07-Str-A3 · May 2013
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
EGBC / Engineers Canada National Examination — Structural Engineering (legacy), 07-Str-A3 Geotechnical Materials and Analysis, May 2013. Three hours; closed book; one Casio or Sharp approved calculator; drawing instruments required; all charts and equations are supplied at the back of the paper. Total value 100 marks over six compulsory questions (20 + 10 + 10 + 20 + 20 + 20). Every question and every sub-part is solved in full below.
Reference texts: Das, Principles of Geotechnical Engineering, 9th ed. (Ch. 2 grain-size distribution and classification, Ch. 6 compaction, Ch. 7 permeability, Ch. 8 seepage and flow nets, Ch. 9 in-situ stresses and capillary rise, Ch. 10 stresses in a soil mass, Ch. 11 consolidation, Ch. 12 shear strength); Knappett & Craig, Craig’s Soil Mechanics, 8th ed. (flow nets, effective stress, heave and piping); Holtz, Kovacs & Sheahan, An Introduction to Geotechnical Engineering, 2nd ed.; Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM), 4th ed. — the governing Canadian practice document for settlement, excavation dewatering and factors of safety; ASTM D2487 (USCS classification) and ASTM D6913 (grain-size analysis).
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. Particle-size distribution curves for four soils A, B, C and D plotted as percentage smaller against log particle size, with the BS sieve sizes and the clay/silt/sand/gravel/cobble fraction boundaries marked.
Find. (i) the uniformly graded soil, its gradation indices and its alternative name; (ii) the soil with the highest saturated shear strength; (iii) the four soils ranked in ascending order of permeability.
[Figure not reproduced: Figure 2 from the May 2013 paper: particle-size distribution curves for soils A, B, C and D. See the official exam paper or the cited reference text.]
Approach. Read $D_{10}$, $D_{30}$ and $D_{60}$ off each curve, form the coefficients of uniformity and curvature, classify each soil against ASTM D2487, and use the fact that both shear strength and permeability are governed by the coarse skeleton and by the proportion of fines that clog its voids.
The uniform soil is soil B. Its curve is very nearly vertical: about 85 % of the material lies between roughly 0.2 mm and 0.6 mm (medium sand), all of it between about 0.08 mm and 2 mm, and it contains practically no fines and no gravel. Reading the curve at the three standard percentages gives $D_{10} \approx 0.22\ \text{mm}$, $D_{30} \approx 0.29\ \text{mm}$ and $D_{60} \approx 0.41\ \text{mm}$, so
$$C_u = \frac{D_{60}}{D_{10}} = \frac{0.41}{0.22} \approx 1.9 \qquad C_c = \frac{D_{30}^{\,2}}{D_{10}\,D_{60}} = \frac{0.29^2}{0.22 \times 0.41} \approx 0.9$$
Under ASTM D2487 (the Unified Soil Classification System) a sand can be called well graded only if $C_u \geq 6$ and $1 \leq C_c \leq 3$ simultaneously; for a gravel the uniformity requirement is $C_u \geq 4$. Anything that fails either test is classed as poorly graded. A genuinely uniform soil therefore has $C_u$ tending towards unity and in practice
$$\boxed{\;C_u < 4\ \text{(and commonly } C_u < 2\text{) for a uniformly graded soil}\;}$$
With $C_u \approx 1.9$ (and $C_c \approx 0.9$, just below the lower limit of 1) soil B fails both well-graded tests, the uniformity test by a wide margin. The other name used in the literature is poorly graded soil (the USCS group symbol SP), also described as single-sized or gap-free uniform sand. The key engineering features that follow from the near vertical curve are a low maximum dry density and a low relative density range (there are no small particles to fill the voids between the large ones), a high void ratio and hence a high permeability, poor stability as a compacted fill, and a marked susceptibility to liquefaction when saturated and loose.
