## 4 Results and Discussions
On the other hand, according to Figure 11, the estimated stiffness \(EI_{3}\) and \(EI_{5}\) are underestimated. While this underestimation means that the estimation accuracy becomes lower than that of Scenario 1 (no stiffness reduction), it can be taken positively because it increases the possibility of reduced stiffness detection. These results imply that the PRE method partially succeeds in stiffness reduction estimation. According to Figure 12, the road unevenness can also be estimated accurately in Scenario 6. This means that the estimation accuracy of the PRE method does not change when the bridge's midspan stiffness decreases. The errors and residuals between the correct and estimated road unevenness functions are also shown in Figure A4, Appendix A. Next, Figure 13 shows the probability density distributions of all estimated mechanical parameters in (a) Scenario 1, (b) Scenario 2, and (c) Scenario 3, respectively. Figure 13a reprints Figure 9. Note that the bridge stiffness \(EI_{i}\) is divided by the standard value \(\overline{EI}\). Thus, when the PRE method accurately estimated the reduced stiffness of the bridge, the normalized values of \(EI_{i}\) in Scenario 2 and Scenario 3 should equal 0.9 and 0.8. In Figure 13b,c, the bridge stuffiness is assumed to be reduced globally by 10% and 20%. It can be confirmed that the estimated values of 20% less stiffness of Scenario 3 tend to be distributed lower than those of 10% less stiffness of Scenario 2. This result implies the feasibility of the PRE method not only to detect but also to estimate the stiffness reductions of bridges. However, \(EI_{1}\) and \(EI_{7}\) are not estimated accurately. The PRE method cannot work well where the amplitude of bridge vibration is small. Figure 10: The estimated road unevenness \(R(x)\): The estimated road unevenness \(\hat{R}_{1}(x)\) and \(\hat{R}_{2}(x)\) shown in this figure are calculated using the optimal particle \(\mathbf{X}_{best}^{(600)}\) minimizing \(J\) in Scenario 1. Figure 11: The application results of the PRE method in the case of Scenario 6 (stiffness decrease section: 3\(L\)/7\(\cdot\)4\(L\)/7, decrease ratio: 60%): these figures indicate the prior and posterior probability density distributions of the estimated mechanical parameters. The MCMC process can react to the reduction of \(EI_{4}\), while the posterior distributions of \(EI_{3}\) and \(EI_{5}\) also shift to lower. Figure 12: The estimated road unevenness \(R(x)\): the estimated road unevenness \(\dot{R}_{1}(x)\) and \(\dot{R}_{2}(x)\) shown in this figure are calculated using the optimal particle \(\textbf{X}_{\text{best}}^{(600)}\) minimizing \(J\) in Scenario 6. The bridge's density \(\rho A\) can be estimated accurately. However, this study assumes that the density \(\rho A\) is constant regardless of location \(x\). If the density \(\rho A\) is divided into elements as well as the stiffness \(EI_{i}\), the estimation accuracy of the density \(\rho A\) and the relationship between the density and stiffness estimation accuracy should be examined in future work. The damping coefficients \(\alpha\) and \(\beta\) cannot be estimated in the shown scenarios. It is believed that these values do not significantly affect the dynamic behavior of the VBI system. According to Figure 13, the vehicle parameters in each scenario can also be estimated accurately. Because the estimation accuracy of bridge vibration decreases, the accuracy of vehicle parameters could be affected. However, the vehicle estimation accuracy in these scenarios does not change after the bridge's stiffness reductions. It can be said that the estimation accuracy of the vehicle parameters is not affected by the bridge's state. Figure 14 shows the probability density distributions of all estimated mechanical parameters in (a) Scenario 4, (b) Scenario 5, and (c) Scenario 6, respectively. According to these figures, the stiffness reduction at midspan can be detected.