## 2 Methodology
### The Original BWBN Model  
A nonlinear single-degree-of freedom (SDOF) system modeled using the BWBN hysteresis model in Figure 1 shows the combination of slip-lock, hysteretic, and linear springs forming the BWBN hysteresis model [25]. The SDOF system excited by a seismic excitation, \(\bar{u}_{g}\), is thus defined [15, 42]:  
\[m\ddot{u}+c\dot{u}+ak_{0}u+(1-\alpha)k_{0}z=-m\bar{u}_{g} \tag{1}\]  
where \(m\), \(c\), and \(k_{0}\) are the mass, viscous damping coefficient, and initial stiffness of the system, respectively; \(\bar{u}\), \(\dot{u}\) and \(u\) are acceleration, velocity and displacement of the mass with respect to the ground; \(\alpha\) is the ratio of post-yield/pre-yield stiffness. The mass normalized form of Equation (1) can be written:  
\[\ddot{u}+2\xi_{0}m\dot{u}+\alpha\omega_{0}^{2}u+(1-\alpha)\omega_{0}^{2}z=- \bar{u}_{g} \tag{2}\]  
here \(\xi_{0}\) is the damping ratio, and \(\omega_{0}\) is the initial natural frequency of the system. The nonlinear restoring force, \(f\), consists of two linear, \(f_{lin}\), and hysteretic, \(f_{phys}\), parts, as shown in Figure 1b and formulated:  
\[f=f_{lin}+f_{phys}=\alpha k_{0}u+(1-\alpha)k_{0}z \tag{3}\]  
where \(z\) is the hysteretic deformation happening in the hysteretic spring of the BWBN model and whose dynamics are defined [15, 42]:  
\[\dot{z}=\frac{h(z,\varepsilon)}{\eta(\varepsilon)}\left[\dot{u}-\
u( \varepsilon)\left[\beta|\dot{u}||z|^{n-1}z+\gamma\dot{u}|z|^{n}\right]\right] \tag{4}\]  
where \(n\), \(\beta\) and \(\gamma\) are hysteretic shape parameters; \(\
u(\varepsilon)\) and \(\eta(\varepsilon)\) are strength and stiffness degradation parameters, respectively, \(\varepsilon\) is the dissipated energy, and \(h(z,\varepsilon)\) is the pinching function obtained using [15, 42]:  
\[h(z,\varepsilon)=1-\xi_{1}(\varepsilon)exp\left(-\left[\frac{z\,sign(\dot{u} )-qz_{max}}{\xi_{2}(\varepsilon)}\right]^{2}\right) \tag{5}\]  
\[z_{max}=\pm\left[\frac{1}{\
u(\varepsilon)\,\left(\beta+\gamma\right)}\right] ^{1/n} \tag{6}\]  
\[\
u(\varepsilon)=1+\delta_{\
u}\varepsilon\text{ and }\eta(\varepsilon)=1+\delta_{ \eta}\varepsilon \tag{7}\]  
\[\xi_{1}(\varepsilon)=\xi_{s}[1-exp(-p\varepsilon)]\text{ and }\xi_{2}( \varepsilon)=\left[\psi+\delta_{\psi}\right][\lambda+\xi_{1}(\varepsilon)] \tag{8}\]where \(\xi_{s}\), \(p\), \(q\), \(\delta_{\varphi}\), \(\psi\) and \(\lambda\) are the parameters defining pinching behavior in the BWBN model; \(\xi_{2}(\varepsilon)\) causes the pinching region to spread, while \(0\leq\xi_{1}(\varepsilon)<1\) controls the pinching severity. The resulting dissipated hysteretic energy is defined as below:  
\[\varepsilon(t)=(1-\alpha)\omega_{0}^{2}\int_{0}^{T}z(u,t)\dot{u}(t)dt \tag{9}\]  
where \(T\) is the total time duration. Importantly, the BWBN model is non-phenomenological, where the curvature terms and nonlinear hysteretic motion, \(z(t)\), define the observed responses, but are not completely mechanically relevant terms. Instead, they capture the observed motion, but not all parameters have direct, concise mechanics definitions. The 12 parameters defining the BWBN model and their roles are summarized in Table 1.  
The stiffness ratio, \(\alpha\), controlling the post-yielding and pinching stiffness values is constant while obtained force-deformation hysteresis loops for experimental columns [42], experimental structure [17], and real-world structure [43] show the stiffness of pinching and post-yielding regions degrade due to damaging external excitations. Moreover, it is apparent the pinching region length increases due to damage [17; 43], where, in contrast, a constant total slip parameter, \(\xi_{s}\), cannot define the pinching region enlargement efficiently. Thus, the model, as defined, has significant flaws hindering its effective use. Finally, the large number of variables in Table 1 requires significant prior information to be uniquely identifiable from the data.