In-situ stress
Rock deep underground is already stressed before any tunnel or cavern is excavated. The vertical stress usually equals the weight of the overlying rock, about 0.025 to 0.027 MPa per metre of depth. The horizontal stresses are less predictable: they depend on tectonic forces, erosion of overburden, topography and the rock's history, and near the surface they are often higher than the vertical stress, especially in old, strong rocks and active mountain belts.
The ratio of horizontal to vertical stress is written K. A simple elastic argument, with the rock laterally restrained, gives K equal to Poisson's ratio over one minus Poisson's ratio, about one third. Measured values are frequently much higher at shallow depth and tend towards one at great depth. Because of this scatter, stress is measured on important projects, by overcoring cells, hydraulic fracturing in boreholes or flat jacks on exposed faces.
When an opening is cut, the stress flows around it and concentrates at its boundary. For a circular opening in elastic rock, the tangential stress at the roof and walls depends on both principal stresses; under equal stresses it is twice the field stress all round. Where the concentrated stress exceeds the rock mass strength, the rock spalls, slabs or squeezes, which is why the orientation and shape of large caverns are chosen with the stress field in mind.
High stress in strong brittle rock can cause rock bursts; in weak rock it causes squeezing, large time-dependent convergence. Both are central concerns in the long Himalayan tunnels.
Align the long axis of a cavern with the major horizontal stress where possible.
Vertical stress σ_v = γ zHorizontal stress σ_h = K σ_v; elastic restrained K = ν / (1 − ν)Circular opening (Kirsch, boundary): roof σ_θ = 3 σ_h − σ_v; side wall σ_θ = 3 σ_v − σ_h
Assuming K = ν / (1 − ν) without measurement at shallow depth.
Problem. At 500 m depth, rock has γ = 27 kN/m³ and measured K = 1.5. Find σ_v, σ_h and the elastic tangential stresses at the roof and side wall of a circular tunnel. Compare K with the elastic restrained value for ν = 0.25.
- σ_v = 27 × 500 = 13 500 kPa = 13.5 MPa; σ_h = 1.5 × 13.5 = 20.25 MPa.
- Roof: σ_θ = 3 × 20.25 − 13.5 = 47.25 MPa.
- Side wall: σ_θ = 3 × 13.5 − 20.25 = 20.25 MPa.
- Elastic restrained K = 0.25 / 0.75 = 0.33, far below the measured 1.5, showing why stress should be measured.
Answer. σ_v = 13.5 MPa, σ_h = 20.25 MPa; roof 47.3 MPa, wall 20.3 MPa
Quick quiz
1What usually sets the vertical in-situ stress?
The weight of the overlying rock.
2Under equal field stresses, what is the boundary stress of a circular opening?
Twice the field stress.
3Name one method of measuring in-situ stress.
Overcoring, hydraulic fracturing or flat jacks.
- BS 5930 (UK): Code of practice for ground investigations.
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References and further reading
Standard texts, listed in no order of merit. We do not copy text, figures, tables or solved problems from any book; our explanations and examples are original.
- R. E. Goodman, Introduction to Rock Mechanics, Wiley, 2nd edition.
- E. Hoek, E. T. Brown, Underground Excavations in Rock, Institution of Mining and Metallurgy, 1980.
Free and official sources
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