Simple Solutions for Lateral Buckling Loads of C-Section Purlins with Two or Three Anti-Sag Bars Under Wind Suction

Abstract

Lateral buckling is the governing failure mode affecting the strength of cold-formed steel purlins. In industrial roofing systems, these purlins are frequently restrained by two or three anti-sag bars within their spans. Previous research by the authors indicated that under wind suction, the buckling behaviour of purlins with multiple anti-sag bars differs significantly from those with fewer restraints, primarily due to the semi-rigid nature of the bracing. This paper investigates the lateral buckling of C-section purlins with two or three anti-sag bars, explicitly accounting for lateral restraints provided by both the roof sheeting and the bars. Simplified analytical solutions are derived to facilitate practical design. Notably, a novel parameter is introduced to identify the controlling buckling mode, which significantly simplifies the calculation procedure. The proposed solutions show excellent agreement with results obtained from both commercial and custom-developed finite element codes.

1. Introduction

Cold-formed steel purlins are widely used in roof systems of industrial buildings to support roof sheeting and transfer wind and gravity loads to the main structural system [1,2]. Due to the low torsional rigidity of their cross-sections (e.g., C-sections), lateral buckling is the primary failure mode controlling the strength of cold-formed steel purlins [3,4,5,6].

In existing design codes (e.g., [7,8]), simplified methods are provided for estimating the lateral buckling loads of general thin-walled beams. However, these methods are not directly applicable to the cold-formed steel purlins examined in this study. One key reason is that roof sheeting, when connected to the top flange of the purlin, can significantly enhance its lateral stiffness. For instance, when self-drilling screws are used to fasten the purlin to the sheeting, the in-plane shear stiffness of the sheeting is often sufficient to restrain lateral translation of the top flange [9]. Nevertheless, lateral buckling remains a critical concern under wind suction, as the compressive bottom flange is laterally unrestrained under such loading conditions.

To improve lateral stability, anti-sag bars are commonly used in medium- to long-span purlin systems. Previous studies on the lateral buckling of cold-formed steel purlins have incorporated the restraining effects of roof sheeting on lateral deflection and/or twist [3,4,10,11,12]. In these studies, anti-sag bars are typically idealized as rigid lateral restraints, and the distance between adjacent bars is taken as the effective buckling length in load calculations [3,4]. However, recent research [6,13,14,15] has demonstrated that the bracing efficiency of anti-sag bars can be considerably reduced due to local web deformations near the connection points. This effect can lead to a significant overestimation of buckling loads when anti-sag bars are assumed to provide rigid restraint [14].

Solutions for predicting the lateral buckling loads of C-section cold-formed steel purlins, with top flanges connected to roof sheeting, are available in most widely adopted design codes [10,11,12]. However, the application of these solutions requires compliance with several restrictive conditions [11,12]; otherwise, full-scale tests are required. Moreover, the treatment of anti-sag bars in these codes is either not explicitly specified [11,12] or is simplistically assumed to provide rigid lateral restraint [10,11,12]. Therefore, developing simplified solutions for cold-formed steel purlins that fully account for the combined restraints from roof sheeting and anti-sag bars is essential for practical engineering applications.

The lateral buckling loads of C-section purlins with a single anti-sag bar at mid-span have been investigated by Zhang and Tong [14], who also proposed corresponding simplified solutions. Nevertheless, installing only one anti-sag bar is insufficient for simply supported purlins in many practical scenarios. For instance, in China, two anti-sag bars are typically recommended for simply supported cold-formed steel purlins with spans exceeding 6 m, and three for those longer than 8 m; this configuration is widely used in industrial construction.

This paper presents an investigation into the lateral buckling of simply supported C-section cold-formed steel purlins equipped with two or three anti-sag bars, based on which simplified solutions for predicting buckling loads are derived. The study also introduces a dedicated parameter to identify the dominant buckling mode for each configuration, which considerably streamlines buckling mode determination compared with the approach proposed by Zhang and Tong [14]. It should be noted that the elastic buckling is only considered in this study.

In the present analysis, both the restraint from roof sheeting and the semi-rigid restraint from anti-sag bars are considered. The steel roof sheeting is assumed to provide rigid lateral restraint to the purlin top flanges, which is applicable to roof systems connected using self-drilling screws. Only wind suction loading is considered in this study (Figure 1).

2. Existing Methods in Design Code

Solutions or methods for predicting lateral buckling loads are provided in many existing design codes [10,11,12]. According to AISI S100 [11], the moment resistance of a C-section loaded in a plane parallel to the web, with the tension flange through-fastened to roof or wall sheeting and the compression flange laterally unbraced, can be calculated in accordance with:

$$M_n=R{S}_e{f}_y$$

(1)

where S_e is the section modulus of the cross-section; f_y is the material yield stress; and the reduction factor R depends on the cross-section depth, loading conditions, and boundary conditions. It should be noted that the solution presented in Equation (1) is essentially based on cross-sectional resistance. In addition, as specified in AISI S100 [11], the use of this solution must satisfy a number of requirements, such as limitations on cross-section dimensions, fastener type, steel yield stress, and span range; otherwise, full-scale testing is required. The influence of anti-sag bars on the buckling strength of the purlins under uplift loading is not explicitly addressed.

A method similar to that in AISI S100 [11] is adopted in AS/NZS 4600 [12] for purlins with one flange through-fastened to sheeting. In AS/NZS 4600 [12], only bridging capable of preventing lateral and torsional deformation at support points is considered.

Different from the methods in AISI S100 [11] and AS/NZS 4600 [12], EC3 [10] adopts a calculation model for predicting lateral buckling loads in which the cold-formed steel purlin is simplified into a stud consisting of the compressive flange and a portion of the web. In this method, the laterally unbraced length of the compressive flange is taken as the stud span, which implicitly assumes rigid bracing provided by anti-sag bars, although this is not explicitly stated.

When anti-sag bars are assumed to provide rigid lateral restraint, cold-formed steel purlins can be treated as simply supported beams with spans equal to the distance between adjacent anti-sag bars when calculating buckling loads. However, a recent study by Zhang and Tong [13] showed that the bracing efficiency of anti-sag bars commonly used in practice may be significantly reduced due to local web deformation near the connection locations. Consequently, anti-sag bars cannot be treated as rigid lateral supports for these cold-formed steel purlins. This implies that the solutions and methods in current design codes (i.e., [10,11,12]) may not be applicable to cold-formed steel purlins laterally braced by anti-sag bars.

