# Dynamic Interaction Analysis of Long-Span Bridges Under Stochastic Traffic and Wind Loads
## 4 Application Examples
### Characteristics of Bridge Dynamics Under Different Traffic Densities
#### 4.2.2 Bridge Vibration Response Under Dense Traffic Flow  
To further investigate the vibration response characteristics of the bridge under the influence of long-term intensive traffic flow, the vertical vibration displacement changes in the bridge were monitored over three distinct time periods, as shown in Figure 11a-c. It is evident that the bridge's vibration response remains consistent across these periods, with a vibration period ranging from 250 to 400 s. The maximum displacement recorded was 1.5 cm during the first time period, which significantly exceeds the maximum displacement observed under long-term sparse traffic conditions.  
Under dense traffic conditions, the interaction between vehicles and the superposition of loads are significantly enhanced, leading to a more complex vibration response and increased amplitude due to the continuous action of vehicles. Monitoring data indicate that under dense traffic conditions, the vertical vibration amplitude of the bridge ranges from 0.3 to 1.5 cm, which is 50% to 114.28% higher than that observed under sparse traffic conditions. This increase is attributed to the cumulative effect of continuous vehicle loads, especially during peak periods when vibration exhibits strong periodicity and a longer duration, making it less likely to decay.  
From the perspective of the frequency domain, the influence trend of frequency on bridge amplitude across the three time periods is roughly the same, as shown in Figure 11d. As the frequency increases, the amplitude continuously decreases and eventually levels off. The amplitude peaked at frequencies near 0.003 Hz, with both peaks near 0.50. Smaller peaks were observed near 0.075 Hz, 0.015 Hz, and 0.032 Hz. Above 0.04 Hz, the amplitude becomes nearly flat and no longer fluctuates substantially. Under the influence of dense traffic flow, the probability of bridge resonance is higher than that of sparse traffic flow and the superposition effect is stronger. Consequently, the peak is less steep, and the peak amplitude is about 42.86% higher than that observed under sparse traffic flow conditions.  
Dense traffic conditions significantly affect the vertical vibration of bridges within a 100 s observation period, as illustrated in Figure 12. The high traffic density results in vehicles passing over the bridge in close succession, leading to a pronounced superposition effect. Consequently, the vertical vibration amplitude can reach up to 0.92 cm, which is 35.29% higher than that observed under sparse traffic conditions. This increased vibration amplitude is due to the cumulative effect of continuous vehicle loads, which results in stronger and more persistent vibrations that are less likely to dissipate quickly. The combined effects of continuous vehicle loading, high vehicle speeds, and dense traffic flow contribute to increased vibration amplitudes. These factors cause the bridge to experience significant changes in displacement and acceleration over a short period.  
In summary, the dynamic increase in bridge vibration observed under dense traffic conditions is largely attributed to the cumulative impact of vehicle loads and their persistent action, which results in a significant buildup of vibration. The interactions among vehicles, along with the possibility of resonance effects, can further intensify these vibrations, particularly at higher vehicle speeds where resonance is more likely to occur, thereby augmenting vertical vibration. Consequently, it is imperative that bridge design, monitoring practices, and safety assessments consider the influence of fluctuating traffic densities on vibrational behavior. This consideration is essential to guarantee the safety and stability of bridges under diverse traffic scenarios.