Analytical solutions for tunnels with any shape excavated sequentially in rheological rock at great depth

The analytical solutions are presented for the sequential excavation of the tunnel with any shape in viscoelastic rock subjected to anisotropic stress conditions. The complex variable method and the Laplace transformation are introduced to express the displacement and stress by the two potentials, therefore the methodology and analytical solutions for displacement and stress are derived for the general viscoelastic cases. The incremental displacement after each excavation step is the summation of (1) the rheological displacement due to the released stresses along the new boundary in this step and (2) the incremental rheological displacement due to the released stresses along excavation boundaries in all previous steps. The specific analytical solution is given for the two step sequential excavations of circular tunnel. A good agreement between the results from finite element simulations and the analytical solutions is observed for the specific top-bottom excavation method. The displacement and stress distributions for two excavation methods, top-bottom and left-right excavations, are analyzed by the proposed solutions. According to the analytical solutions, a fast computational system for predicting the mechanical state during tunnel excavation can be established in the future, which may provide a more convenient way for the preliminary design of the tunnel.