## 3 Design of the Seismic Retrofit  
The case study structures are retrofitted in accordance with the IBC [30] referring to the seismic loads for Life safety limit state (LLS) related to the municipalities of Benevento (Long 14.787\({}^{\circ}\), Lat 41.1305\({}^{\circ}\)) and Tramutola (Long 15.7919\({}^{\circ}\), Lat 40.3176\({}^{\circ}\)), which are two sites of high seismicity in southern Italy. The design is performed assuming a typical residential building use with nominal life V\({}_{\rm N}\) = 50 years and functional class cu = II. A T\({}_{1}\) topography, and soil type B is considered for Benevento resulting in a PGA equal to 2.94 m/s\({}^{2}\), while T\({}_{1}\) topography and soil type C are assumed for Tramutola for a PGA = 3.41 m/s\({}^{2}\). The retrofit design is carried out applying a procedure developed by the authors of work [1,41]. This methodology is implemented in the ADRS space, where the structural system composed of the RC frame and the hysteretic dampers is modelled through an equivalent SDOF system, characterized by its secant stiffness and equivalent viscous damping ratio. It is important to recall that a pre-requisite for this procedure is that the behavior of the building is governed by the first mode, legitimating the condensation of the MDOF structure to the equivalent SDOF system. Therefore, this methodology is valid for low-rise and medium-rise buildings, regular in plan and in elevation, as the case studies analyzed in the present work. The sizing of the dampers is carried out in order to reach a desired structural performance level, which is expressed in terms of target displacement \(d_{p}\), evaluated at the top of the structure. In the present study, \(d_{p}\) is selected with the aim of guaranteeing the Immediate Occupancy performance level [42] by keeping the main frame in the elastic range or by conceiving at most a controlled concrete cracking, limiting structural damage as much as possible. Therefore, the structures are immediately accessible after a main earthquake correspondent to the LLS according to the IBC [30], characterized by return period \(T_{R}\) of 475 years, with a probability of non-exceedance of 10% for nominal life V\({}_{\rm N}\) = 50 years of a residential building (cu = II). Figure 6 shows the capacity curves of the case study buildings and the relative \(d_{p}\)s which correspond to the ending point of the elastic branch, or at most at the beginning of the second branch relevant to the cracked concrete [43]. Figure 6: Capacity curves in X and \(Z\) directions for (a) a 3-story building and (b) a 6-story building with indication of the performance point with target displacement \(d_{p}\). In this study, the two structures are retrofitted by using hysteretic steel dampers installed into diagonal steel braces (Figure 7), which is a common configuration for frame buildings [1; 5; 6; 43; 44; 45]. An elastic-perfectly plastic constitutive behavior is assumed for the dampers, similarly to other studies [1; 6; 41; 43; 44]. Four units on each floor in both X and Z directions are installed in the external bays according to the layout shown in Figure 7.  
The mechanical properties of each diagonal steel brace and the encased hysteretic steel damper form an in-series system can be expressed as a combination of the properties of the linear elastic brace (B) and of the elasto-plastic damper (D), as shown in Figure 8, where \(K_{B}\) and \(K_{D}\) are the elastic stiffnesses of the brace and of the damper, \(V_{y}\) is the maximum force of the damper, \(d_{y}\), \(d_{bd}\) and \(d_{2}\) are the yield displacement, the design seismic displacement calculated at LLS and the ultimate displacement of the damper at CLS, respectively. Damper ductility \(\mu_{D}\), defined as the ratio between \(d_{bd}\) and yield deflection \(d_{y}\), for standard steel hysteretic dampers typically ranges between 4 and 16 [1; 15; 43; 41].