## 5 Conclusions  
In this paper, two possible real-world scenarios of the unsupported track are investigated for the first time. One of the real scenarios is that unsupported track occurs on the bridge. Another real scenario is unilateral and bilateral unsupported sleepers on the same track line.  
An effective method for establishing the spatial vehicle-track-bridge interaction model based on the virtual work principle can avoid directly analyzing the complex equilibrium compatibility and geometric compatibility of the wheel-rail and bridge-rail contact interface. When the support under the track is defective, the original assembled 3-D vehicle-track-bridge system matrix is updated by removing the coupled term of the discrete connection between the bridge and the track.  
This model and method are applied to investigate the wheel-rail contact force, and the dynamic responses of the bridge and the track under the unsupported sleepers. The following conclusions can be drawn from the studies:  
1. The support defects under the track have little effect on the displacement of the bridge, but they increase the acceleration of the bridge. When the sleepers fail to be supported on one side, the acceleration of the bridge is smaller than that when the sleepers fail to be supported on both sides;
2. When the vehicles run smoothly and the track is well supported, the vertical wheel-rail contact force changes periodically. However, the vertical wheel-rail contact force suddenly changes when the train passes through the supporting defect area under the track. Whether it is unilateral or bilateral support failure, the changing trend of the contact force on the unsupported side of the track is the same. The rolling force of the vehicle results in the change of the wheel-rail contact force on the intact side due to the inconsistency of the supporting stiffness of the left-side and right-side rails;
3. When one sleeper is unsupported, the vertical displacement and acceleration of the track change little. When the disease develops into two unsupported sleepers, the dynamic response increases significantly. The effect of driving speed on track  
Figure 16: Effect of different speeds on dynamic response of point A under unilateral unsupported sleeper: (**a**) maximum displacement of the track and (**b**) maximum acceleration of the track.  
acceleration is greater than that of the number of unsupported sleepers, and the effect of the number of unsupported sleepers on track displacement is more significant;
4. For unilateral unsupported sleepers, the dynamic response on the unsupported side is slightly smaller than that of the bilateral unsupported sleeper, while the unsupported side has limited influence on the other supporting side. Therefore, the trackway department should repair the defects under the track in time, and remind the train drivers to pass through the diseased area at a low speed.  
Conceptualization and methodology, N.W.; validation, H.Y.; investigation, W.Z.; writing--original draft preparation, N.W.; writing--review and editing, J.F. and Z.L.; project administration, C.W. and H.Y.; funding acquisition, H.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grants No. 51708429, 52178301), the Open Projects Foundation of State Key Laboratory for Health and Safety of Bridge Structures (Grant No. 2017-04-GF), the Plan of Outstanding Young and Middle-aged Scientific and Technological Innovation Team in Universities of Hubei Province (Grant No. T2020010), the Science and Technology Projects of Fujian Provincial Department of Housing and Urban-Rural Development (Grant No. 2022-K-15), and the 13 th Graduate Education Innovation Fund Project of Wuhan Institute of Technology (Grant No. CX2021116).  
Not applicable.  
The authors declare no conflict of interest.