## 2 Derivation of Functionality Loss Evaluation Model for Railway Network Components
### 2.2. Establishment of Functionality Loss Evaluation Models for Railway Network Components
#### 2.2.2 Derivation of Restoration Curves for Railway Network Components Based on Damage States  
Restoration curves evaluate functionality degradation and recovery times for structures based on their damage states during earthquakes. Restoration models are calculated using the mean and standard deviation of expected recovery times for each damage state of the structures, with Federal Emergency Management Agency [19] values applied for these parameters. Recovery times and standard deviations for bridges, embankments, and tunnels based on damage states are presented in Table 4, and these were used to derive restoration curves. The restoration curves were derived using the mean and standard deviation of recovery times, assuming a normal distribution of recovery times and by calculating probabilities corresponding to specific periods from the cumulative distribution function. The derived restoration curves for railway network components are shown in Figure 5. According to the model, it takes approximately 4 days, 15 days, 261 days, and 716 days for railway bridges to fully recover their functionality after slight, moderate, extensive, and complete damage states, respectively. For embankments, it takes approximately 2 days, 17 days, and 73 days to recover from slight, moderate, and extensive damage states, respectively, while for tunnels, the recovery times are approximately 2 days, 18 days, and 170 days for the respective damage states. The formula for deriving the seismic fragility function is presented in Equation (2). \[\mathrm{Q}[\mathrm{ds}|\mathrm{t}]=\Phi\bigg{(}\frac{\mathrm{lnY-\gamma}}{ \theta}\bigg{)} \tag{2}\]  
where Y represents the time required for recovery, \(\gamma\) denotes the median (day), and \(\theta\) represents the standard deviation (days). 2.3 Derivation of Functionality Curves for Railway Network Components Based on Seismic Damage States  
To develop functionality loss curves for railway network components based on damage states, functionality curves must first be derived. Functionality curves are derived as expected values by combining the probability of damage occurrence for each damage state and the degree of functionality degradation over time when the damage occurs, \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**(a) Railway Bridge**} \\ \hline
**Damage State** & **Mean (day)** & **Standard Deviation \(\sigma\) (days)** \\ \hline Slight & 0.6 & 0.6 \\ Moderate & 2.5 & 2.7 \\ Extensive & 75 & 42 \\ Complete & 230 & 110 \\ \hline \multicolumn{3}{c}{**(b) Railway Embankment**} \\ \hline
**Damage State** & **Recovery Time (day)** & **Standard Deviation \(\sigma\) (days)** \\ \hline Slight & 0.9 & 0.07 \\ Moderate & 3.3 & 3 \\ Extensive/Complete & 40 & 29 \\ \hline \multicolumn{3}{c}{**(c) Railway Tunnel**} \\ \hline
**Damage State** & **Recovery Time (day)** & **Standard Deviation \(\sigma\) (days)** \\ \hline Slight & 0.5 & 0.3 \\ Moderate & 2.4 & 2 \\ Extensive/Complete & 45 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Recovery duration for railway structure [19]. representing the predicted functionality recovery of the structure over time. When there is no functionality loss, the value is 1, while complete functionality loss corresponds to a value of 0. The functionality curves can be calculated using Equation (3). \[Q(t)=\sum_{i=0}^{4}Q[ds_{i}|t]P[ds_{i}|IM] \tag{3}\]  
where \(Q[ds_{i}|t]\) represents the functionality level of a structure over time for each damage state, \(P[ds_{i}|IM]\) represents the probability of each damage state occurring based on earthquake intensity, and \(ds\) represents the damage states. Damage states are categorized into five levels: no damage (\(ds_{0}\)), slight damage (\(ds_{1}\)), moderate damage (\(ds_{2}\)), extensive damage (\(ds_{3}\)), and complete collapse (\(ds_{4}\)). \(P[ds_{i}|IM]\) can be calculated using Equations (4)-(6).