16-undrained shear strengths of anisotropically consolidated clays, brinch hansen

UNDRAINED SHEAR STRENGTHS OF ANISOTROPICALLY CONSOLIDATED CLAYS

by J. BRINCH HANSEN Chief Engineer, Chvistiani and Nielsen, Copenhagen

and R. E. GIBSON Imperial College, London

men saturated non-fissured clays are brought to a state of failure under conditions of no water content change, it is known that they exhibit an angle of shearing resistance + (with respect to applied stresses) equal to zero. This fact forms the basis of the +=O analysis of stability which is now widely used in design problems, such as bearing capacity and earth pressure, where the most critical condition generally occurs immediately after construction before the clay has had opportunity to change its water content appreciably.

In its application the +=O analysis is simple, yet there are many points which require investigation and clarification before all the limitations and implicit errors involved can be fully understood.

These questions have recently been considered by Skempton (1948c) who has shown that in order to resolve the problem it is necessary first to determine the changes of effective stress in a saturated clay when strained under conditions of no volume change and secondly, to derive expressions for the shear strength of a clay when tested and when sheared in the ground, in terms of its fundamental properties.

The calculation of the effective stresses has been made possible by the use of a para- meter known as the compressibility ratio A. This was introduced by Skempton (1948a) and, independently, by Mandel ,(1948). The theoretical treatment involved, in what may be called the “ X theory,” is perhaps not yet fully satisfactory, but it does enable, for the first time, a solution to be obtained to many of the problems implied in the +=O analysis and it will therefore be adopted in this paper.

In the particular case where the clay has been consolidated under an isotropic stress system (the same effective stress in all directions) Skempton has developed an expression for the shear strength as measured in the unconfined compression test and has shown that it approximates closely to the strength mobilized in the ground, when shearing takes place at constant volume. That is to say, he has shown that under these conditions, the &O analysis is a reliable procedure.

In nature, however, clays are not isotropically consolidated except in special circum- s’ aces, and therefore it is essential to investigate the more general case where the clay stratum has been consolidated, and has attained conditions of equilibrium, under vertical and horizontal pressures which are appreciably different from each other ; that is, the case of anisotropic consolidation.

Consideration is first given to the ratio of these pressures in,nature. Then follows a brief discussion of the compressibility ratio h. The fundamental equations concerning the effective stresses and their changes are next dealt with, and an expression is then derived for the resistance to shear mobilized in the ground, along any given slip plane. After this, expressions are derived for the strengths as determined from the vane test and the unconfined compression test, together with certain other laboratory tests, and finally, the results of these-tests are compared with the shear strength in the ground by means of two numerical examples.

It must be emphasized that the present treatment is to be regarded as approximate since its basis, the h theory, is itself not yet fully developed. Nevertheless, it is felt that the results arrived at in the paper represent a further step in our understanding of the behaviour of clays.

190 J. BRINCH HANSEN AND R. E. GIBSON

CONSOLIDATION IN NATURE During the consolidation process in a natural clay stratum the horizontal strains are

zero, while the vertical strain gradually attains a comparatively large value. If, when the clay is fully consolidated the vertical effective pressure at an arbitrary depth is p (equal to the total overburden minus the hydrostatic uplift), and the horizontal effective pressures at the same depth are ph, then we may write :

ph=K p . . . (1) where K is the coefficient of earth pressure at rest. There is a certain amount of experimental evidence, derived from laboratory tests, which indicates that for normally consolidated clays (i.e., clays in which the effective pressures have never exceeded p) K lies between ~0.50 and O-75. A few typical values are quoted below :

Soil Liquid Limit

K Reference

- Blue Marine clay . . . . . . . 58 0.75 Terzaghi (1925) Wiener Tegel . . . . . . . 47 0.67 Hvorslev (I 937) Blue” Leaf” clay ._. . . . . . . . . . 43 0.50 Tschebotarioff (1949) Silty Red clay . . . . . 30 0.50 ,, .I

From this evidence we may expect that in normally consolidated clay strata in nature the magnitude of K will be of the same order as the values quoted above. It is possible that K is a function of more fundamental properties* of the clay, but this question can hardly be solved on the basis of the evidence at present available.

