statics of granular media

STATICS OF GRANULAR MEDIA

BY

V. V. SOKOLOVSKII

Completely revised and enlarged edition

TRANSLATED BY

J. K . L U S H E R

E N G L I S H TRANSLATION EDITED BY

A . W . T . D A N I E L Senior Lecturer in Civil Engineering,

Queen Mary College, University of London

P E R G A M O N P R E S S

O X F O R D . L O N D O N E D I N B U R G H N E W Y O R K P A R I S F R A N K F U R T

1965

P E R G A M O N P R E S S LTD. Headington Hill Hall, Oxford

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Copyright 1965 P E R G A M O N P R E S S L T D .

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 E N G L I S H E D I T I O N

T H I S book differs greatly from the two previous editions of Statika Sypuchei Sredy, in Russian, and consequently from the Enghsh translation 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 problems 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 equihbrium 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 . S O K O L O V S K I I

PREFACE

T H E 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 already 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 equilibrium. The equations of limiting plane equilibrium and their transformation 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 boundary-value problems are formulated for the canonical system and effective methods of numerical integration are suggested.

An important question in the statics of granular mediathe h m iting equilibrium of foundationsis 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 significance^the stability of foundations and slopes. Here again the derivation 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 canonical system. It is shown that for certain particular forms of the limiting conditions the equations of limiting equilibrium have simple integrals. Problems deahng with the compression of strips and rectangles 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 simultaneously zones of limiting and non-limiting equilibrium, together with problems on the equihbrium of embankments, the stabihty of foundations 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 Computer Centre of the Academy of Sciences of the U.S.S.R.

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 observations 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 . S O K O L O V S K H

I N T R O D U C T I O N

I N T H E 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 typeso-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 retaining 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 retaining 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, prismatic 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 equihbrium which makes possible the solution of various problems and the determination of the corresponding slip-hne network. It originates from the works of Ktter (1903), who considered the differential equations of equilibrium and the condition of hmiting equihbrium at each point, formed a set of equations of hmiting euqilibrium and then transformed 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 Novotortsev (1938) to certain particular problems on the stability of foundations, 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 investigations 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 bo th 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 solution 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 resistance 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 experiments, 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.

C H A P T E R 1

L I M I T I N G E Q U I L I B R I U M OF A G R A N U L A R M E D I U M

1. LIMITING CONDITIONS

Let us take some point in a granular medium and consider an element of area passing through this point. On this area there is applied an actual stress forming an angle with the normal and having normal and tangential components cr and (Fig. 1).

F I G . 1

Experiment shows that the resistance to shear over this area in a granular medium with some cohesion can be expressed by the linear relation , , ^

| | = tan -f fc, which applies when the equilibrium is about to be destroyed. This resistance is thus made up of a resistance from internal friction and a resistance from cohesion.

The constants and k are the angle of internal friction and the coefficient of cohesion, but they can be looked upon simply as parameters which characterize the total resistance of the granular medium to shear.

1

2 STATICS OF GRANULAR MEDIA

It is usual to call a granular medium in which cohesion is absent (k = 0) an ideally granular medium, and one in which internal friction is absent ( = 0) an ideally cohesive medium. These media possess certain characteristics which we shall investigate later in considerable detail.

First of all, however, we shall attempt to derive the basic conditions under which the equilibrium of a granular medium is possible at some internal point.

It will be seen that no slip will occur over the element of area under consideration if

| | ^ tan + k, where ^ ^ : co t . The coefficient

= k cot

is the ultimate resistance to uniform three-dimensional tension. This coefficient will be widely used in our future work.

We shall assume also that there is an equivalent stress p' acting on the element of area at an angle ' to the normal and that it has the components + and . The equivalent stress is the resultant of the actual stress and the normal compressive stress H.

The inequaUty which ensures that no sUp takes place now becomes

| T J ^ ( + / O t a n , where ^ -H. In granular media, therefore, in which His small, only small normal

tensile stresses are possible, and in ideally granular media when = 0 only normal compressive stresses are possible. This is the essential difference between granular and continuous media in which is large and in which, consequently, high normal stresses of both signs are possible.

Clearly, the equilibrium of a granular medium at some point will be ensured if the above inequality is valid on any element of area passing through this point.

It is of particular interest to consider the case when the inequality

| J ^ ( + / ) t a n , (1.01) holds on all elements of area and when the equality

| | = ( , + i y ) t a n (1.02) holds on certain elements of area only.

LIMITING EQUILIBRIUM OF A GRANULAR MEDIUM 3

F I G . 2

the normal component of the stress acting on some element of area is measured along the axis of abscissae, and the absolute magnitude of the tangential component \r\ is measured along the axis of ordi-nates.

We shall make use of the well-known transformation formulae

(^n = /2 + 2 + "", + = + o^m^ + of n^ in which / = cosA, m = $, = cosr GM 2

It is usual to call this a state of limiting equiUbrium and the elements of area on which (1.02) applies, slip planes.

The above relations (1.01) and (1.02) can be expressed in a different form by means of the single condition

m a x { | T j - ( + i ) t a n } = 0 . Normally there exist whole zones, at all points of which there

exists a state of limiting equilibrium. They are called zones of limiting equilibrium or limiting zones.

We note that for an ideally cohesive medium in which = 0, instead of (1.01), there exists on all areas the inequality

and on the slip-planes, instead of (1.02), we have the equality

These expressions can also be represented by the single condition

max \\ = k.

An immediate representation of the state of stress at a point in the medium is given by the Mhr stress diagram (Fig. 2). In this diagram


2 = -

4/2 3 1

1

t\].

+ T J - i f ] .

{{On + - tl].

Since the left-hand sides of these expressions are positive and if ^ 2 ^ ^3 or ^ 0, tg ^ 0, ^ 0, the components and on any elements of area passing through the point must satisfy the inequalities

( - ^ i ) ^ + ^ tl ( - ^2)' + tl { - s,y + ^ tl

It is clear that all points with coordinates and , representing the stresses on an element of area, lie within a curvilinear triangle. This

are the direction cosines of the angles A, , between the normal to the element of area and the principal axes 1, 2, 3.

F rom these transformation formulae and the expression /2 + w2 + 2 = 1

we can easily expresses the direction cosines /, m, in terms of the principal normal stresses (, a^, and the components , as follows:

2 ^ fa - ^2) fa - ^3) + ( - ( - ^)

^^2 ^ fa - ^3) JCfn - (Jl) + (^ 2 - 3) (2 - )

^ fa - ) fa - ^2) + (3 - ) (3 - ^)

For convenience we introduce the notat ions:

^1 = 1(0*2 + ^3), 2^ = i(^3 + ) , = ^( + 2), and also

= i(^2 - 0*3), 2 = i(^3 - GTi), 3 = 1( ~ 2) and transform the expressions for the direction cosines to the more convenient form

p = - '


and the elements of area pass through the principal axes 3 or 1 and are inclined to the principal axes 1 or 2 at angles or v. The transformation formulae for these areas are

= 3^ - h cos2 / / , T = + 3 cr = ^1 c o s 2 r , = i s i n2 r .

The hmiting equilibrium of a granular medium at the point can be indicated on the Mhr stress diagram. Indeed, inequality (1.01) shows that if the medium is in equilibrium the semi-circles of stress must not intersect the so-called limiting line

| J = ( + i]Otan

triangle has apices on the axis at the points Pi, 2, with abscissae ffi, ff2, and is bounded by three semi-circles:

{a - s^y + T2 = tl ( - + rl = tl ( - s,y + T5 = tj.

which have centres at the points i> 02> a-On the application of an additional three-dimensional pressure the

radii of the semi-circles remain constant and the whole construction moves in the direction of the axis.

Along one of the semi-circles

{ - s,y + = t\ it is obvious that

cos// = 0 , V + = ,

and the elements of area pass through the principal axis 2 and are inclined to the principal axis 3 at an angle . The transformation formulae for these areas have the much simpler form

= ^2- t2 cos2A, = t2 sin2A. (1.03) By analogy, along the other two semi-circles

( ( T - ^ 3 ) ' + T^ = rf or {a,-s,Y + Tl = tl

it is clear that

cosv = 0 , + = ^ or cosA = 0 , + =

2*


drawn in the plane of the variables and . On the other hand, equality (1.02), which holds on the slip-planes, shows that when a state of limiting equilibrium exists, a point on the stress diagram corresponding to these planes must at the same time lie on the limiting line and in the above-mentioned curvilinear triangle. This is possible only when the large stress semi-circle touches the limiting line at some point R,

For an ideally granular medium, when fc = = 0, the limiting line passes through the origin of coordinates, and for an ideally cohesive medium, when = 0, it is parallel to the axis of abscissae.

