How to Calculate Change in Radius of a Spherical Steel Ball on Loading

It is somewhat more complicated than you imply.

Are the load and reactions applied at points or do they have an area. Young's Modulus certainly applies, but there is likely to be a "shear lag" as the load spreads through the increasing width toward the middle. And then there is Poisson's Ratio to consider.

Yes, the load is applied at a point. But then, once loaded, there will be a small area of contact. This kind of contact area is present in all the point contacts, and are safely neglected, right?

The more I think, the more complicated it gets.

Thanks a lot for this suggestion. I looked up Hertz contact stress and now i can calculate the stress generated in the sphere.

I am not able to upload a sketch right now but the basic setup is:

there are two hollow cylinders, one inside the other. It is required to reduce the friction as the inner cylinder slides against the outer one. We thought of making use of small spherical steel balls (about 1.5mm dia) between these cylinders (effectively a roller bearing). And to ensure pure rolling without sliding (of the steel balls), there has to be a normal force of around 60 Newtons acting on it.

To maintain this force, it was decided to have an interference fit and for this i needed to know the deformation and the stress generated in the ball.

Thanks again.

There is a problem related to your design. If balls can touch each other then you get two balls running one AGAINST the other. You MUST introduce a 3rd cylinder about 1 mm wall with radial holes in which you place the balls. This "separator" should be of a material with a low friction coefficient. You can either use cooper alloys or even high quality (from mechanical point of view) plastics. If your force is radial and does not change direction only the balls in an angle of +/- 45 % from the force acting plane will be loaded (cosine effect). I would try to place the balls in such a way that they will not be on same path in axial movements. Take care to avoid in the pre-loading fit the possible plastic deformations which will compromise your results.

Have you looked at these?

I don't know anything about modulus', equations or the like, but there are some other things that need to be considered in whatever the equation turns out to be.

I used to work in a factory that manufactured ball bearings. They were indeed steel balls, but they came in different grades, some were heat treated, some were not, etc. I would think that the properties of the steel ball itself would play heavily in the equation that you're after.

No help really, just my two cents worth.

Yes indeed the type of heat treatment plays a huge role in determining the strength of the steel ball. Moreover my application here is also similar to ball bearings.

Thanks for the info.

As you may know all steels have same intrinsic stiffness which is called modulus of elasticity or young modulus after the scientific which defined the proportionality between strain and stress. The properties of the steel limit ONLY its loading that means that deformation of all steels will be the same as long as the elasticity limit for CONSIDERED steel will not be passed. This is the reason why a mild steel cannot be loaded as much as a heat treated one. But as long as the load is within limits both will deform the same. You are not the only one to make the error. It is very common to mix strain and stiffness.

Thanks for the info.

4x4 -- I'm having trouble understanding how what you are doing differs from a conventional ball or roller bearing. 34point5 suggested and illustrated a common type of linear/rotational ball bearing. Also you make no mention of lubrication or presence or possible solid contaminants. If this design is to function in the manner of an typical antifriction bearing lubricants an/or contaminants are an important design consideration.

If you want to pursue accurate analysis of the behavior of the structures you are working with there are some useful reference sources in books on the subject of analysis of antifriction bearings. These references tend to be either expensive or hard to find. I don't know if such material, which tends to be under copyright, is readily available on the internet. But with the book and author titles you can mke your own internent searches. Here's some links. The last lengthy link is a useful bibliography listing of many relevant papers and texts.

http://www.amazon.com/roller-bearing-engineering-Arvid-Palmgren/dp/B0007GX7UM

http://www.amazon.com/Rolling-Bearing-Analysis-Tedric-Harris/dp/0471354570

New departure engineering data; analysis of stresses and deflections.

Author: A Burton Jones; General Motors Corporation. New Departure Division.

Publisher: Bristol Conn., New Departure Division, General Motors Corp., ©1946

http://books.google.com/books?id=h6X0NM7ME8IC&pg=PA1092&lpg=PA1092&dq=A+B+Jones+New+Departure&source=bl&ots=JACs1e9C0f&sig=-IaoPn056ZI_NNKU6z3d4aHY2Q4&hl=en&ei=xyiYTbqFGonEsAPMoLzXBQ&sa=X&oi=book_result&ct=result&resnum=7&sqi=2&ved=0CDQQ6AEwBg#v=onepage&q=A%20B%20Jones%20New%20Departure&f=false

Hope this is not too technically complex for your purposes. ....Ed Weldon