© 1983 by Oxford University Press
STABILITY OF THE NATURAL SHAPES OF SINUSOIDALLY LOADED UNIFORM SHALLOW ARCHES
( Division of Engineering, San Francisco State University San Francisco, California 94132, U.S.A. )
The concept of natural structural shapes is based on the simultaneous ‘minimization’ of the strain energy and the mass of a structure—usually a multicriteria control problem. This concept is applied to the optimal design of uniform shallow arches. The design variables are the initial curvature and the axial load which are to be selected in such a way as to obtain Pareto optimal values for the mass and the strain energy of the loaded arch. Necessary conditions for an arbitrarily loaded arch are obtained first. They consist of a fourth-order differential equation to be satisfied by the initial curvature and of an isoperimetric condition to be satisfied by the axial load. Subsequently, a complete solution, including stability implications, is obtained for the sinusoidally loaded arch. It was shown elsewhere that the optimal equilibria (that is, the structural equilibria corresponding to an optimal design) for the minimum mass problem can be stable, stable after snap-through, unstable, unattainable, or nonunique, and that these detrimental aspects cannot be alleviated by the inclusion of the traditional inquality constraints based on compliances or on the axial load. One may, however, work an essentially unconstrained (a restriction to a negative curvature is included) multicriteria problem by including the strain energy as an additional criterion and resolve nearly all of these difficulties. In particular, it is shown that the natural equilibria are stable until asymmetric buckling of the arch becomes a possibility. Furthermore, depending on the range of the applied load, necessary conditions for optimality in the minimum mass problem are identical with the usual critical-point condition, and necessary conditions for the natural shape are identical with the usual sufficient conditions for the stability of the shallow arch.