Mechanics of Non-Homogeneous and Anisotropic Foundations

376 Pages·2001·14.262 MB·English

by  B. Grigori Muravskii (auth.)

Foundations of Engineering Mechanics
B. Grigori Muravskii, Meehanies of Non-Homogeneous
and Anisotropie Foundations
Springer
Berlin
Heidelberg
New York
Barcelona
Hong Kong
London
Milan ONLINE L1BRARY
Engineering
Paris
Singapore
http://www.springer.de/engine/
Tokyo
B. Grigori Muravskii
Meehanies ofNon
Homogeneous
and Anisotropie Foundations
Translated by Boris Krasovitski
With 149 Figures
, Springer
Series Editors:
Vladimir I. Babitsky, DSc J. Wittenburg
Louborough University Universität Karlsruhe (TH)
Department of Mechanical Engineering Institut für Technische Mechanik
LEII 3 TU Loughborough Leicestershire Kaiserstr. 12
United Kingdom 76128 Karlsruhe
Germany
Author:
Grigori B. Muravskii
Geotechnical Department
Faculty of Civil Engineering Technicon
32000 Haifa
Israel
Translator:
Boris Krasovitski
POB7843
36811 Nesher
Israel
ISBN 978-3-642-53602-1
Library of Congress Cataloging-in-Publieation Data
Muravskii, Grigori B:
Meehanics of non-homogeneous and anisotropie foundations I B. Grigori Muravskii. Translated by
Boris Krasovitski. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris;
Singapore; Tokyo: Springer, 2001
(Foundations of engineering meehanics)
ISBN 978-3-642-53602-1 ISBN 978-3-540-44573-9 (eBook)
DOI 10.1007/978-3-540-44573-9
This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is
eoncerned, speeifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication
ofthis publication or parts thereofis permitted only under the provisions ofthe German Copyright
Law ofSeptember 9,1965, in its current version, and permission for use must always be obtained from
Springer-Verlag. Violations are liable for prosecution act under German Copyright Law.
Springer-Verlag is a eompany in the BertelsmannSpringer publishing group
http://www.springer.de
© Springer-Verlag Berlin Heidelberg New York 2001
Softcover reprint of the hardcover 1st edition 2001
The use of general descriptive names, registered names, trademarks, ete. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
Typesetting: Camera ready by author
Cover-Design: de'blik, Berlin
Printed on acid free paper SPIN: 10791920 62/3020/kk -5 43210
Preface
Although realistic soil and rock foundations reveal noticeable deviations in their
properties from homogeneity and isotropy, the model of the homogeneous
isotropie elastic half-space is widely used when studying static and dynamie
interactions between a defonnable foundation and structures. This is explained by
significant mathematieal difficulties inherent in problems conceming mechanies
of anisotropie and heterogeneous elastic bodies. Solving the basic static and
dynamie problems for heterogeneous and anisotropic half-spaces, such as different
contact problems and problems of constructing Green's functions, has become
possible in the last few decades due to the development of computer engineering
techniques and numerical methods.
This book contains the results of investigations in the area of statics and
dynamies of heterogeneous and anisotropic foundations, carried out by the author
in the last five years while working in the Faculty of Civil Engineering at
Technion - Israel Institute of Technology. The book is directed at engineers and
scientists in the areas of soil mechanics, soil-structures interaction, seismology and
geophysics.
