07-Str-A3 · Undated paper

Question 4 of 6: Consolidation test — $e$–log $\sigma'$ curve, preconsolidation pressure and $C_c$

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. Engineers Canada / PEO National Examinations, May 2019 — 07-Str-A3 Geotechnical Materials and Analysis. Closed book, three hours, 100 marks, one approved Casio or Sharp calculator. All six questions are compulsory and are weighted 10 / 10 / 15 / 20 / 15 / 30. The paper carries its own appendix: two sheets of blank semi-logarithmic and squared graph paper (pages 7–9) for the plotted answers, the rectangular-loading $m$–$n$ influence chart (page 10), a Newmark influence chart (page 11) and a two-page formula sheet (pages 12–13) on which the Newmark influence value is printed as $\sigma_z = 0.005\,N q$, i.e. $I_N = 0.005$ over 200 elements. Every chart value quoted below is taken from those appendix sheets.

Sitting. The printed cover reads NATIONAL EXAMINATIONS – May 2019, 07-Str-A3 Geotechnical Materials and Analysis, 3 hours duration, and every interior page repeats it. The sitting is May 2019.

Reference texts.

B. M. Das & K. Sobhan, Principles of Geotechnical Engineering, 9th ed. — Ch. 2–3 (grain size and phase relations), Ch. 6 (compaction), Ch. 7–8 (permeability and seepage, flow nets), Ch. 9 (stresses in a soil mass, the $m$–$n$ chart and Newmark’s chart), Ch. 11 (consolidation, Casagrande’s construction), Ch. 12 (shear strength and the triaxial test).
R. F. Craig / J. A. Knappett, Craig’s Soil Mechanics, 8th ed. — Ch. 2 (seepage and flow nets), Ch. 3 (effective stress), Ch. 4 (consolidation), Ch. 5 (shear strength, the stress-path or “modified” failure envelope and the choice of test), Ch. 6 (stress distribution).
M. E. Harr, Groundwater and Seepage — Ch. 4 (the conformal solution for a single sheet pile in a stratum of finite depth, used here to audit the drawn flow net).
Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM), 4th ed. — Ch. 4 (site investigation and sampling), Ch. 8 (settlement), Ch. 11 (laboratory strength testing). The Canadian reference for practice and terminology.
K. Terzaghi, R. B. Peck & G. Mesri, Soil Mechanics in Engineering Practice, 3rd ed. — Art. 14 (one-dimensional consolidation theory and its assumptions), Art. 16–17 (seepage and piping), Art. 19–20 (shear strength).

Check — two readings of the printed paper, carried as stated.

(1) The printed paper has exactly six questions worth 10 + 10 + 15 + 20 + 15 + 30 = 100 marks, and that numbering is used here.

(2) Question 4 lists the void ratio at $\sigma' = 500$ kPa as $e = 0.925$. Taken with the $400$ kPa point that implies $C_c = 0.57$ over the last increment, against $0.27$–$0.30$ over every earlier virgin increment — the last row is inconsistent with the rest of the test. It is carried exactly as printed, plotted, and excluded from the straight-line fit for $C_c$, with the effect of including it stated where it matters.

Question 4: Consolidation test — $e$–log $\sigma'$ curve, preconsolidation pressure and $C_c$ (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. The six oedometer readings, with $\log_{10}\sigma'$ tabulated for plotting:

$\sigma'$ (kN/m$^2$)	25	50	100	200	400	500
$\log_{10}\sigma'$	1.398	1.699	2.000	2.301	2.602	2.699
Void ratio $e$	1.210	1.195	1.150	1.060	0.980	0.925

Find. (a) the plotted $e$–log $\sigma'$ curve; (b) the preconsolidation pressure $\sigma'_c$ by Casagrande’s construction; (c) the compression index $C_c$.

(a) The laboratory $e$–log $\sigma'$ curve, plotted at one logarithmic cycle to 50 mm and $\Delta e = 0.2$ to 50 mm, with Casagrande’s construction drawn on it: the point T of maximum curvature, the horizontal and the tangent at T, their bisector, and the extended virgin compression line. The bisector cuts the virgin line at $\sigma'_c \approx 75$ kPa.

Approach. Plot $e$ against $\log \sigma'$, identify the flat recompression branch and the straight virgin branch, run Casagrande’s four-line construction at the knee between them for $\sigma'_c$, and take $C_c$ as the slope of the virgin branch.

(a) Plot the curve and read its shape. Plotted on semi-logarithmic paper the six points fall into two clearly different regimes. Between 25 and 50 kPa the curve is nearly flat — the soil is being recompressed along a path it has travelled before. From about 100 kPa onward it straightens into a much steeper line, the virgin compression line, on which the soil is carrying stresses it has never carried. The slope over each increment, $$C = \frac{e_i - e_{i+1}}{\log_{10}(\sigma'_{i+1}/\sigma'_i)},$$ makes the transition explicit:

