Boundary element solutions for plates on elastic foundations

Applications of the boundary element method to plate bending problems are quite sparse. The usual approach has been to treat same specific problems or types of problems. However, many practically important problems have been left without consideration. Plates on elastic foundations is one of these practically important areas and, moreover, it seems to be especially suitable to be treated with boundary element method. The analysis presented in this paper is based on the Kirchhoff plate bending theory and a fundamental singular solution is a displacement field caused by a unit lateral load acting at a point of an infinite plate resting on a linearly elastic foundation, which can be either of Winkler or Pasternak types or an elastic half space or, more generally, a foundation for which an axisymmetric external stress will result in an axisymmetric state of deformation. The derived integral equations base on the so-called direct formulation of the boundary element method. The two solution equations are formulated in terms of displacement rotation, moment and resultant boundary shear in every boundary nodal point. From these four unknown variables in every boundary node at least two has to be prescribed. The considered boundary conditions are free, simply supported and rotationally restrained edges. The developed computer code was designed to include the singular solutions for plates on Winkler, Pasternak and elastic half space foundations. These solutions were plotted and compared with each other and with available solutions in literature. The developed method has the advantage that the treatment of plates with finite dimensions and varying boundary conditions will be greatly facilitated. The analytic solution for these types of problems are rare and, moreover, the evaluating of numerical results from these solutions is a cumbersome task. (orig.)