# Thermal Expansion of Ring

1. Sep 15, 2008

### minger

Hopefully quick/easy question. I am modeling essentially a flat plate under pressure load in ANSYS with a large thermal change. With fixed or simple support at the outer edges, of course the thermal stresses are crazy high.

To try and get a better estimate of stresses without modeling the entire stucture, I'm trying to get a thermal radial deflection of the support structure. Assuming that it's a flat ring or thin shelled cylinder does anyone know of a formula for deflection?

I looked in Roarks and didn't find anything, and in Shigley's I found an equation for a flat plate with fixed supports at the end:

$$\sigma = \epsilon E = \frac{\alpha \Delta T E}{1 - \nu}$$
$$\epsilon = \frac{\alpha \Delta T}{1 - \nu}$$

But then I'm not sure if this applies for radial loads, and furthermore, I'm not sure what length (diameter, radius, etc) to use with strain to get actual deflection.

Thanks for the help,

2. Sep 16, 2008

### FredGarvin

Roark's does have a section on thermal stresses. There is a paragraph that states explicitly what you are looking for in the first section:

Maybe I'm not understanding your question. I am assuming that you are constrained around the entire perimeter. Is this the case?

3. Sep 16, 2008

### minger

No, rather the opposite. I'm trying to get an idea of how much an unconstrained ring will expand due to thermal expansion. That radial deflection will then be modified and used as radial deflection in the part that I'm actually trying to model.

If I hold the outer surface of the inner part fixed, and I raise the temp to 1400°F, the stresses are predictably erroneously high. So, my thought is that if I force the outer edges out, then it will help reduce the stresses by giving it room to move.

So, I see the equations of stress, but my question is can I simply convert that stress to strain, and if so, what length do I multiply by to get deflection?

4. Sep 17, 2008

### FredGarvin

If I hold the outer surface of the inner part fixed, and I raise the temp to 1400°F, the stresses are predictably erroneously high. So, my thought is that if I force the outer edges out, then it will help reduce the stresses by giving it room to move.

So, I see the equations of stress, but my question is can I simply convert that stress to strain, and if so, what length do I multiply by to get deflection?[/QUOTE]

Here are a couple of links to get things going:
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thexp2.html#c2

According to Goodier and Timoshenko, a thin circular disck with a symmetric temperature symmetrical about the center:

$$\sigma_r = \frac{E}{1-\nu^2} [ \epsilon_r +\nu \epsilon_{\theta} - (1+\nu) \alpha T ]$$

and
$$\sigma_{\theta} = \frac{E}{1-\nu^2} [ \epsilon_{\theta} +\nu \epsilon_r - (1+\nu) \alpha T ]$$

5. Sep 17, 2008

### minger

Thanks Fred, but I ended up just modeling the whole damn thing in ANSYS. After getting the temperature distribution it wasn't too bad.

Those equations would have helped though. Always a good answer Fred

6. Sep 18, 2008

### FredGarvin

LOL. I figured by the time I got my butt around to finding that section you had figured things out. Sorry it took so long. I am planning on modelling up a simple ring and running a temperature across it to see what kind of correlation I get. You piqued my curiosity.

7. Sep 18, 2008

### minger

I "think" the radial expansion I found was about 2/3 predicted by:

$$\frac{\alpha \Delta T}{1 - \nu}$$

If I remember correctly...