Approximations Method for Space Frame Synthesis

Abstract

A method is presented for the minimum mass design of three-dimensi onal space frames constructed of thinwalled rectangular cross-sectional members. Constraints on nodal displacements and rotations, material stress, local buckling, and cross-sectional dimensions are included. A high-quality separable approximate problem is formed in terms of the reciprocals of the four section properties of the frame element cross section, replacing all implicit functions with simplified explicit relations. The cross-sectional dimensions are efficiently calculated without using multilevel techniques. Several test problems are solved, demonstrating that a series of approximate problem solutions converge rapidly to an optimal design. HIS paper reports on a new approach and capability for the optimization of three-dimensi onal space frame type structures with nonstandardized element cross sections. The mass of the structure is to be minimized subject to constraints which include nodal displacements and rotations, material stress, local buckling, and design variable side constraints. The structure is represented by a finite element model using a twelve degree of freedom (two end nodes, with three displacements and three rotations per node) frame element. Figure 1 details the rectangular, thin-walled, doubly symmetric cross section for the /th element. A separable approximate problem is formed using reciprocal section properties (RSP's)—the inverse of an element's cross-sectional area and polar and bending moments of inertia (i.e., Xu = l/At; ** = !//,.; Xi3 = l/Izzi; Xi4 = \/Iyyi). These approximations are of high quality when formed in terms of the RSP's, due to the algebraic form of the actual problem statement. Side constraints on the cross-sectional dimensions (CSD's) are reflected into RSP space by means of an approximate relationship between these two sets of variables. The form of this approximate relationship is tailored to take advantage of the strengths of the selected optimizer. One of two optimizers may be selected to solve the approximate problem. The first is NEWSUMT, an extended quadratic interior penalty function optimizer.l The second is DUAL2, a dual space optimizer that maximizes the dual function by a second-order method.2 These problem solving capabilities are combined in a selfcontained Fortran code which allows the user to optimize a general three-dimensional space frame structure modeled with frame elements of thin-walled rectangular cross section. Several example problems are solved, showing that this new approach to the frame problem is useful and efficient.