribed motions at its base {un are derived from the equations Figure 3-9 Structure supported on piles. The motion of the nodes b can be expressed as a function of {u~}. 3-9 for an embedded structure with a rigid basemat and adjacent walls supported on piles. Part of the structure-soil interface can be flexible.5: {un = fum. 3. that of nodes b equals [St6][U~] + [Stb]{Ut}.1 Formulation in Total Displacements 27 Figure 3-8 Structure-soil irregular soil region. defining [A]. For the special case of rigid soil the formulation is also valid. Nodes b lying on the line which subsequently will form the rigid base and nodes b are introduced.
24) The matrix [TSb] represents the quasi-static transformation. 3. only the dynamic-stiffness matrix of the structure appears in the equations. accustomed to work with seismic inertial loads (calculated as the product of the mass of the structure and some suitably chosen earthquake excitation) which are applied in all nodes of the structure. The total displacements can be split into those caused by the so-called kinematic interaction (superscript k) and by the inertial one (superscript i). From the set of equations associated with nodes s. The analysis is thus performed in the following two steps. the mass of the structure is set equal to zero by definition. Introducing this condition in Eq. the displacement amplitudes for kinematic interac- . 3. 3. 3.1 results in (1 [ (1 + + 2'i)[Kss] 2'i)[Kbs] : (1 + 2'i)[Ksb] J {{u:}} i (1 + 2'i)[Kbb] + [Sfb] {un = [Sfb]{Uf} (3. 3.9). The equations expressed in the total motion {u~} follow from Eq. 3.21) As expected.24 in Eq. Substituting Eq.23) Using Eq. In general. 3-1).[KbsUKssrl[KsbD + [SfbD{un = [Sfb]{Uf} = [Stb]{Un (3. but has to be determined prior to the actual dynamic analysis. Each column of [TSb] can be visualized as the static displacements of the nodes s of the structure when a unit displacement is imposed at a specific node b.22b) In the kinematic-interaction part. this excitation is not equal to the free-field motion. The structural engineer is.25) For a general configuration. which is a function of the static-stiffness matrix of the structure. it follows that (3.2.23 leads to «1 + 2'i)([Kbb] .1 Flexible Base For a structure with a flexible base (Fig.8. geometry alone is not sufficient to calculate [TSbJ.5 and using Eq. 3.28 Basic Equation of Motion Chap. however.2 KINEMATIC AND INERTIAL INTERACTIONS In the equations of motion formulated in total displacements (Eq. the loads for earthquake excitation are applied only at the nodes located on the structuresoil interface. 3 corresponding to the nodes of the structure not lying on the base.5 as (3. All other displacements at the nodes b are zero. In general. 3. the nonzero load vector could also be formulated as [Stb]{U{}. and no loads are applied at the nodes s. 3. this procedure results in {u~} = {u:} {un = {un + {u~} + {ui} {{O} } (3.22a) (3.
. y U ~ k .. Figure 3-10 Physical interpretation of kinematic and inertial interactions (surface structure with vertically incident waves). In this case [T.Sec. 3-10).26) The term co2{ Uk} represents the negative-acceleration amplitude of the kinematic motion. 3.. respectively. [Slb] = [Stb]. 3. For vertically incident body waves... I \'f~u li + =u f &. Thus for a surface structure excited by vertically incident waves.23 leads to the equations of motion for the inertial interaction part (3.22 is substituted in Eq._--. Proceeding further with the derivation of the formulation. 3-3)... Eq.bD with this vector results in zero forces.~~&. {un equals {un. 3. V ). _. as the structure moves as a rigid body. 3.. For the inertial-interaction part of the dynamic response.. .. 3-10.. as [Sgb] = [0] (Eq. For a horizontal and vertical uniform earthquake excitation. shown in Fig.5.b] can be established from rigid body kinematics (horizontal and vertical translations uh and U For the sake of conciseness. Multiplying ([Kbb] [Kb. and making use of Eq. 3-10.. {un contains the constant amplitudes of this motion denoted as uh and u".]-'[K.][K. the elements in {un are constant for the nodes b located on the free surface.J. For a structure supported on the surface (Fig.6 and Fig._~ ~ I I . 3. the vector and matrix symbols as well as subscripts have been deleted in Fig.'u . . . . KINEMATIC _.2 Kinematic and Inertial Interactions 29 tion of the nodes located on the structure-soil interface {un differ from those of the scattered motion {un (and also from the free-field {utD. The presence of this familiar load term represents the advantage of this method. the load vector thus consists of the negative inertia loads (mass of structure times acceleration) determined from {Uk} (and not from fum.
this method is appropriate.2 Rigid Base For a structure with a rigid base (Fig. for a flexible basemat. 3-11. KINEMATIC INERTIAL S8 FREE FIELD SCATTERED Figure 3-11 ible base). the equations of motion for the kinematic-interaction part (Eq.23) are just as complicated as those for the total motion (Eq. 3.22.30 Basic Equation of Motion Chap. the inertial interaction part.[S!o]{u!} (3. The latter determines the loading to be applied in the actual dynamic analysis. For a rigid basemat. by omitting the mass of the structure and subjecting the dynamic system to the same load vector (which depends on the free-field motion). 3-5) the total motion is split up into those caused by kinematic and inertial interactions. As. The procedure is illustrated in Fig. 3. the kinematic motion is first calculated.2. 3. Setting the mass of the structure equal to zero for the kinematic-interaction part of the analysis transforms Eq. 3 Summarizing. however. Analogously to Eq. 3.28) . {u~} = {un + {u~} (3. Physical interpretation of kinematic and inertial interactions (flex- 3.20 to (1 [ (1 + + 2Ci)[Kss] + 2Ci)[Kos] (1 (1 + 2Ci)[Kso] ] + 2Ci)[K~0] + [S!o] {{u~}} {{O} } {u~} .9).27b) can be formulated. this procedure should not be used.27a) {u~} = {u~} {u~} (3. as is demonstrated in the following.
30.31) which expresses static equilibrium of the structure. 3. 3. use is made of [K~o] . 3.[KosUKsXl[Kso] = [0] (3. 3.28.Sec. With {un. The matrix [Tso] depends on geometric quantities only and not on the stiffness properties of the structure.29a). 3.30) = {un The quasi-static transformation matrix [Tso] is determined formally from the static part of the equilibrium equations of nodes s in Eq. the latter being determined by apply- . 3.28 leads to the equations of motion of the inertial-interaction part: (3. 3. This is illustrated in Fig. 3-5 is deleted. and inertial In deriving Eq. the displaced structure for kinematic interaction as a dashed line.27 in Eq. analogously to the matrix [A]. It follows from Eq.2 Kinematic and Inertial Interactions 31 Solving this equation leads to {un = [Tso]{u~} with {u~} (3. This matrix is not actually calculated using Eq. This is obvious.29b) (3. Figure 3-12 Kinematic motions. 3. The un deformed structure is shown as a thin line. For the sake of clarity. as the presence of a massless structure does not change the response of the base.29b that the motion of the rigid base in the kinematicinteraction part of the analysis (structure massless) is the same as that of the ground enforcing the rigid-body kinematics along the base.29a) (3. Substituting Eq.29b. the kinematic interaction part of the motion follows throughout the structure using rigid-body kinematics (Eq. and the deformed structure exhibiting the total motion as a solid line. the inner part of the structure shown in Fig. 3.20 and using Eq. 3-12. The symbols for vectors and matrices are omitted.32) The load vector is equal to the negative product of the mass of the structure and of the seismic-acceleration amplitude. but follows directly from rigid-body kinematics.
This is indicated in the figures by showing the base as a dashed line. The kinematic and inertial motions are illustrated in Figs. Physical interpretation of kinematic and inertial interactions (rigid In the formulations derived above. This scheme is not further investigated. This not only affects [S~o] and {un but also the modeling of the actual structure. This physical interpretation of {un is important. Using rigid-body kinematics. it is possible to develop formulations where the mass of the base is included in the subsystem of the soil. 1-2c. For instance.19.32 Basic Equation of Motion Chap. the compatibility constraints of the rigid base are enforced. Reference should also be made to the discussion in Chapter 1 in connection with Fig. the free-field response can be used to define one part. no support motion is applied for the inertial-interaction part. Other possibilities exist to split the total motion into two parts. This method of . and e. ° °fooofo S ° u -SCATTERED FREE FIELD Figure 3-13 base). d. which does not include the mass of the base. The base is assumed to be massless. f@s. the motion of the kinematic interaction {u~}( = fum follows from Eq. 3-12 and 3-13. Alternatively. Figure 3-13 should be compared to the corresponding one illustrating the formulation in total motions using scattered waves (Fig. from the knowledge of the free-field response {un. 3 ing the acceleration amplitude {iin(= -w2{um in point 0 at the base and applying rigid-body kinematics. 3-7). for the substructure of the soil with excavation (system g). the inertia loading to be applied throughout the structure in the inertial-interaction part of the analysis is determined. The inertial-interaction part of the dynamic analysis is modified as well. The structure is supported on the generalized spring. Summarizing. 3.
3. the interaction occurring between the missile and the .. from the starting-up procedure of the machine or from extraordinary conditions. the vertical component R(t) determined as the resultant of the measured vertical reactions of the four supports of a weaver's loom (Sulzer-Ruti Machinery Works Ltd. The amplitudes of the load in the frequency domain P(w) = Re (P) + i 1m (P) follow from Eq. 3.. Switzerland) is plotted over a complete cycle of 360 in Fig.2 and 8.e.2 Impact Vertical periodic load over one cycle. The weight of the machine operating at 9 Hz equals 27. 2.J[Re (P)J2 + [1m (P)J2 and the phase angle arctan [1m (P)/Re (P)] of the first harmonic (at 9 Hz) and of the higher ones up to the twentieth (at 180 Hz) are shown in Table 3-1.3 APPLIED LOADS AND THEIR CHARACTERIZATION 3.18a [multiplied by L\w/(2n)].Sec.J 0 « 29 28 27 26 25 24 It: UJ « o ~ > Figure 3-14 3. for example. In general.1 Rotating Machinery Unbalanced mass in rotating machinery installed in buildings will lead to periodic loads acting on the structure.46 kN. As an example. 8630 Ruti.3 Applied Loads and Their Characterization 33 the so-called quasi-static transmission of the free-field input motion and other procedures are discussed in Sections 8. 3-14...3. the structure) has a large mass and is quite stiff compared to the missile. resulting.3. 0 R(t) [kNJ 30 0 . loads and moments acting in all three directions will be applied to the structure. which is shown as a dashed line. 3. If the target (i. As an example of a short-duration nonperiodic load. The absolute value . as a short circuit.J . the impact forces are discussed. as over a very large number of cycles virtually the same variation of the load with time will occur. Transients also have to be investigated..
and the dispersion of the waves along the travel path.(!). 3-15.052 0.43 131 38 .122 0.lco) (Eq. This applies. For engineering-design aspects. 2..132 0.555 0.81 -103 -144 -165 .695 0.080 0.34 Basic Equation of Motion Chap.029 0. This is caused by the complexities of the source mechanism. In this case of a soft missile.lt) specified in the time domain.063 0.107 0.232 0.052 0.177 0. the reflections and refractions at the irregular interfaces. and on the distribution of its mass and of its buckling load. 3.144 0.86 -137 73 -154 0 0 12 13 14 15 16 17 18 19 20 target during impact can be neglected.. The force-time diagram for a Boeing 707-320 crashing onto a rigid target is shown in Fig.3.061 0.3 Earthquake Earthquake time histories recorded on firm ground are quite irregular.580 0.012 0. the response spectrum that specifies the maximum response of a one-degree-of-freedom dynamic system excited by rit) . For a ground motion r.16 -126 74 -142 -129 .518 1. a load-time relationship for impact on a rigid target can be established which depends on the initial velocity Vo of the missile.l7a) is a complete measure. the Fourier transform u.223 0. 3 TABLE 3-1 Fourier Amplitudes of the Vertical Periodic Load Amplitude Harmonic Number 1 2 3 4 5 6 7 8 9 10 11 Absolute Value (kN) 1. for the postulated aircraft crash on certain structures of nuclear-power plants. for example.016 Phase Angle (deg) 138 59 . as its (inverse) transform recovers .048 0.
