Axisymmetric Problem for an Elastic Cylinder of Finite Length with Fixed Lateral Surface with Regard for its Weight

Abstract

UDC 539.3 We consider an elastic cylinder with regard for its weight. The conditions of sliding fixing are imposed on the lower base of the cylinder, its upper base is subjected to the action of an axisymmetric normal load, and the lateral surface is fixed. The Hankel integral transform is used to reduce the problem to an integral equation of the first kind for normal stresses acting on the fixed cylindrical surface. After finding the singularities of the unknown function, the solution of the integral equation is sought in the form of a series in Jacobi polynomials. The results of numerical evaluation of the normal stresses on the fixed surface of the cylinder are obtained both with regard for its weight and by neglecting its weight. Elastic cylinders of finite length can be regarded as one of the most extensively used types of structural elements. This explains a great number of publications devoted to the investigation of their stressed states. A survey of the main achievements in this field of mechanics prior to 1963 can be found in [1]. A survey of more recent achievements (after 1963) is presented in [7]. Nevertheless, despite a large number of approximate numerical methods aimed at the solution of axisymmetric problems of the theory of elasticity for solid cylinders of finite length, the analytic methods capable of construction of solutions in the form of explicit functional dependences on the type of load and geometric parameters of the cylinder are still insufficient. Attempts to obtain the exact solutions were made in [3] and [4]. In these works, the solution is constructed in the form of expansions in trigonometric, hyperbolic, and Bessel functions, which leads to infinite systems of linear algebraic equations. In [6], the technique of p -analytic functions is used to obtain the exact solutions. However, the numerical realization of the proposed algorithms of solution is not presented in these works. Among recent works, we should especially mention [5] and [14], where the methods of superposition and expansion in Fourier–Bessel series are used not only to reduce the solution of the problems to infinite systems of linear algebraic equations but also to obtain numerical values of stresses in the cylinder by the method of improved reduction. In [17], the solution of the problem of stressed state of a circular cylinder loaded at the end faces and along the cylindrical surface is also constructed in the form of Fourier–Bessel series, which leads to the necessity of solving infinite systems of linear algebraic equations with an aim to find the coefficients of these representations. In [18] and [19], the authors present the numerical results obtained in the case of a cylinder loaded by axisymmetric normal loads imposed on the end faces or along the cylindrical surface. In all works, including the recent results, the action of the bulk forces in the form of the weight of the material is not taken into account, which leads to the solution of inhomogeneous Lame equations. As an exception, we can mention the work [20] devoted to the numerical solution of the problem of stressed state for a hollow cylinder under the action of the weight of its material. An analytic method capable of the solution of these problems is outlined in [11], where it is also shown how to construct the exact solution of the axisymmetric problem for a finite elastic cylinder in the case where the conditions of sliding fixing are imposed on the cylindrical surface.