# Experimental Investigation and Load Capacity of Slender Cold-Formed Lipped Channel Sections with Holes in Compression

Authors: Xingyou Yao
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

The use of cold-formed steel (CFS) channels with circular or rectangular web holes is becoming increasingly popular in building structures. However, such holes can result in sections becoming more susceptible to buckle and display lower load-carrying capacities. This paper presents a total of 42 axial compression tests of CFS lipped channel slender columns with and without circular and rectangular web holes, including different hole sizes and cross sections. The test results show that the axial members with a small ratio of width to thickness were governed by global buckling, while the members with a large ratio of width to thickness were controlled by the interaction of local, distortional, and global buckling. The axial strength decreased maximum by 20.48% and 22.98% for the member with circular holes and rectangular holes, compared to a member without a web hole. Then, a nonlinear elastoplastic finite element model (FEM) was developed, and the analysis results showed good agreement with the test results. The validated FE model was used to conduct a parametric study involving 36 FEMs to investigate the effects of column slenderness, dimension of the hole, and the number of holes on the axial strength of such channels. Furthermore, the formulas to predict the global buckling coefficient and the effective area were modified for such sections with holes by using the verified FEM. Finally, the tests and parametric study results were compared against the design strengths calculated in accordance with the developed method. The comparison results show that the proposed design method closely predicts the axial capacity of CFS channels with circular or rectangular web holes.


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## Body

## 1. Introduction

The CFS lipped channel sections have been widely used in buildings as walls, floors, and ceilings due to their high-strength-to-weight ratio and ease of construction. However, the holes often appear in the web because of the installation of electrical, plumbing, and heating conduits for such lipped channel sections. The occurrence of holes in the web would decrease the cross-section area and stiffness.In the literature, some work has been reported on the buckling behavior and the reduction in compression resistance of channel sections having web openings. Stub column tests conducted by Colberg [1] indicated that the load-carrying capacity decreased with the increasing of the ratio of circular hole diameter-to-depth of the web. The same conclusions have also been reported for stub columns with circular, rectangular, and slotted web holes [2, 3]. The compression tests of 24 short and intermediate CFS lipped channel columns with and without slotted web holes showed that the presence of slotted holes caused a slight decrease in the ultimate capacity [4]. The axially compressed tests of intermediate length CFS lipped channel columns with circular holes indicated that the stress of the plate adjacent to the circular hole was higher than that of other parts and the plate adjacent to the circle hole yielded early [5]. For lipped channel columns with web stiffener, numerical and experimental studies were carried out to analyze the effects of holes on the failure mode and load-carrying capacity by Yao et al. [5]. The results demonstrated that the holes led to the change of bucking mode and decreased the ultimate strength. The modified direct strength method was developed based on the test results by Moen and Schafer [6, 7]. The effects of holes on the buckling behavior of CFS channels under axial compression were studied by Kulatunga et al. [8, 9], and it was found that the ultimate failure load of the channels under compression varied greatly with the presence of holes.In the literature, some other significant work has been reported on the reduction in strength of channel sections having openings covering shear by Pham [10], Pham et al. [11], and Keerthan and Mahendran [12, 13], and the bending of the beam by Moen et al. [14] and Zhao et al. [15, 16].Recently, Chen et al. [17–19] reported experimental and numerical studies on the axial strength of single channels and back-to-back channels with edge-stiffened holes and found that the axial strength of channels with edge-stiffened holes was greater than those of plain channels. Meanwhile, some other work has been reported on the strength of channel sections having edge-stiffened holes covering web crippling by Uzzaman et al. [20, 21].In the literature, limited work has been reported on CFS slender channels with holes in the web. A total of 21 channel specimens test, including C-shaped cross sections with and without web openings conducted by Dung et al. [22]. The flexural-torsional buckling was observed for all slender compressive specimens and the average strength of the specimen was slightly reduced because of the web opening. In order to develop the effective width method to predict the load capacity of a channel with holes, an approximate method to predict the elastic buckling stress of plates with holes was developed [23]. Meanwhile, the simplified methods to predict the local, distortional, and global critical elastic buckling loads of CFS columns with holes were developed based on theoretical and FE analysis for developing the direct strength method to calculate the strength of CFS columns with holes [24–26].In terms of the design standards, the American Iron and Steel Institute (AISI) [27] and the Australian and New Zealand Standards (AS/NZS) [28] do not provide sufficient effective width design rules for determining the axial capacity of CFS channels with holes because the reduced effective width design method is only suitable to members with relatively small holes in these codes. The direct strength method is given in AISI for determining the axial capacity of CFS channels with holes, but the elastic buckling capacity of members with holes needs to be calculated by using the finite element method or complicated formula. Meanwhile, there is no design provision about the perforated plates and members in the Chinese code [29].As mentioned previously in the literature, most of the researches considered the buckling behavior and design method of the stud and intermediate length CFS channels with small holes. Nevertheless, limited work has been reported on slender columns with large holes under axial compression. Thus, this paper reports 42 new experiments on the axial strength of CFS slender lipped channels with circular and rectangular holes in the web. Tensile coupon tests were also conducted to determine the material properties of the channels. A nonlinear elastoplastic FE model was then developed and validated against the test results in terms of ultimate strength and buckling modes. Using the validated FE model, a parametric study involving 36 models was conducted to investigate the effects of column slenderness, hole size, and the number of holes on the axial strength of such columns. The proven FEM was used to develop the global buckling coefficient equation and effective area formula of the CFS lipped channel section under axial compression, considering the effect of holes. Finally, the prediction method of ultimate axial strength of CFS lipped channels with holes in the web was proposed and the design strengths were compared against both the test and FEA results.


