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If you
take a long stick, like a yardstick, put one end on the ground and
lean on top of the other, the stick may bend and then break in the
middle. This is called "buckling".
Leonhard
Euler explored this phenomena in the 1700’s and came up with a formula
to determine how much force you could put on a pole before it buckled.
This force is called the "Critical Load" and one way to calculate it
is shown below. This particular equation works best for
"isotropic" materials. Isotropic materials behave uniformly in
all directions. Since wood has a "grain", it is stronger in one
direction than another and is not an isotropic material.
Critical
Load = π2 x E x I / L2
E = Modulus of Elasticity
L = Unsupported Length of Column with “pinned” ends
I = Least Moment of Inertia for the Column’s Cross section
However, despite this, the equation still tells us some things.
First, since "Fat Fluffy" thing have greater "Least Moment of Inertia"
than "Skinny" things, bridge members with "Fat Fluffy" cross sections bend less easily than
those with "Skinny" cross sections. The picture below shows "Good" versus "Bad"
cross sections.
Second, the
equation tells us that "Length" is more important than "Shape".
Suppose you had a two choices in making your bridge stronger.
The first was to increase a member's Moment of Inertia by a factor of
three. The second was to re-enforce a member at two points so
that it's "Unsupported length" was only 1/3 as long as before.
Since Euler's equation varies directly with "I", but with the square of
"L", that means that the length change would make the member 9
times stronger, versus the 3 time stronger for the moment of Inertia
change.
Summary
We can get two engineering
rules from this. You can fix buckling by:
But of
the two, the best way to keep the top
members in Katherine's championship bridge from buckling is to reduce
the unsupported length by putting in cross bracing. The weight
budget for this would come from reducing the weight of the "End
Posts", which we discussed
here.
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