The influence of a neighboring tunnel on the critical velocity of a three-dimensional tunnel-soil system

This paper proposes an analytical solution for the calculation of vibrations from a twin tunnel embedded in a half-space to assess the influence of a neighbouring tunnel on the ground-borne vibrations due to underground railway traffic. Using the wave function method, the total wave field in a half-space containing two cavities is expanded as the combination of down-going plane waves from the free surface and outgoing cylindrical waves from each cavity. The two tunnels are modelled as hollow cylinders, and the wave field inside them can be represented as the superposition of outgoing and regular cylindrical waves. To solve this multiple scattering problem, three sets of coordinate system defined at the scattering surfaces are used to describe the cylindrical waves in polar coordinates and the plane waves in rectangular coordinates. The application of transformation and translation properties between plane and cylindrical waves makes it possible to represent the waves emanating from one scattering surface in terms of the coordinates defined at the other two surfaces. The boundary conditions at the free surface and two tunnels can then be applied to solve for the expansion coefficients for the vector wave functions of plane and cylindrical waves. The present approach is also valid for a multi-layered ground, and in the layer with the tunnels, the field is composed of outgoing waves from the tunnels, down-going waves from the upper interface of the layer and up-going waves from the lower interface. Numerical results show that there is one critical velocity around shear wave speed of soils for a single/twin tunnel in a homogeneous half-space. For a multi-layered geometry, two global critical velocities may exist for the twin tunnel case, and they are determined by the properties of the soil layer where the twin tunnel is located in and the layer below the tunnels.