## 4 Case Study
### Optimal Separation Gap Distance \(d_{g,min}^{\text{P}}\)  
In this section, the optimal \(\text{d}_{\text{g,min}}^{\text{P}_{\text{t}}}\) is evaluated at a targeted probability \(\text{P}_{\text{t}}\) having as a criterion the combination of POs-\(\delta_{\text{max}}\) and POs-IDR\({}_{\text{max}}\). For the implementation of the proposed decision framework (Decision A), two examples of targeted probability \(\text{P}_{\text{t}}\) are considered: (a) \(\text{P}_{\text{t}}>\text{P}_{\text{t,crit}}\) and (b) \(\text{P}_{\text{t}}<\text{P}_{\text{t,crit}}\). The PO of \(\text{P}_{\text{t,crit}}-\text{d}_{\text{g,crit}}\) is also defined for all the examined performance levels of the \(\text{IDR}_{\text{max}}\), and the results are depicted in Table 1.  
To further discussion, Figure 7 shows the following:  
1. The DHCs-IDR\({}_{\text{max}}\) when the separation distance \(\text{d}_{\text{g}}\) is considered equal to \(\text{d}_{\text{g,min}}^{\text{PO}_{\text{t}}}\) (blue line), \(\text{d}_{\text{g,min}}^{\text{d}_{\text{max,t}}}\) (green line), or \(\text{d}_{\text{g,crit}}\)(red line);
2. The capacity levels of the \(\text{IDR}_{\text{max}}\) at IO, LS, and CP limit states (violet line);
3. The predicted seismic demands of the \(\text{IDR}_{\text{max}}\) at the examined \(\text{P}_{\text{t}}\). So, in the case of \(\text{P}_{\text{t}}=3.03\cdot 10^{-3}\) (\(\text{>P}_{\text{t,crit}}=1.98\cdot 10^{-3}\)), the optimal values of the separation gap distance are \(\text{d}_{\text{g,min}}^{\text{d}_{\text{max,t}}}=4.39\) cm and \(\text{d}_{\text{g,min}}^{\text{PO}_{\text{t}}}=1.17\) cm at the performance level of IO (1.0 %\(\text{h}_{\text{st}}\)) based on the acceptable structural bounding POs-\(\delta_{\text{max}}\) and POs-IDR\({}_{\text{max}}\), respectively. The combination of POs-\(\delta_{\text{max}}\) and POs-IDR\({}_{\text{max}}\) indicates that \(\text{d}_{\text{g,min}}^{\text{PO}_{\text{t}}}=1.17\) cm is the optimal \(\text{d}_{\text{g,min}}^{\text{P}_{\text{t}}}\). Figure 5: Demand hazard (annual probability P) of \(\text{IDR}_{\text{max}}\) (%\(\text{h}_{\text{st}}\)). Figure 6: Acceptable structural bounding POs-\(\delta_{\text{max}}\) and POs-IDR\({}_{\text{max}}\) of the multistory RC structure. Furthermore, as can be observed in Figure 7a, the predicted \(\text{IDR}_{\text{max}}\) demand is less than the capacity level of IO when \(\text{d}_{\text{g}}\) is equal to either \(\text{d}_{\text{g,min}}^{\delta_{\text{max}}}\) or \(\text{d}_{\text{g,crit}}\). In fact, \(\text{IDR}_{\text{max}}\) is equal to 0.87 (%hst) and 0.88 (%hst) in the cases of \(\text{d}_{\text{g,crit}}\) and \(\text{d}_{\text{g,min}}^{\delta_{\text{max}}}\), respectively (Figure 7a). Therefore, the separation gap distances of \(\text{d}_{\text{g,min}}^{\delta_{\text{max}}}=4.39\) cm and \(\text{d}_{\text{g,crit}}=4.99\) cm are also safe decision solutions. In summary, the optimal value of separation gap distance \(\text{d}_{\text{g,min}}^{\text{P}_{\text{t}}}\) is equal to 1.17 cm, and any separation gap distance between 1.17 and 4.99 cm (optimal range) also verifies the \(\text{IDR}_{\text{max}}\) performance level of IO at the structural bounding risk \(\text{P}_{\text{t}}=3.03\cdot 10^{-3}\). On the other hand, when \(\text{P}_{\text{t}}=1.69\cdot 10^{-3}\) (<P\({}_{\text{t,crit}}=1.98\cdot 10^{-3}\)), \(\text{d}_{\text{g,min}}^{\delta_{\text{max}}}\) is equal to 5.24 cm based on the POs-\(\delta_{\text{max}}\). The corresponding solution for the acceptable POs-\(\text{IDR}_{\text{max}}\) at the performance level of IO is \(\text{d}_{\text{g,min}}^{\text{PO}_{\text{t}}}=9.0\) cm. As can be observed in Figure 7b, the predicted \(\text{IDR}_{\text{max}}\) is about 1.05 (%hst) for both \(\text{d}_{\text{g,min}}^{\delta_{\text{max}}}\) and \(\text{d}_{\text{g,crit}}\) and greater than the examined capacity level. Thus, \(\text{d}_{\text{g,min}}^{\delta_{\text{max}}}\) or \(\text{d}_{\text{g,crit}}\) is not an acceptable solution, and the optimal \(\text{d}_{\text{g,min}}^{\text{P}_{\text{t}}}\) is equal to 9.0 cm (Table 1).