Experiment and Design Method of Cold-Formed Thin-Walled Steel Double-Lipped Equal-Leg Angle under Axial Compression

Xingyou Yao; Yafei Liu; Shile Zhang; Yanli Guo; Chengli Hu

doi:10.3390/buildings12111775

Experiment and Design Method of Cold-Formed Thin-Walled Steel Double-Lipped Equal-Leg Angle under Axial Compression

Yao, Xingyou;Liu, Yafei;Zhang, Shile;Guo, Yanli;Hu, Chengli 2022-10-23 00:00:00 buildings Article Experiment and Design Method of Cold-Formed Thin-Walled Steel Double-Lipped Equal-Leg Angle under Axial Compression 1 1 2 , 1 1 Xingyou Yao , Yafei Liu , Shile Zhang *, Yanli Guo and Chengli Hu School of Architecture and Civil Engineering, Nanchang Institute of Technology, Nanchang 330099, China School of Yaohu, Nanchang Institute of Technology, Nanchang 330099, China * Correspondence: yaoxingyoujd@163.com; Tel.: +86-15079190103 Abstract: The cold-formed steel (CFS) double-lipped equal-leg angle is widely used in modular container houses and cold-formed steel buildings. To study the buckling behavior and bearing capacity design method of the cold-formed steel (CFS) double-lipped equal-leg angle under axial compression, 24 CFS double-lipped equal-leg angles with different sections and slenderness ratios the axial compression were conducted. The test results showed that the distortional buckling occurs for specimens with a small width-to-thickness ratio and small slenderness ratio. The buckling interactive with distortional and global ﬂexural buckling was observed for the specimens with small width-to- thickness ratios and large slenderness ratios. The specimens with large width-to-thickness ratios and small slenderness ratios showed interactive buckling with local and distortion buckling. The specimens with large width-to-thickness ratios and large slenderness ratio developed interactive buckling with local, distortional, and global ﬂexural buckling. The ﬁnite element model established Citation: Yao, X.; Liu, Y.; Zhang, S.; by ABAQUS software was used to simulate and analyze the test. The buckling modes and the Guo, Y.; Hu, C. Experiment and load-carrying capacities analyzed by the ﬁnite element model agreed with the test results, which Design Method of Cold-Formed Thin-Walled Steel Double-Lipped showed that the developed ﬁnite element model was feasible to analyze the buckling and bearing Equal-Leg Angle under Axial capacity of the CFS double-lipped equal-leg angles. The experimental results were compared with Compression. Buildings 2022, 12, 1775. those calculated by the direct strength method in the North American standard and the effective https://doi.org/10.3390/ width method in the Chinese standard. The comparisons indicated that the calculated results are buildings12111775 very conservative with maximum value 36% and 51% for direct strength method and effective width Academic Editors: Krishanu Roy, method, respectively. The coefﬁcient of variation was 0.276 and 0.397, respectively. Finally, the Boshan Chen, Beibei Li, Lulu Zhang, modiﬁed direct strength method and the modiﬁed effective width method were proposed based on Letian Hai, Quanxi Ye and the experimental results. The comparison on the ultimate strength between test results and calculated Zhiyuan Fang results by using the modiﬁed method showed a good agreement. The modiﬁed method can be as a proposed desigh method for the ultimate strength of the CFS double-lipped equal-leg angles under Received: 12 September 2022 axial compression. Accepted: 20 October 2022 Published: 23 October 2022 Keywords: double-lipped equal-leg steel angle; axial compression; distortional buckling; global Publisher’s Note: MDPI stays neutral buckling; effective width method; direct strength method with regard to jurisdictional claims in published maps and institutional afﬁl- iations. 1. Introduction Cold-formed thin-walled steel members have been widely used in construction houses Copyright: © 2022 by the authors. because of their high stiffness and strength, lightweight, and convenient machine and Licensee MDPI, Basel, Switzerland. construction. The buckling behaviors and design methods of cold-formed thin-walled steel This article is an open access article channel sections with or without holes have been studied by many researchers [1–9]. In distributed under the terms and recent years, with the cold-formed steel sections becoming common as structural members, conditions of the Creative Commons the cold-formed thin-walled steel double-lipped equal-leg angles as a primary structural Attribution (CC BY) license (https:// member have been widely used in tower structures, truss structures, and cold-formed steel creativecommons.org/licenses/by/ buildings [10,11]. Although the double-lipped equal-leg angle cross-section is simple, the 4.0/). Buildings 2022, 12, 1775. https://doi.org/10.3390/buildings12111775 https://www.mdpi.com/journal/buildings Buildings 2022, 12, 1775 2 of 17 centroid and shear center is inconsistent, the width-thickness ratio of the plate is large, and a partially stiffened element is present. The buckling mode and the effect on ultimate strength are complex. The double-lipped equal-leg angle cross-section is easy to buckle with local buckling, distortional buckling, and global buckling, which can affect the steel angle’s ultimate strength. Al-Sayed et al. [12] performed the buckling tests of the ﬁxed-ended equal leg and unequal leg angles under axial compression. The test and calculated results showed that the specimens with ﬂexural–torsional buckling in the inelastic range were conservative. The experimental results are 10% higher than the theoretical results. Popovic et al. [13] and Young [14] conducted axial compression tests on cold-formed plain angles. Based on the test and speciﬁcation calculations, they suggested ignoring the ﬂexural–torsional buckling but only considering the ﬂexural buckling. The buckling analysis on the cold-formed angles under axial compression conducted by Chodraui et al. Based on the ﬁnite strip method, indicated that the local buckling of elements and global torsional buckling of members was consistent [15]. Landesmann et al. [16], Silvestre et al. [17], and Dinis et al. [18] proposed a new design method for the short to medium-length equal leg angle under axial compression based on the direct strength method through the buckling test and ﬁnite element analysis. The ﬂexural–torsional buckling performance and the load-carrying capacities of the S690 high-strength angle steel column were studied by Zhang et al. based on an experiment and numerical analysis [19]. The predicted results showed that most design codes were too conservative. The modiﬁed design method based on the direct strength method considered the interaction of ﬂexural–torsional buckling about the strong axis and ﬂexural buckling about the weak axis. Zhang et al. [20] and Wang et al. [21] conducted experimental and numerical studies on S690 and S960 high-strength short equal leg angles. The studies showed that Australian and North American speciﬁcations were too conservative. Dinis et al. [22] presented the design method based on the direct strength method for the bearing capacity of short-to-medium length pin-ended hot-rolled steel equal-leg angle columns based on the test results. The experiments about cold-formed lipped angle columns were conducted by Young [23]. The specimens showed the local, ﬂexural, and ﬂexural–torsional buckling and the interactive buckling of these buckling modes. The calculated results indicated that the North American and Australian codes were conservative. Young and Ellobody [24] conducted ﬁnite element analysis on the buckling performance of lipped angles under axial compression. It indicated that the Australian and North American codes were conservative in calculating the ultimate strength of members with a relatively large width-thickness ratio. However, it was not safe to calculate the ulti- mate strength of members with a small width-thickness ratio. Shifferaw et al. [25] conducted theoretical and ﬁnite element research on the global buckling performance ofﬁxed-ended cold-formed thin-walled lipped angle columns. It was shown that the members exhibited signiﬁcant post-buckling strength when they were subjected to global ﬂexural–torsional buckling. Therefore, a direct strength method was proposed to consider the post-buckling strength of cold-formed thin-wall-lipped angle columns. An axial compression test was conducted on 12 cold-formed thin-walled steel columns with unequal leg angle sections by Zhou et al. [26]. The results showed that the direct strength method in the North Ameri- can code is not accurate, and the modiﬁcation for the calculation formula of the ultimate strength based on the direct strength method was given. Ananthi et al. [27] validated the FE model against the experimental test results, which showed good agreement regarding failure loads and deformed shapes at failure. The studies above-mentioned were focus on the equal-leg angle with or without lips. The buckling behavior and design method of the cold-formed thin-walled double- lipped equal-leg angles have not enough investigated. This paper studied the buckling behavior and ultimate strength of 24 cold-formed double-lipped equal-leg angles under axial compression. The analysis model of cold-formed thin-walled double-lipped equal-leg angles under axial compression is developed by using ﬁnite element software. Based on the test results and the predicted results using the direct strength method and the effective Buildings 2022, 12, x FOR PEER REVIEW 3 of 18 Buildings 2022, 12, x FOR PEER REVIEW 3 of 18 Buildings 2022, 12, 1775 3 of 17 under axial under axial c co ompression mpression. The . The an analy alys sis model is model o off col cold d-forme -formed d thi thin n-wa -walle lled do d dou ub ble-l le-lipped ipped equal-leg angles under axial compression is developed by using finite element software. equal-leg angles under axial compression is developed by using finite element software. Base Based d on on the the test test results and the results and the predic predictted ed re resu sults lts us using ing the the d diirec rectt s sttreng rength th me method thod and and width method, a modiﬁed method for calculating the load-carrying capacities of cold- the effective width method, a modified method for calculating the load-carrying capaci- the effective width method, a modified method for calculating the load-carrying capaci- tie tie formed s s of co of cold- thin-walled ld-ffo or rmed thin med thin double-lipped -w -wa alled lled double double equal-leg -l -liipp pped equal-le ed equal-le angles g ang g ang under lles es axial und unde er axial compression r axial compression compression was proposed. was proposed. was proposed. 2. Experimental Program 2 2.. Ex Expe peri rime ment ntal al P Pr ro og gram ram 2.1. Specimen Design 2.1. Specimen 2.1. Specimen Design Design Axial compression test was carried out on 24 cold-formed thin-walled steel double- Axial compression test was carried out on 24 cold-formed thin-walled steel dou- Axial compression test was carried out on 24 cold-formed thin-walled steel dou- lipped equal-leg angles. The cross-section and the geometric parameters of the double- ble-lipped e ble-lipped eq qual-leg ual-leg ang anglle es s. The cross-sectio . The cross-section n and and th the geometr e geometriic p c pa ar ra amete meters of rs of the dou- the dou- lipped equal-leg angle are shown in Figure 1. The nominal section dimensions are shown ble-lipped equal-leg angle are shown in Figure 1. The nominal section dimensions are ble-lipped equal-leg angle are shown in Figure 1. The nominal section dimensions are in Table 1,where a and a are the widths of two legs, respectively, b and b are the widths 1 2 1 2 shown in shown in Tab Tabl le e 1,whe 1,wher re e a a1 1 and a and a2 2 ar are e the the wid widt ths hs of of tw two le o legs, gs, resp respec ectiv tive ely, ly, b b1 1 and and b b2 2 a ar re e t th he e of the ﬁrst lips, respectively, c and c are the widths of the second lips, respectively, and t 1 2 widths of the first lips, respectively, c1 and c2 are the widths of the second lips, respec- widths of the first lips, respectively, c1 and c2 are the widths of the second lips, respec- is the thickness of the section.The length of specimens included 400 mm, 900 mm, 1500 mm, tive tively, ly, and and t is the t is the thickn thickness of the ess of the sec sect tio ion. n.The The leng length of th of specimen specimens included 400 mm, s included 400 mm, 900 900 and 2100 mm.The numbering rules of specimens are shown in Figure 2. For example, mm, 1500 mm, and 2100 mm.The numbering rules of specimens are shown in Figure 2. mm, 1500 mm, and 2100 mm.The numbering rules of specimens are shown in Figure 2. DLA6020-400-1 deﬁnes the specimen as follows: DLA means double-lipped equal-leg For exa For exam mpl ple, e, DLA602 DLA6020- 0-400- 400-1 defi 1 defin nes t es th he speci e specim men as en as f fo oll llows: DLA mea ows: DLA mean ns double- s double-l li ipped pped angle; 60 indicates that the nominal width of the leg is 60 mm; 20 represents that the width equal-leg equal-leg an ang glle; 60 ind e; 60 indiic ca ates tes th that at the n the no ominal minal wi widt dth h of t of th he e lle eg i g is s 6 60 0 mm; 2 mm; 20 0 r re epr pre es se en nts ts of the ﬁrst lip is 20 mm; 400 represents the length of the specimen is 400 mm; 1 means the tha thatt the wid the widtth of the h of the fir first st lip lip i is s 2 20 0 m mm m;; 40 400 0 repre repres sen ents the ts the len leng gth of th of the specim the specimen en is 400 is 400 sequence number of the same specimen. The measured cross-section dimensions and the mm; 1 mea mm; 1 mean ns t s th he seq e sequ uence number of ence number of tth he e sa same me spe spec cimen. The imen. The meas measured ured cro cross-section ss-section dimensions lengths of all and the specimens lengths of all specimens are g are given in Table 2.iven in Table 2. dimensions and the lengths of all specimens are given in Table 2. Figure 1. Figure 1. Figure 1. Sec Sec Section t tio ion of dou n of dou of double-lipped b ble- le-lip lipped equ ped equa aequal-leg l-leg l-leg a an ngle. gle. angle. Figure 2. Labeling rule for steel angle specimen. Figure 2. Figure 2. Label Labeling ing rule for st rule for eesteel l angle angle specim specimen. en. Table 1. Nominal section dimensions of the double-lipped equal-leg angle. Cross-Section a /mm a /mm b /mm b /mm c /mm c /mm t/mm 1 2 1 2 1 2 DLA6020 60 60 20 20 10 10 2 DLA9020 90 60 20 20 10 10 2 DLA12024 120 120 24 24 10 10 2 Buildings 2022, 12, 1775 4 of 17 Table 2. The measured sectional dimensions of the specimens. Specimen a /mm a /mm b /mm b /mm c mm c /mm t/mm L/mm 1 2 1 2 1 2 DLA6020-400-1 61.62 61.54 20.95 20.99 11.63 11.48 1.97 400.00 DLA6020-400-2 61.27 61.59 21.46 21.31 11.35 10.47 1.97 400.00 DLA6020-900-1 61.02 60.71 21.33 20.73 10.32 10.84 1.97 900.00 DLA6020-900-2 61.64 61.52 21.36 21.35 10.84 10.99 1.96 900.00 DLA6020-1500-1 62.20 62.08 21.75 21.90 10.70 10.20 1.97 1500.00 DLA6020-1500-2 61.71 62.00 21.35 21.74 11.28 11.43 1.98 1500.00 DLA6020-2100-1 61.74 61.71 22.37 21.88 10.84 11.22 1.97 2100.00 DLA6020-2100-2 61.44 61.33 21.37 22.29 10.52 11.95 1.96 2101.00 DLA9020-400-1 91.56 91.72 21.77 21.79 10.78 10.17 1.96 400.00 DLA9020-400-2 91.57 91.29 21.88 21.50 11.32 10.27 1.97 400.00 DLA9020-900-1 91.65 91.97 21.34 20.88 11.59 10.10 1.98 900.00 DLA9020-900-2 91.70 91.72 22.15 20.85 10.27 10.62 1.97 900.00 DLA9020-1500-1 92.16 91.42 21.18 21.21 10.90 10.43 1.98 1499.50 DLA9020-1500-2 92.86 92.57 21.70 21.02 11.12 11.09 1.99 1499.50 DLA9020-2100-1 91.77 91.72 21.29 21.97 10.49 10.70 1.96 2101.10 DLA9020-2100-2 92.58 91.84 20.79 21.03 10.82 10.88 2.00 2101.50 DLA12024-400-1 120.15 121.40 25.60 25.72 12.85 13.06 1.98 400.00 DLA12024-400-2 120.61 122.35 25.86 25.29 12.95 13.04 1.98 400.67 DLA12024-900-1 122.29 121.34 24.89 25.08 13.02 12.23 1.98 900.00 DLA12024-900-2 120.64 121.85 25.80 25.40 12.11 13.31 1.98 900.00 DLA12024-1500-1 121.55 121.29 25.16 25.72 12.42 12.46 1.97 1499.90 DLA12024-1500-2 121.44 121.68 25.37 25.69 12.54 12.73 1.97 1499.10 DLA12024-2100-1 122.31 121.93 25.65 25.14 13.92 12.11 1.98 2101.10 DLA12024-2100-2 122.32 120.99 25.47 24.98 12.62 13.40 1.98 2101.10 2.2. Material Properties The zinc-coated steel plate with grade Q550 was used to manufacture the cold-formed steel double-lipped equal-leg angles. Three standard coupon specimens cut at the legs of the specimen were tested to obtain the material properties of the specimens in a 30 kN MTS testing machine based on the Chinese code “Tensile tests of metallic materials Part 1: test methods at room temperature” (GB/T228.1-2010) [28]. The material properties were determined from the stress–strain curves of the coupon specimens. The stress–strain curves of three standard coupon specimens are shown in Figure 3. The average results of the material properties, including the yield strength, the tensile strength, the elastic modulus, Buildings 2022, 12, x FOR PEER REVIEW 5 of 18 and the elongation of the steel obtained from the coupon tests are 403 MPa, 523 MPa, 2.11 10 MPa, and 0.27. Figure 3. Stress–strain curves of tension coupons. Figure 3. Stress–strain curves of tension coupons. 