Environmental Engineering Reference
In-Depth Information
it is equivalent to a sensitivity study. From this perspective, the MCS-based expanded RBD
approach can be considered as a process of performing systematic design sensitivity studies
(i.e., varying design parameters
B
and
D
) and determining the final design by comparing the
sensitivity study results with a predefined target failure probability
p
T
.
7.3.2 Desired sample number in direct MCS
The resolution and accuracy of
P
(
B
,
D
|
F
) and
P
(
F
) from MCS are pivotal for the
P
(
F
|
B
,
D
)
obtained (see
Equation 7.4
)
and depend on the number of failure samples generated in MCS.
As the number of failure samples increases, the resolution and accuracy improve. Since the
target failure probability
p
T
is predefined and generally small (e.g.,
p
T
= 0.001) in RBD, a
large number of failure samples necessitate an increase in the total sample number in MCS.
For example, the minimum MCS sample number (
n
min
) in the drilled shaft design example
is estimated as (Ang and Tang, 2007; Wang et al., 2011a)
((
1
/
p
)
−
1
)
n n
T
B
D
n
=
(7. 5)
min
(
COV
)
2
T
in which
COV
T
is the target coefficient of variation for the failure probability estimated from
bility level of interest (i.e.,
p
T
) decreases, the desired level of accuracy (i.e.,
COV
T
) improves,
or the number (
n
B
n
D
) of
B
and
D
combinations increases. For
p
T
= 0.001,
COV
T
= 30%,
and
n
B
n
D
= 1000,
Equation 7.5
l
leads to a relatively large
n
min
value of 11,100,000. If
p
T
and
COV
T
are further decreased to 0.0001 and 10%, respectively, the
n
min
value increases rap-
idly to 999,900,000. Such a large number of MCS samples require extensive computational
efforts and might lead to computational difficulties (e.g., long computational time and insuf-
ficient computer memory), particularly for design situations with a small target failure prob-
ability, a high-accuracy requirement, and a large number of design parameter combinations.
In this chapter, Subset Simulation is employed to calculate
P
(
B
,
D
|
F
),
P
(
F
), and
P
(
F
|
B
,
D
)
for the expanded RBD and to improve efficiency and resolution at small failure probability
levels, as discussed in the next section.
7.3.3 Integration of expanded rbD approach
with Subset Simulation
Consider designing a drilled shaft by the expanded RBD approach and performing a Subset
Simulation with
m
+ 1 level of simulations to estimate
P
(
B
,
D
|
F
),
P
(
F
), and
P
(
F
|
B
,
D
) in
Equation 7.4
.
Since the design parameters
B
and
D
are artificially treated as uncertain
parameters in the expanded RBD approach, their random samples are generated during
Subset Simulation. The sample space Ω of uncertain parameters (including
B
and
D
in the
expanded RBD of drilled shafts) is divided into
m
+ 1 individual subsets {Ω
i
,
i
= 0, 1, 2, …,
m
}
by the intermediate threshold values {
y
i
,
i
= 1, 2, …,
m
} of the driving variable
Y
(Au and
Wang, 2014). As mentioned before, the driving variable
Y
is a key factor that affects the gen-
eration of conditional samples of interest in Subset Simulation. For the drilled shaft design
problem herein,
Y
is defined as
BD
/
FS
min
, in which
FS
min
is the minimum FS among the
FS
uls
and
FS
sls
for the respective ULS and SLS requirements. The effects of the driving variable on
the expanded RBD will be further discussed in Section 7.6.6.
In Subset Simulation, the intermediate threshold values {
y
i
, i = 1, 2, …,
m
} of
Y
=
BD
/
FS
min
are adaptively determined to generate
m
+ 1 individual subsets {Ω
i
, i = 0, 1, 2, …,
m
}
of
B
and
D
, and samples in different subsets are generated level by level and correspond to
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