Tunnels are rarely driven through uniform, isotropic ground. In reality, rock masses often exhibit anisotropy, and shallow tunnels must additionally cope with stresses introduced by surcharge loads applied at the ground surface. A new analytical study addresses exactly this combination of difficulties, presenting a closed-form solution for the stresses that develop in the rock surrounding a shallow tunnel excavated in a transversely isotropic rock mass subjected to surcharge loadings.
The study, published open access in the journal Geomechanics and Geophysics for Geo-Energy and Geo-Resources, was authored by Guangshang Zeng, Huaning Wang, Fei Song, Mingjing Jiang and Fengyong Lv. The article was shared early as a peer-reviewed, accepted manuscript carrying a permanent DOI, with a final version of record to follow.
The central achievement of the work is the full incorporation of the initial stresses caused by gravity, surcharge loads, support pressure and the orientation of the anisotropy into a single theoretical derivation. This means the solution does not treat surface loading or the directional character of the rock as secondary corrections, but builds them into the mathematical framework from the outset.
To make such a derivation tractable, the authors divided the mechanical model of the shallow buried tunnel into two simply connected domains. One domain is a model with a hole in an infinite plane, representing the tunnel opening, and the other is a model without a hole in a half plane, representing the ground surface region. This decomposition allows each part of the problem to be solved with established techniques before the two solutions are combined.
The analytical solution itself is obtained using complex variable theory together with the Schwartz alternating method.
A key step in validating the new analytical solution is checking that its degenerate solution is consistent with the results of other references. The authors report that the degenerate solution of their formulation is consistent with the results of other references, providing a baseline check on the mathematics. They also compared the full solution against finite element numerical simulations and found good agreement between the analytical predictions and the numerical results, indicating that the closed-form model captures the same stress distribution that a full numerical model produces.
Beyond verification, the paper includes a parameter analysis examining how three factors affect the surrounding rock stress: the location of the surcharge load, the angle of anisotropy, and the depth of the tunnel. Because the solution is analytical, these parameter studies can be carried out quickly, without the computational cost of repeated numerical simulations. The authors state that the analytical model can accurately and quickly predict the ground response, which is precisely the kind of capability useful in the early stages of tunnel design, when many configurations must be screened before committing to detailed numerical analysis.
The authors position the work as significant for anisotropic rock mechanics and for the verification of related numerical simulations, since an independent closed-form benchmark gives modelers a rigorous reference against which to test their codes. They also suggest that the solution provides a new idea for the preliminary design of shallow buried tunnels in transversely isotropic rock mass in the future.
For engineers, the practical appeal is straightforward: a fast, verified tool for estimating stresses around shallow tunnels in directionally dependent ground under surface loads.
Subject of Research: An analytical solution for stresses around shallow tunnels excavated in transversely isotropic rock masses under surcharge loading
Article Title: Analytical solution for stresses around shallow tunnels in transversely isotropic ground subjected to surcharge loadings
Article References: Zeng, G., Wang, H., Song, F., Jiang, M., & Lv, F. (2026). Analytical solution for stresses around shallow tunnels in transversely isotropic ground subjected to surcharge loadings. Geomechanics and Geophysics for Geo-Energy and Geo-Resources. https://doi.org/10.1007/s40948-026-01254-z
Image Credits: AI Generated
DOI: 10.1007/s40948-026-01254-z
Keywords: tunnel engineering, rock mechanics, anisotropy, transversely isotropic rock, analytical solution, complex variable theory, Schwartz alternating method, surcharge loading, shallow tunnels, numerical verification, geomechanics, Analytical
News Source: Denise Maddox. (October 28, 2026). New Analytical Solution Predicts Stresses Around Shallow Tunnels in Anisotropic Rock. Scienmag.
