# Comparison of Damage Indexes for Assessing Seismic Fragility of Bearings in an Offshore Bridge
## 4 Results and Discussion
### Comparison of Various Damage Indexes for Bearings  
We further analyze the characteristics and applicability of the theoretical and empirical method, as well as the impacts of various damage indexes on the performance for bearings throughout the operational lifespan. As representatives, the bearings 1#, 3#, and 5# were selected for comparative analysis of the regression fitting curves for the seismic response, with bearings in service for 0 years as examples. As depicted in Figure 12, a strong linear correlation can be observed between the dynamic response of bearings and PGA. The fitting curves for bearings 1# and 3# are almost identical and are noticeably smaller than the fitting curve for bearing 5#, thus reinforcing the previous observation. Subsequently, a time-dependent fragility analysis was conducted on bearing 5# using different damage indexes throughout the service period, as depicted in Figures 13-15. Figure 12: Regression fitting curves for the seismic response of representative bearings. Figure 13: Time-dependent fragility analysis for bearing 5# from various cases. The probabilities of exceeding the threshold for the bearings in all six cases increase as the service life extends. Throughout the service period, the time-dependent deterioration of materials has a negligible effect on the probabilities subject to minor and medium damage states. However, as the damage worsens, the probabilities show a significant increase as the service period prolongs. Considering the example of the area enclosed by the fragility curve and PGA-axis after 50 years of service, it can be observed that the increase in probabilities is minimal for low damage degrees (minor and medium damage), with an increment of approximately 1%. In contrast, for higher damage degrees, the change in the probabilities is more significant, with an increase of up to 7%. Take the increment for the probabilities subject to minor and serious damage during the service period of 0-50 years as an example (Figures 14 and 15). Focusing on the relative  
Figure 14: Increment for the exceedance probability under minor damage during the service period of 0–50 years for bearing 5#. Figure 15: Increment for exceedance probability under serious damage during the service period of 0–50 years for bearing 5#. displacement, when the degree of damage degree is small and large, the maximum increment for the probabilities obtained by the theoretical method is 168%, and 143% is obtained by the empirical method, respectively. Furthermore, when the relative displacement ductility ratio is considered, the difference in the maximum increment obtained by the theoretical and empirical methods can be ignored when the extent of damage is low. When the extent of damage is significant, the maximum increment of the probabilities using the theoretical method is larger, at approximately 129% of that using the empirical method. Furthermore, when the damage is minor, the maximum increment of the probabilities using the relative displacement ductility ratio (Case 7) is approximately 147% of that obtained by the relative displacement (Case 3), whereas the value is approximately 105% when the damage degree is large. As the damage severity increases, the maximum increment of the exceedance probability shifts towards a higher PGA value. For minor damage, the maximum increment is observed near a PGA of 0.5 g, while for more severe damage, it occurs near a PGA of 0.9 g. It is evident that as the damage severity of the bearing increases, it becomes more sensitive to strong ground motion. Moreover, to further examine the relation between the general cost of the structure and the probability of failure, the overall expenditure is partitioned into construction costs and failure maintenance costs [33]. The construction expenses are considered to be constant, while the failure maintenance cost is determined as the multiplication of the cost of upkeep and the failure probability of the structure. See Equation (23) for details.