Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 111

(a) 35

PDF isoline (Plackett)
PDF isoline (Gaussian)
Limit state I
Limit state II
Limit state III

30

25

20

15

10

0 4 8 12 6 0 4 8

c (kPa)

(b) 35

PDF isoline (Frank)
PDF isoline (Gaussian)
Limit state I
Limit state II
Limit state III

30

25

20

15

10

0 4 8 12 6 0 4 8

c (kPa)

(c) 35

PDF isoline (No.16)
PDF isoline (Gaussian)
Limit state I
Limit state II
Limit state III

30

25

20

15

10

0 4 8 12 6 0 4 8

c (kPa)

Figure 2.10 Comparison of PDF isolines of shear strength parameters associated with different copulas for
ρ = − 0.5, COV c = 0.4 and COV ϕ = 0.2. (a) Plackett copula versus Gaussian copula, (b) Frank copula
versus Gaussian copula, and (c) No.16 copula versus Gaussian copula.

between cohesion and friction angle becomes stronger, the PDF isoline becomes narrower,
and the difference among the joint PDFs associated with different copulas becomes more
significant. Like the results shown in Figure 2.11 , the probabilities of failure will become
smaller although the limit state surfaces remain unchanged.