Optimization of cold-formed c-shaped and z-shaped thin-walled profiles

Optimization of cold-formed c-shaped and z-shaped thin-walled profiles M. J. Smyczynskia,*, M. Rodaka, P. Paczosa a Institute of Applied Mechanics, Poznan University of Technology, ul. Jana Pawla II 24, 60-965 Poznan, Poland *Corresponding author. E-mail address: mikolaj.s[email protected]n.pl Abstract: The paper is devoted to the optimization of thin-walled beam cross-sections subjected to pure bending, with a focus on achieving efficient structural performance through multi-objective optimization. Four non-standard cross-sectional types (C1, C2, Z1, Z2) manufactured by Zaprom Ltd were analyzed at two beam lengths (420 mm and 800 mm). The adopted criteria were the minimization of the cross-sectional area, reflecting material consumption, and the minimization of maximum deflection, indicating structural stiffness. The optimization was conducted using the Normal Constraint method, ensuring a uniform distribution of Paretooptimal solutions. The results revealed distinct trade-offs between mass and stiffness, with C2 consistently outperforming the other variants by offering the most favorable balance. C1 proved less efficient, while Z1 was found to be the least effective in mass utilization. Z2 provided intermediate performance, superior to Z1 but inferior to C2. These findings highlight C2 as the most advantageous solution for achieving optimal mass–stiffness efficiency in thin-walled beam design. Keywords: Thin-walled beam; Optimization, Numerical analysis. 1. Introduction The shape of the cross-section significantly influences the stability and load-carrying capacity of thin-walled profiles. Studies show that different shapes, such as top-hat, channel, and cruciform sections, exhibit varied stability characteristics under compression The findings from these studies have practical implications for optimizing the design of thinwalled profiles in construction and engineering applications. By understanding the stability behavior, engineers can enhance the performance and safety of these structures. Hancock [1] reviewed and summarised the major research developments in cold-formed steel structures published during 1999-2001. Davies [2] outlined recent developments in the practical utilisation of cold-formed steel structures. Messac et al. [3] and Messac and Mattson [4] proposed the normal constraint (NC) method for generating a set of evenly spaced solutions on a Pareto frontier – for multiobjective optimization problems. Magnucki [5] presented optimization of an open cross section of the thin walled beam under strength and stability constraints. Magnucki et al. [6] presented optimal design of an open cross section of a cold formed beam with cosinusoidally corrugated flanges. Magnucki and Magnucka-Blandzi [7] proposed a variational design of open cross-section thin-walled beams. Magnucka-Blandzi and Magnucki [8] reviewed selected problems of buckling and optimal design of thin-walled beams. Magnucka-Blandzi [9] described the effective shaping of cold-formed thin-walled channel beams with double-box flanges subjected to pure bending. Lee et al. [10] used a micro Generic Algorithm to find an optimum cross-section of cold-formed steel beams. Ostwald et al. [11] and Kasperska et al. [12] analysed the problem of optimal design of the beams with selected channel section shapes under bending moment. Ostwald and Rodak [13] presented an analysis of the multicriteria optimal design of cold-formed beams with generalized open shapes under different loads. Stulpinas and Daniunas [14] developed an optimization algorithm for the closed crosssections of portal frames members. Sun et al. [15] proposed a novel topology optimization