From (1.03) we obtain

4^ = c o t 2 | = t an , da^

and introducing the notation

2=- we have that

cot2 |A| = tan or | | = . These relations establish the position of the slip-planes passing

through the principal axis 2. There are two such slip-planes and they are inclined to the principal axis 1 at angles and intersect at an angle 2.

It follows that through every point in the zones of limiting equilibrium there pass two surfaces, the tangential planes to which coincide with the slip-planes. These surfaces form a system of two isogonal families and are usually called slip-surfaces.

The normal and tangential components of stress and on the slip-planes can be expressed in the form

^n = S2 - t2 sin, | | = t2 cos. (1.04) It is not difficult now to derive the limiting condition expressed in

terms of ^2 and ig- Substituting (1.04) into the equaUty (1.02), we immediately find that

= U2I = (^ 2 + )$. (1.05) With different relative magnitudes of the principal normal stresses

(^2^^^ ^ 0*1 or ^3 ^ (Tl ^ Tg the slip-planes pass through the principal axes 3 or 1, and instead of (1.05) we have that

U 3 I = (-^ 3 + ^ ) or \ti\ = (sj, + )$,


F I G . 3

If we discard the limitation that ^ 2 ^ 3, we see that the limiting condition can be represented in the form of a six-sided pyramid formed by three pairs of the following planes

U l i = ( 5 i + ), U^l = (^2 + H)smQ, \h\ = (ss + / ) s i n ,

with axis inclined equally to the principal axes 1, 2, 3 and with apex at the point

cTj = 2 = 3 = / . The intersection of this pyramid with the plane

+ 2 + 3 = 0

Each of these conditions taken separately depends on the relative magnitudes of the principal normal stresses. They should therefore be combined to form the one limiting condition

[ U l i - (s, + H)smQ] [\t2\- + )s in] [\^\- fe + )8] = O, which is of a symmetrical form. (1.06)

We can give a geometrical interpretation of the limiting condition in the form of the so-called limiting surface, constructed in the three-dimensional space of Oi o^.

We first draw the axis o-^= = equally inclined to the principal axes 1, 2, 3 and the normal plane

+ ^ 2 + 3 = 0

passing through the origin of coordinates. We shall denote the projections of the principal axes 1, 2, 3 on this plane by I, II , I II (Fig. 3).

8 808 OF GRANULAR MEDIA

and radius of inscribed circle r = yik. Experiment shows [52] that the resistance to shear over a given plane

in a medium with high cohesion can be expressed by a non-linear relation

\rn\=F(a)

which is valid when limiting equilibrium is reached. It is of particular interest to consider the equilibrium of a cohesive

medium at a point when the inequality

| r j ^ ^ ( ) , (1.07)

is satisfied on all planes, and when the equality

| T j = () (1.08) appUes on certain planes only.

This equilibrium, as before, is called limiting equilibrium, and the planes on which (1.08) is satisfied are called slip-planes.

Expressions (1.07) and (1.08) can now be represented by the single condition

m a x { | T | - i ^ ( a ) } = 0 .

forms a hexagon of side

= 6 VI sino . ^

and radius of inscribed circle _ ]/6HsmQ

1/(3 + sin2) We note that for an ideally cohesive medium the limiting condition

can be represented in the form of a right six-sided prism formed by three pairs of planes:

\h\ = k. \t2\=k. \t,\=k

with axis inclined equally to the principal axes 1, 2, 3 . The intersection of this prism with the plane

(Tl + ^2 + 3 = 0

forms a right hexagon of side

k

LIMITING EQUILIBRIUM OF A GRANULAR MEDIUM

Limiting equilibrium of a cohesive medium can be conveniently represented on the Mhr stress diagram. The inequality (1.07) shows that for normal equilibrium the stress semi-circles must not intersect the limiting curve , , .

\T\=F(a) drawn in the plane of the variables a and T. In addition equality (1.08), which holds on the sUp-planes, indicates that when limiting equilibrium is reached the largest stress semi-circle touches the limiting curve at some point R (Fig. 4).

FIG, 4

The discussion which follows can be conducted in an analogous way to the above discussion provided we introduce an auxiliary quantity , which on the stress diagram forms an angle between the tangent to the limiting curve at the point R and the axis of abscissae. Bearing in mind (1.03), we find as before that

d\r\ = cot2|A| = t an .

These expressions define immediately the position of two slip-planes. Consequently, in the zone of limiting equilibrium there exists a system of two famiUes of slip-surfaces.

The normal and tangential stress components and on the slip-planes can be expressed with the aid of (1.04).

It is easy also to derive the limiting condition expressed in terms of ^2 and / g . Substituting (1.04) in (1.08), and taking into account that F'(a) = tan , we find that

2 = U 2 l = / ( ^ 2 ) . (1.09) With different relative magnitudes of the principal normal stresses

^2(y^ 0*1 or 3 ^ ^ we have, instead of (1.09), that | 3 | = / ( ^ 3 ) or | | = / ( ^ ) .


These conditions also can easily be combined to form the one limiting condition

[UiI - / f e ) ] [\t2\-/fe)] [UsI - / f e ) ] = 0 , (1.10) which is of a symmetrical form.

We shall consider now a granular medium in which the angle of internal friction in horizontal planes is less than the angle of internal friction in other planesf

<

and we shall call this material a lamellar granular medium. It will be seen that no slip will take place along any element of

area with normal passing through this point if

|T| ^ ( + ) t a n ^ {ay + ^ t a n ,

depending on whether the area is inclined or horizontal. The equilibrium of a lamellar granular medium at some point

depends, evidently, on whether the first of the above inequalities is satisfied on any area passing through this point. It is of particular interest to investigate the state of equilibrium when the inequality

| r j ^ ( + ^ ) t a n , (1.11) holds on all inclined planes, and when the equality

| , | = (, + )1 (1.12) holds on horizontal planes.

This state will be called special limiting equilibrium, and horizontal areas on which (1.12) is satisfied, as before, will be called slip-planes.

Usually there exist whole zones in which special limiting equilibrium occurs at every point. These are called zones of special limiting equilibrium or special limiting zones.

In a zone of special limiting equilibrium a slip-plane parallel to the zx plane passes through every point.

It should be remembered that in a rectilinear and rectangular system of coordinates x, the state of stress at a point can be defined by

t Here it is supposed that the ultimate resistance on horizontal planes is H, The more general assumption, that the ultimate resistance differs from has already been examined in the first [61] and second [68] editions of Siatika Sypuchei Sredy.


2. LIMITING PLANE EQUILIBRIUM OF A GRANULAR M E D I U M

We shall define plane equilibrium as the equilibrium of an infinitely long cylindrical or prismatic body under the action of forces perpendicular to the generators and distributed uniformly in the direction of these generators.

In the study of plane equilibrium it is normal to use a rectilinear system of coordinates x, y, with the axis parallel to the generators.

The stress components Ty^ = ^ = 0 and the remaining components ax,ay, a and r^ y^ are independent of the coordinate z. By convention we take a compressive stress to be positive and a tensile stress negative. This is indicated in Fig. 5, in which the stress components are shown in their positive directions.

In the case of plane equilibrium, instead of a three-dimensional stress system, it is sufficient to consider the stress distribution in the ATj-plane, and instead of stresses on some element of area, we have only to consider stresses on some linear element.

three normal and three tangential components of stress a^^Oy, and "^yzi "^2X9 "^xy

The principal normal stresses , ag , are the roots of the cubic equation

Txy ay - at Ty^ = 0 ,

which are always real. It is sometimes convenient to make use also of curvilinear and

orthogonal coordinates. Referred to cyhndrical coordinates r, , z, the state of stress at a

point can be described by the three normal components ^, ^, ^ and the three tangential components of stress Tq^, r^r, ,.^. A corresponding cubic equation for the determination of the principal normal stresses , 2, a^ follows from the equation given above after replacement of subscripts y, z, by r, , z.

The same applies for other systems of curviUnear and orthogonal coordinates.


