## 3 Results  
### Numerical Benchmark: Calibration of a SDoF Bouc-Wen Type Hysteretic System  
The Bouc-Wen model of hysteresis is widely used in structural engineering [29; 30; 31; 32; 33; 34]. The model was proposed by [PERSON] [35], and thereafter modified by [PERSON] [36]. The differential equation governing the motion of a single degree of freedom (SDoF) is written in the form:  
\[m\ddot{u}+c\dot{u}+f_{r}(u(t),z(t))=p(t), \tag{10}\]  
where \(m\) is the mass of the system, \(c\) is the viscous linear damping coefficient, \(u(t)\) is the displacement, \(f_{r}(u,z)\) is the restoring force, and \(p(t)\) is the external excitation. \(\dot{u}(t)\) and \(\ddot{u}(t)\) denote derivations in time. According to the Bouc-Wen model, the restoring force is given by the following expression:  
\[f_{r}(u,z)=f_{r}^{el}(u,z)+f_{r}^{h}(u,z)=\alpha k_{i}u(t)+(1-\alpha)k_{i}z(t), \tag{11}\]  
where \(f_{r}^{el}(u,z)\) represents the elastic component whereas \(f_{r}^{h}(u,z)\) is the hysteretic component which depends on the past history of the system response, \(k_{i}\) is the initial stiffness of the system, \(\alpha\) is the ratio between the final stiffness and initial one and \(z(t)\) is the hysteretic displacement as defined by [PERSON] and [PERSON] [37] to enhance the capacity of the original model to represent hysteretic cycles:  
\[\dot{z}(t)=\frac{A(\epsilon)\dot{u}(t)-\
u(\epsilon)\left[\beta\big{|}\dot{u }(t)\big{|}\big{|}z(t)\big{|}_{N-1}z(t)+\gamma\dot{u}(t)\big{|}z(t)\big{|}^{N }\right]}{\eta(\epsilon)}. \tag{12}\]  
Figure 1: Flowchart of the Bayesian model calibration procedure. In Equation (12) the parameters \(\beta\), \(\gamma\) and \(N\) control the shape of the cycles, while the additional parameters \(A(\varepsilon)\), \(\eta(\varepsilon)\) and \(\
u(\varepsilon)\) are degradation functions in terms of the dissipated hysteretic energy \(\varepsilon^{h}(t)\) and take into account the stiffness and strength degradation:  
\[A(\varepsilon)=A_{0}(\varepsilon)-\delta_{A}\varepsilon^{h}(t),\
u( \varepsilon)=\
u_{0}(\varepsilon)-\delta_{\
u}\varepsilon^{h}(t),\,\eta( \varepsilon)=\eta_{0}(\varepsilon)-\delta_{\eta}\varepsilon^{h}(t), \tag{13}\]  
where the constant values of \(A_{0}\), \(\
u_{0}\), \(\eta_{0}\) are usually set to unity [38]. Whereas the values \(\delta_{A}\), \(\delta_{\
u}\), \(\delta_{\eta}\) are constant terms which specify the amount of stiffness and strength degradation [39]. Finally, the dissipated hysteretic energy \(\varepsilon^{h}(t)\) is given by:  
\[\varepsilon^{h}(t)=\int_{u(0)}^{u(t)}f_{r}^{h}(u,z)du=(1-\alpha)k_{i}\int_{0}^ {t}z(\tau)\dot{u}(\tau)d\tau. \tag{14}\]  
A record of the Montenegro earthquake (1979) was selected from the PEER Ground Motion Database (PEER, 2019) as seismic excitation (Figure 2) for the benchmark. This input excitation was used to simulate the response of a SDoF Bouc-Wen-Baber-Noori (BWBN) nonlinear system with mass of \(m=1200\) kg and monitored by a dynamic monitoring system composed of one accelerometer installed on the system lumped mass. The initial stiffness \(k_{i}\) of the system was calibrated with the objective of obtaining a plausible system frequency for a masonry structure of approximately 4 Hz. A damping ratio of \(\xi=3\%\) was adopted according to relevant literature on masonry structures. Finally, the remaining BWBN parameters used to generate the simulated record are presented in Table 1, while Figure 3 depicts the simulation record data. The Bayesian calibration was then carried out on the simulated velocity response. The latter was chosen because it is simpler to describe such a system by a single ODE with respect to velocity. Deterministic values were set for the following parameters of the BWBN model: \(m=1200\) kg, \(A=1\), \(N=1\), and also for the damping ratio, \(\xi=3\%\). The latter assumption was made to release uncertainties related to the identification of \(\xi\) from the calibration process.