In this paper we shall concern ourselves exclusively with and shall assume that K is a constant for any given clay.

THE COMPRESSIBILITY RATIOS

normally consolidated clays

In considering the effective stresses set up in a saturated clay by a system of applied stresses, it is essential to allow for the fact that the behaviour of a clay is, in general, appreciably different in the three cases represented by compression, expansion and recompression. This difference may be expressed quantitatively in terms of d compressibility ratio X and a recompressibility ratio &. These parameters are defined by the equations :

. . . (2)

cs A,= c

CT . . . (3) where C, is the expansibility, i.e., the volume increase per unit volume per unit decrease in an effective stress, and C, is the compressibility, i.e., the volume decrease per unit volume per unit increase in an effective stress. C, is the recompressibility, defined as is C,, but for the case of recompression.

For normally consolidated clays characterized by point A in Fig. 1, h will generally be much nearer zero than unity, since an initially increasing effective stress will cause a consolidation from A to D, while a decreasing effective stress will bring about a much smaller expansion from A to B.

The recompressibility ratio h,, on the other hand, will be greater than h since, as seen in Fig. 1, reconsolidation from B to A, is more nearly of the same order of magnitude as the expansion from B to C.

*e.g., K may be expressed as h/l -J+ where b is the Poisson’s Ratio of the clay structure if linear stress-strain relations are assumed.

STRENGTHS OF ANISOTROPICALLY CONSOLIDATED CLAYS 191

\

0 I

I I 1 I I 1 I 4 I 1 I I 1 I I I I 1

I I EFFEtTlVL IllE55

FIG. 1 RELATION BETWEEN EFFECTIVE STRESS AND FIG.~ MOHRCIRCLEOFEFFECTNESTRBSSES

STRAIN AT FAILURE

Experimental data on the compressibility ratio is scanty, but three values of A, as determined from pore-water pressure measurements in undrained triaxial tests, are given in the table below. Concerning the recompressibility ratio &, which may be found by means of similar tests with over-consolidatedclays, still less evidence is available, but two experimental values are given in the table.

Reference for experimental

data Clay

A or Ar at failure

~___ Taylor (1948) Marine clay, Maine, normally consolidated. Undisturbed (high x =0.03

sensitivity). Rendulic (1937) Wiener Tegel, normally consolidated. Remoulded. x =0.3 Taylor (1944) Boston clay, normally consolidated. Remoulded. x =0.2 Taylor ( 1944) Boston clay, lightlv over-consolidated. Remoulded. x,=0+5 Taylor (1944j 1 Boston clay, lightli over-consolidated. Undisturbed. 1 x;=o.4

THE EFFECTIVE STRESSES The deformations and conditions of failure of clays are determined, almost exclusively,

by the effective stresses between the grains, whereas they are practically independent of the pressure in the pore water. This has been proved experimentally by Rendulic (1937) and Hvorslev (1937).

Laboratory tests carried out on two widely different clays, Wiener Tegel and Klein Belt Ton, in a remoulded and reconsolidated state, led Hvorslev to propose the following expression for the shear strength along any plane in the clay :

s=ce+(u* 4) i!an +f=ce+oh fan +f . . . (4) where c, and &, the true cohesion and the true angle of internal friction, are, for any given clay, functions of the water content alone ; on being the total normal pressure, oh the normal effective pressure and u the pore water pressure acting on the shear plane at failure.

