Solution of contact problems of anisotropic plates bending on an elastic base using the compensating loads method

¹ Kazan National Research Technical University named after A.N.Tupolev-KAI, 420111 Kazan, Russia
² Kazan (Volga Region) Federal University, 420008 Kazan, Russia

^* Corresponding author: pvelikanov@mail.ru

Abstract

In critical structures used elements consisting of anisotropic materials with concentrated force or moment applied to them are often found. In order to equalize the internal force factors, as well as to ensure the required strength and rigidity of the structure, in places where a local load is applied, it is necessary to strengthen the plate with a rigid lining (stamp). There are several ways to solve contact problems. The way seems simpler if it is possible to determine the exact fundamental solution (FS) for the plate. The exact FS significantly reduces the amount of computational work due to the fact that the boundary conditions and the conditions of conjugation of solutions at the boundary of the contact area are met in advance. It remains only to formulate the conditions for the compatibility of movements in the contact area. With this formulation, the integral equation of the contact condition becomes a singular integral equation of the first kind. Such equations are reduced by regularization to Fredholm equations of the second kind and the problem becomes mathematically and physically correct. The paper considers contact problems without taking into account tangential interaction between objects. For such problems, the following way of simplification is possible: replacing the singular core for a half-space with a core for a layer of constant thickness modeled by an elastic base with a single base coefficient. Such a replacement for thin shells is quite justified and gives solutions that are in good agreement with the solutions of the theory of elasticity. The deflection of the plate is sought using the compensating loads method. To work out the methodology, a test contact problem was solved.