An efficient formulation of Krylov's prediction model for train induced vibrations based on the dynamic reciprocity theorem.

In Krylov's analytical prediction model, the free field vibration response during the passage of a train is written as the superposition of the effect of all sleeper forces, using Lamb's approximate solution for the Green's function of a halfspace. When this formulation is extended with the Green's functions of a layered soil, considerable computational effort is required if these Green's functions are needed in a wide range of source-receiver distances and frequencies. It is demonstrated in this paper how the free field response can alternatively be computed, using the dynamic reciprocity theorem, applied to moving loads. The formulation is based on the response of the soil due to the moving load distribution for a single axle load. The equations are written in the wave-number-frequency domain, accounting for the invariance of the geometry in the direction of the track. The approach allows for a very efficient calculation of the free field vibration response, distinguishing the quasistatic contribution from the effect of the sleeper passage frequency and its higher harmonics. The methodology is validated by means of in situ vibration measurements during the passage of a Thalys high-speed train on the track between Brussels and Paris. It is shown that the model has good predictive capabilities in the near field at low and high frequencies, but underestimates the response in the midfrequency band.