Arab J Sci Eng DOI 10.1007/s13369-014-1282-5

RESEARCH ARTICLE - CIVIL ENGINEERING

Finite Element-Based Investigation on Performance of Intermediate Length Thin-Walled Columns with Lateral Stiffeners M. Anbarasu · S. Sukumar

Received: 27 April 2013 / Accepted: 30 November 2013 © King Fahd University of Petroleum and Minerals 2014

Abstract Distortional buckling of compression members usually comprises rotation and translation of each flange and lip about the flange-web juncture in opposite directions. The present study makes an attempt to possibly delay or prevent the distortional buckling mode by the introduction of lateral stiffeners. The stiffened channel section has been selected based on the elastic buckling analysis using CUFSM software to ensure the occurrence of distortional buckling. The test programme consisted of two pure axial compression tests under hinged–hinged end condition. A finite element model of the tested compression members was then developed and validated with the help of experimental results and also by the column test results conducted by Young and Yan (J Struct Eng 732:728–736, 2002). Geometric, material nonlinearities and initial geometric imperfection were included in the finite element model. An extensive series of parametric analyses was undertaken using the validated finite element model to investigate the effect of varying depth and many lateral stiffeners on the ultimate capacity of the columns. The details of the parametric study results are presented as charts. The lateral stiffeners effect on behaviour and strength is discussed. Keywords Cold-formed steel · Column · Distortional buckling · Lateral stiffener · Thin-walled member

M. Anbarasu (B) · S. Sukumar Department of Civil Engineering, Government College of Engineering, Salem 636 011, Tamilnadu, India e-mail: [email protected]

1 Introduction Until recently, the hot-rolled steel members have been recognized as the most popular and widely used steel group, but in recent times, the use of cold-formed high strength steel members has rapidly increased. However, the structural behaviour of these thin-walled steel structures is characterized by a range of buckling modes such as local buckling, distortional buckling and global buckling. The behaviour of short compression member is well defined in the literature. The global buckling (flexural/flexural torsional) behaviour of cold-formed steel sections have been extensively studied in the past. Distortional buckling plays an important role in the use of open cold-formed steel columns. Distortional buckling occurs at intermediate length columns. Therefore, it is important to eliminate or delay these buckling problems with the introduction of lateral stiffeners intermediate length cold-formed steel column.

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Takahashi [1] was the first researcher to publish a paper describing the distortional buckling mode. Hancock [2] presented a detailed study of a range of buckling modes (Local, distortional, and flexural-torsional) in a lipped-channel section. Lau and Hancock [3] provided simple analytical expressions to allow the distortional buckling stress to be calculated explicitly for any geometry of cross-section of thinwalled lipped-channel section columns. Kwon and Hancock [4] studied simple lipped channels and lipped channels with intermediate stiffener under fixed boundary conditions. They have chosen section geometry and yield strength of steel to ensure that a substantial post-buckling strength reserve occurs in a distortional mode for the test section. Davies and Jiang [5] used the generalized beam theory to analyse the individual buckling modes either separately or in selected combinations. The distortional buckling strength of a few innovative and complex geometrical sections has been studied by Narayanan and Mahendran [9]. For intermediate length pallet rack columns, the distortional strength was studied by providing stiffeners to connect the flanges of upright sections by Talikoti and Bajoria [12]. Kut and Stachowicz [13] have experimentally and numerically studied the sheet metal forming procedure by blanking. The partly closed thin-walled steel columns were studied by Veljkovic and Johansson [14]. Kwon et al. [15] studied the buckling interaction of the channel columns. Shi et. al [16] conducted tests and finite element analysis on the local buckling of 420 MPa steel equal angle columns under axial compression. Theofanous and Gardner [17] studied the effect of element’s interaction and material nonlinearity on the ultimate capacity of stainless steel cross-sections. Anbarasu and Sukumar [18] studied the effect of stiffener ties in the intermediate length cold-formed steel (CFS) columns. Anbarasu and Sukumar [19] studied the connectors interaction on the behaviour and ultimate strength of stiffened channel columns. The review of these papers suggests that buckling studies of cold- formed steel column with lateral stiffeners are limited. Therefore, a detailed parametric study based on finite element analyses was undertaken to fully understand the buckling behaviour of intermediate length cold-formed steel column with lateral stiffeners. The aim of this paper was to investigate the behaviour and strength of intermediate length pin-ended column with lateral stiffeners using finite element analysis. The finite element programme ANSYS was used in the analysis. Lateral stiffeners are the transverse elements of CFS sheet used to connect the lips of the sections using self-drilling screw. For this work, stiffened channel-shaped section with lip was considered. The column length and cross-section dimensions were carefully selected to ensure distortional buckling using CUFSM [7] software. Totally, two columns have been tested under pinned end condition. The finite element model was

