Plastic Zone Radius (Mohr–Coulomb)

When the stress around an opening exceeds what the rock mass can carry, a ring of failed ground forms at the wall and grows outwards until the stresses at its boundary drop back to what the intact mass can hold. How far out it reaches is the question this page answers, for a Mohr-Coulomb material in a hydrostatic stress field. It is the classical convergence-confinement result, and it is the reason the method is a design tool rather than a description.

The critical support pressure comes first. Set $R_p = R$ in the relation and it collapses to $p_i = (2p_0 - \sigma_{cm})/(1+k)$. That is the support pressure at which the plastic zone is on the point of appearing: above it, the ground around the opening stays elastic everywhere and the elastic convergence page is the right one; below it, a failed ring forms and this page is. If $2p_0$ is itself less than the rock mass strength, no plastic zone forms even with no support at all, and the whole question is moot.

Now the contrast that justifies the whole method. Take a 5 m tunnel at 10 MPa in a mass worth 4 MPa unconfined with a friction angle of 30°. At 0.5 MPa of support the failed ring reaches 7.75 m from the centre — 2.75 m of broken ground beyond the wall. At 2 MPa it reaches 6.12 m. Raising the support pressure by 1.5 MPa took 1.63 m of failed rock out of the roof. Compare that with the elastic page, where half a megapascal moved the wall by four tenths of a millimetre. Support does not resist the in-situ stress; it stops the ground failing, and the two cost entirely different amounts. That sentence is the whole of convergence-confinement, and these two numbers are the arithmetic behind it.

Why there is no Hoek-Brown ground reaction curve on this site. It is a fair thing to wonder about, because the neighbouring rock mechanics pages carry the complete 2002 Hoek-Brown set — the criterion, $m_b$, $s$ and $a$, and the modulus — and for a jointed rock mass Hoek-Brown is the more realistic strength model of the two. The reason is structural rather than a judgement about the physics. The Mohr-Coulomb envelope is a straight line, so the equilibrium equation integrates in closed form and the plastic radius falls out in one expression you could check with a calculator. The Hoek-Brown envelope is curved; the same integration has no elementary solution; and every published Hoek-Brown ground reaction curve is produced by numerical integration or by an iterative scheme. This site's rule is that a formula page carries a closed-form relation a reader could work by hand, and that curve fails the rule. It is not absent because it is unimportant. It is absent because it is not a formula. Use a convergence-confinement program for it — and use this page to understand what the program is doing.

What the closed-form solution leaves out, which is a good deal. A hydrostatic stress field, which is uncommon. A circular opening, which is true of a bored tunnel and of very little else. No weight of the broken rock inside the plastic ring, which for a large plastic zone in the roof is a real omission — that material has failed but it has not vanished, and something is holding it up. And no time at all: in squeezing ground the ring keeps growing after the face has passed, and this solution has no clock in it. It also assumes the failed material keeps some residual strength described by the same friction angle, which is a convenient assumption rather than a measured one.

Note that $k$ cannot be solved for on this page. It appears both inside the bracket and in the exponent, which makes recovering it a transcendental problem rather than an algebraic one, and this site does not put an iterative solve behind a formula page. It comes from the friction angle, through $k = (1 + \sin\phi)/(1 - \sin\phi)$: 3 at 30°, 4 at about 37°, and it is exactly the Rankine passive coefficient wearing a different hat.

Two directions worth using. Forwards, it is a design tool: solve for $p_i$ and you get the support pressure that holds the failed ring to a chosen radius, which is a number shotcrete and bolts can actually deliver. Backwards, solve for $\sigma_{cm}$ and you have a back-analysis worth doing far more often than it is done — the extent of the broken ring is observable with a borehole camera or an extensometer, and what falls out is the strength the rock mass is really behaving with. Compare it against the unconfined Hoek-Brown value your GSI implies. If the tunnel says the mass is much weaker than the GSI does, the GSI was optimistic, and it is much better to learn that from the first kilometre than from the last.