## 2 Research Methods
### Torsional Stress Analysis
#### 2.2.3 Restrained Torsional Shear Stress  
According to reference [26], when a box girder is restrained to twist, the total shear flow, \(q\), of a box girder with a closed section is the sum of the free torsional shear flow, \(q_{\text{s}}\), and the secondary shear flow, \(q_{\overline{\omega}}\), which balances the warping normal stress. This may be given as:  
\[q=q_{\text{s}}+q_{\overline{\omega}} \tag{17}\]  
As shown in Figure 3, a micro-element is taken from the cell wall. By balancing the forces, it can be concluded that [17]:  
\[\frac{\partial\sigma_{\overline{\omega}}}{\partial z}t+\frac{\partial q_{ \overline{\omega}}}{\partial s}=0 \tag{18}\]  
Figure 3: Stress diagram of the micro-element.  
From Equation (18), the secondary shear flow is calculated as:  
\[q_{\varpi}=E_{\mathrm{s}}\beta^{\prime\prime\prime}\hat{S}_{\varpi}+q_{\varpi 0} \tag{19}\]  
where \(q_{\varpi 0}\) is the additional shear flow at the fictitious notch of a closed section; \(\hat{S}_{\varpi}\) is the converted sectorial static moment.  
The converted sectorial static moment of the closed section can be obtained by:  
\[\hat{S}_{\varpi}=\begin{cases}\lambda_{1}S_{\varpi}&(\text{Top steel flange})\\ \lambda_{2}S_{\varpi}&(\text{Flat steel web})\\ S_{\varpi}&(\text{Bottom steel flange})\end{cases} \tag{20}\]  
where \(S_{\varpi}\) is the sectorial static moment, \(S_{\varpi}=\int_{0}^{s}\varpi t\mathrm{d}s\).  
Since each cell of the box girder section is closed, and the additional shear flow of each cell at the fictitious notch is not zero, the value should be determined according to the continuity condition of warping displacement. The continuity condition of the warping displacement of a closed section may be expressed as follows [26]:  
\[\oint\gamma_{\varpi}\mathrm{d}s=\oint\frac{q_{\varpi}}{G_{\mathrm{s}}t}\mathrm{ d}s=0 \tag{21}\]  
The deformation coordination condition at the fictitious notch of cell \(i\) may be expressed as:  
\[E_{\mathrm{s}}\beta^{\prime\prime\prime}\left(\oint_{i}\frac{S_{\varpi}}{G_{ \mathrm{s}}t}\mathrm{d}s-\int_{i,k}\frac{S_{\varpi}}{G_{\mathrm{s}}t}\mathrm{ d}s\right)+\oint_{i}\frac{q_{\varpi}u_{i}}{G_{\mathrm{s}}t}\mathrm{d}s-\int_{i,k} \frac{q_{\varpi}u_{k}}{G_{\mathrm{s}}t}\mathrm{d}s=0 \tag{22}\]  
where \(q_{\varpi 0i}\) is the additional shear flow at the fictitious notch of cell \(i\).  
For a multi-cell box girder, multiple linear equations may be obtained according to Equation (22), which can solve the additional shear flow of any cell.  
Let:  
\[q_{\varpi 0i}=E_{\mathrm{s}}\beta^{\prime\prime\prime}S_{i} \tag{23}\]  
where \(S_{i}\) is the additional sectorial static moment of cell \(i\).  
The secondary shear flow in the box girder section is:  
\[q_{\varpi}=-\frac{M_{\varpi}}{I_{\varpi}}S_{\varpi} \tag{24}\]  
where \(M_{\varpi}\) is the secondary torque, \(M_{\varpi}=-E_{\mathrm{s}}I_{\varpi}\beta^{\prime\prime\prime}\); \(\overline{S}_{\varpi}\) is the generalized sectorial static moment, \(\overline{S}_{\varpi}=\begin{cases}S_{i}-S_{k}&(\text{Common cell wall})\\ S_{\varpi}+S_{i}&(\text{Independent cell wall})\\ \lambda_{1}S_{\varpi}&(\text{Cantilever plate})\end{cases}\).  
The total shear stress of the box girder section is calculated as:  
\[\tau=\tau_{\mathrm{s}}+\tau_{\varpi}=\frac{\overline{q}_{\mathrm{s}}}{t}- \frac{M_{\varpi}}{I_{\varpi}t}\overline{S}_{\varpi} \tag{25}\]  
In Equation (25), \(t\) is the thickness of the original section.