Question

The Boussinesq equation representing the vertical stress is ___________

(a) \(σ_z=\frac{3}{2π} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^{5/2}\)

(b) \(σ_z=\frac{3Q}{2π} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^5\)

(c) \(σ_z=\frac{3Q}{2πz^2}\left[\frac{1}{1+(\frac{r}{z})^2} \right]^{\frac{5}{2}}\)

(d) \(σ_z=\frac{3Q}{2π} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^2\)

I have been asked this question in an interview for job.

The origin of the question is Stress Distribution in chapter Stress Distribution of Geotechnical Engineering I

Answer 1

Right option is (c) \(σ_z=\frac{3Q}{2πz^2}\left[\frac{1}{1+(\frac{r}{z})^2} \right]^{\frac{5}{2}}\)

To elaborate: Boussinesq showed that the polar radial stress is given by,

\(σ_R=\frac{3Q}{2π}   \frac{cos⁡β}{R^2} \)

 Boussinesq’s vertical stress σz is given by,

σz=σRcos^2 β

∴ \(σ_z=\frac{3Q}{2πz^2}\left[\frac{1}{1+(\frac{r}{z})^2} \right]^{\frac{5}{2}}\) where, σz is the vertical stress

Q is the point load acting at the ground surface

r is the radial horizontal distance

z is the vertical distance.