rowe 1969.pdf

8/10/2019 Rowe 1969.pdf

1/12

ROWE P.

W. (1969).

Gdolechnique

19, No.

1, 75-86.

THE RELATION BETWEEN THE SHEAR STRENGTH OF SANDS

IN TRIAXIAL COMPRESSION, PLANE STRAIN AND DIRECT

SHEAR

P. IV. ROWE*

SYNOPSIS

A theoretical relation is derived between the peak

Coulomb 4 values for saturated drained sands

measured in the direct shear test and the plane strain

compression test using the stress-dilatancy equation

and the assumption that the directions of principal

strain increment and principal stress coincide.

Limiting dilatancy rates in triaxial compression

allow an overall comparison of the Q values to be

expected in these three types of test. Present

available experim ental data indicate quite close

agreemen t over the range +, = 17-39 for cohesion-

less soils.

On derive une relation theorique entre les valeurs

4 de pointe d e Coulomb pour des sables drain sa-

tures mesurCes dans l essai de cisaillement direct et

l essai d e compression d deformation en plan en util-

isant 16qua tion de contrainte-dilatab ilitk et la sup-

position que les directions de l accroissement de

dhformation principale et de la contrainte principale

coincident.

Des taux limites de dilatabilitb en com-

pression tr axiale perm ettent de faire une comparai-

son densemble des valeurs 4 auxquelles on peut

satten.lre pour ces trois types dessais. Les don es

exp6rimentales disponibles actuelles indiquent une

concordance t s proche dans le cas oh la gam me

des valeurs de 4, = 17-39 pour les sols sans

coh ion.

INTRODUCTION

The number of variables governing the shear strength of sands is so great that any one

report must necessarily be confined to a limited aspect of the subject. In order that the

many contributions from workers in different countries using different sands, apparatus, and

technique may be related it is necessary to separate the strength component of particle struc-

ture from that of inter-particle friction and to relate the strengths derived by means of a

variety of stress systems.

The present contribution derives the relation between the strength of a given sand in the

plane strain and direct shear tests, expressed in terms of the peak effective stress ratio, and

compares theoretical and experimental results with strength limits previously derived for

triaxial compression. It is convenient first to review briefly the essential general findings

which comprise the stress-dilatancy treatment of cohesionless soils subject to effective stress,

when applied to the special boundary conditions of the triaxial test where either (TV (TV 03 or

q1 =(T~ > (TV nd to plane strain compression where o1 > u2 > o3 and e2 =O.

The stress-dilatancy equation is the name given to one particular equation in this treat-

ment, namely

in compression

R=D K . . . . . . . . (la)

in extension

R =K/D . . . . . . . . . (lb)

where

R=o;/oi, D=(l-dvs/dc a,), vs

is the volume decrease per unit volume, Ebb is the major

principal compressive strain

8/10/2019 Rowe 1969.pdf

2/12

76

P. W. ROWE

These relations which apply to the components of strain increments associated with slip

movements, at all stages of deformation to failure, have been derived in three separate ways

by Row e (1962), Rowe, B arden and Lee (1964) and Horne (1965). A fuller treatment of the

original derivation was reported by Barden and Khayatt (1966).

The measured ratio &/de, of total volumetric increment dv and major principal strain

increment de, includes elastic type strain increments prior to failure (Ro we, Barden and Lee,

1964)

yet satisfies equation (1) closely over the major part of a stress path to failure in tests

with increasing R at constant U& s reported by Row e (1962), Lee (1966), Barden and Khayatt

(1966) and Parikh (1967). Wh ere

D=l

prior to peak, the experimental observations by

Kirkpatrick (1961) also give +r values in agreement with the range described below. In the

case of stress paths at constant R, separation of the elastic component from the slip component

has led to equation (1) for the slip components (El-Sohby, 1964). Constant volume tests which

exhibit constant

R

over a major portion of the stress path include marked pre-peak elastic

components.

It may be noted that although the term elastic refers to strain components associated

with stored energy in grain compression, it may not necessarily be recoverable as with a true

elastic component because some of the stored energy is absorbed in slips during unloading.

This is particularly ma rked during unloading in varying R tests where repeated loading and

unloading result in a continual absorption of energy in friction at interparticle slips.

When considering soils at the peak, denoted by the maximum principal effective stress

ratio, or at the critical state when after large strains no further change in stress ratio occurs

with strain, small strain increments take place at constant effective stress and the elastic

incremental strains are zero.

