## 4 Case Study
### Risk Assessment  
To evaluate the impact of corrosion on seismic risk, this case study examined three locations: Haifa, Jerusalem, and Bikat HaYarden. Using acceleration data from maps provided by the Geophysical Institute of Israel [80], a hazard curve was generated for each location. These hazard curves, representing the annual probability of exceedance for various levels of ground motion, were integral to the risk assessment. They facilitated the estimation of potential seismic impacts on the structure throughout its lifespan, thereby informing the subsequent analysis of how corrosion affects seismic risk. Next, an analysis comparing the risk of the corroded structure against the baseline (non-corroded) condition was conducted. To achieve this, the risk ratio was defined as the ratio of the annual seismic risk of a corroded structure to its baseline risk. The risk ratio, RR(t), at a given time \(t\) is defined as  
\[\text{RR}(t)=\frac{\text{R}_{\text{corroded}}(t)}{\text{R}_{\text{baseline}}(t)} \tag{25}\]  
where  
\(\text{R}_{\text{corroded}}(t)\) is the annual seismic risk of the corroded structure at time t;  
\(\text{R}_{\text{baseline}}(t)\) is the annual seismic risk of the baseline (non-corroded) structure at time t.  
Figure 9: A comparison of fragility curves for an RC structure under different corrosion levels at different time intervals (50, 75, 100, and 120 years). The baseline fragility curves are represented by dashed lines, while the corrosion-adjusted curves are shown in solid lines. This metric illustrates the change in seismic vulnerability due to corrosion. The progression of these ratios was plotted over a 120-year period to capture the long-term effects of corrosion on seismic vulnerability. The baseline risk was derived from the structure's original, non-corroded condition, as outlined in the fragility curves. The influence of corrosion was then integrated into these fragility curves using the degradation factors determined from the previous analysis. Figure 10: Cont. The analysis presented in Figure 11 reveals patterns of risk escalation based on corrosion severity:  
* Minor corrosion: the risk ratio remains close to one throughout the 120-year period across all locations, indicating that low-level corrosion has a minimal impact on seismic risk due to gradual material degradation. * Moderate corrosion: A noticeable increase in the risk ratio begins around 40 years, with the risk ratio doubling by 120 years in both Haifa and Bikat HaYarden, and slightly less pronounced in Jerusalem. This indicates that moderate corrosion leads to significant structural compromise over time. * Intensive corrosion: The risk ratio escalates sharply after 40 years in all locations. This underscores the severe impact of the intensive long-term effect of the corrosion on structural vulnerability. However, it should be noted that the high-risk ratio does not indicate a high risk, but a higher risk relative to baseline. Next, to comprehensively assess the impact of corrosion on the seismic vulnerability of the structure, a cumulative risk ratio analysis was conducted. This analysis involved aggregating the annual risk ratios over a 120-year period. The cumulative risk ratio, therefore, provides a measure of the total risk accrued over the lifecycle of the structure due to corrosion, relative to the baseline condition. The cumulative risk ratio (CRR) is calculated as the ratio of the sum of the risk ratios over the entire period:  
\[\text{CRR}=\frac{\sum_{t=0}^{LC}\text{R}_{\text{corrected}}(\text{t})}{\sum_{t= 0}^{LC}\text{R}_{\text{baseline}}(\text{t})} \tag{26}\]  
Figure 10: Fragility curves for different damage states (DS1 to DS4) under minor (**top**), moderate (**middle**), and intensive (**bottom**) corrosion scenarios over time intervals of 50, 75, 100, and 120 years. As illustrated in Figure 12, cumulative risk ratios are presented for three corrosion levels (minor, moderate, and intensive) across three different locations (Bikat HaYarden, Jerusalem, and Haifa).