Keywords: uncertain parameters, parametric stiffness matrix, perturbation methods.

Uncertain structures are characterized by the fact that some of their mechanical and geometrical properties are defined from a probabilistic point of view [1]. The fundamental problem related to the analysis of these structures consists in finding explicit relationships, almost always approximate, between the structural response and the physical uncertainties. The principal difficulty is due to the nonlinear nature of this problem always arising, even for linear structures [2]. All the analysis approaches for uncertain structures are based on some approximating hypotheses regarding the stiffness of the material or the stiffness of the discretized structure and, in some cases, on some corresponding assumptions concerning the response. For example, one of the most used methods, the first order perturbation approach, assumes a linear dependence of the structural stiffness on the quantities describing the uncertainties and, at the same time, furnishes a linear dependence between the corresponding response and the uncertain parameters [3,4]. However, in any approach the hypothesis of linear dependence of the stiffness on the uncertainty quantities could be very appropriate, independent of the response assumptions.

In this work it is shown that, in the case of linear structures, it is always possible to express the structural stiffness as a linear combination of random quantities (fields or variables) representing the physical uncertainties. It will be shown that in some cases, as for the uncertain non-symmetric continuum, this representation is straightforward. In other cases, where this relationship is nonlinear, and this surely happens when both geometrical and mechanical uncertainties are present, it is always possible to introduce a transformation of the random quantities in such a way that the stiffness depends linearly on the new random quantities.

The three-dimensional, two-dimensional and one-dimensional continua cases are treated together with the corresponding finite element discretized structures. The linear representation of the structural stiffness may be very useful for obtaining the response statistics, using both exact and approximate analyses. The use of the perturbation approach could be an example: if the structural stiffness is linear with respect to the uncertain parameters, the application of high order perturbation techniques becomes straightforward and greater accuracy is encountered as shown in the numerical application.

1

H.G. Matthies, C.E. Brenner, C.G. Bucher, C.G. Soares, "Uncertainties in probabilistic numerical analysis of structures and solids - stochastic finite elements", Structural Safety, 19, 283-336, 1997. doi:10.1016/S0167-4730(97)00013-1

2

A. Der Kiureghian, J.B. Ke, "The stochastic finite element method in structural reliability", Probabilistic Engineering Mechanics, 3, 83-91, 1988. doi:10.1016/0266-8920(88)90019-7

3

K.L. Liu, A. Mani, T. Belytschko, "Finite element methods in probabilistic mechanics", Probabilistic Engineering Mechanics, 2, 201-213, 1987. doi:10.1016/0266-8920(87)90010-5

4

H. Contreras, "The stochastic finite element", Computers and Structures, 12, 341-348, 1980. doi:10.1016/0045-7949(80)90031-0

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