## 6 Results and Discussion
### Crowd HSI Analysis  
In crowd HSI analysis it is commonly accepted that the anthropic pedestrian loading \(F_{crowd}\) in the human walking frequency range (1.8-2.2 Hz), following the model of [PERSON] et al. (1978) [35] is equivalent to \(\sqrt{N}F_{p}\) due to the small probability that if there are \(N\) pedestrians on the walkway, all of them are synchronized [41]. However, for the purpose of this study, an HSI analysis of the crowd was considered in which pedestrians are fully synchronized, defined as the worst-case loading scenario, where all pedestrians would march at the same time, applying a constant periodic load with a frequency equal to that of the bridge, for which \(F_{crowd}=NF_{p}\).  
Multiple simulations were performed using the methodology described in Section 5 considering different occupancy conditions (Mass Relation, \(M_{crowd}/M_{br}\)). The peak accelerations and displacements for each occupancy case, considering the HSI model and the model without interaction, are presented in Figure 14. The results suggest that there is an overestimation of the dynamic response of the structure in the model without interaction with respect to that of the HSI model for \(M_{crowd}/M_{br}\) greater than 15%. However, for the same conditions, considering low occupancy rates, the model without interaction underestimated the response of the structure. This phenomenon is mainly due to the crowd-structure frequency detuning (\(\omega_{crowd}/\omega_{br}\)), for which the frequency of the coupled HSI system decreases as the number of pedestrians increases, as shown in Figure 14 (top left).  
Figure 13: Acceleration at Mid-Span \(A_{c}(t)\). (**a**) Vertical direction. (**b**) Lateral direction.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Vertical**} & \multicolumn{2}{c}{**Lateral**} \\ \cline{2-7}  & **Max. HSI** & **Max. without** & **\(\Delta_{Vert}\)[\%]** & **Max. HSI** & **Max. without** & **Max. HSI** & \(\Delta_{Lat}\)[\%]** \\ \hline \(A_{g}\) [m/s\({}^{2}\)] & 1.39 & 1.02 & 26.41 & 0.51 & 0.42 & 19.08 \\ \(A_{c}\) [m/s\({}^{2}\)] & 1.60 & 1.03 & 35.59 & 0.57 & 0.42 & 25.79 \\ \(D_{c}\) [mm] & 11.57 & 8.22 & 28.94 & 13.69 & 10.17 & 25.69 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Peak Response During Gait, With HSI and Without HSI.  
The instantaneous natural frequency of the HSI system with the pedestrian at mid-span decreases less than 30% for occupancy condition levels \(M_{br}/M_{br}\) up to 75.0%, compared to those of the system Without dynamic HSI effects (Figure 14), for vertical and lateral directions. This is especially relevant as it suggests that pedestrian bridges with fundamental natural frequencies greater than walking frequencies, which in principle should not have excessive vibration problems, could display amplification of the dynamic response due the reduction in the natural frequency and variation in the structural damping of the coupled system, compromising the serviceability of these structures.  
To reduce the effect of de-synchronization, simulations were performed by adjusting the step frequency \(\omega_{w}\) so that the ratio of frequencies \(\omega_{crowd}/\omega_{br}\) was approximately one. The simulation results are shown in Figure 15. In this case, it is observed that the peak values of acceleration and displacement are always greater than those of the model without HSI for all occupancy levels. This suggests that the traditional model underestimates the dynamic response of a structure under periodic anthropic loads.  
Figure 14: Peak Accelerations and Displacements at Mid-Span for Different Occupancy Levels.  
Figure 15: Peak Accelerations and Displacements at Mid-Span for Different Occupancy Levels. (Pace Frequency equal to Bridge Natural Frequency).