## 2 Case Study
The self-weight and self-mass of structural elements were calculated by the software on the basis of their geometric and material properties. The FEM analyses are performed considering the load combinations at the ultimate limit state (ULS) provided by [41] for static and seismic load conditions. For the seismic analyses, the life-safety limit state, in Italian \"Stato Limite di salvaguardia della Vita\" (SLV), is considered herein. The seismic action was determined considering a building design service life of 50 yr and a ground of type \(B\), which involves a return period of the seismic actions of 475 yr. In the following, the ULS acronym is used to indicate the static load combination of actions. Table 3 shows the periods and the participating mass ratios of the most significant modes of the OB, i.e., the modes having a participating mass ratio greater or equal to 5% and that, globally, provide a mass participating ratio \(\geq\)85% in each main direction. \(T_{i}\) indicates the period of the \(i\)-th mode, and \(\alpha_{i,j}\) indicates the participating mass ratio of the \(i\)-th mode in \(j\) direction. The values of the participating mass ratio in \(j\) direction which contributes the most for each mode are underlined in Table 3.  
A fast non-linear analysis was performed. This is a modal analysis method, implemented by SAP2000, designed to be used for structural systems that are primarily linear elastic but that only have a limited number of predefined non-linear elements. FNA is well-suited for time-history analysis, which is the analysis used in the case study, because of its computationally efficient formulation. This efficiency is due to the separation of the non-linear-objects vector from the elastic stiffness matrix and the damped equations of motions. At each time increment, forces within the non-linear-objects vector are resolved through an iterative process, while the uncoupled modal equations are solved exactly. This allows the formulation to rapidly converge to the equilibrium solution. Regarding the considered case study, the only non-linear members are the devices used to reproduce the stiffness and the dissipation properties of the TMD, which are specified in the following section. Hence, FNA is a suitable analysis method for the case study. The ground motion is modeled using three different artificial accelerograms acting simultaneously along the horizontal directions \(X\) and \(Y\) and the vertical direction \(Z\). Three independent analyses \begin{table}
\begin{tabular}{c c c c} \hline  & \(g_{1,k}\) [kN/m\({}^{2}\)] & \(g_{2,k}\) [kN/m\({}^{2}\)] & \(q_{k}\) [kN/m\({}^{2}\)] \\ \hline Floor & 3.5 & 2.7 & 2.0 \\ Balcony & 3.5 & 2.7 & 4.0 \\ Rooftop & 3.5 & 1.5 & 2.0 \\ Stairs & 4.3 & 1.0 & 4.0 \\ \hline \end{tabular}
\end{table} 
Table 2: Gravity loads per unit area. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Mode \(i\)** & \(T_{i}\) & \(\alpha_{i,X}\) & \(\alpha_{i,Y}\) & \(\alpha_{i,Rz}\) & \(\Sigma\,\alpha_{i,X}\) & \(\Sigma\,\alpha_{i,Y}\) & \(\Sigma\,\alpha_{i,Rz}\) \\ \hline
1 & 0.39 s & 66.3\% & 0.1  
were conducted, considering different ground motions. The accelerograms were generated by the software SIMQKE_GR [43], matching their spectrum with the site elastic response spectrum provided by the Italian Code for a 5% damping coefficient, which is shown in Figure 5. The 5% damping factor takes account of all phenomena related to the building shaking under horizontal loads contributing to energy dissipation in elastic conditions, such as concrete viscous properties and cracking. The accelerograms have a stationary part that starts at time \(t=2\,\mathrm{s}\) and lasts \(10\,\mathrm{s}\), for a total duration of \(30\,\mathrm{s}\). One of the horizontal accelerograms used in the analyses is shown in Figure 6. The accelerometer record presented in Figure 6 refers to the life-safety limit state according to the Italian Building Code [41], hence with a 10% probability of occurrence in 50 years.