# Seismic Performance of Frame Structure with Hysteretic Intermediate Discontinuity
## 3 Harmonic Analysis
### Gain Maps  
This section analyses how \(\Psi\) affects the dynamical behaviour of the system with intermediate discontinuity. The effects of increasing values of \(\Psi\) are assessed by evaluating the gain coefficient Equations (8) and (9). The results of the parametric analysis are summarised in gain maps that represent the gain coefficients, \(\alpha_{1}\) and \(\alpha_{2}\), plotted in the parameter plane \(\eta-\rho\). Each map refers to a specific value of \(\Psi\). The maps on the left column of Figure 5 are obtained for \(\Psi=0.25\), whereas the maps on the right column refer to \(\Psi=0.75\). Moreover, the maps of Figure 5a show the \(\alpha_{1}\) coefficient, whereas the maps of Figure 5b show the \(\alpha_{2}\) coefficient. The maps are obtained for \(\beta=0.75\) and \(A=0.1\) g; they show a strong dependence on \(\rho\) and, up to a threshold value, on \(\eta\). At higher values of \(\eta\) (where the value of \(\eta\) depends on the considered map), the gain coefficients become independent of \(\eta\) (i.e., in both the \(\alpha_{1}\) and \(\alpha_{2}\) maps, the contour levels tend to become horizontal). It is useful to remark that \(\eta\) directly depends on the yielding displacement; hence, high values of \(\eta\) refer to high values of \(u_{y}\). A frame with intermediate discontinuity and small values of \(\eta\) manifests a stronger non-linear behaviour of the disconnection devices than systems with higher \(\eta\). For very high value of \(\eta\), the disconnection devices tend to manifest a linear elastic behaviour since the high-yielding displacement \(u_{y}\) cannot be reached during the motion. The impossibility to reach \(u_{y}\) during the motion is more evident at higher values of the stiffness ratio \(\Psi\), due to the higher post-elastic stiffness of the disconnection devices. For the chosen values of \(\beta\), \(A\), and \(\Psi\), both \(\alpha_{1}\) and \(\alpha_{2}\) are smaller than unity in the whole parameter plane, which means that the intermediate discontinuity reduces the displacement of the substructure and the drift of the superstructure with respect to those of the original frame structure. Under a harmonic excitation, after a transient time that depends on the damping of the structure, the system undergoes stationary dynamics. To understand the behaviour of the disconnection devices for increasing values of \(\eta\), the stationary hysteretic cycles are shown for points P\({}_{1}\) and P\({}_{3}\) in Figure 5c, and for points P\({}_{2}\) and P\({}_{4}\) in Figure 5d. The hysteretic cycles show the value of the dimensionless force \(\sigma_{h}=F_{h}/k_{d}u_{y}\) as a function of the dimensionless displacement \(\chi_{d}=u_{d}(t)/u_{y}\). By comparing the cycles in the first column of Figure 5c,d, which refer to \(\Psi=0.25\), with the cycles in the second column, which are obtained for \(\Psi=0.75\), it can be observed that a higher value of \(\Psi\) corresponds to smaller hysteresis cycles. Similarly, an increase of \(\eta\) also leads to a reduction of the area of the hysteretic cycles (compare the cycles of Figure 5c,d). Figure 4: Frequency-response curves: (**a**) \(L=3\), \(A=0.6\) g, \(\rho=0.2\), and different values of the parameter \(\psi\); (**b**) \(L=5\), \(A=0.6\) g, \(\rho=0.2\), and different values of the parameter \(\psi\); (**c**) \(L=3\), \(A=0.6\) g, \(\Psi=0.25\), and different values of \(\rho\); and (**d**) \(L=3\), \(\Psi=0.25\), \(\rho=0.2\), and different values of \(A\). Figure 6 shows the gain maps obtained for \(\beta=1\) and \(\Psi=0.25\). The maps in Figure 6a and Figure 6b are obtained for \(A=0.1\) g and \(A=0.6\) g, respectively, assuming that the discontinuity is located at the third level (\(L=3\)). Figure 6c, instead, shows maps obtained for \(A=0.1\) g and a discontinuity located at the fifth level (\(L=5\)). The comparison of Figure 6a and Figure 6b highlights the dependence of the gain maps on \(A\).