# Seismic Performance of Frame Structure with Hysteretic Intermediate Discontinuity
## 1 Introduction  
The seismic protection of structures is one of the biggest challenges in civil engineering. In recent years, the building technical codes increased the requirements in terms of structural capacity that a structure needs to fulfil, both for new buildings and the seismic retrofit of existing buildings. New buildings can meet the needed requirements by means of a proper design. Instead, improving the seismic response of existing buildings may require invasive interventions that may modify the serviceability of such buildings. There is a wide literature available about the seismic retrofit of structures. Base isolation (BI) and Tuned Mass Damper (TMD) are two widely known techniques. The first studies on base isolation and tuned mass damper systems were presented in [1, 2], respectively. [PERSON], in [3], and [PERSON] et al., in [4], showed that a TMD placed on the floor immediately above the isolation layer in a base isolated structure can reduce the base displacement. A more complex version of TMD, named Tuned Mass Damper Inerter (TMDI), was presented in [5] and was obtained by coupling a TMD with an inerter. In [5], such a device was intended to reduce the displacement demand of base-isolated structures, but in [6], instead, the same device was used to reduce the vibrations in tall buildings. Protection systems based on the TMD have been used with limited success in the seismic protection of structures because of the difficulty in tuning the TMD to multiple frequencies, as it is needed for seismic excitations. [PERSON] et al. [7] showed that it is possible to decrease a TMD tendency to detuning by designing a TMD with a high ratio between the mass of the TMD and the mass of the structure to be protected. To achieve such a high mass ratio, some authors purposed the combined use of TMD and BI. This combination is commonly named inter-storey isolation: in inter-storey isolation, base-isolated and tuned-mass-damper behaviours coexist. The inter-storey isolation has become widely used for the seismic protection of structures also because it is usually characterized by a low impact on the aesthetic of the building. Even though there are many works about the use of inter-storey isolation, most of them are related to case studies, and only a few of them are devoted to clarify the general aspects of its mechanical behaviour. For example, [PERSON] et al. added new storeys on the rooftop of an existing 12-storey building [8] and connected the two parts of the structure with an isolation layer, thus making the upper part of the structure work as a TMD to reduce the vibration of the lower part of the structure. [PERSON] et al. [9; 10; 11] presented analytical and experimental analyses of the response of building structures with inter-storey isolation. In [12; 13; 14], the roof of the building was used as a vibrating damper to protect the structure, whereas in [15; 16], entire sections of the studied structures were converted into vibrating masses by means of a uniform disconnection along the height of the building. An analogous technique for the reduction of the seismic displacements is the partial mass isolation (PMI) technique presented in [17], where the performance of the PMI is compared to those of conventional TMD and BI. In order to highlight the general mechanisms behind inter-storey isolation, a suitable option is to analyse its effects on an archetype model of a structure. In the literature, archetype models are often used to highlight the main characteristics of multi-Degree-Of-Freedom (DOF) models. For instance, in [1; 2; 18; 19], the studied systems are represented by archetype 2-DOF models. A similar approach, aimed at studying the general behaviour of the inter-storey isolation with a 2-DOF archetype model of the structure, has been presented in [20; 21; 22]. Specifically, in [20; 21] and in this paper, the inter-storey isolation is named intermediate discontinuity, since it represents a sudden change in the stiffness of the structure.