Foundations of Engineering Mechanics

Series Editors: V.I. Babitsky, J. Wittenburg

For further volumes:http://www.springer.com/series/3582

Sergey M. Aleynikov†

Spatial Contact Problemsin Geotechnics

Boundary-Element Method

With 295 Figures

123

Series Editors:

V.I. BabitskyUniversity LoughboroughDepartment of Mechanical EngineeringLoughborough LE11 3TU, LeicestershireUnited Kingdom

Author:

Sergey M. Aleynikov†

Voronezh State Architecture and CivilEngineering University20-Letya Oktyabrya Street, 84394006, VoronezhRussiae-mail: alasmbkb@box.vsi.ru

Originally published in Russian as “Boundary Element Method in Contact Problemsfor Elastic Spatial-and-Nonhomogeneous Bases”, in 2000 by Publishing House of CivilEngineering Universities Association, Moscow, Russia, ISBN 5-93093-053-8, 601 p.

J. WittenburgUniversität KarlsruheFakultät MaschinenbauInstitut für Technische MechanikKaiserstrasse 1276128 KarlsruheGermany

ISSN 1612-1384 e-ISSN 1860-6237ISBN 978-3-540-25138-5 e-ISBN 978-3-540-44776-4DOI 10.1007/b11479Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2009934850

© Springer-Verlag Berlin Heidelberg 2011This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevant protectivelaws and regulations and therefore free for general use.

Cover design: deblik, Berlin

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Obituary

The author of this monograph – Dr. Sergey M. Aleynikov, a well-known specialistin the field of numerical simulation of soil-foundation contact interaction, passedaway on August 4, 2009.

Sergey M. Aleynikov graduated from the Department of Applied Mathematicsand Mechanics of the Voronezh State University, then he took a Postgraduate Courseat the Heat and Gas Transfer Institute of the USA Academy of Science in the cityof Minsk, the capital of Belorussia. After defending the PhD thesis in 1983, hestarted working at the High Mathematics Chair at the Voronezh State University ofArchitecture and Civil Engineering, and finally, he was the Head of this Chair forfive last years.

At the university he began, in close collaboration with Dr. S.V. Ikonin, activelydeveloping the Boundary Integral Approach (BEM approach) for solving non-classical problem analyses of foundation engineering interaction. In co-authorshipwith Dr. A. A. Sedaev, he generalized the method of duel grids for numerical solu-tion of elastic Hertzian contact problems and suggested the method of generation ofrandom (or irregular) dual grids. On the base of N.K. Snitko’s ideas, Dr. Aleynikovproposed the integrated approach to the definition of dominant function for base-ment soil with a depth variable modulus of elasticity.

He became a recognized specialist in the area of modern method analyses offoundation engineering interaction. More than once he went abroad with lecturesand for conducting mutual research in Denmark, Spain, Poland, Canada, Germany,Belgium, the Netherlands and Croatia.

In 2007 S. M. Aleynikov made the translation of the book ‘Boundary Elements.Theory and Applications’ “written by G.T. Katzikadelis”.

His professional competence was highly appreciated and widely recognized. Hewas a full member of Russian Transport Academy, a member of the panel of Rus-sian Society for Soil Mechanics and Geotechnical Engineering, a member of theInternational Society for Soil Mechanics and Geotechnical Engineering (ISSMGE),and the European Technical Committee ERTC7 “Numerical Methods in Geotech-nical Engineering”. Dr. S.M. Aleynikov a big was a member of the Expert Councilof the Highest Certification Commission of Education and Science Ministry of theRussian Federation.

v

vi Obituary

This survey monograph by S. M. Aleynikov was published in Russia in 2001 andwon a high appraisal of the Russian academic community in the field. There is nodoubt its English translation will be useful for Western readers as well.

The image of this indefatigable and original person will be held in our remem-brance forever.

