07-Str-B1 · May 2015

Question 8 of 9: Consolidation Settlement of a 1.5 m × 2.5 m Footing Over a Normally Consolidated Clay

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

Notes on this paper

National Examinations — May 2015 — 07-Str-B1 Geotechnical Design. Three-hour, OPEN-BOOK exam; any non-communicating calculator permitted (the candidate must record its make and model). Format: Section A carries five discussion questions of 7 marks each, of which any FOUR are to be answered; Section B carries four design problems of 24 marks each, of which any THREE are to be answered — a marked total of 100. The paper instructs candidates to state any interpretive assumptions, to identify the source of every design chart and assumed value, and to exercise sound engineering judgment where data are absent. All nine printed questions are worked below, because the set is intended as a study resource.

Reference texts: Das, B.M., Principles of Foundation Engineering (9th ed., Cengage) — general bearing-capacity equation, pile and pile-group capacity, consolidation settlement of footings, retaining walls; Das, B.M., Principles of Geotechnical Engineering (9th ed., Cengage) — lateral earth pressure, effective stress, consolidation theory; Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM, 4th ed., 2006) — Canadian practice for site investigation, SPT/CPT interpretation, pile design and tolerable settlement; Craig, R.F. / Knappett, J.A., Craig's Soil Mechanics (8th ed., CRC Press) — shear strength and earth-pressure theory; Duncan, J.M., Wright, S.G. & Brandon, T.L., Soil Strength and Slope Stability (2nd ed., Wiley) — fully softened and residual strengths for fissured and expansive clays; Fredlund, D.G., Rahardjo, H. & Fredlund, M.D., Unsaturated Soil Mechanics in Engineering Practice (Wiley) — swelling soils and matric suction.

Note — Figure 2 is printed over a coarse halftone. The soil-property annotations inside the photograph-style Figure 2 (Question 8) are printed over a coarse dot screen. The values used below are read from the printed figure and are: upper sand $\gamma = 15\ \text{kN/m}^3$ over 1.5 m, lower sand $\gamma_{sat} = 18\ \text{kN/m}^3$ over 1.5 m, normally consolidated clay 2.5 m thick with $w = 35\%$ and $LL = 48$, over sand; groundwater table at the underside of the footing.

Assumptions declared once, applied throughout. $\gamma_w = 9.81\ \text{kN/m}^3$; reinforced concrete $\gamma_c = 24\ \text{kN/m}^3$; specific gravity of soil solids $G_s = 2.70$ where a void ratio must be back-figured from water content; loads are vertical and concentric unless stated. Every assumption that changes a numerical answer is repeated in the question where it is used.

Question 8: Consolidation Settlement of a 1.5 m × 2.5 m 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. A rectangular footing carrying a column load of 120 kN, founded at the base of a 1.5 m upper sand layer, with the groundwater table at founding level, a second 1.5 m sand layer beneath, and a 2.5 m normally consolidated clay layer below that (Figure 2).

Given data and assumed values — Question 8
Quantity	Symbol	Value
Footing dimensions	$B \times L$	1.5 m × 2.5 m
Column load	$Q$	120 kN
Founding depth (= GWT)	$D_f$	1.5 m
Upper sand (above GWT)	$H_1,\ \gamma$	1.5 m, 15 kN/m³
Lower sand (below GWT)	$H_2,\ \gamma_{sat}$	1.5 m, 18 kN/m³
Clay layer, normally consolidated	$H_c$	2.5 m
Clay water content, liquid limit	$w,\ LL$	35 %, 48
Specific gravity of solids (assumed)	$G_s$	2.70

Find. The primary consolidation settlement of the clay layer, plus two alternative methods for the same calculation and the extra data each would require.

[Figure not reproduced: Figure 8.1 — Figure 2 redrawn. The footing base coincides with the groundwater table at 1.5 m; the 2:1 dispersion cone is used to carry the footing pressure down to the top, middle and base of the clay layer. See the official exam paper.]

Approach. Establish the clay's index properties ($e_0$, $\gamma_{sat}$, $C_c$) from $w$ and $LL$, compute the existing effective overburden at mid-clay, spread the footing load to the three levels in the clay by the 2:1 method, average with the weighting the paper supplies, and apply the normally consolidated one-dimensional consolidation equation.

