Question 2 of 6: Triaxial Testing of Intact Granite — Mohr-Coulomb and Hoek-Brown Criteria

Question 2: Triaxial Testing of Intact Granite — Mohr-Coulomb and Hoek-Brown Criteria (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.

Given. Five triaxial tests on intact granite specimens, confining pressure σ₃ from 5 to 25 MPa:

2.1 — Mohr-Coulomb parameters and applicability

Find. The Mohr-Coulomb cohesion and friction angle from a least-squares fit of σ₁ vs. σ₃, and a comment on how well the linear criterion represents the data.

Approach. The Mohr-Coulomb failure envelope, expressed in principal-stress space, is the straight line $\sigma_1=\sigma_c+\sigma_3\tan^2\!\left(45^\circ+\tfrac{\phi'}{2}\right)$, where σ_c is the intercept (uniaxial compressive strength) and the slope gives the friction angle. Fit this line to the five points by least squares.

1. Least-squares slope and intercept. With $\bar\sigma_3=15$, $\bar\sigma_1=361.2$, $\sum\sigma_3\sigma_1=29{,}550$, $\sum\sigma_3^2=1375$ ($n=5$): slope $=\dfrac{n\sum\sigma_3\sigma_1-\sum\sigma_3\sum\sigma_1}{n\sum\sigma_3^2-(\sum\sigma_3)^2}=\dfrac{5(29{,}550)-75(1806)}{5(1375)-75^2}=\dfrac{12{,}300}{1250}=9.84$; intercept $=\bar\sigma_1-\text{slope}\cdot\bar\sigma_3=361.2-9.84(15)=213.6$. So $\sigma_1=213.6+9.84\,\sigma_3$ (MPa), giving the uniaxial compressive strength $\boxed{\sigma_c=213.6\ \text{MPa}}$.
2. Friction angle and cohesion. $\tan^2(45^\circ+\phi'/2)=9.84\Rightarrow 45^\circ+\phi'/2=\tan^{-1}\sqrt{9.84}=\tan^{-1}(3.137)=72.32^\circ\Rightarrow\boxed{\phi'=54.6^\circ}$. Cohesion follows from $\sigma_c=\dfrac{2c'\cos\phi'}{1-\sin\phi'}\Rightarrow c'=\dfrac{\sigma_c(1-\sin\phi')}{2\cos\phi'}=\dfrac{213.6(1-0.8154)}{2(0.5789)}=\boxed{34.0\ \text{MPa}}$.
3. Goodness of fit and applicability. The fitted line predicts σ₁ = 262.8, 312.0, 361.2, 410.4, 459.6 MPa against the measured 255, 310, 375, 420, 446 MPa — a coefficient of determination $R^2=0.978$. The fit is good but systematically biased: it over-predicts strength at both ends of the tested range (262.8 fitted vs. 255 measured at σ₃=5 MPa; 459.6 vs. 446 at σ₃=25 MPa) and under-predicts in the middle (361.2 vs. 375 at σ₃=15 MPa) — the classic signature of forcing a straight line through a data set that is genuinely curved (concave-down) in σ₁–σ₃ space, as intact rock strength envelopes almost always are. Mohr-Coulomb is therefore an adequate local linear approximation over this confining-stress range but is not the more fundamentally correct description of intact rock strength — see 2.2.

2.2 — Hoek-Brown criterion and applicability

Find. The Hoek-Brown intact-rock parameters σ_ci and m_i, and a comment on the criterion's applicability relative to Mohr-Coulomb.

Approach. For intact rock (s=1), the Hoek-Brown criterion $\sigma_1=\sigma_3+\sqrt{m_i\sigma_{ci}\sigma_3+\sigma_{ci}^2}$ is linearised by squaring: $(\sigma_1-\sigma_3)^2=m_i\sigma_{ci}\,\sigma_3+\sigma_{ci}^2$, a straight line in $\sigma_3$ vs. $(\sigma_1-\sigma_3)^2$ whose intercept is $\sigma_{ci}^2$ and whose slope is $m_i\sigma_{ci}$.

1. Transform and fit. $(\sigma_1-\sigma_3)^2$ = 62{,}500; 90{,}000; 129{,}600; 160{,}000; 177{,}241 at σ₃ = 5, 10, 15, 20, 25 MPa. Least squares gives slope $=5989.6$, intercept $=34{,}024$, so $\boxed{\sigma_{ci}=\sqrt{34{,}024}=184.5\ \text{MPa}}$ and $\boxed{m_i=5989.6/184.5=32.5}$.
2. Sanity-check against published data. Hoek's tabulated $m_i$ for granite is $32\pm3$ — the fitted $m_i=32.5$ falls almost exactly on the published mean, a strong independent confirmation that the fit is physically sound and not an artefact of only five points.
3. Goodness of fit and applicability. The fitted curve predicts σ₁ = 257.9, 316.5, 366.9, 412.2, 453.7 MPa against measured 255, 310, 375, 420, 446 MPa, $R^2=0.990$ (computed in σ₁ space, on the same basis as the 0.978 quoted for Mohr-Coulomb above, so that the two are directly comparable; the linearised $(\sigma_1-\sigma_3)^2$ vs. σ₃ regression used to obtain the parameters has its own $R^2=0.986$) — a closer match than the Mohr-Coulomb line at both ends of the tested range (σ₃=5 and 25 MPa), because the Hoek-Brown power-law form directly reproduces the concave-down curvature (diminishing strength gain per unit increase in confinement) that a straight line cannot. For intact, unweathered granite tested over a wide confining-stress range, the Hoek-Brown criterion is therefore the more applicable and more widely recommended failure criterion; Mohr-Coulomb remains useful as a locally linearised approximation (e.g. for hand slope-stability calculations at a known, narrow stress range, as used with a single c′/φ′ pair in Questions 3 and 6 of this exam).