Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 356

anchored sheet pile wall, and a rock tunnel) and serviceability limit states (a laterally loaded
pile, and consolidation settlement of soft clay accelerated by vertical drains).
Figure 9.3 uses the Low and Tang (2004) spreadsheet-based FORM procedure by auto-
matically changing the x vector using the constrained optimization routine Solver that
resides in Microsoft Excel. The other two methods, namely the Low and Tang (2007) x -via-
n method and the x -via- n -via- u method, as summarized in Figure 9.2 , will obtain the same
reliability index and design point for this example and for the examples of the following
sections. Each of these three spreadsheet-based FORM computational procedures will be
illustrated in different sections.

9.3 SorM analYSIS on the FounDatIon oF
ForM reSultS For a roCk SloPe

This case is adapted from Low (2008b) and Low (2013).

9.3.1 Constrained optimizational ForM spreadsheet
approach with respect to the u vector
Figure 9.5 shows the FORM and SORM analyses of a two-dimensional rock slope with
five correlated random variables, two of which obey the highly asymmetric truncated

Excel's Solver automatically minimizes β by changing u , which are initially
zeros, subject to the constraint that g ( x )=0.

n=Lu

x i *

Para1 Para2 Para3 Para4 Correlation matrix Rn u

Normal

35 5 31.909 1 -0.5 0 0 0 -0.618-0.618

ϕ

Normal

c 10 2 9.033 -0.5 1 0 0 0 -0.483-0.915

z 14 3 13.591 0 0 1 -0.5 0 -0.136 -0.136

1.051

Normal

Tr_Exp z w /z 0. 501 0.642 0 0 -0.5 1 0 0.978

Tr_Exp

α 0.08 0 .16 0.110 0 0 0 0 1 .104 .104

g ( x )

P f (FORM)

β

H ψ f ψ p γ w T

γ θ

60 50 35.0 2.61 0 0 0.000 1.887 0.0296

Tension crack

SORM formula P f (SORM)
Tvedt 0.0215
Breitung 0.0228
Hohenbichler_Rackwitz 0.0218
Koyluoglu_Nielsen 0.0218
Cai_Elishakoff 0.0215
Hong(P 3 )
Hong(P 4 )

α W

Z

W V Z w

H

θ U

Assumed
water pressure
distribution

0.0215
0.0216
Average 0.0218

ψ f

T

ψ p

Failure surface

Figure 9.5 FORM based on automatically changing the u vector in the Microsoft Excel platform, and SORM
estimation of curvatures and P f on the basis of FORM β and design point for a rock slope with five
correlated random variables, two of which ( z w / z and α ) obey the truncated exponential distribution.