An efficient and modular numerical prediction model is presented to predict vibrations in the free field due to metro trains in the tunnel. The three-dimensional dynamic tunnel-soil interaction problem is solved with a subdomain formulation, using a finite element formulation for the tunnel and a boundary element method for the soil. The periodicity of the tunnel and the soil in the longitudinal direction is exploited using the Floquet transform, limiting the discretization effort to a single bounded reference cell. The Craig-Bampton substructuring technique is used to efficiently incorporate a track in the tunnel. The track-tunnel-soil interaction problem is solved in the frequency-wavenumber domain and the wave field radiated into the soil is computed.
The numerical model can also account for moving loads and various excitation mechanisms, including quasi-static loads, random loads due to the rail and wheel unevenness, impact excitation due to the rail joints and wheel flats, and parametric excitation excitation due to the sleeper periodicity. In this paper, only the excitation due to rail unevenness and a moving harmonic load are considered. A general analytical formulation is discussed to compute the response of three-dimensional invariant and periodic media that are excited by moving loads.
To demonstrate the efficiency of the approach, an invariant tunnel with a concrete lining, embedded in a homogeneous full space is considered. The system is excited by a vehicle moving on an uneven rail. It is emphasized that the wheel/rail interaction strongly depends on the dynamic response of the wheel, the rail and the contact spring. The free field vibrations are predicted, by first computing the contact forces generated by the wheel-track interaction and then solving the dynamic track-tunnel-soil interaction problem.
This numerical model provides a better understanding of wave propagation in the track, the tunnel and the surrounding soil and enables to investigate the inherent physics of underground railway vibrations and to control the vibrations propagating out of the tunnel.