Vibrations of rectangular plates with elastic intermediate line-supports and edge constraints

A set of static beam functions, which are the solutions of an elastically point-supported beam under a Fourier series of static sinusoidal loads distributed along the length of the beam, are developed as the admissible functions to analyze the vibrations of orthotropic rectangular plates with elastic intermediate line-supports using the Rayleigh–Ritz method. Both the elastic rotational and the elastic translational constraints along the edges of the plate are also considered simultaneously. Unlike conventional admissible functions, this set of static beam functions not only can automatically adjust to the stiffnesses of the intermediate line-supports but also can properly describe the discontinuity of shear forces at the line-supports so that higher accuracy and faster convergence can be expected for the dynamic analysis of such plates. The suggested approach is effective even for various limiting cases by letting the corresponding stiffnesses approach their natural limits of zero or infinity. The present method is theoretically sound and mathematically simple, with each of the static beam functions being only a third-order polynomial plus a sine function. A common and efficient computational program can be compiled because of the fact that a change of the line-support parameters (locations, number and stiffnesses) and the boundary conditions of the plate only results in a corresponding change of the coefficients of the polynomial in the static beam functions. Several numerical examples are presented and the results obtained, where possible, are compared with the known solutions in literature. The present method has proved to be extremely effective for solving the aforementioned problems.