What is the required force to bend a rod into a horseshoe shape?

[DeereAdam]

I need to calculate the force necessary to bend a rod 180 degrees around a radius of 1.75 meters. Basically bending the rod into a horseshoe. I have looked at the simple bending calculations and I am sure this must be similar to those but I do not see anything for a constant radius bend. Any help is greatly appreciated. The materials to be bent are copper and aluminum. Thanks

 

[OldEngr63]

If you are talking about permanently forming the rod into the bent form, you will need some information about the yield point for the material. Permanent forming involves large plastic deformation, so you will need to determine the amount of strain you wish to create in the rod, and then see where that puts you on the stress--strain diagram. The required force will be that which develops the force for this stress, but you must allow for some amount of spring back. This is essentially a manufacturing engineering problem, one that cannot be entirely calculated but only estimated.

 

[DeereAdam]

I believe for the diameter I am looking at bending around the bend will be entirely or mostly elastic (depending on the diameter of the rod)... I will research the stress strain tables further though... are there any calculations for an elastic bend? thanks for the help

 

[OldEngr63]

There is a question here that needs to be resolved, even if this can be accomplished with a fully elastic bend. Do you want it to be a fully elastic bend, in which case it will fully spring back on release of the force, or do you want to form it to the bend radius so that it does not spring back? These are two different problems.

Assuming that it will bend to that radius with only elastic deformations, then you are talking about the problem of the elastica, a problem of classical elasticity theory that involves elliptic integrals in the solution.

The forming problem is another matter altogether.

 

[AlephZero]

Assuming that it will bend to that radius with only elastic deformations, then you are talking about the problem of the elastica, a problem of classical elasticity theory that involves elliptic integrals in the solution.

You don't need elliptic integrals for this situation. If you apply equal and opposite moments (and no shear forces) at the ends of the rod, the bending moment will be constant along the length and the deformed shape will be (exactly) an arc of a circle. You can find the curvature from beam bending theory.

Incidentlally this sort of question is a good test of a finite element program that claims to do large displacement small strain analysis of beams. You should be able to deform a straight beam into a complete circle.

You can get an estimate of the force for plastic deformation of the ring, from the force needed to reach the yield stress. But as OldEngr said, a complete solution including the elastic springback of the material to the required shape would be a seriously hard problem.

 

[OldEngr63]

Small curvature theory does not apply to this problem, so simple beam theory goes out the window. It is necessary to use the full expression for the curvature, and that is what gets you into elliptic integrals for the solution. See for example Theory of Elasticity by Timoshenko for the elastica.

 

[DeereAdam]

The bend should be ( I have not checked all diameters of rod ) elastic. I appreciate all the input as I am no expert in this area but I am studying on it to complete this calculation. In the end I hope to enter a formula into excel to calculate the force required based on the diameter of the rod. I hope this clarifies more on this problem. thanks everyone