Keywords

1 Introduction

Cold-formed channel sections have been commonly used in structural applications for years [1]. These sections are very sensitive to instability due to their small thickness. These sections, therefore, have been added stiffeners in the web to increase their stability that results in significant improvements of strength performances, known as SupaCee sections. Also, SupaCee’s lips have been rounded to become safer while installing on-site.

Direct Strength Method (DSM) is a new design method developed by Schafer and Pekoz [2,3,4] currently regulated in the Australian/New Zealand Standard AS/NZS 4600 [5] or North American Specification AISI S100–16 [6] for the design of cold-formed steel structures. Elastic buckling analysis is compulsory in this design method that can be supported by THIN-WALL-2 [7, 8] or CUFSM [9] software programs. The DSM design method has been demonstrated its advances compared to the traditional design method [2] and provides insight understandings into the buckling behaviors of cold-formed steel sections [1]. This method will be used in this investigation.

The presence of stiffeners was found to be the strength improvements of cold-formed steel sections that have been investigated by a variety of past studies. Experiments on cold-formed steel storage racks with edge stiffeners have been investigated by Hancock et al. [10,11,12,13,14] to insightfully understand the distortional buckling behaviors. The behaviors of cold-formed steel channel columns with the edge or intermediate stiffeners were also studied by Wang et al., Yan and Young, Xiang et al., and Manikandan et al., and El-taly et al. [15,16,17,18,19,20,21,22,23]. A variety of stiffeners of cold-formed steel channel sections in bending were investigated by Ye et al., and Chun-gang et al. [24, 25] to demonstrate the beneficial effects of these stiffeners on the sectional capacities. Cold-formed steel built-up I sections in the form of back-to-back SupaCee sections were also tested by Manikandan and Arun [26] or Ganeshkumar et al. [27] to observe their behaviors under compression or bending. A series of SupaCee section beams were tested by Pham and Hancock [28,29,30,31] to provide deep insights into their behaviors under shear or combined bending and shear, followed by the development of numerical studies and design proposals. The past researches presented experiments or numerical studies of cold-formed steel channel sections with stiffeners or SupaCee sections that were then used for design proposals.

The effects of stiffeners on the SupaCee section member capacities under compression or bending were also investigated by Pham and Vu [32] to demonstrate the innovation and the strength improvement of SupaCee section members compared to the traditional channel section members. The shortcoming of these new section members was the reductions of their global buckling strengths that result in the lower capacities for long members. This shortcoming can be solved by using full bracing systems to prevent the occurrence of global buckling. In this case, the member capacities reach the sectional capacities that will be examined in this paper. The sectional capacities of SupaCee sections are determined by using the DSM design equations formulated in AS/NZS 4600 [5], and are then compared to those of channel sections to evaluate the strength improvements of SupaCee sections. The investigated sections are commercial sections from BlueScope Lysaght catalogs [33] with steel grade G450 regulated in AS 1397 [34].

2 Summarize the Investigated Capacities of SupaCee Section Members

SupaCee member capacities were determined in comparison with those of channel members, as fully reported in Pham and Vu [32]. The dimensions of the channel and SupaCee sections are listed in Table 1, and their nomenclatures are illustrated in Fig. 1. The same amount of material is required for channel and SupaCee sections to assess the effectiveness of SupaCee sections. The material properties of investigated steel grade G450 according to AS 1397 [34] include the yield stress fy = 450 (MPa), Young’s modulus E = 200 (GPa) and the Poisson ratio \(\upsilon \) = 0.3. The model configurations for compression and bending are shown in Fig. 2 and Fig. 3 with the bracing at the mid-length of the members. The member lengths varied from 2.0 to 8.0 m. The investigated results are illustrated in percentage diagrams, where the horizontal axis is for the capacities of channel members and the vertical axis is for the deviations (in %) between capacities of SupaCee and channel members (see Fig. 4).

Fig. 1.
figure 1

Nomenclature for channel and SupaCee sections

Full size image
Table 1. The nominal dimensions of the channel and SupaCee sectionsa
Full size table
Fig. 2.
figure 2

Compression configuration model

Full size image
Fig. 3.
figure 3

Flexural configuration model

Full size image
Fig. 4.
figure 4

Compression and flexural capacities of the channel and SupaCee cold-formed steel members [32]

Full size image

The investigated results in Fig. 4 demonstrated that SupaCee member capacities, in general, are higher than those of channel members, but the capacities of several former members are still lower than latter member capacities. Pham and Vu [32] pointed out the reason for these lower capacities of SupaCee members was the reductions of global buckling strengths led to the lower strengths for long members with the governing of the global buckling modes or led to the significant decrease of local buckling strengths due to the interaction between global and local buckling modes. The global buckling failure modes can be avoided using the full bracing models that will be presented in Sect. 4 of this paper.

