State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 99

ds

s a − u a

s 3 − u a

dy

t a

a

s 1 − u a

y

dx

Figure 3.17 Passive-type failure in compacted clay till resulting
from swelling in confining oedometer ring.

x

Figure 3.18 Net normal and shear stresses on inclined plane at
point in soil mass below horizontal ground surface.

principal stress is called the minor principal stress and is
given the symbol σ 3 . The horizontal and vertical planes
constitute the principal planes in the case of a horizontal
ground surface. The vertical net normal stress is often near

planes. The horizontal plane has an infinitesimal length of
dx. Its length can be written in terms of the sloping length
ds and the angle α .

the net major principal stress σ 1 − u a and the horizontal

net normal stress is near to the net minor principal stress

σ 3 −

u a .
If the magnitude and the direction of the stresses acting
on any two mutually orthogonal planes (e.g., the principal
planes) are known, the stress condition on any inclined plane
can be determined. In other words, the net normal stress
and shear stress on any inclined plane can be computed
from the known net principal stresses. The matric suction
u a −

dx = ds cos α (3.36)

The vertical plane has an infinitesimal length of dy :

dy = ds sin α (3.37)

All the planes have a unit thickness in the perpendicular
direction. The equilibrium of the triangular element requires
that the summation of forces in the horizontal and vertical
directions be equal to zero. Summing forces horizontally
gives
− σ α −

u w on every inclined plane through a point is constant
since it is an isotropic tensor. Therefore, only the net nor-
mal stress and shear stress on an inclined plane need to be
considered.
Let us consider an unsaturated soil under at rest condi-
tions beneath a horizontal ground surface. The net normal
stress and shear stress on a plane with an inclination angle
α from the horizontal plane are illustrated in Fig. 3.18. The
inclined plane has an infinitesimal length ds and results in
a triangular free-body element with horizontal and vertical

u a ds sin α + σ 3 − u a dy

+ τ α ds cos α =

0
(3.38)

Summing forces vertically gives
− σ α −

u a ds cos α + σ 1 − u a dx

− τ α ds sin α = 0

(3.39)

Table 3.1 Coefficients of Earth Pressure At Rest

Type of Soil Liquid Limit, w L Plastic Limit, w p Plasticity Index, PI Activity K 0

Loose, saturated sand — — — — 0.46

Dense, saturated sand — — — — 0.36

Compacted residual clay — — 9.3 0.44 0.42

Compacted residual clay — — 31 1.55 0.66

Undisturbed, organic, silty clay 74.0 28.6 45.4 1.2 0.57

Remolded kaolin 61 38 23 0.32 0.66

Undisturbed marine clay 37 21 16 0.21 0.48

Quick clay 34 24 10 0.18 0.52

Source : From test results of Bishop (1957, 1961b) and Simons (1958).