Analytical stress solution
Kirsch tunnel stress: circular opening
Explore the closed-form elastic boundary stress around an unsupported circular opening under biaxial far-field stress, with the convention and scope shown beside the result.
Core relationship
σθ/σ∥ = (1 + k) − 2(1 − k) cos(2θ), where k = σ⊥/σ∥
Inputs & units
- Parallel far-field stress σ∥ [MPa]
- Stress ratio k = σ⊥/σ∥ [–]
- Boundary angle θ from the parallel-load axis [degrees]
Assumptions
- Infinite, homogeneous, isotropic, linear-elastic continuum.
- Circular opening with evaluated point on the excavation boundary.
- No internal pressure or support pressure; far-field principal directions align with the stated axes.
Limitations
- Not suitable by itself for non-circular geometry, anisotropy, joints, plasticity, nearby excavations or construction sequence.
- Boundary stress is not a support design and does not predict deformation or failure on its own.
- In-situ stress orientation and magnitude uncertainty must be assessed independently.
Synthetic worked example
For σ∥ = 10 MPa and k = 0, the idealized springline tangential stress is +30 MPa while the crown value is −10 MPa under a compression-positive convention.
Interactive benchmark
Kirsch boundary-stress example
Educational preliminary example only. It demonstrates one closed-form boundary equation and is not an excavation or support design.
Allowed example range: 0–1000 MPa
Allowed example range: 0–3
Measured from the parallel-load axis, 0–180°
At selected angle
30.00 MPa
Crown (θ = 0°)
-10.00 MPa
Springline (θ = 90°)
30.00 MPa
Convention: compression is positive; k = σ⊥/σ∥; θ is measured from the σ∥ axis. Negative tangential output denotes tension under this convention.
Engineering use boundary
This page is educational and supports preliminary assessment. It does not replace site investigation, competent engineering judgement, final design, independent review or permitting requirements.