Optimized components in frame synthesis

The incorporation of optimized thin-walled beam elements into procedures for the minimum-weight design of indeterminate elastic planar frame structures is examined. It is shown that all frame structures composed of beam elements with thin-walled cross sections will be fully stressed in the sense that each element should be proportioned to be on the verge of local instability under at least one independent load condition. This fact leads to a formulation of the design requirements as an inequality-constrained minimization problem with the only variables being moments of inertia of the elements. Solutions are examined numerically by use of a nonlinear programing technique. The formulation also suggests a simple and efficient direct iterative procedure for obtaining designs which are fully-stressed with respect to both local buckling and yielding. For the examples considered, both procedures are shown to produce identical results. Nomenclature A = cross-sectional area D = depth of beam cross section E = Young's rnodulus / = moment of inertia K = buckling coefficient M = bending moment 9fTl = maximum absolute value of M in element W = weight Z = section modulus kAjki = constants I — length t = thickness p = specific weight ay = yield stress a = Wl/Z