We shall first derive the principal normal stresses from the above cubic equation, which can now be written in the simpHfied form

(o'z - (yd = 0 .

' o r

Oy -

O!

-.

F I G . 5 F I G . 6

Solving this equation, we find the two values

^min

the third, ^, lying between them:

Thus 3 and become ^^^ and ^^, and the quantities

S = S2= i ( + CTmin). t = = \ (^ " )

can be expressed in terms of the stress components as follows:

1 / -f. ^ i^x - (^yY + -^ly

It is often convenient in a study of plane equilibrium to make use of a cylindrical system of coordinates r, , z, the z-axis of which is parallel to the generators. In this case ^^ = ^^ = 0, whilst the remaining components .^, ^, and ^^ are independent of the coordinate z. The sign convention for these components will be clear from Fig. 6, in which the stress components are shown in their positive directions.

By analogy with the foregoing we can easily show that the quantities s and t can be expressed in terms of the stress components as follows:

^ = y (- + )>


Let us consider a point in a granular medium and some element of area passing through this point. On this element there will be applied an actual stress forming an angle with the normal and having normal and tangential components and . The element may also be considered as being subjected to an equivalent stress p' forming an angle ' with the normal and having components -\- and (Fig. 7).

' /

F I G . 7 F I G . 8

In order to determine the components and we apply formulae (1.03) rewritten in the following way

= J - / c o s 2 A , T = /sin2A (1.13) remembering that the angles and/I are related by the expression

By analogy, in order to find the components = S and || = on the slip-lines, we make use of formulae (1.04), namely

.S' = ^ - s i n , = c o s . (1.14) The state of plane stress at some point can be conveniently re

presented by a Mohr ' s circle diagram (Fig. 8). The normal component of the stress applied to an element is measured along the axis of abscissae, and the tangential component is measured along the axis of ordinates. It will be seen that every point with coordinates and r lies on a stress circle

(a - sy + = ^ which has its centre at the point Q.


Limiting plane equilibrium of a granular medium at a point can also be represented on the Mhr stress diagram. The stress circle in this case touches the limiting lines

\'^n\ ( + H)tan or = {S + H)tang drawn in the plane of the variables a and r in two symmetrical points R,

Instead of slip-planes we now consider slip-segments. There are two such segments, inclined to the direction of ^^ at angles and intersecting at an angle 2,

It follows that at every point in the zone of limiting equilibrium we can draws two lines, the tangents to which coincide with the slip-segments. These lines form a system of two isogonal families and are called slip-Hnes.

The limiting condition (1.05) establishes a linear relation between s and t of the form

t = (s + H) sing or an even simpler linear relation between a = s + and t, namely

t = sing. F rom (1.13) the normal and tangential components of stress a

and T on any linear element are given by: = (1 sing cos2A) H, = sing sin2A, (1.15)

and from (1.14) we obtain for the components S and on the slip-segments the expressions

S = acos^Q H, r = a s i n g c o s g . (1.16) We can in addition derive expressions relating and to the equi

valent stress p' and its angle of inclination \ For simplicity we shall discard the accents, i.e. we shall use the same

notation for the equivalent stresses as for the actual stresses, so that

+ = cosa, = sind or = ( + ) t ana . The angle can be expressed without difficulty in terms of . Indeed,

the equations

sing sin2A = (1 - sing cos2A) ta or sin(2A + ) = sing

after simple rearrangement give

= (1 - ) - ^ + ^ ( - ) + / , =1,


- 2 2 ^ ^ ""^s in(z l + ) ' and the other corresponds to = + 1 , when

^ 1 . ^ 1 / ^ sx sinZl ^+ = ^ - T ( ^ - ^ ) ' A+ = - ( z 1 - ) , + = / 7 - ^

These stress states will be called minimal and maximal, since

0+ ^ sin(zl + ) ^ J _ sm{A )

where m is a whole number. We have introduced here the notation suggested by Caquot [7], namely:

. . sin y ^ ^ sing 2

Also, it is clear that the equivalent stress can be expressed in terms of the mean normal equivalent stress a and the angle as follows:

sin(zl - ) , . ,/r 2 2 i\ = a = (T(cos - y[cos^ - cos^g]).

As is to be expected, is independent of the sign of the angle . Thus and are now given by the expressions

A = ( l - . ) ^ + i . ( . z l - a ) + m . , . = ; , - j - | H _ . ( U 7 ) and in the particular case when = 0

A = (l + / , = -; . (1.18) ^ 1 - smg ^ ^

A value must be assigned to w ; it is usual to take m = 0 or w = 1 . The foregoing computations can be replaced by geometrical con

structions on the stress diagram. The actual stress and the angle are represented by the line OP and its angle of inclination to the axis of abscissae. Similarly, the equivalent stress p' and the angle ' are represented by the line O'P and its angle of inclination to the same axis of abscissae. The angles , and A are represented by certain angles on this diagram. All the constructions are obvious and do not require further explanation.

For given values of and there exist two different stress states: one of them corresponds to = - 1, so that

-=-^{+), A_ = - - ( Z l + ) , .=-^


The quantities occurring in the above formulae can be represented on a Mohr ' s diagram. The stress circles passing through the given point have centres at the points Q_ and + with abscissae s. and and radii /_ and (Fig. 9).

y /

F I G . 9

It will be noted that for an ideally cohesive medium the limiting condition shows that

t = k .

All the above relations can be considerably simplified if we consider the case when = 0 and put a = s + H.

The normal and tangential stress components and on any elementary segment then become

On = S k COS2A, Tn = k

and for the components S and Ton the slip-segments we have

S = s, T = k .

Instead of the tangential stress component , it is sometimes convenient to make use of the angle , taking into account that

T = A:sinZl, |zl | ^ y .

It will be seen that the angle can again be expressed in terms of the angle . Indeed, the equation

sin 2 A = sinzl


gives immediately

= ( 1 - ) - ^ + + = 1 ,

where m is any whole number. In addition, it is clear that the normal stress component = rt

must be expressable in terms of the mean normal stress s and the ang leJ . Indeed,

= s xk cosZl

and, of course, is independent of the sign of the angle zl. Finally, therefore, and s assume the form

= {1 ) ^ + ^ + , s = + xk cosA (1.19)

and in the particular case when A = 0

= ( 1 - ) ^ + ^ , s = n + k. (1.20)

A value must be assigned to the integer w ; it is usual to take m = 0 or /w = 1.

For given values of and A there exist two different states of stress; one corresponds to = 1, so that

A . A , A v _ = , ^ - = Y " - y ' s^=n-kcosA,

and the other corresponds to = + 1 , when

A ^ A , ^ + = - , A + = y , s+=n+kcosA.

These stress states will be called minimal and maximal, since

s+ = 2k cosA ^ 0 .

The limiting plane equilibrium of a cohesive medium at some point can also be represented on the M h r stress diagram (Fig. 10). In this case the stress circle touches the limiting curves

| r J = P ( a ) or T=F(S), drawn in the plane of the variables a and in two symmetrical points R.


Clearly, through every point in the zone of limiting equilibrium we can draw two lines which make up a system of two families and are called slip-lines.

F I G . 10

The limiting condition (1.09) establishes a definite non-linear relation between s and t in the form

t=f(s). Also, we must take into account here that

l dS

= F ( 5 ) = t a n e , ^ = / ' ( ^ ) = sing.

and that the angle of internal friction and the coefficient are variable and related by the differential equation

dH = (7 co to . d

Thus, as before, expressions (1.15) or (1.16) are valid and they determine the stress components a and on any elementary segments or on slip-segments. Similarly, expressions (1.17) or (1.18), which give in terms of and , remain unchanged.

The foregoing computations can be replaced by geometrical constructions on the stress diagram. They do not, however, bring out facts which have not aheady been considered and we shall not discuss them further here.

We shall introduce now a rectilinear system of coordinates x, y and denote the angle between the direction of ^ and the x-axis by


, and the angles of inclination of the slip-Unes relative to the x-axis by 9? (Fig. 11).

Making use of the expressions for and , we shall represent the stress components a , a y and r^y in terms of two variables and as follows:

= (1 sing cos29?) - H, x^y = a sing sin29!?. (1.21)

We note that if the value of a is sufficiently large, the coefficient ceases to have any real influence on the stress components. Therefore, as a increases, the limiting equiUbrium tends to the corresponding limiting equilibrium of an ideally granular medium.

