Analytical solution for calculating vibrations from twin circular tunnels

The calculation of vibrations from a single tunnel embedded in a full- or half-space due to underground moving trains has been well studied in the literature. A common assumption made in the majority of vibration prediction models is to neglect the presence of a neighboring tunnel, however, twin tunnels are a particularly preferred construction for urban underground railways with one as the inbound tunnel and the other as the outbound tunnel. The neglect of the interaction between twin tunnels may lead to inaccuracies of the calculation of vibrations from underground railways. The present paper proposes a novel analytical solution, which takes the multiple scattering effects between the two tunnels into account. The two tunnels are modelled as elastic hollow cylinders and the soil surrounding the tunnels as an elastic, homogeneous full-space containing two cylindrical cavities. The wave field in the full-space with two cavities consists of outgoing waves from each tunnel outside the two scattering surfaces while the wave field in the tunnel wall is a combination of outgoing and regular cylindrical waves. The translation of the outgoing cylindrical wave functions is required to satisfy the boundary conditions at the other tunnel. Numerical results show that there are two critical velocities for a twin tunnel in a full-space system, both of which are around the shear wave velocity of the soil. Through comparison with the single tunnel solution, the addition of a second tunnel at different separation distances and angles has a significant influence on the soil response, especially in the high frequency range, but the response at the tunnel with the source is not much affected.