Optimization of cylindrical shells stiffened by rings under external pressure including their post-buckling behaviour

• The inner rings increase the first critical buckling load of a shell of revolution.
• The stabilization of a post-buckling path is obtainable simultaneously.
• The three stiffening rings are enough for all considered cases.
• The material volume of the optimal shell is not greater than the reference one.
• The best results were obtained by varying the internal diameters of the inner rings.

Optimization of a simply supported cylindrical shell stiffened by inner rings is considered in this paper. The shell is loaded by external pressure. The first critical buckling load is maximized. A material volume and a slope of a post-critical path are assumed as optimization constraints. The inner rings are made from the material obtained from the shell by decreasing its thickness. This way the volume of the material remain constant. The structure is modelled and solved by the finite element method (FEM). The linear and nonlinear stability analyses are done in the ANSYS software. An effect of geometric imperfections on a shape of equilibrium path is discussed. The optimization was performed numerically using the modified particle swarm optimization method (MPSO). The results are compared with a reference cylindrical shell with no rings. The single ring placed in the middle of the shell is good enough for stabilization of a post-buckling path for a shorter shell, regardless of its thickness. It is necessary to use three rings optimally distributed along the shell length, for a longer shell. The additional optimization profit was obtained by varying internal diameters of the rings. The proposed concept of shell stiffening by inner rings eliminates a major disadvantage of smooth cylindrical shells, namely an unstable post-buckling path.