## 4 Out-of-Plane Elastic Buckling Load of Vierendeel Truss Arches
The calculated loads incorporating the torsional effects are remarkably larger than that, if we ignore the torsional effects but slightly smaller than the FE results. Moreover, the deviation between the FE results and the theoretical results incorporating the torsional effects decrease with \(L_{c}\) and increase with \(B\). Figure 9d presents the relationship between \(q_{cr}\) and the transverse-to-chord member stiffness ratio (\(i_{t}/i_{c}\)), which represents the combined influence of \(L_{c}\) and \(B\). Generally, \(q_{cr}\) increases rapidly as \(i_{t}/i_{c}\) increases from 0.1 to 3.0 and then increases gradually approaching a plateau as \(i_{t}/i_{c}\) increases beyond 6.0. It is also noted that the \(q_{cr}\) calculated from Equations (13) and (14) considering the torsional effects are consistent with the FE results and much larger than that from these equations if we ignore the torsional effects. This trend is similar to the relationship between the sectional torsional stiffness and member stiffness ratio (\(i_{t}/i_{c}\)), indicating that \(i_{t}/i_{c}\) may play a critical role in determining the out-of-plane buckling loads of the Vierendeel truss arches. In practice, the ends of large-span truss arches are usually fixed for better flexural and torsional resistance. However, the buckling load formulas for the end-fixed arches are too difficult to derive analytically. Here, a fitted formula is proposed based on the out-of-plan static analyses of the end-fixed truss arches conducted by the FE simulations. Based on the simulations, the buckling load formulas for the end-fixed Vierendeel truss arches can be expressed as:  
\[\left\{ \begin{array}{l}q_{cr}=\dfrac{q_{cr0}}{1+q_{cr0}R/K_{V}}\\ q_{cr0}=\dfrac{4\pi^{2}E_{ly}}{S^{2}R}\Big{[}A\Big{(}\frac{\Theta}{\pi}\Big{)} ^{2}+B\Big{(}\frac{\Theta}{\pi}\Big{)}+0.85\Big{]}\end{array} \right. \tag{16}\]  
where  
\[\left\{ \begin{array}{l}A=1258.18\Big{(}\frac{GJ}{E_{ly}}\Big{)}^{2}-90.11 \Big{(}\frac{GJ}{E_{ly}}\Big{)}+1.84\\ B=-1435.94\Big{(}\frac{GJ}{E_{ly}}\Big{)}^{2}+97.89\Big{(}\frac{GJ}{E_{ly}} \Big{)}-2.76\end{array} \right. \tag{17}\]  
Figure 10 shows the buckling loads of a set of Vierendeel truss arches with identical geometric dimensions: \(D_{c}\times t_{c}\) = 121 mm \(\times\) 10 mm, \(D_{t}\times t_{t}\) = 100 mm \(\times\) 10 mm, \(B\) = \(H\) = \(L_{c}\) = 1.0 m, and \(L\) = 50 m. For the truss arches with \(f/L\) ranging from 0.1 to 0.45, the buckling loads calculated by the fitted Equation (16) agree well with the FE simulations (the maximum error is within 5%). These results are much larger than that of the pin-ended \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dimension** & \(L\) **(=50 m)** & \(L_{c}\) & \(B\) & \(i_{t}/i_{c}\) \\ \hline Increment of average \(q_{cr}\) by & 37.93\% & 429.39\% & 52.15\% & 41.45\% \\ Equation (13) & 38.30\% & 442.88\% & 52.61\% & 41.67\% \\ Increment of average \(q_{cr0}\) by & & & & \\ Equation (14) & 38.30\% & 442.88\% & 52.61\% & 41.67\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Increment of the average buckling load over the considered dimensions for the pin-ended Vierendeel truss arches considering \(GI_{pci}\) with respect to that ignoring \(GI_{pci}\). arches calculated by Equation (13), confirming the enhanced buckling resistances in the truss arches with fixed ends. Figure 11 shows the buckling loads for the end-fixed Vierendeel truss arches, with identical geometric dimensions as the corresponding the pin-ended arches shown in Figure 8. In the end-fixed arches, the changing trends of the buckling loads with \(L\), \(L_{c}\), \(B\), and \(i_{t}/i_{c}\) are similar to that of the pin-ended arches. For all cases considered here, the buckling loads of the end-fixed arches calculated by Equation (16) agree well with the FE results and are much larger than that of the pin-ended arches. In addition, the buckling loads incorporating the torsional stiffness of chords are remarkably larger than that ignoring the torsional effects.