2.3. Initial Imperfection The initial imperfection is produced in the manufacturing, transportation, and fab- rication of cold-formed steel double-lipped equal-leg angles. The initial imperfections greatly influencethe buckling mode and ultimate capacities of double-lipped equal-leg angles. Therefore, all the specimens’ initial geometric imperfections were measured, in- cluding the local, distortion, and global imperfections. The measuring positions are shown in Figures 4 and 5. In Figure 4, the initial global buckling imperfection about the weak axis and the initial distortional buckling imperfection of the specimensare meas- ured at positions 1, 2, 3, 4, and 5, respectively. Positions 6, 7, and 8 in Figure 5 measure the initial local buckling imperfection.The measurements of the initial imperfectionsare shown in Figure 6. The numbers of measurements are 11, 10, 11, and 15 along the longi- tudinal direction for the specimens with lengths of 400 mm, 900 mm, 1500 mm, and 2100 mm, respectively. Three sections at 1/2 span and 1/4 span of the specimens are selected to measure the local initialimperfections along the longitudinal direction, and the distance of the measured positions at each cross-section is 10 mm. Figure 4. The measured position of global and distortional initial imperfections. Buildings 2022, 12, x FOR PEER REVIEW 5 of 18 Buildings 2022, 12, 1775 5 of 17 2.3. Initial Imperfection Figure 3. Stress–strain curves of tension coupons. The initial imperfection is produced in the manufacturing, transportation, and fabrica- 2tion .3. Initial Imperfection of cold-formed steel double-lipped equal-leg angles. The initial imperfections greatly inﬂuencethe buckling mode and ultimate capacities of double-lipped equal-leg angles. The initial imperfection is produced in the manufacturing, transportation, and fab- rication of cold-formed steel double-lipped equal-leg angles. The initial imperfections Therefore, all the specimens’ initial geometric imperfections were measured, including greatly influencethe buckling mode and ultimate capacities of double-lipped equal-leg the local, distortion, and global imperfections. The measuring positions are shown in angles. Therefore, all the specimens’ initial geometric imperfections were measured, in- Figures 4 and 5. In Figure 4, the initial global buckling imperfection about the weak axis cluding the local, distortion, and global imperfections. The measuring positions are and the initial distortional buckling imperfection of the specimensare measured at positions shown in Figures 4 and 5. In Figure 4, the initial global buckling imperfection about the weak axis and the initial distortional buckling imperfection of the specimensare meas- 1, 2, 3, 4, and 5, respectively. Positions 6, 7, and 8 in Figure 5 measure the initial local ured at positions 1, 2, 3, 4, and 5, respectively. Positions 6, 7, and 8 in Figure 5 measure buckling imperfection.The measurements of the initial imperfectionsare shown in Figure 6. the initial local buckling imperfection.The measurements of the initial imperfectionsare The numbers of measurements are 11, 10, 11, and 15 along the longitudinal direction for shown in Figure 6. The numbers of measurements are 11, 10, 11, and 15 along the longi- the specimens with lengths of 400 mm, 900 mm, 1500 mm, and 2100 mm, respectively. tudinal direction for the specimens with lengths of 400 mm, 900 mm, 1500 mm, and 2100 Three sections at 1/2 span and 1/4 span of the specimens are selected to measure the local mm, respectively. Three sections at 1/2 span and 1/4 span of the specimens are selected to measure the local initialimperfections along the longitudinal direction, and the distance initialimperfections along the longitudinal direction, and the distance of the measured of the measured positions at each cross-section is 10 mm. positions at each cross-section is 10 mm. Buildings 2022, 12, x FOR PEER REVIEW 6 of 18 Buildings 2022, 12, x FOR PEER REVIEW 6 of 18 Figure 4. The measured position of global and distortional initial imperfections. Figure 4. The measured position of global and distortional initial imperfections. Figure 5. The measured position of local initial imperfections. Figure 5. The measured position of local initial imperfections. Figure 5. The measured position of local initial imperfections. (a) (b) (c) Figure 6. Initial imperfection measurement of the specimens. (a) Initial imperfection of local buck- ling. (b) Initial imperfection of distortional buckling. (c) Initial imperfection of global buckling about weak axis. The measurement values of initial imperfections for some specimens are shown in Figure 7. It can be seen from Figure 7 that the maximum value of initial distortional- buckling imperfection is greater than the maximum values of initial local buckling im- (a) (b) (c) perfection and initial global buckling imperfection. The distributions of initial geometric imperfections of other specimens are similar, and all maximum values of the initial im- Figure 6. Initial imperfection measurement of the specimens. (a) Initial imperfection of local buck- perfectionsof the specimens are less than L/1000. Figure 6. Initial imperfection measurement of the specimens. (a) Initial imperfection of local buckling. ling. (b) Initial imperfection of distortional buckling. (c) Initial imperfection of global buckling (b) Initial imperfection of distortional buckling. (c) Initial imperfection of global buckling about about weak axis. weak axis. The measurement values of initial imperfections for some specimens are shown in Figure 7. It can be seen from Figure 7 that the maximum value of initial distortional- buckling imperfection is greater than the maximum values of initial local buckling im- perfection and initial global buckling imperfection. The distributions of initial geometric imperfections of other specimens are similar, and all maximum values of the initial im- perfectionsof the specimens are less than L/1000. (a) (b) (c) Figure 7. Initial imperfections of the specimens. (a) Global initial imperfection of DLA6020−400−2. (b) Distortion initial imperfection of DLA6020−900−2. (c) Local initial imperfection of DLA6020−400−1. 2.4. Test Setup and Procedure (a) (b) (c) Figure 7. Initial imperfections of the specimens. (a) Global initial imperfection of DLA6020−400−2. (b) Distortion initial imperfection of DLA6020−900−2. (c) Local initial imperfection of DLA6020−400−1. 2.4. Test Setup and Procedure Buildings 2022, 12, x FOR PEER REVIEW 6 of 18 Figure 5. The measured position of local initial imperfections. (a) (b) (c) Figure 6. Initial imperfection measurement of the specimens. (a) Initial imperfection of local buck- Buildings 2022, 12, 1775 6 of 17 ling. (b) Initial imperfection of distortional buckling. (c) Initial imperfection of global buckling about weak axis. The measurement values of initial imperfections for some specimens are shown in The measurement values of initial imperfections for some specimens are shown in Figure 7. It can be seen from Figure 7 that the maximum value of initial distortional- Figure 7. It can be seen from Figure 7 that the maximum value of initial distortionalbuckling buckling imperfection is greater than the maximum values of initial local buckling im- imperfection is greater than the maximum values of initial local buckling imperfection and perfect initial global ion and in buckling itial glob imperfection. al buckling The imperfec distributions tion. The of distr initial ibutions geometric of init imperfections ial geometric imperfections of other specimens are similar, and all maximum values of the initial im- of other specimens are similar, and all maximum values of the initial imperfectionsof the perfectionso specimens ar f the specimen e less than L/1000. s are less than L/1000. (a) (b) (c) Buildings 2022, 12, x FOR PEER REVIEW 7 of 18 Figure 7. Initial imperfections of the specimens. (a) Global initial imperfection of DLA6020−400−2. Figure 7. Initial imperfections of the specimens. (a) Global initial imperfection of DLA6020-400-2. (b) Distortion initial imperfection of DLA6020−900−2. (c) Local initial imperfection of (b) Distortion initial imperfection of DLA6020-900-2. (c) Local initial imperfection of DLA6020-400-1. DLA6020−400−1. All the double-lipped equal-leg angles were axially compressed using a steel frame 2.4. Test Setup and Procedure system and a 500 kN servo-controlled hydraulic testing machine, as shown in Figure 8. All the double-lipped equal-leg angles were axially compressed using a steel frame 2.4. Test Setup and Procedure The specimens were placed directly in the groove of the top bearing plate connected with system and a 500 kN servo-controlled hydraulic testing machine, as shown in Figure 8. The the actuator and the bottom bearing plate. The geometric center of the specimen coin- specimens were placed directly in the groove of the top bearing plate connected with the actuator and the bottom bearing plate. The geometric center of the specimen coincided cided with the geometric center of the upper loading plate and lower plate. The LVDTs with the geometric center of the upper loading plate and lower plate. The LVDTs (linear (linear variable displacement transducers) were set up at the mid-section of specimens, as variable displacement transducers) were set up at the mid-section of specimens, as shown shown in Figure 9 D1, D2, D3, and D4. The distances of all LVDTs to the edge of the plate in Figure 9 D1, D2, D3, and D4. The distances of all LVDTs to the edge of the plate of of double-lipped equal-leg angle were 10 mm. A displacement transducer was arranged double-lipped equal-leg angle were 10 mm. A displacement transducer was arranged at at the top bearing plate to obtain the vertical displacement of the specimen. The YG16 the top bearing plate to obtain the vertical displacement of the specimen. The YG16 static static strain displacement acquisition system automatically collected the load and dis- strain displacement acquisition system automatically collected the load and displacements of the specimen. placements of the specimen. Figure 8. Test setup. Figure 8. Test setup. Figure 9. Instrumentation arrangement. 3. Test Results 3.1. Failure Modes The buckling modes of all double-lipped equal-leg angles are shown in Table 3, where L, D, and F represent local buckling, distorted buckling, and global flexural buck- ling.It can be seen from Table 3 that the distortional buckling occurred for specimens with a small width-thickness ratio and small slenderness ratio. In contrast, the distor- tional and global flexural buckling occurred for specimens with a small width-to-thickness ratio and large slenderness ratio. For the specimens with a large width-to-thickness ratio and small slenderness ratio, the interactive buckling of local and distortional buckling was discovered. In contrast, the interactive buckling of local, dis- tortional, and global flexural buckling was found for the specimens with a large width ratio and large slenderness ratio. Buildings 2022, 12, x FOR PEER REVIEW 7 of 18 All the double-lipped equal-leg angles were axially compressed using a steel frame system and a 500 kN servo-controlled hydraulic testing machine, as shown in Figure 8. The specimens were placed directly in the groove of the top bearing plate connected with the actuator and the bottom bearing plate. The geometric center of the specimen coin- cided with the geometric center of the upper loading plate and lower plate. The LVDTs (linear variable displacement transducers) were set up at the mid-section of specimens, as shown in Figure 9 D1, D2, D3, and D4. The distances of all LVDTs to the edge of the plate of double-lipped equal-leg angle were 10 mm. A displacement transducer was arranged at the top bearing plate to obtain the vertical displacement of the specimen. The YG16 static strain displacement acquisition system automatically collected the load and dis- placements of the specimen. Buildings 2022, 12, 1775 7 of 17 Figure 8. Test setup. Figure 9. Figure 9. Instr Instr umentation arr umentation a arrangement. ngement. 3. Test Results 3. Test Results 3.1. Failure Modes 3.1. Failure Modes The buckling modes of all double-lipped equal-leg angles are shown in Table 3, where The buckling modes of all double-lipped equal-leg angles are shown in Table 3, where L, D, L, D, and Fand F repr represent esen local t loc buckling, al buckling, d distorted istorted buckling, buckling, and and glob global al flexur ﬂexural al buck buckling.It - ling.It can be seen from Table 3 that the distortional buckling occurred for specimens can be seen from Table 3 that the distortional buckling occurred for specimens with a with a small width-thickness ratio and small slenderness ratio. In contrast, the distor- small width-thickness ratio and small slenderness ratio. In contrast, the distortional and tional and global flexural buckling occurred for specimens with a small global ﬂexural buckling occurred for specimens with a small width-to-thickness ratio and width-to-thickness ratio and large slenderness ratio. For the specimens with a large large slenderness ratio. For the specimens with a large width-to-thickness ratio and small width-to-thickness ratio and small slenderness ratio, the interactive buckling of local and slenderness ratio, the interactive buckling of local and distortional buckling was discovered. distortional buckling was discovered. In contrast, the interactive buckling of local, dis- In contrast, the interactive buckling of local, distortional, and global ﬂexural buckling was tortional, and global flexural buckling was found for the specimens with a large width found for the specimens with a large width ratio and large slenderness ratio. ratio and large slenderness ratio. Table 3. Comparison of buckling modes and ultimate strengths between test and finite element analysis. Experiment Finite Element Analysis Specimens P /P Buckling Buckling m t P /kN P /kN t m Mode Mode DLA6020-400-1 106.00 D 107.62 D 1.02 DLA6020-400-2 113.00 D 110.69 D 0.98 DLA6020-900-1 100.10 D + F 99.74 D + F 1.00 DLA6020-900-2 92.50 D + F 95.36 D + F 1.03 DLA6020-1500-1 73.60 D + F 72.11 D + F 0.98 DLA6020-1500-2 69.10 D + F 70.49 D + F 1.02 DLA6020-2100-1 46.90 D + F 46.56 D + F 0.99 DLA6020-2100-2 44.30 D + F 45.71 D + F 1.03 DLA9020-400-1 139.90 L + D 140.02 L + D 1.00 DLA9020-400-2 139.50 L + D 138.84 L + D 1.00 DLA9020-900-1 119.90 D + F + L 120.49 D + F + L 1.00 DLA9020-900-2 127.60 D + F + L 129.24 D + F + L 1.01 DLA9020-1500-1 62.00 D + F + L 65.50 D + F + L 1.06 DLA9020-1500-2 76.10 D + F + L 75.96 D + F + L 1.00 DLA9020-2100-1 56.20 D + F + L 57.00 D + F + L 1.01 DLA9020-2100-2 48.70 D + F + L 47.96 D + F + L 0.98 DLA 12024-400-1 165.89 D + F + L 164.70 D + F + L 1.01 DLA 12024-400-2 164.69 D + F + L 165.10 D + F + L 1.00 DLA 12024-900-1 143.14 D + F + L 148.10 D + F + L 0.97 DLA 12024-900-2 145.24 D + F + L 147.90 D + F + L 0.98 DLA 12024-1500-1 127.76 D + F + L 130.20 D + F + L 0.98 DLA 12024-1500-2 120.44 D + F + L 124.70 D + F + L 0.97 DLA 12024-2100-1 95.75 D + F + L 101.00 D + F + L 0.95 DLA 12024-2100-2 96.64 D + F + L 99.10 D + F + L 0.98 Buildings 2022, 12, x FOR PEER REVIEW 8 of 18 Table 3. Comparison of buckling modes and ultimate strengths between test and finite element analysis. Experiment Finite Element Analysis Specimens Pm/Pt Pt/kN Buckling Mode Pm/kN Buckling Mode DLA6020-400-1 106.00 D 107.62 D 1.02 DLA6020-400-2 113.00 D 110.69 D 0.98 DLA6020-900-1 100.10 D+F 99.74 D+F 1.00 DLA6020-900-2 92.50 D+F 95.36 D+F 1.03 DLA6020-1500-1 73.60 D+F 72.11 D+F 0.98 DLA6020-1500-2 69.10 D+F 70.49 D+F 1.02 DLA6020-2100-1 46.90 D+F 46.56 D+F 0.99 DLA6020-2100-2 44.30 D+F 45.71 D+F 1.03 DLA9020-400-1 139.90 L+D 140.02 L+D 1.00 DLA9020-400-2 139.50 L+D 138.84 L+D 1.00 DLA9020-900-1 119.90 D+F+L 120.49 D+F+L 1.00 DLA9020-900-2 127.60 D+F+L 129.24 D+F+L 1.01 DLA9020-1500-1 62.00 D+F+L 65.50 D+F+L 1.06 DLA9020-1500-2 76.10 D+F+L 75.96 D+F+L 1.00 DLA9020-2100-1 56.20 D+F+L 57.00 D+F+L 1.01 DLA9020-2100-2 48.70 D+F+L 47.96 D+F+L 0.98 DLA 12024-400-1 165.89 D+F+L 164.70 D+F+L 1.01 DLA 12024-400-2 164.69 D+F+L 165.10 D+F+L 1.00 DLA 12024-900-1 143.14 D+F+L 148.10 D+F+L 0.97 DLA 12024-900-2 145.24 D+F+L 147.90 D+F+L 0.98 DLA 12024-1500-1 127.76 D+F+L 130.20 D+F+L 0.98 DLA 12024-1500-2 120.44 D+F+L 124.70 D+F+L 0.97 Buildings 2022, 12, 1775 8 of 17 DLA 12024-2100-1 95.75 D+F+L 101.00 D+F+L 0.95 DLA 12024-2100-2 96.64 D+F+L 99.10 D+F+L 0.98 3.1.1. The Short Angle Columns 3.1.1. The Short Angle Columns The buckling processes of the short double-lipped equal-leg angles with a length of The buckling processes of the short double-lipped equal-leg angles with a length of 400 mm are shown in Figure 10. The deformation was not evident at the initial loading 400 mm are shown in Figure 10. The deformation was not evident at the initial loading stage. With the load increase, the specimens DLA9020 series with a large stage. With the load increase, the specimens DLA9020 series with a large width-to-thickness width-to-thickness ratio appeared the local buckling (Figure 10a). When the load was ratio appeared the local buckling (Figure 10a). When the load was continued, distortional continued, distortional buckling was observed. The angle deformation between the two buckling was observed. The angle deformation between the two legs became larger legs became larger (Figure 10b). When the ultimate bearing capacity is reached, the (Figure 10b). When the ultimate bearing capacity is reached, the specimen fails. specimen fails. DLA9020-400-1 DLA9020-400-1 DLA6020-400-2 DLA9020-400-2 (a) (b) Figure 10. Buckling mode of short angle column with a length of 400 mm. (a) Local buckling. (b) Figure 10. Buckling mode of short angle column with a length of 400 mm. (a) Local buckling. Buildings 2022, 12, x FOR PEER REVIEW 9 of 18 Distortional buckling. (b) Distortional buckling. 