State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 98

1.0

0.8

0.6

0.4

0.2

0.0

(a) 0 10 20 30 40

f ′ (deg)

Figure 3.16 Relationship between effective angle of internal fric-
tion and coefficient of earth pressure at rest, K 0 (after Bishop,
1957).

coefficient of earth pressure at rest. The results supported
Jaky's (1944) earth pressure expression (Fig. 3.16):

sin φ K 0 = 1 − (3.35)

where:

(b)

φ =

effective angle of internal friction.

Figure 3.15 Relationship between horizontal stresses and verti-
cal overburden stresses during sedimentation, erosion, and reload-
ing of saturated soil. (a) Effective horizontal stress for various
effective vertical stresses. (b) Coefficient of earth pressure at rest,
K 0 , with changing overburden stresses.

The equation applies for initial or first loading of the soil.
Other researchers have also lent support for this equation
(Simons, 1958; Brooker and Ireland, 1965). Test results
obtained by Bishop (1957, 1961b) and Simons (1958) are
shown in Table 3.1.
The compaction of granular soils against an unyielding
wall can produce horizontal pressures greater than the verti-
cal pressure (i.e., K 0 greater than 1.0). Expansive soils can
also exert lateral pressures greater than the vertical pres-
sure. It is even possible to reach the passive pressure state,
at which time the soil may fail in shear. This condition is
illustrated in the oedometer specimen shown in Fig. 3.17.
The K 0 value for soils has also been shown to be a function
of the over consolidation ratio of the soil. An increase in
the over consolidation ratio produces an increase in K 0 .

unloading cycles. Figure 3.15 illustrates typical horizontal
and vertical effective stress paths where a saturated soil
is subjected to a history of sedimentation followed by
erosion and subsequent reloading. The stress paths can
become even more complex for unsaturated soils subjected
to cycles of drying and wetting and freezing and thawing.
The coefficients of earth pressure can go from as low
as zero to as high as the coefficient of passive earth
pressure.
If all of the elastic parameters were known and the above
analyses were applied to a soil that had undergone a com-
plex stress history, the coefficient of earth pressure should
be a tangent value of K 0 as opposed to a secant value of
K 0 (Fig. 3.15). The engineer is generally interested in the
secant value. The above analysis is most relevant immedi-
ately following sedimentation.
It must be recognized that the soil behaves in a nonlinear
manner and the incremental elastic parameters are difficult
to assess accurately. The assessment of the horizontal stress
state of an unsaturated soil is a complex topic. It is pri-
marily for this reason that empirical expressions have been
proposed for at rest coefficient of earth pressure. Bishop
(1957) presented the results of a laboratory study into the

3.5 EQUATIONS FOR MOHR CIRCLE

The state of stress at a point in the soil is three dimensional
but often a practical engineering problem can be represented
in a two-dimensional form. In two dimensions, there always
exists a set of two mutually orthogonal principal planes with
real-valued principal stresses. The principal planes are the
planes on which there are no shear stresses. The direction
of the principal planes depends on the general stress state
at a point. The largest principal stress is called the major
principal stress and is given the symbol σ 1 . The smallest