Question

A concentrated vertical load of 3000 kN is applied on a horizontal ground surface. Points P and Q are at depths 1 m and 2 m below the ground, respectively, along the line of application of the load. Considering the ground to be a linearly elastic, isotropic, semi-infinite medium, the ratio of the increase in vertical stress at P to the increase in vertical stress at Q is \(\underline{\hspace{2cm}}\) (in integer).

Hint:

In semi-infinite media, the stress distribution follows an inverse-square law with respect to the depth.

Answer 1

Correct Answer: 4

For a linearly elastic, isotropic, semi-infinite medium, the ratio of the increase in vertical stress at points at different depths can be approximated using Boussinesq's solution for vertical stress distribution. The vertical stress increase at a depth \( z \) is inversely proportional to the square of the distance from the load:
\[ \frac{\Delta \sigma_P}{\Delta \sigma_Q} = \frac{z_Q}{z_P}. \] Thus, the ratio is:
\[ \frac{1}{2} = 4. \] Therefore, the ratio of the increase in vertical stress at P to the increase in vertical stress at Q is \( 4 \).