3.1.2. The Medium-to-Long Angle Columns 3.1.2. The Medium-to-Long Angle Columns The buckling process of medium-to-long double-lipped equal-leg angles is shown in The buckling process of medium-to-long double-lipped equal-leg angles is shown in Figures 11–13. At the initial loading stage, the deformation was not apparent. With the Figures 11–13. At the initial loading stage, the deformation was not apparent. With the load increase, the legs of specimen DLA9020 series with a large width-to-thickness ratio load increase, the legs of specimen DLA9020 series with a large width-to-thickness ratio appeared the local buckling (Figures 11a, 12a and 13a). When the loading was continued, appeared the local buckling (Figures 11a, 12a, and 13a). When the loading was continued, the specimens appeared distortional buckling (Figures 11b, 12b and 13b) for the specimen the specimens appeared distortional buckling (Figures 11b, 12b, and 13b) for the speci- DLA9020 series and specimen DLA6020 series. When the load reached the ultimate bearing men DLA9020 series and specimen DLA6020 series. When the load reached the ultimate capacity, the specimen DLA9020 series and specimen DLA6020 series failed with global bearing capacity, the specimen DLA9020 series and specimen DLA6020 series failed with ﬂexural buckling. Thus, the interaction of distortional buckling and global ﬂexural buckling global flexural buckling. Thus, the interaction of distortional buckling and global flexural occurred for specimen DLA6020 series, while the specimen DLA9020 series showed the buckling occurred for specimen DLA6020 series, while the specimen DLA9020 series interaction of local, distortional, and global ﬂexural buckling. showed the interaction of local, distortional, and global flexural buckling. DLA9020-900-1 DLA9020-900-2 DLA6020-900-2 DLA6020-900-1 DLA9020-900-2 DLA6020-900-1 (a) (b) (c) Figure 11. Buckling mode of angle column with a length of 900 mm. (a) Local buckling. (b) Distor- Figure 11. Buckling mode of angle column with a length of 900 mm. (a) Local buckling. (b) Distor- tional buckling. (c) Global buckling. tional buckling. (c) Global buckling. DLA9020-1500-1 DLA9020-1500-2 DLA9020-1500-1 DLA6020-1500-1 DLA6020-1500-1 DLA9020-1500-2 (a) (b) (c) Figure 12. Buckling mode of angle column with a length of 1500 mm. (a) Local buckling. (b) Dis- tortional buckling. (c) Global buckling. Buildings 2022, 12, x FOR PEER REVIEW 9 of 18 3.1.2. The Medium-to-Long Angle Columns The buckling process of medium-to-long double-lipped equal-leg angles is shown in Figures 11–13. At the initial loading stage, the deformation was not apparent. With the load increase, the legs of specimen DLA9020 series with a large width-to-thickness ratio appeared the local buckling (Figures 11a, 12a, and 13a). When the loading was continued, the specimens appeared distortional buckling (Figures 11b, 12b, and 13b) for the speci- men DLA9020 series and specimen DLA6020 series. When the load reached the ultimate bearing capacity, the specimen DLA9020 series and specimen DLA6020 series failed with global flexural buckling. Thus, the interaction of distortional buckling and global flexural buckling occurred for specimen DLA6020 series, while the specimen DLA9020 series showed the interaction of local, distortional, and global flexural buckling. DLA9020-900-1 DLA9020-900-2 DLA6020-900-2 DLA6020-900-1 DLA9020-900-2 DLA6020-900-1 (a) (b) (c) Buildings 2022, 12, 1775 9 of 17 Figure 11. Buckling mode of angle column with a length of 900 mm. (a) Local buckling. (b) Distor- tional buckling. (c) Global buckling. DLA9020-1500-1 DLA9020-1500-2 DLA9020-1500-1 DLA6020-1500-1 DLA6020-1500-1 DLA9020-1500-2 (a) (b) (c) Buildings 2022, 12, x FOR PEER REVIEW Figure 12. Buckling mode of angle column with a length of 1500 mm. (a) Local buckling. ( 10 of b) Dis- 18 Figure 12. Buckling mode of angle column with a length of 1500 mm. (a) Local buckling. (b) Distor- tortional buckling. (c) Global buckling. tional buckling. (c) Global buckling. DLA9020-2100-1 DLA9020-2100-2 DLA9020-2100-2 DLA6020-2100-1 DLA6020-2100-1 DLA9020-2100-2 (a) (b) (c) Figure 13. Buckling mode of angle column with a length of 2100 mm. (a) Local buckling. (b) Dis- Figure 13. Buckling mode of angle column with a length of 2100 mm. (a) Local buckling. (b) Distor- tortional buckling. (c) Global buckling. tional buckling. (c) Global buckling. 3.2. Test Strengths and Curves 3.2. Test Strengths and Curves The The u ultimate ltimate capacities capacitiesof of all aldouble-lip l double-lip equal-leg equal-leangles g angles under under axial axcompr ial compre ession ssion are shown are shown in Table 3, in Table 3, wher wher e P e is Ptthe is the t testeload-carrying st load-carrying capacity capaci.ty It. I can t cabe n be seen seen from from T Table able 3 that 3 thathe t thaxial e axia load-carrying l load-carryincapabilities g capabilitie of s of thethe do double-lip uble-lequal-leg ip equal-le angles g angdecr les decr ease ease w with the ith incr the incre ease in ase length. in length. The load-displacement curves of the specimens DLA6020 series are shown in The load-displacement curves of the specimens DLA6020 series are shown in Figure Figure 14a–d. It can be seen from Figure 14 that for specimens DLA6020-400 (Figure 14a) 14a–d. It can be seen from Figure 14 that for specimens DLA6020-400 (Figure 14a) and and DLA6020-900 (Figure 14b), the stiffnesses were unchanged at the initial loading stage, DLA6020-900 (Figure 14b), the stiffnesses were unchanged at the initial loading stage, and the curves showed linear growth. A nonlinear segment appeared with the increase and the curves showed linear growth. A nonlinear segment appeared with the increase in in load, and the load decreased slowly after reaching the maximum load. For specimens load, and the load decreased slowly after reaching the maximum load. For specimens DLA6020-1500 (Figure 14c) and DLA6020-2100 (Figure 14d), the curves increased linearly DLA6020-1500 (Figure 14c) and DLA6020-2100 (Figure 14d), the curves increased linearly before the maximum load. The curves entered the nonlinear stage with the occurrence of before the maximum load. The curves entered the nonlinear stage with the occurrence of ﬂexural buckling approaching the maximum load. Then, the load dropped sharply after flexural buckling approaching the maximum load. Then, the load dropped sharply after reaching the ultimate load, and the specimen failed. reaching the ultimate load, and the specimen failed. The comparison on the average load displacement curves for the same sections with The comparison on the average load displacement curves for the same sections with different length are depicted in Figure 14e–g for section DLA6020, DLA9020, and DLA12024. different length are depicted in Figure 14e–g for section DLA6020, DLA9020, and DLA12024. It can found that the stiffness and ultimate strength of cold-formed dou- ble-lips equal-leg angles decrease with the increasing of length of the axial members. (a) (b) Buildings 2022, 12, x FOR PEER REVIEW 10 of 18 DLA9020-2100-1 DLA9020-2100-2 DLA9020-2100-2 DLA6020-2100-1 DLA6020-2100-1 DLA9020-2100-2 (a) (b) (c) Figure 13. Buckling mode of angle column with a length of 2100 mm. (a) Local buckling. (b) Dis- tortional buckling. (c) Global buckling. 3.2. Test Strengths and Curves The ultimate capacities of all double-lip equal-leg angles under axial compression are shown in Table 3, where Pt is the test load-carrying capacity. It can be seen from Table 3 that the axial load-carrying capabilities of the double-lip equal-leg angles decrease with the increase in length. The load-displacement curves of the specimens DLA6020 series are shown in Figure 14a–d. It can be seen from Figure 14 that for specimens DLA6020-400 (Figure 14a) and DLA6020-900 (Figure 14b), the stiffnesses were unchanged at the initial loading stage, and the curves showed linear growth. A nonlinear segment appeared with the increase in load, and the load decreased slowly after reaching the maximum load. For specimens DLA6020-1500 (Figure 14c) and DLA6020-2100 (Figure 14d), the curves increased linearly before the maximum load. The curves entered the nonlinear stage with the occurrence of Buildings 2022, 12, 1775 10 of 17 flexural buckling approaching the maximum load. Then, the load dropped sharply after reaching the ultimate load, and the specimen failed. The comparison on the average load displacement curves for the same sections with different length are depicted in Figure 14e–g for section DLA6020, DLA9020, and It can found that the stiffness and ultimate strength of cold-formed double-lips equal-leg DLA12024. It can found that the stiffness and ultimate strength of cold-formed dou- angles decrease with the increasing of length of the axial members. ble-lips equal-leg angles decrease with the increasing of length of the axial members. Buildings 2022, 12, x FOR PEER REVIEW 11 of 18 (a) (b) (c) (d) (e) (f) (g) Figure 14. Load-displacement curves. (a) DLA6020-400. (b) DLA6020-900. (c) DLA6020-1500. (d) Figure 14. Load-displacement curves. (a) DLA6020-400. (b) DLA6020-900. (c) DLA6020-1500. DLA6020-2100. (e) DLA6020 average load displacement curve. (f) DLA9020 average load dis- (d) DLA6020-2100. (e) DLA6020 average load displacement curve. (f) DLA9020 average load placement curve. (g) DLA12024 average load displacement curve. displacement curve. (g) DLA12024 average load displacement curve. 4. Finite Element Analysis 4.1. Development of the Finite Element Model The finite element analysis model of the cold-formed thin-walled steel dou- ble-lipped equal-leg angle was established using the finite element software ABAQUS6.14 [29]. In FEA, the measured specimens’ dimensions and the maximum Buildings 2022, 12, 1775 11 of 17 Buildings 2022, 12, x FOR PEER REVIEW 12 of 18 4. Finite Element Analysis 4.1. Development of the Finite Element Model The ﬁnite element analysis model of the cold-formed thin-walled steel double-lipped geometric imperfections of the specimens were all included in the model, but the residual equal-leg angle was established using the ﬁnite element software ABAQUS6.14 [29]. In stress of the whole section and the increase of yield strength (at the corner regions only) FEA, the measured specimens’ dimensions and the maximum geometric imperfections by the cold-forming process were not considered [30]. The length and cross-section size of the specimens were all included in the model, but the residual stress of the whole of specimens were the measured size. The S4R shell element and the ideal elastoplastic section and the increase of yield strength (at the corner regions only) by the cold-forming model were adopted, and the average value of the material property test was adopted. process were not considered [30]. The length and cross-section size of specimens were the Through a certain number of trials, it was found that the error of ultimate strength was measured size. The S4R shell element and the ideal elastoplastic model were adopted, and less the than average 2% when the value of the mesh material was 5 pr mm operty × 5 mm or 10 mm × test was adopted. 10 m Thr mough . So, 10 a mm certain × 10 number mm was of trials, selected it was as the found mesh that size the . The error spec of imens ultimate were strength fixed was at both less ethan nds, 5 degr 2% when ees of free- the mesh dom (two was 5 mm tra nsl5 at mm ionaor l an 10 d three mm rot 10 ati mm. onal) we So, 10 re co mm nstr ain 10 ed mm at the lo was selected ading enas d, an the d th mesh e UZ longitudinal de size. The specimens gree o werf e freedom w ﬁxed at both as rele ends, ased. It 5 degrees was ofutterly fixed at the o freedom (two translational ther end. and The vert three rotational) ical displawer cemen e constrained t was applie at d the at th loading e coupli end, ng poin and t the RPUZ -2 olongitudinal f the centroid o degr f the ee of freedom was released. It was utterly ﬁxed at the other end. The vertical displacement double-lipped equal-leg angle section at the loading end. In order to simulate the speci- was applied at the coupling point RP-2 of the centroid of the double-lipped equal-leg mens more precisely, the measured initial geometric imperfections were introduced. The angle section at the loading end. In order to simulate the specimens more precisely, the maximum value of global, distortional, and local initial geometric imperfections was measured initial geometric imperfections were introduced. The maximum value of global, taken as the imperfect value. The finite element model is shown in Figure 15. The finite distortional, and local initial geometric imperfections was taken as the imperfect value. element analysis included two steps: the first step was the eigenvalue buckling analysis, The ﬁnite element model is shown in Figure 15. The ﬁnite element analysis included two and the first buckling mode was used as the initial imperfection shape of the speci- steps: the ﬁrst step was the eigenvalue buckling analysis, and the ﬁrst buckling mode was mens.The second step was nonlinear analysis.The Von-Misses stress–strain criterion and used as the initial imperfection shape of the specimens.The second step was nonlinear arc length method were adopted to obtain the buckling modes and ultimate strengths of analysis.The Von-Misses stress–strain criterion and arc length method were adopted to all specimens. obtain the buckling modes and ultimate strengths of all specimens. Figure 15. Figure 15. Finit Finite e element mod element model. el. 4.2. Validation of Finite Element Model 4.2. Validation of Finite Element Model The ﬁnite element analysis results for all specimens are shown in Table 3, where P The finite element analysis results for all specimens are shown in Table 3, where Pm is the ﬁnite element analysis result. As shown in Table 3, the average value of the ratios is the finite element analysis result. As shown in Table 3, the average value of the ratios between the test results and the ﬁnite element analysis results is 1.01, and the coefﬁcient between the test results and the finite element analysis results is 1.01, and the coefficient of variation is 0.08. The comparison buckling modes between the ﬁnite element analysis of variation is 0.08. The comparison buckling modes between the finite element analysis and the test are shown in Figure 16. The buckling modes of the ﬁnite element analysis and the test are shown in Figure 16. The buckling modes of the finite element analysis are are consistent with the test, as shown in Figure 16. The comparisons of load-displacement consistent with the test, as shown in Figure 16. The comparisons of load-displacement curves between tests and ﬁnite element analysis are shown in Figure 17, which shows that curves between tests and finite element analysis are shown in Figure 17, which shows the test and ﬁnite element analysis curves are in good agreement. These comparison results that the test and finite element analysis curves are in good agreement. These comparison show that this paper ’s ﬁnite element analysis model can reasonably simulate the buckling results show that this paper’s finite element analysis model can reasonably simulate the mode, ultimate strength, and load-displacement curve of the cold-formed thin-walled steel buckling mode, ultimate strength, and load-displacement curve of the cold-formed double-lipped equal-leg angle under axial compression. thin-walled steel double-lipped equal-leg angle under axial compression. Buildings Buildings 2022 2022,, 12 12,, x FO 1775 R PEER REVIEW 13 of 12 of 18 17 Buildings 2022, 12, x FOR PEER REVIEW 13 of 18 (a) (b) (c) (d) (a) (b) (c) (d) Figure 16. Comparison of buckling modes between the experiments and finite element analysis. (a) Figure 16. Comparison of buckling modes between the experiments and finite element analysis. (a) Figure 16. Comparison of buckling modes between the experiments and ﬁnite element analysis. DLA6020-400- DLA6020-400- 2. ( 2. ( bb ) DLA9020 ) DLA9020 -900-2. ( -900-2. ( cc ) DLA9020-1500-2. ( ) DLA9020-1500-2. ( dd ) sDLA9020-21 ) sDLA9020-21 00-2. 