J Elast DOI 10.1007/s10659-015-9534-5 A General Approach to the Solution of Boussinesq’s Problem for Polynomial Pressures Acting over Polygonal Domains Francesco Marmo 1 · Luciano Rosati 1 Received: 24 November 2014 © Springer Science+Business Media Dordrecht 2015 Abstract We outline a general approach for extending the classical Boussinesq’s solution to the case of pressures distributed according to a polynomial law of arbitrary order over a polygonal domain. To this end we exploit a generalized version of the Gauss theorem and recent results of potential theory which consistently take into account the singularities affect- ing the expressions of the fields of interest, an issue which seems to have been overlooked in the existing literature. For linearly varying pressures we derive analytical expressions of displacements, strains and stresses at an arbitrary point of the half-space as a function of the loading function and of the position vectors which define the boundary of the loaded region. We briefly discuss how bilinear and more general pressure distributions can be accommo- dated in our formulation since the paper is mainly motivated by the interest in developing efficient computational tools for solving 3D problems in foundation engineering and con- tact mechanics. Finally, comparisons with existing solutions and numerical examples are discussed. Keywords Boussinesq’s problem · Half-space · Potential theory Mathematics Subject Classification (2000) 31C05 · 74B05 1 Introduction In a celebrated paper [5] Boussinesq determined the displacement and stress field in a lin- early elastic, homogeneous and isotropic half-space subject to a vertical point load. The solution was obtained by reducing the original problem to a boundary value problem in potential theory. Considering the half-space surface subject only to normal tractions, the B F. Marmo f.marmo@unina.it L. Rosati rosati@unina.it 1 Department of Structures in Engineering and Architecture, University of Naples Federico II, Naples, Italy