The train is simulated by a multiple-mass-spring-damper model

and the bridge vibration is solved by incorporating the slab track kinematic equations into the bridge model. Combined with pile-group foundation model

a semi-analytical coupling model of train-rail-bridge-bridge pier-group pile foundation is established. Based on the Fourier transform and the Green function of elastic half-space in the frequency domain

the ground wave field caused by the moving train on viaduct is obtained. The vibration laws of the upper stiff-layer and lower soft-layer ground and the upper soft-layer and lower stiff-layer ground are studied. Moreover

the effects of pile diameter and pile length on ground vibration attenuation are analyzed. The results show that the vibration of the upper stiff-layer and lower soft-layer ground at the center of the track is larger than that of the upper soft-layer and lower stiff-layer ground at lower vehicle speed; but the vibration of the upper soft-layer and lower stiff-layer ground increases significantly when the vehicle speed is close to the shear wave velocity of the surface soft soil

which indicate that the speed of the train should be avoided to approach the shear wave velocity of the surface soil. The vibration at the center of the track increases with the increase of the pile length and pile diameter

especially when the vehicle speed is large. The pile has a good damping effect

and the ground vibration attenuates rapidly due to the geometric damping and pile scattering effects within the pile-group area. The vibration attenuation speed is faster when the pile diameter is larger

but it is less affected by pile length. The ground vibration amplitude of the upper stiff-layer and lower soft-layer ground on the outside of the group pile foundation is smaller than that of the upper soft-layer and lower stiff-layer ground

which indicates that the effect of the upper stiff-layer and lower soft-layer ground on vibration attenuation is better. Rational design of pile foundation can meet the vibration control requirements at the center of the track and effectively reduce the environmental vibration around the viaduct.