Question 6 of 6: CU triaxial strength parameters from the modified failure envelope

Question 6: CU triaxial strength parameters from the modified failure envelope (8 + 8 + 4 = 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. Three CU-with-pore-pressure tests on saturated clay:

Find. The effective-stress parameters $c'$ and $\phi'$ from the modified (stress-point) failure envelope, and answers to parts (i)–(iii).

Approach. Convert each test to its stress point $p' = \tfrac12(\sigma_1'+\sigma_3')$, $q = \tfrac12(\sigma_1'-\sigma_3')$, fit the straight $K_f$ line $q = a + p'\tan\alpha'$, and recover $\phi'$ and $c'$ from the standard transformation.

1. Effective principal stresses. $\sigma_1 = \sigma_3 + (\sigma_1-\sigma_3)$ and, subtracting the measured $u$, $\sigma_3' = \sigma_3 - u$, $\sigma_1' = \sigma_1 - u$. For the three tests $(\sigma_3',\sigma_1') = (70,170),\ (140,340),\ (280,680)\ \text{kPa}$.
2. Stress points. $q = \tfrac12(\sigma_1'-\sigma_3') = \tfrac12(\sigma_1-\sigma_3)$ (the pore pressure cancels), so $q = 50,\ 100,\ 200\ \text{kPa}$; and $p' = \tfrac12(\sigma_1'+\sigma_3') = 120,\ 240,\ 480\ \text{kPa}$.
3. Fit the modified ($K_f$) line. The three points $(p',q)$ plot on a straight line through the origin: $q = 50/120 = 100/240 = 200/480$, so $\tan\alpha' = 0.4167$ and the intercept $a = 0$.
4. Recover $\phi'$ and $c'$. From the standard transformation $\phi' = \sin^{-1}(\tan\alpha')$ and $c' = a/\cos\phi'$: $\phi' = \sin^{-1}(0.4167)$, giving $\boxed{\phi' \approx 24.6^\circ,\quad c' = 0}$.

(i) Advantage of the modified envelope. Each test plots as a single point $(p',q)$ instead of a full Mohr circle, so a straight best-fit line can be drawn through scattered data unambiguously. Fitting a common tangent to several overlapping Mohr circles is imprecise (small errors in any one circle swing the tangent), whereas a linear regression of stress points is objective and averages out scatter.

(ii) The clay is normally consolidated. The modified envelope passes through the origin ($a = 0$), so $c' = 0$ — there is no true cohesion intercept. A cohesionless effective-stress envelope is the signature of a normally consolidated clay; an over-consolidated clay would show a positive $c'$ (and $a\gt 0$) at low stresses. The steadily positive pore pressures generated on shear ($u\gt 0$ throughout, contractive response) confirm normally consolidated behaviour.

(iii) These parameters give the long-term (drained) stability. $c'$ and $\phi'$ are effective-stress parameters, obtained by removing the measured pore pressure. They therefore describe the fully drained condition and must be used with an effective-stress analysis for the long-term stability of the earth dam (steady seepage, when excess pore pressures have dissipated). The short-term (end-of-construction, undrained) stability of a saturated clay is governed instead by the total-stress undrained strength $c_u$ ($\phi = 0$ analysis), not by $c'$ and $\phi'$.