## 3 Framework Development
### Internal Pressure (P)  
Various models are available in the literature for predicting rust production. [PERSON] et al. (1993) [71] introduced a linear relationship integrating [PERSON]'s law, correlating the current density of corrosion (i\({}_{corr}\) in \(\mu\)A/cm\({}^{2}\)) to the decrease in the steel bar diameter. A conversion factor of 0.023 (from \(\mu\)A/cm\({}^{2}\) to mm per year) is used:  
\[D_{rb}=D_{b}-0.023i_{corr}\Delta\text{t}(\text{mm}) \tag{10}\]  
where, \(D_{rb}\), denotes the residual steel diameter, \(D_{b}\) denotes the initial steel diameter, and \(\Delta\text{t}\) denotes the time since the onset of corrosion. \[\Delta\text{V}_{s}=0.023\times 2\pi D_{b}i_{corr}\Delta\text{t}\Big{(}\text{mm }^{3}/\text{mm}\Big{)} \tag{11}\]  
Assuming a uniform corrosion process, the radius decreases from \(R_{b}\) to \(R_{rb}\) as a result of corrosion (as described in Equation (12)). This decrease has been calculated based on the volume of consumed steel per unit length of anodic steel, \(\Delta\text{V}_{s}\) (as detailed in Equation (11)). \[R_{rb}=\sqrt{(R_{b}^{2}-\
icefrac{{\Delta\text{V}_{s}}}{{\pi}}(\text{mm})} \tag{12}\]  
The steel radius, including the rust layer, is given as follows:  
\[R_{r}=R_{rb}+\delta_{o}(\text{mm}) \tag{13}\]  
where \(\delta_{o}\) denotes the thickness of the oxide layer around the steel surface. The total volume of oxide produced is expressed as \(\Delta V_{r}=\Delta V_{g}\rho_{s}/\left(\rho_{r}r_{m}\right)\), and the thickness t\({}_{r}\) can be calculated as follows:  
\[t_{r}=\sqrt{\left(R_{rb}^{2}+\
icefrac{{\Delta V_{r}}}{{\pi}}-R_{rb}(\text{ mm})\right.} \tag{14}\]  
By applying an equivalent pressure around the steel, the expansion of the corroding reinforcement is simulated. The first step in this process is to calculate the effective, non-dimensional mass loss:  
\[\gamma=\frac{\frac{\left(D_{b}+\delta_{o}\times 2\right)^{2}}{D_{b}^{2}}-1}{ \beta-1} \tag{15}\]The strain resulting from the unrestrained expansion of the reinforcement can be determined as follows:  
\[\varepsilon_{s,free}=\sqrt{1+c(b-1)}-1 \tag{16}\]  
This represents the average deformation of the corrosion area relative to the original steel diameter and changes with the corrosion levels. The confinement effect of the concrete on the steel generates internal stress, which is calculated by multiplying the average stiffness by the strain of the corroded system. To determine the average stiffness, it can be based on the volume fractions of both the steel and the layers of corrosion:  
\[E_{s,eq}=\frac{1+c(b-1)}{\frac{1-c}{E_{s}}+\frac{cb}{E_{o}}}(\text{GPa}) \tag{17}\]  
where \(E_{s}\) denotes the modulus of elasticity of steel (200 GPa), and \(E_{0}\) refers to the modulus of the elasticity of the oxide in this work taken as (7 GPa). The internal pressure (P) denotes the confinement effect of the concrete on steel, dependent on the degree of corrosion, calculated in the following equation:  
\[P=E_{s,eq}\times\varepsilon_{s,free} \tag{18}\]  
The mechanical parameters throughout the structure's design life cycle, calculated using Equations (1)-(17), are presented in Figure 4.  
In Figure 4, we show the mechanical parameters that govern the degradation of the structure over time due to corrosion, recalculated every 10 years to reflect increasing  
\[\Delta\text{V}_{s}=0.023\times 2\pi D_{b}i_{corr}\Delta\text{t}\Big{(}\text{mm }^{3}/\text{mm}\Big{)} \tag{11}\]0 Input mechanical parameters for discrete simulations at various stages along structure’s design lifecycle: (**a**) modulus of elasticity, (**b**) cohesive strength, (**c**) tensile stress, and (**d**) friction angle. corrosion severity. After 20 years, corrosion becomes active, significantly decreasing the structure's ability to resist stresses. The modulus of elasticity (Figure 4a) decreases over time, particularly under moderate and intensive corrosion, leading to a loss in stiffness and increased vulnerability to deformation under seismic shear forces.