Rock Mechanics and Rock Engineering

, Volume 52, Issue 10, pp 3987–3997 | Cite as

Functional Catastrophe Analysis of Progressive Failures for Deep Tunnel Roof Considering Variable Dilatancy Angle and Detaching Velocity

  • Kaihang HanEmail author
  • Jiann-Wen Woody Ju
  • Heng Kong
  • Mengshu Wang
Original Paper


The phenomena of progressive failures are very common and important in geotechnical engineering. In this paper, a reliable prediction model is proposed to interpret the progressive failure phenomenon of roof collapse in deep tunnels using the functional catastrophe theory. The progressive collapse mechanisms and collapsing block shapes of deep circular tunnels under conditions of plane strain are investigated. The analytical solutions for the shape curves of the collapsing blocks of circular tunnels are derived based on the nonlinear power-law failure criterion considering variable dilatancy angle and detaching velocity. Moreover, criterions with variable dilatancy angle on progressive failure occurrence for deep tunnels are obtained. Then, the analytical predictions obtained in this paper are compared with experimental testing results, which indicate that the impacts of variable detaching velocity on the shape curves of the several continuous collapsing blocks should be considered to obtain more consistent prediction results with the corresponding experimental testing results.


Deep tunnel Tunnel roof collapse Progressive failure Functional catastrophe theory Nonlinear power-law failure criterion Dilatancy angle Detaching velocity 

List of symbols

Latin symbols


Half-width of the first collapsing block


Intercept in y axis of the first collapsing block


Half-width of the second collapsing block


Intercept in y axis of the second collapsing block


Tunnel radius


Thickness of the plastic detaching zone


Function describing the shape of a circular tunnel


Shape curves of the collapsing blocks


Shape curves of the first collapsing blocks


Shape curves of the second collapsing blocks


Nonlinear coefficient


Initial cohesion of soil at zero stress


Functional of f(x), total potential energy of the studied system


Strain energy of the internal forces on the detaching zone


Applied loads of the detaching surface


Parameter describing the variable detaching velocity


Parameter describing the variable detaching velocity


Overall weight of each collapsing block

Greek symbols


Weight per unit volume of the rock mass


Normal stress on the failure surface


Shear stress on the failure surface


Absolute value of tensile stress when τ = 0


Dilatancy angle


Dilative coefficient


Dilatancy factor


Plastic shear strain


Angle between u and the vertical direction



The authors appreciate the support from the Fundamental Research Funds for the Central Universities of China (Grant no. 2015YJS128).


