## 5 Structural Behavior and Vibrational Analysis
### Modal Calibration
From Table 3, it can be noted that the value of the natural period of the fundamental vertical mode of vibration (\(T\) = 0.219 s) is near that of the torsional mode of vibration (\(T\) = 0.197 s), and consequently, if vertical mode occurs, the torsional one could also simultaneously activate, causing dangerous vibrations in the structure. So, it is suggested to introduce into the model additional steel cables, represented in Figures 10c and 12, in order to make the translational and the torsional modes of vibration asynchronous. Figure 11: Three-dimensional model after the first update. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Model**} & \multicolumn{3}{c}{**Effective Modal Mass**} & \multicolumn{3}{c}{**Modal Participating**} & \multicolumn{3}{c}{**Nat.**} & \multicolumn{3}{c}{**Net.**} \\  & **Mass** & & & & & & & & & **Mass Ratio** & **Freq.** & **Per.** \\ \hline \(n_{e}\) & \(M_{i}\) [kg] & \(m_{eX}\) [kg] & \(m_{eX}\) [kg] & \(m_{eZ}\) [kg] & \(m_{eY}\) & \(m_{eZ}\) & \(\rho_{mX}\) & \(\rho_{mZ}\) & \(\rho_{mZ}\) & **[Hz]** & **T [s]** \\ \hline \(1\) & 32600.03 & 0.00 & 58121.09 & 0.00 & 679.33 & 0.01 & 0.21 & 0.000 & 0.763 & 0.000 & 3.169 & 0.316 \\ \(2\) & 34323.16 & 0.00 & 0.00 & 54911.9 & 0.78 & 118.15 & 0.00 & 0.000 & 0.000 & 0.721 & 4.576 & 0.219 \\ \(3\) & 13929.08 & 0.00 & 6.53 & 0.00 & 466015 & 0.00 & 903.1 & 0.000 & 0.000 & 0.000 & 5.083 & 0.197 \\ \(14\) & 12619.2 & 52992.5 & 0.00 & 0.00 & 0.01 & 1188580 & 1.04 & 0.696 & 0.000 & 0.000 & 15.459 & 0.065 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of the modal analysis after the first model update (\(m_{eX}\), \(m_{eY}\), \(m_{eZ}\): translational masses; \(m_{\#jX}\), \(m_{\#jY}\), \(m_{\#jZ}\): rotational masses). Figure 10: Cross-section of the torsional modal shape: (**a**) original model (2 nd mode of vibration, \(T\)=0.239 s; \(m_{\#jX}\) = 191285.51 kg m\({}^{2}\)); (**b**) after the first model update (3 rd mode of vibration, \(T\) = 0.197 s; \(m_{\#jX}\) = 466015.09 kg m\({}^{2}\)); (**c**) after the second model update (3 rd mode of vibration, \(T\) = 0.17 s; \(m_{\#jX}\) = 421632.16 kg m\({}^{2}\)). The results of the modal analysis after this second model update are reported in Table 4. After the second modal correction, the percentage gap in terms of periods between vertical and torsional modes of vibration increases from 8.78% to 21% with respect to the initial model and from 10% to 21% with respect to the first modal update. The adoption of the above modifications could safeguard the bridge from irregular deformations and from dangerous coupling lateral-torsional vibration modes under external loads such as wind or earthquakes [17; 18; 21], thereby enhancing its service performance. Finally, it can be verified that the final structural configuration of the bridge fulfills the limits imposed by technical codes with regard to the vibrations due to walking pedestrians. In fact, common physical activities such as jumping, running and walking on footbridges may produce dynamic effects, and thus, significant vibrations [19; 20]. The assessment of the compliance of the main required comfort constraints in the vertical and horizontal directions, according to Eurocode 0 [23] and Eurocode 5 [12], is summarized in Table 5.  
The comfort criteria are defined in terms of the maximum acceptable acceleration of any part of the deck. The maximum acceptable values of acceleration are 0.7 m/s\({}^{2}\) for vertical vibrations and 0.2 m/s\({}^{2}\) for horizontal vibrations. Such verification of the comfort criteria should be performed only if the deck fundamental frequency is less than 5 Hz for vertical vibrations and less than 2.5 Hz for horizontal vibrations. In conclusion, assessing the acceptability of these vibrations from the early conceptual design stage could help to avoid both discomfort to pedestrians and deterioration of the structural integrity during the bridge's service life.