Shear Strength Applications in Plastic and Limit Equilibrium - Unsaturated Soil Mechanics in Engineering Practice - page 613

Equation 12.32 for the shear strength of an unsaturated
soil has the advantage that derivations relevant to saturated
soils can readily be modified to accommodate situations
where the soil is unsaturated. It is simply necessary to real-
ize that the cohesion of a soil has two components (i.e.,
effective cohesion and cohesion due to matric suction).
The initial, total vertical stress in a soil mass is equal to
the overburden pressure, ρgy . The total horizontal pressure
is equal to the coefficient of earth pressure at rest multiplied
by the overburden pressure. The assumption is made that the
soil has an initial matric suction equal to (u a −

a

Horizontal movement

u w ) 0 .The
pore-water pressure in the soil is influenced by environmen-
tal conditions but is assumed to remain constant during the
analysis. Changes in total stress are assumed to not influence
pore pressures.

a

Figure 12.37 Stresses on an element in soil mass behind friction-
less wall.

12.3.3 Active Earth Pressure
Let us suppose that the wall a - a in Fig. 12.37 is allowed
to move away from the soil mass. The horizontal stress is
reduced until a limiting value corresponding to the plastic
equilibrium state is attained. Failure is induced by reducing
the horizontal stress. The horizontal stress is the minor prin-
cipal stress while the vertical stress is the major principal
stress. The horizontal stress at any point corresponding to
the active state can be computed from the vertical stress and
the failure criterion for the soil.
Figure 12.38 illustrates how the active and passive pressures
in a soil change as the matric suction changes. The active
pressure is shown to decrease as matric suction increases. In
other words, as the pore-water pressure in the soil becomes
more negative, the soil becomes stronger. Consequently, less
force would be carried by the retaining wall.
Let us consider a vertical plane with the soil having con-
stant matric suction as shown in Fig. 12.37. An element
from a depth y has an overburden stress σ v . The element
of soil is shown in Fig. 12.39 along with the definition of
pertinent variables. If the wall moves away from the soil,
the active earth pressure is developed. The active pressure is
designated as σ h −

12.3.2 Extended Rankine Theory of Earth Pressures
The active and passive earth pressures for an unsaturated soil
can be determined by assuming that the soil is in a state of
plastic equilibrium. Let us first review the stresses in a soil
mass when the surfaces of failure are planar. The major and
minor principal planes at all points are assumed to have similar
directions. The solution is known as the Rankine earth pres-
sure theory. It is necessary to extend the conventional earth
pressure concepts when dealing with unsaturated soils. For
this reason, the earth pressure theories presented herein are
referred to as the extended Rankine theory of earth pressures.
Figure 12.37 shows a vertical, frictionless plane passed
through a soil mass of infinite depth. An element of unsat-
urated soil at any depth is subjected to a vertical stress σ v and
a horizontal stress σ h . These planes are assumed to be prin-
cipal planes and the vertical and horizontal stresses are the
principal stresses. The ground surface is horizontal and the
vertical stress is written in terms of the density of the soil.
The state of stress for an element of unsaturated soil can
be shown on an extended Mohr diagram. The equation cor-
responding to the limiting or failure condition (i.e., shear
strength equation) can be written as follows assuming a lin-
ear shear strength equation:

u a . The horizontal pressure can be writ-
ten in terms of the vertical pressure σ v − u a by considering

the geometrics of the Mohr circle:

c + u a ) tan φ + u w ) tan φ b τ = (σ n − (u a − (12.31)

[ (σ v − u a ) − (σ h − u a ) ] / 2

sin φ = (12.34)

The above linear shear strength equation can also be writ-
ten in a form similar to that used for saturated soils:

[ (σ h − u a ) + (σ v − u a ) ] + c cot φ

where:

u a ) tan φ τ = c + (σ n − (12.32)

c + u w ) tan φ b ].

c = total cohesion [i.e., c = (u a −

Consequently, the shear strength equation for an unsatu-
rated soil can be viewed as a two-parameter shear strength
equation. The total cohesion c has two components and can
be written as

Rearranging the above equation and solving for the net
horizontal stress σ h −

u a gives

sin φ cos φ

u a ) 1

−

σ h − u a = (σ v −

sin φ −

2 c . (12.35)

c + u w ) tan φ b c = (u a − (12.33)

1 + 1 + sin φ