Civil and Structural Engineering Computing: 2001
Chapter 12 G. Muscolino
Department of Construction and Advanced Technology, University of Messina, Italy Keywords: stochastic dynamics, random vibrations, nonlinear dynamic systems, cumulant-neglect closure, quasi-moments neglect closure, Fokker-Planck-Kolmogorov differential equation
In this review some of the most recent developments of importance within the field of stochastic dynamics in structural engineering are discussed. Particularly emphasis is put on the improvement of the calculation of non-Gaussian probability density function. In this area methods for the evaluation of this density for nonlinear systems are treated in detail. The study of engineering problems by stochastic approaches implies that is accepted the idea that it is either impossible or infeasible to devise absolutely sure the structural system or the loads. Here the characterisation of uncertainties of both structural systems and loads is based on probability theory [1,2]. Indeed this theory is suitable for quantitative analysis and useful for a great variety of problem, while other theories, like fuzzy set, is best set for the analysis of qualitative information [3]. Stochastic dynamics, originally called random vibration, was the name given to the body of theory associated with dynamic system responding to random excitations. The ultimate purpose of this theory is to provide a solid basis for improving the reliability of structures, vehicles etc., that must withstand randomly fluctuating loads. Nowadays the stochastic dynamics can be subdivided in five main sections: a) classical stochastic dynamics which treats the analysis of structural linear or nonlinear deterministic systems subjected to random loads by solving differential or algebraic equations; b) Monte Carlo simulation which requires the generation of samples of the input process, the deterministic analysis of structures and the evaluation of the resultant response statistics; c) stochastic finite elements in which the parameters of the linear or nonlinear structural system are assumed uncertain and the loads could be stochastic or deterministic; d) time dependent reliability in which the failure probability can be defined by the first outcrossing probability or by the fatigue models; e) system identification in which information about excitation and response data are used to deduce the properties of structural systems. In the second part of review, in the framework of Classical Stochastic Dynamics, the most common methods able to perform the stochastic analysis of one- dimensional nonlinear structural systems are treated in detail. The probabilistic characterisation of a stochastic process are performed either by means of the probability density function (PDF) or the statistical moments, cumulants or quasi- moments. These methods are based on the so-called non-Gaussian closure techniques or on approximate solutions of the Fokker-Planck-Kolmogorov (FKP) partial differential equation. A procedure able to evaluate the moments of the response for a nonlinear system subjected to external stationary zero-mean Gaussian process will be given. A different approach based on the theory of continuous Markov processes is also illustrated. The state transition PDF for such processes is governed by the Fokker-Planck-Kolmogorov (FPK) differential equation [4], which is a linear partial differential equation. For white noise excitations, the solution of this equation offers a direct approach to the exact treatment of the problem. Unfortunately, to date all known solutions have been obtained only for limited cases of nonlinear stiffness and damping [5]. For this reason approximate methods to solve the FPK equation are needed. In the numerical application section it has been evidenced the advantages of both so-called quasi-moments non-Gaussian closure technique (with respect the cumulant closure one) and of the approximation of the unknown probability density function (PDF) by an exponential form. This approximation, in fact, guaranties the positivity of the PDF and avoids some of the main drawbacks of the usual closure schemes. It is also shown show the effectiveness of the weight residual method to solve the FPK equation by adopting exponential approximation of the PDF. It has been also evidenced that the extension of described procedure to multidimensional nonlinear systems can be easily performed by applying the Kronecker Algebra.
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