## 2. Experimental Investigation

### 2.1. Test Specimens

This test employed 42 CFS lipped channel axially compressed members. There were 6 nonperforated members, 18 members with circular holes, and 18 members with rectangular holes. The definitions of geometric parameters of the specimen and the location, spacing, and types of holes are illustrated in Figure1. Four cross sections 75 × 40 × 10 × 0.8 mm, 75 × 40 × 10 × 1 mm, 90 × 50 × 15 × 0.8 mm, and 80 × 60 × 10 × 0.8 mm (h × b × a × t, h: web depth, b: flange width, a: lip length, t: thickness) were selected. The nominal thicknesses of the specimens are 0.8 mm and 1.0 mm. The diameters of circular holes are 0.3, 0.5, and 0.7 to the web depth. The heights of the rectangular holes are 0.2, 0.4, and 0.6 to the web depth. The ratio of length-to-width (Lh/Hh) of the rectangular hole is equal to 2. Each specimen is 2000 mm and has three holes. The hole spacings are 500 mm and 600 mm for members with circular and rectangular holes, respectively. The labeling rule of all specimens is defined in Figure 2. The nominal and measured dimensions of all specimens are tabulated in Table 1.Figure 1
Section dimensions and hole size of CFS lipped channel columns: (a) lipped channel section, (b) web with circular holes, and (c) web with rectangular holes.
(a)(b)(c)Figure 2
Specimen labeling.Table 1
Nominal and measured section dimensions of specimens.
SpecimenNominal section dimensionsMeasured section dimensionsMaximum of initial geometric imperfectionsh (mm)b (mm)a (mm)t (mm)Slenderness ratioL (mm)h1 (mm)b1 (mm)b2 (mm)a1 (mm)a2 (mm)Δmax (mm)C7508-CH3-17540100.889.912000.0074.7338.5140.359.758.870.927C7508-CH3-22000.0073.9439.8139.349.659.721.062C7508-CH5-11999.9574.6638.9538.9310.319.690.823C7508-CH5-22000.0073.7839.6339.739.5710.061.326C7508-CH7-11999.6074.0139.5839.589.569.841.856C7508-CH7-21999.7073.9238.9539.8310.229.640.902C7510-NH-1754010190.552000.0075.0539.9739.849.989.840.812C7510-NH-22000.1075.1339.9540.009.859.981.856C7510-RH2-12001.2574.5139.5039.699.399.920.925C7510-RH2-22001.1573.9739.2438.9110.0010.500.756C7510-RH4-12001.2073.7040.0939.479.529.931.012C7510-RH4-22001.1074.0439.4539.419.7010.041.201C7510-RH6-12001.2573.9139.3539.689.839.860.856C7510-RH6-22001.1574.1038.8339.839.909.440.914C9008-NH-19050150.872.382000.0589.9649.6949.6714.8314.951.302C9008-NH-21999.8589.9150.0250.0414.8614.841.202C9008-CH3-11999.2589.8248.9550.1314.2114.450.852C9008-CH3-21999.2589.8548.1849.2715.4814.580.962C9008-CH5-11999.5588.2549.6550.8114.0914.710.861C9008-CH5-21999.5089.9750.2649.5214.0214.530.934C9008-CH7-11999.5889.9950.2649.5214.0214.531.023C9008-CH7-21999.6289.9750.2649.5214.0214.530.954C9008-RH2-11999.6089.9649.7949.6013.9614.840.895C9008-RH2-21999.7089.2649.6549.9314.2915.330.697C9008-RH4-11999.9589.0748.4549.0815.5714.521.203C9008-RH4-22000.0589.5849.7648.8614.6014.701.322C9008-RH6-11999.8090.0149.3549.2914.7114.770.891C9008-RH6-21999.8589.9549.5749.3815.0414.781.849C8008-NH-18060100.882.002000.0580.025959.049.849.862.467C8008-NH-21999.879.8960.0360.079.69.581.312C8008-CH3-11999.779.8859.6260.278.889.111.512C8008-CH3-21999.8579.1460.5859.849.628.490.912C8008-CH5-12000.0580.2259.1059.4110.019.321.325C8008-CH5-21999.979.6659.3459.469.879.400.816C8008-CH7-1200079.9058.7159.289.759.810.988C8008-CH7-22000.0580.4358.7059.0110.079.701.324C8008-RH2-12000.1079.2959.3359.409.749.921.624C8008-RH2-22000.1579.4859.0259.8610.229.161.214C8008-RH4-11999.9580.2459.8058.759.619.221.212C8008-RH4-22000.0080.5158.6759.1010.029.390.897C8008-RH6-11999.9078.2559.1059.0210.929.801.245C8008-RH6-12000.0079.8059.4859.269.099.301.311


### 2.2. Material Properties

Steel sheets of grade LQ550 with two kinds of thicknesses were selected to manufacture the specimens. 3 tensile coupon tests of steel sheets with different thicknesses were conducted to determine the material properties based on Chinese specifications of “Tensile tests of metallic materials part 1: test methods at room temperature” (GB/T228.1-2010) [30]. The stress-strain curves of one of the coupons for every kind of steel plate are shown in Figure 3, and the other stress-strain curves are the same as these curves. The measured average material properties, including the static 0.2% proof stress (f0.2), initial Young’s modulus (E), ultimate strength (fu), and ultimate strain (εu), are listed in Table 2.Figure 3
Stress-strain curves of coupon test. (a) Thicknesst = 0.80 mm. (b) Thickness t = 1.00 mm.
(a)(b)Table 2
Mean value of coupon tests.
t (mm)f0.2 (MPa)fu (MPa)E (MPa)εu (%)0.806006022.16 × 1057.191.006206342.15 × 1058.92


### 2.3. Initial Geometric Imperfections

Initial geometric imperfections are usually generated in cold form and transportation, which has a great effect on the buckling behavior of CFS members. Initial geometric imperfections along the longitudinal direction of all specimens were measured before tests. The initial geometric imperfections were measured with an interval of 150 mm along the specimen length. The measuring locations at cross section are illustrated in Figure4(a), which was suggested in reference [31]. The locations 1, 2, 3, and 4 indicate the deviations from flats for the initial local geometric imperfection, the initial distortional geometric imperfection, the initial global geometric imperfection about the weak axis, and the initial global geometric imperfections about the great axis, respectively. The dial gauge was used to measure the initial geometric imperfection, as shown in Figure 4(b). Table 1 lists the maximum magnitudes (Δmax) of four kinds of initial geometric imperfections for all specimens. The initial geometric imperfections along the longitudinal direction for specimens C7510-NH-1, C7508-CH3-1, C8008-NH-1, and C9008-NH-1 are illustrated in Figures 5(a)–5(d), respectively. The initial geometric imperfections of the other specimens have almost the same distribution as these specimens. The measured initial geometric imperfections indicate that the imperfections have no obvious rules along the longitudinal direction, but the initial distortional geometric imperfection magnitudes are more than the initial local geometric imperfections and initial global geometric imperfections, and the specimens with large width-to-thickness ratio have bigger initial geometric imperfection magnitudes.Figure 4
Measure of the longitudinal initial geometric imperfection at cross section. (a) Measured location. (b) Measured setup.
(a)(b)Figure 5
Initial geometric imperfections of specimens. (a) C7510-NH-1. (b) C7508-CH3-1. (c) C8008-NH-1. (d) C9008-NH-1.
(a)(b)(c)(d)


### 2.4. Test Rig and Procedure

The specimens were loaded by a 50 kN Hydraulic Jack and reaction frame system as shown in Figure6. The upper actuator and lower pedestal were fitted with 250 mm × 200 mm × 8 mm steel platen ground flat and parallel. The column specimens were put directly on the steel platen as they were compressed. Friction between the column ends and the steel platens were the only lateral forces that restrained the column cross section under load. The axial compressive loading was obtained from the load cell positioned at the top of the test specimens. To understand clearly the displacement, deformation, and stress of the specimen during the loading process, the position transducers and strain gauges were set up as suggested in reference [31]. For specimens without holes, four-position transducers (Figure 7(a)) and four strain gauges (Figure 7(b)) were arranged at the mid-height of the column and a position transducer is arranged at the upper end plate of the specimen to measure the vertical displacement under load. For specimens with holes, four strain gauges (Figure 7(c)) were adhered at the middle height location and four strain gauges (Figure 7(d)) were adhered at the locations which are higher 100 mm than mid-height. Four-position transducers (Figure 7(e)) were arranged at the middle location to measure the displacement of web adjacent to the hole and flanges. Three-position transducers (Figure 7(f)) were set up at the web, which is higher 100 mm than the mid-height of specimens. All the test data could be obtained by using the YJ16 data acquisition instrument.Figure 6
Test setup.
(a)(b)Figure 7
Position transducers and strain gauge arrangement. (a) Position transducers at midheight for a nonperforated specimen. (b) Strain gauges at midheight for a nonperforated specimen. (c) Strain gauges at midheight for a perforated specimen. (d) Position transducers at midheight for a perforated specimen. (e) Strain gauges at higher 100 mm than midheight for a perforated specimen. (f) Position transducers at higher 100 mm than midheight for a perforated specimen.
(a)(b)(c)(d)(e)(f)


