Certain contact problems for a half-space reinforced by elastic gussets: PMM vol. 36, n≗5, 1972, pp. 770–787

Abstract

Contact problems for bodies with elastic reinforcements in the form of gussets (stringers) of slight thickness are directly connected with questions of load transmission from the gussets to elastic bodies, which are important for engineering practice. Various plane problems have been investigated in many papers. For example, two fundamental problems on the transmission of a load from a gusset infinite in both directions to a semi-infinite and infinite plate have been examined in [1]. The model of a one-dimensional elastic continuum of the gusset is taken as the fundamental physical model. A number of papers devoted to various extensions and modifications of the fundamental Melan problems was later executed within the scope of the physical assumptions in [1]. A sufficiently complete bibliography of these papers is contained in [2, 3], The papers [4–7] are devoted to giving a foundation for the model of the one-dimensional model of the elastic continuum of the gusset and to investigating some other contact problems for a half-plane with elastic gussets. While the domain of plane contact problems for bodies with elastic gussets of slight thickness has been developed sufficiently well, the domain of three-dimensional contact problems for bodies with elastic gussets of slight cross section has hardly ever been investigated, and the authors know of no papers in this area where a rigorous solution of such problems would be presented. In some sense, paper [8], referring to questions of determining the contact stresses on the lateral surface of a cylindrical rod imbedded in an elastic space or half-space, is an exception. Such a situation in the area of three-dimensional problems is explained by the fact that significant mathematical difficulties are encountered in their solution. Moreover, the model of one-dimensional elastic continuum for the gusset in combination with the model of contact along a line is not directly applicable in the formulation of three-dimensional contact problems for bodies with elastic gussets of small cross section. The model of contact along an area, when it is assumed that the stresses in the contact zone are distributed uniformly in the transverse direction, also does not correspond completely to reality. In contrast to the case of plane problems, a new approach is proposed herein to the formulation of three-dimensional contact problems for a half-space reinforced by elastic gussets of small cross section. Three kinds of contact problems are then examined, namely, problems when the elastic half-space is reinforced on some part of its boundary by an infinitely long gusset, a semi-infinite gusset. and a gusset of finite length. In the proposed formulation, the solution of these problems reduces to solving integro-differential equations with kernels expressible by complete elliptic integrals of the first and second kinds, under definite boundary conditions. An effective method of solving these equations is proposed.