## 3 Sectional Stiffness in Vierendeel Truss Arches
### Sectional Torsional Stiffness  
To calculate the sectional torsional stiffness, [PERSON] et al. assumed the web-open truss arch to be an equivalent thin-walled box arch with a hollow rectangular section composed  
Figure 5: Moment diagram of a typical Vierendeel truss segment under the pure shear \(V\). of four thin plates [33]. With the equivalent box cross-section, the torsional stiffness of a typical Vierendeel truss arch can be expressed as:  
\[\left\{ \begin{array}{l}GJ=G\frac{4A_{0}^{2}}{\frac{4}{1}}\\ t_{ri}=\frac{F}{6}\times\frac{L_{z}B}{\frac{L_{z}B}{24I_{c}}+\frac{1}{12I_{t}}L_{ t}^{2}}\\ \end{array} \right. \tag{4}\]  
where \(l_{i}\) is the \(i\)-th side length of the rectangular section, \(t_{i}\) is the thickness of the \(i\)-th box side plate, \(A_{0}\) is the area of the rectangular section. Since the spans (\(L\)) of the considered truss arches (over 20 m) are much larger than the sectional side length (\(B\) or \(H\)), the influence of the arch curvature on the sectional torsional stiffness can be ignored for simplification. An equivalent straight truss model is thus built with a length of 100 m and a square cross-section (\(H=B=1.0\) m). To verify the sectional torsional stiffness, static analyses via the FE program ABAQUS were performed on the equivalent straight truss. In the long-span truss models, boundary conditions were confirmed to have little influence on the sectional torsional stiffness. Herein, the bottom end of each chord is pinned to the ground, and the torsion \(M_{z}\) is applied on the top free section, as shown in Figure 6a. Under the torsional moment \(M_{z}\), the sectional twist around the centroid axis \(z\) is resisted by the shears of chords along the transverse tubes, as shown in Figure 6. Thus, the torsional equilibrium of a typical truss segment can be calculated as:  
\[\left\{ \begin{array}{l}\overrightarrow{V_{1}}+\overrightarrow{V_{2}}+ \overrightarrow{V_{3}}+\overrightarrow{V_{4}}=0\\ \sum\limits_{i=1}^{4}V_{i}r_{i}=M_{z}\end{array} \right. \tag{5}\]  
where \(r_{i}\) (\(i=1\)-\(4\)) is the distance from the torsion centroid (\(O\)) to the \(i\)-th side of the rectangular section, and \(V_{i}\) is the shear along the \(i\)-th transverse tube. In each segment, the shear stiffness of each chord can be expressed as:  
\[K_{V_{i}}=\frac{V_{i}}{r_{i}\beta/L_{c}} \tag{6}\]  
where \(\beta\) is the sectional rotation angle of a Vierendeel truss segment under the external torsion moment \(M_{z}\). Accordingly, the sectional torsional stiffness can be calculated as:  
\[GJ=\frac{M_{z}}{\beta/L_{c}}=\sum\limits_{i=1}^{4}K_{V_{i}}r_{i}^{2} \tag{7}\]  
Further, the sectional torsional stiffness can be written as:  
\[GJ=\frac{H^{2}}{\frac{L_{c}^{2}}{12 EI_{c}}+\frac{L_{c}B}{6 EI_{t}}+\frac{2 nL_{ c}}{BA_{t}G}}+\frac{B^{2}}{\frac{L_{c}^{2}}{12 EI_{c}}+\frac{L_{c}H}{6 EI_{t}}+ \frac{2 nL_{c}}{HA_{t}G}} \tag{8}\]  
When the torsional stiffness of chords is considered, the sectional twist will be resisted by both the shear and torsion of each chord, as shown in Figure 7. Thus, the torsional equilibrium of a typical truss segment can be calculated as:  
\[\left\{ \begin{array}{l}\overrightarrow{V_{1}}+\overrightarrow{V_{2}}+ \overrightarrow{V_{3}}+\overrightarrow{V_{4}}=0\\ \sum\limits_{i=1}^{4}\left(V_{i}r_{i}+M_{i}\right)=M_{z}\end{array} \right. \tag{9}\]  
chords on the sectional torsional stiffness should be investigated in trusses with different transverse tubes and chords. Here, the segmental length (\(L_{c}\)), the external diameter (\(D_{c}\)), and wall thickness (\(t_{c}\)) of the chord tubes are set to be constant at 1000 mm, 121 mm, and 10 mm, respectively; the external diameters (\(D_{t}\)) ranging from 20 mm to 200 mm with a constant wall thickness (\(t_{t}\)) of 10 mm, are set for the transverse tubes. Figure 8a shows the sectional torsional stiffnesses (\(G\)) of a series of models with different segmental lengths.