## 4 Conclusions  
This paper develops a synchronized assessment method on bridge structural damage and moving force. Through transforming structural damage to virtual force, this method unifies the SDI and MFI to a multi-force identification problem. Then the TLSF technique is proposed to improve the calculation efficiency and stability of the inverse problem. A continuous beam and a truss bridge are simulated as examples to validate the proposed method. The effectiveness in respect to the aspects of force type, velocity, number, as well as damage severity, location, and measurement noise are studied. The conclusions are presented as follow:  
1. A harmonic force and a square wave impulse force are identified via the TLSF in the continuous beam example. Both of them are determined with good accuracy. With the enhancement of TLSF, the dimension of the system matrix has been reduced from \((1000\times 1000)\) to \((64\times 100)\), which significantly improves the efficiency. In addition, the L-curves of the TLSF matrix is smooth and contains little residual even in the case of considering 5% white noise, which proves the outstanding robustness of this method to noise. However, a visible error is shown in the result calculated from the response of the right span (the force is applied on the left span). This is because the restrain of the mid-support disturbs the TLSF matrix. Therefore, the optimization of sensor arrangement should be further studied.  
2. The structural damage and 2-axle moving load are synchronized and identified in the truss bridge example. For the MFI, the velocity, damage severity, sensor position, and noise only cause local interference in the results, the RPEs in these cases are kept  
Figure 8: Identified and fitted results of \(\xi_{1}\). (**a**) Case 1; (**b**) Case 2; (**c**) Case 3; (**d**) Case 4; (**e**) Case 5; (**f**) Case 6.  
within tolerable limits. However, in the 2-axle forces cases, the error is increased due to the interplay of bridge-force interactions. Despite this, the error obeys normal distribution, which could be reduced by a certain optimization method. In terms of SDI, results find that the accuracy is low correlated to damage severity and location but is dominated by the precision of the TLSF matrix and is interfered with the singularity generated from multi-force identification. These drawbacks could be improved by choosing appropriate objective functions and optimization algorithms.  
The numerical studies prove the potential of this method in real-world application. The essential problem is to establish an accurate TLSF matrix, which makes a finite element model of the intact bridge necessary. Due to this, the method is better applied on new bridges with complete structure information and the maintenance of subsequent damages. In addition, many factors should be considered before the method is applied. The influences of road profile, multilane, and the vehicle-bridge coupling effect will be studied in the future.  
Conceptualization, methodology, and validation, J.Z. and Z.X.; resources and data curation, C.L.; writing, Z.X. All authors have read and agreed to the published version of the manuscript.  
The authors gratefully acknowledge the support provided by the National Science Foundation of the Jiangsu Higher Education Institution of China (Grant No. 20 KJD560002) and the Nantong Science and Technology Plan Project (Grant No. JC2020124).  
Data Availability StatementData is contained within the article.  
The authors declare no conflict of interest.