## 2 The Proposed Model
### Axial Compression Point
When the corrosion of stirrups is not ignored, the stirrup ratio, considering the corrosion of stirrups \(\mu_{vc}\), can be evaluated as  
Figure 2:2  
Figure 2:3  
where \(A_{\text{st,red}}(t)\) represents the reduced area of the corroded stirrup; \(f_{sc,y}\) is the yield stress of corroded reinforcement bar; \(\eta_{s}\) represents the corrosion rate of reinforcement; and \(a_{y}\) is the influence coefficient of pitting corrosion on bar's yield strength, \(a_{y}=1.18\) when the bar is corroded in the solution, and \(a_{y}=2.10\) in a chloride environment [33]. The reduced area of stirrups caused by uniform and pitting corrosion can be derived by  
Figure 2:4  
where \(A_{\text{P}}(t)\) is the reduced area due to pitting corrosion (see Figure 4), which can be calculated by [31]  
\[A_{\text{P}}(t)=\left\{ \begin{array}{c}A_{1}+A_{2}\\ \frac{\pi\varphi_{0}^{2}}{4}-A_{1}+A_{2}\\ \frac{\pi\varphi_{0}^{2}}{4}\end{array} \right.\quad\left\{ \begin{array}{c} p(t)\leq\frac{\varphi_{0}}{\sqrt{2}}\\ \frac{\varphi_{0}}{\sqrt{2}}<p(t)\leq\varphi_{0}\\ p(t)>\varphi_{0}\end{array} \right., \tag{13}\]  
Figure 4: Pit configuration: \(p(t)\): the pitting depth; \(A_{\text{P}}(t)\): the reduced area due to pitting corrosion; \(\theta_{1}\) and \(\theta_{2}\): the angle parameters; \(\varphi_{0}\): the diameter of the steel bar. where  
\[A_{1}=\frac{1}{2}\Bigg{[}\theta_{1}\Big{(}\frac{\varphi_{0}}{2}\Big{)}^{2}-a \Bigg{|}\frac{\varphi_{0}}{2}-\frac{p(t)^{2}}{\varphi_{0}}\Bigg{|}\Bigg{]}, \tag{14}\]  
\[A_{2}=\frac{1}{2}\Bigg{[}\theta_{2}p(t)^{2}-a\frac{p(t)^{2}}{\varphi_{0}}\Bigg{]}, \tag{15}\]  
\[a=2p(t)\sqrt{1-\Big{(}\frac{p(t)}{\varphi_{0}}\Big{)}^{2}}, \tag{16}\]  
\[\theta_{1}=2\text{arcsin}\Big{(}\frac{a}{\varphi_{0}}\Big{)}, \tag{17}\]  
\[\theta_{2}=2\text{arcsin}\Big{(}\frac{a}{2p(t)}\Big{)}, \tag{18}\]  
where \(p(t)\) is the pitting depth, which can be evaluated as  
\[p(t)=0.0116i_{corr}tR, \tag{19}\]  
in which the pitting factor \(R=\frac{\text{Maximum depth of pit}}{\text{Average depth of pit}}=4\) in this work. It should be emphasized that the failure of reinforcing steels often results from the ductility reduction of the local area caused by pitting corrosion. For longitudinal compressive steel bars, corrosion easily leads to concrete cover peeling, which greatly increases the risk of buckling. In this paper, the maximum allowable stress of compression bars is the minimum between the yield strength and critical stress. The critical stress can be calculated as [34]  
\[\sigma_{cr}=\frac{3.46\sqrt{E_{r}Ik_{1}}}{A_{all}}, \tag{20}\]  
\[E_{r}=\frac{4E_{s}E_{p}}{\big{(}\sqrt{E_{s}}+\sqrt{E_{p}}\big{)}^{2}}=\frac{4 E_{s}\beta_{1}}{\big{(}1+\sqrt{\beta_{1}}\big{)}^{2}}, \tag{21}\]  
with  
\[I=\frac{\pi\varphi_{1}^{4}}{64}, \tag{22}\]  
where \(E_{p}\) is the hardening modulus of the steel bar, \(E_{p}=\beta_{1}E_{s}\), with \(\beta_{1}=1\) in the elastic range or assumed to be 0.03 at the yielding stage [29]; \(A_{all}\) is the area of longitudinal reinforcement; \(E_{r}\) is the elastic modulus of the corroded steel bar; \(E_{s}\) is the elastic modulus of the steel bar; and \(k_{1}\) denotes the stiffness parameter, for a corner bar, which can be calculated by  
\[A_{1}=\frac{1}{2}\Bigg{[}\theta_{1}\Big{(}\frac{\varphi_{0}}{2}\Big{)}^{2}-a \Bigg{|}\frac{\varphi_{0}}{2}-\frac{p(t)^{2}}{\varphi_{0}}\Bigg{|}\Bigg{]}, \tag{14}\]0  
and for a mid-face bar, by  
\[A_{1}=\frac{1}{2}\Bigg{[}\theta_{1}\Big{(}\frac{\varphi_{0}}{2}\Big{)}^{2}-a \Bigg{|}\frac{\varphi_{0}}{2}-\frac{p(t)^{2}}{\varphi_{0}}\Bigg{|}\Bigg{]}, \tag{14}\]1  
where \(I_{st}\) is the inertia moment of a stirrup, \(I_{st}=\frac{\pi\varphi_{st}^{4}}{64}\); \(A_{st}\) is the area of stirrup, \(A_{st}=\frac{\pi\varphi_{st}^{4}}{4}\); and \(\varphi_{st}\) is the diameter of stirrup. The yield stress attenuation coefficient \(\beta\) of longitudinal reinforcement caused by buckling can be defined as \(\beta=\frac{\sigma_{cr}}{f_{y}}\), and \(\beta\leq 1\).