Introduction

Yielding, local buckling, distortional buckling, global buckling and interactive failure of these modes may affect the strength and behaviour of the cold-formed steel (CFS) section under compression. Conventionally, the effective width method (EWM) was more familiar with the design of CFS compression members. However, EWM involves tedious and iterative calculations for the design of CFS compression members. Therefore, the direct strength method (DSM) has been recently included in the North American Specification (AISI-2007) for the design of CFS compression members. The interactive buckling modes decrease the strength of the member to a value below the strength under the independent buckling modes. The aim of this study is to evaluate the accuracy of the DSM for the interactive failure.

Review of literature

A brief review of literature on the ultimate strength and buckling modes of web stiffened cold formed steel channel column with various types of edge stiffeners are presented here. Kwon and Hancock (1992) was the first to publish a paper describing the interaction between local and distortional buckling of the intermediate cold-formed steel thin-walled sections. Very recently, the interactive behaviour of lipped channel columns with and without intermediate web stiffeners has fascinated several researchers. Young and Yan (2002) and Yan and Young (2002) discussed local and distortional buckling of plain channel with upright and complex edge stiffener under axial compression. Experimental investigations on stiffened lipped channels by Yang and Hancock (2004), by Yap and Hancock (2008) on an complex high-strength cold-formed cross-shaped steel column, by Kwon et al. (2009) on an cold-formed simple lipped channels and lipped channels with intermediate stiffeners in the flanges and web, by Yap and Hancock (2011) on an web-stiffened high strength steel cold-formed lipped channel columns. Local-distortional buckling mode interactions of lipped channel columns were discussed by Young et al. (2013), He et al. (2014) Martins et al. (2015) and Manikandan and Arun (2016). Local, distortional and flexural interaction of channels column were reported by Dinis and Camotim (2011) and Wang et al. (2016). From the review of literature it is observed that, results on strength and buckling behaviour of cold-formed steel channel column with intermediate web stiffener is scattered. Hence in this study, web stiffened cold formed steel channel columns with various types of edge stiffeners are selected.

Totally, six cold formed steel columns with different cross-section geometry are tested and results are compared with finite element analysis. The numerical analysis is conducted by using the nonlinear finite element package ANSYS. Numerical models are validated with the experimental results. The validated finite element model is then used to undertake a parametric study. The strength of the column is primarily governed by thickness, yield stress and overall slenderness ratio. Two columns with dissimilar configurations are selected for the parametric study. The dimensions are chosen based on the limitations for pre-qualified sections (AISI-S100 2007). The slenderness ratio varies from 20 to 120 and yield stress also varies from 270 to 550 N/mm2 for the selected profiles. The result predicted by the finite element analysis is compared with the design column strengths determined by Direct Strength Method (DSM). Based on the evaluation of results, an appropriate recommendation on DSM is projected to improve the evaluation on the strength of CFS columns. The efficiency of the proposed model is confirmed against the test results available in the literature for open columns.

Experimental investigations

Material properties

The material properties of the section are directly obtained from the coupon test as per the Indian Standard Specifications [IS 1608-1]. Totally, six coupons are tested and average material properties are presented in Table 1. Typical stress–strain behaviour for the material testing is shown in Fig. 1.

Table 1 Material properties of steel
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Fig. 1
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Stress–strain behaviour of tensile coupons

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Specimen labeling

In this study, four series of cross section are tested as shown in Fig. 2. They are web stiffened channel (WSC), web stiffened channel with inclined edge stiffener (WSC-IE), web stiffened channel with upright edge stiffener (WSC-UE) and web stiffened channel with complex edge stiffener (WSC-CE). Based on the variables in the parametric study, four different types of specimen labeling rules are used and also it is discussed in the appropriate section. The following specimen labels are used for the verification of finite element analysis and evaluate the effectiveness of the proposed cross section as shown in Fig. 2. For an example, the label “WSC-IE-λ50”, the first term defines the type of cross section [web stiffened channel (WSC)], the second term defines the types of edge stiffener [inclined edge stiffener (IE), upright edge stiffener (UE) and complex edge stiffener (CE)] and last term defines the slenderness ratio of the member (λ50; λ is the symbol for slenderness ratio and 50 for magnitude of slenderness ratio).

Fig. 2
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Section geometry with acronym

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Selection of section and test setup

The selected cross-section profile with corresponding nomenclature is shown in Fig. 2. The dimension of the selected cross section is presented in Table 2 and also it is met with the pre-qualified column dimensions of the cold formed steel section. Totally, six specimens are tested with a nominal overall slenderness ratio ranging from 30 to 50. All the specimens are fabricated from locally available 1.6 mm thick cold rolled steel sheets by using the brake-pressed operation. All the specimens are tested under axial loading with subjected to pinned end conditions. To achieve the uniform compressive force over the entire cross section of the column, the thick steel plates are welded at each end of the column. The loads are applied at the C.G of the section by using the load cell and before applying the load, the verticality of the column is verified. Before positioning the data acquisition (LVDT), a pre-quantity of 5 kN load is applied to the specimen. The entire load–deflection response is monitored by using the data acquisition system. Observation of test results is presented in Table 3. The experimental set-up is illustrated in Fig. 3.

