Approximate stress distribution methods

Approximate methods give a quick estimate of the vertical stress below a loaded area without using elastic theory. They assume that the load spreads out with depth over an increasing area.

2:1 method (2 vertical : 1 horizontal)

The load $Q$ is assumed to spread at a slope of 2 vertical to 1 horizontal (about 26.6° from the vertical) on all sides. At depth $z$, the loaded area increases by $z$ in each direction.

     |<-- B -->|
     ▇▇▇▇▇▇▇▇▇▇▇          ground
      \       /
       \     /     2V : 1H
        \   /
   z     \ /
      |<- B+z ->|

• Rectangular footing $B\times L$:

$$\Delta\sigma_z = \frac{Q}{(B+z)(L+z)}$$

• Square footing ($B\times B$): $\Delta\sigma_z = \dfrac{Q}{(B+z)^2}$
• Strip footing (per unit length): $\Delta\sigma_z = \dfrac{Q}{B+z}$
• Circular footing of diameter $D$: $\Delta\sigma_z = \dfrac{Q}{\frac{\pi}{4}(D+z)^2}$

60° distribution (30° method)

The load is assumed to spread at an angle of $30^\circ$ with the vertical (60° with horizontal) on each side, so the width at depth $z$ is $B + 2z\tan30^\circ$:

$$\Delta\sigma_z = \frac{Q}{(B+2z\tan30^\circ)(L+2z\tan30^\circ)}$$

Comments

• The result is the average stress over the width at depth $z$, not the maximum under the centre.
• The 2:1 method agrees reasonably with Boussinesq (under the centre) at depths of about 1 to 4 times the width of the footing. It is useful for quick settlement estimates of layered soils.