3 Determination of Sectional Capacities of Cold-Formed Steel Section Using the Direct Strength Method

Sectional capacities of cold-formed steel sections can be determined using the DSM design based on elastic buckling stresses (fol, fod) and the yield stress (fy). Local and distortional buckling strengths can be directly determined by using these stresses as shown in Eqs. (1)–(3) and (4)–(6) for compression and bending respectively.

The nominal capacity of a cold-formed steel section under compression is less of the nominal strengths of local buckling (Ncl) and distortional buckling (Ncd).

$$ N_{s} \, = \, Min \, (N_{cl} , \, N_{cd} ) $$
(1)
$$ N_{cl} = \left\{ {\begin{array}{*{20}l} {N_{y} } \hfill & {for \, \lambda_{{\text{l}}} \le {0}{\text{.776}}} \hfill \\ {\left[ {1 - 0.15\left( {\frac{{N_{ol} }}{{N_{y} }}} \right)^{0.4} } \right]\left( {\frac{{N_{ol} }}{{N_{y} }}} \right)^{0.4} N_{y} } \hfill & {for \, \lambda_{{\text{l}}} { > 0}{\text{.776}}} \hfill \\ \end{array} } \right. $$
(2)
$$ N_{cd} = \left\{ {\begin{array}{*{20}l} {N_{y} } \hfill & {for \, \lambda_{{\text{l}}} \le {0}{\text{.561}}} \hfill \\ {\left[ {1 - 0.25\left( {\frac{{N_{od} }}{{N_{y} }}} \right)^{0.6} } \right]\left( {\frac{{N_{od} }}{{N_{y} }}} \right)^{0.6} N_{y} } \hfill & { \, for \, \lambda_{{\text{l}}} { > 0}{\text{.561}}} \hfill \\ \end{array} } \right. $$
(3)

where \({\lambda }_{l}=\sqrt{{N}_{y}/{N}_{ol}}\); \({\lambda }_{d}=\sqrt{{N}_{y}/{N}_{od}}\); Ny is the nominal yield capacity of the member in compression; Nol is the elastic local buckling load; Nod is the elastic distortional buckling load.

The nominal moment capacity (Mb) is the smaller of the nominal strengths in local buckling (Mbl), and distortional buckling (Mbd).

$$ M_{s} \, = \, Min \, (M_{bl} , \, M_{bd} ) $$
(4)
$$ M_{bl} = \left\{ {\begin{array}{*{20}l} {M_{y} } \hfill & {for \, \lambda_{{\text{l}}} \le {0}{\text{.776}}} \hfill \\ {\left[ {1 - 0.15\left( {\frac{{M_{ol} }}{{M_{y} }}} \right)^{0.4} } \right]\left( {\frac{{M_{ol} }}{{M_{y} }}} \right)^{0.4} M_{y} } \hfill & {for \, \lambda_{{\text{l}}} { > 0}{\text{.776}}} \hfill \\ \end{array} } \right. $$
(5)
$$ M_{bd} = \left\{ {\begin{array}{*{20}l} {M_{y} } \hfill & {for \, \lambda_{{\text{l}}} \le {0}{\text{.673}}} \hfill \\ {\left[ {1 - 0.22\left( {\frac{{M_{od} }}{{M_{y} }}} \right)^{0.5} } \right]\left( {\frac{{M_{od} }}{{M_{y} }}} \right)^{0.5} M_{y} } \hfill & {for \, \lambda_{{\text{l}}} { > 0}{\text{.673}}} \hfill \\ \end{array} } \right. $$
(6)

where \({\lambda }_{l}=\sqrt{{M}_{y}/{M}_{ol}}\); \({\lambda }_{d}=\sqrt{{M}_{y}/{M}_{od}}\); My is the yield moment; Mol and Mod are the elastic local and distortional buckling moments respectively.

Local and distortional buckling stresses can be determined using numerical tools such as THIN-WALL-2 [7, 8] or CUFSM [9] software packages.

