Reliability Index

The generalized reliability index \beta_{gen} 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 = \int_{g(\vect{X}\,,\,\vect{d}) \le 0} \pdf\, d\vect{x}.

The generalized reliability index is defined as:

\beta_{gen} = \Phi^{-1}(1-P_f) = -\Phi^{-1}(P_f),

where \Phi is the CDF of the normal distribution with zero mean and unit variance.

As \beta_{gen} increases, P_f decreases rapidly.

According to the method used to evaluate P_f, the generalized reliability index differs:

  • when P_f has been obtained from the FORM approximation, then \beta_{gen} is equal to the Hasofer-Lindt reliability index \beta, which is the distance of the design point from the origin of the standard space,

  • when P_f has been obtained from a SORM approximation, then \beta_{gen} is equal to \beta_{Breitung}, \beta_{Tvedt} or \beta_{Hohenbichler},

  • when P_f has been obtained from another technique (Monte Carlo simulations, importance samplings,…), we get the generalized index \beta_{gen}.