## 2.1. Test Specimens

This test employed 42 CFS lipped channel axially compressed members. There were 6 nonperforated members, 18 members with circular holes, and 18 members with rectangular holes. The definitions of geometric parameters of the specimen and the location, spacing, and types of holes are illustrated in Figure1. Four cross sections 75 × 40 × 10 × 0.8 mm, 75 × 40 × 10 × 1 mm, 90 × 50 × 15 × 0.8 mm, and 80 × 60 × 10 × 0.8 mm (h × b × a × t, h: web depth, b: flange width, a: lip length, t: thickness) were selected. The nominal thicknesses of the specimens are 0.8 mm and 1.0 mm. The diameters of circular holes are 0.3, 0.5, and 0.7 to the web depth. The heights of the rectangular holes are 0.2, 0.4, and 0.6 to the web depth. The ratio of length-to-width (Lh/Hh) of the rectangular hole is equal to 2. Each specimen is 2000 mm and has three holes. The hole spacings are 500 mm and 600 mm for members with circular and rectangular holes, respectively. The labeling rule of all specimens is defined in Figure 2. The nominal and measured dimensions of all specimens are tabulated in Table 1.Figure 1
Section dimensions and hole size of CFS lipped channel columns: (a) lipped channel section, (b) web with circular holes, and (c) web with rectangular holes.
(a)(b)(c)Figure 2
Specimen labeling.Table 1
Nominal and measured section dimensions of specimens.
SpecimenNominal section dimensionsMeasured section dimensionsMaximum of initial geometric imperfectionsh (mm)b (mm)a (mm)t (mm)Slenderness ratioL (mm)h1 (mm)b1 (mm)b2 (mm)a1 (mm)a2 (mm)Δmax (mm)C7508-CH3-17540100.889.912000.0074.7338.5140.359.758.870.927C7508-CH3-22000.0073.9439.8139.349.659.721.062C7508-CH5-11999.9574.6638.9538.9310.319.690.823C7508-CH5-22000.0073.7839.6339.739.5710.061.326C7508-CH7-11999.6074.0139.5839.589.569.841.856C7508-CH7-21999.7073.9238.9539.8310.229.640.902C7510-NH-1754010190.552000.0075.0539.9739.849.989.840.812C7510-NH-22000.1075.1339.9540.009.859.981.856C7510-RH2-12001.2574.5139.5039.699.399.920.925C7510-RH2-22001.1573.9739.2438.9110.0010.500.756C7510-RH4-12001.2073.7040.0939.479.529.931.012C7510-RH4-22001.1074.0439.4539.419.7010.041.201C7510-RH6-12001.2573.9139.3539.689.839.860.856C7510-RH6-22001.1574.1038.8339.839.909.440.914C9008-NH-19050150.872.382000.0589.9649.6949.6714.8314.951.302C9008-NH-21999.8589.9150.0250.0414.8614.841.202C9008-CH3-11999.2589.8248.9550.1314.2114.450.852C9008-CH3-21999.2589.8548.1849.2715.4814.580.962C9008-CH5-11999.5588.2549.6550.8114.0914.710.861C9008-CH5-21999.5089.9750.2649.5214.0214.530.934C9008-CH7-11999.5889.9950.2649.5214.0214.531.023C9008-CH7-21999.6289.9750.2649.5214.0214.530.954C9008-RH2-11999.6089.9649.7949.6013.9614.840.895C9008-RH2-21999.7089.2649.6549.9314.2915.330.697C9008-RH4-11999.9589.0748.4549.0815.5714.521.203C9008-RH4-22000.0589.5849.7648.8614.6014.701.322C9008-RH6-11999.8090.0149.3549.2914.7114.770.891C9008-RH6-21999.8589.9549.5749.3815.0414.781.849C8008-NH-18060100.882.002000.0580.025959.049.849.862.467C8008-NH-21999.879.8960.0360.079.69.581.312C8008-CH3-11999.779.8859.6260.278.889.111.512C8008-CH3-21999.8579.1460.5859.849.628.490.912C8008-CH5-12000.0580.2259.1059.4110.019.321.325C8008-CH5-21999.979.6659.3459.469.879.400.816C8008-CH7-1200079.9058.7159.289.759.810.988C8008-CH7-22000.0580.4358.7059.0110.079.701.324C8008-RH2-12000.1079.2959.3359.409.749.921.624C8008-RH2-22000.1579.4859.0259.8610.229.161.214C8008-RH4-11999.9580.2459.8058.759.619.221.212C8008-RH4-22000.0080.5158.6759.1010.029.390.897C8008-RH6-11999.9078.2559.1059.0210.929.801.245C8008-RH6-12000.0079.8059.4859.269.099.301.311


## 2.2. Material Properties

Steel sheets of grade LQ550 with two kinds of thicknesses were selected to manufacture the specimens. 3 tensile coupon tests of steel sheets with different thicknesses were conducted to determine the material properties based on Chinese specifications of “Tensile tests of metallic materials part 1: test methods at room temperature” (GB/T228.1-2010) [30]. The stress-strain curves of one of the coupons for every kind of steel plate are shown in Figure 3, and the other stress-strain curves are the same as these curves. The measured average material properties, including the static 0.2% proof stress (f0.2), initial Young’s modulus (E), ultimate strength (fu), and ultimate strain (εu), are listed in Table 2.Figure 3
Stress-strain curves of coupon test. (a) Thicknesst = 0.80 mm. (b) Thickness t = 1.00 mm.
(a)(b)Table 2
Mean value of coupon tests.
t (mm)f0.2 (MPa)fu (MPa)E (MPa)εu (%)0.806006022.16 × 1057.191.006206342.15 × 1058.92


## 2.3. Initial Geometric Imperfections

Initial geometric imperfections are usually generated in cold form and transportation, which has a great effect on the buckling behavior of CFS members. Initial geometric imperfections along the longitudinal direction of all specimens were measured before tests. The initial geometric imperfections were measured with an interval of 150 mm along the specimen length. The measuring locations at cross section are illustrated in Figure4(a), which was suggested in reference [31]. The locations 1, 2, 3, and 4 indicate the deviations from flats for the initial local geometric imperfection, the initial distortional geometric imperfection, the initial global geometric imperfection about the weak axis, and the initial global geometric imperfections about the great axis, respectively. The dial gauge was used to measure the initial geometric imperfection, as shown in Figure 4(b). Table 1 lists the maximum magnitudes (Δmax) of four kinds of initial geometric imperfections for all specimens. The initial geometric imperfections along the longitudinal direction for specimens C7510-NH-1, C7508-CH3-1, C8008-NH-1, and C9008-NH-1 are illustrated in Figures 5(a)–5(d), respectively. The initial geometric imperfections of the other specimens have almost the same distribution as these specimens. The measured initial geometric imperfections indicate that the imperfections have no obvious rules along the longitudinal direction, but the initial distortional geometric imperfection magnitudes are more than the initial local geometric imperfections and initial global geometric imperfections, and the specimens with large width-to-thickness ratio have bigger initial geometric imperfection magnitudes.Figure 4
Measure of the longitudinal initial geometric imperfection at cross section. (a) Measured location. (b) Measured setup.
(a)(b)Figure 5
Initial geometric imperfections of specimens. (a) C7510-NH-1. (b) C7508-CH3-1. (c) C8008-NH-1. (d) C9008-NH-1.
(a)(b)(c)(d)


## 2.4. Test Rig and Procedure

The specimens were loaded by a 50 kN Hydraulic Jack and reaction frame system as shown in Figure6. The upper actuator and lower pedestal were fitted with 250 mm × 200 mm × 8 mm steel platen ground flat and parallel. The column specimens were put directly on the steel platen as they were compressed. Friction between the column ends and the steel platens were the only lateral forces that restrained the column cross section under load. The axial compressive loading was obtained from the load cell positioned at the top of the test specimens. To understand clearly the displacement, deformation, and stress of the specimen during the loading process, the position transducers and strain gauges were set up as suggested in reference [31]. For specimens without holes, four-position transducers (Figure 7(a)) and four strain gauges (Figure 7(b)) were arranged at the mid-height of the column and a position transducer is arranged at the upper end plate of the specimen to measure the vertical displacement under load. For specimens with holes, four strain gauges (Figure 7(c)) were adhered at the middle height location and four strain gauges (Figure 7(d)) were adhered at the locations which are higher 100 mm than mid-height. Four-position transducers (Figure 7(e)) were arranged at the middle location to measure the displacement of web adjacent to the hole and flanges. Three-position transducers (Figure 7(f)) were set up at the web, which is higher 100 mm than the mid-height of specimens. All the test data could be obtained by using the YJ16 data acquisition instrument.Figure 6
Test setup.
(a)(b)Figure 7
Position transducers and strain gauge arrangement. (a) Position transducers at midheight for a nonperforated specimen. (b) Strain gauges at midheight for a nonperforated specimen. (c) Strain gauges at midheight for a perforated specimen. (d) Position transducers at midheight for a perforated specimen. (e) Strain gauges at higher 100 mm than midheight for a perforated specimen. (f) Position transducers at higher 100 mm than midheight for a perforated specimen.
(a)(b)(c)(d)(e)(f)


