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Boundary Element Methods for Soil-Structure Interaction This page intentionally left blank 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 NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: Print ISBN: 0-306-48387-4 1-4020-1300-0 ©2004 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2003 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at: http://kluweronline.com http://ebooks.kluweronline.com CONTENTS INTRODUCTION W S Hall (Teesside), G Oliveto (Catania) xvii PART SOIL-STRUCTURE INTERACTION TWENTY FIVE YEARS OF BOUNDARY ELEMENTS FOR DYNAMIC SOIL-STRUCTURE INTERACTION J Dominguez (Seville) Introduction Dynamic Stiffness of Foundations 2.1 THREE-DIMENSIONAL FOUNDATIONS 13 2.2 STRIP FOUNDATIONS 16 2.3 AXISYMMETRIC FOUNDATIONS 20 2.4 FOUNDATIONS ON SATURATED POROELASTIC SOILS 24 Seismic Response of Foundations 28 Dynamic Soil-Water-Structure Interaction Seismic Response of Dams 31 4.1 FLUID-SOLID INTERFACES 34 Gravity Dams 35 5.1 DAM ON A RIGID FOUNDATION EMPTY RESERVOIR 36 5.2 DAM ON A RIGID FOUNDATION RESERVOIR FULL OF WATER 37 5.3 DAM ON A FLEXIBLE FOUNDATION EMPTY RESERVOIR 38 5.4 DAM ON A FLEXIBLE FOUNDATION 39 RESERVOIR FULL OF WATER 42 5.5 BOTTOM SEDIMENT EFFECTS vi Arch Dams 6.1 DAM ON A RIGID FOUNDATION EMPTY RESERVOIR 6.2 DAM ON A FLEXIBLE FOUNDATION EMPTY RESERVOIR 6.3 DAM ON A FLEXIBLE FOUNDATION RESERVOIR FULL OF WATER 6.4 TRAVELLING WAVE EFFECTS 6.5 POROELASTIC SEDIMENT EFFECTS References COMPUTATIONAL SOIL-STRUCTURE INTERACTION D Clouteau (Paris), D Aubry (Paris) Introduction 1.1 PHYSICAL MODELS 1.2 NUMERICAL MODELS 1.3 HETEROGENEITIES IN THE BEM 1.4 TIME DOMAIN BEM/ FREQUENCY DOMAIN BEM 1.5 STOCHASTIC APPROACH 1.6 UNBOUNDED STRUCTURES 1.7 GUIDELINES Physical and Mathematical Models 2.1 GEOMETRY 2.2 THE UNKNOWN FIELDS 2.3 LOADS 2.3.1 Incident Fields 2.3.2 Initial Conditions 2.3.3 Applied Forces and Tractions 2.4 LINEAR EQUATIONS 2.4.1 Field Equations 2.4.2 Coupling Equations VARIABILITY ON THE PARAMETERS 2.5 2.5.1 Stochastic Model of the Soil Parameters 2.5.2 Stochastic Model for the Applied Loads 2.6 SUMMARY OF MODELLING SECTION 2.6.1 Wellposedness and Approximation Domain Decomposition 3.1 COUPLING FIELDS 44 45 46 49 51 56 57 61 61 62 63 64 65 65 66 66 66 67 68 68 68 69 69 69 69 70 70 71 73 74 74 74 vii 3.2 3.3 3.4 3.5 LOCAL BOUNDARY VALUE PROBLEMS VARIATIONAL FORMULATIONS THE SFSI EQUATION FEM AND REDUCTION TECHNIQUES 3.5.1 Component Mode Synthesis 3.5.2 Principal Directions Boundary Integral Equations and BEM 4.1 REGULARIZED BOUNDARY INTEGRAL EQUATION IN A LAYERED HALF-SPACE 4.2 REGULARIZING TENSORS 4.3 BOUNDARY ELEMENTS 4.4 COUPLING WITH OTHER NUMERICAL TECHNIQUES 4.5 FEM-BEM COUPLING INSIDE A VOLUME Unbounded Interfaces 5.1 GENERAL SPACE-WAVENUMBER TRANSFORM 5.2 INVARIANT OPERATORS 5.3 DOMAIN DECOMPOSITION ON INVARIANT DOMAINS 5.4 BEM ON INVARIANT DOMAINS 5.5 NON INVARIANT UNBOUNDED INTERFACES 5.5.1 Statistically Homogeneous Random Medium 5.5.2 Weakly Perturbed Invariant Domains 5.5.3 Truncated Invariant Domain Green’s Functions of a Layered Half-Space 6.1 SOLUTION IN THE SLOWNESS SPACE 6.2 FAST INVERSE HANKEL TRANSFORM 6.3 SINGULARITIES Applications 7.1 SOIL-FLUID-STRUCTURE INTERACTION 7.2 MODAL REDUCTION FOR SSI 7.2.1 Selecting Dynamic Interface Modes 7.2.2 Selecting Input Shapes for Static Correction 7.3 SSI ON A RANDOM SOIL 7.4 SFSI FOR PERIODIC