Semianalytical Solutions for Elastic–Brittle–Plastic Surrounding Rock under Biaxial In Situ Stress Field Based on Unified Strength Criterion
Publication: International Journal of Geomechanics
Volume 23, Issue 10
Abstract
Semianalytical solutions for circular tunnels under biaxial in situ stress fields are developed in this study. The surrounding rock is assumed to behave as an elastic–brittle–plastic model and is characterized by the unified strength criterion. For the stress analysis, the perturbation solutions for arbitrary-orders plastic radius for deep-buried tunnels are presented with a newly developed numerical iterative method. Furthermore, a stress renewal algorithm is provided for determining the stress function in elastic zones more accurately. In accordance with the small deformation assumption and nonassociated flow rule, the analytical expressions for radial and circumference displacement in a plastic zone are presented for the first time. Subsequently, the undetermined coefficients in plastic displacement can be solved numerically with the least square method. Based on the proposed semianalytical solutions for circular tunnels, an improved equivalent circular method is developed for estimating the plastic radius and convergence of tunnels with arbitrary shaped cross sections approximately. The reliabilities of the proposed semianalytical solutions and the improved equivalent circular method are verified by comparing with numerical simulation and field data. Meanwhile, the whole calculating process is illustrated in a flow chart for the convenience of programming. From the perspective of practicality, the provided solutions can be employed for predicting plastic radius and ground response curves in tunnel excavation, which can supply important references for the tunnel stability assessment and support design.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data, models, and code generated or used during the study appear in the published article.
Acknowledgments
The first author was supported by the China Scholarship Council (No. 202006370287) to study at the Technical University of Denmark.
References
Alonso, E., L. R. Alejano, F. Varas, G. Fdez-Manin, and C. Carranza-Torres. 2003. “Ground response curves for rock masses exhibiting strain-softening behaviour.” Int. J. Numer. Anal. Methods Geomech. 27 (13): 1153–1185. https://doi.org/10.1002/nag.315.
Barber, J. R. 2002. Elasticity. 2nd ed. Dordrecht, Netherlands: Kluwer Academic Publishers.
Brown, E. T., J. W. Bray, B. Ladanyi, and E. Hoek. 1983. “Ground response curves for rock tunnels.” J. Geotech. Eng. 109 (1): 15–39. https://doi.org/10.1061/(ASCE)0733-9410(1983)109:1(15).
Carranza-Torres, C. 2003. “Dimensionless graphical representation of the exact elasto-plastic solution of a circular tunnel in a Mohr-Coulomb material subject to uniform far-field stresses.” Rock Mech. Rock Eng. 36 (3): 237–253. https://doi.org/10.1007/s00603-002-0048-7.
Carranza-Torres, C., B. Rysdahl, and M. Kasim. 2013. “On the elastic analysis of a circular lined tunnel considering the delayed installation of the support.” Int. J. Rock Mech. Min. Sci. 61: 57–85. https://doi.org/10.1016/j.ijrmms.2013.01.010.
Detournay, E., and C. Fairhurst. 1987. “Two-dimensional elastoplastic analysis of a long, cylindrical cavity under non-hydrostatic loading.” Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 24 (4): 197–211. https://doi.org/10.1016/0148-9062(87)90175-6.
Guo, X. F., Z. Q. Zhao, X. Gao, X. Wu, and N. Ma. 2019. “Analytical solutions for characteristic radii of circular roadway surrounding rock plastic zone and their application.” Int. J. Min. Sci. Technol. 29 (2): 263–272. https://doi.org/10.1016/j.ijmst.2018.10.002.
Hoek, E., C. Carranza-Torres, and B. Corkum. 2002. “Hoek-Brown failure criterion-2002 edition.” Proc. NARMS-Tac 1 (1): 267–273.
Kabwe, E., M. Karakus, and E. K. Chanda. 2020. “Proposed solution for the ground reaction of non-circular tunnels in an elastic-perfectly plastic rock mass.” Comput. Geotech. 119: 103354. https://doi.org/10.1016/j.compgeo.2019.103354.
Lee, Y. K., and S. Pietruszczak. 2008. “A new numerical procedure for elasto-plastic analysis of a circular opening excavated in a strain-softening rock mass.” Tunnelling Underground Space Technol. 23 (5): 588–599. https://doi.org/10.1016/j.tust.2007.11.002.
Leitman, M. J., and P. Villaggio. 2009. “Plastic zone around circular holes.” J. Eng. Mech. 135 (12): 1467–1471. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000062.
Massinas, S. A., and M. G. Sakellariou. 2009. “Closed-form solution for plastic zone formation around a circular tunnel in half-space obeying Mohr–Coulomb criterion.” Géotechnique 59 (8): 691–701. https://doi.org/10.1680/geot.8.069.
Oke, J., N. Vlachopoulos, and M. Diederichs. 2018. “Improvement to the convergence-confinement method: Inclusion of support installation proximity and stiffness.” Rock Mech. Rock Eng. 51 (5): 1495–1519. https://doi.org/10.1007/s00603-018-1418-0.
Pan, Q. J., and D. Dias. 2017a. “Probabilistic evaluation of tunnel face stability in spatially random soils using sparse polynomial chaos expansion with global sensitivity analysis.” Acta Geotech. 12 (6): 1415–1429. https://doi.org/10.1007/s11440-017-0541-5.
