Two Way Bending Flat Plate

[TeemoshenksEIT]

I want to analyze a flat plate, 1/4"x3'x3' that has a 500 lb load over a 4in2 area. The plate is a 50 ksi plate, that sits on beams on 4 sides, this plate is removable and not clamped down. I usually use Roarke's, but I don't know how to account for curling/lifting at the edges. Before I go and FEA the problem I want to know is there somewhere in Roarke's where they account for uplift of edges?

 

[dik]

Check Alex's spreadsheet...

Dik

 

[TeemoshenksEIT]

dik, Alex's spreadsheet considers pinned edges. However, would that not be non-conservative? Since in reality, I wouldn't consider the edges of the plate pinned as it will experience uplift due to point load in the middle. As the edges are not restrained in the vertical or horizontal directions the max deflection would be larger than pinned condition right?

 

[jayrod12]

Pinned edges are more conservative than fixed. Pinned assumes it's free to rotate accordingly causing larger deflections than if fixed at the support.

 

[phamENG]

What you have is essentially a pinned connection. The corners are less so - you'll get uplift over the support (not just at the outer edge) at the corner, but the effect is probably minor given the size of the application.

I'd start with the Roarks for a pin supported plate and see what the deflections are. Set the theoretical pin support at the inside face of the real support (analytical plate will be slightly smaller than the real plate). Then look at the angle of the analytical plate at the support. Then, use similar triangles to find the upward deflection.

 

[phamENG]

For more accurate analysis, you can look into the finite difference method. It's a numerical method that lends itself well to plate analysis.

 

[dik]

Pinned and fixed, but not a point load... Suggestions using Timoshenko are good. The solution is pretty simple and IMHO, will give you a reasonable starting point if you need precision check Timoshenko... or a finite difference method noted above. A yield line solution will give you a pretty good max answer, but deflections are not considered.

Dik

 

[r13]

Use FEM program, and place the pin at the edge of the support beams, then you have a plate with cantilever on ends. There is no simple formula for this.

 

[dik]

As a quick kick at the cat... I'd determine the point load moment from a simple span beam and use the equivalent UDL for this moment... plug the info into Alex's spreadsheet to determine the thickness and deflection and then do a quick yield line check... easy for a point load (I'd neglect corner levers)... if you don't have access to an FEM program or you have hundreds of this design. I'd use the above for a simple one off.

Dik

 

[r13]

dik remind me of something, yes, use one way span and consistent displacement method to determine proportion of load to each direction, then use the proportioned force to calculate support reactions and internal forces. If the concentrate load is in the middle of the square plate, then you only need place 1/2 of the load on the one way simply support beam model.

 

[dik]

With the concentrated load at centre, the moment will be higher than a UDL of the same force... hence determining the equivalent UDL for the point load... not exact, but a good guess and very simple. It's the sort of one handed calculation that I do in seconds. The quick yield line confirms the plate is capable of supporting the point load for centalised loading it's a couple of minutes, for non-central loading time is similar, but would use my SMath program). It's not an exact solution, but a good design without being overly conservative. OK for a one off, but if I had a 100 of these, I would do a more precise analysis.

Dik

 

[rb1957]

but the load is distributed over only a small area of the plate … very close to being a point load

another day in paradise, or is paradise one day closer ?

 

[dik]

An attempt to accommodate this by using the equivalent moment of PL/4 instead of ql^2/8...

Dik

 

[JStephen]

I'd use the closest thing from Roark or the yield line analysis.
If there's just one such item, you can likely double the thickness cheaper than you can spend an extra hour analyzing it.
If there's a bunch of them or it's part of a nuclear reactor, then go with FEM.
If the plate already exists, it wouldn't be too hard to do a load test on it.