boundary element methods for soil structure interaction

Tương tác mềm giữa kết cấu và nền đất

Boundary Element Methods for Soil-Structure Interaction

Edited by

W.S HALL

University of Teesside, Middlesbrough, United Kingdom

and

G OLIVETO

University of Catania, Catania, Italy

KLUWER ACADEMIC PUBLISHERS

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Print ISBN: 1-4020-1300-0

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PART 1 SOIL-STRUCTURE INTERACTION

1 TWENTY FIVE YEARS OF BOUNDARY ELEMENTS FOR DYNAMIC SOIL-STRUCTURE INTERACTION

J Dominguez (Seville)

191316202428

3134353637383942

Seismic Response of Foundations

Dynamic Soil-Water-Structure Interaction Seismic

DAM ON A RIGID FOUNDATION

RESERVOIR FULL OF WATER

DAM ON A FLEXIBLE FOUNDATION

EMPTY RESERVOIR

DAM ON A FLEXIBLE FOUNDATION

RESERVOIR FULL OF WATER

BOTTOM SEDIMENT EFFECTS

STRIP FOUNDATIONS

AXISYMMETRIC FOUNDATIONS

FOUNDATIONS ON SATURATED

POROELASTIC SOILS

49515657

61616263

646565666666676868686969696970707173747474

DAM ON A FLEXIBLE FOUNDATION

RESERVOIR FULL OF WATER

TRAVELLING WAVE EFFECTS

POROELASTIC SEDIMENT EFFECTS

7 References

2 COMPUTATIONAL SOIL-STRUCTURE INTERACTION

D Clouteau (Paris), D Aubry (Paris)

HETEROGENEITIES IN THE BEM

TIME DOMAIN BEM/ FREQUENCY

2.4.1

2.4.2

Field Equations Coupling Equations

2.5 VARIABILITY ON THE PARAMETERS

2.5.1

2.5.2

Stochastic Model of the Soil Parameters Stochastic Model for the Applied Loads

2.6.1 Wellposedness and Approximation

THE SFSI EQUATION

FEM AND REDUCTION TECHNIQUES

REGULARIZED BOUNDARY INTEGRAL

EQUATION IN A LAYERED HALF-SPACE

BEM ON INVARIANT DOMAINS

NON INVARIANT UNBOUNDED

SOLUTION IN THE SLOWNESS SPACE

FAST INVERSE HANKEL TRANSFORM

7.3

7.4

SSI ON A RANDOM SOIL

SFSI FOR PERIODIC SHEET-PILES

74757677787980808182

838487

8789

899092929292939495959696969899100103

HANKEL TRANSFORMRECONSTRUCTION FORMULAE

3 THE SEMI-ANALYTICAL LESS SCALED BOUNDARY FINITE-ELEMENT METHOD

FUNDAMENTAL-SOLUTION-TO MODEL UNBOUNDED SOIL

J P Wolf (Lausanne), C Song (Sydney)

DYNAMIC STIFFNESS MATRIX

HIGH-FREQUENCY ASYMPTOTIC EXPANSION

OF DYNAMIC STIFFNESS MATRIX

Mechanically Based Derivation

Analytic Solution in Frequency Domain

Numerical Solution in Frequency and Time Domains

Extensions

8.1 INCOMPRESSIBLE ELASTICITY

107107109112112114122

122123124

124125125

127129130134134

135136137139140141144

148

149

149

REDUCED SET OF BASE FUNCTIONS

TWO-DIMENSIONAL LAYERED UNBOUNDED

IN-PLANE MOTION OF SEMI-INFINITE WEDGE

IN-PLANE MOTION OF CIRCULAR CAVITY IN

4 BEM ANALYSIS OF SSI PROBLEMS IN RANDOM MEDIA

G D Manolis, C Z Karakostas (Thessaloniki)