  1. Alejano LR, Alonso E (2005) Considerations of the dilatancy angle in rocks and rock masses. Int J Rock Mech Min Sci 42(4):481–507CrossRefGoogle Scholar
  2. Arnold VI, Afraimovich VS (1999) Bifurcation theory and catastrophe theory. Springer, BerlinGoogle Scholar
  3. Atkinson JH, Potts DM (1977) Stability of a shallow circular tunnel in cohesionless soil. Géotechnique 27(2):203–215CrossRefGoogle Scholar
  4. Costa YD, Zornberg JG, Bueno BS, Costa CL (2009) Failure mechanisms in sand over a deep active trapdoor. J Geotech Geoenviron Eng 135(11):1741–1753CrossRefGoogle Scholar
  5. Davis EH, Gunn MJ, Mair RJ, Seneviratine HN (1980) The stability of shallow tunnels and underground openings in cohesive material. Géotechnique 30(4):397–416CrossRefGoogle Scholar
  6. Detournay E (1986) Elasto-plastic model of a deep tunnel for a rock with variable dilatancy. Rock Mech Rock Eng 19:99–108CrossRefGoogle Scholar
  7. Du XF (1994) Application of catastrophe theory in economic field. Xi’an Electronic Science and Technology University Press, Chengdu (in Chinese) Google Scholar
  8. Fraldi M, Guarracino F (2009) Limit analysis of collapse mechanisms in cavities and tunnels according to the Hoek–Brown failure criterion. Int J Rock Mech Min Sci 46(4):665–673CrossRefGoogle Scholar
  9. Fraldi M, Guarracino F (2010) Analytical solutions for collapse mechanisms in tunnels with arbitrary cross sections. Int J Solids Struct 47(2):216–223CrossRefGoogle Scholar
  10. Fraldi M, Guarracino F (2011) Evaluation of impending collapse in circular tunnels by analytical and numerical approaches. Tunn Undergr Space Technol 26(4):507–516CrossRefGoogle Scholar
  11. Fraldi M, Guarracino F (2012) Limit analysis of progressive tunnel failure of tunnels in Hoek–Brown rock masses. Int J Rock Mech Min Sci 50:170–173CrossRefGoogle Scholar
  12. Huang F, Yang XL (2011) Upper bound limit analysis of collapse shape for circular tunnel subjected to pore pressure based on the Hoek–Brown failure criterion. Tunn Undergr Space Technol 26(5):614–618CrossRefGoogle Scholar
  13. Jacobsz S (2016) Trapdoor experiments studying cavity propagation. In: Proceedings of the 1st Southern African Geotechnical Conference, Durban, South Africa, pp 159–165Google Scholar
  14. Klar A, Osman AS, Bolton M (2007) 2D and 3D upper bound solutions for tunnel excavation using ‘elastic’ flow fields. Int J Numer Anal Methods Geomech 31(12):1367–1374CrossRefGoogle Scholar
  15. Li T, Yang X (2017) Limit analysis of failure mechanism of tunnel roof collapse considering variable detaching velocity along yield surface. Int J Rock Mech Min Sci 100:229–237CrossRefGoogle Scholar
  16. Lyamin AV, Sloan SW (2000) Stability of a plane strain circular tunnel in a cohesive–frictional soil. In: Proceedings of the Booker Memorial Symposium, Sydney, NSW, Australia, pp 16–17Google Scholar
  17. Osman AS, Mair RJ, Bolton MD (2006) On the kinematics of 2D tunnel collapse in undrained clay. Géotechnique 56(9):585–595CrossRefGoogle Scholar
  18. Santichaianant K (2002) Centrifuge modeling and analysis of active trapdoor in sand. Ph.D. Dissertation. University of ColoradoGoogle Scholar
  19. Sloan SW, Assadi A (1993) Stability of shallow tunnels in soft ground. In: Proceedings of the Wroth Memorial Symposium, Thomas Telford, London, pp 644–663Google Scholar
  20. Stone KJL, Wood DM (1992) Effects of dilatancy and particle size observed in model tests on sand. Soils Found 32(4):43–57CrossRefGoogle Scholar
  21. Thom R (1972) Structural stability and morphogenesis. Westview Press, ColoradoGoogle Scholar
  22. Wang Z, Qiao C, Song C, Xu J (2014) Upper bound limit analysis of support pressures of shallow tunnels in layered jointed rock strata. Tunn Undergr Space Technol 43:171–183CrossRefGoogle Scholar
  23. Yang XL, Huang F (2011) Collapse mechanism of shallow tunnel based on nonlinear Hoek–Brown failure criterion. Tunn Undergr Space Technol 26(6):686–691CrossRefGoogle Scholar
  24. Yang F, Yang JS (2010) Stability of shallow tunnel using rigid blocks and finite-element upper bound solutions. Int J Geomech 10(6):242–247CrossRefGoogle Scholar
  25. Yang XL, Yao C (2017) Axisymmetric failure mechanism of a deep cavity in layered soils subjected to pore pressure. Int J Geomech 17(8):04017031CrossRefGoogle Scholar
  26. Yang X, Li Z, Liu Z, Xiao H (2017) Collapse analysis of tunnel floor in karst area based on Hoek–Brown rock media. J Cent S Univ 24(4):957–966CrossRefGoogle Scholar
  27. Zeeman EC (1976) Catastrophe theory. Scientific American, New YorkGoogle Scholar
  28. Zhang XJ, Chen WF (1987) Stability analysis of slopes with general nonlinear failure criterion. Int J Numer Anal Methods Geomech 11(1):33–50CrossRefGoogle Scholar
  29. Zhang C, Han K (2015) Collapsed shape of shallow unlined tunnels based on functional catastrophe theory. Math Probl Eng 2015(3):1–13Google Scholar
  30. Zhang J, Wang C (2015) Energy analysis of stability on shallow tunnels based on non-associated flow rule and non-linear failure criterion. J Cent S Univ 22(3):1070–1078CrossRefGoogle Scholar
  31. Zhang C, Han K, Fang Q, Zhang DL (2014a) Functional catastrophe analysis of collapse mechanisms for deep tunnels based on the Hoek–Brown failure criterion. J Zhejiang Univ-Sci A 15(9):723–731CrossRefGoogle Scholar
  32. Zhang C, Han K, Zhang D, Li H, Cai Y (2014b) Test study of collapse characteristics of tunnels in soft ground in urban areas. Chin J Rock Mechan Eng 33(12):2433–2442 (in Chinese) Google Scholar
  33. Zhang R, Xiao HB, Li WT (2016) Functional catastrophe analysis of collapse mechanism for shallow tunnels with considering settlement. Math Probl Eng 2016(11):1–11Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Underground Polis Academy of Shenzhen UniversityShenzhenChina
  2. 2.Department of Civil and Environmental EngineeringUniversity of CaliforniaLos AngelesUSA
  3. 3.Beijing Municipal Construction Co., LtdBeijingChina
  4. 4.School of Civil EngineeringBeijing Jiaotong UniversityBeijingChina

Personalised recommendations