Formula for bending a rod in the elastic range

In summary, there is no formula for calculating the radius that a rod of a given radius can be bent around while staying in the elastic range. This depends entirely on the elastic limit, and that was not in your assumed input.
  • #1
pistorinoj
5
0
Is there a formula to calculate the radius that a rod of a given radius can be bent around while staying in the elastic range?

For example, if I had a rod that was 1/4" in diameter and made of HDPE (which I think has a modulus of elasticity of 0.8 GPa), how would I calculate the minimum radius the rod could be bent around while still staying in the elastic range so that it could be unwound without deformation?
 
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  • #2
The short answer is no. This depends entirely on the elastic limit, and that was not in your assumed input. The modulus of elasticity does not give you the necessary information.
 
  • #3
Dr.D said:
The short answer is no. This depends entirely on the elastic limit, and that was not in your assumed input. The modulus of elasticity does not give you the necessary information.
Sorry, but I am not following you.

Are you saying that even knowing all the characteristics of HDPE, there is no formula for calculating the minimum bending radius?
Or are saying that I did not provide enough information in my question?

I do see places providing the following data for HDPE:
Film Tensile Strength at Yield, MD
21.0 MPa 3050 psi

Film Tensile Strength at Yield, TD
23.0 MPa 3340 psi
 
  • #4
I saw no mention of yield values in your original post, only the modulus of elasticity.

The initials HDPE mean nothing to me, but I presume that you are talking about one of the many plastic type materials in use today. There is some question as to just how well classical failure theories apply to such materials. I suspect you need something in the way of very new information, but I do not have such.
 
  • #5
Dr.D said:
The initials HDPE mean nothing to me, but I presume that you are talking about one of the many plastic type materials in use today.
HDPE is High Density PolyEthylene. Here in the States it is most commonly seen as gallon jugs for milk.
 
  • #6
I think milk jugs are usually made from Low Density PolyEthylene (LDPE) but I will take a formula that works for either one.
HDPE is also commonly used in making robot parts.
 
  • #7
There are simple and more complicated ways of getting an answer to your problem depending on the accuracy required .

Easiest way to get a ball park answer is to consider the bending of a cantilever made from the chosen material .

Work out the local radius of curvature at the fixed end when the maximum fibre stress is just at yield .

To be sure that the rod will unbend properly you would need to make the radius of curvature to be used in practice a little larger than the one as calculated above . This is particularly important for plastic materials like your HDPE .
 
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  • #8
Beam bending theory is based on the relationship between local curvature and local bending moment .

Using this relationship it would be relatively easy to derive a formula relating local radius of curvature to maximum fibre stress for any given cross section geometry .

Please let us know if you want to look at this more analytic method for solving your problem .

Euler Bernoulli beam theory
 
  • #9
Just did a sample problem to see what sort of radius of curvature is involved . Rod is 6,35 mm dia 75 mm long HDPE

HDPE rod.jpg
 
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  • #10
Nidum said:
Beam bending theory is based on the relationship between local curvature and local bending moment .

Using this relationship it would be relatively easy to derive a formula relating local radius of curvature to maximum fibre stress for any given cross section geometry .

Please let us know if you want to look at this more analytic method for solving your problem .

Euler Bernoulli beam theory
Yes, that would be great. Ideally, I could review a list of materials and find one that can be wound in the smallest space balancing stiffness and the overall diameter of the material. For example, I could use acetal (delrin), PTFE, composites, etc. at smaller diameters if those materials could be wound elastically in a smaller space yet be stiff enough for my purposes.

Could you show me how you calculated R156 for the HDPE size you used?

Finally, I see various references to engineering handbooks possibly having tables of this information. Is there one that you know of?

Thanks so much, this is really helpful.
 

1. What is the formula for bending a rod in the elastic range?

The formula for bending a rod in the elastic range is known as the Euler-Bernoulli beam theory. It is expressed as M = -EI(d^2y/dx^2), where M is the bending moment, E is the modulus of elasticity, and I is the area moment of inertia of the cross section of the rod. This formula applies to rods that are experiencing a small amount of bending stress within their elastic limit.

2. How is the modulus of elasticity determined?

The modulus of elasticity (E) for a rod can be determined by conducting a tensile test. This involves applying a known amount of force to the rod and measuring the resulting elongation. The modulus of elasticity is then calculated as the ratio of stress to strain, where stress is the applied force divided by the cross-sectional area of the rod and strain is the change in length divided by the original length of the rod.

3. What is the elastic range of a rod?

The elastic range of a rod is the range of stress or force that a rod can withstand without causing permanent deformation. This range varies depending on the material and the dimensions of the rod, and is typically determined by conducting tensile tests and plotting stress-strain curves. The elastic range is important to consider when determining the maximum bending force that a rod can withstand before permanent deformation occurs.

4. What is the significance of the Euler-Bernoulli beam theory?

The Euler-Bernoulli beam theory is significant because it provides a mathematical model for predicting the bending behavior of rods within their elastic range. This theory is widely used in engineering and design to ensure that rods and other structural elements can withstand the expected bending forces and maintain their shape and structural integrity.

5. Are there any limitations to using the Euler-Bernoulli beam theory for bending rods?

While the Euler-Bernoulli beam theory is a useful tool for predicting the bending behavior of rods, it does have some limitations. For example, it assumes that the rod is homogeneous, isotropic, and has a constant cross-sectional area. In reality, many rods have varying cross sections and may be made of composite materials, which can affect their bending behavior. Additionally, the theory does not account for shear stresses, which may be significant in certain cases. Therefore, it is important to consider these limitations and use the theory with caution when designing and analyzing bending rods.

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