Given. Table 1 — CU triaxial at failure (pore pressure measured; $u_0=0$ after consolidation):
Given data — CU triaxial series (Table 1)
Specimen
$\sigma_3$ (kPa)
$(\sigma_1-\sigma_3)_f$ (kPa)
$u_f$ (kPa)
A
150
103
82
B
300
202
169
Find. (i) $c',\phi'$; (ii) $c_{cu},\phi_{cu}$; (iii) $(\sigma_1-\sigma_3)_f$ at $\sigma_3'=250$ kPa; (iv) the consolidation state.
Modified (stress-path) diagram: total-stress points A, B and effective-stress points A′, B′ plotted as $p=(\sigma_1+\sigma_3)/2$ vs $q=(\sigma_1-\sigma_3)/2$. The effective $K_f$ line passes essentially through the origin.
Approach. Reduce each test to a stress point $(p,q)$; fit the modified ($K_f$) line for both total and effective stresses; convert its slope and intercept to Mohr–Coulomb $c,\phi$; then use Skempton's $A_f$ to predict the deviator stress at a new consolidation stress and to classify the clay.
Stress points. With $q=(\sigma_1-\sigma_3)/2$ (unchanged by $u$) and $p'=p-u$:
$\sigma_1$
$q$
$p=\frac{\sigma_1+\sigma_3}{2}$
$p'=p-u$
$A_f=\frac{u_f}{(\sigma_1-\sigma_3)_f}$
A
253
51.5
201.5
119.5
0.80
B
502
101.0
401.0
232.0
0.84
(i) Effective parameters — modified envelope through A′, B′. The $K_f'$ line $q=a'+p'\tan\alpha'$ has
$$\tan\alpha'=\frac{q_B-q_A}{p_B'-p_A'}=\frac{101.0-51.5}{232.0-119.5}=0.440,\qquad a'\approx0.$$
Converting to Mohr–Coulomb ($\sin\phi'=\tan\alpha'$, $c'=a'/\cos\phi'$):
$$\phi'=\sin^{-1}(0.440)=\boxed{26.1^\circ},\qquad c'\approx\boxed{0}.$$
(ii) Total parameters — analytical fit through A, B. The total-stress $K_f$ line has
$$\tan\alpha=\frac{q_B-q_A}{p_B-p_A}=\frac{101.0-51.5}{401.0-201.5}=0.248,\qquad a\approx1.5\ \text{kPa}.$$
Hence
$$\phi_{cu}=\sin^{-1}(0.248)=\boxed{14.4^\circ},\qquad c_{cu}=\frac{a}{\cos\phi_{cu}}\approx\boxed{1.6\ \text{kPa}\ (\approx0)}.$$
(iii) Deviator stress at $\sigma_3'=250$ kPa. The soil is saturated ($B=1$) and its effective envelope is unique ($c'=0$), so with $N=\tan^2(45^\circ+\phi'/2)$ and $A_f\approx0.82$ (average; consistent with Figure 3 at OCR = 1), a constant-cell undrained shear from $\sigma_{3c}'=250$ kPa gives
$$N=\tan^2\!\Big(45^\circ+\tfrac{26.1^\circ}{2}\Big)=2.57,$$
$$(\sigma_1-\sigma_3)_f=\frac{\sigma_{3c}'\,(N-1)}{(1-A_f)+N A_f}=\frac{250(2.57-1)}{(1-0.82)+2.57(0.82)}=\boxed{\approx 172\ \text{kPa}.}$$
A linear interpolation of the measured deviators against the consolidation stress (A: 150→103, B: 300→202) gives 169 kPa — the same answer, confirming the estimate.
(iv) Consolidation state. Three independent indicators all point to normally consolidated: the effective envelope passes through the origin ($c'\approx0$); the pore-pressure parameter $A_f\approx0.82$ is high and positive (0.5–1.0 range — the soil is contractive, generating positive excess pressure on shearing); and Figure 3 maps $A_f\approx0.8$ to an overconsolidation ratio of about 1. An overconsolidated clay would instead show $c'>0$ and a low or negative $A_f$.
Q6 — results
Part
Result
(i) Effective parameters
$c'\approx0$, $\phi'\approx26.1^\circ$
(ii) Total parameters
$c_{cu}\approx0$, $\phi_{cu}\approx14.4^\circ$
(iii) $(\sigma_1-\sigma_3)_f$ at $\sigma_3'=250$ kPa
