Automatic minimum weight design of elastic redundant trusses under multiple static loading conditions

A method for automated selection of a minimum weight truss, from a subset of configurations obtained by omitting various member combinations from a primary configuration, is presented. A feasible direction algorithm is used to find an upper bound solution for any selected configuration. A dual simplex algorithm is used to rapidly generate lower bound solutions for many subset configurations considering only equilibrium conditions and stress limits. The lower bound solutions guide the selection of subset configurations for which upper bound solutions of reduced weight are sought. Examples of planar and space trusses illustrate the efficacy of the bounding technique presented. I. Mathematical Formulation C ONSIDER an elastic redundant truss with given geometric configuration and support conditions. The truss is to sustain several given alternative systems of static loads applied at the joints. Only q of these load systems which are supposed to govern the design will be considered. For simplicity, the truss is conventionally idealized; that is, the members of the truss are assumed to be joined together by frictionless connections and only axial forces, tension or compression, occur in the members under the loading. The design problem addressed herein is the selection of a minimum weight optimum truss configuration, as well as its member sizing, from a subset of truss configurations obtained by omitting various members from the given primary configuration. The design is subject to the following constraints: the crosssectional area Aj of member j, when it remains in the truss, is restricted to a given continuous range, the upper limit A? of which is finite and the lower limit A! may be, and usually is, greater than zero; the stress ajk, defined as the internal force/^ of the member j under the kih loading condition divided by Aj, has given lower limit — ffjk and upper limit ajk, and these stress limits are assumed independent of the cross-sectional area of the member; and the displacement component uik of a joint under loading condition k also has specified lower — uikl and upper uik" limits (each joint, except for fixed supports, has at least one degree of freedom, the corresponding displacement components are numbered in a fixed order and identified as ut). The problem can thus be formulated as with