Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 143

3.3.15 reliability index for normally distributed factor of safety
If the factor of safety is assumed to be normally distributed, the reliability index is computed
using Equation 3.9 :

F − 1

β =

MLV

(3.9)

Normal

σ

F

where
β Normal is the value of β for the normal distribution assuming the factor of safety is nor-
mally distributed
F M LV is the most likely value of the factor of safety, calculated using the most likely val-
ues of all variables and σ F is the standard deviation of F.

This value of β Normal is shown graphically in Figure 3.11. For this example, with F M LV = 1.50,
and σ F = 0.23, β Normal is equal to 2.17:

F − 1150 . − 1

β =

MLV

= = 217 .

(3.10)

Normal

σ

023 .

F

and the probability of failure calculated using the Excel function NORMSDIST ( Equation
3.8 ) is 1.5%.

3.3.16 reliability index for a lognormally
distributed factor of safety
If the factor of safety is assumed to be lognormally distributed, the reliability index is com-
puted using Equation 3.11 :

F

MLV
COV
COV

ln

1
1

+ ( )

2

(3.11)

F

β Lognormal =

ln( + ( ))

2

F

For the same example (F M LV = 1.50, and σ F = 0.23), COV = 0.23/1.50 = 15.3%, β Lognormal
calculated using Equation 3.11 is equal to 2.58, and the probability of failure calculated
using the Excel function NORMSDIST ( Equation 3.8 ) is 0.5%, one-third of the value of P f
for the normal distribution.
This example shows that the assumption regarding the distribution of factor of safety
has an important effect on the computed probability of failure: P f = 1.5% if the factor of
safety is assumed to be normally distributed and P f = 0.5% if the factor of safety is assumed
to be lognormally distributed. This important factor is further discussed in the following
sections.

3.3.17 effect of standard deviation on estimated value of
probability of failure
Figure 3.12 shows the CDF for two values of standard deviation of factor of safety, σ F 1 = 0.23
and σ F 2 = 0.38. As can be seen in Figure 3.12 , an increase of 60% in standard deviation