Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 142

the relationships among most likely values of factor of safety (F M LV ), standard deviation, and
β. In the figure the F M LV is the average (and in this case mode) of the PDF. The lower bell-
shaped curve represents a case where F M LV = 1.5 and standard deviation = 0.5. For this case
F M LV is one standard deviation above the factor of safety, which is 1.0, which means β = 1.
Table 3.2 shows that for β = 1, the probability of failure is 16%.
The narrow bell-shaped curve in Figure 3.10 represents a case where the most likely fac-
tor of safety is smaller (F M LV = 1.2), but the standard deviation of the factor of safety too
is much smaller (σ F = 0.1). For this case F M LV is two standard deviations above the factor of
safety = 1.0, which means β = 2. Table 3.2 shows that for β = 2, the probability of failure is
2.3%. Thus despite having a lower overall factor of safety, the probability of failure is less
because the standard deviation is smaller.
The shaded areas beneath the PDF curves in Figure 3.10 can be shown to be equal to the
probability of failure. The broader curve has a much greater area left of F = 1.0, correspond-
ing to Pf f = 16%. The narrower curve has a smaller area left of F = 1.0, corresponding to
P f = 2.5%.

3.3.14 Probability of failure on the CDF curve
The ordinate of the CDF is the area beneath the PDF curve, as discussed earlier. The prob-
ability of failure is the intercept of the CDF curve with the F = 1.0 line, as shown in Figure
3.11 . This applies to both normal and lognormal distributions of factor of safety.

P f = 1.5% based on normal distribution of F

100%

90%

80%

Most likely value of factor of safety = FMLV = 1.50
Standard deviation of factor of safety = 0.23

70%

60%

50%

40%

Normal CDF

30%

2.17 standard
deviations

20%

Normal
probabilityof
failure = 1.5%

10%

0%

0.50 1.00 1.50 2.00 2.50

F = factor of safety

Figure 3.11 Reliability index example for FMLV = 1.50, σ F = 0.23, β Normal = 2 .17.