EARTHQUAKE SOURCE MECHANISM

3 Integral Equation Formulation

163165

168

172

175179180180181181182183184184185187187187188190191OUT-OF-PLANE MOTION OF CIRCULAR CAVITY

IN FULL PLANE WITH HYSTERETIC DAMPING

BRIEF REVIEW OF LITERATURE ON

BUILDING STRUCTURES AND SSI

BRIEF REVIEW OF LITERATURE ON

BRIDGES AND SSI

2 Seismic Design of Building Structures Including SSI

RESPONSE SPECTRUM ANALYSIS WITH SSI

NUMERICAL EXAMPLE: BUILDING

STRUCTURE

192192194195195199201201203206207208211213213216216216217222223227228

235235

238238238239246247

ITERATIVE ANALYSIS PROCEDURE

MODELLING ABUTMENT STIFFNESS FOR

3.6 REMARKS AND CONCLUSIONS

4

5

References

Appendix

PART 2 RELATED TOPICS AND APPLICATIONS

6 BEM TECHNIQUES FOR NONLOCAL ELASTICITY

NONLOCAL HYPERELASTIC MATERIAL

LINEAR LOCAL ELASTICITY WITH

A Boundary/Domain Stationarity Principle

Symmetric Galerkin BEM Technique

Nonsymmetric Collocation BEM Technique

Conclusions

249251251

251251253

255257

260261264268269270272

275277279281284284

285287290293294295

7 BEM FOR CRACK DYNAMICS

Time Domain Method (TDM)

Laplace Transform Method (LTM)

Dual Reciprocity Method (DRM)

Cauchy and Hadamard Principal-Value Integrals

8 SYMMETRIC GALERKIN BOUNDARY ELEMENT

ANALYSIS IN THREE-DIMENSIONAL LINEAR-ELASTIC FRACTURE MECHANICS

A Frangi (Milan), G Maier(Milan), G Novati(Trento),

FRACTURES IN INFINITE DOMAINS

EDGE CRACKED BAR

CIRCULAR EDGE CRACK IN A PLATE

QUARTER ELLIPTIC CORNER CRACK

315315316320321324326327328331336

339339341

342343344

9 NUMERICAL SIMULATION OF SEISMIC WAVE

SCATTERING AND SITE AMPLIFICATION, WITH

APPLICATION TO THE MEXICO CITY VALLEY

L C Wrobel (London), E Reinoso (Mexico City),

BEM Formulation for SH Waves

BEM Formulation for P, SV and Rayleigh Waves

Observed Amplification in the Mexico City Valley

One-dimensional Response in the Mexico City Valley

Two-dimensional Modelling Using the BEM

Conclusions

References

345345347349351355359364365369373377

INDEX

CNRS UMR 8370, École Centrale

de Paris, Châtenay Malabry,

France.

D CLOUTEAU

Laboratoire de Mécanique de

Sols-Structures-Matériaux,

CNRS UMR 8370, École Centrale

de Paris, Châtenay Malabry,

Politecnico di Milano, P.za L.

da Vinci 32, 20133 Milan, Italy.

C.Z KARAKOSTAS

Institute of Engineering Seismology and Earthquake Engineering,

P.O Box 53, GR 551 02 Finikas, Thessaloniki, Greece.

G MAIER

Department of Structural Engineering,

Politecnico di Milano, P.za L.

da Vinci 32, 20133 Milan, Italy.

G.D MANOLIS

Department of Civil Engineering, Aristotle University,

P.O Box 502, GR540 06, Thessaloniki, Greece.

Università di Palermo, Viale della

Scienze, 90128 Palermo, Italy.

Civil Engineering Department, National Technical University Of Athens, Greece.

C SONG

School of Civil and Environmental Engineering, University of New South Wales, Sydney, NSW 2052, Australia.

J P WOLF

Institute of Hydraulics and Energy, Department of Civil Engineering, Swiss Federal Institute of Technology, Lausanne, Switzerland.

L C.WROBEL

Department of Mechanical Engineering, Brunel University, Uxbridge, UB8 3PH, UK.

