A flywheel rim bursts when its tip speed reaches the square root of its tensile strength divided by its density, whatever its size: about 120 to 169 m/s for cast iron, 210 to 248 m/s for mild steel and 27 to 34 m/s for plain concrete. A 500 kg steel ring at its burst speed holds about 11 MJ, five times what it holds at 1,500 rpm, so a runaway drive is far more dangerous than a wheel at its working speed, and tripling the size of a cast iron wheel at the same rpm brings it to its burst stress.
Part of Flywheel Energy Storage
Deriving the rim stress of a spinning ring, turning it into a burst speed and a burst energy for the materials in this wiki, and using it to set a safe speed.
Deriving The Hoop Stress
Take a thin ring of mean radius r, cross-section area A and density ρ, turning at ω. An element subtending a small angle dθ has mass ρ × A × r × dθ and a centrifugal force dF = ρ × A × r × dθ × ω² × r directed outward.
Cut the ring along a diameter and consider one half. The outward forces from all elements, resolved along the axis of symmetry, add up to ∫ from −π/2 to π/2 of ρ × A × r² × ω² × cos θ dθ = 2 × ρ × A × r² × ω². Two cut faces, each carrying a tension T, resist it: 2T = 2 × ρ × A × r² × ω², so
T = ρ × A × r² × ω² and σ = T ÷ A = ρ × ω² × r² = ρ × v²
where v = ω × r is the tip speed. Size, thickness and rpm have vanished: only density and the tip speed matter. That is the whole reason a “bigger version” of a working wheel can be a bomb.
For a solid uniform disc the standard elasticity results, quoted from textbook memory and not derived here, are a peak stress at the centre of (3 + ν) ÷ 8 × ρ × v² (0.41 × ρ × v² for ν = 0.3), rising to (3 + ν) ÷ 4 × ρ × v² (0.83 × ρ × v²) at the edge of a small central bore. The ring formula is exact for a thin ring and is used as the baseline everywhere in this series; the disc factors are in Rim Mass vs Solid Disc.
Burst Speed And Burst Energy For The Corpus Materials
Setting σ equal to the tensile strength gives v_burst = √(σ ÷ ρ). Strengths are from Tensile Strength (psi converted at 1 psi = 6.895 kPa); densities are 7,200 (cast iron, corpus), 7,850 (steel, corpus value in another article) and 2,400 kg/m³ (concrete, a recalled typical value the corpus also uses).
| Material | Tensile strength | Burst speed, thin ring | Rpm of a 0.6 m radius ring | Energy of a 500 kg ring at burst |
|---|---|---|---|---|
| Cast iron | 103 MPa | 120 m/s | 1,907 | 3.6 MJ |
| Cast iron | 207 MPa | 169 m/s | 2,697 | 7.2 MJ |
| Mild steel | 345 MPa | 210 m/s | 3,335 | 11.0 MJ |
| Mild steel | 483 MPa | 248 m/s | 3,946 | 15.4 MJ |
| Plain concrete | 1.7 to 2.8 MPa | 27 to 34 m/s | not applicable | 0.18 to 0.29 MJ |
Working, steel at 345 MPa: v = √(344.7e6 ÷ 7,850) = 210 m/s; energy = ½ × 500 × 210² = 11.0 MJ. The energy per kilogram at burst is ½ × σ ÷ ρ: 7.2 to 14.4 kJ/kg for cast iron and 22.0 to 30.7 kJ/kg for steel.
Timber is not in the table. The corpus gives oak a tensile strength of 8,000 to 15,000 psi along the grain, but a wheel rim is built from joined pieces and the corpus gives no joint strength, so no burst speed can be stated for a wooden rim and none should be inferred from the clear-wood figure.
Runaway Is Worse Than Working Speed
The 500 kg steel ring at its working 1,500 rpm holds 2.2 MJ at 94.2 m/s. Taken to its weakest-steel burst speed of 209.5 m/s it holds ½ × 500 × 209.5² = 10.97 MJ. The ratio is (209.5 ÷ 94.2)² = 4.9, which is also the margin on stress. That is the number to keep in mind for drive runaway: a 5-fold energy increase comes with the failure, not before it. In TNT-equivalent terms (4.184 MJ per kilogram), 11 MJ is 2.6 kg, against 0.53 kg for the 2.2 MJ working case; these are energy equivalents, not blast predictions.
Worked Example: Scaling A Working Wheel
A cast iron disc of 0.25 m radius at 1,500 rpm has a tip speed of 39.3 m/s and a ring stress of 7,200 × 39.3² = 11.1 MPa, 9.3 times below the weakest cast iron. Its energy (disc) is 38.6 kJ. Scale every dimension by a factor s and keep the rpm:
| Scale s | Radius | Tip speed | Ring stress | Margin on 103 MPa | Energy |
|---|---|---|---|---|---|
| 1 | 0.25 m | 39.3 m/s | 11.1 MPa | 9.3 | 38.6 kJ |
| 2 | 0.50 m | 78.5 m/s | 44.4 MPa | 2.3 | 1,235 kJ |
| 3 | 0.75 m | 118 m/s | 99.9 MPa | 1.03 | 9,380 kJ |
Stress goes as s², energy as s⁵ (mass s³ times radius squared s² times the unchanged ω²). Tripling the wheel multiplies the energy by 243 and leaves it at the burst stress of the weakest iron. Scale by tip speed, never by rpm.
From Burst Speed To Working Speed
Rim stress goes with speed squared, so a safety factor S on stress is a factor √S on speed. Safety Factors gives 4 to 8 on stress for machine components under repeated loading, so working speed is v_burst ÷ 2 for S = 4 and v_burst ÷ 2.83 for S = 8. The carrier’s rule “operate at 50% of burst speed for a 4x safety factor” is correct, and a four-times factor is the minimum. For a built wheel in unproven iron, use 8.
Which strength do you use? The corpus test procedure is to test at least five specimens of the real material and take the average of the lowest three, not the best. A casting’s flaws make the wheel weaker than a test piece; in tension the corpus calls cast iron brittle and unreliable. Do not pick the high end of a range for a wheel you cannot test.
What Leaves When It Lets Go
A rim breaks into pieces that fly off tangentially at the tip speed. With no air drag the longest range of a piece is v² ÷ g, launched at 45 degrees: 94.2² ÷ 9.81 = 905 m at 1,500 rpm, and 209.5² ÷ 9.81 = 4,475 m at the burst speed. Drag shortens both, and this page has no sourced drag figure, so these are upper bounds, not exclusion distances. A piece of one eighth of the 500 kg ring carries ½ × 62.5 × 94.2² = 277 kJ at working speed.
Common Mistakes
- Quoting a “safe” speed from a rule of thumb. Compute from the strength you can show.
- Assuming energy at working speed is the worst case. An overspeed event is the worst case.
- Assuming a bigger wheel at the same rpm is similar. It is not.
- Forgetting joints, keyways and holes. The ring result is for sound, uncut metal.
Safety
The wheel is limited by its material, not by your drive
Fragment energies of hundreds of kilojoules leave in the plane of the rim, and the longest throw may be hundreds of metres. Never run a wheel above the speed calculated from tested strength with a safety factor of at least 4, preferably 8, never scale a design by eye, and run every wheel where nobody stands in its plane; see Housing and Speed Limiting.
What’s Next
- Learn next: Housing — where the wheel runs
- Combine with: Speed Limiting — keeping the drive below the burst speed