Complex variable solution on asymmetrical sequential shallow tunnelling in gravitational geomaterial considering static equilibrium

Abstract: Asymmetrical sequential excavation is common in shallow tunnel engineering, especially for large-span tunnels. Owing to the lack of necessary conformal mappings, existing complex variable solutions on shallow tunnelling are only suitable for symmetrical cavities, and can not deal with asymmetrical sequential tunnelling effectively. This paper proposes a new complex variable solution on asymmetrical sequential shallow tunnelling by incorporating a bidirectional conformal mapping scheme consisting of Charge Simulation Method and Complex Dipole Simulation Method. Moreover, to eliminate the far-field displacement singularity of present complex variable method, a rigid static equilibrium mechanical model is established by fixing the far-field ground surface to equilibriate the nonzero resultant along cavity boundary due to gravitational shallow tunnelling. The corresponding mixed boundary conditions along ground surface are transformed into homogenerous Riemann-Hilbert problems with extra constraints of traction along cavity boundaries, which are solved in an iterative manner to obtain reasonable stress and displacement fields of asymmetrical sequential shallow tunnelling. The proposed solution is validated by sufficient comparisons with equivalent finite element solution with good agreements. The comparisons also suggest that the proposed solution should be more accurate than the finite element one. A parametric investigation is finally conducted to illustrate possible practical applications of the proposed solution with several engineering recommendations. Additionally, the theoretical improvements and defects of the proposed solution are discussed for objectivity.