Understanding the vertical stress distribution in soil is crucial in geotechnical engineering, especially when dealing with concentrated loads. The question asks for the intensity of vertical stress ($\sigma_z$) directly below a point load (Q) at a vertical distance (Z).

Vertical Stress Calculation: Boussinesq's Theory

The calculation of vertical stress due to a point load on the surface of a semi-infinite, elastic, isotropic, and homogeneous soil mass is typically done using Boussinesq's theory. This theory provides a fundamental solution for stress distribution under such conditions.

Boussinesq's Formula for Point Load

Boussinesq's general equation for the vertical stress ($\sigma_z$) at any point in the soil due to a surface point load (Q) is given by:

\[ \sigma_z = \frac{3Q}{2\pi Z^2} \left[ \frac{1}{1 + \left(\frac{r}{Z}\right)^2} \right]^{5/2} \]

Where:

• \( \sigma_z \) is the vertical stress at depth Z
• \( Q \) is the magnitude of the point load
• \( Z \) is the vertical distance (depth) from the point load to the point where stress is being calculated
• \( r \) is the horizontal radial distance from the line of action of the point load to the point where stress is being calculated

Stress Directly Below the Load Point

The question specifies the intensity of vertical stress ($\sigma_z$) of the soil just below the load point. This means we are considering the point directly beneath the load, where the horizontal radial distance \(r\) is zero.

By substituting \(r = 0\) into Boussinesq's general formula, the equation simplifies significantly:

\[ \sigma_z = \frac{3Q}{2\pi Z^2} \left[ \frac{1}{1 + \left(\frac{0}{Z}\right)^2} \right]^{5/2} \]

\[ \sigma_z = \frac{3Q}{2\pi Z^2} \left[ \frac{1}{1 + 0} \right]^{5/2} \]

\[ \sigma_z = \frac{3Q}{2\pi Z^2} (1)^{5/2} \]

\[ \sigma_z = \frac{3Q}{2\pi Z^2} \]

Numerical Value of the Constant

Now, let's calculate the numerical value of the constant \( \frac{3}{2\pi} \):

We know that \( \pi \approx 3.14159 \).

So, \( \frac{3}{2\pi} = \frac{3}{2 \times 3.14159} = \frac{3}{6.28318} \approx 0.47746 \)

Rounding this to four decimal places, the constant is approximately \(0.4775\).

Therefore, the formula for vertical stress directly below the load point becomes:

\[ \sigma_z = 0.4775 \frac{Q}{Z^2} \]

Comparing with Options

Let's compare this derived formula with the given options:

• Option 1: \(0.4775\frac{Z}{Q}\)
• Option 2: \(0.4775\frac{Q}{Z}\)
• Option 3: \(0.4775\frac{Z}{Q^2}\)
• Option 4: \(0.4775\frac{Q}{Z^2}\)

The derived formula \(0.4775\frac{Q}{Z^2}\) matches Option 4.