Skip to main content
SpringerLink
Log in

A Closed-Form Analytical Solution for Circular Opening in Rocks Using Drucker–Prager Criterion

  • Original Paper
  • Published:
Indian Geotechnical Journal Aims and scope Submit manuscript

Abstract

Wellbore and tunnel problems are of true triaxial stress state, even if the ground is under axisymmetric loading condition. A closed-form analytical solution is proposed using Drucker–Prager failure criterion. The solutions are obtained for rock mass exhibiting elastic–perfectly plastic or elastic–brittle–plastic behaviour. The proposed solution is then compared with the finite element analysis (FE-analysis) results. Parametric studies are also carried out. The results of the proposed analytical solution are found to be in good agreement with the FE-analysis results. The proposed analytical solution can thus be used for predicting the stresses and deformation of underground circular openings considering true triaxial stress state.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Detournay E, Cheng AHD (1988) Poroelastic response of a borehole in a non-hydrostatic stress field. Int J Rock Mech Min Sci 25(3):171–182. https://doi.org/10.1016/0148-9062(88)92299-1

    Article  Google Scholar 

  2. Cui L, Abousleiman Y, Cheng A (1997) Poroelastic solution for an inclined borehole. J Appl Mech 64(1):32–38. https://doi.org/10.1115/1.2787291

    Article  MATH  Google Scholar 

  3. Abousleiman Y, Cui L (1998) Poroelastic solutions in transversely isotropic media for wellbore and cylinder. Int J Solids Struct 35(34–35):4905–4929. https://doi.org/10.1016/S0020-7683(98)00101-2

    Article  MATH  Google Scholar 

  4. Al-Ajmi AM, Zimmerman RW (2006) Stability analysis of vertical boreholes using the Mogi-Coulomb failure criterion. Int J Rock Mech Min Sci 43(8):1200–1211. https://doi.org/10.1016/j.ijrmms.2006.04.001

    Article  Google Scholar 

  5. Liu H, Zou D, Liu J (2007) Particle shape effect on macro-and micro behaviours of monodisperse ellipsoids. Int J Numer Anal Methods Geomech. 2008(32):189–213. https://doi.org/10.1002/nag

    Google Scholar 

  6. Zare-Reisabadi MR, Kaffash A, Shadizadeh SR (2012) Determination of optimal well trajectory during drilling and production based on borehole stability. Int J Rock Mech Min Sci 56:77–87. https://doi.org/10.1016/j.ijrmms.2012.07.018

    Article  Google Scholar 

  7. Zhang W, Gao J, Lan K, Liu X, Feng G, Ma Q (2015) Analysis of borehole collapse and fracture initiation positions and drilling trajectory optimization. J Pet Sci Eng 129:29–39. https://doi.org/10.1016/j.petrol.2014.08.021

    Article  Google Scholar 

  8. Das B, Chatterjee R (2016) Wellbore stability analysis and prediction of minimum mud weight for few wells in Krishna–Godavari Basin, India. Int J Rock Mech Min Sci 2017(93):30–37. https://doi.org/10.1016/j.ijrmms.2016.12.018

    Google Scholar 

  9. Salenҫon J (1969) Contraction quasi-statique d’une cavite ´ a ´ symétrie sphe ´rique ou cylindrique dans un milieu e ´lastoplastique. Annls Ponts Chauss 4:231–236

    Google Scholar 

  10. Detournay E (1986) Elastoplastic model of a deep tunnel for a rock with variable dilatancy. Rock Mech Rock Eng 19:99–108

    Article  Google Scholar 

  11. Brown ET, Bray JW, Ladanyi B, Hoek E (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)

    Article  Google Scholar 

  12. Panet M (1995) Calcul des tunnels par la méthode de convergence-confinement. Press de l’e ´cole Nationale des Ponts et Chausse ´es

  13. Pan XD, Brown ET (1996) Influence of axial stress and dilatancy on rock tunnel stability. J Geotech Eng ASCE 122(2):139–146

    Article  Google Scholar 

  14. 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

    Google Scholar 

  15. Sharan SK (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

    Article  Google Scholar 

  16. Sharan SK (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

    Article  Google Scholar 

  17. Li Y, Cao S, Fantuzzi N, Liu Y (2015) Elasto–plastic analysis of a circular borehole in elastic-strain softening coal seams. Int J Rock Mech Min Sci 80:316–324. https://doi.org/10.1016/j.ijrmms.2015.10.002

    Article  Google Scholar 

  18. Hoek E, Brown ET (1980) Empirical strength criterion for rock masses. J Geotech Eng Div ASCE 106(GT9):1013–1035

    Google Scholar 

  19. Chen R, Tonon F (2011) Closed-form solutions for a circular tunnel in elastic–brittle–plastic ground with the original and generalized Hoek–Brown failure criteria. Rock Mech Rock Eng 44(2):169–178. https://doi.org/10.1007/s00603-010-0122-5

