A General Formulation of the Optimal Frame Problem

Optimal design techniques have been extensively applied to steel structures and, to a lesser degree, to reinforced concrete structures. In the latter case, for given geometry and preassigned stiffnesses, optimal designs have been found which simultaneously satisfy (a) limit equilibrium (plastic limit stage), (b) serviceability (elastic limit stage), and (c) optimality (minimum material consumption). The limitations to these designs are: 1. A subsequent check of plastic compatibility may invalidate the design. 2. The resulting member stiffnesses may differ appreciably from the preassigned values. 3. A different geometry may result in a better solution while still satisfying all design criteria. The present paper attempts to eliminate these limitations through a more general formulation of the optimal frame problem wherein design plastic moments, member stiffnesses, and frame geometry are all treated as variables and are found for simultaneous satisfaction of (a) optimality, (b) limit equilibrium, (c) serviceability, (d) plastic compatibility, and (e) elastic compatibility. With some simplifying assumptions to linearize the problem, the general formulation is illustrated for a reinforced concrete continuous beam example. The resulting optimal design is compared with conventional elastic and plastic designs with respect to safety, serviceability, compatibility, and efficiency.