questions: strength of plywood

Imagine a rough table made of a single 4 x 8 sheet of plywood, with a leg (4x4) at each corner. Imagine a weight placed exactly in the center of this table.
The preliminary question is, how much will the plywood sag under the weight? I should think that it would depend on (a) how thick the sheet is and (b) how heavy the weight is. (Also possibly relevant might be the area occupied by the weight; would a 100-lb weight on a 36 sq. in. base cause more sag than the same weight distributed over, say, 324 sq. in.?)
My real question is this: is there a formula or rule of thumb to calculate how thick the plywood sheet needs to be to support a given weight without sagging beyond a certain limit?
I know I could prevent sagging altogether by putting a fifth leg in the center, but the space beneath the table needs to be completely open.
Thanks in advance for any insights you might have.
cheers,
Henry
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Henry wrote:

This can be calculated fairly easily. I don't have time now but perhaps this evening will. What is the type of plywood? The parameter needed is Young's modulus, or modulus of elasticity. The remaining just depends on the dimensions of the surface.
Phil

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On Tue, 12 Aug 2003 14:57:24 +0300, snipped-for-privacy@eircom.net (Henry) wrote:

To take a 3/4" ply sheet and add 4 legs would be a poor design of a table. The table that size will certainly sag under its own weight. What is lacking is an apron let into the legs, and for a 4x8' sheet, at least six 2x2" legs or possibly some angle iron.
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I've probably got the matnamatical answer to that here somewhere in my reference library but I'd avoid the whole question by either building the top of the table top as a torsion box assembly or provide a skirt and leg base for the table along with additional cross pieces in the middle.
--
Mike G.
Heirloom Woods
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Deflection with a uniform load is determined by the formula
5wL^4/384EI
w = weight per unit length L = length E = modulus of elasticity of the material used I = the bending moment of the member being considered.
Here you can see that since the length is to the 4th power that as the member gets longer the deflection gets much much greater. Make your sheet of plywood 20 feet long and it is likely to touch the ground in the middle under its own weight. People have advocated using steel because the modulus of elasticity (stiffness basically) is much higher than plywood so you can get some advantage there, however the most effective way to minimize defiection is to work on I, the bending moment.
The bending moment is based on the geometry of the member. For a rectangular member the formula is
bh^3/12
where b is the width and h is the height. As the height is cubed you can see that a small change in h means a big change in deflection. This is why a thin deep apron would add more strength than doubling the thickness of the plywood. To maximize the bending moment putting most of the material at the top and the bottom is most effective. This is why steel beams are shaped like an I. You can do the same thing by taking two thinner pieces of plywood and running a series of 1x2 ribs between them. The result will be a strong light top. Maximize the depth of your top and you minimize the bending. Double the thickness and the deflection is decreased by a factor of 8 (2 cubed) Triple it and it decreases by a factor of 27. A sandwich of 1/2" plywood with 1x2s on edge (every 8 inches or so) inside will be strong enough to hold several hundred pound with minimal deflection.
-Jack
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etc.
Thanks, Jack! Exactly what I needed to know.
cheers,
Henry
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Nitpick: I is the moment of inertia of the cross section. A bending moment is a force (what most people would think of as torque).
Mike
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snipped-for-privacy@eircom.net (Henry) writes:

<snip>
I quit doing beam deflection calculations the day I walked out of school so I won't bore you with a lot of math which requires you to make a lot of assumptions anyway.
The short answer is:
You can't get there from here.
Plywood without proper support, will sag of it own weight.
Build a frame using 1x4's doubled around the outside with 1/4 cross members on 16" centers parallel to the 4 ft dimension.
Cover with a piece of 1/2" 4 ply, CDX if for storage, 2 pieces, if it will be an active table.
You will also need 1x4's for the diagonal bracing used to keep the legs in place.
At least that is the way I build them, only I use 2x6's for frame and legs, but then again, It is used for 250 lb rolls of fiberglass.
HTH
--
Lew

S/A: Challenge, The Bullet Proof Boat, (Under Construction in the Southland)
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Actually, this is likely to be a lower bound to the deflection. Since the sheet is significantly wide and fairly flexible, there will be some sagging resulting from bending in both directions (plus or minus anticlastic curvature effects). The deflection of a plate is a lot more complex than that of a beam.
You'll never get better than a rough estimate from basic elastic theory, since wood in general (including plywood) is not a nice isotropic material.
You really want to reinforce this plywood sheet, as so many have pointed out. Wood will be more weight efficient that steel angles. Tables are made like tables for a good reason. The torsion box is a nice alternative.
Mike
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On Tue, 12 Aug 2003 22:22:33 GMT, "Michael Daly"

I should have referenced this formula in my original post. It comes from the AWI spec book and they got the formula and the field testing from the University of West Virginia wood sciences people.
My own estimates of the deflecting strength of a table are more rough and tumble.
If I've a question about a table's strength, I stand in the middle of it. If it adequately resists my two hundred pounds, distributed through two size eleven shoes, I figure it'll hold up the grits.
I miss the old days of American engineering, where three times theory was the norm.
Regards, Tom Tom Watson - Woodworker Gulph Mills, Pennsylvania http://users.snip.net/~tjwatson
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That's interesting - the formula is identical to the one previously posted but with the constants multiplied/divided out ( 5*12/384 = 0.15625) and the variables factored into something simpler. Hence it's the plain old elastic theory solution - no correction for the specific properties/behavior of wood. It may still be reasonable if applied correctly (something that really is a beam), but I'm fairly certain that a wide, thin sheet of ply supported at four corners isn't quite right.
To use this reliably, I'd expect the short sides of the plywood to be fairly well supported so they don't flex much. Four legs bolted on won't do that.
Consider a really thin sheet (one extreme of behavior). It will act as a membrane and it will sag like a "bowl" and the legs would collapse At the other extreme is an infinitely rigid board. The original poster's plywood table is somewhere in between, probably closer to the membrane. Hence the beam formula is probably not reliable.
The real deflection formula would be a fourth order, second degree differential equation (or is that second order, fourth degree?) and you don't want to go there.
The problem with those handbooks is that they don't always tell you the limits of applicability.
I haven't heard anyone say "This is a good idea" so I think that the smart money is on making a stiffer table.
Mike
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SNIP of some stuff about a plywood table

More sniping of actual rlevant info>

Just a nit Tom, it's WVU, not UWV (I wouldn't call yours University of Penn State) :)
Dave Hall
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(David Hall)

Yeah, no doubt :)
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HEY! Watch your mouth! I kinda like my PSU sheepskin, thankyewverymuch. Besides, seems like those ivy league schools hand out diplomas to about anyone ;-) We're much more selective - football players and engineer types (& maybe a few others).
Renata
On 26 Aug 2003 14:20:03 -0700, snipped-for-privacy@nhsd.k2.pa.us (David Hall) wrote:

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