Tools › Structures › Stress Concentrations › Rotating Disk With Hole › Example

Rotating disk with a central hole (Timoshenko and Goodier): hoop stress at the bore and the factor 2 limit

Tool	Rotating Disk With Hole
Source	Timoshenko, S. P., and Goodier, J. N. (1970). Theory of Elasticity, 3rd ed. McGraw-Hill, New York. Chapter 4, "Rotating Disks" (thin disk with a central hole, plane stress).
Level	Graduate

Problem

A thin disk of 6 in outer radius with a 1 in central hole, density 0.1 lb/in³, ν = 0.33 and E = 10 Msi, spins at 10,000 rev/min. Find the hoop stress at the bore, the peak radial stress and where it occurs, the centre stress of the same disk without the hole, and the stress concentration of the hole.

Inputs to Enter

a = 1.0 in, b = 6.0 in, r = 1.0 in (the bore)
ρ = 0.1 lb/in³, ν = 0.33, E = 10 Msi
N = 10000 rev/min

Open the Rotating Disk With Hole tool (loaded with this example)

Answers

σ_θ at the bore = (3+ν)ρω²[b² + (1−ν)a²/(3+ν)]/4 = 8.560 ksi; σ_r = 0 there
Peak radial stress q(b − a)² = 2.956 ksi at r = √(ab) = 2.449 in, q = (3+ν)ρω²/8
Solid disk centre stress qb² = 4.256 ksi; K_hole = 2 + 2(1−ν)(a/b)²/(3+ν) = 2.011
Limits checked at the build: a → 0.01 in gives K → 2.000; a/b = 0.5 gives K = 2.10

Notes

The tool's default inputs are this case (hand values from the closed forms recorded in the changelog). The speed is a plain rev/min number; all stresses are computed in PHP and the formulas printed as text.