What shape will a bent wire take?

[Screwdriver]

Homework Statement

Not homework, but something I'm interested in finding out. The setup is a flexible wire with left endpoint fixed at $x=0$ and right endpoint at $x=L$. You push the right endpoint with some horizontal force directed towards the left endpoint which will move the right endpoint to a position $x=L-a$ and cause the wire to deform and bend into some sort of downwards-facing parabola thing. What I want to know is what function actually describes the shape of the wire.

Homework Equations

Material from classical mechanics, hopefully.

The Attempt at a Solution

I believe that the solution might have something to do with the principle of stationary action. My original idea was to simply find the function of minimum arc length that joins two points, with the constraint that the arc length has to be greater than just the straight line distance between the two points using what little I know about calculus of variations. I couldn't figure out how to actually impose that constraint though, I just kept getting $f(x)=mx+b$. Then I tried to use an approach similar to the derivation of the wave equation where you consider a small chunk of the wire, resolve the tensions, and then use some shrewd approximations to get an equation out of it. This didn't work though, because all I ended up doing was finding the wave equation again. I think there might be other forces besides the tension on a small chunk since the wire will want to return to a straight line so I could maybe throw in a spring-like force but I wouldn't really know where to put it.

Thanks for any suggestions.

 

[PhanthomJay]

Initially, you might want to assume that the wire is taut enough so as to neglect any initial sag (wire length = span length), but in reality it must have some sag since it has weight, and take the shape of a catenary curve.

 

[PhanthomJay]

Initially, you might want to assume that the wire is taut enough so as to neglect any initial sag (wire length = span length), but in reality it must have some sag since it has weight, and it takes the shape of a catenary curve. When the support at one end is moved to the other, the sag increases by an amount depending on the amount of horiz displacement of the support, but the shape is still that of a catenary. If the displacement is small enough, the catenary can be approximated by a parabola for ease of calculation to avoid the messy hyperbolic functions associated with the catenary. Sags must be less than say 10 percent of the span length to make this approximation.

A poor attempt at an edit from my phone

 

[Screwdriver]

Thanks, I understand what you're saying and I'm familiar with the derivation for the shape of a hanging rope, but (and I probably should have mentioned this) the wire is in a horizontal plane. http://postimage.org/image/bw5kwmt37/ what it should look like at first (my sock is on the fixed end and the bent part wouldn't be there). Then after pushing it, it should look something like http://postimage.org/image/a81gq9w6x/.

It certainly seems to be some kind of hyperbolic cosine, but I'm not sure.

 

[Redbelly98]

Initially, you might want to assume that the wire is taut enough so as to neglect any initial sag (wire length = span length), but in reality it must have some sag since it has weight, and it takes the shape of a catenary curve. When the support at one end is moved to the other, the sag increases by an amount depending on the amount of horiz displacement of the support, but the shape is still that of a catenary. If the displacement is small enough, the catenary can be approximated by a parabola for ease of calculation to avoid the messy hyperbolic functions associated with the catenary. Sags must be less than say 10 percent of the span length to make this approximation.

The catenary shape is for ropes or lines with negligible stiffness, i.e. only weight of and tension in the rope is considered.

If it is a stiff wire of uniform cross section, I recall from my brief look into such problems that the shape is some sort of polynomial, perhaps 4th degree or could be lower. But my memory is hazy.

Since bending of stiff beams and wires would be covered in a mechanical engineering curriculum, and not in a typical physics major curriculum, I'll go ahead and move this thread to the Mechanical Engineering forum (was in: Homework & Coursework, Intro Physics); hopefully one of the ME regulars can comment.

 

[256bits]

Screwdriver
What you are referring to is similar to a slender column or strut in compression, but then you do say flexible wire so I am not too sure.
The shape can be of several depending upon your end constraints.
http://www.mitcalc.com/doc/buckling/help/en/buckling.htm

 

[AlephZero]

Most of the previous answers are largely irrelevant, because "simple" beam bending and buckling theory assumes small displacements and strains.

The way I interpret your question you can assume the strains are small, but the displacements (and specifically the rotations) are not. But the good news is that the problem is statically determinate, so for any shape of curve you can find the bending moment at any point along the wire, and hence the curvature, and get a differential equation to solve. But don't make the same approximations as in Euler-Timoshenko beam theory, or you will get the answer to the wrong question!

Whether the equation have an analytic solution is another question, though. http://en.wikipedia.org/wiki/Elastica_Theory has a few references to the literature.

 

[Screwdriver]

Most of the previous answers are largely irrelevant, because "simple" beam bending and buckling theory assumes small displacements and strains.

The way I interpret your question you can assume the strains are small, but the displacements (and specifically the rotations) are not. But the good news is that the problem is statically determinate, so for any shape of curve you can find the bending moment at any point along the wire, and hence the curvature, and get a differential equation to solve. But don't make the same approximations as in Euler-Timoshenko beam theory, or you will get the answer to the wrong question!

Whether the equation have an analytic solution is another question, though. http://en.wikipedia.org/wiki/Elastica_Theory has a few references to the literature.

Interesting. I flipped through this paper in the link and it says that "... the differential equation for the elastica, expressing curvature as a function of arclength, are equivalent to those of the motion of the pendulum..."

An unexpected fact, but disconcerting since the pendulum DE is unsolvable without resorting to aggressive elliptic functions.