Calculation of the stress state of cylindrical bodies based on the three-dimensional theory of elasticity

Bodies with cylindrical and plane-parallel boundaries in the form of disks or rings are considered. To describe their three-dimensional stress state, the general solution of the Navier equations in the cylindrical coordinate system is used. After integrating the stresses over the thickness of the cylindrical plate, the normal and tangential forces are expressed in terms of one two-dimensional harmonic and two biharmonic functions. The boundary conditions on its plane surfaces and the equilibrium equations are satisfied. A closed system of partial differential equations for the introduced two-dimensional functions is constructed without using hypotheses about the geometric nature of the plate deformation. An averaged two-dimensional representation of the general solution in the polar coordi­nate system based on the three-dimensional theory of elasticity is obtained for the first time. New solutions of the equilibrium equations of plates in the polar coordinate system are constructed. It is shown that their averaged stress-strain state is divided into two cases: an axisymmetric stress state, and a non-axisymmetric state, which depends on the polar angle φ. Analytical formulas are given for expressing displacements and stresses in these cases.