The Shear Strength of Unsaturated Soils 1960

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Canadian

Revue

Geotechnical

Journal

Piihlishc~rl

v

THE N A T I O N A L R E S E A R C H C O U N C I L O F C A N A D A

canadienne de

I

d

PiibliP par

L E

C O N S E I L

N A T I O N A L E

R E C H E R C H E S D U

C A N A D A

V olume 15 N u m b e r

3

August 1978 Vol um e 15 nu me so aoGt 1978

The shear strength

of

unsaturated soils

D .

G . F R E D L U N D

Universi ty ofSoskarc he~ vnn, nskatoon, Sask., Canada S7N

OW0

ND

N .

R. M ORGENSTERN

Un iversit y ofAlbe rtrr, Edm onron, A lro., Crrnmrkr

T6G G7

ND

R . A. WIDGER

Department of Highways, Prince Albert , Sask., Can ada S6V5S4

Received June 1, 1977

Accepted Febl-t~a ry , 1978

The shear strength of an unsaturated soil is written in terms of two independent stress state

variables. One form of the shear streng th equation is

=

C

+

( u u , ) tan

4 +

1 1 ,

-

i, tan

+

The transition from a saturated soil to an unsatu rated soil is readily visible.

A

second form of

the shear strength equation is

= C +

( u

( , ) tan

4

+

1 1 ,

u, tan b

Here the independent roles of chan ges in total stress

u

and changes in p ore-water pressure

u ,

are easily visualized.

Published research literature provides limited data. However, the data substantiate that the

shear strength can be described by a planar su ~f ac ef the forms proposed. A procedure is also

outlined to evalua te the pertinent sh ear strength param eters from laboratory test results.

La resistance au cisaillement d un sol non sature est ecrite en fonction de deux variables d eta t

de contrainte indipend antes. Une forme de I iquation de resistance au cisaillement est

=

c

+

( u

- u, tan

4

+

( u ,

-

u,)

tan

+

Le passage de l etat sature 1 Ctat non sature est evident. Une deuxieme forme de cette

equation est

T =

C

+

T

,

tan

+'

+

I ( ,

i,

tan

b

Dans ce c as, les influences reciproques de s changements dan s la contrainte totaleu et des

chang emen ts dan s la pression interstitielle~ i ont facilement observables.

Peu de donn ees sont disponibles dans la l i t terature. Cependant, ce s donne es etablissent que la

resistance au cisaillement peut t tr e decrite par une surface plane ayant les formes propos tes. Une

procedure est egalement decrite pour evaluer les parametres pertinents de la resistance au

cisaillement en partant d e resultats d essais d e laboratoire.

[Traduit par la revue]

Can. Geotech. J., 15,313-321 (1978)



314 C A N .

G E O T E CH .

J.

V O L . 15.

1978

Introduction

During the past two decades there has been an

increasing use of two independent stress variables

to describe the behavior of unsaturated soils

(Coleman 1962; Bishop and Blight 19 63; Burland

196.5; Aitchison 1967; Matyas and Radhakrishna

1968; Barden et al. 1969; Brackley 1971; Fredlund

1974; Fredlund and Morgens tern 1976, 197 7) .

Most consideration has been given to describing

the volume change behavior ( an d suitable constitu-

tive relations) for unsaturated soils. Several in-

vestigations have been m ade into th e shear strength

characterization of unsaturated soils (Bishop

et al.

196 0; Massachusetts In stitute of T echnology

(M IT ) 1963; Sridharan 196 8; Maranha das Neves

197 1 ) however, none has proven completely

successful.

Fredlund and Morgenstern (1977) showed from

a stress field analysis tha t an y two of th ree possible

stress variables can be used to define the stress

state in an unsaturated soil. Possible combinations

are: ( 1 ) ( a u,) and (u:, u,,. ), ( 2 )

cr

u,,.)

and (u,, LL,,.), nd

3 )

(C LL,, ) nd (U ~i , ) ,

where

I=

total normal stress, u,,

=

pore air pres-

sure, and u,,. = pore-water pressure. Null experi-

ments (i.e. A a = Au,, = Au,,.) supported the pro-

posed theoretical stress state variables.

