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COMPUTATIONAL METHODS FOR ENGINEERING SCIENCE
Edited by: B.H.V. Topping
Explicit Solutions for the Static and Dynamic Analysis of Discretized Structures with Uncertain Parameters
G. Muscolino1 and A. Sofi2
1Department of Civil Engineering, University of Messina, Italy
G. Muscolino, A. Sofi, "Explicit Solutions for the Static and Dynamic Analysis of Discretized Structures with Uncertain Parameters", in B.H.V. Topping, (Editor), "Computational Methods for Engineering Science", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 3, pp 47-73, 2012. doi:10.4203/csets.30.3
Keywords: uncertain parameters, interval analysis in structural engineering, static analysis, dynamic analysis in frequency domain.
Standard structural analysis tools are devoted to the numerical evaluation of the system response resulting from external loads for given geometry and material properties. However, in practical engineering, on account of physical imperfections, model inaccuracies and system complexities, almost all structures exhibit physical and geometrical uncertainties to some degree. These sources of uncertainty, which affect to a certain extent the structural response, are usually described following two contrasting points of view, known as probabilistic and non-probabilistic approaches. The first ones are certainly the most widely adopted and can be developed in three main ways: the Monte Carlo simulation method, the perturbation techniques  and the spectral methods . Unfortunately, the probabilistic approaches require a wealth of data, often unavailable, to define the probability distribution density of the uncertain structural parameters. This implies that, when crucial information on a variability is missing, it is not good practice to model it as a probabilistic quantity. Starting from the pioneering study of Ben-Haim and Elishakoff , in recent years several non-probabilistic approaches have been proposed to perform the static and dynamic analysis of structures. These approaches are mainly based on: convex models, interval models and fuzzy sets . Today, the interval model may be considered as the most widely used analytical tool. This model is derived from the interval analysis  in which the number is treated as an interval variable with given lower and upper bounds.
In the framework of probabilistic approaches, explicit solutions have been evaluated in statics [6,7] and dynamics [8,9] by taking into account the properties of the natural deformation modes of the finite element discretized structure. In a non-probabilistic context, explicit solutions have been obtained recently for both static [10,11] and dynamic  analysis.
In this paper, a procedure recently proposed by the authors  to derive in explicit form both the stiffness matrix and the frequency response function (FRF) of linear discretized structures with stiffening uncertain parameters is reviewed. The proposed method is based on the following main steps: i) the decomposition of the deviation of the stiffness matrix (with respect to its nominal value) to obtain a sum of rank-one matrices each associated with one uncertain parameter; ii) the use of an innovative series expansion, referred to as a rational series expansion, to evaluate in approximate but explicit form the inverse of the stiffness matrix as well as the FRF in the modal subspace. The presented procedure allows us to obtain very accurate explicit expressions of both the static response and the frequency domain dynamic response of structural systems with uncertain parameters. Furthermore, in the paper the proposed series expansion is also adopted in conjunction with the so-called improved interval analysis, recently introduced by the authors , to determine the range of the static displacements and of the modulus of the FRF of structures with uncertain-but-bounded parameters.
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