# Shear Lag Effect of Ultra-Wide Box Girder under Influence of Shear Deformation
## 5 Correction of Cantilever Plate Displacement for Flexural Displacement Function
\begin{cases}-y_{s}\Big{(}{b_{1}}^{2}-x^{2}\Big{)}-\frac{2q_{1 }}{t}(b_{1}-x)+k_{1}d&\text{Inner chamber top plate}\\ -y_{s}\Big{[}4{b_{2}}^{2}-(b_{1}+2b_{2}-x)^{2}\Big{]}-\frac{2q_{1}}{t}[2b_{2} -(b_{1}+2b_{2}-x)]+k_{1}d&\text{Outer chamber top plate}\\ -y_{s}\Big{[}{b_{3}}^{2}-(b_{1}+2b_{2}+b_{3}-x)^{2}\Big{]}\varphi+k_{1}d& \text{Cantilever plate}\\ y_{x}\Big{(}{b_{1}}^{2}-x^{2}\Big{)}-\frac{2q_{2}}{t}(b_{1}-x)+k_{1}d&\text{ Inner chamber bottom plate}\\ y_{x}\Big{[}4{b_{2}}^{2}-(b_{1}+2b_{2}-x)^{2}\Big{]}-\frac{2q_{1}}{t}[2b_{2} -(b_{1}+2b_{2}-x)]+k_{1}d&\text{Outer chamber bottom plate}\\ k_{1}d&\text{Innerweb}\\ k_{2}d&\text{Outer web}\end{cases}  \tag{74}\]  
The final form of the flexural displacement function of the bent box girder under the influence of the shear deformation of each plate can be obtained by substituting Equation (74) into Equation (39) and applying the axial equilibrium condition. The detailed process is not described herein. Owing to the influence of the aforementioned parameter \(\varphi\), the axial equilibrium parameter in Equation (42) can be corrected as follows:  
Figure 10: Comparison of the shear lag coefficients of the top wing plate of the section at \(y=1450\) cm for a single-box, three-chamber box girder. \[d_{e}=\frac{2\Big{[}(y_{s}A_{t}-y_{x}A_{h})\Big{(}{b_{1}}^{2}+4{b_{2}}^{2}\Big{)}+ (A_{t}-A_{h})\Big{(}\frac{2q_{2}}{t}b_{1}+\frac{2q_{1}}{t}2b_{2}\Big{)}+y_{s} \varphi A_{c}{b_{3}}^{2}\Big{]}}{3A}. \tag{75}\]  
In this study, the energy variational method was utilized to analyze the flexural displacement function of the box girder. The expressions of the cross-sectional parameters Equations (49)-(59) in the established control differential equations for the corrected cross-sectional parameters, considering the effect of the correction parameter \(\varphi\), are as follows:  
\[ \begin{split} A_{x}=&\int_{A}\left(\frac{\partial \eta_{\zeta}}{\partial x}\right)^{2}dA=\frac{1}{3}\Big{\{}A_{t}\Big{[}(y_{s}2b _{1}+2q_{2}/t)^{2}+(4b_{2}y_{s}+2q_{1}/t)^{2}\Big{]}+\\ & A_{b}\Big{[}(y_{s}2b_{1}+2q_{2}/t)^{2}+(4b_{2}y_{x}+2q_{1}/t)^{ 2}\Big{]}+A_{c}(\varphi y_{s}2b_{3})^{2}\Big{\}}\end{split}  \tag{76}\]  
\[ \begin{split} I_{\zeta}=&\int_{A}\eta_{\zeta}(x,y) ^{2}dA=\Big{(}A_{1}{k_{1}}^{2}+A_{2}{k_{2}}^{2}\Big{)}d^{2}+\\ A_{t}\left\{\begin{array}{l}\frac{8}{15}\Bigg{[}{y_{s}}^{2}{b_{ 1}}^{4}+\bigg{(}\frac{q_{2}b_{1}}{t}\bigg{)}^{2}+{y_{s}}^{2}\Big{(}4b_{2}^{2} \Big{)}^{2}+\bigg{(}\frac{2q_{1}b_{2}}{t}\bigg{)}^{2}+\bigg{(}\frac{2q_{3}b_{2} }{t}\bigg{)}^{2}\Bigg{]}-\\ \frac{4}{3}\Bigg{(}{y_{s}}{k_{1}}{d{b_{1}}^{2}}+{y_{s}}{k_{2}}{d{4{b_{2}}}^ {2}}-\frac{4y_{s}{q_{2}}{b_{1}}^{3}}{t}-\frac{8y_{s}{q_{1}}{b_{2}}^{3}}{t}- \frac{8y_{s}{q_{3}}{b_{2}}^{3}}{t}+\frac{4k_{1}dq_{2}b_{1}}{t}+\frac{2k_{2}dq_ {1}b_{2}}{t}+\frac{2k_{2}dq_{3}b_{2}}{t}\Bigg{)}\end{array}\right\}+\\ A_{b}\left\{\begin{array}{l}\frac{8}{15}\Bigg{[}{y_{x}}^{2}{b_{ 1}}^{4}+\bigg{(}\frac{q_{2}b_{1}}{t}\bigg{)}^{2}+{y_{x}}^{2}\Big{(}4b_{2}^{2} \Big{)}^{2}+\bigg{(}\frac{2q_{1}b_{2}}{t}\bigg{)}^{2}+\bigg{(}\frac{2q_{3}b_{2} }{t}\bigg{)}^{2}\Bigg{]}+\\ \frac{4}{3}\Bigg{(}{y_{x}}{k_{1}}{d{b_{1}}^{2}}+{y_{x}}{k_{2}}{d{4{b_{2}}}^ {2}}-\frac{4y_{x}{q_{2}}{b_{1}}^{3}}{t}-\frac{8y_{x}{q_{1}}{b_{2}}^{3}}{t}- \frac{8y_{x}{q_{3}}{b_{2}}^{3}}{t}+\frac{4k_{1}dq_{2}b_{1}}{t}+\frac{2k_{2}dq_ {1}b_{2}}{t}+\frac{2k_{2}dq_{3}b_{2}}{t}\Bigg{)}\end{array}\right\}+\\ A_{c}\varphi y_{s}\Big{(}\frac{8}{15}{y_{s}}{b_{3}}^{2}-\frac{4}{3}k_{2}d \Big{)}{b_{3}}^{2}\end{split}  \tag{77}\]  
\[I_{n\zeta}=\int_{A}\varphi^{n}\eta_{\zeta}(x,y)dA=\frac{2}{3}{y_{s}}^{n+1}{y_{ s}}^{n+1}\Bigg{[}A_{t}\Big{(}{b_{1}}^{2}+4{b_{2}}^{2}\Big{)}+A_{c}\varphi{b_{3}}^{2}+A_{ b}\Big{(}{b_{1}}^{2}+4{b_{1}}^{2}\Big{)}\bigg{(}\frac{y_{x}}{y_{s}}\bigg{)}^{n+1} \Bigg{]},(n=1,3) \tag{78}\]  
Here, \(A_{t}\) represents the area of the top plate, \(A_{b}\) represents the area of the bottom plate, \(A_{c}\) represents the area of the cantilever plate, \(A_{w}\) represents the area of one side of the web, and \(A\) represents the full cross-sectional area of the box girder.