## 3 Seismic Response of the Reference Structure and Mitigation Target  
A finite element model of the reference structure was generated in the UGS-FEMAP(r) environment [24]. The mesh geometry and the relevant data are reported in Figure 3.  
Linear bar elements were used for the mesh of the pillars (one single element per structural item) and horizontal beams (two elements per structural item), while each floor was modeled as a rigid body. To implement the rigid-body condition, auxiliary nodes were generated at the center of each floor and were then rigidly connected to all the nodes of the horizontal beams that support the floor; in these rigid connections, the degrees of freedom of the floor-beam grids were slaved to those of the auxiliary nodes at the center of the floors. The central floor nodes were also used to define the inertial properties of each floor by means of lumped mass elements. The grids at the base of the model (Grids 1, 7, 13, and 19) were constrained in all six degrees of freedom, and a modal analysis was launched in the NC-Nastran environment to evaluate the first 10 normal modes of the structure, together with their natural frequencies and generalized masses. The Lanczos method [25] was selected for the calculation of the first twenty modes and related parameters (frequency and generalized mass); the obtained results are recapped in Table 2 and Figure 4 with reference to the first ten modes only. Figure 3: Finite element model of the reference structure, grids, and elements. The modal database of the reference structure and the earthquake accelerogram (a\({}_{y}\)(t)) were given as the input to a set of in-house developed MATLAB(r) [26] routines that implement the formulation that is described in Appendix A for the evaluation of the seismic response of the structure. A viscous modal damping of 0.015 (typical value for metallic structures, [27]) was considered for all modes. Only the first six modes that were characterized by displacements along the direction of excitation (Modes 2,3,5,7,8,10) were taken into account since the residual inertia associated \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Mode Number and Name** & **Frequency (Hz)** & **Generalized Mass (kgm2)** \\ \hline
1 & 1 st bending in the XZ plane & 1.41 & 902.43 \\ \hline
2 & 1 st bending in the YZ plane & 1.41 & 902.43 \\ \hline
3 & 1 st torsion & 3.17 & 61.47 \\ \hline
4 & 2 nd bending in the XZ plane & 4.43 & 926.30 \\ \hline
5 & 2 nd bending in the YZ plane & 4.43 & 926.30 \\ \hline
6 & 3 rd bending in the XZ plane & 7.94 & 1271.39 \\ \hline
7 & 3 rd bending in the YZ plane & 7.94 & 1271.39 \\ \hline
8 & 2 nd torsion & 9.63 & 60.92 \\ \hline
9 & 4 th bending in the XZ plane & 11.51 & 773.40 \\ \hline
10 & 4 th bending in the YZ plane & 11.51 & 773.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Modal parameters of the reference structure (first 10 modes only). Figure 4: First 10 modes of the reference structure. with all the remaining modes was equal to only 2% of the overall inertia that was exhibited by the structure when moving along the \(y\)-axis. The definition of \"residual inertia\" and the explanation of its link to the number of selected modes are given in Appendix A. The response of the structure is synthesized here in terms of the time histories of the following parameters:  
* The floor acceleration, which is intended as the acceleration (\(a_{\rm yj}\)) of the j-th floor (j = 1,2,..., 5) with respect to the ground and along the direction of the excitation (y-axis);
* 1)-th floor (j = 1,2,..., 5). When j = 1, the interstory acceleration/speed coincides with the first floor's acceleration (speed) with respect to the ground. The obtained time histories are plotted in Figures 5-19, while the peak values of the relevant parameters (\(a_{\rm yj}\), \(\Delta a_{\rm yj}\), \(\Delta v_{\rm yj}\) (j = 1,2,..., 5)) are recapped in Table 3.  
Figure 8: Seismic response, 4 th-floor acceleration (Node 48) with respect to the ground.