Mathematical model of the pipeline is formulated within the framework of the elastic shell theory, and the basic parameters of this model are determined. For moment and semi-momentless shells, equations of the first approximation with respect to the small curvature parameter are given. The model for calculation by the finite element method in CAE Abaqus is constructed and numerical experiments are performed. Numerical solution to the model problem is found, and stresses and bending moments are calculated on its basis. Stresses and bending moments are compared with their exact values. Numerical analysis is carried out by comparing pipe models within the framework of the shell theory and the rod theory. It is established that the proposed mathematical model makes it possible to find stresses in pipes with high accuracy.