Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 350

β = R / r

Limit state surface: boundary between
safe and unsafe domains

σ ϕ

Safe

One-sigma
dispersion ellipse

β-ellipse

μ c

r

σ c

R

Unsafe

Design
point

Mean-value
point

μ ϕ

Friction angle, ϕ

Figure 9.1 Illustration of the reliability index β in the plane when c and ϕ are negatively correlated.

In FORM, one can rewrite Equation 9.1b as follows (Low and Tang 2004) and regard the
computation of β as that of finding the smallest equivalent hyperellipsoid (centered at the
equivalent normal mean-value point μ N and with equivalent normal standard deviations σ N )
that is tangent to the LSS:

T

x − µ

i N

x − µ

i N i i

(9.2)

β = min

x

r

− 1

σ

i N

σ

i N

∈ F

where µ i N and σ i N can be calculated by the Rackwitz and Fiessler (1978) transformation.
Hence, for correlated non-normals, the ellipsoid perspective still applies in the original coor-
dinate system, except that the non-normal distributions are replaced by an equivalent nor-
mal ellipsoid, centered not at the original mean values of the non-normal distributions, but
at the equivalent normal mean μ N .
Equation 9.2 and the Rackwitz-Fiessler equations (for µ i N and σ i N ) were used in the
spreadsheet-automated constrained optimization FORM computational approach in Low
and Tang (2004). An alternative to the 2004 FORM procedure is given in Low and Tang
(2007), which uses the following equation for the reliability index β:

β= ∈ min

x

nr n

T − 1

(9.3)

F

The computational approaches of Equations 9.1b through 9.3 and associated ellipsoidal
perspectives are complementary to the classical u -space computational approach and may
help to reduce the conceptual and language barriers of FORM.
The two spreadsheet-based computational approaches of FORM are compared in Figure
9.2 . Either method can be used as an alternative to the classical u -space FORM procedure.
A third alternative (illustrated in Lü and Low 2011) is also shown in Figure 9.2 , for which
the Microsoft Excel's built-in constrained optimization routine ( Solver ) is invoked to auto-
matically vary the u vector so that β and the design point are obtained. This requires only
adding one u column to the 2007 procedure, and expressing the unrotated n vector in terms
of u , where u is the uncorrelated standard equivalent normal vector in the rotated space of