Consider that a force P is acting on the surface of a half-space (Boussinesq’s problem). The expression for the vertical stress (σz) at any point (r, z), within the half-space is given as,

$\rm \sigma_z=\frac{3P}{2\pi} \frac{z^3}{_{(r^2+z^2)}\frac{5}{2}}$

where, r is the radial distance, and z is the depth with downward direction taken as positive. At any given r, there is a variation of σ_z along z, and at a specific z, the value of σz will be maximum. What is the locus of the maximum σz ?