Solving Westergaard Half-Space Problems Using Potential Theory

Abstract

 The governing differential equation is derived for an elastic half-space problem with horizontal inextensibility.  The potential theory is applied to solve the Westergaard problem for a point load on the boundary.  The approach adopts first principles to derive the governing equations, highlighting limitations and scope.  Results are validated by comparison with literature sources.The Westergaard half-space problem has been solved using the potential theory in this work. It is a classical theme in elasticity theory that seeks to find the displacements and stresses in the half-space caused by known boundary loads. It has important applications in analyzing stresses and displacement fields in soil due to applied points and distributed loads on the boundary caused by structures placed on the soil. It is governed by stress-strain, strain-displacement, and equilibrium equations. Horizontal inextensibility is assumed in developing the problem, simplifying the displacement formulation to a three-dimensional (3D) Laplace equation.The potential theory is applied to find the vertical displacement. Stressdisplacement equations obtained from the simultaneous use of the kinematic and stress-strain equations are used to obtain the stress fields.The specific problem of point load at the origin was considered and solved. The equilibrium of internal vertical stresses and the external vertical load is used to find the integration constant. Hence, vertical displacements were found. The stress fields were found from the stressdisplacement equations. The expressions for the vertical displacements and stresses were found to be exact within the framework of the theory used, as they satisfied all the governing equations of the problem. However, the solutions become unbounded at the origin due to the singularity of the vertical displacement and stresses. The obtained solutions are identical to previously obtained solutions.