Voronezh, Russia Professor Igor S. Surovtsev(VSUAC)

Foreword

The theory of elasticity occupies a prominent position in the development of geome-chanics and in particular the study of interaction between structural elements andgeomaterials. The general area of contact between geomaterials and structural ele-ments is referred to as soil–structure interaction and solutions based on the theory ofelasticity have been successfully applied for the study of structural foundations, lay-ered soil systems, earth-retaining structures and tunnels. The subject matter relatedto soil–structure interaction also forms an important component in work related tothe mathematical theory of contact problems starting with the pre-eminent work ofBoussinesq and Hertz. The mathematical theory of elastostatic contact problems inparticular attracted the attention of the earlier Russian school of eminent elasticians,including Galin, Ufliand, Muskhelishvili, Shtaerman, Koronev, Popov, and others,with special emphasis on the application of elastic contact problems to structuralfoundations made by Gorbunov-Posadov and colleagues. These contributions wereless well known in the English literature in the middle of the last century and a sys-tematic exposition of the contributions of the Russian researchers to soil–structureinteraction was documented in the treatise by Selvadurai and to contact mechan-ics documented in the comprehensive volume by Gladwell. Since the publicationof these expository volumes in the 1980s, a number of authoritative volumes haveappeared in the area of elastostatic contact problems, where both classical and non-classical contact mechanics problems were discussed; a critical examination of theinfluences of frictional and unilateral contact problems have found applications inthe treatment of traditional interface mechanics problems as well as new develop-ments in materials science and advanced materials modelling.

The present volume is a welcome addition to the literature on elastostatic contactproblems with special reference to geomechanics. The volume commences with asystematic exposition of the fundamental solutions of Kelvin, Boussinesq, Cerrutiand Mindlin, which is the underpinning of many interesting applications of elasto-statics to contact problems in geomechanics. An aspect of spatial non-homogeneity,which is relevant to accreted materials that attain increases in the stiffness withdepth due to gravitational effects, is also included in the presentation. The volumeproceeds to the presentation of the traditional contact problems in elastostatics thatinclude axisymmetric and torsional indentation problems. A chapter in the volumeis devoted to the numerical implementation of the contact problem by appeal to

vii

viii Foreword

boundary elements. This is a useful development that complements the analyti-cal aspects and contains sufficient depth to enable the reader to appreciate thecontributions that numerical schemes can make to the formulation and solutionof contact problems in geomechanics. The treatment of the class of contact prob-lems is not restricted to linear responses; non-linear effects arising from non-lineardeformability of the geomaterial is also presented, with the approaches encom-passing both finite difference and boundary element methods. The conventionalanalytical approaches to the formulation and solution of contact problems in geome-chanics are by necessity restricted to simplified geometries. The author has venturedto include approaches that can be used for the study of complex foundation shapesassociated with pile foundations and other interactions between piles and founda-tion bases. Finally, the volume culminates with the study of the mechanics of con-tact between a structure and a poroelastic material saturated with an incompressiblefluid. The integral equations governing this class of poroelastic contact problem aresummarized and numerical techniques for their solution are presented.

The volume is a very useful contribution to the literature in geomechanics ofcontact problems as applied to practical problems involving the interaction of geo-materials and supporting soils. It contains a balance of analytical and computationalapproaches and this makes it a volume that will be of benefit to the researcher andpractitioner alike.

William Scott Professor and James McGill Professor A.P.S. Selvadurai FRSCMcGill University, Montreal, QC, Canada

Preface

The studies of contact interaction in the mechanics of deformable solids have beencarried out since late 19th century, starting from the works of Winkler (1867), Hertz(1881), and Boussinesq (1885). These studies have been further developed by spe-cialists in the mechanics of deformable solids as well as in structural mechanics,bases and foundations. Thousands of papers on this topic have been published, mostof their authors using simplifying assumptions of theoretical modeling on a flat oraxially symmetrical stressed state of a base under a punch (a foundation model). Itis seen from the detailed analysis of references found in literature that mathematicalmodeling of essentially spatial contact interaction is in its early stage.