## 3. Test Results

### 3.1. Failure Buckling Mode and Load-Strain Curve

Table3 presents the results of 42 axially compressed specimens about buckling modes and load capacities in this experiment, where L, D, FT, and F represent the local buckling, distortional buckling, flexural-torsional buckling, and flexural buckling, respectively; Pt, Pa, Pc are the test result, finite element analysis result, and calculated result by using the proposed method, respectively.Table 3
Comparison of buckling modes and ultimate capacities obtained from test, FEA, and predictions by using the proposed method.
SpecimenTest buckling modeFEA buckling modePt (kN)Decrease of load capacities (%)Pa (kN)Pc (kN)Pt/PaPt/PcC7508-CH3-1L +D + FTL + D + FT16.3916.8215.480.971.06C7508-CH3-2L + D + FTL + D + FT16.5816.7515.940.991.04C7508-CH5-1L + D + FTL + D + FT15.386.7016.4715.150.931.02C7508-CH5-2L + D + FTL + D + FT15.386.7016.1414.970.951.03C7508-CH7-1L + D + FTL + D + FT14.6011.4315.2913.490.961.08C7508-CH7-2L + D + FTL + D + FT14.3213.1315.5313.530.921.06C7510-NH-1FTFT27.7828.0625.610.991.08C7510-NH-2FTFT26.9327.6125.320.981.06C7510-RH2-1FTFT23.9212.5625.0323.570.961.01C7510-RH2-2FTFT24.3710.9126.3224.160.931.01C7510-RH4-1FTFT23.9612.4124.5623.220.981.03C7510-RH4-2FTFT22.6317.2724.9023.220.910.97C7510-RH6-1FTFT21.4521.5922.0821.080.971.02C7510-RH6-2FTFT21.0722.9822.1720.980.951.00C9008-NH-1L + D + FTL + D + FT23.7223.9423.310.991.02C9008-NH-2L + D + FTL + D + FT24.5223.9223.521.021.04C9008-CH3-1L + D + FTL + D + FT22.337.4221.6022.061.031.01C9008-CH3-2L + D + FTL + D + FT22.676.0122.0922.491.031.01C9008-CH5-1L + D + FTL + D + FT21.4611.0321.0221.381.021.00C9008-CH5-2L + D + FTL + D + FT20.5015.0021.2821.450.960.96C9008-CH7-1L + D + FL + D + F19.1820.4819.5819.090.981.00C9008-CH7-2L + D + FL + D + F19.4119.5320.0319.060.971.02C9008-RH2-1L + D + FTL + D + FT23.303.4024.6222.060.951.06C9008-RH2-2L + D + FTL + D + FT24.80x2.8225.3722.520.981.10C9008-RH4-1L + D + FTL + D + FT22.885.1422.8722.721.001.01C9008-RH4-2L + D + FTL + D + FT22.347.3821.9722.721.020.98C9008-RH6-1L + D + FL + D + F19.3519.7819.4920.480.990.94C9008-RH6-2L + D + FL + D + F19.5019.1520.8920.570.930.95C8008-NH-1L + D + FTL + D + FT20.3719.7320.831.030.98C8008-NH-2L + D + FTL + D + FT20.9920.6320.531.021.02C8008-CH3-1L + D + FTL + D + FT19.306.6718.5919.241.041.00C8008-CH3-2L + D + FTL + D + FT19.455.9518.3719.691.060.99C8008-CH5-1L + D + FTL + D + F18.3811.1217.3319.161.060.96C8008-CH5-2L + D + FTL + D + FT18.2011.9917.5119.191.040.95C8008-CH7-1L + D + FL + D + F17.2716.4917.0317.501.010.99C8008-CH7-2L + D + FL + D + F16.9018.2817.2117.510.980.97C8008-RH2-1L + D + FTL + D + FT20.600.3919.9119.681.031.05C8008-RH2-2L + D + FTL + D + FT19.674.8819.7620.001.000.98C8008-RH4-1L + D + FL + D + F19.187.2519.2819.800.990.97C8008-RH4-2L + D + FL + D + F19.406.1918.7719.901.030.97C8008-RH6-1L + D + FL + D + F18.3411.3218.1918.661.010.98C8008-RH6-1L + D + FL + D + F17.1617.0217.7418.100.970.95Average0.9891.008Standard deviation0.0380.040Coefficient of variation0.0390.040As shown in Table3, all specimens with circular holes exhibited an obvious interaction of local buckling (see Figure 8(a)), distortional buckling (see Figure 8(b)), and global buckling (see Figure 8(c)) (flexural buckling or flexural-tortional buckling). For global buckling, section C7508 with small width to thickness exhibited flexural-torsional buckling, section C8008 and section C9008 with the size of holes d/h = 0.3 and 0.5 displaying flexural-torsional buckling, and section C8008 and section C9008 with the size of holes d/h = 0.7 displaying flexural buckling. The interaction buckling progression under load for specimen C8008-CH7-1 is depicted in Figure 8.Figure 8
Buckling modes for specimen C8008-CH3-1. (a) Local buckling. (b) Distortional buckling. (c) Flexure-torsional buckling.
(a)(b)(c)The test buckling mode in Table3 indicates that all specimens with rectangular holes for section C8008 and C9008 with large width-to-thickness ratio exhibited obvious interaction of local buckling, distortional buckling, and global buckling. The section with small holes exhibited flexural-torsional buckling and flexural buckling occurred for the section with large holes. Section C7510 displayed the flexural-torsional buckling because of the small width-to-thickness ratio. Specimen C7510-RH6-1 is depicted in Figure 9 as an example. The buckling deformations for all other specimens with circular and rectangular holes are provided in reference [32].Figure 9
Flexural-torsional buckling for specimen C7510-RH6-1.The influence of the circular and rectangular holes on the buckling load of columns and the stress of the web adjacent to the hole are provided in Figures10 and 11 for section C9008, respectively. The results of Figures 10 and 11 indicate that the buckling stress of the web adjacent to the hole and buckling loads of specimens significantly increased with the increasing of the size of holes and the circular and rectangular holes have a significant impact on the stress concentration and distribution of CFS lipped channel columns.Figure 10
Load-strain curve for section C9008 with different circular hole sizes.Figure 11
Load-strain curve for section C9008 with different rectangular hole sizes.Figures12 and 13 depict the web stress response of section C9008 with circular and rectangular holes at the edge of the hole (strain gauge 4) and the same transverse location but higher 100 mm than mid-height (strain gauge 1), respectively. The comparisons show that the web stress at the edge of the hole is higher than that of the other longitudinal locations because of the stress concentrate led by the hole. Meanwhile, the stress increases obviously with the increase of the size of holes.Figure 12
Load-strain curves for section C9008 with circular holes for No. 1 and No. 4 strain gauges. (a) Specimen C9008-CH3-1. (b) Specimen C9008-CH5-2. (c) Specimen C9008-CH7-2.
(a)(b)(c)Figure 13
Load-strain curve for section C9008 with rectangular holes for No. 1 and No. 4 strain gauges. (a) Specimen C9008-RH2-2. (b) Specimen C9008-RH4-1. (c) Specimen C9008-RH6-2.
(a)(b)(c)


### 3.2. Load Capacities and Load-Displacement Curve

Experimental ultimate capacities for all specimens with circular holes and rectangular holes are summarized in Table3. The load capacities reduce with a decreased value of 6.7% and 13.13% when the ratio of hole diameter-to-web depth increases from 0.3 to 0.5 and 0.7 for section C7508, respectively. The reduction values of load capacity are 7.42%, 15%, 20.48%, and 6.72%, 11.99%, and 18.28% when the ratio of hole diameter-to-web depth increases from 0 to 0.3, 0.5, 0.7 for sections C9008 and C8008, respectively. The load capacities reduce with a reduction value of 12.56%, 17.27%, and 22.98%, 3.4%, 7.38%, and 19.78%, and 4.88%, 7.25%, and 17.02% when the ratio of hole height-to-web depth increases from 0 to 0.2, 0.4, and 0.6 for sections C7510, C9008, and C8008, respectively. The comparison results indicate that the ultimate capacities of the specimens decreased obviously with the increase of the size of circular and rectangular holes.Figures14 and 15 depict the comparison of load-displacement curves of section C9008 with various sizes of circular and rectangular holes, respectively. It can be observed that the load capacity decreases with the increase of the size of holes.Figure 14
Load-displacement curves of section C9008 with circular holes.Figure 15
Load-displacement curves of section C9008 with rectangular holes.


