# Shear Lag Effect of Ultra-Wide Box Girder under Influence of Shear Deformation
## 3 Analysis of Flexural Displacement Function of Box Girder via Energy Variational Method  
According to the flexural displacement function of the flexural thin-walled box girder under the shear deformation of each plate, the energy variational principle [30; 31] was applied to analyze the flexural displacement function of the thin-walled box girder. Using Equation (44), the longitudinal strain equation for each point of the thin-walled box girder section can be obtained as follows:  
\[\varepsilon_{z}=\frac{\partial u}{\partial z}=-y\omega^{\prime\prime}(z)+ \bigg{[}\alpha y-\frac{1}{3}y^{3}+\eta_{\zeta}(x,y)\bigg{]}g^{\prime}(z)=-y \omega^{\prime\prime}(z)+\eta(x,y)g^{\prime}(z). \tag{46}\]  
The shear strain of each wing plate is expressed as follows:  
\[\gamma_{xz}=\frac{\partial u}{\partial x}=\frac{\partial\eta_{\zeta}(x,y)}{ \partial x}g(z). \tag{47}\]  
The shear strain of each web is expressed as follows:  
\[\gamma_{yz}=\frac{\partial u}{\partial s}+\frac{\partial v}{\partial z}= \frac{\partial u}{\partial y}+\omega^{\prime}(z)=\frac{\partial\eta(x,y)}{ \partial y}g(z). \tag{48}\]  
According to the energy variational principle, the positive strain energy of each plate of the box girder under flexural deformation can be expressed as follows:  
\[U_{1}=\frac{1}{2}\iint\sigma_{z}\varepsilon_{z}dAdz=\frac{E}{2}\iint\varepsilon _{z}{}^{2}dAdz. \tag{49}\]  
The flexural shear strain energy of each plate of the box girder is given as  
\[U_{2}=\frac{1}{2}\iint\big{(}\tau_{xz}\tau_{xz}+\tau_{yz}\tau_{yz}\big{)}dAdz =\frac{G}{2}\iint\Big{(}\gamma_{xz}{}^{2}+\gamma_{yz}{}^{2}\Big{)}dAdz \tag{50}\]  
By substituting Equations (46)-(48) into Equations (49) and (50), the total strain energy of the box girder under flexural deformation can be obtained as follows:\[U=\frac{1}{2}{\int_{l}}\int_{A}\Big{(}E\epsilon^{2}+G\gamma^{2}\Big{)}dAdz=\frac{1} {2}{\int_{l}}\Big{[}E{I_{x}}{\omega^{\prime\prime}}^{2}+E{I_{\eta}}g^{\prime}-2E{ I_{\eta\eta}}{\omega^{\prime\prime}}g^{\prime}+G\big{(}A_{x}+A_{y}\big{)}g^{2} \Big{]}dz \tag{51}\]  
where \(I_{\eta}\) represents the cross-sectional warping moment of inertia, \(A_{x}\) represents the shear warping moment of the wing plate, and \(A_{y}\) represents the shear buckling moment of the web. The parameter equation of the shear warping sections defined in this study is as follows:  
\[A_{x}={\int_{A}}\bigg{(}\frac{\partial\eta_{\zeta}}{\partial x}\bigg{)}^{2}dA= \frac{1}{3}\Big{\{}A_{l}\Big{[}\big{(}y_{s}2b_{1}+2q_{2}/t\big{)}^{2}+(4b_{2}y_ {s}+2q_{1}/t\big{)}^{2}\Big{]}+\ . \tag{52}\]  
\[A_{y}={\int_{A}}\bigg{(}\frac{\partial\eta}{\partial y}\bigg{)}^{2}dA= \alpha^{2}A_{w}+I_{4w}-\alpha I_{2w} \tag{53}\]  
\[I_{\eta}={\int_{A}}\eta(x,y)^{2}dA=\alpha^{2}I_{x}+\frac{1}{9}I_{6}+I_{\zeta }-\frac{2}{3}\big{(}\alpha I_{4}-3\alpha I_{1\zeta}+I_{3\zeta}\big{)} \tag{54}\]  
\[I_{y\eta}={\int_{A}}y\eta(x,y)dA=\alpha I_{x}-\frac{1}{3}I_{4}+I_{1\zeta} \tag{55}\]  
\[\gamma_{xz}=\frac{\partial u}{\partial x}=\frac{\partial\eta_{\zeta}(x,y)}{ \partial x}g(z). \tag{47}\]0  
\[\gamma_{xz}=\frac{\partial u}{\partial x}=\frac{\partial\eta_{\zeta}(x,y)}{ \partial x}g(z). \tag{47}\]1  
\[\gamma_{xz}=\frac{\partial u}{\partial x}=\frac{\partial\eta_{\zeta}(x,y)}{ \partial x}g(z). \tag{47}\]2  
\[A_{t}\left\{\begin{array}{l}\frac{8}{15}\bigg{[}{y_{s}}^{2}{b_ {1}}^{4}+\bigg{(}\frac{q_{2}b_{1}}{t}\bigg{)}^{2}+{y_{s}}^{2}\Big{(}4b_{2} \Big{)}^{2}+\bigg{(}\frac{2q_{1}b_{2}}{t}\bigg{)}^{2}+\bigg{(}\frac{2q_{3}b_{ 2}}{t}\bigg{)}^{2}\bigg{]}-\\ \frac{4}{3}\bigg{(}{y_{s}}{k_{1}}{d{b_{1}}^{2}}+{y_{s}}{k_{2}}{d{4{b_{2}}}^{2} }-\frac{4{y_{s}}{q_{2}}{b_{1}}^{3}}{t}-\frac{8{y_{s}}{q_{1}}{b_{2}}^{3}}{t}- \frac{8{y_{s}}{q_{3}}{b_{2}}^{3}}{t}+\frac{4{k_{1}}dq_{2}b_{1}}{t}+\frac{4{k_ {2}}dq_{1}b_{2}}{t}+\frac{4{k_{2}}dq_{3}b_{2}}{t}\bigg{)}\ \ \ \right\}+\\ A_{b}\left\{