F I G . 11 F I G . 12

We turn now to a system of polar coordinates r, , which it is sometimes more convenient to use, and we shall denote the angle between the direction of a^ax and the straight line OP by xp and the inclinations of the slip-lines relative to the same line OP by y> + (Fig. 12).

Making use of the expressions for and we can represent the three stress components ,., Cq and x^q in terms of a and xp in the following way

^0 = (1 sing cos2^) - H, x^o = a sing sin2v^. (1.22)

It will be seen that for an ideally cohesive medium the above expressions can be considerably simplified if we consider the case when g = 0 and put = -h iy. Instead of (1.21) we then find that

s k ^, Xxy = k s in299 . (1.23) GM 3


I = .y : c o s 2 ^ , ^ = k s i n 2 ^ . (1-24)

and instead of (1.22) we have

Let us now consider briefly a lamellar granular medium, which was mentioned above.

From (1.12) the special limiting plane equilibrium of such a medium is given by

T^xyl = (o'y + / f ) t a n . Horizontal elementary segments on which this equation is satis

fied, are called as before, slip-segments. In the zone of special limiting equilibrium a straight slip-line, paral

lel to the X-axis, passes through every point in the :v j -p l ane .

3. EQUATIONS OF LIMITING PLANE EQUILIBRIUM

We shall consider now the equations which define the limiting plane equilibrium of a granular medium, using an ordinary system of rectangular coordinates y and considering, for generality, that the X-axis is inclined to the horizontal at an angle oc.

The basic equations are the differential equations of plane equilibrium

y ^ y ^

which contain the density of the medium . In addition, we have the limiting condition

t = {s - H) sing which can be expressed as,

\ {Ox - ^ + rly = ( + , + 2)\ (1.26)

The set of three equations (1.25), (1.26) contain the three unknown stress components , cr^ and r^^y We can say, therefore, that the problem of finding these stress components when the boundary conditions are statically defined is statically determinate.

We shall now consider the question of mechanical similarity, which is of considerable importance for our future discussions. It enables


' ' '

We transform the set of equations (1.25) and (1.26) by transferring to these non-dimensional variables, and then discard the bars over the letters. We then have

dx and also

sin , ^

- - ^ = c o s a , dy

1'- sin^p 4 (^ + ,. It can be seen that in problems characterized by different values of

, k and y, the latter equations coincide identically if the non-dimensional numbers and / in these problems are the same.

Thus the law of mechanical similarity can be stated as follows: in geometrically similar regions, if the values of , kip and are identical, the stresses at corresponding points are similar if they are similar on the boundaries.

Let us consider, for example, a region of limiting equilibrium and a geometrically similar model, the characteristic length / of which is reduced times. Clearly, at corresponding points in the region and in the model the stress components ^, Oy and r^y will coincide if the density of the model is increased iV times.

A number of ideas in connexion with model analysis of granular media at limiting equilibrium have been suggested by Pokrovskii [40], who applied them in his well-known method of centrifugal model analysis.

The law of mechanical similarity given above can be formulated in a slightly different way: in geometrically similar regions, if the values of and lip are identical, the equivalent stresses at corresponding points are similar if they are similar at the boundaries.

At each point in the model the components of equivalent stress -{ H,Cy + and Txy will be times less than those at the corresponding point in the region, if the equivalent stresses on the boundaries 3*

US to find the conditions for which the limiting stress states of geometrically similar regions are mechanically similar, and for which the slip-lines are geometrically similar.

In an examination of any specific problem it is convenient to introduce a characteristic length / and an equivalent stress and also the non-dimensional variables

_ _ -\- ^ _ Oy + _ _ ^ " ' y ~ ' ~ f Oy - , T;Cy


of the model are also times less. An increase in density by times, with the geometrical dimensions of the region constant, leads to an increase in the components of equivalent stress + H,ay + ^ a n d T^ xy by times, if the equivalent stresses on the boundary are also increased times.

We shall go on now to investigate the set of basic equations of limiting plane equilibrium of a granular medium. If we substitute expressions (1.21), which satisfy identically condition (1.26), into the differential equations of equilibrium (1.25), we arrive at the so-called basic set of equations

^ 3 . , ^ da (1 + 8 ^) + sing ^ -

la s i n g ^ s i n 2 9 5 - ^ - cos29?- |y - j = s i na . da da

sing -^ + (1 - sing cos29!?) +

+ 2a sing^cos299 = cosx.

(1.27)

dx ' "^"^^ dy I

We now transform these equations, making use of the angle

2= -

between the slip-lines; this leads to symmetry of the equations [63]. Multiplying now the first equation by sin (9? ), the second by

- cos ( ) and adding, we have that sinjoc + g) da _ ^ ^ d

- + 2 a t a n g - dx cosg

_ . ^ d cos(a + g) + 2 a t a n g ^ - 7 ^

dy ^ '^"'^ dy ' cosg

Introducing the auxiliary quantities

cos(^ + ) +

s in(^ + ) = 0 . (1.28)

P =

Q =

da _ ^ ^ dw + 2a tang ^

dx da

dx - 7

_ ^ dw + 2 t a n g ^ - y

sin(x + g) cosg

cos (a + g) dy ^ ^ dy

we can rewrite equations (1.28) as follows: + t a n ( 9 ) + ) = 0 .

cosg

EQUILffiRIUM OF A GRANULAR MEDIUM 23

=

sin(99 + e)dX'- $( + e)dy' Rdxcos( + )

sin(99 + s)dx cos(9? 4 - ) rfj; *

If the denominators in the right-hand sides of these equations are non-zero, the values of the derivatives wiU be determined uniquely; if the denominators vanish simultaneously with the numerators, the values of the derivatives are not unique, and the Une y = y(x) is caUed a characteristic; if the denominators vanish, then since the numerators are non-zero, the values of the derivatives are infinite, and the line y = y(x) is caUed a Une of discontinuity.

If we equate to zero simultaneously the numerators and denominators of the right-hand sides of the above equations, we can establish the differential equations of the characteristics. They consist of the set of equations and dy^dxtan( + ) (1.29)

da + 2a tan d = ^ [sin(c + ) rfx + COS( + ) dy]. (1 .30) | cos

t Alternatively, we may say, if dy = dx tan (9? ) as in (1.29) then R = + tan (97 ) = 0, whence (1.30) holds good on the characteristics.

Let US consider the following question: is it possible to determine the values of the first derivatives of the unknown functions a and with respect to the coordinates and y along some line y = y{x) in the xj'-plane? In order to answer this question we make use of the two equations

, da , da , , d , d ,

which are vaUd along the Une in question. Taking the auxiUary quantity

Rdx = da + 2a tSind [sin((x + ) J x + cos(a + ) dy], cos

we transform the previous equations in the foUowing way:

F rom the two equations in and Q it can easily be found that R dx sm( + )


The family of characteristics given by the upper signs will be called the first family, and that given by the lower signs, the second.

Thus the basic set of equations has two real different families of characteristics; these equations are, consequently, of the hyperbolic type.

Clearly, the characteristics are inclined to the x-axis at angles + , i.e. at the same angles as the slip-lines. It follows immediately that the characteristics in the x^-plane are sUp-lines.

In the region under consideration in the >;-plane two characteristics intersecting at an angle 2 pass through every point, and consequently the whole of this region is covered by a network of characteristics.

If we equate to zero solely the denominators in the right-hand sides of the above equations, we find similarly the differential equations of the lines of discontinuity

dy = rfx tan(9P + ).

Thus the lines of discontinuity are incUned at angles 99 =F e to the X-axis. This shows that a line of discontinuity can be a slip-line or a slip-line envelope.

We would point out that the method used here for deriving the equations of the characteristics and the canonical set of equations was suggested for hydraulics problems by Khristianovich [26].

On the lines of discontinuity the derivatives of a and or of the stress components , Oy and x^y with respect to the coordinates x, y become infinite, and a and or the stress components a , a y and Xxy are subjected to finite discontinuities. The initial equations, therefore, describe the limiting equilibrium only as far as these lines.

We shall assume that the first and second famiUes of characteristics are determined respectively by the parameters and \. We shall take a network of characteristics as a system of curvilinear coordinates in the x>;-plane and we shall consider x, y, a, to be functions of and .