00-2. (a) DLA6020-400-2. (b) DLA9020-900-2. (c) DLA9020-1500-2. (d) sDLA9020-2100-2. (a) (b) (a) (b) (c) (d) (c) (d) Figure 17. Comparison of load-displacement curves between experiments and finite element Figure 17. Comparison of load-displacement curves between experiments and finite element Figure 17. Comparison of load-displacement curves between experiments and ﬁnite element analysis. analysis. ( analysis. ( aa ) D ) D LL A9020-400-2. A9020-400-2. (b (b ) DLA9020-9 ) DLA9020-9 00-3. ( 00-3. ( cc ) DLA6 ) DLA6 020-1500-2. ( 020-1500-2. ( dd ) DLA6020-2100 ) DLA6020-2100 -2. -2. (a) DLA9020-400-2. (b) DLA9020-900-3. (c) DLA6020-1500-2. (d) DLA6020-2100-2. Buildings 2022, 12, 1775 13 of 17 5. Assessment and Suggested of the Design Method 5.1. Direct Strength Method The direct strength method in North American speciﬁcation [31] was used to calculate the ultimate strength of cold-formed steel members. The nominal axial strength of cold- formed steel double-lipped equal-leg angle section is the minimum value of the axial strength of local buckling interacting with global buckling P and the axial strength of nl distortional buckling P . nd The axial strength of local buckling interacting with global buckling P is calculated nl according to Equation (1): < P l 0.776 ne 0.4 0.4 P = (1) P P nl crl crl : 1 0.15 P l > 0.776 ne P P ne ne where l = P /P . P is the axial strength of global buckling. P is the critical elastic ne ne ` crl crl local buckling strength. The elastic local buckling stress can be calculated through ﬁnite strip software CUFSM [32]. The axial strength of global buckling P can be calculated by Equation (2): ne P = A F (2) ne g n where A is the gross area of cross-section and F is the global buckling stress, which can g n be calculated according to Equation (3): 0.658 F l 1.5 y c F = (3) 0.877 F l > 1.5 y c where l = is the smallest value of ﬂexural, torsional, and ﬂexural–torsional buckling cre stresses. F can be determined by Equation (4). cre 2 2 x y 0 0 2 2 (F s ) F s (F s ) F F s F (F s ) = 0 (4) cre ex cre ey cre t cre ey cre ex cre cre r r 0 0 In which p E s = (4a) ex (K L /r ) x x x p E s = (4b) ey K L /r y y y " # 1 p EC s = G J + (4c) 2 2 Ar (K L ) t t where s , s , s are the elastic buckling stresses for ﬂexural buckling about the principal ex ey t x-axis, y-axis, and torsional buckling, respectively. x , y are the distances from the shear 0 0 centre to the centroid along the x-axis and y-axis. r is the polar radius of gyration. K , K , x y K are the effective length factor for bending about x-axis in accordance, bending about y-axis in accordance, and twisting determined in accordance. L , L , L are unbraced x y t lengths of members for bending about x-axis, y-axis, and torsion, respectively. r , r are the x y radius of gyration of full unreduced cross-section about the x-axis and y-axis. J, G, E, C are St. Venant torsion constant of cross-section, shear modulus of steel, modulus of elasticity of steel, and torsional warping constant of cross-section, respectively. Buildings 2022, 12, 1775 14 of 17 The axial distorted buckling strength can be determined according to Equation (5). P l 0.561 0.6 0.6 P = (5) P P nd crd crd : 1 0.25 P l > 0.561 P P y y where l = P /P , P = A F , F is the yield stress and P is the critical elastic d y crd y g y y crd distortional buckling strength. The elastic distortional buckling stress can be calculated through ﬁnite strip software CUFSM. 5.2. Effect Width Method The effective width method in Technical Code for Cold-Formed Thin-walled Steel Structures [33] predicts the ultimate strength of cold-formed steel members.The axial strength of cold-formed steel double-lipped equal-leg angle section can be determined according to Equation (6): N = j A f (6) e y where j is the global stability coefﬁcient of axial double-lipped equal-leg angle, which can be determined according to the minimum value of the slenderness ratio l of ﬂexural bucking and the slenderness ratio l of ﬂexural–torsional buckling. A is the effective w e cross-sectional area, A = b t, b is the effective width of the elements of double-lipped e e e equal-leg angle and can be calculated using the Formula (7). c b > 18ar > t t 21.8ar b b b e c 0.1 18ar < < 38ar = t t (7) > 25ar c b 38ar t t 235k For axial compression double-lipped equal-leg angle, b = b, a = 1, r = , k is the j f buckling coefﬁcient of the element of angle. 5.3. Recommendations for the Design of Double-Lipped Equal-Leg Angle The ultimate strength predicted using the direct strength method and effective width method for double-lipped equal-leg angle are shown in Table 4, where P and P are the z y calculated strength using the direct strength method and effective method, respectively. As shown in Table 4, the average ratios of the calculated capacities to test results P /P and z t P /P are 0.641 and 0.494, with a coefﬁcient of variation of 0.276 and 0.397. The comparison y t of ultimate strength between tests and the predicted results shows that the results calculated by the direct strength and effective width methods are conservative. The main reason is that the torsion of the leg with the lip is considered torsional buckling of the angle and distortional buckling of the leg. The torsion is considered repeatedly. Therefore, it is suggested to ignore the effect of torsion and only calculate the ﬂexural buckling when calculating the global buckling of the double-lipped equal-leg angle. For Formula (1) in the direct strength method, F is obtained as the minimum value of the cre ﬂexural buckling of the double-lipped equal-leg angle about the x-axis and y-axis. For Formula (6) in the effective width method, the slenderness ratio of the global buckling is the minimum value of the slenderness ratio of the ﬂexural buckling for the double-lipped equal-leg angle about the x-axis and y-axis. The predicted ultimate strength using the proposed direct strength method and effec- tive width method are shown in Table 4. P and P are calculated using the suggested za ya direct strength and effective width methods. As shown in Table 4, the average ratios of the calculated capacities to test results P /P and P /P is 1.075 and 0.953, with the za t ya t coefﬁcient of variation of 0.052 and 0.124. The ultimate strength calculated by the modiﬁed direct strength and effective width method agrees with the test results. Therefore, the Buildings 2022, 12, 1775 15 of 17 modiﬁed direct strength method and effective width method are accurate and feasible for calculating the ultimate strength of cold-formed steel double-lipped equal-leg angle under axial compression. Table 4. Comparison of ultimate strength between tests and the predicted results by using DSM, EWM, modiﬁed DSM, and modiﬁed EWM. Test DSM MDSM EWM MEWM Specimens P /P P /P P /P P /P y t ya t z t za t P /kN P /kN P /kN P /kN P /kN t z za y ya DLA6020-400-1 106 110.24 108.24 83.87 99.42 1.04 1.02 0.79 0.94 DLA6020-400-2 113 113.45 112.4 83.55 105.06 1 0.99 0.74 0.93 DLA6020-900-1 100.1 73.07 105.43 42.35 88.51 0.73 1.05 0.42 0.88 DLA6020-900-2 92.5 75.2 101.33 44.03 83.21 0.81 1.1 0.48 0.9 DLA6020-1500-1 73.6 35.41 80.36 19.87 63.73 0.48 1.09 0.27 0.87 DLA6020-1500-2 69.1 35.75 75.39 20.35 63.87 0.52 1.09 0.29 0.92 DLA6020-2100-1 46.9 22.49 51.36 12.51 35.31 0.48 1.1 0.27 0.75 DLA6020-2100-2 44.3 22.25 50.56 12.31 37.73 0.5 1.14 0.28 0.85 DLA9020-400-1 139.9 100.25 140.99 91.57 127.97 0.72 1.01 0.65 0.91 DLA9020-400-2 139.5 100.83 141.44 91.19 122.94 0.72 1.01 0.65 0.88 DLA9020-900-1 119.9 68.76 123.5 50.97 113.45 0.57 1.03 0.43 0.95 DLA9020-900-2 127.6 68.68 130.99 52.03 120.07 0.54 1.03 0.41 0.94 DLA9020-1500-1 62 40.14 74.66 23.71 76.89 0.65 1.2 0.38 1.24 DLA9020-1500-2 76.1 38.23 82.05 22.09 77.08 0.5 1.08 0.29 1.01 DLA9020-2100-1 48.7 24.08 54.47 37.13 55.44 0.49 1.12 0.76 1.14 DLA9020-2100-2 49.3 24.69 56.06 38.61 55.46 0.5 1.14 0.78 1.12 DLA12024-400-1 165.89 176.27 185.32 176.27 162.59 1.06 1.12 1.06 0.98 DLA12024-400-2 164.69 175.97 185.09 184.56 162.81 1.07 1.12 1.12 0.99 DLA12024-900-1 143.14 114.77 153.67 151.27 146.35 0.80 1.07 1.06 1.02 DLA12024-900-2 145.24 119.15 156.07 153.67 148.54 0.82 1.07 1.06 1.02 DLA12024-1500-1 127.76 47.59 129.36 100.76 119.81 0.37 1.01 0.79 0.94 DLA12024-1500-2 120.44 47.78 129.85 101.36 120.60 0.40 1.08 0.84 1.00 DLA12024-2100-1 95.75 26.99 98.17 57.24 101.33 0.28 1.02 0.60 1.06 DLA12024-2100-2 96.64 26.66 98.34 56.48 100.69 0.28 1.03 0.58 1.04 Mean value 0.639 1.075 0.625 0.904 Variance 0.234 0.193 0.272 0.160 Coefﬁcient of variation 0.366 0.179 0.435 0.177 6. Conclusions (1) The axial compression test of 24 cold-formed thin-walled double-lipped equal-leg angles showed that the distortional buckling occurred for specimens with a small width-to-thickness ratio and small slenderness ratio. The buckling interactive with distortional and global ﬂexural buckling was observed for the specimens with small width-to-thickness ratios and large slenderness ratios. The specimens with large width-to-thickness ratios and small slenderness ratios showed interactive buckling with local and distortion buckling, while the specimens with large width-to-thickness ratios and large slenderness ratios developed interactive buckling with local, distor- tional, and global ﬂexural buckling. The ultimate strengths of specimens decreased with the increase of the length of the double-lipped equal-leg angle. (2) The ultimate strengths, buckling modes, and axial compression displacement curves of the specimens analyzed by the ﬁnite element method were in good agreement with the test results. The results showed that the developed ﬁnite element model was feasible for the buckling analysis of cold-formed thin-walled steel double-lipped equal-leg angle. (3) The distortional buckling of the leg with lip and the global torsional buckling angle for cold-formed thin-walled steel double-lipped equal-leg angle is consistent. The axial strength of the double-lipped equal-leg angle calculated by the direct strength and effective width methods indicated that the design methods were too conservative. Buildings 2022, 12, 1775 16 of 17 Therefore, the suggested approaches were proposed by ignoring the global torsional buckling. The results obtained by the proposed direct strength method and effective width method were accurate, indicating that the proposed method can be used to determine the ultimate strength of the cold-formed thin-walled steel double-lipped equal-leg angle. (4) Further numerical and experimental studies are needed before the modiﬁed design method can be used in the codes. Meanwhile, the cold-formed thin-walled steel lipped equal-leg angle, unequal-leg angle, and lipped unequal-leg angle should be studied by experiment and numerical analysis. Author Contributions: Conceptualization, X.Y. and Y.G.; methodology, X.Y.; validation, S.Z.; inves- tigation, Y.G.; data curation, S.Z.; writing—original draft preparation, Y.L.; writing—review and editing, C.H.; funding acquisition, X.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Natural Science Foundation of China grant number [51868049]. Data Availability Statement: All the data included in this study are available upon request by contacting the corresponding author. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Yao, X.; Guo, Y.; Li, Y. Effective width method for distortional buckling design of cold-formed lipped channel sections. Thin-Walled Struct. 2016, 109, 344–351. 2. Chen, B.; Roy, K.; Uzzaman, A.; Raftery, G.; Nash, D.; Clifton, C.; Pouladi, P.; Lim, B.P.J. Effects of edge-stiffened web openings on the behaviour of cold-formed steel channel sections under compression. Thin-Walled Struct. 2019, 144, 106307. [CrossRef] 3. Chen, B.; Roy, K.; Uzzaman, A.; Lim, B.P. Moment capacity of cold-formed channel beams with edge-stiffened web holes, un-stiffened web holes and plain webs. Thin-Walled Struct. 2020, 157, 107070. [CrossRef] 4. Chen, B.; Roy, K.; Uzzaman, A.; Raftery, G.; Nash, D.; Lim, B.P.J. Parametric study and simpliﬁed design equations for cold-formed steel channels with edge-stiffened holes under axial compression. J. Constr. Steel Res. 2020, 172, 106161. [CrossRef] 5. Fang, Z.; Roy, K.; Mares, J.; Sham, C.W.; Chen, B.; Lim, J.B. Deep learning-based axial capacity prediction for cold-formed steel channel sections using Deep Belief Network. Structures 2021, 33, 2792–2802. [CrossRef] 6. Fang, Z.; Roy, K.; Chen, B.; Sham, C.W.; Hajirasouliha, I.; Lim, J.B. Deep learning-based procedure for structural design of cold-formed steel channel sections with edge-stiffened and un-stiffened holes under axial compression. Thin-Walled Struct. 2021, 166, 108076. [CrossRef] 7. Fang, Z.; Roy, K.; Chen, B.; Xie, Z.; Lim, J.B. Local and distortional buckling behaviour of aluminium alloy back-to-back channels with web holes under axial compression. J. Build. Eng. 2022, 47, 103837. [CrossRef] 8. Fang, Z.; Roy, K.; Lakshmanan, D.; Pranomrum, P.; Li, F.; Lau, H.H.; Lim, J.B. Structural behaviour of back-to-back cold-formed steel channel sections with web openings under axial compression at elevated temperatures. J. Build. Eng. 2022, 54, 104512. [CrossRef] 9. Chen, B.; Roy, K.; Fang, Z.; Uzzaman, A.; Pham, C.H.; Raftery, G.; Lim, B.P.J. Shear capacity cold-formed steel channels with edge -stiffened hole, un-stiffened hole, and plain web. J. Struct. Eng. ASCE 2022, 148, 3250. [CrossRef] 10. Yang, F.; Han, J.; Yang, J.; Li, Z. Analysis and experimental study on ultimate bearing capacity of structural members of cold-bending angle steel for transmission tower under axial compression. J. Power Syst. Technol. 2010, 2, 194–199. 11. Xu, B. Stability and Bearing Capacity of Doub1e Lipped Channel Cold-Formed Thin-Wall Steel Column; Xi’an University of Architecture and Technology: Xi’an, China, 2017. 12. Al-Sayed, S.H.; Bjorhovde, R. Experimental study of single angle columns. J. Constr. Steel Res. 1989, 12, 83–102. [CrossRef] 13. Popovic, D.; Hancock, G.J.; Rasmussen, K.J.R. Compression tests on cold-formed angles loaded parallel with a leg. J. Struct. Eng. ASCE 2001, 127, 600–607. [CrossRef] 14. Young, B. Tests and design of ﬁxed-ended cold-formed steel plain angle columns. J. Struct. Eng. 2004, 130, 1931–1940. [CrossRef] 15. Chodraui, G.M.B.; Shifferaw, Y.; Malite, M. On the stability of cold-formed steel angles under compression. Rem Rev. Esc. de Minas 2007, 60, 355–363. [CrossRef] 16. Landesmann, A.; Camotim, D.; Dinis, P.B.; Cruz, R. Short-to-intermediate slender pin-ended cold-formed steel equal-leg angle columns: Experimental investigation, numerical simulation and DSM design. J. Eng. Struct. 2017, 132, 471–493. [CrossRef] 17. Silvestre, N.; Dinis, P.B.; Camotim, D. Developments on the design of cold-formed steel angles. J. Struct. Eng. 2013, 139, 680–694. [CrossRef] 18. Camotim, D.; Dinis, P.B.; Landesmann, A. Behavior, failure, and direct strength method design of steel angle columns: Geometrical simplicity versus structural complexity. J. Struct. Eng. 2020, 146, 04020226. [CrossRef] Buildings 2022, 12, 1775 17 of 17 19. Zhang, L.; Liang, Y.; Zhao, O. Flexural-torsional buckling behaviour and resistances of ﬁxed-ended press braked S690 high strength steel angle section columns. Eng. Struct. 