Introduction

In recent years, the cold-formed steel structural members are widely used in building construction. The cold-formed steel members can be formed by press brake or bending brake operations. They are usually formed as open and built-up section. A lot of research is carried out on cold-formed open section by Young and Hancock (1992), Popovic et al. (1999), Dhanalakshmi and shanmugam (2001), Schafer (2002), Yan and Young (2002), Narayanan and Mahendran (2003), Elobody and Young (2005), Young and Ellobody (2007), Zhang et al. (2007) and Young and Chen (2008a). In addition, some research was carried out on cold-formed built-up section such as Stone and Laboube (2005), Sukumar et al. (2006), Young and Chen (2008b), Whittle and Ram seyer (2009), Reyes and Guzmanc (2011), Georgieva et al. (2012), Zhang and Young (2012), Piyawat et al. (2013), Yuanqi Li et al. (2014), Aruna et al. (2015), Ting et al. (2017), Fratamico et al. (2018), and Roy et al. (2018a, b, c, 2019). Very little research is carried out on cold-formed built-up closed I section formed by connecting two open channels back to back and box section formed by connecting the channel sections toe to toe. Still, many test data have not been reported on cold-formed steel built-up closed section formed by connecting two angle sections with edge and intermediate stiffeners.

This paper describes an experimental investigation on the compressive strength and behaviour of cold-formed built-up closed section with intermediate web stiffeners under hinged end conditions. Totally 12 specimens were tested by varying the column length. An accurate and reliable finite-element model was developed using ANSYS. The finite-element model was validated against the test results. The validated finite-element model was used for an extensive parametric study. The strength obtained from the finite-element analysis was compared with the design strengths calculated using AISI specification of cold-formed steel structures. The reliability of the current design equation on the cold-formed built-up closed section with intermediated stiffeners was also investigated.

Experimental investigation

Test specimens

The test specimens were formed by connecting two angle sections with intermediate web stiffeners using self-tapping screws. Figure 1 illustrates the typical cross section of built-up section. The nominal dimensions of the cross section such as width of the top and bottom web element (w1), width of the inclined web element (w2), width of the intermediate web element (w3), width of the flange (Wf), width of the lip (Wl), and total width of the web (Ww) are presented in Table 1. The screws were arranged at a spacing of 200 mm and minimum edge distance of 20 mm, as shown in Fig. 2.

Fig. 1
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Cross section of built-up closed section

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Table 1 Nominal dimensions of the cross section
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Fig. 2
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Arrangements of screw spacing for 440 mm length of column

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The test specimens were categorized into three series based on the width of the intermediate web element and the specimens were labelled, such that BC40, BC50, and BC60 accordingly. For example, the label “BC40L440”, “BC” indicates the built-up closed section, “40” indicates the width of the intermediate web element, and the letter “L” indicates the nominal length of the specimen and follows by the digits “440” showing the length of the column.

Tensile coupon tests were conducted to obtain the material properties of the specimen. The coupon specimens were prepared according to IS 1608-2005 part I. Strain gauge was used to measure the longitudinal strain and the test results are listed in Table 2. Figure 3 shows the stress–strain behaviour for the specimen.

Table 2 Tensile test results
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Fig. 3
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Stress–strain behaviour for the specimen

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Experimental setup and operations

The column test was performed in a 1000 kN capacity self-straining loading frame. All the specimens were tested under axial compression with hinged end conditions (2015). Thick rubber gaskets were placed between the base plate and the platens (thick steel plate) to simulate the hinged–end conditions, at both supports (2006). The verticality of the specimen was also checked. The load was applied at the bottom end of the specimen through a hydraulic jack of 1000 kN capacity. A load cell was mounted above the hydraulic jack to measure the load increments. Three LVDTs were used two at the mid height, one on the flange and the other on the web to measure the lateral deflection and one at the bottom plate of the specimen to measure the axial shortening of the specimen. A data acquisition system was used to record the applied load and readings of the LVDT. The experimental setup is shown in Fig. 4.

