Environmental Engineering Reference
In-Depth Information
Table 7.9
Calculation of
p
(
S
uL
=
13|
F
) in the James Bay Dike design example
Simulation
level i
P (L
i
)
P (F|L
i
)
P (L
i
|F)
n
ji
n
i
p (S
uL
=
13|F)
0
0.9
0
0
0
0
0.269
1
0.09
0
0
0
0
2
0.009
0.1489
0.5727
27
134
3
0.001
1
0.4274
359
1000
The
S
uL
values of the 1134 failure samples are collected and analyzed. The maximum and
minimum
S
uL
values are found to be 21.7 and 8.6 kPa, respectively. Consider, for example,
estimating the failure probability
P
(
F
|
S
uL
= 13) when the
S
uL
value is taken as a deterministic
value of 13 kPa and the other uncertain parameters (i.e.,
D
Till
, γ
Fill
,
S
uM
, ϕ
Fill
, and
T
cr
) remain
as random. An
S
uL
bin of (12, 14] is adopted in the Bayesian analysis.
P
(
L
i
|
F
) is estimated
using
Equation 7.20
,
as summarized in the fourth column of
Table 7.9
.
Among all 134
failure samples that occur at the simulation level '2' (i.e.,
n
f2
= 134), 27 samples are found
to have
S
uL
values that lie within the bin (12, 14] (i.e.,
n
j2
= 27). Similarly, among all 1000
failure samples that occur at the simulation level '3' (i.e.,
n
f3
= 1000), 359 samples are found
to have
S
uL
values that lie within the bin (12, 14] (i.e.,
n
j3
= 359). As shown in the last col-
(
∈ 12 14
(
,
)
is then calculated analytically as
PS
12 14 14 12 , where
CDF
S
u
in
this example is a normal CDF with a mean and standard deviation of 31.2 and 6.31 kPa,
respectively. Finally,
Equations 7.12
and
7.15
are used to estimate the failure probability at
S
uL
= 13 kPa:
(
∈
(
,
)
=
CDF
(
)
−
CDF
(
)
uL
S
S
u
u
0 269 0 234
0 00204
.
×
.
%
P
(
FS
|
=
13
)
≈
PF
(
|
S
∈
(
12 14
,
)
=
=
0 3091
.
(7. 31)
uL
uL
.
analysis results by open squares for various
S
uL
values. Note that the conditional probability
1
0.1
0.01
0.001
Repeated simulation runs
Bayesian analysis
0.0001
12
14
16
18
20
22
S
uL
(kN
/
m
2
)
Figure 7.14
Bayesian analysis results for
S
uL
.
Search WWH ::
Custom Search