Modeling and simulation of bivariate distribution of shear strength parameters using copulas - Risk and Reliability in Geotechnical Engineering - page 106

deterministic parameters can be studied. Second, FS n is closer to the nominal factor of safety
computed in practice, because engineers will use lower bound strength values, rather than
mean values.

2.5.4 reliability results produced by different copulas
The impact of copula selection on probability of failure is systematically studied based
on three factors: (1) geometrical parameters ( a , b ) ( Figure 2.7 ), (2) COV scaling factor, λ
defined as COV c = 0.4/λ and COV ϕ = 0.2/λ, and (3) ρ. In the parametric studies shown in
Figure 2.8 , each factor is varied over a range of values shown in the figure caption while the
other parameters remain unchanged. To facilitate comparisons between the four subplots in
Figure 2.8 , changes in each factor are presented in a uniform way as changes in the nominal
factor of safety. In other words, the horizontal axes are identical, although changes in the
nominal factor of safety are caused by different factors in each subplot.

2.5.4.1 Effect of geometrical parameters on probability of failure
Applying the direct integration method in Section 2.5.2, the probability of failure of the
retaining wall is obtained. The nominal factor of safety is calculated by the steps presented

(a) (b)

10 -1 10 -1

10 -2 10 -2

Gaussian copula
Plackett copula
Frank copula
No.16 copula

Gaussian copula
Plackett copula
Frank copula
No.16 copula

10 -3 10 -3

1.03 1.06 1.09 1.12 1.15 1.18 1.21 1.24 1.27 1.30 1.03 1.06 1.09 1.12 1.15 1.18 1.21 1.24 1.27 1.30

Nominal factor of safety Nominal factor of safety

(c) (d)

10 -1 10 -1

10 -2

10 -2 10 -3

10 -4

10 -3 10 -5

Gaussian copula
Plackett copula
Frank copula
No.16 copula

Gaussian copula
Plackett copula
Frank copula
No.16 copula

10 -6

10 -7 10 -4

1.03 1.06 1.09 1.12 1.15 1.18 1.21 1.24 1.27 1.30 1.03 1.06 1.09 1.12 1.15 1.18 1.21 1.24 1.27 1.30

Nominal factor of safety Nominal factor of safety

Figure 2.8 Probabilities of retaining wall overturning failure produced by different copulas. (a) a varying from
0.40 m to 0.58 m, (b) b varying from 1.40 m to 1.64 m, (c) λ varying from 1.00 to 1.95, and (d) ρ
varying from − 0.50 to − 0.88.