The maximum shear stress is given by,
\(?_{max}=\frac{1}{2} (?_1-?_2 )=\frac{q}{?} sin?\)
Sin? is maximum when ?= ?/2
? Sin ?/2=1
? \(?_{max}=\frac{q}{?}.\)

The maximum value of ?_z on vertical line is obtained at point of intersection of vertical plane with radial line at angle of39�15?. The corresponding value of \(\frac{r}{z}\) is 0.817.
? \(z=\frac{r}{0.817}=1.225r \)
K_B=0.1332
\(?_z=K_B \frac{Q}{z^2} \)
\(?_z=0.1332 \frac{Q}{1.225r^2} \)
? \((?_z)_{max}=\frac{0.0888Q}{r^2} \)

By means of Boussinesq?s stress distribution ory, following vertical pressure distribution diagrams can be prepared,

• stress isobars
• vertical pressure distribution on a horizontal plane
• vertical pressure distribution on a vertical plane.

The velocity with which excess pore water flows is given by Darcy?s law,
\(v=ki, \,where\, i=\frac{1}{?_w}\frac{?\overline{u}}{?z}\)
Therefore, \(v=\frac{k}{?_w}\frac{?\overline{u}}{?z}.\)

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

Let a uniformly loaded circular area is divided into 44 sectors.
q=load
?_z=vertical stress at depth z
The stress at each unit area will be \(\frac{?_z}{44} \)
? \(\frac{?_z}{44}=\frac{q}{44}\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right] \)
The influence factor is given by \(i_f=\frac{1}{44} \left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right] \).

On a horizontal plane, relation between stress component in x-direction in Cartesian coordinates and polar coordinates is,
\(?_x=?_r sin^2?? \,where\, ?_r=K \frac{Q cos??}{r}\)
And \(K=\frac{2}{2?+sin2?}\)
? \(?_x=\frac{2Qx^3}{?(x^2+z^2)^2}. \)

In Terzaghi?s Theory of one dimensional consolidation, as load is applied in one direction only, excess pore water drains out in in one direction and that is vertical direction.

In case of cohesionless soil, no resistance to deformation is offered at outer edges making it zero at outer edges. The pressure is parabolic with maximum value at centre.

For any position of point P subtending angle ? with AB, vertical stress is given by,
\(?_z=\frac{q}{a?}\left[x?-\frac{az}{(x-?)^2+z^2}(x-?)\right] \)
Where, ?_z= vertical stress
x=distance from support A
z=depth from ground surface
q=load applied.