# Earthquake Damage Index and Fragility Analysis of Steel Damper for Seismic Isolation Bridge
## 3 Steel Damper Damage Indexes
### Evaluation Using CIS Indexes  
Equation (5) gives the corresponding calculation method of cumulative inelastic strain (CIS) for various situations in the reciprocating loading process. \[\left(\Delta\varepsilon_{p}\right)_{n}=\left\{ \begin{array}{c|c}0&|\delta_{n }^{+}|\text{ and }|\delta_{n}^{-}|\leq\delta_{y}\\ |g|\delta_{n}^{+}|-g|\delta_{n}^{-}|\text{ }|&|\delta_{n}^{+}|\text{ and }|\delta_{n}^{-}|> \delta_{y}\text{ and }\delta_{n}^{+}\cdot\delta_{n}^{-}>0\\ g(\max(|\delta_{n}^{+}|,|\delta_{n}^{-}|))-\varepsilon_{y}&|\delta_{n}^{+}| \text{ or }|\delta_{n}^{-}|>\delta_{y}\\ |g|\delta_{n}^{+}|+g|\delta_{n}^{-}||-2\varepsilon_{y}&|\delta_{n}^{+}|\text{ and }|\delta_{n}^{-}|>\delta_{y}\text{ and }\delta_{n}^{+}\cdot\delta_{n}^{-}<0\end{array} \right. \tag{5}\]  
where \(g(\delta)\) is a function of strain and peak displacement, \((\Delta\varepsilon_{p})_{n}\) is the CIS of the damper at the nth cycle, \(\delta_{n}^{+}\) is the peak displacement at the nth cycle, and \(\delta_{n}^{-}\) is the trough displacement at the nth cycle. Installation gaps in the experimental setup and the expansion and contraction of relevant components during the loading process can cause a specific deviation between the actual deformation value of the steel damper and the target loading value. Based on the experimental measurement data, the CIS values of the two types of steel dampers were calculated using Equation (5) according to the actual deformation in the dampers. The relevant parameters are shown in Table 2.  
As shown in Table 2, as the loading rate decreases, the steel dampers can withstand more cycles and have a larger CIS value. On the other hand, the dampers can withstand more cycles and have a larger CIS value for steel with a higher elongation rate under similar loading speeds. Therefore, the ductility of the steel should be further considered when evaluating the damage of dampers under the mentioned deformation level. The index of elongation can reflect this capacity to some extent. The ratio of the CIS to the steel elongation is listed in Table 2. It can be seen that the ratio (CIS/\(\lambda\)) of the #1 cylindrical damper and #2 E-shaped damper with faster loading speed is 11.4, while this ratio is close to 14 for the #2 cylindrical damper and #1 E-shaped damper with the slower loading speed. The relationship between strain displacement may differ for different steel dampers, while the corresponding CIS values should be essentially the same for the same ductile capacity of the steel when it is destroyed. The relative movement speed between the pier and the girder under common earthquakes is generally higher than two mm/s or four mm/s. Usually, it ranges from 10 mm/s to 1000 mm/s, and the actual elongation rate of common steel is about 30%. The CIS value is chosen as 3.49. When the cumulative damage exceeds this threshold, it is considered that the damper has suffered significant damage or failure. According to the yield state of steel and the performance state of the damper in operation, combined with the observed conditions in the experiment and the experimental \begin{table}
\begin{tabular}{c c c c c} \hline
**Specimen** & **Maximum Loading** & **Cumulative Inelastic** & **Steel Elongation** & **CIS/\(\lambda\)** \\
**Number** & **Rate (mm/s)** & **Strain (CIS)** & \(\lambda\) **(\%)** & **CIS/\(\lambda\)** \\ \hline Cylindrical \#1 & 250 & 3.95 & 34.5 & 11.4 \\ Cylindrical \#2 & 60 & 4.80 & 34.5 & 13.9 \\ E-shaped \#1 & 10 & 4.13 & 30.5 & 13.5 \\ E-shaped \#2 & 40 & 3.49 & 30.5 & 11.4 \\ \hline \end{tabular}
\end{table} 
Table 2: Steel damper experimental data. results, the steel damper is divided into five performance levels, where their corresponding CIS values are shown in Table 3. The cumulative inelastic strain at the onset of yielding is defined as CIS\({}_{1}\). In contrast, the cumulative inelastic strain after the steel damper completes the loading steps (Figure 8) specified in the Chinese standard [22] is defined as CIS\({}_{2}\).