Shape optimisation of cold-formed steel columns with manufacturing constraints using the Hough transform

Abstract

This paper introduces manufacturing constraints into a recently developed evolutionary algorithm for shape optimisation of CFS profiles. The algorithm is referred to as "self-shape optimisation" and uses Genetic Algorithm (GA) together with the Augmented Lagrangian (AL) method to avoid ill-conditioned problems. Simple manufacturing rules derived from the limitations of current cold-forming processes, i.e. a limited ability to form continuously curved surfaces without discrete bends, are described in the paper and incorporated into the algorithm. The Hough transform is used to detect straight lines and transform arbitrarily drawn cross-sections into manufacturable ones. Firstly, the algorithm is verified against a known optimisation problem and found to accurately converge to a manufacturable optimum solution. Secondly, the algorithm is applied to singly-symmetric CFS columns each of which is subject to an axial compressive load of 75kN and has a uniform wall thickness of 1.2 mm. The strength of the columns is evaluated by the Direct Strength Method (DSM) and all buckling modes are considered. Various column lengths (from 500 mm to 3000 mm) and numbers of roll-forming bends were investigated. The optimised cross-sections are presented and discussed. ABSTRACTThis paper introduces manufacturing constraints into a recently developed evolutionary algorithm for shape optimisation of CFS profiles. The algorithm is referred to as "self-shape optimisation" and uses Genetic Algorithm (GA) together with the Augmented Lagrangian (AL) method to avoid illconditioned problems. Simple manufacturing rules derived from the limitations of current coldforming processes, i.e. a limited ability to form continuously curved surfaces without discrete bends, are described in the paper and incorporated into the algorithm. The Hough transform is used to detect straight lines and transform arbitrarily drawn cross-sections into manufacturable ones. Firstly, the algorithm is verified against a known optimisation problem and found to accurately converge to a manufacturable optimum solution. Secondly, the algorithm is applied to singly-symmetric CFS columns each of which is subject to an axial compressive load of 75 kN and has a uniform wall thickness of 1.2 mm. The strength of the columns is evaluated by the Direct Strength Method (DSM) and all buckling modes are considered. Various column lengths (from 500 mm to 3,000 mm) and numbers of roll-forming bends were investigated. The optimised cross-sections are presented and discussed.2

Shape optimisation of manufacturable and usable cold-formed steel singly-symmetric and open columns

“…The same manufacturing rules published in [10], emerged from the limitations of rolled-forming machines, are applied to the algorithm in this paper. The rules are:…”

“…Manufacturing rules are now introduced as an intrinsic part of the cross-sectional shapes. Specifically, instead of drawing the cross-sections in the first generation with short elements (2 mm long) and aligning them using the Hough transform to ultimately create manufacturable cross-sections as in [10], the cross-sections are initially drawn herein with a fixed number of longer elements of random sizes, directly resulting in manufacturable cross-sections. The cross-over and mutation operators keep the cross-sections manufacturable.…”

“…Recently, manufacturing constraints have been introduced into shape optimisation algorithms for CFS members in [9][10][11][12]. This inclusion resulted in cross-sections with a marginal reduction in the member capacity relative to their unconstrained counter-parts.…”

“…Franco et al [12] developed a grammar-based model to shape optimise CFS open profiles in which the manufacturing features, with given stiffener sizes, were intrinsically considered in the algorithm. Leng et al [9] and Wang et al [10] incorporated manufacturing constraints into CFS shape optimisation algorithms by allowing limited numbers of roll-forming bends per cross-section. Handling constraints, however, may add complexity to the algorithm and cause convergence issues.…”

“…The algorithm is applied to simply-supported singly-symmetric, free-to-warp open section columns. Manufacturing features, defined in [10], are incorporated into the algorithm as an intrinsic part of the cross-sections.…”