Th e objective of this pape r is to present the

shear strength of an unsaturated soil in terms of

two independent stress state variables. The shear

strength, using two possible combinations of stress

state variables, is presented and the relationship

between the two cases is shown. The first stress

state variables used are (a u,,. ) and (u, u , ~ ) .

The advantage of this combination of variables is

that it provides a readily visualized transition from

the unsaturated to the saturated case. The dis-

advantage arises in that, when the pore-water pres-

sure is changed, two stress state variables are being

affected. The relative significance of each variable

must be borne in mind when considering the shear

strength. (This is also the disadvantage associated

with utilizing the ( a u,) and ( a u,,.) combina-

tion of stress state variables.)

Th e second com bination of stress state variables

used is ((7 LL:,) nd (u,, u,,.). T h e advan tage of

this combination is that only one stress variable is

affected when the pore-water pressure is changed.

Regardless of the combination of stress variables

used to define the shear strength, the value of shear

strength obtained for a particular soil with certain

values of a, u,, and u , must be the same.

Research data from the testing program of other

workers are used to demonstrate the use of the

proposed shear strength equations for unsaturated

soils.

Theory

Fo r a saturated soil, a stress circle correspond ing

to failure conditions is plotted on a two-dimen-

sion al plot of effective no rm al stress versus shea r

streng th. series of tests gives a line of failure

(i.e . a Mohr-Coulomb failure envelope).

In .th e case of an unsaturated soil, the stress

circle corresponding to the failure conditions must

be plotted on a three-dimensional diagram. The

axes in the horizontal plane are the stress state

variables and the ordinate is the shear strength.

A series of tests gives a surface defining failure

conditions.

a) Shear Strength in Terms

of

the

(U u,,.)

and

(u,, u,,.) Stress State Variables

The vertical plane of a u,,.) versus shear

strength, with (LL;, u,,) equal to zero, corresponds

to the case where the soil is satura ted. If th e soil

has a positive matri x suction, i.e. (u, u,,.) is

greater than zero, a third dimension is required to

plot the stress circle. It is initially assumed th at the

surface defined by a series of tests, .on unsa turated

soil samples, is planar. Therefore, the equation

defining the surface of failure can be w ritten as an

extension of the conventional saturated soil case:

[ I

=

c (a u,,.) ta n

+'

(u, u,,.) ta n +I

where c = effective cohesion parameter,

+'

= the

friction angle with respect to changes in (a ~ i

wh en (u;, u,,.) is held con stan t, an d 4

=

friction

angle with respect to changes in (u,, u,~.)when

7 u,,) is held const ant. three-dimension al plot

of th e stress state variables versus she ar strength is

difficult to analyze. Therefore, a procedure is out-

lined to assist in obtaining the desired shear

strength parameters. For convenience in handling

the strength data from triaxial tests, the stress

point at the top of each stress circle can be used,

i.e. [+ (< TI (I:,) u,,.] and (u;, u,,.). Th e equa tion

for the plane throu gh th e stress points is

[2] +(al I: , ) =

d

+

[+(a1

+

03) LL,,.] an

+'

+ (u , ,,. tan +

where d

=

the intercept when the two stress points

are zero,

I '

the angle between the stress point

plane and the [ (al + a3) u,,.] axis wh en

( u , , u , ~ )s held constant, $ = the angle between

the stress point plane an d the (u, u , ~ xis when



F R E D L U N D ET A L

-

STRESSES AT MAX

( CONSTANT WATER CONTENT 1

SATURATED SPECIMENS

-

(CONSOLIDATED UNDRAlNED

1

FIG 1. Triaxial tests on a compacted shale (clay fraction

22%)

compacted at

a

water content of 18.6% and

sheared at constant water content (from Bishop

et

al .

