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.]

case # | case 1 | case 2 | case 3 | case 4 |

tube type | square | round | thick round | big round |

dimension | 2.000 | 2.000 | 2.000 | 2.5123 |

wall thick | 0.125 | 0.125 | 0.1624 | 0.125 |

Area, A | 0.9375 | 0.7363 | 0.9375 | 0.9375 |

% wt of sq | 100% | 78.5% | 100% | 100% |

Moment, I | 0.5518 | 0.3250 | 0.3988 | 0.6697 |

% I of sq | 100% | 58.9% | 72.3% | 121.4% |

normalized I | 0.5885 | 0.4414 | 0.4254 | 0.7144 |

% nor I of sq | 100% | 75.0% | 72.3% | 121.4% |

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 2.

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 values.

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: tqp@inel.gov.