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
⎦
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
5
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
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 DonHill
 Slope, Normal Distribution, reliability index, Slope Failure

Click to edit the document details