## 4 Discussion of FE Analysis Results  
The springing in the arch rib has the largest axial force. Taking the analysis results of a FE model as an example, Figure 12a shows the springing axial force versus vault horizontal displacement curves under four conditions: not considering geometric and material nonlinearity, only considering material nonlinearity, only considering geometric nonlinearity, and simultaneously considering geometric and material nonlinearity. Figure 12a shows that geometric nonlinearity has little influence on the springing axial force of steel arch bridges, while material nonlinearity has a great influence, and both material and geometric nonlinearity should be considered when calculating the out-of-plane stable bearing capacity of steel arch bridges. An arch under a central in-plane load may lose its stability in an out-of-plane buckling mode when the load attends a certain value. When the axial force of springing reaches the maximum value (red dot in Figure 12a), out-of-plane instability of the arch rib occurs, and the mode of out-of-plane instability is symmetrical, as shown in Figure 12b. In FE analysis, the maximum axial force of the arch rib in case of out-of-plane instability of the arch bridge is defined as out-of-plane critical axial forces, \(N_{cr}\). Figure 11: Comparison of Test and FE Analysis results. Figure 10: Test Specim and Loading Photo. Since there are three different arch rib cross sections in nine FE analysis cases, and the critical axial force \(N_{\sigma}\) by FE analysis with different sections is not comparable, the \(N_{\sigma}\) of the arch rib is normalized for the purpose of discussion by the yield force \(N_{y}\) at the full-section yield of arch rib, which is the product of yield stress and cross-sectional area of an arch rib. When \(N_{\sigma}/N_{y}=1\), that is, when out-of-plane instability occurs, the full section of the arch springing reaches the yield strength of the material. The influences of rise-to-span ratio, arch rib spacing, and \(\gamma\)-value with different stiffness ratios on normalized critical axial forces for the through-type bridge (as representative of the three types) are shown in Figure 13a-c. The influence of bridge type is studied by translating the location of the girder framework, as already explained, and the critical axial force with different stiffness ratios is shown in Figure 13d. These figures show each arch rib type in a different color with the same fill. Cases with the same lateral bracing are notated by marks with the same shape. Figure 13 shows that \(N_{\sigma}/N_{y}\) ranges from 0.33 to 0.97 under different construction parameters; in other words, when the steel arch bridge occurs out-of-plane instability, the structural parameters have a great influence on whether the arch springing section yield or yield area, especially the parameter of lateral bracing arrangement range. Figure 13a demonstrates that the change in the rise-to-span ratio exerts little influence on normalized critical axial force except in a few cases. This is because a decrease in the rise-to-span ratio leads to reduced critical flexure load but also makes the arch ribs more flat, which leads to an increase in the axial force in the arch rib. These two influences together cause a slight influence on the rise-to-span ratio on the normalized critical axial force. The influence of the rise-to-span ratio is taken into account in all three codes. Figure 13b shows that the normalized critical axial force is slightly affected by arch rib spacing when the stiffness ratio is small. However, it decreases with increasing arch rib spacing when the stiffness ratio is relatively large. Since lateral bracing with the same cross-section becomes weaker as arch rib spacing increases, the deformation of arch ribs becomes  
Figure 12: FE Analysis Result of A FE Model: (**a**) Relation Curve between the Axial Force at the Springing and the Horizontal Displacement at Vault; (**b**) Out-of-plane Instability Mode of Arch Rib. less restrained.