Assumptions. (i) $G_s = 2.70$ and the clay is fully saturated, so $e_0 = wG_s$. (ii) $C_c$ is estimated from Terzaghi and Peck's empirical relation $C_c = 0.009(LL - 10)$, valid for normally consolidated inorganic clays of moderate sensitivity. (iii) The clay is normally consolidated ($\sigma'_c = \sigma'_0$), as the figure states, so the virgin compression line applies throughout and no recompression index is needed. (iv) $Q = 120$ kN is the net load applied at founding level. (v) The 2:1 approximate method is used for stress distribution, as the paper permits. (vi) Only primary consolidation of the clay is computed; elastic settlement of the sands and secondary compression are excluded.

Void ratio and unit weight of the clay. For a saturated soil, $e_0 = wG_s$, and the saturated unit weight follows from the phase relations: $$e_0 = 0.35(2.70) = 0.945 \qquad \gamma_{sat} = \frac{(G_s + e_0)\gamma_w}{1+e_0} = \frac{(2.70+0.945)(9.81)}{1.945} = 18.38\ \text{kN/m}^3$$ so the buoyant unit weight is $\gamma' = 18.38 - 9.81 = 8.57$ kN/m³.
Compression index. From the liquid limit, $$C_c = 0.009(LL - 10) = 0.009(48-10) = 0.342$$
Existing effective overburden at the middle of the clay layer (depth 4.25 m, that is 1.5 m of moist sand, 1.5 m of submerged sand and 1.25 m of submerged clay): $$\sigma'_0 = 1.5(15) + 1.5(18-9.81) + 1.25(8.57)$$ $$\sigma'_0 = 22.50 + 12.29 + 10.72 = \boxed{45.50\ \text{kPa}}$$
Contact pressure and depths to the three levels. The footing pressure is $q = Q/(BL) = 120/(1.5 \times 2.5) = 32.0$ kPa. Measured below the footing base, the top, middle and base of the clay lie at $$z_t = 3.0 - 1.5 = 1.50\ \text{m}, \qquad z_m = 2.75\ \text{m}, \qquad z_b = 4.00\ \text{m}$$
Stress increase by the 2:1 method. The load is assumed to spread over an area that grows by one horizontal in two vertical, so $\Delta\sigma' = Q/[(B+z)(L+z)]$: $$\Delta\sigma'_t = \frac{120}{(1.5+1.5)(2.5+1.5)} = \frac{120}{12.00} = 10.00\ \text{kPa}$$ $$\Delta\sigma'_m = \frac{120}{(4.25)(5.25)} = \frac{120}{22.31} = 5.38\ \text{kPa} \qquad \Delta\sigma'_b = \frac{120}{(5.5)(6.5)} = \frac{120}{35.75} = 3.36\ \text{kPa}$$
Weighted average increase, using the relation supplied with the paper. $$\Delta\sigma'_{av} = \frac{\Delta\sigma'_t + 4\Delta\sigma'_m + \Delta\sigma'_b}{6} = \frac{10.00 + 4(5.38) + 3.36}{6} = \frac{34.87}{6} = \boxed{5.81\ \text{kPa}}$$ This is Simpson's rule applied over the layer, and it is more accurate than simply taking the mid-depth value.
Primary consolidation settlement. For a normally consolidated clay, $$S_c = \frac{C_cH_c}{1+e_0}\log_{10}\!\left(\frac{\sigma'_0 + \Delta\sigma'_{av}}{\sigma'_0}\right) = \frac{0.342(2.5)}{1.945}\log_{10}\!\left(\frac{45.50+5.81}{45.50}\right)$$ $$S_c = 0.4396 \times \log_{10}(1.1277) = 0.4396(0.05221) = 0.02295\ \text{m}$$ $$\boxed{S_c = 22.9\ \text{mm}}$$

Twenty-three millimetres of consolidation settlement is tolerable for an ordinary structure; CFEM's usual limit for total settlement of a footing on clay is 25 to 50 mm, with the differential settlement between adjacent footings the more important criterion. Note also that this settlement will take years to develop: the rate follows from $T_v = c_vt/H_{dr}^2$, and with drainage to sand on both faces $H_{dr} = 1.25$ m.

Two other methods for the same consolidation settlement.