r. and Pg max = 0. the response spectra of an acceleration time history of the 1971 San Fernando earthquake. small deviations of the dynamic properties of the system will have only a negligible influence on the structural response. The duration of the earthquake does affect the shape. 3-16b) are plotted for the indicated damping ratios.224 g (Fig. = c/(2. 3...Sec. 3-16c.8 1. from which all three spectral values can be read. Different frequency ranges will be governed by different sizes and locations of earthquakes. A total of four logarithmic scales are present.6 0. in Fig.2 TIME [SJ Figure 3-15 Force-time diagrams for Boeing 707-320. reflecting the incidentals of a particular ground motion./klm and the damping ratio . To determine the shape of the design-response spectrum. In particular. 3-16a). the response spectrum can be displayed on a so-called tripartite plot. This so-called spectral displacement Sa = r max is used to define two other related measures: the spectral pseudo-velocity S.0 1.jkiii) of the dynamic system. the engineering seismologist can contribute significantly to a realistic shape of the design-response spectrum applicable to a specific site. Z :IE 40 20 W 0 a: e 10 25 4 2 0... represents a more important characterization (Fig. as the longer the shaking. The maximum relative displacement rmax is calculated by varying the natural frequency co = . This processing results in the peaks and dips being smoothed out. which is statistically very close to the maximum relative velocity Tmax (with the exception of very low frequencies).3 Applied Loads and Their Characterization 35 400 200 100 Vo·200m/s .. the larger the chance is of a large response arising. As an example.134 m. max = 0. the response spectra of comparable excitations which can be expected at the site are processed statistically.4 0.2 0. . which from a practical point of view is equal to the maximum total acceleration P:"ax' Because the three spectral values are related by powers of the natural frequency co. and the spectral pseudo-acceleration Sa = co2Sa. As the shape will be a smooth curve. = coSa. the H 115 record with the N lIE component with rg max = 0. by including specific records of potential earthquakes that could occur. The shape is jagged.282 mis..
the specified values are not conservative enough. The shapes of the spectra for the horizontal direction are shown as dashed lines for.. for very flexible structures (which are hardly to be encountered in a nuclear-power plant) with a natural frequency below 1 Hz. the design spectrum can be selected as two-thirds of . (a) One-degree-of-freedom dynamic system.1 1. Nuclear Regulatory Commission (NRC) published in December 1973 the revised Regulatory Guide l." which is applicable for a wide range of sites. "Design Response Spectra for Seismic Design of Nuclear-Power Plants.05. 20.---____1 5. 0: TIME 10.0 [H z l Figure 3-16 Response spectrum.. 0 >- u . 15.J '" '" :> '> >-a: .S. the U.----. [1]). = 0. H 115 record with N lIE component (after Ref.0 UNDAMPED NATURAL FRECUENCY ro 121t 100.10 . [8] c) E w ~ r% t..02.36 Basic Equation of Motion Chap. Possibly. 3 b) 0> a) :L z e::: 2 0..10 in Fig. For instance. 3-17b. For the vertical direction.00 -fwfH"H I0: w _j t3 u -.---. Spectral shapes have been derived which are intended at least for a certain class of structures for general application. . (b) acceleration time history.J 0 C :=J (f'J '" '" '" n. and 0.25 O.60. (c) corresponding response spectra as tripartite plot. 1971 San Fernando earthquake.0.01 .OO t. +------.0 10. excluding unusually soft ones.
25. the maximum relative velocity. u o --' UJ > > UJ . -I----MlI1IWII1II o f- Z a: w et:: ~ -1. (b) horizontal U.0 10.. for a near-field earthquake.0 100. It is determined as ... s ~ .00 >. Once the design-response spectrum is known.Sec. can even be selected as equal.----__.. the maximum force can easily be calculated for a single-degree-of-freedom system.. [8] b) > <f) 1. a: ~ . the value in the horizontal one throughout the frequency range of interest.+------+------.3 Applied Loads and Their Characterization 37 a) :L '" O.IOH~-~ o a. could possibly be used. a: TIME 15. 5.-~--~--~--T_-L--_. This high-frequency acceleration of the response spectrum directly determines the total acceleration loads for structures which are quite stiff. the anchor value follows which is needed to scale the response spectrum.. The anchor value is normally chosen as the maximum (effective) acceleration of the ground motion. NRC design response spectrum. normalized to 1. (a) Artificial acceleration time history. 20. which is a measure of the kinetic energy.0g.1 1. 10.S. U _j O. Based on a seismic hazard study and after selecting the acceptable probability that the earthquake used for design will be exceeded. or.0 UNDAMPED NATURAL FREQUENCY 0) / 21t [Hz] Figure 3-17 Design motion.. In other cases. 3. 30.
4. This also applies in other circumstances. the maximum response also foIIows directly from the design-response spectrum. For a surface structure excited by vertically incident waves. sufficiently reliable earthquake records which fit the situation encountered will be available. This procedure is consistent with today's state of knowledge.. 3-17a.. however.1 Statement of Problem The different aspects of considering soil-structure interaction in the seismic analysis are discussed qualitatively in Chapter 1 (Fig. with its response spectra plotted as solid lines in Fig.e. In this case.1. have to be calculated. 3-17b) kr max = kSa. The other horizontal and the vertical time histories used for practical applications in this text are not shown. a unique time history does not exist. do not exist. In general at least two horizontal time histories and one vertical (scaled to twothirds) which are statistically independent are needed for analysis. Its contribution is. not accurately reproduced. 3-17b. This motion of 30-s duration can be digitized at increments At = 0. only the inertial interaction part of the analysis really has to be analyzed (Fig. Obviously. alternatively. as mSa• For a multi-degree-of-freedom system with classical modes. 2. the location (i. which is based on the theory of random processes. It is also worth mentioning that seismic design criteria will also address other aspects (e. It is to be expected that in the future. 3. In addition. considers the general statistical characteristics of reasonable earthquakes. 3 (Fig. The amplitudes of the various frequencies are adjusted so that the artificial earthquake's response spectrum just envelops the specified design response spectrum for each damping value of interest. or. when a subsystem is decoupled for analysis (Section 4.01 s. methods of analysis.). the control point) in which the motion acts and the wave pattern have to be specified. 1-4. I) or for a nonlinear analysis. assuming a certain combination of the contributions of the individual modes. If 4096 points are selected. contributing to a design criterium which is insensitive to the exact values of parameters difficult to estimate. damping ratios. classical modes. The calculation of artificial time histories.38 Basic Equation of Motion Chap. which are compatible with the design-response spectrum. however. The following examination is restricted to the actual interaction analysis illustrated in Fig. for example. As discussed in Section 6. The latter can be examined qualitatively for a horizontal excitation with an amplitude UII . The highest frequency contained in the motion equals 50 Hz (Eq.g.4 INTRODUCTORY EXAMPLE 3. For a typical soil-structure interaction system.19). This is the case for the artificial time history shown in Fig. etc. about lOs of zero motion can be added. eliminating the need of using artificial earthquakes. 3-10). 1-2). so-caIled artificial time histories. the actual design motion is not sufficient to define the seismic loading case.
as energy is radiated and dissipated in the soil.006. The effective values m. C represents the hysteretic damping ratio of the structure. 3. based on the simple model shown in Fig. and c. and Ch in the horizontal direction and as k. the soil's dynamic stiffness is attached. This is the straightforward structural model of a single-story building frame or of a bridge whose girder can be regarded as rigid longitudinally compared to the (hinged) columns (Fig.33) c=- 2kC W (3. k.5).Sec. and h as well as a characteristic length a of the rigid base are easily determined. h is the distance from the base to the centroid of the inertial forces. (3. 3-19). of a damper. 3-18. and a damper with a coefficient c. The values m. in the rotational (rocking) direction. Methods to calculate these values are specified in Chapter 4 (see Problem 4. The latter consists of a spring and. The structure is modeled with a mass m. a lateral stiffness with a spring coefficient k. The corresponding coefficients are denoted as k. At the other end of the bar.4 Introductory Example 39 k m uf ~ I h . In the lower part of Fig. 3-18. the mechanism of the . In addition. No mass exists at the base of the structure. at the base. which are connected to a rigid bar of height h. the idealized structure can also be interpreted as the model for a multistory structure which responds essentially in the fixed-base condition as if it were a single-degree-offreedom system. The fixed-base frequency of the structure is denoted as Ws.8 ull Ug uf I I I I I I I I I II I-I(j) / / / II 1/ U~I o Figure 3-18 Model with one dynamic degree of freedom. c. and c are associated with the fundamental mode of vibration of the structure built in at its base.34) The damping coefficient c is thus frequency dependent. k. of frequency w.
e. 3 m h k c Figure 3-19 Structures that can be discretized as an equivalent model with one dynamic degree of freedom. 3-18). ll . not considering soil-structure interaction). It is advantageous to split the total displacement amplitudes into their components u' =u ll + u. only one of them is dynamic. 3-18 is thus well suited to identify the key parameters affecting soil-structure interaction and to study their effects. the total lateral displacements of the mass with amplitude u'. This is the same number as that of the structure built in at its base (i.35a) (3. of the basemat with amplitude u~ and the rocking amplitude e/>.. Although the system has three degrees of freedom. as there is only one mass.40 m Basic Equation of Motion Chap. dynamic stiffness is shown in detail.35b) where Uo is the amplitude of the base relative to the free-field motion denoted as u (g for ground) and u represents the amplitude of the relative displacement of the mass referred to a moving frame of reference attached to the rigid base and which is equal to the structural distorsion (Fig. All springs and dampers have a length approaching zero. The model shown in Fig. + he/> + u (3.
For a harmonic excitation (3.36 and using Eqs. Substituting Eq. The ratio ell of the viscous radiation damping of the elastic soil in the rocking direction is .=2 k m (3.Sec. 'g (3. Comparing Eq. approximations are introduced.37) where the subscript x denotes the horizontal direction for a purely elastic soil. the moment amplitude of the soil M.42a) (3.2kx wc - x (3. for the rocking degree of freedom. k. = kx( 1 + iW~:)Uo = kx(l + 2e)uo (3.38) applies. The horizontal-force amplitude of the soil Ph is formulated as (3. At this stage. by multiplying the spring coefficient k .40.4.4 Introductory Example 41 The modeling of the soil is treated rigorously in Chapter 7.43) Analogously. 3.44) The subscript ¢> denotes the rocking direction of the undamped soil. the corresponding equation is written as (3. 3. = 00). = kiP + cr~ = kll(l + 2e i + 2egi)¢> ll (3.36) where. and Ch depend on the frequency of excitation w. As discussed in connection with Eq. only the motion of the base relative to that of the ground results in forces. 3.38 and 3. 3. the corresponding natural frequency co. is specified by W1.37 leads to P. 3. Assuming the structure to be rigid (k = 00) and that the foundation cannot rock (k. 3. as a damped soil is addressed. For a soil without material damping.1). which nevertheless lead to valuable engineeringtype procedures.41 with Eq. as will be shown later on (Section 5. is formulated as M. 3.39) where ex represents the ratio of the viscous radiation damping in the horizontal direction ex -.40) The material damping of the soil is introduced. This results in Ph = kx(1 + 2e) + 2'gi)uo (3.41) where the subscript h is used again. (for frequency w) with the factor (1 + 2egi). where is the hysteretic damping ratio.38 in Eq.42b) follow. in an approximate manner.
