beam/plate in bending, is there a difference? 2

[PEinOHIO]

if i have a plate in bending (simply supported, force applied at the center), 15ft long, 5 ft wide, 1 inch thick, can i use standard beam bending formulas or is there a correction factor if it is more of a plate then a beam?

 

[MiketheEngineer]

Sure - but at 5' wide - you might not get all the section you would expect. Spread the load over the 5' using a small beam or 2x4??

 

[ShaggyPE]

See section 8.11 of the attached link (its from Roark):

http://www.scribd.com/doc/54618141/13/Beams-of-Relatively-Great-Width

My recollection (just from my prep for the PE exam 4 years ago) was that the formula was very similar... there was just a multiplication by a factor that came from the ratio of the width to the thickness or something.



-Dustin
Professional Engineer
Pretty good with SolidWorks

 

[rb1957]

ditto ... but more than just spreading the load ...

what is the load ? distributed ? point ??

also consider membrane loads ... if the plate is deflecting more than 1/2 it's thickness plate bending theory (no axial load on the NA) starts to break-down and the internal membrane loads help react the applied load.

also, if you have a localised load, with a wide plate the edges are probably deflecting so as to increase the load in the remainder of the plate (ie flexing up with a down load).

 

[JStephen]

Bending stress is same as a beam, assuming load conditions are spread across the plate. Allowable stress is generally higher, since you don't have buckling issues as with a beam. There is an adjustment for deflection as noted above. Where that comes from, is that the usual assumption is that the compression and tension flanges are free to expand laterally due to Poisson's effect, but in a wide bar or plate, they are tied together and can't independently expand and contract laterally.

Also note that in certain cases, membrane action must be taken into account- particularly if the ends are both attached rather than free to move toward each other.

 

[ESPcomposites]

Jstephen,

Is the last paragraph any different for a beam though? I think that statement is applicable to both.

Brian

http://www.espcomposites.com

 

[racookpe1978]

"Also note that in certain cases, membrane action must be taken into account- particularly if the ends are both attached rather than free to move toward each other. "

But - at 5 feet wide, unless the total load is (somehow) applied in a UNIFORM "line of force" exactly across the width of the plate at the center of the plate - and ONLY if the this "line of force" will be resisted by ONLY the two ends of the plate (with neither side being fastened!) will the plate "bend" as you are expecting.

Otherwise, you will get a multi-curved "sagging rectangular trampoline" if all four sides are bolted or welded. The "curve" across ANY section of the plate will depend on how the four edges are fastened: If they are free to flex (free to bend up as if tied by ropes around a stretched tarp) then the curve will have no inflection point and continue from the edges down towards the center of mass of the load. This will also tend to happen (but to a lessor degree) if the edges of the plate are fixed to a frame around all four edges, but that frame is not stiff enough and the frame or its legs twists under load.)

If the edges are firmly welded to a frame that has significant rigidity, then the plate will begin flat at all edges, then bend down towards the load, then inflect and bend up across the middle of the load to the other side.

 

[hokie66]

You guys (except for Mike) try to make everything difficult. The OP asked a simple question about a simply supported beam, and it has been turned into a discussion about membrane action. The only thing which needs to be determined is whether the load is a point load or a line load.

 

[JStephen]

I'm assuming the plate is "simply supported" like a beam, supported at the ends only. If that is the case, I think most people would treat it like a beam regardless of whether the load was a point load or distributed across the width. If it is "simply supported" along all four edges, that is different, and refer to Roark's plate formulas (if the situation is even covered).

ESP- the membrane situation would also exist with a beam, but is much more likely with a plate due to the much lower bending stiffness. Note that Roark's criteria for when this becomes significant in a 2-D plate is when the deflection exceeds half the thickness, which wouldn't take much in the situation described- plate weight alone might deflect that much.

 

[corus]

Plates can be treated as beams but as Greg has pointed out, the localised force at the centre will produce localised deflection/bending/stresses etc. I'd refer to Roark or plate bending theory.

Tara

http://tinyurl.com/4ydjg7m

 

[rb1957]

i disagree hokie, plate bending expressions apply only if the deflection is small (something like 1/2 the thickness).

if you apply plate bending and get a deflection greater than this, then the plate will be reacting (at least some of) the load as a membrane (ie tension on the mid-plane).

mind you, i think hokie is right about the discussion of the side edge restraint ... the OP clearly said his plate is supported on the two ends only.

 

[racookpe1978]

15 ft = 180 inch long steel plate, simply supported both ends (Reference problem statement).

Hmmmn. Are you sure you want to do this?

I just supported a 115 inch long 1x1 steel beam at both in the garage, supporting both ends on rollers. Assume 113 between rollers. Zero'ed out the deflection at midpoint (55 inches) with a laser level. (Found 1/16+ some-odd deflection for beam weight alone by the way.)

Put a 68 pound anvil at the midpoint, measured a 11/16 inch deflection at the 55 inch point. If your actual "load" is more than 65-70 pounds per inch of the loaded length of the plate, you can't assume any maximum of "only" 1/2 thickness of the plate deflection.

 

[JStephen]

Treating the plate as a beam, that 1/2 the thickness issue only applies if the ends are supported against tension- otherwise, you just have bending and it's not an issue.

 

[hokie66]

JStephen is correct. Most of us, certainly the structural engineers, would take the OP's description of simply supported to mean there is a pin at one end and a roller at the other...thus no horizontal force component, no membrane action.

 

[rb1957]

well, FWIW, i disagree. i think you can have a plate sitting on two benches (ie not axially restrained) and i don't think plate bending theory will predict the deflection. now, of course, it mihgt deflect so much that it falls off the supports, but that's a different problem.

and i'm not saying it'll deflect only 1/2 the thickness, i'm saying that plate bending theory is valid If it deflects less than 1/2 thickness.

 

[hokie66]

I wonder where you got that idea, rb. The deflection with a concentrated load at centre span is just PL^3/48EI, as long as the plate is bending elastically.

 

[ESPcomposites]

Well, beam theory is only valid for "small deflections" (i.e. sin(theta)=theta. Beyond that, it will start to lose accuracy, though it is not dramatic until the deflection is relatively large. Note that this may still be in the elastic range if it is thin. That is discounting membrane action. But specifically, the PL^3/48EI should be adjusted by (1-v^2) for a plate to account for the added stiffness of a plate.

Brian

http://www.espcomposites.com

 

[desertfox]

Hi PEinOHIO

Beam theory is only valid for small deflectionns I agree with ESPcomposites and rb1957.
See this paper:-

http://hpfem.org/~pavel/public/book2/p_211.pdf

Also rb1957 is correct, if the plate deflects more than approximately half its thickness, then diaphragm stresses cannot be ignored see R.J.Roark's formula for stress and strain 5th edition page 405

desertfox

 

[hokie66]

For a 15' long, simply supported span (no horizontal restraint), I figure (using beam theory) that a 1" thick plate deflects 1.6" under its own weight. What do you guys get using your more complex theory? Structural folks only tend to invoke Timoshenko when the span to depth ratio gets small.

 

[racookpe1978]

For a 15' long, simply supported span (no horizontal restraint), I figure (using beam theory) that a 1" thick plate deflects 1.6" under its own weight.

OK. So why did I measure only about 1/16 inch deflection for a 1x1 bar at 115 inch length, when the bar was also simply supported with o end restraints? It's shorter, obviously, but what am I measuring incorrectly?