## 3 Box Girder's Longitudinal Displacement Functions
### Displacement Functions of the Single-Cell Box Girder  
Considering the warping of webs, longitudinal displacement functions with their origins being on the zero points of the shear flow in the single-cell box girder are established as follows:  
Figure 4: Shear flow distribution of the closed double-cell section. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline points & 1-left & 2 & 3 & 4-left & 4-right & 5 \\ \hline \(q\) & \(-t_{2}b_{31}h_{2}\) & \(t_{2}b_{32}h_{2}\) & 0 & \(-t_{1}b_{21}h_{1}\) & \(t_{1}b_{12}h_{1}\) & \(-t_{1}b_{11}h_{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Shear flow magnitudes of the closed double-cell section at critical points. \[\left\{ \begin{array}{ll}u_{11}=h_{1}\Bigg{[}w^{\prime}(z)+\Bigg{(}\eta_{1}\frac{x ^{3}}{b_{1}^{3}}+d_{1}\Bigg{)}\varphi(z)\Bigg{]}&(0\leq x\leq b_{1})\\ u_{12}=h_{1}\Bigg{[}w^{\prime}(z)+\Bigg{(}-\eta_{2}\frac{(x-b_{1}-b_{2})^{3}}{b_ {2}^{3}}+d_{2}\Bigg{)}\varphi(z)\Bigg{]}&(b_{1}\leq x\leq b_{1}+b_{2})\\ u_{13}=-h_{2}\Bigg{[}w^{\prime}(z)+\Bigg{(}\eta_{3}\frac{x^{3}}{b_{3}^{3}}+d_{3 }\Bigg{)}\varphi(z)\Bigg{]}&(0\leq x\leq b_{3})\\ u_{1u2}=-yw^{\prime}(z)+h_{1}(\eta_{1}+d_{1})\varphi(z)&(-h_{1}\leq y\leq h_{2 })\end{array} \right. \tag{10}\]  
where \(u_{1i}\) represents the longitudinal displacement function of single-cell box girder's slabs: \(i=1\) for the top plate, \(i=2\) for the cantilever plate, \(i=3\) for the bottom plate, and \(i=w_{2}\) for the side web; \(\eta_{i}\) and \(d_{i}\) are introduced coefficients. The term \(\eta_{i}\) can be selected from three different analytical perspectives:  
(1) The first approach for selecting the coefficients for single-cell box girders (\(1 CS\)) is based on the classical cubic parabola formula [17]. The \(\eta_{i}\) can be expressed as follows:  
\[\eta_{1}=\eta_{2}=\eta_{3}=1 \tag{11}\]  
(2) The second approach of selecting coefficients (\(2 CS\)) involves using the relative magnitudes of the shear flow, which can be derived from Table 1. This is mainly due to differences in the maximum shear flow in different flanges. The \(\eta_{i}\) can be expressed as follows:  
\[\eta_{1}=\frac{b_{1}t_{1}h_{1}}{b_{1}t_{1}h_{1}}\qquad\eta_{2}=\frac{b_{2}t_{1 }h_{1}}{b_{1}t_{1}h_{1}}\qquad\eta_{3}=\frac{b_{3}t_{2}h_{2}}{b_{1}t_{1}h_{1}} \tag{12}\]  
(3) The third approach of selecting coefficients (\(3 CS\)) involves using the relative shear deformation of each flange. The \(\eta_{i}\) can be expressed as follows:  
\[\eta_{1}=\frac{b_{1}^{2}t_{1}h_{1}}{b_{1}^{2}t_{1}h_{1}}\qquad\eta_{2}=\frac{ b_{2}^{2}t_{1}h_{1}}{b_{1}^{2}t_{1}h_{1}}\qquad\eta_{3}=\frac{b_{3}^{2}t_{2}h_{2}}{b_ {1}^{2}t_{1}h_{1}} \tag{13}\]  
In addition, because the axial force caused by the shear lag effect is zero [24] and the displacements and stresses are continuous. The expressions for \(d_{i}\) are derived as follows:  
\[\oint\sigma_{1}dA =2E\varphi^{\prime}(z)\cdot\left[ \begin{array}{l}\int_{0}^{b _{1}}t_{1}h_{1}\Bigg{(}\eta_{1}\frac{x^{3}}{b_{1}^{3}}+d_{1}\Bigg{)}dx+\int_{ b_{1}}^{b_{1}+b_{2}}t_{1}h_{1}\Bigg{(}-\eta_{2}\frac{(x-b_{1}-b_{2})^{3}}{b_{2}^{3}}+d_ {2}\Bigg{)}dx\\ -\int_{0}^{b_{3}}t_{2}h_{2}\Bigg{(}\eta_{3}\frac{x^{3}}{b_{3}^{3}}+d_{3} \Bigg{)}dx+\int_{-h_{1}}^{b_{2}}t_{w2}h_{1}(\eta_{1}+d_{1})dx\end{array} \right] \tag{14}\] \[=2E\varphi^{\prime}(z)\cdot\left[ \begin{array}{l}t_{1}h_{1}b_{ 1}\Big{(}\frac{\eta_{1}}{4}+d_{1}\Big{)}+t_{1}h_{1}b_{2}\Big{(}\frac{\eta_{2} }{4}+d_{2}\Big{)}\\ -t_{2}h_{2}b_{3}\Big{(}\frac{\eta_{3}}{4}+d_{3}\Big{)}+t_{w2}(\eta_{1}+d_{1}) hh_{1}\end{array} \right]=0\] \[\left\{ \begin{array}{l}d_{1}=-(A_{1}h_{1}\eta_{1}+4A_{2}h_{1} \eta_{1}-3A_{2}h_{1}\eta_{2}+4A_{3}h_{1}\eta_{1}+3A_{3}h_{2}\eta_{3}+4A_{w}h_{ 1}\eta_{1})/(4h_{1}A)\\ d_{2}=\eta_{1}+d_{1}-\eta_{2}\hskip 71.13189 ptd_{3}=-\frac{h_{1}}{h_{2}}( \eta_{1}+d_{1})-\eta_{3}\end{array} \right.