Soil A will have the highest shear strength when saturated. It is by far the coarsest of the four: nothing is finer than about 0.1 mm, only some 25 % is finer than 2 mm, and the curve runs all the way out to 62 mm, so the material is a medium to coarse gravel with a sand matrix. Reading the curve gives $D_{10} \approx 0.46\ \text{mm}$, $D_{30} \approx 3.4\ \text{mm}$ and $D_{60} \approx 16.5\ \text{mm}$, hence
$$C_u = \frac{16.5}{0.46} \approx 36 \qquad C_c = \frac{3.4^2}{0.46 \times 16.5} \approx 1.5$$
which comfortably satisfies the well-graded gravel criteria ($C_u \geq 4$, $1 \leq C_c \leq 3$). Three effects combine. The wide, well-graded distribution lets small particles wedge into the voids between large ones, so the compacted skeleton is dense, highly interlocked and strongly dilatant, giving peak friction angles of roughly $38^\circ$ to $45^\circ$. The particles are bulky rather than platy, so there is no preferred shear plane and no residual strength collapse. Most importantly for the “saturated” qualifier, the permeability of a gravel is so high that any realistic rate of loading is fully drained: no excess pore pressure can build up, the effective stress stays equal to the total stress minus the static pore pressure, and the full $\tau_f = \sigma'\tan\phi'$ is available.
The others fall short for identifiable reasons. Soil D, a clayey silt, is governed under saturated undrained loading by a small undrained strength $s_u$ and will generate large positive excess pore pressures. Soil C carries roughly 35 % fines, so despite extending to 62 mm its behaviour is controlled by the fine matrix. Soil B is a clean sand with a lower friction angle than a gravel and, being uniform and potentially loose, is the one soil of the four at real risk of flow liquefaction under rapid saturated loading.
Permeability is set by the size of the smallest pore throats along a flow path, and the fine fraction of a soil controls those throats even when it is only a modest proportion of the mass. Ranking the four curves by fines content and effective size gives
$$\boxed{\;k_D \;<\; k_C \;<\; k_B \;<\; k_A\;}$$
| Rank | Soil | Character read from the curve | Indicative k (m/s) |
|---|---|---|---|
| 1 (lowest) | D | Finest of the four — about 30 % finer than 0.002 mm (clay size) and about 95 % finer than 63 µm; a clayey SILT with no coarse skeleton at all | 10−8 to 10−10 |
| 2 | C | Spans clay to coarse gravel but carries roughly 35 % fines, which fill the voids between the coarse particles and control the throats; a well-graded but dirty soil | 10−7 to 10−8 |
| 3 | B | Clean uniform sand, no fines; Hazen with D10 = 0.22 mm gives k ≈ 1.0 × 0.22² = 0.048 cm/s | ≈ 4.8 × 10−4 |
| 4 (highest) | A | Gravel with only a small sand tail; Hazen with D10 = 0.46 mm gives k ≈ 1.0 × 0.46² = 0.21 cm/s | ≈ 2.1 × 10−3 |
Note that C sits below B even though C contains far coarser particles than B does. That is the single most instructive point in this part: a well-graded soil is always less permeable than a uniform soil of comparable maximum size, because the finer fractions occupy the very voids that would otherwise conduct the flow. It is the reason filter design is specified on $D_{15}$ and $D_{85}$ ratios rather than on mean size, and the reason a well-graded granular fill makes a good low-permeability road base but a poor drainage layer.
| Item | Answer |
|---|---|
| Uniform soil | Soil B |
| Gradation indices of soil B | Cu ≈ 1.9, Cc ≈ 0.9 |
| ASTM criterion for uniform (poorly graded) | Cu < 6 for sand (< 4 for gravel); uniform soils commonly Cu < 2 |
| Other name for a uniform soil | Poorly graded (USCS SP; also “single-sized”) |
| Highest saturated shear strength | Soil A (Cu ≈ 36, Cc ≈ 1.5, free draining) |
| Permeability, ascending | D < C < B < A |