The existing methods for the calculation of complex-shaped foundations are, asa rule, based on a bed coefficient hypothesis. This results in the introduction ofempirical coefficients into the calculation methods, thus restricting the range of theirapplication. In the recent years more attention is paid to finite-element approachto mathematical modeling of spatial contact interaction of foundations with bases.However, in such studies the dimensionality of the algebraic analogue of the con-tact problem sharply increases and the problem must be restricted to a number ofpartial problems – for example, by imposing restrictions to shape and size of boththe foundations themselves and the soil massifs around the foundations, by consid-ering loads in assumption of existence of symmetry axes or planes in the calculationscheme etc. Such studies are rather rare and lack proper consideration of loads ofgeneral spatial type (horizontal, vertical forces and moments) and the possibility oftheir combined action. And extremely rare are studies where the complex shape ofvarious foundations, applied in industrial and civil engineering, is fully taken intoaccount and theoretically based calculations are made.

Creation of new progressive foundation structures and solution of current prob-lems of geotechnical engineering result in more complicated problems to be solvedand in the increasing accuracy of the calculation results. The mathematical descrip-tion of the problems has become so complicated that traditional methods are nolonger suitable for their solution. The lack of reliable mathematical methods to acertain extent retards elaboration and implementation of new foundation structuresin engineering. Hence, the development of boundary element method (BEM), a rel-atively new trend in structural mechanics, based on boundary integral equations,seems to be quite promising from the point of view of both theory and application as

ix

x Preface

an efficient tool for solving 3-D problems. The BEM advantages over other methodsof numeric modeling consist in lowering the problem dimensionality (not the wholecalculation domain is subject to discretization, but only the boundary surface), inthe possibility of a detailed analysis of separate stressed areas, in the simplified datapreparation stage etc. This determines the broad application of BEM for solving var-ious problems of structural mechanics, especially the unlimited domains. Simultane-ously, by the present time numerical implementation of BEM to the spatial problemsof structural mechanics in the field of interaction of foundations and bases has notbeen sufficiently elaborated yet and appropriate boundary element algorithms andsoftware are still unavailable. Therefore, there is an urgent need to develop efficientnumeric approaches using the BEM to solve spatial contact problems of interactionof complex-shaped volumetric punches with deformed bases.

The present book is devoted to one of the BEM application areas – numericalmodeling of contact interaction of rigid foundation structures with soil. The mainattention is paid to the specific features of stress-strained states of elastic bases atspatial conditions. Contrary to the finite element method, special literature for theBEM application in mechanics of spatial contact interactions between bases andfoundations is at present unavailable. In recent publications, devoted to the calcula-tion of bases and foundations, BEM is merely mentioned. On the other hand, well-known books, describing theory and application of BEM, do not appropriately coverthe issues of creating calculation models and numerical algorithms for analyzingspatial contact interaction of foundation structures with soil bases.

The whole material is set in six chapters. The first chapter presents some intro-ductive data while reviewing spatial contact models in geotechnics. Classical funda-mental solutions for the spatial theory of elasticity obtained by Boussinesq, Cerruti,Mindlin are quoted as well as their generalizations, suitable for calculating construc-tions on elastic nonclassical bases. The properties of the influence functions are ana-lyzed, required for characterizing elastic bases with nonhomogeneous deformationproperties (connected half-spaces, elastic layers of constant and variable thickness).

In the same chapter a numerical-and-analytical procedure is developed forconstruction of fundamental solutions of spatial elasticity theory for multilayerbases without restrictions on the layer thickness and elastic parameters. Usingthe two-dimensional Fourier transform, the formulae have been derived, enablingthree-dimensional contact problems for complex-shaped structures deepened intospatially nonhomogeneous (layered) soils to be solved in the framework of theBEM numerical algorithm. The final part of the first chapter contains the results onthe formulation of influence functions for elastic bases with variable deformationproperties. The Boussinesq problem is solved for an elastic half-space when thedeformation modulus increases with depth according to a most general law. Properrelations, enabling adequate description of the experimental data, are considered.An efficient numerical-and-analytical procedure is developed for construction of theinfluence functions, taking into account the soil deformation modulus variation withdepth. All the theoretical results for the influence functions were obtained within aunique approach enabling all the main types of nonhomogeneities of natural soilbases to be taken into account.

