Boussinesq?s vertical stress ?_z is given by,
\(?_z=\frac{3Q}{2?z^2} \left[\frac{1}{1+(\frac{r}{z})^2}\right]^{\frac{5}{2}} \)
Substituting \(\frac{r}{z}=1,\)
We get,
\(?_z=\frac{0.0844Q}{z^2}. \)

The Westergaard?s equation is given by,
\(?_z=\frac{1}{\left[1+2(\frac{r}{z})^2 \right]^\frac{3}{2}} \frac{Q}{z^2}\)
? \(?_z=K_W \frac{Q}{z^2}.\)
Where K_W is Westergaard?s influence factor given by,
\(K_W=\frac{1}{?\left[1+2(\frac{r}{z})^2\right]^\frac{3}{2}}. \)

Given,
r/z=0.3
The Boussinesq influence factor is given by,
\(K_B=\frac{3}{2?}\left[\frac{1}{1+(\frac{r}{z})^2}\right]^{\frac{5}{2}} \)
\(K_B=\frac{3}{2?}\left[\frac{1}{1+(0.3)}^2\right]^{\frac{5}{2}} \)
K_B=0.3840.

The influence chart is used to find vertical stress on Westergaard?s equation. Whereas, an isobar is a curve or contour connecting all points below ground surface or ground level of equal vertical pressure. In an isobar, at each and every point vertical pressure is same.

For a triangular and uniformly distributed semi-infinite loads, shear stress ?_xz in plane xz is given by,
\(?_{xz}=-\frac{qz}{a?}\) ? where ? is angle subtended by point P on OA,
z=depth
q=load intensity.

Given,
Z=4m
Q=20 kN
r=1
The Boussinesq?s shear stress ?_rz is given by,
\(?_{rz}=\frac{3Qr}{2?z^3}\left[\frac{1}{1+(\frac{r}{z})^2}\right]^{\frac{5}{2}} \)
? \(?_{rz}=\frac{3*20*1}{2?4^3}\left[\frac{1}{1+(\frac{1}{4})^2}\right]^{\frac{5}{2}} \)
? ?_rz=0.128 kN/m².

Given,
contact pressure=85 kN/m²
load of 212.5 kN
\(Contact\, pressure=\frac{normal\, load}{true \,contact\, area}\)
? \(true \, contact \,area=\frac{normal load}{contact pressure}\)
Area=212.5/85=2.5 m².

On a xz-plane, relation between shear stress component in Cartesian coordinates and polar coordinates is,
\(?_{xz}=?_r sin?cos?? \,where\, ?_r=K \frac{Q cos??}{r}\)
And \(K=\frac{2}{2?+sin2?}\)
?\(?_{xz}=\frac{2Qx^2 z}{?(x^2+z^2)^2}. \)

The Boussinesq?s vertical pressure ?_z due to line load is given by,
\(?_z=\frac{2q?}{?z}\frac{1}{[1+(\frac{x}{z})^2 ]^2} \)
Where q?=line load intensity per unit length
X=horizontal distance from line load
Z= depth of point.

In case of real elastic material, oretical intensity of contact pressure at centre is q/2 and infinite at outer edges. But at outer edges, due to local yielding, causes redistribution of pressure making it finite at edges.