# Comparison of Damage Indexes for Assessing Seismic Fragility of Bearings in an Offshore Bridge
## 2 Methodology
### Time-Dependent Bond-Slip Behavior  
As we know, the reinforcement provides strength and ductility through their anchorage to concrete. However, during the service life, the reliability of this connection will gradually decrease as the reinforcement and concrete materials degrade. In other words, with an increase throughout the operational lifespan of the bridge, the deterioration of the concrete and reinforcement will cause the mechanical properties to continue to decrease, and a large amount of rust accumulation will be formed in the contact interface between the reinforcement and concrete, reducing the bonding effect of the concrete to the reinforcement [18]. Therefore, the bond-slip is selected as the evaluation index for the bond strength degradation, and the \"Strain Penetration Model for Fully Anchored Steel Reinforcing Bars\" is selected to model the bond-slip behavior [19], see Equation (13) for more details. Figure 1 shows the relationship between the yield strength and corrosion rate of the longitudinal reinforcement and rising rate of the bond-slip for C40 concrete (piers 1\(\#\), 2\(\#\), and 5\(\#\)) during its entire life cycle. \[S_{y}(mm)=2.54\times\left[\frac{d_{b}(mm)}{8437}\frac{f_{y}(MPa)}{\sqrt{f_{c^{ \prime}}(MPa)}}\times(2\alpha+1)\right]^{1/\alpha}+0.34 \tag{13}\]  
where \(f_{y}\) and \(f_{c^{\prime}}\) are the reinforcement yield strength and concrete peak compressive stress, respectively; \(d_{b}\) is the longitudinal reinforcement diameter; \(\alpha\) is the bond-slip coefficient, typically assumed to be 0.4. The command for \"Strain Penetration Model for Fully Anchored Steel Reinforcing Bars\" creates a uniaxial material object to capture strain penetration effects at various intersections, such as column-to-footing, column-to-bridge bent caps, and wall-to-footing connections. In these scenarios, bond-slip occurs along a section of the anchorage length due to strain penetration. The model can also be applied to beam end regions where strain penetration may involve bar slippage along the entire anchorage length, provided appropriate model parameters are chosen. This model specifically addresses fully anchored steel reinforcement bars that experience bond-slip within a portion of the anchorage length due to strain penetration effects. Such effects are commonly observed in situations where column and wall longitudinal bars are anchored into footings or bridge joints. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{ \begin{tabular}{c} **Time** \\ **(Year)** \\ \end{tabular}  } & \multicolumn{3}{c}{**Diameter (mm)**} & \multicolumn{3}{c}{**Yield Strength (MPa)**} & \multicolumn{3}{c}{**Elastic Modulus**} \\ \cline{2-13}  & \multicolumn{3}{c}{**C50**} & \multicolumn{3}{c}{**C40**} & \multicolumn{3}{c}{**C50**} & \multicolumn{3}{c}{**C40**} & \multicolumn{3}{c}{**C50**} & \multicolumn{3}{c}{**C40**} \\ \cline{2-13}  & **L** & **S** & **L** & **S** & **L** & **S** & **L** & **S** & **L** & **S** & **L** & **S** \\ \hline
0 & 32.00 & 16.00 & 32.00 & 16.00 & 335.00 & 235.00 & 335.00 & 235.00 & 2.00 & 2.10 & 2.00 & 2.10 \\
30 & 31.92 & 15.70 & 31.76 & 15 As the service life of the bridge increases, Figure 1 indicates that the yield strength of the longitudinal reinforcement deteriorates with the increase in the corrosion rate of the longitudinal reinforcement. Among them, the yield strengths after 30, 50, 70, and 100 years are reduced by 0.51%, 0.98%, 1.56%, and 2.52%, respectively. Additionally, as the operational lifespan of the structure increases, the bond-slip and its rising rate increase with the corrosion rate of longitudinal reinforcement. Apparently, the rising rate in bond-slip is greatest when the bridge is in service for 0-30 years, since the time for C40 concrete to crack to the surface of the longitudinal reinforcement is 27 years. When the service life of the bridge is 30-50 years, the rising rate decreases gradually.