Can anyone help with thoughts on this problem?

What would be the shape of curvature, (approximate will do though precise would be nice if it's simple), of a thin flexible rod compressed at its ends via contact points of a non-constraining type and within elastic deformation limits? (Think Star Wars IV, Hans Solo and team in the trash compactor, trying to use the metal rod to stop the walls moving inwards, or a bow (as in "and arrow") created by bending a straight uniform stick and restraining the ends with a string.)

I want to ignore effects of gravity, it's neither too long nor ultra flexible.

By the contacts at each end I mean holding and compressing the rod with just a plain contact point like putting it internally across the jaws of a vice and closing it rather than clamping each end in a vice so they cannot change angle. Effectively I'm looking for a simple plain curve with no turning points.

The elastic limits restriction means simply for relatively small deformations which allow full returning to its original straight condition. No area of weakness is allowed to create a tighter bend locally.

Is this going to be pretty close to deformation to an arc of a circle?

 

I've bent sticks before and going off my experience I think what you're going to see would be more accurately described as parabola

 

I agree, it will not be an arc. The radius of curvature will be lower in the center.

See here, for example.

 

Thanks both of you for coming in on this, you've given me the answer I want. The analysis Wayne posted is great and very thorough. My maths is very good but lacks the depth in that field nowadays, though I can follow most of what is there. Anyway, it is sufficient to prove what we suspected, it sure isn't a close approximation to an arc for anything beyond the tiniest deformations. Strantor's view that it is closer to a parabola is definitely a better approximation for the small case. Thanks again for the insight and info.