Sensitivity Factors are evaluated under the following context:
denotes a random input vector, representing the
sources of uncertainties, its joint density probability,
a deterministic vector, representing the fixed
variables the limit state function of
the event considered here and
its boundary (also called limit state surface).
In this context, the probability 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 .
A sensitivity factor is defined as the derivative of the Hasofer-Lind
reliability index with respect to the parameter . The
parameter is a parameter in a distribution of the
random vector .
If represents the vector of all the parameters
of the distribution of which appear in the definition
of the isoprobabilistic transformation , and
the design point associated to the event
considered in the -space, and if the mapping of the limit
state function by the is noted
then the sensitivity factors vector is defined as:
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 .
Here, the event considered is explicited directly from the limit state
function : this is the classical
structural reliability formulation.
However, if the event is a threshold exceedance, it is useful to
explicit the variable of interest
, evaluated from the model
. In that case, the event considered, associated to
the threshold has the formulation:
and the limit state function is :
is the threshold exceedance probability, defined as:
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 . They can be seen as a
specific sensitivity analysis technique dedicated to the quantity Z around
a particular threshold rather than to its variance.