Environmental Engineering Reference
In-Depth Information
Table 3.17 Factors of safety for each case using the PEM
Case
tan
δ
P
i
F
FP
i
2
*
γ
eq
γ
bf
i
1
6.01
18.28
0.45
0.125
1.434
0.257
2
8.13
18.28
0.45
0.125
1.075
0.144
3
6.01
19.42
0.45
0.125
1.506
0.284
4
8.13
19.42
0.45
0.125
1.129
0.159
5
6.01
18.28
0.55
0.125
1.753
0.384
6
8.13
18.28
0.55
0.125
1.314
0.216
7
6.01
19.42
0.55
0.125
1.841
0.424
8
8.13
19.42
0.55
0.125
1.379
0.238
Sum
11.43
2.11
F
mean
1.434
Table 3.18
Comparison of the PEM assuming normal distribution of the factor of safety with
the Monte Carlo methods for the retaining wall example
Deterministic
factor of safety
Probability of
failure (%)
Failure mode
Probability method
Sliding on sand
1.40
Monte Carlo
2.4
Point Estimate, normal dist. of F
4.5
Sliding in clay
1.95
Monte Carlo
2.2
Point Estimate, normal dist. of F
5.0
Bearing capacity
1.97
Monte Carlo
1.8
Point Estimate, normal dist. of
F
4.3
Normal Distribution of the Factor of Safety:
1431
0 253
.
−
2
σ
=
211143
.
−
.
=
0 253
.
β
=
=
170
.
P
=
1
−
NORMSDIST
( .)
170
=
44.%
F
.
The same analysis is applied to the factor of safety against sliding on a clay surface and
bearing capacity failure. The standard deviation of the undrained shear strength of the clay is
estimated to be 24 kN/m
2
. The calculated probability of failure for sliding on a clay surface
is 5.00%. The calculated probability of failure for bearing capacity is 4.34%.
The results of the PEM assuming a normal distribution for the factor of safety is com-
pared to the Monte Carlo simulation method, showing how the simplifying assumptions of
the PEM affect the result in
Table 3.18.
3.12 PeM WIth a lognorMal DIStrIbutIon
For the FaCtor oF SaFetY
If a lognormal distribution for the factor of safety is assumed instead of a normal distribu-
tion, these steps are followed:
1. Same as for a normal distribution for the factor of safety
2. Same as for a normal distribution for the factor of safety
3. Same as for a normal distribution for the factor of safety
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