# Comparison of Damage Indexes for Assessing Seismic Fragility of Bearings in an Offshore Bridge
## 2 Methodology
### Time-Dependent Deterioration of Materials
\[t_{i}t=\left(\frac{c}{K}\right)^{2}\times\ 10^{-6} \tag{7}\]  
\[t_{i}=t_{i}^{\prime}+0.2t_{1} \tag{8}\]  
\[t_{\sigma}=t_{i}+t_{c} \tag{9}\]  
\[t_{c}=\frac{\delta_{cr}}{\lambda_{cl}} \tag{10}\]  
\[f_{yc}=(1-0.339\rho)\cdot f_{y} \tag{11}\]  
\[E=(1-1.166\rho)\cdot E_{s} \tag{12}\]  
where \(t_{i}\) and \(t_{i}^{\prime}\) are the corrosion initiation time for the reinforcement (year), taking into account and disregarding the chloride erosion, respectively; \(t_{1}\) the cumulative duration (year) required for chloride ions to reach a stable concentration on the surface of the concrete; \(t_{cr}\) and \(t_{c}\) are the time (year) for concrete cracking and time (year) from the initiation of steel corrosion to concrete cracking; \(c\) is the concrete cover thickness (mm); \(K\) is the coefficient of chloride penetration (m\({}^{2}\)/year); \(E\) and \(E_{s}\) are the elastic moduli of the corroded and non-corroded reinforcement, respectively (MPa); \(f_{yc}\) and \(f_{y}\) are the yield strengths of the corroded and non-corroded reinforcement, respectively (MPa); \(\delta_{cr}\) is the corrosion depth for the steel when the concrete cracks (mm); and \(\lambda_{cl}\) is the annual average corrosion rate for the steel before the concrete cracks (mm/year), as shown in Tables 2 and 3. The Kent-Park model is a bilinear model that captures the behavior of concrete in both the elastic and plastic ranges. The model is based on the stress-strain relationships for concrete in tension and compression. The model is also able to account for the effects of carbonation on the mechanical properties of concrete. The mechanical properties of the confined concrete are determined by the properties of the unconfined concrete and the reinforcement. The reinforcement provides confinement to the concrete, which increases its strength and ductility. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Time**} & \multicolumn{2}{c}{**Peak Stress (MPa)**} & \multicolumn{2}{c}{**Peak Strain (\(\varepsilon\))**} & \multicolumn{2}{c}{**Ultimate Strain (\(\varepsilon\))**} & \multicolumn{2}{c}{**Elastic Modulus (MPa)**} \\ \cline{2-9}  & **C40** & **C50** & **C40** & **C50** & **C40** & **C50** & **C40** & **C50** \\ \hline
0 & 34.00 & 42.50 & \(-0.002000\) & \(-0.002000\) & \(-0.004000\) & \(-0.004000\) & 32,500.00 & 34,500.00 \\
30 & 34.18 & 42.60 & \(-0.001999\) & \(-0.001999\) & \(-0.003988\) & \(-0.003993\) & 32,609.50 & 34,567.39 \\
50 & 34.21 & 42.63 & \(-0.001998\) & \(-0.001999\) & \(-0.003984\) & \(-0.003991\) & 32,640.59 & 34,587.04 \\
70 & 34.24 & 42.66 & \(-0.001998\) & \(-0.001999\) & \(-0.003981\) & \(-0.003989\) & 32,666.78 & 34,602.83 \\
100 & 34.26 & 42.69 & \(-0.001997\) & \(-0.001999\) & \(-0.003978\) & \(-0.003987\) & 32,699.55 & 34,623.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Time-varying effects on the concrete’s material properties. The time-varying durability analysis of materials is used to predict the degradation of the materials over time. The degradation is caused by a number of factors, including environmental factors such as moisture and chlorides, and loading factors such as cyclic loading. The degradation can lead to a decrease in the strength and durability of the materials, which can in turn lead to a decrease in the performance of the structure. The time-varying durability analysis of materials is an important part of the design of reinforced concrete structures. The analysis can be used to identify the critical areas of the structure and to select the appropriate materials and reinforcement for those areas. Tables 2 and 3 present the time-varying effects on the concrete's material properties and the time-varying effects on the reinforcement's material properties, respectively.