Preface xi

The second chapter is devoted to the mathematical formulation of mixed prob-lems of the elasticity theory for a half-space and to the numerical-and-analyticalmethods of their solution. The results obtained in this chapter on developing themathematical means are the reference data for BEM-based numerical modeling ofthe spatial contact interaction. The boundary integral equations of the spatial contactproblem are written for the case when the calculation scheme is accepted in the formof variously deepened punches undergoing the action of the spatial system of forces.It is shown how to reduce the initial integral equation system of the contact problemwith respect to the contact stress function and the punch displacement parametersto the appropriate finite-dimensional algebraic analogue. Much attention is paid tocalculating the matrix coefficients of the resolving system of algebraic equations. Anumerical-and-analytical procedure is given for integrating Mindlin’s fundamentalsolutions over flat triangular and quadrangular boundary elements, arbitrary orientedin the half-space. For convenience, to apply the developed approach in practicalcalculations, the boundary integral equations of the spatial contact problems for anumber of essential special cases are presented. The contact problems at axial load-ing and torsion of absolutely rigid rotation bodies deepened into the half-space, areconsidered. Boundary-element formulations of the contact problems for complex-shaped punches with flat and smooth bases (shallow foundations), situated on spa-tially nonhomogeneous bases of the semi-infinite elastic massif type are presented.

The third chapter deals with practical implementation of the developed numer-ical algorithms and substantiation of the reliability of the numerical solutions. Itpresents the general characteristics and structure of the Rostwerk software pack-age for investigating three-dimensional stress-strained states of elastic bases corre-sponding to the interaction of foundation structures with soil under force factors ofgeneral kind. Procedures for creating input databases are described in detail. Algo-rithms and modules for automatic formation of boundary element grids in plane andin space are presented. An original algorithm for triangulation of flat single- andmultiply connected domains, bounded by straight line segments or circle arcs, isdescribed. An algorithm of generation (according to the given triangulation) of dualpolygonal boundary element grids of Dirichlet cell type is considered. The createdobject library of boundary element modules, partitioned into boundary elements,enabling spatial discretization of complex-shaped surfaces of foundation structures,is described. Specific features of solving the systems of linear algebraic equationswith asymmetric and close-packed matrices, arising in boundary element analysis,are considered. For solving such systems by direct (Gauss type) methods a spe-cial scaling procedure is applied, improving the conditioning of matrices for thefinite-dimensional algebraic analogue of a contact problem. The data about the reli-ability of the numerical solutions are presented. The BEM accuracy and efficiencyare demonstrated by the examples of the solved test problems for flat punches of cir-cular, annular and polygonal shapes. Boundary-element solutions for spatial contactproblems concerning a rigid spherical inclusion and a cylindrical deepened punchin an elastic half-space are obtained. The final part of the chapter gives the resultsfor numerical-and-analytical solution of the spatial contact problem on impressinga deepened conical punch into an elastic half-space. The method of determination

xii Preface

of the deformation modulus from tests for deepened conical indenters with differentangles by static loading is substantiated theoretically.

In the fourth chapter the results of the boundary-element solutions of spatialcontact problems for complex-shaped punches, located on the surfaces of elasticnonclassical bases, are analyzed. The problems under consideration correspond tothe modeling of contact interaction of shallow foundations with elastic nonhomo-geneous bases. Contact pressure fields under punches of various shape under aneccentric load (a contact problem on a strongly inclined punch) are obtained. Theinfluence of non-uniform (over the area) compressibility as well as depth-dependentnonhomogeneity of the base deformational properties on the formation and devel-opment of detachment zones, settlements and slopes of punches with the increaseof the absolute values of overturning moments is shown. An algorithm to calculatethe boundaries of the section core for rigid complex-shaped foundation plates fromthe stress values is described. Some optimization problems are solved for load andshape parameter control in order to provide uniform settlement of rigid foundation.As an example for the application of the developed boundary element method, acontact problem is solved and the elastic base stress-strained state is determinedfor a rigid strip foundation of variable width. In the same chapter a spatial contactmodel of the base is built taking into account nonlinear elastic soil properties. Aprocedure for the model parameter characterization based on the direct punch testdata is considered. Finally, the chapter contains the studies of contact problems ofbending of orthotropic plates situated on elastic nonclassical bases, performed byBEM combined with finite difference method.

