# Earthquake Damage Index and Fragility Analysis of Steel Damper for Seismic Isolation Bridge
## 3 Steel Damper Damage Indexes  
### Relationship between Strain and Displacement  
After choosing the strain-based damage assessment, since the displacement response is easier to obtain than the strain response in practical engineering, the quantitative relationship between the strain and displacement of the steel dampers should be determined. The theoretical relationships between the displacement and strain of the cylindrical and E-shaped steels are obtained through a plane section assumption, as shown in Equation (1) [4] and Equation (2) [21].  
\[\delta_{\text{max}}=\frac{6}{5}\varepsilon_{\text{max}}\bigg{(}\frac{32F}{\pi \sigma_{y}}\bigg{)}^{-1/3}\bigg{|}^{5/3} \tag{1}\]  
Figure 4: Damage phenomenon of steel dampers. (**a**) Cylindrical #1 (**b**) Cylindrical #2 (**c**) E-shape #1.  
Figure 5: Force-displacement curves of steel dampers. (**a**) Cylindrical #1 (**b**) Cylindrical #2 (**c**) E-shape #1 (**d**) E-shape #2.  
where \(F\) is the damping force, \(\delta\) is the displacement of the steel damper, \(l\) is the height of the steel damping element, \(\epsilon_{\text{max}}\) is the maximum strain at the bending side surface of the steel damper, and \(\sigma_{y}\) is the steel's yield strength.  
\[\delta_{\text{max}}=2\frac{hl}{b}\epsilon_{\text{max}}(1+\frac{\beta}{2}\frac{h} {l}\frac{\varepsilon_{y}}{\epsilon_{\text{max}}}) \tag{2}\]  
where \(\beta=2(a/a_{1})^{3}+(a/a_{2})^{3}\), \(\delta_{y}\) is the yield displacement of the steel damper, \(h\) is the height of the middle limb, \(l\) is the length of the cross beam on one side, \(a\), \(a_{1}\), and \(a_{2}\) are the widths of the cross beam, middle limb, and side limb, respectively, \(\varepsilon_{y}\) is the steel's yield strain, and \(\varepsilon_{\text{max}}\) is the maximum strain on the steel damper's surface, as shown Figure 6.  
Theoretical equations only apply to the steel dampers' state before entering the initial yield. After significant yielding of the steel damper, especially after entering large deformation, there is a nonlinear relationship between its corresponding displacement and strain, which can be calculated through numerical models. Figure 6 shows the strain-displacement curves of two types of steel damper obtained by theoretical equations and numerical models, respectively, where the strain values extracted by numerical models are the maximum strain response values in the damper energy dissipation region. As shown in Figure 7, Equations (1) and (2) underestimate the strain response of the damper to varying degrees, leading to an overestimation of the number of horizontal reciprocating loadings that the damper can withstand when using theoretical equations for design. Figure 6 also includes the experimental measurement values of the strain response, which began to fail close to 10,000 microstrains due to factors such as strain gauge adhesion. However, the experimental measurement values were compatible with both the approximate and numerical solutions of theoretical equations.  
Figure 6: Schematic diagram of the E-shaped steel damper.  
Figure 7: Relationship between strain and displacement of the steel damper. (**a**) Cylindrical (**b**) E-shape.  
The strain-displacement curves of the cylindrical and E-shaped steel dampers obtained through numerical modeling in Figure 6 were fitted. Equations (3) and (4) establish the relationship between the two.  
\[\varepsilon_{\text{C}}=0.129\delta^{2}+0.08981\delta+0.00003174 \tag{3}\]  
\[\varepsilon_{\text{E}}=0.1738\delta^{2}+0.1122\delta-0.001506 \tag{4}\]