Fig. 4
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Experimental setup

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Finite-element analysis

General

Finite-element analysis package ANSYS was used for the numerical investigation and it was carried out in two stages. In the first stage, an Eigen buckling analysis was carried out to establish the possible Eigen buckling modes of the specimen, which was used as initial geometric imperfection of the models for the subsequent non-linear buckling analysis. In the second stage, after incorporating the geometric imperfections, a non-linear buckling analysis was carried out using the arc-length method (2007). Centre line dimensions were used to model the cross section of the specimens.

Finite-element model

Shell 181 elements were used in the buckling analysis and structural mass 21 element was used to create the master node which has 6 degrees of freedom. 10 × 10 mm element size was used to model the specimens. The connections between two angle sections were modelled by coupling the translational and rotational degree of freedoms of x, y, and z directions at the screw location. In FEA, the material behaviour was described by a bilinear stress–strain curve. The material properties were taken from the tensile test results such as average yield stress of the material 272 N/mm2, young’s modulus 2.04 × 105 N/mm2, and tangent modulus as 2% of the young’s modulus. The effect of residual stress on the ultimate load was considered to be negligible as recommended by Schaffer and Pekoz (1998). The strain hardening of the corners due to cold forming was neglected. The maximum initial local and overall imperfection was taken as 0.25 times the thickness of the plate element (2002) and 1/1000 of the column length (AISC 2005), respectively. Super position of two possible least different eigen modes was factored by the magnitude of initial local and overall geometric imperfection. The loading end and reaction end were defined as master nodes, which were modelled at the centroid of the section. These master nodes were coupled to each node on the edge of the cross section. The load and boundary conditions were established to the master node. Rotation about y-axis and translations in both x and z directions were restrained at the top end and translation in three directions x, y, z and rotation about y-axis were restrained at the bottom. The modelling of screw connection and boundary conditions is shown in Fig. 5.

Fig. 5
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Modeling of specimen

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Validation of finite-element model

The finite-element analysis results were compared with the experimental results. The main aim of this comparison is to validate and ensure the accuracy of the finite-element analysis. The comparison of ultimate compressive stress and failure modes was obtained from the experimental and finite-element analysis is presented in Table 3. It shows that the FEA results were slightly higher than the experimental results. The mean value of σFEA/σEXP ratio is 1.047 with the corresponding coefficient of variation of 0.019. The failure modes were local buckling and flexural buckling and interaction of local and flexural buckling. The local buckling was observed in the column of length of 440 mm. The interaction of local and flexural buckling was observed in 840 and 1640 mm length of the columns and flexural buckling was observed in 2240 mm length of the column. The comparison of axial compressive stress vs. axial shortening curve and axial compressive stress vs. lateral deflection curve of the specimens was obtained from the FEA and experiments are shown in Figs. 6, 7, 8, 9, 10, and 11. It shows that both column stiffness and behaviour reflects good agreement between experimental and finite-element results. The comparison of deformed shapes observed from the experimental and FEA for BC40L840, BC40L1640, and BC40L2240 is shown in Figs. 12, 13, and 14, respectively. It is shown that the deformed shapes of the specimens obtained from the FEA closely simulate the experimental deformed shapes.

Table 3 Comparison between experiment and FEA results
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Fig. 6
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Axial compressive stress vs. axial shortening curves for BC40 series

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Fig. 7
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Axial compressive stress vs. lateral deflection curves for BC40 series

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Fig. 8
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Axial compressive stress vs. axial shortening curves for BC50 series

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Fig. 9
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Axial compressive stress vs. lateral deflection curves for BC50 series

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Fig. 10
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Axial compressive stress vs. axial shortening curves for BC60 series

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Fig. 11
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Axial compressive stress vs. lateral deflection curves for BC60 series

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Fig. 12
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Comparison of experimental and FEA-deformed shape for specimen BC40L840

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Fig. 13
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Comparison of experimental and FEA-deformed shape for specimen BC40L1640

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Fig. 14
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Comparison of experimental and FEA-deformed shape for specimen BC40L2240

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Parametric study

FEA model was validated by the experimental results. It was shown that the FEA closely predicted the behaviour of stiffened built-up closed section. Hence, parametric study was carried out using validated finite-element model to investigate the effect of all influential parameters such as width of the top and bottom of the web element, width of the intermediate web element, width of the flange, width of the lip, and angle of the inclined web element. Totally, 60 specimens were taken for the parametric studies. Identification label of the specimen is shown in Fig. 15. For example, in the label, “TB30-I80-F50-L15-A45-440” defines the specimens as follows:

Fig. 15
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Identification of the specimen

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  • “TB30” indicates the top and bottom of the web element with the width of 30 m (i.e., w1).

