# Viscoelastic solutions for stresses and displacements around non-circular tunnels sequentially excavated at great depths

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## Abstract

This research study presents analytical solutions for the stresses and displacements around deeply buried non-circular tunnels, taking into account the viscoelasticity of the ground, and the sequential excavation of the tunnels’ cross-sections. General initial far-field stress states are assumed, and the time-dependent pressures exerted at the internal tunnel boundaries are found to account for the support effects or water pressures of the hydraulic tunnels. Then, solutions are derived for tunnels with a time-varying sizes and/or shape, by assuming the time-dependent functions specified by the designers. The analytical solutions for the stresses and displacements around elliptical and square tunnels are specifically presented for linearly viscoelastic models using a Muskhelishvili complex variable method and Laplace transform techniques. For validation purposes, numerical analyses are performed for the excavations of elliptical and square tunnels in rock which are simulated by Poynting–Thomson or generalized Kelvin viscoelastic models. Good agreements are observed between the analytical and numerical results of this study. Then, parametric analyses are carried out in order to investigate the effects of the far-field shear stress, along with the distribution forms of the internal pressures, on the ground displacements and stresses. The proposed analytical solutions can be employed to accurately predict the stress concentrations, as well as the time-dependent displacements around deeply buried elliptical or square-shaped tunnels. Furthermore, it is confirmed that this study’s described methodology may be potentially applied to obtain analytical solutions for other arbitrary shaped tunnels sequentially excavated in viscoelastic rock.

## Keywords

Analytical solutions Non-circular tunnels Sequential excavations Viscoelastic rock## Notes

### Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11572228, 51639008); National Basic Research Program of China (973 Program) with Grant No. 2014CB046901; State Key Lab. of Disaster Reduction in Civil Engineering with Grant No. SLDRCE14-B-11; Fundamental Research Funds for the Central Universities. These supports are greatly appreciated. The authors thank the reviewers for valuable comments and suggestions for greatly improving the presentation of the paper.

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