1960 .

[$ (a l

u,,.] is held constant. The relation-

and obtain the relationship between the various

ship between a plane through the stress points and

angles of friction.

the failure surface is shown in the Appendix along

[51

tan cp' = tan cpL tan

cp

with the corresponding soil param eters. It is neces-

sary to plot the data in a special manner in order

Test ata

to obtain the 4 friction parameters.

Published research literature reveals that few

b) Shear S trength in Terms o f the

(a u,,)

and

tests have been performed where the principal

(u,,

u,,.)

Stress State Variables

stresses and the pore-air and pore-water pressures

The equat ion for the fai lure surface when

have been measured as the specimen is loaded.

( U u,,) and (u:, u,<.) are used as stress vari-

Thr ee sets of data are presented below.

ables is

a ) Compacted Shale Bishop et al.

1960

[3]

C (U u, ,) t an cpn (u, u,) ta n cp

where

c

cohesion intercept when the two stress

variables are zero;

=

the friction angle with

respect to changes in ( a

u.,)

wh en (u, u,,.) is

held constant;

cpil =

friction angle with respect to

changes in (u,, u,,) when ( U u,) is held

constant. The number of parameters, however, can

be reduced. When the matrix suction is zero, the

(U u:,) plane will have the same friction angle

parameter as the (a u,,,) plane when using the

previous com bination, ( a ) , of stress state variables.

Therefore,

cp

is the same as

4'.

Also,

c

is the

same as

c'.

Th e shear st rength equat ion is now

[4]

C'

(U u,) t an cp' ( u , , u , ~ )an cp

Equation

1

or 4 will give the same value for shear

strength. Therefore, it is possible to equate them

Triaxial tests were performed on a shale com-

pacted at a water content of

18.6%.

The clay con-

tent was 22%. All specimens were sheared at a

constant water content. The failure envelope for

tests on saturated specimens gave cp'

=

24.8' and

c' 15.8

kPa (2.3 psi) (Fig.

1) .

The da ta for

tests on unsaturated test specimens are plotted on

Fig.

1

and summ arized in Table

1.

Figure 2 shows

plot of [4 (n l as ) u,~.] nd (11, u,\,) versus

i - (u l

~ 7 : ~ .

t is obvious that this type of plot is

not very useful since it is impossible to visualize

the plane on which the points fall. As proposed in

the Appendix, (AT, cos

+ )

is plotted versus

u;,

u,,.) in order to obtain (Y (F i g . 3 ) . The

graphically obtained angle for the compacted shale

is

-3.9 .

This corresponds to a + value of

-4.2 and a cp value of -4.6 . O ne form of the

shear strength equation would be



C A N . G E O T E C H . J. VOL. 15

1978

[6]

= 2.3 tr

u,,.

tan

24.8'

( u : , I,,.)

tan

4.6

T h e

+

angle is negative since the pore-wate

pressure variable appears in both stress variables

Th e physical significance can b e stated as follows

A decrease in pore-water pressure u,,. is not a

efficient in increasing the

U ~ 1 , ~ )

an

4

com

ponent of streng th as is a correspondin g increas

in total stress

7.

Rather, decreasing the pore-wate

pressure increases the frictional resistance of th

soil by (tan

24.8'

tan

4 . 6 ' ) ,

which is equal t

tan 20.9'. On th e basis of [5] the 20.9 angle

equal to +I . Therefore, another form of the shea

strength equation is

[7]

= 2.3 n

u:,

tan 24.8'

L I : , u,,. tan 20.9'

From this equation, the relative shear strengt

contribution s of the total stress and the pore-wate

pressures are readily inspected.

(13)

Boulder Clay (Bishop t al.