W S HALL

School of Computing and Mathematics, University of

Teesside, Middlesbrough, TS1 3BA UK

G OLIVETO

Division of Structural Engineering, Department of Civil and Environmental Engineering, University of Catania, Viale A Doria 6, 95125 Catania, Italy

Soil-Structure Interaction is a challenging multidisciplinary subjectwhich covers several areas of Civil Engineering Virtually everyconstruction is connected to the ground and the interaction between theartefact and the foundation medium may affect considerably both thesuperstructure and the foundation soil The Soil-Structure Interactionproblem has become an important feature of Structural Engineering withthe advent of massive constructions on soft soils such as nuclear powerplants, concrete and earth dams Buildings, bridges, tunnels andunderground structures may also require particular attention to be given

to the problems of Soil-Structure Interaction Dynamic Soil-StructureInteraction is prominent in Earthquake Engineering problems

The complexity of the problem, due also to its multidisciplinarynature and to the fact of having to consider bounded and unboundedmedia of different mechanical characteristics, requires a numericaltreatment for any application of engineering significance The BoundaryElement Method appears to be well suited to solve problems of Soil-Structure Interaction through its ability to discretize only the boundaries

of complex and often unbounded geometries Non-linear problems whichoften arise in Soil-Structure Interaction may also be treatedadvantageously by a judicious mix of Boundary and Finite Elementdiscretizations The purpose of this state of the art book on “BoundaryElement Methods for Soil-Structure Interaction” is to review progress

made in the applications of the Boundary Element Method in the solution

of Soil-Structure Interaction for the scientific communities of Structuraland Earthquake Engineering The object is to provide these communitieswith a wealth of efficient computational methods for the solution ofproblems which would otherwise require less accurate and/orcomputationally more expensive procedures

The book contains nine chapters from leading European experts

on Boundary Element Methods and Soil-Structure Interaction Its conceptoriginated at the EUROMECH Colloquium 414 on Boundary ElementMethods for Soil-Structure Interaction which took place in Catania, Italyfrom 21 to 23 June, 2000 at which the authors made short presentations

on the state-of-the-art in their particular area of expertise Since that timeeach author has developed these first ideas into a significant contribution

to the subject Scientific papers also presented at the Colloquium havealready appeared as a Special Issue of Meccanica (Advances in BoundaryElement Methods in Soil-Structure Interaction and Other Applications,Volume 36, Issue 4, 2001)

The book is organised into two parts Part 1, containing five of thenine chapters that constitute the book, deals with problems specific toSoil-Structure or Fluid-Structure-Soil Interaction Part 2, containing theremaining four, is devoted to related topics and applications thatnevertheless are of interest to specific aspects of Soil-StructureInteraction

In Part 1 the first Chapter is by Professor J Dominguez of theUniversity of Seville and contains a review of 25 years of dynamic Soil-Structure Interaction The material is introduced from an engineeringpoint of view and after a brief introduction of the Soil-StructureInteraction problem deals with the dynamic stiffness of foundations, theseismic response of foundations and with seismic problems related togravity and arch dams In particular the situations of empty and fullreservoir are covered and the effects of bottom sediments and travellingwaves are considered

The second Chapter by Dr D Clouteau and Professor D Aubry ofthe University of Paris is devoted to formulation and computationalaspects of the Soil-Structure Interaction problem An introduction isprovided to briefly describe the physical and numerical models used inthe treatment of the problem A physical and mathematical formulation

of the problem is provided in the second section Then the concept ofdomain decomposition is introduced, together with several techniques

useful for the reduction of degrees of freedom to be considered in theanalyses The fourth section is specifically dedicated to boundary integralequations, the Boundary Element Method and to coupling with othernumerical techniques, particularly to the FEM-BEM coupling The fifthsection considers unbounded interfaces, invariant operators and invariantdomains in connection with their application to specific problems Thenext section deals with Green functions in layered half-spaces Theseventh and final section of this Chapter is devoted to applications Thetechniques described in the previous sections are therefore applied toSoil-Structure-Fluid Interaction in arch dams and sheet-piles, Soil-Structure Interaction for a nuclear reactor resting on a layered half-spacewith random heterogeneities and to geophysics boreholes Topographiceffects and the characteristic city effect are also described.

