18-Env-A3 Geotechnical and Hydrogeological Engineering · December 2016
Question 6 of 6: Stress Distribution and Consolidation Settlement Under a Large Foundation
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2016 — 04-Env-A3 / Geotechnical & Hydrogeological Engineering. 3 hours duration; open book exam, any non-communicating calculator permitted. FIVE (5) questions constitute a complete exam paper (the first five as they appear in the answer book are marked, 20 marks each, 100 marks total); all six printed questions are solved below for completeness.
Reference texts. Braja M. Das, Principles of Geotechnical Engineering (9th ed.) — weight–volume relations, permeability, seepage/flow nets, stress distribution, consolidation and lateral earth pressure chapters; Craig & Knappett, Craig's Soil Mechanics (8th ed.) — cross-reference for seepage, flow nets and anchored sheet-pile wall design.
Question 6: Stress Distribution and Consolidation Settlement Under a Large Foundation (20 marks)
Given. A 10 m × 20 m raft applying a net pressure of 200 kPa directly onto a 10 m thick clay layer over gravel.
Given data
Quantity
Symbol
Value
Foundation width / length
$B,\,L$
10 m / 20 m
Net applied pressure
$q$
200 kPa
Clay thickness
$H$
10 m
Modulus of volume compressibility
$m_v$
$5\times10^{-5}$ m²/kN
Find. (a) the vertical stress increase $\Delta\sigma_z$ at 2, 4, 6, 8 and 10 m depth below the loaded ground surface, immediately on loading; (b) the primary consolidation settlement of the foundation.
Figure 5 (left) and the resulting stress-increase profile with depth under the centre of the raft (right).
Approach. "Immediately after" a load is applied to a saturated clay, none of it has yet transferred to effective stress (Terzaghi's theory: $\Delta u=\Delta\sigma$ at $t=0$) — so the quantity actually being tested here is the ELASTIC (Boussinesq) vertical stress increase the new load imposes at each depth, i.e. the stress increment that will eventually become effective as consolidation proceeds; the five requested depths at 2 m spacing are exactly the ordinates needed to sum the settlement contribution of five equal 2 m sub-layers spanning the full 10 m clay. $\Delta\sigma_z$ under the raft's CENTRE is found by superposing four $5\,\text{m}\times10\,\text{m}$ corner solutions (Newmark's formula), and $S_c=\sum m_v\,\Delta\sigma_z\,\Delta z$ over the five sub-layers.
Part (a) — stress increase at each depth (Boussinesq, under the raft centre). With $m=(L/2)/z=10/z$, $n=(B/2)/z=5/z$, $\Delta\sigma_z=4qI(m,n)$:
$z$ (m)
2
4
6
8
10
$\Delta\sigma_z$ (kPa)
195.1
174.1
145.5
118.5
96.1
The stress falls from 98% of $q$ near the surface to 48% of $q$ at the base of the clay, as expected for a foundation whose plan dimensions are comparable to the clay thickness.
Part (b) — sum the five 2 m sub-layer contributions. Dividing the 10 m clay into five equal 2 m sub-layers and using the tabulated stress at the base of each as representative,
$$S_c=m_v\,\Delta z\sum_{i=1}^{5}\Delta\sigma_i=5\times10^{-5}\times2\times(195.1+174.1+145.5+118.5+96.1)$$
$$=5\times10^{-5}\times2\times729.4=\boxed{72.9\ \text{mm}}.$$
Check: part (a)'s boxed values are the elastic stress INCREASE the 200 kPa load imposes at each depth (the quantity that eventually becomes effective stress as excess pore pressure dissipates) — at $t=0$ itself the effective stress is by definition still whatever it was before loading, since none of $\Delta\sigma$ has transferred out of the pore water yet; the source gives no clay unit weight, so the pre-load geostatic effective-stress profile cannot itself be computed from the data given, confirming that the elastic Boussinesq increase (independent of unit weight) is the quantity the question is actually after.
