Beam Theory vs. Plate Theory 6

[BobK98]

What criteria do you recommend using to determine whether or not load, stress and deflection calculations should be based on beam or plate theory?

All through college, all I was ever exposed to was beam theory. (ie, simply supported beam, load applied at distance X across the span...) I went on, got my BSME, and joined the workforce. That was back in December '98.

Since that time, about once every few months, I have received requests for calculations to determine the required steel or aluminum plate thickness required to support a specific load over a given span.

And my senior engineer dropped Roark's Formulas for Stress and Strain, with tables for plate calculations, in my lap as a reference.

We since picked up another engineer who is also more experienced than I, but he sticks with the beam approach, examining the condition on a per foot basis.

Is there a way, such as thickness to width ratio that I could use to determine which approach is more applicable?

 

[DaveVikingPE]

My opinion is that the connectivity, in addition to geometry, on a case-by-case basis is the best guide for plates vs. beams.

If you've got a "beam" that's rigidly connected to a structure along three of its sides, you might consider it a plate. Roark and Young is probably the best reference for plate bending formulas (unless you want to get into Timoshenko's books - I'm still reserving that for my Ph.D.). However, the best books that give good, working examples of plate theory (and shell theory, too) vs. beam theory are Blodgett's "Design of Weldments" and "Design of Welded Structures." These are available directly from the Lincoln Electric Welding Company for $10 each.

 

[borac]

sometime, you just have to excersize jour best judgement. Always think in terms of how this member will behave under the service load and remember - there is no hard rule. There are some in AISC steel manual (plates vs. bars) but that doesn't apply in real life. Anyway, as long you don't forget to analyze it, you'll be fine.

 

[Qshake]

I agree, in general, with the previous replies but will throw my $0.02 in anyway...

As you noted it needs to be on a case by case basis. In the public works and government sector that I've been in since the dawn of time, there has been very little need to use plate theory other than to reinforce some techniques that we learned in school and wish to pull them out every now and then. However, this is very short-sighted when considering all areas that engineers practice. Having said that please note...

If you have a small element whose stress or deflection is critical - use plates.

As Daveviking noted if your boundary conditions are such to mandate a look at plate theory then use it. Especially if you have a small element with relatively large boundary effects.

If you are looking at something in the public sector work and by code are required to factor the loads or reduce the stress then plate theory is a little akin to smashing a fly with a mallet while a perfectly good flyswatter is nearby.

I've stated this all with respect to small elements or elements with normal aspect ratios (nearly 1:1). However, if you get out of this range then two things will apply - for large aspect ratios general beam theory may be more applicable do to the nature of plates to bend in the short direction, hence the long direction isn't really a concern. If the element is really large or thick, chances are you have a geometric or materail non-linearity and should be using a more appropriate analysis.

 

[BobK98]

I think I know where you all are pointing me, but maybe it will help if I narrow my application down.

The work probably best described as by code. Most of the calculations are for expansion joint covers, where 8' or 10' plate lengths are simply supported continuously on the long edges by continuous framework on the sides of the joint. By Roark's standards, length of the supported edges divided by the width of the unsupported edges is almost always infinite.

Problem is that if I go by the plate theory via Roark, there are instances where the maximum yield stress of the metal would be exceeded. But the same cover calculated by beam theory says that the stresses are within acceptable limits. Which should I believe?

 

[poldolo]

You should use a FEM software if you want a good model without wasting time in a theretical model. You should use the shell elements (QUAD4 or 8 using Nastran) if you have to model a thin plate with nearly equal length-width.

 

[RevoSU]

If I may, I have a similar problem. A roof of a tank collapsed. This was because the breather on top did not function well which resulted in a vacuum pressure. The roof is cone shaped with a 15 degr. sloop. We decided to just pull it back into shape, but I need to calculate the force needed to insure the welds would hold. It’s a 6mm plate self-supporting roof. The FEM is the only way to go I think and a model software would be best. But you need to be exact when modeling!

 

[PEStructural]

I would not use Roark's formulas for members that exceed the maximum a/b ratio for that given table. For example, if you are using the table for rectangular plates, simply supported on 4 sides, uniform load; the max a/b ratio listed before infinity is 5.0. If your a/b ratio exceeds this, then the member is acting like a one-way system (i.e. beam action). If your member is <= 5.0, then I would use the plate theory.

As a note about FEM's, I usually model plates when there are unusual characteristics about the member. I recently had a concrete suspended slab that was 34'x60'. It had six 6'x6' holes. On the perimeter of those holes were pumps that weighed 24,000 lbs each. I did not feel comfortable doing a simple plate analysis, so I chose to run an FEM. But for the case of a member with no unusual characteristics, it is safe to use the Roark's formulas. I have compared them to FEM's and they are close.

