# 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
#### 2.2.2 Calculation of Coefficient \(k_{g}\)  
According to Equations (15)-(18), the global shear buckling coefficient of CSWs in HCGs \(k_{g}\) is associated with the length to height ratio \(\lambda\) (\(l\)/\(h\)), the rigidity ratios \(\alpha(D_{x}/D_{y})\) and \(\beta(D_{xy}/D_{y})\), the modulus ratio \(\gamma(G_{xy}/(E_{y}-2v_{y}G_{xy}))\), the expression \(h^{2}/(Rd)\) and \(6s/(3a+c)\). 1. Value ranges of \(\alpha\), \(\beta\), \(\gamma\), \(h^{2}/(Rd)\) and \(6s/(3a+c)\) for common HCG bridges with CSWs  
For trapezoidal CSWs that are commonly used in girder bridges, the rigidity ratios \(\alpha\) and \(\beta\) have the relationship: \(\beta/\alpha=2s^{2}/[(1+\
u)q^{2}]\). A statistical analysis of available bridges with CSWs (as shown in Table 1) shows that the rigidity ratio \(\alpha\) varies from 0.0006 to 0.0069, \(\gamma\) varies from 0.38 to 0.44, \(6s/(3a+c)\) is about 6, and \(\beta\) is about (1.67-2.0)\(\alpha\). The following parametric study considers \(\alpha\) ranging from 0.0005 to 0.0070, \(\gamma\) ranging from 0.38 to 0.44, \(6s/(3a+c)\) equal to 6 and \(\beta\) equal to 1.6\(\alpha\), 1.8\(\alpha\), 2.0\(\alpha\) respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Bridges**} & \(a\) & \(c\) & \(d\) & \(t_{min}\) & \(t_{max}\) & \(\frac{\epsilon}{3a+c}\) & \multicolumn{2}{c}{**Corresponding to \(t_{min}\)**} & \multicolumn{2}{c}{**Corresponding to \(t_{max}\)**} & \multicolumn{2}{c}{\(\gamma\)} \\ \cline{2-13}  & mm & mm & mm & mm & mm & mm & \(\alpha\) & \(\beta\) & \(\beta\)/\(\alpha\) & \(\alpha\) & \(\beta\) & \(\beta\)/\(\alpha\) & \\ \hline Cognac & 353 & 353 & 150 & 8 & 8 & 6.00 & 0.0013 & 0.0022 & 1.69 & 0.0013 & 0.0022 & 1.69 & 0.4406 \\ Maurpe & 284 & 284 & 150 & 8 & 8 & 6.00 & 0.0012 & 0.0022 & 1.83 & 0.0012 & 0.0022 & 1.83 & 0.4093 \\ Dole & 430 & 430 & 220 & 8 & 12 & 6.00 & 0.0006 & 0.0010 & 1.67 & 0.0013 & 0.0023 & 1.77 &  
Table 1 shows that the corrugation depth \(d\) varies from 0.15 m to 0.22 m. The upper limit of \(H/d\) does not exceed 133 considering the girder height \(H\) generally is not more than 20 m. For common HCG bridges, the girder height to length ratio \(H/L\) generally ranges from 1/11 to 1/30, and the central angle \(L/R\) is generally no more than \(\pi/2\). So, the upper limit of \(H/R\) of HCG bridges does not exceed 0.143, where \(L\) is the girder span. Thus, the upper limit of \(H^{2}/(Rd)\) of HCG bridges does not exceed 20. Because the CSW height \(h\) is smaller than the girder height \(H\), the upper limit of \(h^{2}/(Rd)\) does not exceed 20. In order to expand the scope of application of the formulas, the following parametric study considers \(h^{2}/(Rd)\) ranging from 0 to 30. 2. Influence of \(\alpha\) and \(\beta/\alpha\) on the global shear buckling coefficient \(k_{g}\)  
Theoretically, the more numbers used in the trigonometric series (as shown in Equations (12)-(14)), the more precise the solution is. If \(m\) and \(n\) increase toward infinity, exact results of shear buckling stress of CSWs can be obtained. However, the calculation effort increases with the increasing numbers \(m\) and \(n\) in the trigonometric series. In the case of the CSW with a length to height ratio \(l/h\) less than 5, the deviation between the results with \(m=30\), \(n=30\) and the results with \(m=25\), \(n=25\) is less than 1%. In what follows, \(m=30\) and \(n=30\) are adopted. Table 2 shows the values of \(k_{g}\) calculated for various values of \(D_{x}/D_{y}\) and \(l/h\), and for \(\beta=1.6\alpha\), \(\beta=1.8\alpha\) and \(\beta=2.0\alpha\) respectively when \(h^{2}/(Rd)=5\) and \(\gamma=0.4\) for a four-edge simple support. The results for \(\beta=1.6\alpha\) and \(\beta=2.0\alpha\), compared to for \(\beta=1.8\alpha\), deviate less than 0.6%. The results show that the parameter \(\beta/\alpha\) has little effect on the coefficient \(k_{g}\) for common bridges with CSWs. From an engineering application point of view, the deviations can be ignored.