Practical reliability analysis and design by Monte Carlo Simulation in spreadsheet - Risk and Reliability in Geotechnical Engineering - page 320

high with a 56-m-wide berm at mid-height. The slope angle of the embankment is about
18.4° (3H:1 V). The embankment is overlying on a clay crust with a thickness T cr . The clay
crust is underlain by a layer of 8.0-m-thick sensitive marine clay and a layer of lacustrine
clay with a thickness T L . The undrained shear strength (i.e., S uM and S uL ) of the marine clay
and the lacustrine clay was measured by field vane tests (Ladd et al., 1983; Soulié et al.,
1990; Christian et al., 1994). The lacustrine clay is overlying on a stiff till layer, the depth
to the top of which is D Till .
Six system parameters have been identified as uncertain parameters in the previous stud-
ies (e.g., El-Ramly et al., 2002; Xu and Low, 2006; Wang et al., 2010), including the friction
angle ϕ Fill and unit weight γ Fill of embankment material, the thickness T cr of clay crust, the
undrained shear strength S uM of the marine clay, the undrained shear strength S uL of the
lacustrine clay, and the depth of the till layer D Till . In the reliability analysis of the design
scenario, these six uncertain parameters are represented by six independent Gaussian ran-
dom variables (El-Ramly, 2001), respectively. Table 7.6 summarizes the statistics (i.e., mean,
standard deviation, and COV) of these six random variables. These statistics are used to
generate random samples for each random variable in an uncertain model worksheet, as
shown in Figure 7.10 . Note that the thickness of the lacustrine clay T L is an uncertain
variable that depends on T cr and D Till and has a mean of about 6.5 m (see Figure 7.9 ) . In
addition to these uncertain parameters, other system parameters are taken as deterministic,
including an undrained shear strength of 41 kPa for the clay crust and unit weights of 19,
19, and 20.5 kN/m 3 for the clay crust, marine clay, and lacustrine clay (El-Ramly, 2001),
respectively. With these soil properties, a circular critical slip surface is used together with
a simplified Bishop method to calculate FS along the critical slip surface in the determin-
istic slope stability analysis (El-Ramly, 2001; Wang et al., 2010). The critical slip surface
is always tangential to the top of the till layer and passes through the point A ( x A = 4.9 m,
y A = 36.0 m), as shown in Figure 7.9 . The x -coordinate of the center is fixed at 85.9 m. For
each set of simulation samples, the critical slip surface is uniquely specified by the value of
D Till , and its corresponding FS is calculated in a deterministic model worksheet, as shown
in Figure 7.11 .
After the uncertain model worksheet (see Figure 7.10 ) and deterministic model work-
sheet (see Figure 7.11 ) are set up, the Subset Simulation Add-In (see Figure 7.12 ) is invoked
for uncertainty propagation. A Subset Simulation run is performed with the highest simu-
lation level m = 3, p 0 = 0.1, and N = 1000 samples per level. The driving variable Y in
Subset Simulation is defined as 1/ FS to drive the sampling space to gradually approach
the failure domain, which is of particular interest in probabilistic failure analysis. After
the simulation, a total of 3700 samples are generated from the Subset Simulation, includ-
ing 900 samples for simulation levels '0', '1', and '2', respectively, and 1000 samples for
simulation level '3'.

Table 7.6 Uncertainty characterization of the James Bay Dike design example
Soil layers

Uncertain parameters a Mean Standard deviation COV (%)

Embankment ϕ Fill (°) 30.0 1.79 6.0

Embankment 20.0 1.10 5.5

γ Fill (kN/m 3 )

Clay crust T cr (m) 4.0 0.48 12.0

Marine clay S uM (kN/m 2 ) 34.5 3.95 11.5

Lacustrine clay S uL (kN/m 2 ) 31.2 6.31 20.2

Till D Till (m) 18.5 1.00 5.4

All uncertain parameters follow normal distributions. The thickness of the lacustrine clay layer T L is an uncertain param-
eter that depends on T cr and D Till and has a mean of about 6.5 m.

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