The third Chapter by Dr J P Wolf of the University of Lausanneand Dr C Song of the University of Sydney presents an alternativeapproximate approach to the solution of the dynamic Soil-StructureInteraction problem which is essentially based on the Boundary ElementMethod but does not require fundamental solutions This method isappealing when the fundamental solutions are not known or when theyare difficult to evaluate After an introduction to the literature on themethod, the dynamic unbounded Soil-Structure Interaction problem isdefined and the unknown quantities are identified The next sectionpresents the main concept on which the scaled-boundary element method

is based In sections 4 and 5 two derivations of this approximate methodare presented with the first being mathematically motivated and thesecond mechanically oriented In section 6 the analytical solution in theradial direction is explicitly provided in the frequency domain while insection 7 the corresponding numerical solutions are formulated in thefrequency and time domains Several possible extensions of theprocedure are discussed in section 8 while a set of numerical applications

is reported in section 9 Section 10 presents some results obtained forbounded media and the final section of this Chapter discusses problemsconnected with the implementation of the method, its advantages and itslimitations

The fourth Chapter by Professor G D Manolis & Dr C ZKarakostas of the University of Thessaloniki addresses the problem ofSoil-Structure Interaction in random media The first section presents anintroduction to the problem and an outline of the presentation of thematerial The second section presents a review of the literature on soil

dynamics and dynamic Soil-Structure Interaction when the soil isconsidered as a random medium Section 3 presents the generalformulation of the problem in the form of a stochastic integral equation, aformal analytical solution and a closure approximation for zero meanforcing function Section 4 addresses the problem of forced vibration inrandom soil media After reporting an analytical solution for a simpleproblem, a BEM based approximate solution making use of theperturbation theory is developed and illustrated by means of a numericalexample The fifth section is used for the formal development of a BEMformulation based on the perturbation theory Two examples concerningunlined circular tunnels are used to show the excellent performance ofthe theory in the presence of small random perturbations For largerandom perturbations a different BEM formulation based on orthogonalpolynomial expansions is developed in section 6 An exampleconsidering the propagation of an SH wave in a random medium showswhy small perturbation theory cannot reliably predict results when themedium randomness is large.

The fifth Chapter by Professor C C Spyrakos of the University ofAthens considers Soil-Structure Interaction as it is currently used inengineering practice with reference to buildings and bridges Initially twobrief literature reviews are given separately for buildings and bridges.Then the second section is dedicated to the seismic design of buildingsincluding the effects of Soil-Structure Interaction Two design proceduresderived from seismic codes and guidelines are presented and applied tothe case of an actual building showing the effects of Soil-StructureInteraction on the structural response and on the resulting design Thethird section is devoted to the seismic analysis of bridges including SSI

It starts by providing information on the modelling of the variousstructural parts, especially soil and abutments, and continues bypresenting two analysis procedures: a linear iterative static procedure and

a non-linear static procedure The section concludes with a numericalapplication illustrating the linear iterative procedure and with aparametric study considering various soil types Seismic codes andguidelines also mainly inspire this Chapter

The sixth Chapter, by Professor C Polizzotto of the University ofPalermo (which is the first of Part 2 covering related topics andapplications of the Boundary Element Method) considers the problem ofnon-local elasticity This problem is of interest in Soil-StructureInteraction because some classical soil models are non-local and because

the singularity problems of local elasticity vanish if the non-localapproach is adopted After the introduction, the Eringen non-local elasticmodel is reviewed in section 2 The third section introduces a non-localhyperelastic material through thermodynamically consistent constitutiveequations The fourth section discusses the static boundary value problemfor such a material proving that, whenever it exists, the solution isunique Moreover the problem is formulated as a classic linear elastic one

of the local type with an unknown initial strain field accounting for thenon-local behaviour In section 5 some variational principles of localelasticity are extended to the non-local model considered while in section

6 a stationary principle is provided in terms of boundary integrals Thesymmetric Galerkin and non-symmetric collocation BEM formulationsfor non-local elasticity are presented in the next two sections 7 and 8

