## 3 Framework Development
### Time-Dependent Parameters  
Concrete compressive strength can decrease when affected by the extent of cracking caused by corrosive products reacting on the concrete surface and the steel bar. Despite extensive research, quantifying this loss and identifying the exact degradation region through analytical methods remains a considerable challenge [59; 60]. We adopted the analytical expression from [PERSON] and [PERSON] (1986) [61]. In this analysis, the time-dependent deterioration of the material's properties is primarily driven by the corrosion penetration depth \(\mathrm{d}_{\mathrm{e}}\) (Equation (5)), which evolves as a function of time \(\mathrm{t}\) (Equation (5)), then influencing the crack width \(w_{\mathrm{c}}\) (Equation (4)). The crack width, in turn, determines the expansion of the beam thickness \(b_{f}\) (Equation (3)), which directly impacts the tensile strain \(\varepsilon_{1}\) (Equation (2)). Finally, the modified compressive strength \(f_{ck}\), as shown in Equation (1), is a function of the strain and therefore indirectly dependent on the corrosion penetration depth \(\mathrm{d}_{\mathrm{e}}\), linking it back to time \(t\). Moreover, other mechanical properties, such as those described in Equations (6)-(9), depend on the modified compressive strength \(f_{ck}\), which deteriorates over time as corrosion advances. The time-dependent degradation of \(f_{ck}\), influences parameters such as elastic modulus, tensile strength, and other material characteristics that evolve as corrosion weakens the structural integrity over long periods. \[f_{ck}^{*}=\frac{f_{ck}}{1+k_{r}\frac{\varepsilon_{1}}{\varepsilon_{co}}}, \tag{1}\]  
where \(k_{r}\) (0.1 for medium-diameter ribbed bars) relates to bar roughness and diameter [62], \(\varepsilon_{co}\) (0.002) is the strain at peak compressive stress as provided by the IS 456 [63]; the material is subjected to compression, but \(\varepsilon_{1}\) represents the tensile strain under this compressive load for a cracked material, given by Equation (2). \[\varepsilon_{1}=\frac{b_{f}-b_{o}}{b_{0}}, \tag{2}\]  
where \(b_{0}\) denotes the width of the cross-section in the absence of corrosion cracks, and \(\mathrm{b}_{\mathrm{f}}\) denotes the thickness expanded due to corrosion-induced cracking. The approximate increase in beam thickness can be determined using Equation (3). \[b_{f}-b_{o}=n_{bars}w_{c}, \tag{3}\]  
where \(n_{bars}\) denotes the no. of bars in each layer, and \(w_{c}\) denotes the width of the crack opening corresponding to a specific level of corrosion over the period of time \(\Delta\mathrm{t}\), as expressed by Equation (4). \[w_{c}=2\pi({v_{rs}}-1)d_{c}, \tag{4}\]  
where \(\upsilon_{rs}\) denotes the volumetric expansion ratio of oxides compared to its intact state, typically assumed to be 2 [64]; \(\mathrm{d}_{\mathrm{e}}\) denotes the penetration depth of the corrosion, expressed by the following determination using Equation (5). \[d_{c}=0.0115i_{co}t, \tag{5}\]  
where \(i_{co}\) denotes the corrosion current density per cm/per year, and \(\mathrm{t}\) denotes the time passed since the onset of corrosion propagation (years). The elastic modulus (\(E_{c}\)) of concrete is related to its compressive strength (\(f_{ck}\)) at each lifetime stage, following the theory proposed by [PERSON] and [PERSON] [65] (Equation (6)). \[E_{c}=2.1\times 10^{5}\Big{(}\frac{\gamma}{2.3}\Big{)}^{1.5}\Big{(}\frac{f_{ck}}{ 200}\Big{)}^{0.5}, \tag{6}\]where the modulus of elasticity of concrete (E\({}_{c}\)) is measured in kgf/cm\({}^{2}\), \(\gamma\) denotes the density of concrete (held constant at 2.5 ton/m\({}^{3}\)), and \(f_{ck}\) denotes the compressive strength characteristics, also in kgf/cm\({}^{2}\). The ACI 318 [66], as also mentioned in [67; 68], provides accurate estimates for direct tensile strength of specimens where size effect is negligible.