Some characteristic features of the book are:
i) Constructing (Chap.l) solutions in a general fonn for the heterogeneous (in
the depth direction) transversely isotropic elastic half-space subjected to
different loadings, hannonic in time. Characteristics of the given half-space
have an influence on functions (of depth z and parameter k of Hankel's
transfonns), which are detennined from a system of ordinary differential
equations.
ii) New dynamic solutions relating to the homogeneous transversely isotropic
elastic half-space subjected to the action of vertical and horizontal forces
applied to the half-space surface or below the surface. Solutions are
presented relating to basic contact problems for this kind ofhalf-space.
iii) Dynamic and static solutions for the linearly heterogeneous half-space,
which, as applied to static problems, was intensively studied in works by
R. E. Gibson and his coworkers. Solutions are presented for a half-space
with exponentially varying stiffuess. In these cases it is possible to find
analytical solutions for the above-mentioned ordinary differential equations;
thus the required amplitudes of vibrations are expressed in the fonn of
integrals containing the detennined functions.
iv) Development ofnumerical-analytical methods for the considered problems,
which consist of a combinations of integral representations of the sought
functions and numerical treatment of ordinary differential equations for the
VI Preface
corresponding Fourier-Bessel transfonnations. The piecewise constant
approximation for varying coefficients of these differential equations results
in the well-known method of thin layers. Employing the general
representations developed in Chap.l and using simple solutions in the
intervals (elements) with constant properties allows the construction of
effective solutions. Another appropriate treatment is the use of Runge-Kutta
method with additional application of Gogunov's method to eliminate
computational difficulties connected with large values of the transfonnation
parameter k.
The book presents, graphically, a large number of results of computations,
which give a clear pieture of the behavior of the mechanical systems considered.
These results can serve for estimating accuracy of different simplified methods,
e.g. combining the division of a half-space into thin layers with setting a fonn of
displacements distribution within the layers.
I express my gratitude to my colleagues - Professors ofTechnion R. Baker and
s. Frydman for useful discussions of geotechnical problems connected with the
material of the book.
Haifa, April 2001. G. B. Muravskii
Contents
Introduction ................................................................................. 1
Chapter 1. General Solutions of Harmonie Vibrations in Heterogeneous
Transversely Isotropie Half-Spaee .................•....................................5
1.1 Basic Relationships of Theory of Elasticity for Transversely Isotropic
Body ......................................................................................... 5
1.2 Particular Solutions for Transversely Isotropie Heterogeneous Elastic
Medium .....................................................................................8
1.3 Statement of Boundary Conditions for Planes z = const ....................... 13
1.4 General Solution for Harmonie Vibration ofHalf-Space Subjected
to Surface Loads ......................................................................... 18
1.5 General Solution for Harmonie Vibrations of Half-Space Subjected
to Loads Applied below its Surface .................................................. 31
1.5.1 Action ofVertieal Force ................................................... .31
1.5.2 Action ofHorizontal Force ................................................ .34
1.6 Application ofFunctions Related to Dilatation and Rotation
ofDisplacement Field .................................................................. .37
1.6.1 Vertieal Force Applied within Half-Space ............................... .40
1.6.2 Horizontal Force Applied within Half-Space ............................ .42
1.7 Application ofSuperposition Principle to Loadings ofRectangular
and Circular Domains .................................................................. .44
1.7.1 Vertical Load Distributed Uniformly over Rectangular Domain ...... .44
1.7.2 Horizontal Load Distributed Uniformly over Rectangular Domain ... 50
1.7.3 Vertical Load Distributed Uniformly over Circular Domain ........... 53
1.7.4 Horizontal Load Distributed Uniformly over Circular Domain ....... 54
1.7.5 Axisymmetric Radial Load Applied to Circular Domain ............... 57
