Geometrically nonlinear numerical procedure for Saint-Venant's problem for a symmetrical elastic shell with a closed contour and circular axis

An original numerical procedure of treating Saint-Venant's problem for toroidal-like shells under pressure and bending moment is suggested. Its peculiarity consists in that the loading and displacement parameters are separated into two groups: the first one describes the ring-like transverse deformation of the shell cross section, while the second the beam-like axial deformation of the latter. One of them is considered separately in the sequential iteration procedure while the other is reckoned to be known. After each step of calculation, the parameters considered to be known are made more exact.A lot of the well-known problems widely presented in the literature are considered. Among them are: (1) Karman's problem, which explains the ovalization of the cross section and the enlarged flexibility of a pipe bend as compared to a straight pipe under bending; (2) the so-called pressure reduction effect when the internal pressure prevents ovalization, which is a demonstration of the influence of the geometrical nonlinearity even at small displacements; (3) Brazier's effect, which deals with nonlinearly enlarged ovalization of an initially straight elastic pipe with an increase in the bending moment.The main advantage of the method is that it allows analyzing a toroidal shell of arbitrary cross section with a variable wall thickness. As an example, a toroidal shell with two long symmetrical axial cracks is considered, where cracks are modeled as the jumps of the contour angular displacement whose values are related to the crack depth.