CU tests on overconsolidated clay

Step 1: Effective principal stresses at failure

$\sigma_3' = \sigma_3 - u$ and $\sigma_1' = \sigma_3' + \sigma_d$.

Step 2: Envelope. Draw the four effective-stress circles and the common tangent. Alternatively, fit a straight line through the points $(p', q)$ (the $K_f$ line), $q = a + p'\tan\alpha$. A least-squares fit gives:

$$\tan\alpha = 0.4421, \qquad a = 18.7\ \text{kPa}$$

Step 3: Parameters

$$\sin\phi' = \tan\alpha = 0.4421 \;\Rightarrow\; \phi' = 26.2^\circ$$ $$c' = \frac{a}{\cos\phi'} = \frac{18.7}{0.8975} = 20.8\ \text{kPa}$$

Answer: $c' \approx 21$ kN/m² and $\phi' \approx 26^\circ$ (in terms of effective stress).