Calculating the Volumetric Strain for a Rectangular Steel Bar

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In summary, a rectangular steel bar with dimensions 250 mm x 50 mm x 25 mm is subjected to compressive and tensile forces on different faces. Using Hooke's Law and the equation for volumetric strain, the change in volume is calculated to be 25.92 mm^{2}, which was found to be incorrect. A different equation for change in volume is suggested and the strain values are confirmed to be correct. The correct change in volume is approximately twice the previously calculated value.
  • #1
cicatriz
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Homework Statement



A rectangular steel bar has length 250 mm, width 50 mm, and thickness 25 mm. The bar is subjected to a compressive force of 450 kN on the 250 mm x 50 mm face, a tensile force of 450 kN on the 250 mm x 25 mm face, and a tensile force of 45 kN on the 50 mm x 25 mm face.

(a) Find the change in volume of the bar under the force system.


Homework Equations



The forces can be assumed to be uniformly distributed over the respective faces.

Take E = 200 kN/mm[tex]^{2}[/tex] and Poissons ratio = 0.26

The Attempt at a Solution



I have calculated [tex]\sigma[/tex]x = -0.036 kN/mm[tex]^{2}[/tex], [tex]\sigma[/tex]y = 0.072 kN/mm[tex]^{2}[/tex] and [tex]\sigma[/tex]z = 0.036 kN/mm[tex]^{2}[/tex].

With these values, using Hooke's Law: [tex]\epsilon[/tex] = 1/E[[tex]\sigma[/tex]1 - [tex]\upsilon[/tex]([tex]\sigma[/tex]2 + [tex]\sigma[/tex]3)] I have calculated:

[tex]\epsilon[/tex]x = 3.6x10[tex]^{-4}[/tex], [tex]\epsilon[/tex]y = -3.204x10[tex]^{-4}[/tex] and [tex]\epsilon[/tex]z = 1.332x10[tex]^{-4}[/tex].

Furthermore, using the equation for volumetric strain: [tex]\nabla[/tex]V/Vo = [tex]\epsilon[/tex](1-2[tex]\upsilon[/tex]) I have calculated the change in volume to be 25.92 mm[tex]^{2}[/tex]. This, according to the answer I have been provided with, appears to be incorrect.

I would appreciate any guidance with this.
 
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  • #2
Hi cicatriz, welcome to PF. Try using this equation for change in volume:

[tex]\Delta V=V_0\left[(1+\epsilon_1)(1+\epsilon_2)(1+\epsilon_3)-1\right][/tex]

I'm not sure how the other one was derived (hydrostatic pressure maybe?) but it doesn't look right. I checked your strain values and they look fine. The change in volume should be about twice what you got previously.
 
  • #3




Your calculations for the strains and stresses appear to be correct. However, the method you have used to calculate the change in volume may not be accurate for this specific scenario.

In order to accurately calculate the change in volume, we need to consider the effects of the different forces on each face of the bar. The compressive force on the 250 mm x 50 mm face will cause a decrease in length, while the tensile forces on the 250 mm x 25 mm and 50 mm x 25 mm faces will cause an increase in length. This will result in a change in volume.

To calculate the change in volume, we can use the following formula:

\Delta V = V_0 \left( \frac{\sigma_x}{E} + \frac{\sigma_y}{E} + \frac{\sigma_z}{E} \right)

Where \Delta V is the change in volume, V_0 is the initial volume of the bar, \sigma_x, \sigma_y, and \sigma_z are the stresses on each face, and E is the Young's modulus of the material.

Plugging in the values from your calculations, we get:

\Delta V = (250 mm \times 50 mm \times 25 mm) \left( \frac{-0.036 kN/mm^2}{200 kN/mm^2} + \frac{0.072 kN/mm^2}{200 kN/mm^2} + \frac{0.036 kN/mm^2}{200 kN/mm^2} \right) = 25 mm^3

This is the correct change in volume of the bar. I would suggest double checking your calculations and equations to ensure accuracy. Additionally, it is always helpful to double check your answer with the provided answer to ensure accuracy.
 

1. What is volumetric strain?

Volumetric strain is a measure of the change in volume of a material due to external forces or stress. It is expressed as a decimal or percentage.

2. How is volumetric strain calculated?

Volumetric strain is calculated by dividing the change in volume (ΔV) by the original volume (V). This can be expressed as (ΔV / V) or as a percentage by multiplying by 100.

3. What are some real-world examples of volumetric strain?

Volumetric strain can be seen in many everyday objects and phenomena such as balloons expanding when filled with air, rubber bands stretching, or even the ground shifting during an earthquake.

4. How does volumetric strain differ from other types of strain?

Volumetric strain specifically measures the change in volume of a material, while other types of strain, such as shear strain, measure the change in shape or size of a material.

5. What factors can affect the volumetric strain of a material?

The volumetric strain of a material can be affected by factors such as the type and magnitude of external forces or stress, the properties of the material itself (such as elasticity), and the temperature and pressure of the environment.

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