## 3.1. Failure Buckling Mode and Load-Strain Curve

Table3 presents the results of 42 axially compressed specimens about buckling modes and load capacities in this experiment, where L, D, FT, and F represent the local buckling, distortional buckling, flexural-torsional buckling, and flexural buckling, respectively; Pt, Pa, Pc are the test result, finite element analysis result, and calculated result by using the proposed method, respectively.Table 3
Comparison of buckling modes and ultimate capacities obtained from test, FEA, and predictions by using the proposed method.
SpecimenTest buckling modeFEA buckling modePt (kN)Decrease of load capacities (%)Pa (kN)Pc (kN)Pt/PaPt/PcC7508-CH3-1L +D + FTL + D + FT16.3916.8215.480.971.06C7508-CH3-2L + D + FTL + D + FT16.5816.7515.940.991.04C7508-CH5-1L + D + FTL + D + FT15.386.7016.4715.150.931.02C7508-CH5-2L + D + FTL + D + FT15.386.7016.1414.970.951.03C7508-CH7-1L + D + FTL + D + FT14.6011.4315.2913.490.961.08C7508-CH7-2L + D + FTL + D + FT14.3213.1315.5313.530.921.06C7510-NH-1FTFT27.7828.0625.610.991.08C7510-NH-2FTFT26.9327.6125.320.981.06C7510-RH2-1FTFT23.9212.5625.0323.570.961.01C7510-RH2-2FTFT24.3710.9126.3224.160.931.01C7510-RH4-1FTFT23.9612.4124.5623.220.981.03C7510-RH4-2FTFT22.6317.2724.9023.220.910.97C7510-RH6-1FTFT21.4521.5922.0821.080.971.02C7510-RH6-2FTFT21.0722.9822.1720.980.951.00C9008-NH-1L + D + FTL + D + FT23.7223.9423.310.991.02C9008-NH-2L + D + FTL + D + FT24.5223.9223.521.021.04C9008-CH3-1L + D + FTL + D + FT22.337.4221.6022.061.031.01C9008-CH3-2L + D + FTL + D + FT22.676.0122.0922.491.031.01C9008-CH5-1L + D + FTL + D + FT21.4611.0321.0221.381.021.00C9008-CH5-2L + D + FTL + D + FT20.5015.0021.2821.450.960.96C9008-CH7-1L + D + FL + D + F19.1820.4819.5819.090.981.00C9008-CH7-2L + D + FL + D + F19.4119.5320.0319.060.971.02C9008-RH2-1L + D + FTL + D + FT23.303.4024.6222.060.951.06C9008-RH2-2L + D + FTL + D + FT24.80x2.8225.3722.520.981.10C9008-RH4-1L + D + FTL + D + FT22.885.1422.8722.721.001.01C9008-RH4-2L + D + FTL + D + FT22.347.3821.9722.721.020.98C9008-RH6-1L + D + FL + D + F19.3519.7819.4920.480.990.94C9008-RH6-2L + D + FL + D + F19.5019.1520.8920.570.930.95C8008-NH-1L + D + FTL + D + FT20.3719.7320.831.030.98C8008-NH-2L + D + FTL + D + FT20.9920.6320.531.021.02C8008-CH3-1L + D + FTL + D + FT19.306.6718.5919.241.041.00C8008-CH3-2L + D + FTL + D + FT19.455.9518.3719.691.060.99C8008-CH5-1L + D + FTL + D + F18.3811.1217.3319.161.060.96C8008-CH5-2L + D + FTL + D + FT18.2011.9917.5119.191.040.95C8008-CH7-1L + D + FL + D + F17.2716.4917.0317.501.010.99C8008-CH7-2L + D + FL + D + F16.9018.2817.2117.510.980.97C8008-RH2-1L + D + FTL + D + FT20.600.3919.9119.681.031.05C8008-RH2-2L + D + FTL + D + FT19.674.8819.7620.001.000.98C8008-RH4-1L + D + FL + D + F19.187.2519.2819.800.990.97C8008-RH4-2L + D + FL + D + F19.406.1918.7719.901.030.97C8008-RH6-1L + D + FL + D + F18.3411.3218.1918.661.010.98C8008-RH6-1L + D + FL + D + F17.1617.0217.7418.100.970.95Average0.9891.008Standard deviation0.0380.040Coefficient of variation0.0390.040As shown in Table3, all specimens with circular holes exhibited an obvious interaction of local buckling (see Figure 8(a)), distortional buckling (see Figure 8(b)), and global buckling (see Figure 8(c)) (flexural buckling or flexural-tortional buckling). For global buckling, section C7508 with small width to thickness exhibited flexural-torsional buckling, section C8008 and section C9008 with the size of holes d/h = 0.3 and 0.5 displaying flexural-torsional buckling, and section C8008 and section C9008 with the size of holes d/h = 0.7 displaying flexural buckling. The interaction buckling progression under load for specimen C8008-CH7-1 is depicted in Figure 8.Figure 8
Buckling modes for specimen C8008-CH3-1. (a) Local buckling. (b) Distortional buckling. (c) Flexure-torsional buckling.
(a)(b)(c)The test buckling mode in Table3 indicates that all specimens with rectangular holes for section C8008 and C9008 with large width-to-thickness ratio exhibited obvious interaction of local buckling, distortional buckling, and global buckling. The section with small holes exhibited flexural-torsional buckling and flexural buckling occurred for the section with large holes. Section C7510 displayed the flexural-torsional buckling because of the small width-to-thickness ratio. Specimen C7510-RH6-1 is depicted in Figure 9 as an example. The buckling deformations for all other specimens with circular and rectangular holes are provided in reference [32].Figure 9
Flexural-torsional buckling for specimen C7510-RH6-1.The influence of the circular and rectangular holes on the buckling load of columns and the stress of the web adjacent to the hole are provided in Figures10 and 11 for section C9008, respectively. The results of Figures 10 and 11 indicate that the buckling stress of the web adjacent to the hole and buckling loads of specimens significantly increased with the increasing of the size of holes and the circular and rectangular holes have a significant impact on the stress concentration and distribution of CFS lipped channel columns.Figure 10
Load-strain curve for section C9008 with different circular hole sizes.Figure 11
Load-strain curve for section C9008 with different rectangular hole sizes.Figures12 and 13 depict the web stress response of section C9008 with circular and rectangular holes at the edge of the hole (strain gauge 4) and the same transverse location but higher 100 mm than mid-height (strain gauge 1), respectively. The comparisons show that the web stress at the edge of the hole is higher than that of the other longitudinal locations because of the stress concentrate led by the hole. Meanwhile, the stress increases obviously with the increase of the size of holes.Figure 12
Load-strain curves for section C9008 with circular holes for No. 1 and No. 4 strain gauges. (a) Specimen C9008-CH3-1. (b) Specimen C9008-CH5-2. (c) Specimen C9008-CH7-2.
(a)(b)(c)Figure 13
Load-strain curve for section C9008 with rectangular holes for No. 1 and No. 4 strain gauges. (a) Specimen C9008-RH2-2. (b) Specimen C9008-RH4-1. (c) Specimen C9008-RH6-2.
(a)(b)(c)


