# Shear Strength of Trapezoidal Corrugated Steel Webs for Horizontally Curved Girder Bridges
## 2 Elastic Global and Local Shear Buckling Stress of CSWs
### Elastic Global Shear Buckling Stress of CSWs  
#### 2.2.1 Critical Buckling Stress under Pure Shear  
According to the theory of thin plates and shells, if the ratio of a shell's height to its short side is less than 0.2, the shell can be analyzed as a shallow shell. For HCGs with vertical CSWs, webs always meet this condition and can be treated as orthotropic cylindrical shallow shells (Figure 2) for the global shear buckling analysis. According to the stability theory of plates and shells, the equilibrium equation and the deformation compatibility equation of an orthotropic cylindrical shallow shell shown in Figure 2 under pure shear force can be expressed respectively as Equations (9) and (10) [30]. In the following equations, the part that appears in bold type shows the difference between a plate and a shell. \[\frac{1}{t}\bigg{(}D_{x}\frac{\partial^{4}}{\partial x^{4}}+D_{xy}\frac{ \partial^{4}}{\partial x^{2}\partial y^{2}}+D_{y}\frac{\partial^{4}}{ \partial y^{4}}\bigg{)}w+\frac{1}{R}\frac{\partial^{2}\Phi}{\partial y^{2}}=2 \tau\frac{\partial^{2}w}{\partial x\partial y} \tag{9}\]  
\[\bigg{[}\frac{1}{E_{y}}\frac{\partial^{4}}{\partial x^{4}}+\bigg{(}\frac{1}{ C_{xy}}-2\frac{v_{y}}{E_{y}}\bigg{)}\frac{\partial^{4}}{\partial x^{2} \partial y^{2}}+\frac{1}{E_{x}}\frac{\partial^{4}}{\partial y^{4}}\bigg{]} \Phi-\frac{1}{R}\frac{\partial^{2}w}{\partial y^{2}}=0 \tag{10}\]  
where \(\Phi\) is the stress function, \(w\) is the out of plane deflection of the shell, \(\tau\) is the shear stress, \(R\) is the curvature radius of the HCG. Substituting Equation (10) into Equation (9), Equation (9) can be expressed as Equation (11). \[\frac{1}{t}\bigg{(}D_{x}\frac{\partial^{4}}{\partial x^{4}}+D_{xy}\frac{ \partial^{4}}{\partial x^{2}\partial y^{2}}+D_{y}\frac{\partial^{4}}{ \partial y^{4}}\bigg{)}w+\frac{1}{R^{2}}\bigg{[}\frac{1}{E_{y}}\frac{\partial ^{4}}{\partial x^{4}}+\bigg{(}\frac{1}{C_{xy}}-2\frac{v_{y}}{E_{y}}\bigg{)} \frac{\partial^{4}}{\partial x^{2}\partial y^{2}}+\frac{1}{E_{x}}\frac{ \partial^{4}}{\partial y^{4}}\bigg{]}^{-1}\frac{\partial^{4}w}{\partial y^{4} }=2\tau\frac{\partial^{2}w}{\partial x\partial y} \tag{11}\]  
It can be assumed that the boundary conditions of CSWs satisfy a four-edge simple support, a four-edge fixed support, or the two edges constrained by flanges fixed and the other two edges simply supported (the edges \(x=0\) and \(x=l\) are simply supported, the edges \(y=0\) and \(y=h\) are fixed supported). The functions of deflection can be expressed respectively as Equations (12)-(14). Figure 2: Corrugated steel web and its equivalent orthotropic cylindrical shallow shell: (**a**) Corrugated steel web; (**b**) Equivalent orthotropic cylindrical shallow shell. For the edges \(x=0\) and \(x=l\) simply supported, and the edges \(y=0\) and \(y=h\) fixed:  
\[\left. \begin{array}{l}\frac{D_{y}}{h^{2}}\frac{\pi^{4}}{4\lambda^{3}} \left\{\begin{array}{l}A_{mm}\Big{\{}am^{4}\Big{[}n^{-2}+(n+2)^{-2}\Big{]}+2 \beta\lambda^{2}m^{2}+\lambda^{4}\Big{[}n^{2}+(n+2)^{2}\Big{]}\Big{]}\\ -A_{m,n+2}\Big{[}am^{4}(n+2)^{-2}+\beta\lambda^{2}m^{2}+\lambda^{4}(n+2)^{2} \Big{]}-A_{m,n-2}\Big{[}am^{4}n^{-2}+\beta\lambda^{2}m^{2}+\lambda^{4}n^{2} \Big{]}\end{array}\right\}\\ +\frac{D_{y}}{h^{2}}\frac{h^{4}}{R^{2}d^{2}}\frac{6\epsilon}{(3a+c)} \frac{\alpha\gamma\lambda^{5}}{4}\left\{\begin{array}{l}A_{mm}\Bigg{\{} \Big{(}\alpha\gamma m^{4}+\alpha\lambda^{2}m^{2}n^{2}+\gamma\lambda^{4}n^{4} \Big{)}^{-1}n^{2}\\ -A_{m,n+2}\Big{[}\alpha\gamma m^{4}+\alpha\lambda^{2}m^{2}(n+2)^{2}+\gamma \lambda^{4}(n+2)^{4}\Big{]}^{-1}(n+2)^{2}\\ -A_{m,n-2}\Big{(}\alpha\gamma m^{4}+\alpha\lambda^{2}m^{2}(n+2)^{2}+\gamma \lambda^{4}(n+2)^{4}\Big{]}^{-1}(n+2)^{2}\\ -A_{m,n-2}\Big{(}\alpha\gamma m^{4}+\alpha\lambda^{2}m^{2}n^{2}+\gamma\lambda^{4 }n^{4}\Big{)}^{-1}n^{2}\\ \end{array}\right\}\\ -8\tau\sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty}A_{ij} \frac{mi}{m^{2}-i^{2}}\bigg{[}\frac{1}{n^{2}-i^{2}}-\frac{1}{(n+2)^{2}-i^{2}} -\frac{1}{n^{2}-(j+2)^{2}}+\frac{1}{(n+2)^{2}-(j+2)^{2}}\bigg{]}=0\end{array}