The equations of the characteristics (1.29) and (1.30) can then be re-written in the form of a convenient canonical system, comprising two equations

t and are new parameters.


and two other equations

da ^ .

cosg

cosg

sm(a - e ) - ^ + cos(


(1.37) + ^0 or = C e x p ( - 2 9 9 tang), X sin(9? - ) y 5{ - ) = /()

contain arbitrary constants or C and an arbitrary function /().

The equations of the characteristics (1.29) remain the same dy = dxtanif + ) , (1.33)

but equations (1.30) assume the very much simplified form da + 2 tang d = 0 .

These equations, after the introduction of a new function

c o t o , a

can be integrated to give

9^ = const. (1.34)

The canonical equations (1.31) and (1.32) now give

f = f t a n ( ^ - . ) , | ^ = | | t a n ( , 4 - e ) . (1.35) since for the parameters and it can be shown that

^-^ V = X- ' (1.36) Such a choice of independent variables is, of course, admissible

only if and are variable. There are, however, particular solutions corresponding to constant and , the so-called integrals of the equations of limiting equiUbrium. These solutions are frequently encountered in practice and must be considered separately.

1. If I is constant and variable, we have from (1.36) that is a, function only of the one variable . Consequently, along the first family of characteristics

dy = dxt^in( ) , = const. Therefore,

X sin(99 ) y cos(99 ) = const., = const. ,

and the required integral is

X sin(99 - ) - y cos{ - ) = /(). Thus the integrals of the equations of Umiting equiUbrium

LIMmNG EQUILIBRIUM OF A GRANULAR MEDIUM 27

X = %'^{ ) ) cos ( ) y = sm^( - )

) sm{ - )

In practice we often encounter the degenerate case, when the straight characteristics pass through one point O and thus form a pencil of rays.

It is convenient to introduce a system of polar coordinates r, with pole at the point O, bearing in mind that

: = r coso, y = r sino.

From the conditions = = 0, it is apparent that the arbitrary function /() = 0 and the integrals (1.37) can be further simplified to

a = C e x p ( - 2 t ang) , ( - ) = or = + (1.38)

and depend only on one arbitrary constant C. The curved characteristics are now extremely simplethey comprise

the logarithmic spirals

r exp( tang) = const., which, with increase in the angle , move further away from the point O.

2. If is constant and is variable, (1.36) shows that is a function of the single variable . As before, the integrals of the equations of limiting equilibrium

(1.39) X - = Vo or a = D &{2 t ang) , ] X sin( + ) - y cos( + ) = g{^)

have arbitrary constants or D and an arbitrary function g{^).

The first family of characterisitics in the xj;-plane consists of the straight lines = const., whilst the second family can be found by integrating the equation

dy = dx tan(99 + ).

This solution is vaUd only up to the envelope of the straight characteristics. In order to find this envelope we differentiate integral (1.37) with respect to :

X cos(99 ) + y { ) = {)

and add the same integral (1.37). Finally, the required envelope is given by the equations


The first family of characteristics in the x>^-plane can be found by integrating the equation

dy = dx tan(9!? ) , whilst the second consists of the straight lines = const.

The envelope of the straight characteristics is now given by the following equations:

X = cos^(9? + ) g() cos (99 + ) y = sin^(9? + ) sin (9? + )

Here again we often meet the degenerate case when the straight characteristics pass through one point O and form a pencil of rays.

As before, we make use of polar coordinates r, with pole at the point O. F rom the condition that = y = 0, it will be seen that the arbitrary function g(^) = 0, and that the integrals (1.39) can be written in the simplified form

a = D exp(299 tang) . tan(9!? + ) = X

or = - (1.40)

and depend only on the one arbitrary constant D, The curved characteristics are logarithmic spirals

r e x p ( 0 t a n g ) = const. which, as increases, approach the point O.

3. Finally, if and are constant, then a and are also constant. The first and second families of characteristics form two isogonal systems of parallel straight lines in the xj^-plane.

At a later stage we shall frequently be required to solve ordinary boundary-value problems for equations (1.35), and to determine the arbitrary functions which appear in the integrals (1.37) and (1.39).

FTG. 13 F I G . 14


'""^^

It should be borne in mind that the characteristics in the can be represented by straight lines parallel to the coordinate axes, and that the equations = const, and a = const, can be represented by straight lines parallel to the bisector of the coordinate angles (Fig. 13).

We will show now how to transform the equations of the characteristics and the canonical equations into new variables u and v. We first replace the coordinates and y by

X = ^ [x sin(a? + ) V cos(w + )], cosg vr / vr /J

y = [y cos(9? - ) - X s in(^ - ) ] , and put

u _ V la sin2g " " 7 \ 2k '

It will be seen that and y are the coordinates of some point relative to axes passing through the origin O and parallel to the directions of the slip-Unes at (Fig. 14).

As a result the equations of the characteristics (1.33) assume the simpler form

v + ^ = 0, + (1.41) cosg cosg

and the canonical equations (1.35) become

- & + ^ = 0^ 4 ^ + ^ = 0 . (1.42) 2 cosg 2 cosg ^ ^

We shaU consider the usual case when the self-weight of the medium is directed along the y-soiis, i.e. when oc = 0.

The equations of the characteristics (1.29) remain unaltered dy = dxtsin( + ) , (1.43)

but equations (1.30) are considerably simplified and become da + 2a tang d = yidy + tang dx), (1.44)

Similarly, the canonical equations (1.31) become

'y=^t.ni


^ . So? I dy ^ dx\ ^ + 2 a t a n g - ^ = y ( 4 - t a n g ^ ) .

(1.46)

Subsequently we shall constantly be required to solve the ordinary boundary-value problems for equations (1.45) and (1.46).

We should, however, remember that the characteristics in the plane are now no longer straight lines parallel to the coordinate axes, as is the case when body forces are absent, but are represented by curves.

F rom equations (1.44) we can easily derive the well-known equations of Ktter [28] containing the radii of curvature and of the characteristics (slip-hnes) of the first and second famiUes. The first equation

^ J /J ^ j \ ydx cos (99 + ) do - la tanp dw = y{dy - tanp dx) = - --- ^

^ ^ ^ ^ cosg cos(99 - ) can be written a

s

do ^ ^ T> cos(99 + ) -3 2 a t a n p = - R^ ^ (1.47)

and the second equation becomes

4 ^ + 2 a t a n g = y ; ? , i 2 f c ^ . (1.48) d ^ f ^ Q g ^

We see that if the shape of the slip-lines is given, i.e. if Rx and R^ are known functions of 99, the above equations can be integrated.

The particular case when the slip-Unes are circles has been investigated by Caquot [7], and another case, when the sUp-lines are logarithmic spirals, has been studied by Golushkevich [14].

Let us suppose, for example, that Rj, = i?exp( \ and let u s integrate equation (1.47) in the foUowing way:

= exp(299 tang) - yRj, ^^^^^ sin(99 + - go), cosg

where by definition tango = 2 tang + n.

and the canonical equations (1.32) can be written as

EQUILIBRIUM OF A GRANULAR MEDIUM 31

Similarly, we shall put = {) and integrate (1.48) as follows:

cosgo ^.^ cosg

a = exp(- 29? tang) + ^^j^^ sin(9? - + go).

These results, especially for = 0, are particularly useful for approximate computations.

We shall now give an approximate method which is quite effective for solving the basic boundary-value problems. This method, which makes it possible to find the required functions at a finite number of nodal points of the network of characteristics, will be based on the differential equations (1.43) and (1.44).

The numerical solution of specific problems should be carried out using non-dimensional variables, which can be taken as

_ x _ V _


Let US consider some nodal point in the network of characteristics and the adjacent points 1 and 2 located respectively on the horizontal and vertical characteristics passing through this node. We will show how to find the values of x, y, a, at this nodal point, if the values of ^ 1 , >, , 9 i and JC2J2 ? j at the neighbouring points 1 and 2 are known.

In order to find approximately the required quantities, we must replace the differentials dx, dy, da, d in the differential equations of the characteristics (1.43) and (1.44) by the finite differences

y - y i . - , 9^-9^1 and

x-x2> y - y i . o - O l , -

Thus, instead of the differential equations (1.43) and (1.44) we obtain

y - y i = ( x - Xi) tan(99i - ) , j 2(99 9^ 1) tang = y yi (x ) t ang , ]

and y - y 2 = ( x - X2) tan(992 + ) , j

- 2 + 22( - ) tang = y - y2 + (x - 2) t ang . J

The first boundary-value problem is that the values of x, y and , are given along the segment We divide AB into a number of parts , construct in the /^-plane a coordinate network of characteristics and draw up a corresponding table.