42a.43. 3.45 lead to k.i + 2ei)cp (3. (1 + 2. 3. 3. the response determined from solving this equation depends on m" oi. the natural frequency co.48a) = mm2ug = mharu. 3.44.48b) (3.48c) Dividing Eqs. e.46b) Similarly.4.(1 + 2'". follows from (3. ''''' . by formulating the dynamic equilibrium of the mass point and the horizontal and rotational equilibrium equations of the total system (Fig.48c by mhco" and using Eqs. (3. and 3.20 or Eq. Using Eqs. 3. i) -1 -1 1 -1 -1 U 1 Uo (3.2 Equations of Motion of Coupled System The equations of motion for this structure with a rigid basemat can be established either by applying Eq. starting from scratch.47 results in the following symmetric equations of motion of the coupled system: m m.44 and 3. and 3.33.45) Equations 3. for a rigid structure (k = 00) whose foundation is only allowed to rock (kh = 00).32 or.47) 3.41.48a and b by mor.46a) (3. 3. 3.42 Basic Equation of Motion Chap. (3. = k". co. 3 defined as (3.46a leads to -mm2(u -mm2(u -mhm2(u + + + Uo U U + hcp) + k(1 + 2'i)u o + hcp) + kil + 2Ci + 2ei)uo o + hcp) + k. Eq. 3-18). 3.49) For a specified m with the corresponding ug. 3.. and eg' It is worth pointing c.
3 Equivalent One-Degree-of-Freedom System To gain further physical insight. Its properties (natural frequency W. The equivalent frequency W is determined by noting that in an undamped system. 3... 3.52) (3.i co2 1 + 2'i ) _ co2 ( 1 + 2..57) .Sec. ug 3. and hcp from Eq.49 leads to Uo '9 = co~ 1 + 2Cxi + 2'giU co.47 yields It follows that the the soil-structure system is always smaller than the fixed-based frequency of the structure cos- w = 1 + kjkh + kh2jk. fundamental frequency w of co.40) and (Eq.43. .55) =- m co k C=- _ 2k{ results.4.45) depend on co.4 Introductory Example 43 out that (even for constant spring and damping coefficients) Cx (Eq. 3. Note that the mass m is the same in the two dynamic systems. A tilde (-) is used to denote the properties of this replacement oscillator. 3.51.. and 3..i (3. Setting the coefficient of u equal to zero with.51) . 3.33.co. 3. = 0 and with co = win Eq.50b) U is expressed as (3.co.1. ratio of hysteretic damping C) are selected such that when excited by the replacement excitation Ug. 1 + 2. the response is infinite at this natural frequency. 3.56) w2 = . For harmonic motion.co~ 1 + 2. the equation of motion of the equivalent onedegree-of-freedom system is equal to (-mco2 or with + icoc + k)u = mco2u" -2 co (3.50a) (3.2 + coh2 + cor2 \'us 'x '9 2 or substituting Eqs.) + 2'gi . 1 + 2'i + 2'i u . an equivalent one-degree-of-freedom system is introduced. 3. Eliminating u.51 leads to 1 1 1 1 (3.54) (3. essentially the same response u results as for the coupled system described by Eq. co2 co2 1 + 2.53) (3. = = = . (3.co.
co. is evaluated at resonance and then used over the whole range of frequency. and are neglected compared to unity.56. This operation transforms Eq.51 and 3. x i . 3. 1 + 2'i . As under normal circumstances will not be smaller than" the equivalent damping ratio . i) g (3. 3. This replacement oscillator is defined by the natural frequency co (Eq. (3.51 to [ e. be somewhat larger than the damping ratio. . will. the corresponding relationship for the equivalent system results: (3.2. t.59) Equating the left-hand sides of Eqs. the same as that of the equivalent onedegree-of-freedom system resting on rigid ground.60) For resonance. For consistency and . 3.62) The equivalent effective seismic input ug will thus always be smaller than ug• Summarizing. 3.44 Basic Equation of Motion Chap. 3 To establish the equivalent damping ratio products of i. from a practical point of view.61) and is subjected to an effective ground-displacement amplitude ug (Eq. using Eq. Finally. of the structure.57) and by the hysteretic-damping ratio' (Eq. . to read = CO. it follows that ex 'g (3.co~ ~ ~(1 2 2 + 2'i - 2. 3. = . this equation can be further simplified. results. the seismic response of the coupled system shown in Fig.56 into the left-hand side of Eq. 3.58 and 3.~= 0) and if the material-damping ratio of the structure is equal to that of the soil (' = 'g).58) Substituting Eq. comparing the right-hand sides of Eqs.55. An alternative derivation based on energy considerations is addressed in Problem 3.10. 3-18 is.62).55. C. as is to be expected. The displacement amplitude U of this equivalent system is equal to . The four different parts contributing to . 3. are clearly separated in this equation. even in the absence of radiation damping. If no radiation damping occurs in the horizontal and rocking directions ('x = . 3. t. co 3.61) The equivalent damping ratio.~ should also be evaluated for co = co. which would apply if soil-structure interaction were not taken into account.59 (which have the same real parts) leads to (3.
63b) hcp = co. This represents a simple procedure applicable to design which takes soil-structure interaction into account. for example. .64a) where c s is the shear-wave velocity of the soil.64c) where p represents the mass density of the soil (shear modulus G = pc. C} can be calculated based on the original seismic input ug. (The last factor reflects the different seismic excitation to be applied. 3. - 2ei. the response of the dynamic system will depend on the properties of the structure compared to those of the soil.). Again. The last equation describes the motion of the mass point relative to that of the free field. The corresponding transverse-shear force in the structure equals k\ u\.(1 + 2'i COh + 2'i 2 s .(1 u + u + hcp = co o (-J.) . 3-18. The mass ratio m=~ pa (3.g. 2.2. relative to the freefield motion and of the rotation cpfollow from Eq. = k \ U \. the radius for a circular basemat).2'if>~)u (3.63a) (3. The ratio of the stiffness of the structure to that of the soil: S=- - cosh Cs (3. Alternatively. 3.64b) where a represents a characteristic length of the rigid base (e.4 Introductory Example 46 that of the structural distortion. It has to be calculated to design. 3.2 . 3.2'if>i. .50. the pseudo-acceleration of the equivalent oscillator (6). CO.+ 2(.2Ci}u -. = CO. For the model illustrated in Fig.4 Dimensionless Parameters For a specific excitation. s increases./6)2). the following dimensionless parameters are introduced: 1. which is of course again the transverse-shear force. neglecting squares of damping terms compared to unity results in u.4. The amplitudes of the horizontal displacement of the base u.63c) CO~ co. g (3. co2 )i(-J.. As for certain tall buildings COs is approximately inversely proportional to h. the gap between neighboring buildings or the movement of a bridge abutment.Sec. For decreasing stiffness of the soil.2'gi}u . The slenderness ratio h h=a (3. resulting in 6)2\ U \ (CO.) MUltiplying this value with m leads to m \ U \ co. cosh will be constant for this type of structure._!_) co co.
which depend on the frequency of excitation.v (3. Expressing the properties ro and. 3-21. where the rocking motion is of paramount importance.66b) 3. k x = 8Ga 2. The methods discussed in Section 7.65c) 4 = 1 _ vpcsa 0. the following expressions. 3.. The latter is thus important for stiff structures with a large mass supported on flexible soil. are the static-stiffness coefficients for a rigid circular basemat of radius a.3.65b) 8Ga3 k~ = 3(1 .5 Parametric Study As expected. 7. .61) in terms of the dimensionless parameters leads to ro2 co. which are frequency independent.4.65d) The values k. and of the soil 5. and C r to be calculated. the ratio co/cos is defined. In Fig. varying the slenderness ratio Ii.4).v) C~ (3. decreasing the stiffness of the soil (s increases) results in a decreasing of ro/cos' Augmenting the mass of the structure also leads to smaller values of ro/cos. The other parameters are specified in the caption.. is larger than for slender structures (ii large). As a crude approximation. 3-20. Poisson's ratio v of the soil. and k".66a) (3. 3 4. For squat structures (ii small). are plotted as a function of S. = 1 + mS2[2 8 1 h2 --: v + 3(1 _ v)] (3. As will be explained in depth in further sections (Sections 5. Extreme cases are also included to emphasize the effects.2 allow the corresponding coefficients of the soil kh' Ch. of the replacement oscillator (Eqs. ro/cos and. ratios of the structure. In this .65a) (3. and 7. the mass ratio m is varied.57 and 3. In Fig. are used for the undamped soil. k.46 Basic Equation of Motion Chap. The variables correspond to a typical reactor building of a nuclear-power plant.1. The ratio ro/cos can be regarded as characterizing the effect of soil-structure interaction.4 (3. for certain sites no radiation damping exists below a specific frequency. whose mode shapes of the structure-soil system will predominantly consist of a translation. These dimensionless parameters are sufficient to characterize the response of the structure founded on a half-space. Hysteretic-damping '8' To introduce the frequency of the excitation co.
'g of equivalent one-degree-of-freedom system (m = 3. . 'g of equivalent one-degree-of-freedom system (ii = 1.. 3._ ii 2 _ .. . '..ii 5 I 0.1 0.. \ "...2 --iii---ffis 1 ."".. \ 0._' .66b.8 ... quency...: '. ' iiI-l0 -.0. " .....". varying slenderness ratio. _ ..-" _.4 0. -ffiol0 -- \ \".. .' <:»: -. = 0...20 0. -..4 --- 0.05)...6 0.:.6 0.. \ \\". (a) Natural fre- ID IDs 10 t--.. \'..'""..' (b) damping.". the structural ...1 Figure 3-20 Properties v = 0.025..8 ID IDs a) 0.. 1 ii -·-ii 2 .:.10 \.. = 0.5 1 O+nmr-"'-rrTTTnr~~~rrrm0. \ \ \ '. a) 0.P". ' -. . \ \ 0.10 I 1:'/ I /. 5 10 o f....:..5 --iii 0 1 .5 1 S 0 +nmr_.._ .2 iioO. only the first two terms involving the material dam pings are present in Eq. fi 1 0 0 0 iioO.5 \ .67 ...../ 0. -ii 5 0 0 0 -- 0..--0..."...25 0.. "".05 O+nmr-"rrrrrrrr~r--o-".---S 0.. iii 0 3 -·-ffio 5 ..-A:::M:.Sec..33 iioO..0 0. The corresponding effective damping ratio.D:.: .. \ .. \ -. is also shown in Figs.05).... \". _.05 0.-~·~rJO ~ADIATION . 0... case.67 / I I I I I / I / b} 0.. (a) Natural frequency..33... varying mass ratio.33 hoO. 3. .25 b} ffi.15 \'. - '. ffi = 3 _.. = 0..33.5 1 5 10 ~:::._. = 0. \ 0..025.G:""rrrm__ S 0.05 0."mr-"rrrrrrrr-r--o-".05 01 0...1 0.15 _.. is generally larger than' and the effective seismic input ug smaller than ug..4 Introductory Example 47 1.20 ---- .. (b) damping.IN. ... ... Although for the equivalent oscillator the damping ratio ...05 0.'" 5 0.ffi 0 5 _ . 3-20 and 3-21. '.:.05 .. 0....-rrrn. 0.5 1 5 10 __ S 10 Figure 3-21 Properties v = 0.