In the fifth chapter BEM is applied to calculate contact interaction of foundationstructures with soil, taking into account the deepening factor. The need for spatially-based calculation of bases of deepened foundations is explained. The principles forfoundation structure calculations from the base deformations are briefly reviewed aswell as the existing problem formulations and solution methods for spatial problemsof contact interaction of deepened foundation structures with soil bases. Solutions ofspatial contact problems for deepened monolithic-type foundation structures mostwidely used in the recent years are also considered, namely for (1) pyramidal piles;(2) foundations made of short vertical or inclined bored piles with caps; (3) boredpile foundations with support extensions; (4) slot foundations with the longitudinalcross-section of various shape. Heterogeneous stress-strained states of the base aretaken into account as well as the formation of cavities between the soil and the foun-dation structures. The effect of the foundation shape on its displacement and slopeat various spatial loading is estimated quantitatively. Numerous examples show theresults of the boundary-element modeling to be in good agreement with the exper-imental measurements performed for spatial foundation structures, in most casesBEM results being closer to the experiment than those obtained by other knowncalculation methods.

Finally, the sixth chapter presents solutions of spatial problems of applied geome-chanics related to variation of pore pressure in the soil. The influence of the porepressure decline on the soil settlement and cracking as well as the induced seis-micity and other environmental hazards due to pumping out gas and oil deposits or

Preface xiii

intense removal of underground water at industrial or civil engineering is discussed.The methods for numerical modelling of soil mass deformations due to the reduc-tion of the pore pressure are described. The approach is based on the application ofintegral representations for displacements in a half-space saturated with liquid (orgas) according to the theory of linear pore-elasticity (filtration consolidation). Spa-tial deformation of the earth surface due to operating horizontal gas-and-oil wells orwater drains is studied with the account of the run-off mode. Finally, the results forboundary-element solutions of the spatial contract interaction of structures with thesoil at reduced pore pressure are presented.

The studies, presented in the book, are of applied character and have beeninitially oriented at geotechnical objects in industrial and civil engineering. Theboundary element methods developed are suitable for wide applications to calculatethe spatial deformation of soil bases. They provide high reliability and efficiencyof design solutions for foundation structures. Moreover, the boundary elementapproach presented here can be helpful for solving other spatially-based problemsof mechanics and mathematical physics.

The book summarizes the studies performed in the recent years in VoronezhState University of Architecture and Civil Engineering. The author is grateful toProf. Viktor N. Nikolaevskiy for his all-round support as well as to Dr. SergeyV. Ikonin and Dr. Alexandr A. Sedaev for fruitful communications and helpful dis-cussions which have enabled the book to be made more substantial.

The preparation of the book for publication was essentially supported bythe Dean of Geotechnical Faculty in Varaždin, University of Zagreb (Croatia),Prof. Mladen Kranjcec and the Vice Dean of the faculty Dr. Božo Soldo. The trans-lation from Russian would not have been possible without the key professional con-tribution of Dr. Yuriy Azhniuk from Institute of Electron Physics, Ukr. Nat Acad Sci(Uzhhorod, Ukraine). The author is truly indebted to all of them.

The author hopes that the work presented in the book can be a helpful studyfor numerical experiments in geotechnical engineering and will be grateful to thereaders for their comments.

Voronezh, Russia Sergey M. Aleynikov

Contents

1 Spatial Contact Models of Elastic Bases . . . . . . . . . . . . . . . . 11.1 Fundamental Solutions of Static Problems of Spatial

Theory of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Concentrated Forces in an Elastic Body . . . . . . . . . . 11.1.2 Green’s Displacement Tensor . . . . . . . . . . . . . . . . 21.1.3 Kelvin’s Tensor of Influence . . . . . . . . . . . . . . . . 3

1.2 Elastic Homogeneous Isotropic Half-Space . . . . . . . . . . . . 51.2.1 Mindlin’s Solution . . . . . . . . . . . . . . . . . . . . . 51.2.2 Boussinesq and Cerruti Solutions . . . . . . . . . . . . . . 6

