Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 362

reliability-based critical slip surfaces are considered, the system failure probability bounds
obtained in Low et al. (2011) agreed well with the MCS and IS results of Ching et al. (2009).
As shown in Figure 9.9 , the upper clay layer is 18 m thick, with undrained shear strength
c u 1 ; the lower clay layer is 10 m thick, with undrained shear strength c u 2 . The undrained
shear strengths are normally distributed and independent. A hard layer exists below the
second clay layer.
Since the shear strengths are characterized by c u 1 and c u 2 , with ϕ u = 0, Bishop's simplified
method and the ordinary method of slices will yield the same factor of safety, and either
method can be used. Also, in this case where the upper clay layer is weaker than the lower clay
layer, it is logical to locate two reliability-based critical slip circles, as shown in Figure 9.9 ,
one entirely in the upper clay layer and the other passing through both layers. The FORM
reliability indices for the two modes are 2.795 and 2.893, respectively. It is interesting to note
that although c u 1 and c u 2 are uncorrelated, there is correlation between the two failure modes
(ρ 12 = 0.4535), because c u 1 affects both slip circles. The bounds on system failure probability,
computed in two cells in Figure 9.9 by efficient implementation (in a ubiquitous spreadsheet
platform) of the Kounias-Ditlevsen bimodal bounds for systems with multiple failure modes,
are 0.432-0.441%, compared with the MCS estimated range of 0.37-0.506% (from Ching
et al.'s reported MCS mean of 0.44% and COV of 15.04%).
The two reliability-based critical slip circles in Figure 9.9 h ave the smallest β values among
all possible slip circles tangent to the bottoms of the upper and lower clay layers, respec-
tively. One can search for more reliability-based critical slip circles corresponding to differ-
ent trial-tangent depths. Alternatively, a series of β values can be obtained as a function of
the x-coordinate values of the lower exit end of critical slip circles, as shown in Figure 9.10 ,
where the existence of two stationary values (“troughs”) of β is obvious. It would be inter-
esting to investigate the effect on the bounds of system failure probability when more reli-
ability-based modes are considered. This is done in Figure 9.11 , which, in contrast to Figure
9.9 , has three additional modes (β 3 , β 6 , β 7 ) adjacent to the mode corresponding to the local
minimum β 1 , and three additional modes (β 4 , β 5 , β 8 ) adjacent to the mode corresponding to
the local minimum β 2 .

Top layer c u 1 , lower layer c u 2 , both lognormally distributed and uncorrelated
For both layers, γ =19 kN/m 3

32 m

μ σ

x* n

β

18 m
Clay1

c u 1
c u 2

50.601 120 36 -2.795 2.795

β 1 = 2.795

24 m

153.25 160 48 0.000

β

c u 1
c u 2

78.194 12036 -1.312
71.888 16048 -2.579

2.893

10 m
Clay2

β 2 = 2.893

4 m

Hard layer

ρ of failure modes
ρ = A T R -1 A

Matrix A

n 1 /β 1 n 2 /β 2

-0.4535

β

-1.000 1 0.4535 2.7948

0.000 -0.8912 0.4535 1 2.8933

System P f bounds
Lower 0.432%

Upper 0.441%

Figure 9.9 FORM results for two reliability-based critical slip circles followed by system reliability analysis.