Environmental Engineering Reference
In-Depth Information
Table 12.8 Comparison of Commonly Used Methods of Slices
Factors of Safety
Method
Moment Equilibrium
Force Equilibrium
Interslice Force Assumption
Ordinary
x
X
=
0,
E
=
0
Bishop's Simplified
x
X
=
0,
E
≥
0
Janbu's Simplified
x
X
=
0,
E
≥
0
E
L
)t
R
/b
a
Janbu's Generalized
x
X
R
=
E
R
tan
α
t
−
(E
R
−
tan
θ
b
Spencer
x
x
X/E
=
Morgenstern-Price
x
x
X/E
=
λf (x )
Lowe-Karafiath
x
X/E
=
Average slope of ground and slip surface
Corps of Engineers
x
X/E
=
Average ground surface slope
a
α
t
=
angle between the line of thrust across a slice and the horizontal.
t
R
=
vertical distance from the base of the slice to the line of thrust on the right side of the slice.
b
θ
=
angle of the resultant interslice force from the horizontal.
There have been a number of optimization techniques that
result in a variety of approaches for the determination of the
shape and the location of the critical slip surface (Celestino
and Duncan, 1981; Nguyen, 1985; Chen and Shao, 1988;
Greco, 1988). Each approach has its own advantages and
shortcomings.
The “dynamic programming” method combines a finite
element stress analysis with an optimization technique to
provide a slope stability solution. Dynamic programming
overcomes some of the difficulties associated with limit
equilibrium methods of slices. The dynamic programming
approach requires the designation of stress-strain soil prop-
erties such as Poisson's ratio
μ
and modulus of elasticity
E
for the soil mass.
The dynamic programming method for a slope stability
analysis is more complex than the method of slices, but the
method lends itself well to computer simulation. In 1980,
Baker introduced an optimization procedure that utilized the
algorithm of the dynamic programming method to determine
the critical slip surface. The factor of safety was calculated
using the Spencer (1967) method of slices. Yamagami and
Ueta (1988) enhanced Baker's (1980) approach by combin-
ing the dynamic programming method with a finite element
stress analysis for the calculation of the factor of safety. The
critical slip surface was assumed to be a chain of linear seg-
ments connecting two
state
points located in two successive
stages
(Fig. 12.80). The resisting and the actuating forces
used to calculate an
auxiliary function
were determined from
stresses interpolated from Gaussian points within the domain
of the problem. Zou et al., (1995) proposed an improved
dynamic programming technique that used essentially the
same method as introduced by Yamagami and Ueta (1988).
The analytical procedure involves the use of a partial
differential solver to compute the stress states throughout
a soil continuum. The shape and location of the critical
slip surface corresponding to a minimum factor of safety is
determined through use of the dynamic programming opti-
mization technique. A number of example problems were
studied by Pham and Fredlund (2003b), and the results were
compared to conventional method-of-slices solutions.
12.6.1 Background to Optimization Theory
for Slope Stability
Bellman (1957) introduced a mathematical method called
the dynamic programming method. One of the objectives
of the dynamic programming method was to maximize or
minimize a function. The dynamic programming method has
been widely used in various fields other than geotechnical
engineering. Baker (1980) applied the optimization tech-
nique to slope stability analysis. The definition for the factor
of safety when using optimization methods can be written
as follows.
For an arbitrary slip surface
AB
, as shown in Fig. 12.80,
the equation for the factor of safety
F
s
can be written as
B
A
τ
f
dL
F
s
=
(12.92)
B
A
τdL
where:
τ
=
mobilized shear stress along the slip surface,
τ
f
=
shear strength of the soil, and
=
dL
an increment of length along the slip surface.
The critical slip surface is approximated by an assem-
blage of linear segments. Each linear segment connects two
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