## 4 Results of Vehicle-Bridge Coupling Analysis
### Impact Analysis of Damage Parameters  
Based on the research above on the dynamic response of different vehicle parameters to the damaged bridge, this section studies the dynamic response of the vehicle-bridge coupling vibration to the bridge with sustained damage. The vehicle speed is 90 km/h, the vehicle weight is 35 t, and the vehicle is a spatial dual-axle model. As the bridge continues to serve after damage, the stiffness of the bridge itself will decrease with the passage of time. Assuming that the stiffness reduction coefficient starts at 0.76, six working conditions are considered: 0.76, 0.7, 0.65, 0.6, 0.55, and 0.5. According to Figure 21, it is obvious that with the continuous damage to the bridge, the stress of each section continues to increase. The stress growth trend is different at different cross-section locations. For the midspan position, the stiffness reduction coefficient is from 0.76 to 0.5, and the change in stress value shows obvious fluctuation. When the reduction coefficient is from 0.76 to 0.7, the stress changes faster, and the growth rate is 12.5%. In the range of 0.7-0.65, the stress changes slowly, and the growth rate is 1.78%. In the range of 0.65-0.6, the stress growth rate is 7.02%, which is faster than the stiffness decay in the second stage. When the reduction coefficient is in the range from 0.6 to 0.5, the curve of the stress change is almost linear, and the average stress growth rate is 8.83%. It can be inferred that with the continuous damage to the bridge, the stress increase rate will be larger and larger. If the stress in the span continues to increase until the tensile strength is reached or even exceeded, the concrete will be damaged. For the position of the one-fourth section of the middle span, the stress value fluctuates to a certain extent. In the range of the stiffness reduction coefficient from 0.76 to 0.5, the slope of each straight line changes as follows: k\({}_{0.6-0.55}\) > k\({}_{0.76-0.6}\) > k\({}_{0.55-0.5}\). It can be seen from the slope that the stress increases slowly and linearly in the range of 0.76 to 0.60. In the range of 0.60 to 0.55, the stress increases rapidly; from 0.55 to 0.50, the rate slows down again. As the damage continues,  
Figure 19: Peak displacement of each section under different vehicle weights. Figure 20: Impact coefficient of each section under different vehicle weights. the growth rate of the stress at the position of the one-fourth section first increases and then decreases for the damaged concrete. For the position of the one-third section of the middle span, the stress growth rate does not change rapidly in the range of 0.76 to 0.6 and always maintains a gentle rate of increase. In the range of 0.6 to 0.55, the stress growth rate has a large change and becomes smaller again after 0.55 to 0.5. It can be inferred that the stress growth rate will increase up to failure at the position of the one-third section as the stiffness continuously attenuates. The results of the dynamic displacement peak value and impact coefficient also show similar rules. As can be seen from Figure 22, in the process of the continuous damage to the bridge, the displacement of each section is increasing, and the displacement increment of the midspan section is the largest. When the stiffness reduction coefficient reaches 0.6, the displacement increases almost linearly. When the stiffness of the one-fourth section reaches 0.6, the displacement growth fluctuates a little. The displacement change is gentler than that in the previous stage when the reduction coefficient is 0.55 to 0.5. When the reduction coefficient is in the range from 0.76 to 0.5, the displacement is relatively gentle at the one-third section compared with the other two places. When the damage coefficient decreases from 1 to 0.76, the displacement of each section increases rapidly, with an increase of 86.36% in the midspan, 114.08% in the one-fourth section, and 192.13% in the one-third section.