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I am wondering if there is a formula (or how to derive one) for calculating the minimum radius a rod (with a given radius/diameter) can be bent around while staying in the elastic range. Obviously, this will depend on the modulus of elasticity for the material.

I am searching for a general formula for this rather than a specific solution for a particular material. With a general formula, I can evaluate trade offs using different materials (with different moduli), different rod radii, and different radii of bending.

Stated more simply, if I wanted to wind a rod of radius X made of a material with modulus of elasticity M around a spool, how would I calculate the minimum radius S of the spool?

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  • $\begingroup$ Using small angle (large radius) approximation is easier to drive compared to the more accurate and realistic large deformation model $\endgroup$ – John Alexiou Jun 25 '17 at 10:06
  • $\begingroup$ @nobody its not clear what you want for your bounty. There is nothing essentially different for a composite. $\endgroup$ – agentp Nov 19 '17 at 23:26
  • $\begingroup$ "Minimum radius" sounds weird it throws me off. lol $\endgroup$ – Zero Nov 20 '17 at 3:19
  • $\begingroup$ @agentp I just wanted an answer. The one which now newly exists looks fine, but because I can't judge the validity of its assumptions I'm going to wait with awarding the bounty in case something more plausible pops up. If nothing changes, that answer is getting the bounty in 4 days. $\endgroup$ – Nobody Nov 21 '17 at 10:54
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    $\begingroup$ @Nobody perhaps I wasn't clear in my last comment to the answer below. The fact that you are bending a rod to a constant radius means that by definition, there is no shear deformation. This makes the assumptions behind Euler-Bernoulli bending valid. Note that this beam theory is used to design 99% (probably more!) of the structures you see everyday so I would say that engineers have a fair bit of confidence in it. Let me know if you want any further clarification. $\endgroup$ – Robbie van Leeuwen Nov 21 '17 at 16:52
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If we're staying within the realms of beam theory, we can go with this approach which is valid for any material that exhibits linear-elastic behaviour before yield:

The curvature of a beam is related to the applied moment and it's flexural stiffness:

$$ \begin{align} \kappa = \frac{M}{EI} \end{align} $$

where, for a cicular rod, $I = \frac{\pi d^4}{64}$. The curvature is also equal to the inverse of the bent radius, giving the following:

$$ \begin{align} R = \frac{1}{\kappa} = \frac{EI}{M} \end{align} $$

For a rod of circular cross-section, the stress at the extreme fibre is related to the applied bending moment and the elastic section modulus:

$$ \begin{align} \sigma = \frac{M}{Z} \end{align} $$

where for a circular rod, $Z = \frac{\pi d^3}{32}$. This gives:

$$ \begin{align} \therefore M = \frac{\sigma \pi d^3}{32} \end{align} $$

Therefore, if a given material has a certain yield strength, $\sigma = f_y$, the minimum radius can be determined through the substitution of the above equations:

$$ \begin{align} R_{min} = E \, \frac{\pi d^4}{64} \frac{32}{f_y \pi d^3} \end{align} $$

$$ \begin{align} \boxed{\therefore R_{min} = \frac{E d}{2 f_y}} \end{align} $$

where $E$ is the elastic modulus of the material, $d$ is the diameter of the rod and $f_y$ is the yield strength of the material.

Refer to this solution to question 6a on page 8 for a similar solution for a rectangular section. Note that the end result is the same for both a rectangle and circle as both have a distance of $D/2$ from the neutral axis to the extreme fibre.

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  • $\begingroup$ "If we're staying within the realms of beam theory" sounds like a caveat. Could you explain it? When would you use Euler–Bernoulli and when Timoshenko beam theory? Or anything else, those are just the two I found when searching for the solution myself. $\endgroup$ – Nobody Nov 18 '17 at 21:12
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    $\begingroup$ For the bending of a rod it is definitely a sound assumption. Typically Euler-Bernoulli beam theory (which is applied to this answer) is used for relatively slender beams and Timoshenko beam theory used for relatively deep beams. The distinction comes about as Euler-Bernoulli theory ignores shear deformation, which can be significant when you have a relatively deep beam (think a concrete wall spanning between two columns). For the bending of a rod to a radius, I would hang my hat on the Euler-Bernoulli formulation presented above. $\endgroup$ – Robbie van Leeuwen Nov 18 '17 at 21:20
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    $\begingroup$ There is a similar formulation (a rectangle not a rod) in the answer to question 6a of this exam, on page 8. Here the curvature is derived and not the radius. $\endgroup$ – Robbie van Leeuwen Nov 18 '17 at 21:29
  • $\begingroup$ Relatively slender/deep compared to the radius of potential bends? Compared to its length doesn't make much sense, does it? That exam/solution might also be a valuable addition to the answer body. $\endgroup$ – Nobody Nov 18 '17 at 21:39
  • $\begingroup$ Typically it is a comparison of depth to span/length for transversely loaded beams. You are right that it doesn't apply to the radius of bend. In fact in your case, bending a rod into a circular shape should only induce bending moment and negligible shear, meaning that the Euler-Bernoulli assumption of negligible shear deformation is completely justified. $\endgroup$ – Robbie van Leeuwen Nov 18 '17 at 21:44

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