Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 141

Table 3.2 Relationship between reliability index and
probability of failure
Reliability index β Normal or β Lognormal

Probability of failure, Pf f

0.50 31%

1.00 16%

1.50 6.7%

2.00 2.3%

2.50 0.62%

3.00 0.13%

4.00 0.003%

5.00 0.00003%

The probability of failure for any value of β can be evaluated using the NORMSDIST
function in Excel, as shown by Equation 3.8

P f =− 1 NORMSDIST (β

(3.8)

Because β is uniquely related to probability of failure, the value of β has sometimes been
used in lieu of the probability of failure as a measure of safety. However, in order to evaluate
β, the distribution of the factor of safety must be assumed. Therefore, whether the value of
β or the value of P f is used as the safety criterion, it is necessary to assume a distribution for
factor of safety. Most geotechnical engineers prefer to use P f rather than β, because P f has an
easily understood relationship to likely performance and β does not. Figure 3.10 i illustrates

4.500

Average = 1.2
St Dev = 0.1
β = 2.0

4.000

3.500

3.000

2.500

2.000

1.500

Average = 1.2
St dev. = 0.5
β = 1.0

1.000

0.500

0.000

0 0.5 1 1.5 2 2.5 3

Factor of safety

Figure 3.10 Factor of safety and reliability index. (After Christian, J.T., Ladd, C.C., and Baecher, G.B., 1994, Journal
of Geotechnical Engineering , Vol. 120, No. 12, p. 2180-2207. Used with permission from ASCE.)