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validated by the experiments conducted and also by the column test results conducted by Ben Young and Jintang yan [8]. A parametric study was done to investigate the behaviour of intermediate length thin-walled columns with lateral stiffeners of varying the number and depth of stiffeners.

2 Test Programme 2.1 Test Specimens The test programme considered two column specimens. The first section is fully opened, and the second section is with one lateral stiffener of 50 mm depth at the centre. The section was carefully selected to have distortional buckling using CUFSM software. CUFSM employs the semianalytical finite strip method (FSM) to provide solutions for the cross-section stability of thin-walled members. A thinwalled cross-section is discretized into a series of longitudinal strips, or elements. The FSM is a variant of the finite element method that has been put to highly effective use in the study of the stability of thin-walled structures. For any thinwalled profile which may effectively be modelled as extruded FSM provides an inevitably powerful simplification to finite element method. The dimensions of the cross-section were chosen by keeping the plate slenderness ratio (b/t) within limits to eliminate local buckling. All specimens were tested in pure axial compression with pinned end conditions. The buckling plot for the selected section is shown in Fig. 1. It can be observed that local buckling occurred at very short half wavelengths of approximately 60 mm and it can be noted that local buckling occurs in all plate elements. The distortional mode has a minimum at 1,067 mm in half wave length. From the buckling plot, intermediate length is chosen to investigate distortional buckling for column length as 1,200 mm to eliminate global buckling effects. The crosssection profile of the selected section is shown in Fig. 2. The specimens are fabricated by press-braking operation. Lateral stiffeners are the transverse elements made up of the same material which is used for specimen, cut into required shape and connect the lips of the section using self-drilling screw. One screw of 6 mm in diameter is used at each interconnection. Table 1 shows the dimensions of the specimen. 2.2 Labelling Figure 3 explains a typical specimen label for parametric study. 2.3 Tension Coupon Tests The tensile coupon tests are carried out in accordance with IS 1608-2005 (Part-1) [11]. The stress–strain curves obtained

Arab J Sci Eng Fig. 1 Buckling plot of a selected section

B

350 300

Stress in N/mm2

A

C

D

250 200 150 100 50 0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

Strain in mm/mm

C

A

Fig. 4 Stress–strain curve of the coupon material

B Fig. 2 Geometry of the column

Table 2 Material properties of steel

Table 1 Dimensions of the specimen

Sl. no. Thickness Yield (mm) stress σy (Mpa)

Ultimate stress σu (Mpa)

Modulus % of of elasticity elongation E (Mpa)

1

310

2.01 × 105

Specimen

A

B

C

D

t

L

Nominal dimensions (mm) Average measured dimensions (mm)

15

60

60

60

1.2

1,200

15.1

59.7

59.8

59.9

1.19

1,198

1.2

248

27

2.4 Test Set-up

SC-S0-d50 Depth of stiffeners Number of stiffeners Stiffened channel Fig. 3 Typical specimen label for parametric study

from the coupon test is shown in Fig. 4. The material properties determined from coupon tests are summarized in Table 2.

The compression tests were carried out using the 400-kN capacity loading frame. The specimens had a 12-mm-thick rectangular steel plate welded to each end. The specimens were mounted between the platens, and its verticality was checked. At either end between the platens and the end plates of the specimen, rubber gaskets were placed to facilitate the pinned end condition at either support [10,18]. The test setup and schematic diagram are shown in Fig. 5a, b. A force control method was used to apply a uniform distributed compression load gradually. Dial gauges were placed at mid-height on the web and flange to measure lateral displacement and one at the lower end to measure the axial deformation. The centroid axis of the end plates and the specimen cross-section were kept the same to simulate a uniformly

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Arab J Sci Eng Fig. 5 Test configuration

Table 3 Test results Specimen ID

Experimental load, KN

Failure mode

Ultimate stress, σu N/mm2

σu /σy

SC-S0

56.2

D+FT

149.87

0.604

SC-S1-d50

66.4

L+D+F

177.06

0.714

due to the provision of lateral stiffener. The provision of stiffeners improves the torsional rigidity of the section. The lateral stiffeners on the section have significant influence on the capacity.

