## 3 Structural Pounding Decision Frameworks
### Decision B: Minimum Acceptable Structural Bounding Risk \(p_{min}^{d_{g,t}}\)  
In this case, both POs-\(\delta_{\max}\) and POs-EDP are combined to determine the minimum acceptable bounding risk \(\mathrm{P_{min}^{d_{g,t}}}\) at a targeted value of separation gap distance \(\mathrm{d_{g,t}}\). Now, the criterion for developing the decision model is to calibrate the structural bounding risk taking into account the capacity level of the EDP. The proposed decision framework is presented in Figure 3. The PO of \(\mathrm{P_{t,crit}}\)-\(\mathrm{d_{g,crit}}\) is also involved. The minimum acceptable bounding risk \(\mathrm{P_{min}^{d_{g,t}}}\) is defined as follows:  
* Case B1: \(\mathrm{d_{g,t}}<\mathrm{d_{g,crit}}\) and \(\mathrm{P_{dg,t}^{PO}}<\mathrm{P_{dg,t}^{S_{max}}}\) (Figure 3a)  
At a targeted separation gap distance of \(\mathrm{d_{g,t}}\), the capacity level of the EDP is reached at \(\mathrm{P_{dg,t}^{PO}}\) (based on POs-EDP), and when the POs-\(\delta_{\max}\) is considered, the seismic risk is equal to \(\mathrm{P_{dg,t}^{S_{max}}}\) (Figure 3a). Once \(\mathrm{P_{dg,t}^{S_{max}}}\) is greater than \(\mathrm{P_{dg,t}^{PO}}\), \(\mathrm{P_{dg,t}^{PO}}\) is adopted as the minimum acceptable level. Nevertheless, the structural bounding seismic risk of \(\mathrm{P_{dg,t}^{S_{max}}}\) is also an acceptable choice since the capacity of the EDP is not exceeded. On the other hand, \(\mathrm{P_{t,crit}}\) cannot be considered as an acceptable level of structural bounding risk once the capacity level of the EDP is exceeded at this value of \(\mathrm{P}\) (meaning \(\mathrm{P_{t,crit}}\)). Consequently, for the targeted separation gap distance \(\mathrm{d_{g,t}}\), the level of structural bounding risk \(\Big{(}\mathrm{P_{dg,t}^{Q_{g,t}}}\Big{)}\) can be calibrated within the range of \(\mathrm{P_{dg,t}^{PO}}-\mathrm{P_{dg,t}^{S_{max}}}\). Thus, the following solutions are provided:  
\[\mathrm{Minimum\ acceptable\ level\ of\ structural\ bounding\ risk\ \mathrm{P_{min}^{d_{g,t}}}:\ \mathrm{P_{min}^{d_{g,t}}}=P_{dg,t}^{PO}} \tag{13}\]  
\[\mathrm{Acceptable\ levels\ of\ structural\ bounding\ risk\ \mathrm{P^{d_{g,t}}}:\ \mathrm{P_{dg,t}^{PO}}\leq\mathrm{P^{d_{g,t}}}\leq\mathrm{P_{dg,t}^{S_{max}}}} \tag{14}\]  
* Case B2: \(\mathrm{d_{g,t}}>\mathrm{d_{g,crit}}\) and \(\mathrm{P_{dg,t}^{PO}}>\mathrm{P_{dg,t}^{S_{max}}}\) (Figure 3b)  
In this condition, at a targeted separation gap distance of \(\mathrm{d_{g,t}}\), the capacity level of the EDP is reached at \(\mathrm{P_{dg,t}^{PO}}\), which is greater than the corresponding annual probability of \(\mathrm{P_{dg,t}^{S_{max}}}\) when POs-\(\delta_{\max}\) is examined (Figure 3b). Values of seismic risk lower than \(\mathrm{P_{dg,t}^{PO}}\) indicate that the EDP's capacity is exceeded, thus indicating that \(\mathrm{P_{dg,t}^{S_{max}}}\) is not an acceptable solution. Also, \(\mathrm{P_{t,crit}}\) is an acceptable structural risk level of the EDP. For separation gap distance equal to \(\mathrm{d_{g,t}}\) the level of structural bounding risk can be calibrated within the range of \(\mathrm{P_{dg,t}^{PO}}-\mathrm{P_{t,crit}}\). So, the solutions in Case B2 are as follows:  
\[\mathrm{Minimum\ acceptable\ level\ of\ structural\ bounding\ risk\ \mathrm{P_{min}^{d_{g,t}}}:\ \mathrm{P_{min}^{d_{g,t}}}=P_{dg,t}^{PO}} \tag{15}\]  
\[\mathrm{Acceptable\ levels\ of\ structural\ bounding\ risk\ \mathrm{P^{d_{g,t}}}:\ \mathrm{P_{dg,t}^{PO}}\leq\mathrm{P^{d_{g,t}}}\leq\mathrm{P_{t,crit}}} \tag{16}\]* Case B3: \(\mathrm{d_{g,t}}=\mathrm{d_{g,crit}}\) In this discrete case of \(\mathrm{d_{g,t}}=\mathrm{d_{g,crit}}\), the minimum acceptable structural bounding risk is defined as follows: \[\mathrm{P_{min}^{d_{g,t}}}=\mathrm{P_{t,crit}}\Big{(}=\mathrm{P_{d_{g,t}}^{PO}}= \mathrm{P_{d_{g,t}}^{S_{max}}}\Big{)}\] (17) Decision B is summarized in the flowchart of Figure 3c. The decision process is further discussed through a numerical example in Section 4.