ABSTRACT
Let us assume that the reliability assessment problem under consideration is governed by a vector X of n basic random variables Xi (i
= 1, 2, … , n), that is,
(19.1)
where is transpose. Assuming, furthermore, that the random variables X have a joint probability density function f (x), then the probability of failure P(F) — that is the probability that a limit state will be reached — is defined by
(19.2)
X = ′( , , , )1 2X X Xn…
P F f d
( ) ( )
=
x x
whereby g(x) is the limit state function that divides the n-dimensional probability space into a failure domain F
= {x : g(x)
≤ 0} and a safe domain S
= {x : g(x)
> 0}. As already mentioned, the computational challenge in determining the integral of Equation 19.2 lies in evaluating the limit state function g(x), which for nonlinear systems usually requires an incremental/iterative numerical approach. The basic idea in utilizing the response surface method is to replace the true limit state function g(x) by an approximation
η(x), the so-called response surface, whose function values can be computed more easily. In this context it is important to realize that the limit state function g(x) serves the sole purpose of
defining the bounds of integration in Equation 19.2. As such, it is quite important that the function
η(x) approximates this boundary sufficiently well, in particular in the region that contributes most to the failure probability P(F). As an example, consider a two-dimensional problem with standard normal random variables X1 and X2 , and a limit state function g(x1, x2)
= 3 – x1 – x2. In Figure 19.1, the integrand of Equation 19.2 in the failure domain is displayed. It is clearly visible that only a very narrow region around the so-called design point x * really contributes to the value of the integral (i.e., the probability of failure P(F)). Even a relatively small deviation of the response surface
η(x) from the true limit state function g(x) in this region may, therefore, lead to significantly erroneous estimates of the probability of failure. To avoid this type of error, it must be ensured that the important region is sufficiently well covered when designing the response surface.