2020, 223, 111180. [CrossRef] 20. Zhang, L.; Wang, F.; Liang, Y.; Zhao, O. Press-braked S690 high strength steel equal-leg angle and plain channel section stub columns: Testing, numerical simulation and design. Eng. Struct. 2019, 201, 109764. [CrossRef] 21. Wang, F.; Zhao, O.; Young, B. Testing and numerical modelling of S960 ultra-high strength steel angle and channel section stub columns. Eng. Struct. 2020, 204, 109902. [CrossRef] 22. Dinis, P.B.; Camotim, D.; Landesmann, A. Design of simply supported hot-rolled steel short-to-intermediate angle columns-design approach based on the Direct Strength Method (DSM). J. Constr. Steel Res. 2019, 12, 278–290. 23. Young, B. Experimental investigation of cold-formed steel lipped angle concentrically loaded compression members. J. Struct. Eng. 2005, 131, 1390–1396. [CrossRef] 24. Young, B.; Ellobody, E. Buckling analysis of cold-formed steel lipped angle columns. J. ASCE J. Struct. Eng. 2005, 131, 1570–1579. [CrossRef] 25. Shifferaw, Y.; Schafer, B.W. Cold-formed steel lipped and plain angle columns with ﬁxed ends. Thin-Walled Struct. 2014, 80, 142–152. [CrossRef] 26. Zhou, Y. Study the Axial Bearing Capacity of Cold-Formed Thin-Wall Steel Columns with Complex Lipped and Unequal Angular Sections; Zhengzhou University: Zhengzhou, China, 2020. 27. Ananthi, B.G.G.; Roy, K.; Chen, B.; Lim, J.B.P. Testing, simulation and design of back-to-back built-up cold-formed steel unequal angle sections under axial compression. J. Steel Compos. Struct. 2019, 33, 595–614. 28. General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China. Metallic Material Tensile Test Part 1: GB/T 228.1—2010; China Standards Press: Beijing, China, 2010. 29. ABAQUS. ABAQUS/Standard User’s Manual Volumes I–III and ABAQUS CAE Manual; Dassault Systemes Simulia Corporation: Johnston, RI, USA, 2014. 30. Zhou, X.; Chen, M. Experimental investigation and ﬁnite element analysis of web-stiffened cold-formed lipped channel columns with batten sheets. Thin-Walled Struct. 2018, 125, 38–50. 31. American Iron and Steel Institute. North American Speciﬁcation for the Design of Cold-Formed Steel Structural Members: AISI S100—2016; American Iron and Steel Institute: Washington, DC, USA, 2016. 32. Schafer, B.W. CUFSM4.05—Finite Strip Buckling Analysis of Thin-Walled Members; Department of Civil Engineering, Johns Hopkins University: Baltimore, MD, USA, 2012. 33. 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Abstract

buildings Article Experiment and Design Method of Cold-Formed Thin-Walled Steel Double-Lipped Equal-Leg Angle under Axial Compression 1 1 2 , 1 1 Xingyou Yao , Yafei Liu , Shile Zhang *, Yanli Guo and Chengli Hu School of Architecture and Civil Engineering, Nanchang Institute of Technology, Nanchang 330099, China School of Yaohu, Nanchang Institute of Technology, Nanchang 330099, China * Correspondence: yaoxingyoujd@163.com; Tel.: +86-15079190103 Abstract: The cold-formed steel (CFS) double-lipped equal-leg angle is widely used in modular container houses and cold-formed steel buildings. To study the buckling behavior and bearing capacity design method of the cold-formed steel (CFS) double-lipped equal-leg angle under axial compression, 24 CFS double-lipped equal-leg angles with different sections and slenderness ratios the axial compression were conducted. The test results showed that the distortional buckling occurs for specimens with a small width-to-thickness ratio and small slenderness ratio. The buckling interactive with distortional and global ﬂexural buckling was observed for the specimens with small width-to- thickness ratios and large slenderness ratios. The specimens with large width-to-thickness ratios and small slenderness ratios showed interactive buckling with local and distortion buckling. The specimens with large width-to-thickness ratios and large slenderness ratio developed interactive buckling with local, distortional, and global ﬂexural buckling. The ﬁnite element model established Citation: Yao, X.; Liu, Y.; Zhang, S.; by ABAQUS software was used to simulate and analyze the test. The buckling modes and the Guo, Y.; Hu, C. Experiment and load-carrying capacities analyzed by the ﬁnite element model agreed with the test results, which Design Method of Cold-Formed Thin-Walled Steel Double-Lipped showed that the developed ﬁnite element model was feasible to analyze the buckling and bearing Equal-Leg Angle under Axial capacity of the CFS double-lipped equal-leg angles. The experimental results were compared with Compression. Buildings 2022, 12, 1775. those calculated by the direct strength method in the North American standard and the effective https://doi.org/10.3390/ width method in the Chinese standard. The comparisons indicated that the calculated results are buildings12111775 very conservative with maximum value 36% and 51% for direct strength method and effective width Academic Editors: Krishanu Roy, method, respectively. The coefﬁcient of variation was 0.276 and 0.397, respectively. Finally, the Boshan Chen, Beibei Li, Lulu Zhang, modiﬁed direct strength method and the modiﬁed effective width method were proposed based on Letian Hai, Quanxi Ye and the experimental results. The comparison on the ultimate strength between test results and calculated Zhiyuan Fang results by using the modiﬁed method showed a good agreement. The modiﬁed method can be as a proposed desigh method for the ultimate strength of the CFS double-lipped equal-leg angles under Received: 12 September 2022 axial compression. Accepted: 20 October 2022 Published: 23 October 2022 Keywords: double-lipped equal-leg steel angle; axial compression; distortional buckling; global Publisher’s Note: MDPI stays neutral buckling; effective width method; direct strength method with regard to jurisdictional claims in published maps and institutional afﬁl- iations. 1. Introduction Cold-formed thin-walled steel members have been widely used in construction houses Copyright: © 2022 by the authors. because of their high stiffness and strength, lightweight, and convenient machine and Licensee MDPI, Basel, Switzerland. construction. The buckling behaviors and design methods of cold-formed thin-walled steel This article is an open access article channel sections with or without holes have been studied by many researchers [1–9]. In distributed under the terms and recent years, with the cold-formed steel sections becoming common as structural members, conditions of the Creative Commons the cold-formed thin-walled steel double-lipped equal-leg angles as a primary structural Attribution (CC BY) license (https:// member have been widely used in tower structures, truss structures, and cold-formed steel creativecommons.org/licenses/by/ buildings [10,11]. Although the double-lipped equal-leg angle cross-section is simple, the 4.0/). Buildings 2022, 12, 1775. https://doi.org/10.3390/buildings12111775 https://www.mdpi.com/journal/buildings Buildings 2022, 12, 1775 2 of 17 centroid and shear center is inconsistent, the width-thickness ratio of the plate is large, and a partially stiffened element is present. The buckling mode and the effect on ultimate strength are complex. The double-lipped equal-leg angle cross-section is easy to buckle with local buckling, distortional buckling, and global buckling, which can affect the steel angle’s ultimate strength. Al-Sayed et al. [12] performed the buckling tests of the ﬁxed-ended equal leg and unequal leg angles under axial compression. The test and calculated results showed that the specimens with ﬂexural–torsional buckling in the inelastic range were conservative. The experimental results are 10% higher than the theoretical results. Popovic et al. [13] and Young [14] conducted axial compression tests on cold-formed plain angles. Based on the test and speciﬁcation calculations, they suggested ignoring the ﬂexural–torsional buckling but only considering the ﬂexural buckling. The buckling analysis on the cold-formed angles under axial compression conducted by Chodraui et al. Based on the ﬁnite strip method, indicated that the local buckling of elements and global torsional buckling of members was consistent [15]. Landesmann et al. [16], Silvestre et al. [17], and Dinis et al. [18] proposed a new design method for the short to medium-length equal leg angle under axial compression based on the direct strength method through the buckling test and ﬁnite element analysis. The ﬂexural–torsional buckling performance and the load-carrying capacities of the S690 high-strength angle steel column were studied by Zhang et al. based on an experiment and numerical analysis [19]. The predicted results showed that most design codes were too conservative. The modiﬁed design method based on the direct strength method considered the interaction of ﬂexural–torsional buckling about the strong axis and ﬂexural buckling about the weak axis. Zhang et al. [20] and Wang et al. [21] conducted experimental and numerical studies on S690 and S960 high-strength short equal leg angles. The studies showed that Australian and North American speciﬁcations were too conservative. Dinis et al. [22] presented the design method based on the direct strength method for the bearing capacity of short-to-medium length pin-ended hot-rolled steel equal-leg angle columns based on the test results. The experiments about cold-formed lipped angle columns were conducted by Young [23]. The specimens showed the local, ﬂexural, and ﬂexural–torsional buckling and the interactive buckling of these buckling modes. The calculated results indicated that the North American and Australian codes were conservative. Young and Ellobody [24] conducted ﬁnite element analysis on the buckling performance of lipped angles under axial compression. It indicated that the Australian and North American codes were conservative in calculating the ultimate strength of members with a relatively large width-thickness ratio. However, it was not safe to calculate the ulti- mate strength of members with a small width-thickness ratio. Shifferaw et al. [25] conducted theoretical and ﬁnite element research on the global buckling performance ofﬁxed-ended cold-formed thin-walled lipped angle columns. It was shown that the members exhibited signiﬁcant post-buckling strength when they were subjected to global ﬂexural–torsional buckling. Therefore, a direct strength method was proposed to consider the post-buckling strength of cold-formed thin-wall-lipped angle columns. An axial compression test was conducted on 12 cold-formed thin-walled steel columns with unequal leg angle sections by Zhou et al. [26]. The results showed that the direct strength method in the North Ameri- can code is not accurate, and the modiﬁcation for the calculation formula of the ultimate strength based on the direct strength method was given. Ananthi et al. [27] validated the FE model against the experimental test results, which showed good agreement regarding failure loads and deformed shapes at failure. The studies above-mentioned were focus on the equal-leg angle with or without lips. The buckling behavior and design method of the cold-formed thin-walled double- lipped equal-leg angles have not enough investigated. This paper studied the buckling behavior and ultimate strength of 24 cold-formed double-lipped equal-leg angles under axial compression. The analysis model of cold-formed thin-walled double-lipped equal-leg angles under axial compression is developed by using ﬁnite element software. Based on the test results and the predicted results using the direct strength method and the effective Buildings 2022, 12, x FOR PEER REVIEW 3 of 18 Buildings 2022, 12, x FOR PEER REVIEW 3 of 18 Buildings 2022, 12, 1775 3 of 17 under axial under axial c co ompression mpression. The . The an analy alys sis model is model o off col cold d-forme -formed d thi thin n-wa -walle lled do d dou ub ble-l le-lipped ipped equal-leg angles under axial compression is developed by using finite element software. equal-leg angles under axial compression is developed by using finite element software. Base Based d on on the the test test results and the results and the predic predictted ed re resu sults lts us using ing the the d diirec rectt s sttreng rength th me method thod and and width method, a modiﬁed method for calculating the load-carrying capacities of cold- the effective width method, a modified method for calculating the load-carrying capaci- the effective width method, a modified method for calculating the load-carrying capaci- tie tie formed s s of co of cold- thin-walled ld-ffo or rmed thin med thin double-lipped -w -wa alled lled double double equal-leg -l -liipp pped equal-le ed equal-le angles g ang g ang under lles es axial und unde er axial compression r axial compression compression was proposed. was proposed. was proposed. 2. Experimental Program 2 2.. Ex Expe peri rime ment ntal al P Pr ro og gram ram 2.1. Specimen Design 2.1. Specimen 2.1. Specimen Design Design Axial compression test was carried out on 24 cold-formed thin-walled steel double- Axial compression test was carried out on 24 cold-formed thin-walled steel dou- Axial compression test was carried out on 24 cold-formed thin-walled steel dou- lipped equal-leg angles. The cross-section and the geometric parameters of the double- ble-lipped e ble-lipped eq qual-leg ual-leg ang anglle es s. The cross-sectio . The cross-section n and and th the geometr e geometriic p c pa ar ra amete meters of rs of the dou- the dou- lipped equal-leg angle are shown in Figure 1. The nominal section dimensions are shown ble-lipped equal-leg angle are shown in Figure 1. The nominal section dimensions are ble-lipped equal-leg angle are shown in Figure 1. The nominal section dimensions are in Table 1,where a and a are the widths of two legs, respectively, b and b are the widths 1 2 1 2 shown in shown in Tab Tabl le e 1,whe 1,wher re e a a1 1 and a and a2 2 ar are e the the wid widt ths hs of of tw two le o legs, gs, resp respec ectiv tive ely, ly, b b1 1 and and b b2 2 a ar re e t th he e of the ﬁrst lips, respectively, c and c are the widths of the second lips, respectively, and t 1 2 widths of the first lips, respectively, c1 and c2 are the widths of the second lips, respec- widths of the first lips, respectively, c1 and c2 are the widths of the second lips, respec- is the thickness of the section.The length of specimens included 400 mm, 900 mm, 1500 mm, tive tively, ly, and and t is the t is the thickn thickness of the ess of the sec sect tio ion. n.The The leng length of th of specimen specimens included 400 mm, s included 400 mm, 900 900 and 2100 mm.The numbering rules of specimens are shown in Figure 2. For example, mm, 1500 mm, and 2100 mm.The numbering rules of specimens are shown in Figure 2. mm, 1500 mm, and 2100 mm.The numbering rules of specimens are shown in Figure 2. DLA6020-400-1 deﬁnes the specimen as follows: DLA means double-lipped equal-leg For exa For exam mpl ple, e, DLA602 DLA6020- 0-400- 400-1 defi 1 defin nes t es th he speci e specim men as en as f fo oll llows: DLA mea ows: DLA mean ns double- s double-l li ipped pped angle; 60 indicates that the nominal width of the leg is 60 mm; 20 represents that the width equal-leg equal-leg an ang glle; 60 ind e; 60 indiic ca ates tes th that at the n the no ominal minal wi widt dth h of t of th he e lle eg i g is s 6 60 0 mm; 2 mm; 20 0 r re epr pre es se en nts ts of the ﬁrst lip is 20 mm; 400 represents the length of the specimen is 400 mm; 1 means the tha thatt the wid the widtth of the h of the fir first st lip lip i is s 2 20 0 m mm m;; 40 400 0 repre repres sen ents the ts the len leng gth of th of the specim the specimen en is 400 is 400 sequence number of the same specimen. The measured cross-section dimensions and the mm; 1 mea mm; 1 mean ns t s th he seq e sequ uence number of ence number of tth he e sa same me spe spec cimen. The imen. The meas measured ured cro cross-section ss-section dimensions lengths of all and the specimens lengths of all specimens are g are given in Table 2.iven in Table 2. dimensions and the lengths of all specimens are given in Table 2. Figure 1. Figure 1. Figure 1. Sec Sec Section t tio ion of dou n of dou of double-lipped b ble- le-lip lipped equ ped equa aequal-leg l-leg l-leg a an ngle. gle. angle. Figure 2. Labeling rule for steel angle specimen. Figure 2. Figure 2. Label Labeling ing rule for st rule for eesteel l angle angle specim specimen. en. Table 1. Nominal