55 for a stiffness ratio s = 0 (soil rigid. . = 0. that is. e Iu I lugl 20 a) 5=0 20 b) 15 i=O Figure 3-22 Influence of soil-structure interaction as a function of excitation frequency (ii = 1. replacing by C. Cs = 00). = 0.025. 3 response U for a specific OJ can be larger or smaIIer than if soil-structure interaction is neglected. (a) Structural distortion. the absolute value of the structural distortion I U I (nondimensionalized with Ug) is plotted for s = 1 and s = 0 as a function of the dimensionless excitation frequency OJ/OJs' The parameters are again listed in the caption. which occurs at the natural frequency of the corresponding dynamic system (OJ = ill. In addition. v = 0. ill by OJ" and ug by Ug' If only the peak response is examined.33.05). 'g . m = 3. 3. In Fig. OJ = OJs).)] is practically always smaIIer than that for the same structure on rigid soil (= ug/2C). then that which takes soil-structure interaction into account [= (ug/2e)(ill2/OJ. The latter foIIows from Eq. 3-22a.48 Basic Equation of Motion Chap. (b) displacement of mass relative to free field. selected values.
normalized to 0.S.49). '6 ..3 (Fig. the artificial time history described in Section 3. (a) Structural distortion... = 0. = </J = O.. The agreement with the results of the equivalent one-degree-of-freedom system is excellent.-. Its corresponding response spectrum closely follows the U. o~~-r-.~. = 0.":":. 3-22b..:-:'..) max [mm] 100 b) 75 50 25 ros/21t=3Hz ros 121t = 4 Hz ros/21t=5Hz J I I I I I • I I . 3.-:-.~. although to a lesser extent. are plotted.. In contrast to the peak response for the harmonic motion.~:":'. 3. (b) displacement of mass relative to free field.1 g. determined from solving the coupled system (Eq.--:.--:. v = 0. artificialtime history (jj = 1.025. Finally. varying fixed-base frequency.~. 3-23a and b.~.~~.Sec..~.' 0. m = 3. ."":.-.. Disregarding soilstructure interaction..-:".' / ::.. 3-17a) is applied. 3. The fixed-base frequency of the structure is varied.. 0. + h</J I (Eq. Taking soil-structure interaction into account again reduces the peak response.4 Introductory Example 49 shown as dots. .05).-.~----... The maxima of the structural distortion Umax and of the displacement of the mass point relative to the free-field motion (u + u. + h</J)max umox (mm] 20 a) 15 ros I 21t = 3 Hz ros 121t = 4Hz ros I 21t = 5 Hz 10 !5 ------- .. + h</J)max are plotted as a function of the stiffness ratio s in Fig.. NRC Regulatory Guide 1.5 5 10 ( u + Uo + h .63c) is plotted for the same parameters in Fig. The motion at the base is examined in Problem 3. the interaction effect is negligible for very small and very large ratios of cak»...5 5 10 Figure 3-23 Maximum response. u.6. (u + u.-.v~ .33. respectively..~.":":~::-:~.. Obviously. The absolute value of the displacement amplitude of the mass point relative to the free-field motion I u + u.-""n-~s 01 "" .60.
The amplitudes of the total displacements are associated with nodes within the structure and on the structure-soil interface. 2. the actual structure and the soil with excavation. The earthquake excitation in other points of the free field is thus not required. It is worth mentioning that the equation of motion of the equivalent onedegree-of-freedom system can also be solved directly in the time domain.7. That end of this spring not connected to the structure is excited by the free-field motion in those nodes which subsequently will lie on the structure-soil interface. 2'6)' r + . SUMMARY 1. The nodes of the structure not in contact with the soil are unloaded.+. 0)2 s rg (3. of the difference of the structure and of the soil. and 3. The structural distortion Umax always decreases. . the load vector depends on this free-field motion. Using frequency-independent spring and damping coefficients and assuming the damping whose ratio is to be viscous. both determined only in those nodes which subsequently will lie on the structure-soil interface. the compatibility constraints are incorporated into the dynamic-stiffness matrices. the following differential equation results: e r . and in the embedded region. The basic equation expressed in total motion results in a simple procedure which can be used to calculate the most general case of soil-structure interaction straightforwardly and economically.13 for an application.. S.. The discretized system consists of the structure.. For a rigid base.67) The structural distortion as a function of time and the prescribed acceleration time history are denoted as rand r. when taking soil-structure interaction into account.8.9).50 Basic Equation of Motion Chap. The corresponding coefficient matrix is formed by assembling the dynamicstiffness matrices of the discretized structure and of the unbounded soil with excavation with degrees of freedom along the structure-soil interface (base). and is supported on a generalized spring characterized by the dynamic-stiffness matrix of the soil (free field). An even simpler model can be selected to analyze the vertical earthquake excitation (see Problems 3.- 6)2 .. No need to determine the seismic-input motion of any other reference soil system exists. 4. the basic equation of motion in the frequency domain is derived. See Problem 3. Also for a structure with a rigid base. respectively. 3. By merging the two substructures. 6)2 _ r. 3 increases for (sufficiently large) increasing s. The corresponding load vector equals the product of the dynamic-stiffness matrix of the soil (free field) and the free-field-motion vector. 3.
As the physical insight gained by splitting the procedure into two steps can be valuable. correctly captures the essential effects of soil-structure interaction. which models the structure. A corresponding artificial time history can be calculated whose response spectrum closely follows the design-response spectrum for the damping values of interest. 3 Summary 51 6. Alternatively. The design motion for earthquake excitation is specified as a design-response spectrum determined from a shape intended for general application and an anchor value. determining the kinematic motion is as complicated as calculating the total motion directly from the basic equation of motion and should thus not be performed. For a general configuration with a flexible base. 9. An analysis with the same load vector as in the basic equation of motion and omitting the mass of the structure results in the kinematic motion. The kinematic motion throughout the structure (and thus also the loads for the inertial interaction) follow from rigid-body kinematics. 14. a viable alternative to solving the basic equation in total motion exists for a rigid base. 10. whereby the compatibility constraints are enforced. 7. The total motion can be split up into that arising from kinematic and that from inertial interactions. The soft-missile impact on a rigid target results in a short-duration nonperiodic load which depends on the initial velocity of the missile and on the distribution of its mass and of its buckling load.Chap. For a rigid base. at a distance equal to the height. the load vector can also be determined as the product of the dynamic-stiffness matrix of the soil with excavation and the vector of the corresponding scattered seismic-input motion. the kinematic motion equals that of the free field. A simple system consisting of a vertical rigid bar with the translational and rocking springs and dash pots representing the soil attached at one end and at the other one. when multiplied by the mass of the structure. 13. The coefficient matrix of the inertial interaction is the same as that of the basic equation of motion. . 8. Unbalanced 12. Even when an embedded structure is subjected only to vertically incident waves. leads to the inertial loads (acting in all nodes of the structure) of the inertial interaction. rotational components arise. For a surface structure excited by vertically incident waves. the kinematic motion at the base equals the scattered motion of the soil with excavation. mass in rotating machines will lead to periodic loading which is characterized by the Fourier amplitudes of the first and higher harmonics. 11. a spring and a dashpot connected to a mass. The corresponding acceleration (constant throughout the structure) multiplied by the mass of the structure leads to the loads of the inertial interaction. The latter forms the seismic acceleration which.
29 and 3. P3-la.32). is always smaller than the fixed-base frequency of the structure. be larger than the hysteretic damping ratio of the structure. 3. A mass m is supported by two massless truss bars of length l. PROBLEMS 3. This surface structure is shown in Fig. which is a function of the three frequencies defined above. For an artificial earthquake time history.52 Basic Equation of Motion Chap. 3. . S~. taking soil-structure interaction into account. reduce the maximum structural distortion. Eqs. of the slenderness ratio. respectively. of the ratio of the mass of the structure to that of the soil. will. Its effective support motion will be smaller than that of the fixed-base structure. Its damping ratio. formulate the equations of motion in total displacements (amplitudes with superscript t. and S~ in the horizontal. The dynamic-stiffness coefficients of the soil are denoted as S~. 18. 3 15. and of the ratios of hysteretic damping of the structure and of the soil. and rotational directions. Taking soil-structure interaction into account will reduce the peak structural distortion for harmonic excitation. 16. 19. will.20) and in kinematic and inertial displacements (amplitudes with superscripts k and i. the factor being equal to the square of the ratio of the equivalent natural frequency to the fixedbase frequency. The response of this system for a prescribed horizontal seismic excitation with a specified frequency is a function of the fixed-base frequency of the structure. of Poisson's ratio of the soil. and damping ratio" which are connected to a rigid massless basemat. The coupled system can be replaced by an equivalent one-degree-of-freedom system. Eq. in general.1. vertical. The scattered motion (which is assumed to be specified) consists of the translational components with the amplitudes u~ and w~ and of a rotation with the amplitude p~. Using the nomenclature indicated in Fig. which can be calculated by adding the contributions of the four damping ratios introduced above. while for a specific frequency of the excitation the result can be either smaller or larger than that of the fixed-base structure. of the frequencies for a rigid structure assuming in addition that either the rocking or the translational spring is rigid. while the maximum displacement of the structure relative to the free-field motion can be increased. the ratios for the viscous radiation damping in the horizontal and rocking directions and the ratios of the hysteretic damping of the structure and of the soil. modulus of elasticity E. in general. P3-la. Its natural frequency. The response is a function of the ratio of the stiffness of the structure to that of the soil. area A. 17.
whereby in [Sbb] the contribution of both bars is present.J3 EA(l 1 ~ 4 3 I + 2'0 4 4 -4 _v'3 4 1 _. (a) Structural system."jLEw=t. P3-1c) matrix [A] of the rigid basemat follows from The matrices [S'o] and [S~o] associated with the structure are calculated using Eqs. Adding the effect of the mass and of the soil.. (c) rigid basemat. 3. u~. 3 Problems 53 l'li 2 ut C) wI L~c:='f. (b) inclined bar. Solution: The dynamic-stiffness matrix (relating the amplitudes of the displacements u'.Chap. 3. P3-1b is equal to 4 .l6a and 3.::=::jt:!_ ut l ut l u& l2 II L i" . 3. the equations of motion in totaldisplacement amplitudes are equal to (Eq. and w~ to those of the loads) of the massless damped bar in Fig.20): . wt.13 as (Fig.l6b.J34 -4 ~ 4 3 3 1 -4 -~4 3 _v'3 -4 4 "4 1 v'3 4 "4 The kinematic-transformation Eq. l i II Figure P3-1 Mass supported by two truss bars on rigid basemat.
Jjl --4- 1 1 ° ° 2 . The derivation of the basic equation of motion (Eq. 3. the free field f (Fig. and CPl.9) is based on substructuring with replacement.Jj/ -4- 3 wt s~.64 Basic Equation of Motion Chap. 3. Adding the excavated part of the soil (the reference system e) to the irregular substructure of the soil with excavation (system g) results in a reference system with regular geometry. w~.29 as Uk J~l wk Uk wk cpk ° ° {.32. as will be . uo.:l Uk The inertial-displacement amplitudes are determined using Eq. Wi.Jj 3[2 [Sl Sg v cpt The kinematic displacement amplitudes follow from Eq.2. The coefficient matrix of the left-hand side is the same as that in the case of total displacements but with the unknowns u'. 3-3). The right-hand side equals m m = !_fl 2 m co2 m wk Uk cpk ° wk ° co2 m G:l 3. The latter's response to the seismic excitation [ut} is relatively easy to calculate.rs): ut wt . 3. 3 Sf S: 2 2 3 1 -2 -2 2 -2 3 3 1 --4- .Jjl -4ut + EA(l + 2ei) / -2 .
taking only the work of the moments into account. P3-2). Figure P3-2 Continuous beam with infinitely many spans with one span having a different moment of inertia. the free-field system/results. i. In particular. Hints: The span A-B represents one of the substructures (denoted as s).. The modulus of elasticity is E. the other substructure being g. The same applies to the dynamic-stiffness matrix [Stb] (Section 7. [Stb]. but an exterior load.5). This concept can be illustrated by solving the following simple static problem. whose response is tabulated. One span of a continuous beam with infinitely many spans of equal length I is loaded with an evenly distributed load p (Fig. one obtains [Stb]' Results: .e. All spans have the same moment of inertia I with the exception of one of the spans adjacent to the loaded one. Determine the moment distribution along this span.Chap. which is a continuous beam with infinitely many identical spans. Replacing the span A-B with Is by one with a moment of inertia I. where it is denoted as Is. the rotations at A and B for the distributed load follow. The load vector of the basic equation of motion is a function of the free-field reference system only. whereby the loading is not a prescribed motion. The moment that leads to a unit rotation at the end of the continuous beam with infinitely many identical spans is equal to the two elements of the diagonal matrix of the stiffness matrix [Sib]' After adding to [Sib] the full 2 x 2 stiffness matrix of the span A-B with I (system e). the beams to the left of A and to the right of B. which correspond to the free-field response (un. For the numerical calculation choose Is = 21. 3 Problems 55 shown in Chapter 6.
