## 4 Performance-Based Fragility Curves
Based on the proposed fragility curves as illustrated in Figures 9 and 10, the vulnerability assessment can be predicted as in the example in Table 5 for five different expected seismic levels with a structure of 1.5% damping located on soil S3. The values presented in Table 5 are the probabilities of exceeding damage levels for 200, 500, 1000, 2400, and 4800 average years of return period for the earthquake. These are equivalent to a 10% probability of occurrence within 20, 50, 100, 250, and 500 years of the life span of the structure, as recommended in the KBC. Table 5 can serve as a set of recommendations for port owners who wish to select a suitable level of risk for their existing cranes, and an appropriate retrofit plan if the cranes suffer an earthquake event in the future. For example, if the owners expect a minor performance level with seismic Level I for their existing cranes at a Korean port [23], and the selected return period of an earthquake is 2400 years, then the probability of exceeding minor damage to the crane structures or facilities is approximately 12.4% under the average spectral acceleration \(S_{a}\) of 0.3\(\times\)\(g\). Simultaneously, the probability of exceeding an uplift is approximately 42.5%. On the other hand, assuming that the earthquake has a return period of 1000 years, the probability of exceeding the minor damage of the structure is just only approximately 0.6%. However, business operations can be interrupted towing to the 7.5% probability of uplift with derailment, because of the operational downtime required for jacking and repositioning the wheels of the crane back to the rail. Figure 11 illustrates the fragility curves that were plotted with four different natural periods of the crane to serve as a preliminary design, and to estimate the probability of exceedance for an expected limit state. Based on Figure 11, designers or port owners can predict the probability of exceeding the limit states of container cranes by calculating a linear interpolation of the fundamental periods ranging from 0.5 to 2.0 s. Table 6 lists the parameters of these fragility curve models. It is assumed that the standard deviation of \(\ln(S_{a})\) is similar to those obtained from Section 4 (see Figure 9 as well). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Verance Levels** & **200** & **500** & **1000** & **2400** & **4800** \\ \hline \hline
**Uplift coupled with/without derailment** & \(\approx\)0.0 & 0.4 & 7.5 & 42.5 & 74.9 \\ Minor & \(\approx\)0.0 & \(\approx\)0.0 & 0.6 & 12.4 & 40.5 \\ Moderate & \(\approx\)0.0 & \(\approx\)0.0 & 0.3 & 7.7 & 30.1 \\ Extensive & \(\approx\)0.0 & \(\approx\)0.0 & 0.1 & 4.7 & 21.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Example of probability of exceeding damage levels (unit %) for five seismic levels (soil S3) of Korean port design requirements. Figure 10: Fragility curves for uplift, minor, moderate, and extensive limit states. Figure 11: Fragility curves with different periods of container crane for (**a**) minor damage state; (**b**) moderate damage state; and (**c**) extensive damage state. \begin{table}
\begin{tabular}{c l c c c} \hline \hline
**T (s)** & \multicolumn{2}{c}{**Parameters of Fragility Curve**} & **IO** & **LS** & **CP** \\ \hline \multirow{2}{*}{0.5} & Median & 1.277 & 1.364 & 1.459 \\  & Standard Deviation & 0.232 & 0.235 & 0.240 \\ \hline \multirow{2}{*}{1.0} & Median & 0.657 & 0.702 & 0.751 \\  & Standard Deviation & 0.232 & 0.235 & 0.240 \\ \hline \multirow{2}{*}{1.5} & Median & 0.426 & 0.455 & 0.487 \\  & Standard Deviation & 0.232 & 0.235 & 0.240 \\ \hline \multirow{2}{*}{2.0} & Median & 0.320 & 0.342 & 0.365 \\  & Standard Deviation & 0.232 & 0.235 & 0.240 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Median and standard deviation of \(\ln(S_{a})\).