Question 5 of 5: Kirsch boundary stress around a deep circular tunnel

Question 5: Kirsch boundary stress around a deep circular tunnel (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A circular tunnel at depth Z, with rock unit weight, UCS and tensile strength, checked at two horizontal-to-vertical in-situ stress ratios k.

Find. The Kirsch tangential boundary stress at the tunnel sidewall and at the roof/floor for k = 0.3 and k = 2.0, and whether either exceeds the rock's compressive or tensile strength.

Approach. Compute the far-field vertical stress from depth, then evaluate the Kirsch closed-form tangential stress at the two boundary points that bound the maximum compressive (sidewall) and maximum tensile (roof/floor) response — the tunnel diameter does not enter the boundary-stress formula itself (only the far-field stress ratio and magnitude govern the boundary concentration), it only sets the physical scale of the opening.

1. Far-field vertical stress. $$\sigma_v=\gamma Z=(25\ \text{kN/m}^3)(700\ \text{m})=17{,}500\ \text{kPa}=17.5\ \text{MPa}$$ $$\boxed{\sigma_v=17.5\ \text{MPa}}$$
2. Kirsch boundary tangential stress. At the wall of a circular opening (r = a), the Kirsch elastic solution gives the tangential stress at the sidewall (aligned with the horizontal far-field stress direction) and at the roof/floor (aligned with the vertical far-field stress direction) as $$\sigma_{\theta,side}=\sigma_v(3-k),\qquad\sigma_{\theta,roof}=\sigma_v(3k-1)$$ these being the two extremes of the general boundary formula σ_θ=σ_v[(1+k)+2(1−k)cos2θ] at θ=0° (side) and θ=90° (roof/floor).
3. Case (a): k = 0.3. $$\sigma_{\theta,side}=17.5(3-0.3)=17.5(2.7)=47.25\ \text{MPa (compression)}$$ $$\sigma_{\theta,roof}=17.5(3(0.3)-1)=17.5(-0.1)=-1.75\ \text{MPa (tension)}$$ Sidewall: 47.25 MPa < UCS = 70 MPa — not exceeded. Roof/floor: the boundary is in tension at 1.75 MPa magnitude, which is less than the tensile strength of 2.8 MPa — also not exceeded. $$\boxed{k=0.3:\ \text{boundary strength NOT exceeded (roof close to, but below, tensile limit)}}$$
4. Case (b): k = 2.0. $$\sigma_{\theta,side}=17.5(3-2.0)=17.5(1.0)=17.5\ \text{MPa (compression)}$$ $$\sigma_{\theta,roof}=17.5(3(2.0)-1)=17.5(5.0)=87.5\ \text{MPa (compression)}$$ Sidewall: 17.5 MPa < UCS — not exceeded. Roof/floor: 87.5 MPa > UCS = 70 MPa — exceeded, by a wide margin (25% over capacity). $$\boxed{k=2.0:\ \text{boundary strength EXCEEDED at the roof/floor (compressive failure)}}$$

The two cases illustrate the two distinct failure modes a Kirsch check must screen for. At low k (horizontal stress well below vertical), the roof/floor goes into tension — here it stays just inside the tensile limit, but a slightly lower k or higher σ_v would open a tensile (roof-slabbing) failure. At high k (horizontal stress well above vertical), the roof/floor instead sees a large tangential compressive concentration (3k−1 grows without bound as k increases) that clearly exceeds UCS here — a compressive (spalling/crushing) failure mode. The sidewall, by contrast, is the location of maximum tangential stress only when k is small; it never governs in either case checked here.