Optimization of postbuckling path for cylindrical shells under external pressure

Abstract

The standard problem of structural optimization under stability constraints is usually formulated as maximization of the instability load for a prescribed volume of a design element. Very often the standard optimal structure has unstable postbuckling behaviour and it is very sensitive to imperfections. That is weakness of the design and it indicates that the combination of geometrically nonlinear analysis with the design becomes necessary, especially from practical point of view. The postbuckling constraints of a special form added to formulation of optimization problem permit to modify the postbuckling path and a stable postbuckling path can be created, even in the case of unstable behaviour of the reference structure. The effect of modification of the postbuckling behaviour in most cases has been obtained by changing sizing variables which are usually dimensions of the design element.

In this paper an alternative concept is applied, namely stabilization of the postbuckling path is obtained by application of additional loadings acting on the structure without changing its shape and sizes of the optimized element. These loadings can be either active ones applied to the structure or passive ones (reactions of the additional supports), or both active and passive forces acting simultaneously. In the paper stabilization of the postbuckling path for a simply supported cylindrical shell under radial compressive pressure is considered. All three types of stabilizing forces are investigated. In the case of the active loadings a shell is axially loaded by the active forces at both ends. We look for the minimum value of the axial force which stabilize the postbuckling path. In the case of the passive force an axial movement of the both ends of a shell is constrained by additional elastic elements. In this case we look for the minimum value of the axial stiffness of these additional elements, which causes stabilization of the postbuckling path. The last case consists two types of loadings. First, an axial pretension of a certain value is applied and next the axial movement of the both ends of a shell is blocked. Then, the external pressure is applied. In this case we look for the minimum pretension, which together with the pressure stabilizes the postbuckling path. Calculations were performed using ANSYS code for elastic and elastic-plastic deformations of shells of different length and thickness. It occurred that in a case of the passive force stabilization of the postbuckling path is not possible for any stiffness of additional element. For a case of the active force and for the case of the mixed variant of loadings such stabilization can take place for elastic and elastic-plastic deformations.