Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Extension Strain and Rock Strength Limits for Deep Tunnels, Cliffs, Mountain Walls and the Highest Mountains

  • 656 Accesses

  • 1 Citations

Abstract

Brittle rock can fail in tension even when all principal stresses are compressive. The culprit is Poisson’s ratio, but marked stress anisotropy due to a neighbouring free surface, and due to a raised principal tangential stress is also necessary. Extension strain-induced failure causes fracture initiation in tension. Propagation in unstable shear may occur if the tunnels or mine openings are deep enough, and if they are located in hard, brittle, sparsely jointed rock. Both in laboratory uniaxial compression test samples with strength σc and in deep tunnels, extension fracturing and acoustic emission begin when the principal applied or induced stress reaches the magnitude of tensile strength divided by Poisson’s ratio σt/ν. The traditionally expected fracture initiation when the principal or maximum tangential stress σ1 or σθ = 0.4 ± 0.1 × σc can actually be explained with arithmetic. Using related logic, cliffs and the near-vertical mountain walls frequented by rock climbers, may have erosional or glacial origin, but extension strain limits their height, including vertical walls of sheeting joints and long continuous fractures. Shear failure seems to be reserved for occasional major rock avalanches. Equations with soil mechanics origin involving Coulomb parameters c and φ and density that may apply to vertical cuts in soil, give greatly exaggerated heights for rock cliffs and mountain walls since rock is brittle and favours failure in tension. Tensile strength, Poisson’s ratio and density are suggested for estimating the maximum heights of rock cliffs and mountain walls, not compression strength and density. However, overall mountain heights are limited by critical state maximum shear strength, or by the slightly lower brittle–ductile transition strength.

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

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
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Abbreviations

σc :

Uniaxial compression strength (of rock)

qc :

Unconfined compression strength (of soil)

σt :

Uniaxial tensile strength (of rock)

ν:

Poisson’s ratio

σh :

Minor horizontal principal stress

σH :

Major horizontal principal stress

σv :

Vertical principal stress

σ1 :

Major principal stress

σ3 :

Minor principal stress

k0 :

Ratio of σh/σv

K0 :

Ratio of σH/σv

σθ :

Maximum tangential stress (also σmax)

SRF:

Stress reduction factor (from Q-value)

Rf :

Depth of failure + excavation radius (a)

FRACOD:

Fracture mechanics numerical code

DDM:

Displacement discontinuity method

NGI:

Norwegian Geotechnical Institute

Q:

Rock mass quality

ε3 :

Lateral extension strain (radial)

εt :

Critical extensional strain

E:

Young’s modulus

E’ = E/(1–ν2):

For plane strain

Hc :

Critical height of vertical cutting in soil

C:

Cohesion of soil (or intact rock)

φ:

Friction angle of soil (or intact rock)

γ:

Density of soil (or intact rock)

JRC:

Joint roughness coefficient

JCS:

Joint wall compression strength

R:

Equivalent roughness of broken rock, screes

S:

Equivalent strength of broken rock, screes

τ:

Shear stress along potential rock-failure plane

T:

Shear resistance, N normal resistance

σxx :

And σzz horizontal and vertical stress components

φr :

Residual friction angle of potential rock-failure plane

φb :

Basic friction angle of flat rock surface

φc :

Friction angle subtended by critical state (tan−1 ½)

σn :

Normal stress acting across potential rock-failure plane

τmax :

Maximum (critical state) shear strength of intact rock

σ3critical :

Confining pressure needed to reach critical state τmax

References

  1. Addis MA, Barton N, Bandis C, Henry JP (1990) Laboratory studies on the stability of vertical and deviated boreholes. In: 65th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, New Orleans, September 23–26, 1990

  2. Aydan Ö, Ulusay R (2003) Geotechnical and geoenvironmental characteristics of man-made underground structures in Cappadocia. Turk Eng Geol 69:245–272

  3. Barton N (1976) The shear strength of rock and rock joints. Int J Rock Mech Min Sci Geomech Abstr 13:No. 9: 255–279

  4. Barton N (1999) General report concerning some 20th Century lessons and 21st Century challenges in applied rock mechanics, safety and control of the environment. In: Proc. of 9th ISRM Congress, Paris, 3: 1659–1679, Balkema, Netherlands

  5. Barton N (2013) Shear strength criteria for rock, rock joints, rockfill and rock masses: problems and some solutions. J of Rock Mech Geotech Engr Elsevier 5:249–261

  6. Barton N, Grimstad E (2014) Q-system—an illustrated guide following forty years in tunnelling. 43 p. http://www.nickbarton.com

  7. Barton N, Infanti N (2004) Unexpected stress problems in shallow basalts at the Ita hydroelectric power project in S.E. Brazil. Proc. ARMS 2004: 3rd Asian Rock Mechanics Symposium, Kyoto

