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

  • Aditya Singh
  • K. Seshagiri Rao
  • Ramanathan AyothiramanEmail author
Original Paper


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.


Drucker–Prager criterion Elastic–brittle–plastic Elastic–perfectly plastic Wellbore Tunnel 



  1. 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. CrossRefGoogle Scholar
  2. 2.
    Cui L, Abousleiman Y, Cheng A (1997) Poroelastic solution for an inclined borehole. J Appl Mech 64(1):32–38. CrossRefzbMATHGoogle Scholar
  3. 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. CrossRefzbMATHGoogle Scholar
  4. 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. CrossRefGoogle Scholar
  5. 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. Google Scholar
  6. 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. CrossRefGoogle Scholar
  7. 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. CrossRefGoogle Scholar
  8. 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. Google Scholar
  9. 9.
    Salenҫon J (1969) Contraction quasi-statique d’une cavite ´ a ´ symétrie sphe ´rique ou cylindrique dans un milieu e ´lastoplastique. Annls Ponts Chauss 4:231–236Google Scholar
  10. 10.
    Detournay E (1986) Elastoplastic model of a deep tunnel for a rock with variable dilatancy. Rock Mech Rock Eng 19:99–108CrossRefGoogle Scholar
  11. 11.
    Brown ET, Bray JW, Ladanyi B, Hoek E (1983) Ground response curves for rock tunnels. J Geotech Eng. 109(1):15–39. CrossRefGoogle Scholar
  12. 12.
    Panet M (1995) Calcul des tunnels par la méthode de convergence-confinement. Press de l’e ´cole Nationale des Ponts et Chausse ´esGoogle Scholar
  13. 13.
    Pan XD, Brown ET (1996) Influence of axial stress and dilatancy on rock tunnel stability. J Geotech Eng ASCE 122(2):139–146CrossRefGoogle Scholar
  14. 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. Google Scholar
  15. 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. CrossRefGoogle Scholar
  16. 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. CrossRefGoogle Scholar
  17. 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. CrossRefGoogle Scholar
  18. 18.
    Hoek E, Brown ET (1980) Empirical strength criterion for rock masses. J Geotech Eng Div ASCE 106(GT9):1013–1035Google Scholar
  19. 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. CrossRefGoogle Scholar
  20. 20.
    Hoek E, Carranza C, Corkum B (2002) Hoek-Brown failure criterion—2002 edition. Narms-Tac. Google Scholar
  21. 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. CrossRefGoogle Scholar
  22. 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. CrossRefGoogle Scholar
  23. 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. CrossRefGoogle Scholar
  24. 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. CrossRefGoogle Scholar
  25. 25.
    Mogi K (1971) Effect of the triaxial stress system on the failure of dolomite and limestone. Tectonophysics 11(2):111–127. CrossRefGoogle Scholar
  26. 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. CrossRefGoogle Scholar
  27. 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 105:18999–19013CrossRefGoogle Scholar
  28. 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. CrossRefGoogle Scholar
  29. 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. CrossRefGoogle Scholar
  30. 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:L09306CrossRefGoogle Scholar
  31. 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. CrossRefGoogle Scholar
  32. 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. CrossRefGoogle Scholar
  33. 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. CrossRefGoogle Scholar
  34. 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):26–47CrossRefGoogle Scholar
  35. 35.
    Drucker DC, Prager W (1952) Soil mechanics and plastic analysis or limit design. Q Appl Math 9(2):157–165MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Papanastasiou P, Durban D (1996) Elastoplastic analysis of cylindrical cavity problems in geomaterials. Int J Numer Anal Methods Geomech 1997(21):133–149zbMATHGoogle Scholar
  37. 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. CrossRefGoogle Scholar
  38. 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. CrossRefGoogle Scholar
  39. 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 12Google Scholar
  40. 40.
    Zhang L, Zhu H (2007) Three-dimensional Hoek-Brown strength criterion for rocks. J Geotech Geoenviron Eng 133(9):1128–1135. CrossRefGoogle Scholar
  41. 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. CrossRefGoogle Scholar
  42. 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. CrossRefGoogle Scholar
  43. 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. CrossRefGoogle Scholar
  44. 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. CrossRefGoogle Scholar
  45. 45.
    Alejano LR, Bobet A (2012) Drucker–Prager criterion. Rock Mech Rock Eng 45(6):995–999. CrossRefGoogle Scholar
  46. 46.
    Kennedy TC, Lindberg HE (1978) Tunnel closure for nonlinear Mohr–Coulomb functions. J Geotech Eng ASCE 104(6):1313–1326Google Scholar
  47. 47.
    An Fritz P (1984) analytical solution for axisymmetric tunnel problems in elasto-viscoplastic media. Int J Numer Anal Method Geomech 8:325–342CrossRefGoogle Scholar
  48. 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. CrossRefGoogle Scholar
  49. 49.
    Ogawa T, Lo KY (1987) Effects of dilatancy and yield criteria on displacements around tunnels. Can Geotech J 24(1):100–113CrossRefGoogle Scholar
  50. 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):551–563CrossRefGoogle Scholar
  51. 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, AustraliaGoogle Scholar
  52. 52.
    Collins IF, Pender MJ, Yan W (1992) Cavity expansion in sands under drained loading conditions. Int J Numer Anal Meth Geomech 16(1):3–23CrossRefzbMATHGoogle Scholar
  53. 53.
    Collins IF, Stimpson JR (1994) Similarity solutions for drained and undrained in soils. Geotechnique 44:21–34. CrossRefGoogle Scholar
  54. 54.
    Collins IF, Yu HS (1996) Undrained cavity expansions in critical state soils. Int J Numer Anal Methods Geomech 20:489–516.;2-V CrossRefzbMATHGoogle Scholar
  55. 55.
    Cao LF, Teh CI, Cang MF (2001) Undrained cavity expansion in modifed Cam clay I: theoretical analysis. Géotechnique 51:323–334. CrossRefGoogle Scholar
  56. 56.
    Chen WF, Mizuno E (1990) Nonlinear analysis in soil mechanics. Elsevier, AmsterdamGoogle Scholar

Copyright information

© Indian Geotechnical Society 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringIndian Institute of Technology DelhiHauz Khas, New DelhiIndia

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