Hertz Contact Solution of Elastic Theory for Concave to Convex Shapes 1

[sponcyv]

For the equation:

contact stress = {(1 / (pi[((1-v1^2)/E1)) + ((1-v2^2)/E2))) ^ 0.5} * {((Fn/b) * (Sum (1/pi)))^0.5}

Where Sum (1/pi) = [(1/p1) - (1/p2)] for concave shapes in contact with convex shapes

Sum (1/pi) approaches 0 as the two radii get closer, however when the two radii equal each other, the second part of the equation equals 0 from multiplication and the entire equation will equal 0. This is confusing to me. How could the stress be 0?

 

[dhengr]

You really need to give more info. for a meaningful answer. I’m not sure exactly where that exact equation came from, or what it is intended to show, but I think I see most of the correct terms in the equation. As the two radii get closer together, the bearing area ultimately goes to infinity, and thus the bearing stress goes to zero. There is yielding, in bearing, taking place: if a sphere in a spherical socket, the bearing surface shape is circular or elliptical; if a round rod inside a pipe, the bearing surface shape is a line load which gets wider as radii get closer; at the limit these become two flat surfaces.

 

[sponcyv]

The equation comes from page 332 on the link:

http://books.google.com/books?id=9T...A#v=onepage&q=contact stresses convex&f=false

The object is a sphere that fits in a concave spherical void. If I understand what you are saying correctly, I believe if both radii are equal that the stress can be distributed fully over the contact area and "contact stresses" can be ignored. I believe this is what the equation is eluding to when I calculate 0 stress. If both radii are equal, then the contact should not be a point or a line contact, but a full surface contact.

 

[dhengr]

The second (your linked) form of the equation is easier to read, and in a little more std. form. It does not mean that you can ignore the contact stress, it just means that as the radii get larger and get closer to each other, Hertz’s formulation breaks down. And, given large enough radii, we say they are flat surfaces, and the bearing stress is P/A, except for the damn bearing plate flexibility. Given a sphere with radius r, I would likely use a contact (bearing) area of say (.6 - .8)(? r ^2). If the two radii ‘could’ be made equal is the operative terminology. Within your manufacturing and machining abilities and tolerances the two radii will not be the same; you still have contact stresses (bearing stresses),wear, small areas of yielding in bearing, work hardening, etc. etc.

 

[sponcyv]

dhengr,

Yeah, I tried to type in the equation and it didn't work so well. Thank you for your reply.

To give more info, the semi-spherical ball (about half of the sphere) sits in a semi spherical socket and there is a hole in the center of the spherical socket. Thus, my potential bearing area is essentially a radial ring. I have found some information on Roark's ball and socket stress equations. I've attached the equations to this post.

I'm getting a higher value using this equation than I did using Hertz' equation for concave to convex contact. I would have expected better results. I do not have the actual reference with text, so I'm not sure what the assumptions are.

I have a static load condition and I'm trying to determine the actual stresses on the ball and socket. I would like to specify equal radii, but I should use some tolerance in my calculations to cover the machining tolerances as you say.

 

[dhengr]

O.K., if we can get back to what we were talking about before all the shooting started....
We still need more info.: the load, the ball and socket dia., the dia. and location of the hole, type of materials, etc. etc. I’m not asking for any proprietary info. here, but if the dia. diff. of the ball and socket can have the affect you see so far, imagine what the hole size can do. Which calc. gave you the higher stress, I can’t see that from here, show me. You must understand where the equations you are using came from, and what their assumptions were to understand your results. And, that may involve tracking down some of the ref’s., and that’s been a long time ago for me. The ring brg. area sound right for a first step, but you’ve not done that calc. have you? And, I don’t know if you’ll find a simple eqn. for that. Your last Roark eqn. looks about right for a first step, but I’m not sure what the assumptions are for the other eqn. which you say gives lower brg. stresses. Note that the denominator in KD changes, do you have Roark’s book? Answer my questions, and post your calcs. and I’ll think on it some more too. Shoot at ya later.

 

[sponcyv]

Thanks dhengr - I'll get you the info first thing tomorrow.

 

[sponcyv]

dhengr,

The attachment shows my calcs for Hertz' contact solution theory for concave to convex shapes and also Roark's ball and socket maximum compressive stresses. I am using steel, but have not specified what type (not needed for calcs.)