The recurrence formulae (1.49) and (1.50), together with the boundary data, enable us to perform the necessary computations at all the nodal points of the coordinate network of characteristics and thus to fill in the corresponding squares in the table.

The method of finding the values of x, y, , at some internal nodal point from the values of X i , J i , , and X2, y29^29 2 at the neighbouring points 1 and 2 is shown in Fig. 15.

As a result of the existence and uniqueness of the solutions it can be stated that with a sufficient density of the coordinate network of characteristics the values found for x, y and , give an approximate solution to the first boundary-value problem.

Second boundary-value problem. The values of x, y and , are given along the segments OA and OB of the characteristics. We divide OA

L I M m N G EQUILIBRIUM OF A GRANULAR MEDIUM 33

and OB into a number of parts, construct a coordinate network of characteristics in the /^-plane and draw up the corresponding table.

The recurrence formulae (1.49) and (1.50), together with the boundary data, enable us to carry out the necessary computations at all the internal nodal points of the coordinate network of characteristics and thus fill in the appropriate squares in the table.

The method of finding the values of x, y, , at some internal nodal point from the values of Xi, ,, and Xg, ^'2, at the neighbouring points 1 and 2 is shown in Fig. 16.

7'

F I G . 15 F I G . 16

As a result of the existence and uniqueness of the solutions it can be stated that the values found for x, y and , represent an approximate solution to the second boundary-value problem.

We often have to deal with degenerate cases, when : = j = 0 along one of the segments of the characteristics. In these cases the segments of the characteristics in the x j -p lane are reduced to a single point.

The third boundary-value problem is that along the segment of the bisector of the coordinate angle two finite or differential relations between x, y and , are known, and along the segment O J? of a characteristic of the second family the values of x, y and , are given. We divide the segment OA into several parts , construct a coordinate network of characteristics in the //-plane and draw up a corresponding table.

Formulae (1.49) and the boundary data enable us to carry out the computations a t each nodal point on the line OA and to fill in the squares on the diagonal column of the table corresponding to this segment.


The method for finding the values of jc, y , , at some nodal point on the line OA from the values X i , j ' l , , and Xa,ya'>Oa^ a at the neighbouring points 1 and a is shown in Fig. 17.

The values of x, y and , found in this way represent an approximate solution to the third boundary-value problem.

The values of x, y and , might be given along the segment OA of the characteristic of the first family, and two finite or differential relations between x, y and , might be known along the segment OB of the bisector of the coordinate angle. In this case formulae (1.50) and the boundary data enable us to carry out the necessary calculations at each nodal point on the segment OB and to fill in the diagonal squares in the table.

F I G . 1 7 F I G . 1 8

The values of x, y, , , at a nodal point on the line OB can be found from the values x^yy^yO^, and Xj,, yb>Oby9b at neighbouring points 2 and b in the same way as before.

The fourth boundary-value problem is that along the segments OA and O are given two finite or differential relations between x, y a n d a , . As before, we divide the segments and (95into a number of parts, construct the coordinate network of characteristics in the -plane and draw up a corresponding table.

The above formulae (1.49) and the boundary data enable us to carry out the computations at all the nodal points of the segment OA, whilst formulae (1.50) and the boundary data are used to find the required values at the nodal points along the segment OB. We can then fill in the squares of the two diagonal rows of the table corresponding to these segments.

LIMmNG EQUILIBRIUM OF A GRANULAR MEDIUM 3 5

4 . LIMITING EQUILIBRIUM OF FOUNDATIONS

Let us commence our study of the stress states in foundations with the simple problem of finding the stresses within some region from their boundary values. GM 4

The method of finding the values of x, y, a, at the nodal points of OA and OB from the values , J i , , and Xa> > at the neighbouring points 1 and a, or from values of J2> ^-g, 2 and x^,


This problem can be conveniently illustrated by a model in the form of a normal spring balance, in which the vertical movement of the weight pan is restrained by friction in the guides.

If a small weight is placed in the pan it remains in equilibrium due to the considerable friction in the guides; if, however, the weight is large enough an increased spring pressure is required to give equilibrium. Limiting equilibrium for such a system occurs when an increase in the weight or increase in the spring pressure, however small, causes the system to move. A spring balance which is light in comparison with the weight corresponds to a weightless granular medium, and a heavy balance corresponds to a granular medium which possesses self-weight.

I F I G . 1 9

1 I 4

FIG. 20

The problem considered below of the limiting equilibrium of foundations is analogous to the following problem with a balance: a sufficiently large weight is placed on the pan ; it is required to find the force F i n the spring required to set up a state of limiting equilibrium in the balance. Obviously this problem has two solutions, one when the force in the spring is less than the weight and the other when it is greater. The displacement of the pan when limiting equilibrium is destroyed, however, takes place in opposite directions, as is shown in Figs. 19 and 20 by the dotted line.

The boundary values of the actual and equivalent stresses will in future be called the actual and equivalent pressures respectively.

We shall first consider the particular case of the limiting equilibrium of a foundation bounded by the x-axis, along which there acts a uniformly distributed normal equivalent pressure p.


1 + sin^ = (tan6)2''

the above formulae and analogous formulae, which we shall meet later, can be represented in a slightly different form.

It follows from the above that the mean normal equivalent stress

2 ^ I - siTiQ

We see that = I gives the smaller and that = +1 gives the larger values of the mean normal equivalent stress. Therefore the stress state corresponding to = 1 will be called a minimal, and that corresponding to = + 1 a maximal stress state. The slip-lines in this case are particularly simple; they are represented by parallel straight lines.

By analogy with the model it is clear that the minimal stress state corresponds to limiting equilibrium, the destruction of which leads to a settlement of the foundation (a lowering of the pan in Fig. 19), and that the maximal stress state corresponds to limiting equilibrium, the destruction of which induces heaving of the foundation (a rise in level of the pan in Fig. 20).

In the first case internal friction resists the settlement and thus reduces the stresses sufficiently to prevent this settlement; in the second 4*

The stress state set up in the foundation will be simple. It will be independent of the coordinate x, and can be described by the differential equations of equilibrium

dy dy ^'

which can be integrated to give

a, + H = p^-yy, r , , = 0 . (1.51)

Thus, we easily find from (1.18) that

^ = -^^' ^ = ( l - . ) ^ - f m . , . = 1 , (1.52)

^ " d ^ l s ^ ^ , , l + ; . s i n ^ ' 1 - 8

Note that as a result of the identity

1 sin^

38 8 0 and p'(x) < 0 respectively.

We shaU make a number of constructions in the |i?-plane, which, in general, we shaU asume to be multi-sheeted. In this particular problem we shaU assume that it is three-sheeted.

On sheets I and III we draw segments ^00 ^ 1 1 and A11A22 of the straight line (1.53) corresponding to like segments of the x-axis. Taking these segments as the hypotenuses, we construct the right-angled triangles ^00 ^10 ^ 1 1 and ^21 ^22 on sheets I and III and the rectangle >4io ^20^21-^11 on sheet II . A single three-sheeted combined region as shown in Fig. 21 can be formed from these regions by joining the sheets along the segments ^10 and ^21

It is important to note that such a three-sheeted region can be unfolded to form a one-sheeted region, shown in Fig. 22, by folding the rectangle -^10 ^20 ^21 -^11 about ^ 1 1 and by folding the right-angled triangle A11A21A22 about A11A21*

EQUILIBRIUM OF A GRANULAR MEDIUM 39

The segments AqqAh and An A 22 on sheets I and III correspond to the like segments of the A:-axis, and along these segments, therefore, we know that , . ^

X = (), 7 = 0 . F rom the data of the first boundary-value problem the solution

to equations (1.35) can be obtained in the right-angled triangles AqqAiqAii and ^ 1 ^ 2 1 ^ 2 2 on sheets I and III . In addition, the values of and y can be established along the two segments of the characteristics AiqA^ and ^21 on sheet II . Similarly, the data of the second boundary-value problem enable us to find the solution to equations (1.35) in the rectangle A i q ^ 20-^21 ^ 1 1

4

F I G . 2 1

Thus, in the three-sheeted combined region in the f ?y-plane the continuous functions : = x{^'r\)^y = >^(|,t?)canbe determined which are the solution to equations (1.35) in each of the one-sheeted regions ^00 -^10 -^ 115 ^ 1 1 -^21 -^22 ^Jid ^20 ^21 -^11 Oil the segment of the characteristics ^ 1 0 ^ 1 1 and ^21 derivatives and :/3?7, have finite discontinuities.

