###### Abstract  
An improved spatial vehicle-track-bridge interaction system considering the unsupported track is proposed using the virtual work principle. When the track fails to be supported due to the defects under the track, the corresponding bridge-rail connection coupled matrix should be removed. Using the proposed dynamic model, a novel numerical analysis of the unilateral and bilateral unsupported sleepers in the bridge and the subgrade is carried out. The results indicate that the wheel-rail contact force changes dramatically when the vehicles pass through the unsupported track. The unsupported track has a little effect on the displacement of the bridge, but it increases the acceleration of the bridge. The displacement and acceleration of the track increase significantly with the expansion of the supported defects under the track and the increase of driving speed. 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 a limited effect on the other side of the normal support. train-track-bridge interaction system; unsupported track; virtual work principle; wheel-rail contact force; displacement; acceleration +
Footnote †: journal: Applied  
0000-0002-2000-2000-20000-20000-20000-20000-20000-20000-20000-20000-20000-20000-20000-20000-20000-20000-20000-20000-20000-20000-200000-20000-200000-20000-20000-2  
The track response caused by the change of subgrade stiffness and settlement has been studied by experiment and numerical simulation. [PERSON] et al. [11] found that the track-supported stiffness needs to be within a reasonable range, and whether it is too large or too small, the track deterioration may take place. Track stiffness is of great significance for judging the structural performance of the track. Based on the test data of the track-supported stiffness and damping, [PERSON] et al. [12] demonstrated that the wheel-rail contact force and fatigue strength of rail fasteners increase significantly under poor support. Excessive strain results in the cracking of the unsupported track at the resonance of the sleepers at 200 Hz and 740 Hz. The clamped beam as the simplified model was proposed by [PERSON] et al. [13] to explain the complex factors of the impact loading. The reason why the low velocity of the train induces a high impact on the track is the rail quasistatic deflection rate. The greater damping support is beneficial for dynamic performance and decelerates the deterioration of the track. [PERSON] et al. [14] conducted a 1:5 scale wheel-rail model test, in which the rail pads were removed to simulate the unsupported track. They found that ballast is an important damping source of railway track, and the vibration amplitude of the sleeper is the largest at low frequency. When the track is entirely unsupported, its vibration is almost undamped. The frequency response function of the track acceleration reaches the maximum at mid-span and has a steep dip above the sleepers. [PERSON] et al. [15] investigated the displacement of the track and wheel-rail contact force under fully supported, partially supported, and poorly supported. The result shows that three consecutive unsupported sleepers significantly affect the track displacement. A three-layer ballasted track model was employed by [PERSON] et al. [16; 17] to study the position and the magnitude of the wheel-rail contact force. They found that the peak force depends on the number of unsupported sleepers, the distance between two groups of unsupported sleepers, and the driving speeds of the train. [PERSON] et al. [18] established a three-dimensional dynamic finite element model considering wheel-rail friction using the Shuohuang heavy-haul trackway line as the research object.