Environmental Engineering Reference
In-Depth Information
Substituting Eq. 12.71 into Eq. 12.73 and replacing the
σ n β term with N gives
surface. The center of moments would appear to be imma-
terial when both force and moment equilibrium is satisfied
(e.g., in the GLE method). The computed factor of safety can
vary with the point selected for the summation of moments
when only moment equilibrium is satisfied.
Let us consider moment equilibrium for a composite slip
surface (Fig. 12.72) with respect to the center of moments:
c β
F s +
N tan φ β
F s
u a tan φ β
F s
W
(X R
X L )
sin α
u w ) tan φ b β
F s
(u a
+
N cos α
=
0
(12.74)
Wx
Nf
S m R
A L a L +
=
0
(12.78)
Rearranging Eq. 12.74 gives rise to the normal
force
equation:
Substituting Eq. 12.71 for S m into Eq. 12.78 and replacing
the n β) term with N yields
c β sin α
F s
β sin α
F s
( tan φ
W
(X R
X L )
+
u a
Wx
Nf
c βR
1
F m
[ N tan φ
β sin α
F s
A L a L +
=
+
tan φ b
tan φ b )
+
u w
u w ) tan φ b β ] R
N
=
u a tan φ β
+
(u a
(12.79)
m α
(12.75)
where:
where
F m =
factor of safety with respect to moment equilib-
rium.
F s / sin α tan φ
m α =
cos α
+
(12.76)
The factor of safety F s in Eq. 12.75 is equal to the moment
equilibrium factor of safety F m when solving for moment
equilibrium and is equal to the force equilibrium factor of
safety F f when solving for force equilibrium. In most cases,
the pore-air pressure u a is atmospheric, and as a result,
Eq. 12.75 reduces to the form
Rearranging Eq. 12.79 yields
c βR
N
u w β tan φ b
+
tan φ
u a β 1
R tan φ
tan φ b
tan φ
F m =
Wx
Nf
(12.80)
c β sin α
F s
β sin α
F s
tan φ b
A L a L +
W
(X R
X L )
+
u w
N
=
m α
In the case where the pore-air pressure is atmospheric (i.e.,
u a =
(12.77)
If the base of the slice is located in saturated soil (i.e.,
soil with positive pore-water pressures), the tan φ b term in
Eq. 12.77 becomes equal to tan φ . The normal force equation
then reverts to the conventional normal force equation used
in saturated slope stability analysis. The angle φ b can be used
whenever the pore-water pressure is negative and the angle
φ can be used whenever the pore-water pressure is positive.
The φ b angle can also be considered to be equal to φ at low-
matric-suction values up to the air-entry value of the soil while
a lower φ b angle is used at high matric suctions (Fredlund
et al., 1987).
The vertical interslice shear forces X L and X R in the nor-
mal force equation can be computed using an interslice force
function.
0), Eq. 12.80 has the form
c βR
N
R tan φ
u w β tan φ b
tan φ
+
A L a L + Wx
Nf
F m =
(12.81)
The φ b value can be set equal to the φ value when the
pore-water pressure is positive. Equation 12.81 can also be
simplified for a circular slip surface as follows:
[ c β
u a β) tan φ ] R
+
(N
u w β
A L a L + Wx
F m =
(12.82)
The radius R is constant for all slices, and the normal force
N acts through the center of rotation (i.e., f
=
0) when a
circular slip surface is considered.
12.5.8 Factor of Safety with Respect to Moment
Equilibrium
Two independent factor-of-safety equations can be derived:
one with respect to moment equilibrium and the other with
respect to horizontal force equilibrium. Moment equilibrium
can be satisfied with respect to an arbitrary point above the
central portion of the slip surface. The center of rotation is
an obvious center for moment equilibrium for a circular slip
12.5.9 Factor of Safety with Respect to Force
Equilibrium
The factor of safety with respect to force equilibrium is
derived from the summation of forces in the horizontal direc-
tion for all slices:
S m cos α
N sin α
A L +
=
0
(12.83)
 
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