The $\Delta e$ method read directly from the laboratory $e$–$\log\sigma'$ curve. Instead of using an empirical $C_c$, plot the oedometer curve for an undisturbed sample from mid-clay, enter it at $\sigma'_0$, read the void ratio $e_0$, enter again at $\sigma'_0 + \Delta\sigma'_{av}$, read $e_1$, and compute $$S_c = \frac{\Delta e}{1+e_0}H_c$$ This is the fundamental form of which the $C_c$ equation is a special case, and it is the only method that handles a non-linear or overconsolidated curve correctly. Additional data required: a one-dimensional oedometer (consolidation) test on a good-quality undisturbed sample from the clay — which in turn means thin-walled Shelby tube or piston sampling — giving the complete $e$–$\log\sigma'$ curve, the preconsolidation pressure $\sigma'_c$ (by the Casagrande construction), the recompression index $C_r$ for the part of the loading below $\sigma'_c$, and the in-situ void ratio measured on the same specimen rather than inferred from $wG_s$.
The coefficient of volume compressibility ($m_v$) method. Over a modest stress increment the compressibility can be treated as linear, so $$S_c = m_v\,\Delta\sigma'_{av}\,H_c \qquad \text{with}\quad m_v = \frac{1}{1+e_0}\cdot\frac{\Delta e}{\Delta\sigma'}$$ This is the standard British and European practice and is convenient when the stress increment is small, as it is here. Additional data required: $m_v$ from the same oedometer test, but evaluated over the specific stress range $\sigma'_0$ to $\sigma'_0 + \Delta\sigma'$ rather than over the whole curve — $m_v$ is strongly stress-dependent, so the increment must match the field increment. If the layer is thick, $m_v$ is needed for several sub-layers.

A third possibility worth naming is the Skempton–Bjerrum approach, which corrects the one-dimensional result for the three-dimensional stress path beneath a footing by a factor $\mu$ that depends on the pore-pressure parameter $A$ and the layer geometry; it requires $A$ from a consolidated-undrained triaxial test with pore-pressure measurement. For a small footing on a relatively thin clay layer $\mu$ is usually between 0.6 and 1.0, so the one-dimensional answer above is conservative.

Final results — Question 8
Quantity	Symbol	Value
Initial void ratio of clay	$e_0$	0.945
Saturated / buoyant unit weight of clay	$\gamma_{sat}$ / $\gamma'$	18.38 / 8.57 kN/m³
Compression index	$C_c$	0.342
Effective overburden at mid-clay	$\sigma'_0$	45.50 kPa
Contact pressure	$q$	32.0 kPa
Stress increase, top / middle / base	$\Delta\sigma'_t,\Delta\sigma'_m,\Delta\sigma'_b$	10.00 / 5.38 / 3.36 kPa
Weighted average stress increase	$\Delta\sigma'_{av}$	5.81 kPa
Primary consolidation settlement	$S_c$	22.9 mm

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

Key relations. $e_0 = wG_s$ (saturated); $\gamma_{sat} = (G_s+e_0)\gamma_w/(1+e_0)$; $C_c = 0.009(LL-10)$; 2:1 stress spread $\Delta\sigma' = Q/[(B+z)(L+z)]$; Simpson weighting $\Delta\sigma'_{av} = (\Delta\sigma'_t + 4\Delta\sigma'_m + \Delta\sigma'_b)/6$; $S_c = \frac{C_cH_c}{1+e_0}\log_{10}\frac{\sigma'_0+\Delta\sigma'_{av}}{\sigma'_0}$; alternatives $S_c = \Delta e\,H_c/(1+e_0)$ and $S_c = m_v\Delta\sigma'H_c$.

Why this works. Saturated clay compresses because water is squeezed out of the voids, and the volume change is a nearly linear function of the logarithm of effective stress along the virgin compression line. The settlement is therefore the layer thickness multiplied by the volumetric strain, which is $\Delta e/(1+e_0)$. Because the stress increment decays rapidly with depth, using a single mid-depth value would misrepresent a thick layer; the parabolic (Simpson) weighting supplied with the paper integrates the increment over the layer with three ordinates.

Common pitfall. Adding the stress increase to a total rather than an effective overburden stress, which here would put $\sigma'_0$ near 68 kPa and understate the settlement by about a quarter; and forgetting that the water table sits at founding level, so both the lower sand and the clay must be entered at their buoyant unit weights. A third slip is using the footing pressure $q = 32$ kPa in the 2:1 formula instead of the load $Q$ — the formula already divides by the spread area.

Source. Das, Principles of Foundation Engineering, 9th ed., §5.9–§5.11 (consolidation settlement of shallow foundations; 2:1 method; average stress increase); Das, Principles of Geotechnical Engineering, 9th ed., Ch. 11 (Compressibility of Soil); CFEM 4th ed., §11 (Settlement).

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