1. Introduction
Cold-formed steel (CFS) is widely used in many countries in structural and non-structural members due to its high strength-to-weight ratio and low cost of material and manufacture [
1]. Buckling refers to a type of structural failure where a compressed member, such as a cold-formed steel section, suddenly bends or collapses under an applied load. As thin-walled sections become increasingly complex, the issue of buckling becomes more serious. An edge stiffener is a structural component that is attached to the edge of a thin-walled steel element to provide additional support and improve its stability. The edge stiffener width−thickness ratio and the edge stiffeners’ form strongly influence the element’s buckling performance and can even affect its buckling mode [
2]. If the edge stiffeners of open cross-section thin-walled steel elements do not have folds, they will have poor local stability. Simple edge stiffeners consisting of a single lip at the free edge provide substantial benefits, but when the lip becomes too long, it may buckle first and cause instability. To improve the elements’ stability, the edge stiffeners must be stiffened moderately. Hence, thin-walled steel elements with complex edge stiffeners have emerged, which can alleviate this instability by providing additional folds to the lip itself [
3]. As shown in
Figure 1, the shear center will be far away from the flange−stiffener juncture when the simple edge stiffeners are folded inward. As the distortional buckling mainly occurs near the flange−stiffener juncture, multiple folds can improve the ultimate bearing capacity of the element [
4]. Therefore, an improvement of the stiffening form at the edge stiffeners is a vital way to limit the buckling of flanges and enhance the mechanical properties of the section.
For a long time, scholars worldwide have done a lot of research on the design of thin-walled sections and have given their opinions on the design of simple edge stiffeners. Starting in 1939, Winter [
5] did a lot of experiments on CFS and explored the performance of simple edge stiffeners. In 1978, Desmond et al. [
6] gave a criterion for determining whether an edge stiffener needs to be stiffened and a simplified formula for calculating the minimum stiffness limits by theoretical analysis. They concluded that stiffener adequacy is assessed using stiffener rigidity, for which the ultimate strength of an edge-stiffened element equals that of an element of similar dimensions and material properties but supported by a web at the stiffener location [
7]. Based on the research in [
8,
9], the post-buckling properties were studied by Sarawit et al. [
10]. The ultimate strength of an element with different degrees of imperfections was found, and the percent difference in strength was introduced to analyze the imperfection sensitivity. Bambach [
11] carried out geometric and material non-linear finite element analysis on about 200 edge-stiffened elements and analyzed the basic properties of elements with three sides simply supported and one side stiffened. It was shown that a buckling coefficient
k ≥ 4 indicates that the element is sufficiently stiffened. Bambach [
12] investigated the buckling behavior of stiffened elements by conducting compression tests on 30 edge-stiffened elements, gave the limiting conditions for simple edge stiffeners, and modified the strength calculation equations. A series of axial compression tests were conducted on simple edge columns constructed from high strength Grade G550 steel sheet of thickness 1 mm by Huang et al. [
13]. By parametric analysis, it was found that the edge width has a great influence on the bearing capacity of specimens because of the significant restriction of edge stiffeners on distortional deformation. Guo et al. [
14] presented a total of 44 axially compressed tests of CFS lipped channel columns including different cross-sections. The validated FE model was used to conduct a parametric study to investigate the effects of the section on the ultimate strength of such channels. To improve the bearing capacity of cold-formed Z-section steel members, Ye et al. [
15] optimized the design of its edge stiffener inclination, suggesting that the design inclination of the edge stiffener should be 100° in practical engineering.
Some studies on complex edge stiffeners have also been carried out. In 1993, Seah et al. [
16] investigated the buckling performance of thin-walled elements with complex edge stiffeners under axial pressure. They found that the complex edges’ cross-sectional performance was superior to simple edges. Schafer et al. [
4] isolated the flanges and complex edge stiffeners as compressive elements and performed finite element analysis with supports instead of web effects to obtain approximate solutions. They concluded that the complex edge-stiffened elements have a more extensive application range than the simple ones and can provide better strength with slightly increased imperfection sensitivity. Xi et al. [
3] conducted the tests about thin-walled elements with complex edge stiffeners and concluded that the load capacity of the inward complex edge stiffeners is significantly higher than the outward ones in elastic buckling. Du et al. [
17] analyzed the buckling performance of thin-walled channel steel columns by SAP2000 and concluded that web-flange width ratios of between 1.1 and 1.4 significantly affected the critical buckling stress. Jiao et al. [
18] used ABAQUS to analyze the buckling performance of thin-walled steel columns with solid struts at both ends. They explored the effects of edge stiffener form, height, and thickness on the buckling mode of steel columns under axial compression. They also concluded that the larger the web width is, the more significant the advantage of complex edge stiffeners is, and the greater the buckling load is when local buckling first occurs in the element. Zhang et al. [
19] carried out a series of numerical and theoretical researches on cold-formed steel elements with complex edge stiffeners and developed a detailed non-linear finite element model (FEM) [
20]. They also proposed a method to determine the critical stress of the complex edge section. Aruna et al. [
21] investigate the axial compression capacity of CFS equal angle sections with complex edge stiffeners. Both experimental and numerical analyses were carried out, and these implied that the existing direct strength method (DSM) rules are over-conservative. Wang et al. [
22] test a total of 16 pin-ended cold-formed steel columns. The ultimate load-carrying capacity and deformation behavior of cold-formed steel members with complicated cross-sections were studied. It was found that the loading efficiency of columns with complex edge stiffeners is higher than columns with simple edge stiffeners. Song et al. [
23,
24] studied columns with complex edge stiffeners and cap-shaped stiffeners. The effects of specimen length, the width−thickness ratio of the plates, section types, and the eccentric distance on stability were carried out. The buckling mode and deflection behavior of channel columns with complex edge stiffeners and cap-shaped stiffeners were investigated.
