# Cantilever cable

Hello, I am a PhD student in textiles. I was wondering whether anyone has any experience with cables and stuff. For my PhD, I have implemented a Cosserat model for these one dimensional structures as described by the equations of motion given in this article. Ofcourse one tries to validate their model, so I took a few basic static and a dynamic problems out of elasticity theory that have an analytical solution - like e.g. what is the deflection curve of a cantilevered beam with a load at its end, a uniformal load, that kind of stuff - and compared the simulation results. The comparison was very good.
However, for one particular problem, the results of my simulations differ significantly from what some other researchers mention. The particular problem is that of a horizontal (= x-direction, vertical = z-direction) cantilevered cable without any twist, solely loaded by its own weight, of which the ends are moved towards each other. Inituitively, one expects that the deformation will occur only in the vertical xz-plane in which the cable is located, as all forces are in that plane. This is also the result of my simulations. However, other researchers seem to say that there will develop a deformation that is out of plane and loop formation will occur. A movie of which can be viewed at this site:
http://seesar.lbl.gov/ANAG/staff/bono/html/sample_results.html [Broken]
What I am asking is this: is there anyone busy with the same kind of things that can tell me whether a cantilever cable hanging under gravity, with two ends approaching each other, will indeed deform out of the xz-plane?

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FredGarvin
I do not have any experience in this area. I will just share some thoughts/opinions.

I would think that the only way to get a deformation out of plane like that would be either annother external force which is not part of the model or the cable develops a spring constant that is a function of it's buckling. Is there any mention of the reaction forces at the supports as a function of the distance the supports are apart? I woul be curious to hear an explaination for this added torque.

Well I do consider it possible that there will be an out of plane deformation even if all forces, speeds and positions are originally in a single 2D plane. After all, a straight beam that only experiences forces perpendicular to its ends will after all eventually buckle out as well, even though all forces, speeds and positions were eventually located in a 1D line, simply because the smallest deformation or offset in another dimension eventually will start the buckling from a critical load on.

So as a possible explanation for this added torque: it may be that a small offset torqua or displacement in the y direction reenforces itself and that such an out of plane deformation is a lower internal energy configuration. But even when I introduce a small offset twist or an offset position in the y-direction, I get none of that dramatic out of plane deformation. This is why I wonder whether these other people are wrong or whether there is an error in my model, despite the good agreements with some practical issues in elasticity theory - although only one dynamic problem has been tested - that have analytical solutions.

I think to have read somewhere Euler gave analytical solutions to the postbuckling behaviour of such beams in terms of elliptic integrals but can't remember where to have read that and Google searches using "Euler" and "postbuckling" result in a lot of hits but no useful ones so far. This was however for only axial loading, although I don't expect the influence of gravity to be that big as to the result...

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PerennialII
Gold Member
Wow ! Talk about 'post-buckling' behavior, the behavior really gets quite complex ('somewhat' of a strain thinking it intuitively) when the displacements & rotations & deformations occur in the range you've under study. Really interesting problem.
Simon666 said:
So as a possible explanation for this added torque: it may be that a small offset torqua or displacement in the y direction reenforces itself and that such an out of plane deformation is a lower internal energy configuration. But even when I introduce a small offset twist or an offset position in the y-direction, I get none of that dramatic out of plane deformation. This is why I wonder whether these other people are wrong or whether there is an error in my model, despite the good agreements with some practical issues in elasticity theory - although only one dynamic problem has been tested - that have analytical solutions.
I think to have read somewhere Euler gave analytical solutions to the postbuckling behaviour of such beams in terms of elliptic integrals but can't remember where to have read that and Google searches using "Euler" and "postbuckling" result in a lot of hits but no useful ones so far. This was however for only axial loading, although I don't expect the influence of gravity to be that big as to the result...
Remember seeing those post-buckling analyses somewhere as well, I'll see what have on my shelf, have a couple of volumes solely about buckling so would think they'd include those as well.

PerennialII
Gold Member
Couldn't come up with the Euler ones was thinking back, did come up with some relevant post-buckling (elastic, elastic-plastic and plastic post-buckling) articles related to beams. Analytical, "semi-analytical" and numerical stuff ... I can hook you up with the e-articles if you still have use for them (do provide quite a bit of references, a few of them are of review type focusing on theoretical basis of post-buckling and as such can be of greater use).

PerennialII said:
Couldn't come up with the Euler ones was thinking back, did come up with some relevant post-buckling (elastic, elastic-plastic and plastic post-buckling) articles related to beams. Analytical, "semi-analytical" and numerical stuff ... I can hook you up with the e-articles if you still have use for them (do provide quite a bit of references, a few of them are of review type focusing on theoretical basis of post-buckling and as such can be of greater use).
I have in the meantime contacted a few authors of relevant publications and have received some interesting answers, but anything that can help is welcome. If you have (the names of) such articles, I'm still interested.

PerennialII
Gold Member

Not exactly what I was looking for, but thanks for the effort. I've in the meantime found some more relevant publications like by Sachin Goyal, http://www-personal.engin.umich.edu/~sgoyal/mypapers/DETC2003-MECH-48322.PDF [Broken], that listen closer to my interest. It's one of his other ones which mentions two interesting things:

1. That in case there is no small disturbance, the planar solution is the one that is obtained.
2. In case there is a small disturbance, the point of transition from planar to nonplanar solution will become later and later if the ends of the rod are approaching each other faster, due to inertia.

Since I am using an explicit formula (Euler integration) for solving the relevant equations, I hence have a limited time step due to stability reasons and hence my loading rate needs to be very high to have a reasonable computation time. I think that hence my model may be accurate but due to the limited time step and fast loading rate and insufficient disturbance, the nonplanar solution doesn't develop in my case.

i am now trying to introduce a small out of plane deformation and see whether the system will then spontaneously evolve to a lower energy nonplanar solution or will - probably wrongly - return to the original planar solution.

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PerennialII