Environmental Engineering Reference
In-Depth Information
Low and Tang (2004) FORM procedure:
minimize β by directly varying x
Low and Tang (2007) FORM procedure:
minimize β by varying n, on which x depends
T
x i - μ i N
x i - μ i N
σ i N
β =min n T R -1 n
β =min
R -1
x ŒF
σ i N
x ŒF
Use Excel's Solver to change the n vector .
ϕ Φ -1 F ( x )
σ N =
f ( x )
Subject to g ( x )=0
For each trial n , get x i = F -1 Φ ( n i )
μ N = x - σ N × Φ -1 F ( x )
Use Excel's Solver to change the x vector.
Subject to g ( x )=0
ird spreadsheet-based FORM procedure:
minimize β by varying u , from which n and x are readily obtainable
β =min u T u
Use Excel's Solver to change the u vector, subject to g ( x ) = 0
x ŒF
For each automated trial u , get n = Lu , and x i = F -1 Φ ( n i )
Figure 9.2 Comparison of the two FORM computational approaches of Low and Tang (2004, 2007), and
the additional u -to- n -to- x approach. All three procedures use the optimization routine Solver
resident in the Microsoft Excel spreadsheet.
the classical mathematical approach of FORM. The vectors n and u can be obtained from
one another, n = Lu and u = L −1 n, as follows (e.g., Low et al. 2011):
β=
min
nr n
T
1
=
min
n
T
(
Lu
)
1
n
=
min(
L nLn
1
)(
T
1
)
(9.4a)
x
F
x
F
x
F
(9.4b)
that is,
β=
min
uu
T
,
where
uLn
=
1
,
and
n
=
Lu
,
x
F
in which L is the lower triangular matrix of r . When the random variables are uncorrelated,
u = n by Equation 9.4 , because then L −1 = L = i (the identity matrix).
The probability of failure can be approximately estimated as follows:
P f ≈− =−
1 ΦΦ
()
β
()
β
(9.5)
where Φ is the cumulative distribution function (CDF) of the standard normal random variable.
Equation 9.5 is exact when the LSS is planar and the parameters follow normal distribu-
tions. Inaccuracies in P f estimation may arise when the LSS is significantly nonlinear. More
refined alternatives have been proposed, for example, the SORM, by Fiessler et al. (1979),
Tvedt (1983, 1988, 1990), Breitung (1984), Hohenbichler and Rackwitz (1988), Koyluoglu
and Nielsen (1994), Cai and Elishakoff (1994), Hong (1999), and Zhao and Ono (1999).
SORM analysis requires the FORM β value and design point values as inputs, and there-
fore is an extension dependent on FORM results. Hence, the SORM results are displayed
alongside the FORM results in some of the examples to follow. In general, the SORM
attempts to assess the curvatures of the LSS near the FORM design point in the dimension-
less and rotated u -space. The failure probability is calculated from the FORM reliability
index β and estimated principal curvatures of the LSS using established SORM equations.
Some files illustrating the Low and Tang (2004, 2007) approaches are available at http://
alum.mit.edu/www/bklow . Step-by-step guidance that enables hands-on appreciations of
 
 
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