Understanding Vertical Stress using Boussinesq's Analysis

This question involves calculating the increase in vertical stress within a saturated clay layer caused by a surface point load. We will use Boussinesq's elastic analysis, a fundamental method in geotechnical engineering to determine stress distribution beneath applied loads on the ground surface.

Key Parameters and Formula

The problem provides the following information:

• Point Load ($P$): 500 kN
• Depth ($z$): 3 m
• Radial distance ($r$): 2 m

Boussinesq's equation for the vertical stress ($\sigma_z$) at a depth $z$ and radial distance $r$ from a point load $P$ applied on the surface of a semi-infinite, homogeneous, and elastic medium is:

$$ \sigma_z = \frac{3P}{2\pi} \frac{z^3}{R^5} $$

where $R$ is the direct distance from the point of load application to the point where stress is being calculated. It is calculated using the Pythagorean theorem:

$$ R = \sqrt{r^2 + z^2} $$

Step-by-Step Calculation

Step 1: Calculate the radial distance $R$.

Substitute the given values of $r$ and $z$ into the formula for $R$:

$$ R = \sqrt{(2 \text{ m})^2 + (3 \text{ m})^2} $$

$$ R = \sqrt{4 \text{ m}^2 + 9 \text{ m}^2} $$

$$ R = \sqrt{13 \text{ m}^2} $$

$$ R \approx 3.606 \text{ m} $$

Step 2: Calculate $R^5$.

$$ R^5 = (\sqrt{13})^5 = (13^{0.5})^5 = 13^{2.5} $$

$$ R^5 \approx (3.606)^5 \approx 609.24 \text{ m}^5 $$

Step 3: Substitute all values into Boussinesq's formula for $\sigma_z$.

$$ \sigma_z = \frac{3 \times 500 \text{ kN}}{2\pi} \frac{(3 \text{ m})^3}{R^5} $$

$$ \sigma_z = \frac{1500 \text{ kN}}{2\pi} \frac{27 \text{ m}^3}{609.24 \text{ m}^5} $$

First, calculate the constant part:

$$ \frac{3 \times 500}{2\pi} = \frac{1500}{2\pi} \approx \frac{1500}{6.283} \approx 238.73 \text{ kN/m}^2 $$

Now, calculate the stress:

$$ \sigma_z \approx 238.73 \text{ kN/m}^2 \times \frac{27 \text{ m}^3}{609.24 \text{ m}^5} \times \text{ m}^3 $$

$$ \sigma_z \approx 238.73 \times \frac{27}{609.24} \text{ kN/m}^2 $$

$$ \sigma_z \approx 238.73 \times 0.04431 \text{ kN/m}^2 $$

$$ \sigma_z \approx 10.579 \text{ kN/m}^2 $$

Since 1 kPa = 1 kN/m$^2$, the vertical stress is approximately 10.58 kPa.

Conclusion

Based on Boussinesq's elastic analysis, the estimated vertical stress at a depth of 3 m and a radial distance of 2 m from the 500 kN point load application is approximately 10.58 kPa.