Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 109

those shown in Figures 2.8a through c, but the probabilities of failure associated with the
Gaussian copula can be several orders of magnitude smaller than those associated with the
other three copulas, especially for a strong negative correlation between cohesion and fric-
tion angle. Furthermore, the probabilities of failure associated with the Gaussian copula
are very sensitive to the change of FS n . When FS n varies from 1.03 to 1.17, the probability
of failure associated with the Gaussian copula decreases from 3.63E-02 to 1.02E-07 (more
than four orders of magnitude!). The relative differences in probabilities of failure associ-
ated with different copulas and nominal factors of safety are listed in the last three columns
of Table 2.11 . The same conclusions as those drawn from the other three cases can also
be made. However, the ratios p f / p f Gaussian associated with FS n = 1.30 are significantly larger
than those for the other three cases. For instance, the ratios p f / p f Gaussian associated with
FS n = 1.30 are 2.45E04, 1.13E04, and 7.54E04 for the Plackett, Frank, and No.16 copulas,
respectively.
2.5.5 Discussions
It can be concluded from the above results that the probabilities of overturning failure for a
retaining wall associated with the four selected copulas differ greatly, especially when small
COVs or large correlation coefficients underlying shear strength parameters are used. In this
section, some discussions are presented to explain such observations following two ways:
(1) a comparison among simulated samples of cohesion and friction angle associated with
various copulas is carried out, and (2) relative locations between the limit state surfaces and
the joint PDF isolines of cohesion and friction angle are investigated.
Figure 2.9 shows the simulated samples of cohesion and friction angle from the selected
four copulas for ρ = −0.5, COV c = 0.4, and COV ϕ = 0.2. The sample size is 1000. The
contour lines of constants FS = 1.0, 1.4, and 2.0 for a representative retaining wall with
H = 6 m, a = 0.4 m, b = 1.4 m, γ soil = 18 kN/m 3 , and γ wall = 24 kN/m 3 are also plotted in
Figure 2.9 . Again, the numbers ( N ) of samples falling in the regions associated with FS ≤ 1.0,
1.0 < FS ≤ 1.4, 1.4 < FS ≤ 2.0, and FS > 2.0 are shown in the corresponding regions. Note
that different copulas characterize different dependence structures between cohesion and
friction angle although the same marginal distributions and correlation coefficient of shear
strength parameters are followed. For example, the numbers of samples falling in the afore-
mentioned four regions for the Gaussian copula are 38, 290, 323, and 349. They are 63, 197,
393, and 347 for the No.16 copula. It is noted that the region associated with FS ≤ 1.0 is of
more significance to practice because the probability of failure is basically derived from this
region. Thus, geotechnical engineers often pay more attention to the differences in this region
associated with different copulas. The numbers of samples falling in this region are 38, 47,
47, and 63 for the Gaussian, Plackett, Frank, and No.16 copulas, respectively. It is evident
from these results that the No.16 copula leads to the largest probability of failure while the
Gaussian copula results in the smallest probability of failure.
To make a better comparison between the Gaussian copula and the other three copulas,
Figure 2.10 shows the joint PDF isolines of shear strength parameters associated with the
four copulas selected. The joint PDF isoline associated with the Gaussian copula is plotted
using a dashed line. For illustration, a typical PDF isoline value of 0.001 is used. This makes
the PDF isoline envelope nearly the whole domain where simulated samples may appear.
Note that the shape of such an isoline is similar to that of the scatter plots of simulated
samples. It is evident that the joint PDFs of the shear strength parameters associated with
different copulas differ considerably, especially between the Gaussian and No.16 copulas.
Such a difference further leads to the difference in probability of failure between Gaussian