Some fundamental solutions for the Kirchhoff, Reissner and Mindlin plates and a unified BEM formulation

Abstract

We derive analytical solutions for the deflection of thin circular plates, which are loaded by centrally located concentrated bending moments and transverse forces. Green's functions for clamped and simply supported plates are presented. Reduction of these Green's functions leads to the corresponding fundamental solution for the Kirchhoff plate bending model (K problem). This fundamental solution reduces to those obtained through the direct simplification of the fundamental solution for the sixth-order Reissner and Mindlin plate bending models (RM problem). This allows to decompose each fundamental tensor of the problem RM into the sum of the fundamental tensor of the problem K and a correction tensor (Sh problem), which contains the contribution of the shear strains, e.g. U ij RM( r)= U ij K( r)+ U ij Sh( r). Within the boundary element analysis this enables the investigation of the contributions of the shear strains to the solutions of Reissner and Mindlin plate bending models, as corrections of the Kirchhoff values, all determined from the same BEM code. This opens up new possibilities for the analysis of plates by the BEM.