Analytical Solution to Cylindrical Cavity Expansion in Mohr–Coulomb Soils Subject to Biaxial Stress Condition
Publication: International Journal of Geomechanics
Volume 21, Issue 9
Abstract
In this paper, a semianalytical approach is presented to formulate an excavated cavity in Mohr–Coulomb soils subject to biaxial initial stresses. The response of cavity expansion within the plastic region was investigated by combining the Mohr–Coulomb criterion with the equilibrium equation. The elastic–plastic (EP) boundary was expressed by a new conformal mapping function with three unknown parameters, which were determined by the stress continuity condition at the EP boundary and the volume-conservation condition. In the elastic zone, the stress function was expanded into Fourier series for convenience of application and the coefficients of Fourier series were determined according to stresses at the EP boundary. An unassociated flow rule with a dilation angle of 0° was adopted in the plastic zone, and Hooke’s law governed the behavior of soil within the elastic zone. Comparisons between the proposed solution and Galin’s solution showed good agreements, which validates the proposed framework. Extensive parametric studies were also performed to explore the effects of the internal friction angle, cohesion, and coefficient of earth pressure at rest on responses of cavity expansion. The results suggested that an increase of the internal friction angle, cohesion, Young’s modulus of soil, and coefficient of earth pressure at rest all leads to a higher expansion pressure at the cavity wall, while the extent of the plastic region around the cavity shrinks with an increase of the aforementioned three parameters.
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Acknowledgments
This study was financially supported by the National Natural Science Foundation of China (Grant No. 41772290).
References
Cao, L. F., C. I. Teh, and M. F. Chang. 2001. “Undrained cavity expansion in modified Cam clay I: Theoretical analysis.” Géotechnique 51 (4): 323–334. https://doi.org/10.1680/geot.2001.51.4.323.
Carter, J. P., J. R. Booker, and S. K. Yeung. 1986. “Cavity expansion in cohesive frictional soils.” Géotechnique 36 (3): 349–358. https://doi.org/10.1680/geot.1986.36.3.349.
Chang, M. F., C. I. Teh, and L. F. Cao. 2001. “Undrained cavity expansion in modified Cam clay II: Application to the interpretation of the piezocone test.” Géotechnique 51 (4): 335–350. https://doi.org/10.1680/geot.2001.51.4.335.
Chen, H. H., L. Li, J. P. Li, and D. A. Sun. 2020. “Elastoplastic solution to drained expansion of a cylindrical cavity in anisotropic critical-state soils.” J. Eng. Mech. 146 (5): 04020036. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001763.
Chen, H. H., H. H. Zhu, and L. Y. Zhang. 2021. “Analytical solution for deep circular tunnels in rock with consideration of disturbed zone, 3D strength and large strain.” Rock Mech. Rock Eng. 54: 1391–1410. https://doi.org/10.1007/s00603-020-02339-1.
Chen, S. L., and Y. N. Abousleiman. 2012. “Exact undrained elasto-plastic solution for cylindrical cavity expansion in modified Cam Clay soil.” Géotechnique 62 (5): 447–456. https://doi.org/10.1680/geot.11.P.027.
Chen, S. L., and Y. N. Abousleiman. 2013. “Exact drained solution for cylindrical cavity expansion in modified Cam Clay soil.” Géotechnique 63 (6): 510–517. https://doi.org/10.1680/geot.11.P.088.
Collins, I. F., and J. R. Stimpson. 1994. “Similarity solutions for drained a undrained cavity expansions in soils.” Géotechnique 44 (1): 21–34. https://doi.org/10.1680/geot.1994.44.1.21.
Cudmani, R., and V. A. Osinov. 2001. “The cavity expansion problem for the interpretation of cone penetration and pressuremeter tests.” Can. Geotech. J. 38 (3): 622–638. https://doi.org/10.1139/t00-124.
Detournay, E. 1985. “An approximative solution of the plastic zone around a circular tunnel subject to a non-hydrostatic far-field stress loading condition.” Proc. French Acad. Sci. 301 (12): 857–886.
Detournay, E. 1986. “An approximate statical solution of the elastoplastic interface for the problem of Galin with a cohesive-frictional material.” Int. J. Solids Struct. 22 (12): 1435–1454. https://doi.org/10.1016/0020-7683(86)90054-5.
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.
Detournay, E., and C. M. S. John. 1988. “Design charts for a deep circular tunnel under non-uniform loading.” Rock Mech. Rock Eng. 21 (2): 119–137. https://doi.org/10.1007/BF01043117.
Galin, L. A. 1947. “Plane elastic-plastic problem: Plastic regions around circular holes in plates and beams.” Div. Appl. Math. 10: 365–386.
Gao, X. L., X. X. Wei, and Z. K. Wang. 1991. “A general analytical solution of a strain-hardening elasto-plastic plate containing a circular hole subjected to biaxial loading—With applications in pressure vessels.” Int. J. Pressure Vessels Piping 47 (1): 35–55. https://doi.org/10.1016/0308-0161(91)90085-G.
Haythem, G., J. Mustafa, and H. Alain. 2020. “Cavity expansion in rock masses obeying the ‘Hoek–Brown’ failure criterion.” Rock Mech. Rock Eng. 53: 927–941. https://doi.org/10.1007/s00603-019-01920-7.