SHEET-PILES 74 75 76 77 78 79 80 80 81 82 83 84 87 87 89 89 90 92 92 92 92 93 94 95 95 96 96 96 98 99 100 103 viii 7.5 TOPOGRAPHIC SITE EFFECTS USING SSI FRAMEWORK 7.6 THE CITY-SITE EFFECT 7.6.1 Spectral Ratios 7.7 SSI IN BOREHOLE GEOPHYSICS Conclusion References 10.Appendix: Mathematical Results and Formulae 10.1 MATHEMATICAL PROPERTIES OF VARIATIONAL BIE 10.1.1 Coupling on a Volume 10.2 PROPER NORM FOR RESIDUAL FORCES 10.3 MATRICES FOR THE REFLECTIONTRANSMISSION SCHEME 10.4 HANKEL TRANSFORM 10.5 RECONSTRUCTION FORMULAE 107 107 109 112 112 114 122 122 123 124 124 125 125 THE SEMI-ANALYTICAL FUNDAMENTAL-SOLUTIONLESS SCALED BOUNDARY FINITE-ELEMENT METHOD TO MODEL UNBOUNDED SOIL J P Wolf (Lausanne), C Song (Sydney) Introduction 127 Objective of Dynamic Soil-Structure Interaction Analysis 129 Salient Concept 130 Scaled-Boundary-Transformation-Based Derivation 134 4.1 GOVERNING EQUATIONS OF ELASTODYNAMICS 134 4.2 BOUNDARY DISCRETISATION WITH FINITE ELEMENTS 135 4.3 DYNAMIC STIFFNESS MATRIX 136 4.4 HIGH-FREQUENCY ASYMPTOTIC EXPANSION OF DYNAMIC STIFFNESS MATRIX 137 4.5 MATERIAL DAMPING 139 4.6 UNIT-IMPULSE RESPONSE MATRIX 140 Mechanically Based Derivation 141 Analytic Solution in Frequency Domain 144 Numerical Solution in Frequency and Time Domains 148 Extensions 149 8.1 INCOMPRESSIBLE ELASTICITY 149 ix 8.2 VARIATION OF MATERIAL PROPERTIES IN RADIAL DIRECTION 149 150 8.3 REDUCED SET OF BASE FUNCTIONS 8.4 TWO-DIMENSIONAL LAYERED UNBOUNDED SOIL 151 8.5 SUBSTRUCTURING 152 Numerical Examples 153 9.1 PRISM FOUNDATION EMBEDDED 153 IN HALF-SPACE 9.2 SPHERICAL CAVITY IN FULL-SPACE WITH SPHERICAL SYMMETRY 156 9.3 IN-PLANE MOTION OF SEMI-INFINITE WEDGE 159 9.4 IN-PLANE MOTION OF CIRCULAR CAVITY IN FULL PLANE 161 OUT-OF-PLANE MOTION OF CIRCULAR CAVITY 9.5 IN FULL PLANE WITH HYSTERETIC DAMPING 163 165 10 Bounded Medium 168 11 Concluding Remarks 12 References 172 BEM ANALYSIS OF SSI PROBLEMS IN RANDOM MEDIA G D Manolis, C Z Karakostas (Thessaloniki) 175 Introduction 179 Review of the Literature 180 2.1 RANDOM LOADING 180 2.2 MONTE CARLO SIMULATIONS 181 2.3 RANDOM BOUNDARIES 181 2.4 SOIL MODELLING 182 2.5 FOUNDATIONS 183 2.6 SLOPE STABILITY 184 2.7 CONSOLIDATION 184 2.8 SOIL-STRUCTURE INTERACTION 185 2.9 EARTHQUAKE SOURCE MECHANISM 187 2.10 PROBABILISTIC RESPONSE SPECTRA 187 Integral Equation Formulation 187 3.1 THEORETICAL BACKGROUND 188 3.2 FORMAL SOLUTION 190 3.3 CLOSURE APPROXIMATION 191 Vibrations in Random Soil Media 396 NL-FEM, see nonlocal Finite Element Method nodal value,329, node,302, coincident ,299, vector,276, non-axisymmetrical load,88, non-harmonic force,11, non-homogeneous half-space,28, soil,11,16, non-invariant unbounded interface,92, non-linear behaviour,9,62,129,238,251, behaviour of bridge,257, contact condition,114, effect,9, FEM,63-4, first order ordinary differential equation,137,148, hysteretic response,180, interaction analysis,236, nonlocal elasticity,285, soil abutment interaction,269, soil behaviour,238, soil dynamics,62, soil response,180,235, spring,260, SSI analysis,236, static analysis,260, structure, 180,186, time domain dynamic analysis,260, nonlinearity,236,284, nonlocal,275, constitutive equation,284,290, correction,291,293-5, elastic medium,275, elasticity,275-7,284, elasticity model,278, elasticity problem,283, field,279, finite element method (NL-FEM),276, hyperelastic material,280-84, 294, hyperelastic material