## 3.2. Load Capacities and Load-Displacement Curve

Experimental ultimate capacities for all specimens with circular holes and rectangular holes are summarized in Table3. The load capacities reduce with a decreased value of 6.7% and 13.13% when the ratio of hole diameter-to-web depth increases from 0.3 to 0.5 and 0.7 for section C7508, respectively. The reduction values of load capacity are 7.42%, 15%, 20.48%, and 6.72%, 11.99%, and 18.28% when the ratio of hole diameter-to-web depth increases from 0 to 0.3, 0.5, 0.7 for sections C9008 and C8008, respectively. The load capacities reduce with a reduction value of 12.56%, 17.27%, and 22.98%, 3.4%, 7.38%, and 19.78%, and 4.88%, 7.25%, and 17.02% when the ratio of hole height-to-web depth increases from 0 to 0.2, 0.4, and 0.6 for sections C7510, C9008, and C8008, respectively. The comparison results indicate that the ultimate capacities of the specimens decreased obviously with the increase of the size of circular and rectangular holes.Figures14 and 15 depict the comparison of load-displacement curves of section C9008 with various sizes of circular and rectangular holes, respectively. It can be observed that the load capacity decreases with the increase of the size of holes.Figure 14
Load-displacement curves of section C9008 with circular holes.Figure 15
Load-displacement curves of section C9008 with rectangular holes.


## 4. Verification of Finite Element Models

The finite element program ABAQUS [33] was used to simulate the buckling modes and behavior of the specimens and perform further parametric analysis on the CFS lipped channel axially compressed columns with circular and rectangular holes. The S4R shell element was selected for the modeling column and the analytical rigid body element was used for modeling the upper and lower end steel plates of the specimens. The dimensions of the cross-sections measured before the tests were used in the FEM. The ultimate capacities obtained by using FEM indicated a maximum error of 1.3 percent for specimen C7508-CH5-1 with a mesh size of 5 mm × 5 mm and 10 mm × 10 mm. So the mesh size of 10 mm × 10 mm was selected to model specimens in order to reduce the analysis time. The ideal elastoplastic model based on the material properties obtained from the coupon tests shown in Table 2 was used in FEM. The end constraint conditions of the test plate were all fixed by constraining the five degrees of freedom of the reference point RP1 at the upper end plate (2 translation degrees of freedom and 3 rotational degrees of freedom, releasing UZ degrees of freedom to control displacement) and the six degrees of freedom of the reference point RP2 at the lower end plate. The measured actual dimensions of the specimens (see in Table 1) were all included in the models. The whole analysis procedure included two phases: Firstly, an eigenvalue buckling analysis was performed to find the most probable elastic buckling mode of the specimen. The measured maximum magnitude of the initial geometric imperfections shown in Table 1 was applied to the first eigenmode to produce the geometric imperfection of FEM. Secondly, nonlinear analysis considering both materials nonlinear and geometric nonlinear was carried out by using the arc-length method to analyze the failure modes and ultimate load capacity of specimens.The failure buckling modes and load-carrying capacities of all 42 specimens obtained by FEA are presented in Table3. The maximum deviation value about the ultimate load capacity between the FEA and test is 10%. The mean value of the test-to-FEA ratio for load capacity is 0.989, with the corresponding coefficient of variation (COV) of 0.039 for all specimens. Specimen C8008-CH7-1 and specimen C7510-RH6-1 are taken as examples to present failure buckling mode comparisons between FEA and test results, as shown in Figures 16 and 17. The comparison of load-displacement curves for specimen C9008-20-CH3-1 between FEA results and the test results is shown in Figure 18. As shown in Figure 18, it should be noted that the load-displacement curves dropped abruptly because of the use of high-strength steel, which leads to the effect of the difference of boundary conditions between test and finite element can be ignored. Although boundary conditions employed in the testing are not perfectly in line with the fix-ended conditions used in the FEA, the above-mentioned comparison indicates that the load-carrying capacities, the failure modes, and load-displacement curves obtained from FEA show good agreement with the test results. Therefore, the FEM developed in this study has been well validated with the tests.Figure 16
Comparison on buckling modes for specimen C8008-CH7-1. (a) Local buckling. (b) Distortional buckling. (c) Flexural buckling.
(a)(b)(c)Figure 17
Comparison of buckling modes for specimen C7510-RH6-1.Figure 18
Comparison of the load-displacement curve for specimen C9008 between FEA and test. (a) Specimen C9008-CH3-1. (b) Specimen C9008-CH5-1. (c) Specimen C9008-CH7-1.
(a)(b)(c)


## 5. Finite Element Parametric Study

FE parametric analyses on 36 CFS lipped channel columns with various cross-section dimensions and sizes of web holes were carried out by using the validated FEM. The web height ranges from 70 to 100 mm and the plate thicknesses are 1.0 and 1.2 mm. The circular hole diameter-to-web depth ratio is 0.3 and 0.7. The rectangular hole height-to-web depth ratio is 0.3 and 0.5. The hole height-to-length ratio is 0.5 for all rectangular holes. The labeling rule for FEA members is the same as the one for test specimens, as shown in Figure2, except that the last digit in FEA represents the number of holes. The results of ultimate load capacity obtained from FEA (Pa) are listed in Tables 4 and 5 for CFS lipped channel columns with circular and rectangular holes, respectively.Table 4
Comparison of ultimate strength between the FEA results and the predicted results by using the modified method for columns with circular holes.
SpecimenL (mm)h (mm)b (mm)a (mm)t (mm)fy (MPa)S (mm)Hole numberd (mm)Pa (kN)Pc (kN)Pc/PaC8910-CH3-630008940131280500626.79.309.130.98C8910-CH3-315008940131280500326.721.3123.271.09C8910-CH3-16008940131280126.731.8730.580.96C8910-CH7-630008940131280500662.37.947.941.00C8910-CH7-315008940131280500362.317.5816.170.92C8910-CH7-16008940131280162.320.9618.750.90C7012-CH3-630007035111.23505006217.717.110.92C7012-CH3-315007035111.235050032127.0324.350.90C7012-CH3-16007035111.235012144.0043.480.99C7012-CH7-630007035111.23505006496.746.901.02C7012-CH7-315007035111.235050034920.0218.070.90C7012-CH7-16007035111.235014927.3624.750.91C10010-CH3-630001005012155050063013.8013.370.97C10010-CH3-315001005012155050033036.6834.590.94C10010-CH3-16001005012155013046.4042.830.92C10010-CH7-630001005012155050067012.7312.971.02C10010-CH7-315001005012155050037030.4828.230.93C10010-CH7-16001005012155017032.8529.880.91Average0.954Standard deviation0.054Coefficient of variation0.057Table 5
Comparison on ultimate strength between the FEA results and predicted results using the modified method for columns with rectangular holes.
SpecimenL (mm)h (mm)b (mm)a (mm)t (mm)fy (MPa)S (mm)Hole numberHh (mm)Lh (mm)Pa (kN)Pc (kN)Pc/PaC8910-RH3-630008940131280500626.753.49.209.070.99C8910-RH3-315008940131280500326.753.420.8522.951.10C8910-RH3-16008940131280126.753.430.0730.031.00C8910-RH5-630008940131280500644.5898.588.841.03C8910-RH5-315008940131280500344.58921.3819.620.92C8910-RH5-16008940131280144.58925.8123.920.93C7012-RH3-630007035111.2350500621427.647.070.93C7012-RH3-315007035111.23505003214226.1724.200.92C7012-RH3-16007035111.23501214242.5542.991.01C7012-RH5-630007035111.2350500635707.196.930.96C7012-RH5-315007035111.23505003357023.0621.040.91C7012-RH5-16007035111.23501357033.7731.280.93C10010-RH3-63000100501215505006306013.5613.300.98C10010-RH3-31500100501215505003306034.1632.050.94C10010-RH3-1600100501215501306043.5242.020.97C10010-RH5-630001005012155050065010013.1413.020.99C10010-RH5-315001005012155050035010031.4029.370.94C10010-RH5-160010050121550015010038.9636.560.94Average0.965Standard deviation0.049Coefficient of variation0.051Tables4 and 5 show that the axial capacities of CFS channels with circular holes and rectangular holes were reduced as the column slenderness increased. For example, the axial capacity was reduced by 69.4% for C8910-RH3 columns when the length of the column increased from 600 mm to 3000 mm. Meanwhile, it can be found that the effect of the hole size on the axial capacity of the CFS lipped channels with holes was significant. For example, when the ratio d/h increased from 0.3 to 0.7, the axial capacity of section C8910-CH-1 decreased by 34.23%.