The functions : = j c ( | , r\)^ y yi^^ v) transform the three-sheeted combined region into a like region in the :vj^-plane, shown in Fig. 23, provided there are no lines of discontinuity in this region. If there are, the transformation exists only as far as the line of discontinuity.

It is not difficult to find the functional determinant of the transformation on the boundary, if we bear in mind that along the x-axis = /2. This determinant is

= -2 1 + sing

cosg tan^g P(x)

P'ipc) < 0 ,

so that the transformation reverses the direction of rotation.


In addition there are clearly no discontinuities in the boundary region. However, lines of discontinuity can, in general, appear within the region constructed.

The direction of the concavity of the characteristics (slip-lines) in the xj -plane can be found from their transforms in the | i / -p lane . Let us consider, for example, the characteristic >4 o o ^ 20 on which = = const. As we follow this characteristic from the point AQO

F I G . 2 3

towards the point AIQ on sheet I increases, but beyond the point AIQ towards A20 on sheet II decreases. If follows that in the x>'-plane the angle of incUnation

of the characteristic AQQ A^Q relative to the x-axis increases with increase in distance from the point ^00 towards the point A-^Q and decreases with increase in distance from the point A^Q towards the point y^ao. Consequently the characteristic in question (a slip-line) has the same direction of concavity over AQ^A^Q and ^10^20 but has a point of inflexion 3XAIQ, Analogous reasoning is of considerable assistance in constructing a network of characteristics (slip-lines) in the x;;-plane.

We turn now to the solution of the same problem for a material possessing self weight, without the requirement that the function pipe) on segments ^00 ^ 1 1 and ^22 be monotonic. We draw in the A//-plane the combined region shown in Fig. 24.


It has already been shown in Section 3 that there is a certain degree of freedom in the choice of parameters and , which enables us to put = X along the boundary formed by the x-axis. Segments ^ o o ^ i i and A-^-Â^^ then correspond in the A//-plane to like segments A^Â^ and AÂî- Along AQQ A^^ A22, therefore, we know from (1.53) that

p(x) ^ x = xo + A, y = 0, 0 = ^ - ^ , = ^ .

From these data for the first boundary-value problem we can obtain solutions to equations (1.45) and (1.46) in the right-angled trian-

FiG. 24

gles AQQ AIQ All and An A21 ^22 At the same time the values of x, y and , along the segments of characteristics AIQ An and ^21 will be determined. These data for the second boundary-value problem enable us to find solutions to equations (1.45) and (1.46) in the rectangle AiQ A2Q A21 All,

In the combined region, therefore, the continuous functions X = x{X^),y = y {, ), a = {,), = (, ) will be determined, and these are solutions to equations (1.45) and (1.46) in each of the regions 1 1 2 and On the segments of characteristics AIQ An and AnA2i the derivatives dxjdX, dyfdX, , and /, y|, , / undergo finite discontinuities.

We can also quite easily find the functional determinant of the transformation at the boundary, bearing in mind that = XQ and = /2. In its final form this determinant will be

S(xsy) ^ 1 + sing (,) 2cosg '

so that, as before, the transformation reverses the direction of rotation.


Although under the influence of self-weight the shape of the characteristics (slip-lines) in the A:>^-plane changes somewhat, the general distribution remains the same as in Fig. 23.

We shall consider now the particular case when the equivalent normal pressure along the segment -^n ^22 of the x-axis is uniformly distributed and equal to its value at the point ^ n .

For a weightless medium we must consider separately the regions A11A21A22 and AiQ A20 A21 All the x j -plane shown in Fig. 25.

F I G . 2 5

In the region An A21 A22 the quantities a and are constant,

a =

2 ' 1 + sing '

and the network of slip-lines consists of two isogonal families of straight lines.

In the region ^10 ^20 ^21 ^ 1 1 the xj^-plane we can use the integrals (1.37) of the equations of limiting equilibrium

a = C e x p ( - 2 tang) , $( - ) - y $( - ) = /(). The arbitrary constant C can be expressed in terms of the values

of and in the region An A21 A22 and the arbitrary function fisp) can be found from the values of = (99) and y - (^) along the segment of characteristic AIQ An, Thus,

a = - ^ . exp [( - 2) t ang] , tan(a? - ) = , 1 + smg ^ ^ X - ()


and the network of slip-lines is formed by a family of non-parallel straight lines and an isogonal family of curves.

The combined region in the |?j-plane can now be considerably simplified and can be located on only one sheet. The right-angled triangle ^21 -^22 is transformed into the single point A^, and the rectangle ^ 1 0 ^ 2 0 ^ 2 1 ^ 1 1 becomes the segment of a characteristic -^10 -^11

Thus the region - ^ 1 1 ^ 2 1 ^ 2 2 the ^rj-plane corresponds in the iiy-plane to the single point An and the region >4io ^20-^21^11 corresponds to the segment of characteristic AIQ An.

For a medium with self-weight we have from (1.52) with = 1 that in the region ^^^ in the x>'-plane

^ = f Z 2 L , ^ = (1.54) 1 + smg ' ^ 2 ' ^ ^

and the slip-line network remains the same as for a weightless material. We can take advantage of the freedom of choice of parameters

and and put = XQ along the characteristic y = ix - Xi)oois,

The segment An ^21 the A:>^-plane will then correspond to a segment of characteristic AnA^i in the /^-plane.

The combined region in the A^-plane does not now contain the right-angled triangle ^^, and along ^ we know that

= + , y = ( - ) cote, a = ^^^^^^ > ^ ^ T *

The solutions to equations (1.45) and (1.46) can be obtained first in the right-angled triangle ^00 ^10 ^ 1 1 the A/^-plane and then in the rectangle ^ 1 0 - ^ 2 0 ^ 2 1 ^ 1 1 from the data along the segments of characteristics AIQ An and ^ 2 1

In this case, in contrast to a weightless material, the slip-lines in the x>^-plane remain straight only in the region ^21 -^22 Iii the other two regions the slip-lines of both families are curves.

We give now a numerical solution to the present problem for o = 30 and

p=Po-hyx in non-dimensional variables with the characteristic length / = pjy.

In order to transfer to these non-dimensional varibles we simply put Po = I and y = 1 in all the formulae. Conversely, in order t o


transfer back to variables having dimensions, we multiply the non-dimensional coordinates and y by ^ and the non-dimensional quantity by PQ.

A numerical solution to this problem by the approximate method of Section 3 involves completing Table 1 | by the method of the first boundary-value problem.

The squares corresponding to the nodal points in the A/-plane are denoted by the two letters ij, the former indicating the number of the column and the latter the number of the row.

F I G . 2 6

In the diagonal squares 0,10, 1 , 9 , . . . , 10,0, corresponding to points on the X-axis, we put = 0 and write down arbitrary values of chosen in ascending order, and also the values of

1 +x a =

1 + sing '

The computation of the values of x, y and , in the internal squares is carried out by making use of the recurrence formulae (1.49) and (1.50).

Figure 26 shows the network of characteristics (slip-lines) for the coordinates of the nodal points given in Table 1.

We shall try to find now the maximal stress state, assuming that loss of hmiting equiUbrium causes the foundation to heave.

t In all tables for the sake of brevity only the first two decimal places are given, although the computations have been carried out to a greater accuracy.

