Finitely loaded area

Vertical line load at the surface

Vertical Point Load at the Surface

Vertical Point Load at the Surface

The solution to the problem of calculating the stresses in an elastic half space subjected to a vertical point load at the surface will be of value in estimating the stresses induced in a deposit of soil whose depth is large compared to the dimensions of that part of the surface that is loaded. ${\displaystyle \Delta \sigma _{z}=-{\frac {3P}{2\pi R^{2}}}\cos ^{3}\theta }$ ${\displaystyle \Delta \sigma _{r}={\frac {P}{2\pi R^{2}}}(-3\cos \theta \sin ^{2}\theta +{\frac {1-2\mu }{1+\cos \theta }})}$ ${\displaystyle \Delta \sigma _{t}={\frac {P}{2\pi R^{2}}}(1-2\mu )(\cos \theta -{\frac {1}{1+\cos \theta }})}$ ${\displaystyle \Delta \tau =-{\frac {3P}{2\pi R^{2}}}\cos ^{2}\theta \sin \theta }$ ${\displaystyle \cos \theta ={\frac {z}{R}}}$, ${\displaystyle R={\sqrt {r^{2}+z^{2}}}}$ ${\displaystyle \Delta \sigma _{z}=-{\frac {3Pz^{3}}{2\pi R^{5}}}=-{\frac {3P}{2\pi }}{\frac {z^{3}}{(r^{2}+z^{2})^{5/2}}}=-{\frac {3P}{2\pi z^{2}\left[1+\left({\frac {r}{z}}\right)^{2}\right]^{\frac {5}{2}}}}}$^[3]

Below centre of uniformly loaded circular area

Below centre of uniformly loaded circular area

${\displaystyle \sigma =q\{1-{\frac {1}{[({\frac {R}{z}})^{2}+1]^{3/2}}}\}}$^[4]