Pan, Q. J., and D. Dias. 2017b. “Upper-bound analysis on the face stability of a non-circular tunnel.” Tunnelling Underground Space Technol. 62: 96–102. https://doi.org/10.1016/j.tust.2016.11.010.
Pan, Q. J., and D. Dias. 2018. “Probabilistic analysis of a rock tunnel face using polynomial chaos expansion method.” Int. J. Geomech. 18 (4): 04018013. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001116.
Park, K. H., and Y. J. Kim. 2006. “Analytical solution for a circular opening in an elastic-brittle-plastic rock.” Int. J. Rock Mech. Min. Sci. 43 (4): 616–622. https://doi.org/10.1016/j.ijrmms.2005.11.004.
Park, K. H., B. Tontavanich, and J.-G. Lee. 2008. “A simple procedure for ground response curve of circular tunnel in elastic-strain softening rock masses.” Tunnelling Underground Space Technol. 23 (2): 151–159. https://doi.org/10.1016/j.tust.2007.03.002.
Qin, C. B., and S. C. Chian. 2018. “Revisiting crown stability of tunnels deeply buried in non-uniform rock surrounds.” Tunnelling Underground Space Technol. 73: 154–161. https://doi.org/10.1016/j.tust.2017.12.006.
Sharan, S. 2005. “Exact and approximate solutions for displacements around circular openings in elastic-brittle-plastic Hoek-Brown rock.” Int. J. Rock Mech. Min. Sci. 42 (4): 542–549. https://doi.org/10.1016/j.ijrmms.2005.03.019.
Sheng, Y. M., J. F. Zou, Y. P. Dong, and G. H. Chen. 2022. “Novel perturbation solutions for deep-buried non-circular tunnels under biaxial in situ stress field based on Mohr-Coulomb criterion.” Appl. Math. Modell. 110: 408–440.
Su, T., H. Peng, and H. Liu. 2018. “Mechanical analysis of the circular tunnel considering the interaction between the ground response curve and support response curve.” Math. Probl. Eng. 2018: 7892010.
Tu, H. L., C. S. Qiao, and Z. M. Han. 2018. “Elastic-brittle-plastic analysis of the radial subgrade modulus for a circular cavity based on the generalized nonlinear unified strength criterion.” Tunnelling Underground Space Technol. 71: 623–636. https://doi.org/10.1016/j.tust.2017.11.004.
Wang, S. L., Z. J. Wu, M. W. Guo, and X. Ge. 2012. “Theoretical solutions of a circular tunnel with the influence of axial in situ stress in elastic-brittle-plastic rock.” Tunnelling Underground Space Technol. 30: 155–168. https://doi.org/10.1016/j.tust.2012.02.016.
Xu, S. Q., and M. H. Yu. 2006. “The effect of the intermediate principal stress on the ground response of circular openings in rock mass.” Rock Mech. Rock Eng. 39 (2): 169–181. https://doi.org/10.1007/s00603-005-0064-5.
Zareifard, M. R., and A. Fahimifar. 2015. “Elastic-brittle-plastic analysis of circular deep underwater cavities in a Mohr-Coulomb rock mass considering seepage forces.” Int. J. Geomech. 15 (5): 04014077. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000400.
Zhang, J. Z., X. P. Zhou, and P. Yin. 2019. “Visco-plastic deformation analysis of rock tunnels based on fractional derivatives.” Tunnelling Underground Space Technol. 85: 209–219. https://doi.org/10.1016/j.tust.2018.12.019.
Zhang, L. 2008. “A generalized three-dimensional Hoek–Brown strength criterion.” Rock Mech. Rock Eng. 41 (6): 893–915. https://doi.org/10.1007/s00603-008-0169-8.
Zhou, H., G. Kong, and H. Liu. 2016. “A semi-analytical solution for cylindrical cavity expansion in elastic-perfectly plastic soil under biaxial in situ stress field.” Geotechnique 66 (7): 584–595. https://doi.org/10.1680/jgeot.15.P.115.
Zhou, H., H. Liu, G. Kong, and Z. Cao. 2014a. “Analytical solution for pressure-controlled elliptical cavity expansion in elastic-perfectly plastic soil.” Geotech. Lett. 4 (2): 72–78. https://doi.org/10.1680/geolett.14.00004.
Zhou, H., H. L. Liu, G. Q. Kong, and X. Huang. 2014b. “Analytical solution of undrained cylindrical cavity expansion in saturated soil under anisotropic initial stress.” Comput. Geotech. 55: 232–239. https://doi.org/10.1016/j.compgeo.2013.09.011.
Zhou, H., B. Sheil, and H. Liu. 2022. “Noncircular cavity expansion in undrained soil: Semi-analytical solution.” J. Eng. Mech. 148 (7): 04022032. https://doi.org/10.1061/(ASCE)EM.1943-7889.0002118.
Zou, J. F., Y. M. Sheng, M. Y. Xia, and F. Wang. 2020. “A novel numerical-iterative-approach for strain-softening surrounding rock incorporating rockbolts effectiveness and hydraulic-mechanical coupling based on three-dimensional Hoek-Brown strength criterion.” Tunnelling Underground Space Technol. 101: 103358. https://doi.org/10.1016/j.tust.2020.103358.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 23 • Issue 10 • October 2023
Copyright
© 2023 American Society of Civil Engineers.
History
Received: Jun 10, 2022
Accepted: Apr 23, 2023
Published online: Jul 28, 2023
Published in print: Oct 1, 2023
Discussion open until: Dec 28, 2023
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.