OPTIMIZATION OF CROSS-SECTIONAL DIMENSIONS FOR COLD-FORMED STEEL LIPPED CHANNEL COLUMNS

A parametric optimization problem of cross-sectional sizes for cold-formed steel lipped channel structural members subjected to central compression has been considered by the paper. An optimization problem is formulated as follow: to define optimum cross-sectional sizes of cold- formed structural member taking into account post-buckling behavior and structural requirements when stripe width and profile thickness as well as type of the cold-formed profile are constant and defined by the designer. Maximization of load-carrying capacity of the cold-formed structural member has been assumed as purpose function. The latter has been presented in the form of linear convolution of the buckling resistance to the central compression taking into account flexural, torsional and torsional- flexural buckling of the thin-walled structural member determined according to the requirements EN 1993-1-3:2012 and EN 1993-1-5:2012. Searching for the optimum cross-sectional sizes of the cold-formed structural member has been performed taking into account the possibility of it post-critical buckling behavior based on the local buckling of the web and flanges and/or distortional buckling of the edge fold stiffeners. As optimization results cold-formed steel lipped channel have been obtained. With the same stripe width optimum profiles have higher load-carrying capacity level taking into account buckling resistance under the central compression comparing with the cold-formed steel lipped channel proposed by Ukrainian manufacturers. Besides, torsional-flexural buckling resistance of the cold-formed steel lipped channel is determinative for all optimum cross-sectional decisions. to the requirements EN 1993-1-3:2012 and EN 1993-1- 5:2012. Searching for the optimum cross-sectional sizes has been performed taking into account a possibility of post-critical buckling behavior of the structural member based on the local buckling of the web and flanges and/or distortional buckling of the edge fold stiffeners. Formulated parametric optimization problem has been solved using software OptCAD. Update gradient method of the purpose function projection on the active constraints hyperplanes with simultaneous liquidations of the residuals in the constraints has been implemented by the software. As optimization results cold-formed C-profiles have been obtained. With the same stripe width optimum profiles have higher load-bearing capacity level taking into account buckling resistance under central compression comparing with the cold-formed C-profiles proposed by Ukrainian manufacturers. Besides, torsional-flexural buckling resistance of the cold-formed C-profile is determinative for all optimum cross-sectional decisions. A parametric optimization problem of cross-sectional sizes for cold-formed steel lipped channel structural members subjected to central compression has been considered by the paper. An optimization problem is formulated to define optimum cross-sectional sizes of cold-formed structural member taking into account post-buckling behavior and structural requirements when stripe width, profile thickness and profile type are constant and defined by the designer. Maximization of load- carrying capacity of the cold-formed structural member has been assumed as purpose function.

Introduction. Previously, the use of cold-formed thin-walled profiles was limited to cases where reducing the weight of the structure was a priority, such as in the aviation or automotive industries. However, due to the development of production technology, corrosion protection, product availability as well as implementation of the design code the use of thin-walled structural elements, including cold-formed profiles is gradually expanding.
Today, various structural systems made from cold-formed steel (CFS) structural members, which are widely used in the construction industry, are actively imported to the Ukrainian market of steel structures. Implementation of steel structures made from thin-walled cold-formed profiles in building practice is relevant and economically reasonable. There are specific fields of application where their efficiency is the highest [9]. However, the widespread application of the structures made from thin-walled cold-formed profiles of the domestic production is delayed due to the lack of domestic experience in economic and reliable design of such structures. A high degree of flexibility in the manufacturing of various cross-sectional shapes provides a unique opportunity to further improve the load-carrying capacity of these structural elements through an optimisation process, leading to more efficient and economical structural systems [1].
The paper [2] reviews the existing studies on the structural optimization of CFS sections and the structural and thermal performance of such CFS structures. The methodologies used in the existing literature for optimizing CFS members have been summarized and presented systematically.
In the paper [1] a practical methodology for the optimum design of coldformed steel beam-column members with different lengths and thicknesses, subject to various combinations of axial compression and bending moment and with constant material use has been proposed by the paper. The optimisation process is carried out using a genetic algorithm and aims to maximise the resistances of CFS members, determined according to the Eurocode 3 design guidelines.
Optimisation technique was employed to enhance the structural performance and to effectively use the given amount of material of CFS members in the paper [3]. Lipped channel, folded-flange, and super-sigma have been optimised using the particle swarm optimisation method. Results showed that the flexural capacity of the optimised sections was improved significantly compared to conventional CFS sections.
