Question 5 of 6: Preload Consolidation of a Saturated Clay Layer

Question 5: Preload Consolidation of a Saturated Clay Layer (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.

Find. (a) ultimate primary settlement $S_c$; (b) time $t_{90}$ for 90% of primary consolidation.

Approach. Compute the effective stress at the clay's mid-depth before and after the fill is placed (the fill is dry, so its full weight is effective stress; the clay was already saturated and unloaded before the fill), then apply the normally-consolidated settlement formula. For part (b), the impermeable rock base means single drainage, so the drainage path equals the full clay thickness.

1. Part (a) — unit weights. The dry fill has void ratio $e_{fill}=n_{fill}/(1-n_{fill})=0.10/0.90=0.1111$, so $$\gamma_{d,fill}=\frac{G_{s,fill}\gamma_w}{1+e_{fill}}=\frac{2.65\times9.81}{1.1111}=\boxed{23.40\ \text{kN/m}^3}.$$ The saturated clay's unit weight, and its buoyant (submerged) unit weight, are $$\begin{aligned} \gamma_{sat,clay}&=\frac{(G_{s,clay}+e_0)\gamma_w}{1+e_0}=\frac{(2.452+3.2)\times9.81}{4.2}=13.20\ \text{kN/m}^3, \\ \gamma_b'&=13.20-9.81=\boxed{3.39\ \text{kN/m}^3}. \end{aligned}$$
2. Effective stress at clay mid-depth, before and after the fill. Before the fill, the clay's own top surface was the (submerged, since it is stated to be fully saturated) ground surface, so at mid-depth ($z=H/2=2.5$ m into the clay), $$\sigma_0'=\gamma_b'\,z=3.39\times2.5=\boxed{8.48\ \text{kPa}}.$$ The 10 m of DRY fill sits entirely above the water table, so its full weight adds directly to the effective stress (no buoyancy reduction): $$\Delta\sigma'=\gamma_{d,fill}\,H_{fill}=23.40\times10=\boxed{234.0\ \text{kPa}}.$$ $$\sigma_1'=\sigma_0'+\Delta\sigma'=8.48+234.0=\boxed{242.5\ \text{kPa}}.$$
3. Apply the normally-consolidated settlement formula. $$\begin{aligned} S_c&=\frac{C_c H}{1+e_0}\log_{10}\!\left(\frac{\sigma_1'}{\sigma_0'}\right)=\frac{0.551\times5}{1+3.2}\log_{10}\!\left(\frac{242.5}{8.48}\right) \\ &=0.656\times1.457=\boxed{0.956\ \text{m}\ (\approx956\ \text{mm})}. \end{aligned}$$ Nearly a metre of settlement out of a 5 m clay layer is large but consistent with the data: the clay starts almost unloaded ($e_0=3.2$ is a very soft, high-void-ratio clay) and the stress ratio $\sigma_1'/\sigma_0'\approx29$ is huge — exactly the kind of large, rapid settlement a preload fill is designed to force to completion before construction, rather than let happen slowly under a permanent structure.
4. Part (b) — time for 90% consolidation. Impermeable rock underlies the clay, so drainage is SINGLE (upward only, into the permeable fill) and the drainage path equals the full layer thickness: $H_{dr}=H=5\ \text{m}=500$ cm. Using the standard time factor for $U=90\%$, $T_{v,90}=0.848$, $$t_{90}=\frac{T_{v,90}\,H_{dr}^2}{C_v}=\frac{0.848\times500^2}{0.002}=1.06\times10^{8}\ \text{s}=\boxed{3.36\ \text{years}}.$$