1.3 Coupled Half-Spaces . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Elastic Layered Bases . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 Constant-Width Elastic Layer . . . . . . . . . . . . . . . . 121.4.2 Variable-Thickness Elastic Layer . . . . . . . . . . . . . . 171.4.3 Multilayer Elastic Half-Space . . . . . . . . . . . . . . . . 25

1.5 Elastic Bases with the Deformation Modulus, Variable with Depth 551.5.1 Variation of Deformation Modulus with Depth . . . . . . . 551.5.2 Normal Concentrated Force Acting on the

Half-Space Surface . . . . . . . . . . . . . . . . . . . . . 581.5.3 Settlement of a Nonhomogeneous Half-Space Surface . . . 63

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2 Static Analysis of Contact Problems for an Elastic Half-Space . . . . 912.1 Boundary Integral Equations of the Contact Problem

for an Absolutely Rigid Punch, Deepened into an ElasticHalf-Space, Under a Spatial Load System . . . . . . . . . . . . . 91

2.2 Finite-Measure Analogue of the Contact Problem UsingDirect Boundary-Element Method . . . . . . . . . . . . . . . . . 96

2.3 Numerical-and-Analytical Method of Integrationof Fundamental Mindlin’s Solutions . . . . . . . . . . . . . . . . 101

2.4 Punch in the Shape of a Rotation Body, Deepenedinto an Elastic Half-Space . . . . . . . . . . . . . . . . . . . . . . 1082.4.1 Axisymmetric Contact Problem . . . . . . . . . . . . . . . 1102.4.2 Torsion of an Axisymmetric Punch in an Elastic

Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . 115

xv

xvi Contents

2.5 Contact Problems for Rigid Punches Located on theElastic Base Surface . . . . . . . . . . . . . . . . . . . . . . . . . 1192.5.1 Indentation of a Punch with a Flat Smooth Base

into an Elastic Half-Space . . . . . . . . . . . . . . . . . . 1212.5.2 Torsion of an Elastic Half-Space by a Rigid Punch . . . . . 126

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3 Computer Implementation of Boundary-Element Algorithms . . . . 1353.1 Software for Solving Spatial Problems of Contact

of Foundations with Soil Bases . . . . . . . . . . . . . . . . . . . 1363.2 Specific Features of Numerical Solutions of Linear

Algebraic Equation Systems with Non-symmetricalMatrices, Arising in Boundary-Element Analysis . . . . . . . . . 146

3.3 Effective Discretization of 2-D Domains of ComplexShape at Numerical Solving of Spatial Contact Problemsof Theory of Elasticity . . . . . . . . . . . . . . . . . . . . . . . 1503.3.1 Algorithm of Triangulation in the Boundary-

Element Method . . . . . . . . . . . . . . . . . . . . . . . 1513.3.2 Dual Grids and Their Application in Boundary-

Element Method . . . . . . . . . . . . . . . . . . . . . . . 1623.4 Automated Construction of Spatial Grids of Boundary

Elements on the Surfaces of Contact of DeepenedFoundation Structures with Soil . . . . . . . . . . . . . . . . . . . 174

3.5 Test Examples of Numerical Modeling of Spatial Problemsof Contact Interaction . . . . . . . . . . . . . . . . . . . . . . . . 1923.5.1 Contact Problems for Flat Punches with a Smooth Base . . 1923.5.2 Contact Problems with the Account of the

Deepening Factor for Axisymmetric Punches,Interacting with an Elastic Half-Space . . . . . . . . . . . 218

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

4 Contact Interaction of Shallow Foundationswith Nonhomogeneous Bases . . . . . . . . . . . . . . . . . . . . . . 2514.1 Spatial Contact Problems for Rigid Flat-Bottom Punches . . . . . 2534.2 Contact Problems for Rigid Rectangular Punches, Resting

on Elastic Nonhomogeneous Bases . . . . . . . . . . . . . . . . . 2784.2.1 Contact Interaction at Central Loading . . . . . . . . . . . 2824.2.2 Contact Interaction at Off-Centre Loading with the

Account of Unilateral Constraints . . . . . . . . . . . . . 2954.3 Control of the Parameters of Loading and Shape to Provide

a Uniform Settlement of Rigid Foundation Plates . . . . . . . . . 3004.3.1 Formulation of the Problem and Its Numerical

