PARAMETRIC INSTABILITY OF POLYGONAL MINDLIN–REISSNER-PLATES SUBJECTED TO HARMONIC IN-PLANE FORCES

In this paper, the dynamic stability analysis of shear-deformable plates of arbitrary polygonal planform is performed within the framework of the Mindlin–Reissner theory. The plates are considered to be subjected to a parametric excitation by harmonic in-plane forces. The influence of plate shear and rotatory inertia is taken into account, a two-parameter Pasternak foundation is chosen, and the more accurate theory of Brunelle and Robertson is included. Considering harmonic in-plane forces yields partial differential equations with time-dependent parameters. Ordinary differential equations for the generalized co-ordinates are derived by expanding the deflection and the cross-sectional rotations of the plate in series representations in terms of normal modes and using Galerkinʹs principle. The normal modes are governed by Helmholtz-eigenvalue problems in the case of simply supported boundaries. Parametric instability of flexural- and thickness-shear motions are studied in more detail. The governing equations enable a number of results to be obtained which reveal the influence of the special shape of the plate domain represented by the Helmholtz eigenvalue, parameters of the foundation and a tracer for the Brunelle and Robertson theory. The main merit of the approach is that the particular shape and mechanical properties of the polygonal plate are represented in these equations in terms of Helmholtz eigenvalues which allows a general analysis for plates of arbitrary polygonal planform to be performed. A vast amount of literature exists on Helmholtz eigenvalues in the context of natural vibrations of membranes, which may be utilized. The boundaries of the principal instability region are calculated and the stability charts of these two motions are represented graphically. These results are finally derived in a non-dimensional form and illustrated by means of numerical examples.