In the seventh Chapter by Professor M H Aliabadi of theUniversity of London the application of the Boundary Element Method

to crack problems in dynamic fracture mechanics is presented After areview of the literature on the subject provided in the introductorysection, a formulation of the dual Boundary Element Method for three-dimensional crack problems in the time domain is presented in section 2together with a numerical solution procedure The third section presents aformulation of the problem in the Laplace transformed domain with theDurbin method used to bring the solution back into the time domain Inthe fourth section the dual reciprocity BEM method is presented leading

to a system of coupled second order ordinary differential equations whichcan be solved by direct time integration methods The fifth section points

to the singularity problems that must be addressed in each of thepreviously presented formulations and provides the necessary lead to therelevant literature on the subject Finally two examples are presented inthe section on numerical applications where the results obtained by thethree previously mentioned methods are compared among themselvesand against solutions available in the literature

The eighth Chapter by Professors A Frangi and G Maier of MilanPolytechnic, Professor G Novati and Dr R Springhetti of the University

of Trento is devoted to the application of the symmetric GalerkinBoundary Element Method (SGBEM) to the solution of three-dimensional linear elastic problems in Fracture Mechanics A briefintroductory section reviews the literature on the subject and focuses onthe particular problem in hand In the second section two relevantboundary integral equations are formulated for displacements and

tractions A special regularising procedure is then applied to remove highorder singularities arriving at a couple of self-adjoint boundary integralequations containing only weakly singular terms The symmetricalstructure of the problem can be maintained also in the discrete boundaryelement formulation if a Galerkin interpolation scheme is used The thirdsection deals with the numerical evaluation of the weakly singularintegrals Specific integration formulae, based on appropriate co-ordinatetransformations, are provided for the cases of coincident elements,adjacent elements with a common edge and elements having in commononly a vertex In the fourth section several numerical applications arecarried out and compared against results available in the literature Thefirst three examples refer to typical fractures in an infinite medium: apenny shaped crack, an elliptical crack and a spherical-cap crack Theresults provided in the form of displacement discontinuities or stressintensity factors compare favourably against analytical or numericalresults available in the literature The other three examples refer to edgecracks in finite domains: an edge crack in a bar, a circular edge crack in aplate, a quarter elliptic corner crack in a plate Once again the resultscompare very well with others available in literature.

The ninth and final Chapter of the book by Professor L C Wrobel

of Brunel University, Dr E Reinoso of the University of Mexico City andProfessor H Power of the University of Nottingham presents animportant application of the BEM to a specific and relevant topic indynamic Soil-Structure Interaction and earthquake engineering, typicallythe problem of site or local amplification effect An introductory sectionexplains how local amplification effects are predicted by one-dimensional theories and why more comprehensive two- and three-dimensional theories may be required in many practical applications Thesame section also gives a review of the subject showing how the problemhas been dealt with in the literature The second section summarises themain results of the theory of wave propagation in an elastic,homogeneous and isotropic half space in a way suitable for the envisagedapplications In particular P, SH, SV and Rayleigh waves are described.Section 3 presents a two-dimensional BEM formulation for SH incidentwaves in canyons and valleys Application of the model to the MexicoCity valley situation shows that the one-dimensional theory predicts goodresults towards the centre of the valley, but is not adequate towards theedges where the response is much more irregular The fourth sectionpresents a similar formulation for P, SV and Rayleigh incident waves

The next three sections deal, respectively, with the observedamplifications in the Mexico City valley, with the amplificationspredicted by the one-dimensional theory and with those predicted by thetwo-dimensional BEM model The conclusion is that, although the one-dimensional theory can often predict the average amplificationbehaviour, in many cases the two-dimensional theory is more adequateand in some cases only a full three-dimensional model can explain thecomplete behavioural pattern.

