Find. The vertical stress increase $\Delta\sigma_z$ at 5 m below the footing centre, by three independent methods, and a comparison.
Footing plan — for the influence-factor method the 4 m square is split into four 2 m × 2 m rectangles that share the corner O directly above the point of interest.
Approach. Superpose Boussinesq solutions for a uniformly loaded area: the exact m–n corner-influence factor (four quadrants), Newmark's influence chart, and the approximate 2:1 spread — then cross-check against the crude point-load idealisation.
Method 1 — m–n influence factors (four quadrants). Split the footing into four 2 m×2 m rectangles with a common corner at O. Each has
$$m=\frac{B}{z}=\frac{2}{5}=0.4,\qquad n=\frac{L}{z}=\frac{2}{5}=0.4.$$
The Boussinesq corner factor is
$$I=\frac{1}{4\pi}\!\left[\frac{2mn\sqrt{m^2+n^2+1}}{m^2+n^2+1+m^2n^2}\cdot\frac{m^2+n^2+2}{m^2+n^2+1}+\tan^{-1}\!\frac{2mn\sqrt{m^2+n^2+1}}{m^2+n^2+1-m^2n^2}\right]=0.0602.$$
Superposing the four identical quadrants,
$$\Delta\sigma_z = 4\,I\,q = 4(0.0602)(100)=\boxed{24.1\ \text{kPa}}.$$
Method 2 — Newmark's influence chart. Placing the scaled footing plan on the chart with the centre over the plot origin, the loaded area covers about $N\approx48$ influence units. With the chart's influence value $I_N=0.005$,
$$\Delta\sigma_z=q\,(0.005\,N)=100(0.005)(48)\approx 24\ \text{kPa},$$
in essentially exact agreement with the influence-factor result (the two methods share the same Boussinesq basis).
Method 3 — 2:1 (trapezoidal) spread. The load is assumed to spread on 2 vertical : 1 horizontal planes, so at depth $z$ it acts over $(B+z)(L+z)$:
$$\Delta\sigma_z=\frac{qBL}{(B+z)(L+z)}=\frac{100(4)(4)}{(4+5)(4+5)}=\frac{1600}{81}=\boxed{19.8\ \text{kPa}}.$$
Cross-check — point-load (Boussinesq) idealisation. Lumping the whole load $Q=qBL=1600$ kN at the centre and evaluating directly below it ($r=0$),
$$\Delta\sigma_z=\frac{3Q}{2\pi z^2}=\frac{3(1600)}{2\pi(5)^2}=30.6\ \text{kPa}.$$
This overestimates, because concentrating a 4 m footing at a point exaggerates the near-axis stress at a depth only slightly greater than the footing width.
Q3 — comparison of methods ($\Delta\sigma_z$ at centre, $z=5$ m)
Method
$\Delta\sigma_z$ (kPa)
Comment
m–n influence factors (exact)
24.1
reference value
Newmark's chart
≈24
same Boussinesq basis
2:1 trapezoidal spread
19.8
underestimates (empirical)
point-load idealisation
30.6
overestimates (footing not a point)
The two rigorous Boussinesq methods agree at $\Delta\sigma_z\approx24$ kPa. The 2:1 rule is about 18% low and the point-load idealisation about 27% high — both expected at $z/B=1.25$, where the footing is neither a point nor deep enough for the 2:1 spread to be accurate.