model, 276,281-2, level,279, material behaviour,276, nonlinear elasticity,282,284, strain,279, stress,278, stress-strain relation,276, theory,277, treatment,275, nonlocality,284, correction strain,283, effect,280, feature,276,295, integral operator,280, residual,276,280, non-planar crack,330, non-symmetric collocation BEM, 293, nonstationary stochastic process, 186-7, normal mode theory, 185,195, unit vector,316, Northridge earthquake,235, nuclear power plant,61,100, reactor building,96,98, numerical evaluation,320, 397 numerical method,297, model,347, procedure,299, simulation,345, O obliquely incident SH wave,30, ocean storm loading, 182, wave, 180,186, opening displacement,329, ordinary differential equation, 128,131-3,148,169, orthotropic bimaterial plate,167, outgoing wave,24, P P-SV problem,94, P-wave,38-9,52-3,96,207, 214,348,350,355,357-60, 367,369,371, incident ,349, panel clustering,399, parabolic valley,354,367, parallel layered geometry, 10, layers, 16, partial differential equation, 128, 130-34,169, partially saturated sediment,43, Pasternak model,275, pdf, see probability density function penny-shaped crack,298,328-9, 331, periodic sheet pile,96,103, periodicity condition,90, perturbation,206,208, expansion,209, method,177,195,201,213, 225,227, pervious solid zone,26, petroleum industry,61, pier,183,269, pile,182-3,244,255,262,265, footing,255, foundation,253,255, foundation stiffness,256-7, group,180,182,253, periodic sheet ,96,103, stiffness,253-4,256,264, pipeline,183, plane crack,298, SH wave, 100, strain,185,227,332,334-6, wave,194,221,348, wave deconvolution,72, plastic rotations,238, plasticity,275, plate,340, platform foundation design,182, foundation stability,182, polarisation,72, polynomial basis,218, chaos,216,220,222, 224-5,227, chaos transformation,179, pore pressure,184, poroelastic material,27, material property,27, medium,25,93, saturated soil,128, sediment effect,56, 398 porous zone,25,26, bottom sediment,56, sediment,33,55, potential,347, power plant,7, spectral density,71, spectral density function (psdf),202-3,205, prediction phase,276, pressure wave,192,213,348, pressurized water reactor building,97, primal domain decomposition,74, primary wave,348, prism foundation embedded in half-space,153, probabilistic response spectrum,187, spectrum,187, probability density function (pdf),177,183,187,218, propagating earthquake motion, 186, propagation,297, velocity, 176, propagator method,93, proper norm for residual forces, 124, properties of the soil,96, psdf, see power spectral density function pyramidal subdomain,321,325, Q quadratic continuous element,299, discontinuous element, 299,302, element,25-6,36,38,45,310, 339,345, shape function,300-01,306, quadrilateral element,45,237,321,330, isoparametric nine-node element,328, nine-node element,328, quarter point element,330, source and field element,320, quarter elliptic corner crack in a plate,339-41, quarter point scheme,328, quay wall,66,103,104, R radial basis function,305, radiation condition,42,49,112,127, 132,138,145-148,168-9, 171,217, damping,12-3,42,78,244, 249-50, raft foundation settlement,182, railway track,65, random boundary,181, boundary condition,200, building distribution,110, city, 109, coefficient,199, collection of scatterers,176, continuum,176,216, density, 194, distribution,176, field, 182,201, 399 random function,185, fundamental