Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 378

Figure 9.21 Initially the ni i values were zeros, and g( x ) exhibits negative values. Hence, the computed β value
must be treated as negative ( β = − 1.485).

P f ≈ Φ (−β), or 93% for the case of β = −1.485 in Figure 9.21 , which means that the limit-
ing final settlement s cf of 2.0 m is almost certainly to be exceeded. This is not surprising
given that the computed expected s cf is about 2.68 m as shown in Figure 9.18 . Besides
estimating the probability of failure for serviceability limit states such as Equations 9.14
through 9.16 for s cf , U , and s r , respectively, it would also be of interest to obtain the CDF
from the β indices, as follows:

g x

1 ()

= Limiting s − s : CDF = Φβ ()

(9.17)

cf cf

g x

2 ()

=− ULimitingU : CDF =− Φβ () (9.18)

g x

3 ()

= Limiting s − s : CDF = Φβ () (9.19)

r r

The reason for Equations 9.17 and 9.19 b eing different from Equation 9.18 is readily appreci-
ated if one notes that, for Equations 9.17 and 9.19, CDF = P [ s < s limit ] = 1 − P [ s > s limit ] = 1 − P f .
In contrast, for Equation 9.18 , CDF = P [ U < U limit ] = P f .