# Flexural Behavior of Corroded High-Speed Railway Simply Supported Prestressed Concrete Box Grider
## 3 Analytical Model
### Calculation of Flexural Behavior
#### 3.2.2 Flexural Bearing Capacity  
The basic assumptions for the flexural bearing capacity of corroded PC beams are as follows: (1) for PC beams having a reliable anchorage system, the deterioration of the bond has little effect on the flexural behavior; (2) the plane section remains satisfied for average concrete strain during bending; (3) the contribution of concrete in the tensile zone can be neglected; (4) the constitutive models of corroded steel strands and concrete are in Yu [12] and the Chinese code [25], respectively; (5) all normal reinforcing bars yield in the ultimate condition. The stress-strain constitutive model of concrete adopts a quadratic parabolic segment followed by a horizontal linear segment. The prestressing strand adopts a bilinear constitutive relationship, with the nominal yield strength set at 0.85 times the ultimate strength. When the strand mass loss is greater than a critical value, the hardened section of the stress-strain curve disappears and the curve becomes a single linear curve. The strain and stress distribution of a normal section of the test beam is shown in Figure 7. The following equation can be obtained:  
\[\frac{\varepsilon_{\text{cu}}}{\varepsilon_{\text{p},\rho}}=\frac{x_{\text{c} }}{h_{\text{p}}-x_{\text{c}}}=\frac{x}{\beta_{1}h_{\text{p}}-x} \tag{15}\]  
where \(\varepsilon_{\text{p},\rho}\) denotes the strain of the corroded prestressing strand; \(\varepsilon_{\text{cu}}\) is the ultimate compressive strain (when \(f_{\text{cu}}\leq 50\) MPa, \(\varepsilon_{\text{cu}}=0.0033\)); \(h_{\text{p}}\) is the distance from the strand to the compressive edge; \(x\) and \(x_{\text{c}}\) are the actual and equivalent compressed zone heights of the concrete, respectively; and \(\beta_{1}\) denotes the height coefficient of the equivalent rectangular compression zone, which is set to 0.8 when the concrete strength is less than 50 MPa. Considering the axial force equilibrium, the equivalent compressed zone height of concrete is calculated from Equation (16). For all test beams, it lies in the flange plate, which can be verified in the crack profiles of the test beams. \[x=\left(\sigma_{\text{p},\rho}A_{\text{p},\rho}+f_{\text{yk}}A_{\text{s}}-f^ {\prime}_{\text{yk}}A^{\prime}_{\text{s}}\right)/\left(\alpha_{1}f_{\text{ck} }b^{\prime}_{t}\right) \tag{16}\]  
where \(\sigma_{\text{p},\rho}\) is the stress of the corroded strand calculated by Equation (17); \(\alpha_{1}\) denotes the stress coefficient of the equivalent rectangular compression zone; \(b_{t}\) denotes the width of the flange plate; \(f_{\text{yk}}\) and \(A_{\text{s}}\) are the standard yield strength and area of a normal tensile steel bar, respectively; and \(f_{\text{yk}}\)' and \(A_{\text{s}}\)' are the standard yield strength and area of a normal compressive steel bar, respectively; and \(f_{\text{yk}}\)' and \(A_{\text{s}}\)' are the standard yield strength and area of a normal compressive steel bar, respectively;  
\[\sigma_{\text{p},\rho}= \begin{cases}\varepsilon_{\text{p},\rho}E_{\text{p}, \rho}&(\varepsilon_{\text{p},\rho}\leq\varepsilon_{\text{py},\rho})\\ 0.85_{\text{fpk},\rho}+0.15_{\text{fpk},\rho}\left(\frac{\varepsilon_{\text{p},\rho}-\varepsilon_{\text{py},\rho}}{\varepsilon_{\text{pu},\rho}-\varepsilon _{\text{py},\rho}}\right)&(\varepsilon_{\text{p},\rho}>\varepsilon_{\text{ py},\rho})\end{cases}  \tag{17}\]The ultimate and actual strain of the corroded strand can be calculated according to Equations (9) and (15), respectively. By comparing these two values, the failure mode can be determined. If \(\varepsilon_{\mathrm{p,\rho}}\leq\varepsilon_{\mathrm{pu,\rho}}\), the test beams fail with concrete crushing; otherwise, rupture of the corroded strand happens.