# Seismic Performance of Frame Structure with Hysteretic Intermediate Discontinuity
## 2 The Mechanical Model  
### Archetype Model  
In the following, two archetype systems are used to compare results with and without inter-storey isolation. The Main System archetype (MS) represents the structure with the introduction of an intermediate discontinuity (Figure 1a), whereas the Auxiliary System archetype (AS) represents the same structure without the intermediate discontinuity (Figure 1b). In the case of the MS, the part of the structure below the discontinuity is called substructure, whereas the part above the discontinuity is called superstructure. The MS and AS are 3-DOF and 2-DOF equivalent models, respectively (Figure 2). In such models, \(m_{1}\) and \(k_{1}\) represent the mass and the stiffness of the substructure, whereas \(k_{2}\) and \(m_{2}\) refer to the superstructure. In the MS, the constitutive behaviour of the disconnection devices is modelled by means of the Bouc-Wen model. There are several variations of such a model, for instance, [PERSON] and [PERSON] [26] proposed a generalized Bouc-Wen model to describe asymmetric hysteresis, whereas [PERSON] [27] extended the six-parameters formulation to an eight-parameter formulation to include the cyclical degradation of the material properties. In this paper, the classical Bouc-Wen formulation [28] is deemed fit to accurately describe the constitutive behaviour of the disconnection devices. Figure 1: Structural scheme of the (**a**) frame structure with intermediate discontinuity; (**b**) original frame structure (without intermediate discontinuity). It is worth observing that no specific reference is made to any new or existing structures. Moreover, it is assumed that the intermediate discontinuity keeps the structure to be protected in the elastic field. To evaluate the effectiveness of the intermediate discontinuity, the (linear elastic) displacements provided by the MS are compared with those of the AS. For the MS, the equations of motion, obtained by a direct approach, are written as  
\[c_{11}\dot{u}_{1}(t)+c_{12}\dot{u}_{d}-F_{h}(t)+k_{1}u_{1}(t)+m_{ 1}\dot{u}_{1}(t)=-m_{1}a_{g}(t), \tag{1a}\] \[c_{21}\dot{u}_{1}+c_{22}\dot{u}_{d}(t)+c_{23}\dot{u}_{2}(t)+F_{h} (t)+k_{2}[u_{d}(t)-u_{2}(t)]+m_{d}\dot{u}_{d}(t)=-m_{d}a_{g}(t),\] (1b) \[c_{32}\dot{u}_{d}(t)+c_{33}\dot{u}_{2}(t)+k_{2}u_{2}(t)-k_{2}u_{ d}(t)+m_{2}\dot{u}_{2}(t)=-m_{2}a_{g}(t),\] (1c) \[\dot{z}(t)=[\dot{u}_{d}(t)-\dot{u}_{1}(t)]\{\tilde{A}-|z(t)|^{n}\{ \beta_{1}+\gamma_{1}\text{sgn}\{z(t)[\dot{u}_{d}(t)-\dot{u}_{1}(t)]\}\}\}u_{ y}^{-1}, \tag{1d}\]  
where the restoring force \(F_{h}(t)\) is defined as  
\[F_{h}(t)=\Psi k_{d}(u_{d}(t)-u_{1}(t))+k_{d}u_{y}(1-\Psi)z(t), \tag{2}\]  
\(z(t)\) is a hysteretic variable that describes the post-yielding behaviour, whereas \(\tilde{A}\), \(\beta_{1}\), \(n\), and \(\gamma_{1}\) are the Bouc-Wen parameters defined in [28]. The Bouc-Wen parameters are redundant; therefore, both [PERSON] [29] and [PERSON] et al. [30] suggested assigning specific values to some of them. In details, they suggested using \(\tilde{A}=1\), so that \(k_{d}=F_{y}/u_{y}\) acquires the meaning of pre-yielding (elastic) stiffness. Additionally, some researchers [29; 31] suggested using \(\beta_{1}+\gamma_{1}=1\), where \(\beta_{1}\) and \(\gamma_{1}\) are parameters that control the shape and dimension of the hysteresis cycle. The choice to adopt these two constraints, \(\tilde{A}=1\) and \(\beta_{1}+\gamma_{1}=1\), provides a physical meaning to \(F_{y}\), \(u_{y}\), and \(\Psi\). Specifically, \(F_{y}\) is the yielding force, \(u_{y}\) is the yielding displacement, and \(\Psi\) is the ratio between post-yielding and pre-yielding (elastic) stiffness. Moreover, the values of the hysteretic dimensionless parameter \(z(t)\) remain inside the range \([-1,1]\). Therefore, in this study, the values adopted for the parameters are \(\tilde{A}=1\), \(\beta_{1}=\gamma_{1}=1/2\), and \(n=2\). The damping coefficients \(c_{ij}\) in Equations (1a)-(1c) can be grouped in matrix form as  
\[\mathbf{C}=\left(