Another fact which emerged from Hvorslev’s experimental investigation was that c, could be expressed as KP,, K being a fundamental constant for any given clay, and 9, being the so-called “ equivalent consolidation pressure ” which, with sufficient exactness for the present analysis, may be assumed equal to the greatest effective pressure to which the clay has been subjected during its history. In our case of normally consolidated clays, PO is equal to 9. Equation 4 can then be written :

S=K p-+dn hW2 +f . . . (5)


_J HORIZON

DIRECTION OF

It is convenient to represent this criterion of failure by means of a Mohr envelope as shown in Fig. 2. When c is the radius of the Mohr stress circle, it will be seen from Fig. 2 that the principal effective stresses at failure are :

C--K p cos $& sin #f fc

SLIDING

SURFACE CT’*=

C---K p co.? #Jf _-c

sin I$f . . . (6)

The shear stress s acting on the failure plane is given by

s=c cos \& . . . (7) Although s is, by definition, the maxi- mum resistance to sliding that the clay is capable of mobilizing along the failure plane, it will be more convenient to characterize this “ strength” by c. This follows from the demonstration by Skempton (194Sc) that in the special case of isotropic consolidation the results of the “ +=O ” analysis are con-

FIG. 3. DIAGRAM SHOWING ORIENTATION OF PRINCIPAL sistent with the laboratory tests when

AXES OF STRESS AND SLIP PLANE AT FAILURE not s but c is used as the shear strength mobilized on the slip surface.

In this paper we shall, for the general case of anisotropic consolidation, derive expressions for c/p, as developed by slides in nature and as measured in various tests, in terms of K, &, K, h and h,, these parameters being regarded as the fundamental constants of the clay.

First, however, we must investigate the laws governing the changes of the effective stresses brought about by a system of applied pressures in the case when no water content change is permitted to occur.

We consider the clay in an initial state with the principal effective stresses alI, u’% and u’*. In order to bring about a state of failure these effective stresses are augmented by A ~‘1, A a’) and A da. Further, we assume that the directions of the principal axes are unchanged, although the principal stresses may have changed places with one another. The condition of no volume change may then be expressed as follows :

A V=C1 A dl+Cz A d2+CI A dg=O

In this expression C must correspond to A a’, i.e., C should be put equal to G, C, or C,, respectively, when A U’ implies an expansion, an initial compression, or a recompression, respectively. In any given case the equation 8 can be divided by either C, or Ccr, making one or two coefficients equal to unity and the others equal to X or X,.

In the special case of plane strain it can be demonstrated, as Skempton has shown, that the condition fp=O, implies A ds=O, i.e., unaltered effective intermediate principal stress. Therefore, in this case the corresponding term in equation 8 should simply be left out.

STRENGTHS OF ANISOTROPICAIiLY CONSOLIDATED CLAYS 193

In the more general case, where the initial principal axes do not coincide with those occur- ring at failure, it is not self-evident along which axes the stress increments to be inserted in equation8 shouldbe measured. We shall be concerned with this case in the following section.

SLIDES IN NATURE

The first case to be analysed is that of a slide in nature, in which we must determine the shearing resistance mobilized along a plane making an angle b with the horizon (Fig. 3). The initial state of stress in the ground may be represented by a

P Mohr circle (Fig. 4) with the radius :

co=*@ (I-K) . . . (9) the principal effective stresses being P vertical and Kp horizontal. Another representation of this stress system is shown in Fig. 5 (curve I), where the normal stress in an arbitrary direction is

FIG. 4. MOHR CIRCLE OF INITIAL STATE OF measured as the length of the corresponding radius EFFECTIVE STRESS IN GROUND vector. Let us consider a change in this state of

stress in the stratum which causes a failure plane to develop, oriented at some point, at.an angle /3 to the horizon. The minor principal stress at failure, it may be seen from Fig. 2, makes an angle :

a=4S0+& 12 . . * w

with the direction of the failure plane.

In order to find the effective stresses at failure we must determine the changes of effective stress necessary to bring this condition about, and further, these changes must satisfy the condition of no volume change (equation 8). This equation, however, only applies, as mentioned previously, when the changes of effective stress d u’~, da’,, and A a’,areprincipal stresses, and it is necessary to determine the orientation of the planes on which this condition is satisfied.

Initially the normal effective stress in a direction making an angle v with the vertical is from the Mohr circle (Fig. 4) found to be :

uv~=p-co+co cos 2v . . . (11)

a,~=p-+Y-c, cos 2v . . . (12)

The effective stresses at failure may be represented by another Mohr circle (Fig. 2) with the unknown radius c. From Fig. 3 it will be seen that the major principal stress at failure makes an angle (a-p) with the vertical. Another representation of this stress system is given by the curve II in Fig. 5.

