Boussinesq (1842 – 1929) evolved equations that can be used to determine stresses at any point P at a depth z as a result of a surface point load.

Vertical pressure within an earth mass

Boussinesq's Formula Equation

At point P of above figure due to a point load Q,

\[Vertical\hspace{.2cm}{stress}, \sigma_{z}=\frac{3Q}{2\pi{z^{2}}}\frac{1}{[1+(r/z)^{2}]^{5/2}}=\frac{Q}{z^{2}}I_{B}\]


r = the horizontal distance between an arbitrary point P below the surface and the vertical axis through the point load Q.

z = the vertical depth of the point P from the surface.

IB = Boussinesq stress coefficient = \[\frac{3}{2\pi}\frac{1}{[1+(r/z)^{2}]^{5/2}}\]


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