## 5 Out-of-Plane Inelastic Buckling Behaviors of Vierendeel Truss Arches
### Cases under the FSUVL  
Under the full-span uniform vertical load (FSUVL), the end-fixed circular truss arches tend to be under the combined actions consisting of compression, bending, and shear. For comparison, the set of the Vierendeel truss arches under the FSUVL, with the same dimensions as the previous cases under the FSURL, was investigated based on the FE inelastic analyses. In these cases, elastic local buckling can also be prevented before global buckling with the same small \(\lambda_{c}\). Figure 15 shows the variations of the uniform vertical load (\(q_{f}\)) with the vault out-of-plane displacement (\(u\)) for the arches (\(L=50\) m). It is seen that the load \(q_{f}\) for a typical arch (\(L=50\) m\({}_{f}/L=0.3\)) increases with the increase in the out-of-plane vault displacement and reaches its ultimate value for the case ignoring the torsional stiffness of chords at 26.8 kN/m, shown as point B\({}_{1}\), which is smaller than that (32.2 KN/m) in the case incorporating the torsional effects, shown as point B. Figure 16 presents the overall stress distribution of the typical arch under the ultimate FSUVL (corresponding to point B). It is seen that the truss arch firstly yields at both feet of chords, and then buckles under the ultimate load. This behavior confirms the findings by [PERSON] et al. [10] that local yielding of chords will occur before global buckling in the planar Vierendeel truss arches with \(i_{t}/i_{c}\) larger than 2.25. Thereafter, with the further increase in \(u\), the yielding spots develop from the feet to the quarters along the arch until it fails eventually.  
Under the FSUVL, the end-fixed truss arches usually bear compression and bending simultaneously. Thus, an interaction strength design equation can be deduced to check the ultimate design strength, which is expressed as [27; 37; 38]:  
\[\frac{N^{*}}{\varphi N_{y}}+\frac{\alpha M^{*}}{M_{y}}\leq 1 \tag{23}\]  
where \(N^{*}\) and \(M^{*}\) are the maximum axial compression and the maximum bending moment of the Vierendeel truss arches based on the first-order elastic analyses, respectively;  
Figure 16: Stress distribution of the typical Vierendeel truss arch under the ultimate FSUVL.  
Figure 15: Out-of-plane vault displacements of the typical Vierendeel truss arch under FSUVL.  
and \(M_{y}\) are the compression and bending moment at yielding of the whole cross-section, respectively; \(\varphi\) is the out-of-plane reduction factor for the buckling compressions of the truss arches under the FSUVL and can be referred from the curve \(b\) from the GB50017-2017. Since the Vierendeel truss arches do not have diagonal tubes to resist shear, they possess small sectional shear stiffness. Thus, the second-order sectional bending moment under shear should be considered by multiplying the \(M^{\bullet}\) with a moment amplification factor \(\alpha\) that can be calculated as:  
\[\alpha=\frac{1}{1-\frac{N^{\bullet}}{q_{cr}R}} \tag{24}\]  
where \(q_{cr}\) is the elastic buckling load of the arches in the FSURL manner given by Equation (14). It has been verified that, when \(\alpha\) is less than 1.4, the first-order moment amplified by \(\alpha\) can provide a good estimation of the second-order moment [4]; when \(\alpha\) is larger than 1.4, the modified bending moment (\(\alpha M^{\bullet}\)) should be replaced by the second-order moment obtained numerically.  
Based on the FE analyses, we obtained the maximum strength pairs (\(N^{\bullet}/(\varphi N_{y})\), \(\alpha M^{\bullet}/M_{y}\)) for the Vierendeel truss arches at different spans (20 m\(\sim\)60 m) and rise-to-span ratios ranging from 0.1 to 0.5. Moreover, the interaction design curve obtained by Equation (23) is presented for comparison, as shown in Figure 17. It is seen that all strength capacity pairs are above the interaction design curve, indicating that the reduction factor calculated by Equation (21) can provide a conservative prediction for the out-of-plane stability design of the end-fixed Vierendeel truss arches.