Environmental Engineering Reference
In-Depth Information
Table 7.5 Summary of failure samples generated by different driving variables
Number
of failure
samples
Percentage of failure samples (%)
D max (m)
Limit
state
Driving
variable Y
B = 0.9 m
B = 1.2 m
B = 1.5 m
B = 0.9 m
B = 1.2 m
B = 1.5 m
ULS
1/FS min
22,253
98.3
1.7
0.0
4.4
2.8
N/A
BD/FS min
12,844
93.4
6.5
0.1
5.2
3.4
2.4
SLS
I/FS min
31,042
90.4
8.3
1.3
6.6
4.6
3.4
BD/FS min
24,708
82.7
15.0
2.3
7.4
5.6
3.8
98.3% have B = 0.9 m, 1.7% have B = 1.2 m, and none of them have B = 1.5 m. Since there
is no failure sample with B = 1.5 m, it is difficult to estimate failure probability for designs
with B = 1.5 m. In contrast, the Subset Simulation with Y = BD/FS min generates 12,844 ULS
failure samples. The percentage of ULS failure samples with B = 0.9 m decreases to 93.4%,
while the percentages of ULS failure samples with B = 1.2 m and B = 1.5 m increase to 6.5
and 0.1%, respectively. The simulation with Y = BD/FS min generates more ULS failure sam-
ples with relatively large B values (e.g., B = 1.2 m and B = 1.5 m) than the simulation with
Y = 1/ FS min . In addition, Table 7.5 shows that the maximum values ( D max ) of D among the
ULS failure samples at Y = BD/FS min are also larger than those at Y = 1/ FS min . Similar obser-
vations are also found for the SLS failure samples summarized in the fourth and fifth rows
of Table 7.5 . It is evident that using Y = BD/FS min leads to failure samples with relatively
large B and D values. It is more appropriate to use Y = BD/FS min than Y = 1/ FS min in the
expanded RBD of drilled shafts.
7.7 IlluStratIVe eXaMPle II: JaMeS baY DIke DeSIgn SCenarIo
In this section, a design scenario of the James Bay Dike in Canada is reanalyzed using the
probabilistic failure analysis approach for identifying the uncertain parameter that is the
most influential to the dike performance and for understanding how the performance (e.g.,
failure probability P ( F )) of the designed dike changes as this important uncertain parameter
changes. The James Bay Dike is a 50-km-long earth dike of the James Bay hydroelectric
project in Canada. Soil properties and various design scenarios of the dike were studied by
Ladd et al. (1983), Soulié et al. (1990), Christian et al. (1994), El-Ramly et al. (2002), Xu
and Low (2006), and Wang et al. (2010). As shown in Figure 7.9 , the embankment is 12 m
56.0 m
45
( x A , y A )
3
35
1
3
12.0 m
γ Fill ; ϕ Fill
Embankment
1
25
Clay crust
T cr
S uM
15
Marine clay
Lacustrine clay S uL
8.0 m
T L
D Till
5
Till
Critical slip circle
-5
0
20
40
60
80
100
120
140
160
Distance (m)
Figure 7.9 Illustration of a James Bay Dike design scenario.
 
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