Sensitivity Factors from FORM method

Sensitivity Factors are evaluated under the following context: \inputRV denotes a random input vector, representing the sources of uncertainties, \pdf its joint density probability, \model the limit state function of the model. We denote by:

(1)\cD_f = \{\vect{x} \in \Rset^\inputDim \, / \, \model(\vect{x}) \le 0\}

the event considered here and {\model(\inputRV) = 0} its boundary (also called limit state surface). The probability content of the event \cD_f is P_f:

(2)P_f = \int_{\model(\inputRV) \le 0} \pdf\, d\vect{x}.

In this context, the probability P_f can often be efficiently estimated by FORM or SORM approximations. The FORM importance factors offer a way to analyze the sensitivity of the probability the realization of the event with respect to the parameters of the probability distribution of \inputRV.

A sensitivity factor is defined as the derivative of the Hasofer-Lind reliability index with respect to the parameter \theta. The parameter \theta is a parameter in a distribution of the random vector \inputRV. If \vect{\theta} represents the vector of all the parameters of the distribution of \inputRV which appear in the definition of the isoprobabilistic transformation T, and \vect{U}_{\vect{\theta}}^{*} the design point associated to the event considered in (1) in the U-space, and if the mapping of the limit state function by the T is denoted by:

h(\vect{U}\,,\,\vect{\theta}) = \model \circ T^{-1}(\vect{U}\,,\,\vect{\theta}),

then the sensitivity factors vector is defined as:

\nabla_{\vect{\theta}} \beta_{HL} = \displaystyle +\frac{1} {||\nabla_{\vect{\theta}} h(U_{\vect{\theta}}^{*})||} \nabla_{\vect{u}} h(U_{\vect{\theta}}^{*}).

The sensitivity factors indicate the importance on the Hasofer-Lind reliability index (refer to ) of the value of the parameters used to define the distribution of the random vector \inputRV.

In other words, the FORM sensitivity factors rank the importance of the parameters of the input components in the realization of the event defined in (1).