16-Civ-A4 Geotechnical Materials and Analysis · December 2017

Question 3 of 6: Vertical Stress Increase Beneath a Frame Footing

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format: National Examinations — 16-Civ-A4 Geotechnical Materials and Analysis, December 2017. Closed book, 3 hours, 100 marks. Answer all six questions. Charts (rectangular-area influence chart, Newmark chart) and a formula sheet are provided at the back of the exam.

Reference texts: R.F. Craig & J. Knappett, Craig’s Soil Mechanics (8th ed.); B.M. Das, Principles of Geotechnical Engineering; R.D. Holtz, W.D. Kovacs & T.C. Sheahan, An Introduction to Geotechnical Engineering (2nd ed.); M. Budhu, Soil Mechanics and Foundations.

Question 3: Vertical Stress Increase Beneath a Frame Footing (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. A square frame footing: a 6 m × 6 m loaded (shaded) area with a concentric 3 m × 3 m unloaded void (1.5 m of loaded soil on every side). Point A is at the outer corner of the 6 m square.

Given data
Uniform contact pressure, q	100 kPa
Outer footing plan	6 m × 6 m
Central void	3 m × 3 m (centred)
Depth below A	z = 1.5 m
Point A	outer corner of the 6 m square

Figure 3 — plan of the frame footing. The shaded ring (6×6 outer, 3×3 void) carries 100 kPa; A is at the outer corner. Stress is sought at z = 1.5 m directly below A.

Find. The increase in vertical stress Δσ_z at 1.5 m below A, by Newmark’s chart and by a second method, with a comparison.

Approach. Because A sits at the outer corner, the loaded frame is treated by superposition of corner rectangles: the full 6×6 corner rectangle minus the influence of the central void (itself expressed as four corner rectangles about A). The rectangular-area corner influence factor supplies the “other suitable method,” and Newmark’s chart supplies the graphical count.

Corner influence factor. For a uniform rectangle of sides B, L loaded at its corner, with m = B/z and n = L/z (interchangeable), $$I=\frac{1}{4\pi}\left[\frac{2mn\sqrt{m^{2}+n^{2}+1}}{m^{2}+n^{2}+1+m^{2}n^{2}}\cdot\frac{m^{2}+n^{2}+2}{m^{2}+n^{2}+1}+\tan^{-1}\!\frac{2mn\sqrt{m^{2}+n^{2}+1}}{m^{2}+n^{2}+1-m^{2}n^{2}}\right].$$
Whole 6×6 area at corner A. With m = n = 6/1.5 = 4: $I_\text{outer}=0.2473$.
Subtract the central void. The void spans 1.5–4.5 m from A in both directions, so its influence is the four-rectangle superposition $$I_\text{void}=I(3,3)-I(1,3)-I(3,1)+I(1,1)=0.0124\quad(m,n=\text{distance}/z).$$
Net stress (influence-factor method). $$\Delta\sigma_z=q\,(I_\text{outer}-I_\text{void})=100\,(0.2473-0.0124)=\boxed{23.5\ \text{kPa}}.$$
Newmark’s chart. Draw the footing to the chart’s depth scale (length AB = z = 1.5 m, so the 6 m side is 4AB and the void starts 1AB from A). Put A over the chart centre and count the influence elements covered by the loaded ring, leaving out those under the void. The analytical result predicts N = Δσ_z/(0.005q) ≈ 47 blocks, and a careful count lands within about ±2 blocks of that. With N ≈ 47 and I_N = 0.005, $$\Delta\sigma_z=0.005\,N\,q=0.005\times47\times100=23.5\ \text{kPa},$$ which agrees with the analytical value to within the counting tolerance (±2 blocks is ±1 kPa).
Quick approximate check. The 2:1-type formula $\sigma_z=qBL/[(B+z)(L+z)]$ applied, very roughly, as (outer − void) gives $100\cdot\tfrac{36}{56.25}-100\cdot\tfrac{9}{20.25}=19.6$ kPa — about 17% lower. This is only an order-of-magnitude check: the 2:1 formula gives the average stress over the spread area, not the stress under a particular point such as a corner, so its closeness here is partly coincidence.

Increase in vertical stress at 1.5 m below A
Method	Δσ_z
Newmark’s chart (N ≈ 47)	23.5 kPa
Rectangular-area influence factor	23.5 kPa
Approximate 2:1 spread	19.6 kPa

Comment. The Newmark chart and the influence-factor method are both rigorous Boussinesq solutions and agree almost exactly (23.5 kPa); any small difference in practice comes only from the human error of counting partial blocks on the chart. The approximate 2:1 method is quicker but differs by roughly 17% here, because it spreads the load uniformly over a widening area and cannot represent the stress at a specific point such as corner A — acceptable for a first estimate, not for design at a specific point.

Topic: Elastic vertical-stress increase beneath a loaded area (superposition of corner rectangles).

Key relations: corner influence factor I(m, n); Δσ_z = q·ΣI (superposition); Newmark Δσ_z = 0.005 N q.

Why this works: Boussinesq stresses are linear in load, so any polygon can be built from corner rectangles taken with + and − signs about the point of interest — a frame is simply “solid outer minus void.” Newmark’s chart is the same solution rendered as equal-influence blocks so it can be evaluated by counting.

Common pitfall: Forgetting to subtract the void, or mis-placing A (if A were at the void’s corner rather than the outer corner the rectangle set changes). Also, drawing the footing at the wrong scale on the Newmark chart (the AB length must equal z).

Source: Das, Principles of Geotechnical Engineering, Ch. 10; Craig & Knappett, Ch. 6 (Newmark’s influence chart).

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