How to calculate deflection for thermal expansion of a ring?

[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,

 

[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:

Stresses Due To External Constraint
3. A solid body of any form is subjected to a temperature change \Delta T throughout while held to the same form and volume; the resulting stress is
\frac{\Delta T \gamma E}{(1-2 \nu)} (compression).

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

 

[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?

 

[FredGarvin]

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?[/QUOTE]

Here are a couple of links to get things going:
http://www.eng-tips.com/viewthread.cfm?qid=99646
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 ]

 

[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

 

[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.

 

[minger]

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

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

If I remember correctly...