1.7.6 Antisymmetric Vertical Load Applied to Circular Domain ............. 58
1.7.7 Horizontal Load Acting in Tangential Direction ......................... 61
1.7.8 Self-Balanced Horizontal Load ............................................ 62
1.7.9 Loading ofInfmite Strip ................................................... 65
Chapter 2. Statie and Dynamic Problems for Homogeneous Transversely
Isotropie Half-Space ..•......•.•........••..•.......•.•.....................••.••....•.....6 8
2.1 Vibrations ofHalf-Space Subjected to Vertical Force Applied
to Half-Space Surface .................................................................. 68
2.1.1 Static Action ofVertical Force ............................................. 72
VIII Contents
2.1.2 Free Vibrations ofHalf-Space .............................................. 76
2.1.3 Forced Vibrations ofHalf-Space ...........................................7 7
2.2 Model ofDefonnable Foundation as Limiting Case ofTransversely
Isotropic Half-Space .................................................................... 88
2.3 Vibrations ofHalf-Space Subjected to Action ofHorizontal Force Applied
to Half-Space Surface ..................................................................9 0
2.3.1 Static Action ofHorizontal Force .......................................... 92
2.3.2 Analysis of Amplitudes ofVibrations ..................................... 94
2.4 Torsional Vibrations ofHalf-Space ..............................................9 6
2.5 Action ofForce Applied in Infinite Space .......................................9 9
2.5.1 Action ofVertical Force .................................................... 99
2.5.2 Action ofHorizontal Force ................................................ 102
2.6 Vibrations ofTransversely Isotropic Half-Space under Action
ofForce Applied within Half-Space ................................................. 106
2.6.1 Action ofVertical Force ................................................... 106
2.6.2 Action ofHorizontal Force ................................................ 111
2.7 Contact Problems for Transversely Isotropic Half-Space .................... 116
2.7.1 Static Stiffnesses for Circular Disk on Transversely Isotropic
Half-Space .......................................................................... 116
2.7.2 Dynamic Stiffnesses for Circular Disk on Transversely Isotropic
Half-Space .......................................................................... 135
2.8 Plane Problems for Transversely Isotropie Half-Space ....................... 147
2.8.1 Action of Vertical Load Distributed Unifonnly along Infinite
Line on Half-Space Surface ...................................................... 148
2.8.2 Action ofHorizontal Load Distributed Unifonnly along Infmite
Line on Half-Space Surface ...................................................... 151
2.8.3. Contact Problems for Strip Stamp ....................................... 154
Chapter 3. Mechanics ofIsotropic Half-Space with Sbear Modulus Varying
Linearly with Depth .•.•....•.•.•••••...••••...••••.•••••.•.•.•.•••••.••••.•.•..•..••••..• 168
3.1 Fundamental Solutions ofSystem ofEquations (1.l30}--(1.133) ........... 168
3.2 Vibrations ofIsotropic Linearly Heterogeneous Half-Space under
Action ofVertical Force Applied to Half-Space Surface .......................... 174
3.3 Vibrations ofIsotropic Linearly Heterogeneous Half-Space under
Action ofHorizontal Force Applied to Half-Space Surface ...................... 179
3.4 Determining Properties ofLinearly Heterogeneous Half-Space Using
Characteristics ofSurface Waves ..................................................... 183
3.4.1 Application ofSolution Related to Vertical Vibrations
of Half-Space Surface under Action of Vertical Force ....................... 184
3.4.2 Application ofSolution Related to Horizontal Vibrations
ofHalf-Space Surface under Action ofHorizontal Force .................... 191
3.5 Some Static Problems for Linearly Heterogeneous Half-Space ............. 194
3.5.1 Displacements ofHalf-Space Surface under Action
of Surface Concentrated F orces .................................................. 194
3.5.2 Static Stiffnesses for Circular Disk Resting on Isotropic
Contents IX
Linearly Heterogeneous Half-Space .............................................2 06
3.6 Dynamic Stiffness ofCircular Disk Resting on Linearly
Heterogeneous Half-Space ............................................................2 14
3.7 Vibrations ofLinearly Heterogeneous Half-Space Subjected
to Force Applied within Half-Space .................................................2 30
