An original numerical procedure for treating the Saint-Venant's problem for toroidal shells of arbitrary cross section under pressure and bending moment is suggested.
The main attention of investigators is given to geometrically nonlinear effects. Two of them have a direct relation to Saint-Venant's problem. The first is so called Brazier's effect, which deals with a nonlinearly increasing ovalization of cross section of an initially straight elastic pipe with growth of bending moment. This phenomenon is of practical importance for a thin-walled shell.
Another manifestation of geometrically nonlinear effects consists in an additional action of external or internal pressure. The external pressure promotes the ovalization while the internal one resists it. The current rules of designing submarine pipelines consider it as a possible limit state that requires due analysis for the substantiation of their integrity.
In the paper, two auxiliary tasks are considered separately: the loading in the plane and the loading in the axial direction of the shell. The plane section hypothesis for both tasks is used. The model proposed in the paper allows taking into account the plasticity effect as well as the distribution of the radial and shear stresses acting in the plane of the cross section.
The procedure implies the fine meshing of the cross section along the circumferential as well as the radial directions. In spite of very large number of unknowns the application of the sweeping (transfer matrix) method allows the procedure to be technically reduced to the solution of 6 linear equations at each step of iteration. The main idea consists in introducing at each iteration the Basic set of looking for parameters which are redefined by taking into account the Correction solution. The last one presents itself the numerical solution of all governing equations (equilibrium, physical, geometrical) based on current Basic geometry and taking into account the Basic stresses and deformations.
The results of comparison of the proposed procedure with known geometrically and physically nonlinear analytical and numerical solutions show the perfect accuracy of the method.