Question 2 of 6: Failure Criteria and Direct Shear Testing

Question 2: Failure Criteria and Direct Shear Testing (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.

2.1 — Hoek-Brown, Mohr-Coulomb and Barton-Bandis criteria compared

Mohr-Coulomb is a two-parameter (c, φ) linear criterion. Its advantage is simplicity: it is easy to fit from a handful of triaxial or shear tests, its parameters have direct physical meaning, and it plugs straight into closed-form limit-equilibrium formulas (e.g. slope wedge FS). For intact rock its disadvantage is that the real failure envelope is curved (strength gain per unit confinement falls off at high σ₃), so a single straight-line fit over- or under-estimates strength outside the tested confinement range. For joints it is a fair approximation of a smooth, planar, non-dilatant surface but cannot represent roughness-driven dilation or the scale effect that governs real discontinuities. For fractured rock masses it has no built-in mechanism to scale strength down from intact-rock values as fracturing/GSI worsens; the analyst must estimate c and φ for the mass by other means (e.g. back-calculating from a Hoek-Brown fit).

Hoek-Brown is a nonlinear empirical criterion originally calibrated for intact rock ($\sigma_1=\sigma_3+\sqrt{m_i\sigma_{ci}\sigma_3+\sigma_{ci}^2}$) and extended via the Geological Strength Index (GSI) to fractured/jointed rock masses. Its advantage for intact rock is that it captures the true curvature of the envelope (including the low-confinement region, where Mohr-Coulomb is weakest) using only two intact-rock constants (m_i, σ_ci) plus GSI for the rock mass extension; it is the most widely used criterion for fractured rock masses for exactly this reason. Its disadvantage is that m_i and GSI are partly subjective/table-based estimates rather than direct measurements, and the criterion is not intended to model a single discrete joint at all — it describes the smeared-out strength of many intersecting discontinuities, not the shear behaviour of one surface.

Barton-Bandis ($\tau=\sigma_n\tan\!\left[\mathrm{JRC}\log_{10}(\mathrm{JCS}/\sigma_n)+\phi_b\right]$) is purpose-built for joints: it captures the roughness (JRC), wall strength (JCS) and scale effects that a linear Mohr-Coulomb line cannot, and reduces smoothly toward the basic friction angle at high normal stress as asperities are sheared through — a real, observed behaviour. Its disadvantage is that JRC and JCS are estimated from roughness profile comparison charts and Schmidt-hammer rebound tests (some subjectivity, scale-dependent), and it applies to a single discontinuity surface, not directly to intact rock or to a heavily fractured rock mass where many joint sets of different orientation interact (that regime is better served by Hoek-Brown/GSI or explicit discontinuum modelling).

In short: Hoek-Brown is preferred for intact rock and GSI-scaled rock masses, Barton-Bandis for individual joints where roughness/wall-strength data are available, and Mohr-Coulomb is the common simplification used everywhere once equivalent (c, φ) values have been back-fitted from either of the other two, because most limit-equilibrium design tools are written in terms of c and φ.

2.2 — Basic friction angle and asperity angle from direct shear data

Find. The basic friction angle φ_b, the asperity (dilation) angle i, and an appropriate failure criterion for this joint.

Approach. Fit Patton's bilinear model: the low-normal-stress segment mobilises both base friction and asperity override (secant slope ≈ φ_b+i), while the high-normal-stress segment approaches the basic sliding friction alone (asperities increasingly sheared through, slope ≈ φ_b), so φ_b is read from the highest-stress segment and i from the excess of the lowest-stress segment over it.

1. Overall least-squares trend (all 5 points). $\tau = 0.1194\,\sigma_n - 24.65\ \text{kPa}$, i.e. an overall secant angle $\phi=\tan^{-1}(0.1194)=\boxed{6.8^\circ}$ — consistent with a single, essentially straight-line trend across the tested range (no strong systematic curvature).
2. Basic friction angle (high-stress segment, 360→420 kPa). $\phi_b=\tan^{-1}\!\left(\dfrac{25-20}{420-360}\right)=\tan^{-1}(0.0833)=\boxed{4.8^\circ}$.
3. Combined (dilatant) angle (low-stress segment, 200→255 kPa). $\phi_b+i=\tan^{-1}\!\left(\dfrac{5-0}{255-200}\right)=\tan^{-1}(0.0909)=5.2^\circ$.
4. Asperity angle. $i=(\phi_b+i)-\phi_b=5.2^\circ-4.8^\circ=\boxed{0.4^\circ}$ — small, consistent with the near-linear trend identified in Step 1: this dataset shows little measurable dilation contribution over the tested stress range.

Because the measured curvature between the low- and high-stress segments is small (0.4°) over this stress range, a single linear Coulomb line through near-zero cohesion is an adequate practical fit for design; Patton's bilinear form (or full Barton-Bandis, if JRC/JCS were characterised) remains the theoretically correct criterion for a rough natural joint in general, and would be preferred if the joint were to be loaded outside the 200–420 kPa range tested here.