1960

These triaxial tests were perform ed o n specimen

compacted at a water content of 11.6 an

sheared at a constant water content. The clay fra

+ ' = 24.8

c . 2.3 psi

y .

22.8

d - 2.1

p s i

0

FIG 2. Three dimensional strength envelope for com

pacted shale.



FR E L UN T

AL

V I 4

12

I 6 ( P S I ) 2 0

I I

3

w 2

10

a

I

FIG. 3 Change in shear strength from saturated strength plane for compacted shale.

I STRESSES T 15 S TR I N

I

FIG. 4. Triaxial tests on a compacted Boulder clay (clay fraction

18 ) compacted at

a

water content of

11.6 and sheared at constant water content (from Bishop

et al

1960).

-

o

Ua - Uw

V

9 8

8 12 16 2 0 ( P S I )

24

I I I

a 3.3

3

5 4

I0

a

6

- 2 0

I

I I

0 25 5 0 75

100

125

1 5 0 ( k P o )

u a

w

FIG.

Change in shear strength from satuiated strength plane for Boulder clay.



C A N . GEOTEC H. J. VOL 15,

1978

-

-

I;

-

-

I;

; I

w m m t - - a m - N

r j r j d d i - d d d

I I I I I I I

- 0 0 -

.

. . -?? -?

D a t - N t - = - o -

-

-

I I I I I I I

= =?-?03?09N1?

N N 0 0 - - 0 0 0

I I I I I I I

tion was 1 8 % . Th e data are shown in Fig. 4 an

summarized in Table 2. The saturated specimen

gave a +' angle equal to 27.3' and an effectiv

cohesion c' = 9.6 kPa (1.4 psi). Figure 5 show

the (AT ,, os $I) versus (u:, u,) plot. Th e be

fit line gives an angle

a

= -3.3 . Th e correspon

ing angles + = -3.6' and + = -4.1 . T h e $

angle is 24.0'. Th e form s of the shea r strengt

cquation for the Boulder clay are :

[8]

= 1.4 T u,,.) tan 27.3

-

u:,

-

u, .) tan 4.1

[9]

= 1.4 + 7

-

L:, tan 27.3'

+ u : ,- u,,) tan 24.0

The interpretation of [8] and [9] is the same

for equations [6] and [7].

(c) Potters Flint and Peerless C l a y M I T 1963

A

series of triaxial tests was run on an artificia

mixture (b y weight) of 8 0% Potter s flint an

20% Peerless clay, at MIT. The samples wer

compacted at a water content of 17 .5% , 3%

dry of optimum . The dry density was

1

Mg/m3 (100 lb/ft3). The specimens were fai le

at a constant water content. The saturated spec

inlens gave an effective angle of interna l frictio

9' = 35.6 and an effective cohesion

c'

= 0.0 kPa

The results are summarized in Table 3. Figure

shows a plot of AT,, os I) ) versus (us,- u,\.) an

yields an angle

a =

2.4'. T he corresponding ang

$"

is 2.8 and If is 3.4 . Th e + angle is 37.8

6

These results show the opposite behavior to tha

-

reported in parts ( a) and (b ) . All samples wer

C

tested at low confining pressures. Th er e is consider

-

able scatter in the data (Fig. 6) and the author

c

are not definite as to its interpretation. Th e resul

.

-

would indicate that a decrease in pore-water pres

-

sure increases the fr ictiona l resistance of the so

w

f

-

more than a corresponding increase in confinin

pressure.

Z

r ummary

The three sets of data are considered to b

a

analyzed in terms of two independent sets of stre

-

variables. Tw o possible form s for the shear strengt

w

z equation are given by [I] and [4].

Z

-

The graphical form of [ I ] and [4] are shown

Fig. 7 using data on the compacted shale from

Bishop et al . (1960). The authors suggest that th

-

-

second form (i.e. [4]) will prove to be the mos

useful form in engineering practice. Further testin

i

using other soils will improve our understandin

of the shear strength of unsaturated soils.



