## 3 Results
### Demonstration on a Case Study
#### 3.2.1 Reference Model  
In this subsection, the model that came as the result of a nonlinear identification previously performed on the Pizzoli Town Hall structure is described for completeness. The nonlinear identification, like in [41; 42], relied on a plane frame model in the direction of minor inertia of the building. Assuming the mass contribution of the raised ground floor in the dynamics of the structure is negligible, the building has been modelled with two lumped masses in correspondence to the first and under-roof floor.  
The identification procedure consisted of using absolute acceleration data measured at the floors to approximate the response with a generalized linear model obtained by expanding the Bouc-Wen model of hysteresis.  
The basic functions of the generalized linear model ended up being nonlinear in terms of the exponential parameters of the BW model. These parameters were therefore identified with specialized optimization algorithms, while for the remaining parameters a direct estimate in the joint time-frequency domain was performed. The identification provided also instantaneous values of the BW model parameters made explicit by the generalized linear model. The main findings of the identification can be summarized as follow:  
* The adopted procedure allowed verification of the consistency of the assumed nonlinear model. This was done by checking the stability of the values of the model parameters over time.
* The resulting model satisfactorily reproduced the experimental response. This was validated in [43] by comparing the model results with the experimental findings obtained by the authors and by other researchers [44] independently,
* The procedure allowed for the collection of timely information on the health of the structure immediately after the occurrence of the earthquake.  
In the identification process, a global box-like behavior was assumed, also based on the results of on-site inspections that led to the verification of the existence of good connections between walls and floor-walls before the occurrence of seismic events. In addition, no strength deterioration was accounted for (i.e., \(\
u\left(\varepsilon^{h}\right)=1\)), while for the stiffness degradation, only the proportional component was maintained (i.e., \(\eta\left(\varepsilon^{h}\right)=1\) and \(A\left(\varepsilon^{h}\right)=1-\delta_{A}\varepsilon^{h}(t)\)). The parameter \(\alpha\) was instead imposed to zero. These choices were made to make the reference model consistent with the model identified in [43], which was calibrated with experimental data.  
The reference two DoFs model has been reconducted to a DoF by imposing the acceleration response of the first floor of the Town Hall (recorded by the monitoring system), \(\ddot{u}_{1}(t)\), as the input signal. The output was obtained for the inter-story drift, \(u(t)\), between the under-roof and the first floor. For further information on the identified model one can refer to [43].  
Thus, the reference model used for the analysis from now on becomes:  
\[m\ddot{u}(t)+f(t)=-m\dddot{u}_{1}(t); \tag{16}\]  
\[\dot{f}_{L}(t)=k\left(\varepsilon^{h}(t),t\right)\dddot{u}(t); \tag{17}\]  
\[k\left(\varepsilon^{h}(t),t\right)=k_{i}\left(1-\delta_{A}\varepsilon^{h}(t) \right)=k_{i}-k_{i}\delta_{A}\varepsilon^{h}(t); \tag{18}\]  
\[\dot{f}_{NL}(t)=-\beta\dot{u}(t)\big{|}f(t)\big{|}^{N}sign\Big{[}\dot{u}(t)f( t)\Big{]}-\gamma\dot{u}\big{|}f(t)\Big{|}^{N}. \tag{19}\]  
The reference model is depicted in Figure 9 with the acceleration input \(\ddot{u}_{1}(t)\), while the corresponding model parameters, identified in a previous study with the Pizzoli Town Hall building responses, are reported in Table 4. For the numerical simulations, a viscous damping term (with a damping ratio of 3% set as a deterministic value in the calibration process) has been added to the model of Equation (16).