## 6 Conclusions  
This work reported the out-of-plane stability of the circular steel tubular Vierendeel truss arches with rectangular cross-sections under different loading cases. Considering both the shear deformation and torsion of chords, the sectional torsional stiffness of the pin-ended Vierendeel truss segment was deduced theoretically using the static equilibrium theorem. The effects of torsional stiffness of chords on the out-of-plane elastic buckling load were investigated for the circular steel tubular Vierendeel truss arches with pinned and fixed ends. Moreover, the out-of-plane elastic and inelastic buckling incorporating torsional effects were discussed for the stability design of the end-fixed Vierendeel truss arches. Some conclusions can be drawn as follows:  
1. The sectional torsional stiffness of the pin-ended circular steel tubular Vierendeel truss arches decreases with the segment length but increases with the transverse-to-chord member stiffness ratio (\(i_{t}/i_{c}\)). Incorporating the torsional stiffness of individual chords can remarkably enhance the sectional torsional stiffness of the Vierendeel truss arches. 2. The out-of-plane elastic buckling loads of the circular Vierendeel truss arches, either the pin-ended cases derived by the equilibrium theorem, or the fix-ended cases derived by the numerical fitting, increase significantly when the torsional stiffness of each chord is taken into account. The dependence of the out-of-plane buckling load, similar to that of the sectional torsional stiffness on \(i_{t}/i_{c}\), is observed in both the pin-ended and the end-fixed Vierendeel truss arches. 3. For the fixed Vierendeel truss arches with large \(i_{t}/i_{c}\), the local yielding of chords occurs before the global buckling. Incorporating the torsional stiffness of chords will remarkably increase the ultimate buckling loads of arches under different loading cases (FSURL, FSUVL, and HSUVL). The curve \(b\) from the design codes (GB50017-2017 or Eurocode 3) can provide a lower bound prediction of the reduction factors for the out-of-plane stability design of the end-fixed Vierendeel truss arches. These conclusions may provide practical guidance for the stability design of circular steel tubular Vierendeel truss arches. However, the proposed equations in this work can only provide a conservative prediction on the out-of-plane buckling loads in the Vierendeel truss arches with rectangular cross-sections at some specific dimensions. For extensive application or optimization design in practice, more research can be conducted in the near future by utilizing the optimization methods, such as the artificial neural network or the fuzzy logic algorithm [39; 40], into the current design procedures to improve the adaptivity of the stability design. Figure 21: Out-of-plane stability design procedure of the Vierendeel truss arches. **Author Contributions:** Q.L. : conceptualization, validation, writing--review and editing, funding acquisition; Y.F. : investigation, writing--original draft preparation, methodology; C.W. : investigation, software; X.W. : writing--review and editing, project administration; X.R. : writing--review and editing. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Natural Science Fund Committee of China (Grant No. 11632014, 51808444), Scientific Research Program of Shanxi Provincial Education Department (18 JK0502), State Key Laboratory for Strength and Vibration of Mechanical Structure open Foundation (Grant No. SV2018-KF-35), Startup Foundation of XUST (Grant No. 2018 YQ2-05). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.