In the case of plane strain compression, writing de, =de 2, =0, equations (1) apply and are

identical, althou gh the boundary strain conditions influence the value of 4r. Expe rimental

verification of equation (1) has been reported by Row e (1964a) and Wightman (1967). Procter

1967)

found similar behaviour in triaxial comp ression, extension and plane strain conditions

using the hollow cylinder test.

The data have been based on measurements of the overall change in dimensions of a sand

element rather than on internal strains but all these experiments have been conducted since

1953 using lubricated ends (Rowe, 1962).

Kirkpatrick and Belshaw 1968) have reported evi-

dence that the internal strains a re sensibly uniform throug hout the right cylinder, maintained

during triaxial compression with lubricated ends.

Cole 1967) and Roscoe 1967) report good agreement with the stress-dilatancy equation on

the basis of more than 170 tests with Leighton Buzzard sand in the Camb ridge Mk . 6 S.S.A.

following important developments in technique which allowed the determination of principal

stress and strain-rate ratios within a uniform central element.

The present data confirm the following summary of statements concerning the approxi-

mate value of & in equation (1) in compression (Row e, 1962, 1963, 1964a, 1964b) and extension

(Barden and Khayatt, 1966).

In triaxial strain,

compression or extension

(b, I +r 5 do7 . f . . . . . . (2)

In the densest state up to peak stress ratio

r=r$rr

. . . . . . . . . .

(3)

In the loosest state at the peak stress ratio, which is the critical state

r f = +,, . . . . . . . . . . (4)

In all cases where &

8/10/2019 Rowe 1969.pdf

3/12

SHEARSTRENGTHOFSANDS 77

In lane strain,

for any packing up to the peak stress ratio, in compression or extension

I = +,, . . . . . . . . . . (5)

Look ing at the range of $t values (equation (2)) it is noted that the insertion of &= &.,

leads to a maximum value of

K

(equation (I)), namely

K,,,

in contrast to the insertion of

$f=+LI giving a minimum value of K, namely K,.

This maximum-minimum range of K

values is, how ever, not to be confused with the minimum energy principle underlying the

derivation of equation (l), nor need there be any confusion betwee n the minimum energy

principle and a minimum energy ratio.

Equation (1) states the minimum absolute energy

absorbe d in interparticle friction

at a given applied stress level and energy input

and with respect

to all possible instantaneous mean particle slip directions and not with resp ect to all possible

$r values. In the latter connexion it may be noted that for the particular case of dv, =0 at

the critical state, D =I and equation (1) is identical to the Rankine equation with +=& =& ,.

In the derivation of the Rankine equation the minimum energy p rinciple is seen even mo re

clearly if the Rankine equation is derived using Coulom bs me thod of determining the mini-

mum boundary force for all possible straig ht slip plane directions as follows. From Fig.

1

whence

alb

o;b

cot /3

=

tan ($+B)

The minimum value of U; for a given uj is given w hen ,6=4 5-$12 , and the insertion of this

value of p into equation (6) gives the Rankine equation u; uj = tan2 (45 + 2).

Consequently

for this special case of no volume change rate, wh ere the stress ratio is identical to the ratio

of energy input to energy output, in both triaxial and plane strain and wh ere K is a maximum,

the energy absorbed is a minimum with respect to all possible mean values of p for the chosen

stress level.

Also the absorbed energy has to be considered at a chosen stress or energy level.

It has

been shown (Ro we , 19 64b) that if the ratio o f incremental energy input in the major principal

direction divided by the incremental energy output in the minor principal direction is

E

where

U; de,

EC-

- uj

de,

in plane strain comp ression

nd

u;

de1

-_-

E = -2ujdE3

E _ 27; dcl

- u; de,

in triaxial strain com pression

in triaxial extension

Fig. 1.

Coulomb minimum energy solution

8/10/2019 Rowe 1969.pdf

4/12

78

I. W.

ROW

then at any stage of a compression test the absolute increment of work absorbed dW is given by

dW = crj e lR

1-g

[

1

In a test one might choose the value of the cell pressure U; and keep it constant, and ch oose

the stress level by selecting a value of R which may or may not be the peak, but wh ich remains

sensibly constant during a chosen small applied strain A,. The applied input energy level

uj de,R is thus chosen. For this applied energy input and chosen stres s level the absolute

energy absorbed d

W

is a minimum when

E

is a minimum.

