Elastic Equilibrium of a Hollow Cylinder of Finite Length Under Axisymmetric Force Loading

We develop a method for solving axisymmetric problems of the theory of elasticity for a hollow cylinder of finite length subjected to force loading of the end faces and also of the inner and outer cylindrical surfaces. As a result of the direct integration of equilibrium equations, we formulate auxiliary problems for the evaluation of components of the stress tensor via a single function, which is called the Vihak function. By using the obtained expressions, the continuity equations are reduced to a governing equation for the determining function and the system of original boundary conditions is equivalently replaced by integral conditions. We construct the complete systems of eigenfunctions and associated functions with an aim to represent the solution of the problem in the form of superposition of the elementary and self-balanced components. We develop a method for solving the governing equation with the corresponding integral conditions for each component. To determine the self-balanced solution, we develop a recurrent algorithm, which guarantees the possibility of complete separation of variables in the governing equation. The solution thus constructed enables us to perform the exact analysis of stresses in elastic cylinders with different ratios of the generatrix length to the radius.