FORM

The First Order Reliability Method is used under the following context: let \vect{X} be a probabilistic input vector with joint density probability \pdf, let \vect{d} be a deterministic vector, let g(\vect{X}\,,\,\vect{d}) be the limit state function of the model and let \cD_f = \{\vect{X} \in \Rset^n \, / \, g(\vect{X}\,,\,\vect{d}) \le 0\} be an event whose probability P_f is defined as:

(1)P_f = \Prob{g(\vect{X}\,,\,\vect{d})\leq 0}= \int_{\cD_f} \pdf\, d\vect{x}

The objective of FORM is to evaluate P_f. The method proceeds in three steps:

Step 1: Map the probabilistic model in terms of \vect{X} thanks to an isoprobabilistic transformation T which is a diffeomorphism from \supp{\vect{X}} into \Rset^n, such that the distribution of the random vector \vect{U}=T(\vect{X}) has the following properties: \vect{U} and \mat{R}\,\vect{U} have the same distribution for all rotations \mat{R}\in{\cS\cO}_n(\Rset). The usual Isoprobabilistic transformations are the Generalized Nataf transformation and the Rosenblatt one. The mapping of the limit state function is h(\vect{U}\,,\,\vect{d}) = g(T^{-1}(\vect{U})\,,\,\vect{d}). Then, the event probability P_f can be written as:

(2)P_f = \Prob{h(\vect{U}\,,\,\vect{d})\leq 0} = \int_{\Rset^n} \boldsymbol{1}_{h(\vect{u}\,,\,\vect{d}\leq 0} \,f_{\vect{U}}(\vect{u})\,d\vect{u}

where f_{\vect{U}} is the density function of the distribution in the standard space: that distribution is spherical (invariant by rotation by definition). That property implies that f_{\vect{U}} is a function of ||\vect{U}||^2 only.

Furthermore, we suppose that outside the sphere tangent to the limit state surface in the standard space, f_{\vect{U}} is decreasing.

Step 2: Find the design point P^* which is the point verifying the event of maximum likelihood : the decreasing hypothesis of the standard distribution f_{\vect{U}} outside the sphere tangent to the limit state surface in the standard space implies that the design point is the point on the limit state boundary that is closest to the origin of the standard space. Thus, P^* is the result of a constrained optimization problem.

Step 3: In the standard space, approximate the limit state surface by a linear surface at the design point P^*. Then, the probability P_f (2) where the limit state surface has been approximated by a linear surface (hyperplane) can be obtained exactly, thanks to the rotation invariance of the standard distribution f_{\vect{U}}:

(3)P_{f,FORM} = \left| \begin{array}{ll} E(-\beta) & \mbox{if the origin fails}, \\ E(+\beta) & \mbox{if the origin is safe} \end{array} \right.

where \beta is the Hasofer-Lind reliability index, defined as the distance of the design point P^* to the origin of the standard space and E the marginal cumulative density function along any direction of the spherical distribution in the standard space (refer to Generalized Nataf Transformation and Rosenblatt Transformation).

Let us recall here that in the Rosenblatt standard space, random vectors follow the standard normal distribution (with zero mean, unit variance and unit correlation matrix), which implies that E = \Phi where \Phi is the CDF of the normal distribution with zero mean and unit variance. In the Generalized Nataf standard space, random vectors follow a spherical distribution, with zero mean, unit variance, unit correlation matrix and whose type \psi is the type of the copula of the physical random vector \vect{X}: in that case, E is the 1D cumulative distribution function with zero mean, unit variance and with type \psi.