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


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.

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σc :

Uniaxial compression strength (of rock)

qc :

Unconfined compression strength (of soil)

σt :

Uniaxial tensile strength (of rock)


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)


Stress reduction factor (from Q-value)

Rf :

Depth of failure + excavation radius (a)


Fracture mechanics numerical code


Displacement discontinuity method


Norwegian Geotechnical Institute


Rock mass quality

ε3 :

Lateral extension strain (radial)

εt :

Critical extensional strain


Young’s modulus

E’ = E/(1–ν2):

For plane strain

Hc :

Critical height of vertical cutting in soil


Cohesion of soil (or intact rock)


Friction angle of soil (or intact rock)


Density of soil (or intact rock)


Joint roughness coefficient


Joint wall compression strength


Equivalent roughness of broken rock, screes


Equivalent strength of broken rock, screes


Shear stress along potential rock-failure plane


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


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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

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