The stressed state of the double-layer rectangular plate under shift. The simplified two-dimensional model

A considerable attention is given to studying the stressed state of adhesive joints due to the structure’s bearing capacity, which is usually determined by the strength of the connections, where the stressed state is irregular. Most of the existing mathematical models of joints are one-dimensional, i.e. they imply an even distribution of stresses across the width of joints. However, there are constructions for which the classical models are not applicable. In order to calculate the stressed state of such joints it’s necessary to take into consideration the irregularity of stresses not only in length but also in width of joints. To solve such problems, a simplified two-dimensional model of the overlapping adhesive joint of the rectangular plates is proposed. The simplification consists in the fact that the displacements of the layers are considered only along one of the axes. The model is actually a two-dimensional generalization of the Volkersen classical connection model. The stresses are considered even distributed over the layer thickness and the adhesive layer works only on a shift. These simplifications allowed us to obtain an analytical solution to the studied problem. The problem of the stressed state of the adhesive jointing of the two rectangular plates is solved, one of which is rigidly fixed along one side, and the second one is loaded with an uneven shear load on the opposite side. The problem is reduced to a system of second-order differential equations in partial derivatives with respect to longitudinal displacements of the two carrier (outer) layers. The solution is constructed using the method of separating the variables, in the form of a functional series consisting of the Eigen functions of the spectral problem. The boundary conditions at the unloaded ends are accurately satisfied. The satisfaction of the boundary conditions on the sides leads to a system of linear equations for the unknown coefficients of the functional series. The convergence of the obtained solution is proved. The model problem is solved and the numerical results are compared with the results of the calculations performed using the finite element method. It is shown that the proposed approach has a sufficient accuracy for engineering tasks.