solution,219, generalized forcing function, 188, heterogeneity,97, integral,190, integral equation, 176,188, kernel,188, Lamé modulus,100, loading, 180, medium,114,175-6,179,191, 198,206,208,222, motion,202,205, operator,190-91, parameter,208,211,214, 216-7, perturbation,92, phase angle,202, process,178,186,193, response,177 response statistics,193, seismic excitation,180, soil,100, soil continuum,185, variables,176,183,202,209, 218, vibration,202, vibration analysis,186, vibration theory,180,364, wave number,194,204, randomly distributed buildings,107, layered half-space,177, layered medium,176, structured ground,194, randomness,185,193,204-5,217, 221-3,225,228, ground ,192, high low ,227, ,227, ray method,64, technique,112, Rayleigh wave,52,107,346,350, 355,357-9,367,369,371, wave velocity,153,156,351, reactor building,66, receiver borehole,112, station,223, reciprocity conditions,75, reconstruction formulae,125, rectangular bar,310,331, plate,298,308,339, surface and embedded foundation,16, reduced set of base functions, 150, reduction basis,98, technique,66,96, reflected P-wave,349, SH-wave,349, SV-wave,349, wave,349, reflection coefficient,350, transmission coefficient,93-4, transmission scheme,124, regular mesh,112, regularization,298, procedure,318, analytical procedure, 318, 400 regularization technique,64, regularized formulation,339, variational direct integral equation, 80, weak form,318-9, regularizing coordinate transformation, 315, coordinates,323,326, reinforced concrete building, 235,247, relaxation time, 110, reservoir,52-5,96-7, geometry,32,45, on rigid foundation,37, residual force, 124, resolvent kernel,176,188-90, resonance frequency,97,107,110, resonant mode,346, response amplification,235-6, matrix,140,148, spectrum,364, spectrum analysis,246, statistics,198,222, retrofit analysis,269, reversible deformation processes, 280, Ricker input signal,108, rigid bedrock,10,18-9,44, block,4,16, body motion,29,82,124, element,263, footing, 10, foundation,3,35,48,54-5, interface,154, massless foundation,29, plate,180, square,98, square foundation,7,180, support,243,250, rigidly supported structure, 240,243, Ritz analysis,78, Galerkin projection,65, road,65, road or railway traffic,7, rock,31,96,193,244,248, formation,272, rocking stiffness,241-2, rod finite element,183, rotational component,348, spring,263, wave,348, rough seabed topography,181, S S coda wave,175, S-wave,56,98,107-8,110,175, 207,345,348,364,367, San Fernando earthquake, 238,260, saturated poroelastic half-space,27, poroelastic soil,24, soil, 149, scalar wave equation,128,178, scaled boundary coordinate,131,133-4,148, finite element equation, 135,140,142, 401 scaled boundary finite element equation in displacement, 144-5,148,150,166-7,169, finite element equation in dynamic stiffness,137,139, 141,147,149,151-2,166, 168-9, finite element method,127, 129,130,132,142,149,150, 153,158,162,164-5,169,171, transformation,132-3, transformation based derivation,129,134, scaling centre,132-4,141,151-3, 165-8,170, equation,133, scattered field,69, wave,181,351, scattering expansion,195, of seismic waves,345,369, of SH waves,346, second moment statistics,177, second order statistics,198, stochastic field,70, secondary wave,348, sediment,36,41,43, fully saturated ,43, partially saturated ,43, porous ,33,55, porous bottom ,56, thickness,346, seepage force,27, seismic behaviour,65,114, seismic