Напружено-деформований стан пластини, яка послаблена круговим вирізом та посилена двома круглими накладками

Abstract

The problem of axisymmetric deformation of a structure was solved in this article. This structure consists of a plate that was weakened by a circular cutout and at the same time reinforced by two concentric round patches. The patches are glued overlapping on both sides of the plate, in the place of the cutout. The patches are joined to the main plate by a thin adhesive layer and perceive shearing and tearing forces. To solve the problem, some hypotheses were used, namely: it was assumed that the stresses were distributed uniformly over the thickness of the adhesive layer. For patches, the Kirchhoff-Love hypothesis is adopted. The adhesive layer is considered an elastic Winkler base. An axisymmetric problem is considered. It is assumed that the displacement of the layers depends only on the radial coordinate and doesn’t depend on the angular coordinate. The main plate is not subjected to bending due to the symmetry of the structure. This problem is a generalization of the classic model of the stress state of the adhesive joint for the rods to an area with radial symmetry. These assumptions have allowed obtaining a solution to the problem in an analytical form. The problem is considered separately in the adhesive joint area and as well as the outside of the adhesive joint area. In the area of gluing, the problem is reduced to a seventh-order differential equation concerning shear stresses. The solution to this problem is presented as an expansion into a functional series in terms of modified Bessel functions of the second kind. The obtained shear stresses make it possible to obtain normal stresses, as well as radial and transverse displacements of the layers in the adhesive joint zone. The displacements outside the adhesive joint zone were obtained from the well-known differential equations for the deformation of round plates in the absence of shear forces. The unknown coefficients in both cases are found from the boundary and conjugation conditions. The model task was solved. A finite element model of an adhesive joint has been developed. The largest size of the adhesive layer element is chosen to be sufficiently small and equal to half the thickness of the adhesive layer because of the enormous stress gradients in the adhesive layer. The results of the analytical model were compared with those results of the finite element model. The comparison showed the high accuracy of the proposed model.