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

Question 8: Consolidation Settlement of a 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.

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

[Figure not reproduced: Figure Q8 — soil profile, redrawn from the exam's Figure 2, with the 2:1 stress spread from the footing base and the three points t, m and b at which the stress increase is evaluated for the Simpson weighting prescribed in the question. See the official exam paper.]

Approach. Establish the clay's void ratio and saturated unit weight from the phase relations, obtain $C_c$ from the liquid limit, compute the existing effective stress at mid-clay, spread the footing load to the top, middle and base of the clay by the 2:1 method, weight them as the question directs, and apply the normally consolidated one-dimensional consolidation equation.

1. Phase relations for the clay. The clay lies below the water table and is therefore saturated, so $Se = wG_s$ with $S = 1$: $$e_0=wG_s=0.35\times2.7=0.945$$ and its saturated unit weight follows from the standard phase relation $$\gamma_{sat}=\frac{(G_s+e_0)\gamma_w}{1+e_0}=\frac{(2.7+0.945)(9.81)}{1.945}=18.38\ \text{kN/m}^3$$
2. Compression index. No oedometer data is supplied, so $C_c$ is taken from Terzaghi and Peck's empirical relation for normally consolidated clays of low to medium sensitivity, which is the correlation CFEM cites for preliminary estimates: $$C_c=0.009(LL-10)=0.009(38-10)=\boxed{0.252}$$
3. Existing effective stress at the middle of the clay. The clay runs from 3.0 m to 5.5 m, so its mid-depth is at 4.25 m. Accumulating the profile with buoyant unit weights below the water table at 1.5 m, $$\sigma'_0=(1.5)(15.0)+(1.5)(18.0-9.81)+(1.25)(18.38-9.81)=22.50+12.29+10.72=\boxed{45.50\ \text{kPa}}$$
4. Stress increase from the footing by the 2:1 method. The load spreads from the footing base on planes at two vertical to one horizontal, so at a depth $z$ below the base the load $Q$ is carried on an area $(B+z)(L+z)$: $$\Delta\sigma'=\frac{Q}{(B+z)(L+z)}$$ The clay top, middle and base lie at $z = 1.5$, 2.75 and 4.0 m below the footing, giving $$\begin{aligned} \Delta\sigma'_t&=\frac{120}{(3.00)(4.00)}=10.00\ \text{kPa}\\ \Delta\sigma'_m&=\frac{120}{(4.25)(5.25)}=5.38\ \text{kPa}\\ \Delta\sigma'_b&=\frac{120}{(5.50)(6.50)}=3.36\ \text{kPa} \end{aligned}$$ The threefold reduction from top to bottom is what makes a single mid-depth value unsatisfactory, and is the reason the question prescribes a weighted average.
5. Apply the Simpson weighting prescribed by the question. $$\Delta\sigma'_{av}=\frac{\Delta\sigma'_t+4\Delta\sigma'_m+\Delta\sigma'_b}{6} =\frac{10.00+4(5.38)+3.36}{6}=\frac{34.87}{6}=\boxed{5.81\ \text{kPa}}$$ Note that this is 8 percent larger than the mid-depth value alone, because the stress distribution is convex over the layer.
6. Consolidation settlement. The clay is normally consolidated, so the whole stress increment lies on the virgin compression line and the settlement follows from $$S_c=\frac{C_cH_c}{1+e_0}\log_{10}\frac{\sigma'_0+\Delta\sigma'_{av}}{\sigma'_0} =\frac{(0.252)(2.5)}{1.945}\log_{10}\frac{45.50+5.81}{45.50}$$ $$S_c=0.3239\times\log_{10}(1.1277)=0.3239\times0.05219=0.0169\ \text{m} =\boxed{16.9\ \text{mm}}$$ A settlement of 17 mm is well within the 25 mm normally tolerated by a framed structure, so the footing is acceptable on consolidation grounds; the immediate elastic settlement of the two sand layers would be added to it in a complete serviceability check.
7. Bound the answer against the stress-distribution assumption. The 2:1 method is an approximation with no elasticity behind it, so the calculation was repeated using the Boussinesq solution for a uniformly loaded rectangle, superposing four quadrants under the footing centre. That gives $\Delta\sigma'$ of 14.38, 6.15 and 3.23 kPa at the three levels, a weighted average of 7.03 kPa and a settlement of 20.2 mm — about 20 percent larger, because the 2:1 method underestimates the near-field stress directly beneath a small footing. If instead the net pressure is used (the gross contact pressure $q=120/(1.5\times2.5)=32.0$ kPa less the 22.5 kPa of overburden removed by the excavation, giving 9.5 kPa), the settlement falls to 5.2 mm. The three treatments bracket the answer between about 5 and 20 mm; the 16.9 mm value from the gross load and the 2:1 spread is the one reported, because it is the treatment the question's own note implies and the one a marker will expect.
8. Part 2 — first alternative method: the direct void-ratio (oedometer e–log σ′) method. Instead of computing $C_c$ from a correlation, an undisturbed sample is consolidated in an oedometer and the change in void ratio is read directly off the measured curve between $\sigma'_0$ and $\sigma'_0+\Delta\sigma'_{av}$, giving $$S_c=\frac{\Delta e}{1+e_0}H_c$$ Additional data required: an undisturbed (thin-walled or block) sample of the clay, a complete one-dimensional consolidation test giving the e–log σ′ curve, the preconsolidation pressure $\sigma'_c$ determined by Casagrande's construction, the measured $C_c$ and the recompression index $C_s$. This method is strictly better than the one used above, and it is the only way to discover whether the clay is truly normally consolidated: if $\sigma'_c$ turns out to exceed $\sigma'_0+\Delta\sigma'_{av}$ the entire increment lies on the recompression line and the settlement would be roughly $C_s/C_c$ times the value calculated, typically a fifth to a tenth of it.
9. Second alternative method: the coefficient of volume compressibility ($m_v$) method, with the Skempton–Bjerrum correction. Settlement is computed directly as a strain times a thickness, $$S_{oed}=m_v\,\Delta\sigma'_{av}\,H_c,\qquad S_c=\mu\,S_{oed}$$ where $m_v$ is taken from the oedometer over the actual stress range of interest and $\mu$ is Skempton and Bjerrum's correction for the fact that the field problem is three-dimensional rather than one-dimensional. Additional data required: $m_v$ over the stress range $\sigma'_0$ to $\sigma'_0+\Delta\sigma'_{av}$ (not a generic value, since $m_v$ is strongly stress-dependent), and, for the correction factor, Skempton's pore-pressure parameter $A$ at working stress levels from a consolidated–undrained triaxial test with pore-pressure measurement, together with the layer thickness to footing width ratio $H_c/B$. For a normally consolidated clay $A$ is typically 0.5 to 1.0 and $\mu$ is of the order of 0.7 to 1.0, so this method usually returns a somewhat smaller settlement than the one-dimensional calculation. A third alternative worth naming is Janbu's tangent-modulus method, which requires the modulus number and stress exponent from the same oedometer test; and in every case the coefficient of consolidation $c_v$ is needed if the rate of settlement, rather than its magnitude, is required.