# Finite Element Models of a Benchmark Footbridge
## 4 Validation of FE Models
### Modal Analysis  
Table 7 summarizes the first eleven natural frequencies extracted through modal analysis from B and S models, with a qualitative description of the dominant deflected shape based on visual inspection of the mode shapes. The experimental modes are extracted with reference to the embedded fiber optic sensors at the south-east leg (discrete strain sensors collecting dynamic response between two fibers in the vertical plane at the top and the bottom of the deck center line). Consequently, flexural vertical modes of the south-east leg are exclusively detected by the SHM system. Therefore, in the following, focus is made on mode shapes with a prevailing vertical component, which are plotted in Figures 12 and 13. Table 8 reports the Modal Assurance Criterion (MAC) values between B and S mode shapes, calculated for the vertical component of the first eight modes. It can be observed that MAC values are very close to unity for modes 3 to 6. Some modes are ordered differently in the B and S models, e.g., mode shapes 1 and 2 present close natural frequencies, and they are reversed in the B model with respect to the S model. This is confirmed by MAC values, which are above 0.9 when mode 1-B and mode 2-S are considered (and vice \begin{table}
\begin{tabular}{l l l l l} \hline \hline  & \multicolumn{3}{c}{**B Model**} & \multicolumn{2}{c}{**S Model**} \\ \hline
**Mode** & **Frequency** & **Qualitative Description** & **Frequency** & **Qualitative Description** \\ \hline
1 & 3.0869 & Flexural-torsional & 3.1197 & Flexural vertical \\
2 & 3.1568 & Flexural vertical & 3.1451 & Flexural-torsional \\
3 & 3.5925 & Flexural vertical & 3.5351 & Flexural vertical \\
4 & 3.708 & Flexural vertical & 3.7129 & Flexural vertical \\
5 & 4.1319 & Flexural vertical & 4.143 & Flexural vertical \\
6 & 4.488 & Flexural vertical & 4.4908 & Flexural vertical \\
7 & 4.6273 & Flexural lateral & 5.2466 & Flexural-torsional \\
8 & 5.3695 & Flexural-torsional & 5.858 & Flexural lateral \\
9 & 5.9998 & Flexural lateral (east legs) & 6.0714 & Flexural-torsional \\
10 & 6.0627 & Flexural lateral & 6.3452 & Flexural-torsional \\
11 & 6.4939 & Flexural vertical & 6.5534 & Flexural vertical \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Summary of first 11 natural frequencies from B and S models [Hz]. Figure 11: Comparison between experimental and numerical displacements. The analysis of frequencies and mode shapes allows outlining the following observations:  
* The B and S models provide consistent results in terms of modal properties;
* Modes 1 and 2 have very close natural frequencies, and in both models, the south-east leg has a significant vertical component. The comparison with experimental natural frequencies is shown in Figure 13. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & \multicolumn{8}{c}{**S Model**} \\ \cline{2-10}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline \multirow{8}{*}{**B Model**} & 1 & 0.062 & **0.9098** & 0.0128 & 0.0055 & 0.0121 & 0.0013 & 0.0591 & 0.0686 \\  & 2 & **0.9658** & 0.0322 & 0.0011 & 0.0038 & 0 & 0.0006 & 0.0084 & 0.0234 \\  & 3 & 0.0002 & 0.0138 & **0.9976** & 0.0346 & 0.0378 & 0.0006 & 0.0016 & 0.003 \\  & 4 & 0.0009 & 0.0312 & 0.0151 & **0.9783** & 0 & 0.0032 & 0.0049 & 0.0398 \\  & 5 & 0.0042 & 0.0131 & 0.0437 & 0.0086 & **0.9912** & 0.0044 & 0.006 & 0.0084 \\  & 6 & 0.0045 & 0.0003 & 0.0015 & 0.0119 & 0.0012 & **0.9911** & 0.002 & 0 \\  & 7 & 0.02 & 0.4102 & 0.0155 & 0.0012 & 0.158 & 0.0586 & 0.2993 & 0.1816 \\  & 8 & 0 & 0.0409 & 0 & 0.0398 & 0.0026 & 0.0072 & **0.9229** & 0.139 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: MAC between B and S mode shapes. Figure 13: Mode shapes 5 to 8 of B model (**left**) and 5 to 7 of S model (**right**). frequencies makes it difficult to definitely identify which of the two numerical modes corresponds to the experimental first mode;
* Mode 3 mainly involves the main span and the south-west and north-east leg.