16-Civ-B3 Geotechnical Design · May 2015

Question 8 of 9: Consolidation Settlement of a Rectangular Footing over a Normally Consolidated Clay

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

Notes on this paper

Paper format. Professional Engineers Ontario / Engineers Canada National Examinations, May 2015 — 98-Civ-B3 Geotechnical Design. Three hours, OPEN BOOK, any non-communicating calculator. Section A carries five discussion questions of 7 marks each (answer any four); Section B carries four design questions of 24 marks each (answer any three); the examinable total is 4 × 7 + 3 × 24 = 100 marks. All nine questions are worked below, because the set is a study resource rather than a timed attempt.

Reference texts (98-Civ-B3 / 16-Civ-B3 Geotechnical Design).

B. M. Das, Principles of Foundation Engineering, 9th ed. — bearing capacity (Ch. 3), stress increase in a soil mass (Ch. 6), retaining walls (Ch. 8), pile foundations (Ch. 11), subsurface exploration (Ch. 2).
B. M. Das, Principles of Geotechnical Engineering, 9th ed. — consolidation (Ch. 11), shear strength (Ch. 12), lateral earth pressure (Ch. 13).
Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM), 4th ed. — the governing Canadian practice document for site investigation, bearing resistance, deep foundations and earth-retaining structures.
R. F. Craig, Craig's Soil Mechanics, 9th ed. — earth pressure theory, effective stress and slope stability.
D. P. Coduto, Foundation Design: Principles and Practices, 3rd ed. — in-situ testing, settlement of shallow and deep foundations.
M. J. Tomlinson & J. Woodward, Pile Design and Construction Practice, 6th ed. — shaft adhesion in clay, pile-group behaviour, load testing.

Sources of design charts and assumed values (page-1 Note 6). Note 6 of this paper requires the candidate to identify the source of every design chart used and of every value assumed in the absence of data. They are named where used and collected here:

Q6 — bearing capacity factors $N_c$, $N_q$, $N_{\gamma}$ from Das, Principles of Foundation Engineering, Table 3.3 (Prandtl–Reissner $N_q$, Vesic $N_{\gamma}=2(N_q+1)\tan\phi'$); shape factors after De Beer (1970) and depth factors after Hansen (1970), Das Table 3.4. Assumed: unit weight of water $\gamma_w = 9.81$ kN/m$^3$; general shear failure; the sand extends at least $2B$ below the base.
Q7 — adhesion factor $\alpha = 1.0$ for soft clay ($c_u \le 50$ kPa) from NAVFAC DM-7.2 Fig. 1 and Tomlinson & Woodward Table 4.6; $\lambda = 0.24$ at an embedded length of 12 m from Vijayvergiya & Focht (1972) as tabulated by Das, Table 11.7; bearing factor $N_c^{*}=9$ for $L/D \ge 4$ (Skempton). Assumed: pile spacing $s = 3d = 1.5$ m centre to centre, driven closed-end concrete piles, clay $\gamma_{sat} = 17$ kN/m$^3$ with the water table at ground level.
Q8 — compression index from the Skempton correlation $C_c = 0.009\,(LL-10)$, Das Principles of Geotechnical Engineering Eq. (11.42); $2{:}1$ stress distribution, Das Eq. (6.31); Boussinesq rectangular influence factor (Das Table 6.6) used as the cross-check.
Q9 — Coulomb active pressure coefficient, Das Principles of Foundation Engineering Eq. (8.13) with the wall friction angle prescribed by the question, $\delta = 0.6\phi' = 18^{\circ}$. Assumed: unit weight of reinforced concrete $\gamma_c = 24$ kN/m$^3$ (CSA A23.3 nominal); passive resistance in front of the toe neglected; the backfill surface is horizontal and carries no surcharge.

Question 8: Consolidation Settlement of a Rectangular Footing over a Normally Consolidated Clay (24 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.

