## 5 Structural Behavior and Vibrational Analysis
### Modal Calibration  
The results of the modal analysis are summarized in Table 2, while the initial bridge configuration is reported in Figure 9. In particular, Table 2 shows the fundamental modes of vibration in the transversal (\(y\)) direction (\(T=0.335\) s), in the vertical (\(z\)) direction (\(T=0.218\) s) and in the torsional direction around the longitudinal axis (\(x\)) (\(T=0.239\) s; \(m@\))\(j\)X=191285.51 kg.m\({}^{2}\)). The torsional mode of vibration around the longitudinal axis is depicted in Figure 10a; it can be observed that the corresponding cross-section deformation is nonhomogeneous and assumes a rhomboidal shape. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & **GL24h** & **GL28h** & **GL32h** & & **S275H** & **S355** \\ \hline \(f_{t,0,k}\) [kN/cm\({}^{2}\)] & 1.92 & 2.23 & 2.56 & \(f_{yk}\) [kN/cm\({}^{2}\)] & 27.5 & 35.5 \\ \(f_{c,0,k}\) [kN/cm\({}^{2}\)] & 2.4 & 2.8 & 3.2 & \(f_{kk}\) [kN/cm\({}^{2}\)] & 43 & 51 \\ \(E\) [kN/cm\({}^{2}\)] & 1150 & 1260 & 1420 & \(E\) [kN/cm\({}^{2}\)] & 21000 & 21000 \\ \(\gamma^{\prime}\) [kN/m\({}^{3}\)] & 4.2 & 4.6 & 4.9 & \(\gamma\) [kN/m\({}^{3}\)] & 78.5 & 78.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of materials (GL24h, GL28h, GL32h: Glued Laminated Timber grades; \(f_{t,0,k}\): characteristic tension strength parallel the grain;\(f_{c,0,k}\): characteristic compression strength parallel the grain; S275H: Steel Grade according to UNI EN 10210-1; S355: Steel Grade according to UNI EN 10025-2;\(f_{yk}\): characteristic yield strength; \(f_{tk}\): characteristic failure strength; \(E\): Young’s modulus; \(\gamma\): specific weight). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{3}{c}{**Modal**} & \multicolumn{3}{c}{**Effective Modal Mass**} & \multicolumn{3}{c}{**Modal Participating**} & \multicolumn{3}{c}{**Nat.**} & \multicolumn{3}{c}{**Nat.**} \\ \multicolumn{3}{c}{**Mass**} & & & & & & & **Mass Ratio** & & **Freq.** & **Per.** \\ \hline \(\mathbf{n_{c}}\) & \(M_{t}\) [kg] & \(m_{eX}\) [kg] & \(m_{eX}\) [kg] & \(m_{eZ}\) [kg] & \(m_{eZ}\) [kg] & \(m_{eZ}\) [kg] & \(m_{eX}\) & \(m_{eZ}\) & \(m_{eZ}\) & \(\mathbf{f_{eZ}}\) & \(\mathbf{T}\) [s] \\ \hline
1 & 21094.16 & 0.00 & 50308.9 & 0.00 & 4158.6 & 0.00 & 2.28 & 0.000 & 0.662 & 0.000 & 2.989 & 0.335 \\
2 & 15913.94 & 0.00 & 6975 & 0.00 & 191285 & 0.00 & 120.83 & 0.000 & 0.092 & 0.000 & 4.178 & 0.239 \\
3 & 34247.83 & 0.00 & 0.00 & 54767.6 & 0.77 & 118.43 & 0.00 & 0.000 & 0.000 & 0.721 & 4.580 & 0.218 \\
14 & 12602.66 & 52917.5 & 0.00 & 0.00 & 0.01 & 1182525 & 0.97 & 0.696 & 0.000 & 0.000 & 15.463 & 0.065 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of the modal analysis of the original model (\(m_{eX}\), \(m_{eY}\), \(m_{eZ}\): translational masses; \(m_{\oplus\rangle|X}\), \(m_{\oplus\rangle|Y}\), \(m_{\oplus\rangle|Z}\): rotational masses). Figure 9: Three-dimensional model of the initial configuration. In order to avoid such a deformed configuration, lateral steel tension rods were added on both sides of the bridge along the entire span, thereby conferring stability to the horizontal section (Figures 10b and 11). The results of the modal analysis of the new model are reported in Table 3; Figure 10b shows the cross-section deformation of the torsional mode of vibration (\(T\) = 0.197 s; \(m\#jX\) = 466015.09 kg m\({}^{2}\)), which was homogeneous after the above adjustment. Moreover, it can be argued that the modal participating mass ratios of the fundamental modes of vibration in the three directions are significant (\(\sim\)70%), indicating that there is no excessive dispersion of mass among oscillation modes, and that the structure is well organized [33; 34].