William J. Hall, Editor
Library of Congress Cataloguing in Publication Data
WOLF, JOHN P. (date) Dynamic soil-structure interaction.
(Prentice-Hall international series in civil engineer- ing and engineering mechanics)
Bibliography: p. Includes index. 1. Soil dynamics. 2. Structural dynamics. 3. Founda­
tions. 4. Earthquake engineering. I. Title. 11. Series: Prentice-Hall civil engineering and engineering mechanics series. TA710.W59 1985 624.1'5136 ISBN 0-13-221565-9
Editorial/production supervision and interior design: Theresa A. Soler Manufacturing buyer: Anthony Caruso
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CONTENTS
PREFACE
1.1 Objective of Soil-Structure Interaction Analysis 1.2 Types of Prescribed Loadings,
in Particular Seismic Excitation 2 1.3 Effects of Soil-Structure Interaction 4 1.4 Assumed Linearity 7 1.5 Types of Problems Encompassed 8 1.6 Direct and Substructure Methods 9 1.7 Organization of Text 11
Summary 11
2. FUNDAMENTALS OF DISCRETE DYNAMIC SYSTEM
2.1 Equations of Motion in the Time Domain 13 2.2 Transformation to Modal Amplitudes 14 2.3 Equations of Motion for Harmonic Excitation 14 2.4 Correspondence Principle 15 2.5 Discrete Fourier Transform 16 2.6 Method of Complex Response 17
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13
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3. BASIC EQUATION OF MOTION
3.1 Formulation in Total Displacements 18 3.1.1 Flexible Base, 18 3.1.2 Rigid Base, 23 3.1.3 Special Foundations, 26
3.2 Kinematic and Inertial Interactions 28 3.2.1 Flexible Base, 28 3.2.2 Rigid Base, 30
3.3 Applied Loads and Their Characterization 33 3.3.1 Rotating Machinery, 33 3.3.2 Impact, 33 3.3.3 Earthquake, 34
3.4 Introductory Example 38 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 50 Problems 52
4. MODELlNG OF STRUCTURE
4.1 General Considerations 69 4.1.1 Frequency Content and Spatial Variation
of Applied Loads, 70 4.1.2 Decoupling of Subsystem, 71
4.2 Spatial Variation of Seismic Loads 72 4.2.1 Free-Field Motion, 72 4.2.2 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
4.3 Direct Discretization 83 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
Contents
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69
Contents
4.4 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 101 Problems 102
5. FUNDAMENTALS OF WAVE PROPAGATION
5.1 One-Dimensional Wave Equation 114 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
5.2 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
5.3 Dynamic-Stiffness Matrix for Out-of-Plane Motion 140 5.3.1 Types of Waves, 140 5.3.2 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
5.4 Dynamic-Stiffness Matrix for In-Plane Motion 146 5.4.1 Types of Waves, 146 5.4.2 Transfer- and Dynamic-Stiffness Matrices
of Layer and of Half-Space, 148 5.4.3 Special Cases, 151 5.4.4 Loaded Layer, 153
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5.5 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 164 Problems 166
6. FREE-FIELD RESPONSE OF SITE
6.1 Definition of Task 179 6.1.1 Three Aspects When Determining Seismic
Environment, 179 6.1.2 Location of Control Point, 180
6.2 Amplification, Dispersion, and Attenuation 182 6.2.1 Dynamic-Stiffness Matrix of Site, 182 6.2.2 Site Amplijicationfor Body Waves, 183 6.2.3 Surface Waves, 185 6.2.4 Amplijications, 186 6.2.5 Apparent Velocity and Decay Factors, 187
6.3 Half-Space 188 6.3.1 Incident SH-Waves, 189 6.3.2 Incident P-Waves, 189 6.3.3 Incident SV-Waves, 191 6.3.4 Rayleigh Waves, 192 6.3.5 Displacements and Stresses versus Depth, 193
6.4 Single Layer on Half-Space 195 6.4.1 SH-Waves, 195 6.4.2 Love Waves, 198 6.4.3 Physical Interpretation of Variables, 200 6.4.4 P- and SV-Waves, 202 6.4.5 Rayleigh Waves, 203
6.5 Parametric Study of Out-of-Plane Motion 204 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
Contents
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Contents
6.6 Parametric Study of In-Plane Motion 219 6.6.1 Scope of Investigation, 219 6.6.2 Vertically Incident SV- and P-Waves, 220 6.6.3 Inclined P- and SV-Waves, 221
Incident P-Waves, 221 Incident SY-Waves, 224 Combinations of Incident P- and SY-Waves, 228
6.6.4 Rayleigh Waves, 232 Dispersion and Attenuation, 232 Disp1acements and Stresses versus Depth, 238 Scattered Motion, 241
