Tangential stress σθ & predicted failure around the opening
Tangential stress σθ — blue envelope (radial extent ∝ σθ); values in MPa:
● elastic,
● spalling,
● severe.
Predicted failure — filled rust zone, spalled ground out to Rf. ━ rock bolt; the ○ marks Rf on the bolt — its final 1 m sits beyond the failure boundary, anchored in stable rock.
Predicted failure — filled rust zone, spalled ground out to Rf. ━ rock bolt; the ○ marks Rf on the bolt — its final 1 m sits beyond the failure boundary, anchored in stable rock.
What the rust zone is · the failure criterion
Rf(θ) / a(θ) = 0.49 + 1.25 · σθ(θ) / σc
clamped to ≥ 1 — no spalling predicted where σθ / σc < ≈ 0.4
The rust zone is the predicted depth of brittle (stress-driven) spalling. At each angle θ around the opening the local boundary tangential stress σθ is divided by the rock strength σc (UCS) and put through eq. 11 of Martin, Kaiser & McCreath (1999): broken ground reaches a radius Rf(θ) from the centre, with a(θ) the wall radius at that angle. Because σθ is taken locally, the zone is deepest where stress concentrates and shallow where it relaxes.
Why the rust depth and the df tile can differ. The tiles (Rf/a, df, bolt length) are driven by the single peak boundary stress σmax, wherever it occurs — the worst-case, design-governing number. The rust zone applies the same relation angle by angle with the local σθ. For circles and ellipses σmax sits on a wall, so the two coincide there — the ○ on the bolt lands exactly on the rust boundary. For square and rectangular openings σmax lives in the corners: the worst-case depth shows up as the corner lobes, while the rust at the flat walls (where the representative bolt is drawn) stays shallower than the tile’s df. Eq. 11 was calibrated on circular tunnels — wrapping it around cornered shapes is an indicative extension, and the tiles remain the conservative numbers for support design.
Reading the depth measures · ratio vs depth
Example for a = 2.5 m. Rf/a is the failure radius from the tunnel centre, normalised by a (a pure ratio). Depth of failure df = Rf − a is the spalled thickness from the wall (metres). Same notch, two readings: df = a (Rf/a − 1).
Maximum tangential stress σmax
90MPa
at roof & floor
Damage index Di = σmax/σc
0.41
minor spalling
Normalised failure radius Rf/a
1.00
Eq. 11: Rf/a = 0.49 + 1.25 (σmax/σc) · from the tunnel centre, dimensionless
Depth of failure df = Rf − a
0.00m
Rf/a = 1.00
First-pass rock-bolt length
≥ 1.0m
failure depth + 1 m anchorage; round up to the next available length
The rust zone is the spalled ground the support must hold; the bolt anchors in stable rock beyond Rf. a is the opening half-width along the worst wall. Stresses use the Kirsch/Inglis closed form (circle & ellipse) or a superellipse approximation (rectangle & square).
Hoek–Brown failure envelope · frictional versus brittle
The red dot is the boundary state (σ3 = 0, σ1 = σmax). When it sits above the dashed brittle envelope, eq. 10 predicts cohesion-loss failure. A frictional criterion would still call the rock stable — this gap is the paper's central observation.
