## 4 Fragility Assessment of Structural Pounding
### Displacement-Based Fragility Curves  
Herein, the five different methodologies presented in Section 2 are used for the probabilistic evaluation of the pounding effect in terms of IDR\({}_{\rm max}\) | PGA and TDR\({}_{\rm max}\) | PGA. The probability data that are directly deduced based on the empirical CDF method are considered as reference values to evaluate the accuracy and the limitations of the other four methods. Three different separation gap distances are examined: (a) d\({}_{\rm g}\) = 0.0 cm; (b) d\({}_{\rm g}\) = 4.5 cm; and (c) d\({}_{\rm g}\) = 9.0 cm, while results based on the free vibration mode (without pounding) of the eight-story RC structure are also presented. Fragility curves are developed considering the damage states--capacity levels of [62]:  
1. Immediate occupancy (IO) that corresponds to a maximum interstory drift (IDR\({}_{\rm max}\)) is equal to 1% of the story height (h\({}_{\rm st}\)), and \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Seismic Excitations**} & **Duration** & **Maximum Acceleration \(\alpha_{\rm max}\)** & **M\({}_{\rm w}\)3** & **R 4** \\  & **(s)** & **(m/s2)** & **M\({}_{\rm w}\)3** & **(km)** \\ \cline{2-6}  & \multicolumn{4}{c}{component FN 1 **} & component FP 2  & & \\ \hline Italy Arienzo, 1980 (EQ283) & 24 & 0.268 & 0.405 & 6.9 & 52.9 \\ \hline Italy Auletta, 1980 (EQ284) & 34 & 0.615 & 0.655 & 6.9 & 9.6 \\ \hline Chi-Chi Taiwan-06, 1999 (EQ3479) & 42 & 0.073 & 0.070 & 6.3 & 83.4 \\ \hline Denali- Alaska, 2002 (EQ2107) & 60 & 0.869 & 0.975 & 7.9 & 50.9 \\ \hline Loma Prieta, 1989 (EQ804) & 25 & 1.090 & 0.509 & 6.9 & 63.1 \\ \hline Chi-Chi Taiwan-04, 1999 (EQ2805) & 60 & 0.096 & 0.075 & 6.2 & 116.2 \\ \hline San Fernando, 1971(EQ59) & 14 & 0.153 & 0.181 & 6.6 & 89.7 \\ \hline \end{tabular}
\end{table} 
Table 1: Main characteristics of seismic excitations. 2. 1% maximum top drift (TDR\({}_{\text{max}}\)) as a function of the total height of the structure (H\({}_{\text{tot}}\)). Figure 5 shows the fragility curves in terms of IDR\({}_{\text{max}}\) | PGA at the performance level of IO. As it can be observed, all the examined procedures are developing fragilities that are in a good agreement with the probability data points of the empirical CDF method. Nevertheless, in the case of analyzing the eight-story RC frame without the bounding effect, the PSDM-based fragility curve is shifted to greater values of PGA in comparison to the deduced fragilities based on the MLE, MM, and IM percentiles procedures. In the case of developing fragilities in terms of TDRmax | PGA, the IM percentiles method gives more conservative results in comparison to the other methodologies (Figure 6). On the other hand, the PSDM-based fragility curves indicate that the corresponding seismic assessment of the RC frame against bounding is less vulnerable than the expected one (probability data of empirical method CDF) when d\({}_{\text{g}}\) = 0.0 cm. Nevertheless, the MM fails to formulate fragility curves in cases of considering separation gap distance d\({}_{\text{g}}\) = 0.0 cm and d\({}_{\text{g}}\) = 4.5 cm. This is attributed to the fact that analyses fail to be carried up to values of PGA that exceed the examined performance level. Figure 5: MLE, MM, IM percentiles, PSDM methodologies for the fragility assessment of the eight-story RC frame in terms of maximum interstory drift (IDR\({}_{\text{max}}\) -%h\({}_{\text{st}}\)) as a function of the PGA. Examined cases: (a) without the bounding effect, (b) d\({}_{\text{g}}\) = 0.0 cm, (c) d\({}_{\text{g}}\) = 4.5 cm, and (d) d\({}_{\text{g}}\) = 9.0 cm. The difference among the fragility curves is owed to the estimated values of medians \(\mu\) (Table 2). The estimated value of median \(\mu\) in the case of PSDM-based fragility curve is greater than the other methods, which explains the shift of the particular curve.