The final normal effective stress in a direction making an angle v with the vertical is, from the Mohr circle (Fig. 2) found to be :

, _ U" I-

C--K p LX%- +f sin C#f

+c cOS2 (v+j3-a) . . .

, C-K p COS +f U” a= -c c0S 2 (v+/3-a) . . .

sin $f (14)

N


the changes of normal effective stress on these planes being :

d (1,‘l=(u,‘1-u,~)

A u,‘5=(uv’l-u”~)

The condition of no volume change requires, in the case of plane strain deformation to which slides in nature generally approximate, as previously indicated, that :

(%‘,--a,~)+~ (%‘r%$ =o . . . WI where w is that particular value of Y between 0” and 90” for which these effective stress changes are principal changes. This condition is satisfied if :

ddu,‘, o ~ = dv

dA ’ 0” a

dV =o . . . (16)

These equations both lead to : C sin 2~ -. = ______

Sit2 2(w+p-a) co . . . (17)

Using this equation to eliminate v from equations 10, 11, 13 and 14, and inserting the resulting equations into condition 15, we are led to :

-LK cos fjf+$ (1-t K) silz C&-sin #f 1d ,p

( l+h)[(i)l-2j % c0S 2 (a-8)+6)‘]*

. . . (18) The positive value of the square root in this equation must be taken, this being implied by the condition that :

Au’,1 > Au,‘,

From this equation the “ strength ” c on the sliding surface (the actual shear stress on the sliding surface being c cos ~#t) may be found, c, being determined from equation 9.

The above equation shows that a clay, which has been normally and anisotropically consolidated in nature, has a strength dependent not only on the consolidation pressure p, but also on the orientation /? of the failure plane.

Three particular cases of failure in nature will now be discussed, numerical examples of which are considered later.

(1) Active E&h Pressure. When an active earth pressure acts against a smooth vertical wall the major principal stress must be vertical. Thus from Fig. 3 it is seen that a-_S=O”, whence from equation 18 :

6 -z P . . . (19)

(2) Passive Earth Pressure. In this case the major principal stress is horizontal and hence u-/9&W. Equation 18 then yields :

CP -= P . . . (20)


(3) Failure on a Horizontal Plane. A further case of some general importance is that in which failure occurs on a horizontal plane. On putting /3=0’ in equation 18, the corresponding value of c @ may be found from :

;=K cos +f+& (1fK) s~n~*-s~n~(~)[(~)‘+2~~sinC+(~)llt

. . . (21) in any given case (see numerical examples later).

VAWE TESTS A recently developed method of measuring the strength of clay is the vane test which

is carried out in situ. This test, which has been described by Carlson (1948) and Skempton (1948b), mobilizes the shear strength of the clay on a vertical.cylindrical* surface (height H) and on the horizontal end surfaces (diameter D). It will be assumed in the following analysis that the vane test is carried out at a point sufficiently far below the bottom of the borehole for the initial state of stress to have remained substantially unaltered by the presence of the boring. By this test an average shear strength sV=cV cos & is measured but, as follows from the preceding discussion, the strengths c, and c, mobilized on the cylindrical and end-, surfaces respectively, will not be equal.

L-v=&

In order to evaluate c, we use the fact that in the undisturbed clay all the effective stresses in the horizontal plane are Kp. At failure we have the principal stresses (Fig. 2) :

It may easily be shown that : ,

=c, F . . . (22)

I./*= Cc-$ COS +l_c

sin+f ’

The condition of no volume change requires :

(u’x-Kp)+h (u’,-Kp)=O

Equations 23 and 24 give :

CC K cos &+K sin 41 -_= P

. . . (24)

. . . w

The strength c, may clearly be found from equation 21.

In the above treatment a condition of plane strain is assumed. Although in the neighbourhood of the vane the conditions of strain are somewhat complex, it is felt, nevertheless,

*We shall assume that failure does, in fact, take place on a cylindrical surface although, as yet, this has not been confirmed experimentally.


that only coamparatively slight error will result in practice from making the above assump- tion.

