Environmental Engineering Reference
In-Depth Information
The probability of failure,
p
f
, is given by the following double integral:
∫∫
p
=
f cc
(, )φφ
dd
f
(2.36)
FS
≤
1
where
f
(
c
, ϕ) is the joint PDF of
c
and ϕ. Applying
Equation 2.6
,
Equation 2.36
can be fur-
ther expressed as
∫∫
1
p
=
f cf
() () ((),
φ
DF cF
();)
φ θφ
dd
c
f
2
1
2
(2.37)
FS
≤
1
It is evident from
Equation 2.37
that the double integral could be time-consuming. For
this reason, the first derivative of a copula function is employed, which is given by
Mu u
(, ;)
θ
=∂
C uu
(, ;)
θ
/
∂
u
12
12
2
(2.38)
By substituting
Equation 2.38
into
Equation 2.37
,
the double integral in
Equation 2.37
can be reduced to a single integral:
∫
p
=
f
() ((),
φ
MF cF
();)
φ θφ
d
f
2
1
2
(2.39)
FS
≤
1
For further derivation, the expression of
c
in terms of ϕ should be available, which can
be obtained based on the limit state function
g
(
c
, ϕ) =
FS
(
c
, ϕ) − 1 = 0. For the retaining
wall example, by applying Shengjin's formulas (Fan 1989) to solve the cubic equation with
respect to
c
,
c
can be expressed as
(
)
6
W
×
rm
+
W
×
rm
1
1
2
2
c
=
05
. γ
K
H
−
(2.40)
3
soil
a
γ
K
soil
a
Substituting
Equation 2.40
into
Equation 2.39
,
one can obtain the probability of retain-
ing wall failure:
φ
0
6
(
W
×
rm
+
W
×
rm
)
∫
1
1
2
2
p
=
f
()
φ
MF
05
.
γ
KH
−
,
F
2
();
φθφ
d
(2.41)
3
f
2
1
soil
a
γ
K
soil
a
0
in which ϕ
0
is calculated by
π
6
(
W
×
rm
+
W
×
rm
)
1
1
2
2
φ
=−
2
arctan
(2.42)
0
2
γ
H
3
soil
Note that the probability of failure shown in
Equation 2.41
is easily solved when the
first derivative of a copula is available. For convenience,
Table 2.10
summarizes the first
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