section dimensions of the double-lipped equal-leg angle. Cross-Section a /mm a /mm b /mm b /mm c /mm c /mm t/mm 1 2 1 2 1 2 DLA6020 60 60 20 20 10 10 2 DLA9020 90 60 20 20 10 10 2 DLA12024 120 120 24 24 10 10 2 Buildings 2022, 12, 1775 4 of 17 Table 2. The measured sectional dimensions of the specimens. Specimen a /mm a /mm b /mm b /mm c mm c /mm t/mm L/mm 1 2 1 2 1 2 DLA6020-400-1 61.62 61.54 20.95 20.99 11.63 11.48 1.97 400.00 DLA6020-400-2 61.27 61.59 21.46 21.31 11.35 10.47 1.97 400.00 DLA6020-900-1 61.02 60.71 21.33 20.73 10.32 10.84 1.97 900.00 DLA6020-900-2 61.64 61.52 21.36 21.35 10.84 10.99 1.96 900.00 DLA6020-1500-1 62.20 62.08 21.75 21.90 10.70 10.20 1.97 1500.00 DLA6020-1500-2 61.71 62.00 21.35 21.74 11.28 11.43 1.98 1500.00 DLA6020-2100-1 61.74 61.71 22.37 21.88 10.84 11.22 1.97 2100.00 DLA6020-2100-2 61.44 61.33 21.37 22.29 10.52 11.95 1.96 2101.00 DLA9020-400-1 91.56 91.72 21.77 21.79 10.78 10.17 1.96 400.00 DLA9020-400-2 91.57 91.29 21.88 21.50 11.32 10.27 1.97 400.00 DLA9020-900-1 91.65 91.97 21.34 20.88 11.59 10.10 1.98 900.00 DLA9020-900-2 91.70 91.72 22.15 20.85 10.27 10.62 1.97 900.00 DLA9020-1500-1 92.16 91.42 21.18 21.21 10.90 10.43 1.98 1499.50 DLA9020-1500-2 92.86 92.57 21.70 21.02 11.12 11.09 1.99 1499.50 DLA9020-2100-1 91.77 91.72 21.29 21.97 10.49 10.70 1.96 2101.10 DLA9020-2100-2 92.58 91.84 20.79 21.03 10.82 10.88 2.00 2101.50 DLA12024-400-1 120.15 121.40 25.60 25.72 12.85 13.06 1.98 400.00 DLA12024-400-2 120.61 122.35 25.86 25.29 12.95 13.04 1.98 400.67 DLA12024-900-1 122.29 121.34 24.89 25.08 13.02 12.23 1.98 900.00 DLA12024-900-2 120.64 121.85 25.80 25.40 12.11 13.31 1.98 900.00 DLA12024-1500-1 121.55 121.29 25.16 25.72 12.42 12.46 1.97 1499.90 DLA12024-1500-2 121.44 121.68 25.37 25.69 12.54 12.73 1.97 1499.10 DLA12024-2100-1 122.31 121.93 25.65 25.14 13.92 12.11 1.98 2101.10 DLA12024-2100-2 122.32 120.99 25.47 24.98 12.62 13.40 1.98 2101.10 2.2. Material Properties The zinc-coated steel plate with grade Q550 was used to manufacture the cold-formed steel double-lipped equal-leg angles. Three standard coupon specimens cut at the legs of the specimen were tested to obtain the material properties of the specimens in a 30 kN MTS testing machine based on the Chinese code “Tensile tests of metallic materials Part 1: test methods at room temperature” (GB/T228.1-2010) [28]. The material properties were determined from the stress–strain curves of the coupon specimens. The stress–strain curves of three standard coupon specimens are shown in Figure 3. The average results of the material properties, including the yield strength, the tensile strength, the elastic modulus, Buildings 2022, 12, x FOR PEER REVIEW 5 of 18 and the elongation of the steel obtained from the coupon tests are 403 MPa, 523 MPa, 2.11 10 MPa, and 0.27. Figure 3. Stress–strain curves of tension coupons. Figure 3. Stress–strain curves of tension coupons. 2.3. Initial Imperfection The initial imperfection is produced in the manufacturing, transportation, and fab- rication of cold-formed steel double-lipped equal-leg angles. The initial imperfections greatly influencethe buckling mode and ultimate capacities of double-lipped equal-leg angles. Therefore, all the specimens’ initial geometric imperfections were measured, in- cluding the local, distortion, and global imperfections. The measuring positions are shown in Figures 4 and 5. In Figure 4, the initial global buckling imperfection about the weak axis and the initial distortional buckling imperfection of the specimensare meas- ured at positions 1, 2, 3, 4, and 5, respectively. Positions 6, 7, and 8 in Figure 5 measure the initial local buckling imperfection.The measurements of the initial imperfectionsare shown in Figure 6. The numbers of measurements are 11, 10, 11, and 15 along the longi- tudinal direction for the specimens with lengths of 400 mm, 900 mm, 1500 mm, and 2100 mm, respectively. Three sections at 1/2 span and 1/4 span of the specimens are selected to measure the local initialimperfections along the longitudinal direction, and the distance of the measured positions at each cross-section is 10 mm. Figure 4. The measured position of global and distortional initial imperfections. Buildings 2022, 12, x FOR PEER REVIEW 5 of 18 Buildings 2022, 12, 1775 5 of 17 2.3. Initial Imperfection Figure 3. Stress–strain curves of tension coupons. The initial imperfection is produced in the manufacturing, transportation, and fabrica- 2tion .3. Initial Imperfection of cold-formed steel double-lipped equal-leg angles. The initial imperfections greatly inﬂuencethe buckling mode and ultimate capacities of double-lipped equal-leg angles. The initial imperfection is produced in the manufacturing, transportation, and fab- rication of cold-formed steel double-lipped equal-leg angles. The initial imperfections Therefore, all the specimens’ initial geometric imperfections were measured, including greatly influencethe buckling mode and ultimate capacities of double-lipped equal-leg the local, distortion, and global imperfections. The measuring positions are shown in angles. Therefore, all the specimens’ initial geometric imperfections were measured, in- Figures 4 and 5. In Figure 4, the initial global buckling imperfection about the weak axis cluding the local, distortion, and global imperfections. The measuring positions are and the initial distortional buckling imperfection of the specimensare measured at positions shown in Figures 4 and 5. In Figure 4, the initial global buckling imperfection about the weak axis and the initial distortional buckling imperfection of the specimensare meas- 1, 2, 3, 4, and 5, respectively. Positions 6, 7, and 8 in Figure 5 measure the initial local ured at positions 1, 2, 3, 4, and 5, respectively. Positions 6, 7, and 8 in Figure 5 measure buckling imperfection.The measurements of the initial imperfectionsare shown in Figure 6. the initial local buckling imperfection.The measurements of the initial imperfectionsare The numbers of measurements are 11, 10, 11, and 15 along the longitudinal direction for shown in Figure 6. The numbers of measurements are 11, 10, 11, and 15 along the longi- the specimens with lengths of 400 mm, 900 mm, 1500 mm, and 2100 mm, respectively. tudinal direction for the specimens with lengths of 400 mm, 900 mm, 1500 mm, and 2100 Three sections at 1/2 span and 1/4 span of the specimens are selected to measure the local mm, respectively. Three sections at 1/2 span and 1/4 span of the specimens are selected to measure the local initialimperfections along the longitudinal direction, and the distance initialimperfections along the longitudinal direction, and the distance of the measured of the measured positions at each cross-section is 10 mm. positions at each cross-section is 10 mm. Buildings 2022, 12, x FOR PEER REVIEW 6 of 18 Buildings 2022, 12, x FOR PEER REVIEW 6 of 18 Figure 4. The measured position of global and distortional initial imperfections. Figure 4. The measured position of global and distortional initial imperfections. Figure 5. The measured position of local initial imperfections. Figure 5. The measured position of local initial imperfections. Figure 5. The measured position of local initial imperfections. (a) (b) (c) Figure 6. Initial imperfection measurement of the specimens. (a) Initial imperfection of local buck- ling. (b) Initial imperfection of distortional buckling. (c) Initial imperfection of global buckling about weak axis. The measurement values of initial imperfections for some specimens are shown in Figure 7. It can be seen from Figure 7 that the maximum value of initial distortional- buckling imperfection is greater than the maximum values of initial local buckling im- (a) (b) (c) perfection and initial global buckling imperfection. The distributions of initial geometric imperfections of other specimens are similar, and all maximum values of the initial im- Figure 6. Initial imperfection measurement of the specimens. (a) Initial imperfection of local buck- perfectionsof the specimens are less than L/1000. Figure 6. Initial imperfection measurement of the specimens. (a) Initial imperfection of local buckling. ling. (b) Initial imperfection of distortional buckling. (c) Initial imperfection of global buckling (b) Initial imperfection of distortional buckling. (c) Initial imperfection of global buckling about about weak axis. weak axis. The measurement values of initial imperfections for some specimens are shown in Figure 7. It can be seen from Figure 7 that the maximum value of initial distortional- buckling imperfection is greater than the maximum values of initial local buckling im- perfection and initial global buckling imperfection. The distributions of initial geometric imperfections of other specimens are similar, and all maximum values of the initial im- perfectionsof the specimens are less than L/1000. (a) (b) (c) Figure 7. Initial imperfections of the specimens. (a) Global initial imperfection of DLA6020−400−2. (b) Distortion initial imperfection of DLA6020−900−2. (c) Local initial imperfection of DLA6020−400−1. 2.4. Test Setup and Procedure (a) (b) (c) Figure 7. Initial imperfections of the specimens. (a) Global initial imperfection of DLA6020−400−2. (b) Distortion initial imperfection of DLA6020−900−2. (c) Local initial imperfection of DLA6020−400−1. 2.4. Test Setup and Procedure Buildings 2022, 12, x FOR PEER REVIEW 6 of 18 Figure 5. The measured position of local initial imperfections. (a) (b) (c) Figure 6. Initial imperfection measurement of the specimens. (a) Initial imperfection of local buck- Buildings 2022, 12, 1775 6 of 17 ling. (b) Initial imperfection of distortional buckling. (c) Initial imperfection of global buckling about weak axis. The measurement values of initial imperfections for some specimens are shown in The measurement values of initial imperfections for some specimens are shown in Figure 7. It can be seen from Figure 7 that the maximum value of initial distortional- Figure 7. It can be seen from Figure 7 that the maximum value of initial distortionalbuckling buckling imperfection is greater than the maximum values of initial local buckling im- imperfection is greater than the maximum values of initial local buckling imperfection and perfect initial global ion and in buckling itial glob imperfection. al buckling The imperfec distributions tion. The of distr initial ibutions geometric of init imperfections ial geometric imperfections of other specimens are similar, and all maximum values of the initial im- of other specimens are similar, and all maximum values of the initial imperfectionsof the perfectionso specimens ar f the specimen e less than L/1000. s are less than L/1000. (a) (b) (c) Buildings 2022, 12, x FOR PEER REVIEW 7 of 18 Figure 7. Initial imperfections of the specimens. (a) Global initial imperfection of DLA6020−400−2. Figure 7. Initial imperfections of the specimens. (a) Global initial imperfection of DLA6020-400-2. (b) Distortion initial imperfection of DLA6020−900−2. (c) Local initial imperfection of (b) Distortion initial imperfection of DLA6020-900-2. (c) Local initial imperfection of DLA6020-400-1. DLA6020−400−1. All the double-lipped equal-leg angles were axially compressed using a steel frame 2.4. Test Setup and Procedure system and a 500 kN servo-controlled hydraulic testing machine, as shown in Figure 8. All the double-lipped equal-leg angles were axially compressed using a steel frame 2.4. Test Setup and Procedure The specimens were placed directly in the groove of the top bearing plate connected with system and a 500 kN servo-controlled hydraulic testing machine, as shown in Figure 8. The the actuator and the bottom bearing plate. The geometric center of the specimen coin- specimens were placed directly in the groove of the top bearing plate connected with the actuator and the bottom bearing plate. The geometric center of the specimen coincided cided with the geometric center of the upper loading plate and lower plate. The LVDTs with the geometric center of the upper loading plate and lower plate. The LVDTs (linear (linear variable displacement transducers) were set up at the mid-section of specimens, as variable displacement transducers) were set up at the mid-section of specimens, as shown shown in Figure 9 D1, D2, D3, and D4. The distances of all LVDTs to the edge of the plate in Figure 9 D1, D2, D3, and D4. The distances of all LVDTs to the edge of the plate of of double-lipped equal-leg angle were 10 mm. A displacement transducer was arranged double-lipped equal-leg angle were 10 mm. A displacement transducer was arranged at at the top bearing plate to obtain the vertical displacement of the specimen. The YG16 the top bearing plate to obtain the vertical displacement of the specimen. The YG16 static static strain displacement acquisition system automatically collected the load and dis- strain displacement acquisition system automatically collected the load and displacements of the specimen. placements of the specimen. Figure 8. Test setup. Figure 8. Test setup. Figure 9. Instrumentation arrangement. 3. Test Results 3.1. Failure Modes The buckling modes of all double-lipped equal-leg angles are shown in Table 3, where L, D, and F represent local buckling, distorted buckling, and global flexural buck- ling.It can be seen from Table 3 that the distortional buckling occurred for specimens with a small width-thickness ratio and small slenderness ratio. In contrast, the distor- tional and global flexural buckling occurred for specimens with a small width-to-thickness ratio and large slenderness ratio. For the specimens with a large width-to-thickness ratio and small slenderness ratio, the interactive buckling of local and distortional buckling was discovered. In contrast, the interactive buckling of local, dis- tortional, and global flexural buckling was found for the specimens with a large width ratio and large slenderness ratio. Buildings 2022, 12, x FOR PEER REVIEW 7 of 18 All the double-lipped equal-leg angles were axially compressed using a steel frame system and a 500 kN servo-controlled hydraulic testing machine, as shown in Figure 8. The specimens were placed directly in the groove of the top bearing plate connected with the actuator and the bottom bearing plate. The geometric center of the specimen coin- cided with the geometric center of the upper loading plate and lower plate. The LVDTs (linear variable displacement transducers) were set up at the mid-section of specimens, as shown in Figure 9 D1, D2, D3, and D4. The distances of all LVDTs to the edge of the plate of double-lipped equal-leg angle were 10 mm. A displacement transducer was arranged at the top bearing plate to obtain the vertical displacement of the specimen. The YG16 static strain displacement acquisition system automatically collected the load and dis- placements of the specimen. Buildings 2022, 12, 1775 7 of 17 Figure 8. Test setup. Figure 9. Figure 9. Instr Instr umentation arr umentation a arrangement. ngement. 