## 3 Structural Pounding Decision Frameworks
### Decision A: Optimal Separation Gap Distance \(d_{g,min}^{\mathrm{P_{t}}}\)  
In this section, a decision framework for the estimation of the optimal separation gap distance \(d_{g,min}^{\mathrm{P_{t}}}\) at a target probability \(\mathrm{P_{t}}\) (targeted structural bounding seismic risk) is proposed. An optimal distance between the adjacent structures is the minimum acceptable value of \(\mathrm{d_{g}}\) that satisfies certain criteria at \(\mathrm{P_{t}}\) as defined below. For this purpose, the acceptable POs-\(\delta_{\mathrm{max}}\) and POs-EDP are used. The proposed framework is implemented as follows (see also Figure 2):  
Acceptable POs-\(\delta_{\mathrm{max}}\): Taking into account the acceptable POs-\(\delta_{\mathrm{max}}\), the optimum separation \(d_{g,min}^{\mathrm{P_{t}}}\) can be straightforwardly predicted, and it is equal to \(d_{g,min}^{\delta_{\mathrm{max}}}\), as presented in Figure 2. In fact, \(d_{g,min}^{\delta_{\mathrm{max}}}\) (or \(d_{g,min}^{\mathrm{P_{t}}}\)) denotes the peak relative displacement demand of the structures at the top level of colliding (\(\delta_{\mathrm{max,t}}\)) when \(\mathrm{P_{t}}\) is examined. Thus, \(d_{g,min}^{\delta_{\mathrm{max}}}\) is the minimum acceptable value of \(\mathrm{d_{g}}\) that satisfies the displacement demand of \(\delta_{\mathrm{max,t}}\) at \(\mathrm{P_{t}}\). Acceptable POs-EDP: The optimum \(d_{g,min}^{\mathrm{P_{t}}}\) at \(\mathrm{P_{t}}\) is equal to \(d_{g,min}^{\mathrm{PO_{t}}}\) based on POs-EDP (Figure 2). This value of \(\mathrm{d_{g}}\) represents the peak relative displacement at the level of the top contact point that causes the capacity level of the EDP to be reached at the target probability \(\mathrm{P_{t}}\). Consequently, the \(d_{g,min}^{\mathrm{PO_{t}}}\) complies with the acceptable EDP's seismic performance due to structural bounding at the targeted level of \(\mathrm{P_{t}}\). Combination of POs-\(\delta_{\mathrm{max}}\) and POs-EDP: The optimum \(d_{g,min}^{\mathrm{P_{t}}}\) at the same \(\mathrm{P_{t}}\) differs between POs-\(\delta_{\mathrm{max}}\) and POs-EDP, as can be observed in Figure 2. So, in the case of considering both POs-\(\delta_{\mathrm{max}}\) and POs-EDP, a decision model should be developed for the evaluation of the optimum \(d_{g,min}^{\mathrm{P_{t}}}\). The criterion for developing the proposed solution is to calibrate the minimum acceptable \(\mathrm{d_{g}}\) based on the capacity level of the EDP at \(\mathrm{P_{t}}\). For this purpose, three different separation gap distances are considered, namely \(d_{g,min}^{\delta_{\mathrm{max,t}}}\), \(d_{g,min}^{\mathrm{PO_{t}}}\), and \(d_{g,crit}\), and three distinct cases are comprised as follows:  
* Case A1: \(\mathrm{P_{t}}>\mathrm{P_{t,crit}}\) and \(d_{g,min}^{\mathrm{PO_{t}}}<d_{g,min}^{\delta_{\mathrm{max,t}}}\) (Figure 2a) In the case of considering \(d_{g,min}^{\delta_{\mathrm{max,t}}}\) as the separation gap distance, once \(d_{g,min}^{\delta_{\mathrm{max,t}}}\) is greater than \(d_{g,min}^{\mathrm{PO_{t}}}\), the DHC-EDP indicates that the demand of the examined EDP will be less than the capacity level of the EDP at the targeted probability of \(\mathrm{P_{t}}\) (Figure 1). Therefore, between \(d_{g,min}^{\delta_{\mathrm{max}}}\) and \(d_{g,min}^{\mathrm{PO_{t}}}\), the optimal separation gap distance \(\left(d_{g,min}^{\mathrm{P_{t}}}\right)\) is \(d_{g,min}^{\mathrm{PO_{t}}}\). Of course, \(d_{g,min}^{\delta_{\mathrm{max,t}}}\) is also a safe decision option, although it is conservative. Similarly, the separation gap distance of \(\mathrm{d_{g,crit}}\) could also be used as a solution for an adequate \(d_{g}^{\mathrm{P_{t}}}\) between the adjacent structures since the capacity level of the EDP is reached at a lower level of \(\mathrm{P_{t}}\) (Figure 2a). The solution of \(\mathrm{d_{g,crit}}\) is more conservative in comparison to \(d_{g,min}^{\mathrm{PO_{t}}}\).