# Seismic Performance of Frame Structure with Hysteretic Intermediate Discontinuity
## 5 Conclusions  
This paper proposes a modelling approach for the study of the intermediate discontinuity. The structure with the discontinuity layer has been modelled by a 3-DOF model and the disconnection devices have been described by using the Bouc-Wen model. In order to evaluate the improvement of the dynamical behaviour obtained by the introduction of the discontinuity, two different systems have been analysed. The first system is the frame after the introduction of the discontinuity, called the main system (MS) and the second system is the reference frame, called the auxiliary system (AS).  
The first part of the paper is focused on the dynamical behaviour of the MS that is analysed by using harmonic excitation. The results of the analysis have been organized in frequency-response curves and gain maps. It has been found that the value of the post-yielding to pre-yielding stiffness ratio \(\Psi\) in the Bouc-Wen model significantly affects the value of the first frequency of the MS. Moreover, by comparing the frequency-response curves of the AS and MS for different values of \(\Psi\), it has been found that lower values of \(\Psi\) may extend the range of frequency where the intermediate discontinuity is effective. The  
Figure 14: Mean gain maps for \(L=3\) and (**a**) \(\Psi=0.10\); (**b**) \(\Psi=0.75\).  
harmonic maps highlighted that small values of \(\Psi\) may reduce the displacement of the superstructure, but also increase the displacement of the substructure.  
The second part of the paper is focused on the seismic response of the models. A preliminary analysis has been carried out considering single seismic records. Five gain coefficients have been defined to evaluate the effects of the introduction of the intermediate discontinuity. Such coefficients have been defined as ratios between the displacements (or the accelerations) of the MS and those of the AS. The results of the analysis have been summarized in gain maps. Such maps show the gain coefficients in a specific parameters plane. The results also show that an increase of \(\Psi\) reduces the displacement of the substructure and increases that of the superstructure. Moreover, higher values of \(\Psi\) reduce the displacement of the discontinuity layer, thus reducing the likelihood of such displacement exceeding a threshold value. It has been found that another possibility to reduce the displacement of the discontinuity layer is to change the level of the discontinuity \(L\). As \(L\) increases, the displacement of the discontinuity layer decreases due to the filtering effect of the substructure. Additionally, the effects of the intermediate discontinuity on the accelerations acting on the structure have been evaluated, and it has been found that when \(\Psi\) decreases, the accelerations may increase.  
Finally, a seismic analysis is carried out considering several seismic records within definite ranges of magnitude and epicentral distance. The results have been organized in gain maps that show the gain coefficients in different parameters planes. Such maps may be a useful auxiliary tool in the understanding of the behaviour resulting from the introduction of an intermediate discontinuity in a frame structure.  
Conceptualization, A.D.E., S.P. and A.C.; Methodology, A.D.E. and A.C.; Software, S.P.; Formal analysis, S.P.; Investigation, A.D.E., S.P. and A.C.; Writing--original draft, S.P.; Writing--review & editing, A.D.E. and A.C.; Supervision, A.D.E. and A.C. All authors have read and agreed to the published version of the manuscript.  
This research was partially funded by CARISPAQ Foundation, year 2022.  
Data Availability Statement: Data, models, and code used during the study are available from the corresponding author by request. Specifically, the following data are available: (_i_) data of the results, and (_ii_) the source code for the numerical integration procedure.  
The authors declare no conflict of interest.