# Flexural Behavior of Corroded High-Speed Railway Simply Supported Prestressed Concrete Box Grider
## 3 Analytical Model
### Impact of Strand Corrosion
#### 3.1.1 Strand Cross-Section Reduction  
In the present study, the seven-wire strand was adopted. As shown in Figure 5, it is generally assumed that strand corrosion can only occur on the external surface [11; 32; 33].  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Specimen** & **L1** & **L2** & **L3** & **L4** & **L5** & **L6** \\ \hline Design corrosion time/h & 0 & 35.25 & 70.50 & 105.75 & 141.00 & 211.50 \\ Actual mass loss/\% & 0 & 2.12 & 4.06 & 6.28 & 7.84 & 12.30 \\ Design mass loss/\% & 0 & 2 & 4 & 6 & 8 & 12 \\ Standard deviation of mass loss/\% & 0 & 0.63 & 1.25 & 0.83 & 1.07 & 2.54 \\ Test cracking load/kN & 72.0 & 68.0 & 62.2 & 65.8 & 58.0 & 52.0 \\ Test cracking moment/kN-m & 23.58 & 22.20 & 20.35 & 21.56 & 18.88 & 17.03 \\ Predicted cracking moment/kN-m & 23.06 & 21.50 & 20.27 & 19.32 & 18.62 & 17.41 \\ Test ultimate load/kN & 232.5 & 230.7 & 222.0 & 218.6 & 196.7 & 163.0 \\ Test flexural bearing capacity/kN-m & 76.14 & 75.55 & 72.71 & 71.59 & 64.43 & 53.38 \\ Predicted flexural bearing capacity/kN-m & 49.43 & 48.48 & 48.11 & 47.18 & 42.90 & 35.32 \\ Yield displacement/mm & 11.357 & 11.035 & 12.316 & 11.903 & 12.100 & 11.024 \\ Ultimate displacement/mm & 36.500 & 34.900 & 38.140 & 35.800 & 35.280 & 31.356 \\ Displacement ductility factor & 3.21 & 3.16 & 3.10 & 3.01 & 2.92 & 2.84 \\ Test failure mode & CC & CC & CC & CC & SR & SR \\ Predicted failure mode & CC & CC & CC & CC & SR & SR \\ \hline \end{tabular} Note: CC = concrete crushing; SR = strand rupture.  
\end{table}
Table 4: Experimental results.  
Figure 4: Four-point bending test: (**a**) site photo; (**b**) schematic diagram.  
For chloride-induced corrosion, there are many suggestions [34; 35; 36] to use the hemispherical corrosion pit model to calculate the remaining area; the predictions are well in agreement with the tests.  
The area reduced by corrosion (\(A_{\text{c}}\)) of an external wire is calculated as follows:  
\[A_{\text{c}}=\left\{\begin{array}{ll}A_{1}+A_{2},\;p_{\text{m}}\leq\frac{ \sqrt{2}}{2}D_{0}\\ \frac{\pi D_{0}^{2}}{4}-A_{1}+A_{2},\;\frac{\sqrt{2}}{2}D_{0}\leq p_{\text{m}} \leq D_{0}\\ \frac{\pi D_{0}^{2}}{4},\;p_{\text{m}}>D_{0}\end{array}\right. \tag{1}\]  
where \(D_{0}\) is the initial nominal diameter of the external wire, which is 5.025 mm in this test; \(p_{\text{m}}\) is the maximum corrosion depth; and \(A_{1}\) and \(A_{2}\) are the areas of the blue and pink parts of Figure 5, respectively.  
The other parameters in Equation (1) are given as follows:  
\[A_{1}=\frac{1}{2}\Bigg{[}\theta_{1}\bigg{(}\frac{D_{0}}{2}\bigg{)}^{2}-a \bigg{|}\frac{D_{0}}{2}-\frac{{p_{\text{m}}}^{2}}{D_{0}}\bigg{|}\Bigg{]};A_{2} =\frac{1}{2}\Bigg{[}\theta_{2}{p_{\text{m}}}^{2}-a\frac{{p_{\text{m}}}^{2}}{D _{0}}\Bigg{]} \tag{2}\]  
\[a=2p_{\text{m}}\sqrt{1-\bigg{(}\frac{p_{\text{m}}}{D_{0}}\bigg{)}^{2}} \tag{3}\]  
\[\theta_{1}=2\text{arcsin}\bigg{(}\frac{a}{D_{0}}\bigg{)};\theta_{2}=2\text{ arcsin}\bigg{(}\frac{a}{2p_{\text{m}}}\bigg{)} \tag{4}\]  
According to [PERSON] et al. [11], there is a ratio (\(R\)) between the maximum corrosion depth and the average corrosion depth of the corroed prestressing strand, ranging from 2.93 to 15.05. Since there is no well-developed model for the stochastic distribution of the corrosion penetration factor of prestressed strands, the parameter in this analytical model is taken similarly to reference [21]; here, it is taken as the mean value \(R=9\).  
\[R=\frac{p_{\text{m}}}{p_{\text{a}}}=\frac{p_{\text{m}}}{D_{0}\big{(}1-\sqrt{1- \hat{p}}\big{)}/2} \tag{5}\]  
where \(p_{\text{a}}\) is the average corrosion depth.  
To simplify the analysis, six external wires share the same corrosion pit depth. The remaining area \(A_{\text{p},\rho}\) is given by Equation (6).  
\[A_{\text{p},\rho}=\frac{\pi}{4}D_{1}^{2}+6\Big{(}\frac{\pi}{4}D_{0}^{2}-A_{ \text{c}}\Big{)} \tag{6}\]  
where \(D_{1}\) is the initial nominal diameter of the middle wire, which is 5.15 mm in this test.