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Abstract: Two alternative formulations based on the concept of simultaneous analysis and design are presented and evaluated for optimal design of framed structures. Different behavior variables, such as nodal displacements and member forces, are also treated as optimization variables in addition to the actual design variables for the problem. With these formulations, the equilibrium equations become equality constraints in the optimization process. The objective and all constraints become explicit functions of the optimization variables. Therefore,their derivatives can be obtained quite easily compared to those for the conventional approach where special design sensitivity analysis procedures must be used to calculate derivatives. It is also easier to use existing analysis software for optimization with the alternative formulations because the sensitivity equations are not formed or solved. A sequential quadratic programming method that5995
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exploits sparsity of problem functions is used to solve sample problems and evaluate the formulations. Implementation of the alternative formulations with an existing analysis program is explained. Advantages and disadvantages of the formulations are discussed. It is concluded that the alternative formulations work quite well for optimization of framed structures and have potential for further development.
DOI: 10.1061/_ASCE_0733-9445_2006_132:12_1880_
CE Database subject headings: Simulation; Optimization; Framed structures; Computer programming; Frames.
Introduction
Three alternative formulations based on the concept of simultaneous analysis and design (SAND) have been presented and applied to optimal design of trusses by Wang and Arora (2005). The formulations involved combinations of displacements, axial forces, and stresses as optimization variables. Introduction of more variables in the formulations changed forms of the constraints and their derivatives. All functions of the formulations became explicit in terms of the optimization variables. Therefore, design sensitivity analysis methods, which require adaptation of structural analysis procedures for optimization, were no longer needed. All the formulations worked quite well and very accurate optimal solutions were obtained. Solutions of sample problems were also compared with those obtained by the conventional formulation. It was concluded that the alternative formulations were more efficient than the conventional formulation in most cases.Also their implementation with the existing analysis software was easier compared to that for the conventional formulation. In the present paper, two of the formulations are extended and evaluated for optimization of framed structures.
An overview of the literature on different formulations for structural optimization has been presented by Kirsch and Rozvany(1994), and Arora and Wang (2005). These references should be consulted for the state-of-the-art on the subject. Here we present an overview of the literature related to the framed structures Optimal design of framed structures has been actively studies in the literature (Chan et al. 1995; Soegiarso and Adeli 1997; Pezeshk et al. 2000; Arora 2000). An extensive list of references on the subject can be found in Burns (2002). Khan (1984), Sadek (1992), and Chan et al. (1995) formulated the problem of optimum design of frames using some approximate relationships between cross-sectional properties and used the optimality criterion method to obtain solutions. Saka (1980) presented a formulation that included the displacements of joints as optimization variables in addition to the member areas. Member stiffness equations based on the displacement method were imposed as equality constraints. The SIMPLEX method was used to solve the problem after move limits were specified for the linearized subproblems. It was concluded that the proposed approach made it possible to avoid the solution of equilibrium equations and the number of iterations involved was smaller. Another class of alternative formulations for optimum structural design is the so-called displacement-based two-phase procedure. Missoum et al. (2002) presented that procedure and applied it to optimize trusses and geometrically nonlinear framed structures. These formulations have severe limitations as noted by the authors and by Arora and Wang (2005)