Implementation . . . . . . . . . . . . . . . . . . . . . . . 3014.3.2 External Load Control . . . . . . . . . . . . . . . . . . . . 3034.3.3 Shape Parameter Control . . . . . . . . . . . . . . . . . . 307

Contents xvii

4.4 Spatial Stress-Strained State of the Base of a Rigid StripVariable-Width Foundation . . . . . . . . . . . . . . . . . . . . . 3114.4.1 Contact Problem for a Variable-Width Strip Foundation . . 3124.4.2 Stress-Strained State of a Strip Foundation Base . . . . . . 3154.4.3 Contact Pressure Distribution in the Area of the

Strip Foundation Width Variation . . . . . . . . . . . . . . 3174.5 Calculation of the Section Kernel Boundary for Rigid

Foundation Plates . . . . . . . . . . . . . . . . . . . . . . . . . . 3234.6 Numerical Algorithms of Solving Boundary Integral

Equations in Spatial Contact Problems for a NonlinearlyDeformable Base . . . . . . . . . . . . . . . . . . . . . . . . . . 3344.6.1 Spatial Contact Model for a Nonlinearly

Deformable Base . . . . . . . . . . . . . . . . . . . . . . 3354.6.2 System of Nonlinear Contact Equations of the

Contact Problem for Absolutely Rigid Punches ofa Complex Shape with a Flat Base . . . . . . . . . . . . . 337

4.6.3 Iterative Processes of Solving a Finite-MeasureAnalogue of the Spatial Contact Problem for aNonlinearly Deformable Base . . . . . . . . . . . . . . . . 339

4.6.4 Contact Problem for a Round Punch on aNonlinearly Deformable Base . . . . . . . . . . . . . . . . 341

4.6.5 Estimation of Nonlinear Deformation Effects fromPunch Test Results . . . . . . . . . . . . . . . . . . . . . 348

4.7 Contact Problem for Orthotropic Foundation Plateswith the Account of the Specific Features of SpatiallyNonhomogeneous Base Deformation . . . . . . . . . . . . . . . . 3514.7.1 Static Calculations of Foundation Plates on Elastic Bases . 3524.7.2 System of Integro-Differential Equations of

Bending of a Plate, Resting on an Elastic Base . . . . . . . 3584.7.3 Calculation of Rectangular Orthotropic Plates

Based on Combining Finite-Difference andBoundary-Element Methods . . . . . . . . . . . . . . . . 361

4.7.4 Examples of Numerical Modelling of the ContactInteraction of Plates with Elastic Bases . . . . . . . . . . . 364

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

5 Calculation of Bases for Rigid Complex-Shaped DeepenedFoundations According to the Second Limiting State in aThree-Dimensional Formulation . . . . . . . . . . . . . . . . . . . . 3855.1 General Information on the Calculation of Bases for

Foundation Structures from the Deformations . . . . . . . . . . . 3905.2 Spatial Problems for Calculation of Foundation Bases with

the Account of the Depth Factor . . . . . . . . . . . . . . . . . . 3965.3 Calculation of Bases for Pyramidal Piles Under Vertical,

Horizontal, and Momental Loads . . . . . . . . . . . . . . . . . . 415

xviii Contents

5.3.1 Existing Approaches to the Calculation of Pileswith a Variable Cross-Section . . . . . . . . . . . . . . . . 416

5.3.2 Calculation for the Vertical Load . . . . . . . . . . . . . . 4205.3.3 Calculation for the Action of a Horizontal Load . . . . . . 4205.3.4 Calculation for the Action of an Inclined Load . . . . . . . 4225.3.5 Calculation for the Combined Action of an Inclined

Force and a Moment . . . . . . . . . . . . . . . . . . . . 4235.4 Interaction of Bases and Rigid Bored Foundations with

Vertical and Inclined Piles . . . . . . . . . . . . . . . . . . . . . 4245.4.1 Structure, Design, and Specific Features of

Calculation of Rigid Pile Foundations with ShortPiles and a Pile Raft . . . . . . . . . . . . . . . . . . . . . 425