  8. Barton N, Shen B (2017) Risk of shear failure and extensional failure around over-stressed excavations in brittle rock. J Rock Mech Geotech Eng 9:210–225. https://doi.org/10.1016/j.jrmge.2016.11.004

  9. Barton N, Warren C (1995) Rock mass classification of chalk marl in the UK Channel Tunnels. Channel Tunnel Engineering Geology Symp., Brighton

  10. Bray JW (1967) A study of jointed and fractured rock. Part I. Fracture patterns and their failure characteristics. Felsmechanik 2–3:117–136

  11. Byerlee JD (1967) Frictional characteristics of granite under high confining pressure. J Geophys Res 72:3639–3648

  12. Byerlee JD (1968) Brittle-ductile transition in rocks. J Geophys Res 73:4741–4750

  13. Davis S (2013) Learning to fly: an uncommon memoir of human flight, unexpected love, and one amazing dog. Simon & Schuster, New York

  14. Drucker DC, Prager W (1952) Soil mechanics and plastic analysis or limit design. Q Appl Math 10:157–165

  15. Florine H, Moye J (2016) On the Nose—A lifelong obsession with Yosemite’s most iconic climb. Rowman & Littlefield, Lanham

  16. Grimstad E, Barton N (1993) Updating of the Q-System for NMT. In: Proc. of Int. Symp. on Sprayed Concrete—Modern Use of Wet Mix Sprayed Concrete for Underground Support, Fagernes, 1993, Eds Kompen, Opsahl and Berg. Norwegian Concrete Association, Oslo, 46–66

  17. Hall J, G Child (2002) Climbing Free. My life in the Vertical World. Harper Collins Publishers, New York

  18. Honnold A, Roberts D (2016) Alone on the Wall. W.W. Norton & Company, New York

  19. Martel SJ (2017) Progress in understanding sheeting joints over the past two centuries. J Struct Geol 94:68–86

  20. Martin CD, Kaiser PK, McCreath DR (1998) Hoek–Brown parameters for predicting the depth of brittle failure around tunnels. Can Geotech J 36:136–151

  21. Melosh HJ (2011) Planetary surface processes. Cambridge University Press, Cambridge

  22. Mogi K (1966a) Some precise measurements of fracture strength of rocks under uniform compressive stress. Felsmechanik Ingenieurgeologie 4:41–55

  23. Mogi K (1966b) Pressure dependence of rock strength and transition from brittle to ductile flow. Bull Earthq Res Inst Tokyo 44:215–232

  24. Shen B, Barton N (2018) Rock fracturing mechanisms around underground excavations. J Geotech Eng Korea. (in press)

  25. Shen B, Stephansson O, Rinne M (2013) Modelling rock fracturing processes: a fracture mechanics approach using FRACOD. Springer, Heidelberg

  26. Shen B, Shi J, Barton N (2018) An approximate Non-Linear modified Mohr–Coulomb Shear strength criterion with critical state for intact rocks. J Rock Mech Geotech Eng. https://doi.org/10.1016/j.jrmge.2018.04.002

  27. Singh M, Raj A, Singh B (2011) Modified Mohr–Coulomb criterion for non-linear triaxial and polyaxial strength of intact rocks. Int J Rock Mech Min Sci 48(4):546–555

  28. Stacey TR (1981) A simple extension strain criterion for fracture of brittle rock. Int J Rock Mech Min Sci Geomech Abstr 18:469–474

  29. Stacey TR, Xianbin Y, Armstrong R, Keyter GJ (2003) New slope stability considerations for deep open pit mines. Afr Inst Min Metall 103(6):373–389

  30. Stock GM, Luco N, Collins BD, Harp EL, Reichenbach P, Frankel KL (2012) Quantitative rock-fall hazard and risk assessment for Yosemite Valley. Yosemite National Park, California

  31. Terzaghi K (1962) Stability of steep slopes on hard unweathered rock. Geotechnique 12:251–263

  32. Verruijt A (2001) Soil Mechanics. Delft University of Technology, The Netherlands

  33. Weber S, Beutel J, Faillettaz J, Hasler A, Krautblatter M, Vieli A (2016) Quantifying irreversible movement in steep fractured bedrock permafrost at Matterhorn (CH). The Cryosphere Discussions Journal

  34. Wolters G, Müller G (2008) Effect of cliff shape on internal stresses and rock slope stability. J Coastal Res 24(1), 43–50

Download references

Author information

Correspondence to Nick Barton.

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

Verify currency and authenticity via CrossMark

Cite this article

Barton, N., Shen, B. Extension Strain and Rock Strength Limits for Deep Tunnels, Cliffs, Mountain Walls and the Highest Mountains. Rock Mech Rock Eng 51, 3945–3962 (2018). https://doi.org/10.1007/s00603-018-1558-2

Download citation

Keywords

  • Extension strain
  • Tensile strength
  • Poisson’s ratio
  • Shear strength
  • Fracturing
  • Tunnels
  • Cliffs
  • Mountain walls
  • Mountains