  • “I80” indicates the intermediate web element with the width of 80 mm (i.e., w3).

  • “F50” indicates the flange of the specimen with the width of 50 mm (i.e., Wf).

  • “L15” indicates the lip of the specimen with the width of 15 mm (i.e., Wl).

  • “A45” indicates the angle of inclined web element with an angle of 45°.

  • “440” mean the column length of the specimen.

Design rule

The main design rules investigated in this study are that specified in the American Iron and Steel Institute (2007). The nominal axial strength Pn is calculated from the following design formula for concentrically loaded compression members using the AISI specifications:

$${P}_{\rm n} = A_{\rm e} F_{\rm n},$$
(1)

where Ae is the effective area and Fn is the critical buckling stress.

The critical buckling stress Fn is calculated as

$$F_{\rm n} = (0.658^{{{\lambda}_{\text{c}}}^{2}})\quad {\text{for}}\, {\lambda}_c \leq 1.5,$$
(2)
$$F_{\rm n} = (0.877 / {{{\lambda}_{\text{c}}}^{2}}) \quad {\text{for}}\, {\lambda}_c > 1.5,$$
(3)

where λc = non-dimensional slenderness ratio calculated as

$${\lambda}_c = \sqrt {\frac{{F_{y} }}{{F_{e} }}} ,$$

where Fy is the yield stress which is equal to the 0.2% proof stress, Fe is the least of the elastic flexural, torsional, and flexural-torsional buckling stress determined in accordance with Sects. C 4.1.1–C 4.1.5 of the AISI Specification. The modified slenderness approach in Sect. D 1.2 of the AISI specification (described in Eq. 4) was used to calculate the critical elastic column buckling load for the built-up compression members:

$$\left( {\frac{KL}{r}} \right)_{m} = \sqrt {\left( {\frac{KL}{r}} \right)_{0 }^{2} + \left( {\frac{a}{{r_{i} }}} \right)^{2} } ,$$
(4)

where \(\left( {\frac{KL}{r}} \right)_{m}\) is the modified slenderness ratio, \(\left( {\frac{KL}{r}} \right)_{o}\) is the overall slenderness ratio of the entire section about built-up member axis, "a"  is the intermediate fasterner spacing, and "ri" is the minimum radii of gyration of full unreduced cross-sectional area of an individual shape in a built-up member.

Reliability analysis

The reliability of the current design method was evaluated using reliability analysis. A target reliability index (β) of 2.5 for cold-formed structural members is recommended by the AlSI Specification (2007). The resistance factor (ϕ) of 0.8 was used in the analysis as specified in the NAS Specification (2007) and AS/NZS Standard (AS/NZS 2005). A load combination of 1.2 DL + 1.6 LL as specified in the American Society of Civil Engineers Standard (2005) was used in the reliability analysis, where DL is the dead load and LL is the live load. The statistical parameters Mm, Fm, VM, and VF are the mean values and coefficients of variation for material properties and fabrication variables. These values are obtained from Table Fl of the AISI Specification [2007] for concentrically loaded compression members, where Mm = 1.10, Fm = 1.00, VM = 0.10, and VF = 0.05. The statistical parameters Pm and VP are the mean value and coefficient of variation of σExp or σFEA/σAISI ratio, as shown in Table 4. The correction factor Cp is used to account for the influence due to a small number of specimens.

Table 4 Comparison of FEA results with design strength of cold-formed built-up closed sections
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Results and discussion

The parametric study was used to investigate the effect of all influential cross-sectional parameters on the strength and behaviour of cold-formed built-up closed section. Local buckling, distortional buckling, and flexural bulking, and interaction of local–distortional, local–flexural, and distortional–flexural buckling were observed from the FEA. Figures 16, 17, 18, 19, and 20 shows the comparison between the ultimate compressive stresses with width of the various elements.

