Sensitivity Factors from FORM method

Sensitivity Factors are evaluated under the following context: \vect{X} denotes a random input vector, representing the sources of uncertainties, \pdf its joint density probability, \vect{d} a deterministic vector, representing the fixed variables g(\vect{X}\,,\,\vect{d}) the limit state function of the model, \cD_f = \{\vect{X} \in \Rset^n \, / \, g(\vect{X}\,,\,\vect{d}) \le 0\} the event considered here and {g(\vect{X}\,,\,\vect{d}) = 0} its boundary (also called limit state surface).
The probability content of the event \cD_f is P_f:

(1)P_f = \int_{g(\vect{X}\,,\,\vect{d}) \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 \vect{X}.
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 \vect{X}.
If \vect{\theta} represents the vector of all the parameters of the distribution of \vect{X} which appear in the definition of the isoprobabilistic transformation T, and U_{\vect{\theta}}^{*} the design point associated to the event considered in the U-space, and if the mapping of the limit state function by the T is noted h(\vect{U}\,,\,\vect{\theta}) =  g[T^{-1}(\vect{U}\,,\,\vect{\theta}), \vect{d}], then the sensitivity factors vector is defined as:

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

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 \vect{X}.

Here, the event considered is explicited directly from the limit state function g(\vect{X}\,,\,\vect{d}): this is the classical structural reliability formulation. However, if the event is a threshold exceedance, it is useful to explicit the variable of interest Z=\tilde{g}(\vect{X}\,,\,\vect{d}), evaluated from the model \tilde{g}(.). In that case, the event considered, associated to the threshold z_s has the formulation: \cD_f = \{ \vect{X} \in \Rset^n \, / \, Z=\tilde{g}(\vect{X}\,,\,\vect{d}) > z_s \} and the limit state function is : g(\vect{X}\,,\,\vect{d}) = z_s - Z = z_s - \tilde{g}(\vect{X}\,,\,\vect{d}). P_f is the threshold exceedance probability, defined as: P_f     =       P(Z \geq z_s) = \int_{g(\vect{X}\,,\,\vect{d}) \le 0}  \pdf\, d\vect{x}. Thus, the FORM sensitivity factors offer a way to rank the importance of the parameters of the input components with respect to the threshold exceedance by the quantity of interest Z. They can be seen as a specific sensitivity analysis technique dedicated to the quantity Z around a particular threshold rather than to its variance.