SKEMPTON (1954) OCR

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i)J Ji 10 • .. zu "• 
THE PORE-PRESSURE COEFF1CIENTS A AND B 
by 
A. w. SKEMPTON, D.Sc;, A.M.I.C.E. 
SYNOPSIS 
In a number ofprobieffiH involving the undrained 
shear strength of soils (especla.lly in the design of 
earth dams)the change in pore pressure .du occurring 
under changes in total stresses mutt be known. 
The equation .du = B[.du, + A.(.do-1 - .dcr,)] is 
derived, and some typical values of the experi­
mentally determined pore-1>ressure coefficients A 
and B are given. Some practical applications of 
these coefficients have been outlined by Bishop 
{1954). 
Pour un certain nombre de problemes compor­
tant la resistance au cisaillement a tcneur en eau 
oonsta.nte des sols (en particuller, pour le calcul des 
barrages en terre}, ii est necessalre de connattre les 
changements dans la pression intentitielle .du qui Ille 
produisent lors des changemenudans les oontraintee 
tota.les. L'equation .du = B[.da, + A.(.da1 - .da1}J est 
derivee et certaines valeurs typiques des coefficients 
A et B de pression interstitielle obtenues ex~ri­
mentalernent sont donnees dans cet article. Certaines 
applications pratiques de ces coefficients ont ete 
exposees par Bi.shop (1954}. 
INTRODUCTION 
In problems concerning the undrained shear strength of soils, it has been found convenient 
to express the pore-pressure change Liu, which occurs under.changes in the principal stresses 
Ja1 and Ju3, by the following equation: 
.du = B[.du~ + A('.do-1 -Jua)], 
where A and B are "pore-pressure coefficients." These coefficients are measured experi­
mentally in the undrained trlaxial test, and the values of Jo-1 and Ju8 are, 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 grow1d or in an earth darn, 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 Ju1 and .:1u8 can be considered 
as taking place in two stages (St'e Fig. 1). 
Firstly, the element is subjected to an 
equal all-round increment Llu8 and, sec­
ondly, it is subjected to a deviator stress 
(.du1 -.du8). Corresponding to each of 
these stages there will be pore-pressure 
changes Ju. and Ju,, where: 
Undrained 
test 
Drained 
test 
THE COEFFICIENT B 
.dVi1 
Fig.1 
The relation between Ju,. and Ju3 for a typical test on a partially saturated soil is shown 
in Fig. 2 (a). Tue increase in effective stress in the test is: 
.:1o-' = Jo-3 - Ju. 
and, if Ce is the compressibility of the soil structure, then the volume change is : 
'JVe =~Ce. V(Llu3-Llua), 
where Vis the original volume of the sample. And, 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 in 
the void space is : 
• The all-round pressure condition is assumed for siuiplicity of· presentation. The case of an element 
consolidated under p and Kp can a.l5o be treated by the pore-pressure coefficients. 
143 
http:V(.dua-.du
I• 
., . 
/ 
144 SKEMPTON 
1°lOD 
Q!!t 
! • ,,· 
"I ... .. 
::, = ... 
f 
I· 
... H IOI 
t>EGI\EE OF V, TUIIATION (per Wit) 
(t.1) (b) 
l'ig. 2 
But these two changes in volume are identical and, hence, 
..du•=B = l 
..dua 1 nC., +­c. 
Now, in saturated soils {zero air voids}, C.o/Co ~s approximately equal to zero, since the 
compressibility of water is negligible compared with that of the soil structure. Consequently. 
for such soils, 
B = l, when the degree of saturation = 1. 
An experimental confirmation of this result, for a· saturated clay, is given below in Table I. 
0 
15 lb/sq. in. 
30 
45 ,, ;, 
60 ,. ,, 
Table 1 
0 
14'.7 lb/sq. in. 
29·5 
45•0 
59·8 :: :: 
B 
0·980 
0·984 
1·000 
0·996 
If, in contrast, the soil is dry, then C.,/C., approaches infinity, since the compressibility of air 
is far greater than that of the soil structure. Hence, for dry soils, 
B = 0, when the degree of saturation = 0. 
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 
The changes in pore pressure during the application of a deviator stress are shown, for 
two compacted clay soils, in Fig. 3. At any time when the increment of deviator stress is 
(..::1u1 -..dus}, the pore pressure due to this increment is ..du, and the corresponding changes in 
the principal effective stresses are : 
..::10-1' = (..::10-1 - ..daa) - .du, 
and 
http:express:i.OI
PORE-PRESSURE COEFFICIENTS A AND 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: 
.::We = - Cc. V. ½(.Jai' + 2.Ja31) 
or AV. = - C •. V. ¼[(.Ja1 - Aas) - 3..du4]. 