damage,235, design codes,242,246,261, design coefficient,245, design of building structures, 238, excitation,53,238,264, experiment,61,92,113, field,62, force,68,76,237,246,269, fragility curve,184, free field,61, incident,235, incident field,65, load,99,238,241,249,269, loading,68,96,249, motion,65,130,180,364, random excitation,180, response,32,45,56,96,111, 235,250, bridge response,269, response of arch dam,33-4,54, response of bridge,236, response of dam,31,36, response of foundation,28,30, response of gravity dam,33, response of structured, retrofit,251, risk zone,249,262, safety ,61,96, signal,64, site response,184, slope failure stability matrix,183, soil-structure-interaction,128, traction,68, wave,51,114,347, wave attenuation,175, wave propagation,63,345, 402 seismic wave scattering,345-6, zone,250,261,364, seismicity,204, seismological issues,96, seismology,63, self-adjoint bilinear form,320, semianalytical fundamentalsolution-less boundary element method based on finite elements,169, discontinuous element, 299,302, elliptical soil deposit,18,31, infinite wedge,159-61, Markovian stochastic model, 186, rock formation,272, semicircular alluvial valley,359, canyon,360, surface crack,336,338, series expansion,189,227, solution,189, SGBEM, see symmetric Galerkin Boundary Element Method SH incident wave,349, obliquely incident wave, 30, plane wave,100, problem,94, wave,52,55,101-2,206, 223,228,346,349,351, 353-4,359,367,369, shallow embedment,253, shape function,25,133,136, 155,291, constant ,86,298, FE ,86, quadratic ,300-01,306 , sharp crack tip,275, shear horizontally polarised wave,179, modulus, 149,156,183,194, 241,250,268, strain,241, strength of soil,184, wall,236,240, wall structure,237, wave,43,103,139,163,175, 184,192,348, wave modulus,241, wave velocity,13,21,103, 153,205,223,240-41,248, 347,359, sheet pile,26,41,87, SIF, see stress intensity factor silty-clayey mixture,272, soil,272, similarity centre,134,141, factor,134, simulation analysis,236, technique,177, singular double surface integral, 316-7,339, integral,63-4,84,127, 168,304,308,316, kernel,288, 403 singular point,323, value decomposition,80,100, singularity,275,287,315, higher order ,308,318 weak ,308,323, site amplification,30,345-6,369, effect,61,96, response,346, seismic hazard,183, six-node triangular element, 52,320, sliding displacement,329, slip rate,185, slope stability,183, analysis,182, slowness space,94, small displacement,276, randomness,177-8,195,225, smooth boundary surface,294, soft alluvial layer,364, clay,272,365, soil,61,102,201,235,237, soil,241,243,266-7,275, abutment interaction,238, backfill stiffness,251, 256-7,260, characteristic,62, damping factor,244, damping,362, deposit, 114,242, dynamics,179-80, elastic modulus,254, element,236, failure,238, flexible ,235, soil fluid interface,74-5,77,105, fluid structure interaction, 61,66,76-8,96-7,103, foundation interface, 10,14-5,28, foundation separation,11, foundation,275, free surface, 14, hard ,359, homogeneous ,181, homogeneous elastic ,30, horizontally layered ,28, impedance,97-9, incompressible ,149, layer, 107,182,242, layered ,10, linear elastic ,183, linear unbounded ,129, material state,182, modelling,181, non-homogeneous ,11,16, non-linear behaviour,238, non-linear dynamics,62, non-linear response, 180,235, parameter,70-1,253, poroelastic saturated ,128, pressure,258,260, profile