Increment (kPa) 25–50 50–100 100–200 200–400 400–500

Slope $C$ 0.050 0.149 0.299 0.266 0.568

The first increment gives the recompression index, $C_r \approx 0.05$; the 100–400 kPa increments give a consistent virgin slope near 0.28; the last increment is discussed in the callout below.
(b) Locate the point of maximum curvature. Casagrande’s construction starts at the point T on the plotted curve where the radius of curvature is smallest. Working on the drawn scale (one log cycle to 50 mm, $\Delta e = 0.2$ to 50 mm) and computing the curvature $\kappa = |y''| / (1+y'^{2})^{3/2}$ of a smooth curve through the data, the maximum falls at $$\sigma'_T \approx 64\ \text{kPa}, \qquad e_T \approx 1.185 .$$ The tangent to the curve at T has slope $\mathrm{d}e/\mathrm{d}(\log\sigma') = -0.118$.
Draw the horizontal, the tangent and their bisector. Through T draw a horizontal line and the tangent found in Step 2; the angle between them is bisected. Because the bisector is defined by an angle on the drawing, it depends on the plotting scale — which is why the construction must always be performed on the plotted sheet and why $\sigma'_c$ is quoted to no better than the nearest 5 kPa.
Extend the virgin line and intersect it. Fitting the straight portion (the 100, 200 and 400 kPa points) by least squares gives the virgin compression line $$e = 1.7131 - 0.2824 \log_{10}\sigma' .$$ Extending it upward and to the left until it meets the bisector locates the preconsolidation pressure: $$\boxed{\sigma'_c \approx 75\ \text{kPa}}$$ (the construction as drawn returns 77 kPa; 75 kPa is the honest precision of a graphical method).
Cross-check the construction. A second, scale-independent estimate is the intersection of the two straight branches: the recompression line through the 25 and 50 kPa points, $e = 1.2796 - 0.0498\log_{10}\sigma'$, meets the virgin line at $\log_{10}\sigma' = 1.864$, i.e. $\sigma' = 73$ kPa. The two estimates agree to 5 %, so $\sigma'_c \approx 75$ kPa is secure. Since a preconsolidation pressure of 75 kPa corresponds to roughly 4 m of overburden, this is a lightly overconsolidated near-surface clay.
(c) Compression index. $C_c$ is the slope of the virgin branch, so it is taken from the straight portion only: $$C_c = \frac{e_1 - e_2}{\log_{10}(\sigma'_2/\sigma'_1)} = \frac{1.150 - 0.980}{\log_{10}(400/100)} = \frac{0.170}{0.6021} = \boxed{0.282}$$ The individual virgin increments bracket this tightly (0.299 over 100–200 kPa and 0.266 over 200–400 kPa), and the least-squares slope of Step 4 is 0.2824, so $C_c = 0.28$ is well determined. For comparison, Skempton’s correlation $C_c = 0.009(LL - 10)$ on the formula sheet would imply a liquid limit near 41 % — an entirely ordinary value for a clay of this compressibility, which is a useful sanity check on the fitted slope.

Increment (kPa)	25–50	50–100	100–200	200–400	400–500
Slope $C$	0.050	0.149	0.299	0.266	0.568

Quantity	Value
Point of maximum curvature T	$\sigma' \approx 64$ kPa, $e \approx 1.185$
Virgin compression line	$e = 1.713 - 0.282\log_{10}\sigma'$
(b) Preconsolidation pressure, Casagrande	$\sigma'_c \approx 75$ kPa
Cross-check: two-straight-line intersection	73 kPa
(c) Compression index (virgin branch, 100–400 kPa)	$C_c = 0.28$
Recompression index (25–50 kPa)	$C_r \approx 0.05$

Check — the last data point. The 400 → 500 kPa increment gives a slope of 0.568, more than twice the 0.266 of the increment before it. A virgin compression line cannot steepen like that, so either $e = 0.925$ or the pressure 500 kPa is a misprint (with $\sigma' = 800$ kPa the same void ratio would give a slope of 0.183, and with $e = 0.960$ at 500 kPa the slope would be 0.206). The point is plotted as printed but excluded from the fit for $C_c$; including it would raise the least-squares slope from 0.282 to 0.308 and would move $\sigma'_c$ only from 77 to 82 kPa, so no conclusion in this answer turns on it.

Topic. One-dimensional consolidation — interpreting the oedometer $e$–log $\sigma'$ curve, Casagrande’s construction for $\sigma'_c$, and the compression and recompression indices.

Key relations. $C_c = \Delta e / \log_{10}(\sigma'_f/\sigma'_0)$; $S_c = \dfrac{C_c H_0}{1+e_0}\log_{10}\dfrac{\sigma'_f}{\sigma'_0}$; $OCR = \sigma'_c/\sigma'_0$; $C_c \approx 0.009(LL-10)$ (Skempton).

Why this works. A clay remembers the largest effective stress it has ever carried. Below that stress it can only recover elastic strain, so the $e$–log $\sigma'$ path is flat; above it the fabric must reorganise and the path steepens onto the virgin line. The transition is a curve rather than a corner because sampling disturbance and the finite duration of each load increment smear it, and Casagrande’s construction is a repeatable rule for putting the corner back: the bisector at the point of tightest curvature splits the difference between the two branches, and where it meets the virgin line is taken as $\sigma'_c$. The two indices then do different jobs in a settlement calculation — $C_r$ for the part of the stress increase that stays below $\sigma'_c$, $C_c$ for the part above it.

Common pitfall. Computing $C_c$ from the first two points, or from the whole data set as a single average. The recompression branch is five times flatter than the virgin branch, so either mistake under-predicts settlement by a factor of two or more. The second pitfall is drawing the construction on a sheet whose vertical scale has been squashed to fit the page: the tangent, and therefore the bisector, rotate with the aspect ratio, so the graph must be plotted at a stated scale before any angle is bisected.

Source. Das & Sobhan, Principles of Geotechnical Engineering, 9th ed., §11.5 (normally consolidated and overconsolidated clays), §11.6 (Casagrande’s procedure), §11.7–11.8 ($C_c$, $C_r$ and settlement); Craig / Knappett, 8th ed., §4.2; CFEM 4th ed., §8.3.

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