## 1 Introduction
### Effect of Crossbeams on Torsional Effects  
When considering the crossbeam of the twin I-girder bridge in Figure 1, [PERSON] et al. [23] proposed that the influence of the crossbeam on the calculation of the non-uniform torsion for such an open section can be handled by equating the stiffness of the crossbeam, as reflected in the torsion differential equation as shown below:  
\[E_{1}I_{\omega}\varphi^{4}(x)-G(K+K_{d})\varphi^{\prime\prime}(x)=m_{T}(x) \tag{8}\]  
where \(K_{d}\) is the equivalent stiffness of the crossbeam, expressed as follows:  
\[K_{d}=\frac{\Omega_{c}^{2}}{l_{c}\cdot b}\cdot\frac{12 EI_{c}F_{c}}{l_{c}^{2}G F_{c}+14.4 EI_{c}} \tag{9}\]  
and where \(\Omega_{c}\) is the difference in the principal sectoral coordinates between the ends of the crossbeam, \(l_{c}\) is the length of the crossbeam, \(b\) is the crossbeam spacing, and \(l_{c}\) and \(F_{c}\) are, respectively, the weak axis moment of inertia and the cross-sectional area of the crossbeam.  
Referring to Equation (8), when the crossbeam is taken into account, the open section non-uniform torsion differential equation takes the same form as it would without it, except that its uniform torsional stiffness is increased. After calculation, one can find the following values \(K\) = 5.32 \(\times\) 108 mm\({}^{4}\), \(K_{d}\) = 2.51 \(\times\) 106 mm\({}^{4}\) for the twin I-girder bridge presented in Figure 1. Added to that, concerning the twin I-girder bridge, \(K_{d}\) is generally smaller with respect to \(K\). Therefore, for analytical simplicity, the effect of the crossbeam on the overall torsional stiffness of the structure can be ignored.  
The application of the above method is illustrated by the example of a two-span continuous girder. Each span of the bridge is 18.5 m long, with an effective span diameter of 18 m and the cross-sectional dimensions shown in Figure 1. The steel girders are made of Q345 steel, and the deck slab is made of C50 concrete. It is possible to calculate the principal sectoral moment of inertia of the section \(I_{w}\) = 6.93 \(\times\) 10 mm\({}^{16}\), the uniform torsional moment of inertia \(K\) = 5.32 \(\times\) 108 mm\({}^{4}\), and the equivalent stiffness of the crossbeam \(K_{d}\) = 2.51 \(\times\) 106 mm\({}^{4}\). A concentrated torque of \(T\) = 6.7 \(\times\) 108 N\(\cdot\)mm is applied in the second span.  
After obtaining the distribution of the bimoment \(B\) along the length of the beam, the stresses at each point of the section can be obtained according to the formula \(\sigma_{\omega}=\frac{B}{L\omega}\omega(s)\). Referring to this formula, it can be found that for a definite section, the warping normal stress is proportional to the principal sectoral coordinate \(\omega\). Figure 7 shows the distribution of the principal sectoral coordinate of the section where point A is the shear centre of the section. The characteristic points are sorted, considering their distribution characteristics and the symmetry of the section. As for Figure 8, it only shows the calculation results of the warping normal stress for characteristic points 1, 3, and 5 (P\({}_{1}\), P\({}_{3}\), and P\({}_{5}\)).  
Figure 7: Distribution of principal sectoral coordinates w and ordering of feature points.