Hankel transformation method for solving the Westergaard problem for point, line and distributed loads on elastic half-space - Ike, Charles Chinwuba (Latin American Journal of Solids and Structures, 2019)
The Hankel transformation method was used in this work to determine the normal and shear stress distributions due to point, line and distributed loads applied to the surface of an elastic media. The elastic media considered in this study was assumed to be inextensible in the horizontal directions, and the only non-vanishing displacement component is the vertical component. Such materials were first considered by Westergaard as models of elastic half-space with alternating layers of soft and stiff materials with the stiff materials of negligible thickness and so closely spaced that the composite material characteristics is idealised as isotropic and homogeneous. The study adopted a displacement formulation. The Hankel transformation was applied to the governing Cauchy--Navier differential equation of equilibrium to reduce the problem to a second order ordinary differential equation (ODE) in terms of the deflection function in the Hankel transform space. Solution of the ODE subject to the boundedness condition yielded bounded deflection function in the Hankel transform space. Equilibrium of the internal vertical forces and the external applied load, was used to obtain the constant of integration, for the deflection function in the Hankel transform space. Inversion yielded the deflection in the physical domain variable. The stress displacement equations were then used to determine the Cauchy stresses. The vertical stress distributions due to line and distributed load over a rectangular area were also determined by using the point load solution for vertical stresses as Green functions, and then performing integration along the line and over the rectangular area of the load. The results obtained for vertical stresses due to point, line and distributed loads were determined in terms of dimensionless influence coefficients which were presented. The results obtained for the deflection, normal and shear stresses due to point, line and distributed load agreed with the solutions originally presented by Westergaard who used a stress function method.