# Assessment of Mechanical Properties of Corroded Prestressing Strands
## 4 Prestressed Concrete Beams with Corroded Strands
### Determination of the Flexural Strength of a Corroded PSC Beam  
To calculate the flexural strength of a PSC beam, the approximate value of \(f_{\text{ps}}\), which refers to the stress of the strand at the moment of the beam's failure, can be obtained from the design codes, or the strain compatibility method can be utilized to derive the value. According to ACI 318-14 [24], Equation (7) is the approximate expression for \(f_{\text{ps}}\). \(f_{\text{cps}}\) and \(\rho_{\text{c}\text{p}}\) refer to the ultimate strength of the corroded strand and steel ratio, respectively. The section loss by corrosion can be calculated from Equations (1)-(3), and the decreased ultimate strength can be calculated using Equations (4) and (5), along with the result of tensile tests given in Figure 17.  
\[f_{\text{cps}}=f_{\text{cpu}}\left[1-\frac{\gamma_{p}}{\beta_{1}}\left\{\rho_{ \text{c}\text{p}}\frac{f_{\text{cpu}}}{f_{\text{c}\text{k}}}+\frac{d}{d_{p}} (\omega-\omega^{\prime})\right\}\right], \tag{7}\]  
where  
\(f_{\text{cps}}\): stress of the corroded prestressing strand when the beam fails;  
\(f_{\text{cpu}}\): ultimate stress of the corroded prestressing strand;  
\(f_{\text{cpu}}\): coefficient according to the type of strand \(\left\{ \begin{array}{l}0.55\text{ \emph{when} }f_{\text{py}}/f_{\text{pu}}\geq 0.80\\ 0.40\text{ \emph{when} }f_{\text{py}}/f_{\text{pu}}\geq 0.85\\ 0.28\text{ \emph{when} }f_{\text{py}}/f_{\text{pu}}\geq 0.90\end{array} \right.\);  
\(\beta_{1}\): ratio between the depth of an equivalent rectangular concrete stress block and the neutral axis depth (= a/c);  
\(\rho_{\text{c}\text{p}}\): corroded prestressing steel ratio \(\left(=A_{\text{c}\text{p}}/bd_{p}\right)\);  
\(d\): depth of tensile reinforcement;  
\(d_{p}\): depth of prestressing strand;  
\(\omega\): tensile reinforcement index \(\left(=\rho\frac{f_{\text{y}}}{f_{\text{x}}},\ \rho=\frac{A_{\text{x}}}{bd}\right)\);  
\(\omega^{\prime}\): compression reinforcement index \(\left(=\rho^{\prime}\frac{f_{\text{y}}}{f_{\text{x}}},\ \rho^{\prime}=\frac{A_{\text{x}^{\prime}}}{bd}\right)\). \(f_{\text{c}\text{p}\text{s}}\) from Equation (7) for each specimen and the corresponding flexural strength are listed in Table 4. The RB, MB1, and MB2 specimens showed reasonable estimations of the strength, while QB showed a value different from the test value. The reason is that the cross-section of corrosion was at 1/4 of the span length, such that a lesser flexural moment was carried than that occurring at the center. Therefore, the governing cross-section for the nominal flexural strength became the center (154.47 kN) rather than 1/4 (211.73 kN), even though the reduced \(f_{\text{c}\text{ps}}\) is considered. The failure mode, however, showed a wire fracture at the 1/4 of the span length. This means that the reduced ultimate strain of the corroded strand could be the governing factor of the flexural strength. To consider this, the strain compatibility method should be used. Although MB2 showed a lower flexural strength of 135.6 kN than that of MB1 at 146.4 kN, the flexural strength calculated by Equation (7) was higher (138.53 kN \(<\) 140.6 kN). This was because MB2's maximum wire-unit section loss of the corroded strand was lower than that of MB1 (18.89% \(<\) 22.24%), making the \(f_{\text{ps}}\) from Equation (7) a lower value than that of MB1 (1289 MPa \(<\) 1327 MPa). This phenomenon was caused by an uncertainty in the tensile behavior as well as an undefined bonding behavior of the corroded strand. For issues regarding the residual service life of an entire PSC bridge with respect to corrosion, the flexural strength prediction method mentioned above is not enough. Section loss and stress concentration due to corrosion results in change of stress conditions in a bridge. The residual service life is estimated by serviceability limit state in terms of stress limits at the joints of post-tensionedbox girders. Ultimate limit state needs to be checked considering strength reduction and stress concentration.