3. Effective Stiffness of Anti-Sag Bar

According to Zhang and Tong [13], the reduction in stiffness caused by local web deformation near the anti-sag bar connection should be considered when calculating the effective stiffness of the anti-sag bar (k_LT). An approximate solution for k_LT was proposed by Zhang and Tong [13]:

$${\displaystyle \frac{1}{k_{LT}}}={\displaystyle \frac{1}{k_w}}+{\displaystyle \frac{1}{k_{sb}}}$$

(2)

where k_sb is the axial tensile stiffness of the anti-sag bar, which can be expressed as:

$$k_{sb}={\displaystyle \frac{E{A}_{sb}}{l_{sb}}}$$

(3)

where E is the Young’s modulus of steel, A_sb is the cross-sectional area of the anti-sag bar, and l_sb is the length of the anti-sag bar. In Equation (2), the effects of local web deformation are accounted for using the parameter k_w, which can be expressed as:

$$k_w={\displaystyle \frac{13.21E{t}^3}{h^2{\left(1-\cos 2\pi \xi \right)}^{0.64\left({\xi }^2-\xi +1.25\right)}}}\left[1-20{\left({\displaystyle \frac{t}{h}}\right)}^{1.4}\right]$$

(4)

where h is the distance between the mid-thicknesses of the top and bottom flanges (Figure 1), and ξ is the normalized distance between the anti-sag bar and the mid-thickness of the bottom flange:

$$\xi =1-{\displaystyle \frac{e_0}{h}}$$

(5)

where e₀ is the distance between the mid-thickness of the top flange and the anti-sag bar (Figure 1). Very good agreement between the predictions from Equation (2) and finite element (FE) analysis results was reported by Zhang and Tong [13].

4. Total Potential for Lateral Buckling of Cold-Formed Steel Purlins

As discussed by Zhang and Tong [6], a number of buckling theories have been developed for the lateral buckling of thin-walled beams. However, the general expressions for total potential based on these theories are not suitable for the lateral buckling analysis of the cold-formed steel purlins considered in this study [14]. This is because the horizontal restraints at the top flange provided by roof sheeting, together with the anti-sag bars connected to the web, may lead to notable differences in the total potential expressions compared with those for general thin-walled beams.

4.1. Expression of Total Potential

Two typical total potential formulations for the lateral buckling of thin-walled beams reported in the literature have been discussed in Zhang and Tong [6,14]. Based on the theory proposed by Zhang and Tong [14], a new total potential formulation was presented for the lateral buckling of C-section purlins with a single anti-sag bar at mid-span.

This total potential consists of the linear potential and nonlinear potentials arising from longitudinal normal stress, shear stress, and transverse normal stress. As demonstrated by Zhang and Tong [14], the contribution of transverse normal stresses induced by the restraint from roof sheeting and anti-sag bars, which has not been considered in existing theories, should be included in the total potential. The proposed total potential was successfully applied by Zhang and Tong [14] to the lateral buckling analysis of C-section purlins with a single anti-sag bar at mid-span.

As shown in Figure 2, a right-handed x–y–z coordinate system is established for the C-section, where the origin o is located at the centroid of the cross-section. Following a procedure similar to that described by Zhang and Tong [14], the total potential for the lateral buckling of C-section purlins with two anti-sag bars located at one-third points of the span can be expressed as:

$$\begin{array}{ll}\prod =& {\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^L\left[E{I}_y{u^{″}}^2+E{I}_x{v^{″}}^2+E{I}_{\omega }{{\theta }^{″}}^2+G{I}_k{{\theta }^{\prime }}^2-2M_y{v}^{″}\theta +2M_x{u}^{″}\theta +2M_y{\beta }_y{{\theta }^{\prime }}^2+\left(q_y{a}_y+q_x{a}_x\right){\theta }^2\right]dz}\\ & +{\displaystyle \frac{1}{2}}{k}_{LT}\left(u_{L/3,e_0}^2+u_{2L/3,e_0}^2\right)+{\displaystyle \frac{1}{2}}{F}_{LT}{a}_{LT}\left({\theta }_{L/3}^2+{\theta }_{2L/3}^2\right)\end{array}$$

(6)

where u and v are the buckling displacements in the x- and y-directions, respectively (Figure 2), and θ is the twist angle of the cross-section. E and G are the Young’s modulus and shear modulus of the material, respectively. I_x and I_y are the second moments of area about the x- and y-axes, respectively; I_xy is the product moment of area of the cross-section; I_k is the free torsion constant; and I_ω is the warping constant of the cross-section. B_ω is the bi-moment of the cross-section. L is the span of the purlin. The terms $u_{L/3,e_0}$ and $u_{2L/3,e_0}$ representing the lateral displacements at the anti-sag bar locations correspond to positions L/3 and 2L/3. β_y and β_ω represent the mono-symmetry parameters of the cross-section and can be expressed as:

$${\beta }_y={\displaystyle \frac{{\displaystyle {\int }_A\left[x\left(x^2+y^2\right)\right]}dA}{2I_y}}-x_0$$

(7)

$${\beta }_{\omega }={\displaystyle {\int }_A\omega \left(x^2+y^2\right)}dA$$

(8)

where x₀ is the x-coordinate of the shear centre S (Figure 2), and ω is the principal sectorial coordinate with respect to the shear centre of the cross-section.

As noted by Zhang and Tong [6,14], M_x, M_y, q_x and q_y in Equation (6) are resultants of the pre-buckling stresses in the cross-section. Among these stress resultants, M_x and M_y are the bending moments about the x and y axes caused by longitudinal normal stresses, while q_x and q_y are the resultants of transverse normal stress. It has been shown that q_y and q_x are equal to the external distributed loads in the y-direction and the reaction forces in the x-direction provided by horizontal bracing at the top flange of the cross-section, respectively (Zhang and Tong [6,14]). In Equations (9) and (10), a_x and a_y are related to the distances between the load application points and the shear centre of the cross-section (see Figure 2).

In this study, the external distributed load along the y-axis, q_y, is assumed to be acted at the same positions as the horizontal restraints at the top flange of cross-sections, indicating that both the external load and the horizontal restraint are transferred from the roof sheeting to the purlin through the fasteners. For the general case considered herein, vertical loads are applied at the mid-width of the top flange, which gives:

$$a_y=-{\displaystyle \frac{h}{2}}$$

(9)

$$a_x=-\left({\displaystyle \frac{b}{2}}+d_s\right)$$

(10)

The values of a_x and a_y defined in Equations (9) and (10) are used in the remainder of this paper unless otherwise stated.

