Westergaard is a British scientist. For vertical stress computation, he had proposed a formula in 1938. The formula is presented below.
Westergaard's Equation for Point Loads
If Q is the point load and σz is the vertical stress due to the point load,
\[\sigma_{z}=\frac{Q}{2{\pi}z^{2}}\times \frac{\sqrt{(1-2\mu)/(2-2\mu)}} {[(1-2\mu)/(2-\mu)+(r/z)^{2})]^{3/2}}= \frac{Q}{z^{2}}I_{w}\]
Here, μ= Poisson's Ratio. If Poisson's Ratio is considered zero for all practical purposes,
\[\sigma_{z}=\frac{Q}{{\pi}z^{2}}\times \frac{1} {[1+2(r/z)^{2})]^{3/2}}= \frac{Q}{z^{2}}I_{w}\]
Where, \[I_{w}= \frac{(1/\pi)} {[1+2(r/z)^{2})]^{3/2}}\]
\[I_{w}\] is known as Westergaard stress coefficient.