method for thin-walled structures with directional straight stiffeners using the material-field series expansion and multifield superposition. Zhang et al. [16] presents a multi-component topology optimization method for thin-walled structures. Liu et al. [17] studied a global optimization approach based on a numerical implementation of the DSM for the objective function of cold-formed steel columns. Vinot et al. [18] described a methodology for optimizing the shape of thin-walled structure with the use of FEM. Magnucki and Paczos [19] presented theoretical shape optimization of cold-formed thin-walled channel beams with drop flanges in pure bending Pawlak et al. [20] analysed experimentally and numerically the strength and resistance to loss of stability of thin-walled channel columns. Pawlak et al. [21] presented current state of knowledge of imperfections in thin-walled steel profiles with modified crosssectional shapes. Mahado [22] analyzed the static non-linear behaviour of thin-walled composite beams with initial imperfections. Jasion et al. [23] conducted numerical and experimental analysis of buckling and post-buckling behaviour of selected cold-formed Cbeams with modified cross-sections and compared them to a classical one. Chu et al. [24] investigated numerically the local and distorsional buckling behaviour of cold-formed steel zedsection beams subjected to uniformly distributed transverse loads. The objective of this study was to investigate the efficiency of selected thin-walled beam crosssections through multi-objective optimization. The research aimed at identifying optimal geometrical configurations that minimize two conflicting criteria: the cross-sectional area, representing material usage and structural mass, and the maximum deflection, reflecting the global stiffness of the element. The optimization process was carried out for different crosssection types (manufactured by Zaprom Ltd) and beam lengths, with the intention of assessing their comparative performance and determining the most effective solutions in terms of the mass–stiffness trade-off. 2. Optimization model – scalar and bi-objective optimization Structural design, which is a process aimed at achieving a specific goal, such as minimal manufacturing cost, is called structural optimization. When designing beams, many constraints related to strength, stiffness, and stability must be taken into account. Therefore, simply selecting a good structure, one that is acceptable given the constraints, can pose a challenge for the designer. Optimization, therefore, enables the selection of not only a good structure but also one that is optimal with respect to the selected criterion or criteria. If the construction optimization model contains one criterion, we talk about single-criterion or scalar optimization, while if there are several objective functions, we talk about multi-criterion or vector optimization. This paper presents a two-criteria optimization of beams C1, C2, Z1, and Z2 (Figure 2 - Figure 5) of the length L subjected to pure bending (Figure 1). The optimization criteria were the beam cross-sectional area and its maximum deflection. Figure 1 Beam loading scheme Figure 2 Cross section C1 Figure 3 Cross section C2 Figure 4 Cross section Z1 Figure 5 Cross section Z2 The parameters related to the geometry of the beam cross-section were chosen as decision variables: 𝑥 = [𝑡,ℎ,𝑏,𝑐,𝑑]𝑇. The objective functions were: • cross-sectional area of the beam: 𝑓1(𝑥)=𝐴, (1) • maximum beam deflection: 𝑓2(𝑥)= √𝑣2+𝑤2, (2) where 𝑣 and 𝑤 are the displacement of the shear center in the vertical and horizontal planes, respectively, and are calculated from the equation [ (𝜋𝐿)2𝐸𝐽𝑧(𝜋𝐿)2𝐸𝐽𝑦𝑧 −𝑀𝑦 (𝜋𝐿)2𝐸𝐽𝑦𝑧 (𝜋𝐿)2𝐸𝐽𝑦0 −𝑀𝑦0 