Fourier Integral Transformation Method for Solving Two Dimensional Elasticity Problems in Plane Strain Using Love Stress Functions

Abstract

The Fourier integral method was used in this work to determine the stress fields in a two dimensional (2D) elastic soil mass of semi-infinite extent subject to line and strip loads of uniform intensity acting on the boundary. The two dimensional plane strain problem was formulated using stress-based method. The Fourier integral was used to transform the biharmonic stress compatibility equation to a fourth order linear ordinary differential equation (ODE) in terms of the stress function. The ODE was solved subject to the boundedness condition to obtain the bounded stress function. Cartesian stress components were obtained using the Love stress functions. Application of the stress boundary conditions for the case of line load of uniform intensity and the cases of uniformly distributed load on a strip of finite width gave the respective unknown constants of the Love stress functions; and hence the complete determination of the Cartesian stress components for the two cases considered. Inversion of the Fourier integral expressions obtained for the normal and shear stresses in the Fourier parameter gave respective expressions for the normal and shear stress fields for line and finite strip loads of finite width in the physical domain variables. The results obtained agreed with the results from previous studies which used displacement based methods.

Solving Westergaard Half-Space Problems Using Potential Theory

“…By coordinate transformation from Cartesian to cylindrical polar coordinates, (37) where J is the Jacobian of the transformation and r is the radial coordinate,  is the angular coordinate, and z is the depth coordinate. The relations between 3D Cartesian coordinates (x, y, z) and cylindrical coordinates (r, , z) are…”

“…Ike [36] used Elzaki transformation method to find closed-form solutions to elastic half-plane problems in polar coordinates using Airy stress function of expressing the 2D elasticity problem. Ike [37] also used Fourier integral transform method to obtain closed-form solutions to 2D elastic half-plane problems using Love stress function approach. Ike [38] used the Hankel transformation method to solve the closed-form Westergaard half-space problem for concentrated and line load.…”