Statics of Granular Media

279 Pages·1965·5.57 MB·English

by  V. V. Sokolovskii and A. W. T. Daniel (Auth.)

STATICS OF
GRANULAR MEDIA
BY
V. V. SOKOLOVSKII
Completely revised and enlarged edition
TRANSLATED BY
J. K. LUSHER
ENGLISH TRANSLATION EDITED BY
A.W.T. DANIEL
Senior Lecturer in Civil Engineering,
Queen Mary College, University of London
P E R G A M ON PRESS
OXFORD . LONDON · EDINBURGH · NEW YORK
PARIS·FRANKFURT
1965
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Copyright © 1965
PERGAMON PRESS LTD.
First edition 1965
Library of Congress Catalog Card Number 63-21098
This is a completely revised and enlarged edition based on
a translation of the second Russian edition of the original volume
Cmamma Cbmyneu cpedhi (Statika sypuchei sredy)
published in 1960 by Fizmatgiz, Moscow
PREFACE TO THE ENGLISH EDITION
THIS book differs greatly from the two previous editions of Statika
Sypuchei Sredy, in Russian, and consequently from the Enghsh trans­
lation of the second edition pubhshed by Butterworths under the title
Statics of Soil Media, so that it can truly be regarded as a new book.
First of all a number of new problems have been included which were
solved in the period between the two editions. These comprise prob­
lems such as stability of slopes, the shape of curvilinear overhang
slopes, solutions of the equations in the boundary layers, curvihnear
retaining waUs, stability of layered foundations and limiting equih­
brium of foundations with curvilinear contours.
Further, new variables have been introduced, which enable us to
transform the basic formulae and equations into a more convenient
form and to achieve greater elegance in the theory.
FinaUy, numerical results have been included for solutions of typical
problems for various values of the mechanical constants, which now
obviates the need for tedious calculations.
The Enghsh translation will undoubtedly help to increase the number
of readers, both theoreticians and those engaged in normal engineering
practice.
The author is indebted to Pergamon Press Ltd., by whose initiative
the translation of this book has been carried out, and would like to
express his sincere appreciation to aU who have taken part in editing
and preparing the manuscript for the press.
V. V. SOKOLOVSKII
PREFACE
THE present book is devoted to the theory of hmiting equihbrium of
a granular medium, and is issued as a third and completely revised
edition. It covers a wide range of subjects, some new and others al­
ready considered in previous editions. The contents of the book are
briefly as foUows:
Chapter 1 describes the theory of hmiting plane equilibrium of a
granular medium on the basis of the usual condition of hmiting equili­
brium. The equations of limiting plane equilibrium and their trans­
formation into the canonical system is investigated in detail. The
question of mechanical similarity is studied, which is of considerable
importance both in calculations and in model analysis. Basic bound­
ary-value problems are formulated for the canonical system and
effective methods of numerical integration are suggested.
An important question in the statics of granular media—the hm­
iting equilibrium of foundations—is also studied. The derivation of
the required solutions is reduced to combinations of the boundary-
value problems for the canonical system.
Chapter 2 deals with problems of considerable practical signifi­
cance—^the stability of foundations and slopes. Here again the deri­
vation of the required solutions leads to combinations of the boundary-
value problems for the canonical system. In aU these problems we
encounter a basic solution with a singular point, which in the plane of
the characteristics corresponds to a whole segment of a characteristic.
Considerable attention is devoted to the problem of the shape of
slopes; overhang slopes, which have discontinuous stress states, are
investigated in detail.
Chapter 3 is devoted to the classical problem of the pressure of a
fill on a retaining wall. The waUs are classified according to the slope
of their rear faces. Problems in which discontinuities occur in the
stress field are also investigated.
The chapter also includes a study of the equations of limiting plane
equilibrium in narrow layers along the rear face of the waU, and the
derivation of approximate integrals.
viii
PREFACE IX
The theory of the hmiting plane equilibrium of a granular medium
with a lameUar structure occupies a special place and is iUustrated by the
extremely interesting problem of the stability of lamellar foundations.
Chapter 4 describes the theory of the limiting plane equUibrium of
an ideaUy cohesive medium in the absence of internal friction. This
theory is analogous to the theory of plane plastic equilibrium and
enables us to derive solutions to a considerable number of problems
on the stabihty of foundations and slopes and the pressure of a fiU
on retaining waUs. Problems are considered in which discontinuities
occur in the stress state.