The paper [4] presents a procedure to obtain optimized steel channel crosssections for use in compression or bending. The cross-sections are optimized with respect to their structural capacity, determined according to the EN1993-1-3 using genetic algorithms. The optimization for compression is carried out for different column lengths and includes the effects of the shift of the effective centroid induced by local buckling.
A methodology that would enable the development of optimised CFS beam sections with maximum flexural strength for practical applications has been provided by the paper [5]. The optimised sections are designed to comply with the Eurocode 3 geometrical requirements as well as with a number of manufacturing and practical constraints. The flexural strengths of the sections are determined based on the effective width method adopted in EC3, while the optimisation process is performed using the particle swarm optimisation method.
Papers [6,7] aim at finding the optimal folding of open cold formed steel cross sections under compression. The design space is searched primarily via a stochastic search algorithm, genetic algorithm. The near-optimal folding of the cross section is then fine-tuned through a few steps of the gradient descent optimization [7]. To arrive at practical designs the optimization problem is augmented with constraints on the geometrical properties of the cross section. The optimal cross sections are found to have compressive capacities that are higher than the original designs [6].
Applied optimum design problems for structures in some cases are formulated as parametric optimization problems, namely as searching problems for unknown structural parameters, which provide an extreme value of the specified purpose function in the feasible region defined by the specified constraints [8]. The mathematical model of the parametric optimization problems includes a set of design variables, an objective function, as well as constraints, which reflect generally non-linear dependences between them. If the purpose function and constraints of the mathematical model are continuously differentiable functions, as well as the search space is smooth, then the parametric optimization problems are successfully solved using gradient projection non-linear methods [9]. The gradient projection methods operate with the first derivatives or gradients only both of the objective function and constraints. The methods are based on the iterative construction of such a sequence of the approximations of design variables that provides convergence to the optimum solution (optimum values of the structural parameters) [10]. Additionally, a sensitivity analysis is a useful optional feature that could be used in scope of the numerical algorithms developed based on the gradients methods [11].
The paper [12] covers optimization problems and algorithms in the design of CFS structural members. A brief review of problem formulation and solution techniques in past research as well as details of optimization algorithms, including gradient-based, stochastic search, artificial neural network, and ant colony optimization, are discussed.
In order to increase the widespread application of the structures made from CFS profiles of the domestic production, effective national ranges of assortments of CFS profiles have to be developed. In this paper, CFS lipped channel structural members subjected to central compression are considered as research object, which investigated for the searching for optimum crosssectional dimensions. The following research tasks are formulated: to develop a mathematical model and a numerical technique to solve an optimization problem for cross-sectional sizes of CFS structural members; to perform numerical investigations in order to obtain optimal solutions for considered research object; to develop a guide for designers relating to the optimum material distribution in the cross-sections of the CFS structural members 1. Problem formulation. Let formulate a parametric optimization problem as follow: to find optimum values of cross-sectional sizes for CFS lipped channel structural members subjected to central compression taking into account structural requirements when the profile perimeter (strip width), profile thickness as well as the type of the CFS cross-section are constant and specified in advance.
The formulated parametric optimization problem can be stated in the following mathematical terms: to find unknown structural parameters: providing the least value of the determined objective function: in a feasible region (search space)  defined by the following system of constraints: where X  is the vector of the design variables (unknown structural parameters); X N is the total number of the design variables; ,  Design sizes of plane cross-sectional elements ( Fig. 1) for CFS lipped channel structural member are considered as state variables of the optimization problem and calculated according to [13] depending on the design variables h , b and c , internal bend radius 1,5 r t  and profile thickness t as presented below: where p h is the design web height; p b is the design flange width; p c is the design length of a single edge fold. Geometrical properties of CFS lipped channel cross section are determined depending on the design variables and the state variables as presented below: where g A is gross cross-sectional area; y I , z I are second moment of inertia relative to the main axis of inertia of the cross-section; y i , z i are radiuses of inertia of the cross-section; z is the distance which defines the center of mass location.
Relative slenderness ph  of the web and relative slenderness pb  of the flanges for CFS lipped channel are calculated according to [13,14] as follow: Relative slenderness pc  of the single edge fold is determined according to [13,14] as follow: is buckling factor calculated according to the polynomial dependency proposed by the papers [15] taking into account that the formulated parametric optimization task is solved using a gradient-based steepest descent method.
