Find. (a) the $e$–$\log\sigma'$ plot; (b) the preconsolidation pressure $p_c$ by Casagrande's construction; (c) the compression index $C_c$.
Laboratory $e$–$\log\sigma'$ curve. The response is flat (recompression) below about 90 kPa and steepens into a straight virgin line above it; Casagrande's construction locates $p_c$ near the break.
Approach. Plot $e$ against $\log\sigma'$; apply Casagrande's construction at the point of maximum curvature to read $p_c$; take the slope of the straight virgin portion as $C_c$.
(a) Plot. Plotting the six points shows the classic two-branch shape: a gently sloping recompression branch from 25–100 kPa ($e$ from 1.210 to 1.150) that bends into a much steeper straight virgin branch beyond 100 kPa (Figure above).
(b) Casagrande's method for $p_c$. At the point of minimum radius of curvature (near $\sigma'\approx70$–90 kPa) draw (i) a horizontal line, (ii) a tangent, and (iii) the bisector of the angle between them. The straight virgin line, projected back, meets the bisector at the preconsolidation pressure:
$$\boxed{p_c \approx 90\ \text{kN/m}^2\ \text{(graphical)}.}$$
(c) Compression index $C_c$. $C_c$ is the slope of the virgin straight line. Reading two well-separated points on that branch ($\sigma'=100$, $e=1.150$ and $\sigma'=400$, $e=0.980$):
$$C_c=\frac{e_1-e_2}{\log(\sigma_2'/\sigma_1')}=\frac{1.150-0.980}{\log(400/100)}=\frac{0.170}{0.602}=\boxed{0.28}.$$
(The 200–400 kPa pair gives 0.27, confirming the virgin-line slope $C_c\approx0.28$.)
Q4 — consolidation parameters
Quantity
Value
Method
Preconsolidation pressure $p_c$
≈ 90 kN/m$^2$
Casagrande construction (graphical)
Compression index $C_c$
0.28
slope of virgin line (100–400 kPa)
Check: $p_c$ from Casagrande's construction is a graphical read and carries roughly ±15 kPa depending on where the maximum-curvature point is judged; the soil is lightly overconsolidated (the recompression branch precedes the virgin line). Empirically $C_c\approx0.009(LL-10)$ would imply a liquid limit near 40, consistent with a medium-plasticity clay.