## 6. Modified Design Method for Columns with Web Holes

The test results and FEA results indicate that the web holes influence the load capacity of CFS lipped channel axially compressed columns. Therefore, the modified equation (1) is used to predict the design strength of CFS lipped channel columns with web holes under axial compression.(1)N=φh⋅Aeh⋅fy,where φh and Aeh are the modified global buckling coefficient and effective section area of compression members with web holes, and fy is the material yield strength. It needs to note that the hole spacings should be more than 2 timing the diameter of the circular hole and 4 timing the height of the rectangular hole.


### 6.1. Modified Global Buckling Coefficient of Compression Members with Web Holes

The global buckling coefficient for CFS axially compressed columns without holes can be determined with the following:(2)φ=PcrePy=121+1λ¯21+ε0−1+1λ¯21+ε02−4λ¯2,where λ¯=λ/πfy/E, λ is the slenderness ratio, λ=L/I/A, L is the effective length of the column, I is the moment of inertia of cross section about the axis of buckling, A is the area of gross cross section, ε0 is the initial eccentricity, Pcre and Py are critical global buckling strength and yield strength, and Pcre can be calculated as follows for global flexural buckling:(3)Pcre=π2EIL2=π2EAλ2.For CFS lipped channel columns with holes, Moen and Schafer [24] proposed the approximate prediction method for critical elastic flexural buckling strength (Pcre), which can be dominated by the weighted average method and the prediction equation is as follows:(4)Pcre.h=π2EL2IgLg+InetLnetL=π2EAλh2,where Ig, Lg are the moment of inertia of nominal gross cross section about the axis of buckling and segment length without holes, respectively, Inet, Lnet are the moment of inertia of net cross section about the axis of buckling and length of holes, respectively, λh is the slenderness ratio considering the effect of holes.Section 75 × 30 × 10 × 2.5 is used as an example to verify the accuracy and applicability of equation (4). The comparisons of the critical elastic flexural buckling strength between predicted results with different methods and FEA results are plotted in Figure 19, where Pcre, Pnet, and Pcre.h are FEA results, the predicted results with the moment of inertia of net cross section, and the predicted results with equation (4), respectively. It can be observed that the predictions using the net section are too conservative and the predictions with formula (4) are unconservative for members with large web holes. Therefore, equation (4) is modified as equation (5) to give more reliable prediction results for the critical elastic flexural buckling strength of CFS lipped channel columns with web hole based FEA.(5)Pmcre.h=π2EL2IgLg+InetLnetL1+0.08Hhh−0.3Hhh21−nLhL2=π2EIhL2=π2EAλmh2,where Ih and λmh are the modified moments of inertia of gross cross section about the axis of buckling and slenderness ratio considering the effect of holes.Figure 19
Comparison of elastic flexural buckling capacity between FE results and predicted results.The comparison of critical global buckling strength between FEA results and predicted results (Pmcre.h) by using equation (5) shown in Figure 19 indicates that formula (5) is accurate.Based on formulas (3) and (5), the following equation can be obtained:(6)λmhλ=PcrePmcre.h=IIh,where Ih=IgLg+InetLnet/L1+0.08Hh/h−0.3Hh/h21−nLh/L2, then the modified global buckling coefficient can be determined using the following:(7)φh=Pmcre.hPy=121+1λ¯mh21+ε0−1+1λ¯mh21+ε02−4λ¯mh2,where λ¯mh=λmh/πfy/E, λmh is the modified slenderness ratio considering the effect of holes which can be calculated by using equation (6).For the global flexural-torsional buckling coefficient of CFS axially compressed columns with holes, equation (7) can also be used. Saint-Venant torsion constant can be calculated using equation (8) and the warping constant can be predicted assuming cross-section thickness is zero at the hole as reported in [34].(8)Ith=ItgLg+ItnetLnetL1+0.08Hhh−0.3Hhh21−nLhL2,where Itg and Itnet are Saint-Venant torsion constants of gross cross section and net cross section.For CFS lipped channel axiallycompressed members with a circular hole, the diameterd can be used to replace Hh and Lh in equations (5) and (8).


### 6.2. Modified Effective Area Equation for Compression Members with Web Holes

The effective area of CFS axially compressed stud columns with holes can be determined by using the modified effective width method in [34]. However, it needs to note that the effect of web holes on the effective area will decrease with the increase of the length of columns. Therefore, the effective areas of CFS columns with different lengths can be predicted by using equation (9), considering the effect of dimensions of holes and the length of columns based on FEA results.(9)Aeh=Ae1,λ¯≤0.561AAnet,1+λ¯−λ¯1λ¯2−λ¯1⋅Ae2−Ae1Ae1Ae1,0.561AAnet<λ¯<1.5AAnet,Ae2,λ¯≥1.5AAnet,where Ae1, Ae2 are the effective area considering the effect of holes using the modified effective width method in [16] and the effective area without considering the effect of holes. Anet is the net sectional area, λ¯ is the normalized slenderness ratio of columns, and λ¯1=0.561A/Anet, λ¯2=1.5A/Anet.


### 6.3. Comparison of Design Capacity with Test and FEA Data

The load capacities determined by the modified design formula (1) were compared with the ultimate capacities obtained from tests and FE analyses, as listed in Tables 3–5. The mean value of the test-to-prediction ultimate strength ratio (Pt/Pc) is 1.008 with the corresponding standard deviation of 0.04, as shown in Table 3. The mean values of prediction-to-FEA ultimate strength ratios (Pc/Pa) are 0.954 and 0.965 with the corresponding standard deviation of 0.054 and 0.049 for columns with circular and rectangular holes as listed in Tables 4 and 5, respectively.Figures20 and 21 show that the modified load capacity curves for section 75 × 30 × 10 × 2.5 with the size of circular holes d/h = 0.3, 0.5, and 0.7 and rectangular holes Hh/h = 0.3, 0.5, and 0.7 give good agreement with the FEA results for CFS lipped channel columns with web holes, respectively, where Pa and Pc are FEA and predicted load capacities, Py is the yield strength.Figure 20
Comparison of the ultimate capacity of columns with circular holes between FEA results and predicted results using the modified method. (a)d/h = 0.3. (b) d/h = 0.5. (c) d/h = 0.7.
(a)(b)(c)Figure 21
Comparison of ultimate capacity of columns with rectangular holes between FEA results and predicted results using the modified method (Hh : Lh = 1 : 2). (a) Hh/h = 0.3. (b) Hh/h = 0.5. (c) Hh/h = 0.7.
(a)(b)(c)The comparison results shown in Tables3–5 and Figures 20 and 21 indicate that the predicted results by using a modified formula based on modified global buckling coefficient and modified effective area are extremely accurate.


