Analytical solution for long lined tunnels subjected to travelling loads

• Analytical solution is derived for dynamic response of long lined tunnels to travelling loads.
• Integration transforms are used to simplify the governing equation to an algebraic equation.
• Both the velocity and frequency of the travelling loads have dramatic effects on tunnel responses.

An analytical solution is derived for dynamic response of long lined tunnels subjected to travelling loads. For the derivation, the long lined tunnel is assumed to be infinitely long with a uniform cross-section resting on a viscoelastic foundation. Fourier and Laplace transforms are utilized to simplify the governing equation of the tunnel to an algebraic equation, so that the solution can be conveniently obtained in the frequency domain. The convolution theorem is employed to convert the solution into the time domain. Final solutions of tunnel responses investigated are deflection, velocity, acceleration, bending moment, and shear force. The proposed solution is verified by providing comparisons between its results and those from the Finite Element program ABAQUS. Further parametric analysis, such as the influence of wave velocity and frequency on dynamic responses of the tunnel is presented with the analytical solution. These relationships can be an effective tool for practitioners.