6.7 Soft Site 242 6.7.1 Description of Site and of Control Motion, 242 6.7.2 Vertically Incident and Inclined SH-Waves, 244 6.7.3 Vertically Incident and Inclined P- and SV-Waves, 245 6.7.4 Love Waves, 249 6.7.5 Rayleigh Waves, 251 6.7.6 In-Plane Displacements and Stresses versus Depth, 253 6.7.7 Scattered Motion, 257
6.8 Rock Site 259 6.8.1 Description of Site and of Control Motion, 259 6.8.2 Love Waves, 259 6.8.3 Rayleigh Waves, 261 6.8.4 Assumed Wave Patterns, 263 6.8.5 Scattered Motion, 265
Summary 266 Problems 271
7. MODELlNG OF SOIL
7.1 General Considerations 273 7.1.1 Dynamic-Stiffness Matrices of Soil, 273 7.1.2 Elementary Boundaries, 275 7.1.3 Local Boundaries, 275 7.1.4 Consistent Boundaries, 279 7.1.5 Sommerfeld's Radiation Condition, 279 7.1.6 Use of Analytical Solution, 281
7.2 Dynamic-Stiffness Coefficients of Surface Foundation 282 7.2.1 Weighted Residual Formulation, 282 7.2.2 Green's Influence Function
for Two-Dimensional Case, 285 7.2.3 Green's Influence Function for Axisymmetric Case, 287 7.2.4 Element Size and Numerical Integration, 292 7.2.5 Nondimensionalized Spring and Damping Coefficients, 293
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7.3
7.4
Three-Dimensional Rigid Basemat (Disk Foundation) 301 7.4.1 Dynamic-Stiffness Coefficients, 301 7.4.2 Two-Dimensional Versus Three-Dimensional
Modeling, 307 7.4.3 Scattered Motion, 310
7.5 Dynamic-Stiffness Coefficients of Embedded Foundation 312 7.5.1 Significance of Boundary-Element Method, 312 7.5.2 Example to Illustrate Basic Concepts
of Boundary-Element Method, 313 Analytical Solution of Illustrative Example, 313 Weighted-Residual Technique, 316
7.5.3 Dynamic-Stiffness Coefficients of Soil Domains Calculated by Boundary-Element Method, 318 Reference Soil System, 318 Generalization to Three Dimensions for System Ground, 319 System Excavated Part, 321 System Free Field, 321
7.5.4 Green's Influence Functions, 322 7.5.5 Pile Foundation, 325
7.6 Embedded Rectangular Foundation 327 7.6.1 Scope of Investigation, 327 7.6.2 Green's Influence Function, 328 7.6.3 Complete Set of Results, 329 7.6.4 Properties of Dynamic-Stiffness Coefficients
of Excavated Part and System Free Field, 329 7.6.5 Parametric Study System Ground, 333 7.6.6 Parametric Study System Free Field, 334
7.7 Dynamic-Stiffness Coefficients of Adjacent Foundations 336 Summary 340 Problems 343
Contents
8. ALTERNATIVE FORMULATION OF EQUATION OF MOTION 369
8.1 Direct Analysis of Total Structure-Soil System 369 8.1.1 Equation of Motion in Time Domain, 369 8.1.2 Equation of Motion in Frequency Domain, 371
8.2 Substructure Analysis with Flexible Base 373 8.2.1 Basic Equation of Motion in Total Displacements, 373 8.2.2 Base Response Motion Relative to Free Field, 375
Contents xi
8.2.4 Quasi-static Transmission of Base Response Motion, 379 8.2.5 Transformation to Modal Amplitudes
of Fixed-Base Structure, 380 8.3 Substructure Analysis with Rigid Base 383
8.3.1 Basic Equation of Motion in Total Displacements, 383 8.3.2 Base Response Motion Relative to Scattered Motion, 384 8.3.3 Quasi-static Transmission of Scattered Motion, 384 8.3.4 Quasi-static Transmission of Base Response Motion, 386 8.3.5 Transformation to Modal Amplitudes
of Fixed-Base Structure, 386 8.4 Approximate Formulation in Time Domain 387
8.4.1 Basic Equation of Motion in Total Displacements, 387 8.4.2 Quasi-static Transmission
of Base Response Motion, 388 8.4.3 Transformation to Modal Amplitudes
of Total System, 388 8.4.4 Transformation to Modal Amplitudes
of Fixed-Base Structure, 389 8.5 Analysis of Nonlinear Structure with Linear Soil
(Far Field) 390 8.5.1 Types of Nonlinearities, 390 8.5.2 Equation of Motion in Time Domain Using
Convolution Integrals, 391 8.5.3 Transformation of Stiffness Matrix, 392 8.5.4 Rod with Exponentially Increasing Area, 393 8.5.5 Disk on Half-Space and on Layer, 395
Summary 396 Problems 398
9.1 Evaluation of Interaction Effects 405 9.1.1 Dimensionless Parameters, 405 9.1.2 Equivalent One-Degree-of-Freedom System, 406 9.1.3 Depth of Layer, 407 9.1.4 Mass of Base, 408 9.1.5 Bridge Structure, 409 9.1.6 Second Mode, 411
9.2 Effects of Horizontally Propagating Waves 412 9.2.1 Investigated Structures, 414 9.2.2 Inclined SH-Waves, 415 9.2.3 Inclined P- and SV-Waves, 419 9.2.4 Rayleigh Waves, 423
