Practical reliability analysis and design by Monte Carlo Simulation in spreadsheet - Risk and Reliability in Geotechnical Engineering - page 318

Table 7.4 Feasible designs for SLS requirement (i.e., β t SlS = 2. and p t SlS = 000 47 )

.

Design approach B = 0.9 m B = 1.2 m B = 1.5 m

Expanded RBD with Subset Simulation D ≥ 6.4 m D ≥ 4.4 m D ≥ 3.2 m

Expanded RBD with Direct MCS D ≥ 6.2 m D ≥ 4.4 m D ≥ 3.4 m

7.6.6 effects of the driving variable
Since the driving variable Y is a key factor that affects the generation of conditional samples
of interest in Subset Simulation, proper selection of Y plays a pivotal role in the integration
of the expanded RBD approach with Subset Simulation. Note that the convention of Subset
Simulation is to define Y as a variable that monotonically increases as the simulation level
m increases. In the expanded RBD approach, the conditional samples of interest are failure
samples conditional on design parameters (e.g., B and D for drilled shaft). Since the failure
is defined as FS uls < 1 or FS sls < 1, failure samples have relative small values of FS . To assure
that Y is a monotonic variable that increases as FS decreases, Y is defined to be proportional
to the reciprocal of FS (i.e., 1 / FS ). As two FS are calculated in the deterministic model, the
minimum (i.e., FS min ) of these two FS is used to define Y .
In addition, the deterministic calculation model (i.e., Equations 7.23 through 7.27 ) for
drilled shaft design shows that the values of FS min decrease as the design parameters B and
D decrease. Using 1 / FS min as a driving variable tends to drive the sampling space to samples
with low FS min values, the overwhelming majority of which correspond to relatively small
values of B and D . On the other hand, failure samples with relatively large values of B and
D are also of interest in the expanded RBD approach. If there is an insufficient number
of failure samples with relatively large B or D values, the resolution and accuracy of the
estimates of their conditional probabilities (e.g., P ( B , D | F ) and P ( F | B , D )) would be poor,
and hence, some feasible designs with relatively large B or D values would not be identified
properly. To assure that feasible designs with a wide range of B and D values are all covered
properly in the expanded RBD approach, it is necessary to define the driving variable as a
combination of failure criterion (e.g., FS min ) and design parameters of interest (e.g., B and D
in drilled shaft design).
This study defines the driving variable as Y = BD/FS min . Subset Simulation generates sam-
ples with increasing values of BD/FS min as the level m increases. The increase of BD/FS min is
attributed to two factors: decrease in the denominator FS min and increase in the numerator
BD . Thus, the effect of driving variable Y = BD/FS min on the sampling process is two-fold.
On the one hand, due to the effect of denominator FS min (i.e., the effect of 1/FS min ), Subset
Simulation drives the sampling space to the failure domain with relatively small FS min values
that usually correspond to relatively small B and D values. On the other hand, because of
the effect of the numerator BD , Subset Simulation generates samples with relatively large
B and D values. The combined effects of the denominator FS min and numerator BD in the
driving variable Y = BD/FS min improve the efficiency of generating failure samples that cover
a wide range of B and D values, particularly those with relatively large values of B and D .
To illustrate the effect of the driving variable, two Subset Simulations are performed
with two different driving variables: one with Y = BD/FS min and the other with Y = 1/FS min .
Defining the driving variable as a function of the failure criterion (e.g., Y = 1/FS min ) is a com-
mon practice in Subset Simulation (e.g., Au et al., 2010). Two Subset Simulations both have
N = 10,000 samples per level, p 0 = 0.2, and the highest simulation level m = 4, resulting in
42,000 samples per simulation.
Table 7.5 summarizes the failure samples generated in these two Subset Simulation runs.
The Subset Simulation with Y = 1/FS min generates 22,253 ULS failure samples, among which,