# Seismic Isolation Devices for Protecting RC Buildings: The Frangipane School in Reggio Calabria
## 4 Finite Element Numerical Analysis
### Vibration Modes of MU System (LMOD Analysis)  
The first 12 vibration modes were first numerically estimated for the Frangipane structural system. The most flexible direction of the system was found to coincide with the longitudinal one (\(x\) direction), with a fundamental vibration period of \(T_{1}\) = 0.55 s and a participant mass in the order of M = 75.02%. It is worth noting that the calculated participating mass also denotes a mostly longitudinal translational vibration mode of the structure as a whole, with minimum rotational mass contribution (around 0.12%).  
Similar trends were observed for the second vibration mode, which is characterized by rather clear translational vibration in the transversal (\(y\)) direction of the system and a rather null mass contribution participating in rotation (less than 0.13%). The third vibration mode, finally, was detected in a purely torsional shape, with around 77% of participating mass in the rotational term. For the fourth (translational in the \(x\) direction, fifth (translational in the \(y\) direction) and sixth (rotational around \(z\) direction) vibration modes, the participating mass was quantified in the order of 10-11% each. This means that with the first six vibration  
Figure 8: Lumped plasticity model approach: (**a**) moment-rotation constitutive law according to EC8; (**b**) quadrilinear moment-rotation constitutive law; (**c**) constitutive shear law.  
modes of the system, the minimum of 85% of participating mass required by EC8 was properly satisfied. Detailed mass contributions for vibration modes are summarized in Table 3, while Figure 9 reports the modal shapes of the system.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Tran-x**} & \multicolumn{4}{c}{**Tran-y**} & \multicolumn{4}{c}{**Rot-x**} & \multicolumn{4}{c}{**Rot-y**} & \multicolumn{4}{c}{**Rot-z**} \\ \hline
**Mode** & **T** & **M** & **Sum** & **M** & **Sum** & **M** & **Sum** & **M** & **Sum** & **M** & **Sum** \\
**Order** & **[s]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** & **[\%]** \\ \hline
1 & 0.5542 & 75.0189 & 75.0189 & 0.0052 & 0.0052 & 0.0002 & 0.0002 & 0.0003 & 0.0003 & 0.1187 & 0.1187 \\
2 & 0.4648 & 0.0045 & 75.0233 & 76.8187 & 76.8239 & 0.0003 & 0.0005 & 0 & 0.0003 & 0.0141 & 0.1328 \\
3 & 0.4515 & 0.1118 & 75.1352 & 0.0277 & 76.8516 & 0.0004 & 0.0009 & 0.0001 & 0.0005 & 76.4454 & 76.5782 \\
4 & 0.2517 & 10.2946 & 85.4298 & 0 & 76.8516 & 0.0015 & 0.0024 & 0.0097 & 0.0101 & 0.0023 & 76.5806 \\
5 & 0.1761 & 0 & 85.4298 & 8.8536 & 85.7052 & 0.106 & 0.1084 & 0 & 0.0102 & 3.7938 & 80.3744 \\
6 & 0.1697 & 0.0001 & 85.4299 & 4.0332 & 89.7384 & 0.0177 & 0.1261 & 0.0001 & 0.0103 & 8.8168 & 89.1912 \\
7 & 0.1516 & 8.5453 & 93.9752 & 0 & 89.7384 & 0.008 & 0.1342 & 0.3111 & 0.3214 & 0.0021 & 89.1933 \\
8 & 0.1114 & 5.798 & 99.7732 & 0.0044 & 89.7429 & 0.0132 & 0.1473 & 0.3608 & 0.6822 & 0.0204 & 89.2137 \\
9 & 0.1052 & 0.0189 & 99.7921 & 4.409 & 94.1519 & 1.0836 & 1.231 & 0.0032 & 0.6854 & 2.9865 & 92.2002 \\
10 & 0.1031 & 0.0066 & 99.7987 & 2.5569 & 96.7088 & 0.0925 & 1.3235 & 0.0073 & 0.6927 & 4.2332 & 06.4334 \\
11 & 0.0771 & 0 & 99.7987 & 0.0007 & 96.7095 & 0.0165 & 1.34 & 0.5196 & 1.2123 & 0.0001 & 96.4335 \\
12 & 0.0738 & 0.0119 & 99.8107 & 0 & 96.7095 & 0.0079 & 1.3479 & 0.1769 & 1.3892 & 0.0465 & 96.4801 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary of vibration periods and participating mass contributions for the first 12 vibration modes of MU system (LMOD analysis, Midas).  
Figure 9: Fundamental vibration modes of the MU system (Midas): top view and axonometric view. (**a**) Mode 1 T\({}_{1}\) = 0.5542 s (M = 75.02%). (**b**) Mode 2 \(T_{2}\) = 0.4648 s (M = 76.82%). (**c**) Mode 3 \(T_{3}\) = 0.4515 s (M = 76.45%).