Geometrically Nonlinear Analysis of Beam-Reinforced Thin Plates Using the Methodology of Groebner Bases

Abstract

This paper illustrates the utility of the methodology of Groebner bases computations combined with the energy method in the analysis of geometric nonlinear beam-reinforced thin rectangular isotropic plates (BRP) for modeling rectangular duct system deflections under internal positive pressure. The governing integro-partial differential equation is derived based on Kirchhoff/von Karman plate theory. With Rayleigh/Ritz methodology, a system of coupled polynomial algebraic equations is generated by using the exact solution of the BRP plate from linear analysis as a shape function. Then Groebner basis methodology is employed to decouple these equations. Under certain conditions, an analytical expression for the lateral displacements of the BRP plate under pressure can be obtained in a fully symbolic form, in terms of such parameters as geometric and material properties of the beams and panels. The analytical solutions have been compared with the results using the finite element software, ANSYS. The comparative study indicates that for commonly used duct panels with the aspect ratio (L:W) less than 1:4, the analytical solutions for displacements are in very close agreement. Finally, the study is found to be a unique alternative, worthy of further investigation, and potentially effective in the analysis of similar problems occurring in a variety of engineering applications.