## 2 Methodology
### Proposed BWBN-VP Model  
In Figure 1b, the stiffness configuration in the original BWBN model shows that the post-yield and pinching region stiffness values can be considered equal for negligible slip-lock stiffness, as shown schematically in Figure 2a. In the original model, the stiffness ratio, \(\alpha\), is constant. Therefore, the values of the post-yield and pinching stiffness are constant, \(k_{p}=\alpha k_{0}\), while experimental results [17; 42] and real observations in structurally damaged RC-frame buildings [43] reveal that the post-yield and pinching stiffness values, \(k_{p}\), degrade due to earthquake-induced structural damage. The peak-oriented hysteretic model [57] shows post-yielding stiffness, \(k_{p}\), degrades from initial stiffness, \(k_{0}\), when the maximum deformation exceeds its previous value [57; 58]:  
\[k_{p}=k_{0}\bigg{[}\frac{d_{y}}{D_{max}}\bigg{]}^{a} \tag{10}\]  
where \(d_{y}\) stands for yield deformation, \(a\) parameter is a constant, and \(D_{max}\) is the maximum deformation experienced by the system. Therefore, to allow the pinching stiffness, \(k_{p}\), to vary in the original BWBN hysteresis model, a varying \(\alpha\) value [47; 48] is defined for the proposed BWBN-VP hysteresis model in this study:  
\[\alpha(D_{max})=\alpha_{0}\bigg{[}C_{1}\frac{D_{max}}{d_{y}}\bigg{]}^{C_{2}}; \qquad C_{1}\geq 0;\ C_{2}\leq 0\quad\rightarrow\quad 0<\alpha\leq\alpha_{0} \tag{11}\]  
The parameter of \(\alpha_{0}\) is the initial stiffness ratio at zero deformation. \(C_{2}\) is a negative value adjusting decay of \(\alpha\) with respect to \(D_{max}\), and \(C_{1}\) parameter is a positive constant defining the decay initiation. For \(C_{1}\geq 1\), decay begins prior to \(d_{y}\), while for \(0\leq C_{1}<1\), damage to post-yield and pinching stiffness begins for deformation greater than \(\mathrm{d}_{y}\). Figure 2a shows that the elastic stiffness, \(k_{e}\), degrades from hysteresis loop 1 to loop 2, while the post-yield and pinching stiffnesses, \(k_{p}\), remain constant in the original BWBN model. In contrast, the proposed BWBN-VP model captures stiffness degradation for all pinching, elastic, and post-yielding parts of the hysteresis loops, as shown in Figure 2b. Moreover, the slip length in the pinching region increases as a function of dissipated energy, \(\epsilon(t)\), in the original BWBN model, as seen in Figure 2a. \begin{table}
\begin{tabular}{c c c c} \hline
**Event** & **Parameter** & **Description** & **Control Type** \\ \hline \(1\) & \(\alpha\) & Ratio of linear to nonlinear response & \\ \hline \(2\) & \(n\) & Yielding sharpness & \\ \hline \(3\) & \(\beta\) & Basic hysteresis shape control & Basic shape \\ \hline \(4\) & \(\gamma\) & Basic hysteresis shape control & \\ \hline \(5\) & \(\delta_{\eta}\) & Stiffness degradation & Stiffness \\ \hline \(6\) & \(\delta_{v}\) & Strength degradation & Strength \\ \hline \(7\) & \(\xi_{s}\) & Measure of total slip & \\ \hline \(8\) & \(p\) & Pinching initiation & \\ \hline \(9\) & \(q\) & Pinching slope & Pinching \\ \hline \(10\) & \(\psi\) & Pinching magnitude & \\ \hline \(11\) & \(\delta_{\psi}\) & Pinching rate & \\ \hline \(12\) & \(\lambda\) & Pinching severity & \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the original BWBN model [56]. SHM results for an RC building show that increases in the pinching region length lead to pinching stiffness degradation [43]. To enhance the consistency between the slip increase and stiffness degradation in the pinching region, this study introduces a modification wherein the parameter \(\xi_{s}\), is modeled to increase as a function of \(D_{max}\)[47, 48], This adjustment results in the refined BWBN-VP hysteresis model:  
\[\xi_{s}(D_{max})=1-\left[C_{3}\frac{D_{max}}{d_{y}}\right]^{C_{4}};\ \ \ \ \ C_{3}\geq 0;\ C_{4}\leq 0\ \ \ \ \rightarrow\ \ \ 0<\xi_{s}\leq 1 \tag{12}\]  
where \(C_{3}\) and \(C_{4}\) have the similar roles of \(C_{1}\) and \(C_{2}\) in Equation (11).