Tubing Selection for Recumbent Frames
"A few more things you didn't care to know about... Tubing Stiffness"
By John Zabriskie 1995
John Zabriskie has elaborated
on one of his Internet postings, to provide us with a tech article explaining
the weight vs strength/rigidity trade offs when using square tubing.
Have you ever wondered whay most bicycles are built of round tubing? Or why the Rans Rocket uses square tubing for a main boom? Of course, there are reasons for everything and tradeoffs to be made. But, in case you ever
wondered what kind of tubing to use on your next generation hpv, the following
discussion about tubing stiffness may help. If you're technically disadvantaged
or impaired, you may want to skip this article, or get help.
The table below summarizes some of the choices we're faced with when looking
at frames. I've taken the liberty of avoiding any discussion of material
(aluminum, steel, titanium, etc.) - we'll leave that to another time. I've
specified 4 geometric cases for our consideration. We want to compare the
relative merits of square versus round tubing. Assume the material, tube
length, applied load, and end support conditions are the same. If you compare
bending a 2" square tube to a 2" diameter tube there are a few
ways to do it.
[A note of explanation: "Normalizing" means to divide some value
by some other value so that everything is on the same playing field, so
to speak. In the cases below, we'll want to evaluate the stiffness relative
to the weight, so we'll divide the moment of inertia by the cross sectional
area. This is OK since we've assumed that the tube material is the same
in all cases, therefore the moment is representative of the stiffness and
the area representative of the weight.]
The areas are indicative of weight (wt = area * length * density).
|% wt of sq
|% I of sq
|% nor I of sq
Case 1 vs 2: The round tube of same outer dimension and wall thickness will
weigh less than a square tube. It is much less stiff (based on the moment),
though not as much if the moment is normalized by the cross-sectional area.
Case 1 vs 3: For a round tube of the same outer dimension as the square
tube to weigh the same, it must have a thicker wall. This does increase
the stiffness over case 2, but it is still less than the square tube. Considering
normalized stiffness (on a per weight basis), it is slightly less than case
Case 1 vs 4: For a round tube of the same thickness and having the same
weight as a square tube, the outer diamer must be larger. Here we see that
for the same weight, this larger diameter tube is significantly stiffer.
Normalizing the stiffness to the weight doesn't matter, since the weights
are the same.
Some of you have asked about torsion. The torsional moment, J is simply
the sum of the moments about each axis. Since the shapes above have symmetry,
Ixx and Iyy are equal, so J = 2 * I. Torsional stiffness has the same relative
Formulas used: D = OD, d = ID
Area of round tube = PI/4 (D^2 - d^2)
Area of square tube = (D^2 -d^2)
Moment of inertia, round tube = (PI/64)*(D^4 - d^4)
Moment of inertia, square tube = (1/12)*(D^4 - d^4)
Summary: If you want a light but stiff structure, use as little material as possible
(thin wall thickness), but place it as far to the outside of the envelope
as possible (maximize the outer dimension). Of course, when carried to the
extreme, you have a soda pop can. If you step on an empty pop can carefully,
it will support your weight. But if you dimple it slightly (starting a local
buckling failure), it will collapse catastrophically. That is why the wall
thickness must be thick enough to resist local dimpling. A practical rule
of thumb is to keep the outer dimension to wall thickness ratio below 50:1.
Square or round, which is best? It depends... actually, it's not a cop-out
answer. There are other factors to consider; most have to do with time and
convenience. Hanging stuff off framed structures is normally easier with
square tubing. There is a lot of time spent mitering round tube joints properly,
straight hacksaw cuts are a lot more convenient and take much less time.
John Zabriskie has been to Minnesota twice that he can remember. One of
the memories involves the Can-Am McLarens at Brainerd International Raceway.
The other concerns spitting across the Mississippi River from shore to shore.
He currently lives in Idaho, halfway between Sun Valley and Jackson Hole.
He likes commuting to work on his SWB Linear except when it snows. Then
he'd rather ski,... to work, if need be. His next hpv project under consideration
is a commuter trike. He can be reached via e-mail: firstname.lastname@example.org.
Practical Considerations By Mark Stonich 1995
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