# Earthquake Damage Index and Fragility Analysis of Steel Damper for Seismic Isolation Bridge
## 1 Introduction  
Using steel dampers has been proven to be one of the most effective methods for protecting bridges from earthquakes. Due to their stable performance, low maintenance costs, and high predictability, steel dampers have been widely utilized in bridge design for seismic isolation [1]. Various types of steel dampers, such as X-shaped [2], triangular [3], tapered [4], C-shaped [5], S-shaped [6], and E-shaped [7], have been developed and improved over time. The energy dissipation of metal dampers mainly depends on the non-elastic deformation of the metal, which may lead to fatigue damage after multiple cycles in an earthquake. In particular, strong aftershocks may occur after large earthquakes. For example, Kyushu Island experienced three strong tremps within three days of the 2016 Kumamoto earthquake [7, 8]. Therefore, steel dampers should have excellent low-cycle fatigue performance, and scholars have paid increasing attention to their damage assessment. The damage process of steel dampers is essentially the accumulation and development of the produced plastic deformation. Evaluating the damage level of steel dampers solely based on the number of cycles in the hysteresis curve is empirical and cannot directly reflect the damage severity. In order to solve this issue, [PERSON] [9] defined a damage index for steel dampers in steel frame structures, considering both the cumulative energy dissipation and maximum deformation effects of the component. [PERSON] [10] defined a strain level-based damage index for steel damper bearing according to the combined effects of cumulative energy dissipation and maximum deformation in the Park-Ang model [11]. [PERSON] [12] evaluated the low-cycle fatigue performance of triangular steel dampers employed in bridge design under different earthquake excitations using the shaking table test of the Sutong Bridge as a background based on the improved strain counting method, cumulative inelastic strain method, and equivalent hysteresis energy method. However, these three methods did not consider the non-cumulative circulatory damage generated by the steel damper when it undergoes unidirectional maximum deformation. Generally, the low-cycle fatigue damage of steel dampers under earthquake is recognized to be the combined effect of cumulative cyclic and maximum deformation effects. The required cumulative inelastic strain (CIS) value generally exceeds 0.7 [13; 14], and the ratio of the area enclosed by the hysteresis curve to that enclosed by the hysteresis curve at the design displacement should exceed 30 [15]. The fragility analysis can effectively evaluate the reliability of bridge components under earthquakes, and some scholars have utilized it to assess the performance of devices in seismically isolated design bridges [16; 17]. [PERSON] [18] adopted a steel continuous girder bridge as an example and demonstrated the effectiveness of viscous damping devices in promoting the bridges' seismic performance by forming fragility curves. [PERSON] [19] utilized a continuous girder bridge with lead-rubber bearings as an example and discussed the seismic performance of bridge structures with four devices, including elastic-plastic cables, viscous damping devices, frictional damping devices, and shape memory alloys. [PERSON] [20] employed a continuous girder bridge to analyze the fragility of non-isolated and seismically isolated design bridges. However, the damaged state grading of friction pendulum bearings was relatively coarse, and further refinement and improvement of quantitative indicators are still required. Scholars have not well-developed the corresponding evaluation system for seismic damage of seismic isolation devices while analyzing their seismic performance, and the performance classification is also relatively coarse. Moreover, there are few studies on the fragility analysis of steel dampers in seismic isolation bridges.