Solution of the plane problem of the theory of elasticity on bending of an articulated fixed multilayer panel with a circular axis

Abstract

This article suggests a method for the solution of the plane problem of the theory of elasticity on the bending of an articulated fixed multilayered panel with a circular axis based on the polynomial approximation of displacements through the thickness of the panel. In contrast to the known solutions of this problem, in this case the coefficients of the approximating polynomials are calculated from the equilibrium conditions and equality of displacements and transverse stresses at the transition across the layer interface and solution of differential equations of equilibrium at several points through the thickness of the layers. Finally, the problem is reduced to the solution of a system of linear equations with respect to the coefficients of approximating polynomials. The validity of the method is confirmed by comparing the results of calculations obtained on its basis and the results obtained with the help of the reference finite element model. The problem is solved in two stages. At the first stage, for a single-layer panel, we investigate the dependence of the polynomial degree on the ratio of the average panel radius to its thickness and the ratio of the transverse shear modulus to the modulus of longitudinal elasticity, which characterize the nonlinearity of displacements. At the second stage, on the example of a three-layered panel, we consider the application of the proposed method for the calculation of multilayered panels. In such case, the results obtained at the first stage are used in selecting the initial degree of polynomials approximating displacements through the thickness of layers. The method proposed in this article makes it possible to obtain an analytical solution without introducing simplifying hypotheses about the nature of displacement of layers and their elastic characteristics in a wide range of variation in geometric dimensions and elastic characteristics of panel layers. This method can be used both for verification of numerical models and for carrying out strength calculations of multilayer panels.