  Three-dimensional effect of tunnel face and gravitational excavation
generally occur in shallow tunnelling, which are nevertheless not adequately
considered in present complex variable solutions. In this paper, a new
time-dependent complex variable solution on quasi three-dimensional shallow
tunnelling in gravitational geomaterial is derived, and the far-field
displacement singularity is eliminated by fixed far-field ground surface in the
whole excavation time span. With an equivalent coefficient of three-dimensional
effect, the quasi three-dimensional shallow tunnelling is transformed into a
plane strain problem with time-dependent virtual traction along tunnel
periphery. The mixed boundaries of fixed far-field ground surface and nearby
free segment form a homogenerous Riemann-Hilbert problem with extra constraints
of the virtual traction along tunnel periphery, which is simultaneously solved
using an iterative linear system with good numerical stability. The mixed
boundary conditions along the ground surface in the whole excavation time span
are well satisified in a numerical case, which is further examined by comparing
with corresponding finite element solution. The results are in good agreements,
and the proposed solution illustrates high efficiency. More discussions are
made on excavation rate, viscosity, and solution convergence. A latent paradox
is disclosed for objectivity.
