Hoek-Brown and Curved Envelopes

Part 3, Part 3: Rock Strength and Failure

Learning objectives

  • Recognize that real strength envelopes curve: the failure line flattens as confinement rises
  • State the Hoek-Brown criterion and its parameters, the material constant m and the intact flag s equal to one
  • Show that Hoek-Brown reduces to the UCS at zero confinement and sits below the Coulomb line at high confinement
  • Choose the criterion by confinement range: Coulomb for a narrow window, Hoek-Brown across a wide one

The Line That Should Be a Curve

Coulomb's straight line is a convenience, not a law. Do the triaxial test across a wide range of confining pressures and the peaks do not fall on a line; they fall on a gentle curve that steepens near zero confinement and flattens as the rock is squeezed harder. A straight Coulomb fit through a narrow confinement window is fine, but extrapolate it far and it over-predicts strength at low confinement and under-predicts the curvature everywhere. The Hoek-Brown criterion, built empirically by Evert Hoek and Edwin Brown from thousands of tests, captures the curve directly: sigma1=sigma3+Cosqrtm,sigma3/Co+s\sigma_1 = \sigma_3 + C_o\sqrt{m\,\sigma_3/C_o + s}osqrtm,sigma3/Co+s, where CoC_oo is the unconfined strength, mm is a material constant (about 17 for sandstone, 9 for limestone, 32 for granite), and ss is a flag that is 1 for intact rock and less for a fractured rock mass.

Hoek Brown And Curved EnvelopesInteractive figure, enable JavaScript to interact.

Fit both criteria to the same data in the figure. At zero confinement Hoek-Brown returns sigma1=Co\sigma_1 = C_oo exactly, anchoring at the UCS just as Coulomb does, so the two agree where the rock is unconfined. Raise the confining pressure and they part company: the Hoek-Brown curve bends below the straight Coulomb line, because real rock gains strength with confinement more slowly than a constant friction angle implies. The gap is small at reservoir confining pressures, which is why Coulomb survives as the working tool, but it grows at the high confinements of deep wells and mine pillars, where Hoek-Brown is the honest choice.

The Deeper Value: Rock Masses

Hoek-Brown's real power is the parameter the intact test cannot supply: the flag ss and its companion reductions describe not a lab plug but a jointed rock mass. Drop ss below 1 and lower mm, following the Geological Strength Index that Hoek built for exactly this, and the criterion predicts the strength of rock cut by joint sets, the strength that actually governs a wellbore wall or a tunnel roof. This is the formal version of the honesty from section 3.1: the lab gives intact strength, the field delivers less, and Hoek-Brown is the accepted machinery for how much less. The course uses intact Hoek-Brown to draw the curve and Coulomb's q=3.12q = 3.12 to run the reservoir-depth calculations, naming the approximation each time. With the compressive curve and the tensile tail both in hand, one shape remains to close the failure surface: the compaction end cap, next.

References

  • Hoek, E., & Brown, E. T. (1980). Empirical strength criterion for rock masses. Journal of the Geotechnical Engineering Division, ASCE, 106(GT9), 1013-1035.
  • Hoek, E., Carranza-Torres, C., & Corkum, B. (2002). Hoek-Brown failure criterion, 2002 edition. Proc. NARMS-TAC, 267-273.
  • Jaeger, J. C., Cook, N. G. W., & Zimmerman, R. W. (2007). Fundamentals of Rock Mechanics (4th ed.). Blackwell.

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