statistics,182, properties,181, random ,100, random continuum,185, reaction modulus,253, response,180, saturated ,148, saturated poroelastic ,24, seismic structure interaction,128, 404 soil shear modulus,263,269, shear stiffness,237, silty-clayey ,272, soft ,61,102,201,235,237, spring,249,261, spring stiffness,249, stiffness,252-3,257,264-5, stochasticity,187,191, stratified ,97, stratum,245, structure interaction,9,29,46, 61,64-5,68,73,96-7,100,102, 107,112,130,150,176,178, 184,191,235-8,240,245-6, 249-50,263,268-9,275,295, non-linear structure interaction analysis,236 structure interface,68,98, 114,130, two phase poroelastic ,24, unbounded ,128,130, 132-3,136-8,141-3,145-7, 149,151,157,162,171, uniform viscoelastic ,33, yielding,259,261, zoned viscoelastic ,19, solid angle,319, body,281, fluid interaction,33, fluid interface,34, mechanics,127, Somigliana equation,317, formulae,287,290-91,293, Sommerfeld radiation condition, 351,355, space correlation function,177, time random field, 186, wavenumber transform,87, spectral approach,96, finite elements,63, ratio,109,111,362,364-7, 369-70,372, spherical cap crack,328,330,332, cavity,157-60, cavity in full-space,156, spread footing,109,261, circular ,100, spring,260, coefficient, 158-9, dashpot mass model,171, stiffness,248-9, square embedded foundation,15-7, prism embedded in halfspace,153, SSI, see Soil-Structure Interaction standard deviation,62,111,365, standing wave,38, state equation,276,280, static external action,282, field,318,320, fundamental solution,307, load,7,276, response,99, stability of slopes,183, stiffness,6,15, stiffness matrix,142-3, 166-7, 405 stationarity principle,276, problem,286, stationary straight crack,298, statistical correlation,205, statistically homogeneous random medium,92, stiff overconsolidated clay, 272,265,269, stiffness,12,61,242,255,260,267, abutment ,255,260,263, coefficient,14,22-3, function,21, inverse, 10, matrix,13,21,253,276, stochastic analysis,64-5,73,84, boundary element method, 178, constitutive parameter,66, deconvolution problem,72, differential equation,176-7, 187-8, dynamic analysis,187, field,70-1, field simulation,201,204, finite element method,63,178, fundamental solution,222, geological medium,178, load,65-6, model for strong ground motions,185, model of the soil parameters, 70, problem, 190,206,209, process,201, realisation,228, time predictable model,185, stochasticity,193,217, storeys, 107, strain energy,99,279, field,277-8,281, plane ,185,227,332,334-6, point,275, rate field,280, stratified half-space,81, soil,97, stress,317, amplitude, 136, analysis,275, distribution,15, field,277,286,297, intensity factor (SIF), 165,168,309,315-6, 328-31,334,336,338-9,341, mode I intensity factor, 331, point,275, recovery, 128, response, 130,275, singularity,128,165,171,275, singularity crack, 170, strain behaviour of soil specimens, 182, strain constitutive relation, 275, strain relation,94,281, wave loadingl85, strip footing,11,16,27-8,30,183, foundation,16,24,250, structural damage,236,269, displacement,102, dynamics,61-2,88,107, 406 structural mechanics,181, response,62,187, stiffness,243, vibration amplitude,102, structure-soil interface, 129,131-6,138-9,141-3, 145-55,157-9,161,168,171, subduction earthquake,362, subregion technique,309, substructure,152,170, method,129, technique,9,87, substructuring,62-4,88,100, 152,170, surface