# 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
\[C_{B}=C_{c}+C_{fm}\cdot P_{f} \tag{23}\]  
where \(P_{f}\) is the probability of failure for the structure, \(C_{B}\), \(C_{c}\), and \(C_{fm}\) are the overall expenditure, construction expenses, and cost of upkeep, respectively. In simpler terms, the exceedance probability plays a crucial role in determining the balance between the economy and safety of the structure's seismic design. In summary, if security is more important, a damage index calculation method with a higher increment of exceedance probability should be selected, while, if the economy is the focus, the damage index calculation method with a smaller increment of exceedance probability should be selected. See Table 9 for more details. Although the exceedance probability of the bearing exhibits the characteristic of continuous increase with the prolongation of service time, the increment is relatively small. Merely examining the change rate of the failure probabilities throughout the operational lifespan of the structure is insufficient for assessing the suitability of the damage index. Figure 16 illustrates the comparison of time-varying fragility for different cases, using bearing 5# as an example after 50 years of service. This analysis aims to further enhance the applicability of various damage indexes. Figure 16 demonstrates a positive correlation between the failure probabilities of the bearing and the PGA. Throughout the operational lifespan of the structure, the exceedance probability for the bearing using the relative displacement is greater than that using the relative displacement ductility ratio, and the difference between them increases with an increase in the damage degree. As an example, the enclosed region between the curve and the PGA-axis calculated by the relative displacement (Case 1) are approximately 120%, 128%, 131%, and 151% larger than those using the relative displacement ductility ratio (Case 5) for the four damage states, respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & **When the Extent of Damage Is Low** & **When the Extent of Damage Is Significant** \\  & **(LS1 and LS2)** & **(LS3 and LS4)** \\ \hline Security & Case 5, Case 6, and Case 7 & Case 2 and Case 6 \\ Economy & Case 1 & Case 1 and Case 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Application of damage index (by the increment of exceedance probability). Focusing on relative displacement, irrespective of the extent of damage, the exceedance probability derived from the empirical method (Case 1) is always greater than that obtained based on the theoretical method (Case 2). Moreover, the failure probability calculated using the theoretical method (Case 2) is even lower compared to the probability obtained by averaging the results of the two methods (Case 3). When the extent of damage is low (minor damage), specifically considering the relative displacement ductility ratio, the probability obtained by empirical and theoretical methods is basically equal. When the damage is slightly aggravated (medium damage), it shows the same trend as the relative displacement, and the probability obtained by the empirical method (Case 5) is greater. When the damage degree continues to increase, the exceedance probability obtained by the empirical method (Case 5) is also greater, but the probability obtained by the average value (Case 7) of empirical and theoretical methods is closer to that obtained by the empirical method. Table 10 presents the discussion for the damage indexes using the ranking of the fragility curves throughout the operational lifespan of the bridge. Tables 9 and 10 reveal that the classification of the damage indexes for bearings based on the order of the exceedance probability and that from the ranking order of its increment are different. That is, the magnitude of the exceedance probability of the bearing itself is not related to its increment throughout the operational lifespan.