Generalized Hoek–Brown Failure Criterion (2002)

Mohr–Coulomb draws a straight line through a set of triaxial results, and for soil that is close enough to the truth to have run the subject for a century. Rock does not oblige. A jointed rock mass gains strength very quickly as confinement is first applied and then far more slowly, so its failure envelope is a curve, and a straight line fitted anywhere on it is wrong everywhere else. The Hoek–Brown criterion is that curve written down: $\sigma_1 = \sigma_3 + \sigma_{ci}(m_b\sigma_3/\sigma_{ci} + s)^{a}$.

Every term earns its place. $\sigma_{ci}$ is the uniaxial compressive strength of the INTACT rock — a laboratory core, not the mass — and it sets the scale of the whole thing. $m_b$ behaves like a frictional term and controls how fast the envelope rises with confinement. $s$ behaves like a cohesive term: it is 1 for genuinely intact rock and falls towards zero as the mass is broken up, and $\sigma_{ci}s^{a}$ is the unconfined strength of the mass itself. $a$ is the curvature, near 0.5 for good rock and rising towards 0.6 for poor.

The number that shocks people is the unconfined one. Take a limestone whose core tested at 50 MPa, put it in a mass at GSI 45, and the rock mass is worth about 2.2 MPa unconfined — four and a half percent of what the laboratory said. That is not an error in the criterion. It is what jointing does, and it is why quoting a UCS as "the strength of the ground" is out by a factor of twenty or more. It is also why the low-confinement end of the envelope is the end that governs a tunnel wall or a slope face, where $\sigma_3$ has fallen to nearly nothing, and why a friction angle fitted to triaxial data at 10 MPa of confinement badly overestimates the rock that is actually about to fail.

Now the trap this whole shard is written around. The criterion has been through several editions — 1980, 1988, 1992, 1995, 1997 and the 2002 recalibration that this site uses — and the definitions of $m_b$, $s$ and $a$ changed between them. The 2002 edition introduced the disturbance factor $D$, which did not exist before. Take an $m_b$ out of a 1997 paper, substitute it into the 2002 equation, and you get an answer with the right units, the right order of magnitude and no meaning whatsoever. Nothing flags it. There is no dimensional error, no absurd number, no warning; the result simply looks plausible and is not. If you did not compute $m_b$, $s$ and $a$ yourself from GSI and $D$, find out which edition they came from before you use them.

Two boundaries. The criterion is written in EFFECTIVE stresses, so pore pressure comes off before anything else. And it assumes the rock mass is HOMOGENEOUS AND ISOTROPIC at the scale of the problem — enough joint sets, closely enough spaced, that the mass has no preferred direction. A slope cut by two dominant sets, or a tunnel in bedded shale, is a structurally controlled problem and belongs in a wedge analysis, not here. Hoek's own advice is that if you can see the individual blocks that will fail, Hoek–Brown is the wrong tool.