Journal of Elasticity

, Volume 122, Issue 1, pp 75–112

# A General Approach to the Solution of Boussinesq’s Problem for Polynomial Pressures Acting over Polygonal Domains

Article

## 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 affecting 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 accommodated 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 contact mechanics. Finally, comparisons with existing solutions and numerical examples are discussed.

## Keywords

Boussinesq’s problem Half-space Potential theory

31C05 74B05

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