Question 6 of 6: Surface Support and Circular Excavation Behaviour at Depth

Question 6: Surface Support and Circular Excavation Behaviour at Depth (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.

6.1 — Surface support techniques

Shotcrete (sprayed concrete), plain or fibre-reinforced. A cement-based mortar or concrete pneumatically sprayed directly onto the exposed rock surface, forming a continuous skin that seals the rock against air/moisture-driven weathering and physically confines and binds small blocks bounded by intersecting joints, preventing raveling. Advantages: rapid application over large, irregular areas immediately after excavation; conforms to any profile with no fabrication; steel- or synthetic-fibre reinforcement gives useful post-crack ductility and impact resistance without a separate mesh-fixing operation. Disadvantages: needs curing time to reach full strength (a window during which support is only partial); can debond from wet, dusty, or poorly prepared surfaces and from surfaces that continue to deform after application; plain (unreinforced) shotcrete is brittle and can spall in a rockburst-prone or highly stressed excavation.

Weld-mesh (or chain-link) panels fixed with rock bolts. Steel mesh panels held against the rock face by the bolt pattern's washer plates, retaining loose surface blocks between bolts without needing to bond to the rock at all. Advantages: low cost and simple, fast installation using the bolt pattern already required for reinforcement; the rock surface remains visible for inspection and mapping through the mesh; flexible enough to accommodate some continuing rock movement without failing outright. Disadvantages: mesh alone provides no continuous confinement or sealing against weathering/oxidation of the exposed rock, only point restraint at the bolt spacing, so unsupported spans between bolts can still ravel; mesh is easily damaged by blasting fly-rock and by corrosion in a wet or acid-generating environment unless galvanised or otherwise protected.

6.2 — Circular excavation stress state at 750 m depth

Given. Depth 750 m; σ_h=0.35σ_v; rock cohesion c=25 MPa, φ=26°.

Find. The predicted elastic boundary-stress state of the circular opening and whether it is safely within the rock's strength.

Approach. Compute the far-field vertical and horizontal stresses, apply the Kirsch closed-form solution for the tangential (boundary) stress at the roof/floor and at the sidewalls of a circular opening, then compare the more highly stressed sidewall condition to the rock's Mohr-Coulomb-equivalent unconfined compressive strength.

1. Far-field stresses. $\sigma_v=\gamma H=25\times0.750=\boxed{18.75\ \text{MPa}}$; $\sigma_h=0.35\sigma_v=\boxed{6.56\ \text{MPa}}$.
2. Kirsch boundary stresses. At the roof/floor (boundary points aligned with the σ_h direction): $\sigma_{\theta,roof}=3\sigma_h-\sigma_v=3(6.56)-18.75=\boxed{0.94\ \text{MPa}}$ (low, compressive, no tension predicted). At the sidewalls (aligned with σ_v): $\sigma_{\theta,wall}=3\sigma_v-\sigma_h=3(18.75)-6.56=\boxed{49.7\ \text{MPa}}$.
3. Equivalent unconfined compressive strength. $\sigma_c=\dfrac{2c\cos\phi}{1-\sin\phi}=\dfrac{2(25)\cos26^\circ}{1-\sin26^\circ}=\dfrac{44.9}{0.562}=\boxed{80.0\ \text{MPa}}$.
4. Factor of safety and prediction. $\mathrm{FS}_{wall}=\sigma_c/\sigma_{\theta,wall}=80.0/49.7=\boxed{1.61}$. The sidewall boundary stress remains below the rock's compressive strength with a moderate margin, and the roof/floor stress is low and positive (no tensile zone predicted), so the excavation is expected to behave elastically and remain stable without progressive spalling or squeezing — though the FS of 1.6 at the sidewall is not large, and stress concentrations at any corners, changes in cross-section, or nearby openings would warrant local support (e.g. the surface support of 6.1) as a precaution.