Boussinesq problem

Remarks

The problem is solved in closed form. With $r = \sqrt{x^2 + y^2}$ and $R = \sqrt{r^2 + z^2}$, the vertical displacement is $$u_z = \frac{P}{4 \pi \mu R} \left( 2 (1 - \nu) + \frac{z^2}{R^2} \right),$$ the radial displacement is $u_r = \frac{P}{4 \pi \mu R} \left( \frac{r z}{R^2} - \frac{(1 - 2\nu) r}{R + z} \right)$, and the vertical normal stress is $\sigma_{zz} = -\frac{3 P z^3}{2 \pi R^5}$. On the surface $z = 0$ this gives the settlement $u_z = \frac{(1 - \nu) P}{2 \pi \mu r}$, which is singular at the loaded point. The vertical stress does not depend on the elastic constants. By superposition, the solution for a distributed normal pressure on the surface is an integral of this one. It is a starting point of the Hertz theory of contact mechanics. It is also used to compute stresses under foundations in soil mechanics.