 

[dpa]

DaveViking,
I only paid $7.50 for my Lincoln Arc Welding Foundation books. I noticed today in the U Dub Book Store that &quot;Weldments&quot; is now $15.00 and &quot;Welded Structures&quot; is now $25.00. At that price you are still only paying for the paper, ink and binding. These are excellent books furnished out of the generosity of the Lincoln Arc Welding Foundation. They skip all the theoretical gobbledygook and just tell you how to do it. For example they don't call it VQ/Ib they call it what it really is Vay/Ib. I spent an entire afternoon once looking through old textbooks mumbling &quot;What the &*$#@ is Q?&quot; Ans: it is a ridulous term dreamed up by some sadist to confuse young engineers. There is a great section on torsional stiffness complete with photographs of actual torsion tests and tabulated results proving that polar moment of inertia is a dangerous way to calculale torsional deflections. Everyone who deals with steel should have these books.

just my 2.4 piasters

 

[tqd]

I'm not sure that you're talking about the difference b/w normal beam and deep beam or b/w slab and normal beam.
If it is the latter subject, I think a plate model for analysis is the precise solution. However in some special condition we can deduce to a simpler problem - beam. For example, a square slab is supported in two opposite side (or a infinitely long slab supported on 4 side) with a uniform load or line load. Something like a plane-strain problem.
Of course in other cases (e.g: a slab supported a concentrated load or a 2-way slab), beam theory is unapplicable.

 

[KALC]

Having spent years chugging around in Roarke, I can recommend: try to model it as a beam any time that you can. Steel likes to act as a beam. Roark's plate analyses are based on beam-like behavior of the plate material, anyway, on a differential-element basis with &quot;Poissons&quot; considerations. Even Blodgett's treatment of plates and beams in the Lincoln books, (often referenced here in the E-tips), is directly taken from Roarke.

Waxing philosophical, for your further consideration:

Steel, beams, and engineers have several things in common: They can be quite simple, easy to analyze. They are often quite predictable in their behavior under load. Their individual contributions to the strength of a design can most often be superimposed to get the composite strength of the final configuration.

What would the world be like without Roarke. I hope he knew what he was doing because it seems everything built today uses Roarke as a basis.

 

[KALC]

Oh, another thing, a pet peeve of mine...don't jump to software for engineering solutions before you understand the model. A good rule is:

If you can't explain it, don't compute it.

That is, I believe an engineer must be able to model his problem, (in his head, on the back of an envelop, on a spread sheet, markerboard) before ever considering going to FEA or other software to resove the details.

Bobk98, you're doing the right thing, asking the questions to resolve the model. I politely disagree with those who suggest you go straight to software for analysis or solution. You've still got, basically, wide thin beams which get some stiffness help from Poisson.

 

[diblazing]

Like BobK98 I am an even younger engineer (mechanical '02) that was only exposed to beam theory in school. I am now faced with a critical design issue that may need plate theory to be correctly modeled. I don't even know where to begin; I have sifted through Kirchoff and Timoshenko and don't really have a good grasp on either. I need to determine the appropriate thickness and the necessary number of bolts needed for a ? x 8&quot; x 5' Steel Flat that will be cantilevered out 3 5/16&quot; (width wise). This flat will have a possible point force at the end of the cantilever of 12000 lbs. Is it possible to use beam theory on a per unit basis or do I need plate theory in order to determine the bending in both directions. I have not seen any plate theory that deals with a cantilever situation either. I would also like to know about any rules of thumb if applicable. Any help or direction would be appreciated.

Thanks,
Nick

 

[desertfox]

Hi BobK98

My 5th edition of &quot;Roarks stress & strain&quot; on chapter 7 (flexure of straight bars)gives span to depth ratio's for beams as follows:-


l/d for metal beams greater than 8 (compact section)

l/d for metal beams with relatively thin webs 15 or more

and greater than 24 for rectangular timber beams

hope this helps

regards desertfox

 

[MotorCity]

Take a close look at the member you are trying to analyze. Technically speaking, if all of the supports are colinear, use beam theory. If all of the supports are coplanar but not colinear, use plate theory.

Judgement and experience comes into play when making analytical assumptions. For example, you may be able to approximate plate theory results by performing beam theory analysis. In some cases you may find that the difference in results provided by either theory is negligible (i.e. using a plate thickness of 0.5&quot; vs. a plate thickness of 0.5564&quot

Blodgett, Roark, FEA, modeling software, etc. can be extremely useful resources, but only if used correctly.

Good Luck.

 

[JoeTank]

BobK98,
It's getting late and I just couldn't bear to read every word of every post. But, from what I've seen it doesn't look like one of your basic questions was answered. One commonly stated criteia for the use of valid plate equations is that the deflection should not exceed one-half the plate thickness. Anything more than that and you need to be looking at large delection theory for a solution.

Steve Braune
Tank Industry Consultants

http://www.tankindustry.com

 

[jheidt2543]

KALC here is another analogy between engineers and steel:

&quot;Men and still have a common trait, neither is any good when they loose their temper.&quot; - Anonymous

Just a note for the good of the order!

PS: I sure see a lot of quotes by that Anonymous guy, he must have been pretty smart!