# STRESSES IN ADHESIVELI BONDED JOINTS OF TWO COAXIAL PIPES. SIMPLIFIED TWO-DIMENSIONAL MODEL

### Abstract

A simplified two-dimensional model of an overlap glue joint between two coaxial pipes is proposed. At the base of the model lies the hypothesis about the high rigidity of the pipes being joined in the circumferential direction. It is assumed that normal stresses and axial displacements are evenly distributed across the thickness of the layers being joined, and displacements occur only in the axial direction. Those. displacements in the radial and circumferential directions are zero. The adhesive layer works only on shear, and the shear stresses in the adhesive are evenly distributed in thickness. It is assumed that the longitudinal displacements or normal forces are given on the ends of the pipes in an arbitrary manner, and not uniformly as in the known classical solutions. Edge conditions, displacements or forces at the ends of pipes, are expanded into Fourier series in the angular coordinate, in the interval from zero to pi. This model can be used to determine the stresses in the joint between the skin of the aircraft and the power frame, to which the load is applied at certain points. The task is reduced to a system of two differential equations for the longitudinal displacements of the outer and inner tubes. This model is a development of the previously proposed model of adhesive bonding of rectangular plates. To construct the solution, the classical method of separation of variables was used. Displacements have the form of superposition of series of eigenfunctions, the coefficients of which are unknown and are found from boundary conditions. Satisfaction of boundary conditions leads to systems of linear equations for unknown coefficients. An estimate for the decay of the coefficients is found and the convergence of the solution is proved. The solution of the problem satisfies the criterion of Saint-Venant, i.e. in the case of a large connection length, the displacements and stresses at a distance from the ends exponentially tend to the classical one-dimensional solution. The model problem is solved and the results of calculations are compared with calculations performed using the finite element method. In the model problem, the inner tube is assumed to be fixed on one of the ends, and a force is exerted on the opposite end of the outer tube in two symmetric sectors, between which lie no load sectors. Graphs of tangential stresses in the glue are given in the article. The high accuracy of the proposed simplified mathematical model is shown. The proposed solution has good convergence and counting speed, so it can be used to build a solution for optimization and design problems.

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*Bulletin of Zaporizhzhia National University. Physical and Mathematical Sciences*, (2), 81-89. Retrieved from http://journalsofznu.zp.ua/index.php/phys-math/article/view/258