The factor F in equation 22 is generally very close to unity. With vanes as used at present, H is usually equal to 1.5 D, and it can then be shown that F may assume values between 1 and about 1.03. Thus it is convenient to write the average vane strength in the form :

c, K cos &+K sin & F -r p 1+ &;

( ) sin 4f . . . (26)

where F is a factor not differing greatly from unity, and which, in any particular case, may be determined numerically from the preceding equations.

EXTRACTION OF SAMPLES Having found the stresses causing failure in the ground, and having analysed the vane

test, we shall now develop expressions for the strength as determined from laboratory tests. As a preliminary to this, however, we must consider the question of sampling.

In a normally consolidated clay stratum the effective stresses are p vertical and Kp horizontal. Prior to taking a sample a borehole is made, and in the immediate region of the bottom of the borehole a complex state of stress exists. It is usual to extract a cylindrical sample from the clay within a depth below the bottom of the borehole varying from zero to a few borehole diameters. In this short distance the changes of stress are considerable.

In order to investigate this problem we shall assume that the total vertical and total horizontal stresses acting at an arbitrary depth below the bottom of the borehole are u, and a=, and the corresponding effective stresses are 0: and a’=. The condition of no volume change leads to the equation :

2 (a’=- KP)+X (~'z-P)=O . . . cm

In forming this expression it has been assumed that (U’X -Kp)>O and (u’~--$) ~0 . . . (28)

FIG. 5. POLAR DIAGRAM SHOWING INITIAL AND FINAL STRESSES IN GROUND

FIG. 6. POLAR DIAGRAM SHOWING CHANGES OF EFFECTIVE STRESSES

DURING SAMPLING


Now the limiting values of u, and uX and (~~-0~) are given by the following cases : (a) Immediately below the bottom of the borehole a state of passive failure is incipient. (b) At an appreciable depth below the bottom of the borehole the effective stresses

remain unaltered at their original values of p and Kp (by St. Venant’s principle), and it can be shown that the conditions stated in equation 28 hold good for these limiting conditions and all intermediate cases. From equation 27 it may readily be found that :

A+2 ufz=P - ( “) A+2 + A&2 b-ux)

dx =p AS2 ( “) A$2 (ur-ux) 1 . . . hi -

(29)

Fig. 6 shows the effective stresses in a vertical plane. Curve I represents the initial effective stresses and Curve II the effective stresses when the borehole has been made (equation 29), using a value of (a,--a,) intermediate between the two limits stated above.

After a sample has been extracted from the bottom of the borehole the applied stresses acting on it are all zero. These changes of applied stress give rise to changes in effective stress and in pore-water pressure. The change of effective stress to a value a”‘, in all dirac- tions, results in a pore-water tension -u', being set up, since the total stress on the sample is zero. It will be assumed that a capillary tension can be mobilized on the surface of the sample sufficient to maintain this pore-water tension. Volume changes due to the absorption of free surface water or the mobilization of capillary tension will be assumed negligible in the following analysis.

There are two possible cases to be considered, depending upon whether :

(i) uZ-uX <O or (ii) ~~--a~>0 . . . (W

The conditions of no volume change are, in these two cases : (i) (u’~ --a’,) +2& (u’~ -u’J =0 . . . (31)

or (ii) A (u’,-u’J+2 (u’~-u’~)=O . . . W)

From equation 29 it therefore follows that :

(33)

(34)

These are the general expressions for the effective stress in a sample. It is of interest, however, to derive the limiting theoretical condition of case (i) which corresponds to incipient passive failure in the clay at the bottom of the borehole. Under this condition :

u’,= C--P cm +t _-c sin cpf

ufr = C-K@ COS +f

sin +f +c . . . (35)

Equation 33 then becomes :

(ia) u’o F=

(h+2K) (1 f is sin +f)-4 K COS +f ‘s

r r . . . (36) (/\+2) + (2--h) sin +f


For the other limiting condition of no stress change in the clay prior to sampling we have, as before :

(iib) Zj = (9?$) . . . These two equations, 36 and 37, represent the extreme theoretical limits of the effective

stizsses in a sample taken from a borehole in a normally consolidated clay. In Fig. 6, Curve III shows the effective stress in a sample for a particular case intermediate between these limits.

