Strain energy for circular plates

[harpreet singh]

I need to calculate the strain energy for a clamped circular plate by the governing equation in polar coordinates.. Load is distributed uniformly over the entire area.. And displacement is in transversal direction to the plate,..
please do help me...

 

[Q_Goest]

Have you looked at "Roark's Formulas for Stress and Strain"? They have an extensive list of formulas for flat, circular plates depending on how the plate is supported and loaded.

For example: flat, circular plate that's either fixed, simply supported, guided or unsupported on the OD. Similarly on the ID. Load can be pressurized over some partial area of the surface, over the entire area, or linear load around some circumference between the ID and OD.

 

[harpreet singh]

Thanx

Actually i don't have this book here in lib. Can u please provide me the link from where I can read it...

 

[Q_Goest]

Not online as far as I know, but you can look.

There's a forum set up specifically for Roark's here:
http://www.roarksformulas.com/

If you can post a picture of your model, it would help (ie: show load, show supports, show dimensions, indicate what you are trying to model, etc...).

 

[harpreet singh]

Thanks a lot bro. I was totally unaware of it..

 

[harpreet singh]

description

I don't have any photo to upload but you i can explain it.

I am modelling a circular plate with a hole at its center. Plate is clamped at its circumference and has uniformly distributed load acting in transverse direction to the plate causing the deflection of plate in transverse direction. I want to calculate strain energy inn that plate on behalf of these conditions.

 

[Q_Goest]

Please clarify "transverse direction". Is that perpendicular to the flat surface of the circular plate or parallel with it?

If perpendicular, case 2e applies and you may be able to glean the necessary information from the Google book review here:
http://books.google.com/books?id=05DHO7VzwnoC

See page 465 which shows "case 2e". Better yet, pick up the book at your library.

 

[harpreet singh]

Ya it is perpendicular to the plate only..
Thanx..

 

[Q_Goest]

Just to clarify: strain energy is not calculated directly from this case. You will need to integrate the deflection as a function of force to determine energy. In other words, the plate acts as a spring and has some force necessary to deflect the plate through some distance. Energy is force times distance, but since force will vary as a function of displacement, you must integrate this.

That isn't so hard really. You don't need to actually integrate the equations Roark's gives you. In fact, it would probably be impossible to actually integrate it anyway. What you would do is determine the spring rate (if linear) or find the force needed to displace the plate at specific states. In other words, apply some small pressure and determine displacement from the equations in Roark. Do this again for a slightly higher pressure and find displacement. Do this numerous times to find displacement as it varies with pressure. Once you do this you can graph displacement as a function of pressure. Then, take the graph and do a curve fit to determine an equation of displacement as a function of pressure and integrate THAT equation to determine strain energy. Note this integration can be done either using calculus OR just doing a numerical analysis on it.

 

[FredGarvin]

If you want to get technical about it, strain energy is the integral of the work done to produce the deflection or:

$$U = \int_{V} \frac{1}{2}\sigma\epsilon dV$$

I would imagine that it is not too much of a stretch to reduce this to a function of radius only (constant thickness and geometric symmetry) and then reduce the integrand to a 1-D problem.

 

[Q_Goest]

If you want to get technical about it, strain energy is the integral of the work done to produce the deflection or:

$$U = \int_{V} \frac{1}{2}\sigma\epsilon dV$$

I would imagine that it is not too much of a stretch to reduce this to a function of radius only (constant thickness and geometric symmetry) and then reduce the integrand to a 1-D problem.

Hi Fred,
I think what you’re getting at is that I’ve oversimplifed this. Strain energy equals the external work done by forces on the plate (which is what I was counting on here), but those forces are distributed (pressure load) so there’s no single force. <argh> This would be much simpler if it were a linearly distributed load so the total load would equate to a single displacement.

Ok, I’ll have to give it a bit more thought. Roark's gives deflection at any point in the plate for any given pressure. Maybe there’s a simple way of taking that equation and integrating.

 

[FredGarvin]

Hi Fred,
I think what you’re getting at is that I’ve oversimplifed this. Strain energy equals the external work done by forces on the plate (which is what I was counting on here), but those forces are distributed (pressure load) so there’s no single force. <argh> This would be much simpler if it were a linearly distributed load so the total load would equate to a single displacement.

Ok, I’ll have to give it a bit more thought. Roark's gives deflection at any point in the plate for any given pressure. Maybe there’s a simple way of taking that equation and integrating.

No. I don't think you've over simplified it at all. I was stating the strict definition for the strain energy because I got the impression that the OP needed to work with the classical equations. I thought about your method and that's the way I would prefer to do it if I had no constraints on the method. I would have taken the pressure and divided by the area and replaced it with a single point source load in the center and then called it good enough. I would MUCH rather do that than deal with a triple integral of the stress-strain tensors. Again though, I think the OP is asking a homework question and I don't think our practical approach would be rigid enough (pardon the pun) for a mechanics of materials class homework question.