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D. M . W o o d , 1 A n d r z e j D r e s c h e r , 2 a n d M u n i r a m

Budhu 3

On the Determination of Stress State in the Simple Shear Apparatus

ni Vector of direction cosines of the outward normal at coordinate xi on the sur-

REFERENCE: Wood, D. M., Drescher, A., and Budhu, M., "On the Determination of Stress State In the Simple Shear Apparatus," Geo-

technical Testing Journal, GTJODJ, Vol. 2, No. 4, Dec. 1979, pp. 211-221.

Pi Pi, Qi, ei

ABSTRACT: The simple shear apparatus is one of the few commonly

available laboratory apparatus that permits the application of controlled rotations of the principal axes of stress and strain to soil samples. However, because of the boundary conditions in the apparatus the soil sample does not respond as a single element, and this should be reflected in the analysis of test results. In the Cambridge University simple shear apparatus, the sample is surrounded by an array of load cells (contact stress transducers) that measure the complete distribution of boundary stresses throughout a test. For simple shear test results to be presented in terms of useful stress parameters, a procedure for computing the stress state from the load cell measurements is required. Such a procedure is described, making use of the concept of an average stress tensor to determine a representative stress state in the central part of the sample, which is least influenced by the ends of the apparatus. Less complex and expensive apparatus exist that can only measure the average normal and shear stresses applied to the top and bottom horizontal boundaries of the sample. Patterns of soil response have been determined from tests on Leighton Buzzard sand in the more elaborately instrumented Cambridge apparatus, and a method is described for using these patterns to deduce the complete stress state in the less complex apparatus.

PL, PR; QL, QR; eL, eR

R

Rcv S

1/2(al + a3)

s

Surface of volume V

t

Ti, ~m V X i or x,y ot

7 6ij

KEY WORLDS: soil tests, shear apparatus, laboratory equipment, shear stress, sands

o

Nomenclature

a,b,c,d,e,f,g,h eo fxi' fyi

Coordinates of points of action of forces on central zone (Zone 2; see Fig. 7b) Initial void ratio Typical forces in x and y directions (i =

a l , 0"2, 0"3

ax, ay, az Ti, o0"

1 . . . 8)

fxi*' fYi* Corrected typical forces in x and y direc-

ryx, rxy

tions (i -- 1 . . . 8) fx, Fy Out-of-balance forces in x and y directions k, k 1 Soil constants 1,:o Coefficient of earth pressure at rest e Width of strip over which forces Pi and Qi act AM Out-of-balance moment

q~cv x ¢

1University lecturer, Engineering Department, Cambridge University, England. 2Research engineer, Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw. 3University lecturer, Department of Civil Engineering, University of Guyana, Georgetown, Guyana (formerly research student, Cambridge University Engineering Department).

face (i = 1, 2, 3) Vector of stress components (i = 1, 2, 3) Normal forces, shear forces, and eccentricities, respectively, measured by boundary load cells (i = 1 . . . 8) Calculated normal forces, shear forces, and eccentricities, respectively, for left and right sides of central zone (Zone 2 in Fig. 7a) Stress ratio = 7yx/Oy Critical state (constant-volume sheafing) value of R

t/2(al

-

a3)

Typical boundary forces (i = 1, 2, 3) Volume Coordinates (i : 1, 2, 3) Measure of shear strain -----tan 0 Shear strain Kronecker delta (i = I, 2, 3 ; j ---- i, 2, 3) Direct strain Angle of rotation of ends of simple shear apparatus Angle between direction of major principal strain increment and vertical Major, intermediate, and minor principal stresses, respectively Normal stresses in x, y, and z directions Stress tensor and average stress tensor, respectively (i : 1, 2, 3 ; j ------1, 2, 3) Shear stress on horizontal (xz) and vertical (yz) planes, respectively Angle of friction Angle of friction mobilized at critical state (constant-volume shearing) Angle between direction of major principal stress increment and vertical Angle between direction of major principal stress and vertical

Introduction

From the early work of Kjellman [1] and Roscoe [2], the simple shear apparatus has been developed at Cambridge University and elsewhere to become a basic apparatus for experimental investigation of the stress-strain behavior of soils. The purpose of the simple