𝐺𝐽𝑠+(𝜋𝐿)2𝐸𝐽𝜔+2𝛽𝐶𝑧𝑀𝑦 ] [𝑣 𝑤 𝜓]=[ 0 4𝜋𝑀𝑦 0] And 𝑀𝑦 – bending moment, 𝐸 – Young's modulus, 𝐺 – Kirchhoff's modulus, 𝐽𝑦, 𝐽𝑦𝑧, 𝐽𝑧, – moments of inertia, 𝐽ω – warping constant, 𝐽𝑠 – St-Venant torsion constant, 𝛽𝐶𝑧=121 𝐽𝑦𝐽𝑧−𝐽𝑦𝑧 2(𝐼𝑧∫𝑦(𝑦2+𝑧2) 𝐴d𝐴+𝐼𝑦𝑧∫𝑧(𝑦2+𝑧2) 𝐴d𝐴)−𝑧𝐶, 𝑧𝐶 – shear center coordinate The following constraints were present in the optimization model: • geometric constraints 𝑡𝑚𝑖𝑛≤𝑡≤𝑡𝑚𝑎𝑥, 4𝑡≤ℎ+𝑡≤𝐻𝑚𝑎𝑥, 3𝑡≤𝑏+𝑡≤𝐷𝑚𝑎𝑥, and in the case of cross-sections C1 and C2 2𝑡≤𝑐≤0,4(ℎ−𝑡), 𝑡≤𝑑≤𝑏−0,5𝑡, while in the case of cross-sections Z1 and Z2 2𝑡≤𝑐≤0,4ℎ, 𝑡≤𝑑≤max{√2(𝑏−𝑡),√2(𝑐−𝑡)}, • strength constraint 𝑀𝑦≤𝑀𝑆𝐶, where 𝑀𝑆𝐶=𝐽𝑦𝐽𝑧−𝐽𝑦𝑧 2 𝑧𝑚𝑎𝑥𝐽𝑧−𝑦𝑚𝑎𝑥𝐽𝑦𝑧σ𝑦 𝑛 and (𝑦𝑚𝑎𝑥, 𝑧𝑚𝑎𝑥) is the point where the maximum normal stress occurs, σ𝑦 – yield point, 𝑛 – safety factor, • global stability constraint 𝑀𝑦≤𝑀𝐺𝐵, where 𝑀𝐺𝐵=𝑀𝑘𝑟 𝑛𝑔 𝑀𝑘𝑟=(𝜋𝐿)2𝐸𝐽𝑦𝐽𝑧−𝐽𝑦𝑧 2 𝐽𝑦{𝛽𝑆𝑧±√𝛽𝑆𝑧 2+𝐽𝑦𝐽𝜔 𝐽𝑦𝐽𝑧−𝐽𝑦𝑧 2[1+𝐺𝐽𝑠 𝐸𝐽𝜔(𝐿𝜋)2]}, 𝑛𝑔 – safety factor, • local and distortional stability constraint 𝑀𝑦≤𝑀𝐿𝐵, where 𝑀𝐿𝐵=𝑀𝑘𝑟 𝑛𝑙 and 𝑀𝑘𝑟 calculated using the finite strip method, 𝑛𝑙 – safety factor,. • maximum beam deflection constraint δ𝑚𝑎𝑥≤δ𝑑𝑜𝑝, where δ𝑚𝑎𝑥 is the maximum deflection of the beam (2), while δ𝑑𝑜𝑝 is the permissible beam deflection. 3. Results of cross-section optimization Numerical calculations were performed for the following input data: • beam length: 𝐿=420; 800 mm, • bending moment: 𝑀𝑦=0,2 kNm • minimum and maximum dimensions: dimension minimum value mm maximum value mm 𝑡 0,0 1,5 𝐻=ℎ+𝑡 0,0 80 for cross-sections C1, C2 120 for cross-sections Z1, Z2 𝐷 = 𝑏+𝑡 0,0 40 • material constants: - Young's modulus - 𝐸=2,0∙105MPa, - Kirchhoff's modulus - 𝐺= 𝐸 2(1+𝑣)=7,6923∙104MPa, - Poisson's ratio - 𝑣=0,3, - yield point - σ𝑦=330 MPa • permissible beam deflection - δ𝑑𝑜𝑝=𝐿 250 • safety factors: - strength 𝑛=1,5, - global 𝑛𝑔=2,7, - local 𝑛𝑙=1,8. The calculations generated a discrete set of Pareto-optimal and non-dominated solutions, which are presented in the tables and figures. The calculations were performed using the normal constraints method proposed by Messac et al. [3], Messac and Mattson [4]. This method involves optimizing a single objective function with additional constraints. It allows for obtaining all non-dominated solutions, and the resulting discrete set of non-dominated solutions is uniformly distributed on the Pareto boundary. For the problem considered in this study, a single objective function was optimized with an additional constraint dependent on parameter w. For w=0,0, the minimum beam cross-sectional area was obtained, while for w=1,0, the minimum beam deflection was obtained. 3.1. Cross-sections C1 Table 1 and Table 2 present the results obtained for C1 beams with lengths L=420mm and L=800mm, respectively. Figure 6 shows the Pareto-optimal boundaries in the criteria space. The selected optimal cross-sections are shown in Figure 7 and Figure 8. Table 1 Set of Pareto-optimal solutions for cross-section C1, My = 0.2 kNm, L = 420 mm w 𝑓1,𝑚𝑖𝑛 mm2 𝑓2,𝑚𝑖𝑛 mm t mm h mm b mm c mm d mm 0,0 77,8 1,114 0,445 49,17 39,55 14,02 5,73 0,1 87,2 0,954 0,444 69,09 39,56 14,64 5,77 0,2 100,9 0,804 0,483 73,70 39,51 17,76 5,82 0,3 119,1 0,665 0,534 78,13 39,45 21,78 5,77 0,4 144,2 0,542 0,603 79,31 39,39 27,94 5,64 0,5 177,9 0,440 0,749 79,25 39,25 28,60 4,09 0,6 219,9 0,357 0,920 79,08 39,08 30,52 2,70 0,7 269,9 0,293 1,141 78,86 38,86 31,09 1,14 0,8 337,3 0,271 1,083 78,92 38,92 31,13 38,37 0,9 393,2 0,221 1,500 78,50 38,50 30,80 14,83 1,0 462,0 0,202 1,500 78,50 38,50 30,80 37,75 Table 2 Set of Pareto-optimal solutions for cross-section C1, My = 0.2 kNm, L = 800 mm w 𝑓1,𝑚𝑖𝑛 mm2 𝑓2,𝑚𝑖𝑛 mm t mm h mm b mm c mm d mm 0,0 93,0 3,200 0,458 66,67 39,54 17,60 6,53 0,1 104,4 2,783 0,478 79,01 39,52 19,09 6,44 0,2 119,9 2,394 0,531 78,53 39,45 22,38 6,29 0,3 139,8 