F K E D L U N D ET AL .

F IG .

6.

Change in shear s t reng th f rom satu rated s t reng th p lane fo r Po t ters f l in t and Peer less c lay .

T A B L E. M IT results for Po tters flint and Peerless clay (M IT 1963)

Calculated Calculated*

Test I(.,

03

(a1

f

03')

0 3 1 ~ 1 1.) 1,

: ( a i

0 3

1,

; ( a ,

03 AT AT cos 4

No. Sample (kg/cm2) (kg/cm2) (kg/cm2) (kg/cm2 ) (kg/cm 2) (kg/cm2) (kg/cm2) (kg/cm2

SB-I

2 SB-2

3 SD-I

SB-4

5 SD-2

Smturoted

6

SA-I

7 SA-2

8

SA-3

9 SA-4

*The ca l cu la t ed va lue re fe rs t o t he s hea r s t re s s

on

the saturat ion fai lure ei ivelope

I

=

2 3

0-

Ua

ton 24.8 ( U a - U

tan

20 9

FIG. 7. G rap h ica l fo rm o f [I] n d [4] o r co m p ac ted

shale data .

cknowledgements

The authors wish to acknowledge the financial

support of the Department of Highways, Govern

ment of Saskatchewan, for their financial suppor

of the research work presented in this paper.

A IT C H IS O N ,

.

D. 1967. Separate roles of site investigation

quantification of soil properties, and selection of operationa

environment in the determination of foundation design o

expansive soils. Proceedings, 3rd Asian Regional Conferenc

on Soil Mechanics and Foundation Engineering, Vol. 2

Haifa, Israel.

B A R D E N ,

.,

M A D E D O R ,. 0.. nd SI DE S, . R. 1969. Volum

change characteristics of unsaturated clay. ASCE Journal o

the Soil Mechanics and Foundations Division, 95(SM1), pp

33-5 1.

BISHOP,A.

W . ,

an d B L IG H T , . E .

1963. Some aspects o

effective stress in saturated and unsaturated soil

Geotechnique, 13 pp. 177-197.

BISHOP, . W . , A L P A N , . , B L IG H T , . E . , an d D O N A L D ,. B

1960. Factors controlling the strength of partly saturate

cohesive soils. Research Conference on ' Shear Strength o

Cohesive Soils, ASCE, University of Colorado, Boulder, CO

B R A C K L E Y ,. J. A. 1971. Partial collapse in unsaturate d exp an

sive clay. Proceedings, 5th Regional Conference on So

Mechanics and Foundation Engineering, South Africa, pp

23-30.

B U R L A N D ,. B. 1965. Some aspects of the mechanical behavio

of partly satura ted soils.I n Moisture equilibriumand moistur

changes in soils beneath covered areas. Butterworth an

Company (Australia) Ltd. , Sydney, Australia.



320

C A N . GEOTECH.

J. VOL. 15. 1978

C O L E M A N ,. D. 1962. Stresslst rain relations for partly satu rated

soil. Corre spon denc e. Geo techn ique, 12(4 ), pp. 348-350.

F R E D L U N D ,. G. 1974. Engineering approach to soil continua.

Proceedings, 2nd Symposium on the Applications of Solid

Mechanics, Ham ilton, Ont. , Vol. I pp. 46-59.

F R E D L U N D ,. G. , and MORG ENSTERN ,. R. 1976. Constitu-

tive relations for volume change in unsaturated soils. Cana-

dian Geotechnical Journ al, 13, pp. 261-276.

1977. Stress state variables for u nsaturated soils. ASCE

Journal of the Geotechnical Engineering Division, 107(GTS),

pp. 447-466.

M A R A N H AAS NE VE S, . 1971. The influence of negative pore

water pressures on the strength characteristics of compacted

soils. Publication No. 386, National Laboratory of Civil En-

gineering, Lisbon, Portugal. (In Portuguese.)

