Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 349

This chapter only deals with certain aspects of reliability, namely methodology and con-
cepts and insights, and not reliability in its broadest sense. Some of the RBD examples in
Sections 9.2 through 9.10 have explicit closed-form performance functions, while other
examples have implicit performance functions based on numerical methods. The spread-
sheet-based reliability computational procedures presented herein can be coupled with
stand-alone numerical geotechnical packages (e.g., FEM, FDM) via bridging techniques
such as response surface or artificial neural network methods.
A brief summary is given next for two relatively intuitive and transparent FORM compu-
tational procedures of Low and Tang (2004, 2007) and its relationship to the u -space com-
putational approach, together with the dispersion ellipsoid perspective in the original space
of random variables, aiming at overcoming the language and conceptual barriers (aptly
noted by Whitman 1984) surrounding reliability analysis.

9.1.1 three spreadsheet ForM procedures and intuitive
dispersion ellipsoid perspective
The matrix formulation (Veneziano 1974, Ditlevsen 1981) of the Hasofer and Lind (1974)
index β is

T − 1

β = min( x − µ ) Cx ( − µ )

(9.1a)

x ∈ F

where x is a vector representing the set of random variables x i , μ is the vector of mean values
μ i , C is the covariance matrix, and F is the failure domain. The notations “ T ” and “ − 1”
denote transpose and inverse , respectively.
The following alternative formulation, which is mathematically equivalent to Equation
9.1a , was used in Low and Tang (1997b), because the correlation matrix r is easier to set up,
and conveys the correlation structure more explicitly than the covariance matrix C :

T

x − µ x − µ

(9.1b)

i i − 1 i i

β= min

x

r

σ σ

∈ F

i i

where r is the correlation matrix, σ i are the standard deviations, and other symbols as
defined for Equation 9.1a . The point denoted by the x i values, which minimize Equation 9.1
and satisfy x ∈ F , is the design point. This is the point of tangency of an expanding disper-
sion ellipsoid with the LSS, which separates safe combinations of parametric values from
unsafe combinations ( Figure 9.1 ) . The one-standard-deviation (1-σ) dispersion ellipse and
the β-ellipse in Figure 9.1 are tilted by virtue of cohesion c and friction angle ϕ being nega-
tively correlated. The quadratic form in Equation 9.1 also appears in the negative exponent
of the established probability density function (PDF) of the multivariate normal distribu-
tion. As a multivariate normal dispersion ellipsoid expands from the mean-value point, its
expanding surfaces are contours of decreasing probability values. Hence, to obtain β by
Equation 9.1 means maximizing the value of the multivariate normal PDF (at the most prob-
able failure combination of parametric values) and is graphically equivalent to finding the
smallest ellipsoid tangent to the LSS at the most probable failure point (the design point ).
This intuitive and visual understanding of the design point is consistent with the more math-
ematical approach in Shinozuka (1983), in which all variables were standardized and the
limit state equation was written in terms of standardized variables.