Quantity	Symbol	Value
Footing plan dimensions	$B \times L$	1.5 m $\times$ 2.5 m
Column load	$Q$	120 kN
Depth of footing base	$D_f$	1.5 m
Upper sand, 0 to 1.5 m (moist)	$\gamma$	15 kN/m$^3$
Lower sand, 1.5 to 3.0 m (saturated)	$\gamma_{sat}$	18 kN/m$^3$
Groundwater table depth	$d_w$	1.5 m
Clay layer, 3.0 to 5.5 m	$H_c$	2.5 m
Clay water content / specific gravity / liquid limit	$w,\ G_s,\ LL$	35 per cent, 2.7, 38

Find. The primary consolidation settlement of the normally consolidated clay layer under the footing load, plus two alternative methods for the same calculation and the extra data each of them needs.

[Figure not reproduced: Figure 2 (redrawn) — footing at 1.5 m depth over 1.5 m of saturated sand and a 2.5 m normally consolidated clay layer. The settlement is computed at the top, middle and bottom of the clay and combined by the weighted average the paper prescribes. See the official exam paper.]

Approach. Establish the clay's index properties from $w$ and $G_s$, compute the existing effective overburden at mid-clay, distribute the foundation pressure to the top, middle and bottom of the clay with the 2:1 method, combine them with the weighted average given in the question, and apply the one-dimensional normally consolidated compression equation.

Contact pressure applied by the footing. The footing area is $A = B \times L = 1.5 \times 2.5 = 3.75$ m$^2$, so $$q = \frac{Q}{BL} = \frac{120}{3.75} = 32.0\ \text{kPa}$$ This is taken as the net pressure increase transmitted at foundation level (see the assumptions note below).
Index properties of the clay. The clay is saturated, so $e_0 = wG_s = 0.35(2.7) = 0.945$, and $$\gamma_{sat} = \frac{(G_s+e_0)\gamma_w}{1+e_0} = \frac{(2.7+0.945)(9.81)}{1.945} = 18.38\ \text{kN/m}^3$$ giving a submerged unit weight of $\gamma' = 18.38 - 9.81 = 8.57$ kN/m$^3$.
Existing effective overburden at the middle of the clay. The middle of the clay lies at $3.0 + 1.25 = 4.25$ m depth, with the water table at 1.5 m: $$\sigma'_0 = 1.5(15) + 1.5(18-9.81) + 1.25(8.57)$$ so $\sigma'_0 = 22.50 + 12.29 + 10.72 = 45.50$ kPa. The clay is normally consolidated, so $\sigma'_c = \sigma'_0$ and the whole stress increment lies on the virgin compression line.
Stress increase at the three levels by the 2:1 method. Measuring $z$ downward from the base of the footing, the top, middle and bottom of the clay are at $z = 1.5$, $2.75$ and $4.0$ m. The 2:1 distribution spreads the load over an area $(B+z)(L+z)$, so $\Delta\sigma' = qBL/[(B+z)(L+z)]$, and $$\Delta\sigma'_t = \frac{32(3.75)}{(3.0)(4.0)} = 10.00\ \text{kPa}$$ with $\Delta\sigma'_m = 120/[(4.25)(5.25)] = 5.38$ kPa and $\Delta\sigma'_b = 120/[(5.5)(6.5)] = 3.36$ kPa.
Weighted average increase prescribed by the paper. Applying Simpson's rule as the note directs, $$\Delta\sigma'_{av} = \frac{\Delta\sigma'_t + 4\Delta\sigma'_m + \Delta\sigma'_b}{6} = \frac{10.00 + 4(5.38) + 3.36}{6} = 5.81\ \text{kPa}$$ The increment is small compared with the existing overburden, which already tells us the settlement will be modest.
Compression index. No oedometer data is given, so the compression index is estimated from the liquid limit using Skempton's correlation for normally consolidated remoulded clays: $$C_c = 0.009\,(LL-10) = 0.009(38-10) = 0.252$$
Primary consolidation settlement. For a normally consolidated layer, $$S_c = \frac{C_c H_c}{1+e_0}\log_{10}\!\left(\frac{\sigma'_0+\Delta\sigma'_{av}}{\sigma'_0}\right) = \frac{0.252(2.5)}{1.945}\log_{10}\!\left(\frac{45.50+5.81}{45.50}\right)$$ The leading coefficient is $0.3239$ m and the logarithm is $\log_{10}(1.1277) = 0.05219$, so $$\boxed{S_c = 0.3239 \times 0.05219 = 0.0169\ \text{m} = 16.9\ \text{mm}}$$ This is comfortably within the 25 mm normally accepted for an isolated footing, so the clay layer is not a problem for this light load.
Cross-check on the stress distribution. Replacing the 2:1 approximation by the exact Boussinesq solution for a uniformly loaded rectangle (the footing divided into four quadrants, each contributing $I_c$ under its corner) gives $\Delta\sigma'_t = 14.38$ kPa, $\Delta\sigma'_m = 6.15$ kPa and $\Delta\sigma'_b = 3.23$ kPa, hence $\Delta\sigma'_{av} = 7.03$ kPa and $S_c = 20.2$ mm. The 2:1 method understates the stress immediately beneath the footing, so it is about 16 per cent unconservative here; both values round to "of the order of 20 mm", which is the useful engineering statement.