The corresponding terms $F_{LT}{a}_{LT}\left({\theta }_{L/3}^2+{\theta }_{2L/3}^2\right)/2$ in Equation (6) represent the transverse stress effects caused by the reaction force from the anti-sag bar (F_LT). This force is derived by considering the term containing q_xa_x, as a concentrated force can be approximated as a uniformly distributed load over an infinitesimal width. a_LT represents the relative horizontal position of F_LT with respect to the shear centre of the cross-section. For the configuration considered in this study, the anti-sag bar is connected to the web of the cross-section (Figure 1).

$$a_{LT}=-d_s$$

(11)

For the C-section purlins with the top flange laterally restrained, the relationship between the buckling displacements u and θ can be expressed as:

$$u=-{\displaystyle \frac{1}{2}}h\theta$$

(12)

Considering the relationship in Equation (12), Equation (6) can be rewritten as:

$$\begin{array}{ll}\prod =& {\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^L\left[E{\overline{I}}_{\omega }{{\theta }^{″}}^2+E{I}_x{v^{″}}^2+G{I}_k{{\theta }^{\prime }}^2-M_{x}h\theta {\theta }^{″}-2M_y{v}^{″}\theta +2M_y{\beta }_y{{\theta }^{\prime }}^2+\left(q_y{a}_y+q_x{a}_x\right){\theta }^2\right]}dz\\ & +{\displaystyle \frac{1}{2}}{k}_{LT}\left(u_{L/3,e_0}^2+u_{2L/3,e_0}^2\right)+{\displaystyle \frac{1}{2}}{F}_{LT}{a}_{LT}\left({\theta }_{L/3}^2+{\theta }_{2L/3}^2\right)\end{array}$$

(13)

where

$${\overline{I}}_{\omega }=I_{\omega }+{\displaystyle \frac{h^2{I}_y}{4}}$$

(14)

4.2. Verification of Total Potential

The total potential formulation for the lateral buckling of C-section purlins with two anti-sag bars proposed in this study is first verified. The buckling loads predicted using the total potential formulation are compared with those obtained from finite element (FE) analysis using shell element modelling. The buckling loads based on the total potential formulation (Equation (6)) are obtained using an in-house FE program employing the eigenvalue method, which is widely used in buckling analysis of structural members. The validity of shell element modelling for verifying total potential formulations of thin-walled beams and cold-formed steel purlins in lateral buckling analysis has been demonstrated in previous studies [6,14].

Comparisons between results obtained from shell element modelling using ANSYS 16.0 [16] and those from the FE program based on the total potential formulation are shown in Figure 3 for C-section purlins. In all analyses, the material properties are taken as: elastic modulus E = 206,000 N/mm², Poisson’s ratio υ = 0.3, and shear modulus G = E/2(1+ υ). These material properties are used throughout this paper. The values of k_LT are calculated using Equation (2) for all cases shown in Figure 3. The dimensions of C-section used in Figure 3 C200t2.0 is given in

Table 1. Dimensions of cross-sections.

ID	Section	H (mm)	B (mm)	C (mm)	t (mm)	Diagram
C140t(1.5–3.0)	C140 × 50 × 20 × (1.5–3.0)	140	50	20	1.5–3.0
C160t(1.5–3.0)	C160 × 60 × 20 × (1.5–3.0)	160	60	20	1.5–3.0
C180t(1.5–3.0)	C180 × 70 × 20 × (1.5–3.0)	180	70	20	1.5–3.0
C200t(1.5–3.0)	C200 × 70 × 20 × (1.5–3.0)	200	70	20	1.5–3.0
C220t(1.5–3.0)	C220 × 75 × 20 × (1.5–3.0)	220	75	20	1.5–3.0
C250t(1.5–3.0)	C250 × 75 × 20 × (1.5–3.0)	250	75	20	1.5–3.0
C300t(1.5–3.0)	C300 × 80 × 20 × (1.5–3.0)	300	80	20	1.5–3.0

.

In Figure 3, the vertical axis M_x,cr represents the absolute value of the maximum bending moment about the x-axis along the span at lateral buckling. The positive directions of M_x and M_y used in this paper are shown in Figure 2. It can be observed that the results obtained using the total potential formulation proposed in this study (Equation (6)) agree closely with those from shell element modelling (ANSYS) for all cases, indicating that the proposed total potential formulation is capable of accurately predicting the lateral buckling behaviour of the C-section purlins considered herein. The buckling loads based on the assumption of rigid lateral restraint provided by anti-sag bars are also shown in Figure 3 (rigid support), which may significantly overestimate the buckling loads, particularly for purlins with relatively short spans.

5. Simple Solutions of Buckling Loads

Explicit analytical solutions are valuable for practical engineering design and for developing provisions in design codes, although buckling loads can be obtained through numerical analysis. Therefore, simplified solutions for the buckling loads of cold-formed C-section purlins with two anti-sag bars are developed in this section. The derivation is based on the total potential formulation given in Equation (13) using the Rayleigh–Ritz method.

Finite element (FE) analyses using shell element modelling (ANSYS 16.0) indicate that the dominant lateral buckling mode of C-section purlins with two anti-sag bars may correspond to one half-wave, two half-waves, or three half-waves (Figure 4). For small values of k_LT (i.e., weak lateral bracing provided by the anti-sag bars), purlins tend to fail in the one-half-wave mode, whereas the two-half-wave and three-half-wave modes may occur for medium and large values of k_LT, respectively.

Similar to Zhang and Tong [14], simplified solutions for buckling loads corresponding to one, two, and three half-wave failure modes are developed in the following sections. The minimum value among these three buckling loads is taken as the governing buckling load for each case.

5.1. Calculation Model for M_y and q_x of C-Section Purlins

To derive simplified buckling load solutions using the Rayleigh–Ritz method, simplified expressions for the stress resultants M_x, M_y, q_x and q_y, are required. The distributions of M_x and q_y can be obtained directly from loading and boundary conditions, whereas M_y and q_x depend on the reaction forces associated with the restraints provided by roof sheeting and anti-sag bars. As noted by Zhang and Tong [6,14], analytical solutions for M_y and q_x are complicated even for C-section purlins with a single anti-sag bar, which makes it difficult to derive simplified buckling load solutions using the Rayleigh–Ritz method. Therefore, a simplified calculation model was proposed by Zhang and Tong [14] for the distributions of M_y and q_x, based on which simplified buckling solutions were successfully derived for C-section purlins with one anti-sag bar. A similar calculation model is adopted here for C-section purlins with two anti-sag bars.

