Strip footing vs 2:1 method

Given

Strip footing $B = 2$ m, load $Q = 500$ kN/m. Contact pressure $q = 500/2 = 250$ kN/m². Depth $z = 5$ m, below the centre (where stress is maximum).

Boussinesq (strip load)

$$\Delta\sigma_z = \frac{q}{\pi}\left[\alpha + \sin\alpha\cos(\alpha + 2\delta)\right]$$

Below the centre, $\delta = 0$, so $\Delta\sigma_z = \dfrac{q}{\pi}\left(\alpha + \sin\alpha\right)$ where $\alpha = 2\tan^{-1}\left(\dfrac{B/2}{z}\right) = 2\tan^{-1}(0.2) = 0.3948$ rad (22.62°).

$\sin\alpha = 0.3846$

$$\Delta\sigma_z = \frac{250}{\pi}(0.3948 + 0.3846) = 79.58\times0.7794 = 62.0\ \text{kN/m}^2$$

2:1 method

$$\Delta\sigma_z = \frac{Q}{B+z} = \frac{500}{2+5} = 71.4\ \text{kN/m}^2$$

Comparison

Answer: Maximum stress at 5 m depth $= 62.0$ kN/m² (Boussinesq); the 2:1 method gives 71.4 kN/m², about 15% higher (conservative).