Quantity	Value
Contact pressure, $q$	32.0 kPa
Initial void ratio, $e_0 = wG_s$	0.945
Clay saturated / submerged unit weight	18.38 / 8.57 kN/m$^3$
Effective overburden at mid-clay, $\sigma'_0$	45.50 kPa
$\Delta\sigma'_t$ / $\Delta\sigma'_m$ / $\Delta\sigma'_b$ (2:1)	10.00 / 5.38 / 3.36 kPa
Weighted average, $\Delta\sigma'_{av}$	5.81 kPa
Compression index, $C_c = 0.009(LL-10)$	0.252
Primary consolidation settlement, $S_c$	16.9 mm
Same calculation with Boussinesq stresses	20.2 mm

Two other methods for the same settlement, and the data they need.

Method A — direct use of the laboratory $e$–$\log\sigma'$ curve, with the clay divided into sublayers. Instead of estimating $C_c$ from the liquid limit and treating the clay as a single layer, undisturbed samples are consolidated in an oedometer and the settlement is computed sublayer by sublayer as $S_c = \sum \Delta e_i H_i/(1+e_{0i})$, reading $\Delta e_i$ straight off the measured curve at each sublayer's own $\sigma'_0$ and $\Delta\sigma'$. This removes two approximations at once: the correlation for $C_c$, whose scatter is easily a factor of $\pm 30$ per cent, and the assumption that a single mid-layer stress represents a layer over which $\Delta\sigma'$ falls by a factor of three. Additional data required: undisturbed (Shelby tube or block) samples from at least three levels in the clay; a complete one-dimensional consolidation test on each, giving the $e$–$\log\sigma'$ curve, the preconsolidation pressure $\sigma'_c$ (Casagrande construction), the recompression index $C_r$ for the part of the increment below $\sigma'_c$, and the coefficient of consolidation $c_v$ so that the time–settlement curve can also be produced; plus measured $e_0$ and $\gamma$ for each sublayer.

Method B — the Skempton–Bjerrum method. The one-dimensional equation used above assumes zero lateral strain, which is true for a wide fill but not for a footing only 1.5 m wide over a 2.5 m clay layer: some of the applied stress is carried immediately by an undrained shear distortion, so the consolidation settlement is less than the oedometer prediction. Skempton and Bjerrum write $S_{c(SB)} = \mu\,S_{c(oed)}$, where the settlement ratio $\mu$ depends on the pore-pressure parameter $A$ and on the geometry ratio $H_c/B$; for a normally consolidated clay ($A \approx 0.5$ to $1.0$) under a square or circular footing $\mu$ typically lies between 0.6 and 1.0. The total settlement is then the sum of the immediate (undrained) settlement $S_i$, this corrected consolidation settlement, and secondary compression $S_s = C_{\alpha}H\log_{10}(t_2/t_1)/(1+e_p)$. Additional data required: the pore-pressure parameter $A$ at failure from CU triaxial tests with pore-pressure measurement on undisturbed samples; the undrained modulus $E_u$ (from the same triaxial tests or from a pressuremeter) and Poisson's ratio for the immediate settlement; the drainage geometry and $c_v$ for the rate; and the secondary compression index $C_{\alpha}$ from the long-term (24-hour-plus) portion of the oedometer curves.