    Article  Google Scholar 

  20. Hoek E, Carranza C, Corkum B (2002) Hoek-Brown failure criterion—2002 edition. Narms-Tac. https://doi.org/10.1016/0148-9062(74)91782-3

    Google Scholar 

  21. Lu A-Z, Xu G, Sun F, Sun W-Q (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

    Article  Google Scholar 

  22. Wang S, Wu Z, Guo M, Ge X (2012) Theoretical solutions of a circular tunnel with the influence of axial in situ stress in elastic–brittle–plastic rock. Tunn Undergr Sp Technol 30:155–168. https://doi.org/10.1016/j.tust.2012.02.016

    Article  Google Scholar 

  23. Li Y, Cao S, Fantuzzi N, Liu Y (2015) Elasto-plastic analysis of a circular borehole in elastic-strain softening coal seams. Int J Rock Mech Min Sci 80:316–324. https://doi.org/10.1016/j.ijrmms.2015.10.002

    Article  Google Scholar 

  24. Feng Zou J, Shuai Li S, Xu Y, Cheng Dan H, Heng Zhao L (2016) Theoretical solutions for a circular opening in an elastic–brittle–plastic rock mass incorporating the out-of-plane stress and seepage force. KSCE J Civ Eng 20(2):687–701. https://doi.org/10.1007/s12205-015-0789-y

    Article  Google Scholar 

  25. Mogi K (1971) Effect of the triaxial stress system on the failure of dolomite and limestone. Tectonophysics 11(2):111–127. https://doi.org/10.1016/0040-1951(71)90059-X

    Article  Google Scholar 

  26. Haimson B, Chang C (2000) A new true triaxial cell for testing mechanical properties of rock, and its use to determine rock strength and deformability of Westerly granite. Int J Rock Mech Min Sci 37(1–2):285–296. https://doi.org/10.1016/S1365-1609(99)00106-9

    Article  Google Scholar 

  27. Chang C, Haimson BC (2000) True triaxial strength and deformability of the German Continental deep drilling program (KTB) deep hole amphibolite. J Geophys Res 105:18999–19013

    Article  Google Scholar 

  28. Tiwari RP, Rao KS (2006) Post failure behaviour of a rock mass under the influence of triaxial and true triaxial confinement. Eng Geol 84(3–4):112–129. https://doi.org/10.1016/j.enggeo.2006.01.001

    Article  Google Scholar 

  29. Tiwari RP, Rao KS (2007) Response of an anisotropic rock mass under polyaxial stress state. J Mater Civ Eng ASCE 19(May):393–403. https://doi.org/10.1061/(ASCE)0899-1561(2007)19:5(393)

    Article  Google Scholar 

  30. Oku H, Haimson B, Song S-R (2007) True triaxial strength and deformability of the siltstone overlying the Chelungpu fault (Chi–Chi earthquake), Taiwan. Geophys Res Lett. 34:L09306

    Article  Google Scholar 

  31. Lee H, Haimson BC (2011) True triaxial strength, deformability, and brittle failure of granodiorite from the San Andreas Fault observatory at depth. Int J Rock Mech Min Sci 48(7):1199–1207. https://doi.org/10.1016/j.ijrmms.2011.08.003

    Article  Google Scholar 

  32. Sriapai T, Walsri C, Fuenkajorn K (2013) True-triaxial compressive strength of Maha Sarakham salt. Int J Rock Mech Min Sci 61:256–265. https://doi.org/10.1016/j.ijrmms.2013.03.010

    Article  Google Scholar 

  33. Ma X, Haimson BC (2016) Failure characteristics of two porous sandstones subjected to true triaxial stresses. J Geophys Res Solid Earth 121(9):6477–6498. https://doi.org/10.1002/2016JB012979

    Article  Google Scholar 

  34. Rukhaiyar S, Samadhiya NK (2017) A polyaxial strength model for intact sandstone based on Artificial Neural Network. Int J Rock Mech Min Sci 31(95):26–47

    Article  Google Scholar 

  35. Drucker DC, Prager W (1952) Soil mechanics and plastic analysis or limit design. Q Appl Math 9(2):157–165

    Article  MathSciNet  MATH  Google Scholar 

  36. Papanastasiou P, Durban D (1996) Elastoplastic analysis of cylindrical cavity problems in geomaterials. Int J Numer Anal Methods Geomech 1997(21):133–149

    MATH  Google Scholar 

  37. Chen SL, Abousleiman YN (2017) Wellbore stability analysis using strain hardening and/or softening plasticity models. Int J Rock Mech Min Sci 93:260–268. https://doi.org/10.1016/j.ijrmms.2017.02.007

    Article  Google Scholar 

  38. Chen SL, Abousleiman Y, Muraleetharan KK (2012) Closed-form elastoplastic solution for the Wellbore problem in strain hardening/softening rock formations. Int J Geomech 12(4):494–507. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000157

