Non Deterministic Approaches

 In most industrial cases, the integrals of Eq.(1) and of Eq.(2) (ref. this section) cannot be evaluated in closed form. Only for simple academic cases $f_{X}$ can be defined analytically. In such a case, the performance function $G_{j}(x)$ is available for each specific failure mode of the structure so that $\Omega_{j}$ can be defined. The boundary of the domain $\Omega_{j}$ is also called limit-state surface and the function $G_{j}(x)$ is called the limit-state function (LSF).

 In the most widely used reliability methods, approximations are made in the space of standard uncorrelated normal variates, Y, obtained from a transformation of the basic variables

$Y=T(X) $

(3)

 where the transformation T is expressed in terms of the distributions of X.

 In the space Y, denoted as the standard normal space, approximations of the probability of failure $P_{f,j}$ of Eq.(1) (ref. this section) are obtained by replacing the limit state surface with first or second order approximating surfaces. These surfaces are fitted to the limit state surface at points with minimal distance to the origin (design points).