3. Test Results 3. Test Results 3.1. Failure Modes 3.1. Failure Modes The buckling modes of all double-lipped equal-leg angles are shown in Table 3, where The buckling modes of all double-lipped equal-leg angles are shown in Table 3, where L, D, L, D, and Fand F repr represent esen local t loc buckling, al buckling, d distorted istorted buckling, buckling, and and glob global al flexur ﬂexural al buck buckling.It - ling.It can be seen from Table 3 that the distortional buckling occurred for specimens can be seen from Table 3 that the distortional buckling occurred for specimens with a with a small width-thickness ratio and small slenderness ratio. In contrast, the distor- small width-thickness ratio and small slenderness ratio. In contrast, the distortional and tional and global flexural buckling occurred for specimens with a small global ﬂexural buckling occurred for specimens with a small width-to-thickness ratio and width-to-thickness ratio and large slenderness ratio. For the specimens with a large large slenderness ratio. For the specimens with a large width-to-thickness ratio and small width-to-thickness ratio and small slenderness ratio, the interactive buckling of local and slenderness ratio, the interactive buckling of local and distortional buckling was discovered. distortional buckling was discovered. In contrast, the interactive buckling of local, dis- In contrast, the interactive buckling of local, distortional, and global ﬂexural buckling was tortional, and global flexural buckling was found for the specimens with a large width found for the specimens with a large width ratio and large slenderness ratio. ratio and large slenderness ratio. Table 3. Comparison of buckling modes and ultimate strengths between test and finite element analysis. Experiment Finite Element Analysis Specimens P /P Buckling Buckling m t P /kN P /kN t m Mode Mode DLA6020-400-1 106.00 D 107.62 D 1.02 DLA6020-400-2 113.00 D 110.69 D 0.98 DLA6020-900-1 100.10 D + F 99.74 D + F 1.00 DLA6020-900-2 92.50 D + F 95.36 D + F 1.03 DLA6020-1500-1 73.60 D + F 72.11 D + F 0.98 DLA6020-1500-2 69.10 D + F 70.49 D + F 1.02 DLA6020-2100-1 46.90 D + F 46.56 D + F 0.99 DLA6020-2100-2 44.30 D + F 45.71 D + F 1.03 DLA9020-400-1 139.90 L + D 140.02 L + D 1.00 DLA9020-400-2 139.50 L + D 138.84 L + D 1.00 DLA9020-900-1 119.90 D + F + L 120.49 D + F + L 1.00 DLA9020-900-2 127.60 D + F + L 129.24 D + F + L 1.01 DLA9020-1500-1 62.00 D + F + L 65.50 D + F + L 1.06 DLA9020-1500-2 76.10 D + F + L 75.96 D + F + L 1.00 DLA9020-2100-1 56.20 D + F + L 57.00 D + F + L 1.01 DLA9020-2100-2 48.70 D + F + L 47.96 D + F + L 0.98 DLA 12024-400-1 165.89 D + F + L 164.70 D + F + L 1.01 DLA 12024-400-2 164.69 D + F + L 165.10 D + F + L 1.00 DLA 12024-900-1 143.14 D + F + L 148.10 D + F + L 0.97 DLA 12024-900-2 145.24 D + F + L 147.90 D + F + L 0.98 DLA 12024-1500-1 127.76 D + F + L 130.20 D + F + L 0.98 DLA 12024-1500-2 120.44 D + F + L 124.70 D + F + L 0.97 DLA 12024-2100-1 95.75 D + F + L 101.00 D + F + L 0.95 DLA 12024-2100-2 96.64 D + F + L 99.10 D + F + L 0.98 Buildings 2022, 12, x FOR PEER REVIEW 8 of 18 Table 3. Comparison of buckling modes and ultimate strengths between test and finite element analysis. Experiment Finite Element Analysis Specimens Pm/Pt Pt/kN Buckling Mode Pm/kN Buckling Mode DLA6020-400-1 106.00 D 107.62 D 1.02 DLA6020-400-2 113.00 D 110.69 D 0.98 DLA6020-900-1 100.10 D+F 99.74 D+F 1.00 DLA6020-900-2 92.50 D+F 95.36 D+F 1.03 DLA6020-1500-1 73.60 D+F 72.11 D+F 0.98 DLA6020-1500-2 69.10 D+F 70.49 D+F 1.02 DLA6020-2100-1 46.90 D+F 46.56 D+F 0.99 DLA6020-2100-2 44.30 D+F 45.71 D+F 1.03 DLA9020-400-1 139.90 L+D 140.02 L+D 1.00 DLA9020-400-2 139.50 L+D 138.84 L+D 1.00 DLA9020-900-1 119.90 D+F+L 120.49 D+F+L 1.00 DLA9020-900-2 127.60 D+F+L 129.24 D+F+L 1.01 DLA9020-1500-1 62.00 D+F+L 65.50 D+F+L 1.06 DLA9020-1500-2 76.10 D+F+L 75.96 D+F+L 1.00 DLA9020-2100-1 56.20 D+F+L 57.00 D+F+L 1.01 DLA9020-2100-2 48.70 D+F+L 47.96 D+F+L 0.98 DLA 12024-400-1 165.89 D+F+L 164.70 D+F+L 1.01 DLA 12024-400-2 164.69 D+F+L 165.10 D+F+L 1.00 DLA 12024-900-1 143.14 D+F+L 148.10 D+F+L 0.97 DLA 12024-900-2 145.24 D+F+L 147.90 D+F+L 0.98 DLA 12024-1500-1 127.76 D+F+L 130.20 D+F+L 0.98 DLA 12024-1500-2 120.44 D+F+L 124.70 D+F+L 0.97 Buildings 2022, 12, 1775 8 of 17 DLA 12024-2100-1 95.75 D+F+L 101.00 D+F+L 0.95 DLA 12024-2100-2 96.64 D+F+L 99.10 D+F+L 0.98 3.1.1. The Short Angle Columns 3.1.1. The Short Angle Columns The buckling processes of the short double-lipped equal-leg angles with a length of The buckling processes of the short double-lipped equal-leg angles with a length of 400 mm are shown in Figure 10. The deformation was not evident at the initial loading 400 mm are shown in Figure 10. The deformation was not evident at the initial loading stage. With the load increase, the specimens DLA9020 series with a large stage. With the load increase, the specimens DLA9020 series with a large width-to-thickness width-to-thickness ratio appeared the local buckling (Figure 10a). When the load was ratio appeared the local buckling (Figure 10a). When the load was continued, distortional continued, distortional buckling was observed. The angle deformation between the two buckling was observed. The angle deformation between the two legs became larger legs became larger (Figure 10b). When the ultimate bearing capacity is reached, the (Figure 10b). When the ultimate bearing capacity is reached, the specimen fails. specimen fails. DLA9020-400-1 DLA9020-400-1 DLA6020-400-2 DLA9020-400-2 (a) (b) Figure 10. Buckling mode of short angle column with a length of 400 mm. (a) Local buckling. (b) Figure 10. Buckling mode of short angle column with a length of 400 mm. (a) Local buckling. Buildings 2022, 12, x FOR PEER REVIEW 9 of 18 Distortional buckling. (b) Distortional buckling. 3.1.2. The Medium-to-Long Angle Columns 3.1.2. The Medium-to-Long Angle Columns The buckling process of medium-to-long double-lipped equal-leg angles is shown in The buckling process of medium-to-long double-lipped equal-leg angles is shown in Figures 11–13. At the initial loading stage, the deformation was not apparent. With the Figures 11–13. At the initial loading stage, the deformation was not apparent. With the load increase, the legs of specimen DLA9020 series with a large width-to-thickness ratio load increase, the legs of specimen DLA9020 series with a large width-to-thickness ratio appeared the local buckling (Figures 11a, 12a and 13a). When the loading was continued, appeared the local buckling (Figures 11a, 12a, and 13a). When the loading was continued, the specimens appeared distortional buckling (Figures 11b, 12b and 13b) for the specimen the specimens appeared distortional buckling (Figures 11b, 12b, and 13b) for the speci- DLA9020 series and specimen DLA6020 series. When the load reached the ultimate bearing men DLA9020 series and specimen DLA6020 series. When the load reached the ultimate capacity, the specimen DLA9020 series and specimen DLA6020 series failed with global bearing capacity, the specimen DLA9020 series and specimen DLA6020 series failed with ﬂexural buckling. Thus, the interaction of distortional buckling and global ﬂexural buckling global flexural buckling. Thus, the interaction of distortional buckling and global flexural occurred for specimen DLA6020 series, while the specimen DLA9020 series showed the buckling occurred for specimen DLA6020 series, while the specimen DLA9020 series interaction of local, distortional, and global ﬂexural buckling. showed the interaction of local, distortional, and global flexural buckling. DLA9020-900-1 DLA9020-900-2 DLA6020-900-2 DLA6020-900-1 DLA9020-900-2 DLA6020-900-1 (a) (b) (c) Figure 11. Buckling mode of angle column with a length of 900 mm. (a) Local buckling. (b) Distor- Figure 11. Buckling mode of angle column with a length of 900 mm. (a) Local buckling. (b) Distor- tional buckling. (c) Global buckling. tional buckling. (c) Global buckling. DLA9020-1500-1 DLA9020-1500-2 DLA9020-1500-1 DLA6020-1500-1 DLA6020-1500-1 DLA9020-1500-2 (a) (b) (c) Figure 12. Buckling mode of angle column with a length of 1500 mm. (a) Local buckling. (b) Dis- tortional buckling. (c) Global buckling. Buildings 2022, 12, x FOR PEER REVIEW 9 of 18 3.1.2. The Medium-to-Long Angle Columns The buckling process of medium-to-long double-lipped equal-leg angles is shown in Figures 11–13. At the initial loading stage, the deformation was not apparent. With the load increase, the legs of specimen DLA9020 series with a large width-to-thickness ratio appeared the local buckling (Figures 11a, 12a, and 13a). When the loading was continued, the specimens appeared distortional buckling (Figures 11b, 12b, and 13b) for the speci- men DLA9020 series and specimen DLA6020 series. When the load reached the ultimate bearing capacity, the specimen DLA9020 series and specimen DLA6020 series failed with global flexural buckling. Thus, the interaction of distortional buckling and global flexural buckling occurred for specimen DLA6020 series, while the specimen DLA9020 series showed the interaction of local, distortional, and global flexural buckling. DLA9020-900-1 DLA9020-900-2 DLA6020-900-2 DLA6020-900-1 DLA9020-900-2 DLA6020-900-1 (a) (b) (c) Buildings 2022, 12, 1775 9 of 17 Figure 11. Buckling mode of angle column with a length of 900 mm. (a) Local buckling. (b) Distor- tional buckling. (c) Global buckling. DLA9020-1500-1 DLA9020-1500-2 DLA9020-1500-1 DLA6020-1500-1 DLA6020-1500-1 DLA9020-1500-2 (a) (b) (c) Buildings 2022, 12, x FOR PEER REVIEW Figure 12. Buckling mode of angle column with a length of 1500 mm. (a) Local buckling. ( 10 of b) Dis- 18 Figure 12. Buckling mode of angle column with a length of 1500 mm. (a) Local buckling. (b) Distor- tortional buckling. (c) Global buckling. tional buckling. (c) Global buckling. DLA9020-2100-1 DLA9020-2100-2 DLA9020-2100-2 DLA6020-2100-1 DLA6020-2100-1 DLA9020-2100-2 (a) (b) (c) Figure 13. Buckling mode of angle column with a length of 2100 mm. (a) Local buckling. (b) Dis- Figure 13. Buckling mode of angle column with a length of 2100 mm. (a) Local buckling. (b) Distor- tortional buckling. (c) Global buckling. tional buckling. (c) Global buckling. 3.2. Test Strengths and Curves 3.2. Test Strengths and Curves The The u ultimate ltimate capacities capacitiesof of all aldouble-lip l double-lip equal-leg equal-leangles g angles under under axial axcompr ial compre ession ssion are shown are shown in Table 3, in Table 3, wher wher e P e is Ptthe is the t testeload-carrying st load-carrying capacity capaci.ty It. I can t cabe n be seen seen from from T Table able 3 that 3 thathe t thaxial e axia load-carrying l load-carryincapabilities g capabilitie of s of thethe do double-lip uble-lequal-leg ip equal-le angles g angdecr les decr ease ease w with the ith incr the incre ease in ase length. in length. The load-displacement curves of the specimens DLA6020 series are shown in The load-displacement curves of the specimens DLA6020 series are shown in Figure Figure 14a–d. It can be seen from Figure 14 that for specimens DLA6020-400 (Figure 14a) 14a–d. It can be seen from Figure 14 that for specimens DLA6020-400 (Figure 14a) and and DLA6020-900 (Figure 14b), the stiffnesses were unchanged at the initial loading stage, DLA6020-900 (Figure 14b), the stiffnesses were unchanged at the initial loading stage, and the curves showed linear growth. A nonlinear segment appeared with the increase and the curves showed linear growth. A nonlinear segment appeared with the increase in in load, and the load decreased slowly after reaching the maximum load. For specimens load, and the load decreased slowly after reaching the maximum load. For specimens DLA6020-1500 (Figure 14c) and DLA6020-2100 (Figure 14d), the curves increased linearly DLA6020-1500 (Figure 14c) and DLA6020-2100 (Figure 14d), the curves increased linearly before the maximum load. The curves entered the nonlinear stage with the occurrence of before the maximum load. The curves entered the nonlinear stage with the occurrence of ﬂexural buckling approaching the maximum load. Then, the load dropped sharply after flexural buckling approaching the maximum load. Then, the load dropped sharply after reaching the ultimate load, and the specimen failed. reaching the ultimate load, and the specimen failed. The comparison on the average load displacement curves for the same sections with The comparison on the average load displacement curves for the same sections with different length are depicted in Figure 14e–g for section DLA6020, DLA9020, and DLA12024. different length are depicted in Figure 14e–g for section DLA6020, DLA9020, and DLA12024. It can found that the stiffness and ultimate strength of cold-formed dou- ble-lips equal-leg angles decrease with the increasing of length of the axial members. (a) (b) Buildings 2022, 12, x FOR PEER REVIEW 10 of 18 DLA9020-2100-1 DLA9020-2100-2 DLA9020-2100-2 DLA6020-2100-1 DLA6020-2100-1 DLA9020-2100-2 (a) (b) (c) Figure 13. Buckling mode of angle column with a length of 2100 mm. (a) Local buckling. (b) Dis- tortional buckling. (c) Global buckling. 3.2. Test Strengths and Curves The ultimate capacities of all double-lip equal-leg angles under axial compression are shown in Table 3, where Pt is the test load-carrying capacity. It can be seen from Table 3 that the axial load-carrying capabilities of the double-lip equal-leg angles decrease with the increase in length. The load-displacement curves of the specimens DLA6020 series are shown in Figure 14a–d. It can be seen from Figure 14 that for specimens DLA6020-400 (Figure 14a) and DLA6020-900 (Figure 14b), the stiffnesses were unchanged at the initial loading stage, and the curves showed linear growth. A nonlinear segment appeared with the increase in load, and the load decreased slowly after reaching the maximum load. For specimens DLA6020-1500 (Figure 14c) and DLA6020-2100 (Figure 14d), the curves increased linearly before the maximum load. The curves entered the nonlinear stage with the occurrence of Buildings 2022, 12, 1775 10 of 17 flexural buckling approaching the maximum load. Then, the load dropped sharply after reaching the ultimate load, and the specimen failed. The comparison on the average load displacement curves for the same sections with different length are depicted in Figure 14e–g for section DLA6020, DLA9020, and It can found that the stiffness and ultimate strength of cold-formed double-lips equal-leg DLA12024. It can found that the stiffness and ultimate strength of cold-formed dou- angles decrease with the increasing of length of the axial members. ble-lips equal-leg angles decrease with the increasing of length of the axial members. Buildings 2022, 12, x FOR PEER REVIEW 11 of 18 (a) (b) (c) (d) (e) (f) (g) Figure 14. Load-displacement curves. (a) DLA6020-400. (b) DLA6020-900. (c) DLA6020-1500. (d) Figure 14. Load-displacement curves. (a) DLA6020-400. (b) DLA6020-900. (c) DLA6020-1500. DLA6020-2100. (e) DLA6020 average load displacement curve. (f) DLA9020 average load dis- (d) DLA6020-2100. (e) DLA6020 average load displacement curve. (f) DLA9020 average load placement curve. (g) DLA12024 average load displacement curve. displacement curve. (g) DLA12024 average load displacement curve. 4. Finite Element Analysis 4.1. Development of the Finite Element Model The finite element analysis model of the cold-formed thin-walled steel dou- ble-lipped equal-leg angle was established using the finite element software ABAQUS6.14 [29]. In FEA, the measured specimens’ dimensions and the maximum Buildings 2022, 12, 1775 11 of 17 Buildings 2022, 12, x FOR PEER REVIEW 12 of 18 4. Finite Element Analysis 4.1. Development of the Finite Element Model The ﬁnite element analysis model of the cold-formed thin-walled steel double-lipped geometric imperfections of the specimens were all included in the model, but the residual equal-leg angle was established using the ﬁnite element software ABAQUS6.14 [29]. In stress of the whole section and the increase of yield strength (at the corner regions only) FEA, the measured specimens’ dimensions and the maximum geometric imperfections by the cold-forming process were not considered [30]. The length and cross-section size of the specimens were all included in the model, but the residual stress of the whole of specimens were the measured size. The S4R shell element and the ideal elastoplastic section and the increase of yield strength (at the corner regions only) by the cold-forming model were adopted, and the average value of the material property test was adopted. process were not considered [30]. The length and cross-section size of specimens were the Through a certain number of trials, it was found that the error of ultimate strength was measured size. The S4R shell element and the ideal elastoplastic model were adopted, and less the than average 2% when the value of the mesh material was 5 pr mm operty × 5 mm or 10 mm × test was adopted. 