    Article  Google Scholar 

  39. Zou J, Zuo S, Xu Y (2016) Solution of strain-softening surrounding rock in deep tunnel incorporating 3D Hoek–Brown failure criterion and flow rule. Math Probl Eng, Article ID 7947036, pp 12

  40. Zhang L, Zhu H (2007) Three-dimensional Hoek-Brown strength criterion for rocks. J Geotech Geoenviron Eng 133(9):1128–1135. https://doi.org/10.1061/(ASCE)1090-0241(2007)133:9(1128)

    Article  Google Scholar 

  41. Xu SQ, Yu MH (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

    Article  Google Scholar 

  42. Zhang C, Zhao J, Zhang Q, Hu X (2012) A new closed-form solution for circular openings modeled by the Unified strength theory and radius-dependent Young’s modulus. Comput Geotech 42:118–128. https://doi.org/10.1016/j.compgeo.2012.01.005

    Article  Google Scholar 

  43. Singh A, Rao KS, Ayothiraman R (2017) Effect of intermediate principal stress on cylindrical tunnel in an elasto–plastic rock mass. Procedia Eng 173:1056–1063. https://doi.org/10.1016/j.proeng.2016.12.185

    Article  Google Scholar 

  44. Zhou XP, Bao XR, Yu MH, Xie Q (2010) Triaxial stress state of cylindrical openings for rocks modeled by elastoplasticity and strength criterion. Theor Appl Fract Mech 53:65–73. https://doi.org/10.1016/j.tafmec.2009.12.006

    Article  Google Scholar 

  45. Alejano LR, Bobet A (2012) Drucker–Prager criterion. Rock Mech Rock Eng 45(6):995–999. https://doi.org/10.1007/s00603-012-0278-2

    Article  Google Scholar 

  46. Kennedy TC, Lindberg HE (1978) Tunnel closure for nonlinear Mohr–Coulomb functions. J Geotech Eng ASCE 104(6):1313–1326

    Google Scholar 

  47. An Fritz P (1984) analytical solution for axisymmetric tunnel problems in elasto-viscoplastic media. Int J Numer Anal Method Geomech 8:325–342

    Article  Google Scholar 

  48. Detournay E, Fairhurst C (1987) Two-dimensional elastoplastic analysis of a long, cylindrical cavity under non-hydrostatic loading. Int J Rock Mech Min Sci 24:197–211. https://doi.org/10.1016/0148-9062(87)90175-6

    Article  Google Scholar 

  49. Ogawa T, Lo KY (1987) Effects of dilatancy and yield criteria on displacements around tunnels. Can Geotech J 24(1):100–113

    Article  Google Scholar 

  50. Charlez PhA, Roatesi S (1999) A fully analytical solution of the wellbore stability problem under undrained conditions using a linearized Cam-clay model. Oil Gas Sci Technol 54(5):551–563

    Article  Google Scholar 

  51. Sheng DC, Sloan SW, Yu HS (1999) Practical implementation of critical state models in FEM. In: Proc., 8th Australia New Zealand Conf. on Geomechanics, Hobart, Australia

  52. Collins IF, Pender MJ, Yan W (1992) Cavity expansion in sands under drained loading conditions. Int J Numer Anal Meth Geomech 16(1):3–23

    Article  MATH  Google Scholar 

  53. Collins IF, Stimpson JR (1994) Similarity solutions for drained and undrained in soils. Geotechnique 44:21–34. https://doi.org/10.1680/geot.1994.44.1.21

    Article  Google Scholar 

  54. Collins IF, Yu HS (1996) Undrained cavity expansions in critical state soils. Int J Numer Anal Methods Geomech 20:489–516. https://doi.org/10.1002/(SICI)1096-9853(199607)20:7%3c489:AID-NAG829%3e3.0.CO;2-V

    Article  MATH  Google Scholar 

  55. Cao LF, Teh CI, Cang MF (2001) Undrained cavity expansion in modifed Cam clay I: theoretical analysis. Géotechnique 51:323–334. https://doi.org/10.1680/geot.2001.51.4.323

    Article  Google Scholar 

  56. Chen WF, Mizuno E (1990) Nonlinear analysis in soil mechanics. Elsevier, Amsterdam

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India

    Aditya Singh, K. Seshagiri Rao & Ramanathan Ayothiraman

Authors
  1. Aditya Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Seshagiri Rao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ramanathan Ayothiraman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ramanathan Ayothiraman.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singh, A., Seshagiri Rao, K. & Ayothiraman, R. A Closed-Form Analytical Solution for Circular Opening in Rocks Using Drucker–Prager Criterion. Indian Geotech J 49, 437–454 (2019). https://doi.org/10.1007/s40098-019-00358-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40098-019-00358-6

Keywords

Search

Navigation