3 Finite Element Modelling distributed load for pin-end conditions. The lateral and axial deformations of the column were recorded for every increment of load. After the ultimate load was reached, two more readings were observed before the specimen became unstable (Fig. 10). 2.5 Test Results and Discussion The test of ultimate loads and failure modes of the sections are summarized in Table 3. The tested specimens are shown in Fig. 6. On observation, it clearly indicates that the predominant mode of failure of the fully opened section is combined combined distortional mode (D) and flexural torsional buckling. The open section failure mode changes to the interference of combined local (L), distortional (D) and flexural buckling (F) buckling mode

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3.1 General The finite element programme ANSYS 13 was used to simulate the ultimate loads, axial deflections and failure modes of the tested columns. The models were based on the centre line dimensions of the cross-sections. The residual stresses and the rounded corners of the sections were not included in the model. The effect of residual stresses on the ultimate load is considered to be negligible as shown by Schafer and Pekoz [6]. The numerical simulation consisted of two stages. In the first stage, an eigen-buckling analysis was performed on a ‘perfect’ geometry to establish probable buckling modes of a column. In the second stage, a non-linear analysis by incorporating both geometric and material non-linearities were then performed using the arc length method to obtain the ultimate load and failure modes of the column.

Arab J Sci Eng

Fig. 6 Tested specimens Fig. 7 Meshed model

3.2 Element Type and Mesh The specimens were modelled using Shell 181 elements. It is a four-node element with six degrees of freedom at each node: translations in the x-, y-, and z-directions, and rotations about the x-, y-, and z-axes. (If the membrane option is used, the element has translational degrees of freedom only). The degenerate triangular option should only be used as filler elements in mesh generation. From the convergence study, it is found that an approximate mesh size of 12 mm × 15 mm (width by length) in all flange and web elements and 7.5 mm × 15 mm in the lip are adopted. Meshed model with single lateral stiffener is shown in Fig. 7. 3.3 Boundary Condition For the unloaded end, all three translations together with the rotation along the longitudinal axis of the cross-section were restrained. At the loaded end, translation along the longitudinal axis was released and other translations were restrained. Rotation about the longitudinal axis restrained and other rotations were released. The boundary conditions introduced to the centroid node and they were distributed to the other nodes through the rigid region which was created at both ends as shows in Fig. 8.

Fig. 8 Column ends modelled using Rigid region

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3.4 Screw Modelling and Method of Loading The self-drilling screws were modelled using spring elements, and in the longitudinal and transverse direction, a linear-elastic behaviour up to 0.8 mm of relative displacement was assumed with a constant stiffness of 5 kN/mm after which a perfectly plastic relationship was assumed at 4 kN. These values were based on tests performed by the supplier. On top of the master node, the load was applied and the corresponding reaction was recorded at the bottom node. The main purpose of the rigid region on both sides was that the displacement at top node being noted due to the reaction at the bottom node along the y-direction.

Fig. 9 Schematic diagrams of initial geometrical imperfections Table 4 Comparisons of results

3.5 Material Properties The material nonlinearity was included in the FEM by specifying the true values of stresses and strains. The engineering stresses and strains obtained from the tensile coupon tests were used to calculate the true stresses and strains.