Due to symmetry of deformation in the pre-buckling state, the reaction forces and deformations at the two anti-sag bar connection sections are identical. Therefore, in developing the equivalent uniformly distributed load in the x-direction (q_Ex), the forces and deformations at either connection section can be considered. The C-section purlin is taken as the representative example in this section.

As illustrated in Figure 5, lateral translation of the C-section purlin results from the combined action of the equivalent uniformly distributed load q_Ex, representing reactions from lateral restraints at the top flange, and concentrated forces at the one-third span locations, corresponding to the reaction forces from the anti-sag bars (F_LT). The reaction force from the anti-sag bar is mainly resisted by two substructures, each consisting of one flange together with the adjacent lip and a portion of the web. The resistance force of each substructure against F_LT is assumed to be inversely proportional to the distance between the loading position of F_LT and the corresponding flange. For the purlins considered in this study, lateral translation of the top flange is restrained. Therefore, the reaction force provided by the lateral restraint at the top flange due to F_LT acting on the corresponding substructure (F_R, see Figure 5a) is equal to the resistance force of this substructure.

$$F_R=-{\displaystyle \frac{h-e_0}{h}}{F}_{LT}$$

(15)

where F_R acts in the opposite direction to F_LT (Figure 5a). Consequently, the resultant concentrated force at each one-third span location, combining F_LT and F_R, becomes:

$$F_{LT,E}=F_R+F_{LT}={\displaystyle \frac{e_0}{h}}{F}_{LT}$$

(16)

Under the actions of F_LT,E and q_EX, the lateral deflection at the one-third span location can be expressed as:

$$\delta ={\delta }_{q_{Ex}}+{\delta }_{F_{LT,E}}$$

(17)

where δ_qEx and ${\delta }_{F_{LT,E}}$ are the lateral deflections caused by q_EX and F_LT,E, respectively. Using classical structural mechanics:

$${\delta }_{q_{Ex}}={\displaystyle \frac{17q_{Ex}{L}^4}{1458E{I}_y}}$$

(18a)

$${\delta }_{F_{LT,E}}=F_{LT,E}{\displaystyle \frac{e_0}{h}}{\displaystyle \frac{5L^3}{162E{I}_y}}$$

(18b)

The lateral deflections given in Equations (18)–(19b) are defined with respect to the shear centre of the cross-section. Meanwhile, F_LT can be determined directly once k_LT is known:

$$F_{LT}=-u_{LT,e_0}{k}_{LT}=-\delta {\displaystyle \frac{e_0}{h/2}}{k}_{LT}$$

(19a)

which can be rewritten as:

$$\delta =-{\displaystyle \frac{F_{LT}h/2}{e_0{k}_{LT}}}$$

(19b)

Substituting Equations (18a), (18b) and (19b) into Equation (17) gives:

$$q_{Ex}=-{\displaystyle \frac{1458E{I}_y{F}_{LT}}{17L^4}}\left({\displaystyle \frac{5L^3{e}_0}{162hE{I}_y}}+{\displaystyle \frac{h}{2k_{LT}{e}_0}}\right)$$

(20)

The reaction force from the anti-sag bar, F_LT, in the pre-buckling state can be obtained from Equation (19a), provided that the lateral displacement at the bracing points $u_{LT,e_0}$ is known:

$$u_{LT,e_0}=-e_0{\theta }_{LT}$$

(21a)

For a simply supported C-section purlin with two anti-sag bars, the twist angle θ can be obtained analytically by solving:

$$E{\overline{I}}_{\omega }{\theta }^{(4)}-G{I}_k{\theta }^{″}=q_y{a}_x$$

(21b)

By applying the boundary and continuity conditions, the solution for θ can be obtained, and subsequently the twist at the bracing locations θ_LT in Equation (21a). Substituting this result into Equation (19a), the analytical solution for F_LT can be expressed as:

$$F_{LT}={\chi }_{C2}{q}_y$$

(22a)

where

$${\chi }_{C2}={\displaystyle \frac{k_{LT}{e}_0{a}_x\left[{\left(\frac{L}{3}\right)}^2+\frac{{\psi }_C-1}{{\lambda }_C^2}\right]}{G{I}_k-k_{LT}{e}_0^2\left(\frac{{\psi }_C}{{\lambda }_C}\sinh {\lambda }_C\frac{L}{3}-\frac{L}{3}\right)}}$$

(22b)

and:

$${\psi }_C=\cosh {\displaystyle \frac{{\lambda }_{C}L}{6}}/\cosh {\displaystyle \frac{{\lambda }_{C}L}{2}}$$

(23a)

$${\lambda }_C=\sqrt{{\displaystyle \frac{G{I}_k}{E{\overline{I}}_{\omega }}}}$$

(23b)

Substituting Equation (22a) into Equation (20), and considering Equations (22b)–(23b) gives:

$$q_{Ex}={\eta }_{C2}{q}_y$$

(24)

where

$${\eta }_{C2}={\displaystyle \frac{-1458E{I}_y{\chi }_{C2}}{17L^4}}\left({\displaystyle \frac{5L^3{e}_0}{162hE{I}_y}}+{\displaystyle \frac{h}{2k_{LT}{e}_0}}\right)$$

(25)

The analytical solution of q_x obtained by solving Equation (21b) together with:

$$M_y=-E{I}_y{u}^{″}=E{I}_y{\displaystyle \frac{h}{2}}{\theta }^{″}$$

(26a)

$$q_x=-M_y^{″}$$

(26b)

Equation (26b) shows that the distribution of q_x along the span is non-uniform. In this study, the approximate solution given in Equation (24) is used to derive simplified buckling load solutions. To reduce the error introduced by using q_Ex instead of q_x, the coefficient η_C2 in Equation (25) is replaced by ${\overline{\eta }}_{C2}$:

$${\overline{\eta }}_{C2}={\displaystyle \frac{-89E{I}_y{\chi }_{C2}}{L^4}}\left({\displaystyle \frac{5L^3{e}_0}{162hE{I}_y}}+{\displaystyle \frac{h}{2k_{LT}{e}_0}}\right)$$

(27)

Consequently, the bending moment about the y-axis under the combined action of q_EX and F_LT,E can be expressed as:

$$\mathrm{For} 0\le z<{\displaystyle \frac{L}{3}}, M_y={\displaystyle \frac{q_{Ex}}{2}}\left(Lz-z^2\right)+{\displaystyle \frac{e_0{F}_{LT,E}}{h}}z=\left[{\displaystyle \frac{{\overline{\eta }}_{C2}}{2}}\left(Lz-z^2\right)-{\displaystyle \frac{e_0{\chi }_{C2}}{h}}z \right]q_y$$