Hill, R. 1998. The mathematical theory of plasticity. New York: Oxford University Press.
Houlsby, G. T., and N. J. Withers. 1988. “Analysis of the cone pressuremeter test in clay.” Géotechnique 38 (4): 575–587. https://doi.org/10.1680/geot.1988.38.4.575.
Lee, Y. S., and H. Gong. 1987. “Application of complex variables and pseudo-stress function to power-law materials and stress analysis of single rigid inclusion in power-law materials subjected to simple tension and pure shear.” Int. J. Mech. Sci. 29 (10–11): 669–694. https://doi.org/10.1016/0020-7403(87)90055-5.
Li, L., H. H. Chen, J. P. Li, and D. A. Sun. 2021. “An elastoplastic solution to undrained expansion of a cylindrical cavity in SANICLAY under plane stress condition.” Comput. Geotech. 132: 103990. https://doi.org/10.1016/j.compgeo.2020.103990.
Li, L., J. P. Li, and D. A. Sun. 2016. “Anisotropically elasto-plastic solution to undrained cylindrical cavity expansion in K0-consolidated clay.” Comput. Geotech. 73: 83–90. https://doi.org/10.1016/j.compgeo.2015.11.022.
Masri, R., and D. Durban. 2007. “Cylindrical cavity expansion in compressible Mises and Tresca solids.” Eur. J. Mech. A Solids 26 (4): 712–727. https://doi.org/10.1016/j.euromechsol.2006.12.003.
Muskhelishvili, N. I. 1963. Some basic problems of the mathematical theory of elasticity. Groningen, Netherlands: Noordhoff.
Rostami, A., Y. L. Yi, and A. Bayat. 2017. “Estimation of maximum annular pressure during HDD in noncohesive soils.” Int. J. Geomech. 17 (4): 06016029. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000801.
Satapathy, S. 2001. “Dynamic spherical cavity expansion in brittle ceramics.” Int. J. Solids Struct. 38 (32–33): 5833–5845. https://doi.org/10.1016/S0020-7683(00)00388-7.
Savin, G. N. 1961. Stress concentration around holes. New York: Pergamon Press.
Sharma, D. S. 2012. “Stress distribution around polygonal holes.” Int. J. Mech. Sci. 65 (1): 115–124. https://doi.org/10.1016/j.ijmecsci.2012.09.009.
Shuttle, D. 2007. “Cylindrical cavity expansion and contraction in Tresca soil.” Géotechnique 57 (3): 305–308. https://doi.org/10.1680/geot.2007.57.3.305.
Staheli, K., D. Bennett, H. W. O’Donnell, and T. J. Hurley. 1998. Installation of pipelines beneath levees using horizontal directional drilling. Technical Rep. CPAR-GL-98-1. Vicksburg, MS: US Army Corps of Engineers, Waterways Experiment Station.
Timosenko, S. P., and J. N. Goodier. 1951. Theory of elasticity. New York: McGraw-Hill.
Wroth, C. P., and D. Windle. 1975. “Analysis of the pressuremeter test allowing for volume change.” Géotechnique 25 (3): 598–604. https://doi.org/10.1680/geot.1975.25.3.598.
Yang, C. Y., J. P. Li, L. Li, and D. A. Sun. 2021. “Expansion responses of a cylindrical cavity in overconsolidated unsaturated soils: A semi-analytical elastoplastic solution.” Comput. Geotech. 130: 103922. https://doi.org/10.1016/j.compgeo.2020.103922.
Yu, H. S., and J. P. Carter. 2002. “Rigorous similarity solutions for cavity expansion in cohesive-frictional soils.” Int. J. Geomech. 2 (2): 233–258. https://doi.org/10.1061/(ASCE)1532-3641(2002)2:2(233).
Yu, H. S., and R. K. Rowe. 1999. “Plasticity solutions for soil behaviour around contracting cavities and tunnels.” Int. J. Numer. Anal. Methods Geomech. 23 (12): 1245–1279. https://doi.org/10.1002/(SICI)1096-9853(199910)23:12%3C1245::AID-NAG30%3E3.0.CO;2-W.
Zhou, H., H. L. Liu, G. Q. Kong, and X. Huang. 2014. “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, X. Y., Y. S. Xu, D. A. Sun, Y. Z. Tan, and Y. F. Xu. 2021. “Three-dimensional thermal-hydraulic coupled analysis in the nuclear waste repository.” Ann. Nucl. Energy 151: 107866. https://doi.org/10.1016/j.anucene.2020.107866.
Zhuang, P. Z., and H. S. Yu. 2019. “A unified analytical solution for elastic–plastic stress analysis of a cylindrical cavity in Mohr–Coulomb materials under biaxial in situ stresses.” Géotechnique 69 (4): 369–376. https://doi.org/10.1680/jgeot.17.P.281.
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Published In
International Journal of Geomechanics
Volume 21 • Issue 9 • September 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Oct 18, 2020
Accepted: Apr 16, 2021
Published online: Jun 17, 2021
Published in print: Sep 1, 2021
Discussion open until: Nov 17, 2021
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