Environmental Engineering Reference
In-Depth Information
A
=
resultant external water forces (the L and R subscripts
designate the left and right sides of the slope, respec-
tively),
α
=
angle between the tangent to the center of the base of
each slice and the horizontal, (when the angle slopes in
the same direction as the overall slope of the geometry,
α is positive, and vice versa), and
β
=
sloping distance across the base of a slice.
The example shown in Fig. 12.72 is typical of a steep
slope with a deep groundwater table. The crest of the slope
is highly desiccated and the tension cracks are assumed to
be filled with water. The tension crack zone is assumed
to have no shear strength and the presence of water in this
zone produces an external water force A L . The assumed slip
surface in the tension crack zone is a vertical line. The depth
of the tension crack is generally estimated (Spencer, 1968,
1973). The weight of the soil in the tension crack zone acts
as a surcharge on the crest of the slope. The external water
force A L is computed as the hydrostatic force on a vertical
plane. An external water force can also be present at the toe
of the slope as a result of partial submergence. This water
force is designated as A R .
(a)
(b)
12.5.6 Shear Force Mobilized Equation
The mobilized shear force at the base of a slice can be written
using the shear strength equation for an unsaturated soil:
Figure 12.73 (a) Acting total and pore-water stresses and
(b) shearing resistance at the base of a slice.
F s c +
u w ) tan φ b
(12.71)
β
where:
u a ) tan φ +
S m =
n
(u a
c
=
total cohesion of the soil, which has two components
[i.e., c +
u w ) tan φ b ].
Combining the cohesion components has the advantage
that the shear strength equation retains the conventional form
used for saturated soils. Consequently, it is possible to use a
computer program written for analyzing saturated soils when
considering the analysis of an unsaturated soil slope. The soil
in the negative pore-water pressure region must be subdivided
into several discrete layers with each layer having a constant
cohesion value. The pore-air and pore-water pressures must
be set to zero in this case. This approach has the disadvantage
that cohesion is not considered as a continuous function and
total cohesion must be manually computed for each layer.
The slope stability derivations directly calculate the shear
strength contribution from negative pore-water pressures.
The mobilized shear force defined using Eq. 12.71 is used
throughout the derivation. The effect of partial submergence
at the toe of the slope, the effect of seismic loading, and
external line loads are not shown in the derivations.
(u a
where:
σ n =
total stress normal to the base of a slice, and
F s =
factor of safety, defined as the factor by which the
shear strength parameters must be reduced in order to
bring the soil mass into a state of limiting equilibrium
along the assumed slip surface.
The factors of safety for the cohesive parameter (i.e., c )
and the frictional parameters (i.e., tan φ and tan φ b )are
assumed to be equal for all soils involved and for all slices.
The components of the mobilized shear force at the base of
a slice are illustrated in Fig. 12.73. The contributions from
the total stress and the negative pore-water pressures are
separated by using the friction angles, φ and φ b .
It is possible to consider the matric suction term as part
of the cohesion of the soil since matric suction can then be
visualized as increasing the cohesion of the soil. As a result,
the conventional factor-of-safety equations take on a form
similar to that of a saturated soil. The mobilized shear force
at the base of a slice, S m , will then have the following form:
12.5.7 Normal Force Equation
The normal force at the base of a slice, N , is derived by
summing forces in the vertical direction:
F s c
u a ) tan φ
β
S m =
+
n
(12.72)
W
(X R
X L )
S m sin α
N cos α
=
0
(12.73)
 
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