Question 4 of 6: Consolidation Settlement and Heave

Question 4: Consolidation Settlement and Heave (25 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.

Profile: 0–4 m sand, 4–10 m clay (6 m thick), water table at ground surface; $\gamma_{sat}=18\ \text{kN/m}^3$ for both soils. Fill: 4 m at $\gamma=20\ \text{kN/m}^3$ over an extensive area.

Find. (a) the ultimate consolidation settlement of the clay; (b) the eventual heave if the fill is later removed.

Approach. Compute the in-situ effective stress at mid-clay, add the (depth-uniform, because the fill is extensive) stress increment, read the void ratios off the virgin loading line for the settlement, then follow the swelling line for the rebound on unloading.

1. Initial effective stress at mid-clay. With the water table at the surface the whole profile is submerged, so use the buoyant unit weight $\gamma'=\gamma_{sat}-\gamma_w=18-9.81=8.19\ \text{kN/m}^3$. Mid-clay is at $z=4+3=7$ m: $$\sigma_0' = \gamma' z = 8.19\times 7 = 57.3\ \text{kPa}.$$
2. Stress increment from the fill. Placed over an extensive area, the 4 m fill acts as a wide surcharge whose vertical stress does not dissipate with depth: $$\Delta\sigma' = 4\times 20 = 80\ \text{kPa},\qquad \sigma_1' = 57.3+80 = 137.3\ \text{kPa}.$$
3. Void ratios and compression index from the curve. Interpolating the loading (virgin) data on a log scale, $e_0=1.211$ at $\sigma_0'$ and $e_1=1.117$ at $\sigma_1'$. The virgin slope is $$C_c=\frac{e\,(107)-e\,(214)}{\log(214/107)}=\frac{1.144-1.068}{\log 2}=0.253.$$
4. Settlement (a). Using the change in void ratio directly, $$s_c = H\,\frac{e_0-e_1}{1+e_0}= 6000\times\frac{1.211-1.117}{1+1.211}= \boxed{255\ \text{mm}}.$$ The compression-index form $s_c=H\,\dfrac{C_c}{1+e_0}\log\dfrac{\sigma_1'}{\sigma_0'}=6000\times\dfrac{0.253}{2.211}\log\dfrac{137.3}{57.3}=260$ mm agrees within the graph-reading tolerance.
5. Heave on unloading (b). Once consolidation under the fill is complete the clay sits at $\sigma_1'=137.3$ kPa on the virgin line ($e_1=1.117$). Removing the fill unloads it back to $\sigma_0'=57.3$ kPa along the swelling line, whose slope from the rebound data is $$C_s=\frac{1.024-0.994}{\log(429/54)}=0.0333.$$ The rebound (upward heave) is $$s_h = H\,\frac{C_s}{1+e_1}\log\frac{\sigma_1'}{\sigma_0'} = 6000\times\frac{0.0333}{2.117}\log\frac{137.3}{57.3}= \boxed{36\ \text{mm}}.$$