Table 2 Details of specimen
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Table 3 Observation of test results
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Fig. 3
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Experimental set-up

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Finite element analyses

Numerical investigation is carried out by using the finite element analysis software package ANSYS. Shell 181 element is used for modeling the specimen. To minimizing the computational time and improving the accuracy of the finite element analysis, a mesh sensitive analysis is carried out. From the mesh sensitive analysis, a mesh size of 100 mm2 is used for entire study. Residual stress (He et al. (2014) and GB 50018-2002 (2002)) and effect of cold forming process (Yan and Young 2002) are ignoring in the finite element (FE) model. Master nodes (MN1 and MN2) are generated by using the Structural 3D element available from the ANSYS material library. Rigid regions are generated through the master node and master node are coincides at the C.G of the section. Load and end boundary conditions are applied through the master nodes. All the specimens are analyzed under axial loading with pinned end conditions. Detailed finite element modeling are illustrated in Fig. 4. Material imperfections are incorporated in the FE model. Material properties are directly obtained from the coupon test and bilinear material properties are used. Before the tests, the section dimensions of each specimen are measured at the mid-height and both ends by using the vernier caliper. During the measurements of imperfection, it is observed that the geometric imperfections and rounded response are not observed. Because, all the specimens are fabricated using the latest computerized cutting and sharp folding press braking machine. The sharp cutting edges provide right angular edge only. Based on the test results, the effect of cold forming process has been ignored in the FE model. Two types of analysis are carried out. First is Eigen buckling analysis and second is non linear large displacement static buckling analysis. The main purpose of the Eigen buckling analysis is to find possible buckling modes.

Fig. 4
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Details of finite element model

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Test verification

From the Fig. 5, it is observed that the strength of the section increases with modifying the types of edge stiffeners. The comparison of load deflection curves and failure modes between test and FEA are shown in Figs. 6 and 7, respectively. In this experimental study, interactive failure modes are observed such as local and distortional buckling and local and flexural buckling and also it is confirmed with FEA (Fig. 7). Totally, six specimens are tested and results are compared with finite element analysis (Table 3). The mean and standard deviation of P EXP and P ANSYS are 0.98 and 0.03, respectively. The similarity of Figs. 5 and 7 demonstrates the consistency of the FEA predictions.

Fig. 5
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Comparison of load deflection curve between test and finite element analysis

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Fig. 6
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Load deflection curve for tested specimens

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Fig. 7
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Comparison of failure modes a WSC-IE-λ50, b WSC-UE-λ50, c WSC-UE-λ40, d WSC-CE-λ50

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Parametric studies

The established FE model is used for an extensive parametric study. From this test results it is observed that, the WSC-UE and WSC-CE series provides a better performance compared to all other specimens. Hence, a detailed parametric study is carried out on the WSC-UE and WSC-CE series of specimens by using the verified finite element FE model. The parametric study included 67 specimens that consisted of 9 series as shown in Table 2. All the results are compared with the Direct Strength Specifications (DSM) for cold formed steel structures.

Theoretical investigation

The nominal capacity of members in axial compression (P n, DSM) shall be least of local buckling (P nl), distortional buckling (P nd) and flexural torsional buckling (P ne).

$$ P_{{n{\text{e}}}} = \left\{ {\begin{array}{ll} (0.658)^{{\lambda c^{2} }} P_{y} & {\text{for }} \lambda_{c} \le 1.5 \\ \left( {\frac{0.877}{{\lambda c^{2} }}} \right)P_{y} & {\text{for }} \lambda_{c} > 1.5 \\ \end{array} } \right. $$
(1)

where λ c  = \( \sqrt {P_{y} /P_{\text{cre}} } \) and P y = Af y .P y is the squash load.

$$ P_{nl} = \left\{ {\begin{array}{ll} {P_{ne} } & {{\text{for }}\lambda_{1} \le 0.776} \\ {\left( {1 - 0.15\left( {\frac{{P_{crl} }}{{P_{ne} }}} \right)^{0.4} } \right)\left( {\frac{{P_{crl} }}{{P_{ne} }}} \right) P_{ne} } & {{\text{for }} \lambda_{1} > 0.776} \\ \end{array} } \right. $$
(2)

where λ 1 = \( \sqrt {P_{\text{ne}} /P_{\text{crl}} } \)

$$ P_{nd} = \left\{ {\begin{array}{ll} {P_{y} } & {{\text{for }} \lambda_{d} \le 0.561} \\ {\left( {1 - 0.25\left( {\frac{{P_{crd} }}{{P_{y} }}} \right)^{0.6} } \right)\left( {\frac{{P_{crd} }}{{P_{y} }}} \right)^{0.6} P_{y} } & {{\text{for }} \lambda_{d} > 0.561} \\ \end{array} } \right. $$
(3)

where λ d  = \( \sqrt {P_{\text{y}} /P_{\text{crd}} }. \)

$$ P_{{n,{\text{DSM }}}} = {\text{Minmum}} \left( { P_{{n{\text{l}}}} ,P_{{n{\text{d}}}} ,P_{{n{\text{e}}}} } \right) $$
(4)

