# Dynamic Interaction Analysis of Long-Span Bridges Under Stochastic Traffic and Wind Loads
## 4 Application Examples
### Characteristics of Bridge Dynamics Under Different Traffic Densities
#### 4.2.1 Bridge Vibration Response Under Sparse Traffic Flow  
In this study, the dynamic response of the bridge under different traffic conditions (sparse and dense) and wind loads is analyzed separately to isolate the individual effects of each factor. This approach enables a clearer understanding of how each factor influences the bridge's vibration response. The combined effects of traffic and wind loads are discussed in Section 4.2.3, where the interactions between these factors are examined in detail.  
To further analyze the vibration response characteristics of the bridge under extended periods of sparse traffic, the vertical vibration displacements were closely monitored and recorded over three distinct time intervals, as shown in Figure 9a-c. A comparative analysis of the bridge's vibration responses across these intervals reveals a consistent pattern, with a cycle duration of approximately 300-350 s. It is important to note that during the second time period, the bridge experienced a peak displacement of 0.78 cm, which is slightly higher than the general displacement range observed.  
Long-term monitoring data indicate that the vertical vibration amplitude typically ranges from 0.1 cm to 0.7 cm, with the noted peak of 0.78 cm being an exception rather  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Lanes**} & **Class 1** & **Class 2** & **Class 3** & **Class 4** & **Class 5** & **Class 6** \\  & **Car (veh)** & **Car (veh)** & **Car (veh)** & **Car (veh)** & **Car (veh)** & **Car (veh)** \\ \hline
1–4 lanes & 5112 & 2903 & 1412 & 856 & 443 & 48 \\
5–8 lanes & 4833 & 2823 & 1325 & 721 & 313 & 33 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The number of vehicles in the eight two-way lanes.  
Figure 13: The horizontal vibration under wind effects.  
than the norm. These vibrations are characterized by smooth fluctuations and lack high-frequency components. Under sparse traffic conditions, the substantial intervals between vehicles result in significant vibration attenuation. The vertical displacement and acceleration are less prone to accumulation, leading to an overall controllable response that tends to decrease over time and eventually levels off.  
When the bridge is affected by the external force of the traffic flow, its vibration frequency may align with the frequency of the external force, potentially causing resonance of the axle, and thus, several peaks are generated in the monitoring process, as shown in Figure 9d. The amplitude peaked at frequencies near 0.003 Hz, both around 0.35. Additionally, small peaks were observed near 0.01 Hz, 0.02 Hz, and 0.04 Hz. Above 0.04 Hz, the amplitude becomes nearly flat and no longer fluctuates substantially.  
The impact of sparse traffic on the bridge is characterized by isolated impact responses occurring over a period of approximately 100 s, as shown in Figure 10. When a single vehicle crosses the bridge, the vertical vibration amplitude may peak at 0.68 cm. However, because of the extended intervals between vehicles, the vibration diminishes quickly without leading to substantial resonance or superposition effects. The transverse vibration amplitude remains low and shows little variation. As a result, the influence of sparse traffic on the bridge is localized and short-lived without any significant build-up of vibration.