Nonlinear random vibrations of beams with in-span supports via statistical linearization with constrained modes

A statistical linearization technique is developed for determining second-order response statistics of beams with in-span elastic concentrated supports. The nonlinearities considered relate both to the support restoring forces, and to the assumption of relatively large beam displacements. A significant novel aspect of the technique is the utilization of constrained modes involving generalized functions in their definition; thus, shear-force discontinuities at the support locations can be readily accounted for. Overall, a set of nonlinear modal equations is derived and replaced by a set of equivalent linear ones based on an error minimization scheme in a mean square sense. This yields a set of algebraic nonlinear equations for the beam response second-order statistics, which can be readily solved in a computationally efficient manner via a simple iterative scheme. It is noted that the technique applies to an arbitrary number of supports yielding accurate and computationally efficient solutions for the second-order statistics of the response. Two illustrative numerical examples are considered for assessing the reliability and accuracy of the technique as compared with pertinent Monte Carlo simulation data. The latter are generated based on a boundary integral solution methodology in conjunction with a Newmark numerical integration scheme.