Massachuse tts Institute of Techn ology. 1963. Engineering be-

haviorof partially saturated so ils. Phase Report No. 1 to U.S .

Army Engineers Waterways Experimental Station, Vicks-

burg, Mississippi. The Soil Engineering Division, Departm ent

of Civil Engineering, Massachusetts Institute of Technology,

Con tract No. DA-22-079-eng-288.

MATYAS,

E . L . , an d R A D H A K R IS H N A ,. S. 1968. Volume

change characteristics of partially saturated soils. Geo-

techn ique, 18(4), pp. 432-448.

S R I D H A R A N ,. 1968. Some studies on the strength of partly

saturated clays. Ph.D. thesis, Purdue University , Lafayette,

IN .

W ID GE R, . A. 1976. Slope stabili ty in unsaturated soils. MS c .

thesis, University of Saskatchewan, Saskatoon, S ask.

Appendix-Stress Point Meth od to Obtain the

Shear Strength Parameters

This ap pendix derives the mathem atical relation-

ship between the stress points a t the to p of a stress

circle and the failure surface for an unsaturated

soil (Widger 19 76 ). Th e stress variables are

[ (w

W

u,,.] and (u, u,\,). Figu re A 1 views

the strength data parallel to the

(u,

u,) axis.

The conversions for the zero matrix suction plane

(Bishop

t al. 1960)

are

a nd

[A21

C

dl/ cos cp'

where q' angle of the line through the stres

point at the top of the stress circles and d' the

intercept formed by the line through the stres

points.

It is also necessary to convert the angle

cp

fo

the un satur ated case. Fr om the definition of cp ,

[A31

tan cp A T ~ / ( u , u , ~ . )

and

[A41

tan q A T ~ / ( U : , u . )

where q angle between the line through th

stress points and the (u,, u,,.) axis,

AT,

chang

in shear strength between the saturated soil failur

envelope and the failure envelope correspondin

to a parti cular (u:, u,,.) value, AT, change i

shear strength between the saturated soil envelop

through the stress points and the envelope throug

the stress points at a particular (u, u , ) value.

The matrix suction is the same value in [A3

and [A41 and therefore

[A51

(t an cp )

/AT,

( t a n

AT^

Considering th e shear strength axis gives

and

[A71

T^ d,' d'

where c,' th e cohesion inter cept of the failu r

envelope at a mat rix sucti on (u, , u,,,) an

d,

the cohesion intercept at a matrix suctio

(u:, u,,.) using the stress point meth od. Sinc

cp'

is constant for all matrix suction values,

[A s] AT,, (c,' ') COS 4'

Substituting [A61 and [AS] into [A51 gives

tan 4

tan

4

os

bl

0

w

FIG.A l . Fai lu re c i rc les look ing paral le l to (u , ,

u,,.

ax is to com pare fa i lu re envelopes .



F R E D L U N D E T A L

FIG A2 Three-dimensional fai lure surface using the

stress point method.

In order to compute + , it is necessary to obtain

I The procedure that appears most convenient

involves the use of the stress point method fail-

ure surface Fig. A 2 ). Imagine viewing along

the failure envelope for a satura ted soil plan e

A B F E ); if the matrix suction variable d oes not

affect the shear strength, the three-dimensional

failure surface as viewed along the matrix suction

axis appears as a line. However, the change in

shear strength

AT,,)

must be multiplied

by

cos

+'

in

order to transform it into a line perpendicular to

the saturation failure plane Fig. A 3 . Let the slope

of the plot of

AT^

cos +') versus matrix suction be

tan

a

FIG. A3 Failure surface showing project ion of shear

strength onto the saturated strength plane.

T^ cos

'

tan

1

u

Substituting [A41 into [A101 gives

tall

tan $

O S

1 1 ~

The value of tan 4 can now be com puted from

[Agl.

The Shear Strength of Unsaturated Soils 1960

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