0149-6115179/0012-0211500.40

© 1980 by the American Society for Testing and Materials 211

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212

GEOTECHNICAL TESTING JOURNAL

shear apparatus is to apply rotations of the principal axes of stress and strain to samples of soil kept under a condition of plane strain. In the Cambridge apparatus these rotations are achieved by applying normal and shear loads or displacements to the top or bottom faces of an initially cuboidal sample, which deforms into a parallelepiped. Parallelepipedal deformation is ensured by allowing two initially vertical sides of the box containing the sample to rotate, following the horizontal displacement of the top or bottom face (Fig. la). An alternative simple shear apparatus, which, because of its simplicity, has been more widely used, has been developed at the Norwegian Geotechnical Institute [3]. In this apparatus a circular cylindrical sample of soil is tested, the section of the sample being maintained approximately constant during shear deformation by containing the sample within a rubber membrane bound with wire (Fig. lb). Knowledge of the total vertical normal load and total horizontal shearing load applied to the top and bottom faces of the sample is not sufficient to determine the stress state in the sample, for two reasons. First, this information can only fix one point on Mohr's circle of stress; knowledge of the normal stress on the vertical plane perpendicular to the direction of shear, and of the intermediate principal stress on the vertical plane parallel to the direction of shear, is needed to complete the description of the stress state. Secondly, and more seriously, it was demonstrated by Roscoe [2] that even for an ideal elastic material the boundary conditions imposed by the apparatus, that is, complementary shear stresses being largely absent from the ends of the sample, were such that the stress state in the sample could not be expected to be uniform. For that ideal material, however, conditions were shown to be rather

a~

./

/

uniform in the central third of the sample, away from the influence of the ends. Pr6vost and H6eg [4] have shown that slip between the elastic sample and the top and bottom boundaries makes the stress distribution still less uniform. The qualitative effect that can be expected to result from the absence of complementary shear stresses on the ends of the sample is shown in Fig. 2. The shear stresses over the top and bottom boundaries will be nonuniform, with zero or small values at the ends of the sample, as shown in the top of Fig. 2. The normal stresses will also be distributed nonuniformly, as shown in the bottom of Fig. 2, in order to preserve moment equilibrium of the sample. In order to be able both to accommodate these nonuniformities and to determine the stress state, load cells are placed around the boundary of the sample in the rectangular Cambridge apparatus (Fig. la). These load cells measure the normal and shear components of applied load, and also the eccentricities of the normal loads. The aim of this paper is not to discuss the technical details of the various simple shear apparatus, but to analyze a suggested procedure for determining the stress state in the central zone of the sample, and to compare two possible methods of implementing this procedure. Then, by making use of the results of simple shear tests on sand that have been analyzed according to the proposed procedure, a method is developed for computing the complete stress state at any stage of a simple shear test on sand conducted in an apparatus in which stresses are measured only on the horizontal boundaries. It is hoped that these analyses may be of value to future investigators drawing conclusions from their experimental measurements.

b.C / wire binding

["

o

. "

Q "

i]

lob[(.] (-r~l I~

FIG. 1--(a) Diagrams of Cambridge simple shear apparatus: the sample is square in plan and enclosed within rigid boundaries containing load cells (contact stress transducers). (b) Diagrams of GEONOR simple shear apparatus: the sample is circular in plan and contained within a rubber membrane with a spiral wire binding.

FIG. 2--Top: nonuniform distribution of shear stress from absence of complementary shear stresses on ends of sample. Bottom: nonuniform distribution of normal stress to preserve moment equilibrium of the sample.

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213

WOOD ET AL ON SIMPLE SHEAR APPARATUS

Proposed Procedure for Calculating Stress State

Although these postulates--the first one in particular--are open to doubt, we shall assume their validity. Of course, any postulate concerning stress distribution in a sample of any material subjected to boundary tractions and displacements can be questioned, but some postulates are required in order to proceed. We will consider now further assumptions required to obtain the stress state from the second and third postulates. We shall limit consideration to the stresses acting in the plane of deformation (the x-y plane in Fig. 3), disregarding the components of stress acting perpendicular to and parallel to that plane (the intermediate principal stress is discussed later). It is, of course, of interest to study the behavior of the soil in terms of plane-strain stress components; however, in circular simple shear apparatus of the type shown in Fig. l b it is not possible to measure the forces on the side walls. However, some information concerning these side forces is necessary if comparisons are to be made between soil behavior in the simple shear apparatus and in other non-planestrain apparatus. The object of the proposed procedure is to determine the stress components Ox, Oy, and Zyx or Ol, 03 and the angle ¢ of inclination of the plane on which the principal stress al acts to the horizontal x axis in Zone 2 (see Fig. 4, left portion). The starting poin:t of our analysis is the set of boundary forces acting on eight segments of the sample boundary (Fig. 3). By means of the load cells the normal and tangential components Pi and Qi, respectively, and the eccentricity e i of the force with respect to the center of each segment are determined. In general, the stress state in a body cannot be determined from the forces acting on its boundary unless the constitutive relations of the material comprising the body are known. We may, however, define an average stress tensor ~(/:

Although there are technical differences among the various models of simple shear apparatus, designated Mk 1 through Mk 7, that have been constructed at Cambridge University, the basic postulates that have been used to determine the stress state in the soil sample have been approximately the same. These postulates are retained in the proposed procedure, and are stated below: 1. In a sample, rectangular in plan, that is sheared in plane strain, as in Fig. la, any stress inhomogeneities that occur are likely to be most pronounced in regions adjacent to the rotating end flaps and the sides of the sarfiple. The stress state in the central zone of the sample (Zone 2 in Fig. 3) is supposed to be sufficiently homogeneous to be identified with the stress state in an infinitesimal element of soil subjected to plane strain simple shear. 2. The stress state in the central zone of the sample may be evaluated from the magnitudes of the stress vectors acting on the sides of the central zone. 3. The stress vectors acting on the sides of the central zone may be deduced from the magnitudes of the resultant forces acting on those sides.

Q

/II

l/1/

~(/---- (l/V) ~

aijdV (1) v where aij is the actual stress state at each material point of the body and V is its volume. Equation 1 implies that we are introducing a fictitious homogeneous stress state representative of the whole d

X

FIG. 3--Pi, Qi, and ei measured by load cells around the sample in the Cambridge simple shear apparatus,

....x~X

strain

Y~ ' ~

0"I

~

6

0

increment

f

stress x

-O.04

-0.02

-20

0.02

~

0.04

-4(

stress increment /

X

......

-60 Xl

x~

X

FIG. 4--Left: definition sketches for angles ~, ~, and X. Right: variations of angles ~, ~, and X with ctfor cyclic simple shear test on dense sand.

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214

GEOTECHNICAL TESTING JOURNAL

body, and thus that the body may be identified with an infinitesimal element. The concept of an average stress tensor is often used in mechanics, particularly when we are interested in the average behavior of a certain region consisting of several materials or of grains and pores [5, 6]. Equation 1 may be transformed as follows: aij : ~ik Ctjk : Xi,k (rkj

(2)

and thus ~ # = (l/V) I

(3)

Xi'ktrkjdV V

where the comma indicates partial differentiation. If the stress state akj satisfies the equilibrium equations akj.j-----O

(4)

then Eq 3 becomes ~/:

(l/V) ~ d

(Xiakj),kdV

(5)

v

Using Gauss's divergence theorem, Eq 5 becomes ~(/=(I/V) I

xiCrkjnkdS:(1/V)f S

pjxidS

(6)

s

Equation 6 shows that to evaluate the components of ~# it is sufficient to integrate the products pj xi over the boundary of the region V. For a discrete distribution of boundary forces Tj over the surface (see Fig. 5), Eq 6 becomes n

~:(]/V)

E

m:l

(Tjmxi m)

(7)

Equation 1 is valid for any shape of the region V, and may thus be directly applied to the determination of the stress state in the soil sample sheared in the simple shear apparatus, or to any part of that sample, for example, the central zone. It is apparent that Eq 7 takes into account the points of action of the boundary forces, and thus makes use of the measured eccentricities. An essential limitation in the application of the concept of the average stress tensor ~O" to the simple shear apparatus is the requirement that the equilibrium equations (Eq 4) and thus the

equilibrium of all boundary forces and their moments must be satisfied. Because there are errors in the measurements of Pi, Qi, and el, and because there may be friction forces at the vertical sides of the sample, the set of measured boundary forces does not in general satisfy horizontal, vertical, or moment equilibrium. Cole [7] was aware of this and proposed that the forces Fx and Fy required to give equilibrium in the horizontal and vertical directions, respectively, should be calculated. Cole assumed that these were the result only of friction on the vertical side walls. He then suggested t h a t F x andFy should be distributed over Zones 1, 2, and 3 in proportion to the magnitude of the sum of the top and bottom normal forces Pi acting on the boundaries of each region. However, making the assumption that the out-of-balance forces Fx and Fy are the result only of frictional forces on the side walls requires that their directions should be opposite to those of shear and vertical displacement of the sample (as shown in Fig. 6a, where dilation of the material is assumed). Fig. 6b, taken from Stroud [8], shows that this is not always correct. In some tests the direction of F x or Fy or both will assist rather than resist the deformation of the material. Further, the assumption that there are tangential forces Fx and Fy in the plane of shear, and thus tangential stresses there, violates our first postulate, since for symmetrical planestrain shear the only admissible stress component acting on the z face of an infinitesimal element is the normal stress az. It would be possible to apply Eq 7 to any equilibrated set of boundary forces, including the balancing forces F x and Fy acting in the plane of shear. However, the point of action of these forces is not known, and it is therefore necessary to correct the measured forces P / a n d Qi so that they satisfy equilibrium. The proposed procedure for determining the stress state in the central zone of the sample is then as follows: 1. Correct Pi and Qi to satisfy the equilibrium equations. Correct e i to satisfy moment equilibrium.

