Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 377

Limit state surface 2

Safe
domain

U L2

β = R / r

Limit state
surface 1

β-ellipse

σ Cv

One-sigma
dispersion
ellipse

U L1

μ Ch

r

R σ Ch

Unsafe
domain

Mean-value point
U L1 < U m < U L2

μ Cv

Coeff. of consolidation, c v

Figure 9.20 Illustration of reliability index in the plane. With respect to LSS1, reliability index β is positive;
with respect to LSS2, β is negative.

In contrast, in Figure 9.20 , the reliability index β with respect to the LSS1 (for which
the limiting degree of consolidation is U L1 ) is positive, while that with respect to LSS2 (for
which limiting U = U L2 ) must be treated as negative, by virtue of U L1 < U μ < U L2 , where U μ
is the average value of the degree of consolidation evaluated using mean values (μ Cv , μ Ch ) of
the coefficients of consolidation c v and c h .

9.10.3 reliability analysis for different limiting state surfaces
The Low and Tang (2007) procedure for FORM can deal with various correlated non-
normal distributions (lognormal, general beta, gamma, type 1 extreme, exponential, …).
For the case in hand, only correlated lognormals are illustrated. The values of compression
ratio C R , recompression ratio C RR , and coefficients of consolidation c v and c h in Figure 9.18
are taken to be the mean values in Figure 9.21 . Assumed values of standard deviations are
used for illustrative purpose. Positive correlations, logical between C R and C RR and between
c v and c h , are modeled.
The deterministic setup of Figure 9.18 and the reliability analysis of Figure 9.21 are cou-
pled easily by replacing the C R , C RR , c v , and c h values (cells H19, H20, H11, and H13) of
Figure 9.18 with the formulas “ = Z28”, “ = Z29”, “ = Z30”, and “Z31” that refer to the x *
values in Figure 9.21 . The performance function g 1 ( x ) is, by Equation 9.14 , “ = W34 − N25,”
where cell W34 has value 2.0 for this analysis. The computed β index is 1.485, treated as
(−1.485) because the mean-value point is in the unsafe zone as indicated by the negative g ( x )
value when the ni i values were initialized to zeros prior to spreadsheet-automated reliability
analysis.
By varying the s limit value (cell W34) between 1.2 and 4.8 at intervals of 0.2, and each time
recomputing the β index, 19 values of β were obtained as shown in Figure 9.22 .

9.10.4 obtaining probability of failure (Pf) f ) and CDF from β indices
Referring to Figure 9.19 , for s L1 = 2.0 m and β = −1.485 from Figure 9.21 , the probabil-
ity of failure P f is the integration of the probability density over the entire unsafe zone
( s > s L1 ). A good estimate of P f can often be obtained from the established Equation 9.5 ,