## 5 Results
### Fragility Curves  
In the previous section, a cyclic pushover analysis was performed to evaluate the cyclic behavior of a reinforced bridge pier with pitting and uniform corrosion. In particular, we only show the fragility curves of the bridge system with pitting corrosion and a corrosion ratio of 20%. The fragility curves under uniform corrosion and pitting corrosion at other corrosion levels were calculated using the same procedure. Based on Equation (3), the probability of failure of the bridge system with a corrosion ratio of 20% was calculated, as presented in Figure 6. Figure 6a indicates that with the increased thickness of the steel plates, the probability of slight damage decreases when the intensity of the mainshock is greater than 0.1 g. However, with the increased thickness of the steel plates, the probability of slight damage decreases slowly when the ground-motion intensity is larger than 1.0 g, and the probability of slight damage almost reaches 1.0 at a larger intensity of ground motion in any scenario. Therefore, the reinforced steel plates are effective in reducing slight damage when the intensity of the ground motion is between 0.1 g and 1.0 g. For the moderate damage state, the same trend can also be found, as shown in Figure 6b. The probability of failure decreased with the thickness of the steel plates when the ground-motion intensity was larger than 0.3 g. The probability of moderate damage increased sharply when ground motions were greater than 0.3 g and smaller than 1.1 g for each case. Figure 6c implies that the probability of major damage decreases with an increase in the thickness of the steel plates. For example, the probability of major damage is approximately 40% for a bridge with 20% corrosion when the ground-motion intensity is 0.8 g, and that of the bridge reinforced by 5 mm steel plates is only approximately 7% under the same intensity of ground motion. Therefore, bonded steel plates are useful for corroded bridge systems to resist mainshock-aftershock sequences. Figure 6d shows that with the increased thickness of the steel plates, the probability of collapse decreases significantly if the intensity of the ground motion is larger than 0.5 g. For example, for an underground motion intensity of 0.8 g, the probability of collapse for the bridge structure with a 20% pitting corrosion ratio is approximately 20%, which is approximately four times that of the bridge structure reinforced with 5 mm steel plates. The findings indicate that steel plates are significantly effective for a bridge system to resist earthquake-load-induced structural collapse. In addition, Figure 6d indicates that the threshold value of the ground-motion intensity that causes collapse increases with the thickness of the steel plates. Therefore, bonded steel plates are useful for enhancing the structural capacity, and reducing the vulnerability of civil structures subjected to mainshock-aftershock sequences. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & \multicolumn{6}{c}{_Fmax (kN)_} \\ \cline{2-8}  & **0.0\%** & **15\%** & **15\%-R1** & **15\%-R2** & **15\%-R3** & **15\%-R4** & **15\%-R5** \\ \hline Pitting & 1321.8 & 1157.3 & 1282.5 & 1433.5 & 1595.0 & 1743.5 & 1905.0 \\ \hline Uniform & 1321.8 & 1213.2 & 1344.5 & 1509.9 & 1656.7 & 1809.5 & 1990.0 \\ \hline Pitting (\%) & 0.00 & \(-\)12.45 & \(-\)2.90 & 8.40 & 20.60 & 31.90 & 44.12 \\ \hline Uniform (\%) & 0.00 & \(-\)8.22 & 1.70 & 14.2 & 25.30 & 36.80 & 50.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Peak force of a bridge pier with 15% corrosion under different thicknesses of steel plates.