## 2 Finite Element Models  
Figure 2 shows the schematic model of a typical circular steel tubular Vierendeel truss arch studied in this work. The truss arch is constructed by four circular steel tubular chords arranged at the four corners of the rectangular cross-section. All chords are segmented by a set of rectangular diaphragms made of four connected transverse tubes that are distributed normal to the arch's sectional centroid axis with a constant interval. In each segment, no diagonal tubes are included.  
Figure 3 presents the geometric dimensions of the global and sectional models for a typical circular steel tubular Vierendeel truss arch. \(L\) is the arch span; \(f\) is the rise of the arch; \(R\) is the curvature radius of the cross-sectional centroid arch axis; \(\Theta\) is the included angle; \(S\) (=2\(\Theta R\)) is the total developed length of the arch; \(B\) and \(H\) are the width and height of the rectangular section, respectively; \(L_{c}\) is the segmental length between the neighboring transverse diaphragms; \(D_{c}\) and \(t_{c}\) are the outer diameter and thickness of the chords, respectively; \(D_{t}\) and \(t_{t}\) are the outer diameter and thickness of the transverse tubes, respectively.  
In this study, the stability analyses of the steel tubular truss arches were performed by using the finite element (FE) software ABAQUS [31]. To consider the effect of the shear deformation on the stability of the Vierendeel truss arches, the Timoshenko beam element B31 was used to model the chord tubes and transverse tubes. All structural members in each model are meshed at a constant mesh size. We calculated the computational error for each model at different mesh sizes from 10 mm to 150 mm and found that the computational errors for models with mesh size less than 150 mm could be negligible. Thus, in this study, all models are meshed at the size of 100 mm for saving computational cost. A typical arch model with a span of 50 m and a segmental length of 1 m has 2834 nodes and 2984 meshes  
Figure 2: Schematic model of a typical circular steel tubular Vierendeel truss arch with rectangular section. (**a**) Elevation. (**b**) Diaphragm.  
in total. Two types of arches are considered with different boundary constraints: the pinned and fixe ends. In the end-fixed arch models, all translations and rotations along the x-, y-, and z-directions are fixed, while in the pin-ended arches, the translations in the y- and z-directions and the rotation along the z-direction are fixed at both ends. With these boundary constraints, circular arches are likely to bear the combined compressive and bending actions. For simplification, the translation in the y-direction of one end is usually set free to ensure the arch's cross-section under a uniform compression [32], as shown in Figure 3a. Thus, the elastic buckling loads for the pin-ended arches can be derived analytically.  
Bilinear constitutive models were adopted for all steel tubular members. Specifically, the yielding stress for the chord tubes is 235 MPa (\(f_{yc}\)) and that for the transverse tubes is 345 MPa (\(f_{yt}\)). For all steel tubular members, the initial elastic modulus (\(E\)) is 2.06 \(\times\) 10\({}^{5}\) MPa, and the Poisson's ratio (\(v\)) is 0.3. The out-of-plane global initial geometric imperfection (with the maximum amplitude of S/500) was introduced to all FE models studied in this work, and the influence of the welding residual stress was neglected. Moreover, three typical loading cases were considered herein such as the full-span radially uniform load (FSRUL) \(q_{r}\), the full-span uniformly vertical load (FSUVL) \(q_{f}\), and the half-span uniformly vertical load (HSUVL) \(q_{h}\), as shown in Figure 4.