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questions: strength of plywood



 
 
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  #1  
Old August 12th 03, 12:57 PM
Henry
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Default 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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  #2  
Old August 12th 03, 01:43 PM
PC
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Default questions: strength of plywood



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


  #4  
Old August 12th 03, 04:21 PM
Mike G
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Default questions: strength of plywood

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



  #5  
Old August 12th 03, 07:21 PM
JackD
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Default questions: strength of plywood


"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



  #6  
Old August 12th 03, 07:47 PM
Henry
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Default questions: strength of plywood

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
  #8  
Old August 12th 03, 11:06 PM
Michael Daly
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Default questions: strength of plywood

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
  #9  
Old August 12th 03, 11:22 PM
Michael Daly
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Default questions: strength of plywood

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
  #10  
Old August 12th 03, 11:39 PM
Tom Watson
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Default questions: strength of plywood

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
 




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