the original structure without the absorber can be regarded as the free-field reference system f. and of a damper with a constant c is installed at this top floor.56 Basic Equation of Motion Chap. --~======~ Figure P3-3 absorber. Derive the equations of motion using substructure concepts.3. a tuned-mass absorber consisting of a mass m.54 0. To reduce the dynamic response.10 8 = pl? 8 3.0105~. of a spring with a constant k.0037 EI pl» pl3 MA M8 = --0. the dynamic degrees of freedom of the (two-dimensional) structure . Assuming the columns to be inextensional and lumping the mass of the structure on the level of the floors. In a multistory building the vibrations at the top floor arising from periodic loads from a machine installed at another floor are unacceptably strong for the employees (Fig. P3-3).l = 1 -0. Structure excited by periodic load from machine with tuned-mass Solution: To establish the coupled equations of motion. 3 with Is = 21 rut} b 0.
and I. Applying a harmonic load of unit amplitude at the top floor and calculating the inverse of the displacement amplitude at the same location leads to the dynamic-stiffness coefficient st. The seismic excitation of the free field consists of a vertically propagating shear wave with a horizontal displacement rf(z. Solution: The first two terms and the third one on the left-hand side correspond to the contributions of the pile [the "structure"(s)] and of the ground with excavation (system g). which is equal to z f a k ± rf ~Ep . The equations of motion of the coupled system then follow from Eq. will often be measured and also S tb will be determined experimentally with an eccentric vibrator varying the frequency to cover the range of interest. respectively. that is. As the original structure is also one of the two substructures g. The base of the structure is assumed to be fixed. t). The other substructure consists of the absorber mechanism. on individual springs whose coefficients k(z) (per unit length) depend on the soil properties and on the radius a of the pile (Fig. . of the two substructures.())2m: -k - icoc ik + icoc + st. Derive the differential equation of motion. The subscripts p and q are used to indicate the properties of the pile and of the soil. p. the method of substructuring with replacement can also be used to establish the differential equations of motion in the time domain. the area. In actual practice. that is. the mass density. -k - icoc ]{U'} u~ = {O} StbUt 3. I. including the loads from the machine. neglecting all damping terms. This is. The corresponding displacement amplitudes at the top floor of the free field then follow. respectively. The modulus of elasticity.9 as ut ut k [ + icac .3). 3. and the moment of inertia are denoted as E. If certain approximations are introduced.4. Pp Eq Pq Figure P3-4 Pile supported on individual springs. the case for a vertical pile supported on a Winkler type of foundation. 3 Problems 57 are equal to the lateral displacements of the floors. The equations of motion (in the frequency domain) of the original structure. can easily be formulated (Section 4. A.Chap. respectively. The right-hand side represents the product of the dynamic stiffness of the free field and its motion. Sfb = st. P3-4). for example.
5.23).025. this node in general being different for each stick. Verify that the maximum and minimum occur approximately at wlws and at wrl. f zz - _ 1 c. 3.05. . respectively.. 'g Results: Solving the equations of motion (Eq. P3-6 follow. Those based on examining the equivalent one-degree-of-freedom system (Eqs. Introducing the wave equation of the soil (which is derived in Section 5.1 for harmonic excitation). For the range of frequencies where IuD + Ug III Ug I is larger than 1. The stick could be a beam.63a and 3. 3. For the coupled system used to analyze the horizontal excitation and which is examined in Section 3. be smaller than that of selecting isolated springs to model the soil.25 leads to that is. V = 0. The equilibrium equations of each stick result in Equation 3. 3-22: S = 1. 3.} of each stick can be eliminated independently of those of the other sticks (Eq. Assume that the model of a structure consists of separate "sticks" (branches). 3.56 with l/wK = 0).33..zzzz +p p A··' r + kr r = krf + p A··f + EqI····f r ci r q The shear-wave velocity of the soil is denoted as c.6. . plot the absolute value of the total-displacement amplitude of the base (nondimensionalized with lUg \). 3 The last two terms on the right-hand side correspond to the contribution of the excavated soil. The latter expression represents the natural frequency of the system with a rigid horizontal soil spring. Use the same parameters as in Fig. 3. the motion of the base with soil-structure interaction is larger than without.49) for Uo the results plotted in Fig. r. + w. The influence of the last two terms on the right-hand side will. [un = [un. (Eq. thus neglecting the coupling and damping effects. that is. which are not connected to each other and which are attached to the flexible base in one node only. the pile replaced by soil (system e). divided by co. 3-18). r··f the right-hand side can be rewritten. resulting in E p Ir' . This example is only used to illustrate the method of substructuring with replacement.Jw.55) are very similar. 3.58 Basic Equation of Motion Chap. Solution: The amplitudes [u.4 (Fig. and = 0. IuD + Ug III Ug I versus the ratio of the excitation frequency to that of the fixed base cok». = 0. in = 3. in general. Show that for such an embedded structure the motion of the kinematic interaction of the base [un is equal to that of the scattered motion [un. jj = 1. The curve exhibits a maximum and a minimum.
--------O. m = 3.4 ~ (Os Total horizontal displacement at base as a function of excitation frequency (ii = 1. v = 0.2 104---=-----1.4 0.6 0.4 1.B 1. Figure P3-6 'g 3. whereby only one of them is dynamic.. the amplitudes of the total vertical displacements of the mass w' and of the massless base wg. = 0. = kvwo . as described in Section 3.2 0+--~--.33. 3 Problems 59 2._-_. Figure P3-7 Model with one dynamic degree of freedom (vertical).2 0. .B 0.0 I.4 for the horizontal excitation.6 1. P3-7 can be used to analyze the vertical one with amplitude Wg' Two degrees of freedom.2 1.6 O.4-_+.--. the (even simpler) system shown in Fig. Derive the equations of motion using as unknowns the amplitudes of the relative displacements wand Wo and establish the properties of the equivalent one-degree-of-freedom system._-_.~ o 0..025.05)._--. = 0.4 0...B 1. is formulated as P.0 1.7.Chap. Solution: w. = Wg w~ = Wg + Wo + W + Wo + cvwo The vertical-force amplitude of the damped soil P. Analogously. are introduced.
the vertical-spring coefficient k. _ zQ. of the undamped soil as . as an approximation. See Problem 3. 1 . As the radiation damping is much larger in the vertical direction than for the other degrees of freedom.)2s 1 + 1 Q.)Cz 2kz where Cz is the corresponding damping coefficient of the elastic soil and introducing the material damping.8..8 for an evaluation. The natural frequency cO.). For the equivalent one-degree-of-freedom system for vertical excitation derived in connection with Fig. it is assumed that products of the damping ratios can be neglected compared to unity. is equal to that of the undamped case kz• Defining the viscous-radiation damping ratio in the vertical direction C. P3-7. 3. 3 where. and' are the fixed-base frequency and hysteretic-damping ratio of the structure.)2 v In the derivation of the properties of the equivalent one-degree-of-freedom sys- tem. the following dimensionless parameters can be defined: 1. respectively. g as leads to The following symmetric equations of motion result: where and Q. The stiffness ratio of the structure and of the soil . this assumption is questionable and introduces restrictions when using these formulas.60 Basic Equation of Motion Chap.7\2 '-" = Q. the ratio of hysteretic damping and the replacement excitation IV of the approximate equivalent one-degree-of-freedom system g are equal to e.
7) as a function of the dimensionless parameters.~+ w.C ~. 1 + -«. 1 + 4(Cz 1 11 + 4(Cz + (g) + (g)2 r _ r 6)2 Cz + (g . Poisson's ratio v of the soil 4. WI Wv 1 + 4(Cz mt 1 (1 . - 1 + . P3-8 as dashed lines. Results: 6)2 w. as the direct solution of the two equations of motion represents a simpler approach. ( = 0.3 _ m pa 3. The equation of motion for w of the coupled system equals + 4": + (2( + w~ -2( + 2(.v) +:. + w. 3 Problems 61 2. selecting v = 0. Wg while that of the equivalent one-degree-of-freedom system equals which results in 1_ 6)2 - w. These more accurate results should not be used in practice. + 2Cg)iJw + Cg) Wv 1 + 4«(. 3-20b for a small value of the slenderness ratio ii. Plot 6)/w.v Express the equivalent properties of the one-degree-of-freedom system 6)2/W]and' (see Problem 3.w~ _ w~1 + 4((.=:.Chap.025. The mass ratio m=. + (g)2 These are plotted in Fig. the following (frequency-independent) formulas for the spring and damping coefficients can be used for a rigid circular basemat of radius a: k z = 4Ga 1.)((g [1 .1 to 10) for m = 3. . Compare the results with those shown in Fig.~) = w~ plotted in Fig. and (g = 0.33. + (g)2 w. Derive a more accurate expression for 6) and' (which is also more complicated) by equating the real and imaginary parts of the equation for w of the coupled system with those of the equivalent one-degree-of-freedom system. The hysteretic-damping ratios of the structure ( and of the soil (g As a crude approximation. P3-8 as solid lines.05.( +(1-:. and' as a function of s (in the range from 0. Note that the radiation damping in the vertical direction is larger than that in the horizontal one. Also calculate the curve for " neglecting the influence of radiation damping.
2 Figure P3-8 Properties of equivalent one-degree-of-freedomsystem (vertical) (m = 3. Compare the curves for s = I with those for s = 0. and setting this value equal to that of the equivalent one-degree-of-freedom system using the same total strain energy leads to the equation rkU2 + r khU~ + r klp2 'o 2 'oh 2 'or 2 k2 k 2 k ¢J2 _!!_ + hUa + _!_2 2 2 e .6 "" 0. (b) damping.4 0. I will W g I and of the displacement amplitude of the mass relative to the free field I W + Wa III Wg I as a function of the ratio of the excitation frequency and that of the fixed-base frequency OJIOJ •• Use the spring and damping coefficients introduced in Problem 3.62 Basic Equation of Motion Chap.8 "" "" <.7.05. = 0. 3-18 exists.33. 'g 3. 0. . = 0.0 a) .33.05). plot the absolute value of the total vertical-displacement amplitude of the base I Wa + Wg III Wg I as a function of OJIOJ s- 'g Results: Solving the two equations of motion of the coupled system leads to the results plotted in Fig.9. = 0. P3-9. that is.10. m = 3. Adding the dissipated energies of the structure and of the soil in the horizontal and rocking directions.= .025. not taking soil-structure interaction into account. It is based on energy considerations.025.8. (a) Natural frequency... and = 0. An alternative derivation of the equivalent damping ratio for the coupled system shown in Fig. plot the absolute values of the vertical structural distortion (nondimensionalized with I W g I). 3 1.. For hysteretic damping. For which range of OJIOJ. 3. the dissipated energy (which is independent of the frequency of excitation) is proportional to the product of the damping ratio and of the strain energy. For the coupled system analyzed in Problem 3. v = 0. The following parameters apply: s = I. does neglecting soil-structure interaction lead to a smaller response? For the same set of parameters. 0. . v = 0.
4 COs Iw+wol 20 ""TWO! b) 15 5=0 10 5 IWo+WIII IWgI 0 2. (b) displacement of mass relative to free field.8 1.4 0.6 0.0 1. v = 0.0 1.2 0.0 0.4 0.2 1.4 0.2 1.2 0 0 0.6 0. = 0.33.2 1. = 0.8 1.4 CO COs CO IDs Figure P3-9 Influence of soil-structure interaction as a function of frequency of vertical excitation (m = 3. '8 .8 1.4 c) 0 0.6 0.025.Chap.05).0 1.2 0.2 0.4 0. (c) total displacement at base.6 1.8 1.6 0. (a) Structural distortion.8 0.2 1. .0 1. 3 Iwi IWIII 20 Problems 63 a) 15 10 5 0 co 0 0.4 1.