Reduction factors h  and b  corresponded to the local buckling of the profile web and flange respectively are determined according to [13,14] as presented below: Reduction factor c  corresponded to the local buckling of the single edge fold is calculated according to [13,14] as follow: Cross-section flanges and web of CFS lipped channel structural member are subjected to post-buckling behavior (when local buckling occurs) in the case when its slenderness exceed limit value, namely 0 Single edge fold of CFS lipped channel cross-section is subjected to postbuckling behavior (when local buckling occurs) in case when it slenderness exceeds limit value ( 0, 748 pc   ).In this case effective single edge fold width eff c is determined according to [13] as follow: is buckling factor calculated according to the polynomial dependency proposed by the papers [15] taking into account that the formulated parametric optimization task is solved using a gradient-based steepest descent method.
Design cross-section of the stiffener (Fig. 2) consists of single edge fold with effective width eff c together with effective adjacent part of the flange with effective width 0.5 eff b . Geometrical properties of the design cross-section of the stiffener are determined according to [13] as follow: where s A and s I are area and second moment of inertia for the design crosssection of the stiffener (Fig. 1.2). Single edge folds in CFS lipped channel structural members ensure partial restraint for plane flanges which can be simulated using a linear spring. In case of the central compression stiffness for such linear spring can be estimated according to [13] as presented below: 3,64 0,5 0,5 0,5 0,5 1, 5 0, 5 0, 5 It should be noted that analytical expression for stiffness of the linear spring presented above is restricted by the case of cold-formed structural members with flanges stiffened by single or double edge folds only and cross-section symmetrical relatively to the main axes of inertia which is perpendicular to the web plane.
Then relative slenderness of the stiffener d  corresponded to the flexural buckling of the stiffener is calculated according to [13] as follow: The reduction factor d  for the flexural buckling of the stiffener (or reduction factor for the distortional buckling cross-section resistance) is determined depending on relative slenderness d  . Taking into account that the formulated parametric optimization task is solved using a gradient-based steepest descent method, reduction factor for the distortional buckling crosssection resistance d  can be calculated depending on d  using polynomial dependency proposed by the paper [15]: We suppose that in the optimum cross-section the compressive stress at the centreline of the stiffener estimated on the basis of the effective cross-section equals to the base yield strength. Therefore, the reduced thickness red t of the design cross-section of the stiffener allowing for reduced stiffener resistance due to flexural buckling of the stiffener is determined according to [13] as follow: where , s red A is the reduced effective area of the stiffener allowing for flexural buckling.
Finally, the area of the effective cross-section of CFS lipped channel structural member subjected to central compression is calculated as presented below: The maximization criterion of the overall design buckling resistance represented as a linear convolution of the corresponded design buckling resistances with the same weight factors can be considered as the purpose function (1.2) and can be written as follow:  [13,16].
Then the purpose function can be rewritten as follow: are the elastic critical forces for the relevant buckling mode calculated taking into account gross cross-section geometrical properties of the structural member according to [16]. System of constraints (1.3) for the formulated optimization problem consists of a constraint on the profile perimeter or on a strip width which can be written as presented below: where max P is the maximum value of the cross-section perimeter for CFS lipped channel.
The constraints reflected design code requirements [13] for the ultimate slenderness of the cross-section elements of the CFS channel with flanges stiffened by single edge folds are also included in the system of constraints (1.3) and presented below: 1 0 50 Additionally, a constraint on the minimum gap between single edge folds ends allowing for providing an access to the internal surface of the CFS lipped channel (for example, in order to organize a bolted connection on the profile flanges) is included to the system of constraints (1.3) as well and written as below: where min d is the minimum gap between single edge folds ends. Thus, the optimization problem of cross-sectional sizes for CFS lipped channel structural member is formulated as follow: to find optimum values of Cprofile cross-sectional sizes (web height h , flange width b and single edge fold length c ) providing the maximum value of the determined objective function (1.4) in the feasible region defined by the system of constraints (1.5) -(1.9) when the profile perimeter (strip width) max P , profile thickness t as well as the type of the cold-formed cross-section are constant and specified in advance.
The formulated parametric optimization problem has been solved using the software OptCAD where the modified method of the purpose function gradient projection onto the surface of the active constraints with simultaneous liquidations of the residuals in the violated constraints has been implemented [17].
