Behavior of Stochastic Shear Beams Under Random Loading via Stochastic Variational Principles

In this study we derive, apparently for the first time in the literature, some exact solutions for the shear beams with stochastic flexibility, when these beams are acted upon by the random loading. The importance of these solutions lies in the fact that they can serve as benchmark solutions, to which the approximate solutions of various nature can be compared. Then we formulate stochastic variational principles, the first principle governs the behavior of the mean displacement, whereas the second principle is satisfied by the displacements covariance function. These variational principles allow us to formulate approximate techniques for the cases in which the exact solution is presently unavailable. In particular, we develop a stochastic version of the Rayleigh–Ritz method. Several examples are evaluated to shed more light on the probabilistic behavior of randomly excited structures possessing random flexibility.