Yl/"................ i-i a.

I0~ N b.

0

0

000 0

0

0

0

• 00

•

•

0 I •

0

0

•

•

I

o •

•

o •

@

@

.I~

c~

0

O O

-IC

//•

oR . Fy

~"surface S -2C

volu/me V

L

-3C

x2

FIG. 5--General distribution of jbrces T 1 and T2 around surface S of volume V.

FIG. 6--(a) Positive directions of out-of-balancefrictional forces F x and Fy expected for dilating sample. (b) Variation of measured forces F x and Fy with shear strain a in test on dense sand (after Stroud [8]).

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215

WOOD ET AL ON SIMPLE SHEAR APPARATUS

2. Determine the forces acting between the central zone and Zones 1 and 3 and determine also their eccentricities using moment equilibrium for regions 1 and 3 (Fig. 7a). 3. Determine the sum in Eq 7.

27

1 t

\

Assuming that the forces acting on the central zone are equilibrated, the explicit formulas for the stress components oy, a~ ~ y = ~ x are: Oy = ( l l a b f ) ( - - P 2 a + QL e . cos 0 -- PLe" sin 0 -- QR h . cos O + P R h - s i n 0 ) Ox = (1/abO(Q2d + Q L f " sin 0 + P L f " cos 0 QR g . Sin O -- PR g . Cos O -- Q s c ) -

-

FIG. 8--Shift of Mohr's circle of stress parallel to r axis to ensure Zyx = Txy •

(8)

7yx ---- rxy = (1/abO(--P2 d + Q L f " cos 0 -- P L f " sin 0 -- Q R g ' C o s O + P g g ' s i n O + P s c )

where a, b, c, d, e, f, and g are distances as shown in Fig. 7b and the directions of Pi and Qi are also as shown in Fig. 7b. It should be noted that Eqs 7 and 8 give positive tensile stresses. Equation t can only be expressed in the forms given in Eqs 6 or 7 if the external forces satisfy the equations of equilibrium. However, whereas force equilibrium must be satisfied, nonequilibrium of moments can be accommodated. This would lead to an asymmetric stress tensor, with 7yx ~ 7xy, but it is possible to make this tensor symmetric by writing

Correction of Measured Forces and Eccentricities Two methods have been used for correcting the measured forces and eccentricities. In Method 1 equilibrium of forces alone is achieved and Eq 10 is used to obtain a symmetric stress tensor. In Method 2 an attempt is made to satisfy also equilibrium of moments so that Eq 7 may be used. Equilibrium of forces (Method 1)

The forces measured on the boundary of the sample shown in Fig. 3 will not generally be in equilibrium. Adding the forces in the x a n d y directions, we find that in order to satisfy equilibrium, outof-balance forces F x and Fy must be supplied where:

(~

F# : (t/2V) t

(9)

(xipJ + xJ pi)dS

Os

Fx = (Q1) + (Q2) + (Q3) + (-Q4) + (-Qs) + ( - Q O + (P7 cos 0) -}- (Q7 sin 0) + ( - P 8 cos 0) + ( - Q 8 sin 0)

(11)

or

Fy = ( - e l )

n

~0 = (1/2v)

Z (T/"xi m + T / " x F )

(lo)

m=l

This is equivalent to shifting the Mohr circle of stress along the r axis (Fig. 8). In order to be able to apply Eq 10, it is still necessary to assign definite points of action to the forces PL, QL, PR, and QR (Fig. 7). While this proposed procedure is based on a well-defined concept of average stress tensor, it leaves open the question of how the measured forces Pi and Qi and ei should be corrected in order to satisfy the equations of equilibrium. In the next section we will discuss some possible methods of making these corrections. Their influence on the magnitudes of the components of ~0 will also be considered.