LIMITING EQUILIBRIUM OF A GRANULAR MEDIUM

Table I t

45

j \ ^ 0 1 2 3 4 5 6 7 8 9 10

X 100 y (\ 000 a

1-33

1-57 X 0-90 0-95 y 1 000 009 1 1-27 1-36

1-57 1-59

0-80 0-85 0-90 y 000 009 0-17 a 1-20 1-29 1-38

1-57 1-59 1-61

0-70 0-75 0-79 0-84 y 3 000 009 0-17 0-26 a 0-13 1-23 1-32 1-41

1-57 1-59 1-61 1-63

0-60 0-65 0-69 0-74 0-78 y 000 009 017 0-26 0-35 a 4 107 6 1-25 1-34 1-43

1-57 1-59 1-62 1-63 1-65

0-50 0-55 0-59 0-64 0-68 0-72 y 000 009 0-17 0-26 0-35 0-43 5 100 1-09 1-18 1-27 1-36 1-45

1-57 1-60 1-62 1-64 1-65 1-67

0-40 0-45 0-49 0-54 0-58 0-62 0-66 y 000 009 017 0-26 0-35 0-43 0-52 6 0-93 103 1-12 1-21 1-30 1-38 1-47

1-57 1-60 1-62 1-64 1-66 1-67 1-68

0-30 0-35 0-39 0-44 0-48 0-52 0-55 0-59 y 000 009 0-17 0-26 0-35 0-43 0-52 0-61 7 0-87 0-96 105 1-14 1-23 1-32 1-41 1-49

1-57 1-60 1-62 1-64 1-66 1-68 1-69 1-70

0-20 0-25 0-29 0-34 0-38 0-41 0-45 0-49 0-52 y 000 009 0-17 0-26 0-35 0-43 0-52 0-61 0-69

0-80 0-89 0-98 107 1-16 1-25 1-34 1-43 1-51

1-57 1-60 1-63 1-65 1-67 1-68 1-70 1-71 1-72

010 015 019 0-23 0-27 0-31 0-35 0-38 0-42 0-45 y

000 009 0-17 0-26 0-35 0-43 0-52 0-61 0-69 0-78 0-73 0-83 0-92 1-01 1-10 1-18 1-27 1-36 1-45 1-53

1-57 1-60 1-63 1-65 1-67 0-69 1-70 1-71 1-72 1-73

000 005 009 013 017 0-21 0-25 0-28 0-31 0-35 0-38 y 1 000 009 0-17 0-26 0-35 0-43 0-52 0-61 0-69 0-78 0-87 0-67 0-76 0-85 0-94 103 1-12 1-21 1-29 1-38 1-46 1-55


We shall consider first the problem for a weightless medium and, as before, we shall assume that on 0 1 and 1^22 P'(x) > 0 and p'(x) < 0 respectively.

We shall make a series of constructions in the | i ; -plane and assume that it is made up of three sheets. These sheets are joined along the segments of the characteristics AQI An and ^ 1 2 to form one three-sheeted combined region shown in Fig. 27 and in developed form, in Fig. 28.

Fio. 2 7 F I G . 2 8

Segments and ^22 on sheets I and III correspond to like segments on the x-axis, and along these segments, therefore, we know that ^ ^ ^^^^^

The functional determinant of the transformation on the boundary is

^ ^ 1 - s i n g 0{,) cosQ

Pix) P'ipc)

> 0 ,

so that the transformation in this case does not alter the direction of rotation.

The remainder of the process is exactly as given previously. The transform in the ^>^-plane of the three-sheeted combined region is shown in Fig. 29.

We shall consider now the same problem for a medium possessing self-weight, without stipulating that the function p{x) be monotonic on the segments AQQ An and AnA^^* As before, we draw the combined region in the A/^-plane as shown in Fig. 30.

Then from (1.55) along AQQ An ^22 we know that

LIMITING EQUILIBRIUM OF A GRANULAR MEDIUM 4 7

The functional determinant on the boundary is

d(x,y) ^ 1 - sing (,) 2cosg '

so that the transformation does not alter the direction of rotation. The remainder of the process is as before.

The general distribution of the characteristics (slip-Unes) in the xy-plane remains the same as in Fig. 29, although their shape is slightly different.

F I G . 3 0

We shall consider now the particular case when the equivalent normal pressure along ^ 2 2 on the x-axis is uniformly distributed and equal to its value at the point An.

For a weightless medium we must consider the regions AnA^^A^^ and 22 in the xj^-plane shown in Fig. 31.

In the region v4i2 ^ 2 2 in the xy-planQ and are constant.

=


and the network of sUp-hnes consists of two isogonal families of parallel straight lines.

In the r e g i o n ^ o i ^ 2 > 4 i 2 - 4 i i in the x j -p lane we must use the integrals (1.39) of the equations of limiting equilibrium. The arbitrary constant D can be expressed in terms of and in the region ^ 1 1 ^ 1 2 ^ 22> and the arbitrary function g{) can be found from the values o = {) and y = () along ^ o i ^ i i - Thus

' = + )^ =

and the network of slip-Unes is formed by a family of paraUel straight lines and an isogonal family of curves.

The combined region in the Iry-plane can be considerably simplified and can be located on one sheet. The right-angled triangle ^12 ^22 is transformed into the point An, and the rectangle A^i A02 A12 An becomes the segment of a characteristic AQI An-

Hence the region ^12^22 in the xj -plane corresponds to the point yi 11 in the f/y-plane, and t h e r e g i o n ^ o i ^ 2 2 1 corresponds to the segment of a characteristic ^01 ^ 1 1

For a medium with self-weight we have from (1.52) with = + 1 that in the region ^12 ^22 in the x>'-plane

and the network of sUp-Unes coincides with that for a weightless medium.

The combined region in the -planQ does not contain the right-angled triangle 1 ^ 1 2 ^ 22, bnt along AnA^z we know

= + , y = (x - )^, ^ = , = 0.

Here, in contrast to the case of a weightless medium, the slip-lines in the xj -plane remain straight only in the region AnA-^^A^^y and in the remaining regions they are curvilinear.

Below is given a numerical solution to the above problem for Q = 30 and

in non-dimensional variables with the characteristic length / = ^-

LIMITING EQUILIBRIUM OF A GRANULAR MEDIUM 4 9

In order to transfer to these non-dimensional variables we put Po = I and = 1 in all the formulae. Conversely, in order to transfer back to dimensional variables we multiply the non-dimensional coordinates X and y by and the non-dimensional quantity hy PQ,

A numerical solution to this problem by the approximate method of Section 3 comprises completing Table 2 according to the method of the first boundary-value problem.

TABLE 2

0 1 2 3 4 5 6 7 8 9 10

X y a

-

0 000 000 200 000

X y a

-

1 0-10 000 2-20 000

005 003 2 16 004

y a

2 0-20 000 2-40 000

0-15 003 2-36 0-04

Oi l 006 2-31 0-07

y a

3 0-30 000 2-60 000

0-25 003 2-56 003

0-21 006 2-51 0-07

0-17 008 2-46 010

y

4 0-40 000 2-80 000

0-35 0-03 2-76 0-03

0-31 006 2-71 006

0-27 009 2-66 0-10

0-23 011 2-60 0-13

y

5 0-50 000 300 000

0-45 0-03 2-96 003

0-41 0-06 2-91 006

0-37 0-09 2-86 009

0-33 Oi l 2-80 0-12

0-30 0-14 2-74 016

y

6 0-60 000 3-20 000

0-55 003 3-16 003

0-51 006 3-11 006

0-46 009 3-06 008

0-43 0-11 300 0-11

0-39 0-14 2-94 0-14

0-36 016 2-87 018

y

-

7 0-70 000 3-40 000

0-65 0-03 3-36 003

0-61 006 3-31 005

0-56 009 3-26 008

0-53 Oi l 3-21 0-11

0-49 0-14 3-15 0-14

0-46 016 3-08 0-17

0-43 019 301 0-20

y

8 0-80 000 3-60 000

0-75 003 3-56 0-02

0-71 006 3-51 005

0-66 0-09 3-46 007

0-62 Oi l 3-41 0-10

0-59 0-14 3-35 013

0-56 0-17 3-28 0-16

0-53 0-19 3-21 0-18

0-50 0-21 3-14 0-22

y

9 0-90 000 3-80 000

0-85 003 3-76 002

0-81 006 3-71 005

0-76 009 3-66 007

0-72 Oi l 3-61 010

0-69 0-14 3-55 0-12

0-65 0-17 3-49 0-15

0-62 0-19 3-42 0-17

0-60 0-21 3-35 0-20

0-58 0-23 3-27 0-23

y

10 100 0-00 400 000

0-95 003 3-96 002

0-91 006 3-91 004

0-86 009 3-86 007

0-82 Oi l 3-81 009

0-78 014 3-75 0-11

0-75 0-17

statics of granular media

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