2,033 0,588 79,41 39,41 26,91 6,14 0,4 165,7 1,712 0,692 79,29 39,31 28,76 4,74 0,5 198,0 1,435 0,828 79,17 39,17 29,94 3,45 0,6 237,2 1,203 0,992 79,01 39,01 31,12 2,12 0,7 283,1 1,017 1,197 78,80 38,80 31,04 1,20 0,8 347,6 0,955 1,117 78,88 38,88 31,11 38,32 0,9 403,0 0,831 1,302 78,70 38,70 30,96 38,05 1,0 462,0 0,733 1,500 78,50 38,50 30,80 37,75 Figure 6 Pareto-optimal edges for beams with a C1 cross-section Figure 7 Optimal cross-sections C1 for L = 420 mm w 𝑓1,𝑚𝑖𝑛 mm2 𝑓2,𝑚𝑖𝑛 mm t mm h mm b mm c mm d mm 0,0 95,3 0,989 0,506 68,11 39,49 27,24 6,98 0,1 107,8 0,854 0,535 77,56 39,44 30,92 7,08 0,2 123,5 0,725 0,568 89,16 39,41 35,56 7,09 0,3 143,9 0,605 0,605 103,72 39,38 41,38 7,07 0,4 170,8 0,496 0,659 119,34 39,34 47,74 6,80 0,5 207,8 0,407 0,816 119,18 39,18 47,67 4,80 0,6 253,9 0,334 1,060 113,94 38,94 45,55 1,11 0,7 307,9 0,276 1,250 118,75 38,75 47,50 1,25 0,8 385,5 0,262 1,098 118,90 38,90 47,56 53,46 0,9 450,0 0,224 1,500 118,50 38,50 47,40 28,54 1,0 521,3 0,198 1,500 118,50 38,50 47,40 52,33 Table 8 Set of Pareto-optimal solutions for cross-section Z2, My = 0.2 kNm, L = 800 mm w 𝑓1,𝑚𝑖𝑛 mm2 𝑓2,𝑚𝑖𝑛 mm t mm h mm b mm c mm d mm 0,0 104,5 3,200 0,517 77,03 39,48 30,81 7,56 0,1 117,7 2,782 0,549 85,97 39,45 34,36 7,58 0,2 134,2 2,384 0,583 97,10 39,42 38,84 7,60 0,3 155,3 2,015 0,622 111,41 39,36 44,46 7,69 0,4 183,0 1,683 0,708 119,29 39,29 47,72 6,51 0,5 221,8 1,417 1,010 99,48 38,99 39,79 1,12 0,6 262,9 1,166 1,075 117,51 38,92 47,00 1,11 0,7 315,2 0,981 1,280 118,72 38,72 47,49 1,28 0,8 391,5 0,938 1,115 118,88 38,88 47,55 53,41 0,9 453,3 0,811 1,500 118,50 38,50 47,40 29,67 1,0 521,3 0,719 1,500 118,50 38,50 47,40 52,33 Figure 12 Pareto-optimal edges for beams with a Z2 cross-section Figure 13 Optimal cross-sections Z2 for L = 420 mm Cross-section Z2 occupies an intermediate position between the C-type variants and Z1. For L=420mm, solutions range from A=95,3 mm²; deflection =0.989 mm to A=521,3mm²; deflection =0,198 mm. Initial solutions are less favorable than C2 and somewhat heavier than C1 at similar stiffness levels, while at high stiffness the area grows sharply. The most rational compromise occurs within w∈[0,3;0,5], where deflection can be reduced by approximately 50% at the cost of a moderate increase in area. At L=800 mm, the same tendencies are observed, with deflections significantly amplified by beam length. In comparative terms, Z2 is less efficient than C2 but superior to Z1, making it a viable alternative when Z-type cross-sections are required. The comparative analysis of all four cross-sections highlights clear differences in structural efficiency. C2 consistently outperforms the other variants, achieving the lowest deflections for a given cross-sectional area. In practical terms, C2 offers the most advantageous mass–stiffness balance, particularly around the Pareto “knee” (w≈0,3–0,5), where a significant reduction in deflection is obtained with only a modest increase in area. C1, while structurally similar, is consistently less efficient than C2 and should be regarded as a secondary option, unless specific design constraints prevent the application of C2. Z1 proves to be the least efficient solution, as it requires markedly larger cross-sectional areas to achieve comparable stiffness. Its application may be justified only in special cases where geometry or manufacturing constraints necessitate Z-type sections. Z2 represents a middle ground, more effective than Z1 but not as efficient as C2. It is therefore best suited as an alternative when Ztype cross-sections are required by design. Overall, the results indicate that C2 is the optimal cross-section for minimizing both mass and deflection, especially for moderate stiffness requirements. Nevertheless, the final choice should also account for manufacturing feasibility, particularly regarding dimensions approaching limiting values. 