After some thought, it may not be feasible to apply either Hertz' or Roark's directly. Both Hertz' and Roark's methods assume normal load being applied on a tangential point. If you can picture the ball and socket having a hole in them and thus the stresses will be distributed based on the contact with the side of the ring. The normal stress will be directed through the center of the hole and the stresses on the side of the ring will be distributed radially to the side of the ring. If the radii of the ball and socket do not match, the contact will be made on the edge of the socket's hole. This means the deformation would differ from the deformation assumed by Hertz' and Roark's methods and the contact area would be different. I may need to look for contact stresses between convex objects and sharp line edges or more precisely radial edges.

You wouldn't happen to know where to find that?

 

[dhengr]

Sponcyv:
It seems to me you are comparing apples and oranges, and trying to get doughnuts.

Reread your first link, that you got off the internet. It’s a Hertz stress calc., but isn’t that calc. for two cylinders of diff. radii, and a contact line of length “b”, and thus for the contact stress G you will have a contact width of “2a”? Your “radial circumf. of center of ring” really doesn’t have any real (or exact) meaning here. Redo your calcs., I haven’t checked them, but there must be something amiss because a contact width of “2a” = 2.298" doesn’t sound right with a 3.5" dia. steel cylinder. Your actual ring width of .719" sounds pretty small too, for your load, and will have to be (backed up) pressed into a much stronger support structure.

Your second calc. (from Roark) is a Hertz stress calc. too, for a true ball and socket condition; with dia’s. of Di, and a load condition factor K for a socket, instead of a sphere on a flat surface or on another sphere; and “2a” is a contact surface dia. for the given max. contact stress.

What it sounds like you have is basically a solid half sphere (ball) with a 3.5" dia. on a doughnut which should have a spherically shaped, upper, inner edge to rec’v. the ball. So, instead of the line load on the cylinders, you have an annular load of some avg. circumf. (7.56") and a width similar to “2a” which gives you your contact area. You really can’t account for your manuf. tolerance in these calcs., but you must be aware of it and account for it in your final thinking. And, it won’t be .01" now that we know about what size you are dealing with, although 3.5" is probably kinda small. I’m going to give up if you don’t provide a sketch, hide your edress in your sketch or its notes. Wake up, I’m not sure what you are trying to accomplish and I can’t really see what you have from here, and I shouldn’t have to keep guessing, I’m not the one asking for the help. Getting sufficient design info. around here is like pulling teeth. If you can’t spend the time, I’ll quit wasting mine. You can actually buy bearings like this, if my guess above it correct. I’ll have to dig a bit to find a catalog, but I’ve used them on heavy equipment design before. They’re called ‘spherical plain bearings’ and Torrington is one supplier. The ME guys will know if they’re still in business, or know of other suppliers.

 

[sponcyv]

dhengr,

I wish I could give you sketches and discuss the actual function of this device, but my company has signed a non-disclosure clause and providing this information would be grounds for legal action.

When I posted this question, I was searching for conceptual information. This is the first I've dealt with Hertz or Roark. I searched the internet for conceptual information on contact stresses and I found the text from Hertz. I came to this forum to get a better understanding of the stress distributions that Hertz touches on. I never intended for anyone to fully design or solve this problem for me.

I do thank you for the time and interest and I am sorry if you feel like I am wasting your time.

You are absolutely right about the Hertz' equations being for cylinders rather than spheres. Thank you for pointing that out. The ball and socket method is more applicable here, but I really don't have a point contact either. The radii must be equal so I do not have a point or line contact. If you can picture the socket having a larger radius than the ball and the hole in the socket is larger than the hole in the ball, the ball would come in to contact with the edge of the socket's hole and you would have an annular line contact. It would be difficult to analyze these stresses and to get the stresses low enough to work.

"What it sounds like you have is basically a solid half sphere (ball) with a 3.5" dia. on a doughnut which should have a spherically shaped, upper, inner edge to rec'v. the ball. So, instead of the line load on the cylinders, you have an annular load of some avg. circumf. (7.56") and a width similar to "2a" which gives you your contact area. You really can't account for your manuf. tolerance in these calcs., but you must be aware of it and account for it in your final thinking."

Your statement above is pretty much accurate. The "doughnut" is actually a square plate with a certain thickness and the semi-sphere socket is formed in to the top of this plate. The half sphere sits down in the socket and is free to rotate to a certain degree. The force acts through the centerline of the hole in the sphere. The rotation point is about the center of the sphere. Another problem I will need to solve is the effect on the stress distribution when the force is angled due to rotation. The stresses can be assumed to be equally distributed over the contact surface when the force is normal to the plate. However, when the force is at an angle to the plate, the stresses will be redistributed. The edress is jvmytouch3g at gmail dot com.