A Closed-Form Elastic Solution of Ground Response Curve for Noncircular Openings
Publication: International Journal of Geomechanics
Volume 21, Issue 12
Abstract
Ground response curves (GRCs) can be used to characterize the evolution law of stress and displacement during tunnel excavation. In this study, an attempt is made to obtain the GRC and find the stress field for a noncircular tunnel using a complex variable method. The analytical solution of an elastic GRC for a noncircular tunnel with a complex geometry configuration is proposed, and the stress and displacement of the rock mass caused by tunnel excavation are provided. The proposed analytical solution for a noncircular tunnel is compared with the known solutions of circular openings by degenerating the proposed model. The proposed solution is compared with a quasi-two-dimensional calculation model using FLAC finite-difference code, as well as with the known solutions of circular openings by degenerating the proposed model. The results indicated good agreement with the analytical and numerical solutions, verifying the correctness of the derivation. Finally, the GRCs are investigated based on three key points. It illustrates that the stresses and displacements of the three key points are linear and that the predicted GRCs of three key points by both models are in general agreement, if the difference in gravity field between the two methods is not considered.
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Data Availability Statement
All data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
The authors acknowledge the financial support from the National Natural Science Foundation for Young Scientists of China (Grant No. 50909056) and the support from the Natural Science Foundation for Youths of Shandong Province (Grant No. ZR2014EEM014).
References
Alejano, L. R., and E. Alonso. 2005. “Considerations of the dilatancy angle in rocks and rock masses.” Int. J. Rock Mech. Min. Sci. 42 (4): 481–507. https://doi.org/10.1016/j.ijrmms.2005.01.003.
Brown, E. T., J. W. Bray, B. Landayi, 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).
Brown, E. T., J. W. Bray, and F. J. Santarelli. 1989. “Influence of stress-dependent elastic moduli on stresses and strains around axisymmetric boreholes.” Rock Mech. Rock Eng. 22 (3): 189–203. https://doi.org/10.1007/BF01470986.
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 (10): 57–85. https://doi.org/10.1016/j.ijrmms.2013.01.010.
Einstein, H. H., and C. W. Schwartz. 1979. “Simplified analysis for tunnel supports.” J. Geotech. Eng. Div. 105 (4): 499–518. https://doi.org/10.1061/AJGEB6.0000786.
Exadaktylos, G. E., P. A. Liolios, and M. C. Stavropoulou. 2003. “A semi-analytical elastic stress–displacement solution for notched circular openings in rocks.” Int. J. Solids Struct. 40 (5): 1165–1187. https://doi.org/10.1016/S0020-7683(02)00646-7.
Exadaktylos, G. E., and M. C. Stavropoulou. 2002. “A closed-form elastic solution for stresses and displacements around tunnels.” Int. J. Rock Mech. Min. Sci. 39 (7): 905–916. https://doi.org/10.1016/S1365-1609(02)00079-5.
Fahimifar, A., H. Ghadami, and M. Ahmadvand. 2015. “An elasto-plastic model for underwater tunnels considering seepage body forces and strain-softening behaviour.” Eur. J. Environ. Civ. Eng. 19 (2): 129–151. https://doi.org/10.1080/19648189.2014.939305.
Han, J. X., S. C. Li, S. C. Li, and W. M. Yang. 2013. “A procedure of strain-softening model for elasto-plastic analysis of a circular opening considering elasto-plastic coupling.” Tunnelling Underground Space Technol. 37: 128–134. https://doi.org/10.1016/j.tust.2013.04.001.
Kargar, A. R. 2019. “An analytical solution for circular tunnels excavated in rock masses exhibiting viscous elastic-plastic behavior.” Int. J. Rock Mech. Min. Sci. 124: 104128. https://doi.org/10.1016/j.ijrmms.2019.104128.
Kargar, A. R., R. Rahmannejad, and M. A. Hajabasi. 2014. “A semi-analytical elastic solution for stress field of lined noncircular tunnels at great depth using complex variable method.” Int. J. Solids Struct. 51 (6): 1475–1482. https://doi.org/10.1016/j.ijsolstr.2013.12.038.
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.