3.7.1 Action ofVertical Force ................................................... 230
3.7.2 Action ofHorizontal Force ................................................ 238
3.8 Plane Problems for Linearly Heterogeneous Half-space .....................2 44
3.8.1 Action ofVertical Load Distributed Uniformly over Infinite ........... .
Straight Line on Half-Space Surface .. , .................................... '" ... 244
3.8.2 Action ofHorizontal Load Distributed Uniformly over Infmite
Straight Line on Half-Space Surface ............................................2 46
3.8.3 Static Surface Green's Functions in Plane Problems
for Linearly Heterogeneous Half-Space ........................................ 249
Chapter 4. Mechanies of Transversely Isotropie Half-Spaee with Stiffness
Varying Exponeotially with Depth •.••••.•..•••••••••••••••••.•.•.•••.....•.••.••..••• 257
4.1 Vibrations ofTransversely Isotropie Half-Space
Having Elastic Coefficients Bounded at Infinite Depth ...........................2 57
4.1.1 Variation ofElastic Parameters ofHalf-Space with Depth .......... .257
4.1.2 Construction of Fundamental Solutions ..................................2 61
4.1.3 Vibrations ofHalf-Space under Action ofVertical Force
Applied to Half-Space Surface ......... '" ....................................... 265
4.1.4 Vibrations ofHalf-Space under Action ofHorizontal Force
Applied to Half-Space Surface ...................................................2 68
4.2 Vibrations ofTransversely Isotropie Half-Space with
Stiffness Increasing without Bounds ............... '" ...............................2 70
4.2.1 Varying ofElastic Parameters ofHalf-Space with Depth ............. 270
4.2.2 Construction ofFundamental Solutions ..................................2 72
4.2.3 Vibrations ofHalf-Space under Action ofVertical Force
Applied to Half-Space Surface ................................................... 276
4.2.4 Vibrations ofHalf-Space under Action ofHorizontal Force
Applied to Half-Space Surface ...................................................2 80
Chapter 5. Applieatioo of Numerieal--Analytieal Methods to Statie and
Dyoamie Problems for Heterogeneous Half-Spaee. •.••••••••.•.••••.••••••••..•.• 281
5.1 Introduction ........................................................................ 281
5.2 Heterogeneous Half-Space Subjected to Vertical Force
Applied to Half-Space Surface ........................................................2 84
5.2.1 Application ofThin Layer Technique for Numerical
Solution of Differential Equations ........................................ '" ....2 85
5.2.2 Application ofRunge-Kutta Method for Numerical
Solution ofDifferential Equations .............................................. .297
X Contents
5.2.3 Parameter Determination for Isotropie Half-Space with Shear
Modulus Varying by Power Law (Action ofVertical Force) ............... 299
5.3 Heterogeneous Half-Space Subjected to Horizontal Force
Applied to Half-Space Surface ............................................ , ......... 306
5.3.1 Application ofThin Layer Technique for Numerical
Solution ofDifferential Equations ............................................. .307
5.3.2 Application ofRunge-Kutta Method for Numerical Solution
of Differential Equations ......................................................... 31 0
5.3.3 Parameter Determination for Isotropie Half-Space with Shear
Modulus Varying by Power Law (Action ofHorizontal Force) ............. 311
5.4 Vibration Problems for Heterogeneous Half-Space Subjected
to Force Applied within Half-Space ................................................ 315
5.4.1 Action ofVertical Force. Application ofThin Layer
Technique for Numerical Solution ofDifferential Equations ............... 315
5.4.2 Action ofVertical Force. Application ofRunge-Kutta
Method for Numerical Solution of Differential Equations .................. 321
5.4.3 Action ofHorizontal Force. Application ofThin Layer
Technique for Numerical Solution ofDifferential Equations ............... 324
5.4.4 Action ofHorizontal Force. Application ofRunge-Kutta
Method for Numerical Solution ofDifferential Equations .................. 327
5.5 Static solutions ..................................................................... .328
5.6 Contact Problems for Circular Disk Resting on Half-Space with
Stiffness Increasing with Depth by Power Law ................................... .341
5.6.1 Static Stiffnesses for Circular Disk. ..................................... .341
5.6.2 Dynamic Stiffnesses for Circular Disk ................................. .350
References ..................................................................................3 56
Index ....................................................................................... 362