i j
!
f THE PORE-PRESSUHE COEFFICIENTS A AND B by A.
w.
S"E:'IPTO:-l, D.Se., A.aU.C.E.
SYNOPSIS Pour un certain nombre de prcblemes compor In a numbcrof problems involving the undrained tant la resistance au cisaillement a teneur en eau shear strength of sous (especially in the design of constante des sols (en particulier, pour Ie calcul des earth
In problems concerning the undrained shear strength of soils, it has been found convenient to express the pore-pressure change iJu. which occurs under changes in the principal stresses .:30'1 and iJ 0'3, by the following equation: \. iJu = B[iJ0'3 + A(:.lO'l-iJO'a)l,
when: A and Bare" pore-pressure coefficients.":' These coefficients arc measured experi
mentally in the undrained t~a."ia~ te.st. ..and the vai;ses .of A?,l and AO'~ ar~, in .general. chosen
to represent the changes In principal stress occurring In the practical problem under consideration, If the sample in the test. or if an element of soil in the ground or in an earth dam. is originally in equilibrium under an all-round • effective pressure p (which may in certain cases be close to zero). then the application of . . the stresses Au) and A0'3 can be considered :IS taking place in two stages (sec Fig. 1). Firstlv, the element is subjected to an equal all-round increment .::l0'3 and. sec ondly. it is subjected to a deviator stress (.:lC7)-AO'~). Corresponding to each of these stages there ....;11 be pore-pressure changes Au.. and .::lUd, where:
G:·.... . QA~I..-,..i~· j~j' v
[}.~'
.·<.I~I
Undrained test Drained ~ test
THE CqEFFIC1F.NT B
The relation between iJu.. and A0'3 for a typical-test on a partially saturated soil is shown . in Fig. 2 (a). The increase in effective stress in the test is : .::lu' = .::l0'3 -:.::lll..
and, if C. is the compressibility of the soil structure, then the volume change is : .dV. = -C•. V(t.l?,3-.::lU..). where V is the original volume. of the sample. !.nd. if C. is the compressibility of the fluid (air and water) in the voids and if n is the porosity of the soil, then the change in volume iri lhe void space is : .. .... .dV. = -C•. dr. AUe • TIle all-round pressure condition is assumed for sin:plicity of presentation .. The case of an element consolidated under p and J\P can also be treated by thr.pore-pressure coefficients, 113
J
ie
;r
\11'.:;
j~lo
..
.:
I
-j -I I
I
J SKEMPTON
144
j
I
.a*.
::: "r-:.-+---"~-,.f--+--+--I--1 :
I
j
I
j
I
(bl
(i1)
But these two changes in volume are identical and, hence, .dUll = B .da:s
=
l
1 11C~
1+ C"
\
I
Now, in saturated soils [zero air voids). C~/C" is approximately equal to zero. since tlu compressibility of water is negligible compared with that of the soil structure. Consequently. (or such soils, B == 1, when the degree of saturation = 1. An experimental confirmation of this result. for a saturated clay. is given below in Table I.
l
Twle 1
a...
.ju_
B ;
Il
II
15 Ib/sq. in. 3/)
"5 60
.... ....
I", 7 Ib/S
S9·t!
"
..
....
0·9~O
1/·984
1·000 0·996
If, in contrast, the soil is dry, then C~/C" approaches infinity. since thc compressibility of air is far greater than that of the soil structure. Hence, for dry soils. B = O. when the degree of saturation = O. For partially saturated soils. 0 < B < 1 and. at the Proctor optimum water content and density, the values of B range typically from about 0'1 to 0·5. The relation between Band the degree of saturation, for a clay gravel, is shown in Fig. 2 (b). THE COEFFICIENT A T~, changes
in pore pressure during the ~pp\ic~tion._Qf. a .~e..Y~.tor _~tre!'s arc shown, [II. two compacted clay soils, in Fig. 3. At any time whcn the increment of deviator stress I~ (.daJ -.Ja:d. the pore pressure due to this increment is ~1tI~ and the corresponding change- in the principal effective stresses arc: and
.dal' = (.dal - .d(3) - .du~ .da3' = - .dUd.
II
II
1 h
jl
\1 ,
I,
1: I
1/
J
I
U
,.I
/OL/
- - _.
---~~--:-:------.
( PORE-PRESSURE COEFFICIE~TS
A~D
A
B
145
If, for the moment, it is assumed that the soil behaves in accordance with elastic theory, the volume change of the soil structure under the increment of deviator stress is : .::l V~ = - C~. V. H.::lul' + 2.au3'r or .::l V~ = - C~. V. ![(.::lul - .::lc73) - 3.::lu~].