(28a)

$$\mathrm{For} {\displaystyle \frac{L}{3}}\le z<{\displaystyle \frac{2L}{3}}, M_y={\displaystyle \frac{q_{Ex}}{2}}\left(Lz-z^2\right)+{\displaystyle \frac{e_0{F}_{LT,E}}{h}}{\displaystyle \frac{L}{3}}=\left[{\displaystyle \frac{{\overline{\eta }}_{C2}}{2}}\left(Lz-z^2\right)-{\displaystyle \frac{e_0{\chi }_{C2}}{h}}{\displaystyle \frac{L}{3}} \right]q_y$$

(28b)

$$\mathrm{For} {\displaystyle \frac{2L}{3}}\le z\le L, M_y={\displaystyle \frac{q_{Ex}}{2}}\left(Lz-z^2\right)+{\displaystyle \frac{e_0{F}_{LT,E}}{h}}\left(L-z\right)=\left[{\displaystyle \frac{{\overline{\eta }}_{C2}}{2}}\left(Lz-z^2\right)-{\displaystyle \frac{e_0{\chi }_{C2}}{h}}\left(L-z\right)\right] q_y$$

(28c)

Predictions from Equations (28a)–(28c) are compared with analytical results obtained using Equation (26a).

The distributions of M_y for purlins with different spans (i.e., L = 4.0 m, 6.0 m, 8.0 m and 12.0 m) are shown in Figure 6a, where C200t2.0 and e₀ = 0.5h are used. Good agreement is observed between Equations (28a)–(28c) and Equation (26a). Similar agreement is also observed for different values of e₀, as shown in Figure 6b (C200t2.0 and L = 8000 mm).

The influence of the transverse stress due to the action of F_R should be considered in the total potential as long as the uniformly distributed load q_Ex is adopted. This influence would be automatically included in the term due to the action of q_x if the analytical results of q_x are used in the development. Therefore, the total potential of Equation (13) becomes:

$$\begin{array}{ll}\prod =& {\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^L\left[E{\overline{I}}_{\omega }{{\theta }^{″}}^2+E{I}_x{v^{″}}^2+G{I}_k{{\theta }^{\prime }}^2-M_{\mathrm{x}}h\theta {\theta }^{″}-2M_y{v}^{″}\theta +2M_y{\beta }_y{{\theta }^{\prime }}^2+\left(q_y{a}_y+q_{Ex}{a}_x\right){\theta }^2\right]}dz\\ & +{\displaystyle \frac{1}{2}}{k}_{LT}\left(u_{L/3,e_0}^2+u_{2L/3,e_0}^2\right)+{\displaystyle \frac{1}{2}}\left[F_{LT}{a}_{LT}+F_R\left(d_s+{\displaystyle \frac{b}{2}}\right)\right]\left({\theta }_{L/3}^2+{\theta }_{2L/3}^2\right)\end{array}$$

(29)

The possible lateral buckling modes shown in Figure 4 indicate that the buckling deformations at L/3 and 2L/3 may be symmetric or anti-symmetric, leading to:

$$u_{LT,e_0}^2=u_{L/3,e_0}^2=u_{2L/3,e_0}^2$$

(30a)

$${\theta }_{LT}^2={\theta }_{L/3}^2={\theta }_{2L/3}^2$$

(30b)

Therefore, Equation (29) can be rewritten as:

$$\begin{array}{ll}\prod =& {\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^L\left[E{\overline{I}}_{\omega }{{\theta }^{″}}^2+E{I}_x{v^{″}}^2+G{I}_k{{\theta }^{\prime }}^2-M_{\mathrm{x}}h\theta {\theta }^{″}-2M_y{v}^{″}\theta +2M_y{\beta }_y{{\theta }^{\prime }}^2+\left(q_y{a}_y+q_{Ex}{a}_x\right){\theta }^2\right]}dz\\ & +k_{LT}{u}_{LT,e_0}^2+\left[F_{LT}{a}_{LT}+F_R\left(d_s+{\displaystyle \frac{b}{2}}\right)\right]{\theta }_{LT}^2\end{array}$$

(31)

5.2. Simple Solutions of Buckling Load of C-Section Purlins

FE analyses presented in previous sections show that the dominant buckling mode of C-section purlins with two anti-sag bars may correspond to one, two, or three half-wave modes (Figure 4), depending on the effective stiffness of the anti-sag bars. Simplified solutions corresponding to these three buckling modes are therefore derived.

The buckling displacement functions for twist angle θ and vertical displacement v are assumed as:

$$\theta =A_{n1}\sin \left({\displaystyle \frac{n\pi z}{L}}\right)$$

(32a)

$$v=A_{n2}\sin \left({\displaystyle \frac{n\pi z}{L}}\right)$$

(32b)

where n (=1, 2 or 3) is the number of half-waves; and A_n1 and A_n2 are constants. Substituting Equations (32a) and (32b) into Equation (31), and requiring the determinant of the coefficient matrix of the resulting equations to be zero:

$${\displaystyle \frac{\partial \mathrm{\Pi }}{\partial A_{n1}}}=0$$

(33a)

$${\displaystyle \frac{\partial \mathrm{\Pi }}{\partial A_{n2}}}=0$$

(33b)

The buckling loads can then be expressed as:

$$\begin{array}{ll}{M}_{xcr,n}=& -C_{n1}{P}_x\left\{C_{n2}\left(a_y+{\overline{\eta }}_{C2}{a}_x\right)+C_{n3}{\beta }_y+C_{n4}\right.\\ & -\sqrt{{\left[C_{n2}\left(a_y+{\overline{\eta }}_{C2}{a}_x\right)+C_{n3}{\beta }_y+C_{n4}\right]}^2+\left[{\displaystyle \frac{P_{\omega n}}{P_x}}+{\displaystyle \frac{3{\left(3-n\right)}^2{k}_{LT}{e_0}^{2}L}{4{\pi }^2{P}_x}}\right]}\end{array}$$

(34)

where n = 1, 2 or 3; the subscript “n” in M_xcr,n, C_n1–C_n4 and $P_{\omega n}$ represents the number of buckling half-waves.

$$P_{\omega n}={\displaystyle \frac{n^2{\pi }^{2}E{\overline{I}}_{\omega }}{L^2}}+G{I}_k$$

(35a)

$$P_x={\displaystyle \frac{{\pi }^{2}E{I}_x}{L^2}}$$

(35b)

Other parameters in Equation (34) are listed in

Table 2. Parameters for C-section purlins with two anti-sag bars.

