Green’s Function for Elastic Half-Strip

Abstract

In the presented work, vertical displacements of the upper face of an elastic half-strip with a hinged lower face are analytically determined under the action of a vertical concentrated force applied vertically to the upper face. In this case, the method of special approximationis used, previously effectively used in the works of I. Sneddon and, later, V. M. Alexandrov. On the lower face of the half-strip, vertical displacements and tangential stresses are assumed to be equal to zero. The desired expression for the displacements consists of the displacements of the infinite strip under the action of two symmetrically applied vertical forces and a self-balanced normal horizontal load applied to the end of the half-strip and equal to the normal horizontal stresses under the action of two symmetrically applied forces to the infinite strip with the opposite sign. The displacements under a self-balanced load according to the Ritz method are presented as a double series in classical orthogonal functions – Hermite poly-nomials with weight and Legendre polynomials with undetermined coefficients, which are determined under the condition of the minimum of the functional of the total energy of deformations of the half-strip and the work of the end self-balanced horizontal load. The obtained expression for displacements contains elementary functions, has a logarithmic singularity at the point of application of the force and decreases at infinity. Graphs of vertical displacements of the upper face of the half-strip are given for different positions of the external vertical force. The accuracy of the adopted special approximation is also shown graphically. The obtained results can be used to solve various contact problems for an elastic half-strip loaded along the upper face.