And the volume change in the void space is :
=-
C". nV . .::lu~. But, as before, these two volume changes are identical and, hence, .::lV"
I
.::lu~ =
·c . !(.::lul
. n "
- 4lu3)
1 ~ C~ .::lu~
or
= B. !(.::lul - .::lu3)'
In general, however, the behaviour of soils is by no means in accordance with clastic theory and the abo v·e expression must be written in the form : .::l,,~
=
B.A(.::lul - .::lU3) ,
where A is a coefficient to be determined experimentally. Combining the expressions for the two co'rnP'?nents of pore pressure, we have: .::lu = B(.::lu3
+ A(.::lul -
.::lu3)].
which is the equation given at the beginning of the Paper. It may be noted .that for the important particular case of fully saturated soils. where B == I, the equation becomes: .::lu
= .::lu3 + A(.::lul
- .::l«13)'
This expression was given by the author in 1948. Test results for a saturated clay are plotted in Fig. 4. '''r---~-~---.,
c
,
'0
.,
AXI.. " STk.AIN t,... u .. t.)
;
'0
A"tAt,. ~T"AIN
Fig. 3.
.,
(per c... d
:"
•
J
AXIA .. ST... ,,,,,, (,""
• CI"~'
;1
;1
Undrained tri&%ial teaU on two compacted clay-litTavel.
i
iI \ I\
.
.
-!
1-16
SKE~IPTO~
For any given soil. the coefficient A varies with the stresses and strains. Its value may be quoted at failure (maximum deviator stress], at maximum effective principal stress ratio. or at any other required point. At failure, the values of A for various clay soils, with positive total stress increments. may be summarized approximately as in Table 2. With decreasing total stresses. A will have different values in general. but the data for this case are scanty, Talll.2
Type of Clay
A'
Clays of high sensitivity ~onnally consolidated clays Compacted sandy clays Lightly over-consolidated clays Compacted clay-gravels Heavily over-consolidated clays
+~
to +l}
+t
to
+t to + I
+ 1
o to +
-! to + -] to
!
0
i
ALTERl"ATIVE FORMS OF THE PORE·PRESSURE EQUATION
The pore-pressure equation: ..du = B[..du3
+ A(Jul -
JU3)]
may be written in several alternative forms, each of which has some particular advantages. In the normal laboratory undrained test, the pore pressures under JU3 and under (Jul - ja~) are measured and, hence, the' coefficients determined directly from the test are those in the following equation: ..du = B ...du3 + A(Jut - JU3)' In evaluating A from A. care must be taken to use a value of B appropriate to the pressure range in the deviator part of the test . For earth-dam problems, it is convenient to write the basic equation in the forms: ,
<
•
CoL •• )
rI.
,,,It, ,1&,,'lnd
.
s: , I
I
,
Il
and 1~lOtlC.~
..
4X'Al STRA'N
Ju = B[Ju} - (I - A) (JUt - L1a~)j
, I
'.100
•
Flr;.
~.
I'
",cr"
u
, of
.c.or
,c.,.....
.,
= B= B
[1 - (1 -
A)
(I - ~~) ].
The" overall .. coefficient B is a useful para meter, especially in stability calculations in v olving rapid draw-down, and it can be measured directly in the laboratory for the relevant values of stress-change in any par ticular problem. From a physical point of view, the pon pressure equation is best written in the form: o
.
A.
'- ,
......"'" _
IS 'b/ICI.....
'PC" "n' f
"
AI' ;aUt
~ IU~.I" .••
~ ;11
-,
'I
f.-
V
/
. . N'" Il"'pl... p. 15 IbftCll.....
e
./
II
15
.".,
,!.
A•.O'j '
~
..
.,
----
UndraiJlec1 triaxial testa 011 two .ampl•• 0' r&lDoulc1.c1. .a~c.ec1 clay
•
Ju =
B [ HAut
- I + 2..du:) + 3A -j-(JUt
] - J0'3) ,
since this shows that, for a material bchavine in accordance with elastic theory. with A = !. the pore pressure depends solely ,,11
,
,
• I
I~
(
PORE-PRESSURE COEFFICIENTS
the mean principal stress, whereas in soils with A inrluencc on the pore pressures.
:;6
A
A~D
B
147
! the pure shear stress has a marked
.~PPLICATIONS
During the past few years a number of practical problems have been encountered in which the pore-pressure coefficients have proved to be helpful. Bishop (1954) has described briefly some of these applications. ....CKNOWLEDGEMENT
The test results given in this article were obtained in the Civil Engineering Department. Imperial College. University of London, and the Author is particularly indebted to :vIr D. J. Henkel who has supervised much of the work on pore-pressure measurement. REFERENCES The Use of Pore-Pressure Coefficients in Practice. Giolr,J&"jl{lI~, 4 : 4 : 148. S"£MPTON. A. W., 194~. The Effective Stresses in Saturated Clays Str.l.ine
Pros,
(
.
