## 3 Framework Development
### Integrating Corrosion Effects into Seismic Fragility Curves  
In this step, the degradation of the reinforced concrete (RC) is integrated into seismic vulnerability assessments. This step is based on the degradation of the RC capacity as found in the previous step. In this research, seismic vulnerability is assessed using seismic fragility curves. A seismic fragility function for a structure or a system is mostly formulated as a lognormal cumulative distribution function (CDF) [73; 74]. In order to define this function, two parameters are required to be determined: the median capacity of the component to resist a given damage state \(\theta_{\text{ds}}\) and the standard deviation of this capacity \(\beta_{\text{ds}}\). The form of this function is shown in Equation (19). \[\text{P}[\text{DS}\geq\text{ds}|\text{IM}=\text{x}]=\Phi\bigg{(}\frac{\ln( \text{x}/\theta_{\text{ds}})}{\beta_{\text{ds}}}\bigg{)};\text{ds}\in\{1,2, \ldots\text{N}_{\text{DS}}\} \tag{19}\]  
where P represents the conditional probability of being at or exceeding a particular damage state (DS) for a given seismic intensity, and x is defined by the earthquake intensity measure (IM) where \begin{tabular}{l l} DS & The damage state of a particular component, \(\{0,1,\ldots\text{N}_{\text{DS}}\}\); \\ ds & A particular value of the DS; \\ N\({}_{\text{DS}}\) & The number of possible damage states; \\ IM & Uncertain excitation, the ground-motion-intensity measure (i.e., PGA, PGD, or PGV); \\ X & A particular value of the IM; \\ \(\Phi\) & The standard cumulative normal distribution function; \\ \(\theta_{\text{ds}}\) & The median capacity of the component to resist a damage state ds measured in terms \\  & of the IM; \\ \(\beta_{\text{ds}}\) & The logarithmic standard deviation in the uncertain capacity of the component to \\  & resist a damage state ds. \\ \end{tabular}   
When multiple damage states are defined, they are arranged in order of severity, from the least severe to the most severe. The fragility function then represents the cumulative probability of reaching or exceeding a specific damage state. \[\text{P}(\text{DS}=\text{ds}_{\text{i}}|\text{IM})=\left\{ \begin{array}{cc}1-P (\text{DS}\geq\text{ds}_{\text{i}}|\text{IM})&\text{i}=0\\ P(\text{DS}\geq\text{ds}_{\text{i}}|\text{IM})-P(\text{DS}\geq\text{ds}_{\text {i}+1}|\text{IM})&1\leq\text{i}\leq\text{n}-1\\ P(\text{DS}\geq\text{ds}_{\text{i}}|\text{IM})&\text{i}=\text{n}\end{array} \right. \tag{20}\]  
1. Baseline fragility curve The baseline fragility curves provide a probabilistic measure of the structure's seismic vulnerability without corrosion and degradation consideration. The baseline curves parameters are based on the parameters that are available in the literature and common databases such as [75; 76; 77]. These curves are set as a starting point before the integration degradation effects;  
2. Adjusted Fragility Curve  
In this step, the baseline seismic fragility curves are adjusted to account for the impact of corrosion on the structure;  
3. Adjustment of Median Capacity \(\theta_{\text{ds}_{\text{i}}}\) Due to Corrosion  
The adjustment is made using the degradation factor, denoted as \(\mathfrak{f}_{\text{CL}}(\mathfrak{t})\), which quantifies the reduction in structural capacity due to corrosion relative to the baseline parameters. The degradation factor depends on the corrosion level (CL) and the time (t) elapsed, and it is determined using analyses conducted in prior steps of this research, where the impact of corrosion on structural elements was thoroughly evaluated (as illustrated in Figure 2). The degradation factor \(\mathfrak{f}_{\text{CL}}(\mathfrak{t})\) is applied to the baseline median capacity \(\theta_{\text{ds}_{\text{i}}}\) to yield the adjusted median capacity \(\theta^{\prime}_{\text{ds}_{\text{i}}}(\mathfrak{t})\). \[\theta^{\prime}_{\text{ds}_{\text{i}}}(\mathfrak{t})=\theta_{\text{ds}_{\text {i}}}\cdot\mathfrak{f}_{\text{CL}}(\mathfrak{t}) \tag{21}\]  
where