Specimen ID

Test load, Ptest KN

ANSYS load, PFEA KN

Ptest /PFEA

SC-S0

56.4

59.5

0.948

SC-S1-d50

66.4

70.51

0.942

Mean

0.945

SD

0.004

3.6 Geometric Imperfections The load-carrying capacity is mainly affected by geometric imperfections which occur due to manufacturing and fabrication. This imperfection can be simulated using buckling analysis of the specimen. Since precise data on the distribution of geometric imperfection are not available, scaled value of linear buckling mode shape is used to create an initial geometric imperfection for nonlinear analysis. The degree of imperfection is assumed as the maximum amplitude of the buckling mode shape and considered as a percentage of the structure thickness. A superposition of the minimum local buckling mode and minimum distortional buckling mode is employed for the imperfection shape that is critical and obtained by conducting the eigen-buckling analysis. The maximum value of distortional imperfection was taken equal to the plate thickness as recommended by Schafer and Pekoz [6]. Local buckling imperfection was taken as 0.25 times the thickness. As shown in Fig. 9, the imperfections due to local and distortional are defined as l and d , respectively. 3.7 Non-Linear Analysis Two types of analysis, first the bifurcation buckling analysis to determine the elastic buckling loads and modes followed by the nonlinear analysis to determine the ultimate loads and deformations, including post-local buckling reserve strength was carried out using ANSYS. Thenon linear analysis was performed considering both geometric and material nonlinearities.

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Table 5 Comparison of finite element and experimental results of thinwalled lipped-channel columns tested by Young and Yan [8] Specimen ID (as per literature)

Ptest (kN)

PFEA (kN)

Ptest /PFEA

L48F0300

111.9

112.134

1.002

L48F01000

102.3

99.357

0.971

L48F01500

98.6

99.625

1.01

L48F02000

90.1

87.895

0.975

L48F02500

73.9

77.792

1.052

L48F03000

54.3

57.29

1.055

Mean

0.990

COV

0.035

4 Validation of Finite Element Model The finite element model has been validated by comparing results with experiments conducted and also with column test results conducted by Young and Yan [8]. The ultimate loads (PFEA ) predicted by the FEA are compared with the experimental ultimate loads (Ptest ) as shown in Tables 4 and 5 that shows good agreement. The mean and SD of the test to FEA ultimate loads for tests conducted and test results from the literature [8] are 0.945, 0.990 and 0.004, 0.035, respectively. As an example, the load versus axial shortening curve obtained in FEA is compared with the test results for SC-S0 section in Fig. 10 and it closely matches with the experimental results. The ANSYS results are slightly higher than

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– The term ‘d/S’ is defined as the ratio of the depth of the stiffener plate-to-the breadth of the stiffener plate.

Load in kN

60 50

The width of the lateral stiffener and the slenderness ratio of the column are 240 and 35 mm, respectively. Four groups of sections are formulated based on the ‘d/S’ ratio are 0.208, 0.312 and 0.417. Since the number of stiffeners varies from 1 to 5, the ‘a/L’ ratio varies from 0.50, 0.33, 0.25, 0.20 and 0.167. The λs /λ ratio varies from 0.017 to 0.208.

40 30 20 10 0 0

1

2

3

4

5

Axial Shortening in mm ANSYS

Test

Fig. 10 Comparison of test and FE analysis load versus axial shortening curves for SC-S0

the test results. The difference in the ANSYS model is more likely due to assumed imperfections of the sections. The deformed shape of the specimen obtained from the FEA as shown in Fig. 11a, b closely simulated the experimental buckling mode. The resemblance of Fig. 11a, b demonstrates the reliability of the FEA predictions. Similar results have been obtained from other specimens also. 5 Parametric Study The validated finite element models were then used to conduct an extensive parametric study to investigate the column behaviour by the addition of lateral stiffeners. This parametric study included the effects of various parameters such as depth and number of lateral stiffeners. The parameters which have direct influence on the response of the column are illustrated in Fig. 12. The parametric study included a total of 20 analysis cases. The two main parameters that were varied during the analysis are 1. Number of lateral stiffeners, varies from 1 to 5. 2. ‘d’ the depth of the stiffener plate, varied as 20, 30, 40 and 50 mm. The term λs /λ which is defined as the ratio of the stiffener plate slenderness to column slenderness  σy d (1) λs = × t E where, λs = Slenderness ratio of the lateral stiffener plate λ = Slenderness ratio of the column d = depth of lateral stiffener plate t = thickness of lateral stiffener plate – The term ‘a/L’ is defined as the ratio of the centre to centre distance (a) between stiffeners to the overall length (L) of the Column.