One might call this an energy ratio

principle but it does not invoke a new principle since only one of the four test variables in the

above expressions is free to vary with E and the particle sliding direction; the other three are

chosen and held constant.

It is seen therefore that equation (1) does not invoke a new principle in respect of mini-

mization of energy, but is merely a generalization of the Coulomb-Rankine expression to

include dilatancy rates other than unity.

It may a lso be noted that the flow rule of plasticity associated with the Mohr-C oulomb

failure criterion constitutes a special case of the stress-dilatancy relation whe re the interparticle

friction angle (b, =0 and in plane strain ~1&Ju $ dr, = - 1.

In an outstanding general treatment of rotund particles in contact, Horne

(1965)

not only

verified equation

(1)

but extended the finding to show that for the case of triaxial compression

the maximum possible value of D =2. Thus at peak

1~0~2 . . . . . . . . (7)

It may be noted that equation (1) was derived originally with the aid of an equation

R =

tan CI an (1$,+/3)

. . . . . . . .

(8)

which applied to any formal p acking or any two particles in contact. This equation was also

derived independently by Biarez (1961 ), Leussink and Wittke (1963) and Parkin (1965).

Insertion of the critical direction for a single pair of contacts and with allowance for deviations

of individual contact directions from the mean critical direction led to the equation

Horne derived

R = tan cc1

8/10/2019 Rowe 1969.pdf

5/12

SHEAR STRENGTH OF SANDS

79

0 08 06 04 02 0

INITIAL RELATIVE POROSITY

ZO

I)

t

0

0

I* n

I5

m

.

I.0

I.0 08 06 04 02 0

.

A

RELATIVE POROSITY AT PEAK

(a)

Fig. 2 (above). Experimental relations between

dilatancy rate and relative porosity at low pressures

Fig. 3 (right).

Relation between , and CCV

I . 01

/

/

- - _

I.0

0.8 0.6 o -4

0.2 0

5(

1

4[

3[

2c

I(

(

)-

)-

)_

)-

I-

)


@I

IO 20 30

+;

-

4c

The experimental relation between 4, and +,, shown in Fig. 3, which is in agreement with

a theoretical prediction by Horne, refers to a free mass of particles under water.

The values of +, and d,, in Fig. 3 were determined from the slope of stress ratio-dilatancy

rate plots of triaxial compression tests, on samples in dense states during pre-peak loading to

give c$, and at large strains using free ends to give &.

In the case of glass ballotini, quartz

and feldspar it was possible to obtain very similar values of $, by means of a direct shear test

of a mass of particles sliding over a polished block of the mineral under water, but no material

in block form was available for the zircon or the crushed glass.

PEAK STRENGTH LIMITS IN TRIAXIAL AND PLANE STRAIN COMPRESSIOS

Tviaxial compression

Taking the upper dilatancy rate D =2 and using equations (la) and (3) the upper limit for

the stress ratio of a sand in the densest state is given by

o~/c$, = 2 tan2 (45+$,/2)

. . . . . . .

(11)

Taking the lower dilatancy rate

D = 1

and using equations (la) and (4) the lower limit for

the stress ratio of a sand in the loosest state is given by

CJ;/IJ~= tan2 (45+&,/2) . . . . . . (12)

8/10/2019 Rowe 1969.pdf

6/12

80

P. W. ROWE

INITIALRELATIVE POROSIN

Fig.

4. Relation between Coulomb ,,,, as

measured in plane strain, triaxial compres-

sion and direct shear

50

I

(b)

$,= 26

,

J

0

05

IO


- - Experiment

v .I

I .,T;;;i*

.

P.S.

I

T Glasr ballotinl

o D.S.

I

b

I

05

INITIAL RELA TIVE PORO SITY

I

I-O

lane strain compression

Using equations (la) and (5) and assuming the same dilatancy

limits in plane strain as for

triaxial strain, namely

dense

$ = 2tan2 (45+&.,/Z)

*3

. . . . . .

*

(13)

loose

$ = tan2 (45+$,,/2)

03

. . . . . .

-

(14)

These upper and lower limits are shown on Figs 4(a), (b)

and (c) in terms of the Coulomb angle

4

max,

using the Rankine equation, plotted at relative porosities of 1 and 0 respectively for

each test type.

The variation of dilatancy rate with relative porosity is not necessarily linear

and is unknown,

Experimental results show a relation between d,,, and relative porosity

which is nearly linear and straight lines are drawn between the theoretical limits in Figs 4(a),

(b) and (c).