Overall the book provides an authoritative guide to the literature

on the subject covered and is expected to be an invaluable tool forpractising engineers, students and scholars in the fields of structural,geotechnical and earthquake engineering Engineers and students mayreadily locate the material or methods available for the solution of theirparticular problem while scholars may discover methods previously notconsidered for the particular application being considered The bookshould also be of interest to the larger community of appliedmathematicians and software developers in seeing a field where theBoundary Element Method can provide a wealth of relevant and efficientsolutions Finally the book can be used as a starting point for researchand for the investigation of unsolved problems in Soil-Structure andFluid-Structure-Soil Interaction, particularly non-linear coupled problemswhich could be advantageously approached by means of BoundaryElement Methods

W S Hall, Middlesbrough

G Oliveto, CataniaFebruary 2003

The editors would like to take this opportunity to thank CRUI, the BritishCouncil, the European Mechanics Society, GNDT-CNR and MIUR fortheir support over a number of years, first for a bilateral research projectbetween the University of Catania and the University of Teesside Thiseventually lead to the Catania EUROMECH Colloquium 414 in June

2000, at which were laid the foundations for the present volume andrelated special issue of the journal Meccanica

SOIL-STRUCTURE INTERACTION

TWENTY FIVE YEARS OF BOUNDARY ELEMENTS FOR

DYNAMIC SOIL-STRUCTURE INTERACTION

J DOMÍNGUEZ

Escuela Superior de Ingenieros, Universidad de Sevilla,

Camino de los Descubrimientos s/n 41092 Sevilla, SPAIN.

1 Introduction

This chapter is intended to show the applicability of the Boundary ElementMethod (BEM) to Dynamic Soil Structure Interaction (DSSI) problems Toshow this a review of the work carried out by the author and his co-workersduring the last twenty five years is presented Reference is made to thework done by many others, however, the chapter is not a state of the artreview of all the work done in the field of numerical dynamic soil-structureinteraction

The behaviour of structures based on compliant soils and subject todynamic actions may depend to a large extent on the soil properties and onthe foundation characteristics The analysis of this behaviour requires amodel which takes into account not only the structure but also the soil andthe dynamic interaction forces existing between them The first DSSIproblems, studied during the late thirties, were related to the vibration oflarge machines mounted on massive foundations The dynamic behaviour

of these machines could only be understood by taking into account thedynamic interaction between the soil and the machine foundation Tallbuildings, or any other structure based on the ground, subject to wind loadsare also examples of problems where DSSI effects may be important andwhere the excitation is directly applied to the structure The analysis ofstructures under the effects of earthquakes leads to a second kind of DSSIproblem where the excitation is transmitted through the soil

To show in a simple manner the important effects of DSSI on thedynamic response of ground based structures the following simple problem

is analyzed Consider a single degree of freedom system consisting of a

1

W.S Hall and G Oliveto (eds.), Boundary Element Methods For Soil-Structure Interaction, 1–60.

© 2003 Kluwer Academic Publishers Printed in the Netherlands.

concentrated mass M which can move horizontally (Figure 1), connected to

an elastic foundation through a flexural bar with stiffness K The foundation

may move horizontally with a stiffness and rotate with a stiffnessAny vertical motion is restricted

The horizontal displacement of the mass under a ground motionexcitation can be written as:

elastic deformation of the flexural member The mass acceleration ü iswritten as:

and the equilibrium equation for the mass M as:

The total mass acceleration can be obtained in terms of the groundacceleration and the acceleration due to the elastic deformation Fromthe two equilibrium equations of the flexural member:

one obtains

and by substitution into equation (2),

The equilibrium equation (3) becomes:

The foundation flexibility can modify the natural frequency of thesystem to an important extent and therefore its response to dynamic loads

of any kind

As mentioned before, design of machine foundations was the firstengineering problem where DSSI effects were considered The basic goal

in this case is to limit the foundation motion amplitudes which allow for

a satisfactory operation of the machine and do not disturb the people inthe vicinity The design rules for machine foundations were based ontradition and rules-of-thumb during the first half of the twentieth century.Those methods were often obtained from a Winkler elastic reaction of thesoil and an added mass corresponding to part of the soil that would bevibrating in phase with the foundation A revision of the classicalmethods may be found in the books by Whitman and Richart (1967) and

by Richart et al (1970)

The basic foundation stiffness problem can be seen in Figure 2.The foundation is assumed to be a rigid block on a half-space

compute the foundation stiffness, its mass is considered to be zero Therelation between the dynamic force applied to the foundation and its

