# Seismic Performance of Frame Structure with Hysteretic Intermediate Discontinuity
## 4 Seismic Analysis
### Gain Maps Obtained from Multiple Seismic Records  
The seismic analysis is carried out by using several seismic records obtained from [32]. The magnitude varies from 4 to 8 and the epicentral distance varies from 10 to 30 km. In the analysis, the target parameters are the mean displacements \(\overline{u}_{i}\) that are defined as the mean of the maximum displacements \(u_{i,j}\) obtained from the integration of the equations of motion with each seismic record; they read  
\[\overline{u}_{i}=\frac{1}{n_{e}}\sum\limits_{j=1}^{n_{e}}\max[u_{i,j}(t)] \tag{11}\]  
with \((i=1,d,2)\). The displacements \(u_{1,j}\), \(u_{d,j}\), and \(u_{2,j}\) are those of the substructure, discontinuity level, and superstructure obtained from the \(j\)th seismic record. The total number of seismic records used in the simulations is \(n_{e}\). The number of seismic records used is sufficient to reach a coefficient of variation (CoV) of \(\overline{u}_{i}\), \(\text{CoV}_{\overline{u}_{i}}\leq 0.2\). Therefore, the predictability of the model to identify the average displacement is higher than 80%. The \(\text{CoV}_{\overline{u}_{i}}\) is defined as  
\[CoV_{\overline{u}_{i}}=\frac{\sigma(\overline{u}_{i})}{\mu(\overline{u}_{i})} \tag{12}\]  
where \(\sigma(\overline{u}_{i})\) and \(\mu(\overline{u}_{i})\) are the standard deviation and mean value of \(\overline{u}_{i}\), respectively. To understand the effectiveness of the disconnection, three coefficients (called mean-gain coefficients in the following) similar to those in Equation (8) are introduced:  
\[\overline{\alpha_{1}}=\frac{\sum\limits_{j=1}^{n_{e}}\max[u_{1,j}(t)]}{\sum \limits_{j=1}^{n_{e}}\max[u_{1,j}^{A}(t)]};\ \ \overline{\alpha_{2}}=\frac{\sum\limits_{j=1}^{n_{e}}\max[u_{2,j}(t)-u_{d,j}(t )]}{\sum\limits_{j=1}^{n_{e}}\max[u_{2,j}^{A}(t)-u_{1,j}^{A}(t)]};\ \ \overline{ \alpha_{3}}=\frac{\sum\limits_{j=1}^{n_{e}}\max[u_{d,j}(t)-u_{1,j}^{A}(t)]}{n_ {e}u_{0}} \tag{13}\]  
Figure 13 shows the gain maps of the mean displacements. Such maps represent the mean-gain coefficients in the parameters plane \(PGA\)-\(\rho\), where \(PGA\) is the peak ground acceleration and \(\rho\) is the stiffness ratio previously defined. All the gain maps are obtained for \(\eta=0.2\). Similarly to previous figures, the regions where \(\overline{\alpha}_{i}<1,(i=1,2)\) are named gain regions and are coloured, whereas those where \(\overline{\alpha}_{i}>1\) are identified by a dark grey colour. The first, second, and third rows of the figure refer to different levels of the discontinuity, \(L=1\), \(L=3\), and \(L=5\), respectively. All maps are obtained for \(\Psi=0.25\). The maps show wide gain regions. As can be observed, \(L=1\) provides the best performances in the reduction of the displacement of the substructure because the values of \(\overline{\alpha}_{1}\) obtained for \(L=1\) are in general smaller than those obtained for other values of \(L\). On the contrary, \(L=5\) provides the largest reduction of the drift of the superstructure since, in this case, \(\overline{\alpha}_{2}\) is smaller than those in the other cases. The maps of \(\overline{\alpha}_{3}\) show values higher than unity only in the zone with high values of \(PGA\) and low values of \(\rho\), specifically in the region where \(PGA>1.1\) g and \(\rho<0.1\). In this region, \(\overline{\alpha}_{1}\) and \(\overline{\alpha}_{2}\) have small values, but the displacement of the discontinuity level exceeds the threshold value \(u_{o}\) (i.e., \(\overline{\alpha}_{3}\) is higher than unity). Nevertheless, such a region is very small and a slight increase of \(\rho\) is sufficient to move in a region of the maps where \(\overline{\alpha}_{3}\leq 1\) and at the same \(\overline{\alpha}_{1}\leq 1\) and \(\overline{\alpha}_{2}\leq 1\). Since all the three coefficients are less than unity the disconnection can be considered a successful protection strategy for the original frame structure.