# Seismic Vulnerability Assessment of a Shallow Two-Story Underground RC Box Structure
## 3 Fragility Curve Development
Figure 8 shows separately the three fragility curves built from the data of three damage states obtained after the tunnel sustained artificial ground motions, which are indicated by the white circle (for minor), triangular (for moderate), and diamond (for extensive) shapes. A group of ten earthquake PGAs were applied to calculate the three damage states including the minor (PGA from 0.02 g to 0.23 g), moderate (PGA from 0.18 g to 0.7 g), and extensive (PGA from 0.4 g to 1.3 g) states, in which each PGA level contains 50 different artificial earthquakes. As a result, for the concentrated load condition, the median values of PGA (50% probability of exceedance) were 0.103 g, 0.330 g, and 0.806 g for minor, moderate, and extensive damage states respectively, while they were 0.116 g, 0.328 g, and 0.818 g for the distributed load condition, respectively. In addition, considering the influence of the degree of earthquake intensity increment, there was a significant difference between both pushover models in the minor state. For example, at a PGA of 0.1 g, only 9 minor damage cases happened after sustaining 50 earth shakes under the distributed load case, whereas 23 minor damage cases were expected by the hypothesis of the shear force acting on the tunnel roof. On the other hand, the number of collapse increased significantly with a slight increase in the seismic PGA, especially in the minor and moderate damage states. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**PGA (g)**} & \multicolumn{3}{c}{**Concentrated**} & \multicolumn{3}{c}{**Distributed**} \\ \cline{2-7}  & **Minor** & **Moderate** & **Extensive** & **Minor** & **Moderate** & **Extensive** \\ \hline  
\hline \(\theta\) & 0.103 & 0.330 & 0.806 & 0.116 & 0.328 & 0.818 \\ \(\beta\) & 0.164 & 0.157 & 0.158 & 0.161 & 0.161 & 0.157 \\ \hline \hline \end{tabular} Note: \(\theta\), \(\beta\): median and standard deviation of ln(PGA).  
\end{table} 
Table 4: Number of failures per each PGA level. Figure 8: _Cont._To compare two hypotheses of concentrated load and invert distributed load, the data from Table 4 and the fitted fragility curves from Figure 8 are broken down into three Figure 9a-c. It is clearly seen that the shapes of fragility curves of both hypotheses are similar. However, at the same PGA level, the probability of exceeding the damage state is larger with the point load comparing to the distributed load, except for moderate damage state, it showed nearly same results. For example, in minor damage state, at the same PGA of 0.11 g, the exceedance probability for the point load hypothesis is double the distributed load, which are 66% and 37%, respectively. For extensive state, at the PGA of 0.8 g, they were 48% for point load and 44% for distributed load. The results are similar with a value of 65% for moderate state at a PGA of 0.35 g as shown in Figure 9.  
Figure 8: Fragility curves for damage states: (**a**) for concentrated load; (**b**) for Invert distributed load. Figure 9: _Cont._  
To determine the required number of earthquakes to be applied in practical structural analysis, two statistical parameters, which are required to define a fragility curve, are obtained using the proposed method and three fragility curves, ranging from 10 to 50 earthquakes per PGA level, are derived gradually. The median value at each damage state is acquired with its corresponding log-normal standard deviation value summarized in Table 5 and plotted in Figure 10.  
Figure 9: Fitted fragility curves of 50 earthquakes per each PGA level: (**a**) Minor; (**b**) Moderate; and (**c**) Extensive damage state.