THE UNCONFINED &MPRESSION TEST

The most widely used undrained test is the unconfined compression test in which a cylindrical sample, under no radial restraint, is compressed axially to failure. Usually the axis of the sample in the test is the same as the vertical direction of the sample in its natural position.

At failure the principal effective stresses are :

O’a== &l--K p cos $r

sin #f +c”

These stresses are represented by Curve IV, in Fig. 6. In the great majority of practical cases we have expansion radially and recompression axially, and then the condition of no volume change yields :

(u’1-u’0)+2 x, (ufJ-O’,,)=o . . . , (39) Inserting equation 38 into 39 we find the following expression for half the unconfined compression strength :

,

cu K cos +f+ Y-0 sin +r

p = 1+ l-2!* .sin&

( )

. .

1+2&

(40)

The value of u’~ /p to be used in this expression theoretically lies between the limits represented by equations 36 and 37. In practice, however, some disturbance will take place on sampling, which will cause a small reduction in strength below the theoretical values.

Thus, with typical good sampling conditions, corresponding to values of (7 )inter-

mediate between the two limits and probably closer to condition (b), the mkasnred strength may well lie in the neighbourhood of the. result given by condition (a) ; although this does not, of course, imply that passive failure has occurred prior to sampling. In that case the strength would be very considerably less than the corresponding theoretical expression.

The immediate triaxial test gives, as one would expect, the same value of c /$J as the unconfined compression test, since an addition of an all-round pressure leaves the effective stresses unchanged (Skempton, 1948a).

QUICK SHEAR Box TESTS

Occasionally the quick box test is used, in which a sample is cut from an undisturbed core so as to fit tightly into the box. After the sample has been put into the box the shear test is carried out immediately so as not to allow time for drainage to take place. In some of the shearing devices it is necessary to apply a vertical load to the sample in order to

STRENGTHS OF ANI$OTROPICALLY CONSOLIDATED CLAYS 199

f $. a :

c--i


maintain stability, and this often results in an appreciable water content change by the time failure conditions have been attained.

This test may be looked upon as the plane strain analogue of the unconfined compression test. At failure the intermediate effective principal stress is unaltered, inrhile the other principal effective stresses are :

U’I= cb-Kp cos C$f

sin +t +cb

IS’*= C&-K@ cos +f

sin +f -cb . . .

AS we have expansion and recompression only, the condition of no volume change is :

(dl-do)+hr (u’3-u’o)=o . . . (42) Inserting equations 41 into 42 we find the following expression for cb (the measured

shear stress is, of course, sb=cb cos #f) :

. . . (43)

This equation should be used together with equations 33 or 34 in order to calculate cb. An uncertainty is associated with this expression, in practice, arising from the difficulty of evaluating do/p with any exactness. As has been mentioned before, the measured strength, if no water content change takes place during the test, would probably approximate to that given by condition (a) (see equation 36).

CONSOLIDATED QUICK TRIAXIAL TESTS

Sometimes an attempt is made to restore the effective stresses in the sample to their initial values in situ by reconsolidating the sample under an applied pressure p before bringing it to failure.

In the consolidated quick triaxial test the sample is first reconsolidated isotropically under a pressure p and then compressed axially to failure under conditions of no water content change. At failure the stresses are :

u‘1= Cm---K@ cm +f

sin +f +cm

u’*=u’*= Cm-Kp COS +f -cc, . . .

sin +f

The condition of no volume change gives in this case :

(u’l-p)+22x (u’a-P)=O . . . (45)

Inserting equations 44 into 45 we find the following expression for half the compression strength :

K cos r$f + Sin +f

. (46)

16-undrained shear strengths of anisotropically consolidated clays, brinch hansen

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