Analytical approach to optimization of columns for postbuckling behaviour

The postcritical analysis performed for a structure optimized only for buckling constraints shows that the behaviour of the designed element after buckling is in many cases unstable. That should be considered as a drawback of the standard design because the applicability of such an unstable optimal structure may be questionable. Recently, the new concept of structural optimization against instability has been proposed. The postbuckling analysis is included directly in the optimization problem and the modified design is formulated. As a result, the modified optimal structure with stable behaviour after buckling is obtained. Many modified design problems for structures exposed to elastic instability have already been formulated and numerically solved. The aim of the present paper is to develop an analytical approach to optimization of columns for stable postbuckling path. A perturbation method combined with Croll’s manoeuvre makes it possible to derive general formulae for the subsequent corrections for the force and hence to analyze stability in the postbuckling range. Based on that, the analytical formulae can be obtained. The constraint imposed on the second correction of the force allows obtaining s stable postcritical designed structure locally that is in the vicinity of the critical point. The relevant optimization problem is then expressed as an isoperimetric problem of the calculus of variations. The optimizations of the Stern column and of a column loaded by a vessel containing liquid are presented as examples.