Environmental Engineering Reference
In-Depth Information
response (e.g.,
FS
, 1/
FS
, or
BD
/
FS
min
) calculated in the deterministic modeling worksheet
are random.
7.5.3 uncertainty propagation
When the deterministic analysis and uncertainty model worksheets are completed and
linked together, MCS or Subset Simulation procedure is invoked for uncertainty propaga-
tion. An Excel Add-In called UPSS (Uncertainty Propagation using Subset Simulation) has
been developed for implementing Direct MCS and Subset Simulation (Au and Wang, 2014;
Au et al., 2010). UPSS can be obtained through the following webpage:
https://sites.google.
com/site/upssvba/
.
Figure 7.4
shows a variant of UPSS that is tailored to the integration of
Subset Simulation with the expanded RBD approach (Wang and Cao, 2013). The Subset
Simulation userform is shown in
Figure 7.4a
.
The upper four input fields of the userform
(i.e., number of Subset Simulation runs, number of samples per level
N
, conditional prob-
ability
p
0
from one level to the next level, and the highest Subset Simulation level
m
) con-
trol the number of samples generated by Subset Simulation. The lower four input fields of
the userform record the cell references of the random variables, their PDF values, and the
cell references of the system response (e.g., driving variable
Y
), and other variables
V
that
will be recorded during the simulation, respectively. After setting up the userform, Subset
Simulation can be performed by clicking the “Run” button.
After each simulation run, the Add-In provides the CCDF of the driving variable versus
the threshold level, that is, the estimate for
P
(
Y
>
y
) versus
y
, into a new spreadsheet, and
a plot is produced. Then, based on the output information from Subset Simulation, failure
probability
P
(
F
) or its conditional counterparts (e.g.,
P
(
F
|
B
,
D
) for the expanded RBD of
drilled shafts and
P
(
F
|
x
i
) for probabilistic failure analysis) are calculated using the proce-
dures and equations described in Sections 7.3 and 7.4.
To facilitate the integration of Subset Simulation with the expanded RBD approach,
Equations 7.4
and
7.6
through
7.9
are implemented in Excel spreadsheet as a VBA Add-In
for calculating the conditional failure probability (e.g.,
P
(
F
|
B
,
D
)) for an expanded RBD.
Figure 7.4b
shows the RBD userform of the Add-In. The checkbox “RBD” at the top of the
userform is used to enable/disenable the input fields of “Failure Modes,” “Values of Design
Parameters,” and “Samples of Design Parameters” for RBD. The Add-In can calculate con-
ditional failure probability for up to five failure modes simultaneously, for example, ULS
and SLS failures in the RBD of drilled shafts. Each failure mode is defined by three input
fields, including the system response (e.g., values of 1/
FS
uls
or 1/
FS
sls
of random samples gen-
erated during Subset Simulation) of interest, the type of the failure criterion (i.e., >, =, or <),
and a critical value (e.g., one for 1/
FS
uls
and 1/
FS
sls
) for judging the occurrence of failure. The
two input fields (i.e., “Values of Design Parameters” and “Samples of Design Parameters”)
at the bottom require cell references of possible values of design parameters (e.g.,
B
and
D
in the RBD of drilled shafts) and their random samples generated during Subset Simulation,
respectively. After setting up the userform, the conditional failure probability is calculated
by clicking the “Run” button. After the calculation, the RBD Add-In generates a worksheet
for each failure mode, which contains the conditional failure probability of designs with
various combinations of design parameters for the failure mode and a plot of the conditional
failure probability versus design parameters.
The next two sections illustrate how the spreadsheet Add-In is used in expanded RBD
with Subset Simulation and probabilistic failure analysis with Subset Simulation, respec-
tively. The three modules of the spreadsheet implementation are illustrated in detail through
a drilled shaft design example in the next section.
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