## 3 Numerical Validations
### Truss Bridge Example  
A three-dimensional truss bridge is established, as shown in Figure 5. The following parameters are designed: the main girders are a 32 b steel I-beam, the truss members are 50 mm diameter steel rods, the bridge deck is a 400 mm thick concrete slab. The Young's modulus of steel and concrete are 210 GPa and 35 GPa; the densities are 7850 kg/m\({}^{3}\) and 2500 kg/m\({}^{3}\). Structural damages are simulated by the decay of [PERSON]'s modulus in the hangers. Three displacement sensors, S1-S3, are utilized on the middle and quartile of the span. In order to simulate a 2-axle vehicle, two moving loads \(F_{1}=30+2\sin(10\pi t)+4\cos(15\pi t)+\cos(3\pi t+\frac{\pi}{5})\) kN and \(F_{2}=80-6\sin(10\pi t)-10\cos(15\pi t)-2\cos(3\pi t\)\(+\frac{\pi}{5})\) kN are applied with the same speed, and the interval distance is 4 m. The sampling rate is set as 50 Hz. Regarding real-word scenarios, the aspects of force speed, damage severity, measurement noise, sensor arrangement, and damage location are considered in the case setting (shown in Table 2) [29]. Based on the mechanical property of the truss bar, there is only one basic nodal force, the axial force, in the member. Hence the virtual force of the damaged hanger \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case No.** & **Velocity (m/s)** & **Damage Severity** & **Damage Location** & **Force No.** & **Noise Level** & **Used Sensor** \\ \hline
1 & 10 & 5\% & Damage 1 & F1 & 5\% & S1 and S2 \\
2 & 10 & 20\% & Damage 1 & F1 & 10\% & S1 and S3 \\
3 & 10 & 20\% & Damage 1 & F1 & 10\% & S2 and S3 \\
4 & 20 & & & & & \\ \hline
5 & 10 & 20\% & Damage 2 & F1 and F2 &
\begin{tabular}{c} 5\% \\ 10\% \\ \end{tabular} & S1, S2 and S3 \\
6 & 10 & 20\% & Damage 2 & F1 and F2 &
\begin{tabular}{c} 5\% \\ 10\% \\ \end{tabular} & S1, S2 and S3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Case setting. Figure 5: A simply supported truss beam under moving load. is dominated by the axial virtual force \(\lambda_{11}^{e}\) according to Equation (4). Therefore, three unknowns \(F1,\ F2,\ \text{and}\ \lambda_{11}^{e}\) should be solved in the identification Equation (17). Following the TLSF procedure that is shown in Section 3.1, the corresponding TLSF coefficients are calculated. Finally, the moving force is reconstructed by Equation (13), while the damage severity \(1-\zeta_{1}\) is calculated via Equation (18). The synchronized identification results and discussion are presented as follow. Result discussion for MFI  
The moving load identification results are shown in Figure 6. The RPEs are shown in Figure 7.  
Influence of damage severity and location  
Comparing the results of Cases 1 and 2 in Figure 6a, it is apparent that the change of damage severity has little effect on the determined accuracy. It is because that small local damage has limited influence on the global stiffness of the bridge, so the dynamic behaviors of intact and damaged are similar. This minor difference is optimized by TLSF and hence does not affect the results. On the other hand, local interference is observed at the damage location. Visible increase is found around 3.6 s when the force passes Damage 1. A similar situation is also found in Cases 5 and 6 (Figure 6d,e). The results show large variation when the forces move through Damage 2, which proves that the damage changes the local stiffness. Effects of force speed and number  
In Cases 3, 4, and 6, the effects of load velocity and number are studied. It can be seen from Figure 6b,c that the change of velocity does not greatly influence the identification precision. The RPEs in Figure 7 also agree that both cases are determined with good accuracy. It is worth noting that significant error occurs at the beginning and end of the time period; it is a typical problem in MFI caused by the vibration instability when the load enters or exits the bridge [30]. When 2-axle forces are applied in Cases 5 and 6, a distinct error is discovered at the wave crest; phase difference is also partially found in the curve.