+ ( - P 2 ) + ( - P 3 ) + (P4) + (es) + (P6)

+ (Q7 cos0) + ( - P 7 sin 0) + ( - Q 8 cosO) + ( P a sin 0)

(12)

If it is supposed that F x and Fy do not arise from friction at the sides of the sample (which, it was shown above, could lead to curious conclusions concerning the direction of these forces), but rather from errors in the measured forces, then it is necessary to distribute the errors among the measured forces in order that equilibrium may be achieved. The simplest method of distributing the errors is to assume that the measured forces are all in error by the same proportion. A typical member of the set of horizontal components of force, each enclosed in parentheses in Eq 11, would b e f x i and the corrected component f xi*:

IAiI

~_I IP2 o2

fxi*

y[I -

-

_.9.J

= fx i -- F x ~

(13)

flail and similarly for the typical vertical component parentheses in Eq 12:

°J

fy,* = A; - Fy ~

Ifyil

rtfwi[

I - -

enclosed in

(14)

b~l

a.

FIG. 7--(a) Normal and shear forces and eccentricities for the central zone (Zone 2). (b) Coordinates of points of action of forces on the central zone.

fYi'

Equilibrium of forces and moments (Method 2)

Although the distribution of errors of force is fairly uncontroversial, there are rather more options for methods of distributing errors of moments. The moment of a force obviously depends on its

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216

GEOTECHNICAL TESTING J O U R N A L

own magnitude as well as on its line of action. Either of these parameters may be in error and produce an error in the resultant moment. Further, the moment of any particular force (though not the moment of the whole set of forces in force equilibrium) will depend on the axis about which the moments are taken. Any method that is considered for balancing the error in moments must not depend on the choice of this axis. For example, to apply a correction to each moment according to its magnitude (as was done with the forces in Eqs 13 and 14 would not be permissible. One approach could be to recognize that both measured forces and measured eccentricities were in error and to change all the eccentricities by a fixed amount and all the forces by a fixed proportion, having made a preliminary decision as to the way in which the error should be assigned to magnitude and to line of action of the force. Provided the forces were all altered by the same proportion, these changes would not affect the force equilibrium. A simpler approach is adopted here to demonstrate the effect of establishing moment equilibrium: each measured eccentricity is changed by the same amount, so that the measured normal forces are all simply displaced clockwise (or counterclockwise) around the boundary of the apparatus in order to give a resultant moment of

wise) then the correction to each eccentricity (also measured clockwise, as in Fig. 3) is Ae =

-

(AM/EPi)

-

(15)

where Pi are the compressive normal forces measured with the load cells.

Comparison of Results Figures 9-13 show the results of a single monotonic loading test conducted by Budhu [9] at a constant average vertical stress of 98 kPa on dry dense 14/25 Leighton Buzzard sand with a constant rate of strain. The average grain diameter of the sand was 1 mm; the minimum void ratio, obtained by pouring, was 0.51 and the maximum void ratio, also obtained by pouring, was 0.79. The test results are displayed in five different forms, and results obtained by Methods 1 and 2 as described above are compared. The results are plotted against a parameter ~ which is a measure of shear strain and is equal to the horizontal displacement of the top of the sample relative to the bottom, divided by the height of the sample. The variation of the calculated stresses ay, Zyx, and ax with shear strain ct is shown in Figs. 9-11. Note that although the average vertical stress was maintained constant at 98 kPa, the computed

zero.

If the out-of-balance moment is AM (measured positive clock-

1.5 -kgf/cm 2

~y

0

0

0

0

0 0 0 0 0 0

•

00

O0

0

0

0

0

0

0

@0

Ill

•

@

II

•

•

0

0

0

1,0c

00

O

O

0

0@

oM1

0.5

oM2

Co

oS5

o11

o~5

,~ o12

o125

oI~-

FIG. 9-- Variation of oy. with t~computed by Method 1 (force equilibrium) and Method 2 (force and moment equilibrium). 1 kgf / c m 2 = 98.06 kPa.

1.5 -kgflcm 2

10-

O

•

•

0

•

0

•

0

II •

0.5

0

• tl

oM1 ql

O,

eM2 I

0

005

I

Orl

I

0.15

I

o~

0.2

I

0.25

++~

0.3

FIG. 10-- Variation of ryx with shear strain a computed by Methods 1 and2. 1 kgf/cm 2 = 98.06 kPa.

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WOOD ET AL ON SIMPLE SHEAR APPARATUS

217

2.0 -kgf/crn 2 ~x

ii10 I I 0 @

1-5

O0

0

0000

O0

0

0

0

0

0

0

0 0

0

•

0

0

1.0

0

oM1 oM2

0.5

CO

I

0!05

O-15

0!I

c