breaking crack,328,336, element,82,168,171,317, axisymmetric element,20, finite element,132,135, foundation,14, load,215, mesh,77,328, rotor,342, rotor operator,319 , stiffness,30, strip footing,18-9, wave,156,346, SV-wave,38-9,52-4,56,96, 100,108,195,227,346,350, 355,357-60,367,369,371, symmetric Galerkin boundary element method (SGBEM), 63,276,287,290,293-4, 315,318,328,336,339,341, synthetic accelerogram,185, seismogram,71, system identification,237, T Taylor series expansion,177,179, 195,204,208, TD-BEM, see time domain BEM formulation, TDM, see time domain method tectonic fault,185, test function ,318, thermodynamic framework, 276,279, thick alluvial deposit,369, shell element,77, shell finite element,96, walled tube,298, thin layer method,151, shell element,77, Thomson-Haskel vector,94, three-dimensional dynamic soil structure interaction,10, foundation,13, surface and embedded foundation,30, three-node line element, 159,162, parabolic element,215-6, tieback,103, time dependent boundary condition,297, fundamental solution,302, problem,87, time domain,33,73,128-30,138, 148,171,186,297,309,362, 364-5, BEM (TD-BEM), 11,30,64,114, formulation,309, 407 time domain method (TDM),298-9,310, problem,87, transient SSI problems,237, time-frequency algorithm,114, time harmonic condition,192, elastic wave,216, time harmonic elastodynamics,225, excitation,4, loading,6, motion,38, plane wave,53, point load solution,20, problem,35, scalar wave propagation,227, time increment,310, time step,300, topographic effect,96, irregularity,345, site effect,107, total displacement,349,352, displacement vector,350, potential energy principal, 276,283, traction,316-7,355, boundary integral equation, 299,302,315, equation,298,302-3,306, 311,317, free boundary,351,355, 357-8, integral equation,314,319, traffic induced vibration,65, on bridges,6, transfer function,362,364,366,368, 370,372, matrix concept,182, transform Floquet ,88-9,91, parameter,208, transformation,320,343-4, transient inelastic response of structures,237, problem,177,297, transition site,362, zone,359, translational pile stiffness,253, spring,263, stiffness,252-3, stiffness parameters ,255, transversal wave number,348, transverse stiffness,251, wave,192,348, travelling wave,30,45-6,50, wave effect,8,51,54, triangular element,45, tube wave,112, tunnel,65,87,130, circular unlined ,213-6, construction,182, unlined ,227, turbulence,186, two node finite element,164, phase elastic material,278, phase poroelastic soil,24, 408 two state conservation integral of elastodynamics,298, U unbounded continuum,228, domain,62,152, elastic domain,63, elastic medium,348, half-space,351, heterogeneity,63, interface,87, medium,63,128,133,170,217, region,10, soil,128,130,132-3,136-8 , 142-3,145-7,149, 152,156,163,171, underground explosion,7, inhomogeneity,33, opening,178, region,31, uniform half-space,29,54, loading,328, stress,278, soil deposit,242, strain,278, viscoelastic half-space foundation,42, viscoelastic soil,33, unit force,288, imposed strain,288, impulse,223, impulse of acceleration, 130,148, impulse response, 171, impulse response matrix, 130,139-40, relative displacement, 288, rigid body motion,14, vector,317, unlined tunnel,227, uplifting,236, urban area,96, V valley,61,352-5,357-9, 363-4,367,369, alluvial ,345,362, parabolic ,354,367, semicircular alluvial ,359, variance,209,212,223,225, amplitude,226, phase