Influence of the Parameterization in the Interval Solution of Elastic Beams

We are going to analyze the interval solution of an elastic beam under uncertain boundary conditions. Boundary conditions are defined as rotational springs presenting interval stiffness. Developments occur according to the interval analysis theory, which is affected, at the same time, by the overestimation of interval limits (also known as overbounding, because of the propagation of the uncertainty in the model). We suggest a method which aims to reduce such an overestimation in the uncertain solution. This method consists in a reparameterization of the closed form Euler-Bernoulli solution and set intersection. 1. Introduction While dealing with real problem solutions, during some structural engineering procedures, uncertainties must be taken into consideration. One example is the presence of epistemic uncertainties due to lack of knowledge for what concerns physical phenomena under study. This is an issue which could be included among the problem equations under the label of uncertain parameters. At this stage it is necessary to introduce a representation of such parameters according to a theory of uncertainty. The classical way of proceeding is to represent aleatory uncertainties as stochastic variables and solving the problem in two different ways: through a probabilistic analysis if equations are allowed to be integrated or through a Monte Carlo approach, if a numerical solution is required. Nowadays the increasing tendency is to represent epistemic uncertainty through means of nonprobabilistic methods with an uncertain interval analysis [1]. In particular interval analysis [2] is fascinating because it gives the possibility to compute uncertain bounded solutions with the certainty, under defined conditions, to include all the possible solutions which can be obtained from our model. All the practical implications of such property can be found both in the formulation of interval global optimization algorithms [3] and in the solution of inverse engineering problems [4]. It is commonly known that the main drawback of interval analysis is the so-called dependency effect, which leads to a deep overestimation (also called overbounding) of the interval solution bounds. This could represent a serious problem since, even though all possible solutions are included with certainty, very large bounds generally make the result meaningless from a physical point of view, and hence useless from an engineering point of view. This is the reason why there is a diffused attempt, together with the community of people who deal with interval analysis based applications, to