To Solution of Contact Problem for Rectangular Plate on Elastic Half-Space

Abstract

Until the present time there is no exact solution to the contact problem for a rectangular plate on an elastic base with distribution properties. Practical analogues of this design are slab foundations widely used in construction. A lot of scientists have solved this problem in various ways. The methods of finite differences, B. N. Zhemochkin and power series do not distinguish a specific feature in contact stresses at the edges of the plate. The author of the paper has obtained an expansion of the Boussinesq solution for determining displacements of the elastic half-space surface in the form of a double series according to the Chebyshev polynomials of the first kind in a rectangular region. For the first time, such a representation for the symmetric part of the Boussinesq solution was obtained by V. I. Seimov and it has been applied to study symmetric vibrations of a rectangular stamp, taking into account inertial properties of the half-space. Using this expansion, the author gives a solution to the problem for a rectangular plate lying on an elastic half-space under the action of an arbitrarily applied concentrated force. In this case, the required displacements are specified in the form of a double row in the Chebyshev polynomials of the first kind. Contact stresses are also specified in the form of a double row according to the Chebyshev polynomials of the first kind with weight. In the integral equation of the contact problem integration over a rectangular region is performed while taking into account the orthogonality of the Chebyshev polynomials. In the resulting expression the coefficients are equal for the same products of the Chebyshev polynomials. The result is an infinite system of linear algebraic equations, which is solved by the amplification method. Thus the sought coefficients are found in the expansion for contact stresses.