405
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9.3 Examples from Actual Practice 426 9.3.1 Through-Soil Coupling of Reactor Building and
of Reactor-Auxiliary and Fuel-Handling Building, 426 9.3.2 Pile-Soil-Pile Interaction, 428 9.3.3 Horizontally Propagating Waves
on Hyperbolic Cooling Tower, 433 9.3.4 Horizontal(v Propagating Waves on Nuclear Island
with Aseismic Bearings, 436 9.4 Concluding Remarks 441
9.4.1 Needfor Adequate Consideration, 441 9.4.2 Modeling Aspects, 442 9.4.3 Recorded Field Performance, 443
Summary 447 Problems 450
INDEX
Contents
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459
PREFACE
In the seismic analysis of a structure founded on rock, the motion experienced by the base is essentially identical to that occurring in the same point before the structure is built. The calculation can thus be restricted to the structure excited by this specified motion. In the case of a soft site, two important modi­ fications arise for the same incident seismic waves from the source. First, the free-field motion at the site in the absence of the structure is strongly affected. Second, the presence of the structure in the soil will change the dynamic system from the fixed-base condition. The structure will interact with the surrounding soil, 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. Soil-structure interaction is also important for other load­ ing cases (e.g., arising from unbalanced mass in rotating machinery).
The effect of soil-structure interaction is recognized to be important and cannot, in general, be neglected. 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, especially nuclear-power plants, very complex analyses are required which are based on recent research results, some of which have not been fully evaluated. 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. This rigorous procedure, based on wave propagation, makes use of such familiar concepts as the direct­ stiffness method of structural analysis and also applies such recent develop-
xiii
xiv Preface
ments as the boundary-element method. The results of vast parametric studies are included, which can be used directly by the analyst. Examples from actual practice demonstrate that these methods are being applied. Besides the rigorous procedure, simple approximate methods are developed which nevertheless capture the essential features of soil-structure interaction. This allows the analyst to perform preliminary calculations with simplified models to determine the key parameters before starting with complicated computations.
Many aspects of this field are so well established that it is difficult to assign credit for them. Credit references have thus been restricted to those necessary for copyright reasons. Sincere apologies are offered to anybody who might feel offended by being left out. The author has been influenced over the years by the research published by many authorities; to name just a few (in alphabetical order): Professors E. Kausel, J. E. Luco, J. Lysmer, J. M. Roesset, and A. S. Veletsos.
At the end of each chapter a summary is included. Then problems are formulated which not only allow the student to corroborate full understanding of the analytical techniques, but which lead to new insights into the various aspects. The problems thus form an important part of the text. The reader who has no intention of solving the problems in detail will find it advantageous to glance through them. As a rule, a detailed solution procedure is provided and the results, in many cases quite important on their own merit, are presented in the form of figures.
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. It is offered for advanced undergraduate and graduate students in civil engineering. As a pre­ requisite some knowledge of structural dynamics is essential which can be acquired in a one-term course.
The important contributions of the author's colleagues and students are gratefully acknowledged. In particular, the author would sincerely like to thank Messrs. G. von Arx, K. Bucher, G. Darbre, P. Obernhuber, P. Skrikerud, D. Somaini, and B. Weber for their dedicated efforts. Finally, the author is indebted to Electrowatt Engineering Services Ltd. for its financial support.
