Concept:
Vertical stress due to concentrated load is given as per Boussinesq’s equation.
Formula:
\({{\rm{\sigma }}_{\rm{z}}}{\rm{\;}} = {\rm{\;}}\frac{3}{{2{\rm{\pi }}}}\frac{{\rm{Q}}}{{{{\rm{z}}^2}}}{\left( {\frac{1}{{1 + {{\left( {\frac{{\rm{r}}}{{\rm{Z}}}} \right)}^2}}}} \right)^{\frac{5}{2}}}\)
where,
σ_{z} = Stress magnitude, q = Vertical point load, r = Radial distance, and Z = Vertical distance.
Given:
Z = 10 m, and Q = 500 kN
R = 0 {∵ directly below the load axis}
\(\therefore {{\rm{\sigma }}_{\rm{z}}} = \frac{3}{{2{\rm{\pi }}}} \times \frac{{500}}{{\left( {{{10}^2}} \right)}} \times {\left( {\frac{1}{{1 + {{\left( {\frac{0}{{10}}} \right)}^2}}}} \right)^{5/2}} = 2.38{\rm{\;kN}}/{{\rm{m}}^2}\)
Important Point:
As per Boussinesq’s stress directly below a point load at any distance ‘z’ is given as
\({{\rm{\sigma }}_{\rm{z}}}{\rm{\;}} = {\rm{\;}}\frac{3}{{2{\rm{\pi }}}}\frac{{\rm{Q}}}{{{{\rm{z}}^2}}}\)
Difference between Boussinesq’s and Westergaard’s equation of vertical stress :
Boussinesq’s |
Westergaard’s |
Soil mass is considered isotropic |
Soil mass is considered anisotropic |
It is used for unstratified soil |
It is used for stratified soil |
Poisson’s ratio in derivation is not assumed zero |
Poisson’s ratio is assumed zero |
\({\sigma _z}\; = \;\frac{3}{{2\pi }}\frac{Q}{{{z^2}}}\frac{1}{{{{\left( {1 + {{\left( {\frac{r}{z}} \right)}^2}} \right)}^{\frac{5}{2}}}}}\) |
\({\sigma _z}\; = \;\frac{1}{\pi }\frac{Q}{{{z^2}}}\frac{1}{{{{\left( {1 + 2{{\left( {\frac{r}{z}} \right)}^2}} \right)}^{\frac{3}{2}}}}}\) |