## 4 Results and Discussion
### Reference Footridge Performance, Including HSI
Lateral structural acceleration in the contact point of the ST1 and frequential response **(top left)**; lateral structural displacement in the contact point of the ST1 and frequential response **(top right)**; lateral structural acceleration in the middle span and frequential response **(bottom left)**; and lateral structural displacement in the middle span and frequential response **(bottom right)**. Figure 9: General vertical structural response of the reference footbridge under human-induced gait loads with HSI effects by ST1. Vertical structural acceleration in the middle span and frequential response **(top left)**; vertical structural displacement in the middle span and frequential response **(top right)**; vertical structural acceleration in the contact point of the ST1 and frequential response **(bottom left)**; and vertical structural displacement in the contact point of the ST1 and frequential response **(bottom right)**. middle span exhibited frequential behavior similar to the fundamental frequency around 2.0 Hz, with some content frequency due to dynamic HSI effects (close to 1.80 Hz) and harmonic behavior (around 4.0 Hz). The free vibration behavior of the footbridge was identified once the pedestrian left the structure, which allowed the identification of the structural damping to be assigned, as presented in Figure 9 (bottom). It is also important to note that the vertical structural response was found to be directly proportional to the gait velocity. Significant differences in both accelerations and displacements were observed, with vertical accelerations increasing by more than 70% and displacements exceeding 100% when comparing tests conducted at different gait velocities. These findings underscore the influence of gait velocity on the dynamic response of the footbridge, reinforcing the critical importance of considering such effects when evaluating structural performance under pedestrian-induced loads. The assessment of the peak dynamic response of the reference footbridge was carried out using a hybrid spider-violin plot for each gait velocity evaluated by TSs, as shown in Figure 10. This schematic analysis allows the identification, for \(n\) independent events (each branch), of the normal probability density function (PDF), using Equation (13), of the peak dynamic responses (\(P_{SR}\)) normalized by the maximum value in the respective event. This equation uses the average (\(\mu_{PsR}\)) and standard deviation (\(\sigma_{PsR}\)) of the normalized peak structural response generated during the gait tests on the reference footbridge, including the dynamic HSI effects using the RTHS technique. \[PDF_{PsR}=f\big{(}P_{SR}\big{|}\mu_{PsR},\sigma_{PsR}\big{)}=\frac{1}{\sigma \sqrt{2\pi}}e^{\frac{-(P_{SR}-P_{SB})^{2}}{2\mu_{PsR}^{2}}},\text{ for }\forall\ P_{SR}\in\mathbb{R} \tag{13}\]  
Figure 10: Assessment of peak dynamic response of the reference footbridge using a hybrid spider–violin plot. Normalized general structural displacements for the three gait velocities: 1.20 m\(\cdot\)s\({}^{-1}\) (**top left**), 1.50 m\(\cdot\)s\({}^{-1}\) (**top middle**), and 1.80 m\(\cdot\)s\({}^{-1}\) (**top right**); normalized general structural acceleration for the three gait velocities: 1.20 m\(\cdot\)s\({}^{-1}\) (**bottom left**), 1.50 m\(\cdot\)s\({}^{-1}\) (**bottom middle**), and 1.80 m\(\cdot\)s\({}^{-1}\) (**bottom right**). Each branch presents the respective PDF shape with punctual events that were composed, and the average structural response was identified. In addition, the colored shadow represents the first quantile of the standard deviation of the data points that compose the specific event. This approach allowed the identification of a higher dispersion in the vertical displacement of the middle span of the structure and contact point of the TSs compared to the same response in the lateral direction, with percentage differences of mean normalized values of less than 10.0% for all gait velocities, as shown in Figure 10 (top).