Random analysis of geometrically non-linear FE modelled structures under seismic actions

In the framework of the finite element (FE) method, by using the “total Lagrangian approach”, the stochastic analysis of geometrically non-linear structures subjected to seismic inputs is performed. For this purpose the equations of motion are written with the non-linear contribution in an explicit representation, as pseudo-forces, and with the ground motion modelled as a filtered non-stationary white noise Gaussian process, using a Tajimi-Kanai-like filter. Then equations for the moments of the response are obtained by extending the classical Itô's rule to vectors of random processes. The equations of motion, and the equations for moments, obtained here, show a perfect formal similarity. By using this similarity a very effective computational procedure for evaluating response moments of any order is proposed. Within the framework of non-Gaussian closure schemes, a technique is here presented based on a truncated Gram-Charlie expansion. To achieve this the Hermite coefficients are evaluated for multi-degree-of-freedom systems, once the multi-dimensional Hermite polynomials have been obtained in compact form.