"Henry" wrote in message
...
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

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