Currently, most of the research on complex edge stiffeners is qualitative, and the design of CFS always focuses on simple edge stiffeners but lacks relevant provisions for complex edge stiffeners. This paper analyzes and optimizes complex edge stiffeners based on the finite strip method (FSM) and related buckling theories. Firstly, the minimum limit criteria of the edge stiffeners were put forward by theoretical analysis. Secondly, the simple edge stiffeners’ limits required by AISI S100 [
25] were used to verify the accuracy of these criteria. Next, as the norm for the limit value of simple edge stiffeners was obtained by considering defects based on experiments, the revision of the criterion was introduced according to the norm and applied to the analysis of complex edge stiffeners. The limits of the width−thickness ratio of the complex edge stiffeners were given based on axial compression elements and the FSM models under different steel components were tested for verification. Finally, the optimal width−thickness ratio and the optimal configuration of the complex edge stiffeners were found through the optimization algorithm, which provided suggestions for the subsequent design.
5. Optimization of Complex Edge Stiffeners by GA
The genetic algorithm (GA) is a random search method based on principles of natural evolution, in which a population of candidate solutions is evolved through special selection rules to optimize the fitness function (optimization target) during the evolution process [
31]. In this paper, Python3.9.2 is used to program the section optimization program, the critical buckling stress is calculated by the FSM, and the optimal section is selected by the GA. The program flowchart is shown in
Figure 11. According to the characteristics of the problem, the program contains five main files constructed with an object-oriented program: using.py provides the user data input interface and displays the output results; cufsm.py calculates the critical buckling stress of the section based on the FSM; genetics.py realizes the GA optimization design of the provided section, and optimizes the width and its proportion of the edge stiffeners; material.py contains the Material class. This module provides the Material class for the typical steel structure. Its subclasses include Q235 class, Q355 class, Q390 class, etc., which can offer a variety of steel properties and provide the function of customizing materials; section.py completes the definition of the geometric properties of the section.
As shown in
Figure 9, the value of
k first increases and then decreases with the increase of
a/t, that is, after
kmax is reached, the critical buckling stress of the edge stiffener decreases with the increase of
a/t. On increasing of the edge stiffeners’ width−thickness ratio, although the edge stiffeners can continue to provide bearing capacity for the member, it also causes the waste of materials when meeting the bearing capacity requirements of the member, and buckling coefficient
k represents the critical stress, which can represent the material utilization efficiency. Hence, considering economic benefits,
a/t under
k =
kmax is the optimal width−thickness ratio for complex edge stiffeners.
Through parameterized analysis, a series of discrete points were obtained, as shown in
Figure 9. The parameter
kmax is the maximum value among these points, as indicated in
Table 6. To obtain a more precise and advantageous optimal edge stiffener width−thickness ratio and configuration, an optimization design was conducted using the GA.
Table 6 illustrates the relationship between
kmax and
a/t obtained through parameterized analysis and those obtained by the GA, respectively, when
a2:
a1 = 1. The
kmax values obtained by GA are not less than those obtained by the parametric analysis. The high coefficient of determination of R
2 = 0.996, the small root mean square error of RMSE = 0.006, and the small mean absolute percentage error of MAPE = 0.09% collectively indicate that the results obtained through GA are comparable to those from the parameterized analysis, without significant deviations. Therefore, the GA employed in this study is a viable and effective approach.
Through the optimization algorithm, the values of
k and
a/t are shown in
Table 7, which are the results of the GA optimization according to the stress values under different width ratios of the primary and secondary edges. According to
Table 7 and
Figure 12a, when
a2:
a1 = 1:1,
kmax is larger and the material utilization ratio is higher.
The results of the optimal values of
a/t and
a2:
a1 obtained by the multi-objective genetic algorithm optimization in
Table 8 also prove that the material function can be better played when the width ratio of the primary and secondary edges is 1:1.
Values of
a/t and
b/t show an approximately linear relationship, as shown in
Figure 12b. Equation (7) can also be used for selection when
a2:
a1 ≥ 0.5. The statistical performance of this equation, for
a2:
a1 ratios of 1, 0.75, and 0.5, respectively, are as follows: R
2 = 1.00, 0.98, 0.98; RMSE = 0.52, 1.04, 1.10; MAPE = 2.03%, 4.05%, 4.29%; and SMAPE = 2.03%, 4.05%, 4.29%.