## 4 Out-of-Plane Elastic Buckling Load of Vierendeel Truss Arches  
For the pin-ended truss arches with diagonal tubes, [PERSON] et al. have derived an out-of-plane buckling load incorporating the shear effect of chords using the static equilibrium approach [29]. Under the full-span uniform radial load (FSURL), the out-of-plane critical buckling load (\(q_{cr}\)) for the truss arches can be expressed as:  
\[q_{cr}=\frac{q_{cr0}}{1+q_{cr0}R/K_{V}} \tag{13}\]  
where \(K_{V}\) is the sectional shear stiffness of the truss arch, \(q_{cr0}\) is the out-of-plane critical buckling load of the truss arch without considering the shear deformation of chords. Figure 8: Sectional torsional stiffness and as a function of **(a)** segment length or **(b)**\(i_{t}/i_{c}\) for the equivalent straight trusses obtained from FE results and derived equations [33]. According to [PERSON]'s equation [19], the elastic out-of-plane buckling load of the pin-ended circular solid-web arch under the FVURL can be derived as:  
\[q_{cr0}=\frac{P_{y}\Big{[}1-(\Theta/\pi)^{2}\Big{]}^{2}}{R\Big{[}1+(EI_{y}/GJ)( \Theta/\pi)^{2}\Big{]}} \tag{14}\]  
where \(P_{y}\) is the equivalent axial buckling compression obtained in the first flexural buckling mode of the truss arch around the \(\omega\)-\(y\) axis. The arch has a developed length of \(S\), an in-plane radius \(R\), and a constant cross-section. For large-span arches, the \(P_{y}\) can be approximately calculated as:  
\[P_{y}=\frac{\pi^{2}EI_{y}}{S^{2}} \tag{15}\]  
Figure 9 shows the buckling loads obtained by the proposed Equations (13) and (14), and the FE simulations for the circular steel tubular Vierendeel truss arches at different spans (\(L\)), segmental lengths (\(L_{c}\)), sectional side lengths (\(B\)), and transverse-to-chord member stiffness ratios (\(i_{t}/i_{c}\)). The dimensions (\(L\), \(L_{c}\), and \(B\)) of the typical Vierendeel truss arch are initially set to be 50 m, 1.0 m, and 1.0 m, respectively. The influence of each dimension on the buckling load can be studied independently by fixing two other dimensions at their initial values. For all cases, the rise-to-span ratio (\(f/L\)) is fixed at 0.2; the external diameter (\(D_{c}\)) and wall thickness (\(t_{c}\)) of each chord are kept at 152 mm and 8 mm, respectively. As observed in Figure 9a-d, the buckling loads in the FVURL manner (\(q_{cr}\)) decrease with the increase in \(L\) (or \(L_{c}\)) but increase with the increase in \(i_{t}/i_{c}\) (or \(B\)). For all cases considered in this study, the buckling loads calculated from Equations (13) and (14), either  
Figure 9: Comparison of buckling loads between FE and theoretical results for the pin-ended Vierendeel truss arches with varying (**a**) spans, (**b**) segmental lengths, (**c**) sectional side lengths, and (**d**) transverse-to-chord member stiffness ratios. incorporating the torsional effects of chords or not, agree well with each other. This indicates that both the Kirchhoff equation and the equation derived by [PERSON] et al. [29] can predict nearly identical out-of-plane buckling loads for large-span truss arches. It is also noted that the buckling load for cases incorporating the torsional effects of chords is generally larger than that for cases ignoring the torsional effects of chords, as shown in Table 1. This indicates that the torsional effects play a significant role in determining the buckling loads (\(q_{cr}\)). However, for the arches with spans larger than 70 m, as shown in Figure 9a, the discrepancy due to the torsional effects becomes negligible. Figure 9b,c show the numerically and theoretically obtained buckling loads of the truss arches as a function of the segmental length and the sectional side length, respectively. Generally, the buckling loads for all cases (\(L=50\) m) decrease with the segmental length (\(L_{c}\)) and increase with the sectional side length (\(B\)). It is also observed that the buckling loads calculated by Equations (13) and (14), incorporating the torsional stiffness of chords or not, are consistent with each other.