Question

A concentrated load of 50 kN acts vertically at a point on the soil surface. If Boussinesq's equation is applied for computation of stress, then the ratio of vertical stresses at depths of 3 m and 5 m respectively vertically below the point of application of load is
1. 0.36
2. 0.60
3. 1.66
4. 2.77

Answer 1

Correct Answer - Option 4 : 2.77

Concept:

According to Boussinesq's Vertical stress at a point is given by;

\({\sigma _z} = {k_B}\frac{Q}{{{z^2}}}\)

Where, \({k_B} = \frac{3}{{2\pi }}{\left[ {\frac{1}{{1 + \frac{{{r^2}}}{{{z^2}}}}}} \right]^{5/2}}\)

Vertical stress below the point at depth z, (menas, r/z = 0)

So, \({k_B} = \frac{3}{{2\pi }} = 0.4775\)

∴ \({\sigma _z} = 0.4775\frac{Q}{{{z^2}}}\)

Calculation:

Given,

Q = 50 kN, Z₁ = 3 m, Z₂ = 5 m

Vertical stress below the point load at depth z₁, \({\sigma _{{z_1}}} = 0.4775\frac{Q}{{z_1^2}}\)

Vertical stress below the point load at depth z₂, \({\sigma _{{z_2}}} = 0.4775\frac{Q}{{z_2^2}}\)

So, \(\frac{{{\sigma _{{z_1}}}}}{{{\sigma _{{z_2}}}}} = \frac{{z_2^2}}{{z_1^2}}\)

⇒ \(\frac{{{\sigma _{{z_1}}}}}{{{\sigma _{{z_2}}}}} = \frac{{{5^2}}}{{{3^2}}} = 2.77\)