Stressed state of a thick-walled pipe under the action of internal pressure and axial force

Abstract

A classical problem of determining the stress state of a thick-walled pipe with fixed bottoms under the action of internal pressure and the axial force generated by it is considered. Under these conditions, the axial strain is zero. The scheme of a perfectly plastic body is accepted. Information Maple technologies for all key stages of solving this problem have been developed. In order to analyse the pipe material for plastic deformation without fracture, the basic differential equation of the problem is formulated and solved with respect to the stress state index, which is equal to the ratio of the first invariant of the stress tensor to the stress intensity. It is shown that the boundary conditions lead to a system of three linear inhomogeneous equations with respect to unknown parameters of the stress equations. It is determined that this system always has a solution, since the determinant of the system matrix is inversely proportional to the square of the radius of the boundary separating the plastic and elastic regions of the pipe and is always different from zero. It is shown that the equation that establishes the relationship between the applied pressure and the radius of the boundary between the plastic and elastic regions cannot be solved directly with respect to an unknown radius. However, it is advisable to use it in a form solved with respect to pressure. The calculated distributions of stress components normalised by the yield strength along the pipe thickness are consistent with the literature data, which confirm that the maximum values of the circumferential load are observed on the outer surface. This is interpreted as a sign of the reliability of the theoretical solution to Bridgman's experimental data, according to which the material fracture of a thick-walled pipe loaded with internal pressure begins from the outer surface. The regularities of changes in the stress state index obtained in this work improve the above justification, since from the standpoint of the theory of deformability of materials during cold deformation, the ultimate deformations of the material usually decrease with the growth of the stress state index, the highest values of which occur at the points of the outer surface