## 6 Conclusions  
The accuracy of provisions for the out-of-plane stability of steel arch bridges in major design codes was evaluated using the results of FE analysis. Summaries of this work are shown as follows.  
(1) According to FE analysis, the influence of the rise-to-span ratio on critical axial force is generally small. The critical axial force decreases with increasing arch rib spacing when the stiffness ratio is relatively large. A smaller ratio of arch rib length provided with lateral bracing (\(\gamma\)-value) significantly reduces the critical axial force, and normalized critical axial force decreases with increasing stiffness ratio. The critical axial force of half-through type arch bridges is lowest when the stiffness ratio is relatively small. A deck-type bridge has a larger critical axial force than a through-type bridge when the stiffness ratio is relatively large, while the results are the opposite when the ratio is small.  
(2) The Chinese code generally overestimates the critical axial force when the \(\gamma\)-value is more than 0.86 and otherwise underestimates it, except in cases with a smaller stiffness ratio. The code also overestimates the positive effect of the \(\gamma\)-value and the negative effects of the rise-to-span ratio and stiffness ratio. The effect of arch rib spacing is estimated to be non-monotonous when the stiffness ratio is small. Otherwise, the negative effects are overestimated. The influence of bridge type is properly evaluated.  
(3) The Japanese code accurately estimates the critical axial force when the stiffness ratio is small and otherwise overestimates it. The positive effects of the rise-to-span ratio and stiffness ratio, and the negative impact of \(\gamma\)-value, are overestimated. The influence of bridge type and arch rib spacing is properly evaluated.  
(4) The Eurocode generally underestimates the critical axial force for deck-type bridges. It accurately estimates the critical axial force for through and half-through type bridges when the stiffness ratio is small and the \(\gamma\)-value is smaller than 0.86, and otherwise mostly overestimates it. The Eurocode also overestimates the negative effects of the rise-to-span ratio, arch rib spacing, and stiffness ratio. The effect of \(\gamma\)-value on through and half-through type bridges is estimated to be non-monotonous, while the positive effect of \(\gamma\)-value is overestimated for deck-type bridges.  
(5) Considering the influence of the rise-to-span ratio, the ratio of lateral bracing, and arch rib spacing with different stiffness ratios, factors to improve the accuracy of the critical axial force obtained by the three codes are proposed for a practical design process. The accuracy of these proposed corrections is verified.  
[PERSON] and [PERSON] conceived the analyses; [PERSON], [PERSON] and [PERSON] performed the finite element analysis and analyzed the data; [PERSON] contributed materials/analysis tools, and [PERSON] wrote the paper. All authors have read and agreed to the published version of the manuscript. This research was funded by the Natural Science Foundation of Fujian Province (No. 2021)011059) and the Research Start-up Fund of Fujian University of Technology (No. GY-Z20015).  
Figure 18: Estimation Accuracy: (**a**) Influence of Rise-to-span Ratio; (**b**) Influence of \(\gamma\)-value; (**c**) Influence of Arch Rib Spacing.  
Data Availability Statement: The data presented in this study are available on request from the corresponding author.  
The authors acknowledge the funding from the Natural Science Foundation of Fujian Province (No. 2021)011059) and the Research Start-up Fund of Fujian University of Technology (No. GY-Z20015). The authors acknowledge the support from [PERSON] from Nagasaki University, Japan.  
The authors declare no conflict of interest.