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

TL;DR

The paper addresses asymmetrical sequential shallow tunnelling in gravitational geomaterials by developing a bidirectional conformal-mapping-based complex-variable solution that accommodates irregular multi-stage cavities and enforces a rigid static equilibrium to eliminate far-field singularities. It leverages a two-step mapping to unit annuli and solves a homogeneous Riemann-Hilbert problem for the complex potentials, iteratively determining coefficients that yield stress and displacement fields, which are then mapped back to physical space. Validation against a finite element model for a four-stage case shows good agreement and, in some aspects, improved accuracy, particularly near convex/corner regions, with a parametric study offering practical design guidance (e.g., effects of lateral stiffness, corner rounding, and far-field extent). The approach extends complex-variable methods to complex, asymmetrical sequential tunnelling without requiring temporary supports, providing a computationally efficient analytical tool for preliminary design and sensitivity analyses, while noting limitations related to topology, 3D effects, and concave geometries.

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.

Figures (14)

• Figure 1: Typical sequential shallow excavation schemes with liners and temporary supports in Ref sharifzadeh13:_desig_niayes
• Figure 2: Geomaterial before excavation and two possible sequential exavation schemes
• Figure 3: Boundary conditions of sequential excavation scheme of example in Fig. \ref{['fig:2']}c-3
• Figure 4: Curvilinear grids of backward conformal mapping of geomaterial for four excavation stages
• Figure 5: Geometry and meshing scheme of the finite element solution