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

Henry wrote:
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?

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


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

On Tue, 12 Aug 2003 14:57:24 +0300, (Henry) wrote:

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

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.

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
www.heirloom-woods.net
"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

"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

JackD wrote:


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.

etc.

Thanks, Jack! Exactly what I needed to know.

cheers,

Henry

(Henry) writes:

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?
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)
Visit: http://home.earthlink.net/~lewhodgett for Pictures

On 12-Aug-2003, "JackD" wrote:

I = the bending moment of the member being considered.

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

On 12-Aug-2003, Tom Watson wrote:

D = 0.1563wl^4
_____________

Ebh^3

Whe

D = deflection (in inches)
w = load per lineal inch of span
l = span (length)
E = modulus of elasticity
b = base (width)
h = depth (thickness)

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

On Tue, 12 Aug 2003 22:22:33 GMT, "Michael Daly"
wrote:

On 12-Aug-2003, Tom Watson wrote:

D = 0.1563wl^4
_____________

Ebh^3

Whe

D = deflection (in inches)
w = load per lineal inch of span
l = span (length)
E = modulus of elasticity
b = base (width)
h = depth (thickness)

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

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

On 12-Aug-2003, Tom Watson wrote:

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.

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

Tom Watson wrote in message
SNIP of some stuff about a plywood table

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.

More sniping of actual rlevant info
Regards, Tom
Tom Watson - Woodworker
Gulph Mills, Pennsylvania
http://users.snip.net/~tjwatson

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

Dave Hall

Tom Watson wrote in message . ..
On 13 Aug 2003 10:34:43 -0700, (David Hall)
wrote:

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

Dave Hall

Akshully, I think the Penn State guys would gladly have their
sheepskins inscribed by the University of Pennsylvania.


Regards, Tom
Tom Watson - Woodworker
Gulph Mills, Pennsylvania
http://users.snip.net/~tjwatson

Yeah, no doubt

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, (David Hall)
wrote:

Tom Watson wrote in message . ..
On 13 Aug 2003 10:34:43 -0700, (David Hall)
wrote:

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

Dave Hall

Akshully, I think the Penn State guys would gladly have their
sheepskins inscribed by the University of Pennsylvania.


Regards, Tom
Tom Watson - Woodworker
Gulph Mills, Pennsylvania
http://users.snip.net/~tjwatson

Yeah, no doubt