Axially symmetric elasticity problems for the hollow cylinder with the stress-free ends. Analytical solving via a variational method of homogeneous solutions

An axially symmetric problem for a hollow cylinder with unloaded bases is considered.  On the inner and outer cylindrical surfaces, the normal and tangential loads are prescribed.  The problem is reduced to a biharmonic equation with corresponding boundary conditions.  Application of the method of variables separation results in a homogeneous boundary value problem for the ordinary differential equation.  Its eigenfunctions have been used to construct an infinite system of homogeneous solutions for the initial biharmonic problem.  Its solution, represented as a series expansion in terms of homogeneous solutions, depends on four infinite sequences of real constants.  To determine them, the variational method has been applied, in which the subordination of the solution to the boundary conditions, given on cylindrical surfaces, is performed in the norm  ${{L}_{2}}$.  It brings to an infinite system of algebraic equations which has been solved by the reduction method. The quantitative studies have confirmed the good convergence of the method.