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.

翻译：隧道掌子面的三维效应与重力开挖通常共存于浅埋隧道工程，然而现有复变解未能充分考虑上述因素。本文推导了重力地质材料中准三维浅埋隧道的时变复变解，通过在开挖全时段固定远场地表位移，消除了远场位移奇异性。引入三维效应等效系数，将准三维浅埋隧道问题转化为沿隧道周边具有时变虚拟牵引力的平面应变问题。固定远场地表与邻近自由段构成的混合边界，结合隧道周边虚拟牵引力的附加约束，形成齐次黎曼-希尔伯特问题，该问题通过具有良好数值稳定性的迭代线性系统同步求解。数值算例表明，开挖全时段沿地表的混合边界条件得到充分满足，与相应有限元解的对比进一步验证了该结论。计算结果吻合良好，所提解法展现出高效性。本文还就开挖速率、粘性系数及解收敛性展开深入讨论，并揭示了客观性存在的潜在悖论。