## 3 TMD Design Method  
The mass of the TMD constituted a RC slab, 19.65 m long and 13.40 m wide, overlooking the entire rooftop and protruding by about 1 m from each facade, as shown in Figure 7. The slab thickness was constant, and it was assumed as a design parameter of the TMD mass. A steel structure composed of braced frames oriented in the \(X\) and \(Y\) directions was installed on the rooftop to support the TMD above the stairwell and the elevator shaft, as shown in Figure 8. The RC slab lay on flat surface sliders (FSSs) anchored to the beam-column joints of the steel structure, which allowed the RC slab to sway out of phase with the building. The correct disposition of LRIs, LDRIs and VLDs in the two proposed solutions will be described in the design process in the next section. In the steel structure, horizontal (Figure 7) and vertical braces (Figure 8) were used to provide in-plane stiffness to the plane at the base of the FSSs and horizontal stiffness to the steel frame, respectively. The steel structure's self-weight is negligible in comparison  
Figure 5: Site elastic response spectrum for the horizontal acceleration component according to [41]. Figure 6: Horizontal accelerogram used in one of the seismic analyses of the building. with that of the building. Therefore, the increase in the building's seismic mass due to this structure is also negligible. The design of the TMD was performed taking into account the availability of a new external living space at the level of the original roof and, consequently, the increase in gravity loads due to the variation in the roof's intended use. In the FEM of the building, the steel structure was modeled through frame elements, while the RC slab was modeled through shell elements. For the FSS modeling, non-linear link elements of 'Friction Isolator' type were used. Since the horizontal stiffness and the energy dissipation contribution of the FSSs are negligible, to provide the TMD system with these properties, two solutions are proposed. In one solution (hereinafter referred to as TMD1), horizontal stiffness and damping are both provided by RIs, which are placed on the perimeter of the steel structure (Figure 7). Figure 8: Elevation of the building equipped with the TMD in \(X\) direction on the left and \(Y\) direction on the right. Figure 7: Top view of the steel structure supporting the TMD, and arrangement of dampers, isolators and sliders. In the other solution (referred to as TMD2), horizontal stiffness is provided by LDRIs and damping by VLDs, which are installed along the perimeter of the steel structure, as can be seen in Figures 7 and 8. The construction of the TMD may follow these steps:  
* Installation of metal anchors to connect the base of the steel frame columns to the top end of the RC columns and walls of the building;
* Assemblance of the steel frame with columns, beams and vertical and horizontal braces;
* Installation of the FSSs, LRIs (in TMD1) and LDRIs (in TMD2) at the intersection nodes of beams, on the top of the steel frame, in the stairwell and in the elevator shaft, using steel plates to support them, if necessary. A way of access must be guaranteed for maintenance and to inspect and replace the isolation devices;
* Positioning of the form and the steel reinforcement for the concrete slab on the top of the steel frame and pouring of the concrete. A slight slope for water collection is required;
* In TMD2, VLDs can be installed after the curing of the concrete, connecting one end to a beam of the steel frame and the other to the under-surface of the RC slab, far enough away from the isolators to let them deform without making contact with the VLDs. In both solutions, the design of the TMD is performed according to the analytical method described by [PERSON] in [PERSON] (2016).