# The math involved in bending a rod

1. ### camacru

13
I have a door lever which has a 90 deg. bend. I need to calculate the length that this lever would have to be if it were in the form of a straight rod, prior to being bent.

I know I have all the input data, I'm just not sure how to express this algebraically. For example, I know that the inner radius of the bend is 0.416", and the outer radius is 0.892". Based on this data, how can I calculate the length of the lever prior to being bent?

2. ### zgozvrm

754
That would depend on how much the rod material can stretch and/or compress. Assuming no stretching, then the outer radius would consist of a quarter-circle having the same radius (0.892"). That arc would then have a length of 1/4 of the circumference of a 0.892" radius circle. Calculate that then add the straight lengths to it.

$$C=2\pi r$$

$$\frac{1}{4}C=\frac{\pi r}{2}$$

If the material stretches, this won't be accurate though.

It would be much easier to take a piece of rod of a known length, bend it, and cut off the amount you don't need. Subtract that amount (that was cut off) from your original length. That will give you the total length needed.

3. ### hamster143

986
It is a nontrivial engineering problem and the answer will depend on properties of the material.

To the lowest approximation, it would probably be most accurate to assume that the outer half of the rod will stretch and the inner half will compress. So, the total length ~ straight lengths + pi r_c/2 where r_c = (0.416+0.892)/2.

My advice would be to do this experimentally, as the previous poster suggested.

4. ### camacru

13
Thanks for the help guys. The lever is actually made out of brass, so there is definitely some degree of stretching and compression. Here's the problem. Basically, what I am doing is a simple exercise in reverse engineering. I have a finished lever sample, and I am trying to duplicate it in Solidworks, but I need to model it in the form of a straight rod because it will be lathed then bent. I modeled the lever, applied a flex modifier and bent the digital rod to 90deg. However, when I supplied the drawings to the shop, the bend was completely off. Perhaps the problem is as simple as not having used the proper bending wheel during the bending process. Nonetheless, I want to re-evaluate my design, and mathematically calculate the dimensions to determine if my model was flawed to begin with. Do you think this can be done without having the exact physical properties of the material?