## 3 FE Model Updating with Experimental Modal Data
### The Updating Results  
Table 3 provides a summary of the numerical modal results compared with the experimental ones. It should be noted that the sensitivity-based parametric updating was also applied to the initial model, but without giving significant improvements in the modal results. As mentioned previously, in comparison to the initial model, the improvement of the manually tuned FE model was significant both in terms of the relative differences of natural frequencies and the MAC values, with the exception of the last mode. It was concluded that the modelling errors due to simplification or idealization from the assumptions, such as those related to the superelevation and boundary conditions, could only be reduced by improvement of the FE model, instead of the parametric updating. It suggested that superelevation and the corresponding transverse slope, even of a small degree, may have a profound influence on the dynamics of the short-span skewed bridge. The sensitivity-based model updating procedure led to the more satisfactory modal outcomes in reference to the experimental modal results. The absolute value of \(\Delta\)f, averaged over six modes excluding Mode 4, is 3.6%, 1.5%, and 1.4% for the initial model, tuned model, and updated model, respectively. The absolute value of \(\Delta\)f, averaged over seven modes, is 1.4% and 1.3% for the tuned model and updated model, respectively. Moreover, the MAC value, averaged over six modes excluding Mode 4, is 0.85, 0.87, and 0.88 for the initial model, tuned model, and updated model, respectively. The MAC value, averaged over seven modes, is both 0.84 for the tuned model and updated model. Although the relative differences of frequencies and the MAC values were not so much different between the tuned and updated models, the improvement was more significant when the modal displacements of the transversal cross section were plotted. As shown in Figure 7, the updated FE model provided an even better match to the experimental results with respect to the cross-sectional deflection than the manually tuned model. It was speculated that the similar MAC values for the tuned and updated models were related to the fact that only one node (No. 34) was measured on the other side of the bridge. Therefore, improvement in the mode shapes, especially those related to the vertical bending in the transversal direction and torsion, were not evident by measure of the MAC values. Figure 5 provides the mode shape plots of the updated model in alignment with the identified experimental modal results. It was evident that most of the modes were associated with a significant cross-section deflection and deformation. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Exp. Mode**} & \multicolumn{1}{c}{**No.**} & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \cline{2-9}  & **fwl3 [Hz]** & 7.54 & 8.04 & 13.08 & 15.28 & 21.99 & 29.24 & 34.96 \\ \hline \multirow{4}{*}{**Initial FE Model**} & **fnl [Hz]** & 7.46 & 8.36 & 12.29 & n/a & 21.41 & 27.64 & 34.23 \\  & **Af [\%]** & \(-1.1\) & 4.0 & \(-6.0\) & n/a & \(-2.6\) & \(-5.5\) & \(-2.1\) \\  & **MAC** & 0.95 & 0.77 & 0.91 & n/a & 0.77 & 0.88 & 0.84 \\ \hline \multirow{4}{*}{**Manually Tuned FE Model**} & **fnl [Hz]** & 7.48 & 7.95 & 13.14 & 15.13 & 22.23 & 28.85 & 33.48 \\  & **Af [\%]** & \(-0.8\) & \(-1.1\) & 0.5 & \(-1.0\) & 1.1 & \(-1.3\) & \(-4.2\) \\  & **MAC** & 0.97 & 0.86 & 0.94 & 0.69 & 0.79 & 0.88 & 0.77 \\ \hline \multirow{4}{*}{**Updated FE Model**} & **fnl [Hz]** & 7.75 & 7.95 & 12.70 & 15.19 & 21.97 & 29.55 & 35.08 \\  & **Af [\%]** & 2.8 & \(-1.1\) & \(-2.9\) & \(-0.6\) & \(-0.1\) & 1.1 & 0.3 \\ \cline{1-1}  & **MAC** & 0.97 & 0.93 & 0.90 & 0.64 & 0.80 & 0.87 & 0.80 \\ \hline \hline \end{tabular} Note: Df = (f\({}_{\rm n}-f_{\rm wl3}\))/fwl3; MAC values are calculated between the experimental and numerical modes.  
\end{table} 
Table 3: Comparison of the experimental and numerical modal results.