Imagine you are designing a circular tunnel at depth. The measured vertical stress is 30 MPa and the horizontal stress is 20 MPa. The average UCS of intact rock is 50 MPa. At first glance, these stresses appear well below the UCS. So why do we still observe spalling and cracking at the tunnel boundary? The problem lies in the redistribution of stresses around the opening after the excavation. To understand where and how much stress concentrates, we use Kirsch Equation. Given: Tunnel radius a = 3 m Vertical stress σᵥ = 30 MPa Horizontal stress σₕ = 20 MPa At the tunnel boundary (r = a): Radial stress: σᵣ = 0 MPa (free surface) Tangential stress: σθ = σₕ + σᵥ − 2(σₕ − σᵥ) cos(2θ) 🔹 Sidewalls (θ = 0°) σθ = 20 + 30 − 2(20 − 30)(1) σθ = 70 MPa 🔹 Crown & invert (θ = 90°) σθ = 20 + 30 − 2(20 − 30)(−1) σθ = 30 MPa What does this tell us? The local induced stress at sidewalls ≈ 70 MPa and it is nearly 2.3 times the in-situ stresses. Hence, if the rock UCS is below 70 MPa, damage initiation is likely immediately after excavation unless confinement or support is provided.