4 Comparisons of Sectional Capacities Between the Channel and SupaCee Sections

As discussed in Sect. 2, the capacities of several SupaCee section members were lower than those of the channel section members due to the significant reductions of global buckling strengths. In order to improve the effectiveness of SupaCee sections, the global buckling modes should be prevented using bracing systems. The bracing systems can be made by the installations of numerous restraint points along the member length to reduce the effective length of the member as illustrated in Fig. 5 and Fig. 6. The bracing systems can be used as the bracing members attached to the webs and/or the cladding plates/ sheet plates attached to the flanges of the sections. As the global buckling modes are prevented, the member capacities reach the sectional capacities of the cold-formed steel sections that can be determined as presented in Sect. 3 using the Direct Strength Method. Elastic buckling analysis, therefore, is compulsory to determine in the application of this method. This analysis is carried out using the THIN-WALL-2 software program [7, 8] to obtain the local and distortional buckling stresses. The THIN-WALL-2 software program was developed at The University of Sydney on the basis of the finite strip method. The outcome is a signature curve to demonstrate the relationship between the elastic buckling stress and the half-wavelength of buckling mode for each section. Figure 7 illustrates the signature curve of a cold-formed steel channel section with two minimum points. The first point provides the local buckling stress with the shortest half-wavelength, while the second point gives the distortional buckling stress with the intermediate half-wavelength. The investigated sections are listed in Table 1, and the elastic local and distortional buckling stresses are presented in Tables 2 and 3. These analysis results show the significant increase of the local buckling stresses fol of SupaCee sections for compression or bending with the appearance of stiffeners in the webs. In terms of distortional buckling stresses of SupaCee sections, a minor reduction is seen for compression by less than 5% whereas they are generally higher for bending compared to those of channel sections with the deviation reaching 10.81% as seen in C25019 section.

Fig. 5.
figure 5

The compression configuration model with bracing systems

Full size image
Fig. 6.
figure 6

The flexural configuration model with bracing systems

Full size image
Fig. 7.
figure 7

A signature curve of a cold-formed steel channel section.

Full size image
Table 2. The sectional buckling stresses of the investigated sections under compression
Full size table
Table 3. The sectional buckling stresses of the investigated sections under bending
Full size table

The sectional capacities of the investigated sections are achieved with two following cases of bracing systems. The investigated results are illustrated in percentage diagrams for the comparisons of sectional capacities between the SupaCee and the channel sections as shown in Fig. 8 and Fig. 9, where the horizontal axis is for sectional capacities of the channel sections and the vertical axis is for the sectional capacity deviations (in %) between two types of investigated sections.

In the first case, only bracing members are used to prevent the global buckling modes (see Figs. 5 and 6). The sectional capacities are determined as Eqs. 1 to 3 and Eqs. 4–6 for compression and bending respectively, as illustrated in Figs. 8(c) and 9(c). The investigated shows that the global buckling failures although are prevented and local buckling strengths of SupaCee sections are generally higher than those of channel sections (see Figs. 8(a) and 9(a)), but the sectional capacities of the former sections do not demonstrate their strength improvements compared to the latter sections with minor strength deviations (see Figs. 8(c) and 9(c)). The reason is the sectional capacities of the investigated sections are governed by distortional buckling modes for almost all sections. Therefore, the innovative capacities of SupaCee sections can be utilized if the distortional buckling failures are prevented as presented in the second case below.

In the second case, both bracing members attached to the webs and cladding plates/ sheet plates attached to the flanges of the sections are used to prevent both the global and distortional buckling modes. The investigated sections, therefore, are only failed due to the combination of yield strengths and local buckling strengths, as illustrated in Eqs. (2) and (5). The sectional capacities are demonstrated in Figs. 8(d) and 9(d). The investigated results show that the sectional capacities of SupaCee sections are significantly higher than those of the channel sections due to the addition of stiffeners in the web. These capacity improvements reach 25% for compression (see Fig. 8(d)) and 15% for bending (see Fig. 9(d)). The investigated results also show that the high effectiveness of stiffeners for local buckling strengths is found in thin and small dimension sections as presented in Pham and Vu [32].

Fig. 8.
figure 8

Comparisons of sectional capacities between the SupaCee and channel sections under compression

Full size image
Fig. 9.
figure 9

Comparisons of sectional capacities between the SupaCee and channel sections under bending

Full size image

5 Conclusion

This paper studies the sectional capacities of SupaCee sections compared to those of channel sections under compression or bending. SupaCee and channel sections were commercial sections from the BlueScope Lysaght catalogs [33]. The sectional capacities are determined using the DSM design regulated in AS/NZS 4600 [5] with the support of the THIN-WALL-2 software program [7, 8]. Based on the investigated results, the remark conclusions are given as follows:

  1. (1)

    The local buckling strengths of SupaCee sections show their strength improvements compared to those of channel sections, but the sectional capacities of former sections are approximate to those of later sections due to the governing of distortional buckling. Therefore, the innovative capacities of SupaCee sections are only utilized if the distortional buckling is avoided.

  2. (2)

    Web stiffeners of SupaCee sections show their high effectiveness for thin and small dimension sections.

  3. (3)

    These remark conclusions provide deep insights into of sectional capacities of the channel and SupaCee sections, this allows the users to use these sections properly in each structural situation.