Lu, A. Z., G. S. Xu, F. Sun, and W. Q. Sun. 2010. “Elasto-plastic analysis of a circular tunnel including the effect of the axial in situ stress.” Int. J. Rock Mech. Min. Sci. 47 (1): 50–59. https://doi.org/10.1016/j.ijrmms.2009.07.003.
Ma, Y. F., H. Qin, C. M. Zheng, M. B. Wang, and T. P. Zhang. 2016. “An elastic-plastic solution for a circular pressure tunnel with liner based on a new model.” Int. J. Earth Sci. Eng. 9 (6): 2354–2360.
Masoud, R., R. Nima, and P. Oreste. 2020. “A new analytical-numerical solution to analyze a circular tunnel using 3D Hoek-Brown failure criterion.” Geomech. Eng. 22 (1): 11–23.
Muskhelishvili, N. I. 1963. Some basic problems of the mathematical theory of elasticity. Groningen, Netherlands: Noordhoff.
Oreste, P. 2015. “Analysis of the interaction between the lining of a TBM tunnel and the ground using the convergence-confinement method.” Am. J. Appl. Sci. 12 (4): 276–283. https://doi.org/10.3844/ajassp.2015.276.283.
Park, K. H. 2014. “Similarity solution for a spherical or circular opening in elastic-strain softening rock mass.” Int. J. Rock Mech. Min. Sci. 71: 151–159. https://doi.org/10.1016/j.ijrmms.2014.07.003.
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.
Sharan, S. K. 2003. “Elastic-brittle-plastic analysis of circular openings in Hoek-Brown media.” Int. J. Rock Mech. Min. Sci. 40 (6): 817–824. https://doi.org/10.1016/S1365-1609(03)00040-6.
Wang, S. L., X. T. Yin, H. Tang, and X. R. Ge. 2010. “A new approach for analyzing circular tunnel in strain-softening rock masses.” Int. J. Rock Mech. Min. Sci. 47 (1): 170–178. https://doi.org/10.1016/j.ijrmms.2009.02.011.
Wang, X. F., B. S. Jiang, Q. Zhang, M. M. Lu, and M. Chen. 2019. “Analytical solution of circular tunnel in elastic-viscoplastic rock mass.” Latin Am. J. Solids Struct. 16 (6): 1–19. https://doi.org/10.1590/1679-78255701.
Wu, X. Z., Y. J. Jinag, and Z. C. Guan. 2018. “A modified strain-softening model with multi-post-peak behaviours and its application in circular tunnel.” Eng. Geol. 240: 21–33. https://doi.org/10.1016/j.enggeo.2018.03.031.
Yang, J. W., Z. Tang, H. E. Feng, X. J. Zhu, and Y. J. Yu. 2015. “Analysis of the complex functions of the surrounding rock stress field in the elastic circular tunnels.” J. Safety Environ. 15 (1): 148–152.
Zhang, Q., B. S. Jiang, X. S. Wu, H. Q. Zhang, and L. J. Han. 2012. “Elasto-plastic coupling analysis of circular openings in elasto-brittle-plastic rock mass.” Theor. Appl. Fract. Mech. 60 (1): 60–67. https://doi.org/10.1016/j.tafmec.2012.06.008.
Zhang, Q., H. Y. Wang, Y. J. Jiang, M. M. Lu, and B. S. Jiang. 2019. “A numerical large strain solution for circular tunnels excavated in strain- softening rock masses.” Comput. Geotech. 114: 103142. https://doi.org/10.1016/j.compgeo.2019.103142.
Zhang, Q., C. H. Zhang, B. S. Jiang, N. Li, and Y. C. Wang. 2018. “Elastoplastic coupling solution of circular openings in strain-softening rock mass considering pressure-dependent effect.” Int. J. Geomech. 18 (1): 04017132. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001043.
Zou, J. F., and Z. H. Qian. 2018. “Face-stability analysis of tunnels excavated below groundwater considering coupled flow deformation.” Int. J. Geomech. 18 (8): 04018089. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001199.
Information & Authors
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Published In
International Journal of Geomechanics
Volume 21 • Issue 12 • December 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Feb 3, 2021
Accepted: Aug 17, 2021
Published online: Sep 30, 2021
Published in print: Dec 1, 2021
Discussion open until: Mar 1, 2022
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