## 6.1. Modified Global Buckling Coefficient of Compression Members with Web Holes

The global buckling coefficient for CFS axially compressed columns without holes can be determined with the following:(2)φ=PcrePy=121+1λ¯21+ε0−1+1λ¯21+ε02−4λ¯2,where λ¯=λ/πfy/E, λ is the slenderness ratio, λ=L/I/A, L is the effective length of the column, I is the moment of inertia of cross section about the axis of buckling, A is the area of gross cross section, ε0 is the initial eccentricity, Pcre and Py are critical global buckling strength and yield strength, and Pcre can be calculated as follows for global flexural buckling:(3)Pcre=π2EIL2=π2EAλ2.For CFS lipped channel columns with holes, Moen and Schafer [24] proposed the approximate prediction method for critical elastic flexural buckling strength (Pcre), which can be dominated by the weighted average method and the prediction equation is as follows:(4)Pcre.h=π2EL2IgLg+InetLnetL=π2EAλh2,where Ig, Lg are the moment of inertia of nominal gross cross section about the axis of buckling and segment length without holes, respectively, Inet, Lnet are the moment of inertia of net cross section about the axis of buckling and length of holes, respectively, λh is the slenderness ratio considering the effect of holes.Section 75 × 30 × 10 × 2.5 is used as an example to verify the accuracy and applicability of equation (4). The comparisons of the critical elastic flexural buckling strength between predicted results with different methods and FEA results are plotted in Figure 19, where Pcre, Pnet, and Pcre.h are FEA results, the predicted results with the moment of inertia of net cross section, and the predicted results with equation (4), respectively. It can be observed that the predictions using the net section are too conservative and the predictions with formula (4) are unconservative for members with large web holes. Therefore, equation (4) is modified as equation (5) to give more reliable prediction results for the critical elastic flexural buckling strength of CFS lipped channel columns with web hole based FEA.(5)Pmcre.h=π2EL2IgLg+InetLnetL1+0.08Hhh−0.3Hhh21−nLhL2=π2EIhL2=π2EAλmh2,where Ih and λmh are the modified moments of inertia of gross cross section about the axis of buckling and slenderness ratio considering the effect of holes.Figure 19
Comparison of elastic flexural buckling capacity between FE results and predicted results.The comparison of critical global buckling strength between FEA results and predicted results (Pmcre.h) by using equation (5) shown in Figure 19 indicates that formula (5) is accurate.Based on formulas (3) and (5), the following equation can be obtained:(6)λmhλ=PcrePmcre.h=IIh,where Ih=IgLg+InetLnet/L1+0.08Hh/h−0.3Hh/h21−nLh/L2, then the modified global buckling coefficient can be determined using the following:(7)φh=Pmcre.hPy=121+1λ¯mh21+ε0−1+1λ¯mh21+ε02−4λ¯mh2,where λ¯mh=λmh/πfy/E, λmh is the modified slenderness ratio considering the effect of holes which can be calculated by using equation (6).For the global flexural-torsional buckling coefficient of CFS axially compressed columns with holes, equation (7) can also be used. Saint-Venant torsion constant can be calculated using equation (8) and the warping constant can be predicted assuming cross-section thickness is zero at the hole as reported in [34].(8)Ith=ItgLg+ItnetLnetL1+0.08Hhh−0.3Hhh21−nLhL2,where Itg and Itnet are Saint-Venant torsion constants of gross cross section and net cross section.For CFS lipped channel axiallycompressed members with a circular hole, the diameterd can be used to replace Hh and Lh in equations (5) and (8).


## 6.2. Modified Effective Area Equation for Compression Members with Web Holes

The effective area of CFS axially compressed stud columns with holes can be determined by using the modified effective width method in [34]. However, it needs to note that the effect of web holes on the effective area will decrease with the increase of the length of columns. Therefore, the effective areas of CFS columns with different lengths can be predicted by using equation (9), considering the effect of dimensions of holes and the length of columns based on FEA results.(9)Aeh=Ae1,λ¯≤0.561AAnet,1+λ¯−λ¯1λ¯2−λ¯1⋅Ae2−Ae1Ae1Ae1,0.561AAnet<λ¯<1.5AAnet,Ae2,λ¯≥1.5AAnet,where Ae1, Ae2 are the effective area considering the effect of holes using the modified effective width method in [16] and the effective area without considering the effect of holes. Anet is the net sectional area, λ¯ is the normalized slenderness ratio of columns, and λ¯1=0.561A/Anet, λ¯2=1.5A/Anet.


## 6.3. Comparison of Design Capacity with Test and FEA Data

The load capacities determined by the modified design formula (1) were compared with the ultimate capacities obtained from tests and FE analyses, as listed in Tables 3–5. The mean value of the test-to-prediction ultimate strength ratio (Pt/Pc) is 1.008 with the corresponding standard deviation of 0.04, as shown in Table 3. The mean values of prediction-to-FEA ultimate strength ratios (Pc/Pa) are 0.954 and 0.965 with the corresponding standard deviation of 0.054 and 0.049 for columns with circular and rectangular holes as listed in Tables 4 and 5, respectively.Figures20 and 21 show that the modified load capacity curves for section 75 × 30 × 10 × 2.5 with the size of circular holes d/h = 0.3, 0.5, and 0.7 and rectangular holes Hh/h = 0.3, 0.5, and 0.7 give good agreement with the FEA results for CFS lipped channel columns with web holes, respectively, where Pa and Pc are FEA and predicted load capacities, Py is the yield strength.Figure 20
Comparison of the ultimate capacity of columns with circular holes between FEA results and predicted results using the modified method. (a)d/h = 0.3. (b) d/h = 0.5. (c) d/h = 0.7.
(a)(b)(c)Figure 21
Comparison of ultimate capacity of columns with rectangular holes between FEA results and predicted results using the modified method (Hh : Lh = 1 : 2). (a) Hh/h = 0.3. (b) Hh/h = 0.5. (c) Hh/h = 0.7.
(a)(b)(c)The comparison results shown in Tables3–5 and Figures 20 and 21 indicate that the predicted results by using a modified formula based on modified global buckling coefficient and modified effective area are extremely accurate.


## 7. Conclusions

This paper presents a total of 42 experimental results on the axial strength of CFS lipped channels with circular holes, rectangular holes, and without holes. Based on the experimental and numerical results presented in this paper, the following conclusions can be drawn:(1)
It was found that all specimens with a large width-to-thickness ratio exhibited the interaction of local buckling, distortional buckling, and global buckling, while specimens with a small width-to-thickness ratio displayed global buckling. The width to thickness of the plate adjacent to the hole decreased with the increase of the hole size, so buckling stress and the buckling load gradually increased. The axial strength of the CFS lipped channel with holes gradually decreased with the increase of the hole size. The axial strength decreased maximum by 20.48% and 22.98% for the same sections with circular holes and rectangular holes, compared to the member without web holes.(2)
A nonlinear FE model was developed, showing a good agreement with the test results both in terms of ultimate strength and buckling mode. Using the validated FEM, a parametric study comprising 36 models was conducted to investigate the effects of column slenderness and size of the hole on the axial strength of such channels. The results obtained from the parametric study show that the axial capacity of CFS lipped channels with holes was greatly decreased due to the presence of holes and the effects of the hole size on the axial capacity of such sections were significant. The axial capacities of CFS lipped channels with holes were decreased with the increase of the column slenderness. The proven FEM was used to develop the global buckling coefficient equation and effective area formula of the CFS lipped channel section under axial compression considering the effect of holes.(3)
An improved design equations based on the developed global buckling coefficient equation and effective area formula were developed in this paper. The results obtained from numerical analysis and tests were compared against the design strengths determined from the developed design method for CFS lipped channels with holes. It was found that the developed design method can closely predict the axial strength of CFS lipped channel columns with holes, being only 4.6% maximum on average to experimental and numerical results.



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*Source: 6658099-2021-07-12.xml*