Plate stresses

[pba]

Folks,
This is a theory question rather than an actual problem.
Some background - The company I work for has an old design guide for design of air ducting. The formula for plate stress under uniform loading is given as: fb = (K1 Q X^2)/t^2

Where fb is plate stress, K1 is a factor relating to the aspect ratio of the plate, Q is the pressure on the plate, X is the distance between stiffeners and t is the plate thickness.

The formula is supposed to be a rough avarage of the stresses generated by fixed and simply supported conditions.

Now. I thought I'd check this by camparison to Pounder's Theory, which gives formulas to calculate stress for fixed and simply supported conditions. It turns out that the stresses given by my companies formula are LOWER than those given by Pounder's theory. Ho hum - good job I've not relied on it!

I then thought I'd do a further comparison by using finite elements. I modelled a number of plates with both fixed and simple supports and with varying aspect ratios. This produced two nice curves of lower stress than either Pounder's theory or my company formula - Just as I'd hoped/expected. BUT closer examination of the curves shows that the simply supported plate has lower stresses than the fixed plate. This is not what I would have expected or what Pounder's theory predicts.

Now for the question(s): Is my F.E analysis wrong? Has anyone else done this type of simple modelling? The models are metric models using the follwing plate sizes (in mm) 1000x200, 1000x400, 1000x600, 1000x800, 1000x1000, 1000x1200, 1000x1400, 1000x1600, 1000x1800 and 1000x2000.

I've applied a uniform load and set the plate at 5mm thick steel.
I need to investigate this myself over the Christmas / New Year period but I'd appreciate any comments people may have.

I'm guessing it's a simple modelling error but can't see it right now.

Everyone - Have a good Christmas...

 

[JStephen]

I would suspect that you are getting large deflections and thus membrane stresses in the plate, which invalidates the normal plate bending equations.

I'm not familiar with the Pounder name. Roark's Formulas for Stress and Strain includes rectangular plate bending, and this may be the design you're referring to. The book gives equations for both edge conditions, and you could average the two cases if desired.

If your finite element model has the edges restrained in the edge of the plate to develop the membrane forces, you might check if the actual plate is adequately restrained as well.

Note that the long thin plate examples (1000x200) should be similar to a 200mm beam near the center.

 

[flamby]

If we think of just the bending stresses, simply supported plate will show more stresses. When you combine the in-plane tensile stresses of the plate in fixed-end condition, I am not so sure. What kind of stresses you are looking at in FEA? Suggest you look at stresses in all XYZ planes.

Ciao.

 

[pba]

Thanks folks.

The large deflection comment is certainly worth looking at. Do you have any idea what would constitute a "large" deflection? I might just change to a very thick plate and re-run the numbers.

Agreed the 1000x200 plate should be like a 200 beam at the centre. I can of course do that as a hand calc, and I will. That should give me lower bending moment (and stress) for the fixed condition...

I'll need to confer with some colleagues here to make sure I'm looking at the right stresses! F.E. is not really my thing - anything more complex than WL/8 is unnecessarily fancy ;-)

I am sure I'm comparing like with like on the two F.E. models.

Most of this will need to wait for the new year as I'm off for the holidays very soon.

 

[miecz]

First, your end panel will behave differently from interior panels. If you shorten up the stiffener spacing for the end panel by 20-25%, it won't govern the design, and you can design for an interior panel.

Next, you need to identify where your maximum stresses are located. For large aspect ratios, with closely spaced stiffeners on a wide duct, I would expect the maximum plate bending stress to be near the center of the long side stiffener.

Interior panels, in the direction of air flow, will behave like a continuous beam over sprung supports. The springs are due to the deflection of the stiffeners. The boundary condition for a single panel in that direction will be close to a fixed end condition. The boundary condition perpendicular to the air flow would be fixed for a square duct, because the side panels have equal forces, and close to fixed for a rectangular duct.

I believe membrane stresses are typically ignored in duct design. The overall effect of the stiffener deflection is to send more of the load to the corners of the duct.

 

[JStephen]

If I remember right, Roark gives the criteria as deflection more than half the plate thickness.

 

[kslee1000]

I am not sure, but I don't think plate theory applies to membrane. Comments?