Environmental Engineering Reference
In-Depth Information
β =2.795
4
β = 2.893
3
β
2
1
-16
-12
-8
-4
04
8 2 6
x
exit
30
β =2.795
β = 2.893
20
Clay layer 1
10
0
Clay layer 2
-10
0
10
20
30
40
50
-10
Figure 9.10
Variation of reliability indices with the x-coordinate of the lower exit point of slip circles, and
the two reliability-based critical slip circles with lower exit points at x
exit
of
−
1.5 m and
+
9 m,
respectively.
It was noted, for example, in Ang and Tang (1984) that the bimodal bounds on failure
probability of systems with multiple failure modes will depend on the ordering of the individ-
ual failure modes. It was suggested, for example, in Madsen et al. (1986), Melchers (1999),
and Haldar and Mahadevan (1999) that ordering the failure modes in decreasing prob-
abilities of failure will lead to closer bounds. This has been done in
Figure 9.11
, yielding
two local minimum modes were considered. A simple Visual Basic for Applications (VBA)
code was also created to investigate the effects of all possible permutations (8!) of the failure
modes on the system failure probability bounds: the same bounds as in
Figure 9.11
were
obtained.
That the system reliability bounds of the eight modes in
Figure 9.11
d
iffer little from the
bounds of the two local minimum modes of
Figure 9.9
can be attributed to the strong cor-
relations among modes 1, 3, 6, and 7, and among modes 2, 4, 5, and 8, as seen from the very
high (≈1.0) intermodal correlation coefficients of ρ
13
, ρ
16
, ρ
17
, and of ρ
24
, ρ
25
, ρ
28
. Physically,
this means that direction vectors (linking the mean-value point and design points of the
failure modes) are nearly parallel for modes 1, 3, 6, and 7, and for modes 2, 4, 5, and 8. The
implied overlapping of the failure probability contents for modes 1, 3, 6, and 7, and also for
modes 2, 4, 5, and 8 means that it is sufficiently accurate to calculate the bounds for the
system failure probability by considering only the two stationary values of reliability index,
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