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Abstract:
In some cases, optimal structural design corresponds to non-analytical points for which the first variation of volume or cost does not necessarily vanish. Such cases may occur, for instance, when optimal design is aimed at maximizing the lowest eigenvalue (natural frequency, buckling load) and a multimodal solution occurs, or when the aim is to minimize the maximum stress intensity. The present paper discusses such cases and provides the relevant optimality conditions.

Abstract:
The application of standard stationarity conditions to the optimal design of a structure against buckling (or other eigenvalue constraints) may actually lead to a reduction of its real strength in the sense that the “optimized” structure has been weakened with respect to higher modes to the extent that these have now become fundamental. In this case a true optimum occurs when the magnitudes of several eigenvalues coalesce, and the corresponding optimality condition is then no longer stationary. This singular condition is derived and applied to several examples of major practical significance, together with a method of obtaining an approximate solution to the resulting set of nonlinear equations.