Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 140

10%

9%

8%

7%

Cumulative normal distribution
Cumulative lognormal distribution

6%

5%

4%

S u = 10 kPa

3%

2.3%

2%

1.1%

1%

0%

0 2 4 6 8 10 12 14

s u — (kPa)

Figure 3.9 Tails of normal and lognormal CDFs.

shown in Figure 3.9. A ccording to the normal distribution, 2.3% of the values of s u would
fall below 10 kPa; whereas according to the lognormal distribution, only 1.1% of the values
of s u would fall below 10 kPa. These differences at the tails of the distributions can result
in significant differences in computed values of probability of failure, as discussed below.

3.3.11 Probability of failure
The “probability of failure” is the probability that failure will occur. In terms of factor of
safety, the probability of failure is defined as the probability that the factor of safety could
be less than 1.0 given adverse values of the variables involved in its calculation.
Estimating the probability of failure is a major motive for applying probability theory in
geotechnical engineering. Several methods for estimating probability of failure are discussed
in a later section of this chapter.

3.3.12 reliability
Reliability is the complement of the probability of failure. For example, if there is a 0.5%
probability that the factor of safety against slope instability may be less than 1.0, the prob-
ability of failure is 0.5% and the reliability is 99.5%.

3.3.13 reliability index
The reliability index (β) is the number of standard deviations between the most likely value
of factor of safety and factor of safety = 1.0. The reliability index (β) is uniquely related to
probability of failure, as shown in Table 3.2. The values of P f in Table 3.2 apply to both
normally distributed and lognormally distributed factors of safety.