Assuming a time harmonic excitation with frequency

the foundation displacement is

where is a complex number The foundation stiffness is also acomplex number:

In order to visualize the meaning of the foundation stiffness onemay consider a simple analogy drawn by Roesset (1980) Assume asingle degree of freedom system as shown in Figure 3 to represent thesoil under the footing

Its equilibrium equation

under time harmonic loading gives

By comparison of equations (16) and (18) one obtains:

dynamic foundation stiffness, respectively These coefficients can berepresented versus the angular frequency as shown in Figure 4

The actual dynamic vertical stiffness coefficients for a rigidcircular massless foundation on an elastic half-space as obtained byVeletsos and Wei (1971) are shown in Figure 5 The qualitativeagreement between both figures is clear

DSSI effects when loads act directly on the structure (wind loads,moving machinery, traffic on bridges, etc) are basically due to thefoundation compliance They can be taken into account by using thefoundation stiffness matrix (a frequency dependent matrix when theanalysis is done in the frequency domain) In many other cases the

dynamic excitation comes from the soil (earthquakes, nearby road orrailway traffic, underground explosions, etc) In those cases, theinfluence of DSSI on the structural response is twofold: on the one handthe excitation due to waves impinging on the structure depends on thesoil properties and the foundation characteristics; on the other hand, theresponse of the structure to the excitation also depends on DSSI effects.

As general statement it can be said that the influence of DSSI onthe response of structures to ground motion is important for large andmassive structures Power plants, bridges, dams and large buildings aretypical examples where this phenomenon is relevant When a largestructure is excited by waves travelling through the soil, as occurs in the

event of an earthquake, two important effects associated to the size of thefoundation and the structure are present The first one is called kinematicinteraction and is associated to the size and geometry of the foundation.The existence of a large massless foundation would produce by itself afiltering of the incident waves in such a way that the foundation response

is a function of its own geometry The phenomenon can be illustrated bythe image of two very light boats on the surface of the sea; one verysmall and the other equally light but large The first one would follow thefree surface motion without any change in it; the second one would haveits own motion and would change the sea motion in its vicinity

The second effect is known as the travelling-wave effect It takesplace when the characteristic length of the structure is of the same order

as the wavelength of the seismic waves For instance, harmonic waveswith a 0.2 s period in a rock with a shear wave velocity of 2500m/s have

a wavelength of 500 m Over a distance of 125 m., which can be thelength of a bridge or a dam, the ground motion changes from itsmaximum value to zero The importance of this effect depends on thesize of the structure and on the type, frequency, and direction of thewaves

Soil-structure interaction is, in most cases, studied assuming linearelastic behaviour Under this assumption the soil-structure system can beanalyzed in the frequency domain using a substructure technique.Foundations are in many cases massive and may be assumed to be rigid.Their dynamic behaviour is characterized by the stiffness (rigidity) matrix,which relates the force vector (forces and moments) applied to thefoundation assumed massless with the resulting displacement vector(displacements and rotations) Once the dynamic stiffness of a foundation isknown, its response including the mass, or that of any structure supported

on it, may be immediately evaluated in those cases where the dynamicexcitation is directly applied to the structure When the system is excited bywaves travelling through the soil, prior to the analysis of the structuremounted on the springs defined by the foundation stiffness, the excitation ofsuch a system must be determined To this end, the forces and momentsneeded to avoid any motion of the massless foundation impinged by thewaves travelling through the soil (kinematic interaction) are computed.Opposite forces and moments are applied to the foundation in the completesoil-foundation-structure model in order to compute the response of thestructure to the incoming waves

The analysis of the seismic response of structures on flexiblefoundations or large structures where the travelling wave effects areimportant require the use of a model where soil and structure are studiedtogether as will be seen in Section 4 of this Chapter Soil-structureinteraction problems where non-linear effects are important require a directtime domain analysis Non-linear contact conditions and non-linearbehaviour of the structure are the most frequent situations for which a timedomain analysis is required

