## 1 Introduction  
Vierendeel truss arches are preferred for obtaining larger spacing, concise appearance, and functional flexibility in long-span spatial structures such as bridges, stadiums, and railway stations [1]. In the classical Pratt or Warren truss arches, chord members bear compression and bending actions transferred from the external loadings, and diagonal members bear most of the shears along the centroid axis of the arches. By contrast, in the Vierendeel truss arches [2, 3], no diagonal members exist and only the chords can be used to resist the sectional shears. The weak shear resistance may determine the strength design and thereby the in-plane and out-of-plane stability design of the Vierendeel truss arches in practical applications. With adequate lateral bracings, arches under compression and bending tend to deform and fail in the in-plane modes. [PERSON] and his coauthors revealed the elastic and inelastic in-plane buckling behaviors for the pin-ended and end-fixed arches and proposed the strength design equation for the I-section steel arches subjected to combined compressive and bending actions [4, 5, 6]. [PERSON] et al. investigated the in-plane failure mechanism of the pin-ended circular steel arches with welded hollow sections numerically and proposed the strength design formula accounting for web local buckling [8]. It was also reported that the sectional shear stiffness played an important role in determining the elastic-plastic stability in circular arches [9]. Further, [PERSON] et al. derived the in-plane elastic buckling loads with the consideration of the effect of shear deformation and proposed an interaction design equation for the circular steel tubular Vierendeel planar truss arche [10]. Moreover, the shear effects of chords were taken into accounted to obtain the in-plane elastic buckling loads for the pin-ended circular Vierendeel-like steel planar arches, i.e., the plate-tube-connected arches and the web-opened steel arches [11; 12; 13]. It can be concluded that considering the sectional shear deformation may produce a much lower in-plane buckling load in truss arches than that calculated by the Timoshenko equation [14]. To meet functional or architectural needs, some arches are designed with sparse lateral bracings [15; 16]. With inadequate lateral supports, the arches may deform in the flexural-torsional modes and fail in the out-of-plane before their in-plane failures if there exist considerable initial out-of-plane geometric imperfections [17; 18]. The timeline for some researches on the out-of-plane stability of circular steel arches is shown in Figure 1. [PERSON] and [PERSON] [19] first derived the out-of-plane elastic buckling load for arches under compression-only or bending-only actions based on the equilibrium theorem. Using the energy-based approach, [PERSON] and [PERSON] derived the elastic flexural-torsional buckling loads for the pin-ended arches [20; 21; 22], and [PERSON] et al. derived the elastic flexural-torsional buckling resistance of the laterally end-fixed circular arches [23; 24; 25]. However, these analytical results were derived from the close-section arches bearing uniform compression or uniform bending. Furthermore, [PERSON] and [PERSON] studied the inelastic flexural-torsional buckling and proposed a design equation for end-fixed steel I-section arches using the finite element method [26]. These out-of-plane buckling behaviors were consistent with the experimental results by [PERSON] et al. [27]. Moreover, for the circular steel tubular truss arches, out-of-plane buckling behaviors were reported to be affected by their end restraints and sectional stiffnesses. The effect of the end restraints was investigated by [PERSON] and [PERSON] on the out-of-plane buckling resistance numerically [28], and the shear stiffness of the chords of the pin-ended circular arches was considered by [PERSON] et al. to derive the out-of-plane buckling loads analytically [29].