Question

A point load of 400 kN is acting on the surface of the ground. The vertical stress directly below the load at 2 m depth as per the Boussinesq’s theory is .........

• A. 4.775 kPa
• B. 47.75 kPa
• C. 477.5 kPa
• D. Infinity

Hint:

Boussinesq’s equation is valid for stress directly beneath a point load in a homogeneous, isotropic, elastic half-space. For calculations, always use SI units and ensure consistent conversion of kN to N and Pa to kPa.

Answer 1

The Correct Option is B

To calculate the vertical stress due to a point load at a certain depth, Boussinesq’s theory provides the formula:
\[ \sigma_z = \frac{3Q}{2\pi z^2} \]
Where:
$\sigma_z$ = vertical stress at depth $z$ directly beneath the point load (in N/m$^2$ or Pa)
$Q$ = point load applied at the surface (in N)
$z$ = depth below the point load (in meters)
$\pi$ = 3.1416 (approximate)
Given:
$Q = 400$ kN = $400 \times 10^3$ N
$z = 2$ m
Substitute values into the formula:
$\sigma_z = \dfrac{3 \cdot 400 \times 10^3}{2 \cdot \pi \cdot (2)^2}$
$\sigma_z = \dfrac{1.2 \times 10^6}{2 \cdot \pi \cdot 4}$
$\sigma_z = \dfrac{1.2 \times 10^6}{8\pi}$
$\pi \approx 3.1416 ⇒ 8\pi \approx 25.1328$
$\sigma_z = \dfrac{1.2 \times 10^6}{25.1328} \approx 47,746.5$ Pa
Now convert to kilopascals (1 kPa = 1000 Pa):
$\sigma_z = \dfrac{47,746.5}{1000} \approx 47.75$ kPa
Thus, the vertical stress at 2 m depth directly under the point load is approximately 47.75 kPa.