## 4 Governing Differential Equations and Boundary Conditions  
The governing differential equations and boundary conditions for box girders can be derived using the principle of minimum potential energy. This principle is expressed as follows [7]:  
\[\delta\Pi=\delta(V+U)=0 \tag{24}\]  
where \(\Pi\) denotes the total potential energy; \(V\) denotes the external load potential energy; and \(U\) denotes the strain energy. Under the vertical load, the \(V\) and \(U\) of each slab can be expressed as follows:  
\[V=\int_{0}^{L}M(z)w^{\prime\prime}(z)dz \tag{25}\]  
\[U=\frac{1}{2}\iint t\Big{(}E\epsilon^{2}+G\gamma^{2}\Big{)}dydz \tag{26}\]  
where \(L\) represents the length of the beam; \(M(z)\) represents the bending moment of the cross section. Substituting Equation (8) into Equation (26), the strain energy can be rewritten as follows:  
\[U=\frac{1}{2}\iint t\Bigg{(}E\bigg{(}\frac{\partial u}{\partial z}\bigg{)}^{2 }+G\bigg{(}\frac{\partial u}{\partial x}\bigg{)}^{2}\Bigg{)}dydz \tag{27}\]  
For the single-cell box girder, substituting Equation (10) into Equation (27), the strain energy of side webs \(U_{1w2}\) is expressed as follows:  
\[U_{1w2}=\frac{1}{2}EI_{1w2}\int_{0}^{L}\Bigg{[}(w^{\prime\prime})^{{}^{2}}- \frac{3h_{1}(h_{2}-h_{1})(\eta_{1}+d_{1})}{h_{2}^{2}-h_{1}h_{2}+h_{1}^{2}}w^{ \prime\prime}\,\varphi^{\prime}+\frac{3h_{1}^{2}(\eta_{1}+d_{1})^{2}}{h_{2}^{ 2}-h_{1}h_{2}+h_{1}^{2}}\left(\varphi^{\prime}\right)^{2}\Bigg{]}dz \tag{28}\]  
where \(I_{1w2}=2t_{1w2}\big{(}h_{2}^{3}+h_{1}^{3}\big{)}/3\) is the moment of inertia of side webs. The strain energy of each flange \(U_{1i}\) is expressed as follows:  
\[U_{1i}=\frac{1}{2}EI_{1i}\int_{0}^{L}\Bigg{[}(w^{\prime\prime})^{{}^{2}}+2w^{ \prime\prime}\,\varphi^{\prime}\Big{(}\frac{\eta_{i}}{4}+d_{i}\Big{)}+\left( \varphi^{\prime}\right)^{2}\Bigg{(}\frac{\eta_{i}^{2}}{7}+\frac{\eta_{i}d_{i}}{2 }+d_{i}^{2}\Bigg{)}+\frac{9G\eta_{i}^{2}}{5 Eb_{k}^{2}}\varphi^{2}\Bigg{]}dz \tag{29}\]where \(I_{1i}=2b_{k}t_{k}h_{k}^{2}\) is the inertia moment of each flange after ignoring its self-inertia moment; \(t_{k}\) is the thickness; \(h_{k}\) is the distance from each plate to the neutral axis; and \(b_{k}\) is the width. For the double-cell box girder, substituting Equation (16) into Equation (27), the strain energy of the middle web \(U_{2w1}\) is expressed as follows:  
\[U_{2w1}=\frac{1}{2}EI_{2w1}\int_{0}^{L}\left[\left(w^{\prime\prime}\right)^{2} -\frac{3h_{1}(h_{2}-h_{1})(\eta_{1}+d_{1})}{h_{2}^{2}-h_{1}h_{2}+h_{1}^{2}}w^{ \prime\prime}\varphi^{\prime}+\frac{3h_{1}^{2}(\eta_{1}+d_{1})^{2}}{h_{2}^{2} -h_{1}h_{2}+h_{1}^{2}}\left(\varphi^{\prime}\right)^{2}\right]dz \tag{30}\]  
where \(I_{2w1}=t_{w1}\left(h_{2}^{3}+h_{1}^{3}\right)/3\) is the moment of inertia of the middle web. The strain energy of side webs \(U_{2w2}\) is expressed as follows:  
\[U_{2w2}=\frac{1}{2}EI_{2w2}\int_{0}^{L}\left[\left(w^{\prime\prime}\right)^{2 }-\frac{3h_{1}(h_{2}-h_{1})(\eta_{2}+d_{2})}{h_{2}^{2}-h_{1}h_{2}+h_{1}^{2}}w^ {\prime\prime}\varphi^{\prime}+\frac{3h_{1}^{2}(\eta_{2}+d_{2})^{2}}{h_{2}^{2} -h_{1}h_{2}+h_{1}^{2}}\left(\varphi^{\prime}\right)^{2}\right]dz \tag{31}\]  
where \(I_{2w2}=2t_{1w2}\left(h_{2}^{3}+h_{1}^{3}\right)/3\) is the moment of inertia of side webs. The strain energy of each flange \(U_{2i}\) is expressed as follows:  
\[U_{2i}=\frac{1}{2}EI_{2i}\int_{0}^{L}\left[\left(w^{\prime\prime}\right)^{2}+2 ww^{\prime\prime}\varphi^{\prime}\left(\frac{\eta_{i}}{4}+d_{i}\right)+ \left(\varphi^{\prime}\right)^{2}\left(\frac{\eta_{i}^{2}}{7}+\frac{\eta_{i}d _{i}}{2}+d_{i}^{2}\right)+\frac{9G\eta_{i}^{2}}{5 Eb_{k}^{2}}\varphi^{2}\right]dz \tag{32}\]  
where \(I_{2i}=2b_{k}t_{k}h_{k}^{2}\) is the inertia moment of each flange after ignoring its self-inertia moment. The total strain energy \(U\) of the system can be obtained by adding the strain energy of each plate.