Environmental Engineering Reference
In-Depth Information
in Section 2.5.3.
Figures 2.8a
and b show the probabilities of failure on log scale produced
by different copulas for various values of
FS
n
as a result of varying
a
and
b
, respectively. It
is clear that the probabilities of failure produced by different copulas differ considerably.
The Gaussian copula results in the smallest probability of failure among the four selected
copulas. On the other hand, the No.16 copula leads to the largest probability of failure.
Therefore, the Gaussian copula, commonly used for modeling the bivariate distribution of
shear strength parameters, will significantly overestimate the retaining wall reliability. As
a
or
b
increases,
FS
n
increases, while the probability of failure decreases. It is interesting to
note that the probabilities of failure for equal values of
FS
n
as a result of varying
a
and
b
are the same, indicating the same sensitivity of
a
and
b
to the safety of the retaining wall.
Table 2.11
summarizes the relative differences in probabilities of failure associated with
different copulas and nominal factors of safety. In
Table 2.11
,
the values are calculated by
p
f
/
p
f
Gaussian
in which
p
f
is the probability of failure produced by the Plackett, Frank, and
No.16 copulas while
p
f
Gaussian
is the probability of failure produced by the Gaussian copula.
It is evident that the differences in probabilities of failure produced by different copulas
are significant, especially for
FS
n
= 1.30. For example, the ratios
p
f
/
p
f
Gaussian
associated with
FS
n
= 1.30 are 3.89, 2.92, and 21.68 for the Plackett, Frank, and No.16 copulas, respec-
tively. These results further indicate that the probabilities of failure of the retaining wall can
differ fairly significantly. Thus, it is important to identify the best-fit copula underlying the
measured data for shear strength parameters. Otherwise, the misuse of copula may cause
unacceptable errors in the probability of failure. The differences in probabilities of failure
increase with increasing
FS
n
or decreasing probability of failure, especially for the No.16
copula. For example, the ratio
p
f
/
p
f
Gaussian
associated with the No.16 copula increases from
1.68 to 21.68 when
FS
n
increases from 1.03 to 1.30.
2.5.4.2 Effect of COV of shear strength parameters on probability of failure
Figure 2.8c
shows the probabilities of failure on a log scale produced by different copulas
for various values of
FS
n
as a result of varying the COV scaling factor, λ. Like the results
shown in
Figures 2.8a
and
b
,
the probabilities of failure produced by different copulas
differ considerably. Again, the Gaussian copula produces the smallest probability of fail-
ure among the selected four copulas. As the COV
c
and COV
ϕ
decrease (or λ increases),
FS
n
increases and the probability of failure decreases. However, the probability of failure
ranges from 1.03 to 1.30, the probabilities of failure for the Gaussian copula are within
[3.63E-02, 1.77E-04] as shown in
Figure 2.8c
,
which is significantly wider than [3.63E-02,
1.19E-03] as shown in
Figures 2.8a
and
b
.
The relative differences in probabilities of failure
produced by different copulas and nominal factors of safety are also listed in
Table 2.11
.
When the same
FS
n
is adopted, the differences in probabilities of failure associated with
different copulas are significant. For the Plackett, Frank, and No.16 copulas, the ratios
p
f
/
p
f
Gaussian
associated with
FS
n
= 1.30 are 7.17, 5.47, and 87.45, respectively. The probability
of failure for the Gaussian copula is 87.45 times smaller than that for the No.16 copula.
In addition, for a specified copula, the differences in probabilities of failure increase as the
FS
n
increases.
2.5.4.3 Effect of correlation between cohesion and
friction angle on probability of failure
Figure 2.8d
shows the probabilities of failure on log scale produced by different copulas
for various values of
FS
n
as a result of varying ρ. The results are qualitatively the same as
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