angle,226, variational approach,336, BEM,315, BEM formulation,63, BIE,122, formulation,75, interaction problem,76, principle,63,285, statement,276, vector wave equation, 128, of elastodynamics, 153, vehicle traffic,61, vertical harmonic force,9, stiffness,241, vibration of large machines, 1, in random soil medium, 191, virtual work,123,139,288, 409 viscoelastic half-plane, 18-9, half-space,10,22,31, material,139, medium,38,52, soil model,10, solid,33, zone,25, viscous damping, 82, Volterra integral equation of the second kind,188, volume integral,306, W water compressibility,33,45, foundation interface,49, foundation rock interaction, 50-51, wave,49, wave acoustic ,177, acoustic propagation, 181, bulk ,93, complex number,359, compressional velocity, 103, elastic ,176,177, elastic propagation,192, electromagnetic propagation,181, energy,182, equation,69,196, focusing,346, front, 138, guided ,92-3,112, harmonic ,45,52, harmonic propagation, 194, wave incident 29,65,103, 105,297,346,349, incoming ,30,99, length,154,156,171, longitudinal ,192,348, Love surface ,355, motion,176,191,205, multiple ,346, number domain,93-4, number has a small fluctuation,193, number,201,205,217-8, 220,222-3,227,351,354, ocean ,180,186, outgoing ,24, plane ,194,221,348, primary ,348, problem,217, propagation constant, 176, propagation in a halfspace,347, propagation model,364, propagation velocity,208,216, propagation,185,192,201, 203,211,214,346, random number, 194,204, Rayleigh 52,107,346,350,355,357-9, 367,369,371, Rayleigh velocity, 153,156,351, reflected ,349, reflected P ,349, reflected SH ,349, reflected SV ,349, reflection coefficient,41, rotational ,348, S coda ,175, 410 wave S,56,98,107-8,110,175, 207,345,348,364,367, scattered ,181,351, scattering of seismic 345,369, scattering of SH ,346, scattering,351-2,357, secondary ,348, seismic ,51,114,347, seismic attenuation, 175, seismic propagation, 63,345, seismic scattering,345-6, SH ,52-5,101-2,206, 223,228,346,349,351, 353-4,359,367,369, shear ,43,103,139,163, 175,184,192,348, shear modulus,241, shear velocity, 13,21, 103,153,205,223,240-41, 248,347,359, speed,217, standing ,38, surface ,156,346, SV ,38-9,52-4,56,96, 100,108,195,227,346,350, 355,357-60,367,369,371, time harmonic elastic 216, time harmonic plane ,53, wave transversal number,348, transverse ,192,348, travelling effect,8,51,54, tube ,112, velocity,155,201,214,347,359 weak impedance,107, singularity,308,323, weakly nonlocal,275, singular integral,86,315,320, singular kernel,320, weighted residual approximation,132, technique,128,134-5,169, weighting function,135, Williams-Westergaard asymptotic formulae,336, wind load,1,6, turbulence,180,186, wing wall,251,253,256,265, Winkler foundation,183, X Xochimilco-Chalco lake,364, Z zoned soil,18, viscoelastic soil,19, .. .Boundary Element Methods for Soil- Structure Interaction This page intentionally left blank Boundary Element Methods for Soil- Structure Interaction Edited by W.S HALL... European experts on Boundary Element Methods and Soil- Structure Interaction Its concept originated at the EUROMECH Colloquium 414 on Boundary Element Methods for Soil- Structure Interaction which... ? ?Boundary Element Methods for Soil- Structure Interaction? ?? is to review progress xviii made in the applications of the Boundary Element Method in the solution of Soil- Structure Interaction for the scientific