n	Cn1	Cn2	Cn3	Cn4
1	$0.85{\displaystyle \frac{1}{16k_{11}}}$	${\displaystyle \frac{1}{4{\pi }^2{k}_{11}}}$	${\displaystyle \frac{k_{12}}{k_{11}}}$	${\displaystyle \frac{1}{16{\pi }^2{k}_{11}}}\left[{\displaystyle \frac{\left(3+{\pi }^2\right)}{3}}h+{\displaystyle \frac{12}{L}}{\chi }_{C2}\left(d_s+\xi a_x\right)\right]$
2	$0.92{\displaystyle \frac{1}{8k_{21}}}$	${\displaystyle \frac{1}{8{\pi }^2{k}_{21}}}$	${\displaystyle \frac{2k_{22}}{k_{21}}}$	${\displaystyle \frac{1}{32{\pi }^2{k}_{21}}}\left[{\displaystyle \frac{\left(3+4{\pi }^2\right)}{3}}h+{\displaystyle \frac{12}{L}}{\chi }_{C2}\left(d_s+\xi a_x\right)\right]$
3	$0.88{\displaystyle \frac{3}{16k_{31}}}$	${\displaystyle \frac{1}{12{\pi }^2{k}_{31}}}$	${\displaystyle \frac{3k_{32}}{k_{31}}}$	${\displaystyle \frac{\left(1+3{\pi }^2\right)h}{48{\pi }^2{k}_{31}}}$

.

The constants k₁₁–k₃₂ used in Table 2 are given by:

$$k_{11}={\displaystyle \frac{1}{72{\pi }^2}}\left[3{\overline{\eta }}_{C2}\left({\pi }^2+3\right)+\left(8{\pi }^2+27\right){\displaystyle \frac{e_0{\chi }_{C2}}{hL}}\right]$$

(36a)

$$k_{12}={\displaystyle \frac{1}{72{\pi }^2}}\left[3{\overline{\eta }}_{C2}\left({\pi }^2-3\right)+\left(8{\pi }^2-27\right){\displaystyle \frac{e_0{\chi }_{C2}}{hL}}\right]$$

(36b)

$$k_{21}={\displaystyle \frac{1}{288{\pi }^2}}\left[3{\overline{\eta }}_{C2}\left(4{\pi }^2+3\right)+\left(32{\pi }^2+27\right){\displaystyle \frac{e_0{\chi }_{C2}}{hL}}\right]$$

(36c)

$$k_{22}={\displaystyle \frac{1}{288{\pi }^2}}\left[3{\overline{\eta }}_{C2}\left(4{\pi }^2-3\right)+\left(32{\pi }^2-27\right){\displaystyle \frac{e_0{\chi }_{C2}}{hL}}\right]$$

(36d)

$$k_{31}={\displaystyle \frac{1}{72{\pi }^2}}\left[{\overline{\eta }}_{C2}\left(3{\pi }^2+1\right)+8{\pi }^2{\displaystyle \frac{e_0{\chi }_{C2}}{hL}}\right]$$

(36e)

$$k_{32}={\displaystyle \frac{1}{72{\pi }^2}}\left[{\overline{\eta }}_{C2}\left(3{\pi }^2-1\right)+8{\pi }^2{\displaystyle \frac{e_0{\chi }_{C2}}{hL}}\right]$$

(36f)

It should be noted that the constants 0.85, 0.92, and 0.88 in C_n1 for n = 1, 2 and 3, respectively, are calibration coefficients used to account for approximations in the assumed M_y distribution and differences between the actual and assumed buckling displacement functions.

6. Buckling Load of C-Section Purlins with Three Anti-Sag Bars

In this section, three anti-sag bars are assumed to be located at the quarter points of the span. Unlike purlins with two anti-sag bars, the dominant buckling mode of C-section purlins with three anti-sag bars may correspond to two-, three-, or four-half-wave modes. The closed-form solutions for the lateral buckling loads are directly presented in this section, as the derivation procedure based on the Rayleigh–Ritz method is similar to that used for the two anti-sag bar cases.

The buckling loads of C-section purlins with three anti-sag bars can be expressed as:

$$\begin{array}{ll}{M}_{xcr,n}=& -C_{n1}{P}_x\left\{C_{n2}\left(a_y+{\overline{\eta }}_{C3}{a}_x\right)+C_{n3}{\beta }_y+C_{n4}\right.\\ & -\sqrt{{\left[C_{n2}\left(a_y+{\overline{\eta }}_{C3}{a}_x\right)+C_{n3}{\beta }_y+C_{n4}\right]}^2+\left[{\displaystyle \frac{P_{\omega n}}{P_x}}+{\displaystyle \frac{\left(4-n\right)\left(11-n\right)k_{LT}{e_0}^{2}L}{18{\pi }^2{P}_x}}\right]}\end{array}$$

(37)

in which n = 2, 3 or 4 correspond to the possible dominant buckling modes; P_x and $P_{\omega n}$ are given in Equations (35a) and (35b). The constants C_n1–C_n4 are summarized in

Table 3. Parameters for C-section purlins with three anti-sag bars.

n	Cn1 *	Cn2	Cn3	Cn4
2	$0.88{\displaystyle \frac{1}{8{\overline{k}}_{21}}}$	${\displaystyle \frac{1}{8{\pi }^2{\overline{k}}_{21}}}$	${\displaystyle \frac{2{\overline{k}}_{22}}{{\overline{k}}_{21}}}$	${\displaystyle \frac{1}{32{\pi }^2{\overline{k}}_{21}}}\left[{\displaystyle \frac{\left(3+4{\pi }^2\right)h}{3}}+{\displaystyle \frac{16{\chi }_{C31}}{L}}\left(d_s+\xi a_x\right)\right]$
3	$0.72{\displaystyle \frac{3}{16{\overline{k}}_{31}}}$	${\displaystyle \frac{1}{12{\pi }^2{\overline{k}}_{31}}}$	${\displaystyle \frac{3{\overline{k}}_{32}}{{\overline{k}}_{31}}}$	${\displaystyle \frac{1}{48{\pi }^2{\overline{k}}_{31}}}\left[\left(1+3{\pi }^2\right)h+{\displaystyle \frac{8\left({\chi }_{C31}+{\chi }_{C32}\right)}{L}}\left(d_s+\xi a_x\right)\right]$
4	$0.82{\displaystyle \frac{1}{4{\overline{k}}_{41}}}$	${\displaystyle \frac{1}{16{\pi }^2{\overline{k}}_{41}}}$	${\displaystyle \frac{4{\overline{k}}_{42}}{{\overline{k}}_{41}}}$	${\displaystyle \frac{\left(3+16{\pi }^2\right)h}{192{\pi }^2{\overline{k}}_{41}}}$

* The values 0.88, 0.72, and 0.82 in C21, C31 and C41, respectively, are adjustment coefficients.

