## 3 Box Girder's Longitudinal Displacement Functions
### Displacement Functions of the Double-Cell Box Girder  
Similarly to single-cell box girders, the longitudinal displacement functions of double-cell box girders are constructed as follows:  
\[ \begin{cases}u_{21}=h_{1}\Bigg{[}w^{\prime}(z)+\Bigg{(}-\eta_{1}\frac{(x-b_{11}) ^{3}}{b_{11}^{3}}+d_{1}\Bigg{)}\varphi(z)\Bigg{]}&(0\leq x\leq b_{11})\\ u_{22}=h_{1}\Bigg{[}w^{\prime}(z)+\Bigg{(}\eta_{2}\frac{(x-b_{11})^{3}}{b_{12}^ {3}}+d_{2}\Bigg{)}\varphi(z)\Bigg{]}&(b_{11}\leq x\leq b_{1})\\ u_{23}=h_{1}\Bigg{[}w^{\prime}(z)+\Bigg{(}-\eta_{3}\frac{(x-b_{1}-b_{21})^{3}}{b _{21}^{3}}+d_{3}\Bigg{)}\varphi(z)\Bigg{]}&(b_{1}\leq x\leq b_{1}+b_{2})\\ u_{24}=-h_{2}\Bigg{[}w^{\prime}(z)+\Bigg{(}-\eta_{4}\frac{(x-b_{31})^{3}}{b _{31}^{3}}+d_{4}\Bigg{)}\varphi(z)\Bigg{]}&(0\leq x\leq b_{31})\\ u_{25}=-h_{2}\Bigg{[}w^{\prime}(z)+\Bigg{(}\eta_{5}\frac{(x-b_{31})^{3}}{b_{3 2}^{3}}+d_{5}\Bigg{)}\varphi(z)\Bigg{]}&(b_{31}\leq x\leq b_{3})\\ u_{2w1}=-y\omega^{\prime}(z)+h_{1}(\eta_{1}+d_{1})\varphi(z)&(-h_{1}\leq y\leq h _{2})\\ u_{2w2}=-y\omega^{\prime}(z)+h_{1}(\eta_{2}+d_{2})\varphi(z)&(-h_{1}\leq y\leq h _{2})\\ \end{cases}  \tag{16}\]  
where \(u_{2i}\) represents the longitudinal displacement function of double-cell box girder's slabs: \(i=1\) for the top plate of length \(b_{11}\), \(i=2\) for the top plate of length \(b_{12}\), \(i=3\) for the cantilever plate of length \(b_{21}\), \(i=4\) for the bottom plate of length \(b_{31}\), \(i=5\) for the bottom plate of length \(b_{32}\), \(i=w_{1}\) for the middle web, and \(i=w_{2}\) for the side web; \(\eta_{i}\) and \(d_{i}\) are introduced coefficients. The term \(\eta_{i}\) can also be set from the same perspectives as the single-cell box girder:  
(1) In the first approach for selecting the coefficients for double-cell box girders (1 CD), the \(\eta_{i}\) can be expressed as follows:  
\[\eta_{1}=\eta_{2}=\eta_{3}=\eta_{4}=\eta_{5}=1 \tag{17}\]  
(2) In the second approach (2 CD), using Table 2, the \(\eta_{i}\) can be expressed as follows:  
\[\eta_{1}=\frac{b_{11}t_{1}h_{1}}{b_{11}t_{1}h_{1}}\quad\eta_{2}=\frac{b_{12}t_ {1}h_{1}}{b_{11}t_{1}h_{1}}\quad\eta_{3}=\frac{b_{21}t_{1}h_{1}}{b_{11}t_{1}h_{ 1}}\quad\eta_{4}=\frac{b_{31}t_{2}h_{2}}{b_{11}t_{1}h_{1}}\quad\eta_{5}=\frac{b _{32}t_{2}h_{2}}{b_{11}t_{1}h_{1}} \tag{18}\]  
(3) In the third approach (3 CD), the \(\eta_{i}\) can be expressed as follows:  
\[\eta_{1}=\frac{b_{11}^{2}t_{1}h_{1}}{b_{11}^{2}t_{1}h_{1}}\quad\eta_{2}=\frac {b_{12}^{2}t_{1}h_{1}}{b_{11}^{2}t_{1}h_{1}}\quad\eta_{3}=\frac{b_{21}^{2}t_{1 }h_{1}}{b_{11}^{2}t_{1}h_{1}}\quad\eta_{4}=\frac{b_{31}^{2}t_{2}h_{2}}{b_{11}^ {2}t_{1}h_{1}}\quad\eta_{5}=\frac{b_{32}^{2}t_{2}h_{2}}{b_{11}^{2}t_{1}h_{1}} \tag{19}\]  
However, when considering the continuity condition, a contradiction emerges:  
\[d_{5}=-\frac{h_{1}}{h_{2}}(\eta_{1}+d_{1})-\eta_{4}\hskip 28.452756 ptd_{5}=- \frac{h_{1}}{h_{2}}(\eta_{2}+d_{1})-\eta_{5} \tag{20}\]  
Therefore, it is necessary to correct \(\eta_{5}\), which is rewritten as follows:  
\[\eta_{5}=\frac{h_{1}}{h_{2}}(\eta_{1}-\eta_{2})+\eta_{4} \tag{21}\]  
The coefficients \(d_{i}\) are derived as follows:\[\frac{\oint\sigma_{1}dA}{2E\varphi^{\prime}(z)} = \left[\begin{array}{c}\int_{0}^{b_{11}}t_{1}h_{1}\Bigg{(}-\eta_{1 }\frac{(x-b_{11})^{3}}{b_{11}^{3}}+d_{1}\Bigg{)}dx+\int_{b_{11}}^{b_{11}+b_{12} }t_{1}h_{1}\Bigg{(}\eta_{2}\frac{(x-b_{11})^{3}}{b_{12}^{3}}+d_{2}\Bigg{)}dx\\ +\int_{b_{11}+b_{12}}^{b_{11}+b_{12}+b_{21}}t_{1}h_{1}\Bigg{(}-\eta_{3}\frac{(x -b_{11}-b_{12}-b_{21})^{3}}{b_{21}^{3}}+d_{3}\Bigg{)}dx\\ -\int_{0}^{b_{31}}t_{2}h_{2}\Bigg{(}-\eta_{4}\frac{(x-b_{31})^{3}}{b_{31}^{3}} +d_{4}\Bigg{)}dx-\int_{b_{31}}^{b_{31}+b_{32}}t_{2}h_{2}\Bigg{(}\eta_{5}\frac{ (x-b_{31})^{3}}{b_{32}^{3}}+d_{5}\Bigg{)}dx\\ +\frac{1}{2}\int_{-h_{1}}^{h_{2}}t_{w1}h_{1}(\eta_{1}+d_{1})dx+\int_{-h_{1}}^{ h_{2}}t_{w2}h_{1}(\eta_{2}+d_{2})dx\\ =\left[