Fig. 16
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Ultimate compressive stress vs. width of the top and bottom of the web element for the FEA specimens

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Fig. 17
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Ultimate compressive stress vs. width of the intermediate web element for the FEA specimens

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Fig. 18
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Ultimate compressive stress vs. width of the flange for the FEA specimens

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Fig. 19
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Ultimate compressive stress vs. width of the lip for the FEA specimens

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Fig. 20
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Ultimate compressive stress vs. angle of the inclined web element for the FEA specimens

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As shown in Fig. 16, the ultimate compressive stress of the columns has closer values when the width increases from 30 to 80 and the compressive stress suddenly decreases beyond this width for the column length 440 mm and 840 mm except for the column length 1640 and 2240 mm. As shown in Fig. 17, the ultimate compressive stress decreases with an increase in the width of the intermediate web element for the all column length. As shown in Fig. 18, the ultimate compressive stress increase when the width of the flange is increase from 40 to 50 and slightly decrease beyond this width for the column length with 440 mm and 840 mm. However, the ultimate compressive stress is increase when the width of the flange is increase from 40 to 80 and the compressive stress suddenly decreases beyond this width for the column length with 1640 mm and 2240 mm. This indicates that the effect of flange width is dissimilar for different failure modes. As shown in Fig. 19, the ultimate compressive stresses have almost same for the lip width with 10, 15, and 30 mm. As shown in Fig. 20, the ultimate compressive stress of the columns has closer values when the angle of inclination increases from 30° to 60°, and beyond this angle, the stress is suddenly decrease for all the column length of the specimen. The parametric study results are concluded that 1. The width of the top and bottom of the web element, width of the intermediate web element, width of the flange, and angle of inclination of the web element having a significant effect on the strength and behaviour of the cold-formed built-up closed section. 2. Section with the lowest W/t ratio has more axial compressive resistance. 3. The variation of lip width does not affect the strength of built-up cold-formed closed columns. 4. The variation angle 30–60° does not influence the strength of built-up cold-formed closed columns.

The results obtained from the experiment and the results of the parametric study from FEA are compared with the nominal unfactored design strengths obtained using the AISI Specification are presented in Table 4. It was observed that, local buckling, flexural buckling was observed for the slenderness ratio ranges from 9.89 to 54.06, 63.21 to 168.56, respectively. Interaction of local–distortional, distortional–flexural, and local–flexural buckling was observed for the slenderness ratio ranges from 24.33 to 55.44, 45.42 to 46.71, and 33.11 to 109.39, respectively. The mean value of the (σExp or σFEA/σAISI) ratio is 1.07, with the coefficient of variation (COV) of 0.081, and the corresponding values of β is 3.19. It is shown that the reliability index is greater than the target value of 2.5. Therefore, the column strengths predicted by the AISI predictions are conservative and reliable.

Conclusions

This paper describes strength and behaviour of cold-formed stiffened built-up closed sections with intermediate web stiffeners. Three series of test were conducted. Totally, 12 specimens were tested under the hinged end conditions. The accurate and reliable finite-element model was created using ANSYS. The ultimate compressive stress and failure modes obtained from the finite-element analysis were compared against those are obtained by the experiment. It was shown that FEA accurately predicts the capacity of the cold-formed built-up closed section with intermediate stiffeners. Therefore, the validated FEA model was used for the parametric study. Totally, 60 specimens were used for the parametric study to investigate the effect of all influential parameters such as width of the top and bottom of the web element, width of the intermediate web element and width of the flange, width of the lip, and angle of inclined web element. It is observed that the element with the lowest width-to-thickness ratio has more load-carrying capacity. The increase of width of lip and variation in angle of inclined element from 30° to 60° does not give any significant effects on the section. The results obtained from the parametric study were compared with the unfactored design strength calculated by the AISI specifications. The reliability of the AISI predictions was assessed by reliability analysis. It is shown that AISI predictions are conservative and reliable.