Vertical point load (Q) on the surface is 500 kN, σz (pressure increment) at 10 m depth (Z = 10 m,) directly under the axis of load will be
2.38 kN/m2
Understanding Stress Distribution Under Point Loads
This question asks us to calculate the vertical stress increment in the soil directly below a concentrated vertical point load applied on the surface. This scenario is a classic application of Boussinesq's theory in soil mechanics.
Boussinesq's Theory for Vertical Point Load
Boussinesq's theory provides a method to determine the distribution of stresses within an elastic, homogeneous, isotropic, semi-infinite medium due to a concentrated point load on its surface. The formula for the vertical stress increment ($\sigma_z$) at a point located at a radial distance $r$ from the axis of the load and at a depth $Z$ below the surface, caused by a vertical point load $Q$, is given by:
$$ \sigma_z = \frac{3Q}{2\pi Z^2} \left[ \frac{1}{1 + (r/Z)^2} \right]^{5/2} $$
Here:
- $Q$ is the magnitude of the vertical point load.
- $Z$ is the vertical depth from the surface to the point where stress is being calculated.
- $r$ is the radial distance from the vertical axis passing through the point load to the point where stress is being calculated.
- $\sigma_z$ is the vertical stress increment at the point $(r, Z)$.
- $\pi$ is the mathematical constant approximately equal to 3.14159.
Calculating Stress Directly Under the Load Axis
The question specifies that we need to find the pressure increment directly under the axis of the load. This means the radial distance $r$ is zero ($r = 0$). Let's substitute $r=0$ into Boussinesq's formula:
$$ \sigma_z = \frac{3Q}{2\pi Z^2} \left[ \frac{1}{1 + (0/Z)^2} \right]^{5/2} $$
Since $0/Z = 0$, the equation simplifies to:
$$ \sigma_z = \frac{3Q}{2\pi Z^2} \left[ \frac{1}{1 + 0} \right]^{5/2} $$
$$ \sigma_z = \frac{3Q}{2\pi Z^2} (1)^{5/2} $$
$$ \sigma_z = \frac{3Q}{2\pi Z^2} $$
This simplified formula is used to calculate the vertical stress directly beneath a vertical point load at a given depth.
Applying Given Values to the Formula
We are given the following values:
- Vertical point load, $Q = 500 \text{ kN}$.
- Depth, $Z = 10 \text{ m}$.
Now, we substitute these values into the simplified formula:
$$ \sigma_z = \frac{3 \times 500 \text{ kN}}{2\pi (10 \text{ m})^2} $$
$$ \sigma_z = \frac{1500 \text{ kN}}{2\pi (100 \text{ m}^2)} $$
$$ \sigma_z = \frac{1500}{200\pi} \text{ kN/m}^2 $$
$$ \sigma_z = \frac{15}{2\pi} \text{ kN/m}^2 $$
Performing the Calculation
Let's calculate the numerical value:
$$ \sigma_z = \frac{15}{2 \times 3.14159} \text{ kN/m}^2 $$
$$ \sigma_z = \frac{15}{6.28318} \text{ kN/m}^2 $$
$$ \sigma_z \approx 2.3873 \text{ kN/m}^2 $$
Comparing with Options
The calculated vertical stress increment is approximately 2.3873 kN/m2. Let's look at the provided options:
- 1.68 kN/m2
- 2.38 kN/m2
- 9.54 kN/m2
- 4.636 kN/m2
The calculated value is very close to 2.38 kN/m2.
| Parameter | Value | Units |
|---|---|---|
| Point Load ($Q$) | 500 | kN |
| Depth ($Z$) | 10 | m |
| Radial Distance ($r$) | 0 (directly under) | m |
| Calculated Vertical Stress Increment ($\sigma_z$) | $\approx 2.3873$ | kN/m2 |
Based on our calculation using Boussinesq's theory for a vertical point load, the vertical stress increment at 10 m depth directly under the 500 kN load is approximately 2.38 kN/m2.
Revision Table: Vertical Point Load Stress Calculation
| Concept | Formula (Directly under load, r=0) | Variables |
|---|---|---|
| Vertical Stress Increment ($\sigma_z$) | $$ \sigma_z = \frac{3Q}{2\pi Z^2} $$ | $Q$: Point load, $Z$: Depth, $\pi$: Pi |
Additional Information: Soil Stress Distribution Concepts
Boussinesq's theory is a fundamental concept in geotechnical engineering for understanding how loads applied on the surface distribute stresses within the soil mass. While useful, it's important to remember its assumptions:
- The soil is treated as an elastic, homogeneous, and isotropic material.
- The soil mass is semi-infinite (extends infinitely downwards and horizontally).
- The load is a true point load, concentrated at a single point on the surface.
In reality, soil is rarely perfectly elastic, homogeneous, or isotropic. Footings are also not true point loads but rather distributed loads. However, Boussinesq's theory provides a reasonable first approximation for many practical problems, especially for loads applied over relatively small areas compared to the depth of interest.
The vertical stress increment decreases rapidly with both depth ($Z$) and radial distance ($r$) from the load axis. Isobar diagrams (lines of equal vertical stress) can be plotted based on Boussinesq's equation to visualize the stress distribution in the soil mass.
Important Questions from Vertical Stress Distribution
In Newmark’s influence chart for stress distribution, there are ten concentric circles and ten radial lines. The influence factor of the chart is
The time-dependent deformation on soil is known as?
Contact pressure in soil body is also called _______.
______ is a curve or cont our connecting all points below the ground surface of equal vertical pressure.
The intensity of vertical stress (σz) of the soil just below the load point is given by ______, where Z-vertical distance between point load and the stress and Q- point load.
