A reasonable solution to the elastic problem of half-plane

Abstract

For the half-plane problem with external loads acting on the surface, Flamant first obtained the elastic solutions when the external load is concentrated force or uniformly distributed load. The stress field obtained decreases with the increase of depth, and each stress component is equal to zero at infinity, which is in line with our expectation. However, the displacement solution obtained does not conform to the actual situation because an abnormal phenomenon appears in which the displacement increases when depth increases. In this paper, we will consider a more complex form of external load and the past concentrated force and uniformly distributed load are only special cases of this kind of load. And the method of conformal transformation and series approximation proposed are adopted to avoid the abnormal phenomenon of displacement field. The analytic functions are obtained by the Cauchy integral of the surface stress boundary condition, and then the analytic functions are transformed by the method of series approximation. The new analytic function can ensure that the displacement at infinity is bounded. The solution proposed is an analytical method. The stress field obtained has a very high accuracy, and the displacement field obtained also conforms to the law of gradual decrease with the increase of depth. By comparing the analytical solution of stress and displacement with the numerical solution of FEM, the correctness of the derivation process in this paper will be verified, and it can be found that the numerical method has some limitations in solving the half-plane problem.