A third possibility worth naming is the stress-path method of Lambe, in which the actual total-stress path followed by a representative element beneath the footing is reproduced in a triaxial cell and the strains are integrated over depth; it is the most realistic of the three but needs a large, specialised testing programme, including the initial $K_0$ state and the in-situ stress history.

Check — assumptions.

The 120 kN is treated as the net load increase at foundation level — that is, $q = 32.0$ kPa is taken as the pressure added to the ground rather than the gross contact pressure. If instead the gross pressure is reduced by the overburden removed during excavation, $q_{net} = 32.0 - 1.5(15) = 9.5$ kPa, and the same calculation returns $\Delta\sigma'_{av} = 1.73$ kPa and $S_c = 5.2$ mm. The figure dimensions no separate footing depth, so $D_f = 1.5$ m (the only dimensioned level) is used; the settlement lies between 5 mm and 20 mm on any reading and the design conclusion — that consolidation settlement is not critical — is unchanged.
The clay is normally consolidated as the figure states, so the entire increment is on the virgin line and no recompression branch applies.
The clay is fully saturated, giving $e_0 = wG_s$ exactly.
Only primary consolidation of the clay is computed; immediate settlement of the two sand layers and secondary compression are additional and would be assessed separately (elastic settlement of the sands is of the order of a few millimetres at 32 kPa).
$C_c$ from Skempton's $LL$ correlation, Das Principles of Geotechnical Engineering Eq. (11.42), as required by page-1 Note 6.

Topic. One-dimensional primary consolidation settlement of a normally consolidated clay beneath a shallow foundation.

Key relations. $S_c = \dfrac{C_cH_c}{1+e_0}\log_{10}\dfrac{\sigma'_0+\Delta\sigma'_{av}}{\sigma'_0}$ (normally consolidated); $e_0 = wG_s$ for a saturated soil; $\gamma_{sat} = (G_s+e_0)\gamma_w/(1+e_0)$; $C_c = 0.009(LL-10)$; 2:1 spread $\Delta\sigma' = qBL/[(B+z)(L+z)]$; Simpson weighting $\Delta\sigma'_{av} = (\Delta\sigma'_t+4\Delta\sigma'_m+\Delta\sigma'_b)/6$.

Why this works. Consolidation settlement is the volume change that follows the slow dissipation of the excess pore pressure set up by the load. Because the $e$–$\log\sigma'$ relationship of a normally consolidated clay is a straight line of slope $-C_c$, the change in void ratio is $C_c\log_{10}(\sigma'_f/\sigma'_0)$ and the strain is that divided by $(1+e_0)$. The logarithm is what makes the answer here so small: the increment of 5.81 kPa on an existing 45.5 kPa is only a 13 per cent stress rise, and the logarithm of 1.128 is 0.052. Simpson's weighting is used rather than a simple mid-layer value because $\Delta\sigma'$ falls steeply and non-linearly through the layer, and a parabola through three points captures that far better than one point does.

Common pitfall. Using total rather than effective stress for $\sigma'_0$, or forgetting to submerge the sand below the water table — both inflate $\sigma'_0$ and so reduce the predicted settlement, which is unconservative. Another frequent error is to compute $\Delta\sigma'$ at the mid-depth of the clay measured from the ground surface rather than from the base of the footing.

Source. Das, Principles of Geotechnical Engineering 9th ed. §11.10–11.13 and Eq. (11.42); Das, Principles of Foundation Engineering 9th ed. §6.10 (2:1 method) and Table 6.6 (Boussinesq influence factors); CFEM 4th ed. Ch. 11.

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