On the kart: the 50 mm rear axle, the 30 mm frame tubes, the torsion bar under the front, the steering column. All hollow. Every one of them is a bending spring, and how stiff it is comes almost entirely from its diameter, raised to the fourth power.
The principle in one sentence
A tube resists bending with the metal farthest from its center, so diameter matters enormously and wall thickness matters much less than you'd think.
The one equation
bending stiffness ~ E x (D^4 - d^4)where E is the material's stiffness (Young's modulus, M09), D the outside diameter, d the inside diameter (D minus twice the wall). The exact constant in front (pi/64) is the same for every round tube, so for comparing two tubes it cancels. "~" means "goes as": double the right side and you double the stiffness.
Math: powers. x^2 means x times x ("x squared"); x^3 is x times x times x ("cubed"); x^4 is x multiplied by itself four times. On a calculator: the x^y or ^ key, or just multiply it out. Powers make small changes big. 1.1^2 = 1.21, 1.1^3 = 1.33, 1.1^4 = 1.46: a 10% bigger diameter is a 46% stiffer tube. That is the single most important fact on this page. Powers also work in reverse: 0.9^4 = 0.66, a 10% smaller diameter is a third less stiff.
Math: subtracting two big powers. D^4 - d^4 with D = 50 and d = 45: 50^4 = 6,250,000 and 45^4 = 4,100,625; the difference is 2,149,375. The two numbers are close, and the answer is their difference, so keep all the digits until the end. Rounding either one early throws the answer off badly. Physically, D^4 is what a solid bar of that diameter would give you, d^4 is what the hollow middle would have given, and the difference is the metal that is actually there doing the work.
Why it works
Bend a tube and the metal on the outside of the curve stretches while the metal on the inside compresses. Metal at the center line does neither: it's along for the ride and contributes nothing. The farther a bit of metal is from the center, the more it stretches for the same bend and the more it fights back, and the effect compounds: twice as far out, it stretches twice as much AND has twice the lever arm, so it resists four times as hard. Add that up over a circle and you get the fourth power. A hollow tube throws away the useless metal in the middle and keeps the useful metal at the edge. That is why tubes are stiff for their weight, and why bicycle frames, kart frames and axles are all hollow.
On our kart, in numbers
The axle: solid vs tube. A solid 50 mm bar has D^4 = 6,250,000. A 50 mm tube with a 2.5 mm wall has an inside diameter of 45 mm: D^4 - d^4 = 2,149,375. The solid bar is 2.9 times stiffer. But the tube has only 19% of the metal (area goes as D^2 - d^2), so per pound the tube is 1.8 times as stiff as the bar. The tube wins on weight; the bar wins on stiffness; the choice depends on which you're short of.
Soft, medium, hard. Kart axles are sold in stiffness grades at the same 50 mm outside diameter. Two ways to do that: change the wall (2.5 mm wall vs 3 mm wall is 16% stiffer), or change the alloy and heat treatment (E barely moves between steels, but the metal's strength and its behavior past the elastic range do). Our axle's grade is unknown; it's a tuning item because a softer axle lets the inside rear lift more easily (lesson 13).
The frame. 30 mm tubes with about a 2 mm wall (typical). If Birel had used 32 mm tubes with the same wall, the frame would be 23% stiffer. Two millimeters. This is why chassis makers argue about tube diameter, and why a "stiff" and a "soft" chassis can look identical.
The torsion bar. A short tube under the front of the frame, bolted in or left out. In: it adds its own D^4 to the front's resistance to twisting. Out: the frame twists on its own. The baseline sheet says "in" for our chassis. Same physics, used as a switch.
Two things to notice
- The fourth power means the outside skin does the work. A dent, a crack, or corrosion pitting on the outside of a tube removes the metal that mattered most. That is why the chassis inspection looks at the tube surface near the welds, and why a cracked axle is replaced, not welded.
- Diameter is the coarse knob, wall is the fine one. Going from a 40 mm to a 50 mm axle is a 2.4x change (1.25^4). Going from 2.5 to 3 mm wall at 50 mm is 16%. When a class standardizes on 50 mm axles, it's saying the fine knob is the only one you get.
Problems
1. The frame tube. Compute D^4 - d^4 for a 30 mm tube with a 2 mm wall (so d = 26 mm), and for 32 mm with the same wall (d = 28). Divide the second by the first: how many times stiffer is the 32 mm frame?
2. Wall vs diameter. Which is stiffer, and by what ratio: a 50 mm axle with a 3 mm wall, or a 52 mm axle with a 2.5 mm wall? Do the fourth powers carefully (keep all digits).
3. Weight of the axle. The tube's metal area goes as D^2 - d^2. Compare a 50 mm / 2.5 mm wall axle with a 50 mm / 3 mm wall axle: how much heavier is the thicker one, as a percentage? And how much stiffer (from the fourth powers)? Which grows faster, the weight or the stiffness, when you thicken the wall?
Go look: measure the axle's outside diameter with a paper strip (circumference / 3.14). Find a cut end of a tube anywhere on the kart or in the garage and look at how thin the wall is compared to the diameter. Push down on the middle of the rear axle with the kart on the stand and feel how little it gives: that's D^4 at work.