6 Parametric Study Results The normalized ratios of the ultimate-to-the yield stress of the column were influenced by various parameters and are graphically represented in the following subsections. The plot for σu /σy versus a/L (Fig. 13) indicates the relationship between the normalized ratio of the centre to centre length between lateral stiffeners to the overall length of the column (a/L), and the normalized ratio of the ultimate stress to the yield stress of the column (σu /σy ) for different values of (d/S). Obviously, the centre to centre distance between stiffeners has a significant effect on the strength of columns. Enhanced column strength values were obtained upon decreasing the stiffeners spacing (a). Increasing the number of stiffeners from 1 to 5 improved the ultimate capacity of the columns. The plot for σu /σy versus d/S (Fig. 14) demonstrates the influence of changing the depth of the lateral stiffener on the column strength for a slenderness ratio of 35. Apparently, as shown in Fig. 13, the column strength increases with the increase in depth of the stiffener plate. The plot for σu /σy versus λs /λ (Fig. 15) demonstrates the influence of stiffener plate slenderness ratio of the column strength for a slenderness ratio of 35. Apparently, as shown in Fig. 14, the column strength increases with the increase in the plate slenderness ratio. The parametric study results are summarized in Table 6.

7 Discussions The results reported in this paper illustrate a number of important points in relation to the performance of structural behaviour and ultimate capacity of an intermediate length column with lateral stiffeners. From the parametric study, it is ascertained that interaction between the parameters like the ratio of the slenderness of stiffener plate to column (λs /λ), the ratio of length (a/L) and ratio of depth to width of stiffener plate (d/S) influence the strength of the columns. With the addition of stiffeners in the columns, the failure mode shifted from combined flexural torsional buckling and distortional mode to the interference of combined flexural buckling and distortional buckling mode. From the ultimate loads of all

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Arab J Sci Eng Fig. 11 a Comparison of experimental and FEA deformed shapes. b Comparison of experimental and FEA deformed shape for Specimen L48F0300

(A)

(a) For Specimen SC-S0

(b) For Specimen SC-S1-d50

(B)

the columns with a number of stiffeners 1, 2, 3, 4 and 5, it is inferred that the improvement in torsional rigidity increases the load-carrying capacity. The percentage increase in the ultimate load of the column with different depth such as 20, 30, 40 and 50 mm stiffeners over fully opened section ranges from 4.13 to 34.01 %. The ultimate load-carrying capacity of the column with lateral stiffeners increases uniformly by the increase in number and depth of lateral stiffeners. The Figs. 13, 14 and 15 provide the stress factor to calculate the ultimate capacity of column with lateral stiffeners of required number and depth of lateral stiffeners.

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8 Summary and Conclusions The paper presents numerical investigations of intermediate length cold-formed steel compression members with and without stiffeners compressed between pin ends. A finite element model including geometric and material nonlinearities has been developed and verified against experiments conducted and test results available in the literature. It is shown that the ultimate loads of the columns predicted using the finite element analysis are generally in good agreement with the experimental ultimate loads. The validated finite element

Arab J Sci Eng

0.90

0.9

0.85

0.85

0.80

0.8

σu /σy

σu /σy

Fig. 12 Geometry of the columns

0.75

0.7

0.70

0.65

0.65 0.60 0.15

0.75

0.6 0.10

0.25

0.35

0.45

0.20

0.25

d/s = 0.207 d/s = 0.345

Fig. 13 σu /σy versus a/L curves

0.30

0.35

0.40

d/S

a/L d/s = 0.138 d/s = 0.276

0.15

0.55

a/L = 0.50

a/L = 0.33

a/L = 0.20

a/L = 0.167

a/L = 0.25

Fig. 14 σu /σy versus d/S curves 0.9

1. Adding the stiffeners to the intermediate length column has a significant effect on the improvement in ultimate load. 2. The ultimate load capacity of the intermediate length column increases with an increase in depth and number of lateral stiffeners. 3. The addition of stiffeners improves the capacity of the intermediate length column by enhancing the torsional rigidity of the section.

0.85 0.8

σu /σy

models are then used to undertake an extensive parametric study by varying the depth and number of stiffeners. The parametric study results are presented in the form design charts. Based on the study, the following conclusions are drawn within the limit of the present investigation.

