Question 2 of 5: Mohr-Coulomb Parameters from Triaxial Compression Data
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2013 — 04-Geol-A5 Rock Mechanics. Three-hour, closed-book exam; one of two approved calculators permitted, plus two sheets of the candidate's own rock-mechanics formulae/notes. Five questions of equal value (20 marks each); the paper instructs candidates to answer only the first 4 of 5 questions appearing in the answer book — all five are answered here as a complete study resource. Selected equations, RMR tables (Bieniawski 1989) and the Modified Lauffer stand-up-time chart are supplied at the back of the exam and are reproduced where used.
Reference texts: Bieniawski, Engineering Rock Mass Classifications (Wiley, 1989) — the RMR system, discontinuity-condition guidelines, and excavation/support tables used in Q1; Hoek, Practical Rock Engineering — Mohr-Coulomb strength parameters from triaxial data, Kirsch stress solutions around circular openings, and thick-wall liner design used in Q2/Q3/Q5; Brady & Brown, Rock Mechanics for Underground Mining (3rd ed.) — tributary-area pillar stress analysis and elastic pillar deformation used in Q4; EGBC Geoscience Professional Practice Guidelines for assumption-disclosure conventions.
Question 2: Mohr-Coulomb Parameters from Triaxial Compression Data (20 marks)
Given. Five triaxial failure pairs $(\sigma_3,\sigma_1)$ in MPa: (16.8, 159.3), (13.2, 154.5), (25.0, 198.0), (9.7, 140.1), (20.0, 168.0). The Mohr-Coulomb straight-line failure locus $\sigma_1=\sigma_3\tan^2\Psi+S_c$, with $\Psi=45^{\circ}+\phi/2$ and $S_c=2C\tan\Psi$.
Triaxial test results
Confining stress $\sigma_3$ (MPa)
Failure axial stress $\sigma_1$ (MPa)
16.8
159.3
13.2
154.5
25.0
198.0
9.7
140.1
20.0
168.0
Find. (a) Cohesion $C$, friction angle $\phi$, failure-plane angle $\Psi$ and UCS $S_c$. (b) Data-quality issues evident from the fit. (c) Failure axial stress at $\sigma_3=5,15,22.5$ MPa. (d) An independent check on part (c).
Approach. The Mohr-Coulomb failure locus $\sigma_1=\sigma_3\tan^2\Psi+S_c$ is linear in $\sigma_3$, so a least-squares regression of the five $(\sigma_3,\sigma_1)$ pairs gives the slope ($\tan^2\Psi$) and intercept ($S_c$) directly; $\phi$ and $C$ follow from the standard relations. The fitted line is then simply evaluated at the three new confining stresses for part (c).
Linear regression of the failure data. With $\bar\sigma_3=16.94$ MPa, $\bar\sigma_1=163.98$ MPa,
$$\text{slope}=\tan^2\Psi=\frac{\sum(\sigma_{3,i}-\bar\sigma_3)(\sigma_{1,i}-\bar\sigma_1)}{\sum(\sigma_{3,i}-\bar\sigma_3)^2}=3.520,\qquad \text{intercept}=S_c=\bar\sigma_1-\text{slope}\times\bar\sigma_3=\boxed{104.3\ \text{MPa}}$$
The coefficient of determination is $R^2=0.940$ — a reasonably tight but imperfect fit to a straight line.
Failure angle and friction angle.
$$\Psi=\arctan\sqrt{3.520}=\boxed{61.9^{\circ}},\qquad \phi=2(\Psi-45^{\circ})=\boxed{33.9^{\circ}}$$
Part (b) — problems evident from the data. The fit is good ($R^2=0.940$) but not exact — the largest residual is at $\sigma_3=25.0$ MPa, where the measured 198.0 MPa sits noticeably off the fitted line, hinting that the true envelope may curve slightly (a common real-rock departure from a strictly linear Mohr-Coulomb locus, especially toward the high-confinement end). More fundamentally, only five specimens were tested with NO repeat tests at any single confining stress, so there is no way to separate genuine specimen-to-specimen strength variability from a real nonlinearity in the failure locus — the $R^2$ of 0.940 is the only available proxy for data quality, not a true measurement-repeatability estimate.
Part (c) — failure axial stress at new confining stresses. Evaluating the fitted line $\sigma_1=3.520\,\sigma_3+104.3$:
$$\sigma_3=5\ \text{MPa}:\ \sigma_1=3.520(5)+104.3=\boxed{122.0\ \text{MPa}}$$
$$\sigma_3=15\ \text{MPa}:\ \sigma_1=3.520(15)+104.3=\boxed{157.1\ \text{MPa}}$$
$$\sigma_3=22.5\ \text{MPa}:\ \sigma_1=3.520(22.5)+104.3=\boxed{183.6\ \text{MPa}}$$
Part (d) — verifying the predicted values. Run the three additional triaxial tests (at $\sigma_3=5,15,22.5$ MPa) and compare the measured failure stresses against the predicted 122.0/157.1/183.6 MPa; independently, an unconfined compression test ($\sigma_3=0$) should return a UCS close to the regression intercept of 104.3 MPa, and a direct-shear or Brazilian tensile test ($S_T=C/\tan\phi\approx18.4$ MPa) provides a cross-check on $C$ and $\phi$ from a wholly different loading path.