= krcp + cr~ = kr{1 + 2'~i + 2'gi)CP applies. 3-18.~+ 'g Neglecting the damping terms when formulating the equilibrium equations results = 'h C+ in U o = = k kh U cP khu kr Derive the equivalent damping ratio as specified in Eq.11. P3-ll). 3 where and" are the (hysteretic) damping ratios of the damped soil in the horizontal and rocking directions. 'h 'g " =. products of damping ratios compared to unity do not have to be formally neglected. . and I/h2m and of the frequencies wh(kr = 00) and of wr(kh = 00). respectively. 3. For horizontal excitation with amplitude Ug. and hcp. The coefficient matrix is a function of C. 3.44. the horizontal-force amplitude Ph of the damped soil is formulated as Ph = khuo + ChUo = kh{1 + 2Ci + 2'gi)uo where' x and' s are the ratios of the viscous radiation damping of the undamped soil in the horizontal direction and of the hysteretic-material damping. 3. A rigid undamped structure of mass m and moment of inertia I (with respect to the center of mass. dynamic degrees of freedom. (a) Derive the symmetric equations of motion for harmonic response using the unknowns u. Analogously for the moment amplitude of the soil M. this system has two mI FigureP3-11 Rigid structure founded on flexiblesoil. The two equations are the same as Eqs. 'g. u.64 Basic Equation of Motion Chap.~. Note that in this derivation. io. and cpo Introducing the same approximation as discussed in connection with Fig.61. w~ = kh m co. which is at a height h) rests on the surface of flexible soil (Fig. respectively. 2_ 1+ h-rn kr .41 and 3.
) [ 1 -F 1 .) + =0 which is approximately satisfied for O)I ~ (1 + h~m)(O)~ 0).) .2 = -i. 1 (d) The structure-soil system does.( 1 +h ) t 1 (O)~ + 0). as in Problem 3. 3. O)~ 0).56? 1 1 O)I = O)~ 0)1 be calculated + 0).O)7[M]){cf>. in general. not have classical modes. + equal to the equation specified above leads to (1 + h2m/I)(0)~ 0)2 + 0).10. I=0 O)I. (c) Under what condition can the fundamental frequency approximately using the equivalent of Eq. the transformation to modal coordinates does not decouple the damping terms.} = 0 {cf>d = 1(0)~rot OI ) Uol U) (c) Setting the approximation 2_ 0)1 - O)~O). 3 Problems 65 (b) Identify the stiffness matrix [K] and mass matrix [M] of the system and calculate the two natural frequencies 0)1 and 0)2 of the coupled system with the corresponding mode shapes {cf>d and {cf>2}.(h2m/I) + O)~O)~ ] 1 (O)~ + 0)~)2 ([K] . 'I Results: ~~(1 [ (a) + 2Ci + 2'gi) -1 -1 (b) (KJ {. that is. Identify the damping matrix [e] and show that for 0) = 0)1 the offdiagonal term of the transformed damping matrix {cf>dT[C]{cf>2} does not vanish.Chap. Determine an approximate equation for the corresponding modaldamping ratio based on energy considerations. ~l Uo [M] = [mm m + {2 m 1 4 with the unknowns I[K] .O)7[M] 2 and hcf>.
- Ch(khU~j2) khu?. of the shape of a triangle with a right angle (width 2b. c.(krrp2/2) + krrp2/2 ex + Cg C'" + Cg formulating equilibrium (neglecting damping terms) rp = hkhuo kr and introducing the approximation discussed in part (c) leads to c.L (co~ .1)(CO~ _ co h cor coi l)(c '" + 2Cgkr) co 7": 0 The modal-damping ratio is determined on the basis of energy consideration: e.5n.1 -v C'" = 0. P3-12a). = O. height Sb/3) rests on the surface of flexible soil which is modeled as a homogeneous half-plane with shear modulus G. the same approximations as discussed in Problem 3. Poisson's ratio v.19n1 _ O.11 are introduced.C".66 Basic Equation of Motion Chap. the coefficients of the springs and dashpots of the undamped half-plane of unit depth are specified as k. = = G O. g 3.72npcsb Gb2 k". whereby this two-dimensional case represents a cross section (Fig. and hrp (h = Sb/9). formulate the symmetric equations of motion for harmonic response using the relative-displacement amplitudes u. As a crude approximation. and material-damping ratio Cg• This could be regarded as a simplified model of a gravity dam supported on poor rock. (a) Making use of the results in Problem 3. = C + cor + cor CO~ex CO. 3 [<1>] = [co~ 1 cor --1 co~ 1 coi --1 1 2 The off-diagonal term of [<I>)T[C][<I>] equals C x + 2 Cgkh + .12. The rigid structure of density p ./2 Ch Cr = = With + C. density p.11. .Sv pcsb3 To determine the dynamic-stiffness coefficients of the damped soil.35n1 _ v O.
0 1.2 0 0 0.4 0. (c) rocking. (a) Triangular structure on half-plane.5 2. 3 Problems 67 a) uo • G P ~Q IUo+ugl lug! 2. .4 1.2 1.8 0.6 0.5 3.5 3.5 2.8 16 1.0 2.5 1.0 3.0 1.5 Cs Figure P3-12 Rigid structure interacting with soil.0 0.8 1.2 1.4 0.6 14 1.Chap.8 0.5 1.0 3.0 0 rob 0 0.0 1.2 b) Ih.0 1.1 lug I 2.5 Cs c) rob 2.6 0. (b) total displacement at base.0 0.
33.08 v . P3-12b and c.13. for the following parameters: v = 0. Plot the ratios I u. 1 ./2n w/2n Ratio < 0. 1. ~= 0.9 g t = 0. < 1.33. oo2b2 1 . Results: Dimensionless parameters: r cob c.lp = I.05.03oob (1 . 0. 3-17b): w system: (= = 1.S. a shearwave velocity c.05. Is soil-structure interaction important in this case? Results: (a) oo~ _ p c~ 1 002 . 3 (b) As the structure is modeled as a rigid body. NRC applies (Fig. Use the response-spectrum method. fixed-base frequency oo.c. + Ug III Ug I and I hrp III Ug I as a function of table.17 ---+ Sa = = 3.05 2. see Fig. effective radius of basemat a = 20 m. ii Equivalent one-degree-of-freedom Response spectrum (Fig.41 p.002b2 1 .12 oo.19. 3..17 ---+ S. m (f). ( = 0. a mass density p = 2. effective mass m = 50 Gg. 3-17b).0 Mg/m".0. (b) For the plots.68 Basic Equation of Motion Chap.4.42 p. and a damping ratio (g = 0.V 002 P c2 1 oo~ = 0.32. Based on the approximations introduced in Section 3. The design response spectrum of the U. and material damping ( = 0. = 200 mis. p. = 0.05. v) 1- v s = 2. calculate the ratio of the relative displacement (which is equal to the ratio of the total accelerations) of the structure founded on the soil described above to that of the one founded on rock.3 Hz.45 g . and (g = 0. it is not possible to evaluate the effects of soil-structure interaction by comparing natural frequencies and damping ratios.5. A structure has the following properties: effective height h = 20 rn.v L = 1./2n = 4 Hz. It is built on the surface of a half-space with Poisson's ratio v = 0.5 = = 4 Hz.
4. does not mean that a very complicated detailed model is always necessary. as the dynamic properties of the structure are better known and the problem of representing the radiation of energy in the unbounded domain does not exist. The structures and the applied loads vary widely. as the computational effort of the latter can be an order of magnitude larger than that of a static analysis working with the same discretization. In many cases an approximate model capturing these essential dynamic properties is sufficient. however. proceeding floorwise. 69 . Selecting a crude model does not mean that the structural engineer is not familiar with sophisticated methods. calculating one part of the structure after the other (e. a large finite-element model can be first established and then reduced systematically by diminishing the number of dynamic degrees of freedom. in general. mostly of low frequency.g. The total dynamic system with a correct representation of the stiffness and the mass has to be modeled. also in view of the many uncertainties of the dynamic load. Procedures to establish such models in a direct way are discussed in Section 4. Economical considerations in general also demand the use of a simple model for a dynamic analysis. which makes modeling a difficult task.MODELING OF STRUCTURE 4.3. Various methods to achieve this are outlined in Section 4. This. Fewer pitfalls are encountered in modeling the structure than the soil. starting at the top). It is well known that many dynamic loads excite only certain modes. which are normally larger than for static ones. it is. not permissible to perform the dynamic analysis in steps.. Alternatively.1 GENERAL CONSIDERATIONS In contrast to the procedure often used in determining the static response.
such as the displacement or the acceleration at a specific location (e. In contrast. the type of result required plays an important role. in general. As the global values correspond to integrals of the mode shapes. The complexity of the model of the same structure excited by a horizontally propagating wave for which the motions at the base of the structure differ from node to node increases by at least one order of magnitude. This is in contrast to the modeling . but not the spatial variation. Besides distinguishing between global and local results. another global result. All modes with nonnegligible generalized modal loads in the range of frequency of the applied loading have to be represented. the contribution of the higher modes will be small and can thus be neglected.1 Frequency of Applied Content and Spatial Variation Loads The applied loading influences the modeling of a structure significantly. In addition to the frequency content and spatial variation of the applied loading. it is essential to realize that modes of high frequency contribute significantly more to the maximum total accelerations than to the maximum relative displacements.g.3. Obviously. a mode of high frequency can significantly influence the local response.2. For instance. a two-dimensional model of the structure can be appropriate. These are proportional to the product of the mass and the mode shape. the base shear of a building modeled as a vertical beam will be in equilibrium with the inertial loads. As will be discussed in Section 4. The resultant of the inertial loads corresponding to the modes of high frequency will thus be small. The shapes of the modes of high frequency with many nodes exhibit zones of positive and negative values. to be restricted to represent even fewer of the modes of low frequency. As discussed in connection with the tripartite representation of a response spectrum (Section 3.1. is the same.. an approximately axisymmetric surface structure (even with a flexible base) can be modeled very simply if excited by an earthquake arising from vertically incident waves.70 Modeling of Structure Chap. 4 Lack of knowledge of the dynamic characteristics of the structure can in many cases be compensated by the choice of a more detailed model.3). In both cases the frequency content of the earthquake. these modes will hardly contribute to the overturning moment at the base. If only global results are to be determined. 4. This means that impact loads with a high-frequency content demand a much finer model than the seismic excitation. at the top of the structure) if the mode shape exhibits a large value at this point. ro2 is larger for modes with high frequencies than for those with low frequencies. Analogously. the dynamic model will tend. If the motion is expected to occur essentially in a plane. The same structure excited by loads differing in frequency content and spatial variation has to be discretized differently. for which only all modes with frequencies smaller than 30 Hz have to be included. the pseudo total acceleration (which is very close to the total acceleration) is equal to ro2 times the relative displacement of a mode with frequency ro.
The acceleration or displacement time history u. For a dynamic load (e. varies significantly with the stress level and the type of the loading. using certain results of the crude model of the total structure-soil system (e. the response u. The matrix [S] of a bounded system depends on the static-stiffness matrix [K]. In the first step. as cylindrical shells and neighboring shear walls can stiffen the base significantly.1).Sec.2).2) and allows a much simpler model of the structure to be used for earthquake excitation (Section 4. which lies on the structure-soil interface.1 General Considerations 71 of the soil (see Section 7. These matrices appear in the corresponding basic equations of motion (Eqs. this motion is applied to the support of the subsystem. Similarly.2). a subsystem with a small mass can be decoupled from the dynamic model. The assumption of a rigid base reduces the number of degrees of freedom of the structuresoil interface (see Sections 3. This procedure is illustrated in Fig. the two systems are assumed to behave independently of one another with no feedback from the subsystem to the dynamic model of the first step. The materialdamping ratio " which is actually a global representation of nonlinear effects.can be regarded as rigid. In many cases the base.. 4-1. results associated with the three-dimensionality of the structure (e. In the second step. 4.] and [S. the matrix symbols are deleted. and a damping ratio is to be determined. This motion will be influenced by the dynamic characteristics of the model of this first step.g. For instance. the result of the first step. [S~o] for the rigid base. In this method. When implementing this procedure. the damping ratio" and the mass matrix [M] (Eq. the time history of the motion at the support point of the 'e . of the component with a static-stiffness matrix [Kcl.15).o].g. an earthquake excitation specified at the surface of the free field denoted as uf' or an impact load acting on the dome of the structure).b]' [Sbb] for the flexible base and [S.1. applying the maximum total acceleration as the loading). at the location where the subsystem is in reality connected to the structure is calculated. This could apply to a component fixed to the structure at one point. the torsional response) cannot be calculated in this way. 3.. 3. The matrix [K] depends on the (dynamic) material properties of the structure.. In most cases it will be necessary to analyze the structural element separately.1 and 3.2 Decoupling of Subsystem The results of the crude dynamic model are in general not sufficient to design the structure directly. Of course.. the stress resultant in an equivalent beam with a vertical axis representing all walls cut by a horizontal section between two adjacent floors of the dynamic model is not suitable for designing a specific portion of the wall.9 and 3. the dynamic model without the decoupled subsystem is analyzed. In the figure. The selected model of the structure determines the dynamic-stiffness matrices [S.4. even if the basemat itself is not necessarily rigid. a mass matrix [Me].g. 4.. resulting in its response u.
