## 4 Case Study
### Minimum Acceptable Structural Pounding Risk \(P_{min}^{\mathrm{d_{g,t}}}\)  
Herein, the structural pounding risk of the eight-story RC frame is calibrated based on the acceptable POs-\(\delta_{\mathrm{max}}\) and POs-IDR\({}_{\mathrm{max}}\) at a targeted value of separation gap distance \(\mathrm{d_{g,t}}\) between the adjacent structures. For the implementation of the proposed decision framework (Decision B), two conditions of \(\mathrm{d_{g,t}}\) are considered: (a) \(\mathrm{d_{g,t}}\) = 3.0 cm less than \(\mathrm{d_{g,crit}}\) and (b) \(\mathrm{d_{g,t}}\) = 8.0 cm greater than \(\mathrm{d_{g,crit}}\). In Table 2, the minimum acceptable structural pounding risk \(\mathrm{P_{min}^{\mathrm{d_{g,t}}}}\) is presented for all the examined cases. Moreover, in Figure 8, a detailed presentation of the results when the performance of the eight-story RC frame due to structural pounding is to be justified at the capacity level of IO in terms of IDR\({}_{\mathrm{max}}\), is shown. So, based on the DHC-\(\delta_{\mathrm{max}}\), the eight-story RC frame reaches \(\mathrm{d_{g,t}}\) = 3.0 cm (=\(\delta_{\mathrm{max}}\)) at probability \(\mathrm{P_{dg,t}^{\delta_{\mathrm{max}}}}\)= 10.65 \(\times\) 10\({}^{-3}\). For this value of seismic risk, the predicted IDR\({}_{\mathrm{max}}\) demand is 0.54 (%h\({}_{\mathrm{st}}\)) in the case with no consideration of bounding and 0.63 (%h\({}_{\mathrm{st}}\)) in bounding conditions (Figure 8b). Also, in Figure 8, it can be observed that due to structural pounding conditions, the DHC-IDR\({}_{\mathrm{max}}\) yields greater demands in comparison to the free vibration case. Taking into account the POs-IDR\({}_{\mathrm{max}}\) at the performance level of IO, the structural pounding risk \(\mathrm{P_{dg,t}^{\mathrm{PO}}}\) is equal to 2.43 \(\times\) 10\({}^{-3}\) when \(\mathrm{d_{g,t}}\) = 3.0 cm(\(<\)d\({}_{\mathrm{g,crit}}\)). Thus, \(\mathrm{P_{dg,t}^{\mathrm{max}}}\) is greater than \(\mathrm{P_{dg,t}^{\mathrm{PO}}}\) and the minimum acceptable level is \(\mathrm{P_{dg,t}^{\mathrm{PO}}}\) (Decision Case B1). Of course, the structural pounding risk of \(\mathrm{P_{dg,t}^{\delta_{\mathrm{max}}}}\) is also a safe decision option since the demand of the IDR\({}_{\mathrm{max}}\) is 0.63 (%h\({}_{\mathrm{st}}\)), lower than the capacity level of IO (1 %h\({}_{\mathrm{st}}\)) (Figure 8b). On the other hand, the annual probability \(\mathrm{P_{t,crit}}\) = 1.98 \(\times\) 10\({}^{-3}\) cannot be considered as an acceptable level of structural pounding risk since the IO capacity level of the IDR\({}_{\mathrm{max}}\) is exceeded (IDR\({}_{\mathrm{max}}\) = 1.07 (%h\({}_{\mathrm{st}}\)) > 1 (%h\({}_{\mathrm{st}}\)))-Figure 8b). So, the acceptable levels of structural pounding risk range between \(\mathrm{P_{dg,t}^{\mathrm{PO}}}\) and \(\mathrm{P_{dg,t}^{\mathrm{max}}}\) (2.43\(\cdot\)10\({}^{-3}\) and 10.65\(\cdot\)10\({}^{-3}\)) at the targeted \(\mathrm{d_{g,t}}\). In this way, the seismic performance of the eight-story RC frame in bounding conditions can be verified by calibrating the levels of the structural pounding risk within the shaded area presented in Figure 8a. Nevertheless, for \(\mathrm{d_{g,t}}\) = 8.0 cm (\(>\mathrm{d_{g,crit}}\)), the IDR\({}_{\mathrm{max}}\) capacity level of IO is reached at annual probability \(\mathrm{P_{dg,t}^{\mathrm{PO}}}\) =1.76 \(\cdot\) 10\({}^{-3}\) when bounding conditions are involved (Figure 8c and Table 2). In the case of considering the POs-\(\delta_{\mathrm{max}}\), the eight-story RC frame reaches \(\mathrm{d_{g,t}}\) = 8.0 cm (= \(\delta_{\mathrm{max,t}}\)) at \(\mathrm{P_{dg,t}^{\mathrm{max}}}\) = 0.42 \(\cdot\) 10\({}^{-3}\), which is lower than \(\mathrm{P_{dg,t}^{\mathrm{PO}}}\) (=1.76 \(\cdot\) 10\({}^{-3}\)). Based on the proposed decision Case B2, values of seismic risk lower than \(\mathrm{P_{dg,t}^{\mathrm{PO}}}\) indicate that the capacity level of the IDR\({}_{\mathrm{max}}\) is exceeded, and thus,\(\mathrm{P_{dg,t}^{\mathrm{max}}}\) is not an acceptable solution (Figure 8c).