0.75 0.7 0.65 0.6 0

0.05

0.1

0.15

0.2

s/

d/s = 0.138

d/s = 0.207

d/s = 0.276

d/s = 0.345

Fig. 15 σu /σy versus λs /λ curves

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Arab J Sci Eng Table 6 Parametric study results Specimen ID

Ultimate load in kN

% increase

a/L

d/S

(λs /λ)

σu /σy

SC-S0

59.30

–

–

–

–

–

SC-S1-d20

61.75

4.13

0.500

0.138

0.017

0.663

SC-S2-d20

66.90

12.82

0.333

0.138

0.033

0.718

SC-S3-d20

68.05

14.76

0.250

0.138

0.050

0.731

SC-S4-d20

70.80

19.39

0.200

0.138

0.067

0.760

SC-S5-d20

76.60

29.17

0.167

0.138

0.083

0.822

SC-S1-d30

64.21

8.28

0.500

0.207

0.025

0.689

SC-S2-d30

68.23

15.06

0.333

0.207

0.050

0.732

SC-S3-d30

69.40

17.03

0.250

0.207

0.075

0.745

SC-S4-d30

72.80

22.77

0.200

0.207

0.100

0.782

SC-S5-d30

77.40

30.52

0.167

0.207

0.125

0.831

SC-S1-d40

67.40

13.66

0.500

0.276

0.033

0.724

SC-S2-d40

69.32

16.90

0.333

0.276

0.067

0.744

SC-S3-d40

70.00

18.04

0.250

0.276

0.100

0.751

SC-S4-d40

74.80

26.14

0.200

0.276

0.133

0.803

SC-S5-d40

78.23

31.92

0.167

0.276

0.166

0.840

SC-S1-d50

70.11

18.23

0.500

0.345

0.042

0.752

SC-S2-d50

71.12

19.93

0.333

0.345

0.083

0.764

SC-S3-d50

71.60

20.74

0.250

0.345

0.125

0.774

SC-S4-d50

76.12

28.36

0.200

0.345

0.166

0.817

SC-S5-d50

79.47

34.01

0.167

0.345

0.208

0.853

4. From this investigation, it is observed that the use of lateral stiffeners at the proper depth and spacing help increase not only the ultimate load capacity but also changes the failure mode. 5. The finite element model developed using ANSYS software is sufficiently accurate in predicting the behaviour and ultimate capacity of the columns. Therefore, the finite element analysis can be used with a high level of confidence in predicting the load capacity of axially loaded cold-formed steel intermediate length columns with lateral stiffeners.

Based on this study, it is recommended that depending upon the structural needs the dimensions of depth and number of lateral stiffeners for the column can be used. This report shows that further research is needed in this area to have more experimental data to add the design codal provisions for the interaction of stiffeners in the ultimate load capacity of intermediate length columns. The findings reported in this paper are based on uniform distribution of stiffeners throughout the length of the column. The non-uniform distribution of stiffeners is under investigation to fully understand the structural behaviour of intermediate length columns.

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Arab J Sci Eng 12. Talikoti, R.S.; Bajoria, K.M.: New approach to improving the distortional strength of intermediate length thin-walled open section columns. Electron. J. Struct. Eng. 5, 69–79 (2005) 13. Kut, S.; Stachowicz, F.: Blanking with pre-bending of steel sheets—experimental investigation and fem simulation. Arab. J. Sci. Eng. 30(1C), 17–26 (2005) 14. Veljkovic, M.; Johansson, B.: Thin-walled steel columns with partially closed cross-section: tests and computer simulations. J. Constr. Steel Res. 64, 816–821 (2008) 15. Kwon, et al.: Compression tests of high strength cold-formed steel channels with buckling interaction. J. Constr. Steel Res. 65, 278– 289 (2009) 16. Shi, G.; et al.: Tests and finite element analysis on the local buckling of 420 MPa steel equal angle columns under axial compression. Steel Compos. Struct. Int. J. 12(1), 31–51 (2011)

17. Theofanous, M.; Gardner, L.: Effect of element interaction and material nonlinearity on the ultimate capacity of stainless steel cross-sections. Steel Compos. Struct. Int. J. 12(1), 73–92 (2011) 18. Anbarasu, M.; Sukumar, S.: Study on the effect of ties in the intermediate length cold formed steel (CFS) columns. Struct. Eng. Mech. Int. J. 46(3), 323–335 (2013) 19. Anbarasu, M.; Sukumar, S.: Effect of connectors interaction in behavior and ultimate strength of intermediate length cold formed steel (CFS) open columns. Asian J. Civil Eng. 14(2), 305–318 (2013)

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