Elastic Tunnel Convergence

This is the straight part of the ground reaction curve, and it is the one piece of convergence-confinement that closes in a single line: the wall of a circular tunnel in an elastic ground moves inwards by $(p_0 - p_i)R/2G$. Excavating removes the support the rock was giving itself, the difference between the in-situ stress and whatever you put back is what drives the closure, and the shear modulus of the ground resists it.

The number that matters is not the convergence; it is how little the support changes it. Take a 5 m radius tunnel at 15 MPa in a rock mass with a shear modulus of 3 GPa. Unsupported, the wall comes in 12.5 mm. Install a substantial shotcrete lining pushing back at 0.5 MPa — a real lining, not a token one — and the closure becomes 12.08 mm. You have bought 0.42 mm out of 12.5, which is three and a third percent. That is not a defect in the lining. It is arithmetic: a rock mass shear modulus is measured in gigapascals and a support pressure in hundreds of kilopascals, so $p_i/p_0$ is small and the support removes a correspondingly small slice of an elastic movement.

So what is support for? It earns its living in the plastic range, and the contrast is the whole argument of convergence-confinement. Once the ground around the opening starts to fail, a modest confinement at the wall suppresses a large volume of failed rock, because the confinement propagates outwards through the failing material. The plastic radius page on this site works the same tunnel: raising the support from 0.5 MPa to 2 MPa pulls the failed ring in by more than one and a half metres. Same order of pressure; utterly different effect. Support is not fighting the in-situ stress — nothing you can build inside a tunnel fights 15 MPa — it is preventing the ground from failing, and those two jobs cost entirely different amounts.

What the relation assumes. A circular opening. An infinite elastic medium. Plane strain. And a hydrostatic stress field, with the same stress in every direction. That last one is the assumption that bites, because real stress fields are rarely hydrostatic: where the horizontal-to-vertical stress ratio is well away from one, the roof and the walls converge by quite different amounts, and this single number describes neither of them. It is a first estimate and a teaching tool, and where the stress field is strongly anisotropic it should be replaced by the Kirsch solution or by a numerical model.

The face effect, which is what makes measured convergence so hard to interpret. This equation gives the movement from the undisturbed state — from the ground as it was before anything was excavated. A substantial fraction of that movement, commonly a quarter to a third, has already happened by the time the face arrives at a given section, because the ground ahead of a tunnel starts deforming before it is exposed. Convergence targets can only be installed after the face has passed, so they measure what is left. Comparing an instrument reading directly with this number will therefore make the ground look stiffer than it is. Proper practice back-extrapolates the convergence curve to estimate the movement lost, and the estimate is never better than approximate.

Back-analysis is where this equation is genuinely valuable. Solve for $G$ and you have one of the very few independent checks on a rock mass modulus estimated from GSI, using the tunnel itself as the test specimen. Two cautions come with it. The lost face movement makes the measured closure too small and therefore the back-analysed modulus too large. And if any part of the ground has yielded, the closure is not elastic at all and the modulus that falls out is fictitious and far too low — which is itself useful information, because a back-analysed $G$ well below anything the rock could plausibly have is good evidence that a plastic zone has formed.

One last observation about geometry. Elastic convergence is directly proportional to the radius, so a tunnel twice as wide closes twice as far in the same ground under the same stress, and the strain $u/R$ is identical. That is the arithmetic behind heading-and-bench and multiple-drift sequences: each smaller opening closes less, and each is closed and supported before the next is taken. It is also why a pilot tunnel flatters a large cavern — the pilot will always look better behaved than the full section ever will.