MODAL SENSITIVITIES OF STRUCTURES TO SIMULTANEOUS IMPRECISE MATERIAL PARAMETERS. B. Lallemand, A. Cherki, T. Tison and P. Level Laboratoire d'Automatique et de Mecanique lndustrielles et Humaines URA CNRS 01775 Universite de Valenciennes, BP311 59304 Valenciennes, France ABSTRACT. In this paper, fuzzy numbers are used to modelize material properties uncertainties and the rules of fuzzy sets are then used to solve the resulting fuzzy eigenvalue problem. The used method is based on a perturbation algorithm. The effectiveness of the present method is demonstrated for plate structure modelized in the case of a dynamic finite element analysis. The results of a such calculation allow to determine the sensitivity of the modal behaviour to multiple simultaneous material parameters. NOMENCLATURE [K]: [M]: A.,: {<1>}; : Nmode: [ I] : [ ... ]a: Jl(X) : Stiffness matrix. Mass matrix. i 1 h eigenvalue. i 1 h eigenvector. Size of the modal basis. Identity matrix. a-level-cut. Membership function of x. 1. INTRODUCTION The knowledge at our disposal about any situation is generally imperfect. Either we have suspicious about their validity, they are in that case uncertain, or we have difficulties in expressing them clearly, then they are imprecise. This two types of imperfection are both intimately mixed up. Our capability to describe accurately a system is a reciprocal function of its complexity. The number of components, the relationships between them, the difficulty to deftne their specifications come into play. The requirement to study or to manage such systems leads necessarily to take vague, imprecise, uncertain data into account. Uncertainty was tackled by the probability notion. However that one doesn't allow to solve the problem formulated by imprecise or vague knowledge. This latter were only taken into consideration with the notion of fuzzy sets. This notion was arisen from the idea of partial membership of a class, of category with ill-defined boundaries, of graduality in the passage of a situation to another one, of a generalisation of the classical theory of set allowing intermediate situation between the all and the nothing. The developments of this concept supply the means to represent and to handle imperfectly described, vague or imprecise knowledge. They draw up equally an interface between the data described symbolically and numerically. The fuzzy logic leads to reason with such knowledge. As to the theory of possibilities, it forms the frame approving to treat concepts of non-probabilistic uncertainty, and to exploit, in a same formality, imprecision and uncertainty. Intuitively, probabilities are more similar to a degree of frequency or of plausibility, when possibilities are associated to our perception of a degree of feasibility or of easiness to realisation. In structural analysis, many factors are the cause of uncertainty or imprecision. They are tied either to exogenous factors, such as boundary conditions or applied loads, or to endogenous factors, such as mechanical or geometric characteristics. These uncertainties have necessarily some repercussions on the structural behaviour. More generally, we can suppose that each structure possess its own sensitivity to different types of uncertainty. There is no suitable technique available for the analysis of all types of imprecision in structural analysis. The stochastic finite element method can be used to handle uncertain parameters that are described by probability distributions. The stochastic finite element method was developed in the 1980's to account for uncertainties in the system parameters, geometry, and external actions. The uncertain quantities were modelized as random variables with known characteristics (see ref. [1-3,8,11,16,17,25]). A large number of papers discussed the application of fuzzy set theory to structural design (ref. [4,5]), in particular in structural optimisation (see ref. [19,20,28]), in random vibration with application to aseismic structures (see ref. [26]), in finite element analysis of engineering systems containing vague information, as boundary conditions (see ref. [6,7]). For uncertainty concerning mechanical parameters, some methodologies are proposed. However they are still numerically expensive, they propagate numerical errors, or they are inefficient for large finite element models parameters (see ref. [15,22-24] for static response and ref. [21] for dynamic response). 1591