## 4 Flexural Strength Evaluation of Corroded PC Beams
### Strain Compatibility Using OpenSEES  
According to a previous study, it was found that the corrosion on strands led to a rapid reduction in elongation. Therefore, using the \(f_{ps}\) approximation has limitations in the evaluation of the PC beams, because it considers only a reduction in the ultimate strength of corroded strands. To consider the effect of reduced elongation, the Open System for Earthquake Engineering Simulation (OpenSEES) [28] analysis program was used for the strain compatibility method in this paper. This program uses the fiber elements assigned with section properties, and the analysis is conducted using the strain compatibility method on the sections. The analysis modeling is shown in Figure 9. Since the parabolic tendon profile could not be reflected in the strain compatibility analysis, the element was divided into length \(t\) (10 mm), and the position of the tendon in each element was changed gradually according to the tendon profile. Each section of the element consisted of four rebars and three strands spaced 20 mm apart, which were then modeled as if bonded perfectly. The cross-sectional area of the strands was not changed according to the corrosion, because the material model included the effect of corrosion. The distance between two supports was 2500 mm. Figure 9: Analysis model for PC beams in OpenSEES. The concrete material model used in this paper was the modified Kent and Park model [29] which has been predefined in OpenSEES [28]--using the name concrete02--as shown in Figure 10a. In this model, the stress-strain curve follows the example of the Kent and Park model [29] up to the peak strength, \(f_{c}^{\prime}\). The linear decrease was then defined in the post peak branch as expressed in Equation (9). Linear tension softening was also considered in this model, and the material properties (shown in Table 5) were defined based on Eurocode 2 [30] and the cylinder test. The material model of reinforcements was defined using the bi-linear curve (as shown in Figure 10b)--predefined and named steel01 in OpenSEES [28]. The yield stress, \(f_{y}\), was 400 MPa. The elastic modulus, \(E\), was 200 GPa. \[f_{c}=\ \left\{ \begin{array}{ll}f_{c}^{\prime}\left[\frac{2\varepsilon_{c}}{ \varepsilon_{cu}}-\left(\frac{\varepsilon_{c}}{\varepsilon_{cu}}\right)^{2} \right]&\varepsilon_{c}\ \leq\ \varepsilon_{co}\\ f_{c}^{\prime}-\left(\frac{\varepsilon_{c}-\varepsilon_{cu}}{\varepsilon_{cu} }-\varepsilon_{co}\right)\left(f_{c}^{\prime}-f_{cu}^{\prime}\right)& \varepsilon_{c}\ >\ \varepsilon_{co}\end{array} \right. \tag{9}\]  
where \(f_{c}\) and \(\varepsilon_{c}\) are the compressive stress and strain, respectively; \(f_{c}^{\prime}\) and \(f_{cu}\) are the maximum stress and crushing stress, respectively; \(\varepsilon_{co}\) and \(\varepsilon_{cu}\) are the strain at the maximum stress and crushing stress, respectively; and \(f_{ct}\) is the tensile strength of concrete. Based on the material properties shown in Table 2, the bi-linear model was used for the non-corroded strands. In the case of \(\eta\leq 5\%\), the material model of corroded strands was defined as shown in Figure 11a. Because the yield properties were not affected significantly by corrosion [18], \(f_{py}\) was unchanged, and the ultimate properties were defined using Equations (6) and (7). The \(f_{c,pu}\) can be obtained by dividing \(F_{c,pu}\) by the gross sectional area of the strand, \(A_{ps}\). \(A_{ps}\) is used to reflect the corrosion effect on the stress-strain curve. In the case of \(\eta>5\%\), the ultimate properties were lower than the yield properties, and brittle behaviors were observed in the tensile tests. Therefore, the material model of corroded strands was built based on the brittle material model (as shown in Figure 11b) considering only the ultimate properties from Equations (6) and (7), with the stiffness changed to the secant stiffness, \(E_{se}\). After the \(f_{c,pu}\), a sharp decrease in strength was modeled in both corroded cases to simulate the wire rupture.