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Application Example 17
(Reliability Analysis)
STABILITY OF NATURAL SLOPES
The possible instability of natural slopes is a major geologic hazard in many regions of the world.
The degree of stability of a slope may be expressed through the factor of safety F, which is the
ratio between a soil resistance measure and the associated applied load. Hence values of F larger
than 1 are associated with stability and values smaller than 1 indicate unstable conditions.
For natural slopes with a shallow soil layer (socalled infinite slopes; see Figure 1), the factor of
safety is given by (Skempton and DeLory, 1957):
'
c
+[γ
z
−
(z
−
z
w
)
γ
]
cos
2
α
tan
φ
'
(1)
w
F
=
γ
sin
z
α
cos
α
Eq. 1 can be simplified to:
F
=
c'
tan
φ
'
−
m
⎛
⎜
⎜
γ
w
tan
φ
'
⎞
(2)
γ
zsin
α
cos
α
+
tan
α
⎝
γ
tan
α
⎠
⎟
⎟
where
'
c
is the cohesion of the soil
φ
'
is the angle of internal friction of the soil
γ
is the unit weight of the soil
γ
w
is the unit weight of water
α
is the inclination of the slope to the horizontal
z
is the depth below the ground surface
z
w
is the depth of the water table below the ground surface
m
is the degree of saturation of the soil, m =
⎡
z
w
−
z
⎤
⎢
⎥
⎣
z
⎦
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0
5
10
15
20
0
5
10
15
20
25
)
)
z
w
z
SURFACE
ROCK
20
15
10
5
x (m
y (m
SOIL BEDROCK
INTERFACE/FAILURE
SOIL
Figure 1.
Infinite Slope Geometry and Definition of Parameters
As indicated above, the failure criterion is F < 1. For example, a slope with the parameters in
Table 1 has F= 1.15 and is stable.
z (m)
5
z
w
(m)
2.5
m
0.5
c' (kN/m
2
)
25
φ
'
30
α
35
γ
(kN/m
2
)
20
γ
w
(kN/m
2
)
9.81
Table 1.
Hypothetical slope parameters
However, if the parameters are uncertain, there may be a nonzero probability of slope failure. Of
the variables in Table 1, the most uncertain ones are the strength parameters
'
c
and
φ
'
, with
typical coefficients of variation
V
'
c
=
20
.
0
and
V
φ
'
=
25
.
0
. Experience shows that they may be
assumed to have normal distribution.
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This note was uploaded on 02/27/2008 for the course PTE 461 taught by Professor Donhill during the Fall '07 term at USC.
 Fall '07
 DonHill

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