Circular footing 1 m dia, 1000 kN

Part 1: General or local shear failure at a site

The mode depends on the stiffness and compressibility of the soil.

• Sand: relative density $D_r > 70\%$ (SPT $N > 30$) gives general shear. $D_r < 20\%$ (loose, $N < 10$) gives punching. In between, local shear.
• Clay: stress–strain curve from an unconfined or triaxial test. A sharp peak at a small strain (below about 5%) means general shear (stiff clay). A strain at failure of about 20% or more with no peak means local or punching shear (soft clay).
• Plate load test: a clear breaking point in the load–settlement curve indicates general shear.
• For local shear, Terzaghi reduces strength: $c' = \tfrac{2}{3}c$ and $\tan\phi' = \tfrac{2}{3}\tan\phi$, and uses $N_c', N_q', N_\gamma'$.

Part 2: Depth of footing

Given: circular footing $B = 1$ m, $Q = 1000$ kN, FS = 2.5, $\phi = 30^\circ$, $c = 10$ kN/m², $\gamma = 18$ kN/m³, $N_c = 37.2$, $N_q = 22.5$, $N_\gamma = 19.7$ (general shear assumed).

$$\begin{aligned} A &= \frac{\pi}{4}(1)^2 = 0.7854\ \text{m}^2 \\ q_{applied} &= \frac{1000}{0.7854} = 1273.2\ \text{kN/m}^2 \\ q_u &= 2.5 \times 1273.2 = 3183.1\ \text{kN/m}^2 \end{aligned}$$

Terzaghi (circular): $q_u = 1.3cN_c + \gamma D_fN_q + 0.3\gamma BN_\gamma$

$$\begin{aligned} 1.3 \times 10 \times 37.2 &= 483.6 \\ 0.3 \times 18 \times 1 \times 19.7 &= 106.4 \\ 3183.1 &= 483.6 + 106.4 + 18 \times 22.5\,D_f \\ D_f &= \frac{2593.1}{405} = 6.40\ \text{m} \end{aligned}$$

Answer: depth of footing $\approx 6.4$ m. Since $D_f/B = 6.4 \gg 1$, the footing is really deep and the shallow-footing equation is only an approximation; a larger diameter (for example 2 m) would give a much shallower foundation, which is the practical alternative.