Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 358

Limit state
surface

β = R/r

Safe domain

Unsafe
domain

β-ellipse

ϕ′

I

1-sigma ellipse

II

r

R

Design point

c II ′ c I ′

c ′

Figure 9.6 Distinguishing negative from positive reliability index. Increasing the magnitude of reinforcing
force T will increase the β value of Case II from negative to zero to target β .

is already in the unsafe domain, and the computed β must be given a negative sign because
it is the distance from the unsafe mean-value point to the safe boundary (the LSS). Case II's
probability of failure is >0.5.
The reinforcing force T required to achieve a target reliability index β (e.g., 2.5) for the
slope of Figure 9.5 can be obtained as follows:

ii. If the performance function g ( x ) is positive at the mean-value point (as in Case I), β is
positive. Perform FORM reliability analysis with increasing T until β = β target .

ii. If the performance function g ( x ) is negative at the mean-value point (as in Case II),
choose a value of T so that g ( x ) is > 0 at the mean-value point, then perform FORM
reliability analysis by increasing T until β = β target .

The RBD of the embedment depth of an anchored sheet pile wall in Low (2005) provides
another example of the need to distinguish negative β from positive β values.

9.4 ProbabIlIStIC analYSeS oF a SloPe
FaIlure In San FranCISCo baY MuD

The failure of a slope excavated underwater in San Francisco Bay has been described in
Duncan and Buchignani (1973), Duncan (2000, 2001), and Duncan and Wright (2005). The
slope was part of a temporary excavation and was designed with an unusually low factor
of safety to minimize construction costs. During construction, a portion of the excavated
slope failed.
Low and Duncan (2013) analyzed the same underwater slope that failed, first determinis-
tically using data from field vane shear and laboratory triaxial tests, Figure 9.7 , then proba-
bilistically, accounting for parametric uncertainty and positive correlation of the undrained
shear strength and soil unit weight.
In the deterministic analysis, the factors of safety were computed with search for critical
noncircular slip surface based on a reformulated Spencer method. The F s values obtained
were 1.20, 1.16, and 1.00, for three types of undrained shear strength data, namely field
vane tests, unconsolidated-undrained (UU) triaxial tests on trimmed 35 mm specimens, and