4. Conclusions The conducted research provides comprehensive insight into the optimization of thin-walled beam cross-sections under pure bending conditions. By employing a bi-objective optimization framework, the study systematically evaluated the balance between cross-sectional area and maximum deflection, thereby quantifying the trade-off between material efficiency and structural stiffness. The application of the Normal Constraint method proved effective in generating a well-distributed set of Pareto-optimal solutions, which facilitated a detailed comparative analysis across the four investigated non-standard cross-section types, manufactured by Zaprom Ltd. The results clearly demonstrate that cross-section C2 consistently exhibits superior performance. For both beam lengths considered, C2 achieved the lowest deflections for a given cross-sectional area. Its Pareto front displayed a distinct “knee” in the region w∈[0.3;0.5], representing highly favorable compromise solutions. In these cases, substantial reductions in deflection could be obtained with only moderate increases in cross-sectional area. This makes C2 the most efficient option when both material consumption and stiffness requirements are considered simultaneously. Cross-section C1, while structurally similar to C2, proved consistently less efficient. Although it provides feasible solutions within the investigated range, the corresponding Pareto front indicates that C1 requires noticeably larger areas to achieve the same stiffness as C2. Consequently, C1 should be regarded as a secondary alternative, particularly in situations where design or manufacturing constraints limit the applicability of C2. Cross-section Z1 emerged as the least effective solution among the analyzed variants. The results showed that Z1 consistently required much larger cross-sectional areas to reach comparable stiffness levels, which translates into significant increases in material usage. This poor mass efficiency makes Z1 a suboptimal choice in most practical applications. Its use could only be justified under specific circumstances, such as geometric or technological requirements that necessitate Z-type geometries. Cross-section Z2 offered intermediate performance, falling between C1/C2 and Z1. While less efficient than C2, Z2 demonstrated better results than Z1, particularly at moderate stiffness levels. Its Pareto front also revealed acceptable compromise solutions in the range w∈[0.3;0.5], where meaningful deflection reductions could be achieved without excessive increases in crosssectional area. Thus, Z2 can be considered a viable alternative when Z-type cross-sections are required by design, even though it cannot match the efficiency of C2. A consistent observation across all cross-sections was the strong dependence of deflection on beam length. When the length increased from 420 mm to 800 mm, maximum deflections approximately tripled, while the cross-sectional areas required to maintain comparable stiffness grew significantly. This finding underscores the importance of length as a governing factor in thin-walled beam optimization and highlights the need to account for scaling effects in practical design. In conclusion, the comparative analysis unambiguously identifies C2 as the optimal crosssection, delivering the best trade-off between mass and stiffness. Its efficiency advantage is robust across both investigated beam lengths and throughout the Pareto frontier. C1, while feasible, is less efficient and should only be adopted in constrained cases. 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