Узагальнення моделі Голанда і Рейсснера на випадок осьової симетрії

Abstract

The purpose of this work is to create a mathematical model of the stress state of overlapped circular axisymmetric adhesive joints and to build an appropriate analytical solution to the problem. To solve the problem, a simplified model of the adhesive bond of two overlapped plates is proposed. The simplification is that the movement of the layers depends only on the radial coordinate and does not depend on the angular one. The model is a generalization of the classical model of the connection of Holland and Reissner in the case of axial symmetry. The stresses are considered to be evenly distributed over the thickness of the layers, and the adhesive layer works only on the shift. These simplifications allowed us to obtain an analytical solution to the studied problem. The problem of the stress state of the adhesive bond of two plates is solved, one of which is weakened by a round hole, and the other is a round plate concentric with the hole. A load is applied to the plate weakened by a round hole. The discussed area is divided into three parts: the area of bonding, as well as areas inside and outside the bonding. In the field of bonding, the problem is reduced to third- and fourth-order differential equations concerning tangent and normal stresses, respectively, the solutions of which are constructed as linear combinations of Bessel functions of the first and second genera and modified Bessel functions of the first and second genera. Using the found tangential and normal stresses, we obtain linear inhomogeneous Euler differential equations concerning longitudinal and transverse displacements. The solution of the obtained equations is also constructed using Bessel functions. Outside the area of bonding, displacements are described by the equations of bending of round plates in the absence of shear forces. Boundary conditions are met exactly. The satisfaction of marginal conditions, as well as boundary conditions, leads to a system of linear equations concerning the unknown coefficients of the obtained solutions. The model problem is solved and the numerical results are compared with the results of calculations performed by using the finite element method. It is shown that the proposed model has sufficient accuracy for engineering problems and can be used to solve problems of the design of aerospace structures.