1 Free-Field Motion Before discussing the seismic load applied to the structure.1 addresses the uncoupling criterion.. 4-1. the free-field response of the site has to be examined. At this stage.2 SPATIAL VARIATION OF SEISMIC LOADS 4. 4-9).72 Modeling of Structure Chap.3 (Fig. 4 Figure 4-1 Uncoupling of subsystem with small mass.3. its maximum response can be calculated straightforwardly with the response-spectrum method of dynamic analysis.and SH-waves propagate with the velocities c and c. This concept is discussed in connection with the design response spectrum of the control motion in Section 3. respectively.2. 4. Assuming the subsystem to have classical modes.3. ur p . the following simplified description illustrated in Fig. The compressional P-wave for which the particle motion coincides with the direction of propagation causes the vertical component of the earthquake with amplitude The two shear waves SV and SH whose particle motion is perpendicular to the direction of propagation lead to the two horizontal components with amplitudes u~ and ut. Vertically incident waves propagate vertically. is often displayed in the form of the socalled in-structure response spectrum. Problem 4. subsystem. the particle motion at a specific time is constant. It is convenient to distinguish between vertically incident waves and horizontally propagating waves. In a horizontal plane. Chapter 6 contains a thorough discussion of the different types of waves. denoted as Us in Fig. This represents the maximum response of a single-degree-of-freedom system characterized by its own natural frequency COc and damping ratio Cc. The P-wave and the SV. 4-2 is sufficient for an understanding of the concept. Examples of such in-structure response spectra are contained in Section 4.
1a) (4. 4.2 Spatial Variation of Seismic Loads 73 HORIZONTALLY PROPAGATING WAVES VERTICALLY INCIDENT WAVES P-WAVE SV-WAVE ~ { SH -WAVE Figure 4-2 waves. (The letters V and H denote that the particle motions take place in the vertical plane and in the horizontal direction. Besides inclined body waves.) As the wave front is inclined. respectively. for SV.Sec.= cos IfI 1 _2_ cos IfI This means that at a specific time.and SH-waves for P-wave (4. The other horizontal motion ut is caused by the SH-wave.and SV-waves will result in the horizontal and vertical freefield motions u. the particle motion at two points on the x-axis will differ. 4-2. horizontally. surface waves (so-called Rayleigh and Love waves) exist. inclined body waves are generated. which also propagate Fig. Free-field motion of horizontally propagating and vertically incident Selecting the angle of incidence IfI (measured from the horizontal x-axis in the vertical x-z plane) to be different from 90°. the inclined body waves propagate horizontally across a horizontal plane in the x-direction with an apparent velocity Ca which is a function of the velocity of propagation of the body wave and of 1fI. and u{. Both the P.1b) c. Projecting the wavelength measured along the direction of propagation onto the horizontal x-axis results in c. These are not shown in .
lead to the loads to be applied in the inertial-interaction part of the analysis (see Fig.3. 3.2. and w%and the three rotations with amplitudes a!. and upward when dealing with the structure. 4.29b). which is the direction of (apparent) propagation. v~ = u{. u! = u~. 3-10.74 Modeling of Structure Chap. multiplied with the mass of the structure. leading also to constant accelerations. 4. 3. 4-3b). in addition. Since for a rigid basemat the scattered motion {un equals that of kinematic interaction (Eq. A surface structure with a rigid basemat is first assumed. The six rigid-body components of {un are the three translations with amplitudes u!. In any horizontal plane of the structure. the motion . for this discussion.2. 4-3c. some "average" vertical displacement with an amplitude w! and. 4-3c. Analogously. The variation corresponding to u{ along the x-axis. the latter being zero for this case. The corresponding motion of the kinematic interaction in the structure follows. The amplitude of the resulting vertical displacement is denoted as u~. and y!. the vertical component of the free-field motion u{ is examined first. Eq. P!. This procedure is presented in Section 3. The amplitude P! generates a horizontal displacement with an amplitude u~. 3-10). A simple transformation of coordinates is then needed when assembling the contributions of the two substructures. which. Addressing the effect of horizontally propagating waves. be determined approximately. v!. To determine {un accurately. The displacements of kinematic interaction are constant throughout the structure for vertically incident waves. As discussed in connection with Fig.19 should be used. The vector {un can. a rocking component turning about the y-axis with an amplitude P! arise. Because in this equation [stbl and its inverse are present. which increases linearly with height. The resulting motion is shown in the vertical x-z plane and in the horizontal x-y plane in Fig.2. the motion of kinematic interaction is equal to that of the free field for a surface structure with vertically incident waves. based on geometric considerations only (see Problems 4. which requires the calculation of the dynamic-stiffness matrix of the soil [Srbl (= [stbl for a surface structure). is shown in Fig. and w! = u{ apply. using rigid-body kinematics. As the basemat is rigid. the influence of the stiffness of the site on {un will be small. the (tangential) horizontal component of the free-field motion u{ will result in a translational component with an amplitude v! and a torsional component turning about the z-axis with an amplitude y! (Fig. and an additional vertical displacement proportional to the horizontal distance from the center of the basemat.4).2 Kinematic Motion of Surface Structure with Rigid Base Turning to the spatial variation of the seismic loading applied to a structure. 4 It is appropriate to select the positive direction of the z-axis as pointing downward when addressing the soil. it is informative to split the total motion into components caused by the kinematic and inertial interactions. and 4.
u%. 3. (a) Horizontal component coinciding with direction of propagation..Sec. (b) horizontal component perpendicular to direction of propagation.19 would show that small additional terms can arise. for the free-field motion uf. Even if this rigorous formulation is used.. assuming the same wave as vertically incident. The corresponding displacement amplitudes u.4): For instance. 4. ut . the (radial) horizontal component of the free-field motion leads only to a translational component with an amplitude u% (Fig.Y I:::::!:==::::I-. however. assuming so-called welded contact (see Sections 7. 4-3a). These values can. of kinematic interaction consists of a constant displacement in the y-direction. the kinematic motion of the rigid base will consist of six components at the most. A rigorous analysis based on Eq. be neglected.3 and 7. The corresponding kinematic motion (displacements and accelerations) throughout the structure follows from rigid-body kinematics.. The important components of motion are properly represented.2 Spatial Variation of Seismic Loads 75 .X Y Y t::===::::::tr-x -r-e-x Figure 4-3 Kinematic motion of surface structure with rigid basemat for horizontally propagating wave. are constant throughout the structure. onto which a constant tangential displacement is superimposed.. Because of the averaging effect resulting in self-canceling. an additional rocking component with an amplitude o:~ arises. The resulting kinematic motion of the rigid base as discussed above is only approximate. Finally. v%. (c) vertical component. and w% for horizontally propagating waves are smaller than the corresponding values.
In Fig. 4-5). 4. 4 The embedded structure with a rigid base (basemat and side walls) is examined next.2. As discussed in Section 3. By the same reasoning. 3-10). 4-3 and 4-4. Using the results discussed in connection with Figs. w!. 4-3). the horizontal component of the free-field motion from a vertically incident wave leads to v: and ex!. ut y: 4. As u~ varies along the vertical wall of the embedment. Only the case where the kinematic motion of the base can be described by the six components u:. This will also affect the modeling. an additional rocking component with an amplitude P! is generated.g. the response of the embedded structure with a rigid base for a horizontally propagating wave can be determined.5 Axisymmetric Structure The spatial variation of the kinematic motion in an axisymmetric structure (e. ut y t Figure 4-4 Kinematic motion of embedded structure with rigid base for vertically incident waves.76 4. the component of the free-field motion propagating horizontally in the x-direction results in the translational component with an amplitude v!.2. . leading to a linear variation of u~ with height and to a vertical component with an amplitude u~.3 Kinematic Motion Modeling of Structure of Embedded Structure with Rigid Base Chap.4 Kinematic Motion of Structure with Flexible Base Examining the influence of a flexible basemat. The amplitude u{ results in w! and no additional rotational term. the kinematic motion throughout such a structure is equal to that of the free field at the surface only if the structure is founded on the surface and only for vertically incident waves (Fig. there is no advantage in splitting the total motion into components caused by kinematic and inertial interaction in this case.. For an embedded structure or for horizontally propagating waves. a solid or a shell) is addressed next (Fig. and in the rocking component with an amplitude ex:.2. v!. 4-4 the kinematic motion resulting from the horizontal component of the free-field motion u~ assumed to arise from a vertically incident wave is shown. the seismic loading case for a flexible base is complex.2. in the torsional component with an amplitude (Fig. For instance.
cylindrical coordinates with the radius r and the circumferential = x y and denoting the amplitudes k Uk and v leads to r cos () (4.xPg Uk = w zoo + yy: (4.2a) u~ = u: z P: - rt:. For this latter case. A horizontal section at the height z is shown in Fig.2b) 0 (4.Sec.2c) Introducing angle (). 4. the rotational components vanish.u~. to a structure with a rigid base even for embedment and even for horizontally propagating waves and to a surface structure with a flexible basemat for vertically incident waves. and u: are (4. P:. In a point with the coordinates x. as discussed above.2 Spatial Variation of Seismic Loads 77 / / Figure 4-5 Spatial variation of kinematic motion in axisymmetric structure.3b) in this coordinate system as = r sin () of the components .zrt: + xy: g + yrtg . and z the components of the kinematic motion u~.3a) (4. at the center of the base 0 is examined. This applies. y. 4-5. and y: u~ = v: .
(4. of the zeroth harmonic. etc. integrated around the circumference (fn UkUI dO.sin 0 zP: + rr: w = w: + r sin 0 IX: .7b) So 2n cosjO sin 10 dO = 0 the dynamic-stiffness matrices for all harmonics of the symmetric and antimetric cases are independent from one another.3 in Eq.6c) . the kinematic motion varies only with the zeroth and first harmonics of a Fourier series in O.4a) (4.sin 0 ZIX: + cos 0 zP: v = -sin 0 u: + cos 0 v: . The inertial loads acting in the inertialinteraction part of the analysis will vary analogously. With respect to the plane 0 = 0°. 4.4b) (4. 4. The discretized loading applied in the finite-element model is proportional to the product of the kinematic and inertial motions in the same direction.32) can also be based on a Fourier expansion of the displacements in the circumferential direction.5c) =- sin 0 U~ + cos 0 u~ Substituting Eqs. A finite-element discretization of the axisymmetric structure for inertial interaction (Eq. Thus the lth symmetric and antimetric loading terms only excite the corresponding Ith displacement term.2 and 4.4 results in u" = cos 0 u: + sin 0 v: . For instance. and IX: as the first antimetric harmonic.6a) (4. only the first harmonic (symmetric to the earthquake component) is nonzero for the horizontal component. u! represents the nth amplitude of the radial displacement and is a function of rand z. the terms in and P: vary as the first symmetric harmonic. cos nO + L: u: sin nO = . only the zeroth and first harmonics are excited for the cases studied. respectively. and i is used to indicate inertial interaction. 3. The same applies to the total motion. respectively.r cos 0 P: k k Circumferentially. as they are calculated as the product of the mass and of the acceleration of the kinematic motion. . This means that in the inertial-interaction analysis.5b) (4. The terms in and vary as the symmetric and antimetric parts. For any axisymmetric surface structure. 4 Uk V k = COS 0 u~ + sin 0 u~ (4. . 4. As the trigonometric functions are orthogonal.78 Modeling of Structure Chap.L: v~ sin nO + L: v: cos nO = . Because of Eq.).7a) (4.5a) (4. • = L: w: cos nO + L: w: sin nO • .cos 0 ZIX: . u: w: r: v: u' Vi Wi L: u. f2n Jo cosjO cos 10 dO = 0 for i= I (4.7 the loading cases also decouple. The superscripts s and a denote the symmetric and antimetric parts.6b) (4.
