Analytical solution of ground-borne vibration due to a spatially periodic harmonic moving load in a tunnel embedded in layered soil

In this study,we propose a novel coupled periodic tunnel–soil analytical model for predicting ground-borne vibrations caused by vibration sources in tunnels.The problem of a multilayered soil overlying a semi-infinite half-space was solved using the transfer matrix method.To account for the interactions between the soil layer and tunnel structure,the transformation characteristics between cylindrical waves and plane waves were considered and used to convert the corresponding wave potentials into forms in terms of the Cartesian or cylindrical coordinate system.The induced ground-borne vibration was obtained analytically by applying a spatially periodic harmonic moving load to the tunnel invert.The accuracy and efficiency of the proposed model were verified by comparing the results under a moving constant and harmonic load with those from previous studies.Subsequently,the response characteristics under a spatially periodic harmonic moving load were identified,and the effects of a wide range of factors on the responses were systematically investigated.The numerical results showed that moving and Doppler effects can be caused by a spatially periodic harmonic moving load.The critical frequency and frequency bandwidth of the response are affected by the load type,frequency,velocity,and wavenumber in one periodicity length.Increasing the tunnel depth is an efficient way to reduce ground-borne vibrations.The effect of vibration amplification on the free surface should be considered to avoid excessive vibration levels that disturb residents.