# Seismic Performance of Frame Structure with Hysteretic Intermediate Discontinuity
## 2 The Mechanical Model
### Dynamic Equivalence between Multiple-DOF Systems and Archetype Models  
This subsection presents the equivalence procedure used to derive the mechanical characteristics of the AS. Such a procedure was developed mainly for low- and medium-rise frame structures and aims at obtaining the masses \(m_{1}\) and \(m_{2}\) and stiffnesses \(k_{1}\) and \(k_{2}\) of the AS starting from the geometrical and mechanical characteristics of the multiple-DOF (M-DOF) frame structure without intermediate discontinuity. If the response of the M-DOF frame has a predominant vibration mode, then the AS can correctly represent the behaviour of such a M-DOF system. The equivalence procedure is described in the details in [21]. It is assumed that \(m_{1}\), \(m_{2}\), \(k_{1}\), and \(k_{2}\) remain the same for the MS. The procedure is based on the following two assumptions:  
* The frequency \(\omega_{m}\) of the main mode of the M-DOF system without discontinuity and of the first mode of the AS (Figure 2b) are the same;
* The modal displacements of the storey of the discontinuity and top storey (i.e., \(L\)-th and \(N\)-th storeys in Figure 1a of the M-DOF system without discontinuity are equal to the modal displacements \(u_{1}\) and \(u_{2}\) of the AS.  
The masses \(m_{1}\) and \(m_{2}\) of the AS correspond to the total mass of the substructure and superstructure, respectively. Assuming storeys of equal mass \(m\) entails  
\[m_{1}=mL;\;\;m_{2}=m(N-L), \tag{4}\]  
where \(m\) is the mass of a storey, \(L\) is the level where the intermediate discontinuity is introduced, and \(N\) is the number of storeys of the frame. The masses of the MS are \(\tilde{m}_{1}\), \(m_{d}\), and \(m_{2}\). In particular, the mass of the substructure reads  
\[\tilde{m}_{1}=m(L-1)+m_{0};\;\;m_{d}=m, \tag{5}\]  
where \(m_{0}=\gamma m\) (\(\gamma=0.5\)) is the mass of additional stiffening elements. Such stiffening elements are those that should be added to the level just below the discontinuity to maintain the same lateral stiffness available before the introduction of the discontinuity (Figure 1a). The value \(\gamma=0.5\) is chosen because only horizontal structural elements are needed to restore the stiffness of the disconnected storey, which, consequently, has a mass smaller than that of an entire storey.  
The equivalence procedure requires the identification of the main mode of the M-DOF frame structure without discontinuity. It is assumed that the main mode of the structure has the highest modal mass participation factor \(\mathcal{L}\). Let \(\Phi\) be the vector that collects the modal displacements along a reference vertical line and along the direction to which corresponds the highest factor \(\mathcal{L}\). It reads  
\[\Phi=\{\phi_{1},\phi_{2},\ldots,\phi_{L},\ldots,\phi_{N-1},\phi_{N}\}^{T} \tag{6}\]  
The following relationships for equivalent stiffnesses \(k_{1}\) and \(k_{2}\) of the AS can be obtained as in [21]:  
\[k_{1}=\frac{\omega_{m}^{2}(m_{1}\phi_{L}+m_{2}\phi_{N})}{\phi_{L}};\;\;k_{2}= \frac{\omega_{m}^{2}m_{2}\phi_{N}}{\phi_{N}-\phi_{L}}, \tag{7}\]  
where \(\phi_{L}\) and \(\phi_{N}\) are the components of the vector \(\Phi\) that refer to the \(L\) and \(N\) storeys of the M-DOF system.  
It is worth remarking that the stiffness values of the superstructure and substructure are assumed to remain unchanged after the discontinuity is introduced (i.e., \(k_{1}\) and \(k_{2}\) obtained by Equation (7) are the same for both AS and MS). The ability of both MS and AS to describe a generic 3D frame structure, without and with intermediate discontinuity, and the limitations of such archetype models were discussed in [23].