## 2 Research Methods
### Torsional Stress Analysis
#### 2.2.4 Solving the Torsional Differential Equations  
According to [PERSON]'s second theory [23], the restrained torsional differential equation about the torsional angle, \(\theta\), is given as follows:  
\[\theta^{(4)}-k^{2}\theta^{\prime\prime}=-\frac{\mu m_{\mathrm{z}}}{E_{\mathrm{ s}}I_{\varpi}} \tag{26}\]where \(k^{2}\) is the characteristic parameter under the restrained torsion, \(k^{2}=\frac{\mu G_{\rm s}I_{\rm d}}{E_{\rm s}I_{\rm dm}}\); \(\mu\) is the restrained coefficient of the section, \(\mu=1-\frac{I_{\rm d}}{I_{\rm\rho}}\); \(I_{\rm\rho}\) is the polar moment of inertia; \(m_{\rm z}\) is the concentration of the torque load.  
The relationship between the generalized warping displacement, \(\beta^{\prime}\), and the torsional angle, \(\theta\), is given by:  
\[\beta^{\prime}=\frac{1}{\mu}\bigg{(}\theta-\frac{M_{\rm z}}{G_{\rm s}I_{\rm \rho}}\bigg{)} \tag{27}\]  
where \(M_{\rm z}\) is the total torque of the box girder, \(M_{\rm z}=M_{\rm s}+M_{\overline{w}}\); \(M_{\rm s}\) is the free torsional torque.  
The initial parameter method is used to solve the differential equation of the constrained torsion. Firstly, four initial parameters \(\theta_{0}\), \(\beta^{\prime}_{0}\), \(B_{0}\), and \(M_{0}\) are defined as the torsional angle, generalized warping displacement, torsional bi-moment, and total torque at the starting end of the box girder, respectively. Let \(m_{\rm z}=0\) in Equation (26), and the initial parameter expressions of the torsional angle, generalized warping displacement, torsional bi-moment, and total torque of any section may be obtained as follows:  
\[\begin{array}{l}\theta(z)=\theta_{0}+\frac{\mu\beta^{\prime}_{0}}{k}\mathrm{ sinh}kz+\frac{B_{0}}{G_{\rm s}I_{\rm d}}(1-\cosh kz)+\frac{M_{0}}{G_{\rm s}I_{\rm d }}(z-\frac{\mu}{k}\mathrm{sinh}kz)\\ \beta^{\prime}(z)=\beta^{\prime}_{0}\cdot\cosh kz-\frac{kB_{0}}{\mu G_{\rm s}I _{\rm d}}\mathrm{sinh}kz+\frac{M_{0}}{G_{\rm s}I_{\rm d}}(1-\cosh kz)\\ B_{\overline{w}}(z)=-\frac{\mu\beta^{\prime}_{0}G_{\rm s}I_{\rm d}}{k}\mathrm{ sinh}kz+B_{0}\cosh kz+\frac{\mu M_{0}}{k}\mathrm{sinh}kz\\ M_{\rm z}(z)=M_{0}\end{array} \tag{28}\]  
The boundary condition [27]:  
(1) The fixed end:  
\[\begin{array}{l}\theta=0\\ \beta^{\prime}=0\end{array} \tag{29}\]  
(2) The simply supported end:  
\[\begin{array}{l}\theta=0\\ \beta^{\prime\prime}=0\end{array} \tag{30}\]  
(3) The free end:  
\[\begin{array}{l}G_{\rm s}I_{\rm d}\theta^{\prime}-E_{\rm s}I_{\rm\overline{ w}}\beta^{\prime\prime}=0\\ \beta^{\prime\prime}=0\end{array}\]  
In Equation (28), the four initial parameters are determined by the boundary condition, and its solution is suitable for the case of no load on the span of the box girder. As shown in Figure 4, when there is an external torque load on the span, the initial parameters are given by:  
\[\begin{array}{lcl}\theta(z)&=\theta_{0}+\frac{\mu\beta^{\prime}_{0}}{k} \mathrm{sinh}kz+\frac{B_{0}}{G_{\rm s}I_{\rm d}}(1-\cosh kz)+\frac{M_{0}}{ KG_{\rm s}I_{\rm d}}(kz-\mu\mathrm{sinh}kz)\\ &-\Big{\|}_{a}\frac{\bar{M}}{k}\mathrm{sinh}k(z-a)-\mu\mathrm{sinh}(kz-a)]- \Big{\|}_{b}f^{2}\frac{\bar{m}}{k}[k(z-\xi)-\mu\mathrm{sinh}k(z-\xi)]\mathrm{ d}\xi\\ \beta^{\prime}(z)&=\beta^{\prime}_{0}\cosh kz-\frac{kB_{0}}{\mu G_{\rm s}I_{ \rm d}}\mathrm{sinh}kz+\frac{M_{0}}{G_{\rm s}I_{\rm d}}(1-\cosh kz)\\ &-\Big{\|}_{a}\frac{\bar{M}}{G_{\rm s}I_{\rm d}}[1-\cosh k(z-a)]-\Big{\|}_{b} \int_{z}^{z}\frac{\bar{m}}{G_{\rm s}I_{\rm d}}[1-\cosh k(z-\xi)]\mathrm{d}\xi \\ B_{\overline{w}}(z)&=-\frac{B_{0}G_{\rm s}I_{\rm d}B_{\rm s}^{\prime}}{k} \mathrm{sinh}kz+B_{0}\cosh kz+\frac{bM_{0}}{k}\mathrm{sinh}kz\\ &-\Big{\|}_{a}\frac{u\bar{M}}{k}\mathrm{sinh}k(z-a)-\Big{\|}_{a}\int_{b}^{z} \frac{u\bar{m}}{k}\mathrm{sinh}k(z-\xi)\mathrm{d}\xi\\ M_{\rm z}(z)&=M_{0}-\Big{\|}_{a}\widetilde{M}-\big{\|}_{b}f^{2}_{\rm z}\,\bar{m} \mathrm{d}\xi\end{array} \tag{30}\]  
where symbol \(\big{\|}_{a}\) and symbol \(\big{\|}_{b}\) respectively indicate that this item is included only when \(z>a\) or \(z>b\), when \(z>c\), and the upper limit of the integral is changed to \(c\).