On a generalized Loveʹs problem

We present explicit expressions for computing the displacements induced in a homogeneous, linearly elastic half-space by uniform vertical pressure applied over an arbitrary polygonal region of the horizontal surface. By suitably applying Gauss theorem and recent results of potential theory we derive formulas which allow one to evaluate the displacements at an arbitrary point of the half-space solely as a function of the position vectors of the boundary of the loaded region assumed to be polygonal. Representative numerical examples referred to geodetically observed elastic displacements of the Earth surface due to water loads show the effectiveness and the flexibility of the proposed approach. Actually, it allows for a more realistic evaluation of displacements distribution and to achieve a considerable simplification in data handling since it is now possible to avoid tiling of complex regions by the simple load shapes, such as circles or rectangles, for which analytical solutions are currently available in the literature.