Hertz Contact pressure when R1=-R2 4

[IJR]

Pals

I have a cylinder to stay fit in a groove. According to text books the effective radius is

1/R=1/R1+1/R2

and the effective bearing width b happens to depend on that value

now when R2=-R1 this one becomes 0. I expect cylinders in groove to reduce contact stress but I can not find the number.

I can not find a way around that


respects

IJR

 

[prex]

The theoretical contact stress is in fact zero under those conditions, as you have a cylinder inside a hole with the same diameter.

prex

http://www.xcalcs.com

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http://www.megamag.it

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: Air bearing pads

 

[IJR]

prex

what would be the practical value to be considered or at least assumed?

respects
ijr

 

[prex]

If you have a round in a hole with nominally the same diameters, then Hertz contact formulae won't give you anything useful.
In structural calculations that situation, that is typical for bolts and other fasteners in shear, is handled with a nominal contact pressure, equal to the shear force divided by the diameter and the thickness of the plate with the hole; this stress is compared to an allowable stress that may be typically two times the normal allowable (that depends on the applicable code).

prex

http://www.xcalcs.com

: Online engineering calculations

http://www.megamag.it

: Magnetic brakes for fun rides

http://www.levitans.com

: Air bearing pads

 

[dinjin]

Can you simply enter
a value for R2 equal to (R1 - x) and input
x=.001, x=.0001, x=.00001, etc to approximate
some value?

 

[IJR]

Thanks prex, dinjin

prex's comment that as R1 approaches -R2 contact stresses gets down to zero is a very good comment. Dinjin, that limit approximation wont be helpful and it will simply bring zero contact stress and some very large contact width.

I found out that when R1 approaches -R2 then we stop talking of contact stresses and start talking of bearing stresses using area of projection of contact surfaces.

And so it becomes a simple bearing problem

Here is a good and simple book(though it discusses elastomeric bearings more than it does mechanical bearings):

Bearings in Structural Engineering by J.E.Long, John Wiley and Sons ISBN 0-470-54456-2

respects
IJR

 

[BWAppliedKinetics]

Depends what your application is, for general strength analysis it is usual to approximate bearing stress as simply the load over the simplified bearing area (hole/pin diameter X hole depth). Anything more than that and you are getting into some nifty equations or FEA.

BW

http://www.akeng.com

 

[prost]

Back to the problem with the negative+positive radii of curvature that balance each other. You may have guessed already that the Hertzian theory has many assumptions in its derivation, and that a key assumption (relative size of the contact area and the radii of curvature of the contacting bodies) is violated when you have a cylinder in contact with a groove, each with approximately the same radius. I looked in Roark's--they show the case of cylinder in a groove, but you don't know from the table that if the two diameters (or radii) are close, that the formulae don't work. There is a nice treatment of the case of a cylinder in contact with a groove or hole or the same diameter in KL Johnson's contact mechanics book--you might not find your answer, but you will find excellent references cited in Johnson to correctly analyzing your particular case.

As far as the suggestion of using load over a simplified bearing area--that will work if you don't care what the stresses around the hole are; you are merely transferring load through the hole or groove. But that won't work if you need to know the stresses around the groove, or you are analyzing the case of say a roller bearing in its groove.

 

[BWAppliedKinetics]

prost you are correct in the applicability, I am assuming that if he is looking at contact stress than he is not interested in other failure modes (or is handling them separately).

For the OP, if you are concerned with failure of the joint remeber that for "pin in plate" there are a half dozen different failure modes that can be checked by hand (or all at once with FEA).

Brad Woodward

http://www.akeng.com

http://www.appliedkinetics.ca

 

[dinjin]

I think the railroad cars used bearings similar to
what you are describing and at one time used fat
to lubricate the journal bearings. They finally went
to tapered roller bearings in these applications.
Interesting problem!!!