Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 110

(a) (b)

40 40

2.0 2.0

35 35

1.4 1.4

30 30

1.0 1.0

25 25

20 20

15 15

10 10

N = 290

N = 38 N = 323 N = 349 N = 47 N = 266 N = 347 N = 340

5 5

04 8 2 6 0 4 8 2 6
c (kPa)

04 8 2 6 0 4 8 2 6

c (kPa)

(c) (d)

40 40

2.0

2.0

35

35

1.4

1.4

30 30

1.0

1.0

25 25

20

20

15 15

10 10

N = 47 N = 272 N = 341 N = 63 N = 197 N = 347

N = 340 N = 393

5

5

04 8 2 6 0 4 8 2 6
c (kPa)

04 8 2 6 0 4 8 2 6
c (kPa)

Figure 2.9 Scatter plots of c and ϕ generated by different copulas with ρ = − 0.5, COV c = 0.4 and COV ϕ = 0.2.
(a) Gaussian copula, (b) Plackett copula, (c) Frank copula, and (d) No.16 copula.

copula and the other three copulas, as shown in Figure 2.8 . To investigate the effect of rela-
tive locations between the limit state surfaces and the joint PDF isolines on reliability, three
limit state surfaces are considered. They are associated with three retaining walls defined by
(1) a = 0.4 m, b = 1.4 m, and FS n = 1.03, (2) a = 0.58 m, b = 1.4 m, and FS n = 1.30, and (3)
a = 0.4 m, b = 1.64 m, and FS n = 1.30. The respective limit state surfaces are referred to as
“Limit state I”, “Limit state II”, and “Limit state III”. It can be seen that the shift of FS n from
1.03 to 1.30 is due to the shift of a from 0.4 to 0.58 m or b from 1.4 to 1.64 m. Similarly,
the shift of FS n from 1.03 to 1.30 is caused by the shift of limit state surfaces from I to II
or from I to III. Consequently, the probability of failure decreases when FS n increases from
1.03 to 1.30 although the joint PDF of the shear strength parameters remains unchanged.
Figure 2.11 shows the joint PDF isolines of the shear strength parameters associated with
different copulas for ρ = −0.5, COV c = 0.2, and COV ϕ = 0.1. Compared with Figure 2.10 , the
boundary of the joint PDF isoline becomes smaller as the COVs of c and ϕ decrease. In other
words, the probability content of the joint PDF over the same failure set of the retaining wall
decreases. As a result, the probabilities of failure of the retaining wall will thus decrease
although the limit state surfaces do not change. Similarly, Figure 2.12 shows the joint PDF
isolines of the shear strength parameters associated with different copulas for ρ = −0.8,
COV c = 0.4, and COV ϕ = 0.2. In comparison with Figure 2.10 , as the negative correlation