# Contact between two rings or two cylinders

Hello,

I would be interrested in comments, references, books, papers and web pages regarding the problem of mechanical contact between two rings.

The attached picture describes the geometry of the problem: a ring (tube) is resting on the bottom of a larger ring (tube). These two rings are in contact in a certain zone, the size of which depends on the relative size of the rings and the forces acting on the system. Mainly the own weight of the rings and the contact pressure are involved. The inner ring is extremely heavy and the outer ring is used to reinforce it. Eventually, the large ring may rest on two rollers (a more complicated problem).

I would like to calculate the deformations of the inner ring assuming a very high rigidity of the outer ring. A more complicated problem might involve the deformation of the outer ring, assuming it is resting on two rollers. I am also interrested in an analytical expression for the contact pressure and if possible for the angle of contact.

PS1: I don't know how to insert a picture in my post. Any suggestion?
PS2: But the attachment is shown! Is there a way to insert it somewhere in the text ?

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Cylinder on Cylinder contact

I recommend "Mechanical Engineering Design" 7th edition by Shigley, Mischke, and Budynas. ISBN: 007-252036-1

Section 4-20 has some great preliminary analysis tools and formulae for contact stresses of spheres on spheres and cylinders on cylinders (which I think is what you have). Let me know if you can get your hands on a copy. If not, I'll give you the formulas.

Cheers...

PerennialII
Gold Member
.... considering coupling the different deformations a pretty simple finite element analysis will do, and in this arrangement attaining a convergent solution using pretty much any commercial etc. code will be tolerably easy, can do with an infinitesimal contact formulation using quadratic bricks in quite a crude mesh.

I agree with you PerennialII.
But I am less interrested in numerical solutions than in theory itself.
This is not a problem for a living but for the fun only.
Numerical solutions maybe useful as a check.

For thin shells, I am even not sure there will be only one contact region. There could be detachment in between two contact zones. For a very low rigidy of the inner ring, clearly, the solution might become quite complicated (complex folding, is it called 'buckling' maybe?)(even if the inner ring fits the outer ring!). I would be happy already to have an analytical understanding in the simplest case: inner ring nearly fitting the outer ring and a 'reasonable' rigidity. It would be even nicer to detect when a detachment might appear, and then would forget about this problem. Introducing the penetration of the external rollers effect up to the inner ring would be interresting too: I would like to understand the impact on the contact forces too.

However, I also have another interrest with this problem which is closer to numerical. I would have fun to express it as a quadratic minimisation problem with constraints expressing the 'positivity' of the contact forces. This is probably easier to develop with some public domain NLP code.

PerennialII
Gold Member
lalbatros said:
I agree with you PerennialII.
But I am less interrested in numerical solutions than in theory itself.
This is not a problem for a living but for the fun only.
Numerical solutions maybe useful as a check.

For thin shells, I am even not sure there will be only one contact region. There could be detachment in between two contact zones. For a very low rigidy of the inner ring, clearly, the solution might become quite complicated (complex folding, is it called 'buckling' maybe?)(even if the inner ring fits the outer ring!). I would be happy already to have an analytical understanding in the simplest case: inner ring nearly fitting the outer ring and a 'reasonable' rigidity. It would be even nicer to detect when a detachment might appear, and then would forget about this problem. Introducing the penetration of the external rollers effect up to the inner ring would be interresting too: I would like to understand the impact on the contact forces too.

However, I also have another interrest with this problem which is closer to numerical. I would have fun to express it as a quadratic minimisation problem with constraints expressing the 'positivity' of the contact forces. This is probably easier to develop with some public domain NLP code.
Yeah, you can get the analytical from most mechanics books along with the accompanying theoretical base treating contact problems like LunchBox suggested. Actually many FE theory manuals have concise & pretty complete descriptions of generalized contact mechanics as well.

Likewise for the supported case, two contact zones with separation in between. I think one could construct a simple model of this one as well by coupling the overall deformation of the tubes to the contact problem, to get an approximate closed form solution. Kind of a bimaterial simply supported beam with contact/foundation thing.

The optimization analysis sounds interesting, although I don't quite gather are you describing a typical constrained modeling approach to contact modeling or it + another optimization problem imposed on top of the numerical contact problem? Probably the former ... got carried away ... so presenting it as an optimization problem for example using a Lagrangian formulation and solving it using nonlinear optimization methods. That would be interesting, taking a somewhat different approach than typically in for example FEA. And of course contact problems involving very hard & soft materials are always interesting, nothing beats a problem where the other material completely "flows", buckles etc. with excessive deformations (other than can be a real joy numerically at times) .