# Earthquake Damage Index and Fragility Analysis of Steel Damper for Seismic Isolation Bridge
## 3 Steel Damper Damage Indexes
### Evaluation Based on Maximum Deformation Index  
The unidirectional maximum displacement capacity of steel dampers is directly related to the ultimate strain of the steel. [PERSON] [23] established a complete stress-strain model by collecting stress-strain curves from 141 sets of tensile tests on carbon steel, and statistically regressed the Equation for the ultimate strain of steel, as shown in Equation (6).  
\[\varepsilon_{u}=0.2416\varepsilon_{0.2}e^{0.0096/\eta} \tag{6}\]  
In the regression equation, \(\varepsilon_{0.2}\) is the strain value at \(\sigma_{0.2}\), \(\sigma_{0.2}\) is the stress value corresponding to the residual strain of 0.2%, \(\varepsilon\) is the ratio of \(\sigma_{0.2}\) to \(E_{0}\), where \(E_{0}\) is the initial elastic modulus, and the recommended range for \(e\) is 0.0017\(-\)0.0035. Thus, the ultimate strain of steel \(\varepsilon_{u}\) is determined to be in the range of 0.008-0.14. Since more ductile steel is preferred for the damper design, the conservative value of the ultimate strain is taken as 0.1.  
Like the CIS-based evaluation, the performance state of steel dampers is divided into five levels when evaluated by the maximum deformation index. Table 4 shows the detailed threshold values and their corresponding performance state descriptions according to the yield condition of the steel and the design requirements at the Chinese standard level [22], as well as the experimental phenomena. Here, \(\varepsilon_{1}\) is the yield strain of general steel, \(\varepsilon_{2}\) is the strain value corresponding to 1.0 times the design displacement of the steel damper, \(\varepsilon_{3}\) is the strain value corresponding to 1.2 times the design displacement of the steel damper, and \(\varepsilon_{4}\) is the ultimate strain of the steel.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Threshold Values** & **Performance Level** & **Performance Status** \\ \hline \(\varepsilon_{1}=0.002\) & 1 & The damper is in an elastic state. \\ \hline \(\varepsilon_{2}=0.039\) & 2 & The damper has little damage, and part of the section enters the yield state. \\ \hline \(\varepsilon_{3}=0.049\) & 3 & The damper enters a stronger nonlinear state, and more sections enter yield. \\ \hline \(\varepsilon_{4}=0.1\) & 4 & The damper enters a strongly nonlinear state, and the material enters the necking phase. \\ \hline \(\varepsilon_{4}=0.1\) & 4 & The damper enters a strongly nonlinear state, and the material enters the necking phase. \\ \hline \(\varepsilon_{4}=0.1\) & 5 & The maximum strain of the damper exceeds the ultimate strain, and the element fails. \\ \hline \hline \end{tabular}
\end{table}
Table 4: Damage limit value of damper under the maximum deformation effect.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Threshold Values** & **Performance Level** & **Performance Status** \\ \hline CIS\({}_{1}=0\) & 1 & The damper is in an elastic state. \\ \hline CIS\({}_{2}=0.81\) & 2 & The damper enters a larger deformation stage, and the residual deformation at the end of loading is not apparent. \\ \hline CIS\({}_{3}=1.38\) & 3 & The damper fully enters the plastic state, the residual deformation is significant, and there are tiny cracks that do not significantly influence the hysteresis curve. \\ \hline CIS\({}_{4}=3.49\) & 4 & Small cracks in the damper gradually develop, and the damping force value decreases significantly when each cycle is loaded. Besides, cracks begin to appear in other places in the energy consumption area. \\ \hline \(\varepsilon_{4}=0.1\) & 5 & The damper breaks at the most severely developed crack, the damping force drops significantly, and the component fails. \\ \hline \hline \end{tabular}
\end{table}
Table 3: Damage threshold value of steel damper under the cumulative hysteresis effect.