4). This means that. a much coarser model with fewer nodes could have been selected (see Section 4. The amplitude u! (resulting from un leads to Uk and vk. an applied torsional moment will only result in v~. for the case shown in Fig. 4. for example. with the exception of the zeroth symmetric one. This so-called shield building of reinforced concrete built in on the level of the basemat (Fig. the out-of-plane displacement u. will vary as cos around the circumference for an applied bending moment acting along the y-axis. w! (arising from vertically incident wt) leads to only a vertical loading wk (Eq.6 Example For the sake of illustration. in a beam. 4. even if the applied load acts only in one or two directions.2). The fact that for the earthquake excitations discussed above.2. 4-3b. e u. Six cross-sectional values are determined for the beam. The corresponding in-plane displacements u~ and v~ are zero. In the theory of strength of materials. a constraint is thus introduced into the analysis which is not present in the theory of elasticity applied to axisymmetric solids and shells. for a beam with a vertical axis. Two dynamic models are used for comparison: a shell model using 36 curved higher-order isoparametric frusta in the meridional direction with a Fourier series (zeroth and first harmonics) in the circumferential direction. plane cross sections are postulated to remain plane. vi. which give rise to nonzero uj.6c).3. (= wD. which excite much higher frequencies (see Section 4.2 Spatial Variation of Seismic Loads 79 and all harmonics. . For instance. but also u~ and v~. all three components of a specific harmonic are excited. Similarly. The other dimensions are specified in Fig. assuming vertically incident waves. Using beam theory. setting u~ = w~ = O. In general.8) has to be independent of e. An applied normal force will lead to only an out-of-plane displacement u. and a beam model with 52 nodes. Additional approximations thus apply when using beams. For the section to translate in-plane as a rigid body. For instance. Consistent mass and mass moment of inertia are introduced in the beam model. For many cases. For the seismic loading case.Sec. only the zeroth and first harmonics are excited for axisymmetric structures is of paramount importance when modeling the structure. The same applies if the structure with a rigid base is embedded. 4. beams with a corresponding annular section can be used as an approximation. corresponding to the term in wi (Eq. = U cos e- v sin e = uj cos- e + vi sin. and excites not only w~.20 m. and wi. This can only be the case if uj = vj.3. the outer containment of a reactor building of a nuclear-power plant is examined.e (4. This discretization is also used for the calculation of impact loads. 4-6. 4-6) consists of a cylindrical and a spherical part with a thickness of 1. the displacement ux. vanish for the vertical component.5) of the zeroth symmetric harmonic. 4. only the zeroth anti metric harmonic and the first antimetric one are excited.
46 13. a TABLE 4-1 Horizontal Natural Frequencies (Hz) (First Harmonic) Mode Number Model Beam Shell 4.38 40. as is shown in Fig.38 2 13.74 22. which stiffens the model. discrepancies arise. the natural frequencies corresponding to the first harmonic of the two models are compared. The same applies to the mode shapes. Excellent agreement exists for the lower modes. In Table 4-1.22 21.64 4 5 28.90m "----- 29.41 4.80 BEAM SHELL Modeling of Structure Chap. where the elevations of the second and fifth are plotted.vl. smaller than the radial displacement with the amplitude u1 of the shell (corresponds to the lateral displacement at (J = 0°) and larger than the circumferential displacement with the amplitude vl (corresponds to the lateral displacement of the shell at 90°).31 3 25.OOm "----- Figure 4-6 Dynamic models of reactor-shield building.06 . 4-7. The lateral deflection of the beam is.41 25. 30m "----- 46. For the higher modes. As the beam theory introduces a constraint. the natural-frequency values of the beam are higher than those of the shell. in general.30m L___ / / O. 4 67. As u1 *.
Even the in-structure response spectra. \ \ \ --~--~~------~-tl I I / . as the generalized modal loads (participation factors) of the lower modes dominate.2 Spatial Variation of Seismic Loads 81 CIRCUMFERENTIAL RADIAL (SHELL) BEAM (SHELL) 2ND MODE BEAM 5TH MODE SHELL .. This is demonstrated in Fig.. calculated. where the horizontal and vertical projections at the indicated level are shown. 4-7b.. 4. .~- .. .. For a horizontal earthquake associated with vertically propagating waves.. the two models will lead to very similar results. (b) vertical and horizontal projections at level 29. in contrast to the beam theory.. for . the deviation from the results of the beam for the fifth mode is larger than for the second.. [2]).. As expected./ I 2ND MODE 5TH MODE Figure 4-7 Horizontal mode shapes (Ref.".. (a) Elevation of mode shapes.Sec. ..30 m.-:-.. distortion of the cross section takes place.
the contribution of the shear force to the strain energy be taken into account. Any stiff floors will tend to force the shell model to behave more as a beam.63 23.10 9. which corresponds to the torsional modes.86 25. 4 example.86 28.45 29.22 4 58. In Table 4-2 the natural frequencies of the zeroth harmonic are compared.28 3 45. at 0 and 90 around the circumference at this level of the shield building. the strain energy of the shear force becomes dominant.65 Torsional Mode Number 2 28. For the vertical modes (symmetric term). TABLE 4-3 Horizontal Natural Frequencies of Beam (Hz) Mode Number Strain Energy Moment only Shear force only Moment and shear force 5.07 18. When modeling an axisymmetric structure as a beam. to a lesser extent.18 13.74 5 85.99 6.15 The agreement is excellent for all indicated modes for the anti metric term.48 40. can hardly be distinguished from that of the beam model. The latter cannot be represented by a beam.65 9.56 28. the second mode already shows large discrepancies. as shown in Table 4-3.82 0 Modeling of Structure Chap. the shield building is obviously not a slender structure. For instance.41 . v~) in the dome which leads to significant local bending behavior.87 44. The rigid basemat and.05 4.72 2 39.21 12. Neglecting its influence results in somewhat larger frequencies (results not shown). the spherical part have this effect in the shell model of the shield building. It is thus imperative that when calculating the static-stiffness matrix of a beam element. TABLE 4-2 Natural Frequencies (Hz) (Zeroth Harmonic) Vertical Mode Number Model Beam Shell 13.45 29. Neglecting the contribution of the bending moment would be just as wrong.54 41. It is interesting to note that for the higher modes with many nodes. This is caused by the "breathing" mode of the shell model (u~.46 3 58. The mass moment of inertia in the beam model is of secondary importance.41 2 28. it is important to make use of all features of beam theory.
Their total number can be increased straightforwardly by selecting more elements. The next example is such a case. the dimensions. To calculate results of acceptable accuracy.4) or by direct discretization. is relatively small. and certain high-rise buildings. but using as many identical elements as possible.3 DIRECT DISCRETIZATION 4. chimneys. Continuity across element boundaries is satisfied. have been developed which express the displacement field of an element in terms of the nodal values of this element. being the same for identical elements. The number of degrees of freedom of the model using sophisticated elements. Interpolation functions. Each part of the structure with the same structural behavior (e.3.3 Direct Discretization 83 4. In all circumstances. a coarse dynamic model is sufficient. by adding mass of non structural parts) for the dynamic one. plate. To determine the dynamicstiffnessmatrix. either no model of the total structure is established for the static loading case. the structural behavior determined from the static analysis should be used when establishing a dynamic model. In other instances. if appropriate. Finally. the static finite-element model of a complex structure is so detailed that it would be uneconomical to use it in a dynamic analysis. In some instances the finite-element model of the total structure used for the static analysis can also be applied with no or only small modifications (e. This can be the case if the structure is of simple geometry (which does not mean that the structural behavior is necessarily easy to determine). with higher-order isoparametric elements. which allows efficient solution procedures to be applied. It is so well established that it does not have to be treated in any detail. bridges modeled with three-dimensional curved beams. These results are especially valuable if the static loads act similarly to the applied dynamic loads .Sec.g. which can be established either by reducing the degrees of freedom (Section 4. A detailed dynamic model then has to be created. certain shell and plate structures. or it cannot capture the essential dynamic response. and the material properties of the elements. if necessary. the displacements throughout the structure have to be expressed as a function of these degrees of freedom. so-called shape functions. varying the shapes. The dynamicstiffness matrix of an element is easily established.1 Finite-Element Model In any discretization. the dynamic degrees of freedom in a structure have to be able to represent all significant inertial loads. The generalized displacements at the nodes located on the boundaries of the elements are the degrees of freedom. shell. Assembling the contributions of all elements results in the banded dynamic-stiffness matrix of the dynamic model.g. 4. or if beam elements are used. solid) is divided into elements.. beam. The finite-element concept represents one approach to achieve this. A few examples are doubly curved arch dams modeled. Many examples are contained in this section..
is analyzed for an impact force.2 Models for Impact Load At first. and fuel-handling buildings for . and all / / Figure 4-8 Dynamic model of reactor-auxiliary impact analysis (Ref. The box-type structure. 4. high frequencies up to 100 Hz are excited. The force-time diagram for an impact on a rigid target is shown in Fig. is shown with its most important dimensions in Fig. The modeling of the structure thus has to be rather detailed.3. the combined reactor-auxiliary and fuel-handling building. Because of the content of high frequencies present in the force-time curve of the airplane crash and the small area of loading. These safetyrelevant structures and the equipment located within have to be designed to resist the postulated crash of a Boeing 707-320 impacting at a velocity of 370 km/h. The local behavior must be adequately represented. The two buildings share the same basemat and surround the reactor building without any connections to it. which is assumed to be built in at its base. Real-world structures for which dynamic analyses have to be performed are used as far as possible. The outer walls and some of the floors are also common to the two buildings. An impact area of 37 m. 4-8. Calculating such additional loading cases in the static analysis can be advisable. 3-15. Some principles of the direct discretization are illustrated in the following. which form one structure of a nuclear-power plant. [2]).84 Modeling of Structure Chap. starting in this text with elaborate models and working toward approximate ones used for preliminary design. 4 (including the inertial ones).is assumed.
Both points are located on the floor that is situated on the same level as the center ofimpact.3 Direct Discretization 85 modes up to at least 100 Hz included. only one-fourth has to be modeled (Fig. [2]). Material damping of 0. the response parallel to the direction of - LNlMf'ED NATLRAL FREOL. stiff components are subjected to high accelerations and small relative displacements._LL:!. a component with a fundamental frequency of 15 Hz located at point A will experience the following maximum response parallel to the impact force in the y-direction: a total acceleration of 4 g and a displacement relative to the floor of 5 mm. are calculated.ENCY [Hz] 1. the acceleration time histories in these two points have to be determined. resulting in thousands of dynamic degrees of freedom for each set of boundary conditions. only two sets of boundary conditions have to be processed. 4. To be able to do this. As is characteristic for impact loads. 4-9. 4-8).-_u. in-structure response spectra..00 -PONT A ------.£u.07 times the critical is selected.. For the horizontal impact force acting on the axis of symmetry as shown in Fig.L___'_-.L_J QOOI QOI QIO 1. 4-8.1. To design components located at points A and B.IIlW>f'ED NATLRAL PERIOD C51 Figure 4-9 In-structure response spectra (1 % damping) in reactor-auxiliary fuel-handling buildings for impact analysis (Ref. By assuming symmetry of the structure. The in-structure response spectra in the two horizontal directions are shown for 1% damping of the component in Fig.--.00 l.2. and . A combination of plate elements with highorder shape functions (in bending and in stretching) for the walls and the floors and of three-dimensional beam elements for the columns is used.00 QIO 10.PONT B QOOI L_~~~. For instance. as discussed in Section 4.Sec.L_~--'--":l.. Moving from point A to point B.(uL~__.
This action might not be possible to undo. Are you sure you want to continue?
We've moved you to where you read on your other device.
Get the full title to continue listening from where you left off, or restart the preview.