Environmental Engineering Reference
In-Depth Information
developed by Filz and Navin (2006) is described here. The full method is presented well in
Low (1996), Low and Tang (1997, 2004), and Low and Phoon (2002).
The Hasofer Lind method is a first-order reliability method (FORM). The application of
the method to geotechnical engineering analysis has been described by Baecher and Christian
(2003). The Simplified Hasofer Lind method presented here uses the factor of safety equal
to 1.0 as the “performance function,” and closed-form equations are not required. The
method determines reliability geometrically as the shortest distance from the mean values of
the variables to the performance function. This distance is essentially the reliability index,
which can be used to find the probability of failure.
Three main stages are involved in the Simplified Hasofer Lind method. Stage one involves
formulating trial values of the variables based on an assumed value of the reliability index.
In most cases the initially assumed value of the reliability index is taken as 1.0 (Filz and
Navin 2006). The factor of safety is calculated using reduced values of the variables cor-
responding to reliability index = 1.0. If the factor of safety calculated using these reduced
values of the variables is not one, a different trial value of the reliability index is used to
calculate new values of the variables. This iterative process is continued until the reliability
index is found that will result in a factor of safety equal to 1.0.
Stage 2 is used to determine the “gradient of the failure function.” This stage determines
the change in the factor of safety with a change of the variables. This step requires N calcu-
lations of the factor of safety where N is the number of variables.
The final stage, stage 3, is similar to stage 1 where trial values of the reliability index are
used to generate new values of the variables and the factor of safety is calculated. However,
in this stage the values of the variables are also based on the gradients calculated in stage
two. Like stage one, an iterative process is used until a factor of safety = 1.0 with reduced
values of the variables is found.
This Simplified Hasofer Lind reliability method will be explained with the use of the same
retaining wall sliding on a granular surface example used for the previous methods.
1. Estimate the standard deviations of the quantities involved in
Equation 3.20.
σ
efp
= standard deviation of the equivalent fluid pressure = 1.06 kN/m
3
σ
tan δ
= standard deviation of tan δ = 0.05
σ
γ
bf
= standard deviation of the unit weight of backfill = 0.565 kN/m
3
2. Assume either normal or lognormal distributions of the variables, and indicate whether
the variables are related to “capacity” or “demand,” or in simple terms, load or resis-
tance. If the variable is related to demand (load), a negative value of the standard devia-
tion is used for the calculations. In this example problem the equivalent fluid pressure
is related to demand, so the standard deviation used in the calculations is −1.06 kN/m
3
.
3. Use an initial trial value of the reliability index to calculate factored values of the
variables, and use these values to calculate the factor of safety. This is accomplished
using
Equations 3.24
or
3.25
.
Then calculate the factor of safety using these values
of the variables. If the factor of safety is greater than one, a higher trial value of the
reliability index should be used. A lower value of the reliability index should be used if
the calculated factor of safety is less than one. This iterative process is continued until
a factor of safety equal to one is calculated as shown in
Table 3.9.
x
i
=−
βσ µ
*
+
(3.24)
xi
xi
σ
µ
2
2
1
2
(*
−
βζ λ
+
)
xi
xi
2
* ζ
xi
xe
i
=
where
ζ
=
ln
1
+
and
λ
=
ln()
µ
−
(3.25)
xi
xi
xi
xi
xi
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