## 3 Structural Pounding Decision Frameworks
### Structural Pounding Acceptable PO  
Based on the knowledge accumulated so far, the most crucial EDPs for the structural bounding are (a) the peak relative displacement demand (\(\delta_{\text{max}}\)) of the adjacent structures at the top level of colliding and (b) other important global or local EDPs such as \(\text{IDR}_{\text{max}}\), \(\text{V}_{\text{max}}\), and \(\theta_{\text{p},\text{max}}\). The structural demand of \(\delta_{\text{max}}\) is estimated considering the seismic performance of structures without bounding. Any other global or local EDP is evaluated with and without bounding conditions. The \(\delta_{\text{max}}\) parameter is evaluated separately from the other EDPs to be directly compared with the seismic separation distance defined based on code provisions.  
Based on the methodology presented in Section 2, the demand hazard curve of \(\delta_{\text{max}}\) (DHC-\(\delta_{\text{max}}\)) can be defined as pairs of P vs. \(\delta_{\text{max}}\) (Figure 1a). Taking the separation gap distance \(\text{d}_{\text{g}}\) between the adjacent structures as a capacity level of \(\delta_{\text{max}}\), the corresponding P (level of seismic risk) can be evaluated, and the acceptable PO (P-\(\text{d}_{\text{g}}\) pair) can be obtained (Figure 1a,b). Different values of \(\text{d}_{\text{g},\text{i}}\) (as capacity levels) indicate different values of seismic risk (\(\text{P}_{\text{i}}\)) and, therefore, different acceptable POs (\(\text{P}_{\text{i}}\)-\(\text{d}_{\text{g},\text{i}}\) pairs) as a function of the \(\text{d}_{\text{g}}\) (violet line in Figure 1b). Therefore, it can easily be understood that the DHC-\(\delta_{\text{max}}\) also depicts the acceptable POs-\(\delta_{\text{max}}\).  
Similarly, the demand hazard curve of an EDP (DHC-EDP) can be defined as pairs of P vs. EDP, and then the structural seismic risk (\(\text{P}_{\text{i}}\)) at the capacity of the EDP can be estimated. The pair of \(\text{P}_{\text{i}}-\text{EDP}_{\text{capacity}}\) is an acceptable PO at a predefined separation gap distance \(\text{d}_{\text{g},\text{i}}\) (Figure 1c). Once the gap distance \(\text{d}_{\text{g},\text{i}}\) is decreased, the \(\text{P}_{\text{i}}\) is increased, as shown in Figure 1c.  
Thus, the determination of the acceptable POs-EDP requires the evaluation of DHCs-EDP at different--predefined \(\text{d}_{\text{g}}\) values, including the case of no bounding. In this way, characteristic POs of the EDP can be deduced as pairs of \(\text{P}_{\text{i}}\)-\(\text{d}_{\text{g},\text{i}}\) (Figure 1d). By connecting the pairs of \(\text{P}_{\text{i}}\)-\(\text{d}_{\text{g},\text{i}}\), a threshold line is laid down, and the acceptable POs of an EDP as a function of the \(\text{d}_{\text{g}}\) are defined (violet line in Figure 1d).  
In Figures 2 and 3, the acceptable POs-EDPs are presented with a blue line, while the green line depicts the POs-\(\delta_{\rm max}\). It can be observed that when these performance objectives are combined (POs-\(\delta_{\rm max}\) and POs-EDP), a distinct PO is defined, namely \(\rm P_{t,crit}-d_{g,crit}\) (Figures 2 and 3). This distinct PO is a critical performance objective since it alters the prediction in the proposed method, as outlined in Sections 3.2 and 3.3.  
Having defined the acceptable POs, the calibration of the structural bounding risk P on the seismic performance of the structures as a function of the \(d_{g}\) can be performed, as described below.  
Figure 1: Demand hazard curves (**a**,**c**) and acceptable POs (**b**,**d**) in structural bounding conditions.  
Figure 2: Proposed method for defining the optimal separation gap distance \(\mathrm{d}_{\mathrm{gmin}}^{\mathrm{P}_{t}}\) at a target probability \(\mathrm{P}_{t}\) (Decision A).  
Figure 3: Proposed method for the estimation of the minimum acceptable seismic risk \(\mathrm{P_{min}^{d_{g1}}}\) at a targeted separation gap distance \(\mathrm{d_{g1}}\) (Decision B).