. Consider a point load P = 25kN at origin (0,0,0). Calculate the vertical stress increase

(40) al z = 2 and 4 m for x = 3 and y=4 m.

P.T.O

To calculate the vertical stress increase due to a point load at a specific location, we can use the Boussinesq equation for vertical stress increase. The formula for vertical stress increase at a point \((x, y, z)\) due to a point load \(P\) at the origin is:

\[ \Delta \sigma_z = \frac{3Pz^3}{2\pi (x^2 + y^2 + z^2)^{5/2}} \]

Given:
- \( P = 25 \text{kN} \)
- \( z \) = 2 m and 4 m
- \( x = 3 \) m
- \( y = 4 \) m

We need to calculate \(\Delta \sigma_z\) for these conditions.

First, let's calculate the vertical stress increase at \(z = 2 \, \text{m}\):

\[ \Delta \sigma_z = \frac{3 \times 25 \times 2^3}{2\pi (3^2 + 4^2 + 2^2)^{5/2}} \]

Step-by-step:

1. Compute the denominator:
\[ x^2 + y^2 + z^2 = 3^2 + 4^2 + 2^2 = 9 + 16 + 4 = 29 \]
\[ (x^2 + y^2 + z^2)^{5/2} = 29^{5/2} \]

2. Compute the numerator:
\[ 3Pz^3 = 3 \times 25 \times 2^3 = 3 \times 25 \times 8 = 600 \]

3. Combine them:
\[ \Delta \sigma_z = \frac{600}{2\pi \times 29^{5/2}} \]

4. Calculate \( 29^{5/2} \):
\[ 29^{5/2} = 29^{2.5} = 23775.5 \]

5. Now compute \(\Delta \sigma_z\):
\[ \Delta \sigma_z = \frac{600}{2\pi \times 23775.5} \approx \frac{600}{149258.24} \approx 0.004 \text{ kN/m}^2 \]

Now, let's calculate the vertical stress increase at \(z = 4 \, \text{m}\):

\[ \Delta \sigma_z = \frac{3 \times 25 \times 4^3}{2\pi (3^2 + 4^2 + 4^2)^{5/2}} \]

Step-by-step:

1. Compute the denominator:
\[ x^2 + y^2 + z^2 = 3^2 + 4^2 + 4^2 = 9 + 16 + 16 = 41 \]
\[ (x^2 + y^2 + z^2)^{5/2} = 41^{5/2} \]

2. Compute the numerator:
\[ 3Pz^3 = 3 \times 25 \times 4^3 = 3 \times 25 \times 64 = 4800 \]

3. Combine them:
\[ \Delta \sigma_z = \frac{4800}{2\pi \times 41^{5/2}} \]

4. Calculate \( 41^{5/2} \):
\[ 41^{5/2} = 41^{2.5} = 222412.04 \]

5. Now compute \(\Delta \sigma_z\):
\[ \Delta \sigma_z = \frac{4800}{2\pi \times 222412.04} \approx \frac{4800}{1398674.29} \approx 0.0034 \text{ kN/m}^2 \]

Thus, the vertical stress increases are approximately:
- At \( z = 2 \, \text{m} \): \( \Delta \sigma_z \approx 0.004 \, \text{kN/m}^2 \)
- At \( z = 4 \, \text{m} \): \( \Delta \sigma_z \approx 0.0034 \, \text{kN/m}^2 \)

Would you like more details or have any questions on these calculations?

Here are 8 related questions to expand on this topic:
1. How does the distance from the load affect the vertical stress increase?
2. What is the significance of Boussinesq's equation in geotechnical engineering?
3. How would the stress distribution change if the point load were at a different location?
4. How does soil type affect the vertical stress distribution?
5. What are the limitations of using Boussinesq’s equation in practical scenarios?
6. How would you calculate the horizontal stress increase due to the point load?
7. What are other methods to calculate stress distribution in soils?
8. How would multiple point loads interact in terms of stress distribution?

**Tip:** Always double-check the units used in calculations to ensure consistency and accuracy.

Learn how to calculate the vertical stress increase at different depths due to a point load of P = 25kN using Boussinesq's equation. Detailed step-by-step calculations for z = 2m and z = 4m, including formula derivation and numerical results.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation