# Shear Lag Effect of Ultra-Wide Box Girder under Influence of Shear Deformation
## 1 Introduction
\begin{array}{ll}u_{2}(z)-y_{s}\Big{(}{b_{1}}^{2}-x^{2}\Big{)} g(z)-\frac{2q_{2}}{I_{1}}(b_{1}-x)g(z)&\text{Plate\#1}\\ u_{6}(z)+y_{x}\Big{(}{b_{1}}^{2}-x^{2}\Big{)}g(z)-\frac{2q_{2}}{I_{1}}(b_{1}-x) g(z)&\text{Plate\#4}\\ u_{3}(z)-y_{s}\Big{[}4{b_{2}}^{2}-(b_{1}+2b_{2}-x)^{2}\Big{]}g(z)-\frac{2q_{1}} {I_{1}}[2b_{2}-(b_{1}+2b_{2}-x)]g(z)&\text{Plate\#2}\\ u_{7}(z)+y_{x}\Big{[}4{b_{2}}^{2}-(b_{1}+2b_{2}-x)^{2}\Big{]}g(z)-\frac{2q_{1} }{I_{1}}[2b_{2}-(b_{1}+2b_{2}-x)]g(z)&\text{Plate\#5}\\ u_{3}(z)-y_{s}\Big{[}{b_{3}}^{2}-(b_{1}+2b_{2}+b_{3}-x)^{2}\Big{]}g(z)&\text{ Plate\#3}\end{array} \right. \tag{18}\]  
According to Equation (18), the flexural displacements of the top plate of the inner chamber (plate#1), top plate of the outer chamber (plate#2) and cantilever plate (plate#3), bottom plate of the inner chamber (plate#4), bottom plate of the outer chamber (plate#5) can be represented by \(u_{2}(z)\), \(u_{3}(z)\), \(u_{6}(z)\), and \(u_{7}(z)\), respectively. As shown in Figure 5, the distribution of the shear flow on the box girder web is a quadratic function. By analyzing the shear flow on the box girder web, the shear flows \(q_{8}(z)\) and \(q_{9}(z)\) at points 8 and 9 (see Figure 3) of the box girder web can be obtained as follows:  
\[q_{8}(z)=q_{08}(z)+(q_{1}-q_{2})=\frac{M^{\prime}(z)}{I_{x}}y_{s}A_{s1}+\frac{ M^{\prime}(z)}{2I_{x}}{y_{s}}^{2}t_{w}+(q_{1}-q_{2}) \tag{19}\]  
\[q_{9}(z)=q_{09}(z)+q_{1}=\frac{M^{\prime}(z)}{I_{x}}y_{s}A_{s2}+\frac{M^{\prime }(z)}{2I_{x}}{y_{s}}^{2}t_{w}+q_{1}, \tag{20}\]  
where \(t_{w}\) represents the web thickness and \(A_{si}\) represents the area of the wing plate corresponding to the open-section web. According to Equations (19) and (20), the flexural shear flow at any point of the box girder web can be expressed as follows:  
\[q(y,z)=q_{8,9}(z)-\frac{M^{\prime}(z)}{2I_{x}}y^{2}t_{w}. \tag{21}\]  
The web shear deformation equation obtained from Equations (5) and (21) is as follows:  
\[\gamma_{sz}=\frac{\partial u}{\partial s}+\frac{\partial v}{\partial z}= \frac{q(z)}{Gt_{w}}=\frac{q_{8,9}(z)}{Gt_{w}}-\frac{M^{\prime}(z)}{2G{I_{x}}}y ^{2}. \tag{22}\]According to the rigid peripheral assumption, if \(w(z)\) is the vertical flexural displacement of the girder body, the relationship between the flexural angle of the box girder and the circumferential displacement of the web plate is as follows:  
\[w^{\prime}(z)=\frac{\partial v}{\partial z}. \tag{23}\]  
Equation (22) is substituted into Equation (23) and \(s\) is replaced with \(y\) to represent the position of any vertical point on the web. After simplification of the integral, the equation for the flexural longitudinal displacement at any point on the box girder web is obtained as follows:  
\[u(y,z)=-yw^{\prime}(z)+(\frac{q_{08,09}(z)}{Gt_{w}}+\frac{q_{i}}{Gt_{w}})y- \frac{M^{\prime}(z)}{6 GI_{x}}y^{3}+u_{8,9}(z). \tag{24}\]  
By simplifying Equations (19), (20) and (24), \(u_{2}(z)\), \(u_{3}(z)\), \(u_{6}(z)\), and \(u_{7}(z)\) are obtained. By combining Equations (18)-(20) and (24), the flexural longitudinal displacements of the box girder represented by \(u_{8}(z)\) and \(u_{9}(z)\) are obtained, the equations for which are presented below. Inner chamber top and bottom plates and inner web:  
\[u(x,y,z)=-yw^{\prime}(z)+\bigg{[}\alpha_{1}y-\frac{1}{3}y^{3}+\eta_{1}(x,y) \bigg{]}g(z)+u_{8}(z) \tag{25}\]  
Outer chamber top and bottom plates, cantilever plate, and outer web plate:  
\[u(x,y,z)=-yw^{\prime}(z)+\bigg{[}\alpha_{2}y-\frac{1}{3}y^{3}+\eta_{2}(x,y) \bigg{]}g(z)+u_{9}(z) \tag{26}\]  
where \(\alpha_{1}=\frac{2\eta_{i}A_{1}}{I_{w}}+{y_{s}}^{2}+\frac{2(q_{1}-q_{2})}{I_{ w}}\); \(\alpha_{2}=\frac{2{y_{s}}A_{2}}{I_{w}}+{y_{s}}^{2}+\frac{2q_{1}}{I_{w}}\). \(\eta_{1}(x,y)\) and \(\eta_{2}(x,y)\); the equation is as follows:  
\[\eta_{1}(x,y)=\left\{ \begin{array}{cc}-y_{s}\Big{(}{b_{1}}^{2}-x^{2}\Big{)} -\frac{2q_{2}}{I}(b_{1}-x)&\text{Plate\#1}\\ y_{x}\Big{(}{b_{1}}^{2}-x^{2}\Big{)}-\frac{2q_{2}}{I}(b_{1}-x)&\text{Plate\#4} \end{array} \right. \tag{27}\]  
\[\eta_{2}(x,y)=\left\{