.

The constants ${\overline{k}}_{21}-{\overline{k}}_{42}$ used in Table 3 are given in Equations (38)–(43).

$${\overline{k}}_{21}={\displaystyle \frac{1}{96{\pi }^2}}\left[\left(4{\pi }^2+3\right){\overline{\eta }}_{C3}+{\displaystyle \frac{3e_0}{hL}}\left(\left(3{\pi }^2+4\right){\chi }_{C31}+2{\pi }^2{\chi }_{C32}\right)\right]$$

(38)

$${\overline{k}}_{22}={\displaystyle \frac{1}{96{\pi }^2}}\left[\left(4{\pi }^2-3\right){\overline{\eta }}_{C3}+{\displaystyle \frac{3e_0}{hL}}\left(\left(3{\pi }^2-4\right){\chi }_{C31}+2{\pi }^2{\chi }_{C32}\right)\right]$$

(39)

$${\overline{k}}_{31}={\displaystyle \frac{1}{288{\pi }^2}}\left[4\left(3{\pi }^2+1\right){\overline{\eta }}_{C3}+\left(27{\pi }^2+8\right){\displaystyle \frac{e_0}{hL}}{\chi }_{C31}+\left(18{\pi }^2+8\right){\displaystyle \frac{e_0}{hL}}{\chi }_{C32}\right]$$

(40)

$${\overline{k}}_{32}={\displaystyle \frac{1}{288{\pi }^2}}\left[4\left(3{\pi }^2-1\right){\overline{\eta }}_{C3}+\left(27{\pi }^2-8\right){\displaystyle \frac{e_0}{hL}}{\chi }_{C31}+\left(18{\pi }^2-8\right){\displaystyle \frac{e_0}{hL}}{\chi }_{C32}\right]$$

(41)

$${\overline{k}}_{41}={\displaystyle \frac{1}{384{\pi }^2}}\left[\left(16{\pi }^2+3\right){\overline{\eta }}_{C3}+{\displaystyle \frac{12{\pi }^2{e}_0}{hL}}\left(3{\chi }_{C31}+2{\chi }_{C32}\right)\right]$$

(42)

$${\overline{k}}_{42}={\displaystyle \frac{1}{384{\pi }^2}}\left[\left(16{\pi }^2-3\right){\overline{\eta }}_{C3}+{\displaystyle \frac{12{\pi }^2{e}_0}{hL}}\left(3{\chi }_{C31}+2{\chi }_{C32}\right)\right]$$

(43)

$${\overline{\eta }}_{C3}={\displaystyle \frac{-112E{I}_y}{L^4}}\left[\left({\displaystyle \frac{h}{2k_{LT}{e}_0}}+{\displaystyle \frac{L^3{e}_0}{48hE{I}_y}}\right){\chi }_{C31}+{\displaystyle \frac{11L^3{e}_0}{768hE{I}_y}}{\chi }_{C32}\right]$$

(44)

$$\begin{array}{ll}{\chi }_{C31}=& {\mathrm{\Omega }}_C\left[-{\psi }_{C4}{\lambda }_C^3\left(3{\lambda }_C^2{l}^2+K{l}^3-2\right)-2{\lambda }_C^3{\psi }_{C2}+K{\lambda }_C^2{l}^2{\psi }_{C1}\left(3{\psi }_{C2}-2\right)\right.\\ & \left.+2K\left({\lambda }_{C}l{\psi }_{C2}-{\psi }_{C1}\right)\left({\psi }_{C2}-1\right)\right]/\left[-{\psi }_{C4}{\lambda }_C^2\left(2{\lambda }_C^4+4{\lambda }_C^{2}Kl+K^2{l}^2\right)+2{\lambda }_C^{3}K{\psi }_{C3}\right.\\ & \left.+2{\lambda }_{C}l{K}^2{\psi }_{C1}\left(2{\psi }_{C2}-1\right)-K^2{\psi }_{C1}^2\right]\end{array}$$

(45a)

$$\begin{array}{ll}{\chi }_{C32}=& {\mathrm{\Omega }}_C\left[-{\psi }_{C4}{\lambda }_C^3\left(4{\lambda }_C^2{l}^2+K{l}^3\right)+{\psi }_{C1}{\lambda }_C^2\left(4{\lambda }_C{\psi }_{C1}-3K{l}^2+4K{l}^2{\psi }_{C2}\right)\right.\\ & \left.+2K\left({\lambda }_{C}l-{\psi }_{C1}\right)\left({\psi }_{C2}-1\right)\right]/\left[-{\psi }_{C4}{\lambda }_C^2\left(2{\lambda }_C^4+4{\lambda }_C^{2}Kl+K^2{l}^2\right)+2{\lambda }_C^{3}K{\psi }_{C3}\right.\\ & \left.+2{\lambda }_{C}l{K}^2{\psi }_{C1}\left(2{\psi }_{C2}-1\right)-K^2{\psi }_{C1}^2\right]\end{array}$$

(45b)

in which

$${\mathrm{\Omega }}_C={\displaystyle \frac{k_{LT}{e}_0{\lambda }_C{a}_x}{G{I}_k}}$$

(46a)

$$K={\displaystyle \frac{k_{LT}{e_0}^2}{E{\overline{I}}_{\omega }}}$$

(46b)

$$l={\displaystyle \frac{L}{4}}$$

(46c)

$${\psi }_{C1}=\sinh \left({\lambda }_{C}l\right)$$

(47a)

$${\psi }_{C2}=\cosh \left({\lambda }_{C}l\right)$$

(47b)

$${\psi }_{C3}=\sinh \left(2{\lambda }_{C}l\right)$$

(47c)

$${\psi }_{C4}=\cosh \left(2{\lambda }_{C}l\right)$$

(47d)

7. Results and Discussions

7.1. Comparisons of Buckling Loads

Comparisons between the predictions of the proposed solutions and the FE results are shown in Figure 7 for C300t2.0 purlins (Table 1) with two anti-sag bars. In Figure 7, M_xcr,1, M_xcr,2 and M_xcr,3, obtained using Equation (34), represent the buckling loads corresponding to the one-, two-, and three-half-wave failure modes, respectively. It can be observed from Figure 7 that the minimum values of M_xcr,1, M_xcr,2 and M_xcr,3 agree well with the FE results (M_cr) for all cases. Here, M_cr denotes the buckling load corresponding to the first buckling mode obtained from FE analysis using the developed FE code. The results based on the assumption of rigid lateral restraints provided by anti-sag bars (M_crR) are also included in Figure 7. These results may significantly overestimate the buckling loads, particularly for purlins with relatively short spans. Acceptable accuracy of M_crR is only observed when the three-half-wave mode is the dominant failure mode, i.e., when M_xcr,3 governs.

