Hoek–Brown Constants s and a (2002)

Two constants share this page because the 2002 paper presents them together and because both are read off the same Geological Strength Index. They behave very differently, and the difference is worth understanding.

$s$ is the cohesion that survived. It is 1 for genuinely intact, undisturbed rock and it falls away exponentially as the mass is broken into blocks: about 0.0039 at GSI 50, about 1.4 × 10⁻⁴ at GSI 25. Small numbers here are normal and they are the whole message — $\sigma_{ci}s^{a}$ is the unconfined strength of the rock mass, and for anything but very good ground almost none of the laboratory strength is left.

$s$ is even more sensitive to GSI than $m_b$ is, because its denominator only runs from 9 down to 6 rather than 28 down to 14. Move GSI by the five points two geologists might casually differ by and $s$ moves by a factor of about 1.75; move it by ten and the factor is over three. Since $s$ enters the strength as $s^{a}$ with $a$ near 0.5, that becomes roughly a factor of 1.3 and 1.75 on the unconfined strength of the mass. Run both ends of your GSI range and quote the band.

The disturbance factor bites here harder than anywhere except the modulus, for the same reason: $9 - 3D$ is a short lever. A generously chosen $D$ does not merely flatter the answer, it changes its order of magnitude, and unlike a laboratory result nobody can check it after the fact.

$a$ is the odd one out, and pleasantly so. $a = \tfrac{1}{2} + \tfrac{1}{6}(e^{-GSI/15} - e^{-20/3})$ contains GSI and nothing else — no rock type, no disturbance factor. It is the one parameter in the 2002 set that an optimistic $D$ cannot improve. It runs from about 0.585 at GSI 10 down to essentially 0.5 for GSI above 60, where the exponential term has all but vanished.

That 0.5 is not a coincidence. The original 1980 Hoek–Brown criterion used an exponent of exactly one half for every rock mass, and the generalized form reduces to the classic one for good rock. The rising $a$ for poor rock flattens the envelope at low confinement, which is exactly where the original was found to overpredict strength — the correction was introduced because real weak rock masses turned out to have less unconfined strength than a square-root envelope allowed.

The awkward consequence is that $s$ and $a$ were introduced together and revised together. An $a$ of 0.5 taken from an older paper, paired with a modern $s$, is a combination that has never been calibrated against anything at all.