Kształtowanie Niekołowych Cylindrów Grubościenych Wskazujących Pełne Uplastycznienie w Stadium Zniszczenia

This paper deals with the problem of seeking the shapes of non-circular thick-walled cylinders which indicate full plasticization upon reaching the limiting load capacity, under a constant external and internal pressure.
A thick-walled cylinder is considered with variable radii a(ϑ) and b(ϑ) in the form of a series of the parameter α which characterizes the non-circularity of the cylinder [formulae (3.1)]. Assuming a plane state of deformation and incompressibility of the material, the state of stress is determined in cylindrical coordinates by means of three components σr, σ and τr0 were expanded into a series of the parameter α), the problem of eigenvalues is obtained in the first approximation. The condition for the existence of non-zero solutions is determined by the double sequence of the ratios of the cylinder dimensions [formulae (3.25)].
In paragraph 3.6 a compilation is given of the obtained formulae with an accuracy to the second approximation. Namely, the formulae (3.46) determine the shape of the cylinder as a function of the angle ϑ, whist formulae (3.48) determine the components of the state of stress as a function of the angle ϑ and radius τ.
Two numerical examples are analysed in detail in p. 4, whilst in 5 the method is given of determining the Hencky-Prandtl nets for the cylinders under consideration, and such a net is shown in one of the particular cases.