10 m Thr mough . So, 10 a mm certain × 10 number mm was of trials, selected it was as the found mesh that size the . The error spec of imens ultimate were strength fixed was at both less ethan nds, 5 degr 2% when ees of free- the mesh dom (two was 5 mm tra nsl5 at mm ionaor l an 10 d three mm rot 10 ati mm. onal) we So, 10 re co mm nstr ain 10 ed mm at the lo was selected ading enas d, an the d th mesh e UZ longitudinal de size. The specimens gree o werf e freedom w ﬁxed at both as rele ends, ased. It 5 degrees was ofutterly fixed at the o freedom (two translational ther end. and The vert three rotational) ical displawer cemen e constrained t was applie at d the at th loading e coupli end, ng poin and t the RPUZ -2 olongitudinal f the centroid o degr f the ee of freedom was released. It was utterly ﬁxed at the other end. The vertical displacement double-lipped equal-leg angle section at the loading end. In order to simulate the speci- was applied at the coupling point RP-2 of the centroid of the double-lipped equal-leg mens more precisely, the measured initial geometric imperfections were introduced. The angle section at the loading end. In order to simulate the specimens more precisely, the maximum value of global, distortional, and local initial geometric imperfections was measured initial geometric imperfections were introduced. The maximum value of global, taken as the imperfect value. The finite element model is shown in Figure 15. The finite distortional, and local initial geometric imperfections was taken as the imperfect value. element analysis included two steps: the first step was the eigenvalue buckling analysis, The ﬁnite element model is shown in Figure 15. The ﬁnite element analysis included two and the first buckling mode was used as the initial imperfection shape of the speci- steps: the ﬁrst step was the eigenvalue buckling analysis, and the ﬁrst buckling mode was mens.The second step was nonlinear analysis.The Von-Misses stress–strain criterion and used as the initial imperfection shape of the specimens.The second step was nonlinear arc length method were adopted to obtain the buckling modes and ultimate strengths of analysis.The Von-Misses stress–strain criterion and arc length method were adopted to all specimens. obtain the buckling modes and ultimate strengths of all specimens. Figure 15. Figure 15. Finit Finite e element mod element model. el. 4.2. Validation of Finite Element Model 4.2. Validation of Finite Element Model The ﬁnite element analysis results for all specimens are shown in Table 3, where P The finite element analysis results for all specimens are shown in Table 3, where Pm is the ﬁnite element analysis result. As shown in Table 3, the average value of the ratios is the finite element analysis result. As shown in Table 3, the average value of the ratios between the test results and the ﬁnite element analysis results is 1.01, and the coefﬁcient between the test results and the finite element analysis results is 1.01, and the coefficient of variation is 0.08. The comparison buckling modes between the ﬁnite element analysis of variation is 0.08. The comparison buckling modes between the finite element analysis and the test are shown in Figure 16. The buckling modes of the ﬁnite element analysis and the test are shown in Figure 16. The buckling modes of the finite element analysis are are consistent with the test, as shown in Figure 16. The comparisons of load-displacement consistent with the test, as shown in Figure 16. The comparisons of load-displacement curves between tests and ﬁnite element analysis are shown in Figure 17, which shows that curves between tests and finite element analysis are shown in Figure 17, which shows the test and ﬁnite element analysis curves are in good agreement. These comparison results that the test and finite element analysis curves are in good agreement. These comparison show that this paper ’s ﬁnite element analysis model can reasonably simulate the buckling results show that this paper’s finite element analysis model can reasonably simulate the mode, ultimate strength, and load-displacement curve of the cold-formed thin-walled steel buckling mode, ultimate strength, and load-displacement curve of the cold-formed double-lipped equal-leg angle under axial compression. thin-walled steel double-lipped equal-leg angle under axial compression. Buildings Buildings 2022 2022,, 12 12,, x FO 1775 R PEER REVIEW 13 of 12 of 18 17 Buildings 2022, 12, x FOR PEER REVIEW 13 of 18 (a) (b) (c) (d) (a) (b) (c) (d) Figure 16. Comparison of buckling modes between the experiments and finite element analysis. (a) Figure 16. Comparison of buckling modes between the experiments and finite element analysis. (a) Figure 16. Comparison of buckling modes between the experiments and ﬁnite element analysis. DLA6020-400- DLA6020-400- 2. ( 2. ( bb ) DLA9020 ) DLA9020 -900-2. ( -900-2. ( cc ) DLA9020-1500-2. ( ) DLA9020-1500-2. ( dd ) sDLA9020-21 ) sDLA9020-21 00-2. 00-2. (a) DLA6020-400-2. (b) DLA9020-900-2. (c) DLA9020-1500-2. (d) sDLA9020-2100-2. (a) (b) (a) (b) (c) (d) (c) (d) Figure 17. Comparison of load-displacement curves between experiments and finite element Figure 17. Comparison of load-displacement curves between experiments and finite element Figure 17. Comparison of load-displacement curves between experiments and ﬁnite element analysis. analysis. ( analysis. ( aa ) D ) D LL A9020-400-2. A9020-400-2. (b (b ) DLA9020-9 ) DLA9020-9 00-3. ( 00-3. ( cc ) DLA6 ) DLA6 020-1500-2. ( 020-1500-2. ( dd ) DLA6020-2100 ) DLA6020-2100 -2. -2. (a) DLA9020-400-2. (b) DLA9020-900-3. (c) DLA6020-1500-2. (d) DLA6020-2100-2. Buildings 2022, 12, 1775 13 of 17 5. Assessment and Suggested of the Design Method 5.1. Direct Strength Method The direct strength method in North American speciﬁcation [31] was used to calculate the ultimate strength of cold-formed steel members. The nominal axial strength of cold- formed steel double-lipped equal-leg angle section is the minimum value of the axial strength of local buckling interacting with global buckling P and the axial strength of nl distortional buckling P . nd The axial strength of local buckling interacting with global buckling P is calculated nl according to Equation (1): < P l 0.776 ne 0.4 0.4 P = (1) P P nl crl crl : 1 0.15 P l > 0.776 ne P P ne ne where l = P /P . P is the axial strength of global buckling. P is the critical elastic ne ne ` crl crl local buckling strength. The elastic local buckling stress can be calculated through ﬁnite strip software CUFSM [32]. The axial strength of global buckling P can be calculated by Equation (2): ne P = A F (2) ne g n where A is the gross area of cross-section and F is the global buckling stress, which can g n be calculated according to Equation (3): 0.658 F l 1.5 y c F = (3) 0.877 F l > 1.5 y c where l = is the smallest value of ﬂexural, torsional, and ﬂexural–torsional buckling cre stresses. F can be determined by Equation (4). cre 2 2 x y 0 0 2 2 (F s ) F s (F s ) F F s F (F s ) = 0 (4) cre ex cre ey cre t cre ey cre ex cre cre r r 0 0 In which p E s = (4a) ex (K L /r ) x x x p E s = (4b) ey K L /r y y y " # 1 p EC s = G J + (4c) 2 2 Ar (K L ) t t where s , s , s are the elastic buckling stresses for ﬂexural buckling about the principal ex ey t x-axis, y-axis, and torsional buckling, respectively. x , y are the distances from the shear 0 0 centre to the centroid along the x-axis and y-axis. r is the polar radius of gyration. K , K , x y K are the effective length factor for bending about x-axis in accordance, bending about y-axis in accordance, and twisting determined in accordance. L , L , L are unbraced x y t lengths of members for bending about x-axis, y-axis, and torsion, respectively. r , r are the x y radius of gyration of full unreduced cross-section about the x-axis and y-axis. J, G, E, C are St. Venant torsion constant of cross-section, shear modulus of steel, modulus of elasticity of steel, and torsional warping constant of cross-section, respectively. Buildings 2022, 12, 1775 14 of 17 The axial distorted buckling strength can be determined according to Equation (5). P l 0.561 0.6 0.6 P = (5) P P nd crd crd : 1 0.25 P l > 0.561 P P y y where l = P /P , P = A F , F is the yield stress and P is the critical elastic d y crd y g y y crd distortional buckling strength. The elastic distortional buckling stress can be calculated through ﬁnite strip software CUFSM. 5.2. Effect Width Method The effective width method in Technical Code for Cold-Formed Thin-walled Steel Structures [33] predicts the ultimate strength of cold-formed steel members.The axial strength of cold-formed steel double-lipped equal-leg angle section can be determined according to Equation (6): N = j A f (6) e y where j is the global stability coefﬁcient of axial double-lipped equal-leg angle, which can be determined according to the minimum value of the slenderness ratio l of ﬂexural bucking and the slenderness ratio l of ﬂexural–torsional buckling. A is the effective w e cross-sectional area, A = b t, b is the effective width of the elements of double-lipped e e e equal-leg angle and can be calculated using the Formula (7). c b > 18ar > t t 21.8ar b b b e c 0.1 18ar < < 38ar = t t (7) > 25ar c b 38ar t t 235k For axial compression double-lipped equal-leg angle, b = b, a = 1, r = , k is the j f buckling coefﬁcient of the element of angle. 5.3. Recommendations for the Design of Double-Lipped Equal-Leg Angle The ultimate strength predicted using the direct strength method and effective width method for double-lipped equal-leg angle are shown in Table 4, where P and P are the z y calculated strength using the direct strength method and effective method, respectively. As shown in Table 4, the average ratios of the calculated capacities to test results P /P and z t P /P are 0.641 and 0.494, with a coefﬁcient of variation of 0.276 and 0.397. The comparison y t of ultimate strength between tests and the predicted results shows that the results calculated by the direct strength and effective width methods are conservative. The main reason is that the torsion of the leg with the lip is considered torsional buckling of the angle and distortional buckling of the leg. The torsion is considered repeatedly. Therefore, it is suggested to ignore the effect of torsion and only calculate the ﬂexural buckling when calculating the global buckling of the double-lipped equal-leg angle. For Formula (1) in the direct strength method, F is obtained as the minimum value of the cre ﬂexural buckling of the double-lipped equal-leg angle about the x-axis and y-axis. For Formula (6) in the effective width method, the slenderness ratio of the global buckling is the minimum value of the slenderness ratio of the ﬂexural buckling for the double-lipped equal-leg angle about the x-axis and y-axis. The predicted ultimate strength using the proposed direct strength method and effec- tive width method are shown in Table 4. P and P are calculated using the suggested za ya direct strength and effective width methods. As shown in Table 4, the average ratios of the calculated capacities to test results P /P and P /P is 1.075 and 0.953, with the za t ya t coefﬁcient of variation of 0.052 and 0.124. The ultimate strength calculated by the modiﬁed direct strength and effective width method agrees with the test results. Therefore, the Buildings 2022, 12, 1775 15 of 17 modiﬁed direct strength method and effective width method are accurate and feasible for calculating the ultimate strength of cold-formed steel double-lipped equal-leg angle under axial compression. Table 4. Comparison of ultimate strength between tests and the predicted results by using DSM, EWM, modiﬁed DSM, and modiﬁed EWM. Test DSM MDSM EWM MEWM Specimens P /P P /P P /P P /P y t ya t z t za t P /kN P /kN P /kN P /kN P /kN t z za y ya DLA6020-400-1 106 110.24 108.24 83.87 99.42 1.04 1.02 0.79 0.94 DLA6020-400-2 113 113.45 112.4 83.55 105.06 1 0.99 0.74 0.93 DLA6020-900-1 100.1 73.07 105.43 42.35 88.51 0.73 1.05 0.42 0.88 DLA6020-900-2 92.5 75.2 101.33 44.03 83.21 0.81 1.1 0.48 0.9 DLA6020-1500-1 73.6 35.41 80.36 19.87 63.73 0.48 1.09 0.27 0.87 DLA6020-1500-2 69.1 35.75 75.39 20.35 63.87 0.52 1.09 0.29 0.92 DLA6020-2100-1 46.9 22.49 51.36 12.51 35.31 0.48 1.1 0.27 0.75 DLA6020-2100-2 44.3 22.25 50.56 12.31 37.73 0.5 1.14 0.28 0.85 DLA9020-400-1 139.9 100.25 140.99 91.57 127.97 0.72 1.01 0.65 0.91 DLA9020-400-2 139.5 100.83 141.44 91.19 122.94 0.72 1.01 0.65 0.88 DLA9020-900-1 119.9 68.76 123.5 50.97 113.45 0.57 1.03 0.43 0.95 DLA9020-900-2 127.6 68.68 130.99 52.03 120.07 0.54 1.03 0.41 0.94 DLA9020-1500-1 62 40.14 74.66 23.71 76.89 0.65 1.2 0.38 1.24 DLA9020-1500-2 76.1 38.23 82.05 22.09 77.08 0.5 1.08 0.29 1.01 DLA9020-2100-1 48.7 24.08 54.47 37.13 55.44 0.49 1.12 0.76 1.14 DLA9020-2100-2 49.3 24.69 56.06 38.61 55.46 0.5 1.14 0.78 1.12 DLA12024-400-1 165.89 176.27 185.32 176.27 162.59 1.06 1.12 1.06 0.98 DLA12024-400-2 164.69 175.97 185.09 184.56 162.81 1.07 1.12 1.12 0.99 DLA12024-900-1 143.14 114.77 153.67 151.27 146.35 0.80 1.07 1.06 1.02 DLA12024-900-2 145.24 119.15 156.07 153.67 148.54 0.82 1.07 1.06 1.02 DLA12024-1500-1 127.76 47.59 129.36 100.76 119.81 0.37 1.01 0.79 0.94 DLA12024-1500-2 120.44 47.78 129.85 101.36 120.60 0.40 1.08 0.84 1.00 DLA12024-2100-1 95.75 26.99 98.17 57.24 101.33 0.28 1.02 0.60 1.06 DLA12024-2100-2 96.64 26.66 98.34 56.48 100.69 0.28 1.03 0.58 1.04 Mean value 0.639 1.075 0.625 0.904 Variance 0.234 0.193 0.272 0.160 Coefﬁcient of variation 0.366 0.179 0.435 0.177 6. Conclusions (1) The axial compression test of 24 cold-formed thin-walled double-lipped equal-leg angles showed that the distortional buckling occurred for specimens with a small width-to-thickness ratio and small slenderness ratio. The buckling interactive with distortional and global ﬂexural buckling was observed for the specimens with small width-to-thickness ratios and large slenderness ratios. The specimens with large width-to-thickness ratios and small slenderness ratios showed interactive buckling with local and distortion buckling, while the specimens with large width-to-thickness ratios and large slenderness ratios developed interactive buckling with local, distor- tional, and global ﬂexural buckling. The ultimate strengths of specimens decreased with the increase of the length of the double-lipped equal-leg angle. (2) The ultimate strengths, buckling modes, and axial compression displacement curves of the specimens analyzed by the ﬁnite element method were in good agreement with the test results. The results showed that the developed ﬁnite element model was feasible for the buckling analysis of cold-formed thin-walled steel double-lipped equal-leg angle. (3) The distortional buckling of the leg with lip and the global torsional buckling angle for cold-formed thin-walled steel double-lipped equal-leg angle is consistent. The axial strength of the double-lipped equal-leg angle calculated by the direct strength and effective width methods indicated that the design methods were too conservative. Buildings 2022, 12, 1775 16 of 17 Therefore, the suggested approaches were proposed by ignoring the global torsional buckling. The results obtained by the proposed direct strength method and effective width method were accurate, indicating that the proposed method can be used to determine the ultimate strength of the cold-formed thin-walled steel double-lipped equal-leg angle. (4) Further numerical and experimental studies are needed before the modiﬁed design method can be used in the codes. Meanwhile, the cold-formed thin-walled steel lipped equal-leg angle, unequal-leg angle, and lipped unequal-leg angle should be studied by experiment and numerical analysis. Author Contributions: Conceptualization, X.Y. and Y.G.; methodology, X.Y.; validation, S.Z.; inves- tigation, Y.G.; data curation, S.Z.; writing—original draft preparation, Y.L.; writing—review and editing, C.H.; funding acquisition, X.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Natural Science Foundation of China grant number [51868049]. Data Availability Statement: All the data included in this study are available upon request by contacting the corresponding author. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Yao, X.; Guo, Y.; Li, Y. Effective width method for distortional buckling design of cold-formed lipped channel sections. Thin-Walled Struct. 2016, 109, 344–351. 2. Chen, B.; Roy, K.; Uzzaman, A.; Raftery, G.; Nash, D.; Clifton, C.; Pouladi, P.; Lim, B.P.J. Effects of edge-stiffened web openings on the behaviour of cold-formed steel channel sections under compression. Thin-Walled Struct. 2019, 144, 106307. [CrossRef] 3. Chen, B.; Roy, K.; Uzzaman, A.; Lim, B.P. Moment capacity of cold-formed channel beams with edge-stiffened web holes, un-stiffened web holes and plain webs. Thin-Walled Struct. 2020, 157, 107070. [CrossRef] 4. Chen, B.; Roy, K.; Uzzaman, A.; Raftery, G.; Nash, D.; Lim, B.P.J. 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Buildings – Multidisciplinary Digital Publishing Institute

Published: Oct 23, 2022

Keywords: double-lipped equal-leg steel angle; axial compression; distortional buckling; global buckling; effective width method; direct strength method

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Experiment and Design Method of Cold-Formed Thin-Walled Steel Double-Lipped Equal-Leg Angle under Axial Compression

Experiment and Design Method of Cold-Formed Thin-Walled Steel Double-Lipped Equal-Leg Angle under Axial Compression

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Experiment and Design Method of Cold-Formed Thin-Walled Steel Double-Lipped Equal-Leg Angle under Axial Compression

Experiment and Design Method of Cold-Formed Thin-Walled Steel Double-Lipped Equal-Leg Angle under Axial Compression

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