# FORM - Sensitivity of FORM results to distribution parameters and limit state function parameters

Hello,

In Reliability FORM analysis,

Is there any function to get sensitivity of reliability index ( \beta ) and probability of failure ( P_F ) with respect to probability density function (joint PDF of all basic random variables, f_X(x) ) parameters and limit-state function g(x) parameters.

Supposedly,

f_\textbf{X} (\textbf{x}) = f_\textbf{X} ( \textbf{x} , \theta_d )
g(x) = g (\textbf{x} , \theta_l )

Where,
\theta_d are the probability distribution parameters of the basic random variables such as mean and variance.
\theta_l are the limit state function parameters such as constant parameter.

Is there any way we can get something like,

\triangledown_{\theta l} \beta or \triangledown_{\theta d} P_F

Regards,
Maitreya Manoj Kurumbhati

Hi Maitreya,

Unfortunately UQLab does not feature a built-in function for the computation of the sensitivity of the reliability index with respect to distribution or limit-state parameters. However, using the by-products of FORM in UQLab, you can actually implement a simple algorithm to obtain those quantities. Please have a look at the Chapter 8 of [1] for more details on sensitivity reliability using FORM results.

If you are interested in those quantities in a more general setting, you may also considering simulation methods. In this regard, I would recommend having a look at the recent work of Iason Papaioannou from TU Munich ([2]) who estimates the sensitivity of the failure probability with respect to deterministic limit-state (and in some case, distribution) parameters using a sequential sampling scheme based on importance sampling. You may also find other interesting references in this paper.

Best,
Moustapha

[1] Ditlevsen O, Madsen HO. Structural reliability methods. Chichester: Wiley; 1996.
[2] Papaioannou I., Breitung K., Straub, D. Reliability sensitivity estimation with sequential importance sampling. Struct. Saf. 2018; 75:24-34

2 Likes

HI Maliki,

Yes, this totally answers my question. Thank you so much.
I will surely take a look at what you have referred.

Regards,
Maitreya Manoj Kurumbhati