Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 174

3. Assign a weighting probability to each of the cases. When the variables are uncorre-
lated, the probability is the same for each case and calculated from

1

P = (3.38)

C

In this equation C is the number of cases.

4. Calculate F mean by averaging the calculated values of the factor of safety from each of
the cases considered.

5. Using the F mean value from step 4 and the results of the factor of safety calculations
from each case, calculate the standard deviation from Equation 3.39 :

∑

2 mean

σ F = PF − F

(3.39) i i

6. Calculate the reliability index and probability of failure using Equations 3.37 or 3.12
assuming a normal or lognormal distribution of the factor of safety and the 'norms-
dist' function in excel.

F − 1

mean

β = (3.37)

NORMAL

σ

FS

F

MLV
COV
COV

ln

(3.11)

1
1

+
+

( )

2

F

β Lognormal =

ln( ( ))

2

F

(3.8)

P f =− 1 NORMSDIST( β

Today, when virtually all calculations of factor of safety are performed using spread-
sheets or other computer programs, the additional calculations for reliability require little
extra effort and little additional engineering time. These calculations can be done about as
quickly as new parameter values can be entered into a spreadsheet or data file.
The bulk of the analysis effort is required to develop the data for the first calculation in
all of the methods. Thus, although additional calculations must be performed, they involve
little time and effort beyond estimating values of the standard deviations of the parameters.
The use of prudent and informed judgment is as important in estimating values of standard
deviations of parameters as it is in estimating most likely values of parameters.
The great advantage of computing P f (the probability that the factor of safety could be
less than 1.0) is that it provides an overall measure of the uncertainty in the results of the
analysis. Computing both F and P f adds little to the time and effort required for the analysis,
but adds greatly to the value of the result.

3.13 CoMMentS on the MethoDS

The methods described in this chapter to compute the probability of failure have their own
advantages and disadvantages. Because use of these methods are in addition to traditional
geotechnical analyses, ease of use is important.
Table 3.20 details the number of calculations required for each method.