Practical reliability analysis and design by Monte Carlo Simulation in spreadsheet - Risk and Reliability in Geotechnical Engineering - page 317

7.6.4 Determination of feasible designs
Figure 7.8a and b shows, by solid lines, the conditional probability P ( F | B , D ) obtained
from a single run of Subset Simulation for ULS and SLS failures, respectively. Note that
P ( F | B , D ) is a variation of failure probability as a function of ( B , D ). The horizontal axis
in Figure 7.8 represents the variation of D , and the values of p f ULS or p f SLS for three dif-
ferent values of B are included in the figure. For a given value of B , p f ULS and p f SLS for
the respective ULS and SLS failures decrease as D increases. Similarly, for a given value
of D , p f ULS and p f SLS decrease as B increases. Figure 7.8 also includes p T ULS = 0 00069 .

and p T SLS = 00. 47 adopted for transmission line foundations in North America (Phoon
et al., 2003a,b), and feasible designs are those that fall below the p T ULS and p T SLS shown
in the figure.
For the ULS requirement ( Figure 7.8a ) , the feasible designs include the drilled shafts with
B = 1.5 m and D ≥ 2.0 m, B = 1.2 m and D ≥ 3.0 m, or B = 0.9 m and D ≥ 4.6 m. For the SLS
requirement ( Figure 7.8b ), the feasible designs include those with B = 1.5 m and D ≥ 3.2 m,
B = 1.2 m and D ≥ 4.4 m, or B = 0.9 m and D ≥ 6.4 m. For a given value of B , the minimum
feasible values (i.e., 3.2, 4.4, and 6.4 m for B = 1.5, 1.2, and 0.9 m, respectively) of D for the
SLS requirement are larger than those (i.e., 2.0, 3.0, and 4.6 m for B = 1.5, 1.2, and 0.9 m,
respectively) for the ULS requirement. The SLS requirement is therefore the critical one that
controls the design, and the feasible designs are the same as those for the SLS requirement.
Phoon et al. (1995) and Wang et al. (2011a) also found in their design calculations for this
example that the SLS requirement is the critical one, which is consistent with the observa-
tion herein.

7.6.5 results comparison

Figure 7.8 also shows, by dashed lines, the values of p f ULS or p f SLS estimated from Direct
MCS with 10,000,000 (Wang et al., 2011a). It is evident that the dashed lines almost over-
lap the solid lines, and the results from Subset Simulation are in good agreement with those
from Direct MCS. Tables 7.3 and 7.4 summarize the respective feasible designs for ULS and
SLS requirements from Subset Simulation and Direct MCS. For ULS requirement, the fea-
sible designs obtained from Direct MCS are those with B = 1.5 m and D ≥ 2.0 m, B = 1.2 m
and D ≥ 2.8 m, or B = 0.9 m and D ≥ 4.6 m, which agree well with those (i.e., B = 1.5 m
and D ≥ 2.0 m, B = 1.2 m and D ≥ 3.0 m, or B = 0.9 m and D ≥ 4.6 m) obtained from
Subset Simulation (see Table 7.3 ). Similar observations are also found from the SLS feasible
designs obtained from Direct MCS and Subset Simulation (see Table 7.4 ) . Such good agree-
ment suggests that the integration of the expanded RBD approach with Subset Simulation
works well. When compared with the expanded RBD approach with Direct MCS, only
42,000 random samples are, however, needed in Subset Simulation, which are much less
than the 10,000,000 samples used in Direct MCS. Subset Simulation substantially improves
the computational efficiency at small probability levels, and it significantly enhances the
expanded RBD approach in design situations with a small target failure probability (e.g.,
p T ULS = 0 000 . 9 and p T SLS = 000 . 47 in this example).

Table 7.3 Feasible designs for ULS requirement (i.e., β t ulS = 3. and p t ulS = 0 000

. 9 )

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

Expanded RBD with Subset Simulation

D ≥ 4.6 m D ≥ 3.0 m D ≥ 2.0 m

Expanded RBD with Direct MCS

D ≥ 4.6 m D ≥ 2.8 m D ≥ 2.0 m