Comparisons between the predictions of the proposed solutions for purlins with three anti-sag bars and the FE results (M_cr) are presented in Figure 8. The minimum values of M_xcr,2, M_xcr,3 and M_xcr,4, calculated using Equation (37), provide accurate predictions of the buckling loads for the considered purlins. The results based on the rigid bracing assumption (M_crR) are close to M_xcr,4 in Figure 8; otherwise, they may significantly overestimate the buckling loads.

Further comparisons are presented in Figure 9a–c for C-section purlins with two anti-sag bars considering different cross-section thicknesses (Figure 9a), cross-section heights (Figure 9b), and vertical locations of anti-sag bars (Figure 9c). The predictions of the proposed solution (Equation (34)) show good agreement with the FE results. In these comparisons, the minimum value among the three buckling loads corresponding to the possible dominant modes is taken as the predicted buckling load. Similar agreement between the proposed solutions (Equation (37)) and the FE results is observed for C-section purlins with three anti-sag bars, as shown in Figure 10a–c.

7.2. Determination of Dominant Buckling Mode

As discussed above, the dominant buckling mode of the considered purlins varies depending on the cross-section properties, span, number of anti-sag bars, and vertical location of the anti-sag bars. Therefore, in the previous sections, the minimum value among the buckling loads corresponding to the possible failure modes was used to predict the buckling load for each case. However, this requires calculation of buckling loads for all possible modes. To simplify the calculation, a parameter, φ, is introduced to determine the dominant buckling mode:

$$\phi ={\displaystyle \frac{k_{\mathrm{LT}}{e}_0^2{L}^3}{E{\overline{I}}_{\mathsf{\omega }}}}$$

(48)

The values of φ for determining the dominant buckling loads of C-section purlins with two or three anti-sag bars are listed in

Table 4. Values of φ for determining dominant buckling modes.

Number of Anti-Sag Bar	Dominant Buckling Mode	Condition
2	one half-wave	φ ≤ 70
	two half-waves	70 < φ < 500
	three half-waves	φ ≥ 500
3	one half-wave	φ ≤ 440
	two half-waves	440 < φ < 1900
	three half-waves	φ ≥ 1900

.

The values of φ for C-section purlins with two anti-sag bars are illustrated in Figure 11, where all C-sections listed in Table 1 are considered. Four vertical locations of anti-sag bars within the web (i₀ = 0.5h, 0.675h, 0.75h and 0.875h) are adopted. The purlin span varies from 4000 mm to 12,000 mm at intervals of 500 mm. All results shown in Figure 11 are obtained from FE analysis using the previously developed FE code.

It can be seen from Figure 11 that the dominant buckling mode of C-section purlins with two anti-sag bars can be accurately determined using the φ values provided in Table 4. In a few cases where φ is very close to the boundary values (e.g., φ = 500 and 70 in Figure 11), misclassification of the buckling mode may occur. However, in such cases, the buckling loads corresponding to the adjacent modes are very close, and the prediction accuracy of the buckling load remains satisfactory. For example, φ = 63.14 (close to one of the boundary values 70) is obtained using Equation (48) for the purlin C220t1.5 with e₀ = 0.5h and L = 4500 mm (shown in Figure 11 using “Solid Triangle”). The dominant buckling mode is the one-half-wave mode according to Table 4, while the FE result indicates a two-half-wave failure mode. However, the difference in predicted buckling loads is negligible (M_xcr,1 = 36.56 kN∙m and M_xcr,2 = 36.80 kN∙m using Equation (34)).

The applicability of φ for determining the dominant buckling mode of C-section purlins with three anti-sag bars is demonstrated in Figure 12, which also shows good agreement.

8. Conclusions

The lateral buckling behaviour of C-section purlins with two or three anti-sag bars is investigated in this study by considering the effective stiffness of the anti-sag bars. The total potential energy expressions, based on the buckling theory recently proposed by the authors, are first established, incorporating the effects of lateral restraints at the top flange and the anti-sag bars. Based on the proposed total potential, closed-form solutions for the lateral buckling loads of C-section purlins are developed, which can be used for hand calculations.

The performance of the proposed solutions is evaluated by comparing the predicted buckling loads with those obtained from the FE code developed in this study. Very good agreement is observed between the two sets of results. In addition, a parameter φ is proposed to determine the dominant buckling mode of C-section purlins with two or three anti-sag bars. Comparisons with FE results demonstrate that φ can accurately identify the dominant buckling mode of the considered purlins, thereby simplifying the calculation of buckling loads using the proposed solutions.

For ease of using the presented solutions, the buckling loads of C-section purlins with two or three anti-sag bars can be calculated following the procedures in

Table 5. Calculation procedure for buckling loads.

Steps	With Two Anti-Sag Rods	Steps	With Three Anti-Sag Rods
1	$L,h,I_{\omega },I_y,I_k,{\overline{I}}_{\omega },d_s,a_x,e_0,{\beta }_y,\xi$	1	$L,h,I_{\omega },I_y,I_k,{\overline{I}}_{\omega },d_s,a_x,e_0,{\beta }_y,\xi$
2	${\chi }_{C2}(\mathrm{Equation} (22\mathrm{b}))$, ${\overline{\eta }}_{C2}(\mathrm{Equation} (27))$	2	${\chi }_{C31},{\chi }_{C32}(\mathrm{Equations} (45\mathrm{a}) \mathrm{and} (45\mathrm{b}))$ ${\overline{\eta }}_{C3}(\mathrm{Equation} (44))$
3	$k_{11}~k_{32}(\mathrm{Equations} (36\mathrm{a})-(36\mathrm{f}))$	3	${\overline{k}}_{11}~{\overline{k}}_{42}(\mathrm{Equations} (38)-(43))$
4	$C_{n1}~C_{n4}(\mathrm{Table} 2)$	4	$C_{n1}~C_{n4}(\mathrm{Table} 3)$
5	$M_{cr1}~M_{cr3}(\mathrm{Equation} (34))$	5	$M_{cr1}~M_{cr4}(\mathrm{Equation} (37))$
6	Buckling load = min(Mcr1~Mcr3)	6	Buckling load = min(Mcr1~Mcr4)

.