5.4.2 Vertical Cylindrical Piles Under an Inclined Load . . . . . 4285.4.3 Foundations with Inclined Piles and a Rectangular

Pile Raft . . . . . . . . . . . . . . . . . . . . . . . . . . . 4355.5 Spatial Contact Problem for a Bored Pile Foundation

with a Widening . . . . . . . . . . . . . . . . . . . . . . . . . . . 4385.5.1 Production and Structures of Bored Pile

Foundations with a Support Widening . . . . . . . . . . . 4395.5.2 Engineering Methods for Calculation of Bored Pile

Foundation Bases from the Base Deformation . . . . . . . 4415.5.3 Calculation of Deformations of the Base of a Bored

Pile Foundation with a Spheroconical WideningUnder a Central Loading (Axisymmetric Contact Problem) 443

5.5.4 Calculation of Displacements and Slopes of aBored Pile Foundation Under an Inclined Load . . . . . . 448

5.6 Calculation of Contact Interaction of Bases with SlottedFoundations of Industrial and Civil Buildings . . . . . . . . . . . 4545.6.1 Slotted Foundations of Various Structural Shapes . . . . . 4545.6.2 Calculation of Slotted Foundations Based on the

Base Deformation . . . . . . . . . . . . . . . . . . . . . . 4575.6.3 Contact Stress on the Lateral Surface of a Slotted

Foundation . . . . . . . . . . . . . . . . . . . . . . . . . 4765.6.4 Slotted Foundations with Lateral Widenings . . . . . . . . 491

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

6 Spatial Contact Problems for Porous Elastic Bases . . . . . . . . . . 5056.1 Soil Mass Deformation Due to the Pore Pressure Decline . . . . . 509

6.1.1 Integral Representation of Displacementsin a Porous Elastic Medium . . . . . . . . . . . . . . . . . 509

6.1.2 Dilatation Relations . . . . . . . . . . . . . . . . . . . . . 5126.2 Distribution of Pressure in a Layer in Case of Functioning

Horizontal Wells . . . . . . . . . . . . . . . . . . . . . . . . . . 5146.2.1 Distributed Sources of Predetermined Intensity . . . . . . 5146.2.2 Account of the Finite Radius of the Well . . . . . . . . . . 516

Contents xix

6.3 Contact Problems for Foundation Structures at a ReducedPore Pressure in the Soil . . . . . . . . . . . . . . . . . . . . . . 5176.3.1 Integral Equations of a Spatial Contact Problem . . . . . . 5176.3.2 Finite-Dimensional Algebraic Analogue of the

Integral Equation System . . . . . . . . . . . . . . . . . . 5196.3.3 Numerical Algorithm of Solution of the Contact Problem . 5206.3.4 Contact Problem for Shallow Foundations . . . . . . . . . 522

6.4 Examples of Numerical Calculations . . . . . . . . . . . . . . . . 5246.4.1 Spatial Deformation of the Land Surface . . . . . . . . . . 5266.4.2 Surface Deformations of the Layer . . . . . . . . . . . . . 5326.4.3 Settlements and Slopes of Rigid Foundation Plates . . . . 533

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

Appendix A Fundamental Solutions of Spatial Theory ofElasticity for a Homogeneous Isotropic Half-Space . . . . . . . . . . 543

Appendix B Numerical Schemes for Surface Integral Calculations . . 555

Appendix C Round Punch on an Elastic Layer of VariableThickness at Central and Off-Centre Load . . . . . . . . . . . . . . 569

Appendix D Foundation Under a Tower-Type Structure on aWedge Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

Appendix E Finite-Difference Equations of Cylindrical Bendof Orthotropic Slabs Located on an Elastic Foundation . . . . . . . 591

Appendix F Calculation of the Base for a Pyramidal PileUnder Vertical Load According to the “Instructions Manualfor Design of Foundations Made of Pyramidal Piles” . . . . . . . . . 603

Appendix G Isolines of Contact Stress on a Lateral Surface ofa Slotted Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

Appendix H Numeric Schemes of Volume Integration . . . . . . . . . 629

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637