2 Dynamic Stiffness of Foundations.

The first study of the stiffness of a foundation representing the soil as alinear elastic half-space was carried out by Reissner (1936) He studied theresponse of a disc on the surface of the soil subjected to vertical harmonicforces A uniform distribution of stresses under the disc was assumed.Knowing that the actual stress distribution was far from being uniform, in

the mid 1950' s, several authors carried out studies assuming certain stress

distributions for circular and rectangular foundations (Arnold et al; 1965;Bycroft, 1956) The mixed boundary value problem, with prescribed

displacements under the rigid footing and zero traction over the remaining

portion of the surface, was studied during the 1960' s and early 1970' s

(Paul, 1967; Veletsos and Wei, 1971; Luco and Westman, 1971) Relaxedboundary conditions were assumed under the footing Several studies werealso made using viscoelastic soil models (Veletsos and Verbic, 1973)

Wong and Luco (1976) computed dynamic compliances (stiffnessinverse) of a surface rigid massless foundation of arbitrary shape on anelastic half-space by dividing the soil-foundation interface into rectangularelements The tractions were considered to be uniformly distributed withinthe elements and a relation between the tractions over an element and the

displacements on the soil surface was obtained by integration of Lamb's

point load solution (1904) This method is, in fact, a Boundary ElementMethod with a half-space fundamental solution However, the integration ofthis fundamental solution is rather involved and only surface foundationsmay be analyzed

The first numerical technique widely used for computation offoundation stiffness was the Finite Element Method (FEM) Thedevelopment of energy absorbing boundaries for 2-D by Waas (1972) andfor axisymmetric problems by Kausel (1974) made possible the analysis offoundations resting on, or embedded in, layered soils The finite elementmodels, however, contain assumptions like the existence of a rigid bedrock

at the bottom, or a parallel layered geometry extending to infinity, that arenot always realistic In addition, 3-D dynamic soil-structure interactionproblems present important difficulties for finite element models due to thelarge number of elements involved in the analysis and the lack of infiniteelements such as those existing for 2-D problems

Boundary Element Methods (BEM) based on boundary integralequations are very well suited for dynamic soil-structure interactionproblems and they have also become a very widely used approach for thesolution of this type of problems Unbounded regions are naturallyrepresented The radiation of waves towards infinity is automaticallyincluded in the model, which is based on an integral representation valid forinternal and external regions

The first BE application for DSSI problems was presented byDomínguez in 1978(a) The direct formulation of the BEM was applied tocompute dynamic stiffness of foundations The frequency domainformulation was used to obtain stiffness of rectangular foundations resting

on, or embedded in, a viscoelastic half-space Otternstrener and Schmid(1981) and Otternstrener (1982), followed the same approach to study,

respectively, dynamic stiffness of foundations and cross-interactionbetween two foundations Non-homogeneous soils have been studied byAbascal and Domínguez (1986) Apsel and Luco (1983) used an indirectBEM in combination with semi-explicit Green's functions to computestiffness of circular foundations embedded in a layered half-space.Dynamic stiffness of circular foundations on the surface or embedded inlayered soils have been computed using the direct BEM by Alarcón et al.(1989) and Emperador and Domínguez (1989) Karabalis and Beskos(1984) computed dynamic stiffness of surface foundations excited by non-harmonic forces using the time domain BEM Also in the time domain,Mohammadi and Karabalis (1990) studied the use of adaptive discretizationtechniques and compared "relaxed" versus "non-relaxed" boundaryconditions The BEM has also been used to compute dynamic stiffness offoundations when soil-foundation separation exists, by Hillmer and Schmid(1988), and Abascal and Domínguez (1990).

In most problems where the soil-structure interaction effect isimportant the foundation is massive and may be studied as a rigid body.When the foundation is a strip footing that may be represented by a planemodel (Figure 8), it has three degrees of freedom corresponding to thehorizontal, vertical and rocking (rotation) co-ordinates For 3-D foundations(Figure 9) each vector has six components: one vertical, two horizontal, tworocking and one torsional

For a harmonic excitation with frequency the dynamic stiffness

matrix relates the vector of forces (and moments) R, applied to the foundation and the resulting vector of displacements (and rotations) u,

when the foundation is assumed to be massless