Environmental Engineering Reference
In-Depth Information
50
= 15 kN/m
3
;
Medium layer:
c
= 0.33;
E
= 15,000 kPa
Weak layer:
c
′
= 0 kPa;
φ′
= 10
°
;
γ
= 18 kN/m
3
;
μ
= 0.45;
E
= 2000 kPa
Hard layer:
c
′
= 100 kPa;
φ′
= 30
°
;
γ
= 20 kN/m
3
;
μ
= 0.35;
E
= 100,000 kPa
′
= 20 kPa;
φ′
= 30
°
;
γ
μ
45
Method
DP
Enhanced method
Morgenstern-Price
Simplified Bishop
X
Y
R
Factor of Safety
40
1.000
1.102
1.140
1.125
35.29 30.76 19.79
35.29 30.76 19.79
35
30
25
20
Medium layer
Weak layer
15
10
Hard layer
5
Dynamic programming, DP
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
x
-Coordinate, m
Figure 12.87
Nonhomogeneous slope with internal weak layer of soil.
of water into a sloping ground surface. As moisture moves
into the slope there is also a tendency for the moisture
to move down the slope toward the toe of the slope. The
infiltration of water into the slope tends to control the shape
and location of the critical slip surface.
The infiltration of rainwater on a sloping surface is depen-
dent upon the water storage capacity and the hydraulic con-
ductivity of the soil. The evolution of the shape and position
of the critical slip surface with respect to time lend them-
selves to the dynamic programming method of analysis.
There are no restrictions placed on the shape and location
of the critical slip surface. Figure 12.88 shows the shape
and location of the critical slip surface with time after the
commencement of rainfall. The initial factor of safety of the
slope was 1.83 and after 4 days the factor of safety was
reduced to 1.12. The boundary condition at the ground sur-
face was given a pore-water pressure value of 0 kPa. In other
words, it was assumed that there was sufficient rainfall over
the entire 4 days to reduce the pore-water pressure at the
ground surface to zero.
The ability to undertake parametric-type analyses provides
the geotechnical engineer with a useful tool in the study of
potential hazards. Comparative studies between the dynamic
programming method and methods of slices show that the
computed results are similar when the shape of the critical
slip surface is known. However, there are situations where
the actual shape of the critical slip surface is not known, such
as in the case of rainfall infiltration at ground surface. The
dynamic programming solution appears to show a critical
slip surface that goes slightly deeper (e.g., below the toe of
the slope). Critical slip surfaces located using limit equilib-
rium methods of slices appear to exit higher on the slope
(Pham and Fredlund, 2003). The stresses computed from a
finite element analysis appear to better reflect stress states
below the ground surface geometry.
All limit equilibrium methods of slices tend toward an
upper bound type of solution because the shape of the slip
surface is controlled by the analyst (Fredlund, 1984). Conse-
quently, it is anticipated that the computed factors of safety
would tend toward being slightly higher than the correct
solution. Various limit equilibrium methods of analysis can
yield slightly different factors of safety as a consequence of
the assumption invoked to render the analysis determinate,
but all results will tend toward an upper bound solution from
a plasticity standpoint.
The shape of the slip surface is not dictated by the ana-
lyst in the dynamic programming method. Consequently, the
solution should tend toward a more correct solution, and it
is expected that the computed factors of safety should be
slightly lower. This behavior was observed in the compara-
tive study undertaken when Poisson's ratio was 0.33. While
the factors of safety computed from the dynamic program-
ming technique are generally lower than (or equal to) those
computed by limit equilibrium analyses, the difference is
relatively small for the simple slope examples. The small
difference is encouraging because the increased flexibility
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