The limiting dilatancy rate of 2 is not necessarily reached by dense packings.

Increase of

mean principal stress, for example, leads to crushin,

5 and a reduction of dilatancy rate to an

extent such that at very high pressures the maximum dilatancy rate is unity. However, for

a given measured value of &,,X

for one sand in one state and mean principal stress at failure

in the triaxial test, the corresponding value of $max

in plane strain may be deduced from a plot

8/10/2019 Rowe 1969.pdf

7/12


81

of the limiting values for the particular values of #, and &.

For this purpose 4, may be

determined during a reloading stress path in the triaxial test and Fig. 3 used to assess &,

or the value of qGcymay be measured directly in the triaxial test using free ends and large

strains.

The value of & may differ slightly in plane strain from that in triaxial strain but

more experimental evidence is required on this point.

PEAK STRENGTH IN DIRECT SHEAR AND PL.4NE STRAIN COMPRESSION

It has been noted (Rowe, 1958; Rowe, Barden and Lee, 1964) that the Coulomb 4 measured

in direct shear must differ from that measured on a principal stress element in a compression

test apparatus. This is in contrast to the result obtained when assuming the Coulomb failure

criterion and the orientation of the principal stress element in a shear box such that the

sliding plane coincides with the plane of applied shear as given for example by Taylor (1918).

The problem was considered by Hansen (1961) who assumed coincidence between the direc-

tions of principal stress and total strain.

De Saint Venant (1870) inferred that for plastic deformations of isotropic materials the

principal directions of stress and strain rate should coincide. Hill (1950), treating a perfectly

plastic material, stated that if the principal axes of stress coincide with the axes of anisotropy,

so do the principal axes of strain increment.

A characteristic of the particulate model is that during a stress path under an increasing

stress ratio a new sample particle structure is formed with a new set of contacts during each

small stress ratio increment. The model is orientated in the applied direction of principal

stress and if that direction changes during an increase in stress ratio, when interparticle slips

predominate, the new orientation of critical contacts must be such as to achieve equilibrium

with the new direction of applied stress. Two points arise. The frrst is that the orientation

of the critical contacts controls the axes of anisotropy of the model, whose properties depend

entirely on the contacts, so that there is conformity with the statement of Hill. Second, the

coincidence of stress and strain rate direction in an element under test with fixed principal

stress directions, prior to any formation of discontinuity or slip bands, ought therefore to

apply equally to an element which undergoes a gradual reorientation of the directions of

principal stress.

Cole (1967) and Roscoe, Bassett and Cole (1967) have reported direct measurements of

normal and shear stress on the boundary of an element of sand in the centre of the Cambridge

S.S.A. Mk. 6 and assuming uniformity of strain throughout the element have shown, for the

first time, coincidence of the directions of principal stress and principal strain rate during small

incremental stress rotations.

In their diagrams for Mohrs circles of stress and strain increment for the stress condition

in simple shear they use the same angle # between the horizontal plane of the simple shear

apparatus and the direction of principal stress on the one hand and between the horizontal

plane and that of the principal strain increment on the other. A necessary consequence is

that Mohrs circles of stress and strain rate are geometrically similar. This together with the

condition of zero strain increment in the horizontal direction in direct shear (Fig. 5) leads to

the following equations. From Mohrs circle of strain increments

dv

-=

2

de, d4

cos

2 + _

2

. . . . . . .

In two dimensional strain

From equations (15) and (16)

dv = dc,+dcg

. . . . . . . .

(16)

cos 2 =

dv

dvldc,

1-D

2 de,-dv = i -dvl dc, = -

+D

8/10/2019 Rowe 1969.pdf

8/12

82

P W ROWE

xis of

A x i s

of

( a ) M o h r r c i r c l e o f s f r e s s

( b ) M o h r r c i r c l e o f s t r a i n i n c r e m e n f

Fig 5

Mohrs circles of stress and strain increments

From LMohrs circle of stress

(T;+uj

-

=a-TCOt2*

. . . .

2

and

U;-U;

n

= 7

cosec2

. . . .

L

From equations (18) and 19)

u; + u;

R+l

-7--7=R_l

cl-u3

= csin 2#--cos 2#

[Ssing equation (17)

R+l (T

-=-

R-l i-

J

&(l-D)2

(1-D) u 20

(I+0)2)=$D))

whence

7

I

i- R-l)

~=&~(R/D+l)

Inserting the flow rule equation (la) into equation (21)

.

