Environmental Engineering Reference
In-Depth Information
to the Total Probability Theorem (e.g., Ang and Tang, 2007), the failure probability can be
written as
m
∑
|
0
PF
()
=
PFLPL
i
(
) ()
(7.18)
i
i
=
where
L
0
= {
Y
≤
y
1
},
L
i
= {
y
i
≤
Y
≤
y
i
+1
} for
i
= 1, …,
m
− 1,
L
m
= {
Y
≥
y
m
},
P
(
F
|
L
i
) is the con-
ditional failure probability given sampling in
L
i
, and
P
(
L
i
) is the probability of the event
L
i
.
P
(
F
|
L
i
) is estimated as the fraction of the failure samples in
L
i
. The failure samples are
collected from samples generated by Subset Simulation and are based on the performance
failure criteria (e.g.,
FS
< 1).
P
(
L
i
) is calculated as
PL
()
()
=−
=− =… −
=
1
p
0
0
i
i
+
PL
pp
1
,
i
1
,
,
m
1
(7.19)
i
0
0
PL
()
p
m
m
0
()
L
When
P
(
F
),
P
(
F
|
L
i
), and
P
(
L
i
) are
obtained, the conditional probability
P
(
L
i
|
F
) is calculated using the Bayes' theorem as
Note that
P
(
L
i
∩
L
j
) = 0 for
i
≠
j
and
∑
m
P
=
1
i
=
0
i
PF
(|
LPL
PF
)( )
()
i
i
(7. 2 0)
PL F
(
|
)
=
i
The conditional PDF
p
(
x
k
|
F
) of an uncertain parameter
X
k
is then given by the Total
Probability Theorem as
m
∑
0
px F
(
)
=
p xL
(
∩
FPLF
) (
)
(7. 21)
k
k
i
i
i
=
where
p
(
x
k
|
L
i
∩
F
) is the conditional PDF of
x
k
estimated from failure samples that lie in
L
i
through the histogram.
x
k
is divided into a number of bins (e.g.,
n
b
bins), and
p
(
x
k
|
L
i
∩
F
) for
x
k
within a bin
j
(i.e.,
Px
(
∈
bin
|
L
∩ , for
j
= 1, 2, …,
n
b
) is estimated as
F
)
k
j
i
n
n
ji
fi
px
(|
L
∩≈
FPx in
)
(
∈
|
L
∩=
F
)
(7. 2 2)
k
i
k
j
i
in which
n
ji
is the number of failure samples at the simulation level '
i
' with the
x
k
value
falling into bin
j
and
n
i
is the total failure sample number at the simulation level '
i
'. Using
Equations 7.18,
7.21
,
and
7.22
,
P
(
F
) and
p
(
x
k
|
F
) can be calculated from Subset Simulation.
P
(
F
|
x
k
) is then estimated using
Equation 7.12
accordingly. This is further illustrated using a
slope stability analysis example later in Section 7.7.
7.5 SPreaDSheet IMPleMentatIon oF MCS-baSeD
relIabIlItY analYSIS anD DeSIgn
This section implements the MCS-based reliability analysis and design procedures (e.g.,
expanded RBD and probabilistic failure analysis described above) in a spreadsheet
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