TABLE OF CONTENTS Thomas Fetz
Institut f r Technische Mathematik, Geometrie und Bauinformatik, Universit t Innsbruck
Summary. This paper is devoted to the construction of sets of joint probability measures for the case that the marginal sets of probability measures are generated by random sets. Different conditions on the choice of the weights of the joint focal sets and on the probability measures on these sets lead to different types of independence such as strong independence, random set independence, fuzzy set independence and unknown interaction. As an application the upper probabilities of failure of a beam bedded on two springs are computed.
Precise probability theory alone often does not suffice for modelling the uncertainties arising in civil engineering problems such as the reliability analysis of structures and much more in soil mechanics. The goal is to have practicable measures for the risk of failure in the case where the material properties are not or only partly given by precise values or probability measures. One should also have the possibility to assess subjective knowledge and expert estimates.
It is usually easy for the planning engineer to provide such information by using random sets (weighted focal sets) to model the fluctuations of the parameters involved. In most cases intervals are used for the focal sets which has the advantage that the computations can be performed by interval analysis. This leads, if the intervals are nested, to fuzzy numbers and possibility measures or, if not nested, to plausibility measures and evidence theory. Fuzzy numbers or...
