Obert–Duvall Pillar Strength

A short, fat pillar is stronger than a tall, slender one made of the same rock, and the reason is confinement. Friction against the roof and the floor stops the pillar from spreading sideways, that restraint reaches a certain distance in from each end, and in a squat pillar those two zones overlap through the whole core. Confined rock is stronger — which is the Hoek–Brown envelope curving upwards, stated in a form a mining engineer can use on a shift.

Leonard Obert and Wilbur Duvall's 1967 expression is the simplest honest version: $S_p = S_1(0.778 + 0.222\,W_p/h)$. The two coefficients sum to 1, so a cubical pillar at $W_p/h = 1$ has exactly the strength of the cubical specimen $S_1$, and everything else is scaled from there.

$S_1$ is not a UCS from a core, and this is the first place people go wrong. It is the strength of a CUBICAL specimen. A standard uniaxial test uses a slender cylinder at about 2:1 height to diameter, and that geometry reads LOWER than a cube of the same rock for exactly the reason this formula exists — the cylinder is less confined by its platens. So using a core UCS directly as $S_1$ is conservative, and using a cube strength where a core value belongs is not. On top of that sits a size effect that is far larger than the shape effect: a pillar tens of metres across contains joints, veins and weak seams that a 50 mm specimen never sampled, and the usual practice is to knock the laboratory strength down substantially — often to a third or less — before calling it $S_1$.

The second mistake is the width. $W_p$ is the pillar's LEAST plan dimension. It is not the room span, and for a long rib pillar it is not the long side. And $h$ is the full mining height floor to roof: if a soft floor has heaved or the roof has spalled since the pillar was cut, the pillar is taller than the plan says, its $W_p/h$ is lower than the plan says, and it is weaker than this number.

The third is extrapolation, and the formula gives no warning at all. Obert and Duvall fitted this line on hard-rock LABORATORY SPECIMENS over roughly $W_p/h = 0.5$ to 4, and it is generally regarded as defensible out to about 8. Beyond that the straight line keeps climbing without limit, which real pillars do not do — a genuinely squat pillar enters a regime where it does not fail in any recognisable sense but simply squashes and accepts more load, and a linear form has no way to represent that. At $W_p/h = 12$ the equation will hand you a number with a straight face, and that number is an unsupported projection. Formulations aimed at squat pillars exist precisely because this one runs out.

What is genuinely valuable, and underused: back-calculating $S_1$ from a pillar that actually failed. A failed pillar of known dimensions under a known tributary load gives one real point on the strength curve for that seam or orebody, and it is worth more than any laboratory programme. Collect several and you can fit your own formula rather than borrowing one fitted on somebody else's rock in 1967.