In-situ Stress & Openings
Rock deep underground is squeezed by the weight above and tectonic forces. Learn vertical and horizontal in-situ stresses, the stress ratio k, stress measurement, the Kirsch solution for stresses around a circular opening, stress concentration, spalling and rockbursts.
At 1 km depth, the vertical stress is about 27 MPa — like 270 atmospheres. When we excavate a tunnel, stresses flow around the opening and concentrate at its boundary, sometimes to 3× the original values. If boundary stress exceeds rock strength, the rock spalls or bursts violently.
An analogy
Stress around a tunnel is like water flowing around a bridge pier: the flow speeds up at the sides — stress concentrates at the walls where it must detour.
Where stress comes from
Gravity gives vertical stress from overburden weight. Horizontal stresses come from Poisson effect, tectonic plate forces and erosion — often greater than vertical at shallow depth (k 1–3), tending toward 1 at great depth.
Measuring stress
Overcoring (CSIRO hollow inclusion cells), hydraulic fracturing in boreholes, borehole breakouts and core disking give magnitude and direction. World Stress Map data guide early studies.
Stress around openings
For a circular opening in elastic rock (Kirsch), the tangential stress at the boundary is 3σ₁ − σ₂ at the point perpendicular to σ₁. Elongating openings in the direction of the major stress reduces concentrations; sharp corners increase them.
Rockbursts and seismicity
In deep, brittle, high-stress rock, stored strain energy releases violently (strainbursts, fault-slip). Management: destressing blasts, yielding support (dynamic bolts, mesh), mining sequences away from pillars, microseismic monitoring and re-entry protocols.
Formulas
- Vertical stress:
σ_v = ρ g z ≈ 0.027 z (MPa, z in m) - Horizontal stress:
σ_h = k σ_v - Roof/floor (Kirsch):
σ_θ = 3σ_h − σ_v - Sidewall (Kirsch):
σ_θ = 3σ_v − σ_h - Spalling indicator:
σ_max / UCS > 0.4 → spalling likely
Worked example: Stresses at 800 m
Given: ρ = 2700 kg/m³, z = 800 m, k = 1.5
Find: σ_v, σ_h and boundary stresses of a circular tunnel.
- σ_v = 2700 × 9.81 × 800 = 21.2 MPa; σ_h = 31.8 MPa
- Roof = 3 × 31.8 − 21.2 = 74.2 MPa
- Wall = 3 × 21.2 − 31.8 = 31.8 MPa
Answer: Roof 74 MPa — highest stress in the roof/floor
Worked example: Spalling check
Given: UCS = 100 MPa, σ_max = 74 MPa
Find: σ_max/UCS.
- 74/100 = 0.74 > 0.4
Answer: Severe spalling likely in roof — needs support and possibly reshaping
Common mistakes
- Wrong: Assuming k = 1 everywhere. — Right: Measure stress; horizontal stress often dominates.
- Wrong: Using static support for burst-prone ground. — Right: Use dynamic, energy-absorbing support.
Applications
- Deep mines: Rockburst-prone mines in Canada, South Africa, Australia.
- Drive orientation: Align drives with major stress to reduce damage.
- Sequencing: Avoid high-stress remnant pillars.
Key takeaways
- σ_v ≈ 0.027 z MPa; σ_h = kσ_v.
- Kirsch: 3σ₁ − σ₂ at boundary.
- σ_max/UCS > 0.4 → spalling.
- Deep mines need burst-resistant support.
Going further
- Numerical modelling: Map3D, FLAC3D, Abaqus for stress in complex geometry.
- Microseismic systems: Locate and characterise seismic events.
- Dynamic support: Yielding bolts absorb burst energy.