# Comparison of Damage Indexes for Assessing Seismic Fragility of Bearings in an Offshore Bridge
## 2 Methodology  
### Time-Dependent Deterioration of Materials  
Reinforced concrete structures consist of concrete and reinforcement, and their performance relies on the properties of both materials. The mechanical behavior of these materials, particularly concrete and reinforcement, must be fully considered using suitable constitutive models. Concrete piers are comprised of both confined and unconstrained concrete. The unconstrained concrete serves as a protective layer for the piers, while the confined concrete in the core area is reinforced by longitudinal reinforcement and stirrups to increase its strength and durability. The constitutive relationship for the concrete adopts the Kent-Park model (Concrete 01) (Kent and Park 1990, [9; 10]), and the mechanical properties of the confined concrete are as shown in Equations (1)-(6) [9; 13; 14]. \[\left\{ \begin{array}{c}x_{c}=k\cdot\sqrt{t}\\ \overline{S}=\frac{A}{A}\end{array} \right. \tag{1}\]  
\[\sigma_{cp}=(1+0.619\cdot\overline{S})\cdot\sigma_{p} \tag{2}\]  
\[\varepsilon_{cp}=(1-0.106\cdot\overline{S})\cdot\varepsilon_{p} \tag{3}\]  
\[\varepsilon_{cu}=(1-0.459\cdot\overline{S})\cdot\varepsilon_{u} \tag{4}\]  
\[E_{c}=(1+0.503\cdot\overline{S})\cdot E \tag{5}\]  
\[G_{c}=\frac{E_{c}}{2\cdot(1+\mu)} \tag{6}\]  
where \(k\), \(\overline{S}\), and \(t\) are the carbonation coefficient, carbonation rate, and carbonation time, respectively. \(A_{c}\) is the area of carbonation. \(A\) is the area of total cross-sectional, \(x_{c}\) is the depth of carbonation, \(\sigma_{p}\), \(\varepsilon_{p}\), \(\varepsilon_{u}\), and \(E\) are peak stress, peak strain, ultimate strain, and elastic modulus of concrete before carbonation, respectively, \(\sigma_{cp}\), \(\varepsilon_{cp}\), \(\varepsilon_{cu}\), \(E_{c}\), and \(G_{c}\) are peak stress, peak strain, ultimate strain, elastic modulus, and shear modulus of carbonated concrete, respectively, and \(\mu\) is Poisson's ratio. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Theme** & **Year** & **Main Contribution** \\ \hline \multirow{8}{*}{Assessment of the seismic performance} & 2012 & [PERSON], [PERSON] [5] proposed damage indexes for elastomeric pads and sliding bearings. \\  & 2000, 2001, and 2005 & [PERSON], [PERSON] [6], [PERSON], [PERSON] [2], and [PERSON] [3] proposed a fragility analysis theory based on traditional reliability theory. \\  & 2014 & [PERSON] and [PERSON] [7] analyzed the fragility curves for bearings along two horizontal directions. \\  & 2015 & [PERSON] and [PERSON] [8] conducted a seismic fragility analysis of the movable steel bearings. \\  & 2020 and 2021 & [PERSON], [PERSON] [9] found that material deterioration leads to an increase in the seismic response and failure probability of bearings. \\  & & [PERSON] and [PERSON] [11] discovered that the peak deformation of bearings increases significantly when an aging factor is taken into account. \\  & 2018 & [PERSON] and [PERSON] [12] highlighted the importance of realistic bearing degradation models for bridges. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. Steel 02 possesses an isotropic strain hardening property and accurately represents both the bi-directional Bauschinger effect and isotropic strengthening effect. In the works of [PERSON], [PERSON] [10], [PERSON], [PERSON] [15], and [PERSON] and [PERSON] [16], the calculations for the corrosion initiation time of the reinforcement, chloride ion concentration on the concrete surface, yield strength, and elastic modulus of reinforcement are provided as shown in Equations (7)-(12). For more details, the reader can refer to [PERSON], [PERSON] [17], and [PERSON], [PERSON] [10].