7

2=

J

I((R-1)

R(K) ...

. .

where with reference to equation (5)

K = tan2 (45++,,/2)

. . .

Substituting R = tan2 (45 +&,,/2) and T/O= tan & equation (22) becomes

tan & = tan & cos $,,

. . . .

.

. . .

18)

. . .

w

. .

20)

. .

21)

. . .

22)

. .

(23)

. .

(24)

Cole (1967) obtained a numerical relation between T/O and R and noted that it must depend

on the value of +, and Roscoe, Bassett and Cole (1967)

concluded that a relation between these

parameters was imminent.

has published the relation

Also, since the original preparation of this Paper, Davis (1968)

7 cos 4 sin +

2

z-7

l-sm$sin+

between

T/U

in direct shear, 4 in plane strain and a parameter denoted here by z+8which is

defined by the identity D =tan2 (45+$/2).

relation, equation (22) is obtained.

Eliminating I$ and using the stress-dilatancy

8/10/2019 Rowe 1969.pdf

9/12


233

Equations (22) and (24) relate the peak stress ratio R or & in the plane strain test to the

peak stress ratio T/U or q&

measured on a horizontal plane in simple shear, as a function of

the critical state angle &,

which in turn is a function of 4,. For the constant volume test as

when

R = I

and & =&, equation (22) simplifies to

tan &s = sin q&

. . .

. (25)

in agreement with Hill (1950) and as noted by Bishop (1954). Only in the imaginary case of

frictionless particles with q5, =& =0 does & =&.

The physical basis for this relation is that the orientation of the plane strain element

within the shear box is governed by the restriction of movement such that the resultant

horizontal strain increment is at all times zero. Consequently the critical slip plane in the

plane strain element in general does not coincide with the horizontal plane of the box. The

peak das applied to the horizontal plane must therefore be less than the &,, associated with

the critical plane in the plane strain element which governs failure, and on the basis of the

Coulomb theory this should lead to slip planes in the shear bos at angles inclined to the plane

of the box.

Some numerical values are given in Fig. 6 in which the difference (& -&) is plotted

against +ds.

One can insert any value of

R

from unity upwards and deduce the corresponding

value of T/O n equation (22), but the peak dilatancy rate range of 1 to 2 determines the range

of R or Q-/Uor any given value of C,,. These are shown by solid lines in Fig. 6 for feldspar,

quartz and glass sands, and it is interesting that for a given material the difference (+,, -&J

is approximately constant over the entire dilatancy rate range.

Working with the conventional shear box and its associated non-uniform strain pattern,

Rowe (1954) observed values of &

for Mersey River Sand between 23 and 42, while the

range recently obtained by Wightman in plane strain compression over the same order of mean

pressure range is 32-46 or some 4 higher, in close agreement with $,,=32 for that sand.

It seemed therefore of interest to carry out conventional direct shear box tests on feldspar

sand and glass ballotini at stress levels comparable with those used in plane strain.

The

results are shown in Fig. 6. These showed some 9 difference between shear box and plane

strain values of dmax

at the dense and loose limits for feldspar and some 2 difference for

glass. Despite the objections to the conventional shear box it would seem that the experi-

ments agree quite closely with theory over the wide range of q5, values investigated. Such

Fig. 6.

4

r

Difference between ax

in plane strain and direct shear for various values of

8/10/2019 Rowe 1969.pdf

10/12

84

P. W. ROWE

experiments need to be repeated in a direct shear apparatus which applies m ore uniform

strain through out the samp le, but two points arise.

First, if the theory is correc t it wou ld

appear that the difference between peak values measured in a conventional shear box and

those in an apparatus which allows much more uniform strain may be expected to be small.

Secon d, there is still a fairly w ide use of the direct shea r tes t using various types of appara tus

and the present results (Fig. 6) transferred to Fig. 4 indicate the relative magnitude of the

Coulomb q5 to be expected from these tests and triaxial and plane strain compression tests.

In this respect it may be noted that the trend for (&raxisl - $d,rect shear) to decrease and change

sign passing from loose to dense states of a medium-fine quartz sand was noted by Nash

(1953).

Considering the direction of the Coulom b slip plane in the direct shear test, point A in

Fig. 7(a) represents the stresses on the horizontal plane and AP is drawn horizontally throug h

A to give the pole of the diagram at P.

