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slenderness ratio
29 nov 1995
from some email to tagdi, after his posting with some some numbers that
indicate that some metals are stronger in compression than tension... 

>>i'm not sure you have the whole picture here, but it's a good start. look up
>>steel, especially piano wire... metals are often used in long thin strands,
>>ie wires or cables or rods, which are strong in tension but very weak in
>>compression, because the strands buckle. look up "slenderness ratio" for
>>a column...
another way to verify that long thin pieces of metal are stronger in tension 
than in compression is to unbend a wire paper clip or coathanger into a
straight piece of wire. then try to pull the wire apart, holding the ends
in your hands. impossible. but, if you push on the wire from each end it
will buckle or fold up easily. this is more of a geometric property of
materials than a matter of the basic strength of the material.

when wood is used as a column or post, in compression, it is considered to be
as strong as the wood itself for a short column, but a lot weaker in a long 
column. in the case of wood, the difference between "short" and "long"
happens when the ratio of the length of the column to the thickness of
the column exceeds about 12:1; the number 12 is the critical "slenderness
ratio." for example, a 1" x 1" column of wood might support 1000 psi in
compression, and it might have a "modulus of elasticity" e, of 1,000,000,
and it might be used to hold up 800 pounds. how long can the column be,
if it has no side-bracing to keep it straight? i have a book that has a
calculation, from euler, who said that if a column is so long that when you
bend it a little bit, the new forces on the column make it bend more,
then the column is too long to be safe. this "euler limit for buckling"
requires that for a column of length l and thickness d, the compressive
stress in the wood, c, be less than 0.3e/((l/d)^2). so l^2 < 0.3ed^2/c, or
l < square root(0.3(10^6)(1)^2/800) = 19.36" above. so even though the wood
itself in this column can support 1000 pounds, it cannot stably support
800 pounds, if the post is more than 19.36" long... if it were 16' long,
it could support about 1000 pounds hanging from the bottom, but it could
only hold up about 8 pounds, before it became unstable in buckling.

materials in tension don't buckle, so they don't have this critical
"slenderness ratio" limit on their strength. bucky talked about this...


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