John P. Wolf
1.1 OBJECTIVE OF SOIL-STRUCTURE INTERACTION ANALYSIS
Structural dynamics deals with methods to determine the stresses and displace­ ments ofa structure subjected to dynamic loads. The dimensions of the structure are finite. It is thus rather straightforward to determine a dynamic model with a finite number of degrees of freedom. The corresponding dynamic equations of motion of the discretized structure are then formulated, and highly developed methods for solving them are readily available. In general, however, the structure will interact with the surrounding soil. It is thus not permissible to analyze only the structure. It must also be considered that in many important cases (e.g., earthquake excitation) the loading is applied to the soil region around the structure; this means that the former has to be modeled anyway. The soil is a semi-infinite medium, an unbounded domain. For static loading, a fictitious boundary at a sufficient distance from the structure, where the response is expected to have died out from a practical point of view, can be introduced. This leads to a finite domain for the soil which can be modeled similarly to the structure. The total discretized system, consisting of the structure and the soil, can then be analyzed straightforwardly. However, for dynamic loading, this procedure cannot be used. The fictitious boundary would reflect waves origi­ nating from the vibrating structure back into the discretized soil region instead of letting them pass through and propagate toward infinity. This need to model the unbounded foundation medium properly distinguishes soil dynamics from structural dynamics.
The fundamental objective of the analysis of soil-structure interaction is
1
2
Figure 1-1 Fundamental objective of analysis of soil-structure interaction.
illustrated in Fig. 1-1. A specified time-varying load acts on a structure embedded in layered soil. The dynamic response of the structure and, to a lesser extent, of the soil is to be calculated, taking into account the radiation of energy of the waves propagating into the soil region not included in the model. The discretized model of the structure and of the soil domain is shown schematically. The semi­ infinite soil domain, represented by a layered half-space, represents an energy sink.
1.2 TYPES OF PRESCRIBED LOADINGS, 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, impact loads [e.g., the crash of an aircraft onto a nuclear power plant (which can govern the design, although the probability of occurrence is small)], blast loadings, and so on. Probably the most important loading, and definitely the most complicated to analyze, is earthquake excitation, which acts primarily on the soil.
Earthquakes are caused by a sudden energy release in a volume of rock lying on a fault. This source is normally located a large distance away and at a significant depth from the site. 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, of course, not the case), it would still be impossible to model all aspects because of the size of the dimensions compared to those of the structure. In any event, the many uncertainties involved make it meaningless to analyze the complete earthquake-excitation problem.
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. The procedure normally followed can be characterized as follows: In the control
Sec. 1.2 Types of Prescribed Loadings, in Particular Seismic Excitation 3
point located at the surface of the so-called free field (site prior to construc­ tion, i.e., without the excavation and without the structure), the earthquake motion (e.g., the acceleration as a function of time) is specified (Fig. 1-1). To be able to do this, the seismic hazard of the region is assessed. 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. In this evaluation the type of the structure will play an important role. For a poten­ tially hazardous structure such as a nuclear power plant this probability will be selected to be very small. 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.g., a hospital or a fire station). This then allows the determination of the most important parameter, that which is assumed to characterize the motion (e.g., the peak ground acceleration). The other param­ eters, such as the duration of the motion and the frequency content, are selected-mostly empirically-by the engineering seismologist, based on past earthquakes of the region. All this should allow the source mechanism, the transmission path, the local geology, and the soil conditions at the site to be taken into account very approximately. These activities leading to the definition of a design motion in a selected control point precede the actual soil-structure interaction analysis. The largest uncertainties arise in these preliminary phases. Quite arbitrary assumptions with far-reaching consequences have to be made. It is outside the scope of this work to discuss these very important aspects of earthquake engineering in any depth.
To analyze soil-structure interaction, it is sufficient to think of the pre­ scribed motion as being derived from an observed record at this very site, or at least at a similar site. Starting from this control motion in one point, the earth­ quake motion throughout the free field (characterized by its spatial and temporal variation) is calculated. As will become apparent when discussing the site response, this can again be achieved only by making quite arbitrary and stringent assumptions regarding the wave pattern in the control point. These assumptions, which will, of course, also be influenced by the opinion of the engineering seismologist as described above, will affect the characteristics of the free-field response as far as the amplitudes and the frequency content away from the control…