## 5 Out-of-Plane Inelastic Buckling Behaviors of Vierendeel Truss Arches  
### Cases under the FSURL  
It has been reported that the buckling behaviors of the truss arches (locally or globally) are dependent of their loading cases [17]. Here, three typical loading cases (FSURL, FSUVL, and HSUVL) were considered to study the elastic-plastic out-of-plane buckling behaviors of the Vierendeel truss arches. Under the full-span uniform radial load (FSURL), the end-fixed circular truss arches bear large compression and small bending. To prevent the elastic local buckling before the global buckling, the Chinese Code for Design of Steel Structures (GB50017-2017) recommended that the slenderness ratio of a single chord \(\lambda_{c}\) should meet the following requirements [34]:  
\[\lambda_{c}<\min\{0.5\lambda_{oy},40\} \tag{18}\]  
Here, \(\lambda_{oy}\) is the equivalent slenderness ratio of the truss segment, which can be expressed as:  
\[\lambda_{oy}=\sqrt{\lambda_{y}^{2}+\frac{\pi^{2}}{12}\lambda_{c}^{2}\bigg{(}1 +\frac{2i_{c}}{i_{t}}\bigg{)}} \tag{19}\]  
where \(\lambda_{y}\) is the out-of-plane slenderness ratio of the truss segment, \(i_{c}\) and \(i_{t}\) are the member stiffnesses of chords and transverse tubes, respectively. Given a set of the Vierendeel truss arches having \(f/L=0.3\), \(L=50\) m, \(B=H=L_{c}=1.0\) m, \(D_{c}\times t_{c}=121\) mm \(\times\) 10 mm, and \(D_{t}\times t_{t}=200\) mm \(\times\) 10 mm, \(\lambda_{c}\) can thus be calculated to be 25.4, which satisfies the Equation (18), meaning that the local buckling can be prevented before the global buckling. According to the FE results, the relationship between the vault displacements (out-of-plane \(u\) and in-plane \(w\)) and the uniform radial load (\(q_{r}\)) can be established, as shown in Figure 12. It is seen that both vault displacements (\(u,w\)) of the arches (\(L=50\) m) increase with the increase in the load \(q_{r}\). Under the increased \(q_{r}\), the arches deform out of the arch plane due to the global out-of-plane geometric imperfection (\(S/500\)), leading to a second-order bending along the \(o\)-\(y\) axis, and results in the out-of-plane buckling. It is noted that the out-of-plane ultimate load (ignoring the torsional stiffness of chords) is 42.1 kN/m, shown as point A\({}_{1}\) in Figure 11, which is much less than that (60.2 kN/m) in cases considering the torsional effects, shown as point A in Figure 11. For comparison, the in-plane ultimate load of a typical arch (\(L=50\) m, \(f/L=0.3\)) is calculated to be 97.3 kN/m by introducing a global in-plane geometric imperfection (\(S/500\)) alternatively. Figure 13 presents the overall stress distribution of the typical arch under the ultimate FSURL (corresponding to point A). Under the combined actions of compression and bending, the initial yielding starts from the four symmetric spots: the right chords at both arch feet and the top-right chords at the 3/8 and 5/8 of the span. After that, the arch fails in a symmetric inelastic global buckling mode, accompanied by the decrease in \(q_{r}\) and the further increase in \(u\). According to [PERSON] et al. [35], the normalized slenderness of an arch can be defined as:  
\[\lambda_{n}=\sqrt{\frac{N_{y}}{N_{cr}}}=\sqrt{\frac{4A_{c}f_{wc}}{q_{cr}R}} \tag{20}\]  
where \(N_{y}\) and \(N_{cr}\) are the yielding and critical compressions on the cross-section of the truss arch, respectively; \(q_{cr}\) is the out-of-plane elastic critical FSURL of the truss arch and can be calculated by Equation (16). Thus, the reduction factor (\(\varphi\)) for the out-of-plane ultimate strength relative to the yielding strength of the Vierendeel truss arches can be defined as:  
\[\varphi=\frac{N_{u}}{N_{y}}=\frac{q_{u}R}{4A_{c}f_{yc}} \tag{21}\]  
where \(N_{u}\) is the ultimate compression on the cross-section of the truss arch and \(q_{u}\) is the ultimate bearing load in the FSURL manner. It is acknowledged that the ultimate bearing capacity of a truss arch is closely related to its \(f\) and \(L\)[17].