Minimum Depth of Foundation:

Let consider a column carrying a load P to be transmitted to the soil through a footing of area (A) as in Figure.

Figure

Hence, the vertical pressure on soil just below the footing is

P1 = P/A

So on a soil element 'M' just below the footing the horizontal pressure p₂ caused by the footing is

 P₂       =  p₁(1 - sin φ/1 + sin φ)

On a wedge 'N' just outside the footing this horizontal pressure p₂ will cause a vertical pressure p₃ such that

p₃ = p₂ (1 - sin φ/1 + sin φ)

Substituting Eq. (9.16) in Eq. (9.17), we get

p₃ = p₁ (1 - sin φ/1 + sin φ)² = P/A(1 - sin φ/1 + sin φ)²

We know that the vertical pressure p₃ = wh, where w is the unit weight of soil and h is the depth of the footing.

∴ wh² = P/A (1 - sin φ /1 + sin φ)²

or the minimum depth of footing for equilibrium that will be

h =   P/wA (1 - sin φ /1 + sin φ)²