Question

What is the Boussinesq's vertical stress at a point \(1\) m directly below a concentrated load of \(200\) kN applied at the ground surface?

• A. \(1\ \mathrm{kN/m^2}\)
• B. \(95.54\ \mathrm{kN/m^2}\)
• C. \(7\ \mathrm{kN/m^2}\)
• D. \(23.85\ \mathrm{kN/m^2}\)

Hint:

For a point directly below a concentrated load, \[ \boxed{\sigma_z=\frac{3P}{2\pi z^2}} \] This is one of the most frequently used forms of Boussinesq's equation.

Answer 1

The Correct Option is B

Concept: According to Boussinesq's theory, the vertical stress directly below a concentrated load is \[ \boxed{\sigma_z=\frac{3P}{2\pi z^2}} \] where \[ P=\text{Point load}, \] and \[ z=\text{Depth below the load}. \]

Step 1: Write the given data. \[ P=200\ \text{kN}, \] \[ z=1\ \text{m}. \]

Step 2: Apply Boussinesq's equation. \[ \sigma_z = \frac{3\times200}{2\pi(1)^2} = \frac{600}{6.283} = 95.54\ \text{kN/m}^2. \] Hence, \[ \boxed{\sigma_z=95.54\ \text{kN/m}^2.} \] Therefore, the correct option is \[ \boxed{(B)\;95.54\ \text{kN/m}^2.} \]