Explicit solutions for the response probability density function of nonlinear transformations of static random inputs

• A new version of the probabilistic transformation method is proposed.
• The probability density functions of single response quantity are evaluated.
• The exact pdf of linear systems subjected to a quadratic combination of Gaussian variables is given.
• Approximated results for some non-linear systems subjected to generic random loads are given.

This work is the second paper of two companion ones. Both of them show the use of a new version of the Probabilistic Transformation Method (PTM) for finding the probability density function (pdf) of a limited number of response quantities in the transformations of static random inputs. This is made without performing multi-dimensional integrals of the response total joint pdf for saturating the non-interested variables. While in the first paper the linear transformations have been considered, in the present one some nonlinear systems are taken into account. In particular, first the case when the loads on a linear structural system are a nonlinear combination of static random inputs is studied. Then the attention is placed on the case of nonlinear structural systems, for which the new version of the PTM allows to determine approximated, but accurate, results.