Question 4 of 5: Instability mechanisms, tunnel damage zone, failure criteria and pillar design

Question 4: Instability mechanisms, tunnel damage zone, failure criteria and pillar design (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.

(a) Stress-controlled vs. material-strength instability

Stress-controlled (structurally-controlled) instability occurs when the rock mass itself is strong enough, but pre-existing discontinuities (joints, bedding, faults) provide kinematically free planes that slide, topple or fall under the redistributed excavation stresses — the rock material never approaches its own strength; failure is governed entirely by discontinuity geometry and shear strength (e.g. a wedge or planar slide as in Question 2). By contrast, material-property (strength) failure occurs when the induced stress at the excavation boundary exceeds the intact rock or rock-mass strength itself — spalling, squeezing or rock burst in massive, sparsely-jointed rock, governed by a strength criterion (Mohr-Coulomb or Hoek-Brown) rather than by joint orientation (e.g. the Kirsch boundary-stress check of Question 5). In practice, most excavations are checked against both mechanisms independently, since either can govern depending on rock mass structure and in-situ stress.

(b) Zone of influence and the excavation-damage zone

Excavating an opening redistributes the pre-existing far-field stress: material removed from the opening can no longer carry load, so stress concentrates around the boundary and decays back toward the far-field value with distance, per the Kirsch (or equivalent) elastic solution — the “zone of influence” is conventionally taken as the region where the redistributed stress still differs from the far-field value by more than a small tolerance (commonly ±5–10%), which for a circular opening extends to roughly 3–5 excavation radii from the boundary (Kirsch stresses fall to within a few percent of far-field by r ≈ 3a, per the reference-section Kirsch equations given on this exam). Within that zone, but much closer to the boundary, lies the smaller excavation damage zone (EDZ) — rock whose strength or stiffness has been measurably degraded by blast-induced micro-fracturing, stress-induced yielding, or spalling, not merely elastically stressed. The EDZ extent is determined by a combination of (i) back-analysis of the elastic stress concentration against the rock's own strength criterion (where induced stress exceeds strength, yielding/damage is predicted), (ii) direct field evidence — borehole camera or acoustic televiewer logging, seismic velocity surveys (damaged rock shows a measurable P-wave velocity reduction), and convergence/extensometer monitoring, and (iii) numerical modelling (boundary-element or finite-element) that couples the elastic stress redistribution to a strain-softening or Hoek-Brown yield criterion so that the yielded (damaged) zone is mapped directly rather than inferred from elastic stress alone.

(c) Major failure criteria and rock capacity

The two failure criteria used throughout this exam's own reference section are the standard pair taught in practice. The Mohr-Coulomb criterion, $$\tau=c+\sigma_n\tan\phi\quad\text{or equivalently}\quad\sigma_1=\sigma_c+\sigma_3\tan^2\!\left(45^\circ+\tfrac{\phi}{2}\right)$$ is a linear envelope in shear/normal-stress (or σ₁/σ₃) space, defined by two constants (cohesion c, friction angle φ, or equivalently UCS σ_c and tensile strength σ_t) fitted from triaxial or direct-shear/point-load test data — used in Question 1 (discontinuity/fault strength) and implicitly for discontinuity strength throughout this exam. The Hoek-Brown criterion, $$\sigma_1=\sigma_3+\sigma_c\sqrt{m\frac{\sigma_3}{\sigma_c}+s}$$ is a non-linear (curved) envelope calibrated from intact-rock triaxial data (constant m_i) and then scaled down to rock-mass strength via GSI/RMR (parameters m, s), and is the standard choice whenever the rock mass, not just a single discontinuity, is the failure surface (e.g. pillar or tunnel-boundary capacity). Rock capacity is determined, in either case, by (i) establishing the criterion's constants from laboratory or index (point-load) testing, scaled to rock-mass conditions via a classification system such as RMR/GSI, then (ii) comparing the induced stress state (from the elastic Kirsch/tributary-area solution, or a numerical model) against the envelope: capacity is exceeded wherever the induced stress state plots outside (above) the envelope.

(d) Optimum pillar width and spacing

Pillar sizing balances two opposing requirements: pillars must be large enough (relative to the mined-out opening) to carry the tributary overburden load with an adequate factor of safety, but no larger than necessary, since every tonne left in a pillar is a tonne of ore not recovered. The standard design sequence is: (i) estimate the average pillar stress by tributary area theory, $$\sigma_{v,avg}=\gamma Z\frac{(W_p+W_o)^2}{W_p^{\,2}}$$ for square pillars of width W_p on centre-opening span W_o at depth Z; (ii) estimate pillar strength from an empirical pillar-strength formula — strength generally increases with pillar width-to-height ratio, since a squatter pillar develops more confinement at its core; (iii) size W_p so that FoS = σ_strength/σ_v,avg meets the operation's target (typically 1.5–2.0 for a permanent pillar, lower for a temporary one); and (iv) iterate W_p and W_o against the required extraction ratio r = 1−(W_p/(W_p+W_o))², since the economics of the operation depend directly on how much ore is left unmined in the pillars. As an illustration, at Z = 150 m, γ = 25 kN/m³, UCS = 80 MPa, W_p = 8 m and W_o = 5 m, tributary theory gives σ_v,avg = 9.90 MPa and FoS = 80/9.90 = 8.1 — comfortably conservative, meaning W_p could be reduced (and recovery increased) before the target FoS band is reached, which is exactly the iterative optimisation an operator would run in practice rather than accepting the first pillar size that merely clears FoS > 1.