When Should You Use Beam Theory vs. Plate Theory for Stress Calculations?

[leon25034796]

Can somebody provide some clarification:
I am calculating the stress in a plate with the following dimensions:
41" long by 30" wide and the plate is 3/8" thick. The plate is simply supported along the 41" length sides, with the short lengths free. I have calculated the stress using beam theory with a point load of 18815N at the centre of the plate as 227N/mm2.
However, I have a Roark book and I'm looking at the tables for 'formulas for flat plates with straight boundaries and constant thickness'. I can't find the exact formula I want but have used a formula for a rectangular plate simply supported at all edges with a uniform load over a central rectangular area (I've assumed this area to be as small as possible to try and best represent a point load)
My question is, when do I use plate theory and when do I use beam theory? And is there a plate theory formula for 2 simply supported sides and 2 free sides?

 

[minger]

If the plate is simply supported on two sides and free on the opposite two sides, then you should be able to treat it simple as a very wide beam - assuming that the plate is uniformly loaded along the width.

You'll notice that all of the cases in the Roark has boundary conditions on all, three (and in one case two adjacent) sides.

 

[FredGarvin]

Did you look under the section 8.11 Beams of Relatively Great Width (pg 169 in my copy)?

 

[minger]

We share the same edition apparently. Good catch,

 

[leon25034796]

Thanks for your responses. I also share the same copy.
There is one other thing that I can't get my head around:
I have treated this as a simply supported beam and calulated the stress at the centre with a point load as shown below and get 227N/mm2; this to me would be a more severe condition than all four sides simply supported??
However for peace of mind (as this is a practical situation) I used the case on Page 503 1c. i.e. Uniform load over central rectangular area. To try and simulate a point load I used the case a=1.4b and used the maximum value in this table i.e.2 (smallest ratio available of centrally loaded area to area of plate) to put back in the stress equation. This gave me a stress of 414N/mm2?? How can this be a worse stress than a simply supported beam across 2 beams?
I have read the respective section for beams of relatively great width and I can't really see how that helps me it also doesn't define what is a 'wide' beam.

Any comments would be appreciated

 

[Nick Bruno]

I thought that beams were just viewed as deflected members in 2-D and plates were viewed in 3-D.

So I think if you can visualize your problem in 2-D with the given constraints, you should use the beam equation.

 

[leon25034796]

But I am looking for the worst case scenario and surely the two theories should give an answer approximately the same; or as in this case I haven't got two exactly comparable formulas; so as detailed below I would still expect the beam theory stress to be the most severe case?

 

[minger]

Not entirely. With plates, the additional boundary conditions can lead to odd things. They certainly won't be equal.

As said, if all of your loads and constraints are constant depthwise (which they are) then you should surely use beam-type equations. Because there is the large depth ratio, use the stiffening factors mentioned in the section that Fred referenced.

 

[nvn]

leon25034796: Solving your given problem as a beam the way you did in posts 1 and 5 will be incorrect and grossly unconservative, because you have a concentrated load, not a uniformly-distributed line load across the width of the plate. The simply-supported plate stress you listed in post 5 will be very close to the correct stress, as it will be only about 3.4 % below the x-direction stress in your actual plate. But what really matters is the von Mises stress in your actual plate, which will be about 7.3 % below the simply-supported plate stress you listed in post 5.

 

[canmalican]

Hi guys

I'm currently reading about theory of plates, and i couldn't identify when shud i use classical theory or reissner mindlin theory for stiffened plate. Is there anything related to finite element for plate theorie and 3d calculation?.Any advise would be appreciated.Thanks