# Vertical Vibrations of Footbridges Due to Group Loading: Effect of Pedestrian-Structure Interaction
## 3 Results and Discussions
### Investigation of Group Loading  
After obtaining the _DLF_ of each test subject, the case of group loading can be investigated. An additional uncertainty is present in this case, which is the potential lack of synchronization among the test subjects, in spite of the use of the metronome. This problem was tackled by employing a video camera to record the crossings. However, there was a drawback: the camera was placed near one of the footbridge supports, and observation of the images to identify phase differences was possible only for part of the crossing due to eventual occlusions and the lack of resolution when the pedestrians were far from the camera. It is also worth mentioning the accuracy of the phase difference calculations: since the camera recorded 30 frames/s, by checking the period of the walking cycle of a typical pedestrian, a total of 29 frames corresponded to an interval of 2 cycles (for instance, the time interval between successive touches of the tip of the right foot of the pedestrian on the ground). This leads to a time interval of 0.966 s, corresponding to a period T of (0.966/2) = 0.4833 s. The corresponding pacing rate \(f_{p}\) is equal to (1/T) = 2.069 Hz, which is in agreement with the value set in the metronome. Therefore, each video frame covers a change in phase angle of (360\({}^{\circ}\)/14.5) = 24.8 degrees, or 0.433 rads. This is the maximum error in the phase by calculating the phase angles using the processing of the videos. It was noted that the majority of the test subjects had a phase difference (with respect to the leading test subject) of either null or 360\({}^{\circ}\), the latter meaning that a test subject was placing the opposite foot on the ground. It should be noted that this does not make a difference when applying Equation (5), in which the load of each foot is considered to be the same. Due to these several factors, six useful crossings of the whole group of pedestrians, identified by numbers, were selected, and the identified phase differences among each test subject and the leader were introduced to the applied loads of the respective simulations. Distances among pedestrians were disregarded, following the line of thought for group  
Figure 8: Third crossing of Test Subject 2: (a) time response; (b) spectrum. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Pedestrian** & **DLF** & **SD** & **Pedestrian** & **DLF** & **SD** \\ \hline
1 & 0.25 & 0.00 & 6 & 0.30 & 0.01 \\
2 & 0.26 & 0.00 & 7 & 0.28 & 0.01 \\
3 & 0.30 & 0.01 & 8 & 0.25 & 0.03 \\
4 & 0.26 & 0.01 & 9 & 0.25 & 0.01 \\
5 & 0.23 & 0.02 & 10 & 0.35 & 0.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculated dynamic load factors (DLFs) of the test subjects. loading in the UK code [14]. A value of 2.4% for the damping ratio was adopted (highest value from Figure 6), and the root mean square (RMS) acceleration during the crossing was compared between measurements and simulations. This metric was preferred since the time response and spectral signatures from measurements and simulations are not expected to match, as can be seen in Figure 9 for all selected crossings. The RMS results obtained for the selected crossings are shown in Figure 10.  
As a remark, it can be noticed from the plots in Figure 9 that the experimental peak accelerations reached during the crossings are lower than the vibration limits recommended for this footbridge, when the codes and guidelines that deal with the vibration serviceability of footbridges are applied [14; 15; 16]. The summation of absolute errors between the values of each numerical simulation and the experimental values confirms the visual impression of Figure 10 that the performance of the interaction model was superior. However, the difference between the results of force and interaction models is not very significant. The association between the level of vibration and the number of test subjects is a factor of relevance in such a difference. Thus, the effect of the number of pedestrians in the group will be further explored.