Erosion of optimum designs by compound branching phenomena

Abstract It is well-known that a straightforward process of structural optimization often demands the simultaneity of two or more failure modes. It has, however, been emphasized by Koiter and others that this can lead to a dangerous situation in which the nonlinear coupling of two quite stable post-buckling modes can generate highly unstable behaviour and associated imperfection-sensitivity. The question then arises as to how the imperfection-sensitivity might invalidate the apparent optimum, and it is a study of the mechanics of this that is presented here. A two-degree-of-freedom buckling model which exhibits this unexpected coupling action is introduced and its primary and secondary branching characteristics are derived analytically and numerically. The simultaneity of the primary buckling loads is shown to arise as the solution of a simple but realistic optimization scheme, and the subsequent erosion of this optimum is observed. It is seen that the compound imperfection-sensitivity can quite seriously modify and perhaps destroy the apparent optimum solution, and the key role of secondary bifurcations is delineated.