Environmental Engineering Reference
In-Depth Information
Table 3.11 Final factor of safety calculations for the
Simplified Hasofer Lind method
Assumed β
1.28
1.58
1.94
1.98
8.11
8.34
8.62
8.66
γ eq
18.76
18.72
18.69
18.68
γ bf
0.46
0.45
0.44
0.44
Tan δ
FS
1.12
1.07
1.01
1.000
)
dg
dx
i
α i
=
(3.28)
)
2
dg
dx
i
x i
′ =−
αβ σµ
*
+
(3.29)
i
xi
xi
σ
µ
2
2
1 2
xi
xi
xe
i
=
(
αβζλ
*
+
)
where
ζ
=
ln
1
+
and
λ
=
ln(
µ
)
*
2
ζ xi
(3.30)
i
i
i
xi
xi
xi
The α I term is a unit vector and g is the failure function, which in this case is the
calculation of the factor of safety.
6. The final value of the reliability index, in this case 1.98, is the reliability index for the
assumed failure mechanism. This can be converted into a probability of failure using
Equation 3.8 . For sliding on the silty sand layer, the Hasofer Lind probability of failure
is 2.4%.
The Hasofer Lind method was also applied to the other failure mechanisms, sliding within
the clay and bearing capacity failure in the clay. The standard deviation of the undrained
shear strength of the clay was estimated to be 24 kN/m 2 in these analyses. The calculated
probability of failure for sliding on the clay surface was 2.20%, and the calculated prob-
ability of bearing capacity failure was 1.80%.
The results of the Simplified Hasofer Lind method for sliding on the granular layer, slid-
ing through the clay, and bearing capacity failure in the clay are compared with the results
of the Monte Carlo simulation method in Table 3.12 . Comparing the Simplified Hasofer
Lind method with the Monte Carlo simulation method shows what effect the simplifying
Table 3.12 Comparison of the Simplified Hasofer Lind and Monte Carlo Simulation methods for the
retaining wall example
Deterministic factor of
safety
Failure mode
Probability method
Probability of failure (%)
Sliding on sand
1.40
Monte Carlo
2.4
Simplified Hasofer Lind
2.4
Sliding in clay
1.95
Monte Carlo
2.2
Simplified Hasofer Lind
2.5
Bearing capacity
1.97
Monte Carlo
1.8
Simplified Hasofer Lind
2.3
 
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