Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 173

Table 3.19 Comparison of the PEM assuming normal and lognormal 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

Point Estimate, lognormal F 2.6

Sliding in clay 1.95 Monte Carlo 2.2

Point Estimate, normal dist.
of F

5.0

Point Estimate, lognormal F 1.6

Bearing capacity 1.97 Monte Carlo 1.8

Point Estimate, normal dist.
of F

4.3

Point Estimate, lognormal F 1.1

4. Same as for a normal distribution for the factor of safety

5. Same as for a normal distribution for the factor of safety

6. If a lognormal distribution of the factor of safety is assumed, then the probability of
failure can be calculated using Equation 3.40 and the 'NORMSDIST' function in
Microsoft's Excel.

Lognormal distribution of the factor of safety:

( ) ( )

ln F 1 + ( COV )

2

ln 1431 0 253 . / + ( . /1143 .)

2

MLV F

β Lognormal = = = 194 .

(3.40)

+

2

+ /

2

ln( 1 ( COV )) ln( 10253 143 ( . . )

F

P f =− 1 NORMSDIST( .) 1 94 = 259 . %

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 24.0 kN/m 2 . The calculated probability of failure for sliding on a clay surface
is 1.58%. The calculated probability of failure for bearing capacity is 1.14%.
The results of the PEM assuming a lognormal distribution for the factor of safety is com-
pared to the Monte Carlo simulation method to see how the simplifying assumptions of the
PEM affect the result as shown in Table 3.19.

3.12.1 Summary of the PeM
The steps involved in using the PEM are

1. Estimate the standard deviations of the parameters that involve uncertainty, using the
methods discussed in this chapter.

2. Compute the factor of safety with each parameter increased or decreased by one stan-
dard deviation. There are 2 N cases where N is the number of parameters whose values
are being varied.