The generalized density evolution equation for the dynamic analysis of slender masonry structures

Structural dynamics deals with the issues of response analysis, reliability evaluation and system control of any structure subj ected to dynamic actions. Generally dynamic actions exhibit significant randomness that stems from complicated physical mechanisms. However, this is not the only cause of uncertainties: even those due to the initial data and physical parameters of the stru cture contribute to the randomness of the dynamics response. This leads to structural analyses involving random parameters, usually known as stochastic structural analysis [1]. Generally, the state equation of a dynamic system, which is typically obtained by discretizing the structure into finite elements, as well as the initial joint probability density function (PDF) of all the considered random variables are assigned. Each solution of the state equation is a stochastic process that is parametrized over t he time and has the Euclidean space as state space. The aim, therefore, is to determine the evolution of the PDF of some quantities of interest. Since the beginning of the last century, this issue has been addres sed by formulating (stochastic) partial differential equations to obtain the desired results. The classical Liouville, Fokker Planck and Dostupov Pugachev equations belong to this class [2]. These equations are usually high - dimensiona l with strong non - linearities in their coefficients . Despite the numerous studies that have allowed a rigorous formulation of these equations and a better understanding of their properties, the available solutions of practical interes t are still very limited. A different way to deal with the problem is to use the so - called principle of preservation of probability [3]. This is a consequence of the transformation law of the probability density function as a result of a (sufficiently regular) transformation of the state space, and leads to writing a linear partial differential equation for any quantity, whose PDF has to be determined. The coefficients of this equation at each instant depend on t he value of the state variables. Therefore, writing this equation requires the (deterministic) solution to the state equation of the system. The application of this method, known as generalized density evolution equation proceeds as follows [1, 3]. Firstly, the representative points in the random parameter space are selected. A cell with a given volume is associated to each representative point, so that all these cells form a partition of the random space ; t hen, the probability measure is assigned to each point. Successively, for each representative point, a deterministic analysis of the dynamical system is carried out, the coefficients of the partial differential equations are determined and the equations ar e numerically solved. Finally, the results obtained in the previous steps are synthesized and the instantaneous value of the desired PDF are determined. The method of generalized density evolution equation has been implemented into the MADY code, which already has routines for nonlinear dynamic analysis of plane, three - dimensional, or beam and shell - based structures [4]. Then, it has been applied to the study of a masonry tower subjected to seismic actions. It was assumed that the structure was made of normal elastic material , with low tensile strength and limited compressive strength [5]. The initial joint PDF of the main mechanical p arameters was specified and the evolution of the PDFs of the displacement on the top of the tower as well as of some structural damage parameters has been determine d.