24-Bld-A6 Geotechnical Materials and Analysis · December 2018
Question 7 of 7
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
07-BLD-A6 Geotechnical Materials and Analysis — National Examinations, December 2018. 3 hours, closed book, 100 marks. Section A (Q1–Q3) is compulsory; Section B directs "answer any three of Q4–Q7," but for completeness this solution answers all four.
Reference texts: B.M. Das, Principles of Geotechnical Engineering, 9th ed.; B.M. Das, Principles of Foundation Engineering, 9th ed.
Figure 5(b) (reproduced): 1200 kN column on a 4 m × 4 m footing at the ground surface; sand to 6 m depth (water level at 1.5 m), clay from 6–9 m, sand-and-gravel bearing stratum below. Mid-clay depth (settlement reference) is 7.5 m.
Check: (1) the sand's unit weight is given as a single value (18.3 kN/m³) spanning both above and below the 1.5 m water table — this solution uses it as moist above the water table and, for lack of a separate saturated value, also as the total unit weight below it (subtracting $\gamma_w$ there for effective stress), consistent with how the source presents it. (2) The clay's $\gamma_b=9.2$ kN/m³ is read as already-buoyant/submerged (needs no further $\gamma_w$ subtraction). (3) Figure 5(a) shows two void-ratio curves; the upper one is used because the exam's own "$e_0$" tick mark sits on it ($e_0\approx0.98$), even though $G_s=2.7$ combined with $\gamma_b=9.2$ back-calculates a lower in-situ void ratio ($\approx0.81$) closer to the second curve — a data inconsistency noted rather than silently resolved.
Given. $Q=1200\text{ kN}$ on a $4\text{ m}\times4\text{ m}$ footing at the surface; sand 0–6 m ($\gamma=18.3\text{ kN/m}^3$, water table at 1.5 m); clay 6–9 m ($\gamma_b=9.2\text{ kN/m}^3$); $G_s=2.7$; Figure 5(a) void-ratio curve.
Find. Primary consolidation settlement of the clay under the footing centre.
Approach. Compute the pre-construction effective stress $\sigma_0'$ at clay mid-depth from the layered profile; compute the footing-induced stress increase $\Delta\sigma$ at that same depth under the centre (four corner-quadrant Boussinesq, cross-checked with the approximate 2:1 formula); read $e_0,C_c$ from Figure 5(a); apply the standard consolidation-settlement formula.
Pre-construction effective stress at mid-clay ($z=7.5$ m). $\sigma_0'=18.3(1.5)+(18.3-9.81)(4.5)+9.2(1.5)=27.45+38.21+13.80=\boxed{79.5\text{ kPa}}$.
Stress increase at mid-clay, under the footing centre. Split the footing into four $2\text{ m}\times2\text{ m}$ quadrants sharing the centre as a common corner: $m=n=2/7.5=0.267\Rightarrow I\approx0.0303$ (matches the Fadum chart provided). $\Delta\sigma=4qI=4(75)(0.0303)=9.10\text{ kPa}$ — in close agreement with the formula sheet's approximate method, $\dfrac{qBL}{(B+z)(L+z)}=\dfrac{75(16)}{11.5^2}=9.07\text{ kPa}$.
Compression parameters (Figure 5a). Reading the curve on which $e_0$ is marked: $e_0\approx0.98$ (flat, over-consolidated branch, ending almost exactly at $\sigma_0'$); the virgin line runs through $(100,\,0.97)$–$(1000,\,0.72)$, giving $C_c=\dfrac{0.97-0.72}{\log_{10}(1000/100)}=0.25$.