2. Optimization results. As optimization results the cold-formed C-profiles with optimum cross-sectional dimensions have been obtained. Characteristics of the obtained C-profiles with optimum cross-sectional sizes have been presented by the Table 1.
With the same stripe width C-profile structural members with optimum cross-sectional sizes have higher design buckling resistance under central compression comparing with the cold-formed C-profiles proposed by Ukrainian manufacturers (group of the companies «Blachy Pruszyński», companies BF Zavod and STEELCO) [18]. It should be noted that torsionalflexural buckling resistance of C-profile cold-formed structural members was determinative for all optimum cross-sectional decisions.
Thus, for example, the initial cold-formed C-profile with the strip width 25. 8 kN which is greater than 9.67% comparing to the one of initial profile (see Table 2).
With the same stripe width CFS lipped channel structural members with optimum cross-sectional dimensions have higher load-carrying capacity taking into account minimum buckling resistance under the central compression comparing with ones proposed by Ukrainian manufacturers.
Conclusion. Searching for optimum cross-sectional sizes of CFS lipped channel structural members taking into account post-buckling behavior and structural requirements has been realized. Presented results of the performed optimization calculation allow developing guides for designers relating to the optimum material distribution in the cross-sections of CFS structural members as well as are base to develop effective national ranges of assortments of coldformed profiles.   Load-carrying capacity of CFS lipped channel structural members subjected to central compression has been determined by the torsional-flexural buckling resistance for all optimum solutions. The single edge fold length improves significantly the torsional-flexural buckling resistance of CFS lipped channel structural members subjected to central compression. For all optimum solutions the optimum size of the single edge fold length has been obtained larger than ones in the initial CFS lipped channels manufactured by the Ukrainian manufacturers (group of the companies «Blachy Pruszyński», companies BF Zavod and STEELCO). At the same time, for all optimum solution the local buckling of the single edge folds has not been occurred.

OPTIMIZATION OF CROSS-SECTIONAL DIMENSIONS FOR COLD-FORMED STEEL LIPPED CHANNEL COLUMNS
Parametric optimization problem of cross-sectional sizes for cold-formed C-profiles subjected to central compression has been considered by the paper. Parametric optimization problem for crosssectional sizes of cold-formed C-profiles has been formulated as follow: to define optimum crosssectional sizes taking into account post-buckling behavior and structural requirements when stripe width and thickness as well as type of the cold-formed profile are constant and defined by the designer.
Criterion of the profile load-bearing capacity maximization has been assumed as purpose function. The latter has been presented in the form of linear convolution of the resistance to central compression taking into account flexural, torsional and torsional-flexural buckling of thin-walled structural member determined according to the requirements EN 1993-1-3:2012 and EN 1993-1-5:2012. Searching for the optimum cross-sectional sizes has been performed taking into account a possibility of post-critical buckling behavior of the structural member based on the local buckling of the web and flanges and/or distortional buckling of the edge fold stiffeners.
Formulated parametric optimization problem has been solved using software OptCAD. Update gradient method of the purpose function projection on the active constraints hyperplanes with simultaneous liquidations of the residuals in the constraints has been implemented by the software.
As optimization results cold-formed C-profiles have been obtained. With the same stripe width optimum profiles have higher load-bearing capacity level taking into account buckling resistance under central compression comparing with the cold-formed C-profiles proposed by Ukrainian manufacturers. Besides, torsional-flexural buckling resistance of the cold-formed C-profile is determinative for all optimum cross-sectional decisions.
В результате оптимизационного расчета получены С-образные холодногнутые профили, которые при одной и той же ширине заготовки характеризуются большей несущей способностью на потерю устойчивости при центральном сжатии по сравнению с холодногнутыми профилями отечественных производителей. При этом, для всех оптимальных решений поперечных сечений определяющей была несущая способность С-образного холодногнутого профиля на потерю устойчивости по изгибно-крутильной форме выпучивания.  A parametric optimization problem of cross-sectional sizes for cold-formed steel lipped channel structural members subjected to central compression has been considered by the paper. An optimization problem is formulated to define optimum cross-sectional sizes of cold-formed structural member taking into account post-buckling behavior and structural requirements when stripe width, profile thickness and profile type are constant and defined by the designer. Maximization of loadcarrying capacity of the cold-formed structural member has been assumed as purpose function. Formulated parametric optimization problem has been solved using a modified gradient method of the purpose function projection on the active constraints hyperplanes with simultaneous liquidations of the residuals in the violated constraints. As