## 4 Results and Discussions
Figure 14c reprints Figure 11. In these scenarios, the applicability of the PRE method to the estimation of local stiffness reduction of the bridge. It is not easy to recognize the 10% stiffness reduction in Figure 14a, while the 30% and 60% decreases can be observed in Figure 14b,c. Estimating the slight ratio of the local stiffness reduction requires the quantitative evaluation of the results, while the global reduction can be observed in Figure 13b. Figure 15 shows the probability density distributions of all estimated mechanical parameters in (a) Scenario 7, (b) Scenario 8, and (c) Scenario 9, respectively. These cases indicate the ranging stiffness reductions and provide a discussion basis for the PRE applicability to the width of the stiffness reduction. The PRE method can recognize two elements with reduced stiffness from these sub-figures. Figure 13: The application results of the PRE method: (**a**) Scenario 1; (**b**) Scenario 2 (stiffness decrease section: 0–\(L\), decrease ratio: 10%); and (**c**) Scenario 3 (stiffness decrease section: 0–\(L\), decrease ratio: 20%). Figure 16 shows the probability density distributions of all estimated mechanical parameters in (a) Scenario 10, (b) Scenario, 11 and (c) Scenario 12, respectively. These cases indicate edge stiffness reduction. According to these figures, the tendency of stiffness reduction cannot be detected from the estimated distributions of \(EI_{7}\). However, the next elements have underestimated values. This underestimation can help detect stiffness reduction at the bridge's edge. Figure 14: The application results of the PRE method: (**a**) Scenario 4 (stiffness decrease section: 3\(L\)/7\(-\)4\(L\)/7, decrease ratio: 10%); (**b**) Scenario 5 (stiffness decrease section: 3\(L\)/7\(-\)4\(L\)/7, decrease ratio: 30%); and (**c**) Scenario 6 (stiffness decrease section: 3\(L\)/7\(-\)4\(L\)/7, decrease ratio: 60%). Figure 15: The application results of the PRE method: (**a**) Scenario 7 (stiffness decrease section: 4\(L\)/7\(-\)6\(L\)/7, decrease ratio: 10%); (**b**) Scenario 8 (stiffness decrease section: 4\(L\)/7\(-\)6\(L\)/7, decrease ratio: 30%); and (**c**) Scenario 9 (stiffness decrease: 4\(L\)/7\(-\)6\(L\)/7, decrease ratio: 60%). Table 7 shows the average and standard deviation of the estimated normalized mechanical parameters. The bold style in this table indicates the average of the estimated stiffness at the elements with reduced stiffness. The obtained average can be used to evaluate the estimation accuracy. On the other hand, the standard deviation indicates the reliability of the obtained distribution. According to the averages in this table, the reduced stiffness of the bridge can be directly detected except in the edge elements. It is also noted that the values of the reduced stiffness are often overestimated. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Scenario**} & \multicolumn{2}{c}{**Scenario**} & \multicolumn{2}{c}{**Scenario**} & \multicolumn{2}{c}{**Scenario**} & \multicolumn{2}{c}{**Scenario**} & \multicolumn{2}{c}{**Scenario**} & \multicolumn{2}{c}{**Scenario**} & \multicolumn{2}{c}{**Scenario**} & \multicolumn{2}{c}{**Scenario**} & \multicolumn{2}{c}{**Scenario**} & \multicolumn{2}{c}{**Scenario**} & \multicolumn{2}{c}{**Scenario**} & \multicolumn{2}{c}{**Scenario**} \\  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline \multicolumn{11}{c}{**ave. std. **} & **ave. std. ** & **ave. std. ** &  
In Table 7, if a standard deviation is more significant, the reliability of the corresponding estimated parameter is lower. The standard deviations of suspensions' and tires' characteristics \(c_{si}\), \(k_{si}\), \(m_{ui}\) and \(k_{ui}\) tend to be larger than those of sprung-mass. The PRE method can estimate the vehicle's body mass \(m_{si}\) more accurately than the other parameters. This tendency can be attributed to the equations of motion of the vehicle.