Direct Shear Box Test on a Planar Discontinuity: 24-MMP-B1 Applied Rock Mechanics, May 2014, Q1

Question 1: Direct Shear Box Test on a Planar Discontinuity (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.

1.1 — Shear stress–displacement response; peak and residual strength

Given. Sample 1 was tested at a constant normal stress σ_n = 200 kPa. The shear-box recorder gives ten (shear displacement, shear stress) pairs over the course of the test.

Direct shear box test — Sample 1 (σ_n = 200 kPa)
Shear displacement (mm)	Shear stress (kPa)
0.05	138
1.21	186
3.52	238
4.32	234
8.78	223
9.48	206
11.56	205
12.64	199
17.54	168
22.11	168

Find. The peak shear strength and the residual (steady-state) shear strength of the discontinuity at this normal stress.

Shear stress vs. shear displacement, Sample 1 (σ_n = 200 kPa). Stress rises to a well-defined peak at small displacement, then softens and levels to a constant residual value at large displacement.

Approach. Read the maximum ordinate as the peak strength and the ordinate of the flat, displacement-independent tail as the residual strength.

Identify the peak. The curve rises steeply from 138 kPa at 0.05 mm displacement to a maximum of $\tau_{peak}=238\ \text{kPa}$ at 3.52 mm displacement, then softens (234, 223, 206, 205, 199 kPa) as the discontinuity dilates and asperities shear off.
Identify the residual. Beyond about 17 mm of displacement the stress has fallen to a constant plateau: $\tau=168\ \text{kPa}$ at both 17.54 mm and 22.11 mm, two consecutive readings identical to three significant figures, so $\boxed{\tau_{residual}=168\ \text{kPa}}$ is taken as the post-peak, fully-softened (residual) shear strength once asperities are worn flat and only frictional sliding on a smooth surface remains.

Question 1.1 — peak and residual shear strength at σ_n = 200 kPa
Quantity	Value
Peak shear strength, τ_peak	238 kPa (at 3.52 mm displacement)
Residual shear strength, τ_residual	168 kPa (plateau beyond ≈17.5 mm)

1.2 — Peak/residual strength vs. normal stress; friction angles

Given. A single normal-stress level was tested, σ_n = 200 kPa, giving one (σ_n, τ_peak) point at (200, 238) kPa and one (σ_n, τ_residual) point at (200, 168) kPa.

Find. The peak and residual friction angles of the discontinuity.

Check: with only ONE normal-stress level tested, a Mohr-Coulomb line ($\tau = c' + \sigma_n\tan\phi'$) cannot be fitted uniquely from the data alone — two unknowns (c′, φ′) need two independent points. The standard assumption for a clean, planar, weathered-granite discontinuity with no significant infill is that cohesion is negligible, $c'=0$, so the strength envelope is a straight line through the origin. This is stated explicitly as the required assumption (per the question's own instruction to "note any assumptions").

Peak and residual shear strength vs. normal stress, with the assumed c′=0 envelope through the origin fitted to the single test point at σ_n=200 kPa.

Approach. With c′=0, the Mohr-Coulomb relation reduces to $\tau=\sigma_n\tan\phi'$, so each friction angle follows directly from $\phi'=\tan^{-1}(\tau/\sigma_n)$ at the single tested point.

Peak friction angle. $\phi'_{peak}=\tan^{-1}\!\left(\dfrac{\tau_{peak}}{\sigma_n}\right)=\tan^{-1}\!\left(\dfrac{238}{200}\right)=\tan^{-1}(1.190)=\boxed{49.96^\circ\approx 50^\circ}$.
Residual friction angle. $\phi'_{residual}=\tan^{-1}\!\left(\dfrac{\tau_{residual}}{\sigma_n}\right)=\tan^{-1}\!\left(\dfrac{168}{200}\right)=\tan^{-1}(0.840)=\boxed{40.03^\circ\approx 40^\circ}$. The 10° drop from peak to residual is consistent with wearing-down of the surface roughness (dilation) component of strength once the asperities have sheared through.

Question 1 — summary
Quantity	Value
Peak shear strength (σ_n=200 kPa)	238 kPa
Residual shear strength (σ_n=200 kPa)	168 kPa
Peak friction angle, φ′_peak (c′=0 assumed)	≈50.0°
Residual friction angle, φ′_residual (c′=0 assumed)	≈40.0°

Topic. Direct shear box testing of a rock discontinuity: peak/residual strength and the Mohr-Coulomb friction angle.

Key relations. Mohr-Coulomb criterion $\tau=c'+\sigma_n\tan\phi'$; with c′=0, $\phi'=\tan^{-1}(\tau/\sigma_n)$.

Why this works. A direct shear test forces failure on a pre-defined plane at a fixed normal stress and records the full stress–displacement history. The initial rise reflects elastic loading plus dilation over surface asperities; the peak marks the point where asperities begin to shear or override; the post-peak softening reflects progressive asperity damage; and the residual plateau is the base sliding friction once the surface has been worn smooth, independent of the original roughness. Plotting strength against normal stress (ideally from several tests at different σ_n) recovers the cohesion intercept and friction angle; with only one test, forcing the line through the origin is the standard fallback for a clean discontinuity.

Common pitfall. Reporting a single-test friction angle without stating the c′=0 assumption it depends on, or confusing the peak strength (governs first movement / serviceability) with the residual strength (governs the ultimate, large-displacement stability that controls back-analysis of an already-failed slope).

Source. Brady & Brown, Rock Mechanics for Underground Mining, 3rd ed., ch. 4 (shear strength of discontinuities).

Question 1 of 6: Direct Shear Box Test on a Planar Discontinuity

Question 1: Direct Shear Box Test on a Planar Discontinuity (20 marks)

1.1 — Shear stress–displacement response; peak and residual strength

1.2 — Peak/residual strength vs. normal stress; friction angles