Effect of thickness variations

In this parametric study, initially effects of thickness variation on all the series of sections are analyzed. The specimens are labeled such that the cross section shapes and thickness of the sections could be identified, as shown in Table 4. For an example, the label “WSC-IE-T1.6-λ50”, the first term defines the type of cross section [web stiffened channel (WSC)], the second term defines the types of edge stiffener [inclined edge stiffener (IE), upright edge stiffener (UE) and complex edge stiffener (CE)], third term defines the thickness (T) of the specimen in mm and last term defines the slenderness ratio of the member (λ50; λ is the symbol for slenderness ratio and 50 for magnitude of slenderness ratio). The column strength obtained from the parametric study is exposed in Table 4. From this study, it is observed that the thickness of the section considerably affects the strength of the section.

Table 4 Effect of thickness variations
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Behaviour of web stiffened channel with upright edge stiffener

Two series of WSC-UE (WSC1-UE and WSC2-UE) with 11 slenderness variation (λ = 20, 30, 40, 50, 60, 70, 80, 90, 100, 110 and 120) gave the 22 cross-sectional geometries. All the specimens are failed by interactive mode of buckling. The observed failure modes are mixed local and distortional buckling, mixed flexural and distortional buckling and mixed local and flexural buckling. The comparison between FEA and DSM are presented in Table 5 and Fig. 8. From this study, it is observed that the DSM is producing the unconservative results for the interactive failure modes of the intermediate slenderness ranges of web stiffened channel with upright edge stiffener.

Table 5 Comparison of FEA and DSM results- Specimen with upright edge stiffener
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Fig. 8
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Comparison of results—specimen with upright edge stiffener a WSC1-UE, b WSC2-UE

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Behaviour of web stiffened channel with complex edge stiffener

Three series of WSC-CE section (WSC-CE-F270-λ, WSC-CE-F350-λ and WSC-CE-F550-λ) with 11 slenderness variation (λ = 20, 30, 40, 50, 60, 70, 80, 90, 100, 110 and 120) gave the 33 cross-sectional geometries. For as an example, load deflection curve and the failure mode of WSC-CE-F270-λ series of specimen are shown in Figs. 9 and 10, respectively. Similar results are obtained for all the cases and results are tabulated in Table 6. The comparison between FEA and DSM are presented in Table 6 and Fig. 11. Except for some of the cases, all the specimens are failed by interactive failure. From the Fig. 11, it is observed that the DSM is conservative for all the series of specimens except for the WSC-CE-F550 series.

Fig. 9
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Load-deflection curve WSC-CE-F270-λ series

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Fig. 10
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Failure modes of WSC-CE-F270-λ series

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Table 6 Comparison of FEA and DSM results- Specimen with complex edge stiffener
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Fig. 11
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Comparison of results. Specimen with complex edge stiffener a WSC-CE-F270, b WSC-CE-F350, c WSC-CE-F550

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New design equation

Figures 8 and 11 shows the variation of P ANSYS and P n, DSM for WSC-UE-λ and WSC-CE-λ series, respectively. It is observed, DSM provides unconservative results for WSC-CE-F550 series. Figure 12 shows the relationship between FEA and DSM results for all the series of specimens considered in this study. To determine the accurate strength of web stiffened cold formed steel channel column, a new design equation is proposed.

Fig. 12
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Relation between FEA and DSM results

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Ultimate strength of web stiffened channel column

$$ P_{\text{Prop}} = 0.793 P_{{n,{\text{DSM}}}} + 14.66 $$
(5)

Accuracy of the proposed design equation is verified with the test results available from the literature as shown in Table 7. The comparisons of the proposed DSM equation with experimental results available from the literature (Yan and Young 2002; Yap and Hancock 2008) are shown in Table 7. The mean and standard deviation of P EXP/Ppro are 1.03 and 0.07, respectively. It is clearly shown that the proposed design equation are able to predict the strength of the web stiffened cold formed steel channel columns more accurately than that the DSM equations.

Table 7 Verification of proposed design equation
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Conclusions

An experimental and numerical investigation of the structural response of the web stiffened cold formed steel channel columns with various types of edge stiffener has been presented in this paper. A nonlinear finite element technique was developed and validate with the experimental results. A good conformity has been achieved between test and FEA. The validated finite element technique was used to analyze a total of 67 web stiffened cold formed steel channel columns with various types of edge stiffener failed by interactive buckling. The result obtained from the parametric studies was compared with the Direct Strength Method (DSM) for cold formed steel structures. The comparison shows that DSM equation was over estimate the interactive modes of buckling. Hence, new design equation was proposed, and also it was validated with the test results available from the literature.