The chapter also studies the theory of the limiting plane equilibrium
of a cohesive medium using a more general form of the condition of
limiting equilibrium. The equations of hmiting plane equihbrium are
investigated in detaU, together with their transformation to the cano­
nical system. It is shown that for certain particular forms of the limit­
ing conditions the equations of limiting equilibrium have simple
integrals. Problems deahng with the compression of strips and rect­
angles are investigated, and their solutions are given in closed form.
Chapter 5 is concerned with the limiting equihbrium of an ideally
granular wedge. The special properties of an ideaUy granular medium,
i.e. one in which cohesion is absent, enable us to find solutions to the
problems encountered here more simply than on the basis of the
general theory. Problems are considered in which there exist simul­
taneously zones of limiting and non-limiting equilibrium, together with
problems on the equihbrium of embankments, the stabihty of founda­
tions and the pressure of a fill on retaining waUs. The solution of aU
these problems is given in closed form, or alternatively is achieved by
integration of the ordinary non-linear differential equations.
Particular attention is devoted to the hmiting equilibrium of an
ideaUy granular wedge with a lamellar structure, and in particular,
to the problem of stabUity of lameUar foundations.
All the chapters are iUustrated by examples, the solutions of which
are presented in graphical or tabular form. In the tables only two
decimal places are given although the calculations were carried out to
a greater accuracy. Some of these examples are intended only to
iUustrate the method of solution, whilst others can be used directly
as a basis for practical calculations.
The tables have been compiled by nimierical integration of the
appropriate diflFerential equations. This was carried out in the Com­
puter Centre of the Academy of Sciences of the U.S.S.R.
χ PREFACE
For convenience, the references are given in a separate hst at the
end of the book and, as usual, reference to any work is indicated by the
appropriate number in square brackets.
In conclusion, the author is grateful for the comments and obser­
vations made by numerous people on the first and second editions of
this book. He conveys his gratitude in particular to A. M. Kochetkov
and Z. N. Butsko for their assistance in compihng the tables and in
preparing the manuscript for the press.
V. V. SOKOLOVSKH
INTRODUCTION
IN THE statics of granular media two types of stress state are studied:
stress states in which a smaU change in body or surface forces wiU not
destroy the equilibrium, and stress states in which a change, no matter
how smaU, in the body or surface forces will cause loss of equilibrium.
Stress states of the second type—so-caUed hmiting stress states—
depend directly on the basic mechanical constants which characterize
the resistance of a granular medium to shear deformation and form
the basis of the theory of limiting equilibrium.
In 1773 Coulomb, the originator of this theory, formulated the
basic theorems of limiting equilibrium and apphed them to determine
the pressure of a fill bounded by a horizontal plane on a vertical retain­
ing waU with an absolutely smooth rear face. His solution was based
on the supposition that there exists a plane surface of rupture. The same
theorems were subsequently used to determine the pressure of a fiU
bounded by an arbitrary surface on inclined and broken-back retain­
ing waUs with rough rear faces. Later, in 1857, Rankine investigated the
limiting equihbrium of an infinite body bounded by an inchned plane,
introduced the concept of slip-surfaces and found the condition of
limiting equilibrium which Pauker subsequently applied in his study
of the stability of foundations. In 1889 Kurdiumov carried out a series
of experiments on the limiting resistance of foundations, which showed
clearly that loss of equilibrium occurs by means of slip of the material
over certain curvihnear surfaces.
New researches in the field of limiting equilibrium have had two
trends.
The first trend has been to create a simplified theory of limiting
equilibrium which makes it possible to solve various problems by
elementary methods. It was developed by Belzetskii (1914), Krey (1918),
Gersevanov (1923), Puzyrevskii (1923) and FeUenius (1926), who made
the assumption of shp-surfaces of various simple shapes—^plane, pris­
matic or circular cylindrical.
This assumption, which means that each problem is reduced to
one of finding the most dangerous position for the shp-surface of the
xi
Xll INTRODUCTION
shape chosen, may not be particularly well-founded, but quite often
gives acceptable results. Therefore this simplified theory, which was
developed further by Prokofev (1934) and Bezukhov (1934) and
summarised in graphical or tabular form, is even now quite widely
used.
