Cosine Integral Transform Method for Solving the Westergaard Problem in Elasticity of the Half-Space

Abstract

The cosine integral transform method is applied to find the expressions for spatial variations of displacements and stresses in the Westergaard continuum under vertical concentrated loading, and distributed loadings acting over lines and geometric areas on the surface. The half-space is considered to be horizontally inextensible and the displacement field reduces to the vertical displacement component. The paper derives a displacement formulation of the equation of equilibrium in the vertical direction. Cosine integral transformation is applied to the formulated equation and the Boundary Value Problem (BVP) is found to simplify to Ordinary Differential Equation (ODE). The general solution of the ODE is obtained in the cosine integral transform space. The requirement of bounded solutions is used to obtain one integration constant. Inversion of the bounded solution gave the solution in the real problem domain space. The stress fields are obtained using the stress-displacement equations. The requirement of equilibrium of the vertical stress fields and the vertical point loading at the origin is used to determine the remaining integration constant, and thus the vertical deflections and the stresses. The solutions obtained are kernel functions employed to derive the expressions for solutions for line, and uniformly distributed loads applied over given geometric areas such as rectangular and circular areas. The vertical stresses are expressed in terms of dimensionless vertical stress influence factors and tabulated. The vertical displacements and stresses obtained are identical with Westergaard solutions obtained by stress function method. The solutions agree with results obtained by Ike using Hankel transform method.