Application of the subset simulation approach to spatially varying soils - Risk and Reliability in Geotechnical Engineering - page 562

Table 15.8 Evolution of the failure threshold C j with different levels j of the SS approach and with the
number of realizations N s per level

Number of realizations N s per level

Failure threshold C j for
each level j 50 100 150 200 250

C 1 0.0086 0.0077 0.0080 0.0076 0.0076

C 2 0.0058 0.0048 0.0050 0.0041 0.0040

C 3 0.0044 0.0015 0.0019 0.0011 0.0011

C 4 0.0017 − 0.0019 − 0.0007 − 0.0020 − 0.0018

C 5 − 0.0015 - - - -

Table 15.9 Values of P e and COV Pe versus the number N s of realizations per level
Number of realizations N s per level

50 100 150 200 250

P e

0.34 × 10 − 4 4.60 × 10 − 4 2.07 × 10 − 4 3.78 × 10 − 4 3.77 × 10 − 4

COV Pe (%) 92 71 60 51 38

As mentioned before, a random field with l ln x = 10 m and l ln y = 1 m was considered herein.
The number M of terms of K-L expansion adopted in the analysis is equal to M = 100. This
corresponds to an error estimate that is smaller than 13%. The intermediate failure prob-
ability p 0 of a given level j ( j = 1, 2, …, m − 1) was chosen equal to 0.1.
The failure thresholds C j of the different levels of the SS were calculated and presented
in Table 15.8 for different values of N s . Table 15.9 presents the P e values and the corre-
sponding values of the coefficient of variation for the different numbers of realizations N s .
As expected, the coefficient of variation of P e decreases with the increase in the number of
realizations N s .
From Tables 15.8 and 15.9 , it was found that P e converges when N s = 200 realizations.
This is because (i) the C j values corresponding to N s = 200 and 250 realizations are quasi-
similar and (ii) the corresponding final P e values are too close (they are equal to 3.78 × 10 −4
and 3.77 × 10 −4 , respectively). Notice that the values of COV Pe for N s = 200 and 250 realiza-
tions are equal to 51 and 38%, respectively, which indicates (as expected) that the COV Pe
decreases with the increase in the number of realizations. It should be mentioned here that
for p 0 equal to 0.1, four levels of SS were found necessary to reach the limit state surface
G = 0 as may be seen from Table 15.8 . Therefore, when N s = 200 realizations, a total number
of N t = 200 × 4 = 800 realizations were required to calculate the final P e value. Remember
that in this case, the COV of P e was equal to 51%. Notice that if the same value of COV (i.e.,
51%) is desired by MCS to calculate P e , the number of realizations would be equal to 12,000
(see Ahmed and Soubra 2012). This means that, for the same accuracy, the SS approach
reduces the number of realizations by 93.3%. On the other hand, if one uses MCS with the
same number of realizations (i.e., 800 realizations), the value of COV of P e would be equal
to 189% (see Ahmed and Soubra 2012). This means that for the same computational effort,
the SS approach provides a smaller value of COV Pe than MCS.

15.6 ConCluSIon

The probabilistic analysis of shallow foundations resting on a spatially varying soil was
generally performed in the literature using MCS methodology. The mean value and the