## 4 Case Study
### Optimal Separation Gap Distance \(d_{g,min}^{\text{P}}\)
Now, considering the acceptable POs-\(\text{IDR}_{\text{max}}\) at the performance level of LS, the following targeted values of probability are adopted: (a) \(\text{P}_{\text{t}}=5.8\cdot 10^{-4}\) (>P\({}_{\text{t,crit}}=4.98\cdot 10^{-4}\)) and (b) \(\text{P}_{\text{t}}=4.7\cdot 10^{-4}\) (<P\({}_{\text{t,crit}}=4.98\cdot 10^{-4}\)). In the case of \(\text{P}_{\text{t}}=5.8\cdot 10^{-4}\), the proposed Decision A yields the following values of \(\text{d}_{\text{g}}\): \(\text{d}_{\text{g,min}}^{\delta_{\text{max}}}=7.26\) cm, \(\text{d}_{\text{g,min}}^{\text{PO}_{\text{t}}}=4.5\) cm, and \(\text{d}_{\text{g,crit}}=7.59\) cm (Figure 7c). Thus, the optimal separation gap \(\text{d}_{\text{g,min}}^{\text{P}_{\text{t}}}\) is equal to 4.5 cm, and an optimal range of \(\text{d}_{\text{g}}^{\text{P}_{\text{t}}}\) between 4.5 and 7.59 cm (Figure 7c) can be set at \(\text{P}_{\text{t}}=5.8\cdot 10^{-4}\). At \(\text{P}_{\text{t}}=4.7\cdot 10^{-4}\), the solution follows the decision Case A2. As can be observed in Figure 7d, in the case of accepting \(\text{d}_{\text{g,min}}^{\delta_{\text{max}}}=7.73\) cm, the \(\text{IDR}_{\text{max}}\) demand at \(\text{P}_{\text{t}}=4.7\cdot 10^{-4}\) is equal to 1.52 (%hst), and thus, the capacity level of LS is exceeded. Similar observations hold when \(\text{d}_{\text{g,crit}}=7.59\) cm is examined. Therefore, the optimal distance is \(\text{d}_{\text{g,min}}^{\text{PO}_{\text{t}}}=8.88\) cm (Table 1). Incorporating the acceptable POs-\(\text{IDR}_{\text{max}}\) at the performance level of CP, the optimal separation gap distance \(\text{d}_{\text{g,min}}^{\text{P}_{\text{t}}}\) is estimated at a targeted level of seismic risk that is equal to \(\text{P}_{\text{t}}=10^{-4}\) (>P\({}_{\text{t,crit}}=0.6\cdot 10^{-4}\)). In this case, \(\text{d}_{\text{g,min}}^{\text{PO}_{\text{t}}}\) is equal to 7.55 cm, and \(\text{d}_{\text{g,min}}^{\delta_{\text{max}},\text{t}}\) is equal to 12.36 cm. The combination of POs-\(\text{IDR}_{\text{max}}\) and POs-\(\delta_{\text{max}}\) indicates that \(\text{d}_{\text{g,crit}}\) is equal to 14.43 cm. So, the optimal value of \(\text{d}_{\text{g}}\) that verifies the \(\text{IDR}_{\text{max}}\) performance level of CP is \(\text{d}_{\text{g,min}}^{\text{P}_{\text{t}}}=7.55\) cm. Structural bounding conditions when the separation distance is equal to either \(\text{d}_{\text{g,crit}}\) or \(\text{d}_{\text{g,min}}^{\delta_{\text{max}},\text{t}}\) yield seismic demands of \(\text{IDR}_{\text{max}}\) (2.15 %hst or 2.26 %hst) that are less than the examined capacity level (see Figure 7e). So, an optimal range of \(\text{d}_{\text{g}}^{\text{P}_{\text{t}}}\) can also be defined as follows: 7.55 cm \(\leq\text{d}_{\text{g}}^{\text{P}_{\text{t}}}\leq\) 14.43 cm. The results in Figure 7f show that \(\text{P}_{\text{t,crit}}=\text{P}_{\text{t,wp}}\) at the performance level of CP (P\({}_{\text{t,wp}}\): corresponding seismic risk P in the case of no pounding). Thus, a target probability \(\text{P}_{\text{t}}\) less than \(\text{P}_{\text{t,crit}}=\text{P}_{\text{t,wp}}=\) 0.60 \(\times\) 10\({}^{-4}\) is an unacceptable level of structural seismic risk since the  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**EDP**} & \multirow{2}{*}{**PL 1**} & \multirow{2}{*}{ \begin{tabular}{c} **P\({}_{\text{t,crit}}\)** \\ (\(\times\)**10\({}^{-3}\))** \\ \end{tabular}  } & \multirow{2}{*}{**P\({}_{\text{t}}\) (\(\times\)10\({}^{-3}\))**} & \multirow{2}{*}{**P\({}_{\text{t}}\) (\(\times\)10\({}^{-3}\))**} & \multirow{2}{*}{ \begin{tabular}{c} \(\text{d}_{\text{g,min}}^{\text{PO}_{\text{t}}}\)** \\ (cm) \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} \(\text{d}_{\text{g,crit}}^{\delta_{\text{max}}}\)** \\ (cm) \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} \(\text{d}_{\text{g,crit}}^{\delta_{\text{min}}}\)** \\ (cm) \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} \(\text{d}_{\text{g,min}}^{\delta_{\text{max}}}\)** \\ (cm) \\ \end{tabular}  } & \multirow{2}{*}{\