And the volume change in the void space is : 
.JV,,= - C,,.nV . .Ju4. 
But, as before, these two volume changes are identical and, hence, 
1 
.Jud = C . ½(Lla1 - Aa3) 
n V 
1 + ... 
C, 
or Aud= B -¼(Ja1 - Aa3). 
In general, however, the behaviour of soils is by no means in accordance with elastic theory 
and' the above expression must be written in the form : 
Aud= B.A(Aa1 - Ja3), 
where A is a coefficient to be determined experimentally. 
Combining the expressions for the two components of pore pressure, we have: 
Liu = B[Lla3 + A(Lla1 - Lias)], 
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 = 1, the equation becomes : 
..du= ..das + A(.Ja1 - .Ja3). 
This expression was given by the author in 1948. Test results for a saturated clay are 
plotted in Fig. 4. 
10 I 21) 
AXIAL STfl.AIN (pu «n,) 
~ 10 '~ u, 
AXIAL STI\AIN ( p~r ren() 
S 10 I~ 
AXIAL SHAIN (ptr etnt) 
Fig. 3. 'Undrained t.riaxial te■tlil on two compacted clay-gravel■ 
:ff(} 
... 
146 SKEMPTON 
For any given soil, the coefficient A varies with the stresses and strb.ins. Its value may 
be quoted at failure (maximum deviator stress), at ~ximum 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 stres~es, A will have different values in general, but the data for this case are scanty. 
Tahle2 
Type of Clay A 
----
Clays oI high sensitivity 
Normally consolidated clays 
Compacted sandy clays 
Lightly over-consolidated days 
Compacted clay-gravels 
Heavily over-consolidated clays 
+i to +lt +t to +1 
+ to+ t 
Oto+ t 
-1 to+. 1 
--l to o 
ALTERNATIVE FORM:S OF .THE PORE-PRESSURE EQUATION 
The pore-pressure equation : 
.du = B[Ao-s + A(Ao-1 - Ao-3)] 
may be written in several alternative forms, each of which ha'> some particular advantages. 
In the normal laboratory undrained test, the pore pressures under .do-8 and under (Ao-1 - Ao-8) 
are measured and, hence, the coefficients determined directly from the test are those in 
the following equation : 
~ 
ll••l Pl.)I 
"'Uy nturatu 
'°' bo<h ,ample, p. 15 l.,tq,1• ~IO!--+-+---+----i 
11. 1.0 
,1.--------.----, 
C 
¾, 
I> 
~ ··-----+---! 
~ 
• •• 
AXIAL STRAIN fp•' ant) 
Val11ti qf A. &•"tn ,U 
-JN1um1i1tn dniator nrcu 
w "21 per cent 
fl i 15 
AXIAL 5TP.AIN (per cent) 
s 
.I:. A; -D· l -0 
' I 
' 
~ 
----
• 
Fis, 4. Undrained t.rialdal ten. on two 
11amplea of remoulded ■atu.rated olay 
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 : 
.du = B[.1(11 ~ .. (1 - A) (Aa1 - .dus)] 
and 
~=J = B = B [ 1 - (I - A)( I - ~::) l 
The " overall " coefficient B is a useful para­
meter,especially in stability calculations in­
volving 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 pore­
pressure equation is best written in the form : 
.du= 
[ 3A - 1 ] B l(..::lu1 + 2Llu3} + 3 (.10-1 - .Lias) , 
sinr.e this shows that, for a material behaving 
in accordance with elastic theory, with 
A = ¼, the pore pressure depends solely on 
PORE•PRESSURE co·EFFICJENTS A AND B 
I 
147 
the mean principal stress, whereas in soils with A =/:- l the pure shear stress has a marked 
influence on the pore pressures. 
APPLICATIONS 
· 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 ~ese applications. 
ACl{NOWLEDGEMENT 
The test ~esults given in this article were obtained in the Civil Engineering Department, 
Imperial College, University of London, and the Author is particularly indebted to Mr D. J. 
Henkel who has supervised much of the work on pore-pressure measurement. 
REFERENCES 
BISHOP, A. W., 1954. The Use of Pore-Pressure Coefficients in Practice. Giotecl111ique, 4 : 4 : 148. 
SKBMPT0N, A. W., 1948. The Effective Stresses in Saturated Clays Strained at Constant Volume. P,,()(;. 
71'/s Im. Conp. App. Mee/&., 1 : 378.