The failure plane of the principal stress element is

then given by PB and the observed slip planes, if they fo rme d at the peak strength , should lie

at angle CLo the horizontal (Fig. 7 (b)).

From the geometry of Fig. 7(a)

I

u3 sin &

sin B

=(a;--oj)cosB

. . . . . .

and inserting p = 0 - &

sin s

cosOsin(O-$,,) = ~_I

. . . . . .

Substituting for tan q& (equation (24)) into equation (27) and rearranging

e COS 1

J-c

I--

sin & co? J~

sin +,, co? f&

1

- sin2 dD s sin2 q5 ,, II-

. . .

(27)

28)

whence from Fig. 7(a)

a =45+-p-e . . . . . . . .

29)

For the special case of no volume change rate at failure, when & =& , equation (28) yields

e=4 5 for all values of 4,.

This is evident immediately from Fig. 7(a) since for this case

Fig. 7.

Slip plane direction in direct shear

8/10/2019 Rowe 1969.pdf

11/12


85

points A and P are both located at C.

Consequ ently a= & as deduced by Hansen (1961),

but with the special value ,,/2.

For sands dilating at the limiting rate D ~2, equ ation (28) yields 0=cos-11/1/3 =5+75,

and using equation (29)

a = %- k-75 - ( ~ + )I

. . . . . ,

(30)

Taking the values of 4, given in Table 1, and deducing &,, from the stress-dilatancy equation

for

D

=2 it is seen that the value of a is about 2 less than q&/2 for quartz sands at the maxi-

mu m dilatancy rate. For intermediate dilatancy rates the value of a may be determined

Table

0

19.5

0

10 28.7

0.40

20 37.3

I.10

30 45.6

1.8

40 53.4

3.0

from equations (28) and (29) using measured values of 4pS, measured & values with equation

(24) or measured peak dilatancy rates with equation (1).

In general for quartz sands

It is clear that for dilating sands the value of a can be considera bly smaller th an +,,/2.

This analysis should apply to drained tests on normally consolidated clays where c=O,

but where a comparison is made betw een m easured and calculated directions of slip surfaces,

e.g. Morgenstern and Tchalenko (1967), it is strictly necessary to maintain reasonably uniform

strain conditions w ith a Cam bridge simple shear apparatus and to record the dilatancy rate

at peak effective stress ratio.

CONCLUDING DISCUSSION

Many earth pressure and stability problems are those of plane strain, but ow ing to pro-

gressive failure throughout the mass the 4 values necessary to fit earth pressures to simple

failure theories for dense sand can vary over a wide range between limits close to the plane

strain peak for an element &,ax

in the active state and close to 4, in the passive state (Rowe,

1967). Average m ass 4 values less than the plane strain peak are more appropriate and this

may be one reason for the app arent success of the use of the triaxial test.

The difference

between the direct shear value and that from the triaxial test for quartz sands in the dense

state is less than the uncertainty associated with use of a singular average Coulom b 4 value

in the mass at failure, whereas in the loose state where m ovements are more serious the

lower direct shear values are conservative.

For sands, therefore, the direct shear test may

still be attractive in practice.

ACKNOWLEDGEMENT

The Author is indebted to Professor M. R. Horne for permission to include his unpublished

theoretical curve in Fig. 3, and to Professor A . W. Bishop and Professor K. H . Roscoe for

improvem ents in presentation.

8/10/2019 Rowe 1969.pdf

12/12

86

I. W. ROWE

REFERENCES

BAR DEN, L. KHAYA TT. A. J. (1966). Incremental strain rate ratios and strength of sand in the triaxial

test. Gdotechnique 16, No. 4, 338-357.

BIA REZ , J. (1961). Contribut ion a letude des properties meca nique sols et des materiau x pulvCru lents.

D.Sc. thesis, Grenoble University.

BISH OP, A. W. (1954). Corre sponde nce on Shear characteristics of a saturated silt measu red in triaxial

compression. G technique 4, No. 1, 43-45.

COL E E. R. L. (1967). Soils in the S.S.A.

Ph.D. thesis, Cambridge University.

DAVIS E. H. (1968).

Theories of plasticity and the failure of soil masses.

(ed. I. K. Lee). Butterw orth.

Soil

mechanics, selected topics

DE SAINT VENA NT, B. (1870). Memoire snr l etablissement des equations differentielles des mouvements

interieurs opCrC s dans les corps solides ductiles an dclB des limites ou lelasticite pourrait les ram ener

a leur premier tat. C. Y.

hebd. Seanc. Acad. Sci., Imis 70, 473-480.