The second trend has been a development of the ideas suggested by
Rankine, and attempts to derive an exact theory of limiting equih­
brium which makes possible the solution of various problems and the
determination of the corresponding slip-hne network. It originates
from the works of Kφtter (1903), who considered the differential equa­
tions of equilibrium and the condition of hmiting equihbrium at each
point, formed a set of equations of hmiting euqilibrium and then trans­
formed them to curvilinear co-ordinates.
The further development of this theory was very much influenced
by Prandtl (1920) who posed and solved a number of problems of
plastic equilibrium. He was the first to use a solution with a singular
point with a pencil of straight slip-lines passing through it. These
results were subsequently applied by Reissner (1925) and Novo­
tortsev (1938) to certain particular problems on the stability of foun­
dations, but only for the case of a weightless granular medium, when
the slip-lines of at least one family are straight and the solutions have
closed form.
Von Karman (1927) and Caquot (1934) adopted a completely
different approach and derived a system of equations of limiting
equilibrium for an ideaUy granular wedge, together with approximate
methods for their solution. They considered a number of interesting
problems on the pressure of a fill on retaining waUs, for which it is
impossible to find simple solutions.
However, due to the absence of a general method, aU these investi­
gations found only a limited application in practice. For example, the
various attempts in the problem of stabUity of foundations to apply
the results obtained for a weightless medium did not meet with any
great success and usuaUy led to distorted results.
The first efforts of the author in 1939 were directed towards the
derivation of a general method which would make it possible to solve
the basic problems for a granular medium when the slip-lines of both
famihes are curves and when the solutions no longer have a simple
closed form. The author was able to formulate and investigate various
problems of limiting equilibrium, and wide use was made of the solu­
tion with a singular point with a pencil of curved slip-lines passing
INTRODUCTION XUl
through it. At a later stage the results of this work were coUected and
presented as the first edition of the present book.
Subsequently the results of many different investigations were
pubhshed of which, for brevity, we shaU mention only a few. In 1948
Golushkevich evolved a graphical method of integrating the equations
of limiting equilibrium in which the slip-line network and a special
polar diagram is constructed. He iUustrated his method mainly by
problems which had already been investigated, both for weightless
media and for those possessing weight. Berezantsev (1948) made a
study of the so-caUed total hmiting equilibrium under conditions of
axial symmetry; he derived a method for solving various problems and
carried out a number of successful experiments on the hmiting resi­
stance of foundations.
Subsequent works by the author (1947-1953) in this field were aimed
on the one hand at finding a general method of approach to problems
of limiting equihbrium for cohesive media, and on the other, at finding
a comparatively simple method for solving the various problems on the
limiting equihbrium of an ideally granular wedge. The results obtained
were combined to form the second edition of this book.
AU these investigations have considerably developed the theory of
limiting equilibrium; the range of problems that can be solved has
been considerably extended, and the effectiveness of the methods used
has been improved, so that the theory can now be used as a reliable
basis for engineering calculations.
There are stiU certain difficulties, of course, which have to be solved,
associated with the complexity and tediousness of the calculations in the
determination of the shp-line networks. However, these difficulties can
be considerably reduced or even ehminated altogether by the use of
graphical or tabular methods, or by the use of various techniques of
approximation. These possibilities for simphfying the calculations
have now begun to be realized to quite a large extent.
The latest works of the author (1955-1957) have been devoted
to two problems on the limiting equilibrium of a medium which
possesses weight, in which discontinuities occur in the stress state.
One deals with the determination of the shape of curved slopes, and
the other is an investigation of the pressure on curvilinear retaining
walls.
The future development of the theory of limiting equihbrium must
certainly be based on experiments which give not only a general picture
of the forms in which loss of equilibrium occurs, but which also give
XIV INTRODUCTION
definite and reliable quantitative results. The main aim of such experi­
ments, which are, of course, extremely important, is to check the
theoretical results and to determine the limits of their apphcability.
This third edition of Statics of Granular Media is devoted to the
theory of limiting plane equihbrium, and contains a general method
for solving the various problems. It does not, however, aim to cover
the whole field of investigations since much of this work has been
pubhshed elsewhere.