### Thomas Fetz

## Modelling uncertainties in limit state functions

### Abstract

In this paper uncertainties in limit state functions $g$ as arising in engineering problems are modelled by adding additional parameters and by introducing parameterized probability density functions which describe the uncertainties of these new additional parameters and of the basic variables of $g$. This will lead to a function $p_f(a,b)$ for the probability of failure depending on parameters $a$ and $b$ corresponding to the two parameterized density functions. Further the parameters $a$ and $b$ are assumed to be uncertain. Using intervals, sets or random sets to model their uncertainty results in upper probabilities $\overline{p}_f$ of failure. In this context we also discuss different notions of independence such as strong independence, epistemic irrelevance and random set independence and present a simple engineering example.

### Keywords

Probability of failure, limit state functions, parameterized probability measures, random sets, random set independence, epistemic irrelevance, strong independence

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### Authors’ addresses

Universität Innsbruck

Institut für Grundlagen der Bauingenieurwissenschaften,

Arbeitsbereich Technische Mathematik

Technikerstraße 13,

A-6020 Innsbruck

### E-mail addresses