EL-SOHBY , M. A. (1964). The behaviour of particulate material under stress.

Ph.D. thesis, Manchester

University.

HANS EN, B. (1961).

Shear box tests on sand. Proc. 5th Int. Conf. Soil Mech. 1 127-131.

HILL R. (1950).

The mathenzatical theorv of plasticitv.

HORNE l . R.(196 5).

Oxford Universitv Press.

The behaviou r of anassembly of rotund, rigid cohesionless particles.

Parts I and

II. Proc. R. Sot. A 286, 62-97.

KHA YAT T, A. J. (1967). Som e increm ental stress-strain relations for sand. Ph.D. thesis, Man chester

University.

KIRKPA TRICK, W. M. (1961). Discussion on Les proprietes des sols et leur mesure.

Proc. 5th Int. Conj.

Soil Mech. 3, 131-133.

KIRKPA TRICK, \V. M. BELSHA W, D. J. (1968).

On the interpretation of the triaxial test.

18 No. 3 336-350.

Geotechnique

KOLBUSZEWSKI,J. J. (1950). Notes on deposition of sands.

Research, Lond. 3, 478-483.

LEE , I. K. (1966). Stress-dilatancy perform ance of feldspar.

J. Soil Mech. Fdns Dio. Am Sot. civ.

Engrs 2,

79-103.

LEUSS INK, H. WITTKE , 1V. (1963).

Differen ce in triaxial and plane strain shear strength. Sym posium

on

laboratory shear testing of soils, 77-89.

Am erican Society of Testing and Materials, S.T. Pub. 361.

MORGENSTERN, N. R. TCHA LENK O, J. S. (1967).

Microscopic structures in kaolin subjected to direct

shear. GCotechnique 17, So. 3, 309-328.

NAS H, K. L. (1953). The shearing resistance o f a fine closely gra ded sand.

Proc. 3rd Int.

Conf. Soil

Mech.

1 160-164.

PARIKH. P. V. (1967). The shearing behaviour of sand under axisymmetric loading. Ph.D. thesis, Man-

Chester University.

PAR KIN. A. K. (1965). The application of discrete un it models to studies of the shear strength of granular

materials.

Ph.D. thesis, -University of Melbou rne.

- _

PROCTE R, D. C. (1967). The stress-dilatancy behaviour of dense sand in the hollow cylinder test.

M.Sc.

thesis, Manchester University.

ROS COE . K. H. (1967). Discussion on Shear strength o f soil other than clay.

Proc. geotechnical Co~f. 2,

188-192.

ROSC OE, K. H., BASSETT, R. H. COLE, E. R. L. (1967). Principal axes observed during simple shear of

a sand.

Pvoc. geotechnical Conj. 1 231-237.

ROW E, P. W. (1954). A stress-strain theory for cohesionless soil with applications to earth pressure at

rest and moving walls. Gtotechnique 4, No. 2, 70-88.

ROW E, P. W. (1958). Closing remark s session 1,

Proc. Conf. Earth Pressures.

ROWE, P. W.

(1962).

The stress-dilatancy relation for static equilibrium of an assembly of particles in con-

tact. Proc. R. Sot. A 269 500-527.

ROW E, P. W. (1963).

Stress-dilatancy, earth pressures and slopes.

Proc. Am. Sot.

civ. Eplgrs

89, 37-61.

ROW E, P. W. (1964a). Discussion on Some experiments on the influence of strain conditions on the strength

of sand. Gdotechnique 14, No. 4, 361-364.

ROW E, P. W. (1964b). Closure to discussion o n Stress-dilatancy, earth pressures and slopes.

Proc. Am.

Sot. civ. Engrs 90, 145-180.

ROWE, P. W. (1967).

Discussion, Session 3. Proc. Geotechnical Conf. 2, 210-211.

ROW E, P. W., BARD EN, L. LEE, I. K. (1964).

Energy components during the triaxial cell and direct

shear tests.

Gdotechnique 14, No. 3, 247-261.

TAYLOR , D. W. (1948).

Fundamentals of soil mechanics.

Wiley.

WIGHTM AN, A. (1967).

The stress-dilatancy of sands during plane strain compr ession. M.Sc. thesis,

Manchester University.

rowe 1969.pdf

Documents