Stress distribution in soil

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

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