Vibrations of oblique shear-deformable plates

Abstract

The proposed numerical analysis of moderately thick plates subject to rather general boundary conditions is based on the direct boundary element method (BEM) in the frequency domain. First order shear-deformation theory of the Reissner–Mindlin-type is considered. A step forward in efficiency is obtained when the force and double force with moment Green's functions of the rectangular simply supported base plate of the same stiffness are applied. The time-reduced equations of hard-hinged polygonal plates correspond to those of a background Kirchhoff plate having frequency-dependent effective parameters like mass, lateral and in-plane load, and is further forced by imposed fictitious curvatures. This analogy holds even for the quasi-static shear forces and bending moments, i.e., when inertia effects become negligible. Furthermore, it can be shown that, in the static case, these stress resultants for certain groups of Reissner-type shear-deformable plates are identical with those resulting from the Kirchhoff theory in the background. Since this analogy is restricted to hard-hinged supports of straight edges, it is necessary to apply, e.g., the direct BEM of analysis to the plate of general planform and boundary conditions. The main effort is thus to study the properties and effective representations of the Green's dyadics and their singularities, in view of their proper integration. Similarly as for Kirchhoff plates, the strong singularity of the infinite domain is identified for the rectangular plate and subject to indirect integration. The resulting direct BEM proves to be efficient, robust and, in connection with proper pre- and post-processors, becomes an effective tool of engineering analyses just within the frequency limits given by the first two of the three spectral branches.