A thin type of steel is to be shrunk on a rigid wheel of 900 mm diameter. What is the internal diameter of the tyre if the hoop stress is limited to 120 N/mm² ? (For the tyre take α = 12 × 10⁻⁶ per °C and E = 2 × 10⁵ N/mm²)
- A899.46 mmCorrect
- B819.46 mm
- C900.54 mm
- D800.54 mm
Solution
Concept
Hoop strain in a shrunk-on tyre
Formula
ε = (D − d) / d = σ / E
Step-by-step solution
1. Hoop strain ε = 120 / 2×10⁵ = 6×10⁻⁴. 2. (900 − d)/d = 6×10⁻⁴ ⇒ d = 900 / 1.0006 = 899.46 mm.
Explanation
The tyre must be slightly smaller than the wheel so that stretching it over the wheel develops the permitted hoop stress.
Why the other options are wrong
900.54 mm is larger than the wheel, so there would be no interference. 819.46 and 800.54 mm are far too small.
Shortcut
d = D / (1 + σ/E).
Common mistake
Adding the strain instead of dividing.
Correct answer: A) 899.46 mm