22-Agric-A2 Soil Physics and Mechanics · December 2019
Question 2 of 6: Consolidation Settlement Beneath a Square Footing on Normally-Consolidated Clay
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. 04-Agric-A2 Soil Physics & Mechanics,
National Exams December 2019 — a three-hour open-book examination;
any non-communicating calculator is permitted. The cover page states that five (5)
questions constitute a complete exam paper and that only the first five as they appear
in the answer book are marked, that each question is of equal value, and that some
questions require a written answer whose clarity and organization matter for marks. All
six printed questions are worked here, because the set is a study resource rather than a
timed attempt; on exam day a candidate submits only the first five, in order.
Check: the printed initial void ratio (e0 = 5.5) is unusually high for a mineral clay, but it is what the paper states. It is carried through the
calculation as given — it drives an unusually low initial effective stress and,
correspondingly, a very large computed settlement (see the discussion after Step 5). If the
original exam intended e0 = 0.55 or 1.5, both σ′0 and Sc
below would scale accordingly; the method is unaffected.
Given.
Quantity
Value
Clay layer thickness, H
12 m
Specific gravity of solids, Gs
2.65
Initial void ratio, e0
5.5
Compression index, Cc
0.45
Footing, B × L
2 m × 2 m
Footing load, Q
10,000 kN
Find. σ′0 and σ′f at the
clay midpoint (a); ultimate primary consolidation Sc (b); Δe (c).
Approach. Because the clay is normally consolidated, the entire stress
increase compresses along the virgin Cc line (Cr is not needed), so the
governing steps are the effective stress profile before and after loading at the layer
midpoint, the Boussinesq stress increase from the square footing at that depth, and the
one-dimensional consolidation formula.
Effective stress at the clay midpoint before loading. Depth to the
midpoint of the 12 m layer, z = 6 m; saturated and buoyant unit weights from e0:
$$\begin{aligned}
\gamma_{sat}&=\frac{(G_s+e_0)}{1+e_0}\gamma_w=\frac{(2.65+5.5)}{6.5}(9.81)=12.30\ \text{kN/m}^3\\
\gamma'&=\gamma_{sat}-\gamma_w=2.49\ \text{kN/m}^3
\end{aligned}$$
$$\sigma_0'=\gamma' z=2.49(6)=\boxed{14.94\ \text{kPa}}$$
Stress increase at the midpoint from the footing (Boussinesq, four-quadrant
corner method). Applied pressure $q=Q/(BL)=10{,}000/4=2500\ \text{kPa}$; at the
centre, split the 2 m×2 m footing into four 1 m×1 m
quadrants with $m=n=(B/2)/z=1/6=0.1667$, giving an influence factor
$I\approx0.0127$ per quadrant:
$$\Delta\sigma=4qI=4(2500)(0.0127)=\boxed{126.8\ \text{kPa}}$$
(the simpler 2:1 approximate method gives
$\Delta\sigma=qBL/[(B+z)(L+z)]=2500(4)/(8)(8)=156.3\ \text{kPa}$, of the same order —
the Boussinesq value is carried forward as the more rigorous of the two.)
Effective stress after loading.
$$\sigma_f'=\sigma_0'+\Delta\sigma=14.94+126.8=\boxed{141.7\ \text{kPa}}$$
b) Ultimate primary consolidation. Normally consolidated clay
compresses entirely along the Cc line:
$$\begin{aligned}
S_c&=\frac{C_cH}{1+e_0}\log_{10}\!\left(\frac{\sigma_f'}{\sigma_0'}\right)\\
&=\frac{0.45(12)}{6.5}\log_{10}\!\left(\frac{141.7}{14.94}\right)\\
&=0.8308(0.9770)=\boxed{0.812\ \text{m}\ (812\ \text{mm})}
\end{aligned}$$
c) Change in void ratio.
$$\Delta e=C_c\log_{10}\!\left(\frac{\sigma_f'}{\sigma_0'}\right)=0.45(0.9770)=\boxed{0.440}$$
The settlement in Step 4 is large (≈6.8% strain over the 12 m layer) precisely
because the flagged e0 = 5.5 makes σ′0 unusually small, so
the footing load increases the effective stress nearly ten-fold at the layer midpoint; a more
typical mineral-clay void ratio (e0 ≈ 0.5–1.5) would give a
proportionally larger σ′0 and a much smaller, more conventional
settlement, as flagged in the check callout above.
